Numerous experimental and theoretical research works have been conducted in recent years on semiconductor quantum well, quantum well lines and quantum dots^[1−4]. Moreover, the limitation of carrier motion in all three spatial dimensions is considered to be an essential property of quantum dots. Special property can be widely used in technological applications^[5−7]. Furthermore, the reduction in the size of a quantum dot is demonstrated to lead to partial or complete quantization of energy levels, which ameliorate the physical properties of these systems. Hence, various quasi-particles in quantum dots have been the focus of numerous researchers^[8−11].

The state of the exciton in semiconductors have been shown to play a key role in the absorption and luminescence spectra understanding the basic photoelectric properties of artificial layered materials^[12−16]. Accordingly, exciton properties in semiconductor quantum devices have become an important topic of research in condensed matter physics. Moreover, studies on the state of exciton in quantum dots have mainly considered the confinement effect of potential well, such as the finite barrier potential^{[17, 18]}, valence-band mixing^[19], dispersion relation^{[20, 21]}, mass^[22−24], dielectric constant mismatch^{[25, 26]}, etc. Qualitative and quantitative results obtained from exciton binding energy and energy level are consistent with the experimental results^[27−29]. In addition, adopting the effective phonon approximation method, the exciton-phonon interaction has been extensively investigated^[30−32]. This is while the linear combination operator method has been rarely utilized in studying excitons in quantum dots. Hence, using the linear combination operator method, the present study aims at investigating the effect of exciton in the Al_xGa_1−xAs crystals.

In addition, Huybrecht variational method was used in the present research to study the ground state energy and ground state binding energy of exciton in quantum dots in Al_xGa_1−xAs crystals. Numerical simulation was also applied to study the relationship between exciton energy level and binding energy with eigenfrequency and Al concentration. Hence, with regards to low-dimensional semiconductor materials, the obtained results are expected to have important theoretical significance.

As can be seen in Fig. 1(a), aluminum gallium arsenide is an atomic semiconductor material formed by doping Al atoms into GaAs. In the above formula, x is a number ranging from 0 to 1, which shows an arbitrary alloy between Ga and Al atoms in Al_xGa_1−xAs. Moreover, due to the different electronegativity and atomic size of Al and Ga atoms in Al_xGa_1−xAs crystal, short range potential is generated in the lattice and, accordingly, bound exciton is formed. The exciton Hamiltonian in this system is formulated as:

1$H=-\frac{{\hslash }^2}{2{\mu }_{\text{e}}}{\nabla }_{\text{e}}^2-\frac{{\hslash }^2}{2{\mu }_{\text{h}}}{\nabla }_{\text{h}}^2-\frac{e^2}{{\epsilon }_{\infty }\left|r_{\text{e}}-r_{\text{h}}\right|}+{\displaystyle \sum _W}\hslash {\omega }_{\text{LO}}{a}_W^{†}{a}_W+{\displaystyle \sum _W}\left[V_W{a}_W\left(\exp \left(iW\cdot r_{\text{h}}\right)-\exp \left(iW\cdot r_{\text{e}}\right)\right)+h\mathrm{.}c\right]+U\left(r_{\text{e}}\right)+U\left(r_{\text{h}}\right)\mathrm{,}$

2$U\left(r_{\text{e,h}}\right)=\frac{1}{2}{\mu }_{\text{e,h}}{\omega }_0^2{r}_{\text{e,h}}^2\mathrm{.}$

While first and the second terms on the right-hand side of Eq. (1) are electron and hole movements, respectively, the third term describes the Coulomb potential between electron−hole interaction. Moreover, the fourth and the fifth terms define the phonon energy and the interaction energy of exciton−phonon coupling, (Fig. 1(b)), respectively. Furthermore, while the parabolic potentials of electrons is expressed by the sixth term, holes are described by the seventh term (Fig. 1(c)). In addition, $V_W$ and $\alpha$ can be written as：

3$V_W=\frac{4ei\pi }{\sqrt{V}}{(\frac{c\hslash {\omega }_{\text{LO}}}{8\pi W^2})}^{1/2}\mathrm{,}c=\frac{1}{{\epsilon }_{\infty }}-\frac{1}{{\epsilon }_0}\mathrm{,}$

4$\alpha =\frac{M{e}^2}{{\hslash }^{2}u}\left(\frac{1}{{\epsilon }_{\infty }}-\frac{1}{{\epsilon }_0}\right)\mathrm{,}$

5$\frac{{\hslash }^2{u}^2}{2M}=\hslash {\omega }_{\text{LO}}\mathrm{.}$

Here ${\epsilon }_{\infty }\left({\epsilon }_0\right)$ denotes the high frequency (static) permittivity of the material.

In order to transform the Hamiltonian into a Hamiltonian in centroid coordinates, the following definition is provided：

6$M={\mu }_{\text{e}}+{\mu }_{\text{h}}\mathrm{,}R={\beta }_e{r}_{\text{e}}+{\beta }_h{r}_{\text{h}}=\frac{{\mu }_{\text{e}}}{M}{r}_{\text{e}}+\frac{{\mu }_{\text{h}}}{M}{r}_{\text{h}}\mathrm{,}$

7$\mu =\frac{{\mu }_{\text{e}}{\mu }_{\text{h}}}{{\mu }_{\text{e}}+{\mu }_{\text{h}}}\mathrm{,}r=r_{\text{e}}-r_{\text{h}}\mathrm{,}$

8${\beta }_{\text{e}}=\frac{{\mu }_{\text{e}}}{M}\mathrm{,}{\beta }_{\text{h}}=\frac{{\mu }_{\text{h}}}{M}\mathrm{,}$

9${\xi }_W=\left(e^{-i{\beta }_{\text{e}}W\cdot r}-e^{i{\beta }_{\text{h}}W\cdot r}\right)\mathrm{.}$

Here $M$ denotes the total mass of exciton, $R$ represents the coordinate of the center-of-mass describing the motion of the center of mass, $\mu$ stands for the reduced mass, and $r$ signifies the relative coordinate relating to the internal relative motion of exciton. Accordingly, the Hamiltonian at center-of-mass coordinates can be formulated as:

10$H^{*}=-\frac{{\hslash }^2}{2M}{\nabla }_R^2-\frac{{\hslash }^2}{2\mu }{\nabla }_r^2-\frac{e^2}{{\epsilon }_{\infty }r}+{\displaystyle \sum _W}\hslash {\omega }_{\text{LO}}{a}_W^{†}{a}_W+{\displaystyle \sum _W}\left[V_W{a}_W\exp \left(iW\cdot R\right){\xi }_W+h\mathrm{.}c\right]+\frac{1}{2}M{\omega }_0^2{R}^2+\frac{1}{2}\mu {\omega }_0^2{r}^2\mathrm{.}$

Subsequently, the following two unitary transformations^[33] are performed for Eq. (10).

11$U_1=\exp \left(-i{\displaystyle \sum _W}W\cdot R{a}_W^{†}{a}_W\right)\mathrm{,}$

12$U_2=\exp \left({\displaystyle \sum _W}{a}_W^{†}{f}_W-a_W{f}_W^{*}\right)\mathrm{.}$

Here $f_W\left(f_W^{*}\right)$ represents the variational parameters of unitary transformation. Meanwhile, the following relations introduce the linear combination operators of the creation (annihilation) operator $B_j^{†}\left(B_j\right)$ of the electron and variational parameter $\lambda$^[34]

13$P_j={\left[\frac{M\hslash \lambda }{2}\right]}^{\frac{1}{2}}\left(B_j+B_j^{{†}_{}}\right)\mathrm{,}$

14$R_j=i{\left[\frac{\hslash }{2M\lambda }\right]}^{\frac{1}{2}}\left(B_j-B_j^{{†}_{}}\right)\mathrm{.}$

15$j=x\mathrm{,}y\mathrm{,}z\mathrm{.}$

The transformed Hamiltonian of the system can be expressed as:

16$\begin{array}{ll}H=\hfill & \frac{\hslash \lambda \left(B_j{B}_j+2B_j^{{†}_{}}{B}_j+B_j^{{†}_{}}{B}_j^{{†}_{}}\right)}{4}+\frac{3\hslash \lambda }{4}-\frac{-2{\displaystyle \sum _W}\hslash W\left(a_W^{{†}_{}}+f_W^{*}\right)\left(a_W+f_W\right){\left[\frac{M\hslash \lambda }{2}\right]}^{\frac{1}{2}}\left(B_j+B_j^{{†}_{}}\right)}{2M}+\hfill \\ \hfill & \frac{{\displaystyle \sum _W}{\hslash }^2{W}^2\left(a_W^{{†}_{}}+f_W^{*}\right)\left(a_W+f_W\right)}{2M}+\frac{{\displaystyle \sum _W}{\hslash }^2{W}^2\left(a_W^{{†}_{}}+f_W^{*}\right)\left(a_W^{{†}_{}}+f_W^{*}\right)\left(a_W+f_W\right)\left(a_W+f_W\right)}{2M}-\hfill \\ \hfill & \frac{{\hslash }^2}{2\mu }\frac{{\partial }^2}{\partial r^2}-\frac{{\hslash }^2}{\mu }\left({\displaystyle \sum _W}{a}_W^{{†}_{}}\nabla f_W-a_W\nabla f_W^{*}\right)\nabla \phi -\phi \frac{{\hslash }^2}{2\mu }\left({\displaystyle \sum _W}{a}_W^{{†}_{}}{\nabla }^2{f}_W-a_W{\nabla }^2{f}_W^{*}\right)-\hfill \\ \hfill & \frac{1}{2!}\frac{{\hslash }^2}{\mu }{\displaystyle \sum _W}\left(a_W^{{†}_{}}{a}_W^{{†}_{}}\nabla f_W\nabla f_W-2a_W^{{†}_{}}{a}_W\nabla f_W^{*}\nabla f_W-\nabla f_W\nabla f_W^{*}+a_W{a}_W\nabla f_W^{*}\nabla f_W^{*}\right)+\hfill \\ \hfill & \frac{e^2}{{\epsilon }_{\infty }r}+{\displaystyle \sum _W}\hslash {\omega }_{\text{LO}}\left(a_W^{{†}_{}}+f_W^{*}\right)\left(a_W+f_W\right)+{\displaystyle \sum _W}\left[V_W\left(a_W+f_W\right){\xi }_W+h\mathrm{.}c\right]-\hfill \\ \hfill & \frac{{\omega }_0^2\hslash }{4\lambda }{\displaystyle \sum _j}\left(B_j^2+B_j^{{†}_2}-2B_j^{{†}_{}}{B}_j-{\delta }_{jj}\right)+\frac{1}{2}\mu {\omega }_0^2{r}^2\mathrm{.}\hfill \end{array}$

Now, let us assume that the ground state normalized wave function $\psi =\phi \left(r\right)|0\rangle$ is chosen, where $|0\rangle$ satisfies $B_j|0\rangle =a_W|0\rangle =0$. $\phi \left(r\right)$ is wave function describing the relative motion of the system. Due to the influence of the parabolic potential, $\phi \left(r\right)$ can be written as:

17$|\phi (r)\rangle ={\left(\pi \right)}^{-\frac{3}{4}}{\lambda }_0^{\frac{3}{2}}\exp (\frac{-{\lambda }_0^2{r}^2}{2})\mathrm{.}$

Here, the expectation value of the total energy can be described as:

18$\begin{array}{ll}E\left(\lambda \mathrm{,}{\lambda }_0\mathrm{,}r\right)=\hfill & \langle \phi (r)\left|H\right|\phi (r)\rangle =\frac{3{\hslash }^2{\lambda }_0^2}{4m}-\frac{2e^2{\lambda }_0}{\sqrt{\pi }{\epsilon }_{\infty }}+\frac{3}{4}\hslash \lambda +2\alpha \hslash {\omega }_{\text{LO}}\left[-1+\text{Exp}\left[\frac{u^2}{4{\lambda }_0^2}\right]\text{Ercf}\left[\frac{u}{2{\lambda }_0}\right]\right]+\hfill \\ \hfill & \alpha \hslash {\omega }_{\text{LO}}\frac{M}{\mu }\left[\frac{1}{2}\left({\beta }_{\text{e}}^2+{\beta }_{\text{h}}^2\right)+{\beta }_{\text{e}}{\beta }_{\text{h}}\left[\frac{-2u+\frac{\text{Exp}\left[\frac{u^2}{4{\lambda }_0^2}\right]\sqrt{\pi }\left(2{\lambda }_0^2+u^2\right]\text{Erfc}\left[\frac{u}{2{\lambda }_0}\right]}{{\lambda }_0}}{2{\lambda }_0\sqrt{\pi }}\right]\right)+\frac{3{\omega }_0^2\hslash }{4\lambda }+\frac{3\mu {\omega }_0^2}{4{\lambda }_0^2}\mathrm{.}\hfill \end{array}$

Here $\text{Erfc}\left[\frac{u}{2{\lambda }_0}\right]\approx 0$ as $\frac{u}{2{\lambda }_0}\gg 2$. Thus, $F\left(\lambda \mathrm{,}{\lambda }_0\mathrm{,}r\right)$ variational is made to $\lambda \mathrm{,}{\lambda }_0$, i.e $\frac{\partial E\left(\lambda \mathrm{,}{\lambda }_0\mathrm{,}r\right)}{\partial \lambda }=0$, $\frac{\partial E\left(\lambda \mathrm{,}{\lambda }_0\mathrm{,}r\right)}{\partial {\lambda }_0}=0$. Accordingly, we can obtain:

19$\frac{3}{4}\hslash -\frac{3{\omega }_0^2\hslash }{4{\lambda }^2}=\mathrm{0,}$

20$\frac{3{\hslash }^2{\lambda }_0}{2\mu }-\frac{2e^2}{\sqrt{\pi }{\epsilon }_{\infty }}+\frac{\alpha {\beta }_{\text{e}}{\beta }_{\text{h}}\hslash M{\omega }_{\text{LO}}}{\mu {\lambda }_0^2\sqrt{\pi }}-\frac{3\mu {\omega }_0^2}{2{\lambda }_0^3}=0.$

The ground state binding energy of the exciton (Fig. 2(a)) is obtained by bringing $\lambda \mathrm{,}{\lambda }_0$ back to $F\left(\lambda \mathrm{,}{\lambda }_0\mathrm{,}r\right)$:

21$E_{\text{b}}=\frac{3{\hslash }^2{\lambda }_0^2}{4m}-\frac{2e^2{\lambda }_0}{\sqrt{\pi }{\epsilon }_{\infty }}+\frac{3}{4}\hslash \lambda -2\alpha \hslash {\omega }_{\text{LO}}+\alpha \hslash {\omega }_{\text{LO}}\frac{M}{\mu }\left[\frac{1}{2}\left({\beta }_{\text{e}}^2+{\beta }_{\text{h}}^2\right)-\frac{u{\beta }_{\text{e}}{\beta }_{\text{h}}}{\sqrt{\pi }{\lambda }_0}\right]+\frac{3{\omega }_0^2\hslash }{4\lambda }+\frac{3\mu {\omega }_0^2}{4{\lambda }_0^2}\mathrm{.}$

The exciton ground state energy (Fig. 2(a)) is:

22$E_0=E_{\text{g}}+\frac{3{\hslash }^2{\lambda }_0^2}{4m}-\frac{2e^2{\lambda }_0}{\sqrt{\pi }{\epsilon }_{\infty }}+\frac{3}{4}\hslash \lambda -2\alpha \hslash {\omega }_{\text{LO}}+\alpha \hslash {\omega }_{\text{LO}}\frac{M}{\mu }\left[\frac{1}{2}\left({\beta }_{\text{e}}^2+{\beta }_{\text{h}}^2\right)-\frac{u{\beta }_{\text{e}}{\beta }_{\text{h}}}{\sqrt{\pi }{\lambda }_0}\right]+\frac{3{\omega }_0^2\hslash }{4\lambda }+\frac{3\mu {\omega }_0^2}{4{\lambda }_0^2}\mathrm{.}$

The numerical results obtained from the analytical research of selecting Al_xGa_1−xAs crystal are discussed. In the following, the exciton parameters in the Al_xGa_1−xAs crystal^[35−37] are shown in Table 1.

Despite that all the exciton energies E_b, E₁ and E_g increased with an increase in the Al concentration of Al_xGa_1−xAs crystals, certain differences were observed. As can be seen in Fig. 3(a), GaAs and AlAs are direct and indirect semiconductors, respectively. Moreover, an increase in the Al concentration leads the Al_xGa_1−xAs crystal to change from a direct to an indirect semiconductor when x > 0.45 (Fig. 3(b)). Furthermore, the comparison of the bandgap energy E_g when x < 0.45 with the bandgap energy E_g when x > 0.45, revealed a rapid increase in the bandgap energy E_g when Al_xGa_1−xAs is a direct semiconductor; this is while the rate of the increase was found to slow down when Al_xGa_1−xAs is an indirect semiconductor. Furthermore, an increase in the Al concentration was observed to uniformly increase the exciton ground state binding energy E_b, which did not vary by the properties of semiconductor. In addition, the exciton ground state energy E₁ was found to rapidly increase when x < 0.45, but remains almost unchange when x > 0.45. In the following, the underlying reasons for this phenomenon are discussed and investigated.

As can be seen in Figs. 4(a) and 4(b), an increase in the Al component was found to lead to an increase in the ground state binding energy and ground state energy of exciton. However, the opposite trend was observed for the ground state energy and the ground state binding energy of exciton. This is to say that while an increase in the eigenfrequency led to a decrease in the ground state binding energy of the exciton, it resulted in an increase in the ground state energy. More importantly, while varying material properties did not change the ground state binding energy, this was not the case for the ground state energy. The ground state binding energy represents the energy that binds electrons and holes together. As can be seen in Fig. 2(b), coupling between the exciton and phonon is enhanced with with an increase in Al concentration. This, accordingly, enhances the exciton self-trapping effect, which in turn inhibits the ability of the exciton to pass through the crystal and raises the binding energy. As can be seen in Figs. 4(c) and 4(d), by an increase in the eigenfrequency, more electrons and holes are bound to their respective positions, which accordingly, causes the exciton to be more stable and more difficult to recombine. This, in turn, leads to an increase and a decrease in the binding energy and the ground state energy, respectively. Using a simple approximation method, Chuu et al.^[36] studied the subband structure and the binding energy of an exciton in the GaAs/AlGaAs superlattice. Both the exciton binding energy and the subband energy were expressed as a function of well width, barrier width, and Al composition. They also considered the influence of the effective-mass mismatch. The energy spacings between the interband or the intersubband transitions were calculated and compared with the observed data, which revealed a good agreement. While the width of the bandgap is considered to be the main difference between the direct and the indirect semiconductors, the ground state binding energy is almost unaffected by this factor. In other words, the binding energy is unaffected by variation in the properties of the material. Furthermore, since the ground state energy of an exciton determines the energy of the exciton recombination, variation in the bandgap is expected to directly affect the ground state energy. Also, in case of a direct semiconductor, an increase in the Al concentration leads to the enhancement of the exciton self-trapping effect. Consequently, the ability of the exciton to pass through the crystal is inhibited. So, the exciton is limited to a smaller range, resulting in the reduction of the energy required for the exciton recombination. Moreover, electrons and holes are located in the conduction and the valence bands, respectively. Therefore, an increase in the eigenfrequency causes them to be more strongly restricted in the conduction and the valence bands (see Figs. 4(c) and 4(d)), resulting in more stable exciton. However, during the exciton recombination process (see Fig. 4(e)), the presence of phonon extinction results in a minimal effect of exciton-phonon interaction on the exciton state. Additionally, the exciton recombination is accompanied with a change in electron-hole momentum, which leads to a weak effect of the confined intensity.

Applying the linear combination operator and unitary transformation methods, the present study investigated the ground state binding energy and ground state energy of exciton in Al_xGa_1−xAs semiconductor. The isotropic parabolic potential and the Al concentration were also studied. Accordingly, the following results are drawn:

(1) The bandgap energy of Al_xGa_1−xAs semiconductor was observed to change by an increase in Al concentration. Moreover, while the Al_xGa_1−xAs semiconductor was found to be a direct semiconductor at Al concentration below 0.45, the Al_xGa_1−xAs semiconductor was observed to be an indirect one at Al concentration above 0.45;

(2) While the ground state binding energy of the exciton was found to always increase with an increase in the Al concentration, it decreased with an increase in the eigenfrequency and, accordingly, not influenced by variations in the properties of the semiconductor;

(3) When the Al_xGa_1−xAs semiconductor was direct, the ground state energy of exciton increased with the increase of Al concentration; however, in the case where the semiconductor was indirect, slight changes were observed in the ground state energy of exciton with an increase in the Al concentration. Hence, the obtained results and properties can significantly contribute to the experimental studies on correlation exciton effect.

Moreover, the findings of the study can potentially be used in adjusting the exciton energy level in band gap engineering of semiconductor doping as well as for studying the luminescent properties of materials.