Transcript GrapheneIntro2 - UMD Physics

Graphene: Scratching the Surface
Michael S. Fuhrer
Professor, Department of Physics and
Director, Center for Nanophysics and Advanced Materials
University of Maryland
Michael S. Fuhrer
University of Maryland
Carbon and Graphene
Carbon
Graphene
- C
-
Hexagonal lattice;
1 pz orbital at each site
4 valence electrons
1 pz orbital
3 sp2 orbitals
Michael S. Fuhrer
University of Maryland
Graphene Unit Cell
Two identical atoms in unit cell:
A
B
Two representations of unit cell:

a1

a2
Two atoms
1/3 each of 6 atoms = 2 atoms
Michael S. Fuhrer
University of Maryland
Band Structure of Graphene
Tight-binding model: P. R. Wallace, (1947)
(nearest neighbor overlap = γ0)
E (k )  EF   0
 3k x a   k y a 
k a
2 y 
 cos
  4 cos 

1  4 cos


 2 
 2   2 
E
kx
ky
Michael S. Fuhrer
University of Maryland
Bonding vs. Anti-bonding
 0
H 
  0
0
0 
E   0
γ0 is energy gained per pi-bond
ψ
“anti-bonding”
anti-symmetric wavefunction
1 1
1 
 
2  1
E1   0
“bonding”
symmetric wavefunction
1
2 
2
Michael S. Fuhrer
1
1

E2   0
University of Maryland
Band Structure of Graphene – Γ point (k = 0)
Bloch states:
FA(r),
or
“anti-bonding”
E = +3γ0
1
 
0
1 1
 
2  1
A
B
Γ point:
k=0
FB(r),
or
0
 
1
A
B
“bonding”
E = -3γ0
1 1
 
2 1
Michael S. Fuhrer
University of Maryland
Band Structure of Graphene – K point
K
FA(r),
or
1
 
0
K 
4
3a

3a
2
FB(r),
or
0
 
1
K
K
Phase:
1
e
e
Michael S. Fuhrer
λ
i
2
3
i
4
3
K
University of Maryland
Bonding is Frustrated at K point
Im
Phase:
e 1
i0
e
e
E2
E1
Re
2
i
3
i
4
3
E3
E
E11  00eeii00d 
d0
 22

i i
d 
 33

EE22 00ee
 44

i i
d 
 33

EE33 00ee
2
4

 2i 0  i  4 i  
 iE
i
d 
d 3 

3i 



e

e

e
 0 d 
 3

 3
   0 id

0
 0e  0
E   0 e
e
e





Michael S. Fuhrer
University of Maryland
Bonding is Frustrated at K point
“anti-bonding”
K
4
K 
3a
FA(r),
1 1
 
2  1
E = 0!
3a

2
or
1
 
0
“bonding”
1 1
 
2 1
E = 0!
FB(r),
0
or  
1
0
5π/3
π/3
4π/3
2π/3
K point:
Bonding and anti-bonding
are degenerate!
π
Michael S. Fuhrer
University of Maryland
Band Structure of Graphene: k·p approximation
Hamiltonian:
 0
vF 
 k x  ik y
k x  ik y  FA (r ) 
 FA (r ) 

   


0  FB (r ) 
 FB (r ) 
K
vF (σ  k )F (r )  F (r )
Eigenvectors:
 i k / 2


1 ik r  ibe

k 
e 
;
i k / 2

e
2


θk is angle k makes with y-axis
b = 1 for electrons, -1 for holes
electron has “pseudospin”
pointsS.parallel
Michael
Fuhrer (anti-parallel) to momentum
K’
Energy:
  bv F k
linear dispersion relation
“massless” electrons
University of Maryland
Visualizing the Pseudospin
0
5π/3
π/3
4π/3
2π/3
π
180 degrees
540 degrees
Michael S. Fuhrer
University of Maryland
Visualizing the Pseudospin
0
5π/3
π/3
4π/3
2π/3
π
0 degrees
180 degrees
Michael S. Fuhrer
University of Maryland
Pseudospin: Absence of Backscattering
K’: k||-x
K: k||-x
bonding
orbitals
K: k||x
anti-bonding
orbitals
bonding
orbitals
real-space
wavefunctions
(color denotes
phase)
anti-bonding
k-space
representation
K’
Michael S. Fuhrer
bonding
K
University of Maryland
“Pseudospin”: Berry’s Phase in IQHE
20
QHE at T=2.3K, B=7.94T
1

  4 n  
2

g s gv  2  2  4
Michael S. Fuhrer
5
0
-80 -60 -40 -20
0
20
40
60
2
e2
 xy  
h
10
xy(e /h)
π Berry’s phase for electron orbits
results in ½-integer quantized Hall
effect
Rxx(k)
15
34
30
26
22
18
14
10
6
2
-2
-6
-10
-14
-18
-22
-26
-30
-34
80
Vg (V)
Berry’s phase = π
University of Maryland