Article pubs.acs.org/JPCC
Ab Initio Calculations of the Transfer and Aggregation of F Centers in CaF2 H. Shi* and L. Chang School of Science, Beijing Institute of Technology, 100081, Beijing, P.R. China
R. Jia Department of Mathematics and Natural Sciences, Bergische Universität Wuppertal, D-42097 Wuppertal, Germany
R. I. Eglitis Institute of Solid State Physics, University of Latvia, 8 Kengaraga Str., Riga LV1063, Latvia ABSTRACT: The F center and R center in CaF2 crystals have been studied by using density functional theory (DFT) with a hybrid B3PW description of exchange and correlation. Our calculations show that the F-center diffusion barrier is equal to 1.67 eV. During the F-center transfer, the trapped electron is more delocalized than that in the regular F-center case, and the gap between defect level and CB in the α-spin state decreases. The surface F-center investigation shows the trend of F centers to locate near the surface. The association energy calculations of R centers indicate stable aggregations of isolated F centers. During the F-center aggregation, a considerable covalency forms between two neighboring fluorine vacancies with trapped electrons. Three incompletely paired electrons trapped in the R center have an up−down−up spin arrangement and induce three defect levels in the gaps between valence bands (VB) and conduction bands (CB) for both the α- and β-spin polarized band structures, respectively. More defect bands lead to more complex electron transitions, which were classified into two F- and four M-like transitions. The DOS calculations clearly reveal the components of defect bands.
I. INTRODUCTION Alkaline-earth fluorides such as CaF2 and BaF2, whose band gaps are larger than 10 eV, are very important for many optical applications. A main industrial application of CaF2 is for 193 nm lithography and in achromatic lens systems for photography and TV lenses.1,2 This wavelength is far shorter than the transparent region of quartz that is the most popular optical material in the ultraviolet (UV) region. Additionally, CaF2 could be chosen as an optical material also due to its cubic crystal structure with perfect optical isotropy, due to its chemical durability and its mechanical properties which make it applicable for lens fabrication.3−5 CaF2 can be used also as a material for radiation detection.6 Irradiation of metals and other crystalline materials with energetic particles, such as electrons, neutrons, or ions, can result in displacement of target atoms from their regular lattice sites to generate Frenkel defects (vacancy and interstitial pairs). Considering the high technological importance of alkaline-fluorides, it is not surprising that, during the last years, they have been the subject of many experimental and theoretical studies.7−32 It is well-known that the optical and mechanical properties of crystals are strongly affected by defects and impurities unavoidably present in any real material. Contemporary knowledge of defects in solids has helped to create a field of technology, namely, defect engineering, which is aimed at © 2012 American Chemical Society
manipulating the nature and concentration of defects in a material so as to tune its properties in a desired manner or to generate different behavior. CaF2 could become an important optical material if one could avoid or, at least, control the photoinduced defect formation, which thus far in applications degrades its optical quality. Recently, high purity single crystals with impurity concentrations below the ppm level have been grown successfully. However, a finite probability for absorption processes remains even in these pure single crystals. Because any absorption process allows for a deposition of energy in the crystal, any energy deposition will in turn lead to a change of material properties. Therefore, it is important to understand the nature of defects in CaF2. One intrinsic color center, named the F center, an electron trapped in an anion vacancy, has been observed in CaF2 by electron spin-resonance techniques. The electron paramagnetic resonance (EPR) spectrum of F centers was identified in CaF2 by Arends33 who carried out his investigations on additively colored crystals. A description of the electron nuclear double resonance (ENDOR) spectrum of the F centers in CaF2 was given by Hayes et al.34 Approximate values for the peak Received: September 13, 2011 Revised: January 10, 2012 Published: January 10, 2012 4832
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position of the F band in CaF2 were obtained by Arends33 by investigating the effects of optical bleaching on the intensity of F-center EPR spectra. More precise values for the peak position of the F band and values for the spin−orbit coupling constant in the excited 2P state of the F band were obtained, using magneto-optical methods, by Cavenett et al.35 Further, the optical absorption energy of the F center (3.3 eV) is well established experimentally.36 F centers in alkaline-earth fluoride have been studied also theoretically with the Xα method,37,38 and Puchina et al., using the Hartree−Fock (HF) embedded molecular cluster method, performed calculations for the ground state of the F center in CaF2 crystal and also for its optical absorption energy.19 Along with the widely studied F center in the CaF2 crystal have been found also other more complex centers, namely, M and R centers composed of two and three F centers, respectively. Hayes demonstrated that F centers in additively colored alkaline-earth fluoride crystals readily aggregate, forming more complex centers,39 and Stenzel et al. by transmission spectroscopy showed that Al Kα X-rays generate a controllable amount of F, M, and R centers in CaF2 single crystals.40 Therefore, as an extension of our single F-center study in CaF2,24 we performed calculations for more complex R centers. The manuscript is organized as follows. Section 2 introduces the computational method and provides details of our firstprinciples calculations. Section 3 is divided into three subsections, dealing with the F-center diffusion, the surface F center, and the R-center calculation results, respectively. The defect atomic structure and relaxation, energetic properties, and the electronic structure, as well as complex electronic transitions, are described in this section. The electron charge and spin density maps, band structures, and DOS plots are illustrated there to help readers to understand the properties of the single F center and their aggregations.
calculations were chosen as a compromise between the accuracy of the calculations and the large computational time for large supercells. They are 7, 7, 7, 7, and 14 for the Coulomb overlap, Coulomb penetration, exchange overlap, the first exchange pseudo-overlap, and the second exchange pseudooverlap, respectively.47 In our calculations, we used the theoretical CaF2 bulk lattice constant (5.50 Å), calculated by us earlier in ref 24. To simulate F and R centers, we started with the 96-atom supercell with one or three of the fluorine atoms removed, respectively. After fluorine atoms are removed, the atomic configuration of the surrounding atoms is reoptimized via a search of the total energy minimum as a function of the atomic displacements from the regular lattice sites. In order to have an accurate description of the F center, a basis set has been added at the fluorine vacancy, corresponding to the ghost atom. For the ghost atom, we used the same basis set as that used for the fluorine atoms of the perfect CaF2 crystal. Additionally, we also optimized a single, diffuse Gaussian s function as its basis set with the exponential coefficient of 0.078, which leads to the minimal total energy of the supercell containing an F center. Our calculations suggest that this single, diffuse Gaussian s function could accurately describe fluorine vacancies.
III. RESULTS AND DISCUSSION A. F-Center Transfer. The F center in CaF2 (an electron trapped in the fluorine vacancy) is created due to removal of the anion by irradiation or by additive coloration. The geometrical structures, atomic relaxations, energetic properties, and electronic structures, as well as complex electron transitions for the F- and M-center systems, have been compiled in our previous work.24,27 As an extension of our previous study, we performed calculations for the F-center transfer between two neighboring atomic sites. According to the crystalline structure of CaF2, it is obvious that the energetically most favorable Fcenter transfer should be the position switch between the two nearest neighboring fluorine sites, as we can see in Figure 1. Our calculations show that the corresponding F-center diffusion barrier is equal to 1.67 eV. For the optimized single basis set, the F-center diffusion barrier is equal to 1.57 eV. The VF site with trapped electron (F center) is surrounded in the ideal lattice by four calcium atoms, forming a tetrahedron, and the next-nearest neighbor shell is formed by six fluorine atoms (see Figure 1). The calculations of the positions of 10 atoms surrounding the F center in CaF2 with symmetry group Td after the lattice relaxation to the minimum of the total energy show that repulsions of the four nearest Ca atoms from the F center by 0.13% of the lattice constant (a0) and inward displacements of the second-nearest neighbor F atoms by 0.27% of a0 are small, since the effective charge of the F center is close to the effective charge of fluorine ions in perfect crystals, indicating that the trapped electron is well localized inside the vacancy. However, the atomic relaxation during the F-center transfer is much stronger than that in the regular Fcenter system. Figure 2a shows the displacements of the surrounding Ca atoms as a function of F0 position. The relaxations increase with the F-center transfer and reach approximately 3.2% of a0 for Ca1, which is much stronger relaxation than that in the regular F-center case. However, the displacements of Ca2, the other two calcium atoms neighboring the F center, are less than 0.5% of a0, close to the regular Fcenter case. Finally, Ca3, the other two calcium atoms
II. CALCULATION METHOD It is a well-known fact that the HF method considerably overestimates the optical band gap and density functional theory (DFT) underestimates it. To study fluorine vacancies (VF) in the CaF2 crystal, we performed our calculations using the hybrid exchange-correlation B3PW functional involving a mixture of nonlocal Fock exact exchange, local-density approximation (LDA) exchange, and Becke’s gradient corrected exchange functional,41 combined with the nonlocal gradient corrected correlation potential of Perdew and Wang.42−44 The first-principles hybrid DFT-B3PW method, according to our previous studies,24 gave the best agreement with experiments for the lattice constant, bulk modulus, and optical band gap for related perfect CaF2, BaF2, and SrF2 crystals. All numerical calculations for F and R centers in the CaF2 crystal were performed by the CRYSTAL-2006 computer code.45 The CRYSTAL-2006 code employs the Gaussian-type functions (GTF) localized at atoms as the basis for an expansion of the crystalline orbitals. In order to employ the LCAO-GTF (linear combination of atomic orbitals) method, it is desirable to have optimized basis sets (BS). In our calculations, we applied the basis sets developed by Catti et al.9 The reciprocal space integration was performed by sampling the Brillouin zone of the 96-atom supercell with the 3 × 3 × 3 Pack−Monkhorst net,46 that provides the balanced summation in direct and reciprocal spaces. The calculation thresholds N (i.e., the calculation of integrals with an accuracy of 10−N) in our 4833
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Figure 2. (a) The displacements of surrounding Ca atoms, (b) effective charges of the F center and F0, and (c) gaps of α-CB (the gap between the defect band and conduction bands in the α-spin state) and β-VB (the gap between the valence bands and defect band in the β-spin state), as a function of the F0 position. Ca1, Ca2, Ca3, and F0 are defined in Figure 1.
Figure 1. A view of the F-center neighbor geometry in CaF2, with the indication of F-center transfer by arrows. The fluorine atom exchanged with the F center is labeled F0, the calcium atoms in different groups are labeled Ca1, Ca2, and Ca3, and VF is labeled VF. Origin is set at the midpoint between the F center and F0.
CaF2 exhibits an absorption band observed experimentally around 3.3 eV.36 Our results for defect levels suggest a possible mechanism for explanation of the optical absorption. In the ground state, the defect band is occupied in the α state but is unoccupied in the β state. The optical absorption corresponds to an electron transition from the F-center ground state to the conduction band (CB). Because of the spin difference between α and β states and the selection rules, the electron transition from the occupied α band to the unoccupied β band is impossible. The experimentally observed optical absorption could be due to an electron transition from the F-center ground state to the CB bottom within the same spin state. Our calculated corresponding value is 4.06 eV, which is reasonable; however, it still is overestimated by around 23% with respect to the experimental result. Additionally, there also should be a possible β-electron transition from the occupied VB top to the unoccupied defect band between the VB−CB gap, and the calculated value of 9.79 eV is considerably larger than the transition in the α-spin case. Figure 2c shows the calculated variation of band gaps with positions of F0 during the F-center transfer. The α-CB gap obviously decreases during the position switching between the F0 and F center. When the F0 shifts to the middle position, the α-CB gap is reduced to around 1.5 eV, being much smaller than that in the regular F-center case. On the other hand, the variation of the VB-β gap is not remarkable, whereas, unlike the α-CB case, it is not monotonic, and the values are around 10 eV, being close to the value of the regular F-center case (around 9.8 eV). We conclude that the upward shift of the α-occupied band is notable and the shift of the βunoccupied band is negligible during the F-center transfer. As
neighboring the F0 atom, have remarkable shifts up to 1.2% of a0 during the F-center transfer. The Mulliken effective charge calculation shows that the electron associated with the removed fluorine atom is well localized (−0.752 e) inside the VF. With exchanging between the F0 and F center, their effective charges decrease, implicating an outward electron transfer, as depicted in Figure 2b. When F0 approaches the midway of the F-center transfer, some part of the trapped electron will be localized on the other side of F0. According to our calculations, the effective charge of −0.040 e is located on the other side of F0, for the F0 location with a coordinate of −0.25 Å. The corresponding values for larger F0coordinate cases are negligible. For the F0-midway case, the trapped electron (the F center) is equally localized (−0.384 e) on both sides of F0 and the effective charge of F0 is −0.787 e, which is much smaller than the fluorine charge of −0.902 e in a perfect CaF2 crystal. We also investigated the BS effect on the VF charge distribution. Our effective charge calculations for each basis set shell show that the inner part of the basis set, describing the 1s (0 e) and 2s (0.013 e) electrons of VF, obviously is useless. Further analysis of effective charges indicates that the unpaired electron inside the VF is more diffuse than the valence electrons in the F− valence shell and p orbitals give just a very minor contribution, as a consequence of both their different symmetry (s for the F center and p for the valence electrons of F−) and lack of nuclear attraction at the vacancy. Eventually, we may predict that the defect wave function could be accurately described by a single, diffuse Gaussian s function. 4834
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center and the nearest Ca atom. The surface F−Ca bond population is 40 me, which is larger than the relevant value (10 me) in the bulk by 30 me. The α energy band corresponding to the surface F center at the Γ point is located 7.13 eV (by 0.37 eV higher than in the bulk F-center case) above the VB top and is separated from the CB bottom by 3.66 eV. Because of the surface effect, this gap is smaller than the corresponding values in the bulk F-center case (4.06 eV). On the other hand, the β band lies 10.23 eV above the VB top (by 0.44 eV higher than in the bulk F-center case). The negligible dispersion effect of the defect band for the 108atom supercell means that the defect−defect interaction is practically eliminated, thus approaching the desired isolated single F-center limit. Here, we should note the effect of the defect band shift with respect to the VB top. As discussed above, the charge movement from 3s to 4s orbitals indicates a more diffuse surface F center, which should cause the defect band to move away (toward the gap) from the VB top. C. R Centers. As a next step dealing with F centers in the CaF2 crystal, we simulated more complex F-center aggregates, namely, the R center, being composed of three neighboring F centers. According to the symmetry Fm3̅m of the CaF2 structure, five possible R-center configurations in which the vacancy−vacancy distances are equal to the first- or secondnearest neighbor lengths, as we can see in Figure 3, named
mentioned before, the electrons trapped in fluorine vacancies form the α-occupied band and the VF effective charge decreases during the F-center transfer; therefore, the upward shift of the α-occupied band could be explained by the fact that the electrons trapped in fluorine vacancies are more delocalized, leading to higher energies. As previously discussed, the defect basis set could be described by a single, diffuse Gaussian s function. We optimized a simple Gaussian function with an exponential coefficient of 0.078 as the VF basis set, which gives the minimal total energy of the supercell containing an F center. For the optimized BS, the formation energy of an F center is equal to 7.69 eV, the repulsion of the nearest four Ca atoms from the F center is around 0.32% of a0, and the attraction of the second-nearest six F atoms is around 0.27% of a0, and the α-CB gap is equal to 4.40 eV. These results all are close to or comparable with the results obtained by using the basis set of fluorine atoms as for the F center. B. Surface F Centers. As an extension of our study dealing with F centers in CaF2 bulks, we performed calculations for CaF2 surface F centers. The atomic and electronic structures of surface F centers are practically unknown, and according to our knowledge have never been addressed in the literature. In the present work, we have studied surface F centers on the (111) surface, which is the most stable one among the (111), (110), and (100) terminated surfaces according to our previous work dealing with the pure CaF2 slabs.24 For a (111) slab of CaF2, there are three sublayers in each layer from the side view, forming a fluorine−calcium−fluorine (F−Ca−F) layer structure. We considered the surface F centers located in the upper sublayer of the first surface layer. We performed our surface Fcenter investigations for a slab containing four layers. According to the results of our calculations, the formation energy of the surface F center is 7.10 eV. The conclusion can be drawn that the defect formation on the CaF2(111) surface is by 0.77 eV smaller than in the bulk (7.87 eV). Similar results have been obtained also for other materials and are due to the reduced coordination at the surface.48 The lower formation energy for the surface F center shows the preference of F centers to locate near the surface. We calculated relaxations of the nearest atoms to the surface F center. Three Ca atoms located at the middle sublayer of the first surface layer are repulsed from the surface F center approximately by 3.05% of a0. Due to the surface effect, the repulsion magnitude from the F center is considerably stronger than that in the bulk (0.13% of a0). Regarding the electronic structure of surface F centers in CaF2, the effective charge analysis shows that the electronic density around it is slightly more delocalized than for the bulk F-center case. The effective charge localized inside the surface F center in CaF2 is −0.722 e, by 0.030 e less than for the bulk F center (−0.752 e). According to our calculations, the decrease of the surface F-center charge is mainly due to the outer p orbitals and the charge movement from 3s to 4s orbitals leads to the surface F center becoming more diffuse. The charges of the three nearest Ca atoms (+1.772 e) located at the middle sublayer of the first surface layer are reduced by 0.021 e in comparison with the charges of the relevant Ca atoms (+1.793 e) near the bulk F center. Additionally, we used a standard Mulliken population analysis to characterize the chemical bonding and covalency effects. Bond populations between the surface F center and the nearest Ca atoms are calculated. The major effect observed here is a strengthening of the chemical bond between the surface F
Figure 3. Schematic sketch of five different R-center configurations. Fluorine vacancies are denoted by squares. Configurations are labeled 1, 2, 3, 4, and 5 from left to right ordinally. The blue circle indicates a calcium atom located at the center of the cubic formed by eight fluorine atoms.
configs 1, 2, 3, 4, and 5 in this paper, are investigated. We concluded that the energetically most favorable configurations for the R center is configs 1 and 2. The energy difference between them is less than 0.002 eV. The total energies of configs 3, 4, and 5 are larger than that of config 1 by 0.29, 0.28, and 0.29 eV, respectively. Configs 1 and 2 have very close total energies and are more stable than other configurations; therefore, we mainly focus our current discussion on configs 1 and 2. As a starting point of our R-center calculations, we computed the formation energy of three neighboring fluorine vacancies. In the supercell calculation for vacancies with trapped electrons, the formation energy of a neutral (the total charge of the supercell is zero) R center in CaF2 could be expressed as follows: (R ) E formation = E(fluorine) × 3 + E(R ) − E(perfect)
(1)
where E(fluorine) is the energy for an isolated fluorine atom and E(R) and E(perfect) are the total energies of the defective crystal containing an R center and the perfect crystal, respectively. Our calculated R-center formation energy for config 1 is equal to 22.98 eV, which is smaller than 3 times of the F-center formation energy (7.87 eV × 3) by 0.63 eV. It means that the formation of the complex of three aggregated F centers is easier than the formation of three separated F centers. 4835
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Here, we can define the association energy of three F centers as the energy difference between the same three remote F centers and the R center, and the corresponding value is +0.63 eV. With this definition, the positive sign indicates stable aggregation. In our previous M-center study,27 a considerable covalent bond forms between two F centers after their aggregation; therefore, we can consider the association energy of two F centers as the F-bond energy. For R-center configs 1 and 2, there are two F-bonds in each R center, so it is reasonable that the F-center-aggregated system is more stable than the system consisting of the separated F centers. The geometry relaxation of atoms surrounding the R center is calculated and depicted in Figures 4 and 5. For config 1, the
Figure 5. A view of the R-center neighbor geometry inCaF2, with the indication of relaxation shifts for config 2. The nearest calcium atoms in inequivalent groups are labeled Ca1, Ca2, Ca3, Ca4, and Ca5, respectively.
config 2, the relaxation of the R-center system is slightly stronger. All nearest Ca atoms are repulsed from the R center, similar to config 1, and the nearest calcium atoms labeled Ca1, Ca2, Ca3, Ca4, and Ca5 shifts backward the R center by 0.25, 0.68, 0.42, 0.22, and 0.26% of a0, respectively (see Figure 5). From our calculated repulsions of the nearest calcium atoms in the F- and R-center systems, we found that the displacements of the calcium atoms with one nearest VF are small and less than 0.3% of a0, the relaxations for the calcium atoms with two nearest VF are about 0.4−0.5% of a0, and for the calcium atom labeled Ca2 in R-center config 2, containing three nearest VF with trapped electrons, the shift reaches around 0.7% of a0. It implicates that the repulsion from fluorine vacancies to the nearest calcium atoms has a quasi-superposition effect. The Ca atoms having more nearest VF exhibit stronger relaxations. Table 1 presents the effective charges and spins of the R center and surrounding calcium atoms for configs 1 and 2. Obviously, the fluorine vacancies VF1 and VF2 are not equivalent for the R-center systems. For config 1, the effective charges of VF1 and VF2 are −0.783 and −0.780 e, respectively. The average charge of three fluorine vacancies is equal to −0.781 e, being more localized inside the R center with respect to the F-center case (−0.752 e). Around 0.068 e from the eight nearest calcium atoms transfers toward to the R center, and according to our calculation, some charges belonging to the nearest fluorine atoms also shift to the R center. For config 2, the average charge of three VF (−0.781 e) is close to the corresponding value of config 1, and also is more localized than that in the Fcenter case. Bond population calculations for the R-center systems show that there is a considerable covalency between VF1 and VF2, 168 me and 176 me for configs 1 and 2, respectively. Nevertheless, they are much weaker than in the Mcenter case (274 me). These weaker covalencies in the R-center systems can be explained by the fact that the electrons localized on VF1 and VF2 are not completely paired. However, we can still
Figure 4. A view of the R-center neighbor geometry in CaF2, with the indication of relaxation shifts for config 1. The nearest calcium atoms in inequivalent groups are labeled Ca1 and Ca2, respectively.
eight nearest Ca atoms are all repulsed by the R center. The four Ca1 atoms move backward the VF1 by 0.52% of a0, and the four Ca2 atoms are displaced by 0.25% of a0 backward the VF2, as we can see in Figure 4. As a whole, according to our calculations, there are no displacements over 0.6% of a0 for atoms surrounding the R center in config 1. However, for 4836
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Table 1. The Effective Charges (Q(e)) of the R Center and Surrounding Atoms in CaF2 for a 96-Atom Supercella config 1
config 2
atoms (shell)
number
Q (e)
ΔQ (e)
spin (e)
VF1 VF2 Ca1 Ca2 VF1 VF2 Ca1 Ca2 Ca3 Ca4 Ca5
1 2 4 4 1 2 1 1 2 2 2
−0.783 −0.780 +1.799 +1.790 −0.766 −0.788 +1.793 +1.808 +1.795 +1.790 +1.791
+0.119 +0.122 −0.004 −0.013 +0.136 +0.114 −0.010 +0.005 −0.008 −0.013 −0.012
−0.385 +0.558 +0.006 +0.030 −0.382 +0.552 −0.022 +0.031 +0.008 +0.032 +0.031
can see that the spin polarizations of the nearest atoms are remarkable and the spin densities between VF1 and VF2 almost disappear, reflecting the presence of paired electron distributions. We define the spin of the R center as the sum of three VF spins. For config 1, the R-center spin is +0.731 e, which is close to that of the F-center case (+0.717 e). The VF1 (−0.385 e) and VF2 (+0.558 e) spins have opposite signs, and the arrangement of VF spins has a (↑↓↑) style. The R-center spin for config 2 is equal to +0.722 e and is smaller than that of config 1. +0.552, −0.382, and +0.552 e for VF2−VF1−VF2, respectively, also have a (↑↓↑) style. According to our calculations, we found that the VF spins in the R-center systems are much smaller than those in the F-center system. Considering that VF1−VF2 covalency consumes some part of unpaired electrons localized inside the R center, the smaller spin densities of fluorine vacancies should be reasonable. Next, we calculated the band structures for the R-center systems (see Figure 7). The optical band gaps for configs 1 and
ΔQ(e) is the charge difference between the defective and perfect crystals (QCa = +1.803 e, QF = −0.902 e in perfect CaF2). Spin is the result of the spin difference of electrons with different spin directions (nα − nβ) also in units of e. a
conclude that there are two considerable covalencies in each R center. The charge density maps of the R center for configs 1 and 2 are displayed in Figure 5, also showing that the VF charges are well localized inside the R center and the deformation of the neighboring ions from their spherical shapes is negligible. Because there are three VF in one R center, not all electrons inside the VF can be paired. The localization of the unpaired electron in the R center is clearly shown in the spin density map and the spin polarization of the nearest neighboring atoms is also appreciable, as we can see in Figure 6. From Figure 6, we
Figure 7. Calculated B3PW band structure of the 96-atom supercell modeling the R center in CaF2 for config 1. α and β denote the upand down-spin bands, respectively. The Fermi energy is shifted to 0 eV.
Table 2. Direct Optical Band Gaps (eV) (Γ → Γ) for RCenter Systems in Configs 1 and 2 α-spin
β-spin
gaps
config 1
config 2
α1 → α3 α2 → α3 α1 → CB α2 → CB β1 → β2 β1 → β3 β1 → CB
3.19 2.72 4.48 4.00 2.66 3.22 4.27
3.31 2.64 4.37 3.69 3.00 3.18 4.12
2 are collected in Table 2. Because of the presence of an unpaired electron in the R center, the band structure is polarized. Three fluorine vacancies induce three α- and three βbands in the VB−CB gaps, two α- and one β-bands labeled α1, α2, and β1 are occupied, and one α- and two β-bands labeled α3, β2, and β3 denote empty vacancy levels, as we can see in Figure 7. The (↑↓↑) spin style of the R center mentioned before also
Figure 6. Electron density (left) and spin density (right) contours in CaF2 with R center, being from 0 to 0.1 e/bohr3 with a linear increment of 0.002 e/bohr3 and from −0.01 to 0.01 e/bohr3 with a linear increment of 4 × 10−4 e/bohr3, respectively. The upper and lower maps indicate configs 1 and 2, respectively. 4837
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IV. CONCLUSIONS We applied the first-principles approach based on the hybrid DFT-B3PW scheme for calculations of F and R centers in CaF2. As an extension of our regular F-center study, we performed calculations of F-center transfers. The structure of CaF2 implies that the energetically most favorable F-center transfer should be the position switch between the two nearest neighbor fluorine sites, and this transfer, according to our calculation, requires overcoming an approximately 1.67 eV high energy barrier. During the F-center transfer, the relaxations of surrounding atoms are stronger than that in the regular F-center system, and the effective charges of the F center and F0 decrease. With the position exchanging between the F center and F0, the gap between the defect level and CB in the α-spin state, namely, the α-CB gap, considerably narrows and reduces to approximately 1.5 eV, which is smaller than the corresponding value in the regular F-center system. However, variation of the gap between the VB and F band in the β-spin state, namely, the β-VB gap, is not notable and monotonic during the F-center transfer. Additionally, we investigated the BS effect on the F-center calculations. We suggest that the single, diffuse Gaussian s function with the exponential coefficient of 0.078 could accurately describe the F center in CaF2. The formation energy of the (111) surface F center in CaF2 (7.10 eV) is by 0.77 eV smaller than in the bulk case. The lower formation energy of a surface F center implicates the preference for F centers to locate near the surface. To further understand F-center aggregation, we studied a more complex case, namely, an R center composed of three neighboring F centers, by means of the hybrid B3PW method. Several R-center configurations were investigated, and we found that configs 1 and 2 are the energetically most favorable configurations for 3-VF systems. The association energy of three F centers in config 1 is +0.63 eV, implicating a trend for the F centers to form aggregates. Bond population analysis shows that there are two considerable covalencies between the VF1 and VF2 in each R center, whereas they are much weaker than in the M-center case. This weaker covalency in the R-center cases can be explained by the fact that the electrons localized on VF1 and VF2 are not completely paired. Because inside the R center there are three well localized electrons, the spin density of the R center and spin polarization of neighboring atoms are appreciable. The R-center spin has a (↑↓↑)-style arrangement. The band structure of the R center indicates that three defect levels are induced by three fluorine vacancies in the gaps between the VB and CB for both α- and β-spin band structures. Several complex electron transition ways were collected as follows: α1 → α3, α2 → α3, α1 → CB, α2 → CB, β1 → β2, β1 → β3, and β1 → CB, classified into F- and M-like transitions. The analysis of DOS calculations clearly reveals that the VF2-s orbitals form the occupied α1 and α2 bands and the unoccupied β2 and β3 bands and the VF1-s orbitals do the major contribution to the occupied β1 band and the empty α3 level.
demonstrates that there should be two and one occupied defect bands in the α and β states, respectively, being in accordance with the band structure calculations. According to the selection rules, the α → β transition is forbidden, and we can summarize that the electron transitions in the R-center systems have the following possibilities: α1 → α3, α2 → α3, α1 → CB, α2 → CB, β1 → β2, β1 → β3, and β1 → CB, and the corresponding values are listed in Table 2. We may classify these electron-transition ways into two categories, the four ways of α1 → α3, α2 → α3, β1 → β2, and β1 → β3 belong to the M-like optical absorption, namely, the transition from the defect occupied band to a defect unoccupied level, similar to the electron-transition way in M-center systems, and the three ways of α1 → CB, α2 → CB, and β 1 → CB express the F-like optical absorption, corresponding to the transition from the defect occupied state to CB. Here, we can consider an R center as a combination of two M centers having one common VF or of three F centers; therefore, the two M centers contribute to the four M-like optical absorptions, and the three F centers introduce the three F-like electron transitions. Thus, from Table 2, we can see that the M- and F-like gaps are comparable to the relevant M- and F-center gaps, respectively, despite that the M-like gaps are larger than the M-center gap of 2.22 eV. To further study the electronic structure and the electron transitions in an R-center system, we calculated the density of states (DOS) of the R-center system. The total and projected DOS of R centers in CaF2 are displayed in Figure 8. We can
Figure 8. Total and partial density of states (DOS) for the R-center in config 1. α and β denote the up- and down-spin states, respectively. The Fermi energy is shifted to 0 eV.
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conclude that the occupied bands labeled α1 and α2 and the unoccupied bands labeled β2 and β3 mainly consist of two VF2-s orbitals, and the VF1-s orbitals do the major contribution to the occupied band named β1 and the empty level named α3, which is in agreement with the former spin density discussion regarding the R-center spin arrangement. Finally, our DOS plot for the β1 band shows that there is also a small contribution from the VF2-s orbitals.
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ACKNOWLEDGMENTS H.S. was supported by NSFC Grant No. 11004008. R.I.E. was supported by ESF Grant No. 2009/0202/1DP/ 1.1.1.2.0/APIA/VIAA/141. 4838
dx.doi.org/10.1021/jp208845w | J. Phys. Chem. C 2012, 116, 4832−4839
The Journal of Physical Chemistry C
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Article
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dx.doi.org/10.1021/jp208845w | J. Phys. Chem. C 2012, 116, 4832−4839
