Chemistry Reference
In-Depth Information
0.2
0.15
600 x 600
0.1
0.05
bulk Landau level + edge states
ε/t
0
-0.05
-0.1
l
c
=20
-0.15
-0.2
2
2.5
3
3.5
4
4.5
5
5.5
k
y
Fig. 5. Energy eigenvalues
n,k
y
, for electrons in graphene computed from the tight
binding model for a hexagonal lattice subjected to a magnetic field
B
=27
.
3Tesla(or
l
c
=20
.The
plot shows
n,k
y
in the units of
t
as function of
k
y
,where
k
y
is the wavevector in the
y
direction. Two sets of horizontal lines are Landau levels corresponding to the two valleys
and
n
= 0 Landau level and the edge states are degenerate.
a
where
a
is the triangular lattice spacing) for a system size of 600
a ×
600
a
We now consider the above system in the presence of an additional
constant electric field applied, in a open circuit geometry, along the
x
-
direction. The single particle Hamiltonian is then given by,
h
=
v
F
α
.
Π
+
1
eEx
(11)
v
F
playing
the role of the speed of light, can be used to solve the problem exactly
[
29
]
.
It is known from special relativity, if
The Lorentz covariant structure of the Hamiltonian, with
, then we can always boost
to a frame of reference where the electric field vanishes and the magnetic
field is reduced. We can then use the finite magnetic field and zero electric
field solution to Dirac equation and boost it to get the exact spectrum in
the presence of crossed electric and magnetic field. Here the boost trans-
formation amounts to doing a transformation on the space-time coordinate
v
F
B>|
E
|
