PHYSICAL REVIEW B 85, 045205 (2012)

Optical and electronic properties of Si3N4 and α-SiO2

G. Kresse,1 M. Marsman,1 L. E. Hintzsche,1 and E. Flage-Larsen2

1University of Vienna, Faculty of Physics and Center for Computational Materials Science, Sensengasse 8/12, A-1090 Vienna, Austria2SINTEF Materials and Chemistry, P.O. Box 124 Blindern, Forskningsveien 1, N-0314 Oslo, Norway

(Received 30 September 2011; revised manuscript received 11 November 2011; published 13 January 2012)

Self-consistent quasiparticle GW (sc-QPGW) calculations are used to calculate the electronic properties of α-Si3N4 and β-Si3N4, as well as α-SiO2 (quartz). The optical properties are evaluated by solving the Bethe-Salpeterequation in the Tamm-Dancoff approximation. For quartz, the predicted dielectric function is in good agreementwith experimental data, with the onset of absorption located about 1.2 eV below the direct quasiparticle gap.For Si3N4, the theoretical dielectric function is fairly structureless and the onset of absorption corresponds toan exciton with a binding energy of 0.6 eV. The calculated sc-QPGW data are compared to more approximatecalculations using G0W0, GW0, and a local multiplicative potential V (r) designed to predict accurate one-electronband gaps. Although these calculations yield similar one-electron energies as the sc-QPGW approach, the bandsare too narrow, leading to a “compressed” optical spectrum with too small excitation energies at higher energies.Finally, we report the absolute shifts of the conduction- and valence-band edges in silicon nitride and silicon tofacilitate the prediction of band alignments at silicon/silicon-nitride interfaces.

DOI: 10.1103/PhysRevB.85.045205 PACS number(s): 71.20.Nr

I. INTRODUCTION

Silicon nitrides are commonly used for high-temperatureand high-endurance applications due to their ability to with-stand high structural loads and their excellent wear resistance.1

Lately, hydrogenated silicon nitride has also been introducedas a passivation layer in silicon solar cells.2 Not only doesthe thin silicon-nitride film protect the silicon solar cellfrom the surrounding gas atmosphere, but it also minimizesthe reflection losses. Despite the technological importanceof silicon nitrides, accurate quasiparticle calculations forcrystalline silicon nitrides have not yet been reported in theliterature, although such calculations are a prerequisite for themodeling of the electronic properties of the interface betweensilicon and (hydrogenated) silicon nitride. In fact, even theexperimental data show a wide spread for the band gap ofcrystalline silicon nitrides, indicating the need for an accuratetheoretical reference. This is what the present work wants tocontribute: accurate quasiparticle band structures for the twomost important phases of silicon nitride, α-Si3N4 and β-Si3N4.

For solids, accurate methods to predict quasiparticle (QP)energies usually rely on the GW approximation proposed byLars Hedin.3 The approach we adopt in the present work isstate of the art: self-consistent quasiparticle GW calculations(sc-QPGW), including corrections for the electrostatic inter-actions between the electrons and holes (sc-QPGWTC−TC).4

Admittedly, there are cheaper approximations, specifically thewidely used G0W0 approximation, which relies on densityfunctional theory (DFT) orbitals and simple perturbationtheory to estimate the difference between the Kohn-Shamone-electron energies and the GW-QP energies.5 But thisapproximation tends to underestimate the QP energies for wideband-gap materials, which is indeed confirmed by our presentcalculations. The sc-QPGW approach with corrections forthe electrostatic interactions between the electrons and holestypically yields results that are accurate to about three percentwith respect to experiment for simple sp-bonded materials.However, few calculations have been reported for complexstructures containing more than four atoms in the unit cell.

Since the uncertainties in the experimental data for Si3N4 arefairly large, it is difficult to assess the accuracy of the presentcalculations by direct comparison with experimental data forSi3N4. We, therefore, decided to include quartz (α-SiO2) in ourstudy as well, mostly as a benchmark system, since accurateexperimental data are available for quartz. Indeed, we findthat the sc-QPGW calculations are in good agreement withprevious GW calculations and in excellent agreement withthe available experimental data, supporting our claim that thepresent QP energies should be reliable to within about threepercent.

Apart from the electronic properties, we present also resultsfor the optical properties of both materials. The opticalproperties are calculated using the Bethe-Salpeter equationwith a static approximation for the screened interaction.6

For SiO2, we again find good agreement with the availableexperimental data, whereas for Si3N4, our results are usefulas a reference for future experimental or theoretical work.Furthermore, we predict an optical gap of 6.1 eV for Si3N4,which is significantly larger than the commonly cited literaturevalue of 5.1 eV.7

The final issue we address in this work is whether accurateband structures and optical spectra can be predicted usingmore-approximate one-electron Hamiltonians with a local,in-real-space multiplicative potential. A simple procedure toconstruct such a local Kohn-Sham-like potential V (r) has beenrecently suggested by Tran and Blaha.8 This MBJLDA method(modified Becke-Johnson combined with the local-densityapproximation for the correlation) yields one-electron bandgaps in excellent agreement with experiment for a wide classof materials. Subsequently, Kim et al. investigated MBJLDAband dispersions for III-V materials and found reasonableagreement with experiment,9 although a tendency toward toonarrow bandwidth was observed. This trend yields too largeeffective masses and is also confirmed in the present work.Unfortunately, this problem precludes the prediction of accu-rate optical spectra from MBJLDA-based BSE calculations,as demonstrated for SiO2.

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KRESSE, MARSMAN, HINTZSCHE, AND FLAGE-LARSEN PHYSICAL REVIEW B 85, 045205 (2012)

TABLE I. Space-group and lattice parameters of the structuresconsidered in the present work. Also shown are the self-consistentMBJLDA parameter csc and the parameter cfit that is needed toreproduce the sc-QPGW band gap using the MBJLDA.

a (A) c (A) csc cfit

α-SiO2a P3221 4.914 5.406 1.41 1.69

α-Si3N4b P31c 7.753 5.619 1.25 1.55

β-Si3N4c P63m 7.607 2.911 1.22 1.5

aReference 12bReference 14cReference 13

II. METHOD

The ab initio calculations presented here were performedusing the VASP code and projector augmented-wave (PAW)10

potentials in the implementation of Kresse and Joubert.11 ThePAW potentials were specifically optimized to yield accuratehigh-energy scattering properties up to 10 Ry above thevacuum level. This was achieved by placing one projectorin the valence band and a second projector typically 5–8 Ryabove the vacuum level. All calculations in this work havebeen performed using a plane-wave energy cutoff of 400 eV.

For SiO2, the quartz structure was investigated and thelattice parameters that were published in Ref. 12 were used.For β-Si3N4 and α-Si3N4, the lattice parameters were takenfrom Refs. 13 and 14, respectively (see Table I for details).The ground-state structure of Si3N4 is the β structure with sixSi and eight N atoms in the unit cell. The α-Si3N4 structureis characterized by a different stacking in the c direction andtwice as many atoms in the unit cell. Since lattice positionsfrom Rietveld analysis usually do not have a high level ofaccuracy leading to significant forces in ab initio calculations(in this case, about 1.0 eV/A), the internal coordinateswere relaxed using first the Perdew, Burke, Ernzerhof (PBE)functional15 and then the hybrid Heyd-Scuseria-Ernzerhof(HSE06) functional.16,17 As the forces were found to befairly small when switching from the PBE to the HSE06functional, we expect that the electronic and optical propertiesare insensitive to the choice of the density functional used forthe relaxation.

In the literature, it is disputed whether the experimentalstructure of β-Si3N4 corresponds to the space group P63 orP63m. To resolve this issue, we performed relaxations startingfrom both the P63 and the P63m structure. VASP consistentlypredicted that the relaxation ends in the more symmetric P63m

structure, regardless of the used functional. This observation isin agreement with other density functional theory studies.18,19

Therefore, our final results for β-Si3N4 are for the P63m

structure.The self-consistent quasiparticle GW calculations were

performed using the method suggested by Faleev et al.4 In thismethod, a Hermitian approximation to the self-energy operatoris constructed by averaging the self-energy operator � betweentwo one-electron orbitals φi and φj at the correspondingone-electron energies Ei and Ej ,

�ij = 1

2[〈φi |�(Ei)

† + �(Ej )|φj 〉]. (1)

Details of the procedure used in the present work can be foundin Ref. 20. Specifically, we also evaluate the Hermitian partof the energy derivative of the self-energy operator � andconstruct a corresponding overlap operator.

In the course of our self-consistency procedure, the con-tributions from particle-hole ladder diagrams are accountedfor in the construction of the dielectric function.20 This can bedone either by solving the Bethe-Salpeter equation21 or by con-structing an approximate many-electron exchange-correlationkernel (Nanoquanta kernel).22–24 The second approach is moreapproximate but also more efficient and, therefore, used in thepresent work (see Ref. 20).

To initialize the iterative sc-QPGW calculations, a set ofone-electron orbitals and one-electron energies is required. Inthe present work, these initial orbitals are obtained by usinga hybrid functional that is based on the HSE constructionscheme. But contrary to the usual HSE scheme, the exactfull exchange is used at large wave vectors, and ε−1

exp of theexact exchange is applied at short wave vectors. Here, εexp

is the ion-clamped (high-frequency) experimental dielectricconstant of the considered material. The semilocal part ofthe functional is chosen such that the total exchange andcorrelation energy is exact for the homogeneous electron gas.This approach is similar to the model-GW approach25,26 andallows one to achieve very rapid convergence in the sc-QPGWcalculations, since the initial orbitals and one-electron energiesare already quite close to the final QP orbitals and QP energies.In practice, it also allows one to restrict the update of theorbitals to a fairly small number. In the present work, thenumber of orbitals included in Eq. (1) is three times largerthan the number of occupied states. The total number oforbitals included in the GW calculations was 512 for SiO2

and β-Si3N4, and 1024 for α-Si3N4. Reduction of the totalnumber of orbitals by a factor of two decreased the QP gapsby less than 100 meV. From this observation, we expect thatthe errors in the predicted QP energies are of that order, oreven slightly smaller.

The calculations for SiO2, α-Si3N4, and β-Si3N4 wereperformed using 6 × 6 × 6, 4 × 4 × 4, and 4 × 4 × 8 k points,respectively. For visualization purposes, the sc-QPGW bandstructures were interpolated to a finer grid using an interpola-tion based on Wannier orbitals implemented in the WANNIER90

code.27

The MBJLDA calculations8 were also performed using theVASP code and the implementation of Kim et al.9 As in Ref. 9,the parameter c was adjusted to fit the sc-QPGW band gaps.A self-consistent evaluation of the c parameter csc

8 yields toosmall band gaps and turned out to be unsatisfactory in thepresent case (compare Table I).

III. RESULTS

A. QP band structure

In Fig. 1, the band structure and electronic density ofstates (DOS) of SiO2 is shown. In SiO2, the valence bandsare dominated by oxygen states. The lowest subband (notshown) is located about 20 eV below the top of the valenceband and dominated by the O 2s orbitals. The second-lowestband (−10 to −5 eV) is made up of bonding O 2p orbitals(one per oxygen) predominantly oriented along the Si-O-Si

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FIG. 1. (Color online) (a) Band structure and (b) electronicdensity of states of α-SiO2. The sc-QPGW results are shown as blacklines, while the MBJLDA results are represented by red (gray) lines.In (b), the crosses correspond to the calculated QP band structure,and the black lines are the interpolation between these points usingthe WANNIER90 code.

bonds. The topmost valence band is nonbonding and madeup of the remaining oxygen 2p orbitals not oriented alongthe Si-O bonds (one per oxygen). For this subband, the GWapproximation yields a bandwidth of about 4 eV, whereas theMBJLDA yields only 3 eV. The top of the bonding O 2p statesat −5 eV, however, agrees between GW and MBJLDA, butthe width of this subband is also smaller for MBJLDA thanGW. The conduction-band states are in reasonable agreementbetween GW and MBJLDA, but with similar reductionsto the bandwidth. Specifically, the dispersion of the lowestconduction band is significantly weaker for MBJLDA than forsc-QPGW. This leads to a sharp resonance in the MBJLDAelectronic density of states (DOS) at 12 eV, which is not presentin the sc-QPGW or PBE calculations.

In summary, whereas the positions of the subbands agreereasonably well for MBJLDA and GW, the width of theindividual subbands is underestimated in MBJLDA and issimilar or even smaller than for PBE (not shown). As wewill see below, this seriously hampers the calculation ofoptical spectra using orbitals and one-electron energies fromMBJLDA.

The situation is qualitatively similar for the nitrides (Figs. 2and 3). In this case, the lowest valence band is made up

A Γ M K Γ-10

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FIG. 2. (Color online) (a) Band structure and (b) electronicdensity of states of β-Si3N4. The sc-QPGW results are shown asblack lines, while the MBJLDA results are represented by red (gray)lines. For details, see the caption of Fig. 1.

by N 2s states (not shown). The second-lowest subband ispredominantly made up by bonding N 2p states. In contrastto oxygen in SiO2, the nitrogen atoms are located in analmost-planar triangular configuration with three Si neighbors(NSi3). The N 2p band is made up of two p orbitals per N atom.The final subband closest to the Fermi level is dominated bythe nonbonding N 2p states (one orbital per N atom), whichare oriented out of the plane formed by the NSi3 triangle. Thebandwidth of the nitrogen subbands is again smaller in theMBJLDA than in the GW calculations, which is particularlyobvious at the lower band edge of the bonding p states (Fig. 2).The conduction bands, however, agree reasonably well forGW and MBJLDA, in particular, close to the conduction-bandedge.

From these results, it is clear that MBJLDA yields goodband topologies close to the conduction-band minimum andvalence-band maximum, but the discrepancies between GWand MBJLDA become larger further away from the bandonsets. We will return to this point later.

Table II summarizes the calculated band gaps. For SiO2, theagreement with the previous calculations by Chang et al. is ex-cellent [Eg(�-�) = 10.1 eV].21 It is unclear what prescriptionwas adopted for the GW procedure, and, in particular, whetherthe orbitals were updated in Ref. 21. However, Chang et al.certainly used a self-consistent procedure in which the QP

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FIG. 3. (Color online) (a) Band structure and (b) electronicdensity of states of α-Si3N4. The sc-QPGW results are shown asblack lines, while the MBJLDA results are represented by red (gray)lines. For details, see the caption of Fig. 1.

energies were determined self-consistently, and electron-holeinteractions were accounted for in the screening. This is similarto the procedure we have adopted here. In Table II, we alsoshow the band gaps evaluated using the standard perturbationalG0W0 approximation, starting from PBE one-electron orbitalsand one-electron energies. As already conjectured in Sec. I,the band gap remains underestimated in this approximation.A much better agreement with the sc-QPGW calculations canbe achieved by iterating the QP energies in the Green functionand using the screening from DFT [GW0(PBE)]. This is inagreement with the observation made in Ref. 28 for simplesemiconductors and insulators.

Furthermore, we have included the band gaps of Si inTable II (compare Refs. 20 and 28) and report the absoluteshift of the conduction- and valence-band edges from PBEto GW0(PBE) for Si and β-Si3N4 in Table III. For Si, 136bands were used corresponding to a similar number of planewaves per unit volume, as in β-Si3N4. As a result of thischoice, the highest unoccupied orbitals have a very similarkinetic energy in our Si and β-Si3N4 calculation, suggestingreasonable error compensation in relative shifts. The calculatedshifts can be used to correct the band alignment for Si/Si3N4

interfaces calculated at the density functional theory level:it is well established that density functional theory yields areasonable lineup of the electrostatic potential at interfaces,

TABLE II. Band gaps of α-SiO2, α-Si3N4, β-Si3N4, and Si inthe cubic diamond structure (cd-Si) for different levels of theory.The table lists the direct quasiparticle band gap at �, Eg(�-�), theminimum QP band gap Eg , and the optical gap Eopt (dominated by�-� transitions). Optical gaps are from BSE calculations on top ofthe corresponding quasiparticle calculations.

Method Eg(�-�) Minimal Eg Optical Eopt

(eV) (eV) (eV)

α-SiO2 sc-QPGW 10.1 9.7 8.85α-Si3N4 sc-QPGW 6.81 6.74 6.21β-Si3N4 sc-QPGW 6.70 6.42 6.18cd-Si sc-QPGW 3.30 1.24

α-SiO2 PBE 6.27 5.96α-SiO2 G0W0(PBE) 9.3 8.9α-SiO2 GW0(PBE) 9.8 9.4 8.55

β-Si3N4 PBE 4.49 4.26β-Si3N4 G0W0(PBE) 6.21 5.92β-Si3N4 GW0(PBE) 6.50 6.21 5.98

cd-Si PBE 2.56 0.61cd-Si G0W0(PBE) 3.23 1.13cd-Si GW0(PBE) 3.32 1.21

however, the valence-band edges and the conduction-bandedges of bulk materials are not very accurate. The simplestapproach to correct for this error is to add the calculated shifts�EVBM and �ECBM for Si and Si3N4 to the DFT valence-and conduction-band edges, respectively.29,30 Since the shiftsfor the valence bands are fairly similar for Si and Si3N4, thecorrection for the band alignment for the valence bands willbe relatively small (0.36 eV), whereas the correction for theconduction-band alignment is predicted to be almost 1 eV.

B. Dielectric properties

The predicted absorption spectrum of α-SiO2 is shown inFig. 4 together with a comparison with experimental data.31

For these calculations, 6 × 6 × 6 k points were used and 24conduction and 18 valence bands were included (2s orbitalsare excluded). The first absorption peak is located at 8.85 eV[8.55 eV for BSE on top of GW0(PBE)]. In both cases, thebound exciton is located roughly 1.2 eV below the smallestdirect QP gap. Chang predicted a slightly larger binding energyof 1.7 eV.21

The experimental data shows three characteristic peaks at10.3, 11.3, and 14 eV, and a broader feature around 17 eV.The theoretical peak positions are located at 10.30, 11.45,14.0, and 17 eV, in very good agreement with experiment.

TABLE III. Shift of the top of the valence band, �EVBM =EVBM[GW0(PBE)] − EVBM(PBE), and bottom of the conductionband �ECBM from PBE to GW0(PBE) calculations.

�EVBM �ECBM

(eV) (eV)

β-Si3N4 −0.98 0.94cd-Si −0.62 −0.02

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FIG. 4. (Color online) (a) Absorption spectra of α-SiO2 predictedusing BSE on top of sc-QPGW (solid line), independent particlespectrum (dotted line), and experiment (squares). The theoreticalspectra have been broadened using a Lorentzian width of 150 meV.To facilitate comparison with experiment, only the response parallelto the hexagonal plane has been calculated; the out-of-plane responseis not shown. (b) Absorption spectra of α-SiO2 predicted usingBSE on top of the MBJLDA (solid line) and scissor-corrected PBEcalculations (dashed red line), and independent particle spectrumusing MBJLDA (dotted line). Experimental data are from Ref. 31.

However, the broadened (150 meV Lorentzian) theoreticalspectrum is somewhat more structured than the experimentalspectrum and the width of the first peak seems to be toosmall, but this can be either related to a too coarse k-pointsampling, finite-temperature effects, which are disregarded inour present calculations, or the Tamm-Dancoff approximation.This approximation neglects coupling between the resonantand antiresonant part of the spectrum, which tends to affectthe transition probabilities somewhat but leaves excitationenergies usually unchanged. Recognizing these restrictions,the agreement is in fact very good, in particular, for thepositions of the peaks, which agree to within 0.1 eV withexperiment. Again, we can compare our results to previouscalculations. Chang et al. predicted a very similar optical-absorption spectrum, with slightly red-shifted absorptionpeaks (peak positions 10.1, 11.3, and 13.5 eV).21 The secondBSE calculation reported in the literature used a scissor-corrected PBE band structure calculation as a starting point.32

This allows one to get exact agreement for the first peak,but since PBE yields a smaller valence- and conduction-bandwidth, the resultant QP spectrum is red shifted for all but thefirst peak (see below). For example, in Ref. 32, the third peakwas located at about 13 eV, clearly at too low frequencies.

This is confirmed by the spectrum calculated using PBEorbitals or MBJLDA orbitals. For the PBE-BSE case, W wascalculated as W = ε−1(ω = 0)v, where the screening tensor

ε(ω = 0) was determined in the random-phase approximationusing PBE orbitals and one-electron energies (PBE-RPA).A scissor correction of 4.0 eV was applied for the BSEcalculations (but not for the determination of W ) in order toshift the first strong absorption peak to the experimentallyobserved strong absorption peak at 10.3 eV. This shift isalready larger than the difference between the PBE andsc-QPGW band gap. However, even this shift is not sufficient torecover agreement at higher energies, since the PBE valence-and conduction-band widths are excessively small and, as aresult, the higher energy excitations occur at too low energies.The MBJLDA-BSE spectrum was determined by solving theBSE equation using the same PBE-RPA W . Here, MBJLDAis only used as an alternative to the conventional scissorcorrection, i.e., instead of opening the one-electron gap usingthe scissor operator, one-electron energies and orbitals are“corrected” using the MBJLDA functional. The MBJLDA-BSE spectrum is even qualitatively wrong, and lacks thecharacteristic four-peak structure observed for the other BSEcalculations. The reason for this error is most likely the weakerdispersion of the conduction bands and valence bands inMBJLDA. Weaker dispersion implies stronger localization,resulting in overestimated excitonic effects. Specifically, inMBJLDA, the first conduction band exhibits a dispersion thatis too weak, causing the already mentioned sharp peak at 12 eVin the DOS [red line in Fig. 4(a)]. This is partly responsiblefor the qualitative errors in the optical-absorption spectrum.When the MBJLDA parameter c is reduced to csc = 1.41,and an additional scissor correction is applied, the spectrumapproaches the PBE spectrum, but PBE plus scissor correctionremains a superior starting point for BSE. For comparison,the independent particle (IP) spectra are also shown forthe sc-QPGW case [Fig. 4(a), dotted line] and MBJLDA[Fig. 4(b)]. Clearly, the oscillator strength is significantlyreduced for MBJLDA, explaining why the spectral weightis reduced for MBJLDA-BSE above 14 eV. However, the IPspectrum does not explain the discrepancies at low energies.

Our present calculations suggest that MBJLDA is not asuitable starting point for the prediction of accurate opticalspectra, and a scissor-corrected PBE calculation remains anoverall better choice. A final answer to that issue, however,requires a more careful evaluation for a larger and morerepresentative materials test set.

For sc-QPGW + BSE, our predicted ion-clamped (static)dielectric constant for SiO2 is 2.37, in perfect agreementwith the experimental value of 2.38.33 We note that for thecalculation of the static dielectric constant, the number ofbands was increased and the Tamm-Dancoff approximationwas not applied. In summary, the agreement of the opticalspectrum suggests that the sc-QPGW is able to describe notonly the band gap but also details of the band structurefurther away from the conduction- and valence-band edgesvery accurately for SiO2. This agreement is achieved withoutany adjustment of parameters.

The theoretical spectra of β-Si3N4 and α-Si3N4 are shownin Fig. 5. For the BSE calculation of β-Si3N4, we have used24 valence-band and 36 conduction-band states, whereas forthe metastable α-Si3N4, only 48 valence-band and conduction-band states were included (the unit cell for α-Si3N4 is twiceas large). This implies that the α-Si3N4 results are slightly

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FIG. 5. (Color online) Absorption spectra of β-Si3N4 (blacksolid line) and α-Si3N4 (red dashed line) predicted using BSE. Thetheoretical spectra has been broadened using a Lorentzian width of150 meV.

less accurate at higher excitation energies. We find that theoptical spectra of Si3N4 are rather featureless, with three broadmaxima between 8 and 11 eV, and a slow decay at higherfrequencies. The ion-clamped dielectric constant is predictedto be 4.18 and 3.97 for β-Si3N4 and α-Si3N4, respectively.

The onset of absorption is located at roughly 6.2 eV, whichis 0.6 eV below the minimum direct QP gap. Comparedto SiO2, this is a much weaker excitonic binding energy,which relates well to the larger dielectric constant of Si3N4.Unfortunately, this value is significantly smaller than thecommonly accepted value for the onset of optical absorption.Goodman measured a value of 5.1 eV in 1968, and this value iscommonly quoted in the literature.7 Independent confirmationof that value is difficult to find. Given the excellent agreementof the optical-excitation spectrum for SiO2, we believe thatour present theoretical value is very accurate. Possible reasonsfor discrepancies with experiment are electron-phonon con-tributions, which can lower the observed optical gap, as wellas impurities (possibly hydrogen) and defects, which mightreduce the observed optical gap even further. We believe thata careful reexamination of the optical spectrum of crystallineSi3N4 and comparison to our theoretical data is required.

IV. SUMMARY AND CONCLUSIONS

The aim of this work was to establish theoretical referencevalues for the QP and optical band gaps of α-Si3N4 and β-Si3N4. At the � point, we found values of 6.75 and 6.83 eV forα-Si3N4 and β-Si3N4, respectively. The smallest fundamentalgaps are, however, indirect and 6.68 and 6.59 eV, respectively.In both materials, the onset of absorption is related to anexciton, with a binding energy of approximately 600 meV,

resulting in optical gaps of 6.2 eV. The excitonic bindingenergy is about half of that in SiO2, where we predict anexcitonic binding of 1.2 eV. This relates well to the largerdielectric constant of Si3N4 of about 4.0 (compared to thedielectric constant of 2.37 in SiO2).

Since highly accurate data for Si3N4 are scarce, wehave evaluated the present approach for SiO2. For SiO2,we find excellent agreement with previous theoretical andexperimental data. In particular, for the optical spectrum, theagreement is very satisfactory. This suggests that sc-QPGWcalculations yield excellent band gaps yielding a very gooddescription of the structure of the valence and conductionbands even far away from the Fermi level, as well as reasonabletransition probabilities.

Furthermore, we have compared the sc-QPGW calculationsto MBJLDA calculations, which are two to three orders ofmagnitude faster than sc-QPGW calculations. If properlyadjusted, MBJLDA can fit the sc-QPGW band gaps. Also,the topology of the conduction-band and valence-band edgesagrees reasonably well with the sc-QPGW calculations.However, we observe that the width of the bonding anion p

and nonbonding anion p subbands, as well as the width of theconduction band, are too small. Only the overall positionsof these bands are in good agreement with the sc-QPGWcalculations. The likely explanation for this observation isthat the anion s, the bonding anion p, and the nonbondinganion p states are located in different regions of space. Hence,MBJLDA is able to place the subbands at energies that differsignificantly from standard PBE calculations, resulting inreasonable agreement with sc-QPGW calculations. However,the width of the subbands is determined by intraband matrixelements and there is not sufficient flexibility in a localpotential V (r) to make those matrix elements different fromthe LDA or PBE matrix elements. In combination with theincreased band gap, the bandwidth of the subbands is evensmaller than in PBE calculations and too narrow compared tothe sc-QPGW calculations. This observation is very similar toour previous observation of reduced bandwidth and increasedeffective masses in III-V semiconductors.9 Unfortunately, thereduced bandwidth has serious consequences for the opticalspectra, which turn out to be nonadequate if calculated usingthe BSE equation and MBJLDA orbitals.

ACKNOWLEDGMENTS

The research leading to these results has received fundingfrom the European Community’s 7th Framework Programmeunder Grant Agreement No. 228513 (HiperSol).

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