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ISSN 1063 7834, Physics of the Solid State, 2009, Vol. 51, No. 11, pp. 23242333. © Pleiades Publishing, Ltd., 2009. Original Russian Text © A.I. Lebedev, 2009, published in Fizika Tverdogo Tela, 2009, Vol. 51, No. 11, pp. 21902198.

MAGNETISM AND FERROELECTRICITY

Ab Initio Studies of Dielectric, Piezoelectric, and Elastic Properties of BaTiO3/SrTiO3 Ferroelectric Superlattices
A. I. Lebedev
Moscow State University, Moscow, 119991 Russia e mail: swan@scon155.phys.msu.su
Received March 2, 2009

Abstract--The phonon spectrum; crystal structure of the polar phase; spontaneous polarization; dielectric constant, piezoelectric, and elastic moduli tensors for free standing and substrate supported superlattices mBaTiO3/nSrTiO3 (with m = n = 14) were calculated within the density functional theory. The simulation of properties of the disordered Ba0.5Sr0.5TiO3 solid solution using two special quasirandom SQS 4 structures and their comparison with the properties of the superlattices revealed a tendency of the BaTiO3SrTiO3 sys tem to superstructure ordering and showed that the superlattices are thermodynamically quite stable. The ground state of the free standing superlattice corresponds to the monoclinic polar phase Cm, which trans forms to the tetragonal polar phase P4mm under in plane compressive strain of the superlattice and to the orthorhombic polar phase Amm2 under in plane tensile strain. With a change in the in plane lattice param eter, in the vicinity of boundaries between neighboring polar phases, some optical and acoustic modes soften and some components of the static dielectric constant, piezoelectric, and elastic moduli tensors diverge crit ically. PACS numbers: 64.60. i, 68.65.Cd, 77.84.Dy, 81.05.Zx DOI: 10.1134/S1063783409110225

1. INTRODUCTION Progress in the growth of ferroelectric superlattices (SLs) with a layer thickness controlled with accuracy of one monolayer has opened up new ways for the preparation of multifunctional ferroelectric materials with high values of the spontaneous polarization, Curie temperature, dielectric constant, and its nonlin earity in the electric field. In view of the fact that the properties of ferroelectric superlattices have been poorly studied experimentally to date, their theoreti cal analysis can enable one to find promising direc tions of investigation and application of these new materials. The study of thin epitaxial films of ferroelectrics with a perovskite structure has demonstrated that their properties differ noticeably from those of bulk sam ples. It has been established that the properties of films are most strongly affected by the mechanical strain produced in them by the substrate. Owing to the strong coupling between the mechanical strain and polariza tion, these strains have a substantial effect on the fer roelectric phase transition temperature and can lead to the appearance of unusual polar states in thin films [14]. Among ferroelectric superlattices, strained BaTiO3/SrTiO3 superlattices have been experimentally investigated most thoroughly to date [520]. The first principles calculations of the properties of these super lattices [18, 19, 2128] have made it possible to reveal

the main effects responsible for the formation of the polar state in these structures. The specific feature of the superlattices under consideration is that the mechanical strain arising in them due to the difference between the lattice parameters in BaTiO3 and SrTiO3 lead to the competition between the polar states in neighboring layers of the superlattice; as a result, the polar phase in the superlattice can appear to be tetrag onal, monoclinic, or orthorhombic depending on the mechanical boundary conditions at the interface with the substrate. However, although the properties of the BaTiO3/SrTiO3 superlattices have been studied in suf ficient detail, a number of questions remain open. In particular, first principles studies of the dielectric properties of these superlattices [23, 24] have revealed only the polar phases P4mm and Cm, whereas the phase Amm2, which is characteristic of stretched BaTiO3 films [2] and PbTiO3/PbZrO3 superlattices [29], has not been found. Fragmentary data on the piezoelectric properties are available only for PbTiO3/PbZrO3 superlattices [30], whereas these data for BaTiO3/SrTiO3 superlattices are absent. Finally, the elastic properties of ferroelectric superlattices and specific features of the behavior of these properties at boundaries between polar phases with different sym metries have not been previously studied at all. This paper reports on the results of the calculations of the phonon spectrum; crystal structure of the polar

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phase; spontaneous polarization; and dielectric con stant, piezoelectric, and elastic moduli tensors for free standing and substrate supported superlattices mBaTiO3/nSrTiO3 (m/n superlattices) with m = n = 1 4. For the 1/1 superlattice as an example, we thor oughly studied the influence of the compression (ten sion) of the superlattice in the layer plane on the struc ture and properties of the polar phase; determined the boundaries between the stability regions of the tetrag onal, orthorhombic, and monoclinic polar phases; and investigated the specific features of the behavior of the static dielectric constant, piezoelectric, and elastic moduli tensors in the vicinity of these boundaries. Moreover, in the present work, we analyzed the impor tant question regarding the thermodynamic stability of the BaTiO3/SrTiO3 superlattices. 2. CALCULATION TECHNIQUE The calculations were carried out within the den sity functional theory with the pseudopotentials and the plane wave expansion of wave functions as imple mented in the ABINIT code [31]. The exchangecor relation interaction was described within the local density approximation according to the procedure proposed in [32]. As pseudopotentials, we used the optimized separable nonlocal pseudopotentials [33] generated with the OPIUM code to which the local potential was added in order to improve their transfer ability [34]. The parameters used for constructing the pseudopotentials, the results of their testing, and other details of calculations are described in [35]. In order to increase the accuracy in the determination of the ori entation of the polarization vector in the monoclinic phase, the relaxation of atomic positions was per formed until the HellmannFeynman forces decreased below 5 в 106 Ha/Bohr. The phonon spec tra and the dielectric, piezoelectric, and elastic prop erties were calculated in the framework of the density functional theory from the formulas obtained from the perturbation theory. The phonon contribution to the static dielectric constant tensor was calculated from the determined frequencies of phonons and their oscillator strengths [36]. The spontaneous polariza tion Ps was calculated by the Berry's phase method [37]. As a rule, epitaxial layers of superlattices are grown on substrates from a material with a cubic structure and the 100 orientation and the structure of atomic layers reproduces the substrate structure. In this respect, the calculations were carried out for pseudot etragonal lattices, in which the translation vectors in the layer plane have identical length and are directed perpendicular to each other. This means that, in the case of the monoclinic and orthorhombic polar phases, an insignificant (smaller than 0.07°) deviation of angles from 90° was ignored in the calculations. It
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was shown that this does not lead to a significant change in the results. 3. RESULTS 3.1. Thermodynamic Properties For the practical use of the ferroelectric superlat tices BaTiO3/SrTiO3, the question of their thermody namic stability is of crucial importance. This charac teristic is determined by the enthalpy of mixing of the superlattice and its relationship to the enthalpy of mixing of the disordered Ba0.5Sr0.5TiO3 solid solution. The most complex part in the first principles calcula tion of the enthalpy of mixing is the determination of characteristics of the solid solution, because the direct simulation of lattices with a large number of randomly located atoms makes this problem almost unsolvable. A fundamentally new approach to the solution of this problem was proposed by Zunger et al. [38], who simulated a disordered solid solution with a special quasirandom structure (SQS), i.e., a short period superstructure, for which, in the determinate filling of the lattice sites by A and B atoms, the statistical char acteristics (the numbers of atomic pairs NAA, NBB, and NAB in several nearest shells) are closest to the corre sponding characteristics of an ideal solid solution. This method has been widely used to calculate the electronic structure and physical properties of semi conductor solid solutions and properties of ordering metal alloys. The properties of ferroelectric solid solu tions have been rarely studied using this method [39, 40]. The structure of the disordered Ba0.5Sr0.5TiO3 solid solution was simulated with two quasirandom struc tures SQS 4 with rhombohedral and monoclinic unit cells constructed with the gensqs program of the ATAT code [41]. The translation vectors, order of filling of atomic layers, and pair correlation functions 2. m (where m is the number of the shell) describing the deviation of the statistical characteristics of these structures from the ideal solid solution are listed in Table 1. It follows from this table that, in the SQS 4a structure, a noticeable deviation for the values of NAA, NBB, and NAB arises only in the fourth shell, and the SQS 4b structure is characterized by small deviations in the second and fourth shells. The enthalpy of mix ing H for the X structures under investigation (super lattices with different periods and SQS 4 structures) was calculated using the formula H = E tot ( X ) [ E tot ( BaTiO 3 ) + E tot ( SrTiO 3 ) ] /2 from the values of the total energy Etot (per formula unit) for completely relaxed nonpolar phases. The val ues of H obtained for these structures are presented in Table 2. An unexpected result of these calculations was that the value of H for the two shortest period superlat
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Table 1. Translation vectors, order of filling of atomic layers, and statistical characteristics of the SQS 4 structures used for simulating the disordered Ba0.5Sr0.5TiO3 solid solution Structure SQS 4a SQS 4b Vectors [ 211 ], [ 112 ], [ 121 ] [ 210 ], [ 210 ], [ 001 ] Direction AABB 111 AABB 120
2, 1

2, 2

2, 3

2, 4

0 0

0 1/3

0 0

1 1/3

tices appeared to be smaller than the value of H for both realizations of the disordered solid solution. This means that the BaTiO3SrTiO3 system is prone to superstructure ordering of the components. Small val ues of H for the 1/1 and 2/2 superlattices indicate that the short period superlattices are thermodynami cally quite stable. Our results differ from the results of the calcula tions of thermodynamic properties of the BaTiO3 SrTiO3 system by Fuks et al. [42]. The analysis of the calculation technique used in this work showed that, in the calculation of the energies of different superstruc tures, the authors of [42] did not perform the relax ation of atomic positions and believed that the lattice remains cubic. Therefore, their values of the enthalpy of mixing for the 1/1 superlattice (42 meV per formula unit) include a large excess energy of elastic stresses. It should be noted that the SQS 4 structures describe rather realistically the local distortions of the solid solution structure. In particular, the predicted interatomic distances in the nonpolar phase are in the ranges 2.7382.782 е (SQS 4a) and 2.7312.782 е (SQS 4b) for the SrO atomic pair and 2.782 2.815 е (SQS 4a) and 2.7762.826 е (SQS 4b) for the BaO atomic pair while the interatomic distances in the reference cubic strontium and barium titanates are equal to 2.750 and 2.809 е. The obtained results are in agreement with the bimodal distribution of bond lengths experimentally revealed for many solid solu tions by the EXAFS method.
Table 2. Enthalpies of mixing for several BaTiO3/SrTiO3 superlattices with different layer thicknesses and two SQS 4 structures simulating the disordered Ba0.5Sr0.5TiO3 solid solution X structure 1/1 SL 2/2 SL 3/3 SL 4/4 SL SQS 4a SQS 4b H, meV 2.9 8.9 11.4 12.6 16.8 11.0

3.2. Ground State of the Strained Superlattice As was noted in Introduction, the sequentially arranged layers in the BaTiO3/SrTiO3 superlattice pro duce mutual mechanical stresses in each other; as a result, the SrTiO3 layers appear to be stretched in the plane and the BaTiO3 layers turn out to be compressed. If the layers would be isolated, these strains should lead to the appearance of the spontaneous polarization with the vector Ps lying in the layer plane (space group Amm2) in the SrTiO3 layers and to an increase in the polarization directed perpendicularly or at an angle to this plane (space group P4mm or Cm) in the BaTiO3 layers [13]. Since the state with a polarization rapidly varying in space is energetically unfavorable, the ques tion of the orientation of the polarization vector in the ground state of the superlattice needs special consider ation. Previous investigations of PbTiO3/PbZrO3 [29] and BaTiO3/SrTiO3 [23, 24] superlattices enable us to expect that the value and orientation of the vector Ps will depend on the substrate induced strain in the superlattice. The ground state of the superlattice was searched as follows. For a specified in plane lattice parameter a0 (varying in the range from 7.35 to 7.50 Bohr), the equilibrium structure of the nonpolar phase with space group P4/mmm was initially determined by minimiz ing the HellmannFeynman forces. Then, for this structure, the frequencies of the phonon spectrum were calculated at the point. The stable state of the superlattice is characterized by positive values of all frequencies at all points of the Brillouin zone. There fore, if there were unstable modes (with imaginary mode frequencies) in the phonon spectrum, small per turbations corresponding to the eigenvector of the less stable mode were introduced into the structure and the equilibrium structure of the distorted phase was calcu lated. The phonon spectrum and equilibrium structure calculations were repeated until the structure with all positive mode frequencies was found. It should be noted that, in the nonpolar phase of the 1/1 BaTiO3/SrTiO3 superlattice, the ferroelectric mode at the point is the only unstable mode. The well known instability of the phonon spectrum of SrTiO3 at the R point of the Brillouin zone disappears in changing over to the superlattice: the phonon energy at the M point of the folded Brillouin zone (into which the R point transforms when doubling the

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period along the c axis) in the free standing 1/1 super lattice is equal to 56 cm1. The calculations demonstrated that, for the 1/1 and 2/2 superlattices supported at the SrTiO3 substrate (the in plane lattice parameter a0 is equal to that of cubic strontium titanate, a0 = 7.3506 Bohr), the ground state corresponds to the tetragonal polar phase with space group P4mm. For the free standing 1/1 and 2/2 superlattices, the P4mm phase appears to be unsta ble, undergoes a distortion, and transforms into the monoclinic polar phase Cm. In the 1/1 and 2/2 super lattices stretched in the layer plane, the orthorhombic Amm2 phase turns out to be most stable. Therefore, the polar state of the structure can be rather finely con trolled by varying the value of a0 (for example, by growing superlattices on different substrates). In order to accurately determine the position of the boundaries between the P4mm and Cm phases and between the Cm and Amm2 phases, the ground state of the 1/1 superlattice was found for a set of lattice parameters a0 and the phonon frequencies for each of these structures were calculated at the point. The dependence of the four lowest frequencies on a0 in phases stable for each lattice parameter is plotted in Fig. 1. It can be seen that, as the boundary between the P4mm and Cm phases is approached from the side of the tetragonal phase, the frequency of the doubly degenerate mode E decreases critically. After the tran sition to the monoclinic phase, there arise two nonde generate soft modes A' and A'' in the phonon spectrum, of which the former mode again softens as the bound ary between the Cm and Amm2 phases is approached. The soft phonon mode on the side of the Amm2 phase has a symmetry of B1. By extrapolating (to zero) the dependence of the square of the frequency of the soft ferroelectric modes E in the P4mm phase, A' in the Cm phase, and B1 in the Amm2 phase on a0, we determined the lattice parame ters corresponding to the boundaries between the polar phases. For the P4mmCm transition, the boundary is located at a0 = 7.4023 Bohr in extrapola tion from the tetragonal phase and 7.4001 Bohr in extrapolation from the monoclinic phase. For the CmAmm2 transition, the boundary is located at a0 = 7.4489 Bohr in extrapolation from the orthorhombic phase and 7.4483 Bohr in extrapolation from the mon

120

E

A'' A'

2

1

, cm1

80 E

A' A''

B

2

40 P4mm 0 7.35 7.40
a0, Bohr

B Cm 7.45

1

Amm2 7.50

Fig. 1. Dependence of the frequencies of four lowest fre quency phonon modes in the polar phase of the 1/1 BaTiO3/SrTiO3 superlattice on the lattice parameter a0. The mode symmetry is shown near the curves. Vertical lines indicate the phase stability boundaries.

oclinic phase. A small difference between the values found through the extrapolation from two sides of the transition enables us to believe that both transitions are close to second order transitions. Since zero strain in the plane of polarized film corresponds to the lattice parameter a0 = 7.4462 Bohr, the superlattice strains corresponding to the phase stability boundaries are equal to 0.605 and + 0.032%. 3.3. Spontaneous Polarization As was revealed in Subsection 3.2, the ground state for the 1/1 and 2/2 superlattices supported at the SrTiO3 substrate corresponds to the tetragonal phase P4mm with the polarization vector directed perpen dicularly to the layer plane. The calculations showed that, for supported n/n superlattices, the spontaneous polarization Ps increases monotonically (from 0.277 to 0.307 C/m2) with an increase in the layer thickness from n = 1 to 4 (Table 3). For the free standing 1/1 and 2/2 superlattices, the ground state corresponds to the monoclinic Cm phase, in which the polarization vec tor is rotated in the ( 110 ) plane from the c axis of the tetragonal nonpolar phase by an angle of 75° for the

Table 3. Spontaneous polarization in the BaTiO3/SrTiO3 superlattices with different layer thicknesses, two SQS 4 struc tures simulating the disordered Ba0.5Sr0.5TiO3 solid solution, and tetragonal barium titanate Structure Orienta tion of Ps [xxz] P s, C/m2 0.241 1/1 SL [001]* 0.277 [xxz] 0.252 2/2 SL [001]* 0.293 3/3 SL [001]* 0.302 4/4 SL [001]* 0.307 SQS 4a [111] 0.225 SQS 4b [001] 0.206 BaTiO3 [001] 0.259

* Data for the superlattices supported at SrTiO3 substrate. PHYSICS OF THE SOLID STATE Vol. 51 No. 11 2009

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4

0.3 Polarization, C/m2 P 0.2 P
z

10

P4mm

Cm

Amm2

10
ij

3

11

0.1 P4mm 0 7.35 7.40
a0, Bohr
Fig. 2. Decomposition of the polarization vector in the polar phase of the 1/1 BaTiO3/SrTiO3 superlattice into the components lying in the layer plane (P) and perpendicu lar to it (Pz) as a function of the lattice parameter a0. Verti cal lines indicate the phase stability boundaries.

33

Cm

Amm2

10

2

33

7.45

7.50

7.35

7.40
a0, Bohr

7.45

7.50

Fig. 3. Dependence of the eigenvalues of the static dielec tric constant tensor ij in the polar phase of the 1/1 BaTiO3/SrTiO3 superlattice on the lattice parameter a0. Vertical lines indicate the phase stability boundaries. Dif ferent symbols represent three different eigenvalues of the dielectric constant tensor.

1/1 superlattice and 63° for the 2/2 superlattice (the values of Ps are given in Table 3). In the 1/1 and 2/2 superlattices stretched in the layer plane, the orthor hombic phase Amm2, in which the polarization vector lies in the layer plane and is aligned with the 110 axis of the tetragonal lattice of the nonpolar phase, appears to be the most stable. Apart from the data for the superlattices, the calcu lated values of the spontaneous polarization for the tetragonal BaTiO3 phase and the disordered Ba0.5Sr0.5TiO3 solid solution simulated with the SQS 4 structures (see Subsection 3.1) are listed in Table 3. It follows from the table that the values of Ps in all super lattices are larger than those in the solid solution and Ps in superlattices supported at the SrTiO3 substrate even exceeds the value for tetragonal BaTiO3. These results agree with the experiment [12]. Figure 2 shows the dependence of the components of polarization, which are parallel and perpendicular to the layer plane, in the polar phase of the 1/1 super lattice on the lattice parameter a0. By extrapolating the dependence of P and P z on a0, we find the position of the boundaries for the P4mmCm and CmAmm2 transitions: a0 = 7.4018 Bohr and a0 = 7.4492 Bohr, which are very close to the values obtained in Subsec tion 3.2 from analyzing the dependence of the fre quency of the ferroelectric mode on a0. 3.4. Dielectric Properties The dependence of the eigenvalues of the static dielectric constant tensor ij (i, j = 1, ..., 3) for the 1/1 superlattice on the in plane lattice parameter a0 is plotted in Fig. 3. In the tetragonal phase, the eigenvec
2 2

tors of the tensor ij are oriented along the crystallo graphic axes and the eigenvalues 11 and 22 coincide with each other. Therefore, the dielectric properties of this phase are described by two nonzero independent quantities 11 and 33. In the monoclinic phase, the polarization vector continuously rotates in the ( 110 ) plane and none of the eigenvectors of the tensor ij coincide with the crystallographic axes of the tetragonal lattice of the nonpolar phase. All three eigenvalues are different. In the situation when all nine components of the tensor ij in the coordinate system of the tetragonal lattice differ from zero, the representation of the eigenvalues of the tensor turns out to be most compact. In the mono clinic phase, the direction of the eigenvector corre sponding to the smallest eigenvalue is close but does not coincide with the polarization direction. In the orthorhombic phase, the eigenvectors of the tensor ij are oriented along the 110, 1 10 , and 001 directions of the tetragonal lattice of the nonpo lar phase. The vector corresponding to the smallest eigenvalue coincides in direction with the polarization vector, and the vector corresponding to the largest eigenvalue is directed along the 001 axis. It follows from Fig. 3 that, with a change in the lat tice parameter a0, at least one of the eigenvalues of the tensor ij is characterized by the divergence at the boundaries between the P4mm and Cm phases and between the Cm and Amm2 phases. As the P4mmCm phase boundary is approached from the tetragonal phase, the components 11 = 22 of this tensor diverge, that is associated with the weakening of the stability of the polarization vector Ps parallel to 001 with respect
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Table 4. Values of the piezoelectric moduli in the monoclinic phase of the freely suspended superlattice BaTiO3/SrTiO3, the tetragonal phase of the same superlattice supported at the SrTiO3 substrate, and the tetragonal phase of barium titanate Structure Orientation of Ps [xxz] e33, (d33, pC/N) 2 e15, C/m (d15, pC/N) C/m2 31.9 (460) 0.09 (19) 1/1 SL [001]* 7.1 (49) 3.2 (31) BaTiO3 [001] 6.3 (42) 2.9 (24)

* Data for the superlattice supported at SrTiO3 substrate.

to its rotation in the ( 110 ) plane. As the boundary between the Cm and Amm2 phases is approached from the orthorhombic phase, the component 33 diverges, that is associated with the decrease in the stability of the polarization vector Ps parallel to 110 with respect to its rotation in the same plane. 3.5. Piezoelectric Properties A high sensitivity of the value and direction of the polarization vector in the superlattices to strains enables us to expect that a strong piezoelectric effect appears in them. It is known that anomalously high piezoelectric moduli characteristic of ferroelectrics of the PbZr1 xTixO3 type in the vicinity of the morpho tropic boundary are associated with the easiness of rotation of the polarization vector in the intermediate monoclinic phase under the lattice strain [43, 44]. A similar situation occurs in the monoclinic phase of the BaTiO3/SrTiO3 superlattice. As far as we know, up to now, the piezoelectric properties of BaTiO3/SrTiO3 ferroelectric superlattices have been studied neither experimentally nor theoretically. The only superlattice for which the piezoelectric properties were calculated was the 1/1 PbTiO3/PbZrO3 superlattice, which was used to simulate the properties of the PbTi0.5Zr0.5O3 solid solution [45]. The results of the calculations of the largest piezo electric moduli ei (i = 1, ..., 3; = 1, ..., 6) for the tet ragonal polar phase of the 1/1 BaTiO3/SrTiO3 super lattice supported at the SrTiO3 substrate and for the monoclinic phase in the same free standing superlat tice are presented in Table 4. It can be seen that the modulus e33 in the tetragonal phase differs little from the modulus for tetragonal BaTiO3. However, for the monoclinic phase, which corresponds to the ground state of the free standing superlattice, the value of e33 appears to be five times larger. A stronger increase in the piezoelectric coefficient d55 expressed in pC/N 6 (di = e S ) is favored by an increase in the = 1 i elastic compliance modulus S33 by a factor of almost 1.5 upon transition to the monoclinic phase (see in more detail in Subsection 3.6).

Changes in the piezoelectric moduli ei in the polar phase of the 1/1 superlattice with a change in the lat tice parameter a0 are shown in Fig. 4. In the tetragonal phase, according to the symmetry principle, the mod uli e31 = e32, e33 and e15 = e24 (a total of three indepen dent parameters) differ from zero. In the superlattices under investigation, only two of these moduli, i.e., e33 and e15, have noticeable values. As the boundary between the P4mm and Cm phases is approached from the tetragonal phase, the value of e33 increases mono tonically and the modulus e15 exhibits a critical behav ior and reaches a record high value of 80 C/m2 in the vicinity of the boundary. In the orthorhombic phase in the coordinate sys tem related to the axes of the tetragonal lattice of the nonpolar phase, the moduli e11 = e22, e12 = e21, e13 = e23, e34 = e35, and e16 = e26 (a total of five independent parameters) are nonzero. Among them, the moduli e11, e12, e34, and e16 are noticeable (Fig. 4). As the boundary between the Cm and Amm2 phases is approached, the critical behavior is observed only for

40

e e e e e

11 12 13 14 15

e16 e31 e33 e34 e36

eiv, C/m2

20

0 P4mm Cm 7.40
a0, Bohr

Amm2 7.45 7.50

20

7.35

Fig. 4. Dependence of the piezoelectric moduli ei in the polar phase of the 1/1 BaTiO3/SrTiO3 superlattice on the lattice parameter a0. Vertical lines indicate the phase sta bility boundaries. 2009

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3 2
S, 10-2 GPa-
1

P4mm

Cm

Amm2

1 0 -1 -2 -3 S11 S12 S13 S14 S15 7.35 S16 S33 S44 S45 S66 7.40
a0, Bohr
Fig. 5. Dependence of the components of the elastic com pliance tensor S in the polar phase of the 1/1 BaTiO3/SrTiO3 superlattice on the lattice parameter a0. Vertical lines indicate the phase stability boundaries.

7.45

7.50

the modulus e34, which reaches a value of 192 C/m2 in the vicinity of the boundary. In the monoclinic phase, the behavior of the piezo electric moduli turns out to be the most complex, because a change in a0 leads to a continuous rotation of the polarization vector in the ( 110 ) plane and all 18 components of the tensor ei in the coordinate system of the tetragonal lattice of the nonpolar phase differ from zero (the number of independent parameters is equal to ten). It follows from Fig. 4 that, in the mono clinic phase, the critical divergence is observed for the moduli e11 = e22, e15 = e24, e12 = e21, e13 = e23, e14 = e25, and e16 = e26 at the boundary between the Cm and P4mm phases and the moduli e33, e34 = e35, e31 = e32, and e36 at the boundary between the Cm and Amm2 phases. It should also be noted that the phase transi tion is accompanied by a small (by ~10%) stepwise change in the modulus e33 (at the CmP4mm bound ary) and the moduli e11 and e12 (at the CmAmm2 boundary). 3.6. Elastic Properties As is known, a ferroelectric phase transition under going in a polar crystal is frequently appears to be fer roelastic; i.e., it is accompanied by the appearance of soft acoustic modes. This takes place if symmetric sec ond rank tensor (strain tensor) and polar vector trans form according to the same irreducible representation [46]. Since the change in the parameter a0 leads to transitions between different polar phases in the BaTiO3/SrTiO3 superlattices, it is of interest to study the influence of these phase transitions on the elastic properties of superlattices. This is especially important

because the elastic properties of superlattices have not been investigated at all to date. The elastic compliance tensor S (, = 1, ..., 6) is characterized by six independent and nine nonzero components in the tetragonal P4mm phase, nine inde pendent and thirteen nonzero components in the orthorhombic Amm2 phase (in our coordinate system, the axes of which coincide with those of the tetragonal lattice of the nonpolar phase), and thirteen indepen dent and twenty one nonzero components in the mon oclinic Cm phase. The results of the calculations of the components of the elastic compliance tensor for the 1/1 superlat tice in the polar state are presented in Fig. 5. It can be seen that, in the vicinity of the boundary between the P4mm and Cm phases, the components of the tensor S undergo either a stepwise change (moduli S13 = S23, S33, S66, S16 = S26, S36, S34 = S35, S46 = S56), or a critical divergence on the side of the monoclinic phase (moduli S11 = S22, S12, S15 = S24, S14 = S25, S45), or a critical divergence on both sides of the boundary between the phases (modulus S44 = S55). Upon transi tion to the monoclinic phase, the moduli S14, S15, S16, S34, S36, S45, and S46 become nonzero. Upon transition to the orthorhombic phase, the moduli S14, S15, S34, and S46 vanish again and the other moduli remain nonzero. At the boundary between the Cm and Amm2 phases, the anomalies of the elastic moduli are less pronounced: the moduli S11, S12, and S66 undergo a stepwise change; the moduli S13, S14, S15, S16, S33, S34, S36, and S46 are characterized by a weak divergence on the side of the monoclinic phase; and the moduli S44 and S45 exhibit a weak divergence on both sides of the boundary. The critical divergence of the modulus S44 at both boundaries P4mmCm and CmAmm2 suggests that the phase transitions occurring under continuous ten sion of the superlattice in the layer plane are ferroelas tic. 4. DISCUSSION OF THE RESULTS The results obtained in this work, which describe the influence of strains on the crystal structure of the BaTiO3/SrTiO3 superlattice in its ground state, are not in complete agreement with the results of the calcula tions performed in [18, 2224]. These results agree with each other in that the ground state of the free standing 1/1 BaTiO3/SrTiO3 superlattice corresponds to the monoclinic Cm phase, the in plane compres sion leads to the transition between the Cm and P4mm phases, and the dielectric constant at the phase boundary exhibits a divergence. However, unlike our results, the authors of [23, 24] did not succeeded in revealing the transition between the monoclinic and orthorhombic polar phases, even though they observed an almost complete rotation of the polarization vector
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to the 110 direction. The absence of this transition was indicated in [24] by the fact that the frequencies of all optical modes in the Cm phase remained real and increased under tension of the superlattice. We believe that our results are more correct because they are in agreement with the results obtained for BaTiO3 [2] and SrTiO3 [47] thin strained films, the data of the recent calculations for the 2/2 BaTiO3/SrTiO3 superlattice [25], and the results of the studies of one more PbTiO3/PbZrO3 superlattice [29]. In all these works the phase sequence P4mmCmAmm2 was observed under tension of superlattices in the layer plane. Small differences between our data for the P4mm Cm transition and the data obtained in [23] lie in the value of the minimum spontaneous polarization, which appeared to be 75% higher in our work, and in a somewhat different position of the boundary between the tetragonal and monoclinic phases. Unfortunately, an insufficient amount of data on the dielectric properties of BaTiO3/SrTiO3 superlat tices in [24] does not enable us to thoroughly compare them with our results. However, a comparison of our data with the results of the calculations of the dielec tric properties of PbTiO3/PbZrO3 superlattices [29] demonstrates that all eigenvalues of the dielectric con stant tensor in the monoclinic phase of the BaTiO3/SrTiO3 superlattices turn out to be higher; i.e., these superlattices can be more promising for practical applications. Before preceding to the discussion of the piezo electric properties of the strained BaTiO3/SrTiO3 superlattices, it should be noted that the situations in which characteristics of superlattices, such as ei and S, exhibit a divergence can be of greatest practical importance. As was shown in Subsections 3.5 and 3.6, these situations arise in the vicinity of the boundaries between the phases. The calculations demonstrate that, in the vicinity of the P4mmCm boundary, the piezoelectric coefficient d11, which reached a value of 2300 pC/N, is the largest in the monoclinic phase, and the coefficient d15, which reached a value of 6200 pC/N, is the largest in the tetragonal phase. In the vicinity of the CmAmm2 boundary, the coeffi cient d33, which reached a value of 9200 pC/N, is the largest in the monoclinic phase, and the coefficient d34 (10 500 pC/N) is the largest in the orthorhombic phase. For comparison, we note that the record value of the piezoelectric coefficient obtained for crystals in the Pb(Zn1/3Nb2/3)O3PbTiO3 system is 2500 pC/N [48]. An anomalous increase in the calculated coeffi cients d15 and d33 to values of ~8500 pC/N in the vicin ity of the P4mmCm boundary was also predicted for isotropically compressed PbTiO3 [49]. Now, we discuss the elastic properties of the super lattices and the results indicating the appearance of ferroelastic phase transitions. According to [46], the phase transition 4mm m in crystals with space
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group P4mm should exhibit ferroelastic properties. Upon this phase transition, lattice distortions are characterized by one or both nonzero strain tensor components u4 and u5, and a soft transverse acoustic phonon with the wave vector directed along the polar axis should appear in the phonon spectrum. The direct calculations of the frequencies of acoustic modes in the tetragonal phase at the point of the Brillouin zone with the reduced wave vector q = (0, 0, 0.05) con firmed this fact: as the P4mmCm boundary is approached, the frequency of the doubly degenerate transverse acoustic phonon with symmetry 5 decreases by a factor of five. The softening of this phonon is directly associated with the divergence of the elastic compliance moduli S44 = S55. The ferroelastic phase transition should also occur at the boundary between the Cm and Amm2 phases. According to [46], the transition mm2 m should be accompanied by the lattice distortion described by one nonzero of three strain tensor components u4, u5, and u6. In the chosen coordinate system with the orienta tion of the polar axis along the 110 direction (unusual for the orthorhombic lattice), the softening of the transverse acoustic phonon with a wave vector directed along this axis should appear. The direct cal culations of the frequencies of acoustic modes in the orthorhombic phase at the point of the Brillouin zone with the reduced wave vector q = (0.035, 0.035, 0) confirmed this fact. Most likely, the specific orienta tion of the polar axis is responsible for the fact that the anomalies in the elastic compliance tensor at the fer roelastic phase transition Cm Amm2 are rather weakly pronounced. For the same reason, some com ponents of the elastic compliance tensor appear to be related to each other: for example, the moduli S44 and S45 that exhibit the divergence in the orthorhombic phase satisfy the expression S44 S45 const in this phase. An intriguing feature revealed in our work is that the character of atomic displacements in unstable fer roelastic modes in the 1/1 and 2/2 superlattices in the P4/mmm phase are quantitatively different: all metal atoms in the 1/1 superlattice move in antiphase with oxygen atoms for all three polarizations of vibrations, whereas, in the 2/2 superlattice, this holds true only for the A2u mode. For the Eu mode, the amplitude of the displacement of different titanium atoms differs by a factor of approximately three and both Ba atoms vibrate in phase with O atoms and antiphase with Ti and Sr atoms. 5. CONCLUSIONS Thus, in the present work, the properties of the mBaTiO3/nSrTiO3 superlattices (with m = n = 1, ..., 4) were calculated within the density functional theory. A comparison of the properties of the superlattices with the properties of the disordered Ba0.5Sr0.5TiO3 solid
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LEBEDEV 12. T. Shimuta, O. Nakagawara, T. Makino, S. Arai, H. Tabata, and T. Kawai, J. Appl. Phys. 91, 2290 (2002). 13. J. Kim, Y. Kim, Y. S. Kim, J. Lee, L. Kim, and D. Jung, Appl. Phys. Lett. 80, 3581 (2002). 14. S. Rios, A. Ruediger, A. Q. Jiang, J. F. Scott, H. Lu, and Z. Chen, J. Phys.: Condens. Matter 15, L305 (2003). 15. A. Q. Jiang, J. F. Scott, H. Lu, and Z. Chen, J. Appl. Phys. 93, 1180 (2003). 16. F. Q. Tong, W. X. Yu, F. Liu, Y. Zuo, and X. Ge, Mater. Sci. Eng., B 98, 6 (2003). 17. T. Harigai, D. Tanaka, H. Kakemoto, S. Wada, and T. Tsurumi, J. Appl. Phys. 94, 7923 (2003). 18. J. Lee, L. Kim, J. Kim, D. Jung, and U. V. Waghmare, J. Appl. Phys. 100, 051 613 (2006). 19. W. Tian, J. C. Jiang, X. Q. Pan, J. H. Haeni, Y. L. Li, L. Q. Chen, D. G. Schlom, J. B. Neaton, K. M. Rabe, and Q. X. Jia, Appl. Phys. Lett. 89, 092 905 (2006). 20. B. R. Kim, T. U. Kim, W. J. Lee, J. H. Moon, B. T. Lee, H. S. Kim, and J. H. Kim, Thin Solid Films 515, 6438 (2007). 21. J. B. Neaton and K. M. Rabe, Appl. Phys. Lett. 82, 1586 (2003). 22. K. Johnston, X. Huang, J. B. Neaton, and K. M. Rabe, Phys. Rev. B: Condens. Matter 71, 100 103 (2005). 23. L. Kim, J. Kim, D. Jung, J. Lee, and U. V. Waghmare, Appl. Phys. Lett. 87, 052 903 (2005). 24. L. Kim, J. Kim, U.V. Waghmare, D. Jung, and J. Lee, Phys. Rev. B: Condens. Matter 72, 214 121 (2005). 25. S. Lisenkov and L. Bellaiche, Phys. Rev. B: Condens. Matter 76, 020 102 (2007). 26. J. H. Lee, U. V. Waghmare, and J. Yu, J. Appl. Phys. 103, 124 106 (2008). 27. Z. Y. Zhu, H. Y. Zhang, M. Tan, X. H. Zhang, and J. C. Han, J. Phys. D: Appl. Phys. 41, 215 408 (2008). 28. S. Lisenkov, I. Ponomareva, and L. Bellaiche, Phys. Rev. B: Condens. Matter 79, 024 101 (2009). 29. C. Bungaro and K. M. Rabe, Phys. Rev. B: Condens. Matter 69, 184 101 (2004). 30. G. SАghi SzabС, R. E. Cohen, and H. Krakauer, Phys. Rev. B: Condens. Matter 59, 12 771 (1999). 31. X. Gonze, J. M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G. M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami, Ph. Ghosez, J. Y. Raty, and D. C. Allan, Comput. Mater. Sci. 25, 478 (2002). 32. J. P. Perdew and A. Zunger, Phys. Rev. B: Condens. Matter 23, 5048 (1981). 33. A. M. Rappe, K. M. Rabe, E. Kaxiras, and J. D. Joan nopoulos, Phys. Rev. B: Condens. Matter 41, 1227 (1990). 34. N. J. Ramer and A. M. Rappe, Phys. Rev. B: Condens. Matter 59, 12 471 (1999). 35. A. I. Lebedev, Fiz. Tverd. Tela (St. Petersburg) 51 (2), 341 (2009) [Phys. Solid State 51 (2), 362 (2009)]. 36. G. M. Rignanese, X. Gonze, and A. Pasquarello, Phys. Rev. B: Condens. Matter 63, 104 305 (2001). 37. R. D. King Smith and D. Vanderbilt, Phys. Rev. B: Condens. Matter 47, 1651 (1993).
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solution simulated with the special quasirandom structures SQS 4 showed that the BaTiO3SrTiO3 sys tem has a tendency to superstructure ordering of the components and that the superlattices themselves are thermodynamically quite stable. The ground state of the free standing superlattice corresponds to the mon oclinic polar phase Cm, which transforms to the tet ragonal polar phase P4mm under the in plane com pression of the superlattice and to the orthorhombic polar phase Amm2 under the in plane tension of the superlattice. All components of the dielectric constant (ij), piezoelectric moduli (ei), and elastic moduli (S) tensors were calculated as a function of the in plane strain of the superlattices. It was demonstrated that the dielectric constant in the monoclinic phase of the superlattices is noticeably higher than that in the same phase of the PbTiO3/PbZrO3 superlattices. The critical behavior of the components of the tensors ij, ei, and S was studied in the vicinity of the bound aries between different polar phases, and it was shown that the transitions between these phases are ferroelas tic. It was demonstrated that uniquely high piezoelec tric parameters can be achieved by finely controlling the superlattice strain. ACKNOWLEDGMENTS This work was supported by the Russian Founda tion for Basic Research (project no. 08 02 01436). REFERENCES
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