## 2 Background Theory  
Let us consider an exciton, \(X(e,h)\), bound to an ionized donor impurity (\(D^{+}\)) confined in a single \(GaAs\) spherical quantum dot (\(SSQD\)) embedded in a glass matrix or host material with large band gap. The system formed \((D^{+},X)\) is similar to the Helium like-atom which constitute a physical three bodies problem. To reduce the number of variables, we suppose that the ionized donor is located at the centre of the \(SSQD\). Within the framework of the effective mass approximation and considering the thermal and strain effects, the Hamiltonian of \((D^{+},X)\) complex can be written as:  
\[\mathcal{H}=-\frac{\hbar^{2}}{2\;m_{e}^{*}(P,T)}\Delta_{e}-\frac{\hbar^{2}}{2 \;m_{h}^{*}(P,T)}\Delta_{h}-\frac{e^{2}}{\varepsilon(P,T)}\bigg{(}\frac{1}{r_{ e}}+\frac{1}{r_{eh}}-\frac{1}{r_{h}}\bigg{)}+V_{w}^{e}+V_{w}^{h}, \tag{1}\]where \(r_{eh}=\big{|}\overrightarrow{r_{e}}-\overrightarrow{r_{h}}\big{|}\) is the electron to hole distance. \(\varepsilon(P,T)\), \(m_{e}^{*}(P,T)\) and \(m_{h}^{*}(P,T)\) are the dielectric constant, electron and hole effective masses, respectively. All these parameters depend on the temperature \(T\), and pressure \(P\). They can be expressed by the following variation laws [33; 34]:  
\[\varepsilon(P,T)=\left\{ \begin{array}{c}12.7\exp(-1.67\times 10^{-3}P)\exp(9.4 \times 10^{-5}(T-75.6))\\ \text{for }0\leq T\leq 200\;K\\ 13.18\exp(-1.73\times 10^{-3}P)\exp(20.4\times 10^{-5}(T-300))\\ \text{for }T>200\;K\end{array} \right. \tag{2}\]  
\[m_{e}^{*}(P,T)=m_{0}\bigg{(}1+7.51\bigg{(}\frac{2}{E_{g}(P,T)}+\frac{1}{E_{g} (P,T)+\Delta_{0}}\bigg{)}\bigg{)}^{-1} \tag{3}\]  
and  
\[m_{h}^{*}(P,T)=m_{0}\bigg{(}0.09-0.2\times 10^{-3}P-3.55\times 10^{-5}T\bigg{)}, \tag{4}\]  
where \(\Delta_{0}\) is the spin-orbit splitting. Here, the band gap \(E_{g}(P,T)\) is given by the Varshni relation [33; 35]:  
\[E_{g}(P,T)=1.519-\frac{5.4\times 10^{-4}\;T^{2}}{T+204}+0.01261\;P+3.77\times 1 0^{-5}P^{2}. \tag{5}\]  
In glass matrix or in other medium with large band gap, the confinement relative to a \(SSQD\) can be described by [12; 13; 14] :  
\[V_{w}^{i}(r_{i})=\left\{ \begin{array}{cc}0&\text{\emph{inside}}\\ \infty\;;&\text{\emph{outside}}\end{array} \right.i=(e,h). \tag{6}\]  
On the other hand, under strain effect, the radius of the \(SSQD\) can be obtained by applying the strain tensor components of spherical dot radius [36]:  
\[R(P)=R(0)\big{[}1-(S_{11}+2S_{12})P\big{]}^{1/3} \tag{7}\]  
where \(R(0)\) is the unstrained \(SSQD\) size and \(S_{11}\) and \(S_{12}\) are the compliance parameters obtained using the elastic constant \(C_{11}\) and \(C_{12}\) such as [16; 37; 38]:  
\[S_{11}=\frac{C_{11}+C_{12}}{(C_{11}-C_{12})(C_{11}+2C_{12})} \tag{8}\]  
and  
\[S_{12}=\frac{C_{12}}{(C_{11}-C_{12})(C_{11}+2C_{12})}. \tag{9}\]  
By using the effective units, \(a_{D}=\varepsilon_{0}\hbar^{2}/m_{e}^{*}\varepsilon^{2}\) for length and \(R_{D}=\hbar^{2}/m_{e}^{*}a_{D}^{2}\) for energy, the Hamiltonian given in Equation (1) can be written as:  
\[\mathcal{H}=\frac{m_{e}^{*}}{2}\bigg{(}-\frac{1}{m_{e}^{*}(P,T)}\Delta_{e}- \frac{1}{m_{h}^{*}(P,T)}\Delta_{h}-\frac{\varepsilon_{0}}{\varepsilon(P,T)} \bigg{(}\frac{1}{r_{e}}+\frac{1}{r_{eh}}-\frac{1}{r_{h}}\bigg{)}\bigg{)}. \tag{10}\]  
In such a situation, the Hylleraas coordinates \((r_{e},r_{h},r_{eh},z_{e},z_{h})\) are the best coordinate system that can be used [39]. Thus, the Laplacian operator can be simply written in the following form:\[\Delta_{\epsilon}=-\frac{\partial^{2}}{\partial r_{e}^{2}}-\frac{\partial^{2}}{ \partial r_{eh}^{2}}-\left(\frac{r_{e}^{2}-r_{h}^{2}+r_{eh}^{2}}{r_{e}r_{eh}} \right)\frac{\partial^{2}}{\partial r_{e}\partial r_{eh}}-\frac{2}{r_{e}}\frac{ \partial}{\partial r_{e}}-\frac{2}{r_{eh}}\frac{\partial}{\partial r_{eh}}. \tag{11}\]  
Note that \(\Delta_{h}\) can be obtained by interchanging \(e\) and \(h\). The Schrodinger equation \(\mathcal{H}\)\(\Psi^{(D^{+},X)}=E_{(D^{+},X)}\)\(\Psi^{(D^{+},X)}\) does not admit an exact solution, so only a numerical approach can determine the lowest energy states.