3.1.Structural properties

The Study of confinement effects and electronic structures in SL _s (InN) _m /(GaN)_n along the (001) growth axes has been considerably focused over the past decade, but the most interesting feature is to know the effect of crystal orientations on their properties and their potential applications ³⁴. There is a wide variety of the growth axes enabling the realization of the SL _s . Two cases will be used: SL _s grown along with the (001) and (110) directions. While the structural properties of (001)-oriented semiconductor SL _s have been studied in great detail by using first-principles methods, there is not a comparable amount of information about the properties of (110)-oriented SL _s ¹². It is reported that (110) oriented SL _s exhibit considerably different characteristics compared to those grown along the (001) direction. We theoretically calculated these variations on ideal cases and to verify the expectation that their properties represent. The calculations in this work are carried out on ideal cases in which all atoms of the bulks or SL _s are located in ideal positions. By choosing the SL _s which have a short period, we can assume that the lattice parameter is constant throughout the crystal as the distance between the successive interfaces is very small. This approximation liberates us from any calculation of relaxation. For the considered structures, we perform the structural optimization by minimizing the total energy with respect to the cell parameters and also the atomic positions.

For the binary compounds, the simplest structure is an eight-atom simple cubic cell. For both directions (001) and (110)-oriented SL _s , the smallest ordered structure is a four-atom tetragonal cell, corresponding to the SL(1,1) and the largest ordered structure is a sixteen-atom tetragonal cell, corresponding to the SL(1,7). The calculated total energies at many different volumes around equilibrium were fitted by the Murnaghan equation of state³⁵ in order to obtain the equilibrium lattice constant a₀ and the bulk modulus B for the binary compounds and all SL _s . It should be noted that the period of our SL _s is D=L×aSL, a_SL being the equilibrium lattice constant of the SL _s in the x direction. The lattice mismatch is found to be 9.76% for InN/GaN. Therefore, it may be crucial for SL systems. The results for the binary compounds are presented in Table I. As shown in Table I, the obtained results in our calculations are well compatible with other previous experimental and theoretical works for both binary compounds InN and GaN. The calculated equilibrium lattice parameters of our binaries InN and GaN in ZB structure are (4.940 Å) and (4.480Å), which differ by only 0.80% and 0.69% compared with the experimental results of 4.98Å ³⁶ for InN and 4.5Å ³⁵ for GaN, respectively. The calculated equilibrium lattice parameters of ZB-GaN is 4.480Å, which differ by only 0.69% of the experimental values of 4.50Å ⁴². The calculated bulk modulus B for ZB InN and GaN shows good agreement with experimental results and previous theoretical studies and it is evident that GaN is less compressible than InN because Ga-N is more tightly bound than In-N and results in a higher covalence for Ga-N than for In-N ³⁷.

The lattice constant and bulk modulus for (InN)/(GaN)n for different values of n in both directions (001) and (110)-oriented SL _s were calculated and are listed in Table II. As expected, the calculations for the (001) and (110) SL(1,1) give the same results since the symmetry is the same. The link between the bulk and the SL direct lattices is shown in the Fig. 1 with ax=ay=aSL=a0/2 and az=L⋅a0 in (001) case (aSL=ax=a0/2, ay=L⋅ax and az=2⋅ax in (110) case). In the first-principles structural calculations, we can also try to obtain this az/ax and ay/ax ratio by minimization. It is found from Table IIthat the lattice constant in all SL _s increases when a few percent of indium atoms is added to the host semiconductor GaN. If we take the case of a (001) and (110) (1.7) SL _s as an example, the host material is GaN with a lattice parameter of 4.48 Å. The incorporation of one InN monolayer increases the lattice parameter to a value of 4.640 and 4.651 Å, respectively. The increase in the lattice parameter is justified by the fact that the atomic size of indium is larger than those of Ga atom.

It is very clear from Table Ithat the bulk modulus for the (InN)/(GaN)_n in both directions SL _s increases with the enhancement of the number of monolayer 𝑛, which suggests the same increasing for the compressibility of each compound. These compounds became harder when the number of monolayers n increase. It represents bond strengthening or weakening effects induced by changing the composition. The stability of (InN)/(GaN)_n SL _s is examined by calculating their formation energies. The formation energy per atom Eform as a function of the number of monolayers n for all superlattices is calculated as follows ⁵¹:

where E_tot is the total energy of the (InN)/(GaN) _n and the total energy of its constituent parts GaN and InN. From Table IIanalysis, it can be found that formation energy for these SL _s have negative values and decrease by a small amount with the number of monolayers. The sign of the formation energy indicates that the present systems are energetically more stable with increasing n and imply that these SL _s are exothermic and also can be synthesized experimentally. The (001) SL _s have formation energy slightly less than the formation energy of (110) SL _s .

3.2.Electronic structure and density of states

In this section, we turn our attention to study the electronic properties of the parents InN and GaN binary compounds and their SL _s via calculating the energy band structure by using our calculated values of the lattice parameter. The results exhibit that cubic GaN and InN are a direct bandgap, where the valence band (VB) maximum and conduction band (CB) minimum are found at the Γ point. The results clearly exhibit that the present calculated bandgaps of 0 eV for InN and 1.916 eV for GaN are, on the whole, underestimated compared to the experimental values of 0.7 eV for InN ⁵² and 3.20 eV for GaN ⁵³, but are in right agreement with other theoretical studies. The large difference in the calculated values of the band gaps as compared to the experimental values can be explained by the well-known fact that, in the electronic band structure calculations within DFT, GGA underestimates the energy gaps in semiconductors, but this will not alter the conclusions of the present work since they are not related to the quantitative estimation of gaps.

The calculated band gap energies of (InN)/(GaN) _n in both (001) and (110) directions as a function of the thickness of the monolayer GaN are investigated and plotted. As an example we show Fig. 2 the band structure of two limit configurations (n = 1 and n = 7), where E _f represents the Fermi level. From the results of the calculated bandgap energies, we find that these materials have a direct bandgap, both valence band maximum and conduction band minimum are lying at Γ point, which is of interest for optoelectronic devices. The symmetry points in Fig. 2 refers to the tetragonal Brillouin zone (BZ) are (X, M Γ, Z, A and R) in (001) direction case and those in (110) direction case are (X, M, Y, Γ, Z, A, R and B). In the case of (001) SL(1,1) the high symmetry points B and Y are identical to R and X, respectively. This remark remains valid for the (110) SL(1,1), but B becomes different from R and Y from X when n increase. It is noticeably indicated that the empty bandgap at the zone center is moved down for these SL _s for the bulk GaN band structure (not presented here). We also notice significant changes in the CB ehavior near R, A, Y and Z for (110) compared with (001). When n increase, the bottom of CB at R and A becomes higher, and it is at Z for (001)- SL _s and Y for (110) SL _s that the bottom of CB starts to lower rapidly. This competition between different high symmetry points of the SL _s is mainly due to the zone folding effect, which is a typical feature for SL _s . In other words, BZ of bulk binaries are folded into a smaller zone in a SL _s . In fact, any wavefunction of the SL _s at any high symmetry point kSL can be written as a linear combination of the wavefunctions of the bulk constituent materials at specific points k_B of the bulk. These points are linked to each other by the following formula: kB=kSL±(0,0,1/(m+n))×2π/a0 ⁵¹. As the period of the SL _s increases along with k, The SL _s BZ volume decreases. This is due to the number of times the bulk energy bands have to be folded (twice, four, six, and eight times according to the value of _n) in order to be contained within the SL _s BZ. For example, in the case of the n = 1, the bulk bands need only be folded in half. We should mention that the number of atoms contained in the SLs2(m+n) is bigger than in bulk. It is for these reasons that the number of valence electrons contained within the SL _s unit cell is 2L times the number of atoms contained within the bulk unit cell, then there are 2L times more energy levels than in the bulk material ⁵⁴. Table IIIsummarizes the results concerning the gaps obtained in our different samples.

Figure 2 The band structure of superlattice (InN)/(GaN) _n with n = 1; 7 for both (001) and (110) cases. 

In order to estimate the effect of n number of monolayers into (InN)/(GaN)_n SL _s on energies band gaps, we have plotted in Fig. 3 the variations of the Γ- Γ gap versus SL thickness in both directions (i.e., versus the number n of monolayers). We remark that the Γ - Γ gap increases with thickness from n = 1 to n = 7 by 0.889 eV and 0.797 eV for the two directions (001) and (110) respectively. We also notice that the Γ - Γ gap in (001) differs by more than 92 meV from the (110)-oriented SL_s; this can be understood by considering the orientation dependence of the ordering potential in the (110) direction, which is given by 2⋅(VGaN-VInN)/n ⁵⁵. For a better understanding of the symmetry of the present SL _s , we advise the reader to look into the very pedagogical paper of Gopalan et al. ⁵⁶. By paying attention to the difference between these SL _s and bulk binaries in details, it is notably visible that the Γ - Γ direct gap of SL _s is lower than the fundamental gaps of GaN Γ - Γ and then is not obtained from the interpolation of their binaries. It is found that the energy band gap Γ - Γ increases slowly with increasing barrier thickness from n = 5, converging to a value near the fundamental gaps of GaN. I. Gorczyca et al. ³ show that (i) the increase of the barrier width (above about 10 monolayers) seems to a have negligible effect on band gap Γ - Γ in the (001) case. Although, it can be seen that the bandgap increase slowly in the (110) than (001) due to the weaker confinement, which allows us to say that the number of monolayers n is bigger in the (110) case and (ii) the contributions to the InN well wave functions arriving from adjacent GaN layers cause a significant increase of the local gap from the value 0.7 eV (appertaining to bulk InN) to about 2.1 eV in the InN layer in these SL _s . Therefore, it appears that the band gaps are most sensitive to the good thickness than to the barrier breadth and that the dependence on barrier thickness is different for SL _s with a various number of good monolayers. Here, the fundamental gap of InN is very important because it reduces the quantum confinement effect in the SL _s .

Such quantum confinement effects will influence the Γ - Γ bandgap values of these SL _s . According to Ref. ⁵⁷, the conventional InN/GaN QW are found to be type I, signification that the band gap values of SL _s are controlled by the InN region, but the major challenges for this conventional system are the largely spontaneous and piezoelectric polarization fields, which significantly reduce the optical gain of the SL _s . However, compared to the (001) case, the direction (110) is predicted to exhibit low quantum confinement effects, i.e., there is a low built-in electric field in the well InN or barrier GaN regions. We can conclude that for a rationally period and low concentrations of In, these conventional systems are very good optical materials.

Figure 3 Direct band gap energies (Γ - Γ) as a function of the number n of monolayers in the superlattice (InN)/(GaN) _n for both (001) and (110)-oriented SL _s  

In the following, we want to comprise what happens in the two cases of SL _s by comparing directly the peaks of the different density of states (DOS) curves and by analyzing the partial densities as a function differents numbers of GaN monolayers in the. For reasons of comparison, the DOS of the bulk materials have been calculated. The VB of binaries was divided into low, intermediate, and high-energy subbands. In both InN and GaN, we have found that VB near the Fermi level E _f was mostly due to orbital p of In and N and orbital d of Ga and In atoms, while the greatest contribution to the bottom of CB was due to orbital p of N and s of In for InN and orbital s of N and Ga and orbital p of N for GaN (Fig. 4). As an example, we show in Fig. 5 the total and partial density of states of two limit configurations (n = 1 and n = 7). The total and partial DOS profiles illustrate the participation in the electronic interactions orbital and their positions. The first little peak, which occurs in the VB of (001) and (110) SL _s , shows a strong contribution from the s orbital of Ga atom. The following VB peaks are all due to a high mixing of orbitals p of the three atoms N, Ga and In and orbital d of Ga and In atoms. The CB peaks are all due to a high mixing of p, s, and d orbital of N, Ga and In atoms for both (001) and (110)-oriented SL ´s. We can see a little difference in the (110)-𝑆 L₈ , where the contribution of the orbitals 𝑑 of In atom and 𝑠 of N atom decrease (increase) considerably from n = 3 to n = 7 in VB near E _f . It is clear from these results that the orbitals contribute almost similar to the higher VB’s and lower CB’s in the (001) and (110)-oriented SL _s . The conclusion given in this section will be confirmed by the investigations of the optical properties reported below.

Figure 4 Total and partial density of states (DOS) for InN and GaN compounds. 

Figure 5 Total and partial density of states (DOS) for (InN)/(GaN) _n of both (001) and (110) oriented SL _s (n = 1 and 7). 

3.3.Optical Properties

It is fundamental to know to acquaint the optical activity of materials for their application in optoelectronic devices. In this consideration, we investigate the optical properties of both directions (001) and (110)-oriented SL _s for different monolayers n. These important properties may be attracted per the complex dielectric function ε(ω)=ε1(ω)+i⋅ε2(ω), which is determined mostly by the transition between the valence and conduction bands. Where ε1(ω)andε2(ω) are the real and imaginary parts of the dielectric function, respectively ⁵⁸. The imaginary part of the dielectric function, ε2(ω) has been obtained directly from the momentum matrix elements between the occupied and the unoccupied wave functions ⁵⁹. The real part ε1(ω) of the dielectric function can be derived from the imaginary part using the Kramers-Kronig relation ⁶⁰. Both real and imaginary parts allow the calculation of important optical functions such as refractive index n(ω) and absorption coefficient a(ω) using the following relations ⁶¹.

Where k is the extinction coefficient, λ is the wavelength of light in the vacuum. To show all the possible optical transitions, we have increased the number of special points in the First BZ up to 600 k-points, with an energy range of up to 20 eV. Figure 6 shows the real and imaginary parts of the dielectric function for SL(1,3), SL(1,5), and SL(1,7) compounds for both growth axis. The occurrence of the peaks in ε2(ω) is principally produced through the inter-band or intra-band optical transitions. The contribution of intra-band transitions is important only in the case of metals ⁶². ε2(ω) spectra represented in terms of the optical transition between the filled and vacant states of various SLs in both (001) and (110) cases are quite dissimilar and reflect various optical transitions. By increasing 𝑛, the peaks move to higher energy, which might be strongly dependent on the ionic polarization of the SL _s due to the large electronegativity of N. The curves of ε2(ω) indicate that the threshold energy (first critical point) occurs at 0.665, 0.833, 0.998 eV in (001) case (0.605, 0.866, 0.899 eV in (110) case) for n = 3,5,7 respectively. These values correspond to the electronic transition value (ΓV→C).

Figure 6 Calculated dielectric functions (real and imaginary parts) for (InN)/(GaN) _n superlattices at direct band gap for the (001) ) growth axis direction. The inset of figure shows the real and imaginary parts plot for (110) ) growth axis direction. 

This point represents the fractionation ΓV→C, which gives the threshold of the direct optical transitions between the highest state of the valence band and the lowest state of the conduction band, which is identified as the fundamental absorption edge. Beyond these points, the curves increase rapidly. As a general observation, all curves of Fig. 6b show three major peaks in our calculations, appearing the resonance behavior. The maximum absorption for the three SL _s is situated at 6.80, 6.366, 8.799 eV in (001) case (6.766, 6.50, 7.01 in (110) case), respectively. Note that some changes in the edge or onset of the states are detected in the dielectric function plots in (110). This could be attributed to the difference between the optical transition states as well as the characteristic asymmetry in the electronic valence charge density distribution and in the bonding charge around the atoms. For both (001) and (110)-oriented SL _s , the most optical transitions contribute to these peaks occur from occupied p orbital of Ga and N atoms and d orbital of Ga atom localized in the highest valence band to unoccupied s orbital of N, Ga, and In atoms localized in lowest conduction band.

The variation of the ε1(ω) according to the energy for the SL _s is represented in Fig. 6a. We have noted that these optical spectrum represented in Fig. 6a are similar to small differences (The position and the height of peaks for both growth axes). The limiting value of the real part of the complex dielectric function obtained at a frequency of irradiation approaching zero is called static dielectric constant ε1(0). Our calculated values of ε1(ω)(0) found in this way are summarized in Table IV. The calculated static dielectric constant ε1(0) decreases with increasing n monolayers, which is consistent with the decrease of the direct bandgap value, indi-cating that the is in an inverse relationship with bandgap accorded to the Penn model . We have concluded that the number of layers used effects the real parts of the dielectric functions for all configurations, and they have an important effect in the (110) case.

The refractive index of (InN)/(GaN)_n SL _s is a valuable tool for the design of photovoltaic and optoelectronic devices. In order to study the transparency of SL _s in response of an incident light the relation of Eq. (3) is used for computed the theoretical refractive index values for all SL _s . The refraction spectra n(x) and refractive index n(0) are depicted in Fig. 7a and reported in Table IV. For the calculated SL _s n(x) is not presenting a large variation for the photon energy in the near infrared or visible radiation. As a result , the static refractive indices n(0) are estimated from the refraction spectra (see Table IV).The maximum refractive index is reached for photon energy around 5.0, 4.89, 6.23 eV for (001) case (5.10, 5.40, 5.30 eV for (110) case), respectively. The spectral plots of the refractive index of Fig. 7a show an increasing from 2.424 (for n =3) to 2.449 (for n = 7) in the (001) SL _s , and its value increases from 2.424 (for n = 3) to 2.457 (for n = 7) for (110)-oriented SL _s . According to the result in both directions, the influence of the indium composition in the refractive index values of SL _s are more important for (001) direction compared to the (110) direction, which allows us to say that the refractive index of SL _s depends on growths axes.

Figure 7 Calculated Refractive indices n(ω)(α) and absorption coefficients α ( ω)(b) for (InN)/(GaN)n superlattices for the (001) growth axis direction. The inset of figure shows the Refractive indices and absorption coefficients plot for (110) growth axis direction. 

The optical absorption spectra are calculated in order to distinguish the optical nature in the SL _s . The dispersion in the optical absorption spectra of the SL(1,n) grown along both directions (n = 3,5,7) are plotted in Fig. 7b. The absorption peak of both directions shifts to higher energy when n increases and the threshold energy increases. We observe from the curves of Fig. 7b that the absorption edge begins from the energy values of 0.600, 0.833, and 1.0 eV for (001) case (0.600, 0.766 and 0.9 eV for (110) case), respectively. It shifts towards lower energy when compared to bulk GaN, whose threshold is located at E _s = 2.05 eV. Two major peaks at approximately 7 eV and 8.9 eV in the absorption coefficient for both growth axes represent the absorption of light at two different wavelengths. The maximum value of a(ω) is approximately 12.5 eV for (001) case (12.8 eV for (110) case) for all values of n, then the absorption coefficient abruptly decreases for the light photon above these energies. The intensity of absorption spectra diminishes with the inclusion of In contents in GaN layers.

As a result, the main conclusion to retain from all these curves is that the growth direction (110) does not ameliorate the optical activity. The similar contribution of the atomic orbitals to the electronic properties is probably the main reason for which the optical properties are expected not to be modified significantly. We suppose that the contribution of great cells will not change the present conclusions, on account of increasing the cell will only make the contribution of the bulk important compared to that of the interface and may not ameliorate the optical activity anymore.