First-principles studies of GaN(0001) heteroepitaxy on $\mathrm{Zr}{\mathrm{B}}_2\left(0001\right)$

We conduct first-principles total-energy density functional calculations to study the heteroepitaxial growth behavior of GaN(0001) on $\mathrm{Zr}{\mathrm{B}}_2\left(0001\right)$ substrates. Using repeated free-surface terminated slabs, our calculated surface energies suggest that the $\mathrm{Zr}{\mathrm{B}}_2\left(0001\right)$ surface is Zr terminated. The relative stability of six different models of the $\mathrm{Ga}\mathrm{N}\left(0001\right)∕\mathrm{Zr}{\mathrm{B}}_2\left(0001\right)$ interface is examined as a function of the chemical potentials of Ga and Zr, and the most favorable interface consists of tetrahedrally coordinated N with one Ga-N bond and three N-Zr bonds. This interface structure gives rise to N-polarity in the GaN epitaxial film. Both of these findings agree with previously reported experimental results.

Figure 1

(Color online) Atomistic representations of the six $\mathrm{Ga}\mathrm{N}\left(0001\right)\text{−}\mathrm{Zr}{\mathrm{B}}_2\left(0001\right)$ relaxed interface models fixed at the stoichiometry of ${\mathrm{Ga}}_6{\mathrm{N}}_6{\mathrm{Zr}}_6{\mathrm{B}}_{12}$. The atoms are represented by spheres: Ga (brown, large), N (dark blue, small), Zr (light blue, large), and B (pink, small). Models 1, 4, and 6 have N-polar GaN surfaces, while models 2, 3, and 5 have Ga-polar GaN surfaces.

Figure 2

(Color online) Schematic representation of the $E_{\text{slab}}$ and $\sigma$ calculations using joined GaN and $\mathrm{Zr}{\mathrm{B}}_2$ slabs and separated slabs.

Figure 3

Comparison of Zr- and B-terminated $\mathrm{Zr}{\mathrm{B}}_2\left(0001\right)$ surface energies $\sigma$ as a function of $\Delta {\mu }_{\mathrm{Zr}}$.

Figure 4

(Color online) 3D plot of interface energies of the six $\mathrm{Ga}\mathrm{N}∕\mathrm{Zr}{\mathrm{B}}_2$ models as bilinear functions of the allowed Ga and Zr chemical potentials, $\Delta \mu$. Model 1 has the lowest interface energy among all six models.

Figure 5

(Color online) 2D plot of interface energy $\Gamma$ as a function of chemical potential $\Delta {\mu }_{\mathrm{Zr}}$ in the Ga-rich condition, i.e., $\Delta {\mu }_{\mathrm{Ga}}=0$.

Figure 6

(Color online) 2D plot of interface energy $\Gamma$ as a function of chemical potential $\Delta {\mu }_{\mathrm{Zr}}$ in the N-rich condition, i.e., $\Delta {\mu }_{\mathrm{Ga}}=-0.813\mathrm{eV}$.

Figure 7

(Color online) 2D plot of interface energy $\Gamma$ as a function of chemical potential $\Delta {\mu }_{\mathrm{Ga}}$ in the Zr-rich condition, i.e., $\Delta {\mu }_{\mathrm{Zr}}=0$.

Figure 8

(Color online) 2D plot of interface energy $\Gamma$ as a function of chemical potential $\Delta {\mu }_{\mathrm{Ga}}$ in the B-rich condition, i.e., $\Delta {\mu }_{\mathrm{Zr}}=-2.952\mathrm{eV}$.

Figure 9

(Color online) Atomic structure of the interface in models 3 and 4 showing the detailed N–B bonding arrangements.

Figure 10

Valence charge density map on the $\left(11\overline{2}0\right)$ plane at the N-B interface model 4. The charge is normalized to 26 electrons per unit cell. The contours are plotted in a range from $0.04\text{to}0.64\text{electrons}∕{\mathrm{\AA }}^3$ in 15 steps. The arrows indicate the position of the interface.