Electronic properties of monolayer and bilayer graphene

Electronic properties of monolayer and bilayer graphene Vladimir Falko (Lancaster) ( ) I & II. II Electrons in monolayer graphene.

A.Geim and K.Novoselov Nature Mat. 6, 183 (2007)

III. Electrons in bilayer graphene, Landau levels and the quantum Hall effect in monolayers and bilayers.

Bilayer graphene

Monolayer graphene

Band structure of bilayer graphene and Berry’s phase 2π, effect of trigonal warping and the Lifshitz transition. transition Landau levels and the quantum Hall effect in bilayer and monolayer graphene.

(valleys)

 p

Bilayer [Bernal (AB) stacking]

Bilayer [Bernal (AB) stacking] /

\

v \

v /

In the vicinity of each of K points

Bilayer [Bernal (AB) stacking] /

\

v \

v /

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)

 1  0.4eV

Fermi level in undoped bilayer graphene h

/

v \

m ~ 0.035me

  p x  ip y

McCann, VF PRL 96, 086805 (2006)

  p x  ip y

 2 2     0 ( )   p 1  ˆ   2m n   H 2  2m  2  0  

 p  ( p cos  , p sin i )

  p x  ip y  pei    p x  ip y  pe i   n ( p )  (cos 2 , sin 2 )

 1     A  p  12  i 2       e    B~   e

  n ( p)

i 22  3 2

  ei 2

Berry phase 2π

 p

/

1

v \

3 \

v

~ Hops between A and B

/

~ A via B

~ v3 Direct inter-layer hops between A and B , ~ 0. 1

v

Berryy phase: p 2π = 3π - π weak magnetic g field

B1 ~ p  mv3 strong magnetic field

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 1 v 4  2 v 2

~ 10 11 cm  2 Lifshitz transition

Bilayer graphene

Monolayer graphene

Band structure of bilayer graphene and Berry’s phase 2π, effect of trigonal warping and the Lifshitz transition. transition Landau levels and the quantum Hall effect in bilayer and monolayer graphene.

2D Landau levels semiconductor QW / heterostructure (GaAs/AlGaAs)

2 p      H   (n  12 )c 2m 4m

    e p  i  c A, rotA  Bl z

  p x  ip i y ;    p x  ip i y 0  0 B

 n 1 

  n n 1

energies / wave functions

2



3





1

2

a -3

  0 ((rr )

2 ge ) ) h

xy (

1 -2

-1 1 -1

g Q QHE integer in semiconductors

2 -2 -3

2

3

n

gh eB

Landau levels and the QHE

    e p  i  c A, rot A  Bl z

  p x  ipp y ;    p x  ipp y

 0     0     0 H1  v  0  0    0    0

2     1  0   0,1    0  H 2  2   2m   0  0 

  0   1   ,   00

 0

4J-degenerate valley zero-energy Landau level index J=1 - monolayer, J=2 - bilayer

also, two also two-fold fold real spin degeneracy

 0   ( ) J    

  

 J

0 0

J

 A  ~  B ~  J  B   0  A

 

      

Monolayer, Berry’s phase 

π

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

0   H 1  v   0  

4-fold degenerate Landau levels

   2n 

v

B

Bilayer, Berry’s phase 2π 

m ~ 0.05me

2   1  0   H2  2m   2 0 

   c n(n  1) 8-fold degenerate ε=0 Landau level McCann, VF - Phys. Rev. Lett. 96, 086805 (2006)

v3 ~ 0.1 v

8-fold degenerate zero-energy gy Landau level

Effect of the trigonal warping term





a

3

3

2

2

1 0 -1 -2

0 -1 -2 -3

-4

-4

E

1L graphene

2L graphene

6

4

xx (k )

6

xx (k )

1

-3

E

c

4

xyy (4e2/h)

xyy (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)

U Unconventional ti l quantum t Hall H ll effect ff t and d Berry’s B ’ phase h off 2π 2 in i bil bilayer graphene h K.Novoselov, E.McCann, S.Morozov, VF, M.Katsnelson, U.Zeitler, D.Jiang, F.Schedin, A.Geim Nature Physics 2, 177 (2006)

QHE in graphene synthesised on SiC

QHE resistance quantisation with accuracy of 3 parts per billion. A. Tzalenchuk, S. Lara-Avila, A. Kalaboukhov, S. Paolillo, M. Syväjärvi, R. Yakimova, O. Kazakova, T.J.B.M. Janssen, V. Fal’ko, S. Kubatkin, Towards T Towards d Quantum Q R Resistance i Standard S d d Based B d on Epitaxial E i i l Graphene, G h arXiv:0909.1220 – to appear in Nature Nanotechnology

Bilayer graphene

Monolayer graphene

Band structure of bilayer graphene, Berry’s phase 2π, effect of trigonal warping and the Lifshitz transition. transition Landau levels and the quantum Hall effect in bilayer and monolayer graphene. Interlayer asymmetry gap in bilayers.

Interlayer asymmetry gap in bilayer graphene



  0

0     

inter-layer asymmetry gap ( can be controlled using electrostatic gate)

McCann, VF - PRL 96, 086805 (2006) McCann - PRB 74, 161403 (2006)

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

Interlayer asymmetry gap 1 2



 12 

Mucha-Kruczynski, Tsyplyatyev, Grishin, McCann, VF, Boswick, Rotenberg Phys. Rev. B 77, 195403 (2008)

T. Ohta et al – Science 313, 951 (‘06) (Rotenberg’s group at Berkeley NL) SiC based highly doped SiC-based bilayer graphene

1 2

U

 12 U

McCann, VF - PRL 96, 086805 (2006) McCann - PRB 74, 161403 (2006)

Gate-controlled interlayer asymmetry gap (transport measurements)

Oostinga, Heersche, Liu, Morpurgo, and Vandersypen, Nature Physics (2007)

How robust is the degeneracy of  0   1 in bilayer graphene?

0

Landau level

  Ez d

3

1

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

 0  1

|  1   0 | c

Inter-layer asymmetry (substrate gate) (substrate,

 ~  ~ 10 2(3) 1 4 2 0

|  1   0 | 

McCann, VF - PRL 96, 086805 (2006)