LO-Phonons and dielectric polarization effects on the electronic properties of doped GaN/InN spherical core/shell quantum dots in a nonparabolic band model

Abstract

The electron energy spectrum of a core/shell spherical quantum dot made of zincblende GaN/InN compounds is investigated taking into account the presence of an off-center donor atom and the influence of band nonparabolicity. The interaction of both the charge carrier and the Coulombic core with longitudinal optical phonons is included through Frö hlich and Aldrich-Bajaj theories, respectively. The ground state energy is determined by solving the resulting conduction band effective mass equation via the variational Ritz principle. A detailed analysis of the features of electron and hole spectra as functions of the core and shell sizes is presented, highlighting the possibility of transitioning between type-I and type-II structures. A detailed discussion about the effects of conduction band nonparabolicity, dielectric mismatch and electron-phonon interaction onto the impurity binding energy is provided. It was found that, in general, nonparabolicity of the conduction band leads to larger impurity binding energy, and that LO-phonon and dielectric mismatch effects tend to reduce the value of the latter quantity.

Appendices

Appendix A: Electrostatic potential created by a source charge placed in spherical nanostructure

This appendix gives the calculation main points of generalized Coulomb potential energy and self-polarization energy in spherical nanostructure/matrix. Those effects are coming from the surface charge densities emergence when two materials touch. The surface charge distributions depend, According to the image charge classical theory, on dielectric constants, sign of source charge and its distance to the interfaces. The very small size of low-dimensional structures reinforce the induced charge effect and make them significant. We note that this analytical calculation is based on calculations performed previously for different geometries [41, 68, 80,81,82].

Let us start by Poisson’s equation describing the electrostatic potential created at $\overrightarrow{r}\equiv (r,\theta ,\varphi )$:

$$\begin{aligned} \nabla ^2 V\left( \overrightarrow{r}-\overrightarrow{s}\right) =-\frac{4\pi }{ \varepsilon _{in}}\delta \left( \overrightarrow{r}-\overrightarrow{s}\right) \end{aligned}$$

(A1)

In spherical coordinates, the right side of Eq. (A1) becomes:

$$\begin{aligned}&-\frac{4\pi }{\varepsilon _{in}}\delta \left( \overrightarrow{r}- \overrightarrow{s}\right) = \nonumber \\&\qquad -\frac{4\pi }{\varepsilon _{in}}\frac{1}{r^{2}} \delta \left( r-s\right) \delta \left( \cos \left( \theta \right) -\sin \left( \theta ^{\prime }\right) \right) \delta \left( \varphi -\varphi ^{\prime }\right) \nonumber \\&\quad =-\frac{4\pi }{\varepsilon _{in}}\frac{1}{r^{2}}\delta \left( r-s\right) \frac{1}{\sin \left( \theta \right) }\delta \left( \theta -\theta ^{\prime }\right) \delta \left( \varphi -\varphi ^{\prime }\right) \end{aligned}$$

(A2)

Using the spherical harmonics completeness relation, Eq. (A2) writes

$$\begin{aligned}&-\frac{4\pi }{\varepsilon _{in}}\delta \left( \overrightarrow{r}- \overrightarrow{s}\right) \nonumber \\&\quad =-\frac{4\pi }{\varepsilon _{in}}\frac{1}{r^{2}} \delta \left( r-s\right) \sum \limits _{l=0}^{+\infty }\sum \limits _{m=-1}^{+1}Y_{l,m}\left( \theta ,\varphi \right) Y_{l,m}\left( \theta ^{\prime },\varphi ^{\prime }\right) \end{aligned}$$

(A3)

Applying the addition theorem for spherical harmonics, Eq. (A3) can be written as:

$$\begin{aligned} -\frac{4\pi }{\varepsilon _{in}}\delta \left( \overrightarrow{r}- \overrightarrow{s}\right) =-\frac{1}{\varepsilon _{in}r^{2}}\delta \left( r-s\right) \sum \limits _{l=0}^{+\infty }\left( 2l+1\right) P_{l}\left( \cos \left( \xi \right) \right) \end{aligned}$$

(A4)

where $P_{l}\left( \cos \left( \xi \right) \right)$ is the Legendre polynomial of l order and $\xi$ is the angle between $\overrightarrow{r} \equiv (r,\theta ,\varphi )$ and $\overrightarrow{s}\equiv (s,\theta ^{^{\prime }},\varphi ^{^{\prime }})$. Supposing that the charge is sited on the z-axis, which means that $\theta ^{^{\prime }}=0$ and $\xi =\theta$, the Poisson equation expression becomes

$$\begin{aligned} \nabla ^2 V\left( \overrightarrow{r}-s\right) =-\frac{1}{\varepsilon _{in}r^{2}} \delta \left( r-s\right) \sum \limits _{l=0}^{+\infty }\left( 2l+1\right) P_{l}\left( \cos \left( \theta \right) \right) \end{aligned}$$

(A5)

Equation (A5), which is invariant with respect to the z-axis, can be solved by developing its solutions in the basis of Legendre polynomials:

$$\begin{aligned} V\left( \overrightarrow{r}-s\right) =\sum \limits _{l=0}^{+\infty }V_{l}\left( r-s\right) P_{l}\left( \cos \left( \theta \right) \right) \end{aligned}$$

(A6)

Expanding calculation using Eqs. (A5) and (A6), we get

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{1}{r^{2}}\frac{d}{dr}\left( r^{2}\frac{dV_{l}\left( r,s\right) }{dr} \right) -\frac{l\left( l+1\right) }{r^{2}}V_{l}\left( r,s\right) =-\frac{1}{ \varepsilon _{in}r^{2}}\left( 2l+1\right) , &{} \quad r=s \\ \frac{1}{r^{2}}\frac{d}{dr}\left( r^{2}\frac{dV_{l}\left( r,s\right) }{dr} \right) -\frac{l\left( l+1\right) }{r^{2}}V_{l}\left( r,s\right) =0, &{} \quad r\ne s \end{array} \right. \end{aligned}$$

(A7)

We search for solutions in the form $V_{l}\left( r,s\right) =C\left( s\right) r^{m}.$ Proceeding with this we find that m may only take l and $-(l+1)$ values. Thence, $V_{l}\left( r,s\right)$ can be given as $V_{l}\left( r,s\right) =C_{1}\left( s\right) r^{l}+C_{2}\left( s\right) r^{-\left( l+1\right) }$. The solutions of Eq. (A7) should respect these boundary conditions:

• I. - $V_{l}^{QD}\left( r,s\right)$ must be regular for $r=0$

• II. - $V_{l}^{matrix}\left( r,s\right) \rightarrow 0$ when $r\rightarrow \infty$

• III. - $\left. V_{l}^{QD}\left( r,s\right) \right| _{r=R^{-}}=\left. V_{l}^{matrix}\left( r,s\right) \right| _{r=R^{+}}$

• IV. - $\left. \varepsilon _{in}dV_{l}^{QD}\left( r,s\right) /dr\right| _{r=R^{-}}=\left. \varepsilon _{out}dV_{l}^{matrix}\left( r,s\right) /dr\right| _{r=R^{+}}$

• V. - Permutation of r and s must be verified i.e. $V_{l}^{QD}\left( r,s\right) =V_{l}^{QD}\left( s,r\right)$

Conditions I and II require that $r^{-(l+1)}$ and $r^{l}~$coefficients in nanostructure and in matrix, respectively, are equal zero. Thus, the solutions of Eq. (A7) in different regions write:

$$\begin{aligned} V_{l}(r,s)=\left\{ \begin{array}{c} V_{l}^{QD}\left( r,s\right) =A\left( s\right) r^{l},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ r<s \\ V_{l}^{QD}\left( r,s\right) =B\left( s\right) r^{l}+\frac{C\left( s\right) }{ r^{l+1}},\ \ \ s<r<R \\ V_{l}^{matrix}\left( r,s\right) =\frac{D\left( s\right) }{r^{l+1}},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ R<r \end{array} \right. \end{aligned}$$

(A8)

The four unknown coefficients A(s), B(s), C(s) and D(s) are determined using the three conditions (3) to (5). The final Poisson equation solution, which contains Coulomb interaction between two charges placed at $\overrightarrow{r}$ and $\overrightarrow{s}$ as well as their interaction with surface charge densities at boundaries, writes:

$$\begin{aligned}&V(r,s) \nonumber \\&\quad =\left\{ \begin{array}{c} V^{QD}\left( \overrightarrow{r},s\right) =\sum \limits _{l=0}^{\infty }h\left( r^{l}s^{l}+\frac{1}{u}\frac{r^{l}}{s^{l+1}}\right) P_{l}\left( \cos \left( \theta \right) \right) ,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ r<s \\ V^{QD}\left( \overrightarrow{r},s\right) =\sum \limits _{l=0}^{\infty }h\left( s^{l}r^{l}+\frac{1}{u}\frac{s^{l}}{r^{l+1}}\right) P_{l}\left( \cos \left( \theta \right) \right) ,\ \ \ \ \ \ \ \ \ \ \ \ \ s<r<R \\ V^{Matrix}\left( \overrightarrow{r},s\right) =\sum \limits _{l=0}^{\infty }h\left( R^{2l+1}+\frac{1}{u}\right) \frac{s^{l}}{r^{l+1}}P_{l}\left( \cos \left( \theta \right) \right) ,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ R<r \end{array} \right. \end{aligned}$$

(A9)

To determine h, we integrate Eq. (A7) from $r-\kappa$ to $r+\kappa$, where $\kappa$ is an infinitesimal value. Doing calculations, we find $h=u/\varepsilon _{in}$ where

$$\begin{aligned} u=\left( l+1\right) \left( \varepsilon _{in}-\varepsilon _{out}\right) /\left( l\varepsilon _{in}+\left( l+1\right) \varepsilon _{out}\right) R^{2l+1}.\qquad \end{aligned}$$

(26)

Coulomb interaction between two particles:

The potential energy describing the interaction between a source charge $q_{1}$ and a target charge $q_{2}$ writes:

$$\begin{aligned} V_{q_{1}\rightarrow q_{2}}=q_{1}q_{2}V\left( \overrightarrow{r},s\right) \end{aligned}$$

(A10)

Doing calculations with $q_{1}=e$; $q_{2}=-e$ ,$~r=r_{e},~s=D$ and $R=b$ it is not hard to find Eq. 8.

Self-polarization energy of single particle

The self-polarization energy $W_{i}(r_{i})$ describing the interaction of a single-particle with surface charge distributions at material boundaries writes:

$$\begin{aligned} W_{i}\left( r\right) =\frac{1}{2}\left[ V_{r>s}\left( \overrightarrow{r} ,s\right) +V_{s>r}\left( \overrightarrow{s},r\right) \right] \end{aligned}$$

(A11)

Removing the terms $r_{l}/s_{l}+1$ and $s_{l}/r_{l}+1$ when $r<s$ and $s<r$ respectively which represent the pure Coulomb interaction term and by setting $q_{1}=q_{2}=e$ and $\theta =0$ we can easily obtain Eq.7.

Appendix B: Polaron - induced screened Coulomb potential

This appendix briefly outlines the derivation of the polaron-induced modification of a Coulombic potential related with an ionized impurity center in a polar crystal. The formulation was originally put forward for excitons by Aldrich and Bajaj [84], following Haken results via Lee-Low-Pines (LLP) variational formalism [83]. Instead, the approach by Aldrich and Bajaj makes use of the free polaron variational wave function proposed by Haga [85]. The system consists of an electron coupled to the screened Coulomb potential, assumed to be located at the origin. The Hamiltonian writes:

$$\begin{aligned} H_{0}=\frac{{p}^{2}}{2m}-\frac{e^{2}}{\varepsilon _{\infty }\left| \mathbf {r }\right| }+\sum \limits _{\mathbf {q}}\hbar \omega b_{q}^{+}b_{q}+\sum \limits _{ \mathbf {q}}\left[ \gamma _{q} e^{i\mathbf {q}.\mathbf {r}} b_{q}+cc\right] \end{aligned}$$

(B1)

where $b_{q}^{+}$ and $b_{q}$ are the creation and annihilation LO phonon operators of wavevector $\mathbf {q}$, $\omega$ is the optical phonon frequency, p is the momentum of the electron and $\mathbf {r}\,$ its position coordinate. Besides, m is the conduction band mass.

The quantity $\gamma _{q}$ is given as:

$$\begin{aligned} \gamma _{q}=-\frac{-i\hbar \omega }{q}\left( \frac{4\pi \alpha }{\beta V} \right) ^{1/2}, \end{aligned}$$

(B2)

where V is the crystal volume, $\alpha =1/2\overline{\varepsilon } \left( e^{2}\beta /\hbar \omega \right)$ is the so-called Frölich constant, with $\beta =\left( 2m\omega /\hbar \right) ^{1/2}$. The quantity $1/ \overline{\varepsilon }=1/\varepsilon _{\infty }-1/\varepsilon _{0}$ is the inverse reduced dielectric constant, with $\varepsilon _{\infty }$ and $\varepsilon _{0}$ being the static dielectric and high-frequency constants, respectively.

To derive the effective potential the model employs the following wavefunction:

$$\begin{aligned} \psi (\mathbf {r}) =\sum \limits _{\mathbf {k}}C_{k}\phi _{k}\left( r,b_{q}^{+}\right) \end{aligned}$$

(B3)

Where $\phi _{k}\left( r,b_{q}^{+}\right)$ is the free polaron wavefunction [85]:

$$\begin{aligned} \phi _{k}\left( r,b_{q}^{+}\right) =e^{i\mathbf {k}.\mathbf {r}} U\left( 1+\sum \limits _{q}c_{q}b_{q}^{+}\right) \left| \phi _{0}\right\rangle \end{aligned}$$

(B4)

where U is the LLP canonical transformation and $\left| \phi _{0}\right\rangle$ is the phonon vacuum state [86], and

$$\begin{aligned} c_q=4i\left( \frac{\pi \alpha \beta ^3}{V}\right) ^{1/2}\frac{qe^{-i\mathbf {q}\cdot \mathbf {r}}}{(1+\alpha /2)(q^2+\beta ^2)^2} \end{aligned}$$

(B5)

The effective Hamiltonian for the poalron interacting with the screened Coulomb center arises from the expression $H=\langle \psi (\mathbf {r})|H_0|\psi ( \mathbf {r})\rangle$:

$$\begin{aligned} H_{AB}=-\alpha \hbar \omega +\frac{p^{2}}{2m^{*}}+V_{AB}\left( r\right) \end{aligned}$$

(B6)

where $m^{*}$, the Haga polaron mass, is defined as:

$$\begin{aligned} m^{*}=\frac{m\left( 1+\alpha /6\right) }{1-\alpha /12} \end{aligned}$$

(B9)

and

$$\begin{aligned} V_{AB}(r)=-\frac{e^2}{\varepsilon _{\infty }}+ \frac{e^{2}\beta }{2 \overline{ \varepsilon }}\frac{e^{-\beta r}}{\left\{ 1+\alpha /\left[ 4\left( 1+\alpha /12\right) ^{2}\right] \right\} \left( 1+\alpha /12\right) } +V_{H}\left( r\right) \end{aligned}$$

(B9)

is the Aldrich-Bajaj modified electrostatic potential with

$$\begin{aligned} V_{H}\left( r\right) =\frac{e^{2}}{\overline{\varepsilon }r}\left( 1-\frac{ e^{-\beta r}}{2}\right) \end{aligned}$$

(B9)

representing the Haken-like potential. Thus, we are led to the Eq. (12) of the article.