Quantum Hall Effects An Introduction Mark O. Goerbig Les Houches - - PowerPoint PPT Presentation

Quantum Hall Effects – An Introduction

Mark O. Goerbig

Les Houches Summer School “Ultracold Gases and Quantum Information” July 2009, Singapore

Outline

• Lecture 1 (Basics)

– History of the quantum Hall effect & samples – Landau quantisation (2D particle in a B field)

⇒ non-relativistic vs relativistic version

• Lecture 2 (Integer quantum Hall effect)

– Landau quantisation in the presence of an external

potential (confinement and weak impurity potential)

– Conductance quantisation

• Lecture 3 (Fractional quantum Hall effect)

– Laughlin’s theory – fractional charge and statistics (anyons) – ...

Metal-Oxide Field-Effect Transistor (MOSFET)

E z

F

E z

F

E z

F

(a) (b) (c)

V V

G G

metal

• xide

(insulator) semiconductor

conduction band acceptor levels band valence

semiconductor

band conduction acceptor levels valence band

metal

• xide

(insulator) metal

• xide

(insulator)

conduction band acceptor levels valence band

II I

V

G

z z E E E

1 metal

• xide

semiconductor

2D electrons

usually silicon-based materials (Si/SiO2 interfaces)

Signature of the Quantum Hall Effect (QHE)

8 12 16 4 Magnetic Field B (T) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ρxy (h/e )

2

0.5 1.0 1.5 2.0 ρ Ω

xx (k )

2/3 3/5 5/9 6/11 7/15 2/5 3/7 4/9 5/11 6/13 7/13 8/15 1 2

/

3 2

/

5/7 4/5 3 4

/ Vx Vy Ix

4/7 5/3 4/3 8/5 7/5 1 2 3 4 5 6

Magnetic Field B[T]

[measurements by J. Smet et al., MPI-Stuttgart]

QHE = plateau in Hall & vanishing longitudinal resistance

GaAs/AlGaAs Heterostructure

dopants

AlGaAs

z

E

F

GaAs

dopants

AlGaAs

z

E

F

GaAs

(a) (b)

2D electrons

Reduced surface roughness (as compared to Si/SiO2) ⇒ enhanced mobility (FQHE)

Graphene

• 2

300 nm SiO graphene (2D metal) (insulator) doped Si (metal) Vg

Exfoliated graphene = 2D graphite Change of carrier density via application of gate voltage Vg

Bandstructure of Graphene

Infrared Transmission Spectroscopy

Grenoble high−field group: Sadowski et al., PRL 97, 266405 (2007) transition C transition B

relative transmission relative transmission Energy [meV] Energy [meV] Transmission energy [meV] Sqrt[B]

selection rules : λ, n → λ′, n±1

Edge States

• ymax

n+1

ν = n ν = n−1

y

ymaxymax

n n−1

n+1 n n−1 (a) (b)

y x

ν = n+1 µ

LLs bended upwards at the edges (confinement potential) chiral edge states ⇒ only forward scattering

ν= n+1 ν= n ν= n−1

Four-terminal Resistance Measurement

I I R ~ 5 6 2 3 4 1 R ~ µ − µ = µ − µ

3

µ − µ = 0

2 5 L H

µ = µ µ = µ

2 L L 3

µ = µ = µ

6 5 R 3 R L

: hot spots [Klass et al, Z. Phys. B:Cond. Matt. 82, 351 (1991)]

IQHE – One-Particle Localisation

n

ε

n

ε

n

ε (b) (c)

(n+1)

ν

NL

(a)

density of states density of states density of states extended states localised states

R R B E

F

R B EF R

h/e (n+1) h/e n

2 2

R R B =n

h/e n

2

F

E

L H L H L H

IQHE – One-Particle Localisation

n

ε

(n+1)

ν

NL

(a)

density of states

R R B =n

h/e n

2

F

E

L H

IQHE – One-Particle Localisation

n

ε

n

ε (b)

(n+1)

ν

NL

(a)

density of states density of states

R R B E

F

R R B =n

h/e n

2

F

E

L H L H

IQHE – One-Particle Localisation

n

ε

n

ε

n

ε (b) (c)

(n+1)

ν

NL

(a)

density of states density of states density of states extended states localised states

R R B E

F

R B EF R

h/e (n+1) h/e n

2 2

R R B =n

h/e n

2

F

E

L H L H L H

IQHE in Graphene

Novoselov et al., Nature 438, 197 (2005) Zhang et al., Nature 438, 201 (2005)

V =15V Density of states B=9T T=30mK T=1.6K ∼ ν ∼ 1/ν Graphene IQHE: R = h/e at = 2(2n+1) at = 2n ν ν

H

ν

2

(no Zeeman) Usual IQHE:

g

Percolation Model – STS Measurement

2DEG on n-InSb surface Hashimoto et al., PRL 101, 256802 (2008) (a)-(g) dI/dV for different values of sample potentials (lower spin branch of LL n = 0) (i) calculated LDOS for a given disorder potential in LL n = 0 (j) dI/dV in upper spin branch of LL n = 0

Percolation Model – Scaling

0.10 1.00

T(K)

1.0 10.0 100.0

(∆B)

−1

(∆B)

−1

N = 1 N = 1 N = 0 N = 1 N = 1

• d

• max
• d

• max

scaling of the plateau width ∆B

Wei et al., Phys. Rev. Lett. 61, 1294 (1988)

⇒ Second-order PT (QPT) critical exponents: 1/zν = 0.42 ± 0.04 and z ≃ 1 ⇒ ν ≃ 2.3 classical percolation: ν = 4/3 special quantum model (numerics): ν = 2.5 ± 0.5

Further Reading

• D. Yoshioka, The Quantum Hall Effect, Springer, Berlin (2002).
• S. M. Girvin, The Quantum Hall Effect: Novel Excitations and Broken

Symmetries, Les Houches Summer School 1998

http://arxiv.org/abs/cond-mat/9907002

• G. Murthy and R. Shankar, Rev. Mod. Phys. 75, 1101

(2003). http://arxiv.org/abs/cond-mat/0205326

• M. O. Goerbig and P

. Lederer, Electrons bidimensionnels sous

champ magn´ etique fort : la physque des effets Hall quantiques

http://www.lps.u- psud.fr/Utilisateurs/goerbig/CoursEHQ2006.pdf

Jain’s Wavefunctions (1989)

Idea: “reinterpretation” of Laughlin’s wavefunction ψL

s ({zj}) = i<j(zi − zj)2sχp=1({zj})

• i<j(zi − zj)2s: “vortex” factor (2s flux quanta per vortex)

χp=1({zj}) =

i<j(zi − zj):

wavefunction at ν∗ = 1

Jain’s Wavefunctions (1989)

Idea: “reinterpretation” of Laughlin’s wavefunction ψL

s ({zj}) = i<j(zi − zj)2sχp=1({zj})

• i<j(zi − zj)2s: “vortex” factor (2s flux quanta per vortex)

χp=1({zj}) =

i<j(zi − zj):

wavefunction at ν∗ = 1 Generalisation to integer ν∗ = p ψJ

s,p({zj}) = PLLL

• i<j(zi − zj)2sχp({zj, ¯

zj}) χp({zj, ¯ zj}): wavefunction for p completely filled levels PLLL: projector on lowest LL (→ analyticity)

Physical Picture: Composite Fermions

CF = electron+”vortex” (carrying 2s flux quanta) with renormalised field coupling eB → (eB)∗

ν = 1/3

pseudo−vortex

electronic filling 1/3 theory CF 1 filled CF level

electron "free" flux quantum

(with 2 flux quanta)

composite fermion (CF)

Physical Picture: Composite Fermions

CF = electron+”vortex” (carrying 2s flux quanta) with renormalised field coupling eB → (eB)∗

ν = 1/3 ν = 2/5

pseudo−vortex

theory CF 2 filled CF levels

electron "free" flux quantum

(with 2 flux quanta)

composite fermion (CF)

electronic filling 1/3 1 filled CF level

At ν = p/(2ps + 1) ↔ ν∗ = nel/n∗

B = p, with n∗ B = (eB)∗/h:

FQHE of electrons = IQHE of CFs

Generalisation: FQHE at half-filling

• 1987: Obervation of a FQHE at ν = 5/2, 7/2 (even

denominator)

• 1991: Proposal of a Pfaffian wave function (Moore & Read;

Greiter, Wilzcek & Wen) ψMR({zj}) = Pf

• 1

zi − zj

i<j

(zi − zj)2 ⇒ quasiparticle charge e∗ = e/4 with non-Abelian statistics

• Further generalisations to ν = K/(K + 2): Read & Rezayi

Multicomponent Systems

Landau levels

|+>

|−>

d ν = 1/2 ν = 1/2 ν = ν + ν = 1

+ + − − T : A sublattice : B sublattice τ τ

2 3 1 2

e

1

e e

spin + isospin : SU(4)

A physical spin: SU(2) two−fold valley degeneracy B bilayer: SU(2) isospin SU(2) isospin C graphene (2D graphite) (doubling of LLs) exciton