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Journal of Advanced Dielectrics Vol. 2, No. 1 (2012) 1250003 (12 pages) ? World Scienti?c Publishing Company DOI: 10.1142/S2010135X12500038

DIELECTRIC, PIEZOELECTRIC, AND ELASTIC PROPERTIES OF BaTiO3/SrTiO3 FERROELECTRIC SUPERLATTICES FROM FIRST PRINCIPLES
ALEXANDER I. LEBEDEV* Physics Department, Moscow State University 119991 Moscow, Russia swan@scon155.phys.msu.ru Received 7 November 2011 Published 27 February 2012
The e?ect of epitaxial strain on the phonon spectra, crystal structure, spontaneous polarization, dielectric, piezoelectric, and elastic properties of (001)-oriented ferroelectric ?BaTiO3 îm = ?SrTiO3 în superlattices (m Ì n Ì 1Ð4) was studied using the ?rst-principles density functional theory. The ground state of free-standing superlattices is the monoclinic Cm polar phase. Under the in-plane biaxial compressive strain, it transforms to tetragonal P4mm polar phase, and under the in-plane biaxial tensile strain, it transforms to orthorhombic Amm2 polar phase. When changing the in-plane lattice parameter, a softening of several optical and acoustic modes appears at the boundaries between the polar phases, and corresponding components of dielectric, piezoelectric, and elastic tensors diverge critically. The comparison of the mixing enthalpy of disordered Ba0:5 Sr0:5 TiO3 solid solution modeled using two special quasirandom structures SQS-4 with the mixing enthalpy of the superlattices reveals a tendency of the BaTiO3 ÐSrTiO3 system to shortrange ordering and shows that these superlattices are thermodynamically quite stable. Keywords : Ferroelectric superlattices; ?rst-principles calculations. PACS number(s): 64.60.-i, 68.65.Cd, 77.84.Dy, 81.05.Zx

1. Introduction
The success in creating of ferroelectric superlattices with a layer thickness controlled with an accuracy of one monolayer o?ers new opportunities for design of new ferroelectric multifunctional materials with high spontaneous polarization, Curie temperature, dielectric constant, and large dielectric and optical nonlinearities. Because of many problems encountered in the growth and experimental studies of ferroelectric superlattices, ?rst-principles calculations of their physical properties can be used to reveal new promising ?elds of investigations and applications of these materials.
*

Earlier studies of thin epitaxial ?lms of ferroelectrics with the perovskite structure have shown that their properties di?er strongly from those of bulk crystals. It was established that substrate-induced strain (epitaxial strain) has a strong inÀuence on the properties of ?lms. Due to strong coupling between strain and polarization, this strain changes signi?cantly the phase transition temperature and can induce unusual polar states in thin ?lms.1Ð4 To date, the most experimentally studied ferroelectric superlattice is the BaTiO3 =SrTiO3 (BTO/ STO) one.5Ð32 Studies of these superlattices from ?rst principles24,25,33Ð44 have established main

Corresponding author. 1250003-1


A. I. Lebedev

factors responsible for the formation of their polar structure. The speci?c feature of the superlattice is that the strains induced in it by the lattice mismatch between BaTiO3 and SrTiO3 and by the substrate result in concurrency of equilibrium polar structures in neighboring layers, so that the polar structure of the superlattice can be tetragonal, monoclinic, or orthorhombic, depending on the mechanical boundary conditions at the interface with the substrate. Although some properties of BTO/STO superlattices have been already studied, a number of problems remain unresolved. For instance, ?rstprinciples study of dielectric properties of these superlattices36,37 have found only P4mm and Cm polar phases, whereas the Amm2 phase, which is characteristic for stretched ?lms of BaTiO3,2 SrTiO3,45 and for PbTiO3/PbZrO3 superlattices,46 was not observed. The piezoelectric properties were calculated only for PbTiO3/PbZrO3 superlattice;47 for BTO/STO superlattices these data are absent. Finally, the elastic properties of ferroelectric superlattices and their behavior at the boundaries between di?erent polar phases have not been studied at all. In this work, ?rst-principles density functional calculations of the phonon spectra, crystal structure, spontaneous polarization, dielectric, piezoelectric, and elastic properties for polar phases of (001)oriented (BTO)m/(STO)n superlattices (SL m=n) with m Ì n Ì 1Ð4 are performed. The inÀuence of compressive and tensile epitaxial strain on the structure and properties of polar phases is studied in details for (BTO)1/(STO)1 superlattice. The stability ranges of tetragonal, monoclinic, and orthorhombic phases are determined. The critical behavior of static dielectric, piezoelectric, and elastic tensors at the boundaries between di?erent polar phases are studied. The ferroelastic type of the phase transitions between the polar phases is established. In addition, an

important question about the thermodynamic stability of BTO/STO superlattices is considered. The remainder of this paper is organized as follows. In Sec. 2, we give the details of our calculations. Next, we present the results for the ground state (Sec. 3.1) and the polarization (Sec. 3.2) of ?BTOîn =?STOîn superlattices. Dielectric, piezoelectric, and elastic properties of (BTO)1/(STO)1 superlattice are described in Secs. 3.3Ð3.5, respectively. The thermodynamic stability of the superlattices is analyzed in Sec. 3.6. The obtained results are discussed in Sec. 4.

2. Calculation Details
The calculations were performed within the ?rstprinciples density functional theory (DFT) with pseudopotentials and a plane-wave basis set as implemented in ABINIT software.49 The local density approximation (LDA) for the exchange-correlation functional50 was used. Optimized separable nonlocal pseudopotentials51 were constructed using the OPIUM software;52 to improve the transferability of pseudopotentials, the local potential correction was added according to Ref. 53. Parameters used for construction of pseudopotentials are given in Table 1; the results of testing of these pseudopotentials and other details of calculations can be found in Ref. 48. The plane-wave cut-o? energy was 30 Ha (816 eV). The integration over the Brillouin zone was performed with a 8 Ò 8 Ò 4 MonkhorstÐPack mesh. The relaxations of the atomic positions and the unit cell parameters were stopped when the HellmannÐFeynman forces were below 5 Ñ 10 Ð6 Ha/ Bohr (0.25 meV/Õ). The lattice parameters calculated using the pseudopotentials were a Ì 7:3506 Bohr (3.8898 Õ) for bulk SrTiO3 and a Ì 7:4923 Bohr (3.9648 Õ), c Ì 7:5732 Bohr (4.0075 Õ) for tetragonal BaTiO3.

Table 1. Parameters used for construction of pseudopotentials.48 Nonrelativistic generation scheme was used for Sr, Ti, and O atoms, and scalar-relativistic generation scheme was used for the Ba atom. All parameters are in Hartree atomic units except for the energy Vloc , which is in Ry. Atom Sr Ba Ti O Con?guration 4s 2 4p 6 4d 0 5s 5s 2 5p 6 5d 0 6s 3s 3p 3d 4s 2s 2 2p 4 3d
0 2 6 0 0 0 0

r

s

r

p

r

d

qs 7.07 7.07 7.07 7.07

q

p

qd 7.07 7.07 7.07 7.07

r

min

rmax 1.52 1.68 1.41 --

Vloc 1.5 1.95 2.65 --

1.68 1.85 1.48 1.40

1.74 1.78 1.72 1.55

1.68 1.83 1.84 1.40

7.07 7.07 7.07 7.57

0.01 0.01 0.01 --

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Dielectric, Piezoelectric, and Elastic Properties of BTO/STO Superlattices

Slight underestimation of the lattice parameters (in our case by 0.4Ð0.7% compared to the experimental data) is a known problem of LDA calculations. The calculations were performed on two structures: supercells of 1 Ò 1 Ò 2n perovskite unit cells for (001)-oriented (BTO)n/(STO)n superlattices (n Ì 1Ð4) and two special quasirandom structures (SQS) for Ba0:5 Sr0:5 TiO3 solid solution; the construction of SQSs is described in Sec. 3.6. Phonon spectra, dielectric, piezoelectric, and elastic properties of the superlattices were calculated within the DFT perturbation theory. Phonon contribution to the dielectric constant was calculated from phonon frequencies and oscillator strengths.54 The Berry phase method55 was used to calculate the spontaneous polarization Ps . As the layers of the superlattices are epitaxially grown on (001)-oriented substrate with a cubic structure, the calculations were performed for pseudotetragonal unit cells in which two in-plane translation vectors have the same length a0 and all three translation vectors are perpendicular to each other. This means that for monoclinic and orthorhombic phases (with Cm and Amm2 space groups) small deviations of the angles between the translation vectors from 90 (which were typically less than 0:07 ) were neglected. As was checked, this does not inÀuence much the results.

3. Results 3.1. Ground state of epitaxially strained superlattices
The lattice mismatch between BaTiO3 and SrTiO3 creates tensile biaxial strain in SrTiO3 layers and compressive biaxial strain in BaTiO3 layers of freestanding BTO/STO superlattice. If the layers were isolated, these strains would result in appearance of the in-plane spontaneous polarization in SrTiO3 (Amm2 space group) and in increase of the out-ofplane polarization in BaTiO3 layers (Cm or P4mm space groups).1Ð3 As the polar state with strong local variations of polarization is energetically unfavorable,33 the structure of the polar ground state of the superlattice requires special consideration. Earlier studies of BTO/STO36,37 and PbTiO3/PbZrO346 superlattices have demonstrated that both the magnitude and orientation of polarization depend also on the substrate-induced (epitaxial) strain in superlattices.

The ground state of BTO/STO superlattice was searched as follows. For a set of in-plane lattice parameters a0 , which were varied from 7.35 to 7.50 Bohr, we ?rst calculated the equilibrium structure of the paraelectric phase with P 4=mmm space group by minimizing the HellmannÐFeynman forces. The phonon frequencies at the Ð point were then calculated for these structures. It is known that the ground state of any crystal is characterized by positive values of all optical phonon frequencies at all points of the Brillouin zone. So, if the structure under consideration exhibited unstable phonons (with imaginary phonon frequencies), the atomic positions in it were slightly distorted according to the eigenvector of the most unstable mode, and a new search for the equilibrium structure was initiated. The phonon frequencies calculation and the search for equilibrium structure were repeated until the structure with all positive phonon frequencies was found. It should be noted that the only unstable mode in the paraelectric P 4=mmm phase of (BTO)1/(STO)1 superlattice is the ferroelectric one at the Ð point. The well-known structural instability of SrTiO3 associated with the R25 phonon mode at the boundary of the Brillouin zone disappears in the superlattice: the frequency of the corresponding phonon at the M point of the folded Brillouin zone (to which the R point transforms when doubling the c lattice parameter) is 55 cm Ð1 for 1/1 free-standing superlattice and 61 cm Ð1 for 1/1 superlattice grown on SrTiO3 substrate (see Fig. 1). The phonon spectra calculations show that in 1/1 superlattice grown on SrTiO3 substrate (compressive epitaxial strain, the in-plane lattice parameter a0 is equal to that of cubic strontium titanate) the tetragonal polar phase with P4mm space group is the ground state (Fig. 1). For free-standing superlattice, the P4mm structure is unstable and transforms to monoclinic Cm polar one. Under tensile epitaxial strain (a0 Ì 7:46 Bohr), the orthorhombic Amm2 polar phase is the most stable one for 1/1 superlattice. This means that the variation of a0 (for example, by growing the superlattice on di?erent substrates) can be used to control the polar state of the superlattice. In order to determine accurately the location of the boundaries between P4mm and Cm phases and between Cm and Amm2 phases for 1/1 superlattice, the ground state was calculated for a set of in-plane lattice parameters a0 , and for each of these

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A. I. Lebedev

(a) Fig. 2. (Color online) Frequencies of four lowest phonon modes at the Ð point for polar phases of (BTO)1/(STO)1 superlattice as a function of the in-plane lattice parameter a0 . The labels near the points indicate the symmetry of modes. Vertical lines indicate the phase boundaries.

(b) Fig. 1. Phonon spectra for (a) the P4mm phase of (BTO)1/ (STO)1 superlattice grown on SrTiO3 substrate (a0 Ì 7:3506 Bohr) and (b) the Cm phase of the same, free-standing superlattice (a0 Ì 7:4461 Bohr).

structures the phonon frequencies at the Ð point were computed. The dependence of four lowest phonon frequencies on the a0 parameter is plotted in Fig. 2. It is seen that the frequency of a doubly degenerate E mode decreases critically when approaching the boundary between P4mm and Cm phases from the tetragonal phase. After transition to the monoclinic phase two nondegenerate A 0 and A 00 soft modes appear in the phonon spectrum; the ?rst of these modes becomes soft again when approaching the boundary between Cm and Amm2 phases. The soft phonon mode in the Amm2 phase has the B1 symmetry. Extrapolation of the squared frequencies of soft ferroelectric modes (E mode in the P4mm phase, A 0 mode in the Cm phase, and B1 mode in the Amm2 phase) as a function of a0 to zero gives the in-plane lattice parameters corresponding to the boundaries between di?erent polar phases. The P4mmÐCm boundary is at a0 Ì 7:4023 Bohr when extrapolating

from the tetragonal phase and at 7.4001 Bohr when extrapolating from the monoclinic phase. The CmÐAmm2 boundary is at a0 Ì 7:4489 Bohr when extrapolating from the orthorhombic phase and at 7.4483 Bohr when extrapolating from the monoclinic phase. Small di?erence between the values obtained from extrapolation from two sides of the boundary means that both phase transitions are close to the second-order ones. Taking into account that a zero in-plane strain in polar superlattice corresponds to the in-plane lattice parameter of a0 Ì 7:4462 Bohr, we get the values of Ð0.605% and ?0.032% for the mis?t strains corresponding to the phase boundaries. Comparison of the energies of di?erent polar phases and calculation of the phonon frequencies at the Ð point for (BTO)2/(STO)2 and (BTO)3/ (STO)3 superlattices shows that the same P4mm, Cm, and Amm2 phases has the lowest energy, respectively, in superlattices grown on SrTiO3, freestanding superlattices, and superlattices grown on the substrate with a0 Ì 7:46 Bohr.

3.2. Spontaneous polarization
The calculated spontaneous polarization for freestanding and substrate-supported superlattices, tetragonal BaTiO3, and disordered Ba0:5 Sr0:5 TiO3 solid solution modeled using SQS-4 structures (see Sec. 3.6) are given in Table 2. As was established in Sec. 3.1, the tetragonal P4mm phase with the polarization vector normal

1250003-4


Dielectric, Piezoelectric, and Elastic Properties of BTO/STO Superlattices Table 2. Spontaneous polarization (in C/m2) for BTO/STO superlattices with di?erent thickness of layers, two SQS-4 structures used for modeling of disordered Ba0:5 Sr0:5 TiO3 solid solution, and tetragonal barium titanate. The in-plane lattice parameters a0 (in Bohr) for superlattices are also presented. Structure Ps orientation Pz Px Ì Py a0
a

SL1/1 Íxxz 0.061 0.165 7.4461 Í001
a

SL2/2 Íxxz 0.113 0.159 7.4432 Í001
a

SL3/3 Íxxz 0.146 0.150 7.4403 Í001
a

SL4/4 Íxxz 0.157 0.144 7.4391 Í001
a

SQS-4a [111] 0.130 0.130

SQS-4b [001] 0.206 0

BaTiO [001] 0.259 0

3

0.277 0 7.3506

0.293 0 7.3506

0.302 0 7.3506

0.307 0 7.3506

Superlattices grown on SrTiO3 substrate.

to the layers is the ground state for BTO/STO superlattices grown on SrTiO3 substrate. The calculations shows that in these superlattices the spontaneous polarization Ps increases monotonically from 0.277 C/m2 to 0.307 C/m2 as the layer thickness is increased from n Ì 1 to 4 unit cells (see Table 2). The obtained Ps values agree well with the value of 0.28 C/m2 estimated from the data of Ref. 33 for the superlattice with equal thickness of BaTiO3 and SrTiO3 layers; the Ps value of 0.259 C/m2 for tetragonal BaTiO3 agrees well with the value of 0.250 C/m2 reported in Ref. 33. As follows from Table 2, for all superlattices the Ps values are larger than those for Ba0:5 Sr0:5 TiO3 solid solution; for superlattices grown on SrTiO3 substrates they are even larger than Ps of tetragonal BaTiO3. These results agree with experiment13 and results of previous calculations.33,42 For free-standing superlattices, the monoclinic Cm phase is the ground state; the components of the polarization vector in this phase are also given in Table 2. It is seen that the polarization is rotated continuously in the (" 10) plane and the magnitude 1 of polarization increases with increasing n. For biaxially stretched superlattices, the Amm2 phase is the ground state and the polarization vector is oriented along [110] direction of the reference tetragonal P 4=mmm structure of the paraelectric phase. The in-plane and out-of-plane components of the polarization for 1/1 superlattice are plotted as a function of the in-plane lattice parameter a0 in 2 Fig. 3. Extrapolation of the P ? and P z2 dependence on a0 to zero gives the positions of the P4mmÐCm and CmÐAmm2 phase boundaries. Their values, a0 Ì 7:4018 Bohr and a0 Ì 7:4492 Bohr, are very close to those obtained in Sec. 3.1 from the frequencies of soft modes.

Fig. 3. The in-plane (P? ) and out-of-plane (Pz ) components of polarization for (BTO)1/(STO)1 superlattice as a function of the in-plane lattice parameter a0 . Vertical lines indicate the phase boundaries.

3.3. Dielectric properties
The eigenvalues of the static dielectric constant tensor "ij (i; j Ì 1; 2; 3) for (BTO)1/(STO)1 superlattice as a function of the in-plane lattice parameter a0 are shown in Fig. 4. In the tetragonal phase, the eigenvectors of the "ij tensor coincide with crystallographic axes and "11 Ì "22 . So, the dielectric properties of this phase are described by two nonzero independent parameters, "11 and "33 . In the monoclinic phase, the polarization vector rotates monotonically in the (110) plane; all three eigenvectors of the "ij tensor are di?erent and do not coincide with crystallographic axes of the reference tetragonal structure. As all nine components of the "ij tensor in this coordinate system are nonzero for the Cm phase, the most compact way to describe the properties of this tensor is to present its eigenvalues. In the Cm phase, the direction of the eigenvector corresponding to the smallest eigenvalue is

1250003-5


A. I. Lebedev

Fig. 4. Eigenvalues of the static dielectric constant tensor "ij for (BTO)1/(STO)1 superlattice as a function of the in-plane lattice parameter a0 . Vertical lines indicate the phase boundaries.

close, but do not coincide with the direction of the polarization. In the orthorhombic phase, the eigenvectors of the "ij tensor are oriented along the [110], [110] and, [001] directions of the reference tetragonal structure. The direction of the eigenvector corresponding to the smallest eigenvalue coincides with the polarization vector and the direction of eigenvector corresponding to the largest eigenvalue is [001]. As follows from Fig. 4, at least one of the eigenvalues of the "ij tensor diverges critically at the P4mmÐCm and CmÐAmm2 boundaries as the inplane lattice parameter a0 is changed. When approaching the P4mmÐCm boundary from the tetragonal phase, the "11 Ì "22 components of this tensor diverge as the polarization vector Ps jj Í001 becomes less stable against its rotation in the (110) plane. When approaching the CmÐAmm2 boundary from the orthorhombic phase, the "33 value diverges as the polarization vector Ps jj Í110 becomes less stable against its rotation in the same plane.

monoclinic phase.56,57 A similar situation appears in the monoclinic phase of BTO/STO superlattice. To our knowledge, the piezoelectric properties of BTO/STO superlattices have not been studied so far neither experimentally, nor theoretically. The only superlattice, for which some piezoelectric properties were calculated, is the PbTiO3/PbZrO3 1/1 superlattice,58 which was used to simulate the properties of PbTi0:5 Zr0:5 O3 solid solution. The largest piezoelectric stress moduli ei (i Ì 1; 2; 3; Ì 1Ð6) calculated for the P4mm phase of (BTO)1/(STO)1 superlattice grown on SrTiO3 substrate and for the Cm phase of the same freestanding superlattice are given in Table 3. It is seen that in tetragonal phases of the superlattice and BaTiO3 the e33 moduli do not di?er much. However in the monoclinic phase, which is the ground state for free-standing superlattice, the e33 value is ?ve times larger. Even stronger e?ect can be seen for P6 the d33 piezoelectric strain coeÁcient (di Ì Ì1 ei S ), which is a result of an 1.5-fold increase in the elastic compliance modulus S33 in the monoclinic phase (see Sec. 3.5). The piezoelectric moduli ei in polar phases of (BTO)1/(STO)1 superlattice as a function of the inplane lattice parameter are shown in Fig. 5. According to the symmetry, in the tetragonal phase the piezoelectric tensor has three independent and ?ve nonzero components: e31 Ì e32 , e33 , and e15 Ì e24 . In our superlattice only two of them have large values: e33 and e15 . When approaching the P4mmÐCm boundary from the tetragonal phase, the e33 value increases monotonically whereas the e15 value diverges critically and reaches the value of 80 C/m2 (not shown). In the orthorhombic phase (in the coordinate system of the reference tetragonal structure) the e11 Ì e22 , e12 Ì e21 , e13 Ì e23 , e34 Ì e35 , and e16 Ì e26 moduli are nonzero, and the total number of
Table 3. Largest piezoelectric moduli for monoclinic phase of free-standing (BTO)1/(STO)1 superlattice, for tetragonal phase of the same superlattice grown on SrTiO3 substrate, and for tetragonal barium titanate. Structure Ps orientation e33 , C/m2 (d33 , pC/N) e15 , C/m2 (d15 , pC/N)
a

3.4. Piezoelectric properties
Due to high sensitivity of both magnitude and orientation of the polarization vector in superlattices to epitaxial strain, we can expect them to be good piezoelectrics. It is known that anomalously high piezoelectric moduli found in some ferroelectrics like PbZr1Ðx TixO3 near the morphotropic phase boundary are due to the ease of strain-induced rotation of polarization in the intermediate

SL1/1 Íxxz 31.9 (460) Ð0.09 (Ð19) Í001
a

BaTiO [001]

3

7.1 (49) 3.2 (31)

6.3 (42) Ð2.9 (Ð24)

Superlattice grown on SrTiO3 substrate.

1250003-6


Dielectric, Piezoelectric, and Elastic Properties of BTO/STO Superlattices

Fig. 5. (Color online) Components of the piezoelectric tensor ei in polar phases of (BTO)1/(STO)1 superlattice as a function of the in-plane lattice parameter a0 . Vertical lines indicate the phase boundaries.

independent parameters is 5. Among them the e11 , e12 , e34 , and e16 moduli have the largest values (see Fig. 5). The only modulus that behaves critically at the boundary between Cm and Amm2 phases is the e34 one, with a maximum value reaching 192 C/m2 (not shown). The most complex behavior of piezoelectric moduli is observed in the monoclinic phase because the change of the in-plane lattice parameter a0 results in monotonic rotation of the polarization vector in the (" 10) plane. In the coordinate system 1 of the reference tetragonal structure, all 18 components of the ei tensor are nonzero (the total number of independent parameters is 10). As follows from Fig. 5, in the monoclinic phase the e11 Ì e22 , e15 Ì e24 , e12 Ì e21 , e13 Ì e23 , e14 Ì e25 , and e16 Ì e26 moduli diverge critically at the boundary between Cm and P4mm phases, and e33 , e34 Ì e35 , e31 Ì e32 , and e36 moduli diverge critically at the boundary between Cm and Amm2 phases. It should be noted that when crossing the boundaries, small additional jumps (about $10%) are observed in the e33 modulus (at the CmÐP4mm boundary) and in e11 and e12 moduli (at the CmÐAmm2 boundary).

primary order parameter. This occurs when the strain tensor and the polar vector transform according to the same irreducible representation of the high-symmetry phase.59 As such phase transitions occur in BTO/STO superlattices when changing the in-plane lattice parameter a0 , it was interesting to study the inÀuence of these phase transitions on the elastic properties of superlattices, especially taking into account that these properties of superlattices have not been studied so far. In the tetragonal P4mm phase, the elastic compliance tensor S (; Ì 1Ð6) has six independent and nine nonzero components. In the orthorhombic Amm2 phase (in the coordinate system of the reference tetragonal structure) the tensor has nine independent and 13 nonzero components, and in the monoclinic Cm phase it has 13 independent and 21 nonzero components. The components of the elastic compliance tensor S for polar phases of (BTO)1/(STO)1 superlattice are plotted as a function of the in-plane lattice parameter a0 in Fig. 6. It is seen that at the boundary between P4mm and Cm phases the components of S tensor exhibit a step-like change (S13 Ì S23 , S33 , S66 , S16 Ì S26 , S36 , S34 Ì S35 , S46 Ì S56 moduli), a critical divergence from the monoclinic side (S11 Ì S22 , S12 , S15 Ì S24 , S14 Ì S25 , S45 moduli), or critical divergences from both sides of the boundary (S44 Ì S55 modulus). In the monoclinic phase, the S14 , S15 , S16 , S34 , S36 , S45 , and S46 moduli become nonzero. In the orthorhombic phase, the S14 , S15 , S34 , and S46 moduli vanish again

3.5. Elastic properties
It is known that ferroelectric phase transitions between two polar phases are often the improper ferroelastic ones, which means that they are accompanied by appearance of soft acoustic modes and spontaneous strain, but the strain is not the
Fig. 6. (Color online) Components of the elastic compliance tensor S for polar phases of (BTO)1/(STO)1 superlattice as a function of the in-plane lattice parameter a0 . Vertical lines indicate the phase boundaries.

1250003-7


A. I. Lebedev

whereas the other moduli remain nonzero. At the boundary between Cm and Amm2 phases the anomalies in elastic moduli are smaller: the S11 , S12 , and S66 moduli exhibit step-like changes, the S13 , S14 , S15 , S16 , S33 , S34 , S36 , and S46 moduli exhibit weak divergence from the monoclinic side, and the S44 and S45 moduli exhibit weak divergence from both sides of the boundary. The critical divergence of the S44 modulus at both P4mmÐCm and CmÐAmm2 boundaries indicates that the phase transitions induced in the superlattice by the increase of the in-plane lattice parameter are indeed the ferroelastic ones.

Fig. 7. (Color online) The unit cell of SQS-4b superstructure contains four primitive cells of the perovskite structure and is constructed using a1 , a 2 , and a 3 translation vectors. The atoms of one of two types, A or B, occupy the sites of one sublattice of the perovskite structure lying on the planes (shown by blue), which are perpendicular to the superlattice axis (shown by red arrow). The stacking sequence of the planes along the superlattice axis is AABB.

3.6. Thermodynamic stability
Thermodynamic stability of ferroelectric superlattices is very important for their possible applications. Thermodynamic stability of BTO/STO superlattice is determined by the mixing enthalpy of the superlattice and its relationship with the mixing enthalpy of the disordered Ba0:5 Sr0:5 TiO3 solid solution. The most complex part of the ?rst-principles calculation of thermodynamic stability is the calculation of the mixing enthalpy for a solid solution because its simulation using supercells with a large number of randomly distributed atoms makes it extremely time-consuming. A conceptually new approach to this problem was proposed by Zunger et al.60 In this approach, a disordered solid solution Ax B1Ðx is modeled using a special quasirandom structure (SQS) -- a shortperiod superstructure, whose statistical properties (numbers of NAA , NBB , and NAB atomic pairs in few nearest shells) are as close as possible to those of ideal disordered solid solution (at the same time the sites of the superstructure are deterministically ?lled with A and B atoms). This method has been widely used to study the electronic structure and physical properties of semiconductor solid solutions and the ordering phenomena in metal alloys. To study

the properties of ferroelectric solid solutions this approach was used quite rare.61,62 The structure of the disordered Ba0:5 Sr0:5 TiO3 solid solution was modeled using two special quasirandom structures SQS-4 constructed with the gensqs program from ATAT toolkit.63 One of these structures is sketched in Fig. 7. The translation vectors, superlattice axis, and stacking sequence of the planes ?lled with the same atoms, Ba or Sr (denoted by A and B), are given in Table 4. The pair correlation functions Õ2;m (m is the shell number), which describe the deviation of statistical properties of these SQSs from those of an ideal solid solution, are also given in this table. For x Ì 0:5 Õ2;m is simply (2NAA =Nm Ð 1), where Nm is a number of neighbors in the mth shell. As follows from this table, for the SQS-4a structure strong deviation from an ideal solid solution appears only in the fourth shell; for the SQS-4b structure deviations appear in the second and fourth shells, but are smaller. The mixing enthalpy ÑH for all studied structures X (superlattices with di?erent periods and SQS structures) was calculated using the formula ÑH Ì Etot ?X îÐ ÍEtot ?BaTiO3 î? Etot ?SrTiO3 î=2 from the values of the total energy Etot (per ?veatom formula unit) for free-standing fully relaxed

Table 4. Translation vectors, superlattice axis, stacking sequence of atomic planes, and correlation functions Õ2;m for SQS-4 structures used for modeling of disordered Ba0:5 Sr0:5 TiO3 solid solution. Structure SQS-4a SQS-4b Translation vectors [211], [112], [121] [210], [210], [001] Axis and stacking sequence [1" ] AABB 11 [" 20] AABB 1 Õ2 0 0
;1

Õ

2;2

Õ2 0 0

;3

Õ2

;4

0 Ð1/3

Ð1 1/3

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Dielectric, Piezoelectric, and Elastic Properties of BTO/STO Superlattices Table 5. The mixing enthalpy (in meV) for ?ve (BTO)n/ (STO)n superlattices with di?erent periods and two SQS-4 structures used for modeling of disordered Ba0:5 Sr0:5 TiO3 solid solution. SL1/1 2.9 SL2/2 8.9 SL3/3 11.4 SL4/4 12.6 SL5/5 13.4 SQS-4a 16.8 SQS-4b 11.0

paraelectric Pm3m and P 4=mmm phases. The obtained values of ÑH for these structures are given in Table 5. An unexpected result of our calculation is the fact that ÑH values for two shortest-period superlattices (1/1 and 2/2) appeared smaller than ÑH values for both realizations of disordered solid solution. This means that a tendency to short-range ordering of components exists in the BaTiO3ÐSrTiO3 system. Low values of ÑH for these superlattices (< 9 meV) indicate that short-period BTO/STO superlattices are thermodynamically stable at 300 K. The tendency to short-range ordering found in (001)-oriented BTO/STO superlattices can be explained by a general tendency of the A cations to order in a layered manner in perovskites, in contrast to the B cations, which prefer a rock-salt ordering.64 One can add that a similar e?ect was observed in our studies of (001)-oriented ?PbTiO3 în =?SrTiO3 în superlattices, where negative values of ÑH for n Ì 1Ð3 and positive values for larger n were observed. Our values of the mixing enthalpy for BTO/STO superlattices are much smaller than ÑH value obtained in Ref. 65 (42 meV per formula unit). Analysis of the calculation technique used in Ref. 65 shows that atomic positions in the superstructures were not relaxed and the superstructures were assumed to be cubic when calculating the energies of di?erent atomic con?gurations. So, the calculated mixing enthalpy in this paper includes a large energy of excess strain.

4. Discussion
Our results on the inÀuence of epitaxial strain on the ground state of BTO/STO superlattice agree only partially with the results obtained in Refs. 24, 34, 36, and 37. The results coincide in that: (1) the ground state for free-standing (BTO)1/(STO)1 superlattice is the monoclinic Cm phase, (2) under the compressive strain, the superlattice undergoes the phase transition from Cm to P4mm phase, and (3)

the dielectric constant diverges at the P4mmÐCm phase boundary. At the same time, in contrast to the results of Refs. 36 and 37, we succeeded to observe the phase transition to the Amm2 orthorhombic phase under the tensile strain (in Refs. 36 and 37 only the rotation of the polarization vector towards the [110] direction was observed under the tensile strain). We consider our results to be more reliable because they agree with the results obtained for strained BaTiO32 and SrTiO345 thin ?lms, the results of recent atomistic calculations of the strainÐtemperature phase diagram for (BTO)2/(STO)2 superlattice,38 and with results obtained for another superlattice, PbTiO3/ PbZrO3.46 In all these systems the same phase sequence, P4mmÐCmÐAmm2, was observed as the in-plane lattice parameter was increased. The increase in the polarization Ps in (BTO)n/ (STO)n superlattices grown on SrTiO3 substrates with increasing the layer thickness n (Table 2) agrees with the results of Ref. 42 in which the explanation of this phenomenon was proposed. In free-standing superlattices, the Pz component of polarization also increased with increasing n, but the Px and Py components decreased with increasing n, in contrast to the results observed for 3/3, 4/4, and 5/5 superlattices with ?xed in-plane lattice parameter equal to 1.01 times the lattice parameter of SrTiO3.34 We attribute these changes to the decrease of the inplane lattice parameter a0 for free-standing superlattices with increasing n (see Table 2). Unfortunately, only a few data points presented in Ref. 37 for the dielectric constant at the P4mmÐCm boundary for (BTO)1/(STO)1 superlattice did not enabled us to make detailed comparison between our results. However, the comparison of our dielectric data with those calculated for ?PbTiO3 î1 =?PbZrO3 î1 superlattice46 shows that in the monoclinic phase all eigenvalues of the "ij tensor for BTO/STO superlattice are higher, and so this superlattice may be more promising for di?erent applications. To obtain large piezoelectric moduli necessary for technological applications, the epitaxial strain in the BTO/STO superlattice should be tuned to a value at which ei and S properties of the superlattice diverge. As was shown in Secs. 3.4 and 3.5, this appears at the phase boundaries. The calculations show that in the vicinity of the P4mmÐCm boundary the d11 piezoelectric coeÁcient reaches a maximum value of 2300 pC/N in the monoclinic phase and the d15 coeÁcient reaches a value of 6200 pC/N in the tetragonal phase. In the vicinity of

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A. I. Lebedev

the CmÐAmm2 boundary, the d33 piezoelectric coeÁcient reaches a value of 920 pC/N in the monoclinic phase and the d34 coeÁcient reaches a value of 10500 pC/N in the orthorhombic phase. For comparison, the maximum piezoelectric coeÁcient obtained experimentally on single crystals of the Pb (Zn1=3 Nb2=3 )O3ÐPbTiO3 system was 2500 pC/N.66 Anomalous increase in the calculated d15 and d33 coeÁcients up to $8500 pC/N was predicted for hydrostatically stressed PbTiO3 in the vicinity of the P4mmÐCm phase boundary.67 Consider now the elastic properties of BTO/STO superlattice and the results indicating the appearance of improper ferroelastic phase transitions. According to Ref. 59, in crystals with P4mm space group the phase transition P 4mm ! m should be of the ferroelastic type. The spontaneous strain at this phase transition is characterized by one or both nonzero u4 and u5 components of the strain tensor, and a soft transverse acoustic (TA) mode with a wave vector parallel to the polar axis should appear in the phonon spectrum in the vicinity of the Ð point. Direct calculations of frequencies of acoustic modes in the tetragonal phase at the point with a reduced wave vector q Ì ?0; 0; 0:05î con?rmed this (see Fig. 8): when approaching the P4mmÐCm boundary, the frequency of a doubly degenerate TA phonon with the Ó5 symmetry decreased by ?ve times. The softening of this mode is directly related to the divergence of the S44 Ì S55 components of the elastic compliance tensor.

At the boundary between Cm and Amm2 phases, there should be another ferroelastic phase transition. According to Ref. 59, the transition mm2 ! m is accompanied by spontaneous strain with one nonzero of three u4 , u5 , u6 components. In our coordinate system with unusual for orthorhombic phase orientation of the polar axis (along the [110] direction) there should be a softening of TA phonon with a wave vector oriented along this axis. This was con?rmed by direct calculations of frequencies of acoustic modes at the point in the Brillouin zone with a reduced wave vector q Ì ?0:035; 0:035; 0î (see Fig. 8). In contrast to the tetragonal phase in which the soft acoustic mode is doubly degenerate, in the orthorhombic phase the only phonon mode that softens at the phase boundary is the TA mode polarized along the [001] axis. In our opinion, this di?erence is the reason why the anomalies in the elastic properties at the ferroelastic CmÐAmm2 phase transition are much weaker than those at the ferroelastic P4mmÐCm phase transition. Unusual orientation of the polar axis in our coordinate system results in coupling of some components of the elastic compliance tensor: for example, the S44 and S45 moduli, which diverge in the orthorhombic phase, satisfy the relation S44 ÐS45 % const. in this phase. Negative value of the acoustic phonon frequency in the Cm phase, which is seen in a narrow wave vector region in Fig. 1, is an artifact of calculations. Computation of the phonon dispersion curves in the vicinity of the Ð point revealed three acoustic branches !?qî, whose frequencies increased monotonically with increasing q, but gave negative values of !?0î (about 8 cm Ð1 ) in the limit q ! 0 because of numerical errors. After application of the acoustic sum rule, !?0î restored its zero value, but the derivative d!=dq near q Ì 0 for the softest mode became negative. So, there is no contradiction between the phonon spectra and the positive de?niteness of the elastic moduli matrix Cij calculated in Sec. 3.5.

5. Conclusion
In this work, the properties of (001)-oriented ?BaTiO3 îm =?SrTiO3 în superlattices with m Ì n Ì 1Ð4 were calculated using the ?rst-principles density functional theory. For free-standing superlattices, the ground state is the monoclinic Cm polar phase. Under the in-plane compressive epitaxial strain, it transforms to tetragonal polar P4mm phase,

Fig. 8. Frequencies of acoustic modes in the vicinity of the Ð point in polar phases of (BTO)1/(STO)1 superlattice as a function of the in-plane lattice parameter a0 . Vertical lines indicate the phase boundaries.

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and under in-plane tensile strain, it transforms to orthorhombic Amm2 polar phase. All components of the static dielectric tensor ("ij ), the piezoelectric tensor (ei ), and the elastic compliance tensor (S ) were calculated as a function of the in-plane lattice parameter for 1/1 superlattice. The critical behavior of some components of "ij , ei ,and S tensors at the boundaries between di?erent polar phases was observed. It was shown that the phase transitions between di?erent polar phases are of the improper ferroelastic type. The possibility of obtaining ultrahigh piezoelectric moduli using ?ne tuning of the epitaxial strain in superlattices was demonstrated. The comparison of the mixing enthalpy calculated for superlattices and disordered Ba0:5 Sr0:5 TiO3 solid solution modeled using two special quasirandom structures SQS-4 revealed a tendency of the BaTiO3ÐSrTiO3 system to short-range ordering of cations and showed that short-period superlattices are thermodynamically quite stable.

Acknowledgment
This work was supported by the RFBR Grant No. 08-02-01436.

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