Coulomb excitations in AA- and AB-stacked bilayer graphites

K.S.Novoselov, A.K.Geim, S.V.Morozov, D.Jiang, Y.zhang, S.V.Dubonos, I.V.Grigorieva Science 306, 666 (2004)

OutlineGeometrical Structure Band structure ( tight-binding method)-Electronic excitations (RPA)Low-frequency and High-frequency electronic excitations Conclusion

Geometrical structure (planar graphenes)

m ono

A A

'1 '

3

0

A B

''1

zigzag

armchaira: ( )

zp Ar r

b: ( )zp Br r

Ic~3.5Å

Tight-Binding Bloch Function1( ) ( ) exp( )

z n nn

nk p R R

R

r r r ik rN

1 1 1 1

2 2 2 2

1 1 1 1

2 2 2 2

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )

mono a a b bk k k k k

a a b bAAk k k k k

a a b bk k k k

a a b bABk k k k k

a a b bk k k k

r C r C r

r C r C r

C r C r

r C r C r

C r C r

a: ( )zp Ar r

b: ( )zp Br r

Monolayer

-8

-4

0

4

8

Enk

e

V

K KM

-1

0

1

E F= 0

MK

m onolay er

M K

-8-4

04

8 (

eV)

00.10.20.30.40.5D O S (States/ atom )

m onolayer

Two linear energy bands intersect at EF

Zero-gap semiconductor (DOS=0 at EF)

Saddle point at M, which cause singularity (log. div.)

AA Stacked

-8

-4

0

4

8

En

ke

V

K KM

-1

0

1

E F= - 4.5 m eV

A A

M K

-8-4

04

8

(eV)

00.10.20.30.4D O S (States/ atom )

A A

Two linear energy band are seperated by 21Carrier density increases

AB Stacked

-8

-4

0

4

8

Enk

e

V

K KM

-1

0

1

E F= 0.14 m eV

A B

M K

-8-4

04

8

(eV

)00.10.20.30.40.5

D O S (States/ atom )

A B

Two linear energy bands change into parabolic bandsThere is some overlap between 1 and *1

Dynamical Screening

e e22

qeV

q

e e

( , )q

eff

VV

q

Vacuum Many-body system

Effective potential

e

e e

2

112 eV

q

2

122

cqIeV eq

1

2e

e e11effV

12effV

1

2Ic

h

e

h e1

2

1

2

(q,)

e

h

(q,)

(q,)

Random Phase Approximation

(1) (1)11 22P P (1) (1)

12 21P P

Random Phase ApproximationRPA Approxmation( , ) ( , )RPAq q

e

h

Dielectric function and Response function

Response Function (monolayer)

0 0.4 0.8 1.2 1.6 2 (eV )

- 0.0004

- 0.0002

0

P (q

,)

q=510-3 Å -1, f =0o; =2 m eVT =0 K

R ePI m P

* and * excitationsSquare-root divergence structure for ImP is caused by excitation from kF to kF+qImP and ReP are related by K-K relation

-1

0

1

E F= 0

Response Function (AA)1 *1 and 1 1 excitations at 1

sp=30bq/2

1 *2 , 2 *1 and 2 1 excitations at 3,2

sp=2 130bq/2

-0.04

-0.02

0

0.02

0.04

R eP (1)

11

I m P (1)

11

A A ; =2 m eVq=510-3 Å -1, f =0o, T =0 K ;

0 0.4 0.8 1.2 1.6 2

eV

-0.04

-0.02

0

0.02

0.04

P(1) (q

,) (

eV-1

. Å-2

)

R eP (1)

12

I m P (1)

12

-1

0

1

E F= - 4.5 m eV

Response Function (AB)

-0.004

-0.002

0

0.002

0.004

P(1) (q

,) (

eV-1

. Å-2

)

R eP (1)

11

I m P (1)

11

A B; =2 m eVq=510-3 Å -1, f =0o, T =0 K ;

0 0.4 0.8 1.2 1.6 2 eV

-0.004

-0.002

0

0.002

0.004

R eP (1)

12

I m P (1)

12

ImP exhibits discontinuous structure due to band edge states

-1

0

1

E F= 0.14 m eV

Loss FunctionLoss function characterizes the dynamics of the power dissipated in the medium due to an external perturbation

( , ) Im[1/ ( , )]P q q

Loss Function (AA)

0 0.4 0.8 1.2 1.6 2 (eV )

0

1

2

3

Im[-1

/]

A A ; =2 m eV

q= 2 Å - 1, f =0o; T =0 K

5

1

5 ; f =30o

5 ; T =300 K

0 0.4 0.8 1.20

0.2

0.4

q= 0.035 0.045 0.055

Intensity of plasmon-1 declines as q↑Intensity of plasmon-2 increases as q↑Intensity of plasmon-3 increases and then decrease as q↑Loss spectra is isotropic and weak temperature dependence

Loss Function (AB)

0 0.4 0.8 1.2 1.6 2 (eV )

0

0.2

0.4

0.6

Im[-

1/]

A B; =2 m eV

q= 2 , f =0o; T =0 K 5

1

5 ; T =300 K 5 ; T =300 K (m onolay er)

No plasmon modeweak temperatue dependence

Plasmon DispersionThree plasmon modes in AA-staced systemOne is acoustic, the others are optical

0 0.02 0.04 0.06 0.08 0.1q (Å -1)

0

0.4

0.8

1.2

1.6

p (e

V)

A A

Response Function (AA and AB)

-0.002

0

0.002

P(1) (q

,) (

eV-1

. Å-2

)

R eP (1)

11

I m P (1)

11

4 5 6 7 8 eV -0.0008

-0.0004

0

0.0004

0.0008

0.0012

R eP (1)

12

I m P (1)

12

A B

-0.001

0

0.001

0.002

R eP (1)

11

I m P (1)

11

A A ; =20 m eVq=0.1 Å -1, f =0o; T =0 K

4 5 6 7 8

eV

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

P(1) (q

,) (

eV-1

. Å-2

)R eP (1)

12

I m P (1)

12

-8 -4 0 4 8 (eV )

0

0.1

0.2

0.3

0.4

DOS (States/atom)

AA

-8 -4 0 4 8 (eV )

0

0.1

0.2

0.3

0.4

0.5DOS (States/atom

)

AB

Loss function (AA and AB)

4 6 8 10 (eV )

0

0.5

1

1.5

2

2.5

Im[-1

/]

A B; =20 m eV

q= 0.1 Å - 1, f =0o; T =0 K 0.2 0.3 0.4 0.5 0.6

4 6 8 10 (eV )

0

0.5

1

1.5

2

2.5

Im[-1

/]

A A ; =20 m eV

q= 0.1 Å - 1, f =0o; T =0 K 0.2 0.3 0.4 0.5 0.6

Plasmon Dispersion

0 0.2 0.4 0.6q (Å -1)5

6

7

8

p (e

V)

A AA Bm onolayerb i layer w ithout in terlayer atom ic in teraction s

Interlayer interaction raise and interlayer atomic interaction raise the -plasmon frequency

ConclusionInterlayer atomic interaction strongly affects the low energy states (near Fermi level) and hence the electronic excitationsWeak dependence on temperature and direction of transferred momentumThree low-frequency plasmon modes in the AA-stacked system but not the AB-stacked systemAA- and AB-stacked system exhibit similar plasmonsThe bilayer graphites differ from the monolayer graphite in the existence of low-frequency plasmons and -plasmon frequency at small momentum