Electronic properties of graphene - II

Electronic properties of graphene - II Vladimir Falko

helped by E.McCann, V.Cheianov K.Kechedzhi, D.Abergel, A.Russell T.Ando, B.Altshuler, I.Aleiner

Bilayer graphene

Monolayer 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]


In the vicinity of each of K points


In the vicinity of each of K points

McCann, VF PRL 96, 086805 (2006)

ARPES: heavily doped bilayer graphene synthesized on silicon carbide T. Ohta et al – Science 313, 951 (2006) (Rotenberg’s group at Berkeley NL)

γ 1 ≈ 0.4eV

Fermi level in undoped bilayer graphene

m ~ 0.05me π = p x + ip y

McCann, VF PRL 96, 086805 (2006)

π = p x + ip y

Hˆ 2 =

−2iϕ

⎛ 0 (π ) ⎞ − p2 ⎛ 0 e ⎞ − p2 r r ⎜ 2 ⎟ = 2m ⎜ −2iϕ ⎟ = 2m n ⋅ σ ⎜π ⎟ ⎜π ⎟ 0 0 ⎝ ⎠ ⎝ ⎠ + 2

−1 2m

r r n ( p ) = (cos 2ϕ , sin 2ϕ ) ψ →e r r n ( p)

i 2×2π σ 3 2

ψ = ei 2πψ

Berry phase 2π

r p

(for a monolayer = π)

Bilayer graphene

Monolayer graphene


γ1

γ3

~ Hops between A and B

~

via A B

~ v3 Direct inter-layer hops between A and B , ~ 0.1

v

Berry phase: 2π = 3π - π

‘trigonal warping’ weak magnetic field

λ−B1 ~ p < mv3 strong magnetic field

λ−B1 ~ p >> mv3

N < NL ~ 1011cm−2 Inter-layer asymmetry (electric field across the structure, effect of a substrate/overlayer)

N L < N < 8N *

~ 4 ×1012 cm−2 K−

K+

NL = 2

( )

v3 2 γ1 v 4 πh 2 v 2

~ 10 11 cm − 2 Lifshitz transition

Summary of band structure: chiral electrons in monolayer and bilayer graphene 2 ⎛0 π+⎞ ⎛ 0 π ⎞ ⎟ + μ⎜ + 2 ⎟ H1 = ςv⎜⎜ ⎟ ⎜ (π ) ⎟ π 0 0 ⎝ ⎠ ⎝ ⎠

valley

‘trigonal warping’ terms

+ 2 ⎛ ⎛ 0 π⎞ 1 0 (π ) ⎞ ⎜ 2 ⎟ + ςv3 ⎜⎜ + ⎟ H2 = ⎟ ⎜ ⎟ 0⎠ 2m ⎝ π 0 ⎠ ⎝π

dominant at a high magnetic field and in high-density structures

⎛A ⎞ ⎜ ⎟ς = +1 ⎜B ⎟ ⎜ ⎟ ⎜ B ⎟ς = −1 ⎜A ⎟ ⎝ ⎠ ⎛A ⎞ ⎜ ~ ⎟ς =+1 ⎜B ⎟ ⎜~ ⎟ ⎜B ⎟ς =−1 ⎜A ⎟ ⎝ ⎠

Bilayer graphene

Monolayer 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)

r2 ππ + + π +π p = ⇒ (n + 12 )hωc H= 2m 4m

r r r r e p = −ih∇ − c A, rotA = Bl z

π = p x + ip y ; π + = p x − ip y πϕ0 = 0 λB

ϕ n +1 =

+ π ϕn n +1

energies / wave functions

ϕ2

π

3

π

+

ϕ1

2 ge ) ) h

2

a -3

r ϕ 0 (r )

σxy (

1 -2

-1 1 -1

integer QHE in semiconductors

-2 -3

2

3

n

gh eB

Landau levels and QHE

r r r r e p = −ih∇ − c A, rot A = Bl z

π = p x + ip y ; π + = p x − ip y

2D Landau levels of chiral electrons

ψ = εψ

J=1 monolayer J=2 bilayer

π ϕ 0 = ... = π ϕ J −1 = 0 J

J

⎛ ϕ 0 ⎞ ⎛ ϕ J −1 ⎞ ⎟⎟ ⇒ ε = 0 ⎜⎜ ⎟⎟,..., ⎜⎜ ⎝0⎠ ⎝ 0 ⎠ 4J-degenerate zero-energy Landau level

also, two-fold real spin degeneracy

⎛0 ⎜ ⎜π J ⎜ ⎜ ⎜ ⎝

(π )

+ J

0 0

(− π )

J

valley index

⎞⎛ A ⎟⎜ ~ ⎟⎜ B ~ + J ⎟⎜ B − π ⎟⎜ 0 ⎟⎠⎜⎝ A

(

)

+⎞ ⎟ +⎟ −⎟ ⎟ − ⎟⎠

↑↓

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)

ε = ± 2n ±

m

~

0

.

05

m

v

λB

e

ψ = εψ ↑↓

Bilayer, J=2, Berry’s phase 2π

m ~ 0.05me

ε ± = ±hωc n(n − 1) 8-fold degenerate ε=0 Landau level for electrons with degree of chirality J=2 McCann, VF - Phys. Rev. Lett. 96, 086805 (2006)

Effect of the trigonal warping term

γ1

γ3

~ Hops between A and B

~

via A B

~ v3 Direct inter-layer hops between A and B , ~ 0.1

v

8-fold degenerate zero-energy Landau level

↑↓

↑↓

a

3

3

2

2

1 0 -1 -2

0 -1 -2 -3

-4

-4

E

1L graphene

2L graphene

6

4

ρxx (kΩ)

6

ρxx (kΩ)

1

-3

E

c

4

σxy (4e2/h)

σxy (4e2/h)

4

p

2

4

p

2

b 0

-4

-2

0

2

n (1012 cm-2)

4

d 0

-4

-2

0

2

4

n (1012 cm-2)

Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene K.Novoselov, E.McCann, S.Morozov, VF, M.Katsnelson, U.Zeitler, D.Jiang, F.Schedin, A.Geim Nature Physics 2, 177 (2006)

How robust is the degeneracy of ε 0 = ε 1 in bilayer graphene?

=0

Landau level

Δ = Ez d

γ3

γ1

~

Direct inter-layer A B hops (warping term, Lifshitz trans.) Distant intra-layer AA,BB hops

ε 0 = ε1

| ε 1 − ε 0 |= δhωc

Inter-layer asymmetry (substrate, gate)

δ ~ γγγ ~ 10 −2(3) 1 4 2 0

| ε 1 − ε 0 |= Δ

McCann, VF - PRL 96, 086805 (2006)

Bilayer graphene

Monolayer graphene


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

⎛ξΔ 0 ⎞ ⎟⎟ + ⎜⎜ ⎝ 0 − ξΔ ⎠

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

inter-layer asymmetry gap (controlled using electrostatic gate)

Interlayer asymmetry gap in bilayer graphene McCann, VF - PRL 96, 086805 (2006) McCann - cond-mat/0608221

⎛ξΔ 0 ⎞ ⎟⎟ + ⎜⎜ ⎝ 0 − ξΔ ⎠

inter-layer asymmetry gap (controlled using electrostatic gate)

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.

Bilayer graphene

Monolayer graphene


2πe < 5% hc 2

Why can one see graphene in an optical microscope? Abergel, VF - PR B 75, 155430 (2007)

Graphene flakes are visible when the oxide layer in SiO2/Si wafer acts as clearing optical film if

λ 2

=

ε s − sin 2 α

visibility,

N+

1 2

s

V = ( R − R0 ) / R0

s = 300nm

δα = 10o

s = 1μm

δα = 10o

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

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