3.1 Structural optimization

On the scale of meV, the energy bands near the energy gap depend critically on those wurzite structural parameters that are not determined by symmetry. We have therefore performed detailed structural optimizations of the unit cell geometries as a function of the external stress by minimizing the total energy.

In the case of hydrostatic pressure or biaxial stress, the wurzite structure has two degrees of freedom, namely the ratio of lattice constants c/a and the bond length d along the hexagonal axis (0001). This defines a parameter u via the relation d = u c. For the ideal unstrained wurzite structure, one has u = 0.375. In spite of the fact that the calculated structural parameters in Table 1 are close to this ideal ratio, the deviations from this ratio alter the valence band splittings by more than 100%. The energy difference between the topmost valence bands at Γ (crystal field splitting) in unstrained GaN and AlN equals 35 meV and -211 meV, respectively, for the optimized values of u (Table 1) instead of 70 meV and -55 meV, respectively, as obtained with the ideal value u=3/8 (see also Reference Suzuki, Uenoyama and Yanase[15]). Thus, accurate structural optimizations are mandatory for predicting fundamental optical transitions in these materials.

We now wish to discuss the band structure of strained nitrides. We have determined the parameters c and u for both small biaxial strain in the (0001) plane and hydrostatic strain as a function of the lateral lattice constant a. For these types of strain, the strain tensor has only diagonal matrix components e_xx = e_yy = e_// = a/a₀ - 1, e_zz = (c/c₀ - 1) (a₀c₀ denote the equilibrium lattice constants). The present ab-initio calculations predict e_zz = σ e_//, with σ equal to -0.4567 (1.0143), -0.5719 (1.1922), and -0.7926 (1.1740) for GaN, AlN, and InN, respectively. The negative values are for biaxial strain, whereas the positive ones in parenthesis are for the case of hydrostatic pressure.

3.2 Three-band k·p Hamiltonian for strained valence band edge energies

Our calculations confirm that all three studied compounds are indeed direct gap semiconductors in the wurtzite structure with the fundamental gap located at the Γ point (k = 0) Reference Strite and Morkoç[1]. The valence band edge states in the strained wurzite structure near k=0 can be represented by a 6 x 6 k·p Hamiltonian Reference Bir and Pikus[16]. Interestingly, this 6 x 6 matrix can be diagonalized analytically for k= 0 and biaxial strain or hydrostatic pressure. The energy eigenvalues, relative to their center of gravity, can be written in the form Reference Chuang and Chang[17]

(1)

(2)

(3)

where the constants D₃ and D₄ are deformation potentials and Δ₂ and Δ₃ are spin-orbit coupling constants. In the absence of spin-orbit interaction and for zero strain, one has E(Γ_9v) = E₊(Γ_7v) = Δ₁/3, E_-(Γ_7v) = -2Δ₁/3. Depending on the sign of Δ₁, the top of the valence band is formed by the E(Γ_9v) state (as in GaN, InN) or the E₊(Γ_7v) state (as in AlN). Therefore, Δ₁ is conventionally termed crystal-field splitting. Let us point out that we have set to zero the average valence band energy since we are only interested in optical excitation energies. The term Θ(e) represents the strain dependence of the crystal field splitting. We also note that this k·p Hamiltonian neglects the influence of strain on the spin-orbit coupling constants.

The shift of the conduction band edge at Γ with strain is governed by two deformation potentials A₁ and A₂: E(Γ_7c) = E(Γ_7c; e =0) + A₁ e_zz + 2 A₂ e_//. Because of the degeneracy of the valence band edge, it is more transparent to measure this strain dependence relative to the center of gravity of the valence band rather than focusing directly on the strain dependence of the gap.

3.3 Validity of linear strain approximation

We have used the results of the ab-initio calculations to determine the valence band parameters Δ_i and deformation potentials D_i and A_i. In order to determine Δ₁independently from Δ₂, Δ₃and separate the strain and relativistic effects from each other, we have also performed nonrelativistic pseudopotential calculations. We find that the numerically determined valence band eigenvalues can be well represented in terms of the model discussed above, provided the strain tensor elements lie in the range of |e_ij| < 0.01 for biaxial strain and |e_ij| < 0.04 for hydrostatic deformation. In the case of larger biaxial strains, Θ(e) shows pronounced nonlinearities. For biaxial strains between 0.01 and 0.04, the discrepancies between the numerical eigenvalues and the eigenvalues given in Equation 1-Equation 3 stem from the nonlinear strain dependence of Θ(e), whereas the spin-orbit interaction parameters are independent of strain in this range.

For a given external biaxial or hydrostatic stress, the total energy minimization relates the strain tensor component e_// to e_zz. In order to determine both pairs of deformation potentials D₃,D₄ and A₁, A₂, therefore we have computed (dΘ(e)/de_//) _e=0 and (dE_gap(e)/de_//) _e=0 both for biaxial strain as well as for hydrostatic pressure. This gives, for each set of deformation potentials, 2 linear equations for the 2 unknowns. One should notice however, that in the case of small gap semiconductor as InN (in our LDA calculations the energy gap equals 0.1 eV in unstrained wurtzite InN) the description of the valence band through the six band model is doubtful. Therefore, the determined valence band parameters for InN have plausibly no physical meaning and they should be treated as formal parameters that reproduce the theoretical valence band splittings and their derivatives with the strain.

3.4 Valence band parameters and deformation potentials

The predicted valence band parameters are summarized in Table 2. The strained valence band eigenvalues as obtained from the full relativistic LDA calculations are depicted in Figure 1. We have compared them with the energies obtained from the model Equation 1-Equation 3 with the parameters from Table 2. As discussed above, the agreement between LDA calculations and the analytic expressions is excellent for moderate strains. For larger strains the discrepancy increases, due to the nonlinearity of Θ(e), but it remains acceptable for strains up to 3%. For unstrained GaN, the present ab-initio calculations predict E(Γ_9v) - E₊(Γ_7v) = 8 meV (6 meV Reference Volm, Oettinger, Streibl, Kovalev, Ben-Chorin, Diener, Meyer, Majewski, Eckey, Hoffman, Amano, Akasaki, Hiramatsu and Detchprohm[7], Reference Dingle, Sell, Stokowski and Ilegems[18], Reference Monemar[19], Reference Pakula, Wysmolek, Korona, Baranowski, Stepniewski, Grzegory, Bockowski, Jun, Krukowski, Wroblewski and Porowski[20]) and E(Γ_9v) - E_-(Γ_7v) = 43 meV ( 22 meV Reference Volm, Oettinger, Streibl, Kovalev, Ben-Chorin, Diener, Meyer, Majewski, Eckey, Hoffman, Amano, Akasaki, Hiramatsu and Detchprohm[7], 18 meV Reference Dingle, Sell, Stokowski and Ilegems[18], 28 meV Reference Monemar[19], 24 meV Reference Pakula, Wysmolek, Korona, Baranowski, Stepniewski, Grzegory, Bockowski, Jun, Krukowski, Wroblewski and Porowski[20]), in fair agreement with the experimental data (in parenthesis) of thin GaN films.

Table 2

Predicted valence band parameters and deformation potentials

Figure 1.

Calculated valence band energies of GaN and AlN (in meV) at Γ as a function of biaxial strain e_//. The top of the valence band is the zero of energy. The lines represent the energies of the analytical model from Equation 1-Equation 3 (full: E(Γ_9v), dotted: E₊(Γ_7v), dashed: E_-(Γ_7v)) with parameters from Table 2. The solid squares are results of the relativistic LDA calculations.

Figure 1 reveals that the energetic ordering of the E(Γ_9v) and E₊(Γ_7v) states changes beyond some critical value of biaxial strain. For GaN and InN, this occurs for 0.32% and 0.37% tensile strain, respectively, whereas in AlN a compressive strain of 1.56% is required.

From the results given in Table 2, one can determine the strain dependence of the energy gap. We find dE_gap/de_// = −6.1 eV (corresponding to 17.4 meV/GPa) for biaxial strain up to 0.32%, and −16 eV for tensile strain larger than 0.32%. This computed values agree very well with the experimental values −8.2 eV Reference Volm, Oettinger, Streibl, Kovalev, Ben-Chorin, Diener, Meyer, Majewski, Eckey, Hoffman, Amano, Akasaki, Hiramatsu and Detchprohm[7] and 24±4 meV/GPa Reference Rieger, Metzger, Angerer, Dimitrov, Ambacher and Stutzmann[21] for compressive strain and −13 eV for tensile strain Reference Volm, Oettinger, Streibl, Kovalev, Ben-Chorin, Diener, Meyer, Majewski, Eckey, Hoffman, Amano, Akasaki, Hiramatsu and Detchprohm[7]. Additionally, we find very good agreement between the predicted and experimental value of the band gap pressure coefficient in GaN, dE_gap/dp = 40.1 meV/GPa (exp: 41 meV/GPa Reference Camphausen and Connell[22], 42 meV/GPa Reference Perlin, Gorczyca, Christensen, Grzegory, Teisseyre and Suski[23]).

In the case of InN, the spin-orbit coupling constants obtained in our pseudopotential calculations (Δ₂ = 3.9 meV and Δ₃ = 5.5 meV) are only moderately smaller than spin-orbit splittings in AlN and GaN (see Table 1). It is in contrast to the recent all-electron calculations, which predict the value of the spin-orbit splitting Δ₀in the zincblende InN to be 3 meV Reference Lambrecht, Kim, Rashkeev and Segall[24] and 6 meV Reference Wei and Zunger[25] (please note that in the cubic model of wurtzite structure Δ₂= Δ₃= Δ₀/3).

In summary, we have predicted the strain dependence of the valence band edge states of wurzite GaN and AlN. We have developed an analytical model for these electronic energies that accurately reproduces the complex interplay between spin-orbit and strain effects.