1. Introduction

There is a growing interest in the study of the III-V Nitride compounds (GaN, InN, A1N and their alloys) due to their technological importance. Particularly, these materials are potential candidates for new generation of light emitters and other optoelectronic devices operating across the visible and UV wavelengths, and most studied of wide bandgap semiconductors have been the III-nitrides. Among them, GaN and its alloys with InN and AIN have attracted great attention since the successful commercialization of bright blue/green light-emitting diodes followed later by the demonstration of injection lasers[1,2]. Researchers considered the III nitrides to be laboratory curiosities, unlikely to be of great practical importance. However, after several breakthroughs in growth and doping technologies, the III nitrides have emerged as the leading semiconductor materials for short-wavelength (green, blue, and violet) visible-light-emitting diodes and laser diodes, as well as solar-blind photodetectors, and show promise for high-power, high-frequency, and high-temperature microelectronics. There are many theoretical studies reported elsewhere for GaN[3,4], for InN[3,5], for AlN[6,7], for InGaN[8,9] and for AlGaN[10,11]. Major developments in wide-gap III–V nitride semiconductors have recently led to the commercial production of high-brightness blue/green light-emitting diodes (LED’s)[12] and to the demonstration of room-temperature (RT) bluish-purple laser light emission in InGaN–GaN–AlGaN-based heterostructures under pulsed currents[13,14] and continuous-wave (CW) operation[15, 16]. These developments are a result of the realization of high quality crystals of AlGaN and InGaN, and p-type conduction in AlGaN[17,18]. The recombination of localized excitons has been proposed as an emission mechanism for this InGaN quantum-well-structure LED’s[19].

In an attempt to give a more systematic understanding of the structural and the electronic properties of these Group-III nitrides and their related alloys, we have carried out our calculations within the local density approximation (LDA) to density functional theory, using the all-electron full potential linear muffin-tin orbital (FP-LMTO) augmented by a plane-wave basis (PLW). The exchange- correlation energy of electrons is described in the local density approximation (LDA) using the parameterization of Vosko et al.[20] and also of Perdew et al.[21]. We have also applied this computational method to AlGaN and InGaN systems greatly used in the heterostructures and the quantum wells to check its transferability to predict the structural and electronic properties from those of their parent compounds.

2. Method of Calculation

The full-potential linear muffin-tin (FP-LMTO) method is a specific implementation of density functional theory within the local density approximation (LDA). In this method there is no shape approximation to the crystal potential, unlike methods based on the atomic-spheres approximation (ASA) where the potential is assumed to be spherically symmetric around each atom. For mathematical convenience, the crystal is divided up into regions inside muffin-tin spheres, where Schrödinger’s equation is solved numerically, and an interstitial region. In all LMTO methods the wave functions in the interstitial region are Hankel functions. Each basis function consists of a numerical solution inside a muffin-tin sphere matched with value and slope to a Hankel function tail at the sphere boundary. The so-called multiple –kappa basis is composed of two or three sets of s, p, d, etc. LMTO’s per atom. The extra variational degrees of freedom provided by this larger basis allow for an accurate treatment of the potential in the interstitial region.

In the present work, we use the Savrasov version of the full potential linear muffin-tin orbital (FP-LMTO) method augmented with a plane wave basis (PLW)[22]. The non overlapping muffin tin spheres potential is expanded in spherical harmonics inside the spheres and Fourier transformed in the interstitial regions. The exchange-correlation energy of electrons is described in the local density approximation (LDA) using the parameterization of Vosko et al. and also of Perdew et al.[21].

It is a well-known that the LDA has a tendency to overestimate the bonding. The improvement by the GGA is in fact mainly due to an improvement in the calculation of the atomic reference energies. Since the free atoms contain regions of very low density and strongly varying electronic density, gradient corrections to the exchange and correlation functional are more important for free atoms than for bulk. The lattice constant is usually underestimated with respect to the experimental lattice constant by up to a few percent by the LDA while the GGA sometimes overestimates it. In this work, the orbital 3d¹⁰, 4s², 4p¹ of Ga, 4d¹⁰, 5s², 5p¹ of In, 2s², 2p³ of N, and 3s², 3p¹ of Al are all included into the self-consistent treatment, i.e. viewed as valence electrons. The present calculation methods are implemented in the available computer code lmtART[23]. Basis functions, charge density, and potential were expanded inside the muffin-tin spheres in spherical harmonic functions with cut-off , and in Fourier series in the interstitial region. In the muffin-tin spheres (MTS) of radius RMT, the upper limit on the angular momentum expansion of the smoothed Hankel functions about a given atomic site is carried out up to [24].

Table 1. The structural parameters of GaN, AlN and InN in the two phases (V0 is the equilibrium volume, a0 the lattice constant, B0 the bulk modulus and B’0 is its pressure derivative). V0 is taken equal to a3/4 for both zinc blende and       for the wurtzite phase for which the volume per unit formula is taken into account. Both FPLMTO results is presented.

3. Results and Discussions

The total energy versus volume for GaN, AlN and InN in both the zincblende (8 atoms) and the wurtzite structures (16 atoms) was calculated in our previous work[24]. Let us discuss now the results of the band structure calculation. The static properties of the phase under study (equilibrium lattice constant, bulk modulus, and its pressure derivative) for the studied compounds are determined using the Murnaghan/Birch equation of state to fit our calculated E-V data. The results compared to experiment and other works are given in Table 1.

In Fig. 1, the band structure of these materials in the Zincblende (B1) phase for the equilibrium volume is shown. The band structure of InN shows a direct band gap at Γ, closely similar to that of GaN. This is very different from what we have found for AlN, where we notice a small indirect Γ–X fundamental band gap. Our calculated band gaps obtained with the LDA for the zincblende form are Eg=3.33eV, Eg=0.6eV and Eg=5.76 eV for GaN, InN and AlN, respectively. These values are in good agreement compared to the experimental bands gaps found by M. Marques et al.[25] (3.3 eV for GaN, 0. 9 eV for InN and 5.94 eV for AlN) and those of Munoz et al.[26]. However, these band gaps are underestimated by other theoretical works as PWPP method and the FP-LAPW method[27]. It is well known that in the wurtzite case of GaN and AlN, the lowest conduction band and the top of the valence bands are situated at the Γ point in the BZ. Without spin-orbit interaction they are related to an s-like Γ_1c state and to p-like Γ_6v and Γ1v states. The two latter states are separated by a crystal-field splitting . Two energy gaps can be defined by and according to the optical transition and . In GaN, is the fundamental gap, whereas in AlN it is the C transition gap due to the negative crystal field splitting[28,29].

In order to characterize the optical gain and the threshold current density of laser diodes for different temperatures (transparency density) and cavity length in our laser structures, we have calculated the energy bandgaps and the refractive index. Optimization of intrinsic and extrinsic parameters such that: absorption coefficient (α), electronic affinity (χ), confinement factor (Γ) and effective mass, represents the principle stage in the design of a laser diode. These optimized parameters are obtained using the combined LMTO method, Mathematica and Mathcad calculations.

The first structure studied is the simple heterostructure based on the studied compounds. Above this transparency threshold, the medium begins to amplify those photons possessing energies which satisfy the Bernard—Durrafourg condition. The gain spectrum is then given by:

(1)

Where is the gain of the semiconductor medium is the empty conduction band absorption (i.e. under zero current) with[43]:

(2)

The Fermi-Dirac functions and are the occupation rates for the levels in the conduction and valence bands satisfying .

Figure 1. The band structure of GaN, InN, AlN, InGaN and AlGaN in the equilibrium phase for the equilibrium volume.

Figure 2. Evolution of the quasi-Fermi level difference as a function of non-equilibrium carrier density in InGaN.

Figure 3. Evolution of the gain curve maximum in InGaN as a function of electron-hole pair density.

Fig. 2 shows how the gain curve evolves as a function of non-equilibrium carrier density in InGaN. Fig. 3 shows the evolution of the maximum gain in InGaN as a function of electron-hole pair density. The curve is linear which is consistent with the fact the optical gain is proportionally to carriers densities. The maximum Gain (which is one of the principal points of these curves since in fact the modes close to will lasing) shows a linear increase with the density of carriers to the above of the threshold of transparency.

In Fig. 4, we have illustrated the evolution of the gain according to the energy of the photons for various densities of injection in the two materials (GaN and In_0.5Ga_0.5N). Once transparency has been achieved, the maximum gain increases linearly with charge density above threshold. This purely phenomenological relationship is very useful in modelling the behaviour of semiconductor lasers.

Figure 4. Evolution of the Gain according to the energy of the photons for various densities of injection in the two materials (a) GaN, (b) (In0.5Ga0.5N).

Figure 5. Evolution of the maximum Gain according to the current of pumping for various values thickness of the two materials (a) GaN, (b) (In0.5Ga0.5N).

4. Laser Diode with Double Heterojunction

To improve the laser diode performances, we must increase the Gain with the minimization of the current. Then we have studied the second structure where the confinement factor plays an important role for satisfying the above condition. From the studied compounds, we have considered the double heterostructure (DH) where at the first stage we started by determining certain sizes characteristic as the losses (parasitic losses which come from the free carriers of the electrical contacts of the diffusion and the losses intrinsic that corresponds to the escape of the photons) for a fixed width of 600 A°. Fig. 5 shows that the Gain increases when the current of pumping decreases in the case of a structure with double heterojunction for two materials with various thicknesses of the cavity. By comparing the variation of the threshold in the DH laser diode type with DH with that of in an heterojunction diode, one observes a difference in the current of threshold (returns decreased the thickness of the cavity of some micrometers) which comes from the weak optical confinement on the one hand, and on the other hand from the lower volume to reverse in an heterojunction. The value of 0.1 μm structural features DH is comparable with the optical wavelengths and consequently allows an optimised confinement of photons. On the other hand, this value is much higher than the electronic wavelengths, so that the confinement of electrons can be improved considerably. It is the role of the quantum well[47-49].

5. Conclusions

Our calculated lattice parameter is found to be in reasonable agreement with other theoretical works as well as the experimental result. From the E-V curves, we found that for the studied compounds, the wurtzite form is slightly lower in energy than that of the zincblende form. Hence, for the low pressure, these compounds show the new equilibrium phase which is the zincblende structure. At high pressure, we can predict that the new phase is probably the rock salt structure. We have also applied this computational method to AlGaN and InGaN alloys to check its transferability to predict the structural and electronic properties from those of their parent compounds. As a function of composition, the alloys of GaN and InN should cover the entire spectral region from the very near UV to the red while AlGaN covers a wide range in the near UV.

We notice that most significant in all this study, is the remarkable quantitative difference found between the two systems studied in this article, i.e., GaN\Al_0.25Ga_0.75N and In_0.5Ga_0.5N\Al_0.25Ga_0.75N. Finally, one can state that the use of a laser diode containing the In_0.5Ga_0.5N\Al_0.25Ga_0.75N\GaN structure leads to better performances.

The results presented in this work were considered as a contribution to the current state of the art, in the development of the lasers diode with double heterojunction. We can say that the laser diode remains a very active subject of research. The works in progress continue to extend their applicability while improving their performance such as for example the spectroscopies for detection of pollution. The difficulties occur when one seeks to join together several extreme characteristics unit[46].