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

\u0124={\u0124}_{0}-\frac{{e}^{2}}{\epsilon |r-{r}_{i}|}

(1)

with

*a) Electric field*

{\u0124}_{0}=\frac{{\widehat{P}}^{2}}{2{m}^{\ast}}+V(\rho ,z)+\left|e\right|\mathit{\text{Fz}}

(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

V(\rho ,z)=\left\{\phantom{\rule{0.3em}{0ex}}\begin{array}{cc}V\left(\rho \right)& \phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\mathrm{z}}_{1}<\mathrm{z}<{\mathrm{z}}_{2},\phantom{\rule{1em}{0ex}}{\mathrm{z}}_{3}<\mathrm{z}<{\mathrm{z}}_{4},\phantom{\rule{1em}{0ex}}{\mathrm{z}}_{5}<\mathrm{z}<{\mathrm{z}}_{6}\\ {V}_{\mathit{\text{II}}}& \phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\text{otherwise}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\end{array}\right.

(3)

where

V\left(\rho \right)=\left\{\begin{array}{cc}0& \phantom{\rule{1em}{0ex}}\rho \le \mathrm{R}\\ {V}_{I}& \phantom{\rule{1em}{0ex}}\rho >\mathrm{R}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\end{array}\right.

(4)

Using the separation of variable, the eigenfunction of{\u0124}_{0}is written as

\psi (\rho ,\varphi ,z)=f\left(\rho \right)h\left(z\right){e}^{\mathrm{im\varphi}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.22144pt}{0ex}}m=0,\pm 1,\mathrm{..}

(5)

The radial wave function of the ground state can be expressed as

f\left(\rho \right)=\left\{\begin{array}{cc}{C}_{w}\phantom{\rule{2.77626pt}{0ex}}{J}_{0}\left({\omega}_{1}\rho \right)& \phantom{\rule{1em}{0ex}}\rho \le \mathrm{R}\\ {C}_{b}\phantom{\rule{2.77626pt}{0ex}}{K}_{0}\left({\omega}_{2}\rho \right)& \phantom{\rule{1em}{0ex}}\rho >\mathrm{R}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\end{array}\right.

(6)

where{\omega}_{1}=\sqrt{\frac{2{m}_{w}^{\ast}}{{\hslash}^{2}}{E}_{\perp}},{\omega}_{2}=\sqrt{\frac{2{m}_{b}^{\ast}}{{\hslash}^{2}}({V}_{I}-{E}_{\perp})}, *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
},

h\left(z\right)=\left\{\begin{array}{cc}{A}_{i}\left(\frac{{\eta}_{1}z-{\eta}_{2}^{2}}{{\left(-{\eta}_{1}\right)}^{2/3}}\right)+{B}_{i}\left(\frac{{\eta}_{1}z-{\eta}_{2}^{2}}{{\left(-{\eta}_{1}\right)}^{2/3}}\right)& \phantom{\rule{1em}{0ex}}\text{well region}\\ {A}_{i}\left(\frac{{\kappa}_{1}z+{\kappa}_{2}^{2}}{{\left(-{\kappa}_{1}\right)}^{2/3}}\right)+{B}_{i}\left(\frac{{\kappa}_{1}z+{\kappa}_{2}^{2}}{{\left(-{\kappa}_{1}\right)}^{2/3}}\right)& \phantom{\rule{1em}{0ex}}\text{barrier region}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\end{array}\right.

(7)

where{\kappa}_{1}=\sqrt{\frac{2{m}_{b}^{\ast}\mathit{\text{eF}}}{{\hslash}^{2}}},{\kappa}_{2}=\sqrt{\frac{2{m}_{b}^{\ast}}{{\hslash}^{2}}({V}_{\mathit{\text{II}}}-{E}_{\left|\right|})},{\eta}_{1}=\sqrt{\frac{2{m}_{w}^{\ast}\mathit{\text{eF}}}{{\hslash}^{2}}},{\eta}_{2}=\sqrt{\frac{2{m}_{w}^{\ast}}{{\hslash}^{2}}{E}_{\left|\right|}}. 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 for\u0124is obtained by variational method,

\Phi =\psi (\rho ,\varphi ,z)\phantom{\rule{2.77626pt}{0ex}}{e}^{-\alpha {\left(\rho -{\rho}_{i}\right)}^{2}-\beta {\left(z-{z}_{i}\right)}^{2}}

(8)

where *α* and *β* are the variational parameters and *ρ*
_{
i
} and *z*
_{
i
} are the position of the impurity.

*b) Magnetic field*The Hamiltonian for the system in the presence of magnetic field is

{\u0124}_{0}=\frac{{\left(\widehat{P}-\frac{e}{c}\overrightarrow{A}\right)}^{2}}{2{m}^{\ast}}+V(\rho ,z)

(9)

where\overrightarrow{A}is the vector potential of magnetic field. For a homogeneous magnetic field\overrightarrow{B}(0,0,B)directed along z-axis, the vector potential is chosen as\overrightarrow{A}=\left(\overrightarrow{B}\times \overrightarrow{r}\right)/2. 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,

f\left(\rho \right)=\left\{\begin{array}{cc}\frac{{C}_{w}^{\prime}}{\rho}\mathit{\text{WM}}\left(\frac{{\gamma}_{1}^{2}}{2\mu},0,\frac{\mu {\rho}^{2}}{2}\right)& \phantom{\rule{1em}{0ex}}\rho \le \mathrm{R}\\ \frac{{C}_{b}^{\prime}}{\rho}\mathit{\text{WW}}\left(\frac{-{\gamma}_{2}^{2}}{2\mu},0,\frac{\mu {\rho}^{2}}{2}\right)& \phantom{\rule{1em}{0ex}}\rho >\mathrm{R}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\end{array}\right.

(10)

where{\gamma}_{1}=\sqrt{{E}_{\perp}/{R}^{\ast}},{\gamma}_{2}=\sqrt{({V}_{I}-{E}_{\perp})/{R}^{\ast}},{R}^{\ast}={m}^{\ast}{e}^{4}/\left(2{c}^{2}{\hslash}^{2}\right)and\mu =\mathrm{\hslash e}{B}_{0}/\left(2{m}^{\ast}c{R}^{\ast}\right). The impurity binding energy, *E*
_{
b
}, is obtained as

{E}_{b}={E}_{0}-\underset{\alpha ,\beta}{min}\frac{<\Phi \left|\u0124\right|\Phi >}{<\Phi |\Phi >}.

(11)