field, and inversely proportional to the electronic density: RH = âB/enel. Here nel is the density per unit surface, and âe is the electron charge. Things are rather ...
1
Introduction to the Quantum Hall Effects
Lecture notes, 2006
Pascal LEDERER
Mark Oliver GOERBIG
Laboratoire de Physique des Solides, CNRS-UMR 8502 Universit´e de Paris Sud, Bˆat. 510 F-91405 Orsay cedex
2
Contents 1 Introduction 7 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 History of the Quantum Hall Effect . . . . . . . . . . . . . . . 9 1.3 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Charged particles in a magnetic field 2.1 Classical treatment . . . . . . . . . . . . . . . . . 2.1.1 Lagrangian approach . . . . . . . . . . . . 2.1.2 Hamiltonian formalism . . . . . . . . . . 2.2 Quantum treatment . . . . . . . . . . . . . . . . . 2.2.1 Wave functions in the symmetric gauge . . 2.2.2 Coherent states and semi-classical motion
. . . . . .
19 19 20 22 23 25 29
3 Transport properties– Integer Quantum Hall Effect (IQHE) 3.1 Resistance and resistivity in 2D . . . . . . . . . . . . . . . . . 3.2 Conductance of a completely filled Landau Level . . . . . . . . 3.3 Localisation in a strong magnetic field . . . . . . . . . . . . . 3.4 Transitions between plateaus – The percolation picture . . . .
33 33 34 37 42
4
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
The Fractional Quantum Hall Effect (FQHE)– From Laughlin’s theory to Composite Fermions. 4.1 Model for electron dynamics restricted to a single LL . . . . . 4.1.1 Matrix elements . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Projected densities algebra . . . . . . . . . . . . . . . . 4.2 The Laughlin wave function . . . . . . . . . . . . . . . . . . . 4.2.1 The many-body wave function for ν = 1 . . . . . . . . 4.2.2 The many-body function for ν = 1/(2s + 1) . . . . . . 4.2.3 Incompressible fluid . . . . . . . . . . . . . . . . . . . . 3
45 46 48 50 51 52 55 58
4
CONTENTS . . . . .
59 62 67 69 70
. . . .
75 75 78 80 82
6 Hamiltonian theory of the Fractional Quantum Hall Effect 6.1 Miscroscopic theory . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Fluctuations of ACS (r) . . . . . . . . . . . . . . . . . . 6.1.2 Decoupling transformation at small wave vector . . . . 6.2 Effective theory at all wave vectors . . . . . . . . . . . . . . . 6.2.1 Approximate treatment of the constraint . . . . . . . . 6.2.2 Energy gaps computation . . . . . . . . . . . . . . . . 6.2.3 Self similarity in the effective model . . . . . . . . . . .
85 86 87 91 95 98 100 103
7 Spin and Quantum Hall Effect– Ferromagnetism 7.1 Interactions are relevant at ν = 1 . . . . . . . . . 7.1.1 Wave functions . . . . . . . . . . . . . . . 7.2 Algebraic structure of the model with spin . . . . 7.3 Effective model . . . . . . . . . . . . . . . . . . . 7.3.1 Spin waves . . . . . . . . . . . . . . . . . . 7.3.2 Skyrmions . . . . . . . . . . . . . . . . . . 7.3.3 Spin-charge entanglement . . . . . . . . . 7.3.4 Effective model for the energy . . . . . . . 7.4 Berry phase and adiabatic transport . . . . . . . 7.5 Applications to quantum Hall magnetism . . . . . 7.5.1 Spin dynamics in a magnetic field . . . . . 7.6 Application to spin textures . . . . . . . . . . . .
109 109 110 112 115 117 118 119 121 123 127 127 128
4.3 5
4.2.4 4.2.5 4.2.6 Jain’s 4.3.1
Fractional charge quasi-particles . Ground state energy . . . . . . . . Neutral Collective Modes . . . . . . generalisation – Composite Fermions The effective potential . . . . . . .
. . . . . . . . . . . . . . . . . . .
. . . . .
. . . . .
. . . . .
Chern-Simons Theories and Anyon Physics 5.1 Chern-Simons transformations . . . . . . . . . . . . . 5.2 Statistical Transmutation – Anyons in 2D . . . . . . . 5.2.1 Anyons and Chern-Simons theories . . . . . . . 5.2.2 Fractional charge and fractional statistics . . .
at ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . .
8 Quantum Hall Effect in bi-layers 131 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8.2 Pseudo-spin analogy . . . . . . . . . . . . . . . . . . . . . . . 133
CONTENTS 8.3 8.4
8.5
5
Differences with the ferromagnetic monolayer case . . . . . . 134 Experimental facts . . . . . . . . . . . . . . . . . . . . . . . . 137 8.4.1 Phase Diagram . . . . . . . . . . . . . . . . . . . . . 137 8.4.2 Excitation gap . . . . . . . . . . . . . . . . . . . . . . 139 8.4.3 Effect of a parallel magnetic field . . . . . . . . . . . . 139 8.4.4 The quasi-Josephson effect . . . . . . . . . . . . . . . . 141 8.4.5 Antiparallel currents experiment . . . . . . . . . . . . . 142 Excitonic superfluidity . . . . . . . . . . . . . . . . . . . . . . 145 8.5.1 Collective modes – Excitonic condensate dynamics . . 148 8.5.2 Charged topological excitations . . . . . . . . . . . . . 150 8.5.3 Kosterlitz-Thouless transition . . . . . . . . . . . . . . 153 8.5.4 Effect of the inter layer tunneling term . . . . . . . . . 154 8.5.5 Combined effects of a tunnel term and a parallel field Bk 155 8.5.6 Effect of an inter-layer voltage bias . . . . . . . . . . . 158
6
CONTENTS
Chapter 1 Introduction 1.1
Motivation
Almost thirty years after the discovery of the Integer Quantum Hall Effect (IQHE, 1980)[1], and the fractional one (FQHE, 1983)[2], two-dimensional electron systems submitted to a perpendicular magnetic field remain a very active field of research, be it experimentally or on the theory level [3]. New surprises arise year after year, exotic states of electronic matter, new materials with fascinating quantum Hall properties keep triggering an intense activity in the field: see for example the number of papers on graphene which appear on cond-mat since the discovery of the QHE in this material in 2005 [4, 5] or the appearance of excitonic superfluidity in quantum Hall bilayers [6]. The continuous increase in sample quality over the years is a key factor in the discovery of new electronic states of two dimensional matter. In heterostructure interfaces such as the GaAs/AlGaAs system, one of the prototypical experimental systems, the electronic mobility µ has now reached values up to µ ≃ 30 × 106 cm2 /Vs, more than two orders of magnitude larger than the values obtained at the time of the first QHE discoveries in the 80’s. The discovery of the quantum Hall effects, in particular that of the FQHE, has taken a large part in a qualitative advance of condensed matter physics regarding electronic fluids in conducting materials. In large band metallic systems, the role of interactions was successfully taken into account until the sixties by the Landau liquid theory [7], which is a perturbation approach: interactions between electrons alter adiabatically the properties of the free electron model, so that the Drude-Sommerfeld model keeps its validity with 7
8
Introduction
renormalized coefficients. The theoretical tools corresponding to this physics have involved sophisticated diagramatic techniques such as Feynman diagrams, which are all based on the existence of a well controlled limit of zero interaction Green’s function. It was realized in the fifties, with the theory of BCS superconductivity [8] that attractive interactions cause a breakdown of the non interacting model, and a spontaneous symmetry breaking (in the superconductivity case a breakdown of gauge invariance), but even in that case Fermi liquid theory seemed an unescapable starting point for conducting systems. The various other hints of the breakdown of perturbation theory, such as the local spin fluctuation problem, the Kondo problem, the Mott insulator problem, the physics of solid or superfluid 3 He in the sixties/seventies or even the Luttinger liquid problem in the eighties did little to suggest the intellectual revolutions which were in store with the discoveries of the fractional quantum Hall effects and of superconducting high Tc cuprates [9]. (The heavy fermion physics is somewhat of a hybrid between the former and the latter electronic systems.) The fundamental novelty of both those phenomena, which involve electron systems in two dimensional geometries, is that the largest term in the Hamiltonian is the interaction term, so that perturbation theory is basically useless. If one tries to do perturbation theory from a limiting case, such as the incompletely filled Landau level in the Hall case, or the zero kinetic energy in the High Tc case, one is faced with the macroscopic degeneracy of the starting ground state. There is no way to evolve adiabatically from this degenerate ground state to the physical one. In both cases, interactions do not alter the quantitative properties of a pre-existing ground state. They are essential at determining the symmetry and properties of the ground state which results from the lifting of a formidable degeneracy. Thus the whole aparatus of perturbation theory turned ou to be inadequate for a theoretical understanding of the QHE, as well as of High Tc superconductivity. New methods have had to be devised, and new concepts emerged to account for unexpected exotic phenomena: new particles, new statistics, new ground states,[11] etc.. The most intuitive method turned out to be very useful and fruitful. It amounts to guessing the many-body wave function for ≈ 1011 particles for the ground state. This is the incredibly original path chosen in 1983 by Laughlin [12]. More or less the same method led P. W. Anderson to propose the RVB wave function as the basic new object to describe High Tc superconductivity in 1987. That wave function is a resonating superposition of singlet pair products involving all electrons. It looks like a BCS wave
1.2. HISTORY OF THE QUANTUM HALL EFFECT
9
function, where strong correlations prevent the simultaneous occupation of any site by two electrons. This proposal has been at the center of active discussions over the last twenty years. In the context of Quantum Hall Effects, new ideas such as Chern-Simons theories, which investigate the formation of composite particles when electrons bind to flux tubes, have been very fruitful [15, 16, 17]. These theories , based on their topological character, have given flesh to the notion of strange phenomena such as charge fractionalization, fractional statistics, and so forth, at work among the elementary excitations of the 2D electronic liquid under magnetic field. A new concept, also emerging over the last twenty years, is that of quantum phase transition and of quantum critical points [18]. Quantum phase transitions occur at zero temperature. They are not controlled by temperature, but by parameters such as pressure, magnetic field, or chemical doping [11, 18]. In heavy fermions for instance, the quantum critical point separates a metallic paramagnetic phase from an insulating antiferromagnetic one. At finite temperature, the “quantum critical regime” involves a broader array of parameter values. Quantum phase transitions, as we shall see, are present in a number of Quantum Hall Effects, as a function of magnetic field or of electron density.
1.2
History of the Quantum Hall Effect
The classical Hall effect The classical Hall effect was discovered by Edwin Hall in 1879, as a minor correction to Maxwell’s “Treatise on Electromagnetism ”, which was supposed to be a final and complete account of the physical properties of Nature. Hall noted that, contrary to Maxwell’s opinion, if a current I is driven through a thin metallic slab in a perpendicular magnetic field B = Bez , an electronic density gradient develops in the slab, in the direction orthogonal to the current. This gradient is equivalent to a transverse voltage V , so that the resulting transverse resistance (the Hall resistance) is proportional to the field, and inversely proportional to the electronic density: RH = −B/enel . Here nel is the density per unit surface, and −e is the electron charge. Things are rather simple to understand with the Drude model, with the electron equation of motion:
10
Introduction
I
−I
gaz d’électrons 2D
résistance longitudinale
résistance de Hall
Figure 1.1: Two dimensional electron system under perpendicular magnetic field. The current I is driven through the two black contacts. The longitudinal resistance is measured between two contacts on the same edge, while the Hall resistance is measured across the sample on the two opposite edges.
dp p p = −e E + ×B − , dt m τ where E is the electric field, m is the electron (band) mass , p its momentum τ the mean diffusion time due to impurities. The stationary solution for this equation, i.e. that for dp/dt = 0, is
py B − m px 0 = −e Ey − B − m
0 = −e Ex +
px , τ py . τ
With the cyclotron frequency ωC ≡ eB/m and the Drude conductivity σ0 = nel e2 τ /m, one gets py px σ0 Ex = −nel e − nel e (ωC τ ), m m py px σ0 Ey = nel e (ωC τ ) − nel e , m m In terms of the current density j = −nel ep/m, in matrix form E = ρ j,
History of the Quantum Hall Effect
11
with the resistivity tensor ρ=
σ0−1 enBel − enBel σ0−1
!
1 = σ0
1 ωC τ −ωC τ 1
!
.
(1.1)
The conductivity follows by matrix inversion, σ=ρ
−1
=
σL −σH σH σL
!
,
(1.2)
with σL = σ0 /(1 + ωC2 τ 2 ) et σH = σ0 ωC τ /(1 + ωC2 τ 2 ). In the limit of a pure metal with infinite τ , ωC τ → ∞, one has ρ=
0 − enBel
B enel
0
!
,
σ=
0 enel B
− enBel 0
!
.
(1.3)
Note that the diagonal (longitudinal ) conductivity is zero together with the longitudinal resistivity. The classical Hall effect, deemed by Hall of purely academic interest, and with no foreseeable application whatsoever is nowadays of current industrial use, and is still useful in condensed matter physics to measure the carrier density in conducting materials, as well as to determine their sign.
Landau quantization Landau was the first to apply quantum mechanics, in 1930, in the study of metallic systems, to the quantum treatment of electronic motion in a static uniform magnetic field. He found that problem to be quite analogous in 2D to that of a harmonic oscillator, with an energy structure of equidistant discrete levels, with a distance h ¯ ωC . Each level is highly degenerate. The surface density of states per Landau level, nB , is nb = B/φ0 per unit area, where φ0 = h/e is the flux quantum , so that nB is the density of flux quanta threading the surface in a perpendicular field B. Because of their fermionic character, electrons added to the plane fill in successive Landau Levels (LL), so that it is natural and useful to define a filling factor ν=
nel . nB
This quantum treatment will be reviewed in chapter 2.
(1.4)
12
Introduction
The Quantum Hall Effect : a macroscopic quantum phenomenon The IQHE, discovered by von Klitzing in 1980 [1] is, at first sight, a direct consequence of Landau quantization, and disorder. In fact, as we shall see, impurity disorder is also a necessary feature: in a tanslationaly invariant system, the Hall resistivity would have the classical value. In fact Hall quantization appears because of the sample impurity potential, not in spite of it. The IQHE appears at low temperature, when kB T ≪ h ¯ ωC , and is defined by the formation of plateaus in the Hall resistance, which become quantized, for certain ranges of values of B, as RH = (h/e2 )1/n, where n is an integer, the integer part of the filling factor: n = [ν]. Each plateau in the Hall resistance coincides with a zero (exponentially small value in fact) of the longitudinal resistance (Fig. 1.2). A remarkable fact about the resistance quantization is that its value is independent of the sample geometry, of its quality (density and/or distribution of impurities, etc.). The Hall resistance is given entirely in terms of fundamental constants, e and h. The accuracy of the determination of the n = 1 plateau value reaches 1 part in 109 , so that it is now used in metrology as a universal resistance standard, the v. Klitzing constant RK−90 = 25812, 807Ω. Another surprise followed shortly after the discovery of the IQHE. In 1983, D. Tsui, H. St¨ormer and A. Gossard found the Fractional Quantum Hall Effect (FQHE) [2]. This occurs for ”magical” values of the filling factor, especially within the lowest LL. The first observed fractional plateaus were at ν = 1/3 and ν = 2/3. Since then, a whole series of plateaux have been detected. The remarkable aspect is that for fractional ν values, there is a huge degeneracy of the N body states. Since, apart from impurities, the only relevant energy is the Coulomb repulsion between particles, one is facing a strongly correlated electron system. Our understanding of the FQHE is still to-day essentially based on a revolutionnary theory put forward by Laughlin in 1983: he proposed, by a series of educated guesses, a wave function for N ≈ 1011 particles, written in the first quantization language, which describes an incompressible electronic liquid state, i.e. one such that elementary and collective excitations are separated from the ground state by a gap[12]. Following the discovery of other families of fractional QH plateaus which are not described by the initial Laughlin wave functions, various generalizations have been proposed. B. Halperin generalised in 1983 the Laughlin wave function to the case of an additional discrete degree of freedom, such as
History of the Quantum Hall Effect
13
3.0 2.0
Ix
Vy
2.5
Vx
1.5
ρxx (kΩ)
2
ρxy (h/e )
2.0
6 54 3
1.5
1
2
1.0
2/3 3/ 2
3/ 4
1/ 2
3/5
3/7
5/9 4/3 5/3 8/5 7/5
0.5
2/5
4/9
4/7
1.0
5/11
6/11
6/13
0.5
5/7
0.0
8/15 4/5
0
0
4
8 Magnetic Field B (T)
7/13
7/15
12
16
champ magnétique B[T]
Figure 1.2: Experimental signature of the quantum Hall effect. Each plateau coincides with a zero longitudinal resistance. The classical Hall resistance curve is the dotted line. Numbers label the filling factor ν = n for the IQHE, and ν = p/q (p and q integers for the FQHE.
14
Introduction
spin [19]. In 1989, Jain generalised the theory to account for observed fractional states with ν = p/(2sp + 1) with s and p integers. He introduced the notion of “Composite Fermions” (CF). The CF theory allows to understand the FQHE as an IQHE of CF. This will be dealt with in chapter 4.
1.3
Samples
The discovery of the IQHE and of the FQHE is intimately connected to the evolution of semiconducting sample preparation to produce 2D electron gases. The order of magnitude of electron densities in thin metallic films was not appropriate for the QHE discovery. The electronic surface density of metallic thin films is of order nel = 1018 m−2 = 1014 cm−2 . As we shall see, the QHE become observable when the electronic surface density is of the order of the magnetic flux density, i. e. nel ∼ nB = eB/h. This would amount to a magnetic field of order ≈ 1000 T, quite out of reach in the laboratory nowadays, when the largest available fields in a dc regime amount to less than 50 T, and less than 80 T for pulsed magnetic fields. More intense fields are available in destructive experiments or nuclear blasts. A useful quantity q which sets a length q scale for the QH physics is the magnetic length, lB = h ¯ /eB = 25, 7nm/ B[T]. The magnetic length is such that the flux 2 which threads a surface equal to 2πlB is the flux quantum φ0 = h/e Lower electronic densities, typically nel ∼ 1011 cm−2 are reached in semiconducting structures. The samples used at the time of the IQHE discovery were MOSFETs, shown schematically in the figure 1.3. In such a device, a metallic film is separated from a semiconductor, which is doped with acceptors, by an oxyde insulating layer. The metal chemical potential is controlled with a voltage bias VG . When VG = 0, the Fermi level EF lies in the gap between the valence band and the conduction band, below the acceptor levels [Fig. 1.3(a)]. Upon lowering the chemical potential in the metal with VG > 0, one introduces holes, which attract electrons from the semiconductor toward the interface with the insulating layer. This results in a downward bending of the semiconductor band close to the interface. Electrons attracted to the interface first fill in acceptor levels, which are below the Fermi level [Fig. 1.3(b)]. By further lowering of the metal chemical potential, the semi conductor conduction band can be bent below the Fermi level close to the insulating layer, so that electrons which occupy states in that part of the
Samples
15
(a) métal
oxyde (isolant)
semiconducteur
I bande de conduction niveaux d’accepteurs
EF
z
métal oxyde semiconducteur
V
G
bande de valence
II z (b)
E1 E0
(c) métal
oxyde (isolant)
semiconducteur
métal bande de conduction
EF
VG
niveaux d’accepteurs
E
oxyde (isolant)
z
électrons 2D
bande de conduction niveaux d’accepteurs
EF VG
bande de valence
bande de valence
z
z
Figure 1.3: Metal-Oxyde Field Effect Transistor (MOSFET). The inset I is a schematic view of a MOSFET. (a) Energy level structure. In the metallic part, the band states are occupied up to the Fermi level EF . The oxyde is an insulating film. The Fermi level in the semiconductor falls in the gap between the valence band and the conduction band. There are acceptor states doped close to the valence band, but above the Fermi level EF (b)The chemical potential in the metal is controlled by a gate bias VG . The introduction of holes results in a band bending in the semiconducting part and (c) when the gate bias exceeds a certain value, the conduction band is filled close to the insulating interface, and a 2D electron gas is formed. The confining potential has a triangular profile with electric subbands which are represented in the inset II.
16
(a)
Introduction AlGaAs
(b)
GaAs
AlGaAs
GaAs
EF
EF
dopants (récepteurs)
dopants (récepteurs)
z
électrons 2D
z
Figure 1.4: Semiconducting (GaAs/AlGaAs) heterostructure. (a) A layer of (receptor) dopants lies on the AlGaAs side, at a certain distance from the interface. The Fermi energy is locked to the dopant levels. The bottom of the GaAs conduction band lies lower than those levels so electrons close to the interface migrate to the GaAs conduction band. (b) This polarisation leads to a band bending close to the interface, and a 2D electron gas forms, on the GaAs side.
conduction band form a 2D gas. Electron motion , in spite of a finite extent of the wave function in the z direction is purely 2D if confinement is such that the energy separation between electronic sub-bands E0 (partially filled) and E1 (empty) is significantly larger than kB T (inset II in Fig. 1.3). The problem with MOSFETS is the small distance between the 2D electron gas and the dopants. The latter also act as scattering centers, so that the mean free path is relatively small, and the electron mobility relatively low. This problem is dealt with by forming a 2D electron gas at the interface of a semiconducting heterostructure, such as for example in the III-V heterostructure GaAs/AlGaAs. The two semi-conductors have different gaps between their valence bands and their conduction bands. When the side with the largest gap, Alx Ga1−x As, is doped, the receptor dopant levels are occupied by electrons, and the Fermi level is tied to receptor levels, which may have a higher energy than the bottom of the conduction band in GaAs. The electrons close to the interface migrate in this conduction band [Fig. 1.4(a)]. This polarisation produces a band bending, now on the GaAs side, which is not disordered by the dopants. This spatial separation between the 2D electron gas and the impurities allows to reach larger mobility values than in MOSFETS. Technological progress in the fabrication of semi-conducting heterostructures along the last twenty years has allowed to increase mobilities
Samples
17 Density of states Graphene IQHE: R H = h/e2ν at ν = 2(2n+1)
Vg =15V T=30mK ∼ 1/ν
Usual IQHE: B=9T T=1.6K
at ν = 2n (no Zeeman)
∼ν
Figure 1.5: Quantum Hall Effect as observed in graphene by Zhang et al (Nature 438, 197 (2005)), and Novoselov et al. (Nature 438, 201 (2005)) by two orders of magnitude: The FQHE was discovered in 1983 in a sample with mobility µ ≃ 0, 1 × 106 cm2 /Vs [2] while samples of the same type reach nowadays a mobility of µ ≃ 30 × 106 cm2 /Vs. The discovery of the QHE in graphene in 2005 opens up a new avenue to experiments and theory in the QHE, because graphene is a qualitatively new 2D material, with original electronic structure. See figure 1.5
18
Introduction
Chapter 2 Charged particles in a magnetic field Our understanding of integer or fractional quantum Hall effects relies mostly on the quantum mechanics of electrons in a 2D plane, or thin slab, when submitted to a perpendicular magnetic field. There is a notable exception, that of the Integer Quantum Hall Effect (IQHE) observed in anisotropic 3D organic salts such as Bechgaard salts. The IQHE may arise in 3D systems under magnetic field provided the electronic structure of the material under magnetic field exhibits the suitable gap structure. However, in the present lectures, I will adress the main stream of quantum Hall effects physics, that of electrons the dynamics of which is restricted to a plane. The topic of this chapter is the single electron quantum mechanics in a plane under magnetic field. I start with a discussion of the classical mechanics, as a limiting case of the quantum mechanical case.
2.1
Classical treatment
The equation of motion of a particle (with charge −e and mass m in a magnetic field B = Bez is as follows: x¨ = −ωC y, ˙
y¨ = ωC x, ˙
(2.1)
This follows from the Lorentz force F = −e˙r×B – By definition, the cyclotron frequency is ωC = eB/m. The equation is solved as: x˙ = −ωC (y − Y ),
y˙ = ωC (x − X), 19
(2.2)
20
Charged particle in a static uniform magnetic field
B
η
R
r Figure 2.1:
Cyclotron motion of an electron in a magnetic field, around the guiding
center R.
where R = (X, Y ) is a constant of motion. With η = (ηx , ηy ) = r − R, one has η¨x = −ωC2 ηx , η¨y = −ωC2 ηy , (2.3) and the solution is
x(t) = X + r sin(ωC t + φ),
y(t) = Y + r cos(ωC t + φ),
(2.4)
where r is the cyclotron motion radius, and φ is an arbitrary angle (constant of motion). The physical meaning of the constant of motion R is transparent: it is the “guiding center”, around which the electron moves on a circle of radius r (Fig 2.1).
2.1.1
Lagrangian approach
Lagrangian mechanics starts from the energy function L and the minimum action principle, which reproduce the equations of motion of the classical system. This function is defined in configuration space (positions qµ and velocities q˙µ ). The minimum action principle results in the Euler-Lagrange equations d ∂L ∂L − = 0, (2.5) dt ∂ q˙µ ∂qµ
Classical approach
21
valid for any index µ. The appropriate function in our case is 1 L(x, y; x, ˙ y) ˙ = m x˙ 2 + y˙ 2 − e [Ax (x, y)x˙ + Ay (x, y)y] ˙ , 2
(2.6)
where A = (Ax , Ay ) is a vector potential which is independent of time. This represents the minimal coupling theory for a charged particle and an electromagnetic field, written in a covariant form, with Einstein’s convention, 1 Lrel = mx˙ µ x˙ µ − ex˙ µ Aµ . 2 The conjugate (or ”canonical”) momenta, which will be needed in the Hamiltonian formulation of clasical or quantum mechanics are px ≡
∂L = mx˙ − eAx , ∂ x˙
py ≡
∂L = my˙ − eAy . ∂ y˙
(2.7)
The Euler-Lagrange equations yield the equations of motion [Eq. (2.1)] m¨ x = −ey(∂ ˙ x Ay − ∂y Ax ),
m¨ y = ex(∂ ˙ x Ay − ∂y Ax ),
(2.8)
where ∂x ≡ ∂/∂x, ∂y ≡ ∂/∂y, and (∂x Ay − ∂y Ax ) = (∇ × A)z = B is the z component of the magnetic field.
Gauge invariance A gauge transformation of the vector potential is defined as A′ = A + ∇χ, where χ is an arbitrary function. The magnetic field is independent of the gauge (it is “gauge invariant”) since ∇ × ∇χ = 0. A usual gauge in non relativistic physics is the Coulomb gauge, ∇ · A = 0.1 The gauge (the gauge function) is not completely determined by the Coulomb gauge condition, which demands only ∆χ = 0, where ∆ = ∇2 is the Laplacian. Gauge transformations in 2D are thus defined as harmonic functions. Two gauge choices are especially useful in the quantum treatment of our problem: the Landau gauge (e.g. for problems defined on a rectangular sample) AL = B(−y, 0, 0) 1
Relativistic mechanics use rather the Lorentz gauge, ∂ µ Aµ = 0.
(2.9)
22
Charged particle in a static uniform magnetic field
and the symmetric gauge(e.g. for problems defined on a disc) AS =
B (−y, x, 0), 2
(2.10)
the function which transforms from one of these two gauges to the other is χ = (B/2)xy. Since velocities x˙ and y, ˙ are also gauge invariant, it is clear that conjugate (or ”canonical”) momenta in equation (2.7) are not. The gauge invariant momenta (or ”mechanical momenta”) are Πx = mx˙ = px +eAx = −mωC ηy ,
Πy = my˙ = py +eAy = mωC ηx , (2.11)
where we used Eq. (2.2).
2.1.2
Hamiltonian formalism
For the quantum treatment of a one particle system, it is often preferred to use the Hamiltonian formalism of classical mechanics. The Hamiltonian is derived from the Lagrangian by a Legendre transformation, H(x, y; px , py ) = xp ˙ x + yp ˙ y − L. It is an energy function defined in phase space (positions/conjugate momenta). One must express velocities in terms of conjugate momenta, using equations (2.7), and one finds for the Hamiltonian H=
i 1 h (px + eAx )2 + (py + eAy )2 . 2m
(2.12)
The Hamiltonian may also be written in a concise fashion, using the ”relative” variables, (ηx , ηy ) (using 2.11), 1 H = mωC2 (ηx2 + ηy2 ), 2
(2.13)
where the ”new” variables are nevertheless defined by the variables in phase space, i.e. (x, y, px , py ).
2.2. QUANTUM TREATMENT
2.2
23
Quantum treatment
The Hamiltonian formalism allows to introduce the canonical quantization, where one imposes the non commutativity of position with its conjugate momenta, in terms of Planck’s constant h ¯, [x, px ] = [y, py ] = i¯ h,
[x, y] = [px , py ] = [x, py ] = [y, px ] = 0.
Since [x, y] = 0, one sees immediately that [ηx , ηy ] = −[X, Y ]. The fact that the guiding center components are constants of motion is expressed by [see also Eq. (2.13)] [X, H] = [Y, H] = 0. (2.14) To compute the commutator between components ηx and ηy , one may use the formula ∂f [A, B]. (2.15) ∂B That formula is valid for two arbitrary operators which commute with their commutator, [A, [A, B]] = [B, [A, B]] = 0. One gets [A, f (B)] =
[ηx , ηy ] =
e m2 ωC2
([px , Ay ] − [py , Ax ])
1 (∂x Ay [px , x] − ∂y Ax [py , y]) eB 2 −i¯ h = eB =
or, in terms of magnetic length lB ≡ 2 [ηx , ηy ] = −ilB ,
q
h ¯ /eB, 2 [X, Y ] = ilB .
(2.16)
The result is of course gauge invariant. A remarkable point is that the dynamics of a charged particle in a magnetic field is perhaps the simplest example of a non commutative geometry. Notice that, without any knowledge on the energy level structure, the latter has to be degenerate. In any level chosen at random, each state must occupy a minimal surface given by the Heisenberg uncertainty principle, 2 σ = ∆X∆Y = 2πlB .
24
Charged particle in a magnetic field
In that sense, the real 2D space looks like the phase space of a 1D particle, where each state occupies a ”surface” 2π¯ h. The level degeneracy may thus be written directly in terms of this minimal surface: the number of states per level and per unit surface being nB = 1/σ = B/φ0 – i.e. the flux density in units of the flux quantum φ0 = h/e. Since electrons follow fermionic statistics, each quantum state is occupied at most by one particle, because of the Pauli principle. When there are many electrons in the system, the filling ν of energy levels is thus described by the ratio between the electron surface density nel and the flux density nB , ν = nel /nB . This ratio is also called the filling factor. The Hamiltonian form (2.13), along with the commutation relations (2.16), is that of a harmonic oscillator – ηx and ηy may be interpreted as conjugate variables. To exhibit explicitly the harmonic oscillator structure, we introduce two sets of ladder operators, (a, a† ) with a = √
1 (ηx − iηy ), 2lB
a† = √
1 (ηx + iηy ) 2lB
lB ηy = √ (a† − a), 2i
lB ηx = √ (a† + a), 2
(2.17)
and (b, b† ) with b = √
1 b† = √ (X − iY ) 2lB ilB Y = √ (b† − b), 2
1 (X + iY ), 2lB
lB X = √ (b† + b), 2
(2.18)
with [a, a† ] = [b, b† ] = 1 et [a, b(†) ] = 0. In terms of ladder operators, the Hamiltonian writes 1 . (2.19) 2 The energy spectrum is thus given by En = h ¯ ωC (n + 1/2), where n is the eigenvalue of operator a† a. In the context of an electron in a magnetic field the equidistant levels of the oscillator are called ”Landau Levels” (LL, see Fig. 2.2). Formally, in fact, the system may be viewed as a system of two harmonic oscillators,
H =h ¯ ωC a† a +
H =h ¯ ωC a† a +
1 1 +h ¯ ω ′ b† b + , 2 2
25
niveaux de Landau
Quantum treatment
4 3 2 1
n=0 m
Figure 2.2: Landau Levels. The quantum number n labels the levels, and m , which is associated to the guiding center, describes the level degeneracy.
where the frequency of the second oscillator vanishes, ω ′ = 0. The second quantum number m is the eigenvalue b† b. The eigenstates are thus determined by the two integer quantum numbers, n and m, associated with the two species of ladder operators, √ √ a† |n, mi = n + 1|n + 1, mi, a|n, mi = n|n − 1, mi (pour n >0); √ √ b† |n, mi = m + 1|n, m + 1i, b|n, mi = m|n, m − 1i (pour m >0). When n = 0 ou m = 0, one finds a|0, mi = 0,
b|n, 0i = 0,
(2.20)
and negative numbers are prohibited. An arbitrary state may thus be constructed with the help of ladder operators, starting from the state |0, 0i, (a† )n (b† )m |n, mi = √ √ |0, 0i. n! m!
(2.21)
The wave functions, which are the state representation in real space, depend on the gauge chosen for the vector potential.
2.2.1
Wave functions in the symmetric gauge
To find the real space representation of eigenstates, φn,m (x, y) = hx, y|n, mi,we must choose a gauge. Here we discuss the symmetric gauge [Eq. (2.10)], A = (B/2)(−y, x, 0); we must translate equations (2.20) and (2.21) in differential equations, using px = −i¯ h∂x and py = −i¯ h∂y . With the help of
26
Charged particle in a magnetic field
equations (2.11) and (2.17), one finds the representation of ladder operators in the symmetric gauge √ z ¯ a= 2 + lB ∂ , 4lB ! √ z∗ + lB ∂ , b= 2 2 4lB
√
z∗ a† = 2 − lB ∂ 2 4lB √ z − lB ∂¯ b† = 2 4lB !
(2.22)
where z = x − iy is the electron position in the complex plane2 , z ∗ = x + iy its complex conjugate, ∂¯ = (∂x − i∂y )/2 et ∂ = (∂x + i∂y )/2. A state in the Lowest LL (LLL) is thus determined by the differential equation
2 ¯ z + 4lB ∂ φn=0 (z, z ∗ ) = 0.
(2.23)
The solution of equation (2.23) is a gaussian multiplied by an arbitrary an¯ (z) = 0, alytic function f (z), with ∂f 2 /4l2 B
φn=0 (z, z ∗ ) = f (z)e−|z|
,
(2.24)
Similarly one finds for the state with m = 0
2 z ∗ + 4lB ∂ φm=0 (z, z ∗ ) = 0,
(2.25)
the solution of which is 2 /4l2 B
φm=0 (z, z ∗ ) = g(z ∗ )e−|z|
,
(2.26)
where the function g(z ∗ ) is anti-analytic, ∂g(z ∗ ) = 0. The state |n = 0, m = 0i must thus be represented by a function which is both analytic and antianalytic. The only function which satisfies both requirements is a constant. With the normalisation, one gets φn=0,m=0 (z, z ∗ ) = hz, z ∗ |n = 0, m = 0i = q 2
1 2 2πlB
2 /4l2 B
e−|z|
,
(2.27)
The sign we chose for the imaginary part is unusual, but is convenient for electrons. For positively charged particles, we would chose the opposite sign, corresponding to the opposite chirality.
Quantum treatment
27
A state corresponding to the quantum number m in the LLL is found from equations (2.20) and (2.22), √ m m 2 z 2 2 ∗ ¯ φn=0,m (z, z ) = q − lB ∂ e−|z| /4lB 2 2πlB m! 4lB = q
and
1
2 m! 2πlB
∗
√
n
2
φn,m=0 (z, z ) = q
2 2πlB n!
= q
2 2πlB n!
1
z √ 2lB
!m
2 /4l2 B
e−|z|
z∗ − lB ∂ 2 4lB z∗ √ 2lB
!n
!n
,
(2.28)
2 /4l2 B
e−|z| 2 /4l2 B
e−|z|
,
(2.29)
for a state centered at the origin m = 0 in LL n. An arbitrary state writes √ m !n m ∗ 2 z z 2 2 ∗ √ e−|z| /4lB (2.30) φn,m (z, z ) = q − lB ∂¯ 2 2lB 2πlB m!n! 4lB
which generates the associated Laguerre polynomials [21]. It is remarkable that, even if functions (2.28) and (2.29) have the same probability density ,3 ∗
2
∗
2
|φn=0,m=j (z, z )| = |φn=j,m=0 (z, z )| ∼
|z|2 2
!j
2
2
e−|z| /2lB , j!
√ with a probability maximum at radius r0 = 2jlB (Fig. 2.3), they do not represent equal energy states. To conclude the discussion of states |n = 0, mi represented in the symmetric gauge, we compute the average value of the guiding center operator. With the help of equations (2.18), one finds that hRi ≡ hn = 0, m|R|n = 0, mi = 0, but h|R|i = 3
D√
E
X 2 + Y 2 = lB
It is a Poissonian distribution.
q
√ 2b† b + 1 = lB 2m + 1.
(2.31)
28
Charged particle in a magnetic field 0.4
n=1 n=3 n=5
(a) 0.35
|φn,m=0(z,z*)|2
0.3 0.25 0.2 0.15 0.1 0.05 0 0
1
2
(b)
3 r/lB=|z|/lB
n=0
4
5
n=1
2
y/l B
0
0
-2
-2
-4
-4 -4
-2
0
2
4
-4
-2
x/l B
0
2
4
x/l B n=3
4
n=5
4
2
y/l B
6
4
2
y/l B
4
2
y/l B
0
-2
0
-2
-4
-4 -4
-2
0
x/l B
2
4
-4
-2
0
2
4
x/l B
Figure 2.3: Probability density for a state |n, m = 0i for various values √ of n. (a) The density depends only on the radius |z| = r and is maximum at r0 = 2jlB . (b) When plotted on the plane, the wave function for n ≥ 1 have a ring shape.
Quantum treatment
(a)
29 p p0
p
x0 x
x
(b)
y y0
y
x
x0 x
Figure 2.4: Coherent states . This means that both√the particle and its guiding center are located on the circle of radius lB 2m + 1, but the phase in undetermined. We may use this to count states, as was done previously, for a disc geometry with 2 radius Rmax √ and surface A = πRmax : as the state with maximum radius has Rmax = lB 2M + 1, this yields the number of states in the thermodynamic 2 limit, M = A/2πlB = AnB , with nB = eB/h, in agreement with the previous argument about the state minimal surface. Similarly one sees that for the state |n, m = 0i in level n the relative variable η is localized on a circle with radius √ RC ≡ h|η|i = lB 2n + 1 (2.32) which is also called the cyclotron radius.
2.2.2
Coherent states and semi-classical motion
To retrieve the classical trajectory, (2.4), one must construct semi-classical states, also called coherent states because they play an important role in quantum optics. For a 1D harmonic oscillator, a coherent state is the eigenstate of the annihilation operator and it is the state with the minimum value
30
Charged particle in a magnetic field
of the product ∆px ∆x. Such as state can be built from the displacement operator in phase space, which displaces the ground state from hxi = 0, hpx i = 0 to (x0 , p0 ) [Fig. 2.4(a)]. The displacement is done with the operator D(x0 , p0 ) = e−i(x0 pˆ−p0 xˆ) ,
(2.33)
where the symbols with hats are operators, not to be confused with variables x0 and y0 . This operator displaces variables, as we can check using formula (2.15) D† (x0 , p0 )ˆ xD(x0 , p0 ) = eix0 pˆxˆe−ix0 pˆ = xˆ + x0 and D† (x0 , p0 )ˆ pD(x0 , p0 ) = e−ip0 xˆ pˆeip0 xˆ = pˆ + p0 . The coherent state writes |x0 , p0 i = D(x0 , p0 )|n = 0i,
(2.34)
where |n = 0i is the 1D harmonic oscillator ground state. Since [D(x0 , y0 ), H] 6= 0 the coherent state is not an eigenstate of the Hamiltonian. Indeed the state changes with time, and that is how we retrieve the trajectory in phase space [Fig. 2.4(a)]. x0 and p0 are not bona fide quantum numbers – this would contradict the fundamental postulates of quantum mechanics, because the associate operators do not commute. The basis |x0 , p0 i is said to be ”overcomplete” [22]. In general, a displacement operator may be constructed from two conjugate operators, which therefore do not commute. In the case of an electron in a magnetic field in a 2D plane, we have two pairs of non commuting con2 2 jugate operators at our disposal, [X, Y ] = ilB et [ηx , ηy ] = −ilB . With the first choice, the displacement operator which acts now in real space, writes −
D(X0 , Y0 ) = e
i l2 B
ˆ (X0 Yˆ −Y0 X)
,
(2.35)
and the coherent state (in the LLL) is |X0 , Y0 ; n = 0i = D(X0 , Y0 )|0, 0i,
(2.36)
where |0, 0i ≡ |n = 0, m = 0i. Since the guiding center is a constant of motion, the displacement operator D(X0 , Y0 ) commutes with the Hamiltonian. The state (2.36) remains an eigenstate of the Hamiltonian, which is why the quantum number n is unchanged.
Quantum treatment
31
The dynamics enters with the second pair of operators, with the displacement operator i (η x ηˆ −η y ηˆ ) ˜ 0x , η0y ) = e l2B 0 y 0 x , D(η (2.37) which generates a displacement to position η 0 = (η0x , η0y ), so that a general semi-classical state may be written as ˜ 0x , η0y )D(X0 , Y0 )|0, 0i. |X0 , Y0 ; η0x , η0y i = D(η
(2.38)
The guiding center is thus centered at R0 = (X0 , Y0 ), and the electron turns around that position on a circle of radius r = |η 0 |. One retrieves thus the motion represented on figure 2.1, in terms of a gaussian wave packet. To prove those dynamic properties, remember that a coherent state is an eigenstate of the ladder operator a, and in our case also of b, with η0x − iη0y √ |X0 , Y0 ; η0x , η0y i, 2lB X + iY0 0 √ |X0 , Y0 ; η0x , η0y i. b |X0 , Y0 ; η0x , η0y i = 2lB
a |X0 , Y0 ; η0x , η0y i =
(2.39)
This can be checked, for example, when expressing the displacement operators in terms of ladder operators (2.17) and (2.18), †
∗
2
†
∗
D(X0 , Y0 ) = eβb −β b = e−|β| /2 eβb e−β b , ˜ x , η0y ) = eαa† −α∗ a = e−|α|2 /2 eαa† e−α∗ a , D(η 0
(2.40)
o` u l’on a d´efini
η0x − iη0y X0 + iY0 , α≡ √ β≡ √ 2lB 2lB where we used the Baker-Hausdorff formula eA+B = eA eB e−[A,B]/2 ,
(2.41)
which is valid when [A, [A, B]] = [B, [A, B]] = 0. The coherent state writes thus 2 2 † † |X0 , Y0 ; η0x , η0y i = e−(|α| +|β| )/2 eαa eβb |0, 0i
and we find with formula (2.15)
2 +|β|2 )/2
a |X0 , Y0 ; η0x , η0y i = e−(|α|
2 +|β|2 )/2
b |X0 , Y0 ; η0x , η0y i = e−(|α|
h
†
i
†
a, eαa eβb |0, 0i = α |X0 , Y0 ; η0x , η0y i, †
h
†
i
eαa b, eβb |0, 0i = β |X0 , Y0 ; η0x , η0y i,
32
Charged particle in a magnetic field
which is nothing but equation (2.39). In order to get the time evolution of the coherent state |α, βi = |X0 , Y0 ; η0x , η0y i, one uses the time evolution operator on state i
|α, βi(t) = e− h¯ Ht |α, βi(t = 0) 2 )/2
= e−(|α|
i
e− h¯ Ht
∞ X (αa† )n
|n = 0, βi(t = 0) n! i (α)n 2 √ |n, βi(t = 0) = e−(|α| )/2 e− h¯ Ht n! n=0 n ∞ X (αe−iωC t ) −(|α|2 )/2 −iωC t/2 √ = e e |n, βi(t = 0) n! n=0 n=0 ∞ X
= e−iωC t/2 |α(t = 0)e−iωC t , βi,
(2.42)
which yields for the eigenvalue time evolution α(t) = α(t = 0)e−iωC t , β(t) = β(t = 0). (2.43) √ √ √ x 2l Re[β], Y = 2l Im[β], η 2lB Re[α(t)] et η0y (t) = (t) = Since X = B 0 B 0 0 √ − 2lB Im[α(t)], we retrieve η0x (t) = η0x (t = 0) cos(ωC t),
η0y (t) = η0y (t = 0) sin(ωC t)
and thus the trajectory given in equation (2.4), identifying r = |η 0 | and R = (X0 , Y0 ), as mentionned earlier.
Chapter 3 Transport properties– Integer Quantum Hall Effect (IQHE) This chapter deals with some aspects of the Integer Quantum Hall Effect (IQHE) physics, using the quantum mechanics of an electron in a constant uniform magnetic field, described in the previous chapter. Two main features allow to understand the IQHE : • each completely filled LL (for ν = n) contributes a conductance quantum e2 /h to the electronic conductivity, • when additional electrons start populating the next LL at ν 6= n, they get localized by the impurities disorder potential in the sample, and they do not contribute to transport. In the absence of impurities, or, more precisely, if translation invariance is not broken in the sample, no plateau can be formed in the Hall resistance, and the classical Hall result is preserved. This last feature seems analogous at first sight to Anderson localization in 2D in the absence of a magnetic field [23]. It happens that localisation is even more relevant in a magnetic field.
3.1
Resistance and resistivity in 2D
Theorists calculate resistivity. Experiments measure resistance. For a classical sytem with the shape of a hypercube of edge length L in d dimensions, 33
34
Transport properties– IQHE
the resistance R and the resistivity ρ are related by the well known equation R = ρL2−d
(3.1)
Thus, in two dimensions, the sample resistance is scale invariant. The product R(e2 /h) is dimensionless. It is an easy exercise to show that in the case of a Hall bar geometry, such as shown in Fig. (1.1), the transverse resistance and the transverse resistivity are equal in 2D, independent of the Hall bar dimensions. This is a basic ingredient to understand the universality of the quantum Hall experimental results. In particular it means that one does not have to measure the physical dimensions of a sample to one part in 1010 in order to obtain the resistivity to that accuracy. The technological progress in semiconductor physics which allowed to manufature 2D Electron Gases (2DEG) with electrical contacts was, in this respect, a decisive one. Even the shape of the sample, or the accurate determination of the Hall voltage probe locations are almost completely irrelevant, in particular, because the dissipation is nearly absent in the QH states.
3.2
Conductance of a completely filled Landau Level
We first discuss the effect of a constant uniform electric field on the Landau level energy structure. We take the electric field along the y direction. It is convenient to deal with a sample with rectangular shape, and to assume in a first step (to be relaxed subsequently) that the system is translation invariant in the x direction. An appropriate gauge in this rectangular geometry is the Landau gauge AL = B(−y, 0, 0), so that the Hamiltonian now writes p2y (px − eBy)2 H= + − eV (y), 2m 2m where the potential is V (y) = −Ey (electric field pointing in the yˆ direction). The system is translation invariant in the xˆ direction, so that px , or equivalently h ¯ k, is a constant of motion, which corresponds to the quantum number m in the symmetric gauge, i.e. to the guiding center eigenvalue. The latter, in a state |n, ki, is delocalized along a straight line in the x direction,
contact
L
35
énergie
Conductance of a filled LL contact
R
µL
µR
NL n
k min
k max y=kl B2
y’
Figure 3.1: LL in the Landau gauge, with a voltage bias between the L and the R contacts, where chemical potentials are respectively µL and µR . Position yk in the direction 2 y is proportionnal to the wave vector k in direction x : yk = klB . 2 with coordinate klB on the y axis. The Hamiltonian is:
H=
p2y 1 + mωC2 (y + klb2 )2 + eEy, 2m 2
which can be re-written, completing the square, as: p2y 1 E + mωC2 (y − yk )2 + h ¯ k + C, H= 2m 2 B 2 where we have set px = h ¯ k, and yk = −klB − eE/mωC2 and C is a constant: 2 E C = − 21 m B . The energy of the state |n, ki is
εn,k = h ¯ ωC n +
1 2 1 + eEyk + m¯ v 2 2
(3.2)
E ~ ∧ B/B ~ 2 and is parallel to the x axis. where v¯ ≡ − B is the drift velocity E This can be derived by deriving explicitly the current:
−e J~ = hn, k|(p + eA)|n, ki . m
D E
Thus there is a net current hJx i along the x axis.
36
Transport properties– IQHE
We conclude that the energy levels follow the electric potential, which adds to the energy in zero electric field. Let us now go one step further by considering a slowly varying electric potential V (y), which we still assume to be translation invariant along x. We can linearize this potential locally, and repeat the previous analysis: the energy eigenvalues will not be linear in k any more, but they will roughly reflect the sum of the LL energy plus the local potential energy. To discuss electrons in a Hall bar, we take into account the sample edges in the y direction, which create a confinement potential. The latter results in an upward bending of Landau levels in the vicinity of the edges, where contacts allow to measure voltage biases as in figure 3.1. This justifies the sketch of the LL in figure 3.1, where the LL energy profile follows the confining potential at the sample edges. The eigenvalues ǫk are not linear in k, but can be linearized locally: it will still reflect the kinetic energy, with the local potential energy added to the LL energy. In order to compute the level contribution to the conductance (along x), we use formula eX In = − hn, k|vx |n, ki, (3.3) L k
where L is the system length along x, and the velocity average value is derived from the energy dispersion relation
1 ∂εn,k 1 ∆εn,k ≃ . h ¯ ∂k h ¯ ∆k In the last line, we assume that ∆k = 2π/L is very small, which is certainly valid if L is very large. Using this, we have L L ∆εn,k = (εn,k+1 − εn,k ). 2π¯ h 2π¯ h Thus vk has opposite signs on the two edges of the sample. This means that in the Hall bar geometry, there are edge currents flowing in opposite directions. This is not surprising, if we remember the semi-classical picture of skipping orbits along an edge. When we sum over vk in equation (3.3, the result depends only on the edge energies, at kmin and kmax , the edge coordinates (see figure ( 3.1)). Thus, provided the electric potential has a slow enough variation in direction y, we may sum over k to get e In = − (εn,kmax − εn,kmin ) . h vk =
3.3. LOCALISATION IN A STRONG MAGNETIC FIELD
37
The energies at kmin and kmax are given by the chemical potential at the contact points, εn,kmin = µL and εn,kmax = µR . Since the difference in chemical potentials is controlled by a voltage bias ∆µ = (µR − µL ) = −eV , we see that the LL conductance is e2 /h, since In =
e2 V. h
(3.4)
When n LL are completely filled, we get a conductance G=n
e2 . h
Since this is a transverse conductance (the current is in direction x, is zero along y, and the difference in chemical potentials is in direction y), the resistance tensor we get is ˆ −1
ˆ=G R
=
0 −RH RH 0
!
,
(3.5)
with the Hall resistance RH = h/e2 n. It is important to realize that this result, although satisfactory –the Hall resistance only depends on universal constants e and h, and an integer n–is not sufficient to explain the occurrence of plateaux. In fact, it is fairly easy to show that the result we have coincide exactly with the classical Hall value at discrete points in the RH curve, corresponding to n filled Landau levels; it is enough to remember that ν = hnel /eB = n and to use equation (3.5)to recover the classical value RH = B/enel . In order for quantized Hall plateaux to be formed, additionnal electrons or holes injected in the system around a density such that n LL are completely filled must be localized, so as to have no contribution to transport properties. This localization phenomenon is described in the next section.
3.3
Localisation in a strong magnetic field
The electric potential Vext (r = R + η) due to impurities is described as a slowly varying function in the xy plane, so that Landau quantization is preserved. We now do not assume any more that the impurity potential preserves translation invariance along the xˆ direction. The potential landscape
38
Localisation in a strong magnetique field
Figure 3.2: Semi-classical motion of an electron in a magnetic field in the presence of an impurity potential. The guiding center follows the landscape equipotentials. The Hall drift of the guiding center, shown by the arrow is a slow motion compared to the fast electronic cyclotron motion. Electronic transport is possible when an equipotential connects the sample edges. If an electronic state is localized within a potential well, it does not contribute to transport.
has hills and valleys and fluctuates in space around an average value which is taken to be zero, with no loss of generality. This potential lifts the LL degeneracy, because the guiding center is not a constant of motion any more. We see this with the Heisenberg equations of motion 2 ∂Vext i¯ hY˙ = −ilB , ∂X (3.6) where we used formula (2.15). We see that the guiding center follows the equipotential lines of the impurity potential (Fig. 3.2). In the case we discussed in the previous section, this led to a Hall current in the direction orthogonal to the electric field. Equation (3.6) is a generalization of this result. The guiding center motion is perpendicular both to the external field and to the local electric field. Quantum states of the LL are thus localized on equipotential lines corresponding to their energies. The wave functions, (in the shape of rings in zero potential as in Fig. 2.3) are deformed to tune to their equipotential lines. Similarly, we get for the ηx et ηy Heisenberg equations of motion 2 ∂Vext i¯ hX˙ = [X, H] = [X, Vext (X, Y )] = ilB ∂Y
i¯ hη˙ x ⇔
η˙x
and
1 = ηx , mωC (ηx2 + ηy2 ) + V (r + η) 2 l2 ∂V = −ωC ηy − B h ¯ ∂ηy
Conductance of a filled LL et
39
η˙y = ωC ηx +
2 lB ∂V . h ¯ ∂ηx
This provides us with a stability criterion for Landau levels in the presence of a disorder impurity potenial, since the first terms on the right in the equations above must remain large compare with the terms due to the impurity potential. The condition reads: *
∂V ∂η
+
≪
h ¯ ωC . lB
(3.7)
This condition is satisfied provided the variation of the potential over a cyclotron radius is small compared to h ¯ ωC . We now have the main ingredients to understand some of the IQHE basic features. The reasoning below is represented in a schematic fashion on figure 3.3. We have seen in the previous section that the Hall resistance for integer ν = n is exactly RH = h/e2 n, while the longitudinal resistance is zero. If the magnetic field intensity is slightly decreased, keeping the electronic density constant, since the number of states per LL decreases, some electrons have to promoted to the LL with n + 1. They occupy preferentially the lowest energy states available, the bottom of basins in the impurity potential landscape. This is a peculiar form of localisation which is induced by the magnetic field. Localized electrons do not contribute to the transport. Both the transverse and longitudinal resistances stay locked at their value for the completely filled level case with ν = n. The fact that the longitudinal resistance is zero shows that transport is ballistic. Indeed, we have seen that contributions to the current from the bulk of the sample compensate, so that transport is due to n edge channels (edge states), one per completely filled LL. The current direction on the edge is determined by the potential gradient, which rises near the sample edge. Edge currents are thus chiral, forward scattering at one edge is dissipation less, and dissipation can only occur if an electron circulating along one edge can be scattered backwards by tunneling to the other edge. This can occur only when edge states trajectories of opposites chiralities happen to be close to each other. When electronic puddles grow because more electrons get promoted to level n + 1, they eventually merge into one another, until equipotentials connect the two edges and eventually an electronic sea extends over the whole sample. When edges get connected by equipotentials, dissipation occurs, the longitudinal resistance is finite, and the transverse resistance varies rapidly as the magnetic field continues
40
Localisation in a strong magnetique field
(a) ε
(b) ε
(c)
EF n
EF n
ε
EF
états localisés
n états étendus densité d’états
densité d’états
densité d’états
NL (n+1)
Rxx R xy
Rxx R xy
Rxx R xy h/e2 n
h/e2 n
h/e2 (n+1) ν =n
B
B
B
Figure 3.3: Quantum Hall effect. In the upper parts of the figure, LL are broadened by the impurity potential. Their filling is controlled by the Fermi level(EF ). In the middle part, samples are seen from above, showing equipotential lines, and the gradual filling of the n-th level (from left to right). The lowest part of the figure is a sketch of the resistance curves, as the LL filling factor varies. This figure is to be read column by column, the filling factor increasing from the first column to the last one. In the first column (a), we have a situation with completely filled LL, ν = n, where the Fermi level sits exactly between LL n and n + 1, the upper level being empty. The Hall resistance is then exactly RH = h/ne2 , and the longitudinal resistance is exponentially small (zero at zero temperature). The second column describes a situation where the LL n + 1 has a low filling factor. Electrons occupy potential wells in the sample and do not contribute to electronic transport. This situation occurs when, at fixed electron density, the magnetic field intensity is slightly decreased from its value for the complete filling of the n-th LL. The resistance values are locked at their value for ν = n. In the last column on the right, the n + 1 LL is half filled: equipotential lines connect the two edges, so that dissipation is allowed through back scattering processes from one edge to the other. The system changes from ballistic regime to a diffusive one, and the Hall resistance varies rapidly towards the next plateau, while the longitudinal resistance reaches it peak value, before decreasing to exponentially small values. Localized states are on the left and on the right of the extended states in the center of the broadened LL. When the highest occupied LL has a filling larger than 1/2, the same reasoning applies in terms of holes.
Conductance of a filled LL
41
R L ~ µ 3− µ 2= 0 µ2 = µL 2
3
µ3 = µL I
I 4
1 6
µ6 = µ5 = µR 5 R ~ µ5− µ3= µR− µL H
Figure 3.4: IQHE measurements at ν = n. The current I is injected through contact 1, and extracted at contact 4. Between those two contacts, the chemical potential µL is constant since (a) there is no backscattering and (b) there are no electrons injected or extracted at contacts 2 and 3 which are used to measure the voltage drop. The chemical potential µR stays also constant along the lower edge between contacts 6 and 5. The longitudinal voltage drop thus vanishes, so that the longitudinal resistance is zero, RL = (µ3 − µ2 )/I = 0. The Hall resistance is determined by the voltage bias between the two edges µ5 − µ3 = µR − µL .
to decrease until a new conducting channel is formed all along the edges.This is reached at half filling of level n + 1. Above that filling ratio, the evolution described so far is reproduced in terms of holes. To understand the IQHE in terms of edge currents, consider the experimental set up with six contacts, as shown on figure (3.4). Electrons are injected through electrode 1 and are extracted through electrode 4. The other contacts 2, 3, 5 and 6 are used for voltage bias measurements, with no electron injected or extracted. Because back-scattering is suppressed when the filling factor is around ν = n, the chemical potentials µR and µL are constant along each edge. The chemical potential varies only along input and output electrode 1 and 4. The longitudinal resistance is measured for instance between contacts 2 and 3, and is found to vanish: RL = −(µ3 − µ2 )/eI = 0. The Hall resistance is determined by the voltage bias between contacts 3 and 5, RH = −(µ5 − µ3 )/eI = −(µR − µL )/eI. This situation is precisely that which was described in the previous section,
42
Transitions between plateaus – percolation
where we computed the resistance for n completely filled LL. We thus find RH = h/ne2 .
3.4
Transitions between plateaus – The percolation picture
The previous section describes a scenario of transitions between Hall plateaus which reminds us closely of a percolation mechanism: the resistance jumps from one plateau to the next one when the electron puddles become macroscopic ones and percolate so as to form an infinite electronic sea which extends to both edges. Percolation transitions are second order transitions, which exhibit critical phenomena, and specific scaling laws for the relevant physical quantities around the critical point. Those quantities do not depend on microscopic details of the system, they are characterized by critical exponents which define a universality class. The transition is controlled by a ”control parameter” K, which could be the temperature, or, in the case of quantum phase transitions, at zero temperature, by another parameter such as pressure or electronic density [18]. In our case, the control parameter for transitions between plateaus is the magnetic field intensity. At the critical field Bc , the correlation length diverges, with a critical exponent ν (not to be confused with the filling parameter) ξ ∼ |δ|−ν ,
(3.8)
where δ ≡ (B − Bc )/Bc . Dynamic fluctuations may be similarly described by a correlation ”time” ξτ ∼ ξ z ∼ |δ|−zν , (3.9)
where z is called the critical dynamic exponent. In a path integral formulation, the characteristic time τ is connected to the temperature T through h ¯ /τ = kB T . A finite temperature may be considered as a finite size in the time direction. [25, 24]. At the critical point, physical quantities follow scaling laws which depend on ratios of dimensionless quantities. One finds for the longitudinal and Hall resistivity
ρL/H = fL/H
h ¯ω τ , kB T ξτ
!
Conductance of a filled LL
43
100.0
dxy
max
dB
dxy
N=0 N=1 N=1
max
10.0
(∆B)
−1
dB
−1
(∆B)
N=1 N=1 1.0 0.10
1.00
T(K)
Figure 3.5: Experiments by Wei et al. [26]. The transition width δB and that of the Hall resistivity derivative ∂ρxy /∂B, measured as a funcition of temperature exhibit a scaling law with exponent 1/zν = 0, 42 ± 0, 04, for transitions between filling factors 1 → 2 (N = 0 ↓), 2 → 3 (N = 1 ↑) and 3 → 4 (N = 1 ↓).
44
Transitions between plateaus – percolation = fL/H
h ¯ ω δ zν , kB T T
!
,
(3.10)
where ω is a characteristic measurement frequency, for example in an ac measurement, and fL/H (x) are universal functions. In the following, we deal only with dc measurements, and ω = 0 properties. For a second order transition, one expects the characteristic width ∆B of the transition as a function of the magnetic field intensity, to vary with temperature as ∆B ∼ T 1/zν .
(3.11)
This scaling law was actually found in the measurements by Wei et al. [26], who found an exponent 1/zν = 0, 42 ± 0, 04 over a temperature interval varying with more than one order of magnitude between 0, 1 and 1, 3K (figure 3.5). The two exponents ν and z may be separetly determined if one takes into account the scaling laws for current fluctuations under applied electric field. One finds h ¯ h ¯ eEℓE ∼ ∼ z, τE ℓE where τE ∼ ℓzE is the characteristic fluctuation time, which is connected to a characteristic length ℓE through equation (3.9). One finds thus ℓE ∼ E −1/(1+z) , and, for the zero frequency resistivity scaling law ρL/H = gL/H
δ
δ
!
, , T 1/zν E 1/ν(1+z)
(3.12)
in terms of universal functions gL,H (x). Other measurements by Wei et al., dealing with the current scaling laws, find that z ≃ 1, which leads to ν ≃ 2, 3 ≃ 7/3 [27]. The critical exponent for classical 2D percolation is νp = 4/3, smaller than the experimental value, close to 7/3. The disagreement is probably due to quantum tunneling effects between trajectories: such processes allow for back-scattering before classical trajectories actually touch each other. Chalker and Coddington take into account quantum tunneling in a transfer matrix approach and find a critical exponentν = 2, 5 ± 0, 5 [29]. Numerical simulations have reproduced the ν = 7/3 exponent [28].
Chapter 4 The Fractional Quantum Hall Effect (FQHE)– From Laughlin’s theory to Composite Fermions. In the previous chapter, we have seen that the IQHE with ν = n is understood on the basis of two main ingredients: (i) Because of LL quantization, there is an excitation gap between the ground state with a number of completely filled levels and the next empty level (ii) Elementary excitations, obtained by promoting an electron to the next LL are localized and do not contribute to electronic transport. Filled Landau levels only contribute each a conductance quantum e2 /h. As emphasized in the Introduction, the observation of the FQHE, first for a fractional filling of the LLL, with ν = p/(2sp + 1), with integer n and p, was a sign of the complete breakdown of perturbation theory, such as diagrammatic analysis based on the knowledge of an unperturbed ground state. For the fractionally filled LLL, the non interacting ground state has a huge degeneracy, which prohibits using theoretical techniques used so far to take electron-electron interactions into account. We know that certain superpositions of ground state configurations must minimize the Coulomb interactions. We know from experiments–the observed activated behaviour of the longitudinal resistance– that there is a gap between the actual ground state and the first excited states. The approach described in the previous chapter allows us to deal with 45
46
FQHE – from Laughlin to Composite Fermions
the FQHE, once we have a mechanism which allows to lift the ground state degeneracy, and to have a gap to the first excited states, be they single quasi-particles or collective excitations. In that case, we may reproduce the piece of reasoning of the previous chapter: excited charged quasi-particles are localized, so that varying the magnetic field intensity around a given exact fraction of the filling factor leads to plateau formation in the Hall resistivity, for the same reason as in the IQHE. The difficult part is to identify the non degenerate ground states, and to characterize their properties, the nature of excited states, etc.. Before actually introducing the Laughlin and Jain trial wave functions which solve this problem, we discuss in the next section the structure of the effective model which describes the electron dynamics when we make the approximation that it is restricted to the states of a single partially filled LL. This model will be the basis for the Hamiltonian theory of the FQHE, which will be developped in the following chapters.
4.1
Model for electron dynamics restricted to a single LL
Since we are interested at first in describing low temperature properties, only the lowest excitation energies are of interest. Because we are dealing with a partially filled LL,(ν 6= n), the relevant excitations are restricted to intralevel dynamics. Furthermore, we consider at first that the spins are fully polarized, and we do not consider spin flip excitations here. Excitations involving intra-LL transitions are forbidden by the Pauli principle when the level is completely filled (Fig. 4.1). They are allowed for a partially filled LL. In that case, the kinetic energy plays no rˆole, since all single electron states are degenerate. We omit this constant in the following. Virtual inter-level excitations may be considered in a perturbative approach, and give rise to a modified dielectric function ǫ(q), which alters the interaction potential between electrons within the same level [30]. In contrast with screening effects in metals, which suppress the long distance part of the Coulomb interaction, electronic interactions screening in the presence of a magnetic field alters the potential only for finite wave vectors: for q → 0 and q → ∞, it vanishes, and ǫ(q) → ǫ, where ǫ is the dielectric constant of the underlying semiconductor. Since the electron spin is not flipped during such processes within a spin branch, it plays no rˆole. Therefore, we deal in the following with spinless
electrons restricted to a single LL (a)
ν= N
47 (b)
ν= N
∆Z h ωC
Figure 4.1: Lowest excitations energies. Each LL is separated in two spin branches because of the Zeeman effect. (a) For filling ν = n, excitations which couple states within the same LL are forbidden because of the Pauli principle. Only inter level excitations are allowed. (b) For fractional filling of the highest occupied LL, (ν 6= n) excitations within the same LL are allowed and provide the lowest excitation energies. Inter LL excitations, with energy h ¯ ωC or ∆z are neglected in the model.
electrons. A more detailed discussion of spin phenomena in the quantum Hall physics–such as the Quantum Hall ferromagnetism– will be given in a later part of these lectures. In second quantized notation, the Hamiltonian restricted to intra LL excitations is 1Z 2 2 ′ † ˆ d r d r ψn (r)ψn (r)V (r − r′ )ψn† (r′ )ψn (r′ ). H= 2
(4.1)
It involves states in the n-th level only, ψn (r) = m hr|n, mien,m et ψn† (r) = P † † m hn, m|rien,m . Operators en,m and en,m are respectively the annihilation and creation operators for an electron in the state |n, mi. They obey the fermionic anti-commutation rules, P
n
o
en,m , e†n′ ,m′ = δn,n′ δm,m′ ,
{en,m , en′ ,m′ } = 0.
(4.2)
Note that the restricted electron fields ψn (r) are not completely localised. Because the sum over states is restricted to m, we have, with rules (4.2) n
o
ψn (r), ψn† (r′ ) =
X m
′ 2 /2l2 B
hr|n, mihn, m|r′ i ∝ e−|r−r |
6= δ(r − r′ ).
(4.3)
The field ψn† (r) creates an electron in the vicinity (4.2) of position r, which is hardly surprising: this is just another manifestation of the position uncertainty when we restrict the dynamics to a single LL. To have a perfect field
48
FQHE – from Laughlin to Composite Fermions
localisation, one would have to sum over n, i.e. to superpose a number of LL. In reciprocal space, the Hamiltonian writes X ˆ = 1 H v(q)ρn (−q)ρn (q), 2A q
(4.4)
with the measure q = A d2 q/(2π)2 and the Coulomb interaction potential v(q) = 2πe2 /ǫq. The operators ρn (q) are the Fourier components of the electronic density operator in the n-th level ρn (r) = ψn† (r)ψn (r), and one has R
P
ρn (q) = =
Z
d2 r
X
m,m′
X
m,m′
hn, m|rie−iq·r hr|n, m′ ie†n,m en,m′
hn, m|e−iq·r |n, m′ ie†n,m en,m′
= hn|e−iq·η |ni = Fn (q)¯ ρ(q),
X
m,m′
hm|e−iq·R |m′ ie†n,m en,m′ (4.5)
where Fn (q) ≡ hn| exp(−iq · η)|ni is the form factor, and we took advantage of the decomposition r = R + η, which allows to factorize matrix elements hn, m|e−iq·r |n′ , m′ i = hn| exp(−iq · η)|n′ i ⊗ hm| exp(−iq · R)|m′ i.
(4.6)
In the last line of equation (4.5), we have defined the projected density operator, X ρ¯(q) ≡ hm|e−iq·R |m′ ie†n,m en,m′ . (4.7) m,m′
4.1.1
Matrix elements
To proceed in practice to actual computations, one needs to compute the matrix elements which enter expression (4.5) for the density operator. The simplest way is to use expressions (2.17) and (2.18) for the operators R and η, in terms of a, a† , b and b† . From now on, we take lB ≡ 1 for simplicity. Using complex notation, with q = qx − iqy and q ∗ = qx + iqy , we have 1 q · η = √ qa + q ∗ a† , 2
1 q · R = √ q ∗ b + qb† , 2
electrons restricted to a single LL
49
so that we get for the first matrix element, with n ≥ n′ , using the BakerHausdorff formula(2.41), − √i (q ∗ a† +qa)
hn|e−iq·η |n′ i = hn|e
2
2 /4
= e−|q|
2 /4
= e−|q|
|n′ i
− √i q ∗ a† − √i qa
hn|e X j
e
2
− √i q ∗ a†
hn|e
2
−|q|2 /4
s
n′ ! n!
−iq ∗ √ 2
−|q|2 /4
s
n′ ! n!
−iq ∗ √ 2
= e
= e
2
|n′ i
− √i qa
|jihj|e
2
|n′ i
n−n′ X n′
n! |q|2 − (n − j)!(n′ − j)!j! 2 j=0
n−n′
n−n′ n′
L
|q|2 2
!
,
!n′ −j
(4.8)
where we have used − √i q ∗ a†
hn|e
2
0 |ji = q n!
pour j > n
1 j! (n−j)!
− √i2 q ∗
n−j
pour j ≤ n
in the third line and the definition of Laguerre polynomials [21], ′ Ln−n (x) n′
′
=
n X
(−x)m n! . ′ ′ m! m=0 (n − m)!(n − n + m)!
Similarly we find for m ≥ m′ − √i (qb† +q ∗ b)
hm|e−iq·R |m′ i = hm|e
−|q|2 /4
= e
2
s
m′ ! m!
|m′ i
−iq √ 2
!m−m′
m−m′ Lm ′
|q|2 . 2 !
(4.9)
Defining functions Gn,n′ (q) ≡
s
n′ ! n!
−iq √ 2
!n−n′
′ Ln−n n′
|q|2 , 2 !
one may also write without the conditions n ≥ n′ et m ≥ m′ , 2 /4
hn|e−iq·η |n′ i = [Θ(n − n′ )Gn,n′ (q ∗ ) + Θ(n′ − n − 1)Gn′ ,n (−q)] e−|q|
(4.10)
50
FQHE – from Laughlin to Composite Fermions
and 2
hm|e−iq·R |m′ i = [Θ(m − m′ )Gm,m′ (q) + Θ(m′ − m − 1)Gm′ ,m (−q ∗ )] e−|q| /4 . (4.11) For the case n = n′ , we find in equation (4.10) the n-th LL form factor: −iq·η
Fn (|q|) ≡ hn|e
4.1.2
|q|2 −|q|2 /4 e . 2 !
|ni = Ln
(4.12)
Projected densities algebra
At first sight, the model defined by (4.4) looks simple. The Hamiltonian is quadratic in density operators. Such models often have exact solutions. It happens that the projection in a single LL generates a non commutative algebra for operators with different wave vectors, which leads to non trivial quantum dynamics. Let us compute the commutator [¯ ρ(q), ρ¯(k)]. For a one particle operator P † A A A in second quantized notation, F (q) = λ,λ′ fλ,λ ′ (q)eλ eλ′ , where fλ,λ′ (q) = A ′ hλ|f (q)|λ i, the commutation rules in second quantized form follow from those in first quantization: h
i
F A (q), F B (q′ ) =
Xh
i
f A (q), f B (q′ )
λ,λ′
λ,λ′
e†λ eλ′ .
(4.13)
The λ index may comprise a number of different quantum indices. This equation follows from the repeated application of [AB, C] = A[B, C]± − [C, A]± B
(4.14)
on electronic operators. Equation (4.14) is valid for commutators as well as anti-commutators. Using equation (2.16), one finds [q · R, q′ · R] = qx qy′ [X, Y ] + qy qx′ [Y, X] = i(qx qy′ − qy qx′ ) = −i(q ∧ q′ ), where we have defined q ∧ q′ ≡ −(q × q′ )z , and one gets, with the help of the Baker-Hausdorff formula (2.41) h
′
e−iq·R , e−iq ·R
i
′
i
′
i
= e−i(q+q )·R e 2 q∧q − e− 2 q∧q q ∧ q′ −i(q+q′ )·R = 2i sin e . 2 !
′
(4.15)
4.2. THE LAUGHLIN WAVE FUNCTION
51
This yields, with equation (4.13), !
q∧k [¯ ρ(q), ρ¯(k)] = 2i sin ρ¯(q + k), 2
(4.16)
for the algebra of projected density operators. This is isomorphous to the magnetic translation algebra . Indeed, operators which describe electronic displacements in the presence of a magnetic field have the same commutation rules. This algebra is closed, and does not depend on the LL n index. With algebra (4.16), the model is completely defined by the Hamiltonian (4.4), which writes, in terms of projected density operators X ˆ = 1 H vn (q)¯ ρ(−q)¯ ρ(q), 2A q
(4.17)
where the form factor has been absorbed in the effective interaction potential in the n-th LL , 2πe2 2πe2 |q|2 vn (q) = [Fn (q)]2 = Ln ǫ|q| ǫ|q| 2 "
!#2
2 /2
e−|q|
.
(4.18)
The model has the same structure for all LL. The information about the level is encoded in the effective potential, which will be discused in the last section of this chapter. The LLL physics, which will be the main topic in the remaining parts of this chapter (except the last section), is thus easily generalized to a LL with higher index: one simply has to take into account the relevant effective potential, and to replace the filling factor ν by the partial filling factor of the n-th level, ν¯ = ν − n.
4.2
The Laughlin wave function
In this section, we discuss the arguments used by Laughlin in 1983 to derive the almost exact ground state for the fractionally filled LLL (Lowest Landau Level), to prove that there is a gap between the ground state and all excited states, and that there exist factionally charged excitations around the fractional filling corresponding to the plateaus observed by Tsui, St¨ ormer and Gossard. Then we will describe Jain’s generalization of Laughlin’s wave functions.
52
FQHE – From Laughlin to Composite Fermions
It is a good training to examine first the many-body wave function for the completely filled LLL. In that case there is a gap to excited state which is, at first sight, a single particle effect, the Zeeman splitting g ∗ µb B (see figure 4.1).1
4.2.1
The many-body wave function for ν = 1
Laughlin exploited a useful property of the single particle Landau Hamiltonian eigenfunctions in the symmetric gauge (see equation 2.28) : ∗
φn=0,m (z, z ) ∝
z √ 2lB
!m
2 /4l2 B
e−|z|
,
so that any analytic function f1 (z)(defined by ∂∂z∗ f1 (z, z∗) = 0 ) in the prefactor of the gaussian belongs to the LLL. All physical results are of course independent of this gauge choice. Turning now to the many-body wave function for the full LLL (i.e. ν = 1), this means that the most general wave function we are looking for has to be of the form X |zi |2 ψν=1 ({zi }) ∝ fN ({zi }) exp − 2 (4.19) 4lB j where {zi } means (z1 , z2 , ...., zN ), and fN is analytic in all variables. N is the total number of electrons, and is equal, since ν = 1 to the total number of states in the LLL. Since we are dealing with a state where all electron spins are identical, the spin wave function is symmetrical under exchange of particles. Since we are dealing with a fermion wave function, the prefactor fN of the orbital part must be totally antisymmetric under exchange of particles. It can only be a single Slater determinant with all LLL single particle states occupied. This determinant reads:
fN = det
1
z10 z11 z20 z21 ... ... 0 1 zN zN
... z1N −1 ... z2N −1 ... ... N −1 ... zN
(4.20)
We will show later on that in fact the gap above the ν = 1 ground state is dominated by exchange effects, and is much larger than the Zeeman gap.
The Laughlin wave function
53
This determinant, called a Vandermonde determinant, is a polynomial in N variables, with N zeros. It has a simple expansion as fN ({zi }) = Πi 1, 2, for a the interaction must be sufficiently short ranged , i.e. V1 /V3 ∼ Laughlin state to be stabilized. Pseudo-potentials with odd indices 8 are plotted in figure 6.4 for electrons in LL n = 0, cf with s = p = 1 and CF2 with s = p = s˜ = p˜ = 1. Notice the difference in scale on the energy axis: the interaction between CF is roughly one order of magnitude smaller than that for electrons. This is easily understood when looking at the effective CF interaction potential (6.52), which is globally reduced by the CF form factor , [FpCF (q)]2 ≃ (1 − c2 )2 , at order O(q 0 ), compared to the potential between electrons. As two factors of this type enter the expression for the CF2 effective interaction potential [see equation (6.58)], the latter is again an order of magnitude smaller than that between CF. In this sense, (1−c2 )2 ≤ 1/9 may be interpreted as the hierachical CF theory small parameter – as discussed earlier, this is a posteriori an indication for the CF LL stability with respect to residual CF interactions. A second remark is about the specific form of the CF (and CF2 ) interaction. Contrary to the electronic case, their pseudo-potentials do not vary monotonically but exhibit a minimum at m = 3. A possible origin of this 8
Remember that only odd index pseudo-potentials matter in the case of fully spin polarised electrons, because of their fermionic statistics (see section 4.2).
108
FQHE Hamiltonian theory
peculiarity is the CF dipolar character, due to their internal structure, as already discussed at the beginning of section 6.2. Since the pair correlation function of the Laughlin liquid (with s = 1) is maximum for that value of the relative kinetic moment, one may expect this pseudo-potential form to stabilise CF Laughlin liquids.
Chapter 7 Spin and Quantum Hall Effect– Ferromagnetism at ν = 1 Until now we have by-passed all questions connected to the electron spin. We have been satisfied with the notion that the Zeeman effect separates LL in two spin branches, the energies of which are separated by a gap ∆z (see figure 4.1). Within this picture, there would be no difference between the IQHE at ν = 2n (both spin branches filled) and that at ν = 2n + 1 (only the lower spin branch filled ) – in both cases a plateau would be due to localisation of additional electrons. The only difference would be the magnitude of the excitation gap. Indeed, since ∆z ≃ h ¯ ωC /70, the two spin branches are not resolved in weak magnetic fields, for which only the IQHE at ν = 2n is observed. In that case, the system behaves as if each state was doubly degenerate, with |n, m; σi (σ =↑, ↓). However this picture turns out to be incorrect. Interactions between electrons have important effects, even for ν = 2n + 1. A new form of quantum magnetism arises. That is the topic of this chapter.
7.1
Interactions are relevant at ν = 1
Let us start with a discussion of the various energy scales. Notice first that , since Landau Level quantization only deals with orbital degrees of freedom, the mass which enters the LL energy separation h ¯ eB/mb is the band mass, mb = 0.068m for GaAs in terms of the bare mass m of the electron. The latter determines the Zeeman gap ∆z = g¯ heB/m, since the Zeeman effect deals 109
110
Spin and Quantum Hall Effect – Ferromagnetism at ν = 1
with the electron spin, an internal degree of freedom. Moreover the effective g factor for GaAs is g = −0, 4, which causes the Zeeman gap in this material to be a factor roughly 70 smaller than the LL separation, as already mentionned earlier. Expressed in Kelvins, the Zeeman gap is ∆z = 0, 33B[T]K, while the LL separation is h ¯ ωC = 24B[T]K, where the magnetic field is measured in Tesla. On the other hand, the characteristic Coulomb energy is e2 /ǫlB = q 50 B[T]K. For a field 6T, which corresponds roughly to filling ν = 1, one finds thus ∆z ≃ 2K ≪ e2 /ǫlB ≃ 120K < h ¯ ωC ≃ 140K. Interactions are therefore of the same order of magnitude as the LL separation, and are more relevant than the Zeeman gap. They must be taken into account when discussing effects connected to the electron spin, in particular at ν = 1, which we will focus on in the remaining parts of this chapter. The first problem is to understand why we observe a Quantum Hall Effect at all at this filling factor. The Zeeman gap is so small that each state is almost degenerate, so we might expect a macroscopic degeneracy at ν = 1 in the kinetic part of the hamiltonian. Just as for the FQHE, interactions are responsible for the lifting of this degenaracy. So we are led to this counter intuitive idea that the IQHE at ν = 1 should rather be looked at as a special case of FQHE.
7.1.1
Wave functions
Let us start with a two spin 1/2 particles wave function, at ν = 1. In the symmetric gauge, the orbital part is built from the single particle wave function, in the symmetric gauge φm (z) = z m , (neglecting normalisation factors) with m = 0, 1 (2.28). As for the spin function, we have four possible states for the coupling of two spin√1/2 particles: an antisymettric singlet , |S = 0, M = 0i = (| ↑↓i − | ↓↑)/ 2, and a symmetric triplet |S =√1, M i, avec |S = 1, M = 1i = | ↑↑i, |S = 1, M = 0i = (| ↑↓i + | ↓↑i)/ 2 and |S = 1, M = −1i = | ↓↓i. We are dealing with a problem without explicit spin-dependent potentials. The interaction is SU(2)invariant, the total spin is a good quantum number. Since the fermionic wave function must be antisymmetric, we have ψS=0 (z1 , z2 ) = φs (z1 , z2 ) ⊗ |S = 0, M = 0i ψS=1 (z1 , z2 ) = φa (z1 , z2 ) ⊗ |S = 1, M i,
and
Interactions are relevant at ν = 1
111
with φs (z1 , z2 ) = z10 z21 + z20 z11 = z1 + z2 and φa (z1 , z2 ) = z10 z21 − z20 z11 = z1 − z2 . The second choice with an antisymmetric orbital wave function is energetically favourable if the interaction is sufficiently strongly repulsive at short range, as is the case for the Coulomb interaction. Thus Coulomb interactions, combined with the Pauli principle, create an exchange force which aligns spins. This is the origin of ferromagnetism in transition metals. It is important to realize that the ν = 1 Laughlin wave function, (4.30) φν=1 ({zi }) =
Y
i. For a spin S = 1/2, an azimuthal angle φ, the same must be true for < S appropriate set of states is: cos 2θ sin 2θ eiφ
|Φθ,φ >=
!
(7.60)
since these obey: θ θ hΦθ,φ |S |Φθ,φ i = h ¯ S cos − sin2 2 2 z
2
and
D
!
=h ¯ S cos θ
E
hΦθ,φ |S x + iS y |Φθ,φ i = Φθ,φ |S + |Φθ,φ = h ¯ S sin θeiφ
(7.61)
(7.62)
~ around axis What is the Berry phase in the case of a slow rotation of ∆ zˆ, at constant θ? γBerry = i =i
Z
0
2π
dφ
cos
θ 2
sin
θ −iφ e 2
Z
0
2π
*
∂ dφ Φθ,φ | Φθ,φ ∂φ
0 i sin 2θ eiφ
!
= −S
+
Z
0
2π
(7.63) dφ(1 − cos θ) (7.64)
7.5. APPLICATIONS TO QUANTUM HALL MAGNETISM = −S
Z
0
2π
dφ
1 X
cos θ
d cos θ′ = −SΩ
127 (7.65)
where Ω is the solid angle subtended by the path as viewed from the origin of the parameter space. This is precisely the Aharonov-Bohm phase one expects for a charge −S particle traveling on the surface of a unit sphere surrounding a magnetic monopole. The degeneracy of the spectrum at the origin is precisely the cause for presence of the magnetic monopole [90] The definition of the connection A implies the existence of a singularity at the south pole, θ = π. A ”Dirac string” (i.e. an infinitely thin solenoid carrying one flux quantum) is attached to the monopole. The singularity would be attached to the north pole if we had chosen the basis e−iφ |Φθ,φ >
(7.66)
In order to reproduce correctly the Berry phase in a path integral for the spin the Hamiltonian of which is given by 7.58, the Lagrangian must be: n
o
L=h ¯ S −m˙ µ Aµ + ∆µ mµ + λ(mµ mµ − 1)
(7.67)
where m is the spin coordinate on the unit sphere, λ is a Lagrange multiplier which enforces the length constraint, and the Berry connection A obeys: ~ ×A=m ∇ ~
(7.68)
This Lagrangian reproduces correctly the equations for the spin dynamics which describe its precession.
7.5 7.5.1
Applications to quantum Hall magnetism Spin dynamics in a magnetic field
In the following, we show that the Lagrangian above allows to describe the quantum spin dynamics in an effective field. The equations of motion are:
Using 7.67 we have
d δL δL = µ dt δ m ˙ δmµ
(7.69)
δL = −Aµ δm ˙µ
(7.70)
128 and
Spin and Quantum Hall effect – ferromagnetism at ν = 1 δL = −m ˙ ν ∂µ Aν + ∆µ + 2λmµ , δmµ
(7.71)
∆µ + 2λmµ = Fµν m ˙ ν,
(7.72)
so that where Fµν = ∂µ Aν − ∂ν Aµ The previous section on the Berry phase shows we must chose: Fµν = ǫαµν mα
(7.73)
~m which is equivalent to ∇ ~ ∧ A[m] = m. The equation of motion becomes: δµ + 2λmµ = ǫαµν mα m ˙ν
(7.74)
Multiplying both members of equation 7.74 by ǫγβµ mβ , then applying on both sides of this equation the identity: ǫναβ ǫνλη = δαλ δβη − δαη δβλ , we get: ~m ~ − ∆∧
γ
=m ˙ γ − mγ (m ˙ β mβ )
(7.75)
The last term vanishes, because of the constraint on the length of m. Using Euler-Lagrange equations, we retrieve the spin precession equations in a magnetic field. Compare 7.67 with the Lagrangian of a particle of mass m, and charge −e in a magnetic field with vector potential A: 1 L = mx˙ µ x˙ µ − ex˙ µ Aµ 2
(7.76)
We see that the Lagrangian in 7.67 is equivalent to a Lagrangian of a zero mass object, with charge −S, placed at m, ~ moving on a unit sphere containing a magnetic monopole. The Zeeman term is analogous to a constant electric ~ which exerts a force S ∆ ~ on the particle. The Lorentz force due to field −∆, the monopole field drives the particle on a constant latitude orbit on the unit sphere. The absence of a kinetic term in m ˙ µm ˙ µ in the Lagrangian indicates that the particle has zero mass, and is in the lowest LL of the monopole field.
7.6
Application to spin textures
Consider a ferromagnet with a local static spin orientation m(r). When an electron is displaced, one may assume that the strong exchange field forces
7.6. APPLICATION TO SPIN TEXTURES
129
the electron spin to follow the local orientation of m(r). If the electron has a velocity x˙ µ , the variation rate of the local spin orientation seen by the electron is m ˙ ν = x˙ µ ∂x∂µ mν . This induces a non trivial Berry phase in the presence of a spin texture. Indeed, the one particle Lagrangian contains an additional term with a time derivative of first order, wich adds to the term due to the field-matter minimal coupling term: L′ = −ex˙ µ Aµ − h ¯Sm ˙ ν Aν [m] ~
(7.77)
L′ = −ex˙ µ (Aµ + aµ )
(7.78)
The first term is the field-matter coupling, the second one gives rise to the Berry phase. We have for the latter ∇m ∧ A = m. ~ That can be re-written, ∂ ν ν µ using m ˙ ≡ x˙ ∂xµ m . Then with
!
∂ ~ mν Aν [m] 2πa = φ0 S ∂xµ µ
(7.79)
a is the Berry connection, a vector potential which adds to the magnetic field vector potential. The curl of a thus contributes a ”Berry” flux which adds to the magnetic field flux: b = ǫαβ
∂aβ ∂xα
(7.80) !
1 ∂ ∂ ~ mν Aν [m] α β ∂x ∂x 2π ! ∂ ∂ ~ mν Aν [m] = φ0 Sǫαβ [ α β ∂x ∂x ! ∂ ∂mγ ∂Aν ν + m ] ∂xβ ∂xα ∂mγ
= (φ0 S/2π)ǫαβ
The first term of the last equation vanishes by symmetry, which results in: b = φ0 Sǫαβ
∂mν ∂mγ (1/2)F νγ ∂xβ ∂xα
(7.81)
with F µν = ǫαµν mα . We used the symmetry ν ↔ γ in the last surviving term . With S = 1/2 one gets b = φ0 ρ˜ (7.82)
130
Spin and Quantum Hall effect – ferromagnetism at ν = 1
with 1 αβ abc a ǫ ǫ m ∂α m b ∂β m c 8π 1 αβ = ǫ m ~ · ∂α m ~ ∧ ∂β m ~ 8π
ρ˜ ≡
(7.83) (7.84)
We recognize in 7.83 the Pontryagin topological density. If now, starting from a uniform magnetization, we deform the ground state magnetization adiabatically into a spin texture, everything happens, for orbital degrees of freedom, as if flux from b(r) was injected adiabatically. In a quantum Hall state with ρii = 0 and ρxy = ν, the Faraday law then causes this spin texture to attract (or repel) at the end of the process a charge density ν ρ˜. Since the skyrmion topological charge is an integer, R Qtop = d2 rρ˜(r) = integer, the charge associated to a skyrmion in the IQHE is δρ = −νe × (integer). We have thus recovered, as a result of the Berry phase, the result obtained earlier by the Hamiltonian approach.
Chapter 8 Quantum Hall Effect in bi-layers 8.1
Introduction
In the previous chapters, we have emphasized the importance of Coulomb interactions in the Quantum Hall Effect physics, including for the ν = 1 filling factor of the LLL. Even if the Zeeman coupling vanishes, Coulomb interactions stabilize a ferromagnetic order, which has important consequences on the excitations spectrum. Instead of a spin degenerate metal at νσ = 1/2, we have a quantum Hall ferromagnetic state with a gap. An analogous effect occurs for bilayers ( a system of two coupled layers), where each layer has filling ν = 1/2. In that case, the rˆole of spin is played by the isospin index of each layer [87, 93, 94]. The analogy with the ferromagnetic monolayer system at ν = 1 will be extensively used in the following. Quantum Hall bilayer physics is quite rich, and involves coupling between layers at various equal, or different, filling factors. This chapter focuses on the particular case of two layers at ν = 1/2, for which exciting results have been obtained in the last few years. Modern MBE techniques have allowed in the recent years to manufacture 2D electron gases with high mobility, in bi-layers or multi-layers structures [95]. As shown on the figure 8.1, a bi layer is a system of two 2D electron gases organized in parallel layers, at a distance d from each other which is comparable with the magnetic length, and to the average distance between electrons in the layer (i.e. d ∼ 10nm ). We know that correlations 131
132
Quantum Hall Effect in bi-layers
W
W
2t
d Figure 8.1: Sketch of the conduction band profile for a two dimensional electrons system in a bi-layer. The order of magnitude for the width, as well as for the distance, of the two layers is W ∼ d ∼ 10nm. In the presence of a tunneling term t, the band splits into a bonding, and an anti-bonding band, with a separation ∆SAS = 2t.
are especially important at high fields, when electrons occupy the LLL only, because the kinetic energy is then frozen out of the problem, and cannot oppose the ferromagnetic polarisation. The FQHE results from gap formation between the ground state and excited states, resulting in an incompressible state. Theory predicts that gaps appear for certain fractional fillings in the bilayer system when inter-layer interactions are strong enough [19, 96]. Such predictions have been backed by experiments [97]. Recently [98], theory has predicted that inter-layer correlations could induce unusual broken symmetry states, with a new type of inter-layer coherence. This new interlayer coherence appears even in the absence of inter-layer tunneling, when the coupling between layers is of purely Coulombic origin. What appears here is excitonic superfluidity, which is the unexpected realization, in the bi-layer physics, of the phenomenon of excitonic superfluidity predicted in 3D semiconductors since 1962 [99]. This phenomenon has been looked for without unquestionable experimental success ever since [100] until it eventually appeared in a spectacular manner in the Quantum Hall bi-layer system [87, 93, 94, 101, 102, 103, 104, 6].
8.2. PSEUDO-SPIN ANALOGY
8.2
133
Pseudo-spin analogy
We assume that the Zeeman effect saturates the “real” spin, so that it does not play any rˆole any more. Each layer is given a “pseudo-spin” label ↑ or + for one layer, and ↓ or − for the other. In the situation we chose to discuss, ν↑ + ν↓ = 1. A state having inter-layer coherence is a state with ferromagnetic pseudospin order, in a direction defined by the polar angle θ, and azimuthal angle φ. In the Landau gauge, such a state writes [see also equation (7.3)] |ψi =
Y k
!
θ θ cos c†k↑ + sin eiφ c†k↓ |0i . 2 2
(8.1)
In the state described by this wave function, each state k is occupied by one electron, and has an amplitude cos(θ/2) to be in the ↑ layer, and amplitude sin(θ/2) exp(iφ) to be in layer ↓. Physically the ratio between the squared amplitude may be altered by applying a voltage between the layers, so as to charge one layer at the expense of the other one, the total filling factor remaining equal to 1. Remember that in the Landau gauge, a state k labels 2 a state localized on a line at guiding center position Xk ≡ klB . We discuss the following cases: Spins along the z axis. For θ = 0 this wave function describes a spin alignment along the zˆ axis, |ψz i = Πk c†k,↑ |0i = Πi v E (q), for all wave vectors, the second term in equation (8.3) creates an easy magnetization plane, perpendicular to zˆ, in the bi-layer plane. The pseudo-spin Hamiltonian symmetry is reduced from SU(2) to U(1).
136
Quantum Hall bi-layers
• A third difference with the mono-layer at ν = 1 arises from the interlayer tunnelling term. This term has the following expression: t X † ck,↑ ck,↓ + c†k,↓ ck,↑ 2 k X S¯kx . = −t
T = −
(8.4) (8.5)
k
It acts as a pseudo-magnetic field applied along xˆ. It stabilises the pseudo-spin orientation in direction xˆ. Indeed the symmetric combination of states in each potential √ well (bonding combination) corresponds −x would to the spinor ψb = (1, 1)/ 2 (bonding state). The direction √ correspond to an anti-bonding state, ψa−b = (1, −1)/ 2
It is feasible experimentally to have widely different values of t between 10−3 et 10−1 × e2 /ǫlB . Note that the tunnel term, since it creates a preferred direction in the xy plane, breaks the U (1) symmetry in the Hamiltonian. • Apply a voltage bias between the layers. This generates a term −e(N↑ − N↓ )V = −2eS z V which is analogous to a magnetic field applied along the z quantization axis of the pseudo-spin. Minimizing the anisotropy term added to the electric term, we see that V creates a charge unbalance between the two layers, which, in pseudo-spin language is a finite value for S z = (N↑ − N↓ )/2 • One may apply a magnetic field Bk parallel to the plane of the bi-layer. Such a parallel field has no effect (except in terms of Zeeman gap) in the monolayer case. One may expect new effects in the bi-layer case, which should be orbital effects (the real spin is expected to be entirely polarised). Chosing a gauge for the associated vector potential, the presence of Bk can be taken into account as follows: Axk = 0 y A|| = Ak = 0 , Azk = Bx
(8.6)
where zˆ is the direction perpendicular to the layers. In that gauge, the gauge invariant tunnel term becomes 2π iφ
t → te
0
R d/2
−d/2
dzAzk
i2π Bxd φ
= te
0
≡ teiQx ,
8.4. EXPERIMENTAL FACTS
137
with Q = 2πBd/φ0 = 2π/Lk , which implies Lk =
φ0 . Bd
in the presence of Bk , the total tunneling term becomes thus T = t
X
eiQx c†k,↑ ck,↓ + e−iQx c†k,↓ ck,↑
k
h
i
= −t eiQx (S x + iS y ) + e−iQx (S x − iS y )
= 2tS cos(φ + Qx)
(8.7)
This term is thus equivalent to a field rotating around the x axis, uniform along y, which is causing the pseudo-spin magnetization to oscillate along x. In fact this term competes with the exchange term, as we shall see later on. • When a Bk field is applied, the tunnelling current between the layers becomes
J↑↓ = it
X k
eiQx c†k,↑ ck,↓ − e−iQx c†k,↓ ck,↑
h
(8.8) i
= it eiQx (S x + iS y ) − e−iQx (S x − iS y ) = 2tS sin(φ + Qx).
8.4 8.4.1
Experimental facts Phase Diagram
As discussed above, the energy difference ∆SAS = 2t between symmetric and antisymmetric superpositions of layer states of the two wells may vary, depending on the sample, from a few millidegrees to a few hundreds of degrees Thus ∆SAS may be much smaller, or much larger than the inter-layer Coulomb interactions. Experiments are able to scan a large array of the characteristic ratio ∆SAS /(e2 /ǫd), from a weak electronic correlation regime to a strong one. When the layers are far from each other (d ≫ lB ), there are no inter-layer correlations, each layer is in the ν = 1/2 metallic ground state, there is no
138
Quantum Hall bi-layers
Figure 8.2: Phase diagram for the bi-layer QHE (after Murphy et al. ??). The samples with parameters below the dotted line exhibit the IQHE and an excitation gap.
QHE. When the inter-layer separation decreases, an excitation gap is found to appear, together with a quantized Hall plateau with σxy = e2 /h [94, 101]. If ∆SAS /Ec ≫ 1 (with Ec = e2 /(ǫd), this is fairly easy to understand, since things look as if the two layers were like a single one, with total filling factor ν = 1. All symmetric states are occupied, we have the usual QHE. A far more interesting situation arises when the ν = 1 QHE is found in the limit ∆SAS → 0. In this limit, the excitation gap is clearly a collective effect, since it may be as large as 20 K while ∆SAS < 1K. The excitation gap survives in this limit because of a spontaneous breaking of the U (1) gauge symmetry associated with the phase degree of freedom –the azimuthal angle φ in expression (8.1) [87, 98]. Figure 8.2 shows the experimental determination of the QHE part of phase diagram, below the dotted line. This change from single particle to collective behaviour is analogous to the ferromagnetic behaviour of a monolayer at ν = 1. In the latter case, the excitation gap remains finite even when the Zeeman effect vanishes, because of the exchange forces connected to the Coulomb interaction. The remarkable fact is that the IQHE at ν = 1 survives when ∆SAS → 0 provided the inter-layer distance between layers is smaller than a critical value d/lB ∼ 2. In that case, the gap is a purely collective effect due to interactions. As we shall see, it is due to a pseudo-ferromagnetic quantum Halll state, which posesses a spontaneous inter-layer coherence.
Experiments
139
Figure 8.3: Experiment by Murphy et al. [94]. The thermal excitation gap ∆ is plotted as a function of the magnetic field tilt angle, in a bilayer with small tunnel term (∆SAS = 0.8K). The black dots correspond to filling factor ν = 1, the triangles to ν = 2/3. The arrow shows the critical angle θc . The continuous line is a guide to the eye. The dotted line is a rough estimate of the tunnel amplitude renormalized by the parallel magnetic field. This single particle effect exhibits a slow negative variation, compared to the observed effect. The inset is an Arrhenious plot of the dissipation, measured by the longitudinal resistance. The low temperature activation energy is ∆ = 8.66K. The gap however decreases sharply at a much lower temperature, roughly 0.4K.
8.4.2
Excitation gap
An additional indication of the collective nature of excitations is provided by the excitation gap variation with temperature, as shown on figure 8.3. The activation energy ∆ at low temperature is clearly larger than ∆SAS . If ∆ was a single particle gap, one would expect an Arrhenius law up to temperatures of the order of ∆/kB . Instead, the gap decreases sharply as soon as A T ∼ 0, 4K. This suggests that the order responsible for the collective excitation gap is vanishing .
8.4.3
Effect of a parallel magnetic field
Another experimental finding suggests strongly a collective order phenomenon: the strong sensitivity of the system to a relatively weak Bk magnetic field, applied in a direction parallel to the layers plane. The figure 8.3 shows that the activation gap decreases rapidly when B|| , the parallel component
140
Quantum Hall bi-layers
Figure 8.4: Example of electronic process in a 2D bi-layer, such that the flux of Bk produces decoherence effects. In this process, an electron tunnels at point A from the upper layer to the lower one. The electron pair thus created moves coherently, then annihilates at point b where the particle tunnels in the other direction. The amplitude for such a process depends on the flux of Bk through the path . of the field, increases. Assume that the electronic gas in each layer is stricly 2D (in other words neglect the physical width of the potential wells). Then the orbital effect of Bk can only be due to electronic processes between the layers with closed loops containing some flux from Bk . Such loops will cause B|| to be felt if there is phase coherence over the whole loop. Such a loop is shown on figure 8.4. An electron tunnels from one layer to the other at point A, travels a distance L|| , tunnels back to the departure layer, then back to point A. The magnetic field parallel component, Bk , contributes to the amplitude of this process a (gauge invariant) Aharonov-Bohm phase factor, exp(2πiφ/φ0 ), where φ is the flux of Bk threading this circuit. Such loops contribute significantly to correlations, since one observes a rapid decrease of the activation gap as a function of Bk : the decrease is by a factor 2 up to a critical field Bk∗ ∼ 0.8T , beyond which the gap remains roughly constant. This value is remarkably small. Let Lk be the length such that the flux through the loop is one flux quantum: Lk Bk∗ d = φ0 ⇔ Lk [˚ A] = 5 ∗ 4, 14×10 /d[˚ A]Bk [T]). With Bk = 0, 8T and d = 150˚ A, one has Lk = 2700˚ A, i.e. roughly twenty times the average distance between electrons in a layer, and thirty times the magnetic length corresponding to B⊥ . A significant decrease of the excitation gap is already observed in a parallel field of 0.1T,
Experiments
141
Figure 8.5: (a) In a standard experiment, the Hall current is transported simultaneously in both layers, without tunnel between layers. (b) It is possible to inject current in one layer and to extract it in another. The tunneling current then behaves as a superfluid current .
which implies enormous coherence lengths. This is again a hint of the strongly collective nature of the observed order in quantum Hall bi-layers.
8.4.4
The quasi-Josephson effect
A spectacular experiment by Spielman et al. [102], confirmed the theoretical ideas about excitonic superfluidity in bi-layers. In the standard transport experiments on bi-layers, a current JHQ is injected in both layers simultaneously, and is also extracted from both layers simultaneously. In the experiment by Spielman et al., a current JHQ is injected in one layer, and extracted from the other one (Fig. 8.5). Qualitative differences arise in the tunnel conductance when the ratio d/lB is varied, for example varying the electronic density at constant filling of the LLL. Below a critical value of the ratio (the critical value which corresponds to the transition line between the QHE regime at ν = 1 and the metallic regime) a giant anomaly appears at zero bias, as shown on fig. 8.6. The qualitative understanding of this experiment is as follows: for d/lB ≫ 1, electronic liquids in different layers are uncorrelated. At zero inter-layer bias, the Coulomb repulsion between electrons in one layer, and an electron in the other will inhibit the inter-layer tunneling process of the latter: the zero bias conductance is strongly suppressed. Only a finite bias, of the order of the Coulomb repulsion e2 /(ǫd), can supersede the latter. When d/lB ≃ 1 a coherent state is established, such that an
142
Quantum Hall bi-layers
electron in one layer is bound to a hole in the other one at the same Landau site. Coulomb repulsion is strongly suppressed by this collective structure for inter-layer tunneling events, and the tunneling conductance increases by two orders of magnitude. At the time of this writing, as far as the author knows, there is yet no general consensus on the intrinsic, or extrinsic character of the zero bias conductance finiteness. Is it impossible, for fundamental reasons, to ever observe a divergent conductance at zero bias, which would be the signature of a complete analogy with the superconducting Josephson junction? Is the conductance finiteness due to experimental limitations, (impurities, etc.), or to the order parameter topological defects at finite temperature? Those questions are still being discussed among specialists.
8.4.5
Antiparallel currents experiment
In order to check the ideas about bosonic superfluid exciton liquid in the bilayer system at ν = 1, one needs an experimental proof of electron-hole pair transport. How can one couple to and detect electrically neutral objects by electric transport? The solution is to notice that electron-hole pair transport in a bi-layer implies an anti-parallel circulation of currents in different layers. Experiments have allowed, in the last few years to get independent electric connections to each layer [108]. It has thus been possible to inject equal intensity currents with opposite flow direction in both layers, to test the contribution of excitons to particle transport [6]. The figure 8.7 is a schematic representation of what one expects from such an anti-aparallel current experiment. The two traces represent the expected Hall voltage in each layer, neglecting all quantum phenomena except the excitonic condensation. Because of the Lorentz force, the Hall voltage is proportional to the magnetic field. In a bi-layer system driven by two oppositely directed parallel currents, the Hall voltage will have opposite signs in the two layers. If the two layers are sufficiently coupled, and the magnetic field has the relevant intensity (so that ν↑ = ν↓ ≃ 1/2 the inter-layer electronhole pairs which form will carry anti-parallel currents. The Hall voltage in both layers must then vanish, as suggested by the figure 8.7. Two experimental groups have confirmed those predictions [104].
143
(a) A)
NT=10.9 D) (d)
B) (b)
NT=6.9
NT=5.4
z
-7
-1
Tunneling =1 /dV (10 W ) Conductance at nTdJ Conductance tunnel
Experiments
0.5
NT=6.4
(c) C)
-5
0
5
-5
0
5
Tension intercouches V (mV) Interlayer Voltage (mV) Figure 8.6:
Quasi-Josephson effect [102]. Plot of the tunneling conductance dJz /dV as a function of voltage bias between the two layers, for various electronic densities, NT in units of 1010 cm−2 . In the samples [from (a) to (c)] with larger electronic density (i.e., smaller lB ) the system does not exhibit any QHE, tunnelling processes are suppressed at zero bias. In the low density sample (d) there is a finite tunneling conductance peak at zero bias. The current at zero bias vanishes, contrary to the superconducting Josephson junction current. Whence the expression ”quasi-Josephson effect”.
144
Quantum Hall bi-layers
tension de Hall
10
+ − + − + − + − + − + − + −
+ − − + − + −
0
+ + − − + + − ν=1 −10
5
10 champ magnétique
Figure 8.7: Antiparallel currents experiment (After ref. [6]). A Hall voltage measurement detects the exciton condensation. The two traces are schematic representations of the Hall voltage in each layer when electric currents flow in opposite directions. Quantum effects other than the excitonic condensation are ignored in this figure. When currents flow in an uncorrelated manner between both layers, one must observe finite Hall voltages, which balance the Lorentz force in each layer. As the currents flow in opposite directions, Hall voltages must have opposite signs in each layer compared to the other. If exitonic condensation occurs, in a certain span of magnetic field values, the opposite currents in the layers will be carried by a uniform exciton current density in one direction. Since excitons are electrically neutral, they are not submitted to Lorentz forces, and the Hall voltage must vanish in both layers, as observed experimentally by Kellogg et coll. [104].
8.5. EXCITONIC SUPERFLUIDITY
8.5
145
Excitonic superfluidity
Within the pseudo-spin analogy(section 9.2), the Coulomb interaction between layers favours a ferromagnetic state with an easy magnetization plane when the inter-layer distance is small enough to stabilize a correlated state. The ground state wave function is then of the form [see equation (8.1)] Y c†k↑ + eiφ c†k↓ √ |ψφ i = |0i . 2
(8.9)
k
In other words, θ/2 = π/4, the magnetization is in the bi-layer plane, and one has hSz i = 0. The amplitude is equal for opposite pseudo-spin states, which means that, for the time being, we consider a situation with zero bias between the two layers. The total occupation of each k state is 1. When the tunnel term t vanishes, φ has any value, provide it is the same all over the bi-layer plane. When the system chooses a particular φ value, among the continuous infinity of choices, the original U (1) symmetry of the Hamiltonian (in the presence of the pseudo-spin anisotropy) is broken by the ground state. In the limit of zero tunneling term, we have thus a one parameter family of equivalent ground states, with the phase φ as parameter. This phase is conjugate to the difference in particle number between the two layers. In equation (8.9), the phase is well defined, but the number of particles of each pseudo-spin (i.e. the number of particles is each layer) is completely undetermined. Similarly, one may construct a state such that the phase is undetermined, while the number of particles in each layer is specified exactly. To do this, integrate 8.9 over the phase. This yields |ψS z i =
Z
dφ −i(N↑ −N↓ )φ e |ψφ i . 2π
(8.10)
We obtain thus a wave function with exactly N↑ particles in the ↑ layer, and N↓ = N − N↑ in the ↓ layer, N being the total number of guiding centers. The angle φ and S z are canonical conjugate variables, [φ, Sz = N↑ − N↓ ] = 1,
(8.11)
whence the uncertainty relation δ(N↑ − N↓ ) × δφ > 1. Since we are dealing with a continuous broken symmetry, there must exist a Goldstone mode the energy of which goes to zero in the limit of infinite
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QHE bi-layers
wavelength. A state such that the phase varies in time and space may be written as i Yh † |ψφ i = ck↑ + eiφ(Xk ,t) c†k↓ |0i , (8.12) k
where φ is the superfluid phase of the system. The long wavelength superfluid mode corresponds to equal intensity currents of opposite signs propagating in the two layers. To understand better why the state described by (8.9) breaks the gauge symmetry associated to the charge difference between layers, consider the θ gauge transformation induced by the unitary operator U− (θ) = ei 2 (N↑ −N↓ ) . This transformation acts on electron creation operators as θ
U−† (θ)c†k↑ U− (θ) = e−i 2 c†k↑
(8.13)
.
(8.14)
U−† (θ)c†k↓ U− (θ)
i θ2
=e
c†k↓
The Hamiltonian is invariant under this transformation, U−† (θ)HU− (θ) = H,
(8.15)
since [H, (N↑ − N↓ )] = 0, in the absence of inter-layer tunneling terms. In contrast, expression (8.9) shows that the coherent phase exhibits a non trivial order parameter. D E D E nel iφ S x (Xk ) ≡ c†k↑ ck↓ = S¯x (Xk ) = e , 2
2 with the total density nel = 1/2πlB . (Here I have defined the xˆ direction as the arbitrary direction of the sponaneous pseudo-spin orientation in the plane x, y). This order parameter is not gauge invariant,
D
E
S x (Xk ) → U−† (θ)c†k↑ ck↓ U− (θ) = eiθ S x (Xk ) .
(8.16)
That is a more formal way to show that the state has less symmetry than the Hamiltonian, and breaks the U (1) symmetry associated with the conservation of the charge difference between layers N↑ − N↓ . 1 In a superconductor, the order parameter χ(r) = hc†↑ (r)c†↓ (r)i transforms in a non trivial way under the gauge transformation associated with the total charge conservation, ˜+ (θ) = exp[iθ(N↑ + N↓ )/2]. The pseudo-spin bi-layer order parameter is invariant under U this transformation: this expresses simply the fact that the total particle number N↑ + N↓ is conserved in the excitonic superfluid. 1
Excitonic superfluidity
147
We can write an expression for the inter-layer tunneling current operator as a function of position in space,
J↑↓,Xk = −it c†k↑ ck↓ − c†k↓ ck↑ , the average of which
(8.17)
hJ↑↓,Xk i = −it [S x (Xk )∗ − S x (Xk )] = t sin(φ). This expression is similar to the Josephson current expression: it depends only on the order parameter phase, not on the inter-layer voltage bias. The pseudo-spin language expresses the conjugate character of phase and charge difference between layers through the commutation relations of the spin density operators. With the order parameter along x, [S y , S z ] = iS x ≃ i. As S y ∝ sin φ ≈ φ, this leads to [φ, S z ] = i. As a consequence, the current associated to the phase gradient Jzz =
2ρE ∇φ h ¯
is indeed the difference of the electric currents in the two layers. An apparent conceptual difficulty is that the wave function (8.9) describes a state where the difference between the layer charges fluctuates, while this difference should be conserved in the limit t = 0 . This is analogous to the superconducting BCS wave function, which has a fluctuating total number of particles, while it is in fact strictly conserved for an isolated sample.The solution of this apparent paradox is that each macroscopic piece of the sample may be subdivided in smaller macroscopic parts, between which particle exchanges are numerous and rapid, so that phase coherence is established in each macroscopic part of the sample, the total particle number remaining constant. Furthermore, in the thermodynamic limit, the ratio δN/N is of order N −1/2 → 0. A similar reasoning holds in the bi-layer case. Examine a slighlty more complicated object than the order parameter, D
GXk ,Xk′ = c†k↑ ck↓ c†k′ ↓ ck′ ↑
E
.
(8.18)
This object conserves the total particle number in each layer. It is equal to hS¯x (Xk )S¯x (Xk′ )i, and it is non zero in the wave function (8.9).
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QHE bi-layers
Notice that the wave function (8.9) is indeed an exciton condensate. To see that, define the state |f erro ↑i as the state where all electrons are in the Q ↑ layer , |f erro ↑i = k c†k↑ |0i. Then the state (8.9) may be re-written as |ψφ i ≡
Y k
1 + eiφ c†k↓ ck↑ |f erro ↑i . √ 2
(8.19)
This can be again re-written in a form reminiscent of a bosonic coherent state, Y exp eiφ b†k |f erro ↑i , (8.20) |ψφ i ≡ k
where b†k = c†k↑ ck↓ is the excitonic boson. This is the reason why one may speak of a ”coherent” state (see section 2.2.2). One also speaks of ”spontaneous phase coherence”, when the tunneling term is absent. Indeed in that case the coherent state is entirely due to Coulomb interactions. On the contrary, when the tunneling term is finite, the symmetric combination of layer states is the most stable, even in the absence of interactions. This is analogous to the magnetization induced by an external magnetic field in the case of ”real” spins.
8.5.1
Collective modes – Excitonic condensate dynamics
As mentionned above, a consequence of the breaking of a continuous symmetry by the phase coherence is the existence of the collective excitation mode (Goldstone mode) the energy of which goes continuously to zero as the wavelength goes to infinity. The Hamiltonian formalism was used above to derive collective mode energies in the ferromagnetic monolayer case. Here we use the Lagrangian formulation, with the inclusion of the Berry connexion term discussed in the previous chapter. The Lagrangian which describes the long wavelength physics, in the absence of applied inter-layer voltage bias, and with zero tunneling term, is ν Z 2 ˙ · A[m] L = d rm (8.21) 2 4πlB Z i ρA ρE h − d2 r β(mz )2 + . |∇mz |2 + |∇mx |2 + |∇my |2 2 2
Excitonic superfluidity
149
Coefficients β, ρA and ρE may be evaluated with a microscopic approach, as we have seen in section 8.3.2 Let us write the Euler-Lagrange equations of motion, d δL δL = . (8.22) µ dt δ m ˙ δmµ Here the ground state is taken with the (vector) order parameter of length 1 aligned along the xˆ axis. For small variations of the order parameter away from xˆ, one may linearize, considering only first order deviations in my and mz . m = [1 − O(m2y + m2z ), my , mz ], and one chooses the Berry connexion A = (0, −mz /2, my /2), which yields ν δL ν mz , = Ay [m] = − 2 2 δ m˙ y 4πlB 4πlb 2 ν m˙ z δL = + ρE ∆my , 2 δmy 4πlB 2
(8.23)
and ν ν my , Az [m] = − 2 2 4πlB 4πlb 2 ν m˙ y + ρA ∆my − 2βmz , = 2 4πlB 2
δL δ m˙ z δL δmz
=
(8.24)
where ∆ = ∇ · ∇ is the Laplacian. In Fourier space, applying 8.22, one finds the system of linear equations iω 4π 2 q ρE ν
4π (2β ν
+ q 2 ρA ) −iω
!
my mz
!
= 0.
(8.25)
So finally the collective mode dispersion relation is given by 4π ω (q) = ν 2
2
(2β + q 2 ρ2A )q 2 ρE .
(8.26)
When d = β = 0, and ρA = ρE = ρ0 one retrieves the collective mode (pseudo-spin wave ) of the ferromagnetic SU (2) phase, ω(q)|B=0 = 2
4π 0 2 ρq . ν
The expansion in gradients of mz is not stricly correct, because the long range nature of he Coulomb interaction induces a non local term which we do not take into account here. The latter term is smaller than the terms considered here.
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QHE bi-layers
Q=+1/2 v=+1
Q=+1/2 v=−1
Q=−1/2 v=+1
Q=−1/2 v=−1
Figure 8.8: Four meron ”flavours”. With two possibilities for the choice of the vorticity, and two additional ones for the choice of the pseudo-spin orientation at the vortex core, merons have a topological charge Q = ±1/2, and exist in four possible ”flavours”.
The mass term β 6= 0 changes qualitatively the collective mode dispersion, which becomes linear in q at small q, lim ω(q)|β6=0
q→0
4π q = 2βρE q . ν
That is analogous to the bosonic superfluid collective mode (with weak repulsive interactions). But here the order parameter represents the condensation of neutral bosons, which carry no charge.
8.5.2
Charged topological excitations
For a system in the same universality class as that of the 2D XY model, there must exist a Kosterlitz-Thouless (KT) transition at TKT = (π/2)ρS /kB . The essence of this transition is the ionisation (dissociation) of vortex-antivortex pairs. In our case, the order parameter symmetry group is U (1), but the pseudo-spin direction is not confined to the xy plane, so that the pseudo-spin vortex is in fact a ”meron” , which may be considered as a half skyrmion. The system order parameter in the presence of a vortex at the origin has the approximate following form q
m = ± 1 − m2z cos θ,
q
1 − m2z sin θ, mz (r)
,
(8.27)
Excitonic superfluidity
151
where the ± sign refers to the vorticity (left or right) and θ is the azimuthal angle of the position vector r. At large distance from the meron center, mz (r) tends to zero to minimise the capacitive energy. At the vortex core, however, we have mz = ±1, mx = my = 0, to avoid the large energy cost of a core singularity. The local topological charge is computed using the Pontryagin density expression [see equation (7.31)] δρ = −
1 ij ǫ (∂i m × ∂j m) · m. 8π
With expression (8.27), this density writes δρ(r) =
1 dmz . 4πr dr
The total charge is Q = d2 rδρ(r) = 21 [mz (∞) − mz (0)]. For a meron, the spin at the core is either ↑ or ↓, and gradually gets oriented in the xy plane as the distance from the core increases. It lies in the xy plane far from the meron core. The topological charge is thus ±1/2 depending on the core spin polarity. The general result for the topological charge is R
Q=
1 [mz (∞) − mz (0)] nv 2
(8.28)
where nv is the vortex winding number. The electric charge is ±νe/2, half that of a skyrmion, which comes as a support of the meron as a half skyrmion, as mentionned above. One may write a meron variational wave function. The simplest one is † † M E Y c + c m,↑ √ m+1,↓ |0i . ψnv =+1,−1/2 =
2
m=0
(8.29)
In this expression, c†m,↑(↓) creates an electron in layer↑ (↓), in the state of angular momentum m in the LLL, and M is the corresponding moment on the sample edge. The vorticity is +1, since far from the core, the spinor is √ χ(θ) = (1/ 2)
eiθ 1
!
,
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QHE bi-layers
where θ is the polar angle of the vector r. The charge is +1/2 because an electron has been suppressed at the center, in the ↓ layer: all states have 1/2 occupation, except m = 0 which is empty. The meron charge can be changed without changing the vorticity, as we see with the wave function † † M E Y c + c † m,↑ √ m+1,↓ |0i . ψnv =+1,+1/2 = c0,↓
2
m=0
This state has charge −1/2, because an electron has been created in the state m = 0 in the ↓ layer. It is useful to examine a meron pair wave function, to check wether the meron is a half skyrmion. Examine the case of a pair of merons with opposite vorticities, but equal charges, placed at points z¯1 and z¯2 . The following wave function seems to obey our requirements, ψλ =
eiφ (zj − z¯1 ) (zj − z¯2 )
Y 1 j
√
2
!
Φf erro ,
(8.30)
j
where φ is an arbitrary angle and ()j is a spinor for the j-th particle. At large distance from z¯1 and z¯2 , the spinor for each particle becomes eiφ 1
zj
!
.
(8.31)
This corresponds to a fixed spin orientation in the xy plane, with an angle φ with the x axis. Vorticity is thus zero. By construction, the spin orientation is purely ↑ for an electron at z¯2 , and purely ↓ for an electron at z¯1 . Moreover, the net charge must be νe since, asymptotically, the factor zj is the same as for the Laughlin quasi-particle in the spin polarised state. For symmetry reasons, one might think that a charge νe/2 is asociated to each localised state near z¯1 or z¯2 . The fact is that this wave function (8.30) is nothing but a different representation for the skyrmion! Choose z¯1 = λ and z¯2 = −λ, and suppose for simplicity that the asymptotic orientation of spins is in the x direction, so that φ = 0. Now rotate all spins by a global rotation around the yˆ axis, with an angle −π/2. Using π 1 exp i σ y √ 4 2
zj − λ zj + λ
!
,
Excitonic superfluidity
153
one finds the variational skyrmion wave function. The previous wave function is well adapted to the U (1) symmetry, because t describes spins oriented mainly in the xy plane.
8.5.3
Kosterlitz-Thouless transition
The presence of topological defects of the vortex type may spoil the phase coherence of the XY ground state. This may happen at zero temperature, because of quantum fluctuations, if the distance between layers exceeds a critical distance d∗ . Here we are discussing thermal effects. The effective model at finite temperature is given by ρS Z 2 E= d r |∇φ|2 . 2 For typical experimental parameter values in the AsGa bi-layers, the HartreeFock estimate of the exchange stiffness ρS goes from 0, 1K to 0, 5K. The Kosterlitz-Thouless transition is due to ionisation of vortices in the XY model, at a temperature TKT approximately given by the exchange stiffness ρS . Free vortices induce a discontinuous renormalisation of the exchange stiffness, which vanishes at TKT . The classical action generates a logarithmic interaction between vortices. A meron gas has an energy of the form E = M Ecore − 2πρS
M X i
