High pressure and polaron effects on optical properties of GaN wedge-shaped quantum dot

In the paper, the optical properties of GaN wedge-shaped quantum dots under the influence of pressure and with electron-phonon (e-p) interaction are taken into account. To this end, we first exert a polarized monochromatic electromagnetic field on our system and theoretically consider intersubband transitions for the electrons. The energy difference and dipole matrix elements are evaluated by considering the effects of e-p and pressure. Then, the refractive index changes (RICs) and absorption coefficients (ACs) are calculated. It is deduced that the energy difference and dipole matrix elements reduce by considering the e-p and pressure effects. It results that the pressure enhances the e-p effect on the energy difference. The ACs and RICs move to lower energies considering the e-p and pressure effects. The RICs and ACs are increased in the presence of the e-p effect, whereas they are reduced with considering the pressure. Considering the e-p and pressure effects on total ACs, it is seen that the peak of AC splits into two peaks.


Introduction
Quantum dots (QDs) have received more attention in electronics and optoelectronics. QDs can be fabricated with various shapes, such as spherical, hexagonal, and so on (Jayabalan et al. 2003;Gaan et al. 2010;Khordad and Vaseghi 2019;Bahramiyan and Servatkhah 2015;Feng and Xiao 2016). The most promising method for fabricating QDs is based on the self-organization of semiconductor nanostructures in heteroepitaxial systems. Smeeton et al. (2007) reported the growth of InGaN QDs by molecular beam epitaxy at Sharp Laboratories of Europe. They have prepared the wedge-shaped QDs by mechanical thinning using diamond lapping films and final thinning using an ion miller. To give more information about experimental data on the wedge-shaped QDs, the reader can refer to Voitsekhovskii et al. (2014); Arapkina and Yur'ev 2010;Valenta et al. 2019).
The study of the physical properties of QDs has been widely performed in the last decades due to their applications in engineering and industry. QDs are 3D structures that can trap charge carriers in quantized energy levels. It should be noted that the optical properties of QDs can be modified by considering different factors such as external fields, pressure, and electron-phonon (e-p) effect (Miao et al. 2020;Sayari et al. 2022;Khordad and Bahramiyan 2014;Servatkhah and Pourmand 2020).
The e-p effect has a key role in the optical, electronic, and thermodynamic properties of QDs. So far, the e-p effect on the optical properties of QDs has been widely studied. For instance, Guo and Chen (1995) have considered the polaron effect on the optical properties of quantum well. Hong et al. (2002) have studied the polaron effect on the optical properties of the quantum disks. The reader can refer to Verzelen et al. (2002); Yu and Wei (2016); Zhang and Shan (2014); Tang et al. (2018).
There are several theoretical methods for investigating e-p interaction in lowdimensional structures. Examples of the approaches are the dielectric-continuum (DC) model, the hydrodynamic (HD) model, and the reformulated-mode (RM) model (Ridley 2009). It is to be noted that the models have some advantages and disadvantages.
It is well known that pressure can influence the effective mass, dielectric constant, and optical properties of QDs (Guo and Chen 1995;Hong et al. 2002;Verzelen et al. 2002). Also, the optical properties can be alerted under the e-p interaction. In the past decades, the optical and thermodynamic properties of QDs considering the pressure effect have been studied (Eshghi 2009;Wei and Alex 1999;Jahromi and Dezhkam 2020). Lu and Guo (2006) have discussed the polaron effect on the optical properties of Qws.  have considered the polaron effect on the optical properties of 2D quantum pseudodot. Wang and Guo (2008) have studied the exciton effect on the optical properties of GaAs/AlGaAs parabolic QDs. The reader can refer to Hashemi et al. (2022); Khordad (2016); Li et al. (2020).
Electron-phonon effect can affect the physical properties of low-dimensional structures. The previous studies show that the physical properties of QDs under the polaron effect differ from those of bulk materials. Polaron effect on the electronic properties of QDs becomes more important considering the pressure effect. For example, Li and Yan (2012) have considered the e-p and pressure effects on surface states in quantum wells. Thang et al. (2019) have studied the effect of confined phonons on the Hall coefficient in a cylindrical QD with infinite potential. The reader can refer to Ngoc et al. (2017); Qin and Gu (1997); Hua et al. (2003); Bouhassoune et al. (2002); Moussaouy et al. (2005); Khordad and Bahramiyan (2015).
In the present work, we consider a wedge-shaped QD. This shape lacks an inversion center and it is a very good approximation for a triangular-shaped QD. In this work, the e-p effect on the optical properties of a wedge-shaped quantum dot (WSQD) is theoretically investigated considering high pressure. First, the effective mass, dielectric constant, and frequency of optical phonons have been considered a function of pressure. Then, the perturbation theory was used to compute the energy spectrum and wave functions considering the e-p effect. Also, the density matrix and iterative procedures have been employed to evaluate the absorption coefficients (ACs) and refractive index changes (RICs).

Theory and model
Consider a WSQD made of GaN (see Fig. 1). The Hamiltonian of the system is expressed by where H e is the electron Hamiltonian as Here, m * is the electron effective mass, and V( , , z) is the confining potential as Considering the finite potential, we should solve the Schrödinger equation numerically. Therefore, we cannot obtain the e-p interaction as an analytical solution. Since we want an analytical solution, we have considered the infinite potential. According to the final results, we can say the infinite potential is approximately correct. To show the potential effect, we calculate the energy levels in a one-dimensional GaAs quantum well (well thickness of 0.5 nm) with finite and infinite potential. The results are presented in Table 1.
The two last terms of Eq. (1) present the Hamiltonian for LO phonons and the e-p effect, respectively.

Hamiltonian for LO phonons
We should obtain the last two terms of Eq. (1). For this goal, we employ the dielectric continuum (DC) model and write the following relations (Chen et al. 2003) (1) (2) Here, D and E are the electric displacement and electric field, respectively. Also, and P are electric potential and polarization, respectively.
Under free oscillations, the charge density is zero [ 0 (r) = 0 ]. Hence, one can find where TO is the frequency of transverse-optical phonon.
Here, we search the solutions of Eq. (5) for two cases: (i) for ( ) = 0 , and (ii) ∇ 2 = 0 . The first solution shows the confined LO phonons. In the first case, we obtain where LO is the frequency of the LO phonon. Equation (7) shows the confined bulk LO modes of frequency LO . It is to be noted that the potential ( , , z) is an arbitrary function of , , z , and it must satisfy the boundary conditions. The solution of the Laplace equation in the whole space by using cylindrical coordinates is given by where A lmn is the normalization constant.
To obtain the third term, H LO , in Eq.
(1), we should use the following relation for the Hamiltonian of the free vibrations (Chen et al. 2003) In the above equation, loc = + 4 3 and = −4π . In Eq. (9), is the reduced mass of the ion pair, is the relative displacement of the positive and negative ions, n * is the number density of ion pairs, and is the electronic polarizability per ion pair. The local field in the macroscopic approach is given by Klein et al. (1990). Now, we can determine the polarization vectors P = 1 4 ∇ for the confined LO phonons by applying Eq. (4) and the condition = 0 . Employing the following relations (Chen et al. 2003) We can obtain, the third term in Eq. (1), the Hamiltonian for confined LO phonons as (Klein et al. 1990) It should be noted that the LO polarizations form an orthonormal set as (Klein et al. 1990) where mm ′ is the Kronecker delta. Using Eq. (13), we can find the constant of A lmn in Eq. (8). Now, by using the orthogonality property, we can write (Klein et al. 1990) The Hamiltonian for the confined LO phonons is given by The electric potential is expanded as: After writing H LO Hamiltonian, Eq. (16), we can now express the Hamiltonian of e-p interaction.
where B LO lnm is a constant, which can be determined by using Eq. (13). In this work, the onephonon absorption is considered. Therefore, we use the perturbation theory as

Pressure effect
In this part, insert the isotropic pressure effect on the Hamiltonian Eq. (1). To this goal, several parameters of the Hamiltonian Eq. (1) have been considered as the function of pressure (Mori and Ando 1989). Our desired parameters are (i) The electron effective mass, (ii) dielectric constant, (iii) lattice constant, (iv) band gap, and (v) optical phonon energy. The pressure dependence of the parameters is given by.
(i) The effective mass (Ting and Chang 1978) where A P = 13.560eV for GaN. (ii) The dielectric constant (Wagner and Bechstedt 2002) where ∞ (0) is the dielectric constant at P=0, B is the material bulk modulus, and f i is a constant. In the paper, f i =0.5 and B=202 GPa are taken for the GaN. (iii) The band gap (Mori and Ando 1989) where E g (0) is the band gap at P = 0 and = 33meV∕GPa for GaN (Bahramiyan and Servatkhah 2015). (iv) The lattice constant (Perlin et al. 1999), where L(0) is the lattice constant at P=0.
(v) The optical phonon energy (Goni et al. 2001) where LO (0) is the bulk optical phonon frequency P =0 and =0.99 for GaN. We should mention that all used relations in Sect. 2.2 correspond to zero temperature.
Since our system is under pressure, the phonons effect is important. According to Eq. (23), the lattice constant is a function of pressure. With changing pressure, the lattice constant is changed, and thereby the phonon effect becomes important. Previously, other authors have studied confined phonons in infinite potentials (Qin and Gu 1997;Hua et al. 2003).

Optical properties
We assume that our system is excited by a linear polarized monochromatic electromagnetic field as The analytical relations of the linear (1) and nonlinear (3) susceptibility are expressed as (Boyd 2003;Cadirci et al. 2020;Heyn et al. 2021;Pokutnyi 2020) where is the carrier density in the dots. The RIC is written in terms of the susceptibility by where n r is the refractive index. Applying the above equations, the RICs are written as (Boyd 2003;Cadirci et al. 2020;Heyn et al. 2021) and where is the permeability, is the electric dipole moment matrix element (Boyd 2003), and E ij = E i − E j . The expression of |Φ > is the wave function of the system. To calculate the matrix elements of M ij , we consider that the electromagnetic field is radiated in the x-direction. Therefore, we should calculate the following integral where Ψ i ( , , z) is the solution of Eq. (19). Also, we write x = sin in the cylindrical coordinates. (25) The dipole matrix element associated with the transition between the two states and its direction gives the polarization of the transition, which determines how the system will interact with an electromagnetic wave of a given polarization.
The total RIC is given by The AC is expressed by The linear and nonlinear ACs are given by Boyd (2003) Applying the above equations, the total AC is written as (Boyd 2003)

Result and discussion
In work, our calculations have been done for a GaN wedge-shaped QD. The geometry of a WSQD has been demonstrated in Fig. 1. The used parameters are: I = 0.4 MW/cm 2 , =10 16 cm −3 , and n r =2.5 (Bairamis et al. 2014). In this work, we consider the low-temperature case. Therefore, only the lowest levels are occupied in each dot. The effective mass of the electrons and the size of the dots depend on the pressure and are determined by Eqs. (20) and (23). Figure 2 shows the energy values of the system as a function of the QD size L for 0 = ∕3 , and L = 4R 0 . In the figure, the ground state, first, and second excited states have been plotted. The energy values reduce with increasing the QD size. Also, the transition energy ( E excitedstate − E groundstate ) decreases with increasing the QD size. Variations in the QD size lead to the apparent variations in the wavefunctions and energies of confined states and hence in the magnitude of inhomogeneous line broadening (Asryan and Suris 1999).
Since the optical properties [Eqs. (28)-(34)] depend strongly on the energy difference and dipole matrix elements, we first calculated the quantities by considering the pressure and polaron effects.
(3) ( , I) = − √ Figure 3 presents the energy difference versus pressure with and without the polaron effect. The energy difference reduces with raising the pressure with and without the polaron effect. The reason is as follows: With increasing pressure, the lengths of the QD [Eq. (23)] reduces, and thereby, the quantum confinement increases. Therefore, the energy levels become close, and their spacing decrease. We see that at a fixed pressure, the energy difference considering the polaron effect is lower than without the polaron effect. The reason is due to the negative contribution of the e-p effect [third term in Eq. (1)]. In Fig. 4, the dipole matrix elements have been displayed versus pressure with and without the polaron effect. The elements reduce by enhancing the pressure. We should note that the matrix elements are computed by using the ground and first excited states which both states alert with pressure. The overlap of wave functions plays an important role in determining the matrix element. By increasing the pressure, the electron wave function is more compressed and localized. Thus, the overlap of wave functions reduces, and thereby, the matrix elements decrease. The matrix elements with the polaron effect are higher than those without the polaron effect. In Fig. 5, the total RICs have been plotted versus photon energy with and without the polaron effect with various pressures. It is well known that the energy difference and the matrix elements determine the shift and magnitude peaks of RICs (Miao et al. 2020). Hence, the total RICs reduce and move to lower energies by considering the pressure. Also, the total RICs enhance and move to lower energies, considering the polaron effect. By raising the pressure, the quantum confinement increases, and thereby, the wave functions overlap, and the matrix elements reduce (see Figs. 3 and 4). Figure 6 shows ACs versus photon energy with and without the polaron effect for various pressures. Figure 6A-C correspond to linear, nonlinear, and total ACs, respectively. The larger linear AC is the opposite sign of the nonlinear AC. Therefore, the total AC will be reduced at this light intensity (I = 0.4 MW/cm 2 ).
Similar to the RICs, the ACs relate to the energy difference and the matrix elements. One can see that the linear and total ACs raise and move to lower energies under the polaron effect. Also, the ACs reduce and move to lower energies by enhancing the pressure. It is to be noted that the total AC considering the pressure and polaron effects, presents two resonance peaks. But, the peaks are not visible due to the energy range. To see the peaks, the total AC has been shown in a small energy range in Fig. 7 for various pressures.
In Fig. 7a-c, we have shown the total AC versus photon energy with various pressures P = 10, 15, and 20 GPa, respectively. The curves in the figures have been plotted with, and without the polaron effect. One can observe that the total AC reduce when the polaron and pressure are simultaneously considered, and we can observe two different peaks. Also, we can see the shift of peak locations in the presence of the polaron and pressure effects.

Conclusion
In the article, the optical properties of a GaN wedge-shaped QD considering the pressure and polaron effects are studied. Energy levels, total RICs and total ACs are evaluated with and without polaron with considering the pressure. It is deduced that the total ACs, and RICs enhance and move to lower energies under the polaron effect. Also, the total RICs and ACs reduce and move to lower energies by raising the pressure. The pressure and polaron effects play important roles in the optical properties of a GaN wedge-shaped QD. It is to be noted that high pressure is employed in the theoretical study. This study can be useful in the optical property applications of QDs.
Author contributions In this work MS has proposed and conceived the main idea as the supervisor (Conceptualization). AS has developed the theory and has done the formal analysis. Programming has been done by AS and RP (Advisor). MS and RP verified the analytical methods and investigated the findings of this work (Methodology and validation). The Original draft of the paper was written by MS and AS, All authors discussed the results and contributed to the review and editing of the final corrected manuscript. All the process has been done under the supervision of MS. The descriptions are accurate and agreed by all authors.
Funding The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.