Electronic correlation in the quantum Hall regime [Elektronische Ressource] / von Marcus Kasner

Sujets

• Physik
• Astronomie

Informations

• Publié par: otto-von-guericke-universitat_magdeburg
• Publié le: 01 janvier 2002
• Nombre de lectures: 26
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Electronic correlation in the quantum Hall regime
Habilitationsschrift
zur Erlangung des akademischen Grades
doctor rerum naturalium habilitatus (Dr. rer. nat. habil.)
genehmigt durch die Fakult at fur
Naturwissenschaften
der Otto-von-Guericke-Universit at Magdeburg
von Dr. rer. nat. Marcus Kasner
geb. am 07.07.1957 in Perleberg
Gutachter:
Prof. Dr. Harald B ottger
Prof. Dr. Bernhard Kramer
Prof. Dr. Ulrich R o ler
Magdeburg, den 09.01.200223
Abstract
Two-dimensional electron systems (2DES) of high-mobility can be experimentally realized in
modern semiconductor devices near interfaces. In particular, the limit of a strong magnetic eld
normal to the interface o ers the opportunity to study an electronic system, which is unique in many
respects. Due to the strong eld, all electrons can be accommodated in the lowest orbital Landau
level of quenched kinetic energy resulting in a macroscopic degeneracy of the non-interacting ground
state. Thus, the only relevant energy scale of the system is set by the interaction energy, which
turns the system into a strongly correlated one.
The existence of energy gaps at certain lling factors is one of the reasons for remarkable
anomalies in magneto-transport, the quantum Hall e ects. These exhibit features that are very
di erent from those in lower magnetic eld experiments. In the case of small occupancy of the
lowest Landau level, the existence of gaps can be traced back to the occurrence of an incompressible
many-particle ground state.
In the Introduction, we review the most important experimental and theoretical developments
of the physics in the quantum Hall regime and derive the appropriate many-particle Hamilto-
nian. Then, we discuss the problem of two interacting particles of equal and opposite sign, which
frequently appears in the many-particle treatment.
In the second part, we study in detail the properties of the many-particle ground state and
its elementary charged excitations, the quasiparticles, in the limit of an in nitely strong magnetic
eld neglecting the spin degree of freedom. This is explicitly done on the basis of a special model
and by means of numerical methods in the case that only one third of the available one-particle
states in the lowest Landau level is occupied, i. e. at lling factor 1 =3. Furthermore, we discuss the
generalizations to lling factors of the form p=q, where p;q are integers and q odd. In particular,
we emphasize Jain’s composite fermion theory and its relation to Chern-Simons eld theories.
The third part is devoted to the consideration of the spin degree of freedom, which recently
received a lot of attention from experiment. It is shown that at certain lling factors the ground
state exhibits spontaneous spin magnetization, which de nes the system as a quantum Hall ferro-
magnet. After the discussion of the ground state properties near lling one, where novel charged
spin texture excitations appear, a many-particle theory is developed to determine thermodynamic
properties at exactly lling factor = 1. The accounts for the spin-wave like excitations
above a completely spin-polarized ground state. The many-particle theory improving the inade-
quate Hartree-Fock theory predicts the temperature dependence of the spin magnetization. The
results are compared with recent experimental data as well as with di erent theoretical approaches.
The spectral function found is used to study other observables as the nuclear spin-lattice relaxation
rate 1=T , the current-voltage characteristic and the conductance for tunneling between two weakly1
coupled parallel layers.
Our considerations are based on a microscopic electronic Hamiltonian. It becomes clear that
this strongly correlated 2DES shares many di culties with theories of metallic ferromagnetism, yet
it is free from the consequences of a complex bandstructure, which have confounded comparison
with experiment.
Eventually, we speculate about possible routes of future research in this lively eld that has
o ered surprises for more than twenty years.4Contents
1 Introduction 7
1.1 Two-dimensional electron systems in a magnetic eld . . . . . . . . . . . . . . . . . 7
1.2 A short historical account of the 2DES in a strong magnetic eld . . . . . . . . . . 9
1.3 Theoretical models of the quantum Hall regime and the lowest Landau level approx-
imation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Two interacting particles in a magnetic eld . . . . . . . . . . . . . . . . . . . . . 16
2 Ground states and quasiparticles of spin-polarized electrons in the lowest Lan-
dau level 21
2.1 The Laughlin function as ground state at lling factor 1 =q (q { odd) . . . . . . . . 21
2.2 The state as a quantum-liquid . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Elementary excitations: quasiholes and quasielectrons as quasiparticles . . . . . . . 28
2.4 The incompressibility of the Laughlin state . . . . . . . . . . . . . . . . . . . . . . 34
2.5 The microscopic understanding of lling factors p=q within the hierarchical theory 36
2.6 Jain’s ground state wavefunctions at lling factors p=q and composite fermions . . 38
3 Ground states and thermodynamics of electrons with spin degree of freedom in
the lowest Landau level 43
3.1 The spin degree of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Ground state and charged excitations at and near lling factor one . . . . . . . . 45
3.3 Exact one-spin ip excitations at lling factor one . . . . . . . . . . . . . . . . . . 55
3.4 Elementary theories of spin magnetization around lling factor one . . . . . . . . . 59
3.5 The failure of the Hartree-Fock approximation . . . . . . . . . . . . . . . . . . . . 63
3.6 Beyond Hartree-Fock: electron scattering on spin-waves . . . . . . . . . . . . . . . 66
3.7 Other theories for the spin magnetization . . . . . . . . . . . . . . . . . . . . . . . 72
3.8 Comparison of theories and experiments: spin magnetization at lling factor one . 75
3.9 of and expts: the nuclear spin-lattice relaxation rate . 80
3.10 Weak interlayer tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.11 Spin-magnetization at fractional lling factors . . . . . . . . . . . . . . . . . . . . . 85
4 Summary and outlook 89
A A particle in a constant magnetic eld 91
B The interaction matrix elements 93
C Thermodynamic Green’s functions and the self-consistent Hartree-Fock approx-
imation (SHF) 95
D List of Symbols 97
E List of Abbreviations 99
56Chapter 1
Introduction
1.1 Two-dimensional electron systems in a magnetic eld
Traditionally, any microscopic description in solid state physics starts from simpli ed models trying
to capture the essential features before systematically extending the model in order to reconcile it
with the often much more complex experimental results. Pursuing such an approach, we describe
electrons of spinS = 1=2 and chargeq = e< 0 moving in a three-dimensional crystal, which are
subject to a spatially constant and time independent magnetic eld B =Be B e (B > 0), inz ? z
a rst approximation as independent particles in a continuum. Due to the separation of the motion
in the x-y-plane and the z-direction, the Hamiltonian can be written in Cartesian coordinates as
21 p2 zH = (p qA) + g BS : (1.1)0 e B
2m 2me e
Here, the kinematical momentum in the plane is given by p = (( h=i)@ qA ; ( h=i)@ qA ),x x y y
whereas the momentum p = ( h=i)@ describes the free motion in the z-direction. The gaugez z
freedom of the vector potential allows to write it as A(r) =B( y; (1 )x; 0) with the parameter
2R. The gauge parameter covers for = 0 and = 1=2 the two mostly used gauges, viz. the
Landau gauge and the symmetric gauge. The last term is the Zeeman energy and is due to the
coupling of the dimensionless electron’s spin S to the magnetic eld ( g { gyromagnetic factor, {e B
Bohr magneton of the electron). The natural length scale is set by the cyclotron or magnetic lengthp p
‘ = h= jqBj, where ‘ [nm] = 25:65= B[T ], which exceeds for accessible elds up to B’ 20 Tc c
the lattice constant by more than a factor of 10 and justi es the application of a continuum model.
The solution of Eq. (1.1) separates in three simple energy eigenvalue problems: rst, the motion
2 2in z-direction for nite extension w with energies E = h k =(2m ), where k / 1=w as longz;i e z;iz;i
as there is no in-plane component of the magnetic eld, second, the spin degree can be simply
described by the two eigenvalues ofS =1=2, which are denoted by the majority spin direction"z
for = +1 and the minority spin direction# for = 1 as long as there is no spin-orbit coupling,
and third, the solution of electrons in the plane that can be reduced to a harmonic oscillator
problem, whose energy scale is h! , and where ! =jqBj=m is the cyclotron frequency.c c e
The problem becomes e ectively two-dimensional if one is able to create well distinguished
energy levels, so-called subbands, by requiring that the temperature T is smaller than the en-
ergy di erence E between the rst excited and the lowest subband energy, i. e. the following01
inequality holds
222 h
k T < E = : (1.2)B 01 2(m w )e
The estimate is based on the assumption of a rectangular quantum well of in nite height in z-
direction. In order to satisfy such a condition for, say, temperatures below 200K, a width smaller