Consistent set of band parameters for the group-III nitrides AlN, GaN, and InN

We have derived consistent sets of band parameters (band gaps, crystal field-splittings, band gap deformation potentials, effective masses, Luttinger and EP parameters) for AlN, GaN, and InN in the zinc-blende and wurtzite phases employing many-body perturbation theory in the G0W0 approximation. The G0W0 method has been combined with density-functional theory (DFT) calculations in the exact-exchange optimized effective potential approach (OEPx) to overcome the limitations of local-density or gradient-corrected DFT functionals (LDA and GGA). The band structures in the vicinity of the Gamma-point have been used to directly parameterize a 4x4 k.p Hamiltonian to capture non-parabolicities in the conduction bands and the more complex valence-band structure of the wurtzite phases. We demonstrate that the band parameters derived in this fashion are in very good agreement with the available experimental data and provide reliable predictions for all parameters which have not been determined experimentally so far.


I. INTRODUCTION
The group III-nitrides AlN, GaN, and InN and their alloys have become an important and versatile class of semiconductor materials, in particular for use in optoelectronic devices and high-power microwave transistors. Current applications in solid state lighting [light emitting diodes (LEDs) and laser diodes (LDs)] range from the visible spectrum 1,2,3,4 to the deep ultra-violet (UV) 5,6 , while future applications as, e.g., chemical sensors 7,8,9 or in quantum cryptography 10 are being explored.
For future progress in these research fields reliable material parameters beyond the fundamental band gap, like effective electron masses and valence-band (Luttinger or Luttinger-like) parameters, are needed to aid interpretation of experimental observations and to simulate (hetero-)structures, like, e.g., optoelectronic devices. Material parameters can be derived from first-principles electronic-structure methods for bulk phases, but the size and complexity of structures required for device simulations currently exceeds the capabilities of first-principles electronic-structure tools by far. To bridge this gap firstprinciple calculations can be used to parameterize simplified methods, like the k·p method, 11,12,13,14 the empirical tight-binding (ETB) method, 15,16,17,18 or the empirical pseudo-potential method (EPM) 19 , which are applicable to large-scale heterostructures at reasonable computational expense.
In this Article we use many-body perturbation theory in the G 0 W 0 approximation, 20 -currently the method of choice for the description of quasiparticle band structures in solids 21,22,23 -in combination with the k · p approach, 11,12,13,14 to derive a consistent set of material parameters for the group-III nitride system. The k · p -Hamiltonian is parameterized to reproduce the G 0 W 0 band structure in the vicinity of the Γ-point. Since the parameters of the k · p method are closely related or, in some cases, even identical to basic band parameters, many key band parameters can be directly obtained using this approach.
The k·p -model Hamiltonian is typically parameterized for bulk structures and is then applicable to heterostructures with finite size (e.g., micro-and nanostructures) within the envelope-function scheme. 24 Ideally, the parameters are determined entirely from consistent experimental input. For the group-III-nitrides, however, many of the key band parameters have not been conclusively determined until now, despite the extensive research effort in this field. 25,26 In a comprehensive review Vurgaftman and Meyer summarized the field of III-V semiconductors in 2001 and recommended up-to-date band parameters for all common compounds and their alloys including the nitrides. 27 Only two years later they realized that it is striking how many of the nitride properties have already been superseded, not only quantitatively but qualitatively. 26 They proceeded to remedy that obsolescence, by providing a completely revised and updated description of the band parameters for nitride-containing semiconductors in 2003. 26 While this update includes evidence supporting a revision of the band gap of InN from its former value of 1.9 eV to a significantly lower value around 0.7 eV, 28,29,30,31,32 they had to concede that in many cases experimental information on certain parameters was simply not available. 26 This was mostly due to growth-related difficulties in producing high quality samples for unambiguous characterization. In the meantime the quality of, e.g., wurtzite InN samples has greatly improved 25 and even the growth of the zinc-blende phase has advanced. 33 Nevertheless, many of the basic material properties of the group-III nitrides are still undetermined or, at least, controversial.
On the theoretical side, certain limitations of densityfunctional theory (DFT) in the local-density or generalized gradient approximation (LDA and GGA, respectively)-currently the most wide-spread ab-initio electronic-structure method for poly-atomic systemshave hindered an unambiguous completion of the missing data.
To overcome these deficiencies we use G 0 W 0 calculations based on DFT calculations in the exact-exchange optimized effective potential approach (OEPx) to determine the basic band parameters. We have previously shown that the OEPx+G 0 W 0 approach provides an accurate description of the quasiparticle band structure for GaN, InN and II-VI compounds 23,34,35,36 . The quasiparticle band structure in the vicinity of the Γ-point is then used to parameterize a 4 × 4 k·p Hamiltonian to determine band-dispersion parameters, like effective masses, Luttinger parameters, etc. This allows us to take the non-parabolicity of the conduction band, which is particularly pronounced in InN 29,37 , and the more complex valence band structure of the wurtzite phases into account properly.
This paper is organized as follows: In Section II we briefly introduce the G 0 W 0 approach and its application to the group-III nitrides, followed by a discussion of certain key parameters of the quasiparticle band structure, such as the fundamental band gaps (and their dependence on the unit-cell volume) and the crystal-field splitting energies (Section III). In Section IV we present our recommendations for the band dispersion parameters (k · p parameters) of the wurtzite and zinc-blende phases of AlN, GaN and InN. A detailed discussion of the parameter sets is given in Section IV B together with a comparison to experimental values and parameter sets obtained by other theoretical approaches. Our conclusions are given in Sec. V.

II. QUASIPARTICLE ENERGY CALCULATIONS
A. GW based on exact-exchange DFT The root of the deficiencies in LDA and GGA for describing spectroscopic properties like the quasiparticle band structure can be found in a combination of different factors. LDA and GGA are approximate (jellium-based) exchange-correlation functionals, which suffer from incomplete cancellation of artificial self-interaction and lack the discontinuity of the exchange-correlation potential with respect to the number of electrons. As a consequence the Kohn-Sham (KS) single-particle eigenvalues cannot be rigorously interpreted as the quasiparticle band structure as measured by direct and inverse photoemission. This becomes most apparent for the band gap, which is severely underestimated by the Kohn-Sham eigenvalue difference in LDA and GGA. For InN this even results in an overlap between the conduction and the valence bands and thus an effectively metallic state, as displayed in Fig. 1. It goes without mentioning that a k·p parameterization derived from this LDA band structure would not appropriately reflect the properties of bulk InN. Many-body perturbation theory in the GW approach 20 presents a quasiparticle theory that overcomes the deficiencies of LDA and GGA and provides a suitable description of the quasiparticle band structure of weakly correlated solids, like AlN, GaN and InN. 21,22,23 Most commonly, the Green's function G 0 and the screened potential W 0 required in the GW approach (henceforth denoted G 0 W 0 ) are calculated from a set of DFT Kohn-Sham single particle energies and wave functions. The DFT ground state calculation is typically carried out in the LDA or GGA and the quasiparticle corrections to the Kohn-Sham eigenvalues are calculated in first order perturbation theory (LDA/GGA + G 0 W 0 ) without resorting to self-consistency in G and W . 38 While the LDA+G 0 W 0 approach is now almost routinely applied to bulk materials, 21,22,23 G 0 W 0 calculations for GaN and InN have been hampered by the deficiencies of the LDA. For zinc-blende GaN the LDA+G 0 W 0 band gap of 2.88 eV 39,40 is still too low compared to the experimental 3.3 eV, 41,42,43 while for InN the LDA predicts a metallic ground state with incorrect band ordering. A single G 0 W 0 iteration proves not to be sufficient to restore a proper semiconducting state and only opens the band gap to 0.02 -0.05 eV, 44,45 which is still far from the experimental value of ∼0.7 eV. 28,29,30 Here we apply the G 0 W 0 approach to DFT calculations in the exact-exchange optimized effective potential approach (OEPx or OEPx(cLDA) if LDA correlation is included). In contrast to LDA and GGA the OEPx approach is fully self-interaction free and correctly predicts InN to be semiconducting with the right band ordering in the wurtzite phase 34 23,34,35,36,47 For wurtzite InN the band gap of 0.7 eV and the non-parabolicity of the conduction band (CB) 34 (shown in Fig. 1) strongly supports the recent experimental findings. 28,29,30,37,49 In addition we have shown that the source for the startling, wide interval of experimentally observed band gaps can be consistently explained by the Burstein-Moss effect (apparent band gap increase with increasing electron concentration in the conduction band) by extending our calculations to finite carrier concentrations. 34

B. Computational Parameters
The LDA and OEPx calculations in the present work were performed with the plane-wave, pseudopotential code S/PHI/nX, 51 while for the G 0 W 0 calculations we have employed the G 0 W 0 space-time method 52 in the gwst implementation. 53,54,55 Local LDA correlation is added in all OEPx calculations. Here we follow the parametrization of Perdew and Zunger 56 for the correlation energy density of the homogeneous electron gas based on the data of Ceperley and Alder. 57 This combination will in the following be denoted OEPx(cLDA). Consistent pseudopotentials were used throughout, i.e. exact-exchange pseudopotentials 58 for the OEPx(cLDA) and LDA ones for the LDA calculations. The cation d-electrons were included explicitly. 23,46 For additional technical details and convergence parameters we refer to previous work. 23,34

C. Lattice Parameters
All calculations are carried out at the experimental lattice constants reported in Tab. I and not at the ab initio ones to avoid artificial strain effects in the derived parameter sets. For an ab initio determination of the lattic constants consistent with the G 0 W 0 calculations the crystal structure would have to be optimized within the G 0 W 0 formalism, too. However, G 0 W 0 total energy calculations for realistic systems have, to our konwledge, not been performed, yet, and the quality of the G 0 W 0 total energy for bulk semiconductors has not been assessed so far. 59 The alternative ab initio choices, LDA and OEPx(cLDA), give different lattice constants 60 and would thus introduce uncontrolable variations in the calculated band parameters that would aggrevate a direct comparison.
The thermodynamically stable phase of InN at the usual growth conditions is the wurtzite phase. Reports of a successful growth of the zinc-blende phase have been scarce. Recently, high-quality films of zb-InN grown on indium oxide have been obtained by Lozano et al. 33 We adopt their lattice constant of 4.98Å, 33 which is in good agreement with previous reports of 4.98Å, 61  Although wurtzite is the phase predominantly grown for InN, reported values for the structural parameters still scatter appreciably. 68 In order to determine the effect of the lattice constants on the band gap (E g ) and the crystal-field splitting (∆ CR ) we have explored the range between the maximum and minimum values of a 0 and c 0 /a 0 reported in Ref. 68 by performing OEPx(cLDA)+G 0 W 0 calculations at the values listed in Tab. II. Since u remains undetermined in Ref. 68 we have optimized it in the LDA. Neither u, E g , nor ∆ CR depends sensitively on the lattice constants in this regime and we have therefore adopted the mean values of a 0 =3.54Å, c 0 =5.706Å(c 0 /a 0 =1.612) and u=0.380 (the LDA-optimized value) for the remainder of this article.
For wz-AlN and wz-GaN the lattice constants are more established. 26,69 For wz-AlN we adopt Schulz and Thiermann's values of a=3.110Å, c=4.980Å, and u=0.382, 70 which are close to those reported by Yim et al. 71 Schulz and Thiermann also provide a value for the internal parameter u, which is identical to the one we obtain by relaxing u in the LDA at the experimental a 0 and c 0 parameters. The same is true for wz-GaN. Schulz and Thiermann's values of a=3.190Åand c=5.189Å 70 are close to those first reported by Maruska and Tietjen, 72 but in addition offer a value of u=0.377, which corresponds to our LDA-relaxed value at the same lattice parameters. Note, that the lattice parameters of wz-InN and wz-GaN have been refined compared to our recently published calculations. 34 The influence of the adjustment on the different band parameters will be discussed where necessary.

III. BAND GAPS, CRYSTAL-FIELD SPLITTINGS, AND BAND-GAP DEFORMATION POTENTIALS
We will now discuss the quasiparticle band structure of AlN, GaN and InN in their zinc-blende and wurtzite phases in terms of certain key band parameters such as the band gap (E g ), the crystal field splitting (∆ CR ) in the wurtzite phase and the band-gap volume deformation potentials α V . At the end of this section we will draw a comparison between LDA and OEPx(cLDA) based G 0 W 0 calculations for AlN.

A. Band Gaps
The OEPx(cLDA)+G 0 W 0 band gaps for the three materials and two phases are reported in Tab. III together with the LDA and OEPx(cLDA) values for comparison. For GaN and InN the OEPx(cLDA)+G 0 W 0 band gaps have been reported previously in Ref. 34. There we have also argued that the wide interval of experimentally observed band gaps for InN can be consistently explained by the Burstein-Moss effect. The OEPx(cLDA)+G 0 W 0 value of 0.69 eV for wz-InN 85 supports recent observations of a band gap at the lower end of the experimentally reported range. For zinc-blende InN, which has been explored far less experimentally, our calculated band gap of 0.53 eV also agrees very well with the recently measured (and Burstein-Moss corrected) 0.6 eV. 79 For GaN the band gaps of both phases are well established experimentally and our OEPx(cLDA)+G 0 W 0 calculated values of 3.24 eV 86 and 3.07 eV agree to within 0.3 eV.
For AlN experimental results for the band gap of the wurtzite phase scatter appreciable, whereas for zinc blende only one value has -to the best of our knowledge -been reported so far. Contrary to GaN, the OEPx(cLDA)+G 0 W 0 gaps for AlN are larger than the experimentally reported values.

B. Crystal-Field Splitting
Experimental values for the crystal-field splitting, ∆ CR , of wz-GaN scatter between 0.009 and 0.038 eV (Tab. III). The OEPx(cLDA)+G 0 W 0 value of 0.033 eV supports a crystal-field splitting within this range.
Theoretical 87 and experimental 73 investigations of wz-AlN agree upon the fact that the crystal-field splitting of AlN is negative. Our calculations also yield a negative value of ∆ CR = −0.295 eV. This result supports a crystalfield splitting in AlN below −0.2 eV, as reported by Chen et al., 73 rather than a small negative value between −0.01 and −0.02, as implied by the results of Freitas et al. 88 For wz-InN a crystal-field splitting between 0.019 and 0.024 eV has been reported recently. 83 This value is significantly smaller than the OEPx(cLDA)+G 0 W 0 value of 0.07 eV.
The crystal-field splitting is known to be sensitive to lattice deformations, such as changes in the c 0 /a 0 ratio or the internal lattice parameter u. 89,90,91 Therefore, the discrepancy between experiment and theory might stem from the uncertainties of the lattice parameters of wz-InN (cf Sec. II C). However, varying the c 0 /a 0 ratio or the unit cell volume within the experimental range discussed in Section II C yields values for ∆ CR which are always larger than 0.06 eV (Tab. II), leaving only the internal lattice parameter u as possible source of error. This parameter is -at least for GaN -known to have a large influence on the crystal-field splitting. 91 Although the LDA-optimized u values are in very good agreement with experimental values for GaN and AlN, experimental confirmation of the u parameter of InN is still pending. We therefore calculated the crystal-field splitting of wz-InN for different values of u (and a 0 and c 0 fixed at the values listed in Tab. I) between 0.377 and 0.383. Generally, ∆ CR decreases with increasing u, but even for u as large as 0.383, the crystal-field splitting is still larger than 0.05 eV The discrepancy between the experimental report and the OEPx(cLDA)+G 0 W 0 calculations can hence not be attributed to the uncertainties in the lattice parameters and has to remain unsettled for the time being.   . All values are given in eV.

C. Band Gap Deformation Potentials
For the hydrostatic band gap deformation potentials the band gaps have been calculated at different volumes (V ) between ±2 % around the equilibrium volume V 0 . In the explored volume range the band gaps vary linearly with ln (V /V 0 ). The linear coefficient is then taken as the hydrostatic volume deformation potential α V . The calculated band gap deformation potentials are listed in Tab. IV for LDA, OEPx(cLDA), and OEPx(cLDA)+G 0 W 0 . We observe that for all compounds and phases the quasiparticle deformation potential is larger in magnitude than that of the DFT Experimentally the band gap deformation potential is usually measured as a function of the applied pressure, which aggravates a direct comparison to our calculated volume deformation potentials. However, since B = −dP/d ln V , where B is the bulk modulus and P the pressure, the pressure deformation potential α P can be expressed in terms of α V according to α P = −α V /B.
Experimentally reported values for the bulk modulus of wz-GaN scatter between 1880 and 2450 kbar. 93,94,95,96,97 Using these values, our volume deformation potential of α V =-7.6 eV would translates into a pressure deformation potential in the range of 3.1 -4.0 meV/kbar, which is comparable to the experimentally determined range of 3.7 and 4.7 meV/kbar. 97,98,99,100,101,102 This large uncertainty has been partially ascribed to the low quality of earlier samples and substrate-induced strain effects. 101 The fact that the pressure dependence of the band gap is sublinear (unlike the volume dependence) further questions the accuracy of linear or quadratic fits for the extraction of the deformation potentials in the experiments. 101 For wz-InN experimentally reported values are sparse. Franssen et al. determined a hydrostatic pressure deformation potential of 2.2 meV/kbar 103 , while Li et al. found 3.0 meV/kbar. 104 This range agrees with our theoretical one of 2.8 -3.3 meV/kbar, using for the conversion of volume to pressure deformation potentials the bulk modulus range of 1260 -1480 kbar 95,96 quoted in the literature.
For wz-AlN we are only aware of one experimental study reporting a pressure deformation potential of 4.9 meV/kbar. 76 With experimental bulk moduli between 1850 and 2079 kbar 105,106,107 the OEPx(cLDA)+G 0 W 0 pressure deformation potential of wz-AlN would fall between 4.7 and 5.3 meV/kbar straddling the experimentally reported value.
To our knowledge, no experimental information on the deformation potential of zb-AlN and zb-InN are available. For zb-GaN our computed volume deformation potential of α V =-7.3 eV translates to a pressure deformation potential range of 3.0 -3.9 meV/kbar using the same bulk modulus range as for wz-GaN. This range is slightly below the experimentally reported range of 4.0 -4.6 meV/kbar. 102

D. Comparison between LDA+G0W0 and
OEPx(cLDA)+G0W0 For the materials presented in this article a meaningful comparison between LDA and OEPx(cLDA) based G 0 W 0 calculations can only be constructed for AlN for reasons given in Section II A. Figure 2 displays the band structure of wz-AlN in the four approaches discussed in this Article. The "band gap problem" has been eliminated from this comparison by aligning the conduction bands at the minimum of the lowest conduction band (ǫ CBM ) and the valence bands at the maximum of the hightest valence band (ǫ VBM ). For this large gap material three main conclusions can be drawn from Fig. 2. First, LDA and both G 0 W 0 calculations yield very similar band dispersions. Or in other words the G 0 W 0 corrections to the LDA in the LDA+G 0 W 0 approach are not k-point dependent shifting bands almost rigidly. A rigid shift between conduction and valence bands is frequently referred to as "scissor operator". Fig. 2, however, illustrates that this shift is not identical for all bands, which cannot be attributed to a single scissor operator. Second, the dispersion obtained in OEPx(cLDA) devi- ates from the other three approaches, which is consistent with the observation made for wz-InN in Fig. 1. We attribute this behaviour to the approximate treatment of correlation in the OEPx(cLDA) approach and the fact that the band structure in OEPx(cLDA) is a Kohn-Sham and not a quasiparticle band structure. While the LDA benefits from a fortuitous error cancellation between the exchange and the correlation part 111 , this is no longer the case once exchange is treated exactly in the OEPx(cLDA) scheme. Using a quasiparticle approach with a more sophisticated description of correlation, like the GW method, then notably changes the dispersion of the OEPx(cLDA) bands. As we will demonstrate in the next Section this will lead to markedly different band parameters not only for the conduction but also for the valence bands (cf Tab. VII). Against common believe OEPx(cLDA) calculations without subsequent G 0 W 0 calculations may therefore provide a distorted picture and we would advise against deriving band parameters from OEPx or OEPx(cLDA) band structures alone. Third, unlike in the LDA+G 0 W 0 case the G 0 W 0 corrections to the OEPx(cLDA) starting point become k-point dependent, a fact already observed for GaN and II-VI compounds. 23 Most remarkably and in contrast to what we observe for GaN and InN (see Sec. IV B 2) the corrections are such that the band dispersion now agrees again with that obtained from the LDA and the LDA+G 0 W 0 approach. Note also that both the band gap and the crystal field splitting still differ slightly between LDA+G 0 W 0 and OEPx(cLDA)+G 0 W 0 for AlN (E g : LDA+G 0 W 0 : 5.95 eV, OEPx(cLDA)+G 0 W 0 : 6.47 eV, ∆ CR : LDA+G 0 W 0 : -0.252 eV, OEPx(cLDA)+G 0 W 0 : -0.295 eV). Unlike for GaN and InN, experimental uncertainties do, at present, not permit a rigorous assessment, which of the two G 0 W 0 calculations provides a better description for AlN (see also Sections III A and III B).

IV. BAND DISPERSION PARAMETERS
We will now turn our attention to band parameters that describe the band dispersion in the vicinity of the Γ-point: the effective masses, the Luttinger(-like) parameters, and the E P parameters. These parameters are obtained by means of the k·p -method. The k·p -method is a well-established approach that permits a description of semiconductor band structures in terms of parameters that can be accessed experimentally. Throughout this paper, we use four-band k·p -theory, which is typically used to describe direct-gap materials, mostly in its spin-polarized form as eight-band k · p -theory. The k·p Hamiltonian and all relevant formulas are given in Appendix A. The k · p -method is a widely accepted technique for, e.g., the interpretation of experimental data 37,112 or modeling of semiconductor nanostructures and (opto-)electronic devices. 12,13,14,113,114,115,116 Its accuracy, however, depends crucially on the quality of the input band parameters, like effective electron masses, Luttinger-parameters, etc., which have to be derived either experimentally or from band structure calculations. As alluded to in the introduction, many important band parameters of the group-III nitrides GaN, InN, and AlN are still unknown. In particular, the band structure of InN is currently the subject of active research in both experiment and theory.
In this paper, we use the k·p -method to derive band dispersion parameters from OEPx(cLDA)+G 0 W 0 band structures. This approach has certain advantages over a simple parabolic approximation around the Γ point. First, the k·p band structure is valid, not only directly at the Γ-point, but also in a certain k-range around it. This allows to extend the fit to larger k's and thereby increases the accuracy of the fitted parameters. Second, the k · p -method is capable of describing non-parabolic bands, such as the CB of InN 34,37 and can therefore also be applied to accurately determine values for the effective electron masses and E P parameters in InN.

A. Computational details
For an accurate fit of the k·p parameters to the quasiparticle band structure a small reciprocal lattice vector spacing is required. Since most GW implementations evaluate the self-energy Σ (the perturbation operator that links the Kohn-Sham with the quasiparticle system) in reciprocal space, the matrix elements with respect to the Kohn-Sham wave functions φ nq |Σ(ǫ qp nq )|φ nq required for the quasiparticle corrections are only available on the k-points of the underlying k-grid. A fine sampling of the Γ-point region would therefore be equivalent to using formidably large k-grids in the computa- In the GW space-time method 52 these problems are easily circumvented, because the self-energy is computed in real-space [Σ R (r, r ′ ; ǫ)]. By means of Fourier interpolation, the self-energy operator can be calculated at arbitrary qpoints. 53 The matrix elements φ nq |Σ q (ǫ)|φ nq are then obtained by integration over r and r ′ . In this fashion the relevant Brillouin zone regions for the band structure fitting can be calculated efficiently without compromising accuracy.
The k·p Hamiltonian and all parameter relations are given in Appendices A and B. To determine the k · p Hamiltonian for a given band structure with band gap E g and crystal-field splitting ∆ CR we fit the parameters m i e , A i , γ i , and E i P . This is achieved by leastsquare-root fitting of the k · p band structure to the OEPx(cLDA)+G 0 W 0 band structure in the vicinity of Γ. Tab. V (wz) and Tab. VI (zb). The resulting k·p band structures are plotted in Figs. 3 and 4 (black solid lines) together with the respective OEPx(cLDA)+G 0 W 0 data (black circles). The excellent agreement of the k·p and OEPx(cLDA)+G 0 W 0 band structures illustrates that the band structures of the wurtzite and zinc-blende phases of all three materials are accurately described by the k·p -method within the chosen k-ranges. Additionally, the k · p band structures based on the parameters recommended by Vurgaftman and Meyer 26,117 (VM '03) are shown (red dashed lines). As alluded to in the introduction, their recommendations are based on available experimental data and selected theoretical values, representing the state-of-the-art parameters up until the year of compilation (2003). We will also compare our results to more recent experimentally and theoretically derived parameters. (see Tab. VII) In the following we will show that the parameters derived from the OEPx(cLDA)+G 0 W 0 calculations match all available experimental data to good accuracy. A comparison to parameters derived by other, theoretical or semi-empirical, methods will be presented thereafter.
Before we proceed, however, we would like to emphasize two points regarding the relation between the VB parameters A i and the effective hole masses in wurtzite crystals: (i) Two different sets of equations, connecting the effective hole masses to the A i parameters, are used in the literature. Reference 118 lists both; one is labeled "Near the band edge (k → 0)" and the other "Far away from the band edge (k is large)". The latter is widely used to calculate the effective hole masses. 119,120,121,122 However, the experimentally relevant effective masses are those close to Γ. Thus, we use the "Near the band edge" equations (see Appendix B) throughout this work. Quoted values differ from the original publications in cases where the original work uses the "Far away from the band edge" equations. (ii) The Luttinger-like parameters, A i , are independent of the spin-orbit and crystalfield interaction parameters ∆ SO and ∆ CR ; the effective hole masses, however, differ for different ∆ SO and ∆ CR parameters. Only the A-band (C-band in AlN) hole masses can be calculated from the Luttinger-like parameters alone. All other hole masses depend additionally on the choice of the spin-orbit and crystal-field splitting energies. 118 Thus, effective Band C-band (Aand B-band in AlN) hole masses derived from different sets of Luttinger-like parameters are comparable, only if the same ∆ SO and ∆ CR values are assumed.

Comparison to experimental values
Experimentally, the band structure of a semiconductor is accessible only indirectly, via band parameters like E g , ∆ SO , ∆ CR , and the effective masses. Angle resolved direct and inverse photoemission experiments, which would, in principle, directly probe the quasiparticle band structure, are not accurate enough, yet, to determine the band structure with sufficient accuracy.
The dispersion of the conduction band around the Γ point depends only on the effective electron masses and E P parameters, which are accessible experimentally. The valence-band parameters, A i , cannot be obtained directly experimentally, but can be related to the effective hole masses (see appendix B), which, in turn, can be measured.
The available experimental values for the wurtzite phases are listed in Tab. VII. For the thermodynamically metastable zinc-blende phases of GaN, AlN, and InN hardly any experimental reports on their band dispersion parameters are available so far. Therefore, we restrict the discussion to the wurtzite phases, for which experimental data on, at least, the effective electron masses are available. For wz-InN also E P has been determined, by fitting a simplified k·p -Hamiltonian to the experimental data. 37,112 For wz-GaN, values for E P 125,126 and also several reports on the effective hole masses are available. 131 Wurtzite

Other parameter sets
Local density approximation (LDA). For means of comparison we have also derived band parameters from LDA and OEPx(cLDA) calculations in the same way as for the OEPx(cLDA)+G 0 W 0 data. LDA band structures are frequently employed for fitting parameter sets, 119,135 but we will demonstrate here, that the LDA is not suitable to consistently determine all parameters for the group-III-nitrides accurately. The parameters derived from the LDA band structures are listed in Tab. VII for the wurtzite phases of GaN and AlN. Since the LDA predicts InN to be metallic, no LDA band parameters could be derived for InN.
The effective electron masses of GaN in LDA are smaller than in OEPx(cLDA)+G 0 W 0 and the experiment. The effective electron masses of a given material are, to a first approximation, proportional to the fundamental band gap. 136 Thus the underestimation of the effective electron masses in LDA is to some degree a natural side effect of the underestimation of the fundamental band gap. Additional factors (e.g. self-interaction) contributing to the deviation of the LDA band structure from the quasiparticle one were alluded to in section II A.
The A-band hole masses in LDA show an increased anisotropy; the deviation from the experimental values increases.
Despite the fact that the band gap of AlN is also significantly smaller in LDA than in OEPx(cLDA)+G 0 W 0 , it is still large, i.e., well above 4 eV. Therefore, an effect on the absolute values of the effective electron masses is not visible, but the LDA predicts an anisotropy of the electron masses, with the opposite sign compared to the OEPx(cLDA)+G 0 W 0 calculations. OEPx(cLDA). As alluded to in section II A, band gaps in the OEPx(cLDA) approach open compared to LDA (cf Tab. III). Following the proportionallity relationship between the direct band gap and the conduction band effective mass, the latter should increase in OEPx(cLDA). This is indeed the case, as Tab. VII demonstrates. They are, however, also larger than the conduction-band effective masses in the OEPx(cLDA)+G 0 W 0 approach, despite the fact that only in InN the OEPx(cLDA) band gap is larger than that in OEPx(cLDA)+G 0 W 0 . We attribute this behaviour to the approximate treatment of correlation in the OEPx(cLDA), which aversely affects the band dispersion as explained in Section III D. We thus do not recommend the use of the OEPx or the OEPx(cLDA) approach alone for the determination of band parameters.

LDA-plus-correction (LDA+C).
In the LDA+C approach 137 delta-function potentials are added at the atomic sites, which artificially push s-like wave functions upwards in energy. As a consequence, the band gaps open, due to the admixture of cation s-states in the conduction band. The potentials have to be fitted to available experimental data, such as the fundamental band gaps and can be applied in an all-electron 87,137 but also in a pseudopotential framework. 138 Carrier and Wei (CW '05) 87 determined the effective electron and hole masses of GaN, AlN, and InN, using this method. Their results are also given in Tab. VII.
Their values for the effective electron masses of all three materials are in good agreement with the OEPx(cLDA)+G 0 W 0 values. The deviations are larger for the effective hole masses. This is not too surprising because the LDA+C approach predominantly affects sderived bands. Since the upper valence bands around Γ are mostly of nitrogen 2p character their description will be closer to the LDA level, whereas the conduction bands feel the additional corrections.

Empirical pseudopotential method (EPM).
A semi-empirical way, often used to calculate band parameters, is the empirical pseudo potential method (EPM). 120,121,124,130 In the EPM the full atomic potentials are replaced by those of pseudo atoms, whose adjustable parameters are fitted to a set of input band parameters, typically taken from experiments. The resulting band structures can then be used analogously to fit the parameters of a k · p Hamiltonian. Since the EPM depends sensitively on the input parameters, appreciable scatter in the reported band parameters is observed. (see Tab. V for a selection)

Vurgaftman and Meyer.
For non of the group-III nitrides a complete set of band parameters has so far been derived from experimental values alone. Therefore, Vurgaftman and Meyer 26 have compiled parameter sets comprising experimental and the most reliable theoretical values in the year 2003.
For wz-GaN, VM'03 recommend the experimental value of the effective electron masses of m e = m ⊥ e = 0.20 m 0 and Luttinger-like parameters derived from EPM calculation by Ren et al. 121 , which yield effective hole masses in good agreement with experimental and the OEPx(cLDA)+G 0 W 0 data (see Tab. VII). The parameter set yields a band structure that agrees well with the OEPx(cLDA)+G 0 W 0 band structure for the CB and the two top VBs (see Fig. 3). It deviates, however, for the C VB (the third valence band counted from the valence band maximum), where the curvatures in the EPM band structure are too large.
Of all the compounds and phases discussed in this article wz-GaN is the best characterized experimentally. The good agreement between our quasiparticle band structures and those based on the parameter set recommended by VM'03 proves the quality of our OEPx(cLDA)+G 0 W 0 band structures.
For wz-AlN the effective electron masses recommended by VM'03 are the averages over several theoretical values; the recommended VB parameters are theoretical values by Kim et al. 135 derived from LDA calculations. These parameters yield a band structure, which is in good overall agreement with the OEPx(cLDA)+G 0 W 0 band structure (see Fig. 3a). The anisotropy of the effective electron masses, however, has the opposite sign, similar to our own LDA calculations. The similarity between VM'03 (i.e. LDA) and OEPx(cLDA)+G 0 W 0 in the valence band region (after adjusting ∆ CR ) is due to the fact that in AlN valence bands are shifted rigidly compared to the LDA, as discussed in Section III D. In OEPx(cLDA) alone, however, the dispersion changes noticeably (similar to what was observed for InN, cf. Fig. 1) giving rise to appreciably different band parameters (Tab. VII).
For wz-InN, VM'03 recommend the experimental effective electron masses by Wu et al. 37 (m e = m ⊥ e = 0.07 m 0 ) and the EPM values from Pugh et al. 122 for the VB. The pseudo potentials used by Pugh et al. were designed to reproduce their LDA calculations, which had been "scissor corrected" to the incorrect band gap of 2.0 eV. These parameters are therefore to no avail from today's perspective.

V. CONCLUSION
We have derived consistent and unbiased band parameters for the wurtzite and zinc-blende phases of GaN, AlN, and InN from accurate OEPx(cLDA)+G 0 W 0 band structure calculations. The band parameters are in very good agreement with the available experimental data, proving the reliability of the method. We also provide reliable val-ues for those parameters which have not been determined experimentally, such as, e.g., the band parameters of the zinc-blende phases of GaN, AlN, and InN or the E p and VB parameters of wurtzite phases. These parameters are essential for understanding the physics of these materials. We have derived complete and consistent parameter sets for the description of the band structures of the group-III nitrides within k·p -theory. The k·p -method is widely used for modeling and simulating (opto-)electronic devices. The parameters presented in this work overcome the apparent lack of consistent band parameter sets for such simulations.
Finally we remark that the combination of the k · pwith the G 0 W 0 method is not restricted to the 4×4 (8×8) k·p Hamiltonians discussed in this work. Since we expect G 0 W 0 to provide the same accuracy for the whole Brillouin zone, the parameters for more complex Hamiltonians can be fitted in the same way.