Copyright American Scientific Publishers All rights reserved Printed in the United States of America Journal of Computational and Theoretical Nanoscience Vol. 9, 667, Effect of Crystal Orientation on Piezoelectric Response of Single-Crystal Piezoelectric Layers Ming Liu and Fuqian Yang Materials Program, Department of Chemical and Materials Engineering, University of Kentucky, Lexington, KY, 456, USA The piezoelectric response of a piezoelectric material is strongly dependent on the relative direction between external loading and the poling direction. In this study, we present a three-dimensional finite element model to analyze the effect of the angle between the loading direction and the poling direction on the deformation state and electric displacement in a piezoelectric layer. The results show that, under the action of electric loading, the apparent longitudinal piezoelectric coefficient can be determined from the ratio of the displacement difference to the potential difference between the top and bottom surfaces, which is a function of the crystal orientation. The deformation-induced electric potential difference between the top and bottom surfaces of the piezoelectric layer is dependent on the expansion/shrinkage of side surfaces when subject to mechanical loading. Keywords: Piezoelectric Response, Thin Films, Simple Compression. indentation, and piezoelectric force microscope. 3 The use of contact mechanics was first proposed by Lefki and Dormans, 8 using continuous charge integration to measure the electric charge along the loading direction, in charactering the piezoelectric coefficients of lead zirconate titanate. Fu et al. 9 used the similar technique to measure the piezoelectric response of PbZr 53 Ti 47 O 3 thin films. Suresh and his co-workers extended the sharp-instrumented indentation technique to characterize the electric response of 3 piezoelectric ceramic-polymer composite, lead zirconate titanate, and barium titanate. Barzegar et al. 3 analyzed the effect of boundary conditions and sample aspect ratio Author to whom correspondence should be addressed.. INTRODUCTION Delivered by Publishing Technology to: onuniversity the longitudinal of Southern piezoelectric California coefficients and developed 3 Jan a modified 3 :55: relationship between the measured piezo- IP: 3.74.55.3 On: Wed, Piezoelectric materials have been widely Copyright used asamerican sensors Scientific electric coefficient Publishersand the theoretical value. All of these and actuators in a variety of applications, such as microcantilever sensors for the detection of proteins and cells is parallel to the symmetrical axis (the polling direction); works were based on the assumption that loading direction in biological fluids and actuators for portable peristaltic and very little work has been done to address the effect micropumps. The electromechanical coupling in piezoelectric materials provides a unique method to control struc- simple compression of piezoelectric materials. of crystal orientation on the piezoelectric response during tural response by electric loading and to sense structural Piezoelectric multilayer structures have been extensively deformation from the change in electric field. They have used in a variety of applications, including sensors and been extensively applied in controlling structural vibration actuators, which generally require that the structures have due to their fast response and structural flexibility. maximum electromechanical response to external stimulus Various techniques have been developed to characterize in an optimal orientation. 4 It is known that piezoelec- piezoelectric properties of materials, including Berlincourt method, laser interferometers, laser scanning vibrometers, tric response of a piezoelectric material depends on the relative direction of electrical/mechanical loading to crystallographic orientation and the poling direction. 5 Thus, it becomes of essence to understanding the effect of the loading direction on the deformation behavior of piezoelectric structures. The purpose of this work is to analyze the piezoelectric behavior of a piezoelectric thin film. The study is focused on the directional dependence of the piezoelectric response when the loading direction is different from the poling direction. Using the tensor transformation, we obtain the piezoelectric coefficients in a new coordinate with ox 3 -axis the same as the loading direction under uniaxial loading. Three-dimensional finite element analysis is used to examine the effect of loading conditions and the relative direction on the piezoelectric response of the thin film. 6 J. Comput. Theor. Nanosci., Vol. 9, No. 546-955//9/6/6 doi:.66/jctn..66
Liu and Yang. PROBLEM FORMULATION and the mechanical boundary condition is Consider simple compression of a piezoelectric thin film in a coordinate system ox x x 3. Normal loading is applied to the surface of the film with the loading direction parallel to the ox 3 -axis. The poling direction or the axisymmetric axis of the film is the same as the ox 3 -axis in the coordinate system ox x x 3 with the ox -axis the same as the ox -axis. The angle between the ox 3 -axis and the ox 3-axis is. For piezoelectric materials in the coordinate system ox x x 3 with the poling direction the same as the ox 3 -axis, the constitutive relations are D i = e ikl kl + ik E k and ij = C E ijkl kl e kij E k i j k l= 3 () where D i are the components of electric displacement vector D, ij are the components of strain tensor, E i are the components of electric field intensity, e ikl are the components of the piezoelectric coefficient tensor measured in To analyze the piezoelectric response of the piezoelectric the possession of a spontaneous electric field, film when subjected to external loading in the ox 3 ik are the direction, we first perform the tensor transformation to obtain components of the dielectric tensor measured at a constant strain and Cijkl E the piezoelectric coefficients of PZT-4 in the coordinate are the components of the elastic stiffness tensor measured at a constant electric field intensity. x x 3 system ox. The transformation law for the elastic stiffness tensor is The repeated indices imply a sum over that index. In the coordinate system ox x x 3, the components of the electric C ijkl = a mi a nj a ok a pl C mnop () field intensity can be calculated from the electric potential of as Delivered by Publishing Technology to: theuniversity transformation of Southern law for California the piezoelectric coefficients is IP: 3.74.55.3 On: Wed, 3 Jan 3 :55: E i = i Copyright American () Scientific Publisherse ijk = a li a mj a nk e lmn () The equilibrium equation describing the piezoelectric behavior of the thin film without the action of body force is and the Gauss s law gives ij i = (3) D i i = (4) Here, a comma denotes the partial derivative with respect to the coordinate x i. The relation between the components of the strain tensor and the components of the displacement vector, u i,is ij = u i j + u j i / (5) The boundary conditions on the bottom surface of the film are u 3 = and = at x 3 = (6) To avoid the rigid body motion, the condition of u = u = is used at the point of (,, ). Two loading cases are considered as below. Case : Electrical loading. For the elastic loading, an electric potential of is applied to the top surface of the film, i.e., = at x 3 = h (7) 33 = at x 3 = h (8) Case : Mechanical loading. For the simple compression, the mechanical boundary condition is u 3 = u at x 3 = h (9) with the electric insulation condition as D 3 = at x 3 = h () Here u is the displacement applied to the top surface of the piezoelectric film, and h is the film thickness. 3. PIEZOELECTRIC COEFFICIENTS OF PZT-4 IN THE COORDINATE SYSTEM ox x x 3 and the transformation law for the dielectric tensor is ij = a ki a lj kl (3) Here, the coefficients with the superposed denote the physical parameters in the coordinate system of ox x x 3. The directional cosine of a ij between the axis ox i and the axis ox j is calculated as a ij = cos sin (4) sin cos with being the angle between the ox 3 -axis and the ox 3 -axis. For the PZT-4 piezoelectric materials in the coordinate system of ox x x 3, the components of the piezoelectric coefficient tensor as represented by a matrix e ij 3 6 is found as e 3 cos e 3 sin e 3 cos sin e 3 sin 3 e 333 sin cos e 3 sin +e 333 sin 3 +e 333 cos sin e 3 sin cos +e 3 cos sin e 3 cos sin +e 3 cos 3 e 3 cos 3 e 3 sin cos e 333 cos sin e 3 cos +e 333 sin cos +e 333 cos 3 e 3 cos sin e 3 sin cos +e 3 sin cos e 3 cos sin (5) J. Comput. Theor. Nanosci. 9, 667, 63
/ Liu and Yang in which e 3 = e 3 = e 3, e 333 = e 33, and e 3 = e 3 = e 5. The following convention on the subscripts is used: ; ; 33 3; 3 4; 3 5; 6. Similarly, one can obtain the components of the elastic stiffness tensor and the components of the dielectric tensor in the coordinate system of ox x x 3, which are not given here. 4. FINITE ELEMENT ANALYSES Consider a piezoelectric layer of m 3 subject to electric or mechanical loading on the top surface. The bottom surface is supported by a rigid substrate and grounded with an electric potential of zero. Uniform finite element meshes are used; and the 3-D finite element model consists of 5 8-node linear piezoelectric elements. The numerical simulation is performed using the commercial nonlinear finite element code ABAQUS. 6 The simulation is focused on the small deformation. Table I lists the material properties of PZT-4 piezoelectric ceramic used in the simulation, in which the values correspond to the coordinate system of ox x x 3 with the poling direction the same as the ox 3 -direction. The PZT-4 piezoelectric ceramic is a transversely isotropic piezoelectric material of the hexagonal crystal class 6 mm. ing direction, in which the results is normalized by the 4.. Electric Loading d 33 at =. For comparison, the d 33 as calculated It is known that an Delivered electric potential by Publishing applied on Technology the top to: from University Eq. (8) of issouthern also included California in Figure. Obviously, the IP: 3.74.55.3 On: Wed, 3 Jan 3 :55: surface of a piezoelectric layer will cause Copyright the normal American displacement of the top surface. The ratio of the normal dis- numerical results are in agreement with the results given Scientific Publishers placement to the applied potential difference represents the capability of the converse piezoelectric effect of a piezoelectric material. For a simple electric loading on the top/bottom surfaces of a piezoelectric layer, there is a simple relationship as, For small deformation, Eq. (6) gives 33 = d 33 E 33 (6) d 33 (θ)/d 33 ().5.5 FEM results Eq. (6) 5 5 5 3 35 Fig.. Dependence of longitudinal piezoelectric coefficient of d 33 on the relative direction of electric loading to the poling direction. components of the piezoelectric coefficient matrix, and C E ij are the components of the elastic stiffness matrix). In crystals having the 4 mm, the orientation dependence of d 33 can be expressed as 8 d 33 = d 33 cos 3 + d 3 + d 5 cos sin (8) Figure shows the simulation results of d 33 as a function of the relative direction of electric loading to the pol- by Eq. (8), which validates the FEM model and suggests that Eq. (7) can be used to measure the longitudinal piezoelectric coefficient. The longitudinal piezoelectric coefficient reaches the maximum value when the loading direction is parallel to the poling direction, as expected. The electric loading will not induce deformation in the ox 3 -direction if the field direction is perpendicular to the poling direction. Figure shows the variation of the displacement components at the center of the top surface with the angle d 33 = u 3 x3 =h u 3 x3 = x3 =h x3 = (7) which can be used to determine the charge constant d 33 from numerical calculation and experimental tests. The longitudinal piezoelectric coefficient of d 33 can be calculated from the elastic stiffness tensor and piezoelectric coefficient tensor, i.e., d ij 3 3 = e ij 3 6 Cij E 6 6 (e ij are the Table I. Material properties of PZT-4 piezoelectric ceramic. 7 Piezoelectric Dielectric Elastic constant constant constant ( N/m (C/m ( 9 C/V m) C E C E C E 3 C E 33 C E 44 e 3 e 33 e 5 33 3.9 7.78 7.43.3.56 6 98 3.84 3.44 6. 5.47 u i h ( 3 ) 3 u 3 u u 3 5 5 5 3 35 Fig.. Dependence of the surface displacement at the center of the top surface on the loading direction ( x3 =h = 5 V). 64 J. Comput. Theor. Nanosci. 9, 667,
Liu and Yang between the loading direction and the poling direction for D x3 =h = 5 V. The displacement component of u at the center is zero independent of the angle between the loading D direction and the poling direction, since the ox coincides with ox. Both u and u 3 change with the angle and there is a / difference in the phase angle. The magnitude of u 3 reaches the maximum value at = and and the minimum value of zero at = / and 3 /. Due to the non-zero angle between the loading direction and the 3 poling direction, electric loading can cause shear deformation. It is interesting to note that the maximum magnitude 4 D 3 of u 3 is larger than that of u, which is controlled by the 5 piezoelectric coupling. 5 5 5 3 35 The change in the material position due to electric loading will introduce local strain in the piezoelectric layer, Fig. 4. Variation of the components of the electric displacement vector which can cause the formation of cracks when local strain with the angle between the loading direction and the poling direction is large enough. Figure 3 shows the variation of the strains ( x3 =h = 5 V). at the center of the top surface with the angle between the loading direction and the poling direction for x3 =h = loading direction and the poling direction for x3 =h = 5 V. The shear strains and 3 are zero independent 5 V. The electric displacement component of D is zero of the angle, which is consistent with u =. For = due to the zero angle between the ox 9 and 7, all the normal strain components are zero, axis. The non-zero component of D 3, as controlled by the while there is non-zero shear component of 3. The electric loading causes shear deformation instead of normal cally with the relative direction of the electric loading to electric potential applied to the top surface, changes cycli- expansion/shrinkage. The normal components of the strain the poling direction. When the surface of the piezoelectric tensor are a function of the angle. The maximum magnitudes of the normal strains and 33 occur at the same ponent D layer is not perpendicular to the poling direction, the com- angle which is different Delivered from those by Publishing for andtechnology 3. The to: University increases with increasing the angle and reaches of Southern California structural damage induced by electric IP: 3.74.55.3 loading likelyon: willwed, the 3 maximum Jan 3 at :55: = 45. Further increasing the angle Copyright American Scientific leads to the Publishers decrease of the component D to the minimum at = 35. The relationship between the electric displacement vector of D and the free charge density of can be expressed as D = (9) be controlled by 33 due to the largest strain magnitude for the same electric loading. Due to the piezoelectric coupling, the deformation induced by the electric loading can cause the change of the distance between positive charges and negative charges and lead to the accumulation of electric charges on the surface of the piezoelectric layer, as represented by electric displacement vector. Figure 4 depicts the dependence of the components of the electric displacement vector at the center of the top surface with the angle between the.5 ε 33 D ij /e 333 ( 3 ) from which, one can find the charge density on the top surface of the layer as and on the side surface as t = D 3 () s = D () ε ij ( 3 ) ε ε.5 ε ε 3.5 ε 3.5 6 8 4 3 36 Fig. 3. Dependence of the strain components at the center of the top surface on the loading direction ( x3 =h = 5 V). According to Figure 4, the maximum surface charge density on the top surface will be created when = 9 and 7, while electric loading will induce the maximum surface charge density on the side surface when = 45 and 5. Comparing Figures 4 to, one can note that the magnitude of D 3 and u reach the maximum at the same. Such a behavior can be used to determine the symmetrical axes of a piezoelectric material. 4.. Mechanical Loading The mechanical loading is achieved by applying the displacement of u 3 on the top surface of the piezoelectric J. Comput. Theor. Nanosci. 9, 667, 65
Liu and Yang Δφ/ h 33 u.5.5.5.5 5 5 5 3 35 Fig. 5. Dependence of the ratio of /u on the angle between the loading direction and the poling direction (u = nm). layer with a frictionless and grounded bottom surface. The mechanical loading causes the deformation of the piezoelectric layer, which induces the electric potential difference across the layer thickness, i.e., the direct piezoelectric behavior, as represented by the ratio of the induced electric potential difference to the applied displacement, /u. Figure 5 shows the variation of the ratio of /u with the angle between the loading direction and the poling direction, in which h 33 (= 53 V/nm) is the component of the piezoelectric stiffness coefficient matrix as calculated between the ox axis and the ox axis, similar to the case Delivered by Publishing Technology to: University of Southern California from the piezoelectric stress coefficient IP: 3.74.55.3 tensor and dielectric constant tensor. Similar to the longitudinal Copyright piezoelec- under the action of electric loading. The component of D 3 On: Wed, 3 Jan 3 :55: is zero as requested by the condition of (). For the com- American Scientific Publishers tric coefficient of d 33, the ratio of /u varies with the angle between the loading direction and the poling direction. For the same applied displacement, the displacementinduced potential difference reaches the maximum value at = and 8, similar to the angle dependence of the longitudinal piezoelectric coefficient of d 33. Due to the piezoelectric effect, the applied displacement of u in the ox 3 direction can cause the in-plane deformation. Figure 6 shows the variation of the in-plane displacement components of u and u at the center of the top 4 u surface with the angle. The in-plane displacement of u is zero, independent of the angle. It is interesting that the applied surface displacement in the ox 3 direction induces the in-plane displacement with the maximum displacement of u occurring at = 45, 35, 5, and 35. Such a behavior is different from the field-induced in-plane deformation, as shown in Figure. The mechanical loading is indistinguishable between = and = 8, in contrast to the polarization effect of electric field. Corresponding to the applied displacement onto the top surface of the piezoelectric layer, mechanical stresses are created in the material. The stress state at the center of the top surface is depicted in Figure 7 as a function of the angle between the loading direction and the poling direction. Due to the piezoelectric coupling, the applied normal surface displacement introduces the non-zero stress component of and 33. The magnitude of the stress component 33 reaches the maximum at = and 8, while reaches the maximum value at = 45 and 35. In general, the stress-induced structural damage is controlled by the normal stress component of 33. Because of 33 >. The variation of the components of the electric displacement vector at the center of the top surface with the angle between the loading direction and the poling direction for u = nm is depicted in Figure 8. The electric displacement component of D is zero due to the zero angle ponent of D, the magnitude reaches the maximum value at = 45 and 35. According to Eq. (), the zero value of D 3 implies that there is no distribution of charges on the top surface. From the constitutive equation of (), one has e 3kl kl + 3k E k = at x 3 = h () For the simple compression of a piezoelectric layer with the surface normal parallel to the poling direction, Eq. () σ, σ, σ 3, σ 3 u i /h ( 4 ) u σ ij /C 33 ( 4 ) 4 6 8 σ σ 33 4 5 5 5 3 35 Fig. 6. Variation of the surface displacement on the angle between the loading direction and the poling direction (u = nm). 5 5 5 3 35 Fig. 7. Dependence of the stress components state at the center of the top surface on the angle between the loading direction and the poling direction (u = nm). 66 J. Comput. Theor. Nanosci. 9, 667,
Liu and Yang D i /e 333 ( 3 ).5.5 D D, D 3 5 5 5 3 35 Fig. 8. Dependence of the components of electric displacement vector at the center of the top surface on the angle between the loading direction and the poling direction (u = nm). reduces to e 3 + e 3 + e 333 33 + 33 E 3 = at x 3 = h (3) The deformation-induced electric field on the top surface of the piezoelectric layer is dependent on the in-plane expansion/shrinkage created by the uniaxial deformation even for the condition of the surface of the piezoelectric layer being perpendicular to the poling direction. Assume that the edge effect ondelivered electric field by is Publishing negligible. Technology There are to: University of Southern California IP: 3.74.55.3 On: Wed, 3 Jan 3 :55: Copyright American Scientific Instrum. Publishers 67, 935 (996). 33 = u /h and E 3 = x3 =h x3 = /h (4) Substituting Eq. (4) into Eq. (3) yields e 3 h + e 3 h + e 333 u 33 x 3 =h x3 = = (5) The ratio of x3 =h x3 = /u is a function of the layer thickness and the strains of and. One needs to be cautious in using the ratio of x3 =h x3 = /u to determine piezoelectric constants from simple uniaxial test. 5. SUMMARY Using the finite element method, we analyzed the effect of the angle between the loading direction and the poling direction on the piezoelectric response of a PZT-4 layer when subjected to either electric or mechanical loading. The results show. Under the only action of electric loading, the longitudinal piezoelectric coefficient can be calculated from the ratio of the displacement difference to the potential difference between the top and bottom surfaces. The longitudinal piezoelectric coefficient is a function of the crystal orientation. Electric loading will induce surface charge on the surfaces of a piezoelectric layer. The maximum surface charge density on the top surface will be created when = 9 and 7, while electric loading will induce the maximum surface charge density on the side surface when = 45 and 35. Mechanical loading will not induce surface charge on the top surface of a piezoelectric layer, when the piezoelectric layer is in contact with an insulated material. The deformation-induced electric field in the piezoelectric layer is dependent on the expansion/shrinkage of side surfaces created by the uniaxial deformation. References. H. S. Tzhou and S. Pandita, J. Robotic Sys. 4, 79 (987).. D. Damjanovic and R. E. Newnham, J. Intelligent Mater. Sys. Struct. 3, 9 (99). 3. Z. Huang, Q. Zhang, S. Corkovic, R. Dorey, and R. W. Whatmore, IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 53, 87 (6). 4. Q. M. Zhang, W. Y. Pan, and L. E. Cross, J. Appl. Phys. 63, 49 (988). 5. J. Li, P. Moses, and D. Viehland, Rev. Sci. Instrum. 66, 5 (995). 6. A. L. Kholkin, C. Wutchrich, D. V. Taylor, and N. Setter, Rev. Sci. 7. F. Q. Yang, Piezoelectric response in the contact deformation of piezoelectric materials, Micro and Nano Mechanical Testing of Materials and Devices, edited by F. Q. Yang and J. C. M. Li, Springer, New York, NY (8). 8. K. Lefki and G. J. M. Dormans, J. Appl. Phys. 76, 764 (994). 9. D. Fu, K. Ishikawa, M. Minakata, and H. Suzuki, Jpn. J. Appl. Phys. 4, 5683 ().. S. Sridhar, A. E. Giannakopoulos, S. Suresh, and U. Ramamurty, J. Appl. Phys. 85, 38 (999).. A. Saigal, A. E. Giannakopoulos, H. E. Pettermann, and S. Suresh, J. Appl. Phys. 86, 63 (999).. A. Barzegar, D. Damjanovic, and N. Setter, IEEE Trans. Ultrason. Ferroelect. Feq. Contr. 5, 897 (5). 3. A. Barzegar, D. Damjanovic, and N. Setter, IEEE Trans. Ultrason. Ferroelect. Feq. Contr. 5, 6 (4). 4. K. Kudo, K. Kakiuchi, N. Endo, N Bamba, K. Hoshikawa, and T. Fukami, Jpn. J. Appl. Phys. 43, 67 (4). 5. S. Trolier-Mckinstry and P. Muralt, J. Electroceramics, 7 (4). 6. ABAQUS User Manual, V. 6.9, Hibbit, Karlsson, Sorensen, Inc., Providence, Rhode Island, USA (994). 7. F. Q. Yang, J. Mater. Sci. 39, 8 (4). 8. J. F. Nye, Physical Properties of Crystals, Oxford University Press, Oxford (957). Received: 4 August. Accepted: 7 October. J. Comput. Theor. Nanosci. 9, 667, 67