Open Access
Add to BibSonomy
Abstract
Based on the first-principle method of density functional theory (DFT), the parameters of the CeV (general term for Ce-related defects) color center of diamonds are optimized and calculated using Vienna Ab-initio Simulation package software in this study. The structure of the diamond CeV2 color center is the most stable when the Ce atom is located at the substitution site with the presence of two vacancies nearby. The electronic structure of the diamond CeV2 color center is calculated on the basis of this stable structure, and the charge transfer between Ce and C atoms is determined. The energy band structure and the density of states verify that the impurity states in the band are mainly composed of the 5d and 4f orbitals of Ce atoms. The ground state of the color center of the diamond CeV2 is located in the 4f orbit, and the excited state is in the 5d orbit. The zero phonon line of the color center of the diamond CeV2 is predicted to be 2.528 eV, and the corresponding fluorescence wavelength is 490.82 nm. This can provide a theoretical basis for preparing the diamond CeV color center for subsequent experiments.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The development of the quantum communication field changes as time progresses. Quantum communication has two types of channels, namely, fiber and free-space channels. However, the quality of single-photon source is an important affecting factor of the security of the entire communication process regardless of which channel is used for experiment. An excellent single-photon source for quantum information processing should satisfy the following requirements: photon emission is sufficiently stable, the photoluminescence (PL) spectrum bandwidth is narrow, the excited state lifetime is low, fully polarized photon absorption and emission processes occur, and no metastable dual levels exist, and photons that cannot be identified by Fourier transform can be produced.
Diamonds have become an excellent color center single-photon source matrix because of its nontoxicity, high biocompatibility, small size and high transparency in the visible light band, high stability, high hardness, acid and alkali resistance, and luminous stability and easy manipulation of electronic energy levels [1,2]. Over 500 types of diamond color centers exist [3]. The most studied diamond color centers are N vacancy (NV) color centers [4–11], Si vacancy (SiV) color centers [12–18], and Ni–N vacancy (NE8) color centers [19–24]. However, these color centers have their own advantages and disadvantages in practical applications. NV color centers have the advantages of stable optical characteristics, electron spins, and ideal solid-state qubits. However, they have low emission probability and a fluorescence spectral bandwidth of 100 nm. They also nearly cover the entire visible light band. When they are applied to the quantum key distribution in free space, they cannot be separated from solar noise and result in limited transmission speed. The excited state of SiV color centers has a long lifetime but does not reflect the quantum spin state clearly. NE8 color centers can work stably at room temperature and have the advantages of short fluorescence lifetime and supporting high-rate single-photon generation. However, their controllable preparation method remains immature, which limits their practical use.
The quantum transition between the electronic states of rare earth ions and nuclear hyperfine energy levels has a high quality factor; thus, rare earth ions have good reference performance in storing and restoring quantum states [25] and quantum-entangled states [26] of single photons. However, few research results are available on the diamond color center of rare earth elements either in laboratory preparation and detection or the use of first-principle calculation. A. Magyar et al. (2014) prepared Eu-doped diamond color centers in a laboratory and performed first-principle calculations, which theoretically and practically proved the existence of Eu color centers [27]. V. S. Sedov et al. (2017) added EuF3 nanoparticles to a diamond matrix and prepared a new diamond-rare earth composite by using microwave plasma chemical vapor deposition equipment. The test showed obvious localized PL at 612 nm due to the presence of EuF3 buried in the diamond matrix with a decay time of 0.34 ms [28]. J. Cajzl et al. (2017) used density functional theory (DFT) and ion implantation to demonstrate the near-infrared spectral luminescence of Er in a single-crystal diamond structure for the first time in theory and experiment [29]. K. Xia and R. Kolesov et al. detected that trivalent Ce ion in YAG (Y3Al5O12) replaced trivalent Y ion to form color center. The sharp zero phonon line (ZPL) was located at 489.15 nm, but the ZPL of Ce3+:YAG was unavailable at room temperature [30–32]. A. Wittlin et al. (2015) proved through the low-temperature infrared absorption spectra of YGG: Ce3+ crystals that at least two Ce3+-related centers existed in YGG in addition to the main Ce3+ sites located at Y ion substitution sites. The 5d to 4f fluorescence of Ce3+ was observed at high pressure, which indicated that a pressure-induced crossing existed between the lowest 5d state of Ce3+ and the conduction band [33]. The emission of Ce3+ usually comes from the transition from 5d to 4f, and this transition is strongly affected by its surrounding environment; the radiation spectrum corresponding to the transition from 5d to 4f energy level in different matrix materials can vary from ultraviolet to visible light because no shielding occurs outside the 5d electron [34]. In practical research, combining the extreme properties of diamond with the unique atomic and optical properties of rare earth ions is an important step to realize quantum information processing and a diamond solid platform of sensor, which provides the doped diamond with excellent luminous properties.
The purposes of this study are to investigate a new type of Ce-doped vacancy (General term for Ce-related defects, CeV) defect structure of diamond and explore its stable and electronic structures through first-principle calculation. This study provides a theoretical basis for whether diamond CeV can become a luminous center, for whether it can be used as a stable single-photon source, and for subsequent experiments.
2. Calculation details
The first-principle method based on DFT is used in this study to optimize the calculation parameters by using Vienna Ab-initio Simulation Package (VASP) software. After optimization, the cutoff energy is 450 eV, the k-point sampling is 5 × 5 × 5, and pseudopotential PAW-PBE is selected. PAW-PBE can accurately reproduce the diamond lattice constant, phonon spectrum, and its dependence on pressure and temperature; it can also be used to calculate the Raman spectrum of diamond close to the experimentally measured value [35]. The diamond supercell model of 128 C atoms is created in the calculation. The atoms are relaxed until the internal forces are within 0.01 eV/Å, and the total energy variation is smaller than 10−5 eV. The calculated optimized diamond lattice constant is a = b = c = 3.568 Å, which is consistent with the experimental value of 3.567 Å [36]. The Ce atom has a unique f orbital electron. The calculation parameters of LDAUU = 7 and LDAUJ = 0.7 are accordingly set in INCAR when the model containing Ce atom is calculated in this study [37].
The crystal cohesive energy is defined as follows:
We provide the formula for calculating the formation energy of defect to determine the stability of the defect structure, namely,
3. Results and discussion
On the basis of the established diamond supercell, three different CeV color center structures are calculated. In these structures, Ce atoms are located at the substitution sites and one, two, and three vacancies exist. The calculation model is shown in Fig. 1, where the yellow sphere represents the Ce atom, the brown sphere represents the C atom, and the gray one represents the vacancy. The stable structure of diamond CeV is determined by analyzing and comparing the cohesive energy, strain energy and formation energy of different vacancy types of diamond CeV color centers. The method for calculating the strain energy is to replace the Ce atom with a single C atom and calculate the difference in the energy with a relaxed C configuration (i.e., in the case of CeV1, the relaxed C coordination state is diamond with only one neutral vacancy, whereas for CeV2 and CeV3, two and three neutral C vacancies, respectively) [40].
The larger the cohesive energy, the more stable the structure. Synthesis of the data in Table 1 shows that the color center structure CeV2, in which the Ce atom is located at the substitution site and two vacancy sites exist, is the most stable. It can be seen from the strain energy in Table 1 that the strain energy decreases significantly with the increase of the number of vacancies. The reason for this is that the atomic radius of Ce is too large, and as the number of vacancies increases, Ce moves to the defect center, resulting in a decrease in strain energy.
Table 1. Cohesive and Strain energy of diamond CeV structures with different vacancy types.
We calculate the formation energy of different valence states to prove the stability of the structure. Given that the calculations are done at static positions (0 K), the Fermi level equals Fermi energy. In the calculation, the chemical potential of C is taken as the supercell energy divided by the number of atoms in the supercell. The chemical potential of Ce is calculated by the total energy of Ce (face-centered cubic structure) divided by the number of Ce atoms. Only the most likely values of Ce charge state in solid, namely, −1, 0 (neutral), 1, 2, 3, and 4, are used to calculate different charged states.
The smaller the formation energy, the more stable the diamond color center structure. Table 2 depicts that the formation energy of the double-vacancy color center structure of CeV2 when the Ce atom is in the substitution site and two vacancies exist in the vicinity is the smallest. This condition indicates that the double-vacancy color center structure of diamond CeV2 is the most stable. Ce atoms with three valence states of 0 valence, 1 valence, and 2 valence are likely to exist in diamond crystals.
Table 2. Formation energy of diamond Ce-defects with different structures.
By comparing the results of cohesive energy, strain energy and formation energy, it can be concluded that although the strain energy of the three-vacancy CeV3 structure is the least, the cohesive energy is smaller than that of the double-vacancy CeV2 structure, and the formation energy is larger than that of the double-vacancy CeV2 structure. So the most stable structure is when Ce atom is in the substitution position, there are two vacant CeV2 structures nearby. The stable structure of diamond CeV2 double-vacancy color center after relaxation is presented in Fig. 2. The structure diagram of CeV2 color center after relaxation shows that the Ce atom is located at the center of three vacancies after relaxation. The subsequent calculations are based on the zero-valent CeV2 stable structure.
The doping of Ce atoms into diamonds will affect the electronic structure. Differential charge density diagram can be used to qualitatively describe the charge distribution between atoms, charge transfer, and bonding between atoms. The charge density differences of diamond CeV2 system are analyzed to understand the changes in electronic structure before and after the formation of diamond CeV2 between pure diamond and doped Ce atoms, as shown in Fig. 3. In the figure, the charge density in the red region is positive due to the increase in charge density, whereas that in the blue region is negative owing to the decrease in charge density.
Figure 3 indicates that a red region appears between the Ce atom and the adjacent six C atoms, which indicates that a common electron exists between the Ce atom and the adjacent six C atoms and exhibits a certain covalent property. The Ce and C atoms appear blue and red, respectively, in the figure; the charge density around the Ce atom decreases, whereas the charge density around the C atom increases. In the diamond CeV2, the Ce atom loses electrons, whereas the surrounding six C atoms obtain electrons. The charge is transferred from the Ce atom to the C atom, which shows certain ionicity characteristics.
Bader charge analysis, which can achieve the purpose of quantifying chemical bonds, is used to calculate the charge transfer in compounds. Therefore, we calculate the Bader charge for the diamond CeV2 color center. The charge transfer amount is shown in Table 3.
Table 3. Amount of charge before and after the formation of the color center of the Ce and C atoms.
The Bader charge analysis indicates the charge transfer between atoms in the system. The Ce atom loses electrons, whereas the C atom obtains electrons. A large amount of charge transfer denotes great ionicity of the system.
In solid-state physics [41], the atomic radius of the Ce atom is 1.71 Å, the atomic radius of the C atom is 0.77 Å, and the sum of their covalent bond radius is 2.48 Å. We measure the bond length and angle between the six C atoms and the Ce atom, as shown in Table 4. The table presents that the Ce element is bonded to the six C atoms. Combined with the charge density diagram and the Bader calculation results, it can be determined that the number of C atoms bonded to the Ce atom is six, namely, C36, C39, C40, C81, C85, and C123. The bonding is shown in Fig. 4.
Table 4. Bond length and angle of diamond CeV2 color center structure.
The energy band of the diamond CeV2 color center structure is shown in Fig. 5. In the Fig. 5(b), the blue dashed line represents the spin-down energy band, and the red solid line represents the spin-up energy band. In the calculation of the energy band, the magnetic moment we get is zero.
Fig. 5. Energy band structure diagram, (a) Diamond supercell band structure diagram, (b) CeV2 color center band structure diagram.
Figure 5 depicts that Ce atoms are doped into the diamond, evident impurity states appear in the band diagram, and the system has a certain spin polarization phenomenon. The diagram of the density of states of the CeV2 color center in Fig. 6 shows that the energy level of the 5d orbit between 2.719 and 10 eV is split due to the action of crystal field and spin orbit. The 5d electron is the outer electron, its energy is regulated by the lattice vibration, and it is distributed within a certain range. The energy level of 4f orbital between −1.5 and 2.719 eV splits under the action of crystal field and spin orbit.
Fig. 6. Diagram of the density of states, (a) Diagram of the density of states of the CeV2 color center, (b) Diagram of the partial wave density of Ce atoms.
The diagram of the density of states of the diamond CeV2 in Fig. 6 exhibits that the degenerate energy level of the splitting between −1.5 and 2.719 eV in the energy level diagram is mainly the 4f orbit of the Ce atom. The degenerate state energy level of splitting between 2.719 and 10 eV is mainly the 5d orbit of Ce atom. Part of the 5d energy level enters the conduction band of the diamond. A good electric shielding effect is formed because the outermost electrons of rare earth ions are 5s2 and 5p6 full-shell structure. As a result, the emission of 4f electrons of rare earth ions maintains the characteristics of ions. The surrounding crystal field weakly affects the rare earth ions. The luminescence originates from the transition among rare earth energy levels, which forms a discrete luminescent center. The diamond lattice slightly affects the emission wavelength, and the luminescence is mainly determined by the luminescence center. We present the energy level diagram of diamond CeV2 in Fig. 7. The ground state is located in the 4f layer, and the electrons in the 4f layer are effectively concealed by the closed 5s and 5p layers. No interaction occurs among electrons. The spin-orbit interaction causes the configuration to split into two energy levels, 2F5/2 and 2F7/2, which are not far apart from each other. This screening results in weak interactions of ions with the surrounding environment. The ground state is split into seven degenerate states due to the interaction between spin orbit and crystal field. The excited state is located in the 5d layer. This state splits into five degenerate states due to the interaction of spin orbit and crystal field. The energy interval between this energy level and the ground state depends on the matrix material.
Fig. 7. Energy level diagram of the CeV2 color center (The upward blue arrow indicates the electronic transition under laser excitation, and the downward green arrow indicates the fluorescence of the Ce).
From the energy level diagram, we can derive the zero phonon energy of the transition of the CeV2 color center from the 5d orbit to the 4f orbit. The zero phonon energy of the diamond CeV2 color center is 2.528 eV, which corresponds to a fluorescence wavelength of 490.82 nm. The fluorescence wavelength is in the blue-green spectral region.
4. Conclusions
The stable structure and electrical properties of CeV color center are studied using VASP software on the basis of DFT. The structural parameters are optimized, and relaxation calculations are performed on three possible color center structures. Through the comparison of cohesive energy, strain energy and formation energy, it was found that the diamond CeV2 color center structure is the most stable when Ce atom is in the substitution site and there are two vacancies nearby. The differential charge density diagram and Bader charge analysis show that the Ce atom loses electrons, whereas the C atoms obtain electrons. The diagrams of energy band and the density of states of the color center of diamond CeV2 are compared. The impurity states in the band diagram are mainly the 5d and 4f orbitals of Ce atoms, the ground state of the CeV2 color center of diamond is located in 4f orbit, and the excited state is in 5d orbit. The ZPL of the diamond CeV2 color center is predicted to be 2.528 eV, and the corresponding fluorescence wavelength is 490.82 nm. This can provide a theoretical basis for preparing the diamond CeV color core for subsequent experiments.
Funding
National Natural Science Foundation of China (61765012); National Key Research and Development Program of China (2017YFF0207200, 2017YFF0207203); Natural Science Foundation of Inner Mongolia (2019MS05008); Inner Mongolia Autonomous Region Science and Technology Innovation Guidance Project (2017CXYD-2, KCBJ2018031).
Disclosures
No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication. I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part.
References
1. L. Robledo, L. Childress, H. Bernien, B. Hensen, P. F. A. Alkemade, and R. Hanson, “High-fidelity projective read-out of a solid-state spin quantum register,” Nature 477(7366), 574–578 (2011). [CrossRef]
2. A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3(12), 706–714 (2009). [CrossRef]
3. M. A. Zaitsev, “Vibronic spectra of impurity-related optical centers in diamond,” Phys. Rev. B 61(19), 12909–12922 (2000). [CrossRef]
4. A. Sipahigil, M. L. Goldman, E. Togan, Y. Chu, and M. D. Lukin, “Quantum Interference of Single Photons from Remote Nitrogen-Vacancy Centers in Diamond,” Phys. Rev. Lett. 108(14), 143601 (2012). [CrossRef]
5. V. M. Acosta, C. Santori, A. Faraon, Z. Huang, K. M. C. Fu, and A. Stacey, “Dynamic Stabilization of the Optical Resonances of Single Nitrogen-Vacancy Centers in Diamond,” Phys. Rev. Lett. 108(20), 206401 (2012). [CrossRef]
6. A. Gruber, “Scanning Confocal Optical Microscopy and Magnetic Resonance on Single Defect Centers,” Science 276(5321), 2012–2014 (1997). [CrossRef]
7. P. Siyushev, H. Pinto, M. Voeroes, A. Gali, F. Jelezko, and J. Wrachtrup, “Scanning Confocal Optical Microscopy and Magnetic Resonance on Single Defect Centers,” Phys. Rev. Lett. 110(16), 167402 (2013). [CrossRef]
8. Y. Dumeige, F. Treussart, R. Alléaume, T. Gacoin, J. F. Roch, and P. Grangier, “Photo-induced creation of nitrogen-related color centers in diamond nanocrystals under femtosecond illumination,” J. Lumin. 109(2), 61–67 (2004). [CrossRef]
9. L. Rondin, G. Dantelle, A. Slablab, F. Grosshans, F. Treussart, and P. Bergonzo, “Surface-induced charge state conversion of nitrogen-vacancy defects in nanodiamonds,” Phys. Rev. B 82(11), 115449 (2010). [CrossRef]
10. F. M. Hossain, M. W. Doherty, H. F. Wilson, and L. C. L. Hollenberg, “Ab Initio Electronic and Optical Properties of the N–V− Center in Diamond,” Phys. Rev. Lett. 101(22), 226403 (2008). [CrossRef]
11. T. Schroder, F. Gadeke, M. J. Banholzer, and O. Benson, “Ultrabright and efficient single-photon generation based on nitrogen-vacancy centres in nanodiamonds on a solid immersion lens,” New J. Phys. 13(5), 055017 (2011). [CrossRef]
12. I. I. Vlasov, A. S. Barnard, V. G. Ralchenko, O. I. Lebedev, M. V. Kanzyuba, and A. V. Saveliev, “Nanodiamond Photoemitters Based on Strong Narrow-Band Luminescence from Silicon-Vacancy Defects,” Adv. Mater. 21(7), 808–812 (2009). [CrossRef]
13. I. I. Vlasov, A. A. Shiryaev, T. Rendler, S. Steinert, S. Y. Lee, and D. Antonov, “Molecular-sized fluorescent nanodiamonds,” Nat. Nanotechnol. 9(1), 54–58 (2014). [CrossRef]
14. U. F. S. D’Haenensjohansson, A. M. Edmonds, B. L. Green, M. E. Newton, G. Davies, and P. M. Martineau, “Optical properties of the neutral silicon split-vacancy center in diamond,” Phys. Rev. B 84(24), 245208 (2011). [CrossRef]
15. L. J. Rogers, K. D. Jahnke, M. W. Doherty, A. Dietrich, L. P. Mcguinness, and C. Müller, “Electronic structure of the negatively charged silicon-vacancy center in diamond,” Phys. Rev. B 89(23), 235101 (2014). [CrossRef]
16. C. Hepp, T. Müller, V. Waselowski, J. N. Becker, B. Pingault, and H. Sternschulte, “Electronic Structure of the Silicon Vacancy Color Center in Diamond,” Phys. Rev. Lett. 112(3), 036405 (2014). [CrossRef]
17. U. D’Haenens-Johansson, A. Edmonds, M. Newton, J. Goss, P. Briddon, and J. Baker, “EPR of a defect in CVD diamond involving both silicon and hydrogen that shows preferential alignment,” Phys. Rev. B: Condens. Matter Mater. Phys. 82(15), 155205 (2010). [CrossRef]
18. E. Neu, C. Hepp, and M. Hauschild, “Low-temperature investigations of single silicon vacancy colour centres in diamond,” New J. Phys. 15(4), 043005 (2013). [CrossRef]
19. E. Wu, J. R. Rabeau, G. Roger, F. Treussart, H. Zeng, and P. Grangier, “Room temperature triggered single-photon source in the near infrared,” New J. Phys. 9(12), 434 (2007). [CrossRef]
20. J. R. Rabeau, Y. L. Chin, S. Prawer, F. Jelezko, T. Gaebel, and J. Wrachtrup, “Fabrication of single nickel-nitrogen defects in diamond by chemical vapor deposition,” Appl. Phys. Lett. 86(13), 131926 (2005). [CrossRef]
21. E. Wu, V. Jacques, F. Treussart, H. Zeng, P. Grangier, and J. F. Roch, “Single-photon emission in the near infrared from diamond colour centre,” J. Lumin. 119-120, 19–23 (2006). [CrossRef]
22. E. Wu, V. Jacques, H. Zeng, P. Grangier, F. Treussart, and J. Roch, “Narrow-band single-photon emission in the near infrared for quantum key distribution,” Opt. Express 14(3), 1296 (2006). [CrossRef]
23. E. Wu, J. R. Rabeau, F. Treussart, H. Zeng, P. Grangier, and S. Prawer, “Nonclassical photon statistics in a single nickel-nitrogen diamond color center photoluminescence at room temperature,” J. Mod. Opt. 55(17), 2893–2901 (2008). [CrossRef]
24. P. Siyushev, V. Jacques, and I. Aharonovich, “Low-temperature optical characterization of a near-infrared single-photon emitter in nanodiamonds,” New J. Phys. 11(11), 113029 (2009). [CrossRef]
25. H. De Riedmatten, M. Afzelius, M. U. Staudt, C. Simon, and N. Gisin, “A solid-state light-matter interface at the single-photon level,” Nature 456(7223), 773–777 (2008). [CrossRef]
26. C. Clausen, I. Usmani, F. Bussières, N. Sangouard, M. Afzelius, and H. De Riedmatten, “Quantum storage of photonic entanglement in a crystal,” Nature 469(7331), 508–511 (2011). [CrossRef]
27. A. Magyar, W. Hu, T. Shanley, M. E. Flatté, E. Hu, and I. Aharonovich, “Synthesis of luminescent europium defects in diamond,” Nat. Commun. 5(1), 3523 (2014). [CrossRef]
28. V. S. Sedov, S. V. Kuznetsov, V. G. Ralchenko, M. N. Mayakova, V. S. Krivobok, and S. S. Savin, “Diamond-EuF 3 nanocomposites with bright orange photoluminescence,” Diamond Relat. Mater. 72, 47–52 (2017). [CrossRef]
29. J. Cajzl, P. Nekvindová, A. Macková, P. Malinsky, D. Sedmidubsky, and H. Michal, “Erbium ion implantation into diamond - measurement and modelling of the crystal structure,” Phys. Chem. Chem. Phys. 19(8), 6233–6245 (2017). [CrossRef]
30. K. Xia, R. Kolesov, Y. Wang, P. Siyushev, R. Reuter, and T. Kornher, “All-Optical Preparation of Coherent Dark States of a Single Rare Earth Ion Spin in a Crystal,” Phys. Rev. Lett. 115(9), 093602 (2015). [CrossRef]
31. R. Kolesov, K. Xia, R. Reuter, M. Jamali, R. Stohr, and T. Inal, “Mapping Spin Coherence of a Single Rare-Earth Ion in a Crystal onto a Single Photon Polarization State,” Phys. Rev. Lett. 111(12), 120502 (2013). [CrossRef]
32. P. Siyushev, K. Xia, R. Reuter, M. Jamali, N. Zhao, and N. Yang, “Coherent properties of single rare-earth spin qubits,” Nat. Commun. 5(1), 3895 (2014). [CrossRef]
33. A. Wittlin, H. Przybylińska, M. Berkowski, A. Kamińska, P. Nowakowski, and P. Sybilski, “Ambient and high pressure spectroscopy of Ce3+ doped yttrium gallium garnet,” Opt. Mater. Express 5(8), 1868 (2015). [CrossRef]
34. N. Kodama, M. Yamaga, and B. Henderson, “Energy levels and symmetry of Ce3+ in fluoride and oxide crystals,” J. Appl. Phys. 84(10), 5820–5822 (1998). [CrossRef]
35. R. J. Nemanich and S. A. Solin, “First- and second-order Raman scattering from finite-size crystals of graphite,” Phys. Rev. B 20(2), 392–401 (1979). [CrossRef]
36. J. Heyd and G. E. Scuseria, “Efficient hybrid density functional calculations in solids: Assessment of the Heyd-Scuseria-Ernzerhof screened Coulomb hybrid functional,” J. Chem. Phys. 121(3), 1187–1192 (2004). [CrossRef]
37. Y. Jiang, J. B. Adams, and M. V. Schilfgaarde, “Density-functional calculation of CeO2 surfaces and prediction of effects of oxygen partial pressure and temperature on stabilities,” J. Chem. Phys. 123(6), 064701 (2005). [CrossRef]
38. S. B. Zhang, “Foreword,” J. Phys.: Condens. Matter 14(24), 000 (2002). [CrossRef]
39. V. D. Walle and G. Chris, “First-principles calculations for defects and impurities: Applications to III-nitrides,” J. Appl. Phys. 95(8), 3851–3879 (2004). [CrossRef]
40. D. E. P. Vanpoucke, S. S. Nicley, J. Raymakers, W. Maes, and K. Haenen, Diam. Relat. Mater. (2019).
41. C. Kittel, Introduction to Solid State Physics, 8th ed. (John Wiley & Sons, Inc, 2005).
