Topological Phases under Strong Magnetic Fields

Mark O. Goerbig

ITAP, Turunc, July 2013

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

Chrystal Electrons and Bloch’s Theorem

• discrete translations T = exp(ip ·Rj)represent a symmetry in a Bravais lattice(described by lattice vectors Rj)

• the operator p (generator ofdiscrete translations) plays therole of a momentum(quasi-momentum or latticemoment)

R j

Chrystal Electrons and Bloch’s Theorem

• discrete translations T = exp(ip ·Rj)represent a symmetry in a Bravais lattice(described by lattice vectors Rj)

• the eigenvalues of p are goodquantum numbers: energy bandsǫl(p)

• Bloch functions:

ψ(r) =∑

p

eip·r/~up(r)

Bravais and Arbitrary Lattices

arbitrary lattice=

Bravais lattice+

basis (of N “atoms”)

M. C. Escher (decomposed)

triangular lattice + complicated basis

There are as many electronic bands as atoms per unit cell

Band Structure and Conduction Properties

I II I II

gap

metal (2D)

energy

Fermi

level

momentum

electron metal hole metal

insulator (2D)Fermi

level

semimetal (2D) Fermi

level

Fermilevel

density

of states

energy

1950-70: Many-Body Theory

• Physical System described by an 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 1: Introduction and Landau quantisation

Mon 2: Landau-level degeneracy and disorder/confinementpotential

Tue 1: Issues of the IQHE

Tue 2: Towards the FQHE, Laughlin’s wave function and itsproperties

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

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

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)

Nobel Prize in Physics 2010 : Graphene

Kostya Novoselov Andre Geim

"for groundbreaking experiments regarding the two-dimensionalmaterial graphene"

What is Graphene?

2s

2p 2p 2p 2px y z zsp spsp

Hybridation sp 2

120o

graphene = 2D carbon crystal(honeycomb)

sp hybridisation2

Graphene and its Family

2D

3D 1D 0D

From Graphite to Graphene

(strong) covalent bondsin the planes

(weak) van der Waalsbonds between the planes

How to Make Graphene: Recipe (1)

put thin graphite chip on scotch−tape

How to Make Graphene: Recipe (2)

~10 : fold scotch−tape on graphite chip and undo :

How to Make Graphene: Recipe (3)

2glue (dirty) scotch−tape on substrate (SiO )

How to Make Graphene: Recipe (4)

lift carefully scotch−tape from substrate

How to Make Graphene: Recipe (5)

place substrate under optical microscope

orientationmarks

thick graphite

graphene ?

less thickgraphite

How to Make Graphene: Recipe (6)

zoom in region where there could be graphene

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

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

Fractional Quantum Hall Effect beyond Laughlin

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

FQHE series: ν = p/(2sp+ 1) = 1/3, 2/5, 3/7, 4/9, ...

Jain’s Wavefunctions (1989)

Idea: “reinterpretation” of Laughlin’s wavefunction

ψLs (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

ψLs (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

ψJs,p(zj) = PLLL

∏

i<j(zi − zj)2sχp(zj, zj)

χp(zj, zj): wavefunction for p completely filled levelsPLLL: 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 1/3theory

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 (evendenominator)

• 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 SystemsLa

ndau

leve

ls

|+>

|−>d

ν = 1/2

ν = 1/2 ν = ν + ν = 1

+

+ −− T

: A sublattice : B sublattice

τ

τ

2

3

1

2

e 1e

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