We consider a cylindrical ZB symmetric I n
x
G a
1−x
N multiple QD of radius R and length L
d
surrounded by two large energy gap materials I n
y
G a
1−y
N in the radial direction and GaN in the z-direction, Figure 1. Within the framework of effective mass approximation, the Hamiltonian description of an electron in the presence of an external field and hydrogenic impurity is given as
(1)
with
a) Electric field
(2)
where m
∗, F and V(ρ,z) are the effective mass of charge carriers, electric field strength and the confinement potential. The confining potential is
(3)
where
(4)
Using the separation of variable, the eigenfunction ofis written as
(5)
The radial wave function of the ground state can be expressed as
(6)
where,, J
0 and K
0 are the zero-order Bessel and modified Bessel functions. The wave function h(z) can be obtained by the linear combination of Airy functions A
i
and B
i
,
(7)
where,,,. Matching the wave functions and their derivatives at the boundaries, the normalization constants and energy eigenvalues, E
0=E
||+E
⊥, are determined.
The wave function foris obtained by variational method,
(8)
where α and β are the variational parameters and ρ
i
and z
i
are the position of the impurity.
b) Magnetic fieldThe Hamiltonian for the system in the presence of magnetic field is
(9)
whereis the vector potential of magnetic field. For a homogeneous magnetic fielddirected along z-axis, the vector potential is chosen as. The wave function for H
0 can be written as Eq. (5), where the ground state radial wave functions f(ρ) are Whittaker functions and h(z) is the linear combination of sin(γ z) and cos(γ z) functions,
(10)
where,,and. The impurity binding energy, E
b
, is obtained as
(11)