CdXO3 (X = C, Si, Ge, Sn, Pb) electronic band structures

Chemical Physics Letters 480 (2009) 273–277

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CdXO3 (X = C, Si, Ge, Sn, Pb) electronic band structures C.A. Barboza a, J.M. Henriques a, E.L. Albuquerque a, E.W.S. Caetano b,*, V.N. Freire c, J.A.P. da Costa c a

Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, 59072-900 Natal, Rio Grande do Norte, Brazil Instituto Federal de Educação, Ciência e Tecnologia do Ceará, 60040-531 Fortaleza, Ceará, Brazil c Departamento de Física, Universidade Federal do Ceará, Centro de Ciências, Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Ceará, Brazil b

a r t i c l e

i n f o

Article history: Received 2 July 2009 In final form 13 September 2009 Available online 16 September 2009

a b s t r a c t Electronic properties for a set of CdXO3 (X = C, Si, Ge, Sn, Pb) crystals were investigated using the density functional theory (DFT) formalism considering both the local density and generalized gradient approximations, LDA and GGA, respectively. Hexagonal CdCO3 and triclinic CdSiO3 have indirect main energy band gaps while orthorhombic CdGeO3 and CdSnO3 exhibit direct interband transitions. Orthorhombic CdPbO3 has a very small indirect band gap. The Kohn–Sham minimum electronic band gap oscillates as a function of the X ns level, changing from 2.94 eV (hexagonal CdCO3 ; LDAÞ to 0.012 eV (orthorhombic CdPbO3 ; LDA). Ó 2009 Elsevier B.V. All rights reserved.

CdXO3 (X = C, Si, Ge, Sn, Pb) crystals exhibit several polymorphs and have a wide range of applications. Starting with CdCO3 , nanoparticles were synthesized through the sonochemical method [1], and shape-controlled hydrothermal synthesis of single-crystalline hexagonal CdCO3 nanowires, nanobelts, nanorolls, and other onedimensional nanostructures was achieved [2], with the possibility of transforming them into oriented nanoporous CdO. An aragonitetype phase was also found for CdCO3 by Liu and Lin [3]. Cadmium silicate (CdSiO3 ), on the other hand, is used as a host matrix for rare-earth doping, allowing for the production of multicolored long-lasting phosphorescence materials [4]. X-ray diffraction data for CdSiO3 samples revealed a phase structure similar to that of pseudowollastonite CaSiO3 (JCPDS 35–0810). A survey on the reported unit-cell parameters and space groups for pseudowollastonite CaSiO3 published by Yang and Prewitt [5] mentions several works favoring either monoclinic or triclinic symmetries. Due to the resemblance between the pseudowollastonite CaSiO3 and CdSiO3 , Lei et al. [6] have argued that the crystal structure of the latter is expected to be an one-dimensional chain of edge-sharing SiO4 tetrahedra. Cadmium germanate ðCdGeO3 Þ at room temperature and atmospheric pressure has a complex structure resembling the pyroxenoids. Four CdGeO3 polymorphs in the temperature range from 600 °C to 1200 °C and varying pressure up to 12 GPa were found by Susaki [7]. The pyroxenoid phase ðCdGeO3 IÞ is transformed successively into garnet ðCdGeO3 IIÞ, ilmenite (CdGeO3 III, hexagonal) and perovskite (CdGeO3 IV, orthorhombic with increasing * Corresponding author. Address: Instituto Federal de Educação, Ciência e Tecnologia do Ceará, Av. 13 de Maio, 2081, Benfica, 60040-531 Fortaleza, Ceará, Brazil. Fax: +55 85 3307 3711. E-mail address: [email protected] (E.W.S. Caetano). 0009-2614/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2009.09.035

pressure. The orthorhombic perovskite phase has Pbnm symmetry, with each germanium (coordination number IV) atom connected to six oxygen atoms, forming a tridimensional array of GeO6 tilted octahedra with cadmium atoms inserted between them [8]. In order to understand the mechanisms behind the stabilization of the various perovskite crystals, CdGeO3 , CaGeO3 , CaSiO3 , MgSiO3 and other ABO3 structures have been investigated [9]. Cadmium stannate ðCdSnO3 Þ is a semiconducting oxide with gas-sensing properties [10–12], and experimental electronic energy band gap of 3.0 eV. Refinement of the CdSnO3 structure confirms that it is a distorted orthorhombic perovskite, isostructural with CaSnO3 and SrSnO3 [13]. By dehydratation of CdPbðOHÞ6 , the double oxide cadmium plumbate CdPbO3 was synthesized, crystallizing in trigonal form with an ilmenite-type structure [14]. Since its parent compound CaPbO3 has also an orthorhombic perovskite phase [15], one can suppose that CdPbO3 crystallizes in a similar fashion. Considering the interest in the CaXO3 (X = C, Si, Ge, Sn, Pb) class of materials [9,15–22], first principles calculations of the structural, electronic, and optical properties were performed by our research group for all the CaCO3 polymorphs (calcite [23], aragonite [24], and vaterite [25]), for triclinic CaSiO3 [26], orthorhombic CaGeO3 [27], CaSnO3 [28], and CaPbO3 [29]. The calculated band structures suggest an overall trend of the minimal band gap energy to decrease, with some oscillation, according to the sequence CaCO3 calcite ð 5:0 eVÞ ! CaSiO3 ð 5:4 eVÞ ! CaGeO3 ð 2:3 eVÞ ! CaSnO3 ð 2:9 eVÞ ! CaPbO3 ð 0:94 eVÞ. Whether this band gap lowering will occur for the CdXO3 class of materials as well still remains an open question due to the lack of similar calculations, with the exception of CdSnO3 , whose calculated electronic band gap for the orthorhombic phase was predicted to be 1.7 eV, a value smaller than the measured gap of 3.0 eV [13]. In this work,

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calculations on the structural and electronic properties of some CdXO3 (X = C, Si, Ge, Sn, Pb) crystals are performed within the density functional theory [30,31] (DFT) approach, following the Kohn– Sham equations [31], allowing us to estimate and compare their main Kohn–Sham electronic band gaps. Due to the availability of X-ray data to build their initial geometries, we sorted out the following crystals for our computations (with references): CdCO3 hexagonal (crystallographic data from ICSD 20181), CdSiO3 triclinic (using the triclinic CaSiO3 structure from ICSD 23567, replacing Ca with Cd), CdGeO3 [7], CdSnO3 [13], and CdPbO3 orthorhombic (using the orthorhombic CaPbO3 crystal from ICSD 87825, replacing Ca with Cd). For all computations we have used the CASTEP code [32]. To improve over the local-density approximation (LDA) for the exchange-correlation (XC) energy [31,33,34], the generalized gradient approximation (GGA) was also taken into account [35,36]. Kohn–Sham electronic wavefunctions were represented using a plane wave basis set with ultrasoft pseudopotentials [38] replacing core electrons. For compounds containing heavy elements (like Pb), relativistic effects are important. It is possible to take into account these relativistic effects through an adequate fitting of the pseudopotential for the heavy species [37], and the CASTEP code generates by default scalar relativistic pseudopotentials. The LDA exchange-correlation functional chosen was the standard parametrization of Perdew and Ceperley [39,40]. For the GGA exchangecorrelation functional we opted for the Perdew–Burke–Ernzerhof (PBE) [41] formula. The calculations proceeded as follows: first, the internal atomic coordinates and unit cell lattice parameters were optimized to a total energy minimum. Following the geometry optimization, the electronic band structures, partial densities of states and optical properties were obtained. Here we present only the results of the band structure calculations, with more detailed data for each CdXO3 crystal being prepared for publication in separate papers. The electronic valence configurations for each 10 atomic species were: Cd 4d 5s2 ; O 2s2 2p4 ; C 2s2 2p2 ; Si 3s2 3p2 ; 10 Ge 4s2 4p2 ; Sn 5s2 5p2 , and Pb 5d 6s2 6p2 . Several tests were carried out varying the plane wave basis set size to ensure well converged structures. We have found that a 600 eV plane wave energy cutoff for the Kohn–Sham wavefunctions and 1500 eV for the charge density were enough to produce good geometries and electronic structures for all crystals, except for CdPbO3 , which required a 700 eV energy cutoff to converge electron eigenenergies. For the sampling used in reciprocal space integrations, we used a Monkhorst–Pack 6  6  4 grid [42] (also following a convergence study). The thresholds to reach an optimal geometry were: total energy variation <0:5  105 eV=atom, maximum force per atom <0.01 eV/Å, maximum atomic displacement <0:5  103 Å, and maximum stress component <0.02 GPa. Self-consistent field steps obeyed a tolerance of 0:5  106 eV in total energy variation. Fig. 1 shows the electronic band structures for hexagonal CdCO3 and triclinic CdSiO3 (both with the same scale along the horizontal

Fig. 1. LDA Kohn–Sham band structures for CdXO3 crystals, with X = C and Si. The Fermi level is set at zero energy.

and vertical directions to help in making comparisons) in the energy range comprising the top of the valence band and the bottom of the conduction band. The Fermi level, defined here as the energy of the highest occupied state at 0 K, was gauged to be zero in all plots. GGA and LDA band structures look very similar in all cases, with the GGA conduction band being well approximated by performing a rigid shift of the LDA conduction band towards smaller energies. Therefore, only LDA band structures were plotted here. For hexagonal CdCO3 one can see that the valence band maximum is at the F point in reciprocal space, and the conduction band minimum appears at the C point, so hexagonal CdCO3 is an indirect band gap material. The same occurs for triclinic CdSiO3 , with main indirect band gap from the Z point (valence band) to the C point (conduction band). The conduction band curve for CdCO3 near its minimum at C displays some degree of asymmetry between the C ! F and C ! Z directions. The dispersion of the uppermost valence band is significant, in contrast with the very small dispersion usually observed for carbonates. For CdCO3 one observes that the highest energy valence bands are formed mainly from O 2p and with a smaller contribution from Cd 4d states (the same is observed for CdSiO3 ). A comparison with previous data from calcite [23] reveals that the O 2p–C 4d hybridization is stronger for orthorhombic CdCO3 than the equivalent O 2p–C 3d hybridization for calcite, which accounts for the large dispersion observed in our results. Hexagonal CdCO3 also has a valence band saddle point at C almost 1 eV below the Fermi level. As we switch from X = C to Si in CdXO3 , we see that the bottom of the conduction band curves near their minima are parabolic and apparently isotropic. The valence band curves near their maxima, on the other hand, look very anisotropic in reciprocal space and are very flat for triclinic CdSiO3 . Indeed, effective masses for holes were calculated through a parabolic fit to the valence band near its maximum for triclinic CdSiO3 and orthorhombic CdGeO3 using the same procedure described in Ref. [26]. These effective masses are inversely proportional to the band curvature at a given point in reciprocal space. For CdSiO3 , holes are heavier than 2.5 free electron masses in all cases, with the valence band along the Z ! Q and Z ! F directions in reciprocal space being so flat that their hole effective masses could not be estimated. CdSiO3 hole masses along the Z ! C and Z ! B lines are very anisotropic, with the first being 41% larger than the second. The electronic states at the top of the valence band, for all crystals in Fig. 1, originate mainly from O 2p electronic states, with a small degree of overlap with Cd 4d levels. Deep Cd 4d bands with high density of states occur at 6.6 eV and 6.0 eV for X = C, Si, respectively. Looking now to Fig. 2 (drawn preserving the same scales of Fig. 1), one can see the LDA Kohn–Sham band structures for CdGeO3 , CdSnO3 and CdPbO3 , all in the orthorhombic phase. The CdGeO3 orthorhombic crystal presents a direct main band gap, from the C point in valence band to the C point in conduction band. In the case of orthorhombic CdGeO3 , both the GGA and LDA effective masses are close, and the valence band curve with the heaviest hole joins the C and Z points to each other, with a pronounced degree of anisotropy for all effective masses. The hole masses along the C ! S and C ! Z directions are 0.580 and 3.977 free electron masses, in this order. For orthorhombic CdGeO3 , the top of the valence band also mixes O 2p with Ge 4s levels with a deep Cd 4d band with high density of states 6.3 eV. Valence s levels from Cd and X dominate the bottom of the conduction band for all crystals. CdSnO3 is a direct band gap material with an almost isotropic conduction band curve and an anisotropic uppermost valence band curve at the C point with small dispersion. CdPbO3 , on the other hand, has its LDA conduction band minimum at the C point only 12 meV (3 meV, GGA) above the Fermi level, which reveals a very small band gap for this material. The valence band maximum

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Table 1 Kohn–Sham energy gaps in eV for the main electronic transitions involving valence band (VB) maxima and conduction band (CB) minima for CdXO3 crystals, with X = C, Si, Ge, Sn and Pb. Material

Transition ðVB ! CBÞ

LDA

GGA

CdCO3 (hexagonal) CdSiO3 (triclinic)

F!C C!C Z!C Q !C C!C C!C C!Y C!X C!C C!Y C!C Along C ! Z

2.94 3.85 2.80 2.81 2.85 1.67 4.08 4.21 1.14 4.01 0.033 0.012

2.70 3.71 2.57 2.60 2.64 0.82 3.28 3.39 0.42 3.14 0.027 0.003

CdGeO3 (orthorhombic) CdSnO3 (orthorhombic) CdPbO3 (orthorhombic)

Fig. 2. LDA Kohn–Sham band structures for orthorhombic CdXO3 crystals, with X = Ge, Sn and Pb. The Fermi level is set at zero energy.

occurs along the C ! Z direction. Even if one uses improved methods to estimate the band gap (GW, screened-exchange, exact-exchange, etc.) for orthorhombic CdPbO3 , taking into account the average error found in band gap estimates using DFT (see next paragraph), it is reasonable to assume an energy underestimation of 100% at most within the LDA framework, so the CdPbO3 must be of a few tens of meV at most. Both the CdSnO3 and CdPbO3 crystals have deep bands originated from Cd 4d levels for an energy of approximately 6.0 eV. The top of the valence band is also formed mainly from O 2p states and Pb 6s levels form the basis of the conduction band of CdPbO3 . Deep Pb 5d levels appear at about 17 eV and 14 eV. The energy band gaps involving interband transitions were calculated for all CdXO3 structures and are presented in Table 1, for both LDA and GGA exchange-correlation functionals. One must note, however, that the Kohn–Sham energy band gaps do not provide correct estimates to excitation energies [43] due to the fact that the exact form of the exchange-correlation functional is unknown, with the general trend of LDA and GGA approximations to underestimate the energy gaps by 40% or more. Notwithstanding this shortcoming, some authors point that a rigid shift in the LDA conduction bands is able to produce an acceptable agreement

with the quasi-particle GW approximation, which predicts optical excitation frequencies in semiconductors with a margin of error of 0.1 eV [43–47]. Results with a methodology similar to our own were presented for Ca2 X (X = Si, Ge, Sn, Pb) compounds by Migas et al. [48]. As the GGA band structure presented here looks very similar to the LDA one, we think that our analysis is useful to detect tendencies of behavior for the different CdXO3 energy gaps as we switch the X atom keeping in mind the need to increase both LDA and GGA band gaps by a few eV to achieve an agreement with the true CdXO3 band gaps. A quick look at Table 1 shows that the GGA band gaps are smaller for the same crystal in comparison with the LDA band gaps. This occurs possibly because the GGA approach underestimates the interatomic interactions and leads to larger unit cell parameters and bond lengths, a factor that for CdXO3 compounds seems to affect the electronic band gaps in a consistent way. The opposite is valid for the LDA exchange-correlation, which overestimates the strength of the interatomic forces and produces smaller unit cells in comparison with experimental data. The hexagonal CdCO3 crystal has an LDA (GGA) indirect band gap of 2.94 eV (2.70 eV) involving a F ! C transition from valence to conduction band. The direct C ! C transition is of 3.85 eV (3.71 eV, GGA) in the LDA approach. Triclinic CdSiO3 has two indirect and very close band gaps also: 2.80 eV (2.57 eV) and 2.81 eV (2.60 eV) using the LDA (GGA) functional involving Z ! C and Q ! C excitation transitions. The LDA direct energy gap C ! C is also close to the indirect gap, being 2.85 eV. The same figure for the GGA functional is 2.64 eV. Cadmium germanate, cadmium stannate, and cadmium plumbate, all orthorhombic, have direct band gaps and similar features in their conduction band structures, in particular, they have the same sequence of maxima and minima in the lowest conduction band: minima at C, Y, along the line connecting the S and T points and maxima at R, S and T. The conduction band lowest energy curve also meets some ‘bifurcations’ along the S ! X ! C and T ! Z ! C paths for the orthorhombic polymorphs. The direct energy gaps of CdGeO3 are 1.67 eV (LDA) and 0.82 eV (GGA), while the secondary C ! Y transition correspond to energies of 4.21 eV (LDA) and 3.39 eV (GGA). For CdSnO3 , the figures are 1.14 eV and 0.42 eV for the direct LDA and GGA gaps, respectively, and 4.01 eV and 3.14 eV for the indirect C ! Y energy gap, LDA and GGA in this order. CdPbO3 , as mentioned before, has a very small indirect band gap of only 12 meV (LDA) or 3 meV (GGA). Its direct band gaps are 33 meV (LDA) and 27 meV (GGA). In Fig. 3 we show how the minimum energy gap and the direct energy gap vary when we switch X from C to Si to Ge to Sn as a function of the X ns energy level. The C 2s state has an energy of about (average between LDA and GGA) 13.7 eV while for the Si 3s level we find an energy of about 10.9 eV. For Ge 4s, Sn 5s, and Pb 6s the figures are approximately 11.8 eV, 10.7 eV and

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C point. Another interesting feature is the strong curvature anisotropy in the uppermost valence band for the CdXO3 polymorphs investigated, which brings very anisotropic hole effective mass tensors. The highest valence states have strong contribution from O 2p levels, while the bottom of the conduction band is formed mainly from Cd and Xs valence states. The band gap trends predicted here, one must note, consider only the influence of X cations and do not take into account the different crystal structures of the different CdXO3 crystalline phases (except for X = Ge, Sn, and Pb, all orthorhombic), which should be an important factor. As we increase the atomic mass of the X atom, both the minimum and direct energy gaps decrease always, a behavior distinct from what is observed for CaXO3 compounds, namely an overall trend of energy gap to decrease, but with some oscillation as the X mass gets larger. The energy gaps. however, oscillate as the ns energy level for the X atomic species changes. Fig. 3. Kohn–Sham minimum band gaps (solid squares and circles) and direct band gaps (open squares and circles) for CdCO3 ; CdSiO3 ; CdGeO3 ; CdSnO3 , and CdPbO3 . Squares-solid lines and circles-dotted lines represent LDA and GGA results, respectively.

12.2 eV, in this order. Thus the X ns eigenenergies oscillate as the atomic mass of the species increases, which means that the corresponding energy band gaps also oscillate when the ns energy level changes. The minimum energy gap decreases with atomic mass in a similar fashion for both the LDA and GGA approaches, with the transition in gap energy from Si to Ge being the sharpest. Since the CdXO3 crystals with X = C, Si have different and unique unit cell types (X = C hexagonal and X = Si triclinic) one must say that any change of energy gap behavior in comparison with X = Ge, Sn, Pb is partially (not completely) due to the presence of distinct atomic species binding with oxygen. The direct band gap also displays a consistent decrease as the mass of X gets larger. So we can contrast the energy gap behavior of CdXO3 for distinct X atoms with the CaXO3 compounds already mentioned, which showed an overall trend of energy gap decrease with larger atomic mass, but with some degree of oscillation: from CaCO3 calcite to triclinic CaSiO3 the minimum energy gap increases from 5.0 eV to 5.4 eV (LDA values), then decreases to 2.3 eV from triclinic CaSiO3 to orthorhombic CaGeO3 , again increases to 2.9 eV for orthorhombic CaSnO3 and then decreases to 0.94 eV for orthorhombic CaPbO3 . For CdXO3 compounds, the bottom of the conduction band originates from Cd and Xs valence states, the Cd 5s states contributing decisively to the formation of the band gap by shaping the dispersion of the lowermost conduction band. However, when one looks to Figs. 1 and 2, one can see that the dispersion at the conduction band minima is not very different for the CdXO3 compounds investigated. Specially for X = Ge, Sn, and Pb, it is easily noted that the decreasing band gap with X goes along with a decrease of the highest valence bands dispersion related with the degree of Cd 4d–O 2p hybridization near the valence band maximum. In summary, we have performed first principles calculations within the density functional theory framework for a series of CdXO3 polymorphs (X = C, Si, Ge, Sn, Pb) obtaining their electronic band structures. Hexagonal CdCO3 has a LDA (GGA) indirect band gap of 2.94 eV (2.70 eV) with valence band maximum at the F point and conduction band minimum at the C point, and a secondary indirect band gap M ! A only 60 meV (80 meV) larger. Triclinic CdSiO3 also exhibits an indirect LDA (GGA) band gap Z ! C of 2.80 eV (2.57 eV) and a very close secondary indirect band gap Q ! C of 2.81 eV (2.60 eV), as well as a direct C ! C transition of 2.85 eV (2.64 eV). Orthorhombic CdGeO3 and CdSnO3 crystals have direct energy gaps within LDA (GGA) of 1.67 eV (0.82 eV) and 1.14 eV (0.42 eV), respectively. CdPbO3 has a very small indirect band gap of 12 meV (3 meV) within the LDA (GGA) approximation. All crystals, therefore, have conduction band minimum at the

Acknowledgements The constructive recommendations made by anonymous referees are thankfully acknowledged. V.N.F., J.A.P.C., and E.L.A. are senior researchers from the Brazilian National Research Council CNPq, and would like to acknowledge the financial support received during the development of this work from the Grants CNPq-CTENERG 504801/2004-0 and CNPq-Rede NanoBioestruturas 555183/2005-0. J.M.H. was sponsored by a graduate fellowship from the Brazilian National Research Council (CNPq) at the Physics Department of the Universidade Federal do Rio Grande do Norte. EWSC received financial support from CNPq-process number 304338/2007-9. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

A. Askarinejad, A. Morsali, Mater. Lett. 62 (2008) 478. Z. Jia, Y. Tang, L. Luo, B. Li, Cryst. Growth Des. 8 (2008) 2116. L.G. Liu, C.C. Lin, Am. Mineral. 82 (1997) 643. Y. Liu, J. Kuang, B. Lei, C. Shi, J. Mater. Chem. 15 (2005) 4025. H. Yang, C.T. Prewitt, Am. Mineral. 84 (1999) 929. B. Lei, Y. Liu, J. Liu, Z. Ye, C. Shi, J. Solid State Chem. 177 (2004) 1333. J.-I. Susaki, Phys. Chem. Miner. 16 (1989) 634. M. Nagai, A. Navrotsky, Phys. Chem. Miner. 14 (1987) 435. B. Magyari-Köpe, L. Vitos, B. Johansson, J. Kollár, Phys. Rev. B 66 (2002) 092103. W. Xing-Hui, W. Yu-DE, Li Yan-Feng, Zhou Zhen-Lai, Mater. Chem. Phys. 77 (2002) 588. Y.-D. Wang, X.-H. Wu, Z.-L. Zhou, Y.-F. Li, Sens. Actuators B 92 (2003) 186. C. Xiangfeng, C. Zhiming, Sens. Actuators B 98 (2004) 215. H. Mizoguchi, H.W. Eng, P.M. Woodward, Inorg. Chem. 43 (2004) 1667. C. Levy-Clément, I. Morgenstern-Badarau, A. Michel, Mater. Rec. Bull. 7 (1972) 35. A. Yamamoto, N.R. Khasanova, F. Izumi, X.-J. Wu, T. Kamiyama, S. Torii, S. Tajima, Chem. Mater. 11 (1999) 747. L.G. Berry, B. Mason, Mineralogy – Concepts Descriptions and Determinations, W.H. Freeman and Company, San Francisco, 1959. C.T. Dervos, J.A. Mergos, A.A. Iosifides, Mater. Lett. 59 (2005) 2042. P. Westbroek, F. Marin, Nature 392 (1998) 861. H. Liao, H. Mutvei, M. Sjöström, L. Hammarström, J. Li, Biomaterials 21 (2001) 457. G. Atlan, O. Delattre, S. Berland, A. LeFaou, G. Nabias, D. Cot, E. Lopez, Biomaterials 20 (1999) 1017. N. Sharma, K.M. Shaju, G.V. Subb Rao, B.V.R. Chowdari, Electrochem. Commun. 4 (2002) 947. Z. Liu, Y. Liu, Mater. Chem. Phys. 93 (2005) 129. S.K. Medeiros, E.L. Albuquerque, F.F. Maia Jr., E.W.S. Caetano, V.N. Freire, J. Phys. D: Appl. Phys. 40 (2007) 5747. S.K. Medeiros, E.L. Albuquerque, F.F. Maia Jr., E.W.S. Caetano, V.N. Freire, Chem. Phys. Lett. 430 (2006) 293. S.K. Medeiros, E.L. Albuquerque, F.F. Maia Jr., E.W.S. Caetano, V.N. Freire, Chem. Phys. Lett. 435 (2007) 59. J.M. Henriques, E.W.S. Caetano, V.N. Freire, J.A.P. da Costa, E.L. Albuquerque, Chem. Phys. Lett. 427 (2006) 113. J.M. Henriques, E.W.S. Caetano, V.N. Freire, J.A.P. da Costa, E.L. Albuquerque, J. Solid State Chem. 180 (2007) 974. J.M. Henriques, E.W.S. Caetano, V.N. Freire, J.A.P. da Costa, E.L. Albuquerque, J. Phys.: Condens. Matter 19 (2007) 106214. J.M. Henriques, C.A. Barboza, E.L. Albuquerque, E.W.S. Caetano, V.N. Freire, J.A.P. da Costa, J. Phys. D: Appl. Phys. 41 (2008) 065405.

C.A. Barboza et al. / Chemical Physics Letters 480 (2009) 273–277 [30] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864. [31] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. [32] M.D. Segall, P.L.D. Lindan, M.J. Probert, C.J. Pickard, P.J. Hasnip, S.J. Clark, M.C. Payne, J. Phys.: Condens. Matter 14 (2002) 2717. [33] R.O. Jones, O. Gunnarsson, Rev. Mod. Phys. 61 (1989) 689. [34] O.V. Gritsenko, P.R. Schipper, E.J. Baerends, J. Chem. Phys. 107 (1997) 5007. [35] A. Dal Corso, A. Pasquarello, A. Baldereschi, R. Car, Phys. Rev. B 53 (1996) 1180. [36] M. Fuchs, M. Bockstedte, E. Pehlke, M. Scheffler, Phys. Rev. B 57 (1998) 2134. [37] P. Schwerdtfeger, J.R. Brown, J.K. Laerdahl, H. Stoll, J. Chem. Phys. 113 (2000) 7110. [38] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [39] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048.

277

[40] D.M. Ceperley, B.J. Alder, Phys. Rev. Lett. 45 (1980) 566. [41] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [42] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [43] R.W. Godby, M. Schlüter, L.J. Sham, Phys. Rev. B 37 (1988) 10159. [44] Z.H. Levine, D.C. Allan, Phys. Rev. B 43 (1991) 4187. [45] U. Schönberger, F. Aryasetiawan, Phys. Rev. B 52 (1995) 8788. [46] S.Q. Wang, H.Q. Ye, J. Phys.: Condens. Matter 15 (2003) L197. [47] D.I. Bilc, R. Orlando, R. Shaltaf, G.-M. Rignanese, J. Íñiguez, Ph. Ghosez, Phys. Rev. B 77 (2008) 165107. [48] D.B. Migas, L. Miglio, V.L. Shaposhnikov, V.E. Borisenko, Phys. Rev. B 67 (2003) 205203.