Electronic properties and the quantum Hall effect in

bilayer graphene

Vladimir Falko

Geim’s group at ManchesterNovoselov et al - Nature 438, 197 (2005)Novoselov et al - Nature Physics 2, 177 (2006)

Kim-Stormer group at Columbia University NYZhang et al - PRL 94, 176803 (2005)Zhang et al - Nature 438, 201 (2005)

Morpurgo’s group at TU-Delft S-Graphene-S Josephson effect transistor Conference ‘Graphene Week’, MPI-PKS Dresden (2006)

Ultra-thin graphitic films: from flakes to micro-devices

Novoselov et al - Science 306, 666 (2004)

Monolayer and bilayer graphene

Berry phase, degeneracy of the zero-energy Landau level, and the QHEMcCann, VF - PRL 96, 086805 (2006); Abergel, VF - PR. B 75, 155430 (2007)

Novoselov, McCann, Morozov, VF, Katsnelson, Zeitler, Jiang, Schedin, Geim - Nature Physics 2, 177 (2006)

Relevance of Fermi surface warping and symmetry-breaking defects for weak localisation and WL magnetoresistance in graphene:

monolayer McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler - PRL 97, 146805 (2006)

bilayer Kechedzhi, McCann, VF, Altshuler – PRL 98, 176806 (2007)

NP junctions: focusing, caustics and Veselago lens for electronsCheianov, VF - PR B 74, 041403 (2006)

Cheianov, VF, Altshuler - Science 315, 1252 (2007)

Random resistor network model for the minimal conductivity of graphene with inhomogeneous charge density

Cheianov, VF, Altshuler, Aleiner (2007)

Specifics of Friedel oscillations in monolayer graphene Cheianov, VF – PRL 97, 226801 (2006)

Berry phase, degeneracy of the zero-energy Landau level, and the QHE in bilayer graphene

McCann, VF - PRL 96, 086805 (2006); Abergel, VF - PR. B 75, 155430 (2007)Novoselov, McCann, Morozov, VF, Katsnelson, Zeitler, Jiang, Schedin, Geim - Nature Physics 2, 177 (2006)

• Tight-binding-model analysis leading to ‘chiral’ electrons characterised by the Berry phase Jπ.

• Landau levels and quantum Hall effect in bilayer graphene.

• Trigonal warping in bilayer graphene.

• FIR magneto-optical properties of bilayer graphene.

- bonds

hybridisation forms strong directed bonds which determine a honeycomb lattice structure.

2sp

C

Carbon has 4 electrons in the outer s-p shell

)(zp orbitals determine conduction properties of graphite

0

Wallace, Phys. Rev. 71, 622 (1947)Slonczewski, Weiss, Phys. Rev. 109, 272 (1958)

0ie 3/2ie

3/2ie

11 p

ARPES: heavily doped graphene synthesized on silicon carbideA. Bostwick et al – Nature Physics, 3, 36 (2007)

Bilayer [Bernal (AB) stacking]

Bilayer [Bernal (AB) stacking]

Closest neighbour intra-layer hops

yx ipp

Bilayer [Bernal (AB) stacking]

Closest neighbour approximation (questions about the effect of the next-neighbour hops are welcome!)

ARPES: heavily doped bilayer graphene synthesized on silicon carbideT. Ohta et al – Science 313, 951 (2006)(Rotenberg’s group at Berkeley NL)

McCann, VFPRL 96, 086805

(2006)

Fermi level in undoped bilayer graphene

eV4.01

yx ipp

McCann, VFPRL 96, 086805

(2006)

emm 05.0~

yx ipp

)( pn

Berry phase Jπ(for a monolayer π

for a bilayer 2π )

iJi

Jee 32

2

Degree of chirality J

Monolayer grapheneBilayer graphene

Tight-binding-model analysis:

‘chiral’ electrons and the Berry phase Jπ.

Landau levels and quantum Hall effect in bilayer and monolayer graphene.

Effect of trigonal warping

Infra-red and FIR magneto-optics in graphene.

2D Landau levels

semiconductor QW / heterostructure

(GaAs/AlGaAs)

xy ( ) )2

h

ge

-1

-2 -1-3

2 31

-3

-2

3

1

2

integer QHEin semiconductors

eB

ghn

a

cnmm

pH

)(42 2

12

yxyx

zce

ippipp

lBArotAip

;

,

nnnB

11

0 0

energies / wave functions

)(0 r

1

2

yxyx

zce

ippipp

lBArotAip

;

,

Landau levels and QHE

0... 10 JJJ

00

,...,0

10

J

2D Landau levels of chiral electrons

J=1 monolayerJ=2 bilayer

A

B

B

A

J

J

J

J

~

~

0

0

0

0

valleyindex

also, two-fold real spin degeneracy

4J-degenerate zero-energy Landau level

McClure, Phys. Rev. 104, 666 (1956)

Haldane, Phys.Rev.Lett. 61, 2015 (1988)

Zheng and Ando Phys. Rev. B 65, 245420 (2002)

McCann and VFPhys. Rev. Lett. 96, 086805 (2006)

4J-degenerate zero-energy Landau level for electrons with degree of chirality J

emm 05.0~

-2-4 40

n (1012 cm-2)

2

2

4

6

0

xx

(k

)

-2-4 40

n (1012 cm-2)

2

2

4

6

0

xx

(k

)

1L graphene 2L graphene

db

EE

pp

xy

(4e2

/h)

1

2

-1

-2

-4

0

-3

4 c

3 x

y (4e2

/h)

1

2

-1

-2

-4

0

-3

4 a3

Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene

K.Novoselov, E.McCann, S.Morozov, V.Fal’ko, M.Katsnelson, U.Zeitler, D.Jiang, F.Schedin, A.Geim Nature Physics 2, 177 (2006)

1

How robust is the degeneracy of Landau level in bilayer graphene?

010

3

Direct inter-layer A hops (warping term, Lifshitz trans.)

B~

10 McCann, VF - PRL 96, 086805 (2006)

Monolayer grapheneBilayer graphene

Tight-binding-model analysis:

‘chiral’ electrons and the Berry phase Jπ.

Landau levels and quantum Hall effect in bilayer and monolayer graphene.

Effect of trigonal warping

Infra-red and FIR magneto-optics in graphene.

1

Hops between A and via B~

BA~

Direct inter-layer hops between A and ,~B 1.0~3

v

v

3

Inter-layer asymmetry (electric field across the structure, effect of a substrate/overlayer)

‘trigonal warping’ term

31 ~ mvpB

strong magnetic field

31 ~ mvpB

weak magnetic field

212104~ cm

KK

211

4

210~2 22

13 cmNvv

vL

21110~ cmNN L

*8NNNL

Lifshitz transition

iyx peipp

8-fold degeneratezero-energy Landau level

1

How robust is the degeneracy of Landau level in bilayer graphene?

010

3

Direct inter-layer A hops (warping term, Lifshitz trans.)

B~

10

Distant intra-layer AA,BB hops )3(210~~ 2

0

41

c || 01

McCann, VF - PRL 96, 086805 (2006)

dEz

Inter-layer asymmetry(substrate, gate)

|| 01

Spontaneous symmetry breaking due to e-e interactions

T. Ohta et al – Science 313, 951 (‘06)(Rotenberg’s group at Berkeley NL)

Interlayer asymmetry gap in bilayer graphene

McCann, VF - PRL 96, 086805 (2006)

inter-layer asymmetry gap

(controlled usingelectrostatic gate)

Lifting degeneracy of Landau levels in bilayer graphene

McCann, VF - PRL 96, 086805 (2006)McCann - cond-mat/0608221

0

0

0

0

02

2

p

p

m

longer than next neighbour in-plane AA and

BB hops (weak)

inter-layer asymmetry

(controlled usinggate voltage)

m2

1

310~~ 20

41

Monolayer grapheneBilayer graphene

Tight-binding-model analysis:

‘chiral’ electrons and the Berry phase Jπ.

Landau levels and quantum Hall effect in bilayer and monolayer graphene.

Effect of trigonal warping

Infra-red and FIR magneto-optics in graphene.

Abergel, VF - PR. B 75, 155430 (2007)

E

E

E

E

E

E

Infrared absorptions due to inter-LL transitions

σ + , Mz=+1 σ - , Mz=-1

Electronic Properties of Graphene-Based Nanostructures ICTP Trieste Italy, 25-29 August 2008

ESF Conference Graphene Week ‘08Obergurgl, Austria, XX March or YY April 2008

(if we and ESF agree on dates)

KK

pp

tt

0)(

0

0

02

2

1

vH

0

0

0

)(0

2

132

2

2

v

mH

Berry phase π

‘trigonal warping’valley symmetry of wave vector K is lower

than the hexagonal crystalline symmetry

Berry phase 2π

1

1

A

B

B

A

1

1

~

~

A

B

B

A

KK

pp

yx

yx

ipp

ipp

''''1 KKKKantisymmKKsymmKK CCCCg

''''2 KKKKantisymmKKsymmKK CCCCg

Weak localisation correction

may be suppressed

by the intervalley scattering

due to atomically

sharp scatterers

or edges

i

can be suppressed

only by decoherence

Berry phase π

killed bytrigonal warping

reflectingthe asymmetry

in each valley

Berry phase 2π

)()( pEpE

KK

pp

Berry phase π

‘slow’ inter-valley scattering:neither WL nor WAL

magnetoresistance

‘fast’ inter-valley scattering: usual WL magnetoresistance cut at

''''1 KKKKantisymmKKsymmKK CCCCg

''''2 KKKKantisymmKKsymmKK CCCCg

Weak localisation magnetoresistance

i

)0()( RBR

i

iB

B

D

B 0~

ii D

B0~

E. McCann, K.Kechedzhi, V.Fal'ko, B.Altshuler,

in preparation

E. McCann, K.Kechedzhi, V.Fal'ko, H.Suzuura,

T.Ando, B.Altshuler, cond-mat/0604015

S.V. Morozov et al, cond-mat/0603826(Manchester group)

Weak localisation magnetoresistance