Choosing the correct hybrid for defect calculations: A case study on intrinsic carrier trapping in $\beta -\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$

Due to its wide band gap and availability as a single crystal, $\beta -\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ has potential for applications in many areas of micro/optoelectronics and photovoltaics. Still, little is as yet known about its intrinsic defects, which may influence carrier concentrations and act as recombination centers. From a theoretical point of view, the problem is that standard (semi)local approximations of density functional theory usually cannot handle wide band-gap oxides, while results of tuned hybrid functional calculations so far have shown little quantitative coincidence with experimental data on $\beta -\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$. Here, we show a method for selecting the optimal hybrid, which reproduces not only the band gap, but also satisfies the generalized Koopmans’ theorem. Unless the screening is strongly orbital/direction dependent in the given material, such an optimal hybrid can reproduce the whole $GW$ band structure quite accurately. With the optimized functional, and introducing a modification into the charge correction process, we are able to give a consistent description of observed carrier trapping by intrinsic defects in $\beta -\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$. With the exception of gallium interstitials, which can act as shallow donors, all other intrinsic defects are deep. Gallium vacancies are the main compensating acceptors in $n$-type samples, while both oxygen interstitials and vacancies act as hole traps, in addition to small hole polarons. Considering the limitations imposed by a medium-sized (160-atom) supercell in an ionic solid, the calculated adiabatic and vertical transitions are in good agreement with available experimental data.

Figure 1

The total energy as a function of defect charge (occupation number of the defect level) for the optimal HSE(0.26,0.00) hybrid. The solid line shows the parabolic fit to the data points.

Figure 2

The band structure of $\beta -\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$, calculated by the HSE(0.26,0.00) hybrid (black solid lines) and by $\mathrm{HSE}(0.25,0.20)+G_0{W}_0$ in Ref. [68]. (red dashed lines). The band gaps are direct at $\mathrm{\Gamma }$: 5.02 eV and 5.04 eV, respectively.

Figure 3

The 160-atom supercell ($\mathrm{Ga}=\mathrm{light}\mathrm{green}\mathrm{spheres}$ and $\mathrm{O}=\mathrm{red}\mathrm{spheres}$), showing the different nonequivalent positions for oxygen (pink) and gallium (dark green). For details, see text.

Figure 4

The Ga-interstitialcy $\mathrm{G}{\mathrm{a}}_{\mathrm{i}}$ (a) and the O split-interstitial ${\mathrm{O}}_{\mathrm{i}}$ (b).

Figure 5

Defect formation energies (eV) as a function of the Fermi-level position between the VBM and the CBM for extreme oxygen-rich (a), intermediate (b), and extreme oxygen-poor (c) growth conditions ($\mathrm{G}{\mathrm{a}}_{\mathrm{i}}=\mathrm{green}, V_{\mathrm{O}}=\mathrm{red}, {\mathrm{O}}_{\mathrm{i}}=\mathrm{blue}, V_{\mathrm{Ga}}=\mathrm{black}\mathrm{lines}$, and the self-trapped hole, $h_{\mathrm{ST}}^{+}$, is indicated by the brown dotted line). The nonequivalent vacancies are numerated as in Fig. 3. The vertical lines (ocher) near the calculated 0 K band edges show their estimated position at room temperature, and the shaded area (ocher) is the estimated range of the Fermi level during high-temperature single-crystal growth (for details see text). The downward-pointing arrows are (room temperature) experimental results for the adiabatic charge transition levels [48, 49], while the upward-pointing ones show calculated values.

Figure 6

Change in the position of the band edges and of the defect level of $V_{\mathrm{O}2}$ upon tuning the HSE parameters from those used in Ref. [50, 51] (on the right) to the present optimal values (on the left). These calculations were carried out in the same 120-atom supercell with fixed geometry as described for Table 2. (The lines are merely guides to the eye.)

Figure 7

The blue lobes show a hole trapped in a small polaron state ($h_{\mathrm{ST}}^{+}$) near an O1 site (a) and between two O2 sites (b). No hole self-trapping was found at O3.