Role of screening in the density functional applied to transition-metal defects in semiconductors

We study selected transition-metal-related point defects in silicon and silicon carbide semiconductors by a range-separated hybrid density functional (HSE06). We find that HSE06 does not fulfill the generalized Koopmans' theorem for every defect, which is due to the self-interaction error in the functional in such cases. Restoring the so-called generalized Koopmans' condition with a simple correction in the functional can eliminate this error and brings the calculated charge transition levels remarkably close to the experimental data as well as to the calculated quasiparticle levels from many-body perturbation theory.

Figure 1

Measured and theoretically predicted charge transition levels of transition metal impurities in $4H$ SiC and Si hosts. In all cases the experimental results[23, 24, 25, 27] are represented with dark gray (first) columns where the thick horizontal line represents the charge transition levels. The results of HSE06 calculations and our correction method with gKT satisfied are represented with red (second) columns and light green (third) columns, respectively, where thin horizontal lines provide the calculated charge transition levels. The bars indicated by dashed lines around the calculated charge transition levels show the inherent uncertainty of the calculation. Chromium needs special consideration regarding the ($0|-$) transition level; see text.

Figure 2

(a) Calculated electron density (with isosurface value of 0.05) of unpaired $d$ orbital of neutral vanadium substitutional defect in $4H$ SiC. (b) Two-dimensional plot of differences between two calculated charge densities using HSE06 functional and corrected HSE06 functional around the vanadium atom. The positive finite density clearly represents the overlocalization error of the HSE06 functional.

Figure 3

Charge-transition-level diagram of two different types of defects of vanadium (V) in Si-rich conditions, as obtained by the HSE06 and $\mathrm{HSE}06+{\mathrm{V}}_{\mathrm{w}}$ functionals (see text). Black vertical lines represent the position of the experimental charge transition levels. The formation energy of the substitutional defect was set to zero as a reference energy.

Figure 4

Partial density of states (PDOS) of substitutional vanadium in $4H$-SiC calculated by (a) HSE06 functional and (b) and $\mathrm{HSE}06+{\mathrm{V}}_{\mathrm{w}}$ functional close to the Fermi level; (c) by HSE06 and $\mathrm{HSE}06+{\mathrm{V}}_{\mathrm{w}}$ functional in a wide energy range. The energy of the valence band edge was set to zero as a reference energy. Up (down) labels the spin-up (spin-down) channel. The spin-up channel is the majority-spin channel. The calculated Fermi level is at 0.37 and 1.27 eV in HSE06 and $\mathrm{HSE}06+{\mathrm{V}}_{\mathrm{w}}$ calculations, respectively.

Figure 5

Radial dependence of charge density of defect state around TM defect upon using various functionals. There are two classes of TM impurities. Members of the first class (I) have delocalized defect states while the defect state is much more localized around the TM impurities in the second class (II). Note for the V${}^{-}$ defect that the V${}_{\mathrm{w}}$ correction with a negative value will reduce the localization of the defect state, as expected. The defect state of Cr${}^0$ appears only with the corrected functional, as explained in the previous section.