Chemistry Reference
In-Depth Information
relativistic boosts and the mixing of electric and magnetic fields in moving
frames of reference.
The dispersion relation in the proximity of the Dirac points is linearly
proportional to
. The low energy modes around these points are de-
scribed by slowly varying fields fermi fields
|
k
|
ψ
rησ
(
R
i
) defined as,
i
K
1
.
R
i
z
rr
ψ
r
1
σ
(
R
i
)+
i
K
2
.
R
i
x
rr
ψ
r
2
σ
(
R
i
)
c
irσ
=
e
α
e
α
(7)
α
x
,α
y
are the Pauli matrices. Here i refers to the unit cell and
r = 1,2 refers to the two types of atoms in each unit cell. The effective
Hamiltonian for the low energy modes is the Dirac Hamiltonian.
Where
d
2
x
Ψ
ησ
α
.
H
=
v
F
p
Ψ
ησ
(8)
ησ
=
√
2
at
where
v
F
is the Fermi velocity. Ψ
ησ
are two component field oper-
ators where
η
(= 1
,
2) is the valley index, corresponds to two Dirac points
and
) is the spin index. The spectrum can be obtained by solving
the one particle equation to get the linear dispersion,
σ
(=
↑, ↓
.In
presence of an external magnetic field perpendicular to the graphene plane
the one particle Hamiltonian,
(
k
)=
±
v
F
|
k
|
h
=
v
F
α
.
Π
,where
Π
=
p
+
e
A
. The energy
eigenvalues are
)
2
|n|
v
F
l
c
n,k
y
=sgn(
n
(9)
=
2
L
y
l
n
is the Landau level index,
k
y
is the quantum number corre-
sponding to translation symmetry along
y
-axis, both
n
and
l
are integers
l
c
=
eB
is the magnetic
length. Unlike the case of the non-relativistic electron in a magnetic field,
where the spectrum has a linear dependence on the magnetic field and the
non-negative integer valued Landau level index, the graphene Landau lev-
els have a square root dependence on both magnetic field and Landau level
index. The degeneracy of each level is given by the number of magnetic
flux quanta passing through the sample. The eigenfunctions are,
(we choose Landau gauge
A
(
r
)=
xB
y
)and
∝ e
ik
y
y
sgn(
n
)
φ
|n|−
1
(
ξ
)
ψ
nk
y
(
x, y
)
(10)
iφ
|n|
(
ξ
)
where
φ
n
(
ξ
)
are the harmonic oscillator eigen-functions and
ξ ≡
x
l
c
k
y
.
l
c
+
