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Electronic calculations on fluorides and oxides of Zr, Hf and Th
Computational Materials Science 18 (2000) 193±198
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Electronic calculations on ¯uorides and oxides of Zr, Hf and Th J.F. Rivas-Silva a,*, J.S. Durand-Nicono a, M. Berrondo b a
Instituto de Fõsica, Benem erita Universidad Aut onoma de Puebla, Apartado Postal J-48, 72570 Puebla, Puebla, Mexico b Department of Physics and Astronomy, Brigham Young University, Provo, UT 84602, USA Accepted 12 May 1999
Abstract The band gaps of the oxides and ¯uorides of zirconium, hafnium and thorium are calculated by means of two quantum chemical methods. Through the ®rst the gap is estimated as a one-electron energy given by the HUMO± LOMO splitting, while through the second it is obtained as the energy dierence between electronic potentials of crystal clusters computed at their experimental con®guration. Doping eects for these compounds are also analyzed via substitutional impurities on Pr4 sites. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Electronic structure; Ab initio methods; Semi-empirical calculations; Optical properties of materials; Detection of radiation
1. Introduction In this work, we present some theoretical calculations performed on zirconium, hafnium and thorium ¯uorides and oxides, aimed at exploring optical properties related to their capability as gamma radiation detectors, as speci®cally induced by praseodymium doping. After the works of Lempicki and Wojtowicz [1] and Blasse [2] some basic rules associated with potentially good radiation detectors can be outlined: 1. By the nature of the common photomultipliers currently in use, the emitted photon should be on the visible frequency range.
* Corresponding author. Tel.: +52-22-45 76 45; fax: +52-2244 89 47. E-mail address: [email protected] (J.F. Rivas-Silva).
2. To allow an ecient outgoing path of the photon across the crystal this ought to be transparent precisely throughout the visible frequency range. 3. The emission of the photon must be a fast enough process so as to make the counting of the arrival events ecient. 4. The material must present a high scattering section to the incoming particles. An essential feature to be considered in these materials is the value of the forbidden band gap, Eg . On one hand, it plays a key role in rule (2) above, where quantitatively Eg should be 3 eV or larger to satisfy the associated requirement. On the other hand, the band gap enters the expression for the eciency accomplished at the ®rst stage of the detection process, i.e. when the arriving radiation produces all the primary defects and ionization, in addition to energy thermalization. At this stage a number of secondary electrons and holes are
0927-0256/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 0 ) 0 0 0 9 5 - 1
194
J.F. Rivas-Silva et al. / Computational Materials Science 18 (2000) 193±198
produced which clear the way for transferring energy through the material until an emitting center is reached (second stage). The process continues upon excitation of this center which subsequently emits the counting photon (third stage). An eciency coecient for each of the involved stages is given according to the performance displayed along the detection process. The total eciency amounts to the product of the three individual coecients. The eciency of the ®rst stage is proportional to the number of secondary electrons and holes, being the maximum value of this number inversely proportional to the gap [3]. It is thus desirable that the latter be as small as possible, but not so small that the associated frequency approaches the visible range. 2. Method and Model Several methods to theoretically calculate a number of properties in solids, such as forbidden band gaps, have been addressed in the literature [4]. However, materials whose complexity stems from possessing dierent atoms per unit cell, where some may be heavy elements or span open shell systems, confronts us with some diculties. In particular, these often emerge due to the slow convergence of the used plane wave or Bloch function expansions. In fact, no precise methodologies applicable to complex cases are available at present. We thus here propose two alternative models to deal with systems that present challenging features. Through those models approximate forbidden band gaps are calculated for the materials mentioned in the previous section. However, for some of these materials better results ± compared to the present ®ndings ± have previously been reported [5]. Through the proposed methods we treat all the materials here considered in a uni®ed manner, where the performance accomplished upon the study of the more dicult cases is analyzed. The ab initio perturbed ion method (aiPI) [6] is based on the theory of electronic separability [7]. The crystalline solid is assumed to be separable in ionic constituents, where each is considered coupled to the crystal lattice through a quantum me-
chanical ion±lattice interaction that includes exchange and correlation eects. By imposing strong orthogonality conditions on the ionic wave functions it is possible to de®ne a total energy that consists of added contributions of each ion. The ionic wave function associated with each constituent is self-consistently obtained in the ®eld of all remaining frozen ions. Equivalently, a complete self consistent ®eld (SCF) atomic process is performed for the selected ion, taking into account in its hamiltonian the interactions due to the rest of the ions. By selecting each non-equivalent ion of the crystal in a stepwise optimization, an iterative procedure is carried out until a global convergence of all wave functions is attained. Through this method energy levels and eective energy of each constituent are obtained. It also allows to deal with the problem of a cluster of atoms introduced in a pure crystal, where an eective energy for the cluster embedded in the sea of ionic constituents within the crystal can be found. This versatility is due to the fact that the solution of the associated quantum equations is carried out in the real space, being an advantage over similar methods working in the reciprocal space [4]. Although the method has been successfully applied to the analysis of dierent materials [8], it was formally developed for strictly ionic crystals. A standard input for the utilized program contains information of the constituents and crystalline data of the compounds. The calculation of properties in a solid can be performed by regarding the latter as a large molecule, where suitable boundary conditions are introduced to represent the crystalline structure involved. Although this goes beyond the viewpoint adopted in solid state physics methods, a reasonable description has previously been accomplished by applying the latter approach to alkaline halide materials [9]. The present results were also obtained along similar lines, albeit the models adopted for the embedding environment were somewhat simpli®ed. Because of the large size of the systems here analyzed, and with the purpose of making a comparison with an ab initio technique ± such as the aiPI ± a semiempirical method was used. The latter corresponds to the relativistically parametrized extended H uckel (REX) [10], which
J.F. Rivas-Silva et al. / Computational Materials Science 18 (2000) 193±198
represents a reliable quantum mechanical method, applicable to a wide variety of molecular systems. For large atomic clusters possessing heavy elements, the latter possibly represents the only computational method with manageable applications. Test calculations were attempted by means of the program package GAUSSIAN, although the size and features of the molecular systems were found to exceed its capability limits.
3. General properties of the analyzed materials In the radiation detection ®eld it is of primary interest to study the Pr4 ion as a substitutional impurity in oxides and ¯uorides containing elements of group IVB. In atomic Pr4 the transition 4f 1 5d0 ! 4f 0 5d1 is dipole allowed, and when this atom is embedded in a Th- or Hf-crystal the doped compound may accordingly represent a gamma radiation detector of potentially good eciency. The ¯uorides and oxides of Zr-, Hf- and Th-crystals at room temperature possess the following symmetries: monoclinic, I2/c (ZrF4 , HfF4 , ThF4 ) cubic, Fm3m HfO2 ; ThO2 , and monoclinic, P21/c ZrO2 . Some of these crystals are of current
195
interest due to their optical properties HfO2 as thin ®lms, and are also of practical interest in the nuclear industry ThF4 [11].
4. Calculations and results In order to use the REX method a cluster model approach was followed. We modeled each crystal by taking three shells of atoms at experimental geometries around a central metal ion. The size of the clusters varied in the range 21±26 atoms. This model is illustrated in Fig. 1. The band gap Eg was calculated in the Koopman's sense: the energy dierence given by the highest occupied molecular orbital (HOMO)±lowest unoccupied molecular orbital (LUMO) splitting. In an extended H uckel scheme the o-diagonal matrix elements of the energy are proportional to the overlap between the corresponding molecular states, and to the average of the atomic ionization energies of such states. They are proportional through the so-called Wolfsberg±Helmholtz constant, which has been empirically estimated in many cases [10]. We used the standard value k 1:75, together with the atomic valence basis set within the program. It is
Fig. 1. (a) Model cluster for the REX calculations on ZrF4 . Bonds are only for illustration. (b) Model cluster for the REX calculations on ThO2 .
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J.F. Rivas-Silva et al. / Computational Materials Science 18 (2000) 193±198
Table 1 Approximate band gaps of the studied crystalsa Crystal
PI-1
PI-2
REX
Experimental
Reference
ZrO2 HfO2 ThO2 ZrF4 HfF4 ThF4
1 1 1 7.77 6.11 7.2
7.31 6.44 3.2 14.78 14.65 11.54
7.86 8.23 4.5 11.53 12.78 7.6
4.7, 4.51 5.38 ÿ ÿ ÿ 4.96
[7,17] [7]
a
[7]
All the entries are in eV.
rather remarkable to note how the method qualitatively reproduces the approximate gaps obtained in the alkali±halide crystals [9]. In the case of thorium crystals, a 5f basis was added in order to improve upon the results obtained by employing the implicit basis 6d7s, which led to values of the gap somewhat too high. The introduction of this basis clearly shows the important role played by the f orbital to analyze the electronic properties of these materials. For the aiPI method, a spherical radius of 15 a.u. for the shell surrounding the central atom was taken, at which distance quantum mechanical interactions with the remaining ions are assumed to occur. For the corresponding classical interactions, a radial distance of 10 a.u. was assumed. The atomic SCF program resorts to Slater basis functions as optimized by Clementi and Roetti [12] and McLean and McLean [13]. A 5f basis was added to the Th atom basis set, as it was done with the REX method. Our modeling closely follows the procedure outlined by Aguado et al. [14] to perform the calculation of the band gap. Two dierent approaches were considered: 1. A small cluster consisting of a positive plus a negative ion of the same class than hosts was built within a pure crystal, like as an impurity. A nearest-neighbor pair of ions was selected for it, preserving the normal geometry of the crystal. This was done after attaining convergence on the pure crystal alone. The small cluster was then embedded in the crystal where an electronic transition from one ion to the other is considered, i.e. M4 Fÿ1 M4 O2ÿ ! M3 F0 M3 O1ÿ . The band gap is thus calculated as the dierence between the eective cluster energies in both states. This transition hence resembles an electron transfer of that in ionic
solids would expectedly occur between anions and cations in the excitation. 2. A simple approximation is made in the Koopman's sense where the band gap amounts to the dierence between the metal ion LUMO and the gas ion HOMO, and where values of the atomic orbitals obtained through an aiPI calculation on the pure crystal are used. The results are compiled in Table 1. Besides obtaining the gap, we also performed a calculation through an REX method on the same crystals but including a substitutional impurity. Accordingly, the lanthanide ion is simply added at the center of the cluster. In Schemes 1 and 2 we illustrate the energy band diagram for the doped crystals with Pr4 , as inferred from the REX calculation. In Scheme 1 typical results for the oxides and ¯uorides of Zr and Hf are depicted, while in Scheme 2 those corresponding to the thorium crystals are shown, where the metal 5f level can be seen. In order to apply the aiPI method to the latter systems ± as in the pure crystal ± the most likely
Scheme 1. Typical band energy diagram for the oxides and ¯uorites of Zr and Hf.
J.F. Rivas-Silva et al. / Computational Materials Science 18 (2000) 193±198
Scheme 2. Band energy diagram for the thorium oxide and ¯uorite calculated with REX, energy in eV.
electronic transition must be known. Unfortunately, this is not possible due to the high intrinsic correlation arising from the f electrons in these systems. A more adequate molecular treatment will be the subject of future investigation. The present results represent a ®rst-step toward improvement where we use the implicit correlation taken into account in the empirical ionization energies involved in the REX approach.
5. Discussion and conclusions The band gaps of the crystal structures here calculated yield in general the right trends as compared to the experimental values. Furthermore, unlike standard methods in solid state physics, the calculations are advantageously inexpensive. For oxides the results obtained through REX method are more accurate than those found by means of the aiPI method, while for ¯uorides rather wide band gaps are predicted by the former method. On the other hand, the aiPI method in modality I yields better results for the ¯uorides, whereas in modality II it improves upon the gaps for the oxides through Koopman's approximation. This can probably be explained if we take into
197
account the strong covalency present in the oxides, as compared to the more ionic character of the ¯uorides. As a consequence, for the latter, a large electronic relaxation can thus be expected in the excitation process in comparison to the oxides. The trend found in these, possessing lower gaps than the ¯uorides, is in line with the theory of the nephelauxetic eect [15]. Interestingly, the addition of a valence 5f basis in the REX calculation for thorium-containing compounds gives better results than when using only the implicit 6d7s basis. For praseodymium-doped crystals, in the Zr and Hf cases, typically ± and remarkably ± the occupied 4f level of Pr4 remains within the forbidden gap of the doped crystals, whereas the unoccupied 5d orbital falls in the conduction band. This means that during the absorption process a 4f electron is promoted to the valence 4d or 5d orbitals of the host metal, Zr- and Hf-crystals, respectively, where such levels constitute vacancies in the ionic state (4+) of these elements ± instead to the 5d level of the praseodymium ion. Hence, as far as the present models are concerned, the excited states can be excitonic or conductiontype and the combined eects resulting thereof makes it very unlikely that the transition follow the selection rules known for the atomic case. In analysis based on schemes such as post-Hartree± Fock, or inclusion of other interactions, the trends here found might substantially change. Meanwhile, the results for the thorium case suggests that the excitation for the doped crystals proceeds from the non-metal 2p level to the 5f level of the metal ion. For the thorium calculation we used the 5f Slater basis from McLean [13] and an energy orbital selected by comparison with the Th4 ion run in vacuo. The qualitative result might seem unusual since the f levels typically yield intrinsic contributions on ground states, while not so on excited states. This feature was checked by performing a calculation of a small cluster of atoms consisting of one central oxygen and eight Th atoms by means of GAUSSIAN, with the basis and pseudopotential parameters of Ermler et al. [16]. Although the obtained absorption values for the doped crystals were not in general satisfactory, the qualitative trends were reasonably consistent. The band diagram for the thorium-doped crystals
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J.F. Rivas-Silva et al. / Computational Materials Science 18 (2000) 193±198
indicates a sensitive character of the impurity valence and that, under speci®c conditions, the ionic state Pr5 might be induced via an electron transfer to one of the neighboring thorium atoms. Finally, according to the results here presented doping through praseodymium entering as a substitutional ion Pr4 in the analyzed crystals is not likely to be a useful detection device, albeit studies of a more sophisticated nature would be necessary to support or refute this ®nding. Acknowledgements Financial support provided by Sistema Nacional de Investigadores (SEP-Mexico) and provision of funds by CONACyT, Consejo Nacional de Ciencia y Tecnologõa (Mexico), through project 1349-PE, are gratefully acknowledged. References [1] A. Lempicki, A.J. Wojtowicz, J. Lumin. 60&61 (1994) 942. [2] G. Blasse, J. Lumin. 61&69 (1994) 930.
[3] J. Robbins, J. Electrochem. Soc. 127 (1980) 2694. [4] R. Dovesi, V. R. Saunders, C. Roetti, CRYSTAL 92, User Manual, University of Turin, Italy, 1997 (and references cited therein). [5] F. Zandiehnadem, R.A. Murray, Physica B 150 (1988) 19. [6] V. Lua~ na, L. Pueyo, Phys. Rev. B 39 (1989) 11093. [7] S. Huzinaga, D. McWilliams, A.A. Cantu, Adv. Quantum Chem. 7 (1973) 187. [8] A. Beltr an Flors, Ph.D. Thesis, Universitat Jaume I, 1993 (and references cited therein). [9] M. Berrondo, J.F. Rivas-Silva, Int. J. Quantum Chem. 57 (1996) 1115. [10] L.L. Lohr, P. Pyyk o, Chem. Phys. Lett. 62 (1979) 333. [11] Beijing JUBO Photoelec. Tech. Co. Ltd. (Web pages, China, 1997); CERAC, Inc. (Web pages, Wisconsin, USA, 1997). [12] E. Clementi, C. Roetti, Atom Data Nucl. Data Tables 14 (1973) 3. [13] A.D. McLean, R.S. McLean, Atom Data Nucl. Data Tables 26 (1981) 198. [14] A. Aguado, A. Ayuela, J.M. L opez, J. Alonso, J.F. RivasSilva, M. Berrondo, Rev. Mex. Fõs 44 (1998) 550. [15] G. Blasse, B.C. Grabmaier, Luminescent Materials, Springer, Berlin, 1994. [16] W.C. Ermler, R.B. Ross, P.A. Christiansen, Int. J. Quantum Chem. 40 (1991) 829. [17] N. Van Vugt, T. Williams, G. Blasse, J. Inorg. Nucl. Chem. 35 (1973) 2601.
www.elsevier.com/locate/commatsci
Electronic calculations on ¯uorides and oxides of Zr, Hf and Th J.F. Rivas-Silva a,*, J.S. Durand-Nicono a, M. Berrondo b a
Instituto de Fõsica, Benem erita Universidad Aut onoma de Puebla, Apartado Postal J-48, 72570 Puebla, Puebla, Mexico b Department of Physics and Astronomy, Brigham Young University, Provo, UT 84602, USA Accepted 12 May 1999
Abstract The band gaps of the oxides and ¯uorides of zirconium, hafnium and thorium are calculated by means of two quantum chemical methods. Through the ®rst the gap is estimated as a one-electron energy given by the HUMO± LOMO splitting, while through the second it is obtained as the energy dierence between electronic potentials of crystal clusters computed at their experimental con®guration. Doping eects for these compounds are also analyzed via substitutional impurities on Pr4 sites. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Electronic structure; Ab initio methods; Semi-empirical calculations; Optical properties of materials; Detection of radiation
1. Introduction In this work, we present some theoretical calculations performed on zirconium, hafnium and thorium ¯uorides and oxides, aimed at exploring optical properties related to their capability as gamma radiation detectors, as speci®cally induced by praseodymium doping. After the works of Lempicki and Wojtowicz [1] and Blasse [2] some basic rules associated with potentially good radiation detectors can be outlined: 1. By the nature of the common photomultipliers currently in use, the emitted photon should be on the visible frequency range.
* Corresponding author. Tel.: +52-22-45 76 45; fax: +52-2244 89 47. E-mail address: [email protected] (J.F. Rivas-Silva).
2. To allow an ecient outgoing path of the photon across the crystal this ought to be transparent precisely throughout the visible frequency range. 3. The emission of the photon must be a fast enough process so as to make the counting of the arrival events ecient. 4. The material must present a high scattering section to the incoming particles. An essential feature to be considered in these materials is the value of the forbidden band gap, Eg . On one hand, it plays a key role in rule (2) above, where quantitatively Eg should be 3 eV or larger to satisfy the associated requirement. On the other hand, the band gap enters the expression for the eciency accomplished at the ®rst stage of the detection process, i.e. when the arriving radiation produces all the primary defects and ionization, in addition to energy thermalization. At this stage a number of secondary electrons and holes are
0927-0256/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 0 ) 0 0 0 9 5 - 1
194
J.F. Rivas-Silva et al. / Computational Materials Science 18 (2000) 193±198
produced which clear the way for transferring energy through the material until an emitting center is reached (second stage). The process continues upon excitation of this center which subsequently emits the counting photon (third stage). An eciency coecient for each of the involved stages is given according to the performance displayed along the detection process. The total eciency amounts to the product of the three individual coecients. The eciency of the ®rst stage is proportional to the number of secondary electrons and holes, being the maximum value of this number inversely proportional to the gap [3]. It is thus desirable that the latter be as small as possible, but not so small that the associated frequency approaches the visible range. 2. Method and Model Several methods to theoretically calculate a number of properties in solids, such as forbidden band gaps, have been addressed in the literature [4]. However, materials whose complexity stems from possessing dierent atoms per unit cell, where some may be heavy elements or span open shell systems, confronts us with some diculties. In particular, these often emerge due to the slow convergence of the used plane wave or Bloch function expansions. In fact, no precise methodologies applicable to complex cases are available at present. We thus here propose two alternative models to deal with systems that present challenging features. Through those models approximate forbidden band gaps are calculated for the materials mentioned in the previous section. However, for some of these materials better results ± compared to the present ®ndings ± have previously been reported [5]. Through the proposed methods we treat all the materials here considered in a uni®ed manner, where the performance accomplished upon the study of the more dicult cases is analyzed. The ab initio perturbed ion method (aiPI) [6] is based on the theory of electronic separability [7]. The crystalline solid is assumed to be separable in ionic constituents, where each is considered coupled to the crystal lattice through a quantum me-
chanical ion±lattice interaction that includes exchange and correlation eects. By imposing strong orthogonality conditions on the ionic wave functions it is possible to de®ne a total energy that consists of added contributions of each ion. The ionic wave function associated with each constituent is self-consistently obtained in the ®eld of all remaining frozen ions. Equivalently, a complete self consistent ®eld (SCF) atomic process is performed for the selected ion, taking into account in its hamiltonian the interactions due to the rest of the ions. By selecting each non-equivalent ion of the crystal in a stepwise optimization, an iterative procedure is carried out until a global convergence of all wave functions is attained. Through this method energy levels and eective energy of each constituent are obtained. It also allows to deal with the problem of a cluster of atoms introduced in a pure crystal, where an eective energy for the cluster embedded in the sea of ionic constituents within the crystal can be found. This versatility is due to the fact that the solution of the associated quantum equations is carried out in the real space, being an advantage over similar methods working in the reciprocal space [4]. Although the method has been successfully applied to the analysis of dierent materials [8], it was formally developed for strictly ionic crystals. A standard input for the utilized program contains information of the constituents and crystalline data of the compounds. The calculation of properties in a solid can be performed by regarding the latter as a large molecule, where suitable boundary conditions are introduced to represent the crystalline structure involved. Although this goes beyond the viewpoint adopted in solid state physics methods, a reasonable description has previously been accomplished by applying the latter approach to alkaline halide materials [9]. The present results were also obtained along similar lines, albeit the models adopted for the embedding environment were somewhat simpli®ed. Because of the large size of the systems here analyzed, and with the purpose of making a comparison with an ab initio technique ± such as the aiPI ± a semiempirical method was used. The latter corresponds to the relativistically parametrized extended H uckel (REX) [10], which
J.F. Rivas-Silva et al. / Computational Materials Science 18 (2000) 193±198
represents a reliable quantum mechanical method, applicable to a wide variety of molecular systems. For large atomic clusters possessing heavy elements, the latter possibly represents the only computational method with manageable applications. Test calculations were attempted by means of the program package GAUSSIAN, although the size and features of the molecular systems were found to exceed its capability limits.
3. General properties of the analyzed materials In the radiation detection ®eld it is of primary interest to study the Pr4 ion as a substitutional impurity in oxides and ¯uorides containing elements of group IVB. In atomic Pr4 the transition 4f 1 5d0 ! 4f 0 5d1 is dipole allowed, and when this atom is embedded in a Th- or Hf-crystal the doped compound may accordingly represent a gamma radiation detector of potentially good eciency. The ¯uorides and oxides of Zr-, Hf- and Th-crystals at room temperature possess the following symmetries: monoclinic, I2/c (ZrF4 , HfF4 , ThF4 ) cubic, Fm3m HfO2 ; ThO2 , and monoclinic, P21/c ZrO2 . Some of these crystals are of current
195
interest due to their optical properties HfO2 as thin ®lms, and are also of practical interest in the nuclear industry ThF4 [11].
4. Calculations and results In order to use the REX method a cluster model approach was followed. We modeled each crystal by taking three shells of atoms at experimental geometries around a central metal ion. The size of the clusters varied in the range 21±26 atoms. This model is illustrated in Fig. 1. The band gap Eg was calculated in the Koopman's sense: the energy dierence given by the highest occupied molecular orbital (HOMO)±lowest unoccupied molecular orbital (LUMO) splitting. In an extended H uckel scheme the o-diagonal matrix elements of the energy are proportional to the overlap between the corresponding molecular states, and to the average of the atomic ionization energies of such states. They are proportional through the so-called Wolfsberg±Helmholtz constant, which has been empirically estimated in many cases [10]. We used the standard value k 1:75, together with the atomic valence basis set within the program. It is
Fig. 1. (a) Model cluster for the REX calculations on ZrF4 . Bonds are only for illustration. (b) Model cluster for the REX calculations on ThO2 .
196
J.F. Rivas-Silva et al. / Computational Materials Science 18 (2000) 193±198
Table 1 Approximate band gaps of the studied crystalsa Crystal
PI-1
PI-2
REX
Experimental
Reference
ZrO2 HfO2 ThO2 ZrF4 HfF4 ThF4
1 1 1 7.77 6.11 7.2
7.31 6.44 3.2 14.78 14.65 11.54
7.86 8.23 4.5 11.53 12.78 7.6
4.7, 4.51 5.38 ÿ ÿ ÿ 4.96
[7,17] [7]
a
[7]
All the entries are in eV.
rather remarkable to note how the method qualitatively reproduces the approximate gaps obtained in the alkali±halide crystals [9]. In the case of thorium crystals, a 5f basis was added in order to improve upon the results obtained by employing the implicit basis 6d7s, which led to values of the gap somewhat too high. The introduction of this basis clearly shows the important role played by the f orbital to analyze the electronic properties of these materials. For the aiPI method, a spherical radius of 15 a.u. for the shell surrounding the central atom was taken, at which distance quantum mechanical interactions with the remaining ions are assumed to occur. For the corresponding classical interactions, a radial distance of 10 a.u. was assumed. The atomic SCF program resorts to Slater basis functions as optimized by Clementi and Roetti [12] and McLean and McLean [13]. A 5f basis was added to the Th atom basis set, as it was done with the REX method. Our modeling closely follows the procedure outlined by Aguado et al. [14] to perform the calculation of the band gap. Two dierent approaches were considered: 1. A small cluster consisting of a positive plus a negative ion of the same class than hosts was built within a pure crystal, like as an impurity. A nearest-neighbor pair of ions was selected for it, preserving the normal geometry of the crystal. This was done after attaining convergence on the pure crystal alone. The small cluster was then embedded in the crystal where an electronic transition from one ion to the other is considered, i.e. M4 Fÿ1 M4 O2ÿ ! M3 F0 M3 O1ÿ . The band gap is thus calculated as the dierence between the eective cluster energies in both states. This transition hence resembles an electron transfer of that in ionic
solids would expectedly occur between anions and cations in the excitation. 2. A simple approximation is made in the Koopman's sense where the band gap amounts to the dierence between the metal ion LUMO and the gas ion HOMO, and where values of the atomic orbitals obtained through an aiPI calculation on the pure crystal are used. The results are compiled in Table 1. Besides obtaining the gap, we also performed a calculation through an REX method on the same crystals but including a substitutional impurity. Accordingly, the lanthanide ion is simply added at the center of the cluster. In Schemes 1 and 2 we illustrate the energy band diagram for the doped crystals with Pr4 , as inferred from the REX calculation. In Scheme 1 typical results for the oxides and ¯uorides of Zr and Hf are depicted, while in Scheme 2 those corresponding to the thorium crystals are shown, where the metal 5f level can be seen. In order to apply the aiPI method to the latter systems ± as in the pure crystal ± the most likely
Scheme 1. Typical band energy diagram for the oxides and ¯uorites of Zr and Hf.
J.F. Rivas-Silva et al. / Computational Materials Science 18 (2000) 193±198
Scheme 2. Band energy diagram for the thorium oxide and ¯uorite calculated with REX, energy in eV.
electronic transition must be known. Unfortunately, this is not possible due to the high intrinsic correlation arising from the f electrons in these systems. A more adequate molecular treatment will be the subject of future investigation. The present results represent a ®rst-step toward improvement where we use the implicit correlation taken into account in the empirical ionization energies involved in the REX approach.
5. Discussion and conclusions The band gaps of the crystal structures here calculated yield in general the right trends as compared to the experimental values. Furthermore, unlike standard methods in solid state physics, the calculations are advantageously inexpensive. For oxides the results obtained through REX method are more accurate than those found by means of the aiPI method, while for ¯uorides rather wide band gaps are predicted by the former method. On the other hand, the aiPI method in modality I yields better results for the ¯uorides, whereas in modality II it improves upon the gaps for the oxides through Koopman's approximation. This can probably be explained if we take into
197
account the strong covalency present in the oxides, as compared to the more ionic character of the ¯uorides. As a consequence, for the latter, a large electronic relaxation can thus be expected in the excitation process in comparison to the oxides. The trend found in these, possessing lower gaps than the ¯uorides, is in line with the theory of the nephelauxetic eect [15]. Interestingly, the addition of a valence 5f basis in the REX calculation for thorium-containing compounds gives better results than when using only the implicit 6d7s basis. For praseodymium-doped crystals, in the Zr and Hf cases, typically ± and remarkably ± the occupied 4f level of Pr4 remains within the forbidden gap of the doped crystals, whereas the unoccupied 5d orbital falls in the conduction band. This means that during the absorption process a 4f electron is promoted to the valence 4d or 5d orbitals of the host metal, Zr- and Hf-crystals, respectively, where such levels constitute vacancies in the ionic state (4+) of these elements ± instead to the 5d level of the praseodymium ion. Hence, as far as the present models are concerned, the excited states can be excitonic or conductiontype and the combined eects resulting thereof makes it very unlikely that the transition follow the selection rules known for the atomic case. In analysis based on schemes such as post-Hartree± Fock, or inclusion of other interactions, the trends here found might substantially change. Meanwhile, the results for the thorium case suggests that the excitation for the doped crystals proceeds from the non-metal 2p level to the 5f level of the metal ion. For the thorium calculation we used the 5f Slater basis from McLean [13] and an energy orbital selected by comparison with the Th4 ion run in vacuo. The qualitative result might seem unusual since the f levels typically yield intrinsic contributions on ground states, while not so on excited states. This feature was checked by performing a calculation of a small cluster of atoms consisting of one central oxygen and eight Th atoms by means of GAUSSIAN, with the basis and pseudopotential parameters of Ermler et al. [16]. Although the obtained absorption values for the doped crystals were not in general satisfactory, the qualitative trends were reasonably consistent. The band diagram for the thorium-doped crystals
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indicates a sensitive character of the impurity valence and that, under speci®c conditions, the ionic state Pr5 might be induced via an electron transfer to one of the neighboring thorium atoms. Finally, according to the results here presented doping through praseodymium entering as a substitutional ion Pr4 in the analyzed crystals is not likely to be a useful detection device, albeit studies of a more sophisticated nature would be necessary to support or refute this ®nding. Acknowledgements Financial support provided by Sistema Nacional de Investigadores (SEP-Mexico) and provision of funds by CONACyT, Consejo Nacional de Ciencia y Tecnologõa (Mexico), through project 1349-PE, are gratefully acknowledged. References [1] A. Lempicki, A.J. Wojtowicz, J. Lumin. 60&61 (1994) 942. [2] G. Blasse, J. Lumin. 61&69 (1994) 930.
[3] J. Robbins, J. Electrochem. Soc. 127 (1980) 2694. [4] R. Dovesi, V. R. Saunders, C. Roetti, CRYSTAL 92, User Manual, University of Turin, Italy, 1997 (and references cited therein). [5] F. Zandiehnadem, R.A. Murray, Physica B 150 (1988) 19. [6] V. Lua~ na, L. Pueyo, Phys. Rev. B 39 (1989) 11093. [7] S. Huzinaga, D. McWilliams, A.A. Cantu, Adv. Quantum Chem. 7 (1973) 187. [8] A. Beltr an Flors, Ph.D. Thesis, Universitat Jaume I, 1993 (and references cited therein). [9] M. Berrondo, J.F. Rivas-Silva, Int. J. Quantum Chem. 57 (1996) 1115. [10] L.L. Lohr, P. Pyyk o, Chem. Phys. Lett. 62 (1979) 333. [11] Beijing JUBO Photoelec. Tech. Co. Ltd. (Web pages, China, 1997); CERAC, Inc. (Web pages, Wisconsin, USA, 1997). [12] E. Clementi, C. Roetti, Atom Data Nucl. Data Tables 14 (1973) 3. [13] A.D. McLean, R.S. McLean, Atom Data Nucl. Data Tables 26 (1981) 198. [14] A. Aguado, A. Ayuela, J.M. L opez, J. Alonso, J.F. RivasSilva, M. Berrondo, Rev. Mex. Fõs 44 (1998) 550. [15] G. Blasse, B.C. Grabmaier, Luminescent Materials, Springer, Berlin, 1994. [16] W.C. Ermler, R.B. Ross, P.A. Christiansen, Int. J. Quantum Chem. 40 (1991) 829. [17] N. Van Vugt, T. Williams, G. Blasse, J. Inorg. Nucl. Chem. 35 (1973) 2601.