Moreover, these parameters are adjusted to obtain a technologically relevant transition energy of 0.8 ev (1.55 µm).

Linear and Nonlinear Optical Absorption Coefficients in wurtzite InGaN/GaN Quantum Dots Switching at 1.55 µm Wavelength: Indium Segregation and Pressure Effects S. Naceur1*, M. Choubani2, H. Maaref3 Laboratoire de Micro-Optoélectronique et Nanostructures, Département de Physique, Faculté des Sciences de Monastir 5019, Avenue de l environnement, Université de Monastir, Tunisia. *Corresponding author: syna1991@gmail.com Abstract-The indium segregation in InxGa1-xN/GaN quantum dots (QDs), leads to a complex potential profile that is hardly considered in the numerical simulations. This phenomenon changes the energy levels, which further influences the nonlinear optical properties related to intersubband transitions in QDs. In this work, the Muraki s model is used to determine the indium content profile in the InxGaN1x/GaN coupled QDs. Also, we explore the effects of the structural dimensions, hydrostatic pressure (P), and the strong built-in electric field (F) due to the spontaneous (Psp) and piezoelectric (Ppz) polarizations on the linear and nonlinear intersubband optical absorption coefficients in vertically coupled QDs. This detailed and comprehensive study is theoretically investigated within the framework of the compact density-matrix approach and the finite difference method (FDM). Obtained results reveal that the resonant peaks of the total absorption coefficient can be blue-shifted or redshifted energies depending on the structural dimensions and the external parameters mentioned above. Moreover, these parameters are adjusted to obtain a technologically relevant transition energy of 0.8 ev (1.55 µm). 1,2,3 Keywords: Built-in electric field, coupled quantum dots, pressure, telecommunication wavelength, indium segregation. I. INTRODUCTION Recently, various experimental and theoretical studies have been developed on the intersubband transitions in III-nitride semiconductor QDs because of their potential application in optoelectronic devices, such as laser diodes (LDs) emitting at 1.55µm [1], photo-detectors [2] and resonant tunneling diodes [3]. On the other hand, it has been found that the band gap energy of InxGa1-xN ternary alloy system can vary from 0.78 ev (for InN) to 3.51 ev (for GaN) [4] in wurtzite phases which covers almost the entire solar spectrum. Moreover, the wurtzite III-nitrides materials have a strong spontaneous macroscopic polarization. Furthermore, a large lattice mismatch between InGaN and GaN (11%) [5] can induce a remarkable piezoelectric polarization [6]. Both the spontaneous and piezoelectric polarizations will induce a strong built-in electric field in the structure under investigation. The internal electric field breaks the structure symmetry and affects obviously the distribution of the carrier wave function in the wurtzite nitride-based coupled QDs. Therefore, we choose to use the wurtzite crystal, because, in order to obtain strong optical nonlinearities, such as optical absorption [7], refractive index change [8], nonlinear optical rectification (NOR) [9] and second harmonic generation (SHG) [10]. From these investigations, it is found that the nonlinear optical properties related to the intersubband transitions in QDs have attracted more attention. Hence, we focus our work on studying the linear and nonlinear absorption coefficients in InxGa1-xN/GaN coupled QDs along the growth direction. As well, due to the coherency different miscibility between GaN and InN materials [11], the indium atoms segregate into the upper layer, which will lead to the non-uniform indium distribution. DOI: 10.23883/IJRTER.2017.3504.WHSGU 143

This phenomenon changes the energy levels and nonlinear optical properties. To our knowledge, the effect of the indium segregation on the nonlinear optical properties has not been studied yet. In this work, we will introduce the indium segregation phenomenon by using the Muraki s model [12]. Also, the resonant energy will be controlled and adjusted to the telecommunication wavelength 1.55 µm. The present paper is organized as follows: in section 2, we describe the theoretical framework. The Hamiltonian, relevant eigenstates, and energy eigenvalues are discussed and numerically determined by using the FDM. Also, the linear, nonlinear and total optical absorption coefficients are given via the compact-density-matrix approach. In section 3, the obtained results and discussions are performed. Finally, in the last section, a conclusion is given. II. THEORETICAL FRAMEWORK 2.1. Spontaneous polarization, piezoelectric polarization and built-in electric field In this work, we consider the hexagonal crystal structure (wurtzite) which is the thermodynamically stable phase. The double QDs system presented in this work is made up of two InxGa1-xN QDs separated and surrounded by GaN barriers: GaN (LB1)/InxGa1-xN (Lw1)/ GaN (LB)/InxGa1-xN (Lw2)/ GaN (LB2). The outer barriers have the same width LB1 = LB2, whereas, Lw1 and Lw2 are the left and right QDs widths, respectively, and LB is the middle barrier thickness. Fig. 1 shows a pictorial view with the z- direction as the growth axis of InGaN QD on GaN. In this study, we go focused only on the effect of the vertical coupling between the QDs along the z direction. z (a) GaN LB2 Lw2 InGaN Lw1 GaN LB1 Energy L LB GaN InGaN (b) GaN InGaN GaN InGaN GaN EC z 0 L Fig. 1. (a) Schematic representation of a multilayer of InxGa1-xN/GaN QD along the growth direction and (b) the associated confinement potential profile. The z-axis is taken to be parallel to c-axis (growth direction) and the x-y plane parallel to the interfaces. In wurtzite nitride structure, strong polarization exists along the c-axis [0001]. This polarization is the sum of two types of polarizations known as spontaneous (Psp) and piezoelectric (Ppz) polarizations. The Psp is due to the non-centrosymmetric property and also the difference in electronegativity of the metal atom (Ga) to that of a nitrogen atom (N). Moreover, under stress condition, the elementary 144

tetrahedrons of the wurtzite structures are deformed which causes the barycenters of the positive and negative charges of the crystal to move away from their usual location, creating an additional electric dipole, polarization from such induced electric dipole is known as piezoelectric polarization (Ppz). In this work, Psp of InxGa1-xN is calculated through the Vegard s law: Psp xpsp (InN) (1 x)psp (GaN) bsp x(1 x) Where the bowing parameter is chosen as bsp GaN and InN are listed in Table I [13]. Parameters C13 (GPa) C33 (GPa) e31 (C/m2) e33 (C/m2) Psp(C/m2) (1) = - 0.037 C/m2 and the spontaneous polarizations for GaN 106 398 InN 92 224-0.49 0.73-0.034-0.57 0.97-0.042 Table I: Elastic constants Cij, piezoelectric constants eij and spontaneous polarization Psp for GaN and InN materials. However, the piezoelectric polarization is given by: a aingan C13 PPz 2 GaN e31 e33 aingan C33 (2) Where, Cij are the elastic constants, eij are the piezoelectric constants and a is the lattice constant. Their values for GaN and InN are listed in Table I. As for the ternary InxGa1-xN material; these parameters can be obtained by the linear interpolation method from the corresponding values of GaN and InN by using the Vegard s law. Thus, the total polarization (Ptot) in strain group III-nitrides crystals can be expressed as the sum of Psp and Ppz polarizations as follows: (3) Ptot Psp PPz Due to the polarizations mismatch, the built-in electric field (F) along the growth direction inside the jth layer is given by [14]: N Pk Pj Lk k F j k 1 (4) N Lk j k 1 k Where Pj is the total macroscopic polarization, j is the dielectric constant and Lj is the width of the jth layer. 2.2. Indium segregation Due to the coherency different miscibility between GaN and InN materials, the indium atoms segregate into the upper layer, this will lead to the non-uniform indium distribution. Due to its fundamental importance, during the past decades, remarkable progress has been achieved on the understanding of the segregation process. To study indium segregation, we have used the phenomenological model proposed by Muraki et al [12]. This model assumes that a fraction R named segregation coefficient, of indium atoms on the top layer, always segregates to the surface layer during the growth of each new layer. However, the rest fraction (1-R) of indium atoms are incorporated into the bulk crystal. According to this model, the indium composition xin in the jth layer is given by: 145

n xin x0 (1 R ) (1 n N; in QDs) (5) N n N (n N; in barriers) xin x0 (1 R )R Where, x0 and N are the nominal indium composition and the total number of InxGa1-xN monolayers in the QDs, respectively. The segregation coefficient introduced by Muraki et al. is written as: d R e (6) where d is half the lattice constant of the InGaN layer and is the segregation length over which the effects of the segregation phenomenon can be effectively sensed. In this paper, the segregation coefficient adopted in our work is fixed at R = 0.9, which is a typical value to fit experimental data [15]. For this value of R and by using Eq. 6, the segregation length is about 1 nm ( 4 monolayers). 2.3. Schrödinger equation In this work, we consider an InxGa1-xN/GaN multiple QDs consisting of N =5 layers, grown along the z-direction. In the framework of effective mass approximation, the Schrödinger equation of an electron confined in this system is given as follows (for in-plane wave vector k11=0) [16]: In the above equation, V(z, P, T) is the confinement potential in the conduction band, Fi is the intrinsic polarization introduced field in the jth layer (j = 1,2,..N) and Cj = Cj-1 +e(fj-1 Fj)Lj-1, with the arbitrary taken C1 = 0. Also, we have considered a pressure and temperature dependent effective mass of the electron given by [17]: 2 EP ( Eg 3 S 0 ) m0 (8) 1 2 m* (P, T ) E ( E ) g g S 0 where m0 is the free electron mass, is the Kane parameter, EP is the energy related to the momentum matrix element, S0 is the spin-orbit splitting and Eg is the energy gap at the -point. For the ternary InxGa1-xN material, the effective mass is calculated by using the Vegard s law with a linear interpolation between the corresponding values of GaN and InN materials. The confinement potential as function of the spatial coordinates, P, and T is given by: 0, inside InGaN QDs (9) V ( z, P, T ) Qc Eg, elsewhere where, QC = 70 % [18], is the conducting band offset parameter and Eg Eg (GaN ) Eg (InxGa1 x N ) is the band gap difference between GaN and InGaN materials at the -point. 146

The energy gaps of InN and GaN semiconductor materials are T and P-dependent quantities and given by: Eg (P,T) Eg (0,0) T 2 g.p d.p2 T (10) where Eg (0, 0) is the energy gap at P = 0 kbar and T = 0 K. (α, β) and (g, d) are the temperature and pressure coefficients, respectively. As for the indium composition-dependent band gap for InxGa1-xN, we have used Vegard s law: (11) Eg xeg (InN) (1 x)eg (GaN) bg x(1 x) Where the bowing parameter is bg = 1.43 ev [19]. All the parameters used and adopted in our calculations are listed in Table II [4, 19]. Parameters GaN InN Eg (0, 0) (ev) 3.51 0.78 α (mev/k) 0.909 0.245 (K) 830 624 g (mev/kbar) 4.2 2.1 d (mev/kbar2) -1.8.10-2 0 ΔS0 (ev) 0.017 0.005 EP (ev) 14 14.6 0 0 r 5.8 8.4 Table II: Recommended parameters for GaN and InN materials (1 kbar = 0.1013 GPa ). 2.4. Finite Difference Method The Schrödinger equation of the considered system cannot be solved analytically. Therefore, the FDM is used in this work to calculate the direct numerical solutions for Eq.7. Thus, the Schrödinger equation which corresponds to the Hamiltonian must be approximated by using the central finite difference approximation schemes. Then, the space under investigation is uniformly discretized into Nz nodes with a mesh size Δz. The finite difference representation of the Schrödinger equation is as follows: Ci 1 i 1 Ai i Bi 1 i 1 E i, i =1, 2, Nz (12) where Ai, Bi+1 and Ci-1 are given by: 147

The index i identifies the grid point on the one-dimensional mesh and Δz is the mesh size between adjacent grid points or nodes. However, the standard boundary conditions still need to be applied which are: i 0 as i (14) Then, there are only (Nz-2) linear equations which are converted into the matrix eigenvalue problem, H =E. The matrix representation of H is given as follows: After diagonalization of the Hamiltonian H, one can calculate the direct numerical solutions for the Schrödinger equation given by Eq. 7, such as eigenenergies and eigenfunctions. 2.5. Linear, nonlinear and total optical absorption coefficients After the energies E and their corresponding envelope wave functions are obtained, the linear α(1), nonlinear α(3)(, I) and total α optical absorption coefficients can be calculated by the density matrix approach. Based on the density-matrix formalism, α(1), α(3)(, I) and α coefficients are given as follows [20]: 148

(, I ) (1) ( ) (3) (, I ) where N is the volume density of carriers in the system. ij i ez j is the dipole matrix element. (16.c) Efi = Ef - Ei is the transition energy between the excited state (f) and the ground state (i). if 1 is called the relaxation rate between excited and ground states. Tif ћ is the incident photon energy. And I is the incident optical intensity. III. RESULTS AND DISCUSSIONS In this work, the parameters adopted in our calculations are as follows [9,20]: Tif = 0.24 ps, N = 5x1024 m-3 and nr = 3.2. Before presenting the outcome of the calculation, we have first presented the segregation and polarization effects on the confinement potential profile for an arbitrary layer widths. As shown in Fig. 2, we have adopted a nominal indium composition x0 = 30% and a segregation coefficient R = 0.9. x0 = 30 % 2,0 With segregation: R = 0.9 With polarisation With segregation and polarisation 1,5 V (ev) 1,0 0,5 0,0-0,5-1,0 0 2 4 6 8 z (nm) 10 12 14 16 Fig. 2. Segregation and polarization effects on the confinement potential profile. 149

From the analysis of this figure, one notices that the segregation changes the QD profile drastically. This change will be followed by a change in the energy levels of the QDs. Thus, it is very important to take into account the segregation effect to study the optical behavior of the coupled InxGa1-xN/GaN QDs correctly. Also, the confinement potential profile is strongly tilted by the built in electric field, which describes the effect of the Psp and Ppz polarizations. In this paper, we search to adjust the intersubband transition wavelength 1.55 µm (0.8 ev) which is needed in the telecommunication domain. For this purpose, we have used the structural dimensions LB1 = LB3 = 10 nm, LB = 1 nm and Lw2 = 2 nm. The nominal indium concentration is fixed at x0 = 30%, P = 0 kbar, T = 300 K and R = 0.9. Fig. 3 presents the variation of the intersubband transition energies (E2-E1), (E3-E1) and (E3-E2) between the three lowest levels versus Lw1, the width of the left QD. Fig. 3. Intersubband transition energies E21, E31 and E32 as a function of the left QD width. It is observed that the suitable width Lw1, which achieves the intersubband transition 1.55 µm is equal to Lw1 = 2.1 nm, and it is obtained only for E31 = E3-E1 transition energy. Also, these results show that the transition energies increases with the increase of the left QD width. Therefore, the obtained results are very important to design a vertically coupled QDs system with adjusted transition energy and manipulating different applications of optoelectronic devices. For the designed structure, the linear α(1), nonlinear α(3)(, I) and total α optical absorption coefficients as a function of the incident photon wavelength are given by Fig. 4. The incident optical intensity (I) is fixed at I = 10 MW/cm2. 150

20 x0 = 30 %, LW1 = 2,1 nm ( ) (3) ( ) ( ) (1) 5-1 ( ), ( ), ( ) (10 m ) 15 10 (1) (3) 5 0-5 1,45 1,50 1,55 1,60 1,65 1,70 ( m) Fig. 4. Linear, nonlinear and total absorption coefficients versus the incident photon wavelength. As shown in this figure, the linear absorption coefficient is positive. Nevertheless, the nonlinear absorption coefficient is negative. Thus, the magnitude of the total optical absorption coefficient is lessened due to the contribution of the nonlinear optical absorption coefficient. Also, we note that for the specific value of Lw1 = 2.1 nm, the desired transition energy is obtained and the structure has a resonant behavior at λ = 1.55 µm (0.8 ev). Also, segregation and built-in electric field (F) effects on the total absorption coefficient are investigated. Obtained results are given by Fig. 5. It is observed that the absorption peak is shifted to higher energy (λ decreases) due to the built-in electric field. This blue- shift can be explained based on the quantum confined Stark effect induced by F. Moreover, considering the indium segregation, the transition energy E31 = E3 - E1 is blue-shifted in comparison with the predicted energy for an ideal square QDs profiles (R = 0). This shift can be explained as follows: if there is indium segregation, we could expect an increase in the transition energy as a consequence of the lower indium content in the layer that leads to a higher energy gap of the material. 151

Fig. 5. Segregation and polarization effects on the total optical absorption coefficient. Thus, it is clearly that the segregation phenomenon changes the energy levels, which further influences the optical performances of semiconductor devices. Therefore, many techniques are used to investigate it, such as transmission electron microscope (TEM) [22], photoluminescence (PL) [23] and so on. For this purpose, the indium composition effect on the total optical absorption coefficient is also investigated. Obtained results are given by Fig. 6. 152

Fig. 6. Total absorption coefficient versus the incident photon wavelength for three different indium compositions. As shown in this figure, the magnitude of the resonant peak of increases with increasing x0 and also a blue- shift (toward higher energies) is obtained. This resonance shift can be understood by the indium segregation effect, which affects the potential confinement profile of the QD. Therefore, the energy levels (first and third levels) become separated from each other. Then, the energy interval of the transition energy E3 - E1 is enlarged, and consequently, the resonant peak of the total absorption coefficient moves to the higher energies. Also, we note from this figure that the optimum indium composition to obtain the desired transition λ = 1.55 µm is equal to x0 = 30% and the optical absorption coefficient increases up to maximum value about 16 105 m-1. Also, based on Eq.16b, we can see that the nonlinear optical absorption coefficient is a function of the incident optical intensity. To show better the influence of I on, in Fig. 7, we set P = 0 kbar, T = 300 K and plot the total optical absorption coefficient as a function of the incident photon wavelength for different values of I. 153

Fig.7. Total absorption coefficient versus the incident photon wavelength for different optical intensities. We can see that the optical absorption coefficient decreases with increasing the incident optical intensity. Also, another important phenomenon observed here is the saturation effect of the absorption which is obtained when the incident optical intensity exceeds a critical value Is = 38 MW/cm2. The saturation effect is due to the nonlinear term which is negative, and its contribution causes a collapse at the center of the total absorption coefficient peaks splitting it into two peaks. Note that the bleaching effect becomes clearer when the incident optical intensity is increased to I = 60 MW/cm2. For all simulations mentioned before, only a blue shift is observed. However, the application of the hydrostatic pressure modifies the barrier height and the effective mass, accordingly, the total absorption coefficient and the associated resonant energy. For this purpose, the pressure effect on the total absorption coefficient is also investigated. In Fig. 8, we plot the variations of α(1), α(3) and α coefficients versus the incident photon wavelength for different hydrostatic pressures. 154

Fig.8. Linear, nonlinear and total absorption coefficients as a function of the incident photon wavelength for three different hydrostatic pressures (1 kbar = 0.1013 GPa ). The calculated results are obtained for different values of P, at a fixed nominal indium composition x0 = 30 % and at T = 300 K. It is clearly seen that the resonant energy E31 and the maximum value of decreases with increasing P. These effects can be easily understood in terms of the reduction energy separation between the first and third states. This red-shift is due to the increase of the electron effective mass and also the barrier height with increasing P, respectively. As a result, the photon energy shifts towards the red (lower energies) with increasing P can be used as a tunable parameter to obtain the desired resonant energy. Obtained results are very important and reveal that the resonant peaks of the total optical absorption coefficient can be blue-shifted or red-shifted energies depending on the vertically coupled QDs structure. Also, this condition can be controlled by changes in structural dimensions, pressure, and indium composition. Hence, one should reduce the influences of these factors on the considered system under investigation to achieve a large absorption coefficient with adjusted resonant energy. IV. CONCLUSION In this paper, the linear, nonlinear and total optical absorption coefficients in InxGa1-xN/GaN vertically coupled QDs are well studied by using the FDM and density matrix approach. Our results suggest that for intersubband transitions in InxGa1-xN/GaN QDs, we can obtain a blue or red-shift by tuning the structural dimensions, hydrostatic pressure, and indium composition. Moreover, the theoretical results reveal that the indium segregation phenomenon has a significant effect on the total optical absorption coefficients. In these materials, the spontaneous and piezoelectric polarizations are taken into 155

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