Self-doping behavior and cation disorder in ${\mathrm{MgSnN}}_2$

Investigations on $\mathrm{II}\text{−}\mathrm{Sn}\text{−}{\mathrm{N}}_2$ ($\mathrm{II}=\mathrm{Mg}$, Ca) have been started very recently compared to the intense research of $\mathrm{Zn}\text{−}\mathrm{IV}\text{−}{\mathrm{N}}_2$ ($\mathrm{IV}=\mathrm{Si}$, Ge, Sn). In this work, we study the phase stability of ${\mathrm{MgSnN}}_2$ and ${\mathrm{ZnSnN}}_2$ in wurtzite and rocksalt phases by first principles calculations. The calculated phase diagram agrees with the experimental observation; i.e., ${\mathrm{MgSnN}}_2$ can form in the wurtzite and rocksalt phases while ${\mathrm{ZnSnN}}_2$ only crystallizes in the wurtzite phase. Due to the higher ionicity of Mg-N bonds compared to Sn-N bonds and Zn-N bonds, wurtzite-type ${\mathrm{MgSnN}}_2$ appears under Mg-rich conditions. The defect properties and doping behavior of ${\mathrm{MgSnN}}_2$ in the wurtzite phase are further investigated. We find that ${\mathrm{MgSnN}}_2$ exhibits self-doped $n$-type conductivity, and donor-type antisite defect ${\mathrm{Sn}}_{\mathrm{Mg}}$ is the primary source of free electrons. The high possibility of forming the stoichiometry-preserving ${\mathrm{Mg}}_{\mathrm{Sn}}+{\mathrm{Sn}}_{\mathrm{Mg}}$ defect complex leads to our study of cation disorder in ${\mathrm{MgSnN}}_2$ by using the cluster expansion method with first principles calculations. It is found that cation disorder in ${\mathrm{MgSnN}}_2$ induces a band-gap reduction because of a violation of the octet rule. The local disorder, namely, forming (4,0) or (0,4) tetrahedra, leads to an appreciable band-gap reduction and hinders the enhancement of the optical absorption.

Figure 1

Band structures of ${\mathrm{MgSnN}}_2$ in the (a) wurtzite phase [space group (SG) $Pna{2}_1$] and rocksalt phase [space group (SG) $P2/c$], respectively.

Figure 2

Phase diagram displaying the chemical potential regions of ${\mathrm{MgSnN}}_2$ in the (a) wurtzite and (b) rocksalt phase, respectively, and ${\mathrm{ZnSnN}}_2$ in the (c) wurtzite and (d) rocksalt phase, respectively. The existence of the gray polygon region represents the stability of the compound. Points $A–E$ in (a) correspond to Mg-poor/Sn-rich/N-moderate (${\mu }_{\mathrm{Mg}}=-2.18\mathrm{eV}$; ${\mu }_{\mathrm{Sn}}=0\mathrm{eV}$; ${\mu }_{\mathrm{N}}=-0.27\mathrm{eV}$), Mg-rich/Sn-rich/N-poor (${\mu }_{\mathrm{Mg}}=-1.23\mathrm{eV}$; ${\mu }_{\mathrm{Sn}}=0\mathrm{eV}$; ${\mu }_{\mathrm{N}}=-0.75\mathrm{eV}$), Mg-rich/Sn-poor/N-rich (${\mu }_{\mathrm{Mg}}=-1.73\mathrm{eV}$; ${\mu }_{\mathrm{Sn}}=-1.00\mathrm{eV}$; ${\mu }_{\mathrm{N}}=0\mathrm{eV}$), Mg-poor/Sn-rich/N-rich (${\mu }_{\mathrm{Mg}}=-2.36\mathrm{eV}$; ${\mu }_{\mathrm{Sn}}=-0.37\mathrm{eV}$; ${\mu }_{\mathrm{N}}=0\mathrm{eV}$), and Mg-moderate/Sn-moderate/N-moderate (${\mu }_{\mathrm{Mg}}=-1.96\mathrm{eV} {\mu }_{\mathrm{Mg}}=-1.96\mathrm{eV}$; ${\mu }_{\mathrm{Sn}}=-0.50\mathrm{eV}$; ${\mu }_{\mathrm{N}}=-0.14\mathrm{eV}$) conditions.

Figure 3

Two-dimensional electron localization function (ELF) contours of (a) ${\mathrm{MgSnN}}_2$ and (b) ${\mathrm{ZnSnN}}_2$ and 1D profile of ELF along the (c) Mg-N and Sn-N bond and (d) Zn-N and Sn-N bond, respectively. Corresponding color code for constituents are shown on the left of (a), (b).

Figure 4

Formation energy of intrinsic point defects in ${\mathrm{MgSnN}}_2$ as a function of Fermi level ($E_F$) under chemical potential point $A$ (a) and point $C$ (b), respectively, which corresponds to Mg-poor/Sn-rich and Mg-rich/Sn-poor conditions.

Figure 5

Formation energy of low-energy intrinsic point defects and defect complexes in ${\mathrm{MgSnN}}_2$ as a function of the chemical potential at points $A–E$ shown in Fig. 2. The most stable charge state of these defects at the calculated Fermi level is shown. Note the formation energy of ${{\mathrm{Sn}}_{\mathrm{Mg}}}^{+}$ is lower than that of ${{\mathrm{Sn}}_{\mathrm{Mg}}}^{2+}$ only at the $A$ point.

Figure 6

Band gaps, degree of global and local disorder of fully ordered (FO), partially disordered, and fully disordered (FD) configurations of stoichiometric ${\mathrm{MgSnN}}_2$.

Figure 7

Total density of states of (a) FO configuration and two cation-disordered ${\mathrm{MgSnN}}_2$ configurations without (4,0) or (0,4) motif [P-1 swap (a) and P-3] and (b) three cation-disordered ${\mathrm{MgSnN}}_2$ configurations with (4,0) motif (P-1, P-2, and FD).

Figure 8

Charge density contours of VBM and CBM for (a) FO, (b) P-1 swap (a), and (c) P-1 configuration. The isosurface values are $0.01e/{\AA }^3$ for the VBM and $0.005e/{\AA }^3$ for the CBM of the FO configurations, and $0.005e/{\AA }^3$ for the VBM and $0.0005e/{\AA }^3$ for the CBM of the P-1 swap (a) and P-1 configurations.

Figure 9

Calculated absorption spectra of the FO, P-1, P-1 swap (a) and P-1 swap (b) configurations.