Photoemission from Optoelectronic Materials and their Nanostructures (Nanostructure Science and Technology)

Nanostructure Science and Technology

A volume in the Nanostructure Science and Technology series. Further titles in the series can be found at: http://www.springer.com/series/6331

Kamakhya Prasad Ghatak · Debashis De · Sitangshu Bhattacharya

Photoemission from Optoelectronic Materials and their Nanostructures

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Kamakhya Prasad Ghatak Department of Electronic Science The University of Calcutta Kolkata, West Bengal 700009 India [email protected]

Debashis De Department of Computer Science and Engineering West Bengal University of Technology Salt Lake City Kolkata 700064 India [email protected]

Sitangshu Bhattacharya Nano Scale Device Research Laboratory Centre for Electronics Design and Technology Indian Institute of Science Bangalore 560012 India [email protected]

ISBN 978-0-387-78605-6 e-ISBN 978-0-387-78606-3 DOI 10.1007/978-0-387-78606-3 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009928695 © Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To Professor Somenath Sarkar for his invaluable inspiration, selfless advice and research-proactive leadership

Preface

The creation of quantized structures like quantum wells (QWs) in ultrathin films (UFs), quantum well wires (QWWs), and quantum dots (QDs) is due to two factors: • reduction of the symmetry of the wave-vector space of the charge carriers in electronic materials having various band structures, and • emergence of modern fabrication techniques like molecular beam epitaxy (MBE), metal organic chemical vapor deposition (MOCVD), fine line lithography (FLL), etc. Quantized structures have garnered much interest in nanoscience because of their promise for unearthing both new scientific revelations and new technological applications. In QWs in UFs, the quantization of the motion of the carriers in the direction perpendicular to the film exhibits the two-dimensional behavior of the charge carriers. Another new structure known as a QWW has been proposed to investigate the physical properties in these materials where the carrier gas is quantized in two transverse directions and they can move only in the longitudinal direction. As the concept of quantization increases from 1D to 3D, the degree of freedom of the free carriers annihilates totally, and the density-of-states (DOS) function changes from Heaviside step function to the Dirac’s delta function forming QDs. An enormous range of important applications of such nanostructures for nanoscience in the quantum regime together with a rapid increase in computing power, have generated considerable interest in the study of the optical properties of quantum effect devices. Examples of such new applications include quantum resistors, resonant tunneling diodes, quantum switches, quantum sensors, quantum logic gates, quantum transistors, optical switching systems, etc. Although many new effects in quantized structures have already been reported, the interest in further research of other aspects of such quantum-confined materials is becoming increasingly important. One such significant property is photoemission, which is a physical phenomenon, and its importance has already been established since the inception of Einstein’s photoelectric effect (for which Einstein won Nobel Prize in 1921), which in recent years finds extensive applications in modern optoelectronics, photoemission spectroscopy and related aspects in connection with nanostructures. It is well known that the photo-emitted current density depends on the DOS function which, in turn, affects significantly the different physical properties of various vii

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materials. Photoemission from different materials having degenerate carrier concentration is determined by their respective energy band structures. It exhibits different values in different materials and varies with the incident photon energy, the carrier concentration, the quantizing electric field, the magnitude of the reciprocal quantizing magnetic field under magnetic quantization, the nano-thickness as in QWs in UFs, QWWs and QDs, superlattice period as in the quantum confined superlattices having various carrier energy spectra, etc. It is noteworthy in this regard that the available books, review articles and monographs on optoelectronics and related topics in general cannot even afford to devote an entire chapter to photoemission from different materials and their nanostructures. This monograph is based on our investigations of photoemission for the last twenty years and an attempt has been made to present the same from a plethora of materials and their nanostructures having different energy-wave vector dispersion relations of the carriers under various physical conditions. In Chapter 1, the fundamentals of photoemission have been investigated from bulk samples of wide gap materials having parabolic energy bands and also under magnetic quantization, cross fields configuration, QWs in UFs, QWWs, QDs, and magneto size quantizations, respectively. Since Iijima’s discovery (S. Iijima, Nature 354, 56 [1991]), carbon nanotubes (CNs) have been recognized as an important quantum material, which has generated new research avenues in the areas of nanoscience and technology in general. In Chapter 2, the elementary theory of photoemission from QWs in UFs and QWWs of nonlinear optical, III–V, II–VI, nGallium Phosphide, n-Germanium, Platinum Antimonide, stressed materials, Bismuth and carbon nanotubes has been presented. Chapter 3 contains an analysis of photoemission from QDs of nonlinear optical, III–V, II–VI, n-Gallium Phosphide, n-Germanium, Tellurium, Graphite, Platinum Antimonide, zero-gap, Lead Germanium Telluride, Gallium Antimonide, stressed materials, Bismuth, IV–VI, II–V, Zinc and Cadmium diphosphides, Bismuth Telluride and Antimony on the basis of their respective carrier energy spectra. Semiconductor superlattices (SLs) enjoy extensive applications in optoelectronics, and in Chapter 4 photoemission has been investigated from III–V, II–VI, IV–VI, and HgTe/CdTe quantum well superlattices (QWSL) with graded interfaces under magnetic quantization. The same chapter also includes photoemission from III–V, II–VI, IV–VI, and HgTe/CdTe quantum well effective mass superlattices under magnetic quantization, together with the quantum dots of the aforementioned microstructures. It is worth remarking that, in the methods as given in the literature, the physics of photoemission has been incorporated in the lower limit of the photoemission integral, and assumes that the band structure of the bulk materials becomes an invariant quantity in the presence of the photo-excitation necessary for photoelectric effect. The basic band structure of optoelectronic materials changes in the presence of external light waves in a fundamental way, which has been incorporated mathematically through the expressions of the DOS function and the velocity along the direction of photoemission respectively, in addition to the appropriate fixation of the lower limit of the photo-emission integral for the purpose of investigating the photoemission from bulk specimens of optoelectronic materials in Chapter 5.

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An important concept highly relevant to the measurement of bandgap in electronic materials in the presence of external photo-excitation has also been discussed in this perspective. The effects of quantizing magnetic field on the band structures of optoelectronic materials are easily observed in experiments. Various important physical features originate from the significant changes in the carrier energy spectra of the materials under magnetic quantization. In Chapter 6, photoemission has been investigated under magnetic quantization from optoelectronic materials on the basis of the concept presented in Chapter 5. Chapter 7 covers the study of the photoemission from QWs in UFs, QWWs, and QDs of optoelectronic materials as an extension of the new dispersion relations of the bulk optoelectronic materials investigated in Chapter 5. In chapters 8 and 9, the photoemission from quantum confined effective mass superlattices of optoelectronic materials and quantum confined superlattices of optoelectronic materials with graded interfaces have respectively been presented in the presence of external photo-excitation. Chapter 10 briefly discusses experimental results, in which we have proposed a single multidimensional open research problem for experimentalists regarding multi-photon photoemission from quantized materials having various band structures under different physical conditions. The book ends with Chapter 11, which contains the conclusion and outlines the scope for future research. To the best of our knowledge, there is no other book devoted totally on the photoemission from optoelectronic compounds and their nanostructures together with the other technologically important macro and micro materials. We earnestly hope that the present book will be a useful reference not only for the present and the next generation of readers but also for researchers in materials science, optoelectronics, and related fields. It is needless to say that the production of an error-free first edition of any book from every point of view is practically impossible, in spite of our joint efforts. We will be grateful to our readers for their constructive criticisms, and will incorporate them in future editions. From Chapter 5 through the end, we have presented 125 open research problems in this important topic, which will be useful not only for alert readers but also for PhD aspirants who wish to contribute to different aspects of photoemission from quantized structures. These open research problems form an integral part of this book, and it will also be useful in graduate courses on modern optoelectronic devices in many universities and institutes. We are confident that our esteemed readers will enjoy the extensive investigations of photoemission from low-dimensional nonlinear optical, III–V, II–VI, Gallium Phosphide, Germanium, Platinum Antimonide, zero-gap, stressed, Bismuth, carbon nanotubes, Gallium Antimonide, IV–VI, Lead Germanium Telluride, Graphite, Tellurium, II–V, Zinc and Cadmium diphosphides, Bismuth Telluride, Antimony, III–V, II–VI, IV–VI, and HgTe/CdTe quantum well superlattices with graded interfaces under magnetic quantization, III–V, II–VI, IV–VI, and HgTe/CdTe effective mass superlattices under magnetic quantization, the quantum dots of the aforementioned superlattices, quantum confined effective mass superlattices and superlattices of optoelectronic materials with graded interfaces on the basis of appropriate respective dispersion relations. The experimental investigations of photoemission from the

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aforementioned nanostructures are relatively less common in the literature, although such studies will provide the key to unlock the band structures of quantized materials which, in turn, control the transport phenomena in such quantum effect devices in the presence of external photo-excitation. Various mathematical analyses and a few chapters of this book are appearing here for the first time. Finally, we hope that our joint efforts will fire the imagination to initiate frontier line researches on this fascinating subject by anyone engaged either in research or in industries connected with nanostructured optoelectronics in general.

Acknowledgments Acknowledgment by Kamakhya Prasad Ghatak I am grateful to G. P. Agarwal for triggering my interest in the active field of nanophotonics while delivering an invited lecture at the University of Calcutta. I express my gratitude to D. Bimberg, V. S. Letokhov, W. L. Freeman, and H. L. Hartnagel for various kinds of academic advice during the last two decades. I am indebted to S. Chatterjee, A. Mallick, and A. Karmakar for numerous academic discussions. I am grateful to S. C. Dasgupta for teaching an engineering student the methods of theoretical physics and inspiring me to solve independently the problems from the two-volume classics of Morse-Feshbach 35 years ago. I simultaneously offer my special thanks to S. S. Baral for creating the passion for theoretical acoustics from the fundamental works of Morse and Ingard. The late P. N. Butcher encouraged me repeatedly to write a research trilogy on the band structure-dependent properties of nanostructured materials. To honor him, we present The Einstein Relation in Compound Semiconductors and Their Nanostructures in the Springer Series in Materials Science, Vol. 116, as the first one; the present book as the second one; and the thermoelectric power in nanostructured materials under strong magnetic field as the third one, which will be published in the Springer Series in Materials Science in 2010, to complete the trilogy as our tribute to Late P. N. Butcher, a thorough scientist in the real sense of the term. I must acknowledge the hidden contribution of my numerous research students who at present hold positions of repute in various academic institutions. My family members deserve a very special mention for really forming the backbone of my long unperturbed research career. I am grateful to my young PhD students not only for confining me in the infinitely deep quantum wells of Ramanujan and Rumi, but also for inspiring me to teach quantum mechanics and related topics from the eightvolume classics of Greiner et al. P. K. Sarkar and S. Bania of my department always tranquilize me at rather difficult moments. In addition, this book has been completed under grant 8023/B0R/RID/RPS-95/2007-08 as sanctioned by the All India Council for Technical Education in their research promotion scheme 2008.

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Acknowledgment by Sitangshu Bhattacharya It is virtually impossible to express my gratitude to all the admirable persons who have influenced my academic and social life from every point of view; nevertheless, a short memento to my teachers S. Mahapatra and H. S. Jamadagni, at the Centre for Electronics Design and Technology, A. K. Sood, at the Department of Physics and M. S. Hegde, at the Solid State and Structural Chemistry Unit and presently the Dean of Science at Indian Institute of Science, Bangalore, for their fruitful academic advices and guidance remains everlasting. I am indebted to my sister Ms. S. Bhattacharya for her belief in me and my work, which still stimulates and amplifies my efficiency for performing in-depth research. I am grateful in the real sense of the term to my teacher K. P. Ghatak for instilling in me the monochromatic idea that the performing good research containing fundamental and innovative concepts is the keystone to excel in creative research activity. The author S. B. is grateful to the DST, India for sanctioning the research grant under the proposal “Fasttrack scheme for young scientist” 2008–2009 having the reference number SR/FTP/ETA-37/08 under which the simulations have been completed. Finally and eternally, I believe our Mother Nature has propelled this joint collaboration in her own unseen way in spite of several insurmountable obstacles.

Acknowledgment by Debashis De I express my gratitude to K. P. Ghatak, S. N. Sarkar, S. Sengupta, P. K. Roy, and A. K. Sen for their constructive academic support. I am highly indebted to my brother S. De for his constant inspiration and mental support. I am grateful to the All India Council for Technical Education for granting me the aforementioned project jointly in their research promotion scheme 2008, under which this book has been completed.

Joint Acknowledgments We are indebted to Dr. K. Howell, Senior Editor, Springer; Dr. D. J. Lockwood, Series Editor, Nanostructure Science and Technology; Dr. A. Greene, Editorial Director, Springer; and C. Balmes, Editorial Assistant, Springer, for their priceless technical assistance. We are grateful to Ms. S. Roy and A. Saha, the two prominent young members of our research team, for their overall help from every point of view without which the writing of this book would be a mere dream. Naturally, the authors are responsible for nonimaginative shortcomings. Kolkata, India Bangalore, India Kolkata, India November 2008

K. P. Ghatak D. De S. Bhattacharya

Contents

1 Fundamentals of Photoemission from Wide Gap Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Theoretical Background . . . . . . . . . . . . . . . . . . 1.2.1 Photoemission from Bulk Semiconductors . . . . 1.2.2 Photoemission Under Magnetic Quantization . . . 1.2.3 Photoemission in the Presence of Cross Fields . . 1.2.4 Photoemission from Quantum Wells in Ultrathin Films of Wide Gap Materials . . . . . . . . . . . 1.2.5 Photoemission from Quantum Well Wires of Wide Gap Materials . . . . . . . . . . . . . . . . 1.2.6 Photoemission from Quantum Dots of Wide Gap Materials . . . . . . . . . . . . . . . . . . . 1.2.7 Photoemission Under Magneto-Size Quantization (MSQ) . . . . . . . . . . . . . . . . . . . . . . . 1.3 Results and Discussions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Fundamentals of Photoemission from Quantum Wells in Ultrathin Films and Quantum Well Wires of Various Nonparabolic Materials . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . 2.2.1 Photoemission from Nonlinear Optical Materials . . . . 2.2.2 Photoemission from III–V Materials . . . . . . . . . . 2.2.3 Photoemission from II–VI Compounds . . . . . . . . . 2.2.4 Photoemission from n-Gallium Phosphide . . . . . . . 2.2.5 Photoemission from n-Germanium . . . . . . . . . . . 2.2.6 Photoemission from Platinum Antimonide . . . . . . . 2.2.7 Photoemission from Stressed Materials . . . . . . . . . 2.2.8 Photoemission from Bismuth . . . . . . . . . . . . . . 2.2.9 Photoemission from (n, n) and (n, 0) Carbon Nanotubes 2.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Fundamentals of Photoemission from Quantum Dots of Various Nonparabolic Materials . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . 3.2.1 Photoemission from Nonlinear Optical Materials . . . 3.2.2 Photoemission from III–V Materials . . . . . . . . . 3.2.3 Photoemission from II–VI Materials . . . . . . . . . 3.2.4 Photoemission from n-Gallium Phosphide . . . . . . 3.2.5 Photoemission from n-Germanium . . . . . . . . . . 3.2.6 Photoemission from Tellurium . . . . . . . . . . . . 3.2.7 Photoemission from Graphite . . . . . . . . . . . . . 3.2.8 Photoemission from Platinum Antimonide . . . . . . 3.2.9 Photoemission from Zero-Gap Materials . . . . . . . 3.2.10 Photoemission from Lead Germanium Telluride . . . 3.2.11 Photoemission from Gallium Antimonide . . . . . . . 3.2.12 Photoemission from Stressed Materials . . . . . . . . 3.2.13 Photoemission from Bismuth . . . . . . . . . . . . . 3.2.14 Photoemission from IV–VI Materials . . . . . . . . . 3.2.15 Photoemission from II–V Materials . . . . . . . . . . 3.2.16 Photoemission from Zinc and Cadmium Diphosphides 3.2.17 Photoemission from Bismuth Telluride . . . . . . . . 3.2.18 Photoemission from Quantum Dots of Antimony . . . 3.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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107 107 109 109 110 120 121 122 124 126 128 129 131 132 137 138 142 146 147 149 150 152 170

4 Photoemission from Quantum Confined Semiconductor Superlattices 173 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . 174 4.2.1 Magneto-photoemission from III–V Quantum Well Superlattices with Graded Interfaces . . . . . . . . . 174 4.2.2 Magneto-Photoemission from II–VI Quantum Well Superlattices with Graded Interfaces . . . . . . . . . 179 4.2.3 Magneto-Photoemission from IV–VI Quantum Well Superlattices with Graded Interfaces . . . . . . . . . 181 4.2.4 Magneto-Photoemission from HgTe/CdTe Quantum Well Superlattices with Graded Interfaces . . . 185 4.2.5 Magneto-Photoemission from III–V Quantum Well Effective Mass Superlattices . . . . . . . . . . . . . 186 4.2.6 Magneto-Photoemission from II–VI Quantum Well Effective Mass Superlattices . . . . . . . . . . . . . 188 4.2.7 Magneto-Photoemission from IV–VI Quantum Well Effective Mass Superlattices . . . . . . . . . . . . . 191 4.2.8 Magneto-Photoemission from HgTe/CdTe Quantum Well Effective Mass Superlattices . . . . . . . . 193

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4.2.9

Photoemission from III–V Quantum Dot Superlattices with Graded Interfaces . . . . . . . 4.2.10 Photoemission from II–VI Quantum Dot Superlattices with Graded Interfaces . . . . . . . 4.2.11 Photoemission from IV–VI Quantum Dot Superlattices with Graded Interfaces . . . . . . . 4.2.12 Photoemission from HgTe/CdTe Quantum Dot Superlattices with Graded Interfaces . . . . . . . 4.2.13 Photoemission from III–V Quantum Dot Effective Mass Superlattices . . . . . . . . . . . . . . . . . 4.2.14 Photoemission from II–VI Quantum Dot Effective Mass Superlattices . . . . . . . . . . . . . . . . . 4.2.15 Photoemission from IV–VI Quantum Dot Effective Mass Superlattices . . . . . . . . . . . . 4.2.16 Photoemission from HgTe/CdTe Quantum Dot Effective Mass Superlattices . . . . . . . . . . . . 4.3 Results and Discussions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Photoemission from Bulk Optoelectronic Materials 5.1 Introduction . . . . . . . . . . . . . . . . . . . 5.2 Theoretical Background . . . . . . . . . . . . . 5.3 Results and Discussions . . . . . . . . . . . . . 5.4 Open Research Problems . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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6 Photoemission under Quantizing Magnetic Field from Optoelectronic Materials . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . 6.2 Theoretical Background . . . . . . . . . . . . . . . 6.3 Results and Discussions . . . . . . . . . . . . . . . 6.4 Open Research Problems . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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Photoemission from Quantum Wells in Ultrathin Films, Quantum Wires, and Dots of Optoelectronic Materials . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 7.2 Theoretical Background . . . . . . . . . . . . . . . . . 7.2.1 Photoemission from Quantum Wells in Ultrathin Films of Optoelectronic Materials . . . . . . . . 7.2.2 Photoemission from Quantum Well Wires of Optoelectronic Materials . . . . . . . . . . . 7.2.3 Photoemission from Quantum Dots of Optoelectronic Materials . . . . . . . . . . . 7.3 Results and Discussions . . . . . . . . . . . . . . . . .

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7.4 Open Research Problems . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Photoemission from Quantum Confined Effective Mass Superlattices of Optoelectronic Materials . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 8.2 Theoretical Background . . . . . . . . . . . . . . . . . 8.2.1 Magneto-Photoemission from Quantum Well Effective Mass Superlattices . . . . . . . . . . . 8.2.2 Photoemission from Effective Mass Quantum Well Wire Superlattices . . . . . . . . . . . . . 8.2.3 Photoemission from Quantum Dots of Effective Mass Superlattices . . . . . . . . . . . . . . . . 8.2.4 Magneto-Photoemission from Effective Mass Superlattices . . . . . . . . . . . . . . . . 8.3 Results and Discussions . . . . . . . . . . . . . . . . . 8.4 Open Research Problems . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9 Photoemission from Quantum Confined Superlattices of Optoelectronic Materials with Graded Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . 9.2.1 Magneto Photoemission from Quantum Well Superlattices . . . . . . . . . . . . . . . . . . . 9.2.2 Photoemission from Quantum Well Wire Superlattices 9.2.3 Photoemission from Quantum Dot Superlattices . . . 9.2.4 Magneto-Photoemission from Superlattices of III-V Optoelectronic Materials . . . . . . . . . . . . 9.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . 9.4 Open Research Problems . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Review of Experimental Results 10.1 Experimental Works . . . . 10.2 Open Research Problem . . References . . . . . . . . . . . .

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Conclusion and Future Research . . . . . . . . . . . . . . . . . . . 11.1 Open Research Problems . . . . . . . . . . . . . . . . . . . . .

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Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Materials Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Symbols

α α0 δ δ || ⊥  o λ ε εsc ε0 ζ (2r)  (j + 1) α0 φ ω0 υ a ac a0 ,b0 − → A B B2 c C0 C1 C2 dx , dy , dz e E EF EFB Eo

Band nonparabolicity parameter Optical absorption coefficient Crystal field splitting constant Dirac’s delta function Spin-orbit splitting constants parallel Spin-orbit splitting constants perpendicular to the C-axis Isotropic spin-orbit splitting constant Interface width in superlattices Wavelength Trace of the strain tensor/energy as measured from the center of the band gap Semiconductor permittivity Permittivity of vacuum Zeta function of order 2r Complete Gamma function Probability of photoemission Work function Cyclotron resonance frequency Frequency The lattice constant Nearest neighbor C–C bonding distance The widths of the barrier and the well for superlattice structures Vector potential Quantizing magnetic field Momentum matrix element Velocity of light Splitting of the two-spin states by the spin orbit coupling and the crystalline field Conduction band deformation potential Strain interaction between the conduction and valance bands Nanothickness along the x, y, and z directions Magnitude of electron charge Total energy of the carrier Fermi energy Fermi energy in the presence of magnetic field Electric field

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xviii E0 Enz EF0D Eg 0 EB EQD E FL Eg EFBL EF2DL Eg EF1D Fj (η) f (E) f0 G0 gv h 

H I ˆi, ˆj and kˆ J k0 I0 J J2D kB k lx ¯l, m, ¯ n¯ L0 m0 m∗ m∗|| m∗⊥ m1 m2 m3 m2 m∗t m∗l m∗⊥,1 , m∗||,1 mr mv

List of Symbols Ground state energy of the electron in the presence of crossed electric and magnetic field Energy of the nth subband 0D Fermi energy Unperturbed band gap Bohr electron energy Totally quantized energy Fermi energy in the presence of light waves Perturbed band gap Fermi energy under quantizing magnetic field in the presence of light waves Fermi energy in QWs in UFs in the presence of light waves Increased band gap Fermi energy in the presence of 2D quantization One parameter Fermi-Dirac integral of order j Fermi Dirac occupation probability factor Optical frequency Nonlinear response from the optical excitation of the free carriers Valley degeneracy Planck’s constant Dirac’s constant Heaviside step function Photocurrent Orthogonal triads photoelectric current density Inverse Bohr radius Light intensity Bulk photoemission current density 2D photocurrent density Boltzmann’s constant Electron wave vector Sample length along x direction Matrix elements of the strain perturbation operator Superlattices period length Free electron mass Isotropic effective electron masses at the edge of the conduction band Longitudinal effective electron masses at the edge of the conduction band Transverse effective electron masses at the edge of the conduction band Effective carrier masses at the band-edge along x direction Effective carrier masses at the band-edge along y direction Effective carrier masses at the band-edge along z direction Effective mass tensor component at the top of the valence band (for electrons) or at the bottom of the conduction band (for holes) The transverse effective masses at k = 0 The longitudinal effective masses at k = 0 Transverse and longitudinal effective electron masses at the edge of the conduction band for the first material in superlattice Reduced mass Effective mass of the heavy hole at the top of the valance band in the absence of any field

List of Symbols N (E) Nc nx , ny , nz n0 N2DT (E) N1DT (E) n1D n2D N0DT (E) n0D n P0 T tc ν (E) v, i υ0 V0 W x, y

xix Density of states in bulk specimens Effective number of states in the conduction band Size quantum numbers along the x, y, and z directions Total bulk electron concentration Total 2D density of states function Total 1D density of states function Electron concentration per unit length Electron concentration per unit area Total 0D density of states function 0D electron concentration per subband Landau quantum number/chiral indices Momentum matrix element Temperature Tight binding parameter Velocity of the emitted electron Integer Threshold frequency Potential barrier Electron affinity Alloy compositions

Chapter 1

Fundamentals of Photoemission from Wide Gap Materials

1.1 Introduction It is well known that the photoelectric effect occupies a singular position in the whole arena of materials science and related disciplines in general together with the fact that the photoemission from the electronic materials is also a vital physical phenomenon from the viewpoint of modern optoelectronics and photoemission spectroscopy [1]. The classical  equation of the photo-emitted current density is [2]  J = 4πem∗ gv (kB T)2 / h3 exp [(hν − φ) / (kB T)], where e, m∗ , gv , kB , T, h, hυ and φ are the electron charge, effective electron mass at the edge of the conduction band, valley degeneracy, the Boltzmann constant, temperature, the Planck’s constant, incident photon energy along z-axis and work function, respectively. The aforementioned equation is valid for both the charge carriers, and in this conventional form it appears that the photoemission changes with the effective mass, temperature, work function, and the incident photon energy, respectively. This relation holds only under the condition of carrier nondegeneracy. The photoemission has different values for different materials and varies with doping and with external fields, which creates quantization of the wave-vector space of the carriers, leading to various types of quantized structures. The nature of these variations has been studied [2–35], and some of the significant features are as follows: 1. The photoemission from bulk materials increases with the increase in doping. 2. The photoemission exhibits oscillatory dependence with inverse quantizing magnetic field because of the Shubnikov–de Haas (SdH) effect. 3. The photoemission changes significantly with the magnitude of the externally applied quantizing electric field in electronic materials. 4. The photoemission from quantum confined Bismuth, nonlinear optical, III–V, II– VI, and IV–VI materials oscillate with nanothickness in various manners which are totally band structure–dependent. 5. The nature of the variations is significantly influenced by the energy band constants of various materials having different band structures. 6. The photoemission has significantly different values in quantum confined semiconductor superlattices and various other quantized structures. K.P. Ghatak et al., Photoemission from Optoelectronic Materials and their Nanostructures, Nanostructure Science and Technology, DOI 10.1007/978-0-387-78606-3_1,  C Springer Science+Business Media, LLC 2009

1

2

1 Fundamentals of Photoemission

In Section 1.2.1 of the theoretical background, an elementary analysis of the photoemission from bulk specimens of wide gap semiconductors having parabolic energy bands is presented. It is well known that the band structure of any electronic material changes fundamentally in the presence of external fields [36]. The effects of the quantizing magnetic field on the energy wave vector dispersion relation of the charge carriers of small gap semiconductors can be observed easily in experiments. It is well known that under magnetic quantization, the motion of the charge carrier parallel to the magnetic field remains unaltered, while the area of the wave vector space normal to the direction of the magnetic field gets quantized in accordance with the Landau rule of area quantization in the wave-vector space of the charge carriers [37]. The physics of the Landau levels is the signature of the concept of singularity of complex variables and the quantized energies are known as Landau sub-bands. The quantizing magnetic field tends to remove the degeneracy and increases the band gap. A semiconductor, placed in a magnetic field B, can absorb the radiative energy with the frequency ω0 (≡ eB / m∗ ). The effect of energy quantization is experimentally noticeable when the separation between any two consecutive Landau levels is greater than kB T. A number of interesting transport phenomena originate from the change in the basic band structure of the semiconductor under magnetic quantization, and these have been widely investigated and have also served as diagnostic tools for characterizing the different materials having different carrier dispersion laws. The discreteness in the Landau levels leads to a whole crop of magneto-oscillatory phenomena, important among which are (i) Shubnikov–de Haas oscillations in magneto-resistance; (ii) de Haas–Van Alphen oscillations in magnetic susceptibility; (iii) magneto-phonon oscillations in thermoelectric power, etc. In Section 1.2.2, the photoemission from the semiconductors having parabolic energy bands under magnetic quantization is explored. The effect of crossed electric and quantizing magnetic fields on the transport properties of different semiconducting compounds has been relatively less investigated compared to the corresponding magnetic quantization, although the cross fields studies are very important with respect to the experimental findings [38– 42]. It is well known that in the presence of electric field (E0 ) along the x-axis and the quantizing magnetic field B along the z-axis, the dispersion relations of the conduction electrons in semiconductors become modified in a fundamental way and the electron moves in both the z and y directions. The motion along the y direction is purely due to the presence of E0 along the x-axis, in the absence of which the effective electron mass along the y-axis becomes infinite, indicating the fact that the electron motion along the y-axis is prohibited. The effective electron mass of the isotropic, bulk semiconductors having parabolic energy bands under cross fields configuration exhibits mass anisotropy which, in turn, depends on the electron energy, the magnetic quantum number, the electric and the magnetic fields, respectively. In Section 1.2.3, the photoemission under cross field configuration from semiconductors having parabolic energy bands is investigated. In recent years, with the advent of fine lithographical methods [43], molecular beam epitaxy [44], metal-organic chemical vapor deposition [45], and other experimental techniques, the restrictions of the motion of the carriers of bulk materials in one (QWs in UFs, NIPI structures, inversion and accumulation layers), two

1.1

Introduction

3

(QWWs), and three (QDs, magneto-size quantized systems, magneto inversion layers, magneto accumulation layers, quantum dot superlattices, magneto quantum well superlattices, and magneto NIPI structures) dimensions have in the last few years attracted much attention not only for their potential in uncovering new phenomena in nanostructured science and technology but also for their interesting quantum device applications [46–48]. In QWs in UFs, the restriction of the motion of the carriers in the direction normal to the film (say, the z direction) may be viewed as carrier confinement in an infinitely deep 1D rectangular potential well, leading to quantization (known as quantum size effect [QSE]) of the wave vector along the direction of the potential well, allowing 2D electron transport parallel to the surface of the film representing new physical features not exhibited in bulk semiconductors [49]. The low-dimensional hetero-structures based on various materials are widely investigated because of the enhancement of carrier mobility [50]. These properties make such structures suitable for applications in quantum well lasers [51], hetero-junction FETs [52], tunneling hot-electron transfer amplifier (THETA) devices [53], high-frequency microwave circuits [54], optical modulators [55], optical switching systems [56], and other devices. The constant energy 3D wave-vector space of bulk semiconductors becomes a 2D wave-vector surface in QWs in UFs because of dimensional quantization. Thus, the concept of reduction of symmetry of the wave-vector space is needed to create low-dimensional structures. In Section 1.2.4, the detailed formulation of photoemission from QWs in UFs of wide gap materials is presented. In QWWs, the restriction of the motion of the carriers along two directions may be viewed as carrier confinement by two infinitely deep 1D rectangular potential wells, along any two orthogonal directions leading to quantization of the wave vectors along the said directions, allowing 1D electron transport. With the help of modern fabricational techniques, such one-dimensional quantized structures have been experimentally realized and enjoy an enormous range of important applications in the realm of nanoscience in the quantum regime. They have generated much interest in the analysis of nanostructured devices for investigating their electronic, optical, and allied properties [57–60]. Examples of such new applications are based on the different transport properties of ballistic charge carriers which include quantum resistors [61–66], resonant tunneling diodes and band filters [67, 68], quantum switches [69], quantum sensors [70–72], quantum logic gates [73–74], quantum transistors and subtuners [75–77], hetero-junction FETs [78], high-speed digital networks [79], high-frequency microwave circuits, optical modulators, optical switching systems, and other devices [80]. In Section 1.2.5, the photoemission from QWWs of wide gap materials is studied. As the dimension of the QWs in UFs increases from 1D to 3D, the degree of freedom of the free carriers decreases drastically and the density-of-states function is changed from a stepped cumulative one to the Dirac’s delta function [81–82], forming QDs. The importance of QDs is already well known in the whole field of nanostructured science and technology. In Section 1.2.6, the photoemission from QDs of wide band gap materials has been formulated. The densityof-states function is also quantized in the presence of magneto-size quantization,

4

1 Fundamentals of Photoemission

and in Section 1.2.7, the photoemission is studied under this condition. Section 1.3 contains the result, and discussions pertinent to this chapter.

1.2 Theoretical Background 1.2.1 Photoemission from Bulk Semiconductors The consequence of the photoelectric effect is the creation of the concept of photoelectric current density (J) which, can, in turn, be written through the photoemission integral (PI ) as [2]: α0 e J= (1.1) (PI ) 4 where α0 is the probability of photoemission, ∞ PI =

    N E νz E f (E) dE

(1.2)

E0

in which, E0 ≡ W − hυ, W is the electron affinity, E ≡ E − E0 , E is the total energy of the electron as measured from the edge of the conduction    band in the vertically upward direction in the absence of any quantization, N E is the DOS   function at E = E ,νz E is the velocity of the emitted electron along the z-axis when E = E , f (E) is the Fermi-Dirac occupation probability factor and can be written as f (E) = [1 + exp ((E − EF )/kB T)]−1 , EF is the Fermi energy as measured from the edge of the conduction band in a vertically upward direction in the absence of any quantization. It appears then that the formulation of J needs an expression of DOS function which can in turn be derived as follows: The DOS function in three dimensions is defined as the number of carrier states per unit volume of wave-vector space per unit energy interval. The generalized formula of the DOS function for bulk specimens is given by: N (E) =

∂ {V (E)} (2π ) ∂E 2gv

3

(1.3)

where V (E) is the volume of k-space which should be determined from the energywave vector dispersion relation of the carriers. For n-type wide gap semiconductors having parabolic energy bands, the energy spectrum of the conduction electrons assumes the form: 2m∗ E (1.4) k2 = 2 where k is the electron wave vector and  (≡ h/ (2π )) is known as Dirac’s constant. Equation (1.4) is basically the equation of an electron parabola in a two-dimensional E-k diagram which can be written as: 2m∗ E . (1.5) kx2 + ky2 + kz2 = 2

1.2

Theoretical Background

5

Therefore, the constant energy surface is a sphere in k-space whose volume can be expressed as: V (E) =

4π 3



2m∗ E 2

3/2 .

(1.6)

Using (1.3) and (1.6), the DOS function assumes the form, N (E) = 4πgv

2m∗ h2

3/2

√ E.

(1.7)

Equation (1.7) is known as the inverted-parabolic dependence of the electronic three-dimensional density-of-states (3D DOS) function for parabolic n-type semiconductors. The expression of the DOS function is very important in semiconductor science in general and is used in investigating the carrier density, the photoelectric current density, Hall coefficient, thermoelectric power, mobility, and almost all transport properties of semiconductor devices. The increasing importance of the DOS function becomes apparent with the advent of nanoscience and technology. In two dimensions, the unit of the DOS function is (1/(eVm2 )), and in one dimension, the unit of the same function is (1/eVm). The plots (a), (b), and (c) of Fig. 1.1 exhibit the plot of the normalized 3D DOS function for bulk specimens of n-type InSb, InAs, and GaAs, respectively, as a function of electron energy, where the real band structure has been neglected for the purpose of simplified presentation, and the numerical values of the effective electron masses at the edge of the conduction band have been taken from Appendix A. Since m∗ of n-InSb is lowest as compared with the other two materials, the 3D DOS function exhibits the lowest value for bulk specimens of n-InSb, indicating the fact that the greatest electron energy states are available for the purpose of electronic conduction in n-InSb as compared with n-InAs and n-GaAs respectively. We observe that the DOS function can be written from (1.7) for E = E as:   N E = 4πgv



2m∗ h2

3/2 √ E .

(1.8)

  The term νz E is the velocity of the emitted electron along the z-axis and can be written as:    1 ∂E  . (1.9) νz E =  ∂kz E=E , kx =ky =0 The use of (1.4) and (1.9) yields the following equation:   νz E =



2 m∗

1/2 √ E .

(1.10)

6

1 Fundamentals of Photoemission

Fig. 1.1 Plot of the normalized 3D density-of-states function in bulk specimens of (a) InSb, (b) InAs, and (c) GaAs, versus electron energy

Combining (1.1), (1.8), and (1.10), it can be written as: 4π α0 em∗ gv J= h3

∞ E0

E dE

. F 1 + exp E−E kB T

(1.11)

Substituting ((E − E0 )/(kB T)) ≡ y and η0 ≡ ((hν − φ)/(kB T)) from (1.11), the expression of the photo-emitted current density from the bulk specimens having parabolic energy bands can be expressed as: J=

4π α0 em∗ gv (kB T)2 F1 (η0 ) h3

(1.12)

where F1 (η0 ) is the special case of the one-parameter Fermi-Dirac integral of order j which assumes the form [83]: Fj (η) =

1  (j + 1)



∞ 0

xj dx , 1 + exp (x − η)

j > −1

(1.13)

1.2

Theoretical Background

7

or for all j, analytically continued as a complex contour integral around the negative x-axis:

 +0 xj dx  (−j) , (1.14) Fj (η) = √ 2π −1 −∞ 1 + exp (−x − η) √ where  (j + 1) = j (j), (1/2) = π, and  (0) = 1. A few important properties of the Fermi-Dirac integral are listed in this context without proof [84] since these are very useful in obtaining direct results without cumbersome mathematics, and are needed for further progress.  d  (1.15a) Fj (η) = Fj−1 (η). 1. dη  (1.15b) 2. Fj (η)dη = Fj+1 (η).

4 3/2 π2 1 + 2 , η > 1.25. (1.15c) 3. F1/2 (η) ≈ √ η 8η 3 π 4. F0 (η) = ln |1 + eη |. 5. Fj (η) ≈ eη ,

η < 0 for all j.

(1.15d) (1.15e)

It appears that the evaluation of J as a function of electron concentration needs an expression of electron statistics, which can be formulated as follows: It is well known that N (E) f (E) dE is the number of electrons in the conduction band per unit volume lying in the E to E + dE energy range. Thus the total electron concentration is given by:  n0 =

ρTop

ρ0

N (E) f (E) dE,

(1.16)

where the lower limit ρ0 can be determined from the equation N (ρ0 ) = 0; and from the nature of variation of f (E), the upper limit ρTop can be replaced by infinity without introducing appreciable error in the subsequent investigations. Therefore, using the expressions of N (E) from (1.7), (1.16) can be written as: n0 = 4πgv

2m∗ h2

3/2 

∞

0

√ E

dE. F 1 + exp E−E kB T

(1.17)

The lower limit of the integral in (1.17) starts with zero of DOS function. Substituting x ≡ (E/kB T) (x is a new variable of normalized energy) and η ≡ (EF /kB T) (normalized Fermi energy) in (1.17) results in the emergence of the following equation: n0 = 4πgv

2m∗ kB T h2

3/2 

∞ 0

x1/2 dx. 1 + exp (x − η)

(1.18)

8

1 Fundamentals of Photoemission

Using (1.13) and (1.18), the electron concentration can be expressed as: n0 = Nc F1/2 (η) ,

(1.19)

where Nc ≡ 2gv

2π m∗ kB T h2

3/2

is known as the effective number of states in the conduction band. A Few Special Cases Case I:: When the Fermi level lies within the band gap, the semiconductor becomes nondegenerate and η (n + (1/2)) ω0 . For E → (n + (1/2)) ω0 , NB (E) → ∞. Thus the DOS function under magnetic quantization tends to infinity with respect to electron energy at the Landau singularity. Converting the summation over n to the integration over n, (1.34) converts to the well-known (1.7). It is worth remarking that the Landau singularity is the basic

12

1 Fundamentals of Photoemission

Fig. 1.4 The plot of the normalized density-of-states function under magnetic quantization as a function of normalized electron energy for n-InSb

cause of oscillations of the electronic properties of electronic materials under magnetic quantization with inverse quantizing magnetic field. Figure 1.4 exhibits the plot of the normalized DOS function under magnetic quantization versus normalized electron energy. The velocity of the photoemitted electron per subband is: √ 

2 1 E− n+ ω0 . vzB (E) = √ 2 m∗

(1.35)

The photoelectric current density JBP per subband is given by:

JBP

α0 gv e = 2

∞

NB (E)νzB (E) f (E) dE,

(1.36)

En

where En ≡ (En + W − hν) and NB (E) is the DOS function per subband. Using (1.34), (1.35), and (1.36), the total photocurrent density can be written as: nmax  nmax α0 e2 Bgv  α0 e2 Bgv kB T  f dE = F0 (η1 ), JB = (E) 2π 2 2 2π 2 2 ∞

n=0 E

n

n=0

(1.37)

1.2

Theoretical Background

13

where η1 ≡

EFB − En . kB T

EFB is the Fermi energy in the presence of a magnetic field as measured from the edge of the conduction band in the absence of any quantization in the vertically upward direction. It is important to note that the velocity of the photo-emitted electron in the nth sub-band and the DOS function of the nth sub-band in this case cancels each other, leaving a constant prefactor. This cancellation effect is very important in QWWs and determines the limit of the maximum resistance offered by one-dimensional systems. Under the condition of extreme degeneracy from (1.37), we can write: nmax   α0 e2 Bgv  EFB − En . (1.38) JB = 2π 2 2 n=0

From (1.38), we observe that JB is independent of temperature under the condition of extreme carrier degeneracy. Under the condition of nondegeneracy from (1.37), we get: JB =

nmax α0 e2 Bgv kB T  exp (η1 ). 2π 2 2

(1.39)

n=0

For bulk materials, in the absence of a magnetic field, converting the summation over n to the corresponding integration over n from (1.39), we can write: ⎤ ⎡



∞

2 Bg k T α e hν − φ −nω −ω 0 v B 0 0 0 exp exp exp dn⎦ . Lim (JB ) = Lim ⎣ B→0 B→0 kB T 2kB T kB T 2π 2 2 0

(1.40) By carrying out elementary integration, we can get the classical photoemission equation as given in the introduction. It appears that the computation of JB requires an expression of electron concentration, which can be written as: √ nmax  ∞ f (E) dE eBgv 2m∗  . (1.41) n0 = √ 2 2 2π  E − En n=0 En Substituting,

EFB − En E − En ≡ x0 and ≡ η2 , from (1.41), one can obtain, kB T kB T n0 = Nc θ

n max n=0

F 1 (η2 ), θ ≡ 2

ω0 . kB T

(1.42)

The decrease of electron concentration with increase in magnetic field under magnetic quantum limit together with the condition of carrier non-degeneracy is known as the magnetic freeze out of the carriers. Under the condition of carrier

14

1 Fundamentals of Photoemission

non-degeneracy, converting the summation over n to the integration over n, (1.42) converts into the well-known (1.20). From (1.42) one can infer that the application of a magnetic field always tends to lower the Fermi energy level with respect to the edge of the conduction band, i.e., it always tends to make the semiconductor less degenerate.

1.2.3 Photoemission in the Presence of Cross Fields From classical electromagnetic theory, one can write: − → − → B =∇× A,

(1.43)

− → where A is the vector potential. In the presence of quantizing magnetic field B along the z direction, (1.43) assumes the form:     ˆ   ∂i ∂j k∂  0 i + 0 j + Bkˆ =  ∂x ∂y ∂z  , A A A  x y z where  i,  j and  k are unit vectors along the x-, y-, and z-axes, respectively. From the above equation, we can write: ∂Az ∂y ∂Ax ∂z ∂Ay ∂x

− − −

∂Ay ∂z ∂Az ∂x ∂Ax ∂y

=0 =0. =B

(1.44)

This particular set is satisfied by Ax ≡ 0, Ay ≡ Bx, and Az ≡ 0. Therefore in the presence of the electric field E0 along the x-axis and the quantizing magnetic field B along the z-axis for the present case we can write: 2

E + eE0 xˆ =

pˆ 2z (ˆpy − eBˆx) pˆ 2x + + , 2m∗ 2m∗ 2m∗

(1.45)

where  x, pˆ x , pˆ y , and pˆ z are operators. We introduce a new operator  θ , which can be expressed as: θˆ = −ˆpy + eBˆx −

m∗ E0 . B

(1.46)

Eliminating the operator  x, between (1.45) and (1.46) the electron energy spectrum in semiconductors having parabolic energy bands in the presence of cross fields configuration can be written as:

E0 ky [kz ]2 m∗ Eo2 1 − . ω0 + − E = n+ ∗ 2 2m B 2B2

(1.47)

1.2

Theoretical Background

15

The photo-emitted current density in this case is thus given by: J (E0 , B) = gv

nmax  ∞ ∂E ∂f (E) −α0 e  I0 dE, ∂kz ∂E 4lx π 2  E¯

(1.48)

n=0

where lx is the sample length along the x direction: E¯ ≡ xl ≡



 xh m∗ E02 1 + W − hν , I ≡ kz dky , n+ ω0 − 0 2 2B2 xl



−m∗ E0 eBlx m∗ E0 , and xh ≡ − . B  B

Using (1.47) and (1.48), we get: J (E0 , B) ≡  where φ¯ ≡ 2B Therefore,



eBlx

−



J (E0 , B) =

−gv α0 e eBlx 2π 2 lx 

2m∗ E0 B

n max



E − E¯ + E0 φ¯

n=0

 ∂f (E) dE, ∂E

α0 e2 Bgv 2π 2 2

.

n max  n=0 0

∞

 eG−η2 dG GkB T + E0 φ¯  2 , 1 + eG−η2

(1.49)

where   EFE0 B − E¯ E − E¯ G≡ , η2 ≡ , and EFE0 B kB T kB T is the Fermi energy in the present case. Differentiating (1.13) with respect to η, one can write: 

∞ 0



  xj ex−η dx 2 =  (j + 1) Fj−1 (η) 1 + ex−η

(1.50)

Using (1.50) and (1.49), the photoemission current density in this case is given by: J (E0 ,B) =

nmax   α0 e2 BkB Tgv  ¯ −1 (η2 ) , F + ψF (η ) 0 2 2π 2 2

(1.51)

n=0

¯

where, ψ¯ ≡ EkB0Tφ . In the absence of electric field (E0 → 0), (1.51) is simplified into the form as given by (1.37).

16

1 Fundamentals of Photoemission

The DOS function in this case is given by: √ nmax  2m∗ m∗ ω0  [E − E3n ]1/2 H (E − E3n ) N (E,E0 ,B) = gv lx π 2 2 eE0 n=0  1/2 − [E − E4n ] H (E − E4n ) ,

(1.52)

  where E3n ≡ (n + (1/2)) ω0 + e2 E02 /2m∗ ω02 − eE0 lx and E4n ≡ E3n + eE0 lx . In Fig. 1.5, we have plotted the normalized DOS function versus the normalized electron energy in the presence of cross fields configuration for three values of electric field. It appears from (1.52) and Fig. 1.5 that the N (E, E0 , B) oscillates with the electron energy.   From (1.47) one can easily observe that the ground state energy E0 of the electron in the  presence of crossed electric and magnetic fields is given by E0 = (1/2)ω0 + e2 E02 /2m∗ ω02 , which, in the absence of an electric field, leads to the ground state energy as ((1/2)ω0 ). The N (E, E0 , B) inthe presence of  cross fields shifts right to the ground state energy, by an amount of e2 E02 /2m∗ ω02 . Figure 1.6 exhibits the plot of the normalized DOS function versus the electric field in this case, and it appears that the DOS function oscillates with the electric field.

Fig. 1.5 Plot of the normalized density-of-states function versus normalized electron energy in the presence of crossed electric and quantizing magnetic fields for n-InSb, where for curve (a) E0 = 105 Vm–1 , for curve (b) E0 = 1.52 × 105 Vm–1 , and for curve (c) E0 = 1.72 × 105 Vm–1

1.2

Theoretical Background

17

Fig. 1.6 Plot of the normalized DOS function versus electric field in the presence of crossed electric and quantizing magnetic fields for n-InSb, where for curve (a) B = 2 tesla and for curve (b) B = 1.25 tesla

Combining (1.52) with the Fermi-Dirac occupation probability factor, the electron concentration can be written as: n0 = C0

n max 

 F 1 (η¯ 1 ) − F 1 (η¯ 2 ) ,

n=0



2

2

(1.53)

e2 E02 kB T eE0 lx , and , η¯ 2 ≡ η¯ q − Nc θ , η¯ 1 ≡ η¯ 2 + C0 ≡ eE0 lx kB T 2m∗ kB Tω02



 1 1 EFE0 B − n + ω0 . η¯ q ≡ kB T 2 In the absence of an electric field, E0 → 0, and applying L’Hospital’s rule, one can show that (1.53) is simplified into the form given by (1.42).

1.2.4 Photoemission from Quantum Wells in Ultrathin Films of Wide Gap Materials The photoelectric current density in QWs in UFs can be written as:

J2D

nzmax ∞ αo egv  = N2D (E)f (E)vz (Enz )dE, 2dz n zmin E nz

(1.54)

18

1 Fundamentals of Photoemission

where dz is the nanothickness along the z-direction, Enz is the energy of the nth subband, N2D (E) is the density-of-states function per subband, vz (Enz ) is the velocity of the electron in the nthz subband, and the factor (1/2) originates because only half of the electron will migrate towards the surface and escape [35]. Therefore, it appears that the formulation of J2D requires an expression of N2D (E) which can in turn be written per subband as: N2D (E) =

2gv ∂A , (2π)2 ∂E

(1.55)

  where A ≡ πks2 is the 2D area of the wave vector space perpendicular to the direction of size quantization and should be obtained from the 2D electron dispersion law of the QWs in UFs. In the presence of size quantization, the wave vector of the electron [85] along the direction of an infinitely deep potential well gets quantized in accordance with the wave vector quantization rule as: ki =

ni π , di

(1.56)

where i ≡ x,y,z,ni and di are the size quantum number and the film thickness in the ith direction, respectively. Thus, combining (1.31) with (1.56), the 2D electron energy spectrum in this case can be written as: 2 ks2 + Enz , E= 2m∗

2 En z ≡ 2m∗



nz π dz

2 .

(1.57)

The vz (Enz ) in this case can be expressed as: 





vz Enz =

2 m∗





Enz .

(1.58)

Using (1.55) and (1.57), the DOS function per subband can be written as: N2D (E) =

m∗ gv . π2

(1.59)

The total DOS function (N2DT (E)) in QWs in UFs of wide gap semiconductors, the dispersion relation of whose bulk electron is defined by the parabolic energy bands can be expressed as: N2DT (E) =

nzmax   m∗ gv  H E − Enz . 2 π nz =1

(1.60)

1.2

Theoretical Background

19

Fig. 1.7 Plot of the normalized 2D DOS function for the quantum wells in ultrathin films of (a) InAs and (b) GaAs as a function of normalized electron energy

Figure 1.7 exhibits the plot of the normalized 2D DOS function versus normalized electron energy for the quantum wells in ultrathin films of n-type InAs and GaAs, respectively, as shown in plots (a) and (b), where the real band structure has been neglected for the purpose of simplified presentation. It appears that the N2DT (E) exhibits step functional dependence with the electron energy. For E < E1 where E1 = (2 /2m∗ )(π/dz )2 , N2DT (E) becomes zero, whereas the 3D DOS function exists which indicates that for E < E1 the surface electron concentration vanishes for 2D systems, and whereas for the corresponding bulk semiconductors, the electron concentration is finite. When E = E1 , the 2D DOS function exhibits a quantum jump having the value (m∗ gv /π 2 ). The 2D DOS function becomes constant for E < 4E1 and when E = 4E1 the said function again jumps with a constant step height (m∗ gv /π 2 ). It is important to note that the upper points of all the steps for the 2D systems must lie on the curve of the corresponding 3D DOS function for bulk semiconductors. The quantization of the 2D DOS function is the basic reason behind the step behavior of many electronic properties of the two-dimensional systems. Thus, combining (1.54), (1.58), and (1.59), we get [25]:

J2D

nzmax αo ekB Tgv  = nz [F0 (η4 )], 2dz2 n zmin

(1.61)

20

1 Fundamentals of Photoemission

where nzmin ≥

dz π

√

2m∗ 



 (W − hν)1/2 , η4 ≡ (EF2D − Enz ) kB T, and EF2D

is the Fermi energy in the presence of size quantization as measured from the edge of the conduction band in the vertically upward direction in the absence of any quantization. The evaluation of J2D in this case requires the expression of electron statistics, which can be expressed as:

n2D

nzmax ∞ 2gv  ∂A = f (E) dE. ∂E (2π)2

(1.62)

nz =1E

nz

Thus, one obtains:

n2D

nzmax m∗ kB Tgv  = F0 (η4 ). π2

(1.63)

nz =1

For bulk semiconductors having parabolic energy bands, converting the summation over nz to the corresponding integration over nz , under the condition of nondegeneracy, we can write from (1.61) that:

J2D

α0 ekB Tgv = e 2dz2

EF kB T

∞

−an2z

nz e

2 dnz , where a ≡ 2m∗

nzmin



π dz

2

(kB T)−1 . (1.64)

Performing the integration, (1.64) converts into the well-known form of the photoemission as given in the introduction of this chapter.

1.2.5 Photoemission from Quantum Well Wires of Wide Gap Materials The expression of 1D dispersion relation for QWWs of semiconducting materials whose bulk energy band structures are defined by the parabolic energy bands can be written as: 2 2 kx2 + Eny nz , Eny nz ≡ E= ∗ 2m 2m∗



ny π dy



2 +

nz π dz

2  .

(1.65)

1.2

Theoretical Background

21

The expression of the total DOS function can be written for the present case as: nymax nzmax  ∂kx   2gv  H E − En y n z N1DT (E) = π ∂E ny =1 nz =1

  nymax nzmax  H E − Eny nz 2gv √ ∗   2m . = h E − Eny nz

(1.66)

ny =1 nz =1

In Fig. 1.8, the normalized 1D DOS function versus normalized electron energy for the QWW of (a) InAs, (b) InSb, and (c) GaAs has been plotted. It appears that the 1D DOS function exhibits oscillatory dependence with energy, and the singularity points are determined by E = Eny nz . The expressions for the vz (Enz ), Enz , Enzmin , and nzmin , respectively, in this case are given by: 

2  Enz , m∗

2 nz π 2 Enz = , 2m∗ dz

vz (Enz ) =

(1.67) (1.68)

Fig. 1.8 Plot of the normalized 1D DOS function for the QWWs of (a) InAs, (b) InSb, and (c) GaAs as a function of normalized electron energy

22

1 Fundamentals of Photoemission



2 nzmin π 2 Enzmin = , 2m∗ dz √ dz 2m∗  (W − hv). nzmin ≥ π 

(1.69) (1.70)

Thus, using the appropriate equations, the 1D photoemission current from QWWs of wide gap materials can be written as [26]:   nymax nzmax ∞   2  nz π gv αo e  2 √ ∗ f (E) dE 2m  J1D = √ 2 m∗ 2m∗ dz h E − Eny nz ny =1 nzmin E ny nz (1.71) √ nymax nzmax  gv α0 e π kB T  = nz F− 1 (η5 ), √ 2 dz 2m∗ n =1 nz y

min

where η5 ≡

EF1D − Eny nz kB T

,

in which EF1D is the Fermi energy in the presence of 2D quantization of the wave vector space as measured from the edge of the conduction band in the vertically upward direction in the absence of any quantization. The electron concentration per unit length can be expressed as: ∞   

nymax nzmax

n1D =

N1D (E)f (E)dE.

(1.72)

ny =1 nz =1E

ny nz

From (1.72), one can write that: ∞

nymax n zmax  f (E) dE 2√ ∗   2m n1D = gv h E − Eny nz ny =1 nz =1E

ny nz

√

(1.73)

nymax nzmax

2gv 2m∗ πkB T   = F− 1 (η5 ). 2 h ny =1 nz =1

1.2.6 Photoemission from Quantum Dots of Wide Gap Materials The electron dispersion relation for the QDs of wide gap materials having parabolic energy bands can be written as: EQD

2 π 2 = 2m∗



nx dx



2 +

ny dy



2 +

nz dz

2  ,

(1.74)

1.2

Theoretical Background

23

where EQD is the totally quantized energy in this case. The total DOS function in this case is given by: nxmax nymax nzmax     2gv  δ  E − EQD , N0DT (E) = dx dy dz

(1.75)

nx =1 ny =1 nz =1

  where δ  E − EQD is the Dirac’s delta function. Figure 1.9 exhibits the plot of the normalized 0D DOS function versus normalized electron energy for QDs of n-GaAs. It appears that the density-of-states becomes a delta function, which is physically correct, since all components of the electron wave vector are quantized in QDs. The expressions of the 0D electron concentration per subband (n0D ), Enz ,nzmin , J0D , and total electron concentration (n0D ), for QDs of wide-gap materials can, respectively, be written as: 2gv F−1 (η) , dx dy dz

2 nz π 2 , E nz = 2m∗ dz

n0D =

(1.76)

(1.77)

Fig. 1.9 Plot of the normalized 0D DOS function for the QDs of GaAs versus normalized electron energy

24

1 Fundamentals of Photoemission

nzmin ≥ J0D

dz π

√

2m∗  (W − hv), 

(1.78)

nxmax nymax nzmax    αo eπgv  nz F−1 (η6 ) , = ∗ 2 m dx dy dz n

(1.79)

nxmax nymax nzmax   2gv  = F−1 (η6 ), dx dy dz

(1.80)

nx =1 ny =1

n0D

zmin

nx =1 ny =1 nz =1

where η6 ≡

EF0D − EQD , kB T

in which EF0D is the Fermi energy in the present case. For bulk specimens of nondegenerate wide-gap materials, converting the summations over nx , ny , and nz to the corresponding integrations over the said variables, (1.79) gets simplified to the well-known expression of the photoemission as given in the introduction.

1.2.7 Photoemission Under Magneto-Size Quantization (MSQ) The dispersion relation of the conduction electrons in QWs in ultrathin semiconducting films in the presence of a quantizing magnetic field B along the z-axis can be written as:



1 2 nz π 2 , ω0 + EMSQE = n + 2 2m∗ dz

(1.81)

where EMSQE is the totally quantized energy. The total DOS function in this case is given by: 

N0DE EMSQE



nnmax nzmax    eBgv  = δ  E − EMSQE . π

(1.82)

n=0 nz =1

It appears that the DOS becomes a delta function, which is physically correct, since all components of the electron wave vector are quantized in MQSE in a different way as compared with the corresponding QDs. In the case of MQSE, the unit of −1 −1   instead of eVm3 as in the case of QDs. Besides, the DOS function is eVm2 quantized energy points on the energy axis in QDs is determined by the equation E ≡ EQD , whereas the same for MQSE is determined by the equation E ≡ EMQSE . The velocity of the electron in the nthz subband is given by: 





2Enz vz Enz = √ , m∗

(1.83)

1.3

Results and Discussions

25

where

2 nz π 2 . 2m∗ dz The electron concentration per unit area is given by: Enz ≡

nMSQE =

nnmax nzmax    eBgv  F−1 ηMSQE , π n =n n=0

z

(1.84)

zmin

where  EFMSQE − EMSQE ≡ and EFMSQE kB T 

ηMSQE

is the Fermi energy in this case. Therefore, the current density can be written as [28]: JMSQE =

nnmax nzmax    e2 Bgv α0  nz F−1 ηMSQE , ∗ 2 m dz

(1.85)

n=0 nz =1

where 

nmax

nzmax

  √

EFMSQE − (W − hν) 2m∗ dz  1 ≤ − ,nzmin ≥ (W − hν) and ω0 2  π √ 

2m∗ dz 1 ≤ EFMSQE − n + ω0 .  π 2

1.3 Results and Discussions Using (1.12) and (1.19), and the table in Appendix A, we have plotted the normalized photoemitted current density as a function of the normalized incident photon energy from bulk samples of n-InSb, assuming parabolic energy bands in Fig. 1.10 by neglecting the nonparabolic behavior for the purpose of elementary presentation. For the fixed value of the electron concentration, it appears that the photoemitted current density increases with increasing incident photon energy for bulk materials. Using (1.37) and (1.42), we have plotted the normalized photoemitted current density in the presence of a quantizing magnetic field as a function of normalized incident photon energy for n-InSb as shown in Fig. 1.11. From Fig. 1.11 it is observed that the photoemission is an increasing function of the incident photon energy.

26

1 Fundamentals of Photoemission

Fig. 1.10 Plot of the normalized photocurrent density from bulk specimens of n-InSb as a function of normalized incident photon energy

Figure 1.12 exhibits the dependence of normalized magneto photocurrent density from n-InSb on the inverse quantizing magnetic field. It appears from Fig. 1.12 that the photoemission current density is an oscillatory function of an inverse quantizing magnetic field. The oscillatory dependence is due to the crossing over of the Fermi level by the Landau subbands in steps, resulting in a successive reduction of the number of occupied Landau levels with the increase in the magnetic field. For each coincidence of a Landau level with the Fermi level, there would be a discontinuity in the DOS function resulting in a peak of oscillation. The peaks should occur whenever the Fermi energy is a multiple of the energy separation between the two consecutive Landau levels. It may be noted that the origin of oscillations in the magneto-photoemission is the same as that of the SdH oscillations. The variations of the photoemission current density are periodic with the quantizing magnetic field. With an increase in the magnetic field, the amplitude of the oscillation will increase; and, ultimately, at very large values of the magnetic field, the conditions for the quantum limit will be reached when the photoemission current density will exhibit a monotonic dependence with an increase in the magnetic field.

1.3

Results and Discussions

27

Fig. 1.11 Plot of the normalized photocurrent density from n-InSb as a function of normalized incident photon energy in the presence of a quantizing magnetic field

Fig. 1.12 Plot of the normalized photocurrent density from n-InSb as a function of an inverse quantizing magnetic field

28

1 Fundamentals of Photoemission

Using (1.51) and (1.53), we have plotted the normalized J (E0 ,B) as a function of the normalized incident photon energy, as shown in Fig. 1.13 for n-InSb; and it appears that the J (E0 , B) increases with the increasing incident photon energy in the presence of crossed electric and quantizing magnetic fields. Using (1.61) and (1.63), in Figs. 1.14, 1.15, and 1.16, we have plotted the normalized photoemission current density as a function of dz , the normalized incident photon energy, and normalized electron degeneracy for the quantum wells in ultrathin films of n-GaAs, n-InAs, and n-InSb. It appears that photoemitted current density increases with decreasing film thickness, increasing photon energy, and increasing electron statistics in a step-like manner exhibiting the signature of 1D quantization of the wave vector space of the conduction electron. Using (1.71) and (1.73), the normalized photocurrent density from the QWWs of the said materials has been plotted in Figs. 1.17, 1.18, and 1.19 for all the cases of Figs. 1.14, 1.15, and 1.16, respectively. The variations of the said variable in QWWs with respect to film thickness, normalized incident photon energy, and the normalized electron degeneracy are more or less similar to the corresponding plots for QWs in UFs, as given in Figs. 1.14, 1.15, and 1.16; only the numerical magnitudes and the dimensions of the appropriate x- and y-axis are different. This is a consequence of 2D quantization of the wave vector space of the conduction electrons. Using (1.79) and (1.80), in Figs. 1.20, 1.21, and 1.22, we have plotted the normalized photoemission from the QDs of the aforementioned materials for all the cases mentioned in Figs. 1.14, 1.15, and 1.16, respectively. For QDs, the numerical values of the photoemitted current

Fig. 1.13 Plot of the normalized photocurrent density from n-InSb as a function of normalized incident photon energy under cross fields configuration

1.3

Results and Discussions

29

Fig. 1.14 Plot of the normalized photocurrent density from quantum wells in ultrathin films of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of film thickness

Fig. 1.15 Plot of the normalized photocurrent density from quantum wells in ultrathin films of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of normalized incident photon energy

30

1 Fundamentals of Photoemission

Fig. 1.16 Plot of the normalized photocurrent density from quantum wells in ultrathin films of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of normalized electron degeneracy

Fig. 1.17 Plot of the normalized photocurrent density from QWWs of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of film thickness

1.3

Results and Discussions

31

Fig. 1.18 Plot of the normalized photocurrent density from QWWs of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of normalized incident photon energy

Fig. 1.19 Plot of the normalized photocurrent density from QWWs of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of normalized electron degeneracy

32

1 Fundamentals of Photoemission

Fig. 1.20 Plot of the normalized photocurrent density from QDs of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of film thickness

Fig. 1.21 Plot of the normalized photocurrent density from QDs of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of normalized incident photon energy

1.3

Results and Discussions

33

Fig. 1.22 Plot of the normalized photocurrent density from QDs of (a) n-GaAs, (b) n-InAs, and (c) n-InSb as a function of normalized electron degeneracy

Fig. 1.23 Plot of the normalized photoemission current density from n-InSb in the presence of magneto-size quantization as a function of normalized incident photon energy

34

1 Fundamentals of Photoemission

densities change due to 3D quantization of the wave vector space of the conduction electrons for all the plots, as given in Figs. 1.14, 1.15, and 1.16. Using (1.84) and (1.85), we have plotted the oscillatory normalized photoemitted current density as a function of the normalized incident photon energy for n-InSb under magneto-size quantized conditions, as shown in Fig. 1.23. Moreover, the photoemission from quantum-confined compounds can become several orders of magnitude larger than that of bulk specimens of the same materials, which is also a direct signature of quantum confinement. The photoemitted current density under magneto-size quantized conditions increases with increasing photon energy in an oscillatory manner. Finally, we note that this oscillatory dependence will be less and less prominent with increasing film thickness; and ultimately, for bulk specimens of the same material, the photoemission will be found to increase continuously with increasing electron degeneracy in a nonoscillatory manner.

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36

1 Fundamentals of Photoemission

52. P. M. Solomon, Proc. IEEE, 70, 489 (1982); T. E. Schlesinger, T. Kuech, Appl. Phys. Lett. 49, 519 (1986). 53. H. Heiblum, D. C. Thomas, C. M. Knoedler, M. I. Nathan, Appl. Phys. Letts. 47, 1105 (1985). 54. O. Aina, M. Mattingly, F. Y. Juan, P. K. Bhattacharya, Appl. Phys. Letts. 50, 43 (1987). 55. I. Suemune, L. A. Coldren, IEEE J. Quant. Electron. 24, 1778 (1988). 56. D. Miller, D. Chemla, T. Damen, T. Wood, C. Burrus, A. Gossard, W. Weigmann, IEEE J. Quant. Electron. 21, 1462 (1985). 57. F. Sols, M. Macucci, U. Ravaioli, K. Hess, Appl. Phys. Lett. 54, 350 (1980). 58. C. S. Lent, D. J. Kirkner, J. Appl. Phys. 67, 6353 (1990). 59. C. S. Kim, A. M. Satanin, Y. S. Joe, R. M. Cosby, Phys. Rev. B, 60, 10962 (1999). 60. S. Midgley, J. B. Wang, Phys. Rev. B 64, 153304 (2001). 61. T. Sugaya, J. P. Bird, M. Ogura, Y. Sugiyama, D. K. Ferry, K. Y. Jang, Appl. Phys. Lett. 80, 434 (2002). 62. B. E. Kane, G. R. Facer, A. S. Dzurak, N. E. Lumpkin, R. G. Clark, L. N. Pfeiffer, K. N. West, Appl. Phys. Lett. 72, 3506 (1998). 63. C. Dekker, Physics Today, 52, 22 (1999). 64. A. Yacoby, H. L. Stormer, N. S. Wingreen, L. N. Pfeiffer, K. W. Baldwin, K. W. West, Phys. Rev. Lett. 77, 4612 (1996). 65. Y. Hayamizu, M. Yoshita, S. Watanabe, H. Akiyama, L. N. Pfeiffer, K. W. West, Appl. Phys. Lett. 81, 4937 (2002). 66. S. Frank, P. Poncharal, Z. L. Wang, W. A. de Heer, Science 280, 1744 (1998). 67. I. Kamiya, I. Tanaka, K. Tanaka, F. Yamada, Y. Shinozuka, H. Sakaki, Physica E 13, 131 (2002). 68. A. K. Geim, P. C. Main, N. La Scala, Jr., L. Eaves, T. J. Foster, P. H. Beton, J. W. Sakai, F. W. Sheard, M. Henini, G. Hill, M. A. Pate, Phys. Rev. Lett. 72, 2061 (1994). 69. A. S. Melnikov, V. M. Vinokur, Nature 415, 60 (2002). 70. K. Schwab, E. A. Henriksen, J. M. Worlock, M. L. Roukes, Nature 404, 974 (2000). 71. L. Kouwenhoven, Nature 403, 374 (2000). 72. S. Komiyama, O. Astafiev, V. Antonov, T. Kutsuwa, H. Hirai, Nature 403, 405 (2000). 73. E. Paspalakis, Z. Kis, E. Voutsinas, A. F. Terzis, Phys. Rev. B 69, 155316 (2004). 74. J. H. Jefferson, M. Fearn, D. L. J. Tipton, T. P. Spiller, Phys. Rev. A 66, 042328 (2002). 75. J. Appenzeller, Ch. Schroer, Th. Schapers, A. v. d. Hart, A. Fröster, B. Lengeler, H. Lüth, Phys. Rev. B 53, 9959 (1996). 76. J. Appenzeller, C. Schroer, J. Appl. Phys. 87, 3165 (2000). 77. P. Debray, O. E. Raichev, M. Rahman, R. Akis, W. C. Mitchel, Appl. Phys. Lett. 74, 768 (1999). 78. P. M. Solomon, Proc. IEEE 70, 489 (1982); T. E. Schlesinger, T. Kuech, Appl. Phys. Lett. 49, 519 (1986). 79. D. Kasemset, C. S. Hong, N. B. Patel, P. D. Dapkus, Appl. Phys. Letts. 41, 912 (1982). 80. K. Woodbridge, P. Blood, E. D. Pletcher, P. J. Hulyer, Appl. Phys. Lett. 45, 16 (1984). 81. D. Bimberg, M. Grundmann, N. N. Ledentsov, Quantum Dot Heterostructures (John Wiley and Sons, USA, 1999) 82. T. Tsuboi, Phys. Stat. Sol. (b), 146, K11 (1988) (and the references cited therein). 83. K. P. Ghatak, S. Bhattacharya, S. Pahari, D. De, S. Ghosh, M. Mitra, Annalen der Physik, 17, 195 (2008). 84. J. S. Blakemore, Semiconductor Statistics (Dover Publications, USA, 1987). 85. W. Zawadzki, In: Two Dimensional Systems, Hetrostructures and Superlattices, Edited by G. Bauer, F. Kuchar, H. Heinrich (Springer-Verlag, Germany, 1984).

Chapter 2

Fundamentals of Photoemission from Quantum Wells in Ultrathin Films and Quantum Well Wires of Various Nonparabolic Materials

2.1 Introduction In chapter 1, the photoemission from wide-gap materials having parabolic energy bands under different physical conditions has been studied. For the purpose of indepth study, in this chapter, the same has been investigated from QWs in UFs and QWWs of non-parabolic materials having different band structures. The journey towards the knowledge temple known as the photoelectric effect begins with the non-linear optical compounds which find applications in non-linear optics and light emitting diodes [1]. The quasi-cubic model can be used to investigate the symmetric properties of both the bands at the zone center of wave vector space of the same compound [2]. Including the anisotropic crystal potential in the Hamiltonian, and special features of the nonlinear optical compounds, Kildal [3] formulated the electron dispersion law under the assumptions of the isotropic momentum matrix and the isotropic spin orbit splitting constant, respectively, although the anisotropies in the two aforementioned band constants are the significant physical features of the said materials [4]. In Section 2.2.1, the photoemission from QWs in UFs and QWWs of nonlinear optical materials is investigated by considering the combined influence of the anisotropies of the said energy band constants together with the inclusion of the crystal field splitting. The III–V compounds finds extensive usage in infrared detectors [5], quantum dot light-emitting diodes [6], quantum cascade lasers [7], quantum well wires [8], optoelectronic sensors [9], high electron mobility transistors [10], etc. The III–V, ternary, and quaternary materials are called Kane-type compounds since their electron energy spectra are defined by the three-band model of Kane [11]. In Section 2.2.2, the photoemission from QWs in UFs and QWWs of III–V materials has been studied and the simplified results for the two-band model of Kane and that of wide gap materials have further been demonstrated as special cases. The II–VI compounds are being extensively used in nanoribbons, blue green diode lasers, photosensitive thin films, infrared detectors, ultra high-speed bipolar transistors, fiber optic communications, microwave devices, photovoltaic and solar cells, semiconductor gamma-ray detector arrays, and semiconductor detector gamma camera; and they allow for a greater density of data storage on optically addressed compact discs [12–14]. The carrier energy spectra in II–VI materials 37 K.P. Ghatak et al., Photoemission from Optoelectronic Materials and their Nanostructures, Nanostructure Science and Technology, DOI 10.1007/978-0-387-78606-3_2,  C Springer Science+Business Media, LLC 2009

38

2 Fundamentals of Photoemission from Quantum Wells

are defined by the Hopfield model [15], where the splitting of the two-spin states by the spin-orbit coupling and the crystalline field has been taken into account. Section 2.2.3 describes the investigation of the photoemission from QWs in UFs and QWWs of II–VI compounds. The n-Gallium Phosphide (n-GaP) is being used in quantum dots, light-emitting diodes [16], high efficiency yellow solid state lamps, light sources, and high peak current pulse for high gain tubes. The green and yellow light-emitting diodes made of nitrogen-doped n-GaP possess a longer device life at high drive currents [17]. In Section 2.2.4, the photoemission from QWs in UFs and QWWs of n-GaP has been studied. The importance of Germanium has been well known since the inception of transistor technology, and in recent years it has been used in memory circuits, single photon detectors, single photon avalanche diodes, ultrafast all-optical switches, THz lasers, and THz spectrometers [18–19]. The investigation of photoemission from QWs in UFs and QWWs of Ge has been presented in Section 2.2.5. The Platinum Antimonide (PtSb2 ) finds application in device miniaturization, colloidal nanoparticle synthesis, sensors and detector materials, and thermophotovoltaic devices [20]. It may also be noted that stressed materials are being widely investigated for strained silicon transistors, quantum cascade lasers, semiconductor strain gauges, thermal detectors, and strained-layer structures [21]. The photoemission from QWs in UFs and QWWs of PtSb2 and stressed materials (taking stressed n-InSb as an example) has, respectively, been investigated in Sections 2.2.6 and 2.2.7. In recent years, Bismuth (Bi) nanolines have been fabricated, and Bi also finds use in arrays of antennas, which leads to the interaction of electromagnetic waves with such Bi-nanowires [22]. Several dispersion relations of the carriers have been proposed for Bi. Shoenberg [23] experimentally verified that the de Haas–Van Alphen and cyclotron resonance experiments supported the ellipsoidal parabolic model of Bi, although the magnetic field dependence of many physical properties of Bi supports the two-band model [24]. The experimental investigations on the magneto-optical [25] and the ultrasonic quantum oscillations [26] support the Lax ellipsoidal nonparabolic model [24]. Kao [27], Dinger and Lawson [28], and Koch and Jensen [29] demonstrated that the Cohen model [30] is in conformity with the experimental results in a better way. Additionally, the hybrid model of bismuth developed by Takoka et al. also finds use in the literature [31]. McClure and Choi [32] devised a new model of Bi and they showed that it can explain the data for a large number of magneto-oscillatory and resonance experiments. In Section 2.2.8, we formulate the photoemission from QWs in UFs and QWWs of Bi in accordance with the aforementioned energy band models for the purpose of relative assessment. It has further been demonstrated that under certain limiting conditions all the results for all the models of 2D and 1D system reduce to the well-known result of the photoemission from bulk samples of non-degenerate wide gap materials. This above statement stands for the compatibility test of the theoretical analysis. With the discovery of carbon nanotubes (CNs) in 1991 [33], novel electronic properties pertaining to their particular tubular structures have found wide applications in modern quantum effect devices [34–39]. Such devices include nanooscillators, ultra-fast optical filters, nano bearings, nano probes, nano cantilevers, field

2.2

Theoretical Background

39

emission displays, nano gears, nano motors, and other nanoelectronic devices [40]. The CNs can be tailored into a metal or a semiconductor depending on the diameter and the chiral index numbers (m,n) [41], where the integers m and n denote the number of unit vectors along two directions in the honeycomb crystal lattice of graphene. For armchair and zigzag nanotubes, the chiral indices are given as m = n [41]. Another class of CN called as chiral CN has distinct integers m and n. Additionally, a CN can be a metallic if m − n = 3q, where q = 1,2,3 . . .; otherwise it is a semiconductor. Single-walled nanotubes (SWN) are a very important class of CNs and can be used as excellent conductors [42], thus miniaturizing electronics beyond the micro-electromechanical scale. One useful application of SWNs is in the development of the first intramolecular field effect transistors (FETs) [43]. Modern industrial technology predicts that traditional interconnects between devices will be a major performance and reliability bottleneck as size reduces. Metallic single-wall CNs (SWCNs) have received considerable attention as potential substitutions for traditional interconnect materials like Cu due to their excellent inherent electrical and thermal properties. Since the carriers are confined in a metallic SWCN [44], the inclusion of the subband energy owing to Born–Von Karman (BVK) boundary conditions [45] for their unique band structure becomes prominent. The quantization of function due to van Hove singularity (VHS) [2.46] of the wave vectors. The investigation of photoemission from carbon nanotubes has been presented in Section 2.2.9. The last section 2.3 contains the results and discussions pertaining to this chapter.

2.2 Theoretical Background 2.2.1 Photoemission from Nonlinear Optical Materials Considering the anisotropies of the effective electron mass and the spin orbit splitting constant, and including the influence of crystal field splitting, the electron dispersion law in bulk specimens of nonlinear optical materials within the framework of k.p formalism can be written as [47]: γ (E) = f1 (E) ks2 + f2 (E) kz2 ,

(2.1)

where 

       1 E + Eg0 E + Eg0 + || + δ E + Eg0 + || γ (E) ≡ E E + Eg0 3 

  2 + E E + Eg0 2|| − 2⊥ , 9 E is the electron energy as measured from the edge of the conduction band in the vertically upward direction in the absence of any quantization, Eg0 is the band gap

40

2 Fundamentals of Photoemission from Quantum Wells

in the absence of any external field, || and ⊥ are the spin orbit splitting constants parallel and perpendicular to the crystal axis, respectively, δ is the crystal field splitting constant,

−1 ! 

2 1 Eg0 + ⊥ δ E + Eg0 + || f1 (E) ≡  Eg0 Eg0 + ⊥ 3 3 



  2 2 + E + Eg0 E + Eg0 + || + 2|| − 2⊥ , ks2 ≡ kx2 + ky2 , 3 9 

−1 !   2 2 ∗ f2 (E) ≡  Eg0 Eg0 + || 2m|| Eg0 + || 3 

   2 E + Eg0 E + Eg0 + || , 3 2







2m∗⊥



and m∗|| and m∗⊥ are the effective masses of the electron at the edge of the conduction band parallel and perpendicular to the direction of the c axis, respectively. The 2D electron energy spectrum for QWs in UFs of nonlinear optical materials in the presence of size-quantization along the x-direction can be expressed as:

πnx γ (E) = f1 (E) dx

2 + f1 (E) ky2 + f2 (E) kz2 .

(2.2)

The subband energy (Enx ) is given by:     γ Enx = f1 Enx (π nx / dx )2 .

(2.3)

The condition for photoemission in this case assumes the form Enxmin + hv ≥ W,

(2.4)

where Enxmin can be expressed through (2.3) as:



 2 γ Enxmin = f1 Enxmin πnxmin / dx .

(2.5)

The total photoelectric current density from QWs in UFs of nonlinear optical materials in this case can, in general, be written as:

J2D

nxmax ∞   αo egv  = N2D (E) f (E) vx Enx dE, 2dx n xmin E nx

(2.6)

2.2

Theoretical Background

41

where nxmin is the nearest integer of the following inequality: nxmin

dx ≥ π



γ (W − hv) f1 (W − hv)

1/ 2 .

(2.7)

It appears that the formulation of J2D requires an expression of N2D (E) which can in turn be written in this case using (2.2) as: N2D (E) =

gv ∂ {φ1 (E,nx )} , (2π) ∂E



−1/2

(2.8)

where φ1 (E,nx ) ≡ f1 (E) f2 (E)





π nx γ (E) − f1 (E) dx

2  .

  The vx Enx in this case can be expressed as:   1   vx Enx = 1 Enx ,  ⎡ "   #  $ #  $ "   ⎤−1 f1 Enx γ Enx γ Enx f1 Enx     ⎦ f1 Enx . "   "   1 Enx ≡ ⎣ − 2 γ Enx 2 f1 Enx

(2.9)

Thus combining the appropriate equations, we get: J2D =

nxmax   αo egv  1 Enx [φ1 (EF2D ,nx ) + φ2 (EF2D ,nx )] , 2hdx n

(2.10)

xmin

where φ2 (EF2D ,nx ) ≡ mation,

%so

r=1 Zr2 [φ1 (EF2D ,nx )]

 Zr,Y ≡ 2 (kB T)

2r



1−2

1−2r

, s0 is the upper limit of the sum-

 ∂ 2r ζ (2r) , ∂EFYD 2r

Y = 2 and ζ (2r) is the Zeta function of order 2r [48]. The evaluation of J2D in this case requires the expression of electron statistics, which can be expressed using (2.8) as: n2D =

nxmax gv  [φ1 (EF2D ,nx ) + φ2 (EF2D ,nx )]. 2π nx =1

(2.11)

42

2 Fundamentals of Photoemission from Quantum Wells

In this context, the photocurrent from QWWs of nonlinear optical materials can be formulated in the following way: For electron motion along the x-direction only, the 1D electron dispersion law in this case can be written, following (2.1), as: 2  γ (E) = f1 (E) kx2 + f1 (E) πny / dy + f2 (E) (π nz / dz )2 .

(2.12)

  The subband energy E is given by the equation:   2     γ E = f1 E π ny / dy + f2 E (π nz / dz )2 .

(2.13)

The 1D DOS function per subband is given by: N1D (E) =

2gv ∂kx . π ∂E

(2.14)

The velocity of the emitted electrons along the x-direction can be written as: vx (E) =

1 ∂E .  ∂kx

(2.15)

Therefore the photocurrent is given by:

nymax nzmax ∞

 1 ∂E 2 ∂kx αo egv  f (E) dE, I= 2 π ∂E  ∂kx

(2.16)

ny =1 nz =1

1

where 1 ≡ E + W − hν. Using (2.16), one can write: nymax nzmax  αo egv kB T  F0 (η6 ) ,where η6 ≡ I= π ny =1 nz =1



  EF1D − E + W − hν . (2.17) kB T

This is the general expression of the photocurrent from QWWs, where EF1D and E are the two-band structure–dependent quantities. Thus, it appears that the evaluation of J1D requires an expression of carrier statistics which can, in turn, be written combining (2.12), (2.14), and the Fermi-Dirac occupation probability factor as: n1D =

2gv π

n ymax nzmax  ny =1 nz =1

     t1 EF1D ,ny ,nz + t2 EF1D ,ny ,nz ,

(2.18)

2.2

Theoretical Background

43

where 1/ 2    2  t1 EF1D ,ny ,nz ≡ γ (EF1D ) − f1 (EF1D ) π ny / dy − f2 (EF1D ) (π nz / dz )2  −1 2 f1 (EF1D ) / 



t2 EF1D ,ny ,nz ≡

S0 

   Zr,Y t1 EF1D ,ny ,nz , and Y = 1.

r=1

2.2.2 Photoemission from III–V Materials (a) Under the conditions || = ⊥ =  (the isotropic spin splitting constant), δ = 0, m∗|| = m∗⊥ = m∗ , (2.1) assumes the form: 2 k 2 2m∗

   E E + Eg0 E + Eg0 +  Eg0 + 13 

. = I (E) , I (E) ≡   Eg0 Eg0 +  E + Eg0 + 23 

(2.19)

Equation (2.19) describes the dispersion relation of the conduction electrons in III–V materials and is well known in the literature as the three-band model of Kane [11]. The dispersion relation of the 2D electrons in this case is given by: 2 ky2 2m∗

+

2 kz2 2 + (πnx / dx )2 = I (E) . ∗ 2m 2m∗

(2.20)

The subband energy can be written as:   2 I Enx = (πnx / dx )2. 2m∗

(2.21)

The velocity of the electron in the nth x band can be determined from the equation: 





vx Enx =

    2 2 Enx , 2 Enx ≡ ∗ m

"  &   I Enx I  Enx .

(2.22)

The photoemission in this case can be expressed as:

J2D

egv αo = π 2 dx



m∗ 2

1 n xmax 2 nxmin

  2 Enx [T3 (EF2D ,nx ) + T4 (EF2D ,nx )] ,

(2.23)

44

2 Fundamentals of Photoemission from Quantum Wells

where √ nxmin ≥

2m∗ 



dx π



T4 (EF2D ,nx ) ≡



2 I (W − hv),T3 (EF2D ,nx ) ≡ I (EF2D ) − 2m∗

sv 



π nx dx

2 

Zr,Y T3 (EF2D ,nx ) , and Y = 2.

r=1

The investigation of J2D requires an expression of electron statistics, which can be written as: n2D =

nxmax m∗ gv  [T3 (EF2D ,nx ) + T4 (EF2D ,nx )] . π2

(2.24)

nx =1

The photocurrent from QWWs of III–V materials has been discussed in accordance with the three-band model of Kane as follows: The one-dimensional electron dispersion law is given by:   2 kx2 + G ,n n = I (E) , 2 y z 2m∗       2 where G2 ny ,nz ≡ 2 π 2 /2m∗ ny / dy + (nz / dz )2 .

(2.25)

The subband energyE can be written as:     G2 ny ,nz = I E .

(2.26)

The photocurrent is given by (2.17), in which E is given by (2.26) and EF1D should be determined from the following equation:

n1D

√ nymax nzmax      2gv 2m∗  = t3 EF1D ,ny ,nz + t4 EF1D ,ny ,nz , π

(2.27)

ny =1 nz =1

where     1 2 t3 EF1D ,ny ,nz ≡ I(EF1D ) − G2 ny ,nz / , 



t4 EF1D ,ny ,nz ≡

S0 

   Zr,Y t3 EF1D ,ny ,nz , and Y = 1.

r=1

(b) Under the inequalities  >> Eg0 or