Validity of the Slater-Janak transition-state model within the $\text{LDA}+U$ approach

Known with the name of the Slater-Janak transition-state model, Janak’s theorem allows a calculation of charge transition levels by analyzing the Kohn-Sham eigenvalues of the density-functional theory without the need of explicitly comparing differently charged systems. Unfortunately, the usual local-density approximation (LDA) and its gradient extensions fail in describing the Kohn-Sham eigenvalues sufficiently well. In this work we show that the Slater-Janak transition state becomes a powerful tool if applied self-consistently within an $\text{LDA}+U$ approach. We first explain this fact analytically and then present a numerical validation, calculating the Slater-Janak transition state for a selection of representative examples in GaN. The formalism is found to be valid for all the investigated examples, which are, besides oxygen donors $\left({\text{O}}_{\text{N}}\right)$ and carbon acceptors $\left({\text{C}}_{\text{N}}\right)$, also systems with negative-$U$ effect (nitrogen vacancies, $V_{\text{N}}$) and strongly correlated electrons (europium substitutionals, ${\text{Eu}}_{\text{Ga}}$).

Figure 1

(Color online) Lower part: The exact total-energy profile of a system of $N$ electrons as a piecewise linear curve, never provided in actual DFT implementations such as LDA/GGA (solid line), where a spurious curvature is a consequence of the incorrect treatment of the self-interaction. The bottom curve is the difference between the other two (see also Ref. 15). Upper part: The LDA/GGA eigenvalue depend roughly linearly on the occupation numbers. The $\text{LDA}+U$ corrections (dotted) are also a linear function of the occupation number. The resulting $\text{LDA}+U$ eigenvalues are in the best case (for the ideal $U$ value) piecewise constant.

Figure 2

(Color online) Highest occupied one-particle level as a function of its occupation for the ${\text{O}}_{\text{N}}$ donor (upper box) and ${\text{C}}_{\text{N}}$ acceptor (lower box) in hexagonal GaN (DFTB-LDA calculations). The dotted lines are linear interpolations of the calculated points. The zero of the energy scale corresponds to the VBM.

Figure 3

(Color online) Highest occupied one-particle level as a function of its occupation for the different charge states of the isolated $V_{\text{N}}$ in hexagonal GaN (DFTB-LDA calculations). The lines are linear interpolations of the calculated points. Note that the KS state in the first two boxes an $s$-like $a_1$ orbital is filled up step by step while the last two boxes are characterized by the $p$-like orbital derived from the splitting of the $t_2$ state in a doublet and a singlet in wurtzite GaN. The zero of the energy scale corresponds to the VBM.

Figure 4

(Color online) Highest occupied one-particle level as a function of its occupation for the ${\text{Eu}}_{\text{Ga}}$ substitutional in cubic GaN. In the first picture DFTB-LDA and in the second $\text{DFTB-LDA}+U$ calculations. The lines are linear interpolations of the calculated data. In the second picture upper bunch of curves refer to calculations where the relaxation of the atomic positions is not considered and lower bunch of curves are calculated taking in account the atomic relaxation.