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15
Triangular lattice
Graphene
Graphene is an atomic-scale honeycomb lattice carbon atoms monolayer.
Conduction via p-orbitals of carbon. Tight binding.
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Artificial Graphene
Weak coupling
Consider a 2D electron gas in a potential U(r) with hexagonal (triangular)
symmetry and spacing L.
p2
H
 U (r )
2m
The lattice translation vectors are
L1 = (L, 0)
L2 = (L/2,√3L/2)
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There are two independent reciprocal lattice vectors in the Brillouin zone
G1 
2p
(3, 3 ),
3L
G2 
2p
(0,2 3 ),
3L
G3  G1 - G2
The points K1, K2, K3 are connected by vectors Gi
K′i are obtained from the Ki by reflection.
 
G1  L1  2p
 
G1  L2  2p
 
G1  L1  0
 
G2  L2  2p
 4p
1 , 0
K1 
3L

4p
K2 
1/ 2 , 3 / 2
3L

4p
K3 
1/ 2 , - 3 / 2
3L




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Below I measure energy in units of the bandwidth:
K 2 (4p / 3L) 2
E0 

2m
2m
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We assume a periodic potential with a single Fourier component
U (r )  2W cosG1  r   cosG2  r   cosG3  r 
This potential has nonzero matrix elements only between states |k> and
|k ± Gi> with matrix elements given by W.
Perturbation Theory. Practically works up to W = W/E0 =2
A state close to the Dirac point, q ≪ 1, is
described by degenerate perturbation theory as
 q  c1 | 1  c2 | 2  c3 | 3 
| j  e
i (K j  q )r
Kinetic energy:
(K i  q) 2
Ki  q 
p2


 ik   E0 
 ik
2m
2m
m 

 0 1 1


U
(
r
)

W
1
0
1
Potential energy:


1 1 0 


Kinetic energy is a weak
perturbation on the potential
energy.
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Diagonalize potential energy
 W
U   W 
W
W
W  (  W ) 2 (  2W )
W 
The eigen-energies of the U-matrix are (-W, -W, 2W).
There is the double degenerate subspace!!!!
In order to project in the double degenerate subspace of U we define:
0



| a  1 / 2 



1
/
2


 2/3 


| b    1 / 6 


 1/ 6 


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Projecting the kinetic energy to this basis and shifting the zero
energy level to E0 one finds:
(K
 a | H | a 
 K3) q
 vqx
2m
( 4 K1  K 2  K 3 )  q
 b | H | b 
 vqx
2m
(K 3  K 2 )  q
 b | H | a 
 vqy
2 3m
 0 1
2
0 -i 
v
K
2p

2m 3Lm
is the Fermi-Dirac velocity
1 0 
 ,  y  
 ,  z  

 x  
1 0 
i 0
 0 - 1
Hence, in the Pauli matrix (pseudo-spin) representation the effective low
energy Hamiltonian reads:
H  v( z qx   x q y )
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One can perform the unitary transformation H → T †HT , where T represents two
subsequent π/2 rotations around x- and y-axes in the pseudo-spin space:
T  xT   y
T  yT   z

1 1  i - 1 - i 

T  
2  - 1  i - 1  i 
T  zT   x
This transforms the Hamiltonian to the conventional form of a 2D Dirac Hamiltonian
H  vσ  q
However, in what follows we will use the “blue” form as it is slightly more convenient
for the study of the edge states.
H  v( z qx   x q y )
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Numerical diagonalization of the Hamiltonian is straightforward.
The hole dispersion along a particular
contour in the BZ, W=W/E0 =1
Two Dirac points of opposite parity.
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Map of the total charge density (in units 1/L2) at
the chemical potential tuned to the Dirac point.
The average hole density is <n> = 8/3L2.
At L = 50nm the average density is 1.1 × 1011cm−2.
Even when the potential is strong, namely W = W/E0 = 1−2, the dispersion is
rather close to the result obtained by perturbation theory. The charge density
plot is fully connected with empty spots at positions of the potential maximums.
So, in clear contrast to natural graphene at W < 2 the system is much closer to
the nearly free electron regime than to the tight-binding one.
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Chiral edge states in toplogical insulator
Topological insulator
Let us switch on the spin orbit interaction Hso.
Whatever is the microscopic mechanism of the interaction the interaction
must satisfy the following conditions
1)The interaction depends on spin s. This is true spin, not pseudospin.
2)The interaction is time reversal invariant.
3) The intercation is space inversion invariant
Therefore matrix element of Hso between two plane wave must be of the
following form
 p 2 | H so | p1   i([p1  p 2 ]  s)
An additional condition follows from the Bloch’s theorem.
Since the spin-orbit interaction has period of the lattice the matrix
element of Hso is nonzero only if
p 2  p1  Gi
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Near the Dirac cone
 2 | H so | 1   i[K1  K 2 ]  s  i sz
Hence the effective 3x3 spin-orbit Hamiltonian reads
 0 i i


H so    i 0 i 
 i  i 0


Projecting this to the degenerate pseudospin states |a> and |b> we find
H so  2sz y
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Total Hamiltonian
H  v( z qx   x q y )  2sz y
This is the K-Dirac cone, for the K’-Dirac cone we have to replace v→-v (opposite
parity).
One can perform the unitary transformation H → T †HT to transform to the
conventional form
H  vσ  q-2ηsz z
The energy is
 2  v 2 q 2  (2s z ) 2
   v 2q 2  2
Due to

the gap is opened at Dirac points
If chemical potential is inside the gap this is insulator
What is topological about this?
Why this is different from usual band insulator?
Topological chiral edge states
Lateral confinement: let us limit the 2D crystal by the infinite wall potential
0 if y  y 0
U conf  
 if y  y 0
The envelope wave function of
the edge state at y > y0
   iq x x y
  A  e e
 
Solution of Dirac Eq. H gives
    2  v 2 q x2  v 2 2
 1
v  2s z
 i
vq x  
H  v( z qx   x q y )  2sz y
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In the explicit coordinate form this is
  eiq x e  y  | a    | b 
x


1 iK 2 r iK 3 r
| a 
e
e
2
1
| b 
2eiK 1r - eiK 2 r  eiK 3 r
6


 2p

| a   eiK 2 r  eiK 3 r  sin 
y0   0
 3L 
Let us tune the position of the wall
2p
y0  p n , n is integer
3L

Hence, to satisfy (x,y0)=0 we need only to
impose
sz  1 / 2,   /v,   -vqx

v  2s z
 i
0
vq x  
For the opposite parity Dirac cone the edge solution has different chirality
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sz  1 / 2,   /v,    vq x
Topological protection:
The edge states are found at a special position of the confining wall. An explicit
calculation at a different wall position/shape is more involved since the calculation
must include admixture of high momentum states to the wave function. However, it
is obvious that a variation of the wall position/shape does not influence the edge
states since they are topologically protected.
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The edge states support the spin polarized current at the edge of system.
x
TI
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