Download Transport study on two-dimensional electrons with controlled short-range alloy disorder

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
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