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ISSN 1063 7834, Physics of the Solid State, 2011, Vol. 53, No. 12, pp. 2463­2467. © Pleiades Publishing, Ltd., 2011. Original Russian Text © A.I. Lebedev, 2011, published in Fizika Tverdogo Tela, 2011, Vol. 53, No. 12, pp. 2340­2344.

FERROELECTRICITY

Quasi Two Dimensional Ferroelectricity in KNbO3/KTaO3 Superlattices
A. I. Lebedev
Moscow State University, Moscow, 119991 Russia e mail: swan@scon155.phys.msu.ru
Received December 6, 2010; in final form, April 20, 2011

Abstract--First principles density functional theory is used to calculate the phonon spectrum in the paraelectric phase, the ground state structure and polarization distribution in the polar phase, and energies of ferro and antiferroelectrically ordered phases of free standing (KNbO3)1(KTaO3)n ferroelectric superlat tices with n = 1­7. It is established that quasi two dimensional ferroelectricity with polarization oriented in the layer plane, which weakly interacts with polarization in neighboring layers, appears in potassium niobate layers with a thickness of one unit cell in the superlattices. The possibility of using of such ferroelectric super lattices as a medium for three dimensional information recording is shown. DOI: 10.1134/S1063783411120122

The problem of critical size of ferroelectric parti cles and critical thickness of ferroelectric thin films, at which ferroelectricity disappears in them, is of funda mental and practical importance. The first studies of ferroelectric nanoparticles showed that ferroelectricity disappears at a particle diameter of ~100 å. However, further experiments on ultrathin films [1] and first principles theoretical calculations [2­6] showed that a nonzero polarization normal to the film surface is retained in films with a thickness up to 3­6 unit cells if the depolarizing field is compensated. In films of PVDF TrFE copolymer, an organic ferroelectric, switchable polarization was observed in films of two monolayers (10 å) thick [7]. The ferroelectric state with polarization parallel to layers was established in BaTiO3 and PbTiO3 films with a thickness of three unit cells [8, 9] and zirconate­titanate films with a thick ness of one unit cell [10]. An unstable ferroelectric mode with polarization in the layer plane is also char ' '' acteristic for Pb B 1/2 B 1/2 O 3 solid solution films with a thickness of one unit cell [11]. In all the above cases, the transition to two dimensional ferroelectricity resulted from the physical reduction of the sample size in one direction. In this work, we demonstrate that arrays of nearly independent ferroelectrically ordered quasi two dimensional layers--the structures which can be used as a medium for three dimensional infor mation recording--can be formed in KNbO3/KTaO3 superlattices. Ferroelectric phenomena in KTa1­ xNbxO3 solid solutions have been studied for a long time. Experi ments show that a polar phase appears in samples at x > xc = 0.008 [12], the properties of which near the critical concentration xc are characterized by manifes

tations of quantum effects and local disorder (dipolar glass). To explain the peculiarities of the system behav ior near xc, a number of models were proposed [13, 14], including the hypothesis of off centering of Nb atoms in KTaO3 [15­19]. However, as shown in [20], the single well shape of the local potential of Nb atom in the solid solution excludes the possibility of its off centering (this was confirmed by the present calcula tions). Properties of ferroelectric superlattices (SLs) in the KNbO3/KTaO3 system have been studied in [21­ 30]; the manifestations of antiferroelectricity in them were observed in [27, 29]. Calculations of phonon fre quencies at the point in the paraelectric phase of (KNbO3)1(KTaO3)7 SL [31] detected weak ferroelec tric instability with polarization in the layer plane at the lattice parameter equal to the KTaO3 lattice parameter. However, the nature of this instability and the possibility of its observation in other conditions (in particular, in SLs with other periods) were not ana lyzed in that paper. The objective of this work is to study the evolution of ferroelectric instability in free standing (KNbO3)1(KTaO3)n SLs with increasing n. These studies were motivated by the disagreement between the results of our previous calculations of the ground state structure of SLs with n = 1 and 3 [30] and the results of [29]. Furthermore, the conclusion about the tendency of Nb atoms in KTa1­ xNbxO3 solid solution to clustering, based on the calculations of [31], seemed to be somewhat strange because the energy gain was observed only in structures with Nb atoms ordered in planes and was absent in structures with Nb atoms ordered in chains.

2463


2464 100 50 A 0 , cm-
1 2u

LEBEDEV 12 10 E, meV E
u

8 6 4 2

-50

-100 -150 1 2 3

4 n

5

6

7

0

0.1

0.2

0.3 1/L

0.4

0.5

Fig. 1. Dependence of the energy of polar optical A2u and Eu modes in the paraelectric P4/mmm phase on the thick ness of potassium tantalate layer in (KNbO3)1(KTaO3)n superlattices.

Fig. 2. Ferroelectric ordering energy per niobium atom as a function of the inverse period L = n + 1 of the (KNbO3)1(KTaO3)n superlattices.

The (KNbO3)1(KTaO3)n superlattices considered in this work were periodic structures grown in [001] direction and consisted of a KNbO3 layer with a thick ness of one unit cell and a KTaO3 layer with a thickness of n unit cells (1 n 7). These structures were simu lated on supercells of 1 â 1 â (n + 1) unit cells. In addi tion, properties of 2 â 2 â 2 and 4 â 2 â 1 supercells containing one niobium atom were studied. The calculations were performed using first princi ples density functional theory with pseudopotentials and wave functions expanded in plane waves as imple mented in the ABINIT program [32]. As in [30], the exchange correlation interaction was described in the local density approximation (LDA). Optimized sepa rable nonlocal pseudopotentials constructed using the OPIUM program and complemented by a local potential to improve transferability were used as pseudopotentials. The parameters used to construct pseudopotentials and other calculation details are given in [30, 33]. The maximum energy of plane waves was 40 Ha (1088 eV); integration over the Brillouin zone was performed using Monkhorst­Pack grids of 8 â 8 â 4, 8 â 8 â 2, and 6 â 6 â 2 size for SLs with n = 1 and 2, n = 3 and 4, n = 5 and 7, respectively. Atomic positions and lattice parameters were relaxed until the Hellmann­Feynman forces become smaller than 5 â 10­6 Ha/Bohr (0.25 meV/å). Phonon spectra were calculated within the density functional perturbation theory. Spontaneous polarization was calculated using the Berry phase method. The KNbO3 and KTaO3 structure calculated using the described approach is in good agreement with experiment. For example, the lattice parameters of cubic KNbO3 and KTaO3 are equal to 3.983 and 3.937 å and differ from the experimental values (4.016 and 3.980 å) by 0.81 and 1.07%, respectively (the

small underestimate of the lattice parameters is typical for LDA approximation). The calculated ratio c/a in the KNbO3 tetragonal phase is 1.0197 whereas the experimental value is 1.0165; the calculated spontane ous polarization in the tetragonal phase of KNbO3 is 0.372 C/m2 (the experimental values are 0.37­ 0.39 C/m2 [34]). The calculated frequencies of two lowest energy optical modes in the paraelectric P4/mmm phase of free standing (KNbO3)1(KTaO3)n SLs with n = 1­7 are shown in Fig. 1. In the SL with n = 1, both ferro electric modes with A2u and Eu symmetry, which genetically arise from unstable TO phonon at the cen ter of the Brillouin zone of the perovskite structure, are unstable. As n increases, the A2u mode becomes stable, and only one unstable Eu phonon remains in the phonon spectrum. The search for an equilibrium structure shows that the ground state for SL with n = 1 is the Cm phase [30]; for other SLs (despite the exist ence of unstable A2u mode in the P4/mmm phase for SLs with n 3) the Amm2 phase is the ground state.1 The polarization vector in this phase is in the layer plane and is oriented along the [110] direction. As follows from Fig. 2, the energy gain resulting from the ferroelectric distortion of the structure does not decrease to zero with increasing n. Along with the existence of unstable phonons in the paraelectric phase, this indicates the stability of the ferroelectric ground state in KNbO3 layers of minimum possible
1

As follows from calculations of the phonon spectra in the paraelectric P4/mmm phase of SLs with n 3, along with the ferroelectric instability, unstable phonons at Z, X, and R points at the boundary of the Brillouin zone are observed. Among cor respondingly distorted phases, the lowest energy is inherent to the polar Amm2 phase in which all phonons are stable at all points of the Brillouin zone. Vol. 53 No. 12 2011

PHYSICS OF THE SOLID STATE


QUASI TWO DIMENSIONAL FERROELECTRICITY 10

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10- Ps, C/m2

1

2Wint, meV -4 -2 0 N 2 4

1

10-

2

0.1

1

2 n

3

Fig. 3. Polarization profiles in (KNbO3)1(KTaO3)n super lattices with n = 3, 5, and 7. N = 0 corresponds to the potassium niobate layer.

Fig. 4. Energy difference of ferroelectric and antiferroelec tric ordering as a function of the thickness of potassium tantalate layer in [(KNbO3)1(KTaO3)n]2 superlattices.

thickness (one unit cell) enclosed between thick KTaO3 layers. The polarization profiles in superlattices were cal culated by the approximate formula Ps = (e/)iwi Z i* ui, where is the unit cell volume of the layer under consideration, ui is the displacement of the ith atom with respect to its position in the nonpolar structure, Z i* is the effective charge of this atom,2 and wi is the weight factor equal to unity for Nb(Ta) and O atoms lying in the Nb(Ta)­O layer, equal to 1/2 for K and O atoms lying in two nearest K­O planes, and equal to zero for other atoms. The dependences of Ps on the layer number in SLs with n = 3, 5, and 7 are shown in Fig. 3. One can see that in the region between KNbO3 layers the polarization decreases approxi mately exponentially with a characteristic decay length of ~3 å. This result indicates strong polariza tion localization in the potassium niobate layer and proves the quasi two dimensional nature of ferroelec tricity in the structures under consideration. The polarization in the KNbO3 layer monotonically decreases from 0.279 to 0.257 C/m2 when going from SL with n = 3 to SL with n = 7 (for comparison, in bulk orthorhombic KNbO3 the calculated Ps value is 0.418 C/m2). The total polarization of SL calculated using the Berry phase method coincides within 3% with the sum of polarizations of the layers. To determine the interaction energy Wint between neighboring polarized KNbO3 layers separated by KTaO3 layers, the energy of [(KNbO3)1(KTaO3)n]2 superlattices with doubled period and antiferroelectric ordering of neighboring potassium niobate layers was
2

calculated. The energy difference of ferroelectrically and antiferroelectrically ordered structures per nio bium atom is shown in Fig. 4. One can see that the dependence of the difference between these energies on the KTaO3 layer thickness is exponential with a characteristic decay length of 2.9 å. The criterion of polarization stability in quasi two dimensional layer with respect to spontaneous reversal of polarization is the condition 2Wint < U, where Wint is the interlayer interaction energy and U is the height of the potential barrier between different orientational states of polarization in the layer. The factor 2 in the formula corresponds to the worst case where the polar ization in the layer under consideration is antiparallel to the polarization direction in neighboring layers. In the studied structures, the easiest way to reorient polarization is its rotation in the layer plane; thus, U is equal to the energy difference between structures polarized along the [110] and [100] directions. According to calculations, U = 1.84 meV for SL with n = 2 and 1.69 meV for SL with n = 3. A comparison of U and 2Wint (Fig. 4) shows that the polarization stability criterion is satisfied at n 2. A comparison of the data shown in Figs. 2 and 4 enables to conclude that the interaction energy between spontaneously polarized layers in superlattices with n 3 is less than 10% of the ordering energy. This gives reasons to consider such superlattices as arrays of nearly independent spontane ously polarized layers, with satisfied conditions of quasi two dimensional ferroelectricity in each of them. A comparison of energies of 2 â 2 â 2, 4 â 2 â 1, and 1 â 1 â 8 supercells with the same concentration but different niobium atom ordering patterns (they corre spond to the absence of pairs of nearest Nb atoms, lin ear chains of Nb atoms, and planes of Nb atoms) show any evidence of niobium clustering in planes: the energies of relaxed paraelectric phases of 4 â 2 â 1 and
2011

In the calculation, the effective charges obtained for the P4/mmm phase of SL with n = 1 were used; the effective charges in SL with n = 2­7 differed slightly from the used ones. PHYSICS OF THE SOLID STATE Vol. 53 No. 12


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LEBEDEV

Frequencies of unstable phonons at high symmetry points of the Brillouin zone for P4/mmm phase of (KNbO3)1(KTaO3)n superlattices with n = 1, 2, and 3 Phonon frequencies, cm­1 n 1 2 3 (0, 0, 0) Z (0, 0, 1/2) X (1/2, 0, 0) R (1/2, 0, 1/2) 143i 118i 114i 126i 114i 113i 71i 50i 46i 61i 50i 45i

1 â 1 â 8 supercells (eight perovskite molecules) are higher than the energy of the 2 â 2 â 2 supercell (con taining no nearest Nb atom pairs) by 3.0 and 8.9 meV, respectively. The tendency to the short range ordering was observed only in structures in which Nb and Ta atoms alternated in chains: e.g., the energy of the paraelectric phase of (KNbO3)1(KTaO3)1 and (KNbO3)1(KTaO3)2 SLs grown in the [111] direction was lower by 14.0 and 18.8 meV, respectively, than the energy of the paraelectric phase of SLs of the same composition, but grown in the [001] direction. The results obtained in this study are in agreement with the results of atomistic simulation of KNbO3/KTaO3 superlattices [24, 25]. According to these works, the polarization component parallel to layers decreased by a factor of 3­4 at a distance of one lattice period (in contrast to the polarization compo nent normal to layers). Our results on the instability of phonon with A2u symmetry in SL with n 3 agree with the data of [29] only partially: in [29], the stable polar phase with the P4mm symmetry was obtained only in SLs with n 2. However, despite the qualitative agree ment of results of this and previous studies, the physi cal conclusions of this work differ significantly. First, as shown above, the P4mm phase is not the ground state structure in any considered KNbO3/KTaO3 SLs. The cause why the Cm or Amm2 phases are the ground state in considered SLs is the discovered in [30] tendency of the polarization rota tion toward the layer plane, which enables to lower the electrostatic and mechanical energy in the structure. Second, the retention of stable polarization in KNbO3 layer with a thickness of one unit cell and highly inhomogeneous distribution of polarization, with the interaction between neighboring polarized layers that exponentially decreases with increasing dis tance between them, leads to a new previously unknown feature of the ground state structure of (KNbO3)1(KTaO3)n SLs--the formation of an array of nearly independent quasi two dimensionally polar ized layers in the bulk of the superlattice. As for the above mentioned simultaneous presence of unstable phonons at the center of the Brillouin zone and at its boundary (with antiferroelectric character of the eigenvector) in the phonon spectra of paraelectric

phase of SL, from the Table it follows that the frequen cies of two pairs of unstable modes (at points and Z, X and R) converge as the potassium tantalate layer thickness n increases. This means that when the inter action between layers weakens, the tendency of the system to ferroelectric ordering in planes cease to depend on whether the polarization in neighboring layers is oriented parallel or antiparallel. This supports the conclusion about the independence of quasi two dimensional ferroelectrically ordered planes in SLs with thick KTaO3 layers. We discuss now some possible applications of the discovered phenomenon. At first sight, the fact that the polarization in considered SLs is in the layer plane may seem uninteresting. So far, when discussing the ferroelectricity in thin films, the main attention was paid to films in which polarization is normal to their surfaces, as they are more suitable for practical appli cations. However, the physical interactions responsi ble for the formation of quasi two dimensional ferro electric ground state in KNbO3/KTaO3 SLs are the factors which transform the superlattice into the array of nearly independent polarized planes. These struc tures can be used as a medium for three dimensional information recording. Accepting that the lateral size of ferroelectric domains that have long term stability with respect to spontaneous reversal of polarization is 250 å [35], the potential volume information density is ~1018 bit/cm3 at an interlayer distance of 16 å (cor responding to the period of the SL with n = 3). This value exceeds the volume information density achieved in modern optical storage devices by six orders of magnitude. It should be added that the two component order parameter, which describes polarization in KNbO3 layers and enables the polarization rotation by 90°, offers one more interesting opportunity. In ferroelec tric memory devices (FeRAM) with polarization nor mal to the film surface the surface charge screening resulting from leakage currents and film conductivity is the main obstacle for implementing the nondestruc tive read out method. The systems with the polariza tion rotation by 90° have a property which in principle can help to overcome the disadvantage: nondestructive read out using the anisotropy of dielectric constant in the layer plane can be implemented in them. For example, in (KNbO3)1(KTaO3)n SL with n = 2 calcu lated dielectric constants in the plane parallel and per pendicular to polarization are 142 and 188; in SL with n = 3 these values are 225 and 281. Finally, we note that similar phenomena were observed in two more ferroelectric superlattices, BaTiO3/BaZrO3 and BaTiO3/BaSnO3. In contrast to the KNbO3/KTaO3 SL considered in this work, the ferroelectric interaction in BaTiO3 layers with a thick ness of one unit cell and the interlayer interaction in titanate based superlattices are complicated by the structural instability with respect to octahedra rota
Vol. 53 No. 12 2011

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tion, which competes with the ferroelectric instability. In one more studied SL, BaTiO3/SrTiO3, the interlayer interaction is too strong and prevents the formation of the quasi two dimensional ground state. The results of the study of these superlattices will be published in a separate paper. In conclusion, the ground state structure and prop erties of ferroelectric (KNbO3)1(KTaO3)n superlattices with n = 1­7 were calculated within the density func tional theory. It was shown that the tendency to ferro electric ordering is retained in KNbO3 layers of mini mum thickness (one unit cell) whereas the interaction energy between neighboring layers exponentially decreases with increasing n. An analysis shows that quasi two dimensional ferroelectricity with polariza tion oriented in the layer plane, which weakly interacts with polarization in neighboring layers, appears in potassium niobate layers in studied superlattices. In superlattices with n 3, the array of nearly independent ferroelectrically polarized planes becomes the ground state. The use of such arrays as a medium for three dimensional information recording enables to achieve the volume information density of the order of 1018 bit/cm3. The results obtained in this work show that in complex systems the polarization calculated using the Berry phase method should not always be interpreted as a bulk property. Changes in polarization in a supercell can be so large that the ferroelectric sys tem acquire properties of two dimensional systems. ACKNOWLEDGMENTS calculations presented in this work were per on the laboratory computer cluster (16 cores) "Chebyshev" SKIF MGU supercomputer. work was supported by the Russian Founda Basic Research, project no. 08 02 01436. REFERENCES
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Translated by A. Kazantsev
2011