Critical evaluation of the LDA + U approach for band gap corrections in point defect calculations: The oxygen vacancy in ZnO case study

Abstract

The LDA þ U approach can be used essentially as a shift potential to open up the band gap of a semiconductor. This approach was previously applied to the oxygen vacancy in ZnO by Paudel and Lambrecht (PL) [Phys. Rev. B 77, 205202 (2008)]. Here, we review the results of that approach and introduce additional refinements of the LDA þ U model. Good agreement is obtained with recent hybrid functional calculations on the position of the "ð2 þ =0Þ transition state. A comparison of various approaches on the oxygen vacancy in ZnO is provided. The relevance of the one-electron levels to the experiments is discussed. 1 Introduction The local density approximation (LDA) is well known to underestimate band gaps in semiconductors. In a recent paper, Paudel and Lambrecht [1] (PL), discussed this problem for the oxygen vacancy V O in ZnO. The problem appears to be rather dramatic in this case as different authors do not even agree on whether the relevant defect level, the 2þ/0 transition level lies in the upper or the lower half of the band gap. Discrepancies also exist between different authors on the magnitude of the energy of formation of the defect and on the positions of the one-electron levels. At the time that paper was written, previous work had addressed the gap corrections mostly in a posteriori fashion [2][3][4][5][6][7]. The point of the PL paper was to use an adjusted Hamiltonian and total energy functional that gave the correct band gap for the host and then apply it to the defect transition levels. In particular, they used the LDA þ U approach with U Coulomb interactions not only for the d states of Zn but also the s orbitals. The idea behind this unorthodox application of LDA þ U is explained below. Since that work and even before, the problem has been investigated by several others by a variety of approaches, hybrid functionals [8][9][10], the GW method [9], and screened exchange [11,12]. Here, we critically re-examine the results of PL and explore the LDA þ U model further with additional U i parameters, which overcome some of the shortcomings of the previous U d þ U s

Which electronic structure method for the study of defects: A commentary

“…Another is to add simply non-local external potentials that shift the appropriate states [57,84], or to use modified pseudopotentials [77,85]. The success of such approaches depends significantly on the details of the implementation [74]. While their main advantage is simplicity and computational efficiency, it is undesirable that one needs to adjust these potentials on a case by case basis.…”

“…We already mentioned that the LDA þ U approach at least partially opens the gap. An extension of this approach is applying it to the states that dominate the CBM, typically cation s-like states [73,74]. Another is to add simply non-local external potentials that shift the appropriate states [57,84], or to use modified pseudopotentials [77,85].…”

“…This situation for example occurs for the oxygen vacancy in ZnO [73], a defect that was discussed by several contributors at the workshop and has become a sort of benchmark [68,[74][75][76][77][78].…”

Structural stability and defect energetics of ZnO from diffusion quantum Monte Carlo

“…Similarly, the transition level 2+/0 ranges from 0.3 to 2.8 eV. 47 In this work, we present the application of DMC to study the structural stability and the energetics of defects in transition-metal oxides. As a benchmark, we have evaluated the equation of state of Zn crystal and ZnO in various phases and studied the energetics of the oxygen vacancy in ZnO.…”

Quantum Spin Hall and Quantum Anomalous Hall States Realized in Junction Quantum Wells

“…As is well known, the band gap of InSb is much underestimated by generalized gradient approximation of density functional theory, but a Coulomb repulsion U s (12 eV) on the s orbital of In with the LDA+U method [18] can be used to obtain the experimentally observed band gap. [19] The LDA+U method is also employed to do the supercell calculations of the magnetic doping with 4 eV U d on the d orbitals of 3d transition metals. We use the virtual crystal approximation [20] to simulate the p-n doping which is assumed to be homogeneous.…”