Probing configurational disorder in ${\mathrm{ZnGeN}}_2$ using cluster-based Monte Carlo

${\mathrm{ZnGeN}}_2$ is sought as a semiconductor with comparable lattice constant to GaN and tunable band gap for integration in optoelectronic devices. Configurational disorder on the cation sublattice of ${\mathrm{ZnGeN}}_2$ can strongly modify the electronic structure compared to the ordered material, and both ordered and disordered forms of ${\mathrm{ZnGeN}}_2$ are candidates for light-emitting diodes and other emitters. The nonisovalent character of the disordered species (${\mathrm{Zn}}^{2+}$ and ${\mathrm{Ge}}^{4+}$) subjects the cation ordering to strong short-range order effects. To model these effects, we use Monte Carlo (MC) simulations utilizing a cluster expansion to approximate formation enthalpy. Representative disordered configurations in 1024-atom supercells are relaxed in density functional theory calculations. From the MC structures, we extract a short-range order parameter (the N-cation coordination motif), and two long-range order parameters (Bragg-Williams and stretching parameters), and examine their correlations. We perform a thermodynamic integration to determine the mixing entropy and free energy. ${\mathrm{ZnGeN}}_2$ exhibits a first-order phase transition with pronounced discontinuities in enthalpy and entropy, as well as in the structural order parameters. We discuss the relationship between the effective temperature used in the MC simulation and the growth temperatures in experiment in relation to the crossover from the nonequilibrium to the equilibrium growth regime. This work expands on current models of site disorder in ${\mathrm{ZnGeN}}_2$ and provides atomic structure models with a systematic variation of the degree of cation disorder.

Figure 1

(a) Octet rule-conserving (left) and breaking (right) motifs in wurtzite-derived ${\mathrm{ZnGeN}}_2$. Orthorhombic primitive cells of (b) $Pna{2}_1$ (SG33), 16 atoms and (c) $Pmc{2}_1$ (SG26), 8 atoms are outlined for ${\mathrm{ZnGeN}}_2$.

Figure 2

Energies calculated via cluster expansion compared to those from DFT for basis set p10t4 and fitting set 1 (of 3) colored to show (a) distribution of supercell sizes and (b) test versus train data.

Figure 3

(a) Contribution of entropy and formation enthalpy to configurational free energy in ${\mathrm{ZnGeN}}_2$ as a function of the reciprocal effective temperature. (b) Free energies from MC calculations compared to those of ordered and fully random structures. The gray bars highlight the first-order transition.

Figure 4

Motif fractions as a function of effective temperature. Solid dots show data from constant-temperature MC equilibration; continuous lines show a simulated anneal. Black crosses $(\times )$ indicate concentration of motif fractions given a fully random distribution.

Figure 5

Long-range order parameter ($\eta$) of supercells as a function of (a) $T_{\mathrm{eff}}$ with dashed line as a guide to the eye and (b) SRO parameter (motif fraction). The color scale indicates enthalpy $\mathrm{\Delta }{H}_{\mathrm{f}}$ of the individual configurations calculated from DFT. The red cross ($\times$) corresponds to a statistically random configuration. Stretching parameter as a function of (c) the LRO parameter $\eta$ and (d) motif fraction of $\mathrm{N}\text{−}{\mathrm{Zn}}_2{\mathrm{Ge}}_2$. Dashed lines show the stretching parameter of fully ordered ($\alpha =1.018$) and fully disordered ($\alpha \equiv 1$) structures.

Figure 6

Schematic illustration of the dependence of $T_{\mathrm{eff}}$ (measuring the degree of disorder) on the growth or deposition temperature $T_{\mathrm{dep}}$. Experimentally, the transition proceeds from disorder to order at $T_{\mathrm{dep}}\approx 1100$ K [10] and in Monte Carlo simulations from order to disorder at $T_{\mathrm{eff}}=2525$ K. Vertical lines indicate the nonequilibrium (ne) and the hypothetical thermodynamic (th) order-disorder transition.

Figure 7

Supercell size effects on convergence of formation enthalpy at (a) 1000 K, (b) 2000 K, and (c) 3000 K, comparing sizes of $16000$ (long dashed purple), 1024 (short dashed green), and 128 (solid orange) atoms. (d) Convergent behavior of formation enthalpy during equilibration at 3000 K for a 1024-atom simulation cell from random (dashed green) and ordered (solid blue) seed configurations.