Landau level properties of two prototypical bilayer graphenes

邱裕煌，何彥宏，林明發

成功大學 物理系

Introduction

Tight-binding Model

Results and Discussions

Conclusions

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hybridization of atomic orbitals

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various dimensionality

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achievement of few-layered sheets

Mechanical exfoliation of highly oriented pyrolytic graphite Science 306, 666 (2004)Epitaxial growth on silicon carbide Science 312,1191 (2006) 4/33

stacking consequence

Bernal stacking5/33

AA

Simple hexagonalstacking

low-energy bands: monolayer & bilayer

Science 313, 951 (2007)

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angle-resolved photoelectron spectroscopy (ARPES)

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Landau quantization

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2DEG

experimental observation of Landau levels in graphene

1. Quantum Hall conductivity2. Scanning tunneling spectroscopy3. Magnetooptical transmission (excitations between Landau levels)

Nature 438, 197 (2005)

Nat. Phys. 3, 623 (2007) PRL 97, 266405 (2006)9/33

Tight-binding model Phys. Rev. 109, 272 (1958)

γ0

γ1 γ3 γ4

γ0

bilayer Bernal graphene

monolayer graphene

10/33γ6

zero-gap semiconductor

semimetal

1

0

1, exp

R

r

RR

R

R

k r ik r r RN

G A d r R

ei G

A R

G

r R d

The Bloch function

Ref: J. M. Luttinger, Phys. R

: Peierls phas

s in a static m

ev. 84, 8

agneti

1

c fiel

e

4

d

~8 1

:

17 ( 9

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51)

2

2

,

1exp

2

1

2

R R

R

k k k k

i k R k R ie G G

RRRR

R

RR

H k r

Pik R V r R

mN

H H

e HN

PH r R V r

i

m

G

R

e

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1

0

1 2

0

0

50 4.1356 10

For a graphene monolayer, 79000 T through

a hexagon area.

Define 79000 / , is related to the dimensi

The flux quantum

R R

B B

G G R R A R R R d

hTm

e

R T B R

����������������������������������������������������������������������������������������������������������������

onality

of the Hamiltonian matrix.

For monolayer graphene, the dimensionality is 4 .

For bilayer graphene, the dimensionality is 8 .B

B

R

R

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ˆB ; A (0, ,0)Bz Bx

Monolayer graphene

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γ0

nE B

14/33

15/33

18/33

Optical absorption function

Velocity matrix element

', is derived by gradient approximation.h hM

',

2 ', , , , ', ' 1

, 1,2

[ ] . .B

h h

R h hi M j M k i Mk B j M kM M

i j

M

H h c

2', '', ' 1

',0

', ', ', ',0 3 4

', ', ',0 3

' , '

Monolayer:

[ ] . .

= ( )

AB-bilayer:

( )+ ( )+ ( )

AA-bilayer:

( )+ ( )

LL wavefunction:

BRh h h hk Mk B M kM M

h h

h h h h h h h h

h h h h h h

n n n n

M A B A H B h c

M

M M M M

M M M

Monolayer

1, 1

c vn n n

AB-bilayer

AB-bilayer

Selection rules

1 1

2 1

2

2

1

2 1,1

0,

,

0

1

2

1

,2

c

c v

v

v

v

c

c

n n

n

n n

n

n n

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Selection rules

'1

'2 2

1

1

1,

,1

1h

h

h

hn n

n n

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Conclusions:

1. Two prototypes of bilayer graphenes, AA- and AB-stacking bilayer graphenes, exhibit different Landau level spectra, which reflect the fact that the magneto-electronic properties strongly depend on the stacking configurations.2. The Landau wave functions of AA- and AB-stacking bilayer graphenes show dissimilar features and the dissimilarities are reflected in the magneto-optical absorption spectra, such as the distinct selection rules and different field-dependent absorption frequencies.

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Thanks for your attention!