Strain gradient induced polarization in SrTiO3 single crystals
P. Zubko, G. Catalan, A. Buckley, P. R. L. Welche, J. F. Scott
aa r X i v : . [ c ond - m a t . m t r l - s c i ] O c t Strain gradient induced polarization in SrTiO single crystals P. Zubko, ∗ G. Catalan, † A. Buckley, P. R. L. Welche, and J. F. Scott
Centre for Ferroics, Department of Earth Sciences,University of Cambridge, Cambridge CB2 3EQ, United Kingdom (Dated: February 11, 2013)Piezoelectricity is inherent only in noncentrosymmetric materials, but a piezoelectric responsecan also be obtained in centrosymmetric crystals if subjected to inhomogeneous deformation. Thisphenomenon, known as flexoelectricity, affects the functional properties of insulators, particularlythin films of high permittivity materials. We have measured strain-gradient-induced polarizationin single crystals of paraelectric SrTiO as a function of temperature and orientation down to andbelow the 105 K phase transition. Estimates were obtained for all the components of the flexoelectrictensor, and calculations based on these indicate that local polarization around defects in SrTiO mayexceed the largest ferroelectric polarizations. A sign reversal of the flexoelectric response detectedbelow the phase transition suggests that the ferroelastic domain walls of SrTiO may be polar. Flexoelectricity is the coupling between dielectric po-larization and strain gradient. Because strain gradientsbreak inversion symmetry, flexoelectricity allows extract-ing charge from deformations even in materials that arenot piezoelectric. The historical development of this phe-nomenon has been summarized in the reviews by Tagant-sev [1] and more recently by Cross [2] and Sharma et al. [3]. For decades after its proposal in the early 1960s [4, 5],there were very few studies of flexoelectricity, which mayseem surprising given that it is a more general propertythan piezoelectricity. The reason is the comparativelysmall magnitude of the flexoelectric coefficient f ∼ e/a (where f is the constant of proportionality between po-larization and strain gradient, e the electronic charge and a the lattice parameter) in typical dielectrics which, com-bined with the small size of strain gradients in bulk sam-ples, means that the flexoelectric effect is generally mi-nuscule. Two recent developments, however, are chang-ing the perceived importance of flexoelectricity. First,a predicted proportionality between flexoelectric coeffi-cient and dielectric constant [5, 6] has inspired Cross andcoworkers to conduct a series of seminal experiments inwhich they have measured the flexoelectric coefficients ofhigh permittivity materials such as relaxors and ferro-electrics [2, 7], and found them to be very large indeed.In addition, Fousek et al. [8] realized that this largeflexoelectric response could be used to produce piezo-electric composites from centrosymmetric materials bysimply tailoring their shapes, an idea which was recentlydemostrated experimentally by Zhu et al. [9]. Second,recent works by Catalan et al. [10] have shown that fer-roelectric thin films can sustain large strain gradients dueto lattice mismatch with the substrate, and that the flex-oelectric effect of such gradients can have an importanteffect on their functional properties, particularly the fre-quently discussed lowering of permittivity in ferroelectricthin films.In spite of its growing relevance, our understanding offlexoelectricity is still rather incomplete. On a theoret-ical level, there are still no first-principles studies. On an experimental level, only ceramic samples have beenanalyzed, with the problem that grain boundaries cancontribute to measured charge due to their possibly po-lar nature [11], through tribological effects or via sur-face piezoelectricity [1]. Also, because of the need for ahigh dielectric constant, only materials with ferroelectricphases (or polar nanoregions in the case of relaxors) havebeen studied so far. This has the drawback for interpreta-tion that ferroelectricity (and thus piezoelectricity) maypersist above the nominal phase transition temperaturedue to local strain effects, and contribute piezoelectricallyto the measured charge.We have chosen to work instead with single crystalsof the paraelectric strontium titanate. At room tem-perature SrTiO (STO) has the cubic perovskite struc-ture and remains centrosymmetric even in the tetragonalphase below 105 K. The absence of piezo- and ferroelec-tric contributions, combined with relatively high dielec-tric permittivity, makes STO a natural choice for study-ing flexoelectricity. The use of single crystals of differentorientations also allows all the flexoelectric tensor com-ponents of STO to be determined.High purity STO single crystals of various dimensions(typically 3–5mm wide, 5–15mm long and 50–500 µ mthick) were obtained from PiKem Ltd. Impurity levels (inparts per million) determined by the supplier are: Ni=3,Fe=2, Cr <
2, Ba, Na and Si each <
1. Ca impurities at < ions above which a ferroelec-tric phase transition can be induced at low temperatures[12]. The transparent colourless crystals were suppliedwith a surface roughness of a few ˚A. Top and bottomAu electrodes of area 10–30mm (depending on crystalsize) were deposited by sputtering. Pt wires (50 µ m indiameter) were attached to the Au electrodes with sil-ver paste which was then annealed at 130 ◦ C to improveconductivity and mechanical robustness.The experimental setup is sketched in figure 1. A dy-namical mechanical analyzer (DMA) with an insulatingquartz probe was used to induce oscillatory bending (typ-ically driven at 30-40Hz) and measure its amplitude. Astatic stress was applied simultaneously to hold the sam-ple in place. For temperature dependent measurements,heating and cooling could be achieved by competitiveaction of a resistive heater and a liquid N bath. Thedisplacement current I due to the induced polarizationwas measured using a Signal Recovery 7265 dual phaselock-in amplifier. The maximum strains (approx. 10 − )achieved by the bending during the experiments were farbelow those which, according to thermodynamic [13, 14]and ab-initio [15] calculations, are capable of inducing apolar phase in the investigated temperature range. Thecrystals are therefore expected to be neither ferro- norpiezoelectric. A direct measurement of piezoelectricityshowed no signal above the noise level placing an upperlimit on the piezoelectric coefficient d of 0.03pC/N. Inaddition, no signs of second harmonic generation (SHG)could be detected. lock-in ~ heaterliquid N Au SrTiO x x x L FIG. 1: Experimental setup for flexoelectric measurements.
The strain gradient was derived from the usual equa-tion for a bent beam (e.g. see [16]) as ∂ǫ ∂x = 3 z (cid:18) L (cid:19) − (cid:18) L − x (cid:19) , (1)where L is the distance between the knife edges (in ourcase L =10, 7.5, or 5mm), z is the displacement at thecentre as measured by the DMA and the distances x i aremeasured from the centre of the crystal.Flexoelectricity is described by a fourth rank tensor f ijkl : P i = f ijkl ∂ǫ kl ∂x j , (2)where P is the flexoelectric polarization and ǫ kl is thesymmetrized elastic strain tensor. If ω/ π is the fre-quency of the applied mechanical stress and A is theelectrode area, the average out of plane polarization canbe computed by measuring the ac current produced bybending and using P = I/ωA . The effective flexoelectriccoefficient µ was calculated from the measured average -250 -200 -150 -100 -50 0 500102030405060 0.00.51.01.52.02.53.03.54.0 -10 -10 -10 -10 I/ z ( p A / m ) Temperature ( o C) C apa c i t an c e ( n F ) (111)(101) Strain gradient (m ) | P | ( C / m ) (100) FIG. 2: Temperature dependence of the ratio
I/z (whichis proportional to µ ) and of the capacitance for a samplepolished down to 100 µ m using diamond impregnated lappingfilm. The flexoelectric response increases with the dielectricconstant, showing an anomaly at the ferroelastic phase tran-sition. The absolute temperature values shown may be a fewdegrees too low due to some thermal lag between the thermo-couple and the sample. Inset shows the linear dependence offlexoelectric polarization on strain gradient for 300 µ m thickSTO crystals of different orientations at room temperature. polarization and strain gradient using P = µ ∂ǫ ∂x and ∂ǫ ∂x = 12 z L ( L − a ) , (3)where a is the half-length of the electrodes.From phenomenological arguments, the flexoelectriccoefficient is expected to be proportional to the dielectricconstant [5, 6]. To test this, the flexoelectric current wasmeasured as a function of temperature (figure 2). As ex-pected, the current increases upon cooling, qualitativelyfollowing the dielectric permittivity. On approaching theferroelastic transition, however, an anomaly, not presentin the dielectric constant, is seen in the flexoelectric re-sponse.Previous studies [17, 18] have shown that ferroelas-tic domains appear below 105K and are responsible forthe large softening of the elastic modulus in STO. Therelaxation of strain gradients due to domain wall read-justments may therefore be expected to reduce the flex-oelectric response, as indeed observed. This gradientrelaxation by domain readjustment can be studied bychanging the static force applied to the crystal. Withlarge static force, the crystal is very bent and the do-main walls impinge on each other, so that they can nolonger move under the dynamic load. Thus, the mechan-ical and flexoelectric response should approach that of amonodomain crystal. For low static forces, the domainwalls are quite free to move and thus the relaxation ofthe strain gradient is maximum. Accordingly, one wouldexpect the flexoelectric current to drop. Instead, how-ever, we observed the current to drop through zero to a -200 -190 -180 -170 -160 -150 -140-40-20020406080 C u rr en t ( p A ) T ( o C) FIG. 3: Temperature and static force dependence of the flex-oelectric current for the 100 µ m sample in figure 2. As thestatic force is increased, the domain walls become less mobiledue to impingement and a recovery of the positive flexoelec-tric current is observed. Similar behaviour was observed forthe original thicker crystals. Domains with the c-axes along x and x are labeled + and − respectively. negative value (bottom curves in figure 3). The fact thatthe change of sign in the current is only seen when do-main walls can move leads us to believe that the domainwalls may be charged with a polarization of opposite signto that of bulk flexoelectricity, so that their motion un-der the periodic stress produces a current of the oppositesign to the flexoelectric response.The possible polarization of domain walls in STO (anon-polar material) is an unexpected result. At present,we can think of three explanations for the observed do-main wall charge: (i) the local strain gradient at thewalls polarizes them through flexoelectricity, (ii) order-parameter coupling between the ferroelastic distortionand the (suppressed) ferroelectric polarization inducespolarization at the domain wall [19], or (iii) the domainwalls trap charged deffects such as oxygen vacancies [20]. x x x µ (C/m)[100] [010] [001] +6 . × − [10¯1] [010] [101] − . × − [11¯2] [¯110] [111] − . × − TABLE I: Orientation dependence of the flexoelectric re-sponse.
We have also measured the flexoelectric response of[001], [101] and [111]-oriented samples (inset of figure 2).The corresponding in-plane orientations were determinedby X-ray diffration and are shown in table I, togetherwith the average measured flexoelectric coefficients. Allthe coefficients were found to be of the same order ofmagnitude (ranging between about 1 and 10nC/m), buthave different signs. While there was some inter-sample variation in the magnitudes of µ for each of the orienta-tions, the signs were robust.For any material belonging to one of the cubic pointgroups there are only five independent components ofthe flexoelectric tensor f ijkl . In the case of STO, whichbelongs to the O h group, the 4-fold rotation symme-try further reduces this number to three: f , f and f (= f ). For different crystal orientations,the measured polarization arises from different combina-tions of the three flexoelectric tensor components, i.e.,the calculated values of µ are effective coefficients ratherthan the flexoelectric tensor components defined in (2).In addition we must not forget the contributions to P from the gradients of ǫ and ǫ . For a bent plate [16] ǫ = − c c ǫ − c c ǫ where c ij are the elastic moduli.In our bending geometry, the anticlastic strain ǫ is as-sumed to be negligible, giving for [001] oriented samples µ = f − ν − ν f . (4)The relevant anisotropic Poisson ratios ν ij can be ob-tained from the known elastic moduli of STO [21]. Forsamples whose edges are not aligned with the crystal-lographic axes ˆ e i the f ijkl above should be replced by f ′ ijkl = L ip L jq L kr L ls f pqrs with L ij = ˆ x i · ˆ e j and ν ij bythe corresponding ν ′ ij .Inserting the values for µ and ν ′ ij for the different orien-tations of the crystals and the corresponding expressionsfor f ′ ijkl in terms of f ijkl into equation (4) leads to threesimultaneous equations. However, it turns out that theseare not independent and hence cannot be solved to findthe individual tensor components, leaving us instead withthe following relations: f − αf = µ (001)
12 (1 − β )( f + f ) − (1 + β ) f = µ (101) where α = c c and β = c + c − c c + c +2 c . Nevertheless, theinterdependence of the equations can be used to checkthe self consistency of our analysis since we can usethe measured µ values for the [001] and [101] to cal-culate the expected µ for [111] which turns out to be − − ǫ = − ν ǫ and ǫ = − ν ǫ [16]. Using this, rough estimates of f ≈ − f ≈ f ≈ et al. and Nagarajan et al. [23] report strains at dislocationsof order 0.05 relaxing over several nm thus giving riseto strain gradients of order 10 m − . Flexoelectric coef-ficients of order 10 − –10 − C/m will therefore give localpolarizations of about 1–10 µ C/cm at room temperature,and up to 100 times more at low temperatures due to theincrease in dielectric permittivity. This implies that thelocal polarization around defects in non-polar materialssuch as STO can be bigger than the ferroelectric polar-ization in the best ferroelectrics. Obviously, this is only alocal effect, but given high enough density of dislocations(as can happen in strained thin films) we should expectthe impact of flexoelectric polarization on the functionalproperties of dielectrics to be very large.Before concluding we should briefly address the possi-bility of artefacts. For high purity STO we do not expectany bulk piezoelectric contribution to the measured cur-rent. The lowering of the symmetry at the surface how-ever, introduces the possibility of a contribution fromsurface piezoelectricity [6]. Experimental data and firstprinciples calculations suggest that the perturbed layerin only a few lattice constants thick, with no consensusas to whether it is polar or not [24, 25]. Our estimatesshow that to mimic the measured response the piezo-electric modulus of the surface has to be approximatelythe same as that of a good ferroelectric such as BaTiO which seems unlikely. Finally, whatever the nature of thesurface, there will in general always be a surface flexo-electric contribution f sf ∼ e/a [6]. However unlike thebulk effect, f sf is not expected to scale with the dielec-tric permittivity [6, 7] and thus it should be about twoorders of magnitude lower than the bulk effect in STO.Nonetheless, recent works have revealed extended nearsurface skin regions up to 100 µ m deep with local fluc-tuations of the ferroelastic phase transition [26]. Suchregions were found to be inhomogeneously strained andthus may possess flexoelectric polarization even in the ab-sence of external stress. This may play some role in oursamples, but even more so in fine-grained ceramics dueto their higher density of surfaces, which may contributeto the very high values of the flexoelectric coefficient ob-tained in ferroelectric ceramics. [7].To conclude, we have measured the dielectric polar-ization induced by bending in single crystals of SrTiO .Measurements of samples with different crystallographicorientations have allowed all components of the flexo-electric tensor to be estimated. These are of the orderof 10 − –10 − C/m, producing, around dislocations or de-fects, local polarizations of several µ C/cm and higherat low temperatures. The analysis of the behaviour ofthe flexoelectric current as a function of static bendingin the low-temperature phase also suggests that the do-main walls of STO are polarized, either intrinsically dueto local gradient coupling, or extrinsically through defectaccumulation.The authors thank Prof. S. A. T. Redfern for his ex-perimental collaboration, Dr. M. Vopsaroiu at the Na-tional Physical Laboratory for help with piezoelectricity measurements, Prof. P. Thomas for SHG measurements,Prof. E. Artacho and Dr. M. 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