Electron in Bilayer Graphene with Magnetic Fields Associated with Solvable Potentials
267
parameter and γ 1 = t ⊥ for the hopping from an atom in the sublattice A1 to an atom
in the sublattice A2.
2.2 Effective Hamiltonian with an Applied Magnetic Field
Let us suppose now that a magnetic field
B orthogonal to the graphene surfaces
is applied, which points along z-direction. For simplicity, we suppose that
B just
changes along a certain given direction on the surface. Working in the Landau gauge
we can write
A(x) = A(x) ˆ
y,
B(x) = B(x)ˆ z, with B(x) = dA(x)/dx. According
with the minimal coupling rule, a term proportional to A(x) must be added to the
operator π [4]. Thus, we get the new operator Π = p x − ip y − i(e/c)A(x), while
the effective Hamiltonian which takes into account the magnetic field effects looks
like
H =
1
2m ∗
0 Π 2
(Π † ) 2 0
.
(2)
The next step is to determine the eigenvectors and eigenvalues of H , which is a
non-trivial but sometimes solvable problem.
2.3 Stationary States of H
The eigenvalue equation for H can be written as
H Ψ (x, y) = EΨ (x, y).
(3)
Due to the gauge chosen, this equation is invariant under translations along ydirection, i.e., in this direction the motion is of free particle type. Thus, it is natural
to propose Ψ (x, y) as follows:
Ψ (x, y) =
e iky
√
2
ψ (2) (x)
ψ (0) (x)
,
(4)
where k is the wavenumber in y-direction and 1/
√
2 is a normalization factor. Since
p j = −i ¯
h∂ j , j = x, y, after plugging equation (4) into (3) the next coupled system
of equations is obtained:
L 2 ψ
(0)
= −εψ
(2) , L
†
2 ψ
(2)
= −εψ
(0) .
(5)