Electronic properties of graphene - II

Vladimir Falko

helped byE.McCann, V.Cheianov

K.Kechedzhi, D.Abergel, A.RussellT.Ando, B.Altshuler, I.Aleiner

Monolayer grapheneBilayer graphene

Band structure of bilayer graphene, ‘chiral’ electrons and Berry’s phase 2π.

Effect of trigonal warping and the Lifshitz transition.

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

Interlayer asymmetry gap in bilayers.

Graphene optics.

Bilayer [Bernal (AB) stacking]

Bilayer [Bernal (AB) stacking]

In the vicinity of each of K points

Bilayer [Bernal (AB) stacking]

In the vicinity of each of K points

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 bilayergrapheneeV4.01 ≈γ

yx ipp +=π

McCann, VFPRL 96, 086805

(2006)

emm 05.0~

yx ipp +=π

)( pn rr

ψψψ πσπ 2222 3 i

i

ee =→×

prBerry phase 2π

(for a monolayer = π)

σππ

πϕ

ϕ rr⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛= −

−

−−

+− n

eH m

pi

i

mp

m 22

2

22

2

21

2

22

00

0)(0ˆ

)2sin,2(cos)( ϕϕ=pn rr

Monolayer grapheneBilayer graphene

Band structure of bilayer graphene, ‘chiral’ electrons and Berry’s phase 2π.

Effect of trigonal warping and the Lifshitz transition.

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

Interlayer asymmetry gap in bilayers.

Graphene optics.

1γ

Hops between A and via B~ BA~

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

vv

3γ

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

‘trigonal warping’

31 ~ mvpB >>−λ

strong magnetic field

31 ~ mvpB <−λ

weak magnetic field

212104~ −× cm

+K−K

( ) 2114

210~2 22

13 −= cmNvv

vL hπ

γ

21110~ −< cmNN L

*8NNNL <<

Lifshitz transition

Berry phase:2π = 3π - π

Summary of band structure:chiral electrons in monolayer and bilayer graphene

valley ‘trigonal warping’ terms

dominant at a high magnetic fieldand in high-density structures

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

+

+

0)(0

00

2

2

1 ππ

μπ

πςvH

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛= +

+

00

0)(0

21

32

2

2 ππ

ςπ

πv

mH

1

1

−=

+=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

ς

ς

AB

BA

1

1

~

~

−=

+=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

ς

ς

AB

B

A

Monolayer grapheneBilayer graphene

Band structure of bilayer graphene, ‘chiral’ electrons and Berry’s phase 2π.

Effect of trigonal warping and the Lifshitz transition.

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

Interlayer asymmetry gap in bilayers.

Monolayer and bilayer graphene optics.

2D Landau levels

semiconductor QW / heterostructure

(GaAs/AlGaAs)

σxy ( ) )2

hge

-1

-2 -1-3

2 31

-3

-2

3

1

2

integer QHEin semiconductors

eBghn

a

cnmm

pH ωππππh

r

)(42 2

12

+⇒+

==++

yxyx

zce

ippipp

lBArotAip

−=+=

=−∇−=+ππ ;

,rrr

hr

nnnB ϕπϕ

πϕλ +++ =

=

11

0 0

energies / wave functions

+π

π

)(0 rrϕ

1ϕ

2ϕ

yxyx

zce

ippipp

lBArotAip

−=+=

=−∇−=+ππ ;

,rrr

hr

Landau levels and QHE

0... 10 === −JJJ ϕπϕπ

εψψ =

00

,...,0

10 =⇒⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ − εϕϕ J

2D Landau levels of chiral electrons

J=1 monolayerJ=2 bilayer

( )

( )( ) ⎟

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−++

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−− +

+

ABBA

J

J

J

J

~~

00

00

ππ

ππ

valleyindex

also, two-fold real spin degeneracy

4J-degenerate zero-energy Landau level

Monolayer, J=1 , Berry’s phase π

McClure, Phys. Rev. 104, 666 (1956)Haldane, Phys.Rev.Lett. 61, 2015 (1988)

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

Bilayer, J=2, Berry’s phase 2π

8-fold degenerate ε=0 Landau level for electrons with degree of chirality J=2

McCann, VF - Phys. Rev. Lett. 96, 086805 (2006)

↑↓

εψψ =

emm 05.0~

B

vnλ

ε 2±=±

)1( −±=± nncωε hemm 05.0~

↑↓

1γ

Hops between A and via B~ BA~

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

vv

3γ

Effect of thetrigonalwarping

term

8-fold degeneratezero-energy Landau level

↑↓

↑↓

-2-4 40

n (1012 cm-2)2

2

4

6

0

ρ xx

(kΩ

)

-2-4 40

n (1012 cm-2)2

2

4

6

0

ρ xx

(kΩ

)

1L graphene 2L graphene

db

EE

pp

σ xy

(4e2

/h) 1

2

-1-2

-4

0

-3

4 c

3

σ xy

(4e2

/h) 1

2

-1-2

-4

0

-3

4 a3

Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer grapheneK.Novoselov, E.McCann, S.Morozov, VF, M.Katsnelson, U.Zeitler, D.Jiang, F.Schedin, A.Geim

Nature Physics 2, 177 (2006)

1γ

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

010 == εε

3γ

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

B~ 10 εε =

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

0

41 −γγγδ

cωδεε h=− || 01

McCann, VF - PRL 96, 086805 (2006)

dEz=Δ

Inter-layer asymmetry(substrate, gate)

Δ=− || 01 εε

Monolayer grapheneBilayer graphene

Band structure of bilayer graphene, ‘chiral’ electrons and Berry’s phase 2π.

Effect of trigonal warping and the Lifshitz transition.

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

Interlayer asymmetry gap in bilayers.

Graphene optics.

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

Interlayer asymmetry gap in bilayer grapheneMcCann, VF - PRL 96, 086805 (2006)

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ−

Δ+

ξξ0

0 inter-layer asymmetry gap

(controlled usingelectrostatic gate)

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

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ−

Δ+

ξξ0

0 inter-layer asymmetry gap

(controlled usingelectrostatic gate)

Interlayer asymmetry gap in bilayer graphene

Band mini-gap in bilayer graphene can be controlled electrically, so that a bilayer graphene transistor can be driven into a

pinched-off (insulating) state.

Monolayer grapheneBilayer graphene

Band structure of bilayer graphene, ‘chiral’ electrons and Berry’s phase 2π.

Effect of trigonal warping and the Lifshitz transition.

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

Interlayer asymmetry gap in bilayers.

Graphene optics.

Why can one see graphene in an

optical microscope?Abergel, VF - PR B 75, 155430 (2007)

%52 2

<ceh

π

00 /)( RRRV −=

Abergel, Russell, VF - Appl. Phys. Lett. 91, 063125 (2007)

Graphene flakes are visible when the oxide layer in SiO2/Si wafer acts

as clearing optical film if

sN

s

21

2sin2 +

−=

αελ

visibility,

o

nms10

300=

=

δα

o

ms10

1=

=

δα

μ

Blake, Hill, Castro Neto, Novoselov, Jiang, Yang, Booth, Geim - Appl. Phys. Lett. 91, 063124 (2007)