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Transport study on two-dimensional electrons with controlled short-range alloy disorder Wanli LI A dissertation presented to the faculty of Princeton University in candidacy for the degree of Doctor of Philosophy Recommended for acceptance by the Physics Department September 2007 c Copyright by Wanli LI, 2007. All rights reserved. Abstract Disorder plays an important role in almost all aspects of solid state physics. However, due to the complexity in the nature of disorder, it is usually hard to experimentally study its effect in a controllable way. We present in this thesis the first systematic study on disorder-related physics in two-dimensional electron systems (2DES). Our samples are modulation doped Alx Ga1−x As − Al0.3 Ga0.7 As heterostructures with a broad range of Al impurity concentration x. We have shown that the alloy potential fluctuation in these systems has a range of atomic dimension and amplitude of 1.13eV. The relative weight of short-range alloy disorder increases in samples with larger x. With the 2DES samples of controlled alloy disorder, we have investigated the quantum Hall plateau-to-plateau transition in the integer quantum Hall regime. We have unambiguously verified that the plateau-to-plateau transition is a universal quantum phase transition, and built up a framework to understand the role that disorder plays in this transition. When the disorder in the system is dominated by short-range alloy potential fluctuations, we have found a perfect power-law temperature scaling dRxy | dB B=Bc ∝ T −κ with a universal exponent κ = 0.42 over two full decades of temperature. The inelastic scattering exponent p is identified to be 2 by an experiment on samples of various sizes. The localization length exponent ν = 2.4 is therefore verified by the experimentally measured values of κ and p. In systems with disorder being dominated by long-range Coulomb potential fluctuations, a semi-classical exponent κ = 0.58 is observed at high temperatures. Below a iii crossover temperature Tc , the universal exponent κ = 0.42 is restored, as the quantum phase coherence length becomes much longer than the Coulomb disorder range. For samples with very high Al concentrations, alloy clustering is likely, and the effective sample size is determined by a hidden length scale related with the cluster size. As a result, the exponent κ=0.58 persists into low temperatures until dRxy | dB B=Bc saturates at 65mK, a relatively high temperature. We have further investigated the physics in the Fractional quantum Hall effect (FQHE) and Wigner crystal regime at high magnetic fields. We have found that alloy disorder does not affect the FQH gaps while it enhances the formation of Wigner crystals. As a result the terminal FQH state in systems with alloy disorder has been shifted to ν = 1 . 3 More excitingly, we have observed the ν = 1 reentrant integer quantum Hall effect, which is a direct manifestation of the particle-hole symmetry in the Wigner crystal phase of the lowest Landau level. In both regimes, we have therefore found that the range of disorder plays critical roles. Our experimental methods with controlled short-range alloy disorder have been proved to be powerful in the investigation of disorder-related physics. iv Acknowledgements As I look back the seven years of my life as a graduate student in Princeton, I feel blessed to have interacted with so many wonderful people. First, I would like to express my deepest gratitude to my thesis advisor Prof. Daniel Tsui. I have benefited so much from his patience as a teacher and from his great insight as a scientist. He introduced me into the world of semiconductor physics, and brought me the opportunity to conduct exciting scientific research in the frontiers of physics. With his guidance, I have not only learned the scientific methods, but also progressed in English writing. What Dan taught me and the wisdom he shared with me in the completion of this thesis will definitely continue to benefit me in my future career. I thank Prof. Shivaji Sondhi for being the reader of my thesis. Discussions with Shivaji have always been enlightening, and his leading expertise is critical for me to have this thesis prepared. I thank Prof. Nai Phuan Ong and Prof. Chiara Nappi for being in my thesis defense committee. Prof. Ong has been taking care of me ever since I came to Princeton, and I have truly benefitted from all his suggestions from English speaking to academic research. Prof. Nappi, as the director for graduate study of Physics Department, has always been helping me and overseeing my progress. I am also deeply grateful to my first-year advisor, Prof. Bob Austin. Bob introduced me the nano-biophysics, as well as the American culture. It was fun working with Bob and I published my first paper in Princeton under his guidance. Bob keeps on being a great support of my career, and I truly appreciate that. v I am also very fortunate to have a great friend Dr. Gabor Csathy who was my mentor when I joined the Tsui group. Actually almost all my knowledge of cryogenics was learned from Gabor. He helped me from the very first stage to engage my thesis projects, and kept on discussing with me when I grew more experienced. A big part of my thesis work was done at ultra-low temperatures in the milliKelvin lab of the University of Florida at Gainesville, FL. This would not be possible without the Cordial help of Dr. Jian-Sheng Xia. Working with Jian-Sheng and living in Gainesville was such a unique experience that I will never forget. Some experiments in Gainesville were carried out with Dr. Carlos Vicente, and I thank Carlos for the help and the fun he brought to our work. I thank Dr. Loren Pfeiffer and Mr. Ken West for providing us their great samples, without which it would be unthinkable to conduct my thesis work. The research works presented in this thesis have been discussed with many faculty members of Princeton. I thank Prof. David Huse, Prof. Duncan Haldane, Prof. Paul Chaikin of the Physics Department of Princeton, Prof. Ravin Bhatt, Prof. Steve Lyon of the EE Department of Princeton for their suggestions. During my thesis research, I have truly benefited from all the members in our group. I thank our former post-doctors Dr. Wei Pan, Dr. Leonid Rokhinson, Dr. Hwayong Noh, Dr. Amlan Majumdar, Dr. Jian Huang, Dr. Tao Zhou and Dr. Mike Hilke for their great helps and suggestions in my research. Dr. Jinjin Li, a great officemate, as well as a close personal friend, helped me not only in work but also in life. I also thank the former students in our group: Ravi Pillarisetty, Yong Chen, Rob Ellis, Zhihai Wang and Keji Lai, for the mutual supports in pursuing our degrees. I have special thanks to Ravi for the popular music he introduced to me - they have vi accompanied me for many late nights at work. Our current post-doctor Dr. Dwight Luhman and current students Han Zhu and Tzu-Ming Lu, have all proved to be great coworkers, and I thank them for the help they offered during my last years in Princeton. I have also benefited a lot from people in Prof. Austin’s group. I thank Dr. Jonas Tegenfeldt for his selfishless help. I also thank my friend Yuexing Zhang for his warm encouragements. In the last two years I have traveled a few times to the National high magnetic field lab (NHMFL) at Tallahassee, FL. I here thank the NHMFL staff Bruce Brandt, Eric Palm, Tim Murphy and Glover Jones for their assistance in my experiments. I also thank Dr. Lloyd Engel, Dr. Zhigang Jiang and Dr. Murthy Ganapathy for the great conversations and suggestions in my every trip to NHMFL. Many personal friends from physics department and EE department have proved to be very resourceful. I thank Chenggang Zhou, Weida Wu and Xin Wan for their great help and suggestions, and I owe them so much for their encouragements. Finally, I thank my family for their constant love and support, which is the ultimate force on me to complete this thesis. My parents Jianxiang Li and Zhihui Gao, have always been proud of me, and I am proud of them. My dearest wife Qing Wang, is the greatest companion I can have. She stands by my side and lights my darkest hour. To them I dedicate this thesis. vii Publications Resulting from this Thesis 1. “Direct observation of alloy scattering of two-dimensional electrons in Alx Ga1−x As,” Wanli Li, G. A. Csathy, D.C. Tsui, L. N. Pfeiffer, K. W. West, Appl. Phys. Lett. 83, 2832 (2003). 2. “Alloy scattering and scaling in the integer quantum Hall plateau-to-plateau transitions,” Wanli Li, G. A. Csathy, D.C. Tsui, L. N. Pfeiffer, K. W. West, Inter. J. of Modern Phys. B 18, 3569 (2004). 3. “Scaling and universality of integer quantum Hall plateau-to-plateau transitions,” Wanli Li, G. A. Csathy, D.C. Tsui, L. N. Pfeiffer, K. W. West, Phys. Rev. Lett. 94, 206807 (2005). 4. “Plateau-to-plateau transition in random-alloy scattering dominated 2DEGs of different densities,” Wanli Li, C. L. Vicente, J. S. Xia, D.C. Tsui, L. N. Pfeiffer, K. W. West, Physica E 34, 217 (2006). 5. “Quantum Hall plateau-to-plateau transition at ultra-low temperatures,” Wanli Li, C. L. Vicente, J. S. Xia, D.C. Tsui, L. N. Pfeiffer, K. W. West, submitted to Phys. Rev. Lett. 6. “Observation of particle-hole symmetry in high magnetic field solid phases of two-dimensional electrons with short-range disorder,” Wanli Li, D. L. Luhman, D.C. Tsui, L. N. Pfeiffer, K. W. West, Manuscript in preparation. viii Thesis Outline This thesis begins by providing the reader with relevant background materials, which is presented in Chapter 1. We begin by reviewing the science of semiconductor heterostructures and point out the novelty of our samples. We emphasize the importance of preparing samples of Alx Ga1−x As − Al0.3 Ga0.7 As heterostructures, which allow us to systematically study disorder-related physics in two-dimensional electron systems (2DES). We then review the specific physics problems that we will investigate in later chapters of the thesis using these novel samples. In Chapter 2, we characterize all the 2DES samples, and look into the nature of the alloy disorder itself. We have measured the electron density, mobility and scattering rate in each sample. Since alloy impurities are neutral, the range of alloy potential is as short as the atomic dimension. The amplitude of the alloy potential fluctuation is found out in this chapter to be as large as 1.13eV. With good knowledge about our samples, we investigate two long-standing problems in the quantum Hall regime in Chapter 3 and Chapter 4. Chapter 3 concentrates on the integer quantum Hall regime, and we have studied the integer quantum Hall plateau-to-plateau transitions. We have unambiguously verified in this chapter that the plateau-to-plateau transition is a universal quantum phase transition, and built up a framework to understand the role that disorder plays in this transition. Prospectives to future works is proposed in the end of this chapter. Chapter 4 concentrates on the regime of fractional quantum Hall effect and Wigner crystals. We investigate the effect of alloy disorder on the competition between the Fractional quantum Hall liquids and the Wigner crystals. We have observed the ix particle-hole symmetry in the Wigner crystal phase, and conclude that the shortrange disorder enhances the formation of Wigner crystals. For reference, detailed description of the experimental techniques and some theoretical results are provided in the appendices. x Contents Abstract iii Acknowledgements v Publications Resulting from this Thesis viii Thesis Outline ix List of Tables xv List of Figures xvi 1 Introduction 1 1.1 2DES in a semiconductor heterostructure . . . . . . . . . . . . . . . . 1 1.2 Fundamental characteristics of 2DES . . . . . . . . . . . . . . . . . . 6 1.3 Disorder in 2DES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.1 Coulomb disorder . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.2 Non-Coulomb disorder . . . . . . . . . . . . . . . . . . . . . . 11 1.3.3 Disorder and electron localization . . . . . . . . . . . . . . . . 14 1.3.4 Important length scales in 2DES . . . . . . . . . . . . . . . . 15 The Quantum Hall physics . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 xi 1.4.1 1.4.2 The integer quantum Hall effect and the quantum Hall plateauto-plateau transitions . . . . . . . . . . . . . . . . . . . . . . . 17 Fractional quantum Hall effect and Wigner Crystals . . . . . . 24 2 Fundamental characteristics of 2DES with short-range alloy disorder at zero and low magnetic fields 29 2.1 Introducing alloy disorder into 2DES . . . . . . . . . . . . . . . . . . 30 2.2 Characterization of samples - density, mobility and scattering rate . . 33 2.3 Alloy scattering rate is temperature independent . . . . . . . . . . . . 34 2.4 Amplitude of the alloy potential fluctuation . . . . . . . . . . . . . . 38 2.5 Possible alloy clustering in samples with high alloy concentrations . . 41 2.6 Lifetime of 2D electrons with alloy disorder . . . . . . . . . . . . . . . 43 2.6.1 Quantum lifetime and transport lifetime . . . . . . . . . . . . 44 2.6.2 Quantum lifetime of 2D electrons with alloy disorder – mea- 2.7 sured by SdH oscillations . . . . . . . . . . . . . . . . . . . . . 46 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3 Investigation on the quantum Hall plateau-to-plateau transition 50 3.1 The universality is called into question . . . . . . . . . . . . . . . . . 50 3.2 Range of disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 Samples and experimental techniques . . . . . . . . . . . . . . . . . . 53 3.4 Critical exponent κ depends on the nature of disorder . . . . . . . . . 54 3.4.1 Three disorder regimes, one optimal window . . . . . . . . . . 56 3.4.2 Measurement on samples with different densities . . . . . . . . 61 3.4.3 On the non-universal exponents . . . . . . . . . . . . . . . . . 65 xii 3.5 Power-law scaling over two full decades of temperature . . . . . . . . 66 3.6 Termination of the power-law scaling at ultra-low temperatures . . . 67 3.7 Experiment on samples of various sizes . . . . . . . . . . . . . . . . . 70 3.7.1 Smaller samples, higher saturation temperatures – identification of the quantum phase coherence length LΦ and inelastic 3.8 exponent p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.7.2 Discussion on the quantum phase coherence length . . . . . . 74 3.7.3 More complicated scaling . . . . . . . . . . . . . . . . . . . . . 76 Outside of the optimal window at ultra-low temperatures . . . . . . . 77 3.8.1 Sample with x = 0 – crossover effect in temperature scaling and the range of Coulomb disorder . . . . . . . . . . . . . . . . . . 3.8.2 77 Evolution of temperature scaling from Regime I to the optimal window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.8.3 Crossover exponents . . . . . . . . . . . . . . . . . . . . . . . 81 3.8.4 Sample with x = 4.1% – A hidden length scale in clustered alloy systems of Regime III . . . . . . . . . . . . . . . . . . . 83 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.10 Perspective of future works . . . . . . . . . . . . . . . . . . . . . . . . 86 3.10.1 Rxx measurement . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.10.2 Correlated alloy disorder . . . . . . . . . . . . . . . . . . . . . 87 4 New physics brought out by alloy disorder in high magnetic fields 88 3.9 4.1 Fractional quantum Hall gaps in 2DES with alloy disorder . . . . . . 89 4.2 Particle-hole symmetry in the Wigner crystal phase . . . . . . . . . . 93 4.2.1 Reentrant insulator between ν = xiii 1 3 and 2 5 . . . . . . . . . . . . 94 4.2.2 Reentrant integer quantum Hall effect (RIQHE) between ν = and 4.2.3 3 5 2 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Particle-hole symmetry . . . . . . . . . . . . . . . . . . . . . . 101 4.3 Alloy disorder and the reentrant insulators . . . . . . . . . . . . . . . 102 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 A The van der Pauw method 105 B Calculation of alloy scattering rate in 2DES 108 B.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 B.2 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 B.3 The potential fluctuation U in our samples . . . . . . . . . . . . . . . 110 C Removal of parallel conductance 111 D Sample preparation recipes 116 E Miscellaneous experimental projects during PhD research 119 E.1 The quantum Hall insulator. . . . . . . . . . . . . . . . . . . . . . . . 119 E.2 A quantum Hall spin filter. . . . . . . . . . . . . . . . . . . . . . . . . 121 E.3 Anomalous Hall effect in a Si-doped quantum well. . . . . . . . . . . 127 Bibliography 132 xiv List of Tables 2.1 Fundamental characteristics of the first series of samples . . . . . . . 33 2.2 Fundamental characteristics of the second series of samples . . . . . . 34 2.3 Fundamental characteristics of the third series of samples after illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4 Lifetimes and scattering rates for the first four samples of the first series 46 3.1 Sample properties and measurement results. The Al concentration x, the electron density ne and mobility µ, the ratio θ between the alloy and the background scattering rates at 0.3K, and the scaling exponent κ of four plateau-to-plateau transitions. There are two wafers with x = 0.85%, and three pieces (A, B, C) are cut from the first wafer. 4.1 Characteristics of the first series of samples after illumination xv . 56 . . . . 90 List of Figures 1.1 A simplified illustration on the mechanism of a semiconductor heterostructure. (a)Two materials A and B are grown together. (b) Band structures of A and B. χA and χB are the electron affinities. (c) With electrons migrate from A to B, a triangular potential well is formed. 1.2 A simplified illustration of the AlGaAs-GaAs-AlGaAs quantum well structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 3 5 Computer simulation of the Coulomb potential in a typical GaAsAlGaAs heterostructure. The bright areas represent high potentials and the dark areas represent lower potentials. The potential difference between the peaks and valleys is 0.4meV. 1.4 . . . . . . . . . . . . . . . 10 Cartoon illustration of the alloy potential fluctuation for a binary alloy Ax B1−x . The system is characterized by the alloy concentration x and the potential difference between the two pure components (EA − EB ). 1.5 13 The integer quantum Hall effect. (a) Plot of a well developed integer quantum Hall effect in a typical 2DES. (b) The Landau levels in 2DES subjected to various magnetic fields. xvi . . . . . . . . . . . . . . . . . . 19 1.6 Temperature dependence of Integer quantum Hall effect in a typical 2DES. It appears that the plateau-to-plateau transitions are sharper at lower temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 22 Fractional quantum Hall effect in a 2DES of very high mobility. Various series of FQH states are observed. This figure is taken from (W. Pan et al, Phys. Rev. Lett. 88, 176802 (2002)). 1.8 . . . . . . . . . . . . . . 24 Magneto-transport around the terminal FQH state. The terminal FQH state has filling factor ν=1/5. A reentrant insulator is observed at ν=0.21. This figure is taken from (H. W. Jiang et al, Phys. Rev. Lett. 65, 633 (1990)). 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Schematic view of the sample structure. Layers I and II are both Al0.3 Ga0.7 As (for the first and second series of samples) or Al0.1 Ga0.9 As (for the third series of samples), and layer III is Alx Ga1−x As. There are δ-dopants between layers I and II, and electrons accumulate in layer III close to the II-III interface. The thicknesses of layers I,II, and III are 80 nm, 100 nm, and 1 µm, respectively. Between the GaAs substrate and layer III, there are 400 periods of superlattice of 3 nm of GaAs and 10 nm of Al0.3 Ga0.7 As. 2.2 . . . . . . . . . . . . . . . . . . . . . . . 32 The T -dependence of the scattering rate for the first four samples of the first series. In the 0.3-4.2 K temperature range all four curves are parallel to each other. 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 36 The T -dependence of the scattering rate for the four samples of the third series. Again, all four curves are approximately parallel to each other. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii 37 2.4 The dependence of τ −1 on x(1 − x) at 0.3 K for the first four samples of the first series. The dotted line is a linear fit to the data. 2.5 . . . . . 40 The dependence of τ −1 on x(1 − x) at 0.3 K for all the samples of the first series. The dotted line is a linear fit to the data from the first four samples. Large deviations from this line are observed for the last two samples. Even after a wave function form correction (the stars), the deviations are still large. . . . . . . . . . . . . . . . . . . . . . . . . 42 2.6 Electron scattering by an impurity in a solid. . . . . . . . . . . . . . 44 2.7 Shubnikov-de Haas oscillation of a sample at various temperatures. . 47 2.8 Extraction of the quantum lifetime of the electrons by Dingle formula. 48 3.1 (a), (b)The longitudinal resistance Rxx and Hall resistance Rxy at different temperatures for the sample with x = 0.85%. In this plot, ν denotes the Landau level filling factors. (c), (d)The transition between the plateaus of ν=4 and ν=3. A critical magnetic field Bc =1.40T is 3.2 observed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dRxy | dB B=Bc vs T for the 4-3 transition in various samples. From down 55 to up, x = 0, 0.85%, 4.1% respectively. Data of different x has been shifted vertically in log-log scale for a clear comparison. Scaling exponents κ are obtained from the linear fits. . . . . . . . . . . . . . . . . 3.3 57 Dependence of the exponent κ on the Al concentration x for the 4-3 transition. In the second regime, the alloy scattering rate τa−1 is from 2.5 times to 6.5 times of the background long-range scattering rate τb−1 , and thus scattering is dominated by alloy disorder. In this regime the exponent κ is 0.42. . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii 59 3.4 Temperature scaling down to 30mK of the 4-3 transition for the sample with x = 0.85%. Data taken in the dilution fridge (up-triangles) and that from the 3 He system (circles) fall on the same straight line in the log-log plot. The slope of both curves in (a) and (b) give the critical exponent κ=0.42 with a high precision. 3.5 . . . . . . . . . . . . . . . . 60 The Hall resistance of the sample with x=0.8% and n=6.8×1010 /cm2 around the 4-3 transition. Within the shown temperature range, a critical exponent κ=0.42 is obtained. . . . . . . . . . . . . . . . . . . 3.6 63 Dependence of the exponent κ on the Al concentration x for the 4-3 transition. The dots represent data from samples of the first two series, and the crosses represent samples from the third series. Data obtained from samples of different densities agrees fairly well with each other. 3.7 Perfect temperature scaling dRxy | dB B=Bc 64 ∝ T −0.42 of the 4-3 transition over two decades of temperature between 1.2K and 12mK. Data from three different experimental cryostats have temperature ranges overlapping with each other and fall on each other at the overlapping temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 The saturation of dRxy | dB B=Bc 68 at low temperatures. The saturation tem- perature Ts =10mK is obtained from the cross point between extrapolations of the higher temperature data (power law (dRxy /dB)|Bc ∝ T −0.42 ) and the lower temperature saturated data (horizontal dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix 69 3.9 dRxy | dB B=Bc of the 4-3 transition with different excitation currents at the base bath temperature 1mK. With excitation current I below 2nA, dRxy | dB B=Bc is constant. Current heating is observed at I=5nA and dRxy | dB B=Bc is reduced substantially from the value of I=2nA. . . . . . 71 3.10 Temperature scaling for samples of various sizes. The dotted straight lines represent the power-law exponent 0.42. Although the data from different samples sample do not fall on each other, the exponent κ = 0.42 is agreed upon by all samples. The power law scaling with κ = 0.42 is terminated at various temperatures. . . . . . . . . . . . . . . . . . 3.11 The sample size dependence of the saturation temperature Ts of 73 dRxy | dB B=Bc . The length-width ratio of all samples is kept to be 4.5:2.5. The value of Ts is inversely proportional to the sample width W within the error. 74 3.12 The temperature scaling of the sample with x=0 over three decades of temperature. Three different temperature scaling behaviors have been observed: dRxy | dB B=Bc saturates in the lowest temperature decade below 15mK; power law scaling with κ=0.58 in the highest temperature decade; power scaling with the universal exponent κ=0.42 in the middle temperature decade. The crossover temperature between the regions with κ=0.58 and κ=0.42 is obtained to be 120mK by extrapolations. xx 78 3.13 Evolution of the crossover effect. The temperature scaling of 4-3 transition in Alx Ga1−x As − Al0.3 Ga0.7 As heterostructures with various x values. Data in the lowest temperature decade has been removed since dRxy | dB B=Bc saturates. (a) x = 0; (b) x = 0.21%; (c) x = 0.85%. Crossover effect between temperature regions κ=0.42 and κ=0.58 has been observed in (a) and (b). Crossover temperature Tc is obtained to be 120mK in (a) and 250mK in (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.14 The zoom-in of the area squared by the dotted lines in 3.13(a). By power-law fitting over a relatively small temperature range, an intermediate exponent κ=0.49 is obtained. . . . . . . . . . . . . . . . . . 82 3.15 Temperature scaling of the 4-3 Transition for 2DES embedded in a Alx Ga1−x As − Al0.3 Ga0.7 As heterostructure with x = 4.1%. A saturation temperature of dRxy | dB B=Bc is observed to be Ts =65mK. The exponent κ=0.58 persists into lower temperatures until Ts is reached. 4.1 Rxx data for the sample with x = 0.85% between filling factors ν =1 and 2. The FQH states, ν = 4.2 5 3 and 43 , are the focus of this plot. . . . 91 Fit of the Rxx data into the exponential formula. Values of the FQH gap are obtained for the FQH states ν = 4.3 84 5 3 and 34 . . . . . . . . . . . 91 Independence of the FQH gap on the alloy concentration x. Both the ν= 5 3 and 4 3 gaps are constants within the experimental uncertainty. xxi 92 4.4 Rxx data of the sample with x = 0.85% over the full range of magnetic field from 0 to 32T. A high resistance peak is observed at 27T (ν=0.37) between the ν = 1 3 and 2 5 FQH states, and is identified to be a reentrant insulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 95 Rxx and Rxy data of the sample with x = 0.85% at 60mK up to 18T. One additional minimum is observed on Rxx at ν=0.63, between the ν= h e2 4.6 2 3 and 3 5 FQH states. At this field, Rxy falls on the quantized value of the ν=1 plateau. . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Temperature evolution of the RIQHE. (a)Temperature evolution of Rxy ; (b) Temperature evolution of Rxx . The minimum on Rxx and the reentrant Rxy at ν=0.63 both develop at lower temperatures. 4.7 . . . . 99 Evolution of the RIQHE with increased alloy disorder. (a) Rxy evolution; (b) Rxx evolution . The RIQHE is not observed in the sample with x=0.21%, but becomes prominent in the sample with x=0.85%. 4.8 The full spectrum of particle-hole symmetry between the corresponding FQH states, and between the RIQHE and the reentrant insulator. 4.9 100 . 101 Rxx data for the third series of sample at 0.3K. For samples with more alloy disorder, the high field part is more insulating. A reentrant insulator state is observed between the ν = samples except for the one with x=0. 1 3 and 2 5 FQH states in all . . . . . . . . . . . . . . . . . 103 A.1 Typical contact locations in the measurements. . . . . . . . . . . . . 106 C.1 Magneto-transport data of a typical sample from the second series before the parallel conductance is removed. . . . . . . . . . . . . . . 112 C.2 Parallel conductance layers and the back gate. xxii . . . . . . . . . . . . 113 C.3 Magneto-transport data of a typical sample from the second series after the parallel conductance is removed. . . . . . . . . . . . . . . . . . . 114 E.1 Hall resistance in the quantum Hall insulator regime. . . . . . . . . . 121 E.2 Sample geometry and the quantum Hall edge current picture. E.3 Magneto-transport of samples with various leg dimensions. E.4 Magneto-transport for a sample with a smaller g factor. . . . . 122 . . . . . 124 . . . . . . . 126 E.5 Magnet-transport data for sample 6-3-03-2 with the full range of field. 128 E.6 Rxx in the mT magnetic field regime. . . . . . . . . . . . . . . . . . 129 E.7 Anomalous Hall effect. . . . . . . . . . . . . . . . . . . . . . . . . . . 131 xxiii Chapter 1 Introduction 1.1 2DES in a semiconductor heterostructure Semiconductor heterostructures have made it possible to create two dimensional electron systems (2DES) of high quality. The growth technology has been so advanced that the electron mean free path in the best GaAs − Alx Ga1−x As heterostructures can reach a length scale of millimeters. These spectacular achievements in material science have made it possible to prepare 2DES in a wide range of electron density and mobility, and create the opportunity for the study in this thesis. All the samples investigated in this thesis are based on the GaAs − Alx Ga1−x As heterostructure, and we start the introduction by giving a brief review on 2DES in a semiconductor heterostructure. A 2DES is an electron system confined in one direction (namely, the Z-direction) and kept free in the other two directions (the X − Y plane). Before the invention of semiconductor heterostructures, 2DES had been realized in Si-MOSFET, which had made the foundation of the modern electronics. In a typical n–type Si-MOSFET, amorphous SiO2 is grown on Si, and a metallic gate is deposited on the insulating 1 1.1: 2DES in a semiconductor heterostructure 2 SiO2 layer. A gate voltage introduces a potential confinement near the Si − SiO2 interface, and creates an inversion carrier layer of electrons, thus a 2DES. In this structure, 2D electrons experience strong Coulomb scattering by the charged impurities near the rough Si − SiO2 interface, and the mobility of electrons is limited. The motivation of growing a semiconductor heterostructure is to realize 2D electrons in a much cleaner system. 2DES is to be physically separated from charged ions and a higher electron mobility can be achieved. This is made possible by the technology of Band engineering. Since we only study semiconductors with low electron (or hole) density and shallow donors in this thesis, the complicated features of the crystalline semiconductor band structure can be largely ignored, and only the energy and effective mass at the extremity of the valence and conduction bands are relevant [Sin ]. Fig. 1.1 illustrates the mechanism of realizing 2DES in a semiconductor heterostructure. Assume that one crystalline material A is grown on top of another crystalline material B. The conduction band bottom of B is lower than that of A, while the valence band top of B is higher than that of A. Electron donors are put into A during the sample growth. If electron donors are ionized in the region of A, the electrons will tend to migrate to the B region to enjoy the lower conduction band energy. Since electrons are charged particles, their displacement alters the band energy configuration as well. As a result, a triangular-shaped potential well [Sin , Dav ] is formed on the B side near the interface between the two materials, and 2D electrons are trapped there. It is then apparent that there is a spacer between the 2DES and the donor. Since the potential confinement is accomplished by two crystalline materials, the system is much cleaner. Moreover, in a heterostructure the 2DES and 1.1: 2DES in a semiconductor heterostructure 3 Figure 1.1: A simplified illustration on the mechanism of a semiconductor heterostructure. (a)Two materials A and B are grown together. (b) Band structures of A and B. χA and χB are the electron affinities. (c) With electrons migrate from A to B, a triangular potential well is formed. the donors are physically separated, therefore the scattering on the 2D electrons by the Coulomb potential of the donor ions is highly reduced and the electrons have a higher mobility. To realize 2DES in a heterostructure, the component materials need to have substantial difference in their band energies. In addition, to form a single crystal, the crystalline structures of the two materials need to be the same, and the difference in lattice constant should be very small. These requirements limit the choice of materials to grow a heterostructure. A Table of the lattice constants and band gaps of various 1.1: 2DES in a semiconductor heterostructure 4 III-V semiconductors is provided in [Gow ], and it can be found that the lattice constants of GaAs and AlAs are the closest while their band energies are substantially different. The lattice constant of GaAs and AlAs is 5.653Å and 5.660Å, respectively. In practice, the lattice mismatch can be reduced even more by using AlGaAs alloy and GaAs as the component materials for a heterostructure. For an alloy, characteristics such as lattice constant and band energy can be obtained by the linear interpolation between that of the pure materials (Vegard’s law). The lattice constant of Alx Ga1−x A is then given by a = x × aAlAs + (1 − x) × aGaAs . For x=0.3, a =5.658Å, and the lattice mismatch between Al0.3 Ga0.7 As and GaAs is only about 0.03%. On the other hand, the band energies of Al0.3 Ga0.7 As and GaAs are quite different. The electron affinity χ (the energy required to excite an electron from the bottom of the conduction band to the vacuum level) of GaAs and Al0.3 Ga0.7 As is 4.07eV and 3.74eV , respectively, and the difference is 0.33eV . Therefore, the GaAs − Al0.3 Ga0.7 As heterostructure can make a good system to realize high-quality 2D electrons. In a typical heterostructure of this type, Si atoms are doped into the Al0.3 Ga0.7 As side as electron donors, and the 2DES resides in the GaAs side since GaAs has higher electron affinity. An alternative design to realize 2DES is the quantum well. Fig. 1.2 illustrates a typical quantum well structure made by GaAs and Al0.3 Ga0.7 As. In this structure, a thin layer of GaAs is sandwiched between two thick layers of Al0.3 Ga0.7 As, and a rectangle-shaped potential well is naturally formed. Electrons accumulate in GaAs while donors are put in Al0.3 Ga0.7 As. These semiconductor heterostructures are usually grown with the state-of-art technology of Molecular Beam Epitaxy (MBE) to achieve the highest quality. The MBE process works in an ultra high vacuum chamber with pressure lower than 5 × 10−11 1.1: 2DES in a semiconductor heterostructure 5 Figure 1.2: A simplified illustration of the AlGaAs-GaAs-AlGaAs quantum well structure. 1.2: Fundamental characteristics of 2DES 6 mbar. Due to this high vacuum, molecules emerge from the furnace do not diffuse like gas, but form a molecular beam. The molecular beam travels in straight lines without collision until it reaches the growing substrate. Growth of each material or dopants is controlled by shutters on the furnaces, and the beam flux can be adjusted by the furnace temperature. MBE is a slow process, and the growing speed is normally 1 monolayer per second. This layer-by-layer growth makes perfect crystals, and produces the best electronic materials up-to-date. The heterostructures with the highest electron mobility in the world are the single GaAs − AlGaAs heterostructures grown by our sample providers in the Bell Labs. Besides the high-mobility GaAs − AlGaAs heterostructures, many other heterostructures have been grown with different materials for various purposes. These include the Si-SiGe, the AlAs-AlGaAs, the InGaAs-InP, and so on. However, when it comes to electron mobility, 2DES residing in GaAs − AlGaAs heterostructures is always the best. All the samples we study in this thesis are based on the clean GaAs − AlGaAs heterostructures, and alloy impurities are intentionally introduced into the 2DES in a controllable manner. 1.2 Fundamental characteristics of 2DES Most of the experiments in this thesis are measurements on the transport properties of 2DES. The most fundamental characteristics are the electron density and mobility. Electron density is the number of electrons in a unit area. A convenient way to measure the electron density n is through the Hall Effect at low magnetic fields. Since the Hall resistance Rxy = B , ne the electron density n can easily be obtained from the slope of the Rxy − B dependence. 7 1.2: Fundamental characteristics of 2DES The definition of electron mobility can be introduced by the simple Drude model. In this model, the electron loses its momentum once it is scattered, and the drifting velocity of an electron can be described by dv dt = eE m∗ − v τ , where v is the electron velocity, E is the electric field, m∗ is the electron effective mass, and τ is the momentum relaxation time (time between two scattering events of the electron by impurities or other type of potential disorder). At large times the steady electron velocity is v = eEτ . m∗ The mobility µ is defined as µ = eτ , m∗ which gives v = µE. On the other hand, the current density j = nev, where n is the number density of 2D electrons. Therefore the conductivity can be written as σ = obtained by µ = j E = neµ, and mobility can be σ . ne At zero magnetic field, the conductivity σ is just the inverse of the resistivity ρ (with unit of Ω for 2D systems). The van der Pauw method [van der Pauw 58a, van der Pauw 58b], can be used in measuring ρ for 2DES samples of any shape. If we have four Ohmic contacts (labeled 1, 2, 3, 4) to the 2D electrons, we first drive current through contacts 1, 2 and measure voltage across contacts 3,4 to obtain a resistance R12,34 ; then we drive current through 1, 3 and measure voltage across 2, 4 to obtain another resistance R13,24 . The resistivity ρ of the sample can be calculated by solving the equation e −πR12,34 ρ +e −πR13,24 ρ = 1. When R12,34 and R13,24 are very close to each other, an approximate solution can be obtained as ρ = π ln 2 × R12,34 +R13,24 . 2 To get precise resistivity value from the van der Pauw method, we have to follow a few tips in the measurements. These tips are listed in Appendix A. Both Hall measurements and Van der Pauw measurements requires at least four Ohmic contacts on the sample. In practice, measurements on a sample subjected to 1.3: Disorder in 2DES 8 magnetic field require at least five Ohmic contacts to monitor the Hall and longitudinal resistances simultaneously. The Ohmic contacts to our 2DES are realized by thermally diffusing InSn from the surface of the samples. The samples studied in this thesis have electron densities ranging from 4×1010 /cm2 to 2.6 × 1011 /cm2 , and mobilities ranging from 1 × 105 cm2 /V.s to 1.2 × 107 cm2 /V.s. The large mobility difference in the samples is mainly caused by various amounts of potential disorders that are intentionally added into the 2DES. Beyond reducing electron mobility, disorder also leads to a lot other interesting phenomena in physics, which is the focus of this thesis. 1.3 Disorder in 2DES If the potential in a solid is perfectly periodic, Bloch’s theorem rules, and the electrons can be considered as free electrons with an effective mass m∗ . However in the real world, the periodicity of the potential in a solid is always imperfect. This imperfectness, or disorder, makes the physics much more complicated and brings in a lot of interesting effects. Random potential fluctuation in a 2DES has contributions from various origins, such as phonons, ionized impurities, alloy impurities and surface roughness. The amplitude of these potential fluctuations varies in different systems. For example, as we have mentioned in the previous section, 2D electrons in a typical Si-MOSFET are strongly scattered by the Coulomb potential of the charged impurities near the Si − SiO2 interface, which leads to a relatively low electron mobility. However, for a heterostructure, the whole sample is a single crystal and there is no trapped 1.3: Disorder in 2DES 9 charge near the potential confinement. Moreover, Coulomb scattering from the ionized donors is largely reduced since the 2DES is physically separated from the dopants by a spacer. Therefore the Coulomb potential fluctuation is weak in a 2DES residing in a heterostructure, and the disorder can be dominated by other types of potential fluctuations that are introduced into the sample intentionally or unintentionally. In this section, we will first provide a review on different types of disorder in a heterostructure, then briefly discuss the electron localization effect which is caused by disorder. 1.3.1 Coulomb disorder In a heterostructure, the 2D electrons are physically separated from the dopants. However, the Coulomb disorder is not totally removed. Since the Coulomb potential is a long range potential (decays by 1/r), the 2D electrons can still ”feel” the random potential of the donor ions although they are remote from each other. To make things more complicated, this Coulomb potential is screened by the 2D electrons as well [Efros 89]. The screening is not as effective as that in a metal, but still smoothes the random potential substantially. The method of 2D Thomas Fermi screening offers a relatively simple recipe to compute the remote Coulomb potentials[Dav ]. We have performed such a computation in a typical 2DES residing in a GaAs − AlGaAs heterostructure, and the potential configuration is demonstrated in Fig. 1.3 to help understanding the nature of Coulomb disorder. In this sample, the electron density is 1×1011 /cm2 , the mobility is about 2×106 cm2 /V.s, and the spacer thickness (distance from the donors to 2D electrons) is 1000Å. On the potential configuration plot, the bright areas represent regions with higher potential while the dark areas have lower 1.3: Disorder in 2DES 10 Figure 1.3: Computer simulation of the Coulomb potential in a typical GaAs-AlGaAs heterostructure. The bright areas represent high potentials and the dark areas represent lower potentials. The potential difference between the peaks and valleys is 0.4meV. potential. The peak-to-valley potential difference is about 0.4meV , and it appears that the range of the potential fluctuation is larger than 0.1µm. In typical GaAs − AlGaAs heterostructures, the demonstrated potential fluctuations from the remote ionized impurities dominate the disorder in 2DES. For 2DES in GaAs − AlGaAs heterostructures with ultra-high mobilities (higher than 107 cm2 /V.s), the spacer thickness is usually much larger than 1000Å, and the potential fluctuation from remote ionized impurities can be reduced to a negligible 1.3: Disorder in 2DES 11 level. What limits the electron mobility in this regime is the background ionized impurity in the GaAs. These background ions are usually believed to be carbon and come into the system from the stainless steel of the MBE chamber, and the density of these impurity ions in GaAs is normally less than 1014 /cm3 . The potential fluctuation from the background ionized impurities, as a type of Coulomb potential fluctuation, also has a long range. In this thesis, we will not distinguish the remote and background ionized impurity disorders, and they will both be referred as ”longrange ionized impurity disorder” or ”Coulomb disorder”. 1.3.2 Non-Coulomb disorder Longitudinal acoustic phonons, which originate from the thermal vibration of the lattice, drive the potential away from being perfectly periodic and provides a common type of non-Coulomb disorder for the 2DES. In 2D, it has been shown that the electron scattering rate from acoustic phonons is proportional to temperature T [Sin ]. At temperatures lower than 1K, phonon scattering quickly diminishes since most of the phonon modes are frozen [Sin ]. In this thesis, we concentrate on physics in the low temperature regime below 1K, and the contribution to disorder from phonons is usually neglected. Two other types of non-Coulomb disorder, however, do not have strong temperature dependence and can play important roles at low temperatures. These are the surface roughness disorder and the alloy disorder. The surface roughness scattering is strong in structures of narrow quantum well. The potential confinement of the quantum well and the electron wave function is illustrated in Fig. 1.2. The imperfectness of the lattice structure near the interface of 1.3: Disorder in 2DES 12 the two component materials (surface roughness) creates a local potential fluctuation. In contrary to the Coulomb disorder, the potential fluctuation from surface roughness in a semiconductor heterostructure usually has range of atomic size[Sin ], and is a type of short-range disorder. In a single heterostructure, since the 2DES predominantly resides on one side, the probability for an electron to appear near the interface is very small, and the surface roughness disorder is negligible. In a narrow quantum well, however, the electron wave function has large value near the interface, and the surface-roughness scattering is significant. The alloy disorder comes into the picture when electrons appear in a material of random alloy. In an alloy system, different types of atoms are placed randomly on the lattice, and electrons are scattered by the random local potential fluctuations. Fig. 1.4 offers a cartoon illustration of the alloy potential fluctuation in a binary alloy crystal Ax B1−x . In a virtual crystal model, assuming the potential of elements A and B is EA and EB respectively in the crystal, the averaged virtual crystal potential is given by Vegard’s law EAverage = xEA + (1 − x)EB . The scattering potential experienced by electrons at atoms A and B is then EA − EAverage = (1 − x)(EA − EB ) and EB − EAverage = x(EB − EA ), respectively. Since the alloy potential fluctuation is neutral, it is limited within the range of the atomic dimension, and is a type of short-range disorder. In typical GaAs − AlGaAs heterostructures, the 2DES predominantly resides on the GaAs side, and only a small part of the wave function penetrates into the AlGaAs alloy. The alloy disorder in these systems is then negligible. In a narrow AlGaAs − GaAs − AlGaAs quantum well, the penetration of the electron wave function into the AlGaAs is larger and the contribution of alloy potential fluctuation in disorder is 1.3: Disorder in 2DES 13 Figure 1.4: Cartoon illustration of the alloy potential fluctuation for a binary alloy Ax B1−x . The system is characterized by the alloy concentration x and the potential difference between the two pure components (EA − EB ). 1.3: Disorder in 2DES 14 more significant. However, the surface roughness potential fluctuation increases even more in a narrow quantum well and dominates the disorder. Alloy potential fluctuation dominates only when the 2DES mainly resides in an alloy. One example is the InGaAs − InP heterostructure. Unlike the GaAs − AlGaAs heterostructures, 2DES in an InGaAs − InP heterostructure resides on the alloy side since InGaAs has a higher electron affinity. The concentrations of both alloy components are high in InGaAs (48% and 52%), and the 2D electrons are strongly scattered by alloy disorder at each atomic site. The InGaAs − InP heterostructure has low electron mobility, and is normally grown only for purposes of optical applications. In this thesis, we introduce alloy disorder into the GaAs − Al0.3 Ga0.7 As heterostructures in a controllable manner. Small portions of Al atoms are put into the GaAs side, resulting in Gax As1−x − Al0.3 Ga0.7 As heterostructures. In these heterostructures, the 2D electrons can be strongly scattered by the alloy potential fluctuations. The nature of disorder varies in samples with different Al concentration x. The physics related with controlled alloy disorder will be the main theme of this thesis. 1.3.3 Disorder and electron localization Of all the interesting disorder-related topics in solid state physics, the most fundamental one is the Anderson localization[Anderson 58]. The model of Anderson localization considers the diffusion of electron wave functions in solids with random potential fluctuations. Provided that the randomness of the potential is large enough, the electron wave function is localized, and can only 1.3: Disorder in 2DES 15 diffuse through a finite length scale - the localization length ξ. Anderson localization is actually a general wave phenomenon that also applies to electromagnetic waves and all types of quantum waves in systems of various dimensions. Following this picture, a highly successful approach of scaling was put forward for non-interacting electrons in 1979 [Abrahams 79]. This scaling theory of localization suggests that a metal-insulator transition (MIT) exists for non-interacting electrons in three dimensions (3D) with zero B field, but there are no extended states thus no MIT in 1D and 2D. However, since 2 is the lower critical dimension of this localization problem, states in 2D are just marginally localized if the random potential is weak, and the localization length can be quite large. If the localization length is larger than the sample size, the sample is effectively in a metallic state with finite conductivity when temperature approaches absolute zero. 1.3.4 Important length scales in 2DES It is then very important to evaluate and compare different length scales to study a system in 2D. We here give a brief overview on the length scales in 2DES that are under concern in this thesis. First, we have the sample size L. The actual sample size varies in different experiments. Usually a piece of specimen with width and length of a few millimeters is cut off from a wafer for measurements. Technology of micro/nano-fabrication has made it possible to prepare for 2DES samples as small as a few µms in both length and width. Most work in this thesis is based on direct-cut samples of mms in length and width, and L is usually much larger that any other concerned length scales. However, 1.3: Disorder in 2DES 16 a smaller length scale of effective sample size is usually considered in the study of localization-related physics. Since the Anderson localization focuses on the wave nature of electrons, this picture only applies within a small range in which the quantum mechanical features of an electron are kept. The quantum phase coherence length LΦ of the electron, therefore is the effective sample size in the consideration of electron localization. The quantum phase of electron is lost when the electron is scattered −1 inelastically, and LΦ can be estimated from the inelastic scattering rate τinel . In a semiconductor system, the inelastic scattering to an electron is dominantly from phonons and other electrons. Since we concentrate on the low temperature regime −1 where most phonon modes are frozen, the electron-electron scattering rate τe−e has −1 the dominant contribution to τinel . At lower temperatures, the number of available −1 states an electron can be scattered to decreases, therefore τe−e decreases and LΦ is larger. The actual length of LΦ will be estimated at various temperatures in chapter 3 of this thesis. With the scaling theory of localization, conductivity σ in a system is determined by comparing the localization length ξ and the effective sample size LΦ , and the ratio LΦ /ξ is usually considered in most experimental studies. Another important length scale is the range of disorder d. As is briefly mentioned in the previous parts of this section, Coulomb potential fluctuation is generally considered as a type of long-range disorder, while the surface roughness and alloy potential fluctuations are types of short-range disorder. The term ”long” or ”short” is relative, and is usually determined by comparison with other length scales. Since quantum mechanics only prevails within the range of quantum phase coherence length LΦ , it 1.4: The Quantum Hall physics 17 is natural to assume that d LΦ is a prerequisite for the picture of Anderson localization to be valid. In a typical clean semiconductor system, LΦ is in the order of µm at T =1K. The potential from ionized impurities is then a type of long-range disorder since its range is comparable to LΦ . The alloy impurity potential, on the other hand, has a range close to the lattice constant 0.56nm, and is definitely a type of short-range disorder. The long-range and short-range disorders lead to very different physics, which will be investigated in this thesis. In this thesis, we study 2DES residing in the Gax As1−x −Al0.3 Ga0.7 As heterostructures with various Al concentration x. The relative weight of short-range alloy disorder increases in samples of larger x values, and the disorder in the system is dominated by short-range alloy potential fluctuations with x being large enough. With these systems, we are able to systematically investigate disorder-related physics in Chapter 3 and Chapter 4. 1.4 1.4.1 The Quantum Hall physics The integer quantum Hall effect and the quantum Hall plateau-to-plateau transitions The idea that electrons can be localized by disorder has led to a significant advance in the understanding of electron transport, and provides model for a type of metalinsulator transition (MIT) - the Anderson localization-delocalization transition. In two dimensional systems, the scaling theory of non-interacting electrons developed in 1979 predicts that any 2DES is Anderson localized and there cannot be any MIT in 2D[Abrahams 79]. However extended electron states and MIT can exist with strong 18 1.4: The Quantum Hall physics magnetic field in the so-called quantum Hall regime. In this section, we will give a brief introduction on the quantum Hall effect, the extended states, and the MIT in the quantum Hall regime - the quantum Hall plateau-to-plateau transition. The quantum Hall effect is a quantum mechanical version of the classical Hall effect, and is observed in 2DES at low temperatures under high magnetic fields. Fig 1.5 (a) shows a well developed quantum Hall effect in a typical sample. With a sweeping magnetic field B, the Hall resistance Rxy is quantized into successive plateaus. The longitudinal resistance Rxx is zero in the Hall plateau regions and reaches maximums in the regions of plateau-to-plateau transitions. The physics of the integer quantum Hall effect lies on the formation of Landau levels under high magnetic fields, and this picture is illustrated in Fig. 1.5 (b). At zero magnetic field and low temperatures, the density of states of a 2DES is a constant, and electrons uniformly fill up all the states below the Fermi energy. However, under strong magnetic field, this continuous spectrum of kinetic energy is split into separated Landau levels, and band gaps are formed between these levels. Standard argument in quantum mechanics shows that the energy of the N th Landau levels is EN = ~ωc (N + 1/2), where ωc is the cyclotron frequency eB . m∗ The band gap between two Landau levels is ~ωc , which increases at higher magnetic field. By considering the Zeeman energy of electrons, each Landau level is further split into two branches with electrons spin up and down, and the degeneracy of each spin-polarized Landau level is eB . h The filling of electrons into the Landau levels depends on both the electron density n and the magnetic field B. A Landau level filling factor ν = is widely used to describe the number of filled levels. nh eB 1.4: The Quantum Hall physics 19 Figure 1.5: The integer quantum Hall effect. (a) Plot of a well developed integer quantum Hall effect in a typical 2DES. (b) The Landau levels in 2DES subjected to various magnetic fields. 20 1.4: The Quantum Hall physics With the electrons being scattered by disorder, the singular energy spectrum of Landau levels spread into a band-like structure, and extended electron states exist in the center of each band. All the other states are more or less localized. In the center of the gap between two Landau levels, electrons are most strongly localized and the localization length ξ is much smaller than the effective sample size. When the Fermi level lies in the gap, Rxx diminishes and Rxy is quantized to the plateau value. The plateau value of Rxy around an integer filling factor N is h N e2 (roughly 21812.8/N Ω), and the quantization is accurate to 10−4 Ω. This perfect quantization can be well understood in the edge current picture of quantum Hall, which will not be discussed in this thesis. We concentrate on the fact that the quantized Hall plateau represents energy regions of localized states and study the localization-delocalization transition in the quantum Hall regime. The delocalized states locate in the center of each Landau level. Theoretical studies have shown that there is only one singular energy level Ec of extended states between two plateau regions[Pruisken 88, Ono 82, Aoki 85, Chalker 87, Huo 92]. Approaching Ec from the plateaus in both sides, the localization length ξ diverges with a power law ξ ∝ |E − Ec |−ν with ν representing the localization length exponent [Pruisken 88, Wei 88, Huckestein 01, Sondhi 97, Struck 06]. Since the Landau levels are shifted by sweeping magnetic field in experiments, this power law can be rewritten as ξ ∝ |B − Bc |−ν where Bc is the critical magnetic field between two plateaus and corresponds to Ec in the energy spectrum. The quantum Hall plateau-to-plateau transition is then a transition from one localized region to another localized region through an energy level of extended states. This localization-delocalization transition is a quantum phase transition, and becomes more prominent when the temperature 21 1.4: The Quantum Hall physics approaches absolute zero. The phases of this transition are the localized regions represented by the Hall plateaus, the critical point is the energy level of extended states represented by the critical field Bc , and the correlation length is the localization length ξ. The order parameter of this transition is not well defined. The localization length ξ is usually difficult to measure directly in experiments, so it is not feasible to extract the critical exponent ν by fitting experimental data into the scaling form ξ ∝ |B − Bc |−ν . A practical way to obtain ν with experiments is offered by the finite size scaling theory[Pruisken 88, Wei 88, Huckestein 01, Sondhi 97, Thouless 77], which we briefly review in the next paragraph. According to the finite size scaling theory of localization, the conductance of a sample is determined by the relative length scale of sample size L in comparison with the localization length ξ. The resistance tensor can then be written as a function of their ratio L/ξ as Rµν = R(L/ξ). Approaching to the critical point of the quantum Hall plateau-to-plateau transition, the localization diverges with a power law ξ ∝ |B − Bc |−ν . On the other hand, the effective sample size L, which is set by the quantum phase coherence length LΦ , increases at lower temperatures. Assuming p the temperature exponent of inelastic scattering is p, LΦ diverges as LΦ ∝ T − 2 . p Substitute ξ ∝ |B − Bc |−ν and LΦ ∝ T − 2 into the resistance function Rµν = R(L/ξ), we obtain the resistance tensor as a function of magnetic field B and temperature T : p p Rµν = R(|B − Bc |−ν · T − 2 ) = f (|B − Bc | · T − 2ν ). The derivative of this formula over magnetic field gives a temperature scaling form at the critical field Bc : p T − 2ν . For Rxy , we have obtained a temperature scaling form κ= p . 2ν dRxy | dB B=Bc dRµν | dB B=Bc ∝ ∝ T −κ with A similar argument on Rxx indicates that the half-width of the Rxx peak in the plateau-to-plateau transition region follows ∆B ∝ T κ . 22 1.4: The Quantum Hall physics Figure 1.6: Temperature dependence of Integer quantum Hall effect in a typical 2DES. It appears that the plateau-to-plateau transitions are sharper at lower temperatures. Besides the temperature scaling for DC conductivity, a dynamic temperature scaling has been introduced [Kivelson 92, Engel 93] to relate the microwave transmission to the quantum Hall plateau-to-plateau transition. It is deduced in a similar fashion 1 that the half-width of the microwave transmission spectrum |∆B| ∝ T zν , where z is the dynamic exponent. These temperature scaling forms offer a way to measure the localization length exponent ν. In this thesis, we concentrate on the transport experiments of scaling. The temperature scaling dRxy | dB B=Bc ∝ T −κ reflects the experimental observation that the plateau-to-plateau transition is sharper at lower temperatures, and this experimental fact is demonstrated in Fig. 1.6. The temperature exponent κ can be obtained directly by fitting experimental data of Rxy into the power-law scaling form. 23 1.4: The Quantum Hall physics If p is identified, the localization length exponent ν can be obtained from the measured value of κ. In the low temperature regime, electron-electron scattering is the dominant mechanism for inelastic scattering to an electron. It is then usually assumed that p = 2 from electron-electron scattering, and κ = ν1 . The critical exponents κ and ν has been the focus of both theoretical and experimental works. While most theories agree with each other on a universal exponent ν = 7 3 [Huckestein 01, Sondhi 97, Chalker 88, Mil’nikov 88, Huckestein 90, Lee 93, Gammel 94], which implies κ = 3/7 ≈ 0.42 with assuming p=2, diverse experimental values of κ have been observed in a wide range from 0.1 to 1 [Wei 88, Wakabayashi 89, Sem , Koch 91a, Balaban 98] in various systems. Although most of these scaling experiments were carried out within only one decade of temperature, the observation of various values of the exponent is puzzling. The universality of the plateau-to-plateau transition has therefore been called into question. In chapter 3 of this thesis, we have solved this problem of quantum Hall plateauto-plateau transition by studying systems with controlled alloy disorder. Our result shows that the range of disorder is a deterministic factor for the measured critical exponent. In systems with disorder being dominated by short-range potential fluctuations, we have observed a perfect power-law temperature scaling dRxy | dB B=Bc ∝ T −κ with the universal exponent κ = 0.42. We further experimentally confirm in Chapter 3 the value of p to be 2, and obtain the localization length exponent ν = 2.4. The plateau-to-plateau transition is then verified to be a universal quantum phase transition. 1.4: The Quantum Hall physics 24 Figure 1.7: Fractional quantum Hall effect in a 2DES of very high mobility. Various series of FQH states are observed. This figure is taken from (W. Pan et al, Phys. Rev. Lett. 88, 176802 (2002)). 1.4.2 Fractional quantum Hall effect and Wigner Crystals While the integer quantum Hall effect can be understood within a picture of the localization-delocalization of non-interacting electrons, it is essential to take the electronelectron interaction into account to appreciate the novel physics with stronger magnetic fields. For high mobility 2DES subjected to strong magnetic field, a rich spectrum of novel electron phases has been discovered beyond the integer quantum Hall effect. The most prominent of these discoveries are the fractional quantum Hall effect (FQHE)[Tsui 82, Laughlin 83, Jain 89] and the Wigner crystals[Wigner 34, Lozovik 75, Chen 04]. In this section, we give a brief review about the various phases in the high magnetic field regime. The roles of disorder in this regime will be the focus of Chapter 4 of this thesis. 25 1.4: The Quantum Hall physics Fig. 1.7 shows the magneto-transport data of a typical high mobility 2DES. Besides the standard integer quantum Hall states with Rxy quantized to states have been discovered with Rxy quantized to tional filling factors i = p 2p±1 n ie2 n , N e2 novel around several series of frac- + N . This fractional quantum Hall effect, first being discovered in 1982, has attracted a major interest from the community of condensed matter physics. Many series of fractional numbers have been discovered in magnetotransport experiments from then on. In the high magnetic field end, the series of fractional quantum Hall states terminate beyond a FQHE state with filling factor ν = 51 . Fig. 1.8 shows the magnetotransport data of a typical 2DES sample around the terminal FQHE state. In some samples like the 2D holes, the terminal FQHE state can be shifted to ν = 1 . 3 A high field insulating phase and a reentrant insulating phase are always observed around this terminal FQHE state, and Data from microwave resonance experiments [Goldman 90, Engel 97, Ye 02, Chen 04]has shown strong evidence that these insulating states represent 2D electron crystals of triangular lattice - the Wigner crystal. In this section, we will give a brief introduction to the understanding of FQHE and Wigner Crystal, and both electron-electron interaction and disorder play very important roles in the physics of this high-field regime. Since electrons in the fully filled Landau levels contribute little to the interaction, the FQHE physics in the high Landau levels is usually considered to be the same as the physics in the lowest Landau level with N=0, where most works in the literature focus on. The principal series of FQHE with filling factors p , 2p+1 such as 1 , 52 , 37 , 3 ... , was first explained by Laughlin[Laughlin 83] with a variational multi-electron wave 1.4: The Quantum Hall physics 26 Figure 1.8: Magneto-transport around the terminal FQH state. The terminal FQH state has filling factor ν=1/5. A reentrant insulator is observed at ν=0.21. This figure is taken from (H. W. Jiang et al, Phys. Rev. Lett. 65, 633 (1990)). 27 1.4: The Quantum Hall physics function. It is shown that at these filling factors, the Laughlin states of fractionally charged quasi-particles are the ground states of the many-electron system. With a particle-hole symmetry, the FQHE series with filling factors p 2p−1 = p−1 2(p−1)+1 are simply interpreted as the quasi-holes with regard to the lowest Landau level. More recently, a composite Fermion (CF) model[Jain 89] is introduced to explain the FQHE and has been widely accepted. In this model, a composite fermion consists of an electron (or hole) bound to even number 2p of magnetic flux quanta Φ0 = he . Formation of these CFs has accounted for all the many body interactions, so only single particle effects need to be considered afterwards. Consequently, the effective field felt by the composite fermions is the field that is left over after taking off the attached flux quanta, and can be written as B ∗ = B − 2pΦ0 n, with n being the electron density. The original filling factor of the electron was ν = the composite fermions is simply given by ν ∗ = nΦ0 B∗ nΦ0 , B and the filling factor ν ∗ of (B ∗ can be negative, i.e., anti- parallel to B). Combining these two equations, one can obtain the fractional filling factors ν = ν∗ , 2pν ∗ ±1 where the minus sign is chosen in the denominator if B ∗ is anti- parallel to B. Therefore the fractional quantum Hall effect can be considered as the integer quantum Hall effect of the composite fermions. For example, the fractional quantum Hall states at ν = 1 3 can be viewed as the ν = 1 state of the composite fermions with two flux quanta attached to each electrons (p = 1). Similar to the integer quantum Hall effect, the plateaus of the fractional filling factors show the localization of the composite fermions. In the CF model, each composite fermion still carry one electron charge, and because they move in an effective magnetic field B ∗ they appear to have a fractional 28 1.4: The Quantum Hall physics topological charge. The composite fermion picture correctly predicts almost all the observed fractional states and their relative intensities, and shows particle-hole symmetry between fractional series p 2p+1 and p . 2p−1 The FQHE originates from the interaction between electrons. However, in the limit of very high magnetic and very strong electron-electron interaction, electrons tend to form a triangular crystalline structure which is known as Wigner crystal[Wigner 34]. In the range of intermediate magnetic fields, the fractional quantum Hall states (the Laughlin liquid) and the Wigner crystal compete to be the ground state of the system. The winner of the competition between the fractional quantum Hall liquid and the Wigner crystal at a certain magnetic field is determined by the profound nature of electron-electron interaction and disorder. Usually the FQHE states cease to appear when the filling factor is smaller than a terminal value ν = 1 5 [Willett 88, Jiang 90]. In Chapter 4, we will show that the short-range disorder does not affect the FQHE states, but promotes the formation of Wigner crystal. As a result, the terminal FQHE state is shifted from ν = 1 5 to ν = 1 3 in 2DES with short-range alloy disorder. More excitingly, perfect particle-hole symmetry of Wigner crystal, together with the particle-hole symmetry of the fractional quantum Hall liquid, has been observed for the first time. Chapter 2 Fundamental characteristics of 2DES with short-range alloy disorder at zero and low magnetic fields Disorder plays an important role in almost all aspects of the physics in 2DES. In this thesis, we present the first systematical study to the effect of disorder on the transport properties of 2DES. To achieve this, we need to have a way to introduce a certain type of disorder in a controllable manner. The ideal type of disorder should have a short potential fluctuation range, which is essential to study the Anderson localization-related physics. We chose the alloy disorder as our focus of research. In this chapter, we first present our method of creating 2DES with controlled alloy disorder, then characterize these systems at zero and low magnetic fields. 29 2.1: Introducing alloy disorder into 2DES 2.1 30 Introducing alloy disorder into 2DES It is well known that alloy potential fluctuation dominates the disorder in an InGaAs− InP heterostructure because the 2D electrons reside on the InGaAs alloy side. However, the concentration of both alloy components of InGaAs has to be high to create the quantum confinement for 2DES. In reality, the concentration of the alloy components are usually fixed to be 48% and 52% [Sin ], and the InGaAs − InP heterostructures are only grown for purposes of optical applications. We aim at studying a series of samples with various amount of alloy disorder, from zero to a high level. The other properties of the samples, such as the quantum confinement and the electron density, should be kept unchanged to single out the role of disorder. The advanced technology of material science has made this possible, and we have had three series of samples with controlled alloy disorder grown in Dr. Pfeiffer’s MBE (Molecular Beam Epitaxy) lab at the Bell labs by L. N. Pfeiffer and K. W. West [Pfeiffer 89]. The first series of samples are based on modulation doped GaAs − Al0.3 Ga0.7 As heterostructures. To study the alloy disorder systematically, a small amount of Al impurities is introduced into the GaAs side during the MBE growth, resulting in a Alx Ga1−x As − Al0.3 Ga0.7 As heterostructure. The Al content x is determined by controlling the growth rates of Ga and Al, which are calibrated in a high precision by RHEED oscillations. The relative error of x values in our samples is less than 1%. Other component parts of the samples, such as the doping and the spacer layers, are all kept unchanged. Electrons from the δ doping layer accumulate on the Alx Ga1−x As side of the interface. A schematic description of the samples is shown in Fig. 2.1. 2.1: Introducing alloy disorder into 2DES 31 The Al impurity concentration x has values 0, 0.21%, 0.33%, and 0.85% in the first four samples of this series. Since x is very small, the band structure of Alx Ga1−x As is almost the same as that of GaAs, and the quantum confinement is the same for these samples. The other two heterostructures in this series has much higher Al impurity concentration x = 4.1% and x = 8.5% . For these two samples, the electron affinities in Alx Ga1−x As is substantially lower than that in GaAs, and the quantum confinements are shallower than those of the first four samples. The second series of samples aim at filling the gaps of x values in the first series. The growth design is exactly the same as that of the first series. Four samples have been grown with x = 0.85%, 1.4%, 1.9% and 2.6%. The third series of samples aim at lower electron density and have much shallower quantum confinements. These samples are the Alx Ga1−x As − Al0.1 Ga0.9 As heterostructures with x = 0, 0.4%, 0.8% and 1.2%. Due to the shallow quantum confinement and the low dopant level, there is no 2D electrons accumulation at low temperatures in dark. LED illumination is required for these samples to induce carriers thus 2DES. Most of the works presented in this thesis were carried out with the first series of samples. Additional results from the second and third series of samples are always consistent with those from the first series. 2.1: Introducing alloy disorder into 2DES 32 Figure 2.1: Schematic view of the sample structure. Layers I and II are both Al0.3 Ga0.7 As (for the first and second series of samples) or Al0.1 Ga0.9 As (for the third series of samples), and layer III is Alx Ga1−x As. There are δ-dopants between layers I and II, and electrons accumulate in layer III close to the II-III interface. The thicknesses of layers I,II, and III are 80 nm, 100 nm, and 1 µm, respectively. Between the GaAs substrate and layer III, there are 400 periods of superlattice of 3 nm of GaAs and 10 nm of Al0.3 Ga0.7 As. 33 2.2: Characterization of samples - density, mobility and scattering rate Table 2.1: Fundamental characteristics of the first series of samples Sample # 7-30-97-2 8-21-97-1 8-6-97-1 7-31-97-2 9-5-97-1 9-17-97-1 2.2 x [%] 0 0.21 0.33 0.85 4.1 8.5 n[1011 /cm2 ] 1.13 1.32 1.25 1.16 0.87 0.66 µ[106 cm2 /V.s] 3.70 2.05 1.62 0.89 0.2 0.14 τ −1 [ns−1 ] 7.08 12.8 16.2 29.3 130.8 186.9 Characterization of samples - density, mobility and scattering rate For each sample, we have measured the Hall resistance in a perpendicular magnetic field and obtained the areal electron density n. We also measured the sheet resistivity ρ through the Van der Pauw method, and obtain the mobility by µ = σ . ne Finally, the total scattering rate of electrons is deduced by τ −1 = e/µm∗ . For the effective mass m∗ in this formula, we used its value in GaAs [Sin ] which is 0.067 times the mass of a bare electron. The measurement results of the first series of samples at 0.3K are summarized in Table 2.1. The last two samples with x = 4.1% and x = 8.5% have much lower electron densities because the quantum confinements are much shallower due to the large x values. Although the second series of samples have exactly the same structure design as the first series, they were grown six years after the growth of the first series, and the MBE system was not in the best condition for their growth. We have found in the second series of samples parallel conductance layers through the magneto-transport data, which will be discussed in Chapter 3. A back gate of -200V was applied on 34 2.3: Alloy scattering rate is temperature independent Table 2.2: Fundamental characteristics of the second series of samples Sample # 12-10-03-1 12-12-03-1 12-15-03-1 12-15-03-2 x [%] 0.85 1.4 1.9 2.6 n[1011 /cm2 ] 1.18 1.14 1.26 1.22 µ[106 cm2 /V.s] 0.91 0.56 0.46 0.34 τ −1 [ns−1 ] 28.7 46.6 56.7 76.7 Table 2.3: Fundamental characteristics of the third series of samples after illumination Sample # 10-25-04-2 12-03-04-1 12-06-04-1 12-06-04-2 x [%] 0 0.4 0.8 1.2 n[1011 /cm2 ] 0.65 0.66 0.65 0.64 µ[106 cm2 /V.s] 9.2 3.2 2.04 1.5 τ −1 [ns−1 ] 2.84 8.2 12.9 17.5 these samples to remove the parallel conductance, and the sample characteristics of the second series are listed in Table 2.2. The third series of samples require illumination to induce carriers. We use a LED with the excitation current as small as 1nA, and we are able to control the density of the samples by changing the time duration of the illumination. We found the sample qualities are the best when the electron density is around 6.5 × 1010 /cm2 . Characteristics of the illuminated samples at 0.3K are listed in Table 2.3. 2.3 Alloy scattering rate is temperature independent We have carried out the characterization at various temperatures, and the temperature dependence of the scattering rate for the first four samples in the first series is shown in Fig. 2.2. It is most remarkable that curves corresponding to different 2.3: Alloy scattering rate is temperature independent 35 samples are parallel to each other. Because the only difference between the samples is the Al impurity content that leads to alloy scattering, the alloy scattering rate for a sample with a given Al concentration x is inferred from Matthiessen’s rule to be τal−1 (x) = τ −1 (x) − τ −1 (x = 0), where τ −1 (x) is the total scattering rate of that sample and τ −1 (x = 0) is the total scattering rate of the sample with no intentional Al impurities. Since all the curves of Fig. 2.2 are parallel, we conclude that in the temperature range of our measurements the alloy scattering rate is T-independent. This observation is consistent with theoretical expectations [Ando 82a, Fu 00, Basu 83, Bastard 84, Chattopadhyay 85] for two dimensional electron systems. As explained above, the scattering rate dependence on temperature is similar for different samples and the displacement of these curves along the vertical axis is due to alloy scattering. The different temperature dependences for different regions of the curves can be associated with other scattering mechanisms present. Above 1.5 K, the total scattering rate for each sample τ −1 (x) increases linearly with the temperature, which can be explained by the acoustic phonon scattering from both the deformation potential and the piezoelectric coupling [Walukiewicz 84, Lin 84]. Below 0.7 K, the scattering rates are temperature independent because phonons are frozen out. The total scattering rate does not extrapolate to zero at T = 0 even for the sample with no intentional Al impurities. This is due to the residual scattering, which is thought to be caused by the background impurities, the ionized charge in the doping layer, and may be partly from the surface roughness and tail scattering as the result of the electron wave function penetrating into the spacer layer[Ando 82a]. Therefore −1 , where the total scattering rate can be expressed as τ −1 (x) = τal−1 (x) + τr−1 + τph −1 τr−1 and τph are the residual and the phonon scattering rates, respectively. We note 2.3: Alloy scattering rate is temperature independent 36 Figure 2.2: The T -dependence of the scattering rate for the first four samples of the first series. In the 0.3-4.2 K temperature range all four curves are parallel to each other. 2.3: Alloy scattering rate is temperature independent 37 Figure 2.3: The T -dependence of the scattering rate for the four samples of the third series. Again, all four curves are approximately parallel to each other. that the scattering rate due to residual disorder is the same as that due to alloy scattering with approximately 0.24% Al impurities, which indicates that above this concentration alloy scattering dominates at low temperatures. The same pattern of τ −1 − T dependence has been observed in the third series of samples. As is shown in Fig. 2.3, all curves are parallel to each other. The residual scattering rate is much lower in this series because of the better sample quality and lower dopant level. Our discussion above can be exactly replicated although samples of the third series have much lower electron densities. 2.4: Amplitude of the alloy potential fluctuation 2.4 38 Amplitude of the alloy potential fluctuation Although the AlGaAs alloy has been studied for a long time, there was no agreement on the amplitude of the alloy potential fluctuation in this material. Previous works of measuring the potential fluctuation primarily focused on three-dimensional carrier systems in bulk materials. The scattering rate in a binary Ax B1−x alloy is predicted within the virtual crystal framework to be τal−1 ∝ x(1 − x)U 2 T 1/2 [Tietjen 65, Makowski 73, Harrison 76a, Harrison 76b], where T is the temperature and U is the amplitude of alloy potential fluctuation. Not surprisingly, the experiments focused exclusively on the temperature dependence of the charge transport and U was extracted from fitting the T dependent mobility data. Using Matthiessen’s rule, the alloy scattering rate was obtained from the total scattering rate by subtracting scattering rates due to other mechanisms, such as phonon and ionized impurity scattering. These other processes not only have considerable contribution to the total scattering rate but also have strong T dependence, leading to large uncertainties in the alloy scattering rate and therefore the extracted values of U . In the case of AlGaAs, one of the most important alloy semiconductors widely used in material structures for device applications as well as fundamental physics research, the value of U in the literature varies widely from 0.12 to 1.56eV[Sin , Ferry 97, Chandra 80, Saxena 81, Saxena 85, Look 92]. Our systems of 2DES with alloy disorder bring a good opportunity to solve this puzzle. We here present our high-accuracy experimental determination of the alloy scattering potential U as inferred from measurements on two dimensionally confined electrons in Alx Ga1−x As. The alloy scattering rate of two dimensional charge carriers 39 2.4: Amplitude of the alloy potential fluctuation is also proportional to x(1 − x)U 2 but, in contrast to that for three dimensional systems, it is expected to be independent of the temperature [Ando 82a, Fu 00, Basu 83, Bastard 84, Chattopadhyay 85]. This temperature independence of τal−1 has been verified in our experimental results in Fig. 2.2, and is theoretically related with the fact that the 2D density of states is a constant (see appendix A). Therefore the dependence of the alloy scattering rate τal−1 on the Al concentration x yields the scattering potential U as a fitting parameter of this dependence. Since we do not need to subtract a large T -dependent background, the uncertainty in U is greatly reduced as compared to its values determined in bulk samples. In Fig. 2.4 we show the dependence on x(1 − x) of the total scattering rate measured at 0.3 K for the first four samples of the first series. This dependence is found to be linear with a slope of 35 ns−1 per 1% Al concentration. Since at this temperature τ −1 (x) = τal−1 (x) + τr−1 , with τr−1 independent of x, the alloy scattering rate τ −1 (x) is also linear with x(1 − x). From the slope of this linear dependence we can extract the alloy potential U . Following a calculation in Appendix A, for spherically symmetric square well potentials around each scattering center we find τal−1 = 4V02 m∗ R∞ 0 u a3 ~ 3 4 (z)dz U 2 x (1 − x), where u (z) is the projection of the electronic wave function in the Z direction, a = 0.566 nm is the lattice constant of the compound crystal and V0 is the volume of the short range square scattering potential around each scattering center. Following the widely used spherical square potential well model with the radius r equal to the nearest neighbor atoms separation, the zinc blende structure of Alx Ga1−x As gives r = √ 3 a 4 and V0 = 34 πr 3 , which was used in the previous experiments[Harrison 76a, Chandra 80, Saxena 81, Saxena 85, Chattopadhyay 85]. The proper way to obtain u (z) is through 2.4: Amplitude of the alloy potential fluctuation 40 Figure 2.4: The dependence of τ −1 on x(1 − x) at 0.3 K for the first four samples of the first series. The dotted line is a linear fit to the data. 2.5: Possible alloy clustering in samples with high alloy concentrations 41 numerically solving the Poisson’s Equation with the boundary condition set by the sample structure. To simplify the problem, we use a good approximation – the Fang1 1 Howard variational wave function [Fang 66] u (z) = 21 b3 2 z · e− 2 bz , with the only 1 33me2 n 3 variational factor b being determined by the electron density n as b = . 8~2 Plugging all the parameters into the fitting formula of τal−1 , we obtain the value of the scattering potential to be U = 1.13 eV . The same result can be verified by analyzing the third series of samples. 2.5 Possible alloy clustering in samples with high alloy concentrations Although the above described simple virtual crystal model accounts for all features of our data for the first four samples in the first series and all the samples in the third series, it fails for the last two samples in the first series (9-5-97-1 and 9-17-97-1). Fig. 2.5 shows the τ −1 − x(1 − x) dependence at 0.3K for all the samples in the first series, and it appears that the samples with x = 4.1% and x = 8.5% have large deviations from the linear dependence set by the first four samples. We have considered the effect that the Z-direction wave function u (z) is different in these two samples due to the changed quantum confinement by high alloy concentration, and we have made a correction on Fig. 2.5 to address this effect. However, the deviations are still large. It is then natural to check the assumptions of our model used above. In the virtual crystal model, the alloy scattering centers locate randomly throughout the sample and the scattering events are independent. Thus scattering events at different 2.5: Possible alloy clustering in samples with high alloy concentrations 42 Figure 2.5: The dependence of τ −1 on x(1 − x) at 0.3 K for all the samples of the first series. The dotted line is a linear fit to the data from the first four samples. Large deviations from this line are observed for the last two samples. Even after a wave function form correction (the stars), the deviations are still large. 2.6: Lifetime of 2D electrons with alloy disorder 43 scattering centers are incoherent. At a high Al alloy concentration, the Al atoms could cluster, and this could undermine the assumption of random and independent scattering centers. Therefore, the assumption of random independent short-range alloy disorder is only valid in the dilute Al alloy impurity regime. In the high alloy impurity concentration regime, although we do not have direct evidence of alloy clustering at this point, it is almost certain that the nature of alloy disorder is different. 2.6 Lifetime of 2D electrons with alloy disorder All the fundamental characteristics at zero magnetic fields have been presented in the previous sections. In this section, we introduce weak magnetic field into the experiments, and gain further understandings to the nature of alloy disorder. As is shown in the previous sections, disorder is calibrated by the electron- disorder scattering rate. The total scattering rate for an electron is the inverse of the lifetime τ . The lifetime of an electron, namely, is the time scale that an electron can keep its original state before being scattered. In 2DES, various lifetimes have been introduced. In this section, we briefly introduce the concepts of quantum lifetime and transport lifetime through the discussion of electron-disorder scattering, then we present our measurement results on these lifetimes in the alloy systems. Our experimental results show undoubtedly that the alloy potential fluctuation is a type of short-range disorder. 44 2.6: Lifetime of 2D electrons with alloy disorder Figure 2.6: Electron scattering by an impurity in a solid. 2.6.1 Quantum lifetime and transport lifetime When we consider an electron being scattered by potentials that are constant in time, such as an impurity, Fermi’s golden rule is the proper way to calculate the scattering rate. Fig. 2.6 illustrates a scattering process of an electron by a potential 1 ~ V . Assume the initial and final states of the electron are plane waves φi = A 2 eik·~r and 1 ~ φf = A 2 ei(k+~q)·~r , where A is the area, ~k is the initial wave vector, q~ is the change on the wave vector by scattering, and θ is the angle change. Following the standard recipe of Fermi’s Golden Rule, one can sum up the scattering probability to all directions R d2 q~ and obtain the scattering rate τ −1 = ns 2π |Ṽ (~q)|2 δ[ε(~k + q~) − ε(~k)] (2π) 2 , where ns is ~ the density of scattering centers, Ṽ (~q) is the 2D Fourier transform of the scattering potential. The delta function δ[ε(~k + q~) − ε(~k)] shows that only energy-conserving scattering processes are counted by the golden rule. This scattering rate describes the probability that an electron is scattered away from its original state within unit time. The inverse of this rate is called the single-particle lifetime or the quantum lifetime τq . 45 2.6: Lifetime of 2D electrons with alloy disorder However, τq−1 is not the same scattering rate we have discussed in the previous section with conductivity and mobility measurement. The mobility-related scattering rate τtr−1 = e m∗ µ is called the transport scattering rate, which is associated with the transport lifetime τtr . The difference between τq−1 and τtr−1 lies in the weighting of different collisions. The quantum scattering rate τq−1 contains a sum over all scattering processes, equally weighted. This means small-angle scattering in which θ is tiny counts as much as backscattering events where θ = π and the electron’s direction is reversed. However, backscattering has a much larger effect on current than small angle scattering. Given that the component of electron’s motion parallel to its original direction is proportional to cos θ, one can put a weighting factor (1 − cos θ) on the R |Ṽ (~q)|2 δ[ε(~k + ~q) − integral and obtain the transport scattering rate as τtr−1 = ns 2π ~ d2 q~ ε(~k)](1 − cos θ) (2π) 2 [Ando 82b]. The ratio between the quantum scattering rate τq−1 and the transport scattering rate τtr−1 depends on the nature of the scattering centers. Assume the range of disorder is d, we have |Ṽ (~q)| ∝ e−|q|d . If the scattering potential is from short-range disorder, like the alloy potential fluctuation, d is only of atomic dimension, and the scattering is nearly independent of q for q < 109 /m. The scattering therefore can be treated as isotropic and the amplitudes of τq−1 and τtr−1 are of the same order. However, for long range disorder like the ionized impurity potential fluctuation, most of the scattering events has small q, which are not taken into account of τtr−1 , and τq−1 can be a few orders of magnitude larger than τtr−1 . Therefore the ratio estimate the range of disorder in a system. τq−1 −1 τtr or τtr τq offers a way to 46 2.6: Lifetime of 2D electrons with alloy disorder Table 2.4: Lifetimes and scattering rates for the first four samples of the first series Sample # 7-30-97-2 8-21-97-1 8-6-97-1 7-31-97-2 2.6.2 x [%] 0 0.21 0.33 0.85 τq [ps] 1.73 1.62 1.66 1.60 τq−1 [ns−1 ] 578 617 602 625 τtr [ps] 140 78 61 34 τtr−1 [ns−1 ] 7.08 12.8 16.2 29.3 τtr τq 81 48 37 21 Quantum lifetime of 2D electrons with alloy disorder – measured by SdH oscillations For 2DES, the quantum lifetime τq can be determined through the amplitude of the Shubnikov-de Haas (SdH) oscillations. Fig. 2.7 shows a typical plot of the SdH oscillation in our experiment (x = 0.85% for the sample this plot). Under low magnetic field, the amplitude of the SdH oscillations is given by the Dingle formula χ ∆ρ = 4ρ0 sinh e χ − ω πτ c q [Ando 82a, Coleridge 91], where ρ0 is the zero-field resistivity, ωc is the cyclotron frequency, and χ = 2π 2 kT . ~ωc For a given temperature, Data on the maximum or the minimum of the SdH oscillations is fit into the Dingle formula, and τq can be obtained as a fitting parameter. Fig. 2.8 shows the fitting of the data in Fig. 2.7, and τq = 1.6ps is obtained at all temperatures. We have measured the value of τq for the first four samples of the first series in a temperature range 0.3K < T < 1.2K. τq is found to be temperature-independent for all samples, and the values of τq and τq−1 are listed in Table 2.4. In these samples, the Al alloy concentration varies from 0 to 0.85%, and the transport scattering rate differs in a factor of 4. However, the quantum scattering rate among these samples vary less than 10%. As a result, the ratio τtr τq decreases from 81 to 21 when the Al concentration increases from 0 to 0.85%. The change of τtr τq 2.6: Lifetime of 2D electrons with alloy disorder Figure 2.7: Shubnikov-de Haas oscillation of a sample at various temperatures. 47 2.6: Lifetime of 2D electrons with alloy disorder 48 Figure 2.8: Extraction of the quantum lifetime of the electrons by Dingle formula. 2.7: Conclusions 49 indicates a change in the nature of disorder. For the sample with x = 0, long-range Coulomb potential fluctuation dominates, and most scattering events are of small angles, which are not counted in τtr−1 . Therefore, τtr−1 is very small in this sample although τq−1 is large. Adding short-range alloy impurities in effectively changes the ratio, because the isotropic short-range scattering centers add equally to τtr−1 and τq−1 [Fang 77]. These experimental results therefore offer a new verification that the alloy potential fluctuation is a type of short-range disorder. 2.7 Conclusions In this chapter, we have investigated the properties of 2DES residing in the Alx Ga1−x As− Al0.33 Ga0.67 As heterostructures at zero and low magnetic fields. The alloy disorder in these heterostructures have been characterized. The alloy scattering rate in 2D is shown to be temperature-independent. The range of the alloy potential fluctuation is close to the lattice constant (0.56nm), and the amplitude is extracted to be 1.13eV . In comparison with the conventional Coulomb potential fluctuation, which has a range of µm and amplitude of meV , the alloy potential fluctuation is a type of short-range disorder with strong amplitude. Chapter 3 Investigation on the quantum Hall plateau-to-plateau transition In the previous chapter, we have demonstrated that the alloy scattering rate in 2DES is temperature-independent and depends on the alloy concentration only. The alloy potential fluctuation has been characterized to be a type of short-range disorder with strong amplitude. By changing the Al concentration x in Alx Ga1−x As − Al0.3 Ga0.7 As heterostructures, we have had various samples with different amounts of alloy disorder. This gives us the opportunity to systematically investigate the quantum Hall plateau-to-plateau transition, which is a localization-delocalization transition. 3.1 The universality is called into question The plateau-to-plateau transition in the quantum Hall regime has been intensively studied [Pruisken 88, Wei 88, Huckestein 01, Sondhi 97] since the discovery of the integer quantum Hall effect (IQHE). In the IQHE, the Hall resistance Rxy has quantized values h N e2 over a wide range of the magnetic field B around integer Landau level filling factors N . The successive Hall plateaus correspond to separated energy regions 50 3.1: The universality is called into question 51 of localized states, and in between them are extended states [Laughlin 81, Aoki 81, Halperin 82]. It was shown that between two plateaus there is only one such extended state at a critical energy Ec [Pruisken 88, Ono 82, Aoki 85, Chalker 87, Huo 92]. As the Fermi energy approaches this critical energy, the localization length is supposed to diverge following a power law ξ ∝ |E − Ec |−ν with a universal critical exponent ν [Pruisken 88, Wei 88, Huckestein 01, Sondhi 97, Struck 06]. In the case the extended state is approached through sweeping magnetic field B, the critical divergence can be written as ξ ∝ |B − Bc |−ν , with the critical field Bc corresponding to Ec . As is discussed in the first chapter, the finite size scaling theory [Pruisken 88, Wei 88, Huckestein 01, Sondhi 97, Thouless 77] is invoked to extract ν from experimentally measured quantities, and a temperature scaling form has been established by this theory. Approaching zero temperature, the derivative of the Hall resistance Rxy taken at Bc diverges as a power law dRxy | dB B=Bc ∝ T −κ , while the half-width for the longitudinal resistance Rxx vanishes as ∆B ∝ T κ , where the exponent κ is expressed as κ = p/2ν with p being the temperature exponent of inelastic scattering. The first experiment on electrons confined to the interface of InGaAs-InP heterostructures found κ=0.42 [Wei 88]. Considering p=2 [Huckestein 99, Wei 94], the exponent ν is extracted to be 2.4, a value independently obtained by subsequent theoretical calculations [Huckestein 01, Sondhi 97, Chalker 88, Mil’nikov 88, Huckestein 90, Lee 93, Gammel 94]. However later studies in various other experimental systems raised doubts about the universality of the critical exponent. In the Si-MOSFET systems, κ was measured to range from 0.16 to 0.65 [Wakabayashi 89, Wakabayashi 92]; in GaAs − AlGaAs heterostructures, κ was found to vary from 0.28 to 0.81 [Sem , 3.2: Range of disorder 52 Koch 91a] or totally absent [Balaban 98]. These measurements show that κ is sampledependent and even varies for different transitions in a single sample. The universality of the quantum Hall plateau-to-plateau transition is therefore called into question. 3.2 Range of disorder It has been long appreciated that the nature of disorder is fundamentally different in the various systems mentioned above [Wakabayashi 89, Koch 91a, Wei 92, Hwang 93]. While the disorder in Si-MOSFET and GaAs − AlGaAs systems is dominated by long-ranged ionized impurity potentials [Nixon 90, Fogler 04], the alloy potential fluctuation plays a major role in the InGaAs − InP system [Sin ]. Since the plateauto-plateau transition is a localization-delocalization transition, the range of disorder can play a critical role. In this chapter, we introduce a new approach to the fundamental problem of the plateau-to-plateau transition in IQHE, an approach that is focused on the nature of the disorder potential. In the first half of this chapter, we present measurements of the temperature scaling on samples with various amounts of alloy disorder. We have found that the scaling exponent κ varies depending on the range of disorder in the system. The universal exponent κ = 0.42 is found in samples with disorder being dominated by short-range alloy potential fluctuations. In the second half of this chapter, we report our experiments at ultra-low temperatures down to 1mK. For a sample with dominant short-range disorder, a perfect power law scaling with κ = 0.42 has been established over two full decades of temperature. We have also experimentally identified the inelastic scattering exponent p = 2, which together with the value of κ determines the localization length exponent ν = 2.4. The ultra-low 3.3: Samples and experimental techniques 53 temperature also leads to a very long quantum phase coherence length LΦ , which we have identified to be a few millimeters at 10mK. The “long” range of Coulomb potential fluctuation is short in comparison with this length scale of LΦ , and we have observed that κ = 0.42 is restored at ultra-low temperatures for samples with dominant Coulomb disorder. 3.3 Samples and experimental techniques All three series of samples are studied in this chapter. Since samples of the first and second series share the same structure design and cover a wide range of Al impurity concentration, they are the focus in our investigation. Samples of the second series have parallel conductance layers besides the 2DES. This is removed by a back gate of -200V, and the detail is described in Appendix C. The complete characteristics of these samples are described in Table 2.2 and 2.3. They are all based on the GaAs − Al0.3 Ga0.7 As heterostructure, a two-dimensional electron system of high mobility (ne = 1.2 × 1011 /cm2 , µ = 3.7 × 106 cm2 /V.s). By introducing a small amount of Al into the GaAs during the growth process we obtain Alx Ga1−x As − Al0.3 Ga0.7 As heterostructures [Li 03]. A total of 9 samples were grown with different Al concentration x by exactly the same MBE process. As is shown in chapter 2, the alloy scattering rate is proportional to x(1 − x), and exceeds the residual scattering rate when x is larger than 0.24%. For samples with x > 2%, the scattering rate has a large deviation from the linear dependence on x(1 − x), and the deviation is believed to be from the effect of alloy clustering, which introduces correlations in the alloy scattering centers and thus renders the model of uncorrelated δ-function-like potential invalid [Sin ]. For samples in which 3.4: Critical exponent κ depends on the nature of disorder 54 the linear dependence of the scattering rate on x(1 − x) still holds, the ratio between the alloy scattering rate 1/τa and the background (residual) scattering rate 1/τb is a good measure of the dominance of alloy disorder. For each sample the longitudinal resistance Rxx and the Hall resistance Rxy are measured simultaneously in a 3 He system from 0.3K to 1K by using two lock-in amplifiers with a current excitation of 1nA and frequency of 5.7Hz. The sweeping rate of the magnetic field is kept sufficiently small to acquire at least 5 data points within 1mT. Figure 3.1 (a) and (b) shows the plots of Rxx and Rxy vs B at different temperatures for the sample with x = 0.85% and Figure 3.1 (c) and (d) shows the zoom-in of the transition between the plateaus around Landau level filling factors 4 and 3 (4-3 transition). 3.4 Critical exponent κ depends on the nature of disorder The measurements at various temperatures in a 3 He system offer us the opportunity to extract an temperature scaling exponent for each of the samples. Critical exponent κ is obtained from the power law fit of dRxy | dB B=Bc versus the temperature. Figure 3.2 shows the fitting of κ of the 4-3 transition for the samples with x = 0, x = 0.85% and x = 4.1%. The Rxy vs B data was smoothed by averaging within 1mT before the derivative was taken. We found that the exponents are all the same for the various transitions in the same sample, but vary from 0.42 to 0.59 for different 55 Ω Ω 3.4: Critical exponent κ depends on the nature of disorder Figure 3.1: (a), (b)The longitudinal resistance Rxx and Hall resistance Rxy at different temperatures for the sample with x = 0.85%. In this plot, ν denotes the Landau level filling factors. (c), (d)The transition between the plateaus of ν=4 and ν=3. A critical magnetic field Bc =1.40T is observed. 3.4: Critical exponent κ depends on the nature of disorder 56 Table 3.1: Sample properties and measurement results. The Al concentration x, the electron density ne and mobility µ, the ratio θ between the alloy and the background scattering rates at 0.3K, and the scaling exponent κ of four plateau-to-plateau transitions. There are two wafers with x = 0.85%, and three pieces (A, B, C) are cut from the first wafer. x ne µ θ κ 11 2 6 2 % 10 /cm 10 cm /Vs 6-5 5-4 4-3 3-2 0 1.13 3.7 0 0.58 0.58 0.57 0.21 1.32 2.05 0.8 0.57 0.56 0.58 0.33 1.25 1.62 1.3 0.49 0.50 0.49 0.85 A 0.43 0.42 0.42 0.41 B 1.16 0.89 3.3 0.42 0.41 0.42 0.42 C 0.42 0.42 0.42 0.41 0.85 1.18 0.91 3.2 0.41 0.42 0.42 0.42 1.4 1.14 0.56 5.6 0.43 0.43 0.42 0.42 1.9 1.26 0.46 0.49 0.49 0.50 0.51 2.6 1.22 0.34 0.58 0.60 0.59 0.58 4.1 0.83 0.20 0.58 0.57 samples. The fitting error is ±0.01. The measured values of κ for different plateauto-plateau transitions in each sample are shown in Table 3.1. All the integer plateauto-plateau transitions we studied are around Landau levels where spin splitting is already resolved at 1.2K. Some high Landau level transitions (10-8, 12-10, 14-12) are observed spin-unresolved at 1.2K, but spin-splitting occurs at about 0.5K, therefore the spin-unresolved transitions are not studied. The transitions at lower Landau levels are also not studied in this chapter because there are fractional quantum Hall states between those integer plateaus. 3.4.1 Three disorder regimes, one optimal window The dependence of the critical exponent κ on x is plotted in Fig. 3.3 over a large range of x from 0 to 4.1%. Values of κ determine three regimes. In the first regime, 3.4: Critical exponent κ depends on the nature of disorder 57 xy Figure 3.2: dR | vs T for the 4-3 transition in various samples. From down to dB B=Bc up, x = 0, 0.85%, 4.1% respectively. Data of different x has been shifted vertically in log-log scale for a clear comparison. Scaling exponents κ are obtained from the linear fits. 3.4: Critical exponent κ depends on the nature of disorder 58 when x is very small, κ is as large as 0.58, and decreases with increasing x. For the second regime x is between 0.65% and 1.6% and the alloy scattering rate is from 2.5 times to 6.5 times of the background ionized impurity scattering rate. In this regime, κ is 0.42 for all samples. Finally, with x larger than 1.6%, the system is driven into the third regime and κ increases with x. From the earlier characterization of scattering mechanisms we observe that, as shown by the large values of θ, in the second regime the disorder is dominated by the short-range alloy potential fluctuations. We have measured five pieces of samples from three different wafers grown in two different years in this regime. As is listed in Table 3.1, all the results show consistently κ = 0.42 within the fitting error ±0.01. Therefore we found that the exponent κ is sample and x-independent only in the short-range disordered regime. Its value is the same as for the InGaAs − InP system [Wei 88] and it is consistent with theoretical calculations [Huckestein 01, Sondhi 97, Chalker 88, Mil’nikov 88, Huckestein 90, Lee 93, Gammel 94]. The universality observed in this second regime is further confirmed by our observation in a larger temperature range from 1K down to 30mK. Figure 3.4 shows The T -dependence of ∆B and dRxy | dB B=Bc at the 4-3 transition of the sample with x = 0.85% in this T range. The data taken in the dilution refrigerator and the data taken from the 3 He system fall on top of each other where they overlap in temperature. Both the power-law of ∆B vs T and that of dRxy | dB B=Bc vs T yield a critical exponent κ=0.42. These power-laws over the large range of temperature confirm the scaling and define the exponent with a higher precision. The deviation of the exponent κ from the universal value 0.42 in the first and the third regimes shows that the nature of the transition is indeed affected by the 3.4: Critical exponent κ depends on the nature of disorder 59 Figure 3.3: Dependence of the exponent κ on the Al concentration x for the 4-3 transition. In the second regime, the alloy scattering rate τa−1 is from 2.5 times to 6.5 times of the background long-range scattering rate τb−1 , and thus scattering is dominated by alloy disorder. In this regime the exponent κ is 0.42. 3.4: Critical exponent κ depends on the nature of disorder 60 Figure 3.4: Temperature scaling down to 30mK of the 4-3 transition for the sample with x = 0.85%. Data taken in the dilution fridge (up-triangles) and that from the 3 He system (circles) fall on the same straight line in the log-log plot. The slope of both curves in (a) and (b) give the critical exponent κ=0.42 with a high precision. 3.4: Critical exponent κ depends on the nature of disorder 61 nature of the disorder. The plateau-to-plateau transition is viewed as a localizationdelocalization transition, while the physics of quantum localization [Anderson 58] applies only within the range of the quantum phase coherence length LΦ , which is usually identified to be the inelastic scattering length lin [Thouless 77]. In the scaling theories, it is assured that the range of disorder is below the length scale of lin by assuming the disorder to be uncorrelated δ-function-like potential fluctuation [Pra a]. However, for samples in the first regime where x is small, the disorder of the system is dominated by the potential of the ionized impurities. Being screened by the 2D electrons, the Coulomb potential fluctuation becomes slowly varying with a large correlation length of the order of µm [Nixon 90, Fogler 04]. With the disorder range comparable with or even larger than lin , the quantum localization crosses over toward the classical percolation. In the second regime, where the disorder is dominated by the short-range potential fluctuations, the transport is quantum in nature and the universality of the plateau-to-plateau transition is restored. In the third regime, the likely clustering of Al atoms introduces correlations in the sample that may change the nature of the disorder and destroy the universal scaling. Therefore, only the second regime gives systems dominated by short-range disorder, and it is our “optimal window” of Al concentration x. 3.4.2 Measurement on samples with different densities Most samples of the first and second series share similar electron densities. To make sure that the nature of disorder is the only factor that determines the critical exponent, we have carried out temperature scaling experiments on the third series of samples. The third series of samples are Alx Ga1−x As − Al0.1 Ga0.9 As heterostructures 3.4: Critical exponent κ depends on the nature of disorder 62 with various x, and the 2DES is induced by LED illumination. In the temperature scaling experiment, the electron densities are around 6.8 × 1010 /cm2 . We have selected two samples, with x = 0.8% and x = 0.4%, from the third series. Magneto-transport measurement was carried out on these samples both in a 3 He system and in a dilution refrigerator. Standard lock-in technique is used to measure the longitudinal resistance Rxx and the Hall resistance Rxy with a current excitation of 1nA and frequency of 5.7Hz. Fig. 3.5 shows the Hall resistance of the sample with x = 0.8% and electron density n = 6.8 × 1010 /cm2 . Around the 4-3 plateau-to-plateau transition, a critical magnetic field of 0.75T is observed for this transition, and the critical exponent is obtained to be κ = 0.42 over the temperature range shown on the figure. The Hall resistance of each sample was measured at various temperatures, and the critical exponent of the plateau-to-plateau transition is obtained by fitting the data to dRxy | dB B=Bc ∝ T −κ . We found the critical exponent is constant for different plateau-to-plateau transitions of the same sample. The obtained exponents of these samples are plotted on Fig. 3.6 together with those from the first and second series of samples. For the sample with x=0.8%, an exponent κ=0.42± 0.01 is obtained; the other sample with x=0.4% falls into the first regime, and a larger exponent κ=0.46± 0.01 is obtained. The results from the third series are consistent with those from the first two series. Therefore the universality of the plateau-to-plateau transition in short-range disordered system has been tested with different electron densities. The nature of disorder, seems to be the deterministic factor for the scaling exponent. 3.4: Critical exponent κ depends on the nature of disorder 63 Figure 3.5: The Hall resistance of the sample with x=0.8% and n=6.8×1010 /cm2 around the 4-3 transition. Within the shown temperature range, a critical exponent κ=0.42 is obtained. 3.4: Critical exponent κ depends on the nature of disorder 64 Figure 3.6: Dependence of the exponent κ on the Al concentration x for the 4-3 transition. The dots represent data from samples of the first two series, and the crosses represent samples from the third series. Data obtained from samples of different densities agrees fairly well with each other. 3.4: Critical exponent κ depends on the nature of disorder 3.4.3 65 On the non-universal exponents In the theoretical calculations, the universal critical exponent ν=7/3 results from a network model of quantum percolation, where the quantum phase coherence is kept in the transport [Chalker 88, Mil’nikov 88, Huckestein 90, Lee 93, Gammel 94]. On the other hand, an exponent ν=4/3 was obtained with theories of classical percolation[Trugman 83, Lee 93]. Using these values of ν and the κ=p/2ν relationship, we infer that the quantum-classical crossover effect increases the exponent κ from 0.42 up towards the classical value of 0.75. The κ values we obtained in the first and third regimes are still well below 0.75, showing that the system is still away from an ideal classical percolation regime. The deviations of exponent κ from the universal value was explained by putting uncertainty on the temperature exponent p [Koch 91b]. p was obtained in the Fermi liquid theory to be 2, which was confirmed by excitation current scaling experiments [Huckestein 99, Wei 94]. However it was proposed that, in samples with a high concentration of ionized impurities, the attractive Coulomb potential of the ions may be attributed to non-Born scattering and leads to a value of p that is larger than 2 [Koch 91a, Koch 91b, Haug 87]. This is not the case in our samples since the alloy potential fluctuations are neutral and cannot give rise to any inelastic scattering. In Section 3.7, our experimental results on samples of various sizes directly show that p = 2, and it is ensured that the deviation of κ is from a change of ν due to a fundamental crossover effect from quantum localization toward classical percolation. 3.5: Power-law scaling over two full decades of temperature 3.5 66 Power-law scaling over two full decades of temperature Since the plateau-to-plateau transition is a quantum phase transition, we hope to investigate the temperature scaling to the zero temperature limit. We first select a sample in the optimal window – sample 7-31-97-2 with 0.85% Al impurity. The electron density in this experiment is 1.2×1011 /cm2 and the mobility is 8.9×105 cm2 /V.s. A rectangle shaped specimen of 4.5mm×2.5mm is cut from the wafer, and diffused Indium Ohmic contacts are made on the edges for the transport measurements. Our ultra-low temperature experiment was carried out in a nuclear demagnetization /dilution refrigerator in collaboration with Dr. Jian-Sheng Xia at the MicroKelvin Lab of the University of Florida. The base bath temperature (Tb ) of the cryostat is below 1mK. Previous works have shown that the electron temperature (Te ) of 2DES can be cooled below 4mK using the specially designed cold contacts in this setup [Pan 99b, Xia 00, Pan 01]. Standard lock-in technique is used to measure the longitudinal magneto-resistance Rxx and the Hall resistance Rxy with a current excitation of 1nA and frequency of 5.7Hz. To minimize disturbance of the measurements to the sample, only one lock-in amplifier was used in the experiment, therefore only one of Rxx and Rxy is measured during each magnetic field sweep. The sweep rate of the magnetic field is as slow as 0.05T/Hour to allow long average time of the lock-in (20s) and to avoid disturbance to the measurements. It is well accepted that a temperature scaling of the plateau-to-plateau transition can be obtained either from dRxy | dB B=Bc ∝ T −κ or from the half-width of Rxx by 3.6: Termination of the power-law scaling at ultra-low temperatures 67 ∆B ∝ T κ . Since the measurement on Rxx results in the same exponent [Pruisken 88, Huckestein 01, Sondhi 97, Wei 88], we here concentrate on the Hall resistance Rxy . Rxx is checked at a few temperatures, and is always consistent with the Rxy measurement. We have obtained the values of the dRxy | dB B=Bc dRxy | dB B=Bc at different temperatures, and plotted − T dependence of the 4-3 transition on a log-log scale in Figure 3.7. The data taken in three different measuring cryostats falls on top of each other where they overlap in temperature. In the temperature range from 1.2K down to 12mK, we have observed a perfect power law scaling (dRxy /dB)|Bc ∝ T −κ with κ=0.42±0.01. We have also checked the half-width ∆B of Rxx at various temperatures, and found that ∆B decreases from 0.28T down to 0.05T as the temperature decreases from 0.8K to 12mK. The scaling of ∆B ∝ T κ results in an exponent κ = 0.41, and is consistent with the Rxy scaling. In the previous study of quantum phase transitions, there was no example [Sondhi 97] that is nearly as clean as the remarkable case of the classical lamda transition in superfluid liquid Helium. This experiment demonstrates the first example that the power law critical behavior of a localization-delocalization quantum phase transition can be observed in two full decades of temperature. 3.6 Termination of the power-law scaling at ultralow temperatures As we lower the temperature below 10mK, dRxy | dB B=Bc is observed to saturate instead of diverging. The saturation is shown in Figure 3.8. We have checked a few temperatures, and found that the half-width ∆B of Rxx also saturates at this temperature. 3.6: Termination of the power-law scaling at ultra-low temperatures 68 xy Figure 3.7: Perfect temperature scaling dR | ∝ T −0.42 of the 4-3 transition dB B=Bc over two decades of temperature between 1.2K and 12mK. Data from three different experimental cryostats have temperature ranges overlapping with each other and fall on each other at the overlapping temperatures. 3.6: Termination of the power-law scaling at ultra-low temperatures 69 xy Figure 3.8: The saturation of dR | at low temperatures. The saturation temperdB B=Bc ature Ts =10mK is obtained from the cross point between extrapolations of the higher temperature data (power law (dRxy /dB)|Bc ∝ T −0.42 ) and the lower temperature saturated data (horizontal dotted line). 70 3.7: Experiment on samples of various sizes Before any further consideration, one has to rule out the possibility that the saturation is due to heating of the electrons by the applied excitation current or by external noise, i.e., the electron temperature Te can not be cooled below 10mK. To investigate the internal heating due to the excitation current, we have measured Rxy with different excitations at the base bath temperature Tb =1mK. The values of dRxy | dB B=Bc that with different excitations are displayed in Fig. 3.9, and we have found dRxy | dB B=Bc is constant for excitations below 2nA. Since the excitation current applied in our experiments is 1nA, the saturation of dRxy | dB B=Bc cannot be from current heating. Furthermore, a high mobility sample with a prominent fractional quantum Hall feature around filling factor 5/2 has tested the system and shown that external noise by itself does not heat the electrons beyond 4mK when the cryostat is at Tb [Pan 99b, Xia 00, Pan 01]. We infer from the arguments above that the saturation of dRxy | dB B=Bc 3.7 below 10mK is not an effect from electron heating. Experiment on samples of various sizes We review the origin of the temperature scaling to understand its termination at low temperatures. The temperature scaling form dRxy | dB B=Bc ∝ T −κ is obtained by the finite size scaling theory[Huckestein 01, Sondhi 97]. In this theory, the transport properties are determined by the ratio between the localization length ξ ∝ |B − Bc |−ν and the effective sample size which is usually considered to be the quantum phase coherence length LΦ . As the temperature approaches zero, LΦ increases following p LΦ ∝ T − 2 , with p being the temperature exponent of inelastic scattering. However, the actual sample size L is the limit for LΦ , and it is anticipated that at low enough temperature LΦ would saturate at L, thus terminates the temperature scaling. This 3.7: Experiment on samples of various sizes 71 xy Figure 3.9: dR | of the 4-3 transition with different excitation currents at the dB B=Bc xy is conbase bath temperature 1mK. With excitation current I below 2nA, dR | dB B=Bc dRxy stant. Current heating is observed at I=5nA and dB |B=Bc is reduced substantially from the value of I=2nA. 72 3.7: Experiment on samples of various sizes kind of finite-size saturation had been observed by Koch et al in mesoscopic samples of size ranging from 10µm to 64µm [Koch 91a]. However our sample has the size of 4.5mm×2.5mm, and LΦ of this macroscopic length scale has never been reported. 3.7.1 Smaller samples, higher saturation temperatures – identification of the quantum phase coherence length LΦ and inelastic exponent p We have fabricated rectangle-shaped samples of various sizes to study the saturation of the temperature scaling. The width of these samples ranges from 500µm down to 100µm, with the length-to-width ratio being kept to 4.5:2.5. Fig. 3.10 demonstrates the data of dRxy | dB B=Bc vs T for all samples with various sizes. Although the data from different samples do not fall on each other, the exponent κ = 0.42 is agreed upon by all samples. For all these samples, dRxy | dB B=Bc saturates at low temperatures, and the saturation temperature Ts varies with sample size. The dependence of Ts on the sample width W is plotted in Fig. 3.11. Within the experimental uncertainty, Ts is found to be inversely proportional to W . While W is reduced from 2.5mm to 100µm, Ts is increased from 10mK to 320mK. The strong size-dependence of Ts clearly demonstrates that the saturation is a finite-size effect. If we assume LΦ reaches the actual sample size W at Ts , the Ts ∝ W −1 dependence implies that LΦ is inversely proportional to the temperature, which confirms the inelastic scattering temperature exponent p=2. The experimental identification of p=2, together with the temperature scaling exponent κ = 0.42, has then determined the localization length exponent ν = 2.4 3.7: Experiment on samples of various sizes 73 Figure 3.10: Temperature scaling for samples of various sizes. The dotted straight lines represent the power-law exponent 0.42. Although the data from different samples sample do not fall on each other, the exponent κ = 0.42 is agreed upon by all samples. The power law scaling with κ = 0.42 is terminated at various temperatures. 3.7: Experiment on samples of various sizes 74 Figure 3.11: The sample size dependence of the saturation temperature Ts of dRxy | . The length-width ratio of all samples is kept to be 4.5:2.5. The value of dB B=Bc Ts is inversely proportional to the sample width W within the error. from the relationship κ = p/2ν. We have therefore verified unambiguously that the quantum Hall plateau-to-plateau transition is a universal quantum phase transition. 3.7.2 Discussion on the quantum phase coherence length The millimeter length scale of LΦ at low temperatures is rather surprising. In the literature, LΦ is expected to be large only along the sample edge due to the suppression of electron-electron scattering in the quantum Hall edge channels [Machida 98]. In the region around the quantum Hall plateau-to-plateau transition, physics of the 75 3.7: Experiment on samples of various sizes bulk dominates, and our observation suggests that quantum phase coherence can be kept over a long distance in the bulk as well. To understand the large bulk LΦ around the plateau-to-plateau transition, we estimate the LΦ of the bulk 2DES at zero magnetic field. For clean 2DES at low temperatures, electron-electron inelastic scattering dominates the dephasing mechanism. Following the method in [Yacoby 94], we esti−1 mate the electron-electron scattering rate τe−e [Yacoby 91, Yacoby 94, Menashe 96, 1 Giuliani 82], and obtain the value of LΦ by LΦ = (Dτe−e ) 2 , with D being the electron diffusion coefficient. At the temperature Ts =10mK, LΦ is estimated to be 1.4mm, which is of the same order of our sample size. This estimation suggests that the millimeter-size LΦ in the bulk is a result from the high sample quality (thus a large diffusion coefficient D) and the low temperature that significantly reduces the electronelectron scattering. One elegant way to visualize the physics underlying the quantum Hall effect is the edge channel picture [Halperin 82, Streda 87, Buttiker 88]. In this picture, the current carrying states consist of edge channels analogous to classical ”skipping orbits” in the quantum Hall plateau regions, and LΦ along the edge is very long due to the perfect reflection of the skipping orbits from the edge potential. We have considered this picture as a different way to qualitatively understand the saturation of dRxy | . dB B=Bc In the plateau-to-plateau transition region, electrons from one edge channel can travel to the opposite-propagating edge channel on the other side via resonant tunneling [Jain 88, McEuen 90], which smears out the sharpness of the plateau-to-plateau transition. The saturation of dRxy | dB B=Bc shows that the probabil- ity of inter-channel tunneling saturates below Ts when the whole sample is phase coherent. 3.7: Experiment on samples of various sizes 76 The millimeter scale LΦ in our samples at ultra-low temperatures has presented an example that quantum mechanics prevails in a macroscopic regime in semiconductor systems. We have therefore observed the quantum localization length ξ in the millimeter length scale around the plateau-to-plateau transition, and tested the physics of Anderson localization in 2DES in a millimeter length scale over a wide temperature range. Although we anticipate that LΦ reaches the sample size at temperature Ts , we did not find any feature of the universal conductance fluctuation (UCF) on either Rxy or Rxx . We suggest that the absence of UCF results from a thermal averaging effect. The thermal length LT is given by LT = (hD/kT ), and is only about 20µm in our samples at Ts =10mK. Since LT is much smaller than the sample size L, the UCF is thermally smeared out even though the electrons are dynamically phase coherent all over the sample [Lee 85, Lee 87]. 3.7.3 More complicated scaling Data from the samples of various sizes has shown more complexity into the scaling of the quantum Hall plateau-to-plateau transitions. As is shown in Fig. 3.10, the data from different sized samples do not fall on each other. This suggests that the actual sample size might set a prefactor to complicated. dRxy , | dB B=Bc which makes the scaling more 77 3.8: Outside of the optimal window at ultra-low temperatures 3.8 Outside of the optimal window at ultra-low temperatures The ultra-low temperature has brought us novel perspectives in our investigation on the sample with x = 0.85%. In this section, we introduce the ultra-low temperatures to the experiments on samples outside of the optimal window as well. We have selected 3 more samples with x = 0, 0.21% and 4.1% from the first series. The samples with x = 0 and 0.21% belong to Regime I, while the sample with x = 4.1% belongs to regime III. 3.8.1 Sample with x = 0 – crossover effect in temperature scaling and the range of Coulomb disorder For the sample with x = 0, we have measured the Hall resistance around the 4-3 transition at various temperatures. The critical field Bc is resolved to be 1.35T and the temperature dependence of dRxy | dB B=Bc is shown in Fig. 3.12. Over three decades of temperature, three different scaling behaviors have been observed. First, in the lowest temperature decade of T < 15mK, dRxy | dB B=Bc saturates. This type of saturation was identified in the previous sections to be a finite size effect when the quantum phase coherence length LΦ reaches the sample size at the saturation temperature Ts . To the high temperature end, in the decade of T > 120mK, we have found a power law scaling dRxy | dB B=Bc ∝ T −κ with κ=0.58. This exponent is consistent with the value measured in the 3 He system, which is presented in section 3.4. The most striking feature of the plot locates in the middle temperature decade, and the universal critical exponent κ=0.42 is restored. On the log-log plot of dRxy | dB B=Bc vs T , we extrapolate 3.8: Outside of the optimal window at ultra-low temperatures 78 Figure 3.12: The temperature scaling of the sample with x=0 over three decades of temperature. Three different temperature scaling behaviors have been observed: dRxy | saturates in the lowest temperature decade below 15mK; power law scaling dB B=Bc with κ=0.58 in the highest temperature decade; power scaling with the universal exponent κ=0.42 in the middle temperature decade. The crossover temperature between the regions with κ=0.58 and κ=0.42 is obtained to be 120mK by extrapolations. 3.8: Outside of the optimal window at ultra-low temperatures 79 the linear parts of κ=0.42 in the middle decade of T and κ=0.58 in the high decade of T , and a crossover temperature Tc = 120mK is obtained from the cross of the extrapolations. The restoration of the universal exponent κ=0.42 indicates that below Tc the potential fluctuation of system can be considered to be short-range disorder. Disorder in the sample with x = 0 is mostly from the Coulomb potential fluctuations. The disorder range d is determined to be long or short by being compared with the quantum phase coherence length LΦ . Since LΦ increases at lower temperatures, it is anticipated that the condition d LΦ is fulfilled at some point when the temperature is low enough. The crossover temperature Tc represents a length scale at which such a transition happens. 3.8.2 Evolution of temperature scaling from Regime I to the optimal window We have further investigated the crossover effect in a different system with x = 0.21%. Adding alloy scattering centers into the 2DES decreases the relative weight of the background long-range Coulomb potential fluctuation in the disorder of the system, thus reduces the effective disorder range. The temperature scaling data of the samples with x = 0, 0.21% and 0.85% is displayed in Fig 3.13 (the saturation parts have been removed for clarity), and a trend of evolution is clearly observed. For the sample with x = 0.21%, the alloy scattering rate τal is about 0.8 times as large as the Coulomb scattering rate τi from ionized impurities. As is presented in section 3.4, the system is in Regime I with a high temperature scaling exponent κ=0.58. However, the universal scaling exponent κ=0.42 is observed below a crossover 3.8: Outside of the optimal window at ultra-low temperatures 80 Figure 3.13: Evolution of the crossover effect. The temperature scaling of 4-3 transition in Alx Ga1−x As − Al0.3 Ga0.7 As heterostructures with various x values. Data in xy the lowest temperature decade has been removed since dR | saturates. dB B=Bc (a) x = 0; (b) x = 0.21%; (c) x = 0.85%. Crossover effect between temperature regions κ=0.42 and κ=0.58 has been observed in (a) and (b). Crossover temperature Tc is obtained to be 120mK in (a) and 250mK in (b). 3.8: Outside of the optimal window at ultra-low temperatures 81 temperature Tc = 250mK. Although this is a similar crossover effect as what we have observed in the sample with x = 0, Tc is substantially higher. The higher Tc indicates a reduction of the effective disorder range by the alloy impurities, and a less LΦ is required to fulfill d LΦ . The sample with x = 0.85% is in the optimal window and τal is 3.3 times of τi . We have observed a perfect power law scaling dRxy | dB B=Bc ∝ T −0.42 over the temperature range from 10mK all the way up to 1.2K, which indicates that d LΦ is always fulfilled in our experiment. If there is a crossover effect, it happens well beyond the temperature range of our experiments. 3.8.3 Crossover exponents The large temperature range in our experiments is essential to observe the the crossover effect. If our measurements are limited in a small temperature range around the crossover temperature, we might obtain a “crossover exponent” which has an intermediate value between the two “true” exponents. Fig. 3.14 shows the zoom-in of the temperature scaling around the Tc in the sample with x = 0, and an intermediate exponent κ=0.49 is obtained by power law fitting over the relatively small temperature range from 55mK to 280mK. As a matter of fact, any scaling exponent between 0 and 0.42 can be obtained if the experiment is performed over a small temperature range around the saturation temperature. Since most experiments in the literature are carried out within in one decade of temperature to measure κ, we suggest that the crossover and saturation effect should be accounted for most of the various values of κ obtained in experiments. 3.8: Outside of the optimal window at ultra-low temperatures 82 Figure 3.14: The zoom-in of the area squared by the dotted lines in 3.13(a). By power-law fitting over a relatively small temperature range, an intermediate exponent κ=0.49 is obtained. 3.8: Outside of the optimal window at ultra-low temperatures 3.8.4 83 Sample with x = 4.1% – A hidden length scale in clustered alloy systems of Regime III We have measured the sample with x = 4.1% down to the base temperature of our dilution refrigerator. The sample in this experiment in 3mm long and 2mm wide. The temperature scaling of the 4-3 transition is shown in Fig. 3.15. We have found that the exponent κ=0.58 persists all the way down to Ts = 65mK. The higher saturation temperature and the persistent exponent κ=0.58 suggest a very different disorder regime in comparison with the other two. The exponent κ=0.58 is the same as the high temperature scaling exponent for the samples in Regime I. It is still unknown at this stage if κ=0.58 represents an alternative universal value or only happens coincidentally. Exponent 0.58 is recently obtained in some works on the quantum Hall plateau-to-insulator transition [van Schaijk 00]. It is widely believed that the plateau-to-plateau transition and plateau-to-insulator transition are governed by same physics, and this coincidence might hide some universality behind. The exponent 0.58 is still well below the classical percolation value of 0.75, therefore it is likely that κ=0.58 represents a picture in the regime between quantum localization and classical percolation. A semi-quantum percolation picture has been recently developed to understand the physics with clustered alloy impurities. In this picture, it is proposed that the overcrowded alloy scattering centers block most of the tunneling paths and an exponent κ=0.56 is obtained [Xin ]. The high saturation temperature Ts =65mK is a surprise. From the finite size experiment in Section 3.7, the phase coherence length LΦ reaches the actual sample size at Ts . Since the mobility of this sample is much lower than those of samples in 3.8: Outside of the optimal window at ultra-low temperatures 84 Figure 3.15: Temperature scaling of the 4-3 Transition for 2DES embedded in a Alx Ga1−x As−Al0.3 Ga0.7 As heterostructure with x = 4.1%. A saturation temperature xy is observed to be Ts =65mK. The exponent κ=0.58 persists into lower | of dR dB B=Bc temperatures until Ts is reached. 85 3.9: Conclusions the other two regimes, it is impossible that LΦ reaches the sample dimension of 2mm at a much higher temperature. For this sample at 65mK, LΦ is estimated to be 20µm by considering the electronelectron scattering rate, and is much smaller than the sample width. We propose that there is a hidden length scale Lh in this sample as the effective sample size, and LΦ =Lh at the temperature Ts . Since alloy clustering is likely, Lh could be related with the cluster size. 3.9 Conclusions In this chapter, we have investigated the quantum Hall plateau-to-plateau transition in the integer quantum Hall regime by measuring samples with controlled alloy disorder. We have verified that the plateau-to-plateau transition is indeed a universal quantum phase transition and built up a framework to understand the role of disorder in this transition. When the disorder in the system is dominated by short-range alloy potential fluctuations, we have found a perfect power-law temperature scaling dRxy | dB B=Bc ∝ T −κ with a universal exponent κ = 0.42 over two full decades of temperature. The inelastic scattering exponent p is identified to be 2 by an experiment on samples of various sizes. The localization length exponent ν = 2.4 is therefore verified by the experimentally measured values of κ and p. At ultra-low temperatures, dRxy | dB B=Bc is found to saturate, and the phase coherence length reaches the sample size at the saturation temperature Ts . In systems with disorder being dominated by long-range Coulomb potential fluctuations, a semi-classical exponent κ = 0.58 is observed at high temperatures. Below 86 3.10: Perspective of future works a crossover temperature Tc , the universal exponent κ = 0.42 is restored, as the quantum phase coherence length becomes much longer than the Coulomb disorder range. We suggest that the various measured exponents in the literature arise from this crossover effect. For samples with very high Al concentrations, alloy clustering is likely, and the effective sample size is determined by a hidden length scale related with the cluster size. As a result, the exponent κ=0.58 persists into low temperatures until dRxy | dB B=Bc saturates at 65mK, a relatively high temperature. 3.10 Perspective of future works 3.10.1 Rxx measurement Due to the experimental constraints, we only concentrated on the the Hall resistance Rxy in our experiments. However, the longitudinal resistance Rxx is also informative. First, a temperature scaling of the plateau-to-plateau transition can be established by analyzing the temperature dependence of Rxx . The half width ∆ of the Rxx peak in the transition region obeys a power law ∆ ∝ T κ , which offers another way to study the plateau-to-plateau transition. Second, Rxx , together with Rxy , gives the conductivity tensor components σxx and σxy . The value of σxx itself contains information about the plateau-to-plateau transition, and is predicted by some theories to be universal[Huckestein 01, Sondhi 97]. Moreover, there is an alternative approach to the universality of the plateau-toplateau transition from the hopping conductivity σxx [Polyakov 93, Hohls 02] away from the critical magnetic field. In this theory, information of the localization length ξ 87 3.10: Perspective of future works can be acquired by the T -dependence of the hopping conductivity σxx =σ0 exp(−(T0 /T )1/2 ) and the localization length scaling ξ ∝ |B − Bc |−ν can be carried out directly. It would be helpful to try this scaling approach with the measurements on Rxx . Therefore, additional measurements of Rxx will greatly improve our understanding of the plateau-to-plateau transition in the framework built by this thesis. 3.10.2 Correlated alloy disorder Another possible experimental direction is in regime III of alloy samples. To explain the experimental data in this regime, we have assumed the existence of alloy clustering and correlation between the alloy scattering centers. In the scaling experiments, a hidden length scale related with alloy clustering is proposed to understand the high saturation temperature of dRxy | . dB B=Bc Future experiments on samples with various sizes will help to determine this length scale. Additionally, with more samples in regime III, we will be able to measure the scaling of the plateau-to-plateau transition with various Al concentration x, and observe the contiuous evolution from independent alloy impurities to correlated alloy impurities. This will help to build a complete picture for correlated disorder. Chapter 4 New physics brought out by alloy disorder in high magnetic fields In the previous chapter, we have investigated the quantum Hall plateau-to-plateau transition of the integer quantum Hall regime. We concentrated on the high flling factors in order to avoid the complication from the fractional quantum Hall effect. In this chapter, we expand our study into the first Landau level (ν < 2) and investigate the novel physics at high magnetic fields. For a clean 2DES subjected to high magnetic fields, the single electron picture, which offers a good understanding to the integer quantum Hall effect, does not work. In this regime, both the electron-electron interaction and the electron-disorder interaction have to be taken into account to understand the novel electron phases such as the fractional quantum Hall (FQH) liquids [Tsui 82, Laughlin 83, Jain 89] and the Wigner crystals [Wigner 34, Lozovik 75, Chen 04]. While various experiments have been carried out to investigate systems with different interaction parameter rs (ratio of electron-electron Coulomb energy and Fermi 88 4.1: Fractional quantum Hall gaps in 2DES with alloy disorder 89 energy) [Csathy 04, Csathy 05], the effect of disorder has not been studied systematically in understanding the competition of FQH liquid and Wigner crystal. For 2DES embedded in a conventional GaAs − AlGaAs heterostructure, disorder is mainly from the static ionized impurities. Being screened by the 2DES, the Coulomb impurity potential has a long range that can be comparable with the quantum phase coherence length of the system; moreover, the amplitude of the screened Coulomb potential has a strong dependence on magnetic field [Shklovskii 86, Efros 93]. This complicated nature of the Coulomb disorder makes it very difficult to be used to study the effect of disorder in magneto-transport. Alloy potential fluctuations, as neutral scattering centers, are not screened by the electrons and its amplitude has no dependence on the magnetic field. Our samples with controlled alloy disorder then offer an opportunity to study the effect of disorder under the condition of higher B-filed and low filling factors. In this chapter, we report the first transport measurements at the lowest Landau level on 2DES with various amount of alloy disorder. 4.1 Fractional quantum Hall gaps in 2DES with alloy disorder First, we study the FQHE between filling factors ν =1 and 2. Since the FQHE originates from electron-electron interactions, and is only observed in relatively clean 2DES, it is appreciated that the existence of disorder disrupts the formation of the composite fermions, thus reduces the FQH gap. 4.1: Fractional quantum Hall gaps in 2DES with alloy disorder 90 Table 4.1: Characteristics of the first series of samples after illumination Sample # 7-30-97-2 8-21-97-1 8-6-97-1 7-31-97-2 x [%] 0 0.21 0.33 0.85 n[1011 /cm2 ] 2.35 2.32 2.37 2.41 µ[106 cm2 /V.s] 12.1 2.74 1.91 0.83 τ −1 [ns−1 ] 2.16 9.55 13.7 31.4 The first four samples of the first series are selected for this experiment. The samples are 2DES residing in Alx Ga1−x As − Al0.3 Ga0.7 As heterostructures with Al concentration x = 0, 0.21%, 0.33% and 0.85%. For these samples, the Al impurity is in the dilute regime, and the scattering rate is proportional to x(1 − x). To enhance the quality of the sample, we have illuminated the samples with an LED at low temperatures. As a result, the electron densities in these samples are increased to around 2.4 × 1011 /cm2 . The fundamental characteristics of these samples after illumination is shown in Table 4.1. In comparison with data in Table 2.1, the mobility of the sample with x = 0 has enhanced a lot, while the mobility of the sample with x = 0.85% changes very little. Therefore the illumination reduces the residual scattering rate, which is mainly from the ionized impurity Coulomb potentials, but does not affect the alloy scattering rate. We have measured the Rxx of all these samples in the magnetic field region between filling factors ν =1 and 2 at various temperatures from 100mK to 500mK. Fig. 4.1 shows the magneto-transport data for the sample with x = 0.85%. Two FQH states, ν= 5 3 and 43 , are observed to develop at lower temperatures. The conductivity in an energy gap is activated by thermal excitations. When the Fermi level lies in the fractional quantum Hall gap ∆, the longitudinal resistance obeys a exponential thermal excitation form Rxx ∝ e − 2k∆ T B [Platzman 88], with kB 4.1: Fractional quantum Hall gaps in 2DES with alloy disorder 91 Figure 4.1: Rxx data for the sample with x = 0.85% between filling factors ν =1 and 2. The FQH states, ν = 35 and 34 , are the focus of this plot. Figure 4.2: Fit of the Rxx data into the exponential formula. Values of the FQH gap are obtained for the FQH states ν = 35 and 34 . 4.1: Fractional quantum Hall gaps in 2DES with alloy disorder 92 Figure 4.3: Independence of the FQH gap on the alloy concentration x. Both the ν = 5 and 34 gaps are constants within the experimental uncertainty. 3 being the Boltzmann constant. Fitting the Rxx data into this exponential formula, as is shown in Fig. 4.2, we obtain the ν = 5 3 FQH gap ∆ 5 =0.43±0.01K, and the ν = 3 4 3 FQH gap ∆ 4 =0.48±0.01K. 3 We have extracted ∆ 5 and ∆ 4 for all the four samples, and the values are displayed 3 3 in Fig. 4.3. Surprisingly, the amplitude of the FQH gaps is independent on x within the experimental error. We then conclude that the form of composite fermion and the FQHE is not disrupted by dilute short-range alloy disorder. Part of this experiment was carried out in the SCM1 system of the National High Magnetic Field Lab (NHMFL) in Tallahassee, FL. 93 4.2: Particle-hole symmetry in the Wigner crystal phase 4.2 Particle-hole symmetry in the Wigner crystal phase In this section, we move into the regime of stronger magnetic fields with ν < 1, where Wigner crystal becomes a strong candidate as the ground state of the 2DES. In the lowest Landau level, pinned Wigner crystal is usually observed around the lowest FQH state with Landau level filling factor ν = 1 5 for most high mobility GaAs − AlGaAs heterostructures [Willett 88, Jiang 90]. Two insulating states, one high field insulator with ν < 1 5 and one reentrant insulator with ν slightly larger than 15 , are separated by the terminal ν = 1 5 FQH liquid. Both these insulating states are proved to be Wigner crystals[Goldman 90, Engel 97, Ye 02, Chen 04]. In 2D hole samples [Santos 92] and in a 2D electron sample of narrow quantum well [Yang 03], the terminal FQH state can be shifted to ν = 1 3 due to profound changes of the energies of the FQH states and the Wigner crystal with different electron-electron interactions. In the Composite Fermion picture of FQHE physics, there exists a “particle-hole” symmetry [Pra b, Jain 89] that connects two FQH states ν = p 2p+1 and ν = 1 − p 2p+1 (p is an integer) and makes each state the mirror state of the other. This particle-hole symmetry has been observed in all series of FQH states (as is shown in Fig. 1.7), and theories expect its existence for Wigner crystals as well. There is indeed evidence from the microwave resonance experiments that the high field end of the ν = 1 Hall plateau can represent a “hole” crystal which is symmetrical to the high field Wigner crystal [Chen 03]. However, the mirror state of the reentrant Wigner crystal has never been observed. 94 4.2: Particle-hole symmetry in the Wigner crystal phase We intend to study the effect of alloy disorder to the Wigner crystal in the lowest Landau level. Two Alx Ga1−x As − Al0.3 Ga0.7 As heterostructures from the first series, with x = 0.85% and 0.21%, are selected to be measured in this experiment. For the sample with x = 0.85%, the terminal FQH state are shifted to ν = insulating state appears between ν = 1 3 1 3 and a reentrant and 52 . Furthermore, we have observed in this 2 3 sample a novel reentrant integer quantum Hall effect (RIQHE) between ν = 3 . 5 and This reentrant state can be considered as a reentrant “hole” crystal, and a perfect particle-hole symmetry is then established for the Wigner crystal phase. Our samples are measured in the in a 3He-4He dilution refrigerator at NHMFL, Tallahassee, FL. A LED is placed above the sample to provide illumination at low temperatures. After illumination, the electron density of the samples is 2.4×1011 /cm2 , and the mobility values are 2.7 × 106 cm2 /V.s and 0.83 × 106 cm2 /V.s for the samples with x = 0.21% and 0.85%, respectively. The 35T resistive magnet system of NHMFL enabled us to reach beyond Landau level filling factor ν = 1 3 at 30T. The magnetic field sweep rate is about 1T/min in the experiments. Standard lock-in technique is used to measure the longitudinal magneto-resistance Rxx and the Hall resistance Rxy simultaneously with a current excitation of 10nA and frequency of 7.4Hz. 4.2.1 Reentrant insulator between ν = 1 3 and 2 5 Fig. 4.4 shows the Rxx data of the sample with x = 0.85% in a wide magnetic field range up to 32T at various temperatures. The fractional quantum Hall states ν = 31 , 2 5 and ν = 1 3 3 7 are all well developed, and a reentrant insulating state is observed between and 25 . At 60mK, the longitudinal resistance Rxx reaches a peak value that exceeds 600kΩ at ν=0.37 and drops back to zero as the system enters the ν = 1 3 FQH 4.2: Particle-hole symmetry in the Wigner crystal phase 95 Figure 4.4: Rxx data of the sample with x = 0.85% over the full range of magnetic field from 0 to 32T. A high resistance peak is observed at 27T (ν=0.37) between the ν = 13 and 25 FQH states, and is identified to be a reentrant insulator. 96 4.2: Particle-hole symmetry in the Wigner crystal phase state. At higher field, the 2DES becomes a high field insulator and Rxx diverges. It is rare in 2DES based on single GaAs − AlGaAs heterostructure that the FQH series terminate at the ν = 1 3 FQH state. Since both the reentrant insulator and the high field insulator are believed to represent pinned Wigner crystals, the shift of the terminal FQH state to ν = 1/3suggests that the chance of Wigner crystal formation is enhanced in this sample with short-range disorder. 4.2.2 Reentrant integer quantum Hall effect (RIQHE) between ν = 2 3 and 3 5 The more remarkable feature of the magneto-transport data is observed at lower magnetic field between ν = 2 3 and 35 . In Fig. 4.5, we show the data of both Rxx and Rxy up to B=18T at 60mK for the sample with x = 0.85%. Besides the FQH states ν = 32 , 3 5 and 47 , which are represented by a series of minimums on Rxx , we have observed one additional minimum on Rxx at ν=0.63, between the ν = FQH states. Around this field, Rxy is non-monotonic and falls from the ν = down to the quantized value h e2 2 3 2 3 and 3 5 plateau of the ν=1 plateau. To rule out the possibility that the reentrant feature of Rxy is from the mixing of Rxx into Rxy in the experiments, we have carried out measurements on this sample with the magnetic field being reversed. The same feature has appeared with reversed field, and the effect is confirmed not to be a mixing effect. In the literature, the only reported Rxx minimum between the ν = states is the ν = 7 11 2 3 and 3 5 FQH fractional quantum Hall state in a sample of ultra-high mobility (3.1 × 107 cm2 /V.s), and is regarded as the FQHE of composite fermions [Pan 03]. However the reentrant behavior of Rxy has ruled this possibility out. What we have 4.2: Particle-hole symmetry in the Wigner crystal phase 97 Figure 4.5: Rxx and Rxy data of the sample with x = 0.85% at 60mK up to 18T. One additional minimum is observed on Rxx at ν=0.63, between the ν = 32 and 35 FQH states. At this field, Rxy falls on the quantized value eh2 of the ν=1 plateau. 98 4.2: Particle-hole symmetry in the Wigner crystal phase observed is then a reentrant integer quantum Hall effect (RIQHE). A similar type of reentrant effect had been found in high Landau levels around ν = 7 2 and 52 , as the electrons form a bubble solid [Pan 99a, Eisenstein 02]. Since the mechanism of bubble solid is theoretically prohibited in the lowest Landau level [Fogler 96, Fogler 97], the RIQHE we have discovered was never expected in this picture. To further investigate the nature of the reentrant state, we have carried out measurements at various temperatures. Fig. 4.6 summarizes the T -dependence of the magneto-transport around the RIQHE. Data shown in the upper panels are taken at 800 mK, and two FQH states ν = 2 3 and 3 5 are observed. As the temperature goes lower, the RIQHE starts to develop, and becomes well established at the base temperature of 60mK. From these observations, it is apparent that the RIQHE is a low temperature effect, and the reentrant state represents the ground state in the magnetic field region between ν = between ν = 3 5 2 3 and 53 . The 60mK Rxy data is also non-monotonic and 47 , and suggests that more reentrant states will develop at lower temperatures. Since the RIQHE is observed in an alloy sample, it is anticipated that alloy disorder is essential in its formation. We have measured another sample with less alloy disorder (x = 0.21%), and Fig. 4.7 compares the results from the two samples at 60mK. Since there is a small density offset between these two samples, the transport data is plotted versus filling factor ν. The reentrant effect does not exist for the sample with x = 0.21%. We then conclude that the reentrant state is the ground state only when the disorder in the system is dominated by short-range alloy potential fluctuations. 4.2: Particle-hole symmetry in the Wigner crystal phase 99 Figure 4.6: Temperature evolution of the RIQHE. (a)Temperature evolution of Rxy ; (b) Temperature evolution of Rxx . The minimum on Rxx and the reentrant Rxy at ν=0.63 both develop at lower temperatures. 4.2: Particle-hole symmetry in the Wigner crystal phase 100 Figure 4.7: Evolution of the RIQHE with increased alloy disorder. (a) Rxy evolution; (b) Rxx evolution . The RIQHE is not observed in the sample with x=0.21%, but becomes prominent in the sample with x=0.85%. 4.2: Particle-hole symmetry in the Wigner crystal phase 101 Figure 4.8: The full spectrum of particle-hole symmetry between the corresponding FQH states, and between the RIQHE and the reentrant insulator. 4.2.3 Particle-hole symmetry The Reentrant integer quantum Hall state appears at ν=0.63, and forms a nice symmetry with the reentrant insulator (RI) at ν=0.37. RI represents the pinned Wigner crystal of electrons, and the reentrance occurs because the downward cusp in the energy of the Laughlin liquid makes it ground state in a narrow range of filling factor around the terminal FQH state [Pra b, Willett 88, Jiang 90]. The particle-hole symmetry between the RIQHE and RI suggests that the RIQHE represents a “hole” crystal with respect to the ν=1 integer quantum Hall state. This is the first time that particle-hole symmetry is observed for Wigner crystals, together with that for the FQH liquids. The complete spectrum of particle-hole symmetry is demonstrated in Fig. 4.8. 4.3: Alloy disorder and the reentrant insulators 102 As we have pointed out in Section 4.1. The dilute alloy disorder does not affect the FQH liquid. However, it appears in this section to enhance the formation of Wigner crystals. Therefore Wigner crystals win more ground competing with the FQH liquid in a 2DES dominated by short-range alloy disorder, and particle-hole symmetry of the Wigner crystal phase can be established in such a system. The microscopic mechanism that alloy disorder enhances Wigner crystal is still unknown at this point. 4.3 Alloy disorder and the reentrant insulators In this section, we present some preliminary experimental results on the reentrant insulators. This experiment is carried out with the third series of samples. Since the densities of these samples are as low as 6.5 × 1010 /cm2 , we have the opportunity to investigate the magnetic field induced Wigner crystals at a relatively low field. We have carried out measurements on these samples in a 3 He system. The samples have Al concentrations x = 0, 0.4%, 0.8% and 1.2%. The densities of all 4 samples are adjusted to be around 6.5 × 1010 /cm2 with well tuned LED illuminations. All the other fundamental characteristics of these samples are listed in Table 2.3. The measurements on ρxx (longitudinal resistance per square) at 0.3K is summarized in Fig. 4.9. It appears that the samples with more alloy disorder is more insulating at high magnetic field, and the terminal FQHE state has been shifted to ν = 1 3 even for the sample with x = 0.4%. Therefore we have shown directly that the magnetic field induced Wigner crystal is enhanced by short-range alloy disorder, although we do not have enough resolution of x at this stage to observe the gradual shift of the terminal FQHE state. With more samples in the future, we would be able 4.3: Alloy disorder and the reentrant insulators 103 Figure 4.9: Rxx data for the third series of sample at 0.3K. For samples with more alloy disorder, the high field part is more insulating. A reentrant insulator state is observed between the ν = 31 and 52 FQH states in all samples except for the one with x=0. 104 4.4: Conclusions to find the critical x value in a sample that the terminal FQH state is just shifted to 31 , and this information might be helpful to understand the formation of Wigner crystals. Before we finish this section, we point out that Fig. 4.9 looks qualitatively similar to the plots shown in a recent work[Csathy 05] that reports the enhancement of Wigner crystal by stronger electron-electron interactions. This striking similarity might offer some insight on the interplay between electron-electron interaction and disorder in the high magnetic firld regime. 4.4 Conclusions We have investigated the physics in the first Landau level for 2DES with controlled alloy disorder. The studies concentrate on the influence of short-range alloy disorder to the competition between the FQH liquids and the Wigner crystals. We have found that the amplitude of the fractional quantum Hall gaps is independent on the amount of alloy disorder in the system. On the other hand, alloy disorder enhances the formation of Wigner crystals. As a result, the terminal FQH state in systems with alloy disorder has been shifted to ν = 31 . More excitingly, alloy disorder has induced a novel reentrant integer quantum Hall effect between filling factors ν = 2 3 and 53 . This reentrant state can be considered as a reentrant “hole” crystal with respect to the ν=1 integer quantum Hall state. For a 2DES with disorder being dominated by short-range alloy potential fluctuation, a complete particle-hole symmetry is then established for both Wigner crystals and FQH liquids. Appendix A The van der Pauw method In this appendix, we provide details of the van der Pauw measurements, as well as tips in the real experiments. The van der Pauw method[van der Pauw 58a, van der Pauw 58b], applies for sheet samples of arbitrary shapes in the resistivity measurements. For a typical 2DES sample with Ohmic contacts illustrated in Fig. A.1, we can obtain the resistivity ρ via the following process. First, we drive current through contacts 1, 2 and measure voltage drop across contacts 3,4 to obtain a resistance R12,34 ; then we drive current through 1, 3 and measure voltage across 2, 4 to obtain another resistance R13,24 . The resistivity ρ of the sample can be calculated by solving the equation e −πR12,34 ρ +e −πR13,24 ρ = 1. When R12,34 and R13,24 are very close to each other, an approximate solution can be obtained as ρ = π ln 2 × R12,34 +R13,24 . 2 For a real experiment, several tips have to be followed in the van der Pauw method to obtain ρ with a high precision. 1. The Ohmic contacts must be at the boundary of the sample. 105 106 Figure A.1: Typical contact locations in the measurements. 2. The Ohmic contacts should be as small as possible. Any errors given by their non-zero size will be of the order d/L, where d is the diameter of the contacts and L is the distance between two contacts. 3. The van der Pauw method discussed above is only for isotropic samples. All samples studied in this thesis are grown in the (100) direction of GaAs, and are isotropic. For anisotropic samples such as heterostructures grown in the (311) direction of GaAs, there are two different resistivities ρx and ρy in different directions x and y, and what the van der Pauw method yields is the geometric average of these √ two resistivities ρx ρy . The values of ρx and ρy in an anisotropic sample can be obtained by measuring a L-shaped Hall bar or by using an extended version of van der Pauw method[Montgomery 71, Price 72]. 4. The van der Pauw method is only good when the sample is homogeneous. The homogeneity of a sample can be checked by reciprocal measurements. Theoretically the reciprocal resistances R12,34 and R34,12 should be identical. In practice, there can be a small difference between these two resistances due to the inhomogeneity in 107 the sample. Usually a difference smaller than 10% between these two resistances is required to obtain a meaningful value of resistivity. 5. For a better precision, one should select contacts combinations carefully so that R12,34 and R34,12 are close to each other. 6. For high mobility samples, the resistivity can be as small as a few Ohms, and it is important to ensure the external magnetic field to be zero. If there is a finite magnetic field, the Hall voltage might mix into the resistance measurement. A field as small as a few milli-Teslas can generate a Hall voltage drop comparable to the voltage drop caused by the sample resistance. Appendix B Calculation of alloy scattering rate in 2DES In this Appendix, we present the details of our calculation to obtain the alloy scattering rate for 2DES. The result of this appendix is used in Chapter 2. For Convenience, everything in this calculation is expressed in unit sample areas. B.1 Assumptions ~ First, the wave function of the 2D electrons is written in the form Φ = eik·~r u (z), where u (z) is the projection of the wave function in Z direction (the direction in which electrons are confined), and ~k and ~r are 2D wave number and position vectors. From Fermi’s Golden Rule, the scattering rate can be calculated via: P M~ ~ 0 2 δ E~ − E~ 0 . Scattering Rate W ~k = 2π kk k k ~ ~k 0 Now consider the alloy Alx Ga1−x As. We assume that the atomic potentials are constant around each atom with a radius r0 . Let U be the difference between the Ga or Al atom potentials. From section 1.3.2, the scattering potential at a Ga atom is ∆U Ga = xU , and the scattering potential at an Al atom is ∆U Al = (1 − x) U . The 108 109 B.2: Calculations signs of the scattering potentials are ignored because they do not matter with Fermi’s Golden rule. To write the two cases of scattering rate together, for a as Ga or Al, the scattering potential ∆U a = ∆U0a is a constant in the space around an a atom with a radius r0 , and is 0 outside. Therefore r0 can be called a “scattering radius”, and defines a scattering potential volume V0 . B.2 Calculations R i“~k−~k0 ”·~r 2 The Matrix element M~k~k0 = e u (z) ∆U a (x, y, z) dxdydz R i“~k−~k0 ”·~r 2 u (z) dxdydz. = ∆U0a V0 e As an approximation, r is small (r → 0) and e R M~k~k0 = ∆U0a V0 u2 (z) dxdydz. ” “ 0 i ~k−~k ·~r → 1, we have Now consider an atom at the depth z0 , R and we have M~k~k0 (z0 ) = ∆U0a V0 (z0 ) u2 (z) dxdydz, because u (z) expands very long (tens of nms) comparing to the scale of the lattice constant, u (z) ≈ u (z0 ) in the sphere around the atom at z0 . So M~k~k0 (z0 ) = ∆U0a u2 (z0 ) · 34 πr03 = ∆U0a u2 (z0 ) V0 . So as to the atom at z0 , (∆U0a )2 u4 (z0 ) V02 · we have W ~k, z0 = 2π ~ 1 (2π)2 R 0 δ E~k − E~k0 d2~k — here we have changed “Sum” to ”Integral” by the standard way. Therefore W ~k, z0 = 2π (∆U0a )2 u4 (z0 ) V02 N (Ek ), where N (Ek ) is the density ~ of states and N (Ek ) = m 2π~2 is the 2D density of states, and m is the effective mass of electrons. Spin degeneracy is not taken into account in the density of states, because electron spin does not change during alloy Scattering. For the whole sample, the total scattering rate is obtained by integral over the unit area: B.3: The potential fluctuation U in our samples W ~k = R U nitArea dxdy R∞ 0 110 R ∞ V02 1 W ~k, z0 = 0 u4 (z) dz· VLattice · ~13 x (1 − x) mU 2 dz0 Vlattice The Volume V0 is the “Scattering Volume” for each atoms, and the VLattice is the average volume in the lattice for one Ga or Al atoms. VLattice = a3 4 in the Alx Ga1−x As V2 0 lattice, where a is the lattice constant. Just Define VLattice = Vef ; R∞ Let I = 0 u4 (z) dz, and we have obtained the alloy scattering rate τ −1 = W = IVef m x (1 ~3 − x) U 2 . The mobility of the sample can also be deduced through the relationship µ = B.3 eτ . m The potential fluctuation U in our samples We assume that the 2D electrons have the Fang-Howard wave function [Fang 66, Sin ] 1 1 in Z direction: u (z) = 12 b3 2 z · e− 2 bz . The only variational factor b can be determined by the density of the 2DEG as b = 1 33me2 n 3 . Since the density of our samples (the first four samples of the first series) 8~2 is around n = 1.22×1011 /cm2 = 1.22×1015 /m2 , we obtain b = 1.823×108 /m. ConseR∞ quently, we can obtain I = 0 u4 (z) dz = 3.4179×107 . For GaAs samples with ZincBlende lattice structure, we take the scattering volume [Harrison 76a, Chandra 80, √ 3 3 4 Saxena 81, Saxena 85, Chattopadhyay 85] V0 = 3 π 42 a and VLattice = a4 , where a is the lattice constant 0.56nm. Plugging these parameters, together with the electron permittivity = b 0 = 13.20 , and the effective mass m = 0.067me into the scattering rate formula, we obtain a linear dependence of τ −1 on x (1 − x) with the prefactor proportional to U 2 . Fitting the formula with our experimental data in Section 2.4 yields U = 1.13eV . Appendix C Removal of parallel conductance In this appendix, we show how the parallel conductance layers in the second series of samples are removed, and demonstrate the difference in the magneto-transport data before and after the removal of parallel conductance. The second series of samples were grown in the Bell labs with exactly the same design as those of the first series. However, the first series of samples were grown in 1997, and the second series were grown after six years in 2003. The MBE was not in the best condition during the growth of the second series. Fig. C. 1 shows the magneto-transport data of a typical sample in the second series. In this figure, it appears that Rxx does not reach zero at integer filling factors, and Rxy is not quantized very well in the plateau region. In fact, a parabola-shaped background is observed in the Rxx data. These types of magneto-transport features can usually be attributed to parallel conductance layers in the heterostructure[Grayson 05]. As is shown in Fig. C. 2, a layer of charge carriers resides in parallel with the 2DES. The parallel charge carriers can be positive or negative, and we have applied different back gate voltages to test it. A positive gate voltage up to +200V has no effect on the magneto-transport. However, when we switch the polarity of the gate voltage to 111 112 Figure C.1: Magneto-transport data of a typical sample from the second series before the parallel conductance is removed. 113 Figure C.2: Parallel conductance layers and the back gate. negative, we found the parallel charge carriers can be removed. At a gate voltage of −200V , all the parallel charge carriers are removed, and we observe perfect magnetotransport data shown in Fig. C.3. We conclude that the parallel conductance layer has negative charge carriers. Since the parallel conductance layer is removed with a gate voltage of -200V, we expected that the 2D electron density decreases with additional negative gate voltage. From a simple capacitance model of the heterostructure, we expected that the 2DES can be totally exhausted with a gate voltage of −400V . However, this does not happen. The 2DES density keeps unchanged even when the gate voltage is −400V . The reason behind this fact is still not understood at this stage. Although we do not have a definite answer about the nature of the parallel conductance layers in the second series of samples, we enjoy the fact that they can be removed in a perfect way with −200V gate voltage. As is presented in Chapter 3, 114 Figure C.3: Magneto-transport data of a typical sample from the second series after the parallel conductance is removed. 115 scaling experiments on the quantum Hall plateau-to-plateau transitions have been carried out in these samples with the parallel conductance layers being removed. Appendix D Sample preparation recipes In this appendix, we provide the processes of preparing a typical DC transport sample that involves photolithography. All the clean room processes are done in the small clean room of Tsui Labs. Photolithography 1. Clean the sample surface with acetone, isopropanol and methanol in this order. If the sample is polluted with a lot of grease, supersonic-clean it with the help of trichloroethylene (TCE). Handling TCE has to be extremely cautious because it evaporates and is dangerous. 2. Blow dry sample with the N2 gun. 3. Place a small portion of old photoresist as glue on the back of a piece of cover slip glass. 4. Drop sample into the center of the glue photoresist. 5. Put the glass cover slip on a hotplate at 1100 C for 5 minutes to dry the glue. 6. Place glass slide on the vacuum chuck used for spinning. 116 117 7. Spin on HMDS for 40 seconds at 4000 RPM. 8. Spin on AZ5214 resist for 40 seconds at 4000 RPM. 9. Prebake on hotplate for 4 minute at 1100 C on the hotplate. 10. Align the photomask with the sample and expose for 15 seconds. 11. Develop in 1:1 H2 O : MIF312 for 25 seconds. 12. Rinse sample in DI water and inspect under microscope. Etching 1. Put the sample into the solution of H2 SO4 : H2 O2 : H2 O = 1 : 8 : 80 for 1-2 minute. 2. Wash out the photo resist with acetone, then rinse it with isopropanol. 3. Blow dry sample with the N2 gun. 4. Drop sample (upside down) into the center of the wax. 5. Wait a few minutes until sample is flat and then remove slide from hotplate. 6. Blow sample with the N2 gun. 7. Inspect the sample under microscope. Ohmic Contacts 1. Carefully put InSn shots to the proper positions on the sample with a soldering iron. 118 2. Anneal the contacts in a forming gas environment (N2 + H2 ) for 14 minutes at 440 degrees. 3. Patiently wait until the alloy annealing station cools down and take the sample out. 4. Wire the sample up. Appendix E Miscellaneous experimental projects during PhD research In this appendix, I present a few experimental projects I carried out during my PhD research. These projects are also about the physics of 2DES. However, they are not within my PhD thesis on the effect of alloy disorder. I here present a brief sketch for each of the projects. E.1 The quantum Hall insulator. The quantum Hall insulator is a novel phase in the quantum Hall regime and has been under intensive study. For a quantum Hall insulator, while Rxx diverges with increasing magnetic field, Rxy is kept to its value h e2 for the first quantum Hall plateau[Hilke 98]. The quantum Hall insulator is usually identified with a very long ν = 1 Hall plateau and is understood within a classical percolation picture[Shimshoni 99]. It is anticipated that such a percolation picture will eventually break down at higher magnetic fields and quantum localization will occur. As a result, the Hall resistance is expected to diverge as well at high magnetic fields[Zulicke 01]. We try to observe this 119 E.1: The quantum Hall insulator. 120 crossover from classical percolation to quantum localization, and carried out transport experiments. The sample is a AlGaAs−GaAs−AlGaAs quantum well of a narrow width 15nm. The electron density is 4.6 × 1010 /cm2 and the mobility is 1.2 × 105 cm2 /V.s. Fig. E. 1 shows the Hall resistance in the insulating regime. The Hall plateau is long at low temperatures. However, Rxy has a trend to diverge at large magnetic field and curves at different temperatures tend to cross at B = 4.8T . Our measurement is incomplete due to various experimental difficulties: 1. the Ohmic contacts tend to break down in the insulating regime. 2. The longitudinal resistance brings a lot of mixing into the Hall resistance when the sample is in deep-insulating regime. Averaging data from direct and reversed magnetic fields helps to remove the mixing. However, since the mixing is very sensitive to both magnetic field and temperature, a complete removal is hard. I have proposed a few tricks to overcome these difficulties: 1. The Ohmic contact areas and the rest areas should be separately gated so that the contact area can have a higher electron density. This way even at high magnetic field the contacts can stay out of the insulating phase and keep being Ohmic. 2. Introduce a Hall probe into the measurement as a precise meter for magnetic fields. The removal of Rxx mixing can therefore be done in a much better way. Transport experiments in the insulating regimes are extremely hard. We hope future study will eventually overcome all the experimental difficulties and demonstrate a clear picture of physics. 121 E.2: A quantum Hall spin filter. Figure E.1: Hall resistance in the quantum Hall insulator regime. E.2 A quantum Hall spin filter. The integer quantum Hall effect arises from the formation of the Landau levels. When the Zeeman energy of the electrons are taken into account, each Landau level is split into two branches with electrons spin up and down. This split has an amplitude of gµB B, and is experimentally observable when it is larger than kB T , where g is the Lander factor, µB is the Bohr magneton and kB is the Boltzmann constant. The amplitude of the spin split is usually smaller than that of the Landau level split, and the ratio between the Zeeman and Landau splits is m∗ g. me In GaAs, the g factor is −0.44 and m∗ = 0.067me . In transport experiments the spin split is represented by the split of one Rxx peaks into two at low temperatures. Therefore each spin-resolved Rxx peak represents a specific choice of electron spin. E.2: A quantum Hall spin filter. Figure E.2: Sample geometry and the quantum Hall edge current picture. 122 E.2: A quantum Hall spin filter. 123 We have studied the sample 7-31-97-2 with 0.85% Al alloy. However, the alloy is not relevant in this experiment, and what matters is the shape we have fabricated the sample into. We have made two Hall-bars with geometry shown in Fig. E. 2 (a). The width of the Hall bar is 250µm, the length is 1mm. The dimension of the contact legs varies in samples. In the first sample, each of the legs is 1mm long and 100µm wide, while the leg dimension is 50µm by 10µm in the second sample. We have measured the magneto-transport of these two samples and found striking features in comparison with the data from a sample with no contact legs (Ohmic contacts directly alloyed to the sample edge). The transport data of all three samples is shown in Fig. E. 3 with magnetic field being swept from 0 to 3T. Rather to our surprise, we have found that the Rxx peaks associated with electrons spin down have been significantly cut smaller for the two samples with contact legs. For the sample with 10µm-wide legs, spin-down Rxx peaks are totally cut off. As to the Hall resistance Rxy , non-monotonic features have been observed. Although it had been discovered before that the quantum Hall features can be altered in samples of smaller sizes [Zheng 85, McEuen 90, van Wees 91], a spin-selecting cut-off effect by smaller contact legs is never expected or reported. However, A survey of recent literature shows that this type of effect may have been observed in a less dramatic way[Vakili 05, Lai 06]. Samples in [Vakili 05, Lai 06] have similar geometries as our sample with 100µm-wide legs, and it is striking that it is found that the spin-down Rxx peaks are always suppressed, even the g factors of their materials (Si and AlAs) are both positive. It is striking that the same type of spin selection has been observed in different materials with various signs of g-factor. We intend to test this in a sample of E.2: A quantum Hall spin filter. Figure E.3: Magneto-transport of samples with various leg dimensions. 124 E.2: A quantum Hall spin filter. 125 Al0.085 Ga0.915 As − Al0.3 Ga0.7 As heterostructure (#9-17-97-1). In this sample 2DES resides in Al0.085 Ga0.915 As and the g factor is only about −0.12. The electron density is 2.1 × 1011 /cm2 and the mobility is 2 × 105 cm2 /V.s. As is shown in Fig. E. 3, the same spin-filtering effect is observed as well although it is much weaker due to the smaller g-factor or low mobility. The spin selection of this effect is then tested again. The filtering against spin-down electrons therefore seems universal for this effect, and we here present a possible explanation within the quantum Hall edge current picture (illustrated in Fig. E. 2(b)). In the edge current picture, the electrons travel through dissipationless edge channels. Due to a screening effect [Chklovskii 92], the width of the inner edge channel in high mobility samples is widened to an order of tens of micrometers, which is much larger than the magnetic length. In the narrow leg, if the two opposite-propagating inner edge currents A and B are very close to each other, the backscatterings between them effectively cancel them both thus cut off the corresponding edge channel’s conductivity. The cut-off of the spin-down Rxx peaks then suggests that the backscattering is stronger when the inner edge channel is spin-down. We propose a novel spin Hall effect in the edge channel to understand this selection. In this proposed spin-Hall effect, assume the current is ~j, and the spin direction is ~s. The electrons accumulate in the direction defined by −~s × ~j. As is shown in Fig. E. 2(b), when the leg edge channel is spin-down, the leg edge currents A and B both accumulate towards the center of the leg and become closer to each other, therefore the scattering is stronger. when the leg edge channel is spin-up, the leg edge currents accumulate towards the edge of the leg the scattering is much weaker. While this proposed spin Hall effect is only a speculation at this stage, it E.2: A quantum Hall spin filter. 126 Figure E.4: Magneto-transport for a sample with a smaller g factor. qualitatively explains the observed selection. More experiments are to be done in the future to verify the nature of this spin-filtering effect. In GaAs materials, the g-factor can be changed from negative to positive by applying a hydrostatic pressure[Chen 03]. If we can measure our sample with various hydrostatic pressures, we might be able to observe a continuous switch of the suppressed Rxx peaks while the sign of g factor is switched. This experiment is to be completed in the future. E.3: Anomalous Hall effect in a Si-doped quantum well. E.3 127 Anomalous Hall effect in a Si-doped quantum well. In this section, we briefly present our observations in a novel quantum well structure. In this structure, Si atoms are added into the confinement of AlGaAs − GaAs − AlGaAs quantum wells. The width of the quantum potential well is 20nm. The sheet density of Si-dopants in the quantum well is 1 × 1011 /cm2 and 2 × 1011 /cm2 for each of the two samples, respectively. Si donors are also implanted outside the quantum wells to offer additional electrons. When Si atoms are added into GaAs and substitute Ga atoms, a Mott transition [Mot ] is appreciated with increasing impurity level since Si has one extra valence electron. When the Si impurity level is low, the extra electrons from Si fill the lower Hubbard band and the system is a Mott insulator. In our systems, the Si donors outside the quantum wells offer additional electrons into the quantum well and these electrons will have to be placed into the upper Hubbard band. It is well know that the lower Hubbard band is anti-ferromagnetic, however, little has been resolved for the upper Hubbard band. With these samples of Si-doped quantum wells, we have the opportunity to study the electronic properties in the upper Hubbard band. We have carried out experiments at both high and low magnetic fields very carefully on the sample with Si density 2 × 1011 /cm2 . The magneto-transport data for this sample over a full range of field is shown in Fig. E. 5, and the electron density E.3: Anomalous Hall effect in a Si-doped quantum well. 128 Figure E.5: Magnet-transport data for sample 6-3-03-2 with the full range of field. E.3: Anomalous Hall effect in a Si-doped quantum well. 129 Figure E.6: Rxx in the mT magnetic field regime. appears to be smaller at high magnetic fields due to strong localizations. More remarkable features have been discovered in the low field regime, and we have found a weak anomalous Hall effect. In the low field regime, we used a bipolar magnet power supply to ensure that the magnetic field passes through zero smoothly. Fig. E. 6 shows our observation of Rxx in the mT field regime. A sharp resistance peak has been observed, and can be attributed to weak localization. More excitingly, we have found a weak anomalous Hall effect, which is shown in Fig. E. 7. Careful Hall measurement have been performed to ensure that no Rxx component is mixed into the Hall resistance Rxy , and a Hall probe is introduced into the system to measure the magnetic field with the precision of 0.01mT. A wiggle on E.3: Anomalous Hall effect in a Si-doped quantum well. 130 the Hall resistance is observed. After subtracting a linear background of Rxy , it is clear that an anomalous Hall effect has been observed around the zero field. The anomalous Hall effect is totally absent in the other sample with less Si impurities. We conclude that the Si impurities play an critical role in this effect. The origin of the anomalous Hall effect is still unknown at this stage, and it might show that the upper Hubbard band has a ferromagnetic structure. Future works on samples of different densities can be helpful to solve this puzzle. E.3: Anomalous Hall effect in a Si-doped quantum well. Figure E.7: Anomalous Hall effect. 131 Bibliography [Abrahams 79] E. Abrahams, P. W. Anderson, D. C. Licciardello & T. V. Ramakrishnan. Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions. Phys. Rev. Lett., vol. 42, page 673, 1979. [Anderson 58] P. W. Anderson. Absence of Diffusion in Certain Random Lattices. Phys. Rev., vol. 109, page 1492, 1958. [Ando 82a] T. Ando. Self-consistent results for a GaAs/Alx Ga1−x As heterojunction .2. Low temperature mobility. J. Phys. Soc. Jpn, vol. 51, page 3900, 1982. [Ando 82b] T. Ando, A. Fowler & F. Stern. Electron properties of twodimensional systems. Rev. Mod. Phys., vol. 54, page 437, 1982. [Aoki 81] H. Aoki & T Ando. Effect of localization on the Hall conductivity in the two-dimensional system in strong manetic fields. Solid State Commun., vol. 38, page 1079, 1981. [Aoki 85] H. Aoki & T. Ando. Critical localization in 2-dimensional Landau quantization. Phys. Rev. Lett., vol. 54, page 831, 1985. 132 133 BIBLIOGRAPHY [Balaban 98] N. Q. Balaban, U. Meirav & I. Bar-Joseph. Absence of scaling in the integer quantum Hall effect. Phys. Rev. Lett., vol. 61, page 4967, 1998. [Bastard 84] G. Bastard. Self-consistent variational calculations and alloy scattering in semiconductor heterojunctions. Surf. Sci., vol. 142, page 284, 1984. [Basu 83] P. K. Basu & B. R. Nag. Estimation of alloy scattering potential in tenaries from the study of two-dimensional electron-transport. Appl. Phys. Lett., vol. 43, page 689, 1983. [Buttiker 88] M. Buttiker. Absence of backscattering in the quantum Hall effect in multiprobe conductors. Phys. Rev. B, vol. 38, page 9375, 1988. [Chalker 87] J. T. Chalker. Anderson localization in quantum Hall systems. J. Phys. C, vol. 20, 1987. [Chalker 88] J. T. Chalker & P. D. Coddington. Percolation, quantum tunnelling and the integer Hall effect. J. Phys. C, vol. 21, page 2665, 1988. [Chandra 80] A. Chandra & L. F. Eastman. A study of alloy scattering in Alx Ga1−x As. J. Appl. Phys., vol. 51, page 2669, 1980. [Chattopadhyay 85] D. Chattopadhyay. Electron mobility of Alx In1−x As. Phys. Rev. B, vol. 31, page 1145, 1985. [Chen 03] Y. Chen, R. M. Lewis, L. W. Engel, D. C. Tsui, L. N. Pfeiffer & K. W. West. Microwave resonance of the 2D Wigner crystal 134 BIBLIOGRAPHY around integer Landau fillings. Phys. Rev. Lett., vol. 91, page 016801, 2003. [Chen 04] Y. Chen, R. M. Lewis, L. W. Engel, D. C. Tsui, L. N. Pfeiffer & K. W. West. Evidence for two different solid phases of twodimensional electrons in high magnetic fields. Phys. Rev. Lett., vol. 93, page 206805, 2004. [Chklovskii 92] D. B. Chklovskii, B. I. Shklovskii & L. I. Glazman. Electrostatics of edge channels. Phys. Rev. B, vol. 46, page 4026, 1992. [Coleridge 91] P. T. Coleridge. Small-angle scattering in two-dimensional electron gases. Phys. Rev. B, vol. 44, page 3793, 1991. [Csathy 04] G. A. Csathy, D. C. Tsui, L. N. Pfeiffer & K. W. West. Possible observation of phase coexistence of the ν=1/3 fractional quantum Hall liquid and a solid. Phys. Rev. Lett., vol. 92, page 256804, 2004. [Csathy 05] G. A. Csathy, H. Noh, D. C. Tsui, L. N. Pfeiffer & K. W. West. Magnetic field induced insulating phases at large rs . Phys. Rev. Lett., vol. 94, page 226802, 2005. [Dav ] J. Davies The Physics of Low-dimensional Semiconductors : An Introduction (Cambridge University Press 1997). [Efros 89] A. L. Efros. Metal-non-metal transition in heterostructures with thick spacer layers. Solid State Commun., vol. 70, page 253, 1989. 135 BIBLIOGRAPHY [Efros 93] A. L. Efros, F. G. Pikus & V. G. Burnett. Density of states of a 2-dimensional electron gas in a long-range random potential. Phys. Rev. B, vol. 47, page 2233, 1993. [Eisenstein 02] J. P. Eisenstein, K. B. Cooper, L. N. Pfeiffer & K. W. West. Insulating and fractional quantum Hall states in the first excited Landau level. Phys. Rev. Lett., vol. 88, page 076801, 2002. [Engel 93] L. W. Engel, D. Shahar, C. Kurdak & D. C. Tsui. Microwave frequency-dependence of integer quantum Hall effect - evidence for finite-frequency scaling. Phys. Rev. Lett., vol. 71, page 2638, 1993. [Engel 97] L. W. Engel, C. C. Li, D. Shahar, D. C. Tsui & M. Shayegan. Microwave resonance in low-filling insulating phases of twodimensional electron system. Solid State Commun., vol. 104, page 167, 1997. [Fang 66] F. F. Fang & W. E. Howard. Negative field-effect mobility on (100) Si surfaces. Phys. Rev. Lett., vol. 16, page 797, 1966. [Fang 77] F. F. Fang, A. B. Fowler & A. Hartstein. Effective mass and collision time of (100) Si surface electrons. Phys. Rev. B, vol. 16, page 4446, 1977. [Ferry 97] D. K. Ferry. Alloy scattering in tenary III-V compounds. Phys. Rev. B, vol. 17, page 912, 1997. 136 BIBLIOGRAPHY [Fogler 96] M. M. Fogler, A. A. Koulakov & B. I. Shklovskii. Ground state of a two-dimensional electron liquid in a weak magnetic field. Phys. Rev. B, vol. 54, page 1853, 1996. [Fogler 97] M. M. Fogler & A. A. Koulakov. Laughlin liquid to chargedensity-wave transition at high Landa levels. Phys. Rev. B, vol. 55, page 9326, 1997. [Fogler 04] M. M. Fogler. Non-linear screening and percolative transition in a two-dimensional electron liquid. Phys. Rev. B, vol. 69, page 121409, 2004. [Fu 00] Y. Fu & M. Willander. Alloy scattering in GaAs/Alx Ga1−x As quantum well infrared photodetector. J. Appl. Phys., vol. 88, page 288, 2000. [Gammel 94] B. M. Gammel & W. Brenig. Scaling of the static conductivity in the quantum Hall effect. Phys. Rev. Lett., vol. 73, page 3286, 1994. [Giuliani 82] G. F. Giuliani & J. J. Quinn. Lifetime of a quasiparticle in a two-dimensional electron gas. Phys. Rev. B, vol. 26, page 4421, 1982. [Goldman 90] V. J. Goldman, M. Santos, M. Shayegan & J. E. Cunningham. Evidence for 2-dimensional quantum Wigner crystal. Phys. Rev. Lett., vol. 65, page 2189, 1990. 137 BIBLIOGRAPHY [Gow ] J. Gowar Optical communication systems 2nd ed. (Englewood cliffs, NJ.: Prentice Hall 1993). [Grayson 05] M. Grayson & F. Fischer. Measuring carrier density in parallel conduction layers of quantum Hall systems. J. Appl. Phys., vol. 98, page 013709, 2005. [Halperin 82] B. I. Halperin. Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a twodimensional disordered potential. Phys. Rev. B, vol. 25, page 2185, 1982. [Harrison 76a] J. W. Harrison & J. R. Hauser. Alloy scattering in tenary III-V compounds. Phys. Rev. B, vol. 13, page 5347, 1976. [Harrison 76b] J. W. Harrison & J. R. Hauser. Theoretical calculation s of electron mobility in ternary 3-5 compounds. J. Appl. Phys., vol. 47, page 292, 1976. [Haug 87] R. J. Haug, R. R. Gerhardts, K. von Klitzing & K. Ploog. Effect of repulsive and attractive scattering centers on the magnetotransport properties of a two-dimensional electron gas. Phys. Rev. Lett., vol. 59, page 1349, 1987. [Hilke 98] M. Hilke, D. Shahar, S. H. Song, D. C. Tsui, Y. H. Xie & D. Monroe. Experimental evidence for a two-dimensional quantized Hall insulator. Nature, vol. 395, page 675, 1998. 138 BIBLIOGRAPHY [Hohls 02] F. Hohls, U. Zeitler & R. J. Haug. Hopping conductivity in the quantum Hall effect: Revival of universal scaling. Phys. Rev. Lett., vol. 88, page 036802, 2002. [Huckestein 90] B. Huckestein & B. Kramer. One-parameter scaling in the lowest Landau band - precise determination of the critical behavior of the localization length. Phys. Rev. Lett., vol. 64, page 1437, 1990. [Huckestein 99] B. Huckestein & M. Backhaus. Integer quantum Hall effect of interacting electrons: Dynamical scaling and critical conductivity. Phys. Rev. Lett., vol. 82, page 5100, 1999. [Huckestein 01] B. Huckestein. Scaling theory of the integer quantum Hall effect. Rev. Mod. Phys., vol. 67, page 357, 2001. [Huo 92] Y. Huo & R. N. Bhatt. Current carrying states in the lowest Landau level. Phys. Rev. Lett., vol. 68, page 1375, 1992. [Hwang 93] S. W. Hwang, H. P. Wei, L. W. Engel, D. C. Tsui & A.M. M. Pruisken. Scaling in spin-degenerate Landau levels in the integer quantum Hall effect. Phys. Rev. B, vol. 48, page 11416, 1993. [Jain 88] J. K. Jain & S. A. Kivelson. Quantum Hall effect in quasi one-dimensional systems - resistance fluctuation and breakdown. Phys. Rev. Lett., vol. 60, page 1542, 1988. [Jain 89] J. K. Jain. Composite fermion approach for the fractional quantum Hall effect. Phys. Rev. Lett., vol. 63, page 199, 1989. 139 BIBLIOGRAPHY [Jiang 90] H. W. Jiang, R. L. Willett, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer & K. W. West. Quantum liquid versus electron solid around 1/5 Landau level filling. Phys. Rev. Lett., vol. 65, page 633, 1990. [Kivelson 92] S. Kivelson, D.-H. Lee & S.-C. Zhang. Global phase diagram in the quantum Hall effect. Phys. Rev. B, vol. 46, page 2223, 1992. [Koch 91a] S. Koch, R. J. Haug, K. von Klitzing & K. Ploog. Experiments on scaling in Alx Ga1−x As/GaAs heterostructures under quantum Hall conditions. Phys. Rev. B, vol. 43, page 6828, 1991. [Koch 91b] S. Koch, R. J. Haug, K. von Klitzing & K. Ploog. Size-dependent analysis of the metal-insulator transition in the quantum Hall effect. Phys. Rev. Lett., vol. 67, page 883, 1991. [Lai 06] K. Lai, T. M. Lu, W. Pan, D. C. Tsui, S. A. Lyon, J. Liu, Y. H. Xie, M. Muhlberger & F. Shaffler. Valley splitting of Si/Si1−x Gex heterostructures in tilted magnetic fields. Phys. Rev. B, vol. 73, page 161301, 2006. [Laughlin 81] R. B. Laughlin. Quantized Hall conductivity in 2 dimensions. Phys. Rev. B, vol. 23, page 5632, 1981. [Laughlin 83] R. B. Laughlin. Anomalous quantum Hall effect - an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett., vol. 50, page 1395, 1983. 140 BIBLIOGRAPHY [Lee 85] P. A. Lee & A. D. Stone. Universal conductance fluctuations in metals. Phys. Rev. Lett., vol. 55, page 1622, 1985. [Lee 87] P. A. Lee & A. D. Stone. Universal conductance fluctuations in metals - effects of finite temperature, interactions, and magnetic field. Phys. Rev. B, vol. 35, page 1039, 1987. [Lee 93] D.-H. Lee, Z. Wang & S. Kivelson. Quantum percolation and plateau transitions in the quantum Hall effect. Phys. Rev. Lett., vol. 70, page 4130, 1993. [Li 03] Wanli Li, G. A. Csathy, D. C. Tsui, L. N. Pfeiffer & K. W. West. Direct observation of alloy scattering of two dimensional electrons in Alx Ga1−x As. Appl. Phys. Lett., vol. 83, page 2832, 2003. [Lin 84] B. J. F. Lin & D. C. Tsui. Mobility of the two-dimensional electron gas in GaAs-Alx Ga1−x As heterostructures. Appl. Phys. Lett., vol. 45, page 695, 1984. [Look 92] D. C. Look, D. K. Lorance, J. R. Sizelove, C. E. Stutz, K. R. Evans & D. W. Whitson. Alloy scattering in p-type Alx Ga1−x As. J. Appl. Phys., vol. 71, page 260, 1992. [Lozovik 75] Y. E. Lozovik & V. I. Yudson. Crystallization of a two- dimensional electron gas in a magnetic field. JETP lett., vol. 22, page 11, 1975. 141 BIBLIOGRAPHY [Machida 98] T. Machida, H. Hatta & S. Komiyama. Phase coherence of edge states over macroscopic length scales. Physica B, vol. 251, page 128, 1998. [Makowski 73] L. Makowski & M. Glicksman. Disorder scattering in solidsolutions of III-V semiconducting compounds. J. Phys. Chem. Solids, vol. 34, page 487, 1973. [McEuen 90] P. L. McEuen, A. Szafer, C. A. Richter, B. W. Alphenaar, J. K. Jain, A. D. Stone, R. G. Wheeler & R. N. Sacks. New resistivity for high-mobility quantum Hall conductors. Phys. Rev. Lett., vol. 64, page 2062, 1990. [Menashe 96] D. Menashe & B. Laikhtman. Quasiparticle lifetime in a twodimensional electron system in the limit of low temperature and excitation energy. Phys. Rev. B, vol. 54, page 11561, 1996. [Mil’nikov 88] G. V. Mil’nikov & I. M. Sokolov. Semiclassical localization in a magnetic field. JETP Lett., vol. 48, page 536, 1988. [Montgomery 71] H. C. Montgomery. Method for measuring electrical resistivity of anisotropic materials. J. Appl. Phys., vol. 42, page 2971, 1971. [Mot ] N. F. Mott Metal insulator transitions (Taylor & Francis 1993). [Nixon 90] J. A. Nixon & J. H. Davies. Potential fluctuations in heterostructure devices. Phys. Rev. B, vol. 41, page 7929, 1990. [Ono 82] Y. Ono. Energy dependence of localization length of two- dimensional electron system moving in a random potential under 142 BIBLIOGRAPHY strong magnetic fields. J. Phys. Soc. Jpn, vol. 51, page 2055, 1982. [Pan 99a] W. Pan, R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer & K. W. West. Strongly anisotropic electronic transport at Landau level filling factor ν=9/2 and ν=5/2 under a tilted magnetic field. Phys. Rev. Lett., vol. 83, page 820, 1999. [Pan 99b] W. Pan, J. S. Xia, V. Shvarts, D. E. Adams, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin & K. W. West. Exact quantization of the even-denominator fractional quantum Hall state at ν=5/2 Landau level filling factor. Phys. Rev. Lett., vol. 83, page 3530, 1999. [Pan 01] W. Pan, J. S. Xia, D. E. Adams, R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer & K. W. Baldwin. New results at half fillings in the second and third Landau level. Physica B, vol. 298, page 113, 2001. [Pan 03] W. Pan, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer & K. W. West. Fractional quantum Hall effect of composite fermions. Phys. Rev. Lett., vol. 90, page 016801, 2003. [Pfeiffer 89] L. N. Pfeiffer, K. W. West, H. L. Stormer & K. W. Baldwin. Electron mobilities exceeding 107 cm2 /V.s in modulation doped GaAs. Appl. Phys. Lett., vol. 55, page 1888, 1989. 143 BIBLIOGRAPHY [Platzman 88] P. M. Platzman. DC conductivity and gap energies in the fractional quantum Hall effect. Phys. Rev. B, vol. 39, page 7985, 1988. [Polyakov 93] D. G. Polyakov & B. I Shklovskii. Variable-Range hopping as the mechanism of the conductivity peak broadening in the quantum Hall regime. Phys. Rev. Lett., vol. 70, page 3796, 1993. [Pra a] A. M. M. Pruisken, in The Quantum Hall Effect, edited by R. Prange and S. Girvin (Springer-Verlag, Berlin, 1986). [Pra b] The quantum Hall effect, edited by R. E. Prange and S. M. Girvin (Springer, New York, 1990), 2nd ed. [Price 72] W. L. V. Price. Extention of van der Pauw’s theorem for measuring specific resistivity in discs of arbitrary shape to anisotropic media. J. Phys. D, vol. 5, page 1127, 1972. [Pruisken 88] A. M. M. Pruisken. Universal Singularities in the integeral quantum Hall effect. Phys. Rev. Lett., vol. 61, page 1297, 1988. [Santos 92] M. B. Santos, Y. W. Suen, M. Shayegan, Y. P. Li, L. W. Engel & D. C. Tsui. Observation of a reentrant insulating phase near the 1/3 fractional quantum Hall liquid in a 2-dimensional hole system. Phys. Rev. Lett., vol. 68, page 1188, 1992. [Saxena 81] A. K. Saxena. Electron mobility in Alx Ga1−x As alloys. Phys. Rev. B, vol. 24, page 3295, 1981. 144 BIBLIOGRAPHY [Saxena 85] A. K. Saxena & A. R. Adams. Determination of alloy scattering potential in Alx Ga1−x As. J. Appl. Phys., vol. 58, page 2640, 1985. [Sem ] M. D’Iorio, V. M. Pudalov and S. M. Semenchinsky, p. 56, High magnetic Field in semiconductor III, Quantum Hall Effect, Transport and Optics (Springer, Berlin, 1992). [Shimshoni 99] E. Shimshoni. Classical versus quantum transport near quantum Hall transitions. Phys. Rev. B, vol. 60, page 10691, 1999. [Shklovskii 86] B. I. Shklovskii & A. L. Efros. State-density oscillations of twodimensional electrons in a transverse magnetic field. JETP Lett., vol. 44, page 669, 1986. [Sin ] J. Singh Physics of Semiconductors and Their Heterostructure (McGraw-Hill 1993). [Sondhi 97] S. L. Sondhi, S.M. Girvin, J. P. Carini & D. Shahar. Continuous quantum phase transitions. Rev. Mod. Phys., vol. 69, page 315, 1997. [Streda 87] P. Streda, J. Kucera & A. H. MacDonald. Edge states, transmission matrices, and the Hall resistance. Phys. Rev. Lett., vol. 59, page 1973, 1987. [Struck 06] A. Struck & B. Kramer. Electron correlations and single particle physics in the integer quantum Hall effect. Phys. Rev. Lett., vol. 97, page 106801, 2006. 145 BIBLIOGRAPHY [Thouless 77] D. J. Thouless. Maximum metallic resistance in thin wires. Phys. Rev. Lett., vol. 39, page 1167, 1977. [Tietjen 65] J. J. Tietjen & L. R. Weisberg. Electron mobility in GaAsx P1−x alloys. Appl. Phys. Lett., vol. 7, page 261, 1965. [Trugman 83] S. A. Trugman. Localization, percolation, and the quantum Hall effect. Phys. Rev. B, vol. 27, page 7539, 1983. [Tsui 82] D.C. Tsui, H. L. Stormer & A.C. Gossard. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett., vol. 48, page 1559, 1982. [Vakili 05] K. Vakili, Y. P. Shkolnikov, E. Tutuc, N. C. Bishop, E. P. De Poortere & M. Shayegan. Spin-dependent resistivity at transitions between integer quantum Hall states. Phys. Rev. Lett., vol. 94, page 176402, 2005. [van der Pauw 58a] L. J. van der Pauw. A method of measuring specific resistivity and Hall effect of discs of arbitrary shape. Phillips Tech. Rev., vol. 13, page 1, 1958. [van der Pauw 58b] L. J. van der Pauw. A method of measuring the resistivity and Hall coefficient on lamellae of arbitrary shape. Phillips Tech. Rev., vol. 20, page 220, 1958. [van Schaijk 00] R. T. F. van Schaijk, A. de Visser, S. M. Olsthoorn, H. P. Wei & A. M. M. Pruisken. Probing the plateau-insulator quantum 146 BIBLIOGRAPHY phase transition in the quantum Hall regime. Phys. Rev. Lett., vol. 84, page 1567, 2000. [van Wees 91] B. J. van Wees, L. P. Kouwenhoven, E. M. M. Willems, C. J. P. M. Harmans, J. E. Mooij, H. van Houten, C. W. J. Beenakker, J. G. Williamson & C. T. Foxon. Quantum ballistic and adiabatic electron-transport studied with quantum point contacts. Phys. Rev. B, vol. 43, page 12431, 1991. [Wakabayashi 89] J. Wakabayashi, J. M. Yamane & S. Kawaji. Experiments on the critical exponnet of localization in Landau subbands with the quantum numbers 0 and 1 in Si-MOS inversion layers. J. Phys. Soc. Jpn., vol. 58, page 1903, 1989. [Wakabayashi 92] J. Wakabayashi, S. Kawaji, T. Goto, T. Fukase & Y. Koike. Localization in Landau subbands with the Landau quantum number 0 and 1 of Si-MOS inversion layers. J. Phys. Soc. Jpn., vol. 61, page 1691, 1992. [Walukiewicz 84] W. Walukiewicz, H. E. Ruda, J. Lagowski & H. C. Gatos. Electron mobility in modulation-doped heterostructures. Phys. Rev. B, vol. 30, page 4571, 1984. [Wei 88] H. P. Wei, D. C. Tsui, M. A. Paalanen & A. M. M.Pruisken. Experiments on delocalization and universality in the integeral quantum Hall effect. Phys. Rev. Lett., vol. 61, page 1294, 1988. 147 BIBLIOGRAPHY [Wei 92] H. P. Wei, S. Y. Lin, D. C. Tsui & A. M. M. Pruisken. Effect of long-range potential fluctuations on scaling in the integer quantum Hall effect. Phys. Rev. B, vol. 45, page 3926, 1992. [Wei 94] H. P. Wei, L. W. Engel & D. C. Tsui. Current scaling in the integer quantum Hall effect. Phys. Rev. B, vol. 50, page 14609, 1994. [Wigner 34] E. Wigner. On the interaction of electrons in metals. Phys. Rev., vol. 46, page 1002, 1934. [Willett 88] R. L. Willett, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. West & K. W. Baldwin. Termination of the series of fractional quantum Hall states at small filling factors. Phys. Rev. B, vol. 38, page 7881, 1988. [Xia 00] J. S. Xia, D. E. Adams, V. Shvarts, W. Pan, H. L. Stormer & D. C. Tsui. Ultra-low-temperature cooling of two-dimensional electron gas. Physica B, vol. 280, page 491, 2000. [Xin ] Xin Wan, private communication. [Yacoby 91] A. Yacoby, U. Sivan, C. P. Umbach & J. M. Hong. Interference and dephasing by electron-electron interaction on length scales shorter than the elastic mean free path. Phys. Rev. Lett., vol. 66, page 1938, 1991. 148 BIBLIOGRAPHY [Yacoby 94] A. Yacoby, M. Heiblum, H. Shtrikman, V. Umansky & D. Mahalu. Dephasing of ballistic electrons as a function of temperature and carrier density. Semicon. Sci. Tech., vol. 9, page 907, 1994. [Yang 03] I. Yang, W. Kang, S. T. Hannahs, L. N. Pfeiffer & K. W. West. Vertical confinement and evolution of reentrant insulating transition in the fractional quantum Hall regime. Phys. Rev. B, vol. 68, page 121302, 2003. [Ye 02] P. D. Ye, L. W. Engel, D. C. Tsui, R. M. Lewis, L. N. Pfeiffer & K. W. West. Correlation lengths of the Wigner-crystal order in a two-dimensional electron system at high magnetic fields. Phys. Rev. Lett., vol. 89, page 176802, 2002. [Zheng 85] H. Z. Zheng, K. K. Choi, D. C. Tsui & G. Weimann. Observation of size effect in the quantum Hall regime. Phys. Rev. Lett., vol. 55, page 1144, 1985. [Zulicke 01] U. Zulicke & E. Shimshoni. Quantum breakdown of the quantum Hall insulator. Phys. Rev. B, vol. 63, page 241301, 2001.