Semiconductors and Semimetals A Treatise
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Semiconductors and Semimetals A Treatise
Edited by R . K. Willardson WILLARDSON CONSULTING SPOKANE, WASHINGTON
Albert C. Beer BATTELLE COLUMBUS LABORATORIES COLUMBUS, OHIO
SEMICONDUCTORS AND SEMIMETALS VOLUME 25 Diluted Magnetic Semiconductors Volume Editors JACEK K. FURDYNA DEPARTMENT OF PHYSICS UNIVERSITY OF NOTRE DAME NOTRE DAME, INDIANA
JACEK KOSSUT INSTITUTE O€ PHYSICS POLISH ACADEMY OF SCIENCES WARSAW, POLAND
ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers
Boston San Diego New York Berkeley L Tokyo Toronto
COPYRIGHT @ 1988 BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL., INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC.
l2SO Sixth Avenue, San Diego. CA 92101
United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24-28 Oval Road, London NWI 7DX
LIBRARYOF CONGRESS CATALOG CARD NUMBER:65-26048
ISBN 0-12-752125-9 PRINTED 1N THE UNITED STATES OF AMERICA
88899091
9 8 7 6 5 4 3 2 1
To the memory of Professor Jerzy Mycielski (1930-1986)
Contributors Numbers in parentheses indicate the pages on which the authors’ contributions begin.
W. M. BECKER, Department of Physics, Purdue University, West Lafayette, Indiana 47907 (35)
J . K. FURDYNA,Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556 (1) J. A. GAJ, Institute of Experimental Physics, University of Warsaw, 69 Hoza, 00-681 Warsaw, Poland (276) T. GIEBULTOWICZ, Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, and National Bureau of Standards, Reactor Division, Bld. 235, Gaithersburg, MD 20899 (125) W. GIRIAT,Centro de Fisica, Instituto Venezolano de Investigaciones Cientifcas (IVIC), Caracas IOIOA, Venezuela (1) T. M. HOLDEN,Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada (125) PIETERH. KEESOM,Department of Physics, Purdue University, West Lafayette, Indiana 47907 (73) J. KOSSUT,Institute of Physics, Polish Academy of Sciences, Al. Lotnikow 32/46, 02-668 Warsaw, Poland (1 83) J. MYciELsKI, Institute of Theoretical Physics,. Warsaw University, 00-681 Warsaw, Poland (31 1) SAULOSEROFF, Department of Physics, San Diego State University, San Diego, California 10775 (73) A. K. RAMDAS,Department of Physics, Purdue University, West Lafayette, Indiana 47907 (345) C . RIGAUX,Groupe de Physique des Solides de I’Ecole Normale Supgrieure, 24 rue Lhomond, 75231 Paris Cedex 05, France (229) S . RODRIGUEZ, Department of Physics, Purdue University, West Lafayette, Indiana 47907 (345) P. A. WOLFF,Francis Bitter National Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (413) xi
Foreword
Diluted magnetic semiconductors have in recent years attracted considerable attention of the scientific community. The present book consists of 10 chapters, each describing a different facet of the physics of these materials. In defining what constitutes the family of diluted magnetic semiconductors (DMS) one might say, to be on the safe side, that any known semiconductor with a fraction of its constituent ions replaced by some species of magnetic ions (i.e., ions bearing a net magnetic moment) is a legitimate member of this group. More realistically, the majority of DMS studied in an extensive way so far involved Mn+' ions embedded in various A"BV' hosts. There is a rationale for this particular choice: (i) Mn+' can be incorporated in sizable amounts in the A"BV' host without affecting substantially the crystallographic quality of the resulting material (e.g., up to nearly 80% of Mn can be accommodated in CdTe); (ii) Mn" possesses a relatively large magnetic moment (S = 5/2), characteristic of a half-filled d shell (from this point of view Euf2, with S = 7/2, might seem to be even more attractive); (iii) Mn+2 is electrically neutral in A"BV' hosts, i.e., it constitutes neither accepting nor donating centers. There are three aspects of DMS properties that make these alloys interesting subjects for scientific investigation (and also give them the potential for future device applications). The first of these concerns semiconducting properties per se: the parameters determining the band structure of the material (e.g., the energy gap) can be varied by changing appropriately the Mn+' mole fraction in the crystal. Similarly, the lattice parameters can be "tuned" by varying the composition of the DMS, this aspect being particularly important in the age of monolithic semiconductor heterostructures, where lattice matching is of obvious importance. In other words, Mn+' affects various properties of the host semiconductor in a manner analogous to, say, Cd in Hgl-,Cd,Te. Secondly, purely magnetic properties of DMS encompass a very broad spectrum of behavior, including paramagnetic, spin-glass and antiferromagnetic properties. It is thus possible to study within one system-say, Cdl-xMnxTe-the development of these magnetic features as a function of the concentration of Mn+'. Spin-spin correlations, for example, and their xiii
xiv
FOREWORD
dependence on the Mn mole fraction, were studied in detail by neutron scattering methods and are described in Chapter 4. Alternatively, DMS offer a possibility of a comparative study of the magnetic behavior of Mn+2spins in different hosts. One is then able to draw conclusions concerning, say, the role played by the anion in the process of mediating the superexchange coupling between two magnetic moments (see, e.g., Spalek et al., 1986, and Samarth, 1986). Thirdly, the interaction which exists between the localized magnetic moments of Mn+2and the conduction and/or valence band electrons (which we shall refer to as the sp-d interaction) results in a series of features which are unique to DMS. The best known (and probably the most spectacular) of these are the huge Faraday rotation of the visible and near-infrared light in wide-gap DMS (see Chapter 7 for details) and the giant negative magnetoresistance in the vicinity of the semiconductor-semimetal transition in Hgl-xMnxTe (dealt with in Chapter 8). As mentioned above, these spectacular features have their common origin in the sp-d exchange interaction. As a result, the band structure of DMS is much more sensitive to the strength of an external magnetic field than in ordinary semiconductors. For this reason DMS are sometimes referred to as “spin amplifiers”. Thus, by means of magnets readily available in the laboratory, one is able to observe phenomena that normally would require the application of megagauss magnetic fields. Since the sp-d interaction causes the semiconducting properties to be strongly sensitive to the details of the magnetic behavior of the Mn+* subsystem, it is thus possible to study the latter indirectly by observing, for example, the optical properties of DMS. The general organization of the present volume reflects these three aspects of DMS physics. The first two chapters describe the crystal structures, methods of preparation, and semiconducting properties (mostly optical) of DMS in the absence of the magnetic field. This is followed by two chapters devoted to the magnetic properties displayed by these materials. The remaining chapters deal with various aspects of the physical properties of DMS that are due to the sp-d interaction of the band electrons with localized magnetic moments. So much for the scope. What about the purpose of the volume? The number of papers published in the literature dealing with DMS, starting with the pioneering works by Delves (1963) and Delves and Lewis (1963), must now exceed 1,000. It is then difficult to make oneself familiar with the field without going to many different sources, scattered throughout theliterature. Thus, the first objective of the present volume is to provide a reasonably full account of the achievements in this field. Secondly, it was thought that, at least in several areas, the stage has been reached warranting formulation of conclusions of a character more general than is appropriate for a typical journal article.
FORE WORD
xv
The work on the book spanned a considerable period of time (as is perhaps unavoidable in the case of any publication involving a large number of contributing authors). The last several years have witnessed a particularly rapid growth of the number of publications concerning DMS. In particular, several important new contributions emerged when the book was already in the middle of its preparation process. For this reason, although they are of unquestionable importance, it was not possible to properly represent them here. To do them partial justice without delaying the appearance of the book, we shall call attention to some of the new developments at least in the Foreword in the form of the list below. (1) Considerable progress is to be noted in the experimental and theoretical studies concerning the relative position of Mn-derived states and various density-of-states features in the valence and conduction bands. The experimental part has made extensive use of the synchrotron radiation photoemission technique (Franciosi etal., 1985a,b, and Wallet al., 1986a,b). The theoretical effort (consisting of a series of papers by Ehrenreich, Hass, Larson, and collaborators) was recently reviewed by Ehrenreich et al. (1987). Based on these studies it was, for example, possible to determine that the dominant mechanism of the Mn-Mn exchange coupling is that of superexchange, and to calculate theoretically the values of the relevant exchange constants. (2) Successful growth by molecular beam epitaxy of superlattices and quantum well systems involving DMS opened a vast field of entirely new opportunities (for a recent review of the work on DMS superlattices, see Furdyna et al., 1987). (3) MaterialsinvolvingMn+2inA1VBV1andA11BV hosts (e.g., Pbl-,Mn,Te and (Cdl-,Mn,)2As3, respectively) were prepared in an attempt to enlarge the family of DMS. Some of the references to these works are given in Chapter 1 in the section devoted to new materials. (4) DMS materials containing substitutional Fe ions instead of the traditional Mn+2(for reviews, see Mycielski, 1986a, and Reifenberger and Kossut, 1986) proved to possess features absent in Mn-based DMS. A considerable enhancement of the low-temperature mobility of electrons in Hgl-,Fe,Se (0.0003 5 x 5 0.01) is the most spectacular example of these new properties. ( 5 ) Studies of the conductivity in the weakly localized regime and in the vicinity of the nonmetal-metal transition have shown that the electronic spin plays an important role in the related processes (see Stankiewicz el al., 1986a,b, Shapira et al., 1985, Wojtowicz et al., 1986, and Sawicki et al., 1986). (6) Investigations of the magnetization induced in DMS by circularly
xvi
FOREWORD
polarized radiation incident on the sample (Krenn et al., 1985) and the “magnetic spectroscopy” (Awschalom and Warnock, 1987) open new possibilities, particularly in the context of time-resolved studies of magnetic systems. Expecting that the book would be read at a rate of one chapter at a time, it was the editors’ decision to give the contributing authors a free hand in their choice of units, notation, etc., provided that they were consistent within their own chapter. For the same reason, a certain degree of overlap between the material contained in some chapters was not frowned upon inshe hope that the contributions would, in this way, be more self-contained. Finally, let us comment on the terminology used in connection with DMS. Quite often these materials also are referred to as semimagnetic semiconductors. Being aware that this is predominantly a matter of taste, we nevertheless prefer the term “diluted magnetic semiconductor”, since this term conveys the physical nature underlying the properties of the material. It refers to a magnetic semiconductor (a name already established, as for EuSe) that has been diluted by the intervening non-magnetic constituent, giving DMS all their physical characteristics. The authors of the chapters contained in this volume were, as a rule, actively involved in the research in the areas which they respectively describe. One of the authors in particular, Professor Jerzy Mycielski, had a very special impact on the development of the physics of DMS. Over and above the contents of his own chapter in this book, his contributions consisted of formulation of many original and important ideas throughout the entire period of DMS studies. The papers of his authorship were among the first that put DMS “on the map”, while his hypothesis concerning a spatial ordering of charges within the donor system in Hg, -xFexSe (Mycielski, 1986b) will remain his last. It was with profound sorrow that we learned of his death in February, 1986, before he could see his contribution in print. To his memory, we would like to dedicate this volume. References Awschalom, D. D., and Warnock, J. (1987). DilutedMagnetic (Semimagnetic)Semiconductors, edited by R. L. Aggarwal, J. K. Furdyna, and S. von Molnar (Materials Research Society Symposia Proceedings, Pittsburgh, PA) Vol. 89, p. 71. Delves, R. T. (1963). J. Phys. Chem. Solids 24, 885. Delves, R. T., and Lewis, B. (1963). J. Phys. Chem. Solids 24, 549. Ehrenreich, H . , Hass, K. C., Larson, B. E., and Johnson, N. F. (1987). Diluted Magnetic (Semimagnetic) Semiconductors, edited by R. L. Aggarwal, J . K . Furdyna, and S. von Molnar (Materials Research Society Symposia Proceedings, Pittsburgh, PA) Vol. 89, p. 187. Franciosi, A . , Chang, S., Caprile, C., Reifenberger, R., and Debska, U. (1985a). J. Vac. Sci. Techno/. A3. 926.
FOREWORD
xvii
Franciosi, A., Reifenberger, R., and Furdyna, J . (1985b). J. Vac. Sci. Technol. A3, 124. Furdyna, J. K., Kossut, J., and Ramdas, A. K. (1987). OpticalPropertiesofNarrow-GapLowDimensionalStructures, edited by C. M. Sotomayor Torres, J. C. Portal, J. C. Maan, and R. A. Stradling (Plenum Press, New York) p. 135. Krenn, H., Zawadzki, W., and Bauer, G. (1985). Phys. Rev. Lett. 55, 1510. Mycielski, A. (1986a). Diluted Magnetic (Semimagnetic) Semiconductors, edited by R. L. Aggarwal, J. K. Furdyna, and S. von Molnar (Materials Research Society Symposia Proceedings, Pittsburgh, PA) Vol. 89, p. 159. Mycielski, J. (1986b). Solid State Commun. 60, 165. Reifenberger, R., and Kossut, J. (1987). J. Vac. Sci. Technol. AS, 2995. Samarth, N. (1986). Ph.D. Thesis, Purdue University, unpublished. Sawicki, M., Dietl, T., Kossut, J., Igalson, J., Wojtowicz, T., and Plesiewicz, W. (1986). Phys. Rev. Lett. 56, 508. Shapka, Y., Ridgley, D. H., Dwight, K., Wold, A., Martin, K. P., Brook, J. S., and Lee, P. A. (1985). Solid State Cornmun. 54, 593. Spalek, J., Lewicki, A., Tarnawski, Z., Furdyna, J. K., Galazka, R. R., and Obuszko, Z. (1986). Phys. Rev. B33, 3407. Stankiewicz, J., von Molnar, S., and Giriat, W. (1986a). Phys. Rev. B33, 3573. Stankiewicz, J., von Molnar, S., and Holtzberg, F. (1986b). J. Magn. Magn. Muter. 54-57, 1217. Wall, A., Caprile, C., Franciosi, A,, Reifenberger, R., and Debska, U. (1986a). J. Vac. Sci. Technol. A4, 818. Wall, A., Caprile, C., Franciosi, A., Vaziri, M., Reifenberger, R., and Furdyna, J. K. (1986b). J. Vac. Sci. Technol. A4, 2010. Wojtowicz, T., Dietl, T., Sawicki, M., Plesiewicz, W., and Jaroszynski, J. (1986). Phys. Rev. Letters 56, 2419.
SEMICONDUCTORS AND SEMIMETALS. VOL. 25
CHAPTER 1
Crystal Structure, Composition, and Materials Preparation of Diluted Magnetic Semiconductors W . Giriat CENTRO DE FISICA INSTITUTO VENEZOLANO D E INVESTIGACIONES CIENTIFICAS (IVIC) CARACAS, VENEZUELA
and J. K. Furdyna DEPARTMENT OF PHYSICS UNIVERSITY OF NOTRE DAME NOTRE DAME, INDIANA
I. INTRODUCTION. . . . . . . . . . . . . . . . . . . . 11. CRYSTAL STRUCTURE AND COMPOSITION OF DMS . . . . . . 1. Overview. . . . . . . . . . . . . . . . . . . . 2 . The A"BV' "Host "Lattice. . . . . . . . . . . . . . 3 . Crystal Structure and Lattice Parameters of DMS . . . . 4. Microscopic Structure of the DMS Lattice . . . . . . . 111. PREPARATION OF DMS CRYSTALS. . . . . . . . . . . . . 5. Preparation of Starting Materials. . . . . . . . . . . 6 . Crystal Growth of DMS Alloys . . . . . . . . . . . IV. THE"NEW"DMS. . . . . . . . . . . . . . . . . . . 1. DMS Films and Superlattices. . . . . . . . . . . . . 8 . A1',FexBV' Alloys . . . . . . . . . . . . . . . . 9. Manganese- and Rare-Earth-Based AIVBV'Alloys. . . . . 10. MisceIIaneous New DMS Systems. . . . . . . . . . . REFERENCES.. . . . . . . . . . . . . . . . . . . .
1
3 3 4 7 11 14
14 17 21
28 28 30 31
32
I. Introduction During the past decade a great deal of attention has been given to semiconductor compounds whose lattice is made up in part of substitutional magnetic ions (Galazka, 1979; Gaj, 1980; Furdyna, 1982; Brandt and Moshchalkov, 1984). Most of these materials are based on A"BV' compounds, with a fraction of the group-I1 constituent replaced by manganese. Cdl-,Mn,Se and Hgl-,Mn,Te are examples of such systems. These ternary
1 Copyright 0 1988 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-752125-9
2
W . GIRIAT AND J . K . FURDYNA
alloys (or "mixed crystals") are of interest for several reasons. Firstly, their semiconducting and structural properties, such as the energy gap, the lattice parameter, etc., can be varied in a controlled fashion by varying the composition (much as in non-magnetic ternary semiconductors, e.g., Hgl-,Cd,Te or Gal-,Al,As). Secondly, as dilute magnetic alloys, these materials are of interest for their magnetic properties, e.g., the spin-glass transition (Galazka, 1982; Oseroff, 1982), antiferromagnetic cluster formation (Dolling, 1982), magnon excitations (Ramdas, 1982), and other magnetic effects of current interest. Thirdly, the presence of magnetic ions in the lattice leads to spin-spin exchange interaction between the localized magnetic moments and the band electrons (Gaj, 1980). This interaction affects the energy band and impurity level parameters of these materials (e.g., by enhancing the electronic g-factors), resulting in new physical effects. Such novel and potentially important phenomena as the magnetic-fieldinduced overlap between valence and conduction bands occurring in Hgl -,MnxTe and Hgl-,Mn,Se (Galazka and Kossut, 1980), extremely large Faraday rotation in Cdl-,Mn,Te (Gaj et al., 1978), giant negative magnetoresistance associated with hopping conduction in Hgl -,Mn,Te (Mycielski and Mycielski, 1980), and the magnetic polaron observed in Cdr-,Mn,Se (Nawrocki et al., 1981; Diet1 and Spalek, 1983) are all consequences of the spin-spin exchange interaction. The latter two features-magnetic properties as such, and the exchange interaction with band electrons-distinguish these materials from other semiconductors. For this reason, this group of alloys has been collectively labeled as "semimagnetic semiconductors" or "diluted magnetic semiconductors" (DMS). So far, practically all research on DMS involved A"BV' semiconductor compounds alloyed with manganese, although recently there has been a gradual increase of research activity on A1'BV'-Fe alloys (e.g., Hgl-,Fe,Te, Cdl-,Fe,Te) and AIVBV'-Mn alloys (e.g., Pbl-,Mn,Te). The investigations of these latter materials are still at a relatively early stage. In the present chapter we shall therefore concentrate on the A?,Mn,BV1 alloys. The material is presented as follows. We first review the crystal structure and composition of the various A:'_,Mn,BV' alloys. We show that this can be done in a unified way for both the zinc blende and wurtzite members of this group of materials. We then present methods of preparation of bulk DMS crystals, including materials purification, synthesis of starting binary compounds, and crystal growth. Finally, we discuss-in a much briefer fashion-the more recent developments in the field of DMS, such as A:'_,Mn,BV' epitaxial films and superlattices, A:'-,Fe,BV' alloys, and AIVBV'-based DMS containing substitutional manganese or rare earth ions.
1.
CRYSTAL STRUCTURE AND MATERIALS PREPARATION
3
11. Crystal Structure and Composition of DMS
1. OVERVIEW The entire family of ternary A:’_,MnxBv’ alloys, along with their crystal structures, is presented in schematic form in Fig. 1. The heavy lines in the figure show the composition ranges in which ternary alloys can be formed, with “cub” and “hex” indicating the crystal structure (zinc blende and wurtzite, respectively) of the stable phases. For example, Cdl-,MnxTe forms a ternary alloy of zinc blende structure with x up to 0.77, while Z ~ I - , M ~ , S ~ exhibits zinc blende structure for x < 0.30 and wurtzite structure for 0.30 c x c 0.55. At values of x beyond the heavy lines mixed phases occur. Given the fact that the stable crystal structures of MnTe, MnSe, and MnS differ from the crystal structure of the A”BV’ compounds (Pajaczkowska, 1978), it is truly remarkable that the range of the A1/-,MnxBV1 solid solutions
CdTe
ZiS FIG.1. A diagrammatic overview of the A:’,Mn,BV’ alloys and their crystal structures. The bold lines indicate ranges of the molar fraction x for which homogeneous crystal phases form. “Hex” and “Cub” indicates wurtzite and zinc blende, respectively. Cd,-,Mn,S and Cdl-,Mn,Se also form homogeneous crystalline phases near x = 1.0 in the NaCl structure, but these are not relevant to the present article.
4
W . GIRIAT AND J. K . FIJRDYNA
is so wide. For example, MnTe itself crystallizes in the NiAs structure, yet in the case of Cdl-,Mn,Te the structure of the "host" CdTe (zinc blende) survives for x as high as 0.77. Both crystal structures of the DMS-zinc blende and wurtzite-are very closely related in spite of the differences in symmetry, in that they are both formed with tetrahedral (s-p3) bonding, involving the two valence s-electrons of the group-I1 element and the six valence p-electrons of the group-VI element. Manganese is a transition metal with valence electrons corresponding to the 4s' orbital. Although manganese differs from the group-I1 elements by the fact that its 3d shell is only half-filled, it can contribute its 4s2 electrons to the s-p3 bonding, and can therefore substitutionally replace the group-I1 elements in the A"Bv' wurtzite or zinc blende structures. These properties of manganese are apparently a necessary but not a sufficient condition for forming stable ternary phases over a wide range of compositions, since other transition metal elements-even those which readily form divalent ions-are not nearly as miscible with A"BV' compounds. For example iron (a transition metal with ionic radius comparable to that of manganese) is about an order of magnitude less miscible, and the other transition metal elements even less than that. It would appear, therefore, that the ease with which Mn substitutes for the group-I1 elements in the zinc blende and wurtzite structures (and the well-known predisposition of this element for the divalent state) results from the fact that the 3d orbitals of Mn++ are exactly half-filled. By Hund's rule, all five spins are parallel in this orbital, and it would require considerable energy to add an electron with opposite spin to the atom. In this sense, the 3d5 orbit is "complete," and thus the Mn atom resembles a group-I1 element.
2. THE A"BV' "HOST" LATTICE The physical properties of the A"BV'-based DMS resemble the properties of the "parent" A"BV' binary compounds. It is therefore instructive to review briefly the structural properties of the A1'Bv' lattice before considering their ternary DMS derivatives. Crystallographic properties of the A"Bv' semiconductors are remarkably systematic. With the one exception of HgS, whose stable phase is cinnabar, they all form in either the zinc blende or the wurtzite phase. As has been pointed out, both structures form as a result of tetrahedral s-p3 bonding. Zinc blende is a cubic structure comprised of two interpenetrating fcc sublattices-one consisting of the znions, the other of the cations-shifted with respect t o each other by one-fourth of the body diagonal of the fcc cubic unit cell. The wurtzite structure consists of two interpenetrating hcp sublattices shifted along the c-axis.
1.
CRYSTAL STRUCTURE AND MATERIALS PREPARATION
5
The similarity of the two structures is based on the following: (a) In either structure, each member of one sublattice is tetrahedrally bonded to four members of the other sublattice. Thus Cd is surrounded by four Te atoms in the zinc blende CdTe, and by four Se atoms in the wurtzite CdSe, and within this local configuration it 'is impossible to distinguish between the two crystal structures. (b) The other similarity arises from the fact that wurtzite and zinc blende crystals are based on the hcp and fcc structures, respectively, both of which derive from the two alternate ways to obtain equivalent close packing of spheres. Thus, in both structures a given atom is surrounded by 12 equidistant like atoms. From the above it is obvious that the distance between like atoms d (e.g., the nearest cation-cation distance) and the bond length b (i.e., the nearest cation-anion distance) are identically related in both the zinc blende and the wurtzite structures. This feature will allow us to discuss both structures in a unified way (Yoder-Short et al., 1985). From purely geometrical considerations, we have d=
Gb.
(1)
For the zinc blende case, then, the lattice parameter a is
The wurtzite structure is based on two interpenetrating hcp structures, and is characterized by two lattice parameters, a and c. It is important to realize that for the ideal hcp situation, based on the standard packing of spheres, a and c are not independent: a is just the separation between the centres of touching sphere in the hexagonal plane (i.e., the separation between like atoms of the sublattice in that plane), and c is the separation between repeating planes (which is obviously also a function of the radius of the spheres). In terms of d , it is easily shown that for Wurtzite structure a = d,
(3)
c = &d.
(4)
Now it is truly remarkable how close is the behavior of the wurtzite Ax*Bvx compounds to the ideal case. For example, in the wurtzite crystal CdSe, a = 4.2985 A and c = 7.0150 A (Pajaczkowska, 1978), giving C
- = 1.632, a
6
W. GIRIAT AND J. K . FURDYNA
a
which is very close to the ratio = 1.633 predicted by the ideal close packing of spheres. Furthermore, in those materials which can exist in both cubic and hexagonal crystal phases, the bond length is the same. For example, for cubic ZnS (Pajaczkowska, 1978) a = 514093
and for hexagonal ZnS
A
= f i x 3.825
A,
a = 3.820 A,
c = 6.260 A
=
fi x 3.833 A .
It is useful to express the length of the tetrahedral bond-which underlies the above lattice properties of the tetrahedrally-bonded crystals-in terms of the covalent radii of the participating group-I1 and group-VI elements (Pauling, 1960; van Vechten and Phillips, 1970). The bond length b is then given as the sum of the two radii,
b = rII
+ rvI.
Such a radius is associated with a specific atom independent of the environment; for example, the same covalent radius of Se predicts rather accurately the bond lengths (and thus all other lattice parameters) in the zinc blende HgSe and ZnSe, and in the wurtzite CdSe. We list in Table I the covalent radii for the group-I1 and group-VI elements. It is remarkable that the bond lengths-and thus other lattice parameters-for all zinc blende and wurtzite A1*Bvlcompounds can be immediately generated, to a close approximation, from Table I. In Table I1 we compare the bond lengths calculated from Table I with those measured experimentally. TABLE I TETRAHEDRAL COVALENT R m n (IN S Se Te
Zn
Cd
Hg
Mn
1.127 1.225 1.405 1.225 1.405 1.403 1.326
A)
van Vechten and Phillips, van Vechten and Phillips, van Vechten and Phillips, van Vechten and Phillips, van Vechten and Phillips, van Vechten and Phillips, Yoder-Short el al., 1985
1970 1970 1970 1970 1970 1970
We shall show next that the covalent radius remains meaningful in the context of the Ai'-,Mn,BV1 ternary alloys, providing a very useful guideline to the variation of the lattice parameter with composition in all DMS compounds.
1.
CRYSTAL STRUCTURE AND MATERIALS PREPARATION
7
TABLE I1
BOND LENGTHSb IN METALCHALCOGENIDES Bond length in
A
Compound
Calculated
Observed*
ZnS ZnSe ZnTe CdS CdSe CdTe HgS HgSe HgTe
2.352 2.450 2.630 2.532 2.630 2.810 2.530 2.628 2.808
2.342 2.454 2.631 2.532 2.633 2.806 2.534 2.634 2.797
*R. W. G . Wyckoff, Crystal Structures, 2nd Edition(Interscience, New York,1963),Vol. 1 , p. 112.
3. CRYSTAL STRUCTURE AND LATTICE PARAMETER OF A:’_,Mn,BV’ DMS Crystal structure of ternary alloys formed by substituting Mn for the group-I1 element in the A1’BV’ lattice is indicated in Fig. 1, where “cub” refers to the zinc blende, and “hex” to the wurtzite structure. As a rule, these DMS retain the crystal structure of the “parent” A1’Bv’ compound. Exceptions to this rule are Znl-,Mn,S and Znl-,Mn,Se: these exhibit the structure of the cubic A1’BV’ host for low Mn content, but above a certain value of x , they form in the wurtzite structure. A natural upper limit on the Mn mole fraction x in DMS is imposed by the fact that the MnBV’binary compounds do not crystallize in the zinc blende or wurtzite structures. It is interesting to note that, although the stable phase of HgS itself is the cinnabar structure (HgS is the only A”BV’ compound to crystallize in this form), the presence of Mn in Hgl-,MnxS appears to stabilize the sphalerite structure of this system (Pajaczkowska, 1978). At this time, relatively little is known about this alloy. The remarkably systematic behavior of lattice parameters observed in the A1’Bv’ compounds, as described in the preceding section, carries over to their A:’-,Mn,BV1 derivatives. Two aspects are particularly worthy of note. First, in all DMS crystallizing in the wurtzite structure, the a :c relationship predicted by the ideal hexagonal close packing of spheres is obeyed. Thus, for Zno.dsMno.ssSe, the observed value is c/a = 1.635, as compared to the = 1.633 (Yoder-Short et al., 1985). Since in the ideal case both ideal
a
W . GIRlAT AND J . K. FURDYNA
a and c are determined by the nearest neighbor distance d of the anion (or cation) sublattice, one concludes that d is the one key parameter which determines the structural dimensions of the ternary DMS lattice. Second, the lattice parameters for all DMS obey Vegard’s law very closely. This is illustrated by Fig. 2, showing the dependence of the three cubic tellurides on composition (Furdyna ef a/., 1983). It is gratifying that an extrapolation to x = I-which indicates the lattice parameter for the hypothetical zinc blende MnTe-is the same for the three alloys. Thus, in spite of the fact that zinc blende MnTe does not exist in nature, one may regard Cdl-,Mn,Te as a true pseudobinary alloy of CdTe and MnTe, and one can write its lattice parameter in the form a = (1 - x)aIr-vI
+ XaMn-VI.
(5)
The behavior illustrated in Fig. 2 for the cubic lattice parameter a is actually true for d independent of structure, so that the Vegard law behavior can be presented in a unified picture for the entire DMS family, wurtzite and zinc blende alike. We show this in Fig. 3. The usefulness of Fig. 3 is made clear by considering, for example, the linear behavior of d observed for the selenides. Note that Hgl-,MnxSe is cubic, Cdl- ,Mn,Se is hexagonal, and ZnI-,Mn,Se changes its crystal structure “in stride” as x increases. Thus, once having established the parameter d for MnSe by extrapolating, e.g., the lattice constant data for the hexagonal Cdl-,Mn,Se to x = 1, we can predict the lattice constant for the cubic Hgl-,Mn,Se, or for either of the two phases 6
.
5
0
5
0.6 0.8 I .o Mole fraction X FIG.2. Lattice parameter as a function of Mn mole fraction x for telluride DMS. Note the Vegard law behavior, and the convergence of the extrapolated lines at a single point (6.334 A ) , which determines the lattice parameter for the zinc blende phase of MnTe. [After Furdyna et a/.(1983), with Hgl-,MnxTe data taken from the paper by Delves and Lewis (1963).]
0
0.2
0.4
1. CRYSTAL STRUCTURE AND MATERIALS PREPARATION
9
MnTe
Mn Se
Mn S
MOLE FRACTION X FIG.3. Mean cation-cation distances d as a function of Mn mole fraction x for A!~,MnxBvl alloys. Analytic expressions correspondingto the figures are given in Table 111. [After YoderShort et at. (1985).]
of Znl-,MnxSe. Two additional points are worthy of note in Fig. 3. First, it is interesting that the mutual relationship of d for HgSe and CdSe in this figure closely resembles that for HgTe and CdTe, despite the difference in structure. This again focuses on the fact that, ultimately, the tetrahedral radii are the meaningful parameters, independent of the lattice environment. Second, we observe that the value of d for Znl-,Mn,Se increases smoothly through the cubic-to-hexagonal transition, suggesting that it is the increase in the behavior of the lattice spacing that forces the onset of the hexagonal structure, rather than the other way around. Analytical expressions for the cation-cation distances d, from which respective lattice parameters can be
10
W. GIRIAT A N D J. K . FURDYNA
TABLE 111
MEANCATION-CATION DISTANCES d IN A:'-,Mn,BV1 ALLOYS
Material
Upper limit of x
Znl-,Mn,S Cdl-,Mn,S Hgl-,MnxS Znl-xMn,Se Cdl-,Mn,Se Hgl-,Mn,Se Znl-,Mn,Te Cdl-,Mn,Te Hgl-xMnxTe
0.60 0.50 0.37 0.57 0.50 0.38 0.86 0.77 0.75
Mean cation-cation distaace d (A) 3.830 4.123 4.139 4.009 4.296 4.301 4.315 4.587 4.568
+ 0.139~
- 0.151~ -
0.167~
+ 0.164~
- 0.123~ - 0.123~
+ 0.168~ - 0.105~ -
0.080~
obtained via Eqs. (2)-(4), are listed in Table I11 for all A:~,Mn,B'" alloys (after Yoder-Short et af., 1985). In analogy with Table 11, we list in Table IV the extrapolated bond lengths b for zinc blende and/or wurtzite MnTe, MnSe, and MnS, along with the covalent radius of manganese obtained from these values and.from the data listed in Table I. The covalent tetrahedral radius for Mn could in principle be obtained from measurements on zinc blende and/or wurtzite phases of MnSe and MnS. These tetrahedrally bonded phases, however, are difficult to prepare, and only limited data are available on the binary compounds. We believe therefore that the value of d for the three manganese chalcogenides obtained by extrapolation of the DMS data to x = 1, as illustrated in Figs. 2 and 3, are more reliable than the literature values obtained from measurements made on the binary compounds MnS and MnSe themselves (Pajaczkowska, 1978). The internal consistency of the extrapolated DMS values further corroborates the reliability of the values listed in Table IV, which ultimately provides the covalent tetrahedral radius of Mn, rMn = (1.326 f 0.018)A (Yoder-Short et al., 1985). It should be noted that, apart from its fundamental importance, the precise knowledge of the lattice parameter is of considerable practical interest in that it provides a convenient determination of crystal composition in ternary alloys. It is a fortunate circumstance in this respect that in the case of all A:'_,Mn,Bv' alloys the linear variation of the lattice parameter with x is very large (contrast this with, e.g., Hgl-,Cd,Te), making this approach to the determination of composition quite reliable. Typical x-ray techniques yield the lattice parameter with precision of 1 part in lo4 or better, thus giving the composition within 1% for most DMS.
1.
CRYSTAL STRUCTURE AND MATERIALS PREPARATION
11
TABLE IV BONDLENGTHS b AND Mn COVALENT RADIUSrMn IN Mn CHALCOGENIDES’ Compound
b (A)
rMn(A)
MnS MnSe MnTe
2.432 2.557 2.746
1.305 1.335 1.341
Average covalent radius of Mn:
1.326
“After Yoder-Short et al., 1985.
4. MICROSCOPIC (LOCAL)
STRUCTURE OF THE
DMS LATTICE
In view of the very consistent adherence to Vegard’s law obtained from x-ray diffraction data described above, one might automatically assume that the crystal structure of a ternary “zinc blende” alloy is strictly zinc blende, i.e., the anions and the cations are distributed over two interpenetrating, geometrically identical fcc sublattices, each with a lattice constant varying linearly with x. Under this assumption, the picture would be identical to the microscopic structure of the parent AxlBvlbinary compound, with the scale adjusted by the Vegard law behavior, and with the cation lattice sites occupied randomly by two kinds of ions. An analogous picture suggests itself for the wurtzite structures. This picture constitutes the foundation of the virtual crystal approximation, frequently used in interpreting the physical properties of ternary alloys. In reality, the microscopic situation is considerably more complicated. Recent extended x-ray absorption fine structure (EXAFS) studies of DMS by Balzarotti et al. (1984) shed important light on the detailed microscopic structure of a disordered DMS lattice (and, by extension, of other tetrahedrally bonded ternary alloys), and we shall examine them in some detail. The EXAFS technique provides detailed information on the immediate atomic environment of a given lattice site. In the case at hand, it is sensitive to the local spatial distribution of the four cations tetrahedrally bound to a given anion. In reporting the results and conclusions which follow from the EXAFS data, we shall, for specificity, refer to Cdl-xMn,Te, on which the measurements were actually performed. The conclusions, however, must apply equally well to other zinc blende ternary alloys (not just DMS) and, indeed, to ternary wurtzite systems as well. Briefly, the EXAFS results carried out on Cdl-xMnxTe for 0 < x < 0.7 indicate that the Cd-Te and the Mn-Te bond lengths in this material remain practically constant throughout the entire range of compositions studied.
12
W. GIRIAT A N D J. K . FURDYNA
The Cd-Te bond length is approximately unchanged from its value in pure CdTe, and the Mn-Te bond length is approximately that predicted for the hypothetical zinc blende MnTe (for which the lattice parameter is predicted in Fig. 2). The actual bond length values are shown as a function of x in Fig. 4. These results are not in contradiction with x-ray diffraction data, which show Vegard’s law behavior for the ternary lattice. X-ray diffraction is not sensitive to the topological details surrounding an individual lattice point, but determines the lattice parameter averaged over all anion and cation sites. It is precisely this averaged a(x) which varies linearly with x. The EXAFS results, however, cannot be reconciled with the interpretation that Vegard’s law is obeyed on the microscopic scale by each of the two interpenetrating fcc lattices (the one occupied by Te atoms, the other bya random distribution of Cd and Mn). Rather, the two sublattices must adjust locally to accommodate the requirement of keeping the respective Cd-Te and Mn-Te bond lengths as nearly constant as possible. While in reality both the anion and the cation lattices are thus distorted from fcc, it is realistic to represent the microscopic behavior in terms of a model proposed by Balzarotti et al. (1984) as follows. The cation sublattice, occupied randomly by Cd and Mn, is to afirst approximation not distorted. It may be regarded, to first order, as an fcc sublattice, obeying Vegard’s law in the spirit of the virtual crystal approximation. The atoms on the anion sublattice, on the other hand, adjust in order to accommodate the requirement that the Cd-Te and the Mn-Te bond lengths remain as constant as possible. The anion sublattice, then, is distorted, and is no longer fcc.
2.80t
-
Cd-Te
T
-
2.75 -
t
n
Q Mn-Te
0.1
0.3
Q
0.5 Mole Fraction X
0 0
0.7
FIG.4. Actual Mn-Te and Cd-Te bond lengths as a function of composition, determined by EXAFS. [After Balzarotti ef al. (1984).]
1. CRYSTAL STRUCTURE AND MATERIALS PREPARATION
13
The justification for the above picture (i.e., for the fact that the two sublattices behave differently) is as follows. In Cdl-,Mn,Te, a given Te site can be surrounded by 0, 1, 2, 3, or 4 Mn nearest neighbors, the remaining ions of the tetrahedral quartet being Cd, respectively. When a given anion is surrounded by four like cations, it will remain in its equilibrium position (assuming nearest neighbor interactions to be dominant). In the three cases of a mixed Cd and Mn environment, however, the anion will move from its equilibrium position so as to adjust its distance as closely as possible to the Cd-Te and Mn-Te bond lengths. Since the Mn-Te bond length is shorter than the Cd-Te bond, the Te ion will be displaced from its equilibrium position towards Mn and away from Cd. The assumption of the model that, unlike the anions, the cation sublattice is not distorted is plausible in that each cation is always surrounded by a tetrad of like atoms (Te in Cdl-,Mn,Te). If a cation were to move to a closer position with respect to one anion, it would automatically move further away from other identical anions. In reality, some distortion of the cation sublattice must also occur, but on the basis of the above argument it is likely to be considerably smaller than the distortion of the anion sublattice and may be neglected at this stage of our understanding. Theoretical calculations for the proposed model carried out by Balzarotti et a/. (1984) are in excellent agreement with observation. The authors show that if all possible configurations of all possible tetrahedra are considered, the average anion sublattice obtained in this way turns out to be exactly the fcc sublattice, with a(x) given by Vegard’s law. X-ray diffraction, by averaging over all possible sites existing in the crystal, measures precisely such an average, Vegard-law-obedient behavior. The ideas advanced by Balzarotti et al. are expected to apply to other random ternary zinc blende alloys of the A:’-,Mn,BV1 type, as well as to analogous wurtzite systems. The constant value of the anion-cation distance that emerges from this picture, gives added strength to the concept of the tetrahedral radius (see earlier discussion) as a physically meaningful concept. While important from a directly crystallographic point of view, these results are also expected to bear on a number of other physical phenomena. For example, in light of these results, the microscopic environment of atoms in the zinc blende and wurtzite ternaries are no longer strictly cubic and hexagonal, respectively, and many optical transitions-particularly those involving local mode excitations-that are forbidden by the ideal symmetry (as manifested in the binary “parent” compound) may be relaxed in these random alloys. Furthermore, magnetic properties of DMS are expected to be dominated by super-exchange interaction, i.e., by Mnf+-Mn++ interactions that are mediated through the intervening anion (Larson et al., 1985; Spalek et al., 1986). The anion-Mn distance, as well as the angle subtended
14
W. GIRIAT AND J . K. FURDYNA
by the Mn++ pair and the anion, are expected to bear directly on this exchange process, and thus the new insights into the bond length are likely to play an important role in understanding the magnetic properties of DMS. 111. Preparation of Bulk DMS Crystals
The literature on the synthesis and crystal growth of A:'_,Mn,BV' ternary alloys is quite extensive. The systems Znl-xMnxS and Cdl-,Mn,S were obtained by Schnaase as early as 1933 by precipitation from dilute aqueous solutions of the binary constituents. Since that time (although mostly after 1960), a large number of studies on synthesis and crystal growth of the A:'_,MnxBv' solid solutions were carried out. We shall not attempt to present a historic survey of the various methods of preparation of these materials. This has been done with admirable completeness in the excellent review by Pajaczkowska (1978). Rather, we shall focus on those techniques which appear to be most successful and practical. In the subject of crystal growth, it is of course difficult, if not impossible, to be entirely objective. It must therefore be stated at the outset that the choice of methods of preparation that we describe are largely flavored by the authors' own experience and their particular assessment of the practical aspects of the procedure, taking into account accessibility, cost, and convenience. 5 . PREPARATION OF STARTING MATERIALS
a. Purification of starting elements
To prepare any semiconducting material, it is of course absolutely necessary to start with high purity elements. Commercially available spectrally pure materials are, unfortunately, very often oxidized to a degree which makes them unsuitable for this purpose. For this reason, we shall first briefly describe a typical method of purification for each of the elements relevant to the A:'_,Mn,Bv' DMS, before discussing the preparation of the compounds themselves. Normally, purity of 6N to 7N is sufficient for preparation of good quality crystals (i.e., adequate for most optical and electrical studies). Mercury can be readily purified to a level of 6 or 7N by double or triple vacuum distillation in quartz tubes. Tellurium and selenium are usually vacuum distilled (twice) and zone refined to attain desired purity. In the case of sulphur, purity better than 6N can be obtained by triple distillation in vacuum. Commercially available cadmium and zinc, even of high nominal purity, are usually highly oxidized as they are delivered, and one must also distill these elements two to three times to make them usable. For storage, materials should be sealed in glass tubes in vacuum.
1. CRYSTAL STRUCTURE AND MATERIALS PREPARATION
15
FIG.5. Arrangement for vacuum sublimitationof Mn: (1) furnace; (2) external ceramic tube; (3) inner ceramic tube; (4) manganese before sublimation; (5) manganese after sublimation; (6) exit to vacuum system.
Manganese constitutes a special case. The commercially available material is typically in the y-Mn form, prepared electrolytically. This form, even with relatively high nominal purity, cannot be used as it stands because it is invariably very heavily oxidized. Even though the oxide layer can be removed by etching (the easiest method is to etch in HNO3), the material oxidizes so rapidly that it is generally difficult to transfer the manganese from the etch to the crucible without prohibitive oxidation. Removal of oxidation by reduction in hydrogen is not practical for the reason that-although higher order Mn oxides (MnO2, Mn203 , and Mn304) can be reduced to MnO at readily accessible temperatures and pressures-the reduction of MnO in H2 is practically impossible, requiring temperatures of about 2000°C and pressure of 150 atmospheres. The most successful method of Mn purification is by sublimation. This not only leaves the troublesome MnO behind, but converts the easily oxidized y-Mn phase to a-Mn, which is much more resistant to oxidation (Kaniewski et al., 1978). A simple arrangement used for vacuum sublimation of Mn is shown in Fig. 5 . Two concentric ceramic tubes are positioned in the furnace, with pieces of manganese placed inside the inner tube, at the hottest zone (marked 4 in the figure). Sublimation is carried out under high dynamic vacuum at a temperature of about 1OOO"C. After sublimation, pure manganese collects on the walls (marked 5 in the figure), and only the MnO powder remains in the hot zone (marked 4) in the inner tube. Such sublimation is generally carried out twice. Immediately after sublimation, the purified manganese is sealed in evacuated glass tubes for storage.
b. Preparation of starting binary compounds In preparing DMS alloys, it is more convenient to start with binary A"BV' and MnB"'compounds than with theelemental constituents. We will therefore briefly describe the preparation of the binary systems relevant to DMS. The preparation of the A"BV1 is relatively easy, largely because the constituent elements have low melting points (see Table V) and high vapor
16
W . GIRIAT A N D J . K . FURDYNA
pressures at relatively low temperatures. Because of these comparatively low temperatures, quartz tubing can be used for the preparation of most of the A"BV' compounds. Stoichiometric amounts of high purity elemental materials are placed in a quartz ampoule, which is then pumped to a vacuum of the order of Torr. The tube with the material is subsequently sealed and heated in such a way as to ensure that there always exists a position within the ampoule where the temperature is less than the lower of the boiling points of the two constituents. After reaction occurs, it is possible to raise the temperature to the melting point of the compound. Melting points of the binary compounds, along with their crystal structure and lattice parameters, are listed in Tabel V. TABLE V MELTING POINTS,LATTICE CONSTANTS, AND CRYSTAL STRUCTURE COMMONMn(Te, Se, S), Hg(Te, Se, S), Cd(Te, Se, S), AND Zn(Te, Se, S) PHASES
OF
m.p. ("C)
4A)
Structure
MnTe
1165
a = 4.1475 c = 6.710
NiAs
MnSe MnS HgTe HgSe HgS CdTe CdSe
1510 1530 670 779 825 1098 1260
NaCl NaCl zinc blende zinc blende cinnabarhinc blende zinc blende wurtzite
CdS
1405
ZnTe ZnSe ZnS
1293 1526 1722
5.462 5.223 6.460 6.084 5.851 6.481 u = 4.2985 c = 7.0150 u = 4.1368 c = 6.7163 6.1037 5.6687 5.4093
wurtzite zinc blende zinc blende zinc blende
Thus, HgTe can be prepared by heating an evacuated sealed ampoule containing Hg and Te so that the highest temperature in the ampoule is ca. 700°C and the lowest is ca. 420°C. This situation is maintained for about 24 hours, which is sufficient for the materials to react. The temperature of the whole ampoule is then raised to 700°C in order to melt the HgTe. Preparation of HgSe proceeds in an analogous manner, except that the melting point of HgSe is 779°C. Reaction of HgS requires higher temperatures, and becomes dangerous because of the high vapor pressure of the elements. Very thick quartz, with external counter pressure, is then recommended. CdTe is
1. CRYSTAL STRUCTURE AND MATERIALS PREPARATION
17
prepared by raising the temperature of the stoichiometric amounts of the constituent elements in an evacuated sealed ampoule from room temperature to 1100°C during 2 to 3 hours. CdSe is prepared by heating the ampoule containing the elements so that the lowest temperature in the ampoule is at 700°C, the reaction occurring below the melting point of CdSe. In the case of CdS, the minimum temperature should be maintained below 440"C, with the high temperature in the ampoule at about 1000°C. Zinc telluride, selenide, and sulfide are prepared like CdS, where now the minimum temperature is 440°C for ZnS, 700°C for ZnSe, and 900°C for ZnTe, and the maximum temperature in the ampoule is held near 1000°C. The A"BV' compounds have the highly desirable property that the vapor pressure of the elements is much higher than that of the compounds. Any unreacted elements can thus be driven off simply by heating the vacuumreacted binary. During this process, unreacted elements evaporate and Can be removed by pumping, so that only the pure stoichiometric compounds remain in the ampoule. This convenient behavior does not occur in other semiconductors, such as the A1"BV, A'"BV1, or Mn compounds. In these materials some of the elements have very low vapor pressures, so that it is not possible to remove them simply by evaporation. To obtain MnS or MnSe, purified Mn is first ground in mortar and pestle to a fine powder and then placed in a quartz ampoule with stoichiometric amounts of Se or S. The ampoule is then sealed off in vacuo, and is placed in a furnace in such a way that the cooler end is at 700°C for MnSe and 450°C for MnS, and the hot end at about 1000°C for both cases. After several days the reaction is complete. MnTe can be prepared much more quickly. Stoichiometric amounts of Mn and Te are placed in a carbonized quartz ampoule in vacuo. The ampoule is inserted into a hot furnace (ca. 1000°C) for about 5 minutes, and is then quickly removed, yielding reacted polycrystalline MnTe.
6 . CRYSTAL GROWTH OF DMS ALLOYS Although similarities exist between the methods of preparation of specific DMS alloys, there are also significant differences because of the wide range of melting points and vapor pressures spanned by this family of materials. For this reason, it is more convenient-at the risk of some repetition-to present the preparation procedure for each alloy system separately. (a) Cdl-,Mn,Te Cdl-,MnxTe crystallizes in the zinc blende structure in the range 0 < x I0.77 (Triboulet and Didier, 1981). For xexceeding this limit, mixed phases occur. The phase diagram for this system is shown in Fig. 6 . The
18
W. GIRIAT AND J. K. FURDYNA
I 150-
Il5Ot Liquid
1100Liquid+solid
I
? w 1050-
I
[r
Solid
3
L
904
Oll
012 013 014 015 d.6 017 018 019
1 3
near-coincidence of the solidus and liquidus curves indicates that this system is ideally suited for growth by the Bridgman method. The melting point, which for pure CdTe occurs just below llOO°C, decreases slightly as the Mn content increases. For example, for x = 0.77 the melting point of Cdl-xMnxTeis about 30°C below that of pure CdTe. Crystal growth by the Bridgman method is typically carried out with a starting temperature of about 1120°C,using a temperature gradient of about 50°C/cm in a vertical furnace. The charge is lowered at a speed of 4 mm/h. High quality monocrystals, with typical dimensions of 15 mm in diameter and 10 cm in length, can be obtained in this manner. The macroscopic quality and homogeneity of the Cdl-*Mn,Te crystals is, at this stage, the best of the DMS alloy family, probably because of the unique coincidence of the solidus and liquidus curves in the phase diagram (Fig. 6). Also, because of this latter feature, the composition of the Bridgman crystals remains close to the starting (i.e., “nominal”) value. In their mechanical properties, single crystals
1. CRYSTAL STRUCTURE AND MATERIALS PREPARATION
19
of Cdl-,Mn,Te in the whole range of compositions strongly resemble those of CdTe, with a tendency to cleave along the (110) faces. Like CdTe, crystals of Cdl-,Mn,Te frequently exhibit twinning (Wu and Sladek, 1982). In terms of their electronic properties, as-grown crystals tend to be p-type, with a high resistivity. One can obtain n-type single crystals, for example, by doping with Ga or In, and by subsequent annealing in the presence of Cd. (b) Cdl-,Mn,Se The phase equilibria of Cdl-,Mn,Se were first investigated by Cook (1968). The phase diagram of this system was determined by lattice constant measurements and differential thermal analysis (DTA). Results of those investigationsare presented in Fig. 7. CdI-,Mn,Se forms a single-phase solid solution for x up to about 0.50, the crystal structure being wurtzite throughout this range of x . The melting point of the solid solution is nearly constant, starting with the value of 1260°C for pure CdSe, and decreasing very slightly as x is increased. The vertical Bridgman technique can be used successfully to grow Cdl-,Mn,Se crystals. The 1260°C melting point is still sufficiently low to permit the use of quartz ampoules, but thick-walled ampoules are recommended. Pre-reacted pure CdSe and MnSe binary compounds are placed in the silica ampoule in desired proportions, and the tube is sealed in good vacuum. The tube is then placed in a furnace at a temperature of 1270°C. The ampoule is lowered at a speed of, typically, 4 mm/h. Single crystals of several cubic centimeters in size can be obtained. All compositions in the range 0 < x < 0.5 can be grown routinely in this manner. Because the solidus L
- I500
Liquid
-
rn
1
I
-
0
- ,I
m
-
-
---
-1200
e
L
-1100
Wurtzite ond rock solt
Wurtzite
Ya-
-1000
I F g
-
- 900 E
-
-
g
1.0
Cd Se
I
I
0.8
I
I
0.6
0 I
0.4
I
I
0.2
I
000
I - 700
0.0
MnSe Mole Fraction X FIG.7. Phase diagram for CdSe-MnSe. [After Cook (1968).]
20
W . GIRIAT AND J. K . FURDYNA
and liquidus lines in the phase diagram are very close, the effect of segregation is not very serious (although it is somewhat more pronounced than in Cdl-,Mn,Te), and the crystals display only a small composition gradient along the direction of growth. (c) Cdl-,Mn,S Single crystal growth of this important system is at a much more preliminary stage than that of the preceding two alloys. The phase diagram of CdS-MnS solid solutions was determined by Cook (1968) and Wiedemeier and Khan (1968). Finely powdered MnS and CdS samples were thoroughly mixed in desired molar ratios and compressed into pellets at a pressure of approximately 13.5 K-bar. The pellets were sealed under high vacuum in quartz tubes, and annealed for approximately 100 hours at each of the following temperatures: 1000", 800°, 700°, and 600°C. The tubes were then taken out of the furnace, and the samples quenched in air. This procedure proved to be sufficient to maintain the equilibrium established at the high temperature, resulting in stable sintered polycrystalline specimens. To ensure that equilibrium was established, selected samples were subjected to additional repeated annealing cycles at 800, 700, and 600"C, for a total of several hundred hours at each temperature. Lattice constants of samples subjected to repeated annealing cycles were identical, within the limits of error, with values for alloys obtained by the original annealing sequence. The temperature-composition phase diagram of the CdS-MnS system is presented in Fig. 8. As can be seen in the figure, the liquidus and solidus are practically indistinguishible on the CdS-rich side of the diagram. In the case at hand, the melting points of the constituent binaries are very high: roughly 1400°C for CdS and about 1600°C for MnS. Because of this, silica can no longer be used in the crystal growth of Cdl-,Mn,S from the melt. Single crystal growth of Cdl-,Mn,S by the Bridgman method was reported by several authors using carbon crucibles, under high inert gas pressure (Ikeda et al., 1968; Komura and Kando, 1975). Compressed pellets were used as starting charges in these investigations. The pellets were first repeatedly annealed at 1000°C in nitrogen/HZS atmosphere. The carbon crucible containing the annealed powder was then placed in a furnace filled with dry Ar gas. The pressure of Ar was between 40 and 100 kg/cm2. After holding the temperature slightly above the melting point of the alloy for approximately one hour, the crucible was lowered out of the furnace at a rate of 30 mm/h. The fusing temperature of the mixed powder was set at 1450°C for the Cd-rich samples (0 5 x I 0.30), and at 1620°C for the Mn-rich composition (0.7 Ix 5 1). This procedure demonstrated that Cdl-,Mn,S forms in the wurtzite structure for Cd-rich compositions, and that the rocksalt phase of this material may form at very high values of x (see Fig. 1).
1.
CRYSTAL STRUCTURE AND MATERIALS PREPARATION
Liquid
II
-
-
,*
..
-
,.'
.,,*-'__,'- -
-
/
/
/
~
0
0
21
;-"I600 - 1500
/
Wurtzite
-
Wurtzite a n d
I
I
l
l
1
I
\
I
I
I
:
,
700
FIG. 8. Phase diagram for CdS-MnS. [After Cook (1968).]
A mixture of both these structures occurs for the intermediate region of x. The size of the crystals obtained in this manner were of the order of several millimeters on the side. It is significant that all the specimens contained more Mn than the starting material, especially the specimens with low Mn content, possibly due to the fact that the volatility of CdS is much higher than that of MnS. The Bridgman crystals were quite homogeneous along the growth directions, as might be expected from the nature of the phase diagram, particularly for the lower values of x . Single crystals of Cdl-,Mn,S in the range of 0 Ix 5 0.50 have also been obtained by chemical transport methods. Pure binary components were mixed in the desired ratio as the starting step. Iodine was used as the transport agent in the amount of about 10 mg/cm3. After one week in the furnace at about 95OoC, small crystals of good quality were obtained, typically in the form of platelets of 5 x 10 x 1 mm in size. The composition of the platelets was usually very close to that of the starting material. Finally, Cdl -,Mn,S crystals have also been grown by the vertical Bridgman method using induction heating and self-sealing graphite crucibles (Debska el al., 1984). By this method crystals of 1 cm3 in size were obtained. While the runs carried out so far were preliminary in nature, it appears that the induction-heated self-sealing approach is highly promising, and warrants further exploration. (For a detailed description of the method, see the discussion of Znl-,Mn,Se, below.)
22
W. GIRIAT A N D J. K . FURDYNA
(d) Znl -,Mn,Te Znl-,MnxTe is reported to form single phase solid solutions over a wider range of Mn concentration than any other member of the DMS family (Pajaczkowska, 1978). Since MnSe and MnS themselves are known to form (albeit with difficulty) stable zinc blende and/or wurtzite phases, while MnTe does not, this wide range of miscibility of Mn in the zinc blende Znl-,Mn,Te is somewhat surprising. The phase diagram of the ZnTe-MnTe system is not exactly known. The melting point of ZnTe is 1295"C, and the melting points of the ternary alloys are expected to be close to that value. As in the earlier examples of DMS growth, the Bridgman method can be used for crystallization of Znl-,Mn,Te, with binary ZnTe and MnTe in appropriate quantities as starting materials. Silica tubes may still be used as ampoules, but-as in the case of Cdl-,Mn,Se-the tubes must be thick-walled because of the high temperatures required. Good quality large crystals have been obtained for low values of x (x < 0. lo), but usually the crystal quality deteriorates as the Mn content increases. For high values of x the ingots consist of relatively small crystals, which are frequently twinned. Typical single crystal size is then about 0.5 cm3, with occasionally larger grains (Furdyna et al., 1983). (e) Znl-,MnxSe
As can be seen from Table 111, MnSe is soluble in ZnSe for x I0.57. In the composition range 0 < x < 0.30 this solid solution crystallizesin the zinc blende structure, and for 0.35 5 x I 0.57 in the wurtzite form. For x > 0.57 mixed phases occur, including various selenides of Mn. Considerable twinning is observed in the zinc blende phase as x approaches the transition value of 0.30. As in most other DMS, pre-reacted binary ZnSe and MnSe are used as starting materials. Taken in desired proportions, they are ground to a powder, mixed, pressed into pellets, and reacted for several days in the temperature range of 1OOO-11OO"C. The resulting sintered compounds are then used as starting material for crystallization. The phase diagram of ZnSe-MnSe is not known, but it is clear that the melting point is slightly over 1500°C throughout the entire range 0 < x < 0.57. Chemical transport is the simplest method to grow single crystals of Znl-,Mn,Se. For this purpose, a tube of inner diameter of about 15 mm and 12to 15 cm long can be used, in which the sintered material is placed, together with iodine (10 mg/cm2) as the transport agent. The tube is then sealed in vacuum and put in a furnace, where the temperature is at 1000°C and 980"C, respectively, at the two ends of the tube. Single crystals can be harvested after one week, size of the crystals being several millimeters in each direction.
1.
CRYSTAL STRUCTURE AND MATERIALS PREPARATION
23
4 6
7 1 C C C
C C
8-
1
5
3 2
FIG.9. Self-sealing graphite crucible used in the growth of Zn, -xMnxSe:(1) graphite crucible; (2) threaded plug; (3) thread; (4) outside silica tube; (5) induction heating coils; (6) powdered material; (7) molten zone: (8) crystallized material; (9) exit to dynamic vacuum. [After Debska et a/. (1984).]
The composition of the crystals is typically quite close to the composition of the original starting sintered materials. The Bridgman method (Twardowski et af., 1983) is made difficult by the high melting point of this system (ca. lSOO"C), where quartz is already too soft to be useful. This can be circumvented by the use of external counterpressures. Normally, argon at about 100 atmospheres is used as the pressurizing gas. Using this method, crystals of Znl-,Mn,Se were grown in the composition range up to x = 0.10. Recently a new and highly promising method of crystal growth of Znl-,Mn,Se was developed (Debska et al., 1984), involving the use of RF induction heating, and utilizing a self-sealing graphite crucible. This method eliminates entirely the need of fused silica ampoules. The crucible assembly consists of a closed graphite tube, into which is fitted a long threaded plug, as shown in Fig. 9. After loading the powdered charge, the graphite crucible
24
W . GIRIAT A N D J. K . FURDYNA
is inverted so that the plug is at the bottom. The crucible is placed at the center of an induction coil, with the bottom half of the plug extending beyond the coil. After the system is evacuated to about Torr, the temperature is slowly raised beyond the melting point of the charge, the graphite acting as the susceptor in the heating process. Some of the material then flows down, and ultimately solidifies in the space between the crucible wall and the plug, thereby sealing the crucible. The center of the crucible is RF-heated to about 100°C above the melting point of the material, and the whole assembly is then lowered out of the stationary induction coil at the rate of about 10 mm/h. The whole process is conducted under dynamic vacuum of 10-4-10-6 Torr. The induction heater in the specific system described by Debska et al. utilized a 20 KW generator operated at about 300 kHz, and produced a hot zone with a sharp thermal gradient. The narrowness of the heater zone is important in view of the fact that graphite has a high thermal conductivity, which can act to “smear out” the temperature gradient. Since the alloy itself has a lower vapor pressure than the constituent elements, it is advisable to pre-synthesize the material at the desired composition prior to the actual crystal growth. A number of good quality Znl-,Mn,Se single crystals have been produced over the entire solid solution range, 0 5 x 5 0.57. Ingots are typically 9 mm in diameter by about 70 to 90 mm, composed of crystals of about 5 to 10 mm in length, with occasional monocrystals as long as 5 to 8 cm. (f) Znl-,Mn,S Znl-xMn,S alloys form single phase zinc blende crystals for x up to = 0.1, and wurtzite crystals for 0.1 < x 5 0.60, as indicated in Table Ill. The phase diagram for this alloy is shown in Fig. 10 (Sombuthawee et al., 1978).
-- -.:--z:--..----
2000@50
2 1600u-
LIQUID
--e--
\
1610
d , ’A /
FIG. 10. Phase diagram for ZnS-MnS. [After Sombuthawee et al. (1978).]
1.
CRYSTAL STRUCTURE AND MATERIALS PREPARATION
25
ZnS has a melting point of 1722"C, the highest of the A"BV' compounds. The melting point for MnS (rocksalt phase) is 1530"C, also the highest among the Mn-chalcogenides . Because of the high melting points involved, chemical transport is commonly preferred as the crystal growth technique of Znl-,Mn,S. To grow single crystals of this alloy, sintered material (prepared from prereacted binaries ZnS and MnS, in a manner similar to that described for Znl-,Mn,Se) is placed in quartz tubes with inner diameter of about 15 mm, 12 to 15 cm long. Iodine(l0 mg/cm3)is used as the transport agent. The tubes are sealed in vacuum and located in the furnace such that Tm,, = 1000°C and Tmin= 980"C, for a period of one week. For samples with a small Mn content (x < 0.15), monocrystals of the size of 0.5 cm3 have been obtained in this manner. For higher values of x the crystals tend to be smaller (several millimeters in each direction). The composition of the crystals is typically quite close to that of the starting sintered material. (g) HgI-,Mn,Te The preparation of this system over a wide range of compositions has been first investigated in the classical work of Delves and Lewis (1963). As with the other tellurides, the Hg, -,Mn,Te system shows that a surprisingly high fraction of Hg atoms (about 80%) can be replaced by Mn, with the zinc blende structure retained. The phase diagram obtained by Delves and Lewis for the HgTe-MnTe system is shown in Fig. 11. From this figure it is clear that the liquidus-solidus separation is considerable, and standard crystallization is therefore not expected to yield highly homogeneous crystals. In spite of the high segregation coefficient, the majority of Hgl-,Mn,Te monocrystals prepared to date have been grown by the vertical Bridgman method, because of its simplicity. Bridgman crystals are typically prepared as follows. Pre-reacted HgTe and MnTe (or HgTe and elemental Mn and Te in proper amounts) are ground and placed into a thick-walled quartz ampoule with an inner diameter of 10 to 15 mm (Kaniewski et al., 1982). For crystals containing more than 5% Mn, the inner wall of the ampoule is coated with carbon. The ampoules, evacuated to approximately Torr and sealed, are placed in the furnace at ca. 450°C. After 24 hours, the temperature is increased to 820°C and is held at this value over a period of 48 hours to achieve complete solution of manganese. The Bridgman growth is then started. The recommended growth speed is slow, typically about 1 mm/h. Higher growth rates result in a high degree of polycrystallinity. As can be expected from the phase diagram, Hgl-,Mn,Te crystals grown by the Bridgman method have a significant composition gradient along the growth direction due to a large segregation coefficient. The segregation
26
c
W. GIRIAT AND J. K . FURDYNA
600
0.0
HgTe
735%
(
Mole Fraction X
Mn ‘e
FIG.1 1 . Phase diagram for HgTe-MnTe. [After Delves and Lewis (1963).]
effects in the growth of Hg,-,Mn,Te are qualitatively similar to those occurring in the well-known Hg, -,Cd,Te ternary alloy. It should therefore be possible to reduce the composition gradient in the former by employing some of the crystal growth techniques used for preparation of high homogeneity Hg,-,Cd,Te, for example, the traveling zone method (Dziuba, 1979), or the rapid quench-solid state recrystallization approach. The success of these latter techniques in preparation of homogeneous Hgl -,Mn,Te still awaits a systematic study. As-grown Hgl-,Mn,Te (like its sister compound Hgl-,Cd,Te) tends to crystallize with a slight Hg-deficiency. The Hg-vacancies then act as acceptors, rendering the as-grown crystals heavily p-type. Since control of electronic properties is particularly important in the Hg-compounds, this “native” nonstoichiometry is an undesirable property. Stoichiometry can be restored by post-growth annealing of Hgl -,Mn,Te in saturated Hg-vapor. Prescriptions for annealing parameters vary from experimentalist to experimentalist, but typically involve annealing between 18O-22O0C for very long times (usually at least 200 hours for every millimeter of sample thickness). (h) Hgl-,Mn,Se The phase diagram and lattice constants of Hgl -,Mn,Se were investigated by Pajaczkowska and Rabenau (1977a). The solubility limit of the solid
1.
CRYSTAL STRUCTURE A N D MATERIALS PREPARATION
I
950-
I-
Heating
/
Mn,Hg,-,Se
21
I
Mn,Hg,-,Se+MnSe
I I
HgSe
Oll
0 012 013 6!4 d.5 016 017 d.8 019 MnSe Mole Fraction X
FIG.12. Phase diagram for HgSe-MnSe. [After Pajaczkowska and Rabenau (1977a).]
phase, determined from lattice parameter measurements, is x 2: 0.38 at 700°C. The phase diagram for the system is shown in Fig. 12. In general, crystallization of Hgl-,Mn,Se is similar to that of the Hgl-,Mn,Te system. Again, it is convenient to use the vertical Bridgman method. Since the working temperature does not exceed 9OO”C, thick-walled silica tubes may be used. After crystallization, several large monocrystalline grains are usually obtained, much as for Hgl-,MnxTe. (i) Hgl-,MnxS We know very little about this system. HgS itself occurs in two phases: the cinnabar (a-HgS) and zinc blende (P-HgS) structures. The cinnabar phase is stable below 300°C. Above this temperature a-HgS transforms to the zinc blende P-HgS phase. Pajaczkowska and Rabenau (1977b) found that at 600°C Hgl-,Mn,S forms single-phase solid solutions in the zinc blende structure for x s 0.37 and, using the hydrothermal process, they succeeded in obtaining single zinc blende crystals of this alloy as large as 2 m m in diameter. This system is particularly interesting, since it is the only one among the DMS for which the “parent” A”BV’ compound has a stable room temperature structure other than zinc blende or wurtzite. Thus the presence of Mn appears to stabilize the tetrahedral coordination in this alloy. IV. The “New” DMS
This chapter has been devoted primarily to the now well-established A:?,Mn,BV’ alloys in bulk crystal form. It is important to remember, however, that other DMS systems are steadily gaining the interest of the scientific community. While these new DMS materials have been studied
28
W. GIRIAT A N D J. K . FURDYNA
much less thoroughly than the bulk A:'-,MnxBv' alloys, progress is rapidly being made, particularly in the layered A:!.,Mn,BV' (e.g., MBE films and superlattices), in bulk A:t,FexBV' alloys, and in the A:!!,MnxBV' systems. In concluding this chapter, we therefore wish to mention these new DMS systems briefly, so as to identify the evolving materials trends and to provide a brief literature guide for these emerging DMS activities. 7. DMS FILMSAND SUPERLATTICES
One of the most important-and most rapidly evolving-recent developments in the area of new DMS systems is the successful preparation of DMS films, heterostructures, and superlattices. The first reported DMS film structures involved the quaternary epitaxial layers of Hgl -x-yCdxMnyTe prepared by the close-spaced isothermal vapor transport growth technique (Debska et af., 1981). The field has taken a major step forward when it was demonstrated that Cdl -,Mn,Te films and Cdl -,Mn,Te/Cdl-,Mn,Te superlattices could be readily grown by molecular beam epitaxy (MBE) on a variety of substrates (Kolodziejski et a/., 1984b; Bicknell et a/., 1984; Kolodziejski et a/., 1984a). Since 1984, the field has made truly remarkable progress, including successful preparation by MBE of films and superlattices of Znl-,Mn,Se (Kolodziejski etal., 1984) and of Hgl-,Mn,Te (Harris et al., 1987; Faurie, 1987). In parallel with these developments, atomic layer epitaxy (ALE)-an equilibrium growth method, in contrast to MBE, which involves nonequilibrium growth-has also been shown as a viable technique for the growth of DMS films (Herman et a/., 1984: Tammenmaa et a/., 1985). It is clear that the ability to grow high quality DMS films, superlattices and heterostructures, coupled with the novel properties of DMS materials, opens a rich spectrum of possibilities in basic and applied science. A survey of these opportunities is beyond the scope of this chapter. It should be mentioned, however, that apart from exciting electronic, magnetic, and optical applications (Datta et a/., 1984), epitaxy also opens the way to the growth of entirely new systems that do not form in the bulk. The recent preparation by MBE of zinc blende Znl-,Mn,Se for x > 0.30 and, indeed, of zinc blende binary MnSe layers in thin film form (Kolodziejski et al., 1986)are examples of such new systems. This aspect alone-i.e., the MBE growth of metastable alloys and compounds-is extremely important in its potential to advance the understanding of the crystal growth of magnetic semiconductor alloys, and of semiconductor compounds in general. 8. A:'_,Fe,Bv'
ALLOYS
While bulk growth and the intrinsic properties of the A:'-,Mn,Bv' DMS are now reasonably well understood, the situation is quite different in the
1.
CRYSTAL STRUCTURE AND MATERIALS PREPARATION
29
case of their Fe-based counterparts, such as Hgl-,Fe,Se or Cdl-,Fe,Se. Literature on these materials has only recently begun to emerge and is already reporting rather unique and striking effects. There are two principal features which make these materials especially interesting, both arising from the presence of substitutional Fe in the A"BV' lattice. First, the magnetism of the Fe++ sub-system is quite different from that of Mn++,owing largely to the fact that the permanent magnetic moment of the Fe++ion vanishes at low temperatures. Specifically, it has been shown that in the dilute limited (x < 0.01) A:'_,FexBv' alloys exhibit Van Vleck paramagnetism (Guldner et ai., 1980; Serre et al., 1982; Lewicki et al., 1986). The second extremely important feature is the influence of the Fe atoms on the electrical properties of Hgl-xFexSe and related alloys. It has been a striking feature of the measurements on Hgl-,Fe,Se that the electron concentration n in this material is surprisingly independent of annealing, of x , and of other factors typically affecting n, and remains fixed at n = 5 x 10l8cm-3 in all samples investigated (Serre el al., 1982; Vaziri and Reifenberger, 1958). In particular, electron concentration in this material is remarkably stable in time (Vaziri et al., 1985), quite unlike the case of HgSe or Hgl-,Mn,Se, which can be strongly affected by shelf-life "passive" annealing and/or thermal cycling. This has recently been explained as follows (Mycielski et al., 1986). The Fe2+level lies at about 230 meV above the bottom of the conduction band of HgSe, with Fe2+ acting as a resonant donor. The position of the Fe2+ thus establishes the Fermi level. In this situation, the electron concentration is determined by the band structure and its corresponding density of states (like in a true seminetal, e.g., Bi) rather than by doping or nonstoichiometric defects. This remains true as long as x (the Fe concentration) exceeds the number of electronic states below the Fe2+ level (5 x 10l8cm-3 corresponds to x 2: 0.0003), and accounts for the stability of n. When the Fe concentration x is below this value, n will increase with x. An interesting approach to probing the role of Fe2+ as a resonant donor is to perform studies on the quaternary alloy Hg,-,-,Fe,Cd,Se, as has been recently reported by Mycielski et al. (1986). While the detailed discussion of the electronic structure is beyond the intended scope of this chapter, we stress that in this context the role of iron is primarily to provide electrons to the conduction band (Furdyna, 1986). Hgl -,FexSe also displays remarkably high mobilities at low temperatures, much higher than would occur in HgSe for comparable levels of electron concentrations, indicating a considerable reduction of ionized impurity scattering. This is, further, accompanied by comparatively low values of the Dingle temperature Td (Vaziri and Reifenberger, 1985). It has been suggested by Mycielski (1986) that the ionization of resonant Fe2+ donors occurs in such a way as to maximize the separation in space between the ionized sites,
30
W . GIRIAT AND J. K. FURDYNA
which in turn leads to (at least partial) space ordering. Such an ordered, periodic array of ionized impurities can then be shown to lead to reduced scattering rates as compared to a random distribution of ionized impurities (Pool et al., 1986). . The preparation of A!'_,Fe,Bv' materials is generally similar to their sister Mn alloys (Mizera et al., 1980). The range of x in which Fe can be incorporated into the A"BV' lattice is, however, considerably less than in the case of Mn alloys. For example, the highest values of x reported so far for various recently investigated alloys are -0.06 for Hgl-,Fe,Te (Guldner et al., 1980), 0.12 for Hgl-,Fe,Se (Vaziri et al., 1985), and 0.15 for Cdl-,Fe,Se (Lewicki et al., 1986). The pattern which thus appears to em&+ is that, generally, the higher the melting point of the A"BV' host, the higher is the amount of Fe which can enter substitutionally at the group-I1 sites. The important problem of establishing the limits of miscibility of Fe in specific A1'BV' lattices, and the determination of the corresponding phase diagrams, still awaits a systematic investigation. 9. MANGANESEAND RARE-EARTH-BASED A1"Bv' ALLOYS
Activity has also been recently increasing in the area of A:?,M,Bv' alloys, where M indicates a substitutional magnetic element (either Mn or a rareearth). Lead salts with a fraction of the Pb-sublattice replaced by Mn are the most thoroughly studied members of this group (Pbl-,Mn,S: Karczewski et al., 1982, 1985; Pbl-,Mn,Se: Kowalczyk and Szczerbakow, 1984; Pbl -,Mn,Te: Niewodniczanska-Zawadzka and Szczerbakow, 1980; Pascher et al., 1983; Anderson and Gorska, 1984). Crystal growth methods employed in preparing these alloys were, for the most part, Bridgman growth for bulk crystals and hot wall epitaxy for single crystal films, with growth parameters similar to those which proved successful in the preparation of the parent lead salt binaries. The miscibility of Mn in these systems appears to be smaller than in the A:'_,Mn,BV' alloys, the highest values being those reported for Pbl-,Mn,Te (x = 0.2). No comprehensive study of this aspect has as yet been described for this class of alloys. In addition to the development of the A!!,Mn,BV' alloys, there is increasing activity in lead salt alloys involving rare earths, for example, Pbl -,Eu,Te and Pbl -,Gd,Te. The methods of successful preparation include Bridgman growth (Golacki and Gorska, 1985), hot wall epitaxy of thin film specimens (Krost etal., 1985), and MBE (Partin, 1983,1984). While most of the non-MBE growth has been restricted to x < 0.10, the MBE films showed very high rare-earth miscibility in the lead salt lattice (e.g., x = 0.40 for Pbl-,Yb,Te (Partin, 1983)). Much of the interest in these materials is stimulated by their photovoltaic properties and their potential for diode laser
1. CRYSTAL STRUCTURE AND MATERIALS PREPARATION
31
applications. By comparison, at this stage we know considerably less about their magnetic properties and, in particular, about the nature of exchange interaction between the localized magnetic ions and band electrons in these materials. 10. MISCELLANEOUS DMS ALLOYS
There are two additional cases of DMS alloys that are potentially highly interesting. (Cdl -,MnX)3As2 resembles, in the absence of an external magnetic field, its parent Cd3As2: it is understood to be a zero-gap semiconductor, and can exhibit relatively high electron mobilities. The introduction of Mn brings into play spin-spin exchange interaction between the band electrons and localized magnetic moments (Neve et al., 1981), and in this respect the system should resemble qualitatively the behavior of doped Hgl -,MnxSe and Hgl-,Mn,Te. Comparatively little research has been done on this and related systems (e.g., Zn3Asz-based DMS), but the feasibility of such alloys has been demonstrated and the initial research (Neve et al., 1981) indicates a highly interesting line of investigation. Finally, we wish to point out the very promising DMS opportunities offered by the chalcopyrite structure. Chalcopyrites are tetrahedrally bonded semiconductor compounds which, in their structure and many physical properties, resemble A"BV1 or A"'BV compounds, as discussed in the excellent monograph by Shay and Wernick (1975). CuInSez and CdGeAsz are examples of such systems. The chalcopyrite crystal structure is shown in Fig. 13, side by side with the zinc blende structure, showing the close relationship of the two crystallographic forms. Considering the predisposition of Mn to act as a group-I1 atom, and the close relationship between the chalcopyrite structure and the simple A"BV' tetrahedrally bonded semiconductors, one is immediately tempted to think of incorporating Mn (and other magnetic atoms) into the chalcopyrite lattice. In particular, structures of the form A:'_,Mn,BIVCx (where Mn occupies substitutionally the group-I1 sites of the original A"B"C; lattice); or A:-,B:'f,Mn2,Cx1 (where Mn can occupy either the group-I or the groupI l l sites of the parent cation lattice) immediately suggest themselves. It should also be borne in mind that Fe is readily incorporated into the chalcopyrite structure. Indeed, the original term "chalcopyrite" refers, specifically, to the compound CuFeS2 , the mineral prototype from which the structure class gets its name. Thus Fe, which can be incorporated to only a limited degree into the A:!-,Fe,Bv' alloys, could play a more important role in the chalcopyrite DMS that may be prepared in the future. There are thus vast new possibilities in this family of materials which-like the "established" A:!-,Mn,BV' alloys-is based on the tetrahedral bond. Initial
32
W . GIRIAT A N D J . K . FURDYNA
Zinc Blende
C hatcopyrite
FIG. 13. Comparison of the chalcopyrite structure (one unit cell, shown on right) with zinc blende structure (two unit cells, left).
successes-and surprises-have already emerged. It is, for example, possible to form glassy DMS systems of Cd~-~Mn,GeAsz(Greene ef af., 1986). Given the similarities-as well as the differences-between the A"BV1 and the chalcopyrite lattice, and given the "natural" way of incorporating Fe into the latter, one can expect truly novel and exciting developments involving DMS chalcopyrite materials in the coming decade.
References Anderson, J. R., and Gorska, M. (1984). Solid State Commun. 51, 115. Balzarotti, A., Czyzyk, M., Kisiel, A . , Motta, N. Podgorny, M., and Zimnal-Starnawska, M. (1984). Phys. Rev. B30, 2295. Bicknell, R. N., Yanka, R. W., Giles-Taylor, N. C., Blanks, D. K., Buckland, E. L . , and Schetzina, J . F. (1984). Appl. Phys. Lett. 45, 92. Brandt, N. B., and Moshchalkov, V. V. (1984). Advances in Physics 33, 193. Cook, W.R. (1968). J. American Ceramic Society 51, 5 1 8 . Datta, S., Furdyna, J. K., and Gunshor, R. L. (1985). SuperlatticesandMicrostructures 1,327. Debska, U., Dietl, M., Grabecki, G., Janik, E., Kierzek-Pecold, E., and Klimkiewicz, M. (1981). Phys. Staf. Solidi (a) 64, 707. Debska, U., Giriat, W., Harrison, H . R., and Yoder-Short, D. R. (1984). J . Cryst. Growth 7 0 , 399.
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Delves, R. T., and Lewis, B. (1963). J. Phys. Chem. Solids 24, 549. Dietl, T., and Spalek, J. (1983). Phys. Rev. B28, 1548. Dolling, T.,Holden, T. M., Sears, V. F., Furdyna, J. K., and Giriat, W. (1982). J. Appl. Phys. 53, 7644. Dziuba, 2. E. (1969). J. Electrochemical Society 116, 104. Faurie, J. P. (1987). Proc. NATO Advanced Research Workshop on Optical Properties of Narrow Gap Low Dimensional Structures, St. Andrews (U.K.), July 29-August 1, 1986, edited by C. M. Sotomayor Torres et al. (Plenum, New York), p. 25. Furdyna, J. K. (1982). J. Appl. Phys. 53, 7637. Furdyna, J. K., Giriat, W., Mitchell, D. F., and Sproule, G. (1983). J. Solid State Chem. 46,349. Gaj, J. A. (1980). J. Phys. SOC.Japan 49, Suppl. A, 797. Gaj, J. A., Galazka, R. R., and Nawrocki, M. (1978). Solid State Commun. 25, 193. Galazka, R. R. (1979). Znst. Phys. Conf. Ser. 43, 133. Galazka, R. R. (1982). Proc. Znt. Conf. on Narrow Gap Semiconductors, Linz, Sept. I981, Lecture Notes in Physics 152 (Springer-Verlag. Berlin), p. 294, and references therein. Galazka, R. R., and Kossut, J. (1980). Lecture Notes in Physics 132 (Springer-Verlag, Berlin), p. 245. Golacki, Z., and Gorska, M. (1985). Acta Phys. Polon. A67, 379. Greene, L. H., Orenstein, J., Wernick, J. H., Hull, G. W., and Berry, E. (1986). Bull. Am. Phys. SOC.31, 383, Abstract EN 15. Guldner, Y., Rigaux, C., Menant, M., Mullin, D. P., and Furdyna, J. K. (1980). Solid State Commun. 33, 133. Harris, K. A., Hwang, S., Burns, R. P., Cook, J. W., Jr., and Schetzina, J. F. (1987). Diluted Magnetic (Semimagnetic) Semiconductors, edited by R. L. Aggarwal, J. K. Furdyna, and S. von Molnar (Vol. 89, Materials Research Society Symposia Proceedings, Pittsburgh, PA), p. 255. Herman, A., Jylha, 0. J., and M. Pessa, M. (1984). J. Crystal Growth 66, 480. Ikeda, M., Itoh, H., and Sato, H. (1968). J. Phys. SOC. Japan 25, 455. Kaniewski, J., Witkowska, B., and Giriat, W. (1982). J. Crystal Growth 60, 179. Karczewski, G., Klimkiewicz, M., Glas, I. Szczerbakow, A., Behrendt, R. (1982). Appl. Phys. A29, 49. Karczewski, G., von Ortenberg, M., Wilamowski, Z., Dobrowolski, W., andNiewodniczanskaZawadzka, J. (1985). Solid-State Commun. 55, 249. Kolodziejski, L. A., Sakamoto, T., Gunshor, R. L., and Datta, S. (1984). Appl. Phys. Lett. 33, 799. Kolodziejski, L. A., Bonsett, T. C., Gunshor, R. L., Datta, S., Bylsma, R. B., Becker, W. M., and Otsuka, N. (1984). Appl. Phys. Lett. 45, 440. Kolodziejski, L. A., Gunshor, R . L., Bonsett, T. C., Venkatasubramanian, R., Datta, S., Bylsma, R. B., Becker, W. M., and Otsuka, N. (1985). Appl. Phys. Lett. 47, 169. Kolodziejski, L. A., Gunshor, R. L., Otsuka, N., Gu, B. P., Hefetz, Y., and Nurmikko, A. V. (1986). Appl. Phys. Lett. 48, 1482. Komura, H., and Kando, Y. (1975). J. Appl. Phys. 46, 5294. Kowalczyk, L., and Szczerbakow, A. (1985). Acta Phys. Polon. A67, 189. Krost, A., Harbeck, B., Faymonville, R., Schlegel, H., Fantner, E. J., Ambrosch, K. E., and Bauer, G. (1985). J. Phys. C. 18, 2119. Larson, B. E., Hass, K. C., Ehrenreich, H., and Carlsson, A. E. (1985). Solid State Commun. 56, 347. Lewicki, A., Mycielski, A., and Spalek, J. (1986). Acta Phys. Polon. A69, 1043. Mizera, E., Klimkiewicz, M., Pajaczkowska, A., and Godwod, K. (1980). Phys. Stat. Solidi(a) 58. 361.
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W. GIRIAT AND J. K. FURDYNA
Mycielski, J. (1986). Solid State Commun. 60, 165. Mycielski, A., and Mycielski, J. (1980). J. Phys. SOC.Japan 49, 809. Mycielski, A., and Dzwonkowski, P., Kowalski, B., Orlowski, B., Dobrowolska, M., Arciszewska, M., Dobrowolski, W., and Baranowski, J. M. (1986). J. Phys. C. 19,3605. Nawrocki, M., Planel, R., Fishman, G . , and Galazka, R. R. (1981). Phys. Rev. Lett. 46,735. Neve, J . J., Bouwens, C . J. R., and Blom, F. A. P. (1981). Solid State Commun. 38, 27. Niewodniczanska-Zawadzka, J., and Szczerbakow, A. (1980). Solid State Commun.34, 887. Oseroff, S. B. (1982). Phys. Rev. 25, 6584. Pajaczkowska, A. (1978). Prog. Crystal Growth Charact. 289. Pajaczkowska, A., and Rabenau, A. (1977a). Mat. Research Bull. 12, 2. Pajaczkowska, A., and Rabenau, A. (1977b). Solid State Chemistry 21, 1. Partin, D. L. (1983). J. Vac. Sci. Technol. B1, 174. Partin, D. L. (1984). J. Electron Mater. 13, 493. Pascher, H., Fantner, E. J., Bauer, G., Zawadzki, W., and von Ortenberg, M. (1983). Solid State Commun. 48, 461. Pool, F., Kossut, J., Debska, U., and Reifenberger, R. (1987). Phys. Rev. B35, 3900. Ramdas, A. K. (1982). J. Appl. Phys. 53, 7649 and references therein. Schnaase, H. (1933). Z.Phys. Chem. B20, 89. Serre, H.,Bastard, G., Rigaux, C . , Mycielski, J., and Furdyna, J. K. (1982). Proc. 4th Int. Conf. on the Physics of Narrow Gap Semiconductors, Linz 1981, Lecture Notes in Physics 152 (Springer, Berlin), p. 321. Shay, J. L., and Wernick, J. H. (1975). Ternary Chalcopyrite Semiconductors: Growth, Electronic Properties, and Applications (Pergamon, Oxford). Sombuthawee, C., Bonsall, S. B., and Hummel, F. A. (1978). J. Solidstate Chemistry25,391. Spalek, J., Lewicki, A., Tarnawski, Z., Furdyna, J. K., Galazka, R. R., and Obuszko, Z. (1986). Phys. Rev. B33, 3407. Tammenmaa, M., Koskinen, T., Hiltunen, L . , Niinisto, L., and Leskela, M. (1985). Thin Solid Films 124, 125. Triboulet, R., and Didier, G. (1981). J. Crystal Growth 52, 614. Twardowski, A., Dietl, T., and Demianiuk, M. (1983). Solid State Comm. 48, 845. Vaziri, M., and Reifenberger, R. (1985). Phys. Rev. B32, 3921. Vaziri, M., Debska, V., and Reifenberger, R. (1985). Appl. Phys. Lett. 47, 407. Wiedemeier, H.,and Khan, A. (1968). Trans. Metall. See. A.I.M.E. 242, 1969. Wu, A. Y., and Sladek, R. J. (1982). J. Appl. Phys. 53, 8589. Yoder-Short, D. R., Debska, U., and Furdyna, J. K . (1985). J. Appl. Phys. 58, 4056.
SEMICONDUCTORS AND SEMIMETALS, VOL. 25
CHAPTER 2
Band Structure and Optical Properties of Wide-Gap A:!-,MnxBv' Alloys at Zero Magnetic Field W. M . Becker PHYSICS DEPARTMENT, PURDUE UNNERSITY WEST LAFAYETTE, INDIANA, USA
I . INTRODUCTION . . . . . . . . . . . . . . . . . . . . 1 . General Discussion. . . . . . . . . . . . . . . . . 11. VALENCEAND CONDUCTION BANDSIN WIDE-GAPDMS ALLOYS. 2. Band Structure at Zone Center. . . . . . . . . . . . (a) Basic Features . . . . . . . . . . . . . . . . . (b) Variation of EBwith Mn Concentration. . . . . . . (c) Variation of E8 with Temperature and Pressure. . . . 3 . Band Structure A way From Zone Center . . . . . . . . 111. OPTICAL PROPERTIES OF d-ELECTRONS. . . . . . . . . . . 4. General Considerations . . . . . . . . . . . . . . . 5 . Mn Transitions in Specific DMS Alloys. . . . . . . . . IV. CONCLUDING REMARKS. . . . . . . . . . . . . . . . . 6 . Summary and Discussion . . . . . . . . . . . . . . REFERENCES.. . . . . . . . . . . . . . . . . . . .
35 35 36 36 36 38 50 56 51
51 59
68 68
I0
I. Introduction 1. GENERAL DISCUSSION
In the A~'_,Mn,B"' alloys, manganese substitutes for the cation in A"BV1 semiconducting compounds, and is in the Mn2'3d' configuration. This conclusion stems from studies on both zero-gap compounds such as HgTe, and on wide-gap materials such as ZnS (McClure, 1963; Langer and Ibuki, 1965) and ZnSe (Langer and Richter, 1966). At very low values of x (typically well below 1 atomic percent), the influence of Mn on the band structure can be disregarded, and Mn interactions with carriers can be neglected. When larger mole fractions of Mn (of the order of a few atomic percent or more) are alloyed with the A"BV' host material, however, the situation is drastically altered. For example, the presence of such Mn concentrations leads to new and interesting electrical and optical behavior due to spin-spin exchange interactions between band electrons and localized moments of the magnetic
35 Copyright B 1988 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-752125-9
36
W . M . BECKER
ions, a subject reviewed in detail elsewhere in this volume. Even in the absence of magnetic field, striking alterations in the properties of DMS occur with alloying as a result of large changes in the band structure, or as a consequence of the introduction of Mn-related transitions in the semiconductor at less than band gap energies, For example, the introduction of Mn at large mole fractions may (1) induce a change from semimetal to semiconductor behavior in initially “zero-gap” or “negative gap” materials or (2) move the intrinsic edge through the visible region in wide-gap DMS. In addition to the changes mentioned above, which are traceable to zone center shifts of band edges, variation of properties due to band structure effects throughout the Brillouin zone may be expected. The importance of Mn in the wide-gap materials is particularly striking, since the localized transitions of the Mn ion tend to dominate the optical properties at high Mn concentrations, a complication which must be considered in the use of DMS alloys for device applications. The systematics of effects in narrow-gap DMS are treated elsewhere in this volume. Here we review the properties of the wide-gap A:’-,Mn,BV’ alloys (A = Cd, Zn; B = Te, Se, S). These latter materials are found to exhibit extremely high resistances at low temperatures, thus inhibiting the study of their properties by galvanomagnetic techniques. In fact, nearly all the present experimental knowledge of the band structure of wide-gap DMS alloys is gleaned from investigation of optical effects, e.g., absorption, reflectivity, photoluminescence, and excitation spectroscopy. Here we review a variety of such experimental results that reveal the behavior of these alloys. In Part 11 of this chapter, the dependence of various band structure features on Mn concentration, on temperature, and on pressure are discussed, and tables are provided summarizing the available data. Part I11 presents results which relate the strong optical features seen at less than band gap energies with various Mn intra-ion transitions. Our concluding remarks are given in Part IV. There we pose a number of problems related to the presence of Mn in DMS that are suggested by the investigations summarized in this review and in related work. 11. Valence and Conduction Bands in Wide-Gap DMS Alloys
2. BANDSTRUCTURE AT ZONE CENTER
a. Basic Features For wide-gap DMS, such as Cdl-xMn,Te, Cdl-,Mn,Se, etc., crystallographic studies indicate that single phase regions-either zinc blende or wurtzite-exist over wide ranges of x [see Giriat and Furdyna, this volume]. All results to date suggest that in these regions of solid solution, the band
2.
31
BAND STRUCTURE AND OPTICAL PROPERTIES
gap is always direct. Therefore, the zone center band structure of these materials should be characteristic either of zinc blende or wurtzite crystals, and experimental data, i.e., optical absorption, excitonic emission, etc., have been interpreted accordingly. To understand these data, we first give a brief review of the two types of band structure to be encountered at k = 0. These can be conveniently introduced in terms of the bonding of the A"BV1 host binaries. (For an extended treatment, see Phillips, 1973.) In the A"BV' compounds, the two valence electrons of the group I1 element and the six electrons of the group VI elements are distributed according to the so-called s-p3 orbital bonding configuration. In this picture, the highest-lying valence band is triply degenerate (in the absence of spin-orbit splitting), and the states at the center of the Brillouin zone are bonding combinations of functions which are plike about the nuclei. The lowest conduction band is s-like. Figure 1 gives the results of a band structure calculation using the pseudopotential method; the structure, here obtained for ZnTe (Cohen and Bergstresser, 1966), may be taken as representative of the cubic (i.e., zinc blende) Cd and Zn chalcogenides. The introduction of the spin-orbit interaction alters the bands, principally by splitting certain degeneracies. At k = 0 in the zinc blende structure, the six-fold degenerate (three-fold not counting the two spin states) T1sv state splits into a four-fold r8(J = %)state and a lower, two-fold r7(J = 3)state, with the splitting indicated in Fig. 2 by A,, , the spin-orbit splitting energy. In the "quasi-cubic" model of the valence and conduction band in wurtzite
a4
- 0 W
-I -2
-3
L
r
k
X
K
r
FIG. 1 . Energy band structure of cubic ZnTe as calculated by the empirical pseudopotential method (Cohen and Bergstresser, 1966).
38
W. M. BECKER
2 INCBLENDE
ZINCBLENDE
Aso = Acr = 0
Ac,= 0
(a)
FIG.2. Zinc blende band structure for (a) Aso = A,,
(b)
=
0 , and (b) Aro # 0 , Acr
1
0.
crystals, the valence band states are considered to be derived from the Tlszinc blende valence band state through the action of two perturbations: crystal field splitting and spin-orbit splitting. The quasi-cubic model may be viewed as a reasonable zeroth-order approximation, since the atoms in the zinc blende and wurtzite structures are in nearly similar positions in the unit cell. The model predicts the splitting of the TSstate by the hexagonal crystal field into a rg and r7 state, each state being two-fold degenerate. Figure 3 gives the details of the resulting band structure around k = 0 and also shows the selection rules for optical transitions near the band edge. b. Variation of Eg with Mn Concentration
A review of the literature reveals distinct trends in the behavior of the fundamental gap in DMS. These are, (1) a general increase in gap energy with Mn mole fraction x at fixed temperature T, and (2) approximate agreement of gap energies when results are extrapolated to the same hypothetical x = 1 zinc blende or wurtzite MnB"' compound, dependent only on the particular chalcogenide in the alloy. This behavior is illustrated in Fig. 4 for the two ternaries involving Cd and Zn and the common anion Te, and, as expected, for intermediate quaternary alloys in the CdxZnyMn,Te (x + y + z = 1) system (Brun Del Re et al., 1983). The linear behavior exhibited in Fig. 4 is, of course, only a first order guideline to what may be expected in the various DMS alloys. Based
2.
BAND STRUCTURE AND OPTICAL PROPERTIES
39
WURTZ ITE
FIG. 3 . Band structure and selection rules at k = 0 for wurtzite structure.
on results for many semiconductor ternary alloy systems, a nonlinear dependence of the band gap, namely a downward "bowing" in E, vs x (Thompson and Woolley, 1967), would be anticipated with refined measurements on crystals of good quality. Such bowing is thought to arise from the nonlinear dependence of the band gap on the crystal potential, and on short range fluctuations of the potential in a random substitutional alloy (Van Vechten, 1970). Surprisingly, such nonlinearities, a characteristic of the complete alloy range, are noticeably absent in the wide-gap DMS. Instead, striking downward bowing near x = 0 is seen in several DMS alloys, an unexpected consequence of magnetic fluctuations (Bylsma et al., 1986). Recent results also indicate a further complication in establishing the functional relationship of Eg vs x close to the boundary region between zinc blende and wurtzite phases due to the presence of polytypism. In such cases, Eg may depend on both x and the particular polytype (Brafman and Steinberger, 1966). These refinements, while sometimes clearly observable, are nevertheless small. Experimental determinations of the energy gap rely strongly (1) on the observation of sharp features at the band edge, i.e., excitonic absorption or emission, or (2) on the analysis of the energy dependence of the absorption edge in terms of specific models. For (l), the identity of the transition is obscured at high x due to alloy broadening. In (2), because of extremely strong Mn-related absorption bands, absorption edge data must be obtained on very thin samples and are not often reliable for highly quantitative analysis. For these reasons, the specific transition marking the band gap, e.g., free exciton, bound exciton, etc., although known in the A"BV' host
40
W. M. BECKER
3.0
I
I
I
I
I
I
I
I
I
FIG.4. Variation of the energy gap in Cd,Zn,Mn,Te (x + y + z = 1) with manganese concentration z for fixed x : y ratios. Experimental values: 0 x = 0,. x = 3y, 0 3x = y , M y = 0, A x = y . Line fits ___ are discussed by the authors (Brun Del Re ef al., 1983).
compound, may often not be identified in the DMS. In the wurtzite systems the transition mechanisms may also be obscure, but the dichroism associated with the hexagonal symmetry of these crystals does provide a useful caliper of valence band splittings in the alloys. Although qualitative similarities in behavior are seen in all DMS alloys, it is convenient to present the results alloy-by-alloy, a strategy which will be carried forward through the remainder of this chapter. Table I and Table I1 summarize some representative data for the Cdl-,Mn,BV' and Znl-xMnxBV' alloys (B = Te, Se, or S), respectively. Comments are given below on these and related results in the literature. It should be noted that the column in Tables I and I1 designated as Eg represents the fundamental gap based on strong experimental features, but the specific electronic transition associated with the features may be obscure for x # 0. In the wurtzite region, the (r9, - r7c)and (r7"- r7c) transition energies (see Fig. 3) are designated in the tables by the superscripts I and 11, respectively, where they have been specifically reported in the literature. (a) Cdl-,Mn,Te We begin with Cdl-,Mn,Te, the most extensively studied and probably best understood of the wide-gap DMS. Absorption edge measurements by
TABLE I OF BANDGAPENERGY VERSUS MOLEFRACTION x IN Cdl-xMnxBv' ALLOYS SUMMARY
Eg in eV
Alloy
1.585 + 1.595 + 1.50 + 1.586 + 1.53 + 1.528 + 1.586 + 1.595 +
Cdl-,Mn,l'x
Cdi -,Mn,Se
1.50~
1.592~
+ 1.08~ + 1.23~ Ell 1.725 + 1 . 2 3 ~ E l 1.800 + 1.33x Ell 1.829 + 1 . 3 3 ~ 1.807 +'1.318x 1.821 + 1 . 5 4 ~ EL 1.74 + 1 . 1 6 ~ Ell 1.756 + 1 . 1 3 ~ E L 1.807 + 1 . 3 4 ~ Ell 1.833 + 1 . 3 1 ~ E L 1.818 + 1 . 4 2 ~ 1.70
E L 1.705
Ell
Cdl -,Mn,S
1.51~ 1.587~ 1.44~ 1.393~ 1.26~ 1.316~
1.842
+ 1.40~
Range of x
77 LHe 300 76 300 300 80 10
0 < x < 0.5 0 < x < 0.2 0 < x 5 0.6 0 < x .c 0.7
R. E.M. E.R. P. E.L. P.R.
Gaj ef al. (1978) Twardowski et a/. (1979) Bottka et al. (1981) Vecchi et a/. (1981) Lautenschlager et a/. (1985) Lee and Ramdas (1984) Lee and Ramdas (1984) Lee and Ramdas (1984)
300 297 297 86 86 LN2 2 300 300 80 80 10 10
0
< x c 0.5 0 < x < 0.3 0 < x < 0.3
D.R. & O.T. O.A.
P. R. & O.A. P.R.
Wiedemeier and Sigai (1970) Antoszewski and Kierzek-Pecold (1980) Antoszewski and Kierzek-Pecold (1980) Antoszewski and Kierzek-Pecold (1980) Antoszewski and Kierzek-Pecold (1980) Giriat and Stankiewicz (1980) Wisniewski and Nawrocki (1983) Lee et al. (1986a) Lee ef al. (1986a) Lee et al. (1986a) Lee et al. (1986a) Lee etal. (1986a) Lee et a/. (1986a)
O.A.
Ikeda etal. (1968)
2.45 298 Minimum at x 0.03 dE,/dX 0.9 x 2 0.1
-
Measurement Methods'
T(K)
0 < x < 0.7 0 < x 5 0.7 0 < x I 0.7 0 < x I 0.7
0 < x < 0.3 0 < x < 0.3 0 < x < 0.5 0 I x I0.277 0 < x < 0.5 0 < x < 0.5 0 < x < 0.5 0 < x < 0.5 0 < x < 0.5 0 < x < 0.5
x=o
Reference
-
t Measurement Methods: R. = Reflectivity; E.M. = Excitonic Magnetoabsorption; E.R. = Electroreflectance; P. = Photoluminescence; E.L. = Ellipsometry; P.R. = Piezomodulated Reflectivity; D.R. = Diffuse Reflection; O.T. = Optical Transmission; O.A. = Optical Absorption; M. = Magnetoreflection; M.R. = Modulated Reflectivity.
TABLE I1 SUMMARY OF BANDGAP ENERGYVERSUS MOLEFRACTION x in Znl-xMnxBV' ALLOY.
Eg in eV
Alloy Znl -xMnxTe
2.377 2.381
+ 3.656~ + 0.68~
Minimum for x < 0.01
2.271 2.365 2.376
Zn,-,Mn,Se
+ 0.518~
+ 0.721~
+ 0.820~
2.80
Minimum at x 2.80
- 0.01
M.inimum for 0 < x < 0.2 dE/dx = 0.58
Znl -,Mn,S
3.8 - 3.9
TW) 1.6 2.2 300 80 10
Range o f x 0.01 -c x
< 0.15
x > 0.02
0 5 x I 0.6 0 5 x < 0.6 0 5 x < 0.6
Measurement Methods'
Reference
M. E.M.
Twardowski (1982) Twardowski e? al. (1984)
P.R.
Lee et al. (1986a) Lee et al. (1986a) Lee e f al. (1986a)
2.2
x=o
M.
Twardowski et al. (1983b)
6.5
x=o
R. & P.
Bylsma et al. (1986)
x > 0.3 x=o
M.R.
2.
43
BAND STRUCTURE AND OPTICAL PROPERTIES
Nguyen The Khoi and Gaj (1977) provided one of the first indications that the energy gap increases with x in this material. These authors also showed that the pinning of the edge occurring near 2 eV at high values of x was directly attributable to the onset of Mn2+ intra-ion transitions. This same pinning effect was also seen by Sundersheshu (1980) using photoconductivity were published by Gaj et techniques. The first quantitative results on Eg(x) al. (1978) using reflectivity data. Linear increases in Egus x were recorded by Twardowski et al. (1979) using magnetoabsorption measurements, but in this case, the data was confined to low x values, 0 < x < 0.2. Modulated electroreflectance results by Bottka et af. (1981) provided a survey of Eg(x, T) over nearly the complete single phase region of Cdl-,Mn,Te, and confirmed the linear increase of Eg with x seen by previous investigators. Photoluminescence results by Vecchi et af. (1981) demonstrated that crystal quality was sufficient to give band edge emissions for all x. Finally, Lee and Ramdas (1984), using piezo-modulated reflectivity, were able to separate band edge features from spectra due to the presence of Mn2+ intra-ion transitions (to be discussed in Part 111). Figure 5 gives their results for various Mn concentrations and temperatures. In the figure, the peak labeled A is identified as the free exciton. No indications of band gap bowing are seen in the entire range of solid solution. In particular, there is an absence of the band gap bowing noted in several other DMS at low x (see below). A summary of measured variations of Egwith x is given in Table I. Based on the many experimental results available on Cdl-,Mn,Te, reasonable estimates (to within k0.02 eV) can be given for Egat x = 1 (Le., for zinc blende MnTe), and these are listed in Table 111 for various temperatures. TABLE I11
ESTIMATES OF ENERGY GAP OF HYPOTHETICALX = 1 MnB”‘ ZINC BLENDE (2.b.) OR WURTZITE(w.) SEMICONDUCTORS
E* (x = 1)
Alloy
Cdl-,Mn,Te
Temperature
in eV
2.86 3.05 3.18
(2.b.)
Cd1-,MnrSe ( w . )
I
2.92 3.15 3.24
Znl-,Mn,Te (2.b.) Znl-,MnxSe (2.b.)
II
2.87 3.13 3.30 2.79 3.09 3.19 -3.5 (2.b)
R.T. LNz LHe R.T. LN2 LHe R.T. LN2 LHe
44
W . M. BECKER
G-4
28
C.-.-
t-
1.4 i
0.0
1
Cd ,+MnxTe
300K 80 K
I 0.2
I
I
I
0.4
I
0.6
MANGANESE MOLE FRACTION ( X ) FIG.5 . Variation of the energy gap and Mn transition with Mn concentration in Cdl-,Mn,Te for 10 K, 80 K, and 300 K (Lee and Ramdas, 1984). In the figure, the peak labeled A is identified as the free exciton. The concentration-independent feature Mn2+ is associated with the leading edge of the Mn 6A1(6S)+ 4T,(4G) absorption band.
(b) Cdl-,Mn,Se
The results of powder and single crystal measurements (diffuse reflectance and optical transmission, respectively) by Wiedemeyer and Sigai (1970) gave the first indications of a linear increase in Eg with x in Cdl-,Mn,Se. Optical absorption work by Antoszewski and Kierzek-Pecold (1980) on single crystal samples over a limited range of Mn concentration showed the dichroic effects expected for the wurtzite structure. Stankiewicz (1983) employed modulated
2.
BAND STRUCTURE AND OPTICAL PROPERTIES
45
2
2.6-
C d , - x MnxSe 10 K
2
Y
m
2.4-
0
z a a
u)
w U
c'a
2.2-
w
LL LL
0
>
(3
Q
5w
2.0-
FIG.6. Variation of A and B exciton energies with Mn concentration in Cdl-,Mn,Se at 10 K, 80K,and300K(Leeetal., 1986a). Notshownisanadditionalpointat2.34eVseeninan.r 3 0.5 sample at 10 K, that is due to a MnZ+transition.
electroreflectance to survey Eg(x,T ) for the complete alloy range of Cdl-,Mn,Se. Giriat and Stankiewicz (1980) demonstrated a linear dependence of Eg on x from photoluminescence results in this material. Lee el a/. (1986a) used piezomodulated absorption (see Fig. 6) to separate band edge and Mn-related transitions. Their technique allowed measurement of the (Fv-),?I energy separation, and also gave Eg(x,2") results. Table IV separately lists value of (r7v - Tsv) and As,, including earlier data from measurements confined to the host A"BV' compound alone.
46
W. M. BECKER
Antoszewski and Kierzek-Pecold’s (1980) optical absorption results indicated that the ( r 7 v - r9,) gap is independent of x. In contrast, work by Stankiewicz (1983) and by Morales ei al. (1985) suggested that this gap decreases with x. Finally, Stankiewicz (1983) found that Aso also decreases withx. Values of Eg extrapolated to x = 1 are listed in Table I11 for both polarization directions of the incident light relative to the c-axis. These estimates are probably correct to not more than k0.05 eV, since in the case of Cdl-,Mn,Se, only the range 0 Ix 5 0.5 is accessible for measurement. TABLE IV SUMMARY OF TRANSITION ENERGIES ASSOCIATED WITH THE VALENCEBANDIN VARIOUSDMS ALLOYS WITH WURTZITE STRUCTURE ~
~~~~~~
~
r7v- rgv Alloy
AS0
in eV
in eV
2 6 lo-’ ~ 20 x 10-3
0.436
Cd, -,Mn,Se
(29 t 5 ) x 10-3 (27 k 3) x lo-’ 14 x
x=O O<x 0.1. According to these authors, alloying effects are ruled out as the mechanism for bowing near x = 0. Recent work by Bylsma et al. (1986) on Znl-,Mn,Se at low values of x (see Fig. 8), and by Diouri et al. (1985) on Cdl-,Mn,Te suggest that the bowing in Cdl-xMn,S may be related to s-d and p-d exchange interactions with the Mn magnetic moments. Crystal field and spin-orbit splitting energies are known for CdS, but still remain to be investigated in Cdl-,Mn,S ternary alloys.
2.
41
BAND STRUCTURE A N D OPTICAL PROPERTIES I
I
-
I
I
I
-
2.9
-
-
T = 114’K
-
-_
2 -
-
rn
W
2.3
I
I
I
I
1
K and 298 K
FIG. (Ikeda et al., 1968).
T-6.5K
- PL energy 3,10-
->
4
-Other PL
3.001
0
f”
Y
z W
.* ’ ,
* *
2.90
2.60/
*
I
2.50 0.00 0.10
I
0.20
I
I
I
I
0.30
040
0.50
0.60
O
CONCENTRATION, X FIG. 8. Band gap energy versus Mn mole fraction x in Znl-xMnxSe, taken from photoluminescence and reflectivity maxima at 6.5 K (Bylsma et al., 1986).
48
W . M. BECKER
2.8 Zn I
-
-
0
Mnx Te
300K BOK
+
I
I
I
0.2
I
"
0.4
I
0.6
MANGANSE MOLE FRACTION ( X I
FIG.9. Variation of the energy gap and Mn transition with Mn concentration in Znl-,Mn,Te for 10 K, 80 K, and 300 K (Lee et a/., 1986a). In the figure, the peak labeled A is identified as the free exciton. The concentration-independent feature, labeled Mn2+,is associated with the leading edge of the Mn 6A~(6S) 4T~(4G)absorption band. -t
(d) Znl -,Mn,Te Deviations from linear behavior and possible bowing effects have been seen only in the region x < 0.1 in Znl-,Mn,Te (Twardowski et al., 1984). Linear behavior at higher x is well established in this material from a variety of experimental results (see Table 11), an example of which is given in Fig. 9. (e) Zn, -,Mn,Se Twardowski et al. (1984) found bowing effects in Znl-,Mn,Se near x = 0 from reflectivity data. Bylsma et al., (1986) used photoluminescence and reflectivity measurements in samples grown by the Bridgman technique to establish a minimum in Eg vs x for 0 c x c 0.2, and a linear increase with
2.
49
BAND STRUCTURE AND OPTICAL PROPERTIES
x for x 2 0.3 (see Fig. 8). Apparent scatter in the data for 0.2 Ix I0.3 was traced to the presence of various polytypes found in the vicinity of the zinc blende-wurtzite structural phase transition (x = 0.3). Recently, metastable zinc blende Znl-,Mn,Se epilayers with (100) orientation, free of polytypes, have been grown on GaAs substrates over the complete alloy range by molecular beam epitaxy (Kolodziejski et al., 1986). In these samples (see Fig. lo), photoluminescence results give both a shallow minimum in Eg near x = 0, and a smooth transition to a linear dependence on x at higher Mn concentrations. Extrapolation of Eg to x = 1 gives the same value for both the bulk crystals and the epilayers.
2.701 , 0
,
.
,
,
0.1 0.2 0.3 0.4 0.5 0.8
X
.
,
1
0 . 7 0.8 0.9 1.0
M n PracLion
FIG. 10. Energy of dominant near-band edge features versus Mn mole fraction for MBEgrown zinc blende Znl-,Mn,Se epilayers (Kolodziejski e t a / . , 1986a). (Cross-hatched points are data obtained on bulk crystals by Twardowski el a/., 1983b).
50
W. M. BECKER
(f) Znl-,Mn,S Crystal growth studies show that Znl-xMnxS is zinc blende for 0 5 x 5 0.1 and wurtzite for 0.1 Ix 5 0.43 (Pajaczkowska, 1978). ZnS itself exhibits both the zinc blende and wurtzite modifications, as well as “rotation twinning” and polytypism (Steinberger, 1983), so that sorting out small band structure changes associated with these crystallographic anomalies as a function of x may be quite complicated. As yet, such changes remain unexplored. c. Variation of Eg with Temperature and Pressure
The study of Egin DMS as a function of temperature and pressure has been stimulated by (1) the need to distinguish between band-to-band transitions and Mn2+ intra-ion (d d) or Mn2+-to-band transitions, and (2) the possibility of observing anomalous thermal shifts which may be related to the onset of magnetic phase transitions (Sundersheshu and Kendelewicz, 1982). For (I), the problem arises because of the strength of the Mn-related absorption features near the energy of the fundamental gap. Pronounced differences in the temperature coefficients of the respective transitions then help to identify them separately. With regard to (2), in the usual semiconductor, the temperature variations of the band gaps are well described by an equation given by Varshni (1967), +
aT2 Eg(T) = Eg(0) - _ _ T + 6’ where Eg(T)and Eg(0)are the band gap energies at T and 0 K, respectively, and a and b are constants of the material. This equation predicts (1) that dEg/dT 0 as T 0 K, and (2) that JdE,/dTI approaches a constant value at high temperatures. Anomalous behavior would then consist of departures from this functional dependence on T near the magnetic ordering temperature. Listed in Table V is a summary of results obtained on several DMS alloys. In each case, Egdecreases approximately linearly with Tabove liquid nitrogen temperature, and results are presented in Table V in terms of an average value of the coefficient dE,/dT. In a number of recent observations, however, departures from the Varshni-like behavior (namely, an extra shift to higher energies) below the characteristic spin freezing temperatures have been reported in DMS. The same effect had been seen earlier in magnetic semiconductors such as MnS (Chou and Fan, 1974) and MnTe (Kendelewicz, 1980a). Temperature anomalies of this type are also cited in Table V, where seen in various DMS alloys. +
+
2.
51
BAND STRUCTURE AND OPTICAL PROPERTIES
TABLE V AVERAGED TEMPERATURE COEFFICIENTS dEg/dTv s x ~ TEMPERATURE o~ RANGE77 - dEg/dT
Alloy
in
Cdl-rMn,Te
3.7 3.7
Cd,-,Mn,Se
eV/K
-
8.9 6.6
Range of x 0 0
-
Measurement Methods’ M.E. O.A.
0.6 0.4
4.6 - 7.5 6.8 9.1 3.5 8.5 3.3 - 10
0 - 0.4 0.4* 0.73* 0.1 0.7* 0 - 0.7
4.5
0
-
x
< 0.1
O.A. O.A. O.A. P.R. P.R. O.A. & R.
Reference Bottka et al. (1981) Sundersheshu and Kendelewicz (1982) Abreu et al. (1981) Diouri et a/. (1985) Diouri et al. (1985) Lee and Ramdas (1984) Lee and Ramdas (1984) Bucker et al. (1985)
3.9 3.9 - 7.3 7.3 5.3 - 8.6 6.8 - 7.9
0.28 x > 0.28 0 - 0.35 0.3 - 0.45’
M.E. M.E. M.E. O.A. O.A.
Antoszewski and Kierzek-Pecold (1980) Stankiewicz (1983) Stankiewicz (1983) Stankiewicz (1983) Abreu et al. (1983) Morales et al. (1985)
Cd,-,Mn,S
4.4 - 9.4
0 - 0.4
O.A.
Ikeda e t a / . (1968)
Znl-xMn,Te
4.14 - 4.61 4.3 - 9.8 6 - 9.4
0 - 0.071 0 - 0.6 0.1 - 0.6*
O.A. P.R. O.A.
Twardowski et al. (1984) Lee et al. (1986a) Morales et a/. (1984)
Znl-,MnxSe
4.7 8.5 9.7
0.066 0.36* 0.55*
R. &. P. R. & P.
-
5.5
0.1
R.
< T < 300 K
0.3
-
R. &. P
Bylsma et al. (1986) Bylsma et al. (1986) Bylsma et al. (1986)
~
*Anomalous extra shift to higher energies (“blue shift”) observed below 77 K. +Measurement Methods: M.E. = Modulated Electroreflectance; O.A. = Optical Absorption; P.R. = Piezomodulated Reflectivity; R. = Reflectivity; P. = Photoluminescence.
Pressure-induced changes in the optical properties of DMS materials have also been pursued, (1) in order to establish various features of the band structure, and (2) again, in order to separate band structure effects from those due to localized Mn-related transitions. Results related to the variation of Eg with pressure are listed in Table VI. For measurements at low pressure, linear pressure coefficients suffice to describe the results. Fitting of the data for much higher pressures, that may involve phase transitions (rendering the samples opaque), would require inclusion of higher order terms in P. In such
52
W. M. BECKER
cases, an expression usually used is
E,(P) = Eg(0)+ yP
+ 6Pz,
(2)
where Eg(0)is the band gap at P = 0. Values of the coefficients y and 6 are given in Table VI for a number of DMS. We now examine the temperature and pressure dependence of Eg for specific DMS alloys. (g) Cdl -,Mn,Te The temperature dependence of the band gap of Cdl-,Mn,Te was first investigated systematically by Bottka et al. (1981). These researchers found that for T > 100 K, (dEg/dTIis constant for x < 0.15,but increases linearly with x thereafter. The authors suggested that the sharp change in slope at x = 0.15 is related to lattice mode softening at the percolation threshold of the Mn spin system. Sundersheshu and Kendelewicz (1982) carefully examined Eg vs T for anomalous optical behavior expected to accompany a magnetically induced shift in the band gap. While such effects were not detected, they found instead a sharp drop in the values of b [see Eq. (I)] above x = 0.3, and took it as indicative of a magnetically induced widening of the gap. Later, Bucker et al. (1985)used reflectivity to survey dEg/dTfor all x. In contrast to the electroreflectance results of Bottka et al. (1981),they found only a uniform increase in the magnitude of the temperature coefficient with x. Lee and Ramdas (1984)noted that (dEg/dT(for 10 4 T I80 K is greater than IdEg/dT)for 80 IT 5 300 K for an x = 0.7 sample, a first indication of the sought-for anomalous effect presumably associated with the magnetic phase transition in this material. Stronger evidence for this behavior was obtained by Diouri et al. (1985)from optical absorption measurements (see Fig. 11). Working with very thin layers ground and polished from the bulk, these investigators found shifts of 45 meV and 17 meV to higher energies in x = 0.7 and x = 0.4 samples, respectively, as compared to Eg(0) values predicted from higher temperature results using Eq. (1). Turning now to pressure effects, Ambrazevicius et al. (1984)were able to observe the pressure variation of Eg by the use of samples thin enough to allow measurements at the level of absorption coefficients greater than lo3cm-'. With these precautions, positive values of the pressure coefficient dEg/dPwere recorded for all x . At lower levels of the absorption coefficient (smaller than lo2cm-I), the sign of the coefficient becomes negative between x = 0.4 a n d x = 0.5.This latter result is interpreted in terms of intra-ion d-d transitions of Mn2+. Similar sign changes were observed by Wei Shan et al. (1985)when comparing behaviors of x = 0.1 and x = 0.5 samples at low levels of absorption.
TABLE VI OF EBAT ROOMTEMPERATURE FROM ABSORPTION EDGEMEASUREMENTS PRESSURE DEPENDENCE
Alloy Cdt-,Mn,Te
dE,/dP eV/MPa 8 1.9 1.3 5.9 4.6
Ea(0)' in eV
y' in
eV/MPa
1.483 1.618 2.161 2.115
8.3 7.1 - 4.9* -5.1'
Znl-,Mn,Te
2.21 2.265 2.28 2.37
10.4 10 10.6 8.2
Znl-,Mn,Se
2.688 2.648 2.615 2.756
7.12 6.1 6.30 6.34
dt in eV/GPaZ
- 4.0
- 3.9
- 28 - 26 - 54 - 80 - 15 - 11 - 18 - 27
Fitting parameter in Eq. (2). *Negative values of y are indicative of absorption effects involving Mn2' 3d5 levels.
X
Reference
0
0.4 0.52 0.6 0.7 0 0.1 0.5 0.615
Babonas ef al. (1971) Ambrazevicius et al. (1984) Arnbrazevicius et al. (1984) Ambrazevicius et al. (1984) Arnbrazevicius et al. (1984) Wei Shan et a/.(1985) Wei Shan et a/. (1985) Wei Shan el al. (1985) Wei Shan et al. (1985)
0
Ves et al. (1986a) Ves el a/.(1986a) Ves e f al. (1986a) Ves et al. (1986a)
0
Ves et al. Ves et al. Ves et al. Ves ef at.
0.01 0.1 0.3 0.1 0.15 0.25
(1985) (1986b) (1986b) (1986b)
54
W.
1
I
I
M. BECKER
1
I
I
I
FIG. 11. Absorption edge as a function of temperature for (a) Cdo.Z,Mno.73Te, and (b) Cdo.sMno.4Te (Diouri et af., 1985). The dotted curves show the extrapolation of Eg obtained from high temperatures according to Eq. (1).
2.
BAND STRUCTURE AND OPTICAL PROPERTIES
55
(h) Cdl -,MnxSe As in Cdl-,Mn,Te,
the temperature coefficient- of the energy gap,
- dEg/dT, of Cdl-,MnxSe increases with increasing x . In Stankiewicz’s
(1983) survey of this phenomenon over the concentration range 0 5 x S 0.4, the coefficient changes rapidly between x = 0.1 and x = 0.3, but is otherwise constant at the ends of the interval. An interesting bowing of Eg(T)occurs in an x = 0.42 sample at low temperatures and is possibly related to an interference effect between band-to-band transitions and excitations of Mn” states. Changes in - dEg/dTwith x were seen by Abreu et al. (1983) using isoabsorption measurements at low values of absorption coefficient. An abrupt drop in the magnitude of the coefficient athigh x reported by these authors is traceable to the onset of Mn2+ d-d transitions, now known (Morales et al., 1985) to have a weaker temperature dependence than bandto-band transitions. Morales et ul. (1985) noted that IdEg/dTI for for 77 < T < 300 K for an x = 0.5 10 < T < 77 K is greater than 1-1 sample, an indication of an anomaly similar to that observed in Cdl -,Mn,Te, as discussed earlier. (i) Cdl-,MnxS Optical absorption measurements carried out on Cdl-,Mn,S by Ikeda et al. (1968) showed that absorption coefficients as large as lo3cm-’ must be achieved in order to measure intrinsic behavior. Results were obtained only at 114 K and 298 K; linear behavior is presumed between these temperatures in listing the results given in Table V for this material. (j) Znl -,MnxTe
Twardowski et al. (1984) found little variation in -dE,/dT for x < 0.1. The piezomodulated reflectivity results of Lee et al. (1986a), obtained from data taken at 80 K and 300 K, show that IdEg/dTI increases by a factor of two between x = 0 and x 5 0.5. Similar to the observations on Cd,-,Mn,Te and Cdl-,MnxSe, Morales Tor0 et ul. (1984) showed that in Zn1-,MnxTe 1-1 for 10 < T < 77 K is greater than 1-1 for 77 < T < 300K at high x, with reverse behavior seen at low x . Ves et al. (1986a) carried out optical absorption measurements on Znl-,Mn,Te under hydrostatic pressure in a diamond anvil cell. As expected, Eg increases with increasing pressure, but strong nonlinearities were observed. Results were limited to low-x material (x 5 0.3) because of the deterioration of the absorption feature associated with Eg at high x and P i n Znl-,Mn,Te, and also because in this material the pressure at which the phase transition occurs decreases with increasing x.
56
W. M. BECKER
(k) Znl-,Mn,Se Bylsma et al. (1986) found a monotonic increase in - dEg/dTwith x for Znl-,Mn,Se samples in the range 0 < x < 0.55. A deviation from Varshnilike behavior, namely, an extra shift of E,(T) to higher energies at low temperatures and large Mn concentrations, was also seen to occur in this material. Ves et al. (1986b) carried out optical absorption measurements at high pressure on samples with x values lying in the zinc blende region (0 < x < 0.3). As in Znl-,Mn,Te, Eg was found to increase with pressure. Pronounced nonlinearities were observed, but values of 6, the quadratic pressure coefficient [see Eq. (2)], were smaller than in Znl-,Mn,Te. (1) Znl-,Mn,S At the time of this writing, no published information on the temperature and pressure dependence of Eg in Znl-,Mn,S is available.
3. BANDSTRUCTURE AWAYFROM ZONECENTER Reflectivity measurements for energies above the fundamental edge have proved indispensable in probing the electronic structures of semiconductors deep into the bands or away from k = 0. In DMS materials, the technique has been applied to several alloys over the available solid solution regions. In this work, there was the expectation of being able to detect sharp spectral features at high x. This was based on earlier findings that these features persist over large alloy ranges in other A”BV’ mixed crystals, such as Hgl -,CdxTe and Znl -,Cd,Te. First indications of reflectivity features in DMS associated with transitions away from k = 0 were seen by Kendelewicz and Kierzek-Pecold (1978) in their work on the narrow-gap alloy Hgl-,Mn,Te. In contrast to the behavior in non-DMS A”BV’ ternary alloys, however, a blurring of the reflectivity spectral structures occurred with growing Mn concentration. This effect was also noted in studies on the wide-gap DMS Cdl-,Mn,Te (Kendelewicz, 1980b; Kendelewicz, 1981; Zimnal-Starnawska et al., 1984; Bucker et al., 1984) and on Znl-,Mn,S and Znl-,Mn,Se (Zimnal-Starnawska et al., 1984). Except for the persistence of the fundamental edge reflectivity feature, other details, such as the El and El + A1 transitions, disappear for x z 0.3. The rate of deterioration of the reflectivity spectra with increasing Mn content is highest for Znl-,Mn,S and lowest for Cdl-,Mn,Te, according to ZimnalStarnawaka et al. (1984). These authors explain the blurring of the spectra in terms of localized Mn2+3ds “spin down” states lying at higher energies (spin splitting of = 6 eV; see also Podgorny and Oleszkiewicz, 1983).
2.
BAND STRUCTURE AND OPTICAL PROPERTIES
57
111. Optical Properties of d-Electrons
4. GENERAL CONSIDERATIONS
The introduction of Mn not only changes the band structure of the host A"BV' semiconductor, but it also introduces new optical transitions. These include prominent absorption and emission features, which can be correlated with the expected splittings of Mn multiplets in a crystalline field. Below is given a brief summary of the origin of the multiplets and their splittings, leading to the hierarchy of states usually observed in wide-gap A:t,Mn,BV' alloys. The reader is referred to standard references (e.g., Tanabe and Sugano, 1970; Abragam and Bleaney, 1970; Griffith, 1971; Ballhausen, 1961; McClure, 1959) for more complete details. Significant understanding of this aspect of DMS can be gained by considering an isolated Mn ion in an A"BV' lattice. A freeMn atom has the 3d54s2 ground state configuration for the outermost shells. When the atom is put into a DMS, it substitutes for the cation. The two 4s electrons form the bonds with the surrounding ions, and the five 3d electrons interact strongly with the electrical potential of the nearest neighbor anions. The total Hamiltonian is H = Ho + V + H,,, where Ho is the many-electron Hamiltonian of a free ion without spin-orbit interaction, Vis the contribution of the crystal field, and H,, is the spin-orbit interaction. Since the magnitude of V satisfies the inequalities H,, < V < H,, V can be treated as a perturbation. We now review very briefly the spectroscopic notation used in describing the ground and excited states of both the free ion, and the ion when in a ligand field of an appropriate symmetry. For the free ion, the state is given by ""L, where S is the total spin, and L is the total orbital angular momentum. In the ground state of a free Mn2+ ion, the electron spins are aligned due to Hund's rule, giving S = f; this state is designated as 6S ( L = 0). When the Mn2+ion is put into a crystal field that is not strong enough to break Hund's rule, the 'S state remains as the ground state. Since it is spherically symmetric, it transforms as A1 (in group theoretical notation), and is therefore unsplit by the crystal field. In the free ion state, spin reversal of a single 3d electron gives rise to four quartet states corresponding to S = 5; 4G ( L = 4), 4P ( L = l), 4D ( L = 2), and 4F ( L = 3). The 4Gstate has nine-fold orbital degeneracy (2L + 1 = 9). This degeneracy, when lifted by a crystal field of cubic symmetry, leads to four states labeled (in group theoretical notation) 4T1,4T2, 4A1, and 4E. The degeneracies of the states are as follows: 4Tl-three fold; 4Tz-three fold; 4A~-one fold; 4 E - t ~ o fold. Using the same method, it is found that the 4D state splits into 4E and 4T2, and the 4P state does not split, but transforms as 4T1. State 4Fsplits into 4T1, 4T2,and 4A2; 4A2 is non-degenerate, and the other states have the degeneracies indicated earlier.
58
W. M. BECKER
Using these eigenstates as basis, the total Hamiltonian matrix for the Mn2+ ion in an A"BV1host (without spin-orbit interaction) can be written and , 4A1, and 4Estates. diagonalized, yielding all the energy levels of the 4 T ~4T2, The energy associated with each state is usually given as a function of the crystal field parameter, Dq, and the Racah parameters B and C (Tanabe and Sugano, 1954). The parameter Dq is related to the orbital splitting of a single d electron in a cubic crystal potential, and B and Care combinations of radial integrals over the interelectronic distance. Tanabe and Sugano (1954) carried out this calculation for a cubic field, and all the energy matrices were collected by McClure (1959). The solution of the secular equation of these matrices gives the appropriate energies of the states. Shown in Fig. 12 is the energy level diagram appropriate for splitting of states of the d5 configuration in a cubic field (Tanabe and Sugano, 1970). For the usual values of Dq/B, the 4Gstates are the lowest excited states, and are therefore the first to become optically accessible in DMS alloys as the band gap widens. Transitions between the states are governed by strict selection rules. For example, in the free ion case, transitions between the 6Sground state and the 4Gexcited states are forbidden both by the AS = 0 selection rule and by the
E/B
,
2::
'A2 'TI
4 T ~ 6A,(dc3 d'yz) 4T1( d r 4 d y )
10 -
-
'Tp(dr5)
6A I
6S
6s
0
obs (Mn IU)
1
2
3
4 Dq/B
FIG. 12. Splitting of states of the dS configuration by a crystal field of oh symmetry (Tanabe and Sugano, 1970). Similar splitting is expected for a crystal field of Td symmetry.
2.
BAND STRUCTURE AND OPTICAL PROPERTIES
59
parity selection rule. The A S = 0 rule is, however, broken by spin-orbit interaction, and the parity rule fails in a crystal field which lacks inversion symmetry, such as Td ,leading to observation of relevant features in both zinc blende and wurtzite DMS alloys. Electric dipole transitions will therefore occur through this and other perturbations of the environment of the ion. Details of Mn2+optical transitions in A"BV' semiconductors have been the subject of intense investigation for many years, but results were predominantly confined to samples with Mn in extreme dilution, In this regime, details of vibrational interactions, Jahn-Teller effects, and other phenomena requiring high resolution could be studied because of the rich structure of the spectra. Such structure disappears rapidly with alloying for x z 0.01, and only broad excitation and emission features remain as the Mn concentration is further increased. Another limitation for study of these transitions in some DMS is that the spectroscopic "window" for observation of Mn-related transitions-the fundamental gap-may open only for high x values or for high pressure. As a result of both these limitations, identification of observed absorption and excitation features associated with Mn has been somewhat controversial in DMS. 5 . Mn TRANSITIONS IN SPECIFIC DMS ALLOYS
In Table VII are listed Mn-related absorption/excitation bands seen in various DMS, the intra-ion transitions from which they derive and, in several selected cases, also crystal field parameters employed in the analysis of the data. In all A:'_,Mn,B"' DMS where Eg exceeds 2.2eV, a broad (FWHM = 0.1 eV) photoluminescence band is seen in the yellow region of the visible part of the spectrum. This transition is generally accepted as the 4T1(4G) 6A~(6S)transition, and is the emission associated with the excitation bands listed in Table VII. Below, we briefly comment on the results listed in the table. Several high pressure measurements that confirm the identity of various d-d transitions seen at zero pressure, and in some cases reveal higher energy transitions not previously accessible by optical techniques, are also discussed. +
(m) Cdl-,Mn,Te The first optical effects seen in Cdl-,Mn,Te related to Mn transitions were those of Nguyen The Khoi and Gaj (1977), who associated an absorption edge at =2.0eV with the onset of the 6A1(6S)+ 4T1(4G)transition. From photoluminescence results on thick samples (=1 mm), Tao et af. (1982) and Moriwaki et af. (1982) suggested that both the 6A~(6S) 4T~(4G)and 6A~(6S) 4T2(4G)excitation features were present at high x . However, it was not until samples thinned to the micron range were employed that the -+
+
TABLE VII MnZ+ABSORPTION/EXCITATION BANDMAXIMA(IN eV)
IN
DMS ALLOYS
_______~
Alloy Cdl-xMn,Te
6Ai
+
4T1
- 2.2
- 2.2
Cdl-,Mn,S Znl -,MnxTe
Znl-,Mn,Se
Znl-,Mn,S
-2.5
- 2.43* - 2.4*
6Ai
+
4Al, 4E
Other
- 2.63*
-2.63*
- 0.48 - 0.4
-2.3 (Threshold) 2.43
2.58
- 2.3
-2.3 -2.33
2.34 2.38 2.4 2.42
- 2.36 -2.38 2.40
-
B(cm-l) 720 713 713
~(cm-') 750 749 75 1
2.22(z.p.) 2.35
2.48 2.57 2.5 2.54
2.44(z.p) 2.53
X
0.4 5 x I0.7 0.4 Ix 5 0.7 0.7 0.7 0.7 0.4 5 x 5 0.7
- 2.6*
- 2.43*
-2.2
Cdl-,Mn,Se
6A1 + 4T2
2.72 -2.576 - 2.605 -2.745 C(cm-') Dq(cm-l) 2664 743(Td), 330(0h) 1638 750(Td), 333(0h) 2637 913(G), 406(0h) 2.655 2.68 2.70 2.9 2.70 2.93 C(cm-') Dq(cm-') 2730 836(%), 372(0h) 2772 830(G), 369(0h) 2782 795(G), 353(0h) 2.635 (z.P.) 2.69
T a o et al. (1982) Moriwaki et al. (1982) Rebmann et a/. (1983) Lascaray et at. (1983) Diouri et al. (1985) Lee and Ramdas (1984) Morales et al. (1985) Ikeda et al. (1968)
-0.1 -0.5 0.6
-
Morales Toro et al. (1984) Morales Toro et al. (1984) Morales Toro et al. (1984)
-0.1 0.5 -0.6
-
Morales Toro et al. (1984) Morales Tor0 et al. (1984) Morales Toro et al. (1984)
0.001 0.23 0.35 0.5
Langer and Richter (1965) Morales (1985) Morales (1985) Morales (1985)
0.23 0.35 0.50
Morales (1985) Morales (1985) Morales (1985)
2.815 (z.P.), 3.13 (z.P.) 0.001 GPa have an arbitrary shift of the ordinate scale.
x = 0.25 sample of Znl-xMnJSe (Ves ef a / . , 1986b). All curves for p
Departures from Varshni-like behavior of the energy gaps below the characteristic spin-freezing temperature seem to be present in several DMS. Unfortunately, some-of the clearest results related to this problem (Diouri et al., 1985) depend on isoabsorption measurements on very thin samples, rather than on observation of the thermal shift of sharp spectral features. Further experimental work on this effect by less ambiguous techniques is needed to clarify the status of this result. Both in the work on the energy gap, and on states away from k = 0, the blurring of experimental features remains a strongly limiting factor in the investigation of the band structure of these materials. The blurring effect needs to be better understood, and new results should be sought on samples prepared by crystal growth methods other than the Bridgman technique. The presence of Mn-related absorption/excitation bands and an associated emission feature at =2eV appears to be well established in all wide-gap DMS. Further, the identification of the three lowest lying absorption/ excitation bands and the 2eV emission in terms of ligand theory seems satisfactory, in spite of the disappearance of spectral detail with increasing
70
W. M. BECKER
Mn content. Much less certain is the origin of the transitions seen at higher energy. This problem, that arose early in the study of Mn at very low values of x (McClure, 1963), remains an active area of theoretical research (Richardson and Janssen, 1986). More experimental work is certainly needed, but present techniques, such as optical absorption at high pressure, seem unlikely to impinge on this problem. Strong evidence exists for the hybridizing of the ground state of the Mn ion with valence band states at energies well below the band edge. Such hybridization does not appear to affect the existence and observation of the lowest energy localized d-d transitions of the ion. Further experimental and theoretical work is needed to understand the coexistence of these two effects. Finally, charge transfer states, such as Mn'+3d6 levels, have been predicted recently for DMS materials (Ehrenreich et al., 1986), but experimental evidence for such states is slight. Recent results by Lee et al. (1986b) using reflectivity at high magnetic fields rule out this mechanism as being responsible for the 2 eV absorption/excitation feature in CdI-,Mn,Te. The possibility exists that short-lived charge transfer states are located in the gap; these will probably require investigation by time-resolved spectroscopy to verify their presence. References Abragam, A., and Bleaney, B. (1970). Electron Paramagnetic Resonance of Transition Ions (Clarendon Press, Oxford). Abreu, R. A., Stankiewicz, J., and Giriat, W. (1983). Phys. Stat. Sol. (a) 75, K153. Ambrazevicius, G., Babonas, G., Marcinkevicius, S., Prochukhan, V. D., and Rud, Yu V. (1984). Solid State Commun. 49, 65 1. Antoszewski, J., and Kierzek-Pecold, J. (1980). Solid Sate Commun. 34, 733. Babonas, G. A., Bendoryus, R. A., and Shileika, A. Yu.(1971). Sov. Phys.-Semicond. 5,392. Ballhausen, C. J. (1962). Introduction to Ligand Field Theory (McGraw-Hill Book Company, Inc., NY.) Bottka, N., Stankiewicz, J., and Giriat, W. (1981). J. Appl. Phys. 52, 4189. Brafman, O.,and Steinberger, I. T. (1966). Phys. Rev. 143, 501. Brun Del Re, R., Donofrio, T., Avon, J., Majid, J., and Woolley, J. C. (1983). I1 Nuovo Cimento 2D, 1411. Bucker, R., Gumlich, H. E., and Krause, M. (1985). J. Phys. C: Solid State Phys. 18, 661. Bylsma, R. B., Becker, W. M., Kossut, J., Debska, U., and Yoder-Short, D.(1986). Phys. Rev. B 33, 8207. Cardona, M., Shaklee, K. L., and Pollak, F. H. (1967). Phys. Rev. 154, 696. Chou, H . H., and Fan, H. Y. (1974). Phys. Rev. B 10, 901. Cohen, M. L., and Bergstresser, T. K. (1966). Phys. Rev. 141, 789. Diouri, J., Lascaray, J. P., and El Amrani, M. (1985). Phys. Rev. B. 31, 7995. Ehrenreich, H . , Hass, K. C., Johnson, H. F., Larson, B. E., and Lempert, R. J. (1986). Proceedings of the 18th International Conference on the Physics of Semiconductors, Stockholm, Sweden. Gai, J. A., Galazka, R. R., and Nawrocki, M. (1978). Solid State Commun. 25, 193. Giriat, W.,and Stankiewicz, J . (1980). Phys. Stat. Sol. (a) 59, K79.
2.
BAND STRUCTURE AND OPTICAL PROPERTIES
71
Griffith, J. S. (1971). The Theory of Transition-Metal Ions (Cambridge University Press, London). Gumlich, H.-E. (1981). Journal of Luminescence 23, 73. Ikeda, M.,Itoh, K., and Sato, H. (1968). J. Phys. SOC.Japan 25, 455. Kendelewicz, T., and Kierzek-Pecold, E. (1978). Solid State Commun.25, 579. Kendelewicz, T. (1980a). Proceedings of the Xth Polish Seminar on Semiconductor Compounds, Jaszowiec, Poland. Kendelewicz, T. (1980b). Solid State Commun. 36, 127. Kendelewicz, T. (1981). J. Phys. C: Solid State Phys. 14, L407. Kolodziejski, L. A., Gunshor, R. L., Venkatasubramanian, R., Bonsett, T. C., Frohne, R., Datta, S., Otsuka, N., Bylsma, R. B., Becker, W. M., andNurmikko, A. V. (1986). J. Vac. Sci. Technol. B4, 583. Langer, D., and Ibuki, S. (1965). Phys. Rev. 138,A809. Langer, D. W., and Richter, H. J. (1966). Phys. Rev. 146, 554. Lascaray, J. P., Diouri, J., El Amrani, M., and Coquillat, D. (1983). Solid State Commun. 47,709. Lautenshlager, P., Logothetidis, S., Vina, L., and Cardona, M. (1985). Phys. Rev. B32,3811. Lee, Y. R., and Ramdas, A. K. (1984). Solid State Commun. 51, 861. Lee, Y. R., Ramdas, A. K., and Aggarwal, R. L. (1986a). Proceedings of the 18th International Conference on the Physics of Semiconductors, Stockholm, Sweden. Lee, Y. R., Ramdas, A. K., and Aggarwal, R. L. (1986b). Phys. Rev. B 33, 7383. McClure, D.S. (1959). “Electronic Spectra of Molecules and Ions in Crystals” in Solid State Physics, Vol. 9 (Academic Press, NY). McClure, D. S. (1963). J. Chem. Phys. 29, 2850. Morales, J. E., Becker, W. M., and Debska, U. (1985). Phys. Rev. B 32, 5202. Morales Toro, J. E., Becker, W. M., Wang, B. I., Debska, U., and Richardson, J. W. (1984). Solid State Commun. 52, 41. Morales Toro, J. E. (1985). Ph.D. Thesis, Purdue University, unpublished. Moriwaki, M. M., Becker, W. M., Gebhardt, W., and Galazka, R. R. (1982). Phys. Rev. B 26, 3165. Moriwaki, M. M., Tao, R. Y.,Galazka, R. R., Becker, W. M., and Richardson, J. W. (1983). Physica 117B & 118B, 467. Muller, E., Gebhardt, W., and Rehwald, W. (1983). J. Phys. C: Solid StatePhys. 16,L1141. Neumann, E. (1971). Thesis, Technische Universitat Berlin, D83, unpublished. Nguyen The Khoi and Gaj, J. A. (1977). Phys. Stat. Sol. (bJ83, K133. Pajaczkowska, A. (1978). Prog. Cryst. Growth Charact. 1, 289. Parsons, R. B., Wardzynski, W., and Yoffe, A. D. (1961). Proc. Roy. SOC.A 262, 120. Phillips, J. C. (1973). Bonds and Bands in Semiconductors (Academic Press). Podgorny, M., and Oleszkiewicz, J. (1983). J. Phys. C: Solid State Phys. 16, 2547. Richardson, J. W., and Janssen, G. J. M. (1986). Proceedingsof theMaterials Research Society Fall Meeting, Boston, Massachusetts. Rebmann, G., Rigaux, C., Bastard, G., Menant, M., Triboulet, R., and Giriat, W. (1983). Physica 117B & 118B,452. Skowronski, M., Baranowski, J. M., and Ludwicki, L. J. (1978). Polska Akademie Nauk (PAN) Nr. 75, 127. Stankiewicz, J. (1983). Phys. Rev. B 27, 3631. Steinberger, I. T. (1983). “Polytypism in Zinc Sulphide,” in Crystal Growth and Characterization of Polytype Structures (Pergamon Press). Sundersheshu, B. S . (1980). Phys. Stat. Sol. (a) 61,K155. Sundersheshu, B. S., and Kendelewicz, T. (1982). Phys. Stat. Sol. (aJ 69, 467.
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Tanabe, Y . , and Sugano, S . (1954). J. Phys. Soc. Japan 9, 753. Tanabe, Y . , and Sugano, S. (1970). Multiplets of Transition Metal Ions in Crystals (Academic Press, New York). Tao, R . Y . , Moriwaki, M. M., Becker, W. M., and Galazka, R. R. (1982). J. Appl. Phys. 53, 3772. Theis, D. (1977). Phys. Stat. Sol. (b) 79, 125. Thompson, A. G., and Woolley, J. C. (1967). Can. J. Phys. 45, 255. Twardowski, A., Nawrocki, M., and Ginter, J. (1979). Phys. Stat. Sol. (b) 96, 497. Twardowski, A. (1983a). Phys. Lett. 94A, 103. Twardowski, A.,Dietl, T., and Demianiuk, M. (1983b). Solid State Commun.48,845. Twardowski, A., Swiderski, P., von Ortenberg, M., and Pauthenet, R. (1984). Solid State Commun. 50, 509. Van Vechten, J . A. (1970). Proceedings of the Tenth International Conference on the Physics of Semiconductors, Cambridge, Massachusetts, 602. Varshni, Y. P. (1967). Physics 34, 149. Vecchi, M. P., Giriat, W., and Videla, L. (1981). Appl. Phys. Lett. 38, 99. Ves, S.,Strossner, K., Christensen, N. E., Chul Koo Kim, and Cardona, M. (1985). SolidState Commun. 56, 479. Ves, S . , Strossner, K . , Gebhardt, W., and Cardona, M. (1986a). Phys. Rev. B 33, 4077. Ves, S.,Strossner, K., Gebhardt, W., and Cardona, M. (1986b). SolidState Commun. 57,335. Wei Shan, Shen, S. C., and Zhu, H. R. (1985). Solid State Commun.55, 475. Wiedemeier, H.,and Sigai, A. G. (1970). J. Electrochem. SOC. 117, 551. Wisniewski, P.,and Nawrocki, M. (1983). Phys. Stat. Sol. (b) 117, K43. Zimnal-Starnawska, M., Podgorny, M., Kisiel, A., Giriat, W., Demianiuk, M., and Zmija, J . (1984). J. Phys. C: Solid State Phys. 17, 615.
SEMICONDUCTORS AND SEMIMETALS. VOL. 25
CHAPTER 3
Magnetic Properties: Macroscopic Studies Saul Oseroff DEPARTMENT OF PHYSICS, SAN DIEGO STATE UNIVERSITY SAN DIEGO, CALIFORNIA, USA
and Pieter H. Keesom DEPARTMENT OF PHYSICS, PURDUE UNIVERSITY WEST LAFAYETTE, INDIANA, USA
INTRODUCTION. . . . . . . . . . . BACKGROUND. . . . . . . . . . . RESULTSAND ANALYSIS. . . . . . . 1. Electron Paramagnetic Resonance. 2. Magnetic Susceptibility . . . . . 3 . Specsfic Heat . . . . . . . . . 4. Remanent Magnetization . . . . IV. CONCLUDING REMARKS.. . . . . . REFERENCES.. . . . . . . . . . . I.
11. 111.
. . . .
. . . .
. . . .
. . . .
. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 14 80 80 91 107 112 117 119
I. Introduction The purpose of this work is to review the magnetic properties of diluted magnetic semiconductors (DMS). These are ternary compounds of the form AI-,Mn,C, where A = Zn, Cd, Hg are group-I1 elements of the periodic table, C = S , Se, Te are group-IV elements and xis the mole fraction of the magnetic ion MnZ+.The properties of interest will be electron spin resonance, magnetization, and the contribution to the specific heat due to the magnetic Mn ions incorporated substitutionally in the lattice (neutron scattering data, that are also of importance as far as magnetic properties of DMS are concerned, are discussed elsewhere in this volume). These properties can be calculated when the energy level scheme of the Mn ions is known. Vice versa, when the above properties are known from the measurements, they can be used to deduce the energy level scheme. The divalent manganese ions have a 3d5 configuration and 6S5/2 ground state. A spherically symmetric S state cannot be affected by a crystalline electric field. However, it is known that a crystal field splitting of the ground state of Mn2+ does occur. This is because it is not a pure 6Sstate, but contains admixtures of higher lying levels.
73 Copyright 0 1988 by Academic Press, Inc. All rights ot reproduction in any form reserved. ISBN 0- 1 2 - 7 9 125-9
74
SAUL OSEROFF AND PIETER H. KEESOM
The mechanisms of such mixing involve higher order processes that take into account crystal fields, spin-orbit coupling, spin-spin interaction, relativistic effects, etc. (Watanabe, 1957; Gabriel et al., 1961). However, the admixtures in question are relatively small and often can be neglected. We shall return to this problem later in this chapter. 11. Background
In this section, we shall try to give a list of general problems that are encountered when studying magnetic properties of DMS with increasing number of magnetic ions incorporated in the host semiconductor matrix. Thus, we start with an examination of the properties of an isolated Mn2+ ion in the 11-VI lattice. We then proceed to describe the interation of the ion with other Mn2+ ions. The latter interaction will be the dominant effect determining the magnetic properties of concentrated systems. We shall also give expressions appropriate to describe various quantities of interest to us in this chapter for each regime of Mn2+ concentration. For concentrations of Mn below 0.1 atomic percent, x < 0.001, the interaction between the Mn ions can probably be neglected. Thus, in any analysis, only isolated paramagnetic Mn ions with a ‘ S m ground state need to be considered. The presence of a crystal electric field will remove part of the sixfold degeneracy of the ground state. A crystal field cannot remove all the degeneracy of a 6 S ~ / 2state; at most it will split it into three Kramers doublets. Because of the complexity of the mechanisms involved in the splitting, it is difficult to perform direct calculations. Therefore, spin Hamiltonian method is usually used, Abragam and Bleaney (1970). The crystal field Hamiltonian for Mn2+ can be written as an even polynomial of fourth degree in the projection of the spin operators &, and $. For example, for an orthorhombic lattice, the crystal field Hamiltonian for Mn2+ is:
sy,
where D , E , and a are crystal field parameters to be determined from the experiment. The first term accounts for the tetragonal field with thezdirection taken as the axis of symmetry; the second term accounts for deviations from the axial symmetry, and the last term gives the contribution of the cubic field. Most of the DMS discussed here have a zinc blende structure, two fcc lattices displaced from each other by one quarter of a body diagonal, or a wurtzite structure composed by two hcp lattices. For a zinc blende structure only the last term on Eq. (1) is different from zero. In such a case, the 6S5/2 ground state splits into a doublet and a quadruplet separated by 3a. The order of magnitude of the splitting in a typical DMS is smaller than 0.1 K.
3.
MAGNETIC PROPERTIES : MACROSCOPIC STUDIES
75
When an external magnetic field H is applied, the degeneracy is completely removed. The EPR spectra for the Mn*+,then, consist of five resonance lines which correspond to the fine structure transitions AM = + 1. Their energy level scheme depends on the intensity of the crystal field and external magnetic field H, as well as on the angle between the magnetic field and the crystal axis. Because the Mn nucleus has a magnetic moment Z = 5/2, the interaction of the electron magnetic moment and nuclear magnetic moment will split each of the fine structure energy levels (labeled by M ) into six additional levels. A simplified spin Hamiltonian for Mn2+ in a cubic host, that includes the hyperfine interaction is:
where the first term accounts for the electronic Zeeman interaction, the second for the cubic crystal field, and the third for the hyperfine interaction. In Eq. (2), we have ignored smaller contributions, such as the superhyperfine interaction and the nuclear Zeeman interaction. For a more complete discussion on the subject, see Abragam and Bleaney (1970). When the energies of the Mn levels are known, the magnetization M , susceptibility x and specific heat C follow directly. For low concentrations, the uncoupled Mn ion levels in zero magnetic field will only show up at temperatures such that the value of k T is smaller or comparable than their crystal field and or hyperfine splitting, less than 0.1 K. If a sufficiently strong magnetic field (compared with the crystal field splitting) is applied, H 2 lOkOe, then the Mn ions give a Schottky contribution to M , x, and C characteristic of a system with essentially six evenly spaced levels. The susceptibility for the temperatures above 1 K and low concentration follows a Curie law:
where N Mis~the number of Mn ions/unit volume, g is the Lande factor, PB is the Bohr magneton, S is the total spin, and kg is the Boltzmann constant. When the Mn concentration increases above x = 0.005, the probability that Mn ions will form interacting clusters increases rapidly. The probability for the occurrence of clusters of various sizes can be calculated as a function of concentration, assuming a random distribution of Mn atoms for a given structure of the lattice. Probability tables for the clusters, and corresponding spin Hamiltonians and eigenvalues for cases of nearest, next-nearest, and third nearest-neighbor interactions can be found in the literature (Kreitman and Barnett, 1965; Okada, 1980, Nagata et al., 1980). We show these probabilities for singles, pairs, and triples for the fcc and hcp lattices in
76
SAUL OSEROFF AND PIETER H . KEESOM
TABLE I PROBABILITY THAT
FOREIGN ATOMIS I N A SINGLE, DOUBLE, AND TRIPLE CLUSTER ASSUMING ONLY THE NEAREST-NEIGHBOR INTERACTIONS‘
A
fcc
Cluster Type
hCP
(1 - x)l2 12x(l - X ) l 8 18?(1 - ~)’~[5(1- x) 24?(1 - x)”
S D‘ Tb
z1
(1 12X(1 - X ) ’ 8 i8X2(1 - Xy3[2 + 5(1 - X)I 3x2(1 - ~ ) ~ ‘ + [ 16(1 - X) + (1 - x ) ~ ]
+ 21
~
“After Kreitman and Barnett. 1965.
TABLE I1 PROBABILITY THAT A FOREIGN ATOMIS IN A SINGLE, DOUBLE, OR TRIPLE CLUSTER ASSUMING THE NEAREST-NEIGHBOR AND NEXT-NEAREST-NEIGHBOR INTERACTIONS‘ fcc
Cluster Type
(1 - x)’8 12x(1 - X)26 6x(1 - x ) ~ ’ 902(1 - x ) ’ ~ 92(1 - x)41[4 72u2(1 - x)37[1 24x2(1 - x ) ~ ’
+ (1 - x)]
+ (1 - x ) ~
0
36x2(1 - x)34 0
(1 - x)l8 12x(l - x)26 6x(1 - x ) ~ ’ 9x2(1 - x)”[l 9(1 - x)] 9 2 ( 1 - x)39[2(1 - x)2 + 2(1 - X ) 3 + 11 18x2(1 - ~ ) ~ ’ [+3 5(1 - x)] 3x2(1 - xI2’[l + 6(1 - x)’ + (1 - x ) ~ ]
+
0 36x2(1 - x ) ~ ~
0
‘After Kreitman and Barnett, 1965.
Tables, I, 11, and 111. The corresponding Hamiltonians and eigenvalues for singles, pairs, and triples are given in Table 1V. The susceptibility and specific heat C for a system consisting of clusters of various sizes can be written as:
x
where xi and Ci are the contributions from each cluster type including isolated ions, and Ni is the number (per unit volume) of occurrence of clusters of the i-th type. The susceptibility and the specific heat contributions of each
3.
77
MAGNETIC PROPERTIES: MACROSCOPIC STUDIES
TABLE I11 PROBABILITY THAT A FOREIGN ATOMIS IN A SINGLE, DOUBLE, OR TRIPLE CLUSTER ASSUMING INTERACTIONS UP TO 3RD-NEAREST-NEtGHBORS FOR fcc LATTICE' Cluster Type
Probability
S
(1 - x)42 12x(l - x)5' 6x(1 - x)62 24x(1 182(1 - x)74 9x2(1 - x ) ~ + * 36x2(1 - x)" 36x2(1 - x)'~+ 144x2(1- x ) ~ ' + 252x2(1 - x)93+ 180x2(1- x)94 72x2(1 - x ) ~ ' 144x2(1 - x)" + 144x2(1- x)" + 216x2(1- x)'4 72x2(1 - x ) ' ~+ 144x2(1- x)*' 24x2(1 - x)~' 24x2(1 - x)'~ 36x2(1 - x)~' 72x2(1 - x)" 72x2(1 - x)" 72x2(1 - x)71 72x2(1 - x)'3
Dl L? D3
z' T,'
z3
After Okada, 1980.
of the clusters are given by X , ( T )= 1
s%B
c
3k~T
s
w + 1)(2S + l)exp(-Es/ksT) cs(2s + l)exp(-&/kBT)
d ES exp(- E s / k T ~) Cj(T) = a T exp( - &/kB T ) '
I
(6) (7)
where Es is the energy for the case of zero magnetic field. In the presence of a magnetic field H, Es in Eqs. (6) and (7) should be replaced by E, = Es - g,uBmH where m is the magnetic quantum number. In that case, an additional summation over rn (- S 5 m IS) must be performed. As the concentration increases the mean cluster becomes more extended until, at a certain critical concentration x = x, , its size becomes comparable with the size of the sample. The value of xc , known as the critical concentration for the site-percolation problem, depends on the lattice structure and the number of neighboring magnetic ions for which the exchange interactions are significant. For example, if interactions up to the lst, 2nd, or 3rd neighbors are taken into account in an the fcc lattice, we have xc = 0.195,
78
SAUL OSEROFF AND PETER H. KEESOM
TABLE IV HAMILTONIANS AND EIGENVALUES FOR SINGLE, DOUBLE, AND TRIPLE CLUSTERS'?
Type
Hamiltonian
Eigenvalue
After Okada, 1980.
0.136, and 0.061 respectively (see e.g., Domb and Dalton, 1966; Frisch et al., 1961).
DMS systems usually form a simple crystallographic phase with a zinc blende or wurtzite structure for a wide range of concentrations. In Table V, we give the crystal structures and approximate ranges of composition in which good quality single-phase samples can be obtained. The DMS can sometimes be obtained for higher values of x than that given in Table V, but the crystals tend to be of poorer quality (see e.g., Pajaczkowska, 1978; Furdyna, 1982). In all the systems given in Table V, the maximum concentration of Mn is well above the nearest-neighbor percolation critical value (xc = 0.2). Thus in principle, a long magnetic ordering can be expected below certain temperatures, In these materials it was found that the nearest-neighbor interaction between Mn ions is antiferromagnetic (AF) (Brumage et al., 1964; Kreitman et al., 1966; Davydov, 1980; Nagata et al., 1980; Galazka et al., 1980;
3.
MAGNETIC PROPERTIES: MACROSCOPIC STUDIES
79
TABLE V CRYSTAL STRUCTURE AND RANGEOF COMPOSITION FOR THE DMS STUDIED Material Znl-,Mn,S
Znl-,Mn,Te Cdl-,Mn,S Cdl-,Mn,Se Cdl -,MnxTe Hgl-,MnxS Hgl-,Mn,Se Hgl-,MnxTe
Crystal Structure
Range of Composition
Zinc blende Wurtzite Zinc blende Wurtzite Zinc blende Wurtzite Wurtzite Zinc blende Zinc blende Zinc blende Zinc blende
x < 0.10 0.10 s x s 0.43
x s 0.30 0.30 < x < 0.55 x s 0.75 x s 0.45 x s 0.50 x s 0.70 x ~0.37 x s 0.30 x 5 0.50
Shapira et al., 1984). A spin-glass phase with only AF interactions was predicted for an fcc lattice owing to the frustration of AF ordering (DeSeze, 1977; Aharony, 1978; Villain, 1979). The mechanism of the frustration can be visualized as follows: consider three spins forming an equilateral triangle and assume that a pair of spins is aligned antiparallel. This orientation results in a lowering of the energy of that pair. What about the third spin? If it is aligned antiparallel-and hence favorably coupled-to one of the spins in the original pair, it is of necessity aligned parallel to the other one. This impossibility of satisfying strict antiferromagnetic configuration amongst all nearest neighbors is referred to as frustrated antiferromagnetism. It is currently believed that some form of frustration is necessary for a system to form the spin-glass state. Thus, two main ingredients for a system to show a spin-glass (SG)behavior are the frustration of long magnetic ordering and a random distribution of impurities. By similar arguments, frustration is also expected for an hcp lattice. Most of the systems exhibiting the SG behavior studied so far are diluted solutions of transition ions in metallic matrices (Cannella and Mydosh, 1972; Fischer, 1976; Anderson, 1979; Mydosh, 1981). However, in a metallic SG, it is not always easy to separate the contribution of the conduction electrons from that of the localized moments, e.g., in the case of specific heat (Kondo, 1965). Therefore, for a better understanding of SG it is desirable to study simpler systems. One possibility is to use frustrated systems with only AF interactions. As was mentioned above, DMS are appropriate candidates for this purpose. In the case of DMS with large electronic energy gaps, no free electrons are available at low temperatures to mediate the
80
SAUL OSEROFF AND PIETER H. KEESOM
Ruderman-Kittel-Kasuya-Yosida interaction between the magnetic ions. Then if the SG phase is observed in these systems it must be explained in terms of a mechanism with predominantly short ranged interactions. It is the current opinion that these interactions in DMS are mainly due to the super exchange mechanism (Larson et al., 1985; Lewicki et al., 1985; Spalek et al., 1986). For zero-gap or narrow-gap DMS, another mechanism was suggested leading to long range indirect AF interactions between Mn2+ions. This is the so-called Bloembergen-Rowland mechanism, where the interaction is mediated by carriers from the valence band virtually excited to the conduction band (Bloembergen and Rowland, 1955; Bastard and Lewiner, 1979a,b; Lewiner et al., 1980; Liu, 1982). Summing up the introductory remarks, magnetic properties of DMS may show many facets: starting from that due to the isolated ion, through the isolated cluster, the spin-glass phase and, ultimately-for very concentrated DMS-long range ordered system (see also Giebultowicz and Holden, this volume). Sometimes, as in the case of Cdl-xMnxTe, it is possible to observe all these phases in one DMS system. This situation offered by DMS is rather unique: one is then able to follow the development of the magnetic behavior with increasing concentration of the magnetic constituent in a matrix that is relatively simple from the viewpoint of the electronic band structure. In the following sections, we shall give a review of existing experimental data on spin resonance, magnetic susceptibility, magnetization and specific heat obtained for various DMS, along with their detailed analysis aimed at conclusions of a more general nature. 111. Results and Analysis
This section is divided into several parts, each devoted to a single physically measureable phenomenon or quantity. We start with electron paramagnetic resonance, to be followed by magnetic susceptibility, specific heat, and magnetization. In each case, we first review the available experimental data in the given field, and then give a detailed theoretical analysis.
PARAMAGNETIC RESONANCE 1. ELECTRON a. Experimental Results
The EPR spectra of Mn2+in various II-VI compounds for x < 0.001 have been studied in detail by several authors (Matarrese and Kikuchi, 1956; Dorain, 1958; Matamura, 1959; Lambe and Kikuchi, 1960; Hall er al., 1961; Kikuchi and Azarbayejani, 1962; Schneider et al., 1963; Title, 1963a,b, 1964; Estle and Holton, 1966; Falkowski, 1967; Deigen et al., 1967; Leibler et al., 1970a,b, 1971, 1973). The main features of the spectra are: the gyromagnetic
3.
MAGNETIC PROPERTIES : MACROSCOPIC STUDIES
81
factor g is nearly equal to that of the free electron, 1.99 < g c 2.02; a well resolved isotropic hyperfine structure is observed consisting of six lines, possibly due to configurational interaction between the 3s23p63d54sconfiguration with the 3s23p63d5configuration (Abragam, 1950); and a fine structure that results from an anisotropic contribution. The values of the gyromagnetic factor g, the hyperfine structure constant A , and the crystal field parameters are obtained by fitting the data to the energy spectrum corresponding to a spin Hamiltonian analogous to that given by Eq. (2). As the concentration of Mn increases, the spectrum changes its shape. For x > 0.005, the hyperfine lines initially broaden, eventually becoming a structureless single line for x 1 0.02 (Ishikawa, 1966; Deigen et al., 1967; Leibler et al., 1977; Oseroff et al., 1979, 1980a,b; Andrianov et al., 1980; Oseroff, 1982; Manoogian et al., 1982; Webb et al., 1983; Sayad and Bhagat, 1985). The EPR linewidth AH,, first narrows as x increases and passes through a minimum at x = 0.03. Van Wieringen (1955), after calculating the second and fourth moments of the spectrum, suggested that the variation of the shape of the spectrum may result from the narrowing effect of the hyperfine splitting due to the exchange interaction with other Mn2+ions. The effect of exchange interaction on the linewidth has also been discussed by Bleaney (1970). After reaching the minimum at x = 0.03, the line broadens monotonically with concentration (Leibler et al., 1976, 1977; Oseroff, 1982; Manoogian et ai.,1982; Webb et al., 1983; Sayad and Bhagat, 1984). Different mechanisms which contribute to the broadening of the line with concentration have been discussed by Kittel and Abrahams (1953); Swarup (1959); Deigen et al. (1967); Abragam and Bleaney (1970); and Sayad and Bhagat (1985). Values of the gyromagnetic factor g and of the linewidth AH,, at room temperature, defined as the peak-to-peak in the first derivative of the absorption line, are given in Fig. 1 for Cdl-,MnxTe as a function of concentration and frequency. Similar behavior was found for other DMS studied. At high temperatures, the linewidth and the gyromagnetic factor g were found to be independent of microwave frequency used (Oseroff, 1982; Webb et al., 1983; Sayad and Bhagat, 1985). For concentration above x = 0.03, a significant increase in the linewidth with decreasing temperature was observed for all the DMS studied so far by EPR (Leibler et al., 1976, 1977; Grochulski et al., 1979; Oseroff et al., 1979, 1980a,b; Mullin et al., 1981; Oseroff, 1982; Manoogian et al., 1982; Webb et af., 1983; Sayad and Bhagat, 1985; Kremer and Furdyna, 1985a,b). This can be understood as an increase (as the temperature is lowered) of the internal field due to the presence of finite clusters. A typical dependence of the linewidth AH,, on the temperature can be seen in Figs. 2 and 3 for Cd,-,Mn,Te and Cdl-,Mn,Se. When the linewidth AH,, is of the order of
82
SAUL OSEROFF AND PIETER H. KEESOM
I"""' I
"/ 1 40.05
P
P
i
I
6'
-0.05
X
FIG. 1. Room temperature peak-to-peak linewidth AH,, and g-factor as a function of concentration. Linewidth (0)35 GHz and (A) 11 GHz, from Webb et al. (1983); (0)9.2 GHz, from Oseroff (1982); g-factor: (0)35 GHz and (A)11 GHz, from Webb et a[. (1983). [After
Webb er af. (1983).]
the resonance field, H R,special care must be taken to obtain meaningful data for the width and the position of the line. Because the radio frequency (r.f.) field is linearly polarized in a typical microwave experiment, the experimental signal contains, in addition to the resonance signal induced by the resonanceactive circularly polarized component of the r.f. field, also a contribution due to the resonance-inactive-circular component. Thus, to obtain the true resonance line shape from the raw data, the unwanted component has to be subtracted. Such a correction was made when preparing Figs. 2 and 3. It has been also found that the resonance field H R of the EPR line in DMS shifted to lower magnetic fields with decreasing temperature (Oseroff, 1982; Manoogian et al., 1982; Kremer and Furdyna, 1983, 1985a,b). As the shift is found to be almost independent of the microwave frequency, it is attributed to be due to an internal field rather than to a change in the gyromagnetic factor itself (Oseroff, 1982). The observed change in the position of the resonance line, Hi = H , - HR(whereN, is the magnetic field corresponding to g = 2) is shown in Fig. 4 for Cdl-,Mn,Te and Cdl-,Mn,Se. The difficulty to perform EPR measurements when o n < 1, where o is the microwave frequency and T is the relevant relaxation time, was successfully overcome by using the microwave Faraday rotation technique based on
3.
MAGNETIC PROPERTIES : MACROSCOPIC STUDIES
83
9000-
7000 n
Q)
0
Y
P P
I
a
FIG. 2. Temperature dependence of the peak-to-peak EPR linewidth for Cdl-,MnxTe (0.15 5 x s 0.60). The solid lines are least-squares fits to the data using Eq. (10). [After Oseroff (1 982).]
the dispersion associated with EPR, as discussed by Kremer and Furdyna (1983, 1985b). With this method, Kremer and Furdyna (1983, 1985a,b) performed measurements of the dynamic magnetic susceptibility at 35 GHz down to W T < 0.2. They measured AH,, and H R for concentrations well above the percolation limit, at temperatures below the spin-glass temperature (as revealed by the cusp in the zero field cooled susceptibility, c.f., further in this Chapter). It is to be noted that no anomaly was observed either in AH,, or HR at the spin-glass temperature Tg.
b. Analysis (a) Low Mn Concentrations (x s 0.03) For extremely small concentrations of Mn, x < 0.001, the interaction between the Mn ions is negligibly small. For the wide gap DMS, the gyromagnetic factor and the crystal field contribution were found to be larger for Mn2+ in the zinc than in the cadmium compounds with the same anion, whereas the hyperfine structure parameter A was found to be smaller in the zinc compound (see, for example, Kikuchi and Azarbayejani, 1962, and Title, 1963b). Kikuchi and Azarbayejani (1962) noted that the ionic radius of the Mn2+ ion was intermediate in size between the ionic radii of Zn2+and Cd2+.Then
84
SAUL OSEROFF AND PIETER H. KEESOM
j 2/
3.
MAGNETIC PROPERTIES : MACROSCOPIC STUDIES
85
T( K) FIG. 4. Temperature dependence of the internal magnetic field Hi for Cdl-,Mn,Se, and Cdl-,Mn,Te measured at 9 and 35 GHz. Solid lines are least-squares fits to the data using Eq. (12). H i is defined as Hi = Ho - H E , where Ho is the resonance field corresponding to g = 2, and H R is the measured resonance field. [After Oseroff (1982).]
anions than occurs in cadmium compounds. This results in a larger value of the crystal field parameter, a, and a smaller value of hyperfine constant, A , for the zinc compounds. This means that the bonds in the Mn2+-doped zinc compounds are more covalent than in cadmium compounds. A change in the resonance parameters was also observed, when going from Te to Se to S (Woodbury and Ludwig, 1961; Kikuchi and Azarbayejani, 1962; Title, 1963a,b). The gyromagnetic factor and the crystalline field parameters decreased, and the hyperfine structure constant increased when going from Te to Se to S (Title, 1963b). These tendencies appear to be related to changes in the covalent character of the bonds for different anions (Van Wieringen, 1955; Matamura, 1959; Fidone and Stevens, 1959; Hall et al., 1961; Kikuchi and Azarbayejani, 1962; Title, 1963a,b). A decrease of the g-factor and an increase of the hyperfine structure constant was also observed when the lattice spacings were reduced in the narrow gap DMS (Leibler et al., 1971, 1973). The g-factor for the narrow gap DMS was also found to be smaller than the g-factor for free electrons (Leibler et al., 1970a,b, 1971, 1973). As the concentration of Mn2+ increases, the EPR spectra change their shape. Deigen ef al. (1967) studied the concentration dependence of Cdl-,Mn,Te for lo-' 5 x 5 0.03, and found that the profile of the EPR line changed from almost Gaussian to Lorentzian when going from x = 0.005
86
SAUL OSEROFF AND PIETER H . KEESOM
to x 3 0.03. They found that within experimental error, the values of the hyperfine structure constant, A , and the crystal field parameter, a, were independent of concentration, at least forx < 0.008. Van Wieringen (1959, who studied EPR of ZnS:Mn, calculated second and fourth moments of the spectrum, and suggested that the variation observed in its shape was due to the narrowing of the hyperfine splitting by the exchange interaction between Mn2+ions. Ishiwaka (1966) analyzed the EPR spectra of exchange-coupled Mn2+ ions in ZnS and CdS using a spin-Hamiltonian of the form:
X = gpBusHSz
+
A
N
+ C JijSi i,j
*
Sj,
(8)
zzl
where S, and Z, are the z-components of S = S; and I = xEl I ; , respectively, N is the number of Mn2+ ions coupled together by exchange interaction and J i j is the exchange integral. The first, second, and third terms in Eq. (8) represent the Zeeman, hyperfine and isotropic exchange interaction, respectively. Ishikawa (1966) concluded that the EPR spectra for an exchange-coupled pair of Mn2+ions are described satisfactorily by the spin Hamiltonian given by Eq. (8) and that anisotropic terms due to the crystal field can be neglected. Oseroff (1984) analyzed the spectrum for Cdl -,Mn,Te with 0.002 s x 5 0.01, in the context of the overlap of isolated Mn2+ ions and coupled pairs of Mn2+. It turned out that the EPR results are not sufficiently sensitive to separate unambiguously the contributions due to singles and pairs of MnZ+. To study the effect caused by the distortion on the crystal lattice due to replacing Cd (ionic radius = 0.97 A)by Mn (ionic radius 3 0.80 A),Koh et al. (1984) simulated this system by studying the EPR of alloy Cdl-,Hg,Te:Mn. As the ionic radii of Cd and Hg (ionic radius 1.10 A) differ almost as much as that of Cd and Mn, these authors argued that the lattice distortion in the case of Cdl -,Mn,Te can be expected to be of the same order as in Cdl-,Hg,Te:Mn, with the advantage that in the case of latter-because of the low Mn concentration-the effect of the distortion can be investigated in the highly resolved EPR spectrum of nearly isolated Mn2+. Koh et al. (1984) found similar values for the crystal field splitting for Cdl-,H&Te:Mn to those found previously for very diluted Mn systems in the 11-VI compounds, and on this basis concluded that relatively large concentrations of Mn do not produce sufficient lattice distortions to cause a significant change in the crystal field splitting. (b) High Mn Concentrations (x 2 0.03) For x above =0.03, the room temperature EPR linewidth broadens with increasing concentration, as shown in Fig. 1. Webb et al. (1983) and Sayad
3.
MAGNETIC PROPERTIES: MACROSCOPIC STUDIES
87
and Bhagat (1985) suggested that the broadening of the resonance lines arises from a distribution of internal fields, or rather their projections along the applied field. The observed broadening of the linewidth with decreasing temperature was analyzed by several authors. Leibler et al. (1977), Grochulski et al. (1979), Oseroff et al. (1979) and Manoogian et al. (1982) analyzed the temperature dependence of AH,, in terms of a theory first used for antiferromagnets above the Nee1 temperature (see Huber, 1972 and Seehra, 1972). The broadening of the linewidth was described using Huber's expression: AH,, = A T( - 2T , 7
+ B,
(9)
where a denotes a critical exponent, B the temperature-independent linewidth, and T, the temperature of the order-disorder transition. Equation (9) gives a good fit to the data for temperatures well above TC . However, the predicted divergence at T = T , was not observed when AH,, was measured by using the Faraday rotation technique (Kremer and Furdyna, 1983,1985b). The EPR data analyzed with Eq. (9) could be separated into three regions of concentration. For x < 0.15, the values of T , were found to be smaller than 1 K. For x > 0.25, T, was found to be substantially larger and increased with the concentration. In the transition region, 0.15 S x 5 0.25, there occurred a large and rapid change in T , for all DMS samples studied so far. As can be noticed, the percolation critical point xc calculated for nn falls into this region of concentration. The values of T , determined from Eq. (9) were found to be several times larger than the temperature G where the cusp in the ZFC (zero field cooled) susceptibility was observed. For Cdl-,Mn,Te, Oseroff et al. (1979) found that the critical exponent CY increased with the concentration until x = 0.2, and then it became almost independent of the concentration. They also observed that B, the high temperature linewidth, increased with concentration. Instead, for Cd,Zn,Mn,Te (x + y + z = 1)) Manoogian et al. (1982) obtained the best fit with B = 0 and a = 0.33 f 0.08 for the whole range of concentrations studied. Oseroff et al. (1980a,b) and Oseroff (1982) analyzed their linewidth data using an equation similar to Huber's expression (Dormann and Jaccarino, 1974): AHp, = A [
'AH
IaAH
T - GAH
+ B[; + 11,
where GAHis the order-disorder temperature, CYAH is the critical exponent, 0denoting the Curie-Weiss temperature, and B ( W T + 1) is the high temperature linewidth that becomes dominant for T %- GAH(see Figs. 2 and 3).
88
SAUL OSEROFF AND PIETER H . KEESOM
Using Eq. (lo), the authors obtained values of GAHthat were jn good agreement with those that follow from the dc susceptibility measurements at the same field, = 3.3 kOe. The values of T ~ A Hbecame substantially larger for x > 0.29 than for the low concentration region, and increased (within the large experimental error, linearly) with concentration. The values of T f a ~ were calculated by using only the low temperature data. Oseroff et al. (1980a,b) and Oseroff (1982) justified their analysis by arguing that the high temperature data bear no relation with the critical phenomena, the high temperature behavior being accounted for by the second term of Eq. (10). These authors found that the critical exponent increased with concentration for x < 0.30, and for larger concentration, it became less dependent on it. It is not clear at this point if the behavior observed for C X A His just an artifact of the model or not. The solid lines in Figs. 2 and 3 are a least-squares fit of the experimental results with Eq. (10). Values of &H and CXAHfor a wide range of concentration are given in Tables VI and VII for Cdl-,Mn,Te and Cdl -,Mn,Se, respectively. Webb et af.(1983) and Sayad and Bhagat (1985) used adifferent expression to adjust the data for AHpp(T).Theirs is an empirical expression used first by Bhagat et al. (1981) to fit the temperature dependence of AH in metallic spin-glasses and in re-entrant ferromagnets. They expressed AHpp(T ) as
A H ~ ~ (=T ro ) + rl exp(-T/&), where To is the high temperature linewidth, and
r 1
(1 1)
and To are empirical
TABLE VI OF THE CURIE TEMPERATURE 8; THE EXTRAPOLATED TEMPERATURE FOR THE MAXIMUM OF x, f i ( H 0) AFTER ZERO-FIELD COOLING; THE TEMPERATURES fimAND E H ~AND , CRITICAL EXPONENTS CYANAND C X HOBTAINED ~ FROM THE TEMPERATURE DEPENDENCE OF THE EPR LINEWIDTH’
VALUESAS A FUNCTION OF CONCENTRATION FOR Cdl-,Mn,Te
+
1
2 5 10 15 20 30 40 53 60
3 8 22 35 I0 100 170 230 310 350
“ A f t e r O s e r o f f , 1982.
e l
el
2 + 0.5 8+1 12 + 2 18 ? 2 23 3
*
0.01 0.35 0.85 4+2 9 2 2 13 k 3 20 f 3 25 + 3
0.05 0.2
0.3 0.4 0.5 0.6 1.5 1.5 1.6 1.5
1.7 2.2
3.
89
MAGNETIC PROPERTIES : MACROSCOPIC STUDIES
TABLE VII VALUESAS A FUNCTION OF CONCENTRATION FOR Cd,-,Mn,Se OF THE CURIE TEMPERATURE 0, THE EXTRAPOLATED TEMPERATURE FOR THE MAXIMUM OF x, E(H 0) AFTER ZERO-FIELD COOLING; THE TEMPERATURES f i AND ~ &,yi, ~ A N D CRITICAL EXPONENTS CYAHA N D CYH~OBTAINED FROM THE TEMPERATURE DEPENDENCE OF THE EPR LINEWIDTH' +
(at. To)
-e(+io) (K)
1
3
2 5 11 15 20 23 25 30 35 41 45
10
E{H+O) (K)
26
2 K, DMS behave approximately according to the original suggestion of DeSeze (1 977) regarding the existence of insulating spin-glasses with only antiferromagnetic interactions.
104
2l ;$$
SAUL OSEROFF AND PIETER H. KEESOM I
Cd,-,Mn,Te
10
0
Zn
-Y,
20-
t=
10-
P
0
/
/d
40
M n SeB’
-
/
/
20
f
/
9’
2ol
/
20 40 x(at.96)
rZn,-,Mn,S
-
10
SG
//
SG 0
20 40 x( at .%)
1 0 ° L x(at.%)
FIG. 18. Paramagnetic (P)-spin glass (SG) phase diagram, Tr vs. x, for several DMS. from Galazka et al. (1980), (x) from Oseroff (1982) and (0)from Escorne Cdl-,Mn,Te (0) and Mauger (1982); Cdl-,Mn,Se (+) from Oseroff (1982) and (0)from Amarasekara et al. (1983); Cdl-,Mn,S (0)from Yang et al. (1983); Hgl-,Mn,Se (0)from Khattak et al. (1981); Znl-,Mn,Se (0)from Akbarzadeh et al. (1983); Znl-,Mn,Te (0)from McAlister et al. (1984); Znl-xMn,S (0)Yang et al. (1983).
As mentioned in the previous subsection, Novak et al. (1981, 1984, 1985) measured the field cooled susceptibility for Cdl-,Mn,Te and Cdl-,Mn,Se for 0.005 5 x 5 0.15, and found that the field cooled susceptibility shows a distinct kink, or knee for these low concentrated samples, as illustrated in Fig. 14. Brandt et al. (1981, 1982, 1983) measured the susceptibility for Hgl-,Mn,Te with 0.02 s x s 0.075 down to T = 0.04 K and observed a definite cusp in the susceptibility, as shown in Fig. 13. These results appear to indicate that for temperatures such that IJfn/k~Iz T the interaction between farther Mn ions becomes increasingly important. If the data reported by these last authors are indeed associated with a spin-glass transition, the phase diagrams for Cdl-,Mn,Te, Cdl-,Mn,Se, and Hgl-,Mn,Te can be extended below x = 0.17 for low temperatures. Such diagrams are shown in Figs. 19-21. For x z 0.17, the mechanism suggested t o account for the SG transition is a short range mechanism (probably superexchange: see Spalek et al., 1985 and Larson et al., 1985), where the main contribution comes from the
3.
MAGNETIC PROPERTIES: MACROSCOPIC STUDIES
105
25 Te
Cd,-,Mn,
20 -
15-
5 c-10-
5-
0
0.40
0.20
0.60
X
FIG. 19. Paramagnetic (P)-spin glass (SG) phase diagram, f i vs. x, for Cdl-,Mn,Te with 0.05 < x < 0.60. The solid line is a guide for the eyes. The values of f i were obtained from data reported by Novak et al. (1984), Oseroff (1982), Galazka et al. (1980), and Oseroff and Gandra (1985).
antiferromagnetic interaction between nearest-neighbors. If interactions with farther neighbors are present, the concentration at which the mean cluster becomes infinite is highly reduced. When interactions with 2nd or 3rd neighbors are included in an fcc lattice, the values calculated for x, are reduced from 0.195 to 0.136 and 0.061, respectively (Domb and Dalton, 1966). Under these conditions, it appears reasonable to expect a spin-glass behavior for concentrations below = 0.17. Another exchange mechanism besides the short-range mechanism is also possible in the open-gap semiconductors (Bloembergen and Rowland, 1955). In this mechanism, the localized spins interact via virtual excitation of an electron from the filled valence band to the empty conduction band. Bloembergen and Rowland have shown that the effectiveness of this interaction decreases exponentially with the inter-spin distance and with the square root of the energy of the forbidden gap. Bastard and Lewiner (1979a,b), Lewiner et a!. (1980), and Ginter et al. (1979) studied the BloembergenRowland mechanism for symmetry-induced zero-gap semiconductors,
106
SAUL OSEROFF AND PIETER H. KEESOM
X
FIG. 20. Paramagnetic (P)-spin glass (SG) phase diagram, for Cdl-,Mn,Se with 0.05 < x < 0.50. The solid line is a guide for the eyes. The values of Tr were obtained by Novak e f ai. (1985), Oseroff (1982), and Amarasekara e f a/. (1983).
251
I
I
I
X
FIG.21. Paramagnetic (P)-spin glass (SG)phase diagram, for Hgl -,Mn,Te. (A) from Brandt e t a / . (1983), (x) from Mycielski et a/. (1984) and Otto e t a / . (1980), a.c. magnetic susceptibility data; (0)from Mycielski et a/. (1984), Faraday rotation data. [After Mycielski ef al. (1984).]
3.
MAGNETIC PROPERTIES: MACROSCOPIC STUDIES
107
especially the case of Hgl-,MnxTe at low Mn concentration, and found that it gave rise to an indirect long-ranged exchange interaction of antiferromagnetic sign. Because of the exponential damping, this long-range indirect interaction can be expected to be small in wide-gap semiconductors. For x < 0.20, the temperatures of the cusp for narrow-gap Hgl-,Mn,Te are systematically larger than for Cdl-,Mn,Te or Cdl-,Mn,Se. This may be understood as follows: for Hgl-,Mn,Te, both mechanisms, the Bloembergen-Rowland and the short-range cluster mechanism, contribute. Instead, for the wide gap DMS, only short-range mechanism will result in a significant contribution. 3. SPECIFIC HEAT a. Experimental Results
The specific heat, C , of DMS has been measured in the temperature range between 0.3 K and 50 K and in magnetic fields up to 30 kOe (Nagata et al., 1980, 1981; Galazka et al., 1980). In the case of the pure 11-VI host (i.e., x = 0), only the lattice vibrations contribute to the specific heat, and an extrapolation of the low temperature specific heat to T = 0 determines the Debye temperature. Values of the Debye temperature for different 11-VI systems are collected in Table X. TABLE X VALUESOF THE DEBYE TEMPERATURE FOR
Zn Cd
Hg
DIFFERENT DMS (IN KELVINS) S
Se
Te
260
223
202
174
158 140
1i a 144
144
The low temperature specific heat in DMS is substantially greater than that in non-DMS. The data for Cdl-,Mn,Te at zero magnetic field are given in Fig. 22 as an example. The excess specific heat (Cex= C - CII-”I)has its origin mainly in the magnetic properties of the Mn ions. Small changes in the lattice vibrations spectrum can be also expected, but with the lattice contribution being very small below 2 K (as compared to C,), these changes can be safely neglected. As shown in Fig. 23, for Mn concentrations greater than x = 0.2, C e x behaves very similarly to the magnetic contribution found in metallic spinglasses (Wenger and Keesom, 1975, 1976); that is, C e x is linear in T i n the
108
SAUL OSEROFF AND PIETER H . KEESOM
0.1 0.5 1
3 5 10 30 T( K)
FIG. 22. Specific heat vs. Tfor Cdl-,Mn,Te at H
0.25
/.
-0
a20kG 028kG
a',
=
0. [After Galazka el a/. (1980).]
d
A'
FIG.23. Specific heat for Cdl-,Mn,Te, x = 0.2, 0.3, and 0.5, and pure CdTe. Dashed lines show the linear behavior of C vs. T for x = 0.2 and 0.3. [After Galazka et a/. (1980).]
3.
MAGNETIC PROPERTIES: MACROSCOPIC STUDIES
109
region below and slightly above the spin-glass freezing temperature T f . No anomalies are detected in the vicinity of Tf . In addition, only a weak magnetic field dependence is observed for these samples. Figure 24 shows an additional feature observed in Cdo.sMn0.2Se:Cex is history dependent (Amarasekara et al., 1983). The value of Cexmeasured in zero magnetic field depends on whether the sample was cooled in the absence of the magnetic field, or cooled in a magnetic field and then demagnetized at the lowest temperature. The lower inset of Fig. 24 shows two such sets of results. For very high Mn concentration, Cexshows a broad maximum. For example, for x = 0.70, such a maximum occurs around 36 K. The appearance of such a maximum is interpreted as a paramagnet-antiferromagnetic phase transition. Theantiferromagnetism which makes itself evident through the Cex maximum may be associated with antiferromagnetically ordered clusters. Finally, at the lowest temperatures, a T-' term can be distinguished in the samples with x > 0.3. It is magnetic field independent and corresponds to a nuclear specific heat. This arises when the 3d electrons polarize the 1s core electrons, that in turn produce a field of = 400 kOe at the Mn nuclei. I
Cd,-,MnxSe
0.3-
++
X=O.2O 9 H=O aH=O(demagnetized) f ' + H=POkG +
+
.
-
. .
+A
+-
f
T( K) FIG.24. Excess specific heat C,, vs. T for Cdo.8oMno.zoSe in fields of 0 and 20 kOe. Also shown are measurements in zero field after demagnetization from 20 kOe. The insert shows the zero-field data at low temperature. [After Arnarasekara et af. (1983).]
110
SAUL OSEROFF AND PIETER H. KEESOM
FIG.25. Excess specific heat C,, vs. Tfor Zno.99Mn0.01Se in H = 0, 10 kOe, and 20 kOe. The solid lines indicate the calculated values for Cex.[After Keeson (1986).]
As an example of the excess specific heat for low concentration samples (x < 0. lo), the results for Zn0.99Mn0.01Seareshown in Fig. 25 (Keesom, 1986). In zero magnetic field, C,, decreases monotonically with T ( T 2 0.3 K). In magnetic fields H = 10 kOe and 20 kOe, the excess specific heat Cexshows a pronounced maximum at about 1 . 1 K and 2.2 K respectively.
b. Analysis ( f ) Low Mn Concentrations
As mentioned above, the spin-glass paramagnetic phase boundary originally indicated the concentration of x = 0.17 as a lower limit for the existence of the spin-glass phase (Nagata et al., 1980; Galazka et af., 1980; Oseroff, 1982; McAlister et al., 1984). This is close to the critical concentration for the fcc (= 0.195) and hcp (= 0.204) lattices when calculated for the nearest-neighbors interactions. Therefore, in various attempts to understand Cex, only nn interactions were assumed. In one approach, it was assumed that the presence of large concentrations of Mn would result in small distortions of the lattice which would greatly increase the crystal field splitting (Amarasekara et al., 1983). However, recent EPR measurements by Oseroff (1984) and Koh et al. (1984) showed that this approach is incorrect and that crystal splitting is within the experimental error, independent of
3.
MAGNETIC PROPERTIES: MACROSCOPIC STUDIES
111
the concentration of Mn. Also, as mentioned before, in an attempt to fit the < 1 K, a non-random distribution of Mn ions data with a value of (J,,n/k~( was assumed (Nagata et al., 1980; Galazka et al., 1980). It was shown, however, by Shapira et al. (1984) that the exchange integral for the DMS is Jnn/kB = -lOK, and that the data can be fitted satisfactorily with a completely random distribution. It may be concluded, therefore, that the ground state splitting of cluster systems involving only nn is so large that they cannot contribute to the excess specific heat Cexbelow 3 K for fields H < 100 kOe. When the interacting ions are farther apart than the nearest neighbor distance, a spectrum of different values of the interaction energy has to be used. This, together with the different types of clusters, pairs, triples, etc., makes it nearly impossible to do a meaningful quantitative analysis of the excess specific heat, and a qualitative approach becomes necessary. The value of C,, decreases monotonically above 0.3 K. The value at 0.3 K allows us then to deduce the lower limit of the number of ions forming of a cluster. Moreoever, the maximum of C,, at a given field allows us to estimate the number of single ions present in these samples. In Fig. 25 are shown the experimental data for Zn0.99Mno.olSe, along with the values of C e x calculated using these estimates. The agreement is quite good. The value of the interaction energy is about 0.2 K, much smaller than that expected for the nearest-neighbor interaction, and in agreement with the values estimated by Kreitman et al. (1966). The probability of an ion to be completely isolated (a “single”) is given by Eq. (18). For the Zn and Cd-based DMS an estimate of n gives about 50, while for the Hg-systems n is about 100. While the two former compounds are wide-gap semiconductors, the Hg compound in these low concentration ranges is a zero-gap semiconductor. This fact may be responsible for the difference in the values of n. The values of n estimated from the low temperature susceptibility (Kreitman et al., 1966; Davydov et al., 1980; Novak et al., 1981, 1984, 1985) and specific heat agree very well. (g) High Mn Concentrations For x > 0.2, Cexshows a linear dependence on Tin the region T 5 r f ,and shows no anomaly around Tf (Galazka et al., 1980; Nagata et al., 1980; Amarasekara et al., 1983). A similar behavior was found in metallic spinglasses (Wenger and Keesom, 1975, 1976; Nagata et al., 1979). It was found for many DMS systems that the field dependence of C,, decreases as the concentration x increases. This can be understood by assuming the presence of loosely bound spins, that contribute to C e x when they break away from Mn clusters and whose number decreases as the concentration is increased. For Cdo.~1Mno.20Se, Amarasekara et al. (1983) observed that C e x measured at low temperature and in zero external field depended on how the sample
112
SAUL OSEROFF AND PIETER H. KEESOM
was cooled. That is, it depended on whether the cooling took place in zero field or whether it was done in the presence of a magnetic field and then, after the field was turned off, the actual measurement was performed. The differences between the two cases are shown in the inset of Fig. 24. The authors interpreted this effect as resulting from loose (uncoupled) spins that were bound to large clusters by the magnetic field and then remained frozen after the field was turned off. This suggestion is supported by the fact that a smaller effect was observed as x increased, until for x = 0.40 the effect was not observed at all, agreeing with an almost complete absence of loosely bound spins. 4. REMANENT MAGNETIZATION
One of the characteristic features of SG systems is the presence of a timedependent remanent magnetization ( M R)below a certain temperature TR (Tholence and Tournier, 1974; Guy, 1978; Gray, 1980; Oseroff et al., 1982). In metallic SG, TRwas found to be within a few percent of the temperature Tf ,where a cusp in the low field susceptibility is observed (Chamberlin et al., 1981; Oseroff et al., 1982, 1983). The value of MR depends on the history of the sample. If the sample is cooled from T > TR to Tz < TR in the presence of an external magnetic field, and the remanence is measured after the magnetic field is turned off, the M R is known as the thermoremanent magnetization (TRM). On the other hand, if the sample is cooled from 5 > TRto Tz < TRin zero field, and a field is turned on and after a certain time turned off again, then the MR obtained is known as the isothermal remanent magnetization (IRM) (see Tholence and Tournier, 1974). The remanent magnetization M Rin DMS was measured as a function of time t , magnetic field, temperature and concentration of Mn ions by two different techniques. One set of data was obtained by the EPR method, where the magnetization M and the EPR spectrum were measured simultaneously by observing the EPR spectra of two thin p-doped silicon spin EPR markers glued to the sample (see Schultz et al., 1980 and Oseroff, 1982). In Fig. 26 the increase and decay of TRM and IRM with time, obtained by the EPR method, is given for Cdo.6oMno.~oSe.The other set of data for MRin DMS was obtained by using conventional magnetometers. While there remains no doubt about the existence of remanence phenomena in DMS the results of M R reported by different authors are in considerable, quantitative disagreement. The M R data reported by Oseroff (1982) and Oseroff and Gandra (1985) show behavior similar to that observed in other metallic or non-metallic spin-glasses. However, the MR data reported by Escorne et al. (1981) and Escorne and Mauger (1982) suggest that the freezing process of the Mn spins in DMS can be analyzed in terms of
3.
113
MAGNETIC PROPERTIES: MACROSCOPIC STUDIES
h
rr, (u
Y
0.3
0 L N
I
I
d
d
0.2 I
I
?
0 L
r
I
0.1 d
I 2
1
C I 0;
10
/Y
L _ _ L
25 50 100
---1 I
500 1000
5000
t(s)
FIG. 26. EPR measurements of the increase (i) and the decreae (d) of TRM and IRM as a function of time for Cdo.aoMno.40Seat 4.2 K, normalized to the difference in linewidth between H F C obtained , after FC in the resonance field and HZFC(25), measured after ZFC with the field on for 25 seconds. [After Oseroff (1982).]
clustering, supporting the idea that the DMS behave like mictomagnetic compounds rather than spin glasses. Figure 27 shows the time dependence of TRM for Cdo.60Mn~.4~Te. The evolution with time of the ZFC susceptibility for Cdo.60Mno.40Teis given in Fig, 28. The temperature dependence of TRM in Cd, -,MnxTe is given in Fig. 29. This figure shows that MRvanishes at TRwhich is close to r f . The results shown in Figs. 27 and 29 are in contradiction with the temperature dependence of M R reported for Cdl -xMnxTe by Escorne and Mauger (1982). They observed the remanence well above T f and found that MR did not depend on time on the scale of one-half hour. Also, Galazka et al. (1980) did not detect, within spans of 30 min, any time dependences for the zero-field-cooled magnetization. Possible explanations for this disagreement will be discussed later in this section. As mentioned above, the remanent magnetization depends also on the strength of the magnetic field applied. An example of such dependence is presented in Fig. 30 for Cd0.6Mn0.4Te. The study of the remanent magnetization MRmay provide further insights into whether the ZFC cusp observed in DMS should be associated with a SG transition or whether their behavior is rather more typical of mictomagnets, where the cusp results from cluster freezing. As mentioned above, only few measurements of M R in DMS were reported so far, and the data disagreed. Oseroff (1982) and Oseroff and Gandra (1985) fitted the time dependence of MR for Cdl-,Mn,Te and Cdl-,Mn,Se with a log(t) law, MR(t) = M R ( t o ) k s l o g t / t o ,
(19)
114
SAUL OSEROFF AND PIETER H. KEESOM
0.7 6 -
I
cdO.60Mn0.40Te 0)
3 0.74-
-
9
z
$. 0.72-
z oc !-
&
0.70-
7
0.68
FIG.28. Time dependence of the ZFC susceptibility for Cdo.60Mno.40Te,measured in 18.5 Oe at 5.5 K . [After Oseroff and Gandra (1985).]
where to was the initial time and S the slope of the logarithmic decay. Figure 31 gives S vs. H in Cdo.soMn0.40Tefor the thermo-remanent magnetization (TRM) and the isothermal remanent magnetization (IRM). A similar behavior was observed in metallic and insulating SG (see Guy, 1978, Ferre et al., 1980, and Oseroff et al., 1982). Kinzel(l978) and Dasgupta et al. (1979) performed Monte Carlo calculations of a SG system consisting of an Ising model with random interactions
3.
MAGNETIC PROPERTIES: MACROSCOPIC STUDIES
115
T( K) FIG.29. TRM vs. Tfor Cdl-xMnxTe, obtained by FC to T = 1.8 K in 100 Oe and measured after the field was turned off for 3 minutes. The arrows indicate the maximum of ZFC susceptibility for the three concentrations measured. [After Oseroff and Gandra (1985).]
and approximated the decay of MR by a power law of the following form: MR(f) = M R ( ~ O ) ~ - ~ ,
(20)
where CY is an exponent to be determined from experimental data. Oseroff and Gandra (1985) attempted to describe their data using Eq. (20), and Fig. 32 gives CY vs. Nobtained in this manner for Cdo.60Mno.40Te. The observed dependence of (Y on magnetic field for TRM and IRM agreed well with the calculations of Kinzel (1978) and Dasgupta et al. (1979), and also with the results obtained for metallic spin glasses (Tovar et af., 1985). However, Escorne et af. (1981) and Escorne and Mauger (1982) measured MR for CdI-,MnxTe and reported that it did not depend on time on the scale of onehalf hour. Another conflicting point is that Oseroff (1982) and Oseroff and Gandra (1985) measured MRvs. Tand found that MRvanished at a temperature close to r f . Corresponding TRM vs. Tfor Cdl-,Mn,Te is given in Fig. 29, similar to the dependence found in canonical SG. Instead, Escorne et af. (1981) and Escorne and Mauger (1982) reported that M Rfor Cdo.,oMno.3oTe was almost
116
SAUL OSEROFF AND PIETER H. KEESOM I
I
I
,
I
I
Cd0.60Mn0,40Te
-
TRM + IRM
I
I
h
t
T=6K Cd0.60 Mn0.40Te
0.3
"0
10
20
30
H (kOe)
40
50
FIG. 31. Variation of the slope S for the log(t) decay of TRM (0)and IRM (0)for Cdo.soMno.aoTeas a function of magnetic field. The slope was obtained from the change of M R between 1 0 0 s and 1ooOs. [After Oseroff and Gandra (1985).)
3.
MAGNETIC PROPERTIES: MACROSCOPIC STUDIES 1
I
I
I
I
117
I
CdO 60M "0 40Te
T = 6K
.I
*
u
' 0
10
20
30
40
50
H (kOe) FIG. 32. Field dependence of the exponent a of the power law M ( t ) = M(t,)t-" for Cdo.mMno.4oTemeasured at T = 6 K, (0)IRM data and (0)TRM data. [After Oseroff and Gandra (1985).]
temperature independent below T f and did not disappear at T f , but a measurable remanence persisted up to 30 K, well above T f . So, on the one hand, the MR data reported by Oseroff (1982) and Oseroff and Gandra (1985) supports the idea that DMS behave like spin glasses. But, on the other hand, Escorne et al. (1981) and Escorne and Mauger (1982) data suggest that the whole freezing process of the Mn spins can be analyzed in terms of clustering, and that the DMS behave as mictomagnetic compounds and not as SG. Neglecting experimental errors in taking or analyzing the data, possible reasons for the differences can be: the use of inhomogeneous samples by Escorne et al. (1981) and Escorne and Mauger (1982), or that the data was taken under different physical conditions, like cooling in different magnetic fields or measuring after different times once the magnetic field was turned off. This point, however, requires further clarification. IV. Concluding Remarks The bulk of the data concerning magnetic susceptibility, specific heat and EPR accumulated so far provides a fairly thorough understanding of magnetic properties of DMS. It is now quite firmly established that the distribution of Mn ions throughout the host semiconductor lattice obeys random statistics. Also, the values of nearest neighbor exchange constants J n n are known rather accurately. It is quite likely (although this fact requires further confirmation) that superexchange is the dominant mechanism
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SAUL OSEROFF AND PIETER H. KEESOM
leading to the coupling of Mn magnetic moments. Correlations between values of Jnnand lattice parameters noted in various DMS (Spalek et al., 1986) are consistent with the above notion. The fact that DMS are available in rather wide ranges of Mn2+concentration enabled one to trace the development of specific features in various magnetic properties with increasing numbers of magnetic constituent incorporated in the host lattice. In particular, for dilute samples (x c 0.001), the data are quite well understood. In this range of concentrations, the changes of the values of the gyromagnetic factor, hyperfine coupling constant and crystal field parameters can be explained within one of the models proposed: distortions caused by the different ionic radii of Zn2+, Mn2+,Cd2+,and Hg2+;changes in the covalent character of the bonds for Te, Se, and S, and variations of the band structure. The high temperature susceptibility for these diluted samples showed a Curie-Weiss behavior with B < 1 K. For temperatures below 0.1 K, a departure from a Curie-Weiss law was found for CdI-,Mn,Te and Cdl-,Mn,Se for x as small 0.0005. This may be a consequence of the presence of hyperfine and crystal field splittings. As x increases, the interaction between the Mn becomes evident, a change in the shape of the individual resonances lines in the EPR spectra beginning to manifest itself for x > 0.002. The spectrum becomes a structureless single line with a narrower width for all the DMS with x = 0.02. Contradictory values of 0 for x < 0.05 reported on the basis of the high temperature susceptibility data seem to result from the use of inhomogenous samples or incorrect values for the host diamagnetic susceptibility. More recent studies indicate rather consistently that the exchange interaction between the Mn2+ ions is antiferromagnetic for all values of concentration. This conclusion is confirmed by a very direct determination of Jnn by Shapira et al. (1984) and Aggarwal et al. (1984, 1985). The low concentration data (x 5 0.1) can be analyzed in terms of a spincluster model. It is now clear that a good fit to the specific heat and susceptibility data can be obtained for x s 0.10 and T z 2 K by taking into account only the interactions between Mn nearest-neighbors, with JJn,/kb)= 1 0 K for all the DMS, and assuming a random distribution of Mn2+ ions in the lattice. For lower temperatures, interactions with farther neighbors (fn) should be included. Even accepting that the value of ( J f , , k ~5l 1 K, a systematic analysis of the data including farther-neighbors is missing in the literature. (Recently, an attempt to do such an analysis in the context of magnetization steps was done by Larson et af., 1985.) The broadening and shift observed for the EPR line with x > 0.03 and decreasing temperature can be understood qualitatively by an increase of an internal field due to the presence of finite clusters. However, a quantitative theory of these phenomena is still lacking.
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119
Above x k 0.17 with T 2 1.3 K the EPR, susceptibility and specific heat data show several features that can be associated with spin glass transition. All DMS show a cusp for the ZFC susceptibility measured at low field. The temperature dependence of the maximum with magnetic field is similar to other systems exhibiting a spin glass behavior. The specific heat shows a linear temperature dependence for T < Tf and no anomaly around 3.For x < 0.17, the low temperature magnetization results appear to indicate a similar behavior to those observed in the high concentration region. However, at this point there is not enough data to assure that DMS show a spin-glass behavior in the whole range of concentrations in which these materials exist. Conspicuous in this field is our lack of understanding of the phenomena related to remanent magnetization in the spin glass phase. Also, the question of the nature of the antiferromagnetic ordering in very concentrated samples (x 5: 0.60) requires still further investigation (for discussion of this problem, see the contribution by Giebultowicz and Holden in this volume). It is to be hoped that from the current intense activity a deeper and more thorough understanding of magnetic properties of DMS and of mechanisms underlying these will shortly emerge.
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SEMICONDUCTORS AND SEMIMETALS, VOL. 25
CHAPTER 4
Neutron Scattering Studies of the Magnetic Structure and Dynamics of Diluted Magnetic Semiconductors T. M . Giebultowicz DEPARTMENT OF PHYSICS, UNIVERSITY OF NOTRE DAME NOTRE DAME, INDIANA, USA AND REACTOR DIVISION, NATIONAL BUREAU OF STANDARDS GAITHERSBURG, MARYLAND, USA
and T. M. Holden ATOMIC ENERGY OF CANADA LIMITED, CHALK R N E R NUCLEAR LABORATORIES, CHALK RN ER, ONTARIO, CANADA
INTRODUCTION.. . . . . . . . . . . . . . . . . . . 1 . Motivafion and Introductory Remarks . . . . . . . . . 2 . Theory of Neutron Scattering in Systems with Short-Range Magnetic Order . . . . . . . . . . . . . . . . . . 3 . Theoretical Models of Spin Correlation Phenomena and Spin Dynamics in DMS Systems. . . . . . . . . . . . 11. POWDER DIFFRACTION EXPERIMENTS ON Znl-,Mn,Te. . . . . 4. Elernentsof the Experimental Technique. . . . . . . . . 5 . Results of Experiments . . . . . . . . . . . . . . . 111. NEUTRONSCATTERING STUDIES OF Cdl -,Mn,Te. . . . . . . 6 . Samples and Elements of Experimental Technique. . . . . 7 . Dvfraction Studies of Cdl -,Mn,Te Single Crystals. . . . . 8 . Inelastic Neutron Scattering in Cd0. 35M no.ssT e . . . . . . IV. CONCLUDING REMARKS. . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . I.
125 125 128 135 140 140 141 155 155 157 173 177 179
I. Introduction 1 . MOTIVATION AND INTRODUCTORYREMARKS
Thermal neutron scattering provides an extremely powerful method for investigating a number of important properties of solids. Since the de Broglie wavelength of thermal neutrons ( A 10-8cm) is of the same order of
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125 Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-752125-9
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T. M. GIEBULTOWICZ AND T. M. HOLDEN
magnitude as the interatomic spacings in condensed matter, pronounced interference effects take place in the scattering of neutrons from solids, and such effects can yield information about the structure of the scattering specimen. Moreover, the energy of thermal neutrons (10-100 meV) is comparable to typical energies of excitations in solids. The relative energy changes undergone by neutrons during inelastic scattering events involving creation and/or annihilation of these excitations are therefore large, and thus readily measurable. This makes neutron scattering an excellent tool also for studies of dynamic phenomena in condensed matter. In any material medium, the neutrons are scattered by nuclear forces, giving information about the structure and dynamics of the crystal lattice. In addition, however, the neutron also possesses a magnetic moment y = 1.913 ,UN (nuclear magnetons) which can interact with atomic magnetic moments, if such moments are present in the material. Elastic scattering arising due to this interaction depends on the static correlations between the magnetic moments of the atoms. Measurements of this effect can therefore be used as a method for studying the magnetic order (such as spin structures in magnetic crystals, or short-range ordering phenomena in spin-glasses). Inelastic magnetic scattering, on the other hand, depends in general on the spin-spin correlations both in space and in time. This mode of the scattering can yield information concerning collective magnetic excitations, such as the energies and the dispersion relations for the magnon modes, or the various spin relaxation processes occurring in disordered magnetic systems. Often neutron scattering is the only direct means of probing the magnetic structure in the case of antiferromagnets, or of determining the dispersion relations for the spin waves over a broad range of wave vectors. Neutron scattering studies in diluted magnetic semiconductors (DMS) are important for two reasons. First, as discussed in other chapters in this volume, many novel physical effectsseen in these materials arise because the conduction and/or the valence band electrons are strongly coupled via the spin-spin exchange interactions to the localized moments of the magnetic atoms. A detailed knowledge of the properties of the magnetic ion subsystem, and especially the knowledge of the distribution of the local exchange fields and their spatial and temporal fluctuations, are therefore essential for the understanding of the microscopic mechanisms underlying these effects. Here, neutron scattering methods are indeed quite helpful and, as mentioned above, are sometimes the only way of acquiring the necessary information in this context. Secondly, DMS are in their own right interesting magnetic materials, often referred to as “frustrated” antiferromagnets. This term describes a system where the energies of all pairs of antiferromagnetically bonded spins cannot be minimized simultaneously. A prototypical example of a frustrated
4. NEUTRON DIFFRACTION STUDIES
127
antiferromagnet is the face centered cubic (FCC) array of spins, each spin interacting antiferromagnetically only with its nearest neighbors. The A:'_,Mn,BV' DMS are very close realizations of such a system. The consequences of the frustration of the spin lattice are far reaching in the context of magnetic properties of these materials. For example, as a consequence of frustration, the ground state of the system is highly degenerate, there is no long range order in three dimensions, and the mechanism of spin excitations becomes very complex. Frustration may also result in a spin-glass behavior of the system. Frustrated FCC systems were subjects of many recent theoretical analyses and modelling studies (see, e.g., Villain, 1978; Grest and Gabl, 1979; de Seze, 1979; Alexander and Pincus, 1980; Ching and Huber, 1981, 1982a,b, 1984; Fernandez et al., 1983), where various aspects of this general problem are discussed. Since the cubic members of the DMS family, Cdl-,Mn,Te and Znl-,Mn,Te, resemble closely the prototype FCC frustrated antiferromagnets considered in these theoretical works, the results obtained in both elastic and inelastic neutron scattering experiments on these crystals can therefore be directly compared with the predictions of the models, providing valuable feedback for further development of the theories. In this chapter, we give a review of neutron scattering studies performed on Cdl-,Mn,Te and Znl-,Mn,Te. Since some readers of this book may be unfamiliar with neutron scattering methods, certain basic elements of neutron scattering theory are given in Sec. 2, together with a brief explanation of the data analysis procedures, in order to establish a common language. We then follow, in Sec. 3, with a short description of the models of diluted antiferromagnets used in the interpretation of the experimental results. Cdl-,Mn,Te can be obtained in single crystal form for Mn concentrations 0 s x 4 0.7. On the other hand, Znl-,Mn,Te samples used in the experimental studies described in this chapter were available only as polycrystals. Thus, different experimental techniques and data processing procedures had to be applied in these two cases. Although the materials are quite similar as far as their magnetic properties are concerned, we chose to discuss them in separate sections, hoping that this will contribute to the clarity of the presentation. Part I1 of this chapter is devoted to ZnI-,Mn,Te, where the experimental findings obtained by neutron scattering on powder samples are presented. The analysis of the short-range antiferromagnetic order observed in this material yielded direct information about the spin-spin correlation coefficients between magnetic neighbors as a function of their separation, concentration of Mn ions, and the temperature. Then, in Part 111, we discuss neutron diffraction experiments on single crystals of Cdl-,Mn,Te. In addition to the information similar to that presented in Part 11, these experiments led to the observation of the anisotropy of the magnetic
128
T. M . GIEBULTOWICZ AND T. M. HOLDEN
correlation length and permitted the study of the dynamics of the spin system by inelastic scattering methods. 2. THEORYOF NEUTRONSCATTERING IN SYSTEMS WITH SHORTRANGEMAGNETIC ORDER
a. Basic Definitions The number of neutrons counted in a detector in an experiment to measure both the energy and the angular distributions of neutrons scattered by a sample is proportional to the double differential scattering cross section, dza/dQdc'. The incident (scattered) neutrons are characterized by wave vector k(k') and energy The solid angle into which the neutrons are scattered is designated by Q. The momentum transfer, hQ, and energy transfer, h o , to the sample are given by &(&I).
hQ = hk - hk',
and h o = E - E'.
The number of neutrons counted by a detector in a diffraction experiment, to measure only the angular distribution of scattering by a sample, is proportional to the differential scattering cross section, do/dQ, which is an integral over all scattered neutron energies,
The measured double differential scattering cross section is related to the physical properties of the sample through the scattering function S(Q, o), d2a k' -- - S(Qt a ) . gad&' k
(4)
The scattering function is the time Fourier transform of the correlation function ( F ~ F Q ( ~that ) > is , determined explicitly by the Hamiltonian for the sample, as follows (Marshall and Lovesey, 1971; Sears, 1978):
' j+m
s ( Q , 0)= 277
(F&(O)FQ(t))=
v'a'ua
-m
dt e-'"''(F&O)&(t))
pyPa((v', ( Y ' I F Q I Y , O!)12E?iwa'mf.
(5)
(6)
The states of the sample are specified by a and the energy transfer to the sample is the difference of the initial and final states, h a , , , = Em 0.5,
HM= -4SJ
+ 2SJ(1 - x).
(43)
There is no evidence in the distribution of local fields that the Heisenberg system, unlike the Ising system (Grest and Gabl, 1979), freezes into a spinglass phase.
4.
NEUTRON DIFFRACTION STUDIES
139
The calculations of the dynamics show that the excitations fall in energy as the dilution increases, with an accompanying increase in lifetime. The average spin-wave energy is nearly equal in magnitude to the frequency of the type-I11 AF structure after taking the cubic average and scaling linearly with concentration. Ching and Huber (1984) have also calculated S(Q) numerically for diluted FCC antiferromagnets with J2 = 0.1 J1 . They find that S(Q) is a maximum at wave vectors like (1, i,0), which is an ordering wave vector of the type-I11 A F structure, and rapidly decreases with wave vector offset from (1, &, 0). Close to the characteristic wave vector, t, they found that S(Q) could be closely represented by the following analytical form
rather than by the Ornstein-Zernicke form alone
The first term in Eq. (44) corresponds to a spatial exponential decrease of the form e-x'', and the second to a spatial form (l/r)e-"I'at large values of r.
c. Numerical Calculations of the Correlations Our understanding of the spin structure and dynamics of disordered materials has improved with the recognition of the multicritical character of the approach to long range order. For ferromagnetism or antiferromagnetism on lattices where the nearest neighbors of a given site are not nearest neighbors of each other, long range order may be approached along the concentration axis, at the percolation concentration, or along the temperature axis. Experiments on CrFe alloys (Burke et al., 1983) and on Znl-,Mn,F2 crystals (Cowley et al., 1980) have verified the power laws in concentration and temperature that emerge from this theory. The picture becomes more complex for an FCC lattice where nearest neighbors of a given site are nearest neighbors of each other, and where there are competing second neighbors interactions. In the percolation case, clusters are isolated by nonmagnetic sites, whereas for the FCC lattice, clusters are isolated by frustrated links at the boundaries. That is, there are situations where the spin at a given site would like to point up in the molecular field of another, and the net molecular field at the site is zero. Numerical calculations showed that the ferromagnetic transition was pushed away from the percolation concentration by the frustrated links (Binder et al., 1979), Eu,Srl-,S appearing to be a physical realization of this situation (Maletta ef al., 1982). A spin-glass
140
T. M. GIEBULTOWICZ AND T. M. HOLDEN
state may, however, be established above the percolation concentration, i.e., once an infinite cluster comes into existence. The concentration at which long range order occurs may be pushed to even higher concentrations when both the first and second neighbors interactions are antiferromagnetic. The physical realizations of this scheme of exchange interactions are probably Cdl-,MnxTe and Znl-xMnxTe. Again a spin-glass state appears to be established above the percolation concentration (Escorne and Mauger, 1982; Galazka ef al., 1980). At the present time, there is no definitive evidence for true long range order in these two DMS alloys, at least for x I0.75. 11. Powder Diffraction Experiments on Znl-,Mn,Te
4. ELEMENTS OF
THE
EXPERIMENTAL TECHNIQUE
The powder diffraction measurements on Znl-,Mn,Te were made with the L3 triple-axis spectrometer at the NRU reactor, Chalk River, in the diffractometer mode, with no analyzing crystal. The (113) plane of a squeezed germanium crystal served as monochromator at a fixed take-off angle of 48.69", corresponding to a neutron wavelength A = 1.406 A or an incident energy E = 41.4 meV (10 THz). The mosaic spread of the monochromator was 0.2"and the angular collimation before and after the sample was 0.32". It was desirable to have good resolution in these experiments in order to be able to extract information from the widths of the nuclear Bragg peaks about sample homogeneity via the spread in d-spacings. It was also important to minimize the resolution correction to the characteristic widths of the sharp peaks, that develop as a function of concentration and temperature near the ordering wave vectors for the type-I11 AF structure. In these experiments, it is vital to keep EO high and to achieve good wave vector resolution by collimation. If the high wave vector resolution is achieved by decreasing the neutron energy, the diffractometer integrates less adequately over the whole energy spectrum; what is then measured is the-scattering within a certain energy window which is only part (and an unknown part) of S ( Q ) . Typical low angle nuclear Bragg peaks had widths (full width at half maximum, FWHM) of 0.4".Measurements were made over a range of scattering angles (20) from 5 to 100" to establish the coherent scattering. The powder samples were packed in thin-walled A1 cans (inner diameter 0.7 cm, length 5.0 cm) and weighed to find the effective density. A typical packing fraction was about 0.6. Each sample was placed in a variable temperature cryostat, permitting measurements to be made between 2 and 300 K. An empty-can run in the cryostat was made in order to measure the background. The Bragg scattering from the A1 can give a useful measure of the instrumental width of the powder peaks and permited extraction of
4.
NEUTRON DIFFRACTION STUDIES
141
information about sample homogeneity from the widths of the Bragg peaks of Znl-xMn,Te. The observed counts must be corrected for multiple scattering and self-shielding. Following Sears (1975a, b), the observed counts C are related to the true cross section by
where Kis the instrumental normalization factor, H(k0, k') is the absorption factor (= 10%) which is independent of angle for our experimental arrangement, and m is the multiple-scattering correction which is =1.6% for Znl-xMnxTe samples. With improved accuracy of the coherent and incoherent scattering cross sections (Koester, 1977) due to the individual components of the alloy, the analysis of the Bragg peak intensities permits the calculation of the concentration of Mn atoms in the samples as well as an effective Debye-Waller factor and the normalization factor to put the observed diffuse scattering on an absolute scale, The relative intensities of the three families of peaks in the zinc-blende structure (h + k + 1 = 4n, 4n k 2, and 4n -t 1, where n is an integer) are very sensitive to concentration, since the scattering lengths of zinc and manganese have opposite signs, The accuracy in measurement of concentration is in fact about +- 1 at.9'0. The effective Debye-Waller factor permits an estimate to be made of the thermal diffuse scattering via Eq. (26). The agreement between intensity calibrations at different temperatures was k 5 % . The normalization factor was also measured with a cylindrical vanadium sample of known diameter and irradiated length, correcting the observed counts for self-shielding and multiple scattering (8.7%). Agreement between the internal Bragg scattering normalization and with vanadium was within 5%. 5. RESULTSOF EXPERIMENTS The diffraction pattern for Zno.435Mno.565Te taken with neutrons of wavelength 1.4064 A at 4.2 K (Holden et al., 1982) for scattering angles between 5 and 75" is shown in Fig. 2. We see the expected pattern of Bragg peaks for the zinc-blende structure together with A1 powder peaks from the sample can, superposed on a diffuse background. The principal feature of the diffuse scattering is a broad peak centered on 14.5", which may be indexed as (1, i,0) in the zinc-blende structure. This peak is clearly much wider than the experimental resolution in reciprocal space given by the width of the Bragg peaks. Two other diffuse peaks are identifiable. The peak near 23.3", which is partly overlaid by the adjacent (111) nuclear peak of the zinc-blende structure, may be indexed as (1, *, 0) and the peak near
142
T. M. CIEBULTOWICZ AND T. M . HOLDEN
50001 i I
v, I-
z
3
4000
3000
z
0 IT
'r w
z
'
2000
:
I
5
1
I
.
x = 0.565 T=42K
(Ill)'
A = I3863 6
(220)
II
1
.1
I
3
I
10
I
15
I
20
SCATTERING
I
25
I
30
I
35
40
ANGLE 2 8
FIG.2. Neutron diffraction pattern for Zn,~.43~Mno.s6sTe at 4.2 K, showing Bragg scattering from the zinc-blende structure, the aluminum sample can, and strong magnetic diffuse scattering [After Dolling et al. (1982).] which peaks at 14.5' (Q = 1.1 k').
30.5" may be indexed as (1, *, 2). Similar diffraction patterns were recorded for Zno.6~4Mno.376Te, Zn0.406Mno.s94Te (Dolling el d., 1982), and Zno.32Mno.68Te. For the x = 0.68 sample, there was evidence in the diffraction pattern for small amounts of other phases, such as MnTez and hexagonal MnTe. The manganese concentration, Debye-Waller factor, lattice spacing and intensity scaling factors, obtained by fitting to the integrated powder diffraction intensities and angles are collected in Table I. The table also contains estimates, derived from the (220) and (442) nuclear reflections of Znl-,Mn,Te, of the homogeneity of the samples based on the widths of these Bragg peaks. Excess width, above the experimental resolution, was assumed to originate from a spread of lattice parameters in the sample. The homogeneity of the three samples with lowest c is excellent by this criterion. However, there appear to be concentration fluctuations in the x = 0.68 sample, which may be connected with the appearance of impurity phases. The temperature dependence of the diffuse scattering between 5 and 33" for the concentrations x = 0.376,0.594, and 0.68 is shown in Figs. 3, 4, and 5 . In each case the dashed-dot lines give the nuclear diffuse scattering,
TABLE I MANGANESE CONCENTRATION, TEMPERATURE, LATTICEPARAMETERS, DEBYE-WALLER FACTOR,AND BRAGGPEAKDATAINDICATING HOMOGENEITY OF THE Zn,-,Mn,Te SAMPLES STUDIED.
37.6 k 0.3 59.4 f 0.2 59.3 k 0.3 68.0 & 0.7 56.7 f 0.05 56.3 k 0.6 56.5 k 0.6
77.6 240 4 300 4 300 17
6.176 6.236 6.230 6.263 6.218 6.234 6.218
0.0051 t O.OOO4 0.0131 t 0.0006 0.0047 f 0.0008 0.0156 t 0.0021 0.0035 f 0.0017 0.0162 t 0.0026 0.0051 0.0021
*
"The instrumental width for this reflection is 0.435 f 0.02". bThe instrumental width for this reflection is 0.903 k 0.04".
0.47 f 0.02
0.90
0.04
0.06 rt 0.04
0.0 t 0.15
0.45 k 0.02 0.53 k 0.02
0.92 f 0.04 0.95 f 0.04
0.04 0.05 0.097 f 0.20
0.03 t 0.06 0.048 f 0.050
_+
...
I600 1200 -
X =0.376
4.2 K M=3
. -. . '--0.5
000 400 -
-
- 1.0
/ -
*
-
-
--- - ----
- - - - - - - - - - - - - - - - - - - - - - - - - - - -_ -
-
c
& 0 u
-
c
C
3 I
0
ln
1200
1
I
I
I
*... ...
000-
0 V
400 -
0
1200 -
800 400-
0
Q
77 K
-
I-
z 2
I
1
. . .. - . , - ... _ :. ---- I 5
-----
-
I 10
1
_ . . . a
I
1
1
I
I
- .. - .
I-
r0.5 u w
1
e.
--
-.
ul ul ul
0
I
- 1.0
300 K
....
.*
- . - - -... . *
0.5
- - - - - - - - - - - - - - - - - --- - - - - - -_ _ I
I
15
20
I 25
I 30
-2 z 0
- - - - - - - - - - - - - - - - - - - - - - - - - -___
-- --------1
* -
-1.0
7;
0
50
cl
?
..800 -
.?
+-\
X *0.!594 M=I
400
- 2.0
4.2 K
t
-
Y
-- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - I
0 In
I
1
I
I
s
.-
C
(200)
I I
I
I
a
-2.0 7.
L
78 K
-
I-
z
(111)
u
c
.*
400-
*
m
4
.*
1.0
-- - - _- - - - - - - - _ _ - _ - - -- - - _ - -- - - - - - - - - - - - - - - - - - - 0.
I
I
I I
I
I
I
I
400 -
I
I
.. - - .
I
~*
-.. .
-- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 0-
I
5
I 10
-2.0
240 K ------v
I 15
I 20
1 25
z
0
~
I 30
-1.0
V W
In In
g a u
: 2000 z 3
g
I
1000
1
1000
78.0K
.... - _T1.0
. . - * - -
-
. - .'-Pp,/E~A and x/3<Sz)Eg):
where 2 P2 A=-3 A2'
at
= [E f
a!'
- &f
-
(34) - P'
[
U(z)] Eg +
&+
7+ E f
-
w].
(35)
In Eqs. (34) and ( 3 9 , Pdenotes the interband momentum matrix element and a' = + X a ! ( S Z ) ,
(36)
8' = +x/3<S,>,
(37)
while E denotes the eigenvalue of energy that we are looking for. Plus and minus indices in Eqs. (33) and (35) refer, respectively, to solutions for predominately spin-up and spin-down states in the conduction band. The parameter cf is defined by
with
where n = 0, 1, ... denotes the Landau quantum number, and s = (hc)/(eH). The WKB quantization condition can be written in our case as p Z d z = hn(r
+ $),
r = 0, 1 , 2 , ...
(40)
where za = 0 and q5 = $ for the case of the surface layer potential, while zo = - zo and $ = for the layer at the grain boundary. Here zo denotes the classical turn point of the motion in a confining potential well, i.e., it is
5.
QUANTUM TRANSPORT IN DMS
223
defined by PZ(Z0) =
0
The smallest root of Eq. (41) corresponds to solutions for the conduction band 20’ =
f ayl - Ef
E
uo
where we have explicitly used the form of the potential assumed
for the surface layer, and
U(z) = Uolzl
(44)
for the grain boundary potential. The integration indicated by Eq. (40) can be carried out analytically, giving the following equations
where
a*
=E 7
a’ - ~ f ,
b* = Eg + and
G=
E
+ ~f F B’ -, 3
[
1 for the surface layer .2 for the grain boundary layer
Equation (45) has now to be solved numerically for the energies of the n-th Landau level with spin (+) or (-) in the r-th electric subband. A similar calculation can be carried out when no external magnetic field is applied. The result is formally very similar to that obtained for InSb, with the quantity ~f redefined (cf., Zawadzki 1983):
where m3 = h2(4P2/3Eg)-’ and kll is the momentum within the 2d layer. Solving Eq. (45) with the definition (49) of e f , we may calculate the position
224
J. KOSSUT
of the Fermi level ~ f related . to the electron concentration ns in the r-th subband. 1 nd = -kl;. 2n By calculating the derivative of ~ fwith . respect to kll, we may calculate the effective mass at the Fermi level. Theoretical values of the effective mass listed in Table I were calculated in this manner. By choosing a proper value of the potential slope U o ,one can now fit the electron concentrations in each of the observed subbands and the effective masses to the experimental data, assuming a single Fermi level. A fair agreement can be reached indicating a proper choice of the shape of the potential (and its slope) at the grain boundary. By solving Eq. (45) with given by Eq. (38) for a fixed magnetic field, one can finally determine the value of the g-factor of the nth Landau level as
The g-factors calculated in this way give a fairly good description of the values observed in Hgl- x-yCdxMnyTe.At lower temperatures, they turn out to be positive due to the large contribution of the exchange interaction terms. As T increases (and ( S , ) decreases), the value of the g-factor drops, becoming nearly zero at the highest temperatures at which the oscillations were studied. Let us mention also that the values of the g-factor for the 2d gas of electrons at the grain boundary are slightly different from those expected for bulk electrons. The above results indicate that the influence of the localized magnetic moments also manifest itself in the properties of 2d electrons in diluted magnetic semiconductors. 8 . FUTURE POSSIBILITIES
Other 2d electronic systems in diluted magnetic semiconductors, e.g., those in quantum wells produced by heterostructures and/or superlattices, seem to offer quite interesting possibilities. One of those, noted by von Ortenberg (1982), makes use of the temperature and magnetic field variation of the band edges. Thus, if a superlattice is grown with alternating layers of nonmagnetic (e.g., Hgl- .Cd,Se) and diluted magnetic (e.g., Hgl- xMnxSe) semiconductors, then it should in principle be possible to change the relative position of the band edges in adjacent layers by means of varying the field and/or the temperature. It is argued that the resulting minigap of a superlattice could be tuned by these external factors. However, such possibilities are still be be explored.
5.
QUANTUM TRANSPORT IN DMS
225
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Kaplan, T. A., and Lyons, D. H. (1963). Phys. Rev. 129, 2072. Karczewski, G., and Kowalczyk, L. (1983). Solid State Commun. 48, 653. Kasuya, T. (1955). Progr. Theor. Phys. 16, 45, 58. Kasuya, T. (1966). In “Magnetism” (G. T. Rado, and H. Suhl, eds.) vol. 2B, p. 215. Kasuya, T., and Lyons, D. H. (1966). J. Phys. SOC. Japan 21, 287. Kildal, H. (1974). Phys Rev. B10, 5082. Kossut, J. (1976). Phys. Stat. Sol. (b) 78, 537. Kossut, J . (1978). Solid State Commun. 27, 1237. Leung, W., and Liu, L. (1973). Phys. Rev. BS, 3811. Liu, S. H. (1961). Phys. Rev. 121, 451. Luttinger, J. M. (1956). Phys. Rev. 102, 1030. Luttinger, J. M., and Kohn, W. (1955). Phys. Rev. 97, 869. Lyapilin, I. I., Ponornarev, A. I., Kharus, G. I., Gavaleshko, N. P., and Maryanchuk, P. D. (1983). Zh. Eksp. Teor. Fiz. 85, 1638. (Sov. Phys. JETp58, 953.) Lyapilin, I. I., and Karyagin, V. V. (1980). Fiz. Tverdogo Tela 22,206. (Sov. Phys. SolidState 22, 118.) Mitchell, A. H. (1957). Phys. Rev. 105, 1439. Morrissy, C. (1973). Ph.D. Thesis, Oxford, unpublished. Mycielski, A., Kossut, J., Dobrowolska, M., and Dobrowolski, W. (1982). J. Phys. C15,3293. Neve, J. J., Bouwens, C. J. R., and Blom, F. A. P . (1981). Solid State Commun. 38, 27. Neve, J . J., Kossut, J., van Es, C. M., and Blom, F. A. P. (1982). J. Phys. C15, 4895. Niewodniczanska-Zawadzka, J . , Kossut, J . , Sandauer, A., and Dobrowolski, W. (1982). In “Physics of Narrow Gap Semiconductors” (E. Gornik, H. Heinrich, and L. Palmetshofer, eds., Proc. Internat. Conf. Linz, 1981)LectureNotesinPhysics,vol. 152, p. 326, SpringerVerlag. Niewodniczanska-Zawadzka, J., Elsinger, G., Palmetshofer, L . , Lopez-Otero, A., Fantner, E. J., Bauer, G., and Zawadzki, W. (1983). In “Physics of Semiconductors” (M. Averous, ed., Proc. 16th Internat. Conf. Montpellier, 1982) part I, p. 458, North Holland (Physica 117-118 B + C). von Ortenberg, M. (1982). Phys. Rev. Lett. 49, 1041. Pascher, H., Fantner, E. J., Bauer, G., Zawadzki, W., and von Ortenberg, M. (1983). Solid State Commun. 48, 461. Pastor, K., Grynberg, M., and Galazka, R. R. (1979). Solid State Commun. 29, 739. Pidgeon, C. R . , and Brown, R. (1966). Phys. Rev. 146, 575. Ponikarov, B. B., Tsidilkovskii, I. M . , and Shelushinina, N. G . (1981). Fiz. Tekh. Poluprov. 15, 296. (Sov. Phys. Semicond. 15, 170.) Reifenberger, R., and Schwarzkopf, D. A. (1983). Phys. Rev. Lett. 50, 907. Roth, L., and Argyres, P. (1966). In “Semiconductors and Semimetals” (R. K. Beer, and A. C. Willardson, eds.) vol. 1, p. 159, Academic Press. Sandauer, A. M., and Byszewski, P. (1982). Phys. Stat. Sol. (b) 109, 167. Sandauer, A. M., Dobrowolski, W., Kossut, J., and Galazka, R. R. (1983). In “Physics of Semiconductors” (M. Averous, ed., Proc. 16th Internat. Conf. Monpellier, 1982) part I, p. 455, North Holland (Physica 117-118 B + C). Sawicki, M., Dietl, T., Plesiewicz, W., Sekowski, P., Sniadower, L., Baj, M., and Dmowski, L. (1982). In “Application of High Magnetic Fields in Physics of Semiconductors” (G. Landwehr, ed., Proc. Internat. Conf., Grenoble, 1982). Springer-Verlag. Schrieffer, J. R.,and Wolff, P. A. (1966). Phys. Rev. 149, 491. Seager, C. H. (1982). In “Grain Boundaries in Semiconductors” (H. J. Leamy, G . E. Pike, C. H. Seager, eds.) North Holland. Semenov, Yu. G . , and Shanina, B. D. (1981). Phys. Stat. Sol. (b) 104, 631.
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Stankiewicz, J., Giriat, W., and Bien, M. V. (1975). Phys. Stat. Sol. (b) 68, 485. Story, T., Lewicki, A., and Szczerbakow, A. (1985). Acta Phys. Polon. A67, 317. Takeyama, S.,and Galazka, R. R. (1979). Phys. Stat. Sol. (b) 96, 413. Toth, G., and Leloup, Y. Y. (1970). Phys. Rev. B1, 4573. Uchida, S., Landwehr, G., and Bangert, E. (1983). Solid State Commun. 45, 9869. Vonsovskii, S. V., and Svirskii, M. S. (1964). Zh. Eksp. Teor. Fiz. 47, 1354. (Sov. Phys. JETP 20, 914.) Vonsovskii, S. V., and Turov, E. A. (1953). Zh. Eksp. Teor. Fiz. 24, 419. Vul, B. M., and Zavaritskaya, E. T. (1979). Zh. Eksp. Teor. Fiz. 76, 1089. (Sov. Phys. JETP 49, 551.) Wallace, P. R. (1979). Phys. Stat. Sol. (b) 92, 49. Watson, R. E., and Freeman, A. J. (1966). Phys. Rev. 152, 566. Watson, R. E., and Freeman, A. J. (1969). Phys. Rev. 178, 725. Wittlin, A., Knap, W., Wilamowski, Z., and Grynberg, M. (1980a). Solid State Commun. 36, 233. Wittlin, A., Grynberg, M., Knap, W., Kossut, J., and Wilamowski, Z. (1980b). J. Phys. SOC. Japan 49 (suppl. A), 635. (Proc. 15th Internat. Conf. Phys. Semicond. Kyoto, 1980.) Yosida, K. (1957). Phys. Rev, 106, 893. Zawadzki, W. (1983). J. Phys. C16, 229. Zener, C. (1951a). Phys. Rev. 81, 440. Zener, C. (1951b). Phys. Rev. 82, 403. Zener, C. (1951~).Phys. Rev. 83, 299.
SEMICONDUCTORS A N D SEMIMETALS, VOL. 25
CHAPTER 6
Magnetooptics in Narrow Gap Diluted Magnetic Semiconductors C. Rigaux GROUPE DE PHYSIQUE DES SOLIDES DE
L’ECOLE NORMALE SUPERIEURE PARIS, FRANCE
INTRODUCTION. . . . . . . . . . . . . . . . . . . . THEORETICAL DESCRIPTION OF ELECTRONIC STATESIN NGDMS 1 . Band Structure . . . . . . . . . . . . . . . . . . 2. Landau Levels . . . . . . . . . . . . . . . . . . 3. Magnetooptical Transitions . . . . . . . . . . . . . 111. INTERBANDMAGNETOOPTICS. . . . . . . . . . . . . . . 4. Zero Gap Hg,-,Mn,Te. . . . . . . . . . . . . . . 5. Narrow Gap H g l - , M n x T e . . . . . . . . . . . . . . 6 . Other Semimagnetic Compounds. . . . . . . . . . . IV. INTRAEIANDMAGNETOOPTICS . . . . . . . . . . . . . . 7. Far IR Magnetospectroscopy in Zero Gap DMS. . . . . 8 . Harmonics of the Cyclotron and Combined Resonances in Narrow Gap Semiconductors . . . . . . . . . . . . V. RESULTS OF MAGNETOOPTICAL STUDIES.. . . . . . . . . . 9. Band Parameters . . . . . . . . . . . . . . . . . 10. Exchange Parameters . . . . . . . . . . . . . . . VI. CONCLUDING REMARKS. . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . I. 11.
229 23 1 232 233 237 240 24 1 248 252 257 257 264 267 267 269 27 1 272
I. Introduction Diluted Magnetic Semiconductors (DMS) are characterized by the presence of magnetic ions of a transition element, that enter substitutionally into the semiconductor lattice. The interest in this class of magnetic semiconductors arises from the possibility of controlling the dilution of magnetic ions and the well known band structure of the host semiconductor (typically a 11-VI compound). Thus DMS offer an unique opportunity of gradually incorporating the exchange phenomena, characteristic of magnetic semiconductors, into the electronic properties of classical semiconductors. Narrow-gap DMS are ternary random alloys obtained by substituting magnetic ions (Mn”, Fez+) for Hgz+ in mercury chalcogenides (HgTe, HgSe). Among them, the system Hg1-xMnxTe has been studied particularly 229 Copynght 8 1988 by Academic Press. Inc. All rights o f reproduction in any form reserved. ISBN 0-12-752125-9
230
C. RIGAUX
thoroughly. Both groups of compounds exist in zinc blende structure, forming a single crystallographic phase up to x = 0.5 for Hgl-,Mn,Te and x = 0.3 for Hgl-,Mn,Se. The Te (or Se) atoms occupy the sites of one of the fcc sublattices, whereas mercury and manganese atoms are more or less randomly distributed over the second fcc lattice sites. The delocalized electronic states responsible for the semiconducting properties of these compounds originate from s-(cations) and p-(anions) type orbitals. In the absence of magnetic field, the band structure of Hgl-,Mn,Te or Hgl-,MnxSe is very similar to that of non-magnetic ternary alloys such as Hgl-,Cd,Te or Hgl-,Cd,Se, displaying either a zero gap or a narrow gap semiconductor band structure, with the interaction gap depending on composition. In Hgl -,Mn,Te, the interaction gap separating s(I-6)- and p(rg)-like bands varies from -0.3 t o 1 . 1 eV in the range of composition 0 Ix I0.5. The semiconducting properties of these alloys may be controlled by varying the composition. The essential peculiarities of DMS originate from the unfilled d shells of transition elements. The d electrons tightly bound to the Mn2+ions give rise to localized magnetic moments that are responsible for the magnetic properties of these compounds: the magnetic susceptibility of Mn-based DMS follows a Curie-Weiss law which evidences antiferromagnetic interactions (Spalek et al., 1986) between magnetic ions, even in diluted alloys (Galazka, 1982). The existence of a spin-glass phase, at low temperature, is established from several experiments in HgMnTe (Nagata et al., 1980; Brandt et al., 1983; Mycielski et a/., 1984) and HgMnSe (Khattak et a/., 1981). In the presence of an external magnetic field, the electronic properties of DMS are strongly influenced by the exchange coupling between d electrons and mobile carriers. In NGDMS, the direct effect of magnetic field on the delocalized carriers leads to the Landau quantization, as these compounds are characterized by small effective masses. Moreover, the magnetic field orients the localized magnetic moments, creating a non-vanishing average component of the Mn spins along its direction. The exchange interactions between a delocalized electron and the Mn spins drastically affect the spin properties of mobile carriers in quantizing magnetic field. The effect of magnetic field on electronic states in NGDMS has been analyzed theoretically (Bastard et al., 1978; Jaczynski et a/., 1978; Galazka and Kossut, 1980) in terms of a model where, using the virtual crystal and molecular field approximations, the exchange interactions are incorporated into the framework of the Pidgeon-Brown formalism (Pidgeon and Brown, 1966). Energies of Landau spin sublevels are then modified by exchange contributions involving the d electron magnetization and become temperature dependent.
6.
MAGNETOOPTICS IN NARROW GAP DMS
231
The role of exchange interactions on electronic properties of DMS has been discussed in several review articles (Galazka, 1978; Bastard et al., 1980; Galazka and Kossut, 1980; Gaj, 1980; Mycielski, 1981; Dietl, 1981; Furdyna, 1982; Brandt and Moshchkalkov, 1984). Magnetooptics provide a particularly useful technique for investigating the exchange phenomena and the localized spin ordering in wide gap (Gaj et al., 1978) and narrow gap DMS (Bastard et al., 1980). Near and far infrared (IR) spectroscopy involving interband and intraband transitions between Landau levels have revealed striking magnetooptical anomalies in zero and narrow gap semimagnetic semiconductors. These are related to the effect of exchange on delocalized electronic states: the occurrence of anomalously large and temperature-dependent gyromagnetic factors of conduction electrons (positivein the case of narrow gap semiconductors) and the existence of a magnetically induced overlap between the valence and conduction bands in zero gap HgMnTe are among the most spectacular manifestations of these exchange phenomena. The theoretical analysis of the magnetooptical data not only yields the fundamental band structure parameters (energy gap, effective masses)-similarly as in non-magnetic semiconductors-but also provides access to the magnetization of the subsystem of localized spins. In the latter context, for example, the Faraday rotation, which is directly related to the magnetization, has been used to observe the spin-glass phase transition in Hgl-,Mn,Te (Mycielski et al., 1984). In this chapter the most important magnetooptical experiments in zinc blende NGDMS are reviewed, with emphasis on the striking manifestations of exchange phenomena. The theoretical description of electronic states in the presence of a magnetic field and the determination of magnetooptical transitions are presented in Part 11. Interband magnetooptics in zero and narrow gap DMS are reviewed in Part 111, with emphasis on the differences between magnetic and non-magnetic ternary alloys of similar band structure. FIR magnetospectroscopy, leading to the observation of intraband and interband r8 r8 transitions in zero gap DMS, are described in Part IV. The results of magnetooptical investigations (band and exchange parameters) are summarized in Part V and are compared with those obtained from quantum transport (Kossut, this volume). +
11. Theoretical Description of Electronic States In semimagnetic semiconductors, such as HgMnTe, one may distinguish two sets of electronic states: -Localized d states originating from unfilled d shells of the transition element are characterized by spatially localized wave functions centered around the magnetic ions. The d electrons give rise to localized magnetic
6 .I
232
C . RIGAUX
moments. The 3d5 electrons of Mn form a ‘SW multiplet, whose degeneracy is not lifted by the cubic ligand field. This configuration, corresponding to L = 0, S = 5, J = 8, and the Land6 factor gMn = 2, were experimentally confirmed by electron paramagnetic resonance (Leibler et al., 1978), helicon enhanced spin resonance (Holm and Furdyna, 1977; Mullin et al., 1981) and static magnetic susceptibility measurements (Savage et al., 1973; Andrianov et al., 1976; Spalek et al., 1986). -Extended states giving rise to wide bands originate from outer shells, i.e., 6s2(Hg) or 4sZ(Mn) and 5p4(Te). In random substitutional alloys, the electronic states are described within the virtual crystal approximation (VCA), which replaces the crystalline potential of Hg (6s2) and Mn (4s’) by the weighted average of these atomic constituents. The translational periodicity of the lattice is then restored, and the electrons are described in terms of Bloch states. In practice, the VCA consists of describing the electronic band structure of ternary random alloys in terms of k p models developed for pure crystal components, with the band parameters depending on composition.
-
1. BAND STRUCTURE In II-VI semiconductors, the band edges of relevance are located at the Brillouin zone center (r point), where the states transform according to the representation of the Td crystal symmetry group. The six p-like states are decoupled by the spin-orbit interaction into a quadruplet Ts (J = 4) and a doublet r7 (J = the latter forming a lower-lying valence band, with the energy A = Er, - Er, determined by the spin-orbit splitting. In HgTe, the s-like band, transforming as r6 (J = *), lies below the Ts quadruplet, leading to a symmetry-induced zero-gap semiconducting configuration; the Ts levels then form the degenerate valence and conduction band edges, separated from the r6 valence band by the interaction gap EO = Er6 - Ers = -0.3eV at 4.2 K (Fig. 1). CdTe and (hypothetical) cubic MnTe are wide gap semiconductors. The Tsquadruplet forms two degenerate valence bands and the conduction band edge belongs to r6 representation. The fundamental energy gap (1.6 eV in CdTe) is estimated to be 3.2 eV in cubic MnTe (Gaj et al., 1979). The band structure of ternary alloys Hg, -,Cd,Te and Hg, -,Mn,Te evolves continuously from the zero gap configuration to an open-gap configuration, as shown in Fig. 1. The interaction gap EO , defined as a difference between energies of the r.5and rs states, increases with the alloy composition. These systems undergo a zero gap to semiconductor transition that occurs, at 4.2 K, for x = 0.075 in Hgl-,Mn,Te and x = 0.165 in Hgl-,Cd,Te. In the absence
i),
6.
233
MAGNETOOPTICS IN NARROW GAP DMS
-
F
T
\
/
\
“positive gop”
$i
v
\‘“ I
A
9
‘0
4-
I
I
I
I
I
1
0.02 0.04 006 0.08 0.10 0.12 Mn content ( X I
I
*
0.14
FIG. 1 . Band structure of Hgl-,Mn,Te at the center of the Brillouin zone.
of magnetic field, the electronic states in NGDMS may be described in terms of the Kane band structure model (Kane, 1957).
2. LANDAULEVELS In the presence of an applied magnetic field, the electronic states in DMS are obtained by including the exchange interaction as an additive term X,, in the proper effective mass Hamiltonian XO. For zinc blende narrow gap semiconductors, the most accurate effective mass Hamiltonian XO is provided by the eight-band Pidgeon-Brown model (1966). In this model the k p interactions between closely spaced r6, r7, Ts bands are treated exactly and the interactions between these states, and the more remote higher bands are included up t o order k2. The magnetic Hamiltonian XO, in the absence of exchange, consists of an 8 x 8 matrix D,that reduces to two 4 x 4 matrices Da and Db ,at the r point when inversion asymmetry and warping in Tg bands are neglected. The matrix D has the form:
-
in the basis of the band edge Bloch functions ur(J, MJ) associated with
r 6
234
(f
= 1,2),
C . RIGAUX
Ts (f
= 3,
..., 6 ) , r7(I
i.;(;,-;)
rU2(;-;,)
= 7,8) levels:
= -&(Xi
iY)t
+ Zl)
= ijS1)
Here S , X , K 2 are periodic functions that transform like atomic functions s, p x , pr , p E ,respectively, under the operations of the Td group at the
r
point. XOinvolves the following band parameters: the interaction gap EO = Er, - Er,, the spin-orbit splitting energy A = Er, - Er,, the energy of the s-p interaction Ep = (2/mo)I ( S ( p , l X )12, and modified Luttinger parameters yl, 7, K describing the interaction between Ts and distant bands. The exchange interaction between 3d electrons and delocalized electrons may be described in the vicinity of the r point (Kossut, 1976) by an Heisenbergtype Hamiltonian
where Si is a localized spin at the site Ri (Si = 5);0 is the spin of the mobile electron, and J(r - Ri) is an exchange integral rapidly varying over a unit cell. Since the mobile electron has a spatially extended wave function, it interacts simultaneously with a larger number of Mn ions. This allows one to replace the Mn spin operators by their thermal average ( S ) . In the absence of magnetic field ( S ) vanishes. When the magnetic field H is applied along the z direction, ( S ) has only a non-vanishing component (S,) .
6.
MAGNETOOPTICS IN NARROW GAP DMS
235
Thus, by using the VCA and the molecular field approximation, the exchange perturbation becomes:
where R denotes the sites of the cation sublattice. ( S , ) is directly related to the magnetization M of Mn ions: M
=
-NOxgMnpB(Sz),
(4)
where No is the number of unit cells per unit volume, x is the Mn molar fraction and p~ is the Bohr magneton. For 3d5 electrons of Mn, ( S , ) = -Sfnorm;where S = 5 and fnom denotes the normalized magnetization. For non-interacting magnetic moments,
where B5/2(X)is the Brillouin function for a spin S = 5: B,(x) =
2S+1
2
~
coth
2S+1 ~
2
x
-
1 x -coth-. 2 2
In the basis (l), the exchange Hamiltonian, Eq. ( 3 ) , consists of two 4 x 4 matrices :
where /3Aa/P
0
0 -3AdP
Mb=[
0 0 0
0
-2Afi
(7)
0 0 A 0 0 -3A 2Afi 0
The off diagonal r7 - TSterms in Eq. (7) may be neglected in the case of large spin-orbit splittings. Here M, and Mb involve the s-d and p-d exchange integrals (Y = (SIJ(r)lS),B = (XIJ(r)IX) = (YIJ(r)lY)= ( Z I J ( r ) l Z ) ,that are the only non-vanishing matrix elements of J(r) involving S , X , X Z
236
C . RIGAUX
functions. The exchange parameter appearing in Eq. (7), A = &NoPx(S,) = - &No/3xfnorm, is directly proportional to the magnetization M , Eq. (4). The Landau levels are obtained by solving the two decoupled matrix equations:
(DO- Eaf))lva = 0 ,
(Db
-
Ebf))Wb =
0,
(8)
where Da = D, + Ma and D b = Db + Mb include both band structure and exchange contributions. The wave functions v / o , V b are given in terms of envelope functions f n ( r ) and band edge Bloch functions, Eq. (1):
wa(n) = a i nf n u l vb(n)
=
+ a 3 n f n - i ~ + a 5 n f n + i ~ 5+ a 7 n f n + i ~ 7 , + h n f n - 1 ~ 4 + b6nfn+lU6 + b8nfn-lu8.
(9)
b2nfn~2
For n > 0 , f n ( r ) is the Landau function fn(r) =
exp(i(k,y + kzz))+n(x + A2k,>,
where +n is the n-th harmonic oscillator eigenfunction of characteristic length A = (hc/eH)"2. If n < 0, fn 0. For each quantum level n L 1, there exist four solutions in each a and b ladder that describe r6, TS (light and heavy), and r 7 Landau levels. The eigenvalues and eigenvectors are obtained by solving numerically Eq. (8) for a given set of band and exchange parameters, i.e., at fixed temperature and magnetic field. For the levels n = -1 the solutions are analytical:
where hoo is the free electron cyclotron energy. The exchange contribution induces an energy shift of the levels which is three times larger for b( - 1) than for a( - 1). Approximate expressions of the Landau level energies obtained in the parabolic limit are useful for understanding the influence of exchange phenomena. In the r 6 band, the energies of a and b ladders corresponding to nt (or a) and n l (or b) levels, respectively, are
where m?6 and g?6 are the band-edge effective mass and gyromagnetic factor
6.
MAGNETOOPTICS IN NARROW GAP DMS
237
resulting from the k * p interactions:
g *r , -2
(
z.,:*>*
I------
The last term in Eq. (11) represents the s-d exchange contribution. The exchange effect leads to a modification of the effective g-factor, that can be expressed in the form:
For the l-8 bands, the energies of a and b ladders are given in the parabolic limit by the analytical expressions of the Luttinger model (Luttinger, 1959), in which the parameter K is replaced by R = K - 2A/hw0. In the classical limit (i.e., large n), the l-8 band-edge effective masses are
and the light l-8-band gyromagnetic factor is
As indicated by the above expressions (12)-( 16), the exchange perturbation does not change the effective masses, but contributes an additional term to the effective gyromagnetic factors that depend on temperature and magnetic field through fnorm. The effect of exchange on the magnetic sublevels thus results essentially in a temperature and field dependent shift, the orbital motion in quantizing magnetic field remaining unaffected. 3. MAGNETOOPTICAL TRANSITIONS The interaction between the electromagnetic radiation of frequency o and the band electron is described by the perturbation Hamiltonian (Roth el al., 1959):
238
C. RIGAUX
Here EOis the amplitude and E the unit vector in direction of the electric field of the radiation, and ll is the operator - ih V + e A / c , where A is the vector potential due to the external magnetic field (H 11 d ) . The probability for a direct transition of an electron from an initial state IZ) to a final state IF) is proportional to the square of the modulus of the matrix element nt = (FIE* IIlZ). The absorption coefficient is given by:
where I/ is the crystal volume, q is the refractive index, and g(E1) and (EF) denote the occupation factors for initial and final states, respectively. We consider direct transitions between Landau levels described within the modified Pidgeon-Brown model by wave functions, Eq. (9), of the form:
#I(m) =
C Ajffrn;, ujr ;
j'
yF(n) =
Cj p j f n j u j .
As the envelope function f(r) and the vector potential A(r) are slowly varying over the unit cell, after taking into account the orthogonality of the bandedge Bloch functions the matrix element connecting initial and final states is:
The dominant contribution to the transition results from the first term. The absorption coefficient is finally obtained by summing Eq. (17) over initial ( m , ky ,k,) and final ( m , ky , k,) states:
with the condition
ha
=
EF(n, k,) - EI(m, kz).
(19)
Magnetoabsorption peaks occur for the k,-values corresponding to the minima of the energy distance EF@,k,) - EI(m, k,), usually at k, = 0 for interband transitions. By expressing the transition operator E ll in the form &-TI+ + &+lT-+ &,lT,, where E* = ( E ~f k y ) / f i , lT* = (n, f i r I y ) / f i , and using the Landau level wave functions (9), the selection rules for E + , E - , cZ radiation polarizations are derived. These are listed in Table I .
-
6.
239
MAGNETOOPTICS IN NARROW GAP DMS
TABLE I SELECTION RULESFOR INTERBAND MAGNETOOPTICAL [m. n DENOTELANDAUQUANTUM TRANSITIONS a , b CORRESPOND TO A GIVENSET OF NUMBERS; SOLUTIONS, EQ.(9)]
Polarization
Selection rules
E-
E+
o(m) b(m)
+
+
b(n) n a(n) n
= =
m m
+1 -
1
All allowed interband magnetooptical transitions are included in Table I: l-8 and Ts" l-; transitions in zero gap semiconductors (the latter, i.e., to transitions resulting entirely from the s-p coupling, which induces an s-type component in the TS electronic wave function), and l-8,ighl + r6 and rghcavy-+ r6 in open gap semiconductors. At a given field, the relative transition probabilities are determined by IEjnmnlZ, with the dominant contribution, resulting from the first term in Eq. (18), proportional to E p . The relative strengths of the magnetooptical transitions depend on the Landau level eigenvectors, and can be estimated (for k, = 0) by solving numerically Eq. (8) for a given set of band and exchange parameters. For r6 TS transitions in zero gap semiconductors (and TS r6 in open gap configuration), there exist only one series of dominant transitions for each polarization of the incident radiation. The intraband transitions whose matrix elements do not vanish at k, = 0 are also contained in Table I. The cyclotron resonances, allowed for E polarization, correspond to intra-set transitions (a a or b b) with An = + 1. Even for cyclotron resonance, the intraband part of the matrix element, Eq. (18), is always negligible as compared to the first term resulting from the s-p interband coupling. The combined resonances, allowed for E /I H, correspond to inter-set transitions (a b) with An = + 1. These "spin flip" transitions arise from both spin-orbit effects (mixing spin up and down states in the TS band-edge Bloch functions) and s-p interband coupling in wo and vb wave functions. The mechanism also accounts for the occurrence of the electric dipole spin resonance (EDSR corresponding to a + b with An = 0) in the E + polarization. As this transition takes place at k, # 0, the full eight-band Hamiltonian r6
-+
-+
+
--t
-+
+
240
C . RIGAUX
(Weiler, 1981) or the simplified three band model (Kacman and Zawadzki, 1971) are required to describe the Landau levels away from the r point and the electric dipole spin resonance absorption (McCombe, 1969). The inversion asymmetry of the zinc blende lattice leads to additional that have been neglected in the above terms in the magnetic Hamiltonian KO, formulation as these terms are usually very small. The inversion asymmetry effects lead to induce the electric dipole spin resonance and the combined resonance for any polarization. Of the two possible processes for the excitation of intraband spin-flip resonances, the s-p interband coupling was, until recently, thought to be the dominant mechanism in HgCdTe compounds (McCombe et al., 1970,1971). However, more recently, it was clearly demonstrated in the case of InSb by Chen et al. (1985) that the lack of inversion symmetry in the zinc blende lattice is indeed the dominant mechanism which allows the transitions involving spin-flip resonances by electric dipole excitation.
111. Interband Magnetooptics
Interband Magnetooptical studies in NGDMS were carried out by low temperature IR magnetoabsorption experiments (typically between 200 and 370 meV) performed on very thin unoriented samples, for both Faraday and Voigt geometries using 0 and l 7 polarizations, respectively. Oscillatory magnetotransmission spectra observed at photon energies larger than the interaction gap are associated with interband r6 T8 (zero gap) and l-8 r 6 (open gap) magnetooptical transitions. Several semimagnetic compounds were investigated. The most extensive interband experiments were performed on the Hgl -,MnxTe system, which has been studied in both zero and narrow gap configurations, over a wide range of compositions (0.001 Ix I0.17), at helium temperatures (Bastard et al., 1977,1978) and between 10 and 65 K (Dobrowolska and Dobrowolski, 1981). The r6 r S magnetoabsorption data obtained at 4.2 and 2 I( for dilute HgMnTe alloys ( x I0.015) were the first experimental results to demonstrate striking magnetooptical anomalies characteristic of NGDMS. These were correlated with exchange phenomena by comparison with the spectra of non-magnetic Hgl -,CdxTe alloys of similar band structure (Bastard et al., 1978). Far IR magnetooptics carried out in the same + T pC composition range have enabled the observation of interband transitions in the low energy spectrum (Pastor et al., 1979). Interband magnetoabsorption data have been also reported for zero and narrow gap Hgl-,MnxSe compounds (Dobrowolska et al., 1981), as well as for very dilute semimetallic Hgl-,Fe,Te mixed crystals (Serre et al., 1981, 1982). +
+
-+
6.
MAGNETOOPTICS IN NARROW GAP DMS
241
4. ZEROGAPHgl-,MnxTe
a. Experimental Manifestations of Exchange Interactions Interband r6 Ts magnetoabsorption spectra in zero gap HgMnTe alloys are illustrated in Figs. 2-4 by plots of transition energies vs. magnetic field. The experimental lines are identified with the dominant transitions according to the selection rules: +
&-(or 0 - ) : ar6(n) ar,(n +
+ 1);
&+(oro+):br6(n + 1) 4 brs(n); E
(1 H: br6(n +
1)
+
(20)
ar8(n).
The line identification is indicated in the figures. Only for very diluted alloys (x = 0.0014), the relative ordering of the transitions is identical to that observed in the spectrum of HgTe (Guldner et al., 1973) or Hgl-,Cd,Te alloys (Guldner et al., 1977; Rigaux, 1980). For x L 0.004, a change of regular transition sequence appears, resulting in an inversion of the relative positions of the circular polarization transitions ar6(n) ar,(n + 1)(0-) and br,(n) 4 brs(n - l)(a+)(Figs. 2-4). Comparison between Hgl-,MnxTe and Hgl -,Cd,Te alloys of identical interaction gaps shows that this modification originates from anomalous spin effects in the semimagnetic compounds, whereas the energy separation between consecutive transitions for a given polarization (i.e., the period of oscillations of a given spectrum) remains +
H( kG 1 FIG. 2. Energies of r6 Ts transitions vs. magnetic field for Hgl-,Mn,Te of Mn content x = 0.004 at 4.2 K . Symbols are experimental. Solid lines represent theoretical fits using the modified Pidgeon-Brown model. The quantum number n identifying the transitions according to Eq. (20) is indicated for each line. [After Bastard et 01. (1978).] +
242
C . RIGAUX
0
10
20
30
50
40
H(kG)
FIG. 3. Energies of rs
--f
l-8
transitions vs. magnetic field for Hg,-,Mn,Te of Mn content
x = 0.004 at 2 K. Symbols are experimental. Solid lines represent theoretical fits using the
modified Pidgeon-Brown model. The quantum number n identifying the transitions according to Eq. (20) is indicated for each line. [After Bastard et al. (1978).]
360
I
I
I,.
I
I
'
3 20
2 v
W
280
HW) FIG. 4. Energies of r,5 + Tg transitions vs. magnetic field for Hgl-,Mn,Te of Mn content x = 0.008 at 2 K . Symbols are experimental. Solid lines represent theoretical fits using the modified Pidgeon-Brown model. The quantum number n identifying the transitions according to Eq. (20) is indicated for each line. [After Bastard ef 01. (1978).]
identical for both compounds. These features are illustrated in Fig. 5 by plots of the period ha and the spin splitting = Eb(l) - Ea(l)of the n = 1 conduction Landau level, directly measured form the energy difference at a given field, between the r7+ and E 11 H transitions, c.f., Eq. (20). The quantity ha, representing in the parabolic limit the sum of the cyclotron energies in r' and Ta conduction bands, has comparable values for HgMnTe and HgCdTe, demonstrating the similarity of the band parameters in both families. The strong enhancement of the spin splitting in the semimagnetic
6. I
243
MAGNETOOPTICS IN NARROW GAP DMS
H=20kG
I
I
20
2 E
Y
UJ
10
0
I
I
I
-300
-250
-200
I
e, ( meV 1
- 150
I
-100
FIG.5. Spin and cyclotron splittings, Sc(l) and An; respectively, as a function of EO in HgCdTe (open symbols) and HgMnTe (black symbols) at 4.2 and 2 K, for H = 20 kG. [After Bastard et al. (1978).]
compounds provides direct evidence of the modification of electron spin sublevels induced by the exchange interaction. Another manifestation of exchange interaction in zero gap HgMnTe is the 0 spectrum observation of a new and intense magnetoabsorption line in the ' of alloys for x 2 0.004 (Fig. 6). This line, quantitatively interpreted as being 0 polarization, provides direct due to br6(0) br,(-l) allowed for ' evidence for the emergence of the br8(- 1) heavy hole level above the Fermi energy. One of the most remarkable properties of r6 * Ts magnetooptical spectra is their strong temperature dependence in striking contrast to HgCdTe alloys. On decreasing the temperature, an energy shift is observed for the circular ' , negative for 0-), but E 1) H transitions are polarization lines (positive for o unaffected. The temperature behavior is illustrated in Fig. 6 by the shift of the resonance field of the inter-valence transition br6(0) br,(- 1). The observed temperature dependence of l-6 Ts transitions results in an enhancement of the conduction electron spin splitting between 4.2 and 2 K (Fig. 5). This effect originates entirely from the exchange contribution, which induces an (&)-dependent term in the gyromagnetic factor (11.2, Eq. (16)). A remarkable peculiarity of the spectra is the opposite curvature exhibited by the u+ and o- transitions in the energy diagrams E ( H ) (Figs. 3, 4). This bowing effect, particularly pronounced at 2K, gives rise to an unusual crossing of the lines associated with br,(n) * br,(n - I )(o') and ar,(n) ar,(n + l)(a-)transitions. All these manifestations of the exchange interaction are particularly pronounced at very low temperature (4.2 and 2 K). In the temperature range -+
+
-+
+
244
C. RIGAUX
I
0 10
I
I
20
30
1 LO
I
50
H (kG)
FIG. 6 . Magnetotransmission spectrum at hw = 243.3 meV for HgMnTe of Mn content 0' polarization, observed at T = 4.2 and 2 K . [After
x = 0.015, in the Faraday geometry and Bastard et a / . (1978).]
10-65 K, the results of Dobrowolska and Dobrowolski (1981) show that the magnetooptical transitions conserve the same sequence as in the spectrum of HgTe, and the inter-valence transition is not observed. The main effect of the exchange interaction consists in the temperature dependence of the spin splitting Sc(n), which is seen to decrease with increasing temperature.
b. Theoretical Interpretation Interband magnetooptical spectra were successfully interpreted within the framework of the Pidgeon-Brown model generalized to include the s-d and p-d exchange contributions (Part 11, section 2 ) . A numerical fitting procedure of the r6 + Ts transition energies to the experimental lines was achieved at a fixed magnetic field. The variables of the minimization procedure were restricted to Ep(or P = ( - ih /m )(Sl p x l X ))and the exchange parameters. The spin-orbit splitting and the Luttinger parameters were fixed to the values obtained for HgCdTe alloys of low composition: A = 1 eV; y1 = 3; p = 0; K = -1.65 (Guldner et al., 1977). The interaction gap EO was determined for each composition from the extrapolated zero field energy of the r6 r8 transitions and then adjusted to obtain the best theoretical fits. For dilute HgMnTe compounds ( x < 0.015), EPwas found between 18.3 and 18.8 eV ( T = 4.2 and 2 K), in excellent agreement with the results +
6.
MAGNETOOPTICS IN NARROW GAP DMS
245
obtained for HgCdTe, which demonstrates the similarity of the band structure in both alloy systems. By fitting low temperature magnetoabsorption data, Bastard et al. (1978) determined the ratio r = C Y /of~ exchange integrals and the relative magnetization A ( H, T ) . The parameter r was found to be independent of composition and equal to -0.5 for k 5 0.015. The magnetic field dependence of A ( H ) obtained from the theoretical fits for 4.2 K is shown in Fig. 7. For extremely dilute alloys only (cO = -293 meV, x = 0.0014), the variation A ( H ) follows in relative magnitude the Brillouin function BS/Z(gMnpUgH/kB T ) , which enabled the evaluation of exchange integrals NoP = 1.5 eV and NOCY = -0.75 eV for this particular composition. Using these values, the normalized magnetization fnorm(H, T ) for diluted compounds was deduced from A ( H , T ) . The results, presented in Fig. 16 (see Part 111, section 6b), agree quantitatively with direct magnetization measurements at 4.2K for x = 0.01 and with low field d c magnetic susceptibility data for x = 0.012 (Nagata et al., 1980). The comparison between theory and experiments, at 4.2 and 2 K , is illustrated in Figs. 2, 3, and 4. Excellent agreement is found at both temperatures for a common set of parameter (except A , which is a function of temperature and magnetic field) (Fig. 7). In particular, the curvatures of 'rc and cr- lines originating from the effect of magnetization on the Landau level energies are well accounted for by the model, and the experimental position of the inter-valence transition is quantitatively interpreted (Figs. 2-4). Band-edge effective masses and gyromagnetic factors of conduction electrons for very dilute HgMnTe and HgCdTe alloys (in the parabolic approximation) are shown in Fig. 8. The results show the similarity of
OL 0
'
I
I WG)
I
I
]
50
FIG.7. Relative magnetizationA as a function of the magnetic field, observed at T = 4.2 K for Hg,-,MnxTe of different compositions. [After Bastard et a/. (1978, 1981).]
246
C . RIGAUX
0.035
1 -I
0.0251.
i-"
0 -300
, &@",j -250
-200
FIG.8. Band edge effective masses and gyromagnetic factors for the conduction band in zerogap HgCdTe (crosses) and HgMnTe alloys. Open circles are for T = 4.2 K, and black circles for T = 2 K. [After Bastard (1978).]
effective masses and the strong enhancement of the gyromagnetic factor induced by exchange interactions in the semimagnetic system.
c. MagneticaIIy Induced Semimetal-to-Semiconductor Transition The T8 Landau level spectrum ( k =~0) calculated from the band and exchange parameters is reproduced in Fig. 9 for several Mn compositions. The magnetic field dependence of the energy levels exhibits at 4.2 and 2 K quite interesting peculiarities originating from exchange phenomena: the spin-spin interaction induces a magnetic-field-dependent shift of the l-8 levels (positive for b(n),negative for a(n)),leading to an enhancement of the spin splitting Sc(n) = b(n) - a(n), which may exceed the cyclotron energy. For x > 0.004, the levels b(n) overlap a(n + l), and the usual sequence of the Landau levels becomes modified. The strong upwards shift AE = -3A of the heavy hole level brs(-l) predicted by the model (11.2, Eq. (10)) is responsible for the occurrence of the transition br,(O) br8(-l). The overlap of the lowest electronic level ar,(O) by the uppermost valence state -+
6.
MAGNETOOPTICS IN NARROW GAP DMS
241
brs(- 1) leads to a semimetallic configuration induced by the magnetic field L 0.004). In the high field region, the exchange contributions to ars(0) and brs(- 1) energies reach a saturation, and the field behavior of these levels is entirely governed by the band structure contribution, i.e., ar,(O) increases 1) decreases with field. For some critical field a crossing appears and hS(between the uppermost valence and lowest conduction levels, and a magnetic-field-dependent gap is opened between the Ta bands. Thus a magnetically induced semimetal-to-semiconductor transition occurs in zero gap Hgl-xMnxTe alloys. For x = 0.004 and 2 K, this transition takes place at about 37 kG (Fig. 9). (x
20
2 E
10
Y
w
0
20
10
0 0
10
20
30 40
0
H (tG)
10
20
30 40
FIG. 9. Magnetic field dependence of Ts Landau level energies at kz = 0, for HgMnTe of several Mn compositions, for 4.2 and 2 K. [After Bastard et al. (1978).]
248
C. RIGAUX
5 . NARROW GAP Hgl-,Mn,Te
a. TS
+
r 6
Magnetoabsorption
Semiconducting Hgl-xMnxTe alloys of narrow but positive band gap (100 IEg I300meV) were investigated by low temperature interband magnetoabsorption (Bastard et al., 1981; Dobrowolska and Dobrowolski, 1981). The TS-, rb magnetooptical spectra again show novel features, that contrast sharply with the results obtained in non-magnetic zinc blende semiconductors (Rigaux, 1980). A reduction of the energy gap appears in applied magnetic field, since the lowest energy transition-observed for 0-polarization-decreases in energy with increasing magnetic field (Fig. lob). Semiconducting HgMnTe alloys are characterized by the near-coincidence of 0-and E 1) H transitions, except for the two lowest-lying transitions (labelled - 1 , O in Fig. 10). This feature is illustrated in the magnetotransmission spectra shown in Fig. 11 for an alloy of energy gap E, = 219 meV for 4.2 K. The TS + r6 magnetooptical transitions depend weakly on temperature, in contrast to the behavior observed in zero gap HgMnTe: the lowest-lying transitions are only slightly shifted to lower energies with decreasing temperature whereas the other lines (nearly coincident in 0- and E I( H spectra) are practically unaffected. These characteristic features of narrow gap semimagnetic semiconductors are explained by the effect of exchange on TSand r6 Landau levels. Quantitative analysis of the transition probabilities (that depend on the exchange contribution) indicates that the transition from heavy hole Landau levels are expected to be much weaker for the o+ polarization than for 0- and E I( H polarizations. For E 1) H, transitions from heavy hole ladders are the most intense, the dominant ones corresponding to: ar,(n)
+
bs(n
+
l),
TI 2
-1.
(21)
For the 0- polarization, transitions from heavy and light valence ladders are expected to be strong, the dominant series corresponding to:
br,(n) --* br6(n + l),
n
L
-1.
(22)
Experimental lines observed for the E ( J Hand 0- polarizations are identified according to Eqs. (21) and (22), respectively (Fig. 10). The nearcoincidence (for n > 1) of the 0 - and E (( H transitions demonstrates the fact that the final states in both polarizations are identical, whereas the initial states belong to heavy hole ladders ahh(n) and bhh(n), which lie close to each other. The lowest-lying transitions originating from bra(- 1) and ar,(- 1) are most sensitive to the exchange mechanism, as expected from the model (11.2,
6.
MAGNETOOPTICS IN NARROW GAP DMS
249
.i‘ T=2K
27
li Y
230
210 10
0
20
30
40
50
H(kG)
(4
2w
0
10
20
30
40
50
H(kG)
(b) FIG. 10. Energies of Ts + Ts transitions vs. magnetic field for Hgl-,MnxTe of x = 0.128 (EO = 219 meV), at 2 K, (a) E I( H polarization, and (b) 0 - polarization. Points are experimental. Solid lines represent theoretical fits obtained within the modified Pidgeon-Brown model. The quantum number n identifying the transitions according to Eqs. (21) and (22) is indicated for each line. The line marked “Imp” corresponds to the acceptor -+ br,(O) transition. [After Bastard er a/. (1981).]
250
C. RIGAUX
I
I
I
I
H(kG)
FIG. 1 1 . Magnetotransmission spectra for 0 - and E )I H polarizations, observed at the photon energy hw = 265.3 meV at 4.2 K for Hgl-xMn,Te with x = 0.128. [After Bastard et a/. (1 98 I).]
Eq. (10)). This explains the observed temperature and magnetic field dependence of bra(- 1) -+ br,(O)(a-)and ars(- 1) br,(O)(E (1 H) transitions (Fig. 10). Comparison between theoretical and experimental transition energies vs. magnetic field is shown in Fig. 10 for x = 0.128 (EO = 219 meV). The band parameters eo ,E p , and the exchange parameters r, A(H, 7')were determined at 4.2 and 2 K in the range of composition 0.12 5 x 5 0.17. The results are presented in Part V. The magnetization parameter A ( H ) varies almost linearly with the magnetic field (Fig. 7) and is weakly temperature dependent in this range of composition. Hgl-,Mn,Te semiconductors are usually p-type. The relatively high density of acceptors (Nu = 5 x 10'6cm-3) present in the investigated compounds makes possible the observation of acceptor-to-conduction Landau level transitions. An intense absorption line satellite of the interband br,(O) br,(l) transition of weaker intensity (labelled Imp in Fig. lob), extrapolates at zero field to about 5 meV below the energy gap. This line may result from the transition between an acceptor state (bound to br,(O))and the conduction band Landau level br,( 1). +
-+
6.
MAGNETOOPTICS IN NARROW GAP DMS
b. Effect of Exchange on
r 6
251
Landau Levels
The exchange interaction affects the Landau level spectrum of the narrow (positive) gap HgMnTe in a striking fashion. The magnetic field dependence of the conduction Landau level edges, for a semiconductor of EO = 219 MeV shown in Fig. 12, illustrates the new important features originating from the exchange phenomena. In the conduction band, the relative positions of the levels ar,(n) and br,(n), corresponding, respectively, to nt and nl electron spin states, are inverted with respect to the usual ordering in zinc blende semiconductors, so that in the present case E&) > Eb(n). This effect is a direct consequence of the s-d exchange interactions, which shift the spin sublevels upward for a(n) and downward for b(n). Thus, in semimagnetic alloys with positive energy gap, the gyromagnetic factor of conduction electron is positive. This is in complete contrast to zero gap HgMnTe and to non-magnetic narrow gap semiconductors, such as InSb or HgCdTe, where the electron gyromagnetic factors are always negative. The origin of the sign reversal of the g-factor occurring in these materials can be understood from the analytical expression (14) for the effective gyromagnetic factor in the parabolic limit. The first term gc* resulting from the k * p contribution is negative, whereas the second term, representing the s-d exchange contribution, is positive (a < 0). Thus, with increasing magnitude of the exchange interactions, the effective gyromagnetic factor
301
s
2 w
I
0
€
20[1
0
.--+a
b
FIG. 12. Magnetic field dependence of Landau levels of the conduction band at k, = 0 for HgMnTe with Mn content x = 0.128 calculated at T = 2 K . The labels a, b correspond to two sets of solutions of Eq. (8); the Landau quantum numbers are given at each curve. [After Bastard et al. (1981).]
252
C . RIGAUX
g& will go through zero and subsequently reverse sign when the exchange
contribution becomes dominant. Large andpositive conduction g factors are observed in open gap semimagnetic semiconductors, e.g., for EO = 219 meV, g?6 = 100 at 2 K. This is a unique situation for narrow gap semiconductors, associated exclusively with exchange interactions. As one can see in Fig. 12, the electron spin splitting becomes comparable to the cyclotron energy. In the valence band, the most spectacular effect caused by the exchange interactions is the strong positive energy shift exhibited by the uppermost heavy hole level br&- 1). This feature, accompanied by a downward shift of the first conduction electron level br,(O), leads to the reduction of the energy gap in an applied magnetic field. 6. RESULTSFOR OTHERSEMIMAGNETIC COMPOUNDS a. HgMnSe
Interband magnetooptics in HgMnSe was carried out by Dobrowolska et al. (1981) for several Mn compositions (0.011; 0.075; 0.115). Magnetotransmission measurements were made at temperatures between 10 and 40 K , in the spectral region 230-370meV, on n-type samples with electron concentration ne = 10” ~ m - Interband ~ . r6 Tg transitions observed for (T and ll polarizations in zero gap Hgo.9g9Mno.ollSeare identified according to the selection rules (20), by referring to the spectrum of HgSe (Dobrowolska et al., 1980; Mycielski ef al., 1982). Due to the high carrier concentrations in HgMnSe alloys, the first conduction subband edges are fully populated, precluding the observation of the lowest-lying transitions br,(O) bra(- I), ar,(O) ar8(l), that are the most sensitive to the exchange effect (Fig. 13). The experimental spectra obtained at different temperatures were quantitatively interpreted on the basis of the modified Pidgeon-Brown model: for x = 0.01 1, the magnetization was calculated in terms of a model of isolated and nearest-neighbor interacting Mn pairs. The values of the exchange integrals for r6 and TSbands, the exchange coupling constant JO between pairs, and the band parameters (EO ,P ) were evaluated by fitting r6 Tg transition energies as a function of magnetic field. Higher band parameters and the spin-orbit splitting energy A = 387 meV were taken to be equal to the values obtained for HgSe (Dobrowolska et al., 1980a). Comparison between the calculated and the experimental transition energies vs. magnetic field is illustrated in Fig. 13, at 10 and 40 K, for the values of the exchange integrals NOD = 0.7 eV; NO(Y= -0.4 eV; JO = 2.3 K. The observed temperature dependence of the transitions shows a decrease of the conduction spin splitting with increasing temperature. Figure 14 shows the gyromagnetic factor g? of the n = 1 conduction Landau level vs. magnetic field at several temperatures for HgSe and for Hgo.9g9Mno.ollSe. -+
-+
-+
6.
z
z
o
~
l
20I
'
MAGNETOOPTICS IN NARROW GAP DMS
I 40
'
I 60
253
'
HlkGI -
FIG. 13. Magnetic field dependence of Ts Ts transitions for Hgl-,MnxSe of Mn content x = 0.01 1. Symbols are experimental, full circles, open circles, and crosses denoting the 0-,oi, and E ( 1 H polarizations, respectively. (a) T = 10 K. (b) T = 40 K. Solid lines are theoretical fits obtained using the modified Pidgeon-Brown model. Transitions are identified according to the notation of the three-band model (n' = a(n); n- = b(n)). [After Dobrolowska el at. (1981).] -+
H/kGl--
FIG.14. Magnetic field dependence of the electron g* factor calculated for the n = 1 Landau level for HgSe (dashed lines) and Hg0.9~Mno.ollSe(solid lines). [After Dobrolowska et al. (1981).]
254
C . RIGAUX
The temperature dependence of gf in HgSe results solely from the energy gap variation, whereas the decrease of gf observed between 10 and 4 0 K in the semimagnetic compound is the consequence of the exchange interaction. In the semiconducting region, the magnetoabsorption spectra of Hgl-*MnxSe (x = 0.075 and 0.115) are weakly temperature dependent. The TS+ r6 magnetooptical spectra are interpreted by assuming that the exchange integrals a,p are independent of composition and equal to the values obtained for x = 0.011. The values of ( S z ) deduced by fitting l-8 rs magnetoabsorption spectra considerably differ from those obtained from susceptibility measurements on samples of similar composition (Pajaczkowska et al., 1979; Dobrowolska et al., 1980b). This disagreement indicates that further and more extensive study of Hgl-,Mn,Se crystals is necessary in order to determine reliable values of a and p in this material. -+
b. HgFeTe New DMS containing substitutional Fe2+ions were recently synthetized (for x I 0.02) at Purdue University. The first magnetooptical study carried out by low temperature interband magnetoabsorption in zero gap Hg, -,Fe,Te alloys have emphasized the semimagnetic character of these compounds (Serre et al., 1981, 1982). The r6 l-8 magnetooptical spectra obtained for the rn and ll polarizations show (similarly as in HgMnTe) exchange-enhanced spin splittings, as well as the occurrence of the inter-valence transition br,(O) -+ h 8 ( - 1 ) in the rn+ spectrum of alloys with ~ E O I > 260meV. In complete c6ntrast to HgMnTe, however, the magnetooptical spectra are independent of temperature between 4.2 and 2 K. The r6 r8 magnetooptical spectra of very dilute Hgl -*FexTe alloys were quantitatively interpreted in the framework of the modified Pidgeon-Brown model and the exchange parameters r = -0.3 and A ( H ) were determined. The negative values of r and A imply that a < 0 and fl > 0, similarly as in the case of HgMnTe, leading to exchange-enhanced gyromagnetic factors in both the r6 and the Ts conduction bands. The exchange effects in Hgl-,Fe,Te are, however, much weaker than in Hgl-xMn,Te alloys of comparable interaction gap, as indicated by the values of A ( H ) obtained for both alloy systems: e.g., for EO = -230 meV, at H = 2T and 2 K, A is about twice as large in HgMnTe as it is in HgFeTe. The most striking peculiarities of HgFeTe compounds are the linear field dependence of A ( H ) and its invariance with temperature. The results obtained for several values of the Fe content also show a linear dependence of the magnetization on composition, as illustrated in Fig. 15 by plotting - A / H vs. the relative interaction gap energies. This implies that in dilute HgFeTe alloys, ( S , ) is independent of composition, in sharp contrast with -+
+
6.
--
255
MAGNETOOPTICS IN NARROW GAP DMS I
I
I
I
1
1-
-
-
I
%r E c 0
u-
0.3
0 1 0
I
10
1
20
I
30
H(kG)
I
40
I
50
I
60
FIG. 16. Normalized magnetization &,, vs. magnetic field determined at 2K for Hg,-,Mn,Te alloys of several Mn concentrations. (A: x=0.0014; B: x=0.004; C: x = 0.008; D x=O.Ol; E x = 0.015. Dots are experimental data obtained form interband magnetooptics. [After Bastard ef a/. (1978).] Continuous lines represent theoretical fits obtained by using the generalized Brillouin function given in Eq. (6) with Treplaced by T + r0. The value of the fitting parameter %(x) for each composition is indicated in the figure. [After Bastard and Lewiner (1980).]
the pronounced decrease of ( S , ) observed in HgMnTe with increasing values of x-see Fig. 16 (cf. also, Oseroff and Keesom, this volume). The most important property of Hgl-,Fe,Te alloys is the linear variation of ( S , ) with magnetic field, and ips independence of temperature between 2 and 4.2 K. These observations are corroborated by dc magnetic susceptibility (Mullin, 1980) which is found independent of temperature in Hg, -,Fe,Te alloys of comparable Fe content. This important feature
256
C. RIGAUX
drastically contrasts with the results obtained in dilute Hg, -*Mn,Te alloys, where magnetization manifests a strong temperature dependence. The behavior observed on Hgl-,FexTe alloys originates from paramagnetism of Fe2+ions in the zinc blende lattice (Serre et al., 1982). The d shell of the Fe2+ ions has 6 electrons, and in vacuum the ground state corresponds to S = 2 and L = 2. The cubic crystal field lifts the resulting 25-fold degeneracy into a 10-fold-degenerate level and a 15-fold-degenerate excited state separated by A = 0.3 eV from the ground state (Mahoney et al., 1970). The spin-orbit coupling AL .S (with A = 12 meV) further lifts the degeneracy of the ground state into 5 equidistant levels with an energy separation equal to 6A2/A, the degeneracy of the levels being 1, 3, 2, 3, and 1, respectively. Finally, the Zeeman termpB(L, + gFeSz)Hisintroduced. The resulting spectrum of the d level of an Fe2+ ion is shown in Fig. 17. The ground state is a singlet, that is, there is no permanent moment at zero magnetic field. Consequently, there is no linear Zeeman shift. The quadratic Zeeman shift
Eo(H) = EO - 4-PB2 A H 2 3 A physically corresponds to a field-induced magnetic moment, that varies linearly with magnetic field and arises from the virtual transitions between the ground state and the excited states. The magnetic susceptibility x = 83 p2~ ( A / A 2 is) therefore independent of magnetic field and of temperature, as long as the contribution of the first excited triplet can be neglected 1
b
1,
\
/ /
2
IL
1
L
'I
-
1
H
c
6.
257
MAGNETOOPTICS IN NARROW GAP DMS
(i.e., kT Q 6A2/A). Thus at low temperatures, in contrast with the CurieWeiss paramagnetism of Mn2+ions, the Fe2+magnetism is of the Van Vleck type and arises from the lack of a permanent moment at zero field. IV. Intraband Magnetooptics 7. FARIR MAGNETOSPECTROSCOPY IN ZEROGAPDMS In non-magnetic zero gap semiconductors, FIR spectroscopy has been successfully applied in carrying out detailed studies of the Fg valence and conduction bands. Magnetotransmission experiments performed in the submillimeter and FIR region on unoriented n-type samples of HgTe (Tuchendler et al., 1973) and Hgl-,CdxTe alloys (Kim and Narita, 1976; Guldner et al., 1977) with low electron densities ( h a > EF) led to the observation of intraband electronic and interband Ti' Ti transitions. Measurements performed on oriented p-type samples with magnetic field applied along the principal crystallographic directions were especially useful for determining the Fg band anisotropy in HgTe (Pastor et al., 1978). HgSe is characterized by large electron densities (n, = 10'' ~ m - leading ~ ) to an opacity of the crystal at long wavelengths, due to plasma cut-off. FIR measurements carried out on HgSe at frequencies larger than the plasma frequency have provided observation of intraband electronic transitions (plasma-shifted cyclotron resonance for E I H, combined resonance for E I( H, and electric dipole spin resonance for both E 11 H and E I H). The interband Ti' -+ Fitransitions requiring photon energies larger than the Fermi energy (typically 14 meV) could also be excited in the investigated spectral region (7.61-12.85 meV) (Pastor et al., 1980, 1981). In zero gap DMS, FIR magnetooptics provides one of the best experimental methods for studying the behavior of Tg Landau levels in the presence of magnetic exchange. Pastor et al. (1979) observed at 4.2K a strong influence of the exchange effects on the positions of the magnetotransmission minima associated with intraband and interband Ti --* r; transitions in dilute Hgl -,Mn,Te compounds with low electron densities. Similar features were reported by Kim et al. (1982) in quaternary Hgl-,-,Cd,Mn,Te compounds for several Mn compositions. In zero gap Hgl-xMn,Se, FIR magnetooptical experiments were particularly difficult due to the high electron concentration (typically 10'' cmP3), leading to a plasma edge cutoff at wavelengths about 100pm. Using samples of exceptionally low electron , by appropriate annealing, concentration (5 x 1OI6~ m - ~ ) prepared Witowski et al. (1982) performed a detailed study of intraband spin-flip resonances in zero-gap Hgo.97Mn0.03Se.The strong influence of exchange on the conduction spin sublevels is experimentally manifested by a dramatic +
258
C . RIGAUX
temperature dependence of the spin-flip transitions between 1.9 and 6.5 K. Theoretical analysis of these results yields conduction band and exchange parameters for dilute HgMnSe alloys.
a. Influence of the Exchange Interactions on the TS Magnetoabsorption Spectrum
+
TS
(a) Zero Gap Hgl-,MnxTe Alloys Using molecular gas lasers (HCN, 3.68meV; DCN, 6.36meV; HzO, 10.45meV) as light sources, Pastor et al. (1979) have performed FIR magnetotransmission measurements at 4.2 K in the Voigt geometry, using linearly polarized radiation E I(H and E IH, on oriented Hgl-,Mn,Te crystals of low Mn content (x I 0.018). The evolution of the magnetotransmission spectra with composition, shown in Fig. 18 at the photon energy ho = 3.68 meV, shows a shift of the transmission minima to lower magnetic field as x increases. It was possible to identify the resonances by comparison with similar data obtained by HgTe (Pastor et al., 1978). The electric dipole spin resonance a(0) -,b(0)and the combined resonance (CB) a(0) -+ b(1) are identified in Fig. 18 in the E 11 H spectrum. Figure 19 shows the observation of cyclotron resonance for E I H. Several interband Ti + Titransitions were also observed for E 11 H and E I H. The assignment of the lines made according to the selection rules (Table I), is indicated in Figs. 18 and 19 using the notation of the Luttinger model (Luttinger, 1956). Since the resonance
0
rb
m
30
io
50
MAGNETIC FIELD (kG1
FIG. 18. Magnetotransmission spectra at frw = 3.68 meV for Hgl-,Mn,Te alloys of low x observed in the E 1) H polarization. Experimental lines are identified as follows according to the notation of the Luttinger model: CB: combined resonance ~ ( 1 + ) &2(2);SR: spin resonance &](I).+ &$(I); E: d ( 2 ) EZ(~); 8:$(2) &](I);A : d ( 2 ) E I ( ~ ) .[After Pastor ef a/. (1979).1 -+
+
-+
6 . MAGNETOOPTICS IN
0
1.o
0.5
259
NARROW GAP DMS
15
2.0
x (”/) FIG.19. Comparison of transmission minima observed at I = 118.6pm in several oriented Hgl-,MnxTe samples (H 11 (1 11)) with the predictions of the theory. (a:transitions observed for E 11 H; 0: transitions observed for E I H; 0 : transitions observed for both E (1 H and E 1 H.) [After Pastor et d.(1979).1
field varies linearly with the photon energy for the three different laser wavelengths, the non-parabolicity effects in r s bands are expected to be negligible. The experimental data were thus analyzed on the basis of the Luttinger model, with the effect of anisotropy of the Ts bands ( y 2 # y 3 ) included. In this model, the warping contribution introduces a mixing of the Ts Landau levels at kH = 0, and the k p Hamiltonian D thus becomes an infinite matrix. The exchange interaction between mobile carriers and Mn2+ ions was introduced into the model by using an additional Heisenberg-type Hamiltonian Xex,(Eq. (2)). In the representation of the matrix D,the
-
260
C. RICAUX
exchange interaction X,, has the form of an infinite matrix that depends only on the exchange parameter A = - & Z V ~ f l x f ~The ~~~ effective . Hamiltonian D + X,, was solved numerically, and the calculated transition energies were fitted to the experimental data, treating NODas the only fitting parameter. In the analysis, the normalized magnetization was taken from interband magnetooptical data, and the Luttinger parameters were estimated from those obtained for HgTe (Pastor et al., 1978), taking into account the composition dependence of the energy gap. Comparison of the experimental resonances for different compositions with the prediction of the Luttinger model is shown in Fig. 19, obtained from theoretical fits of the data that yielded the value of Nofl as 650 meV. (b) Hgl-,-,Cd,Mn,Te The quaternary system Hgl -,,Cd,Mn,Te offers the interesting and important possibility of independently varying the energy gap and the magnetic ion concentration in DMS. The first magnetooptical study of quaternary alloys of low Mn content ( y < 0.01) was reported by Kim et al. (1982). FIR magnetotransmission measurements were carried out at 4.2 and 1.6 K, in both Faraday and Voigt geometries, on n-type samples of low electron concentration (8 x 1014~ m - ~Energies ). of the resonance lines vs. magnetic field are shown in Fig. 20 for alloys with interaction gaps EO = - 26 and - 56 meV, respectively. The FIR spectra consist of several Ts" TsC transitions, as well as the spin, combined, and cyclotron resonances. In the low field region, the crossing of the cyclotron and spin resonances lines directly demonstrates crossing of a(1) by b(0) conduction levels. The band edge effective masses and gyromagnetic factors of conduction electrons are estimated from these experiments for several compositions. The g factors are found to be twice as large in the quaternary semimagnetic alloys as those in Hgl -,Cd,Te alloys ofidentical interaction gap, whereas the effective masses remain comparable in both alloys. -+
b. Spin-Flip Resonances in Zero Gap Hgl-,Mn,Se Intraband spin-flip transitions have been studied in zero gap Hg, -,Mn,Se alloys (x = 0.032) by FIR magnetospectroscopy carried out with the use of an optically pumped FIR laser (Witowski et al., 1982). Transmission measurements were performed in magnetic fields up to 2.5T in both Voigt and Faraday geometries, at wavelengths between 96.5 and 170pm, in the temperature range between 1.9 and 6.5 K. In the Faraday configuration, two clearly resolved absorption lines, shown in Fig. 21 for A = 119 pm, are identified as the electric dipole spin resonance (EDSR) transitions a(n) b(n). A strong absorption line observed in the Voigt E 1) H spectrum (Fig. 22) is attributed to the combined resonance allowed for this polarization. -+
FIG. 20. Energies of transmission minima vs. magnetic field for quaternary Hg,-,-,Cd,Mn,Te alloys of interaction gaps = -26meV (a) and EO = -56meV (b). CR, CO, SR, and 11.2.3 denote cyclotron resonance, combined resonance, spin resonance, and interband transitions, respectively. [After Kim et al. (1982).]
EO
262
C . RIGAUX
FARADAY I
L
1
I
I.o
0.5
I.5
MAGNETIC FIELD (7) FIG. 21. FIR magnetotransmission spectrum obtained for Hg,-,Mn,Se (x = 0.032) in the Faraday configuration. ( T = 4.2K; 1 = 118.8pm.) [After Witowski et al. (1982).] I c:
50-
1
I
I
119pm
I
-
?
MAGNETIC FIELD (TI FIG. 22. FIR magnetotransmission spectrum obtained at 118.8pm on Hgl-,Mn,Se ( x = 0.032) in the Voigt E 11 H configuration, at several temperatures. [After Witowski et al. (1982).]
As pointed out previously, the electric dipole spin-flip transitions in zinc blende narrow gap semiconductors are allowed, in the presence of spin-orbit interaction, as a consequence of either s-p interband coupling or of inversion asymmetry. One of the most spectacular features of these experiments is the strong temperature dependence exhibited by the spin-flip transitions. The temperature behavior is illustrated in Fig. 22 by the shift of the combined resonance between 2 and 6.5 K. For EDSR, the effects are even stronger, the resonance
6.
MAGNETOOPTICS IN NARROW GAP DMS
263
field shifting by about 50% in the same temperature range. This feature is evident in Fig. 24, which shows the temperature variation of the spin-flip resonance at 119pm. The strong temperature shift of the spin-flip transitions represents a direct influence of the (&)-dependent term on the energy scheme of the T8 spin sublevels due to the p-d exchange interaction. The experimental data were quantitatively interpreted in terms of the modified Pidgeon-Brown model. A generalized Brillouin function for an effective temperature T(see, Gaj et al., 1979) was used to describe the thermal average of the Mn spins. The parameter To, as well as the fundamental band structure parameters EO and P , were treated as adjustable parameters in the numerical fitting procedure of the transition energies vs. magnetic field. The values of the remaining band parameters were taken from those determined earlier for HgSe (Pastor et al., 1981; Dobrowolska et al., 1980a) and the values of the exchange integrals a,p were taken from the quantum transport results on dilute HgMnSe alloys (Byszewski et al., 1980). The best theoretical fit of the transition energies vs. magnetic field to the experimental line positions at 4.2K is shown in Fig. 23. Two groups of transitions are distinguished: at higher energies are combined resonances, corresponding to a@’) b(n’ + l), and at lower energies occur the EDSR transition a(n) b(n). The quantum numbers n, n’, ascribed to each transition, are indicated in the figure. Since the Fermi level lies at about 16 meV above the r point, several spin-flip transitions are possible in the investigated spectral region. From the fitting procedure, Witowski et a/. determined(at4.2K)~o= -83.1 meV,P= 5.58 10-8eVcm,and%= 7.23K for x = 0.032. The temperature dependence of the resonance positions for several wavelengths was subsequently quantitatively interpreted by fitting %(T) as the only fitting parameter. Comparison between the calculated and experimental line positions for I = 119pm is shown in Fig. 24 indicating excellent agreement. From theoretical fits performed at different wavelengths, the temperature dependence of the parameter %( T ) was estimated. The results show that the parameter To( T ) increases from 6 K to about 11 K between 2 and 6.5 K. Conduction band Landau levels at the r-point, obtained from the analysis of spin-flip transitions in Hgo.97Mno.o~Se at T = 1.6 and 6 K, are depicted in Fig. 25. The b(n)levels are the most strongly affected by the temperature, their energies increasing when the temperature is lowered, similarly as in zero gap HgMnTe alloys. The behavior of the T8 Landau levels explains the important shift of the EDSR at I = 119pm (a(0) b(0);Fig. 22), as well as the transformation of the transition 4 2 ) b(2) into a(1) b(1) observed at L = 163pm when the temperature is raised from 1.6 to 6 K. As one can see in Fig. 23, the spin splitting b(n) - a(n) exceeds the cyclotron splitting a(n + 1) - a(n) in the investigated temperature range.
+
+
+
+
-+
+
264
C. RIGAUX
I
O.8.5
I.o
1.5
MAGNETIC FELD
20
2.5
3bO
(TI
FIG.23. Transition energies vs. magnetic field for EDSR and combined resonance at 4.2 K observed in Hgl-,MnxSe (x = 0.032). Symbols are experimental. Solid lines represent the best fits to the experimental data. Unprimed and primed integers refer to EDSR and combined resonance transitions, respectively. [After Witowski et at. (1982).]
The temperature dependence of the parameter To obtained from the analysis of spin-flip transitions was used to calculate the magnetic susceptibility. A rather satisfactory agreement is found between zealcand direct dc susceptibility measurements (Khattak er al., 1981). 8. CYCLOTRON AND COMBINED RESONANCE HARMONICS IN n-TYPE HgMnTe SEMICONDUCTORS
Intraband magnetooptical studies in n-type narrow band gap Hgl -,Mn,Te (x = 0.11, eg = 165 meV) were reported by Grynberg et al., 1982). Using C02 laser lines (1 15-135 meV), magnetotransmission and magnetophotoconductivity experiments have been carried out at 1.8 Kin fields up to 18Tesla,
6.
265
MAGNETOOPTICS IN NARROW GAP DMS 1
2.2
-
2.0
-
I
1
I
1
1
2 0
-
-I-
1.8-
-
Q
1.6-
-
1.4-
-
z 1.2-
-
J
wLL
9 I-
w
0
1 I .o
I
0.8: I
I
I
1
L
I
1
2
3
4
5
6
7
in the Voigt configuration with unpolarized radiation. The measurements were performed on In-doped samples of electron concentration ne = 8 x 1015cm-3 obtained after appropriate annealing. Pronounced oscillatory magnetotransmission minima, which coincide with magnetophotoconductivity minima, are associated with intraband magnetooptical transitions within the conduction band Landau levels. Using the modified PidgeonBrown model, with the band parameters obtained from interband magnetooptics on very similar HgMnTe crystals (except for EO and A ( H ) ) , the observed resonances were identified as the harmonics of the cyclotron and combined resonances. For magnetic field lower than 8 Tesla, transitions take place from the lowest-lying Landau level b(0) to high quantum levels b(5) and b(6) (cyclotron resonance harmonics) and to a(4), 43,and 4 6 ) (combined resonance harmonics). At very high fields (H L 8 T ) , the magnetic freezing of the conduction electrons on donor states has been experimentally manifested by the observation of transitions from the donor ground state bound to the lowest Landau level to high quantum levels a@),
266
C . RIGAUX
r
I
I
I
I
I
MAGNETIC FIELD ( T I
FIG. 25. Magnetic field dependence of Tg conduction Landau level energies at k, = 0, calculated for 1.9 K (solid lines) and 6 K (dashed lines). The broken arrows show how the magnetic field of the EDSR transition shifts between 1.9 and 6 K for I = 118.8 pm.The solid arrows show the transformation of the EDSR from 4 2 ) -,b(2) at 1.9 K to a ( ] ).+ b(1)at 6 K for I = 163pm. [After Witowski e t a / . (1982).]
b(n) (or to excited donor states bound to these levels). The sensitivity of the combined resonance harmonic transitions b(0) a(n) to the exchange parameter A has made it possible to determine the magnetic field variation A ( H ) up to 18 Tesla. Satisfactory agreement is also found with the interband magnetoabsorption data obtained at lower fields for HgMnTe of similar composition. Low temperature magnetoabsorption and stimulated spin-flip Raman scattering experiments in n-type Hgo.g9Mno.l1Tewere also reported by Geyer and Fan (1980). Weak magnetotransmission structures observed at photon energies smaller than the energy gap were attributed to cyclotron resonance harmonic transitions coupled with LO phonon emission. A line observed in scattering measurements under applied magnetic field was found to be due to spin-flip scattering, and the magnitude of the Land6 factor of conduction electron was estimated as [gel = 45. The observed spin-flip scattering was determined to be stimulated on the basis of its intensity dependence on the +
6.
MAGNETOOPTICS IN NARROW GAP DMS
261
laser excitation intensity and the calculated photon occupation number of the scattered radiation. V. Results of Magnetooptics 9. BANDPARAMETERS
The variation of the interaction gap EO with composition obtained from magnetooptics and other experiments is shown in Figs. 26 and 27 for HgMnTe and HgMnSe, respectively. A non-linear composition dependence is found for Hgl-,MnxTe, with EO(X) varying twice as fast as it does in Hgl-,Cd,Te: thus, to obtain a positive energy gap of 100 eV at 4.2 K, one would require x = 0.22 in Hgl-,CdxTe while in the case of Hgl-,Mn,Te the same energy gap occurs for x = 0.11. The zero gap-to-semiconductor transition for HgMnTe and HgMnSe takes place at x 5: 0.075 and x 3 0.06, respectively, at cryogenic temperatures. Finally, the temperature coefficient d d d T for HgMnTe is positive, twice as large as in HgCdTe, and nearly linear in x up to x = 0.15 (Dobrowolska and Dobrowolski, 1981).
*O 0 X
*
2100 oOl x
*
u
0
5
x
Hg,,
Mn, TI .
10
15
(%I
FIG.26. Interaction gap vs. composition in Hg,-,MnlTe alloys for T = 4.2 K. Black circles are from Bastard et a1 (1978, 1981), open circles from Dobrowolska et at. (1981), crosses from Jaczynski et at. (1978), and triangles from McKnight et at. (1978).
268
C. RIGAUX
-200 -070
005
--"'OO
1
5
X
FIG. 27. Energy gap vs. composition in Hgl-,Mn,Se for T = 10 K . Open circles are from interband magnetooptical data (Dobrowolska et al., 1981 ) ; full circles are from quantum transport (Takeyama and Galazka, 1979).
- I
u
-I -300 -200 -100
o
100
200
300C&meV
FIG.28. The momentum matrix element P v s . the interaction gap for Hgl-,Mn,Te alloys for T = 4.2 K. 0 are from Bastard et al. (1978, 1981); 0 are from Dobrowolska el al. (1981); 3 are from Jaczynski er al. (1978). and C
The Kane matrix element P = - i h / m ( S IpzlZ> was determined from theoretical fits of interband and intraband experiments for several compositions. The results are presented in Figs. 28 and 29 together with those obtained from Shubnikov-de Hass measurements. In both Mn-based systems, the interband momentum matrix element P shows a systematic decrease with x , unlike the behavior observed for Hgl-,Cd,Te (Guldner et al., 1978).
6.
269
MAGNETOOPTICS IN NARROW GAP DMS
0 h
5 2
80
0
-
0
90
0
c
U
a
470,
t
.
..
I
.
I
8
.
I
I
I
0
++4I
0
-
10. EXCHANGE PARAMETERS
The energies of magnetic levels depend on the exchange parameters a,p, and ( S , ) through thequantities a(&) andp(S,) (Section 11.2, Eq. (7)). Thus a typical theoretical analysis of magnetooptical data does not usually permit a separate determination of exchange integrals and of the average component of the localized spins. From interband magnetooptics, one obtains the ratio r = a / p of the exchange integrals and A ( H ) ,the relative magnitude of (S,). There are indications that in the HgMnTe system exchange integrals depend on composition. The values of r obtained in the composition ranges x I0.015 and 0.12 Ix 5 0.17 are significantly different (r = -0.5 & 0.05 and -0.8 +. 0.1, respectively; cf. Bastard et al., 1978 and 1981). Only in the very dilute region (x = 0.0014), where the experimental field variation A ( H ) follows the Brillouin function, was it possible to extract the values of Noa and NO^ from the values of A and r. For larger Mn content, A ( H )departs from the Brillouin law, and the exchange parameters and (S,) cannot be estimated without additional information. Jaczynski et a/. (1978) and Dobrowolska et al. (1981) assumed that ( S z ) may be described by a model of isolated spins and nearest-neighbor-interacting spin pairs, with a statistically random spatial distribution over the fcc lattice, and estimated the exchange coupling constant JObetween the pairs by fitting the magnetization data. Theoretical fits of quantum transport and interband magnetooptics
TABLE I1 EXCHANGE CONSTANTS IN NARROWGAP DMS. (IBMO: INTERBAND MACNETOOPTICS; FIRMO: FARINFRARED MAGNETOOPTICS.) THE CONSTANTS MODEL(PB) OR THE LUTTINGER MODELOF THE BANDSTRUCTURE WEREDERIVEDUSINGTHE PIDCEON-BROWN
T (K)
r
NOD(ev)
NOCU (eV)
0.0014 0,004-0.02 0.12-0.17 0.02; 0.03; 0.04 0.007; 0.01 1 x < 0.02
2-4.2 2-4.2 2-4.2 4.2 10-65 4.2
-0.5 -0.5 -0.8
1.5
- 0.75
1.4 0.6 0.65
- 0.7
Hg,-,Mn,Se
0.011; 0.075; 0.115 0.01 0.016 0.066 0.026; 0.047 0.011; 0.013
10-40 1.6-30 1.6-30 1.6-30 4.2 4.2
0.7 1.54 1.35 0.92 0.9 0.9
- 0.4 - 0.9 - 0.96 -0.32 - 0.35 -0.35
Hgl-,Fe,Te
5 0.016
2-4.2
Compound Hgl-,Mn,Te
Composition x
- 0.3
- 0.4
Experiments IBMO (PB) Bastard et al. (1978) IBMO (PB) Bastard et a / . (1978) IBMO (PB) Bastard et al. (1981) Quantum transport (PB) Jaczynski et a / . (1978) lBMO (PB) Dobrowolska & Dobrowolski (1981) FIRMO (Luttinger) Pastor et al. (1981) IBMO (PB) Dobrowolska et a / . (1981) Quantum transport (PB) Takeyama & Galazka (1979) Quantum transport (PB) Takeyama & Galazka (1979) Quantum transport (PB) Takeyama & Galazka (1979) Quantum transport (PB) Byszewski et al. (1980) Quantum transport (PB) Byszewski et al. (1980) IBMO (PB) Serre et al. (1982)
6.
MAGNETOOPTICS IN NARROW GAP DMS
271
performed with this method lead to quite different values of exchange integrals for dilute HgMnTe. The values of r, Nos, and NoP deduced from different experiments for several semimagnetic compounds of various compositions are collected in Table 11. The origin of the large discrepancies was not yet elucidated. This between the different determinations of (Y and /I points out the necessity of determining ( Sz)from magnetization measurements on samples investigated by magnetooptical experiments, to obtain accurate values of exchange integrals in NGDMS. VI. Concluding Remarks
Magnetooptical experiments in NGDMS demonstrate the str6ng influence of paramagnetic ions on electronic properties of these materials when an external magnetic field is present. Most of the experimental data obtained from magnetooptics and quantum transport (Kossut, this volume) agree very well with the predictions of the theoretical model, justifying the applicability of VCA and the molecular field approach used in treating carrier-ion interactions in random substitutional diluted magnetic alloys. Another important aspect of NGDMS, considered in this volume by Mycielski, is the magnetic behavior of impurity states evidenced by IR and submillimeter magnetospectroscopy . A particularly interesting property of DMS is the proportionality of the exchange-induced spin splitting to the magnetization, allowing an optical access to the magnetic properties of the localized spins. As is well known, the antiferromagnetic interactions between Mn2+ions lead to the formation of a spin-glass phase due to the frustration mechanism of the fcc lattice (de Seze, 1970; see also Oseroff and Keesom, this volume). The large interband Faraday rotation, directly related to the magnetization in DMS, permits the observation at weak fields of the formation of the spin-glass phase in Mnbased compounds (Kierzek-Pecold et al., 1984; Mycielski et al., 1984). In these experiments, the temperature dependence of the Faraday rotation, measured at fixed photon energies below the fundamental absorption edge exhibits characteristic kinks, may be related to the paramagnetic-to spinglass transition in Hgl -xMnxTe and quaternary Hgl -x-y MnxCdyTealloys. The transition temperatures Z(x) observed by means of the low field Faraday rotation and susceptibility measurements are in good mutual agreement. The phase diagram Z(x) obtained from both types of experiments is shown in Fig. 30. Faraday rotation experiments demonstrate the fact that the spin-glass transition occurs at lower temperature in quaternary alloys of larger energy gap than in HgMnTe compounds of identical Mn composition. This may indicate that the antiferromagnetic interaction between Mn2+ions due to the indirect exchange (Bloembergen-Rowland process) could be significant in
272
C. RIGAUX
2o
t
I
"0
010 0.20 0.30 0.40 0.50 X
FIG. 30. Magnetic phase diagram of Hgl-xMnxTe, P and S denoting the paramagnetic and spin-glass regions, respectively. The origin of the data points is as follows: crosses: ac susceptibility (Mycielski et al., 1984); open circles: Faraday rotation experiments (Mycielski et al., 1984); open triangles: dc susceptibility (Nagata et al., 1980); full triangles: Brandt et al. (1983); stars: Faraday rotation data for Hgo.s~Mn0.~5Cdo.zoTe and Hgo.~oMno.&do.~oTe(Mycielski et al., 1984).
the mechanism responsible for the spin-glass formation. This unique opportunity of having access to magnetic properties by studying the magnetooptical properties of the materials, as exemplified by the above remarks, is particularly attractive, and undoubtedly will be pursued further in the future.
References Andrianov, D. G., Gimelfarb, F. A., Kushnir, P. I., Lopatinskii, I . E., Pashkovskii, M. V., Savelev, A. S . , and Fistul, V. I . (1976). Sov. Phys. Semicond. 10,66. (Fit. Tekh. Pafuprov. 10, 111.) Bastard, G., Rigaux, C., and Mycielski, A. (1977). Phys. Stat. Sol. (b) 79, 585. Bastard, G . , Rigaux, C., Guldner, Y., Mycielski, J., and Mycielski, A. (1978). J. Physique 39, 87. Bastard, G. (1978a). These de Doctorat d'Etat, Paris (unpublished). Bastard, G. (1978b). (14th Int. Conf. Phys. Semicond.) Institute ofphysics Con$ Series 43 (Wilson, London), 231. Bastard, G . , and Lewiner, C. (1979). Phys. Rev. B 20, 4256. Bastard, G., Gaj, J. A,, Planel, R., and Rigaux, C. (1980). J. Physique 41, C5, 247. Bastard, G., Rigaux, C., Guldner, Y . , Mycielski, A., Furdyna, J. K., and Mullin, D. P. (1981). Phys. Rev. B 24, 1961. Brandt, N. B., and Moshchalkov, V. V. (1984). Adv. Phys. 33, 193. Brandt, N . B., Moshchalkov, V. V . , Orlov, A. O . , Skrbek, L., Tsidil'kovskii, I. M., and Chudinov, S. M. (1983). Sov. Phys. JETP57, 614. (Zh. Eksp. Teor. Fiz. 84, 1050.)
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Byszewski, P., Cieplak, M. Z., and Mongird-Gorska, A. (1980). J. Phys. C 13, 5383. Chen, Y .-F., Dobrowolska, M., Furdyna, J . K., and Rodriguez, S. (1985).Phys. Rev. B32,890. Davydov, A. B., Noskova, L. M., Ponikarov, B. B., andugodnikova, L. A. (1980). Sov. Phys. Semicond. 14, 869. (Fiz. Tekh. Poluprov. 14, 1461.) De Seze, L. (1970). J. Physique 10, L353. Dietl, T. (1981). Solid State Science 24, 344. Dobrowolska, M., Dobrowolski, W., and Mycielski, A. (1980a). SolidState Commun. 34,441. Dobrowolska, M., Dobrowolski, W., Otto, M., Dietl, T., and Galazka, R. R. (1980b). (15th Int. Conf. Phys. Semicond.) J. Phys. SOC.Japan 49, Suppl. A 477. Dobrowolska, M., and Dobrowolski, W. (1981). J. Phys. C 14, 5689. Dobrowolska, M., Dobrowolski, W., Galazka, R. R., and Mycielski, A. (1981). Phys. Stat. Sol. (b) 105, 477. Furdyna, J. K. (1982). J. Vac. Sci. Technol. 21, 220. Gaj, J. A., Byszewski, P., Cieplak, M. Z., Fishman, G., Galazka, R. R., Ginter, J., Nawrocki, M., Nguyen The Koi, Planel, R., Ranvaud, R., Twardowski, A. (1978). (14th Int. Conf. Phys. Semicond.) Institute of Physics Conf. Series 43 (Wilson, London), p. 1113. Gaj, J . A., Planel, R., and Fishman, G. (1979). Solid State Commun. 29, 435. Gaj, J. A. (1980). (15th Int. Conf. Phys. Semicond.) J. Phys. SOC. Japan 49, Supple. A, 747. Series 43 (Wilson, Galazka, R. R. (1978). (14th Int. Conf. Semicond.) Instituteo~PhysicsCon~. London), 133. Galazka, R. R., and Kossut, J. (1980). In “Narrow Gap Semiconductors: Physics and Applications,” Lecture Notes in Physics 133, p. 245. Springer, Berlin. Galazka, R. R., Nagata, S., and Keesom, P. H. (1980). Phys. Rev. B 22, 3344. Galazka, R. R. (1982). In “Physics of Narrow Gap Semiconductors” (4th Int. Conf.), Lecture Notes in Physics 152, p. 294. Springer, Berlin. Geyer, F. F., and Fan, H. Y. (1980). IEEE J. of Quantum Electron. QE16, 1365. Grynberg, M., Martinez, G., and Brunel, L. C. (1982). Solid State Cornmun. 43, 153. Guldner, Y., Rigaux, C., Grynberg, M., and Mycielski, A. (1973). Phys. Rev. B 8, 3875. Guldner, Y., Rigaux, C., Mycielski, A., and Couder, Y. (1977). Phys. Stut. Sol. (b) 81,615 and Phys. Stut. Sol. (b) 82, 149. Holm, R. T., and Furdyna, J. K. (1977). Phys. Rev. B 15, 844. Kacman, P., and Zawadzki, W. (1971). Phys. Stat. Sol. (b) 47, 629. Jaczynski, M., Kossut, J., and Galazka, R. R. (1978). Phys. Stat. Sol. (b) 88, 73. Kane, E. 0. (1957). J. Phys. Chem. Sol. 1, 249. Khattak, G. D., Amarasekara, C. D., Nagata, S., Galazka, R. R., and Keesom, P. H. (1981). Phys. Rev. B 23, 3553. Kim, R. S., and Narita, S. (1976). Phys. Stat. Sol. (b) 73, 741. Kim, R. S., Mita, Y., Takeyama, S., and Narita, S. (1982). I n “Physics of Narrow Gap Semiconductors” (4th Int. Conf.), Lecture Notes in Physics 152, p. 316. Springer, Berlin. Kierzek-Pecold, E., Szymanska, W., and Galazka, R. R. (1984). Solid State Commun. 50,658. Kossut, J. (1976). Phys. Stat. Sot (b) 78, 537. Leibler, K., Sienkiewicz, A., Checinski, K., Galazka, R. R., and Pajaczkowska, A. (1978). I n “Physics of Narrow Gap Semiconductors” (3rd Int. Conf.), p. 199. Polish Scientific Publishers, Warsaw. Luttinger, J. M. (1953). Phys. Rev. 102, 1030. Mahoney, J., Lin, C., Brumage, W., and Dorman, F. (1970). J. Chem. Phys. 53, 4286. McCombe, D. B. (1969). Phys. Rev. 181, 1206. McCombe, D. B., Wagner, R. J., and Prinz, G. A. (1970). Phys. Rev. Lett. 25, 87. McCombe, D. B., and Wagner, R. J. (1971). Phys. Rev. B 4 , 1285. McKnight, S. W., Amirtharaj, P. M., and Perkowitz, S. (1978). Solid State Commun.25,357.
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Mullin, D. P. (1980). Ph.D. Thesis, Purdue University (unpublished). Mullin, D. P., Galazka, R. R., and Furdyna, J. K. (1981). Phys. Rev. B 24, 355. Mycielski, A., Kossut, J., Dobrowolska, M., and Dobrowolski, W. (1982). J. Phys. CIS, 3293. Mycielski, A., Rigaux, C., Menant, M., Dietl, T., and Otto, M. (1984). Solid State Commun. 50, 257. Mycielski, J. (1981). In “Recent Developments in Condensed Matter Physics,” p. 725. Plenum Press, New York. Nagata, S., Galazka, R. R., Mullin, D. P., Akbarzadeh, H., Khattak, G. D., Furdyna, J. K., and Keesom, P . H. (1980). Phys. Rev. I3 22, 3331. Pajaczkowska, A., and Pauthenet, R. (1979). J. Magn. Mat. 10, 84. Pastor, K., Grynberg, M., andCouder, Y. (1978). In “Physicsof Narrow GapSemiconductors,” p. 97. Polish Scientific Publishers, Warsaw. Pastor, K., Grynberg, M., and Galazka, R. R. (1979). Solid State Commun. 29, 739. Pastor, K., Jaczynski, M., and Furdyna, J. K. (1980). J. Phys. SOC.Japan 49, Suppl. A, 779. Pastor, K., Jaczynski, M., and Furdyna, J. K. (1981). Phys. Rev. B 24, 7313. Pidgeon, C. R., and Brown, R. N. (1966). Phys. Rev. 146, 575. Rigaux, C.(1980). In “Narrow Gap Semiconductors: Physics and Applications.” Lecture Notes in Physics 133,p. 110. Springer, Berlin. Roth, L. R., Lax, B., and Zwerdling, S. (1959). Phys. Rev. 114, 90. Savage, H., Rhyne, J. J., Holm, R. T., Cullen, J . R., Carroll, C. E., and Wohlfarth, E. P . (1973). Phys. Stat. Sol. (b) 58, 685. Serre, H. (1981). These de 3eme cycle, Paris (unpublished). Serre, H., Bastard, G., Rigaux, C., Mycielski, J., and Furdyna, J. K. (1982). In “Physics of Narrow Gap Semiconductors” (4th Int. Conf.), p. 321, Lecture Notes in Physics 152. Springer, Berlin. Sondermann, U., and Vogt, E. (1977). J. Magn. Magn. Mat. 6, 223. (Physica 86-88B,419.) Sondermann, U. (1979). J. Magn. Magn. Mat. 13, 113. Spalek, J., Lewicki, A., Tarnawski, Z., Furdyna, J. K., Galazka, R. R., and Obuszko, Z. (1986). Phys. Rev. B. 33, 3407. Takeyama, S., and Galazka, R. R. (1979). Phys. Stat. Sol. (b) 96, 413. Tuchendler, J., Grynberg, M., and Couder, Y. (1973). Phys. Rev. B 8, 3884. Weiler, M. (1981). In “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), vol. 16,p. 119. Academic Press, New York. Witowski, A., Pastor, K., and Furdyna, J. K. (1982). Phys. Rev. B 26, 931.
SEMICONDUCTORS AND SEMIMETALS. VOL . 25
CHAPTER 7
Magnetooptical Properties of Large-Gap Diluted Magnetic Semiconductors J . A . Guj INSTITUTE OF EXPERIMENTAL PHYSICS. WARSAW UNIVERSITY WARSAW. POLAND
I . INTRODUCTION:POSSIBILITIES OPENEDBY STUDIES OF LARGE GAPDMS . . . . . . . . . . . . . . . . . . . . . 1 . Optical Access to Magnetic Properties. . . . . . . . . 2 . Studies of Excitons in Magnetic Field . . . . . . . . . 3 . Precise Investigation of Selection Rules . . . . . . . . TECHNIQUES . . . . . . . . . . . . . . I1 . EXPERIMENTAL 4 . Faraday Rotation . . . . . . . . . . . . . . . . . 5 . Reflectivity . . . . . . . . . . . . . . . . . . . 6 . Transmission Measurements . . . . . . . . . . . . I . Photoluminescence . . . . . . . . . . . . . . . . 8 . Other Techniques. . . . . . . . . . . . . . . . . 111. EXCHANGE SPLITTING OF THE FREEEXCITON GROUND STATE . 9 . Splitting Valuesand Selection Rules . . . . . . . . . . . 10. Relation of Splittings to Magnetization of the Ion System 1 1 . Transitions from the Spin-Orbit Split-Off Component of the Valence Band . . . . . . . . . . . . . . . . . IV . THEORETICALDESCRIPTIONOF BASIC MAGNETOOPTICAL PROPERTIES OF DMS . . . . . . . . . . . . . . . . . 12. Hamiltonian of the Problem and Typical Approximations 13 . Band EIectron States Split by Exchange . . . . . . . . 14. Zeeman Splitting of the Free Exciton Ground State . . 15 . Faraday Rotation . . . . . . . . . . . . . . . . . 16. Analysis of the Mean Field Approximation in Terms of the Magnetic Polaron . . . . . . . . . . . . . . . . . V . DETAILED MAGNETOABSORPTION STUDIES OF EXCITON STATES . . 17. Experiment Results . . . . . . . . . . . . . . . . 18 . TheoreticalAnalysis . . . . . . . . . . . . . . . . VI . EXCITON LUMINESCENCE . . . . . . . . . . . . . . . . VII . TRANSITIONS AWAYFROM THE BRILLOUIN ZONECENTER. . . 19. Experimental Data . . . . . . . . . . . . . . . . 20 . Attempts of Theoretical Description . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . .
.
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293 296 291 300 300 303 303 305 305 306 308
Copyright 0 1988 bv Academic Press. Inc . All rights of reproduction in any form rexrved . ISBN 0-12-752125-9
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I. Introduction: Possibilities Opened by Studies of Large Gap DMS Diluted magnetic semiconductors known also as semimagnetic semiconductors, offer a wide range of new possibilities in studies of magnetism and semiconductor physics. From among those possibilities we shall mention some proper to large gap semimagnetic semiconductors. 1.
OPTICAL
ACCESSTO MAGNETIC PROPERTIES
As we shall see below, magnetooptical effects such as Zeeman splitting or Faraday rotation in large gap DMS are often proportional to the magnetization of the crystal. This property enables one to study magnetic properties of DMS by means of optical measurements. A beautiful example has been supplied by Komarov et al. (1977), who measured Faraday rotation in Mndoped CdTe under microwave excitation, as a function of magnetic field. As the magnetic field reached the resonance value for Mn++ ions, the microwaves heated efficiently the Mn++spin system, causing a strong decrease of its magnetization and-as a consequence-a sharp drop in Faraday rotation (Fig. 1). That experiment is in fact a form of optical detection of magnetic resonance (ODMR). Another important example of such possibilities is the application of Faraday rotation to spin-glass phase studies. It has been used by Kett et al. (1981) and by Kierzek-Pecold et al. (1984) in Cdl-,Mn,Te.
FIG. 1 Faraday rotation of Cdl-xMn,Te versus magnetic field with microwave excitation for two wavelengths (1 and 2). Note a drop in rotation at 3.5 kOe (resonance field). [After Komarov et al. (1977).]
7.
MAGNETOOPTICAL PROPERTIES
277
2. STUDIES OF EXCITONS IN MAGNETIC FIELD
Exciton studies in DMS may contribute to the solution of a very general problem: that of a hydrogen atom in an arbitrary magnetic field. The problem in its general form has received little attention among atomic physicists, since in order to obtain a magnetic characteristic energy ( A o , ) comparable to the Coulomb characteristic energy (Rydberg), a magnetic field unavailable on earth would be needed. A perturbation treatment is thus sufficient (except in the case of recently studied highly excited Rydberg states). In an exciton in a semiconductor, due to small values of effective masses and to high values of the dielectric constant, these two energies can be made comparable in a field available in the laboratory. However, the problem is usually more difficult than in the hydrogen atom because of the complicated structure of the energy bands involved. Semimagnetic semiconductors, on the other hand, due to large exchange splittings, offer an advantageous possibility to create excitons of holes and electrons from simple nondegenerate bands. We shall consider this in more detail in Part V. 3. PRECISE INVESTIGATION OF SELECTION RULES
Due to high values of the Zeeman splittings in semimagnetic semiconductors, absorption lines observed in different polarizations are sufficiently separated to resolve and study weak lines due to forbidden transitions (they are, typically, lo2times weaker than the allowed transitions). This possibility has been used by Nguyen The Khoi et al. (1983), who explained the observed asymmetrical splitting of the absorption edge of Cdl-,Mn,Te in a magnetic field by a small contribution from forbidden transitions. This contribution has been measured as a function of magnetic field. The origin of the deviation from strict selection rules remains still an open question. 11. Experimental Techniques 4. FARADAY ROTATION
Standard Faraday rotation measurements in DMS are greatly facilitated by the fact that, due to the strong exchange field of the magnetic ions acting on charge carriers of the host semiconductor, Zeeman splittings of optical transitions are very large under typical experimental conditions, giving rise to a strong influence of the magnetic field on the real part of the dielectric function. Thus in a typical experiment, polarization direction of the incident light may rotate many times (Gaj, 1981) before leaving the sample. In such a case, it is sufficient to use a simple transmission setup (Fig. 2) and, without
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bothering to turn the polarizers, to plot the intensity of the transmitted light as a function of magnetic field. Subsequent minima of the resulting curve correspond to multiples of of the rotation angle. Furthermore, varying the wavelength at a fixed magnetic field one obtains also an oscillating curve, permitting a quick determination of the whole Faraday rotation spectrum from a single recorder trace (Fig. 3). In some experiments, however, it is desirable to measure small rotation angles, of the order of minutes of arc. This is the case in the spin-glass phase magnetooptical studies, where use of a small magnetic field is of a great importance. Small rotations occur also in reflection of polarized light from a crystal placed in a magnetic field-the so-called magneto-optical Kerr effect. In small angle measurements various modulation techniques prove useful. An example is shown in Fig. 4. An oscillating plate consisting of two pieces of polarizing foil with mutually perpendicular polarization directions is placed in the optical system in such a way that the outgoing light passes alternatively through two pieces of the foil. If the incident light is polarized along a direction at n / 4 to polarization directions of both pieces of foil and for zero Faraday rotation, no modulation will arise. For small deviations of the outgoing light polarization from the equilibrium direction, a modulation Md
sc
P
FIG. 2. Experimental setup for Faraday rotation measurements in semimagnetic semiconductors. The elements of the setup are designated as follows: Sc-source, P-polarizers, C-superconducting coil, S-sample, Md-modulator, M-monochromator, D-detector, E-electronics.
0.77
078 0.79 2 (pm)
0
FIG. 3. Intensity versus wavelength in a Faraday rotation measurement in Cd,-,Mn,Te, at a fixed magnetic field of about 1 T.
7.
MAGNETOOPTICAL PROPERTIES
219
-
FIG.4. Oscillating plate of a polarization modulator. Polarization directions in two parts of
the plate marked by arrows. The incident light polarization is shown by the bottom arrow.
signal will be proportional to the deviation angle. In such a manner rotations up to a few degrees can be measured with an accuracy depending on the mechanical stability of the system. Accuracy of 1’ is achieved without any special precautions. With Verdet constants of the order of a few deg/cm.Gs and in samples a couple mm thick, it is possible in this technique to use fields of one Gs or less, values quite adequate in spin-glass studies. 5 . REFLECTIVITY
Reflectivity measurements are commonly used in DMS to determine Zeeman splittings of the free exciton ground state. Since only the positions of the reflectivity structures are usually of interest, absolute values of the reflection coefficient are not determined. Relative reflectivity measurement does not present any difficulties. The only precaution to be taken is to avoid photoluminescence excitation. This may be easily done with suitable filters (usually it is convenient to place the monochromator after the sample). A typical experimental setup is shown in Fig. 5 . Figure 6 shows an example of magnetoreflectivity results in Cdl -,Mn,Te. 6. TRANSMISSION MEASUREMENTS
The transmission configuration usually allows the experimenter to obtain more detailed and precise information on optical transitions than the reflectivity. In a transmission spectrum of Cdl-,Mn,Te shown in Fig. 7, besides the ground exciton state, an excited (2s) state is clearly visible in the absence of a magnetic field. Application of a field of 5T (Twardowski, 1981b) produces a great number of absorption peaks, invisible in a straight-forward reflectivity spectrum. In spite of its informative value, the transmission technique is used less frequently, because it requires much more effort. Since
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FIG.5 . Experimental setup for magnetoreflectivity measurements (Sc-source, L-lenses, F-filter, S-sample, C-superconducting coil, Q-quarter wave plate, P-linear polarizer, M-monochromator, D-detector, and E-electronics).
I64
I63
I62
I61
PHOTON ENERGY ( e V )
FIG.6. Reflectivity spectra of Cdl-,MnxTe at 1 . 1 Tin the Faraday configuration (a' and upolarizations) and in the Voigt configuration (n polarization). [After Gaj el of. (1979).]
the absorption coefficient in the excitonic region of adirect gap semiconductor is of the order of 105-106cm-', samples with a thickness below 1pm are necessary. Such samples have been obtained by local etching by Twardowski et al. (1979), using a method known in electron microscopy, as follows. A 200pm slice of the crystal is placed in an etching solution, with both faces protected by plastic masks that cover the faces of the slice except for two small circular areas, available for etching. By carefully controlling the process of etching, a spot of about 20-50pm in diameter and a thickness sufficiently small to allow light to pass can be obtained. Figure 8 shows the etching arrangement and the resulting sample form.
7.
MAGNETOOPTICAL PROPERTIES
281
c+ B=5T
1.61
1.62
1.63 1.64 1.65 1.66 1.67 ENERGY [eV]
FIG.7. Transmission spectra of Cdl-,Mn,Te at zero magnetic field and at 5T.Temperature 1.8 K. [After Twardowski (1981b).] Besides the principal magnetically split absorption peaks A, B, C, D, a rich structure is visible.
FIG.8. Sample etching arrangement. The optical system is used to interrupt etching when the sample becomes sufficiently thin to transmit light (Sc-source, L-lens, E-etching solution, B-bar of a magnetic stirrer, S-sample, P-plastic screens, M-microscope of low magnification).
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M1
sc
L
-
FIG. 9. Experimental setup for magnetotransmission measurements (Sc-source, L-lens, S-sample, C-superconducting coil, M1, M2--monochromators, Q-quarter wave plate, P-polarizer, D-detector, E-electronics).
Inhomogeneous thickness of samples obtained in this manner does not permit quantitative absorption measurements, but it allows to determine positions of the absorption peaks. Results of Fig. 7 have been obtained on such a sample. In such experiments, due to the small size of the transmitting region of the sample, special care must be taken to avoid photoluminescence excitation. A convenient way to do this is to use two monochromators. The first one, with its slits open wide, transmits from the light emitted by the source a band corresponding to the width of the spectrum to be taken. The second one, placed after the sample, is used to scan the spectral range. Figure 9 shows schematically such an experimental setup. 7. PHOTOLUMINESCENCE
In photoluminescence experiments on mixed crystals, it is of particular importance to know exactly the energy gap of the excited region of the sample. Therefore in photoluminescence measurements on semimagnetic semiconductors, it is desirable to do a complementary measurement of absorption or reflectivity on the same sample, under the same experimental conditions. As for photoluminescence itself, a number of methods have been used with pulsed or CW-excitation, using dye, Argon ion, or Helium-Neon lasers. 8. OTHERTECHNIQUES
Apart from the principal experimental methods discussed above, and Raman scattering described in detail in Chapter 9 in this volume, a number of techniques were used in magnetooptical studies of wide gap DMS. Among them must be mentioned optical detection of magnetic resonance (ODMR) in photoluminescence (Komarov et al., 1980; Malyavkin, 1983) and Faraday rotation (Komarov et al., 1977), modulated reflectivity (Ginter et at., 1983) and photoconductivity (Lindstrom e f al., 1983).
7.
MAGNETOOPTICAL PROPERTIES
283
111. Exchange Splitting of the Free Exciton Ground State
9. SPLITTING VALUESAND SELECTION RULES Zeeman effect on the free exciton ground (1s) state has been studied in a number of semimagnetic semiconductors. Results vary depending on crystal structure of the semiconductor.
a. Cubic Structure Magnetoreflectivity and magnetotransmission have been studied in a number of cubic (zinc blende) large gap DMS, such as Cdl-,Mn,Te (Komarov et al., 1977; Twardowski et al., 1979; Gaj et al., 1978, 1979; Rebman et al., 1983), Znl-,Mn,Te (Komarov et al., 1978; Twardowski etaf., 1984) or Znl-,MnxSe (Komarov et af., 1980b; Twardowski et al., 1983). The exciton ground state in those alloys splits in six components: four (denoted A through D) visible in the Q polarization (electric field of light perpendicular to the external magnetic field) and two (E and F) visible in the 72 polarization (electric field of light parallel to the external magnetic field). In the Faraday configuration (light propagation along the external magnetic field), the four a-components exhibit a' (A and B) or a- (C and D) circular polarizations (see also Fig. 16). The whole splitting pattern is approximately symmetric relative to the zero field position. Components B and C are significantly weaker than A and D. The lowest-lying component A is much sharper (especially at high fields) than the remaining ones. The features described above are visible in Figs. 6 and 7 for reflectivity and transmission, respectively, of Cdl -,Mn,Te. For a given semiconductor in a given magnetic field and temperature, the Zeeman pattern can be characterized by splitting values of two pairs of acomponents: strong (A and D) and weak ones (B and C). The splitting of the two n components, E and F, is not independent. It equals one half of the difference between splittings A-D and B-C. The ratio of the two latter splittings has been found to be independent of magnetic field or temperature for a given semiconductor. The polarization selection rules are quite strong. However, absorption measurements allow one to observe very small deviations from perfect selection rules. Nguyen The Khoi el al. (1983) have observed a small (of the order of 1%) admixture of Q- polarization in some a+-polarized Zeeman components in Cdl-,Mn,Te. Figure 10shows the positions of the Zeeman components of the free exciton ground state in a sample of Cdl-,Mn,Te as a function of magnetic field. Splittings increase linearly at low fields and tend to saturate at higher field values. That behavior can be examined in detail in Fig. 11, where the splitting
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of the strong D components A and D is plotted vesus magnetic field for various compositions. In crystals of low Mn mole fraction x , the splitting increases with x and the saturation is clearly visible, while at higher x values the curves show only a slight deviation from linearity. At higher temperatures the values of the splittings becomes smaller and the tendency to saturate is weaker.
FIG. 10. Energy of the Zeernan components of the exciton versus magnetic field for CdI-,Mn,Te in the Faraday configuration. Continuous lines are drawn to guide the eye. [After Twardowski (1981b).]
FIG. 11. Splitting of the two strong u components of the exciton Zeernan structure of Cdl-,Mn,Te versus magnetic field at 1.4 K. Manganese concentration values are given in percentage. [After Gaj e t a / . (1979a).]
7.
285
MAGNETOOPTICAL PROPERTIES
b. Hexagonal Structure Magnetoreflectivity and magnetoabsorption measurements have also been reported on hexagonal (wurtzite) semimagnetic semiconductors, such as Cdl-,Mn,Se (Arciszewska and Nawrocki, 1982; Aggarwal et al., 1983; Komarov et al., 1980c) or Cdl-,Mn,S (Gubarev, 1981). The results show a behavior generally more complicated than in the cubic case. A relatively simple situation can be obtained by applying the magnetic field along the hexagonal axis of the crystal. The resulting splitting pattern and selection rules are then analogous to those in cubic crystals', except for a displacement of the two strong o components relative to the remaining ones due to the crystal axial anisotropy (see Fig. 12). The variation of the effect with composition and temperature are also like in cubic semimagnetic semiconductors. For an arbitrary direction of the magnetic field, the results are more complicated.
-2
2.31
I
I
I
MAGNETIC FIELD f T ) FIG. 12. Energy of the Zeeman components of the exciton versus magnetic field for Cdl-,Mn,Se in the Faraday configuration. Magnetic field is parallel to the crystal hexagonal axis. [After Arciszewska and Nawrocki (1984).] CdS seems to be an exception. The strong u+ component, being narrower (as in other alloys) is placed above its u- counterpart (Gubarev, 1981).
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10. RELATIONSHIP OF THE SPLITTINGS TO THE MAGNETIZATION OF THE IONSYSTEM
The behavior of the Zeeman splittings as a function of the magnetic field, temperature and composition resembles (particularly at low compositions) the magnetization of a paramagnet. Therefore it is natural to look for a relationship between the magnetization of the ion system present in the crystal and the Zeeman splittings. Experiments performed on a number of wide gap DMS show a simple proportionality of the splittings to the magnetization. Figure 13 shows that relationship for Cdl-,Mn,Te. The plot of Fig. 13 has been obtained from data measured on many samples, with compositions varying from x = 0.005 up to x = 0.30. Expressing the magnetization of the ion system by a mean value of the spin component along the magnetic field per cation x ( S , ) , we can write for the splitting: AE
=
AX( S z ) .
The coefficient L determined for Cdl-,Mn,Te from the results presented in Fig. 13 is 1.10eV. In other wide gap semimagnetic semiconductors similar values of that coefficient have been obtained. It must be pointed out that the proportion described above is only approximate. A more detailed discussion will follow in Part IV I
I
1
3
I
,
x 0.5 -0
0 +
I
2 5 0 10 A 20
30
I -10'~
-10-
-10-
SPIN PER UNIT CELL
FIG. 13. Splitting of the two strong u components of the exciton Zeeman structure of Cdl-,Mn,Te versus magnetization, expressed in terms of the mean value of the Mn spin (per cation). The straight line corresponds to a proportionality with a slope of 1.10 eV. Manganese concentration is indicated in percentage. [After Gaj et a/. (1979b).]
7.
MAGNETOOPTICAL PROPERTIES
11. TRANSITIONS FROM THE SPIN-ORBIT SPLIT-OFF OF THE VALENCEBAND COMPONENT
An exciton state can also be created with a hole from the lower component of the valence band, split off by the spin-orbit interaction. Such a state is more or less difficult to observe depending on the value of the spin-orbit splitting A. It is quite easy to see in CdS (A = 0.065 eV), fairly easy in CdSe (A = 0.45 eV) whereas in CdTe (A = 0.95 eV) it takes some effort to observe it. In an alloy, disorder introduces an additional broadening, making its observation more difficult. In CdTe an exciton absorption line associated with transitions from the r7 valence band has been observed by Twardowski et al. (1980). Substituting 2% of Cd by Mn broadens the structure enough to make the absorption peak disappear (only a threshold remains, as seen in Fig. 14). Any magnetooptical measurements on such an ill-defined structure must be inaccurate. Within
I
1
I
I
2.550
2.600
I
[eV]
ENERGY FIG. 14. Transmission versus photon energy for CdTe ( B = 0) and Cdo.9~Mn~.o~Te (B = 0 and 4 T ) . Straight lines were drawn to determine transition energies (marked by arrows). [After Twardowski ef al. (1980).]
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that poor accuracy, Faraday configuration measurements show a splitting of the two components occurring in the g+ and 0- polarizations, respectively, with the value of the splitting proportional to the magnetization in Cdl-,Mn,Te (Twardowski et al., 1980) and in CdMnSe for magnetic field parallel to the hexagonal axis (Arciszewska and Nawrocki, 1984). IV. Theoretical Description of Basic Magnetooptical Properties of DMS
12. HAMILTONIAN OF THE PROBLEM AND TYPICAL APPROXIMATIONS We shall look for solutions of the eigenproblem of a band electron in a given external magnetic field and a given state of the magnetic ion system. Such an approach can be justified if the band electron wave function is sufficiently delocalized to make an influence of the electron on each of the magnetic ions negligible. The Hamiltonian for a band electron will have the form,
H
=
Ho
+ HB + HC + Hex + Heh
(1)
where the five terms in the Hamiltonian correspond to various interactions influencing the band electron state. Relative contributions of each of the terms may vary depending on the crystal in question and on experimental conditions. H, is here a (one electron) Hamiltonian of the electron in a perfect crystal:
H,
P2
=-
2m
+ V(r)
The term
e HB = - - p ' A mc
+ g/fBb'B
(3)
describes (in a linear approximation) the direct influence of the magnetic field on the carrier state. This term is responsible for Landau quantization and spin splitting of Landau levels (this gives, of course, only a small fraction of the total spin splitting in DMS). The Coulomb term HC represents energy of the carrier interacting with an impurity or with another carrier (e.g., electron-hole interaction producing exciton states). The term Hex represents the fundamental interaction in a semimagnetic semiconductor: the carrier-ion exchange. Following Kossut (1976) and Bastard el al. (1978) we shall express it in a Heisenberg form:
Hex =
-c J(r n
- R,)s~
-G
(4)
7.
MAGNETOOPTICAL PROPERTIES
289
where the index n runs over all the magnetic ions. The term Heh represents the electron-hole exchange interaction and may be of importance in exciton problems. The difference between the present case and a non-magnetic semiconductor lies in the presence of the ion-carrier exchange Hamiltonian Hexin Eq. (1). Unfortunately Hex does not possess the translational symmetry of a perfect crystal. To avoid that difficulty, one commonly involves the molecular field approximation. Strictly speaking, this is combined with a virtual crystal approximation: to every cation site we attribute a fraction of the magnetic ion spin equal to the fraction of cation sites really occupied by magnetic ions. Those spins, instead of being treated properly as operators, are represented by numbers, i.e., by thermal average values. Thus in Eq. (4),all the manganese spin operators S n are replaced by their (common) average value ( S n ) and the sum over the magnetic ions is replaced by the sum over all the cation sites multiplied by the magnetic ion mole fraction x. Considering a magnetically isotropic crystal placed in a magnetic field (taken along the z-axis), we have ( S ” ) = ( S y ) = 0, and Eq. (4),then becomes
He, =
-Ci
J(r - R i ) x a Z ( S Z ) .
(5)
It must be pointed out that, within the framework of the effective mass theory, the above Hamiltonian has the form identical to the spin part of HB (Zeeman energy of the electron spin). Therefore the introduction of the exchange term He, does not in principle create any new formal difficulties (within the approximation outlined above) compared to the non-magnetic case. Let us make now another essential approximation, neglecting the term HB. This is justified in wide gap DMS when the temperature is not excessively high, i.e., when the magnetic field effects due to carrier-ion exchange are strong compared to “nonmagnetic” effects described by H B . Now the Landau quantization is absent from the problem, and we may speak of the electron wave vector k in the presence of a magnetic field. In what follows we present a k * p calculation of the energy bands, using the following further approximations: (1) The k * p term is included as a second order perturbation (Luttinger model). In the absence of exchange, this leads to parabolic bands; (2) Warping of the valance band is neglected, as well as effects due to lack of inversion symmetry of the crystal; (3) Spin-orbit splitting of the valence band is assumed to be large compared to the ion-carrier exchange energy.
290
J. A . GAJ
13. BANDELECTRON STATES
SPLIT BY
EXCHANGE
Let us consider a cubic zinc blende crystal. Under the above assumptions, we can treat separately the conduction band and each component of the valence band. Following Bir and Pikus (1972), we shall use as basis vectors for the conduction (r6)band:
I),
+)C
= st
In the chosen bases, matrices of Hexare diagonal:
(Wv3/2IHexlWu3/2)
where
=
3 O 0 0
B
O
O B O 0 - B 0
0
O O 0
-3B
7.
MAGNETOOPTICAL PROPERTIES
291
Here No denotes the number of unit cells per unit volume, and a = (slJls) and p = ( X l J l X ) are exchange integrals for the conduction and valence bands, respectively. Within the assumed approximations the total Hamiltonian will now consist of on!y two terms, H = Ho + H e x . (13) In the k * p calculation the term H , will be represented by
-
Let us point out that at k = 0, where the k p term vanishes, the presence of exchange produces two-fold or four-fold and equal splittings of bands, with the splitting values proportional to x( S z ) and therefore (if diamagnetic effects are neglected) to the magnetization of the crystal. The solutions of the problem follow immediately for the conduction, as well as the lower valence bands. In those cases the entire Hamiltonian matrices (Hexand k * p terms) are diagonal, where the energies Ec = E,"
h2k2 ++- 3A 2mc
and
correspond to the wavefunctions y/,+(k) = sfeik' y;(k) = sleikr y&(k)
=
1
- [(X + iY)l
d3
-i vui/2(k) = - [(X - i Y ) t
\r5
+ Zt]eIb - Zl]eih
for the two bands, respectively. In the case of the TSvalence band, the k p Hamiltonian matrix is
-
(18)
292
J. A. GAJ
where in the spherical approximation
F = Mk2 + *N(k? + k,?) G
=
(A4+ f N ) k 2 - SN(k:
+ k,?)
N
I=JTZ (kx - iky)2, M and N being combinations of interband momentum matrix elements (Bir and Pikus, 1972). Although in a general case the diagonalization of the total Hamiltonian is rather complicated, some particular situations are simple to solve, e.g., the band structure in the vicinity of k = 0 for a given non-zero magnetization. The k p term may then be treated as a perturbation, producing four equally split anisotropic bands:
-
Ei,4 = F I+- 3B, E2,3 = G f B ,
or, in terms of light and heavy hole masses,
(--
h2k2 + 3 h2 - Gh2) ( k : E2,3 = -2mlh 4 2mrh
+ k,?) f B.
\--I
We thus obtain four bands whose constant energy surfaces are ellipsoids: cucumber-(E1,4) or disc-like (E2,3). The diagonalization of the k p Hamiltonian alone produces the well known spherical and parabolic light and heavy hole bands:
-
h2k2 Ehh(k) = E," + Mk2 = E," - ---, 2mhh
The wavefunctions corresponding to Eq. (23) and (24) can be easily obtained by a suitable rotation of the basis functions. Note that for k along the z-axis the Hamiltonian given by Eq. (19) is diagonal, considerably simplifying the calculations (see, e.g., Kane, 1957).
7.
MAGNETOOPTICAL PROPERTIES
293
t 0
- 10 -20
3 +lC;
2
1
0
1
2
3
Z("= 100)-
FIG. 1s. Energy band structure of Cdl-,Mn,Te in dimensionless energy ( E ) and wave vector
(K)coordinates at two values of the angle between the wave vector and the magnetization: 90" (KJ and 10". IAfter Gaj e t a / . (1978).]
In a case of a small magnetic field [and hence a small exchange term B, Eq. (12)] He, can be treated as a perturbation. A straightforward first order calculation produces highly anisotropic splittings of light and heavy hole bands: Elh(k,
B) = Eih(k) f B d 4
Ehh(k,
-
3 COS2 8
B) = E l h ( k ) f 3 8 C O S 6
(25)
(26)
where B is the angle between the wave vector k and the magnetization direction. A numerical diagonalization of the Hamiltonian matrix in a general case produces a highly nonparabolic and anisotropic band structure. This is shown in Fig. 15 for two directions of the wave vector: at angles 90" and 10" to the magnetization direction. 14. ZEEMANSPLITTING OF FREEEXCITON GROUNDSTATE
The problem of exciton states in the presence of an exchange field has been treated, e.g., by Cieplak (1980), who followed an approach developed by Altarelli and Lipari (1973). We shall not enter here into the details of the complete exciton calculation, but will simply note that in the effective mass approximation the Coulomb potential of the electron-hole interaction HC does not mix the basis functions, enabling us to think of an exciton with a hole from one of the exchange-split valence subbands and an electron from one of the conduction subbands, if the electron-hole exchange is to be neglected. This allows us to describe excitonic magnetooptical splittings in terms of interband transitions. Remembering that direct exciton creation
294
J . A. GAJ
occurs practically at k = 0 we can, according to Eqs. (4)and (21), work out a Zeeman scheme presented in Fig. 16 and in Table I. The relative intensities are, of course, proportional to the corresponding interband matrix elements. The data of Table I are in good agreement with the experimental results obtained in Cdl-,Mn,Te if the exchange parameters are: N,a = 0.22 eV and N,P = -0.88eV. For excitons involving holes from the r7 band, the experimental results are not very precise, but within experimental error they agree with the above model (Twardowski et al., 1980). In other cubic semimagnetic semiconductors similar agreement has been found. Table I1 gives exchange parameter values for various large gap DMS, obtained from a comparison of magnetooptical splittings with magnetization. In hexagonal semiconductors the situation is more complicated. In the first rough approximation (the so-called quasi-cubic model), one can use the same wavefunctions as for a cubic crystal, adding a diagonal term corresponding to a crystal (axial) field. If the magnetic field is parallel to the crystal c-axis and the spin-orbit splitting is large compared to the crystal field and to the exchange splittings, the model outlined above can be applied, the only difference consisting in a displacement of the ,;I +*> states relative to the I$, k $>ones due to the crystal field splitting. However, experimental results of Arciszewska and Nawrocki (1982) show clearly that the quasi-cubic approximation is inadequate in Cdl-xMnxSe. In particular, the relation between splittings A-D, B-C, and E-F mentioned in Section 9 is not satisfied (see Fig. 12). To improve the model, an anisotropy of the spin-orbit interaction may be introduced. That additional parameter allows for a reasonably accurate description of the experimental data in Cdl-,Mn,Se, using a single
V
cond. G band
valence bands
G r,
FIG.16. A schematic representation of band splittings in a cubic (zinc blende) semimagnetic semiconductor. Transitions allowed in the Faraday configuration are marked by arrows.
7.
295
MAGNETOOPTICAL. PROPERTIES
TABLE I CALCULATED ENERGIES AND RELATIVE INTENSITIES OF THE ZEEMAN COMPONENTS OF THE FREEEXCITON IN A ZINCBLENDE SEMIMAGNETIC SEMICONDUCTOR States Valence
Conduction
Energy (Relative to zero field position)
3B B B -B -B -3B -B -B B B
Polarization
- 3A
U+
+ 3A -
U+
3A
R
+ 3A
R
- 3.4 + 3A + 3A - 3A + 3A - 3A
U U
-
U+
R
n 0
Relative intensity
Symbol
3 1 2 2 1
3 2 1 I 2
A B
E F
C
D
-'
A = --x NO(Y(S') B = sxNoP(S'> Attention: (S') is normally negative!
TABLE I1 EXCHANGE CONSTANTS IN VARIOUS LARGEGAPDMS DETERMINED FROM MAGNETOREFLECTNITY OR MAGNETOABSORPTION COMBINED WITH MAGNETIZATION MEASUREMENTS
~~
~
Crystal structure
Naa
NaP
Compound
[evl
[evl
Cdl-,Mn,Te
zinc blende
0.22
-0.88
Gaj el al. (1979)
Znl-xMnxTe
zinc blende
0.19
- 1.09
Twardowski et a/. (1984)
Znl-xMnxSe
zinc blende
0.29
- 1.4
Twardowski et al. (1983)
Cdl-,MgSe
wurtzite
0.23
-
Cdl-,Mn,S
wurtzite
Reference
1.27
Arciszewska and Nawrocki (1982)
5.6"
Gubarev (1981)
'A positive value was obtained as a consequence of defining the Heisenberg Hamiltonian without a minus sign [see Eq. (4)]. Both the sign and the unexpectedly high absolute value must be regarded with caution (see also the footnote in Section 9).
296
J . A. GAJ
exchange constant 3/ for all the valence band functions (see Table 11). In principle, two different values px= ( X I J I X ) and pz = ( Z l J l Z ) should be allowed, since the X and Z functions are no longer equivalent. For an arbitrary direction of the magnetic field, the exchange splittings are no longer proportional to the magnetization, since the two perturbations, crystal field and exchange field, are now represented by non-commuting Hamiltonians of comparable magnitude. The situation at k = 0 resembles the eigenproblem in a cubic semiconductor k # 0, which has rather complicated non-linear solutions, as mentioned above. In general, magnetooptical studies of hexagonal DMS also permit one to find (by extrapolation) parameters of the band structure of the host nonmagnetic semiconductor. The assumption that px = pz can be discarded if transitions from the spin-orbit split-off component of the valence band are also studied. 15. FARADAY ROTATION Using the results of the previous sections, one can now calculate Faraday rotation spectra. This has been done using the excitonic absorption model developed by Elliott (1957). The model predicts a series of discrete hydrogenic lines corresponding to creation of consecutive exciton bound states of energies En = Eg - R*/nZ (R*-exciton binding energy) and of oscillator strength varying as l / n 3 . Above the energy gap, absorption is a continuous function of the photon energy, with zero slope at Eg,and slowly rising above that value to approach a square-root curve at high energies ( A o - Eg S= R*). The value of the absorption coefficient at ho = Eg is equal to the value obtained by an infinitesimal broadening of the discrete lines at A o E g . In reality a broadening must always be expected, so that only a limited number of states of the hydrogenic series can be distinguished (two states are observable in Fig. 7). There is no particular difficulty in calculating the Faraday rotation spectra using the full Elliott model. However, a very simplified approach, consisting in the calculation of two first nonvanishing moments of the circular dichroism, gives quite an adequate description of the experimental rotation spectra near the band gap (Gaj, 1981). In the calculation, performed in a linear (low field) approximation, the line broadening, as well as any dichroism above the energy gap, has been neglected. The resulting solution for the rotation angle 8 is:
7.
MAGNETOOPTICAL PROPERTIES
297
0
c 0
c 0
a
035
0.20 log ( E l l eV1
FIG. 17. Faraday rotation spectrum of Cdl-,Mn,Te fitted to Eq. (4): x = 0.05,T = 77 K, = 0.12, Eg = 1.66eV.
KO
where
KO
d(i 4)
is the extinction coefficient at tzo = E g ,
Eo = Eg - 2R*[
f
n=l
4 n
-
2
n = n ~
-
i 41 n
n=1
= Eg - 1.14R* (28)
I is the sample thickness and c is the velocity of light. The resulting solution should, of course, not be used too close to the exciton structure. In Fig. 17 an experimental rotation spectrum has been fitted with Eq. (27) by adjusting two parameters, KO and Eg,producing good agreement and reasonable values of these parameters. (Reflection data give the same value of Eg). The values of A and B were calculated from Eq. (12) using magnetic susceptibility data of Oseroff (1982). 16. ANALYSIS OF THE MEAN FIELDAPPROXIMATION IN TERMSOF THE MAGNETIC POLARON
In a detailed analysis of magnetooptical and magnetoreflectivity data, Zeeman splittings show small deviations from strict proportionality to magnetization. In order to explain those deviations we must go beyond the molecular field approximation and admit a mutual interaction between the ions and the band electrons. The simplest way to do this is by looking for eigenfunctions (in a magnetic field) of an electron interacting equally strongly with N magnetic ions. We take a limited (several tens) number of ions and we assume that the
298
J . A. GAJ
= I
N.32
gpB/ kT=0
n
A
..........
-----
ll 0-
(b) FIG. 18. Exciton absorption spectra calculated by Gaj and Golnik (1985): (a) zero field, no broadening; (b) broadening included.
7.
MAGNETOOPTICAL PROPERTIES
299
electron wave-function is localized. Formally the problem has much in common with the model of the bound magnetic polaron used extensively in physics of magnetic semiconductors (see Chapter 10 in this volume). Attempts to apply that model to magnetoabsorption of semimagnetic semiconductors have been done by von Ortenberg (1984) and by Gaj and Golnik (1985). We shall follow here the approach of the latter work. The physical meaning of the model is represented by the splitting of an electron (or exciton) energy by the exchange field of spin fluctuations of magnetic ions. Even in the absence of a net macroscopic magnetization, those fluctuations will lead to a non-zero splitting of band electron states. Figure 18a shows a diagram of the density of states (that corresponds to optically allowed transitions) for an exciton in a CdTe-type semiconductor interacting with N = 32 ions, each with a spin s = $ (the ion-ion interaction is neglected). Each configuration of the ion system (given by quantum numbers of the total spin Sand its projection M o n a quantization axis) generally gives rise to six eigenstates of the coupled system. It is reasonable to assume that the state of magnetic ions does not change during the optical transition (a “magnetic Frank-Condon principle”). Without an external magnetic field the optical absorption spectrum will be identical with the density of states curve in Fig. 18a. The presence of a magnetic field, on the other hand, will favor states with a large total spin S of the ion system. It may be interesting to note that within this model the position of the center of gravity of the absorption spectrum for each polarization follows exactly the molecular field result. The above model gives a qualitative explanation of the experimentally observed deviation from the mean field behavior in wide gap DMS, when supplemented by a phenomenological broadening large enough to remove the zero-field splitting that is not observed in those compounds. Results (shown in Figs. 18b and 19b) are consistent with experimental data (Fig. 19a).
FIG. 19. Splittingof the two strong acomponents of the excitonic Zeeman structure measured in Cd,-,Mn,Te (a), compared with a calculation for N = 32 Mn ions interacting with the exciton (b). [After Gaj and Golnik (1985).]
300
J. A. G k l
The model discussed above includes a localization of the exciton. The nature of this localization remains an open question. One of the possible explanations is to view it as due to alloy composition fluctuations. V. Detailed Magnetoabsorption Studies of Exciton States 17. EXPERIMENTAL RESULTS
Transmission studies on thin samples reveal many more details than the energy of the exciton ground state and its Zeeman components. We shall present here a survey of experimental results obtained mainly on Cdl-,MnxTe by Twardowski and coworkers (1979, 1981, 1982). In the magnetoabsorption spectra shown in Fig. 7, above the principal components of the exchange split exciton ground states A1, . .., D1, a number of weaker lines are visible that can be denoted A2, ..., As, Bz,..., B4, etc. We have thus four sets of absorption lines Ai through Di. Evolution of all those lines with magnetic field is shown in Fig. 20. As discussed above, the energy of the ground state of each set (A,, ...,DI) is determined by the magnetization of the sample and by the exchange parameters.
1,700
-2 Y
>
0
a W
z W
1,650
1
2
3
4
5
MAGNETIC FIELD (TESLAS)
FIG.20. Positions of the features of the magnetoabsorption of the exciton in Cdl-xMnxTe versus magnetic field. [After Twardowski (1981a).]
7.
MAGNETOOPTICAL PROPERTIES
301
Let us limit our discussion to the case of strong exchange effects (x 2 0.02 at liquid He temperatures). It turns out that, within each set, the positions of the lines related to the principal (i = 1) line depend only on the magnetic field, and not on magnetization. These two types of behavior, governed by magnetization (ground states of sets) or by magnetic field (relative position within sets) can be seen in Figs. 21 and 22. In Fig. 21, where magnetization is varied by changing the temperature at a given magnetic field, constituent lines of the sets A and B follow the variation of the corresponding ground states A1 and BI. The magnetization dependent behavior is eliminated in Fig. 22, where the relative positions within sets A and B are plotted as a function of magnetic field. The data from various compositions (and thus various magnetization values) coincide well in the plot. Such a distinction is not possible in CdTe or in very diluted crystals ( x I0.01), where the exchange energy is comparable to, or smaller than, the cyclotron energy.
FIG. 21. Positions of magnetoabsorption peaks in Cdt-,Mn,Te at a fixed magnetic field B = 5T,versus the splitting of the strong u components A I and D1. Magnetization is varied by changing the temperature and the composition. [After Twardowski (1981a).]
302
J . A. GAJ I
I
1
I
I
B
B
5'10 *
2
1Oo/o
6
3
4
5
B(TJ
(b) FIG.22. Line positions within sets Ai (a) and Bi (b) versus magnetic field. Continuous curves correspond to a calculation described in Section 18. [After Twardowski (1981a).]
7.
MAGNETOOPTICAL PROPERTIES
303
18. THEORETICAL ANALYSIS
The magnetotransmission data reviewed above support the idea that the exchange and Coulomb terms may be considered independently, as brought up in Section 14. In order to include excited excitonic states, we can no longer continue neglecting the orbital part of H B , that must be now added to the Coulomb terms HC . We are thus left with the problem of a simple exciton created from a pair of non-degenerate bands. It should be pointed out that due to a difference in form between the valence subbands of mj = +* and those of mj = +*, two types of solutions may be expected in this case, consistently with the experiment. A calculation along these lines has been performed by Twardowski and Ginter (1982), who used an improved version of the adiabatic method. Comparison of their theoretical results with experimental data is shown in Fig. 22, where continuous lines represent results of the calculation. The agreement is quite good, except for a certain difference in slope for line A1 . Line A2, interpreted as originating from a 3d-2 exciton state, is not included in the calculation. VI. Exciton Luminescence
Most of the luminescence measurements on semimagnetic semiconductors have been performed in order to study bound magnetic polaron (BMP) effects. Those studies are considered in this volume in Chapter 10. However, there exists a phenomenon particularly suited to be considered in this chapter, i.e., the destabilization of exciton complexes in a magnetic field. This effect has been reported by Planel et al. (1980), Ryabchenko et al. (1981), and Golnik et al. (1983). In Fig. 23 photoluminescencespectrain the excitonicregion of Cdl -,Mn,Te are presented, as measured by Planel et al. (1980). The principal line has been identified as originating from the recombination of an exciton bound to a neutral acceptor-an AoX complex. A free exciton line X is also visible. Application of a magnetic field causes a shift of the A'Xline towards lower energies-much slower than that of the lowest free exciton component observed in emission or reflectivity (Fig. 24). At a certain field the two lines meet. When the magnetic field continues to increase, the A'Xline follows the X line as it shifts. This behavior has been interpreted as a destabilization of the bound exciton state by exchange effects. At zero field, the A'Xenergy is lower then the energy of a dissociated system A' + Xby a binding energy that consists of a Coulomb part and an exchange (magnetic polaron) part. Application of a magnetic field, lowers the energy of the ground hole of both the bound and the dissociated systems by affecting the exchange contributions.
304
J. A. GAJ
-.-
H = I IT H = 5.6T
I .oo
L
0 00 ENERGY ( e V )
FIG. 23. Luminescence spectra of Cdo,9~Mno.~Te at 1 . 4 K observed at three values of the magnetic field. The low energy shoulder on the 5.6Tline appears at high fields ( B > 4T) and is not explained. [After Planel et al. (1980).] I 6700,
k., \,'"\-.. ."-a,\ I
I
I
I
XMn=
5%
0
X%
*@.
-2 1650 > c3
LL
a '.
I605
-
0
V
O
W Z W
I630
x-x-x-+x_x
0 . 0 ' 0 ' 0-0-0
I
I
I
I
I
-
7.
MAGNETOOPTICAL PROPERTIES
305
Now, the A' + X system is affected more strongly by that additional energy, since both its holes are free to adjust the direction of their spins to the magnetic field direction, the orbital parts of their wavefunctions being different. The two holes in A'X have a symmetrical orbital wavefunction, and the Pauli principle makes such a free adjustment impossible. In addition, the presence of an external magnetic field reduces magnetic polaron effects. Therefore, if the exchange effects are strong enough, at a certain field value the net binding energy will vanish and the bound state will disappear. Detailed studies of this phenomenon can supply information on excitonic wave functions, on the values of the exchange constant for bound states and, of course, on the magnetic polaron effects. At this moment no precise theoretical interpretation is as yet available. VII. Transitions Away from the Brillouin Zone Center 19. EXPERIMENTAL DATA
The only magnetooptical results off the Brillouin zone center in a large gap semimagnetic semiconductor are those obtained at the L point in Cdl-xMnxTe. Those transitions give rise to reflectivity structures at energies denoted commonly by E I = 3.5 eV and El + A1 = 4 eV. Figure 25 shows a relevant fragment of the reflectivity spectrum of Cdl -xMnxTe after ZimnalStarnawska et af. (1984). The first magnetoreflectivity measurements reported by Dudziak et af. (1982) have already shown that the Zeeman splitting of the El structure is much smaller than the splittings at the r point. In fact it was not possible to detect any Zeeman structure at the L point; there was only a small difference in the structure position when measured with the (T' or the (T- polarizations indicating that the splitting (if any) was much smaller than the structure width. To resolve it, a modulation technique has been applied. Figure 26 shows an experimental setup with an elasto-optical modulator used in a later work (Ginter et af., 1983, see also Coquillat et af., 1986). Results obtained on Cdl-,Mn,Te samples can be summarized as follows: (1) At low (pumped liquid He) temperatures, the structures El and
+ A1 exhibit small splittings of opposite sign. (2) The splitting of the El structure is of the same sign and is reduced in magnitude in comparison with the strongest splitting (between the two strong (T components) observed at the point by a factor of = 15 common for all compositions and temperatures. (3) The splitting of El + A1 (opposite in sign to that of El) is also reduced in magnitude (probably even more) as compared to the splitting at the r point. This result is only qualitative.
El
306
J. A. GAJ
00
0 05
0 12 0 2
03 04
......- ............. I
I
2.0
I
05
......................... I I I
4.0
-,.-
06
"'1
0.7
b.
I
6.0
8.0
Energy ( e V )
FIG 25. Reflectivity curves of Cdl-,MnxTe at room temperature. El and EI + At peaks are marked by arrows. [After Zimnal-Starnawska er af. (1984).]
There is clearly a need for systematic and precise magnetooptical measurements of these transitions in other materials and for a better theoretical model.
20. ATTEMPTS OF THEORETICAL DESCRIPTION It must be said that up to the present moment there exist no satisfactory explanation of the observed strong reduction of the splitting at the L point. Certain features of the observed effects have been explained by Ginter et al. (1983) who used a simple tight binding model to find a relationship between the exchange splittings and the selection rules at the L point, and those at the point. The energy bands in the model are developed from the atomic 5s
7.
L
MAGNETOOPTICAL. PROPERTIES
O
307
PM
FIG. 26. Experimental setup for modulated reflectivity measurements (Sc-source, L-lenses, Q-fused quartz plate used to eliminate the linear polarization of the light emitted by the source, PR-reflecting prism, S-sample, C-superconducting coil, PM-photoelastic modulator, P-polarizer, M-monochromator). [After Ginter ef al. (1983).]
states of Cd (or 4s states of Mn) and 5p states of Te. Besides a periodic potential of the (virtual) crystal, spin-orbit interaction and the ion-carrier exchange are taken into account. The calculation predicts a rather complicated structure due to the existence of four, generally non-equivalent, band extrema. In the experiment this structure cannot be resolved because of the breadth of the reflectivity peaks. An averaged splitting found in the calculation is reduced only four times compared to the splitting at the r point, in contrast to the experimental value of 16. The signs of the splitting obtained in the calculation agree with the experiment. It is clear that the presented model lacks some essential physical features to account for the experimental observation. However, the paper by Ginter eta/.(1983) points out a number of reasons that lead to the splitting reduction at the L point. These reasons, which should remain valid even in a new improved theory, are as follows: (1) A mixing of the s-like and the p-like states occurring in the conduction band produces an effective exchange integral equal to an average (possibly weighted) of a and p and, due to their opposite signs, leads to their partial cancellation. (2) The component of the magnetic field perpendicular to the k vector produces no splitting of the heavy hole states. This effect has already been found in a k p calculation (Gaj et al., 1978),valid near the rpoint. Combined with the necessity of averaging over the four equivalent band extrema, it leads to a reduction of the exchange splittings of the valence states involved.
-
308
J. A. GAI
(3) r he selection rules are much less strict at the L-point than they are at the center of the Brillouin zone.
Acknowledgments It is the author’s pleasure to thank Dr. J. Ginter for many valuable discussions, as well as Ms. M. Arciszewska, Drs. M. Nawrocki, and A. Twardowski for permission to reproduce their unpublished results.
References Aggarwal, R. L., Jasperson, S. N., Stankiewicz, J., Shapka, Y., Foner, S., Khazai, B., and Wold, A. (1983). Phys. Rev. B28, 6907. Altarelli, M., and Lipari, N. 0. (1973). Phys. Rev. B7, 3798. Arciszewska, M., and Nawrocki, M. (1982). In “Physics of Semiconducting Compounds” (1 1th Conf.), p. 225,. Pol. Acad. Sci., Warsaw, Poland. Arciszewska, M., and Nawrocki, M. (1984). Unpublished. Bastard, G., Rigaux, C., Guldner, Y., Mycielski, J., and Mycielski, A. (1978). J. Physique 39, 87. Bir, G. L., and Pikus, G. E. (1972). Simmetria i deformatsionnye effekty vpoluprovodnikakh, Nauka, Moscow, Sections 23 anll 24 (Symmetry and Strain-induced Effects in Semiconductors, Wiley, 1974). Cieplak, M. Z. (1980). Phys. Stat. Sol. (b) 97. Coquillat, D., Lascaray, J. P., Dejardins-Deruelle, M. C., Gaj, J. A., andTriboulet, R. (1986). Solid State Commun. 59, 25. Dudziak, E., Brzezinski, J., and Jedral, L. (1982). In “Physics of SemiconductingCompounds” (11 Conf.), p. 166. Pol. Acad. Sci., Warsaw, Poland. Elliott, R. I. (1957). Phys. Rev. 108, 1384. Gaj, J. A., Ginter, J., and Galazka, R. R. (1978). Phys. Stat. Sol. (b) 89, 655. Gaj, J. A., Byszewski, P., Cieplak, M. Z., Fishman, G., Galazka, R. R., Ginter, J., Nawrocki, M., NguyenTheKhoi, Planel, R., Ranvaud, R., and Twardowski, A. (1979a). In “Physics of Semiconductors” (14th Int. Conf.), p. 1113, Inst. Phys. Conf. No. 43. Gaj, J. A., Planel, R., and Fishman, G. (1979b). Solid State Commun. 29, 435. Gaj, J . A. (1981). In “Exchange Interaction of Paramagnetic Ions With Band Electrons in Cdl-,MnxTe,” Editions of University of Warsaw (in Polish). Gaj, J . A., and Golnik, A. (1985). Acta Phys. Polon. A67, 307. Ginter, J . , Gaj, J. A., and Le Si Dang (1983). Solid State Commun. 48, 849. Golnik, A., Ginter, J., and Gaj, J. A. (1983). J. Phys. C16, 6073. Gubarev, S. I. (1981). Zh. Eksp. Teor. Fiz. 80, 1174. (Sov. Phys. JETP53, 601.) Kane,. E. 0. (1957). J . Phys. Chem. Sol. 1 , 249. Kett, H., Gebhardt, W., Krey, U., and Furdyna, J. K. (1981). J. Magn. Magn. Muter. 25,215. Kierzek-Pecold, E., Szymanska, W., and Galazka, R. R. (1984). Solid State Commun. 50,658. Komarov, A. V., Ryabchenko, S. M., Terletskii, 0. V., Zheru, I. I., and Ivanchuk, R. D. (1977). Zh. Eksp. Teor. Fit. 73, 608. (Sov. Phys. JETP46, 318.) Komarov, A. V., Ryabchenko, S. M., and Vitrikhovskii, N. I. (1978), Pis’mz v Zh. Eksp. Teor. Fiz. 27. (JETP Letter 27, 413.) Komarov, A. V., Ryabchenko, S. M., Terletskii, 0. V., Ivanchuk, R. D., and Savitskii, A. V. (1980a). Fiz. Tekh. Popuprovod. 14, 17. (Sov. Phys. Semicond. 14, 9.)
7.
MAGNETOOPTICAL PROPERTIES
309
Komarov, A. V., Ryabchenko, S. M., and Terletskii, 0. V. (1980b). Phys. Stat. Sol. (b) 102, 603. Komarov, A. V., Ryabchenko, S. M., Semenov, Y. G., Shanina, B. D., and Vitrikhovskii, N. 1. (1980~).Zh. Eksp. Teor. Fiz. 79, 1554. (Sov. Phys. JETP52, 783.) Kossut, J. (1976). Ph.D. Thesis, Institute of Physics, Polish Academy of Sciences, Warsaw. Lindstrorn, M., Kuivalainen, P., Heleskivi, J., and Galazka, R. R. (1983). Physica 117B and 118B, 479. Malyavkin, A. V. (1983). Phys. Stat. Sol. (b) 115, 353. Nguyen The Khoi, Ginter, J., and Twardowski, A. (1983). Phys. State Sol. (b) 117, 67. Ortenberg, M. von (1984). Solid State Commun. 52, 111. Oseroff, S. B. (1982). Phys. Rev. B25, 6584. Planel, R., Gaj, J., and Benoit a la Guillaume, C. (1980). J. Physique 41, C5-39. Rebman, G., Rigaux, C., Bastard, G., Menant, M., Triboulet, R., and Giriat, W. (1983). Physica 117B and 118B, 452. Ryabchenko, S. M., Terletskii, 0. V., Mizetskaya, I. B., and Oleinik, G. S. (1981). Fiz. Tekh. Poluprovodn. 15, 2314. (Sov. Phys. Semicond. 15, 1345.) Twardowski, A. (1981a). Ph.D. Thesis, University of Warsaw. Twardowski, A. (1981b). Unpublished. Twardowski, A,, Nawrocki, M., and Ginter, J. (1979). Phys. Stat. Sol. (b) 96, 497. Twardowski, A., Rokka, E., and Gaj, J. A. (1980). Solid State Commun. 36, 927. Twardowski, A., Dietl, T., and Demianiuk, M. (1983). Solid State Commun. 48, 845. Twardowski, A., Swiderski, P., Ortenberg, M. von, and Pauthenet, R. (1984). Solid State Commun. 50, 509. Twardowski, A., and Ginter, J. (1982). Phys. Stat. Sol. (b) 110, 47. Zimnal-Starnawska, M., Podgorny, M., K i d , A., Giriat, W., Demianiuk, M., and Zmija, J. (1984). J. Phys. C17, 615.
SEMICONDUCTORS AND SEMIMETALS. VOL. 25
CHAPTER 8
Shallow Acceptors in DMS: Splitting, Boil-Off, Giant Negative Magnetoresistance J. Mycielskif INSTITUTE OF THEORETICAL PHYSICS, WARSAW UNIVERSITY WARSAW, POLAND
. .
.
I. INTRODUCTION . . . . . . , . . . . . . . . . . 1. Shallow and Resonant Acceptors in DMS in the Absence of Magnetic Field . . . . . . . . . . . . . . . , . 2. Effect of Exchange Interactionon Band Structure. . . . 11. THEORY OF SHALLOW ACCEPTORS IN THE ABSENCE OF MAGNETIC FIELD . . . .. . . . . ... .. . .. . . ... 3. Acceptor Hamiltonian . . . . . . . , . . . . . . . 4. Variational Procedure . . . . . . . . . . . . . 111. ACCEPTORS IN WIDE-GAP DMS IN A MAGNETIC FIELD.. . . . 5. Exchange Splitting. . . . . . . . . , . . . . 6. ExperimentalResults. . . . . . . . . . . IV. ACCEPTORS IN NARROWGAP DMS IN A MAGNETIC FIELD. . . . 7. Acceptor Hamiltonian in Ultra-Quantum Limit. . . . 8. Acceptor States in Ultra-QuantumLimit; Boil-Off Effect. . 9. Arbitrary Magnetic Fields. . . . . . . . . . . . . . 10. Experimental Binding Energies. .,.... . ., . IN p-TYPENARROW-GAP DMS V. IMPURITYMAGNETOCONDUCTIVITY 1 1 . Hopping Magnetoconductivityand Giant NegativeMagnetoResistance in Hgl-,Mn,Te . . . . . , . . . 12. Nonmetal-to-Metal Transition . . . . . . . . . . , . VI. RESONANT ACCEPTORS IN ZERO-GAP DMS . . . . . . . . 13. MagnetoopticalData. . . . . . . . . . . . . . . . 14. MagnetotransportData. . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
. .
. . . . . .
.
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. . . .
311 312 313 3 16 316 318 322 323 324 325 325 321 329 330 333 333 331 338 339 341 342
I. Introduction
Probably the best known diluted magnetic semiconductors (DMS) that can be made p-type are Cdl-xMnxTe and Hgl-,MnxTe (see the review articles: Galazka, 1979; Galazka and Kossut, 1979; Gaj, 1980; Bastard et al., 1980; Mycielski, 1981; Dietl, 1981; Galazka, 1982; Furdyna, 1982a,b; Grynberg, i ( 1930- 1986)
311 Copyright 0 1988 by Academic Press. Inc. All rights of reproduction in any form IeSeNed. ISBN 0-12-752125-9
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J . MYCIELSKI
1983; Mycielski, 1983 and 1984). Both are A:'_,Mn,BV' systems of the zincblende cubic structure, and both have wide conduction and valence bands. However, Cd, -,Mn,Te is a wide-gap semiconductor, while Hgl -,MnxTe is a narrow-gap semiconductor (for x > 0.075), or a zero-gap semiconductor (for x c 0.075). For specificity, in this chapter we will only consider acceptors in the above materials. We limit our considerations to shallow .acceptors, and ignore more complex systems like excitons bound to acceptors (for excitons see: Gaj, this volume). Taking into account the effect of the exchange interaction of a hole with magnetic ions on the acceptor states, we will neglect the feedback: ordering of spins of magnetic ions by their interaction with the hole, i.e. the bound magnetic polaron effect (for magnetic polarons see: Wolff, this volume). 1. SHALLOW AND RESONANTACCEPTORS IN DMS IN THE
ABSENCE OF
A
MAGNETIC FIELD
In the absence of magnetic field the spins of magnetic ions are not ordered. Thus there is no net exchangeinteraction of these ions with a hole. The acceptor states in DMS are then similar to those in non-magnetic semiconductors. The theoretical ionization energy EA of a shallow acceptor in CdTe is 87.4meV (Baldereschi and Lipari, 1973), and should be the same in Cdl-,Mn,Te, as the parameters of the valence band (Twardowski and Ginter, 1982) and the static dielectric constant (Gebicki and Nazarewicz, 1980) were found to be independent of composition in this material. In a photoluminescence experiment values of 57.8, 58.8, 108, and 147 meV were found for EA in CdTe and were interpreted as corresponding to the substitutional impurities Li, Na, Ag, and Cu, respectively (Molva et al., 1982). In Cdl-,MnxTe the absorption due to acceptor photoionization was measured and analyzed by Wojtal et a/. (1979). The photoionization spectrum was fitted using a formula derived by the quantum defect method (Bebb, 1969). This gave a rather high quantum defect for the ground state, and ionization energies of 165 and 170meV for x of 0.05 and 0.10, respectively. Thus, the acceptor is not very shallow. In the narrow-gap Hgl -,Mn,Te, the theoretical ionization energy of a shallow acceptor increases withx, and is about 9 meV for x = 0.15 (Mycielski and Mycielski, 1980). From transport measurements the value 9.4 meV was obtained for x = 0.15 (Wojtowicz and Mycielski, 1984), while the photoconductivity experiment revealed-for the same value of x-a deeper acceptor, with the ionization energy about 30 meV (Wrobel et al., 1984). A second acceptor level-with an ionization energy about 25 meV-was also reported in Hgl-,Mn,Te with a wider gap (Anderson et al., 1983). Acceptors are believed to be due to mercury vacancies or foreign impurities.
8.
SHALLOW ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS
313
In the zero-gap Hgl-,Mn,Te resonant acceptor states do exist in the conduction band. Two levels were found in magnetooptical experiments, with the zero-field energies 0.9 meV-observed at very low x-and 2.2 meV-observed at x r 0.008 (Bastard et al., 1979). They were identified as Ao (impurity) and A1 (mercury vacancy) levels, respectively, found also in non-magnetic zero-gap Hgl-,CdxTe (Guldner et af., 1977). A level with the energy 1.5meV was also reported by Dobrowolska et al. (1980) and Dobrowolska and Dobrowolski (1981). From magnetotransport measurements at low fields and for x = 0.06 (i.e., close to the zero- to open-gap transition), the resonant level was observed at 4 meV or 7 meV (Sawicki et al., 1983 and Sawicki and Dietl, 1983). 2. EFFECT OF EXCHANGE INTERACTIONON BANDSTRUCTURE
In the absence of a magnetic field (i.e., also of a net exchange interaction), the DMS we consider have the band structure of non-magnetic semiconducting alloys such as Hgl-,Cd,Te. They have a direct band gap (increasing with x) at the r-point of the Brillouin zone. In the open-gap case, the symmetry of the conduction band and upper (light and heavy holes) and lower valence bands is given by the double Td point group representation r6, I‘s , and r7,respectively. In the zero-gap case (Hgl-,Mn,Te for low x), the positions of r6 and r8 levels are reversed, and the light-hole TS band of the open-gap structure becomes the conduction band. In any case, the r7 level is substantially split off from the r8 level due to a strong spin-orbit coupling. The basis for the rs level (in which we are primarily interested in this chapter) for the total angular momentum component ranging from to -9, are a1 = I+,$) = 2 - 1 / 2 ( ( x + i Y ) T ) , (1) @3
I$, $) = i6-”21(X + iY)1 - 2Zt), = It, -+) = 6-’”1(X - iY)T + 2ZJ.),
a 4
=
=
I+,
-+> = i 2 - 9 x - i Y ) l ) ,
(2) (3) (4)
where the position-dependent functions X, Y, Z transform under the operations of 7A point group as p-type functions, and have average square modulus equal to 1. We will now take into account the exchange interaction of the band electrons with the 3ds-electrons of magnetic Mn2+ ions. This interaction is of Heisenberg form (Kossut, 1976)
314
J. MYCIELSKI
where s and S; are the spin operators (in units of h ) of the band electron and of the i-th magnetic ion, respectively, and the summation is over all lattice sites occupied by magnetic ions. The exchange constant J is a short-range function of distance. Using the mean-field and virtual-crystal approximations we can write Eq. (5) in the form (Bastard et a/., 1978 and Jaczynski et al., 1978)
which has the periodicity of the crystal lattice, as the summation is now over all metal lattice sites. Here x is, as before, the molar fraction of the magnetic component, and ( S ) a v is the spin of the magnetic ion, averaged thermally and over all magnetic ions. Thus, ( S ) a v determines the macroscopic magnetization of DMS. In the absence of an external magnetic field (S), = 0. In the presence of a magnetic field ( S ) a v increases with decreasing temperature. In the following, we take the external magnetic field parallel to the z-direction. Thus, (S)avhas a non-vanishing z-component (Sz)av that tends to -5 for strong fields and low temperatures. The exchange Hamiltonian, Eq. (6), written in the basis given by Eqs. (1)-(4) has the form 3Bexch 0 0 Hexch
with
=
k
0
Bexch
0
-Bexch
0
Bexch
P
0
= ;XNoP(Sz)av
0 -3Bexch
,
j
(7)
(8)
(9) where NOis the number of unit cells in unit volume. It is worth noting that &xch may be modified to include the Luttinger parameter K of the valence band (Luttinger, 1956). It follows from Eq. (7) that the effect of exchange interaction results in splitting the I-" level into equally spaced singlets. This splitting dependsthrough (S,)av-On the external magnetic field and the temperature, and can approach values as large as 100 meV. = (ZIJIZ),
a. Wide-Gap DMS In a wide-gap DMS like Cdl-xMnxTe, because of the high effective masses of holes, the exchange splitting is much higher than the Landau and Zeeman splittings. It is, therefore, justified to perform the k p calculation using Eq.(7) while neglecting the direct effect of the external magnetic field on the
8. SHALLOW ACCEPTORS
IN DILUTED MAGNETIC SEMICONDUCTORS
315
spin splitting and the orbital motion of the valence electrons (Gaj eta/., 1978 and Gaj, this volume). One then obtains a dramatic dependence of the splitting of the valence bands on both the direction and the absolute value of the electron wave-vector k. Of particular interest for the following are dispersion relations close to the r point, i.e., for
where mh is the light-hole mass. Neglecting warping, the four non-degenerate valence bands are:
where mhh is the heavy-hole mass and k, is the wave-vector component perpendicular to the magnetic field. Thus, the constant energy surfaces are rotational ellipsoids, cucumber-like for the highest and the lowest bands (f3)and disc-like for the intermediate bands (fts).For mhh S- mm,the mass anisotropy is particularly strong for the bands -t+, being equal to 3mhh/4mm. Since > 0 for both DMS under consideration, i.e., &xch c 0, E-3/2(k) corresponds to the highest-lying valence band.
b. Narrow- and Zero-Gap DMS In a narrow- or zero-gap DMS, like Hgl-,Mn,Te, one cannot neglect the direct effect of the external magnetic field on the spin splitting and the orbital motion of the valence electron, as the low effective masses yield high cyclotron frequencies and high g-factors. The exchange splitting then has to be treated together with Landau and Zeeman splittings. This is done by adding the exchange interaction, Eq. (7)-written in a larger basis, including also r6 and r7functions-to the Pidgeon-Brown k p Hamiltonian (Pidgeon and Brown, 1966) of a zinc-blende type semiconductor in a magnetic field. Inversion asymmetry and warping are neglected in this model. The numerical solution of the modified Pidgeon-Brown Hamiltonian (Bastard et al., 1978; Jaczynski et al., 1978; Bastard et al., 1981) fits very well the observed positions of the Landau levels in the narrow- and zero-gap DMS. In what follows, one of these Landau levels- referred to as bv(- 1)-will be of particular interest. This level originates from Ts and its energy has an analytic form, &,(-I) = -3hUJ~3/2 + 3 f i W o h . - 3Bexch, (13)
-
316
J. MYCIELSKI
where coo is the free-electron cyclotron frequency and ~ 1 3 / 2is the cyclotron frequency corresponding to the transverse effective mass of the bands E,3/z(k) given by Eq. (1 1). The wave function of bu(-1) is a ground-state Landau envelope multiplied by @4. For the usual values of parameters (i.e., p > 0), bu(-l) is the highest valence band level in both the open- and zerogap DMS and is well-separated from other valence levels. It is the uppermost heavy-hole Landau subband. From Eq. (13) and from the form of the wave function, one can see that bu(-l) is the first Landau level originating from E-3/z(k) valence band. In spite of the fact that it corresponds to a heavy-hole Landau subband (in the z-direction), it is well-separated from other valence levels due to a low transverse effective mass of the E-3/2(k) valence band, and to the high value of &ch. It should be stressed, however, that other Landau levels cannot be calculated in a narrow- or zero-gap DMS from the simple expresssion, Eq. (1 1). Because of the -33Bexch term in Eq. (13), &,(-I) increases with the magnetic field (even for x = lop3) up to the saturation of (at a few T ) and only then starts to decrease slowly. In a zero-gap DMS at moderate fields, it may overlap in energy with the lowest conduction-band levels. 11. Theory of Shallow Acceptors in the Absence of a
Magnetic Field We will present here the theory of shallow acceptors in a diamond or zincblende-type open-gap semiconductor in the absence of an external magnetic field, following the variational method of Schechter (1962). Being less accurate than other methods (see Baldereschi and Lipari, 1973), it is, however, more analytic. We present here Schechter’s theory in a simplified and more explicit (while equivalent) version due to Mycielski and Rigaux (1983). 3, ACCEPTOR HAMILTONIAN
In the absence of magnetic field, a shallow acceptor has a four-fold degenerate ground state of Ts symmetry, with the wave functions of the form 4
v/; =
C mj4j;, j= 1
i = 1, ..., 4,
where 4j;(r)are slowly-varying envelope functions. The functions orthonormal:
(14) v/i
are
8.
SHALLOW ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS
317
We are using the valence band Hamiltonian as derived by Luttinger and Kohn (1955), adding to it the Coulomb term. The expectation value of the acceptor Hamiltonian (i.e., an eigenvalue if wi is an eigenstate) for the wave function of the form given by Eq. (14) (say, wl) is
where
E
is the dielectric constant of the crystal and Dii
Di2
Di3
D = ( Dt2
D22
0
D?3
0
0
DT3
-Diz Dii
D22
-D:2
in which the Di, are 3 x 3 matrices given below: - A - )B Dii
=
-A D22
0 -A - iB 0
+ )B
0 0
=
-ti(3B2
~
1
=3
i
+B
0
-A
=
+ )B 0
D12
-A
+ C2)'/'
- (31/2/2)B -443~' + 0
-A -B
-ti(3B2 + C2)'/2 -$(3B2 + C2)'/2 - i ( 3 B 2 + C2)'/2 0 0 0
- )i(3B2 + C2)"2 (31/2/2)B
0
")
.
(21)
0
Here A , B, and Care material constants (see Dresselhaus et al., 1955). Signs of the elements of D12 and D13 involving C 2 correspond to the case of the negative value of the constant N defined by Dresselhaus et al. (1955). If N > 0, the signs should be opposite. The physical meaning of A , B, and C can be seen from the dispersion law of electrons in the valence band: E(k)
=
Ak2 -+ [B2k4+ C*(k;k;
+ k,k: + k,"kx')]"2.
(22)
A and B are real but C may be imaginary (i.e., C 2may be negative). In order
318
J. MYCIELSKI
to have E(k) real and negative for all k f 0, the following conditions must be satisfied: C 2 L -3B2,
A < - [ B +~ f max(0, c ~ ) ] " ~ .
(23) (24)
If C = 0, there is no warping of the valence band. From Eq. (22) we then obtain
+ IBI),
(25)
= h2/2(IAI - 1B1).
(26)
m/i, = A2/2(IAI mhh
4. VARIATIONAL PROCEDURE
We assume now that the wave functions of the acceptor ground state are of the form (see Schechter, 1962 and Kaczmarek, 1966):
where CI, c2, c3, rl , and r2 are real parameters, and rl and r2 are positive. The functions defined by Eq. (27)-(30) fulfill the orthonormalization condition, Eq. (15), if 2 3 mlr 1 + 9 n c X + (:)ncfr? = 1. (31)
8. SHALLOW
ACCEPTORS IN DILUTED MAGNETIC SEMICONDUCTORS
319
As one can see from Eqs. (27)-(30), the envelope functions are composed from an s-type part (that proportional to CI) and two d-type parts (those proportional to cz and c3). The latter parts do mix different @ j in a given v / i . In order to determine the parameters CI, CZ, c3, r l , and r2, and the ionization energy EA, we have to maximize expression (16). From Eqs. (16)-(21)and Eq. (27) we get-after some algebra-E[yl] as a quadratic form of dimensionless quantities a1 , at, and a3, with coefficients depending on dimensionless PI and pz , where
ez
pz = -
EJAI
” *
The quantities d , a$,and a: are the norms of the s-type and the two d-type parts of the envelope, respectively (c.f., condition (31)). Dividing E [ ~ / by I ] d + a: + a: and maximizing the ratio with respect to a2/a1 and 0 3 / 0 1 , we obtain
where the dimensionless quantities f,F,and
r;l
are given by
320
J. MYCIELSKI
The sign of adal is that of B , and a3/al is positive. If the constant Ndefined by Dresselhaus et al. (1955) is positive, nothing is changed in our results except for adal being negative. It follows from Eq. (23) and Eq. (24) that (43)
O-
5z
1.6-
32.0 K-
1.2-
-
30.0 K
W
k
z
20.0 K
-
0.8 0.4
\
-
01
5.0 K \L-44
I
I
-p-----,--~5.0 I
I
K I
I
The equations of motion of M I and M2 are dMi - ~ M xI (H - AM2 dt
+ Ha")
dMz - - - yMz x (H - AM1 dt
+ Ha")
--
and
where H is an externally applied magnetic field. Supposing, for simplicity, that H and Hf) = - Ha) are parallel to a direction that we take as the z-axis, taking the exchange fields - AMz, 1 as approximately + AMs = +HE where
9.
393
RAMAN SCATTERING
Ms is the saturation value of M I we find
-M ;
d dt
=
d -M? dt
=
iy(H -t HA-k H E ) M 7 ~
M$
(83)
r i y ( H - HA - HE)M+ f iyHEMt
(84)
where M t and M? are MI, f iM1, and MzX f iMzy, respectively. Equations (83) and (84) have non-trivial solutions of frequency u& given by
a&
=
fyH + I y l ( H i
+ 2HAHE)’”.
(85)
Thus when H = 0, a long wavelength magnon of frequency Iyl(H2 + 2 H A H ~ ) ’ ’ occurs, * manifests itself in Raman scattering and is the M line in Fig. 19. The polarization features of this line are those predicted by Eq. (64) and illustrated in Fig. 20. We have here the Raman-antiferromagnetic resonance (Raman-AFMR). We assume that HAlike HEis proportional to the saturation values of M I or MZ given by
M S = (hgpBs/a3)BS(r),
(86)
where pug is the Bohr magneton, a the lattice constant, and Bs(y)the Brillouin function with
The Nee1 temperature, TN,is then given by kBTN = (2x/3a3)g2p$(S
+1)~.
(88)
A numerical solution of Eq. (86) yields M S as a function of temperature, T, and in turn the variation of COM with T. The best fit for the data shown in Fig. 21 for x = 0.7 is given by TN= 4 0 K . The magnon feature in the presence of an external magnetic field of 60 k G is shown in Figs. 22a and 22c for Cdl-,Mn,Te, x = 0.70. The spectrum shown in Fig, 22a was recorded at T = 5 K with 21. = 5682 A in the (8-,2) polarization while the spectrum in Fig. 22c was observed in the (8+,2) configuration. In the following, we discuss the Stokes components of these Raman features. The Raman shifts of the peaks of the features in Fig. 22 are OM- = 8.5 cm-’, COM = 12 cm-’, and OM+ = 15.5 cm-’. It can be shown that a one magnon Raman line in an antiferromagnet should split into two components of equal intensity separated by 2gPB H at T = 0 K, if H is along the anisotropy field, HA. The observed spacing between M+ and M - of 7 cm-* for 60 k G is significantly smaller than 2 g p ~ H= 11 cm-‘. The
394
A . K . RAMDAS A N D S. RODRIGUEZ I
-
I
1
I
I
Cd ,+ MnxTe A x.0.7
TEMPERATURE (K)
FIG.21. Temperature dependence of the magnon peak frequency in the magnetically ordered phases of Cdl-,Mn,Te. Triangles: x = 0.7, antiferromagnetic phase; circles: x = 0.4, spinglass phase. The solid curve passing through the points for x = 0.7 is calculated for TN = 40 K solving Eq. (87) for MS as a function of T 5 TN and assuming WM is proportional to MS and equal to 12.5 cm-’ at T = 0 K. [After Venugopalan el a/. (1982).]
polarization characteristics of OM+ and O M - are those expected. These results are independent of the crystal orientations with respect to the applied field suggesting that HA is small compared to the applied field. In Venugopalan et al. (1982), HEfor x = 0.7 was calculated from the observed transition temperature TN = 40 K to be 208 kG,while HAwas determined to be 36 kG as deduced from h c o ~= gpB(H2 + HAH HE)'", strictly applicable to an antiferromagnet exhibiting long range order. While the value for HE is reasonable, that for HA must be viewed as too large in the context of the experimental results (Petrou et al., 1983). We have described the Raman features associated with magnetic excitations appearing in the paramagnetic state as well as in the magnetically ordered phase. The exchange interaction between MnZ+ions is negligible compared to k s T at high temperatures becoming more important as the temperature is lowered. It is of interest to investigate the effects of temperature on the OPM line, particularly the effect of lowering the temperature below TN,the transition temperature characterizing the magnetically ordered phase. The temperature evolution of the OPM line is shown for Cdl-*Mn,Te, x = 0.70, in Fig. 23. The spectra were recorded for a magnetic field of 60 kG using the (&+ ,2) configuration. As the temperature is lowered, the OPM line
-
9. 1
AS
(a)
(&.a H=GOkG
(C)
($.
395
RAMAN SCATTERING I
S
M-
AS
2)
H=GOkG
M-
24
I
I
12 0 12 24 RAMAN SHIFT (cm-')
FIG. 22. Effect of theommagnetic field on the magnon feature of Cd,-,Mn,Te, x = 0.70, at T = 5 K with A L = 5682 A. x , y , andz are along [110], [TlO], and [Ool], respectively: (a) (6-, t), H = 60 kG; (b) (6+, 2). H = 0; (c) (a+,2). H = 60 kG. Owing to imperfect polarizationresults, 2) configuration. leakage of the fairly strong featureM+ appears as a small shoulder in the (6-, [After Petrou et a/. (1983).]
initially broadens and then moves towards higher Raman shifts as a consequence of the increased importance of the exchange interaction. An increase in Raman shift is observed at temperatures well above TN 40 K. As the temperature is lowered through and below TN, the line becomes the magnon component observed in the (a+,2) configuration.
-
396
A. K . RAMDAS A N D S. RODRIGUEZ
I I
OOK
300K I
xi6
6K
I
I
xi6
56c4 ~ 6 c i
10
0
10
10
20
1 -
10
20
RAMAN SHIFT (crn-')
FIG.23. Evolution of the Raman line at WPM of Cdl-xMn,Te, x = 0.70, into the magnon feature as the temperature is lowered from room tempefature to below the Ntel temperature. The spectra were recorded with H = 60 kG, I L = 6764 A in the (a,, 2) polarization; x , y , and z are along [IlO],[TlO], and [@I], respectively. [After Petrou et a/. (1983).]
Finally, the conclusion that the magnetic feature observed in the magnetically ordered phases is a one magnon excitation was initially deduced from its polarization characteristics and temperature behavior (Venugopalan et al., 1982); this is supported by the results of Ching and Huber (1982a, 1982b). The fact that this feature shows a splitting in the presence of a magnetic field and that the OPM line of the paramagnetic phase, clearly associated with a single ion excitation, evolves smoothly into the higher energy component of the magnon provides a strong confirmation of this interpretation. In the same spirit, one might expect a two magnon feature associated with the 2 0 p M line; however, given the intensity of the 2 0 p line ~ compared to that of the OPM line, the intensity of such a feature would preclude its observation. A two magnon feature similar to that seen in MnF2 by Fleury and Loudon (1968) would have symmetric polarization characteristics; such a feature has also not been observed in Cdl -,Mn,Te. In the light of these experimental results, the two magnon interpretation advanced by Grynberg and Picquart (1981) to explain the feature at OM is clearly excluded. 7. SPIN-FLIPRAMANSCATTERING Since the first observations of electron spin-flip Raman scattering in the narrow band gap semiconductor InSb by Slusher et al. (1967) followed by
9.
RAMAN SCATTERING
397
that in the wide band gap semiconductor CdS by Thomas and Hopfield (1968), there has been a continuing interest in this magneto-optical phenomenon. An important contribution to this field is the first demonstration of the spin-flip Raman laser by Patel and Shaw (1970). The success of the InSb spin-flip Raman laser as a practical source of magnetic field tuned, coherent radiation in the 5 to 15 pm spectral region (Patel and Shaw, 1970; Mooradian et al., 1970) can be traced to the large effective g-factor of the electron, 1g*I = 50. In contrast, the wider band gap II-VI semiconductors CdS, CdSe, and CdTe have rather small spin splittings (Thomas and Hopfield, 1968; Walker et al., 1972) characterized by 1g*I 5 2, since their conduction and valence band spin-orbit interactions are small in comparison to their energy band gaps. When detectable, spin-flip Raman scattering provides a practical means of probing the electronic structure of semiconductors, as dramatically illustrated in DMS. The large Raman shifts associated with spin-flip scattering from electrons in DMS were first observed in the narrow gap Hgl-,Mn,Te by Geyer and Fan (1980). The first evidence of a finite spinsplitting of the electronic level in the absence of a magnetic field was reported by Nawrocki et al. (1980, 1981) in the wide gap diluted magnetic semiconductor Cdl-,Mn,Se. The effects of Mn concentration and the antiferromagnetic coupling among the Mn*+ ions on the spin-flip Raman shifts were first observed (Peterson et al., 1982) in a study of Cdl-,Mn,Te. These studies established the nature of Raman scattering associated with the spinflip transitions of electrons bound to donors in DMS. The large Raman shifts depend not only on the applied magnetic field, but also on temperature and manganese concentrations. The far-infrared absorption spectra (Dobrowolska et al., 1982) of Cdl-,Mn,Se also provided evidence for these spin-flip transitions. Following these initial reports, there have been several investigations (Alov et al., 1981, 1983; Heiman et al., 1983a,b, 1984a,b; Douglas et al., 1984; Peterson et al., 1985b) of spin-flip Raman scattering in DMS. The extensive results currently available allow a detailed comparison with the theory of Diet1 and Spalek (1982, 1983). The Raman spectra of Cdl-,Mn,Te(Ga), x = 0.03, are shown in Fig. 24 for the (6+,2) and (6-,2) polarization configurations with T = 40 K and H = 60 kG. The two Stokes features labeled PM and SF are present only in (a+,i),while the corresponding anti-Stokes features appear only in the (6-, 2) configuration. The observed width of the PM line is instrument limited, while that of the SF feature is 23 cm-'. The PM feature is associated with the spin-flip transitions. The SF feature of Fig. 24 is attributed to spin-flip Raman scattering from electrons bound to gallium donors. It has the same polarization characteristics as those of the PM line appearing (Peterson et al., 1982)in the (a+,2)
398
”0
20
I0
0
-10
RAMAN SHIFT ( cm-’
-20
-3
I
FIG. 24. Raman spectra of Cdl-,Mn,Te(Ga), x = 0.03, showing the Ams = f 1 transitions within the Zeeman multiplet of MnZf(PM) and the spin-flip of electrons bound to Ga donors (SF). kcps = 10’ countslsec. [After Peterson et a/. (1985b).]
or (&a_)polarizations for Stokes scattering and in (8-, 2) or (2, a+)for antiStokes. As illustrated in Fig. 25, the peak Raman shift of the spin-flip feature, 6,exhibits a strong .dependence on both temperature and magnetic field. The primary source of the spin splitting of the electronic level is the exchange coupling with the Mn2+ ions (first term of Eq. (72)) with the Zeeman effect (second term of Eq. (72)) making a relatively small contribution. Hence, the Raman shift should be approximately proportional to the magnetization of the Mn2+ ion system, which amplifies the effect of the magnetic field on the electron. As can be seen in Fig. 25, a finite Raman shift is observed for zero magnetic field. This effect is attributed by Diet1 and Spalek (1983) to the “bound magnetic polaron” (BMP). The electron localized on a donor in a diluted magnetic crystal polarizes the magnetic ions within its orbit, creating a spin cloud that exhibits a net magnetic moment.
9.
I- Cd,-,Mn,Te
(Ga 1
X = 0.03
-
A
'E
0
Y
90 -
+
z rn
+ + +
+
0
0
0
-
+
I-
LL
399
RAMAN SCATTERING
0 0
+
60-
A
0
A
+
1
A
0 A
0
A
LL +
o A
0
0
x
X
x
20
;
1
x
:40Ki
x
40
60
MAGNETIC FIELD ( k G ) FIG.25. Magnetic field and temperature dependence of the Raman shift associated with the spin-flip of electrons bound to donors in Cdl-,Mn,Te(Ga), x = 0.03. [After Peterson el uf. (1985b).]
An additional effect on the binding energy of the electron bound to the donor originates from thermodynamic fluctuations of the magnetization and the resulting spin alignment of the magnetic ions around the donor. According to Eq. (72), the spin splitting of the donor energy levels in DMS arises from the combined effect of the magnetization of the Mn2+ions and the external field H. Due to the strong s-d coupling, the effect due to the magnetization dominates. The Raman shift associated with spin-flip scattering from the donor states has the form
where MOis the macroscopic magnetization. The magnetization is proportional to the thermal average of the Mn2+spin projection along H multiplied
400
A . K . RAMDAS A N D S. RODRIGUEZ
by the density of Mn2+ ions contributing to the magnetization, yielding Ao
=
X C Y N O ( S+~ g*pBH. ~)
(90)
Here R is the concentration of Mn2+ ions that contribute to the magnetization. For small x, the crystal is paramagnetic and the thermal average of the Mn" spins is
where B5/2 is the Brillouin function BJ for J = 5. The compositional dependence of spin-flip Raman scattering has two sources. Within a DMS system, such as Cdl-,Mn,Te, the properties of the spin splitting should show a strong dependence on the Mn concentration. For a givenx, these properties should also vary from one DMS system to another. The spin-flip Raman shifts for Cdl-,Mn,Te at T = 1.8 K are shown in Fig. 26 as a function of magnetic field and composition. The results for x = 0.01 show the saturation behavior characteristic of the paramagnetic phase. As the Mn concentration is increased to x = 0.03 and x = 0.05, the Raman shifts increase and the effects of saturation are still clearly evident, I
160 C d
I
,-, MnT,e
I
I
A
'E 120 0
LL
I
+
+s
80
z a
o
A
O
A
+
I
Q
I
I
0 O
Ox:o.05
A
A
A
+
+
+
+
--
+
+x=o
A A
x=o.3
A A
0
A
A A
+O +O
I
+
oX=0.20 A
__
I
+
+
A
X = 0.03
A
I
+
+
__ h
I
+X=O.Q- Cd,-,Mn,Te T=1.8K
n +n
z
I
+
+
I-
V,
+
+
v
I
+ +
T = 1.8K
-
I
A
40
0
i
i
20
i
l
40
l
l
6
0
I
0
l
l
I
20
l
l
40
I
l
60
l
MAGNETIC FIELD ( k G ) FIG.26. Magnetic field and composition dependence of the peak spin-flip Raman shift in the Cdl-,Mn,Te samples at T = 1.8 K . [After Peterson et a/. (1985b).]
9.
RAMAN SCATTERING
401
but less pronounced. For x = 0.10, the deviation from the paramagnetic behavior is quite evident. For H = 60 kG, the Raman shift for x = 0.10 is only four times that for x = 0.01. As x exceeds 0.10, the Raman shifts for a given field actually decrease; note that the shifts for the x = 0.20 sample lie below those for the x = 0.10 sample. And the Raman shifts for the Cdl-,Mn,Te(Ga), x = 0.30, sample are significantly smaller than those for the x = 0.10 and x = 0.20 samples. These trends have their origin in the decrease of the mean magnetic field due to the increasing antiferromagnetic pairing of MnZ+neighbors. In order to discuss the experimental results on the bound magnetic polaron, it is useful to review the theory for it as developed by Dietl and Spalek (1982, 1983). The electron is assumed to be bound to a shallow donor interacting through an s-d coupling with a paramagnetic subsystem of localized magnetic moments. Only the large polaron case is considered, allowing the continuous-medium, effective-mass, and molecular-field approximations. This model assumes that the donor electron interacts with a large cloud of spins behaving classically. Neglecting field induced anisotropy, Dietl and Spalek derive an effective BMP Hamiltonian given by
where ED(^) is the binding energy of the electron arising from the Coulomb potential, A is the magnitude of the spin-splitting with A, parallel to the effective local magnetic field, defining the direction of the spin quantization of the electron,
is the magnetic field induced component of A and
is the characteristic BMP energy for an s-type wave function with an effective Bohr radius a. Here Mo is the magnetic field induced magnetization and x is the magnetic susceptibility. At a finite temperature, a range of A beyond the minimum of H ( A )is accessible to the system. The probability of a thermal fluctuation of the magnetization giving rise to a specific A is
I : [: :
P(A) = Cexp --
(95)
402
A. K . RAMDAS A N D S. RODRIGUEZ
where C is a normalization constant. The probability distribution for A = 1 A ( ,P(A), is given by integrating (95) over all angles, yielding
In the absence of a magnetic field, P(A) reduces to
where
The theory of Dietl and Spalek results in an intensity distribution characterizing the spin-flip Raman line, Z(A), to be P(A), given by Eqs. (96) or (97), multiplied by the probability that the donor electron has its spin aligned parallel (antiparallel) to the effective field A(H, T ) .
where Cis a constant related to the scattering cross section, and f refers to the Stokes and anti-Stokes components of the line. The peak position of the Raman line, 6, satisfies 'F
("4"'
- dAocoth - & p k ~ T
) - 4&pk~T=
0,
(100)
which shows an asymmetry in 6 between the Stokes and anti-Stokes components. The final expression for the Raman intensity as calculated by Dietl and Spalek (1983) and Heiman et al. (1983a) does not include the effects of the magnetic field and temperature on the matrix elements of the Raman tensor. Since Raman scattering proceeds through intermediate states, the Raman matrix elements exhibit an angular dependence such that the cross section for scattering from a donor electron with a given A depends not just on \A1 as implicit in Eq. (99), but also on the angle between A and A0 11 H. This angular dependence has a small effect on the peak position 6. However, in the light of the approximations that made the problem tractable and led to the result of Eq. (92), this small correction can be neglected and Eq. (100) serves as a good approximation for 6 .
9.
RAMAN SCATTERING
403
In order to calculate 6 , the magnetic field induced spin-splitting, A0 and the characteristic energy of the bound magnetic polaron, c P , must first be determined. For a weakly antiferromagnetic system,
where we use the notation of Heiman et al. (1983a) 35 R(cYNo)~ w:=---.-
96 na3No
As demonstrated by Diet1 and Spalek (1983), the dependence of the effective Bohr radius (a) on X, TAF,H , and T can be determined using a variational technique. The values of the effective mass m* and the static dielectric constant K are needed in order to evaluate the radius a using this procedure. Since the effects of composition on these parameters are not known, only an approximate value for a can be deduced. Therefore, in generating the theoretical fits discussed here, it is convenient to treat WO,and hence a, as a third adjustable parameter along with R and TAF.The resulting values of a may then be compared with the known Bohr radius for x = 0. The low field data and the associated theoretical curves are plotted in Fig. 27 for Cdl-,Mn,Te(Ga), x = 0.03, 0.05, and 0.10. As can be seen in Fig. 27(a), a zero-field shift of 3.5 cm-' was observed in Cdl-,Mn,Te(Ga), x = 0.03, at T = 1.8 K. The data for the other two samples, x = 0.05 and x = 0.10, give evidence of zero-field shifts of - 4 and -6cm-'. The magnetic field and temperature dependence of the data, particularly for x = 0.10, are well described by these curves. The zero-field spin-flip Raman spectra of the Cdl-,Mn,Se, x = 0.10, sample are shown in Fig. 28 for T = 1.8, 5 , 10, and 20 K. As can be seen in the figure, the Raman shift of the spin-flip feature decreases for higher temperatures. Except for the temperature, the experimental conditions were identical for the four scans. The four theoretical curves are best fits generated from Eqs. (97), (98), and (99) for TAF= 1.28 K and WO= 0.63 meV, using the same value for Cin Eq. (99). The predicted variation of the peak intensity with temperature agrees quite well with the experimental results.
a 0 a
? ?;I
*wg u &
*
l3
?
0
0
3 15
0
5
10
15
0
5
10
15
MAGNETIC FIELD ( k G ) FIG. 27. Magnetic field and temperature dependence of the peak spin-flip Raman shift in the Cdl-,Mn,Te(Ga), x = 0.05, 0.10, and 0.20, samples at low field. [After Peterson et al. (1985b).]
a N
9. RAMAN SCATTERING
405
406
A. K. RAMDAS A N D S . RODRIGUEZ
VI. Vibrational, Electronic, and Magnetic Excitations in Superlattices During the past decade or so the fabrication of heterostructure of semiconductors by techniques such as molecular beam epitaxy (MBE), or metaloorganic chemical vapor deposition (MOCVD) has been a major technological breakthrough. Single quantum well, multiple quantum wells, superlattices and such heterostructures possess properties and exhibit phenomena not encountered in the bulk. Such “synthetic,” “modulated” or “engineered” structures are fascinating in the context of fundamental physics just as much as for their technological importance. Electronic and optical properties of multilayer structures have brought out several phenomena unique to superlattices, e.g., Brillouin zone folding effects, plasma dispersion in layered electron gases and multiple quantum well effects. Much of the focus till recently has been on 111-V semiconductors and their ternary alloys (Ploog and Dohler, 1983; Dingle, 1975; Klein, 1986). In the past few years heterostructures of DMS-superlattices-have been successfully fabricated (Kolodziejski et al., 1984; Bicknell et al., 1985; Datta et al., 1985). DMS superlattices offer the exciting prospect of tuning the electronic potential within the individual layers, after fabrication, using external parameters such as temperature and magnetic field. It has also been pointed out that (Ortenberg, 1982) by “spin doping” a superlattice, a tunable electronic energy gap can be created at the zone boundary. Here we present illustrative results that demonstrate the effectiveness of the Raman scattering technique for investigating various structural aspects of DMS superlattices (Venugopalan et al., 1984). Figure 29 shows the low frequency spectrum of a Cdl-,Mn,Te/ Cd, ,Mn,Te superlattice recorded at 295 K . The experimental results discussed here are on films grown on a GaAs (001) substrate. A buffer layer of either CdTe or Cdl-,Mn,Te, -2pm in thickness, was first deposited on the substrate; this was followed by the epitaxial growth of the superlattice layers of Cdl -,MnxTe, with. alternating Mn concentrations in adjoining layers. The experiments were performed with samples containing 270 alternating layers of Cd0.5Mn0.5Teand Cdo.g9Mno.] lTe; transmission electron microscopy indicates a layer thickness of 59A. In this spectral region, bulk crystals corresponding to the composition of either layer show no discrete Raman lines due to phonons (Venugopalan et al., 1982). However, in the superlattice we observe very distinct Stokes and anti-Stokes components at 9.2, 1 1 , 19, and 20.7 cm-’. These new signatures, characteristic of the superlattice, can be attributed to the Brillouin zone folding effect resulting from the new period (0) imposed along the growth direction of the sample. Further, the point group symmetry of the superlattice with its growth axis along 11111 (denoted hereafter as 2) is reduced to C3”, whereas bulk
9.
501
-2b
407
RAMAN SCATTERING
-10
10
0
20
RAMAN SHIFT (cm-')
FIG. 29. Stokes (S) and anti-Stokes (AS) components of the folded longitudinal acoustic x = 0.5 and y = 0.11. [After branch of the superlattice Cdl-,Mn,Te/Cdl,Mn,Te, Venugopalan et al. (1984).]
crystals of Cdl-xMnxTe possess the higher symmetry Td, for 0 Ix 5 0.75. Hence phonons propagating along L with wave vectors 191 = 2nN/D, (N = 1,2,3, ...) within the extended zone are now mapped onto the center of the folded Brillouin zone. In addition, the lower symmetry of the superlattice causes the splittings observed at q 0 for these phonons. Similar Raman scattering results on GaAs-A1As and GaAs-Gal -,ALAS superlattices grown along [lo01 have been reported by Colvard et a/. (1980, 1985) and Sapriel et al. (1983). We consider the superlattices as an elastic continuum containing two alternating layers with densities p1 and p2, their bulk longitudinal acoustic (LA) velocities along L being 01 and 02, respectively. Then, the angular frequency ( 0 )of LA phonons traveling along L and the magnitude of their wave vector (4)are related (Rytov, 1956) by
-
where dl and d2 are the respective thicknesses of the adjacent layers, = dl + d2, and a = (PIu1/p2u2). In the backscattering geometry
D
408
A . K . RAMDAS A N D S. RODRIGUEZ
A.
employed in this experiment, q = 5.437 x lo5 cm-'; dl = d2 = 59 Using the interpolated values of the densities and the elastic constants for crystals with x = 0.11 and 0.5, we find LY = 1.064, u1 = 3.445 x lo5cm/sec and u2 = 3.466 x lo5 cm/sec. The solutions of Eq. (105) then predict that the first pair ( N = 1) of lines due to the folded LA phonon branch should be seen at 8.8 and 10.8 cm-'. The excellent agreement with the experimental values, viz., 9.2 and 1 1 .O cm-', demonstrates the sensitivity of this technique to the growth parameters of the superlattice. The higher frequency pair of lines in Fig. 29 at 19 and 20.7 cm-' are assigned to the second fold ( N = 2) of the LA phonon branch. For 59 A layer thickness, from Eq. (105), these are expected at 18.5 and 20.5cm-', respectively. Since AL here is very close to the energy gap of the superlattice, the presence of the N = 2 nodes indicates a resonant enhancement as noted by Colvard et al. (1985). Striking resonance effects observed for longitudinal optic (LO) phonons are illustrated in Fig. 30 where a magnetic field of 60 kG shifted the onset of the photoluminescence peak to the region beyond 600cm-'. The two prominent peaks in the 160-210 cm-' range arise from the fundamental LO modes characteristic of the "two-mode" behavior of Cdl-,Mn,Te alloys
1
-
I
I
I
I
SL
1500
I
I
I
325 - 4
H=60KG
v)
CL
0
v
>
c
1000
v)
z W
k
z
500
0
200
400
600
RAMAN SHIFT ( cm-' FIG. 30. Resonant scattering by longitudinal optic phonons, their overtones and combinations in the superlattice of Cdl-,MnxTe/Cdl,Mn,Te, x = 0.5 and y = 0.11. The line labels denote the following assignments: (1) L o ] , (2) L o z , (3) 2 L o 1 , (4) Lo1 + L o z , ( 5 ) 2L02, (6) 3LO,, (7) 2LO1 + L02, (8) LO1 + 2L02, (9) 3LOz, (10) 4J-01, (11) 3 L 0 1 + LOz, (12) 2L01 + 2LOz. LO1 and LOz are, respectively, at 167 and 199 cm-'. [After Venugopalan et al. (1984).]
9.
409
RAMAN SCATTERING
(Venugopalan et al., 1982). Their respective positions are 167 (LOI) and 199
(L02) cm-I. No zone-folding effects were seen for the LO phonons,
presumably due to the flatness of the dispersion curves of these modes. The additional groups of peaks seen here, centered at 370, 530,and 700 cm-', originate from strong resonant scattering by overtones and combinations of the two fundamental LO phonons. The clear observation of second and third overtones in a very thin (1.6,um) sample, where the effective scattering volume is necessarily small, attests to the resonant enhancement. It is probable that the frequencies of all these features are influenced by any strains present within the superlattice. We further note that although transverse optic modes are also allowed in the experimental geometry, their intensity was too weak to be detected. With 7525A excitation-that lies below the energy gap of the superlattice-none of the features, including LO modes, could be seen. Based on these facts and that the excitation employed for the spectrum shown in Fig. 30 is above the energy gap, the selective enhancement seen for the LO modes and their overtones suggests an underlying Frohlich mechanism. The sloping background beyond 640 cm-' is due to the onset of the photoluminescence peak. Under an applied magnetic field, the position of the luminescence peak shifts towards lower photon energy with an effective g-factor of 100 at 5 K; these data, demonstrate that the very large electronic g-factors first reported by Gaj et al. (1978a) for Cdl-,Mn,Te alloys are also realized in DMS superlattices. We finally discuss the question of magnetic excitations within a DMS superlattice. As noted earlier, Galazka et af., (1980) have shown that Cdl -,Mn,Te alloys possess a spin-glass phase at low temperatures for the composition range 0.17 Ix 5 0.60. In single crystals with x = 0.5 and 0.4, Venugopalan et af. (1982) detected, in the absence of an external magnetic field, a distinct magnon Raman line characteristic of the spin-glass phase. In the superlattice, there appears to be no evidence for such a zero-field excitation. Instead, with an applied field of 60 kilogauss, one observes an intense peak originating from the Raman-paramagnetic resonance (OPM) between the Zeeman sublevels of the ground state of MnZ+ions (Petrou et af.,1983). In addition, the first and second overtones of OPM are also easily detectable. Venugopalan et af. (1984) attribute this to the possibility that even at 5 K the 59 A thick layers (with x = 0.5) of the superlattice may lack the spin-glass ordering, although it does occur for the same composition of the bulk alloy. If so, this would also signify that the dimensional constraints created in a superlattice can exert a decisive influence on magnetic phase transitions. In summary, this study illustrates that Raman scattering can serve as a sensitive, diagnostic probe of diverse structural aspects of DMS superlattices.
-
-
410
A. K . RAMDAS AND S. RODRIGUEZ
Acknowledgments The support from the National Science Foundation during the preparation of the chapter is gratefully acknowledged (Grant No. DMR-84-03325).
References Alov, D. D., Gubarev, S. I., Timofeev, V. B., andshepel, B. N. (1981). Pis’maZh. Eksp. Teor. Fiz. 34, 76; JETP Lett. 34, 71. Alov, D. L., Gubarev, S. I., and Timofeev, V. B. (1983). Zh Eksp. Teor. Fiz. 84, 1806; Sov. PhyS.-JETP 57, 1052. Barker, A. S . , and Loudon, R. (1972). Rev. Mod. Phys. 44, 18. Barker, A. S., and Severs, A. J. (1975). Rev. Mod. Phys. 47, Suppl. 2. Bhagavantam, S., and Venkatarayudu, T. (1969). “Theory of Groups and Its Application to Physical Problems.” Academic Press, New York. Bicknell, R. N., Giles-Taylor, N. C., Blanks, D. K., Yanka, R. W., Buckland, E. L., and Schetzina, J. F. (1985). J. Vac. Sci. Technol. B3, 709. Born, M., and Huang, K. (1968). “Dynamical Theory of Crystal Lattices.” Oxford University Press. London. Brun del Re, R., Donofrio, T., Avon, J., Majid, J., and Woolley, J. C. (1983). I1 Nuovo Cimento 2, 1911. Cardona, M. (1975). “Light Scattering in Solids,” Vol. 1. Springer, Berlin. Cardona, M., and Giintherodt, G. (1982a, b). “Light Scattering in Solids,” Vols. 2,3.Springer, Berlin. Cardona, M., and Giintherodt, G. (1984). “Light Scattering in Solids,” Vol. 4.Springer, Berlin. Ching, W. Y., and Huber, D. L. (1982a). Phys. Rev. B25, 5761. Ching, W. Y., and Huber, D. L. (1982b). Phys. Rev. B25, 6164. Colvard, C., Merlin, R., Klein, M. V., and Gossard, A. C. (1980). Phys. Rev. Lett. 45,298. Colvard, C., Cant, T. A., Klein, M. V., Merlin, R., Fischer, R., Morkoc, H., and Gossard, A. C. (1985). Phys. Rev. B31, 2080. Datta, S., Furdyna, J. K., and Gunshor, R. L. (1985). SuperlatticesandMicrostructures 1,327. Dietl, T., and Spalek, J. (1982). Phys. Rev. Lett. 48, 355. Dietl, T.,and Spalek, J. (1983). Phys. Rev. B28, 1548. Dingle, R. (1975). “Advances in Solid State Physics” (H. J. Queisser, ed.), Vol. 15, p. 21. Pergamon Vieweg, Braunschweig. Dobrowolska, M., Drew, H. D., Furdyna, J. K., Ichiguchi, T., Witowski, A., and Wolff, P. A. (1982). Phys. Rev. Lett. 49, 845. Douglas, K., Nakashima, S., and Scott, J. F. (1984). Phys. Rev. B29, 5602. Economou, E. N., Ruvalds, J., and Ngai, K. L. (1972). Phys. Rev. Lett. 29, 110. Fleury, P. A., and Loudon, R. (1968). Phys. Rev. 166, 514. Frohlich, H. (1958). “Theory of Dielectrics,” 2nd Edition. Oxford University Press, London. Furdyna, J. K., Giriat, W., Mitchell, D. F., and Sproule, G. I. (1983). J. Solid State Chem. 46,349. Gaj, J. A., Ginter, J., and Galazka, R. R. (1978a). Phys. Stat. Solidi (b) 89, 655. Gaj. J. A., Galazka, R. R., and Nawrocki, M. (1978b). Solid State Commun. 25, 193. Gaj, J. A., Planel, R., and Fishman, G. (1979). Solid State Commun. 29, 435. Galazka, R. R., Nagata, S., and Keesom, P. H. (1980). Phys. Rev. B22, 3344. Genzel, L., Martin, T. P., and Perry, C. H. (1974). Phys. Status Solidi B62, 83. Geschwind, S., and Romestain, R. (1984). “Light Scattering in Solids IV” (M. Cardona and G. Giintherodt, eds.). Springer, Berlin.
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Geyer, F. F., and Fan, H. Y. (1980). IEEE J. Quantum Electron. QE-16,1365. Grimsditch, M. H., Ramdas, A. K., Rodriguez, S., and Tekippe, V. J. (1977). Phys. Rev. B15, 5869. Grynberg, M., and Picquart, M. (1981). J. Phys. C14, 4677. Hathaway, C. E. (1971). “The Raman Effect” (A. Anderson, ed.), Vol. 1, Chap. 4. Marcel Dekker, Inc. Hayes, W., and Loudon, R. (1978). “Scattering of Light by Crystals.” John Wiley&Sons, New York. Heiman, D., Wolff, P. A., and Warnock, J. (1983a). Phys. Rev. B27, 4848. Heiman, D., Shapira, Y., and Foner, S. (1983b). Solid State Commun. 45, 899. Heiman, D.,Shapira, Y., and Foner, S. (1984a). Solid State Commun. 51, 603. Heiman, D., Shapira, Y., Foner, S., Khazai, B., Kershaw, R., Dwight, K., and Wold, A. (1984b). Phys. Rev. B29, 5634. Imaino, W., Ramdas, A. K., and Rodriguez, S. (1980). Phys. Rev. B22, 5679. Klein, M. V. (1986). IEEE Journal of Quantum Electronics (to be published). Kolodziejski, L. A., Sakamoto, T., Gunshor, R. L., and Datta, S. (1984). Appl. Phys. Lett. 44,799. Lambe, J., and Kikuchi, C. (1960). Phys. Rev. 119, 1256. Landau, L. D., and Lifshitz, E. M. (1960). “Electrodynamics of Continuous Media,” p. 331. Pergamon Press, Oxford. Lee, Y. R., and Ramdas, A. K. (1984). Solid State Commun. 51, 861. Lee, Y. R., Ramdas, A. K., and Aggarwal, R. L. (1987). Proceedings of the 18th International Engstrom, ed.), p. 1759. World Conference on the Physics of Semiconductors (0. Scientific, Singapore. Lifshitz, I. M., and Kosevich, A. M. (1966). Rep. Prog. Phys. 29, Part I, 217. Loudon, R. (1963). Proc. Roy. SOC.(London) A275, 218. Loudon, R. (1964). Advances in Physics 13,423. Loudon, R. (1978). J. Roman Spectrosc. 7 , 10. Maradudin, A. A., and Oitmaa, J. (1969). Solid State Commun.7, 1143. Mooradian, A., and Wright, G. B. (1968). “Proceedings of the Ninth International Conference on the Physics of Semiconductors, MOSCOW”(S. M. Ryvkin, ed.), p. 1020, Nauka, Leningrad. Mooradian, A., Brueck, S. R. J., and Blum, F. A. (1970). Appl. Phys. Lett. 17, 481. Nawrocki, M., Planel, R., Fishman, G., and Galazka, R. (1980). “Proceedings of the XV International Conference on the Physics of Semiconductors” (S. Tanaka and Y. Toyozawa, eds.); J. Phys. SOC.Jpn. 49, Suppl A, 823. Nawrocki, M., Planel, R., Fishman, G., and Galazka, R. (1981). Phys. Rev. Lett. 46, 735. Oka, Y., and Cardona, M. (1981a). Phys. Rev. B23, 4129. Oka, Y., and Cardona, M. (1981b). J. Phys. (Paris) Colloq. 42, C6-459. Ortenberg, M. v. (1982). Phys. Rev. Left. 49, 1041. Patel, C. K. N., and Shaw, E. D. (1970). Phys. Rev. Lett. 24, 451. Peterson, D. L., Petrou, A., Dutta, M., Ramdas, A. K., and Rodriguez, S. (1982). Solidstate Comrnun.43, 667. Peterson, D. L. (1984). Ph.D. Thesis, Purdue University (unpublished). Peterson, D. L., Bartholomew, D. U., Ramdas, A. K., and Rodriguez, S. (1985a). Phys. Rev. B31, 1932. Peterson, D. L., Bartholomew, D. U., Debska, U., Ramdas, A. K., and Rodriguez, S. (1985b). Phys. Rev. B32, 323. Peterson, D. L., Petrou, A., Giriat, W., Ramdas, A. K., and Rodriguez, S. (1986). Phys. Rev. B33, 1160.
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Petrou, A., Peterson, D. L., Venugopalan, S., Galazka, R. R., Ramdas, A. K., and Rodriguez, S. (1983). Phys. Rev. B27, 3471. Pine, A. S., and Dresselhaus, G. (1969). Phys. Rev. 188, 1489. Pine, A. S., and Dresselhaus, G. (1971). Phys. Rev. B4, 356. Planel, R., Gaj, J., and Benoit a la Guillaume (1980). J. Phys. (Paris) Colloq. 41, C5-39. Ploog, K., and Dohler, G. H. (1983). Advan. Phys. 32, 285. Rodriguez, S, and Rarndas, A. K. (1985). “Inelastic Light Scattering in Crystals,” Highlights of Condensed Matter Theory, Proceedings of the International School of Physics “Enrico Ferrni” (Course LXXXIX) (F. Bassani, F. Furni, and M. P. Tosi, eds.), pp. 369-420. North Holland, Amsterdam. Rowe, J . M., Nicklow, R. M., Price, D. L., and Zanio, K . (1974). Phys. Rev. B10, 671. Ryabchenko, S. M., Terletskii, 0. V., Nizetskaya, I. B., and Oleinik, G. S. (1981). Fiz. Tekh. Poluprovodn. 15, 2314; Sov. Phys. Semicond. 15, 1345. Rytov, S. M. (1956). Akust. Zh. 2, 71; Sov. Phys. Acoust. 2 , 68. Sandercock, J. R. (1982). “Light Scattering in Solids” (M. Cardona and G. Giintherodt, eds.), Vol. 3, p. 173. Sapriel, J., Michel, J. C., Toledano, J. C., Vacher, R., Kervarec, J., and Regreny, A. (1983). Phys. Rev. B28, 2007. Selders, M., Chen, E. Y., and Chang, R. K. (1973). Solid State Commun. 12, 1057. Sennett, C. T., Bosomworth, D. R., Hayes, W., and Spray, A. R. L. (1969). J. Phys. C2,1137. Slusher, R. E., Patel, C. K . N., and Fleury, P. A. (1967). Phys. Rev. Lett. 18, 77. Thomas, D. G., and Hopfield, J. J. (1968). Phys. Rev. 175, 1021. Twardowski, A., Nawrocki, M., and Ginter, J. (1979). Phys. Stat. Solidi. (b) 96, 497. Venugopalan, S., Petrou, A., Galazka, R. R., Ramdas, A. K., and Rodriguez, S. (1982). Phys. Rev. B25, 2681. Venugopalan, S., Kolodziejski, L. A., Gunshor, R. L., and Ramdas, A. K. (1984). Appl. Phys. Lett. 45, 974. Walker, T. W., Litton, C. W., Reynolds, D. C., Collins, T. C., Wallace, W. A., Gorrell, J . H., and Jungling, K. C. (1972). “Proceedings of the XI International Conference on the Physics of Semiconductors, Warsaw, 1972,” p. 376. Elsevier, New York. Weinstein, B. A. (1976). “Proceedings of the XI11 International Conference on the Physics of Semiconductors, Rome, 1976” (F. A. Fumi, ed.), p. 326. Tipografia Marves, Rome. Zanio, K. (1978). “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 13. Academic Press, New York.
SEMICONDUCTORS AND SEMIMETALS, VOL. 25
CHAPTER 10
Theory of Bound Magnetic Polarons in Semimagnetic Semiconductors P. A . Worff FRANCIS BITTER NATIONAL MAGNET LABORATORY MASSACHUSETTS INSTITUTE OF TECHNOLOGY CAMBRIDGE. MASSACHUSETTS
I. INTRODUCTION . . . . . . . . . . . . . POLARONS . . 11. FREEVERSUSBOUNDMAGNETIC 111. THEBMP MODEL. . . . . . . . . . . .
IV.
V.
VI . VII. VIII.
IX. X.
. . . . . . . . . . . .
. . . . . .
THE BMP PARTITION FUNCTION . . . . . . . . . . . . 1. Zero Magnetic Field Case . . . . . . . . . . . . . 2 . Finite Field Case. . . . . . . . . . . . . . . . . 3. Soluble Polaron Model . . . . . . . . . . . . . . THERMODYNAMIC PROPERTIES OF BMP. . . . . . . . . . 4. Internal Energy . . . . . . . . . . . . . . . . . 5 . Magnetization . . . . . . . . . . . . . . . . . 6 . Spin Correlation. . . . . . . . . . . . . . . . . THEDIETL-SPALEK FORMALISM. . . . . . . . . . . . . TIMEDEPENDENT SPIN-SPINCORRELATION FUNCTION.. . . COMPARISON OF THEORY WITH EXPERIMENT. . . . . . . . I . Brief Review of Magnetic Semiconductor Work . . . . 8. Optical Evidence f o r BMP in Semimagnetics. . . . . . 9. Analysis of Spin-Flip Raman Scattering Experiments . . 10. Analysis of Acceptor-BMP Experiments. . . . . . . FOR FREEMAGNETIC POLARONS . . . . . . . . EVIDENCE . . . . . . . . . . . . . . . . . . . . CONCLUSION 11. Current Status of BMP Theory. . . . . . . . . . . 12. Directions f o r Future Work . . . . . . . . . . . . REFERENCES.. . . . . . . . . . . . . . . . . . .
413 415 417 420 420 423 424 425 42.5 428 430 430 433 436 436 438 440 44s 449 45 1 45 1 45 1
452
1. Introduction In semiconductors containing magnetic ions, there is generally a sizable exchange interaction between carrier spins and those of the ions. Exchange causes novel spin-dependent phenomena in such materials including giant 413 CopYnEht 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-752125-9
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spin-splittings of the bands, large Faraday rotations, and magnetic polarons. Free magnetic polarons-magnetization clouds associated with unbound carriers-have not yet been conclusively demonstrated. On the other hand, there is abundant evidence that carriers localized at impurities can induce sizable magnetizations in their vicinity. These ferromagnetic complexes, termed bound magnetic polarons (BMP), often have net moments as large as 25 Bohr magnetons. BMP control many properties of magnetic and semimagnetic semiconductors; they are responsible, for example, for prominent features in the luminescence spectra of semimagnetics and for the spectacular metalinsulator transition observed in EuO. They were first studied in the europium chalcogenides, mainly via transport and magnetization measurements; there is now an extensive literature on that subject (see Kasuya and Yanase, 1968; Yanase, 1972; Kuivalainen et al., 1979; Nagaev, 1983 for bibliographies). None of these experiments, however, unambiguously determines microscopic parameters-such as binding energy, moment, or size-of the BMP. Direct observation of BMP in Eu-chalcogenides is impeded by their complicated band structure, relatively poor electronic and optical properties, and by competing magnetic interactions. The situation is more favorable in the semimagnetic semiconductors with large, direct bandgaps and simple band structures; examples are Cdl -,Mn,S, Cdl-,Mn,Se, and Cdl-,Mn,Te. These crystals, in contrast to the EuX compounds, have highly structured luminescence spectra whose features can be associated with well-known centers; a study of the temperature, composition, and magnetic field variation of the features gives microscopic information concerning BMP interactions in such complexes as the acceptor, the acceptor-bound exciton (A'X), and the donor-bound exciton (D'X). Semimagnetic alloys also favor light scattering experiments since, in direct gap crystals, carrier scattering cross sections are enhanced by large factors as the laser frequency approaches the gap. This resonance facilitates spin-flip Raman scattering (SFRS) studies, which provide key evidence concerning magnetic fluctuations in donor-BMP. Two optical experiments (Golnik et al., 1980; Nawrocki et a/., 1980) were the stimulus for recent advances in the theory of BMP. The theoretical work, by Diet1 and Spalek (DS; 1982, 1983) and by Heiman, Wolff, and Warnock (HWW; 1983b), has given new insight into the behavior of BMP. The crucial point was first enunciated by DS. They showed that, at high temperature, magnetic fluctuations determine the behavior of BMP. As temperature is lowered, the BMP continuously evolve from a fluctuation-dominated regime to a collective regime in which the carrier and magnetic ion spins are strongly correlated with one another; the system then behaves like a large classical magnetic dipole whose moment is many Bohr magnetons. The transition
10. THEORY
OF BOUND MAGNETIC POLARONS
415
cannot be described by mean field theory, which predicts a phase transition in the finite BMP system, as well as other unrealistic behavior. This article discusses the theory of BMP in semimagnetics, with emphasis on the interpretation of DS. The formalism used will be that of HWW which is rather different from that of DS. DS approach the BMP problem microscopically but use the continuous medium approximation in the calculation of the partition function; the central ingredient in their analysis is a classical, Landau free energy functional. HWW, on the other hand, base their calculations on a microscopic Hamiltonian, with suitable approximations to evaluate the partition function and correlation functions. Despite their dissimilarities, the two theories give identical results in the continuum limit where they can legitimately be compared. The HWW approach is chosen here because it is somewhat simpler than that of DS, and because it naturally handles the problems of spin saturation and centers containing a small number of spins. Detailed comparisons of theory with experiment will be made wherever possible. For the simpler centers, the agreement is excellent; it fully substantiates the DS picture of a gradual transition from the fluctuationdominated to the collective regime. Other, untested, predictions of the theory are also strengthened by its overall agreement with experiment. Yet, despite these successes, the theory is not complete. There is another class of BMP problems-those of spin relaxation, spin diffusion, and spin kinetics generally-that are just beginning to be addressed. These problems cannot be fully solved in the semiclassical approximation common to the DS and HWW theories; instead they require a quantum mechanical theory. The development of such a theory, and of experiments to test it, is an important next step in the BMP problem. 11. Free Versus Bound Magnetic Polarons
The simplest possible magnetic polaron-the free magnetic polaronconsists of a carrier trapped, via the exchange interaction, in a magnetic potential well created by locally aligning the spins of magnetic ions. The stability of such complexes was investigated by Kasuya et al. (1970). Their analysis shows that the polaron energy is determined by a balance of two terms of opposite sign: a positive kinetic energy due to carrier localization, and a negative energy due to the exchange well. Free polarons are stable in certain regions of the polaron parameter space, unbound in others. The existence of free magnetic polarons in a given material thus depends, in detail, on the parameters that characterize it. Calculations suggest that they are generally not stable in dilute (x < 0.05) semimagnetic alloys. The energetics of BMP are quite different from those of free magnetic
416
P . A. WOLFF
polarons. As described in the introduction, BMP involve a carrier (or carriers) trapped at an impurity by electrostatic forces. The kinetic energy of localization is balanced by the Coulomb attraction of the impurity. Spin alignment of nearby magnetic ions is favored by the exchange interactionbut this effect is not the primary mechanism of localization, as it is in the free polaron. Nevertheless, significant spin alignment can occur in BMP, even when the strength of the exchange interaction is well below that required to stabilize the free magnetic polaron. The preceding argument suggests that carriers localized at impurities in magnetic or semimagnetic semiconductors will generally show BMP effects, whereas free magnetic polarons should be rarer. Experiments on semimagnetics confirm this view; BMP effects have been observed in several ways in a number of semimagnetic crystals. It is only recently, however, that evidence for free polarons in semimagnetics has been presented (Golnik et al., 1983). The evidence is suggestive but not yet conclusive; we will return to it in Part IX. The analysis of Kasuya et al. (1970) implies another important difference between free and bound magnetic polarons. The former, if stable, must be spin saturated; BMP, on the other hand, need not be. The requirement of spin saturation in free polarons is implied by the expression for the polaron free energy. Kasuya et al. (1970) assume an exchange interaction of the form
where (r,s) are the coordinate and spin of the trapped carrier, and ( R j , Sj) those of a magnetic ion at site R j . The free energy is calculated in the molecular field approximation, assuming a variational form for the carrier wave function. For the semimagnetic case, the free energy becomes:
where a is the radius of the carrier wave function, x the susceptibility of the magnetic ion system, and NO = a,' is the unit cell density of the crystal. This expression is valid until the spin correlation function, ( s - S j ) , begins to saturate, i.e., approaches its maximum value, of $ for the Mn2+ ion case. Equation (2) becomes negative for sufficiently small values of a. If saturation has not occurred prior to this crossover point, the polaron is bound and will continue to increase its binding energy by decreasing its radius until saturation occurs. Though approximate-since it relies on mean field theory-this argument is qualitatively correct.
10.
THEORY OF BOUND MAGNETIC POLARONS
417
111. The Bound Magnetic Polaron Model
The simplest bound magnetic polaron is the donor-BMP in a semimagnetic semiconductor of moderate (say x c 0.1) magnetic ion concentration. The single electron in such a complex moves in a nondegenerate conduction band, and has an extended orbit that is well-described by the effective mass approximation. Moreover, for small x, the magnetic ion susceptibility in such crystals as Cdl-,MnxS and Cdl-,Mn,Se is nearly Curie-like in the temperature range of interest. Both of these facts simplify the analysis. Thus, the donor case will be used to develop the theory; these ideas will later be extended, in a less rigorous way, to acceptor-BMP. Donor-BMP have been extensively studied via SFRS in CdMnS (Alov et al., 198 1 , 1983; Douglas et al., 1984; Nawrocki et al., 1984; Heiman et al., 1983a), CdMnSe (Nawrocki et al., 1980; Heiman et al., 1983b; Plane1et al., 1984), and ZnMnSe (Douglas et al., 1984; Heiman et al., 1984), which are n-type as grown; there has also been one report (Peterson et al. , 1982) of SFRS by donor-bound electrons in normally p-type CdMnTe. These experiments were an important stimulus to the theory, by providing the first clear evidence for spin fluctuation effects in BMP. The donor-BMP in semimagnetics is not, however, sufficiently strongly coupled to exhibit the full range of BMP behavior; even at 2 K there is only moderate spin correlation in these complexes. Complete correlation, i.e., spin saturation, is illustrated by the more complicated, but strongly interacting, acceptor-BMP case. The model Hamiltonian of the donor-BMP is:
where m* is the effective mass, EO the dielectric constant, and a the exchange constant (Gaj, 1980) for the conduction band. In applying Eq. (3) to wurtzite crystals, such as CdMnS or CdMnSe, we will ignore the small ( = 5 % ) anisotropy of m* and E O . A more serious approximation is the neglect of direct Mn2+-Mn2+ spin interaction in such materials. Experimentally, Cdl -,Mn,X crystals with small x-values have nearly Curie-like magnetic susceptibilities (Kreitman et al., 1966; Nagata et al., 1980; Galazka et al., 1980) in the temperature range 4-40 K, but with magnitudes corresponding to effective Mn2+concentrations (2)somewhat smaller than x . For example, for x = 0.05 one finds R = 0.03. This behavior, which has caused some confusion in the literature, has recently been shown (Shapira et al., 1984; Aggarwal et al., 1984) to be the result of antiferromagnetically-coupled Mn2+ spin clusters (singlets, doublets, triplets, etc.). Isolated Mn2+spins, with no nearest neighbor Mn2+ ions, give a full Curie contribution to the susceptibility; MnZ+pairs (doublets) are antiferromagnetically coupled and make no
418
P . A. WOLFF
contribution at low temperatures; and there is a small contribution from the two types (open and closed) of spin triplets. This model quantitatively explains the observed variation of R with x , if Mn2+ ions are assumed randomly distributed on the cation sublattice. From the BMP viewpoint, it implies that Mn2+doublets and triplets are essentially “locked out” of the problem by strong, nearest neighbor, Mn2+-Mn2+interactions. The BMP sees only the isolated Mn2+ ions, whose concentration is R. This argument is valid for small x (x ?. 0. l), where most Mn2+ions are in clusters of small size. In that range, however, it enables one to treat the BMP problem-via the R trick-as if there were no direct Mn2+-Mn2+interactions. The resulting model Hamiltonian [Eq. (3)] contains the essential physics of the BMP in its purest form and, at the same time, quantitatively explains the SFRS experiments in CdMnS, CdMnSe, and ZnMnSe. It should always be borne in mind, however, that Eq. (3) involves important approximations; in particular, one cannot expect it to be valid for x > 0.1. The Schrodinger equation, X F = EF, determines the modulating function in the effective mass approximation; the complete wave function is v/ = FUO, where uo is the band edge Bloch function. The exchange potential in the effective mass Schrodinger equation mixes orbital and spin degrees of freedom; for example, exchange can scatter an electron from one orbital state to another while flipping its spin. In practice, such scattering occurs infrequently, as will be demonstrated below. It is then reasonable to seek a solution of the Hartree form:
In this approximation the orbital wave function, q5(r),is determined by the combined actions of the Coulomb potential and the coherent average of the exchange potential; fluctuations of the exchange potential about its average value, which cause scattering, are ignored. To test this Hartree-like approximation, it is useful to make estimates of the spin flip scattering rate. A Born approximation calculation for a free electron in a semimagnetic crystal gives the following expression for the spin flip mean free path:
_1 - S(S + 1) R((oIN0)2(rn*)2
A-
6n
A4No
(5)
With R = 0.1 and parameters appropriate to CdMnSe NO = 0.26 eV and m* = 0.12), Eq. (5) predicts A = 2 x loM4 cm. This value is two orders of magnitude larger than the circumference of the Bohr orbit in CdSe (or CdMnSe). Thus, in a semimagnetic crystal an electron in a donor-BMP circles the
10. THEORY
OF BOUND MAGNETIC POLARONS
419
impurity 100 times-establishing a well-defined orbit-before experiencing an exchange scattering. This estimate is the main justification for Eq. (4). Comparison of theory with experiment also suggests that the Hartree approximation is accurate. With the aid of Eq. (4)one can derive separate, but coupled, Schrodinger equations for q5 and x by integrating Eq. (3) against x* or +*. They are:
and
where and
Eqs. (6) and (7) could, in principle, be solved self-consistently to determine the individual spin states of the BMP. The number of such states is enormous, however, of order 6NwhereN 100 is the number of Mn2+spins within a donor orbit. Many of these states are accessible to the system, even at low temperatures; the properties of the BMP are thus determined by an ensemble average over low lying states. Self consistent equations for the ensemble averaged, orbital wave function are derived by minimizing the free energy with respect to $(r). One requires
-
subject to the normalization condition jJ+(r)IZd3r = 1, where G - kTln ( Z ) and
=
where = l/kB T . The resulting Schrodinger equation for +(r)has the same form as Eq. (6). However, (s * S j ) is now defined as the ensemble average:
420
P. A. WOLFF
over the spin Hamiltonian:
Eqs. (6) and (12) determine 4(r) self consistently. Their solution is usually calculated-or estimated-variationally by assumin a simple form for W ) , such as the hydrogenic wave function +(r) = e-r'u/ na ,calculating G from Eq. (1 l), and minimizing it with respect to a.
$-J
IV. The BMP Partition Function
1. ZEROMAGNETIC FIELDCASE
The spin partition function [Eq. (1 l ) ] determines the thermodynamic properties of BMP. Evaluation of this formula is a primary task of the theory. Throughout the calculation the wave function 4(r) will be assumed known. In donor-BMP, where the exchange interaction is relatively weak, 4(r) is nearly the same (+1-2%) as the donor wave function of the binary host compound. On the other hand, for acceptors, a variational calculation is required to determine the optimum size of the BMP. Subsequent analysis is simplified by rewriting the spin Hamiltonian in the form: %spin
= -a
C [ ( ~ * S j ) l 4 ( R j )=I ~- C] j
j
[Kj(s.Sj)]E
-(S*r),
(14)
The expression for the coupling constant, Kj, involves the exchange interaction and the square of the BMP wave function. Both factors are smaller in donor-BMP than in acceptor-BMP; when combined they imply that polaron interactions in donors are an order of magnitude weaker than in acceptors. The quantity l- is proportional to the effective field experienced by the electron in the BMP. It determines the Zeeman splitting of the complex. r is a vector operator whose components do not commute with one another; that fact prevents an exact evaluation of the trace in Eq. (1 1). On the other hand, if r were a classical variable, the trace over spin variables could be calculated via the standard identity:
10.
THEORY OF BOUND MAGNETIC POLARONS
421
This identity is not correct when r is an operator. In the Mn2+case, however, the individual spins are fairly large (S = 8) and combine in the BMP to give a net moment of many Bohr magnetons. Thus, it should be a good approximation to treat r as a classical variable. An exactly soluble model of the BMP, discussed in Part IV, Section 3, suggests that this approximation is correct to order N-l”, where Nis the number of Mn2+spins within the BMP orbit. The classical approximation gives the following expression for the partition function:
where 55/2(x)
= i [ e5/2x
+
e 3 / 2 ~ + e l / 2 ~+ e-1/2X
+
e-3/2~
+ e-5’2x].
(19)
5 5 / 2 ( ~ )is an even, analytic function. This fact can be used to calculate 2. After performing the angular integrations, the expression is rewritten in the form:
and evaluated by displacing the A-contour upwards in the complex plane, above the line 1 = i/3/2, to make the y-integration convergent. Integration over y gives the relation:
Finally, the A-integral is calculated by moving the contour back to the real
422
P. A. WOLFF
axis. Only the pole at A = iB/2 contributes. The result is:
It is interesting to note that the first factor in this expression is the partition function for an Ising model of the BMP-with Hamiltonian XIsing
=
C[~j(szsjz)l. j
Equation (22) is a fairly complicated function of temperature because the alignment of each MnZ+ion varies nonlinearly with (PKj/2), and saturates when the exchange field exceeds thermal energies (BKj S 1). Saturation occurs at low temperatures in acceptor-BMP, but not in donor-BMP. In discussing the latter, it is often convenient to simplify the expression for 2. For small y , the distribution of internal fields appearing in Eq. (18) is Gaussian:
where
wd'=
(;:)
- C(K,?). j
-
Equation (23) is valid well beyond the fluctuation range ( y WO),but fails before y reaches its maximum value, $X(CYNO). Substitution of Eq. (23) into Eq. (18) yields the following formula for the partition function in the Gaussian approximation:
(25)
Equation (25) is similar to that of DS, and will later be used to make a connection between the DS and HWW theories.
10.
423
THEORY OF BOUND MAGNETIC POLARONS
2 . FINITEFIELDCASE Eqs. (18)-(25) can easily be extended to the finite magnetic field case. When fields are present, the Hamiltonian takes the form: Xspin
=-
c [Kj(s Sj)] - /.fBg*(s Bo) - PBgMn c (sj Bo). *
*
*
j
(26)
j
The partition function becomes:
(27) where b*
pBg*Bo
and
bMn
PBgMnBo.
(28)
The angular integration in Eq. (27) is complicated, but simplifies in the Gaussian limit. One then finds:
where
The effective field (beff)experienced by the carrier is the sum of the external field (b*)and the internal, exchange field defined by:
424
P. A. WOLFF
At low temperatures, the exchange field is often much larger than the applied field. In CdMnSe, for example, aNo = 0.26 eV; if x = 0.05 and T = 4 K, the dimensionless factor in brackets in Eq. (31) then has the value 100. Field enhancement factors of that size are not unusual in semimagnetics. Such materials act as amplifiers of external magnetic fields, at least insofar as their effect on carrier spin degrees of freedom is concerned. It should be clearly understood, however, that the exchange interaction does not affect carrier orbital motion. 3. SOLUBLE POLARON MODEL Several authors (Planel, 1982; Golnik et al., 1983; Ryabchenko and Semenov, 1983) have discussed an exactly soluble model of BMP. Though unrealistic, it provides a valuable test of the classical approximation. The model assumes that a donor-bound electron interacts with N Mn2+ ions through an exchange interaction of constant strength: N
x = -K C
j= 1
=
(s.Sj)
K 2
- - [(S * s)2 - S 2 - s2],
(32)
where S = Cj”,,(S,). This Hamiltonian would follow from Eq. (13) if NMn2+ ions were distributed on a spherical shell surrounding the impurity. The operators J = ( S + s), J z , and S 2 commute with the model Hamiltonian. Thus, it has eigenvalues
E(J, J z , S ) = - - J(J K2 [
+ 1) - S(S + 1) - -431 ,
(33)
that are independent of J z . For a given S , the allowed values of the total angular momentum are J = ( S f i).The corresponding energies are: KS E ( + , J , , S ) = --, 2
In these two groups of levels the electron spin is aligned parallel and antiparallel to the internal field, respectively. SFRS causes transitions between them, with S conserved. The partition function for the model is:
z
=2
C(D(S)[(S+ 1)eflKS/’ + Se-”K/2(S+1) 1 1 9
(35)
10.
THEORY OF BOUND MAGNETIC POLARONS
425
where D ( S ) is the number of Mn2+spin states, of fixed S,, with total spin S . Yanase and Kasuya (1968) describe a technique for calculatingD (S);when 1 Q S Q 5/2N, it can be shown that: 3/2
D(S) = -!(L)(2s + 1)e-6S2/35N. \/?T 35N The partition function then becomes:
+
~ ( 2 s l)e-PKS/2(S+l) 6S*/3SN 1e1. +
(37)
This summation over S is similar in form to the integral over y in Eq. (25). We now compare the classical partition function for the model to the exact result [Eq. (35)]. If S is assumed to be a classical variable, the trace over the carrier spin in the partition function can be evaluated with Eq. (17); one finds Z = Tr[e°K(S'S)] I :
[
b(S)(ZS + 1)cosh b K Y ] j .
The function of S under the summation sign in this formula differs from that appearing in Eq. (35) by terms of order S - ' . Since the main contribution to the sums comes from terms with S N"' , or larger, we conclude that the classical approximation to the partition function is accurate to order N-"2.
-
V. Thermodynamic Properties of BMP 4. INTERNALENERGY
Much of BMP behavior is determined by statistical mechanics, and can be calculated from the partition function [Eqs. (22), (25) and (29)l. Thermodynamic functions demonstrate, in particular, the gradual transition from the fluctuation-dominated regime to the collective regime mentioned above. Consider first the internal energy, U,of a BMP. It is an especially important thermodynamic variable, directly measured by luminescence studies of acceptor-BMP. U can be calculated from the usual statistical mechanical relation:
Differentiation of Eq. (22) produces a complicated, not-too-meaningful formula. The result simplifies, however, in the unsaturated limit. One then
426
P. A. WOLFF
finds [from Eq. (25)l:
u = - pWo2 -[ 4
1.
12 + (pwo)2 4 + (pwo)2
This expression can be seen to be identical to that of DS [their Eq. (4.11)] by making the replacement p W,’ + 4&,,and assuming a Curie susceptibility. Equation (40) has different temperature variations in the high (pWOe 1) and low ( ~ WBO 1) temperature regimes. To understand the meaning of these two regimes, we note that there are two ways in which the BMP can lower its internal energy to increase binding. At high temperature, though (r) = 0, there are sizable fluctuations of r about its mean value-of size because the BMP contains a finite number of Mn2+ spins. As temperature is reduced the carrier spin, which relaxes rapidly compared to those of the Mn2+ions, gradually aligns with the instantaneous fluctuations of r, causing a decrease in U. Throughout this spin alignment process, as will be shown below, the magnitude of r remains essentially unchanged, determined solely by the statistics of Mn2+ spin orientation. Hence the term “fluctuation regime”. When alignment of the carrier spin is nearly complete-for B WO= 2-the system can further reduce its energy by forcing the Mn2+ spins to adopt statistically unfavorable, but energetically favorable, configurations with gradually increases throughout this BMPlarger values of formation regime, where fl WO> 1. Ultimately, at sufficiently low temperatures, the Mn2+ spins may saturate, i.e., ( s - s,) 5/4. This interpretation of the temperature variation of U can be tested by rewriting the expression for the internal energy as a product of two factors, one describing the carrier spin alignment process, the other the gradual lengthening of r:
a-
m.m
--.)
(r2)is calculated from Eq. (25) with the aid of the classical approximation r. One finds
for
10.
427
THEORY OF BOUND MAGNETIC POLARONS
This expression implies = when pW0 = 0 (high temperature remains essentially constant until regime). AS ~ W Oincreases, ~ W=O1; thereafter it increases approximately as pW0/2. Conversely, the spin orientation factor, 2(s T ) / m i n Eq. (41), grows linearly with (pW0) throughout the fluctuation regime (0 c ~ W 1) range. Thus, though the transition is a gradual one, it has two distinct regimes-a higher temperature one dominated by magnetic ion spin fluctuations, whose internal fields align the carrier spin; and a lower temperature one controlled by true BMP interactions that gradually increase the net moment of the complex. Not surprisingly, in the high temperature (pWo 6 1) regime, there are large (percentagewise) fluctuations of the BMP energy about its mean value. In the Gaussian limit,
-
4 =“:[1+
(
(4+/32WgZ) - 4+pZwgZ 4pW0
>’I
’
(43)
and
On the other hand, in the BMP regime the energy is quite sharply defined since
In donor-BMP, the exchange interaction is barely strong enough to force such complexes into the collective regime at T = 4 K. A Cd0.9oMn0.1oSe sample (Heiman et al., 1983b), for instance, has Wo = 0.6 meV, implying PWo = 1 at T = 8 K. It is not surprising, therefore, that SFRS experiments on CdMnSe (discussed in Part VIII, Section 3) show only faint indications of BMP formation in the temperature range T 2 2 K. If the Mn2+susceptibility of such systems were strictly Curie-like one could, in principle, reach the BMP regime by going to lower temperatures. In practice, this idea does not work because next nearest neighbor Mn2+-Mn2+exchange interactions cause the Mn2+susceptibility to saturate. Phenomenologically, this effect is described by making the replacement T + ( T + TAF)in the Curie law susceptibility, where TAF= 2.3 K in Cdo.goMno.loSe. Since magnetization fluctuations of a thermodynamic systems are proportional to its susceptibility (Landau and Lifshitz, 1977), this change also requires a modification
428
P . A. WOLFF
of the parameter, W,” -+ W 2 = W,”(T/T + TAF),that controls them in the Gaussian limit [Eq. (25)l. With that replacement, there is little further increase in donor-BMP bindng energy or spin correlation below T = 2 K . The situation is quite different in acceptor-BMP, where Wo = 5 meV and BMP formation commences when T = 30-50 K. In many cases, the Mn2+ spins near the center of the hole orbit are fully saturated at low temperatures. Such complexes behave like a single, giant magnetic dipole whose moment is 2 0 - 5 0 ~ Their ~ . properties will be discussed, in connection with experiments on acceptor-BMP, in Part VIII, Section 10. 5 . MAGNETIZATION Magnetization is another important thermodynamic property of BMP; in the Gaussian limit it is determined by the formula
X
where [from Eq. (30)]
The second factor in Eq. (44) is the BMP contribution to the magnetization. At low temperatures, the BMP “moment” is much larger than that of the carrier (geffP g*). The extra moment is provided by Mn2+ spins in the vicinity of the BMP, whose alignment with the magnetic field is enhanced by the exchange interaction. In the low field limit, the BMP has susceptibility:
This expression can be shown to be identical to Eq. (4.8) of DS, via the replacements discussed after Eq. (40). Note that XBMP is a smooth function
10.
THEORY OF BOUND MAGNETIC POLARONS
429
of temperature, without singularities at finite T. We will see below that, in contrast, a mean field theory of BMP predicts a spurious, finite-temperature divergence of X B M P . That prediction is incorrect, since a finite system cannot undergo a phase transition. Experimentally, there is no evidence of singularities in X B M P . As BO increases, Eq. (44) describes the gradual saturation of BMP magnetization. The general formula is complicated, but has simple and interesting limits when /3 WO6 1 and j?WO% 1. For p WO6 1. One finds:
This result follows directly from mean field theory (MFT), which assumes that each type of spin aligns in the external field plus the average exchange field created by the other type. The MFT equations for the BMP are:
and <Sjz>
=
SP[~M~ + Kj(sz)I,
when Mn2+ spins are not saturated. Elimination of formula for ( s z ) :
(49) (Sj,) gives
the MFT
When ~ W6O1, the second term in the argument of the hyperbolic tangent is small compared to the first. If it is neglected, Eqs. (48) and (49) are equivalent to Eq. (47). The approximations s (Sj) = (s Sj) = (s) Sj ,used in deriving Eqs. (48) and (49), eliminate the spin-spin correlations responsible for BMP formation. Thus, Eq. (50) cannot give correct results for ~ W O F )l . In fact, it predicts a divergence in K B M f~ o r b WO= 2 and finite (sz) in zero applied field if ~ W >O2. Neither effect is physical. To correctly describe the BMP susceptibility in the BMP regime ~ Ws Ol), we must return to Eq. (44). When ~ W sO1, Eq. (44) implies:
-
-
where S(q) =
[
I:
ctnh(q) - -
430
P. A, WOLFF
is the Langevin function that determines the spin alignment of a classical moment in a field. In our case, the classical moment is that of the BMP, whose spins are aligned with one another via the exchange interaction.
6. SPINCORRELATION Finally, we consider the spin correlation function, (s * Sj), for the BMP. Note that (s * Sj) = - aX/aK,. For such an operator-the derivative of the Hamiltonian with respect to a c-number parameter-one can use the cyclic property of the trace to prove that:
Here B5/z(x)is the Brillouin function defined by 5/2&/2($X)
5bz(X)/55/2(X).
(54)
This formula is based on Eq. (22) which includes the effects of spin saturation. When PKj a 1, the second term, in Eq. (53) is small and (s S,) + $ 8 5 / 2 -, $ as anticipated. In the fluctation regime, where PWO Q 1, one finds a small correlation: ( s - Sj) = 105/48(pKj) 4 1. This statement seems, at first glance, to contradict our earlier assertion that there can be considerable alignment of the carrier spin (with the fluctuating internal field) in the PWo e 1 regime. Actually, both statements are true. To reconcile them, remember that the fluctuating moment of the BMP involves of the total number of Mn2+ spins within only a small fraction, 0 (1/N1”), it. Thus, even full alignment of the carrier spin does not represent, on average, a high degree of spin correlation. For pWo Q 1, one expects ( s - Sj) = l/N’/’. VI. The Dietl/Spalek Formalism
Equations (40) and (46) of the preceding section are identical-assuming a Curie-law MnZ+ suceptibility-to the corresponding formulas in the Dietl-Spalek paper [their Eqs. (4.11) and (4.8)]. This fact suggests that, in the Gaussian limit, the two theories are equivalent. T o confirm that
10. THEORY
OF BOUND MAGNETIC POLARONS
431
relationship, which is obscured by their different formalisms, we recast our expression for 2 into a form that makes apparent its connection to the DS formulas. The DS theory uses a continuous medium approximation (Spalek, 1980) to describe MnZ+ magnetization. To make contact with their work, we partition the space surrounding the BMP into cells, each small compared to the volume of its orbit. We will also assume that each of the cells contains a sizable number of MnZ+ions. The two conditions can only simultaneously be met if there are many ions within the BMP orbit, i.e., if (4na3N0/3)XP 1. This inequality is fairly well satisfied in donor-BMP systems. For example, in n-Cdl-,Mn,Se with x = 0.05 (X = 0.03),one finds (4na3N0/3)X= 180. The continuum approximation should work well under such conditions. However, it is a poor approximation for acceptor-BMP, since their orbits usually contain about ten Mn2+ions. Equation (18) for Zcan be rewritten as a product of traces over the various cells:
Here yc(c = 1,2, ..., N ) is the contribution of Mn2+spins in cell “c” to the internal field. In the last line of the equation we have made the approximation Kj = Kc = constant for all j E c. To evaluate the cell spin traces in Eq. ( 5 9 , we assume (i) that Mn2+spins are not saturated and (ii) that each cell contains sufficiently many spins so that its spin statistics are Gaussian. One then finds:
432
P. A. WOLFF
=
=
! !
d3A
-( [ 5 5 / 2 ( - iAKc)]“ceiX’Yc)
(27CI3
Here W? = 35/12ncK:, and nc is the average number of Mn2+ spins in cell “c”. yc is related to the magnetization density in cell “c” via the expression
With this change of variables, Eq. (55) takes the form Z =
where
P(y)= (const.)!
5
d3M(rl)
!
P(y)d3y.
(57)
d3M(rN)[cosh(PTY)
uc is the volume of cell “c”, and x = 35/12[(pBgMn)/kT]xNO is the Curie susceptibility of the unpaired Mn2+ spins. In the continuum limit, Eq. (58) becomes a functional integral over suitably-weighted “paths” of M(r). With the change of variables y A (to match DS’s notation) one finds: +
p[M(r)l4A - A[M(r)llBM(r), where
and
10.
THEORY OF BOUND MAGNETIC POLARONS
433
These formulas are identical to Eqs. ( 3 . 2 ) , (3.3) and ( 3 . 5 ) of the DS paper for the case of a Curie-law, MnZ+susceptibility. The DS theory and the preceding analysis rely on the Gaussian approximation. Neither is correct for systems such as acceptor-BMP, in which spin saturation occurs. However, for donor-BMP, which do not saturate, they provide an accurate description of the behavior of the complex. DS also point out an interesting relationship between the mean field theory of the BMP and the Gaussian approximation to its partition function [Eq. (25)l:
The integrand in this expression has a maximum at the value of y determined by the formula:
The maximum is relatively sharp at low temperatures (PWO>> I), where BMP formation occurs. In that limit, @ym) % 1 and the factor 2 on the right hand side of Eq. (63) become unimportant; the formula is then equivalent to the mean field theory (MFT) result [Eq. (SO)] in zero magnetic field. At higher temperatures, on the other hand, the full equation is required to avoid the spurious phase transition predicted by MFT. In this regime, the distribution of y-values is broad and dominated by fluctuations. To summarize, MFT predicts the low temperature spin alignment of BMP, and the high temperature response to weak fields. It is totally inadequate in describing fluctuations or the gradual transition from the fluctuationdominated to the collective regime. VII. The Spin-Spin Correlation Function
Thermodynamic variables, such as the internal energy or magnetization discussed in Part V, can be used to interpret certain experiments concerning BMP. Others, however, require a knowledge of time-dependent properties. For example, the spin flip Raman scattering (SFRS) spectrum, S ( o ) , is the Fourier transform of the time-dependent, spin-spin correlation function (Wolff et al., 1977):
434
P. A. WOLFF
Here (with A = 1)
s(t) = ei"rs(0)e-ixt,
(66)
and 2 is given by Eq. (18) or Eq. (27). The unit vector (Y is determined by the geometry of the experiment and by the laser frequency. For our purposes, a can be viewed as a constant-involving neither the electron nor the Mn2+ spin variables. In the semiclassical approximation d3y6(y -
r - b*)eflz(Sj'bMd
The time dependent operator in this equation can be evaluated by rewriting (s * a) in terms of stepping-up and stepping-down operators relative to the quantization direction y. One finds:
This identity can now be used to evaluate the electron spin trace in Eq. (67). A tedious, but straightforward, calculation yields the result:
10. THEORY OF
BOUND MAGNETIC POLARONS
435
Finally, the trace over (Sj) is evaluated after substituting Eq. (69) into Eq. (67). In the Gaussian approximation, one finds (Heiman et al., 1983)
where
[1:
]
C = Z-'(6) exp -iVN(pb~~)' .
The integral in Eq. (70) is similar to that determining the partition function in the Gaussian approximation [Eq. (29)]. This fact substantitates the DS hypothesis that the integrand of the partition function determines the spin flip spectrum, with the two terms in the cosh(By/2) factor corresponding to Stokes and anti-Stokes scattering, respectively. This statement will become clearer in Part VIII, where Eqs. (69) and (70) are compared with SFRS experiments. We will also see there that the factor T [ t ; y , p l in Eq. (69) predicts striking polarization effects in SFRS. Their observation provides a stringent test of the theory. The Fourier transform of Eq. (70) determines the spin flip spectrum:
We are primarily interested in the second and third terms of this formula, that describe Stokes and anti-Stokes SFRS. The first, zero-frequency term can be shown (Romestain et al., 1975) to cause Faraday rotation. Equation (71) contains no damping. The electron spin precesses freely, without spin relaxation, in an effective field (beff)that is the sum of the external field and an internal, exchange field of the MnZ+ions [Eqs. (30) and (31)]. Damping is eliminated by the classical approximation which ignores the (s'S7) terms in the exchange interaction. In practice, Eq. (71) reproduces the measured SFRS spectra almost perfectly. The theory succeeds because the Mn2+spins relax slowly at low temperatures; thus, the electron spin has ample time to align with the slowly fluctuating effective field.
436
P. A. WOLFF
VIII. Comparison of Experiment With Theory 7. BRIEFREVIEWOF MAGNETIC SEMICONDUCTOR WORK Research on magnetic semiconductors was stimulated nearly twenty years ago by the discovery of novel electronic, optical, and magnetic phenomena not encountered in conventional semiconductors. These effects are caused by the carrier-local moment exchange interaction. They include: (i) Giant red-shifts of the band gap in ferromagnetic semiconductors such as EuO (Busch et al., 1964; Busch and Wachter, 1966a; Schoenes and Wachter, 1974a) and CdCrzSe4 (Harbeke and Pinch, 1966; Busch et al., 1966b). (ii) Giant Faraday rotations in ferromagnetic semiconductors (Ahn and Shafer, 1970; Tu et al., 1972; Schoenes et al., 1974b). (iii) A metal-insulator transition in Eu-rich EuO (Oliver et al., 1970; Oliver et al. , 1972; Torrance et ai., 1972). (iv) Large resistivity maxima in the vicinity of magnetic phase transitions, accompanied by negative magnetoresistance (Heikes and Chen, 1964; von Molnar and Methfessel, 1967). (v) Substantial variation of the paramagnetic Curie temperature with x in Eul-xGdxSe and Eul-,Gd,Se (Holtzberg et al., 1964, 1965). The first two phenomena listed result from the rigid, exchange-induced shift of the energy bands. No polaron effects need by invoked to explain them. On the other hand, the behavior of the conductivity and susceptibility suggest carrier localization by magnetic polaron formation. Susceptibilities and magnetization fluctuation amplitudes are known to diverge at second order magnetic phase transitions. The large susceptibilities favor polaron creation and carrier trapping. Spatial fluctuations of the exchange potential can scatter or localize carriers even when polaron formation is not possible. The divergences are suppressed by magnetic fields, that usually produce a negative magnetoresistance. It is important to realize that these conductivity effects are huge. For example, at the metal-insulator transition in EuO the resistivity changes by ten orders of magnitude in a small temperature interval. The mobility of Eul-,Gd,Se decreases by about a factor 1000 near its antiferromagnetic transition temperature. Magnetic fields in the 10T range increase the conductivity by comparable factors. Theorists have developed a variety of magnetic polaron models to explain these remarkable results. Kasuya and Yanase (1968; see also Yanase and Kasuya, 1968) attribute the unique magnetic and transport properties of dilute Eul-,Gd,Se and Eul-xLa,Se alloys to BMP formation at the Gd3+or
10.
THEORY OF BOUND MAGNETIC POLARONS
431
La3+ donors. Experimentally, the binding energy of the polaron is about 0.5 eV; this fact implies that its wave function is localized, mainly confined to the central Gd3+(or La3+)ion and its twelve nearest neighbor Eu2+ions. Kasuya and Yanase call this complex a “giant spin molecule”. It is rather different from the more extended BMP found in semimagnetics. In the latter, most of the BMP binding energy is provided by the Coulomb interaction with corrections due to exchange, whereas in the former exchange is dominant. Yanase and Kasuya (1968) calculate the magnetic energy, statistical weights, and magnetization of such BMP in detail, assuming tight binding wave functions. The spin problem can be solved exactly because there are only two exchange constants (Kj in our language)-that of the carrier with the central Gd3+ ion, and the carrier-nearest neighbor Eu2+ coupling. In other respects, the analysis is similar to that of Part IV, Section 1 . It gives a good fit to the susceptibility data (Holtzberg et al., 1964, 1965) with reasonable choices of the exchange constants, and also explains the resistivity maximum (von Molnar and Methfessel, 1967) as a transition from band conduction to hopping conduction between impurities. Though fluctuation effects are implicit in the theory, they are not readily apparent from the data, nor are they emphasized by Yanase and Kasuya. Unfortunately, there is no measurement that gives microscopic information concerning fluctuations, comparable to that provided by SFRS in n-type semimagnetics, for the BMP in EuI-,Gd,Se or Eul-xLa,Se. Development of such experiments could elucidate a fascinating, tightly-coupled BMP complex. The BMP in Eu-rich EuO are less well understood. In that case, it is believed (Oliver et al., 1970, 1972) that oxygen vacancies which bind two electrons are responsible for doping of the crystal. For impurity concentrations in the 1-5 x lOI9 cm-3 range, there is an abrupt transition from band conduction to an activated conductivity regime. The transition occurs at T = 50K, below the Curie point (69 K) of the undoped material. The resistivity for T < K can exceed that for T > Tt by factors larger than 10”. Transport experiments show that the resistivity change is caused by a large decrease in free carrier concentration. Theories and models of the metal-insulator transition (Torrance et al., 1972; Leroux-Hugon, 1972; Nagaev and Grigin, 1974; Kubler and Vigren, 1975; Leroux-Hugon, 1976; Kuivalainen et al., 1979; Mauger, 1983) agree in ascribing it to electron localization via BMP formation at oxygen vacancies. However, the detailed electronic structure of these centers is still in doubt. It is not yet known, for example, whether the ground state is a singlet or a triplet. Analysis of the structure of the BMP is hindered by the exceedingly complicated, many-body nature of the metal-insulator transition.
438
P . A. WOLFF
8. OPTICAL EVIDENCE FOR BMP
IN
SEMIMAGNETICS
Optical experiments have given the most convincing evidence for the existence of BMP in semimagnetic semiconductors. Two techniques are used: spin flip Raman scattering and luminescence. When observable, SFRS gives detailed information concerning the BMP whose interpretation is relatively straightforward since the center under study involves a single carrier plus associated Mn2+ spin cloud. Luminescence lines that exhibit BMP effects are, in contrast, the result of recombination by complexes, such as excitons bound to neutral impurities (a three body problem) or donor-acceptor pairs (whose energy of emission depends on the separation of donor and acceptor). Neither spectrum is easy to interpret. Unfortunately, to date SFRS has only been seen in donor-BMP where polaron effects are weak. The observation of SFRS by more strongly coupled acceptor-BMP may be precluded, in cubic crystals, by valence band degeneracy. Here, nature conspires against us since the only wide gap semimagnetic that naturally occurs p-type is cubic p-CdMnTe; (CdMn)S and (CdMn)Se, with simpler, wurtzite valence band structures, are invariably n-type. The luminescence spectra of lightly-alloyed Cdl-,Mn,S, Cdl-,Mn,Se, and Cdl-,Mn,Te crystals with x I0.05 contain features that are similar, though less sharp, to those observed in the parent binary compoundsincluding free exciton lines, bound exciton lines, and donor-acceptor pair (DAP) lines. The bound exciton lines are generally those of excitons bound to neutral acceptors (AOX) or neutral donors (D'X). These identifications are made by comparing alloy luminescence spectra to that of the binary. In Cdl-,Mn,Te, for example, the spectra contain a prominent line that continuously evolves (Plane1 et al., 1980) into the known bound exciton (A'X) line of pure CdTe as x -,0. Thus, in the dilute alloys, at least, one can be sure that this feature is caused by A'Xrecombination. Its position determines the binding energy of the exciton to the neutral acceptor. Measurements of this energy were performed by Golnik et al. (1980). They found, in contrast to the CdTe case, that in the semimagnetic alloys, the A'X binding energy is a rapidly varying function of temperature; at low temperatures, it exceeds that in CdTe by a sizable factor. The data are shown in Fig. 1. Large temperature variations of impurity energies are not ordinarily observed in semiconductors. Golnik et al. attribute them to BMP formation around the bound exciton complex, and give a qualitative interpretation of the results. This important experiment was one of the first to give convincing evidence for BMP in semimagnetics. Unfortunately, the complexity of the three body problem has so far prevented the development of a quantitative theory of the AOX-BMP.
There have also been studies (Huber et al., 1983) of D'X luminescence in
10.
THEORY OF BOUND MAGNETIC POLARONS
,
C31-, Mn,Te
o
x
0 I
0
1
439
l
=0.05
I
z0.2 zo.3
i
aa
a 4
30
10
I
I
I0
I
I
30 T(K)
I
50
FIG. 1 . Variation of AoX binding energy with temperature in (Cd,Mn)Te alloys. Solid line: theory of GGNPB; dashed line gives AoX binding in pure CdTe. [After Golnik et al. (1980).]
(CdMn)Se. Here, again, one faces a three body problem in interpreting the data. Experimentally, (Cdl-,Mn,)Se samples with x = 0.05 have a DoX binding energy larger than that of CdSe, and nearly temperature independent. The increased binding is attributed to carrier spin alignment in the exchange field caused by Mn2+ spin fluctuations. On the other hand, in crystals with x = 0.10 the DoX spectrum abruptly shifts and broadens as temperature is lowered from T = 10 K to 2 K, an unexpected result ascribed to polaron formation. The complexity of the AoX and DoX centers has forced investigators to seek luminescence techniques for studying isolated donor- or acceptorBMP. That goal can be achieved via DAP luminescence, though at a considerable experimental price. In DAP emission the energy of radiation from a given donor-acceptor pair varies, because of the residual Coulomb interaction between their charges, with the distance R, between them. The complete DAP line is the superposition of radiation from all pairs. Usually it is too broad to give information concerning BMP formation. This convolution can be unfolded, experimentally, by studying DAP spectra as a function of delay time after pulsed excitation. The effective distance at
440
P. A. WOLFF
which pairs recombine is related to the delay time (t) by the approximate expression:
e2 R ( t ) = -ln(t/to), EO ED
(72)
where ED is the donor binding energy and to = 10-”-lO-’o sec. Nhung and Plane1 (1983) have performed a remarkable series of measurements of this type in CdMnTe, inferred acceptor-BMP binding energies from them, and interpreted the results via the DS theory. We will discuss their work in some detail in Part VIII, Section 4. SFRS is the other important optical method for studying BMP. In the (CdMn) chalcogenides, it was first studied by Nawrocki et al. (1980). Their observation of a finite, zero-field spin splitting was unambiguous evidence for BMP effects. It motivated recent theoretical work on BMP and stimulated further experiments. We will see below that the SFRS data clearly demonstrate the role of fluctuations in BMP. 9.2 ANALYSIS OF SPINFLIP RAMAN SCATTERING EXPERIMENTS
SFRS in wide gap semimagnetics has been observed in CdMnSe, CdMnS, CdMnTe, and ZnMnSe. In most cases, the scattering is caused by electrons bound to donors. The spin flip line is identified by its variation with magnetic field; typical data showing that variation in (CdMn)Se are illustrated in Fig. 2. The large initial slope and large saturation value of the spin splitting imply that electrons in CdMnSe have g-values at low temperatures in excess of 100, as compared to a g* below two in CdSe. Figure 3, which shows the low field data on an expanded scale, clearly indicates a finite spin flip frequency in zero field, whose value is nearly independent of temperature. This effect is unique to semimagnetics. In conventional semiconductors the spin flip Raman frequency shift varies linearly with field near BO = 0. The existence of this splitting implies that electrons in donor-BMP feel an internal field, due to Mn2+spin fluctuations, even when BO = 0. Though, on average, the Mn2+ magnetization within the donor orbit vanishes, it does so by fluctuating in time about zero. The electron spin, whose relaxation is rapid, aligns with the fluctuating Mn2+ magnetization to produce a net spin-spin correlation responsible for the zero field splitting. To compare SFRS data with theory, it is necessary to perform the integrals in Eq. (71). The result depends on the geometry of the experiment (through the vector a)and is complicated. However, it has simple limits for large and small Bo.They are
10.
THEORY OF BOUND MAGNETIC POLARONS
441
These formulas interpolate smoothly into one another, and the latter can be shown to give a good approximation to the spectrum over the whole field range below saturation (0 < b~~ < k T ) . Equation (74) is essentially the formula used by DS [their Eq. (3.25)] in analyzing.spin flip spectra.
FIG.2. Variation of spin-flip energy with magnetic field in (Cd,Mn)Se. [After Heiman et al.
(1 983 b).]
442
P. A. WOLFF I
>
0
T=l 9 K
0
3 4 69 I28
x A
0
0
I
2
I
I 4
6
+ o 8
-
18 0 283 I0
B (kG) FIG. 3. Low field spin-flip energies in (Cd,Mn)Se. [After Heiman et ul. (1983b).]
The theoretical curves in Figs. 2 and 3 were calculated from Eq. (74). They illustrate the magnetic field variation of the Stokes peak of that spectrum. Figure 4 compares the complete spectrum at zero field [Eq. (73)] with experiment. The agreement is excellent in both cases. DS achieve comparable fits to the data of Nawrocki et al. (1980). The fits determine two parameters:
and the temperature TAFused to correct the Mn2+ susceptibility for next nearest neighbor interactions (Part V, Section 2). For Cdo.~oMno.loSe,the parameters are WO= 0.66 f 0.07 meV and T A =~ 2.3 K. The value of WO Calculated from Eq. (75) is WO= 0.56 f 0.06meV and is in reasonable agreement with experiment. At the lowest temperature ( T = 2 K) achieved in these measurements, ~ W =O4. The system is then just beginning to enter the polaron regime. At higher temperatures fluctuations control its behavior. In particular, the zero field splitting is temperature independent as implied by the discussion after Eq. (42). Temperature independent energy shifts are characteristic of the fluctuation regime.
10. THEORY OF
443
BOUND MAGNETIC POLARONS
C d , _ , Mnx S e
'
O
-
r
-
0
-
x = o 10
-
2
1
3
4
STOKES SHIFT ( m e V ) FIG. 4. Zero field SFRS spectrum of (Cd,Mn)Se. Points are experimental; solid line the theory. [After Heiman el al. (1983b).]
A further test of the theory is provided by studies of polarization ratios in SFRS. Two groups (Alov et al., 1983; Plane1 et al., 1984) have pointed out that, in a particular geometry, polarization of SFRS measures the fieldinduced spin alignment of BMP in wurtzite crystals. The appropriate configuration is one in which BolltllZn, the incident laser beam propagates normal to t with Eo 11 t,and one studies the circular polarization of scattered light propagating parallel to P. Since light polarized parallel to 2 does not couple to the A-valence band, the main channel for light scattering in this geometry is via virtual transitions to the B-valence band. This fact simplifies the analysis and provides optimal polarization signals. To estimate the polarization, we make use of Eq. (71). It can be shown that the vector a = (2i ij) for right- and left-circularly polarized scattered light. The angular integrals in the expression for the Stokes spectrum become:
"
x4
)at12
11
ly*a*(2
i y - ( a * x a**)
Y
Y2
--
+
where 6' is the angle between the vector y and the &axis. The polarization
444
P . A. WOLFF
ratio is:
Here the average is over the angular weighting function in Eq. (76). For example: ex@pdp=
sinh(x)
2( 4 X
9
where x = abed W?, and C(x) is the Langevin function. Similarly 1 2
- (1
+ cos2#) =
ex@(l+ p 2 )d p = X
Hence
Alov et al. (1983) have derived a formula of precisely this form, but with argument x = /3MeffBoin a more intuitive way, and shown that it fits the field dependence of their polarization data. The BMP moments (Meff) they infer vary, with SFRS frequency shift, from 2 5 , to ~ ~5 6 ~ This ~ . frequency variation agrees with that predicted by Eqs. (78)-(80), and thus fully accounts for the polarization data. The experiment also implies that the SFRS line is inhomogeneously broadened with different frequency shifts corresponding to BMP with different effective moments and internal fields. It is illuminating to briefly review the arguments of Alov et al. (1983) and Plane1 et al. (1984) leading to Eq. (80). Both groups emphasize that the donor-bound electron experiences an effective field that, because of Mn2+ spin fluctuations, need not be parallel to the applied field. As a consequence, the electron spin wave functions take the form: XT = cos(8/2))t) - sin(B/2)Jl), XJ = sin(8/2)lt)
+ cos(8/2)11),
(81)
where the spinors It), 1.1) are referred to the c-axis. These states are then used, in conjunction with B-valence band edge wave functions
10.
THEORY OF BOUND MAGNETIC POLARONS
445
to calculate spin flip matrix elements. One can easily show that the US,+1/2 channels generate negatively or positively polarized SFRS, with amplitudes cos2(O/2) or sin2(O/2). Equation (80) follows if it is assumed that the angular probability distribution has the classical form, e-’’BMeffBo. These polarization experiments clearly indicate that BMP have large ( 2 5 - 5 0 ~ ~moments ) associated with them, whose alignment by a field follows the classical law-but with a different effective moment at each point in the SFRS spectrum. Thus, in the fluctuation regime, where the spectrum is relatively broad [Eq. (43)], it is not correct to speak of the BMP moment as if it were a rigid, time invariant entity. Whenp WO4 1, each BMP can have a wide range of moments and gradually fluctuates, in time, over many values. The situation is different in the collective regime. When ~ W%-O1, the distribution of effective fields is sharply peaked about the value The BMP then has a well-defined moment, whose average alignment is truly described by the classical Langevin formula [Eq. (51)].
a.
10. ANALYSIS OF ACCEPTOR-BMP EXPERIMENTS
Acceptor-BMP are potentially more interesting than donor-BMP because they can exhibit the full range of possible BMP behavior, from the fluctuation-dominated regime at high temperatures to the fully saturated collective regime at low temperatures. In a typical wide gap semimagnetic crystal, the coupling constant for acceptor-BMP is WO= 6meV, as compared to 0.7 meV in the corresponding donor-BMP. Thus, in the acceptor case, the transition from the fluctuation-dominated regime to the collective regime occurs at T = 3 0 K , where the MnZ+ susceptibility is Curie-like and the theoretical ideas of Parts I11 and IV should apply. Unfortunately, it is not easy to study the properties of the simple acceptorBMP. SFRS of holes has not yet been seen in semimagnetics, and most luminescence features are produced by three-body complexes whose wave functions are exceedingly difficult to calculate. The donor-acceptor pair (DAP) line is an exception; it involves the recombination of electrons and holes bound to fairly well separated donors and acceptors. Moreover, since the coupling constant of acceptor-BMP is much larger than that of donorBMP, any polaron effects observed in DAP luminescence can be attributed to the acceptor.
446
P. A. WOLFF
The time resolved DAP luminescence technique, developed by Nhung and Planel (NP, 1983) has made possible direct optical measurements of the acceptor-BMP binding energy in CdMnTe. Their results for a series of alloys are shown in Fig. 5 . These data have two important features: (1) acceptorBMP binding energies are substantially larger, at high temperatures, than that of the acceptor in CdTe and (2) there is a gradual increase in binding energy with decreasing temperature. NP ascribe the high temperature energy difference to Mn2+ spin fluctuations and suggest that BMP formation is responsible for the increase in binding energy at lower temperatures. In their original publication, NP tested this picture with a calculation that made several assumptions concerning the acceptor-BMP, namely: (1) Mn2+ magnetization is a continuous function of position, M(r). (2) The hole wave function is hydrogenic. (3) Mn2+ magnetization is linear in effective field to saturation, and constant thereafter. (4) The magnetic entropy can be calculated in the classical, Gaussian approximation.
Since then two groups (Wolff and Warnock, 1984 and Nhung et al., 1984) have developed formalisms which do not require assumptions (l), (3), and
I0 010 A
8
0
400
0
I
10
I
20
T(K)
-
I
30
5 '10
2 010 0 '10
I
40
FIG.5. Acceptor-BMP energies vs. temperature in (Cd,Mn)Te alloys. Dashed line: theory of Nhung and Planel; solid line: theory of Warnock and Wolff. [After Nhung and Planel (1983).]
10. THEORY OF
441
BOUND MAGNETIC POLARONS
(4). The analysis parallels that of Part IV. For the valence band one postulates, in the single band hydrogenic approximation, a spin Hamiltonian of the form:
x = - J Cj [(s
*
Ci [ K j ( s
~ j ) ~ $ ( ~ j z= ) l~ ]
*
~ j ) ] .
(83)
where the hole spin s = and $(r) = e - ' / " / G is the acceptor wave function. The radius a is determined variationally. The spin partition function, in the classical approximation, is then: 2 = Tr {s.sjl
5
-1 - Tr
S[y -
S[y -
C (KjSj)]eacs")d3y
C (KjSj)][&20Y
+ e1/20Y + e-'/2@Y+ e - 3 / 2 8 Y ] d 3 y
{SJI
= ( 6 ) N j d '(2n)3 dd'yeiX.ynIss/z(-UKj)][e3/20y
+ e1/20Y + e-1/2@r + e-3/2i37
11.
(84)
This expression is the analogue, for a spin$ carrier, of Eq. (18). The integrals can again be evaluated as outlined in Part IV, Section 1. The result is:
a formula quite similar to Eq. (22). Wolff and Warnock (1984) calculated BMP energies from this expression by making the continuum approximation, though in principle the discrete sum could have been evaluated. Their results are indicated in Fig. 5 along with earlier calculations of NP. The only adjustable parameter (a = 13 A ) in this fit was determined from the acceptor binding energy measured by N P in CdTe. Other quantities @NO,X) are known from independent measurements. Both of the calculations illustrated in Fig. 5 imply that Mn2+ spins within the acceptor orbit are fully aligned, with the hole spin, at low temperatures. The net moment of the BMP is then . radius of the acceptor wave function could, in principle, about 20 p ~ The vary with temperature via the changing exchange interaction. However, this effect is found to be small. The very recent calculations of Nhung et al. (1985) are compared with their experiments in Fig. 6. These results were obtained by numerically integrating an expression for the internal energy derived from Eq. (85):
448
P. A. WOLFF
u=--a(ln z) aP = -
S ~ i ~ y y ~ ( y ) [ + e +~ /+e’/2’r ~ ’ y - )e-1/28r
- )e-3/28r]
jd3y~(y)[e3/2P +~e 1 / 2 6 ~+ e-l/Wr + e - 3 / 2 8 ~ ~
(86)
A finite number of Mn2+ions was randomly distributed throughout the BMP to give the correct x-value, and the sum over j was cut off at R, = 1.5a = 15 A. Physically, R , is the distance beyond which the hole exchange interaction is no longer strong enough to break up antiferromagnetic, Mn2+-Mn2+ spin alignments. Though the calculations outlined above differ in detail, they contain similar physics. It is remarkable, and gratifying, that first principles calculations can predict theEvs. Tcurve for a center as complicated as the acceptorBMP. The detailed shape of this curve is sensitive to the cutoff R,;in this regard, the Nhung et al. (1985) calculation is the more realistic of the two. Both calculations use hydrogenic wave functions for the acceptor. Errors in the binding energy caused by this approximation are minimized by choosing the acceptor radius to give the correct energy in CdTe. However, other properties of the acceptor-BMP may be poorly described by a wave function that does not incorporate the degeneracy of the valence band.
I
I
0
I
I
I
I
10
20
30
40
Temperature ( K ) FIG. 6. Comparison of theory and experiment for acceptor-BMP energies. The solid curve is the exact, numerical calculation; the triangles are experimental points for Cd0.9sMno.osTe.The dashed curve corresponds to a truncated Gaussian approximation. [After Nhung et al. (1985).]
10.
THEORY OF BOUND MAGNETIC POLARONS
449
It would be interesting to repeat the Nhung-Planel experiments in CdMnS, where the valence band exchange parameter, PNo, is reported to be 5 eV (Gubarev, 1981). This coupling is anomalously large compared to other semimagnetics, where PNo = 1 eV. The large interaction should produce striking polaron effects in CdMnS.
IX. Evidence for Free Magnetic Polarons in Semimagnetics Exciton luminescence in semimagnetic crystals with x < 0.05 is known to be caused by excitons bound to neutral impurities (DoXor AOX). Recently, Golnik et al. (1983) have done an extensive series of exciton luminescence experiments in Cdl-,Mn,Te samples with x ranging to larger values. They show that the luminescence spectrum changes dramatically between x = 0.05 and 0.10. The AoX emission, that dominates the spectrum for x < 0.05 becomes weak and broad for x > 0.10. A new line nearer the band edge, labelled L2, emerges at x = 0.05 and becomes the strongest feature in the spectrum for x 2 0.10. Golnik et al. suggest that the L2 line is caused by magnetically localized excitons. Localization could result from Mnz+ spin fluctuations, polaron formation, or a combination of the two effects. Several trends in the data suggest magnetic localization of excitons: (1) As temperature increases, the L2 line evolves into the free exciton line. This result is implied by Eq. (2), which shows that the binding energy of a free polaron decreases with increasing temperature. (2) With increasing magnetic field, which smooths out magnetic fluctuations, the L2 line evolves into the free exciton line. (3) Curves of L2 energy vs. temperature have considerable structure at the spin-glass transition in crystals with x = 0.2, 0.3, and 0.4. In addition, the magnetically localized exciton hypothesis can explain the different variations with temperature of the intensities of the L1 (AOX) and L2 line. These arguments are not conclusive, but make a persuasive case for free polarons. It is not easy to think of other mechanisms for the anomalies Golnik et al. observe. Other experiments also give hints of free polaron formation. Warnock et al. (1984) have seen large optical pumping signals by exciting CdMnSe and CdMnTe crystals with a tunable laser whose energy was a few meV below the free exciton, but little polarization with higher energy excitation. Their measurements suggest that the spins of free excitons relax rapidly, by exchange scatterings, whereas those of excitons created in magnetic fluctuations are prevented, energetically, from relaxing by the sizable local exchange field. They infer, from the energy of the subsequent Stokes-shifted radiation, that the magnetic spin “trap” is most probably a free polaron.
450
P. A. WOLFF
This interpretation is supported by a magnetic field experiment, which narrows the exciton luminescence line by a factor of six, and shifts it towards the free exciton. The stability of free polarons can be tested with the analysis of Kasuya et al. (1970).As they indicate, the polaron becomes stable when its free energy vanishes: AF=
[
2mh:a2
]
35 R(JNol2 = o. 48 (8na3No)kT
Here we have used Eq. (40),in the collective limit, to evaluate the exchange energy where the wave function has been assumed hydrogenic. Equation (87) for AF remains valid until Mn2+ spin saturation occurs. That condition is not precisely defined since Mn2+spins at different positions in the BMP orbit saturate at different temperatures. Roughly speaking, however, we may consider saturation to occur when PKj > 1 for a Mn2+ ion at the center of the BMP. To avoid saturation we require:
If the radius a is eliminated from these equations, one obtains the inequality:
Note that this criterion, for the stability of free polarons, is quite sensitive to temperature and x-value. Electrons have small masses and small exchange constants (JNo= 0.25 eV). For such carriers, Eq. (89) cannot be satisfied at any temperature above T = 1 K. Thus, free electrons do not form polarons in the known semimagnetics. Holes (or excitons) have larger exchange constants and larger masses; both factors favor polaron formation. With (JNo) = 1 eV, m* = 1, and T = 10 K, Eq. (89) is satisfied for R > 0.05. This critical value of R agrees surprisingly well, considering the crudeness of the approximations, made in deriving Eq. (89), with the R determined by Golnik et al. from the L1 -,L2 crossover of the luminescence spectra. We conclude, therefore, that there is considerable circumstantial evidence for free magnetic polarons, and that the evidence is not inconsistent with the criterion of Kasuya et al. for their stability.
10.
THEORY OF BOUND MAGNETIC POLARONS
451
X. Conclusion
11. CURRENT STATUS OF BMP THEORY The theory outlined in Parts IV and V predicts the equilibrium properties
of BMP in detail. Excellent agreement with experiment confirms the Dietl-
Spalek picture of their behavior, whose key feature is the gradual transition from a high temperature, fluctuation-dominated regime to a low temperature, collective one. The fluctuation regime is exemplified by donor-BMP and has been thoroughly studied via SFRS in CdMnS and CdMnSe. On the other hand, only BMP involving holes have exchange interactions strong enough to produce the complete range of possible BMP effects, culminating in spin saturation at low temperatures. Unfortunately, the centers containing holes that are easy to study experimentally (AOX,D o X ) involve three carriers-two holes and one electron or vice versa. To date, the complexity of the resulting theoretical problem has prevented quantitative comparisons of theory with experiment for such complexes. A theoretical advance in this direction could provide several interesting, new tests of the BMP model. An exception to these statements is the elegant, but complicated, time resolved DAP luminescence technique of Nhung and Plane1 (1983). Their measurements determine binding energy versus temperature for the simple acceptor-BMP in Cdl -xMnxTe,and clearly demonstrate BMP formation at low temperatures. Though theory, including the effects of spin saturation, gives energies in surprisingly good agreement with their measurements, there are still some uncertainties in the analysis. The DS and HWW theories of BMP use a classical approximation to calculate Mn2+spin statistics. Carrier spins are assumed to follow the slowly varying internal field of the Mn2+ions. This approach accurately predicts the equilibrium properties of BMP, but is inadequate for treatment of Mn2+or carrier spin relaxation processes. 12. DIRECTIONS FOR FUTURE WORK
Recent advances in our understanding of polaron structure and energetics have begun to stimulate interest in other properties of polarons in semimagnetic semiconductors, including: (i) Spin dynamics. (ii) Free polarons. (iii) Polarons in the spin-glass regime. Harris and Nurmikko (1983) have used picosecond optical techniques to explore the polaron formation process in CdMnSe. They observe a sizable frequency shift of the DoX luminescence feature about 400 psec after
452
P. A. WOLFF
excitation, and propose that this delay is the BMP formation time. Optical pumping experiments (Warnock et af., 1984) show that free carrier spin relaxation rates are much faster than those of polarons in CdMnSe. These preliminary, but tantalizing, results suggest a thorough study of BMP spin dynamics; the problem seems naturally suited to recently-developed picosecond optical spectroscopy techniques. On the theoretical side, a fully quantum mechanical theory of BMP is required. The work of Yanase and Kasuya (1968) suggests that such a theory may not be inordinately difficult, since they have rigorously solved a quantum mechanical BMP problem with spatially varying exchange constants. Hopefully, a complete quantum mechanical theory will clarify the roles of spin relaxation and spin diffusion in the BMP formation process. The luminescence studies of Golnik et af. (1983) demonstrate a dramatic change in the spectrum of Cdl-,MnxTe as x increases beyond x = 0.07. The properties of the new L2 feature they observed are consistent with those expected from recombination of a magnetically self-trapped exciton. That interpretation is supported by optical pumping experiments (Warnock et al., 1984). However, further measurements to confirm this hypothesis , and better theoretical estimates of the free polaron energy, are needed. In particular, for crystals with larger x-values, competition between the strong carrier-Mn2+ exchange interaction, which tends to align Mn2+spins parallel to that of the carrier, and the direct Mn2+-Mn2+interaction which favors antiferromagnetic or spin-glass alignment, will play an important role in determining the polaron binding energy. Cdl-xMnxTecrystals with x 5; 0.2 have a spin-glass phase at low temperatures. Golnik et al. (1983) see substantial structure, in their measurements of the energy of the L2 feature versus temperature, at temperatures near the spin-glass transition in samples withx = 0.2,0.3, and 0.4.These remarkable observations imply that a localized exciton, whose center-of-mass wave function may be no larger than 1OA in radius, can sense the Mn2+spin-glass phase. It would be interesting to exploit this opportunity, by using the strong (exchange fields approaching 100T), localized magnetic probe that nature here provides us, as a tool to study the spin-glass phase in semimagnetics. References Aggarwal, R . L., Jasperson, S., Shapira, Y . , Foner, S., Sakikabara, S., Goto, T., Miura, N., Dwight, K., and Wold, A. (1985). Proc. XVII Intl. Conf. Phys. of Semiconductors, Sun Francisco, I984 (J. D. Chadi and W. A. Harrison, eds.), p. 1419. Springer. Ahn, K., and Shafer, M. (1970). J. Appl. Phys. 41, 1260. Alov, D., Gubarev, S., Timofeev, V., and Shepel, B. (1981). JETP Letfers 34, 71. (Pis’ma v Zh. Eksp. Teor. Fiz. 34, 76.)
10.
THEORY OF BOUND MAGNETIC POLARONS
453
Alov, D., Gubarev, S., and Timofeev, V. (1983). Sov. Phys. JETP57, 1052. (Zh. Eksp. Teor. Fiz. 84, 1806.) Busch, G., Junod, P., and Wachter, P. (1964). Phys. Letters 12, 11. Busch, G., and Wachter, P. (1966a). Phys. Condensed Matter 5, 232. Busch, G., Magyar, B.. and Wachter, P. (1966b). Phys. Letters 23, 438. Dietl, T., and Spalek, J. (1982). Phys. Rev. Letters 48, 355. Dietl, T., and Spalek, J. (1983). Phys. Rev. B28, 1548. Douglas, K., Nakashirna, S., and Scott, J. (1984). Phys. Rev. B29, 5602. Gaj, J. (1980). Proc. X V Intl. Conf. Phys. of Semiconductors, Kyoto; J. Phys. SOC. Japan, Suppl. A49, 797. Galazka, R., Nagata, S., and Keesom, P. (1980). Phys. Rev. B22, 3344. Golnik, A., Gaj, J., Nawrocki, M., Planel, R., and Benoit a la Guillaume, C. (1980). Proc. X V Intl. Conf. Phys. of Semiconductors, Kyoto; J. Phys. SOC. Japan, Suppl. A49, 819. Golnik, A,, Ginter, J., and Gaj, J. (1983). J. Phys. C16, 6073. Gubarev, S. (1981). Sov. Phys. JETP53, 601. (Zh. Eksp. Teor. Fiz. 80, 1174.) Harbeke, G., and Pinch, H. (1966). Phys. Rev. Letters 17, 1090. Harris, J., and Nurrnikko, A. (1983) Phys. Rev. Letters 51, 1472. Heikes, R., and Chen, C. (1964). Physics 1, 159. Heiman, D., Shapira, Y., and Foner, S. (1983a). Solid State Commun. 45, 899. Heiman, D., Wolff, P., and Warnock, J. (1983b) Phys. Rev. B27, 4848. Heiman, D., Shapira, Y., and Foner, S. (1984). Solid State Commun. 51, 603. Holtzberg, F., McGuire, T., Methfessel, S., and Suits, J. (1964). Phys. Rev. Letters 13, 18. Holtzberg, F., McGuire, T., Methfessel, S., and Suits, J. (1965). Proc. Intl. ConJ Magnetism, Nottingham (The Institute of Physics and the Physical Society, London, 1965). Huber, C., Nurmikko, A., Gal, M., and Wold, A. (1983). Solid State Commun. 46, 41. Kasuya, T., and Yanase, A. (1968). Rev. Mod. Phys. 40, 684. Kasuya, T., Yanase, A., and Takeda, T. (1970). Solid State Commun. 8, 1543. Kreitman, M., Milford, F., Kenan, R., and Daunt. J. (1966) Phys. Rev. 144, 367. Kiibler, J., and Vigren, D. (1975). Phys. Rev. B11, 4440. Kuivalainen, R., Sinkkonen, J., Kaski, K., and Stubb, T . (1979). Phys. Stat. Solidi(b)94, 181. Landau, L., and Lifshitz, E. (1977). Statistical Physics (Pergamon Press, New York), Chap. 12. Leroux-Hugon, P. (1972). Phys. Rev. Letters 29, 939. Leroux-Hugon, P . (1976). J. Magn. Magn. Muter. 3, 165. Mauger, A. (1983). Phys. Rev. B27, 2308. Nagaev, E., and Origin, A. (1974). Phys. Stot. Solidi (b)65, 457. Nagaev. E . (1983). Physics of Magnetic Semiconductors (MIR Publishers, Moscow). Nagata, S., Galazka, R., Mullin, D., Akbarzadeh, H., Khattak, G., Furdyna, J., and Keesom, P. (1980). Phys. Rev. B22, 3331. Nawrocki, M., Planel, R., Fishman, G., and Galazka, R. (1980). Proc. X V I n t l . Conf. Phys. of Semiconductors, Kyoto; J. Phys. SOC.Japan Suppl. A49, 823. Nawrocki, M., Planel, R., Molot, F., and Kozielski, M. (1984). Phys. Stat. Solidi (b)123, 99. Nhung, T., and Planel, R. (1983). Proc. XVI Intl. Conf. of Semiconductors, Montpellier; Physica 117B-l18B, 488. Nhung, T., Planel, R., Benoit A la Guillaume, C., and Battacharjee, A. (1985). Phys. Rev. B31, 2388. Oliver, M., Kafalas, J., Dirnmock, J., and Reed, T. (1970). Phys. Rev. Letters 24, 1064. Oliver, M., Dimmock, J., McWhorter, A., and Reed, T. (1972). Phys. Rev. B5, 1078. Peterson, D., Petrou, A., Datta, M., Ramdas, A., and Rodriguez, S. (1983). Solid State Commun. 43, 667.
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Planel, R., Gaj, J., and Benoit a la Guillaume, C. (1980). Proc. ist Intl. Meeting on Magnetic Semiconductors, Montpellier; J. Phys. CON.41, C5-39. Planel, R. (1982). Proc. XVI Intl. Conf. on Appl. of High Magnetic Fields in Semiconductor Physics, Grenoble; Lecture Notes in Physics 111, p. 309. Springer, Berlin. Planel, R., Nhung, T., Fishman, G., and Nawrocki, M. (1984). J. de Physique 45, 1071. Romestain, R., Geschwind, S., and Devlin, G. (1975). Phys. Rev. Letters 35, 803. Ryabchenko, M., and Semenov, Yu. (1983). Sov.Phys. JETP57, 825. (Zh. Eksp. Teor. Fiz. 84, 1419.) Schoenes, J., and Wachter, P . (1974a). Phys. Rev. B9, 3097. Schoenes, J., Wachter, P., and Rhys. F. (1947b). Solid State Commun. 15, 1891. Shapira, Y., Foner, S., Ridgley, D., Dwight, K., and Wold, A. (1984) Phys. Rev. B30,4021. Spalek, J. (1980). J. Mag. Magn. Muter. 15-18, 1289. Torrance, J., Shafer, M., and McGuire, T . (1972). Phys. Rev. Letters 29, 1168. Tu, K., Ahn, K., and Suits, J. (1972). IEEE Trans. on Magnetic 8 , 651. von Molnar, S., and Methfessel, S. (1967). J. Appl. Phys. 38, 959. Warnock, J., Heiman, D., Wolff, P., Kershaw, R., Ridgley, D., Dwight, K., Wold, A., and Galazka, R. (1985). Proc. XVII Intl. Conf. Phys. of Semiconductors, Sun Francisco ( J . D. Chadi and W. A. Harrison, eds.), p. 1407. Springer. Wolff, P., Ramos, J., and Yuen, S. (1977). Theory of Light Scattering in Condensed Matter, edited by Bernard Bendow, Joseph L. Birman, and Vladimir M. Agranovich (Plenum, New York). Wolff, P., and Warnock, J. (1984). J. Appl. Phys. 55, 2300. Yanase, A. (1972). Internat. J. Magnet. 2, 99. Yanase, A., and Kasuya, T. (1968). J. Phys. SOC.Japan 25, 1025.
Index A
B
Acceptors, see also Valence band, in DMS alloys acceptor-bound magnetic polaron, in Cd,-,MnxTe, 445-449 luminescence, from acceptor-donor pairs, 439-440 resonant acceptor states, 312 shallow acceptors, 316-322 Hamiltonian, 3 16-318 variational treatment, 3 18-322 Cdl -,Mn,Te, acceptor parameters, 321-322 Hgl-,Mn,Te, acceptor parameters, 322 Acceptors, in magnetic field, see also Valence band, in DMS alloys narrow-gap DMS alloys, 325-333 arbitrary magnetic fields, 329-330 binding energy, Hgl-,Mn,Te, 330-333 hopping conduction, p-type Hgl-,Mn,Te, 333-339 magnetic boil off, 327-328 resonant states, zero-gap DMS alloys, 338-342 ultra-quantum limit, 325-329 wide-gap DMS alloys Cdl-,Mn,Te, 325 spin splitting, exchange-induced, 322-325 Antiferromagnetic order, see also Correlation length, magnetic; Exchange interaction, d-d; Neutron scattering, magnetic; Short-range order, magnetic in FCC lattices, 136 long range, absence of in Cdl-,Mn,Te, 163 in Znl-xMnxTe, 147 short range in Cdl-,MnxTe, 159-173 in Znl-,MnxTe, 141-155
Band structure, see also Energy gap; Exchange interaction, sp-d; Overlap of bands, exchange-induced; Spin splitting; Valence band, in DMS alloys anisotropy of valence band, exchangeinduced, 292-293, 315, 318-320, 327 band parameters effective mass, in narrow-gap DMS alloys, 237, 246 g-factor, effective, 202, 221, 224, 237, 246, 253, 428 for Hgl-,MnxSe, 253, 263, 267-270 for Hgl-,MnxTe, 246, 267-270 momentum matrix element, 196, 268-269, 292 for 2-dimensional electron gas, 220-224 narrow-gap DMS alloys, in magnetic field Landau levels, 199-204, 206-207, 247, 251, 266 Pidgeon-Brown model, 193-199, 233-237 quasi-Ge model, 192 three band model, 192 narrow-gap DMS alloys, in zero magnetic field, near r-point, 190-191, 232-233 wide-gap DMS alloys, in magnetic field, 288-295, 313-315 wide-gap DMS alloys, in zero magnetic field, 36-39 near r-point, 38-39 away from zone center, 37, 56, 305 Bonding, tetrahedral, 4, 6 Bond length A"Bvl compounds, 7 ternary DMS alloys, 11-14 tetrahedrally bonded MnBV' compounds,
455
11
456
INDEX
Bound magnetic polaron acceptor-bound polaron, 445-449 in Cdl-,MnxSe, 405, 440-445 in Cdl-,Mn,Te, 399, 404, 446-449 Dietl-Spalek theory, 401-403, 430-432 donor-bound polaron, 398-405, 440-445 effect on exciton spin splitting, 297-299 in Eul-,Gd,Se, Eu~-,ta,Se, 436 in EuO, 437 Hartree approximation, 418-419 internal energy, 425-428 magnetization, 428-430 partition function, 420-424 Raman scattering, spin-flip, 398-405, 440-445
soluble model of, 424, 425 spin fluctuation effects, 299, 401, 425, 439,444-445
spin-spin correlation in, 430, 433-435 C
Cation-cation distance, see Lattice parameters (Cd1-,MnJ3Asz, 3 1 exchange integral, sp-d interaction, 215 Shubnikov-de Haas effect, 215 Cdl-,Mn,S crystal growth, 20-21 crystal structure, 3, 7-10, 79 energy gap, 46-47, 55 exchange integral, d-d interaction, 100 exchange integral, sp-d interaction, 295 exciton, spin splitting, 285 lattice parameter, 8-10 magnetic phase diagram, 104 Mn d-electron transitions, 60-62 spin-flip Raman scattering, 417 Cdl -,Mn,Se bound magnetic polaron, 405, 440-445 crystal growth, 19-20 crystal structure, 3, 7-10, 79 Curie-Weiss temperature, 89 electron paramagnetic resonance, 80, 84 energy gap, 44-45, 55 exchange integral, d-d interaction, 100 exchange integral, sp-d interaction, 295 exciton, spin splitting, 285 free magnetic polaron, 449-450 lattice parameter, 8-10
magnetic phase diagram, 104, 106 magnetic susceptibility, 92 Mn d-electron transitions, 60-62 specific heat, 109, 111 spin-flip Raman scattering, 403, 405, 414, 417, 440-445
spin-glass transition, 89, 94, 97, 104 Cdl-,Mn,Te acceptors, in magnetic field, 325 antiferromagnetic short range order, 159-173
correlation length, 164-170 band edge, spin splitting, 277, 280-281, 293-294
bound magnetic polaron, 399, 404, 446-449
crystal growth, 17-19 crystal structure, 3, 7-10, 79 Curie-Weiss temperature, 88 electron paramagnetic resonance, 80, 82-86
energy gap, 40, 44, 52 EXAFS, 11-13 exchange integral, d-d interaction, 100, 168, 177-178
exchange integral, sp-d interaction, 294-295
exciton, spin splitting, 281, 283-284, 294, 300
Faraday rotation, 276, 297 free magnetic polaron, 449-450 lattice parameter, 8-10 magnetic phase diagram, 104-105 magnetic susceptibillity, 91 magnons, 173-177, 390-396 Mn d-electron transitions, 59-61 Mn++ pairs, 388-389 neutron scattering, 155-172 inelastic, 155, 173-177 phonons, 365-373 specific heat, 107-108 spin dynamics, 173-177 spin-flip Raman scattering, 397-401, 404
spin-glass transition, 88, 94, 97, 105 superlattice, 28, 406-409 twinning, 156, 165, 175 Chalcopyrite DMS alloys, 31 Correlation length, magnetic, 133 anisotropy, 135, 164-169
457
INDEX
dependence on composition Cdl-,Mn,Te, 167 Zn,-,Mn,Te, 150-155 temperature dependence Cdl-,MnrTe, 170 Znl -xMnr.Te, 150- 155 Covalent radii of elements, 6, 11 Crystal structure, A:L,MnxBV1 alloys, 3-4, 7-14, 79
Curie-Weiss temperature, 88-89, 99-100 Cyclotron resonance, see Magnetooptical transitions, intra-band
D Debye temperature, 107 Diamagnetic susceptibility, A"BV' compounds, 91, 98 Dingle temperature, 209, see ufso Shubnikov-de Haas effect Donor-acceptor pair luminescence, 439-440
E Electron paramagnetic resonance, in DMS alloys, 80-90, see also Raman scattering, magnetic excitations in Cdl-,Mn,Se, 84-85, 89 in Cdl-,Mn,Te, 82-86, 88-90 g-factor, of Mn++ ions, 82 internal field, 82, 89 linewidth, effect of d-d exchange interaction, 82-90 in quaternary DMS alloys, 90 Energy gap bowing effects, 38-39 Cdl-,Mn,S, 47 sp-d exchange contribution, 39, 47-49 Znl-xMn,Se, 47, 49 Znl-,MnxTe, 48 concentration dependence, 41, 42 Cdl-,Mn,S, 46, 47 Cdl-xMnxSe, 44-45 Cdl-,Mn,Te, 40, 44 Hgl-,Mn,Se, 268 Hgl-,Mn,Te, 192, 267 Znl-xMnxSe, 47-49 Znl-,Mn,Te, 48
MnB"' compounds, 43 pressure dependence, 51-53 Cdl-,Mn,Te, 52 Znt-,MnxSe, 56, 68-69 Znl-xMn,Te, 55 temperature dependence, 50-51 Cdl-,Mn,S, 55 Cdt-,Mn,Se, 55 Cdl-xMn,Te, 52 Znl -xMnxSe, 56 Znl-,Mn,Te, 55 Epitaxy, A:'_,Mn,Bv' alloys, 28 EXAFS, in ternary DMS alloys, 11-13 Exchange integral, nearest-neighbor d-d interaction Cdl-,Mn,S, 100 Cdl-,MnxSe, 100 Cdl-,Mn,Te, 100, 168, 177-178 Hgl-,Mn,S, 100 Hgl-,MnxSe, 100 Hgl-,Mn,Te, 100 Znl.,Mn,S, 100 Znl-,Mn,Se, 100 Znl_,Mn,Te, 100, 154 Exchange integral, sp-d interaction, 188-198, 296, 375 (Cdl-,Mnx)3As2, 215 Cdl-,Mn,S, 295 Cdl-,Mn,Se, 295 Cdl-,Mn,Te, 294-295 definition, 188, 198, 235, 291, 296, 314 Hgl-,Mn,Se, 213-214, 270 Hgl-,Mn,Te, 213, 270 Pb1-,MnxBV' alloys, 217 physical interpretation, 188-189 Znl-,MnxSe, 295 Znl-,Mn,Te, 295 Exchange interaction, d-d, see also Antiferromagnetic order; Exchange integral, nearest-neighbor d-d interaction; Raman scattering, magnetic excitations; Spin-glass transition Bloembergen-Rowland mechanism, 80, 105, 107, 271
Curie-Weiss temperature, 99-100 effective Mn concentration, 101-103, 417-41 8
electron paramagnetic resonance, effect on, 81-85, 88-90
458
INDEX
Hamiltonian, 86 for specific cluster forms, 78 interaction beyond nearest neighbor, 77, 103, 118 superexchange, 13, 80, 84 Exchange interaction, ion-carrier, see Exchange interaction, sp-d Exchange interaction, sp-d, see also Spin splitting; Bound magnetic polaron Hamiltonian, 185-186, 197, 234-235, 288-289, 313-314, 375, 378, 416-417, 420 basis vectors, 193, 196, 234, 290, 313 diagonalization, 194-198, 235-236, 290-291, 314 eigenvalues, 197-198, 235-236, 290-293, 314-315 Pidgeon-Brown model, modification of, 197-199, 233-237 valence band, effect on, 292-293, 313-318 Exchange splitting, see Exchange interaction, sp-d; Spin splitting Excitons, in magnetic field in Cdl-,MnxS, 285 in Cdl-,Mn,Se, 285 in Cdl-,Mn,Te, 281, 283-284, 294, 300 complexes, in Cdl-,Mn,Te, 303-305 excited states, 300-303 luminescence, in Cdl-,Mn,Te, 303-305 magnetic polaron, effect of, 297-299 selection rules, magnetooptical transitions, 283, 294-295 from spin-orbit split-off valence band, 287 spin splitting, ground state, 277, 279, 283, 286, 293-296 in Znl-,Mn,Se, 283 in Znl-xMnxTe, 283 Extended x-ray absorption fine structure, 11-13
F Faraday rotation Cdl-,MnxTe, 276, 297 experimental technique, 277-279 Hgl-,Mn,Te, 271
magnetic properties, measurement of, 271-272, 276 theoretical model, 296-297 Fe-based DMS, 28-30, 254-257 Free magnetic polaron, 415-416, 449-450
G g-factor, effective, 202, 237, 428 Hgl-,Mn,Se, 253 Hgl-,MnxTe, 246 2-dimensional electron gas, 221, 224
H Hgl-,-,Cd,Fe,Se, 29-30 Hg~-,-,Cd,Mn,Te magnetooptical studies, 260-261 spin-glass transition, 27 1 thin film, 28 2-dimensional electron gas, 217, 219, 221 Hgl-,FexSe, 29-30 Hgl-,Fe,Te energy gap, 254 magnetooptics, 254-257, 270 Van Vleck paramagnetism, 256-257 Hgl-,Mn,S crystal growth, 27 crystal structure, 3, 7-10, 79 exchange integral, d-d interaction, 100 lattice parameter, 8-10 Hgl-,Mn,Se band parameters, 253, 263, 267-269 combined resonance, 264 conduction-valence band overlap, exchange-induced, 21 1-212 crystal growth, 26-27 crystal structure, 3, 7-10, 79 energy gap, 268 exchange integral, d-d interaction, 100 exchange integral, sp-d interaction, 213-214, 270 g-factor, 253 lattice parameter, 8-10 magnetic breakdown, 210-21 1 magnetic phase diagram, 104 momentum matrix element, 269 Shubnikov-de Haas effect, 204-21 1 spin splitting of Landau levels, 205
459
INDEX
Hgl-xMnxTe acceptors, in magnetic field, 330-333 band parameters, 246, 267-269 band structure, near r-point, 191-192, 233, 267 combined resonance, 258-264 conduction-valence band overlap, exchange-induced, 199-200, 21 1-212 crystal growth, 25-26 crystal structure, 3, 7-10, 79 cyclotron resonance, 264 effective mass, 246 energy gap, 192, 267 exchange integral, d-d interaction, 100 exchange integral, sp-d interaction, 213, 270 Faraday rotation, 271 g-factor, 246 hopping conduction, 333-338 lattice parameter, 8-10 magnetic phase diagram, 104, 106, 272 magnetic susceptibility, 92 momentum matrix element, 268 negative magnetoresistance, 333-338 nonmetal-metal transition, 338 resonant acceptor states, 339-342 shallow acceptors, 322 Shubikov-de Haas effect, 204-209 spin-glass transition, 95, 106, 272 spin splitting of Landau levels, 199-206 stimulated spin-flip Raman scattering, 266 2-dimensional electron gas, 217-219 Hopping conduction, in magnetic field in Hgl-,Mn,Te, p-type, 333-339 magnetoresistance, giant negative, 334-339 anisotropy of, 336-337
I Impurity band conduction, see Hopping conduction
K Kerr effect, magnetooptical, 278
L Landau levels, see Band structure Lattice parameters A"BV' compounds, 4-6 A'l',Mn,BV' alloys, 7-10 Lattice vibrations. see Phonons
M Magnetic breakdown, 210-21 1 Magnetic excitations, see also Spin dynamics magnons, in Cdl-,Mn,Te, 173-177, 390-396 neutron scattering, inelastic, 173-177 Raman scattering, 373-375, 390-396 Magnetic phase diagram, see specific DMS alloys Magnetic polaron, see Bound magnetic polaron; Free magnetic polaron Magnetic susceptibility, see also Spin-glass transition Cdl-,Mn,Se, 92 Cdl-,Mn,Te, 91 Curie-Weiss form, 91, 97 Hgl-,Mn,Te, 92 irreversible effects, 93-94, 96 of specific cluster configurations, 76-77 Magnetization, 198, 235, 314, 400-401, 428 Brillouin function, 235 effective Mn concentration, 101-102, 403, 418 modified, 101 equations of motion, for magnetic excitations, 373-375 in Hgl-,Fe,Te, 254-255 magnetic polaron correction, 297-299 mean field approximation, 198, 289, 314, 400 remanent, 112-117 spin splitting, relation to, 198, 235, 286 Magnetooptical transitions, see a h Excitons, in magnetic field narrow-gap DMS, 237-240 inter-band, 241-250 intra-band, 257-266 selection rules, 239 wide-gap DMS away from zone center, 305-308 selection rules, 283, 294-295
460
INDEX
Magnetoresistance, negative, in Hgl-,Mn,Te, 333-337, see also Hopping conduction, in magnetic field Magnons, see also Magnetic excitations; Raman scattering; Spin dynamics neutron scattering, CdI-,Mn,Te, 173-177 Raman scattering, Cdl-,Mn,Te, 390-396 Manganese covalent radius, 6, 11 distribution in A:L,Mn*BV1 lattice, 76-77, 101-102, 111, 147, 157 frustrated antiferromagnetic lattice, 79 percolation, nearest-neighbor, 77-78, 93, 103, 110 probability, specific cluster forms, 76-77 effective concentration, for magnetization, 101-102, 400, 418 similarity to group-I1 elements, 4 Manganese d-states charge transfer states, 70 hybridization, p-d, 70 intra-ion transitions, 58-68 pressure shifts Cdl-xMnxTe, 61 Znl-,Mn,Se, 65, 68, 69 Znl-*Mn,Te, 63 Racah parameters, 58-60 Znl-,Mn,Se, 60 Znl-*Mn,Te, 60 splitting and degeneracy, 57 Tanabe-Sugano diagram, 58 Mean field approximation, 198, 289, 314 magnetic polaron correction, 297-299 Mn++-Mn++interaction, see Exchange interaction, d-d MnS bond length, 11 lattice parameter, tetrahedrally bonded phases, 8-10 synthesis, 17 MnSe bond length, 11 lattice parameter, tetrahedrally bonded phases, 8-10 synthesis, 17 thin films, zinc blende, 28 MnTe bond length, 11
lattice parameter, hypothetical zinc blende phase, 8-10 synthesis, 17 Mott transition, see Nonmetal-metal transition, exchange-induced
N Neutron scattering, magnetic Cdl-,Mn,Te diffuse scattering, 155-172 inelastic scattering, 155, 173-177 in short-range-ordered systems, 128-135 theory, for magnetic short range order powder, 131-135 single crystal, 130, 135-140 Znl-xMn,Te, diffuse scattering, 140-155 Neutron scattering, nuclear Bragg scattering, 130 Cdl-,MnxTe, 157 Znl-xMnxTe, 141-143 diffuse, 130, 157 Nonmetal-metal transition, exchangeinduced, 337-338, see also Hopping conduction, in magnetic field
0 Optical absorption, in zero magnetic field Varshni equation, 50 Mn d-electron transitions, 59-69 Overlap of bands, exchange-induced, 199-201, 2 11, 246-247
P Pbl-,Mn,BV1 alloys, 30 exchange integrals, sp-d interactions, 217 Shubnikov-de Haas effect, 216 Percolation, see Manganese, distribution in A{~,Mn,B"' lattice Phonons, see also Raman scattering in Cd,-,Mn,Te, 365-373 in Znl-,MnxTe, 366-373 Preparation of ATTBV' compounds, 15-17 of Mn chalcogenides, 17 Purification of elements, 14-15
INDEX
Q Quantum oscillations, see Shubnikov-de Haas effect
R Raman scattering cross section, 347 experimental technique, 350-352 magnetic excitations magnons, in Cdl-,Mn,Te, 390-396 Raman antiferromagnetic resonance, 393 Raman electron paramagnetic resonance, 376-389 theory, 373-376 phonons mixed crystals, 357-365 perfect crystals, 353-357 selection rules, 348-350 spin-flip, 396-405, 414, 417, 440-445 Cdl-,Mn,S, 417 Cdl-,Mn,Se, 403, 405, 414, 417, 440-445 Cdl-,Mn,Te, 397-401, 404, 414 Hgl-,Mn,Te, 266 Znl-,Mn,Se, 417 tensors zinc-blende structure, 356-357 wurtzite structure, 356-357 theory, 346-350 Rare-earth-based DMS alloys, 30 Remanent magnetization, 93-94, 112-1 17 Resonant states, see Acceptors
S
Semiconductor-semimetal transition, see Overlap of bands, exchange-induced; Zero-gap Hg, -xMnxBV'alloys Short-range order, magnetic anisotropy, 135, 164-169 correlation length, 133 in Cdl-,Mn,Te, 157-173 in Znl-,Mn,Te, 141-155 models, for DMS alloys, 139, 168 neutron scattering from, 128, 135 phenomenological description, 133
461
Shubnikov-de Haas effect, 204-21 1 in (Cdl-,Mnx)3As2, 215 in Hgl-,Mn,Se, 204-211 in Hgl-,Mn,Te, 204-209 in Pb,-,Mn,Bv' alloys, 216 temperature dependence, of peak positions, 205-208 thermo-oscillations, in Hg, -,Mn,Te, 207 in 2-dimensional electron gas, 218 Specific heat, in DMS alloys, 107-112 Spin-dependent scattering, 212 Spin dynamics in Cdl-,Mn,Te, 173-177 models, for DMS, 138, 174 Spin-flip Raman scattering, see Raman scattering, spin-flip Spin-flip transitions, see Magnetooptical transitions Spin fluctuations, 299, 401, 427, 439, 444-445, see also Bound magnetic polaron Spin-glass transition below percolation concentration, 95-96, 103-106 Cdl-,Mn,Se, 92-96, 106 Cdl-,MnxTe, 94-96, 105 determination by Faraday rotation, 271-272, 276 effect of magnetic field, 97 frustrated antiferromagnetic lattice, 79 Hgl-,Mn,Te, 95, 106, 271-272 irreversible effects, 93-94, 96, 112-1 17 magnetic phase diagram for specific DMS alloys, 104-106, 272 percolation, nearest neighbor, 77-78, 103, 110 remanent magnetization, 112-1 17 Znl-,Mn,Te, 93 Spin-orbit splitting, valence band CdS, 287 CdSe, 287 CdTe, 287 Spin splitting, see also Bound magnetic polaron; Exchange interaction, sp-d of band edge, in DMS alloys, 292-296 in Cdl-,Mn,Se, 285 in Cdl-,Mn,Te, 277, 280-281, 283-284 relationship to magnetization, 286 of excitons, in DMS alloys, 277, 279, 283. 296
462
INDEX
excited state, 300-303 ground state, 293-296, 384 of Landau levels, 199-206 Hgl-,Mn,Se, 205 Hg -,Mn,Te, 199-206 2-dimensional electron gas, in DMS alloys, 219 of valence band, in DMS alloys, 314-315, 323 away from zone center, in Cdl-,Mn,Te, 305-308 Spin waves, in type-Ill antiferromagnets, 136 Stimulated spin-flip Raman scattering, 266 Superexchange, in A:'-,Mn,BV' alloys, 13, 80, 104 Superlattices, A!!,Mn,BV'-based, 28 Cdl -,Mn,Te/Cdl -,,Mn,Te, 406-409 Raman scattering, 406-409 Rytov formula, 407
T Tetrahedral bond, 4, 6 Tetrahedral radius, 6, 11 Twinning Cdl-,MnxTe, 156, 165, 175 zinc-blende crystals, 156 Two-dimensional electron gas, in DMS alloys, 217-224
V Valence band, in DMS alloys, see also Acceptors effect of sp-d exchange narrow-gap DMS, 315-316 wide-gap DMS, 314-315 anisotropy, exchange-induced, 292-293, 315, 318-320, 327-329 Van Vleck paramagnetism, in A!'-,FexBV' alloys, 257
2
Zeeman splitting, see Spin splitting Zero-gap Hg,-,Mn,BV' alloys conduction-valence band overlap, exchange-induced, 199-201, 21 1, 246-247 magnetooptical studies, 241-247, 260-265 resonant acceptor states, 312, 338-342 transition to open-gap, 190-192, 232-233, 267 Znl -,Mn,S crystal growth, 24-25 crystal structure, 3, 7-10, 79 exchange integral, d-d interaction, 100
exciton, spin splitting, 283 lattice parameter, 8-10 magnetic phase diagram, 104 Mn d-electron transitions, 60, 67 Znl-,Mn,Se crystal growth, 22-24 crystal structure, 3, 7-10, 79 energy gap, 47-49, 56, 68-69 exchange integral, d-d interaction, 100 exchange integral, sp-d interaction, 295 exciton, spin splitting, 283 lattice parameter, 8-10 magnetic phase diagram, 104 Mn d-electron transitions, 60, 65, 67-69 specific heat, 110, 111 superlattice, 28 thin film, 28 Zn 1-xMn,Te crystal growth, 22 crystal structure, 3, 7-10, 79 energy gap, 48, 55 exchange integral, d-d interaction, 100, 154 exchange integral, sp-d interaction, 295 lattice parameter, 8-10, 143 magnetic correlation length, 150-155 magnetic phase diagram, 104 Mn d-electron transitions, 60, 63 neutron scattering, 140-155 phonons, 366-373 short-range antiferromagnetic order, 141-155