Revisiting the neutral C-vacancy in diamond: Localization of electrons through DFT+U

Highlights

• DFT+U parametrisation for defective diamond can provide hybrid functional quality electronic structures at the cost of DFT calculations.

• For PBE+U the optimum parameter pair for the neutral vacancy is (U,J) = (8.56;15.06) eV.

• The same parametrisation can also successfully be used for the ⟨001⟩ split interstitial defect.

Abstract

The neutral C-vacancy is investigated using density functional theory calculations. We show that local functionals, such as PBE, can predict the correct stability order of the different spin states, and that the success of this prediction is related to the accurate description of the local magnetic configuration. Despite the correct prediction of the stability order, the PBE functional still fails predicting the defect states correctly. Introduction of a fraction of exact exchange, as is done in hybrid functionals such as HSE06, remedies this failure, but at a steep computational cost. Since the defect states are strongly localized, the introduction of additional on-site Coulomb and exchange interactions, through the DFT+U method, is shown to resolve the failure as well, but at a much lower computational cost. In this work, we present optimized U and J parameters for DFT+U calculations, allowing for the accurate prediction of defect states in defective diamond. The transferability of the U and J parameters is tested through the study of the ⟨001⟩ split-interstitial.

Introduction

Defects play an important role in the properties and performance of semiconductor devices. Depending on the application, they can change the physical and chemical properties (e.g. the introduction of luminescent centres [1], [2]) or need to be avoided (e.g. because they deteriorate conductivity [3]). In the case of diamond, intrinsic defects have been studied at increasing levels of theory, in lock-step with the advancing state of the art [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. In recent years, the use of hybrid functionals has become more and more frequent in the solid state community as a result of ever growing computational resources, and the continuous improvement of methodologies. Some authors even used hybrid functionals in their simulation of infrared and Raman spectra, this of course in combination with small localized basis-sets [15], [16]. Although it is getting quite common to use such hybrid functionals to accurately determine the electronic structure and local spin polarization of complicated systems [17], [18], [19], [20], [21], their computational cost is still prohibitively high for extensive usage [22]; e.g. structure optimization of large super cells required for studying single defects. For most structure optimizations, pure density functional theory (DFT) functionals (LDA and GGA) perform admirably well. However, in strongly correlated systems in which the specific electronic structure is tightly related to the atomic structure, problems occur. The neutral C-vacancy in diamond is such a system.

The formation of a neutral vacancy in diamond gives rise to four dangling bonds. This structure has a T_d symmetry at this level of consideration. Each dangling bond is occupied by a single electron, which can have either up or down spin. As the other electrons of the C atoms neighboring the vacancy are involved in strong covalent bonds, the four electrons in the dangling bonds can only couple to each other, giving rise to three possible spin states for the vacancy defect: S_z = 0,1, and 2. The corresponding defect states are expected to be localized at the defect site, with an energy in the band gap of the host material. Of the three spin states only the S_z = 2 state (with all four electrons having the same spin) has the same T_d symmetry as the host lattice. In case of the S_z = 0 and 1 spin states, the magnetic configuration has a D_2d and a C_3v symmetry, respectively. In case of the S_z = 0 spin state Jahn-Teller distortion will further lower the symmetry to C_2v [14], [23]. As such, one may expect the modeling of this defect to be problematic.

Recently, Zelferino and co-workers argued that due to the open shell nature of the defect, only functionals including a fraction of the exact exchange term can adequately describe this defect [14]. They observe that pure DFT functionals present a qualitatively incorrect electronic structure and fail to indicate the correct spin ground state. As vacancies play a crucial role in many technologically interesting defects in diamond, such as the NV-center [24], [25], [26], [27], it is important to understand how and why pure DFT fails for the vacancy defect. Furthermore, as the solution of hybrid functionals is too computationally expensive for large scale usage, an alternative post-DFT approach is desirable, albeit only as a means of providing a more accurate defect geometry for subsequent analysis using a hybrid functional.

In this work, we will therefore revisit the neutral C vacancy in diamond. After introducing the computational methods used, results are presented and discussed in Section 3. We first examine the behavior of the defect using a pure DFT functional, and show how using knowledge of the magnetic configuration can help us avoid getting stuck in a local minimum. The qualitative failure with regard to the defect states is recognised as the well-known band gap problem of DFT. Hybrid functional calculations are performed to provide reference data, and subsequently we present a DFT+U study of the defect. We show how the on-site Coulomb and exchange parameters, U and J, influence different parts of the electronic structure independently. An optimum value of (U,J) = (8.56, 15.06) eV is obtained after fitting to the reference hybrid functional electronic structure obtained. After validating the (U,J)-pair for the neutral vacancy defect, we also address the ⟨001⟩ split-interstitial defect to test the transferability of the U and J parameters. The conclusions are presented in the final section of the manuscript.