GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
SELF-CONSISTENT INVESTIGATION OF
EXCHANGE AND CORRELATION IN THE
TWO-DIMENSIONAL ELECTRON GAS UNDER
STRONG PERPENDICULAR MAGNETIC FIELDS
by
G¨
on¨
ul B˙ILGEC
¸ AKY ¨
UZ
March, 2011 ˙IZM˙IR
EXCHANGE AND CORRELATION IN THE
TWO-DIMENSIONAL ELECTRON GAS UNDER
STRONG PERPENDICULAR MAGNETIC FIELDS
A Thesis Submitted to the
Graduate School of Natural and Applied Sciences of Dokuz Eyl¨ul University In Partial Fulfillment of the Requirements for the Degree
of Doctor of Philosophy in Physics
by
G¨
on¨
ul B˙ILGEC
¸ AKY ¨
UZ
March, 2011 ˙IZM˙IR
I would like to express my sincere thanks to all the people who have supported me over my Ph.D. study.
I owe my deepest gratitude to my supervisor Prof. Dr. ˙Ismail S ¨OKMEN, who gave me the opportunity to make this thesis and great privilege of working with him. I indebted to my supervisor for his invaluable advises and guidance, endless patience and suggestions throughout this work. Particular thanks go to my second supervisor Assoc. Prof. Dr. Afif SIDDIK˙I for creating a group ”Nano-electronic and Nano-transport” for valuable colleagues and giving me to opportunity to be a part of it and for critical suggestions and endless patience.
I would express my thanks to Assist. Prof. Dr. Kadir AKG ¨UNG ¨OR and Assist. Prof. Dr. Hande ¨UST ¨UNEL TOFFOL˙I for their helps and technical supports.
I am very grateful to all my friends for the support and their good humor. I enjoyed very much working with them.
Finally but not least, I would like to thank my husband Cenk AKY ¨UZ and my family whom without their continuous support, motivation and encouragement, I would not to be able to complete this work.
G¨on¨ul B˙ILGEC¸ AKY ¨UZ
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Anneme ...
ivCORRELATION IN THE TWO-DIMENSIONAL ELECTRON GAS UNDER STRONG PERPENDICULAR MAGNETIC FIELDS
ABSTRACT
In this thesis, the emergent role of the exchange and the correlation effects on a two dimensional electron gas in the presence of a strong perpendicular mag-netic field was studied within the Thomas-Fermi-Dirac-Poisson approximation. The effective Lande-g factor of the two-dimensional system was calculated. In our studies, the exchange functional within a framework of the local spin den-sity approximation derived for the homogeneous two-dimensional electron gas and the correlation functionals, which correspond this exchange functionals were constructed by using quantum Monte Carlo methods, were used.
After investigating the effects of the exchange and the correlation on the sys-tem, the magnetoconductivity in a two-dimensional electron system including the exchange and correlation interaction was also calculated. In the magneto-conductivity calculations, the short-range disorder was investigated within the Self-consistent Born approximation. This approximation is an appropriate form of broadened Landau density of states for strong, quantizing magnetic fields to calculate conductivities in the quantum Hall regime.
In the second part of the thesis, the total ground state energy of a semiconduc-tor quantum dot which is defined by the “trench gate method” was calculated. In our ground state energy calculations, we used a recently developed energy func-tional called “orbital-free energy funcfunc-tional”, Thomas-Fermi approximation and local density approximation. We compared the total energies obtained with three different approximations for the square quantum slab geometry with different particle number.
Keywords: Quantum Hall effect, Lande-g Factor Thomas-Fermi-Dirac-Poisson Approximation.
ELEKTRON GAZINDA DE ˘G˙IS¸-TOKUS¸ VE KORELASYON ETK˙ILER˙IN˙IN ¨OZ-UYUMLU OLARAK ˙INCELENMES˙I
¨ OZ
Bu tezde, g¨u¸cl¨u dik bir manyetik alan altındaki iki boyutlu elektron gazı ¨uzerinde olduk¸ca etkisi oldu˘gu bilinen de˘gi¸s-toku¸s ve korelasyon etkileri, Thomas-Fermi-Dirac-Poisson yakla¸sımı kullanılarak incelendi. ˙Iki-boyutlu sistemin etkin Lande-g ¸carpanı hesaplandı. Hesaplarımızda, de˘gi¸s-toku¸s potansiyeli olarak ho-mojen iki-boyutlu elektron gazı i¸cin t¨uretilen yerel spin yo˘gunluk yakla¸sımını ve korelasyon fonksiyonelleri i¸cin ise bu de˘gi¸s-toku¸s fonksiyonellerine kar¸sılık gelen ve quantum Monte Carlo y¨ontemleri ile elde edilen korelasyon potansiyelleri kul-lanıldı.
De˘gi¸s-toku¸s ve korelasyon etkilerinin sistem ¨uzerindeki etkileri incelendikten sonra, de˘gi¸s-toku¸s ve korelasyon potansiyellerinin de yer aldı˘gı etkin potansiyel altında sistemin manyeto iletkenli˘gi hesaplandı. ˙Iletkenlik hesaplarında, ¨oz-u yumlu Born yakla¸sımı kullanılarak kısa-erimli yapı bozukluklarının bulundu˘gu ¨ozel durumlar ara¸stırıldı. Bu yakla¸sım, quantum Hall rejimindeki, g¨u¸cl¨u kuantize manyetik alanlar ile genle¸smi¸s Landau durum yo˘gunlu˘gu i¸cin olduk¸ca uygun bir yakla¸sımdır.
Tezin ikinci kısmında, “trench gate y¨ontemi” ile tanımlanan yarıiletken bir kuantum nokta’nın toplam taban durum enerjisi hesaplandı. Taban durum enerji hesaplarında, “orbitalsiz enerji fonksiyoneli” olarak isimlendirilen yeni geli¸stirilen bir enerji fonksiyoneli, Thomas-Fermi yakla¸sımı ve yerel yo˘gunluk yakla¸sımı kul-lanıldı. Farklı elektron sayısına sahip, kare kuantum dot’daki elektronların taban durum enerjileri, yukarıdaki ¨u¸c farklı y¨ontemle hesaplanarak kar¸sıla¸stırıldı. Anahtar s¨ozc¨ukler: Kuantum Hall Olayı, Lande-g C¸ arpanı, Thomas-Fermi-Dirac-Poisson Yakla¸sımı.
Ph.D. THESIS EXAMINATION RESULT FORM . . . ii
ACKNOWLEDGEMENTS . . . iii
ABSTRACT . . . v
¨ OZ . . . vi
CHAPTER ONE - INTRODUCTION . . . 1
CHAPTER TWO - FUNDAMENTALS OF THE QUANTUM HALL EFFECT AND 2DEG . . . 3
2.1 The quantized Hall effect . . . 3
2.2 Two-Dimensional Electron Gas . . . 6
2.3 2DEG in a Homogeneous Magnetic Field . . . 9
2.3.1 Classical Treatment . . . 9
2.3.2 Quantum Mechanical Treatment . . . 12
2.4 Characterization of Electron Transport . . . 17
2.4.1 Magnetotransport in the Classical Regime . . . 17
2.4.2 Magnetotransport in the Quantum Regime . . . 19
2.4.3 Quantum Transport Models of the 2DEG . . . 25
CHAPTER THREE - ELECTROSTATIC SELF-CONSISTENCY AND MANY BODY APPROXIMATIONS . . . 31
3.1 Many Body Effects and Some Approximations . . . 31
3.1.1 Thomas-Fermi Approximation . . . 31
3.1.2 Exchange-Correlation Effects in 2D and Local Density Ap-proximation . . . 34
3.1.3 Orbital Free Energy Functional . . . 37
3.2 2D-Quantum Hall System and Self-Consistent Thomas-Fermi-Poisson Approximation . . . 38
3.2.1 In-plane charges and gates . . . 38
CHAPTER FOUR - THE SELF-CONSISTENT CALCULATION OF EXCHANGE & CORRELATION ENHANCED ODD IN-TEGER QUANTIZED HALL PLATEAUS WITHIN
THOMAS-FERMI-DIRAC APPROXIMATION . . . 49
4.1 Introduction . . . 49
4.2 Model . . . 54
4.3 Results . . . 57
4.3.1 Tanatar-Ceperley Parametrization . . . 57
4.3.2 Attacalite Correlation Parametrization . . . 63
CHAPTER FIVE-MAGNETOTRANSPORT CALCULATIONS FOR TWO-DIMENSIONAL ELECTRONS WITH EXCHANGE AND CORRELATION INTERACTIONS . . . 65
5.1 Introduction . . . 65
5.2 Model . . . 65
5.3 Results . . . 67
CHAPTER SIX-ENERGY CALCULATIONS OF THE REALIS-TIC QUANTUM SLAB . . . 77
6.1 Introduction . . . 77
6.2 Model . . . 78
6.3 Numerical Procedure and Results . . . 79
CHAPTER SEVEN - CONCLUSION . . . 86
REFERENCES . . . 88
APPENDIX A . . . 98
A.1 Exchange and Correlation Potentials . . . 98
APPENDIX B . . . 107
B.2 Orbital Free Energy Functional . . . 107
ABBREVIATIONS and SYMBOLS . . . 112 viii
INTRODUCTION
The quantum Hall effect (QHE) is one of the most remarkable condensed-matter phenomena which occur at low temperatures (≤ 4 K) in high mobility two dimensional electron gas system (2DEG) under strong perpendicular mag-netic fields (B ∼ 1 − 30 T). The basic experimental observation of the QHE is the nearly vanishing longitudinal conductance σxx → 0 and the quantization
of the transverse conductance (σxy = νe2/h, where ν is a quantum number) of
a real transistor-like device containing a 2DEG subjected to a strong magnetic field (Klitzing, Dorda, & Pepper, 1980). This quantization is independent of all microscopic details such as the type of semiconductor material, the purity of the sample, the precise value of the magnetic field, etc. Therefore, the QHE is used the standard of electrical resistance (ρxy = h/ie2), where i define the number of
fully occupied Landau Levels (LLs). The QHE has been observed in two different forms, called as integer quantum Hall effect (IQHE) and fractional quantum Hall effect (FQHE). In the IQHE, which has been discovered by von Klitzing in 1980, the quantum number ν is a simple integer with a precision of about 10−10 and
an absolute accuracy of about 10−8. Second form of the QHE, which is called as
FQHE, has been found by Tsui, St¨ormer and Gossard (Tsui, St¨ormer, & Gos-sard, 1982) by pushing the experiments further with even cleaner samples, lower temperatures and stronger magnetic field. In this phenomena, plateaus occur also at filling fractions ν that were not integers but simple fractions. In 1983, Laughlin has proposed a phenomenological theory of the FQHE, for the filling factors, ν = 1
2k+1, with k a positive integer (Laughlin, 1983) and this theory was
also confirmed by experiments (de-Picciotto et al., 1997). The FQHE is the result of quite different underlying physics involving strong Coulomb interactions and correlations among the electrons.
The QHE continues to draw attention both in theoretical and experimental 1
physics, although a basic theoretical understanding has been established some years after its discovery. Many new features have been experimentally investi-gated, however a unified theory for the QHE is still missing and research on the theory of the QHE continues today. Also measurements on to the QHE are very important for the developments of the devices since these measurements include analysis and characterization of the 2DEG.
In this thesis, the many body effects and enhanced Lande-g∗ factor by
us-ing Thomas-Fermi-Dirac-Poisson Approximation (TFDPA) has been investigated. The thesis is structured as follows: In chapter 2, the essential ingredients of the QHE, the self-consistent theory and many-body effects of a Hall bar under quan-tum Hall conditions have been introduced. In chapter 3, the TFDPA approach is implemented to Hall bar geometry with plane gate in order to define general method in this work. Also the exchange and correlation effects in local spin den-sity approximation (LSDA) have been investigated on the incompressible strips (ISs) at integer quantum Hall region. The self consistent calculation to inves-tigate the many body effects has been done within TFDPA, in chapter 4. In chapter 5, the calculations of the magnetoconductivity in a 2DEG including the exchange and correlation interaction have been shown. In chapter 6, we calcu-lated the total energy of a semiconductor quantum dot, which is defined by the trench gate method by using recently developed energy functional called orbital-free energy functional (OFEF) have been calculated. In this last chapter, also the total energies obtained by Thomas-Fermi approximation (TFA), OFEF and standard local-density approximation (LDA) for the square quantum slab geom-etry compared. Finally, the conclusions of the complete thesis are presented in chapter 7.
FUNDAMENTALS OF THE QUANTUM HALL EFFECT AND 2DEG
2.1 The quantized Hall effect
The QHE was discovered in 1980 during an experiment at the High Mag-netic Field Laboratory in Grenoble. The research is about the characterization of the electronic transport of silicon field effect transistors to improve the mo-bility of these devices. For this research, specially designed Hall devices were provided which allow direct measurements of the resistivity tensor, as shown in Figure 2.1. This important experiment is performed at liquid helium temperature
Figure 2.1 Typical silicon MOSFET device used for measurements of the xx- and xy-components of the re-sistivity tensor. For a fixed source-drain current be-tween the contacts S and D, the potential drops bebe-tween the probes P-P and H-H are directly proportional to the resistivities ρxx and ρxy. A positive gate voltage
increases the carrier density below the gate (K. von Kl-itzing)
(∼ 4.2 K) in order to suppress the disturbing scattering processes originating from the electron-phonon interactions. In the ideal case, the energy spectrum of a 2DEG in strong magnetic fields consists of discrete energy levels. However,
if the Fermi energy is located in the gap of the electronic spectrum and if the temperature is sufficiently low, that excitations across the gap are impossible and the QHE phenomena is observed. This new quantum mechanical phenomena is first observed by K. von Klitzing, Dorda and Pepper (1980). The experimental curve, which led to the discovery of the QHE is shown in Figure 2.2.
Figure 2.2 Hall resistance and longitudinal resistance (at zero magnetic field and at B = 19.8 Tesla) of a silicon MOSFET at liquid helium temperature as a function of the gate voltage. The quantized Hall plateau for filling factor 4 is enlarged (K. von Klitzing).
This curve include the electrical (longitudinal) Rxx and the Hall resistances
RH of the silicon field effect transistor as a function of the gate voltage. The
elec-trical resistance becomes monotonically smaller while the electron concentration increases linearly with increasing gate voltage. The Hall voltage decreases with increasing gate voltage, in a constant magnetic field, because the Hall voltage
inversely proportional to the electron concentration. The plateaus in the Hall resistance which is the ratio of the Hall voltage divided by the current through the sample are observed at the special gate voltages, where the longitudinal resis-tance becomes zero. In this interval, the density of state of mobile electrons at the Fermi energy is vanish. The longitudinal (ρxx) and Hall (ρxy) resistivity
compo-nents remain unchanged in finite gate voltage interval values. These unchanged resistivity values indicate that the electrons in these intervals do not contribute to the electronic transport so, they are localized. The role of localized electrons was not clear in the Hall effect measurements and it is generally believed by the experimentalists that the Hall effect measures only delocalized electrons. These assumptions are also partly supported by the theory (Aoki, & Kamimura, 1977) and formed the basis of the analysis of QHE data published in 1977 (Englert, & Klitzing, 1978). The plateau values of the Hall resistance (ρxy) are not influenced
by amount of localized electrons and can be expressed with high precision by the equation ρxy = h/ie2. Here h is the Planck’s constant (h = 2π~), e is the
ele-mentary charge and i is the number of fully occupied LLs. The most important equation in connection with the quantized Hall resistance is given by
VH = h/e2· I, (2.1.1)
and the most important aspect of the QHE for applications is the fact that the quantized Hall resistance has always fundamental value of h/e2 = 25812.8... Ohm.
This value does not depend on material, geometry and microscopic details of the semiconductor. The experiment about the discovery of this resistance standard value has been repeated by the many metrological institutes and it is confirmed that their effect is extremely stable and reproducible. The data published until 1988 for the fundamental value of the quantized Hall resistance are summarized in Table 2.1. However, the fine-structure constant cannot be directly determined with high accuracy using this high reproducibility and stability of the quantized Hall resistance. The combination with other experiments like high precision
mea-surements of the anomalous magnetic moment of the electron or gyromagnetic ratio of protons lead to a least square adjustment of the value of the fine-structure constant with an uncertainty of only 3.3 × 10−9 resulting in a value for the von
Klitzing constant of RK = 25812.807449 ± 0.000086 Ohm (CODATA 2002). Table 2.1 Summary of high precision data for the quantized Hall resistance up to 1988 which led to the fixed value of 25 812.807 Ohm recommended as a reference standard for all resistance calibrations after 1.1.1990.
Hall Resistance RH = V /I PRL 45,494 (1980) 25812.68(8) Ω BIPM (PARIS) 25812.809(3) Ω PTB (D) 25812.802(3) Ω ETL (JAPAN) 25812.804(8) Ω VSL (NL) 25812.802(5) Ω NRC(CAN) 25812.814 (6) Ω EAM (CH) 25812.809(4) Ω NBS (USA) 25812.810(2) Ω NPL (GB) 25812.811(2) Ω 1.1.1990 25812.81280700 Ω
2.2 Two-Dimensional Electron Gas
A 2DEG exhibits the QHE when it is placed under a strong perpendicular magnetic field. Therefore the two-dimensionality is a prerequisite for the occur-rence of the QHE. By two-dimensional, it is meant that the electrons have well separated quantized energy levels for one spatial dimension, but are quasi-free to move on the other two spatial dimensions. That is, if only lowest subband of a quantum well is partially occupied and the distance between the chemical poten-tial and the next higher subband is much larger than the thermal energy kBT ,
it is called the electrons in the quantum well a two-dimensional electron system. These systems can be observed in many material systems, e.g. layer compounds, at oxide interface. The most famous example of a semiconductor-oxide interface is the Si − SiO2 interface, the core of the silicon MOSFET (Metal
Oxide Semiconductor Field Effect Transistor), on which the QHE was originally discovered (Klitzing, Dorda, & Pepper, 1980). However, the most commonly used a semiconductor heterostructure is the GaAs/AlGaAs heterostructure, the core of the GaAs-MODFET (Modulation Doped Field Effect Transistor), on which the FQHE was discovered (Tsui, St¨ormer, & Gossard, 1982). Considerably higher mobilities can be achieved in GaAs-based heterostructure. The 2DEG is gen-erated by electrochemical. To produce atomically smooth layers of GaAs and
AlxGa1−xAs superlattice Moleculer Beam Epitaxy (MBE) is used. Figure 2.3
shows the typical layer structure of such a heterostructure.
Figure 2.3 A typical modulation doped GaAs/AlGaAs heterostructure and the sketch of its conduction band. The electrons are provided by the silicon donors + of the doped AlGaAs layer, and trapped at the potential well in the undoped GaAs. The spacer layer separates the electrons from the donors and therefore decreases the Coulomb interaction between them. This improves the electron mobility of the 2DEG. EF is the Fermi energy.
By means of MBE, a buffer layer is composed of alternating layers of AlGaAs and GaAs, which helps to define atomically smooth surfaces for the next growth layers. On top of the buffer layer, undoped AlGaAs layer is grown, which is called as spacer. A silicon doped n-type AlGaAs layer and finally above that, a thin
GaAs cap layer is grown. Thin cap layer is grown to prevent the wafer from oxida-tion. 2DEG system develops at the heterojunction between the spacer and GaAs layer, by means of electron transfer from the silicon donors to the AlGaAs through the spacer layer into the GaAs region. Therefore, the spacer layer separates the electrons in the conduction band of the GaAs from positively charged donors. This technique is called modulation doping which results in drastic reduction of the impurity scattering and therefore in a great increase of the electron mobility. The conduction band energy difference between GaAs and AlGaAs is 0.3 eV and therefore, when an epitaxially thin and clean layer AlGaAs is grown on GaAs, there is a formation of triangular potential. At low temperatures, conduction elec-trons from donor impurities cool down to triangular potential barriers. In this triangular potential well, energy quantization occurs in one direction. Therefore, electrons have mobility only in two dimensions. The corresponding energy band diagram can be obtained by solving the Schr¨odinger and Poisson’s equations self-consistently. Such a solution is sketched in Figure 2.3 for the conduction band of the modulation doped GaAs/AlGaAs heterostructure. At the heterojunction there exists a quasi-triangular potential well, whose effective thickness is compa-rable or smaller than the de Broglie wavelength of the electrons. The energy of the electrons is quantized and grouped in the energy subbands. At low tempera-tures (T < 4 K) and with small carrier densities only the lowest energy subband is occupied with electrons. As a result, the electrons are strictly confined in this potential well in the z-direction, but they are free to move in the x − y plane and therefore total energy is the energy of electrons grouped in the lowest energy subband (E0
z) plus kinetic energy, such that
Ez = Ez0+
~2(k2
x+ ky2)
2m∗ , (2.2.1)
where m∗ is the effective electron mass and k
x and ky are the wavevector
compo-nents in the momentum space. Because of the strong asymmetry of the potential well, the wave function of the electrons in the z-direction is mainly localized in
Table 2.2 Characteristic quantities for a 2DEG in GaAs with spin degeneracy 2.
Kinetic Energy DOS Fermi Wave Number Fermi Energy
E = ~2k2 2m D = m ∗ π~2 kF = √ 2πn EF = Dn = π~ 2 m∗n
GaAs. Therefore m∗ can be taken as the effective electron mass in GaAs, which
is m∗ = 0.066 m
0.
2.3 2DEG in a Homogeneous Magnetic Field
2.3.1 Classical Treatment
A classical electron which is confined in a two-dimensional plane (x, y) subject to a constant magnetic field Bˆz, which is perpendicular to this plane, will move according to Newton’s equation. The newtonian equations of this classical motion due to Lorentz force are given as,
x¨ ¨ y = eB m − ˙y ˙x . (2.3.1)
The general solution of these equations corresponds to motion in a circle of arbi-trary radius R,
~r = R(cos(ωct + δ), sin(ωct + δ)). (2.3.2)
Here ωc = eB/mc is the classical cyclotron frequency and δ define an arbitrary
phase for the motion. A fast particle travels in a large circle but returns to the starting point in the same time as a slow particle which travels in a small circle. This motion is isochronous much like that of a harmonic oscillator whose period is independent of the amplitude of the motion. It must be reviewed the formal Lagrangian and Hamiltonian approaches to this problem because of some subtleties involving distinctions between canonical and mechanical momentum in
the presence of a magnetic field. Lagrangian equation of this classical motion given as, L = 1 2m ˙x µ˙xµ− e c˙x µAµ, (2.3.3)
here ~A is the vector potential and µ = 1, 2 define x and y directions, respectively.
The Euler-Lagrange equation of the motion becomes
m¨xν = −e c[∂νA µ− ∂ µAν] ˙xµ, (2.3.4) by using ∂L ∂xν = − e c˙x µ∂ νAµ and ∂L ∂ ˙xν = m ˙x ν − e cA ν, (2.3.5) where B = ∇ × A and Bα = ²αβγ∂
βAγ shows that Equation 2.3.4 is equivalent
to Equation 2.3.1. Lagrangian can be deduced the canonical momentum
pµ ≡ ∂L
∂ ˙xµ = m ˙x µ− e
cA
µ, (2.3.6)
and the Hamiltonian;
H[p, x] ≡ ˙xµpµ− L( ˙x, x) = 1 2m à pµ+e cA µ | {z } ! ³ pµ+e cA µ´. (2.3.7) pµ mec
Hamilton’s equations of motion
˙xµ = ∂H ∂pµ = 1 mp µ mec, (2.3.8) ˙pµ= −∂H ∂xµ = − e mc ³ pν + e cA ν´∂ µAν (2.3.9)
show that it is the mechanical momentum, which is equal to the usual expression related to the velocity,
pµ
mec = m ˙xµ. (2.3.10)
Using Hamilton’s equations of motion, one can retrieve Newton’s Law for the Lorentz force (Equation 2.3.4) by taking a time derivative of ˙xµ in Equation 2.3.8
and then using Equation 2.3.9. The classical equations of motion only involve the curl of the vector potential and so the particular gauge choice is not so important of the classical level.
Newton’s Equation changes into following form for 2DEG in a perpendicular magnetic field, since electric and magnetic fields are perpendicular to each other,
m ˙v = −eE −e
c(v × B). (2.3.11)
To solve this equation, the motion is described in a coordinate system moving with the drift velocity:
vD ≡ E × B
c
B2. (2.3.12)
The drift velocity is constant in time and has the dimension of velocity. One can separate the velocity in two parts v ≡ v0 + v
D. Substituting this velocity into
the equation of motion, the following equation is obtained
mdv
dt = Lorentz f orce = q(E + v × B), (2.3.13)
or
mdv0
dt = q(v
0 × B), (2.3.14)
which has same form as Equation 2.3.1. The motion can be regarded as superpo-sition of the motion in a uniform magnetic field and a drift of the cyclotron orbit with the constant velocity (vD) as given Equation 2.3.12. This motion is shown
E and B:
j = nqc(E × B)
B2 (2.3.15)
where n is the number density of the electrons and the current is called the Hall current. This equation turns out to be a practically important property. The measurement of the Hall effect gives information about the type of the charge carrier (”electron” or ”hole”) and the number density of carriers.
Figure 2.4 An electron in this figure spirals under the electric (E) and magnetic (B) fields, which are perpen-dicular to each other.
2.3.2 Quantum Mechanical Treatment
The most important quantum effect in the QHE is the quantization of the cyclotron motion. To see this, it is needed to calculate the energy levels of an electron in a constant magnetic field B and for this reason, one has to solve the Schr¨odinger’s equation for stationary wavefunctions. Since we will be dealing with the Hamiltonian, we must choose a gauge for the vector potential. One convenient choice is the so-called Landau gauge, A(r) = xBˆy which obeys ∇ × A = Bˆz. In this gauge, the vector potential points in the y-direction but varies only with the x-position. Hence the system has translation invariance in the y-direction. The canonical momentum py = −i~∂/∂y is a constant of motion. In the Landau
gauge, the Hamiltonian of this system can be written as H = 1 2m µ p2x+ (py+ eB c x) 2 ¶ , (2.3.16)
and the wavefunction can be written as,
Ψk(x, y) = eikyfk(x), (2.3.17)
taking advantage of the translation invariance in the y-direction. This has ad-vantage that it is an eigenstate of py and hence we can make the replacement
py → ~k in the Hamiltonian. After separating variables we have the effective
one-dimensional Schr¨odinger equation,
hkfk(x) = ²kfk(x), (2.3.18) where hk≡ 1 2mp 2 x+ 1 2m µ ~k + eB c x ¶2 , (2.3.19) hk = 1 2mp 2 x+ 1 2m µ eB c ¶2µ x +~ck eB ¶2 , (2.3.20) hk = 1 2mp 2 x+ m 2ω 2 c(x − Xc)2. (2.3.21)
This is simply a one-dimensional displaced harmonic oscillator problem for an effective parabolic magnetic potential with center coordinate Xc= −ky`2B which
determined by the y momentum quantum number. Where `B =
p
~/mωc is the
magnetic length. The boundary conditions fk(x) → 0 for |x| → ∞ yield the
will be an entire group of energy eigenvalues for the y-direction,
²kn = (n + 1/2)~ωc
which depend only on n. Also this group are completely independent of y mo-mentum ~k. The corresponding (unnormalized) eigenfunctions are
Ψnk(r) = 1 √ Le ikyH n(x + k`2B)e −2`21 B (x+k`2 B)2 , (2.3.22)
here Hn is the nth Hermite polynomial and these harmonic oscillator levels are
called Landau levels (LLs). The degeneracy of each level is enormous due to the lack of dependence of the energy on k. In the Hall system, we assume periodic boundary conditions in the y-direction and because of the vector po-tential, it is impossible to simultaneously have periodic boundary conditions in the x-direction. However since the basis wave functions are harmonic oscillator polynomials multiplied by strongly converging gaussians, they rapidly vanish for positions away from the center position Xc. Let us suppose that the sample is
rectangular with dimensions Lx, Ly and that the left hand edge is at x = −Lx
and the right hand edge is at x = 0. The values of the wavevector k for which the basis state is substiantly inside the sample run from k = 0 to k = Lx/`2B. Hence,
periodic boundary conditions are impossible since the states at the left edge and the right edge differ strongly in their k values. The total number of states in each LL is N = Ly 2π LxZ/`2B 0 dk = LxLy 2π`2 B = Nφ, (2.3.23)
where Nφ ≡ BLΦ0xLy is the number of flux quanta penetrating the sample and
φ0 = hc/e is the flux quantum. Dividing the number of electrons by the number
of states per landau level (and spin), we obtain the ”filling fraction” (or ”filling factor”) of LLs
ν = Nel Nφ
= 2π`2
here Nel is the number of electron and nel is the electron density. ν is an even
integer if spin splitting is not resolved, and an odd integer if it is resolved when the all LLs are completely filled or empty. The number of states per unit area with energy eigenvalues below E is called as integrated density of states Z(E),
Z(E) = 1 2π`2 B X n,σ Θ(E − En,σ), (2.3.25)
where En,σ is the sum of the Landau energy En= ~ωc(n + 1/2) and the Zeeman
spin energy ²σ = (σ2)g∗µBB with the spin quantum number σ = ∓1, the effective
band Lande-g factor (g∗) and the Bohr magneton µ
B = 2m0e~c.
To obtain the Landau density of states (DOS), one performs the derivative
dZ/dE = D(E) D(E) = 1 2π`2 B X n,σ δ(E − En,σ), (2.3.26)
where the δ-function of energy expresses the macroscopic degeneracy of the Lan-dau energies, because of their independence of the center coordinate X. In real systems, these δ-functions are broadened by disorder effects. Disorder is indis-pensible in semiconductor systems containing a 2DEG. For instance, in GaAs sys-tems, the 2DEG is usually generated by a doping layer behind a spacer, and the ionized donors act as randomly distributed scattering potentials for the electrons. Therefore, disorder lifts the degeneracy of the LLs and leads to a broadening of the δ-functions into spectral functions An,σ(E) of finite widths Γ,
DΓ(E) = 1 2π`2 B X n,σ An,σ(E). (2.3.27)
one is the Gaussian model (G¨uven, & Gerhardts, 2003) for the disorder broadening AG n,σ(E) = (2πΓ2n,σ)−1/2exp à −1 2 · E − Enσ Γn, σ ¸2! , (2.3.28)
which is proved to be useful for the understanding of many experiments and can be motivated by microscopic considerations. The line width Γnσ is related
to the density and the potential of the impurities. It is usually assumed to be independent of the spin quantum number (σ) and landau quantum number n. Second frequently used model for DOS is that of the so called Self-consistent
Born approximation (SCBA) (Ando, Fowler, & Stern, 1982),
ASCBAn,σ = 1 πΓn,σ Ã 1 − · E − En,σ 2Γn,σ ¸2!1 2 . (2.3.29)
In this model, the spectral functions of the individual LLs are half ellipses instead of gaussians. The DOS can be derived by a systematic, perturbative treatment of the average effect of electron scattering by randomly distributed impurities for both models. Their difference are due to the fact that different classes of scat-tering processes are considered in both approximations. In the SCBA, coherent scattering of an electron by more than one impurity is neglected. Such coherent scattering processes include the scattering from arbitrarily strong effective poten-tials due to superpositions of nearby impurity potenpoten-tials. However, the coherent multi-center scattering which is included in the Gaussian model, leads to soft but rapidly decaying tails of the spectral functions.
2.4 Characterization of Electron Transport
2.4.1 Magnetotransport in the Classical Regime
The Drude Model
The Drude model is the basic theoretical model for electrical transport. This model gives a reasonably good description of transport at high temperature al-though a very simplified model. From the classical equation of motion for an electron in a magnetic field parallel to the z axis and with the relaxation time approximation for the momentum distribution one finds the steady state current density j = σE, σ = σ0 1 + ω2 cτ2 1 ωcτ −ωcτ 1 , (2.4.1) with σ0 = e 2n elτ
m , relaxation time τ and the cyclotron frequency ωc= eB/m∗. In
this model, only two parameters are needed for describing the transport proper-ties. The first one is the mobility (µ) and the second one is the electron density (n). Inverting conductivity gives the resistivity tensor,
ρ = ρxx ρxy ρyx ρyy ρ−1 = 1 ρxxρyy− ρxyρyx ρyy −ρxy −ρyx ρxx , (2.4.2)
with the following diagonal and off-diagonal elements. Apparently, the formula for the components of the resistivity are somewhat simpler than these for the conductivity components. Note that the ρxx component
ρxx = ρ0 = 1 σ0 = m e2n elτ , (2.4.3)
is independent of B, while the ρxx component, ρxy = ρH = ωcτ σ0 = B ecnel , (2.4.4)
is independent of τ . Equation 2.4.3 is a special case of the general result that yields no effect of the magnetic field on the resistance and Equation 2.4.4 says that the off-diagonal Hall resistivity does not depend on the scattering mechanism, which determine the relaxation time τ . The diagonal component of the resistivity tensor (ρxx) vanishes if and only if the diagonal component of the conductivity
tensor (σyy) also vanishes,
ρxx = 0 ⇔ σxx = 0 if ρxy 6= 0. (2.4.5)
since the Hall resistivity is finite for B 6= 0 due to the properties of two dimen-sional inversion. We see that ρxx → 0 means τ → ∞ from Equation 2.4.3. In
this situation, the scattering and thus of dissipation vanish. Therefore, in the presence of a perpendicular magnetic field, the longitudinal resistivity ρxx, and
simultaneously the longitudinal conductivity σxx of a 2DEG can be non-zero only
if the electrons undergo scattering processes. In the no-scattering case, we have
ρxx = σxx = 0 σH0 =
1
ρxy
= ec
Bnel. (2.4.6)
Only in this dissipationless case, the Hall conductivity, which we define in general as σH = σxy, has this simple form. This dissipationless situation represents the
idealized form of the Hall effect, in the limit τ → ∞. It says that the electrostatic
Hall field,
EH =
1
ecnel
j × B, (2.4.7)
compensates the effect of the Lorentz force on the electrons. These results cor-respond to the usual experimental situation where a current is drawn through a Hall bar. The Lorentz force drives the electrons towards one of the sample
boundaries and the charge imbalance produces an electrostatic field EH, which
in stationary state just compensates the effect of the Lorentz force.
2.4.2 Magnetotransport in the Quantum Regime
At low magnetic fields (∼ 0.4 T for GaAs) the two-dimensional electron system behaves classically and the Hall resistivity increases linearly with magnetic field. This is the classical Hall effect regime. As the magnetic field is increased above 0.4 T, the 2DEG leaves the classical regime and shows a unique remarkable be-havior. In this magnetic field values, the longitudinal resistivity ρxx shows strong
oscillations while the Hall resistivity ρxy starts deviating from the linearity. This
is the Shubnikov-de Haas (SdH) effect, which was discovered by Shubnikov and de Haas in 1930 in a Bismuth crystal. As the magnetic field is further increased, the 2DEG enters the QHE regime, where ρxy develops plateaus and ρxx drops
to zero. The transitions from one regime to the other has smooth borders. The application of a uniform magnetic field normal to the plane of the 2DEG leads to the Landau quantization of the energy spectrum. As a result of this quantization, LLs are formed and SdH oscillations are observed in the ρxx measurements. To
explain the SdH oscillations which arise from high magnetic fields, it is needed to return to the concept of Landau quantization and LLs, because it is a pure quantum phenomenon. When the spin splitting is taken into account the density of states are also split. The LLs split partly into two as the field is increased and they become completely separate at high magnetic fields. At ν = 1 the chemical potential lies below the Fermi energy (unlike at ν = 3), because the spin levels are now completely separated. All of the electrons lying in the lowest LL have their spins aligned in the same direction. This is called the magnetic quantum limit. Below this limit, where ν < 1 the electron-electron interaction dominates and other interesting physical effects like the FQHE (Tsui, St¨ormer, & Gossard, 1982) are observed. For a perfect system with LLs in the form of δ-functions,
the jumps are abrupt. But in reality they are rounded, because the LLs are broadened due to scattering and at very low temperature, kBT ¿ EF, only
im-purity scattering is effective. At even integer filling factors the magnitude of the jumps is ~ωc− g∗µBB, and for odd integers, it is g∗µBB. Using the self-consistent
Born Approximation (SCBA) (Ando, Fowler, & Stern, 1982) the scattering time
τ is τ ∝ 1/D(EF)2, where D(EF) is the density of states at the Fermi energy.
When the Fermi energy lies exactly in between two LLs, D(EF) is minimum and
therefore τt is maximum. This leads to the minima in the SdH oscillations at
integer filling factors. In other words, when the highest filled LL is completely filled, chemical potential (µ) will be in the gap between the highest occupied and the lowest unoccupied level, such that there will be no empty states available at energies close to µ. This is analogous to an insulator, and the conductivity will be a minimum. From the definition of filling factor
ν = nel nL
= hnel
eB , (2.4.8)
gives the filled LLs. The period of the SdH oscillations can easily be written as
∆ µ 1 B ¶ = e hnel . (2.4.9)
Thus, a plot of the conductivity as a function of (1/B), allows to read off the density of the 2DEG.
The integer quantum Hall effect
The minima of the SdH oscillations, which correspond to a fully occupied number of LLs (integer filling factor), tend to decrease and become zero ρxx = 0,
as the magnetic field is increased. The vanishing of the longitudinal resistivity occurs in a certain range of magnetic field values around these integer filling factors, corresponding to a dissipationless current flow. At these magnetic field
values, the Hall resistivity ρxy exhibits plateaus with the quantized values ρxy =
h/ie2, where h is the Planck constant, e is the elementary electron charge, and i
is the integer filling factor and this phenomena is the quantized Hall effect (QHE). When a whole number of LLs is fully occupied, the filling factor (ν) becomes an integer, thus we have,
ν = hns
eB =⇒ ns =
ieB
h . (2.4.10)
If we put this in equation ρxy = nBse, the Hall resistivity becomes,
ρxy = 1 i h e2 = 25812.807 i Ω. (2.4.11)
Although this simple calculation predicts the value of the quantized Hall re-sistance, it gives this value only at one magnetic field value at which an integer number of LLs is completely filled. However, in realistic systems, the plateaus extend over finite widths of magnetic field and a real Hall device has always a finite width and length with metallic contacts and even high mobility devices contain impurities and potential fluctuations, which lift the degeneracy of the LLs which are broadened due to scattering of electrons. There two main models which try to explain the QHE namely the bulk (disorder) (Laughlin, 1981) and the edge pictures (Halperin, 1982). These two models also describe the transport properties of the current carrying states.
Disorder, Localization and Edge Picture
The bulk model of the QHE starts from the resistivity to include a certain amount of disorder to understand why broad plateaus appear in the Hall resis-tance. There are extended and localized states in the infinite two-dimensional electron system. Scattering centers which are composed of the impurities or the
positive charged donors are located randomly in the doping layer and cause energy fluctuations at the LLs. The average magnitude of the fluctuations is equal to the broadening of the LLs and so it is two classes of states. First one is called extended (delocalized states) at the centers of LLs allow the electrons to move through the 2DEG. Second class of the states is called as localized states and these states are located in the tails of the LLs. When the chemical potential (µ) is in the localized states between the LL centers, both the longitudinal resistivity and conductivity becomes zero σxx = ρxx = 0. The Hall resistivity ρxy is the quantized. But if
µ is in the delocalized states, both σxx and ρxx are finite, ρxy is not quantized.
The universality of the transport properties in the quantum Hall regime occurs because of the random disorder and uncontrolled imperfections which materials contain. Localization in the presence of disorder plays an essential role in the quantization and this localization is strongly modified by the strong magnetic field. In two dimension for zero magnetic field and non-interacting electrons, all states are localized even for arbitrarily weak disorder.
The idea of the edge channels were introduced firstly by Halperin (1982). In this picture, the QHE is understood by the imperfect cancelation of the current of the up-going and down-going edge channels. The edge channel picture of the IQHE, elaborated by B¨uttiker (1988), proved to be very successful for the description of the resistances measured on complicated samples with many gates and contacts. There are many experimental indications that, in the plateau regime of the IQHE, the interior of the sample is not important: it can be partly removed or, by suitable gates, tuned to another electron density, without changing the quantized value of the measured Hall-resistance. Also the exact arrangement of the contacts plays a minor role. This has been interpreted as indication that the relevant currents flow near the sample edges. In a real system, no energy gap at the Fermi energy exists under quantum Hall condition even if no disorder due to impurities is included. The energy of the LLs is plotted across the width of the device in Figure 2.5. The Fermi energy in the inner part of the sample is
assumed to be in the gap for ν = 2. Close to the edge (within a characteristic depletion length of about 1 µm) the carrier density becomes finally zero. This corresponds to an increase in the LL energies at the edge so that these levels become unoccupied outside the sample. All occupied LLs inside the sample have to cross the Fermi energy close to the boundary of the device. At these crossing points partially-filled LLs with metallic properties are present. Self-consistent calculations for the occupation of the LLs show that not lines but metallic stripes with a finite width are formed parallel to the edge (Chklovskii, Shklovskii, & Glazmann, 1992).
Figure 2.5 Ideal LLs for a device with boundaries. The fully oc-cupied LLs in the inner part of device rise in energy close to the edge forming compressible stripes which are close to the crossing points of the LL with the Fermi energy EF.
The experimental results confirm qualitatively the picture of incompressible stripes (ISs) close to the edge (Larkin, & Davies, 1995) which move away from the edge with increasing magnetic field until the whole inner part of the device becomes an incompressible region at integer filling factor. At slightly higher magnetic fields this bulk incompressible region disappears and only incompressible
stripes with lower filling factor remain close to the boundary of the device.
Compressible and Incompressible Regions
D.B.Chklovskii and coworkers point out that, in real 2DEG system with lateral confinement, in which the electron density decreases from a finite value (bulk) to zero near the edges, electronic screening effects become extremely important in high magnetic fields (Chklovskii, Shklovskii, & Glazmann, 1992). At high magnetic fields, the magnetic length `B =
q
~
eB is much smaller than the length
scale on which electron density and confinement potential vary. The Landau bands show a spatial dispersion given by the external confinement potential when one neglects screening effects under these conditions. If in the bulk of the sample several LLs are occupied, density profile drops like a step function towards zero at the edges with wide plateaus. These plateaus correspond to the integer filling factor of occupied bands and they are separated by steep steps of a width given by the extent of the Landau wavefunctions. In the idealized case of the small collision broadening and low temperature, the thermodynamic density of states (TDOS) (∂nel/∂µ) is extremely high when the chemical potential falls onto a
Landau energy. However if the chemical potential falls into a gap between such energies TDOS is nearly zero. In the first situation, screening is nearly perfect since it costs no energy to change the position of electrons. In the second case, no screening is possible since occupied and empty electron states are separated by the large (as compared with kBT ) energy gap. In an inhomogeneous sample with
sufficiently high bulk density one meets both situations. There are compressible
regions (CSs) in which screening is nearly perfect, so that the total, screened
potential is flat and one of the LLs is pinned to the Fermi level. There are also
incompressible regions (ISs), in which µ falls between adjacent Landau bands, so
that no redistribution of electrons is possible and the density is constant, since the filling factor of the LLs has a fixed integer value.
2.4.3 Quantum Transport Models of the 2DEG
The model to be used for the resistivity tensor should show the characteristic behavior known from the quantum Hall regime. So, in the case of odd or even integer local filling factors ν(x) = 2π`Bnel(x), the longitudinal conductivity σl(x)
must nearly vanish. Moreover, the approximations for the conductivity tensor and the density of states (DOS) should satisfy certain consistency relations. However, in the presence of the scatterers each LL is broadened. So, for strong quantizing magnetic fields, a suitable form of a broadened landau DOS should be used,
D(E) = 1 π`2 B ∞ X n=0 δ(E − En) −→ D(E) = gs 2π`2 B ∞ X n=0 An(E), (2.4.12)
with a spectral function An(E) of the nth LL, where gs = 2 accounts for spin
de-generacy. Many different approximations have been proposed for the calculation of such level broadening and transport, till now. Among these, local Ohm’s law (LOL) and SCBA, are used in our investigation and is introduced the following.
Local Ohm’s Law
If a stationary net current is imposed on the Hall bar, this leads to position-dependent current densities and electrical fields. It is assumed that the electro-static potentials and the thermodynamic variables vary on a scale much larger then quantum lengths and also assumed that the current density j(x) and the electric field ˆE(x) satisfy the local version of Ohm’s law (LOL),
j(x) = ˆσ(x) ~E(x), ˆσ(x) = σ(nel(x)). (2.4.13)
Where the position dependent conductivity tensor has the same form as for ho-mogeneous sample, however homogenous density replaced by the local electron
density nel(x). The components jx and Ey of current density and electric field is
assumed to be constant due to the translation invariance in the y-direction
jx(x) ≡ 0, Ey(x) ≡ Ey0. (2.4.14)
The other components (jy and Ex) are described as,
jy(x) = 1 ρl(x) Ey0, Ex(x) = ρH(x) ρl(x) Ey0, (2.4.15) with the longitudinal component ρl = ρxx = ρyy and the Hall components ρH =
ρxy = −ρyx of the resistivity tensor ˆρ = ˆσ−1. For a given applied current I, one
obtains the constant electric field component along the Hall bar
E0 y = I d Z −d dx 1 ρl(x) −1 , (2.4.16)
and for Hall voltage across the sample
VH = d Z −d dxEx(x) = Ey0 d Z −d dxρH(x) ρl(x) , (2.4.17)
with the usual normalization of the resistance to a square-shaped conductor which yields the Hall and the longitudinal resistance;
RH = VH I , Rl(x) = 2dE0 y I . (2.4.18)
Hence, with given the local conductivity tensor elements, one can obtain the current density in x-direction and thereby the global resistances, here one can grasp the essence of the local model. Any reasonable model for the conductivity of a high-mobility 2DEG at zero temperature will give simple results for the conductivity components at integer filling factors. For σyx(x) = −σxy(x) = σH(x)
one can assume the form σH(x) = µ e2 h ¶ ν(x), (2.4.19)
which yields the correct values at integer filling factors. To simulate the behavior of σxx = σyy = σl, a simple approximation of the conductivity model can be given
as σl(x) = σH[² + (1 − ν(x))2/4] 0 < ν(x) < 1.5, σH[² + (2 − ν(x))2/4] 1.5 ≤ ν(x) < 2.5, (2.4.20)
with a small but finite positive value ε to avoid divergences. This definition de-scribes correctly that ρl(x) ∝ σl(x) becomes very small at the ISs with integer
local filling factors. This scheme enclosures the simplest set of assumptions to provide a qualitative understanding of the current distribution and the calculation of the global resistances. This simple model is able to reproduce characteristic features observed in the experiment (Ahlswede, Weitz, Weis, Klitzing, & Eberl, 2001). The current flows preferably along the IS and ρl is the smallest and thus
the gradient of electrochemical potential (µ∗) is the largest. The total potential
V (x) is forced to follow µ∗(x) closely because of the Thomas-Fermi-Poisson self
consistency requirement, so that the current-induced charge of the electron den-sity profile is small. The induced ∆V (x) follows closely µ∗(x) and varies mainly in
the region of ISs. These considerations are general and do not depend on details of the conductivity model. If we want to calculate the resistances between the plateau regions, we need to specify a conductivity model.
Self-Consistent Born Approximation
Self-consistent Born approximation (SCBA) (Ando, Fowler, & Stern, 1982) is known to be the most established one among others which allows consistent models of longitudinal and Hall conductivities. Furthermore, this approximation
is the simplest one free from various difficulties of divergence caused by the sin-gular nature of the DOS. At zero temperature, the integrals in Equations 2.4.16 and 2.4.17 become singular, because longitudinal resistivity vanishes (ρl(x) = 0)
and these equations are not integrable because of singularity. However, in the SCBA to be considered below, they can be integrated unlike the above simplest model and gaussian model (G¨uven, & Gerhardts, 2003). In fact, one can still use the local version of Ohm’s law but now with an averaged conductivity tensor, to simulate qualitatively the expected effects of a nonlocal treatment.
¯ σ = 1 2λ λ Z −λ dξσ(x + ξ), (2.4.21)
where λ is on the scale of Fermi wavelength (λF). Hence, the resistivity
compo-nents in Equation 2.4.3 have to be calculated by tensor inversion of ¯σ(x). SCBA is
more realistic model to calculate the Hall and the longitudinal conductivities and also allows the consideration of anisotropic scattering by randomly distributed finite-range impurity potentials. As origin of finite resistance and broadening of LLs, it is assumed that the electrons are scattered by randomly distributed, iden-tical impurities of density NI. The effect of such scattering in strong magnetic
fields (non-overlapping LLs) has been worked out by Ando and coworkers, in the 1970’s (Ando, & Uemura, 1974) within the SCBA. SCBA is similar to the lowest order cumulant approximation and allow to treat the collision broadening of LLs and transport coefficients in a consistent manner. In the SCBA, the effects of scattering from each scatterer are taken into account in the lowest Born approxi-mation. The level broadening due to scatterers in a two dimensional system, are considered in a self-consistent way within a framework of the Green’s function formalism. In the absence of a magnetic field (in the weak scattering limit) this approximation becomes equivalent to the Boltzman transport theory. In our sys-tem, it is assumed that the considered scatterers are charged donors distributed randomly in a plane parallel to that of 2DEG. In the earlier studies, it has been
generally approximated the impurity potentials (Siddiki, & Gerhardts, 2004) by Gaussian potentials V (r) = V0 πR2exp µ −r 2 R2 ¶ , (2.4.22)
with a range R of the order of the spacing between the 2DEG system and the doping layer. Consistency with the transport coefficients requires the replacement the δ-functions of the Landau DOS by the semielliptical spectral functions
D(E) = gs 2π`2 B ∞ X n=0 An(E), (2.4.23)
with the spectral function An(E) defined in Equation 2.3.29. These functions are
centered around the Landau energies ²n= ~ωc(n + 1/2), so the DOS for each LL
has a semielliptical form with the width Γn. In the case of short-range scatterers
(R < `B), one can get Γn= Γ with
Γ2 = 2
π~ωc
~
τf
, (2.4.24)
where τf is the relaxation time for B = 0 obtained assuming the same scatterers.
The level broadening is the lifetime of broadening and is independent of LL. In the case of long-range scatterer (R > `B), one gets
Γ2
n= 4h(V (r) − hV (r)i)2i, (2.4.25)
where V (r) is the local potential energy. Thus, inhomogeneous type level broad-ening is determined by the fluctuation of the local potential energy. In the case of short range scatterers and sufficiently strong magnetic fields, the longitudinal conductivity of the isotropic system is given by
σxx = e2 π2~ Z dE · −∂f ∂E ¸ · −Γ xx N ΓN ¸ " 1 − µ E − EN ΓN ¶2# . (2.4.26)
Here N should be chosen in such a way that, EN is the closest E and (ΓxxN)2 =
(N + 1/2)Γ2. At zero temperature, the peak value of σ
xx depends only on the LL
index N,
(σxx)peak= e
2
π2~2(N + 1/2). (2.4.27)
In the case of long range scatterers, LL broadening width is just the fluctuation of the gradient of the local potential energy and Hall conductivity σxy(= −σyx)
is given by, σxy = − Nsec H + ∆σxy, (2.4.28) ∆σxy = e 2 π2~ Z dE · −∂f ∂E ¸ (ΓxyN)4 ~ωcΓ3N · 1 − (E − EN ΓN )2 ¸3/2 . (2.4.29)
In the case of short range scatterers one obtains,
(ΓxyN)4 = (N + 1/2)Γ4 (∆σ
xy)peak=
Γ ~ωc
(σxx)peak. (2.4.30)
In the case of long-range scatterers, one has
(ΓxyN)4 = 4(N + 1/2)h(`B∆V (r))2i2 (2.4.31)
ELECTROSTATIC SELF-CONSISTENCY AND MANY BODY APPROXIMATIONS
3.1 Many Body Effects and Some Approximations
3.1.1 Thomas-Fermi Approximation
The Thomas-Fermi theory which based on a semi-classical approximation is a local density functional which has been put on a mathematically rigorous basis (Lieb, 1981) and also has been analyzed in 2D in detail by Lieb and group (Lieb, Solovej, & Yngvason, 1995). The theory provides a functional form for the kinetic energy of a non-interacting electron gas in some known external potential V (r) as a function of the density and has been successfully applied in the electronic-structure calculations of, e.g., quantum Hall systems, where the importance of e-e interactions has been addressed (Siddiki, & Gerhardts, 2003; Siddiki, 2007).
This model is a statistical consideration which can be used to approximate the distribution of electrons in an atom. The assumption stated by Thomas (1927) are that: ”Electrons are distributed uniformly in the six-dimensional phase space for the motion of an electron at the rate of two for each h3 of volume, and there
is an effective potential field that is itself determined by the nuclear charge and this distribution of electrons” (Parr, & Yang, 1989) (p.47). Therefore, Thomas-Fermi formula for electron density can be derived from these assumptions. In this approximation one sets
U[n] ≈ UH[n] = q2 2 Z d3r Z d3r0n(r)n(r0) r − r0 , (3.1.1)
i.e., approximates the full interaction energy by the Hartree energy and the kinetic
energy term is defined as,
T [n] ≈ TLDA[n] = Z
d3rthom(n(r)), (3.1.2) where thom(n) is the kinetic energy density of a homogeneous interacting system
with constant density n. This formula is also first example of a local density approximation (LDA). In this approximation, the real inhomogeneous system is divided into many small cells in each of which volume ∆V = l3, each containing
some fixed number of electrons ∆N. In each cell (i.e., locally) one can then use the per volume energy of a homogeneous system to approximate the contribution of the cell to the real inhomogeneous one. Making the cells infinitesimally small and summing over all of them yields Equation 3.1.2. For a noninteracting system the function thom
s (n) is known explicitly and total kinetic energy defined as,
TT F[n] ≈ TLDA[n] = Z d3rthom s (n(r)), thom s (n) = 3~2 10m(3π 2)2/3n5/3. (3.1.3)
Here the subscript s define the single-particle and the limit ∆V → 0, with
n = ∆N/∆V = n(r) finite, has been taken to give an integration instead of
a summation. This is the famous Thomas-Fermi kinetic energy functional. In addition, where TLDA
s [n] is the LDA to Ts[n], the kinetic energy of
noninteract-ing electrons of density n. Equivalently, it may be considered the noninteractnoninteract-ing version of TLDA[n]. Hence the TFA (Laird, Ross, & Ziegler, 1996) then consists
in combining Equation 3.1.1 and Equation 3.1.3
E[n] = T [n] + U[n] + V [n] ≈ ET F[n] = TLDA
s [n] + VH[n] + V [n]. (3.1.4)
This approximation is not very good for the kinetic energy term T [n]. A more accurate scheme for treating the kinetic energy functional of interacting electrons,
of noninteracting particles of density n;
T [n] = Ts[n] + Tc[n]. (3.1.5)
Where the subscript c stand for correlation and Ts[n] is not known exactly as a
functional of n and using the LDA to approximate it leads one back to the TFA. However it is easily expressed in terms of the single-particle orbitals φi(r) of a
noninteracting system with density n as
Ts[n] = − ~2 2m N X i Z d3rφ∗ i(r)∇2φi(r), (3.1.6)
because the total kinetic energy is just the sum of the individual kinetic energies for noninteracting particles. Since all φi(r) are functionals of n, this expression
for Ts is an explicit orbital functional but an implicit density functional, Ts[n] =
Ts[φ[n]], where the notation indicates that Ts depends on the full set of occupied
orbital φi each of which is a functional of n. The exact energy functional is now
rewritten as,
E[n] = T [n] + U[n] + V [n] = Ts[{φi[n]}] + UH[n] + Exc[n] + V [n]. (3.1.7)
The definition Exc contains the differences T − Ts (i.e. Tc) and U − UH. This
definition shows that a significant part of the correlation energy (Ec) is due to the
difference Tc between noninteracting and the interacting kinetic energies.
Equa-tion 3.1.7 is formally exact, however Exc which is called the exchange-correlation
(XC) energy is unknown. It is often decomposed as Exc = Ex + Ec; where Ex
is due to Pauli principle (exchange energy) and Ec is called correlation energy
which is the calculation of the error in the energy. The exchange and correla-tion potentials are defined as the funccorrela-tional derivates of the XC-free energy with respect to the densities.
3.1.2 Exchange-Correlation Effects in 2D and Local Density Ap-proximation
The electron-electron interactions have important effects and the transport in the 2D electron gas at the semiconductor should be treated as an interacting electron gas. In the local spin density approximation (LSDA), the exchange and the correlation effects have been taken into consideration using an ideal 2DEG system. However, the electrostatic and the exchange components act oppositely. The Hartree screening reduces the energy dispersion of the Landau bands, increas-ing the DOS, while the negative exchange energy of the occupied states broadens the bands, decreasing the DOS (Manolescu, & Gerhardts, 1995).
The electron-electron interaction can affect various properties of the 2DEG. The quasi particle properties, for instance the effective mass and the g∗ factor,
attracted much attention since they are directly observable in the experiments. Hartree approximation is valid when the electron concentration is sufficiently high, i.e., when the average kinetic energy of electrons is much larger than the average interaction energy. Many-body effects such as exchange and correlation can play an important role when the interaction energy has become important. These many body effects can be treated as perturbation to the Hartree energy or can be calculated in the density functional method (DFT). In the DFT, Hohen-berg, Kohn, and Sham have shown that the density distribution of an interacting electron gas under an external field can be obtained by a one-body Schr¨odinger-type equation containing exchange and correlation potential (vxc), in addition
to the usual Hartree potential and the external potential (Hohenberg, & Kohn, 1964; Kohn, & Sham, 1965). The exchange and correlation potentials are given by the functional derivatives of the exchange and correlation part of the ground state energies with respect to the number density n(r) of electrons, respectively.
n(r)²xc(n(r)) in the usual local approximation, here ²xc(n) is the XC energy
per electron of uniform electron gas with the density n. In this approxima-tion, exchange-correlation potential becomes the exchange-correlation part of the chemical potential µxc of the uniform electron gas. This theory has been used
and known to be successful in number of different problems and has become the most used computational tool in the study of strongly inhomogeneous in-teracting systems (Parr, & Yang, 1989). Traditionally, Exc has been evaluated
using density-dependent xc-functionals, either of local (LDA) or semi-local (GGA, meta-GGA, . . .) character (Perdew, & Kurth, 2003). The 2D-LDA consists of the exchange functional derived for the homogeneous 2D electron gas (Rajagopal, & Kimball, 1977) and the corresponding correlation functional constructed using quantum Monte Carlo methods (Tanatar, & Ceperley, 1989; Attaccalite, Maroni, Gori-Giorgi, & Bachelet, 2002). The exchange energy reads
²ex = −R∗y
4√2 3πrs
[(1 + ξ)3/2+ (1 − ξ)3/2], (3.1.8)
here, ξ(y) = (n↑ − n↓)/n is the local spin-polarization, (n↑ + n↓) is the total
electron density, and rs is the dimensionless density parameter which is defined
in terms of the effective Bohr radius a∗
0 and defined as,
rs = 1 a∗ 0 √ πn, a ∗ 0 = εr mef f a0. (3.1.9)
Here bohr radius a0 = 0.529 × 10−9 m and Ry∗ = mef f/ε2rRy generalizes
Tanatar-Ceperley (TC) results for the case of an arbitrary effective electron mass (m∗ =
mef fme) and relative dielectric constant (εr), and converts TC expressions (Tanatar,
& Ceperley, 1989) into SI units. The correlation energy for the unpolarized case (ξ = 0) and for the fully polarized case (ξ = 1) is approximated in the form
²c(ξ) = −Ry∗C0
1 + C1ω
1 + C1ω + C2ω2+ C3ω3
, (3.1.10)
