Perspectives in quantum Hall effects: novel quantum liquids in low-dimensional semiconductor structures

PERSPECTIVES IN QUANTUM HALL EFFECTS PERSPECTIVES IN QUANTUM HALL EFFECTS Novel Quantum Liquids in Low-Dimensional Se...

Author: Sankar Das Sarma | Aron Pinczuk

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PERSPECTIVES IN QUANTUM HALL EFFECTS

PERSPECTIVES IN QUANTUM HALL EFFECTS Novel Quantum Liquids in Low-Dimensional Semiconductor Structures

Edited by

Sankar Das Sarma Aron Pinczuk

WILEYVCH

Wiley-VCH Verlag GmbH & Co. KGaA

Cover illustration Sample of silicon on which the Quantum Hall Effect was verified by Klaus von Klitzing in 1980 (courtesy of Deutsches Museum, Bonn) All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Library of Congress Card No.: Applied for British Library Cataloging-in-Publication Data: A catalogue record for this book is available from the British Library Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at . 0 1997 by John Wiley & Sons, Inc. 0 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form - nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printed in the Federal Republic of Germany Printed on acid-free paper

Printing and Bookbinding buch biicher dd ag, Birkach ISBN-13: 978-0-471-11216-7

ISBN-10: 0-471-1 1216-X

CONTENTS xi xiii

Contributors Preface 1 Localization, Metal-Insulator Transitions, and Quantum Hall Effect

1

S. Das Sarma

Introduction 1.1.1. Background 1.1.2. Overview 1.1.3. Prospectus 1.2. Two-Dimensional Localization: Concepts 1.2.1. Two-Dimensional Scaling Localization 1.2.2. Strong-Field Situation 1.2.3. Quantum Hall Effect and Extended States Scaling Theory for the Plateau Transition 1.2.4. 1.2.5. Disorder-Tuned Field-Induced Metal-Insulator Transition 1.3. Strong-Field Localization: Phenomenology 1.3.1. Plateau Transitions: Integer Effect 1.3.2. Plateau Transitions: Fractional Effect 1.3.3. Spin Effects 1.3.4. Frequency-Domain Experiments 1.3.5. Magnetic-Field-InducedMetal-Insulator Transitions 1.4. Related Topics 1.4.1. Universality 1.4.2. Random Flux Localization References 1.1.

2 Experimental Studies of Multicomponent Quantum Hall Systems J . P. Eisenstein

2.1. 2.2.

Introduction Spin and the FQHE 2.2.1. Tilted Field Technique Phase Transition at v = 815 2.2.2. The v = 512 Enigma 2.2.3.

1 1 2 4 5 5 7 9 12 16 18 18 21 22 23 23 28 28 30 31

37 37 38 39 40 45 V

vi

CONTENTS

FQHE in Double-Layer 2D Systems 2.3.1. Double-Layer Samples 2.3.2. The v = 1/2 FQHE 2.3.3. Collapse of the Odd Integers 2.3.4. Many-Body v = 1 State 2.4. Summary References

2.3.

3 Properties of the Electron Solid

49 50 51 56 58

66 67 71

H. A. Fertig

3.1.

Introduction 3.1.1. Realizations of the Wigner Crystal 3.1.2. Wigner Crystal in a Magnetic Field 3.2. Some Intriguing Experiments 3.2.1. Early Experiments: Fractional Quantum Hall Effects 3.2.2. Insulating State at Low Filling Factors: A Wigner Crystal? 3.2.3. Photoluminescence Experiments Disorder Effects on the Electron Solid: Classical Studies 3.3. 3.3.1. Defects and the State of the Solid 3.3.2. Molecular Dynamics Simulations 3.3.3. Continuum Elasticity Theory Analysis 3.3.4. Effect of Finite Temperatures 3.4. Quantum Effects on Interstitial Electrons 3.4.1. Correlation Effects on Interstitials: A Trial Wavefunction 3.4.2. Interstitials and the Hall Effect 3.5. Photoluminescence as a Probe of the Wigner Crystal 3.5.1. Formalism 3.5.2. Mean-Field Theory 3.5.3. Beyond Mean-Field Theory: Shakeup Effects 3.5.4. Hofstadter Spectrum: Can It Be Seen? 3.6. Conclusion: Some Open Questions References

4 Edge-State Transport

71 72 73 74 74 75 79 81 81 82 86 90 91 92 95 97 97 99 100 103 104 105 109

C. L. Kane and Matthew P. A . Fisher

4.1 4.2.

Introduction Edge States 4.2.1. IQHE 4.2.2. FQHE

109 114 114 119

CONTENTS

Randomness and Hierarchical Edge States The v = 2 Random Edge Fractional Quantum Hall Random Edge Finite-Temperature Effects 4.4. Tunneling as a Probe of Edge-State Structure 4.4.1. Tunneling at a Point Contact 4.4.2. Resonant Tunneling 4.4.3. Generalization to Hierarchical States 4.4.4. Shot Noise Summary 4.5. Appendix: Renormalization Group Analysis References 4.3.

4.3.1. 4.3.2. 4.3.3.

5 Multicomponent Quantum Hall Systems: The Sum of Their Parts and More

vu

126 127 132 135 136 138 145 151 152 154 156 157 161

S. M . Girvin and A. H . MacDonald

5.1. 5.2 5.3. 5.4. 5.5.

Introduction Multicomponent Wavefunctions Chern-Simons Effective Field Theory Fractional Charges in Double-Layer Systems Collective Modes in Double-Layer Quantum Hall Systems 5.6. Broken Symmetries 5.7. Field-Theoretic Approach 5.8. Interlayer Coherence in Double-Layer Systems Experimental Indications of Interlayer Phase 5.8.1. Coherence Effective Action for Double-Layer Systems 5.8.2. 5.8.3. Superffuid Dynamics 5.8.4. Merons: Charged Vortex Excitations 5.8.5. Kosterlitz-Thouless Phase Transition 5.9. Tunneling Between the Layers 5.10. Parallel Magnetic Field in Double-Layer Systems 5.11. Summary References 6 Fermion Chern-Simons Theory and the Unquantized Quantum Hall Effect B. I . Halperin 6.1. 6.2. 6.3. 6.4.

Introduction Formulation of the Theory Energy Scale and the Effective Mass Response Functions

161 165 169 169 172 180 185 192 193 196 199 203 206 209 213 216 218 225 225 227 230 233

viii

CONTENTS

Other Fractions with Even Denominators Effects of Disorder Surface Acoustic Wave Propagation Other Theoretical Developments 6.8.1. Asymptotic Behavior of the Effective Mass and Response Functions 6.8.2. Tunneling Experiments and the One-Electron Green’s Function 6.8.3. One-Particle Green’s Function for Transformed Fermions 6.8.4. Physical Picture of the Composite Fermion 6.8.5. Edge States 6.8.6. Bilayers and Systems with Two Active Spin States 6.8.7. Miscellaneous Calculations 6.8.8. Finite-System Calculations 6.9. Other Experiments 6.9.1. Geometric Measurements of the Effective Cyclotron Radius R: 6.9.2. Measurements of the Effective Mass 6.9.3. Miscellaneous Other Experiments 6.10. Concluding Remarks References 6.5. 6.6. 6.7. 6.8.

7 Composite Fermions

238 24 1 243 247 247 249 25 1 252 253 254 254 254 255 255 256 257 258 259 265

J . K . Jain

7.1. 7.2

7.3

7.4

Introduction Theoretical Background 7.2.1. Statement of the Problem 7.2.2. Landau Levels 7.2.3. Kinetic Energy Bands 7.2.4. Interactions: General Considerations Composite Fermion Theory 7.3.1. Essentials 7.3.2. Heuristic Derivation 7.3.3. Comments Numerical Tests 7.4.1. General Considerations 7.4.2. Spherical Geometry 7.4.3. Composite Fermions on a Sphere 7.4.4. Band Structure of FQHE 7.4.5. Lowest Band 7.4.6. Incompressible States 7.4.7. CF-Quasiparticles

265 267 267 268 269 270 270 270 273 275 278 278 279 280 28 1 282 283 284

CONTENTS

ix

7.4.8. Excitons and Higher Bands 7.4.9. Low-Zeeman-Energy Limit 7.4.10. Composite Fermions in a Quantum Dot 7.4.11. Other Applications 7.5. Quantized Screening and Fractional Local Charge 7.6. Quantized Hall Resistance 7.7. Phenomenological Implications 7.7.1. FQHE 7.7.2. Transitions Between Plateaus 7.7.3. Widths of FQHE Plateaus 7.7.4. FQHE in Low-Zeeman-Energy Limit 7.7.5. Gaps 7.7.6. Shubnikov-de Haas Oscillations 7.7.7. Optical Experiments 7.7.8. Fermi Sea of Composite Fermions 7.7.9. Resonant Tunneling 7.8. Concluding Remarks References

285 288 290 292 293 294 295 295 297 297 297 297 298 298 298 299 300 302

8 Resonant Inelastic Light Scattering from Quantum Hall Systems

307

A . Pinczuk

8.1. 8.2. 8.3.

Introduction Light-Scattering Mechanisms and Selection Rules Experiments at Integer Filling Factors 8.3.1. Results for Filling Factors v = 2 and v = 1 8.3.2. Results from Modulated Systems 8.4. Experiments in the Fractional Quantum Hall Regime 8.5. Concluding Remarks References 9

Case for the Magnetic-Field-Induced Two-Dimensional Wigner Crystal

307 311 317 319 326 331 337 338

343

M. Shayegan

9.1. 9.2.

Introduction Ground States of the 2D System in a Strong Magnetic Field 9.2.1. Ground State in the v 1)

14

LOCALIZATION. METAL-INSULATOR TRANSITIONS

Several different groups [20,21,35,39-421 have carried out finite-size scaling calculations using somewhat differentmodels of the random disorder potential. The agreement among the numerical results of different groups is excellent, and the localization critical exponent for short-range white-noise disorder is found to be (neglecting Landau level coupling) ~=2.3fO.1 w 5.5 k 0.5

for N = O for N = 1

with E, N (N + 1/2)hw,. For white-noise disorder potential with a finite range, the critical exponent in the lowest Landau level remains essentially unchanged, whereas xN = is found to decrease appreciably,from 5.5 to roughly 2.8, when the disorder range equals or exceeds the magnetic length [21,40-42). Calculations including Landau level coupling [21] also reduce the critical exponent xN= 1. While the possible Landau level (in)dependenceof x will be discussed later in the chapter, it should be emphasized that finite-size scaling calculations in higher Landau levels,and/or for finite range disorder potential, and/or in the presence of Landau level coupling are necessarily less accurate than the uncorrelated (zero range) white-noise disorder calculations in the lowest Landau level, which yield a value of x N 2.3. In fact, the localization calculations for higher Landau levels can be carried out only up to a maximum M / t 10, whereas the lowest Landau level calculations extend to M/( 100. The maximum system sizes one can use for higher Landau level/finite range disorder/Landau level coupling situations are typically M 16 - 64, which is substantially less than M 512, the maximum possible system size for short-range lowest-Landau-levelcalculations.Thus the calculated critical exponent xo=2.3 in the lowest Landau level is more reliable than the calculated exponent x1 = 5.5 in the first excited level. Systematic studies [20,21] of the dependence of the critical exponent on the strength and type (attractive/repulsive,etc.) of disorder have been carried out. These studies are somewhat incomplete, and their results are consistent with the conclusion that the localization exponent x is universal and independent of the strength or type of disorder potential. In practice, however, a weak dependence of the calculated exponent on the details of disorder potential is found in numerical simulations [21] and attributed (rather uncritically) to crossover effects. The localization critical exponent for the plateau transition has also been calculated using a quantum percolation network model [43,44], where the random potential is assumed to vary slowly on the scale of the magnetic length and the electron guiding centers follow the semiclassicalequipotential contours of the smooth disorder potential. If quantum tunneling effects are neglected,the problem reduces to a two-dimensional classical percolation problem, where in the symmetric situation, there is exactly one energy at the percolation threshold, where the critical percolation cluster encompasses the entire system. If one identifies [36,43,44] this percolation transition as the delocalizationtransition at E,, onegets x = 4/3. It turns out that quantum tunnelingeffectscan be included in this percolation picture through a quantum network model where the nodes of

-

-

-

-

TWO-DIMENSIONALLOCALIZATION CONCEPTS

15

the network represent saddle points in the disorder potential and the links represent the equipotentials. Transfer matrix calculations [43,44] based on such a quantum percolation network model yield x N 2.5 k 0.5, in agreement with direct calculation of the lowest-Landau-level Lyapunov exponent in the uncorrelated white-noise disorder potential. There are theoretical argument [37], not entirely convincing, which also claim to show that the classical two-dimensional percolation exponent x = 4/3 is modified by quantum tunneling effects to a quantum percolation exponent of x = 4/3 + 1 = 7/3 = 2.33.. .,in agreement with the numerical simulation results both for the quantum percolation network model and the uncorrelated white-noise disorder Anderson model. The numerical results [443 for the percolation network model give an exponent x N 1.3 x 4/3, in agreement with the theoretical [36] classical percolation exponent in the limit of vanishing quantum tunneling. This classical fixed point is, however, unstable to quantum effects, which change the exponent to x x 2.5, as discussed above. The fact that two seeminglyvery different models of disorder (i.e., uncorrelated white noise and smooth long range) lead to identical critical behavior [45] is extremely interesting and argues in favor of the localization exponent x being universal in the quantum Hall plateau transition. It should be noted that the numerically obtained scaling functions in the lowest Landau level are also consistent in the uncorrelated white-noise disorder and quantum percolation network models. This lends additional support for universality in the Landau level localization problem, at least for the lowest Landau level. It should be mentioned that in a different context, in three-dimensional situations involving zero magnetic fields, it has been claimed [46] that the quantum percolation transition belongs to a different universality class than that of white-noise nonlinear sigma model localization. Quantum critical phenomena are characterized [25] by the inherent mixing of statics and dynamics due to quantum fluctuation effects, and the dynamical exponent z enters quantum critical problems in a natural way, describing the dispersion of the critical-mode fluctuations. At finite temperatures/frequencies the scaling behavior as a function of temperature or frequency involves the exponent K = (xz)- l , which is a composite of the localization exponent x and the dynamical exponent z. The easiest way to see this is to consider a finitetemperature experiment where the magnetic field B is being varied to make the chemical potential move through E, to cause delocatization-localization transition. The scaling part of the resistivity p(B, T) behaves in two dimensions as p(B, T )= p(b''X 6B, bZT)

where b is a scale factor and 6 B = ( B - BJB, is the scaling variable. The B and T dependence are combined by the choice b = T - ' I z , leading to p(B, T ) = p(GB/T"),

where K = (xz)-

Thus the temperature scaling occurs according to the exponent IC = l/xz.

16

LOCALIZATION,METAL-INSULATOR TRANSITIONS

Currently, there is no true theory for the dynamical exponent z in the quantum Hall plateau transitions. It has been argued [33] that z = 1 in all quantum phase transitions involving Coulomb systems, based essentially on the fact that Coulomb interaction scales as the inverse of a distance. No real calculations exist establishing z = 1 in quantum Hall plateau transitions. It should be noted that taking z = 1 and x = 7/3, one obtains K = 1/xz = 317 = 0.43, which agrees with temperature (and frequency) scaling experiments discussed in Section 1.3, 1.2.5. Disorder-Tuned Field-Induced Metal-Insulator Transition

As has been emphasized throughout this chapter, all two-dimensionalsystems(in orthogonal and unitary ensembles) are localized in the absence of any magnetic field according to the scaling theory of localization and the associated nonlinear sigma model-based field theory. For weak disorder (k,l>> 1, where k, and 1 are the electron Fermi wavevector and mean free path, respectively)the localization is very weak (only logarithmic correction to the mean-field resistance), with a crossover to strong localization as disorder becomes strong, k,l 1 and k , l c 1) is only a disorder-tuned crossover effect; however, operationally and from a practical experimental viewpoint [28,47], increased disorder causes a rather sharp metal-to-insulator transition in two-dimensional semiconductor-based heterostructures, where the mobility drops abruptly to zero at low temperatures, and the conduction on the insulating strongly localized (i.e., high disorder/low density k,l > 1 to k , l c 1)looks very much like a finite-temperature quantum phase transition in both two- and three-dimensional systems. The usual samples in which quantized Hall effect experiments are carried out are metallic (low disorder, weakly localized) at zero field, with their effective scaling theory-based localization length being many orders of magnitude larger than the effective system size. Thus the fact that the application of an external magnetic field gives rise to delocalized states at Landau level centers, while being very interesting and intriguing theoretically, is not a great experimental surprise because the system even at zero magnetic field has effectiveextended states at the Fermi level. A far more interesting situation would be to pick a system that is strongly localized (k,l L), the experimental scaling of the

'.

-

STRONG-FIELD LOCALIZATION: PHENOMENOLOGY

21

peak width AB against the system size produced [67,68,75] a best fit x x 2.3, essentially independent of the Landau level (01,l and 11) index and disorder (mobility 15,000 to 50,000 cm2V - s- ') provided that the spin splitting was resolved. Careful temperature-dependent scaling analysis in the same samples at higher temperatures (so that L, < L), where L, introduces the finite-size cutoff, also produced values of K = 4/x = l/z,x, so that 4(z), and x can be determined independently. The resulting values of 4 = z; show some variations with Landau level index and disorder with 4 x 1.2 to 1.7, which is quite different from the value 4 x 1 obtained in the earlier [65,66] temperature-dependent studies. The main difference is that K x 0.64 in these experiments [67,68,75] as against K x 0.43 in the earlier ones [65,66]. The reason for this discrepancy is not understood and is attributed uncritically to variations in the dynamical exponent zp There are two possible problems with the length-dependent study [75]. First, the scaling IABI ILI-''X is obviously very limited because there are only four values of L available. Second, the phase-coherent regime L, > L should exhibit fluctuations in transport properties because of the lack of ensemble averaging, which could complicate considerably the scaling analysis of the plateau transitions. For reasons not completely understood, conductance fluctuations in the experiments (in the L, > L regime) were small. Finally, the measured u,","^at the conductivity peak was substantially lower (in the range 0.1 to 0.2e2/h)than the theoretically expected 0.5e2/h value. (The basis of this theoretical expectation, except for some limited numerical simulations [34,76], is essentially a dimensional argument [15- 171.) Temperat ure-dependent scalinganalysis of strong-field magnetotransport has recently been extended [77] to the temperature dependenceof the u!$ima value in the quantized Hall plateau regime. In the activated transport regime, this [77] analysis gives a value of K x 0.3 to 0.7, which is consistent with the K values obtained by studying the plateau transitions.

r,

-

-

1.3.2. Plateau Transitions: Fractional Effect There has so far been only one reported experimental study [78] of temperature scaling in the fractional Hall plateau transition. The study was carried out for the plateau transition between v = 1/3 and v = 2/5 fractional quantized Hall states in high-mobility GaAs/AlGaAs systems with a reported value of K x 0.4 very close to that in the integer plateau transition. Because of the order-of-magnitudesmaller energy gap in the fractional state compared with the integral effect, the temperature range for fractional scaling is necessarily quite limited, and the value of K x 0.4 is not as reliable as it is for the integer effect. It should also be emphasued that temperature scalingin the GaAs/AlGaAsinteger plateau transition has not yet led to a decisive universal value of K, presumably due to long-range disorder fluctuations inherent in high-mobility modulation-doped structures. Thus the universality in K between integer and fractional plateau transitions is not really unambiguously established because the low-mobility samples [66],

22


which show excellent integer scaling, do not exhibit any fractional quantized Hall state. More experiments are clearly needed to clarify this unsatisfactory state of affairs. If the universality in K between integer and fractional effects holds up and leads to universality in the localization exponent x (assuming z to be the same for both effects), then interaction seems to be an irrelevant perturbation for the quantum Hall localization transition (assuming, of course, that the value of the exponent x is indeed around 2.3, as given by numerical simulations based on noninteracting models). This would be a rather remarkable result because in most (if not all) field theories [25] of the quantum localization problem, interaction is found to be a relevant perturbation which nontrivially changes the noninteracting localization expoents. There has been one numerical calculation [79] of the exponent 1, including interaction effects in a mean-field HartreeFock theory, which, not surprisingly, finds x = 2.3, the same exponent as for the corresponding noninteracting theory. This result is expected because the Hartree-Fock theory is an effective one-electron theory that should not change the exponent. There have also been some heuristic corresponding states arguments supporting the universality of the localization transition in integer and fractional cases, based on the composite fermion picture [80]. 1.3.3. Spin Effects The experimental results discussed so far are all for spin-split Landau levels. When the spin splitting is not resolved, the experimental value of K as obtained from temperature scaling of the plateau transition seems to be roughly half of that in the spin-split situation in all the experimentalstudies,(i.e., AB seems to scale as TK12in the spin-degeneratesituation) [Sl,821.Assuming the dynamical exponent z to be spin independent, this implies that the localization exponent x is twice in the spin-degenerate situation compared with the spin-resolved situation. This is very difficult to understand theoretically because within the noninteracting localization picture it is very hard to see how electron spin could be a relevant perturbation. While the experimental situation for temperature scaling in the spin-split case still remains a puzzle, a theoretical resolution has recently been suggested. It has been proposed [21] that the spin-splitting AE* between the neighboring spin levels in the spin-degeneratesituation is small but finite with A E , < k,T as well as AE* < r,where r i s typical Landau level broadening. Thus there are actually two critical energiesE,, in the spin-degeneratecase which are experimentallyunresolved because their separation, AEi, is much smaller than the experimental temperature. It is easy to see that this will lead to an effective temperature exponent ~ / if2the two unresolved transitions are interpreted as being a single localization transition. Direct numerical simulations [83] support this theoretical idea. To check this idea one needs to do temperature-dependent scaling experiments down to low temperatures (k,T < A&), whence the effective exponent ~ / should 2 change to K.

STRONG-FIELD LOCALIZATION PHENOMENOLOGY

23

1.3.4. Frequency-Domain Experiments In a remarkable recent experiment [84], the idea of quantum localization and critical scaling as applied to the plateau transition has been verified by carrying out finite-frequency( -GHz) low-temperature (-0.05 to 0.5 K) measurements of the dynamical magnetoconductance in the transitions between integer Hall plateaus in low-mobility GaAs/AlGaAs heterostructures. In these experiments the effective cutoff is provided by hw (provided that k,T k,T/h) (ABIccwY

(24)

instead of (LIB(K T" as in the temperature scaling experiments (which are recovered here in the k,T > hw regime). In analogy with the finite-temperature measurement, one has

Y = (xz,)where z , is the frequency-domain dynamical exponent. For quantum localization phenomena, temperature and frequency should behave similarly and one expects z,

=ZT =z

(26)

where z is the dynamical exponent for the localization transition. All these expectations were verified spectacularly in this important experiment [84], where the frequency scaling exponent y = 0.41 f 0.04 was found in the spinresolved situation and an exponent y/2 N 0.21 was found in the spin-degenerate situation, in excellent agreement with the temperature scaling exponent K z 0.43. If the localization exponent 1, which should be the same in temperature and frequency scaling experiments, is taken to be the accepted value of 7/3, this implies that z N 1 for quantum Hall plateau transitions.

1.3.5. Magnetic-Field-Induced Metal-Insulator Transitions There has been a great deal of recent experimental activity in studying the direct transition between a zero-field two-dimensionalinsulating phase and a finite field quantum Hall phase in highly disordered low-density samples. Because of the essential requirement of the existence of extended states for the quantized Hall effect phenomenon, these experiments can be thought of as a field-induced two-dimensional delocalization transition [IS). Several groups have carried out such experimentsin the last few years, and the interpretation of these experiments is a matter of some controversy at the present time [50-56,61-641. The most spectacular and convincing experiments [50-53,641 of this type start with a zero-field highly disordered and low-density strongly localized insulating system which eventually exhibits a quantum Hall phase at finite magnetic

24


fields before becoming strongly insulating again at very high fields. There are, therefore, at least two nonuniversal critical fields, B,, and B,, > B,,,in these experiments: For B < B,, and B > B,, the system is strongly insulating, as manifest in the temperature dependence of pxx(T), with pxx(T) increasing (exponentially) strongly with decreasing temperature (typically in the T 0.05 to 1K regime, but higher temperatures have also been used), usually the high-field regime (B> B,,) being even more strongly localized (with very high pxx values) than the low-field regime (B < B,,); for B,,< B < BE,, the magnetoresistance pxx(T)shows a distinct minimum at a particular magnetic field value, with the minimum becoming sharper with decreasing temperature as pxx(7')increases at higher temperature, and p,,(T) shows a quantized Hall plateau corresponding to the pxx(T)minima. The intermediate quantum Hall phase (for B,,< B < B,,), which clearly arises from field-induceddelocalization at B = Bcl,is novel and has not been seen in three-dimensional systems. The nonuniversal critical fields B,, and B,, which define the delocalized state locations depend on the disorder strength, and pxx(T)is essentially temperature indepedent at B = Bcl,B,,. The quantization of the Hall resistance pxyin the B,, < B < B,, regime is quite good, but perhaps not as perfect as in samples of better quality. These field-induced metal-insulator transition experiments are important because they establish beyond any reasonable doubt that nonperturbative delocalization (not found within the prevailing nonlinear sigma model localization theories) indeed occurs (even) in a (strongly) disordered two-dimensional system, due to the application of an external magnetic field. Unambiguous observation of a quantum Hall phase straddled on both low- and high-field sides by strongly insulating phases is a rather spectacular phenomenon, sometimes referred to as a reentrant insulating transition. Such reentrant insulating transitions have been observed in both integer [50,53,64] and fractional [63,85-871 quantum Hall states. Developing a unified quantitative theory for a field-induced metal-insulator transition (or, more precisely, quantum Hall conductor-Anderson insulator transition) has been difficult and controversial, partly because the transport measurements described above allow for competing and conflicting physical interpretations. A popular and physically appealing explanation [15-17] is based on the idea of the floating of strong-field extended states as the magnetic field is lowered. In particular, the extended-state E," associated with the Nth Landau level is postulated [16] to float up in energy according to the heuristic formula

-

E," = EN[l + ( ~ , z ) - ~ ] = hw,[l

+(WCz)-2](N

+ 1/2)

(27)

where z is an effective scattering time related to the effective Landau level broadening r = h/22. No microscopic derivation of Eq. (27) is available, but presumably a strong Landau level coupling effect in low magnetic fields (particularly at high disorder, whence q z v > 1, measured [20] with a 2DES sample having a lowtemperature mobility of about 7 x lo6cm'/V.s and density N, = 2.3 x 10" cm- '. For these data the magnetic field is perpendicular to the 2D plane. The location of the v = 8 / 5 state (at B = 5.95 T) is indicated by the dashed lines; numerous other FQHE states are also evident in the data. Upon tilting the sample, the 8/5 state initially weakens. At about 0 = 25", a weak satellite resistivity minimum appears about 1% higher in magneticfield than the main 8/5 p,, minimum. Increasing the angle further, to about 0 = 30", results in two minima of comparable strength whose field positions straddle the location of v = 8/5. At this angle, the resistivity precisely at v = 8/5 exhibitsa local maximum. Increasing the angle further reverses these trends. The high-field component of the doublet becomes dominant and gradually centers on v = 8/5. Beyond about 0 = 37" a single 8/5 minimum exists and becomes steadily stronger as 0 is increased further. In both the low- and

41

SPIN AND THE FQHE

.1000

1

T r

Y

W

800

0

z a

-

c

E

L 0

5/8

>. 600

t >

W

t;; z

;

a J

112

i

400

513

I

I

a 200

6 6.5 ? MAGNETIC FIELD (tesla)

7.5

Figure 2.1. Diagonal and Hall resistivity data, taken at T = 25 mK,in the filling factor range 2 > v > 1. Dashed lines indicate the v = 8/5 FQHE state. (After Ref. [20].)

high-angle regimes, the Hall resistance exhibits a plateau at pxy= 5h/8e2. The qualitative behavior of the 815 resistivity minimum is depicted in the left panels of Fig. 2.2 (where pxx is plotted against filling factor v rather than magnetic field). The qualitative interpretation [20] of these data is that for this sample, the v = 815 state is making a phase transition from a nonfully polarized ground state at small tilt angles to a polarized state at larger angles. This is just the scenario, discussed at the beginning of the section,which was predicted for the v = 2/5 state. In fact, the 815 and 215 states are closely connected since v = 815 215 = 2 represents the fully filled lowest Landau level. Provided that Landau level mixing can be ignored, particle-hole symmetry dictates that states at v and 2 - v are in fact “identical.” To make the case for a spin phase transition more convincing, a well-defined model for the tilted field behavior of both phases needs to be given and compared quantitatively to experiment. For this we turn to measurements of the energy gap of the quantized Hall state, determined from the activated temperature dependence of the resistivity. Figure 2.2 shows representative Arrhenius plots [log(p,,) versus 1/T] at v=8/5 in both the low- and high-angle phases. The conventional interpretation of the linear portion of these plots is that pxx= const. x exp( - A/27‘), with A being the fundamental energy gap of the FQHE state (i.e., the energy required to excite a well-separated quasielectron-quasihole

+

42

MULTICOMPONENT QUANTUM HALL SYSTEMS

1000

1

FILLING FACTOR

TEMPERATURE" ( K-'

1

Figure 2.2. (Left)Expanded views of p,, versus filling factor v in a narrow range around v = 8/5 at T = 30 mK. Note the splitting of the 8/5 minimum at 8 = 30"(Right) Arrhenius plots for 8/5 minimum at various angles. Curve a, 8 = 0";b, 8 = 18.6"; c, 8 = 42.4"; d, 8 = 49.5'. (After Ref. [20].)

pair). Figure 2.3 shows the observed tilted field dependence of A. The perpendicular magnetic field is fixed at B, = 5.95 T for these data; the gap has been plotted against the total magnetic field B,,, in the figure. This is the natural choice since the Zeeman energy is proportional to B,,,. At small tilt angles A falls linearly with Btot,while in the high-angle phase it rises linearly with B,,,. The magnitudes of the slopes I aA/aB,,, I are nearly equal in the two phases. In essence, the tilted field dependence of the excitation gap is determined by the difference in spin AS (in units of h) of the quantum liquid state before and after excitation of a quasielectron-quasihole pair. In the simplest model we write

where A. is the gap in the absence of the Zeeman energy, p B = 0.67 K/T is the Bohr magneton, and lgl- 0.44 is the bulk GaAs g factor [28]. The quantity A. is in general unknown, but in this simple model it is assumed to depend only on the perpendicular magnetic field (i.e., on the filling factor v), and this is kept fixed in a tilt experiment. We assume further that the quasiparticles are fundamentally spin-1/2 objects [29]. For a maximally spin-polarized quantum Hall ground

SPIN AND THE FQHE

0 15

-

c

Y

>.

0.6

-

8 (degrees)

30

43

45

B i t 5.95T

f9 0:

w

Obr,

I

I

7

8

I

9

Btot (teslo)

Figure 2.3. Results of the tilted field study of the energy gap A of the v = 8 /5 FQHE. (The difference between the symbols is relevant only in the regime where the v = 8/5 resistivity minimum is split. Solid and open symbols refer to low- and high-field components of the doublet, respectively.)(After Ref. [20].)

state, there are four cases to be distinguished [6,7]: (1)both the quasielectron and quasihole spins are polarized just like the parent fluid; (2)the quasielectron spin is reversed but the quasihole remains polarized;(3) the reverse of case 2; and (4) both quasiparticle spins are reversed. For case 1, AS = 0 and the energy gap remains constant in a tilt experiment. In both cases 2 and 3 the net spin change is AS = - 1 and the energy gap contains the positive Zeeman term + Ig I pBBtot.The gap will increase in a tilt experiment. Finally, for case 4 we have AS = - 2 and the gap again increases with tilt, but twice as fast as in case 2 or 3. Hence for a polarized quantum Hall liquid, the energy gap A can either remain the same under tilt (spin-aligned quasiparticles)or it can increase (spin-reversedquasiparticles).The observation of afalling energy gap in the tilt experiment is therefore inconsistent with a polarized ground state, unless the added parallel magnetic field couples to the system in some way other than through the Zeeman energy. On the other hand, such an observation is easily explained if the quantum liquid is less than fully spin polarized, since then it is possible for the net spin to increase upon quasiparticle excitation. In particular, for an unpolarized quantized Hall state such as v = 2/5, the lowest-energy gap will obtain when both quasiparticle spins polarize along the field direction. Excitation of such a triplet state [7] will give AS= + 1. In this case the energy gap will have a negative Zeeman term - lglpBBtot and will decrease with tilt.

44


With this model in hand, the data in Fig. 2.3 are readily interpreted. For total magnetic fields less than about 7 T (i.e., tilt angles below 30") the falling energy gap is consistent with an unpolarized ground state and AS = + 1. From the slope aA/aB,,, in this regime, we deduce a g factor of about 0.4, remarkably close to the bulk GaAs value of g 0.44.Similarly, the rising energy gap for B,,, > 7 T suggests a polarized ground state. The measured slope aA/dB,,, again gives g 0.4, assuming that AS = - 1. It seems quite reasonable that on approaching the critical point from either direction, the spin alignment of the lowest-energy quasiparticle excitations favors the polarization of the phase about to be entered. These experimental results concerning the ground-state spin polarization of the v = 8/5 state are in good agreement with theoretical studies [5,7,30-351 of the particle-hole conjugate state at v = 2/5.Recent exact diagonalization studies [33,35] of systems containing up to eight electrons show that the energetic advantage (per electron and neglecting the Zeeman energy) of the unpolarized over the polarized v = 2/5 state extrapolates to about 0.006e2/d, in the thermodynamic limit. Setting this equal to the Zeeman energy per electron lglp,B/2 in the polarized phase allows us to estimate the critical perpendicular magnetic field B,,, of the transition; for GaAs B,,, x 5 T. Provided that particle-hole symmetry is operative, the same critical magnetic field applies to the v = 8/5 state. Thus in theory at least, an ideal 2D sample with density less than N,,, = 8eB,:,/ 5h x 2 x 10'' cm-2 will possess an unpolarized v = 8/5 ground state, while in higher-density samples the same state will be spin polarized. For a sample with N , < N,,, tilting may be used to induce the phase transition, and it will do so at a total magnetic field less than B,,,. This follows since tilting enhances the Zeeman energy while leaving the Coulomb energies unchanged. Starting in the unpolarized phase at some perpendicular field B , B,,,, tilting should induce the transition at B,,,,, = (BlBl,c)''2. For the sample being discussed here, the data in Fig. 2.3 show that B,,,,c x 7 T,while B, = 5.95 T. Thus the experimentally determined B,,, is approximately 8.2 T. Although this is significantly larger than the theoretical estimate ( x 5 T), we find the agreement satisfactory, considering the simplicity of the theoretical model (infinitely thin 2 D systems, no disorder or Landau level mixing,etc.), and the uncertainty in extrapolating finite-sizecalculations to the thermodynamic limit. Theory and experiment are also in agreement on the nature of the energy gap in the two phases. Both find that the spin change AS upon excitation of a quasielectron-quasihole pair is AS = + 1 in the unpolarized phase and - 1 in the polarized state. (Numerical calculations [33] suggest that in the polarized 2/5 state, the relevant excitation at low fields has only the quasielectron spin reversed while the quasihole remains polarized; this was case 2 above. Reversing the quasihole spin yields comparatively little Coulomb energy advantage, and the possibility the both spins are reversed (i.e., AS= -2) simply costs too much Zeeman energy. Similar results were obtained earlier for the v = 1/3 case [6].) Finally, the experimental finding that there is no noticeable discontinuity in the energy gap at the transition has also been corroborated numerically [33,34].

-

-

-=

SPIN AND THE FQHE

45

There are, of course, aspects of this general picture that are not understood. For example, the splitting of the v = 8/5 resistivity minimum in the vicinity of the transition remains mysterious. A similar splitting was subsequently observed in tilted field studies of the v = 2/3 FQHE [23,24]. One speculative possibility is that these splittings reflect an inhomogeneous state of the quantum Hall liquid. Phase separation of the electron liquid into polarized and unpolarized components is quite plausible near the critical point. On approaching the transition from the unpolarized phase, the satellite p,, minimum first appears on the high-field side of the main minimum. This is at least consistent with the satellite being due to condensed “droplets” of polarized fluid. Conversely, above the transition, in the polarized phase, the satellite eventually disappears off the low-field side ofthe p,, minimum;now the satellite might be due to droplets of the unpolarized fluid. Even if the quantum fluid does actually phase separate into stable domains of polarized and unpolarized liquids, it might simply fluctuate between the two phases. The resistivity splitting might be a sign of enhanced dissipation arising from such fluctuations. Another feature of these results that is not understood is why the transition is observed at v = 8/5 but has not yet [22] been detected at v = 2/5. The theoretical picture that we have applied was developed specifically for v = 2/5; its extension to the 8/5 state is made solely on the grounds of particle-hole symmetry. We can only remark that this symmetry principle connects 2/5 and 8/5 states at the same magnetic field. Experimentally, of course, this means comparing two samples whose densities differ by a factor of 4. Such samples would be quite different; for example, the “thickness” of the 2DES, as measured by the extent of the subband wavefunction,would be considerably larger in the low-density sample than in the high-density sample. This thickness softens the short-range part of the Coulomb interaction and therefore affects the critical magnetic field of the spin phase transition. Other things being equal, the 2/5 critical field would be suppressed relative to the 8/5 case.

2.2.3. The u = 512 Enigma As mentioned above, the first evidence for a spin-unpolarized FQHE state came with the discovery [151, at very low temperature, of a fragile quantized Hall effect in the second Landau level, at the even-denominator fraction v = 5/2. This unusual state remains the only even-denominator FQHE ever found in a single-layer 2DES [36]. Figure 2.4 shows the transport coefficientsp,.. and p,,, around v = 5/2 from the original paper of Willett et al. [l5]. Despite earlier suggestions [37,38] that even-denominator states might exist (at v = 3/4, 9/4, 11/4, and 5/2), these definitive new results were greeted with considerable surprise. This was certainly justified given the total adherence to the odd-denominator rule by all previously discovered FQHE states. The restriction to odd-denominator filling fractions for the primitive Laughlin states at v = l/m stems from the requirement of exchange antisymmetry of the many-body wavefunction. Since the spins are assumed to be fully polarized, this

46

MULTICOMPONENT QUANTUM HALL SYSTEMS I

I

I

I

1

0.1

s!

CU-

c

0.' e Q

2

0.3

MAGNETIC FIELD

(t]

Figure 2.4. Diagonal and Hall resistivity data showing the even-denominator FQHE at v = 5/2. Note the low temperatures involved. (After Ref. [lS].)

antisymmetry requirement forces the exponent m in the Jastrow (orbital) part of the wavefunction to be an odd integer. In the early hierarchical generalizationsof the Laughlin theory, the odd-denominator restriction was found to propagate down from the primitive v = l/m states to the daughter states at v = p/q, encompassing all FQHE states known prior to the v = 5/2 discovery. Thus it was natural to speculate that the existence of an even-denominator state might reflect a breakdown of the assumption of full spin polarization. The relatively low magnetic fields (B x 5 T) at which the v = 5/2 state was first observed made this seem quite plausible. This possibility seemed even more likely after Haldane and Rezayi [39] showed that for a certain hypothetical interaction (the hollow-core

SPIN AND THE FQHE r

47

8 = 23 deg.

3

P

I

J

MAGNETIC FIELD (tesla) Figure 25. Collapse of the v = 5/2 state with tilt. Arrows mark the location of the 5/2 filling fraction. These data were all taken at T = 25 mK. (After Ref. [16].) model) a FQHE could exist at half-filling. The wavefunction they proposed for

this state was explicitly unpolarized. Experimental support for these ideas came from the tilted field studies of Eisenstein et al. [16]. Figure 2.5 shows resistivity data in the vicinity of v = 5/2 for a number of different tilt angles 8. The data show clearly that the 5/2-state collapses as the tilt angle is increased. As discussed earlier for the v = 8/5 case, this behavior suggests that the ground state at v = 5/2 is not fully spin polarized. But again, a detailed study of the energy gap of the 5/2 state is required to strengthen the case. While the relatively low quality of the sample used in the original experiments [l5, 161 and the extreme fragility of the 5/2 state precluded such a study, better samples eventually became available and the necessary data were obtained [40]. Figure 2.6 shows the tilted field dependence of the energy gap A, extracted from the activated temperature dependenceof pxxat v = 5/2, from one such better sample. The gap is plotted against total magnetic field B,,,; the perpendicular field at v = 5/2 is fixed at B , = 3.75 T. The figure shows that A falls, roughly linearly, with B,,,. Just as with v = 8/5, this observation is inconsistent with a fully spin polarized ground state, provided that the parallel magnetic field has coupled to the system only through the Zeeman energy. On the other hand, the linearly decreasing gap is suggestive of an unpolarized (or partially polarized) ground state and a net increase in spin polarization upon quasiparticle excitation (i.e., AS > 0). Assuming that AS = + 1, the slope aA/dB,,, gives a g-factor of 191 x 0.56,

48


loo

' 0

0 (degrees) 15'

I

I

A =A0'QPe BTOT slope

a

BIZ 50

=+g.0.56

3.75 T

-

3.75

1

3.80

I

3.85

I

3.90

3 I5

B m (TI Figure 2.6. Tilted field study of the energy gap A of the v = 5/2 state (from a different sample than shown in Fig. 2.4).The slope of the straight line gives a g factor of g z 0.56. (After Ref. [40].)

some 30% larger than the bulk GaAs value [28]. Note that even for the untilted case, the measured gap at v = 5/2 (A x 0.11K) is only about one-tenth of the Zeeman energy at B = 3.75 T. This suggests that if the Zeeman energy were absent, an unpolarized 5/2 state would be robust, exhibiting an energy gap in the 1 K range. Although the evidence that the v = 5/2 ground state is not fully spin polarized is suggestive, it is not as compelling as that obtained at v = 8/5, 4/3, and 2/3. The reentrant tilted field behavior observed at those odd-denominator states is hard to explain via a non-Zeeman effect of the parallel magnetic field (ie., subband/Landau level mixing). At v = 5/2, however, there is only a monotonic gap suppression on tilting (albeit with a magnitude consistent with an unpolarized ground state). There remains a possibility that the ground state is actually spin polarized, and the tilted field behavior is due to some other effect of the parallel field. Interest in this has been renewed by recent experiments [41] showing that the 5/2 state is observable in (perpendicular) magnetic fields as high as B , 9 T. Given the rapid tilted field collapse of the 5/2 state shown in Fig. 2.6, for which B, = 3.75 T, the existence of the state at B, 9 T seems incongruous within the unpolarized ground-state scenario. There is, however, no obvious inconsistency between this observation and the evidence that the ground state is unpolarized. Indeed, one can construct a simple model [42] that includes sample disorder in a rudimentary way and show that the sample used for Fig. 2.36 could support an unpolarized 5/2 state at B, > 9 T if its density were increased (by a gate, for example). Nevertheless, these new findings [41] do not enhance the case for

-

-

-

FQHE IN DOUBLE-LAYER 2D SYSTEMS

49

a spin-reversed ground state at v = 5/2 and suggest that further work is needed. If, in fact, the 5/2 state is spin polarized, a new explanation for the tilted field experiments has to be constructed. While the experimental debate has centered, thus far, on the ground-state spin configuration, on the theoretical front the very existence of a v = 5/2 FQHE is still not well understood. The excitement over the early work of Haldane and Rezayi [39] largely dissipated when it became apparent that the hollow-core model they employed was a poor representative of real Coulomb interactions, even in the second Landau level [43]. Various modifications to the hollow-core model (e.g., inclusion of Landau level mixing [44]) have been tried, but so far without compelling success; Alternative candidate wavefunctions have also been proposed [43,45-481, but none have so far been shown to be close to the ground state for Coulomb interactions. On the numerical side, exact diagonalization calculations [43,49, SO], while suggesting that the ground state at v = 5/2 is actually spin polarized, reveal at best [SO] only a very weak cusp in the total energy. Such a cusp is the essential ingredient for a quantized Hall effect. It seems, therefore, that after several years of study, a key piece of the 5/2 puzzle is still missing. 2.3. FQHE IN DOUBLE-LAYER 2D SYSTEMS Interestingly, while the v = 5/2 FQHE in the single-layer 2D electron system is hard to observe and even harder to explain, even-denominator states in doublelayer 2D systems are experimentally robust and theoretically well understood. What accounts for the difference? Superficially,one might have expected that the two-spin and the two-layer problems would be very similar. In fact, there are two essential differences between the two cases. In a single-layer system the coulombic repulsion between any two electrons is independent of their relative spin orientation and depends only upon their lateral separation in the plane. For the bilayer case this is obviously not true; the physical separation of the layers makes interlayer interactions weaker than intralayer ones. This asymmetry of the Coulomb interaction is the main reason why v = 1/2 states are readily found in double-layer systems. The second critical difference between the two systems is the existence of interlayer tunneling in the bilayer case. Tunneling hybridizes the quantum states of the two layers and creates an additional energy gap not present in a single-layer system. Imagine first a bilayer system consisting of two identical 2DESs which are separated by a potential barrier sufficiently thick and high that both tunneling and interlayer Coulomb interactions are negligible. Assuming that the layers are electrically connected in parallel, such a system will exhibit a spectrum of QHE states that is identical to that of the individual layers, except that the resistance of each Hall plateau will be one-half of its single-layer value. With v representing the total filling factor of the double-layer system, integer Hall states will appear at v = 2,4,6,. . . and fractional ones at v = 2/3, 4/5, 6/7,. . . No Hall plateaus will be observed at odd-integer or odd-numerator fractional values of v.

50


Now adjust the barrier so that tunneling becomes relevant but interlayer Coulomb effects remain unimportant. In the absence of tunneling each singleparticle state is doubly degenerate;turning on this interlayer coupling hybridizes the individual quantum well states into symmetric and antisymmetric components. These new states are split by an energy gap ASAS which depends on the width and height of the separating barrier. The existence of this new energy gap profoundly affectsthe spectrum of observablequantum Hall states. For example, in the uncoupled case, no v = 1 QHE appears because this would imply the existence of a v = 1/2 QHE in each layer separately. With tunneling, however, odd-integer states appear each time the Fermi level becomes pinned in the symmetric-antisymmetric (SAS) gap of some Landau sublevel. A v = 1 state is then a full Landau level ofsymmetric bilayer states, not two distinct v = 1/2 states in the individual layers. (We are assuming, and will continue to throughout this chapter, that the spin splitting of the Landau level exceeds the tunneling gap ASAS ~511.) If the barrier is at once narrow but very high, tunneling may be unimportant, while interlayer Coulomb effects are essential. It was first shown theoretically by Haldane and Rezayi [52] and Chakraborty and Pietilainen [53] that in such a situation new QHE states, which are intrinsically bilayer in nature, can exist. Subsequently, Yoshioka et al. [54] systematically investigated a wide class of bilayer FQHE states using Halperin’s two-component wavefunctions [3]. These bilayer states depend critically on a delicate balance of inter- and intralayer Coulomb interactions. This balance is conveniently parameterized by the ratio d/Io, where d is the interlayer separation and I, is the magnetic length. Since for a given filling factor I, is proportional to the mean separation between electrons in the same layer, the ratio d/l, is proportional to the ratio of intra- to interlayer Coulomb energies. This ratio needs to be of order unity for the new states to be stable (at least for ideal systems with infinitesimally thin 2D layers.) Although it seems obvious that such states cannot survive at large d/I, values, it is interestingly also true that some (e.g., v = 1 /2) collapse in the opposite limit, d/l, +0. For these, the asymmetry of the Coulomb interaction produced by the finite layer separation is essential. In real samples, of course, tunneling and interlayer Coulomb effects are usually simultaneously operative. In general, an interesting phase diagram can be constructed, with interlayer Coulomb energy on one axis and symmetric-antisymmetric tunneling gap on the other. In some cases the two effects are at odds, and in others they act cooperatively.Indeed, their complex interplay has yielded some of the most interesting QHE phenomena observed in recent years. In this section I discuss the present experimental status of the QHE in double-layer systems, concentrating entirely on the v = 1/2 and v = 1 states. 2.3.1. Double-Layer Samples

Two main approaches have been taken to fabricate bilayer 2D electron systems. Both employ GaAs/AlGaAs heterostructures grown by molecular beam epitaxy.

FQHE IN DOUBLE-LAYER 2 D SYSTEMS

51

In a double-quantum-well (DQW)structure, two thin GaAs layers are emkedded in the alloy Al,Ga, -,As. The GaAs quantum wells are usually 150 to 200.A wide, while the (undoped) alloy barrier between them is typically only 30 A thick. Electrons transfer into the quantum wells from Si doping sheets which are placed well above and below the DQW [55]. Alternatively,bilayer 2D electron systems have also been realized in wide single quantum wells (WSQW)[56]. Electrons in the well accumulate at the upper and lower boundaries, being attracted there by the electricfields of the positively charged donor ions which are placed symmetrically above and below the well. If the well is wide enough, the charge distribution in the well has a distinct dumbbell shape. Unlike the DQW case, where the two 2DES layers are separated by a true semiconductor barrier, in the WSQW case the “barrier” results solely from the Hartree potential of the electrons themselves. One of the main advantages of the DQW geometry is the freedom it provides for tailoring the details of the barrier separating the two quantum wells to a specific experiment. Since both the thickness and the height (via the A1 concentration, x) of the barrier can be adjusted, independent control of tunneling and interlayer Coulomb interactions is possible. The 2D electron layers in DQWs are typically “thinner,” more closely spaced, and yet much more weakly tunneling than comparable WSQW structures. On the other hand, the WSQW geometry yields considerably “cleaner” bilayer 2D systems; this evident from the better developed FQHE spectra seen in such samples. DQW samples suffer from the rather “dirty” Al,Ga, - 4 s barrier (often pure AlAs) between the quantum wells. Most bilayer QHE experiments to date have been performed with equal 2D densities in the layers. This condition is established either by careful choice of the doping layer setback distances or by in situ tuning of the densities using gate electrodes on the bottom or top of the heterostructure sample. In addition, it is generally desirable for the two 2D electron systems to be comparably disordered, but this is usually difficult to assess directly.Transport studies on weakly coupled DQW systems, in which separate electrical connections [57] to the individual layers have been made, show that the two layers can be made to have very nearly the same “quality.” Such separate contacts have not yet been achieved with the strongly coupled systems of interest for bilajler QHE studies. (This is more a question of the tunneling resistancethan of layer spacing per se.) Indeed, all the results discussed in this chapter have been obtained from samples having ohmic contacts that connect to the two layers simultaneously.

2.3.2. The u = 112 FQHE At the qualitative level, the most dramatic discovery made so far in the study of double-layer quantum Hall systems was the observation of the even-denominator v = 1/2 state reported simultaneously by Suen et al. [SS] and Eisenstein et al. [59]. In both cases a deep minimum in the diagonal resistivity pxxwas found at total filling factor v = 1/2 along with a well-defined plateau in the Hall resistance at p,,, = 2h/e2. Figure 2.7 shows relevant resistivity data from both groups.

52


Figure 2.7. Observationsof the v = 1/2 FQHE state in double-layer 2D electron systems. The upper panel shows the data of Suen et al. [SS] from a wide single quantum well; the lower panel shows data from Eisenstein et al. [59], who employed a double-quantum-well structure.


53

A well-developed theoretical picture of a v = 1/2 FQHE state in ideal doublelayer 2D systems was in existence prior to the experimentsnoted above. After the initial prediction by Haldane and Rezayi [52], Yoshioka et al. [54] discussed a specific Halperin [3] trial wavefunction appropriate to v = 1/2 (the so-called “331” state) and established its large overlap with the exact ground state of a few-electron system when d/l, 1. As expected,Yoshioka et al. [54] found that this intrinsically bilayer FQHE state collapses at large layer separation where interlayer Coulomb correlations become negligible. Subsequently,He et al. [60] performed additional numerical calculations for realistic sample geometries, thereby allowing closer contact with experiment. The variational wavefunction widely believed to capture the essential physics of the v = 1/2 state in double-layer systems is a member of the broad class of generalized Laughlin-Jastrow functions introduced by Halperin [3] in the two-spin context. Letting z and w represent the coordinates of electrons in each of the 2D planes, the orbital part of the “331” wavefunction is (aside from the ubiquitous Gaussian factors)

-

The first and last products reflect the correlations within the z and w layers separately, while the middle product term contains the interlayer correlations. The intralayer terms are the same as that of the Laughlin 1/3 state [l]; as two electrons in the same layer approach one another, the wavefunction vanishes as the cube of their separation. The interlayer term, however, makes the wavefunction vanish when two electrons in different layers are directly opposite one another. In a crude sense, YJ31represents two Laughlin 1/3 states locked together so that the electrons in one layer are opposite correlation holes in the other. In fact, one may view each layer as a 1/3 state plus one extra quasihole for each electron (i.e., each layer is actually at filling factor 1/4). The quasiholes in layer z are “bound” onto the electrons in layer w, and vice versa. The coulombic attraction between these quasiholes and electrons reduces the total energy of the system when the two layers are close together. Thus the \y331 state contains not only a commensurability between the number of flux quanta and electrons in each layer, but also one between the number of quasiholes in one layer and the electrons in the other. It is these special commensurability conditions that produce the cusp in the total energy required for a quantized Hall effect. To show that the v = 1/2 state they observed depended critically upon interlayer correlations, Eisenstein, et al. [59] examined the effect in a quartet of samples having different densities and quantum well separations. Figure 2.8 displays resistivity data, taken at 300 mK, for samples A, B, C, and D. Samples A, B, and C are structurally identical (180-&wide wells separated by a 31-A AlAs barrier) but have different densities (A lowest, C highest). Note that while the v = 2/3 pxxminima are all similar, the v = 1/2 feature weakens monotonically as the density increases. This observation is contrary to the usual enhancement of (spin-polarized)FQHE states with increasingmagnetic field found in single-layer

54


!

8

12 6 MAGNETIC FIELD

10

8

10

12

(Tesla)

Figure 2.8. Collapse of the v = 1/2 state in four samples with increasing values of d/lo. Data taken at T = 300mK. These data demonstrate that interlayer correlations are essential to the existence of the v = 1/2 state in double-layer systems. (After Ref. [SS].)

systems. In the present double-layer case, however, the ratio of intra- to interlayer Coulomb energies, which is just d/lo, is critical. Setting d equal to the quantum well center-to-center distance (210A for samples A, B, and C) the ratio d/l, at v = 1/2 increases from 2.4 for sample A to 2.9 for sample C. Sample D has a thicker barrier (99A) and a ratio d/l, = 3.6. This last sample has a density comparable to that of sample B and is arguably the cleanest of the quartet (as evidenced by it exhibiting the largest v = 2/3 activation energy), yet it shows no v = 1/2 FQHE. Taken together, the data in Fig. 2.8, showing the v = 1/2 state to weaken and eventually collapse as d/l, increases, provide compelling evidence that this even-denominator bilayer FQHE does indeed depend essentially upon interlayer Coulomb interactions. Numerical calculations by He et al. [61] for DQW geometries equivalent to the samples just described show that the origin of the v = 1/2 FQHE in those samples is well understood and consistent with the properties of the ’I-”,,, variational wavefunction.Their calculations reveal that the finite thickness of the quantum wells leads to stability of the v = 1/2 state at larger values of d/l, than in


0.8

-

0.6

-

55

-

Q Y

0.4

0.2

-

0.00

100

200 tc

(h

300

Figure 2.9. Calculations by He, Das Sarma, and Xie [61] of the energy gap of the bilayer v = 1/2 state versus the magnetic length 1, in double-quantum-well structures consisting of two 180-A-wide GaAs wells separated by an AlAs barrier of width D,= 31 or 99 A. The solid dots represent the coordinates of the four samples shown in Fig. 2.8. The calculated energy gap shows, qualitatively, the same collapse of the v = 1/2 state that was observed experimentally [59] via the pxxminimum.

the ideal, infinitely thin case [54]. The reason for this is simply that the finite thickness softens the short-range part of the intralayer Coulomb interaction. Consequently, the interlayer interactions can be weaker as well and the v = 1/2 state still exist. Figure 2.9 shows the calculated [61] energy gap at v = 1/2 versus magnetic length I, for DQWs with 180-A quantum wells and either a 31- or 99 A-wide AlAs barrier. The solid dots give the calculated gaps for the various samples of Eisenstein et al. [59]. Although the v = 1/2 activation energies were not measured for these samples, the calculated gaps show the same qualitative collapse as that shown by the resistivity minima in Fig. 2.8. The sample used by Suen et al. [SS] to observe the v = 1/2 FQHE was a 680-Awide single quantum well. This sample differs significantly from those used by Eisenstein et al. [59]. Not only are the “individual” 2D systems much thicker in the Suen sample,but the tunnelingstrength is about an order of magnitudelarger. Setting the layer spacing d equal to the distance between the two (self-consistently calculated)maxima of the charge distribution, Suen et al. [58] concluded that the v = 1/2 state in this sample occurs at d/l, N 7.These conditions are sufficiently far

56


from what was originally expected [54,60] that the possibility of a non-Y33 origin for the v = 1/2 state in the Suen sample had to be considered. Fueling this speculation was the existence of another candidate wavefunction, the Pfaffian state, advanced originally by Moore and Read [46]. This singlecomponent state was originally considered as a possible candidate wavefunction for a spin-polarized v = 5/2 FQHE. Its connection to the present v = 1/2 FQHE in wide single quantum wells derives from the work of Greiter et al. [47], wherein it was suggested that if the Coulomb interaction were sufficiently softened at short distances, a Pfaffian-like v = 1/2 FQHE might exist. With this in mind, Suen et al. [62] performed an extensive study of the v = 1/2 state in wide single wells, varying the well width and sheet density. They concluded that the l/Zstate they were observing was fundamentally two-component in nature and that the Pfaffian was not relevant. Numerical work by He et al. [61] showed that in fact the v = 1/2 ground state in Suen’s sample was basically Y331-likein character, despite the large thickness and substantial tunneling strength. Interestingly, Suen et al. [62] showed that wide single quantum wells can support not only twocomponent states v = 1/2, but also thick variants of the familiar single-component states such as v = 1/3. Which “hat” the sample chooses to wear depends on a subtle balance among tunneling, thickness, and density, and this balance changes from one fraction to another.

2.3.3. Collapse of the Odd Integers It was actually the integer quantized Hall effect (IQHE) that first illustrated the importance of interlayer Coulomb interactions in double-layer 2D systems. In the experiments of Boebinger et al. [63], the IQHE in DQW samples with relatively large amounts of tunneling was investigated. As mentioned above, in the absence of tunneling and interlayer interactions, the IQHE is restricted to even filling factors, v = 2,4,6,. . . . These states correspond to the ordinary IQHE in a single-layer system. If there is tunneling and the symmetric-antisymmetric gap ASAscan be resolved, odd-integer quantized Hall states appear. Provided that the spin Zeeman splitting (gIpBBexceeds ASAS, each of these odd integers corresponds to the Fermi level lying in the SAS gap within a particular Landau spin sublevel [Sl]. What Boebinger et al. [63] and later Santos et al. [64] found was that the lowest odd integers were systematically suppressed. Very strongly tunneling DQWs would exhibit all the odd integers; with slightly weaker tunneling the v = 1 QHE would be missing; weaker still and both v = 1 and 3 were gone; and so on. Representative resistivty data from Boebinger et al. [63] are shown in Fig. 2.10. This suppression of the small odd-integer QHE states in DQWs has been attributed [60,65,66] to a collapse of the SAS tunneling gap induced by interlayer Coulomb interactions. To understand this, imagine first a DQW containing just two electrons. Ignoring the Coulomb interaction, each electron resides in the ground symmetric DQW quantum state. Ifwe now turn on the Coulomb repulsion while holding fixed the in-plane separation of the pair, the interaction energy is


57

Figure 2.10. Missing integer quantized Hall states in a strongly tunneling double quantum well. While v = 5,7,9,... are plainly visible, v = 1 and 3 are absent. (After Ref. [63].)

reduced if the two electrons hop into opposite quantum wells, since this increases their net separation. This hopping requires a complete mixing of the symmetric and antisymmetric DQW states and thus costs tunneling energy in the amount A S A P If A S A S is not too large, this penalty can be outweighed by the Coulomb advantage provided that the layers are far enough apart. Now consider the v = 1 QHE state, which is simply a full Landau level of (spin-polarized)symmetricstate electrons.The hopping scenario described above can still occur if, crudely speak-

58

MULTICOMPONENTQUANTUM HALL SYSTEMS

ing, nearest-neighbor electrons conspire to occupy opposite quantum wells, creating a charge density wavelike interlayer correlated state. The precise conditions for the transition to this state depend on the tunneling gap ASAS,the layer separation d, and the sheet density N,.For the v = 1 QHEin a DQW with a given layer separation and tunneling strength, there exists a critical density, and thus magnetic field, above which this new correlated state is favorable. According to theory [60,65,66] the Coulomb interactions in the DQW serve to convert the simple single-particle tunneling gap into a dispersive collective excitation that goes soft at a finite wavevector q = 1; at the critical magnetic field. Owing to this soft mode, the quantized Hall effect is destroyed. Analogous arguments apply to the higher odd-integer QHE states (v = 3,5,...). Ignoring for the moment couplings across the cyclotron and spin Zeeman gaps, only those electrons in the uppermost filled Landau level of symmetricstates can mix with unfilled antisymmetric states. As there are exactly N o = eB/h such electrons per unit area, in a DQW sample with fixed total density there will be a maximum odd-integer v above which the interactions are not strong enough to kill the QHE. Thus, as the tunneling strength is reduced, first the v = 1 state dies, then v = 3, and so on. This simple scenario is in qualitative agreement with experiment [63,64]. 2.3.4. Many-Body u= 1 State

In the preceding section the focus was on the v = 1 QHE in the strong tunneling limit where the main contributor to the quasiparticle energy gap was a singleparticle effect. What of the opposite limit, when the tunneling is very weak? Is there still a QHE, and if so, what is its character? These questions, and their analogs in the single-layer,two-spin v = 1 QHE, have been the subject of considerable recent interest. Indeed, it is becoming increasingly apparent that that this quantized Hall state exhibits physics far more interesting and subtle than originally thought. In the simplest view, the v = 1 QHE arises from a mundane single-particle effect: tunneling in double layers, spin splitting in single-layer systems. Electron-electrun interactions are often described as merely “enhancing” the energy gap. It is now recognized that this is a very incomplete picture. The resurgence of interest in the v = 1state derives primarily from the fact that in both the single- and double-layer cases a QHE should exist at v = 1 euen in the complete absence of the single-particle energy gap. In the single-layer case, recent theoretical work [67] has shown that not only does the v = 1 quantum Hall state exist in the limit of vanishing Zeeman energy, but its charged elementary excitations are not merely exchange-enhanced singlespin flips. They are, instead, large vortexlike distortions of the spin field called skyrmions (after their relatives in nuclear physics),which possess a large total spin. A number of recent experiments have uncovered evidence for these unusual excitations [68]. Double-layer systems exhibit an even richer spectrum of phenomena at v = 1 than do their single-layer counterparts, owing to the physical separation of the


59

two layers. If the layer spacing is sufficiently small, a v = 1 QHE is predicted to exist even in the absence of tunneling [53,54,69]. The low-lying charged excitations of the bilayer v = 1 state are also vortexlike topological defects (called merons); only the real spin is replaced by a layer index or pseudospin [70,71]. On increasing the layer spacing the v = 1 QHE must collapse [60,65,69] as the system becomes, in effect, two uncoupled layers, each at v = 1/2. This phase transition is obviously not present in the single-layer case. Finally, the separation between the two layers offers another possibility. By applying an additional magnetic field component parallel to the layers, a Bohm-Aharonov phase can be injected into the many-body problem. As we shall see below, this can have dramatic consequences. In this section I draw on the analogy between double-layer quantum Hall systems and 2D ferromagnetism,which has been developed most thoroughly by the Indiana University theory group [70,71]. My discussion is limited to qualitative aspects of this analogy and how they relate to experiment. The reader interested in a deeper understanding of this physics is referred to Chapter 5. Before discussing the experiments, it is useful to introduce the pseudospin description of double-layer 2D systems. In this language an electron in the “upper” layer of a DQW is an eigenstate of ozwith eigenvalue + 1. Similarly, an electron in the “lower” layer is an eigenstate of ozwith eigenvalue - 1. Recalling the properties of the Pauli spin matrices,it is clear that an electron in a symmetric DQW state, which is just (lupper > + llower > )/& is an eigenstate of ox with eigenvalue + 1. Similarly, an antisymmetric state electron has ox = - 1. In the presence of tunneling, the symmetric and antisymmetric states are split by the energy gap ASAS. Therefore, tunneling may be viewed as a pseudomagnetic field lying along the x direction and ASASas the associated pseudo-Zeeman energy. Carrying this a bit further, imagine an electron placed at t = 0 into the upper quantum well. For t > 0 this electron will begin to tunnel back and forth coherently between the layers. In the pseudospin picture this is equivalent to the Larmor precession, in the z-y plane, of an electron with spin initially along 2 in a uniform magnetic field along 2. The Larmor frequency is just AsA$h. Tunneling alone is sufficient to produce a QHE at v = 1 in a double-layer system. This occurs when the Fermi level is in the lowest symmetric/antisymmetric gap and the system is just a filled Landau level of symmetricstate electrons. In the pseudospin picture, the ground state of the system is fully pseudospin polarized along 2. But as mentioned above, a v = 1 QHE should exist even in the absence of tunneling, owing to interaction effects. This many-body integer QHE, first predicted on numerical grounds by Chakraborty and Pietilainen [53], is similar to the bilayer v = 1/2 FQHE insofar as it collapses at large layer spacing. In contrast to the v = 1/2 FQHE, the v = 1QHE also exists in the theoretical limit d/l, +O. The question for experiment is: To what extent can one decide if the v = 1 QHE observed in a bilayer sample derives from single-particletunneling or many-body effects.? To clarify this issue, Murphy et al. [72] performed a systematic study of th? v = 1 QHE in a series of DQW samples, most of which consisted of two 180-A

60


BIB (v = 1) Figure2.11. Existence and lack thereof of the v = 1 QHE in two very similar DQW samples. These samples are represented in the phase diagram in Fig. 2.12 by the leftmost open and solid stars. (After Ref. [72].)

GaAs quantum wells separated by a 31-A Al,Ga, -,As barrier. Within these geometrical constraints, the calculated [73] tunneling gap was varied from a minimum of about A,, x 0.5 K to a maximum of roughly 8.1 K, by tailoring the A1 concentration within the barrier (in both magnitude and profile). These samples had total sheet densities ranging from N,, x 0.8 x 10" to 3.2 x 10" equally split between the two layers. Figure 2.1 1 illustrates resiostivity data from two samples that differ only in barrier thickness (31 versus 40 A) and slightly in density (N,,, = 1.26 versus 1.45 x 10" cm-2. Both show clear quantum Hall states at v=2, 213, and even 415. At v = 1, however, the two samples differ qualitatively. The narrower-barrier, lower-density sample exhibits a strong v = 1 QHE, while the other sample shows no such state. Figure 2.12 contains the phase diagram for v = 1 that Murphy et al. [72] constructed. On the horizontal axis is the tunneling gap [73] ASASin units of the basic Coulomb energy e2/d,, while the vertical axis is d/lo, the ratio of the quantum well center-to-center spacing to the magnetic length, this being the ratio of intra- to interlayer Coulomb energies. The solid symbols denote samples that

FQHE IN DOUBLE-LAYER 2D SYSTEMS I

61

6

NO QHE

.05 &&(e2/e psi

.1

Figure 2.12. Phase diagram for the v = 1 QHE in double-layer 2D electron systems. Solid symbols denote samples that exhibit a v = 1 QHE, while open symbols denote those that do not. (After Ref. [72].)

show a quantized Hall effect at v = 1, while the open symbols indicate those that do not. The figure demonstrates that there exists a well-defined boundary between a QHE and a non-QHE phase. The dashed line estimates the location of that boundary. The phase diagram shows that the QHE is destroyed as the layer separation (or, more properly, d/lo)is increased,even if the tunneling gap is held fixed. In the strong tunneling limit, this is just the conclusion reached earlier by Boebinger et al. [63] and discussed in the preceding section. In this limit the phase transition results from a competition between tunneling and Coulomb interaction effects [60,65,66]. Figure 2.12 contains two important additional results. First, the boundary separating the QHE and non-QHE phases appears to intercept the vertical axis at a finite value of d/l, ( w 2). This is compellingevidence that a v = 1 QHE does exist even in the limit of zero tunneling, as predicted [53]. In this case the phase transition as d/lo increases is driven entirely by Coulomb effects [54,65,69]. Indeed, this would appear to be the case for the samples used in Fig. 2.1 given their coordinates in the phase diagram of Fig. 2.12 (leftmost open and solid stars). Finally, the distribution of points in Fig. 2.12 suggests that the v = 1 QHE evolves continuously from a regime dominated by single-particletunneling to one where many-body effects are paramount. No compressible phase appears in between, and the two regimes cannot be cleanly distinguished. The pseudospin language provides an attractive way to view these results. Consider first the limit of strong tunneling and extremely small layer spacing. In

62


this case, each electron is in a symmetric DQW state. The system is fully pseudospin polarized along A, and the quasiparticle activation gap A at v = 1 is just ASAS, ignoring electron-electron interactions. Turning on the Coulomb interaction can be expected to enhance the gap by adding in an exchange term. If the tunneling is now removed, the QHE survives simply because the exchange energy likes to keep the pseudospins of adjacent electrons parallel. The ground state at v = 1 remains fully pseudospin polarized and the activation gap A is the exchange energy. The crucial point is that the net polarization can point in any direction in the three-dimensional pseudospin space. The v = 1 QHE in this ASAS = d/l, = 0 limit is indifferent to whether all the electrons are in symmetric DQW states or all are in just one of the two layers or in any other combination. The ground state at v = 1 is thus highly degenerate. Tunneling merely breaks the symmetry [30,74], forces the pseudospin to lie along A, and increases the net energy gap. This is completely consistent with the experimental observation that the v = 1 QHE evolves continuously between a purely many-body state in the ASAS +0 limit to a basically single-particle effect at large ASAS. This scenario is very similar to that which occurs at v = 1in a single-layer 2DES [67]. In the limit that d/l, +0 the two problems are in fact identical. When the layer separation is not small (as is the case in the experiments)the picture is modified [70,71]. For sufficiently small d/l, (less than about 2 in the aforementioned experiments [72]), an incompressible many-body QHE state still exists in the ASAS -+O limit. In the pseudospin picture, there are two important differences relative to the d/l, = 0 case. First, the total pseudospin is forced to lie near the x-y plane. The reason for this is simple; if ( S , ) is locally finite, then in that region there are more electrons in one layer than in the other. Owing to the finite layer separation, this produces a large capacitive energy and thus is not favorable. The second important consequence of the layer separation is that the total pseudospin is no longer a good quantum number, owing to the reduced symmetry of the Coulomb interaction. This leads to quantum fluctuations of the pseudospin,which if the layer spacingis large enough, are sufficient to destroy the incompressible quantum Hall state. These fluctuations are presumably responsible for the experimentally observed [72] transition to a compressible phase at d/l, x 2 in the small ASAS limit. In their experiments, Murphy et al. [72] also examined the behavior of the v = 1 QHE in tilted magnetic fields. The motivation for this was not to access the real spin degree of freedom (which was assumed to be fully polarized),but rather, to manipulate the tunneling gap ASAS. The basis for this stems from momentum conservation in the tunneling process. As can readily be shown [75], owing to the altered relationship between wave vector and guiding center for lowest Landau level eigenstates that is induced by an in-plane magnetic field B,,, ASAS is exponentially suppressed:

with a = d/21, and tan 8 = B,,/B,.Murphy et al. [72] planned to use this suppres-

63


121

.

103

a

p:

21j

10' ri

.I.

',

3

\2

4

a-

I

\

01 0

I

1

10

I

20

I

30

Tilt Angle (0)

I

40

1

50

\

. 60

Figure 2.13. Extreme sensitivity of the v = 1 activation gap A (solid dots) to tilting in a weakly tunneling sample (AsASz0.5 K). In contrast, the v = 2/3 state (open triangles) shows little angular dependence.The dashed line is the calculated angular dependence of ASAS,arbitrarily normalized to equal A at 0 = 0. Inset: Typical Arrhenius plot for v = 1 QHE. (After Ref. [72].)

sion to sweep a single sample through the phase diagram from the strong to the weak tunneling regime. What was observed instead was a dramatic sensitivity to tilt at small B,,that was clearly not due to this matrix element effect. Figure 13 shows the measured activation gap A (not to be confused with the tunneling gap ASAS!)at v = 1 versus tilt angle 8. These data were obtained using a sample that is positioned deep in the many-body region of the phase diagram of Fig. 2.12 (the leftmost solid star). For this sample, d/l, = 1.88 at v = 1 and the calculated tunneling gap is only ASAS = 0.5 K.The figure shows that A exceeds A,,, by almost a factor of 20 in the perpendicularfield case, clearly demonstrating the dominance of interaction effects in this integer QHE state. Further evidence for this is provided by the Arrhenius plot in the inset. There it is apparent that the activated behavior collapses for temperatures above about T* w 0.4 K, some 20 times lower than the measured gap itself [76]. As the magnetic field is tilted away from normal to the plane, the measured activation gap A drops rapidly. By 8, w 8" the gap has been roughly halved. At this tilt angle, the applied parallel magnetic field is only B,,x 0.7 T. Interestingly, the gap was found to be rather insensitiveto further increases in the tilt angle. The dashed curve in the figure is proportional to the expected suppression of the single-particle tunneling gap ASAS,due to the in-plane magnetic field. The curve

64


has been normalized so as to match the gap A observed at 8 = 0. Clearly, the observed tilted field behavior of the measured activation gap is inconsistent with the tunneling gap suppression. Arguing against the possibility that their results were due to a crossover of two different branches of the v = 1 excitation spectrum, Murphy et al. [72] suggested instead that a ground-state phase transition was taking place. The tilted field behavior of the v = 1 QHE in samples which are more strongly tunneling than that in Fig. 2.13 was also examined [72]. A similar transition between two distinct incompressible v = 1 states was observed; only the critical tilt angle was found to be larger. Figure 2.14 illustratesthis effect with two samples, one having ASAS = 3.6 and the other ASAS = 8.1 K. Figure 2.15 displays the dependenceof tan 8,upon AsAs/(ez/d0). [For the open symbols the 8 = 0 of ASASis employed,while for the solid symbols the calculated parallel field suppression of the tunneling gap ASAS has been included by evaluating Eq. (3) at 8 = 8,. This correction has a negligibleeffect on the most weakly tunneling sample but is quite significant for the other two.] Figures 2.13 to 2.15 make clear that tunneling plays a fundamental role in the tilted field transition at v = 1. The data suggest a competition between two ground states, one of which, at 8 < 8,,takes advantage of tunneling by forming a many-body condensate out of symmetric state electrons. The competing state apparently ignores tunneling, but in the presence of a sufficiently large in-plane magnetic field, has the lower Coulomb energy.

0.0 1 0

I

I

I

10

20

30

I

40

Tilt Angle (0)

I

50

I

60

70

Figure 2.14. Angular dependence of the v = 1 activation gap A in two more strongly tunneling samples. Dots, A,,, = 3.6 K, A(O = 0) = 5 K; squares, A,,, = 8.1 K, A(O = 0) = 14.6 K. Arrows indicate assigned critical angles. (After Ref. [72].)


1.c

65

tE/ ......... ' i I

/

***.A

0

8 5 4-

/

0.5

/

0.0 0.00

'

/

f

/

IT!

0.05

0.10

AsAs/(e2/e.td

Figure 2.15. Comparison of the experimentally determined [72] critical angle 6, for the v = 1 tilted field transition with the theory of Yang et al. [70]. Open and solid symbols connected by dotted lines refer to the same sample, but with the calculated [73] ASAS evaluated at 6 = 0 and 6 = 6, respectively. F h e angular suppression of ASAS is calculated using Eq. (3). For the datum in the lower left corner the suppression is negligible.] Error bars reflect only the uncertainty in determining the critical angle. The dashed line is the theoretical prediction, evaluated at d/l, = 1.85.

Yang et al. [70] have offered a pretty explanation, based on a textural phase transition within the quantum ferromagnetismmodel, for these tilted field results. Adopting a particular gauge, they first note that owing to the in-plane magnetic field, the tunneling matrix element is not only suppressed in magnitude, but also acquires a spatially varying phase: t = toei#.The phase 4 advances linearly across the 2D plane in the direction perpendicular to B,,. The distance over which 4 advances by 2a is L = h/eB,,d,with d the interlayer spacing. This is just the distance required for the in-plane field B,,to thread one flux quantum between the two 2D planes. The consequence of this phase is that the local direction in pseudospin space which defines symmetric DQW eigenstates rotates as one moves across the plane. For B,,= 0 the v = 1 ground state is fully pseudospin polarized, and whatever tunneling there is orients the polarization along A and thereby lowers the total energy by an amount ASAs/2 per electron. If the system is to maintain this energetic advantage, then, for B,,> 0, the pseudospin field must

66


twist in order to track the phase of the tunneling matrix element. This twisted pseudospin texture maintains the energetic advantage of tunneling, but since neighboring pseudospins are no longer parallel, it costs Coulomb (exchange) energy. As B,,is increased, the pseudospin field winds up more and more tightly and this Coulomb penalty rises. At a critical parallel field B,,,,, Yang et al. [70] predict that the system abandons tunneling and makes a transition to a new uniformly polarized state. The B,,, (or tilt angle, tan 8, = B,,,c/Bl)calculated is proportional to &, at least for small d/l, and ASAS. The dashed line in Fig. 2.15 is the prediction of Yang et al. [70], obtained by evaluating their expression for the pseudospin stiffness at d/l, x 1.85 as appropriate to the experiment. The reader is reminded that the values of ASAS used by Murphy et al. [72] are calculated ones. There is room for significant error in ASAS, owing to uncertainties about the structural parameters (band offsets,effectivemasses in the barrier, etc.) of the samples [73]. [The error bars shown in Fig. 2.15 reflect only the uncertainty in assigning the critical angle ( f 2"); this affects both tan 8, and ASASat 8 = 6,.] With these remarks in mind, the agreement between theory and experiment seems quite good. It should be kept in mind, however, that the theory curve in Fig. 2.15 does not account for quantum fluctuations arising from the relatively close proximity of the samples to the compressible phase boundary at large d/lo. Such fluctuations will probably reduce the pseudospin stiffness and thereby increase the critical angle. Aside from determining the critical angle itself, there is also the issue of why the measured activation gap behaves as it does. This has recently been addressed by Read [77] and by the Indiana group [78]. Both offer arguments for why the gap falls rapidly in the low-angle phase and is roughly constant in the high-angle phase, for 8 >> 8,. Read also gives analytic expressions for the shape of the gap near the transition angle and predicts a sharp downward cusp at 8 = 8,. No evidence for such a cusp has yet been found in experiment [79], but Read [77] remarks that the cusp will probably be wiped out by randomness in the tunneling matrix element. such randomness is surely present in actual samples, fluctuations in the quantum well and tunnel barrier widths being one obvious source. There are several other fascinatingpredictions of the quantum ferromagnetism model of the v = 1 QHE in bilayer systems. These include finite-temperature Kosterlitz-Thouless phase transitions [70,71], linearly dispersing collective modes [69,74], superfluidity in the counterllow transport channel [70,71], possible Josephson-liketunneling behavior [80], and others, all of these represent challenging avenues for future experimental work. 2.4. SUMMARY

In this chapter I have tried to convey a sense of the current excitement surrounding quantum Hall phenomena in multicomponent systems. This subject has yielded some of the most interesting results in the field of low-dimensional

REFERENCES

67

electronic systems in recent years. Despite the highly developed state of the field, much of this new physics was simply not appreciated only a few years ago. In the single-layer two-spin case, the excitement over skyrmion excitations illustrates this point well. The very recent theoretical work of Sondhi et al. [67] and others [Sl] appeared just before the experimental finding by Barrett et al. [68] that the spin polarization of the 2DES decreases both above and below v = 1. Similarly,in double-layer 2D systems the unexpected tilted field phase transition at v = 1 discovered by Murphy et al. [72] spurred the development by Yang et al. [70] of a fruitful analogy between bilayer quantum Hall systems and quantum ferromagnets. This recent theoretical work has led to a number of fascinating predictions worthy of experimental investigation. It will be particularly interesting if experimental probes other than conventional magnetotransport can be brought to bear on multicomponent quantum Hall systems. The application of optically pumped NMR [68] to the spin polarization of the 2DES is a good example of the effectiveness of a new approach. In the double-layer case the development [57] of reliable means for establishing separate ohmic contacts to the individual layers has already opened new avenues for the study of weakly coupled bilayer systems. These include measurements of Coulomb drag [82], interlayer tunneling [83], and electronic compressibility [84]. Extension of this contacting scheme to the kind of strongly coupled bilayer systems that support interlayer quantum Hall states should allow testing some of the dramatic recent predictions [70,71, SO] about such systems. Other nontransport probes, such as optical and surface acoustic wave methods, have so far not been applied with force to the multicomponent problem, but there is little doubt that they could be profitably pursued. Multicomponent quantum Hall systems are still relatively young; I think they’re a good bet for the devoted miner of strongly correlated electron systems.

ACKNOWLEDGMENTS I am indebted to a very long list of collaborators and friends for their help in learning the physics discussed in this chapter. I would like to thank in particular Loren Pfeiffer and Ken West, for growing the heterostructure samples; Sheena Murphy, for her excellent work on the bilayer v = 1 problem; Greg Boebinger, Horst Stormer, and Bob Willett, for several fruitful collaborations; and Steve Girvin, Song He, and Allan MacDonald, for many helpful discussions.

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4. For a review of the early experiments on the FQHE, see the chaptar by A. M. Chang in The Quantum Hall Eflect, edited by R. E. Prange and S. M. Girvin, Springer-Verlag, New York, 1987). 5. T. Chakraborty and F. C. Zhang, Phys. Rev. B 29,7032 (1984); F. C. Zhang and T. Chakraborty, Phys. Rev. B 30,7320 (1984). 6. T. Chakraborty, P. Pietililinen, and F. C. Zhang, Phys. Rev. Lett. 57, 130 (1986). 7. E. H. Rezayi, Phys. Rev. B 36,5454 (1987). 8. There are, however, some new techniques emerging. See Ref. [68]. 9. F. F. Fang and P.J. Stiles, Phys. Rev. 174,823 (1968). 10. D. A. Syphers and J. E. Furneaux, Suif Sci. 196,252 (1988);Solid State Commun. 65, 1513 (1988). 11. R. J. Haug et al., Phys. Rev. B 36,4528 (1987). 12. V. Halonen, P. Pietilainen, and T. Chakraborty, Phys. Rev. B 41, 10202 (1990). 13. J. D. Nickila and A. H. MacDonald, to be published. 14. J. Hampton, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Solid State Commun. 94, 559 (1995). 15. R. L. Willett et al., Phys. Rev. Lett. 59, 1776 (1987). 16. J. P. Eisenstein et al., Phys. Rev. Lett. 61,997 (1988). 17. P. Maksym et al., in High Magnetic Fields in Semiconductor Physics 11, edited by G. Landwehr, Springer-Verlag, Berlin, 1989. 18. J. E. Furneaux, D. A. Syphers, and A. G. Swanson, in High Magnetic Fields in Semiconductor Physics 11, edited by G. Landwehr, Springer-Verlag, Berlin, 1989. 19. R. G. Clark et al., Phys. Rev. Lett. 62, 1536 (1989). 20. J. P. Eisenstein,H. L. Stormer, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 62,1540 (1989). 21. J. E. Furneaux, D. A. Syphers, and A. G. Swanson, Phys. Rev. Lett. 63,1098 (1989). 22. J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer,and K. W. West,Phys. Rev. B41,7910(1990). 23. R. G. Clark et al., Sut$ Sci. 229, 25 (1990); and in Localization and Conjinement of Electrons, edited by F. Kuchar, H. Heinrich, and G. Bauer, Springer-Verlag, Berlin, 1990. 24. L. W. Engel, et al., Phys. Rev. B 45, 3418 (1992). 25. A. Buckthought et al., Solid State Commun. 78, 191 (1991). 26. A. G. Davies et al., Phys. Rev. B 44, 3128 (1991); Sut$ Sci. 263, 81 (1992). 27. N. G. Morawicz et al., Semicond. Sci. Technol. 8, 333 (1993). 28. M. Dobers, K. von Klitzing, and G. Weimann, Phys. Rev. B 38,5453 (1988). 29. We are thus ignoring the possibility of skyrmion-like large-spin excitations. See Section 2.3.4. 30. M. Rasolt, F. Perrot, and A. H. MacDonald, Phys. Rev. Lett. 55,433 (1985);M. Rasolt and A. H. MacDonald, Phys. Rev. B 34,5530 (1986). 31. D. Yoshioka, J. Phys. SOC.Jpn. 55,3960(1986). 32. P. Maksym, J. Phys. C 1,6299 (1989). 33. X. C. Xie, Y. Guo, and F. C. Zhang, Phys. Rev. B 40,3487 (1989). 34. T. Chakraborty, Sut$ Sci. 229, 16 (1990).

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35. P. Beran and R. Morf, Phys. Rev. B 43,12654 (1991). 36. Studies of the 3 state have also been reported by T.Sajoto et al., Phys. Rev. B 41,8449 ( 1990). 37. G. Ebert et al., J . Phys. C 17,1775 (1984). 38. R. G.Clark et al., Surf. Sci. 170,141 (1986). 39. F. D. M. Haldane and E. H. Rezayi, Phys. Rev. Lett. 60,956(1988). 40. J. P. Eisenstein et al., Surf: Sci. 229,31 (1990). 41. R. Du et al., to be published. 42. The effect of disorder can be roughly approximated simply by reducing the energy gap by a constant amount r. For an unpolarized ground state with AS = + 1 and no tilt, we write A =A, - J g J p B B r. After choosing a specific value for r, the proportionality constant between A, and e’/do can be determined by fitting the formula to the measured gap value for the sample’s as-grown density. This fixes the overall density dependence, provided that r remains constant. (For gate-induced density changes, this is a good assumption; see Ref. [84].) For state-of-the-art heterostructures r x 2K.Combining this value with the data in Fig. 2.6leads to the conclusion that the v = 4 gap would increase with density out to about B = 6 T and remains positive to beyond B = 9T. 43. A. H. MacDonald, D. Yoshioka, and S. M. Girvin, Phys. Rev. B 39,8044(1989). 44. E. H.Rezayi and F. D. M. Haldane, Phys. Rev. B 42,4532(1990). 45. X. C. Xie and F. C. Zhang, Mod. Phys. Lett. B 5,471(1991). 46. G. Moore and N. Read, Nucl. Phys. B 360,362(1991). 47. M. Greiter, X. G. Wen, and F. Wilczek, Phys. Rev. Lett. 66,3205(1991). 48. L.Belkhir and J. Jain, Phys. Rev. Lett. 70,643 (1993). 49. T.Chakraborty and P. Pietilainen, Phys. Rev. B 38,10097 (1988). 50. R. Morf,unpublished. 51. Ignoring interaction and level mixing effects, this assumption obviously fails at low magnetic field. 52. F. D. M. Haldane and E. H. Rezayi, Bull. Am. Phys. SOC.32,892(1987). 53. T.Chakraborty and P. Pietiliiinen, Phys. Rev. Lett. 59,2784(1987). 54. D.Yoshioka, A. H. MacDonald, and S . M . Girvin, Phys. Rev. B 39,1932(1989). 55. L.N.Pfeiffer, E. F. Schubert, K. W. West, and C. Magee, Appl. Phys. Lett. 58,2258 (1991). 56. Y. W. Suen et al., Phys. Rev. B 44,5947 (1991). 57. J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Appl. Phys. Lett. 57,2324 (1990). 58. Y. W. Suen et al., Phys. Rev. Lett. 68, 1379 (1992). 59. J. P. Eisenstein et al., Phys. Rev. Lett. 68, 1383 (1992). 60.S.He, X. C. Xie, S. Das Sarma, and F. C. Zhang, Phys. Rev. B 43,9339(1991);S . He, X. C. Xie, and S. Das Sarma, Surf. Sci. 263,87(1992). 61. S.He, S.Das Sarma, and X . C. Xie, Phys. Rev. B 47,4394(1993). 62. Y. W. Suen et al., Phys. Rev. Lett. 72,3405 (1994). 63. G. S.Boebinger, H. W. Jiang, L. N. Pfeifer, and K. W. West, Phys. Rev. Lett. 64, 1793 (1990);Phys. Rev. B 45,11391 (1992).

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64. M. B. Santos, L. W. Engel, S. W. Hwang, and M. Shayegan, Phys. Rev. B 44,5947 (1991). 65. A. H. MacDonald, P. M.Platzman, and G. S . Boebinger, Phys. Rev. Lett. 65, 775 (1990). 66. L. Brey, Phys. Rev. Lett. 65,903 (1990). 67. S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi, Phys. Rev. B 47, 16419 (1993). 68. S.E. Barrett,et al., Phys. Rev. Lett. 74,5112(1995);A.Schmeller,et al.,Phys. Rev. Lett. 75,4290 (1995); and E. Aifer, et al. Phys. Rev. Lett. 76,680 (1996). 69. H. A. Fertig, Phys. Rev. B 40,1087 (1989). 70. Kun Yang et al., Phys. Rev. Lett. 72, 732 (1994). 71. K. Moon et al., Phys. Rev. B 51, 5138 (1995). 72. S. Q. Murphy et al., Phys. Rev. Lett. 72,728 (1994). 73. The tunneling gap AsAsis calculated self-consistently, at zero magnetic field, in the Hartree approximation. The values given here differ slightly from those given by Murphy et al. [72], owing to the use of a revised r-point conduction band offset between GaAs and AlAs (1.05 eV instead of 0.83 eV). These calculations represent a significant source of possible systematicerror. In Ref. [72], for only the most strongly tunneling sample (AsAsx 8.1 K) could this calculation be verified experimentally (via Shubnikov-de Haas analysis). 74. X.G. Wen and A. Zee, Phys. Rev. Lett. 69, 1811 (1992); Phys. Rev. B 47,2265 (1993). 75. J. Hu and A. H. MacDonald, Phys. Rev. B 46, 12554 (1992). 76. The unusual temperature dependence of the bilayer v = 1 QHE has been studied more thoroughly by T. S . Lay, Phys. Rev. B 50,17725 (1994). 77. N. Read, Phys. Rev. B 52,1926 (1995). 78. See Chapter 5, this volume. 79. S. Q. Murphy et al., unpublished. 80. Z . F. Ezawa, Phys. Rev. B 51, 11152 (1995) and the references therein. 81. H. A. Fertig, L. Brey, R. C&b, and A. H. MacDonald, Phys. Rev. B 50, 11018 (1994). 82. T. J. Gramila et al., Phys. Rev. Lett. 66, 1216 (1991). 83. 3. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 69,3804 (1992), Surf: Sci.305, 393 (1994); Phys. Rev. Lett. 74, 1419 (1995). 84. J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Reu. Lett. 68,674 (1992); Phys. Rev. B 50, 1760 (1994).

PERSPECTIVES IN QUANTUM HALL EFFECTS: Novel Quantum Liquids in Low-DimensionalSemiconductorStructures Edited by Sankar Das Sarma, Aron Pinczuk Copyright0 2004 WILEY-VCH Verlag GmbH & Co. KGaA

3

Properties of the Electron Solid H. A. FERTIG Department of Physics and Astronomy and Center for Computational Sciences, University of Kentucky, Lexington, Kentucky

3.1. INTRODUCTION

The ground state of a collection of noninteracting electrons in a structureless background is well known to be a degenerate Fermi gas. Remarkably, this simple model accounts for much of our understanding of the properties of electrons in metals. One reason this simple model works is that the effect of the periodic potential due to the ionic lattice in which the electrons move can be accounted for approximately by using a renormalized electron mass. More complex questions arise when the electron-electron interaction is taken into account; however, in metallic systems many system properties still behave like those of noninteracting electrons, because of screening effects. The cohesive energy of the degenerate electron gas Egasin a jellium model (i.e., electrons moving in a uniform neutralizing charged background) has a wellknown expansion that is valid at high densities,whose first two terms are just the result of the Hartree-Fock approximation. In three dimensions this takes the form [1J 2.21 0.916 E,,, z -- rf

rs

RY

In this expression, r, is a unitless measure of the average interelectron distance, defined as r, = (rne2/h2)(3/47rp)1/3, where p is the electron density. The energy unit is defined in the usual fashion as 1 Ry = rne4/2h2 = 13.6 eV. This model can also be used for the cohesive energy of electrons in solids by replacing m with the electron effective mass m* and the electron charge with e/&, where E is the dielectric constant of the host crystal. For semiconductors, this typically lowers the effective rydberg value by two to three orders of magnitude. The first and second terms in this expansion represent, respectively, the kinetic energy and the

Perspectives in Quantum Hall ENects, Edited by Sankar Das Sarma and Aron Pinczuk.

ISBN 0-471-11216-X 0 1997 John Wiley & Sons,Inc.

71

72

PROPERTIES OF THE ELECTRON SOLID

Hartree-Fock approximation of the potential energy of the system. Higher-order corrections in r, may be computed systematically as well [2]. Long ago, it was pointed out by Wigner [3] that a degenerate Fermi gas is really not the only possible state of electrons in the jellium model. Wigner hypothesized that at low densities, the electrons might crystallize; the energy of such a state has been estimated [4] to have the form E,,x

1.792 2.26

--

rS

b

+pT+z s

Ry

for a BCC lattice. The first term represents the Coulomb energy and is negative because of the interaction of the electrons with the uniform neutralizing background; the second term represents the energy due to zero-point motion of the phonons in this Wigner crystal(WC),and the third term is the first correction due to anharmonicities in the crystal.The constant b has been estimated to be slightly less than unity [4]. Clearly, for very low densities, Eq. (2) will give an energy below that of Eq. (1), indicating a phase transition from the gas to a crystal phase. Physically, the transition may be understood as being driven by quantum fluctuations: As the density of the crystal is increased, the electrons are confined to ever-smaller regions (i.e., the unit cell of the crystal decreases in volume), and the uncertainty principle requires the admixing of higher momentum states to do this. This contribution to the WC energy is contained in the second and third terms of Eq. (2), and it is clear that for small enough r, (high density),these terms will overwhelm the first term. Thus, due to the uncertainty principle, one pays a quantum-mechanical kinetic energy cost to form the crystal. For small enough r,, it becomes favorable for the system to go over to the gas state, which in fact has the minimum possible kinetic energy for the system. The precise value of r, at which the transition takes place is difficult to estimate, as it requires an accurate knowledge of the higher-order terms in both Eqs. (1) and (2) [S]. 3.1.1. Realizations of the Wigner Crystal

Because of the necessity of obtaining low-density electrons to realize a WC, metals are usually not good candidates for observing it. The first convincing evidence of an experimentally realized electron crystal came much after its first prediction, and was two-dimensional rather than three-dimensional.Looking at a system of electrons adsorbed on a helium surface, Grimes and Adams [6] measured the density response function using a radio-frequency technique. They found a series of resonances that could be identified with the predicted dispersion of phonons of the WC [7,8] of electrons adsorbed on a helium surface. This was regarded as strong evidence that the electrons had indeed crystallized in this system. The areal density of electrons in this system is generally extremely small (typically,n, lo7cm-’), so that for all intents and purposes the system may be regarded as classical (i.e., for such large distances between the electrons, the

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INTRODUCTION

73

energy due to the zero-point motion of the phonons is negligiblecompared to the potential energy of the lattice). While there are a number of interesting properties associated with the WC in this classical limit, particularly relating to its behavior in a disorder potential (which we discuss in detail below),one disadvantage of the helium system is that the densities that may be realized practically are far too low to allow an observation of the transition from the crystal to the gas state. A more favorable system to study in this regard is electrons in a semiconductor environment. Semiconductors offer a particular advantage because one can control the density of electrons over several orders of magnitude by doping the system at an appropriate level. However, for bulk doped semiconductors, an immediate problem arises: The dopants that provide the electrons, once ionized, become strong scattering centers, completely changing the character of the ground state. At zero temperature, the electrons fall back on their donors rather than localizing at crystal sites. The resulting ground state is highly disordered and not collective in nature. Clearly, the problem with bulk semiconductors is that the electrons are free to move in the region of the donors, which inevitably imposes a strong disorder potential on them. A major achievement in the last 15 years that allowed this problem to be circumvented was the invention of modulation doping [9, lo]. These systems use high-quality interfaces between different semiconductors, typically GaAs and AlGaAs, to trap a two-dimensional layer of electrons, provided by dopants that are setback some distance d from the interface. By separating the electrons from the dopants, the effect of the resulting disorder potential is substantially smaller, and the prospects of observing the WC are greatly enhanced. The strength of the disorder can be characterized by a setback ratio d/a,, where a, = J.s/.is the typical interelectron spacing in two dimensions. One wishes to achieve very large values of this ratio to minimize the effects of disorder. One way to do this is to work with high-density samples, for which ratios of d/a, 4 to 6 can in practice be achieved. However, the price one pays for such favorable setback ratios is that the resulting densities are typically above the Wigner transition, so that one is in the gaseous rather than the crystal regime.

-

3.1.2. Wigner Crystal in a Magnetic Field It was recognized early on that the prospect of achieving crystallization in these systems can be greatly enhanced by applying a strong perpendicular magnetic field.The reason is that magnetic fields quench the kinetic energy of the system, so that zero-point motion effects-which tend to favor the gaseous state-can be suppressed. In the presence of a magnetic field, the kinetic energy of the electrons has the form [ll] En = (n 1/2)Aw,, where n is an integer, o,= eB/mc is the cyclotron frequency, B is the magnetic field, and m the effective mass of the electrons. For each value of n, there is a vast degeneracy: The number of states is proportional to the area of the system. These energy states are known as Landau levels, and because of the huge degeneracy within a Landau level, it is possible to form linear combinations of states that represent localized Gaussian orbitals. If

+

74


one uses just states in the lowest Landau level, working in the symmetric gauge vector potential [l 11, these orbitals have the form

where 1; = hc/eB is the square of the magnetic length and R represents a site near which the electron is localized. As can be explicitly seen, in the limit of very large magnetic fields, I , *0, so that the electron becomes highly localized while maintaining the minimum possible kinetic energy. Thus in the limit of large fields, the energy of the system can be minimized by putting the electrons in states of the form in Eq. ( 3 ) and then finding the set of R s that is optimal for the potential energy. In the high-field limit, this becomes completely equivalent to finding the ground state of a set of classical charged particles [ l 2 ] , so that one expects a crystalline ground state for the system. Thus, for any density of two-dimensional electrons, a large enough magnetic field should allow the electrons to form a crystalline ground state.

3.2. SOME INTRIGUING EXPERIMENTS In this section we describe some of experimental results that have led to renewed interest in the Wigner crystal in the past few years. This is not meant to represent a complete review of the experimental situation. Rather, it is a sampling of results which may or may not indicate that the W C has been realized in semiconductor systems, but definitely indicates that something interesting is happening to the electrons in these two-dimensional systems in very strong magnetic fields.

3.2.1. Early Experiments: Fractional Quantum Hall Effect The earliest experiments on modulation doped semiconductorsfocused on their transport properties. Placing the two-dimensional electron gas (2DEG)perpendicular to an applied magnetic field in the 2 direction and imposing a current density jxA, one may measure the resulting electric field parallel to the current (Ex), yielding the diagonal resistivity pxx= Ex/jx, as well as the electric field perpendicular to the current (Ey),yielding the Hall resistivity pyx= Ey/jx.According to the simplest semiclassicalmodels of electrons [ 1 3 ] , which ignore electronelectron interactions as well as many quantum effects,the expected values of these quantities are pxx= m/ne2z,pxy= - B/nec, where T is a phenomenological scattering time and n is the two-dimensional density of electrons. It was already known [ 1 1 ] from experiments on related MOSFET systems that the lowtemperature behavior of these transport coefficients was dramatically different than the semiclassical analysis suggested. Near filling fractions v E 27tl;n in the vicinity of an integer, the pxyturns out to be quantized in units of h/e2,and pxx

SOME INTRIGUING EXPERIMENTS

75

nearly vanishes for low-enough temperatures. This important discovery is generally known as the integral quantum Hall efect (IQHE). The filling fraction v may be thought of as a measure of how full the Landau levels are for a given density of electrons; it is equivalent to the ratio of the total number of electrons in the system to the number of states available in a Landau level. Thus, if one wishes the electron to crystallize, one needs to reach the limit v 1/5. To test this idea properly, careful calculations of both the liquid-state energy and the WC energy are necessary. The basic form for the liquid ground state is the well-known Laughlin

78


wavefunction[11,27], which describes the liquid precisely at v = l/m for rn an odd integer, whose energy may be evaluated quite accurately [11,27-291. As suggested by Eq. (2), the first correction to the Hartree-Fock energy of the WC ground state would be due to zero-point motion of the phonons. This was computed using a trial wavefunction method [30], with the resulting estimate for the transition between FQHE and the WC taking place near v = 1/7. An important improvement to both the liquid-state energy [31] and the WC energy [32] is inclusion of Landau level mixing. This not only shifts the transition to near v = 1/5 for the electron systems, but also predicts a transition in the vicinity of v = 1/3 for hole systems. This impressive agreement with experiment strongly favors the interpretation of the insulating state as a WC. Many other experimental probes have been employed to probe the insulating state. Some early experiments [33] used radio-frequency techniques similar to those employed for electrons on helium, looking for the shear modes associated with the crystal. The interpretation of these data proved to be controversial [34], however, as it was unclear whether the electric field probe coupled to phonon modes of the underlying semiconductor structure in addition to, or instead of, the phonon modes of the WC. However, this early data did show the reentrant behavior near v = 1/5 that was found in subsequent dc transport experiments. There are also a number of more recent experimentsprobing the collectivemodes of the 2DEG in its insulating phase by measurements of the frequency-dependent conductivity via surface acoustic wave methods 135,361, and the dielectric response using capacitive coupling methods [37]. The former measurements reveal broad resonances at low temperatures and low filling factors near 1 GHz, and the latter a remarkably large response at very low frequencies. Both results may be qualitatively understood as a collective response of a pinned WC, although the correct quantitative analysis of the data-in particular, what they say about the positional correlation length of the crystal-remains a matter of some controversy [38-401. Some of the transport data in the insulating phase are not clearly consistent with a WC interpretation. Narrowband noise in the sliding state [22,23], which arises in analogous CDW systems [17], has not been observed in the insulating state. Narrowband noise arises in CDW systems because the lattice should deform in a periodic way as it slides across a disorder potential. This leads to a noise spectrum with a peak at a frequency proportional to the velocity of the sliding electrons. (Very recent experimental work on the WC has found some nontrivial coupling between ac and dc electric fields, which may be a precursor of narrowband noise [37].) However, the absence of narrowband noise in these systems should not be construed as evidence that the electron state is extremely disordered, as one expects to observe this phenomenon only if the positional correlation length is on the order of the system size [17]. Computer simulations of the classical electron crystal [26] suggest that such long correlation lengths are unlikely to be achieved for presently available samples. A perhaps more troubling experimental observation has been that the Hall resistivity pxyof these systems appears to be finite in the low-temperature limit,

SOME INTRIGUING EXPERIMENTS

79

and equal to its classical value to within a few percent [41,42]. This is in contrast to what one might naively think For example,if the current at finite temperatures is carried by point defects (as occurs for bulk ionic solids) such as vacancies and interstitials, one expects the effective number of carriers neff to vanish exponentially with temperature. It has already been remarked that the diagonal resistivity, which classically is proportional to the inverse of the carrier density, behaves precisely as if this is occurring. However, classically pxycc he;,', so that this model predicts activated behavior. The distinction between the behaviors of pxx and p x yis so dramatic that one group [43] has postulated that the ground state of these systems is not a WC at all, but rather is a disorder-dominated state dubbed the Hall insulator. However, as described below, when quantum fluctuations are accounted for, one can understand how a model with finite temperature conduction due to point defects would lead to a classical Hall resistivity [44,45].

3.2.3. PhotoluminescenceExperiments Another promising line of experimental research is photoluminescence (PL). These experiments use holes to probe the electron gas and are generally introduced into the system in one of two ways. In localized hole experiments [46-491, a layer of acceptors is intentionally grown into the crystal relatively close to the electron plane. In their ground state, the acceptors are negatively charged; however, by shining light on the sample at appropriate wavelengths an electron can by ionized out of the core of the acceptor, leaving behind a neutral object which presumably has little effect on the state of the 2DEG. By observing recombination of electrons in the 2DEG with the core hole, a characteristic spectrum consisting of a doublet is typically observed.This doublet structure has interesting temperature and magnetic field dependence: at low temperatures and most fields, the greatest portion of the oscillator strength is found in the lower portion of the doublet. At temperatures of order 1 to 2 K, the oscillator strength is transferred to the upper PL line (Fig. 3.3). This behavior has been interpreted as a signal of melting for the WC, and is supported by the observation of very similar behavior at very low temperature when the magnetic field is tuned such that the filling factor is 1/5, where it is expected the system forms a correlated quantum liquid (i.e., the Laughlin state). Another interesting experimental observation is that similar behavior occurs near v = 1/7, 1/9, where no FQHE is observed in transport; it is thus an open question precisely what is happening at these very small filling fractions. The second class of PL experiments uses valence band holes, generally in wide quantum well structures [SO-SZ]. The phenomenology of these data is in many respects quite similar to that of the localized hole experiments as illustrated in Fig. 3.4. The advantage of such geometries is that the holes may in principle by itinerant, allowing them to be sensitive to global properties of the lattice. Localized holes, by contrast, can only probe local properties. The disadvantage to valence band hole experimentsis that not much is currently understood about the initial quantum states of the holes [52], making interpretation of currently

80

PROPERTIES OF THE ELECTRON SOLID I

I

I

0 s = 6xlOlocm-2

H = 26T

l2=0 I

-L I

1 2 Temperature (k)

I

3

Figure3.3. Intensity ratio for lines in photoluminescence doublet as a function of temperature in an insulating state, as measured in localized hole experiments. (From Ref. [46].)

1.8 c, .-

1.41

v)

c

W 4-

0

W

a

-c

m

1.0

\

1.2

0

0.4

?c2

E 2.2

.3

0

c W V

8 W .-c

- 1.8 4-

C

5

-I

0 1.521

0.4

1.4

1.523 Enegy (eV)

0

1

2 T (K)

3

4

Figure 3.4. Photoluminescence spectra observed in an insulating state as measured in valence hole experiments:(a)temperature dependence of the spectrum; (b) intensity ratio of lines labeled S and B' in part (a)as a function of temperature. The ratio drops markedly in the range of temperatures between T,, and Tc2.(From Ref. [Sl].)

DISORDER EFFECTS ON THE ELECTRON SOLID

81

available data difficult. There is also a class of related experiments using recombination of electrons in higher electric subbands of a heterostructure with valence band holes to probe the Wigner crystal structure [53]. One exciting possibility of itinerant hole PL experiments is that one might observe some features of the Hofstadter spectrum [54,.55]. This is the energy spectrum of a charged particle moving in a uniform magnetic field and a periodic potential and is very sensitive to the precise value of the magnetic field. Specifically, for a rational filling fraction v = p / q , the spectrum is expected to have q bands with q - 1 gaps. An observation of this behavior would constitute strong evidence that the electrons have indeed crystallized. The PL spectrum expected from real samples in the WC states requires a detailed treatment of the coupling between the hole and the WC, and some features of the Hofstadter spectrumparticularly the smallest gaps-are in practice dificult to observe [56]. This problem is compounded by the effects of disorder. Nevertheless, the possibility of seeing some of the basic structure of the Hofstadter spectrum in PL experiments remains a tantalizing possibility. 3.3. DISORDER EFFECTS ON THE ELECTRON SOLID: CLASSICAL STUDIES From the discussion in Section 3.2 it is clear that interpretation of almost any experimental data on the WC in semiconductor environments requires some understanding of the sources of disorder in these systems, as well as what effect they have on the ground state of the electrons. The importance of disorder is further emphasized by very general arguments due to Imry and Ma [57] that one cannot have long-range positional order at zero temperature in a classical system in an arbitrarily weak random potential in two dimensions. It thus becomes relevant to study whether crystallization of electrons in the heterostructure systems is in any sense possible even if quantum fluctuations are ignored. Such studies are directly relevant to electrons at the very highest magnetic fields (very low filling fractions),where quantum exchange effects are believed to be small [ 5 8 ] , as well as to electrons on helium surfaces. In principle, classical states may be used as a starting point from which quantum effects and corrections may be studied. 3.3.1. Defects and the State of the Solid Since two-dimensional solids in a random potential cannot exhibit long-range positional order, they cannot be classified as crystals in the sense of classical crystallography. Nevertheless, it is known that if the disorder is not strong enough to induce isolated topological defects of the crystal, specifically dislocations, the system can possess long-range orientational order [59,60]. Furthermore, the positional correlations in this state fall off as a power law with distance rather than exponentially,so that the state may be characterized as having quasilong-range order. Thus, one usually thinks of this as a crystal state.

82


In the absence of disorder, it is known that temperature can destroy the crystal above the Kosterlitz-Thouless transition temperature [61,62] via the unbinding of dislocation pairs with equal and opposite Burgers vectors. (In two-dimensional crystals, a dislocation may be thought of as a line of particles that comes to an abrupt end somewhere inside the crystal. The Burgers vector, which characterizes the strength of the dislocation, is the amount by which a circuit of steps that should close on itself in a perfect lattice fails to do so when the circuit surrounds a dislocation core.) The resulting state has short-range (exponential) order and quasi-long-range orientational order; because of the latter it is not a true liquid and is known as a hexatic state [63]. This state is destroyed at a still higher temperature by disclination defects. Disclinations are defect sites that have the wrong number of nearest neighbors relative to the number expected for a perfect lattice; for triangular lattices, dislinations typically have five or seven nearest neighbors. Such defects may be said to have disclination charge k 1. (However, this should not be confused with electric charge.) When bound together, a net neutral pair of disclinations separated by a single lattice constant forms the core of a dislocation. The hexatic state is destroyed when the temperature is high enough that these disclinations can unbind and be found in isolation in the resulting liquid state.

3.3.2. Molecular Dynamics Simulations It is natural to look for an analog of this phenomenology at zero temperature as a function of disorder strength. Simulations [25,26] of classical electrons moving in a random potential modeled after the heterostructure systems reveal that only half this phenomenology is actually reproduced. The model used contains N electrons in a plane setback from a plane of N positively charged ions by a distance d (Fig. 3 . 9 , and low-energy states are generated by a moleculardynamics simulated annealing method, in which the electrons are assumed to follow Newton’s equations, given an initial temperature well above the melting

d 0

0

Q 0

o

0

0

Figure3.5. Model system of N electrons and N quenched positively charged ions, in planes separated by a distance d. (From Ref. [26].)

DISORDER EFFECTS ON THE ELECTRON SOLID

83

transition, and then are slowly cooled by removing kinetic energy from the electrons in small discrete amounts. Details of these simulations may be found in Ref. [26]. Figure 3.6 shows some typical low-temperatureconfigurations,well below the freezing transition, for different values of the parameter d/u,, where here a, is the lattice constant of a perfect WC at the electron densities used in the simulation. The electric forces in these simulations were screened dielectrically with E = 13 (the dielectric constant of GaAs), and the electron densities shown in the figures were taken to be 5.7 x 10'0cm-2. Periodic boundary conditions were assumed for these systems. For all values of d/a,, disclinations appear in the lowtemperature state. These are denoted in the figures as + for sites with five nearest neighbors, and as x for sites with seven nearest neighbors. For large values of d/a, (Fig. 3 . 6 ~isolated )~ dislocations (denoted as bound pairs of + and x in the figures) may clearly be seen in the configuration. As d is increased further, the density of dislocations decreases until the average spacing between them exceeds the sample size. It is interesting to note that grain boundaries do not appear in these configurations.Some of the present literature on the WC has assumed that the effect of disorder is to introduce well-ordered microdomains, separated by sharp boundaries. This would be reflected in these configuration if the dislocations collected together to form grain boundaries [62]. Apparently, this is not energetically favorable for this type of disorder. However, one cannot rule out that other disorder potentials could introduce these defects. As the ratio d/a, is decreased (Fig 3.6b and c), the ground state of the WC undergoes a zero-temperature phase transition, from a hexatic state to an isotropic state. Both states must be characterized as glasses, because there is only short-range order in the positional correlation functions. In Fig. 3.6b, one may see disclination pairs that are separated by more than a single lattice constant but are still clearly bound together. Above a critical setback distance (Fig. 3.6c), isolated disclinations may be found. Figure 7 illustrates the orientational correlation functionsfor several values of d/a, for low-temperature states generated by the MD method. They can be well fitted by exponential forms in the strong disorder limit (small d/a,), and the resulting correlation length is found to be a slowly varying function for d/a, < 1. In the interval 1.1 < d/a, < 1.2, the correlation length rises rapidly, suggesting a possible divergence (and an associated phase transition). However, once the correlation length exceeds the system size, fits to either an exponential form or a power law become possible, and it becomes difficult to identify precisely what value of d/a, would be the critical one in an infinite system.One can estimate from Fig. 3.7, however, that the transition between the states occurs near d,la,.= 1.15. Several comments are in order. It must be noted that these simulations include neither the finite thickness of the layer nor the finite value of the magnetic length I, when a magnetic field is present. Both these effects tend to soften the electronelectron interaction relative to the electron-ion interaction, so the actually value of d, is expected to be somewhat higher than the simulation value in real heterojunction systems. Experimentally, the best such samples have d/a, x 6, so

l/m), and d is the antisymmetrization operator. The lattice sites in $wc are chosen such that the origin is an interstitial site, so that the location of maximum probability for the interstitial is its expected location for a classical crystal. We work here in units of the magnetic length I, = (hc/eB)1'2, where B is the applied magnetic field. We note that if one uses a Hartree approximation for t j W c , the case m = O corresponds to an antisymmetrized Hartree approximation for the interstitial. Such approximate forms for wavefunctions of the WC at low enough fillings have been argued to be quite good, since one may show that exchange corrections to the WC energy at low fillings are quite small ~581. To understand the effect of the correlation factor on the interstitial, one may use the plasma analogy introduced by Laughlin [27]. Dropping the antisymmetrization in Eq. (9),and using a Hartree approximation for r(/wc, we write the square

a:,

QUANTUM EFFECTS ON INTERSTITIALELECTRONS

93

modulus in the form 1$12 = e-@, where for f l = 1,

This represents the energy of a classical two-dimensional Coulomb particle with charge - 1, interacting with a neutralizing backgrund of charge density ab= 1/2n, and a lattice of charges - m with charge density a, = - m/u2,where a is the WC lattice constant. On a coarse scale, the interstitial is thus interacting with a system of net charge density a = (I - mv)/2x (v is the filling in the absence of the interstitial),and thus spreads out in the center of the disk to an area of l/a. We note that precisely at the filling v = l/m, the interstitial will spread out uniformly over the disk, and for higher fillings the interstitial electron will have its highest probability at the system edge, essentially being ejected from the system. This is the reason the wavefunctions are physically reasonable only for fillings below v = l/m. This behavior of the interstitial wavefunction can be demonstrated quite directly, by computing the probability p( ii) of finding the interstitial at some position ii.Using the Hartree approximation for $wc in Eq. (9)and neglectingthe antisymmetrization, one has

For simplicity, we choose the lattice sites Rf to form a square lattice. The interstitial electron density distribution

can be evaluated directly from the expression above since the integrations over the lattice electron coordinates may be computed analytically. Figure 11 shows the result of this calculation along a crystal symmetry axis. One can clearly see that p( i7)becomes quite spread out as v + l/m. We are now in a position to understand why the correlation factor might lower the energy of the interstitial, especially close to v = l/m. In essence, the Jastrow factor allows the interstitial to sample a large region of the system and thereby can take advantage of quantum fluctuations in the WC. When there are configurations in t,bWc that have large holes in them, the interstitial has a high probability of being located there, especially when v l/m, for which the interstitial density (Fig. 3.1 1) samples a large region of the crystal. Conversely, when the crystal forms regions of high density, the interstitial has a low probability of approaching that region. In this way, the interstitial electron does an excellent job of minimizing its energy.

-

94


m=5

0

0

u = 0.19 u = 0.17 u = 0.10

x/a Figure 3.11. The probability distribution for the interstitial electron in a correlated interstitial state described by Eq. (11) with m = 5 and v = 0.19, 0.17, 0.1. The distance is along they y = 0 axis in a Cartesian coordinate reference frame, where the square lattice R t is expressed as ( i - 0.5,j - 0.5).The distance is in unit of the lattice constant a,. (From Ref. [a].)

The energy of this state may also be computed; details may be found in Ref. [MI. The result is displayed in Fig. 3.12. The energy shown here is defined as the differencebetween the energies to add a particle to the lattice as an interstitial and as a regular lattice electron (i.e,, the chemical potential). Alternatively, one may think of this as the energy to remove an electron from the lattice, rearrange the particles into a perfect lattice with a slightly larger lattice constant, and then replace the electron at an interstitial site. As may be seen, the energy is lowered dramatically by the correlation factor (see inset of Fig. 3.12). This is in fact consistent with experiment, for which the activation energies measured [191 are significantly smaller than expected from simple classical [8] or Hartree-Fock calculations [75] of the vacancy-interstitial creation energies. Furthermore, the energy becomes negative when the filling factor closely approaches v = l/m [76]. This indicates that the Hartree (and probably Hartree-Fock) Wigner crystal ground states are unstable with respect to these excitations if one is too close to v = l/m for m = 5,7; similar results were found for m = 9.

;*'I

QUANTUM EFFECTS ON INTERSTITIAL ELECTRONS

95

1

a

0.1

n 0 rl

cu' 23 W

3 23 W

c

m

V

Figure 3.12. The energy of the correlated interstitial as a function of the Landau level filling factor v for m = 5 and m = 7.The inset shows the energy E at v = 0.19 as a function of the exponent m of the Jastrow factor (i.e., as a function of the strength of the correlation). (From Ref. [MI.)

Although the prediction of an instability of the crystal near v = 1/5 is consistent with experiment,one must not jump to the conclusion that the resulting state is a liquid. This is especially highlighted by the fact that similar results are found for m = 7,9 near fillingfractions 1/7,1/9, where the fractional quantum Hall effect is not observed experimentally.There are indications that something interesting happens near these small filling fractions in photoluminescence experiments [46,47], and there are unexplained anomalies in transport data [7nnear v = 1/7. Since the results above indicate clearly that correlations become important for v = l/m even for large m,the nature of the insulating state at these fillings remains an interesting open question. 3.4.2. Interstitials and the Hall Effect

Another indication that Laughlin-Jastrow correlations are important for interstitial electrons is that they dramatically change their transport properties. In particular, if the current at finite temperatures is indeed carried by correlated interstitials, one finds that the Hall resistivity pxyis equal to its classical value for a noninsulating state, B/nec, where n is the full two-dimensionalelectron density.

96


The key to seeing this is to consider a generalization of Eq. (9) of the form

which allows the correlation factor between the interstitial and the other electrons to be site dependent. This new degree offreedom is useful if we consider systems with weak disorder, such that the WC density is not perfectly uniform throughout the sample (e.g., Fig. 3.6a).By adjusting the values of mi to avoid the regions of higher energy for the interstitial, one can create a state with particularly low energy. Furthermore, by choosing values of the exponents such that (mi)= l/v, where (...) denotes an average over all sites, the interstitial can sample the entire system and take best advantage of the low-energy regions. We thus expect such choices for Eq. (12) would give particularly low energies; this is supported by the observation that the energy in the inset of Fig. 3.12 is a monotonically decreasing function of m. To understand the effect of the correlation factors on the transport properties of the interstitial,it is useful to think of them as being due to magnetic flux tubes of magnitude m,40 attached to the electrons on each lattice site, where 4' is the magnetic flux quantum [78]. This flux is directed to oppose the actual applied magnetic field Be"' = n&,/v. In a mean-field description, the localized electrons will generate a fictitious magnetic field BEL = nL40/v, where n, is the density of lattice electrons, which partially cancels Be"'. The excited conduction electrons are then traveling in a net field of B""' = Be"' - BCL= nC4dv, where n, is the density of interstitial (i.e., conduction) electrons. To the lowest order in the disorder potential, the motion of the conduction electrons can then be described in the Drude model. Suppose that there is a current j , = n,eu with px, = l/n,ep, where p is the effective mobility. A Hall field E, must be present to balance the Lorentz force, so that the measured Hall resistivity would be pxy= Ey/jx= (vB/c)/ n,eu = B/n,ec. For the strongly correlated systems B = B""' = (nc/n)Bex',so that pxy= B/nec, regardless of the number of electrons that are localized. Remarkably, this implies that pxyremains finite and equal to its classical value even at very low temperatures, for which the number of electrons actually conducting becomes vanishingly small. This behavior is precisely what is observed in experiment for the insulating phase [41,42]. One final detail of this model is that it assumes that when lattice electrons are thermally ionized to become interstitials, the vacancy that is left does not carry a flux tube, so that some magnetic field is left behind to produce a small Lorentz force on the interstials when they move. This is sensible since if the flux tubes remained, the interstitials would be avoiding sites that are actually particularly low in energy. This means that the effectivemagnetic field seen by the electrons is highly nonuniform; it essentially looks vanishingly small except at vacancy sites, where it is quite large. However, it may be shown using a Boltzmann approximation that even this very strong inhomogeneity in the field does not alter the Hall resistivity [79].

PHOTOLUMINESCENCE OF THE WIGNER CRYSTAL

97

3.5. PHOTOLUMINESCENCE AS A PROBE OF THE WIGNER CRYSTAL As discussed in the introduction, one probe of the electronic insulating state in semiconductor systems in strong magnetic fields that has generated considerable excitement over the last few years is photoluminescence (PL). In such experiments, either a valence band hole [SO] or a hole bound to an acceptor [46-481 recombines with an electron in the 2DEG, producing a characteristic photon spectrum. We will see that a mean-field analysis of the PL spectrum has, in principle, characteristic signatures of the W C a Hofstadter butterjy [54] spectrum for the case of weak interactions between the electrons and the hole, and a characteristic shift in the PL spectrum upon melting of the crystal. When the possibility of exciting phonons and other collective excitations of the WC in the recombination process-shakeup effects-is included, the Hofstadter spectrum is lost in localized hole experiments and is replaced by a spectrum with several collective mode satellites. For itinerant holes, coupling between the hole and the WC phonons has important effects, but some of the features of the Hofstadter spectrum are likely to survive. 3.5.1. Formalism

We begin by describingcalculations for the localized hole geometries. To find the PL spectrum expected from a WC, it is convenient to employ a supercell method, in which one approximates the initial state of the system as an electron lattice that is commensurate with a superlattice of holes. The supercells contain a single hole per unit cell, and one tries to accommodate as many electrons in them as possible. This approach is sensible since the actual density of holes in real systems is extremely small, so that correlations among the holes are not expected to be important. The advantage of this approach is that it allows one to take advantage of the symmetriesof the electron lattice. Working in the lowest Landau level (as is appropriate in strong magnetic fields),the PL intensity may be written in the form [SS] P(o)K Im [ R ( o id)], where

+

R(G,o)e-G21i/4 R(o)= 7 nh 2x10

c

and we have approximated the core-hole wavefunctions as delta functions, nh is the density of holes, R the volume of the system, and the vectors G are the reciprocal lattice vectors of the superlattice. The magnetic length I , = (hc/eB)’/’ will be set to unity in the remainder of this chapter. R(G, o)is defined as

where g is the Landau level degeneracy, Xis the guiding center quantum number,

98


and

with a& creating an electron in state X and c i creating a hole in the unit cell i, and fi is the inverse temperature. The quantity R, should be interpreted as a Green’s function for the electron lattice, and formally has poles at the quasihole excitations of the WC. We will see that these poles merge together into narrow bands and form well-defined peaks in the PL spectrum. The equation of motion for Rij(G,o)may be written as

- nh

c V(G)eic”G/2-G”/4R.(G - G’,z) 0

C’

-

JOB

d z ’ c (G, G’; z - z’)Rij(G,z’)

where the self-energy is defined implicitly by

In Eqs. (13) and (14), v(q) and V(q) are the Fourier transforms of the electronelectron and electron-hole interactions, respectively, t o is the energy of the localized hole, and the sum over G’ is only over reciprocal lattice vectors, while the sums over q are over all wavevectors.Ri specifiesthe position of the hole in the ith unit cell, and (p(G))e-”14 is the expectation value of a Fourier component of the electron density [SS, 801.Finally, the correlation function Cij(pl,pz)isjust the Fourier transformation of

Equations (13) and (14) represent the first in an infinite series of equations relating an n-particle Green’s function to an (n + 1)-particleGreen’s function [81]. To make practical the computation of the PLYone must find a sensible approximation for the self-energy in Eq. (14). We discuss next two schemesfor doing this.


99

3.5.2. Mean-Field Theory

The simplest approximation possible for the self-energy in Eq. (14) is to use a Hartree-Fock (HF) decomposition of Eq. (15). This yields the result C (G, G , T - z’) = CHF(G,G)6(z - z’), with CHF(G,G ) = W(G - G’)( p(G G ))eiC c”z + (1/2x)C,Y(q)(p( - q))e-2’4-q‘L6,,G., where W is the sum of the direct and exchange Coulomb potentials [55,80]. To understand the results qualitatively, two limiting cases are of interest: (1) no electron-hole interaction (which is appropriate for a hole very far from the electron plane), and (2) with a Coulomb electron-hole interaction. For case 1 one finds a characteristic spectrum for rational filling fractions v = p / q , where p and q are integers, of p narrow peaks in the PL. An example of this is illustrated in Fig. 3.13. This may be understood in terms of the single particle density of states for the WC in the mean-field approximation (MFA): in this case, an individual electron moves in the periodic potential of the other electrons and a magnetic field with q / p flux quanta penetrating each unit cell. The spectrum for this situation is the wellknown Hofstadter butterfly of q bands, and in the MFA, p of these are filled. The

600

01

Y

(a) T = O.o045T,,,,~t

- 13.0

= 217,

= OmeV I

I

I

- 12.0 E

- w(meV)

-11.0

- ...

-10.0 .

Figure 3.13. Example of a photoluminescencespectrum for v = 2/7 with no electron-hole interaction,with T below the melting temperature Tme,,. The PL spectrum has two peaks because the numerator of the rational filling factor is 2. Inset: Same, for T above melting temperature. (From Ref. [SS].)

100


PL spectrum just reflects the recombination of electrons occupying these bands. Thus PL is a weighted measure of the electron density of states. One unfortunate characteristic is that at the lowest filling fractions, for which the WC might be realized in real systems(v < 1/5), the gaps between the bands are extremely narrow (on the order 0.01 MeV); thus in practice one is likely only to resolve a single PL peak. In any case, these gaps (for the localized hole case) do not survive either electron-hole interactions or shakeup effects, so that one should not expect to observe the Hofstadter spectrum in this way. For case 2 (electron-hole interaction), only a single PL line is present, which turns out to be the contribution from a single electron in a bound state with the hole. PL from recombination of other electrons in the crystal is in principle present, but its magnitude is very small. For both cases 1 and 2, there is an interesting finite-temperature effect: When the crystal is brought to melting, there is a sudden shift in the spectrum to higher energies, as may be seen in the inset of Fig. 3.13. The interesting property of this effect is its suddenness,which is a result of the first-order nature of the transition in the Hartree-Fock approximation. Thus, in real, weakly disordered samples, one might expect to see two peaks over a range of temperatures, one each from regions of melted and crystalline electrons, which coexist for some temperature range due to disorder effects [25]. This is in qualitative agreement with experiment [46,47,50], where a shift in oscillator strength between two peaks is observed over a temperature range of about 1 K.

3.5.3. Beyond Mean-Field Theory: Shakeup Effects This general formalism allows one to go beyond mean-field theory, to include the excitation of phonons and other collective modes in the electron-hole recombination process. The strength of this approach is that it treats both the tunneling electron and the lattice electrons on the same footing (i.e., the formalism does not choose out a single electron and specify it as the one that will recombine with the hole). This is contrast to (albeit simpler)independent boson models, in which one electron is specified to recombine with the hole. This acts as a suddenly switched potential for the remaining electrons, whose motion is treated in the harmonic approximation [82]. Such methods do not allow a description of the PL spectrum when the temperature approaches the melting temperature. Furthermore, quantum treatments of the density response [80] of the WC have indicated that there are sharp collective modes with large wavevectors; these cannot be described within a harmonic approximation. The equation-of-motion formalism allows both these effects to be included in a very natural way. The general approach is to carry the hierarchical set of equations described above to one more level. Specifically,rather that write down a H F decomposition of C ,, one derives an equation of motion for it that includes the collective excitations. The result is then inserted into Eqs. (13) and (14). The actual calculation is quite arduous [83], but the result contains the collective modes of the system in a very clear way. One finds that Cij may be approximately

PHOTOLUMINESCENCEOF THE WIGNER CRYSTAL

101

Photoluminescence Power I

0

- 15.5 E

,

- 13.5 - E, - Eh (mew

I

-11.5

Figure 3.14. PL spectra including shakeup effects for (a) v = 1/5, T = 0, no electron-hole interaction; (b) v = 1/5, T = 0, with electron-hole interaction; (c) v = 2/7, T = 0, no electron-hole interaction. Inset: PL spectra for (a) and (b), with Tjust above the melting temperature. To distinguish different spectra, they are separated by 250 units. (From Ref. [84].)

expressed in terms of the density response function [84] x(G, + q, G , + q; t)E x&q; T) = - g ( Tp(G, + q, t)p( - G , - q; 0)). The Fourier transform of this response function contains poles at the collectivemode frequenciesof the system; it is thus through x that shakeup effects are introduced. Examples of the PL spectra calculated in this way are shown in Fig. 3.14 for filling fractions v = 1/5 and v = 2/7. For the case of no electron-hole interaction, at low temperature, a well-defined shakeup peak may be seen approximately 1 MeV below the main PL peak; a second, very weak satellite is observed approximately 1.6 MeV below the main peak. The origins of these peaks may be understood in terms of the collectivemode density of states, which is illustrated in Fig. 3.15. A van Hove singularity, arising from zone-edge phonons, appears as a strong double peak near 0.4 MeV. Two other peaks may be seen above this. There are weak sidebands associatedwith each of these peaks in the PL spectrum. The precise interpretation of these peaks is unclear; however, it has been speculated that these represent vacancy-interstitial excitations [80]. It is crucial to use a fully quantum mechanical treatment of the collective excitations of the

102


100

Phonon Density of States

F

Ee . f!50 3 G 5

0 C Figure 3.15. Collective-mode DOS for v = 1/5, 7' = 0. (From Ref. [84].)

lattice to observe these higher-order satellites; classicaltreatments of the phonons do not produce these unusual excitations. As in the mean-field calculation, there is a sudden shift in the PL spectrum upon melting. The inset to Fig. 3.14 illustrates the PL spectrum in the melted state both without and with an electron-hole interaction. In the former case, there is no phonon sideband present. This is necessarily so, because in the melted phase, the density is uniform and there are no collective modes in the LLL [80]. By contrast, when an electron-hole interaction is present, there is a nonuniform electron density near the hole, allowing some local collective modes to persist even above the melting temperature. With further increase in temperature, or increased setback between the hole and the 2DEG, the oscillator strength of this mode will decrease significantly. Figure 3 . 1 4 ~illustrates results for v = 2/7 in the absence of electron-hole interactions. As can be seen, except for a change in energy scale caused by changing the magnetic field, the lineshape is essentially identical to the case of v = 1/5. This contrasts sharply with the results found in mean-field theory, where without electron-hole interactions, a filling v = p / q generally yields p distinct lines for a localized hole. While the splittings are so small in that situation that they are difficult in practice to resolve, evidently shakeup effects wipe out this structure even in principle.


103

3.5.4. Hofstadter Spectrum: Can It Be Seen?

Is it possible that any remnant of the Hofstadter spectrum is observable in photoluminescence?A possible avenue for this may lie in itinerant hole experiments, in which the hole can hop between sites and is not forced to be in a single initial quantum state before the recombination process. In particular, if the hole is mobile, it then has a density of states of a positively charged particle moving in a periodic potential-which itself will have a Hofstadter spectrum. The higher Hofstadter states will in general be occupied by holes at finite temperatures, so that considerable structure could arise due to the hole density of states. Figure 3.16 illustrates a spectrum for itinerant holes as evaluated using the mean-field approximoation, under the assumption that the holes are in a narrow quantum well 250A from the electron layer, at low and intermediate temperatures. The new structure on the high-energy side of the main PL line comes from thermal excitations to the higher Hofstadter levels in the hole density of states. Observation of a Hofstadter structure in the hole spectrum would indeed give dramatic evidence that crystalline order is present in the electron system at low filling factors. However, as in the case of localized holes, one must consider which features of the Hofstadter spectrum survive the coupling of the hole with distortions in the electron lattice. There are two important considerations that arise in this context: 1. The hole must not be too close to the electron layer, or it will become

energetically favorable for an intersitial to be introduced in the lattice, to

i?

8 1 600

h

E

m

a,

400

-

200 -

-

Y

0 E-E,-Eh-A(mev) Figure 3.16. Photoluminescence for itinerant hole at electron filling v = 2/11, for T z

0.005Tm,,, (solid line) and T z O.OST,,,, (dotted line). Electron and hole cyclotron frequen-

cies given here by o,and oh, respectively,and A is the conduction band-valence band gap. (From [55].)

104

PROPERTIESOF THE ELECTRON SOLID

which the hole will become tightly bound. Hartree-Fock calculations [56,85], as well as classical treatments, give an estimate of the distance below which this occurs to be approximately 0 . 5 ~ ~ . 2. Above the critical setback distance between the hole and the electrons,there is still a polaron effect, in which the lattice is distorted as the hole moves through the system. This will certainly narrow the Hofstadter bands, and can close up the smaller gaps in the spectrum. However, preliminary work [56] has shown that at least some of the bands do remain well separated, with energy gaps very close to those found in the mean-field approximation. Further theoretical work is necessary to sort out this complicated issue. 3.6. CONCLUSION SOME OPEN QUESTIONS Many interesting open questions remain regarding the Wigner crystal. In both the helium system and the semiconductor systems, it is found in the sliding state that the noise spectrum is broadband, and large at low frequencies (possibly following a llf”law [23]). Since the frequency range of this noise lies well below the characteristic phonon frequencies, it is unclear precisely what the source of this noise is and what it might say about the motion of the electrons in the sliding state. One possibility is that the system undergoes a collective creeping motion, which may involve a distribution of waiting times for different patches of the crystal to move. Such possibilities have been discussed in the context of charge density wave [86] depinning, and it has been pointed out that such motion may be described as a self-organized critical phenomenon [87]. There is some indication that this sort of creep motion is relevant to the sliding state in molecular dynamics simulations [26], although it is difficult to glean information about the noise spectrum at very low frequencies from finite time simulations. For the heterostructure systems in a strong magnetic field, where the electrons must choose between a liquid (fractional quantum Hall) state and some other insulating state, the question of the nature of the phase transition between these two states is still unresolved. Are there quantum fluctuations in the crystal state that hint at the liquid states that are nearby in energy, and if so, how do they affect the crystal properties and the transitions between states? The study of correlated interstitials above suggests that such fluctuations are indeed important, particularly for filling fractions in the vicinity of v = l/m for large m, but it remains unclear how they should be incorporated in the crystalline ground state. Nevertheless, photoluminescenceexperiments suggest that something interesting does occur in the insulating ground state near v = 1/7, 1/9,. .. . For heterostructure systems, the most important development one could hope for would be an experiment that leaves a “smoking gun”: data that clearly distinguish between a disordered insulating state and a crystal state with some large correlation length. The observation of a Hofstadter spectrum would certainly give such dramatic evidence, and there is some possibility that itinerant hole photoluminescencecould observe this. However, current experiments [SO]

REFERENCES

105

of this sort are still limited by a lack of knowledge of the initial state of the hole, and further refinements of samples will be necessary before such effects become clearly accessible. On the theoretical side, one must also account for the effects of lattice distortions on the initial states of the hole, which is likely to make only the largest gaps in the Hofstadter spectrum accessible [SS] (at least for low filling factors), thus blowing away some of the smoke in such experiments before it is observed. Nevertheless, the observation of even the broadest features of the Hoftstadter spectrum could yield valuable information about the degree of order in these systems. ACKNOWLEDGMENTS The author has benefited greatly from numerous conversations and collaborations related to this subject. Luis Brey, Min-Chul Cha, Renk Cbte, Sankar Das Sarma, E. Kolomeisky, Dongzi Liu, Allan MacDonald, J. P. Straley, D. C. Tsui, and Lian Zheng have had particular influence in the development of this perspective. Some of the work described was supported by the NSF through Grant Nos. DMR-92-02255and DMR95-038 14, as well as by the Sloan Foundation, the Research Corporation, and NATO. REFERENCES 1. 2. 3. 4. 5.

6. 7. 8. 9.

C. Kittel, Quantum Theory ofsolids, Wiley, New York, 1963. M. Gell-Man and K. Brueckner, Phys. Rev. 106,364 (1957). E. P. Wigner, Phys. Rev. 46, 1002 (1934). W. J. Carr, Phys. Rev. 122, 1437 (1961). In two dimensions, the critical separation has been estimated using quantum Monte Carlo methods to be r, = 37, where r, = (1/fi)(rne2/h2) for an areal density of n, in two dimensions. See B. Tanatar and D. Ceperly, Phys. Rev. B 39,5005 (1989). C. C. Grimes and G. Adams, Phys. Rev. Lett. 42,795 (1979). L. Bonsall and A. A. Maradudin, Phys. Rev. B 15, 1959 (1977). D. S. Fisher, B. I. Halperin, and P. M. Platzman, Phys. Rev. Lett. 42,798 (1979). R. Dingle, H. L. Stormer, A. C. Gossard, and W. Wiegmann, Appl. Phys. Lett. 33,655

(1978). 10. J. Singh, Physics of Semiconductors and Their Heterostructures, McGraw-Hill, New York, 1993. 11. R. E. Prange and S. M. Girvin, eds., The Quantum Hall EfSect, Springer-Verlag, New York, 1987. 12. A possibleobjection to this argument is that the states &are not precisely orthogonal,

so that when one antisymmetrizes the ground-state wavefunction (to impose Fermistatistics on the electrons), there will be nonvanishing corrections to the energy. However, in the limit of large enough magnetic fields, one has l,,/ao 1. Then A = 1only when uint= 0. We shall now argue that as the v = 2 case, the presence of a random edge potential can drive the system to the decoupled fixed point with uint = 0. Thus the conductance is quantized even when channels move in opposite directions. Consider again the decoupled point, uinl = 0. Since p 2 = - 2, the neutral sector (72)and (74) is identical to the neutral sector (64) for v = 2 studied in the preceding section (up to a sign that determines the direction of propagation). Thus the exact solution obtained there by refermionizing the problem may be applied. Moreover, we may use exactly the same arguments to show that uint is irrelevant. We thus establish that the decoupled fixed line is stable. The fixed point is characterized by an exact SU(2) x U(1) symmetry. Having established the stability of the decoupled fixed line, we must also consider the possibility of other stable fixed points that could describe different phases. Consider treating the randomness (74) perturbatively. As in (59), we may analyze the relevancy of weak disorder by considering the leading-order RG for the variance, W, of 5, :

dW dl

-= (3 - 2A)W

(75)

Here A is the scaling dimension of the operator exp(i4,) and may be computed explicitly using the action (71)-(73). When uint= 0, then A = 1, but A > 1 is nonuniversal when uin1# 0. Indeed, A is the same quantity that entered into the conductance above. As seen from (75), when uint is tuned so that A exceeds 312, there will be an edge phase transition to a phase in which (weak) disorder is irrelevant. For filling v = 2/3, this transition was analyzed in Ref. [23]. There it was shown that the transition is of Kosterlitz-Thouless type, with the RG flows shown in Fig. 4.6.

134

EDGE-STATETRANSPORT

0

1

312

A

Figure 4.6. Renormalization group flow diagram for a v = 2/3 random edge as a function of disorder strength Wand the scaling dimension A of the tunneling operator. For A c 3/2 all flows end up at the exactly soluble fixed line A = 1. For A > 3/2 there is a KosterlitzThouless-likeseparatrix separating the disorder-dominated phase from a phase in which disorder is irrelevant.

We thus conclude that there are two possible phases for the random v = 2/3 edge. For sufficientlylarge uinl there is a phase in which disorder is irrelevant. At zero temperature, this phase is characterized by a nonuniversal conductance and nonuniversal tunneling exponents as shown in Section 4.4. The more generic phase, favored by electron interactions, is characterized by charge-spin separation, an exact U(1) x SU(2) symmetry,and a quantized conductance. Unlike the case v = 2, however, the charge ad neutral (or spin) modes propagate in opposite directions. 2. v = 215. The v = 215 edge is described by p1 = 3 and p2 = 2. In this case, sincep, > 0, both modes propagate in the same direction, as seen from (72).Thus, as shown in Section 4.2, even in the absence of edge impurity scattering, the conductance is quantized: G = (2/5)ez/h.Nonetheless,when impurity scattering is present, the edge restructures and exhibits spin-charge separation. To see this, simply repeat the argument for v = 2/3, which shows that the charge and neutral sectors decouple at low energies, uint-+0.The fixed point with U(1) x SU(2) symmetry is stable. Both the charge and neutral modes move in the same direction, generally with different velocities. Since the scaling dimension of the tunneling operator in (74) is A = 1 for any interaction strength, the disorder free edge is always perturbatively unstable to impurity scattering, in contrast to the v = 213 case. 3. v = 4/5. The Hall fluid with v = 415 is described by p1 = 1 and p , = - 4, and the two edge modes move in opposite directions. In this case, since 1 p , I # 2, the neutral sector is no longer identical to the neutral sector of v = 2, so the exact solution employed there can no longer be used. Weak impurity scattering can be analyzed perturbatively, however, by computing the scaling dimension of the tunneling operators (74) in the clean theory. When uin, = 0 it can be shown that A = 2. Including uint, we find that A > 2. Thus A > 312, and from (75) weak

RANDOMNESS AND HIERARCHICAL EDGE STATES

135

impurity scattering is always irrelevant. Thus the only low-energy fixed point is the clean edge, (71)-(73). Since the channels move in opposite directions, the zero-temperature conductance is thus predicted to be nonuniversal. At finite temperatures, however, as we see in Section 4.3.3, quantization is restored. SU(n) Generalizations. We have shown that the edges of disordered v = 2/3 and 2/5 fluids are described by a stable T = 0 fixed point, with an exact U(1) x SU(2) symmetry. This will also be the case for any state with Ip21 = 2. These filling factors can be written as v = 2/(2p k l), with p an even integer. Within Jain’s construction [40], these are precisely the states that can be obtained by attaching p flux quanta to each electron and filling two Landau levels. This suggests that the foregoing results can be generalized to the quantum Hall states derivable from n full Landau levels, which occur at filling factors v = n/(np + 1). This case was studied in detail in Ref. [25], where it was shown that a random potential drives the edge to a stable fixed point characterized by an exact U(1) x SU(n) symmetry. Again, the stable fixed point is characterized by spincharge separation, but now with a more general SU(n)spin. In this case, there is a single charged mode, and n - 1 neutral modes. The neutral modes are related by the exact SU(n)symmetry, and it follows that they all move at the same velocity. In general, the charge mode moves at a different velocity,and when p < 0 it moves in the opposite direction of the n - 1 neutral modes. Only the charge mode contributes to the conductance, which is appropriately quantized, G = ve2/h.

4.3.3. Finite-Temperature Effects The exact solution of the random edge that describes a stable zero-temperature fixed point can also be used to extract physical properties of the edge at low but nonzero temperatures.These properties will be determined by the structure of the fixed point itself and the leading irrelevant operators, such as qntin (3.18). At low but nonzero temperatures these operators have not had “time” to fully renormalize to zero and can then have an important effect on physical observables. Although one can show that the irrelevant operators do not modify the quantized Hall conductance itself, they do dramatically effect the propagation of the neutral modes at finite temperature. To see why, we first note that the existenceof the propagating neutral modes is tied intimately to the exact SU(n) symmetry in the neutral sector at the fixed point. But at finite temperatures, this symmetry is no longer exact, due to the presence of irrelevant operators, so that the neutral modes should no longer be strictly conserved. Thus one expects that at finite temperatures the neutral modes should decay away at a nonvanishing rate, 1/7,. Equivalently, one expects a finite decay length, or inelastic scattering length, I , = ~ ~ 7On , . scales Lmuch larger than I,, the neutral modes should not propagate. Since the fixed point is approached as T-0, however, the decay length should diverge in this limit. By analyzing the leading irrelevant operators [such as (73)] that control the flows into the zero-temperature fixed point, it was shown in Ref. [25] that the

136

EDGE-STATE TRANSPORT

decay rate vanishes algebraically with temperature: 1

-K

=,

TZ

In contrast, the charge mode cannot decay, even at finite temperature, since electriccharge is always conserved. However,due to irrelevant operators, such as (73), which couple the charge and neutral sectors, the charge mode can scatter off the neutral modes. This leads to a charge mode that propagates with a dispersion o = opq iDq2,with a “diffusion”constant D that is temperature independent at low temperatures. This implies a diffusive spreading of a charge pulse as it propagates along an edge. For v = 4/5 we arrived at the striking conclusion that impurity scattering at the edge was ineffective at equilibrating the two edge modes, leading to a nonquantized conductance at T = 0. But a quantized plateau is seen at v = 4/5, albeit with less vigor than might have been expected given estimates for the bulk energy gap. This apparent conflict is resolved when one considers finite temperature effects at the v = 4/5 edge. Although disorder, W, is formally irrelevant, since A > 2 in ( 7 9 , at finite temperatures W has not had “time” to scale all the way to zero. In fact, by cutting off the RG flows with temperature, it was shown in Ref. [25] that there is a characteristic inelastic scattering length, which diverges at low temperatures as

+

On scales longer than this length, equilibration takes place and charge does not propagate upstream. Provided that this length is shorter than the distance between sample probes, a quantized conductance is recovered, as discussed in Ref. [25]. However, this does raise the interesting possibility of observing deviations from quantization in short Hall bars at v = 4/5 and low temperatures. A very clean sample would be favorable for observing such deviations.

4.4. TUNNELING AS A PROBE OF EDGE-STATE STRUCTURE The rich physics “hidden” at the edge of FQHE fluids is not easily revealed via bulk transport measurements. An ideal way to access this edge physics is via laterally confined samples in which two edges of a given sample are brought into close proximity, allowingfor interedgetunneling. The simplest situation is a point contact in an otherwise bulk quantum Hall fluid, as depicted in Fig. 4.2. The point contact can be controlled electrostatically by a gate voltage. When the constriction is open, the two-terminal conductance is given by its quantized value. However, as the channel is pinched off and the top and bottom edges are brought into close proximity, charge will begin to backscatter between the right and left moving edge channels. Such backscattering reduces the two-terminal

TUNNELING AS A PROBE OF EDGE-STATE STRUCTURE

137

1.0

0.8

;u'

u5l

0.6

Y

C 3 0.4

0.2 0.0

-

-0

Figure 4.7. Two-terminal conductance as a function of gate voltage of a GaAs quantum Hall point contact taken at 42mK. The two curves are taken at magnetic fields that correspond to v = 1 and v = 1/3 plateaus. (From Ref. [29].)

conductance. Ultimately, as the gate voltage is increased, the Hall bar will be pinched off completely, and the two-terminal conductance will be zero. In Fig. 4.7, the two-terminal conductance as a function of gate voltage is shown for a GaAs quantum Hall point contact [29] taken at 42mK. The two curves are taken at magnetic fields that correspond to v = 1 and v = 1/3 plateaus. To the right the point contact is open, and the conductance is quantized, whereas to the left the point contact is pinched off. Between there are many resonant structures resulting from random impurities in the vicinity of the point contact. Note the qualitativedifference between the behavior for v = 1 and v = 1/3. For v = 1/3,the valleys between the resonances are deeper and the resonances are sharper. How are we to understand this qualitative difference? In this section we present a theory of tunneling and resonant tunneling at a point contact that answers this question. We begin in Section 4.4.1 with a discussion of tunneling at a point contact. We show that in contrast to the IQHE, the conductance of a FQHE point contact vanishes in the limit of zero temperature. We study resonant tunneling in Section 4.4.2, where we show that in the fractional quantum Hall effect resonances have a temperature-dependent width and a universal, non-Lorentzian lineshape at low temperatures. Finally, in Section 4.4.3, we discuss low-frequency shot noise at a quantum Hall point contact and suggest a method for the direct observation of the fractional charge of the Laughlin quasiparticle. We confine our attention initially to the Laughlin states, v = l/m, for which the edge states have a single branch. A discussion of tunneling in hierarchical quantum Hall states is deferred to Section 4.4.3.

138

EDGE-STATETRANSPORT

4.4.1. Tunneling at a Point Contact

A point contact in an IQHE fluid at v = 1, is isomorphic to a barrier in a onedimensional interacting electron gas. As discussed in Section 4.2, the right- and left-moving edge states, which feed the point contact, are equivalent to the right and left Fermi points of a one-dimensionalnoninteracting electron gas. According to Landauer-Buttiker transport theory [111, the two-terminal conductance through the point contact is proportional to the transmission probability, 1 t 12, for an incident wave-the edge state-to propagate through the point contact: e2 G=-lt12

h

For an IQHE state at v = n, the transmission will involve the transmission probabilities of all n of the edge channels. For a FQHE fluid at v = l/m point contact is isomorphic to a barrier in a onedimensionalinteracting electron gas-a Luttinger liquid. How is the relation (78) for the point-contact conductance modified in this case? To answer this question we use the chiral boson description of FQHE edges described in Section 4.2. The effects of the point contact can be analyzed perturbatively in the two limits depicted schematically in Fig. 4.8: (1) a pinched-off channel with weak tunneling, and (2) an open channel with weak backscattering. These limits are discussed next, after which we piece these two limits together into a unified description. Weak Tunneling Limit. Consider a point contact that is pinched off almost completely. As shown in Fig. 4.8~1,this may be described by quantum Hall fluids

Figure 4.8. Quantum Hall point contact in the (a)weak tunneling limit and (b)weak backscattering limit. The shaded regions represent the quantum Hall fluid, with edge states depicted as lines with arrows. The dashed line represents a weak tunneling matrix element connecting the two edges.


139

on the left- and right-hand sides, which are coupled by a weak perturbation that tunnels electrons between them. For v = l/m the low-energy physics will be described by an edge-state model of the form

s = s; + s::+ s,,,

(79)

where the left and right edges are described by 4n

+ a&,

dx, dr aX4,(iar

with a = L, R. The tunneling between these two edge states at the point contact can be expressed in terms of the edge creation and annihilation operators and has the form

s

+

S,,, = dr teim(4L-4R)C.C.

where 4, is evaluated at the point contact, x, = 0. Here t is the amplitude for the tunneling process [not to be confused with real time, which appears in (85) below]. The two-terminal conductance through the point contact can now be computed perturbatively for small tunneling amplitude t. In the presence of a voltage V across the junction, the tunneling rate to leading order can be obtained from Fermi's golden rule:

Here HI,, is the tunneling Hamiltonian corresponding to (81). The sum on n is over many-body states in which an electron has been transferred across the junction in the s, = f 1 direction. It is straightforward to reexpress this as 2nh dECGL(E)G;(E- eV) - G,'(E

- eV)G,'(E)]

(83)

where G,' and G: are (local)tunneling-in and tunneling-out densities of states for the edge modes, related by G c ( E )= G'( - E). These can be expressed as G:(E) = 2n

=s

n

1 (nleim@EIO) I26(E, - E ,

dt eiEt (eim4&)e

where

- imnC(0)

)

-E )

(84) (85)

+,, is evaluated at x = 0. The tunneling density of states is related to the

140


imaginary-time Green’s function

via analytic continuation, G’(t) = G(T-,it). Since the Euclidean action (80) is quadratic, G(z) may readily be computed, giving G(z) =

(F) + m

TI

zc

(87)

where z, is a short-time cutoff. Upon analytic continuation and Fourier transformation, the tunneling density of states is thereby obtained:

Form = 1, corresponding to the IQHE at v = 1, the tunnelingdensity of states is a constant at zero energy (the Fermi energy).From (83) this gives an ohmic I-V characteristic, with a tunneling conductance, Z/V, proportional to t2.This is the expected result, consistent with Landauer transport theory (78). However, for m > 1 the tunneling density of states uanishes at zero energy, giving rise to a nonohmic Z- V characteristic [17,28]:

The linear conductance is strictly zero! At finite temperatures the density of states is sampled at E x kT, and a nonzero (linear) conductance is expected. Generalizing Fermi’s golden rule to T # 0 gives the expected result for the conductance [17,28]: G oc t2TZrn-’

(90)

For v = 1/3 the predicted conductance vanishes with a large power of temperature, G cc T 4 . In striking contrast to the IQHE, the FQHE point-contact conductance, vanishes identically at zero temperature. What is the physical origin of this difference?At the edge of an IQHE fluid the electrons behave at low energies as if they were not interacting-the edge state is a Fermi liquid. Thus an electron can be added or removed from the edge without appreciably disturbing the other electrons. In contrast, the electrons at the edge of a Laughlin FQHE fluid are in a highly correlated state. Indeed, after removal of an electron from the edge, the remaining electrons are not left in the ground state of the edge with one less electron. Rather, the resulting state contains a “shakeup” spectrum of many low-energy edge excitations and is orthogonal to the ground state. It thus follows that the many-body matrix element for tunneling in (82) vanishes. This is a nice example of a general phenomenon known as the orthogonality catastrophe [41], which arises in such contexts as the Kondo problem and the x-ray edge problem.


0.20

I

141

1

I

V’c=-O.853 & -0.899 V

T (mK) (0)

% 3 040%. Y

0

0.00 0.0

0.2

0.4

0.6

T4 (K4)

0.8

1.0

(b) Figure 4.9. Conductanceofa quantum Hall point contact as a functionof temperature for (a) v = 1 and (b) v = 1/3. (From Ref. [29].)

This orthogonality catastrophe is directly accessible to measurement. Figure 4.9 shows data for the conductance as a function of temperature through a point contact in an IQHE fluid at v = 1 and a FQHE fluid at v = 1/3. The difference in behavior is striking. For v = 1 the conductance approaches a constant at low temperatures, whereas for v = 1/3 the conductance continues to decrease upon cooling. Moreover, the low-temperature behavior for v = I/3 is consistent with the T4 dependence predicted in (90). In our view, these data provide the first compelling experimental evidence for the Luttinger liquid, a phase discussed theoretically over 30 years earlier. It is instructive to recast the result (90)in the language of the renormalization group (RG) [28]. Specifically,the vanishing conductance in the FQHE indicates

142


that the tunneling perturbation, t, is irrelevant. As shown in the appendix, the lowest-order RG flow equation can be obtained from the scaling dimension, A, of From the power-law behavior of te Green's the tunneling operator eimn(+L-4R). function in (87), we may deduce that the scaling dimension of ei"4LR is m/2. It then follows that A = m. The RG flow equation is then simply

dt dl

- = (1 - m)t

For FQHE states with m > 1, t is irrelevant as expected. The perturbative results (89) and (90) can be ontained by integrating this RG flow equationa until the cutoff is of order kT (or eV), giving tcff tm-' and G tZff.

-

N

Weak Backscattering Limit. Having established that the conductance of a FQHE point contact vanishes at T = 0 for weak tunneling, we now turn to the opposite limit, in which the point contact is almost completely open. Consider a bulk quantum Hall fluid in which the top and bottom edges are weakly coupled together, as depicted in Fig. 4.8b. As before, the low-energyphysics is at the edges and can be described by the action

s = so,+ s: + S,", Here SOT and S: describe the top and bottom edge modes, respectively, and are given by the chiral boson action (80). In contrast to the weak tunneling limit, the charge that tunnels between the two edges is now tunneling through the quantum Hall fluid. It is therefore possible that a single Laughlin quasiparticle, with fractional charge e/m, could tunnel between the edges. This process can be described by a term of the form

where u is the tunneling amplitude. In addition, higher-order processes involving tunneling of multiple quasiparticles (or electrons) are also possible. However, as shown below, such processes are less "relevant" at low energiesand temperatures. Consider now applying a voltage V between the source and drain electrodes. In the absence of any coupling, the top and bottom edges would then be in equilibrium at chemical potentials differing by the voltage I/. This results in the flow of a net edge current, I = (l/rn)(e'/h)V. Quasiparticle tunneling between the top and bottom edges backscatters charge and will tend to reduce this current. The reduction may be computed perturbatively in u, also using the golden rule. In fact, the backscattering current will be given by the golden rule expression (82), with two differences. First, the charge e in (82) must be replaced by the quasiparticle charge, e* = e/m. Second, the electron tunneling operator (81) must be replaced by the quasiparticle tunneling term in (93). This difference is crucial,


143

repiacing the electron tunneling DOS (88) with the density of states for the addition of a quasiparticle. This follows from the quasiparticle Green's function

which has the same form as the electron Green's function (87) with rn -+ l/m. The backscattering current can thus be obtained from (89) by replacing m with l/m, giving at zero temperature c17.281

Similarly, at temperature T, the backscattering contribution to the (linear) conductance is given by

Once again, for the IQHE with rn = 1, a temperature-independent correction to the two-terminal conductance is obtained, as expected from Landauer transport theory. However, for the FQHE the perturbation theory (97) is divergent at low temperatures. The quasiparticle tunneling rate grows at low energies, in contrast to electron tunneling. Rather than an orthogonality catastrophe, quasiparticle tunneling causes quite the opposite-an overlap catastrophe. The divergent perturbation theory indicates that the quasiparticle tunneling operator is a relevant perturbation. This may be seen directly by noting from (95) that the scaling dimension of the quasiparticle creation operator ei4 is equal to 1/2rn. The leading-order RG tlow equation for the quasiparticle tunneling amplitude is then simply

dv = (1 dl

);

-

v

For the FQHE, u grows upon scaling to lower energies, flowing out of the perturbative regime where (98) is valid. The behavior in this limit is discussed in the next section. It is instructive to consider, in addition, backscattering processes involving multiple quasiparticles. The operator that tunnels n-quasiparticles is of the form u,e'n(or-4B). It is straightforward to show that the leading-order RG flow equation for u, is

dl

(99)

144


Notice that for v = 1/3 (rn = 3), the single-quasiparticle backscattering process is the only relevant perturbation. In contrast, for m = 5 7 , . . .more than one operator is relevant. In all cases, however, the single-quasiparticleterm is the most relevant. Crossover Between the Two Limits. The preceding results can now be pieced together to form a global picture of the behavior of a point contact in the FQHE. The perturbative results above describe the stability of two renormalization group fixed points. The “perfectly insulating” fixed point, with zero electron tunneling t =0, is stable, whereas the “perfectly conducting” fixed point, with zero quasiparticle tunneling v = 0, is unstable. Provided that these are the only two fixed points, it follows that the RG flows out of the conducting fixed point eventually make their way to the insulating fixed point. This is a very striking conclusion, since it implies that an arbitrarily weak quasiparticle backscattering amplitude v will cause the conductance to vanish completely at zero temperature. Of course, for u very small, very low temperatures would be necessary to see this. In this scenario, the conductance as a function of temperature will behave as shown in Fig. 4.10. At high temperatures, the system does not have “time”to flow out of the perturbative regime, so the conductance is given by G z ( l/m)(e2/h) v2T2/mn-2 . A s the temperature is lowered below a scale T* a ’), perturbation theory breaks down. Eventually, the system crosses over into a low-temperature regime in which the conductance vanishes as T2’”-’. The validity of this scenario rests on the assumption that no other fixed points intervene. This assumption has been verified both by quantum Monte Carlo simulations [27], and more recently by exact nonperturbative methods based on the thermodynamic Bethe ansatz [31]. As seen from (99),for v = 1/3 the single-quasiparticlebackscattering operator, with amplitude v = vlr is the only relevant perturbation about the conducting fixed point. All higher-order processes are irrelevant and hence less important at

T*

Figure 4.10. Schematic plot of the crossover from the weak backscattering limit to the weak tunneling limit as the temperature is lowered. At high temperatures,weak backscattering leads to a small correction to the quantized conductance. As the temperature is lowered below T* cc urn/(”’-’), the system crosses over to the insulating limit.


145

low temperatures. Indeed, for small v and T, the conductance will depend on these parameters only in the combination u / T ' / ~In. this limit the conductance can be expressed in terms of a universal crossover scaling function [27,30,43]

where c is a nonuniversal dimensionful constant. The limiting behavior of the scaling function, G X ) , may be deduced from the perturbative limits. For smallargument X, the perturbation theory result (97) implies that

G ( X )= 1 - x 2

(101)

For large argument, corresponding to the limit T-0, the scaling function must match onto the low-temperature regime (90), which gives a T4 dependence for nu = 1/3. This implies that for X -,a,

G(x) a x - ~

(102)

The exact scaling function, computed by Fendley et al. [31], indeed reduces to (100) and (101) for small and large arguments, respectively. The universal crossover scaling function G is of particular interest because it determines the experimentally accessible lineshape for resonant tunneling, as we describe next.

4.4.2. Resonant Tunneling We now consider the phenomenon of resonant tunneling through a point contact in the FQHE. Resonancesin the conductance are expected when the energy of the incident edge mode coincides with a localized state in the vicinity of the point contact. As a point of reference, we first review resonant tunneling theory for a noninteracting electron gas, which should be applicable to a point contact in the IQHE. As the chemical potential p of the incident edge mode sweeps through the energy of the localized state, E ~ the , conductance will exhibit a peak described by

Here Tr.and r R are tunneling rates from the resonant (localized)state to the left and right leads and r = (r,+ rR)/2.The Fermi function is denoted f (E). At high temperatures, T > r, the resonance has an amplitude T / T and a width T. At low temperatures the lineshape is Lorentzian, with a temperature-independent width. Moreover, when the left and right barriers are identical, the on-resonance transmission at zero temperature is perfect, G = c2/h. How is this modified for tunneling through a FQHE point contact? Since arbitrarily weak quasiparticle backscattering causes the zero-temperature con-

146


ductance through the point contact to vanish, one might expect that resonances simply are not present at T = 0. As we now show, this is not the case. Rather, perfect resonances are possible, but in striking contrast to (103)for noninteracting electrons, they become infinitely sharp in the zero-temperature limit. As before, it is useful to consider perturbatively two limits, the weak tunneling limit and then the opposite limit of weak backscattering. Weak Tunneling Limit. Consider tunneling through a localized state separating two FQHE fluids. Focusing once again on the FQHE edge modes, we consider the model

s = s; + s:: + s,,, + s,,,

(104)

where S; and S;, given in (80), describe the edge modes in the two FQHE fluids, and r S,,, = dz E,dtd

J

describes the localized state with energy c0. The edge modes are coupled to the localized state via a tunneling term,

s

+

S,,, = dz t(ei4L+ eibR)d h.c.

with the tunneling amplitude, t, taken to be the same for left and right edge modes. Once again, the boson fields c$L,R are evaluated at x = 0. Consider first computing the rate, r, for an electron to tunnel from the localized state into the edge modes, perturbatively in t. From Fermi's golden rule, this will depend on the density of states for tunneling into the edge, which is given in (88). We thus find that

At finite temperatures and p x c0, we thus have

Once again at zero temperature there is an orthogonalitycatastrophe that prevents tunneling to FQHE edge states, rn > 1. However, it is only half as severe as that for tunneling between two edge modes (go), since only a single mode is being disturbed. For FQHE states (m2 3), (108) implies that r ho,.In fact, it is possible to prove that interactions (of either sign) always increase the frequency of this mode, as we note below. By expanding the expressionfor f,"' at small k it can be shown that f,"' k4, whereas N

c

N k2 f,"- = - - A 2 4

q 2 V?hLR(q)

+ O(k4)

The n = 0 difference mode is dipole active. It is interesting that for the difference mode, the interaction contributions to the dipole (aIkI2) portions of the n = 0 and n = 1 oscillator strengths are identical. By definition,f,"-is positive definite, so that A-(k = 0) - hw, must also be positive. In the single-mode approximation, the n = 1 (cyclotron) difference mode is always shifted to higher energy by

178


electron-electron interactions. The situation is similar to that for the effect of disorder on the vibration modes of the Wigner crystal at strong magnetic fields, where pinning of the crystal shifts both intra-Landau-level and inter-landaulevel modes upward by the same amount [Sl]. To determine whether or not the n = 0 mode is gapped, it is necessary to determine how *:s behaves at small k. From general properties of wavefunctions in the lowest Landau level, it is possible [65] to conclude that [82]':s k4, whereas the behavior of the n = 0 difference-mode structure factor depends on the difference between interlayer and intralayer correlation functions.It is possible to prove [65] that s,"- k2 at small k, provided that intralayer and interlayer correlation functions separately vanish at large spatial separations. In general it is easy to show from the plasma arguments outlined in Section 5.4 that this is a property of the Ym,m,n wavefunctions.However, an exceptinon occurs for n = m. In this case there is no distinction between the effective plasma interactions of particles in the same layer and particles in different layers. The weighting of particle configurations depends only on the total charge densities of the two layers, and only correlations in the total charge densities of the two layers go to ,,' it is possible [65] zero at large distances. For the broken-symmetry state Y to show explicitly that -:s = exp (- Ikl 12/2), which goes to a constant for k +0. The SMA collectivemode then goes like k2 for this special case ifthe ground state is approximated by Y,. (The italics above are pregnant, as we will see later.) For the {m, m, rn} states, the n = 0 difference mode (intra-Landau level) is gapless at long wavelengths. It is often the case that gapless collective modes can be identified as Goldstone modes associated with a broken symmetry in the ground state, and that is indeed the case here, as we shall see. We emphasize that the results discussed above follow from general sum rules and are independent of the approximate many-body wavefunction (Ym,m,,n) in terms of which we have framed our discussion so far. Since f,"' k4 and f,"- k2 independent of the long-wavelength behavior of S:*, it follows quite generally that the n = 0 sum mode has a gap and that the n = 0 difference mode has a gap except in the case where long-range order is present, which results in correlation functions that do not vanish at large distances. In Fig. 5.1 we show results obtained for the collective-mode energies of a double-layer system with a total Landau level filling factor vT = 1/2 and a layer separation d = 1.51, close to the effective layer separation value for which novel double-layer fractional Hall effects have recently been observed [83]. Numerical calculations [84,85] have established that the ground state at this value of d/l is accurately approximated by the {m,m,n> = {3,3, l } Halperin [21] wavefunction, and we have used the correlation functions [86,87] of that wavefunction to evaluate the oscillator strengths and structure factors. Fork -rO the n = 1 sum mode (the Kohn mode) is unshifted by interactions while the n = 1 difference mode is shifted to higher energies as discussed above. The shift, which is a direct measre of interlayer correlations, should [88] be observable in cyclotron resonance experiments in double-layer systems. Note also that both sum and difference n = 0 modes have a finite gap, as expected from the discussion above.

-

-

-

-

COLLECTIVE MODES IN DOUBLE-LAYER QUANTUM HALL SYSTEMS

179

0.20

0.15

0.10

0.05

'

0.00 0.0

I

8

1.o

4

2.0

3.0

Figure 5.1. Collective-modedispersionfor a double-layersystem at vT = 112 and d/l = 1.5. The energies of the inter-Landau-levelmodes are measured from ha,. The ground state is approximated by the (3,3,1) Halperin state. The plotting symbols refer to the following modes: triangles (n = 1 sum mode),circle (n = 1 difference mode),square (n = 0 sum mode), diamond (n = 0 difference mode).

We are now able to compare our results for the collective modes with the Chern-Simons Landau-Ginzburg (CSLG) theory of the double-layer system C42-441. In the CSLG random-phase approximation, the sum and difference density response functions are given by

where the collective-mode frequencies are given by a+= o,= eB/m*c, 0-= w,(m-n)/(m+n), and m* is the effective band mass of the electrons. For the single-layer case the CSLG random-phase-approximation predictions are correct for the dipole active mode and we might have expected the same to be true in double-layer systems. The sum mode for double-layer systems can clearly be identified with the Kohn mode [64]. However, there are difficulties in identifying the density difference modes in this theory. From Eq. (41) we see that the n = 0 difference mode is dipole active and should have a dipole oscillator strength

180


proportional to VLRhLR. One might therefore be tempted to identify w - with the n = O difference mode. Then, for the case of the {m,m,m}-state random-phase approximation, w - = 0, and it would then be tempting to identify this mode with the gapless n = 0 Goldstone mode. Unfortunately, the single density-difference mode calculated within the double-layer CSLG theory saturates the full dipole oscillator strength Nq2/m*A.This is not acceptable since an excitation within the lowest Landau level cannot contain explicit dependence on the band mass m*. The second possibility is to interpret the difference mode obtained in Eq. (42) as the n = 1 difference mode. In this case one is faced with the difficulty that the mode energy is shifted downward from the cyclotron energy by an amount proportional to w,,whereas the SMA calculations show that it should be shifted upward by an amount proportional to the interlayer Coulomb energy. It has been suggested [89] that these difficulties can be resolved by including the mixing of the vortex excitations with the Gaussian fluctuations in the CSLG theory. Similar difficulties arise in the ferminon-Chern-Simons theory of the single-component v = 1/2 state and can be resolved in that case by taking sufficient care with the Landau parameters of the composite-fermion Fermi liquid [59].

5.6. BROKEN SYMMETRIES

Ferromagnetic states break spin rotation symmetry since they are defined by an order parameter (S), which gives the magnitude and orientation of the magnetization. For the SU(2) symmetric case with no Zeeman term, this orientation is arbitrary. As we will see below, the case of a double-layer system is described by a pseudospin with easy-plane anisotropy [U(l) or X Y symmetry]. Here the magnetization vector is forced to lie in the X Y plane in the ground state. The origin of ferromagnetismin all these systemsis the Coulomb interaction,just as it is for itinerant ferromagnetssuch as iron. Exchangeeffectsare particularly crucial in a 2DEG in a large magnetic field because the kinetic energy is quenched into highly degenerate Landau levels. It is advantageous to follows Hund’s rule and maximize the spin in order to make the spatial wavefunction fully antisymmetric, thereby lowering the Coulomb energy. Since the Landau level is degenerate, this spin alignment can in some cases be complete since it costs no kinetic energy as it does in iron [90]. Before considering the physical consequences of this broken symmetry, let us return to Table 5.1 to consider how the total spin quantum number S for a state can be determined. A portion of our discussion hare follows that of Ref. [22]. As already mentioned, states of the form {m, m, m} are fully ferromagnetically aligned and have total spin S = N/2. To derive the spin quantum numbers for the other states in Table 5.1, we write

BROKEN SYMMETRIES

181

and use the fact that N k.1

We see that

where e(i,k)@(...,Zir...,z k , . ..) = @(. . .,z k , . . .,Zi,...) is a label exchange operator. In Eq. ("I) S* , = X k S t and S,+ and S; are spin raising and lowering operators, and the prime on the sum in Eq. (44) indicates that k = 1 is excluded. In Eq. (45) S is the total-spin quantum number. To identify Halperin {m, m', n} states that are total spin eigenstates, we start with states where m = 1, n = 0, and m' has any odd value. Since j(Zi - Z j )is a Vandermonde determinant, these orbital wavefunctions have the up-spin Landau level full (see Table 5.1). They are eigenstates of S' with eigenvalue

ni,

( N ,-N,) - Nl(m' - 1) - N(m'- 1) = S,. 2 2 2(m' 1)

+

(46)

Moreover, S+Yl,m,,o = 0 since the up-spin Landau level is already full and there are no wavefunctions with a larger value of N,. It follows that @l,m,OIZ] satisfies Eq. (45) with S = (N, - N1)/2,and hence that

(This result can also be established by an explicit algebraic proof.) Since e(i, [j])Q[ZJ@[Z] = Q[Z]e(i, [j])@[Z] for any symmetric polynomial Q[Z], we have from Eqs. (47) and (45) that

+ Zp,m*+ z p , z p C z : ~ I= Srn*(Srn,+ ')@I + z p . m * + z p . z p C z : ~ I

+

(48)

In addition, it follows from Eq. (45) that S2,Y[Z:x] = N ( N / 2 1)/2@[Z:x] for any completely antisymmetric function @[Z], and in particular for generalized Laughlin states with m = m' = n. These states are merely the S' = 0 members of the set of the (N + 1) fully polarized Laughlin states, which are degenerate in the absence of the Zeeman term. We return now to the question of the physical consequences of the spontaneously broken symmetry of ferromagneticstates. We focus initially on the SU(2)

182


tt tttttt tttttttttttttttttttttt ttt Figure 5.2. (a) Simple spin-flip excitation which creates a widely separated particle hole pair; (b) skyrmion spin configuration (shown in cross section). The spins gradually and smoothly rotate from up at the perimeter to down at the origin in a circularly symmetric spin textural defect. For the case of Coulomb interactions,this object costs only 1/2 the energy of the simple spimple spin flip.

invariant case of “real” spins with zero g factor. Consider a state with v = 1and all spins up. Because the up Landau level is maximally filled, the Pauli principle forces us to flip a spin if we are to move an electron to create a pair of charged excitations. This is illustrated in Fig. 5.2a and shows that spin and charge are intimately connected in this case. Transport dissipation measures the thermal activation of charged excitations. In the absence of interactions, the energy cost of charged excitations is zero and there will be no v = 1 quantum Hall plateau because there is no gap. In the presence of Coulomb interactions a flipped spin particle-hole pair causes a loss of exchange energy of [46]

where E is the bulk dielectric constant. This occurs because the electron spatial wavefunction is no longer perfectly antisymmetric.This energy gap is quite large (- 150K at B = 10T) and is vastly larger than the bare Zeeman splitting. Hence Coulomb interactions and the associated ferromagnetism play a dramatic role in producing the charge gap at v = 1 (and because of the spontaneous magnetic ordering will continue to do so, even if g is strictly zero). The exchange energy cost of particle-hole pair excitations is so large that it is worth searching for a modified form of the excitation that is less costly. A prescient analysis of smooth spin textures by Sondhi et al. [24] yielded the exciting idea that “skyrmion” [60]-like spin textures (shown in cross section in Fig. 5.2b) can have relatively low energy and carry a fermion number proportional to their topological charge (Pontryagin index) [24,46]: d2reNvm(r)* [d,m(r) x d,m(r) J

(50)

where m(r) is the unit vector field representing the local spin orientation. The

BROKEN SYMMETRIES

183

fermion number AN is an integer multiple of v because it is the number of times the unit sphere is wrapped around by the order parameter. That is, it is the winding number of the spin texture [91]. For the Laughlin parent states v = l/m, elementary spin textures carry the same fractional charge as the quasiparticles discovered by Laughlin [S] for spinless electrons. As we discuss below, the fact that the charges are the same follows from very general considerations. Actually, the spin texture states we have defined must contain precisely the same number of particles as I$o)since the spin-rotation operator does not change the total electron number. However, the spin density may contain a number of well-separated textures with well-defined nonzero topological charge densities and hence well-localized charges; only the net charge in the spin-texture states defined above will be zero. The system clearly has states with locally nonzero net charge in the spin textures. A simple variational wave function for a skyrmion of size A centered on the origin and carrying p units of topological charge is given by

where Qmmm is defined in Eq. (5), (.) refers to the spinor for the jth particle, and the variational parameter II is a fixed length scale. This is a skyrmion because the 2-j component of the spin has a vortex centered on the origin and the &component is purely down at the origin (whereZ j = 0) and purely up at infinity(where lZjl >>A), as shown in Fig. 5.2. The parameter A is simply the size scale of the skyrmion [24,92]. Notice that in the limit A - 0 [where the continuum effectivefield theory is invalid (see Section 5.7), but this microscopic wavefunction is still sensible], we recover a fully spin-polarized filled Landau level with p Laughlin quasiholes at the origin. Hence the number of flipped spins associated with the presence of the skyrmion interpolates continuously from zero to infinity as A increases. To analyze the skyrmion wavefunction in Eq. (51),we use the Laughlin plasma analogy [S]. In this analogy the norm of $A, Tr(,,JD[zJ IY [z] 12, is viewed as the partition function of a Coulomb gas. To compute the density distribution, we simply need to take a trace over the spin (we specialize heie to the case p = 1 for simplicity)

z=

s

~ [ z ~ e ( 2 / m ) I ~ , , , l o ~ ~ z ,i -+z( m , /2)~~10g(iZk~2+~z)-(m/4)~~iZt~21

(52)

This partition function describes the usual logarithmically interacting charge m Coulomb gas with uniform background charge plus a spatially varying impurity background charge Apb(r), 1 A2 Apb(r)3 ---V2V(r) = 2n n(r2+ 22)2

(53)

184


For large enough scale size 1 >> 1, local neutrality of the plasma [69] implies that the excess electron number density is precisely ( l/m)Apb(r), so that Eq. (54) is in agreement with the standard result for the topological density [92], and the skyrmion carries electron number l/m (for p = 1) and p / m in general. These objects are roughly analogous to the Laughlin quasihole. Explicit wavefunctions for the corresponding quasielectron objects (“anti-skyrmions”)are more difficult to write down, just as they are for the Laughlin quasielectron, due to the analyticity constraint [S]. Sondhi et al. [24] have shown that for the case of pure Coulomb interactions (i.e., with no finite inversion-layer thickness corrections), the optimal skyrmion configuration costs precisely half the energy of the simple spin flip [93]. The reason for this is simply that the skyrmion keeps the orientation of spins close to that of their neighbors and so loses less exchange energy.This is discussed in more detail from a field-theoreticpoint of view in Section 5.7. Optical [94] and standard transport [95] experiments show that the charge excitation gap is indeed much larger than would be expected if interactions were neglected and has approximately the correct Coulomb scale, although there is not yet precise agreement between the observed gap and the best estimates, including the finite g factor [24,25]. The main source of error is probably neglect of finite thickness effects. Calculations including finite thickness corrections do not exist at present. Quantum fluctuation effects (i.e., corrections to HartreeFock) may also be important. It should be noted that the uniform ground state of the v = 1 ferromagnet does not have quantum fluctuations (Hartree-Fock is exact here). However, for finite g factor, the length scales associated with the skyrmion are small and quantum corrections could well become important. The idea that skyrmions are the lowest-energy excitations has received very strong and unequivocal support from numerical simulations, which show that quite remarkably, adding a single electron to a v = 1 system (with g = 0)suddenly changes it from a fully aligned ferromagnet (S = N/2) to a spin singlet (S = 0) due to the formation of a skyrmion [24,96]. (It should be noted that this occurs in the spherical geometry. Things are slightly more complicated on the torus [46].) The notion of charges being carried by skyrmion textures has received additional dramatic experimental confirmation in recent optically pumped NMR measurements by Barrett et al. [26,27] Their Knight shift measurements (see Fig. 5.3) indicate that the electron gas spin polarization has a maximum at filling factor v = 1 and falls off sharply on each side. The rate of fall-off indicates that each charge added or removed from the Landau level turns over about four spins. This is consistent with the charge being carried by skyrmions of finite size. The size of the skyrmion is determined by a competition between the Zeeman coupling,which wants to minimize the number of flipped spins, and the Coulomb self-energy,which wants to expand the skyrmion to spread out the excess charge density over the largest possible area. As mentioned above, Rezayi found numerically that a single charge destroys the spin polarization of a system entirely (if g = 0).Using an effective field-theoreticapproach, Sondhi et al. [24] have estimated the skyrmion size in the regime of small g factor. Microscopic

FIELD-THEORETIC APPROACH

I

0 ’

0.6

185

i i \

0.8

1.0

1.2

v

1.4

1.6

I

1.8

Figure 53. Knight shift measurements of the electron spin polarization of a ZDEG in the vicinity of filling factor v = 1. (After Ref. [26].)

Hartree-Fock calculations expected to be more accurate in the physically accessible Zeeman energy regime have been performed by Fertig et al. [25]. These estimates are roughly consistent with the experimental value of x 3.5 spins per unit of charge. It should be noted that for finite g factor, the energetic advantage of the skyrmion over the simple flipped spin is considerably reduced [25]. Wu and Sondhi have shown that in higher Landau levels, skyrmions have higher energy than other charged excitations 1973. For the SU(2)symmetriccase discussed in this section,the existence of a vector order parameter (S) is in some sense trivial because the magnetization commutes with the Hamiltonian. For the case of double-layer systems we will see that the pseudospin Hamiltonian has only U(1) symmetry and the fact that a nonzero expectation value of (S) appears is highly nontrivial. We will discuss this in more detail in Section 5.7. 5.7. FIELD-THEORETZC APPROACH

In this section we study the ferromagneticbroken-symmetry ground state and its excitations from the point of view of (quantum) Ginzburg-Landau effective field theory. We will begin with the SU(2)invariant case of real spin and then move on to the pseudospin analogy in double-layer systems. We give here an introductory qualitative discussion of the physics. Part of our discussion follows fairly closely the presentation in Ref. [46]. The technical details of the calculations can be found there and elsewhere [24,25,47].

186


The standard first step in this procedure is always to identify the slowly fluctuating order parameter field, which in this case we have reason to believe is the local magnetization. We believe that on long length scales the (coarsegrained) magnetization fluctuates very slowly, because we know that the zerowavevector component of the spin operator (i.e., the total spin) commutes with the Hamiltonian and so is a constant of the motion. We focus on slow tilts of the spin orientation, ignoring variations in the magnitude of the coarse-grained magnetization and so define the order in terms of a local unit vector field m(r). General symmetry arguments can now be used to deduce the form of the Lagrangian. We cannot have any terms that break spin rotational symmetry, and thus the leading term that is an allowed scalar is I

f

where ps is a phenomenologicalspin stiffnesscoefficient and the energy is relative to the ground-state energy. It expressesthe cost due to loss of Coulomb exchange energy when the spin orientation varies with position. For the SU(2) invariant v = 1 case, this stiffnessmay be computed exactly [46]. For Coulomb interactions (with no finite thickness corrections) this calculation yields PS=-

e2/d 1 6 6

In two dimensions the stiffness has units of energy and is approximately 4 K at a field of 10T. Numerical estimates for v = 1/3 yield a value that is about 25 times smaller [46]. As usual, there is a linear time derivative term in the Lagrangian that can be deduced from the fact that each spin processes under the influence of its local exchange field. Equivalently, we may note that when the orientation of a spin is moved around a closed loop, the quantum system picks up a Berry’s phase [98] proportional to the solid angle R enclosed by the path w of the tip of the spin on the unit sphere,as shown in Fig. 5.4.Noting that a charged particle moving on the surface of a unit sphere with a magnetic monopole at the origin also picks up a Berry’s phase proportional to the solid angle subtended by the path [98], we may express the Berry’s phase for a spin S as eiy

= eiSIS

or equivalently, eiy

= ,i

-e

s

i

fa

dm.A(m)

dl(dnt/at).A(m)

(57)

(58)

where A(m) is the vector potential of a unit monopole [24,46,91] at the center of

FIELD-THEORETICAPPROACH

187

Figure 5.4. Path o of a spin Sm on a unit sphere. When viewed from the origin of spin space, the path subtends a solid angle Q, so the path contributes a Berry's phase SR.

the sphere evaluated at point m. That is, V, x A = m. This phase is correctly reproduced in the quantum action by adding the following total derivative term to the Lagrangian for the spin

am L, = S--.A(m) at

(59)

Using the fact that the electronic density is v/2d2, the analog for the present problem for a large collection of electrons with S = 1/2 at filling factor v may be written simply

which yields the Lagrangian V

d2r[VmN(r)].[VmN(r)]

at

(61)

We shall see shortly that higher gradient terms can be unexpectedly significant, but this Lagrangian is adequate to recover the correct spin-wavecollective mode. Taking the spins to be aligned in the 2 direction and looking at small transverse oscillations at wavevector q, we obtain from this Lagrangian the following equation of motion:

dm 4np,q2 A=- 2 x m, dt

hv

188


This yields the dispersion relation hw=-q 471Ps

2

V

which agrees with the long-wavelength limit of exact results obtained by a variety of means [99,100]. At this point we have expanded the Lagrangian to lowest order in gradients and we have correctly found the neutral collective spin-wave modes. Their dispersion is quadratic in wavevector just as it is for the Heisenberg ferromagnet on a lattice. However, here we have an itinerant magnet and we have so far seen no sign of the charge degrees of freedom. It turns out that we have to go to higher-order gradient terms in the action to see charged objects. We have already seen in the discussion of Fig. 5.2 in Section 5.6 that for a filled Landau level, the Pauli principle forces there to be a connection between charge excitations and flipped spins. It turns out that the existence of a finite Hall conductivity in this itinerant magnet causes smooth spin textures to carry charge proportional to their topological density. One can derive this result from a Chern-Simons effective field theory [24,26], or from microscopic considerations involving the fact that the spin density and charge density operators do not commute when projected onto the lowest Landau level [46], or from macroscopic considerations connecting the Berry phase term to the Hall conductivity [46,101]. The latter is the least technical and the most instructive, so we shall pursue it here. Imagine that the order parameter of the ferromagneticsystem is distorted into a smooth texture, as illustrated in Fig. 5.5. As an electron travels around in real space along a path ar which is the boundary of the region r,the spin is assumed to follow the orientation of the local exchange field b(r) and hence traces out a path in spin space labeled w in the (schematic) illustration in Fig. 5.4. That is, given any sufficientlysmooth spin texture, we can write a Hartree-Fock-like Hamiltonian for the electrons which will reproduce this texture self-consistently:

H=

N

-

C b(r).Sj

j= 1

If we drag one electron around in real space along X a n d its spin follows the local

Figure 5.5. Smooth spin texture. An electron moving along the boundary of the region r in real space has its spin follow the path in spin space along the boundary of the region labeled w in Fig. 5.4.

FIELD-THEORETICAPPROACH

189

b(r) adiabatically (which we expect since the exchange energy is so large), the electron will acquire a Berry’s phase Q/2, where Q is the solid angle subtended by the region w shown in Fig. 5.4. In addition to the Berry’s phase from the spin, the electron will acquire a Bohm-Aharonov phase from the magnetic flux enclosed in the region r.At least in the adiabatic limit, the electron cannot distinguish the different sources of the two phases [loll. The electron would acquire the same total phase in the absence of the spin texture if, instead, an additional amount of flux

n

A@=-@o 47l

(where a0is the flux quantum) were added to the region r. We know, however, that adding flux to a region in a system with a finite Hall conductivity changes the total charge in that region [102]. To see this, let @(t) be the time-dependent flux inside r. Then the electric field along the perimeter obeys (from Faraday’s law)

$arE*dr =

1 dQ c dt

Because of the Hall conductivity (and the fact that axx=O), the field at the perimeter induces a current obeying

2.J x dr = axyE-dr

(67)

Integrating this expression along the boundary and using the continuity equation, we have that the total charge inside r obeys

dQ -= dt

+--a,d@ c dt

or

AQ AQ = - evQO

where we have used the fact that the Hall conductivity is quantized (and is negative for B = IBI2): bXY =

ye2 h

--

Thus v electrons flow into the region for each quantum of flux added to the region. This makes sense when we recall that there is one state in each Landau level per quantum of flux penetrating the sample.

190


Figure 5.6. Infinitesimal circuit in spin space associated with an infinitesimalcircuit in real space.

From Eq. (65) we see that the spin texture thus induces an extra charge of

A Q = --v-

n 4R

The solid angle R is, of course, a functional of the spin texture in the region r.For simplicity of analysis of this functional let us consider making up r out of a set of infinitesimal square loop circuits in real space of the form

The corresponding circuit in spin space illustrated in Fig. 5.6 is

+

m(x,y ) m(x dx, y ) -+ m(x + dx, y -+

+dy)

-+

m(x,Y + d y ) -,m(x,y )

(73)

Approximating this circuit as a parallelogramas shown in Fig. 5.6, the solid angle subtended is (to a sufficient approximation)

This may be rewritten in a suggestive form, which tells us the curl of the Berry “connection” [98]:

dw = $&,,m.a,,mx a,mdxdy We can now add up all the infinitesimal contributions to obtain =

dxdyf~,,,m.a,mx a,m

(75)

FIELD-THEORETIC APPROACH

191

which yields a total charge of

or a local-charge-densitydeviation of ev

Sp = --&,,,,m.~,,m x d,m 871

The expression on the right-hand side of Eq. (78) is simply the Pontryagin topological charge density of the spin texture. Its integral over all space is an integer and is a topologically invariant winding number known as the Pontryagin index. The spin textures, which have nonzero Pontryagin index, are the skyrmion configurations illustrated in Fig. 5.2b. A microscopic variational wavefunction for these spin textures was discussed in Section 5.6. The charge density in Eq. (78) can be viewed as the timelike component of a conserved (divergenceless) topological three-current, which results in the following beautiful formula: V

j a = - -&a~Y&,bcm“(r)~smb(r)a,mc(r) 871

(79)

Using the fact that m is a unit vector, it is straightforward to verify that a,,jr = 0. We note that the fact that the expression for the topological current is not parity invariant is a direct reflection of the lack of parity symmetry in the presence of the external magnetic field. The mechanism we have seen here that associatescharge with flux is the reason that quantum Hall fluids are describedby Chern-Simons theories [6,60,63,64,66] and is the same mechanism that causes Laughlin quasiparticles (which are topological vortices) to carry quantized fractional charge proportional to the quantized value of bxY[S]. Having established that the electron charge density is proportional to the topological density of the spin order parameter field, we must now return to our Lagrangian to see what modifications this implies. We have already taken into account the long-range Coulomb force, but it led only to the local spin gradient term, whose coefficient is the spin-wave stiffness. There are, however, additional effects of the charge (topological) density fluctuations which we must take into account: 4711’

- Ap 2 sId’r[Vm*(r)].[Vm’’(r)]

192


Here 6 p , is the Fourier transform of the charge density in Eq. (78).Note that it is second order in spin gradients. The first of the new terms in Eq. (80)representsthe coupling of the charge fluctuations to the external and random disorder potentials V(-q) and is second order in spin gradients. The second new term represents the mutual interaction of the charge fluctuations via the Coulomb potential. Note that this is fourth order in gradients (and so is not a duplicate of the p, term, which also comes from the Coulomb interaction). In general, there will be additional fourth-order terms allowed by symmetry, but we do not bother to write them down since they will not have the divergent Coulomb interaction coefficient 2n/~q,which makes the term we have kept effectively third order in q. We can immediately conclude several interesting things from the rather peculiar nature of our itinerant ferromagnet. First, unlike the case of a regular ferromagnet,a scalar potential can induce the formation of charged skyrmions in the ground state. Thus sufficientlystrong disorder would have the effect of greatly reducing the net spin polarization of the ground state, something that should be directly observable experimentally. Second,we note that (at the classical level)the energy of a skyrmion due to the gradient term is scale invariant:

because we have two spatial integrations and two derivatives. [The quantity E , is defined in Eq. (49).] Now, however, the Coulomb self-energywill want to expand the size of the skyrmion. In real spin systems this effect competes against the small but (usually) nonzero Zeeman coupling, which wants to minimize the number of flipped spins. This competition has been studied in some detail and appears to be essential to explain the experiments of Barrett et al. [24-271.

5.8. INTERLAYER COHERENCE IN DOUBLE-LAYER SYSTEMS The details of the double-layer experiments of Murphy et al. [39] are described by Eisenstein [38]. Here we introduce the main ideas briefly. Double-layer quantum Hall systems (and wide single-well systems [40]) exhibit a variety of nontrivial collective states at different filling factors. Here we focus on the case of total filling factor v = 1(i.e., 1/2 in each layer), which is most closely analogous to the fully ferromagnetic broken-symmetry state for v = 1 with real spins that we have discussed in previous sections. There are many other interesting states that we do not have room to discuss here. One example is the state at total filling factor v = 1/2 (i.e., 1/4 in each layer), which is believed to be described by Halperin’s {3,3, l } wavefunction [3,21,22,84,85]. This state is more nearly like a gapped spin liquid state, although, as we have already seen, it does not satisfy the Fock cyclic condition and so is not a true spin singlet.

INTERLAYER COHERENCE IN DOUBLE-LAYER SYSTEMS

n

Figure 5.7. Schematic conduction-band edge profile for a double-layer two-dimensional electron gas system. Typical widths and separations are W d - lOOA and are comparable to the spacing between electrons within each inversion layer.

193

I

..._._____. ,_____.._..

N

-d-

The schematic energy-level diagram for the growth direction degree of freedom in the double-layer system is shown in Fig. 5.7. For simplicity we assume that electrons can only occupy the lowest electric subband in each quantum well. If the barrier between the wells is not too strong, tunneling from one side to the other is allowed. The lowest energy eigenstates split into symmetricand antisymmetric combinations separated by an energy gap ASASwhich can, depending on the sample, vary from essentially zero to hundreds of kelvin. The splitting can therefore be much less than or even greater than the interlayer interaction energy scale, E , = e 2 / d The analogy with the spin systems studied in previous sections is that within the approximations just mentioned, the electrons in a two-layer system have a double-valued internal quantum number-the layer index. The tunnel splitting in the double well plays the role of the Zeeman splitting for spins. In addition to double quantum wells, there are wide single quantum wells in which the two lowest electric subband states are strongly mixed by Coulomb interactions [40]. These systems exhibit very similar physics and can also be modeled approximately as a double-layer system. Throughout our discussion we assume that the real spins are aligned and their dynamics frozen out by the small Zeeman energy. This is not necessarily a good approximation in experimentally relevant cases but simplifies matters greatly. Dynamics of real spins in double layers is currently a topic of investigation [103,104].

5.8.1. Experimental Indications of Interlayer Phase Coherence Here we review very briefly the main experimental indications that double-well and wide single-well systems at v = 1 can show coherent pseudospin phase order over long length scales and exhibit excitations that are highly collective in nature. When the layers are widely separated, there will be no correlations between them

194


4.0 1

I

Figure 5.8. Phase diagram for the double-layer QHE system. Only samples whose parameters lie below the dashed line exhibit a quantized Hall plateau and excitation gap. (After Ref. [39].)

and we expect no dissipationless quantum Hall state [1051 since each layer has v = 1/2. For smaller separations, it was predicted theoretically [33,85,106] and subsequently observed experimentallythat there is an excitation gap and a quantized Hall plateau [39,40,107]. The resulting phase diagram is shown in Fig. 5.8 and discussed in more detail by Eisenstein [38]. The existence of a gap has either a trivial or a highly nontrivial explanation, depending on the ratio ASAdEc.For largeAsAsthe electronstunnel back and forth so rapidly that it is as if there is only a single quantum well. The tunnel splitting ASASis then analogous to the electric subband splittingin a (wide)single well. All symmetric states are occupied and all antisymmetric states are empty and we simply have the ordinary v = 1 integer Hall effect. Correlations are irrelevant in this limit and the excitation gap is close to the single-particle gap ASAS (or ho,, whichever is smaller). What is highly nontrivial about this system is the fact that the v = 1 quantum Hall plateau survives even when ASAS 0) in terms of the field 8. Since the extra term induced by Q represents a total derivative, the optimal form of the soliton solution is unchanged. However, the energy per unit length of the soliton, which is the domain-line string tension, decreases linearly with Q and hence B,,: T = To( 1-2) where To is the tension in the absence of parallel B field given by Eq. (121), and B: is the critical parallel field at which the string tension goes to zero [125]. We thus see that by tuning B,,one can conveniently control the chemical potential of the domain lines. The domain lines condense and the phase transition occurs (in mean field theory) when the string tension becomes negative. Recall that the charge excitation gap is given by the energy of a vortex pair From Eq. (122)we have that the separated by the optimal distance R, = energy gap far on the commensurate side of the phase transition is given by

Jm

A = 2Em,+ = 2Em

+

(

p[( :)‘I2

1-

$)]lI2

As B,,increases the reduced string tension allows the Coulomb repulsion of the two vortices to stretch the string and lower the energy. Far on the incommensurate side of the phase transition, the possibility of interlayer tunneling becomes irrelevant. From the discussion of the preceding section one can argue that the ratio of the charge gap at B,,= 0 to the charge gap at B,,-+ oc) should be given (very roughly) by

-

Putting in typical values o f t and ps gives gap ratios sin the range 1.5 to 7 in qualitative agreement with experiment. According to the discussion of the preceding section, gap ratios as large as (tmax/tcr)1/40.07(e2/d)/p, can be expected in the regime where the pseudospin texture picture applies. Here t,,, is

-

-

216


the hopping parameter at which the crossover to single-particle excitations occurs. Thus gap ratios as large as an order of magnitude are easily possible. Of course, all the discussion here neglects orbital effects (electric subband mixing) within each of the electron gas layers, and these will always become important at sufficiently strong parallel fields. (Note also that for d near d*, the system will have enhanced sensitivity to renormalization of parameters by electric subband mixing in tilted fields.) It should be emphasized that only this highly collectivepicture involving large length-scale distortions of topological defects can possibly explain the extreme sensitivity of the charge gap to small tilts of the B field. Recall that at Bf, the tumbling length L,, is much larger than the particle spacing and the magnetic length. Simple estimates of the cost to make a local one-body type of excitation (a spin-flip pair, for example)shows that the energy decrease due to B,,is extremely small since l/L,,is so small. Numerical exact-diagonalization calculations on small systems confirm the existence of this phase transition and show that the fermionic excitation gap drops to a much smaller value in the incommensurate phase [47]. The collective excitation modes of the system in the commensurate and incommensurate phases appear to be quite interesting [114]. As mentioned previously, the tumbling of the order parameter may produce a self-grating effect that will allow one to tune the wavevector which couples to optical probes. All of our discussion of the phase transition in a parallel field has been based on mean-field theory. Close to the phase transition, thermal fluctuations will be important. At finite temperatures there is no strict phase transition at Bf in the PT model. However, there is a finite-temperature KT phase transition at a nearby B,, > B f . At finite temperatures translation symmetry is restored [123] in the incommensurate phase by means of dislocations in the domain string structure. Thus there are two separate KT transitions in this system, one for t = 0, the other fort # 0 and B,,> Bf. Read [126] has studied this model at finite temperatures in some detail and has shown that just at the critical value of B,,there should be a square-root singularity in the charge gap. The existing data does not have the resolution to show this however. Fisher [127] and Read [126] have pointed out that at zero temperature the commensurate-incommensurate phase transition must be treated quantum mechanically. It is necessary to take account of the world sheets traced out by the time evolution of the strings which fluctuate into existencedue to quantum zero-point motion. They have also pointed out that the inevitable random variations in the tunneling amplitude with position, which we have not considered at all here, cause a relevant perturbation. 5.11. SUMMARY

We have discussed the origin of spontaneous ordering in multicomponent fractional quantum Hall effect systems. For real spin at filling factors v = 1 this spontaneous ferromagnetism induces a very large charge excitation gap even in

SUMMARY

217

the absence of a Zeeman gap. The charged excitations are interesting topological skyrmion-like objects. Magnetic resonance experiments [26,27] have confirmed this remarkable picture which was developed analytically by Sondhi et al. [24] to explain the numerical results of Rezayi [96]. We have discussed in detail a pseudospin analogy which shows how spontaneous interlayer coherence in double-layer quantum Hall systems arises. This coherent XY phase order occurring over long length scales is essential to explain the experimental observations of Murphy et al. [39] described in Chapter 2. The essential physics is condensation of a charge-neutral bosonic order parameter field (pseudospin magnetization). This condensation controls the charge excitation gap and is very sensitive to interlayer tunneling and parallel magnetic field. We summarize a portion of this rich set of phenomena in the schematic zero-temperature phase diagram of double-layer systems shown in Fig. 5.12. First consider the plane with ASAS= 0 (zero tunneling). We have argued that the system develops spontaneous interlayer phase coherence despite the fact that the tunneling amplitude is zero. However, if the layer spacing d exceeds a critical value d*, the system is unable to support a state with strong interlayer correlations and the spontaneous U(1) symmetry breaking is destroyed by quantum fluctuations [41]. At this same point we expect the fermionic gap Ap to collapse. Little is known about the nature of this quantum transition, which can be viewed as arising from proliferation of quantum-induced vortices (merons). This is similar to the quantum XY model, but in the present case, the merons are fractionalstatistics anyons which will presumably change the universality class of the transition [46]. At finite tunneling, the U(1) symmetry is destroyed and the quantum fluctuations are gapped and hence stabilized. This causes the critical layer spacing to increase, as shown in Fig. 5.12. The third axis in the figure is the tilt of the magnetic field. Magnetic flux between the layers causes the order parameter to want to tumble. For small tilts, the system is in a commensurate phase, with the

Fig. 5.12. Schematic zero-temperature phase diagram (with d/l increasing downward). The lower surface is d*, below which d > d* and the interlayer correlations are too weak to support a fermionic gap, Ap. The upper surface gives I??, the commensurate- incommensurate phase boundary. As d approaches d*, quantum fluctuations soften the spin stiffness and therefore increase B;.

218


order parameter tumbling smoothly. However, above a critical value of the parallel field, this tumbling costs too much exchange energy, and the system goes into an incommensurate phase which spontaneously breaks translation symmetry (in the absence of disorder). This phase transition has been observed by Murphy et al. [38,39] through the rapid drop in the charge gap as the field is tilted. We have presented arguments that the charge gap is determined by the cost of creating a highly collective object: a pair of fractionally charged vortices connected by a string. It is the decrease of the string tension with tilt that causes the extremesensitivity to small tilts. In addition to all this, there is (forzero tunneling) a finite-temperature Kosterlitz-Thouless phase transition. If observed experimentally, this would represent the first finite-temperature phase transition in a quantum Hall system. We have tried to keep the present discussion as qualitative as possible. The reader is directed to the many references for more detailed discussion of technical points. We close by noting that there are still many open questions in this very rich field concerning such things as edge states in multicomponent systems, proper treatment of quantum fluctuations for the highly collective excitations that appear to exist in these systems, and the nature of the phase transition at the critical value of the layer separation. ACKNOWLEDGMENTS The work described here is the result of an active and ongoing collaboration with our colleagues L. Brey, R. Cat&, H. Fertig, K. Moon, H. Mori, K. Yang, L. Belkhir, L. Zheng, D. Yoshioka, and S.-C. Zhang. It is a pleasure to acknowledge numerous useful conversations with D. Arovas, S. Barrett, G. Boebinger, J. Eisenstein, Z. F. Ezawa, M. P. A. Fisher, I. Iwazaki, T.-L. Ho, J. Hu, D. Huse, D.-H. Lee, S . Q. Murphy, A. Pinczuk, M. Rasolt, N. Read, S. Renn, M. Shayegan, S. Sondhi, M. Wallin, X.-G. Wen, Y.-S. Wu, and A. Zee. The work at Indiana University was supported by NSF Grant DMR-9416906. REFERENCES 1. R. E. Prange and S. M. Girvin, eds., The Quantum Hall Effect, 2nd ed., SpringerVerlag, New York, 1990. 2. A. H. MacDonald, ed., Quantum Hall Effect: A Perspective, Kluwer, Boston, 1989. 3. T. Chakraborty and P. Pietiliiinen, The Fractional Quantum Hall Effect: Properties of an incompressible quantumjuid, Springer Series in Solid State Sciences 85, SpringerVerlag New York, 1988, and references therein. 4. M. Stone, ed., The Quantum Hall Effect, World Scientific, Singapore, 1992. 5. R. B. Laughlin, Chapter 7 in The Quantum Hall Effect, 2nd ed., edited by R. E. Prange and S. M. Girvin, Springer-Verlag, New York, 1990.

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51. A. H. MacDonald, Science 267,977 (1995). 52. Xiu Qiu, R. Joynt, and A. H. MacDonald, Phys. Rev. B 40, 11943 (1989). 53. C. B. Hanna and A. H. MacDonald, Bull. Am. Phys. SOC.40,645 (1995); Phys. Rev. B, to appear (1996). 54. Numerical tests and further discussion of the competition between the hollow-core

state and the Pfaffian state can be found in L. Belkhir, Ph.D. thesis, SUNY Stonybrook, 1993, unpublished. 55. J. P. Eisenstein, H. L. Stormer, L. Pfeiffer, and K. W. West, Phys. Rev. Lett. 62,1540 (1989).

56. For simplicity we write the wavefunction for the essentially equivalent v = 1/2 case 57. 58. 59. 60. 61. 62. 63.

rather than dealing with the complications of the higher Landau level wavefunctions needed to describe v = 512. T.-L. Ho, Phys. Rev. Lett. 75, 1186 (1995). B. I. Halperin, Surf. Sci. 305, 1 (1994). Chapter 6, this volume. D.-H. Lee and C. L. Kane, Phys. Rev. Lett. 64,1313 (1990). S. R.Renn has recently considered mutual drag effects in mmm states of double well systems: Phys. Reu. Lett. 68,658 (1958). J.-M. Duan, Eur. Phys. Lett. 29,489 (1995). S. C. Zhang, H. Hansson, and S. Kivelson, Phys. Rev. Lett. 62, 82 (1989); 62, 980 (1989).

64. D. H. Lee and S. C. Zhang, Phys. Rev. Lett. 66,1220 (1991). 65. A. H. MacDonald and S.-C. Zhang, Phys. Rev. B 49,17208 (1994). 66. A. Lopez and E. Fradkin, Phys. Rev. Lett. 69, 2126 (1992); Phys. Rev. B 47, 7080 (1993); Phys. Rev. B 51,4347 (1995).

67. 68. 69. 70. 71. 72. 73. 74.

75. 76.

77. 78.

J. Zang, D. Schmeltzer, and J. L. Birman, Phys. Rev. Lett. 71,773 (1993). N. E. Bonesteel, Phys. Rev. B 48, 11484 (1993); Bull. Am. Phys. SOC.40,645 (1995). T.-L. Ho, Phys. Rev. Lett. 73,874 (1994). J. Dziarmaga, Vortices and hierarchy of states in double-layer fractional Hall effect, preprint (cond-mat/9407085). A. Zee, in Field Theory, Topology and Condensed Matter Physics, edited by H. B. Geyer, Springer lecture Notes in Physics, Springer-Verlag, New York, 1995. J. Froehlich, T. Kerler, U. M. Studer, and E. Thiran, A classification of quantum Hall fluids, Nuc. Phys. B 453,670 (1995). D.-H. Lee and X.-G. Wen, Phys. Rev. B 49,11066 (1994). See, for example, A. H. MacDonald in Proc. Les Houches Summer School on Mesoscopic Physics, edited by E. Akkermans, G. Montambeaux, and J.-L. Pichard, Elsevier, New York, 1996. P. J. Forrester and B. Jancovici, J . Phys. (Paris) Lett. 45, L583 (1984). A. H. MacDonald, H. C. Oji and S. M. Girvin, Phys. Rev. Lett. 55,2208 (1985). The SMA is not reliable for compressible states or for hierarchical incompressible states. See A. Pinczuk, Chapter 8,this volume.

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79. W. Kohn, Phys. Rev. 123, 1242 (1961);C. Kallin and B. I. Halperin, Phys. Rev. B 31, 3635 (1985). 80. R. Renn and B. W. Roberts, Phys. Rev. B 48,10926 (1993). 81. H. MacDonald, unpublished notes. 82. An exception occurs for the Yl,l,l wavefunction, for which v = 1. In this case there are no states in the lowest Landau level, to which Yl*l,lis coupled by the total charge-density operator. The reasons for this will become clearer later in the chapter. 83. Y. W. Suen et al., Phys. Rev. Lett. 68, 1379 (1992); J. P. Eisenstein et al., Phys. Rev. Lett. 68, 1383 (1992). 84. E. H. Rezayi and F. D. M. Hddane, Bull. Am. Phys. SOC.32,892 (1987); S. He, S. Das Sarma, and X. C. Xie, Phys. Rev. B 47,4394 (1993). 85. T. Chakraborty and P. Pietilainen, Phys. Rev. Left. 59,2784 (1987). 86. The correlation functions were obtained by using a hypernetted chain approximation as described in Refs. [lOO]. 87. A brief discussion of these modes has been given: A. H. MacDonald, M. Rasolt, and F. Perrot, Bull. Am. Phys. SOC.33,748 (1988). 88. The absorption strength of this mode will be reduced by a factor of [d/cos(0)112, where 1 is the wavelength of the infrared light and O is the angle between the propagation direction and the normal to the 2D layers. 89. S.-C. Zhang, private communication, 1995. 90. On the other hand, we remind the reader that because of analyticity and Pauli principle constraints on the Landau level wavefunctions, Hund's rule does not always apply. We saw, for instance in Section 5.2, that in some cases, particularly those with even-denominator filling factors such as v = 5/2, that the interaction energy can be optimized by a spin liquid singlet(-like) state. 91. E. Fradkin, Field Theories of Condensed Matter Systems, Addison-Wesley, Reading, Mass., 1990. 92. R. Rajaraman, Solitons and Instantons, North-Holland, Amsterdam, 1982. 93. The precise definitions of the skyrmion and antiskyrmion energies in the presence of the long-range Coulomb interaction is subtle. For a more complete discussion, see Ref. [46]. 94. I. V. Kukushkin, N. J. Pulsford, K. von Klitzing, R.J. Haug, K. Ploog, H. Buhmann, M. Potemski, G. Martinez, and V. B. Timofeev, Europhys. Lett. 22,287 (1993). 95. A. Usher, R. J. Nicholas, J. J. Harris, and C. T. Foxon, Phys. Rev. B 41, 1129 (1990). 96. E. H. Rezayi, Phys. Rev. B 36,5454 (1987);Phys. Rev. B 43,5944 (1991). 97. X.-G. Wu and S. L. Sondhi, Phys. Rev. B 51, 14725 (1995). 98. M. V. Berry, Proc. Roy. SOC.London Ser. A 392,45 (1984). 99. C. Kallin and B. I. Halperin, Phys. Rev. B 31,3635 (1985). 100. M. Rasolt, F. Perrot, and A. H.MacDonald, Phys. Rev. Lett. 55, 433 (1985); M. Rasolt and A. H. MacDonald, Phys. Rev. B 34,5530 (1986);M. Rasolt, B. 1.Halperin, and D. Vanderbilt, Phys. Rev. Lett. 57, 126 (1986). 101. Kun Yang, L. K. Warman, and S . M. Girvin, Phys. Rev. Lett. 70, 2641 (1993).

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102. We require oxx= 0 so that the flux can be added adiabatically and reversibly. 103. T. Nakajima and H. Aoki, Phys. Rev. B 52, 13780 (1995); Phys. Rev. B 51, 7874 (1995). 104. A. Karlhede, unpublished. 105. A single-layer system at Landau level filling factor Y = 1/2 has no charge gap but does show interesting anomalies which may indicate that it forms a liquid of composite fermions. For a discussion of recent work, see Chapters 6 and 7 in this volume. 106. H. A. Fertig, Phys. Rev. B 40,1087 (1989). 107. G. S. Boebinger, H. W. Jiang, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 64, 1793 (1990); G. S. Boebinger, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 45, 11391 (1 992). 108. X. M. Chen and J. J. Quinn, Phys. Rev. B 45,11054 (1992);R. C k e , L. Brey, and A. H. MacDonald, Phys. Rev. B 46,10239 (1992). 109. This phrase was the original title of Ref. [24] (prior to the editorial process). 110. S. Datta, Phys. Lett. 103A, 381 (1984). 11 1. X. G. Wen and A. Zee, Sideways tunneling and fractional Josephson frequency in double layered quantum Hall systems, preprint (condmat/9402025), 1994. 112. M. B. Pogrebinskii, Fiz. Tekh. Poluprovodn. 11,637 (1977)[Sou. Phys. Semicond. 11, 372 (1977)l;P. M. Price, Physica (Amsterdam) 117B, 750(1983);P. M. Solomon, P. J. Price, D. J. Frank, and D. C. La Tulip, Phys. Rev. Lett. 63, 2508 (1989); T. J. Gramila, J. P. Eisenstein, A. H. MacDonald, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 66, 1216 (1991) and references therein. 113. B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B47,7312 (1993);V. Kalmeyer and S.-C. Zhang, Phys. Rev. B 46,9889 (1992). 114. R. Cate, L. Brey, H. Fertig, and A. H. MacDonald, Phys. Rev. B 51, 13475 (1995). 115. J. R. Schrieffer,in The k s s o n ofQuantum Theory,edited by J. De Boer, E. Dal, and 0. Ulfbeck, Elsevier, New York, 1986. 116. For simplicity we ignore the fact that different flavors may have different core energies. 117. I. Tupitsyn, M. Wallin, and A. Rosengren, Phys. Rev. B 53, R 7614 (1996). 118. H. Mooij, in Percolation, Localization, and Superconductivity, edited by A. M. Goldman and S. A. Wolf, Plenum Press, New York, 1984. 119. K. Yang and A. H. MacDonald, Phys. Rev. B 51,17297 (1995). 120. Note the order of limits here. The fact that the order matters really defines what we mean by spontaneously broken symmetry. 121. See, for example, George Griiner, Density Waves in Solids, Addison-Wesley, Reading, 1994, Chapter 7. 122. J. Yang and W.-P. Su, Phys. Rev. B 51, 4626 (1995); J. Y., Phys. Rev. B 51, 16954 (1995). 123. P. Bak, Rep. Prog. Phys. 45,587 (1982);M . den Nijs in Phase Transitions and Critical Phenomena, Vol. 12, edited by C. Domb and J. L. Lebowitz, Academic Press, New York, 1988, pp. 219-333. 124. Murphy et al. have recently revised slightly downward their estimates of Asas in their samples. Hence the numbers presented here are slightly different than those in

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previous publications. See Chapter 2 in this volume for further discussion of this point. 125. We are considering for the moment the case of zero temperature so that B,, is unrenormalized by thermal fluctuations.We also treat the problem classically. 126. N. Read, Phys. Rev. B, 52, 1926 (1995). 127. M. P. A. Fisher, private communication.


6

Fermion Chern-Simons Theory and the Unquantized Quantum Hall Effect B. I. HALPERIN Physics Department, Harvard University, Cambridge, Massachusetts

6.1. INTRODUCTION

During the 1980s, experimental and theoretical studies of two-dimensional electron systems in strong magnetic fields were focused on the integer and fractional quantized Hall effects. More recently, however, it has become clear that these systems are also extremely interesting in the vicinity of even-denominator Landau level filling fractions,such as v = 1/2, where the quantized Hall effect does not occur. The theoretical approach that has proved most useful for understanding the unquantized systems is the fermion Chern-Simons theory [l-81, itself an outgrowth of the composite fermion picture developed by J. K. Jain to describe the most prominent features of the fractional quantized Hall effect [9, lo]. However, many of the computational-techniques were originally developed by R. B. Laughlin and others to explore the properties of anyon systems advanced as a model for high-temperature superconductivity [ll-161. In the fermion Chern-Simons approach, the electron system is subjected to a mathematical transformation which converts it into a new system of fermions that has, in addition to the ordinary Coulomb interaction, an interaction via a fictitious vector potential, known as the Chern-Simons field. The transformation is sometimesdescribed by saying that an even number of fictitious magnetic flux quanta are attached to each electron. In many circumstances it appears that a mean-field approximation to the transformed Hamiltonian embodies much of the physics of the original electron problem and is an excellent starting point for further perturbative calculations. A remarkable consequence of the fermion Chern-Simons theory is that for v = 1/2, and for certain other even-denominator fractions, the ground state and Perspectives in Quantum Hall EfJects, Edited by Sankar Das Sarma and Aron Pinczuk. ~~

~

ISBN 0-471-11216-X

0 1997 John Wiley & Sons, Inc.

225

226

FERMION CHERN-SIMONS THEORY

the low-energy excitations can be described by a modified Fermi-liquid theory, similar in many ways to the theory of electrons in zero magnetic field [S, 61. For a magnetic field B which deviates by a small amount AB from the field that corresponds to an even-denominator filling fraction such as v = 1/2, the elementary fermion excitations no longer move in straight lines but rather follow semiclassical circular orbits. The radius R,* of these orbits is just the cyclotron radius in an effective field B* proportional to AB. Specifically,near v = 1/2, we have

where the Fermi wavevector kF is related to the electron density by

kF = (4xne)'/'

(2)

and B* is just equal to AB in this case. This prediction of the theory has received dramatic support from several recent experimentsthat have essentially measured the effectivecyclotron diameter 2R:, as will be discussed in Sections 6.7 and 6.9.1. If the magnetic field B is chosen so that the mean-field ground state of the transformed fermions contains an integer number of filled Landau levels, the mean-field theory has an energy gap for excitations. If the energy gap is larger than the temperature and larger than any smearing effects due to scattering from impurities, the electron system should exhibit a quantized Hall plateau. As was noted originally by Jain [9], integer fillings for the transformed fermions correspond to electron filling fractions of the form v=-

P

2np

+1

where p and n are integers. These are precisely the filling fractions where the most prominent fractional quantized Hall plateaus have been seen experimentally. The mean-field theory, as noted above, is only a first approximation to a complete analysis. Interactions via the Chern-Simons field are very important at long wavelengths, and they must be taken into account, at least at the level of the random phase approximation (RPA) if one wishes to predict transport properties, dyanmic response functions, or collective excitation spectra at long wavelengths [1,5,7,12,17]. Fluctuations beyond the mean-field theory are also important, in a more subtle way, in determining the overall energy scale for excitations,including the values of the energy gaps at the principal fractional Hall states given by Eq. (3) [ 5 ] . The purpose of this article is to give an overview of results obtained using the fermion Chern-Simons theory of a partly filled Landau level, together with a very brief review of some of the experimentssupporting the theory. No attempt will be made to give details of the calculations leading to the theoretical results; the reader is referred to the original literature for these details.

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227

In Section 6.2 we begin with the mathematical formulation of the fermion Chern-Simons theory, and we obtain the mean-field results for the Fermi-liquid states and for the principal fractional quantized Hall states. In Section 6.3 we review the results of Halperin, Lee, and Read (HLR) [S] for the renormalization of the energy scale and the effective mass, due to fluctuations in the ChernSimons gauge field. Response functions obtained from the RPA and related approximations are reviewed in Section 6.4. Extensions of the theory to even-denominator filling fractions more complicated than v = 1/2 or v = 1/4 are discussed in Section 6.5. In Section 6.6 we discuss the effects of weak disorder, due to impurities, on the Fermi-liquid states of a partially filled Landau level, and we discuss the question of the magnitude of the longitudinal resistivity pxx.Applications to the theory of surface acoustic wave propagation are discussed in Section 6.7 and compared to the experiments of Willett and co-workers [18-201. In Section 6.8 we discuss other developments in the fermion Chern-Simons theory, including recent investigations of the validity of the HLR theory for the asymptotic low-energy behavior of the Fermi-liquid states, the one-electron Green’s function and applications to tunneling experiments, the one-particle Green’s function for the transformed fermions, bilayer systems,and inhomogenous geometries such as occur at the boundary of a fractional quantized Hall system. Some other experiments that bear on the validity of the fermion ChernSimons theory are reviewed briefly in Section 6.9, and concluding remarks are contained in Section 6.10.

6.2. FORMULATION OF THE THEORY We begin by considering a two-dimensional interacting electron system, in a uniform positive background, in a uniform external magnetic field B. We assume that all the electron spins are polarized parallel to B, so that we may ignore the spin degree of freedom. We then make a unitary transformation, originally employed by Leinaas and Myrheim in 1977, in the paper that developed the concept of fractional statistics in two-dimensional systems [21,22]. In first quantized notation, the transformation is written as

where Yeis the many-body wavefunction of the electrons, Ytris the transformed wavefunction, is an even integer, and zj is the position of the jth particle in complex notation:

6

Note that if the electron wavefunction obeys Fermi statistics so that Ye

228


6

changes sign under the interchange of two particles, and if is even, Ytrwill have the same property. If were chosen as an odd integer, the transformation (4) would convert fermions to bosons, or vice versa. A transformation from fermions to bosons was employed by Girvin and MacDonald and others to explore analogies between the quantized Hall effect and superfluidity [17]. In the present chapter, however, we restrict our attention to the case where the transformed particles are fermions. As a conceptual tool, we may extend our discussion to odd-integer values of 6, or even to fractional values of but this would then imply that the initial particles (“electronsy’)obey Bose statistics or fractional statistics (“anyons”). The unitary transformation (4)of the wavefunction,which may be described as a singular gauge transformation, requires a corresponding transformation of the Hamiltonian. Using second quantized notation, where ++(T)is the creation operator for a transformed fermion at point 7,we find

4

4,

where V is the potential energy operator and K is a kinetic energy of the form .

c

z(7)

Here mb is the electron band mass, is the electromagneticvector potential, satisfyingv x A’ = B, and Z(r) is a vector potential that depends on the positions of the particles and is given by

-a (-r ) = 4

1.

dzQ“7-F’)p(7)

and p ( 7 ) is the operator, which measures the density of particles at point 7. The vector potential Z(7) is referred to as a Chern-Simons field because in a Lagrangian formulation, it is obtained by including a term of the ChernSimons form in the field Lagrangian [22]. Throughout this chapter we use units where h = e/c = 1, and the electron charge is -e. The electron creation operator @L(T)is related to the transformed fermion operator ++(T)by

where arg(7 -7’) is the angle that vector (7-7’) forms with the x axis. The

FORMULATION OF THE THEORY

229

potential energy Y has the same form whether expressed in terms of $ or $e, as does the density p(F):

It follows from Eq. (8) that the Chern-Simons magnetic field, defined by

is related to the particle density by

In a translationally invariant system, the mean-field Hamiltonian H, may be obtained by replacing b ( 3 by its mean value,

where n, is the average electron density, and by ignoring the potential energy Y. Thus we have H , =2% where

V

x A*

- iV + A*(7)]2$(r)d2r

(15)

= B* = B - 2n&,

If we define an effective filling factor p by p=-

2zn, B*

the electron filling factor v = 2znJB is related to p by

If p is a positive or negative integer, the mean-field ground state has ( p i filled Landau levels, and the fraction v has the special form stated in Eq. ( 3 ) [1,9]. Within the mean-field approximation, we would identify the energy gap of the fractional quantized Hall state at filling fraction v with the cyclotron energy in the field B*:

230


In the case when v is an even denominator fraction of the form v = 1/2n, if we choose = 2n, we find, according to (16),B* = O-the external magnetic field B is precisely canceled by the average Chern-Simons field 6. Thus the mean-field ground state is a filled Fermi sea, with a Fermi momentum k, given by Eq. (2) [5,6]. The value of k, differs by a factor of from that of electrons in zero magnetic field, because we assume that only one spin state is occupied here. Of course, the mean-field approximation is at best a first estimate of the properties of the actual interacting Fermi system. What one hopes is that the mean-field state has the correct quantum numbers and symmetry properties, and that the exact ground state may be obtained (in principle) by a convergent perturbation expansion starting from the mean-field state. The perturbation Hamiltonian is the difference H - H,. In fact, there are two contributions: the Coulomb interaction V, which has been omitted from H,, and the difference between the actual Chern-Simons vector potential Z(i(r),which is an operator depending on the positions of all the particles, and the average value incorporated in Eq. (16). There is no obvious small parameter in this perturbation expansion; indeed, the coupling constant to the gauge field is the coefficient which is 2 2 in physical applications. However, there is at least some reason for optimism. In the case of fractional quantized Hall fractions, there is an energy gap separating the ground state from the lowest excited state; this is obviously favorable for the possible convergence of perturbation theory. For even-denominator fractions, such as v = 1/2, there is no energy gap in the mean-field theory; however, the existence of a Fermi surface is still very helpful. In general, perturbations about the mean-field theory produce intermediate states in which there are at least two holes below the Fermi energy and two particles above it. The density of states for such excitations tends to zero at low energies, which is the reason perturbation theory can be used to discuss the properties of an ordinary Fermi liquid, such as liquid 3He.Thus, if the matrix elements of the pertrbation are not too divergent at long wavelengths, some type of Fermi-liquid description may be valid for the two-dimensional electron system at even-denominator filling fractions.

4

fi

4,

6.3. ENERGY SCALE AND THE EFFECTIVE MASS As mentioned Section 6.1, one important effect of the gauge-field fluctuations is a large renormalization of the fermion effective mass, and consequently of the energy scale for excitations [S]. Indeed, the mean-field expression (19) for the energy gaps at the principal quantized Hall fractions is in clear violation of an important physical principle. In the limit where the electron-electron interaction is weak compared to the electron-cyclotron frequency, B/mb, the low-energy states have electrons entirely in the lowest Landau level. Then the energy gap must be independent of the electron band mass and must be linearly proportional to the Coulomb energy scale e2/& where E is the dielectric constant of the surrounding medium and lB = IBI- 1/2 is the magnetic length.

ENERGY SCALE AND THE EFFECTIVE MASS

231

If conventional Fermi-liquid theory applied to this system, we would expect that at least for large values of IpI, Eq. (19) should be replaced by [5,23]

where m* is the effective mass of the composite fermions. Thus, in the limit mbe2 2/1, while for q d,-' because of the setback d,. If d, is large compared to k; fluctuations in the impurity potential lead only to small-angle scattering of electrons at the Fermi surface, which is further reduced by electron-screening effects. This is the reason why the mobility of these samples is so high in the absence of a magnetic field. The situation for composite fermions at v = 1/2 (or at other even-denominator fractions) is rather different. A fluctuation in p,(agives rise to a screening fluctuation p ( 3 in the electron density, which is equal to - p , ( 3 in the longwavelength limit. The fluctuation in p ( q ) gives rise to static fluctuations in the Chern-Simons magnetic field b ( 3 , and thus in the vector potential Z(3.

',

242


Specifically, one finds that

The scattering effect of the vector potential is much larger than the scattering caused directly by the screened Coulomb potential, which can account for the experimental result that p, at v = 1/2 is very much larger than the resistivity of the same sample at B = 0. (The importance of this scattering mechanism was noted independently by Kalmeyer and Zhang [6] and by HLR [5].) To estimate the magnitude of p,.. HLR consider a simple model where the number of charged impurities is equal to the number of electrons in the ZDEG, and there is assumed to be no correlation in the positions of the charged impurities. If the scattering rate is calculated in the first Born approximation, the transport mean free path 1 for the compositefermions at v = 1/2 is found to be just equal to the setback distance d,. The conductivity c,, is then determined by Eq. (29). Experimental values of the resistivity at v = 1/2 in high-quality remotely doped samples have typically been smaller by a factor of 3 or more than the estimate obtained from this simple theoretical model [19,521. This implies that the actually mean free path may be significantlylarger than the estimate. Part of this discrepancy probably arises from a tendency of the Born approximation to overestimate scattering effects in the regime of interest. Also, there is probably a significant anticorrelation in the locations of the ionized donors in the actual samples, which should reduce the amount of scattering. (The total number of donors is significantlylarger than the number that are eventually ionized, and it is reasonable that the ionized sites are distributed in a way that tends to minimize charge fluctuations.) In some cases, however, values of pxx have been reported which are as much as 25 times smaller than that predicted by the simple model [53]. It would seem surprising that anticorrelations could be so effective as to account for a discrepancy of this magnitude. Analysis of resistivity measurements is further complicated by the fact that density fluctuations even at very long length scales can lead to an increase in the macroscopic resistivity of a system with (pxyl>>pxx.In fact, the measured resistivity might be largely determined by such inhomogeneities[54-581. Density fluctuations on multiple length scales have been suggested as a mechanism for explaining, at least in part [56], the remarkable linear magnetoresistancethat has been observed in two-dimensionalelectron systems over a wide range of magnetic fields at temperatures high enough so that the quantized Hall plateaus are suppressed [52]. A detailed understanding of the mechanism responsible for electrical resistivity in actual samples is still lacking, however. At any given even denominator v, for sufficiently high levels of disorder, the metallic Fermi-liquid phase should be destroyed because of localization effects. It is an open question whether in principle, even in samples with very low disorder, the metallic behavior should be destroyed in the limit of extremely large length

SURFACE ACOUSTIC WAVE PROPAGATION

243

scales and low temperatures because of logarithmic renormalization effects analogous to weak localization in conventional two-dimensionalsystems [59]. If one treats the composite fermions as a system of noninteracting particles subjected to a random Chern-Simons magnetic field and a random potential, one finds that the leading logarithmic correction, which is responsible for weak localization at B = 0, is suppressed at v = 1/2 due to the broken time reversal symmetry [S, 61. When interactions are taken into account, it appears that there is again alogarithmic correction to pxxsimilar to the case of B = 0 [S]. In practice, however, for high-mobility samples at v = 1/2, the logarithmic term will be very small for any realistic sample size or temperature, so that the metallic phase is not destroyed by weak disorder. If disorder is large enough so that the metallic phase at an even-denominator fraction vo is destroyed by localization, then as the magnetic field or electron density is varied so that v passes through yo, we expect to find a sharp direct transition between quantized Hall states corresponding to odd denominator fractionson either side of vo [ 5 , 6,10,60]. Considerable progress has been made in analyzing these disorder-dominated transitions [61], but this subject will not be discussed further here.

6.7. SURFACE ACOUSTIC WAVE PROPAGATION Experimental studies of surface acoustic wave (SAW) propagation in GaAs samples containing a 2DEG provided the first strong evidence that something quite unusual was happening near filling factor v = 1/2 [lS]. Subsequent refinements of these experiments have provided some of the strongest evidence supporting the fermion Chern-Simons theory [19,20]. We discuss here the theory of these effects. It is expected that interaction between the SAW and the 2DEG occurs by means of the longitudinal electric field due to the piezoelectriceffect in GaAs. The response of the 2DEG may be characterized by a longitudinal conductivity axk7j,a)at the frequency w and wavevector 7 j (parallel to the surface) of the acousticwave. The finite conductivityleads to an acoustic attenuation and a shift Av, in the velocity v, of the SAW, relative to the case of a 2DEG with infinite conductivity, which may be expressed as [18]

Here a is a coupling constant proportional to the piezoelectric coefficient,K is the amplitude attenuation coefficient, and 6, is given by

am=-Us& 27t

244


'

If q - is smaller than the distance between the 2DEG and the free surface,then is essentially the dielectric constant of GaAs, and a, is found to be x7 x S2- '. At larger wavelengths, the electric field arising from the 2DEG penetrates partly into the vacuum, and a weighted average of the GaAs and vacuum dielectric constants must be used. In this case, a, can be as small as x 4 x 10-7S2-'. In general, a, is a function of? whose value can be calculated with reasonable accuracy if the geometry is known. The longitudinal conductivity ox&, w ) was evaluated by HLR using a semiclassical approximation, closely related to the RPA. The resistivity tensor pup(?,a), which is the matrix inverse of a,&, a),is related to a compositefermion resistivity P4&% w ) by

E

P = Pes + P

(43)

where pesis the Chern-Simons resistivity P c s 2= 7T 4( l 0

-

;)

[The origin of the Chern-Simons resistivity is the induced self-consistentChernSimons electric and magnetic fields, given by Eqs. (24) and (25).] The composite fermion conductivity, d,,(q,

0)

= CP(% 41;l

(45)

is approximated, as in the RPA, by the corresponding response function of a set of noninteracting charge fermionsin a uniform static field B*. Now, we use a further semiclassical approximation, which should be valid for q > 2/1. On the other hand, for fixed q 2 2/1, if B is varied sufficiently far away from the value at v = 1/2 so that the effective cyclotron diameter 2R: becomes smaller than the wavelength 2n/q, the enhanced nonlocal conductivity disappears and a,&) returns to the long-wavelength value axx(4= 0), which is independent of B* in the semiclassicalapproximation and is therefore given by (29).

SURFACE ACOUSTIC WAVE PROPAGATION

245

t

According to Eq. (41),an increase in a,,(q) should give rise to a decrease in Avr Thus one would predict that at a given SAW wavevector q, if the magnetic field B is varied, there should be a dip in the SAW velocity near v = 1/2 whose amplitude should become more pronounced with increasing q, and whose width in magnetic field AB should also be proportional to q. These features are in good agreement with the characteristics of the SAW anomaly originally observed by Willett et al. [lS] and confirmed by subsequent experiments [19,20]. Note, however, that the curves for a,&) shown in Fig. 6.3 are not monotonic functions 6f IB* I for ql > 2. In particular, we see a series of maxima in a&) which becomesmore pronounced for increasingql. These maxima can be understood as geometric resonances that occur when the effective cyclotron diameter 2R,* is related to the sound wavelength 1 = 2n/q by 2R,* x (n + 1/4)1

(46)

where n is a positive integer. (More precisely, the resonances occur at qR,* = X,,

246


where X, is the position of the nth zero of the Bessel function J1.)The most prominent peaks, at n = 1, correspond to an effective field B* =

hC

3.83 -qk,

(47)

e

It may be remarked that the semiclassical calculation for the composite fermion conductivity tensor 6(q,w)is very similar to calculations by Cohen et al. in 1960, of geometric resonance effects in ultrasonic attenuation in threedimensional metals [62]. However, the presence of the large off-diagonal ChernSimons contribution in Eq. (43) greatly changes the numerical values, and even the sign of the anomaly in ox, in the present case. Anomalies at the fields corresponding to (47) were not seen in the SAW data available prior to 1993. As SAW measurements have been extended to larger wavevectors q, however, the predicted structure has clearly emerged. In the 1993 data of Willett et al. [20], shown in Fig. 6.4, this is manifested by the appearance of double minima in the SAW velocity curves for frequencies 2 3.7GHz.

1

t " ' " " " ' J

3 .Ox10-4 2.5 2.0

/

2.4GHz 5.4 8.5

1

1.0 0.5

E

55

60 Magnetic field (kG)

65

Figure 6.4. Relative shift AuJus in the surface acoustic wave velocity, as a function of magnetic field, in a GaAs sample containing a two-dimensional electron layer for four values of the SAW frequency. Filling fraction v = 1/2, occurs at 57 kG.(Courtesy of R. L. Willett; data reported by Willett et al. in Ref. [20].)

OTHER THEORETICAL DEVELOPMENTS

247

The splitting between the SAW velocity minima observed experimentallywas found to be in excellent agreement with the predictions of Eq. (47), near v = 1/2, for a range of SAW wavevectors q. More recent experiments, extended to 10.5GHz, have seen the predicted splitting of the velocity minima at filling fraction v=3/2, and have seen the secondary n = 2 minima at v = 1/2 [63]. A strong SAW anomaly has been observed near v = 1/4 and a weak anomaly near v = 3/4, but peak splitting has not yet been seen in these cases. Since the wavevector q of the sound wave is accurately known, the peak splitting of the velocity minima near v = 1/2 and v = 3/2 may be taken as a direct measure of the Fermi wavevector k , of the composite fermions. A noted in than k, in a zero Section 6.5, the value of k, at v = 1/2 is larger by a factor magnetic field, while at v = 3/2 it is reduced by (2/3)’/’ from the zero-field value. Since the effectivefield B* near v = 3/2 also differs by a factor of 3 from the actual field deviation AB, the field splitting near v = 3/2 is reduced by a factor of 33/2 relative to that at v = 1/2. Willett et al. have compared the experimental values of AuJu, near v = 1/2 with the quantitative predictions of the semiclassical calculation of the fermion Chern-Simons theory. They were able to obtain good agreement after making several adjustments. The theoretical predictions were convoluted with a Gaussian of width 1.5% FWHM in the magnetic field to account for presumed density inhomogeneities [20]. In addition, to obtain good agreement between the SAW measurements far from v = 1/2 (or at low frequencies at v = 1/2) and macroscopic dc resistivity measurement made on similar samples, it was necessary to use a value of the material parameter om that was several times larger than that expected from the simple theory discussed above [5, 20,633. It was then necessary to adjust the theoretical values of o,,(q) near v = 1/2 at large values of q by a factor of x 2 to get a good fit to the experimental data. The reasons for these discrepancies in the absolute magnitudes of o,(q) are not understood.

fi

6.8. OTHER THEORETICAL DEVELOPMENTS Many aspects of the fermion Chern-Simons theory are the subject of continuing theoretical investigations. One active area concerns the mathematical implications of the theory in the asymptotic “infrared”limit of long wavelengths and low energies, for an ideal system with no disorder, in the limit B* +O. Other areas concern applications of the theory to more complicated situations, such as bilayer systems or systems with two active spin states, and behavior near a sample boundary. We discuss a few of these issues below. 6.8.1. Asymptotic Behavior of the Effective Mass and Response Functions

There has been considerable interest in the question of whether the divergence of the effective mass m* obtained in the analysis of HLR is truly the correct asymptotic behavior, when all terms in the perturbation expansion are correctly

248


summed. As discussed in Section 6.3, HLR found a logarithmic divergence in m* for the case of Coulomb interactions, and a power-law divergence for shorterrange interactions. Specifically,they found an energy-dependenteffective mass of the form m*(w)

-

(48)

w-”

where w = )E-E,I, and y = 1/3 for short-range interactions. For electronelectron interactions that fall off at large separations as

V(r)

-

l/rq

(49)

with 1< q 1/1,. Figure 8.6 shows that the three structures resolved in the spectra have energies close to the positions of the critical points in the calculated mode dispersions (where the densities of states peak because dw/dq = 0). Particularly striking in the data is the doublet in the energy range of the charge-density excitations, which agrees with the prediction of a characteristic pair of critical points. Since the critical points occur at wavevectors q 2 1/l, lo6 cm- ',much larger than the inplane component of the scattering wavevector k, the results in Fig. 8.6 imply a massive breakdown of wavevector conservation. Breakdown of wavevector conservation in these experiments has been attributed to the loss of translational symmetry due to unscreened residual disorder in the insulating states of the integer quantum Hall effect [181. In the high-quality quantum structures employed in these studies, residual disorder, even if unscreened, is relatively weak. Thus we anticipate that its impact on resonant inelastic light-scattering processes can be considered in the context of perturbation theory [54,55,57]. Within this conceptual framework, resonant lightscattering spectra could be interpreted as a superposition of features due to processes that conserve wavevector (q = k) and structures that arise from processes with breakdown of wavevector conservation (q # k). Figures 8.5 and 8.6 illustrate resonant inelastic light-scattering studies of gap excitations in the quantum Hall regimes. These results offer direct evidence of magnetorotons in the collective-mode dispersions. The rotons, as we have seen,

-

-

-

-

322

RESONANT INELASTIC LIGHT SCATTERING

are major predictions of theories of electron interactions in two-dimensional systems. There are relatively small (about 0.5 meV) differences between measured and calculated critical points in the mode dispersions. The discrepancy could be explained by the approximation ho,>> E, that is used in the calculation. This approximation ignores the coupling to higher inter-Landau-level transitions (AZ = 2) that at these relatively small magnetic fields could reduce the energies of q # 0 magnetoplasmons [13]. Figure 8.7 shows the positions of the peaks of profiles of resonant enhancement of light scattering at o,as a function of magnetic field. The four points correspond to the incoming - resonance for filling factors v = 1,2,3, and 4. The energies in Fig. 8.7 are written as E,(B) = ELO) + (r + 1/2)hwi

c Filling Factor

1570

-

!s

/'c2-6mev'T -

E

Y

' eE01= 1542.4meV

EXPERIMENTS AT INTEGER FILLING FACTORS

323

where 0I

=-eB

p*c

is the effective cyclotron energy for the virtual optical transitions. The slope of the line, 2.6 meV/T, yields I' = 1 for the Landau level number and a value of p* = 0.064m0 for the reduced effective mass in the optical excitations. An interpretation of the results in Fig. 8.7 is offered by three-step processes shown in the upper inset to the figure. In this diagram we require that EAO) = Eel. The intermediate transitions take place in the sequence indicated by the numbers. Steps 1 and 3 are the virtual optical transitions. The inter-Landau-level excitations are created in step 2. The requirement of wavevector conservation is relaxed for optical transitions that are broadened by residual disorder. The impact of breakdown of wavevector conservation in light-scattering spectra was evaluated in the integer quantum Hall regime [61,79]. For q 2 l/lo, when S(4,o)represents the scatteringcross section for charge-density excitations, Marmorkos and Das Sarma write [61]

In Eq. (1 5) a Lorentzian function a

f (k, 47 a) = ( k - q)2 + a2 replaces the conventional wavevectorconservation condition f ( k , q, 0) = 6(k -4). The adjustable parameter a is a measure of the extent of breakdown of wavevector conservation. The effect of disorder on the collective-mode energies is incorporated by a phenomenological broadening parameter r and the replacement o2-,w(o + ir)in S(q, a). Figure 8.8 displays evaluations of Marmorkos and Das Sarma for filling factor v = 2 [61]. In agreement with experiment, the calculation predicts the major spectral features near the positions of the critical points. The relative intensities of the two maxima in the calculated spectra depend on values of a and r. While a detailed comparison with experiment requires consideration of the effects of the outgoing resonance, it emerges from Fig. 8.8 that values (r/oc) =5 x and al, N 0.1 would explain measured spectra. Of particular interest is the adjusted value of (a)-' lWA, which is close to the width of the spacer layer that separates the doping layer of ionized impurities in the AlGaAs barrier from the electron layer in the GaAs quantum well. This result is consistent with the interpretation of residual disorder as the cause of breakdown of wavevector Conservation. N

324


16

I

1

I

1

I

1

I

14

w

- wc[e2/q,hl

Figure8.8. I ( o , k , a ) calculated with Eqs. (15) and (16).The results are for the chargedensity inter-Landau-level excitations (magnetoplasmons) at v = 2. The lines shown are four sets of values of the broadening parameters tl and r.The heavy solid line is for a\, = 1 and T/o, = lo-’, the dotted line has a\, = 0.1 and T/w, = 5 x the dashed line is for or\, = 2 and T/w, = 5 x and the light solid line is for tll, = 2 and T/o, = The inset shows the calculated magnetoplasmon dispersion. (After Ref. [61].)

We consider next the results for the spin-polarized case at v = 1. Figure 8.9 compares measured spectra with the theoretical prediction for the dispersive collective modes [60]. In the interpretation of these results we should highlight the fact that in the presence of spin polarization the exchange self-energiesin Eq. (12)are different for spin-flip and non-spin-flip (or magnetoplasmon)excitations [12,19,29]. The difference in exchange self-energies,AX(@ >> E,, results in a splitting ASF between the two 4 = 0 excitations that can be written as

Figure 8.9a displays spectra of inter-Landau-level excitations with A1 = 1. The incident photon energies are in resonance with excitonic transitions of higher Landau levels (I’ = 2) of the lowest confinement states c, and h, of the 250A quantum well [E,(O)= E,,]. Figure 8.9b is a rendition of the dispersive collective modes calculated within the time-dependent Hartree-Fock approximation. The

EXPERIMENTS AT INTEGER FILLING FACTORS

325

Single Quantum Well ( 250A )

A

n -2.49 x lof1 0 = 10.67T

* = 3.5 x l@cm2Nsec T = 0.5K v = 0.97

Figure 8.9. Resonant inelastic light scattering of inter-Landau-level excitations at v = 1:

(a) spectra measured with four incident photon energies ha,; (b)calculated mode disper-

sion (private communication of C. Kallin). (Ref. [59].)

strongest peaks in the spectra are the q-0 modes. They are the magnetoplasmon = 22.3 meV. at 0,-17.7 meV and the spin-flip inter-Landau-level mode at asp The weaker structures centered at o = 19.8 meV are interpreted as critical points of magnetoplasmons, observed here, as in the case of v = 2, because of breakdown of wavevector conservation due to residual disorder. The q = 0 results in Fig. 8.9 are a direct spectroscopicmeasurement of ASP The observations reveal the unique exchange interactions in the spin-polarized electron gas. These results also display the excellent agreement of the prediction of Hartree-Fock theory with the value of ASFdetermined from the experiment. In the measurements at v = 1 the light-scattering intensities have a marked dependenceon temperature. At T = 5 K the intensity has decreased by a factor of 10 from its value at 1.8 K and the spectral features are no longer observable above 10 K. It is interesting that such temperature dependence is similar to that of N

326


the occupation of the two lowest spin-split Landau levels as determined by interband optical absorption [85]. The similaritiescan be considered as evidence that the temperature dependence of the light-scatteringintensity is related to the loss of spin polarization in the electron gas.

8.3.2. Results from Modulated Systems The wavevector dispersions of gap excitations and other collective modes are among the major predictions of current theories of electron fluids in systems of reduced dimensions. The novel insight that could be reached from the comparison of theoretical predictions with experimental results stimulates great interest in direct measurements of mode dispersions. However, in typical light-scattering experiments the in-plane scattering wavevectors are too small (kli 1/3 the mode dispersions are vastly different from those at v = 1/3. These evaluations predict at v = 1/3 a single principal roton minimum, while at FQHE states with v > 1/3 the calculations indicate the existence of several roton minima in the dispersions of gap excitations [25,27,41,44,47]. It is conceivable that the multiplicity of roton minima at incompressible states with 1/2 > v > 1/3 may prevent the formation of sharp long-wavelength bound states below a continuum of multiroton excitations. The measurement of collective gap excitations of FQHE states with v > 1/3 remains one of the major goals of inelastic light-scattering research of the twodimensionalelectron gas in the extrememagnetic quantum limit. The use of novel methods to access excitations with wavevectors larger than the scattering wavevector, perhaps as extensions of the modulations used in Refs. [80] and [86],

CONCLUDING REMARKS

337

as well as other forms of modulated resonant inelastic light scattering may be required. Further advances will also require imaginative experimental approaches to enhance the contrast between inelastic light-scattering signals and strong backgrounds due to luminescence. These methods might help in gaining insights beyond the striking results already obtained at v = 113. In the case of the weaker fractional state with v > 113, the ability to extract weak scattering intensities that overlap strong luminescence backgrounds is essential for successful measurements of the low-lying collective gap excitations.

8.5. CONCLUDING REMARKS In this chapter we have reviewed novel applications of the resonant inelastic light-scattering method to research of collective excitations in the quantum Hall regimes. We anticipate increasing interest in studies of collective modes of the electron liquid states in the regime of the FQHE. In this area great progress has been achieved in studies of the state with v = 113, and much work remains to be done to measure gap excitations of other incompressible states such as those at 215 and 317. Resonant inelasic light-scattering measurements offer unique insights into dispersive collective excitations. Although these experiments are difficult and time consuming, there is a strong incentive to search for new methods that could enhance the sensitivity of resonant inelastic light scattering in the presence of strong background due to luminescence. Further light-scattering experiments could reveal fundamental behaviors of collective gap modes of fractional quantum Hall states. Areas of further research include studies of the low-lying excitations of the intriguing compressible state at v = 112 [14,26,36-381. I also envision applications of resonant light-scattering methods to studies of the diverse electron quantum liquid phenomena in modulation-doped semiconductor structures created by state-of-the-art technologies. Experimental access to a broad group of collective excitations by light-scattering methods could offer unique insights into the beautiful physics manifested in quantum liquid states. Studies of the intriguing states of electrons in coupled double quantum wells, in which low-lying gap excitations are expected to display unstable behaviors [Sl, 98,100-104], are among current light-scattering experiments [lOS, 1063. ACKNOWLEDGMENTS Measurements in the FQHE regime benefited from the extraordinary efforts of Brian S. Dennis. Brian’s dedication and insights played a key role in the success of the experiments.The unique samples from Loren N. Pfeiffer and Ken W. West are also essential to this work. The research reviewed here is the result of the many collaborations cited in the list of references. I would like to acknowledge, with

338


gratitude, discussions with L. Brey, S. Das Sarma, J. P. Eisenstein, B. I. Halperin, Song He, D. Heiman, J. K. Jain, A. H. MacDonald, P. M. Platzman, L. L. Sohn, H. L. Stormer, and C. Tejedor. I am very grateful to Professor Elias Burstein, with whom, over a long period of time, I had numerous conversations on resonant light-scattering mechanisms by two- dimensional electron plasmas.

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work in progress.


9

Case for the Magnetic-FieldInduced Two-Dimensional Wigner Crystal M. SHAYEGAN Department of Electrical Engineering, Princeton University, Princeton, New Jersey

9.1. INTRODUCTION

One of the most exciting aspects of the physics of two-dimensional (2D) electron systems in semiconductors concerns termination of the fractional quantum Hall (FQH) effect at low Landau-level fillings, v ( v = n$,/B, where n is the electron density, 4, = h/e is the flux quantum, and B is the magnetic field). It is intuitively clear that strong disorder will terminate the FQH effect by the magnetic freeze-out of the carriers in the random disorder potential. However, in a pure system, transition to an ordered array of electrons [Wigner 19343, namely the Wigner crystal (WC), is expected to occur at sufficiently low v and low temperature, T. Thanks to the availability of very low-disorder, dilute 2D carrier systems, research on this subject has intensified in recent years and has been fueled by new experimentalresults as well as controversy. The purpose of this presentation is to provide a review of the experimental progress in this area during the past decade. The emphasis will be on the results of magnetotransport measurements on remotely doped GaAs/AlGaAs heterostructures which comprise the bulk of the experiments so far. Moreover, GaAs/AlGaAs heterostructures provide the cleanest (lowest-disorder)2D carrier systems yet available in any semiconductor. At the outset it may be helpful to give a summary of the most important results of the transport studies. Magnetotransport experiments on 2D electron systems (2DES) in GaAs/AlGaAs heterostructures have established that at v = 1/5 the ground state is a FQH liquid. This is evidenced by the vanishing of the diagonal resistivity p,, at v = 1/5 and the quantization of the Hall resistivity p,, at 5h/e2 (Fig. 9.1). At v slightly above and below 1/5, however, p,, diverges as T+O, indicating an insulating ground state. Distinctivecharacteristics of this insulating Perspectives in Quantum Hall Eflects, Edited by Sankar Das Sarma and Aron Pinczuk.

ISBN 0-471-11216-X 0 1997 John Wiley &Sons, Inc.

343

344

MFI TWO-DIMENSIONAL WIGNER CRYSTAL

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,

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10 15 20 25 30

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Figure9.1. (Left)dc diagonal and Hall resistivities (pxx and pxy) as a function of the magnetic field (B) for a low-disorder 2D electron system at the GaAs/AlGaAs interface with n = 6.6 x 10'0cm-2. Also shown are the conductivities, determined from the measured p,, and pxyrvia B , = p,J(p;, + &)and oXy= p,,,/(~:~+ &). The vertical arrows indicate representative integer (v = 1)and fractional (v = 1/3 and 1/5) quantum Hall states, while the vertical dashed lines mark the fillings near which the insulating phase (IP) is observed. (Note the scale change by a factor of 100 in the pxx plot around 9 T.) (Right) Temperature dependences of p,, minimum at v = 1/5, p,, maximum at v = 0.212, and pxx at v = 0.191 are shown; these data are for a 2DES with similar quality but lower density (n = 5.8 x 10'ocm-2). (From [Sajoto 1993a,b] and [Li 19941.)

INTRODUCTION

345

phase (IP) are its normal pxy ( B/ne), frequency-dependent conductivity, very large dielectric constant, strongly nonlinear I- V, and the generation of broadband noise above the threshold electric field. These characteristics, which all (except the normal pXy)disappear at sufficiently high temperatures, resemble what is observed for the pinned charge-density wave (CDW) in one-dimensional Peierls conductors. Although these observations do not provide direct and conclusive evidence for the transition to a WC, they are very suggestive of, and collectivelymake a cogent case for, the formation of a reentrant, pinned electron WC near v = 1/5. A new development has been the recent observations of strikingly similar reentrant IPS in two other high-quality 2 D systems in GaAs-AlGaAs heterostructures: the 2D hole system (2DHS) and the bilayer electron system in a wide single quantum well. The IPS in these systems, however, occur at much larger v; for 2DHS an IP reentrant around the v = 1/3 FQH liquid has been observed (Fig. 9.2) while the bilayer system shows such phases around the v = 1/3 and even the v = 1/2 liquid states (Fig. 9.3). These observations can be qualitatively understood in terms of the profound effect of Landau level mixing (effective diluteness)in the case of the 2D holes and of the interlayer interaction in the case of the bilayer system, both of which significantlymodify the ground-state energies of the FQH and WC states of the system and shift the liquid-to-solid transition to large v. These results are again very suggestive and provide further credibility to the interpretation of the IP as a pinned WC.

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Figure9.2. Magnetotransport data for a low-disorder, dilute (n = 4.1 x lO"cm-*) 2D hole system at the GaAs/AIGaAs interface. Note that here the IP is observed near the v = 1/3 FQH state. (From [Santos 1992al.)

346


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Figure 9.3. Magnetotransport data for a low-disorder bilayer electron system (n = 1.26 x 10" cm-') in a modulation-doped 750A-wide GaAs quantum well. Here we observe an IP reentrant around the v = 1/2 FQH state. The inset shows the calculated electron charge distribution (dotted curve) and the confinement potential (solid curve). (Data from [Suen 19933 and [Manoharan 1996a,b].)

This chapter is organized in the following manner. In Section 9.2 a general description of the ground states of a 2D system in a strong magnetic field is given. In Section 9.3 we describe the sample structure briefly, and some relevant experimentaldetails are given in Section 9.4. In Section 9.5 we provide a brief and historical review of the magnetotransport results reported for the 2DESs in modulation-doped GaAs/AlGaAs heterostructures. This is followed by a summary and general discussion of the most important features of the data in Section 9.6. Sections 9.7 and 9.8 deal with the results for the 2D hole and the interacting bilayer electron systems, respectively. We concludein Section 9.9 with remarks on the current status and future perspective of the field. It is worth clarifying that except in Sections 9.2 and 9.5, the emphasis of this presentation is on Princeton University data on low-disorder GaAs/AlGaAs heterostructures, with which the author is most familiar. There are recent reports of Ips reentrant around higher (integer) filling factors in three other 2D systems: the low-mobility 2DES in GaAs/AlGaAs, where the amount of disorder is intentionally increased [Jiang 1993, Wang 19941; the 2DES at the Si/DiO, interface [DIorio 1990, Pudalov 1993, Kravchenko 19951; and the 2DHS in Si/SiGe heterostructures [Fang 19951. No discussion of these is given here. Neither do we discuss the results of many other exciting and illuminating new experiments in this subject. These include magnetooptics experiments in both

GROUND STATES OF THE 2D SYSTEM IN A STRONG MAGNETIC FIELD

347

the visible [Turberfield 1990, Goldberg 1990, Buhmann 1991, Goldys 1992, Kukushkin 1993, Butov 1994) and far infrared [Besson 1992, Summers 19931, as well as low-temperature thermopower measurements [Bayot 19941 on lowdisorder GaAs/AlGaAs heterostructures in the low filling range. The results of most of these experiments have also been interpreted as consistent with the localization of electrons into a pinned WC, although these results cannot prove the presence of a WC either. Finally, there have been numerous recent theoretical developments in this area [see, e.g., Chui 19941. We do not discuss these in any detail except when directly related to discussion of the data presented. 9.2. GROUND STATES OF THE 2D SYSTEM IN A STRONG MAGNETIC FIELD The purpose of this section is to provide an overview of two particular ground states of the 2D system in a strong magnetic field, namely the WC and FQH liquid states. We first discuss the WC state, the influence of disorder, and some properties of a partially disordered WC that have been addressed. This is followed by a very brief description of the FQH effect and the competition between the FQH and WC states, and the important role of Landau level mixing on this competition. 9.2.1. Ground State in the v T , N 150mK it is very small and Tindependent. Note that the two temperatures, To and T,, correlate closely with the temperatures T, and T,, above which the noise and nonlinear I-V disappear in the dc data, respectively (see Fig. 9.6). As we discuss in Section 9.6.6, these data can be interpreted in terms of a two-step melting of the 2D WC [Li l994,1995a,b]. Li et al. also measured the dependence of Re a,, and Im a, on the ac excitation level. The data show a nonlinear behavior with Re a,, increasing and Im nXxdecreasing substantially above a threshold excitation field. The overall behavior of the data is consistent with the nonlinear dc data on similar samples (Fig. 9 3 , although the threshold field is somewhat larger than E , 0.1 V/m observed in the dc measurements. While this discrepancy may be understandable given the differences in sample disorder and also in measurement technique, such

-

-

’

-

SUMMARY AND DISCUSSION OF 2D ELECTRON DATA

365

small values of ET are puzzling when compared to ET expected, in a weak pinning model, from the measured pinning frequecy. Using wo = 3 x 10’ rad/s from Eq. (6)we obtain E , 200 V/m, three orders of magnitude larger than observed ET! Note that an ET 200 V/m value in the weak pinning model [Eq. (5)] leads to a WC coherence length ( on the order of the lattice constant a, implying that the application of the weak pinning model to this system is not justified. As discussed in Section 9.2.2, this puzzle led Li et al. to conclude that the strong pinning of the WC by residual impurities in the vicinity of the 2DES is a more appropriate picture. In such a model, Li et al. estimated ( -(nJa)-”’- 10a, where ni 1 x 10’4cm-3 is the concentration of the residual impurities in GaAs. Finally, the discrepancy between the directly measured ET and the one expected from Eq. (6)using the experimentallydeduced wocan be understood if it is assumed that the depinning is caused by the generation and motion of dislocation pairs [Chui 1993a1. In most ordinary crystalline solids, the consecutive motion of the atoms via a moving dislocation can lead to a slip at very low values of shear stress: This is why the observed critical shear strength (the elastic limit) in crystals is smaller than expected from the value of the shear modulus by typically more than two orders of magnitude [Kittel 1986). Associating the depinning of the WC with the dislocation motion, the observation of very small E , is thus not surprising.

--

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9.6.5. Washboard Oscillations and Related Phenomena Li et al. have measured the ac conductivity of the IP in the presence of an applied dc current I in samples with rectangular [Li 1994,1995~1and Corbino [Li 1994, 19961 geometries. In both geometries, when I exceeds the depinning threshold value, a broad resonance whose frequency is proportional to I is observed. The resonance is interpreted to be a manifestation of the WC washboard oscillations. The data taken on a sample with Corbino geometry are summarized in Fig. 9.8 [Li 19961. When the sample is dc biased above the nonlinear conduction threshold, the 2DES shows an inductive ac response (current lags the voltage)and Im crxx becomes negative. Moreover, Im ox, exhibits a broad resonance near a frequency f* which increases with increasing I. The dependence off* on I is shown in Fig. 9.9, together with the washboard frequency expected [from Eq. (8)] for this sample. In Fig. 9.8, note also that the broad resonance disappears at sufficiently large T. The data shown in Figs. 9.8 and 9.9 overall have a striking resemblance to what is observed in the case of sliding CDW systems [Gruner 1988,Bhattacharya 19911and arguably provide the strongest evidence for the WC interpretation of the IP. In the case of CDWs, an “inductive anomaly” has been reported near the washboard frequency and has been attributed to an excess dissipation because of the increased local distortions of the CDW [Coppersmith 19853. By analogy, we may expect similar phenomena for a depinned WC. In Fig. 9.9 the increase off* with I and the magnitude off* are qualitatively consistent with the washboard oscillations. Quantitatively, however, f* increases sublinearly with I and the

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Figme 9.8. Ac response of the 2DES in the presence of an applied dc current, I; (left) Re and Im parts of u , measured at a frequency f = 500 kHz and T N- 25 mK,as a function of B for I = 0 and 4 nA. When I exceeds the threshold value (1.0.1 nA for this sample) the ac response in the IP reentrant around v = 1/5 becomes inductive (negative Im a,J Also shown is the Im Q, for the IP at v = 0.177 as a function off at different T (center) and differentI (rioht).At low T. Im u--exhibits a broad minimum (inductiveanomalv)near an I-dewndent freauencv

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367

Figure 9.9. The dependence of the frequency f* near which Im u, shows a minimim (Fig.9.8) on dc current, I . Also shown is the expected washboard frequency. (From [Li 1994, 19961.)

magnitude off* is about three to four times larger than the expected washboard frequency. These discrepanciesmay be a result of a nonuniform depinning process [Li 1994,1995c, 19961. At small I , only a fraction of the WC domains, those with the smallest depinning threshold fields, may be depinned. The current density for this sliding fraction is therefore larger than expected for uniform depinning and results in a larger washboard frequency, as observed. For larger I , domains with larger threshold fields will be depinned, making the sliding channels effectively wider and thus lowering the washboard frequency and also broadening the resonance. This is consistent with the experimental results shown in Figs. 9.8 and 9.9. Also consistent with this picture is the disappearance of the resonance above a T close to To,the T above which the long-range positional order of the WC is presumably lost (see Section 9.6.6). Other phenomena related to washboard oscillations include the narrowband noise and ac-dc interference effects which can manifest as Shapiro steps. These have been reported for CDWs [Gruner 19881 but have not been observed unambiguously for the reentrant IP. Engel et al. reported the observation of Shapiro-like resonan& in ac-dc measurements [Engel 19923, but these resonances were not reproduced in subsequent coolings of the same sample. They may be a result of timedependent oscillations (switching),another phenomenon that has been observed in the reentrant IP [Li 19931. Given that the observation of

368


narrowband noise and Shapiro steps requires the highest degree of coherence and positional order, it is not surprising that these have not yet been demonstrated conclusively.On the other hand, the inductive anomaly requires the least degree of coherence, consistent with its observation in these 2DES samples which have an estimated coherence length of only about 10a (Section 9.6.4). 9.6.6. Melting-Phase Diagram and the WC-FQH Transition

One of the curious aspects of the transition to a WC state at low v and T has been the possibility of experimentally determining a melting-phase diagram. Several such T versus v phase diagrams have been deduced for GaAs 2DESs from the T dependence of various phenomena, such as nonlinear I-Y [Goldman 1990, Williams 19911, RF absorption [Glattli 1990) and conductivity [Paalanen 1992a], and kinks in dc log p,, versus T - data [Paalanen 1992b1.These diagrams are in reasonable agreement with each other, but there are some quantitative differences. A theoretical phase diagram, calculated using the Kosterlitz-Thouless melting criterion, has been reported to be in fair agreement with these experimental data [Chui 1991~1.Melting-phasediagrams have also been reported from the results of magnetooptics experiments [Goldys 1992, Kukushkin 1993, Summers 19933, although the melting temperatures are much higher than those deduced from transport measurements. Finally, a phase diagram has been reported for 2D holes in GaAs based on thermopower experiments [Bayot 19943. Here we present experimentaldata from the study of the T dependence of the dielectric constant by Li [Li 1994, 1995a,b], which can be associated with a two-stage melting of the 2D WC. Recall (Fig. 9.7) that Re E,, is large and roughly constant up to a temperature To,decreases with increasing T for Toc T c T , and becomes very small and T independent for T > T,. Figure 9.10 shows the B dependence of the measured To and TI. As mentioned before, Toand T, agree well with two other characteristic temperatures observed in similar samples: Tois close to T,, above which the noise and the sharp I-V threshold vanish, and T , correlates with T,, above which the conduction becomes effectively linear (Fig. 9.6). The magnitude and v dependence of both Toand T , also qualitatively agree with most of the phase diagrams deduced from magnetotransport measurements by other groups. The data of Fig. 9.10 can be interpreted qualitatively [Li 19941 in a model originally proposed for the melting of a classical 2D WC [Halperin 1978,Young 1979, Strandburg 19881. In this two-stage melting model, the WC loses its longrange positional order above the classical (first) melting T while its short-age orientational order persists up to a higher (second) melting T. In between these temperatures, the 2D system is in a hexatic (liquid crystal)phase. Associating To and T , with the two melting temperatures, the present data can be understood as follows. Below To, E,, is large and nearly constant, signaling the presence of a pinned WC with a large collective polarization. For T > To,the long-range order is lost and E,, gradually decreases as the orientational order slowly diminishes, until above T , the orientational order disappears completely. Such

'


369

B (TI

Figure 9.10. Two temperatures To and T , deduced from the temperature dependence of the small-signal Re E,, data (see, e.g., Fig. 9.7) versus B. (From [Li 19941.)

a model also explains why the sharp I-V threshold, noise, random switching, and the phenomena attributed to the washboard oscillations (the inductive anomaly and the ac-dc resonances)all disappear above To;these require positional order that is lost for T To.It is worth mentioning that a two-step melting model has also been used to explain the temperature dependence of the magnetooptics data, although the temperatures reported are substantially larger than To and TI [Goldys 1992, Kukushkin, Summers, 19931. The reason for the discrepancy remains unclear. Finally, an issue of fundamental importance is the nature of the T=O transition between the IP and the v = 1/5 FQH state as a function of B. The experimental results so far suggest a continuous IP-FQH transition, although there is no conclusive evidence. For example, Li et al. have observed that the magnitude of the low-T Re E , gradually increases as the v = 1/5 liquid state is approached from the I P [Li 19941. The increase fits reasonably well a IB - B,I dependence,where B, is the transition magnetic field. For the sample of Figs. 9.7 and 9.10, B, ‘Y 12.6T and 13.1Tare deduced for the transition from the high- and l o w 4 side of v = 1/5, respectively;these B, are shown in Fig. 9.10 by vertical lines. Note that the characteristic temperatures Toand T , approach zero near these B,. The gradual increase of the dielectricconstant can be interpreted as a signature of a gradual softening of the WC as the liquid state is approached and suggests a continuous IP-FQH transition [Li 19943.

=-

370


9.7. SUMMARY AND DISCUSSION OF 2D HOLE DATA

In 1991 Santos et al. studied the magnetotransport properties of a very low disorder, dilute 2D hole system (2DHS) at the GaAs/AlGaAs heterojunction. They observed a reentrant IP around the v = 1/3 FQH liquid (Fig. 9.2),which was strikingly similar to the IPS seen in GaAs-AlGaAs 2DES with comparable density near v = 1/5 [Santos 1992a1. They interpreted the IP as manifesting a pinned hole WC and suggested that its observation at such markedly larger v results from Landau level mixing (LLM), which in the case of much heavier GaAs holes, significantlymodifies the ground-state energies of the FQH and WC states of the system. In this section we review the original 2DHS data as well as later experimental results that provide further support for the WC interpretation of the IP. These include (1) the disappearance of the IP near v = 1/3 and its moving to lower v in much higher density 2DHSs in which LLM is substantially reduced [Santos 1992b, Rodgers 1993, Manoharan 1994b1; and (2) the observation of a direct transition from a well-developed high-order FQH state at v = 2/5 to the IP at 1/3 c v < 2/5 [Manoharan 1994a1. Insofar as much a transition is forbidden in the global (disorder versus B) phase diagram of the 2D system proposed by Kivelson, Lee, and Zhang [Kivelson 19921,its observation rules out the disorder-induced Hall insulator as the explanation of the IP. 9.7.1. Sample Structures and Quality

Early work on 2DHS at GaAs/AlGaAs heterojunctions was done on samples for which GaAs(100) substrates were used and Be was employed as a dopant [Stormer 19831.However, Be is not an ideal dopant for GaAs or AlGaAs because of its diffusion and migration, and the early samples were not of very high quality. An alternative is to grow the structure on a GaAs(31l)A substrate and use Si, which is incorporated as an acceptor on the (31l)A surface [Wang 19863. In this manner, using structural parameters somewhat similar to those for 2DESs, very high quality 2DHSs can indeed be fabricated [Davies 1991, Santos 1992a,b]. The 2DHSs discussed in this section are confined to either a GaAs/AlGaAs heterojunction or a GaAs quantum well with AlGaAs barriers. They are grown by MBE on GaAs(311)A substrates and modulation doped with Si. The lowtemperature mobility in these structures is typically 2 3 x lo5cm’/V.s and can exceed 1 x lo6cm2/V-s[Santos 1992c,Heremans 19923. Given that the effective mass of holes in GaAs is about five times larger than that of electrons, such high mobility values and also the strong FQH states observed in these samples [see, e.g., Manoharan 1994bl attest to the unprecendentedly high quality of these 2DHSs. 9.7.2. Insulating Phase Reentrant Around v = 113 Figure 9.2 shows the magnetotransport data of a dilute (n N 4.1 x lO’Ocrn-’), low-disorder ( p N 4 x lo5cmZ/V.s) 2DHS [Santos 1992a1. The resemblance

SUMMARY AND DISCUSSION OF 2D HOLE DATA

371

of the IP observed around the v = 1/3 FQH liquid to the IP in 2DES around v = 1/5 (Fig. 9.1) is remarkable. Although the characteristics of the IP in 2DHS have not been studied in as much detail as for the IP in 2DES, its nonlinear Z-V data [Santos 1992a1 and large dielectric constant [Li 19941 have been established. Why such a similar behavior at markedly different?A main differencebetween the 2D electrons and holes in GaAs is their effective mass: for electrons m: = 0.067me, while for holes mt = 0.37m, [Hirakawa 19931 a factor of about 5 larger. The large mt substantially reduces the Landau level separation ( = h a , = heB/m*), so that at moderate B ( N 5 T), ho,is only a small fraction of the Coulomb energy E, = e2/4na0(nn)- and is comparable to the FQH liquid gap energy (1/3AN 0.1e2/4nsc,IB) for an ideal 2D system. As a result, the energies of the ground states of the system are modified by LLM. In particular, 1/3A and the energy difference between the FQH liquid and WC states are reduced [Yoshioka 1984, 1986; Zhu 1993; Price 19931. It is possible then that while at v = 1/3 the FQH state is the ground state, near v = 1/3 the WC has lower energy; this is analogous to the WC interpretation of IP near v = 1/5 in the GaAs 2DES. Note that the measured 'I3A 1:400 mK for this 2DHS is much smaller than the ideal 1 / 3 A 11K ~ or the 'I3A2:4K that is measured for a 2DES with comparable density and quality, consistent with this picture [Santos 1992a1. An-instructive point of view, which helps bring the parameters and results of the GaAs 2D electron and hole systems into perspective is to consider the T = 0 solid-gas (liquid)transition in a degenerate system of 2D particles characterized by the parameter r, = (7112)- '/'/ax, where a; = 4ncc0h2/m*e2(r, is the average interparticle separation measured in units of the effective Bohr radius ax). In the absence of B, the ground state of a sufficiently dilute system is expected to be a WC; calculations [Tanatar 19891 give a critical r,=37 for the solid-gas transition. [Note that r, is equal to the ratio of the Coulomb energy E, = e2/ 4nccO(nn)-1/2 and the kinetic energy, which is equal to the Fermi energy, E, = E, = nnh2/m*]. The ground state is also expected to be a WC for any r, at sufficiently large B (i.e., for v c vw = 1/6.5) [Lam 19841. We may then expect a phase diagram in the r, versus v plane for the solid-gas transition, as shown schematically in Fig. 9.1 1. Such a diagram can qualitatively explain why the signatures of a WC ground state for a dilute 2DHS (m* =0.37m, n ~4 x 10'0cm-2, r, = 15) are observed near v = 1/3, much larger than v = 1/5, where similar signatures are found in 2DES (m* = 0.067me,n N 10" cm-2, r, 2 3). Two points are worth commenting on. First, note that r, E J h o , and therefore scales with the degree of LLM. In this context, a 2D system with more LLM is effectively more dilute. Second, the exact form of the phase boundary in Fig. 9.1 1 can be complicated because of the unknown effects of disorder and also the presence of the FQH liquid states, which are particularly stable at special v. Nevertheless, ignoring the FQH effect and disorder, a phase boundary that is qualitatively similar to the boundary schematically shown in Fig. 9.1 1 has been calculated based on a simple Lindemann melting criterion [Chui 1991bl.

-

372


Figure 9.11. Schematic phase diagram for the zero-temperature solid-to-gas transitions in a 2D system subjected to a perpendicular B. the shaded areas indicate the ranges of rs for 2DES and 2DHS in GaAs with a density of 4 x 1010cm-2 to 2 x 10" cm-2. Since the effectivemass is much larger for holes ( N 0.37 me) than for electrons ( ~ 0 . 0 6 7 m , )r,, for the 2DHS (6.8 6 r, 5 15) is significantly larger than for the 2DES (1.2 6 r, 5 2.8) in the same density range.

9.7.3. Disappearance of the Reentrant Insulating Phase at High Density The foregoing WC interpretation of the IP near v = 1/3 in a dilute 2DHS implies distinct behavior in a 2DES with lower n or 2DHS with higher n. In particular, if a very dilute (n 2: 2 x lo9 cm-2, r, N 15) 2DES with sufficiently high quality can be realized,it should show an IP near v = 1/3 FQH liquid. Such high-quality,very dilute 2DES is yet unavailable: Although a very low-density (nN 4 x lo9 cm-2) 2DES showing integer quantum Hall effect has been reported [Sajoto 19903, the quality of the system quickly deteriorates when n is decreased below about 2 x 10'0cm-2. On the other hand, in a 2DHS, as n is increased, the onset of the insulating behavior should move to smaller v and the system is expected eventuallyto behave like a 2DES. This has been observed experimentally[Santos 1992b, Rodgers 1993, Manoharan 1994b1. The data of Manoharan et al. are shown in Fig. 9.12 for a 2DHS with n = 2.1 ~ ' 1 0 ' 'cm-2 (r, N 6.7). Note that the R,, peak between v = 1/3 and 2/5 is only about 5 kR at T N 30 mK and does not show any appreciable T dependence [Manoharan 1994b1. 9.7.4. Wigner Crystal Versus Hall Insulator Based on their proposed disorder versus B global phase diagram for the quantum Hall effect, Kivelson et al. have argued that IPS arising primarily from disorder

n

SUMMARY AND DISCUSSION OF 2D HOLE DATA

;IDHole System

4 -

B

n

2.1 x 10" 4omK

C d 2

3-

U

&I2

373

-

1 -

€3

[TI

Figure 9.12. R,, versus B for a high-density 2DHS with n = 2.1 x 10" cm-' (rs = 6.7), showing the disappearance of the IP at 1/3 < v < 2/5. (Data from [Manoharan 1994bl.)

can exist around integral as well as principal FQH states at v = 1/(2i + l), where i is an integer [Kivelson 19921. These IPS were named the Hall insulator (HI) since they are expected to have a normal (nondiverging)pxyas T+O [Viehweger 1991, Kivelson 19921. Consistent with the HI picture are the IPS we have discussed so far, as they are indeed observed around a principal FQH state (such as v = 1/3 or v = 1/5), and they have a normal pxy(see, e.g., Fig. 9.1). Manoharan and Shayegan, however, have recently reported the observation of a direct transition from a well-developed high-order FQH state at v = 2/5 to an IP at 1/3 < v < 2/5 in a very high-quality 2DHS, which rules out the disorderinduced HI as theexplanation [Manoharan 1994al.Thedata, shown in Fig. 9.13, establish that the ground state at v = 2/5 is the FQH liquid, while the system shows an insulating behavior for v N 0.37. Now, a direct transition from a v = 2/5 liquid state to a HI is explicitly forbidden in the global phase diagram of Kivelson et al., the relevant portion of which is shown as an inset to Fig. 9.13. According to the topology of this diagram, the only quantum liquid-to-insulator phase transitions occur between the v = 1/3 state and the HI state; all the high-order FQH states (such as the v = 2/5 state) are separated from the HI by at least one other (metallic)boundary. Therefore, while the low-T normal pxyis consistent with the HI, the observation of a direct transition from the v=2/5 FQH state to the 1/3 < v < 2/5 IP dissociates this IP from any disorder-induced HI. On the other

500

400

100

0 MAGNETICFIELD B [ T I

1o3 1o5

n

1o4

U

10’

0

5

10

15

20

1/T [R’]

25

1o3

0

5

10

15

20

25

1/T [ K ’ ]

Figure 9.13. (Top) pxx and pxyversus B for a 2DHS with n = 1.06 x 10” cm-2 (rs = 9.4). The insulating phase regions are near v = 0.37, v = 0.30, and v < 2/7 and are marked “IP.” The inset schematicallyshows the theoretical disorder versus v- global phase diagram for v centered around 1/3 according to [Kivelson 19921. FQH liquids are marked by fractional quantum number v, and the Hall insulator is denoted “HI.” A bold arrow marks the direction and region traversed in the phase diagram as we increase B. (Bottom) Temperature dependence of pxxfor the reentrant IP at v = 0.37 and 0.30 (right)and for the FQH states at v = 2/5 and 1/3 (left). Lines through the FQH data represent fits to the activated regions, from which the quasiparticle excitation gaps “A are determined. (From [Manoharan 1994alJ

BILAYER ELECTRON SYSTEM I N WIDE QUANTUM WELLS

375

hand, the observed IPS,and their properties, including a normal Hall coefficient at finite T, are consistent with pinned WC states. Finally, since the reentrant IP around v = 1/3 in 2DHS is quite similar to the IP around v = 1/5 in low-disorder 2DES, one may expect that a similar violation of the HI selection rules should also occur in the case of 2DESs. Such a violation has indeed been observed recently: Du et al. have observed a direct transition from the high-order v = 219 FQH liquid to a 1/5 < v < 2/9 IP in a very high-quality 2DES [Du 19951. This observation supports the WC interpretation. 9.8. BILAYER ELECTRON SYSTEM IN WIDE QUANTUM WELLS The introduction of an additional spatial degree of freedom (perpendicular to the 2D plane) can also modify the ground-state energies of the system. There is indeed experimental and theoretical evidence that the finite layer thickness of a real 2D system leads to a weakening and eventual collapse of the FQH effect [Shayegen 1990, He 19901. When the effective layer thickness becomes comparable to or larger than the magnetic length, the short-range component of the Coulomb interaction, which is crucial for the stability of the FQH liquid, is softened. This weakens the FQH liquid and can favor the competing WC state. A closely related phenomenon is the effect of (electric)subband mixing which is inevitably present in real 2D systems with finite layer thickness. This can also weaken and destroy the FQH effect [Halonen 19931. No systematic experimental study of the role of finite layer thickness on the competition between the FQH and WC stateshas been reported. However, the observation of an IP reentrant around the v = 1/3 FQH liquid state in a highdensity, thick hole system in a wide parabolic AlGaAs well can be associated with the lowering of the WC energy relative to that of the v = 1/3 FQH state [Santos 1992bl. There is also recent theoretical work on this subject [Price 19953. In this section we present data for a somewhat different system with an additional degree of freedom, namely, the electron system in a wide GaAs quantum well which has a flat bottom in the absence ofelectrons. When electrons are introduced in a wide well, their electrostatic repulsion forces them to pile up near the walls of the quantum well (Fig. 9.14) and form a bilayer electron system (BLES) [Suen 1991). Magnetotransport data for this system reveal new correlated electron states such as even-denominator FQH states at v = 1/2 and 3/2 which have no counterparts in standard, single-layer 2DESs [Suen 1992a,b, 1994a,b]. (The v = 1/2 FQH state is also observed in BLESSin double quantum wells [Eisenstein 19921).The wide quantum well system also exhibits IPSsimilar to the ones we have discussed so far, but remarkably, the position in v of the IP increases with increasing density in the well. We argue that these IPS are consistent with the condensation of electrons into a pinned, bilayer WC [Suen 1992b, 1993,1994b; Manoharan 1996a,b;Shayegan 19961.

376


t

n = 6.6 x lOl0 cm-2

ASAS P 16.1 K

7

-3 Y

0.2

4

Oe3

0

Figure 9.14. Results of self-consistent calculations of the conduction-band edge, E,, (solid curves), and the electron charge distribution function (dashed curves) are shown for n = 6.6 x 10'' cm-2 (lef) and n = 1.5 x 10' cm-2 (right) electrons in a 750-A-wide GaAs quantum well. (From [Suen 19931 and [Manoharan 1996a,b].)

9.8.1. Details of Sample Structure The structure discussed here is grown by MBE and consists of a 750 A-wide GaAs quantum well, bounded on each side by an undoped AlGaAs spacer layer and Si &doped layers. The samples have typical n 1 x 10" and low-temperature mobility 1 x lo6cm2/V*s.The experiments include measurements of the magnetotransport coefficients of a given sample as a function of n for both symmetric and asymmetric charge distributions in the well. Both n and the charge distribution symmetry are controlled via front- and back-side gates; details can be found elsewhere [Suen 1991, 19931. A relevant parameter in these systems is the energy difference between the lowest two subbands, which for a symmetric charge distribution in the well, corresponds to the symmetric-to-antisymmetricenergy gap (ASAS)and is a measure of the coupling between the two layers. Also relevant is the interlayer distance, defined by parameter d in Fig. 9.14. An important property of the electron system in a wide single quantum well is that for a given well width, both ASAS and d depend on n: Increasing n makes d larger and ASASsmaller, so that in effect the system can be tuned from a BLES at high n to a (thick) single-layer-like system by decreasing n (Fig. 9.14). This evolution with density has a dramatic effect on the correlated electron states and therefore the magnetotransport properties.

-

N

BILAYER ELECTRON SYSTEM I N WIDE QUANTUM WELLS

377

9.8.2. Reentrant Insulating States Around v = 113 and 112

Figure 9.15 shows such evolution. At the lowest n the data exhibit the usual FQH effect at odd-denominator fillings,while at the highest n the strongest FQH states are those with euen numerators, as expected for a system of two 2D layers in parallel. For intermediate densities, an euen-denominator FQH state at v = 112 is observed. Concomitant with the evolution of the FQH states in data of Fig. 9.15, we observe an IP that moves to higher v as n is increased. For very low n, the IP appears near v = 1/5, while at the highest n, there is an IP for v 5 1/2. The IP in the intermediatedensityrange(l.10 x 10" < n,< 1.32 x 10"cm-2)ismost remarkable, as it shows a reentrant behavior around a well-developed v = 1/3 FQH state and, at slightly larger n, around the v = 1/2 FQH state. The data shown in Fig. 9.3 for n = 1.26 x 10" cmP2reveal the striking resemblance of the IP observed in

I

5

10

Bcr)

15

5

10

Bcr)

Figure 9.15. Evolution of the FQH states and insulating phase observed in a 750-Awide GaAs quantum well with increasing density, n. Selected fractional filling are marked by vertical arrows. Note the R,, scale change at high B for most traces. In the IP regions, marked by dashed curves, R,, shows an increase with decreasing T (see, e.g., Fig. 9.3). (Data from [Suen 1993,1994bl and [Manoharan 1996a,b].)

378


this BLES to the 2DES and 2DHS data (Fig. 1 and 2) except that here the IP is reentrant around the v = 1/2 FQH state! The IPS presented in Fig. 9.15 cannot be explained by single-particlelocalization. First, in the case of standard, single-layer 2DESs, it is well known that as n is lowered, the quality of the 2DES deteriorates and the sample shows a disorder-induced IP at progressivelylarger v [Sajoto 19903.This is the opposite behavior observed here: With decreasing n, the mobility decreases and the quality worsens as expected,but the IP moves to smaller v. Second,the observation of Ips that are reentrant around correlated FQH states, particularly around the very fragile v = 1/2 state [Suen 1994a,b], strongly suggests that electron interactions are also important for the IP. To illustrate that the behavior of these IPSwith n is indeed consistent with the WC picture, it is instructive first to discuss the evolution of FQH states in an electron system confined to a wide quantum well [for a review, see Shayegan 19963. This evolution has been studied in detail [Suen 1994a,b] and can be understood by considering the competition between ASAS and the in-plane correlation energy ( -Ce2/4xee,lB, where C is a constant -0.1). When ASAS > Ce2/4nee0lE,for v 1, the electron system occupies the lowest Landau level of the symmetric subband and exhibits “one-component”FQH states such as 1/3, 2/3,3/5, etc., similar to a standard 2DES but with reduced strength because of the large thickness of the electron layer. If the in-plane correlation energy is sufficiently strong (ASAS 5 Ce2/4necolE), on the other hand, the antisymmetricstate can mix into the correlated ground state to lower its energy and a “two-component’’ system ensues, the components being the (nearly degenerate) symmetric and antisymmetric states. Now depending on the strength of the interlayer Coulomb interaction ( e2/4nee,d), two types of two- component FQH states are possible. When e2/4n~,dis sufficiently small (d sufficiently large), the system behaves as two independent layers in parallel, each with a density equal to 42. An example of a two-component FQH state in such a system is the $330 state [Halperin 1983, Yoshioka 1989, He 19933, which has v = 2/3 (1/3 filling for each layer). Note that such a FQH state always has a v with even numerator (because of the even number of layers), and since each layer is a single-component 2DES, an odd denominator. If e2/4ne&,d is large enough and comparable to the intralayer correlation Ce2/4n&&,IB,a second type of FQH state, one with strong interlayer correlation, is possible. Such a FQH state is unique to a two-component system and can be at even-denominator v; a special exampleis the $ 3 3 FQH state, which has filling equal to 1/2 and is associated with the v = 1/2 FQH effect observed in BLESS with appropriate parameters [Suen 1994a,b, Eisenstein 19921. The study by Suen et al. has revealed that the evolution of the FQH states in a wide quantum well (Fig. 9.15) can be understood from the picture above [Suen 1994a,b]. In particular, both ASASand 1, (for a FQH state at a given v ) and therefore the ratio [ E A,A~(e2/4n&&o~E) decrease with increasing n, consistent with the observation that the one-component FQH states get weaker while the two-component ones become stronger. Quantitatively, plots of the measured

-=

-

BILAYER ELECTRON SYSTEM IN WIDE QUANTUM WELLS

379

FQH state energy gaps versus ( reveal a two-to-one-component transition as [ exceeds ~ 0 . 0 7[Suen 1994a,b]. Such a plot is particularly noteworthy for the v = 2/3 state; its measured energy gap shows a minimum near [ = 0.07 as it makes

a two-to-one-component transition. Based on this understading of the FQH states in a wide quantum well, we now discuss an explanation for the evolution of the IPSobserved in Fig. 9.15 in terms of WC formation.

1. At low n, where the system is essentially a single-layer (albeit thick) 2DES, the IP is observed near v = 1/5, similar to the standard single-layer 2DES. 2. At the highest n, where the system effectively contains two layers in parallel, the IP is present for v S 1/2, (i.e., v 5 1/4 for each layer). This is reasonable considering that even at the largest n interlayer interactions are present in this system, as evidenced by the observation of a correlated v = 1 quantum Hall state at high n in the same well [Lay 19941. Such interaction can move the WC ground state to v 1/4 (for each layer), somewhat larger than the v = 1/5 expected if there is no interaction between the two layers (see below). 3. The IP moves very quickly from v = 1/5 to v = 1/2 in the intermediatedensity range, where the FQH states (in the same range of v) make a two-toone-component transition. This observation clearly points to the fact that the IP is influenced by electron-electron interactions and also by the strength of the nearby FQH states [Manoharan 1996a,b]. 4. Consistent with this conclusion are the results of experiments in which the charge distribution in the wide well is made asymmetric(via the application of back and front gates) while keeping the total n in the well fixed. With increased asymmetry, the one-component FQH states gain strength and the two-component states become weaker, while the IP moves to smaller v [Suen 1992b, 1993,1994b; Manoharan 1996b1. 5. Finally, it is plausible that interlayer interactions can modify the groundstate energies so that for appropriate parameters a crossing of the liquid and solid states occurs at such large v (e.g., v N 0.54, i.e., v > 1/4 in each layer; see Fig. 9.3). Calculations [Oji 19871 indicate that the effect of interlayer coupling can be particularly strong near the magnetoroton minimum and lead to the vanishing of the FQH liquid gap. This vanishing can be associated with an instability toward a ground state in which each of the layers condenses into a 2D WC [Oji 19871.

-

In summary, while there is no quantitative understanding of the role of interlayer interactions on the WC versus FQH states of an interacting BLES, the data presented in this section are certainly suggestive of a pinned bilayer

wc.

380

MFI TWO-DIMENSIONAL WlGNER CRYSTAL

9.9. CONCLUDING REMARKS

A distinct state of the low-disorder 2DES at the GaAs/AlGaAs interface at high B and low T is the IP reentrant around the v = 1/5 FQH liquid. Many properties of the IP have been probed by magnetotransport measurements in recent years, and collectively,they make a cogent case for the identification of this state with the long-expected solid phase of 2D electrons at high B. The data on dilute GaAs 2DHSs with much larger effective mass, showing very similar behavior at much larger filling (v = 1/3), provide additional strength for the WC interpretation. The results on the electron systems in wide GaAs quantum wells add a new twist and are consistent with the observation of a bilayer WC. Insofar as the IPSobserved in all these systems can be associated with a WC that is pinned by the disorder potential and has a coherence length of only 10 times the WC lattice constant, there is much room for improvement in sample quality. Realization of samples with lower disorder and larger coherence length, which may show some of the more intrinsic properties of an ordered electron crystal, such as the magnetophonon modes or narrowband noise, remains an experimental challenge.

-

ACKNOWLEDGMENTS This presentation is based primarily on the work of my colleagues L. W. Engel, Y. P. Li, H. C. Manoharan, T. Sajoto, M. B. Santos, Y. W. Suen, and D. C. Tsui. I thank them for many illuminating discussions and a critical reading of this manuscript. I am also indebted to them, particularly to Y. P. Li and H. C. Manoharan for providing the data and figures, some of which are yet unpublished. Finally, I thank B. Steward for her patience and care, and for excellent typing of the manuscript. REFERENCES Ando, T., Fowler, A. B., and Stern, F. (1982).Rev. Mod. Phys. 54,437. Andrei, E. Y., Deville,, G., Glattli, D. C.,Williams, F. I. B., Pans, E., and Etienne,B. (1988). Phys. Rev. Lett. 60,2765. Andrei, E. Y., Deville, G., Glattli, D. C., Williams, F. I. B., Paris, E., and Etienne, B. (1989). Phys. Rev. Lett. 62,973 and 1926. Bayot, V., Ying, X., Santos, M. B., and Shayegan, M. (1994). Europhys. Lett. 25,613 Besson, M., Gornik, E., Engelhardt, C. M., and Weimann, G. (1992). Semicond. Sci. Technol. I, 1274. Bhattacharya,S., Stokes, J. P., and Higgins, M. J. (1991).Phys. Rev. B 43, 1835. Bonsall, L., and Maradudin, A. A. (1977).Phys. Rev. B 15, 1959. Buhmann, H., Joss, W., von Klitzing, K., Kukushkin, I. V., Plaut, A. S., Martinez, G., Ploog, K., and Timofeev, V. B. (1991).Phys. Rev. Lett. 66,926.

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10

Composite Fermions in the Fractional Quantum Hall Effect H. L. STORMER Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey

D. C. TSUI Department of Electrical Engineering,Princeton University, Princeton, New Jersey

10.1. INTRODUCTION

The fractional quantum Hall effect (FQHE) is a deceptively simple physical phenomenon [l] remarkably rich in many-body physics. Discovered in 1982, it has been subject to more than one decade of intense experimental and theoretical studies [2,3]. The unexpected features observed at low temperatures in electronic transport on high-mobility two-dimensional (2D) electron systems were quickly identified as being of many-particle origin, brought about by the correlated motion of many electrons. Energy gaps in the excitation spectrum of the carriers were concluded to be responsible for the characteristic transport features of the FQHE. The origin of these energy gaps could not be attributed to any singleparticle mechanism such as Landau quantization or commensurability with an inherent periodicity of the system. Landau quantization of electron orbits and Zeeman splitting of their spin levels gave rise to the integral quantum Hall effect (IQHE) which exhausted all integer quantum numbers in the characteristic quantized Hall resistance [4]. The FQHE introduced rational, fractional quantum numbers, incompatible with any single-particleinterpretation. Laughlin [S] made a decisive theoretical advance in proposing a very elegant many-particle wavefunction for the most prominent of FQHE states with quantum number 1/3. This remarkably succinct wavefunction has been confirmed by a large number of few-particlenumerical calculationsto be an extraordinarily good approximation for the correlated motion of some 10' carriers in the presence of a very high magnetic field [3]. Fractional charge of the quasiparticle excitation [S], finite gap in the excitation spectrum [S], a roton minimum in the dispersion relation [6], hierarchical sequence of FQHE states

'

Perspectives in Quantum Hall Efects, Edited by Sankar Das Sarma and Aron Pinczuk. ISBN 0-471-11216-X 0 1997 John Wiley & Sons, Inc.

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COMPOSITE FERMIONS IN THE FRACTIONAL QUANTUM HALL EFFECT

[7,8], spin dependence of ground and excited states [9, lo], FQHE in higher Landau levels [ll], and fractional statistics [S, 121-all such intricate behavior was deduced and ultimately derived in ingenious ways from this initial concept. Although, early on, arguments by analogy (in the opening of gaps, carrier localization, etc.) have been employed between IQHE and FQHE to rationalize the transport behavior in the FQHE, the conceptual frameworks for both phenomena are distinctly different. In particular, the energy spectra are of vastly different origins: band electrons of given mass and charge, Landau level splitting, cyclotron transitions, and spin gaps on the one hand, and condensed quantum liquid, many-particle gap, quasiparticle excitations, and complex many-particle spin effects on the other hand. Yet, to the uninitiated, the experimental transport featuresof IQHE and FQHE are extraordinarilyalike:plateaus in the Hall resistance and vanishing diagonal-resistance.Furthermore, the experimental data pattern contains an abundance of self-similarities providing a link, at least in a purely geometrical sense, between the features of the FQHE and the IQHE. Although obvious in retrospect, only recently have such self-similaritiesbeen recognized. A string of related theoretical investigations [13-171 during the 1980s and a set of pointed publications at the end of the decade and into the 1990s [18-201 have decisively altered the conceptual framework by which to categorize the experimentalobservations of the FQHE. To be sure, the wonderful FQHE edifice erected by experimentalists and theorists since 1982 is solidly intact. However, the novel theoretical standpoint has opened up a new vista from which position the initially complex structure assumes a pattern of extreme simplicity. It runs under the name of composite fermions [18] and Chern-Simons gauge transformation [19,20]. The system of electrons, interacting with each other in a complex fashion due to the carriers’Coulomb repulsion, can be transformed to a system of new particles. These new particles, which have the same density as the original electrons, seemingly incorporate part of the external magnetic field, have a mass differentfrom the electron mass, and are moving in the apparent absence of their companions. From this vantage point, the FQHE becomes the IQHE of such bizarre new particles. They are termed composite fermions (cFs). Their mass [21-251 is a measure of the Coulomb interaction between electrons and they carry with them an even number of fictitious magnetic flux quanta. Their orbits are quantized into Landau levels by an effective magnetic field and, according to the most recent observations,each of these levels split into two Zeeman levels: CF spin-up and CF spin-down [26]. At specific applied magnetic fields the ChernSimons flux of the CFs exactly compensate the external magnetic flux [20] and CFs move in an apparently zero effectiue magnetic field despite the presence of an extremely high, true external magnetic field. A degenerate Fermi liquid of CFs ensues. Its Fermi wavevector is related in a simple manner to the Fermi wavevector for the electrons of the same system at zero magnetic field. The transport and scattering behavior of CFs mimics the transport of regular electrons at B = 0 [27]. Most surprisingly yet, at such fields CFs follow semiclassical ballistic trajectories which can be detected in a billiard-type experiment [28-311; all of this in the presence of an external magnetic field of many tesla.

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The theory of CFs has been reviewed in Chapter 7 by J. K. Jain and Chapter 6 by B. I. Halperin. In this chapter we review the experimental evidence for this remarkable state of matter. 10.2. BACKGROUND

The fractional quantum Hall effect (FQHE) is widely known as an electron condensation phenomenon which occurs at low temperatures (7") and in the presence of high magnetic fields (B) in two-dimensional (2D)electron systems of very low disorder. When a Landau level of degeneracy d = eB/h is filled to a rational fraction v = p / q , the electronic transport coefficients of the specimen under study develop features not unlike those observed in the integral quantum Hall effect [4] (IQHE),seen at integer filling v = i: The diagonal resistance, R,,(also called magnetoresistance,R), shows deep minima approachingzero resistance and the Hall resistance, R,, (sometimes R,,) forms plateaus having resistance values of R , = h/ve2(see Fig. 10.1).Disregarding spin effects in higher Landau levels [111, only fractions with odd-denominator q have been observed, the strongest of which occurs at v = 1/3.

:iaLl 1

10 20 MAGNETIC FIELD ( l a )

Figure 10.1. Overview of the low-temperature transport features of IQHE and FQHE The Hall resistance (RH) is quantized to R , = h/ve2,where v is either an integer (IQHE) or a rational fraction (FQHE). Concomitantly, the magnetoresistance R exhibits a minimum which in many cases converges toward R +O.

388


The transport features and their temperature dependence are a clear indication for the existence of energy gaps at any such filling factor v. Different from the IQHE, where the gaps are of single-particle origin and coincide with the gaps between Landau and spin levels, the gaps in the FQHE are of many-particle origin, resulting from the gain in Coulomb energy obtained from the correlated motion of the electrons. At v = 113 and, in fact, for all v = l/q, Laughlin [5] has provided us with a remarkably succinct many-body wavefunction. Experimentally, features at many more filling factors have been observed, ranging to fractions as high as v = 9/19. The most prominent of such sequences appears symmetricallyaround v = 112 at filling factors [111 v = p/(2p rt 1). These so-called higher-order fractions [7,8] have been understood as developing from the primary fractions at v = l/q by “spawning”:As the filling factor deviates from one of the primary fractions, quasiparticles that carry an exact fractional charge e/q are created above the energy gap. When these quasiparticles have reached appropriate densities, they themselves can condense into liquids ofquasiparticles in complete analogy to the process by which electrons of charge e condensed into the primary fractions at v = l/q. In this fashion, a hierarchical succession from v = 113 +215 --+ 311 +419. . . (equivalently, v = 213 + 315 +417.. .) evolves which, taking all possibilities of “spawning” into account, eventually covers all odddenominator fractions. The hierarchical construction [7,8] of FQHE states at a general rational filling factor v = p / q has not been as satisfactory as the description of the primary

0

5

10 MAGNETIC

15

FIELD [TI

20

-

25

Figure 10.2. Magnetoresistance of a 2D electron system with mobility 9.7 x 106cm2/V.s at two relatively high temperatures. The strong minimum at 14T(fillingfactor v = 1/2) shows a roughly linear T dependence (inset: normalized resistance at v = 1/2 versus T) quite distinct from the exponential T dependence of the FQHE features (e.g., at v = 1/3). (From Ref. [35].)

BACKGROUND

3.5x 10-4 I

I

I

I

I

I

389

I

magneticfield (kG)

Figure 10.3. Velocity shift (Au/u) and transmitted amplitude for 3.4-GHz surface acoustic waves at 120mK versus magnetic field. The dotted lines represent Au/v and amplitude calculated from the dc conductivity (offset for clarity). While Av/u and amplitude agree qualitativelywith the calculation over most of the field range (not shown),there are clear discrepancies at v = 1/2 and v = 1/4. (From Ref. [37].)

states at v = l/q. For one, the primary states and the derived states are placed on an unequal footing, although experimentally they behave equivalently. Furthermore, the reason for the predominant appearance of certain FQHE sequences [such as v = p / ( 2 p f l)] does not find an intuitive interpretation within this model. Finally, approximate wavefunctions [32-341 for higher-order FQHE states turn out to be much more complex than those at v = l/q. At the same time the states at even-denominator fractional filling remained totally enigmatic. (We exclude states such as v = 5/2, which are assumed to be true FQHE states caused by spin effects [3].) An early experimental paper by Jiang et al. [35] clearly exposed our lack of understanding of the electronic state at v = 1/2. In a very high quality sample, a deep minimum appeared in the diagonal resistivity, R,,, at v = 1/2 whose temperature dependence was quite distinct (see Fig. 10.2 and also Sajoto et al. [36]). Different from the exponential Tdependence of the FQHE states, the state at v = 1/2 showed a roughly linear T dependence converging toward a nonzero R,, value as T+O. The features appeared not to be explainable in terms of an

390


independent electron model effect, and there was speculationas to the existenceof a novel electron-electron correlation phenomenon at half-filling [35]. Experiments on surface-acoustic-wave propagation by Willett et al. [37] uncovered anomalies in the attenuation and velocity shifts at v = 1/2 that also could not be interpreted as a traditional FQHE state (Fig. 10.3).The two apparently unrelated areas of an unsatisfactory state of affairs in the FQHE, the lack of a simple description of the higher-order FQHE states, and the absence of any understanding of the state at v = 1/2 ultimately turned out to be intimately connected. The interpretation of the FQHE in terms of CFs gives us expert guidance for solving both enigmas. 10.3. COMPOSITE FERMIONS

There have been many theoretical efforts [12-171 over the years exploiting the transmutability of the statistics in 2D in order to develop a more general picture of condensation phenomena in such systems. In pursuit of such goals, Jain [181 has shed new light on the higher-order fractions.He proposed a generalization of Laughlin’s wavefunction which formally invokes the wavefunction of a higher Landau level. In this way, the states at v = p/(2p + l), for example, are being mapped onto electron Landau levels at v = p (spin neglected),and FQHE states at v = 1/3, 2/5, 3/5,. . . resemble electron states at v = 1,2,3,.. . . Different from regular Landau levels, however, the relevant particles are not conventional electrons but strange entities termed composite fermions (CFs). They consist of electrons to which two magnetic flux quanta have been attached by virtue of the electron-electron interaction. The new wavefunctions compare very favorably with numerical few-particle calculations. In this picture the FQHE at v = p/(2p & 1) around v = 1/2 becomes the IQHE of CFs. Most of the many-particle interactions have been incorporated into the new particle by the process of fictitious flux quanta attachment. The resulting CFs can be treated as independent fermions,forming Landau levels of CFs which correspond to the FQHE states around v = 1/2. This scheme can be applied to all FQHE states by generalizing to the attachment of 2,4,6,. .. flux quanta. Jain’s model provides a rationale for the succession of experimentally observed fractions in 2D electron systems of increasing quality. Flux quanta attachment, once accepted, transforms the FQHE into an IQHE of new particles. This allows us, to a degree, to reuse our traditional pictures of Landau quantization and excitation across such Landau gaps even for the FQHE. Experimentally, the most accessibleFQHE parameter is the associated energy gap. Simple thermal activation energy measurements and subsequent Arrhenius plots determine its value. Jain’s model, although involving the analogy to the IQHE as far as the structure of the wavefunction is concerned, does not provide us with a simple schemefor the value of the associated gap energies.Those can be calculated, in principle [38,39], but is a tedious one-by-one process that does not lend itself to easy generalization and straightforward comparison with experi-

COMPOSITE FERMIONS

391

ments. As to the state at v = 1/2, Jain’s model may give a clue as to its structure. Extrapolating the relatedness between the FQHE state at v = p/(2p f 1) and the IQHE state of CFs at v = p, we find that the FQHE state approaches v = 1/2 and the IQHE state approaches the Fermi-liquid state at B = 0 as p + 00. Is there a connection between v = 1/2 and the state at B = O? Halperin, Lee, and Read [20] (HLR), as well as Kalmeyer and Zhang [19], have investigated the state at v = 1/2. Applying a Chern-Simons gauge tranformation, they were able to transform the system of electrons in high magnetic field at exactly Y = 1/2 into a system of Chern-Simons gauge-transformedfermions at exactly B = 0. A system of degeneratefermions ensues whose particles are moving in the apparent absence of a magnetic field. They have wavevectors k and fill a Fermi sea up to a maximum Fermi wavevector k, = (47~1)’’~. As in Jain’s model for the higher-order FQHE fractions, HLRs model has most of the electronelectron interaction incorporated into the particles. The attachment of two fictitious flux quanta in a singular gauge transformation, which exactly cancels the external magnetic field, is the essential step of the procedure. Although arrived at in a different spirit, we identify here the Chern-Simons gauge-transformed fermions [19,20] and composite fermions [l8] with the same particle and, for brevity, refer to both of them as CFs. As it stands today, the main features of the C F model of the FQHE can be summarized as follows: A system of highly interacting 2D electrons in a high magnetic field can be transformed into another system consisting of particles to which an even number of fictitiousmagnetic flux quanta has been attached. These particles are termed composite fermions. In this way the majority of the electronelectron interaction is incorporated into the CFs, which are now moving in an effective magnetic field Be, which equals the external magnetic field reduced by the mean field of the fictitious flux attached to the CFs. At exactly even fractional filling factor such as v = 1/2 (but in principle also at v = 3/8,1/4,5/10, etc.), the external magnetic field is exactly compensated by the attachment of an even number of fictitious flux quanta, and the resulting system is a degenerate Fermi liquid of CFs [19,20,40,41] with well-defined Fermi wavevector k, = ( 4 ~ n ) ’ ’As ~ . the magnetic field deviates from v = 1/2, the CFs experience the effective field Bcff= B and are quantized into CF-Landau levels. Around v = 1/2, the fractional filling factor v = p/(2p k 1) for electrons corresponds to integer filling factor i = + p for CFs, and the FQHE at these positions is equivalent to the IQHE of CFs. The gaps of the FQHE become the gaps between Landau levels of CFs. They reflect the cyclotron splitting ha, = heB,,/m:, of a particle of mass rn& attributed to the C F and having purely electron-electron interaction as its origin. To first order, around each evendenominator fraction such as v = 1/2, the mzF is approximately constant. The equivalence of the FQHE gaps at, for example, v = 1/3 and v = 2/3, requires a dependence of m?,. This should become apparent for large Bcff.As v + 1/2, logarithmic divergencies in the CF mass are theoretically expected [20]. But overall, around v = 1/2 and other even-denominator fractions, a simple Landau level picture, similar to the one around B = 0 for electrons, albeit caused by CFs,

fi

392


should emerge. At exactly even-denominator filling some of the traditional Fermi-liquid phenomena may be observable.

10.4. ACTIVATION ENERGIES IN THE FQHE Ever since the discovery of the FQHE, activation energy measurements have been the prime tool to determine its gap energies. In case of the most prominent of states at v = 1/3, a detailed comparison with theoretical calculations could be performed yielding remarkably good agreement once finite thickness effects of the 2D electron system and Landau level mixing was taken into account [42]. The remaining discrepanciesof 10% are usually attributed to disorder, which remains difficult to assess experimentally as well as theoretically. Gaps from higher-order fractions such as v = 2/5, 3/5, 2/7 were much more difficult to determine. Their smaller size, the resulting smaller dynamic range in an experiment, the relatively larger impact of disorder, and the much less satisfactory theoretical situation for calculating their energy gaps-all these were reasons for reduced experimental and numerical activity in this area. Only recently has the sample quality reached a level that allows activation energy measurements to be performed on several representative fractions of a given sequence. Du et al. [21] have performed such measurements on a modulation- doped GaAs/AlGaAsheterostructure of density n = 1.1 x 10" cmand mobility p = 6.8 x lo6cm2/v.s. Figure 10.4 shows an overview of the FQHE in the lowest Landau level at T = 40 mK. From the slope of log R,, versus 1/T, one can deduce the energy gap A" for the fraction at filling factor v. It is most illuminating to plot the so-determined gap energies A" versus the applied magnetic field (Fig. 10.5). Plotted in this fashion, a striking linearity between gap energies and magnetic field becomes apparent, reminiscent of the opening of the cyclotron energy gap for electrons around B=O. One can characterize this finding phenomenologically in very simple terms once one establishesBlI2at v = 1/2 as a new origin for B and adopts its deviation Befffrom this value as the effective magnetic field. Just as in a simple fermion system of density n, the minima in R,, due to the primary sequences of the FQHE occur at Beff= BJ2p 1)- Bl12= hn/e [(Zp i-l)/p - 21 = & hn/ep. Since the gap energies are roughly linear in Be,,, they can be characterized empirically by an effective mass mzF via hw, = heB,,,/m,*,. The values for mzF (in units of the free electron mass, me) are indicated in Fig. 10.5. They are about one order of magnitude bigger than the band electron mass mb N 0.07me of GaAs. The mass depends on carrier density. Measurements on a specimen with twice the density of the one shown in Fig. 10.5 results in mzF = 0.75m0 and m& = 0.92m0 for the v > 1/2 and v < 1/2 slopes, respectively. These values are bigger by almost exactly a factor than the masses shown in Fig. 10.5. This is a simple consequence of the dependence of the size of the primary gaps at v = 1/3 and v = 2/3:

-

>f i

ACTIVATION ENERGIES IN THE FQHE

10 r

I

s5

I

I

I

I

6-

Q

I

-25

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#

I

40 mK

8-

Y

I

I

393

4

-49

I

-1 3

$ 4 -

3 a 2-

0-

I

8

10 Magnetic Field 6 (T)

I 14

Figure 10.4. FQHE states in the vicinity of v = 1/2 in a sample with n = 1.1 x 10” cm-’. Landau level filling factors are indicated. (From Ref. [21].)

&,

Hence mT13cc cc and equivalently for v = 2/3. ( E is the dielectric constant of GaAs, C , = C2/3 is a prefactor determined from numerical calculais the magnetic length.) In fact, HLR [20] assert that the tions, and I , = CF mass has an overall @ dependence. This leads to a “bowing” in the relationship between energy and Bertin Fig. 10.5. Within the uncertainty of the data a @-dependent mass (dashed line) fits equally well. In this case the mass near v = 1/2 becomes unique, m& = 0.57me. Activation energy measurements on the sequence of higher-order FQHE states around v = 1/2 have also been performed on 2D hole systems by Manoharan et al. [24] (Fig. 10.6).The conclusions are similar to those from 2D electron systems. These authors are certain they have observed the slight “bowing” in their data and hence use a modified abscissa that compensates for this effect. Their mass mT12= 1.4me at v = 1/2 far exceeds C F masses in 2D electron systems at comparable densities. In a simple picture C F masses are expected only to depend on n and should be independent of carrier type. The probable origin of the unexpectedly high mass is mixing between Landau levels of the rather heavy holes, which considerably complicates any kind of theoretical treatment. Under such complex circumstances it is astonishing that the @ dependence of the mass survives.

&

394


Filling Factor v

9.25

4.63

13.88

Magnetic Field B (T) k

l

-4

I

I

I

I

I

2 Effective Magnetic Field Bert(T) -2

0

I

I

,

4

Figure 10.5. Gap energies for various filling factors in the vicinity of v = 1/2 in sample of Fig. 10.4 plotted versus magnetic field. Solid straight lines are a linear approximation of the data. The associated effective masses in units of me are indicated.The dashed curves represent a fit to the data which includes a slight bowing, expected from theory due to the

requirement for electron-hole symmetry. In this case the mass at v = 1/2 is unique,

m* = 0.57~1, (From Ref. [21].)

Independent of whether one approximates the data in Figs. 10.5 and 10.6 by dependence, in both cases a constant masses or whether one includes a negative intercept is obtained as Beff*O. In analogy to the case of electrons around B = 0, the negative extrapolated value can be identified as the broadening r of the C F levelscaused by disorder. Such an identification remains theoretically controversial but shows good agreement with C F scattering times (r A/r) deduced from the resistivity at v = 1/2 and from the Dingle temperature [21,24]. Of course, the geometrical extrapolation of the data in Figs. 10.5 and 10.6 implies that r is independent of Beff.This may not be correct. Being aware that finite thickness of the 2D carrier systems, Landau level mixing (even for electrons), and field dependence of the disorder effects all contribute to the data in Figs. 10.5 and 10.6, one needs to be careful with detailed compari-

fi

-

SHUBNIKOV-DE HAAS EFFECT OF COMPOSITE FERMIONS

395

E

d

-80

-40

0

( ~ ~ &(2PQ+ 1) / [K)

40

80

Figure 10.6. Gap energies for various filling factors in the vicinity of v = 1/2 in a 2D hole sample. The horizontal axis is roughly proportionalto Bcffbut includes a slight distortion that absdrbs the expected dependence of GF. In this regard, the straight lines in this figure are equivalent to the dashed (curved) lines in Fig. 10.5. (From Ref. 1241.)

fi

sons between experiment and theory, particularly since some finer detail of the CF theory at this stage relies on highly educated guesses. Despite such caveats, the roughly linear field dependence of the FQHE gaps around v = 1/2 Landau level filling factor finds a very elegant interpretation in terms of the CF model. 10.5. SHUBNIKOV-DE HAAS EFFECT OF COMPOSITE FERMIONS

It its most simplistic interpretation, the FQHE states around v = 1/2 can be regarded as the consequence of Landau quantization of CFs exposed to an effectivemagnetic field Beff= B - B1,2.The analogy of this scenario with the state of regular electrons around B = 0 lets one speculate as to whether tools that have been developed to derive conventional electron parameters from their magnetic field behavior can also be applied to CFs. In the realm of transport experiments, the Shubnikov-de Haas [43-451 (SdH) effect, which has been such a powerful tool to unravel regular electron and hole masses and scattering times, comes to mind and one wonders whether this technique may lead to sensible results when applied to the FQHE states around v = 1/2. The conceptual background for an interpretation of data such as in Fig. 10.4 in terms of a SdH effect [43,44] is considerably different from the activation energy analysis presented in the preceding section. The model underlying activation energy measurements is a quantum liquid picture in which thermally activated fractionally charged quasiparticles are generated across an energy gap. Similarly, the experimental procedure follows the traditional Arrhenius log R,, versus 1/T data reduction of the FQHE, where R , is the T-dependent value of the resistance at one of the minima. In contrast,

396


Filling Factor, v

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Figure 10.7. Temperature dependence of the magnetoresistance, R (bottom), and masses, m*, deduced from the Shubnikov-de Haas formalism (in units of bare electron mass me = m,,,top) as a function of magnetic field, B, and effective magnetic field, Beff,around v = 1/2 Landau level filling factor. The mass appears to increase as v + 1/2, although this increase is still within the error bars of the measurement. (From Ref. [22].)

by employingthe standard SdH formalism to the oscillationsaround v = 1/2, one implicitly invokes a single-particle picture of a Fermi sea with a given Fermi energy E, and an initially smooth density of states (DOS). The application of a magnetic field introduces periodic oscillations in the DOS whose amplitude at E , is probed by magnetotransport. A successful interpretation of the data, in terms of SdH oscillation, would provide further support for the CF approach. Figure 10.7 shows magnetotransport data [22] for four different temperatures around v = 1/2 of the same sample as used for Fig. 10.5. The oscillations display the familiar pattern seen from regular electrons around B = 0 Apart from minor distortions, the data are symmetricaround Bcff= 0, the spacing of the oscillations is proportional to l/B,,,, the oscillations have a finite onset at IBcfrl 0.7 T, and their amplitude shrinks with increasing T The temperature dependence of the amplitude follows extraordinarily well the traditional sinh dependence of the

-


397

SdH formalism [MI, from which carrier masses and scattering times can be deduced. The carrier masses and their Beff dependence are shown in the top part of Fig. 10.7. To within the accuracy of the data, the mass is roughly constant with a value -0.7me. This is similar, although somewhat higher, than the masses deduced from the activation energies of Fig. 10.6.The scattering time, extracted from the standard Dingle plot [45], amounts to 7 4.4 x 10- l 2 s, equivalent to r = h/7 1.7 K and hence is in magnitude similar to that of r extrapolated in Fig. 10.5. These findings demonstrate the internal consistency of the SdH formalism when applied to FQHE data and the mutual consistency between SdH and activation data. It is remarkable that the formalism of the SdH effect, when applied to this correlated electron system, seems to lead to meaningful results. There appears to be a slight increase of the effective mass, although comparable to the error bars, in the data of Fig. 10.7 as Beff+O. This mass enhancement has been observed more clearly in samples of higher density [25] (Fig. 10.8)and in a 2D hole specimen [24] (Fig. 10.9). The theory of HLR actually predicts such a mass enhancement in the vicinity of v = 1/2, which for the case of Coulomb interactions, takes on the form of a logarithmic correction [20,46-481. The divergence in the data appears to be much more rapid than logarithmic and is best described by a Be;: dependence [25] or by an exponential dependence [24]. This discrepancy between experiment and theory remains unexplained. On the one hand, it may be due to an incomplete theoretical model. On the other hand, it may be caused by an uncritical application of the SdH formalism to CFs around v = 1/2. Considering the effect of Chern-Simons gauge-field fluctuations on the CF Landau levels, Aronov et al. [49] conclude that the SdH formalism needs to be modified for application to CFs where random magnetic field fluctuations,rather than random potential fluctuations, scatter the carriers. These authors find the magnetic field dependence of the amplitude of the oscillations to be proportional to exp[ - (Z/O,T)~] rather than the traditional exp( - 740,~)behavior. Coleridge et al. [SO] arrive at a similar conclusion based on a Gaussian spread in electron density. Indeed, the former dependence fits the field-dependent amplitude in Fig. 10.7 better than the latter dependence, assuming a constant CF mass. However, if the field-dependent CF mass (top of Fig. 10.7 and top of Fig. 10.8)is taken into account, the simple ( 0 ~ 7 ) - dependence also provides an acceptabledescription of the data [25]. In any case, the temperature dependence of the oscillations, which determines the C F mass, is hardly modified in both of these approaches as compared to the regular SdH formalism [44]. This was also concluded from extensivecomputer modeling [25]. Therefore,the traditional data reduction used to generate the masses in Figs. 10.7 and 10.8 is, after all, expected to hold. It remains to be seen whether the apparent strong divergence ( - Beif or exponential) [21,24] of rn& as v + 1/2 is truly a many-particle phenomenon or whether it is the result of the different character of the potential scattering of electronsversus the magnetic scattering of CFs.

-

-

-

398


Figure 10.8. Effective C F mass as determined from the temperature dependence of the Shubnikov-de Haas oscillarions (top and left),and energy gap(solid square) as determined from activation evergy measurements (bottom and right) on higher-order fractional quantum Hall states in the vicinity of Y = 1/2. The data points (solid diamond) at negative energies represent the broadening factor r drived from mass and energy gap data. The lines indicate the roughly linear dependence of the energy gap, its extrapolation to Be, = 0, and the level of agreement of this extrapolated value of r,with the value derived from the gap and mass data. In this higher-density sample the divergence of m& for v + 1/2 as determined by the SdH effect is clearly visible. (From Ref. [25].)

SdH measurementssimilar to those above were reported by Leadley et al. [23] around v = 1/2 as well as around v = 1/4, where four flux quanta are expected to attach to each electron to form a CF (Fig. 10.10).Their conclusionsregarding the applicabilityof the SdH formalism to the FQHE data are similar to the above but they differ in detail. Surprisingly,in these data the CF's mass does not depend on


399

1.8 1.6

1.o 0.8

Figure 10.9. Effective CF mass around v = 1/2 for a 2D hole system also diverges. The masses are determined by SdH analysis. B* = Beff and p-' is determined from v = p/(2p 1). (From Ref. [24].)

carrier density as one would expect from simple theoretical arguments and as observed in Refs. [21,22,24,25]. In contrast, the mass appears to increase as one moves away from v = 1/2 in both directions. Leadley et al. find a universal CF mass, m&he = 0.51 + 0.074Beffindependent of electron density. These authors also do not observe the divergence of m& as B,,,+O (equivalent to v+ 1/2), although this is probably due to the limited range of high-denominator FQHE states they investigate. Such differenceswill need further attention but should not distract from the remarkable result, common to all the reports, that the FQHE data around v = 1/2 (and v = 1/4) can be sensibly interpreted within a standard SdH formalism. The success of such a conventional analysis and in terms of a simple, noninteracting carrier model further strengthens the case for the description of the FQHE in terms of new particles around even-denominator filling factors. A quantitative comparison between theory and experiment is difficult to pursue. One needs to remind oneself that the theoretical model is based on an ideal, infinitely thin 2D electron system in the absence of disorder and Landau level mixing, whereas the 2D electron gas in GaAs/AlGaAs heterostructures has a nonzero thickness, j n i t e carrier scattering time, and is exposed to a j n i t e magnetic field.

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A measurement of the thermodynamic properties of CFs would be very desirable. In particular, a determination of the specific heat of the CF system at evendenominator filling could provide a reliable measure for the density-of-states mass of the CFs at the Fermi energy [20]. Unfortunately, specific heat measurements on 2D systems are extraordinarily difficult to perform [Sl] due to the dominance of the lattice specificheat and are so far unavailable. However, several thermoelectricpower (TEP)measurements have been performed on 2D electron systems [52-541 and 2D hole systems [SS, 56) which shed further light on the applicability of the CF model. Thermoelectriceffects are caused by the application of a temperature gradient VT tocan, -electron system, which leads to a thermally induced electron current j , , = E V T, where%' is the thermoelectric tensor. In equilibrium1this current is canceled by an electric current Tel,="E, where %' is the conductivity tensor. The resulting relationship %%= '?V Tis expressed as 2 = YVT, with Y= (7)'7 being the thermopower tensor-Measurements usually focus on determining the diagonal component S,, of S , which indicates the thermopower along the temperature gradient. In a rectangular 2D geometry in a cubic material and in the

-

THERMOELECTRIC POWER MEASUREMENTS

401

absence of a magnetic field, S,, = So and S , = 0. In the presence of a magnetic field, S,, (the Nernst-Ettinghausen coefficient) becomes finite. In the context of CFs this coefficient has received comparably minor attention. In general, the TEP of a 2D electron system is the sum of two components: (1) carrier diffusion in the electron system, caused by the temperature gradient and (2) phonon drag, caused by phonon diffusion, which “drags” the electrons along. The contribution from these processes are identified as Sd and S g , respectively, so that S = Sd Sg. At low temperatures Sd dominates, whereas at higher temperatures Sg takes over. There are four separate regimes of S that are of relevance to the interpretation of the data.

+

1. Case S: ( B = 0, difision. Here the Mott formula is employed [ S S ] which leads to Sd, = n2/3 x ( p + 1) x k2T/eE,, where p is the exponent in the energydependent relaxation time of the carriers [ t ( E )cc EP],generally put to p = 1. 2. Case S:, (finite B, difision). For kT

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