Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://scon155.phys.msu.su/publ/PhysSolidState_52_1448.pdf Äàòà èçìåíåíèÿ: Mon Jun 21 11:27:41 2010 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 20:29:24 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 63

ISSN 1063 7834, Physics of the Solid State, 2010, Vol. 52, No. 7, pp. 1448­1462. © Pleiades Publishing, Ltd., 2010. Original Russian Text © A.I. Lebedev, 2010, published in Fizika Tverdogo Tela, 2010, Vol. 52, No. 7, pp. 1351­1364.

MAGNETISM AND FERROELECTRICITY

Ground State and Properties of Ferroelectric Superlattices Based on Crystals of the Perovskite Family
A. I. Lebedev
Moscow State University, Moscow, 119991 Russia e mail: swan@scon155.phys.msu.ru
Received November 25, 2009

Abstract--The crystal structure of the ground state of ten free standing ferroelectric superlattices based on crystals with the perovskite structure (BaTiO3/SrTiO3, PbTiO3/SrTiO3, PbTiO3/PbZrO3, SrZrO3/SrTiO3, PbZrO3/BaZrO3, BaTiO3/BaZrO3, PbTiO3/BaTiO3, BaTiO3/CaTiO3, KNbO3/KTaO3, and KNbO3/NaNbO3) was calculated from first principles within the density functional theory taking into account criteria for stability of the structures with respect to acoustic and optical distortions. It was shown that the ground state in all the considered superlattices corresponds to the ferroelectric phase. It was found that the polarization vector has a tendency toward a tilt to the plane of the superlattice layers, which makes it possible to decrease the electrostatic and elastic energy in the superlattices consisting of materials with differ ent ferroelectric properties. The importance of the inclusion of structural distortions due to unstable phonons at the Brillouin zone boundary, which, in a number of cases, lead to significant changes in ferroelectric and dielectric properties of the superlattices, was demonstrated. DOI: 10.1134/S1063783410070218

1. INTRODUCTION Ferroelectric superlattices, which have been stud ied for a significantly shorter period of time than semi conductor and magnetic superlattices, have attracted much attention. It has appeared that, for these artifi cial structures, many physical parameters, such as the Curie temperature, spontaneous polarization, dielec tric constant, and nonlinear dielectric and optical sus ceptibilities, exceed considerably those of bulk crystals and thin films of solid solutions of the corresponding composition. The search for new approaches that will enable one to improve noticeably the characteristics of ferroelectrics is of great scientific and applied impor tance. Since many properties of ferroelectric superlat tices are still poorly studied experimentally, theoretical calculations occupy a significant place in the modern investigations of these structures. The reliable prediction of the physical properties of superlattices from first principles and the correct interpretation of the experimental results require knowledge of the ground state of the superlattices. A number of calculations of the ground state of the fer roelectric superlattices based on crystals with the per ovskite structure are available in the literature; how ever, in the majority of cases, the calculations have been restricted to the determination of a phase (among several ferroelectric phases with a specified symmetry) in which the relaxation of atomic positions and the unit cell parameters results in a structure with the low est total energy. Unfortunately, this approach cannot be considered as quite correct, because the found solu

tions can correspond not to the true energy minimum (the ground state) but to the saddle point only. An example of such calculations can be provided by cal culations of the properties of the KNbO3/KTaO3, PbTiO3/BaTiO3, PbTiO3/SrTiO3, PbTiO3/PbZrO3, SrZrO3/SrTiO3, BaTiO3/CaTiO3, and KNbO3/NaNbO3 superlattices, which will be dis cussed below. It is well known that, in addition to the ferroelectric instability, perovskite crystals often exhibit a structural instability associated with the presence of phonons at the Brillouin zone boundary. Therefore, the inclusion of the competition between the ferroelectric and struc tural instabilities is of fundamental importance for the correct prediction of the crystal properties. A substan tial influence of this competition on the crystal prop erties can be demonstrated using CaTiO3 as an exam ple. In the parent cubic phase Pm3m, calcium titanate exhibits a ferroelectric instability (the frequency of the 15 mode is equal to 165i cm­1 [1]); however, a stron ger structural instability with respect to phonons at the R and M points of the Brillouin zone results in a reduc tion of the symmetry of the structure to orthorhombic and in a complete suppression of the ferroelectric instability (the frequency of the two softest modes with symmetry B1u and B3u in orthorhombic CaTiO3 is equal to 82 cm­1 [2]). In the present work, we investigated properties of ten short period ferroelectric superlattices based on crystals of the perovskite family, for which experimen tal data or results of theoretical calculations are avail

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Table 1. Electronic configurations of atoms and parameters used for constructing pseudopotentials: rs, rp, rd, and rf are the radii of the pseudopotential core for the s, p, d, and f projections, respectively; qs, qp, qd, and qf are the boundary wave vectors used for optimizing the pseudopotential; and rmin, rmax, and Vloc are the range and depth of the correcting local potential (parameters are given in Hartree atomic units, and the energy is given in Ry) Atom Na K Zr Nb Ta Configuration 2s 3s23p63d04s04p0 4s24p64d05s0 4s24p64d05s0 4f145s25p65d06s0
22p63s03p0

r

s

r

p

r

d

r

f

q

s

q

p

q

d

q

f

r

min

r

max

V

loc

1.18 1.38 1.58 1.76 1.74

1.60 1.42 1.74 1.76 1.74

­ 1.44 1.78 1.76 1.74

­ ­ ­ ­ 1.58

7.07 7.47 7.27 7.07 7.07

7.87 7.27 7.07 7.07 7.07

­ 7.07 7.07 7.07 7.07

­ ­ ­ ­ 8.5

0.01 0.01 0.01 0.01 ­

1.08 1.40 1.30 1.48 ­

1.2 0.65 4.04 2.05 ­

able in the literature: BaTiO3/SrTiO3 (see references in [3] and also [4­16]), PbTiO3/SrTiO3 [17­22], PbTiO3/PbZrO3 [23­30], SrZrO3/SrTiO3 [15, 31­ 34], PbZrO3/BaZrO3 [35, 36], BaTiO3/BaZrO3 [15, 37, 38], PbTiO3/BaTiO3 [39­42], BaTiO3/CaTiO3 [43, 44], KNbO3/KTaO3 [45­53], and KNbO3/NaNbO3 [54]. By sequentially performing the relaxation proce dure for positions of atoms and the unit cell parame ters with the subsequent analysis of the stability of the obtained structures for each of the aforementioned superlattices, we determined the ground state, calcu lated main ferroelectric and dielectric properties, and compared the results of the calculations with the data available in the literature. The purpose of this work was to understand whether the structural data known for many solid solutions can be used for predicting the structure of the ground state of the corresponding superlattices and to reveal how strongly the structural instability manifest ing itself in the superlattices will affect their ferroelec tric properties. The data obtained can be used as a guide in interpreting the results of future experimental studies of superlattices. 2. CALCULATION TECHNIQUE The calculations were carried out from first princi ples within the density functional theory with the pseudopotentials and the plane wave expansion of wave functions as implemented in the ABINIT code [55]. The exchange­correlation interaction was described within the local density approximation according to the procedure proposed in [56]. As pseudopotentials, we used the optimized separable nonlocal pseudopotentials [57] generated with the OPIUM code [58] to which the local potential was added in order to improve their transferability [59]. The pseudopotentials for Sr, Ba, Ca, Pb, Ti, and O atoms were taken from [1]; the parameters used in constructing the pseudopotentials for new atoms are presented in Table 1. Among these atoms, the special place is occupied by tantalum, for which the pseudo potential initially constructed without taking into account the 4f shell led to appreciably underestimated
PHYSICS OF THE SOLID STATE Vol. 52 No. 7

values of the lattice parameter for KTaO3 and metallic tantalum. Consequently, the completely filled 4f shell of the tantalum atom was taken into account in the pseudopotential used in our calculations. The maxi mum energy of plane waves was equal to 40 Ha for the KNbO3/KTaO3 superlattice and 30 Ha for other superlattices.1 The integration over the Brillouin zone was performed using the 8 â 8 â 4 k point mesh gener ated for ten atom unit cells and using the 6 â 6 â 4 k point mesh generated for twenty atom unit cells according to the Monkhorst­Pack scheme [60]. The relaxation of atomic positions was performed until the Hellmann­Feynman forces decreased below 5 â 10 6 Ha/Bohr. The phonon spectra and the dielectric and elastic properties were calculated in the frame work of the density functional theory according to 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 effective atomic charges. The spontaneous polarization Ps was calculated by the Berry phase method. Since the unit cell volume in the considered superlattices was 2­4 times larger than that in the parent crystals with the perovskite structure and the value of the polarization in the Berry phase calculations was determined with an accuracy of the "polarization quantum" eR/ (where R is the lat tice vector [61]), the polarization was calculated for both the equilibrium polar structure and the structures in which the polar displacements of atoms (with respect to their positions in the paraelectric phase) were equal to one half and one fourth of the maxi mum polar displacements. This enabled us to recon struct the "trajectory of motion" of the Berry phase and to find the correct value of the polarization. The search for the ground state of the superlattice was performed as follows. First, the equilibrium struc ture of the nonpolar phase with space group P4/mmm was determined by minimizing the Hellmann­Feyn man forces. Then, for this structure, the frequencies of the phonon spectrum were calculated at the point.
1

In this paper, some parameters used in the calculations are given in Hartree atomic units.

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The ground state of the superlattice, which has the lowest total energy, is characterized by positive values of all frequencies of optical phonons at all points of the Brillouin zone and the structure itself should be mechanically stable. The search for the ground state was started with the determination of the structure for which frequencies of all optical phonons at the point would be positive. Since in the phonon spectrum of the P4/mmm phase of all the investigated superlattices, there were unstable modes at the point (with imagi nary mode frequencies), small perturbations corre sponding to the eigenvector of the least stable of these modes were introduced into the structure and the equilibrium structure of the low symmetry phase was determined by means of the relaxation of atomic posi tions and the unit cell parameters. In the case of the doubly degenerate mode Eu, we considered two types of distortions described by the order parameters (, 0) and (, ).2 The analysis of the phonon spectrum and the search for a new low symmetry structure were continued until the structure with positive frequencies of all optical phonons at the point was found. For this structure, we calculated the elastic moduli tensor C and checked whether the structure is mechanically stable (the determinant of the 6 by 6 matrix of the elastic moduli in the Voigt notation and all leading principal minors should be positive). This structure will be referred to as the "ferroelectric ground state." For the "ferroelectric ground state," the existence of other unstable modes in the structure under consid eration was checked by calculating the phonon fre quencies at high symmetry points at the Brillouin zone boundary. If such an instability was revealed, then, in accordance with the position of the least sta ble mode among these unstable modes (usually, it was the M(S) point3), a new unit cell with the double vol ume was constructed, small atomic displacements corresponding to the least stable of the modes under consideration were introduced into this unit cell, and the equilibrium structure was determined by means of the relaxation of atomic positions and the unit cell parameters. The analysis of the phonon spectrum and the search for new low symmetry structures were con tinued until the structure with positive frequencies of all phonons at the center and high symmetry points at the boundary of the Brillouin zone was found. For this structure, we calculated the elastic moduli tensor and checked whether the structure is mechanically stable.
2

This structure corresponded to the structure of the true ground state. Up to now, the described complete scheme of the search for the ground state of ferroelectric superlat tices has been used only two times: for the PbTiO3/SrTiO3 superlattice in [21] and the BaTiO3/SrTiO3 superlattice in the author's previous work [3]. 3. RESULTS OF THE CALCULATIONS The search for the ground state in superlattices with taking into account all possible instabilities of the phonon spectrum and with the simultaneous inclusion of the influence of the substrate induced strain is a very laborious task. In this paper, we will restrict our consideration to the search for the ground state for the shortest period superlattices which are grown in the 001 direction and consist of alternating layers of two components, each having the thickness equal to one unit cell. We will mainly consider properties of free standing superlattices, i.e., superlattices free of sub strate induced strains; however, in a number of cases (for comparison with results of investigations of other authors), we will consider superlattices with other periods and substrate supported superlattices. The importance of research on the properties of free standing superlattices is associated with the fact that the coherently strained state in superlattices is sta ble only in very thin layers; in films with the layer thickness larger than a certain critical thickness, there arise misfit dislocations and the strained state relaxes. The characteristic value of the critical thickness for the considered superlattices is equal to ~100 å; therefore, we can suppose that the majority of the experimentally studied superlattices with the thickness lying in the range 320­4500 å are in a relaxed state close to the state of the corresponding free standing superlattice. We consider first the results obtained from calcula tions of the structure of low symmetry phases without regard for their distortions described by phonons at the Brillouin zone boundary. The phonon spectrum of one of the investigated free standing superlattices, i.e., PbTiO3/BaTiO3, in the paraelectric phase (P4/mmm) and in the ground state (the Pmm2 phase) is shown in Fig. 1 (the dispersion curves were obtained by the extrapolation of the data calculated at 12 and 18 irre ducible points of the Brillouin zone, respectively). It is seen that, in the phonon spectrum of the paraelectric phase, there are two unstable phonons at the point. The frequencies and symmetry of the unstable phonons at the point in the P4/mmm phase for ten superlattices studied in this work are presented in Table 2. It follows from the table that the phonon spec tra of all the investigated superlattices are character ized by unstable ferroelectric modes with symmetry A2u (with the polarization of vibrations along the tet ragonal axis) and Eu (with the polarization in the xy
Vol. 52 No. 7 2010

When the fourth degree invariants in the expansion of the energy in powers of the atomic displacements are taken into account, the energy minimum is reached for one of the afore mentioned order parameters. 3 For the points of the Brillouin zone of the low symmetry phases, we will use the notations of the points of the tetragonal recipro cal lattice for base centered orthorhombic and monoclinic lat tices and its own notations for the simple orthorhombic lattice. The M point of the tetragonal lattice is equivalent to the S point of the orthorhombic lattice.

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1451 (a)

plane) that arise from the unstable mode 15, which is characteristic of many perovskites. With the exception of the BaTiO3/BaZrO3 superlattice in which only one mode, namely, Eu, is unstable, in the other superlat tices, both modes are unstable simultaneously, so that the Eu mode is always less stable. Therefore, one can expect that the symmetry of the polar phase, most likely, will be lower than tetragonal. In phonon spectra of three superlattices, the appearance of three unstable modes with symmetry Eg and B2u at the point indi cates other possible distortions of the structure; how ever, the instability of these modes is weaker and the phases distorted in a corresponding manner (with space groups P2/m, P 1 , and P 4 m 2 ) have a higher energy as compared to the polar phase. The symmetry of the "ferroelectric ground state" and the orientation of the polarization vector with respect to the axes of the tetragonal lattice of the paraelectric phase according to the results obtained from calculations for all superlattices are presented in Table 2. It can be seen that the "ferroelectric ground state" has symmetry Amm2, Pmm2, Cm, P4mm, and P4mm. We think that, in the PbZrO3/BaZrO3 super lattice, the symmetry of the "ferroelectric ground state" is Pmm2, even though the minimum of the total energy was reached by the rotation of the polarization vector through an angle of 0.02° in the xz plane with respect to the polar x axis of the Pmm2 phase (in the setting P2mm). It should be noted that the criterion for mechanical stability is actually important for the verification as to whether the structure under investigation is the ground state. In particular, for the SrZrO3/SrTiO3 superlat tice, the "ferroelectric ground state" is the P4mm phase, which has the lowest energy and is character ized by positive frequencies of all optical phonons at the point. It was surprising to reveal that the Amm2 phase of this superlattice, which has a higher energy, is

800 Frequency, cm­1 600 400 200 0 Z R X MA P4/mmm

­200 800 Frequency, cm­1 600 400 200 0

R (b)

Pmm2

Z

U

X



S

R

U

Fig. 1. Phonon spectrum of the free standing superlattice (PbTiO3)1(BaTiO3)1 in (a) the paraelectric phase P4/mmm and (b) the ground state, i.e., the ferroelectric phase Pmm2.

also characterized by positive frequencies of all phonons at the point. The only indication of the instability of this phase lies in the fact that it is mechanically unstable: the quadratic form, which expresses the strain energy of the crystal in terms of the strain tensor components, was not positive definite (C12 > C11 = C22). The criterion for mechanical stability turned out to be important for other systems: the

Table 2. Relative difference in the lattice parameters a/ a for the starting components, enthalpy of mixing H, unstable optical phonons at the point in the paraelectric phase P4/mmm, space group, and orientation of the polarization vector in the "ferroelectric ground state" of the investigated superlattices Superlattice BaTiO3/SrTiO3 KNbO3/KTaO3 KNbO3/NaNbO PbTiO3/SrTiO3 PbTiO3/PbZrO3 SrZrO3/SrTiO3 PbZrO3/BaZrO3 BaTiO3/BaZrO3 PbTiO3/BaTiO3 BaTiO3/CaTiO3 a/ a , % 2.09 1.16 0.91 0.76 5.12 5.54 0.98 4.76 1.33 3.35 H, meV +2.9 +3.2 ­1.5 ­3.3 +78 +110 ­1.9 +72 ­0.4 +6.8
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Unstable phonons at the point, cm­1 102i 143i 191i 119i 201i 190i 111i 248i 135i 145i
No. 7

Space group Cm (xxz) Cm (xxz) Amm2 (xx0) Pmm2 (x00) Cm (xxz) P4mm (00z) Pmm2 (x00) Amm2 (xx0) Pmm2 (x00) Pmm2 (x00)

3

(Eu), (Eu), (Eu), (Eu), (Eu), (Eu), (Eu), (Eu) (Eu), (Eu),

97i (A2u) 106i (A2u) 189i (A2u), 125i (Eg) 107i (A2u) 181i (A2u), 153i (B2u), 97i (Eu), 33i (Eg) 162i (B2u), 148i (A2u), 100i (Eu) 79i (A2u) 123i (A2u) 106i (A2u)

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Table 3. Frequencies of all unstable phonons or the lowest frequency stable phonon at different points of the Brillouin zone for the "ferroelectric ground state" of the investigated ferroelectric superlattices (the space groups corresponding to the true ground state are printed in boldface) Superlattice BaTiO3/SrTiO3 KNbO3/KTaO3 KNbO3/NaNbO PbTiO3/PbZrO3 BaTiO3/BaZrO3 SrZrO3/SrTiO3 PbTiO3/SrTiO3 PbZrO3/BaZrO3 PbTiO3/BaTiO3 BaTiO3/CaTiO3 Space group Cm Cm Amm2 Cm Amm2 P4mm Phonon frequencies, cm­1 54 76 61 30 91 58 Pmm2 Pmm2 Pmm2 Pmm2 63 51 53 122 Z 67 49 61 17 66 29 Z 43 40 39 62 X 30 41 51 57 X 97 66 33i 21i 87 112i Y 49 28 53 106 M 55 70 73i, 35i, 20i 96i, 15i 60i 119i, 116i S 55i, 33i, 20i 37i, 37i, 28i 57 59i, 40i, 21i U 50 38 52 71 R 97 66 78 28 89 47 T 51 33 54 105 A 96 103 50i, 28i 77i 54i 118i R 29i 7i 60 55i, 17i

3

Amm2 phase for the PbTiO3/BaTiO3 superlattice was also characterized by positive frequencies of all optical phonons at the point but manifested a mechanical instability (C12 > C11 = C22). Table 3 presents the results obtained from calcula tions of the phonon frequencies at the center of the Brillouin zone and at five high symmetry points at its boundary for the "ferroelectric ground state" in all the investigated free standing superlattices. It follows from the table that, for only three of these superlat tices, namely, BaTiO3/SrTiO3, PbTiO3/BaTiO3, and KNbO3/KTaO3, the frequencies of all modes at all points of the Brillouin zone are positive (see also Fig. 1b). The verification of the mechanical stability of the determined structures for these three superlattices confirms that they correspond to the true ground states. For the other seven superlattices, a few modes in phonon spectra at the boundary of the Brillouin zone are unstable simultaneously and the search for the true ground state requires consideration of more complex structures. As follows from Table 3 (see also Fig. 2b), the strongest instability in the "ferroelectric ground state" of the seven remaining superlattices is associated with phonons at the M(S) and A(R) points of the Brillouin zone. The instability of these phonons is usually attrib uted to the instability of the perovskite structure with respect to the rotation of the oxygen octahedra. Therefore, the search for the ground state for the seven remaining superlattices was continued. The found "ferroelectric ground states" served as a starting point; however, the unit cell volume was doubled and distor tions that corresponded to the rotation of the oxygen octahedra and were described by phonons at the M point were added to the structure. The translation vec

tors in the new unit cell with the double volume and the vectors of the parent lattice are related by the ' ' ' expressions a 1 = a1 + a2, a 2 = a1 ­ a2, and a 3 = a3. For this new unit cell, the volume of the Brillouin zone decreases by a factor of two and the M, A, X, and R points of the initial Brillouin zone are mapped into the , Z, M, and A points of the "folded" Brillouin zone, respectively; the Z point in this case remains in place. Since the structural relaxation, as a rule, leads to an increase in the stability of phonons, it is sufficient to trace the change in the frequency of the phonons unstable in the "ferroelectric ground state" during the search for the ground state. After the relaxation of atomic positions and the unit cell parameters, the polar Amm2 phase for the free standing superlattices KNbO3/NaNbO3 and BaTiO3/BaZrO3 transforms to the phase with space group Pmc21 (in the setting P21am) in which the polar ization vector is oriented along the 21 axis (or in the 110 direction of the parent tetragonal lattice). The Cm phase for the PbTiO3/PbZrO3 superlattice trans forms to the phase with space group Pc (in the setting Pa) with the polarization vector oriented in the xxz direction of the parent tetragonal lattice at an angle of 28.4° with respect to the xy plane. It follows from Table 4 that, in the aforementioned phases, the fre quencies of all phonons at all points of the Brillouin zone are positive and the structures themselves are mechanically stable. The structure of the other super lattices after the "inclusion" of the structural distor tions undergoes more complex transformations. It turned out that, in the free standing superlattices PbTiO3/SrTiO3 and PbZrO3/BaZrO3, the structure with space group Amm2, which is formed when rota tions of the oxygen octahedra are added to the polar
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phase of the "ferroelectric ground state" Pmm2, even though has a lower energy, it is not stable: in the phonon spectrum of this structure, the unstable ferro electric mode is observed at the point with the dis placements perpendicular to the current direction of the polarization. The search for the structures with a lower energy leads to the polar Pmc21 phase, which has the lowest energy for the considered superlattices. It follows from Table 4 and Fig. 2c that, in this phase, the frequencies of all phonons at the Brillouin zone center and high symmetry points at the Brillouin zone boundary are positive, and the analysis of the elastic moduli indicates that this phase is mechanically sta ble. Therefore, the Pmc21 structure is the true ground state of these two superlattices. For the free standing superlattices PbTiO3/SrTiO3 and PbZrO3/BaZrO3, it is unusual that the most stable phase in the "ferroelectric ground state" is the phase with the polarization vector Ps parallel to the 100 axis of the parent tetragonal lattice, whereas the polariza tion vector Ps in the true ground state is parallel to the 110 axis; i.e., the polarization vector rotates in the xy plane through an angle of 45°. This suggests that, with the inclusion of the structural distortions, the polar ization direction in the true ground state is not neces sarily inherited from the "ferroelectric ground state." Even more complex change in the polarization direc tion with the inclusion of the structural distortions is observed in the SrZrO3/SrTiO3 and BaTiO3/CaTiO3 superlattices. The search for the ground state of the free standing superlattice SrZrO3/SrTiO3 turned out to be the most complicated problem. The P4bm phase formed from the polar P4mm phase of the "ferroelectric ground state" after the inclusion of the distortions described by phonons at the M point of the Brillouin zone relaxes to the nonpolar P4/mbm phase. In the phonon spectrum of this phase, two unstable phonons with symmetry Eu and Eg are observed at the point of the "folded" Brillouin zone (Table 4). The unstable mode with a frequency of 119i cm­1 is a ferroelectric mode and is characterized by the polarization of vibrations in the xy plane. After the addition of the distortions corresponding to the eigenvector of this mode to the structure and the structural relaxation, we obtain the phase with space group Pm, in which the direction of the polarization vector is close to the 110 direction of the parent lattice (the phase with space group Pmc21, in which the polarization vector exactly coincides with the 110 direction, has a somewhat higher energy). The calculations of the phonon spectrum of the Pm phase revealed two unstable phonons with symmetry A'' at the point (Table 4). After the addition of the distortions corresponding to the least stable among these unstable modes to the structure with space group Pm and the structural relaxation, we obtain the phase with space group Pc, for which the frequencies of all phonons at all points of the Brillouin zone are positive
PHYSICS OF THE SOLID STATE Vol. 52 No. 7

800 Frequency, cm­1 600 400 200 0 Z R X MA P4/mmm

­200 800 Frequency, cm­1

R (b)

Pmm2 600 400 200 0 800 Z U X S R U (c) Pmc21

Frequency, cm­1

600 400 200 0 Z U X

S

R

U

Fig. 2. Phonon spectrum of the free standing superlattice (PbTiO3)1(SrTiO3)1 in (a) the paraelectric phase P4/mmm, (b) the "ferroelectric ground state," i.e., the Pmm2 phase; and (c) the ground state, i.e., the Pmc21 phase.

(Table 4) and the structure itself is mechanically sta ble. Therefore, the Pc phase corresponds to the true ground state. The polarization vector in this phase has the xxz direction and makes an angle of 12.2° with the xy plane; i.e., with the inclusion of the structural distortions, the vector Ps is strongly tilted with respect to the plane. The free standing superlattice BaTiO3/CaTiO3 is one more system that appeared to be difficult to ana lyze. In this superlattice, the "ferroelectric ground state" is the Pmm2 phase. The phonon spectrum of this phase indicates that the structure is unstable with respect to the distortions described by phonons at the S and R points (Table 3). However, neither the Amm2

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Table 4. Frequencies of optical phonons at different points of the "folded" Brillouin zone of the superlattices in which the "ferroelectric ground state" is characterized by the pres ence of unstable optical phonons at the Brillouin zone boundary (the space groups corresponding to the true ground state of the superlattices are printed in boldface) Superlattice KNbO3/NaNbO PbTiO3/SrTiO3 PbTiO3/PbZrO3 SrZrO3/SrTiO3
3

Space group

Phonon frequencies, cm Z 63 42 40 ­ ­ 55 41 66 ­ 61 M 71 53 47 ­ ­ 50 49 86 ­ 93

­1

the polarization vector deviates from the xy plane and makes an angle of 20.1° with it. In this phase, the fre quencies of all phonons at all points of the Brillouin zone are positive (Table 4), and the structure itself is mechanically stable. Therefore, the Pc phase corre sponds to the true ground state of the free standing superlattice BaTiO3/CaTiO3. It is interesting that the true ground state in all the ten superlattices studied in this work is a polar state irrespective of whether the constituent components are paraelectrics, ferroelectrics, or antiferroelectrics. The influence of structural distortions on the physical properties of the superlattices is illustrated in Table 5. It can be seen that the inclusion of structural distortions in most of cases leads to a decrease in the largest eigenvalue of the dielectric constant tensor and makes this tensor more "isotropic." However, the most radical change is observed for the spontaneous polarization, which not only decreases in magnitude but also changes its direction for four of the studied superlattices. For all the considered superlattices, Table 2 pre sents the values of the enthalpy of mixing H, which was estimated as the half sum of the difference between the total energy of the P4/mmm phase of the superlattice and the sum of the total energies of the cubic phases (space group Pm3m) of the constituting components, i.e., the quantities reduced to the five atom formula unit. It can be seen that, except for three systems in which the B sites are alternately occupied by Ti and Zr atoms, the enthalpy of mixing H for the other seven superlattices is small, which suggests that, at 300 K, these superlattices will be thermodynami cally stable. In the titanate­zirconate systems, the enthalpy of mixing H is considerably higher and cor relates well with the relative difference between the lat tice parameters a/ a of the constituent components

A 92 52 48 ­ ­ 74 47 89 ­ 91

PbZrO3/BaZrO3 BaTiO3/BaZrO3 BaTiO3/CaTiO3

Pmc21 65 Pmc21 56 Pc 43 P4/mbm 119i, 70i Pm 71i, 52i Pc 17 Pmc21 41 Pmc21 84 Pmc21 40i Pc 59

phase (which was obtained after the addition of rota tions of the octahedra to the "ferroelectric ground state") nor the Pmc21 phase (in which the polarization vector was additionally tilted in the xy plane by an angle of 45° with respect to its direction in the "ferro electric ground state") proved to be stable. The phonon spectrum of the Pmc21 phase with a lower energy as compared to the Amm2 phase was character ized by the presence of the unstable ferroelectric mode with a frequency of 40i cm­1 (polarized in the z direc tion of the parent tetragonal lattice) at the point (Table 4). The addition of the distortions correspond ing to the eigenvector of this mode to the structure with space group Pmc21 leads to the Pc phase, in which

Table 5. Eigenvalues of the static dielectric constant tensor and the magnitude and orientation of the polarization vector (with respect to the axes of the parent tetragonal phase) in the "ferroelectric" and true ground states of the investigated fer roelectric superlattices (the magnitude of Ps is given in C/m2) "Ferroelectric ground state" Superlattice BaTiO3/SrTiO3 KNbO3/KTaO3 KNbO3/NaNbO PbTiO3/SrTiO3 PbTiO3/PbZrO3 SrZrO3/SrTiO3 PbZrO3/BaZrO3 BaTiO3/BaZrO3 PbTiO3/BaTiO3 BaTiO3/CaTiO3
1

True ground state P
s

2 204.3 122.2 62.6 206.0 110.7 281.4 78.8 44.5 223.5 92.8



3



1



2



3

P 0.241 0.253 0.502 0.483 0.661 0.217 0.394 0.259 0.563 0.419
No. 7

s

3

105.3 82.5 25.0 41.0 23.3 22.4 28.2 34.7 36.2 24.5

907.3 197.5 534.4 207.0 123.3 281.4 89.7 79.7 336.1 95.2

0.241 0.253 0.489 0.585 0.819 0.502 0.480 0.259 0.563 0.559

(xxz) (xxz) (xx0) (x00) (xxz) (00z) (x00) (xx0) (x00) (x00)

105.3 82.5 28.2 60.1 43.7 46.8 38.1 33.7 36.2 45.9

204.3 122.2 41.2 130.5 104.1 72.5 49.9 42.4 223.5 74.7

907.3 197.5 334.7 197.0 104.3 85.7 60.2 71.0 336.1 117.3
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(xxz) (xxz) (xx0) (xx0) (xxz) (xxz) (xx0) (xx0) (x00) (xxz)
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(calculations were carried out using the lattice param eters a for the cubic phases with space group Pm3m). This clearly indicates that high enthalpy of mixing in these cases is due to strong internal strains in the superlattices. It is interesting to note that, despite the large ratio a/ a for the BaTiO3/CaTiO3 superlattice, the enthalpy of mixing H remains rather small. As follows from the calculations performed in [25], one of the methods used to decrease the enthalpy of mixing in superlattices of the titanate­zirconate sys tems can be their growth in the 111 direction. According to our data, the Fm3m phase of the (PbTiO3)1(PbZrO3)1 superlattice grown in the 111 direction has the enthalpy of mixing H = ­20.8 meV per formula unit. However, in the phonon spectrum of this phase, apart from the unstable ferroelectric mode 15 (132i cm­1) at the point, there exist two more ' ' unstable phonons 15 (161i cm­1) and 25 (51i cm­1). Nonetheless, the ground state of this superlattice (the R3 phase) remains polar (|Ps | = 0.620 C/m2). The investigation of the properties of the other superlat tices grown in the 111 direction requires additional calculations, and they will not be considered in the present paper. 4. DISCUSSION OF THE RESULTS We compare now the results of our calculations with the available experimental and calculated data for ferroelectric superlattices and the corresponding solid solutions. At present, the BaTiO3/SrTiO3 superlattice is the most experimentally studied superlattice. It is known from the phase diagram of the BaTiO3­SrTiO3 system [62] that the low temperature phase in the Ba0.5Sr0.5TiO3 solid solution has symmetry R3m. The ground state, i.e., the Cm phase, found for the free standing superlattice (BaTiO3)1(SrTiO3)1 in this work is in agreement both with the results of calculations performed by other authors [3, 12, 13, 63, 64] and with the experimental data obtained for solid solutions: with taking into account the tetragonal perturbation determined by the geometry of the superlattice, the polar R3m phase should transform to the Cm phase. The monoclinic polar distortion of the structure of the BaTiO3/SrTiO3 superlattice, which agrees with the results of our calculations, was observed in [9]. A small splitting of the frequencies Eu ­ A2u in the phonon spectrum of the P4/mmm phase of the superlattice under consideration (Table 2) indicates a sufficiently weak perturbation caused by the mechanical strain of the layers, as well as by the nonequivalence of the envi ronment of the titanium atoms. Nonetheless, the polarization vector appears to be deviated from the 111 direction of the cube toward the xy plane through an angle of 20.6°. The weak instability of the phonon with symmetry R25 in the phonon spectrum of
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SrTiO3 is compensated by a quite stable same phonon in the phonon spectrum of BaTiO3 [1, 65]; conse quently, the ferroelectric instability at the point in the P4/mmm phase is the only instability of the super lattice [3]. According to the results of our calculations, the ground state of the free standing superlattice KNbO3/KTaO3 is the Cm phase. This result is in agree ment with the experimental data for the KNb0.5Ta0.5O3 solid solution, in which the symmetry of the low tem perature phase is R3m [62]. As for the BaTiO3/SrTiO3 superlattice, the found symmetry of the ground state (Cm) with taking into account the tetragonal perturba tion determined by the geometry of the superlattice corresponds to the experimentally observed symmetry in the solid solution, even though the splitting of the modes A2u ­ Eu for this superlattice is larger. The ori entation of the polarization vector in this superlattice is considerably closer to the 111 direction of the cube as compared to the BaTiO3/SrTiO3 superlattice: the deviation of the polarization vector toward the xy plane is equal to 7.2°. The stability of the structures of both constituent components of the superlattice, i.e., KNbO3 and KTaO3, with respect to the rotation of the oxygen octahedra, makes the superlattice structure stable to these distortions. In order to compare with the results of earlier cal culations of the properties of the KNbO3/KTaO3 superlattices [53], we additionally calculated and ana lyzed the properties of these superlattices supported on the KTaO3 substrate and superlattices with a larger period. It turned out that our results differ strongly from the results of previous calculations: according to our data, the P4mm phase of the (KNbO3)1(KTaO3)1 superlattice supported on the KTaO3 substrate is unstable (its phonon spectrum contains the ferroelec tric mode with symmetry E and frequency of 90i cm­1) and the ground state of this superlattice is the Cm phase. Moreover, our calculations for the (KNbO3)1(KTaO3)3 superlattice predict the existence of the polar state in both free standing superlattice (the Amm2 phase, the polarization vector is parallel to 110) and the superlattice supported on the KTaO3 substrate (the Cm phase, the polarization vector is directed along xxz). In [53], the polar state in the (KNbO3)1(KTaO3)3 superlattice supported on the KTaO3 substrate was not found (for space group P4mm). As regards the antiferroelectric phase discussed in [53] for the (KNbO3)1(KTaO3)1 superlattice, it is not clear from the results obtained in [53], in which phase (high symmetry or ferroelectric) the unstable phonon was found at the X point. According to our calcula tions for the free standing superlattice, the phonon spectrum of the Cm phase does not contain this insta bility (Table 3) and the phonon spectrum of the P4/mmm phase, apart from the ferroelectric instabil

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ity, also contains two unstable phonons at the X point (71i, 34i cm­1), one unstable phonon at the R point (61i cm­1), and one unstable phonon at the Z point (126i cm­1). However, the instabilities associated with these phonons at the boundary of the Brillouin zone are weaker than the ferroelectric instability (143i, 106i cm­1). Our results indicate that the formation of the intermediate antiferroelectric phase in the (KNbO3)1(KTaO3)1 superlattice is unlikely, as com pared to the conclusion made in [51] from the increase in the dielectric constant in the electric field in the temperature range 140­230°C. Our results for the PbTiO3/BaTiO3 superlattice appeared to be unexpected. It is known that the sym metry of the low temperature phase in the Pb0.5Ba0.5TiO3 solid solution is P4mm [62]. Our calcu lations predict that the true ground state of this super lattice has a Pmm2 structure with the in plane orienta tion of Ps along the x axis (Table 2); the energies of the Amm2 and P4mm phases with the orientation of Ps along the 110 and 001 directions are higher than the energy of the Pmm2 phase by 3.7 and 26.2 meV, respectively. We can suppose that the Pmm2 phase is the tetragonal polar phase that is observed in solid solutions but in which, for some reason, the polar axis appears to be rotated to the layer plane. Then, taking into account the tetragonal perturbation determined by the geometry of the superlattice, the symmetry of the structure should be lowered exactly to Pmm2. This is evidenced by the relationship between the calcu lated lattice parameters of the Pmm2 phase: a = 7.6366 Bohr b = 7.4353 Bohr c/2 = 7.4221 Bohr. In experiments, this orientation of the polarization vec tor in the PbTiO3/BaTiO3 superlattice is confirmed by the results of Raman scattering investigations [39, 40]; one can add that, in this case, the superlattice remains polar up to 600°C [41]. The structural stability of the (PbTiO3)1(BaTiO3)1 superlattice with respect to the rotation of the oxygen octahedra is most likely associated with the high sta bility of the R25 and M3 modes in BaTiO3, which com pletely compensates for the instability of these modes in the phonon spectrum of PbTiO3. The change in the polarization as a function of the ratio between the thicknesses of PbTiO3 and BaTiO3 layers in the PbTiO3/BaTiO3 superlattices supported on the SrTiO3 substrate was studied from first princi ples in [42]. In this case, it was assumed that the polar phase has symmetry P4mm. According to our calcula tions, the P4mm phase in the (PbTiO3)1(BaTiO3)1 superlattice supported on the SrTiO3 substrate is unstable: its phonon spectrum contains the unstable ferroelectric mode of symmetry E with a frequency of 23i cm­1. The Pm phase is the ground state of this superlattice. In the (PbTiO3)2(BaTiO3)2 superlattice, the mode E is even less stable (its phonon frequency is 50i cm­1) and the Cm phase, whose energy is only 0.26

meV lower than the energy of the Pm phase, becomes the ground state. In [39, 40], it was found that, in the PbTiO3/BaTiO3 superlattice, the local lattice parame ters in the PbTiO3 and BaTiO3 layers differ substan tially (by 0.11 å) in the direction of the z axis perpen dicular to the layer plane, which, in the authors' opin ion, indicates an unusual matching of layers (a PbTiO3/c BaTiO3). However, strong variations in the magnitude and direction of the polarization vector in this matching should necessarily lead to a strong increase in the electrostatic energy, which makes this matching energetically unfavorable. In order to cal culate the difference between the lattice parameters in uniformly polarized superlattices, for free standing (PbTiO3)2(BaTiO3)2 and (PbTiO3)3(BaTiO3)3 we cal culated the structure of the P4/mmm, Pmm2, Amm2, and P4mm phases and determined the lattice parame ters in individual layers along the z axis. It turned out that the difference between these lattice parameters, which was estimated from the difference between the Pb­Pb and Ba­Ba distances in the layers, is equal, on average, to 0.10 å and changes from one polar phase to another polar phase by no more than 0.015 å. From this it follows that the experimentally observed differ ence between the lattice parameters is associated not with the unusual matching of layers but is a simple consequence of different sizes of barium and lead atoms. The result for the "ferroelectric ground state" in the PbTiO3/SrTiO3 superlattice appeared to be very similar to the result for the PbTiO3/BaTiO3 superlat tice. The symmetry of the low temperature phase in the Pb0.5Sr0.5TiO3 solid solution is P4mm [62]. Our cal culations predict that the "ferroelectric ground state" of this superlattice has a structure Pmm2 (Table 2); the energies of the Amm2 and P4mm phases are higher than the energy of the Pmm2 phase by 5.8 and 21.3 meV, respectively. As for the PbTiO3/BaTiO3 superlattice, the results of the calculations for the PbTiO3/SrTiO3 superlattice are in agreement with the experimental data for the solid solutions supposing that, for some reason, the polar axis of the tetragonal structure appears to be rotated to the layer plane and taking into account the tetragonal perturbation deter mined by the geometry of the superlattice. The rela tionship between the lattice parameters in the Pmm2 phase (a = 7.5324 Bohr b = 7.3473 Bohr c/2 = 7.3608 Bohr) agrees with this supposition. The instability of the R25 and M3 phonons in phonon spectra of both PbTiO3 and SrTiO3 [1] is responsible for the instability of the phonon spectrum of the PbTiO3/SrTiO3 superlattice of the Pmm2 phase at the S point of the Brillouin zone (Fig. 2b), and the Pmc21 phase appears to be the true ground state of this superlattice (Table 4, Fig. 2c). This conclusion regard ing the symmetry of the ground state of the free stand
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ing superlattice (PbTiO3)1(SrTiO3)1 is in agreement with the results of the calculations performed in [21]. With the aim of comparing with the results of the calculations carried out in [18, 19, 21] for the PbTiO3/SrTiO3 superlattices supported on the SrTiO3 substrates, in the present work, we performed a series of calculations for similar structures. In the analysis of the properties of these superlattices, the authors of [18, 19] restricted themselves to consideration only of the P4mm phase, and among the polar phases neglecting the rotation of the oxygen octahedra, only the P4mm and Amm2 phases were examined in [21]. In none of these papers, the stability of the found structures was checked. Our calculations showed that, in the phonon spectrum of the (PbTiO3)1(SrTiO3)1 superlattice sup ported on the SrTiO3 substrate, the unstable ferroelec tric mode of symmetry E with a frequency of 60i cm­1 occurs at the point. Among the phases neglecting the rotation of the oxygen octahedra, the Cm phase with the polarization vector oriented in the xxz direction has the lowest energy and the Pm phase with the x0z orientation of the polarization vector has an energy that is higher by only 0.03 meV. For the four layered superlattices (PbTiO3)1(SrTiO3)3, (PbTiO3)2(SrTiO3)2, and (PbTiO3)3(SrTiO3)1 supported on the SrTiO3 sub strate, the P4mm phase also turned out to be unstable (the frequencies of the modes of symmetry E are equal to 76i, 70i, and 19i cm­1, respectively). The "ferro electric ground state" for the (PbTiO3)1(SrTiO3)3 superlattice is the Pm phase (Ps is directed along x0z), and the "ferroelectric ground state" for the other two superlattices is Cm (Ps is directed along xxz). For the subsequent discussion, it is important that, as the number of PbTiO3 layers in the four lay ered superlattices PbTiO3/SrTiO3 increases, the angle between the direction of the polarization vector and the xy plane in the Pm phase monotonically increases from 32.9° for the (PbTiO3)1(SrTiO3)3 superlattice to 78.0° for the (PbTiO3)3(SrTiO3)1 superlattice and this angle in the Cm phase increases from 34.0° to 73.6°. The obtained results enable us to explain the cause of the nonmonotonic dependence of the ratio c/a (the measure of the polarization in the direction perpen dicular to the substrate), which was experimentally observed with increasing thickness of PbTiO3 layers in the PbTiO3/SrTiO3 superlattices grown on the SrTiO3 substrate [18]. In our opinion, this dependence is asso ciated with the manifestation of the structural relax ation and the nonmonotonic dependence of the strains on the ratio between the numbers of PbTiO3 and SrTiO3 layers in this superlattice. In thin structures (the (PbTiO3)1(SrTiO3)3 superlattice), the substrate produces low strains in the superlattice (Table 6), the critical thickness exceeds the superlattice thickness, and superlattice layers are in the state coherently matched to the substrate. As was shown above, this state is characterized by a noticeable z component of
PHYSICS OF THE SOLID STATE Vol. 52 No. 7

Table 6. Tangential strains in the film and the z component of the spontaneous polarization in the (PbTiO3)m(SrTiO3)n superlattices and the PbTiO3 film supported on the SrTiO3 substrate m/n 1/3 2/2 3/1 PbTiO3 (11 + 22)/2, GPa ­1.17 ­1.77 ­0.37 +0.21 Pz, C/m2 0.183 0.301 0.619 0.878

polarization. As the number of PbTiO3 monolayers in the superlattice increases, the strains increase, the critical thickness decreases, and, since the total thick ness of the structure increases too, the region of the structure the most distant from the substrate begins to relax. In this case, the z polarization component aver aged over the volume decreases because, as was dem onstrated above, the polarization vector in the relaxed superlattice lies in the film plane. With a further increase in the number of PbTiO3 monolayers, the strains in the superlattice weaken (Table 6), the struc ture again becomes coherently matched to the sub strate, the tendency toward an increase in the angle of rotation of the polarization vector with respect to the layer plane starts to dominate, and the z component of polarization again increases. The PbTiO3/PbZrO3 superlattices were among the first experimentally investigated superlattices [23]. The interest to these superlattices is determined, to a large extent, by the search for the reasons of unique ferroelectric and piezoelectric properties of PbZr1 xTixO3 solid solutions [24]. In these solid solu tions, the morphotropic boundary between the tetrag onal (P4mm) and rhombohedral (R3m) phases with a thin "layer" of the intermediate monoclinic phase is located at x = 0.5 [66]. Our calculations showed that the "ferroelectric ground state" of the (PbTiO3)1(PbZrO3)1 superlattice is the Cm phase, in which the polarization vector deviates from the 111 direction of the cube toward the xy plane by an angle of 11.9°. The instability of the PbTiO3 and PbZrO3 structures with respect to rotations of the oxygen octa hedra [1, 65] results in the fact that, in the Cm phase, there arise unstable phonons at the M, A, and X points at the boundary of the Brillouin zone (Table 3). The Pc phase (Table 4) with the polarization in the xxz direc tion of the parent tetragonal lattice appears to be the true ground state of the superlattice. The results obtained agree with the results of the calculations per formed in [27], according to which the Cm phase is the "ferroelectric ground state" of this superlattice; unfor tunately, the authors of [27] did not check the stability of the found structure with respect to the phonons at the boundary of the Brillouin zone, even though they admitted the possibility of manifesting this instability.

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In the five superlattices considered above, we observed the same phenomenon: in the ground state, the polarization vector Ps either deviates from the 111 direction of the cube toward the layer plane or was completely located in this plane. Now, we discuss the reason of this phenomenon. In our opinion, the tendency to the tilt of the polarization vector toward the layer plane is associated with the fact that this leads to the disappearance of the necessity to "maintain" close values of Ps in the neighboring layers of the superlattice, which are required for decreasing the electrostatic energy of the bound charge b = ­ · Ps. This can be easily verified by comparing the values of Ps in the neighboring layers of the superlattice for two polarization orientations. The estimates obtained for Ps from the displacements of atoms and their effective charges according to the technique described in [10] show that, in the P4mm phase, the values of Ps in the neighboring layers coincide within accuracy of 1% for the (PbTiO3)1(SrTiO3)1 superlattice and 7.5% for the (PbTiO3)1(BaTiO3)1 superlattice, whereas, in the Pmm2 phase, they differ by a factor of 1.56 and 1.75, respectively. A decrease in the energy of superlattices with the tilt of the polarization vector toward the layer plane can be understood using a simple electrostatic model. When the polarization vector is oriented perpendicu lar to the layer plane of a superlattice, at the bound aries of two ferroelectrics with different spontaneous polarizations, there appears a surface charge which induces an electric field in both layers so that the elec tric displacement fields in them are equal to each other. The field induced change in the polarization in the layers occurs as a result of the displacements of ions from their "equilibrium" positions for these materials. This means that, for such a polarization direction, the total energy of the crystal will be increased by the sum of the energies of the electric field in the layers and the energy of local mechanical strains. The rotation of the polarization vector to the layer plane leads to the disappearance of the afore mentioned electric fields and eliminates the necessity of additional ionic displacements, which results in a decrease in the total energy of the system. An important sequence of the "strained" state in superlattices polarized perpendicular to the layer plane is a decreased polarization in the polar phase: in particular, the values of Ps in the P4mm and Pmm2 phases for the free standing superlattices (PbTiO3)1(SrTiO3)1 are equal to 0.420 and 0.585 C/m2, respectively, and the values of Ps in these phases for the (PbTiO3)1(BaTiO3)1 superlattices differ even more strongly: 0.332 and 0.563 C/m2. The same strong difference between the values of Ps in the P4mm and Pmm2 phases is also observed for the (BaTiO3)1(CaTiO3)1 superlattices: 0.335 and 0.559 C/m2, respectively.

It should be noted that the above values of Ps for the layers of the (PbTiO3)1(SrTiO3)1 superlattice in the Pmm2 phase enable us to draw the conclusion regard ing relatively strong correlations of the transverse polarization component in the neighboring layers: their values noticeably exceed the correlations deter mined for the KNbO3/KTaO3 and BaTiO3/SrTiO3 superlattices by the atomic simulation [48, 49]. The probable factor responsible for this difference can be a significant role played by the displacement of lead atoms in the soft mode in PbTiO3. The SrZrO3/SrTiO3 superlattices are of interest because ferroelectricity was first revealed in these superlattices formed from nonpolar constituent com ponents [15, 32, 34]. At low temperatures, the stron tium titanate has a structure with space group I4/mcm and the strontium zirconate has a structure with space group Pbnm. The SrTi0.5Zr0.5O3 solid solution at 300 K has a structure I4/mcm [67]. According to the results of our calculations, the "ferroelectric ground state" of this superlattice is the P4mm phase. Since both con stituent components SrTiO3 and SrZrO3 are charac terized by the instability with respect to the rotation of the oxygen octahedra, the superlattice itself also exhibits the same instability, and the P4mm phase with the inclusion of these distortions transforms to the Pc phase, which is the true ground state (Table 4). The results of these calculations differ from those obtained in [33, 34], in which the properties of the free standing superlattices SrZrO3/SrTiO3 were calculated from first principles but only the structure with space group P4mm was considered. It follows from our calculations that, in the phonon spectrum of the P4mm phase of the free standing superlattice (SrZrO3)1(SrTiO3)1, all phonons at the point are stable but several modes at the boundary of the Brillouin zone turn out to be unstable (Table 3). The value of Ps in the P4mm phase is actually large (0.502 and 0.427 C/m2 according to our data and data obtained in [34], respectively); how ever, the inclusion of structural distortions results in a decrease in this value by a factor of 2.3 (Table 5). We consider now the BaTiO3/BaZrO3 superlattice. It is believed that the barium zirconate retains the cubic structure Pm3m down to 2 K, even though the calculations reveal a weak instability at the R point of the Brillouin zone in its phonon spectrum [68]. The phase diagram of the BaTiO3­BaZrO3 system has been studied to the composition x = 0.3 [62]; with an increase in x, the phase transition becomes strongly diffuse and the extrapolation of the phase diagram suggests that the BaTi0.5Zr0.5O3 solid solution at T 0 has a cubic or polar rhombohedral structure. At 300 K, the BaTi0.5Zr0.5O3 samples have a cubic structure [69]. The results of our calculations, which predict the ferroelectric phase with space group Pmc21 as the true ground state of the free standing superlattice BaTiO3/BaZrO3, differ strongly from the results for
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solid solution. An unexpected property revealed for this superlattice in the calculations was the instability of the "ferroelectric ground state" (the Amm2 phase) with respect to the rotation of the oxygen octahedra, because this instability is absent in BaTiO3 and weak in BaZrO3. The existence of the found ground state with the in plane polarization is confirmed by the experi ments in which the hysteresis loops in superlattices were observed for a planar arrangement of electrodes on the film surface [15] but were almost absent in short period superlattices for a "vertical" arrangement of electrodes [38]. In [38], the authors consider the problem of the mechanical strain relaxation in the BaTiO3/BaZrO3 pair superlattices with thick layers. According to their estimates, the critical thickness for the BaTiO3/MgO (a/ a 5.2%) is approximately equal to 15 å. Since all titanate­zirconate systems are characterized by close mismatches of the lattice parameters (Table 2), one can expect that the critical thickness in these superlattices should be close to the above value. The BaTiO3/BaZrO3 superlattices in which the periods varied from 32 to 80 å were investigated in [38]. It seems likely that the thinnest films remain coherently strained and the polarization in them lies in the xy plane. With an increase in the layer thickness, the strains begin to relax and the lattice parameters of the layers tend to values characteristic of bulk samples. With strain relaxation, the polarization in the BaZrO3 layers disappear and the polarization vector in the BaTiO3 layers deviates from the film plane. This explains the increase in the z component of polariza tion with an increase in the period of the BaTiO3/BaZrO3 superlattice, which was experimen tally observed in [38]. Now, we consider the PbZrO3/BaZrO3 superlat tice. According to the data obtained in [62], the Pb0.5Ba0.5ZrO3 solid solution has a cubic structure. Our calculations predict that the Pmm2 phase is the "ferroelectric ground state" of this superlattice. The structural instability of both constituent components with respect to the rotation of the oxygen octahedra leads to the instability of the Pmm2 phase, in which the symmetry with the inclusion of rotations of octahedra is lowered to Pmc21 (the true ground state). The theo retical calculations of the properties of this superlat tice have never been carried out previously. The KNbO3/NaNbO3 superlattice was theoreti cally investigated only in one work [54], according to which the Cm phase is the ground state of the unstrained superlattice. The stability of this phase with respect to the distortions described by the phonons at the boundary of the Brillouin zone was not checked. At the same time, it is known that NaNbO3 is one of the most structurally complex crystals, which under goes at least five structural and one ferroelectric phase transition in the temperature range 80­920 K [62].
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The instability of its phonon spectrum, which is asso ciated with the phonons at the M and R points of the Brillouin zone, was investigated using inelastic scatter ing of neutrons [70]. For this reason, it could be expected that the instability of the phonon spectrum at the boundary of the Brillouin zone will be observed in the KNbO3/NaNbO3 superlattice. Indeed, as was confirmed by our calculations (Table 3), the instability of the "ferroelectric ground state" (the Amm2 phase according to our data) occurs at the M, A, and X points of the Brillouin zone. The phase with space group Pmc21 is the true ground state of the free standing KNbO3/NaNbO3 superlattice. These results differ substantially from the experimental data for the K0.5Na0.5NbO3 solid solution, which at a low temper ature has a rhombohedral structure R3m [62] or struc ture R3c in which rotations of octahedra are superim posed on polar displacements [71]. No experimental data on the properties of this superlattice are available in the literature. According to our data, the Pmm2 phase is the "fer roelectric ground state" of the free standing (BaTiO3)1(CaTiO3)1 superlattice. As a result of strong instability of cubic CaTiO3, this phase is also unstable and, with the inclusion of the structural distortions, transforms to the Pc phase with the polarization vector that makes an angle of 20.1° with the xy plane. Unfor tunately, the Ba0.5Ca0.5TiO3 solid solution is not single phase [62] and the found ground state cannot be com pared with data for solid solutions. The results of our calculations differ from the results of theoretical cal culations for the BaTiO3/CaTiO3 superlattices sup ported on the SrTiO3 substrate [43], in which the authors considered only the polar P4mm phase with out check of its stability. Our calculations for the (BaTiO3)1(CaTiO3)1 superlattice supported on the SrTiO3 substrate showed that the unstable ferroelectric mode of symmetry E with a frequency of 92i cm­1 is observed in the P4mm phase and the "ferroelectric ground state" is the Pm phase with the polarization vector that makes an angle of 12.4° with the xy plane. The calculated value of the z component of polariza tion (0.083 C/m2) agrees well with the quantity Ps = 0.085 C/m2 determined from measurements of the parameters of the dielectric hysteresis loops for the electric field applied perpendicular to the layer plane [44]. In [21], the authors discussed the mechanism of appearance of improper ferroelectricity in (PbTiO3)1(SrTiO3)1 superlattices supported on the SrTiO3 substrate. This mechanism consisted in appearing the z component of polarization due to the simultaneous condensation of two unstable modes of ' symmetry M2 and M 4 at the M point of the Brillouin zone of the paraelectric phase P4/mmm. This conclu sion was based on the fact that the direct product of the irreducible representations of these modes transforms

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according to the same irreducible representation as the z component of polarization vector (A2u at the point). The analysis of our results obtained for the (PbTiO3)1(SrTiO3)1 superlattice demonstrates that, despite the presence of the unstable phonons of sym ' metry M2 and M 4 at the M point in the P4/mmm phase of the free standing superlattice, no polarization along the z axis arises in the ground state. For the same superlattice supported on the SrTiO3 substrate, the z component of polarization appears before the inclu sion of rotations of the oxygen octahedra and, hence, conclusive arguments in support of the proposed mechanism in this case are absent too. The origin why the discussed mechanism of appearance of the z com ponent of polarization may not work is most likely associated with strong competition between different instabilities, owing to which the structural distortion according to one of them results in that phonons cor responding to other instabilities become stable. In our calculations, there was only one situation, i.e., the BaTiO3/CaTiO3 superlattice, when the inclu sion of rotations of octahedra resulted in the appear ance of the z component of polarization (Table 5). The analysis shows that the unstable modes of symmetry ' M 4 and M2 at the M point of the P4/mmm phase of the free standing superlattice with frequencies of 113i and 74i cm­1 remain unstable in the "ferroelectric ground state," namely, the Pmm2 phase (the modes with fre quencies of 59i and 21i cm­1 at the S point, Table 3). In essence, the transition to the Pmc21 phase is accom panied by freezing the distortions described by the mode M2 of the P4/mmm phase. The eigenvector of the only unstable phonon in the Pmc21 phase involves antiphase rotations of octahedra in the neighboring ' layers (corresponding to the mode M 4 of the P4/mmm) phase and polar displacements of Ca and Ba atoms along the z axis. After the structural relaxation, strong antiphase rotations of octahedra in the neigh boring layers arise in the structure and the polarization vector deviates from the xy plane. Therefore, the dis tortions observed in the structure of the (BaTiO3)1(CaTiO3)1 superlattice after the inclusion of structural distortions are consistent with the mecha nism proposed in [21]. Summing up the results obtained, we can argue that the experimental data available in the literature on the structure of solid solutions cannot serve as a reli able basis for the prediction of the ground state of fer roelectric superlattices. The strain in layers, nonequiv alence of the environment of atoms, and tendency of the system toward a decrease in the electrostatic energy of the boundary of two different ferroelectrics are responsible for the discrepancy between the ground states in superlattices and solid solutions and for different orientations of the polarization vector in

these states. The structural distortions that arise as a result of the existence of unstable phonons at the boundary of the Brillouin zone can also lead to a sub stantial change in the properties of superlattices, including the change in the direction of the polariza tion vector. The observed tendency toward the tilt of the polar ization vector to the layer plane in superlattices, whose origin was explained above, can complicate the use of superlattices in applications in which the polarization is switched by an electric field applied perpendicular to the film plane. The solution that can help in this case consists in growing thin superlattice films on sub strates that induce biaxial compression in films. As fol lows from the calculations and is confirmed by exper iments, the polarization vector under these conditions is predominantly oriented in the direction perpendic ular to the substrate. 5. CONCLUSIONS Thus, in this work, the crystal structure of the ground state of ten ferroelectric superlattices based on crystals with the perovskite structure (BaTiO3/SrTiO3, PbTiO3/SrTiO3, PbTiO3/PbZrO3, SrZrO3/SrTiO3, PbZrO3/BaZrO3, BaTiO3/BaZrO3, PbTiO3/BaTiO3, BaTiO3/CaTiO3, KNbO3/KTaO3, and KNbO3/NaNbO3) was calculated from first principles within the density functional theory. The ground state of all the considered superlattices was found to be fer roelectric. It was revealed that the polarization vector has a tendency toward a tilt to the plane of superlattice layers; this is associated with the decrease in the elec trostatic and elastic energy of the system consisting of materials with different ferroelectric properties for this orientation of the polarization vector. The importance of the inclusion of structural distortions associated with unstable phonons at the Brillouin zone boundary, which, in a number of cases, lead to significant changes in ferroelectric and dielectric properties of the superlattices, was demonstrated. The results of the calculations are in good agreement with available experimental data. Calculations in this work were performed on a lab oratory computer cluster (16 cores) and the SKIF­ MGU Chebyshev supercomputer. ACKNOWLEDGMENTS This work was supported by the Russian Founda tion for Basic Research (project no. 08 02 01436). REFERENCES
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Translated by O. Borovik Romanova

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