Thermodynamics of boron incorporation in BGaN

We study the thermodynamics of boron (B) incorporation into gallium nitride (GaN) using first-principles calculations. In the dilute limit, we have calculated the formation energies of different configurations of the B impurity in GaN and found that substitution on the cation site is favored over substitution on the anion site. Under $p$-type conditions, interstitial boron can become the more favorable configuration and will ultimately limit the $p$-type conductivity. At higher B concentrations we use the generalized quasi-chemical approximation to elucidate the thermodynamic stability of boron gallium nitride (BGaN) alloys. We also investigate the effects of strain, which will be present if BGaN alloys are grown pseudomorphically on a GaN substrate. Without strain, B incorporation at typical growth conditions is limited to about 1.4% at $800{}^{\circ }\mathrm{C}$. Pseudomorphic strain raises the limit to 3.0% at the same temperature, close to experimentally observed levels of B incorporation.

Figure 1

Formation energies of ${\mathrm{B}}_{\mathrm{Ga}}$, ${\mathrm{B}}_{\mathrm{N}}$, and ${\mathrm{B}}_{\mathrm{i}}$ in GaN under (a) N-poor and (b) N-rich conditions.

Figure 2

The configuration coordinate diagram for the ($+$/0) transition of the ${\mathrm{B}}_{\mathrm{Ga}}$ defect. The absorption energy ($E_{\mathrm{abs}}$), emission energy ($E_{\mathrm{em}}$), and zero-phonon line ($E_{\mathrm{ZPL}}$) are shown.

Figure 3

(a) The atomic configuration of the ${\mathrm{B}}_{\mathrm{Ga}}$ impurity in the neutral charge state, along with the change in charge density (as defined in the text). Ga atoms are shown in blue, N in gold, and B in magenta. The red isosurface bounds a region of increased charge, and the green isosurface a region of decreased charge. (b) The atomic configuration of the ${\mathrm{B}}_{\mathrm{Ga}}$ impurity in the positive charge state. An isosurface of the polaronic state is shown in cyan.

Figure 4

Sixteen-atom cluster of the wurtzite nitride structure. The gold sites represent the nitrogen atoms, and the blue cation sites can be occupied by either a B or Ga atom. Periodic images of atoms are shown in gray.

Figure 5

(a) Schematic of the free energy, at a given temperature, as a function of the alloy concentration $x\in [0,1]$. The end points $\left[\mathrm{\Delta }F\right(0),\mathrm{\Delta }F(1\left)\right]$ are both zero by definition. Between the binodal boundary points, the solid solution can globally lower the free energy by decomposing into two different phases. Between the spinodal points, the solid solution is locally unstable against decomposition. The inflection points mark the limits of spinodal decomposition. (b) $\mathrm{\Delta }F\left(x\right)$ curves for unstrained BGaN alloys calculated using the GQCA at various temperatures. Note that the stable region is negligibly small.

Figure 6

Phase diagram of BGaN alloys obtained using the GQCA approach. The spinodal boundary of bulk BGaN is shown with the dashed line, while the spinodal boundary of BGaN strained to the in-plane lattice constant of GaN is shown with the solid line. The experimental limits of B incorporation are shown as reported in the following studies: (A) Vezin et al. [9], (B) Williamson et al. [10], (C) Cramer et al. [11], and (D) Malinauskas et al. [14].