Theoretical framework

Let us consider an exciton X(e,h) positioned in a AlN/GaN/AlN SCSQD. Using the effective mass approximation and ignoring any induced polarization change at the surface, the Hamiltonian of this system under an external hydrostatic pressure P can be given by:

(1)$$H_X = - \displaystyle{{\hbar} ^2 \over {2m_{\rm e}^{\ast} (P)}}\Delta _{\rm e} - \displaystyle{{\hbar} ^2 \over {2m_{\rm h}^{\ast} (P)}}\Delta _{\rm h} - \displaystyle{{e^2} \over {\varepsilon (P)r_{{\rm eh}}}} + V_{\rm w}^{\rm e} + V_{\rm w}^{\rm h},$$

with $r_{{\rm eh}} = \left\vert {{\vec r}_{\rm e} - {\vec r}_{\rm h}} \right\vert$, where $\vec r_{\rm e}$ and $\vec r_{\rm h}$ are the positions of the electron and hole, respectively. ε(P) is the pressure-dependent dielectric constant, and $m_i^{\ast} (P)(i = {\rm e},{\rm h})$ are the effective masses of the electron and hole, also pressure dependent.

The effect of hydrostatic pressure on the dielectric constant can be obtained by the following equation^[Reference Ha and Ban^11, Reference Wagner and Bechstedt^15, Reference Duque, Morales, Mora-Ramos and Duque^16]:

(2)$$\varepsilon (P) = 1 + [\varepsilon (0) - 1]{\rm exp}\left[ { - \displaystyle{5 \over {3B_0}}(0.9 - f_{\rm i})P} \right],$$

where f _i is the Phillips ionicity parameter of GaN, B ₀ is given by:

(3)$$B_0 = \displaystyle{{(C_{11} + C_{12})C_{33} - 2{(C_{13})}^2} \over {C_{11} + C_{12} + 2C_{33} - 4C_{13}}}.$$

The hydrostatic pressure's effect on the effective masses of the electrons and holes can be calculated by^[Reference Eshghi^17]:

(4)$$m_{i\;} ^{\ast} = \displaystyle{{m_0} \over {1 + C_i/E_{\rm g}(P)}},\quad \; i = ({\rm e,\; h}),$$

where m ₀ is the mass of free electron and C _i is a constant for GaN. E _g(P) is the pressure-dependent band gap of GaN at the Γ center of the Brillouin zone, which can be calculated (in meV) with the following expression^[Reference Zhang and Shi^18]:

(5)$$E_{\rm g}(P) = E_{\rm g}(0) + \alpha P + \beta P^2.$$

It should be noted that in all the relations above the pressure GPa are used. Because of the considerable band offsets between AlN(E _g = 6.2 eV) and GaN(E _g = 3.4 eV), the probability density (therefore the wave function) is only important in the GaN region, and the charge carriers (electrons and holes) are therefore considered confined in the GaN well. As a consequence, it is a good approximation to assume an infinite potential barrier to represent the confining potentials:

(6)$$V_{\rm w}^i (r_i) = \left\{ {\matrix{ {0, \; \;\ {\rm if} \,a \lt r_i(P) \lt b}\hfill \cr {\infty, \; \; {\rm if}\; r_i\left( P \right) \lt a\; \,{\rm and}\,\; r_i\left( P \right) \gt b \quad} \cr} i = ({\rm e,\; h})} \right..$$

It is well established that the hydrostatic pressure significantly influences the volume of solid materials. Yu and Cordona^[Reference Yu and Cordona^10] have reported that low-dimensional semiconductor systems are also affected by hydrostatic pressure. They demonstrated that the volumes as well as energy states of such systems under pressure undergo deformation caused by changes in the elastic constants. Such behavior can be described by assuming the core/shell radii following the equation below:

(7)$$\eqalign{& R_i(P) = R_i(0)[1 - 3P(S_{11} + 2S_{12})]^{1/3}, \cr & \quad \; i = ({\rm core,}\,{\rm shell}).}$$

R _i(0) is the radius when no hydrostatic pressure is applied (i.e., P = 0), and S ₁₁ and S ₁₂ are the compliance constants of GaN given by the following expressions of the elastic constants C ₁₁ and C ₁₂^[Reference Culchac, Porras-Montenegro and Latge^19–Reference Barseghyan, Mora-Ramos and Duque^21]:

(8)$$S_{11} = \displaystyle{{C_{11} + C_{12}} \over {(C_{11} - C_{12})(C_{11} + 2C_{12})}}$$

and

(9)$$S_{12} = \displaystyle{{C_{12}} \over {(C_{11} - C_{12})(C_{11} + 2C_{12})}}\;. $$

Taking the excitonic units $a_X = \hbar ^2\varepsilon /\mu e^2$ for length and $R_X = \hbar ^2/2\mu a_X^2 $ for energy, where the reduced excitonic mass is given by $1/\mu {\rm = 1/m}_{\rm e}^{\ast} {\rm + 1/m}_{\rm h}^{\ast} $, and the Hamiltonian (1) is written as:

(10)$$\eqalign{H_X & = - \displaystyle{1 \over {1 + \sigma}} \left[ {\displaystyle{{m_{\rm e}^{\ast} (0)} \over {m_{\rm e}^{\ast} (P)}}} \right]\Delta _{\rm e} - \displaystyle{\sigma \over {1 + \sigma}} \left[ {\displaystyle{{m_{\rm h}^{\ast} (0)} \over {m_{\rm h}^{\ast} (P)}}} \right]\Delta _{\rm h } \cr & - \displaystyle{{2\varepsilon (0)} \over {\varepsilon (P)r_{{\rm eh}}}},}$$

where the electron-to-hole mass ratio comes from σ = $ m_{e}^{\ast}(0) /m_{h}^{\ast}(0) $.

Hylleraas coordinates (r _e, r _h, r _eh, z _e, z _h) are often chosen for convenience, in a two-body system like exciton, for variational integral calculations.^[Reference Dujardin, Feddi, Assaid and Oukerroum^22–Reference El Khamkhami, Feddi, Assaid, Dujardin, Stébé and Diouri^24] The Laplacian transformation for such a system can be given as following:

(11)$$\eqalign{\Delta _{\rm e} = & \displaystyle{{\partial ^2} \over {\partial r_{\rm e}^2}} + \displaystyle{2 \over {r_{\rm e}}}\displaystyle{\partial \over {\partial r_{\rm e}}} + \left( {\displaystyle{{r_{\rm e}^2 - r_{\rm h}^2 + r_{{\rm eh}}^2} \over {r_{\rm e}r_{{\rm eh}}}}} \right)\displaystyle{{\partial ^2} \over {\partial r_{\rm e}\partial r_{{\rm eh}}}} \cr & + \displaystyle{2 \over {r_{{\rm eh}}}}\displaystyle{\partial \over {\partial r_{{\rm eh}}}} + \displaystyle{{\partial ^2} \over {\partial r_{{\rm eh}}^2}} + \displaystyle{{2z_{\rm e}} \over {r_{\rm e}}}\displaystyle{{\partial ^2} \over {\partial z_{\rm e}\partial r_{\rm e}}} \cr & + 2\left( {\displaystyle{{z_{\rm e} - z_{\rm h}} \over {r_{{\rm eh}}}}} \right)\displaystyle{{\partial ^2} \over {\partial z_{\rm e}\partial r_{{\rm eh}}}} + \displaystyle{{\partial ^2} \over {\partial z_{\rm e}^2}} \;,} $$

Δ_h can be obtained by switching the indices “e” to “h”.

The fundamental energy E _X and its corresponding envelope wave function of the exciton X(e,h) are solutions of the Schrödinger equation H _XΨ_X(r _e, r _h, r _eh, z _e, z _h) = E _XΨ_X(r _e, r _h, r _eh, z _e, z _h). This energy is then computed by minimizing the mean value of H _X:

(12)$$E_X = \min \displaystyle{{\langle \psi _X \vert H_X \vert \psi _X\rangle} \over {\langle \psi _X \vert \psi _X\rangle}}. $$

The trial wave function can be chosen this way:

(13)$$\psi _X = \Phi _{\rm e}(r_{\rm e})\Phi _{\rm h}(r_{\rm h}){\rm exp}( - \gamma r_{{\rm eh}}),$$

where $\Phi _i(r_i) = \displaystyle{{sin\{ [{\pi / {(b - a}})]\,(r_i - a)\}} / {r_i}}$ (i = e,h). Coulombic spatial correlation between the electron and the hole is given by exp(−γr _eh), and the non-linear variational parameter, γ, has be calculated so as to obtain the minimized mean value of energy.

The elementary volume dv in Hylleraas coordinates is given by the following expression:

(14)$$dv = \displaystyle{{8\pi r_{\rm e}r_{\rm h}r_{{\rm eh}}dr_{\rm e}dr_{\rm h}dr_{{\rm eh}}dz_{\rm e}dz_{\rm h}} \over D}.$$

The denominator D is written as

(15)$$D = \sqrt {4\left( {r_{\rm e}^2 - z_{\rm e}^2} \right)\left( {r_{\rm h}^2 - z_{\rm h}^2} \right) - {(r_{\rm e}^2 + r_{\rm h}^2 - r_{{\rm eh}}^2 - 2z_{\rm e}z_{\rm h})}^2}. $$

The integrations over z _e and z _hwere carried out following the generalization of Atanasoff integrals^[Reference Atanasoff^25] and performed in the intervals −r _e ≤ z _e ≤ r _e and Z ₁ ≤ z _h ≤Z ₂, where Z ₁ and Z ₂ are the roots of the above expression D.

Results and discussions

The aim of this work is the investigation of the influences of hydrostatic pressure and core/shell sizes of SCSQD on the binding energy of an exciton confined within, whose physical parameters used in the calculation are listed in Table I.

Table I.

Physical parameters of wurtzite GaN.^[Reference Ha and Ban^11, Reference Zhang and Shi^18]

Firstly, let us take the case when a → 0 (HQD); and to investigate the influence of the pressure P on the excitonic binding energy, we present in Fig. 2 the excitonic binding energy for (P = 0, 2, and 4 Gpa) as a function of shell radius b. We remark that the binding energy is defined as the energy difference between uncorrelated and correlated electron hole pair calculated by minimizing the mean value of the Hamiltonian H _X: E _B = E _e + E _h−E _X. From Fig. 2 we remark that for a specific hydrostatic pressure, the binding energy increases as shell radius b decreases due to the confinement effect and tends toward the three-dimensional (3D) binding energy limits of 1R _X for P = 0 Gpa. We also remark that under the strong confinement regime (b≤1.5a _X), the excitonic binding energy is less sensitive to the hydrostatic pressure P because of the prominent strong confinement causing the correlated pair is not influenced by this hydrostatic pressure. But by increasing the HQD size, we find that the pressure effect becomes more pronounced because the excitonic orbital is more expansive near the surface of the HQD and therefore becomes more sensitive to the pressure effect. This increase of binding energy with P is attributed to reduction in the lattice parameters, which is affected by the pressure following the formula: a _GaN = a _GaN(1 − P/3B ₀),^[Reference Ha and Ban^11] where B ₀ is given in equation (3); as we can see, the lattice parameters depend on the pressure and decrease when the pressure is applied and therefore a confinement reinforcement appears, and also due to an increase in the excitonic Rydberg energy since the applied pressure increases the reduced excitonic mass (equation 4) and meanwhile decreases the dielectric constant (equation 2).

Figure 2.

Excitonic binding energy variation as a function of the shell radius for a → 0 and (P = 0, 2, and 4 GPa).

To analyze the size effect of SCSQD on the binding energy, we plot in Fig. 3(a) the excitonic binding energy as a function of the radii ratio a/b for b = 0.8a _X, 1a _X, and 2a _X, our results indicate that the binding energy for b = 0.8a _X is more important than that obtained for b = 1a _X that is also larger compared with b = 2a _X due to the confinement effect, and for the situation (a/b = 0), we observe that the binding energy decreases when the radius b increases and eventually tends to 1R _X, which corresponds to the binding energy of the bulk material (3D limit); this situation is clearly remarked in Fig. 2. We also find that when a/b tends to unity, the binding energy tends to the well-known 2D excitonic binding energy limit of 4R _X. Indeed, we remark that for a given critical value a/b _cri, the binding energy presents a minimum. This minimum moves toward the larger values of a/b when b decreases and vanishes for the large values of b. When a/b varies from critical a/b _cri to unity (minimal value to the 2D limit), the electron and hole constitute a free rotation on the GaN well spherical surface and therefore the coulombic energy becomes less important compared with the kinetic energy, which leads to an increase of the excitonic binding energy. Indeed the increasing of the biding energy beyond the a/b _cri means that the spherical surface acts as 2D materials formed by few layers of GaN. This critical value a/b _cri permits us to delimit between spherical surface confinement and 3D geometrical.

Figure 3.

(a) Excitonic binding energy variation as a function of core-to-shell radii ratio a/b for b = 0, 0.8a _X, 1a _X, and 2a _X; (b) excitonic binding energy variation as a function of core-to-shell radii ratio a/b for b = 1a _X and (P = 0, 2, and 4 GPa).

Now let us discuss the hydrostatic pressure effects. In Fig. 3(b), we plot the excitonic binding energy as functions of the core/shell radii ratio a/b for b = 1a _X and (P = 0, 2, and 4 GPa). We observe that the binding energy increases with the increasing pressure, i.e., that the pressure effects induce an additive confinement (the wave function of the exciton becomes more strongly localized), which will be added to the geometrical one. This is because of a decrease in the lattice parameters of GaN under pressure, as well as an increase of the reduced excitonic mass and a decreased dielectric constant. When the pressure increases, the variation of these material parameters leads to increased Rydberg energy and hence an increase of the excitonic binding energy; in other words, the pressure keeps the electron and hole close and even the SCSQD size becomes larger.

For more physical interpretation of the binding energy behavior discussed above, we have plotted in Figs. 4(a) and 4(b) the mean distance between the hole and electron 〈r _eh〉 defined as 〈r _eh〉 = 〈Ψ_X|r _eh|Ψ_X〉/〈Ψ_X|Ψ_X〉. In Fig. 4(a), we present the 〈r _eh〉, without hydrostatic pressure, for different shell thickness b = 0.8a _X, 1a _x, and 2a _x. Firstly, we remark that the inter-particle distances increase when the shell radii b increase, which is justified by the decreasing of the binding energy. We can also clearly see that for a fixed shell thickness the mean distance increases when the a/b ratio value ranges between 0 and 0.54, for b = 0.8a _X, between 0 and 0.44 for b = 1a _X, and between 0 and 0.14 for b = 2a _X. In these ranges, the electron and hole move away from each other, which leads to diminish the overlapping of the wave functions and therefore a significant drop of the binding energy is remarked. From a/b = 0.54 for b = 0.8a _X, 0.44 for b = 1a _X, and 0.14 for b = 2a _X, the inter-particle distance begins to decrease and tends to 0.5a _X for all shell radii values when a/b tends to unity. This means that the electron and hole become close now, and consequently the wave function of the exciton is more strongly localized, which approves the binding energy increase remarked in these ranges of a/b values. Figure 4(b) represents the variation of the mean value of the electron–hole distance for a fixed shell thickness b = 1a _X and for different hydrostatic pressure values P = 0, 2, and 4 GPa, we found that the electron–hole distance decreases when the pressure increases indicating that the applied pressure acts as a supplemental confinement and strengthens the interactions between the charges carriers, and therefore the binding energy increases as applied pressure increases.

Figure 4.

(a) Average distance between electron and hole 〈r _eh〉 versus core-to-shell radii ratio a/b for different shell radius values: b = 0.8, 1, and 2a _X; (b) average distance between electron and hole 〈r _eh〉 versus core-to-shell radii ratio a/b for different hydrostatic pressure (P = 0, 2, and 4 GPa) and for b = 1a _X.