Piezoelectric Response from Rotating Polarization

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1 Piezoelectric Response from Rotating Polarization Huaxiang Fu and R. E. Cohen Carnegie Institution of Washington, 5251 Broad Branch Road, N.W., Washington D.C Abstract. Piezoelectric response induced by rotating the polarization, which occurs when the external field is not in the same direction as the internal spontaneous polarization, is studied using a first-principles approach. The piezo-response of the rhombohedral BaTiO 3 to the field along the <001> direction is considered. The response is found to be path-dependent, and along the lowest free-energy path the strain increases significantly with the field. Furthermore, the strain response (collinear response) to a field along the polarization direction (i.e., without polarization rotation) is also calculated for comparison, and the piezo-responses with and without rotating polarization are found to be very different. External electric fields drive atomic displacements and strain when applied to piezoelectrics. The strain develops to relieve the interatomic repulsions. The field is often applied in the same direction as the internal polarization. However, for the ground-state phase, if the energyprofile along the internal spontaneous polarization direction is too steep to be overcome bythe field along the same direction, it could be possible to obtain a larger strain response byapplying a field oblique to the polarization. For BaTiO 3, this is indeed the case. BaTiO 3 is rhombohedral with its polarization along the <111> direction in the low temperature. A field along the <001> direction facilitates a large piezo-response via polarization rotation in rhombohedral BaTiO 3, and a much smaller collinear response is found for the field along the <111> direction. Polarization rotation was originallysuggested [1] as the reason for the giant piezoelectric response in the new single-crystal piezoelectrics Pb(Zn 1/3 Nb 2/3 )O 3 -PbTiO 3 (PZN-PT) and Pb(Mg 1/3 Nb 2/3 )O 3 -PbTiO 3 (PMN-PT). The field dependence of piezoelectric properties was reported byliu et al. [2] for PZN-PT, and bypark et al. [3] for BaTiO 3. In both cases, it was shown that for rhombohedral single crystals, a field oblique to the polarization causes an enhanced piezo-response. Garcia and Vanderbilt [4] used an effective Hamiltonian approach for BaTiO 3 under finite fields, and found a field-induced phase transition, but the rotation mechanism was not emphasized. Ramer et al. [5] studied the tetragonal-rhombohedral

2 phase transition of PZT induced bya stress, where the polarization rotation is expected but not manifested in the calculation. Uchino et al. discussed the large anisotropyof manypiezoelectrics seen experimentally[6], and this analysis is consistent with the importance of polarization rotation. First-principles calculations [7] show that rotating the polarization from the rhombohedral to tetragonal phases is indeed able to drive an enhanced piezoelectric effect in single crystal BaTiO 3. The underlying reason for this enhanced effect is due to the small internal energy barrier and the large polarization variation caused bythe polarization rotation. Though computations of polarization rotation have not yet been done for the new generation single-crystal PMN-PT and PZN-PT, which are complex solid solutions with complex structure [8], the same principles mayalso be applied in PMN-PT and PZN-PT. Pure PZN and PMN are relaxors, but when alloyed with a small amount of PT, PMN and PZN become ferroelectrics with a rhombohedral ground state. In PMN- PT and PZN-PT single crystals with the ground state polarization along <111>, and a field along the cube axis <001> gives a verylarge piezoelectric response, whereas a field along <111> gives a small, ordinarystrain [1,2]. Ceramic samples also show ordinarypiezoelectric coefficients. Supercell computations for ordered PMN-PT show intrinsic piezoelectric coefficients to be much larger than PZT, bya factor of 2.7 [9], but not large enough to explain the experimental values obtained for single crystal PZN-PT and PMN-PT. Here, we compare the strain response caused byrotating the polarization with the collinear response in BaTiO 3 using a first-principles approach. Currently, there is no practical method for treating an infinite insulator in a finite macroscopic electric field. In principle this will require a new energyfunctional including both charge densityand polarization in the framework of densityfunctional theory[10], and the exchange-correlation functional will have a field dependence [11]. Furthermore, a finite electric field gives an infinite potential change across the infinite system, and thus band theoryis no longer valid [12]. Band theorywould predict macroscopic charge flow, i.e., an insulator in a field would behave as a metal within band theory. Thus we use an approximate approach based on the internal energy U at zero field and the free energy F = U E P, wheree is the external field and P is the polarization. The computations were performed using the linearized augmented plane-wave method [13] as follows: We chose different polarization directions (actuallyti displacement directions), and for each direction we optimized the internal energy U, the atomic positions, and the cell shape, while keeping the volume fixed at experimental value (64.2 Å 3 ). Ideallythe free energyf should be optimized, but this is not currentlytractable since the polarization would have to be computed at each optimization step, and forces would not be available. We used RK max =8.0 forthe size of basis set and Monhorst-Pack s meshes for K point sampling. The polarization was calculated using the Berry s geometric phase [14,15] from the wave functions of optimized structure. The polarization directions we considered are shown in Figure 1. Two possible

3 e d c b a g f z y x FIGURE 1. Polarization directions considered in the calculation. Two possible paths for polarization rotation are: a b c d e (Path 1) and a f g e (Path 2). paths are of particular interest for polarization rotation, i.e., path a b c d e ( Path 1 ) and path a f g e ( Path 2 ). For each polarization direction, the internal energyand the polarization component along the field direction are shown in Table 1, which gives the relative stabilityof three possible phases of BaTiO 3. The energydifference between the ordered rhombohedral phase and the tetragonal phase is about 6.4 mev, comparing with 7.8 mev obtained from our pseudopotential calculation [16]. The magnitudes of spontaneous polarization per unit volume (0.44, 0.43, 0.45 C m 2 ) agree with other pseudopotential calculation [17] (0.36, 0.43, and 0.43 C m 2 ), respectively, for the tetragonal, orthorhombic and rhombohedral phases. We note however that the strain ɛ 33 (2%) of tetragonal phase calculated from LAPW is quite large comparing the pseudopotential result (1%). In experiments, about 1% strain is found for tetragonal phase, however, this phase is predominantlya disordered one for BaTiO 3 (rather than a purelydisplacive phase in the theoretical calculation). For PbTiO 3, LAPW-LDA also gives [18] c =1.12 a compared to 1.06 obtained from the pseudopotential calculations within LDA. It is not fullyclear what causes this difference, though we found that for BaTiO 3 the energydifferences of the tetragonal phases with strains ranging from 1 to 2.5% are quite small after the atoms are fullyrelaxed. Our LAPW calculation using generalized gradient approximation (GGA) gives a strain of 1.1% for BaTiO 3. The electric field needed to drive the polarization into a specific direction is obtained from the free energy. In principle, a large number of polarization directions are needed to obtain an accurate curve of transition field since a free energybarrier could occur during the rotation (as it does in our Path 1), whereas we consider onlya few directions because of the large computation demands in LAPW total energyminimization. Also, we have considered onlypolarization rotation (which is

4 A,B C a e b Strain (%) g f b 10 path 2 path 1 o Electric field (mv/a) c A B C Electric field (KV/cm) FIGURE 2. Strain responses under the electric field: (a) theoretical results for BaTiO 3 along two different paths. Symbols connected by dotted lines are from first-principles calculations while solid curves are obtained from analytically fitted internal energies as a function of polarization. (b) experimental measurements for PZN-8%PT (see Ref.1). the dominant effect), but ideallythe free energyand strain should be mapped out as functions of polarization magnitude as well as direction. The strain-vs-field results are shown in Figure 2a for the two paths. We see that the strains respond verydifferentlyfor two paths. Along Path 2, a relativelysmall field can drive a large strain in the low field region, after which the strain increases quite slowly. For this path, the strain response is quite similar to what found [1] for PZN-8%PT as shown in Figure 2b. Our theoretical piezo-coefficient of Path 2 in the low field region is about 293 pc N 1, comparing with the experimental value [3] of 350 pc N 1 obtained for BaTiO 3 at the low temperature. As a sharp contrast to Path 2, Path 1 shows quite slow strain increase driven bythe field. The strain response is about 5 times smaller than that along Path 2 in the low field region. Figure 2 also shows that along Path 2, a field of 22 mv/å is needed to transform the rhombohedral phase into the tetragonal phase, compatible with 25 mv/å obtained from the effective Hamiltonian [4]. In order to find out which path is more favorable, we analytically fit the internal energy as a function of polarization, and minimize the analytically obtained free energy for a given electric field. We found that for a given field Path 2 has a lower energy. To compare the piezo-response resulting from rotating the polarization with the normal response without polarization rotation, we calculated the normal strain-vsfield response for the rhombohedral phase (with the field along the <111> direction) and for the tetragonal phase (with the field along the <001> field). These results are shown in Figure 3 in comparison with the piezo-response from the field rotating the polarization. We see that the strain response is verysmall for the ground-state

5 BaTiO3 <001> (Rhom) Strain (%) <001> (Tetr) <111> (Rhom) Electric field ( mv/a ) FIGURE 3. Strain responses of the rhombohedral (solid line) and tetragonal (dotted line) phases to the <111> and <001> fields, respectively. The strain response (Path 2 in Figure 2) of the rhombohedral phase to the rotating field along the <001> direction is also shown for comparison (solid line with square symbols). rhombohedral phase if the field is along the <111> direction, i.e., the direction of its spontaneous polarization. However, the situation changes dramaticallywhen a polarization-rotating field is applied, and a large strain response is achieved. We also note that we need to be cautious about the tetragonal phase because its energy changes with respect to strain are verysmall. The basic idea that an enhanced piezo-response maybe caused bya polarization-rotating field is indeed born out by Figure 3. To analyze what determines the different strain responses in Figure 2 (between different paths) and in Figure 3 (among different-oriented fields), we look into the internal energyand polarization change listed in Table 1. For the comparison of Path 1 and Path 2, we see that near the rhombohedral phase Path 2 has a quite flat internal energysurface, and also the polarization variation resulting from rotating the polarization is quite large, both of which will facilitate the polarization rotation bylowering the free energy. This is the reason whya large piezo-response occurs for this particular path. From a microscopic perspective, rotating polarization along Path 2 can minimize the repulsion between the Ti atom and the O atoms at the base of the oxygen octahedron. This will be in favor of the flat energy surface. Meanwhile, this rotation path will increase the repulsion between the Ti atom and the O atom at the top of the oxygen octahedron. In order to ease such a repulsion, a

6 TABLE 1. Internal energy difference U = U U ref (in mev), the polarization variation P α = P α Pα ref (in e Bohr) along the field direction α relative to a reference state, and strain ɛ αα (in percentage). Note that for the non-rotation strain response of the tetragonal phase listed in the third panel, the tetragonal phase is taken as reference. Field direction Ref. state direction of P U P α ɛ αα <001> Rhom a b c d e f g <111> Rhom a a a <001> Tetr e e e tetragonal strain is needed, which is the reason whya large strain can be obtained. As to the different piezo-responses from different-oriented fields, we see that for the rhombohedral phase the energyprofile along its spontaneous polarization direction has a verysteep increase awayfrom its equilibrium state. This indicates a large field is needed. Furthermore, the polarization change along the <111> direction, which favors the strain increase, is quite small. We note that our calculations are at zero temperature. At finite temperature, BaTiO 3 is largelydisordered, and our first-principles calculations are not efficient to address the issue. The finite temperature studysuch as the effective Hamiltonian approach [4,19] will be more suitable. Thus we obtain the following model for the large electromechanical coupling in the new single crystal piezoelectrics PMN-PT and PZN-PT. Two ingredients are necessary. First there should be a rhombohedral phase (with small strain) and a tetragonal phase with large strain. Second, the energychange between rhombohedral and tetragonal via polarization rotation should be small. A simple picture is to consider PbTiO 3 which has a verylarge strain of 6% in the tetragonal phase. There is no rhombohedral phase in PbTiO 3, otherwise one could drive a huge strain bydriving it from the rhombohedral to the tetragonal phase. This implies that perovskites with up to 6% strain might be possible. The second ingredient needed is the energychange must be small for polarization rotation. In principle this energychange could go to zero, and the response could become huge. In fact, the energychanges are probablyverysmall in solid solution such as PMN-PT [20,21]. In this case, a verysmall field can cause the polarization

7 rotation, thus a huge piezoelectric response. The piezoelectric constant e 33 from internal strain can be written as (see, for example, Ref. [22]), e 33 = k ec Ω Z k 33 du k 3 dη 3, (1) where Ω is the cell volume, e is the electron charge, c is the c-axis length, Z k 33 are the effective charges, u is the atomic displacement, and η is the strain. Piezoelectric coefficient d 33 is related with e 33 via the elastic constants, which are related with the internal energydifference as we mentioned. More details remain to be worked out, for example, to understand the lack of a large piezoelectric response in relaxors such as PMN, and a better understanding of why a large response is not seen in polycrystalline samples of the new piezoelectrics. Apparentlythe anisotropymust be such that the large strain effect cancels in an aggregate, but exactlyhow this happens is not yet clear. Results presented in this volume show that the large piezoelectric effect in ceramic PZT can be understand as a large d 15 caused essentiallybypolarization rotation [22]. Perhaps large coupling ceramics can be made if theyhave the proper texture. Acknowledgements. This work was supported bythe Office of Naval Research. The authors would like to thank H. Krakauer and D. Singh for sharing their LAPW codes and discussions. We thank S.-E. Park, K. Rabe, N.J. Ramer, A.M. Rappe, G. Saghi-Szabo, T. Shrout, W. Smith, S. Stolbov, and D. Vanderbilt for discussions. Computations were performed on the CraySV1 supported bynsf and the Keck Foundation. REFERENCES 1. S.-E. Park and T.R. Shrout, J. Appl. Phys. 82, 1804 (1997). 2. S.-F. Liu, S.-E. Park, T.R. Shrout, and L.E. Cross, J. Appl. Phys. 85, 2810 (1999). 3. S.-E. Park, S. Wada, P.W. Rehrig, S.-F. Liu, L.E. Cross, and T.R. Shrout, J. Appl. Phys. 86, 2746 (1999). 4. A. Garcia and D. Vanderbilt, Appl. Phys. Lett. 72, 2981(1998). 5. N.J. Ramer, S.P. Lewis, E.J. Mele, and A.M. Rappe, in First-principles Calculations for Ferroelectrics: Fifth Williamsburg Workshop (ed. R.E. Cohen) P156 (AIP, Woodbury, New York, 1998). 6. X.-H. Du, J.-H. Zheng, U. Belegundu, and K. Uchino, Appl. Phys. Lett. 72, 2421 (1998). 7. H. Fu and R.E. Cohen, Nature (London) 403, 281(2000). 8. T. Egami, W. Dmowski, M. Arbas, and P.K. Davies, in First-principles Calculations for Ferroelectrics: Fifth Williamsburg Workshop (ed. R.E. Cohen) 1-10 (AIP, Woodbury, New York, 1998). 9. L. Bellaiche and D. Vanderbilt, Phys. Rev. Lett. 83, 1347 (1999). 10. R.M. Martin and G. Ortiz, Phys. Rev. B 56, 1124 (1997). 11. X. Gonze, Ph. Ghosez, and R.W. Godby, Phys. Rev. Lett. 74, 4035 (1995).

8 12. R.W. Nunes and D. Vanderbilt, Phys. Rev. Lett. 73, 712 (1994). 13. S.H. Wei and H. Krakauer, Phys. Rev. Lett. 55, 1200 (1985); D.J. Singh, Planewaves, Pseudopotentials and the LAPW Method (Kluwer Academic Publisher, Boston, 1994). 14. R.D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993). 15. R. Resta, Rev. Mod. Phys. 66, 899 (1994). 16. H. Fu, R. E. Cohen and O. Gulseren, (unpublished). The energy difference between the rhombohedral and tetragonal phases is 5.87 mev (using Kmeshes), 7.84 mev (using K meshes), and 5.35 mev (using Kmeshes).The energy cut-off for pseudopotential calculations is 65 Ry, and is large enough after convergence check. 17. Ph. Ghosez, J.-P. Michenaud, and X. Gonze, Phys. Rev. B 58, 6224 (1998). 18. G. Saghi-Szabo, R.E. Cohen, and H. Krakauer, Phys. Rev. Lett. 80, 4321(1998). 19. S. Stolbov, H. Fu, R.E. Cohen, L. Bellaiche, and D. Vanderbilt, this volume. 20. T. Egami, Ferroelectrics 222, 163 (1999). 21. T. Egami, S. Teslic, W. Dmowski, P.K. Davies and I.W. Chen, J. Korean Phys. Soc. 32, S935-S938 (1998). 22. L. Bellaiche, A. García, and D. Vanderbilt, this volume.

First-principles calculations of piezoelectricity and polarization rotation in Pb Zr 0.5 Ti 0.5 O 3

First-principles calculations of piezoelectricity and polarization rotation in Pb Zr 0.5 Ti 0.5 O 3 Zhigang Wu and Henry Krakauer Department of Physics, College of William and Mary, Williamsburg, Virginia