† Corresponding author. E-mail:

Project supported by the Ministry of Higher Education and Scientific Research in Iraq, Ibnu Sina Institute and Physics Department of Universiti Teknologi Malaysia (UTM RUG Vote No. 06-H14).

Based on the Schrödinger equation for envelope function in the effective mass approximation, linear and nonlinear optical absorption coefficients in a multi-subband lens quantum dot are investigated. The effects of quantum dot size on the interband and intraband transitions energy are also analyzed. The finite element method is used to calculate the eigenvalues and eigenfunctions. Strain and In-mole-fraction effects are also studied, and the results reveal that with the decrease of the In-mole fraction, the amplitudes of linear and nonlinear absorption coefficients increase. The present computed results show that the absorption coefficients of transitions between the first excited states are stronger than those of the ground states. In addition, it has been found that the quantum dot size affects the amplitudes and peak positions of linear and nonlinear absorption coefficients while the incident optical intensity strongly affects the nonlinear absorption coefficients.

Low-dimensional semiconductor systems (LDSS) and their nonlinear optical properties are of great interest because of their promising utilization and relevance in making new optical devices for the switching^{[1]} and antenna^{[2,3]} applications. It is found that higher optical transition energies do exist in LDSS. By reducing the dimensionality of nano-heterostructures, the optical, electrical, and transport properties of this system could be enhanced due to the geometrical confinement effects.

The exceptional development in the microfabrication makes it possible to produce LDSS in many laboratories and to play a critical role in microelectronic and optoelectronic devices. Some studies also reveal the atomic properties such as discrete energy levels and shell structures motivate the study of the electronic and optical properties of these nanostructures.^{[4,5]}

The electronic and optical properties of bulk materials also depend on the electron–photon interaction that plays a significant role and also affects the LDSS. Therefore, LDSS especially quantum dots (QDs) have been extensively investigated both theoretically and experimentally. Discrete energy levels are formed by the confinement of charge carriers (electrons and holes) of QDs in three dimensions (3D) that changes the optical properties and due to this QDs are also known as artificial atoms.^{[6]}

There have been many theories discussing the electronic properties of QDs. Quantum genetic algorithm,^{[7]} configuration interaction,^{[8,9]} perturbation,^{[10]} variational,^{[11,12]} density functional,^{[13,14]} Hartree–Fock Roothaan method,^{[15]} exact solution,^{[16]} and other theoretical methods^{[17–19]} have been employed to study such properties. According to these theories, electronic properties mainly depend on the shapes of these semiconductor nanocrystals, which have been observed experimentally, but precise calculations of the quantum dot (QD) structure have not yet been realized due to manufacturing imperfections resulting from the growth methods. The k.p. theory gives a complete description of the electronic structure, but its high computational requirements make it difficult to know the structure parameters. According to the literature, many analytical and numerical models have been used to research different shape QDs. For example, Jungho and Chuang^{[20]} offered the quantum disk model, Zhang *et al.*^{[21]} discussed cylindrical QDs, Nenad *et al.*^{[22]} presented QDs in the form of a truncated hexagonal pyramid, and cone or truncated cone dots were discussed in many studies.^{[23–25]} Many researchers have studied QDs in lens or dome shapes,^{[26–29]} the results of which are in good agreement with the experimental results (see Figs. *ρ* and *z* directions,^{[26,27,29]} such as in the cylindrical problem. Another study used one direction in the solution and ignored the other direction.^{[23,28]} None of these studies incorporated the effect of 3D geometry for lens or dome shaped QDs.

This study deals with the inspection of confinement effects of QDs on the electronic and optical properties for III–nitride (III–N) materials. III–N semiconductors (InN, GaN, AlN, and their alloys) have exhibited considerable potential in developing the optoelectronic devices working in the ultra-violet and visible region, for example lasers, light emitting diodes, optical amplifiers, solar cells, and photodetectors.^{[30–32]} A few recent results are indicating the small band-gap (∼ 0.7 eV) of InN.^{[33–35]} This shows that the bandgap of ternary alloy InGaN could cover the whole solar spectrum, i.e., from infrared to ultraviolet. This exclusive bandgap span of nitride-based materials is very crucial in fabricating high-efficiency (50%) multijunction solar cells and full-spectrum devices.^{[36]} Currently, the devices based on III–nitrides are fabricated by using metal–organics chemical vapor deposition (MOCVD), metal–organic–vapor phase epitaxy (MOVPE), and molecular beam epitaxy (MBE) techniques.^{[37–40]}

The InGaN alloy is a major player in the recent development of optoelectronics and high-power as well as high-temperature devices. Particularly, this material exhibits promising applications for future multi-layers solar cells with high efficiency,^{[38,41]} and has also shown a higher degree of resistance to radiation damage compared to other photovoltaic materials.^{[37]} It offers a flexible choice of the band gap energies and a number of the component junctions. Technologically, this would be very important as relatively few precursors are needed to grow different junctions. From III–N group, researchers have not only significantly investigated GaN but also have reasonably developed it, whereas, with reference to the photovoltaic application, the lower band gap InGaN is more practical and yet is a subject of fundamental research.

The optical interband and intersubband (intraband) transitions are the most interesting characteristics of QDs. Furthermore, the linear and the third-order nonlinear optical absorption coefficients (ACs) greatly affect the optoelectronic device applications, such as optical amplifiers, photodetectors, and solar cells. Hence, in recent years, the nonlinear optical properties of QDs are the center of interest for both fundamental and applied research due to large values of the dipole moment matrix elements that attain both linear and nonlinear optical properties, and large value of oscillator strength function. Bouzaïene *et al.*^{[27]} have studied nonlinear optical rectification of lens-shaped InAs QDs inside a large cylinder of GaAs. Within the same assumptions, Khaledi–Nasab *et al.*^{[28]} have investigated the effect of Woods–Saxon (WS) potential on the electronic structure of InAs QDs, which leads to strong alterations in linear susceptibility and group velocity.

All the previous studies were in 2D and unable to explain the real system. To model an appropriate optical device close to reality, the investigation of electronic and optical properties of the 3D system is imperative. To the best of our knowledge, up till now, the linear and third-order nonlinear optical absorption coefficient for interband and intraband transitions of 3D lens-shaped QDs has not been studied.

The focus of this article is to investigate linear and nonlinear optical ACs in different sizes of lens-shaped QDs. Firstly, we calculated the ground state and the first excited state energies and their corresponding wave functions of the QD system with and without strain effect. Calculations were performed using 3D finite element method (FEM). The linear, third-order nonlinear, and total optical ACs as a function of incident photon energy with size and In-composition effect were also studied.

In order to probe semiconductor QDs’ properties with lens-shaped QDs, the effective mass approximation (EMA) of one-band Schrödinger equation for electrons (holes) in conduction (valence) bands has been employed as

*π*,

*m** is the effective mass of electron or hole,

*V*is the potential energy determined by the conduction or valence band offset,

*Ψ*(

*x*,

*y*,

*z*) and

*E*are the wave functions and their corresponding quantized energy levels, respectively.

Figure *Ψ* = 0) is applied on the top and bottom boundaries (I and II). Meanwhile, the Neumann boundary condition (*n̂*, · *Ψ* = 0) is used to the four side boundaries (III). Finally, the periodic boundary conditions

In this study, we are interested in wurtzite (W) structure as its hexagonal structure is more stable than the zinc-blend. The strain effect is calculated to figure out the changes in the confinement energy for both electrons and holes. Due to strain, there is an energy shifting for the conduction and valence band edges as^{[30]}

*a*

_{c}is the conduction band hydrostatic deformation potential,

*D*

_{1},

*D*

_{2},

*D*

_{3}, and

*D*

_{4}are the valence band deformation potentials,

*C*

_{13}, and

*C*

_{33}are the elastic constants, and

*ε*is the elastic strain that comes from the lattice mismatch, it is defined as

*a*

_{QD}and

*a*

_{WL}are the lattice constants for QD and WL, respectively. After getting the energy levels and their corresponding wave functions, one can obtain the linear and the third-order nonlinear optical ACs. Based on the density-matrix method,

^{[44,45]}within any two-level system approach, the expression for total, linear and nonlinear optical ACs for the transitions between initial and final state energies are

*α*(

*ω*,

*I*),

*α*

^{(1)}(

*ω*), and

*α*

^{(3)}(

*ω*,

*I*) are the total, the first-order linear and the third-order nonlinear ACs, respectively, and

*μ*is the permeability of the system taken as

*μ*= 1/

*ε*

_{o}

*c*

^{2}. Here,

*ε*

_{o}, and

*c*represent the electrical permittivity and the speed of the light in vacuum, respectively, and

*ε*

_{r}represents the electrical permittivity of the semiconductor, which is equal to

*n*

_{r}is the refraction index of the semiconductor,

*I*is the incident optical intensity,

*ρ*is the electron density in the QD defined as

*ρ*=

*N*/

*V*

_{QD}, where

*N*is the number of electrons inside QD (in the present work

*N*= 1), and

*V*

_{QD}is the volume of QD, ℏ

*ω*is the incident photon energy,

*Γ*

_{fi}= 1/

*τ*is the relaxation rate of the

*f*and

*i*states (where

*f*≠

*i*),

*τ*is the relaxation time, and

*E*

_{fi}and

*M*

_{fi}are the energy difference and the dipole moment matrix between the two states, respectively. Here,

*Ψ*

_{f}and

*Ψ*

_{i}are envelope functions for initial and final states, respectively, which are calculated by using the numerical model.

Firstly, conduction and valence subbands are treated for In_{x}Ga_{1−x}N/GaN lens-shaped QD with a base diameter ranging from 8 nm to 30 nm, height variation from 2 nm to 13 nm, and in all calculations 2-nm thickness of WL is also assumed, as in the previous experiments.^{[39,43,46,47]} The other related parameters (at room temperature) used in the calculation are listed in Table

The Schrödinger equation (Eq. (_{x}Ga_{1−x}N QD on GaN WL as a function of In-composition is shown in Table _{x}Ga_{1−x}N increases enormously due to compressive strain and causes a change in the confined energy levels of QDs. The difference between strained and unstrained band gap energy also reduces with the decrease of the In-fraction in In_{x}Ga_{1−x}N material. This is accredited to the QDs’ materials and unstrained GaN becoming similar, and thus reduces the strain effect. As it is shown in Table *x* = 0.3 and due to strain the energy level at both s- and p-states increases nearly 0.3 eV. It is important to mention here that Coulomb-coupling interaction, piezoelectric and spontaneous polarization energy are not incorporated in the present calculations because they do not have a dramatic influence on the QDs as their QDs size is very small.^{[50]}

Cross-section distributions for s- and p-state envelope functions are illustrated in Figs.

A comparison between Figs.

After calculating conduction and valence subbands as well as the envelope functions corresponding to the s- and p-state configurations, the optical ACs of s–s, p–p, and all transitions between these energy states are shown in Figs. *ε*_{r} = 9.6 + 5.4*x* that is the electrical permittivity of the semiconductor In_{x}Ga_{1−x}N QD, the relaxation time *τ* = 0.045 ps. In order to investigate the influence of In-composition on ACs, we plotted the linear, nonlinear, and total ACs of the s–s, p–p, and all transitions between two states as a function of photon energy for four different values of *x*-mole fraction (Figs. _{x}Ga_{1−x}N/GaN with x ≥ 0.5 is a good LDSS for solar cell devices, and the best for lasers at longer wavelengths. The theoretical predictions obtained here are in good agreement with the experimental results.^{[51,52]}

The change in the peak positions of photon energy and amplitudes of ACs is obvious when the height of QD is varied from 4 nm to 12 nm (Fig. *V*_{QD}). In addition, we found that the absorption peaks shift towards the red as the QD height increases. With the QD height changing, there is a large overlapping between s- and p-energy states that means that the change in the state energy is small (Fig.

To understand the relation between the incident optical intensity (*I*) and the absorption spectrum, ACs with four values of optical intensity are illustrated in Fig. ^{[53]} However, there is no shift in the peak position with the change of the incident optical intensity. This leads to taking into consideration the third-order nonlinear term in the calculation of the total optical AC, especially with high optical intensity.

In this work, electronic structures of InGaN/GaN lens-shaped QDs were numerically investigated within the framework of Schrödinger equation for envelope function in the effective mass approximation by using FEM. The linear and nonlinear optical ACs were calculated. Our results showed that by decreasing In-concentration of InGaN QDs, the amplitudes of linear and nonlinear absorption coefficients increase. In addition, it has been found that increasing QD size causes a red shift and reduces the amplitudes of total, linear and nonlinear ACs, especially QD height is more effective. Furthermore, we have calculated the total optical AC, and found that it is essential to consider the nonlinearity part particularly with high values of optical intensity. The present results are of great impact and can be used for the improvement of optical devices. These results would be important for applications based on the nonlinear optical properties. This theoretical work may provide the basis for profound outcomes for practical applications of optoelectronic devices.

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