Virtual piezoforce microscopy of polycrystalline ferroelectric films

Transcription

1 JOURNAL OF APPLIED PHYSICS 100, Virtual piezoforce microscopy of polycrystalline ferroelectric films R. Edwin García a School of Materials Engineering, Purdue University, West Lafayette, Indiana Bryan D. Huey Institute of Materials Science, University of Connecticut, Connecticut John E. Blendell School of Materials Engineering, Purdue University, West Lafayette, Indiana Received 29 December 2005; accepted 16 July 2006; published online 21 September 2006 An innovative methodology is presented that utilizes the experimental results of electron backscattered diffraction to map the crystallographic orientation of each grain, the finite element method to simulate the local grain-grain interactions, and finally piezoforce microscopy to infer the local properties of polycrystalline ferroelectric materials by comparing the output of the numerical calculation s with the experimental results. The proposed combined method resolves the local hysteretic and electromechanical interactions in polycrystalline ferroelectric films, thus quantifying the effects of grain corners and boundaries on the polycrystal s macroscopic response. For a polycrystalline lead zirconate titanate sample, a finite range of crystallographic orientations and epitaxial strains is found to enhance the out-of-plane electrical response of the film with respect to its single-crystal, stress-free counterpart. Results show that 111 oriented grains parallel to the normal of the surface of the film yield the largest polarization magnitude enhancement, compressive stresses, and built-in electric fields, as well as an asymmetry in the quasistatic coercive field. In the absence of epitaxial strains, 001 oriented grains will be enhanced in their out-of-plane hysteretic response through the in-plane compressive stresses provided by the local neighboring grains. For the studied sample, grain corners and boundaries become favorable sites for pinning or nucleation of ferroelectric domains, depending on the local state of stress and polarization American Institute of Physics. DOI: / I. INTRODUCTION Ferroelectric lead zirconate titanate PZT films display physical behavior that makes them an important candidate for random access memory applications. In such devices, ferroelectric domains are switched into or out of the plane of the film by the application of an electric field, thus fixing the state of a memory cell. Today s technological advancement, however, demands ever higher memory densities. Therefore, as the device size shrinks, the microstructural features become increasingly important and the spatial variation of the hysteretic behavior increases, making the memory unit potentially unreliable. The local crystallographic orientation and the local grain-grain interactions play an important role in determining the switching of domains. In particular, in ferroelectric thin films, large spatial variations of the fields arise as a combined result of the stresses that develop due to the thermal expansion and lattice mismatch of the filmsubstrate system, the anisotropy of the material properties, and the processing conditions. An approach to enhance the hysteretic response of ferroelectric memories is strain engineering. 1 For single-crystal thin films, in-plane strains can lead to a poor electrical response, but if judicious selection of the substrate and processing conditions is made, the local state of polarization can be dramatically enhanced. In polycrystalline materials the state of strain is not homogeneous. The electrical and mechanical fields are spatially distributed and nontrivially coupled through the direct and converse piezoelectric effects, the local spatial compatibility of the grains, and the anisotropy of the underlying dielectric and elastic properties. In particular, grain corners and boundaries are sites where the polarization switching behavior undergoes the largest variations with respect to the bulk of the grains and deserves special attention. To understand, and ultimately better engineer, the local ferroelectric response of these topologically complex materials, emerging characterization tools and modeling capabilities must be combined. Such approaches should lead to a careful selection of materials and processing conditions in order to enhance the hysteretic response without compromising the life expectancy of the device. The quality and electrical reliability of a memory unit are determined by the shape of the produced hysteresis loop. In order to distinguish between two possible out-of-plane polarization states, a large remnant polarization is desirable. Simultaneously, a smaller coercive field decreases the energy spent to switch a ferroelectric domain and is thus a design ideal. In ferroelectric materials, however, it is not trivial to determine the way in which microstructure couples to the local state of stress and polarization, and such effects are not simple to model. In the continuum limit, many theoretical descriptions are based on Taylor series expansions of the free energy as a function of the controlling variables and accurately describe the average thermodynamic state of singlea Electronic mail: redwing@purdue.edu /2006/100 6 /064105/10/$ , American Institute of Physics

2 García, Huey, and Blendell J. Appl. Phys. 100, crystal and polycrystalline materials. 2 6 Other approaches incorporate the crystallographic compatibility of the stable ferroelectric variants in terms of phase field or other generic energy-minimizing frameworks and effectively predict the appearance of different types of ferroelectric domains for single-crystal materials Further approaches focus on describing the switching phenomenology of a ferroelectric by defining empirical constitutive relations or using mean field methods All of these approaches have provided valuable information to describe the physics of ferroelectrics, specifically single-crystal materials; however, none of them explicitly incorporate the effects of the microstructure. In the present work, a method is presented to quantify the effects of crystallographic orientation, microstructure, and stresses on the local hysteretic response of polycrystalline ferroelectric films. Experimental measurements and numerical methods are combined in order to describe the behavior of a microstructurally realistic ferroelectric film. In particular, the local state of polarization is experimentally determined by performing piezoforce microscopy PFM experiments followed by electron backscattered diffraction EBSD to measure the crystallographic orientation on an area of interest. A model is built to describe the local thermodynamic state of a ferroelectric film based on a variational principle that simultaneously satisfies the mechanical equilibrium and Maxwell s equations. The proposed approach incorporates the effects of microstructure by implementing the finite element method. The effects of microstructure, local crystallographic orientation, grain corners, and boundaries, as well as processing-induced fields on the local hysteretic behavior for a specific PZT film instance are discussed. II. THEORETICAL FRAMEWORK A. Thermodynamics of ferroelectric materials If a nonpiezoelectric phase undergoes a transition to a piezoelectric phase, the material s symmetry changes by the loss of an inversion center. In some cases, the piezoelectric phase transition induces a transformation strain and electric polarization. If crystallographically equivalent remnant states of strain and polarization can be set by applying an electric field, the material is called a ferroelectric. 6,7 In particular, for materials that transform from a cubic to tetragonal phase, such as PZT, the local state of polarization is directly correlated to the degree of tetragonality of the system. At equilibrium, in a stress-free ferroelectric solid, the direction of the polarization order parameter vector, P, lies along the c-axis of the unit cell. Furthermore, the polarization vector possesses several stable states, i.e., ferroelectric variants, which can be reversed in orientation by the application of a sufficiently large electric field or strain. The energetics of a ferroelectric are modeled by expanding in Taylor series the Helmholtz free energy density, f =u Ts, as a function of the independent degrees of freedom of the system. The coefficients of the expansion can be extracted from the experimental quasistatic hysteretic response of a single crystal. Experimentally it is found that if the direction of the polarization order parameter, the c-axis, possesses components out of the plane of the film, the application of an electric field through the film s thickness will enable hysteretic switching, while for a c-axis completely contained within the plane of the film, the energy penalty necessary to induce a polarization reversal is greater than the built-in breakdown electric field. The described hysteretic behavior is mathematically described by assuming that the energetic penalty to perform 90 switch in P is infinitely large compared to the energy necessary to reverse the direction of the polarization vector, i.e., to perform a 180 polarization reversal. The proposed situation is equivalent to having only two thermodynamically stable variants, each 180 away from each other. For such system, a free energy density representation where polarization and strain are the controlling variables is 4,22 f P 3, T ij,t,e 3 = a 2 P b 4 P c 6 P 3 6 P 3 E 3 + q 33kl T 2 kl P ij T ij, where P 3 is the out-of-plane component of the polarization vector, E 3 is the out-of-plane component of the externally applied electric field, T ij is the ij-th component of the total strain tensor, ij is the ij-th component of the stress tensor, and q ijkl is the ijkl-th component of the electrostrictive tensor for a thermodynamic representation in which strain and polarization are the controlling variables. The first four terms of Eq. 1 correspond to the classic Landau expansion. 2,6 The fifth term corresponds to the coupling of strain with the polarization order parameter. The sixth term is the energy penalty from increasing the elastic energy. Higher order terms are neglected in the current formulation. The total Helmholtz free energy is embodied by the expression 22 F P 3 ; T ij,e 3,T f P 3, J = T,E 3,T d. For a single-crystal thin film subjected to a fixed biaxial state of stress due to epitaxial stresses and thermal expansion differences between substrate and film, and by neglecting domain wall energy contributions, 11 the film is at global equilibrium when the variational derivative of Eq. 2, F = ap 3 + bp 3 P 3 + cp 5 3 E 3 + 2q 33kl T kl P 3, 3 equates to zero at all times. 23 The electrostatic and elastic fields of a ferroelectric film are determined by solving for each volume element of material the mechanical equilibrium equation and Coulomb s law in its differential form, 24,25 J = 0 and D =, where D is the spatially dependent total polarization vector and is the local charge density. The local stress and total polarization are coupled through a set of constitutive relations

3 García, Huey, and Blendell J. Appl. Phys. 100, TABLE I. Physical properties utilized for PZT model Ref. 44. Symbol Value Units Symbol Value Units Symbol Value Units C GPa Q m 4 /C F/m C GPa Q m 4 /C F/m C GPa Q m 4 /C 2 M P S/m C GPa a J m/c /K C GPa b J m 3 /C /K C GPa c 0 J m 9 /C 6 D i = ij E j + e ijk T jk, ij = C ijkl T kl e kij E k C ijkl kl T, where ij is the dielectric permittivity, e ijk is the piezoelectric tensor in its e-form, C ijkl is the elastic stiffness tensor, and kl is the kl-th component of the thermal expansion tensor. In the most general case, dielectric, elastic, thermoelastic, and piezoelectric properties are a function of the thermodynamic variables. 26 In particular, piezoelectric properties are coupled to the polarization order parameter P through the relations and d mij = Q ijkl ml P k e ijk = C jklm d ilm, which are satisfied at every instant. ml is the dielectric susceptibility and d mij is the piezoelectric tensor in its d-form. Q ijkl is the electrostrictive tensor. B. Single-crystal film behavior In what follows, the reference system of the film single crystal or polycrystalline is defined as a right-handed Cartesian coordinate system, where the x and y axes are contained within the plane of the film, and the z-axis is the coordinate direction normal to the film s surface. For such a setup, when a single-crystal layer of material of thickness h f is deposited on a substrate of thickness h s, stresses develop in the film as a result of the lattice parameter mismatch epitaxial stresses and material properties discontinuities. 24 These stresses, in the limit of infinitely thin films, i.e., h f h s, can be approximated as independent of the out-of-plane sample reference system direction, z direction in all figures. 27 Moreover, in such a limit, can be assumed to be a constant. For a single-crystal ferroelectric thin film deposited on a thick substrate the crystallographic orientation-dependent sum of the principal stresses is described through the relation f = = E f T = E cos sin 2 T, 8 where E is the effective in-plane stiffness constant of the film. Because of the low anisotropy of the single-crystal properties of PZT, E is taken as a constant in the current single-crystal analysis. f is the orientation-dependent, inplane thermal expansion coefficient, and T=T T o, where T o is the stress-free temperature. T o is set to the Curie temperature of PZT in this paper. is the angle that the polar axis the c-axis makes with the normal of the plane of the film and will be referred to as the misorientation angle. is the substrate or epitaxial strain and sums up all the processing contributions to strain, including those due to epitaxially depositing the film and those imposed by the substrate, i.e., =, s T. In general, the individual components of the thermal expansion tensor of the film, 11 and 33, and the substrate contributions to strain are functions of the independent degrees of freedom of the system e.g., temperature, state of polarization, composition, etc.. Moreover, its values determine the local state of stress and, via Eq. 3, the polarization of the film. For example, for a perfectly rigid substrate zero thermal expansion, s =0, infinitely stiff, a negative in-plane thermal expansion tensor will yield a very different state of polarization compared to a film with positive value of inplane thermal expansion. In this paper, in the absence of detailed experimental information, the values of thermal expansion for PZT are assumed to be independent of temperature, as cited in Table I. For the material parameters described in Table I, the outof-plane hysteretic response as a function of misorientation and substrate strain is summarized in Fig. 1. Here, three main types of out-of-plane polarization response are predicted and are denoted by ferroelectric response, nonferroelectric or electrostrictive phase, and piezoelectric or unswitchable response. For =0, the anisotropy of the thermal expansion tensor contributes to enhance or suppress the out-ofplane remnant polarization and effective electric coercive field. For a single-crystal whose c-axis is perfectly aligned with the z-axis, the remnant polarization and out-of-plane hysteretic response will be less than the equivalent stress-free material because the tensile in-plane stresses that result from cooling the film lowers the effective Curie temperature of the film. However, if the c-axis of the film makes an angle with the z-axis, compressive stresses develop, thus increasing the out-of-plane remnant polarization at the expense of an increase of the out-of-plane coercive field. Furthermore, at a specific misorientation, o =34.38, tensile and compressive contributions to stress cancel each other f =0, making it effectively stress-free. At misorientations greater than o, the film switches from a tensile to a compressive state of stress thus enhancing the hysteretic behavior with respect to its stress-free counterpart. As expected, no hysteretic behavior is observed when the c-axis is completely contained within the plane of the film.

4 García, Huey, and Blendell J. Appl. Phys. 100, FIG. 1. Phase diagram summarizing the predicted out-of-plane hysteretic response for a single-crystal ferroelectric film. The horizontal axis corresponds to the misorientation angle and the vertical axis to the substrate or epitaxial strain. The film has been undercooled by T= 300 K below its Curie deposition temperature. Built-in strains and misorientation values can enhance or suppress both the ferroelectric phase transition and out-ofplane hysteretic response. Moderate compressive values of strain improves the hysteretic response for a finite range of misorientation values. The region corresponds to a solid in its ferroelectric phase, i.e., switchable in its polarization, to the nonferroelectric or electrostrictive phase, and to the piezoelectric nonswitchable phase. In each inset, the response of a stressfree, perfectly oriented PZT crystal is shown as a reference. The dashed line corresponds the locus of points where optimal hysteretic response is predicted to occur. Substrate contributions to in-plane strains impact on the misorientation dependence of the hysteretic response. Figure 1 illustrates that additional tensile contributions of strain to a film already in an initially tensile state of stress, i.e., 0, further decreases the out-of-plane remnant polarization. Moreover, as the tensile strain induced by the substrate increases, the ferroelectric tetragonal phase is completely suppressed for a finite range of misorientations. However, if the c-axis of the film rotates beyond a critical orientation with respect to the film s normal a misorientation, compressive contributions to stress due to the anisotropy of the thermal expansion tensor balances out the tensile contributions, thus restoring the ferroelectric phase. Instead, if compressive built-in strains result during film deposition, i.e., 0, the remnant polarization and coercive field will be enhanced. However, if the misorientation of the c-axis is increased to a value greater than a critical value specified by the ferroelectric to piezoelectric phase boundary, or the epitaxial strain becomes increasingly compressive, the out-of-plane coercive field will become larger than the inherent breakdown electric field of the film, making the film unswitchable, even if the out-of-plane polarization has been enhanced with respect to a stress-free system. For the assumed material parameters, the out-of-plane hysteretic behavior is optimized by taking advantage of the orientation dependence of the stress in the film. Films with small misorientations and negative values of epitaxial strain or large misorientations and positive epitaxial strain will be reliably switched. Moreover, the optimum response is obtained for the substrate strain-misorientation pairs that lie along the FIG. 2. Predicted single-crystal out-of-plane remnant polarization as a function of misorientation. Each curve corresponds to a fixed value of built-in strain. The continuous line corresponds to a stress free single-crystal PZT film. In each inset, the resultant electromechanical response is shown, as well as the stress-free, perfectly oriented PZT crystal butterfly loop. Intermediate values of compressive strain and misorientation enhances the electromechanical behavior, while samples in a highly compressive state of strain provide a linear piezoelectric response no switching. Films in a tensile state of stress will result on electrostrictive response only. dashed curve for the region where the ferroelectric phase is stable. Note, however, that small deviations from the optimum value of strain will lead to memory units that lie in the piezoelectric region, where the film is polarized but unswitchable. The electromechanical behavior of a single-crystal film is linked to its hysteretic response. Figure 2 illustrates such relationship as a function of misorientation, for different values of substrate strain. In films where the state of stress is increasingly tensile, small misorientations favor electrostrictive quadratic contributions to the local electromechanical behavior, while films subjected to a compressive state of stress contribute to an increasingly linear piezoelectric response. Values of substrate strain and misorientation where hysteretic switching is possible will lead to a butterflylike behavior with combined intrinsic and extrinsic contributions to electromechanical strain. The tetragonal to cubic phase transition that occurs for small angle misorientations and positive substrate strains leads to a strictly electrostrictive response of the film. For very large misorientations and films in a highly compressive state of stress, the response will become strictly linear piezoelectric due to the inability to reverse the polarization state of the film. In the context of actuator devices, the out-of-plane electromechanical response of a single-crystal thin film can be engineered by carefully selecting the substrate strain and c-axis misorientation. In polycrystalline ferroelectric films, each grain occupies a position in the phase diagram shown in Fig. 1. Its position is determined by the inherent texture of the film and the local in-plane electrical and mechanical interactions. These interactions shift the form of the hysteresis curve into more tensile or compressive regimes. Furthermore, the local hysteretic behavior is affected by grain corners and boundaries,

5 García, Huey, and Blendell J. Appl. Phys. 100, FIG. 3. Color Lower-left inset shows the entirety of the studied PZT microstructure. Upper-right inset shows subregion where the PFM experiments and simulations were performed. Grains are labeled from a to l. The color scheme corresponds to the normal of the crystallographic plane parallel to the sample reference system s z-axis. The key to the color scheme is illustrated on the lower right inset. 001 planes correspond to red grains, 010 planes to green, and 110 planes to blue grains. The sample reference system is shown on the upper-right extreme of the figure. For the present material system, local ferroelectric domain switching is induced by applying a focused out-of-plane electric field along the z-axis. inducing pinning of ferroelectric domains along the edges of some microstructural instances and favoring nucleation of domains in others. In the following sections, the effect of grain misorientations, corners, and boundaries on the out-ofplane hysteretic behavior is assessed by performing a set of virtual PFM measurements on a simulated PZT film. The simulated film was assembled by combining the direct measurement of the geometry microstructure and crystallographic orientation of each individual grain. The details of the built-in fields in the microstructure are resolved by numerically implementing the formulation described in Sec. II. The effects of grain corners and boundaries are compared against its single-crystal counterpart. The predicted results are validated by comparing them against experimental PFM measurements, for the same region in the microstructure. III. EXPERIMENT A sample of polycrystalline PZT was characterized through two experimental techniques: piezoforce microscopy and electron backscattered diffraction. PFM has been applied widely for assessing the properties of ferroelectric thin films and domain studies. 33,34 In PFM, a small oscillating voltage is applied to the film surface through a conductive atomic force microscopy AFM cantilever in contact with the film. A platinum layer at the film-substrate interface is used as the bottom electrode. The resulting sinusoidally varying surface actuation due to converse piezoactuation is measured, and the difference in phase between the applied signal and the deformation yields the domain polarization. The amplitude of the out-of-plane surface actuation is used to determine the angular dependence of the polarization. Inplane surface actuation is detected for domain orientation studies in three dimensions. 35,36 As the measurement of the surface actuation uses the same detector as topography measurement during scanning of the tip, AFM maps are obtained simultaneously for both topography and electromechanical couplings. Recent work has identified additional experimental artifacts in PFM, allowing quantitative property measurements for out-of-plane piezoactuation. 37 For the studied PZT film, PFM was performed with standard AFM conductive cantilevers. The Pt electrode beneath the film was grounded and a 1 V peak to peak sinusoidally varying voltage was applied directly to the tip at a frequency of 3.2 khz. The resulting amplitude and phase of the tip/ lever system was mapped during scanning. dc voltage offsets of up to ±9 V were applied during scanning to induce domain switching. For piezoactuation hysteresis measurements, the tip was fixed at a position of interest, a sinusoidally varying voltage was applied, and a bias voltage was stepped at approximately 1 Hz while the amplitude and phase were monitored. The actual field applied to the sample is estimated to be on the order of 1 MV/m or less due to voltage drops at the surface layer and at the film tip interface. The laser used to measure the piezoactuation was positioned on the cantilever to minimize experimental artifacts. 38 Hysteresis loops were measured at several points along lines spanning several grains. The loops within a grain were averaged and the loops in the grain boundary regions were compared to those within the grains. EBSD is a technique which allows crystallographic information to be obtained from samples in the scanning electron microscope SEM. In EBSD a stationary electron beam strikes a tilted crystalline sample. Backscattered electrons diffract from the near surface region and the resulting patterns are used to identify the crystal structure and orientation of the diffracting region from which it was generated. EBSD has been demonstrated to have a spatial resolution below 20 nm. In this work the patterns were generated by a 20 kev beam in a field emission scanning electron microscope FESEM. A region that contained the area that was measured using PFM was scanned with a 0.5 m step size and the diffraction patterns were collected. The patterns were indexed as PZT based upon ICSD Pattern No , which corresponds to space group P4mm. Grain boundaries were identified using an 8 misorientation criteria, and within a

6 García, Huey, and Blendell J. Appl. Phys. 100, grain the orientations were averaged to give a single orientation for the entire grain. 90 and 180 twin boundaries were not included as grain boundaries. Figure 3 shows the crystallographic orientation of grains in the region of interest measured by EBSD. In this figure, color indicates the orientation of the crystallographic plane aligned with the normal of the surface of the film. The region examined by PFM is shown in the enlarged view. The grains that were studied are identified by letters. IV. NUMERICAL SETUP The instantaneous electrical and mechanical equilibrium described by Eqs. 4 7 was implemented numerically by modifying the object oriented finite element analysis program, OOF. 24,39 The ferroelectric switching behavior determined by Eqs. 1 3 was added to the numerical description of the system by using the resultant equilibrium strain and electric fields as input parameters to solve for Eq. 3. The value of the local polarization order parameter was determined by using relaxation kinetics, which was implemented through a Runge-Kutta algorithm. The time step is determined by using a predictor-corrector approach. At every tentative time step, the global solution is accepted if the deviation between a long time step and a short time step solutions is smaller than a user-specified tolerance for all the elements. Once a solution for a time iteration is accepted, the time step for the next iteration is set as a function of the largest difference between the long and short time step solutions, for the set of all the elements. The time step dynamically adapts to minimize the number of numerical steps to reach the final simulated instant. 40 The normalized tolerance of the predictor-corrector Runge-Kutta algorithm was set to The system is evolved in time for a fixed value of applied out-of-plane electric field until the state of polarization reaches a steady state value for all volume elements of material. The resultant steady state polarization order parameter is then used to recalculate the mechanical and electrical equilibrium of the solid. The crystallographic orientation information collected for every grain by the EBSD experiment was incorporated into the numerical model by assigning crystallographic orientations to every grain. The previous was accomplished by converting the position-dependent measured Euler angles to PPM2OOF commands, which automatically assigns orientations to a group of pixels representing a two-dimensional grain. 41,42 After assigning orientations to every group of pixels in the digitized image of the studied microstructure, an initial mesh was laid out. The mesh was then refined and annealed multiple times, until the smallest grain in the microstructure see Fig. 3 was resolved. 41,42 This process resulted in linear systems with degrees of freedom, which took on the order of 772 iterations to converge to a normalized tolerance of , for every time step. The geometrical constraints imposed on a film by a substrate with zero thermal expansion tensor are modeled by noting that such interactions are equivalent to a thermodynamic process where a two-dimensional film is allowed to expand or contract freely on a perfectly lubricated surface, FIG. 4. Out-of-plane hysteretic response for grains a, b, c, and l. The location of each probed area is illustrated in the inset of Fig. 3. The hysteretic response of each probed region is a result of the local built-in electromechanical fields and the misorientation of the c-axis of each grain with respect to the sample reference system s z-axis. followed by a thermodynamic process where the macroscopic boundaries are forced back to its original shape. This situation is modeled by mechanically clamping the external boundaries of the film. Finally, the electrostatic interactions induced by a PFM tip were modeled by applying an out-of-plane sinusoidally varying electric field with an amplitude of V/m, at a frequency of 0.1 rad/s on a circular region of 2 m of diameter. This numerical setup corresponds to an ideal PFM experiment with perfectly conducting, perfectly charged, infinitely compliant, and spherically shaped tip. Such a device focuses all of its electrostatic energy on the tip and is void of cantilever-film electrostatic or mechanical interactions. The virtual piezoforce microscopy results provided by the described numerical setup do not include nonidealities found in a typical PFM experiment, such as Hertzian-type mechanical interactions, 43 film surface depolarization due to dipoleinduced dipole interactions, cantilever bending and torsions, 37 harmonic and anharmonic resonant modes, thermal and capacitance effects, spurious voltages, etc. The out-of-plane polarization was sampled at every hundredth of a cycle for user-specified positions. Each simulation took on the order of 72 h in a PowerBook G4 of Macintosh. V. ANALYSIS AND DISCUSSION The out-of-plane hysteretic response was determined through the virtual piezoforce microscopy technique on a set of grains whose out-of-plane hysteretic behavior was experimentally measured Fig. 4. For the scanned area, the polar

7 García, Huey, and Blendell J. Appl. Phys. 100, FIG. 5. Comparison between experimentally and numerically determined hysteresis loop areas for grains a, b, c, and l. Open circles correspond to normalized predicted areas and filled circles to experiment. Note that the predicted hysteretic behavior on the right side of the plot deviates from the experimental behavior because the crystallographic orientation of such region was misdetermined. The corrected crystallographic orientation is shown in Fig. 6. axis for grains a, b, and c is nearly contained within the plane of the film, thus resulting in poor out-of-plane hysteresis behavior, compared to a stress-free single crystal whose c-axis is parallel to the normal of the plane of the film. The predicted hysteresis loop associated with grain b embodies contributions from adjacent grains a and c because the size of the AFM tip is larger than grain b. As a result, the effective hysteretic response undergoes a gradual transition from one polarization state to another, passing through a macroscopic set of metastable intermediate values. Similarly, the hysteretic response associated with the data represented by the triangles pointing up in Fig. 4 integrates contributions from grains l and c. The c-axis of grain c lies within the plane of the film and is thus unswitchable, adding a constant polarization charge to the effective hysteresis loop of the scanned region. A comparison between areas of the predicted and experimentally determined hysteresis loops is illustrated in Fig. 5. For grains a and b, the numerical method produces results whose trends are consistent with those of the experimentally measured PFM. For such grains, the small differences between the numerical and experimentally determined results correspond to spurious voltages and other nonidealities of the experimental setup that are not captured by the simulation. Also, the simulation scans over an area that is an order of magnitude larger than the one measured by the experiment, thus averaging out the spatial variations of the thermodynamic fields. For grains l and c, the large difference in behavior suggests that the crystallographic orientation of the grains was misdetermined, i.e., misindexed. In tetragonal PZT systems such as this one, misindexing the orientation of a grain by 90 is a common occurrence, for the degree of tetragonality of every grain is very low, i.e., c/a=1.031, and affected by the local built-in strains of the analyzed grain. In order to verify the proposed hypothesis, an additional calculation was performed where the crystallographic orientation FIG. 6. Improved comparison between experimentally and numerically determined hysteresis loop areas for grains a, b, c, and l. The crystallographic orientation of the c-axis of the misindexed grain has been rotated 90 to the optimum orientation. Open circles correspond to normalized predicted areas and filled circles to experiment. Original, as measured, comparison can be seen in Fig. 5. of the c-axis of grain l was rotated by 90 with respect to its initial, measured orientation. The crystallography of the system allowed two possible orientations to be tested. Upon inspection it was found that only one crystallographic orientation, the one whose c-axis points out of the plane of the film, is capable of delivering an out-of-plane hysteretic response closer to the one experimentally determined. The remaining crystallographic orientation leaves the c-axis of the grain essentially contained within the plane of the film and thus delivers no improvement on the hysteretic response. Figure 6 compares the hysteretic response of the experimentally determined behavior and the numerical model for the corrected grain. As expected, the rotation of a c-axis changes the predicted out-of-plane hysteretic behavior and brings it close to the measured experimental behavior. The spatial distribution of the calculated polarization fields of the film is illustrated in Fig. 7. The distribution of the magnitude of the polarization vector is consistent with the orientational response predicted in Sec. II B. However, important deviations of the built-in polarization enhancement and suppression, with respect to the equivalent singlecrystal film are predicted to occur at grain corners and boundaries because of local property discontinuities and grain corner effects, combined with the built-in stresses and electrostatic fields. In untextured polycrystalline ferroelectric films this effect can lead to poor or unreliable out-of-plane electrical behavior. Extremal values of the magnitude of the polarization vector occur at specific crystallographic grain orientations, particularly in those grains whose 111 planes are parallel to the normal of the plane of the film z-axis. However, the location of the maximum magnitude of the remnant state of polarization and maximum out-of-plane remnant polarization does not necessarily coincide. The local state of polarization is determined by the spatial distribution of the crystallographic axes for each grain, as well as the local electromechanical interactions. The spatial interlocking of the grains

8 García, Huey, and Blendell J. Appl. Phys. 100, FIG. 7. a Predicted magnitude of polarization vector parameter for PZT film under compression. Note that every grain has a nonzero polarization. The maximum value of polarization corresponds to those grains whose normals of the 111 family of planes are aligned with the normal of the surface of the film see Fig. 3 for comparison. The simulations show that the largest variations of the magnitude of the polarization vector occur at grain corners and boundaries. sc marks the location of the stress-free single-crystal value of remnant polarization vector. b Predicted out-of-plane component of polarization order parameter. Note that several domains can be found within a grain. Regions whose out-of-plane polarization are zero correspond to grains whose c-axis lies within the plane, i.e., grains whose normal of the 110 and 010 planes are parallel to the sample reference system s z-axis. induces local changes on the in-plane electric fields and stresses via the polarization dependent piezoelectric effect see Figs. 7 and 8. These contributions are activated when the film is cooled from its stress-free temperature. Here, the anisotropy of the thermal expansion tensor induces stress concentrations at the grain interfaces, which via the direct piezoelectric effect induces polarization changes. The induced changes in polarization in turn lead to deviations in stress via the converse piezoelectric effect. The thermally induced built-in electrical and mechanical fields are responsible for the appearance of ferroelectric domains see Fig. 7. During undercooling of the film, those regions where the self-induced electric field is the largest in magnitude will be locally self-poled in directions closer to the direction of the self-induced electric field. Furthermore, those regions where the electrical field is the largest in magnitude and are mechanically tensile will be more susceptible to ferroelectric reversals during the film undercooling because of the local energy penalty to reverse the polarization direction is smaller compared to a stress-free region i.e., phase in Fig. 1. Similarly, highly compressive regions with large built-in electric fields will be favored sites for domain FIG. 8. Top inset shows the predicted hydrostatic stress sum of the in-plane principal stresses for a film in a macroscopic compressive state of stress. As a consequence of the high anisotropy of the film s thermomechanical properties, a small fraction of the grains enters a tensile state of stress. Bottom inset shows the corresponding predicted magnitude of electric field vector. The largest values of electric fields occur at grain corners and boundaries, as well as in those grains that are most compressive. Note the correlation between highly stressed regions to the local enhancement of polarization order parameter vector Fig. 7 and the correlation to the crystallographic orientation of each grain see Fig. 3. pinning because the local effective Curie temperature and coercive field will be the largest phase in Fig. 1. Because stress variations and electric fields reach extreme values at grain corners and boundaries, these sites are locations where both domain formation and switching initiates, and where domain pinning is more likely to occur. The numerical model predicts that the magnitude of the forward and backward coercive fields for 111 oriented grains is asymmetric. The predicted asymmetry in the response is due to the local built-in electric field. For an externally applied electric field, the effective local coercive field decreases in one direction because the built-in electric field reinforces the applied field. However, if the externally applied electric field is of the opposite polarity, the local built-in electric field has to be overcome in order to reverse the local state of polarization. The effect of built-in electromechanical fields on the local hysteretic response is further explored in Fig. 9, where a second set of PFM simulations is performed. Here, 111 type grains and a 001 grain grain j are found to locally interact. Grain j becomes essentially stress-free upon cooling, and its out-of-plane polarization switching behavior corresponds to that of a freestanding single-crystal layer. The out-of-plane ferroelectric response is an improvement with

9 García, Huey, and Blendell J. Appl. Phys. 100, each microstructural feature depends on the details of the local grain misorientations and their associated electromechanical interactions. In the absence of epitaxial strains, for chemically homogeneous PZT materials, 111 oriented grains have the largest value of built-in polarization, compressive stresses, and inplane electric fields, as a result of the local grain-grain interactions that arise through the direct and converse piezoelectric contributions to the thermodynamic fields, via the anisotropy of the thermomechanical properties. Finally, the analysis suggests that, in the absence of epitaxial strains, 001 oriented grains can be enhanced in their ferroelectric response if the tensile stresses induced by the underlying substrate are relaxed by favoring populations of grains that contribute with an effective in-plane compressive stress to the local state of the system. However, microstructural designs where 001 populations of grains are favored become more susceptible to mechanical fatigue due to the tensile stress concentrations that arise, particularly at grain corners. FIG. 9. Predicted out-of-plane hysteretic response for grains h, i, j, and k. Circles correspond to grain h, squares to grain i, diamonds to grain j, and triangles to grain k. Calculation shows that 001 grains can be enhanced at the expense of local compressive grain-grain interactions. The continuous line hysteresis loop corresponds to a stress-free, single-crystal film. respect to the equivalent single-crystal film because in the later tensile stresses suppress the out-of-plane polarization response. In grain j, however, the surrounding grains relieve the mechanical tensile constraints imposed by the substrate by interacting with the surrounding neighboring compressive grains. Therefore, industrially processed polycrystalline films can be improved in their out-of-plane electrical behavior by engineering textures orientation probability distributions that favor 111 and 001 orientation populations. Moreover, by carefully adding a small amount of compressive epitaxial or substrate strain, the out-of-plane response can be further improved, as shown in Sec. II. VI. SUMMARY AND CONCLUSIONS For single-crystal PZT films, the out-of-plane hysteretic and electromechanical behaviors can be tailored to a specific application. Here, the ferroelectric, electrostrictive, or linear piezoelectric response of a volume element of material is entirely determined by the orientation of the c-axis of the film and the mechanical constraints imposed by the substrate, thus defining a finite window of processing parameters for memory device optimization. For polycrystalline films, the window of processing parameters is further reduced by the local microstructural interactions. 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