## 3 Results and Discussions
In order to analyse the effect of the confinement on these properties, our calculations are developed for two dot sizes \(R=2~{}a_{D}\) and \(3~{}a_{D}\). We see clearly that these components of AC are strongly dependent on the dot size. As it is well known, the \((D^{+},X)\) complex energy decreases when the dot size increases, due to the decrease of the Coulomb interactions which leads to a diminishing of the transition energy \(E_{fi}=E_{1p}-E_{1s}\). For this reason the curves of the AC show a shift to higher (lower) terahertz frequencies region when the dot size decreases (increases). We remark also that the peaks amplitude increases for strong confinement because the energy difference \(E_{fi}\) dominates the tendency of strength of \(AC\). In Figure 2b, the three parts of RI are drawn against the incident photon energy, as in the case of the AC, all components of RI exhibit the same behaviour, they shift to the higher (lower) energies, for large (small) QD sizes. Finally, we should notice that the peaks resonance position are about 6.38 THz and 2.6 THz for \(R=2~{}a_{D}\) and \(3~{}a_{D}\) respectively, the importance of these results is that the absorption thresholds are located in the range of the low terahertz frequencies. Thus, the control of the frequencies can be realized by tuning the size of the dots. These results can be exploited in the conception of new optoelectronic devices [51; 52; 53], where successful deposition of multi AlAs/GaAs/AlAs spherical quantum dots on substrates (and specifically on graphene) is a key milestone toward the goal to use spherical quantum dots and graphene quantum dots for in situ ultrasensitive terahertz spectroscopy [3; 54; 55; 56]. To give a good picture of the temperature and pressure effects on the \((D^{+},X)\) complex optical responses, we draw in Figure 3a,b the linear and third non-linear and total AC and RI, respectively, versus the incident photon energy \(\hbar\omega\) for different values of \(P\) and \(T\) (\(P=0\), \(P=25\) kbar and \(T=10\), \(T=250\) K). In the case of the pressure effect, we remark that the applied pressure causes an AC shift toward the high value of therahrtz frequencies with an intensities enhancement (Figure 3a). This behaviour was expected because the pressure reinforces the confinement by reducing the \(GaAs\) lattice parameter \(a_{0}\). This dependence of the lattice parameter of \(GaAs\) can be modeled by the following formula: \(da/dP=-2.6694\times 10^{-4}a_{0}\)[57]; so when the pressure is applied the dot size decreases, i.e., the pressure strengths the spatial confinement by shrinking the orbital  
Figure 2: The absorption coefficients (**a**) and the refractive index changes (**b**) as function of incident photon energy \(\hbar\omega\) for \(R=2\) and \(3~{}a_{D}\). wave functions of the three particles (hole, electron and ionized donor) and behaves as an additive confinement and therefore the transition energy \(E_{fi}\) increases and consequently a shift of the peaks toward the high value of therahrtz frequencies is observed. On the other hand, and by looking the AC and RI expressions, we see that AC depends on the electric dipole matrix element \(|M_{fi}|\), the transition energy \(E_{fi}\) and the dielectric constant \(\varepsilon(P)\), which are related to the pressure in a different way, so a competition between these quantities appears. Our calculations indicate that the pressure application causes a big change in \(\varepsilon(P)\) compared to \(|M_{fi}|\) and \(E_{fi}\)[26], so the intensities of AC are dominated by the dielectric constant dependent of pressure. Concerning the RI variation (Figure 3b), the same behaviour of AC was found for all components of RI. Regarding the RI amplitudes, we have found a \"stability\" of intensities when pressure is applied, because RI is independent of \(\varepsilon(P)\)[28].