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ISSN 1063 7834, Physics of the Solid State, 2013, Vol. 55, No. 6, pp. 1198­1206. © Pleiades Publishing, Ltd., 2013. Original Russian Text © A.I. Lebedev, 2013, published in Fizika Tverdogo Tela, 2013, Vol. 55, No. 6, pp. 1110­1118.

FERROELECTRICITY

Properties of BaTiO3/BaZrO3 Ferroelectric Superlattices with Competing Instabilities
A. I. Lebedev
Moscow State University, Moscow, 119991 Russia e mail: swan@scon155.phys.msu.ru
Received December 18, 2012

Abstract--Properties of (BaTiO3)1/(BaZrO3)n ferroelectric superlattices (SLs) with n = 1­7 grown in the [001] direction are calculated from first principles within the density functional theory. It is revealed that the quasi two dimensional ferroelectricity occurs in these SLs in the barium titanate layers with a thickness of one unit cell; the polarization is oriented in the layer plane and weakly interacts with the polarization in neighboring layers. The ferroelectric ordering energy and the height of the barrier separating different orien tational states of polarization in these SLs are sufficiently large to provide the formation of an array of inde pendent polarized planes at 300 K. The effect of the structural instability on the properties of SLs is consid ered. It is shown that the ground state is a result of simultaneous condensation of the 15 polar phonon and phonons at the M point (for SLs with even period) or at the A point (for SLs with odd period); it is a polar structure with out of phase rotations of the octahedra in neighboring layers, in which highly polarized layers are spatially separated from the layers with strong rotations. The competition between the ferroelectric and structural instabilities in biaxially compressed SLs manifests itself in that the switching on of the octahedra rotations leads to an abrupt change of the polarization direction and can cause an improper ferroelectric phase transition to occur. It was shown that the experimentally observed z component of polarization in the SLs can appear only as a result of the mechanical stress relaxation. DOI: 10.1134/S1063783413060218

1. INTRODUCTION Ferroelectric superlattices (SLs) represent a new class of artificial materials whose properties, such as the Curie temperature and spontaneous polarization, often exceed those of bulk ferroelectrics [1, 2]. An analysis of properties of ten ferroelectric SLs with the perovskite structure grown in the [001] direction has shown that they are ferroelectrics at low temperatures, no matter whether they are composed of two ferro electrics, a ferroelectric and a paraelectric, or even two paraelectrics [3]. One of the possible applications of SLs can be non volatile ferroelectric random access memory (FRAM) devices in electronics. The recent studies of the ferroelectric instability in free standing (KNbO3)1(KTaO3)n superlattices grown in the [001] direction showed that, with increasing n, the tendency to the ferroelectric ordering is retained in KNbO3 layers with a thickness of one unit cell while the interaction energy between neighboring KNbO3 layers exponentially decreases [4, 5]. It was shown that at n 2 the ground state of the SLs is an array of ferro electrically polarized planes whose polarization is concentrated in the KNbO3 layer, is parallel to the [110] direction, and weakly interacts with polarization in neighboring layers. The use of such arrays of quasi two dimensional ferroelectric planes as a medium for the three dimensional data recording can make it pos

sible to achieve a volume recording density of ~1018 bit/cm3. Unfortunately, the implementation of the above described ground state in the (KNbO3)1(KTaO3)n SLs requires very low temperatures (~4 K) because of the low ferroelectric ordering energy and low height of the potential barrier separating different orientational states of polarization. That is why one of the aims of this work was to study the feasibility of similar quasi two dimensional structures and the ways to improve their characteristics in BaTiO3/BaZrO3 ferroelectric superlattices with an active layer of barium titanate. In BaTiO3/BaZrO3 superlattices, the tension of the BaTiO3 layers as a result of the epitaxial matching with the BaZrO3 layers having a larger lattice parameter should stabilize the Amm2 ground state necessary to implement the quasi two dimensional structure [4] in the BaTiO3 layers. However, since the simultaneous compression of the BaZrO3 layers can cause the polar ization normal to the interface to occur in them [6], the search for the ground state structure in these SLs requires a detailed analysis. Furthermore, the instabil ity of the BaZrO3 structure with respect to the octahe dra rotation [7­9] can result in the competition of the ferroelectric and structural instabilities in the SLs, and this problem also calls for a detailed study.

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The properties of BaTiO3/BaZrO3 SLs have been studied previously both experimentally [10­15] and theoretically [3]. The initial interest to these SLs was caused by giant dielectric constants observed in them [10]. Room temperature observations of ferroelectric hysteresis loops in the SLs [11, 13, 15] suggested that the ferroelectric transition temperature is them exceeds 300 K. The spontaneous polarization non monotonically varied with the period of SLs and depended on the electrode configuration (in short period SLs with electrodes on both sides of the film, hysteresis loops were absent [13]). These features have already been discussed in [3] and were explained by the rotation of the polarization to the layer plane. The recent study of Raman spectra and dielectric proper ties of BaTiO3/BaZrO3 SLs revealed the appearance of the z component of polarization in barium titanate layers in the SLs grown on MgO substrates [15]. Inter pretation of these results requires the study of the influence of substrate induced strain on the ground state structure of the SLs. 2. CALCULATION TECHNIQUE The (BaTiO3)1(BaZrO3)n superlattices considered in this work are periodic structures grown in the [001] direction and consisting of the BaTiO3 layer with a thickness of one unit cell and the BaZrO3 layer with a thickness of n unit cells (1 n 7). These SLs were modeled on supercells of 1 â 1 â L unit cells, where L = n + 1 is the SL period; in modeling the structures generated by instabilities at the M and A points at the boundary of the Brillouin zone, the primitive cell vol ume was doubled. In this work, particular attention was paid to free standing SLs. In addition, a number of calculations were performed for biaxially com pressed (BaTiO3)1(BaZrO3)1 and (BaTiO3)2(BaZrO3)2 SLs and for free standing (BaTiO3)n(BaZrO3)n SLs with n = 2, 3, and 4. The calculations were performed using the density functional theory with pseudopotentials and wave function expansion in plane waves as implemented in the ABINIT code [16]. As in the earlier study of these SLs [3], the exchange­correlation interaction was described in the local density approximation (LDA). As pseudopotentials, the optimized separable nonlo cal pseudopotentials constructed using the OPIUM program [17] were used; the local potential correction was added to them to improve the transferability. The parameters used for constructing the pseudopotentials and other calculation details are given in [3, 18]. The cut off energy for the plane waves was 30 Ha (Hartree) (816 eV); the integration over the Brillouin zone was performed using the 8 â 8 â 4 Monkhorst­Pack mesh for SLs with n = 1 and 2 and the 8 â 8 â 2 one for SLs with n = 3­7. The relaxation of the lattice parameters and atomic positions was performed until the Hellmann­Feynman forces became less than 5 â
PHYSICS OF THE SOLID STATE Vol. 55 No. 6

10 6 Ha/Bohr (0.25 meV/å). In the case of biaxially compressed SLs, the in plane lattice parameter was fixed equal to 0.97a0, 0.98a0, and 0.99a0, where a0 = 7.7182 Bohr (4.0843 å) is the in plane lattice param eter for the P4/mmm phase of the free standing (BaTiO3)1(BaZrO3)1 SL. The phonon spectra were calculated using the equations obtained from the den sity functional perturbation theory. The total sponta neous polarization in the SLs was calculated by the Berry phase method; its distribution over layers was calculated using the formula P = i w i Z i* u i, from , displacements ui of atoms obtained in the polar phase and tensors of their effective Born charges Z i* in the , paraelectric phase; here wi are the weight factors equal to unity for atoms in the Ti­O or Zr­O layer under consideration, 1/2 for atoms in neighboring Ba­O layers, and zero for other atoms. The properties of BaTiO3 obtained using the described approach were published previously [18] and are in good agreement with experiment.



3. RESULTS 3.1. Structure of BaZrO3 The structure of BaZrO3 calculated using the described approach is in good agreement with experi ment and results of earlier calculations [7­9]. For example, the lattice parameter in cubic BaZrO3 is 7.8724 Bohr (4.1659 å) and differs from the experi mental value (4.191 å at 2 K [7]) by 0.60% (the small underestimate of the lattice parameters is characteris tic of the used LDA approximation). As in earlier stud ies [7­9], the calculations of the phonon spectrum of cubic BaZrO3 reveal an unstable phonon mode at the R point at the boundary of the Brillouin zone with a frequency of 81i cm­1 (Fig. 1). The unstable phonon at the M point observed in [9] is absent in our calcula tions. However, the instability of the phonon spectrum of BaZrO3 at the R point has not yet been confirmed experimentally. One of the possible causes of disagree ment between experiment and theory can be quantum fluctuations [7] which can destroy the long range order in rotations of the oxygen octahedra. According to our calculations, the ground state structure of BaZrO3 is I4/mcm. This result differs from predictions of [8], in which the P 1 structure was considered to be equilibrium. In the present study, it was shown that the P 1 structure is nonequilibrium and slowly relaxes to
1

1

EXAFS measurements of the Debye­Waller factor for Ba­O bonds in BaZrO3 revealed its anomalously high values at 300 K, corresponding to the amplitude of the local rotations of ~4 degrees. In our opinion, this gives an evidence of the instabil ity under consideration. The results of these studies will be pub lished elsewhere.

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1200 , cm- 800 600
1

LEBEDEV , cm- 800 600 400
1

(a)

400 200 0 -200

200 0 -200 Z , cm-1 R M 800 600 400 200 0 RX (b) MA R

X

M

Fig. 1. Phonon spectrum of the BaZrO3 crystal in the cubic phase.

the I4/mcm structure. The calculated static dielectric constant of barium zirconate in the ground state is xx = yy = 58.8, zz = 53.3 (the experimental value is 47 [7]). 3.2. BaTiO3/BaZrO3 Superlattices The phonon spectra of different phases of the free standing (BaTiO3)1(BaZrO3)3 superlattice are shown in Fig. 2. One can see that two types of instability are observed in the phonon spectrum of the paraelectric P4/mmm phase (Fig. 2a): the ferroelectric instability and the structural instability associated with the oxy gen octahedra rotation. The phonon frequencies at high symmetry points of the Brillouin zone in the paraelectric phase of superlattices with n = 1, 2, and 3 are given in Table 1. The instability region appearing as a band of imag inary phonon frequencies along the path ­Z­R­X­ (imaginary frequencies are presented as negative numbers in the figure) is caused by the ferroelectric instability of ...­Ti­O­... chains, which was estab lished for the first time in [19]. An analysis of the eigenvectors of phonons related to this instability region shows that out of phase transverse Ti and O atomic displacements in the xy plane in chains propa gating in the [100] and [010] directions dominate in these eigenvectors at all points of the Brillouin zone; at the Brillouin zone center, this displacement pattern corresponds to the doubly degenerate ferroelectric Eu mode. For out of phase atomic displacements in the chains propagating in the [001] direction and consist ing of alternating titanium and zirconium atoms (... Ti­O­Zr­O­...), the lowest energy ferroelec tric A2u mode at the point is always stable; its fre quency increases from 37 cm­1 for n = 1 to 85 cm­1 for n = 3. A similar instability region along the path

Z , cm-1

RX

(c)

MA

R

800 600 400 200 0 X S Y Z U R T

Fig. 2. Phonon spectra of the free standing (BaTiO3)1(BaZrO3)3 superlattice in (a) P4/mmm, (b) Amm2, and (c) Pmc21 phases.

Z­R­X­ was observed in KNbO3/KTaO3 super lattices [5]. Among the possible polar phases resulting from the condensation of the unstable Eu mode at the point, the Amm2 phase with polarization along the [110] direction has the lowest energy in (BaTiO3)1(BaZrO3)n SLs with n = 1­7. However, the phonon spectra of this phase (Fig. 2b) show that phonons at the M and A points at the boundary of the Brillouin zone remain unstable, which indicates the instability of the Amm2 structure with respect to the oxygen octahedra rota
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Table 1. The lowest frequencies of optical phonons at high symmetry points of the Brillouin zone in P4/mmm and Amm2 phases and in the ground state (Pmc21) of free standing (BaTiO3)1(BaZrO3)n superlattices with n = 1, 2, and 3 Phonon frequencies, cm­1 n (0, 0, 0) 1 Z (0, 0, ) 2 243i 268i 279i 66 46 36 Z (0, 0, 1 ) 2 66 35 1 X( , 0, 0) 2 233i 257i 268i 87 84 68 X/Y 1 1 R( , 0, ) 2 2 11 M( , , 0) 22 90i 88i, 65i 92i, 66i, 54i, 53i 60i 69i, 39i 79i, 58i, 46i, 38i, 25i S 111 A( , , ) 222 86i 90i, 63i, 21i 91i, 68i, 51i, 34i 54i 72i, 41i 79i, 53i, 20i, 16i R

1 2 3 1 2 3

248i 269i 279i 91 76 61 (0, 0, 0)

P4/mmm phase 228i 257i 268i Amm2 phase 89 79 77 U /T Pmc21 phase 78/82 68/74

1 3

84 60

70/76 67/73

86 80

89 81

tion. This instability of the polar Amm2 phases is characteristic of SLs with all periods (see Table 1). Thus, the Amm2 phase is not a true ground state struc ture for the SLs under consideration. As follows from Table 1, the frequencies of two unstable phonons at the M and A points for the Amm2 phase of the superlattices with different periods are very close. Therefore, to search for the true ground state, it is necessary to compare the energies of all structures resulting from the condensation of these two phonons. An analysis of the eigenvectors of unstable phonons at the M point shows that the most unstable phonons are characterized by out of phase rotations of octahe dra in neighboring layers. In SLs with even period (L = 2, 4), the rotation angle is small in the BaTiO3 layers and large in the BaZrO3 layers (Figs. 3a and 3c); in SLs with odd period (L = 3), the rotation angle is zero in the BaTiO3 layers and is large in the BaZrO3 layers (Fig. 3b). Modes with lower instability (Table 1) include modes with other combinations of rotations in layers (in particular, in phase rotations in BaZrO3 lay ers and strong out of phase rotations in every second BaZrO3 layer) and a doubly degenerate mode with a complex combination of distortion and tilting of the oxygen octahedra (for L = 4). An analysis of the eigenvectors of unstable phonons at the A point shows that in SLs with even period (L =
2

2

2, 4), the most unstable phonons are characterized by the out of phase rotations in neighboring BaZrO3 lay ers and the absence of rotations in the BaTiO3 layers

(a)

(d)

(b)

(e)

(c)

(f)

­8 ­4 0 4 8 Rotation angle, deg

­8 ­4 0 4 8 Rotation angle, deg

Hereafter, we shall use the notation of the points in the Brillouin zone of the tetragonal paraelectric structure for their notation in the low symmetry structures. PHYSICS OF THE SOLID STATE Vol. 55 No. 6 2013

Fig. 3. Octahedra rotation angles in the ground state of (BaTiO3)1(BaZrO3)n superlattices corresponding to the phonon condensation at points (a­c) M and (d­f) A. (a, d) n = 1 (L = 2), (b, e) n = 2 (L = 3), (c, f) n = 3 (L = 4). BaTiO3 and BaZrO3 layers are denoted by open and filled squares, respectively. Vertical lines with arrows indicate the physical periods of the resulting structures.


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Table 2. Lattice parameters in the paraelectric P4/mmm phase of free standing (BaTiO3)1(BaZrO3)n superlattices with 1 n 7, the energy of the ferroelectric ground state E1 (Amm2 phase), the energy E2 of the I4/mcm or P4/mbm phases with pure rotations, the energy of the true ground state E3 (Ima2 and Pmc21 phases), and the polarization in the ferroelectric ground (Ps1) and true ground (Ps3) states n Parameter 1 a0, Bohr c0, Bohr E1, meV E2, meV E3, meV Ps1, C/m2 Ps3, C/m2 7.7182 15.3298 ­72.7 ­21.3 ­78.3 0.2755 0.2591 2 7.7747 23.1923 ­101.0 ­34.6 ­116.8 0.2024 0.1859 3 7.8009 31.0604 ­115.1 ­46.7 ­142.3 0.1568 0.1445 4 7.8160 38.9304 ­123.6 ­57.7 ­161.5 0.1273 0.1113 5 7.8258 46.8013 ­129.3 ­68.3 ­177.8 0.1070 0.0984 7 7.8379 62.5443 ­136.2 ­ ­ 0.0809 ­

(Figs. 3d and 3f). In SLs with odd period (L = 3), out of phase rotation angles are large in the BaZrO3 layers and small in the BaTiO3 layers (Fig. 3e). In both cases, the physical period of SLs is doubled in comparison with the period L specified by the layer sequence. Modes with lower instability (Table 1) are qualitatively similar to the above described modes at the M point. An abrupt change in the rotation angle in going from the BaTiO3 layer to the BaZrO3 layer and its weak dependence on the period of SL correlate with the small change of the unstable phonon frequency along the M­A line, which indicates a strong localization of these rotations in the layers. To search for the ground state of SLs, the oxygen octahedra rotations corresponding to the least stable
PS, arb. units 1 n=1 n=3 n=5

phonons at the M and A points of the Brillouin zone were added to the polar Amm2 structure. A compari son of the total energies of the structures resulting from simultaneous condensation of the "rotational" and ferroelectric modes shows that the ground state structure depends on the superlattice period L: for odd L, it is described by the space group Ima2 and results from the condensation of the unstable "rotational" mode at the A point; for even L, it is described by the space group Pmc21 and results from the condensation of the unstable "rotational" mode at the M point. The energy difference of the structures resulting from the condensation of phonons at the M and A points is rather small (1.6­1.9 meV). The energies of the struc tures corresponding to the ferroelectric ground state and to the true ground state, as well as to nonpolar structures with pure octahedra rotations (space group I4/mcm for odd L and P4/mbm for even L) are given in Table 2. An increase in all three energies with increas ing n is due to two effects: (i) an increase in the BaZrO3 volume fraction in the structure, which results in an increase of the instability of SLs with respect to the octahedra rotation; (ii) an increase in biaxial tension of the BaTiO3 layers (as a consequence of an increase in the in plane lattice parameter), which enhances the ferroelectric instability. The corresponding change in the frequencies of unstable phonons at the point can be seen in Table 1. The polarization profiles in the obtained ground states for SLs with n = 1, 3, and 5 are shown in Fig. 4. One can see that the polarization is concentrated in the barium titanate layer and almost exponentially decreases with distance with a characteristic length scale of ~2.0 å. It is interesting that the spontaneous polarization in the true ground state is lower than the polarization in the ferroelectric ground state by only
3

3

10-

1

10-

2

10-

3

-3

-2

-1

0 N

1

2

3

Fig. 4. The polarization distribution between the layers in the ground state structure of (BaTiO3)1(BaZrO3)n super lattices with n = 1, 3, and 5. The layer number N = 0 cor responds to the BaTiO3 layer.

The ferroelectric ground state is a structure with a minimum energy obtained by taking into account only the ferroelectric unstable mode. Vol. 55 No. 6 2013

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6­12% (Table 2). This indicates the weak effect of the octahedra rotations on the spontaneous polarization in SLs. We explain this by the fact that regions with high polarization in the SLs under consideration are spatially separated from regions with strong octahedra rotations, and thus the competition between the struc tural and ferroelectric instabilities becomes signifi cantly decreased. This feature essentially distinguishes the superlattices from such crystals as CaTiO3 and PbTiO3 with competing instabilities in the cubic phase. In the former case, the structural distortions completely suppress ferroelectricity; in the latter case, an opposite effect occurs. To test the possibility of forming the arrays of quasi two dimensional polarized planes in the ground state of (BaTiO3)1(BaZrO3)n SLs, we estimated the inter layer interaction energy. The energy 2Wint (the energy corresponding to two domain walls) was calculated as the difference of the total energies of ferroelectrically and antiferroelectrically ordered SLs with a doubled period in a similar way as in [4]; in this calculation, the octahedra rotations were neglected. The value of 2Wint was 7.46 meV for n = 1, 1.505 meV for n = 2, and 0.420 meV for n = 3, and decreased almost exponen tially with increasing n. For all n, this energy was much lower than the energy gain resulting from the ferro electric ordering E1 (Table 2), which ensures almost independent polarizations in the quasi two dimen sional layers and the formation of the arrays of inde pendent polarized planes similar to those observed in [4]. The potential barrier height U for reorientation of the polarization, which occurs in the SLs under consideration by rotating the polarization vector in the layer plane, was 22.9 meV (per Ti atom) for n = 1, 33.2 meV for n = 2, and 38.3 meV for n = 3. The obtained potential barrier heights and ferroelectric ordering energies are much higher than those for the KNbO3/KTaO3 SLs considered in [4, 5]. In our opin ion, these values are sufficiently large to provide stable polarization in the arrays of quasi two dimensional polarized planes at 300 K. 4. DISCUSSION We start the discussion with an analysis of possible polar structures in (BaTiO3)1(BaZrO3)n SLs. The fer roelectric instability in the ...­Ti­O­... chains can lead not only to the ferroelectric ordering, but also to the formation of antiferroelectric structures resulting from the condensation of unstable phonons at Z, R, and X points at the boundary of the Brillouin zone. Three types of the polarization ordering in the chains, which result from the condensation of these unstable
4

z (a)

4

(b)

y

y

x z (c)

x z (d)

y

y

x

x
Fig. 5. Schematics of the polarization ordering in the ... Ti­O­... chains propagating along the y axis which accompanies the condensation of unstable phonons at points (a) , (b) X, (c) R, and (d) Z.

An additional cause of the decrease in the polarization in free standing SLs when the octahedra rotations are switched on can be a systematic decrease (by 0.02­0.03 Bohr) in the in plane lattice parameter. PHYSICS OF THE SOLID STATE Vol. 55 No. 6

phonons in the paraelectric P4/mmm phase, are shown in Fig. 5. It is seen that these structures differ only in the relative orientation of polarization in neighboring chains. For example, for the phonon con densation at the X point, the antiferroelectric Pmma phase appears in the SLs; the modulation wave vector in this phase is directed along the x axis, whereas the out of phase displacements of Ti and O atoms form ing the chains are directed along the y axis (Fig. 5b). It is remarkable that this phase still exhibits the ferro electric instability with respect to the polar displace ments along the x axis and finally transforms to the polar Pm (Pmc21 for n = 1) phase being ferrielectric. Similarly, the antiferroelectric Cmma phase with the modulation described by the phonon at the R point and with out of phase displacements along the y axis (Fig. 5c) also is ferroelectrically unstable and finally transforms to the ferrielectric Abm2 phase. In contrast to non degenerate unstable phonons at the X and R points, the unstable phonon at the Z point is doubly degenerate, and its condensation can lead to two anti ferroelectric phases, Pmma (Fig. 5d) and Cmcm, with displacements in one or both ...­Ti­O­... chains,

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Table 3. Energies of the antiferroelectric phases appearing as a result of the condensation of unstable phonons at the boundary of the Brillouin zone and of the ferroelectric Amm2 phase in free standing (BaTiO3)1(BaZrO3)n superlattices with n = 1, 2, and 3 (the energy of the paraelectric P4/mmm phase is taken as the energy reference) Unstable phonon X R Z n=1 phase P mm a Pmc21 Cm m a Abm2 P mm a Cmcm Amm2 E, meV ­37.3 ­68.1 ­33.7 ­64.9 ­41.9 ­68.9 ­72.7 phase Pmma Pm Cmma Abm2 Pmma Cmcm Amm2 n=2 E, meV ­52.1 ­95.2 ­51.1 ­94.3 ­60.7 ­100.3 ­101.0 phase Pmma Pm Cmma Abm2 Pmma Cmcm Amm2 n=3 E, meV ­59.8 ­109.0 ­59.7 ­108.8 ­69.6 ­114.9 ­115.1

respectively. These displacements result in the in plane microscopic polarization directed along [100] and [110]. It was the latter of these two phases which was considered above in the calculation of the inter layer interaction energy. The ferroelectric instability of the intermediate Pmma and Cmma phases is easily understandable: in these two phases, the structural relaxation occurs in only one of two ferroelectrically unstable ...­Ti­O­... chains propagating in the [100] and [010] directions, and so the structure retains the ferroelectric instability in the second chain. In the Pmc21, Pm, Abm2, and Cmcm phases, the structural relaxation occurs simulta neously in both chains; this is why these structures are stable (metastable). In all cases, the obtained interme diate and ferrielectric phases have higher energies in comparison with the ferroelectric Amm2 phase (Table 3). Now we discuss the structure of rotations in the SLs under consideration. A comparison of the frequencies of unstable phonons corresponding to various combi nations of rotations in the layers of SLs with the same period shows that phonons with out of phase rota tions in neighboring layers are energetically least sta ble. These rotations are fully consistent with the insta bility at the R point in the parent cubic BaZrO3 (Fig. 1). The main difference between the two rotation systems corresponding to unstable phonons at the M and A points is the relation between rotations in the neighboring periods of the superlattice: for the phonon condensation at the M point, when the structure is translated by one SL period, the octahedra are rotated in phase; in the case of the phonon condensation at the A point, they are rotated out of phase. It can be seen that the ground state of the SL always corre sponds to such a rotation system in which the octahe dra are rotated out of phase in any pair of neighboring layers, i.e., the M phonon is condensed in the SLs with even period and the A phonon is condensed in the SLs with odd period.

One more subject for discussion is the comparison of the obtained results with the experimental data of [15]. According to the X ray data from this work, the spontaneous polarization in SLs is predominantly ori ented in the layer plane; however, the line narrowing and an increase in the frequency of the soft E mode in Raman spectra indicate the monoclinic distortion of the structure and the appearance of the z component of polarization in BaTiO3 layers. The z component of polarization determined from the hysteresis loops nonmonotonically depends on the superlattice period. The authors of [15] supposed that the appearance of the polarization component normal to the film is caused by the ferroelectric phase transition predicted in [6] in biaxially compressed BaZrO3 layers. The calculations performed in the present work showed that the A2u phonon is indeed unstable in biax ially compressed BaZrO3 (space group P4/mmm) with the in plane lattice parameter equal to the lattice parameter of the free standing (BaTiO3)1(BaZrO3)1 SL (a0 = 7.7182 Bohr) (according to our data, the crit ical lattice parameter at which the instability of this phonon appears is ~7.735 Bohr and notably exceeds the value of 7.425 Bohr specified in [6]). However, the instability of the A2u phonon in biaxially stretched BaTiO3 with the lattice parameter equal to a0 is absent (the phonon frequency is 45 cm­1). Therefore, the appearing of the z component of polarization in SLs needs a more detailed analysis. As we have shown previously [3], the A2u mode (which is responsible for the appearance of the z com ponent of polarization) is stable (37 cm­1) in the paraelectric P4/mmm phase of the free standing (BaTiO3)1(BaZrO3)1 SL. The calculations of the phonon frequencies for the P4/mmm phase of (BaTiO3)n(BaZrO3)n SLs with n = 2, 3, and 4 per formed in this work showed that the frequency of this mode rapidly decreases with increasing n (28.5 cm­1 for n = 2 and 17 cm­1 for n = 3) and becomes unstable at n = 4 (7i cm­1). However, the instability of the A2u
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phonon in the paraelectric phase does not mean that the z component of polarization will appear in the fer roelectric ground state. The in plane spontaneous polarization, which appears due to the strong instabil ity of the Eu phonon, abruptly increases the frequency of the corresponding mode (B1 or B2) in the Amm2 phase, and this phase becomes stable with respect to the appearance of the z component of polarization. This stability reduction of the A2u phonon with increasing n recalls the results obtained for the (BaTiO3)m(BaO)n Ruddlesden­Popper SLs [20]. In these SLs, a similar decrease in the A2u phonon fre quency with increasing the BaTiO3 layer thickness was observed; at m 8, it became unstable. However, already at m = 4, the instability at the X point appear ing in the SLs resulted in the formation of domain like structure, since it was energetically more favorable to form a structure with a z component of polarization periodically varying in space, rather than to uniformly polarize the BaO layer having a low dielectric con stant. The macroscopic z component of polarization is absent in such a structure. Another possible cause for the appearance of the z component of polarization can be a substrate induced strain in the superlattice (in [15], the SLs were grown on MgO substrate). Calculations show that in (BaTiO3)n(BaZrO3)n SLs with n = 1 and 2, for the in plane lattice parameter equal to a = 0.99a0 and 0.98a0 (a0 is the in plane lattice parameter of the free stand ing SL) and the disabled octahedra rotations, the polarization vector is rotated by ~38° and ~67° from the layer plane (the space group of the ferroelectric ground state is Cm); for a = 0.97a0 and disabled rota tions, the vector is rotated by 90° (the space group of the ferroelectric ground state is P4mm). However, when the octahedra rotations are enabled, in all SLs we see an unexpected result: the z component of polarization disappears in SLs with a = 0.99a0 and 0.98a0 (space group of the ground state is Pmc21 for n = 1 and Pnc2 for n = 2), whereas in the SL with a = 0.97a0, the polarization rotates toward the layer plane by ~24° (the space group of the ground state is Pc). The competition between the ferroelectric and structural instabilities in crystals with the perovskite structure has been known for a long time [21, 22]; however, the fact that the octahedra rotations can have such a strong effect on the polarization seems to be observed for the first time. This finding can be impor tant for the following reason. In the SLs under consid eration, the energy gain resulting from the structural distortions is less than the energy gain resulting from the ferroelectric ordering (compare the energies E1 and E2 in Table 2). This leads us to expect that, as the temperature increases, the structural phase transition in the SLs will occur at a lower temperature than the ferroelectric phase transition. Then, since the disap pearance of the octahedra rotations results in the onset
PHYSICS OF THE SOLID STATE Vol. 55 No. 6

of the z component of polarization, the situation is possible that an improper ferroelectric phase transi tion accompanied by the appearance of a nonzero z component of polarization can occur in the SLs when the structural phase transition temperature is approached from below (when the octahedra rotation angle rapidly decreases). The abrupt rotation of the polarization vector as the temperature varies can result in the appearance of a number of physical property anomalies resembling the anomaly in the piezoelectric coefficients near the morphotropic phase boundary. However, currently, it is not clear whether the struc tural phase transition occurs in BaZrO3. Nevertheless, even if the establishment of the long range order in the octahedra rotations in BaZrO3 is impossible for some reason, the changes in the local angles of the rotations with temperature can have a significant effect on the z component of polarization. The above considered attempts to explain the appearance of the z component of polarization, which were based on the assumption of a uniformly strained BaTiO3/BaZrO3 SL, were not successful (the strong biaxial substrate induced in plane compression of the SL cannot be retained in a thick SL). It seems that the only way to explain this phenomenon is to consider the stress relaxations occurring in SLs with thick layers. This idea has already been used to explain the appear ance of the z component of polarization in BaTiO3/BaZrO3 SLs with large periods in [3, 13]. Our calculations show that the critical in plane lattice parameter at which the z component of polarization appears in BaTiO3 is ~7.565 Bohr; in this case, the out of plane lattice parameter is c 7.471 Bohr. If we take into account the systematic error (­0.7%) in the pre diction of the lattice parameters for BaTiO3, the exper imental parameter c = 3.981 å in the BaTiO3 layer should correspond to this situation. According to the X ray data [15], such a relaxation of mechanical stress already appears in the SL with a period of L = 32 å. So, this relaxation explains the appearance of the z component of polarization in SLs with the specified and larger periods. The first principles calculations allow us to under stand why the quality of short period BaTiO3/BaZrO3 and BaTiO3/Ba(Ti,Zr)O3 SLs grown in the [001] direction is not very good [15, 23]. This is because of the tendency of BaTi1­ xZrxTiO3 solid solutions to three dimensional chessboard type ordering of cat ions at the B sites. The calculations show that the energy of the paraelectric phase of the (BaTiO3)1(BaZrO3)1 SL with the elpasolite structure (space group Fm3m), which appears when growing the SL in the [111] direction, is by 83 meV (per formula unit) lower than the energy of the same SL but grown in the [001] direction. It is possible that defects gener ated during the stress relaxation in SLs grown in the [001] direction become the nucleation centers of three dimensional chessboard type ordered inclu

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LEBEDEV 2. P. Ghosez and J. Junquera, in Handbook of Theoretical and Computational Nanotechnology, Ed. by M. Rieth and W. Schommers (American Scientific, Valencia, California, United States, 2006), Vol. 9, p. 623. 3. A. I. Lebedev, Phys. Solid State 52 (7), 1448 (2010). 4. A. I. Lebedev, Phys. Solid State 53 (12), 2463 (2011). 5. A. I. Lebedev, Phys. Status Solidi B 249, 789 (2012). 6. O. DiÈguez, K. M. Rabe, and D. Vanderbilt, Phys. Rev. B: Condens. Matter 72, 144101 (2005). 7. A. R. Akbarzadeh, I. Kornev, C. Malibert, L. Bellaiche, and J. M. Kiat, Phys. Rev. B: Condens. Matter 72, 205104 (2005). 8. J. W. Bennett, I. Grinberg, and A. M. Rappe, Phys. Rev. B: Condens. Matter 73, 180102(R) (2006). ´ 9. A. Bili c and J. D. Gale, Phys. Rev. B: Condens. Matter 79, 174107 (2009). 10. T. Tsurumi, T. Ichikawa, T. Harigai, H. Kakemoto, and S. Wada, J. Appl. Phys. 91, 2284 (2002). 11. T. Harigai, S. M. Nam, H. Kakemoto, S. Wada, K. Saito, and T. Tsurumi, Thin Solid Films 509, 13 (2006). 12. T. Harigai and T. Tsurumi, Ferroelectrics 346, 56 (2007). 13. P. R. Choudhury and S. B. Krupanidhi, Appl. Phys. Lett. 92, 102903 (2008). 14. P. R. Choudhury and, S. B. Krupanidhi, J. Appl. Phys. 104, 114105 (2008). 15. M. E. Marssi, Y. Gagou, J. Belhadi, F. D. Guerville, Y. I. Yuzyuk, and I. P. Raevski, J. Appl. Phys. 108, 084104 (2010). 16. X. Gonze, B. Amadon, P. M. Anglade, J. M. Beuken, F. Bottin, P. Boulanger, F. Bruneval, D. Caliste, R. Car acas, M. CÒtÈ, T. Deutsch, L. Genovese, P. Ghosez, M. Giantomassi, S. Goedecker, D. R. Hamann, P. Hermet, F. Jollet, G. Jomard, S. Leroux, M. Man cini, S. Mazevet, M. J. T. Oliveira, G. Onida, Y. Pouil lon, T. Rangel, G. M. Rignanese, D. Sangalli, R. Shal taf, M. Torrent, M. J. Verstraete, G. Zerah, and J. W. Zwanziger, Comput. Phys. Commun. 180, 2582 (2009). 17. Opium­Pseudopotential Generation Project. URL http://opium.sourceforge.net/ 18. A. I. Lebedev, Phys. Solid State 51 (2), 362 (2009). 19. R. Yu, H. Krakauer, Phys. Rev. Lett. 74, 4067 (1995). 20. E. Bousquet, J. Junquera, and P. Ghosez, Phys. Rev. B: Condens. Matter 82, 045426 (2010). 21. W. Zhong and D. Vanderbilt, Phys. Rev. Lett. 74, 2587 (1995). 22. D. Vanderbilt and W. Zhong, Ferroelectrics 206­207, 181 (1998). 23. F. De Guerville, M. El Marssi, I. P. Raevski, M. G. Karkut, and Y. I. Yuzyuk, Phys. Rev. B: Con dens. Matter 74, 064107 (2006).

sions. Furthermore, the possible electrical activity of these defects can be a factor which changes the elec trostatic boundary conditions at the heterointerface and stabilizes the polar state in the BaTiO3 layer with out inducing an appreciable electric field and polar ization in the BaZrO3 layer. 5. CONCLUSIONS The properties of BaTiO3/BaZrO3 superlattices with competing ferroelectric and structural instabili ties were calculated from first principles within the density functional theory. It was established that the quasi two dimensional ferroelectricity with the polar ization oriented in the layer plane, which weakly inter acts with the polarization in neighboring layers, occurs in SLs grown in the [001] direction with one unit cell thick barium titanate layer. It was shown that the energy gain from the ferroelectric ordering and the height of the potential barrier separating different ori entational states of polarization are sufficiently large to observe the formation of an array of independent polarized planes at room temperature. The effect of the structural instability on the properties of SLs was considered. It was shown that the ground state is a result of simultaneous condensation of the ferroelec tric 15 mode and phonons at the M point (for SLs with even period) or at the A point (for SLs with odd period). Thus, the ground state is a structure with out of phase rotations in neighboring layers, in which highly polarized layers and layers with strong octahe dra rotations are spatially separated. This significantly reduces the influence of structural distortions on the spontaneous polarization. It was shown that the switching on of the octahedra rotations in biaxially compressed BaTiO3/BaZrO3 SLs with an in plane strain of 1­3% results in an abrupt change in the polarization direction. This suggests that an improper ferroelectric phase transition accompanied by the appearance of a nonzero z component of polarization can occur in these SLs with increasing temperature. ACKNOWLEDGMENTS The calculations presented in this work were per formed on the laboratory computer cluster (16 cores) and the SKIF MGU "Chebyshev" and "Lomonosov" supercomputers. REFERENCES
1. K. M. Rabe, Curr. Opin. Solid State Mater. Sci. 9, 122 (2005).

Translated by A. Kazantsev

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2013