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BacSr1−cTiO3 perovskite solid solutions: Thermodynamics from ab initio electronic structure calculations
Microelectronic Engineering 81 (2005) 478–484 www.elsevier.com/locate/mee
BacSr1cTiO3 perovskite solid solutions: Thermodynamics from ab initio electronic structure calculations S. Piskunov
a,*
, S. Dorfman b, D. Fuks c, E.A. Kotomin
a
a
c
Institute of Solid State Physics, University of Latvia, Kengaraga 8, LV- 1063 Riga, Latvia b Department of Physics, Technion-Israel Institute of Technology, Haifa 32000 Israel Materials Engineering Department, Ben-Gurion University of the Negev, P.O. Box 653 Beer-Sheva, Israel Available online 7 April 2005
Abstract We suggest theoretical prediction for BacSr1cTiO3 perovskite solid solutions (BST) combining ab initio DFT/B3PW calculations and alloy thermodynamics. This approach is based on calculations of a series of ordered super-structures in Ba–Sr simple cubic sublattice immersed in the rest TiO3 matrix. Although these structures are unstable with respect to the decomposition, the results of total energy calculations allow us to extract the necessary energy parameters and to calculate the phase diagram for the solid solutions (alloys). A novel approach applied to the BST system enables to predict that at T > 400 K Ba and Sr atom distribution is random. But below this temperature at small c Ba atoms aggregate into nanoclusters, thus leading to the formation of Ba-rich complexes of ‘‘almost pure’’ BaTiO3 (BTO) in mostly SrTiO3 (STO) matrix. At large c the formation of analogous SrTiO3 complexes in BaTiO3 is predicted. Ó 2005 Elsevier B.V. All rights reserved. Keywords: BacSr1cTiO3 solid solutions; Hybrid density functional calculations; Ab initio thermodynamics; Electronic structure calculations
1. Introduction Complex ABO3-type perovskite solid solutions continue to attract a growing attention because of numerous applications. It is well recognized nowa*
Corresponding author. Tel.: +371 718 7480; fax: +371 713 2778. E-mail address: [email protected] (S. Piskunov).
days that the dielectric and piezoelectric properties, response on external excitations, etc. in these alloys are entirely linked to the structural properties including compositional ordering and formation of complicated heterostructures. In particular, BST is considered as one of the most promising candidate for memory cell capacitors in dynamic random access memories with extremely high-scale integration [1]. Experimental results show that for
0167-9317/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2005.03.050
S. Piskunov et al. / Microelectronic Engineering 81 (2005) 478–484
the BST solid solutions in the Ba-rich region, the dielectric anomalies are associated with the fluctuations of the order parameter [2]. The dielectric and ultrasonic studies in Sr-rich BST were reported [3], demonstrating that a small addition of Ba to STO matrix leads to a formation of a glassy state at very low Ba concentrations c, and thus complicates significantly the sequence of phase transitions near the point c = 0.15. Structural evolution and polar order in BST was also reported in Ref. [4], on the basis of 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 phase exists up to a Ba concentration of 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 in two regions, one with a sole antiferrodistortive phase transition (c < ccr) and another with a sequence of three BTO-type ferroelectric phase transitions (c > ccr). Thus, there exists great interest in a study of this system in the whole 0 < c < 1 range of substitution at low temperatures, where phase transformations occur. Unfortunately, the experimental phase diagram for BST solid solutions is known for high temperatures [5], in the interval 1538– 1703 K. The complete picture of the whole phase diagram is lacking, although numerous experimental data confirm the differences in the low temperature phase transformations in BST, when composition is varied in a wide range. In this study we show that a statistical thermodynamics approach combined with ab initio DFT-B3PW calculations (see [6] for details) allows predict the main features of the quasi-binary phase diagram for the BST alloys in a wide range of concentrations and shed some light on the complicate picture of the sequence of phase transformations in this system.
2. Results and discussions Table 1 gives the results of calculations for BTO and STO, which are necessary for the further analysis. Using the data from Table 1, and the definition:
479
Table 1 Total energies, Etot, stoichiometric compositions, and equilibrium lattice parameters, aeq for the structures (a–i) from Fig. 1 ˚) Structure xst Etot (a.u.) aeq (A a b c d e f g h i BTO STO
1/2 1/2 1/2 1/4 3/4 1/4 3/4 1/8 7/8
2497.06046 2497.06005 2497.05988 2507.49056 2486.63722 2507.49030 2486.63705 2512.70707 2481.42596 2476.21745 2571.92863
STO DU ¼ Etot ðc EBTO tot þ ð1 cÞ E tot Þ;
3.9631 3.9655 3.9505 3.9445 3.9772 3.9394 3.9756 3.9262 3.9917 4.0045 3.9030
ð1Þ
the formation energies DU have been calculated for all ordered phases in Fig. 1. These energies are positive, i.e. the states represented by the phases considered in Fig. 1 have a higher energy than the reference state and thus formation of the considered phases is energetically unfavorable at T = 0 K, with respect to the heterophase mixture cBTO + (1 c)STO. A total solubility or decomposition in absolutely disordered BST solid solution should occur. The obtained data allow calculate the energy parameter needed to describe the situation at T > 0. Solving the set of equations for the energies of the phases a–c and i from Fig. 1, expressed in terms of Fourier transforms of the mixing potential V~ 1 , V~ 2 , V~ 3 , and V~ ð0Þ, concentrations and order parameters [6], 1 1 DU a ¼ V~ ð0Þc2 þ V~ 1 g21 ; 2 8 1~ 1 2 DU b ¼ V ð0Þc þ V~ 2 g22 ; 2 8 1~ 1 DU c ¼ V ð0Þc2 þ V~ 3 g23 ; 2 8 1~ 3 ~ 2 3 ~ 2 1 ~ 2 V 1 g1 þ V 2 g2 þ V 3 g3 DU h ¼ V ð0Þc2 þ 2 128 128 128 ð2Þ we obtain V~ ð0Þ ¼ 0:149 eV per atom in the quasi-binary solid solution BST. c was taken equal to the stoichiometric composition of the corresponding phases. DUa, D Ub, DUc, and DUh were obtained from the data given in Table 1 using
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Fig. 1. Superstructures in quasibinary BacSr(1c)TiO3 solid solutions that are stable with respect to the formation of anti-phase boundaries.
Eq. 1, they are equal to 0.3422 eV, 0.3534 eV, 0.3581 eV, and 0.2087 eV, respectively (per cell of the BST solid solution). The free energy of the disordered solid solution is
From simple thermodynamic considerations, it follows that an equilibrium phase diagram remains unaffected if the free energy given by Eq. 3 is replaced by
1 F ðcÞ ¼ V~ ð0Þ c2 2 þ kT ½c ln c þ ð1 cÞ lnð1 cÞ:
1 F ðcÞ ¼ V~ ð0Þ cð1 cÞ 2 þ kT ½c ln c þ ð1 cÞ lnð1 cÞ:
ð3Þ
ð4Þ
S. Piskunov et al. / Microelectronic Engineering 81 (2005) 478–484
This expression includes the chemical potential term, and is more convenient because of its symmetry with respect to c = 1/2. The phase diagram of the BST quasi-binary disordered solid solution calculated with Eq. 4 is given in Fig. 2. It has the miscibility gap, and a decomposition reaction takes place because V~ ð0Þ < 0. The calculated phase diagram represents the case of the limited solid solubility in this alloy. The solubility curve (bimodal) is shown in Fig. 2 by the bold line, and the dashed line describes the spinodal. The solubility curve is determined by the necessary minimum condition dF(c)/dc = 0, whereas the spinodal curve by the equation d2F(c)/dc2 = 0. According to the suggested model, the two-phase region is symmetric with respect to the concentration c = 0.5. This follows from the assumption that the energy parameter V~ ð0Þ is concentrationindependent. To analyze the decomposition in the BST solid solution, let us start from the point 1 in Fig. 2. This point represents the high-temperature state of a perovskite alloy with an equilibrium Ba atom concentration c0 at the temperature T0. This is a single-phase state, corresponding to a disordered solid solution in the alloy, when Ba and Sr atoms
randomly occupy the sites of the simple cubic lattice immersed in the field of the rest crystalline lattice with Ti and O atoms. Cooling the system to temperature T 0 brings the system to the state 2, below the spinodal. After annealing at this new temperature T 0 the equilibrium two-phase state of the solid solution on this simple cubic lattice occurs. The thermodynamic mechanism of the formation of this state is the decomposition of single-phase state into twophase state. This two-phase state is a mixture of two random solid solutions in the Ba–Sr sub-system. One phase is an extremely dilute solid solution of Ba atoms, randomly distributed on the lattice sites with the equilibrium concentration c1 (phase 1), the second phase is also a random solid solution of the same type, but with extremely high concentration of Ba atoms, c2 (phase 2). Thus, the two-phase state represents the mixture of the phases: one is highly enriched with Ba, whereas the second one is depleted of Ba atoms. The relative fraction of the phase 2 in a two-phase mixture is defined by the ‘‘lever rule’’, and is equal to (c0 c1)/(c2 c1), whereas the fraction of the phase 1, with its smaller concentration of Ba atoms, is much higher, and is equal to (c2 c0)/
spinodal solvus
500
400 T0
Temperature, in K
481
1
o
300
T"
o
o
o
200 T' o
o
o
2
100
C1
C0
C2
0 0.0
0.2
0.4
0.6
0.8
1.0
Atomic fraction of Ba in (BaxSr1-x)TiO3 Fig. 2. The phase diagram of the quasi-binary disordered solid solution BST.
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S. Piskunov et al. / Microelectronic Engineering 81 (2005) 478–484
(c2 c1). If the solubility regions are narrow, we have only a very small fraction of phase 2, but nevertheless, it has to exist. The two-phase state that corresponds to the temperature T 0 and atomic fraction c0 is characterized therefore as Ba-rich regions (with Ba atomic fraction c2) that are immersed in a Sr-enriched lattice with few Ba atoms randomly distributed on its sites. These small Ba-rich regions are also random solid solutions, but the concentration of Ba therein is very large and the number of sites occupied by Sr correspondingly small. For the temperature T00 > T 0 the atomic fraction of Ba atoms in Ba-rich regions decreases, while the fraction of Sr on the sites increases in these regions. Let us consider now the case when, after cooling from the temperature T01 (points 1 or 3 in Fig. 3) to the temperature T 01 , the system comes to the region of the phase diagram between the binodal and the spinodal (points 2 or 4 in Fig. 3). It is easy to see from Eq. 4 that the condition d2F(c)/dc2 > 0 is satisfied in this region of the phase diagram. For all points c 0 inside this interval the curve F(c 0 ) is concave, and this condition means that the homogeneous solid solution is stable with respect to infinitesimal heterogeneity. In-
spinodal solvus
500
400 T01
Temperature, in K
deed, if d2F(c)/dc2 > 0 it is always possible to choose an infinitesimal region of concentrations c01 < c0 < c02 in the vicinity of the point c 0 , where d2F(c)/dc2 > 0, i.e. where the curve F(c) is concave. This curve lies below the straight line connecting the points ðc01 ; ðF ðc01 ÞÞ and (c02 , F ðc02 Þ). Therefore the homogeneous single-phase alloy is more stable than a mixture of two phases having infinitesimally different compositions. If a homogeneous alloy characterized by the condition d2F(c)/dc2 > 0 at the point c is unstable with respect to the formation of a two-phase mixture with ca and cb phase compositions that are substantially different from the alloy composition, the alloy is nevertheless stable with respect to infinitesimally small composition heterogeneity. This is a metastable alloy, and the described situation corresponds to the points 2 and 4 in Fig. 3. The decomposition reaction in this case should involve the formation of a finite composition heterogeneity. The morphology of this two-phase mixture is presented by small well-separated BTO clusters in STO (point 2) or analogous STO clusters in BTO (point 4). The system will remain in singlephase state if the temperature T is changed, in order to bring the ‘‘alloy’’ to the state above the
1
3
o
o
300
4
2 o
T' 1
o
o o
200
100
C3 0 0.0
0.2
0.4
0.6
0.8
Atomic fraction of Ba in (BaxSr1-x)TiO3 Fig. 3. Phase diagram, the same as for Fig. 2.
1.0
S. Piskunov et al. / Microelectronic Engineering 81 (2005) 478–484
binodal. Thus, one can formulate a simple thermodynamic rule, how to get nanoparticles of BTO in STO even if the Ba atomic fraction in BST is very small (Sr-rich side of the phase diagram): The Barich clusters will be obtained at low Ba composition if the cooling process is such that at the end the BST system comes into the region of the phase diagram between the binodal and spinodal with subsequent decomposition into a two-phase state. A more complicated wave-like or percolation structure will be obtained if at the end of cooling the system finds itself in the region below the spinodal on the phase diagram. At very high temperatures, alloying by Ba atoms will leave the system in a one-phase state, namely a disordered Ba–Sr quasi-binary solid solution immersed as a simple cubic Ising lattice in the lattice of the rest, i.e. Ti and O, atoms. It is difficult to reach the thermodynamic equilibrium in this single-phase state at low temperature, because the solubility region at rather low temperature T 01 in Fig. 3 is extremely narrow. The decomposition reaction for low concentration of Ba in BST involves the formation of a finite composition heterogeneity. Particles of the Ba-rich phase that are formed in this region of phase diagram are well separated. They have low connectivity and may be considered as isolated BTO clusters. The number of Sr atoms in these clusters is extremely small. This situation is typical for decomposition of a binary dilute solid solution with limited solubility [7,8]. The analogous decomposition occurs at the Ba-rich side of the phase diagram. The obtained decomposition in BST solid solution means that at low temperatures and small atomic fraction of Ba there exist clusters in the old phase which consist of large number of Ba atoms, i.e. there are the clusters of ‘‘almost pure’’ BTO in ‘‘almost pure’’ STO. When atomic fraction of Ba in BST is large, one may obtain at low temperatures the clusters of ‘‘almost pure’’ STO in ‘‘almost pure’’ BTO. Lastly, we may compare our results with experimental data. The fact that the top of the solubility curve in Figs. 2, 3 corresponds relatively low temperature means that at higher temperatures the total solubility in quasibinary BTO-STO phase diagram exists in correspondence with the experimental data [5]. Our calculations of the energies of ordered phases definitely show that
483
all considered phases are unfavorable with respect to decomposition of the solid solution, and they are actually not observed in the phase diagram. Finally, the high quality of our calculations is confirmed by comparison with experimental data for the lattice parameters of pure compounds BTO ˚ [9]) and STO (aexp = 3.905 A ˚ [9]), (aexp = 3.996 A see Table 1. Also our calculations of bulk modulus, B fit well the experimental data; BBTO = 178.2 GPa (170 GPa [3]) and BSTO = 192.4 GPa (180 GPa [3]).
3. Conclusions A novel thermodynamic approach applied to the BST system, enables to predict the conditions when the Ba and Sr atom distribution are random, or when Ba atoms aggregate into clusters leading to the formation of Ba-rich complexes of ‘‘almost pure’’ BTO. As follows from our study, such nanoregions may be formed in extremely dilute BST, when the temperature is lowered down in such a way that the spinodal decomposition of the perovskite alloy occurs. This decomposition is dictated by the general thermodynamic properties of the considered system. Similar decomposition in Barich region of BST allows to predict the formation of ‘‘almost pure’’ STO nanoregions when the temperature decreases. The effects of changing the morphology of solid solution as the temperature and/or composition in the alloy is varied, control the total pattern of ferroelectric or ferrodistortive phase transformations in BST. A novel theory could be applied to many perovskite systems, which would permit the prediction of the conditions for a random A–B atom distribution, or for A (or B) atoms to aggregate into clusters to form A/B-rich complexes with corresponding ferroelectric properties even if the atomic fraction of these atoms in the alloy is small. In fact, in this study a new physical mechanism explaining the effect of external conditions on the ferroelectric phase transformations is suggested. For a particular BST perovskite alloy it is definitely shown that this is a spinodal decomposition. This demonstrates a nontrivial situation observed experimentally: Depending on the temperature and concentration
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of Ba and Sr atoms, the picture of ferroelectric or ferrodistortive phase transformations is complicated by the spinodal phase separation and by the formation of specific alloy morphologies above or below the spinodal line.
References [1] K. Abe, S. Komatsu, J. Appl. Phys. 77 (12) (1995) 6461– 6465. [2] N. Singh, A.P. Singh, C.D. Prasad, D. Pandey, J. Phys.: Condens. Matter 8 (1996) 7813–7827. [3] V.V. Lemanov, E.P. Smirnova, P.P. Syrnikov, E.A. Tarakanov, Phys. Rev. B 54 (5) (1996) 3151–3157.
[4] C.M.J.M. Kiat, B. Dkhil, M. Dunlop, H. Dammak, O. Hernandez, Phys. Rev. B 65 (2002) 224104. [5] M. McQuarrie, J. Am. Ceram. Soc. 38 (1955) 444. [6] S. Piskunov, The electronic structure of perfect and defective perovskite crystals: Ab Initio hybrid functional calculations, Ph.D. thesis, Universita¨t Osnabru¨ck, Osnabru¨ck. Available from: (Dec. 2003). [7] J.W. Cahn, J. Chem. Phys. 42 (1) (1965) 93–99. [8] C.N.R. Rao, K.J. Rao, McGraw-Hill, New York, 1978. [9] T. Mitsui, S. Nomura, M. Adachi, J. Harada, T. Ikeda, E. Nakamura, E. Sawaguchi, T. Shigenari, Y. Shiozaki, I. Tatsuzaki, K. Toyoda, T. Yamada, K. Gest, Y. Makita, M. Marutake, T. Shiosaki, K. WakinoLandolt–Bornstein, New Series, Group III: Crystal and Solid State Physics, Ferroelectrics and Related Substances, Subvolume A: Oxides, 16, Springer-Verlag, Berlin, 1981, pp. 400–449.
BacSr1cTiO3 perovskite solid solutions: Thermodynamics from ab initio electronic structure calculations S. Piskunov
a,*
, S. Dorfman b, D. Fuks c, E.A. Kotomin
a
a
c
Institute of Solid State Physics, University of Latvia, Kengaraga 8, LV- 1063 Riga, Latvia b Department of Physics, Technion-Israel Institute of Technology, Haifa 32000 Israel Materials Engineering Department, Ben-Gurion University of the Negev, P.O. Box 653 Beer-Sheva, Israel Available online 7 April 2005
Abstract We suggest theoretical prediction for BacSr1cTiO3 perovskite solid solutions (BST) combining ab initio DFT/B3PW calculations and alloy thermodynamics. This approach is based on calculations of a series of ordered super-structures in Ba–Sr simple cubic sublattice immersed in the rest TiO3 matrix. Although these structures are unstable with respect to the decomposition, the results of total energy calculations allow us to extract the necessary energy parameters and to calculate the phase diagram for the solid solutions (alloys). A novel approach applied to the BST system enables to predict that at T > 400 K Ba and Sr atom distribution is random. But below this temperature at small c Ba atoms aggregate into nanoclusters, thus leading to the formation of Ba-rich complexes of ‘‘almost pure’’ BaTiO3 (BTO) in mostly SrTiO3 (STO) matrix. At large c the formation of analogous SrTiO3 complexes in BaTiO3 is predicted. Ó 2005 Elsevier B.V. All rights reserved. Keywords: BacSr1cTiO3 solid solutions; Hybrid density functional calculations; Ab initio thermodynamics; Electronic structure calculations
1. Introduction Complex ABO3-type perovskite solid solutions continue to attract a growing attention because of numerous applications. It is well recognized nowa*
Corresponding author. Tel.: +371 718 7480; fax: +371 713 2778. E-mail address: [email protected] (S. Piskunov).
days that the dielectric and piezoelectric properties, response on external excitations, etc. in these alloys are entirely linked to the structural properties including compositional ordering and formation of complicated heterostructures. In particular, BST is considered as one of the most promising candidate for memory cell capacitors in dynamic random access memories with extremely high-scale integration [1]. Experimental results show that for
0167-9317/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2005.03.050
S. Piskunov et al. / Microelectronic Engineering 81 (2005) 478–484
the BST solid solutions in the Ba-rich region, the dielectric anomalies are associated with the fluctuations of the order parameter [2]. The dielectric and ultrasonic studies in Sr-rich BST were reported [3], demonstrating that a small addition of Ba to STO matrix leads to a formation of a glassy state at very low Ba concentrations c, and thus complicates significantly the sequence of phase transitions near the point c = 0.15. Structural evolution and polar order in BST was also reported in Ref. [4], on the basis of 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 phase exists up to a Ba concentration of 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 in two regions, one with a sole antiferrodistortive phase transition (c < ccr) and another with a sequence of three BTO-type ferroelectric phase transitions (c > ccr). Thus, there exists great interest in a study of this system in the whole 0 < c < 1 range of substitution at low temperatures, where phase transformations occur. Unfortunately, the experimental phase diagram for BST solid solutions is known for high temperatures [5], in the interval 1538– 1703 K. The complete picture of the whole phase diagram is lacking, although numerous experimental data confirm the differences in the low temperature phase transformations in BST, when composition is varied in a wide range. In this study we show that a statistical thermodynamics approach combined with ab initio DFT-B3PW calculations (see [6] for details) allows predict the main features of the quasi-binary phase diagram for the BST alloys in a wide range of concentrations and shed some light on the complicate picture of the sequence of phase transformations in this system.
2. Results and discussions Table 1 gives the results of calculations for BTO and STO, which are necessary for the further analysis. Using the data from Table 1, and the definition:
479
Table 1 Total energies, Etot, stoichiometric compositions, and equilibrium lattice parameters, aeq for the structures (a–i) from Fig. 1 ˚) Structure xst Etot (a.u.) aeq (A a b c d e f g h i BTO STO
1/2 1/2 1/2 1/4 3/4 1/4 3/4 1/8 7/8
2497.06046 2497.06005 2497.05988 2507.49056 2486.63722 2507.49030 2486.63705 2512.70707 2481.42596 2476.21745 2571.92863
STO DU ¼ Etot ðc EBTO tot þ ð1 cÞ E tot Þ;
3.9631 3.9655 3.9505 3.9445 3.9772 3.9394 3.9756 3.9262 3.9917 4.0045 3.9030
ð1Þ
the formation energies DU have been calculated for all ordered phases in Fig. 1. These energies are positive, i.e. the states represented by the phases considered in Fig. 1 have a higher energy than the reference state and thus formation of the considered phases is energetically unfavorable at T = 0 K, with respect to the heterophase mixture cBTO + (1 c)STO. A total solubility or decomposition in absolutely disordered BST solid solution should occur. The obtained data allow calculate the energy parameter needed to describe the situation at T > 0. Solving the set of equations for the energies of the phases a–c and i from Fig. 1, expressed in terms of Fourier transforms of the mixing potential V~ 1 , V~ 2 , V~ 3 , and V~ ð0Þ, concentrations and order parameters [6], 1 1 DU a ¼ V~ ð0Þc2 þ V~ 1 g21 ; 2 8 1~ 1 2 DU b ¼ V ð0Þc þ V~ 2 g22 ; 2 8 1~ 1 DU c ¼ V ð0Þc2 þ V~ 3 g23 ; 2 8 1~ 3 ~ 2 3 ~ 2 1 ~ 2 V 1 g1 þ V 2 g2 þ V 3 g3 DU h ¼ V ð0Þc2 þ 2 128 128 128 ð2Þ we obtain V~ ð0Þ ¼ 0:149 eV per atom in the quasi-binary solid solution BST. c was taken equal to the stoichiometric composition of the corresponding phases. DUa, D Ub, DUc, and DUh were obtained from the data given in Table 1 using
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Fig. 1. Superstructures in quasibinary BacSr(1c)TiO3 solid solutions that are stable with respect to the formation of anti-phase boundaries.
Eq. 1, they are equal to 0.3422 eV, 0.3534 eV, 0.3581 eV, and 0.2087 eV, respectively (per cell of the BST solid solution). The free energy of the disordered solid solution is
From simple thermodynamic considerations, it follows that an equilibrium phase diagram remains unaffected if the free energy given by Eq. 3 is replaced by
1 F ðcÞ ¼ V~ ð0Þ c2 2 þ kT ½c ln c þ ð1 cÞ lnð1 cÞ:
1 F ðcÞ ¼ V~ ð0Þ cð1 cÞ 2 þ kT ½c ln c þ ð1 cÞ lnð1 cÞ:
ð3Þ
ð4Þ
S. Piskunov et al. / Microelectronic Engineering 81 (2005) 478–484
This expression includes the chemical potential term, and is more convenient because of its symmetry with respect to c = 1/2. The phase diagram of the BST quasi-binary disordered solid solution calculated with Eq. 4 is given in Fig. 2. It has the miscibility gap, and a decomposition reaction takes place because V~ ð0Þ < 0. The calculated phase diagram represents the case of the limited solid solubility in this alloy. The solubility curve (bimodal) is shown in Fig. 2 by the bold line, and the dashed line describes the spinodal. The solubility curve is determined by the necessary minimum condition dF(c)/dc = 0, whereas the spinodal curve by the equation d2F(c)/dc2 = 0. According to the suggested model, the two-phase region is symmetric with respect to the concentration c = 0.5. This follows from the assumption that the energy parameter V~ ð0Þ is concentrationindependent. To analyze the decomposition in the BST solid solution, let us start from the point 1 in Fig. 2. This point represents the high-temperature state of a perovskite alloy with an equilibrium Ba atom concentration c0 at the temperature T0. This is a single-phase state, corresponding to a disordered solid solution in the alloy, when Ba and Sr atoms
randomly occupy the sites of the simple cubic lattice immersed in the field of the rest crystalline lattice with Ti and O atoms. Cooling the system to temperature T 0 brings the system to the state 2, below the spinodal. After annealing at this new temperature T 0 the equilibrium two-phase state of the solid solution on this simple cubic lattice occurs. The thermodynamic mechanism of the formation of this state is the decomposition of single-phase state into twophase state. This two-phase state is a mixture of two random solid solutions in the Ba–Sr sub-system. One phase is an extremely dilute solid solution of Ba atoms, randomly distributed on the lattice sites with the equilibrium concentration c1 (phase 1), the second phase is also a random solid solution of the same type, but with extremely high concentration of Ba atoms, c2 (phase 2). Thus, the two-phase state represents the mixture of the phases: one is highly enriched with Ba, whereas the second one is depleted of Ba atoms. The relative fraction of the phase 2 in a two-phase mixture is defined by the ‘‘lever rule’’, and is equal to (c0 c1)/(c2 c1), whereas the fraction of the phase 1, with its smaller concentration of Ba atoms, is much higher, and is equal to (c2 c0)/
spinodal solvus
500
400 T0
Temperature, in K
481
1
o
300
T"
o
o
o
200 T' o
o
o
2
100
C1
C0
C2
0 0.0
0.2
0.4
0.6
0.8
1.0
Atomic fraction of Ba in (BaxSr1-x)TiO3 Fig. 2. The phase diagram of the quasi-binary disordered solid solution BST.
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(c2 c1). If the solubility regions are narrow, we have only a very small fraction of phase 2, but nevertheless, it has to exist. The two-phase state that corresponds to the temperature T 0 and atomic fraction c0 is characterized therefore as Ba-rich regions (with Ba atomic fraction c2) that are immersed in a Sr-enriched lattice with few Ba atoms randomly distributed on its sites. These small Ba-rich regions are also random solid solutions, but the concentration of Ba therein is very large and the number of sites occupied by Sr correspondingly small. For the temperature T00 > T 0 the atomic fraction of Ba atoms in Ba-rich regions decreases, while the fraction of Sr on the sites increases in these regions. Let us consider now the case when, after cooling from the temperature T01 (points 1 or 3 in Fig. 3) to the temperature T 01 , the system comes to the region of the phase diagram between the binodal and the spinodal (points 2 or 4 in Fig. 3). It is easy to see from Eq. 4 that the condition d2F(c)/dc2 > 0 is satisfied in this region of the phase diagram. For all points c 0 inside this interval the curve F(c 0 ) is concave, and this condition means that the homogeneous solid solution is stable with respect to infinitesimal heterogeneity. In-
spinodal solvus
500
400 T01
Temperature, in K
deed, if d2F(c)/dc2 > 0 it is always possible to choose an infinitesimal region of concentrations c01 < c0 < c02 in the vicinity of the point c 0 , where d2F(c)/dc2 > 0, i.e. where the curve F(c) is concave. This curve lies below the straight line connecting the points ðc01 ; ðF ðc01 ÞÞ and (c02 , F ðc02 Þ). Therefore the homogeneous single-phase alloy is more stable than a mixture of two phases having infinitesimally different compositions. If a homogeneous alloy characterized by the condition d2F(c)/dc2 > 0 at the point c is unstable with respect to the formation of a two-phase mixture with ca and cb phase compositions that are substantially different from the alloy composition, the alloy is nevertheless stable with respect to infinitesimally small composition heterogeneity. This is a metastable alloy, and the described situation corresponds to the points 2 and 4 in Fig. 3. The decomposition reaction in this case should involve the formation of a finite composition heterogeneity. The morphology of this two-phase mixture is presented by small well-separated BTO clusters in STO (point 2) or analogous STO clusters in BTO (point 4). The system will remain in singlephase state if the temperature T is changed, in order to bring the ‘‘alloy’’ to the state above the
1
3
o
o
300
4
2 o
T' 1
o
o o
200
100
C3 0 0.0
0.2
0.4
0.6
0.8
Atomic fraction of Ba in (BaxSr1-x)TiO3 Fig. 3. Phase diagram, the same as for Fig. 2.
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binodal. Thus, one can formulate a simple thermodynamic rule, how to get nanoparticles of BTO in STO even if the Ba atomic fraction in BST is very small (Sr-rich side of the phase diagram): The Barich clusters will be obtained at low Ba composition if the cooling process is such that at the end the BST system comes into the region of the phase diagram between the binodal and spinodal with subsequent decomposition into a two-phase state. A more complicated wave-like or percolation structure will be obtained if at the end of cooling the system finds itself in the region below the spinodal on the phase diagram. At very high temperatures, alloying by Ba atoms will leave the system in a one-phase state, namely a disordered Ba–Sr quasi-binary solid solution immersed as a simple cubic Ising lattice in the lattice of the rest, i.e. Ti and O, atoms. It is difficult to reach the thermodynamic equilibrium in this single-phase state at low temperature, because the solubility region at rather low temperature T 01 in Fig. 3 is extremely narrow. The decomposition reaction for low concentration of Ba in BST involves the formation of a finite composition heterogeneity. Particles of the Ba-rich phase that are formed in this region of phase diagram are well separated. They have low connectivity and may be considered as isolated BTO clusters. The number of Sr atoms in these clusters is extremely small. This situation is typical for decomposition of a binary dilute solid solution with limited solubility [7,8]. The analogous decomposition occurs at the Ba-rich side of the phase diagram. The obtained decomposition in BST solid solution means that at low temperatures and small atomic fraction of Ba there exist clusters in the old phase which consist of large number of Ba atoms, i.e. there are the clusters of ‘‘almost pure’’ BTO in ‘‘almost pure’’ STO. When atomic fraction of Ba in BST is large, one may obtain at low temperatures the clusters of ‘‘almost pure’’ STO in ‘‘almost pure’’ BTO. Lastly, we may compare our results with experimental data. The fact that the top of the solubility curve in Figs. 2, 3 corresponds relatively low temperature means that at higher temperatures the total solubility in quasibinary BTO-STO phase diagram exists in correspondence with the experimental data [5]. Our calculations of the energies of ordered phases definitely show that
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all considered phases are unfavorable with respect to decomposition of the solid solution, and they are actually not observed in the phase diagram. Finally, the high quality of our calculations is confirmed by comparison with experimental data for the lattice parameters of pure compounds BTO ˚ [9]) and STO (aexp = 3.905 A ˚ [9]), (aexp = 3.996 A see Table 1. Also our calculations of bulk modulus, B fit well the experimental data; BBTO = 178.2 GPa (170 GPa [3]) and BSTO = 192.4 GPa (180 GPa [3]).
3. Conclusions A novel thermodynamic approach applied to the BST system, enables to predict the conditions when the Ba and Sr atom distribution are random, or when Ba atoms aggregate into clusters leading to the formation of Ba-rich complexes of ‘‘almost pure’’ BTO. As follows from our study, such nanoregions may be formed in extremely dilute BST, when the temperature is lowered down in such a way that the spinodal decomposition of the perovskite alloy occurs. This decomposition is dictated by the general thermodynamic properties of the considered system. Similar decomposition in Barich region of BST allows to predict the formation of ‘‘almost pure’’ STO nanoregions when the temperature decreases. The effects of changing the morphology of solid solution as the temperature and/or composition in the alloy is varied, control the total pattern of ferroelectric or ferrodistortive phase transformations in BST. A novel theory could be applied to many perovskite systems, which would permit the prediction of the conditions for a random A–B atom distribution, or for A (or B) atoms to aggregate into clusters to form A/B-rich complexes with corresponding ferroelectric properties even if the atomic fraction of these atoms in the alloy is small. In fact, in this study a new physical mechanism explaining the effect of external conditions on the ferroelectric phase transformations is suggested. For a particular BST perovskite alloy it is definitely shown that this is a spinodal decomposition. This demonstrates a nontrivial situation observed experimentally: Depending on the temperature and concentration
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of Ba and Sr atoms, the picture of ferroelectric or ferrodistortive phase transformations is complicated by the spinodal phase separation and by the formation of specific alloy morphologies above or below the spinodal line.
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[4] C.M.J.M. Kiat, B. Dkhil, M. Dunlop, H. Dammak, O. Hernandez, Phys. Rev. B 65 (2002) 224104. [5] M. McQuarrie, J. Am. Ceram. Soc. 38 (1955) 444. [6] S. Piskunov, The electronic structure of perfect and defective perovskite crystals: Ab Initio hybrid functional calculations, Ph.D. thesis, Universita¨t Osnabru¨ck, Osnabru¨ck. Available from: