General procedure for the calculation of accurate defect excitation energies from DFT-1/2 band structures: The case of the ${\mathrm{NV}}^{\ensuremath{-}}$ center in diamond

General procedure for the calculation of accurate defect excitation energies from DFT-1/2 band structures: The case of the ${NV}^{-}$ center in diamond

Bruno Lucatto, Lucy V. C. Assali, Ronaldo Rodrigues Pela, Marcelo Marques, and Lara K. Teles

Phys. Rev. B 96, 075145 – Published 23 August 2017

Abstract

A major challenge in creating a quantum computer is to find a quantum system that can be used to implement the qubits. For this purpose, deep centers are prominent candidates, and ab initio calculations are one of the most important tools to theoretically study their properties. However, these calculations are highly involved, due to the large supercell needed, and the computational cost can be even larger when one goes beyond the Kohn-Sham scheme to correct the band gap problem and achieve good accuracy. In this work, we present a method that overcomes these problems and provides the optical transition energies as a difference of Kohn-Sham eigenvalues; even more, provides a complete and accurate band structure of the defects in a semiconductor. Despite the original motivations, the presented methodology is a general procedure, which can be used to systematically study the optical transitions between localized levels within the band gap of any system. The method is an extension of the low-cost and parameter-free DFT-1/2 approximate quasiparticle correction, and allows it to be applied in the study of complex defects. As a benchmark, we apply the method to the ${NV}^{-}$ center in diamond. The agreement with experiments is remarkable, with an accuracy of 0.1 eV. The band structure agrees with the expected qualitative features of this system, and thus provides a good intuitive physical picture by itself.

Received 29 May 2017

DOI:https://doi.org/10.1103/PhysRevB.96.075145

Physics Subject Headings (PhySH)

Band gap Defects Density functional theory Electronic excitation & ionization First-principles calculations Nitrogen vacancy centers in diamond Quantum information with solid state qubits

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

Bruno Lucatto^1,*, Lucy V. C. Assali², Ronaldo Rodrigues Pela¹, Marcelo Marques¹, and Lara K. Teles¹

¹Grupo de Materiais Semicondutores e Nanotecnologia (GMSN), Technological Institute of Aeronautics (ITA), 12228-900 São José dos Campos, SP, Brazil
²Institute of Physics, University de São Paulo (USP), 05315-970 São Paulo, SP, Brazil

^*brunolucatto@gmail.com; gmsn@ita.br

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Vol. 96, Iss. 7 — 15 August 2017

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Figure 1
Schematic representation of the defect states and their occupation on the ground state of the ${NV}^{-}$ center in diamond. Spin-up levels are on the left of the figure, while spin-down levels are on the right. There are two distinct levels with $a_{1}$ symmetry, which are labeled in crescent order of energy with a number on the left. The double-degenerate level $e$ is indicated with a broken line.
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Figure 2
Schematic configuration coordinate diagram for the ${NV}^{-}$ center. The curves represent the energy of the defect as a function of the displacement of the atoms, measured by a generalized coordinate q, for both its many-body electronic ground and excited states. The minima of these curves correspond to the relaxed geometry of each case. The vertical transitions (green and red arrows) correspond to the peaks in the optical absorption and emission curves, respectively. The transition between the minima is called the zero-phonon line (ZPL), and is the same for both the emission and absorption. The Stokes $(E_{S})$ and anti-Stokes $(E_{a S})$ shift energies are also indicated.
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Figure 3
(a) Schematic representation of a Kohn-Sham band structure with the Valence Band Maximum and Conduction Band Minimum with half-occupation, as considered on the DFT-1/2 method; and (b) Extension of the DFT-1/2 scheme for defect levels within the band gap.
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Figure 4
(a) 215-atom supercell used to simulate the defect avoiding spurious interactions among images. (b) ${NV}^{-}$ center and its surrounding atoms, representing the vacancy as a gray shadow. In both images, brown, bluish-gray, and red circles represent, respectively, the host carbon atoms, the nitrogen atom, and the three carbon atoms neighboring the vacancy. The images have been produced with help of the vesta software [29].
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Figure 5
CUT determination for the DFT-1/2 corrections. The corrections have been performed sequentially for each geometry, in the following order: $C_{Bulk}$ , N, $C_{Defect}$ . (a) Band gap of pure diamond as a function of the CUT for the 0.25 electron removal from the $C_{Bulk}$ atoms' $2 p$ orbital. (b) Transition energy on the ground state's geometry as a function of the CUT of the nitrogen atom, with $C_{Bulk}$ corrected. (c) Transition energy on the ground state's geometry as a function of the CUT of the $C_{Defect}$ atoms, with $C_{Bulk}$ and N corrected. (d) Transition energy on the excited state's geometry as a function of the CUT of the nitrogen atom, with $C_{Bulk}$ corrected. (e) Transition energy on the excited state's geometry as a function of the CUT of the $C_{Defect}$ atoms, with $C_{Bulk}$ and N corrected.
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Figure 6
Supercell band structures, around the gap region and along some special high symmetry directions in the cubic BZ, for the structural geometries of the ${NV}^{-}$ center in the (a) ground state and (b) excited state. The blue and red lines represent, respectively, the spin-up and spin-down energy levels. The occupied states related to the defect are indicated by $↑$ and $↓$ arrows. Both results have been obtained with bulk and local corrections. The direct band gap is a consequence of the supercell band folding.
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Physical Review B

covering condensed matter and materials physics

General procedure for the calculation of accurate defect excitation energies from DFT-1/2 band structures: The case of the ${NV}^{-}$ center in diamond

Bruno Lucatto, Lucy V. C. Assali, Ronaldo Rodrigues Pela, Marcelo Marques, and Lara K. Teles

Phys. Rev. B 96, 075145 – Published 23 August 2017

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Physical Review B

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General procedure for the calculation of accurate defect excitation energies from DFT-1/2 band structures: The case of the NV− center in diamond

Bruno Lucatto, Lucy V. C. Assali, Ronaldo Rodrigues Pela, Marcelo Marques, and Lara K. Teles

Phys. Rev. B 96, 075145 – Published 23 August 2017

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General procedure for the calculation of accurate defect excitation energies from DFT-1/2 band structures: The case of the ${NV}^{-}$ center in diamond