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AB INITIO STUDY OF ABSORPTION AND EMISSION PROPERTIES OF BGO AND BSO
PII: S0022-3697(98)00054-7
Pergamon
J. Phys. Chem Solids Vol 59, No. 9, pp. 1627–1631, 1998 0022-3697/98/$ - see front matter 䉷 1998 Elsevier Science Ltd. All rights reserved
AB INITIO STUDY OF ABSORPTION AND EMISSION PROPERTIES OF BGO AND BSO J. F. RIVAS-SILVA a and M. BERRONDO b a
Instituto de Fı´sica ‘‘Luis Rivera Terrazas’’, Beneme´rita Universidad Auto´noma de Puebla, Apartado Postal J-48, Puebla, Pue. 72570, Mexico b Department of Physics and Astronomy, Brigham Young University, Provo, UT 84602, USA (Received 6 June 1997; accepted 25 March 1998) Abstract—Bismuth ortho-germanate (BGO) and bismuth ortho-silicate (BSO) crystals are described within an atomic cluster model. Electronic ground and first excited state for both crystal structures are calculated through unrestricted Hartree–Fock and configuration interaction methods, where the associated cluster geometries are optimized. The theoretical transition energies corresponding to the absorption and emission processes are very revealing. The displacement of one of the oxygen ions away from the bismuth in the excited state, together with the distributed spin density clearly shows that the excitation process cannot be understood as a deformed bismuth ion excitation. Instead, a molecular-like excitation is proposed. 䉷 1998 Elsevier Science Ltd. All rights reserved Keywords: A. optical materials, B. ab initio calculations, C. luminescence
1. INTRODUCTION
There has recently been a renewed interest concerning the study of optical properties of bismuth ortho-germanate, Bi 4Ge 3O 12 (BGO) and bismuth ortho-silicate Bi 4Si 3O 12 (BSO). Aspects not fully understood relating to the nonlinear optical properties of these crystals and the wellknown ability of BGO as a scintillator [1], have an important bearing in various scientific applications involving these compounds. For example, due to its high density, large atomic number and fast decay time, BGO is specifically useful as a g-ray detector [2]. We are thus here aimed at understanding some of the common and distinctive features of BGO and BSO through a theoretical analysis based on a quantum cluster approach. In this work cluster models are utilized to represent BGO and BSO crystal structures. Therefore, selected sets of atoms, or shells, surrounding a central atomic ion in the crystal, are suitably chosen and quantum-mechanically described. In first approximation, the cluster can be treated as being in vacuum, but this represents an oversimplified model for the crystal. Taking into account the full structure theoretically poses a challenging task. A purely ionic crystal can be reasonably modeled as an atomic cluster embedded [3] in a sea of point charges located at the ion positions in the crystal. Through the latter model, we were able to calculate band gaps of alkali halides, as well as Stokes shifts for doped alkali halides [4]. Embedding procedures at a more sophisticated level have been proposed in the literature, such as those including interatomic potentials to describe the ions outside the quantum cluster, or the well-known shell model,
to account for polarization effects [5]. In the present case, BGO and BSO represent semi-ionic rather than ionic crystals, where the Ge–O and Si–O bonds possess a high degree of covalency. We have thus adopted a model where atomic clusters are centered about the Bi ion and the oxygen dangling bonds are saturated with hydrogen atoms [6, 7]. The number of ions included in the cluster is commensurate with the number of electrons and basis functions that can appropriately be handled to represent the crystal structure, and it is also dictated by the computational limits posed by the quantum chemical approach at hand. Typically, ab initio methods perform well for molecular systems involving a few tens of atoms. By using the unrestricted Hartree–Fock (UHF) and configuration interaction (CI) methods for critical geometries, the ground singlet and lowest triplet state cluster energies have been calculated. For both states the structure geometries were optimized by relaxing the position of the bismuth, as well as the first shell of oxygen ions. By letting the atoms move in an independent manner, symmetric and nonsymmetric lattice relaxations are thereby allowed. Such flexibility proved essential for an adequate description of the excited state, responsible for the emission. Once the optimized energies are obtained, absorption and emission properties can be calculated, along with the Stokes shifts for BGO and BSO. In addition, the triplet state relative lifetimes for the clusters were calculated by treating spin–orbit effects in a perturbative fashion [8, 9]. Section 2 includes a description of the cluster models developed and Section 3 describes the specific calculational methods employed. The main conclusions are presented in Section 4.
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J. F. RIVAS-SILVA and M. BERRONDO 2. CLUSTER MODELS
BGO is a cubic crystal possessing ¹ 43m-symmetry with four formula units per unit cell [1]. BSO is a crystal with similar characteristics, containing Si instead of Ge atoms. Both of these compounds are known to be high performance scintillators. BSO yields a higher emission power over BGO by a factor of 3, but one fifth of the light output by comparison, so BGO is a more popular scintillator [10]. In recent investigations reported by the present authors on alkali halides [3, 4], atomic clusters comprising a number of atoms embedded in point-charge shells were used as models. Analyzing the effect stemming from the rest of the crystal on the central cluster would thus be the next natural step. Here, because Bi-germanate and Bisilicate are normally classified as ionic crystals, a first attempt was made accordingly, by proposing such ionic embedding. However, the results obtained thereby were far from satisfactory, leading to an incorrect charge balance throughout the atoms entering the cluster. Considering the rather large concentration of oxygens in the crystals, especially those around the central Bi atom, one may anticipate the relevant role of covalent bonds in these compounds. Effects on the charge distribution due to such covalent properties of oxide ions are discussed for example in Duffy [11]. In the search for a correct handling of the covalent character of oxygens and germaniums (silicons) in the more external part of the clusters, different schemes were tested. The final model was obtained by saturating the dangling bonds of the oxygen atoms, as well as those of the Ge and Si atoms, by attaching hydrogens on each. This was particularly important because if this action is not performed, double and triplet bonds can be created on these atoms, transforming incorrectly their distribution of charge and
Fig. 1. Space filled perspective of the first coordination shell and central Bi ion in the cluster modeling BGO and BSO.
producing a wrong influence over the central part of the cluster. This is specially true for our small size models. Accordingly, the best cluster model arrived at was the following: a Bi central atom sorrounded by six oxygens and the nearest three germaniums (silicons), plus two Cs atoms at the corresponding Bi sites. On top of these, 15 hydrogen atoms were added to saturate the bonds as explained above (see Figs 1 and 2). The resulting cluster net charge is þ 2. Through this model the central Bi þ3 atom is expected to interact strongly with the three nearest-neighbor oxygens. These, in turn, are bonded to the nearest germaniums (silicons), their dangling bonds being saturated by adding three hydrogen atoms to each at the interatomic distances where the actual neighboring atoms should be. This way of saturating dangling bonds has been reported in the literature for other compounds [7]. The other three oxygen atoms surrounding the bismuth were saturated by placing two hydrogen atoms farther out, with a bond length equivalent to that of H–O in a water molecule. This choice of H–O bond length, which is substantially shorter than that between the oxygens and their next neighboring atoms in the actual crystal, should not be a source of spurious effects when analyzing the electronic properties of the cluster. Cesium atoms were added in order to represent the next neighboring bismuth atoms, since using the latter would pose a computationally demanding task, serving also the purpose of balancing forces that would manifest themselves over the oxygens placed around along direction (1,1,1). The choice to use cesium atoms rests on the fact that each represents a simpler atom than bismuth, while exhibiting a similar valence shell. Besides, since these atoms enter the electronic (molecular) Hamiltonian in the form of pseudopotentials, the cluster simulation is not expected to be particularly sensitive to this substitution. By using this model a reasonable representation of the coordination around the central Bi atom is accomplished.
Fig. 2. Perspective of the total cluster model for the ab initio study of the BGO and BSO crystals.
Absorption and emission properties of BGO and BSO
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Table 1. Theoretical geometry (HF approximation) for the central coordinate complex in the BGO and BSO models. Ground state ˚) (distance in A
BGO Bi 01 02 03 04 05 06 BSO Bi 01 02 03 04 05 06
x
y
z
0.0887 ¹ 0.1766 0.5408 2.0198 ¹ 2.3883 ¹ 0.0427 1.2186
0.0981 0.5276 2.0412 ¹ 0.167 1.2047 ¹ 2.3703 0.0248
0.1021 2.0385 ¹ 0.1349 0.5495 ¹ 0.0381 1.1932 ¹ 2.2155
0.1151 ¹ 0.2031 0.6277 2.021 ¹ 2.3512 ¹ 0.0949 1.2868
0.1116 0.6397 2.0277 ¹ 0.1809 1.2229 ¹ 2.3135 ¹ 0.2153
0.1275 2.0333 ¹ 0.2173 0.6622 ¹ 0.0706 1.2198 ¹ 2.1542
3. CALCULATIONAL METHODS AND RESULTS
In the present work ab initio calculations with Hay–Wadt basis sets to describe valence shells were used, including their effective core potentials [12, 13] as given in the Gaussian program package [14]. The latter include relativistic effects, which is an important ingredient for atoms of high atomic number such as bismuth and cesium. Geometries were optimized by varying the central Bi and the six surrounding oxygen atom distances, while the rest of the atom positions were kept fixed. Ground (singlet) and first excited (triplet) states were optimized at the HF level. For the ground state, the calculated ˚ off the geometry of the minimum is less than 0.20 A experimental value. The optimized geometries for both BGO and BSO for the central shell of the clusters are presented in Table 1. Through an HF calculation of the first excited state we obtained the optimized geometries presented in Table 2. DSCF values for the absorption are shown in Table 3. DSCF calculations were performed at the corresponding
minima for the excited states for the emission energies. Very small energy differences were found between the states, less than 1 eV in both cases. However, at the single excitation configuration interaction (CIS) level, calculations performed at the HF-obtained geometries led to the values for emission presented in Table 3. Finding such a good agreement for BSO in contrast to that of BGO was surprising. Energy was thus fully optimized at the CIS level finding a better transition energy for emission of 1.84 eV for BGO. For this reason, this last geometry (very similar to the first) was used in the rest of the calculations. Meanwhile, significant changes at this level of calculation for the BSO case were not detected. The most remarkable result is that the first triplet excited states calculated at their optimal geometries for both crystals show a salient feature reflected in their Mulliken spin densities, which is presented in Table 4. This indicates a strongly delocalized character for this state. For BGO, similar figures are also consistent with a triplet state of delocalized nature. Regarding the excited state geometry, one of the
Table 2. Theoretical geometry (HF approximation) for the central coordinate complex in the BGO and BSO models. First excited state ˚) (distance in A
BGO Bi 01 02 03 04 05 06 BSO Bi 01 02 03 04 05 06
x
y
z
0.0216 ¹ 0.1888 0.5321 2.6909 ¹ 2.503 0.0027 1.3971
¹ 0.0267 0.5301 1.9047 0.7221 1.1768 ¹ 2.4389 ¹ 0.4417
0.0631 2.0056 ¹ 0.3394 0.9246 0.0397 1.1799 ¹ 2.178
0.0044 ¹ 0.1758 0.6423 2.7429 ¹ 2.5680 ¹ 0.0351 1.3387
0.0545 0.6419 1.9872 0.0658 1.1741 ¹ 2.3852 ¹ 0.3365
0.0583 2.0281 ¹ 0.2386 0.3038 0.0721 1.1918 ¹ 2.1568
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J. F. RIVAS-SILVA and M. BERRONDO
Table 3. Absorption and Emission values (in eV) of BGO and BSO. Theoretical results from a CIS. For BGO a CIS optimized geometry is also used. Experimental, Refs. [1, 2, 10] BGO
BSO
Theoretical
Experimental
Theoretical
Experimental
5.89 1.45
4.90 2.40
6.15 2.62
4.20 2.70
Absorption Emission
oxygen atoms is displaced away from the central Bi atom, stretching out the O–Bi distance by roughly 42%. Also, the central bismuth gets slightly shifted towards the center of the cluster. This deformation of the excited state structure is intimately linked to the enhanced Stokes shift displayed by these systems, which has also been found in some AX 3 systems [15]. A surprising difference between BGO and BSO is the energy ordering of the states, attending to the symmetry of the states. This is shown in Table 5. These distinct orderings may be related with the different optical behavior that these two materials exhibit, although a more precise connection calls for a deeper study. In order to estimate the transition time for emission, we used the perturbed ion scheme developed by Valle´e [8], and simplified by us [9] in applications to Tl-doped alkali-halides. This scheme is utilized to calculate transition times involving emission between the excited triplet and ground singlet states. Due to dipole selection rules, a transition occurring between such states is spinforbidden. Some additional interaction must thus be considered to account for a non-vanishing transition probability. In particular, the spin–orbit interaction leads to a mixture of singlet and triplet states, thereby allowing an otherwise forbidden transition. In the quantum-mechanical perturbative approach on which the calculation of transition time for emission is based, the central atom in the cluster is looked upon as being perturbed by the crystal environment surrounding it. The spin–orbit coupling constant enters the calculation as a parameter. However, for complex systems, the latter can be very difficult to obtain, and since it is generally not available, an empirical ansatz is instead utilized to calculate an effective parameter by averaging over the coordination sphere around the central atom, where empirical spin–orbit parameters are assigned to the constituent atoms. For ionic systems this calculational Table 5. Excited state energies (in eV) from a CIS calculation for the BGO and BSO. For the first, the CIS optimized geometry is used. T and S mean triplet and singlet states, respectively Excited state 1 2 3 4
BGO symmetry
BSO symmetry
1.84 T 3.26 S 3.28 T 3.95 S
2.62 T 3.36 T 3.91 S 4.87 S
Table 4. Mulliken spin density for the first excited state of BGOand BSO in the geometries of Table 2
Bi O O O O O O Ge/Si Ge/Si Ge/Si
BGO
BSO
0.848913 ¹ 0.004916 0.002671 1.131611 ¹ 0.002422 ¹ 0.001387 ¹ 0.005482 0.000498 ¹ 0.085175 ¹ 0.003923
0.955852 ¹ 0.012509 ¹ 0.004744 1.058866 0.000159 ¹ 0.000983 ¹ 0.004953 0.002993 ¹ 0.171960 ¹ 0.005789
scheme has proved to yield very good results [9]. The transition time for emission within this scheme has been thus calculated. Accordingly, by selecting the empirical values for the spin–orbit parameter constants for Bi and O that best fit one of the experimental data, lifetimes shown in Table 6 were found. This gives a theoretical ratio (BGO/BSO) of 1/5 between the structures, as compared to the experimental one of 1/3, speaking about the last values. 4. DISCUSSION AND CONCLUSIONS
The results here presented, obtained through quantummechanical methods as applied to the analysis of optical properties of BGO and BSO crystals, provide some insight in materials science research, especially regarding theoretical foundations amenable to the experimental design of radiation detectors, and they put forward a number of interesting aspects: firstly, the feasibility of analyzing electronic structure which was shown to lead to reasonably good values as compared with the experimental data for absorption and emission processes in these materials; secondly, crucial differences in optical properties found in these materials, as inferred from the calculated emitting state lifetimes. In particular, the predicted elongation of one of the oxygens at the excited state optimal geometry provides a physical basis for the experimentally observed Stokes shift in optical spectroscopy. This explanation differs from the usual atomic model [16], assuming that the pronounced Stokes shift of these materials is due to the displacement of the central Bi þ3 toward the center of the tetrahedra formed with the neighboring oxygens. In BSO the central ion is shifted ˚ , whereas the oxygen moves 0.53 A ˚ . The about 0.13 A Table 6. Semi-empirical lifetimes of the excited states of BGO and BSO (in ns). Experimental, Refs [1, 2, 10] BGO Theoretical – 75.0 444.6 586.0
BSO
Experimental
Theoretical
Experimental
4.5 46.0 277.0 –
– 40.1 100.1 –
2.4 26.0 99.0 –
Absorption and emission properties of BGO and BSO
latter is thus expected to be a more crucial feature to sustain sensitive behavior in the crystal than that prompted by the motion of Bi þ3 atoms. The oxygen elongation can also be related to the stabilization of self-trapped excitons in these materials, analogous to the distortion of some atoms in the excited state, as explained by Shluger and Stefanovich [17] in bcristobalite. A detailed analysis of this aspect will be the subject of future investigation. A strong limitation in the present methods is posed by the cluster size when modeling a crystal. The most advanced computational codes currently available for molecular calculations can efficiently handle a few tens of atoms only, and introducing numerous heavy atoms may turn the problem numerically intractable. Clearly, either further progress in these computational techniques ought to be pursued, or else more approximate models for these systems should be applied. The manner in which atoms are saturated within our cluster model, together with the fact of dealing with systems that possess a rather pronounced degree of covalency, probably represents a somewhat unconventional idea for solid state physiscists in the sense that, very often, materials such as BGO, and BSO, are treated as though they were ionic. In particular, when working with oxygen-containing materials, it is essential to bear in mind the implications of a reduced ionic character in their crystal structure. When the computational techniques progress, in the near future, enough to handle bigger clusters in the ab initio approach, it will be possible to check the kind of saturation mentioned. Here, due to the small size of our models, it was fundamental to prevent annoying multiple bonds. Without the use of this saturation, and so permitting the formation of these bonds on the covalent atoms of the boundary, their spurious influence on the central part possibly could become less and less important when the size of the cluster is grown, for example in the case of materials where polarization effects are not important.
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Though the large distortion of the central shell found here for the excited state can be shared for more atoms around if a big cluster is employed, experimental work supports the central localization of this distortion [2,18]. Acknowledgements—One of us (JFRS) gratefully acknowledges financial support provided by SNI (Sistema Nacional de Investigadores, Mexico). This work was partially supported by CONACyT (Mexico) under project 1349 P-E, and by NIH under grant number R01-CA48002-04, subcontract number 4591710, with the Lawrence Berkeley Laboratory. REFERENCES 1. Weber, M. J. and Monchamp, R. R., J. Appl. Phys., 1973, 44, 5495. 2. Weber, M. J., West, A. C., Dujardin, C., Rupp, B., Berrondo, M. and Rivas-Silva, J. F., in Inorganic Scintillators and Their Applications, ed. P. Dorenbos et al. University Press, The Netherlands, 1996, p. 234. 3. Berrondo, M. and Rivas-Silva, J. F., Int. J. Quantum Chem., 1996, 57, 1115. 4. Berrondo, M. and Rivas-Silva, J. F., Int. J. Quantum Chem., 1995, 29, 253. 5. See e.g. the special issue: Catlow, C.R.A. and Stoneham, A.M., ed., J. Chem. Soc. Farad. Trans. II, 1989, 85. 6. Lasaga, A. C., Phys. Chem. Minerals, 1982, 8, 36. 7. Cunha, C., Canuto, S. and Fazzio, A., Phys. Rev., 1993, 48, 17806. 8. Valle´e, O., Minh, N. T. and Chapelle, J., in Spectral Line Shapes, ed. B. Wende. Walter de Gruyter, Berlin, 1981. 9. Rivas-Silva, J. F. and Berrondo, M., J. Phys. Chem. Solids, 1996, 57, 1705. 10. Ishii, M., Kobayashi, M. and Yamaga, I., in Heavy Scintillators for Scientific and Industrial Applications, ed. F. de Notaristefani et al. Editions Frontier, Gif-sur-Yuette, 1993. 11. Duffy, J. A., Bonding, Energy Levels and Bands in Inorganic solids. Longman Sci. and Tech., 1990, p. 156. 12. Wadt, W. R. and Hay, P. J., J. Chem. Phys., 1985, 82, 284. 13. Hay, P. J. and Wadt, W. R., J. Chem. Phys., 1985, 82, 299. 14. GAUSSIAN 92 Programmer’s Reference. Gaussian, Inc., 1992. 15. Berrondo, M., Rivas-Silva, J. F. and Recamier, J. to be published. 16. Timmersmans, C. W. M. and Blasse, G., J. Solid State Chem., 1984, 52, 222. 17. Shluger, A. and Stefanovitch, E., Phys. Rev., 1990, 42, 9664. 18. Dujardin, A. C., private communication.
Pergamon
J. Phys. Chem Solids Vol 59, No. 9, pp. 1627–1631, 1998 0022-3697/98/$ - see front matter 䉷 1998 Elsevier Science Ltd. All rights reserved
AB INITIO STUDY OF ABSORPTION AND EMISSION PROPERTIES OF BGO AND BSO J. F. RIVAS-SILVA a and M. BERRONDO b a
Instituto de Fı´sica ‘‘Luis Rivera Terrazas’’, Beneme´rita Universidad Auto´noma de Puebla, Apartado Postal J-48, Puebla, Pue. 72570, Mexico b Department of Physics and Astronomy, Brigham Young University, Provo, UT 84602, USA (Received 6 June 1997; accepted 25 March 1998) Abstract—Bismuth ortho-germanate (BGO) and bismuth ortho-silicate (BSO) crystals are described within an atomic cluster model. Electronic ground and first excited state for both crystal structures are calculated through unrestricted Hartree–Fock and configuration interaction methods, where the associated cluster geometries are optimized. The theoretical transition energies corresponding to the absorption and emission processes are very revealing. The displacement of one of the oxygen ions away from the bismuth in the excited state, together with the distributed spin density clearly shows that the excitation process cannot be understood as a deformed bismuth ion excitation. Instead, a molecular-like excitation is proposed. 䉷 1998 Elsevier Science Ltd. All rights reserved Keywords: A. optical materials, B. ab initio calculations, C. luminescence
1. INTRODUCTION
There has recently been a renewed interest concerning the study of optical properties of bismuth ortho-germanate, Bi 4Ge 3O 12 (BGO) and bismuth ortho-silicate Bi 4Si 3O 12 (BSO). Aspects not fully understood relating to the nonlinear optical properties of these crystals and the wellknown ability of BGO as a scintillator [1], have an important bearing in various scientific applications involving these compounds. For example, due to its high density, large atomic number and fast decay time, BGO is specifically useful as a g-ray detector [2]. We are thus here aimed at understanding some of the common and distinctive features of BGO and BSO through a theoretical analysis based on a quantum cluster approach. In this work cluster models are utilized to represent BGO and BSO crystal structures. Therefore, selected sets of atoms, or shells, surrounding a central atomic ion in the crystal, are suitably chosen and quantum-mechanically described. In first approximation, the cluster can be treated as being in vacuum, but this represents an oversimplified model for the crystal. Taking into account the full structure theoretically poses a challenging task. A purely ionic crystal can be reasonably modeled as an atomic cluster embedded [3] in a sea of point charges located at the ion positions in the crystal. Through the latter model, we were able to calculate band gaps of alkali halides, as well as Stokes shifts for doped alkali halides [4]. Embedding procedures at a more sophisticated level have been proposed in the literature, such as those including interatomic potentials to describe the ions outside the quantum cluster, or the well-known shell model,
to account for polarization effects [5]. In the present case, BGO and BSO represent semi-ionic rather than ionic crystals, where the Ge–O and Si–O bonds possess a high degree of covalency. We have thus adopted a model where atomic clusters are centered about the Bi ion and the oxygen dangling bonds are saturated with hydrogen atoms [6, 7]. The number of ions included in the cluster is commensurate with the number of electrons and basis functions that can appropriately be handled to represent the crystal structure, and it is also dictated by the computational limits posed by the quantum chemical approach at hand. Typically, ab initio methods perform well for molecular systems involving a few tens of atoms. By using the unrestricted Hartree–Fock (UHF) and configuration interaction (CI) methods for critical geometries, the ground singlet and lowest triplet state cluster energies have been calculated. For both states the structure geometries were optimized by relaxing the position of the bismuth, as well as the first shell of oxygen ions. By letting the atoms move in an independent manner, symmetric and nonsymmetric lattice relaxations are thereby allowed. Such flexibility proved essential for an adequate description of the excited state, responsible for the emission. Once the optimized energies are obtained, absorption and emission properties can be calculated, along with the Stokes shifts for BGO and BSO. In addition, the triplet state relative lifetimes for the clusters were calculated by treating spin–orbit effects in a perturbative fashion [8, 9]. Section 2 includes a description of the cluster models developed and Section 3 describes the specific calculational methods employed. The main conclusions are presented in Section 4.
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J. F. RIVAS-SILVA and M. BERRONDO 2. CLUSTER MODELS
BGO is a cubic crystal possessing ¹ 43m-symmetry with four formula units per unit cell [1]. BSO is a crystal with similar characteristics, containing Si instead of Ge atoms. Both of these compounds are known to be high performance scintillators. BSO yields a higher emission power over BGO by a factor of 3, but one fifth of the light output by comparison, so BGO is a more popular scintillator [10]. In recent investigations reported by the present authors on alkali halides [3, 4], atomic clusters comprising a number of atoms embedded in point-charge shells were used as models. Analyzing the effect stemming from the rest of the crystal on the central cluster would thus be the next natural step. Here, because Bi-germanate and Bisilicate are normally classified as ionic crystals, a first attempt was made accordingly, by proposing such ionic embedding. However, the results obtained thereby were far from satisfactory, leading to an incorrect charge balance throughout the atoms entering the cluster. Considering the rather large concentration of oxygens in the crystals, especially those around the central Bi atom, one may anticipate the relevant role of covalent bonds in these compounds. Effects on the charge distribution due to such covalent properties of oxide ions are discussed for example in Duffy [11]. In the search for a correct handling of the covalent character of oxygens and germaniums (silicons) in the more external part of the clusters, different schemes were tested. The final model was obtained by saturating the dangling bonds of the oxygen atoms, as well as those of the Ge and Si atoms, by attaching hydrogens on each. This was particularly important because if this action is not performed, double and triplet bonds can be created on these atoms, transforming incorrectly their distribution of charge and
Fig. 1. Space filled perspective of the first coordination shell and central Bi ion in the cluster modeling BGO and BSO.
producing a wrong influence over the central part of the cluster. This is specially true for our small size models. Accordingly, the best cluster model arrived at was the following: a Bi central atom sorrounded by six oxygens and the nearest three germaniums (silicons), plus two Cs atoms at the corresponding Bi sites. On top of these, 15 hydrogen atoms were added to saturate the bonds as explained above (see Figs 1 and 2). The resulting cluster net charge is þ 2. Through this model the central Bi þ3 atom is expected to interact strongly with the three nearest-neighbor oxygens. These, in turn, are bonded to the nearest germaniums (silicons), their dangling bonds being saturated by adding three hydrogen atoms to each at the interatomic distances where the actual neighboring atoms should be. This way of saturating dangling bonds has been reported in the literature for other compounds [7]. The other three oxygen atoms surrounding the bismuth were saturated by placing two hydrogen atoms farther out, with a bond length equivalent to that of H–O in a water molecule. This choice of H–O bond length, which is substantially shorter than that between the oxygens and their next neighboring atoms in the actual crystal, should not be a source of spurious effects when analyzing the electronic properties of the cluster. Cesium atoms were added in order to represent the next neighboring bismuth atoms, since using the latter would pose a computationally demanding task, serving also the purpose of balancing forces that would manifest themselves over the oxygens placed around along direction (1,1,1). The choice to use cesium atoms rests on the fact that each represents a simpler atom than bismuth, while exhibiting a similar valence shell. Besides, since these atoms enter the electronic (molecular) Hamiltonian in the form of pseudopotentials, the cluster simulation is not expected to be particularly sensitive to this substitution. By using this model a reasonable representation of the coordination around the central Bi atom is accomplished.
Fig. 2. Perspective of the total cluster model for the ab initio study of the BGO and BSO crystals.
Absorption and emission properties of BGO and BSO
1629
Table 1. Theoretical geometry (HF approximation) for the central coordinate complex in the BGO and BSO models. Ground state ˚) (distance in A
BGO Bi 01 02 03 04 05 06 BSO Bi 01 02 03 04 05 06
x
y
z
0.0887 ¹ 0.1766 0.5408 2.0198 ¹ 2.3883 ¹ 0.0427 1.2186
0.0981 0.5276 2.0412 ¹ 0.167 1.2047 ¹ 2.3703 0.0248
0.1021 2.0385 ¹ 0.1349 0.5495 ¹ 0.0381 1.1932 ¹ 2.2155
0.1151 ¹ 0.2031 0.6277 2.021 ¹ 2.3512 ¹ 0.0949 1.2868
0.1116 0.6397 2.0277 ¹ 0.1809 1.2229 ¹ 2.3135 ¹ 0.2153
0.1275 2.0333 ¹ 0.2173 0.6622 ¹ 0.0706 1.2198 ¹ 2.1542
3. CALCULATIONAL METHODS AND RESULTS
In the present work ab initio calculations with Hay–Wadt basis sets to describe valence shells were used, including their effective core potentials [12, 13] as given in the Gaussian program package [14]. The latter include relativistic effects, which is an important ingredient for atoms of high atomic number such as bismuth and cesium. Geometries were optimized by varying the central Bi and the six surrounding oxygen atom distances, while the rest of the atom positions were kept fixed. Ground (singlet) and first excited (triplet) states were optimized at the HF level. For the ground state, the calculated ˚ off the geometry of the minimum is less than 0.20 A experimental value. The optimized geometries for both BGO and BSO for the central shell of the clusters are presented in Table 1. Through an HF calculation of the first excited state we obtained the optimized geometries presented in Table 2. DSCF values for the absorption are shown in Table 3. DSCF calculations were performed at the corresponding
minima for the excited states for the emission energies. Very small energy differences were found between the states, less than 1 eV in both cases. However, at the single excitation configuration interaction (CIS) level, calculations performed at the HF-obtained geometries led to the values for emission presented in Table 3. Finding such a good agreement for BSO in contrast to that of BGO was surprising. Energy was thus fully optimized at the CIS level finding a better transition energy for emission of 1.84 eV for BGO. For this reason, this last geometry (very similar to the first) was used in the rest of the calculations. Meanwhile, significant changes at this level of calculation for the BSO case were not detected. The most remarkable result is that the first triplet excited states calculated at their optimal geometries for both crystals show a salient feature reflected in their Mulliken spin densities, which is presented in Table 4. This indicates a strongly delocalized character for this state. For BGO, similar figures are also consistent with a triplet state of delocalized nature. Regarding the excited state geometry, one of the
Table 2. Theoretical geometry (HF approximation) for the central coordinate complex in the BGO and BSO models. First excited state ˚) (distance in A
BGO Bi 01 02 03 04 05 06 BSO Bi 01 02 03 04 05 06
x
y
z
0.0216 ¹ 0.1888 0.5321 2.6909 ¹ 2.503 0.0027 1.3971
¹ 0.0267 0.5301 1.9047 0.7221 1.1768 ¹ 2.4389 ¹ 0.4417
0.0631 2.0056 ¹ 0.3394 0.9246 0.0397 1.1799 ¹ 2.178
0.0044 ¹ 0.1758 0.6423 2.7429 ¹ 2.5680 ¹ 0.0351 1.3387
0.0545 0.6419 1.9872 0.0658 1.1741 ¹ 2.3852 ¹ 0.3365
0.0583 2.0281 ¹ 0.2386 0.3038 0.0721 1.1918 ¹ 2.1568
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J. F. RIVAS-SILVA and M. BERRONDO
Table 3. Absorption and Emission values (in eV) of BGO and BSO. Theoretical results from a CIS. For BGO a CIS optimized geometry is also used. Experimental, Refs. [1, 2, 10] BGO
BSO
Theoretical
Experimental
Theoretical
Experimental
5.89 1.45
4.90 2.40
6.15 2.62
4.20 2.70
Absorption Emission
oxygen atoms is displaced away from the central Bi atom, stretching out the O–Bi distance by roughly 42%. Also, the central bismuth gets slightly shifted towards the center of the cluster. This deformation of the excited state structure is intimately linked to the enhanced Stokes shift displayed by these systems, which has also been found in some AX 3 systems [15]. A surprising difference between BGO and BSO is the energy ordering of the states, attending to the symmetry of the states. This is shown in Table 5. These distinct orderings may be related with the different optical behavior that these two materials exhibit, although a more precise connection calls for a deeper study. In order to estimate the transition time for emission, we used the perturbed ion scheme developed by Valle´e [8], and simplified by us [9] in applications to Tl-doped alkali-halides. This scheme is utilized to calculate transition times involving emission between the excited triplet and ground singlet states. Due to dipole selection rules, a transition occurring between such states is spinforbidden. Some additional interaction must thus be considered to account for a non-vanishing transition probability. In particular, the spin–orbit interaction leads to a mixture of singlet and triplet states, thereby allowing an otherwise forbidden transition. In the quantum-mechanical perturbative approach on which the calculation of transition time for emission is based, the central atom in the cluster is looked upon as being perturbed by the crystal environment surrounding it. The spin–orbit coupling constant enters the calculation as a parameter. However, for complex systems, the latter can be very difficult to obtain, and since it is generally not available, an empirical ansatz is instead utilized to calculate an effective parameter by averaging over the coordination sphere around the central atom, where empirical spin–orbit parameters are assigned to the constituent atoms. For ionic systems this calculational Table 5. Excited state energies (in eV) from a CIS calculation for the BGO and BSO. For the first, the CIS optimized geometry is used. T and S mean triplet and singlet states, respectively Excited state 1 2 3 4
BGO symmetry
BSO symmetry
1.84 T 3.26 S 3.28 T 3.95 S
2.62 T 3.36 T 3.91 S 4.87 S
Table 4. Mulliken spin density for the first excited state of BGOand BSO in the geometries of Table 2
Bi O O O O O O Ge/Si Ge/Si Ge/Si
BGO
BSO
0.848913 ¹ 0.004916 0.002671 1.131611 ¹ 0.002422 ¹ 0.001387 ¹ 0.005482 0.000498 ¹ 0.085175 ¹ 0.003923
0.955852 ¹ 0.012509 ¹ 0.004744 1.058866 0.000159 ¹ 0.000983 ¹ 0.004953 0.002993 ¹ 0.171960 ¹ 0.005789
scheme has proved to yield very good results [9]. The transition time for emission within this scheme has been thus calculated. Accordingly, by selecting the empirical values for the spin–orbit parameter constants for Bi and O that best fit one of the experimental data, lifetimes shown in Table 6 were found. This gives a theoretical ratio (BGO/BSO) of 1/5 between the structures, as compared to the experimental one of 1/3, speaking about the last values. 4. DISCUSSION AND CONCLUSIONS
The results here presented, obtained through quantummechanical methods as applied to the analysis of optical properties of BGO and BSO crystals, provide some insight in materials science research, especially regarding theoretical foundations amenable to the experimental design of radiation detectors, and they put forward a number of interesting aspects: firstly, the feasibility of analyzing electronic structure which was shown to lead to reasonably good values as compared with the experimental data for absorption and emission processes in these materials; secondly, crucial differences in optical properties found in these materials, as inferred from the calculated emitting state lifetimes. In particular, the predicted elongation of one of the oxygens at the excited state optimal geometry provides a physical basis for the experimentally observed Stokes shift in optical spectroscopy. This explanation differs from the usual atomic model [16], assuming that the pronounced Stokes shift of these materials is due to the displacement of the central Bi þ3 toward the center of the tetrahedra formed with the neighboring oxygens. In BSO the central ion is shifted ˚ , whereas the oxygen moves 0.53 A ˚ . The about 0.13 A Table 6. Semi-empirical lifetimes of the excited states of BGO and BSO (in ns). Experimental, Refs [1, 2, 10] BGO Theoretical – 75.0 444.6 586.0
BSO
Experimental
Theoretical
Experimental
4.5 46.0 277.0 –
– 40.1 100.1 –
2.4 26.0 99.0 –
Absorption and emission properties of BGO and BSO
latter is thus expected to be a more crucial feature to sustain sensitive behavior in the crystal than that prompted by the motion of Bi þ3 atoms. The oxygen elongation can also be related to the stabilization of self-trapped excitons in these materials, analogous to the distortion of some atoms in the excited state, as explained by Shluger and Stefanovich [17] in bcristobalite. A detailed analysis of this aspect will be the subject of future investigation. A strong limitation in the present methods is posed by the cluster size when modeling a crystal. The most advanced computational codes currently available for molecular calculations can efficiently handle a few tens of atoms only, and introducing numerous heavy atoms may turn the problem numerically intractable. Clearly, either further progress in these computational techniques ought to be pursued, or else more approximate models for these systems should be applied. The manner in which atoms are saturated within our cluster model, together with the fact of dealing with systems that possess a rather pronounced degree of covalency, probably represents a somewhat unconventional idea for solid state physiscists in the sense that, very often, materials such as BGO, and BSO, are treated as though they were ionic. In particular, when working with oxygen-containing materials, it is essential to bear in mind the implications of a reduced ionic character in their crystal structure. When the computational techniques progress, in the near future, enough to handle bigger clusters in the ab initio approach, it will be possible to check the kind of saturation mentioned. Here, due to the small size of our models, it was fundamental to prevent annoying multiple bonds. Without the use of this saturation, and so permitting the formation of these bonds on the covalent atoms of the boundary, their spurious influence on the central part possibly could become less and less important when the size of the cluster is grown, for example in the case of materials where polarization effects are not important.
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Though the large distortion of the central shell found here for the excited state can be shared for more atoms around if a big cluster is employed, experimental work supports the central localization of this distortion [2,18]. Acknowledgements—One of us (JFRS) gratefully acknowledges financial support provided by SNI (Sistema Nacional de Investigadores, Mexico). This work was partially supported by CONACyT (Mexico) under project 1349 P-E, and by NIH under grant number R01-CA48002-04, subcontract number 4591710, with the Lawrence Berkeley Laboratory. REFERENCES 1. Weber, M. J. and Monchamp, R. R., J. Appl. Phys., 1973, 44, 5495. 2. Weber, M. J., West, A. C., Dujardin, C., Rupp, B., Berrondo, M. and Rivas-Silva, J. F., in Inorganic Scintillators and Their Applications, ed. P. Dorenbos et al. University Press, The Netherlands, 1996, p. 234. 3. Berrondo, M. and Rivas-Silva, J. F., Int. J. Quantum Chem., 1996, 57, 1115. 4. Berrondo, M. and Rivas-Silva, J. F., Int. J. Quantum Chem., 1995, 29, 253. 5. See e.g. the special issue: Catlow, C.R.A. and Stoneham, A.M., ed., J. Chem. Soc. Farad. Trans. II, 1989, 85. 6. Lasaga, A. C., Phys. Chem. Minerals, 1982, 8, 36. 7. Cunha, C., Canuto, S. and Fazzio, A., Phys. Rev., 1993, 48, 17806. 8. Valle´e, O., Minh, N. T. and Chapelle, J., in Spectral Line Shapes, ed. B. Wende. Walter de Gruyter, Berlin, 1981. 9. Rivas-Silva, J. F. and Berrondo, M., J. Phys. Chem. Solids, 1996, 57, 1705. 10. Ishii, M., Kobayashi, M. and Yamaga, I., in Heavy Scintillators for Scientific and Industrial Applications, ed. F. de Notaristefani et al. Editions Frontier, Gif-sur-Yuette, 1993. 11. Duffy, J. A., Bonding, Energy Levels and Bands in Inorganic solids. Longman Sci. and Tech., 1990, p. 156. 12. Wadt, W. R. and Hay, P. J., J. Chem. Phys., 1985, 82, 284. 13. Hay, P. J. and Wadt, W. R., J. Chem. Phys., 1985, 82, 299. 14. GAUSSIAN 92 Programmer’s Reference. Gaussian, Inc., 1992. 15. Berrondo, M., Rivas-Silva, J. F. and Recamier, J. to be published. 16. Timmersmans, C. W. M. and Blasse, G., J. Solid State Chem., 1984, 52, 222. 17. Shluger, A. and Stefanovitch, E., Phys. Rev., 1990, 42, 9664. 18. Dujardin, A. C., private communication.