Quasiparticle self-consistent GW electronic band structures of of Be-IV-N{}_{2} compounds

Quasiparticle self-consistent electronic band structures of of Be-IV-N compounds

Sai Lyu and Walter R. L. Lambrecht Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106-7079, USA
Abstract

The electronic band structures of BeSiN and BeGeN compounds are calculated using the quasiparticle self-consistent method. The lattice parameters are calculated for the wurtzite based crystal structure commonly found in other II-IV-N compounds with the space group. They are determined both in the local density approximation (LDA) and generalized gradient (GGA) approximation, which provide lower and upper limits. At the GGA lattice constants, which gives lattice constants closer to the experimental ones, BeSiN is found to have an indirect of 6.88 eV and its direct gap at is 7.77 eV, while in BeGeN the gap is direct at and equals 5.03 eV. To explain the indirect gap in BeSiN comparisons are made with the w-BN band structure. The effective mass parameters are also evaluated and found to decrease from BeSiN to BeGeN.

I Introduction

BeSiN and BeGeN form part of a larger family of II-IV-N nitrides, which can be viewed as derived from the III-N family by replacing two of the group-III atoms in each tetrahedron surrounding the N by a group-II atom and two by a group-IV atom. By expanding the family of group-III nitrides including these heterovalent ternaries, significant new opportunties for band structure engineering and materials property design are opened. The occurrence of two different valence cations in these structures presents new challenges in terms of stoichiometry control, understanding the possible disorder effects, and more complex defect physics but also offers new possibilities in combination with existing nitrides exploiting band-offsets between lattice matched pairs of compounds. For an overview of this materials faily and related recent work, see Refs. Lambrecht and Punya, 2013; Martinez et al., 2017; Lyu et al., 2018.

BeSiN and BeGeN are expected to have among the highest band gaps in this family of materials and could thus be useful for optical devices with wavelengths in the ultraviolet region. Synthesis of BeSiN and its crystal structure were reported by Eckerlin et al. in 1967.Eckerlin et al. (); Eckerlin () The structural properties and electronic band structures at the level of density functional theory (DFT) of BeSiN and BeGeN were reported by Huang et al. Huang et al. (2001) and Shaposhnikov et al. Shaposhnikov et al. (2008). However, DFT in the local density approximation (LDA) is well known to underestimate the band gaps for semiconductors. A more accurate method is needed to predict the electronic properties of the Be-IV-N compounds, which have not yet been determined experimentally. Here we present a study of the band structures using the quasiparticle self-consistent method, van Schilfgaarde et al. (2006) which provides excellent agreement with experimental band gaps for most tetrahedrally bonded semiconductors. The goals of this work are to predict accurate band gaps, valence band fine structure and effective masses of these materials and to provide and understanding of the direct or indirect nature of the gaps. The details of the computational methodology used here can be found in section II. The structural properties are given in section III.1, and the discussion on stability is in III.2. The electronic band structures are shown in section III.3, and the effective masses in section III.4. We summarize the main results in section IV.

Ii Computational method

The structural properties are obtained via the Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization method Schlegel (1982) using the ABINIT packageGonze et al. (0211) in both local density approximation(LDA) and generalized-gradient approximation in the Perdew-Burke-Ernzerhof form(PBE-GGA)Perdew et al. (1996) within the general context of density functional theory. The BFGS method allows one to optimize the cell shape and atomic positions simultaneously because it makes use of the stress tensor and hence gradients are available for the total energy with respect to atomic displacements as well as the lattice constants defining the shape and volume of the unit cell. The interaction between valence electrons and ions are described using a pseudopotential. To be more specific, we used the Hartwigsen-Goedecker-Hutter (HGH) Hartwigsen et al. (1998) pseudopotentials in LDA and Fritz-Haber-Institute (FHI)Fuchs and Scheffler (1999) pseudopotentials in GGA. The wave functions are expanded in a plane wave (PW) basis set with energy cutoff of 100 Hartree and a k-point mesh sampling of the Brillouin zone is used. The forces are relaxed to be less than Hartree/Bohr.

Next, the density functional theory (DFT) Kohn and Sham (1965) band structures are calculated using the full-potential linearized muffin-tin orbital (FP-LMTO) all-electron method.Methfessel et al. (2000); Kotani and van Schilfgaarde (2010); lms () This provides a check on the pseudopotential band structures. In the FP-LMTO calculations, we used a double- basis set where denotes the decay length of the spherical wave basis function envelopes, which are smoothed Hankel functions. These are then augmented inside the spheres in spherical harmonics up to times radial functions and their energy derivatives corresponding to the actual all-electron potential inside the spheres. Specifically, on Be atom and on other atoms are included in the basis set. The Ge- were treated as bands using local orbitals. The Brillouin zone was sampled using a k-point mesh. This DFT band structure is used as the starting point for the quasiparticle self-consistent (QS) method.van Schilfgaarde et al. (2006); Kotani et al. (2007) The method is a many-body perturbation theoretical method first introduced by Hedin.Hedin (1965); Hedin and Lundqvist (1969) In this method, the complex and energy dependent self-energy describes the dynamic interaction effects beyond the DFT level self-consistent field. In the approximation it is given in terms of the one-electron Green’s function and the screened Coulomb interaction, , as , where is a short hand for the position, spin and time of particle 1 and indicates a time infinitesimally after . The screened Coulomb interaction is itself obtained from the Green’s function via and . All of these equations are in fact solved after Fourier transformation to k-space and energy and in a basis set of auxiliary Bloch functions, which consists of product functions of muffin-tin orbitals inside the spheres and interstitial plane waves. But the above real-space and time notation just provides the most concise way of stating the method’s approximations.

In the QS method, the energy dependent self-energy matrix in the basis of the independent electron Hamiltonian (LDA) eigenstates is replaced by an energy averaged and Hermitian matrix

(1)

This then replaces the starting LDA (or GGA) exchange-correlation potential and defines a new independent particle Hamiltonian whose eigenvalues and eigenstates provide a new . The process is then iterated till the self-energy is self-consistent at which point the Kohn-Sham eigenvalues of coincide with the quasiparticle excitation energies of the many-body theory. Hence the name quasiparticle self-consistent . In the QS calculation, the is calculated up to Ry, and is approximated by a diagonal average matrix when it is above 2 Ry before calculating the quasiparticle shifts in the next step. A cut-off of 3.5 Ry is used for the interstitial plane-waves and for the Coulomb interactions auxiliary basis set. The k-point mesh over which the self-energy is evaluated was set as . Expanding the eigenstates in muffin-tin orbitals, the is then obtained in real space and Fourier transformed back to the k-point set used for the charge self-consistency iterations of and eventually also to the k-points along symmetry lines of the Brillouin zone for the band structure plots. This effective interpolation scheme allows us to obtain the bands at the level at any k-point and hence also accurate effective masses.

Finally, we note that in the QS approximation, the screening of the Coulomb interactions is usually found to be underestimated by about 20 %,Deguchi et al. (2016); Bhandari et al. (2018) resulting in band gaps being overestimated. This can be corrected by including only 80% of the and is referred to as the approximation

Iii Results

iii.1 Crystal structure

Compound method (Å) (Å) (Å) )
BeSiN LDA 5.67 4.92 4.62 128.9 1.74 1.63
GGA 5.75 4.98 4.68 134.0 1.73 1.63
Expt.111From EckerlinEckerlin () 5.747 4.977 4.674 133.7 1.73 1.63
LDA222From Shaposhnikov et al. Shaposhnikov et al. (2008) 5.697 4.939 4.639 133.7 1.73 1.63
GGA 5.772 4.999 4.699 135.6 1.73 1.63
BeGeN LDA 5.81 5.07 4.77 140.5 1.75 1.64
GGA 5.93 5.17 4.86 149.0 1.74 1.64
LDA 5.856 5.105 4.803 143.6 1.74 1.64
GGA 5.972 5.204 4.894 152.1 1.74 1.64
Table 1: Calculated lattice parameters of Be-IV-N compared with experimental values.

The calculated lattice constants for BeSiN and BeGeN in both LDA and GGA are given in Table 1. The space group is and the order of the lattice constants is chosen so that . The relation of the lattice constants in the idealized orthorhombic structure to those in the wurtzite structure is given by , and . The optimized ratio indicates how far the structure is from the idealized wurtzite structure. From the results in Table 1 we can see that BeSiN is closer to the idealized wurtzite structure than BeGeN. The GGA lattice constants are systematically larger than the LDA ones which is commonly found. We also compare our relaxed lattice constants with those reported in literature and with the experimental values for BeSiN. Good agreement is found. Only the experimental lattice constants of BeSiN are available. In this case, we can see that GGA lattice constants are more accurate than LDA ones. Both LDA and GGA produce nearly the same and ratios in both materials studied here. The crystal structure of BeSiN is shown in Fig. 1 and BeGeN has a similar structure.

Figure 1: Crystal structure of BeSiN.

The atomic positions of the atoms in the unit cell are given in terms of Wyckoff 4a positions in Table 2. The origin is set at the position of the 2-fold screw axis in the plane. We may note that the nitrogen atoms are sitting nearly on the top of Be and group-IV atom which shows that BeSiN has its orthorhombic crystal structure very close to the supercell of wurtzite without much distortion.

Compound atom
BeSiN Be 0.125 0.083 -0.013
Si 0.625 0.086 -0.014
N 0.127 0.084 0.365
N 0.633 0.080 0.632
BeGeN Be 0.125 0.080 -0.013
GE 0.624 0.091 -0.013
N 0.115 0.072 0.351
N 0.635 0.091 0.375
Table 2: Wyckoff positions of Be-IV-N in the GGA approximation .

iii.2 Formation energy and stability

We next check the thermodynamic stability. The energies of formation are calculated from the cohesive energies as given in Table 3 which are calculated in the GGA approximation by subtracting the atomic energies from the solid’s total energy. The atomic energies include the spin-polarization energy. The N molecule was calculated in a large unit cell using the FP-LMTO method and including additional augmented plane wave basis functions to well represent the region outside the molecule, which is important to obtain total energies which are not sensitive to the choice of muffin-tin radii. The energies of formation of both compounds is negative, meaning that they are stable relative to the elements in their respective phases at standard conditions (room temperature and atmospheric pressure). They are also found to be stable relative to the competing binary compounds. The formation energies of the latter were calculated for the phases already found to be the minimum energy structures in the Materials Project mp () and compared with the data in that database in Table 3. They show that the BeSiN is stable against the reaction

(2)

by 0.136 eV/atom and similarly for the Ge case by 0.176 eV/atom.

ours MP333Material Projectmp ()
Be -3.44
Si 4.49
Ge 3.39
N 5.17
BeN -1.24 -1.225
SiN -1.04 -1.312
GeN -0.08 -0.26
BeSiN -1.26 -1.412
BeGeN -0.74 -0.813
Table 3: Cohesive energies and formation energies (in eV/atom).

iii.3 Energy bands and density of states

Figure 2: Band structures and partial densities of states of BeSiN and BeGeN in the QS approximation at the GGA lattice constants.
Figure 3: Band structures of BN plotted in Brillouin zone of wurtzite and orthorhombic structure.
Figure 4: Band structures near CBM projected on muffin-tin orbital basis functions.

The band structures and partial densities if states (PDOS) in the 0.8 QS approximation at GGA lattice constants are shown in Fig. 2. BeGeN is found to be a direct band gap semiconductor while BeSiN has an indirect band gap. In BeGeN, both the CBM and VBM occur at the point. An indirect band gap was previously also found for ZnSiN,Punya et al. (2011) MgSiNJaroenjittichai and Lambrecht (2016) and CdSiNLyu and Lambrecht (2017). However, in those cases, the CBM is at and the VBM near one of the Brillouin zone edges. On the other hand, in BeSiN, the VBM is at but the CBM is located between the and points. This is more similar to cubic and wurtzite BN. To further investigate this, we show the band structure of wurtzite BN in Fig. 3. We first show it in the standard wurtzite hexagonal Brillouin zone and then in the Brillouin zone corresponding to the orthorhombic supercell.

We can understand these relations in terms of band folding. The relation between the hexagonal Brillouin zone and the orthorhombic Brillouin zone is shown in Fig. 3 of Ref. Lambrecht et al., 2005. The wurtzite line is folded in two along the -direction and becomes the direction in the orthorhombic structure. The is also folded along the direction but the point lies at along the direction and the orthorhombic about which we fold the bands lies at . So, in the wurtzite w-BN the conduction band minimum ends up at 2/3 . This is very close to the CBM location also in BeSiN.

The band structure of w-BN in the CB is rather complex when folded in the orthorhombic zone. Even in z-BN the lies below the . In wurtzite, the splits into and . So at in w-BN, the conduction band ordering of states is , , and only the third band is the -like . Now after folding into the orthorhombic BZ, the lowest three conduction bands at all arise from the folding of the and bands, the next two, a degenerate and nondegenerate band near about 10-11 eV are derived from the wurtzite B--like states and the band a little below 12 eV is the B--like state.

Comparing this with BeSiN, it appears that here the derived states lie higher than the -like . The CBM at is a folded state but the next band already has a strong -like character. This can be seen in Fig. 4. In this figure we show the band colored according to their weight on muffin-tin orbital basis functions of specific angular momentum character centered on the different atoms. We recognize the typical strongly dispersing mixed cation -like conduction band in both materials but while this is the lowest conduction band in BeGeN it lies just a bit higher in the BeSiN case above the folded bands discussed above.

In Ref. Shaposhnikov et al., 2008, the CBM for BeSiN occurs between and which is simply because of the difference in labeling. Their is actually labeled as in the present work because the and axes are interchanged. So, this location of the CBM is in agreement between LDA, GGA and QS calculations. The CBM position is at the 2/3 of the line at either the LDA or GGA lattice constants.

In BeGeN, the conduction band has a stronger dispersion near its minimum at , while the valence bands are quite flat. Nonetheless the lowest conduction band dispersion near its minimum is less pronounced than in MgGeN, ZnGeN or CdGeN. This is because the Be- states lie less deep on an absolute scale. The N- atomic orbitals form the main set of the valence bands between 0 and 9 eV. The lower set of 8 bands correspond to the N- orbitals. This is seen clearly in the partial density of states decompositions in Fig. 2.

Figure 5: Band structures near VBM region with symmetry labeling at .

A zoom-in of the electronic band structure near the VBM region for BeSiN and BeGeN is shown in Fig. 5. Here, the energy levels at point are labeled by their related irreducible representation of point group . The states , , , correspond to basis function , respectively. These symmetry labels are helpful to understand whether the optical transitions from the valence bands to the conduction band minimum at point are dipole-allowed or not. In the case of BeGeN, the CBM has symmetry, so the dipole-allowed optical transitions are from valence states when , when , and when , respectively. The crystal field splitting of the energy levels near the VBM at the point are given in Table 4. In both cases, the top three valence bands have and symmetries but they occur in different order in the two compounds.

The band gaps of BeSiN and BeGeN calculated by DFT and 0.8 QS at LDA and GGA lattice constants are summarized in Table 5. For comparison, the band gaps in w-BN are 7.09 eV (indirect ) and 10.76 eV (direct at ). For BeGeN, the change in the gap from LDA to GGA lattice constants is as expected for tetrahedrally bonded materials: larger lattice constants relaxed in GGA gives rise to smaller band gaps. The change in gap with respect to the unit cell volume can be quantified by the deformation potential, . The deformation potential of BeSiN is eV from finite difference calculation, and for BeGeN is eV. The large difference between the two reflects the fact that the orbital character of the CBM is different in both materials. In BeGeN, we see that the smaller lattice volume of LDA gives rise to a larger gap already at the DFT level and this is maintained in the QS approximation. For BeSiN, we give both the lowest indirect gap and the direct gap at . In this case, surprisingly, the QS gaps at the GGA lattice constants are slightly larger than those at the LDA lattice constants, even though the lattice constants are larger in GGA. This results from the different orbital character of the CBM in this material. Apparently the deformation potentials of the indirect CBM and even the CBM at states are different from the usual (or -symmetry) CBM in BeGeN or other II-IV-N materials.

To compare the band gaps of these materials with related III-N and other II-IV-N semiconductors, it is useful to show them in a band gap versus lattice constant diagram as shown in in Fig. 6. The lines here are just guides to the eye and do not include alloy band gap bowing effects. We can see from this plot that these materials fall somewhere in between w-BN and the other III-N or II-IV-N materials.

The direct band gap of BeGeN of 5.03 eV corresponds to a wave length of the emitted or absorbed light of 246 nm in the UV region. The gap is still somewhat lower than in AlN (6.3 eV) but still possibly useful to push LEDs toward deep UV compared with GaN or ZnGeN. The gap however is only slightly larger than in MgGeN, another direct gap material but occurs at a much smaller lattice constant, so it will be more difficult to integrate with GaN or ZnGeN in heterostructures. The direct band gap of BeSiN of 7.77 eV corresponds to 159 nm and is even higher than that in AlN but the indirect nature makes it somewhat less attractive for such applications. It might still be a useful material for UV detectors.

BeSiN BeGeN
Sym. Sym.
Table 4: Energy levels in meV at relative to the VBM at including their symmetry label.
BeSiN BeGeN
indirect direct direct
Ref. Huang et al.,2001 DFT LDA 5.08 5.7444estimated from the band structure figure 5.24555Their latttice constants are smaller than that in this work
Ref. Shaposhnikov et al.,2008 DFT LDA 4.95 5.82 3.69
DFT GGA 5.19 5.93 3.37
This work DFT LDA 4.96 5.80 4.12
DFT GGA 5.13 5.95 3.45
GW LDA 6.82 7.74 5.88
GW GGA 6.88 7.77 5.03
Table 5: Band gaps (in eV) calculated in the 0.8 and DFT at the LDA and GGA lattice constants.
Figure 6: Band gaps of II-IV-N and III-N compounds.

iii.4 Effective masses

BeSiN BeGeN
0.54 0.44
0.68 0.54
0.36 0.33
2.59 2.00
2.91 2.03
0.25 0.22
0.37 0.26
4.52 2.09
2.33 1.89
5.77 2.15
0.33 0.27
2.54 1.88
Table 6: Effective mass ( in the unit of free electron mass )
BeSiN BeGeN
-4.00 -4.55
3.59 4.01
0.02 -0.001
-0.36 -0.50
-1.17 -1.63
-0.07 0.04
-0.02 -0.004
0.12 -0.03
1.33 1.65
Table 7: Parameters of effective Hamiltonian : inverse mass ()

Finally, we determined the effective masses. As mentioned in the compuational section, these include the corrections to the bands. The CBM and VBM effective masses are given in Table 6. These correspond to the actual CBM between X and in BeSiN and to the CBM at in BeGeN. The effective masses here can be seen to be significantly larger than in ZnGeN or CdGeN or even MgGeN. As already mentioned, this corresponds to the lower dispersion of these bands and the less deep Be- atomic energy levels on an absolute scale. The VBM is nearly degenerate but we give here the masses in the three Cartesian directions for the three highest VB states. We can see that for both materials and for each state, there is one direction with a small mass and two with a large mass . The direction of the small mass is different in the two materials. In this case of orthorhombic crystal structure, the Cartesian axes coincide with crystal axes, i.e., . The effective masses decrease from BeSiN to BeGeN which is consistent with the trend found in other II-IV-N compounds. The three upper valence bands can be best described by a Luttinger-like effective Hamiltonian.Punya et al. (2011) The inverse mass parameters in such a Hamiltonian for three top valence bands in BeSiN and BeGeN are derived from the masses given in Table 6 and given in Table 7.

Iv Conclusion

In this paper we considered the Be-IV-N compounds, BeSiN and BeGeN, of which only the former has been synthesized in the past. This study should be viewed in the context of the until now underexplored broader family of II-IV-N semiconductors. We determined their optimized lattice parameters and internal structural parameters and found them to be in good agreement with previous DFT calculations and the limited available experimental data. Their band structures were determined with the accurate and predictive QS method and indicate these materials have gaps in the deep UV. The indirect nature of the band gap of BeSiN is confirmed at the QS level. Effective masses were determined as well as band gap deformation potentials and details of the valence band splittings, which are necessary for future exploitation of these materials in heterostructure devices. In a band gap versus lattice constant plot, they fall in a quite well separated area from the other III-N or II-IV-N semiconductors, with a lattice volume closer to those of the ultra wide band gap and extremely hard materials, diamond and tetrahedrally bonded cubic and wurtzite BN. We discussed the relation of their band structure to that of wurtzite BN in terms of band folding, which helps to explain the indirect nature of the gap of BeSiN ad the location of its CBM at 2/3 .

Acknowledgements.
This work was supported by the National Science Foundation, Division of Materials Research under grant No. 1533957 and the DMREF program. Calculations made use of the High Performance Computing Resource in the Core Facility for Advanced Research Computing at Case Western Reserve University.

References

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