Pressure dependence of energy band gaps for AlxGa1 - xN, InxGa1 - xN and InxAl1 - xN

Using a first-principles method, we study the effect of pressure on the band gap energy of wurtzite AlxGa1 - xN, InxGa1 - xN, and InxAl1 - xN. Starting with the binaries, GaN, InN and AlN, the direct band gap is found to increase linearly with pressure but becomes indirect for AlN at 13.88 GPa. The direct band gap pressure coefficients are 31.8 meV GPa-1 for GaN, 18.8 meV GPa-1 for InN and 40.5 meV GPa-1 for AlN, which are in good agreement with other calculations. For the ternary alloys, the fundamental band gaps energy are direct and increase rapidly with pressure. The pressure coefficients vary in the range of 31.9-34.5 meV GPa-1 for AlxGa1 - xN, 19.8-24.8 meV GPa-1 for InxGa1 - xN and 16.7-20.7 meV GPa-1 for InxAl1 - xN; they depend on alloy composition with a strong deviation from linearity. The band gap bowing of InGaN increases linearly with pressure, but those of AlGaN and InAlN strongly decrease when the AlN band gap becomes indirect.


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included the Ga 3d and In 4d as valence states. No shape approximations are employed for either the potential or the charge density. Basis functions were expanded in combinations of spherical harmonic functions inside non-overlapping spheres surrounding the atomic sites (muffin-tin (MT) spheres) and in Fourier series in the interstitial region. In the MT spheres, the l-expansion of the non-spherical potential and charge density was carried out up to l max = 10. In order to achieve energy eigenvalue convergence, the wavefunctions in the interstitial region were expanded in plane waves with a cutoff of k max = 8/R mt (where R mt is the average radius of the MT spheres). In the following calculations, we have distinguished the Al (1s 2 2s 2 2p 6 ), Ga (1s 2 2s 2 2p 6 3s 2 3p 6 ), In (1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 6 ) and N (1s 2 ) inner-shell electrons from the valence electrons of Al (3s 2 3p 1 ), Ga (3d 10 4s 2 4p 1 ), In (4d 10 5s 2 5p 1 ) and N (2s 2 2p 3 ) shells. For GaN and AlN we have adopted the values of 1.9, 1.8 and 1.6 Bohr for gallium, aluminium and nitrogen, respectively, as the MT radii. In the case of InN, 2.05 and 1.75 Bohr for indium and nitrogen, respectively, are used. For the ternary alloys, we have chosen the MT radii values of 1.75 Bohr for gallium, aluminium and indium, and 1.65 Bohr for nitrogen.
To model the Al x Ga 1−x N, In x Ga 1−x N and In x Al 1−x N random wurtzite alloys we have used a small 32-atom X n Y 16−n N 16 supercell (X = Al or In and Y = Ga or Al), which corresponds to a 2 × 2 × 2 supercell which is twice the size of the primitive wurtzite unit cell in both directions of the basal plane and along the c-axis. For a given number n = 0, . . . , 16 of X atoms, different atomic configurations have been optimized structurally. However, it is impossible to treat all different atomic configurations. Therefore, for a given number n of X atoms we usually study only a small number of different configurations in which the X atoms are not really randomly distributed. For each configuration and each atomic number n, the fundamental physical properties (total energy, and band gap) are determined. The configurationally averaged quantity is computed using the Conolly-Williams approach [32] for each given x. The composition-dependent weights are determined for an ideal solid solution. We have used only the X 4m Y 4(4−m) N 16 clusters (m = 0, . . . , 4) to calculate the quantities for the entire composition region. In spite of the small size of our supercell, the calculations are sufficiently converged, and the obtained results for the band gaps are in good agreement with those of Schilfgaarde et al [33].
The k integration over the Brillouin zone is performed using the Monkhorst and Pack mesh [34]. A mesh of eight special k-points was taken in the irreducible wedge of the Brillouin zone for the binary cases; seven special k-points were used for the supercell calculations.
The structural optimization of the wurtzite phase was performed by calculating the total energy as function of the three variables u, c/a and V . The two-dimensional minimization of the total energy versus (u, c/a) for a fixed volume requires that each of the self-consistent calculations is converged, so the iteration process was repeated until the calculated total energy of the crystal converged to less than 1 mRyd. A total of seven iterations was necessary to achieve self-consistency.
For the geometric equilibrium determination of the wurtzite phase we proceeded as follows: we first determined the internal parameter u for a specific volume V and c/a, then by using it we optimized the c/a ratio to obtain (c/a) eq at (V , (c/a) eq , u eq ). Then, using the two parameters u eq and (c/a) eq , we optimized the volume. The equilibrium lattice constants and bulk modulus are calculated by fitting the total energy versus volume according to Murnaghan's equation of state [35].

Results
The calculated structural properties (lattice constants a and c, internal parameters u, bulk moduli B) and energy gaps E g of the binaries are summarized in table 1. We have an underestimation of the lattice parameters and the energy gaps, and an overestimation of the bulk moduli in comparison to those of experiment (table 1), due to the use of the LDA. The pressure coefficient of an interband transition i in a semiconductor is easily calculated. It is given by In the cubic structure, this quantity is related to the volume deformation potential dE/d ln V and the bulk modulus B by Since the wurtzite structure has two structural degrees of freedom (u, η = c/a), the effect of changes in the structural parameters u and η with volume on the band gap yields a generalization of equation (2) to The variation of the energy gap versus volume was calculated using the equilibrium u and η parameters (at p = 0). In fact, for GaN Wagner and Bechstedt [36] and Serrano et al [37] using pseudopotential calculations have shown that the pressure dependence of both c/a ratio and internal parameter u are negligible. Wagner and Bechstedt [36] reported ∂η ∂p and ∂u ∂p as −4.7 × 10 −5 and 5 × 10 −6 GPa −1 , respectively. This result was confirmed by experiment for the c/a ratio [38]. For AlN, Wagner and Bechstedt [36] reported a slope of ∂η ∂p = −5.6×10 −4 GPa −1 and ∂u ∂p = 1.08 × 10 −4 GPa −1 . This is an order of magnitude larger than that of GaN. For InN, the LAPW investigation of Wei and Zunger [24] reported linear pressure coefficients of ∂η ∂p = −6.7 × 10 −6 GPa −1 and ∂u ∂p = 2.02 × 10 −4 GPa −1 (estimated from their results: ∂η ∂ ln V = −0.001 and ∂u ∂ ln V = 0.03, B = 148 GPa). Therefore, although the variation of the band gap energy with c/a ratio and internal parameter u is important, the ∂η ∂p and ∂u ∂p slopes make the second and third terms of equation (3) much smaller than the first.
By the use of our calculated values of the bulk moduli, B, and their first pressure derivatives B , the volume change with applied pressure was calculated using the following equation [35]: The behaviour of the lowest conduction band energies with pressure at a number of highsymmetry points of the Brillouin zone for GaN, InN and AlN is reported in figures 1-3. All the quantities are calculated near equilibrium. For GaN (figure 1) and at pressures up to 28 GPa, the fundamental band gap remains direct. However, at p = 16.1 GPa, there is a band gap crossing of the K c and A c conduction bands. For InN (figure 2), the fundamental band gap stays direct for a pressure applied up to 21.5 GPa. In contrast, for AlN (figure 3), at pressures up to 21 GPa, the fundamental band gap becomes indirect (K c − Γ v ) at p = 13.88 GPa. This is due to the rapid increase of the Γ c conduction band under pressure, while the K c conduction band remains nearly constant.
At high pressure, GaN, InN and AlN present a phase transition from wurtzite to rocksalt structure. The experimental data for the transition of GaN range from 37 to 53.6 GPa [38]- [41]. The theoretical investigations reported values from 42.9 to 55 GPa [20,21,37,42]. For InN, the transition pressure is from 10 to 14.4 GPa experimentally [38,41,43], and from 11.1 to 21.6 GPa 94.6
So for AlN, a modification of the fundamental band gap from Γ c to K c is expected at a pressure of about 13.88 GPa, indicating that AlN probably becomes an indirect band gap material before reaching the phase transition, at least with reference to known experimental data.
In order to calculate the pressure coefficients, we have fitted E g (p) to a quadratic function: where E is in eV, p the pressure in GPa, and α and β the first-and second-order pressure derivatives respectively, which are given in (i) the treatment of the Ga 3d in GaN and In 4d in InN as core or as valence electrons; (ii) the accuracy of the method used, e.g. pseudopotential and full potential; (iii) the choice of the functional for the exchange-correlation energy of the electrons; (iv) the computational parameters such as the energy cutoff (which determines the number of plane waves used in the pseudopotential approaches and in LAPW approaches which are used in the interstitial region).
For the other band gaps, when we increase the cation atomic number, e.g. going from AlN, GaN to InN, the linear pressure coefficient of the M c conduction band increases; it decreases for the A c conduction band, but remains nearly unchanged for the K c conduction band. This trend is in good agreement with the calculations of Christensen and Gorczyca [21] for GaN and AlN.
Experimentally, for GaN, Perlin et al [16] showed that the presence of sapphire substrate leads to an energy gap pressure coefficient reduction of approximately 5% in comparison to free-standing GaN, due to a compressive-biaxial strain [47]. They reported 38.9 meV GPa −1 for GaN on sapphire, and 41.4 meV GPa −1 for free-standing GaN [16]. For InN, only one study recently reported a value of 6 meV GPa −1 [4], with a fundamental band gap energy (at p = 0) of 0.8 eV. For AlN, to our knowledge, only one value has been reported, 49 meV GPa −1 [18].
For the second-order pressure derivatives β of the different band gaps (table 2), our results agree with the LMTO calculations of Christensen and Gorczyca [21] and the pseudopotential results of Van Camp et al [19,20] for GaN and AlN.
In the ternary alloys, the lattice parameters a and c for Al For the pressure coefficients of the ternary alloys, we followed the same procedure as for the binaries. Near equilibrium and at each composition, we calculated the electronic band structures at different values of the hydrostatic pressure. The pressure behaviour of the fundamental band energy is shown in figure 4 for Al x Ga 1−x N, in figure 5 for In x Ga 1−x N and in figure 6 for In x Al 1−x N. For Al x Ga 1−x N (figure 4), the fundamental band gaps of Al 0.75 Ga 0.25 N are larger than those of Al 0.5 Ga 0.5 N which are larger than those of Al 0.25 Ga 0.75 N. In the investigated pressure range (0 to ∼29 GPa) and for all Al composition (25, 50 and 75%), the fundamental band gap (Γ c − Γ v ) remains direct. It increases rapidly with pressure, from 2.68 to 3.41 eV for Table 2. First-and second-order pressure derivatives of Γ, M, K and A conduction bands with respect to the top valence band at (a) Γ for GaN, (b) Γ for InN and (c) Γ for AlN. The linear term α = dE dp (meV GPa −1 ) is given on the first line, and the quadratic term β = d 2 E dp 2 (meV GPa −2 ) on the second line when available.
GaN  28 PL and absorption [4] 6 we calculate a pressure coefficient bowing parameter of 15.34 meV GPa −1 . From experimental data, our results are slightly smaller than those of optical absorption measurements at room temperature [15] on samples with Al composition range 0.12 ≤ x ≤ 0.6, and near-band-edge PL study of single-crystal Al 0.05 Ga 0.95 N and Al 0.35 Ga 0.65 N [25]. However, the results of these studies do not exhibit significant dependence of pressure coefficient on alloy composition; the near-band-edge PL study [25] Table 3. Energy band gaps, E g , first-order α and second-order β pressure derivatives of the fundamental band gap, bulk moduli, B, and their pressure derivatives, B , of Al x Ga 1−x N alloys.  In our case, the introduction of In reduces the pressure coefficient significantly and does not lead to linear variation with alloy composition. We have a bowing parameter of 15.1 meV GPa −1 . The recent work of Perlin et al [29] reported pressure coefficients of the In x Ga 1−x N alloys using FP-LMTO and PL. They showed a dependence of the pressure coefficient of the fundamental band gap (Γ c -Γ v ) on alloy composition with a significant deviation from those of linear interpolation, in agreement with our calculations. For In x Al 1−x N (table 5), there is a strong dependence of the pressure coefficients on alloy composition. It varies from 20.7 meV GPa −1 for In 0.25 Al 0.75 N, to 17.6 meV GPa −1 for In 0.5 Al 0.5 N, and to 16.7 meV GPa −1 for In 0.75 Al 0.25 N, with a strong deviation from linearity. It appears that introducing In decreases the pressure coefficient significantly: for 50 and 75% of In, the values are lower than that of InN. From the variation of the fundamental band gap energy with alloy composition for different pressures (between 0 and 20 GPa), and using equation (7) for the energy band gap, we calculated the band gap bowing parameter at each pressure. This is shown in figure 7 for Al x Ga 1−x N, In x Al 1−x N and In x Ga 1−x N. For Al x Ga 1−x N and In x Al 1−x N, the bowing parameter increases with pressure until a pressure ∼14 GPa, after that it decreases rapidly. The reason is the change of the fundamental band gap in AlN from Γ (Γ c -Γ v ) to K (K c -Γ v ); the Γ c -Γ v band gap increases quickly in contrast to that of K c -Γ v which remains almost constant and smaller than that of Γ c -Γ v , after the crossing of the two bands at p = 13.88 GPa. For In x Ga 1−x N, the band gap bowing parameter increases continuously with pressure. Table 4. Energy band gaps, E g , first-order α and second-order β pressure derivatives of the fundamental band gap, bulk moduli, B, and their pressure derivatives, B , of In x Ga 1−x N. The experimental data are given from PL measurements.

Discussion and conclusion
Using the FPLAPW method, we have studied the behaviour of the wurtzite nitride ternary alloys with pressure, starting with the pressure coefficients of the binaries. In the experimental studies, many parameters contribute to the determination of the pressure coefficients and this is probably the reason for scattered values. Such parameters are, among others, the technique of measurement of the band gap (PL and absorption), the effect of the temperature, the number of experimental points [16], the substrate, and the approximation used to fit E g (p). Perlin et al [16] showed that the presence of sapphire substrate reduces the pressure coefficient by 5%. The approximation used to fit the E g (p) also plays an important role, since the linear approximation is observed to be not suitable above 10 GPa [16,24]. For the ternaries, the reported values of pressure coefficients do not exhibit a dependence on alloy composition. For Al x Ga 1−x N, Shan and his co-workers reported 39 meV GPa −1 for GaN (PL) [15], 40 meV GPa −1 for x(Al) = 0.05 at 295 K (PL) [25], 37.3 meV GPa −1 for x(Al) = 0.12 at 295 K (by absorption) [15], 37.2 meV GPa −1 for x(Al) = 0.2 at 295 K (by absorption) [15] 36 meV GPa −1 for x(Al) = 0.35 at 10 K (PL) [25], 37.6 meV GPa −1 for x(Al) = 0.4 at 295 K (by absorption) [15], and 37.2 meV GPa −1 for x(Al) = 0.6 at 295 K (by absorption) [15]. Further, Shan et al in their study [15] corrected the measured values by taking into account the difference of compressibility between the epitaxial films and the sapphire substrate: the reported values are given in table 3 and are larger. For In x Ga 1−x N, Shan et al [26] using PL and photomodulation spectroscopy reported 39 meV GPa −1 for GaN at 10 K, 39 meV GPa −1 for x(In) = 0.04 at 10 K, 35 meV GPa −1 for x(In) = 0.08 at 10 K (with a band gap energy at p = 0 GPa of 3.08 eV), 36 meV GPa −1 for x(Al) = 0.08 at 295 K (with a band gap energy at p = 0 GPa of 3.04 eV), and 40 meV GPa −1 for x(In) = 0.11 at 295 K. In a previous work Shan et al [27] using PL reported pressure coefficients of 35 meV GPa −1 for x(In) = 0.14 and 39 meV GPa −1 for x(In) = 0.08 with a band gap energy at p = 0 GPa of 3.249 eV which differ from those of photomodulation spectroscopy. In the case of In x Al 1−x N, no experimental results on pressure coefficient are available.
In our work, using the same method, we calculated the pressure coefficient of the binaries and the ternaries over a wide range of composition. We see that the introduction of Al in Al x Ga 1−x N increases and that of In in In x Ga 1−x N and In x Al 1−x N decreases the pressure coefficient. However, this variation is not linear. We also report the variation of the fundamental band gap bowing with pressure. It increases continuously with pressure for In x Ga 1−x N, and has the same variation for Al x Ga 1−x N and In x Al 1−x N until a pressure of ∼14 GPa-after that it decreases significantly. This pressure is in the range where we see that the fundamental band gap of AlN becomes indirect.