Introduction to the Quantum Hall Effect andTopological Phases

Mark O. Goerbig

Ecole du GDR “Physique Mesoscopique, Cargese, November 2016

Historical Introduction

What is the common point between

• graphene,

• quantum Hall effects

• and topological insulators?

... and what is it?

The 1920ies: Band Theory

• quantum treatment of (non-interacting) electrons in aperiodic lattice

• bands = energy of the electrons as a function of aquasi-momentum

1950-70: Many-Body Theory

• Physical System described by a local order parameter(a) ∆k = 〈ψ†

−k,↑ψ†k,↓〉 (superconductivity)

(b) Mµ(r) =∑

τ,τ ′〈ψ†τ (r)σ

µτ,τ ′ψτ ′(r)〉 (ferromagnetism)

• Ginzburg-Landau theory of second-order phase transitions(1957)

∆ = 0(disordered)

↔ ∆ 6= 0(ordered)

• symmetry breaking(a) broken (gauge) symmetry U(1)(b) broken (rotation) symmetry O(3)

• emergence of (collective) Goldstone modes(a) superfluid mode, with ω ∝ |k|(b) spin waves, with ω ∝ |k|2

The Revolution(s) of the 1980ies

3 essential discoveries:

• integer quantum Hall effect (1980, v. Klitzing, Dorda,Pepper)

• fractional quantum Hall effect (1982, Tsui, Störmer,Gossard)

• high-temperature superconductivity (1986, Bednorz, Müller)

Integer Quantum Hall Effect (I)

8 12 160 4Magnetic Field B (T)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

ρ xy

(h/e

)2

0

0.5

1.0

1.5

2.0

ρΩ

xx(k

)

2/3 3/5

5/9

6/11

7/15

2/53/74/9

5/11

6/13

7/13

8/15

1 2/3 2/

5/7

4/5

3 4/

Vx

VyIx

4/7

5/34/3

8/57/5

123456

Magnetic Field B[T]

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

QHE = plateau in Hall res. & vanishing long. res.

Integer Quantum Hall Effect (II)

Quantised Hall resistance at low temperatures

RH =h

e21

n

h/e2: universal constantn: quantum number (topological invariant)

• result independent of geometric and microscopic details

• quantisation of high precision (> 109)

⇒ resistance standard: RK−90 = 25 812, 807Ω

Fractional Quantum Hall Effect

partially filled Landau level → Coulomb interactions relevant

1983: Laughlin’s N -particle wave function

• no (local) order parameter associated with symmetrybreaking

• no Goldstone modes• quasi-particles with fractional charges and statistics

1990ies : description in terms of topological (Chern-Simons)field theories

The Physics of the New Millenium

• simulation of condensed-matter models with optical lattices(cold atoms)

• 2004 : physics of graphene (2D graphite)

• 2005-07 : topological insulators

Graphene – First 2D Crystal

• honeycomb lattice =two triangular (Barvais) lattices

AB

B

B

e3

e1

2e

band structure

Band Structure and Conduction Properties (Bis)

I II I II

gap

metal (2D)

energy

Fermi

level

momentum

electron metal hole metal

Fermi

levelinsulator (2D)

semimetal (2D) Fermi

level

graphene (undoped)Fermi

level

Fermilevel

energy

density

of states

Topological Insulators

generic form of a two-band Hamiltonian:

H = ǫ0(q)1+∑

j=x,y,z

ǫj(q)σj

• Haldane (1988): anomalous quantum Hall effect → quantumspin Hall effect (QSHE)

• Kane and Mele (2005): graphene with spin-orbit coupling• Bernevig, Hughes, Zhang (2006): prediction of a QSHE in

HgTe/CdTe quantum wells• König et al. (2007): experimental verification of the QSHE

⇒ 3D topological insulators (mostly based on bismuth):surface states ∼ ultra-relativistic massless electrons

Outline of the Classes

Mon : Introduction and Landau quantisation

Tue : Issues of the IQHE; Introduction to the Berry phase

Thu : Simple models for topological insulators

Further Reading

• D. Yoshioka, The Quantum Hall Effect, Springer, Berlin (2002).

• S. M. Girvin, The Quantum Hall Effect: Novel Excitations and BrokenSymmetries, Les Houches Summer School 1998http://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, Quantum Hall Effectshttp://arxiv.org/abs/0909.1998

• B. A. Bernevig, Topological Insulators and TopologicalSuperconductors, Princeton UP (2013).

1. Introduction To the Integer Quantum Hall

Effect and Materials

Classical Hall Effect (1879)

B

I

longitudinal Hallresistance resistance

C1

C4

C2 C3

C5C6

2D electron gas_ _ _ _ _ _

++ + + ++

Quantum Hall system :2D electrons in a B-field

Hal

l res

ista

nce

magnetic field B

RH(b)

Hall resistance:

RH = B/enel

Drude model (classical stationary equation):

dp

dt= −e

(

E+p

m×B

)

− p

τ= 0

Resistivity and Conductivity

σ0Ex = −enel

mpx −

enel

mpy(ωCτ)

σ0Ey =enel

mpx(ωCτ)−

enel

mpy

Ohm’s law : E = ρj with current density j = −enelv = −enelp/m

⇒ resistivity/conductivity tensor

ρ =1

σ0

(

1 ωCτ−ωCτ 1

)

σ = ρ−1 =σ0

1 + (ωCτ)2

(

1 −ωCτωCτ 1

)

Link with mobility µ = eτ/m: ωCτ = µB

Hall resistivity : ρH = ωCτ/σ0 = B/enel

Shubnikov-de Haas Effect (1930)H

all r

esis

tanc

e

magnetic field B

long

itudi

nal r

esis

tanc

e

Bc

(a)

Den

sity

of s

tate

s

EnergyEF

hωC

(b)

oscillations in longitudinal resistance→ Einstein relations σ0 ∝ ∂nel/∂µ ∝ ρ(ǫF )→ Landau quantisation (into levels ǫn)

σ0 ∝ ρ(ǫF ) ∝∑

n

f(ǫF − ǫn)

Quantum Hall Effect (QHE)

8 12 160 4Magnetic Field B (T)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

ρ xy

(h/e

)2

0

0.5

1.0

1.5

2.0ρ

Ωxx

(k)

2/3 3/5

5/9

6/11

7/15

2/53/74/9

5/11

6/13

7/13

8/15

1 2/3 2/

5/7

4/5

3 4/

Vx

VyIx

4/7

5/34/3

8/57/5

123456

Magnetic Field B[T]

QHE = plateau in RH & RL = 0

1980 : Integer quantum Hall effect (IQHE)1982 : Fractional quantum Hall effect (FQHE)

Metal-Oxide Field-Effect Transistor (MOSFET)

conductionband

acceptorlevels

valenceband

conductionband

acceptorlevels

valenceband

conductionband

acceptorlevels

valenceband

E

z

FE

z

F

E

z

F

(a)

(b) (c)

VVG

G

metal oxide(insulator)

semiconductor

metal oxide(insulator)

semiconductor metal oxide(insulator)

II

I

VG

z

z

E

E

E

1

0

metaloxide

semiconductor

2D electrons

usually silicon-based materials (Si/SiO2 interfaces)

GaAs/AlGaAs Heterostructure

dopants

AlGaAs

z

EF

GaAs

dopants

AlGaAs

z

EF

GaAs(a) (b)

2D electrons

Impurity levels farther away from 2DEG (as compared toSi/SiO2)

⇒ enhanced mobility (FQHE)

Electronic Mesurement of Graphene

SiO

Si dopé

V

2

g

Novoselov et al., Science 306,p. 666 (2004)

2. Landau Quantisation and Integer

Quantum Hall Effect

Landau quantisation (reminder of first class)

• 2D electrons in continuum limit (|q|a≪ 1)

H(px, py) p = ~q

• Peierls substitution : p → Π = p+ eA(r) for a/lB ≪ 1

H(px, py) → HB(Πx,Πy)

• Quantum mechanics

[x, px] = [y, py] = i~ → [Πx,Πy] = −i~2

l2B

⇒ Ladder operators [a, a†] = 1

Πx =~√2lB

(

a† + a)

Πy =~√2ilB

(

a† − a)

Simple Landau levels

Schrödinger fermions :

HB|n〉 = ǫn|n〉, a†a|n〉 = n|n〉, ǫn = ~ωC(n+ 1/2)

Dirac fermions (graphene) :

HB =√2~vFlB

(

0 aa† 0

)

, ψn =

(

unvn

)

, ǫλ=±,n = λ~vFlB

√2n

Simple Landau levels

Schrödinger fermions :

HB|n〉 = ǫn|n〉, a†a|n〉 = n|n〉, ǫn = ~ωC(n+ 1/2)

Dirac fermions (graphene) :

HB =√2~vFlB

(

0 aa† 0

)

, ψn =

(

unvn

)

, ǫλ=±,n = λ~vFlB

√2n

Eigenstates :

ψn=0 =

(

0|n = 0〉

)

ψλ,n= =1√2

(

|n− 1〉λ|n〉

)

Infrared Transmission Spectroscopy

10 20 30 40 50 60 70 80

0.96

0.98

1.00

B

E

2L3L

2L

3L

0L

1L

Be2cE1 ~1L

1E

1E

A

B

C

D

B

E

2L3L

2L

3L

0L

1L

Be2cE1 ~1L

1E

1E

A

B

C

D

(D)(C)

(B)

Rel

ativ

e tra

nsm

issi

on

Energy (meV)

(A)

0.4 T1.9 K

0.0 0.5 1.0 1.5 2.00

10

20

30

40

50

60

70

80 )(32 DLL )(23 DLL

)(12 CLL )(21 CLL

)(01 BLL )(10 BLL

)(21 ALL

Tran

sitio

n en

ergy

(meV

)

sqrt(B)

10 20 30 40 50 60 70 80 900.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

Rel

ativ

e tra

nsm

issi

on

Energy (meV)

1 T

0.4T

2T4T

10 20 30 40 50 60 70 80 90

0.99

1.00

0.7T

0.2T

0.3T

0.5T

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

transition C

transition B

rela

tive

tran

smis

sion

rela

tive

tran

smis

sion

Energy [meV]

Energy [meV]

Tra

nsm

issi

on e

nerg

y [m

eV]

Sqrt[B]

selectionrules :

λ, n→ λ′, n±1

Edge States

ymaxn+1

ν = n ν =

n−1

yymaxymaxn n−1

n+1

n

n−1

(a)

(b)

y

xν = n+1

µ

LLs bended upwards atthe edges (confinementpotential)

chiral edge states⇒ only forward scattering

ν= n+1 ν= n ν= n−1

Four-terminal Resistance Measurement

I I

R ~

56

2 3

41

R ~ µ − µ = µ − µ

3µ − µ = 02

5

L

H

µ = µµ = µ2 LL3

µ = µ = µ6 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+1)

ν

NL

(a)

density of states

RxyxxR

B=n

h/e n2

FE

IQHE – One-Particle Localisation

n

ε

n

ε(b)

(n+1)

ν

NL

(a)

density of statesdensity of states

RxyxxR

B

EF

RxyxxR

B=n

h/e n2

FE

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

RxyxxR

B

EF

Rxy

B

xx

EF

R

h/e (n+1)

h/e n2

2

RxyxxR

B=n

h/e n2

FE

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 ν2

(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 spinbranch 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

Towards topological band theory

• Berry-ology :

– Berry connexion : ~An(k) = iψ†n(k)∇kψn(k)

– Berry curvature : ~Bn(k) = ∇k × ~An(k)

– Berry phase : γn =∮

dk · ~An(k)

→ Chern number : Cn = 1

2π

∮

BZdk · ~An(k)

Towards topological band theory

• Pseudo-Chern number of a (massive) Dirac point :

C =1

2ξsgn(m)

Towards topological band theory

• Pseudo-Chern number of a (massive) Dirac point :

C =1

2ξsgn(m)

• Remarks :– half integer (not a true toplogical invariant) due to

non-compact support k ∈ R2

– each Dirac point contributes ±1/2 to total Chern number→ Dirac points (on a lattice) come in pairs to get integer

Chern numbers ! (fermion doubling)– Haldane model (CK = CK′) : C = CK + CK′ = ±1– Kane-Mele model : C↑ = −C↓

Non-local transport in QSHE (I)

CdTe/HgTe quantum wells [Roth et al., Science 2009]

I1

2 3

4

56

V

I1

2 3

4

56

V

-0.5 0.0 0.5 1.0 1.5 2.00

5

10

15

20

25

30

35

40

R (

kΩ)

V* (V)

(1 x 0.5) µm2

(2 x 1) µm2

R14,14=3/2 h/e2

R14,23=1/2 h/e2

Non-local transport in QSHE l’EHQS (II)

CdTe/HgTe quantum wells [Roth et al., Science 2009]

Fig. 4

0.0 0.5 1.0 1.5 2.00

5

10

15

20

25

R (

kΩ)

V* (V)

I: 1-4

V: 2-3

1

3

2

4

R14,23=1/4 h/e2

R14,14=3/4 h/e2

3D topological insulators (I)

Frist generation based on Bi1−xSbx alloys [Hasan and Kane, RMP 2010]

→ band inversion above critical Sb concentration xc ≃ 0.04

3D topological insulators (II)

• closing of (∼ mass) gap at band inversion

⇒ Dirac fermions at surface of 3D toplogical insulator (∼ edgestates in 2D) :

Hsurface = vp · σ

p : momentum in surfaceσ : characterises true spin

⇒ single Dirac point (contrary to graphene with 4)

3D topological insulators (III)

2nd generation : Bi2Se3, Bi2Te2, Sb2Te3 [Zhang et al., 2009]

(ab initio calculations)

3D topological insulators (IV)

ARPES measurements of de Dirac fermions at a Bi2Se3 surface[Hsieh et al., 2009]

→ change of Fermi level by chemical doping (absorption of NO2)