(microsoft powerpoint - graphene and quantum hall - hjlee [\310

KIAS-SNU Winter School,

Feb. 29, 2011; 휘닉스 파크

Observation of Quantum Hall Effect in Graphene

Pohang University of Science and Technology

(POSTECH)

Quantum Transport and Superconductivity Laboratory

이후종

Introduction; Integer Quantum-Hall Effect

- in 2DEG, monolayer graphene, bilayer graphene

- edge conducting states

Observed QHE in Graphene

Overview

Observed QHE in Graphene

- Half-integer quantum-Hall effect in graphene

- Edge state equilibrium

- Lifting the degeneracy in high magnetic fields

- Fractional quantum Hall effect in high-mobility graphene in high fields

Summary

Carbon Allotropes : in Diverse Dimensions

Two dimension Three dimension

One dimensionZero dimension

Patterning

Functionality

1a

2a

Graphene

2.46 A°

‘A’ sublattice

‘B’ sublattice

graphene lattice SP2 covalent hybrid

orbital of a carbon atom

π

σ

2.46 A

(real space)

Two equivalent sublattices

Two atoms per unit cell

π-orbital

σ-bond

Band Structure and Low-Energy Dispersion

FE kυ=r

h

KM

Γ

K’ EF

• Dirac cone

• Linear dispersion at zero energy

- Massless relativistic Dirac fermions

- But moving with Fermi velocity

- Carrier type and density are easily controlled by gatingFυ

Chirality or Pseudospin in Graphene Lattice

KK’ kx

ky

kx

EE

electron-like

pseudospin or chirality

Chirality – momentum-locked phase value of a carrier in graphene when the carrier

moves along a Dirac cone (or the sublattice index)

hole-like

Carrier Mobility

VBGSiO2 300nm

Si

negraphene

E

υµ =

j E neσ υ= =

}

I11-2-V10-9

I = 20 nA, L=W=7 µm

22 22

2

2

@ =0,

= ( )2

F FF

F mfp

T

e Ev e v ne g E

ve n

τ τσ τ

π π

σ πτ

= =

= =

h h

hl

Boltzmann theory;

natural graphite

Graphene Preparation – Mechanical Exfoliation

exfoliated graphene on Si sub

1

32

1

10 µm

thin graphite on tape transferring graphene onto Si substrate

Graphene Preparation – Mechanical Exfoliation

10 µm 10 µm

10 µm

10 µm

Introduction; Integer Quantum-Hall Effect

- in 2DEG, monolayer graphene, bilayer graphene

- edge conducting states

Observed QHE in Graphene

Overview

Observed QHE in Graphene

- Half-integer quantum-Hall effect in graphene

- Edge state equilibrium

- Lifting the degeneracy in high magnetic fields

- Fractional quantum Hall effect in high-mobility graphene in high fields

Summary

Classical Hall Effect

1H

H

ne

R Bσ = = − σH is linearly dependent on n with a slope of e/B

Picture by Dr. Dong Su Kim

(Integer) Quantum-Hall Effect

- LLs - 2D carriers in high-enough H field and low T

HH=0 Fermi edge

VSD

S D

(Integer) Quantum-Hall Effect

VSD

VL

von Klitzing, 8.., PRL, 1980

S D

VH

; 2DEG

Halperin, PRB 25, 2185 (1982)

- Dissipationless and chiral edge state carries the current at QH plateaus

- Arising as Landau levels are pushed up by the confining edge potential

von Klitzing, Dorda, and Pepper,

PRL 45, 494 (1980)

Quantum-Hall Edge States

EF

ν=1

BeB

=h

l

Halperin, PRB 25, 2185 (1982)

von Klitzing, Dorda, and Pepper,

PRL 45, 494 (1980)

Quantum-Hall Edge States

ν=2

EF

Halperin, PRB 25, 2185 (1982)

von Klitzing, Dorda, and Pepper,

PRL 45, 494 (1980)

Quantum-Hall Edge States

ν=3

EF

Integer QHE in 2D Electron Gas

6

2-folddegeneracy

1( )

2n cE nω= +h

1 22− 1− 0

0n= 11−2−

nEstrong H

S D

VS

VH

2

xy

e

hσ ν= 0, 2, 4, ...ν = ± ±

0-2-4 2 4

filling factor ν (=eB/h)

2DEG

2

4

-2

-4

0

6

-6

Unconventional QHE in Monolayer Graphene

2DEG4

610 Monolayer

1 22− 1− 0

2-folddegeneracy

1( )

2n cE nω= +h

0n = 11−2−

nE

Landau level

4-folddegeneracy

11− 02− nE

0n = 11−2− 2

2

2n FE e B nυ= ± h

Integer QHE

2 0, 2, 4, ...nν = = ± ±

0-2-4 2 4

filling factor ν (=eB/h)

2DEG

2

4

-2

-4

0

-6

Half integer QHE

4 degenerate zero mode

14( ) 2, 6, 10, ...

2nν = + = ± ± ±

2

6

10

-2

-6

-10

0-4-8 4 8

filling factor ν (=eB/h)

Monolayer

42

xy

e

hσ ν=

Due to the special status of the ν=0 Landau level : half of its

states are hole states, and the other half are electron states.

Unconventional QHE in Bilayer Graphene

1 22− 1− 0

2-folddegeneracy

1( )

2n cE nω= +h

0n = 11−2−

nE

4-folddegeneracy

8-folddegeneracy

( 1)n cE n nω= ± −h

1 22− 1− 0 nE

0,1n =

21−2− 3

2DEG4

610 Monolayer

12Bilayer

Landau level

4-folddegeneracy

11− 02− nE

0n = 11−2− 2

2

DOS

2n FE e B nυ= ± h

2

xy

e

hσ ν=

Integer QHE

0-2-4 2 4

filling factor ν (=eB/h)

2DEG

2

4

-2

-4

0

-6

2 0, 2, 4, ...nν = = ± ±

Half integer QHE

4 degenerate zero mode

-10

2

6

10

-2

-6

0-4-8 4 8

filling factor ν (=eB/h)

Monolayer

2+2

14( ) 2, 6, 10, ...

2nν = + = ± ± ±

Integer QHE

8 degenerate zero mode

4 4, 8, 12, ...nν = = ± ± ±

4

8

-4

-8

-12

0

0-4-8 4 8

filling factor ν (=eB/h)

4+2

Bilayer

0n =except for

Introduction; Integer Quantum-Hall Effect

- in 2DEG, monolayer graphene, bilayer graphene

- edge conducting states

Observed QHE in Graphene

Overview

Observed QHE in Graphene

- Half-integer quantum-Hall effect in graphene

- Edge state equilibrium

- Lifting the degeneracy in high magnetic fields

- Fractional quantum Hall effect in high-mobility graphene in high fields

Summary

Half-integer QHE in Monolayer Graphene

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

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

610 14

-2

-6-10

(K) 420 (T)LLE H∆ =

Room Temperature QHE

0n= 11−2− 2

LLE∆

Novoselov et al., Science 315, 1379 (2007)

2n FE e B nυ= ± h

11− 02−nE2

xp

yp

E

Massive Dirac Fermions

Unconventional QHE in Bilayer Graphene

4-folddegeneracy

8-folddegeneracy

1 22− 1− 0 nE

0,1n=

21−2− 3

DOS

Introduction; Integer Quantum-Hall Effect

- in 2DEG, monolayer graphene, bilayer graphene

- edge conducting states

Observed QHE in Graphene

Overview

Observed QHE in Graphene

- Half-integer quantum-Hall effect in graphene

- Edge state equilibrium

- Lifting the degeneracy in high magnetic fields

- Fractional quantum Hall effect in high-mobility graphene in high fields

Summary

Quantum-Hall Conduction in Bi-polar Junction

H

VBG(V)

VLG (V)

(Two-terminal Studies)

Iin

Iout

H

[ Abanin & Levitov, Science 317, 641 (2007) ]

[ Williams, DiCarlo, Marcus,

Science 317, 638 (2007) ]

ν2

ν1

- For the incident current , only a fraction of transmits.

Quantum-Hall Conduction in p-n-p Junction

(Two-terminal Studies)

[ Ozyilmaz et al,

PRL 99, 166804 (2007) ]

..

11 2

Our Graphene p-n-p Junction (4-terminal Studies)

EF

top gate

S

D

1

2

1

half-integer QHE

EF

H= 0EF

EF

Ki and Lee, PRB 79, 195327 (2009)

half-integer QHE

Longitudinal QH Resistance

H = 10 T

VL

PRL 99, 166804 (2007)

VLG(V)- Consistent with two-terminal results

- Inversion symmetry

- Zero RL ; full transmission of edge states when ν1=ν2- Fractionally quantized RL ; partial transmission of edge states for ν1=ν2

reflection of a certain portion of edge states

Ki and Lee, PRB 79, 195327 (2009)

Diagonal QH Resistances

VD

RD ;

- No inversion symmetry w.r.t. (ν1=0, ν2=0), but inversion symmetry between the two

VLG(V)

Introduction; Integer Quantum-Hall Effect

- in 2DEG, monolayer graphene, bilayer graphene

- edge conducting states

Observed QHE in Graphene

Overview

Observed QHE in Graphene

- Half-integer quantum-Hall effect in graphene

- Edge state equilibrium

- Lifting the degeneracy in high magnetic fields

- Fractional quantum Hall effect in high-mobility graphene in high fields

Summary

K

E

x

- Countercirculating edge states

- Spin is 100% polarized in each edge state

ν=+1

ν= -1ν= 0

ν=+2ν=+4∆E

∆E

∆E

∆En=0LL

n=+1LL

Level Splitting in High Fields

n=0LL

[기동근, 이후종, 물리학과 첨단기술July/August 2009]

magnetic-field-induced

spontaneous symmetry

breaking mediated by e–e

interactions

K’

x

bulk edge

QH ferromagnet

spin valley∆ > ∆

K

K’

E

x

bulk edge

w/o any symmetry

breaking

Real-spin symmetry

broken first

H

ν= -1

n=-1LL

1ν = valley splits

LL

K

K’

E

x

bulk edge

K

E

x

w/o any symmetry

breaking

Level Splitting in High Fields

n=0LL

ν=0

QH inuslator

- No edge state itself,

- Insulating - longitudinal resistance is thermally activated

K’

x

bulk edge spin valley∆ < ∆

0ν = valley splits

Pseudospin symmetry

broken first

LL

QH inuslator for E=0

Level Splitting in High Fields

K

E

K

K’

E

x

bulk edge

spin valley∆ < ∆(a)

(b)

magnetic-field-induced

spontaneous symmetry

breaking mediated by e–e

interactions

[기동근, 이후종, 물리학과 첨단기술July/August 2009]

Pseudospin symmetry broken first

QH ferromagnet for E=0

K’

x

bulk edge

K

K’

E

x

bulk edge

bulk edge

spin valley∆ > ∆

(c)

Incresing H field

w/o symmetry breaking

Real-spin symmetry broken first

Metallic Character of ν=0 State

H=30 T

For high HE

Abanin et al., PRL 98, 196806 (2007)

countercirculating edge

states with opposite spin

K

K’

x

bulk edge

Insulating Character of ν=0 State

2 2

xy

xy

xy xx

ρσ

ρ ρ=

+

K

E

Checkelsky, Li, and Ong, PRL 100, 206801 (2008)

20

-2

-6

ν=6

ν=0

ν=-2

2

K

K’

x

bulk edge

Level Splitting in High Fields

B=45 T

1

2

-1

-2

Y. Zhang et al., Phys. Rev. Lett. 96, 136806 (2006)

4-4

9 T 30 mK

25 T 1.4 K

30 T 1.4 K

37 T 1.4 K

42 T 1.4 K

45 T 1.4 K

25 T 1.4 K

1

2

4

6

-1-2

-4

0

Level Splitting in High Fields

K

K’

E

x

bulk edge

- ν=0 QH plateau ; resolved at B>11 T

- Many-body electron correlation within the LL - an alternative origin for the lifting

of the degeneracy at the Dirac point.

- Weaker features of ν= 3 a hierarchy exists in lifting the degeneracy of LLs

strength of e–e interactions

2

, / , 1B

B

eeB ε

ε∝ = ≈l h

l

±

9 T 30 mK

11.5 T 30 mK

17.5 T 30 mK

-4

-60

Introduction; Integer Quantum-Hall Effect

- in 2DEG, monolayer graphene, bilayer graphene

- edge conducting states

Observed QHE in Graphene

Overview

Observed QHE in Graphene

- Half-integer quantum-Hall effect in graphene

- Edge state equilibrium

- Lifting the degeneracy in high magnetic fields

- Fractional quantum Hall effect in high-mobility graphene in high

fields

Summary

Enhancing Carrier Mobility - Suspending Graphene

- Graphene exfoliation and e-beam patterning electrodes

- Etching away SiO2 in HF solution (DI water:HF = 6:1)

- Immersing in DI water and IPA, and critical-point drying

- Ar/H2 annealing or current annealing to remove water or organic residue

gas

liquid supercritical fluid

Du, G., Andrei,

Nature Nanotechnology 3, 491 (2008)

Enhancing Mobility – Exfoliation on BN Substrates

Dean et al., Nature Nanotechnology 5, 722 (2010)

Substrate-supported geometry while retaining the quality of suspended graphene

- smooth surface, relatively free of dangling bonds and charge traps

- better match of the lattice constant (1.7% mismatch)

- large electrical band gap (~6 eV)

Mechanical Transfer

h-

Fractional QHE in High-mobility Graphene

- QH plateaus at ν=0, 1, 4 for H>25 T as interaction effects lifting the degeneracy

new integer plateaus outside the usual sequence

- Observable only when the energy scale > energy fluctuations induced by external sources

- High quality (ballistic transport and low carrier density) graphene is required to clarify ;

Role of correlations in the low density phases

Whether graphene can support an FQHE

± ±

X. Du, GG E. Y. Andrei, Nature 462, 192, (2009)

X. Du, GG E. Y. Andrei, Nature 462, 192, (2009)

Fractional QHE in High-mobility Graphene

T=1.2 K

FQHE;

- Strongly correlated fractional QH liquid in high H field minimize its energy for the filling factors

(with m and p integers)

- Electrons and magnetic flux quanta bind to form complex composite state with fractionally

charged quasiparticles for elementary excitations

- ν=1/3 plateau ; corresponds to composite particles of one electron and two flux lines, with

fractionally charged quasiparticles of q*=e/3 and an excitation gap, ∆1/3

- Insulating state at | ν |=0.1

2 1

m

pmν =

±

In low magnetic field

Fractional QHE in High-mobility Graphene

K. Bolotin,GPhilip Kim, Nature 462, 196 (2009)

10

6

ν=2

10

- QH plateaus even at 0.3 T2n FE e B nυ= ± h

ν=0.3

Magnetotransport in high magnetic fields

Fractional QHE in High-mobility Graphene

ν=1

ν=2

ν=-1

ν=-2

0.46v =0.68v =0.32v =

K. Bolotin,GPhilip Kim, Nature 462, 196 (2009)

− ν=0, 1 plateaus appear as e–e interactions among Dirac quasiparticles lift the pseudospin

and spin degeneracy of the zeroth Landau level

− ν=1/3 plateau persists up to 10 K for H=6 T, much robust than in 2DEG due to strong e-e

interaction for small ε (=1)

±

onset of an insulating

state at low density

ν=0

- B; not a FQH plateau

- Arising from two-term meas.

Insulating State in Graphene near Zero Density

n=0.17x1011 cm-2

K. Bolotin,GPhilip Kim, Nature 462, 196 (2009)

Insulating state near n=0 stems from the symmetry

breaking of the zeroth Landau level by e–e interactions.

µ∼30,000 cm2/Vs

Fractional QHE in Graphene – 4 Probe Meas.

- A complete lifting of the four-fold degeneracy in n = 0, 1 Landau levels

n=0

Fractional QHE in Graphene – 4 Probe Meas.

n=1n ν

1

2

Strongly interacting

electrons in a high H

2 1

m

pmν =

±

A system of weakly interacting CFs

consisting of an electron bound to

2p magnetic flux vortices

FQHE State

mapping

(i) all degeneracies are explicitly broken,

for example, by coupling to external fields;

(ii) only one of spin or valley isospin degeneracy

is broken, preserving an SU(2) symmetry in the

ν

Rxx

0 1-1

is broken, preserving an SU(2) symmetry in the

remaining degenerate space;

(iii) the full degeneracy is preserved, leading to

an emergent SU(4) symmetry in the combined

spin-isospin space

ν0 1-1

ν0 1-1

Introduction; Integer Quantum-Hall Effect

- in 2DEG, monolayer graphene, bilayer graphene

- edge conducting states

Observed QHE in Graphene

- Half-integer quantum-Hall effect in graphene

Overview

- Half-integer quantum-Hall effect in graphene

- Edge state equilibrium

- Lifting the degeneracy in high magnetic fields

- Fractional quantum Hall effect in high-mobility graphene in high fields

Summary

FQHE - suggesting the possibility of observing novel spin textures with no

analog in other single-layer quantum Hall systems