Quasiparticle self-consistent $GW$ band structures of Mg-IV-N$_2$ compounds: the role of semicore $d$ states

TL;DR

This study improves quasiparticle self-consistent $GW$ calculations of Mg-IV-N$_2$ compounds by including semicore states, revealing significant band gap reductions and implications for optoelectronic applications.

Contribution

It introduces the inclusion of semicore $d$ states in $GW$ calculations for Mg-IV-N$_2$ compounds, enhancing accuracy of band gap predictions.

Findings

Semicore $d$ states significantly reduce the band gap.

MgSnN$_2$ has a band gap in the visible green spectrum.

Materials stability and electronic properties are characterized.

Abstract

We present improved band structure calculations of the Mg-IV-N$_2$ compounds in the quasiparticle self-consistent $GW$ approximation. Compared to previous calculations (Phys. Rev. B 94, 125201 (2016)) we here include the effects of the Ge-3$d$ and Sn-4$d$ semicore states and find that these tend to reduce the band gap significantly. This places the band gap of MgSnN$_2$ in the difficult to reach green region of the visible spectrum. The stability of the materials with respect to competing binary compounds is also evaluated and details of the valence band maximum manifold splitting and effective masses are provided.

Figures

Tables

Table 1. Table 1: Energy levels in meV at Γ Γ \Gamma relative to the VBM at Γ Γ \Gamma including their symmetry label.

MgGeN2		MgSnN2
Sym.	$E$	Sym.	$E$
$a_1$	$0$	$a_1$	$0$
$b_2$	$-81.6$	$b_2$	$-115.7$
$b_1$	$-238.1$	$b_1$	$-144.2$

Table 2. Table 2: Effective mass (in the units of the free electron mass m e subscript 𝑚 𝑒 m_{e} )

	MgGeN2	MgSnN2
$m_x^c$	0.26	0.20
$m_y^c$	0.25	0.20
$m_z^c$	0.24	0.19
$m_x^{a_1}$	2.46	2.60
$m_y^{a_1}$	3.53	2.81
$m_z^{a_1}$	0.22	0.18
$m_x^{b_1}$	0.29	0.21
$m_y^{b_1}$	4.56	3.54
$m_z^{b_1}$	2.54	2.84
$m_x^{b_2}$	2.33	2.83
$m_y^{b_2}$	0.25	0.21
$m_z^{b_2}$	3.39	3.05

Table 3. Table 3: Cohesive energies and formation energies (in eV/atom) calculated in the GGA-PBE approximation.

	This work	Expt.111From Gschneider et al. Gschneider Jr. (1964), except for Mg taken from KaxirasKaxiras (2003), N2 taken from Huber and HerzbergHuber and Herzberg (1979)	MP222Material Projectmp
Mg	1.47	1.51
Ge	3.39	3.84
Sn	2.94	3.12
N	5.17	4.96
Mg3N2	-0.75		-0.90
Ge3N4	-0.08		-0.26
Sn3N4	0.22		0
MgGeN2	-0.60		-0.77
MgSnN2	-0.33		-0.54

Equations

$$3{\mathrm{MgGeN}}_2\to {\mathrm{Mg}}_3{\mathrm{N}}_2+{\mathrm{Ge}}_3{\mathrm{N}}_4,$$

$$3{\mathrm{MgSnN}}_2\to {\mathrm{Mg}}_3{\mathrm{N}}_2+3\mathrm{Sn}+2{\mathrm{N}}_2$$

Full text

The family of II-IV-N2 semiconductors is recently receiving increasing attention. They are related to the group-III nitrides semiconductors by replacing the group-III ion by a pair of a group-II and a group-IV ion in an ordered and octet-rule preserving manner. They share many of their opto-electronic properties with the group-III nitrides which are well known for their applications in light-emitting diodes (LED), laser diodes, UV-detectors and so on. Another reason for their interest is that they many of these compounds provide alternatives to the In and Ga in the form of sustainable and earth-abundant elements, such as Mg, Si, Sn. For an overview of these compounds, see Refs. Lambrecht and Punya, 2013; Martinez et al., 2017; Lyu et al., 2019. Among this family, the Mg-IV-N2 compounds were recently studied in a few papers.Jaroenjittichai and Lambrecht (2016); Quirk et al. (2014); Arab et al. (2016); Bruls et al. (2000, 2001, 1999); Råsander et al. (2017); Kanchiang et al. (2018); Kaewmeecha et al. (2018) In particular, quasiparticle self-consistent band structure calculations of the Mg-IV-N2 were presented by Jaroenjittichai et al. Jaroenjittichai and Lambrecht (2016) The main purpose of the present short note is to point out that with an improved basis set, including Sn-$4d$ and Ge-$3d$ semicore orbitals the band gaps of these materials are significantly reduced. At the same time, we analyze the stability of these materials in further detail than previously studied by comparison with the competing binary nitrides.

Computational approach: The quasiparticle self-consistent $GW$ methodvan Schilfgaarde et al. (2006); Kotani et al. (2007) is used for the band structures in a full-potential linearized muffin-tin orbital implementation.Methfessel et al. (2000); Kotani and van Schilfgaarde (2010); que Here, $W$ is the screened Coulomb interaction and $G$ the one-electron Green’s function and their convolution provides the self-energy $\Sigma$ which incorporates the dynamical effects of the electron-electron interactions beyond the density functional theory (DFT). This approach has been shown to yield accurate band structures, to better than 0.1 eV accuracy in comparison with experiment, in particular when the self-energy is scaled by a factor $0.8$. This scaling factor corrects for the underestimated screening of the screened Coulomb energy $W$ in the random phase approximation which neglects electron-hole interaction diagrams.Deguchi et al. (2016); Bhandari et al. (2018) Essentially the same approach was previously used by Jarroenjittichai et al. Jaroenjittichai and Lambrecht (2016). These results were also found to be in good agreement with hybrid functional calculations,Kanchiang et al. (2018) although a rather high fraction of exact exchange (0.5) was required to obtain good agreement with the previous QS$GW$ results. Such a large mixing fraction is unusual and already an indication that there might be a problem. The accuracy of the QS$GW$ approach still depends strongly on the completeness of the basis set and other convergence parameters. The present calculations improve on the previous work by using an improved basis set. As in previous work, a $3\times 3\times 3$ k-point mesh is used for the calculation of the $GW$ self-energy and subsequently interpolated to a $6\times 6\times 6$ mesh for self-consistent charge density calculations and to the k-points along symmetry lines. However, while previous work only considered Mg-$2p$ orbitals as semi-core states, we here include also Sn-$4d$ and Ge-$3d$ as local orbitals. The levels in fact lie at higher energy (Sn-$4d$ at -1.89 Ry and Ge-$3d$ at -2.15 Ry) than the Mg-$2p$ at (-3.43 Ry) and will be shown to have an important impact in shifting the valence band maximum (VBM) upwards thereby reducing the gap.

Results:

In Fig. 1 we show the band structure of MgGeN2 and MgSnN2 both with and without the inclusion of the local orbitals. The zero reference energy is placed at the valence band maximum (VBM) in both cases. The band effect of these semicore states is clearly visible. Because the $d$-like core levels interact more strongly with the VBM $p$-like states than with the CBM $s$-like states, and hybridization of the band edges with core levels would shift the band edges up, it is actually the VBM which is shifted up rather than the CBM shifting down. The band gaps of MgGeN2 and MgSnN2 are found to be 4.11 eV and 2.28 eV, respectively. Both of them are direct gap semiconductors.

This reduction of the gap is important in particular for MgSnN2. While previous results suggested a gap close to that of ZnGeN2 and GaN in the UV region of the spectrum, it is now found to be 2.28 eV which is in the green region of the spectrum. This is important because it provides additional flexibility to design heterostructures and materials with a band gap in the difficult to reach green region of the spectrum. The gap deformation potential $dE_{g}/d\ln{V}$ can be used to quantify the change in band gap versus the relative change in unit cell volume. From finite-difference calculations, the deformation potential for MgGeN2 is -8.9 eV and for MgSnN2 is -5.4 eV.

Next, a zoom in of the band structure near the VBM is shown in Fig. 2. The splittings of the VBM, their symmetry labeling which dictates the allowed optical transitions and the corresponding effective masses are all summarized in Tables 1 and 2 providing similar but updated information as in Ref. Jaroenjittichai and Lambrecht, 2016.

Finally, we discuss the stability of these compounds using GGA-PBE calculations. First we consider the stability of MgGeN2 against binary compounds. With respect to the reaction,

[TABLE]

MgGeN2 is found to have a lower energy than Mg3N2 plus Ge3N4 by 241 meV/atom. Regarding the stability of MgSnN2, the Sn3N4 itself is unstable relative to Sn and N2 which means it lies above the convex hull. So instead of Sn3N4, we need to consider the stability with respect to the following reaction

[TABLE]

We find that MgSnN2 is stable with respect to this reaction and the corresponding reaction energy is 18 meV/atom. So, in principle both MgGeN2 and MgSnN2 lie on hte convex hull and it should be possible to synthesize them.