Numerical Evaluation of Strain Rate Effect on Mechanicaland Electromechanical Coupling Responses in BaTiO3

Single-Crystal Nanofilm

XIAO BAO TIAN,1,2 XIN HUA YANG,1,3 and WEI ZHONG CAO1

1.—School of Civil Engineering & Mechanics, Huazhong University of Science & Technology,Wuhan 430074, China. 2.—e-mail: txb@hust.edu.cn. 3.—e-mail: yangxinh@hust.edu.cn

The mechanical and electromechanical coupling responses of a ferroelectricsingle-crystal nanofilm under displacement loading at different strain rateshave been simulated using the molecular dynamics method based on the shellmodel. While the linear stress–strain relation is independent of the strainrate, strong strain rate dependence is exhibited in the electromechanicalcoupling response for strain rates between 0 ns�1 and 0.5 ns�1. There is anapproximate semilogarithmic linear relationship between the polarizationstability strain and the strain rate. With increasing strain rate, local 180�domain switches take place sequentially from inside to outside in the stabledomain structure evolution, and the number of domain walls increases.However, after the strain rate exceeds 0.5 ns�1, it has almost no effect onthe domain structure. This work is helpful for improving ferroelectric devicedesign and expanding ferroelectric application fields.

Key words: Ferroelectric nanofilm, strain rate, polarization domainevolution, shell model

INTRODUCTION

Ferroelectric materials, due to their excellentelectromechanical coupling performance, are widelyapplied in nanoscale energy sources and nanosen-sors,1 especially in pressure nanosensors employed inexplosion impacts or other extreme environments. Inthese environments, nanosensors must endure highstrain rates caused by high-speed impact, and thehigh strain rate may result in abnormal or evenerroneous measured data from a ferroelectric pres-sure or acceleration sensor. Therefore, to design fer-roelectric devices for use in such dynamicenvironments, it is very important to understand theform of the strain rate effect on the mechanical andelectromechanical coupling responses.

The strain rate effect is very considerable in metalmaterials under dynamic loading conditions.2 It iswell known that the origin of the strain rate effect

on the mechanical response is inhomogeneous lat-tice defects, such as dislocations. A high strain ratecan accelerate defect movement and evolution. Inferroelectric materials, there can be not only latticedefects,3 but also unevenly distributed polarization,such as domain walls4 and vortexes.5 In general, anuneven polarization structure is extremely sensitiveto mechanical loading; For example, in a polariza-tion structure with vortexes, both the size andnumber of vortexes may vary with increasingstress.6,7 When the strain increases to a certainextent, the vortex domain may evolve into a stable90� or 180� domain structure.7,8

It is very important to choose a suitable methodfor investigating the high strain rate effect inthe mechanical and electromechanical couplingresponses of ferroelectrics. Experiments can enabledirect real-time observations of the external sur-faces of specimens, but cannot reveal the internaldomain structure evolution. Although the phase-field method based on phenomenology can includethe stress, strain, and polarization electric field, thedifficulty of modeling is greatly increased when

(Received April 3, 2013; accepted October 7, 2013;published online November 19, 2013)

Journal of ELECTRONIC MATERIALS, Vol. 43, No. 2, 2014

DOI: 10.1007/s11664-013-2859-6� 2013 TMS

479

strain rate parameters are added. The first-princi-ples method based on quantum mechanics also hassome restrictions, such as the limitation on thesimulation temperature to near 0 K, so that it isdifficult to set dynamic boundary conditions. Inrecent years, the molecular dynamics (MD) methodhas been successfully combined with the shell modelto reproduce the electromechanical couplingbehaviors of ferroelectric single crystals, such asstrain effects on the polarization distribution andhysteresis,9,10 and size dependence in the polariza-tion configuration of PbTiO3 film.11 These prece-dents motivated us to apply this method to simulatethe mechanical and electromechanical couplingresponses of ferroelectric nanofilm under displace-ment loading conditions at different strain rates,and then to analyze the effects of strain rate on thestress–strain relation, polarization configurationevolution, and polarization stability strain.

MOLECULAR DYNAMICS METHOD BASEDON THE SHELL MODEL

In the present MD simulations, a BaTiO3 single-crystal nanofilm was modeled using the shell mod-el.12,13 In the shell model, an ion is separated intotwo charged parts, namely an ion core with the atommass and positive charge, and a massless ion shellwith negative charge. Three kinds of interaction,namely the core–shell interaction inside an ion, thelong-range interactions between charged particlesbelonging to different ions, and the short-rangeinteractions between different ion shells, are con-sidered, so that the shell model can phenomenolog-ically describe the internal deformation caused byinteractions between different particles after ionicpolarization.

The core–shell interaction can generally bedefined by a harmonic spring potential as

V1ðrÞ ¼ k2r2�

2; (1)

where k2 is the core–shell harmonic constant and ris the relative core–shell distance. This potentialfunction is applied at all ions except O ions. For Oions, there are different interactions in the direc-tions perpendicular and parallel to the O–Ti bond.V1ðrÞ applies for the directions perpendicular to theO–Ti bond, but not for the direction parallel to theO–Ti bond. The following fourth-order core–shellinteraction potential is employed in the directionparallel to the O–Ti bond:

V2ðrÞ ¼ k2r2�

2þ k4r4�

24; (2)

where k4 represents the fourth-order effect of core–shell distance on the interaction, caused by thenonsymmetrical oxygen site. The variation of thecore–shell distance can be used to characterizethe electronic polarization. The different potentialsin the directions perpendicular and parallel tothe O–Ti bond reflect the intense polarization

anisotropy in the O-ion interactions caused by O–Tidistance variation. The long-range potential is rep-resented by the Coulomb interaction between coresand/or shells from different ions. The short-rangeinteraction acts only between ion shells. The Buck-ingham potential is employed as the short-rangeinteraction potential for the Ba–O, Ti–O, and O_Ointeractions,

V3ðrÞ ¼ a exp �r=qð Þ � c�

r6; (3)

where a, q, and c are material parameters. The shellmodel parameters were taken from Sang et al.,14

being obtained from first-principles calculations.Accordingly, compared with the first-principlesmethod, although the MD method based on the shellmodel has slightly lower computational accuracy, itcan capture the electronic polarization of each ion,save much computational cost, and extend thetemperature range from zero to nonzero Kelvin. Ithas been validated in previous studies for, e.g.,reproduction of the size-induced topological trans-formations in PbTiO3 nanoparticles,11 cubic–tetrag-onal phase-transition simulation of PbTiO3 bulk,15

and evaluation of the interface effect of ultrathin filmon SrTiO3.16

In this study, a BaTiO3 single-crystal nanofilmunder compressive displacement loading at differ-ent strain rates with an ideal open-circuit electricalboundary was considered as the computationalsystem. It consists of Nx �Ny �Nz unit cells, whereNx, Ny, and Nz are the numbers of Ti atoms alongthe three Cartesian coordinate directions x, y, and z.It is assumed that Nx ¼ Ny ¼ Nz ¼ 10. The Carte-sian coordinate system is located with the h1 0 0i, h01 0i, and h0 0 1i lattice directions as the x-, y-, andz-axis directions, respectively. In the system, all theoutside surfaces are Ba–O planes and the initialspontaneous polarization is zero. The temperatureis fixed at 5 K, near absolute zero, so that all theparticles are still active and can move easily toproper positions.10 Periodic boundary conditions areimposed in the x- and y-axis directions. To investi-gate the rate dependence of the mechanical andelectromechanical coupling responses, uniaxialcompressive displacement loading at differentspeeds of 0.05 ns�1, 0.124 ns�1, 0.187 ns�1, 0.5 ns�1,0.75 ns�1, and 1.5 ns�1 is imposed along the z direc-tion. When the strain reaches a predetermined value,a relaxation process with sufficient time is performedto ensure system equilibrium.

STRAIN RATE EFFECTON THE MECHANICAL RESPONSE

The stress–strain curves for different loading ratesare shown in Fig. 1a. It can be seen that all the curvesnearly overlap each other, which indicates that thestrain rate has a very slight effect on the mechanicalresponse of BaTiO3 single-crystal nanofilm. Whenthe strain is between 0% and �3%, the stress–strainrelation is approximately linear, although it was ex-

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pressed more accurately as a quadratic function byZhang et al.9 This means that the material is notmechanically yielding. According to experimentalobservations on SrTiO3 ferroelectric single crystal,Yang et al.17 also concluded that material yield oc-curs only when the strain is larger than 3%. Theelasticity modulus can be approximately estimatedas 275 GPa from the presented curves.

It is well known that the origin of the high strainrate effect in metal materials is material defectinitiation with subsequent defect evolution ormovement.2 The average lattice deformation isshown in Fig. 1b as the atomic position changesoccurring in the cell. It is shown that the inter-atomic spacing changes uniformly as the strainincreases from 0% (solid circles) to �1.25% (hollowcircles with solid-line circumference) and �3%(hollow circles with dotted-line circumference).There exist no lattice defects in the initial configu-ration of the analyzed single-crystal thin nanofilm,and the linear stress–strain curve and uniforminteratomic spacing change in the range of com-pressive strain less than 3% also reveal no disloca-tion generation. Therefore, these is no obviousorigin inducing the strain rate effect in the aboveloading process.

STRAIN RATE EFFECTON THE ELECTROMECHANICAL COUPLING

RESPONSE

Effect on Domain Structure Stability

The material polarization configuration canevolve with mechanical loading.6 To investigate thestrain rate effect on the electromechanical couplingbehavior, the relationship between the polarizationconfiguration and the strain under differentstrain rate conditions was analyzed. The polarization

toroid moment is generally used to describe domainstructure stability.18–20 In this paper, it is utilized toquantitatively characterize the stability of polariza-tion configuration evolution. The polarization toroidmoment is defined as the polarization dipole momentper unit volume,18 i.e.,

g ¼ 1=ð2NtÞX

i

ri � pi; (4)

where pi is the local dipole of cell i located at posi-tion ri, N is the number of cells in the simulation,and t is the average cell volume. The local dipole pican be used to quantitatively characterize thematerial polarization configuration evolution. Nau-mov et al.19 pointed out that the energy change inthe polarization evolution could be expressed by thepolarization toroid moment g as

U ¼ �tg � ðr �EÞ; (5)

where �r�E represents the coercive field. Whenthe applied external electric field reaches or exceedsthe coercive field, polarization evolution will occur.Therefore, it can be concluded that, the higher thepolarization toroid moment, the greater the energychange and the more unstable the polarizationconfiguration.

The variation of the polarization toroid momentwith the strain under different strain rate condi-tions is shown in Fig. 2a. It can be seen that all thecurves are stable when the strain is larger than�2%, although the curves fluctuate considerablywhen the strain lies between 0% and �0.5%. Formore careful observation, the curves between strainof �0.5% and �2% are magnified in Fig. 2b. Thestrain value when the polarization configurationjust becomes steady is called the polarization sta-bility strain and is marked by dashed lines in

Fig. 1. Response of the film to loading: (a) stress–strain curves for different loading rates, and (b) average lattice deformations for differentstrains.

Numerical Evaluation of Strain Rate Effect on Mechanical and Electromechanical Coupling Responses inBaTiO3 Single-Crystal Nanofilm

481

Fig. 2b. For strain rates from 0.05 ns�1 to 0.5 ns�1,the polarization toroid moment has a large range offluctuation, and the polarization stability strainincreases from �0.75% to �0.82% and �0.92% withincreasing strain rate. When the strain rate isgreater than 0.5 ns�1, the fluctuation range of thecurve becomes narrow, and the polarization stabil-ity strain remains at �1.13%. When the strain rateincreases from 0.05 ns�1 to 1.5 ns�1, the modulus ofthe polarization toroid moment ascends, as indi-cated by the bold arrow in Fig. 2b. This manifeststhat, the greater the strain rate, the larger thepolarization toroid moment, and the more stable thepolarization domain.

According to the results of experimental andtheoretical analyses under no electric loading, theinternal relaxation speed for equilibrium isapproximately on the order of 10�4 nm ps�1.21,22

The film thickness is 4 nm in the calculated system,so the deformation is 4e nm when the film is sub-jected to a strain of e. The displacement loadingspeed can be calculated as v ¼ 4e=t ¼ 4_e, where _e is

the strain rate and t is the loading time. It is notedthat, corresponding to the strain rates of 0.5 ns�1

and 0.05 ns�1, the displacement loading speeds are2 9 10�3 nm ps�1 and 2 9 10�4 nm ps�1, respec-tively. Therefore, it can be easily concluded that thedisplacement loading speed corresponding tothe strain rate of 10�2 ns�1 order coincides with thedomain evolution speed of 10�5 nm ps�1 to10�4 nm ps�1 order caused by internal relaxation ofthe domain structure.

Why does a ferroelectric exhibit a strain rateeffect on polarization stability? Some explanationscan be given as follows: Firstly, when the strain rateis no more than 10�2 ns�1, the ferroelectric relaxa-tion is synchronized with the displacement loading,so that the loading can be regarded as quasistatic.The equilibrium between loading and relaxation inthe particles can be maintained during the loadingprocess. Secondly, as the strain rate increases, theequilibrium between the ion displacement caused bythe external mechanical load and that required bythe internal ferroelectric relaxation is destroyed.According to the speed analysis, the ionic displace-ment change caused by the strain rate loading isfaster than the internal relaxation speed requiredby equilibrium, so that ferroelectric relaxation hys-teresis takes place. The more intense the relaxationhysteresis, the greater the effect of strain rate onthe polarization configuration evolution, and thelarger the baseline of the polarization toroid mo-ment modulus. Therefore, it can be concluded thatthe strain rate affects not only the speed of polari-zation domain evolution but also the strain for theformation of stable polarization domains. In addi-tion, when the strain rate is larger than 0.5 ns�1,the ion displacement change speed of 2 9 10�3 nmps�1 caused by the loading is much larger than theinternal relaxation speed required by equilibrium,so that the influence of molecular thermal motioncan be neglected. Therefore, the stable polarizationconfiguration strain is independent of the strainrate, and the curves also become smooth.

Effect on Polarization Stability Strainand Polarization Configuration

To further analyze the strain rate effect ondomain evolution, four polarization configurationsA, B, C, and D are displayed in Fig. 3. They corre-spond to the yz plane when Nx ¼ 5 and correspondto strain rates of 0.05 ns�1, 0.124 ns�1, 0.187 ns�1,and 0.5 ns�1. It is apparent that the polarizationconfiguration evolves from A to B, C, and D as thestrain rate increases, and local 180� domainswitches take place from inside to outside in thedomain structure. The number of domain walls in-creases from two to three, and finally remains atfour. The domain wall thickness is always about onelattice spacing. When the polarization configurationevolves from A to B, the width of the blue domain in

Fig. 2. Variation of polarization toroid moment with strain: (a) forstrain from 0% to �3%, and (b) for strain from �0.5% to �2%.

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the middle of the configuration varies from onelattice spacing to two lattice spacings. In themeantime, the 180� domain wall in the upper halfmoves upward about one lattice spacing. Then, fromB to C, a 180� domain switch occurs near the top,and a single domain with width of three latticespacings is separated into two domains with a 180�domain wall. Finally, from C to D, a 180� domainswitch occurs near the bottom, and a single domainwith width of three lattice spacings is divided intotwo domains with a 180� domain wall. The polari-zation configuration eventually evolves into amultidomain structure with four 180� domainwalls. Analogous polarization configuration evolu-tion was also observed by Sang et al.14 and Shi-mada et al.23 In addition, the curve of the variationof the polarization stability strain with the strainrate is shown in Fig. 3. It can be seen that there isan approximate semilogarithmic linear relationshipbetween the polarization stability strain and thestrain rate when the strain rate is less than0.5 ns�1, and the polarization stability strainremains unchanged when the strain rate exceeds0.5 ns�1.

Additional information can be obtained based onfurther investigation and analysis. For strain ratesless than 0.5 ns�1, it is obvious from Fig. 2a, b thatall particles in the simulated ferroelectric systemcould fully relax if the strain rate was small enough,e.g., 0.05 ns�1. As the strain rate increases, theferroelectric relaxation exhibits an increasing hys-teresis effect, so that local polarization domainevolution occurs. When the strain rate is larger than0.5 ns�1, instead of the strain rate, the strain playsan important role in the domain configuration evo-lution, so that the polarization stability strain cor-responding to a stable polarization configurationremains unchanged. Accordingly, it can be con-cluded that the stable ferroelectric domain forma-tion is not only related to the temperature, size,11

and load,6 but also depends on the strain rate.

CONCLUSIONS

While the linear stress–strain relation is inde-pendent of the strain rate, the electromechanicalcoupling response of ferroelectrics exhibits strainrate dependence in a certain strain rate range.When the strain rate is less than 0.5 ns�1, there isan approximate semilogarithmic linear relationshipbetween the polarization stability strain and thestrain rate. The polarization domain evolves frominside to outside in the ferroelectric nanofilm withincreasing strain rate, and the number of domainwalls increases. However, when the strain rate islarger than 0.5 ns�1, both the polarization stabilitystrain and the polarization domain configurationremain unchanged. The above understanding of thestrain rate effect on the electromechanical couplingresponse of the ferroelectric single crystal is helpfulfor improving the design of ferroelectric devices andexpanding ferroelectric application fields.

ACKNOWLEDGEMENTS

The authors are grateful for support from StateKey Laboratory of Materials Processing and Die &Mould Technology (Grant No. 2011-P01) and theNational Natural Science Foundation of China(Grant No. 11072082).

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Fig. 3. Evolution of polarization configuration and variation ofpolarization stability strain with strain rate.

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