Band structure of graphene and CNT

Band structure of graphene and CNT

Graphene :

Lattice : 2- dimensional triangular lattice

Two basis atoms

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y 축

1A

3A

2A

1B3B

2B

Bloch State of the π bands

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y 축

1A

3A

2A

1B3B

2B

Three nearest neighbor

Nearest Neighbor Approximation

Multiply on both sides ( )zP r

RHS =

Tight Binding Approximation 에서 Nearest Neighbor Approximation 이라는 것은

Within Nearest Neighbor on site only

B AHC C

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y 축

1A

3A

2A

1B3B

2B

의 nearest neighbor 1A 2A 3A1B

Multiply on both sides

Left=

Right=

A BH C C

3 32 2 3 36 6

2 2

1/ 2

2 20

2 cos( ) 2 cos( )2 2

31 4cos ( ) 4cos( )cos( )

2 2 2

3( , ) 1 4cos( )cos( ) 4cos ( )

2 2 2

y yy y

a aa ai k i ki k i k

x x

x y x

graphenex y y x x

a aH e e k e e k

a a ak k k

a a aE k k k k k

3,

2 3 2x y

a ak k

2 2

,3 3

x yk ka a

2 1,1

3 3k

a

Pass the Fermi point(Dirac point)

( ) 0grahpeneE k

2 1( ,1)

3 3Fermik

a

1 2

2 1

3 3Fermik b b

1a((((((((((((((

2a((((((((((((((

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y 축

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y 축

1b((((((((((((((

2b((((((((((((((

Lattice Reciprocal Lattice

Band Structure of Graphene Dirac point

1 2

4 10 , ( , )

3 3b b fixed

2 1( , )3 3

Band Structure of Graphene

1 2

4 10 , ( , )

3 3b b fixed

( ) ( )graphene grapheneE k E k G ((((((((((((( (

“ - 공간 에서의 주기함수” 를 확인하시기 바랍니다 .k

CNT =

wrapped graphene ribbon

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c

경계조건

y 축

For example,

1/ 2

2 20

3( , ) 1 4cos( )cos( ) 4cos ( )

2 2 2x y y x x

a a aE k k k k k

Subband (n=0) (9,0)0

3( ) 1 4cos( ) 4

2CNT

yE k ak

Subband (n=1)(9,0) 2

0

3( ) 1 4cos( )cos( ) 4cos ( )

2 9 9CNT

yE k ak

0n1n

2n

3n

4n

8n

5n

6n

7n

Low energy effective Hamiltonian near K and K’

Tight-binding π bands, again.

Near K or K’

x x y yH k k

Near K or K’spinor of pseudospin

( )

x x y yH k k

A AH kB B

Mahmut, you have the solution for the spinor

( )

( )when ( ) , Mahmut write the sol.

( )

when ( ) , Mahmut write the sol.

A AH kB B

A kk k

B k

Ak k

B

Bands are doubly degenerate in real spin

With SOC

0

ˆ

ˆ0 0

0 ˆ0

ˆ ˆ = '

x x y y x z

x y z

x y z

H k k S

k ik S

k ik S

H H

In this low-energy Cone region, how and why the

SOC is represented this way?

ˆ 0ˆˆ0

zSOC

z

SH

S

Min et al., PRB74,165310(2006), Kane and Mele, PRL, 95, 226801(2005)

Full 4 component or 8 component solution A bit complicated

Diagonalize in real spin space

Min et al., PRB74,165310(2006), Kane and Mele, PRL, 95, 226801(2005)

ˆ is diagonalized

and as

x x y y x zH k k S

A A

B B

Diagonalize in real spin space

ˆ ˆ =

0 0

0 0

x y

x y

x y

x y

A AH H E

B B

k ikH

k ik

k ik A AE

k ik B B

Diagonalize in real spin space

2 2 2 2

2 2 2

x y

x y

E k k

E k k

Effective Hamiltonian Including the two Fermi point

K and K’ (K’=-K)

, near

, near '

x x y y

x x y y

k k KH

k k K

Without SOC it is not very meaningful

Effective Hamiltonian Including the two Fermi point

K and K’ (K’=-K)

ˆ , 1( ( '))x x y y z zH k k s K K

Why do we need this ? ????

ˆ , 1( ( '))x x y y z zH k k s K K

Why do we need this ?

Near K

ˆ ˆ =

0 0

0 0

x y

x y

x y

x y

A AH H E

B B

k ikH

k ik

k ik A AE

k ik B B

Near K’=-K

ˆ ˆ =

0 0

0 0

x y

x y

x y

x y

A AH H E

B B

k ikH

k ik

k ik A AE

k ik B B