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Ab initio calculations of the atomic and electronic structure of layered Ba0.5Sr0.5TiO3 structures
Materials Science and Engineering B 118 (2005) 15–18
Ab initio calculations of the atomic and electronic structure of layered Ba0.5Sr0.5TiO3 structures S. Piskunova , E.A. Kotomina,b,∗ , D. Fuksc , S. Dorfmand a Institute of Solid State Physics, University of Latvia, Kengaraga 8, LV-1063 Riga, Latvia Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany Materials Engineering Department, Ben-Gurion University of the Negev, POB 653, Beer-Sheva, Israel d Department of Physics, Technion- Israel Institute of Technology, Haifa 32000, Israel b
c
Abstract Understanding of the atomic and electronic structure of Bac Sr1 − c TiO3 (BST) solid solutions is important for several applications including the non-volatile ferroelectric memories (dynamic random access memory, DRAM). We present results of ab initio calculations of several spatial arrangements of Ba0.5 Sr0.5 TiO3 solid solutions based on DFT-HF B3PW hybrid method. We calculate the atomic and electronic structure, the effective charges, interatomic bond populations, the electronic density distribution, and densities of states for three layered structures with the same composition. The suggested method reproduces experimental lattice parameters of both pure BaTiO3 and SrTiO3 . The calculated optical band gaps for the pure SrTiO3 and BaTiO3 are in a good agreement with experimental data, much better than those from the standard LDA or HF calculations. In the studied BST structures with the equiatomic composition (c = 0.5) the gap is reduced by ca. 0.2 eV. The electron density maps demonstrate the covalency effects in the Ti–O bonding. The electron density near the Sr atoms is stronger localized, as compared with the Ba ions. © 2004 Elsevier B.V. All rights reserved. Keywords: Atomic structure; Electronic structure; Ba0.5 Sr0.5 TiO3 ; Perovskite solid solutions
1. Introduction Unusual properties in complex ABO3 -type perovskite solid solutions with a common formula (e.g. piezoelectric and ferroelectric properties, non-linear response on external excitations, etc.) are entirely linked to the structural properties including compositional ordering and formation of complicated heterostructures. The most promising representative of this class of materials, Bac Sr1 − c TiO3 (BST), is considered as a good candidate for memory cell capacitors in dynamic random access memory (DRAM) with extremely high scale integration [1]. In the Ba-rich region of the BST solid solutions the dielectric anomalies are associated with the fluctuations of the order parameter [2]. The dielectric and ultrasonic study in Srrich BST was reported [3], where it was shown that the small addition of Ba to SrTiO3 (STO) leads to the formation of a ∗
Corresponding author. fax: +37 17 112583. E-mail address: [email protected] (E.A. Kotomin).
0921-5107/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.mseb.2004.12.012
glassy state at very low Ba concentrations, which complicates significantly the sequence of phase transitions near the point c = 0.15. The structural evolution and polar ordering in the BST was reported [4], based on a combination of diffraction and diffusion of neutron and high-resolution X-ray experiments as well as dielectric susceptibility and polarization measurements. It is shown that the STO-type antiferrodistortive (AFD) phase exists up to Ba concentration ccr = 0.094, the progressive substitution of Sr by Ba leads to a monotonic decrease and to a vanishing of the oxygen octahedral tilting. The critical concentration ccr separates the phase diagram into the two regions: one with a sole AFD phase transition (c < ccr ) and another with a succession of three BaTiO3 (BTO)-type ferroelectric phase transitions (c > ccr ). Moreover, inside the non-ferroelectric AFD phase a local polarization is observed, with a magnitude comparable to the values of spontaneous polarization in the ferroelectric phases of the Ba rich compounds. The results of Raman study of BST films with the thickness ∼1 m and with Ba atomic fraction c = 0.05, 0.1, 0.2, 0.35, and 0.5 show [5] its striking similarity with relaxor
16
S. Piskunov et al. / Materials Science and Engineering B 118 (2005) 15–18
Fig. 1. Three selected layered geometries of the BST supercells with equatomic concentration of Sr and Ba atoms. The layer orientations are: (a) [0 0 1], (b) [1 1 0], (c) [1 1 1].
ferroelectrics. This was explained by the existence of polar nanoregions in the BST thin films. To describe and explain the relation between the structural and dielectric properties of these materials, significant efforts were employed. Simple purely ionic model that accounts electrostatic interaction was presented [6], in order to reproduce the compositional long-range order observed in a large class of perovskite alloys. To go beyond the ground state behavior and to make conclusions on the thermodynamic behavior as a function of temperature the Metropolis Monte Carlo simulations were applied, with the energy defined as an excess electrostatic energy in heterovalent binaries. This model automatically does not allow ordering in homovalent binary alloys. Account for the charge transfer may be performed by a direct modeling, in the framework of the electrostatic model [7], or may be carried out by ab initio calculation. A rather short but comprehensive review was published [8] on the very recent use of first-principle-derived approaches in investigation of piezoelectricity in simple and complex ferroelectric perovskites. It is interesting to note that most of these investigations were performed for heterovalent solid solutions whereas the case of homovalent alloys, such as BST, is much less studied. In this sense, it worth to mention ref. [9] where the molecular dynamics calculations have been performed. The interatomic pairwise potential used in these calculations included a Coulomb interaction, Born-Mayertype repulsive interaction, and van der Waals attractive interaction. Although the giant dielectric constant in BST for c = 0.7 was explained, other mentioned above fine features of the phase transformations in this system were not reproduced. This clearly demonstrates the importance of ab initio calculations for homovalent perovskite alloys. In this paper, we discuss the electronic structure of layered BST solid solutions with c = 0.5 (Sr0.5 Ba0.5 TiO3 ). We consider three key arrangements of Ba–Sr sublattice layers placed in the (0 0 1), (1 1 0), and (1 1 1) crystallographic planes (Fig. 1). In order to obtain more comprehensive analysis, properties of three BST structures are compared with those obtained for pure bulk STO and BTO perovskites.
2. Computational details To perform ab initio atomic and electronic calculations, we use the CRYSTAL-98 computer code [10,11]. This is
the periodic-structure computer program based on the linear combination of atomic orbitals (LCAO) expanded into the Gaussian basis set (BS). The main advantage of this code is its ability to calculate the electronic structure with both HartreeFock (HF) and Kohn-Sham, i.e. density functional theory (DFT), as well as their various hybrid approximations, using identical BS and computational parameters. The appropriate BSs for Ba, Sr, Ti, and O have been constructed and optimised in ref. [12]. We performed calculations of the BST equiatomic supercells (Fig. 1) using the so-called hybrid B3PW functional involving a hybrid of non-local Fock exchange and Becke’s gradient corrected exchange functional [13] combined with the non-local gradient corrected correlation potential by Perdew and Wang [14,15]. Twenty percent of non-local Fock energy have been utilized in this scheme. This particular B3PW hybrid technique was chosen due to its ability to give the proper description of basic bulk properties and the electronic structure of pure BTO and STO perovskite crystals [12]. The selected supercell is characterized by the layered heterostructure of Sr and Ba atoms. To model the BST structure (as well as pure STO and BTO), we extended standard ABO3 -type primitive unit cell to the supercell with translation vectors obtained by a simple relation: Aj =
3
lij ai
(1)
i=1
where ai are the translation vectors of the primitive cell. In our case lij is the diagonal matrix 2 0 0 l = 0 2 0 (2) 0 0 2 The constructed supercell consists of eight primitive STO unit cells and thus contains 5 × 8 = 40 atoms. Different substitutions of some Ba atoms for Sr atoms allow to construct BST solid solutions with different concentrations and different Ba impurity arrangements, e.g. as shown in Fig. 1. We consider high temperature cubic phase of the BST. The cubic lattice constants were optimized for each particular structure using the computer code which implements conjugated gradients optimization technique [16] with a numerical computation of derivatives. Under the lattice constant opti-
S. Piskunov et al. / Materials Science and Engineering B 118 (2005) 15–18
17
Table 1 Calculated lattice constants a0 , bulk moduli B, Mulliken effective charges Q, bond populations P (in milli e), and optical band gap for three layered BST superstructures (see Fig. 1a–c) as well as for the pure BTO and STO Structure
˚ a0 (A) B (GPa) Q(Ti) (e) Q(O) (e) Q(Ba) (e) Q(Sr) (e) P(O–Ti) (me) P(O–Ba) (me) P(O–Sr) (me) Band gap (eV)
Fig. 1a
Fig. 1b
Fig. 1c
BTO
STO
3.963 181 2.353 −1.396 1.788 1.878 96 −38 −8 3.54
3.965 180 2.354 −1.396 1.792 1.877 96 −40 −6 3.53
3.951 189 2.348 −1.393 1.788 1.876 94 −40 −8 3.56
4.004 (3.996 [17]) 178 (170 [18]) 2.356 −1.384 1.795 – 100 −34 – 3.46 (3.2 [19])
3.903 (3.905 [17]) 192 (180 [18]) 2.342 −1.405 – 1.872 88 – −10 3.56 (3.25 [20])
Experimentally obtained values and corresponding references are given in brackets. Negative bond population means atomic repulsion.
mizations, all atoms were held fixed in their positions in the supercell.
3. Results and discussion As is shown in Table 1, the suggested hybrid calculation scheme allows to reproduce the experimental lattice parameters for the pure BTO and STO in their cubic phases ˚ The relevant bulk moduli only with accuracy of 2 × 10−3 A. slightly overestimate their experimental values. Analysing lattice constants and bulk moduli obtained for three layered BST structures, one can see quite similar behaviour of the two structures with Ba–Sr layers arranged in the (0 0 1) and (1 1 0) planes (structures a and b in Fig. 1, respectively). Their lattice parameters and bulk moduli are practically equal, whereas the lattice constant of the structure c is reduced with respect to the a and b structures, and, on the other hand, its bulk modulus is increased. This could be explained by enlarged distances between neighbouring Ba(Sr) atoms in this structure. The difference electron density maps (calculated with respect to a superposition of ionic densities) are shown in Fig. 2. (Due to the limited article length, only the structure c is shown, but the density maps for other structures look very similarly.) These maps demonstrate the covalency effect in
the Ti–O bonding, which is well known for the pure STO and BTO perovskites. Both the deviation of the calculated effective Mulliken charges from formal ionic charges and the positive O–Ti bond populations (Table 1) agree well with an idea of weak covalency in all three BST structures. We would like also to mention that the electron density near the Sr ions is stronger localized in a comparison with the Ba ions. These ions do not participate in the chemical bonding in the BST. Lastly, the calculated total and projected density of states (DOS) for the BST structure shown in Fig. 1c is plotted in Fig. 3. (Again, the DOS calculated for all three BST structures are quite similar.) The BST upper valence band consists of O 2p atomic orbitals with a small admixure Ti 3d AOs. The conduction band bottom consists, in contrast, mainly of Ti 3d orbitals with a small contribution of O 2p AOs. The Sr (Ba) AOs make a significant contribution to the higher part of the conduction band. All partial atomic contributions of the conduction band are rather narrow. The optical band gaps for the considered BST heterostructures as well as for the pure STO and BTO calculated using the hybrid DFT-HF B3PW method are also collected in Table 1. The experimental values for the pure BTO and STO are only slightly overestimated. Band gaps for all three BST heterostructures, as shown in Fig. 1a–c. Fig. 1c is very close
Fig. 2. The difference electron density maps for BST structure shown in Fig. 1c plotted with respect to the formal ionic charges. Left side is the cross-section in the (1 1 0) plane, right side cross-section in the (0 0 1) plane. Isodensity curves are drawn from −0.05 to +0.05 e a.u.−3 with an increment 0.005 e a.u.−3 .
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S. Piskunov et al. / Materials Science and Engineering B 118 (2005) 15–18
the STO matrix or form clusters triggering phase transitions mentioned in Section 1.
Acknowledgements Authors thank E. Heifets, G. Borstel and R. Eglitis for stimulating discussions. EK and SD were supported by the German–Israeli (GIF) grant Nr G-703.41.10.
References
Fig. 3. Total and projected density of states (DOS) calculated for BST structure shown in Fig. 1c. Note also the enlarged scale for the DOS projected onto Ba and Sr AOs, in fact their contribution in the valence band region is much smaller than that of Ti 3d.
to that for the pure STO, being higher than the linear interpolation of the gap versus composition.
4. Conclusions Summing up, our DFT-HF hybrid calculations reproduce lattice parameters of pure cubic BTO and STO and overestimates their bulk moduli and band gaps no more than 10%. The obtained agreement for the gap values are much better than in the standard HF or DFT-LDA approach [12]. The BST heterostructures in three selected geometries are characterized by the same type of Ti–O covalent bonding as the pure perovskite constituents with their band gaps close to that of the STO. We continue now calculations for a number of the BST heterostructures with different geometries and fractional atomic concentrations of Ba and Sr, in order to calculate the phase diagram of the system [21] and thus predict under which conditions Ba atoms are either randomly distributed in
[1] K. Abe, S. Komatsu, J. Appl. Phys. 77 (1995) 6461. [2] N. Singh, A.P. Singh, C.D. Prasad, D. Pandey, J. Phys. Condens. Matter 8 (1996) 7813. [3] V.V. Lemanov, E.P. Smirnova, P.P. Syrnikov, E.A. Tarakanov, Phys. Rev. B 54 (1996) 3151. [4] C. M`enoret, J.M. Kist, B. Dkhil, M. Dunlop, H. Dammak, O. Hernandez, Phys. Rev. B 65 (2002) 224104. [5] D.A. Tenne, A. Soukiassian, M.H. Zhu, A.M. Klark, Phys. Rev. B 67 (2003) 12302. [6] L. Bellaiche, D. Vanderbilt, Phys. Rev. Lett. 81 (1998) 1318. [7] Z. Wu, H. Krakauer, Phys. Rev. B 63 (2001) 184113. [8] L. Belaiche, Curr. Opin. Solid State Mater. Sci. 6 (2002) 19. [9] H. Tanaka, H. Tabata, K. Ota, T. Kawai, Phys. Rev. B 53 (1996) 14112. [10] Quantum-mechanical ab-initio calculations of the properties of crystalline materials, in: C. Pisani (Ed.), Lecture Notes in Chemistry, 67, Springer, 1996. [11] V.R. Saunders, R. Dovesi, C. Roetti, M. Causa, N.M. Harrison, R. Orlando, C.M. Zicovich-Wilson, CRYSTAL’98 User’s Manual, Universita di Torino, Torino, 1998. [12] S. Piskunov, E. Heifets, R.I. Eglitis, G. Borstel, Comp. Mater. Sci. 29 (2004) 165. [13] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [14] J.P. Perdew, Y. Wang, Phys. Rev. B 33 (1986) 8800. [15] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. [16] E. Heifets, private communication; W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in Fortran77, second ed., Cambridge University Press, Cambridge, MA, 1997. [17] T. Mitsui, S. Nomura, M. Adachi, et. al., in: Landolt-B¨ornstein, Numerical Data and Functional Relationships in Science and Technology, New Series, Group III: Crystal and Solid State Physics, vol. 16, Ferroelectrics and Related Substances, sub-volume a: Oxides, Springer-Verlag, Berlin-Heidelberg-New York, 1981, pp. 400– 449. [18] V.V. Lemanov, E.P. Smirnova, P.P. Syrnikov, E.A. Tarakanov, Phys. Rev. B 54 (1996) 3151. [19] S.H. Wemple, Phys. Rev. B 2 (1970) 2679. [20] K. van Benthem, C. Elsasser, R.H. French, J. Appl. Phys. 90 (2001) 6156. [21] D. Fuks, S. Dorfman, Yu. Zhukovskii, E.A. Kotomin, A.M. Stoneham, Surf. Sci. 499 (2002) 24.
Ab initio calculations of the atomic and electronic structure of layered Ba0.5Sr0.5TiO3 structures S. Piskunova , E.A. Kotomina,b,∗ , D. Fuksc , S. Dorfmand a Institute of Solid State Physics, University of Latvia, Kengaraga 8, LV-1063 Riga, Latvia Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany Materials Engineering Department, Ben-Gurion University of the Negev, POB 653, Beer-Sheva, Israel d Department of Physics, Technion- Israel Institute of Technology, Haifa 32000, Israel b
c
Abstract Understanding of the atomic and electronic structure of Bac Sr1 − c TiO3 (BST) solid solutions is important for several applications including the non-volatile ferroelectric memories (dynamic random access memory, DRAM). We present results of ab initio calculations of several spatial arrangements of Ba0.5 Sr0.5 TiO3 solid solutions based on DFT-HF B3PW hybrid method. We calculate the atomic and electronic structure, the effective charges, interatomic bond populations, the electronic density distribution, and densities of states for three layered structures with the same composition. The suggested method reproduces experimental lattice parameters of both pure BaTiO3 and SrTiO3 . The calculated optical band gaps for the pure SrTiO3 and BaTiO3 are in a good agreement with experimental data, much better than those from the standard LDA or HF calculations. In the studied BST structures with the equiatomic composition (c = 0.5) the gap is reduced by ca. 0.2 eV. The electron density maps demonstrate the covalency effects in the Ti–O bonding. The electron density near the Sr atoms is stronger localized, as compared with the Ba ions. © 2004 Elsevier B.V. All rights reserved. Keywords: Atomic structure; Electronic structure; Ba0.5 Sr0.5 TiO3 ; Perovskite solid solutions
1. Introduction Unusual properties in complex ABO3 -type perovskite solid solutions with a common formula (e.g. piezoelectric and ferroelectric properties, non-linear response on external excitations, etc.) are entirely linked to the structural properties including compositional ordering and formation of complicated heterostructures. The most promising representative of this class of materials, Bac Sr1 − c TiO3 (BST), is considered as a good candidate for memory cell capacitors in dynamic random access memory (DRAM) with extremely high scale integration [1]. In the Ba-rich region of the BST solid solutions the dielectric anomalies are associated with the fluctuations of the order parameter [2]. The dielectric and ultrasonic study in Srrich BST was reported [3], where it was shown that the small addition of Ba to SrTiO3 (STO) leads to the formation of a ∗
Corresponding author. fax: +37 17 112583. E-mail address: [email protected] (E.A. Kotomin).
0921-5107/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.mseb.2004.12.012
glassy state at very low Ba concentrations, which complicates significantly the sequence of phase transitions near the point c = 0.15. The structural evolution and polar ordering in the BST was reported [4], based on a combination of diffraction and diffusion of neutron and high-resolution X-ray experiments as well as dielectric susceptibility and polarization measurements. It is shown that the STO-type antiferrodistortive (AFD) phase exists up to Ba concentration ccr = 0.094, the progressive substitution of Sr by Ba leads to a monotonic decrease and to a vanishing of the oxygen octahedral tilting. The critical concentration ccr separates the phase diagram into the two regions: one with a sole AFD phase transition (c < ccr ) and another with a succession of three BaTiO3 (BTO)-type ferroelectric phase transitions (c > ccr ). Moreover, inside the non-ferroelectric AFD phase a local polarization is observed, with a magnitude comparable to the values of spontaneous polarization in the ferroelectric phases of the Ba rich compounds. The results of Raman study of BST films with the thickness ∼1 m and with Ba atomic fraction c = 0.05, 0.1, 0.2, 0.35, and 0.5 show [5] its striking similarity with relaxor
16
S. Piskunov et al. / Materials Science and Engineering B 118 (2005) 15–18
Fig. 1. Three selected layered geometries of the BST supercells with equatomic concentration of Sr and Ba atoms. The layer orientations are: (a) [0 0 1], (b) [1 1 0], (c) [1 1 1].
ferroelectrics. This was explained by the existence of polar nanoregions in the BST thin films. To describe and explain the relation between the structural and dielectric properties of these materials, significant efforts were employed. Simple purely ionic model that accounts electrostatic interaction was presented [6], in order to reproduce the compositional long-range order observed in a large class of perovskite alloys. To go beyond the ground state behavior and to make conclusions on the thermodynamic behavior as a function of temperature the Metropolis Monte Carlo simulations were applied, with the energy defined as an excess electrostatic energy in heterovalent binaries. This model automatically does not allow ordering in homovalent binary alloys. Account for the charge transfer may be performed by a direct modeling, in the framework of the electrostatic model [7], or may be carried out by ab initio calculation. A rather short but comprehensive review was published [8] on the very recent use of first-principle-derived approaches in investigation of piezoelectricity in simple and complex ferroelectric perovskites. It is interesting to note that most of these investigations were performed for heterovalent solid solutions whereas the case of homovalent alloys, such as BST, is much less studied. In this sense, it worth to mention ref. [9] where the molecular dynamics calculations have been performed. The interatomic pairwise potential used in these calculations included a Coulomb interaction, Born-Mayertype repulsive interaction, and van der Waals attractive interaction. Although the giant dielectric constant in BST for c = 0.7 was explained, other mentioned above fine features of the phase transformations in this system were not reproduced. This clearly demonstrates the importance of ab initio calculations for homovalent perovskite alloys. In this paper, we discuss the electronic structure of layered BST solid solutions with c = 0.5 (Sr0.5 Ba0.5 TiO3 ). We consider three key arrangements of Ba–Sr sublattice layers placed in the (0 0 1), (1 1 0), and (1 1 1) crystallographic planes (Fig. 1). In order to obtain more comprehensive analysis, properties of three BST structures are compared with those obtained for pure bulk STO and BTO perovskites.
2. Computational details To perform ab initio atomic and electronic calculations, we use the CRYSTAL-98 computer code [10,11]. This is
the periodic-structure computer program based on the linear combination of atomic orbitals (LCAO) expanded into the Gaussian basis set (BS). The main advantage of this code is its ability to calculate the electronic structure with both HartreeFock (HF) and Kohn-Sham, i.e. density functional theory (DFT), as well as their various hybrid approximations, using identical BS and computational parameters. The appropriate BSs for Ba, Sr, Ti, and O have been constructed and optimised in ref. [12]. We performed calculations of the BST equiatomic supercells (Fig. 1) using the so-called hybrid B3PW functional involving a hybrid of non-local Fock exchange and Becke’s gradient corrected exchange functional [13] combined with the non-local gradient corrected correlation potential by Perdew and Wang [14,15]. Twenty percent of non-local Fock energy have been utilized in this scheme. This particular B3PW hybrid technique was chosen due to its ability to give the proper description of basic bulk properties and the electronic structure of pure BTO and STO perovskite crystals [12]. The selected supercell is characterized by the layered heterostructure of Sr and Ba atoms. To model the BST structure (as well as pure STO and BTO), we extended standard ABO3 -type primitive unit cell to the supercell with translation vectors obtained by a simple relation: Aj =
3
lij ai
(1)
i=1
where ai are the translation vectors of the primitive cell. In our case lij is the diagonal matrix 2 0 0 l = 0 2 0 (2) 0 0 2 The constructed supercell consists of eight primitive STO unit cells and thus contains 5 × 8 = 40 atoms. Different substitutions of some Ba atoms for Sr atoms allow to construct BST solid solutions with different concentrations and different Ba impurity arrangements, e.g. as shown in Fig. 1. We consider high temperature cubic phase of the BST. The cubic lattice constants were optimized for each particular structure using the computer code which implements conjugated gradients optimization technique [16] with a numerical computation of derivatives. Under the lattice constant opti-
S. Piskunov et al. / Materials Science and Engineering B 118 (2005) 15–18
17
Table 1 Calculated lattice constants a0 , bulk moduli B, Mulliken effective charges Q, bond populations P (in milli e), and optical band gap for three layered BST superstructures (see Fig. 1a–c) as well as for the pure BTO and STO Structure
˚ a0 (A) B (GPa) Q(Ti) (e) Q(O) (e) Q(Ba) (e) Q(Sr) (e) P(O–Ti) (me) P(O–Ba) (me) P(O–Sr) (me) Band gap (eV)
Fig. 1a
Fig. 1b
Fig. 1c
BTO
STO
3.963 181 2.353 −1.396 1.788 1.878 96 −38 −8 3.54
3.965 180 2.354 −1.396 1.792 1.877 96 −40 −6 3.53
3.951 189 2.348 −1.393 1.788 1.876 94 −40 −8 3.56
4.004 (3.996 [17]) 178 (170 [18]) 2.356 −1.384 1.795 – 100 −34 – 3.46 (3.2 [19])
3.903 (3.905 [17]) 192 (180 [18]) 2.342 −1.405 – 1.872 88 – −10 3.56 (3.25 [20])
Experimentally obtained values and corresponding references are given in brackets. Negative bond population means atomic repulsion.
mizations, all atoms were held fixed in their positions in the supercell.
3. Results and discussion As is shown in Table 1, the suggested hybrid calculation scheme allows to reproduce the experimental lattice parameters for the pure BTO and STO in their cubic phases ˚ The relevant bulk moduli only with accuracy of 2 × 10−3 A. slightly overestimate their experimental values. Analysing lattice constants and bulk moduli obtained for three layered BST structures, one can see quite similar behaviour of the two structures with Ba–Sr layers arranged in the (0 0 1) and (1 1 0) planes (structures a and b in Fig. 1, respectively). Their lattice parameters and bulk moduli are practically equal, whereas the lattice constant of the structure c is reduced with respect to the a and b structures, and, on the other hand, its bulk modulus is increased. This could be explained by enlarged distances between neighbouring Ba(Sr) atoms in this structure. The difference electron density maps (calculated with respect to a superposition of ionic densities) are shown in Fig. 2. (Due to the limited article length, only the structure c is shown, but the density maps for other structures look very similarly.) These maps demonstrate the covalency effect in
the Ti–O bonding, which is well known for the pure STO and BTO perovskites. Both the deviation of the calculated effective Mulliken charges from formal ionic charges and the positive O–Ti bond populations (Table 1) agree well with an idea of weak covalency in all three BST structures. We would like also to mention that the electron density near the Sr ions is stronger localized in a comparison with the Ba ions. These ions do not participate in the chemical bonding in the BST. Lastly, the calculated total and projected density of states (DOS) for the BST structure shown in Fig. 1c is plotted in Fig. 3. (Again, the DOS calculated for all three BST structures are quite similar.) The BST upper valence band consists of O 2p atomic orbitals with a small admixure Ti 3d AOs. The conduction band bottom consists, in contrast, mainly of Ti 3d orbitals with a small contribution of O 2p AOs. The Sr (Ba) AOs make a significant contribution to the higher part of the conduction band. All partial atomic contributions of the conduction band are rather narrow. The optical band gaps for the considered BST heterostructures as well as for the pure STO and BTO calculated using the hybrid DFT-HF B3PW method are also collected in Table 1. The experimental values for the pure BTO and STO are only slightly overestimated. Band gaps for all three BST heterostructures, as shown in Fig. 1a–c. Fig. 1c is very close
Fig. 2. The difference electron density maps for BST structure shown in Fig. 1c plotted with respect to the formal ionic charges. Left side is the cross-section in the (1 1 0) plane, right side cross-section in the (0 0 1) plane. Isodensity curves are drawn from −0.05 to +0.05 e a.u.−3 with an increment 0.005 e a.u.−3 .
18
S. Piskunov et al. / Materials Science and Engineering B 118 (2005) 15–18
the STO matrix or form clusters triggering phase transitions mentioned in Section 1.
Acknowledgements Authors thank E. Heifets, G. Borstel and R. Eglitis for stimulating discussions. EK and SD were supported by the German–Israeli (GIF) grant Nr G-703.41.10.
References
Fig. 3. Total and projected density of states (DOS) calculated for BST structure shown in Fig. 1c. Note also the enlarged scale for the DOS projected onto Ba and Sr AOs, in fact their contribution in the valence band region is much smaller than that of Ti 3d.
to that for the pure STO, being higher than the linear interpolation of the gap versus composition.
4. Conclusions Summing up, our DFT-HF hybrid calculations reproduce lattice parameters of pure cubic BTO and STO and overestimates their bulk moduli and band gaps no more than 10%. The obtained agreement for the gap values are much better than in the standard HF or DFT-LDA approach [12]. The BST heterostructures in three selected geometries are characterized by the same type of Ti–O covalent bonding as the pure perovskite constituents with their band gaps close to that of the STO. We continue now calculations for a number of the BST heterostructures with different geometries and fractional atomic concentrations of Ba and Sr, in order to calculate the phase diagram of the system [21] and thus predict under which conditions Ba atoms are either randomly distributed in
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