Theoretical framework

Determination of the energies of 1s and 1p states

To calculate the energies of 1s and 1p states, we consider a confined exciton X(e,h) in a SCSQD (AlN/GaN). In the EMA framework, the Hamiltonian of the system, considering the contribution from applied pressure P, can be expressed as:

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

In Eq. (1), $r_{{\rm eh}} = \vert {\overrightarrow {r_{\rm e}} - \overrightarrow {r_{\rm h}}} \vert $, where $\overrightarrow {r_{\rm e}} $ and $\overrightarrow {r_{\rm e}} $ are the positions of an electron and hole, respectively. $m_i^* \lpar P \rpar $ (i = e,h) are the effective masses (EMs) of the electron and hole, respectively. The dielectric constant ε(P), as the function of the applied pressure, can be written as, according to Refs.^{[Reference Ha and Ban34–Reference Duque, Morales, Mora-Ramos and Duque36]}:

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

where f _i is the Phillips ionicity parameter of GaN, and B ₀ can be written as:

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

The EMs of the electron and hole as the function of the applied pressure can be determined by^{[Reference Eshghi37]}

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

where m ₀ is the free electron mass and C _i is a fixed value for GaN. The band gap of GaN at the center of the Brillouin zone (point Γ), E _g(P), which is pressure-dependent, can be written as^{[Reference Zhang and Shi38]}:

(5)$$E_{\rm g}\lpar P \rpar \;\lpar {{\rm meV}} \rpar = E_{\rm g}\lpar 0 \rpar \;\lpar {{\rm meV}} \rpar + \vartheta P + \beta P^2.$$

In the above relations, the pressure is given in the unit of GPa. The band offset between AlN (E _g = 6.2 eV) and GaN (E _g = 3.4 eV) is very large, and therefore the presence probability of the carriers is important only in the GaN region (shell) and therefore the charge carriers' wave functions stay confined in the shell. As a result, an infinite potential barrier outside the shell is a good approximation to describe the problem:

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

where a and b are the core/shell radii, respectively.

Yu et al.^{[Reference Yu and Cordona39]} have demonstrated that the QD volume is strongly affected by the hydrostatic pressure effects. It can be generalized to obtain the following equation of the core/shell radii:

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

where R _i (0) is the radius when external pressure is zero. S ₁₁ and S ₁₂ are the compliance parameters of GaN defined by the elastic constants C ₁₁ and C ₁₂ in the following relations^{[Reference Baghramyan, Barseghyan, Kirakosyan, Restrepo and Duque20,Reference Culchac, Porras-Montenegro and Latge40,Reference Barseghyan, Mora-Ramos and Duque41]} :

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

and

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

Adopting the excitonic units $a_X = \hbar ^2{\it \varepsilon} /\mu \; e^2$ for length and $R_X = \hbar ^2/2\mu \; a_X^2 $ for energy, where $\mu ^{ - 1} = (m_{\rm e}^*)^{-1}+(m_{\rm h}^*)^{-1}$ is the excitonic reduced mass, the Hamiltonian (1) can also be written as:

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

where $\sigma = m_{\rm e}^* /m_{\rm h}^* $ is the electron to hole mass ratio.

In fact, such a two-body system is well-suited to be described by the Hylleraas coordinates (r _e, r _h, r _eh, z _e, z _h).^{[Reference Dujardin, Feddi, Assaid and Oukerroum42–Reference El Khamkhami, Feddi, Assaid, Dujardin, Stébé and Diouri44]} So the Laplacian, for the electron (positron), can be written as:

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

For positron, Δ_h can be obtained by substituting the index e by h.

Therefore, in such a coordinate system, the Schrödinger equation is given by 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). The eigenvalue energy E _X can be obtained by the minimization of the mean value of H _X:

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

As mentioned earlier, our study is devoted to the excitonic transition from the ground state 1s (n = 1,  l = 0,   m = 0) to the first excited state 1p (n = 1,   l = 1,   m = 0). For this reason, the trial wave functions can be chosen as:

(13)$$\psi _{1s} = N_0J_0\lpar {r_{\rm e}} \rpar J_0\lpar {r_{\rm h}} \rpar Y_0^0 \lpar {\theta_{\rm e},\varphi_{\rm e}} \rpar Y_0^0 \lpar {\theta_{\rm h},\varphi_{\rm h}} \rpar {\rm exp}\lpar { - \gamma r_{{\rm eh}}} \rpar $$

(14)$$\psi _{1p} = N_1J_1\lpar {r_{\rm e}} \rpar J_1\lpar {r_{\rm h}} \rpar Y_1^0 \lpar {\theta_{\rm e},\varphi_{\rm e}} \rpar Y_1^0 \lpar {\theta_{\rm h},\varphi_{\rm h}} \rpar r_{{\rm eh}}{\rm exp}\lpar { - \delta r_{{\rm eh}}} \rpar $$

where J ₀(r _i) and J ₁(r _i) are the zero order and the first order spherical Bessel functions, respectively, considering the core and shell radii.^{[Reference Aghoutane, El-Yadri, El Aouami, Feddi, Dujardin, El Haouari, Duque, Nguyen and Phuc23,Reference El Haouari, Talbi, Feddi, El Ghazi, Oukerroume and Dujardin30]} N ₀ and N ₁ are the normalization constants. $Y_{\rm l}^{\rm m} \lpar {\theta, \varphi} \rpar $ represent the spherical harmonics. exp( − γr _eh) and exp( − δr _eh) describe the coulombic spatial overlapping between the electron and hole for the ground and first excited states respectively. γ and δ are the nonlinear variational parameters, which must be determined in order to minimize the energy mean value E _X.

Optical properties

To discuss optical properties, we recall that the total refractive index (TRI) change is given by, based on the compact density matrix approach^{[Reference Vahdani and Rezaei45–Reference Abraham, Mark and John Peter47]}:

(15)$$\displaystyle{{\Delta n\lpar \omega \rpar } \over {n_{\rm r}}} = \displaystyle{{\Delta ^{\lpar 1 \rpar }n\lpar \omega \rpar } \over {n_{\rm r}}} + \displaystyle{{\Delta ^{\lpar 3 \rpar }n\lpar {\omega, I} \rpar } \over {n_{\rm r}}}$$

Δ⁽¹⁾n(ω) and Δ⁽³⁾n(ω, I) are the linear RI (LRI) and third-order nonlinear RI (NLRI), respectively, given by the following equations:

(16)$$\displaystyle{{\Delta ^{\lpar 1 \rpar }n\lpar \omega \rpar } \over {n_{\rm r}}} = \displaystyle{1 \over {2\varepsilon _0n_{\rm r}^2}} \displaystyle{{\sigma _1{\vert {M_{{\rm fi}}} \vert } ^2\lpar {E_{{\rm fi}} - \hbar \omega} \rpar } \over {{\lpar {E_{{\rm fi}} - \hbar \omega} \rpar }^2 + {\lpar {\hbar \Gamma_{{\rm fi}}} \rpar }^2}}$$

and

(17)$$\eqalign{\displaystyle{{\Delta ^{\lpar 3 \rpar }n\lpar {\omega, I} \rpar } \over {n_{\rm r}}} = & - \displaystyle{{\mu _1cI\sigma _1{\vert {M_{{\rm fi}}} \vert } ^4} \over {\varepsilon _0n_{\rm r}^3}} \displaystyle{{\lpar {E_{{\rm fi}} - \hbar \omega} \rpar } \over {{\lsqb {{\lpar {E_{{\rm fi}} - \hbar \omega} \rpar }^2 + {\lpar {\hbar \Gamma_{{\rm fi}}} \rpar }^2} \rsqb }^2}} \cr & \times \left[ 1 - \displaystyle{{{\lpar {M_{{\rm ff}} - M_{ii}} \rpar }^2} \over {4{\vert {M_{{\rm fi}}} \vert }^2(E_{{\rm fi}}^2 + {\lpar {\hbar \Gamma_{{\rm fi}}} \rpar }^2}}\right. \cr & \times \left\{ \vphantom{\left[ 1 - \displaystyle{{{\lpar {M_{{\rm ff}} - M_{ii}} \rpar }^2} \over {4{\vert {M_{{\rm fi}}} \vert }^2(E_{{\rm fi}}^2 + {\lpar {\hbar \Gamma_{{\rm fi}}} \rpar }^2}}\right.} E_{{\rm fi}}\lpar {E_{{\rm fi}} - \hbar \omega} \rpar - {\lpar {\hbar \Gamma_{{\rm fi}}} \rpar }^2\right. \cr & \left. \left. \vphantom{\left[ 1 - \displaystyle{{{\lpar {M_{{\rm ff}} - M_{ii}} \rpar }^2} \over {4{\vert {M_{{\rm fi}}} \vert }^2(E_{{\rm fi}}^2 + {\lpar {\hbar \Gamma_{{\rm fi}}} \rpar }^2}}\right.} \times - {\lpar {\hbar \Gamma_{{\rm fi}}} \rpar }^2\displaystyle{{2\lpar {2E_{{\rm fi}} - \hbar \omega} \rpar } \over {(E_{{\rm fi}} - \hbar \omega )}} \right\} \right]} $$

In the previous expressions [(16) and (17)], E _fi = E _ff − E _ii denotes the transition of energy between the states 1p and1s. $M_{{\rm fi}} = e\psi _i\vert {\overrightarrow {r_{\rm e}} - \overrightarrow {r_{\rm h}}} \vert \psi _{\rm f} = e\psi _i\vert {r_{{\rm eh}}} \vert \psi _{\rm f}$ denotes the electrical dipole moment of the transition 1s to 1p under the selection rule (Δl = ± 1). c is the speed of light in vacuum. σ ₁ is the density of electron given by: σ ₁ = n/V(P), where V(P) is the volume of QD under pressure, I is the incident light intensity, μ ₁ is the permeability, n _r is the RI of the GaN, ε ₀ denotes the vacuum dielectric constant under zero pressure, $\hbar \omega $ is the energy of photon, Γ_fi is the non-diagonal matrix element, equal to 1/τ _fi, where τ _fi is defined as initial and final state relaxation rates. Following the same steps one can get the TAC:

(18)$$\alpha \lpar {\omega, I} \rpar = \alpha ^{\lpar 1 \rpar }\lpar \omega \rpar + \; \alpha ^{\lpar 3 \rpar }\lpar {\omega, I} \rpar $$

The linear and third-order nonlinear parts of the AC (LAC and NLAC) are written as:

(19)$$\alpha ^{\lpar 1 \rpar }\lpar \omega \rpar = \omega \sqrt {\displaystyle{{\mu _1} \over \varepsilon}} \displaystyle{{\sigma _1\hbar \Gamma _{{\rm fi}}{\vert {M_{{\rm fi}}} \vert } ^2} \over {{\lpar {E_{{\rm fi}} - \hbar \omega} \rpar }^2 + {\lpar {\hbar \Gamma_{{\rm fi}}} \rpar }^2}}$$

and

(20)$$\eqalign{\alpha ^{\lpar 3 \rpar }\lpar {\omega, I} \rpar =& - \omega \sqrt {\displaystyle{{\mu _1} \over \varepsilon}} \left( {\displaystyle{I \over {2\varepsilon_0n_{\rm r}c}}} \right)\; \displaystyle{{4\sigma _1\hbar \Gamma _{{\rm fi}}{\vert {M_{{\rm fi}}} \vert } ^4} \over {{\Big[ {{\lpar {E_{{\rm fi}} - \hbar \omega} \rpar }^2 + {\lpar {\hbar \Gamma_{{\rm fi}}} \rpar }^2} \Big] }^2}} \cr& \left[ {1 - \displaystyle{{{\lpar {M_{{\rm ff}} - M_{ii}} \rpar }^2} \over {4{\vert {M_{{\rm fi}}} \vert }^2}}\left\{ {\displaystyle{{3E_{{\rm fi}}^2 - 4\hbar\!\! \omega E_{{\rm fi}} + \hbar^2\lpar {\omega^2 - \Gamma_{{\rm fi}}^2} \rpar } \over {E_{{\rm fi}}^2 + {\lpar {\hbar \Gamma_{{\rm fi}}} \rpar }^2}}} \right\}} \right]}$$

Results and discussion

The purpose of this paper is to investigate how hydrostatic pressure and core/shell sizes affect optical properties, such as the changes of the optical responses (LAC, NLAC, LRI, and NLRI) of a confined exciton in SCSQDs, the physical parameters of which are listed in Table I.

Table I.

Physical parameters of GaN material.^{[Reference Ha and Ban34,Reference Zhang and Shi38]}

Before we start our discussion, we should note that the energy difference E _fi = E _ff − E _ii, the electric dipole moment M _fi, and incident electromagnetic field strength I are the main parameters affecting the optical properties, where the AC and RI expressions depend strongly on these parameters.

To show the energy difference contribution to the optical responses, and to indicate the confinement effect on these energies, shown in Fig. 1(a), the transition of the energy E _fi = E _1p − E _1s, for (b = 2 and 3a _x) versus the core/shell radii ratio a/b and for different pressure values. It clearly shows that, the E _fi increases for small radii due to the size reduction effect. For the strong confinement, (a = 2a _x for example) the energy difference includes two intervals: one between (a/b = 0 and a/b = 0.6), where it decreases and the other between (a/b = 0.6 and a/b = 1), where it increases. Whereas for larger radii (weak confinement), the energy difference is always an increasing function. We remark also that when a tends to b (a/b = 1), the difference E _1p − E _1s is much larger because the confinement becomes very strong; therefore, the kinetic energies of the uncorrelated particles dominate, in this case, the total energy of exciton. Concerning the effects of hydrostatic pressure on E _fi [dashed line in Fig. 1(a)], we remark that E _fi increases when the pressure is applied, i.e., the pressure behaves as an additive confinement; however, the orbitals waves functions are less sensitive to pressure when a → b, because of the excitonic orbital is less expansive, and therefore the correlated pair (electron–hole) is not influenced by this applied pressure.

Figure 1.

(a) Energy difference as a function of the radii ratio a/b: (b = 2a _X and 3a _x) and for different pressure values (solid line: P = 0 GPa and dashed line: P = 4 GPa). (b) The electric dipole moment as the function of the radii ratio a/b: (b = 2a _X and 3a _x) and for different pressure values (solid line: P = 0 GPa and dashed line: P = 4 GPa).

In Fig. 1(b) we plot the variation of the electric dipole moment M _fi against the radii ratio a/b for b = 2a _x and 3a _x (solid line) and under different pressure values (dashed line). We notice that the electric dipole moment increases with decreasing SCSQD size; however on the other hand, the M _fi reduces when the core/shell radii ratio a/b increases and tends to 0 for a/b → 1. The pressure (dashed line) has a little effect, where the M _fi slowly diminishes when the pressure increases. Another remark can be added: the influence of pressure is less pronounced for strong confinement (a → b) than for weak confinement.

Now let us analyze how the SCSQD sizes influence the behaviors of the optical properties. We plot in Fig. 2(a) the LAC, NLAC, and TAC versus $\hbar\hskip-2.3pt \omega $ for two shell size values: b = 2a _x and 3a _x and for a = b/2. First, we remark from Eqs. (16) and (19), that the LAC and LRI constitute the major contribution to the TAC and TRI, and the NAC and NRI vary oppositely with respect to the LAC and LRI terms, so the NAC and NRI terms reduce the TAC and TRI. From Fig. 2(a), we observe that the two components of the TAC have a prominent peak corresponding to the threshold frequency of the incident light. This threshold is equal to 38 and 22 meV for b = 2a _x and b = 3a _x respectively, so the optical AC are strongly influenced by the shell sizes where, the peaks move to the region of higher energies (blueshift) when the shell radius decreases. This behavior is related to an increase of the excitonic difference energy [Fig. 1(a)] when the radius decreases, due to the confinement effect. Indeed, the confinement effect reinforces the coulombic particle attraction, and therefore the excitonic energies become dominating, and consequently the E _fi increase as the size decreases. Furthermore, we can see that the AC amplitudes decrease with the reduction of the size. This behavior can be justified by the decrease of the M _fi [Fig. 1(b)], when the shell radius diminishes. In Fig. 2(b), we have presented the RI against $\hbar \omega $ for b = 2a _x and 3a _x and for a = b/2. As we found in the optical AC cases, the RI peaks follow the same behavior as that of the AC, they move to the low energies (redshift), and peak amplitudes increase when the confinement becomes weak.

Figure 2.

(a) The LAC, NLAC, and TAC versus the incident $\hbar\hskip-2.3pt \omega $: (b = 2a _X and 3a _x) and a = b/2. (b) The LRI, NLRI, and TRI versus the incident $\hbar \hskip-2.3pt \omega $: (b = 2a _X and 3a _x) and a = b/2.

To describe the effects of hydrostatic pressure (P) on the optical responses of a SCSQD, we plot in Fig. 3(a), the LAC, NLAC, and TAC versus $\hbar \hskip-2.3pt \omega $ for b = 2a _x, a = b/2 and under different pressure values. As expected, we see that the different curves of the optical AC are deeply affected by pressure application, and they shift to the higher energies (blueshift: shifted from 38 meV for P = 0 to about 42.5 meV and 46.5 meV for P = 2 and 4 GPa respectively). This is due to, on the one hand, the pressure enhances the confinement by reducing the lattice parameter of GaN material. The pressure-dependent behavior of GaN material can be generalized by the formula a _GaN = a _GaN(1 − P/3B ₀)^{[Reference Haa, Liua and Zhu32]}; and on the other hand, the applied pressure strongly influences the dielectric constant [Eq. (2)] and the reduced excitonic mass [Eq. (4)]. All these parameters lead to an increase of the E _fi [Fig. 1(a)], and consequently a shift toward high energies of the optical AC can be remarked. In Fig. 3(b), we present the variation of the LRI, NLRI, and TIR versus $\hbar \hskip-2.3pt \omega $ for b = 2a _x, a = b/2, and for P = 0, 2 and 4 GPa. The same behavior of the optical AC showing a blueshift with increasing pressure was found for the optical RI. Concerning the AC and RI intensities, we observe an increase and “stability” of the AC and RI peaks respectively. These behaviors can be explained based on the meaning of the AC and RI expressions. By analyzing Eqs. (19) and (20) we see that AC is related to the transition energy E _fi, the dielectric constant ε(P), and the electric dipole element M _fi. These parameters are functions of the pressure with different ways, which induce a competition between them. The calculus shows that the applied pressure influences ε(P) more than M _fi and E _fi. For the AC, the amplitudes are dominated by a pressure dependent dielectric constant, and therefore an intensity strengthening occurs. Eqs. (16) and (17) show that RI does not depend on ε(P), and therefore, their amplitudes are dominated by the electric dipole element which depends on P [Fig. 1(b)], which explains the “stability” of RI intensities when pressure increases.

Figure 3.

(a) The LAC, NLAC, and TAC versus the incident $\hbar \hskip-2.3pt \omega $: (b = 2a _X and a = b/2) and for different pressure values. (b) The LRI, NLRI, and TRI versus the incident $\hbar\hskip-2.3pt \omega $: (b = 2a _X and a = b/2) and for different pressure values.