Defect Energy Levels in Density Functional Calculations: Alignment and Band Gap Problem

For materials of varying band gap, we compare energy levels of atomically localized defects calculated within a semilocal and a hybrid density-functional scheme. Since the latter scheme partially relieves the band gap problem, our study describes how calculated defect levels shift when the band gap approaches the experimental value. When suitably aligned, defect levels obtained from total-energy differences correspond closely, showing average shifts of at most 0.2 eV irrespective of band gap. Systematic deviations from ideal alignment increase with the extent of the defect wave function. A guideline for comparing calculated and experimental defect levels is provided.

Figure 1

Schematic illustration of the alignment between energy levels obtained with a semilocal and a hybrid density functional. The charge transition levels $\mu$ and $\overline{\mu }$ are referred to the respective valence band maxima (VBM) and to a common reference level, respectively. The conduction band minima (CBM) are also shown.

Figure 2

Comparison between charge transition levels calculated with the semilocal (${\overline{\mu }}^{\mathrm{semiloc}}$) and hybrid (${\overline{\mu }}^{\mathrm{hyb}}$) functionals for a variety of defects in Si, SiC, ${\mathrm{HfO}}_2$, and ${\mathrm{SiO}}_2$. The energy levels corresponding to the valence band maximum (VBM) and conduction band minimum (CBM) are also shown (squares). All energies are referred to a common reference level $\varphi$ (see text), shifted to coincide with the VBM in the hybrid scheme for convenience. For each material, $\Delta$ is the r.m.s. error with respect to the ideal alignment (dashed line).

Figure 3

Optimal slopes derived from linear regressions of the data in Fig. 2 as a function of (a) experimental band gap and (b) average spread of the defect wave functions.