ABSTRACT. Professor Alexander. L. Roytburd Dept. of Materials Science and Engineering

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1 ABSTRAT Title o Document: ORIENTATION DEPENDENE OF THE PIEZOELETRI PROPERTIES OF EPITAXIAL FERROELETRI THIN FILMS Jun Ouyang, Doctor o Philosophy, 25 Directed By: Proessor Alexander. L. Roytburd Dept. o Materials Science and Engineering There are both intrinsic piezoelectric response and extrinsic piezoelectric response in erroelectric materials. The intrinsic piezoelectric response is due to the lattice deormation o a single-domain crystal, which can be characterized by tensors o piezoelectric constants. The extrinsic piezoelectric response depends on extrinsic sources o displacement under the electric ield, which can be the movement o domain walls, phase boundaries, or even deects like grain boundaries or dislocations. Due to the elastic interaction between an epitaxial erroelectric thin ilm and a substrate, the piezoelectric properties o an epitaxial erroelectric ilm are dierent rom those o bulk erroelectric materials. This work is the irst study on the general orientation dependence o the piezoelectric properties o epitaxial erroelectric thin ilms, which includes both theoretical and experiment work on intrinsic and extrinsic piezoelectric properties o epitaxial erroelectric ilms. A complete theoretical analysis o intrinsic piezoelectric responses in a single domain erroelectric ilm, which are characterized by eective longitudinal, transverse and shear piezoelectric coeicients, is presented in this dissertation.

2 On the part o extrinsic piezoelectric response, our recent work on the piezoelectric properties o epitaxial thick lead titanate zirconate (Pb(Zr x Ti 1-x )O 3 with x=.52) ilms with tetragonal distorted structures will be presented as an example. It is shown that (11) oriented epitaxial ilms had much enhanced piezoelectric responses as compared with those o (1) and (111) oriented ilms. Detailed structure analysis showed that instead o an interconnected 3-domain (3-D) architecture that is usually ound in a (1) oriented thick ilm, the (11) ilms consisted o a dominant 2-domain (2-D) architecture, by which the pinning between neighboring domain walls is much reduced. This study demonstrate the possibility o achieving high extrinsic piezoelectric responses by optimizing the epitaxial relationship between the ilm and substrate with respect to the domain mobility, and should also be instructive to the design o erromagnetic and erroelastic thin ilm devices used or transducer applications..

3 ORIENTATION DEPENDENE OF THE PIEZOELETRI PROPERTIES OF EPITAXIAL FERROELETRI THIN FILMS By Jun Ouyang Dissertation submitted to the Faculty o the Graduate School o the University o Maryland, ollege Park in partial ulillment o the requirements or the degree o Doctor o Philosophy 25 Advisory ommittee: Proessor, Alexander Roytburd, hair Proessor, John Melngailis Proessor, Manred Wuttig Associate Proessor, Ichiro Takeuchi Adjunct Proessor, James R. ullen

4 opyright by Jun Ouyang 25

5 Dedication To My Grandma, Heguan Sheng, a great hinese woman who best embodies the traditional oriental virtues o diligence, patience, bravery, and dedication to amily. ii

6 Acknowledgements I would like to express my sincere gratitude to my advisor, Pro. Alexander L. Roytburd, or giving me the opportunity to work on this interesting and challenging project, and more importantly, his guidance and support throughout the course o my graduate study and research. I beneited rom all the discussions we had. It was his wide knowledge in materials science especially on thermodynamics o elastic domains, creative thoughts and sparkling o resh ideas, dedication to scientiic excellence, and most important, his pure love and enthusiasm toward scientiic research, that had led to every progress in my Ph.D. research. The same gratitude goes to Pro. R. Ramesh, who served as my co-advisor beore and ater he let or University o aliornia, Berkeley. It is a great honor or me to have been a part o his team at the Advanced Thin Film Laboratory, which is at the oreront o research in oxide thin ilm unctional materials. His enthusiasm and energy, great instinct on picking up cutting-edge scientiic topics, dedication and the drive or academic excellence have been primary motivating actors in this dissertation. I thank both o them or providing me with numerous opportunities to participate in conerences, or these are essential in building conidence and networks. I would like to thank Pro. hang-beom Eom and Dr. Dong Ming Kim at University o Wisconsin, Madison or their support on providing excellent quality epitaxial PZT thick ilm samples by magnetron sputtering. I am also very grateul to Pro. Susan Trolier-McKinstry and Pro. Wenwu ao at Penn State University or inspiring and ruitul discussions with them. I would also like to express my gratitude to Dr. Igor Levin and Dr. Julia Slutsker in National Institute o Standards and Technology, or providing numerous discussions iii

7 and suggestions, which has proved to be an indispensable part o this dissertation work. Dr. Levin s expertise in structure analysis and Dr. Slutsker s thermodynamic knowledge and modeling skills have been especially helpul to my understanding o the project. I would like to thank Pro. John Melngailis, Pro. Manred Wuttig Pro. I. Takeuchi and Pro. James R. ullen or taking time out o their busy schedules to review my work. I specially thank Dr. V. Nagarajan and S. Prasertchoung or leading me into the electrical characterization by scanning probe microscopy; Dr. T. Zhao and Dr. Junling Wang or helping me with the PLD thin ilm preparation; Ladan Mohaddes-ardabili or introducing me structural analysis by XRD; Zhengkun Ma or numerous times o ocus ion beam milling on my samples; S.-Y. Yang or teaching me using erroelectric tester and impedance-gain phase analyzer; Dr. Haimei Zheng or TEM sample preparation, etc. All the riends and collaborators at University o Maryland, thank you or your support and riendship. Finally, I want to thank my parents and my iancée or their love and support. I owe everything I achieved to them. This dissertation is dedicated to them. iv

8 TABLE OF ONTENTS DEDIATION ii AKNOWLEDGEMENT iii TABLE OF ONTENTS......v LIST OF FIGURES vi LIST OF TABLES... xi HAPTER 1, INTRODUTION Piezoelectricity and erroelectricity Intrinsic v.s. Extrinsic piezoelectric response Bulk erroelectrics v.s. thin ilm erroelectric ilms onclusion HAPTER 2, INTRINSI PIEZOELETRI RESPONSE IN THIN EPITAXIAL FERROELETRI FILMS onverse piezoelectric properties or erroelectric thin ilms Direct piezoelectric properties or erroelectric thin ilms Summary o the intrinsic piezoelectric properties in erroelectric thin ilms Theoretical prediction or various thin ilm material systems Lead Zirconate Titanate solid solution Barium Titanate Relaxor erroelectric systems Experiment measurement on longitudinal piezoelectric coeicient d, Piezoelectric orce microscopy Lead Zirconate Titanate thin ilms Lead Magnesium Niobate-Lead titanate thin ilms onclusion HAPTER 3, EXTRINSI PIEZOELETRI RESPONSE IN THIN EPITAXIAL FERROELETRI FILMS Orientation dependence o 9 o domain pattern Anisotropic misit in a (11) oriented epitaxial ilm Experiment discovery o highly mobile 9 o domain walls and ultra high extrinsic piezoelectric responses due to 9 o domain wall movement Theoretical analysis, and the importance o elasticity onclusion..137 ONLUSIONS AND FUTURE WORK BIBLIOGRAPHY v

9 LIST OF FIGURES Fig. 1.1 (a) onverse piezoelectricity and (b) direct piezoelectricity. S - mechanical strain, E - electric ield strength, P - electrical polarization, T - Mechanical stress 1 Fig Schematics o a piezoelectric crystal (a) under a tensile stress (b).3 Fig. 1.3 (a) Schematic illustration o the unit cell o perovskite lead titanate erroelectric crystals. Figure adapted rom [4]. (b) Switchable polar states under electric ield enable applications or memory devices...7 Figure 1.4. Interrelationship o piezoelectrics and subgroups on the basis o symmetry. Figure adapted rom [1]...8 Figure 1.5. Ferroelectric materials used as actuator (top) and sensor (bottom). Figures adapted rom [5]..9 Fig Equations o converse piezoelectric eects in a tetragonal erroelectric crystal. P is the sel-polarization o the crystal aligned along the tetragonal c-axis (axis 3).1 Fig. 1.7 Eective piezoelectric constants d o PZT with various compositions.13 Fig. 1.8 Orientation dependence o longitudinal piezoelectric constants or single crystals...14 Fig. 1.9 (a) [111] poled 4mm crystal BaTiO 3 has measured d (at room temperature) much larger than d o [1] poled crystal; [9] (b) alculated orientation dependence o d (at room temperature) or 4mm crystal BaTiO Fig Ultra-high piezoelectric behavior discovered in PZN-8%PT single crystal in a engineered domain coniguration: (a) Schematic diagram o domain conigurations in <1)> oriented rhombohedral crystals under bias (step A-piezoelectricity, step B- induced phase transition). (b) Strain vs E-ield behavior or <1> oriented PZN-8%PT crystal. Maximum ield is limited by voltage limit o the apparatus...18 Fig (a) Orientation dependence o eective piezoelectric constant d o PMN %PT single crystal with single domain. (b) ross section plot o (a) in [1 1 1] [1 1 2] plane. Figures adapted rom [17].19 Fig 1.12 (a) ubic to tetragonal phase transormation in BaTiO 3 crystal; (b) The tetragonal erroelectric phase has six dierent domains...2 Fig. 1.13: Eect o 18 o degree domains on longitudinal d piezoelectric response. is the raction o domains with polarization aligned along the applied electric ield ( + domain), 1- is the raction o domains with polarization aligned opposite to the applied electric ield ( - domain). The level o the dashed line is the surace o the crystal when vi

10 there is no electric ield applied...21 Fig. 1.14: Eect o 9 o degree domain wall movement on longitudinal piezoelectric response. It can be seen that the extrinsic contribution dominates the total strain...22 Fig First principle calculations or the longitudinal piezoelectric response o BaTiO 3. (a) Possible poling paths indexed by letters or a BaTiO 3 single crystal in a rhombohedral ground state, the electric ield is applied along the pseudo-cubic [1] direction {e} while the polarization is aligned along [111] {a}. (b) Piezoelectric strain curve associated with the evolution o poling states with lowest energy..26 Fig (a) Microsensor based on the direct piezoelectric eect o erroelectric ilms. (b) longitudinal, (c) shear microacuators based on the converse piezoelectric eect o erroelectric ilms. The blue arrows inside the ilms indicate the alignment o the polar vectors in the ilm..31 Figure A single domain tetragonal erroelectric ilm with normal n under electric ield E (E//n). x 1, x 2, x 3 are the crystalline coordinates. P is the polarization vector, P is parallel to x 3. The deormation is exaggerated here or illustration.37 Figure 2.2. The epitaxial relation between a erroelectric ilm and substrate with arbitrary orientations. x i and x i,sub are the artesian coordinates o the crystalline structures o the ilm and substrate, respectively.47 Fig. 2.3 Direct piezoelectric eect: (a) d 31 /e 31 mode- a longitudinal charge is produced by applying in-plane stresses/strains to the substrate. (b) d mode- a longitudinal charge is produced in the ilm by applying a stress normal to the ilm/substrate couple.56 Fig. 2.4 onverse piezoelectric eect A longitudinal strain and a shear strain are produced by applying an electric ield normal to the ilm [44] ; in-plane stresses are also produced under the electric ield as a result o substrate clamping. The deormation is exaggerated here or illustration 57 Figure 2.5 (a) Orientation dependence o the intrinsic longitudinal converse piezoelectric coeicient d, o PbTiO 3 ilms; (b) The cross section curve when the igure in (a) is cut by the (1 ) plane 64 Figure 2.6 (a) Orientation dependence o the intrinsic longitudinal converse piezoelectric coeicient d, o Pb(Zr.48 Ti.52 )O 3 ilms; (b) The cross section curve when the igure in (a) is cut by the (1 ) plane.65 vii

11 Figure 2.7. (a) Orientation dependence o the intrinsic longitudinal converse piezoelectric coeicient d, o Pb(Zr.52 Ti.48 )O 3 ilms; (b) The cross section curve when the igure in (a) is cut by the (1-1 ) plane 66 Figure 2.8. (a) Orientation dependence o the intrinsic longitudinal converse piezoelectric coeicient d, o Pb(Zr.6 Ti.4 )O 3 ilms; (b) The cross section curve when the igure in (a) is cut by the (1-1 ) plane 67 Figure 2.9 (a) or [1], [11], [111] oriented PZT epitaxial ilms as unction o ilm composition; (b) d, or [1] oriented epitaxial and polycrystalline ilms as unction o ilm composition.68 D e, Fig. 2.1 (a) Eective piezoelectric coeicient 31 o a tetragonal PZT 5/5 ilm as unction o ilm orientation. (b) The cross section curve o (a) cut by the (1 ) plane D e, Fig (a) Eective piezoelectric coeicient 31 o a rhombohedral PZT 52/48 ilm as unction o ilm orientation.. (b) The cross section curve o (a) cut by the pseudo-cubic (1-1 ) plane.72 Figure 2.12 Eective piezoelectric coeicient e 31, o PZT ilms with various compositions.73 D Figure (a) Longitudinal piezoelectric coeicient ; (b) transverse piezoelectric D D coeicient d 31, and (c) transverse piezoelectric coeicient e 31, or PZT ilms grown on Si 74 Fig (a). The cross section curve (cut by the (1) plane) o the longitudinal piezoelectric coeicients or a tetragonal Pb(Zr.5 Ti.5 )O 3 ilm as unction o ilm orientation. (b) The cross section curve (cut by the pseudo-cubic (1-1 ) plane) o the longitudinal piezoelectric coeicients or a rhombohedral Pb(Zr.52 Ti.48 )O 3 ilm as unction o ilm orientation...77 Figure (a) alculated piezoelectric coeicients and (b) e 31, by using dierent sets o electromechanical data 83 Figure 2.16 (a) alculated piezoelectric coeicients and (b) e 31, o single domain tetragonal BaTiO 3 ilms as unction o crystalline orientation o the ilm normal...84 viii

12 Figure 2.17 (a) ross section curves o piezoelectric coeicients and (b) e 31, o single domain tetragonal BaTiO 3 ilms when Fig (a), (b) are cut by the (1 ) plane...85 Figure (a) alculated piezoelectric coeicients e 31, and (b) d, o a single domain rhombohedral.67pb(mg 1/3 Nb 2/3 )O 3 -.PbTiO 3 ilm as unction o crystalline orientation o the ilm normal. (c) ross section curves o piezoelectric coeicients e 31, and (d) d, o a single domain rhombohedral.67pb(mg 1/3 Nb 2/3 )O 3 -.PbTiO 3 ilm when Fig (a), (b) are cut by a pseudo-cubic (1-1 ) plane...91 Figure (a) alculated piezoelectric coeicients e 31, and (b) d, o a single domain tetragonal.58pb(mg 1/3 Nb 2/3 )O PbTiO 3 ilm as unction o crystalline orientation o the ilm normal. (c) ross section curves o piezoelectric coeicients and (d) d, o a single domain tetragonal.58pb(mg 1/3 Nb 2/3 )O PbTiO 3 ilm when Fig (a) (b) are cut by a pseudo-cubic (1 ) plane..93 Figure 2.2. (a) alculated piezoelectric coeicients e 31, and (b) d, o various relaxor erroelectric ilms..95 Fig (a) Schematics o the set-up o piezoelectric orce microscopy (PFM); Figure adapted rom Re. [8]. (b) Measurement o d amplitude under a uniorm tip ield by covering the ilm with Pt top electrode pads. (c) local piezoelectric response measurement (imaging). 98 Fig Recording the amplitude o a piezoelectric response by a position-sensitive detector (PSD).. 1 Fig (a) Detection o 18 o domains by inding the phase angle between the longitudinal piezoelectric response and the tip ac signal. (b) Out-o-plane; (c) in-plane PFM image o a 4nm PZT(2/8)/LSO/STO erroelectric ilm showing both 18 o (by writing) and 9 o domain (grown in-situ) patterns. 11 Figure 2.24 X-ray spectrum and -scans or (1)-(a), (11) - (b) and (111)- (c) or Pb(Zr.2 Ti.8 )O 3 ilms...14 Figure 2.25 Eective longitudinal piezoelectric constants as unction o D Bias or (1), (11) and (111) oriented Pb(Zr.2 Ti.8 )O 3 ilms...15 Figure X-ray spectrum and -scans or (1) (top), (11) (middle) and (111) (bottom) oriented.67pb(mg 1/3 Nb 2/3 )O 3 -.PbTiO 3 ilms Figure Measured eective longitudinal piezoelectric constants as unction o D Bias or (1), (11) and (111) oriented.67pb(mg 1/3 Nb 2/3 )O 3 -.PbTiO 3 ilms e 31, ix

13 Fig (a) topography, (b) out-o -plane piezoelectric orce microscopy (PFM) or illustration o the domain structure o a tetragonal (1) ilm: (PZT 2/8, 5 µm 5 µm, the bright matrix areas are c domains while the orthogonal gray strips are a domains which have zero longitudinal piezoelectric response); (c) topography, (d) in-plane PFM or illustration o the domain structure o a tetragonal (11) ilm (PZT 52/48, 1 µm 1 µm, the areas with grey contrast are c domains which have small piezoelectric shear deormation; while the strips with white and dark contrasts are a domains which have large shear piezoelectric deormation. The opposite contrasts in a domains indicate the opposite signs o the in-plane polarizations) Fig (a)-(c) X-ray result or (11) PZT 52/48 ilm showing a tetragonal distorted structure Fig. 3.3 (a) ross-sectional TEM image (twin traces are // <111>); (b) Plain-view TEM image (dark-ield image near <11> zone axis.; the twin traces are // <111>) or (11) oriented PZT 52/48 ilm. (c) Schematics o 9 o domain pattern in a (11) ilm.125 Fig. 3.4 Piezoelectric Force Microscopy (PFM) results o (11) PZT 52/48 ilm (6µm 6µm), Top: topography, center: out-o plane PFM, bottom: in-plane PFM: (a) Original; (b) Ater writing 1V in a 3µm 3µm area; (c) Poling back using +1V in the same 3µm 3µm area.126 Fig. 3.5 (a) Polarization hysteresis (b) Longitudinal strain measurement under A ield Subcoercive polarization measurements or (1), (11) and (111) oriented PZT 52/48 ilms.131 Fig. 3.6 Schematics o a ield-induced 9 o domain movement in the (11) PZT 52/48 ilm x

14 LIST OF TABLES Table 2.1 alculated orientation preactors or (1), (11) and (111) oriented ik 2 single-domain tetragonal ilms. = ( )( + 44) ( ), = ( )[( ) * 2 ( ) 2( + ) ] Table 2.2. alculated orientation preactors ik or epitaxial single-domain rhombohedral ilms with pseudo cubic orientations o (1), (11) and (111). = ( )( ) [ 2 ( ) 14 = ( )( + 2 ) [2 + 2 ( ] )]..4 Table 2.3. The longitudinal converse piezoelectric coeicients or dierent orientations o constrained tetragonal ilms d and ree standing ilms d nn nn. d ij are the piezoelectric coeicients o single crystals in pseudo cubic coordinates. (11) (11) (11) (11) = + 22, = (111) (111) (111) (111) (111) = = 1 6 ( ) ( ) 44 (111) (111) (111) = + + = 1 ( )( + ) Table 2.4. The converse longitudinal piezoelectric coeicients or dierent orientations o constrained rhombohedral ilms ( d ) and bulk crystals ( d nn nn ). d ij are the piezoelectric coeicients o single-domain rhombohedral single crystals deined in rhombohedral m coordinates. d are the pseudo-tetragonal piezoelectric coeicients o a (1) oriented ij multi-domain rhombohedral single crystal. [15] q 1 = (1) (1) 2 22, 32 q 2 = (1) (1) 2, k 1 = (11) (11) +, k 2 = (11) (11) Table 2.5. Eect o the ilm-substrate interace in-plane stresses on the piezoelectric behavior o a clamped erroelectric ilm 58 Table 2.6. The eective piezoelectric coeicients or a clamped erroelectric ilm with an orientation shown in Fig. 2.2, in comparison with those o a single crystal plate with the same orientation...59 Table 2.7. Experiment results or Pb(Zr.2 Ti.8 )O 3 ilms. values are the equilibrium nn ones at zero electric ield. Film thickness is kept at 13 nm or all ilms..13 Table 2.8. Experiment results or.67pb(mg 1/3 Nb 2/3 ) O 3 -.PbTiO 3 ilms. d values nn are the equilibrium ones at zero electric ield. Film thickness is kept at 3.3 µm or all ilms.11 d xi

15 HAPTER 1, INTRODUTION 1.1 Piezoelectricity and erroelectricity Piezoelectricity Piezoelectricity was irst discovered by urie brothers in 188 on the observation o surace charges induced by mechanical deormation o a quartz crystal (direct piezoelectric eect). The converse piezoelectric eect was veriied by urie brothers one year later, ollowing Lippmann s prediction based on the thermodynamic principle o reversible processes. Fig. 1.1 (a) onverse piezoelectricity and (b) direct piezoelectricity. S - mechanical strain, E - electric ield strength, P - electrical polarization, T - Mechanical stress. A piezoelectric crystal may be deined as a crystal in which electricity or electric polarity is produced by mechanical deormation, or as one that becomes deormed when in an electric ield. Piezoelectricity is dierent rom electrostriction on that the sign o an induced electric polarity/mechanical strain becomes reversed as the sign o an applied mechanical stress/electric ield does so. From this point o view, a piezoelectric crystal 1

16 must have certain one-wayness in its internal structure; in other words, it must have a structural bias that determines whether a given region on the surace shall show a positive or negative charge on compression. In the converse eect, the same onewayness determines the sign o the deormation when an electric ield is applied to the crystal. All crystals can be divided into 32 classes or point groups (rom 7 basic crystal systems). The 32 point groups can be urther classiied into (a) 11 point groups having a centre o symmetry and (b) 21 point groups that do not possess a centre o symmetry. A lack o a centre o symmetry is all-important or the presence o piezoelectricity. As shown in Fig. 1.2, the lack o a centre o symmetry means that a net movement o the positive and negative ions with respect to each other as a result o stress produces an electric dipole, whereas, or the centro-symmetric crystals, the centres o charges o dierent polarity will still coincide even ater the stress inducing a deormation. O the 21 non-centrosymmetric point groups, 2 are piezoelectric (one class, although lacking a centre o symmetry, is not piezoelectric because o the combination o other symmetry elements) [1]. The basic equations that describe these two eects in regard to electric and elastic properties are: (1.1) Here D is the dielectric displacement vector, E is the electric ield vector, T is the mechanical stress tensor, S is the mechanical strain tensor, d is the tensor o piezoelectric 2

17 constants, is the tensor o dielectric constants (superscript T means under constant stress, i.e. mechanically ree condition), s is the tensor o mechanical compliance (superscript E means under constant E ield, i.e. short-circuit condition) [2]. T enter o positive charge T enter o negative charge Fig Schematics o a piezoelectric crystal (a) under a tensile stress (b). Pyroelectricity Out o the 2 point groups which show the piezoelectric eect, 1 point groups have only a unique polar axis. In absence o any load, they develop a polarization spontaneously and orm a permanent dipole within the unit cell. Such kinds o crystals are called polar crystals. The spontaneous polarization (P spon ) is deined as the value o dipole moment per unit volume or the value o charge per unit area on the surace perpendicular to the axis o spontaneous polarization. The value o spontaneous polarization depends on the temperature (), this phenomenon is called pyroelectric 3

18 eect and the crystals with this eect are called pyroelectrics [3]. The pyroelectric eect can be described in terms o pyroelectric coeicient p as shown in (1.2) Ferroelectricity A subgroup o the spontaneously polarized pyroelectrics is a very special category o materials known as erroelectrics. By deinition, erroelectrics are polar materials that possess at least two equilibrium orientations o the spontaneous polarization vector in the absence o an external electric ield, and in which the spontaneous polarization vector may be switched between those orientations by an electric ield. Among all the erroelectric materials, the most extensively studied and widely used are the perovskite erroelectrics. A perect perovskite structure has a general ormula o ABO 3, where A represents a divalent or trivalent cation, and B is typically a tetravalent or trivalent cation. The origin o erroelectricity in this amily o materials can be explained using the well-known example o Barium titanate (BaTiO 3 ). As shown in Figure 1.3, the Ba 2+ cations are located at the corners o the unit cell. A dipole moment occurs due to relative displacements o the Ti 4+ and O 2- ions rom their symmetrical positions. A good introduction to the theories o erroelectrics and the Laudau-Ginzburg-Devonshire (LGD) equation can be ound in [6] and will not be repeated here. The interrelationship o piezoelectrics and subgroups can be seen in Fig All erroelectrics are piezoelectrics, but not all piezoelectrics are erroelectrics. Ferroelectric materials with high piezoelectric constants are widely used in transducer applications. 4

19 The direct eect is suitable or sensor applications, and the inverse eect can be exploited or actuator, as shown in Fig Depending on the symmetry o the crystal, Eq. (1.1) may have a simple or complex ull expression. For example, or a tetragonal distorted erroelectric crystal, there are only three independent piezoelectric constants, d, d 31, and d 15. Fig 1.6 illustrates these piezoelectric constants in the converse piezoelectric eects. For a single domain erroelectric crystal, these piezoelectric constants are correlated with the selpolarization as calculated rom LGD phenomenological theory in Eq. (1.3) [6]. Where Q ij and ij are the electrostrictive and dielectric constants o the crystal, P s is the spontaneous polarization, and is the vacuum dielectric constant. d =2 Q 11 P d 31 =d 32 = 2 Q 12 P s s d 15 =d 24 = 11 Q 44 Ps (1.3) From Eq. (1.3), it can be seen that any process involving a change in polarization or dielectric constant (or example, movement o erroelectric domain walls, poling o erroelectric ceramics, phase transition occurred in erroelectrics which varies the polarization, etc) may aect the magnitude o piezoelectric constants in erroelectric crystals. In act, it is where the piezoelectric properties o erroelectrics dier rom those o non-erroelectric piezoelectric crystals (like quartz). The ollowing sections o this chapter will discuss those extrinsic eects on piezoelectric properties o erroelectric materials. Due to the equivalence between direct and converse piezoelectric eects, the discussions o piezoelectric eects in bulk materials are limited to converse piezoelectric 5

20 eects (section 1.2 and 1.3), while or thin ilm erroelectric materials, this equivalence will be broken apparently, and we will discuss briely in this chapter on both direct and converse piezoelectric eects in erroelectric thin ilms (section 1.3). 6

21 Ba 2+ Ba 2+ Ti 4+ O 2- Ti 4+ O 2- Polarization Voltage 1 Fig. 1.3 (a) Schematic illustration o the unit cell o perovskite lead titanate erroelectric crystals. Figure adapted rom [4]. (b) Switchable polar states under electric ield enable applications or memory devices. 7

22 Figure 1.4. Interrelationship o piezoelectrics and subgroups on the basis o symmetry. Figure adapted rom [1]. 8

23 Figure 1.5. Ferroelectric materials used as actuator (top) and sensor (bottom). Figures adapted rom [5]. 9

24 E P 2 Strain induced by E 3 = 1 3 d 2 = d = d E E E Strain induced by E 2, E 1 : = d E 5 = d15e1 3 3 E 2 P 2 P 2 E 1 Fig Equations o converse piezoelectric eects in a tetragonal erroelectric crystal. P is the sel-polarization o the crystal aligned along the tetragonal c-axis (axis 3). 1

25 1.2 Intrinsic v.s. extrinsic piezoelectric response Intrinsic piezoelectric response reers to the piezoelectric deormation o the lattice o a single-domain crystal, which can be characterized by tensors o piezoelectric constants. In this case, the piezoelectric constants described in Eq. 1.3 may be called intrinsic piezoelectric constants, which describe piezoelectric eect as an electrostriction biased by the polarization (linearized electrostriction) [6]. The extrinsic piezoelectric response, on the other hand, are rom extrinsic sources o displacement under the electric ield, which can be movements o domain walls, phase boundaries, or even deects like grain boundaries or dislocations. Intrinsic piezoelectric response -Orientation dependence v.s. engineered domain coniguration The expression or a piezoelectric strain o a erroelectric crystal is presented by Eq. (1.4) in pseudo cubic coordinates. Where d kij are piezoelectric constants o a bulk crystal, E k are components o the applied electrical ield E, E k =El k, and l k are the direction cosines (the summation is taken over repetitive suixes). ij =d kij E k =d kij l k E (i, j, k=1, 2, 3) (1. 4) The length change along a normal n is determined by a normal strain ij n i n j and it can be characterized by a piezoelectric constant: d = d l n n (i, j, k=1, 2, 3) (1.5) ln kij k i j 11

26 It is reduced to the longitudinal piezoelectric constant d (n)= d = d n n n o a single crystal i the ield is applied along the same normal n (Fig. 1.8). The orientation dependence o intrinsic piezoelectric properties or PZT bulk materials near the morphotropic phase boundary (MPB-PZT) was phenomenologically calculated in [7], [8]. As shown in Fig. 1.7, it was ound that or tetragonal PZT materials, the eective piezoelectric constant d has the maximum value in the spontaneous polarization direction [1]. However, or rhombohedral PZT materials, the maximum values o d can be obtained in the direction 56.7 canted rom the polarization direction [111], which is very close to perovskite [1] directions. nn kij k i j 12

27 Fig. 1.7 Eective piezoelectric constants d o PZT with various compositions. 13

28 n=[n 1 n 2 n 3 ] E x 3 =[1] u u n x 2 =[1] rystal beore deormation rystal ater deormation x 1 =[1] ~ ~ ~ E u E nn n d = d nn = = = d kij n n n k i j Fig. 1.8 Orientation dependence o longitudinal piezoelectric constants or single crystals 14

29 Method o domain engineering, which aims at enhancing the piezoelectric properties o erroelectric materials by applying the electric ield in a non-polar direction across the crystal, has been investigated intensively or the past decade. [9]-[12] The key points o the engineering domain conigurations are: [1] (1) There are crystallographically equivalent domains with inclined polarization to the applied electric ield. (2) Domain wall motion is inhibited, thereore the piezoelectric response is intrinsic. The resulting -E curve is hysteresis-ree. (3) Enhanced d along ield direction than along polar direction. (4) The composition or temperature o the erroelectric crystal is close to phase transition. S. Wada et al. [1] investigated piezoelectric properties o barium titanate single crystals at room temperature as a unction o crystallographic orientation, and they ound that a (111) oriented tetragonal BaTiO 3 single crystal, which is in an engineered domain coniguration, had a higher longitudinal piezoelectric constant d (=23pm/V) than that o a (1) oriented one ( d =125pm/V) (Fig. 1.9 a). This has been explained by D. Damjanovic et al. [12], [13] by the contribution to d rom piezoelectric shear eect. Our recent calculation on the orientation dependence o d or tetragonal BaTiO 3 single crystal showed good agreement with the aorementioned experiment results (Fig 1.9 b). [14]. This veriies that the piezoelectric response o a domain engineered tetragonal erroelectric crystals is intrinsic. 15

30 [111] E-ield d [111] ~23pm/V [1] [1] d [1] ~125pm/V [1] (a) 15 d [1] ~16pm/V [ 1] d (pm/v) [ ] 1 d [111] ~224pm/V [ 1 ] (b) Fig. 1.9 (a) [111] poled 4mm crystal BaTiO 3 has measured d (at room temperature) much larger than d o [1] poled crystal; [9] (b) alculated orientation dependence o d (at room temperature) or 4mm crystal BaTiO 3. 16

31 For relaxor based erroelectrics on the rhombohedral side o their morphotropic phase boundaries, such as xpb(mg 1/3 Nb 2/3 )O 3 -(1-x)PbTiO 3 (x<.35), and xpb(zn 1/3 Nb 2/3 ) O 3 -(1-x)PbTiO 3 (x<.9), giant piezoelectric constants (>25 pm/v) can be achieved by poling the crystal in a pseudo-cubic (1) direction (Fig. 1.1). [9] Recently, ao et al. studied both multi-domain (<1> oriented crystal) and single domain (<111> oriented crystal) properties o.67pb (Mg 1/3 Nb 2/3 )O 3 -.PbTiO 3 single crystals. [15]-[17] They attributed the origin o the superior electromechanical properties in the (1) direction to the large d 15 o the single-domain properties, as they reproduced the large piezoelectric constants in multi-domain (1) crystals rom single domain properties in the rotated coordinated system (Fig 1.11). [17] Further theoretical study [18], [19] showed that the low ield piezoelectric constants or the engineered coniguration are very close to those obtained or the corresponding single domain state and the domain wall inluence is not signiicant. This result proves that the piezoelectric response o a domain engineered rhombohedral erroelectric crystal is intrinsic. From the above two examples, it proves to be possible to estimate the eective piezoelectric constants o a multi-domain single crystal with an arbitrary orientation by knowing those calculated in a rotated coordinated system o a single crystal, providing that the domains constitute an engineered domain coniguration, i.e, all domains are equivalent under the applied electric ield. 17

32 (a) a a (b) (b) Fig Ultra-high piezoelectric behavior discovered in PZN-8%PT single crystal in a engineered domain coniguration: (a) Schematic diagram o domain conigurations in <1)> oriented rhombohedral crystals under bias (step A-piezoelectricity, step B- induced phase transition). (b) Strain vs E-ield behavior or <1> oriented PZN-8%PT crystal. Maximum ield is limited by voltage limit o the apparatus. 18

33 (a) Measured d =282 p/n along (1) (b) Fig (a) Orientation dependence o eective piezoelectric constant d o PMN %PT single crystal with single domain. (b) ross section plot o (a) in [1 1 1] [1 1 2] plane. Figures adapted rom [17]. 19

34 Extrinsic piezoelectric response 18 erroelectric domains As a consequence o minimization o the depolarization energy [2], erroelectric 18 o domains are ound in erroelectric crystals with uncompensated surace charges or in unpoled erroelectric ceramics. On the other hand, elastic non-18 o domains are ormed as a consequence o the release o energy o internal stresses. [21] For example, in a tetragonal erroelectric crystal like BaTiO 3, it has six domain variants when it transorms rom a high temperature cubic phase (Fig 1.12). Ba Ti O (a) High temperature ubic phase Low temperature tetragonal phase (b) Fig 1.12 (a) ubic to tetragonal phase transormation in BaTiO 3 crystal; (b) The tetragonal erroelectric phase has six dierent domains. 2

35 In Fig. 1.13, it is illustrated how the surace displacement o an partially poled erroelectric crystal is aected by the change o raction o 18 o domains. It can be seen that when there are equivalent ractions o the two sets o 18 o domains (=.5), which deorm in opposite directions under an applied electric ield, the apparent d is zero. When the volume raction o 18 o domains decrease by the movement o domain walls ( goes to or 1), the apparent d will increase and approach the intrinsic d value. A detailed investigation on the eect o 18 o degree domains on piezoelectric properties o erroelectric materials (both bulk crystal and ilm) will be ound in [22]. 18 degree twins E Fig. 1.13: Eect o 18 o degree domains on longitudinal d piezoelectric response. is the raction o domains with polarization aligned along the applied electric ield ( + domain), 1- is the raction o domains with polarization aligned opposite to the applied electric ield ( - domain). The level o the dashed line is the surace o the crystal when there is no electric ield applied. 21

36 non-18 o elastic domains Although the switching o 18 o domains will aect the apparent piezoelectric constants, it will not substantially increase the piezoelectric strain since the lattice parameters o opposite 18 o domains are the same. On the other hand, or non-18 o elastic domains, switching between dierent domain variants will bring large extrinsic piezoelectric strains as shown in Fig In the case o a tetragonal erroelectric crystal, the extrinsic piezoelectric strain can be as high as the tetragonality i the 9 o switching between in-plane a-domains and out-o-plane c- domains takes place. This type o extrinsic piezoelectric response dominates in sot-erroelectrics like MPB-PZT ceramics. 9 degree twins 9 o Twin Boundary intrinsic E I II III Extrinsic > 7% Fig. 1.14: Eect o 9 o degree domain wall movement on longitudinal piezoelectric response. It can be seen that the extrinsic contribution dominates the total strain. 22

37 Electric ield induced phase transitions/polarization rotations Field-induced phase transitions will signiicantly change the piezoelectric properties o erroelectric materials. One such example is illustrated in Fig 1.1. A rhombohedral 3m erroelectric crystal with its engineered domain coniguration (poling along pseudo-cubic <1>) will ultimately transorm into a tetragonal 4mm symmetry under an increasing electric ield. The piezoelectric strain curve is then characterized by three stages, the initial stage o engineered domain coniguration under small electric ield, the transition stage, and the inal stage o induced tetragonal single domain coniguration. While the irst and third stage o the piezoelectric response are well characterized by intrinsic piezoelectric constants, the transition stage is not. M. Iwada and Y. Ishibashi s work based on LGD type phenomenological theory has shown the enhancement o d or engineered domain conigurations in both tetragonal and rhombohedral phases, as well as the critical electric ields at which the engineered domains lose their stabilities and phase transitions take place. [23]-[24] However, in order to obtain analytical results, the LGD equation used had to be truncated at 4 th order terms, which had undermined the applicability o these results. Another important piece o research work in this theoretical ramework was by M. Budimir and D. Damjanovic et al. [12] They studied the piezoelectric anisotropy phase transition relations in perovskite single crystals. It was shown that a presence o the erroelectric erroelectric phase transitions in BaTiO 3 leads to enhanced eective d along non-polar directions, while or PbTiO 3 which does not exhibit erroelectric erroelectric phase transitions, eective d has its maximum along the polar axis at all temperatures. This work illustrated the correlation between a large eective d in a non- 23

38 polar direction and a sotened lattice in the shear direction. When the erroelectric lattice is in the vicinity o a phase transition which rotates the polar vector, the shear direction corresponding to this rotation will become sot, which means a larger compliance or easier shear deormation, no matter this deormation is caused by a temperature driven process or a piezoelectric orce. In this case, the shear piezoelectric coeicient will be large enough to vary the orientation dependence o d (BaTiO 3 ). On the other hand, i the erroelectric lattice is ar rom a phase transition, the shear directions are all sti, which means easy piezoelectric deormation is not possible and the shear piezoelectric coeicients are low. For PbTiO 3 material, it is ar rom its morphotropic phase boundary and is a very stable tetragonal structure. onsequently, it has very low shear piezoelectric coeicients and the longitudinal piezoelectric coeicients will dominate the eective d. Recent study on lead zirconate titanate (Pb[Zr 1-x Ti x ]O 3, abbreviated as PZT) bulk materials with a composition near its morphotropic phase boundary (abbreviated as MPB, x~.5, which has a wide variety o applications, such as sensors, actuators and dielectrics, due to its superior electromechanical properties) revealed the presence o a monoclinic phase on the morphotropic phase boundary, which separates the tetragonal phase region on Ti-rich compositions and the rhombohedral phase region on Zr-rich compositions [25]. To account or the unusually high electromechanical constants associated with the morphotropic phase boundary, it was proposed that the monoclinic phase has a polarization vector aligned in a direction between those o the tetragonal phase (pseudocubic [1] direction) and the rhombohedral phase (pseudo-cubic [111] direction), and the polarization vector is apt to change its orientation under poling, which can lead to 24

39 large piezoelectric responses. According to the general piezoelectric anisotropy phase transition relations demonstrated in Re. [12], this special monoclinic phase capable o rotating its polarization vector may be understood as an intermediate phase between tetragonal and rhombohedral phases, which has a very sot shear modulus and consequently, large shear piezoelectric coeicients. Fu and R. ohen et al. proposed a polarization rotation mechanism [26] by irst principle calculations to explain the ultrahigh electromechanical response discovered on Pb(Zn 1/3 Nb 2/3 )O 3 -PbTiO 3 (PZN-PT) single crystals [9]. In their calculations, a pevoskite BaTiO 3 single crystal was used as the model system to illustrate the rotation o the polarization vector under a canted poling ield (Fig 1.15). The key points in this model are: (1) The ground state o the crystal is rhombohedral phase. (2) Strain level is high in tetragonal phase; (3) (4) Large elastic compliance (especially s 44 ); Large eective charges. They suggested that it might be a general mechanism or other perovskite erroelectrics to have enhanced piezoelectric response by applying a electric ield in a non-polar axis. Fu s theoretical work and the experiment work on MPB-PZT were supported by the ab initio calculation on the poling paths in PbZr 1-x Ti x O 3 single crystals. [27] Evolutions o ield-induced low energy phases in domain engineered MPB-PZT erroelectrics were revealed by L. Bellaiche et al. in Re. [27], which cleared showed that monoclinic phases act as an intermediate poling state between tetragonal and rhombohedral phases. Further theoretical investigations were carried out by R. ohen et 25

40 [28], [29] al. on elastic properties o MPB-PZT materials. It was ound that the sotness o MPB-PZT single crystals with respect to polarization rotation (large shear compliances) is directly responsible or its high electromechanical coupling. [29] Similar conclusions also made or relaxor erroelectric materials near their MPBs. [28] [1] [111] (a) Ground state: Rhombohedral (b) Fig First principle calculations or the longitudinal piezoelectric response o BaTiO 3. (a) Possible poling paths indexed by letters or a BaTiO 3 single crystal in a rhombohedral ground state, the electric ield is applied along the pseudo-cubic [1] direction {e} while the polarization is aligned along [111] {a}. (b) Piezoelectric strain curve associated with the evolution o poling states with lowest energy. 26

41 1.3 Bulk erroelectrics v.s. thin ilm erroelectric ilms In bulk erroelectrics (single crystal or ceramics), the direct and converse piezoelectric eects have the same characteristic constants, which are equivalent thermodynamic potential derivatives. [3] It is convenient or people to characterize the piezoelectric properties o erroelectric materials based on one o the two eects and use the obtained coeicients in applications based on the other eect. Recently, there is an increasing number o research works on how to integrate erroelectric ilms into MEMS devices, taking advantage o their excellent piezoelectric properties. [31]-[34] There are two major types o piezoelectric-based MEMS devices, microsensors and microactuators. Piezoelectric sensors usually utilize the direct piezoelectric eect o erroelectric ilms, namely, to detect an environmental stress or strain by monitoring the piezoelectric charges (Fig 1.16 a). On the other hand, piezoelectric actuators usually utilize the converse piezoelectric eect, namely, to produce strains or stresses under an electric ield (Fig 1.16 b, c). Although it is well known that the direct and converse piezoelectric eects in erroelectric ilms can have dierent characteristic constants, [35]-[4] there is no reported work on why and how the piezoelectric properties o a ilm dier rom those o a single crystal, except or the partial case o a (1) oriented tetragonal ilm. [35]-[4] The primary goal o this section is to show how clamping o a substrate has broken the equivalence between direct and converse piezoelectric constants in an erroelectric thin ilm. 27

42 In various applications o erroelectric-based transducers, as well as in characterization o piezoelectric properties o erroelectric ilms, either an electric stimulus or a electric response is required. On the other hand, since the substrate is usually a piezo-inactive insulator or semiconductor, the ilm has to be sandwiched by top and bottom electrode layers in order to apply an electric ield to the ilm or have an electric charge output rom the ilm. As compared with a erroelectric single crystal, which can be considered as a closed system both mechanically and electrically, a erroelectric ilm is an electrically closed but mechanically open system to the substrate. This is the reason why the characteristic constants o direct and converse piezoelectric properties are dierent or a erroelectric ilm. To illustrate how the mechanical interaction between ilm and substrate has changed the eective piezoelectric constants, we will start with a discussion on the longitudinal piezoelectric constants. onsider a piezoelectric single crystal plate with a normal n placed in an electric ield E 3, and at the same time subjected to a mechanical stress ( 3 here is the normal direction o the plate n). Then there are a longitudinal electric polarization p 3 and a strain induced in the crystal. The change o ree energy due to small variations o electric ield E 3 and stress can be written in a orm o an exact dierential: F (crystal) =-p 3 E (1.6) For a thermodynamically reversible process, we can write the Maxwell equation: 28

43 p * * ( d. Where d is the eective longitudinal piezoelectric 3 ) E = ( ) = 3 E3 constant, which has a unit o p/n in the direct eect (change in polarization p 3 per applied stress ) and a unit o pm/v in the converse eect (change in ilm strain * per applied electric ield E 3 ). d can be expressed by Eq. (1.7), [41] where d kij are components o the piezoelectric tensor, which are deined in the artesian coordinates o the crystal. [3] n n, and n j are components o the normal vector n (the k, i summations are taken over repetitive suixes in this article, i not otherwise noted). d = d n n n * kij k i j (1.7) onsider a piezoelectric ilm with normal n under an electric ield E 3 and at the same time subjected to a mechanical stress. Due to the mechanical clamping rom the substrate, the in-plane elastic energy contributes to the total ree energy, which can be expressed by Eq. (1.8). F (ilm) =-p 3 E (1.8) Where 11, 12 and 22 are in-plane stresses due to clamping o substrate, 11, 22, and 12 are the elastic strains, and hence are the in-plane elastic energy. All these quantities aorementioned ater Eq. (1.8) are unctions o applied stress and electric ield E 3. In this case, p 3 and are not partial derivatives o the ree energy. Thereore, the Maxwell equation p ( 3 )E 3 = valid or the case o a single * ( ) d E = 3 crystal plate does not hold true here, which explains why the eective direct 29

44 piezoelectric constant D = p ( 3 ) E3 3 (subscript denotes ilm ) is dierent rom the eective converse constant = ( ) E3. Here subscript D and denote direct and converse piezoelectric eects, respectively. In some o the characterization methods or converse longitudinal piezoelectric constant, such as piezoelectric orce microscopy [42] and single beam intererometer, [43] the total longitudinal displacement o the ilm/substrate couple is measured or calculation o the ield induced strain. In these cases, a contribution rom substrate deormation is added to the apparent longitudinal piezoelectric constant o the ilm, d ( ) E3 =. [44] h Here = is the eective piezoelectric strain with h being the total surace t displacement and t the ilm thickness. Taking the ilm/substrate couple as the objective thermodynamic system, the change o ree energy o this system due to piezoelectric eect can be written in a orm o an exact dierential: =-p 3 V 3 +h, where is the ree energy per area or the ilm/substrate couple, p 3 is the induced longitudinal polarization, V 3 is the voltage drop across the ilm thickness t (V 3 =E 3 t), h is the displacement along the normal direction, and is the stress applied along the normal o the system. The Maxwell relation gives d P h ( = d, D 3 = ) V = ( ) 3 = ( ) V3 E3,. It should be noted that the non-epitaxial erroelectric ilms like polycrystalline and textured ilms behave dierently in piezoelectric properties. Factors other than clamping o substrate, like grain size and distributions, degrees and orientations o 3

45 texture, etc., will require additional concerns in quantiying the eective piezoelectric coeicients. There have been long discussed in literature, and thereore are not included in this dissertation. Instead, intrinsic and extrinsic piezoelectric eects in epitaxial erroelectric ilms are the main ocus o this dissertation. Film Film substrate E (b) substrate (a) Film substrate E (c) Fig (a) Microsensor based on the direct piezoelectric eect o erroelectric ilms. (b) longitudinal, (c) shear microacuators based on the converse piezoelectric eect o erroelectric ilms. The blue arrows inside the ilms indicate the alignment o the polar vectors in the ilm. 31

46 1.4 onclusion The main points in this chapter are summarized here: (1). Piezoelectricity originates rom lack o centro-symmetry in a crystalline structure. Ferroelectric materials is a sub-group o piezoelectric materials. It is characterized by a switchable remnant polarization. (2). Due to their large piezoelectric coeicients, many erroelectric materials are being used in piezoelectric transducer applications. The converse piezoelectric eects are utilized in piezoelectric actuators, where a output mechanical strain can be produced and controlled by an input electric ield. On the other hand, a piezoelectric sensor will utilize the direct piezoelectric eects to transorm input mechanical stimulus to output electric signals. (2). Intrinsic piezoelectric response reers to the piezoelectric deormation o the lattice o a single-domain crystal, which can be characterized by tensors o piezoelectric constants. Intrinsic piezoelectric constants in erroelectric materials describe piezoelectric eect as an electrostriction biased by the polarization (linearized electrostriction) [6]. (3). The extrinsic piezoelectric response, on the other hand, are rom extrinsic sources o displacement under the electric ield. Processes involving changes in polarization or dielectric constant (or example, movement o erroelectric domain walls, poling o erroelectric ceramics, phase transition occurred in erroelectrics which varies the polarization, etc) will add extrinsic piezoelectric response to the intrinsic one. 32

47 (4) The domain engineering method, which aims at enhancing piezoelectric response by poling the erroelectric material in a non-polar direction, will lead to two possible consequences. One is that the domains are equivalent under the poling ield and the piezoelectric response is intrinsic, the enhanced piezoelectric responses can be evaluated by calculating the eective piezoelectric constants o a single-domain single crystal - d = d kijl k ni n (Eq. 1.5), in the domain- engineered orientation (pseudo cubic <1> or ln j rhombohedral crystals and pseudo cubic <111> or tetragonal crystals). The other possible consequence is a ield-induced phase transition, which may lead to large extrinsic piezoelectric responses. (5) Another important source o extrinsic piezoelectric response comes rom domain wall movement. Both 18 o domain wall and non-18 o domain wall movements can aect the piezoelectric properties. However, while 18 o domain wall movement does not substantially increase the piezoelectric strain, switching between non-18 o domains will bring large extrinsic piezoelectric strains. (6). Due to clamping rom the substrate, a erroelectric ilm has dierent apparent direct and transverse piezoelectric constants.

48 HAPTER 2, INTRINSI PIEZOELETRI RESPONSE IN THIN EPITAXIAL FERROELETRI FILMS 2.1 onverse piezoelectric properties or erroelectric thin ilms It is well known that the ield-induced strain o piezoelectric ilms is strongly modiied by their elastic interaction with substrates. The relation between an intrinsic converse piezoresponse o a single crystal or a ree standing ilm and that o a ilm clamped by a substrate is o a great interest or many applications o piezoelectric thin ilms. However, this relation is obtained only or one partial case: a (1) oriented tetragonal ilm clamped by a rigid or a thick substrate under electric ield normal to the ilm. [37] The goal o this section is to calculate the eective piezoelectric coeicients o a substrate-constrained single domain ilm as a unction o its crystallographic orientation and a direction o applied electric ield. This inormation will set up a ramework o reerence or evaluating converse piezoelectric properties o clamped thin ilms growing in arbitrary orientation and help to separate intrinsic properties rom extrinsic eects [45] (domain wall motion, incomplete clamping, substrate bending, etc.). Knowledge o the orientation dependence o piezoelectric coeicients o constrained ilms will allow one to optimize their electromechanical perormance. In the section, the general expressions or characterization o the piezoelectric strain o a clamped ilm with an arbitrary orientation is obtained, and, as typical examples, converse longitudinal piezoelectric coeicients o ilms with tetragonal and rhombohedral crystallographic structure are calculated or three orientations- (1), (11) and (111). We start with an expression or a piezoelectric strain o a ree standing ilm, which is presented by Eq. (2.1) in pseudo cubic coordinates. Where d kij are piezoelectric 34

49 coeicients o a bulk crystal, E k are components o the applied electrical ield E, E k =El k, and l k are the direction cosines (the summation is taken over repetitive suixes). ij =d kij E k =d kij l k E (i, j, k=1, 2, 3) (2. 1) The change o the thickness o a ilm with a normal n is determined by a normal strain ij n i n j and it can be characterized by an eective piezoelectric coeicient: d = d l n n (i, j, k=1, 2, 3). It is reduced to the longitudinal piezoelectric coeicient ln kij k i j d = d n n n o a single crystal or a ree standing ilm i the ield is normal to the nn kij k i j ilm. I the ilm is clamped by a rigid substrate or i a substrate is much thicker than the ilm, then the components o the strain in the ilm plane are equal to zero. The piezoelectric strain is so-called an invariant plane strain: all planes parallel to ilm remain undistorted and its displacement is proportional to the distance o the plane rom the ilmsubstrate interace. The invariant plane strain is expressed by a dyadic tensor = s n, ij i j where s i is a characteristic vector. There is no displacement normal to the plane containing the vectors s and n. Thereore the piezoelectric strain o a completely clamped ilm is characterized by only two parameters: a normal piezoelectric coeicient dln = sini / E and a tangential piezoelectric coeicient dl = si i / E, where is a unit vector along the projection o s on a ilm plane and can be determined by = n (s n)/ s. 35

50 To determine the eective piezoelectric coeicients d ln and dl, it is necessary to ind the vector s. Taking into account that the stress in the ilm is = ( ) and ij ijkl kl kl there is no orce normal to the ( ij n j =), it is possible to show that: s i = n t kt d l E (2.2) ik where kt is the tensor o elastic moduli o the ilm, 1 = ik ( ik ), a Green tensor = ik n m mikp n p. [46]-[48] Then d ln and dl can be expressed as ollows: d ln = kt d ik n t n i l (2.3a) d = kt d l ik n t i l (2.3b) Equations (2.3a) and (2.3b) allow one to compute the eective piezoelectric coeicients or arbitrary orientations o a ilm and the electric ield, i the intrinsic piezoelectric coeicient and the elastic moduli are known. As two typical examples, we calculate the intrinsic longitudinal piezoelectric coeicients or tetragonal and rhombohedral ilms in an usual set up with bottom and top electrodes. In this case, the electrostatic ield is directed along the ilm normal n. The eective longitudinal piezoelectric coeicient can be written as: [41], d nn = ik d = kt d n t n i n (2.4) where kt andd are the tensor o elastic moduli and piezoelectric coeicients o the ilm; = ik ( 1 ik ), a Green tensor ik = n m mikp n p ; [46]-[48] n t and n i are the direction cosines o the ilm normal n. The electric ield E is applied along n. 36

51 x 3 =[1] x 2 =[1] E//n x 1 =[1] n Film P//x 3 s n s s Substrate Figure A single domain tetragonal erroelectric ilm with normal n under electric ield E (E//n). x 1, x 2, x 3 are the crystalline coordinates. P is the polarization vector, P is parallel to x 3. The deormation is exaggerated here or illustration. In a tetragonal phase with 4mm symmetry, P 1 =P 2 =, P 3 =P. There are 7 nonzero piezoelectric coeicients, out o which only 3 are independent. d 311 =d 322 =d 31, d 3 =d, d 223 =d 232 (= 2 1 d24 )=d 113 =d 131 = 2 1 d15. The coordinates o the tetragonal phase are pseudo-cubic coordinates. For a rhombohedral PZT phase, there are 11 non-zero piezoelectric coeicients, out o which only 4 are independent due to the 3m symmetry: d 113 = d 131 = 2 1 d15, d 112 =d 121 = 2 1 d16 =-d 22, d 211 =-d 22, d 222 =d 22, d 223 =d 232 = 2 1 d24 = 2 1 d15, 37

52 d 311 =d 322 =d 31, d 3 =d. It should be noted that, the above piezoelectric coeicients are reerred to a rhombohedral coordinate system, which is deined as ollows: [49] Pseudo-cubic axis [1 1 ] rhombohedral axis x 1 [1] The tensors Pseudo-cubic axis [11 2 ] rhombohedral axis x 2 [1] Pseudo-cubic axis [111] rhombohedral axis x 3 [1] are shown in Table 2.1 and Table 2.2 or three typical ik orientations (1), (11) and (111) in tetragonal and rhombohedral ilms, respectively. The longitudinal piezo-electric coeicients d and nn presented in Table 2.3 and Table 2.4 or comparison. d o a ree standing ilm are nn 38

53 39 Table 2.1 alculated orientation preactors ik or (1), (11) and (111) oriented single-domain tetragonal ilms ) ( ) )( ( =, * ) 2 )[( ( = ] ) 2( ) 2 ( Orientations n ik =( ik ) -1 =(n m mikp n p ) -1 (1) ( 2 / 1 )(11) ) ( ) ( ) ( ) ( ) ( ( 3 / 1 )(111) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) 2 ( ) ( 3

54 Table 2.2. alculated orientation preactors ik or epitaxial single-domain rhombohedral ilms with pseudo cubic orientations o (1), (11) and (111). = = ( )( 2 44 ) [ 2 ( ) 14 ] ( )( 2 44 ) [ ( )] Orientations n (pseudo-cubic) =( ) -1 =(n m mikp n p ) -1 ik ik (1) ( 1 / 2) (11) [ 3( 44 2( ) + ) ] [ 2( ) 14 ] 3[ ] ( 1 / 3) (111) [2 3( ) 2( )] 44 3[2 14 3[ ( )] ]

55 Table 2.3. The longitudinal converse piezoelectric coeicients or dierent orientations o constrained tetragonal ilms coeicients o single crystals in pseudo cubic coordinates. (11) (11) (11) = +, = + = (11) (111) (111) (111) d and ree standing ilms d nn nn. d ij are the piezoelectric 23 (111) = ( + ) ( + ) (111) (111) (111) (111) = + + = ( )( + ) Orientation n Bulk d Film d nn nn (1) d d + 213d 31 / 1 / 2 (11) ( d + d + 15 ) /(2 2) + ) d + ( + ) d + [ ( + ) + 2 ] }/ d {( d31 1 / 3 (111) d + 2d + 2 ) /(3 3) ( 31 d15 {( 13+ ) d + ( + 2) 44d 15+ [ ( ) ] d31}/(3 3) 41

56 Table 2.4. The converse longitudinal piezoelectric coeicients or dierent orientations o constrained rhombohedral ilms ( d ) and bulk crystals ( d nn ). d ij are the piezoelectric nn coeicients o single-domain rhombohedral single crystals deined in rhombohedral coordinates. m d are the pseudo-tetragonal piezoelectric coeicients o a (1) oriented ij multi-domain rhombohedral single crystal. [15] q 1 = (1) (1) 2, q 2 = (1) (1) 2, k 1 = (11) (11) +, k 2 = (11) (11) Orientation n Bulk d Film d nn nn (1) 1 [2( d d d + d 31 ) ) 1 (1) {( 3 3 q q ] d q ] d 31 + [( 13 q 1 + [2( q ) d ) q + [ ( ) q + q ] d } ) 2 m m 13 m d d m 31 ( 2) + 1 / 2 (11) 1 [ 2( d15 + d d + 2 2d ) ) 1 {( [( k ] d k [( ) k 1 k ) d [ 2 + k ] d } 15 2( 14 ) k ) k k ] d / 3 (111) 2 d + 13 d 31 d 42

57 2.2 Direct piezoelectric properties or erroelectric thin ilms The orientation dependence o the direct piezoelectric coeicients D d,, D d 31, and D e, 31 ( D denotes direct piezoelectric eect, denotes ilm ) are generally ormulated in this chapter, which can help to predict and optimize the perormance o piezoelectric MEMS sensor devices based on erroelectric thin ilms. Numerical results are obtained and discussed or Pb(Zr x Ti 1-x )O 3 thin ilms grow on Si substrates with various compositions and structures. Recently, there is an increasing number o research works on erroelectric ilms as sensor components in micro-electric-mechanical system (MEMS), which aims at realizing direct piezoelectric unctions by capacitive sensing in a reduced length scale. [32], [34], [5], [51] In a general coniguration or a piezoelectrically driven MEMS sensor, a erroelectric thin ilm is grown on top o a thick substrate, a stress or a strain ield rom the surrounding environment is applied to the substrate and sensed by the erroelectric ilm through direct piezoelectric eect (via the lateral constraint imposed by the substrate). The primary goal o this section is to calculate the eective direct piezoelectric coeicients D d,, D e, 31 and D 31 o a substrate-constrained single domain erroelectric ilm as a unction o its crystallographic structure and orientation. This inormation will provide a ramework o reerence or evaluating direct piezoelectric properties o clamped erroelectric ilms. The widely used piezoelectric coeicients characterizing piezoelectric sensing processes are the longitudinal D coeicient and the transverse D 31 and D e 31, coeicients. A piezoelectric sensor is said to be in a longitudinal sensing mode when a 43

58 longitudinal orce is applied along the normal o the device and a charge is generated on the surace o the erroelectric ilm and then is collected and analyzed. On the other hand, a transverse sensing mode is when in-plane stresses or strains are applied to the device to induce the charge. For a (1) oriented tetragonal ilm, D d,, D 31 and D e 31, coeicients are expressed in Eq (2.5): [36],[37], D E E, sub E, E, E, d = Q / = d + 2( s s ) d /( s + s ) (2.5.1),,D E sub sub E, sub E, sub E, E, d = Q /( + ) = d ( s + s ) /( s + s ) (2.5.2) , D E sub sub E, E, e = Q /( + ) = d /( s + s ) (2.5.3) Where E Q 3 is the piezoelectrically induced charge ( E denotes at constant electric ield ), is the longitudinal stress rom the surrounding environment applied to the device, d ij (i, j=1~3) are the piezoelectric coeicients o the erroelectric single crystal. sub sub 11, 22 and, are the in-plane stresses and strains applied to the sub 11 sub 22 substrate in transverse sensing processes. E sub s, ij ( sub denotes substrate ) and E s, ij (i, j=1~3) are the elastic compliances under constant electric ield or substrate and ilm material, respectively. As can be seen rom Eq. (2.5), the direct piezoelectric coeicients or a erroelectric ilm are unctions not only o its own electromechanical properties, but also o the substrate s mechanical properties. D e, 31 is always larger than the single crystal piezoelectric coeicient e 31, while D and D 31 can be smaller or larger than single crystal coeicients depending on the mechanical properties o the substrate 44

59 with respect to the ilm material. However, to quantitatively evaluate the direct piezoelectric eect, Eq. (2.5) is good only or [1] tetragonal or pseudo-tetragonal epitaxial ilms. Thereore, in order to ind optimal orientations and structures or piezoelectric sensing perormance, it is important to know how much the eective direct piezoelectric coeicients will change as we vary the orientation and/or structure o the ilm. We start with a general expression or the in-plane stress components o an epitaxial ilm with a normal n in the longitudinal sensing mode, where a longitudinal stress NN is applied to the device (the subscript N denotes the longitudinal direction n, it is NOT a suix number throughout this article). The strain tensors induced due to are = NN pq E s, pqij sub *, sub n n and pq = spqij NN ni n j (p, q, i, j=1~3; the summations are NN i j taken over repetitive suixes i not otherwise noted), or the ilm and substrate, respectively. sub pqij sub s *, = s egh a peaq aiga jh is the eective elastic compliance tensor o the substrate deined in the artesian coordinates o the ilm and sub s egh is the elastic compliance tensor o the substrate deined in its own artesian coordinates ( x a lm is the direction cosines between x l and sub x m (l, m=1-3). [49] sub k, k=1-3), The dierence between the two strain tensors is the sel strain pq = pq sub pq = ( s s ) n n. The in-plane stresses in the ilm are expressed in Eq. (2.6): [46]-[48] E, pqij *, sub pqij NN i j, = - G E n ) = E, E, E, *, sub E, ( lmpq lm n n ) ( spqij spqij) NN ni n pq j (2.6) lm lmpq ( pq 45

60 , Where G E lmpq ( n) is the planar elastic modulus or a ully clamped ilm with a, normal n at constant electric ield. G E ( n) lmpq = E, lmpq E, lm E, pq n n. [43]-[45] E, ( E, E,,, and lmpq lm E, pq ) is the tensor o elastic moduli o the ilm at constant electric ield, = ( 1 ), a Green tensor = n E, n. [43]-[45] Thereore, the piezoelectric E charge induced by the stress NN and lm are: Q N = d NN NN + dklm lmnk = NN[ dnn + d G *, sub E, ( n) ( s s ) n n n. The eective direct piezoelectric coeicient E, klm lmpq pqij pqij k i j ] D d, is shown in Eq. (2.7): D E E, *, sub E, (3) (3) = QN / NN = dnn + dklmg lmpq ( n) ( spqij s pqij ) ni n j E, lmpq = [ d d G ( n) kij + klm s s n n n (2.7) *, sub E, (3) (3) (3) ( pqij pqij )] k i j Where d NN = kijnk ni n j d is the intrinsic longitudinal piezoelectric coeicient or the ilm. [41] The ilm s in-plane axies are labeled as n (1), n (2) and normal as n (3) (previous as n) in the case o transverse piezoelectric eects (Fig. 2.2). In a D 31 sensing mode, a stress ' ield UV is applied to the substrate laterally (the prime in the superscript denotes that the stress components are expressed in the coordinate system o the ilm where the subscript U, V stand or direction o n (U) and n (V), U, V=1,2). In the crystalline coordinate system deined by symmetry o the ilm structure, this stress ield can be written as 46

61 = ij ' UV n n ( U) ( V) i j (i, j=1-3). [49] The sel strain pq = sub s *, pqij *, sub ' ( U) ( V) = s n n and the ij pqij UV i j in-plane stresses in the ilm are shown in Eq. (2.8): [46]-[48], (3) = - G E n ) = E, E, (3) (3) E, *, sub ' ( U) ( V) ( n n ) s n n (2.8) pq lm lmpq ( pq lmpq lm pqij UV i j x 3 =[1] n (3) //n (3,sub) x 2 =[1] x 1 =[1] n (2) //n (2,sub) Film n (1) //n (1,sub) P//x 3 Substrate x sub 1 =[1] x sub 3 =[1] x sub 2 =[1] Figure 2.2. The epitaxial relation between a erroelectric ilm and substrate with arbitrary orientations. x i and x i,sub are the artesian coordinates o the crystalline structures o the ilm and substrate, respectively. 47

62 Thereore, the longitudinal piezoelectric charge induced by lm is: E Q 3 = d klm n (3) lm k E, (3) = d G ( n ) *, sub klm lmpq - pqij s ' UV n n n ( 3) ( U ) ( V ) k i j. Usually, the stress ' UV has two components ' 11 ' and, which are applied along n (1) and n (2), respectively. Then 22 E E, (3) Q 3 =- d G ( n ) klm lmpq s *, sub pqij n n ' (1) (1) [ 11 i j + n n n ' (2) (2) (3) 22 i j ] k. The eective direct piezoelectric coeicient D d 31, can be expressed as in Eq. (2.9): E ' ' E, (3) *, sub ' (1) (1) ' (2) (2) (3) ' ' 31 = Q3 /( ) = dklmg lmpq ( n ) spqij [ 11 ni nj + 22 ni nj ) nk ]/( ) D E, (3) *, sub (1) (1) (2) (2) (3) = d klmg lmpq ( n ) spqij [ ni nj + ni nj ] nk /2 ( = ' 22 ) (2.9) ' 11 On the other hand, in the case o a 31 sensing mode, biaxial in-plane strains 11 D e, and 22 are applied to the substrate and hence the substrate-constrained ilm. The total (1) (1) (2) (2) (3) (3) 11 p q 22 p q p q ilm strain can be written as = n n + n n + n n in the artesian pq coordinates o the ilm (p, q=1-3), where is the normal strain due to Poison eect., (3) The in-plane stresses in the ilm induced by the applied strains are lm = G E lmpq ( n ) * E, (3) (1) (1) (2) (2) pq= Glmpq ( n )[ 11 np nq + 22 np nq ] ( does not contribute to in-plane stresses since G E, lmpq ( n (3) ) n =). [43]-[45] The induced longitu-dinal charge is (3) (3) p n q E (3) lm k Q 3 = d n = klm d G ( n E, klm lmpq (3) ) (1) (1) (2) (2) (3) [ 11 np nq 22np nq ] nk +. The eective direct piezoelectric coeicient D e, 31 is expressed in Eq. (2.1): [52] E E, (3) (1) (1) (2) (2) (3) 31 = Q3 /( ) = dklmg lmpq( n )[ 11n p nq + 22np nq ] nk /( ) D e, 48

63 E, (3) (1) (1) (2) (2) (3) = d klmg lmpq ( n )[ np nq + np nq ] nk /2 (when 11 = 22 ) (2.1) Equations (2.7) (2.9) and (2.1) will allow one to compute the eective direct piezoelectric coeicients or an arbitrary orientation o a ilm with a variable crystalline structure, i the intrinsic piezoelectric coeicients o the ilm and the elastic moduli o the ilm and substrate are known. It can be seen that Eqs. (2.7) (2.9) and (2.1) will reduce to Eqs. (2.5), i we take n (1) =[1], n (2) =[1], n (3) =[1], and the crystalline structure o the ilm is tetragonal. In summary, the eective piezoelectric coeicients or epitaxial erroelectric ilms as components o piezoelectric MEMS sensors are generally ormulated in this article. It is demonstrated how the direct piezoelectric properties o a erroelectric ilm are aected by the ilm/ substrate interace stresses, which are induced by clamping o substrate. 2.3 Summary o the intrinsic piezoelectric properties in erroelectric thin ilms In this section, the eective piezoelectric coeicients o an epitaxial erroelectric ilm are generally ormulated by taking into account o clamping rom a substrate. It is illustrated how the equivalent converse and direct piezoelectric coeicients in a single crystal dier rom each other in an epitaxial ilm, which may serve as an example showing how the ilm-substrate interactions have modiied properties o unctional thin ilm materials. Piezoelectric materials can be used as unctional components in a wide variety o applications, such as sensors, actuators, ultrasonic and solar systems. Piezoelectric properties can be either characterized by direct piezoelectric eects, where electric 49

64 responses (induced charge/voltage) are induced by mechanical stimulus (stress /strain ield), or converse piezoelectric eects, where mechanical response (stresses/strains) are induced by electric stimulus (electric ield). In single crystals, these two kinds o eects have the same characteristic coeicients, which are equivalent thermodynamic potential derivatives. [3] For example, the eective * longitudinal piezoelectric coeicient which has a unit o p/n in the direct eect (change in polarization p 3 per applied stress ) and a unit o pm/v in the converse eect (change in ilm strain per applied electric ield E 3 ), can be expressed by Eq. (2.11), [44] where d kij are components o the piezoelectric tensor, which are deined in the artesian coordinates o the crystal. [49] (3) (3) n,, and are components o the normal vector n (3). k n i (3) n j d = d n n n (2.11) * (3) (3) (3) kij k i j However, or erroelectric ilms, rom the discussion presented in Section 1. 3 and the results obtained in Sections 2.1 and 2.2, the direct and converse piezoelectric coeicients are characteristically dierent rom each other. For a single domain (1) oriented tetragonal piezoelectric ilm, D, is larger than. They can be characterized by the ollowing ormula: [36]-[37], D sub E, E, E, d = d + 2( s s )* d /( s + s ) = d + d (2.12.1) E, E, E, = d 2 s13 d31 * ( s11 + s12 ), d = s / ( s + s ) (2.12.2) 12 sub 2 13 d31, E, 11 E, 12 5

65 Where d 31, d are the piezoelectric coeicients and sub s ij, E s, ij (i, j=1~3) are the elastic compliances deined in a matrix notation. [3] Subscript sub and denote substrate and ilm, respectively, while subscript E denotes at constant electric ield. d is the dierence between the direct coeicient D, and the converse coeicient. The in-plane piezoelectric coeicients are ound to be: [35], d (due to in-plane clamping) 31 =, D E, sub E, sub E, E, d = P /( + ) = d ( s + s )/( s + s ) (2.12.3) , D E, E, e = P /( + ) = d /( s + s ) (2.12.4) e , E, E, E3 = d31 /( s11 + s12 ) = / = e (2.12.5), D In a general case, an epitaxial erroelectric ilm with a normal n (3) bounded by a thick substrate is shown in Fig It is assumed that the substrate is a piezoelectrically inactive single crystal. The substrate is much thicker than the ilm and it has a ixed bottom surace. [53] The epitaxial relationship or the ilm/substrate couple are n (1) //n (1, sub), n (2) //n (2, sub) in plane and n (3) //n (3, sub) along the normal, where n (i) and n (i, sub) (i=1-3) are unit vectors representing the three orthorhombic axes o the ilm/substrate couple, and they are indexed in the artesian coordinates o the crystalline structure o the ilm (x k ) and substrate (x sub k ), respectively. [49] Since a substrate usually has dierent electromechanical properties rom those o a ilm, stresses will arise on the ilm/substrate interace when the couple undergoes a piezoelectric driven process. As illustrated in the previous chapter, it is these in-plane stresses that will modiy the piezoelectric properties o a erroelectric ilm and make them deviate rom those o a single crystal. The in-plane stresses will be concentrated on the ilm i it is grown on a thick 51

66 substrate. It can be written as: [46-48] = - G n ) (2.13) lm E, (3) lmpq ( pq, G E (3) Where lmpq ( n ) is the planar elastic modulus tensor at constant electric ield o a ilm with a normal n (3), which is presented by Eq. (2.14). pq is the sel-strain between ilm and substrate which is induced by the applied mechanical or electrical stimulus. E, (3) lmpq ( n ) = lmpq G E, E, lm (3) (3) (3) E, pq n ( n ) n (2.14) E, E, lmpq, lm, and E, pq are the tensor o elastic moduli o the ilm (at constant (3) electric ield), ( n ) = ( (3) 1 (3) ( n )), a Green tensor ( n ) = n (3) E, n (3). [46-48] The in-plane stresses will induce a longitudinal polarization p3 in a direct piezoelectric mode or a longitudinal strain in a converse piezoelectric mode: p 3 = d n or klm lm (3) k = E s, ijpq (3) pqni (3) j n (2.15) Where E s, ijpq is the tensor o elastic compliance o the piezoelectric crystal Direct piezoelectric coeicients or a clamped erroelectric ilm As can be seen in Fig 2.3 (a), in a transverse piezoelectric mode, in-plane stresses ( 11, 22 ) or strains ( 11, 22 ) are applied to a substrate. An in-plane stress ield will arise in the ilm due to clamping rom the substrate. A longitudinal polarization p in 3 the orm o Eq. (2.15) will be induced, and it can be characterized by a direct transverse 52

67 piezoelectric coeicient 31 = p 3 /( ) or D 31 = p 3 /( ). For a D e, longitudinal piezoelectric mode ( D ) as is shown in Fig 2.3 (b), a normal stress is applied to the ilm/ substrate couple, and in-plane stresses will arise in the ilm due to the discontinuity o elastic properties across the ilm/substrate interace. A polarization induced by these in-plane stresses in the orm o Eq. (2.15) will add to the longitudinal charge d * characterized by Eq. (2.11). The eect o in-plane stresses on the direct piezoelectric responses and the resulting eective piezoelectric coeicients are tabulated in Table 2.5 and Table 2.6, respectively onverse piezoelectric coeicients or a clamped erroelectric ilm On the other hand, or a ilm under an normal electric ield E 3 (Fig. 2.4), there are three converse piezoelectric eects-a transverse eect characterized by e, 31 = 11 ( + 22 ) /2E 3, a longitudinal eect characterized by = /E 3, and a shear eect characterized by 3 = 3 /E 3 ( is the shear direction). It should be noted here that all these strains calculated here are strains o the ilm. The in-plane stresses are: pq = - G = G ( n ) dklm n E 3. Which is E, (3) lmpq( n ) lm E, lmpq (3) (3) k ( u) ( v) = n n = G ( n ) d n n n E 3 when expressed in the orthorhombic uv pq p q E, lmpq (3) klm (3) ( u) ( v) k p q coordinates o n (1), n (2) and n (3). The eective converse piezoelectric coeicient e, 31 can be obtained by taking the average o the in-plane stresses 11 and ield E 3 : 22 over the applied 53

68 31 = [( )/2]/ E 3 = dklm nk Glmpq ( n )[ np nq + np nq ]/ 2 (2.16) e, (3) E, (3) (1) (1) (2) (2) The longitudinal and shear converse piezoelectric coeicients have been ormulated in Section 2.1. Here we can ind another identical expression or by considering the contribution to a longitudinal strain by the in-plane ilm stresses, which is characterized by Eq. (2.15) as = E s, ijpq (3) pqni n (3) j. By doing this, we can ind a general quantitative relationship between the single crystal piezoelectric coeicient * d (which has equivalent values or direct and converse coeicients), the direct piezoelectric coeicient D and the converse piezoelectric coeicient, which is presented in Eq. (2.17). D = * d + D = * d + + D d = + D d (2.17) Where * (3) (3) k i (3) j d = d n n n is the longitudinal piezoelectric coeicient or a single crystal kij plate with normal n (3), D = d G n n s s n n is the change o klm E, (3) (3) E, *, sub (3) (3) lmpq ( ) k ( pqij pqij ) i j direct longitudinal piezoelectric coeicient due to substrate clamping, sub s *, pqij is the eective elastic compliance tensor o the substrate deined in the artesian coordinates o the ilm, = E, d klm s pqij E, (3) (3) (3) (3) lmpq( n ) nk ni n j G is the change o converse longitudinal piezoelectric coeicient due to substrate clamping, D d = D d - d = d n (3) klm k G E, lmpq ( n (3) ) *, sub pqij (3) i (3) j s n n is the characteristic o the dierence between D and. The eect o in-plane stresses on the converse piezoelectric responses and the 54

69 eective converse piezoelectric coeicients are tabulated in Table 2.5 and Table 2.6, respectively. From Table 2.6, it is not diicult to see that expressions or D, D D 31, e, 31 and e, 31 will reduce to Eq. (2.12), or the case o an epitaxial tetragonal ilm grown on a (1) cubic substrate. This set o ormula in Table 2.6 will allow one to compute the eective piezoelectric coeicients or arbitrary crystalline structures and orientations o a ilm and its supporting substrate, i the intrinsic piezoelectric coeicients o the ilm and the elastic moduli o the ilm and substrate are known. In summary, the eective piezoelectric coeicients or an epitaxial erroelectric ilm were generally ormulated in this section. It was demonstrated how the piezoelectric behavior o a erroelectric ilm is aected by the ilm/substrate interace stresses, which are induced by the clamping o substrate. Our result or the eective piezoelectric coeicients o a clamped erroelectric ilm will be equally applicable to piezoelectric ilms which are not erroelectrics. 55

70 n (3) Film! P p 3 22 //n (2) or 22 //n (2) In-plane stresses induced in the ilm Substrate! 11 //n (1) or 11 //n (1) (a) //n (3) n (3) p 3 P Film! In-plane stresses induced in the ilm Substrate (b) Fig. 2.3 Direct piezoelectric eect: (a) d 31 /e 31 mode- a longitudinal charge is produced by applying in-plane stresses/strains to the substrate. (b) d mode- a longitudinal charge is produced in the ilm by applying a stress normal to the ilm/substrate couple. 56

71 E 3 //n (3) n (3) Film P! s n s s In-plane stresses induced in the ilm Substrate Fig. 2.4 onverse piezoelectric eect A longitudinal strain and a shear strain are produced by applying an electric ield normal to the ilm [44] ; in-plane stresses are also produced under the electric ield as a result o substrate clamping. The deormation is exaggerated here or illustration. 57

72 Table 2.5. Eect o the ilm-substrate interace in-plane stresses on the piezoelectric behavior o a clamped erroelectric ilm. Piezoelectric Mode Stimulus Response Sel-strain between ilm and substrate induced by the piezoelectric response due to inplane stresses (3) G( n ) * applied stimulus Direct eect under constant electric ield d d 31 Longitudinal stress In-plane stresses 11, 22 Longitudinal polarization p 3 Longitudinal charge p 3 pq = s s E, *, sub (3) (3) ( pqij pqij ) i j pq = s + n n *, sub (1) (1) pqij [ 11 ni n j (2) (2) 22 n i n j ] p = d G n ) 3 E, (3) (3) klm lmpq ( pq nk will add to intrinsic response 3 (3) (3) (3) dkijnk ni n j E, lmpq (3) (3) ( pqnk p = d G n ) klm e 31 In-plane strains 11, 22 Longitudinal charge p 3 = n n + n pq (1) (1) 11 p q 22 n (2) (2) p q p = d G n ) 3 klm E, lmpq (3) (3) ( pqnk onverse eect under constant mechanical stress d e 31 Longitudinal electric iled E3 Longitudinal electric iled E3 Longitudinal strain In-plane stresses 11, 22 (3) lm = dklmnk E3 (3) lm = dklmnk E3 E, E, (3) (3) (3) = ijpq Glmpq( n ) lmni nj s will add to intrinsic response d (3) (3) kijnk ni n (3) j E E, (3) (1) (1) 11 = lmpq( n ) lmnp nq G E, (3) (2) (2) 22 = lmpq( n ) lmnp nq G 3 58

73 Table 2.6. The eective piezoelectric coeicients or a clamped erroelectric ilm with an orientation shown in Fig. 2.2, in comparison with those o a single crystal plate with the same orientation. Piezoelectric coeicients Longitudinal * d = Transverse Shear Single crystal d = d Direct coeicients Film (3) (3) (3), D * E, (3) E, *, sub (3) (3) (3) d kijnk ni n j d = d dklmglmpq ( n )( s pqij s pqij ) nk ni n j n n n * (3) (1) (1) 31 kij k i j d = d n n n * (3) (2) (2) 32 kij k i j e * = " * * 31 3i i1 i= 1 * 34 e * = " 6 32 i= 1 d =2 d * 35 d = * 36 d = kij n d * * 3i i2 (3) (2) k ni n (3) j (3) (1) (3) 2dkijnk ni n j (3) (1) (2) 2dkijnk ni n j, D E, (3) *, sub (1) (1) d = d G ( n ) s [ n n + n ] n (2) (2) (3) 31 klm lmpq pqij i j i j k (when 11 = 22 ), D E, (3) (3) (1) (1) e = d G ( n ) n [ n n + n (2) (2) 31 klm lmpq k p q p q (when 11 = 22 ) N/A n n ]/ 2 / 2 = * onverse coeicients E, lmpq (3) d - d G ( n ) s klm E, pqij n (3) (3) k ni (measured by total surace D displacement) = d, 31 = e, E, (3) (3) 31 = klmglmpq ( n ) nk 3 (1) (1) p q n (2) (2) p q (3) j d [ n n + n n ]/ 2 (3) E, ( 3) (3) [14] = ik ( n ) kt d n t n i : * ij are the eective elastic constants o the ilm. I ( ) ( ) (# ) n h nu nv, where ghuv * # is the ull tensor notation o * ij, it can be obtained by * # = is the elastic tensor o the single crystal. [49] n (k) (k=1-3) are the orientation vectors o the ilm (Fig. 1) ( ) ghuv n g 59

74 2.4 Theoretical prediction or various thin ilm material systems Based on the available piezoelectric coeicients and mechanical properties or bulk materials, the orientation dependences o the converse longitudinal piezoelectric coeicient d, and piezoelectric orce coeicient D e, 31 are calculated or erroelectric ilm materials like lead zirconate titanate solid solution system, barium titanate, and some relaxor erroelectrics. To compare the direct piezoelectric coeicients with converse ones, direct piezoelectric coeicients D and D 31 are also calculated or lead zirconate titanate materials with silicon and strontium titanate substrates Lead Zirconate Titanate solid solution Lead zirconate titanate (PbZr x Ti 1-x O 3, abbreviated as PZT) bulk ceramics have a wide range o applications in electromechanical systems due to their excellent piezoelectric properties. Recently, there is an increasing number o research works on PZT in thin ilm orm due to the success o its epitaxial growth and its potential or micro-electro-mechanical applications. By using dierent measurement techniques, people obtained piezoelectric coeicients d, ranging rom ~5pm/V (p/n) to ~8pm/V (p/n) or PZT ilms. [38], [45], [54], [55] Despite the substrate clamping eect, which deteriorates the piezo-electric properties o the ilm, some o these reported values were even larger than phenomenologically predicted bulk values. [6] Thereore, a clear understanding o the intrinsic piezoelectric properties in a clamped PZT ilm will help us to tell i a measured eective piezoelectric coeicient has extrinsic contributions, such as domain wall motion, incomplete clamping, substrate bending [37],[43], or even experiment raud. 6

75 Based on the phenomenologically calculated piezoelectric coeicients and available mechanical properties or bulk materials, the orientation dependences o the converse longitudinal piezoelectric coeicient d, and piezoelectric orce coeicient D e, 31 are calculated or epitaxial Pb(Zr x Ti 1-x )O 3 ilms with dierent compositions. The calculations indicate that both tetragonal and rhombohedral Pb(Zr x Ti 1-x )O 3 ilms have their maximum and D e, 31 values along an axis close to the pseudo-cubic [1] direction, which are similar to the orientation dependence results or Pb(Zr x Ti 1-x )O 3 bulk materials. However, the calculated maximum intrinsic d, value (~pm/v), which is in a [1] oriented Pb(Zr.52 Ti.48 )O 3 ilm (on the rhombohedral side o the morphotropic phase boundary), is only about hal o the result or bulk crystals (624pm/V); On the other hand, the calculated maximum intrinsic D e, 31, which is also in a [1] oriented rhombohedral Pb(Zr.52 Ti.48 )O 3 ilm, is about twice as much as that or the corresponding bulk material. This calculation showed that the substrate clamping plays an important role on determining the piezoelectric properties o lead zirconate titanate erroelectric thin ilms. In principle, we can calculate the longitudinal piezoelectric coeicients or arbitrary orientations and here or three typical orientations (1), (11) and (111). The expressions or ik and Table 2.1 through 2.4. d, o these three orientations are calculated and shown in The elastic compliance data or PZT ilm near the PT composition (x=,.2) were taken rom J. D. Freire et al. [56] For PZT ilms with compositions near the MPB (x~.5), the elastic compliance data were taken rom T. Mitsui et al., [57] since there is no single 61

76 crystal data reported. The quantitative values o these three piezoelectric coeicients were given by phenomenological calculations. [6] For tetragonal PbZr x Ti 1-x O 3 ilms with x= (PbTiO 3, d 31 =-23.1pm/V, d =79.2pm/V, d 15 =56.1pm/V) and x=.48 (PZT 48/52, d 31 =-135pm/V, d =313pm/V, d 15 =53pm/V), the orientation dependence o presented in Fig. 2.5 and Fig 2.6, respectively. The quantitative values o the piezoelectric coeicients were given by d, are phenomenological calculations in the pseudo-cubic coordinate system. [6] By the standard transormation law or tensor quantities given by J. F. Nye, [49] the piezoelectric coeicients deined in a rhombohedral coordinate system can be obtained or rhombohedral PbZr x Ti 1-x O 3 material (x$.52). For rhombohedral PbZr x Ti 1-x O 3 ilms with x=.52 (PZT 52/48, d 31 =-62pm/V, d =17pm/V, d 15 =1244pm/V, d 22 =-273pm/V) and x=.6 (PZT6/4, d 31 =-11.3pm/V, d =7.8pm/V, d 15 =359.9pm/V, d 22 =-75.2pm/V), the orientation dependence o The calculation results showed that: d, are presented in Fig. 2.7 and Fig 2.8, respectively. (1) Both tetragonal and rhombohedral Pb(Zr x Ti 1-x )O 3 ilms have similar orientation dependence as those o their bulk materials. [7] The maximum d, is in a direction close to pseudo cubic (1) in both tetragonal and rhombohedral structures. (2) In ilms with a rhombohedral structure, d, values are much more reduced in non-polar directions than in the polar direction [111], as compared with the bulk properties. It is the opposite in tetragonal ilms where the reduced in the polar direction [1] than in a non-polar direction; d, value is much more 62

77 (3) The calculated maximum intrinsic d, (327.5pm/V) or Pb(Zr x Ti 1-x )O 3 material, which is or a [1] oriented Pb(Zr.52 Ti.48 )O 3 ilm (on the rhombohedral side o the morphotropic phase boundary), is only about hal o the result or bulk crystals (624pm/V). This calculation showed that the substrate clamping plays an important role on determining the piezoelectric properties o erroelectric thin ilms. In addition, Fig. 2.9 (a) shows unction o the composition and Fig 2.9 (b) compares d, or [1], [11] [111] oriented PZT ilms as d, or a [1] oriented epitaxial ilm and that o a [1] poled polycrystalline ilm (calculated by using material constants rom T. Mitsui et al. [57] ). It is seen that the order o magnitude or d, is [1]>[11] >[111] in PZT epitaxial ilms with compositions x%.6, and an epitaxial ilm always has larger intrinsic d, than that o a polycrystalline one at the same composition. 63

78 single crystal clamped ilm 12 [ 1] 9 d (pm/v) 8 6 (a) [1 1 1] [ 1 ] Figure 2.5 (a) Orientation dependence o the intrinsic longitudinal converse piezoelectric coeicient o PbTiO 3 ilms; (b) The cross section curve when the igure in (a) is cut by the (1 ) plane. (b) 64

79 single crysal clamped ilm 12 [ 1] 9 d (pm/v) 4 6 (a) [1 1 1] [ 1 ] Figure 2.6 (a) Orientation dependence o the intrinsic longitudinal converse piezoelectric coeicient o Pb(Zr.48 Ti.52 )O 3 ilms; (b) The cross section curve when the igure in (a) is cut by the (1 ) plane. (b) 65

80 single crystal clamped ilm 12 [1 1 1] d (pm/v) (a) # max =15 o 15 [ 1] [1 1-2] (b) 27 Figure 2.7. (a) Orientation dependence o the intrinsic longitudinal converse piezoelectric coeicient o Pb(Zr.52 Ti.48 )O 3 ilms; (b) The cross section curve when the igure in (a) is cut by the (1-1 ) plane. 66

81 Single crystal lamped ilm [ 1] 12 [1 1 1] 9 d (pm/v) (a) # max =145 o [1 1-2] (b) 27 Figure 2.8. (a) Orientation dependence o the intrinsic longitudinal converse piezoelectric coeicient o Pb(Zr.6 Ti.4 )O 3 ilms; (b) The cross section curve when the igure in (a) is cut by the (1-1 ) plane. 67

82 d, (pm/v) 45 d, 3 15 or PZT epitaxial ilms (1) (11) (111) T MPB R d, (pm/v) PZ % d, or [1] oriented PZT ilms Epitaxial polycrystalline MPB T R PZ % (a) (b) Figure 2.9 (a) or [1], [11], [111] oriented PZT epitaxial ilms as unction o ilm composition; (b) or [1] oriented epitaxial and polycrystalline ilms as unction o ilm composition. 68

83 alculations or transverse piezoelectric coeicients D e, 31 (= e, 31 ) based on the results in Table 2.6 are perormed or MPB-PZT thin ilms - x=.5 or a tetragonal structure and x=.52 or a rhombohedral structure. The piezoelectric coeicients (in a pseudo cubic coordinate system) or PZT bulk materials were calculated by using a phenomenological method developed by Haun et al. [6] The elastic modulus o PZT materials were taken rom T. Mitsui et al. [57]. The orientation dependence o D e, 31 with respect to the ilm normal n (3) or a tetragonal PZT 5/5 ilm is visualized in Fig 2.1 (a), where the distance between the surace o the graph and the origin represents the magnitude o D e, 31 or that orientation. Fig. 2.1 (b) shows the cross section curve o (1) plane or D e, 31 and 31 e (or single crystal) o PZT 5/5. It shows that both e 31 and ilm orientation n (3) =[1]. D e, 31 have their maximum values or a Similarly, the orientation dependence o D e, 31 in a rhombohedral PZT 52/48 ilm is visualized in Fig (a), and Fig (b) shows the cross section curve o a pseudocubic (1-1 ) plane (rhombohedral (1) plane) or D e, 31 and 31 e. It shows that both e31 and D e, 31 has minima or the pseudo-cubic [111] orientation, which is the polarization direction. The maximum value o D e, 31 (~3/m 2 ), as well as that o e 31, is in an orientation n (3) very close to [1] axis, which is 54.7 o away rom [1 1 1]. Fig shows the eective piezoelectric coeicient e 31, in [1], [111] oriented epitaxial PZT ilms with respect to compositions (denoted as e 31, [1] and e 31, [111], respectively). For both tetragonal and rhombohedral PZT ilms, 31, [1] e is 69

84 larger than e 31, [111]. However, the dierence between e 31, [1] and e 31, [111] is remarkably large or rhombohedral PZT ilms and it increases as the composition approaches the morphotropic phase boundary. On the rhombohedral side o the MPB (x=.52), e 31, [1] has the maximum value o -3.2 /m 2 while e 31, [111] is only -5.7 /m 2. It should be noted that the highest e 31, reported or PZT materials is around -26 /m 2 in a (1) orientation [58], which agrees well with our calculation. As a comparison, e, 31 values or PZT polycrystalline ilms ( e, poly 31, calculated by using Eq. 2.1 with material constants taken rom Re. [57] and experiment measured 31 or textured PZT ilms ( e e, 31, textu. rhombohedral side o the MPB (x$.52), ) [35] are also plotted in Fig On the e, 31 have the maximum value at x=.52, which is about -1/m 2 and agrees well with the data reported in Re. [34]. For the (1) poly oriented ilms on the same side, it can be seen that e 31, [1] > e (1) > 31, textu. e 31,. poly While or the (111) oriented ilms, it is e, [111] < 31 e (111) < 31, textu. e, poly 31. This can be understood i e 31, increases as the ilm orientation is canted away rom the polar direction [111] toward the [1] orientation, just like in the case o d [8]. Thus, even in a ully poled poly-crystalline ilm, some PZT grains canted away rom the poling direction will enhance e, 31,. This may explain why e 31, or polycrystalline ilms are larger than e 31, [111], which is calculated by assuming that the [111] ilm has a single domain structure. 7

85 [1 ] [ 1 ] (a) -e31, -e31 (single crystal) [ 1] e31 (/m2) 6 [1 1 1] 3 18 [ 1 ] 21 (b) Fig. 2.1 (a) Eective piezoelectric coeicient D e, 31 o a tetragonal PZT 5/5 ilm as unction o ilm orientation. (b) The cross section curve o (a) cut by the (1 ) plane. 71

86 [1 1 1] [1-1 ] [1 1-2] -e31, -e31 (single crystal) 12 [1 1 1] 9 -e31 (/m2) 4 6 (a) 3 # max =59 o 15 [ 1] [1 1-2] (b) 27 Fig (a) Eective piezoelectric coeicient D e, 31 o a rhombohedral PZT 52/48 ilm as unction o ilm orientation.. (b) The cross section curve o (a) cut by the pseudo-cubic (1-1 ) plane. 72

87 -e 31, (/m 2 ) 3 15 MPB Epitaxial [1] Epitaxial [111] Polycrystalline Exp. [1] textured 18 Exp. [111] textured 18 Motorola PZT HD T Phase R phase PbTiO 3 % Figure 2.12 Eective piezoelectric coeicient compositions. e 31, o PZT ilms with various 73

88 D d,, D 31 and D e, 31 values are also numerically calculated or three typical orientations (1), (11) and (111) in Fig as unctions o composition x or Pb(Zr x Ti 1-x )O 3 single domain erroelectric thin ilms epitaxially grown on Si substrate. Piezoelectric and elastic constants or Pb(Zr x Ti 1-x )O 3 materials necessary or calculations are taken rom Re. [6] and [56]-[57], respectively. The elastic constants or single crystal Si are taken rom Re. [59]. From Figures 2.13, it can be seen that Pb(Zr x Ti 1-x )O 3 ilms have their maximum direct piezoelectric coeicients in a pseudo cubic [ 1] orientation, regardless o the ilm composition and structure. Optimum D D d,, 31 and D e, 31 values or Pb(Zr x Ti 1-x )O 3 ilms as MEMS sensor components maybe realized in a pseudo-cubic [1] orientation on the rhombohedral side o the morphotropic boundary (x=.52). The calculated d D, and D e, 31 are about ~45p/N and -3 /m 2, respectively, or this (1) oriented MPB- PZT ilm grown on a Si substrate, which agree well with the reported experiment data in Re [36] and Re [58]. The results suggested that PZT epitaxial thin ilms used or MEMS sensor applications should adopt the rhombohedral composition on the morphotropic phase boundary with a pseudo-cubic [1] orientation. 74

89 d, (p/n) 4 2 (1) ilm (11) ilm (111) ilm T phase R phase -d 31, (p/n) x - PZ% (1) ilm (11) ilm (111) ilm T phase R phase (a) -e 31, (/m 2 ) x - PZ% (1) ilm (11) ilm (111) ilm T phase x - PZ% Figure (a) Longitudinal piezoelectric coeicient D R phase (b) (c) ; (b) transverse piezoelectric coeicient Si. D 31 and (c) transverse piezoelectric coeicient D e, 31 or PZT ilms grown on 75

90 It is more convenient to look at the cross section curves or comparison o substrate-clamping and orientation eects between D and. Fig 2.14 shows the cross section curves o D and as a unction o the ilm orientation n (3), in comparison with that o the intrinsic piezoelectric coeicient d given by Eq. (2.11). In the calculation, Si and SrTiO 3 (STO) were chosen as the substrates and their elastic constants were also taken rom Res. [59] and [6], respectively. Both ilms have their maximum longitudinal piezoelectric coeicients in a direction along or very close to the * pseudo cubic [ 1] direction. It can be seen that D is always larger than. Moreover, as the converse coeicient does not have substrate dependence, the direct coeicient D shows a larger value or a ilm on a relatively sot substrate (Si) than the same ilm on a relatively rigid substrate (SrTiO 3 ). The results indicate that, PZT ilm based transducers can have enhanced longitudinal and transverse piezoelectric responses by choosing a rhombohedral composition near the morphotropic phase boundary, a pseudo-cubic [1] orientation, and a sot substrate. 76

91 d (single crystal) d, (direct) on Si d, (direct) on STO 12 d, (converse) 15 [ 1] d (pm/v or p/n) [1 1 1] [ 1 ] 21 d (single crysal) d, (direct) on Si 12 d, (direct) on STO d, (converse) 6 [ 1] [1 1 1] 9 d (p/n or pm/v) (a) 18 [1 1-2] (b) Fig (a). The cross section curve (cut by the (1) plane) o the longitudinal piezoelectric coeicients or a tetragonal Pb(Zr.5 Ti.5 )O 3 ilm as unction o ilm orientation. (b) The cross section curve (cut by the pseudo-cubic (1-1 ) plane) o the longitudinal piezoelectric coeicients or a rhombohedral Pb(Zr.52 Ti.48 )O 3 ilm as unction o ilm orientation. 77

92 In summary, both tetragonal and rhombohedral PZT epitaxial ilms have largest piezoelectric coeicients along an axis close to the pseudo-cubic [1] direction, which are similar to the orientation dependence results or PZT bulk materials. [7], [8] However, the calculated maximum intrinsic d, (327.5pm/V) or PZT material, which is in a [1] oriented Pb(Zr.52 Ti.48 )O 3 epitaxial ilm on the rhombohedral side o the morphotropic phase boundary, is only about hal o the result or its bulk crystals (624 pm/v); while the maximum intrinsic D e, 31 (~3/m 2 ) is about twice as much as that or the corresponding bulk material (~15/m 2 ). Our calculations showed quantitatively how the clamping o a substrate can vary the intrinsic longitudinal converse piezoelectric coeicients and gives numerical results or PZT ilms with various compositions Barium Titanate material The orientation dependences o the converse longitudinal piezoelectric coeicient e,, and the in-plane converse piezoelectric coeicient 31,, D ( e 31 e31 = e31, = ), are calculated or tetragonal barium titanate epitaxial ilms. The calculations demonstrate that both e 31, and have their maximum values along an axis close to the (111) direction o the pseudo-cubic system, which are similar to the orientation dependence results or a tetragonal BaTiO 3 single crystal. The calculated piezoelectric coeicients or a (111) oriented BaTiO 3 epitaxial ilm ( e, 31 =-23/m 2, =124pm/V) suggest that it is a good candidate material or lead-ree MEMS applications. 78

93 Barium Titanate (BaTiO 3 ) is one o the most intensively studied erroelectric materials. It has an ABO 3 type perovskite cubic structure at high temperature, which is paraelectric, and a erroelectric tetragonal structure at room temperature. Recently, due to the discovery o the large piezoelectric eect in relaxor-erroelectric single crystals with non-polar orientations, [9], [15],[61] there has been a renewed interest on the investigation o piezoelectric properties in classic erroelectrics with simpler structures [6], [1], [12], [13], [62]-[64] such as BaTiO 3. S. Wada et al. investigated piezoelectric properties o barium titanate single crystals at room temperature as a unction o crystallographic orientation [1], and they ound that a (111) oriented tetragonal BaTiO 3 single crystal had a higher longitudinal piezoelectric coeicient d (=23pm/V) than that o a (1) oriented one ( d =125pm/V). This has been explained by D. Damjanovic et al. [12],[13] by the contribution to d rom piezoelectric shear eect. In an eort to realize variable unctions o BaTiO 3 in integrated devices, there have been several methods in growing epitaxial BaTiO 3 ilms with good qualities. [65]-[7] However, despite the act that BaTiO 3 is a lead-ree material with large bulk piezoelectric coeicients, there have been very ew experiment results on its piezoelectric properties in thin ilm orm. In this section we will calculate the intrinsic piezoelectric coeicients o tetragonal BaTiO 3 epitaxial ilms as unction o crystalline orientation and ind the optimal conditions or its piezoelectric perormance. On evaluating piezoelectric properties o erroelectric ilms or applications in MEMS transducers, [35] the two most important coeicients are the longitudinal piezoelectric coeicient e 31,., and the in-plane piezoelectric coeicient is the ratio between a longitudinally generated mechanical strain and a longitudinally applied 79

94 electric ield E, while 3 e 31, is the ratio between an induced longitudinal polarization P 3 (electric charge density) and the sum o laterally applied mechanical strains and (subscripts 1 and 2 indicate the in-plane axies while 3 the longitudinal axis). Excellent piezoelectric transducers either can produce large electric charge signal at small mechanical strains (large e 31, ) or large mechanical strains at a small electric ield (large ), thereore e 31, and are essential quantities describing the perormance o erroelectric-ilm based MEMS transducers. Due to the clamping rom substrate, these two piezoelectric coeicients or erroelectric ilms can be very dierent rom their [44], [52] counterparts in bulk materials. The previous results on evaluation o these two piezoelectric coeicients in lead zirconate titanate (abbreviated as PZT) ilms [44], [52], [71] have shown that a (1) oriented ilm have larger piezoelectric coeicients than those o a (111) oriented ilm o the same material in both tetragonal and rhombohedral erroelectric phases; thereore, it is interesting to see i the orientation dependences o match those o PZT ilms. e 31, and in BaTiO 3 ilms can From Eq. 2.4 and 2.1, it is seen that and e, 31 are unctions o elastic moduli and piezoelectric moduli, as well as the crystallographic orientation o the epitaxial ilm. In literature, there are a ew sets o elastic moduli and piezoelectric moduli data obtained by dierent research groups, [72]-[74] which all give qualitatively similar results when used to calculate e, 31 and, as can be seen in Fig Here we employ the electromechanical constants reported by Z. Li et al., [74] which is the most 8

95 recent experiment data obtained on high-quality BaTiO 3 single crystals and gives about average values in calculated piezoelectric coeicients among the three sets o data. The orientation dependence o in a single domain BaTiO 3 ilm is visualized in Fig (a), where the distance between the surace o the graph and the origin represents the magnitude o in that orientation. Fig (b) shows the orientation dependence o e 31, in a similar manner as Fig 2.16 (a), while Fig (a) (b) shows the cross sectional curve o a (1) plane or and e, 31, respectively. The orientation dependence o single crystal d and e 31 are also shown in Fig 2.17 (a) and (b) or comparison with thin ilm data and evaluation o clamping eect. It can be seen that and e, 31 each has a similar orientation dependence as that o its bulk counterpart. The calculated orientations or maximum and e, 31 are 5 o and 48 o away rom the polar direction [1] (denoted in Fig as 4 o and 42 o away rom the [1] direction), close to those calculated or bulk d (5 o ) and e 31 (52 o ). However, the maximum piezoelectric coeicients or and e, 31 are pm/v and /m 2, respectively, while those or d and e31 are 228 pm/v and -13./m 2, respectively. The dierences in piezoelectric coeicients demonstrate the inluence o substrate clamping on the piezoelectric properties o thin ilm barium titanate materials. The piezoelectric anisotropy relected on Fig and Fig may be used to explain the large dierences observed in piezoelectric properties o [1] and [111] oriented BaTiO 3 ilms or bulk materials. It can be seen rom Fig that all the piezoelectric coeicients ( d, e 31, and e, 31 ) have their maximum in an orientation 81

96 close to [111] (which is 54.7 o away rom the polar direction - [1]). For BaTiO 3 single crystals, d along [111] direction is about twice large (224pm/V) as that along the polar direction [1] (16pm/V), which agrees well with the experiment observations [1] (125 pm/v and 23pm/V or [1] and [111] oriented BaTiO 3 single crystals, respectively). While or the in-plane piezoelectric coeicient, a (111) oriented BaTiO 3 single crystal has a e 31 value (=-12.9 /m 2 ) about our times large as that o a (1) oriented crystal ( /m 2 ). Moreover, the calculations on those o the single crystal. The calculated and e, and e, 31 showed similar anisotropies as 31 are 124 pm/v and -23/m 2 or a (111) oriented epitaxial BaTiO 3 ilm, while they are only 34.8 pm/v and -7.8 /m 2 or a (1) oriented ilm. From our calculations, it is suggested that a (111) oriented tetragonal BaTiO 3 epitaxial ilm has superior piezoelectric properties, as compared with those o a (1) oriented tetragonal BaTiO 3 epitaxial ilm. The calculated (124 pm/v) and e 31, (-23/m 2 ) or a (111) oriented epitaxial BaTiO 3 ilm are even comparable to those or PZT ilms near the morphotropic phase boundary [71], which makes it a good candidate material or lead-ree transducer applications. In summary, two important parameters characterizing piezoelectric perormance o erroelectric ilms, and e, 31, have been calculated or epitaxial tetragonal BaTiO 3 ilms. It is ound that although and e, 31 are quantitatively dierent rom those o a single crystal due to clamping o substrate, they have qualitatively similar orientation dependences as those o a single crystal. A (111) oriented epitaxial BaTiO 3 ilm has (124 pm/v) and e 31, (-23/m 2 ) much larger than those o a (1) oriented ilm, and may serve as an alternative to PZT erroelectric ilms in MEMS applications. 82

97 Z. Li et al., 1991 A. Schaeer et al., 1986 D. Berlincourt et al [ 1] 9 d, (pm/v) [ 1 ] Z. Li et al A. Schaeer et al D. Berlincourt et al [ 1] 9 -e31, (/m2) 3 6 (a) [ 1 ] (b) Figure (a) alculated piezoelectric coeicients and (b) e 31, by using dierent sets o electromechanical data. 83

98 [1 ] (a) [ 1 ] (b) Figure 2.16 (a) alculated piezoelectric coeicients and (b) e 31, o single domain tetragonal BaTiO 3 ilms as unction o crystalline orientation o the ilm normal. 84

99 d (single crystal) d, [ 1] d (pm/v) # max =4 o [1 1 1] 3 18 [ 1 ] 21 (a) e31, -e31 (single crystal) [ 1] -e31 (/m2) # max =42 o [1 1 1] 3 18 [ 1 ] 21 (b) Figure 2.17 (a) ross section curves o piezoelectric coeicients e 31, o and (b) single domain tetragonal BaTiO 3 ilms when Fig (a), (b) are cut by the (1 ) plane. 85

100 2.4.3 Relaxor erroelectric ilms Recently, there have been intensive research works on relaxor-lead titanate (abbreviated as PT) type erroelectric single crystals, among which xpb(mg 1/3 Nb 2/3 )O 3 - (1-x)PbTiO 3 (abbreviated as PMN-PT) and xpb(zn 1/3 Nb 2/3 )O 3 -(1-x)PbTiO 3 (abbreviated as PZN-PT) are centers o attentions due to their unusually high piezoelectric coeicients and electromechanical coupling actors. Like PZT solid solutions, these erroelectric systems have morphotropic phase boundaries (MPB) separating stable erroelectric phases o tetragonal and rhombohedral structures. They have superior piezoelectric properties on compositions o the rhombohedral side o the MPB. Speciically, giant piezoelectric coeicients (>25 pm/v) have been achieved on (1) oriented xpb(mg 1/3 Nb 2/3 )O 3 -(1-x)PbTiO 3 (x%.) [11] and xpb(zn 1/3 Nb 2/3 )O 3 -(1-x)PbTiO 3 [9], [61] (x %.9) single crystals. Ater the discoveries o these superior piezoelectric crystals, it is desirable to integrate them into MEMS devices. For a better design and perormance o MEMS devices, the electrical and mechanical engineers need to have e 31, and values instead o bulk properties o these materials and they seek answers rom materials engineers. The availability o complete sets o electromechanical constants or relaxor- PT single crystals [15], [16], [61],[75] -[77] has made it possible a theoretical study o e 31, and on relaxor erroelectric ilms. There are two types o single crystal database in literature. One is or singledomain single crystal, which may be used to predict the orientation dependence o physical properties or bulk and ilm materials. [16], [76] The other is or single crystals with speciic multi-domain conigurations (or so called Engineered domain structure ), 86

101 which may only be used to predict the properties o bulk or ilm materials under the same domain conigurations. [15], [61], [75], [77] The available single domain single crystal data or rhombohedral.67pb(mg 1/3 Nb 2/3 )O 3 -.PbTiO 3 [16] and tetragonal.58pb(mg 1/3 Nb 2/3 ) O PbTiO 3 [76] are used here or a complete study o e 31, and as unctions o crystalline orientation. In addition, piezoelectric properties or multi-domain [1] ilms o PMN-PT [15], [75], PZN-PT [61] and BiScO 3 -PbTiO 3 [77] are also calculated or comparison. The orientation dependence o e 31, in a single domain.67pb(mg 1/3 Nb 2/3 )O 3 -.PbTiO 3 ilm is visualized in Fig (a), where the distance between the surace o the graph and the origin represents the magnitude o e 31, in that orientation. Fig (b) shows the orientation dependence o in a similar manner as in Fig 2.18 (a), while Fig.2.18 (c) and (d) shows the cross section curve o a pseudo-cubic (1-1 ) plane or e 31, and, respectively. The orientation dependence o single crystal e 31 and d are also plotted in Fig 2.18 (c) and (d) or comparison with the ilm data and evaluation o the clamping eect. It can be seen that e 31, and each has similar orientation dependence as the bulk coeicients. The calculated orientation or maximum e 31, is in a direction 65 o degree away rom the polar direction [1 1 1] (155 o away rom [1 1-2], as denoted in Fig. 1 c), coinciding with the direction calculated or maximum d, (Fig. 1 d) while in the bulk single crystal the orientation or maximum piezoelectric coeicients are calculated to be 71 o (161 o away rom [1 1-2]) and 63 o (153 o away rom [1 1-2]) or e 31 and d, respectively. However, the maximum piezoelectric 87

102 coeicients or e 31, and are -2.4 /m 2 and 169 pm/v, while those or bulk e 31 and d are -21/m 2 and 2411 pm/v, respectively. These dierences relect the important role clamping plays on the eective intrinsic piezoelectric coeicients o relaxor erroelectric ilms. In addition, the piezoelectric anisotropy relected on Fig 2.18 may be used to explain the large dierences observed between piezoelectric properties o [ 1] and [1 1 1] oriented relaxor erroelectric ilms or bulk materials. W. ao et al [17] has shown that a [ 1] oriented.67pmn-.pt single crystal can have much higher d than that o a [1 1 1] oriented.67pmn-.pt single crystal. They attributed this dierence to the large shear piezoelectric coeicient, which can contribute to longitudinal piezo-electric response when the single crystal is rotated to a non-polar orientation. Here their conclusion on single crystals is veriied by our calculation and is urther extended to epitaxial ilms. It can be seen rom Fig (d) that is ~14pm/V along a [ 1] direction (close to the direction o maximum d, ), while is only ~7pm/V along the polar direction [1 1 1]. Moreover, calculations on in-plane piezoelectric coeicients e 31 and e 31, showed that they have a similar anisotropy as that o the longitudinal piezoelectric coeicients. The calculated e 31, is -269 /m 2 or a [ 1] oriented single domain ilm, while it is only -1.8 /m 2 or a [1 1 1] oriented single domain ilm. From our calculations, it is suggested that a [ 1] oriented.67pb(mg 1/3 Nb 2/3 )O 3 -.PbTiO 3 epitaxial ilm will have superior electromechanical properties, as compared with those o a [1 1 1] oriented ilm. Similarly, the orientation dependence o e 31, and or a.58pb(mg 1/3 Nb 2/3 ) 88

103 O PbTiO 3 epitaxial tetragonal ilm, are shown in Fig As can be seen rom Fig 2.19 (c) and (d), the orientations or maximum e 31, and coincide in the polar direction [ 1], just as those or bulk e 31 and d. e 31, and are -11.7/m 2 and 116pm/V or a [ 1] oriented ilm, while or a [1 1 1] oriented ilm they are -2.4 /m 2 and 51pm/V, respectively. Both e 31, and the orientation away rom the polar direction [ 1]. In Fig. 2.2, calculated e, decrease monotonously with the angle o 31 and d, values or some relaxor ilms are plotted as [15],[16], [75]-[77] unction o urie temperature by using available single crystal data. It shows that: (1) For the same relaxor erroelectric material, a (1) oriented ilm have higher e, 31 and d, values than those o a (111) oriented ilm. (2) In general, a (1) oriented relaxor erroelectric ilm has superior piezoelectric properties ( e, 31 ~-2/m 2 and d, >15pm/V). (2) Single domain (1) oriented PMN-%PT has much larger e 31, (-269/m 2 ) and (139pm/V) than those o a multi-domain ilm ( e 31, =-23/m 2 and d, =261pm/V), probably because the inter-clamping o domain variants reduces the contributions rom shear piezoelectric eect in a multi-domain ilm. [78] (4) BiScO 3 -PbTiO 3 (BSO-PT) material has much higher urie temperature (~4 o ) than those o PMN-PT and PZN-PT (<2 o ). The piezoelectric coeicients e, 31 and d, o BSO-PT are comparable with those o PMN- 89

104 PT and PZN-PT, which makes it a promising candidate or high-temperature electromechanical applications. In summary, two important parameters characterizing the piezoelectric perormance o erroelectric ilms, e 31, and, have been calculated or various relaxor erroelectric systems, taking into account the eect o substrate clamping. It is ound that as (1) oriented relaxor erroelectric ilms generally have sound piezoelectric properties (calculated e 31, ~ 2/m 2 and >15pm/V), a (1) oriented single domain rhombohedral ilm on the morphotropic phase boundary o PMN-PT system can have very high piezoelectric coeicients (calculated e, 31 =-269/m 2 and = 139 pm/v), which suggests an optimal design or relaxor erroelectric-based MEMS devices. 9

105 (a) (b) 91

106 -e31, -e31 (single crystal) 15 # max =155 o 12 [ 1] [1 1 1] -e31 (/m2) [1 1-2] (c) # max =155 o single crystal clamped ilm [ 1] [1 1 1] d (pm/v) [1 1-2] (d) 27 Figure (a) alculated piezoelectric coeicients e, 31 and (b) d, o a single domain rhombohedral.67pb(mg 1/3 Nb 2/3 )O 3 -.PbTiO 3 ilm as unction o crystalline orientation o the ilm normal. (c) ross section curves o piezoelectric coeicients and (d) d, o a single domain rhombohedral.67pb(mg 1/3 Nb 2/3 )O 3 -.PbTiO 3 ilm when Fig (a), (b) are cut by a pseudo-cubic (1-1 ) plane. e 31, 92

107 (a) (b) 93

108 -e31, -e31 (single crystal) 12 [ 1] 9 -e31 (/m2) [1 1 1] [ 1 ] (c) 27 Single crystal lamped ilm 12 [ 1] d (pm/v) [1 1 1] 3 18 [ 1 ] (d) Figure (a) alculated piezoelectric coeicients e, 31 and (b) d, o a single domain tetragonal.58pb(mg 1/3 Nb 2/3 )O PbTiO 3 ilm as unction o crystalline orientation o the ilm normal. (c) ross section curves o piezoelectric coeicients and (d) e 31, d, o a single domain tetragonal.58pb(mg 1/3 Nb 2/3 )O PbTiO 3 ilm when Fig (a) (b) are cut by a pseudo-cubic (1 ) plane. 94

109 -e 31, (/m 2 ) /m 2 PMN-%PT PMN-%PT multi-domain (1) ilm single-domain (1) ilm single-domain (111) ilm PMN-PT PZN-PT PMN-42%PT BSO-57%PT 2 4 urie Temperature ( o ) (a) d, (pm/v) PMN-%PT Multi-domain [1] ilm single-domain [1] ilm single-domain [111] ilm PMN-PT PMN-42%PT 15pm/V PZN-PT BSO-57%PT PMN-%PT 2 4 urie Temperature ( o ) Figure 2.2. (a) alculated piezoelectric coeicients relaxor erroelectric ilms. e, (b) 31 and (b) d, o various 95

110 2.5 Experiment measurement on longitudinal piezoelectric coeicient d, Piezoelectric orce microscopy Piezoelectric orce microscopy is a voltage-modulated scanning probe microscopy technique. [79]-[8] In this technique, a conductive tip is biased by an ac ield in contact mode, [8] and the tip ield can be written as: E tip =E dc +E ac sin(&t) (2.18) When & is the voltage modulation requency, which is much smaller than the cantilever resonance requency. The tip delection as a unction o applied bias can be written as: [79] =A +A 1 sin(&t+' 1 )+A 2 sin(2&t+' 2 ) (2.19) Where A is static response, and A 1, A 2, ' 1, ' 2 are amplitudes and phase shits o irst and second harmonic responses. Magnitudes o A o, A 1, and A 2 are relatively small and only the latter two components along with corresponding phase shits can be determined by lock-in technique. [81] Separation o the irst and second harmonic response will allow quantiication o the electromechanical properties o the materials being tested. I the longitudinal tip delection signal is recorded and separated, the irst harmonic signal will be the longitudinal piezoelectric response. A 1, the amplitude o irst harmonic response, is in proportion with longitudinal piezoelectric constant d at small ac ield, and can be used to measure d indirectly. For example, or a ully poled tetragonal erroelectric plate with polarization aligned along the longitudinal direction 3, the longitudinal piezoelectric displacement can be written as: (E)=(E)h=Q 11 [P(E) 2 -P 2 ]*h= Q 11 h[p(e) 2 -P 2 ]= Q 11 h[(p +(E) 2 -P 2 ] =2Q 11 P (Eh+ Q 11 h((e) 2 (2.2) 96

111 Where Q 11 is the eective eletrostrictive coeicient, P(E) and P are the polarization at ield E and in the remnant state. h is the thickness o the plate and ( is the dielectric constant. Substitute E with E tip in Eq. (2.18) and keep only items which depend on requency & (thereore can be detected by the lock-in technique), then we have: 1 (&)= 2Q 11 P ( E sin(&t)+ Q11 ( 2 E cos(2&t) (2.21) 2 ac 2 ac ompare Eqs (2.19) and (2.21), it is ound that A 1 =2Q 11 P ( E =d E ac (Eq. 1.3). It should be noted that this technique or measuring d is good only at small ac electric ield when piezoelectric response is strictly a linearized electrostriction. [6] Fig (a) schematically shows the set-up o piezoelectric orce microscopy. The output o the circuit can be either a topography (contact mode imaging by Z (height)- eedback o the piezo scanner [79], [8] ), a d -D bias loop, or a piezoelectric orce microscopy image (local measurement o d ). As shown in Fig (b), the d -D bias loop was made by detecting the Dbiased piezoelectric response o a erroelectric thin ilm capacitor under an alternating ac ac [82], [83] ield. A X-cut quartz (d 11 =2.3pm/V) coated with top and bottom Au electrodes was used to calibrate the out-o-plane tip displacement at a given A amplitude and thereore the d o a erroelectric ilm. In this setup, a erroelectric ilm (in this igure, PZT ) is sandwiched between a uniorm layer o bottom electrode and top electrode pads. A direct current (dc) ield is applied to the top electrode through the AFM tip to write the domain state while the applied ac signal is used to read the piezoelectric response o the domain state by the lock-in technique. [81] A hysteresis loop is traced out by plotting the calibrated d coeicients as a unction o the dc ield. 97

112 Topography Amplitude detection (a) Phase detection Pt (b) PZT Bottom electrode Substrate PZT Bottom electrode Substrate (c) Fig (a) Schematics o the set-up o piezoelectric orce microscopy (PFM); Figure adapted rom Re. [8]. (b) Measurement o d amplitude under a uniorm tip ield by covering the ilm with Pt top electrode pads. (c) local piezoelectric response measurement (imaging). 98

113 Fig (c) shows the piezoelectric imaging technique. In this technique, an acbiased AFM probe is scanning on the surace o a naked erroelectric ilm, the amplitude and phase inormation o local piezoelectric responses are recorded during the scan to simultaneously orm contours o domain patterns. The amplitude o piezoelectric response is recorded by a position-sensitive detector (PSD), [79], [8] which is shown in Fig The PSD is a our-quadrant (A, B,, D) laser photo detector. The dierence in laser throughput intensity between the two upper detectors (A+B) and the two lower detectors (+D) is in proportion with the vertical displacement o the tip (thereore the longitudinal piezoelectric response o the ilm since the tip is in contact with ilm), while the dierence in laser throughput intensity between the two let-side detectors (B+D) and the two right-side detectors (A+) is in proportion with the lateral displacement o the tip (thereore the shear piezoelectric response o the ilm). At the same the amplitude inormation is detected, the phase inormation is also detected and separated by the lock-in ampliier (Fig 2.23 a). When the polarization vector inside a erroelectric domain is aligned along the electric ield direction, the phase angle between the piezoelectric response and the ac ield is zero, and this domain is said to be in phase with the ac signal applied to the tip. On the other hand, i the polarization vector inside a erroelectric domain is aligned against the electric ield direction, the phase angle between the piezoelectric response and the ac ield is 18 o, and this domain is said to be out-o-phase with the ac signal. There is a strong contrast between neighboring 18 o domains as shown in Fig 2.23 (b) and (c). In a 9 o domain with polarization vector aligned in-plane (usually called a domain), the longitudinal piezoelectric response is zero while its shear piezoelectric 99

114 response reaches a maximum (Fig b and c). On the other hand, or a 9 o domain with polarization vector aligned out-o-plane (usually called c domain), the amplitude o longitudinal piezoelectric response is maximum while its shear piezoelectric response is zero (Fig b and c). z(t) * h F ) d E(z, t)dz # * 1 2t F t F ) d 15 E A ( t) dt Out-o-plane In-plane # piezo z piezo Laser Photo-detector Out o plane signal = (A+B) (+D) In plane signal = (B+D) (A+) Fig Recording the amplitude o a piezoelectric response by a position-sensitive detector (PSD). 1

115 time + + h z z z z # = time # = 18 time z(t) * h F ) d E(z, t)dz (a) 1. µm (b) (c) Fig (a) Detection o 18 o domains by inding the phase angle between the longitudinal piezoelectric response and the tip ac signal. (b) Out-o-plane; (c) in-plane PFM image o a 4nm PZT(2/8)/LSO/STO erroelectric ilm showing both 18 o (by writing) and 9 o domain (grown in-situ) patterns. Figure adapted rom [8]. 11

116 2.5.2 Lead zirconate titanate thin ilms Longitudinal piezoelectric constant were measured experimentally or erroelectric Pb(Zr.2 Ti.8 )O 3 thin ilms, which were prepared by liquid delivery metalorganic chemical vapor deposition (MOVD) on SrTiO 3 substrate (a layer o SrRuO 3 bottom electrode was pre-deposited on SrTiO 3 by pulsed laser deposition). The details o MOVD and pulsed laser deposition processes are described in Re [84] and [85], respectively. The composition o the erroelectric ilm is chosen to obtain best matching o lattice parameters o Pb [Zr x Ti (1-x) ]O 3 and SrRuO 3 /SrTiO 3 heterostructures. In our experiments,.3 mol/l Pb(thd) 2 in THF,.3 mol/l Ti(OiPr) 2 (thd) 2 in THF and.3 mol/l Zr(dmhd) 4 in THF (all rom Mitsubishi Materials orporation) were used as precursors. Pt was deposited as the top electrode on 32 µm diameter circular pads patterned by standard lithography processes. Film thickness was 13 nm or all ilms. X-ray results showed very good cube on cube epitaxy (Fig. 2.24). All ilms demonstrated small FWHM widths (Table 2.7), which indicates good crystalline quality. There was no evidence o multi-domain structures or all ilms, as veriied by piezoresponse microscopy on image mode. [86] Quantitative d, ( d )-E loops (Fig. 2.25) were obtained by recording the eedback Z output when nn [82], [83] applying a D sweep bias. For each ilm, ten capacitors were tested and results averaged. Special precautions (ixing exterior surace o substrates and using ull coverage o substrates by the ilms) were undertaken to minimize eects o substrate deormation on measuring d. The remnant nn d values ( d at zero ield) are listed in Table 2.7 and compared with the theoretical calculations. Piezoelectric and elastic nn nn 12

117 constants necessary or calculations are taken rom Res [6] and [56], respectively. It is seen that the theoretical calculations agree well with experiment results. Table 2.7. Experiment results or Pb(Zr.2 Ti.8 )O 3 ilms. d nn values are the equilibrium ones at zero electric ield. Film thickness is kept at 13 nm or all ilms. Orientation (1) (11) (111) Bulk Theory d (pm/v) nn Experiment d (pm/v) nn d (pm/v) 67±7 35±4 25±3 nn FWHM or X-ray.26 o.5 o.71 o peaks (in parenthesis) (2) (11) (111) 13

118 Intensity (arbi. unit) (1) PZT (1) SRO (1) STO In te n s ity (a rb i. u n it) (11) scan (degrees) (2) PZT (2) SRO (2) STO (a) Intensity (arbi. unit) Intensity (arbi. unit) SRO (11) STO (11) PZT (11) Intensity (arbi. unit) (1) scan STO (22) SRO (22) PZT (22) STO (111) PZT (111) Intensity (arbi. unit) (1) scan # (degrees) PZT (222) SRO (222) STO (222) (b) (c) Figure 2.24 X-ray spectrum and -scans or (1)-(a), (11) - (b) and (111)- (c) or Pb(Zr.2 Ti.8 )O 3 ilms. 14

119 d nn (pm/v) (1) (11) (111) E (kv/cm) Figure 2.25 Eective longitudinal piezoelectric constants as unction o D Bias or (1), (11) and (111) oriented Pb(Zr.2 Ti.8 )O 3 ilms. 15

120 2.5.2 Lead Magnesium Niobate lead titanate erroelectric ilms Method o domain engineering, which aims at enhancing the piezoelectric properties o erroelectric materials by applying the electric ield in a non-polar direction across the crystal, has been investigated intensively or the past decade. [8]-[12] For relaxor based erroelectrics on the rhombohedral side o their morphotropic phase boundaries, such as xpb(mg 1/3 Nb 2/3 )O 3 -(1-x)PbTiO 3 (x<.35), and xpb(zn 1/3 Nb 2/3 )O 3 -(1-x)PbTiO 3 (x<.9), giant piezoelectric constants (>25 pm/v) can be achieved by poling the crystal in a pseudo-cubic (1) direction. [9] Recently, ao et al. studied both multidomain (1 oriented crystal) and single domain (111 oriented crystal) properties o.67pb (Mg 1/3 Nb 2/3 )O 3 -.PbTiO 3 single crystals. [15]-[17] They attributed the origin o the superior electromechanical properties in the (1) direction to the large d 15 o the single-domain properties, as they reproduced the large piezoelectric constants in multidomain (1) crystals rom single domain properties in the rotated coordinated system. [17] Further theoretical study [18], [19] showed that the low ield piezoelectric constants or the engineered coniguration are very close to those obtained or the corresponding single domain state and the domain wall inluence is not signiicant. Thereore, it is possible to estimate the piezoelectric constants o a multi-domain single crystal with an arbitrary orientation by knowing those calculated in a rotated coordinated system. Our recent theoretical work [44] on lead zirconate titanate thin ilms (PbZr x Ti 1-x O 3, x=.2) completed the orientation dependence o the intrinsic converse piezoelectric constants in epitaxial single domain tetragonal ilms, which was initially proposed in Re. [36] or a partial case o a (1) oriented tetragonal ilm. Here we calculate the orientation dependence o the intrinsic converse longitudinal piezoelectric constant or a 16

121 rhombohedral ilm, and apply the theoretical results to.67pb (Mg 1/3 Nb 2/3 )O 3 -.PbTiO 3 relaxor erroelectric ilms. The tensor ik or epitaxial rhombohedral ilms o (1), (11) and (111) orientations are shown in Table 2.2. Expressions or the intrinsic converse longitudinal piezoelectric constants d ( nn ) o these three orientations are list in Table 2.4, in comparison with those or orientation, a calculated value o d o single crystals. It should be noted here that or (1) nn d or a multi-domain ilm (which has a macroscopic nn symmetry o pseudo-tetragonal 4mm [15] ) is also listed or comparison. Since all the necessary single crystal properties were given in Re. [15] and [16] (note that the d 22 in Re. [16] should take a negative sign by using the current coordinate system), we calculated d as a unction o orientation and the results are visualized in Fig (b) nn (d). It can be seen that in both ilm and single crystal case, the [1] pseudo cubic direction (the dash-dot line with an arrow in the igure) is close to the direction o maximum longitudinal converse piezoelectric constants, which accounts or the large piezoelectric response in a [1] oriented single crystal or ilm. The converse longitudinal piezoelectric constants were measured experimentally or epitaxial.67pb(mg 1/3 Nb 2/3 )O 3 -.PbTiO 3 ilms, which were prepared by an on-axis radio-requency magnetron sputtering technique on SrTiO 3 substrates with 3 dierent pseudo cubic orientations, namely, (1), (11) and (111). Prior to the ilm deposition, 2nm thick epitaxial SrRuO 3 bottom electrodes were deposited on the substrate. The details o the sputtering can be ound in Re [87]. The cube on cube epitaxy and good crystallinity o the ilms are conirmed by the X-ray results (#-2# scan and -scan results 17

122 in Fig 2.26, FWHM results in Table 2.8, all results were obtained by using a Siemens D 5 our-circle diractometer). For the purpose o electric and piezoelectric measurements, Pt was deposited as the top electrode on 54 µm diameter circular pads patterned by standard lithography processes. [88], [89] Film thickness was 3.3 µm or all ilms. Quantitative d -E loops (Fig. nn 2.27) were obtained by piezoelectric orce microscopy [82], [83] in a commercial Dimension 3 TM AFM (Veeco Metrology). For each ilm, ten capacitors were tested and results averaged. Test capacitors were pre-poled in order to obtain d o stabilized domain nn structures. Special precautions (ixing exterior surace o substrates and using ull coverage o substrates by the ilms) were undertaken to minimize eects o substrate deormation on measuring d [43]. The remnant nn in Table 2.8 and compared with the theoretical calculations. d values ( d at zero ield) are listed It is seen that the piezoelectric constants o (11) and (111) oriented ilms agreed airly well with our calculations based on single-domain bulk properties. However, while the large piezoelectric constants in multi-domain (1) crystals can be reproduced rom the single domain piezoelectric properties in a rotated coordinated system, it is not the case in a clamped ilm- the intrinsic converse longitudinal piezoelectric constant o the (1) oriented ilm agreed well with that calculated or a clamped multi-domain ilm, which was several times less than that calculated or a clamped single-domain ilm. This can be explained as ollows: In an engineered multi-domain single crystal or ree ilm, since the deormation is ree o bound, shear piezoelectric deormation o each individual domain variant can equally contribute to the longitudinal piezoelectric strain. Thereore a multi-domain single crystal or ree ilm deorms like a single-domain crystal. nn nn [17], [18] 18

123 However, when the ilm is constrained by a thick substrate, the shear deormations o neighboring individual domain variants are interlocked due to the clamping o the substrate. For (1 1 1) oriented ilm, it has only one domain variant ater poling, thereore its d should be close to that o a single-domain ilm. For a (1) oriented ilm, it has nn our equivalent <1 1 1> domains ater poling ([1 1 1], [-1 1 1], [1-1 1] and [-1-1 1]), thereore its small d values as compared with that o a single domain ilm may be nn explained by the interlocking o shear piezoelectric deormations among domain variants. For the (11) orientation, its d value (152pm/V) or a single domain ilm is much nn smaller than its bulk value (937 pm/v), which indicates a much reduced contribution to longitudinal piezoelectric response rom the ilm s shear piezoelectric deormation. In addition, a poled (11) oriented ilm only has two domain variants ([1 1 1] and [1 1-1]), thereore the interlocking o shear deormation may be much less eective in the poled (11) ilm, as compared with that o a (1) ilm. In summary, the general expression or the intrinsic converse longitudinal piezoelectric constant ( d ) o a substrate-clamped single-domain rhombohedral ilm is obtained as unction o the ilm orientation. The nn d or epitaxial.67pb(mg nn 1/3 Nb 2/3 )O 3 -.PbTiO 3 ilms o three dierent crystallographic orientations, (1), (11), and (111), are calculated and compared with measured results using piezoelectric orce microscopy. It is shown that the orientation or maximum direction. The measured d is close to the pseudo cubic [1] nn d or a.67pb(mg nn 1/3 Nb 2/3 )O 3 -.PbTiO 3 ilm o this orientation is about 25pm/V, which makes it a promising candidate or MEMS devices. 19

124 It should be notiied that, as the (111) and (11) oriented ilms showed d values nn close to calculated results or single domain ilms, the (1) oriented ilm showed a much lower value than that o a single domain ilm. This discrepancy may be understood on the basis o an interlocking mechanism o shear deormations among dierent domain variants. Table 2.8. Experiment results or.67pb(mg 1/3 Nb 2/3 ) O 3 -.PbTiO 3 ilms. d nn values are the equilibrium ones at zero electric ield. Film thickness is kept at 3.3 µm or all ilms. Orientation n (1) (11) (111) Bulk d (pm/v) nn Theory Experiment d (pm/v) 1395/261 * nn d (pm/v) 251±47 13±1 73±7 nn FWHM or X-ray.28 o.4 o.3 o peaks (in parenthesis) (2) (11) (111) * alculation based on the piezoelectric and mechanical properties o a multi-domain (1) oriented rhombohedral single crystal. 6 11

125 PMN-PT on SRO/(1)STO Intensity (arbi. unit) Intensity (Arbi. Unit) (11) scan (degree) (2) PMN-PT (2) SRO (2) STO Intensity (arbi. unit) # PMN-PT on SRO/(11)STO Intensity (Arbi. unit) (1) scan (degrees) (11) PMN-PT (11) SRO (11) STO # PMN-PT on SRO/(111)STO Intensity (arbi. unit) Intensity (arbi. unit) (1) scan (degrees) (111) PMN-PT (111) SRO (111) STO # Figure X-ray spectrum and -scans or (1) (top), (11) (middle) and (111) (bottom) oriented.67pb(mg 1/3 Nb 2/3 )O 3 -.PbTiO 3 ilms. 111

126 d nn (pm/v) PMN-.PT on SRO/STO (1) (11) (111) E (kv/cm) Figure Measured eective longitudinal piezoelectric constants as unction o D Bias or (1), (11) and (111) oriented.67pb(mg 1/3 Nb 2/3 )O 3 -.PbTiO 3 ilms. 112

127 2.6 onclusion The ollowing points summarize this chapter: (1) The converse piezoelectric property o a erroelectric ilm is characterized by a ield induced invariant-plane strain. There are longitudinal and shear piezoelectric strain components, which are characterized by two eective piezoelectric coeicients: d ln = kt d ik n t n i l (2.3a) d = kt d l ik n t i l (2.3b) Equations (2.3a) and (2.3b) allow one to compute the eective piezoelectric coeicients or arbitrary orientations o a ilm and the electric ield, i the intrinsic piezoelectric coeicient and the elastic moduli are known. For tetragonal and rhombohedral epitaxial ilms with electric ield applied along the normal, exact solutions o the longitudinal piezoelectric coeicients d ( d = nn d nn =, kt d ik n t n i n ) are provided and compared with experiment measured results by piezoelectric orce microscopy. (2) The direct piezoelectric coeicients are dierent rom their corresponding converse ones, not only on that it has a non-zero d 31 component, but the apparent d value are quantitatively dierent rom the converse d coeicient. It is shown that these two d coeicients are equivalent only when the ilm/substrate couple is studied as a whole, instead o measuring the ilm s physical quantities alone. 113

128 (3) The eective piezoelectric properties, which are characterized by direct and converse eective piezoelectric coeicients - D, D D 31, e, 31, (measured by ilm strain), (measured by total strain), e, 31 and 3 are studied in a general theoretical ramework by studying the ilm-substrate in-plane stresses induced by piezoelectric stimuli (mechanical or electrical). The modiication on piezoelectric responses o a erroelectric ilm by in-plane stresses are a longitudinal polarization p3 in a direct piezoelectric mode and a longitudinal strain in a converse piezoelectric mode: p 3 d klm lm n = or (3) k = E s, ijpq (3) pqni (3) j n (2.15) It is shown that eective piezoelectric coeicients can be obtained by taking into account o the above additional terms in the apparent piezoelectric responses. (3) For a single domain erroelectric ilm, the clamping rom a rigid substrate will make the converse longitudinal piezoelectric coeicient dierent rom the direct coeicient D. is also reduced by clamping as compared with the bulk value. Experiment measurements o d, by piezoelectric orce microscopy (PFM) on tetragonal lead titanate zirconate (Pb(Zr x Ti 1-x )O 3 with x=.2) and rhombohedral lead magnesium niobate-lead titanate (xpb(mg 1/3 Nb 2/3 )O 3 -(1-x)PbTiO 3 with x=.67) epitaxial erroelectric ilms with various crystalline orientations have shown good agreement with theoretical prediction on the reduction o d, by the substrateclamping eect. While it is shown that a pseudo cubic (1) orientation in Pb(Zr x Ti 1- x)o 3 (rom x= to x=.6) and xpb(mg 1/3 Nb 2/3 ) O 3 -(1-x)PbTiO 3 (x=.58 and.67) 114

129 epitaxial single domain ilms will have the maximum d, coeicient, it is a (111) oriented epitaxial ilm that have the maximum d, in BaTiO 3 system. It should be noted that all piezoelectric coeicients calculated and compared in this dissertation reer to those at room temperature. (5) On the other hand, the transverse piezoelectric coeicients (e 31, ) will be enhanced in thin ilms as compared with those in bulk materials, which is good or an erroelectric ilm to be used in transverse piezoelectric applications. e 31, has a similar orientation dependence to that o d,, i.e., a (1) pseudo-cubic orientation or maximum e 31, in Pb(Zr x Ti 1-x )O 3 (rom x= to x=.6) and xpb(mg 1/3 Nb 2/3 ) O 3 -(1- x)pbtio 3 (x=.58 and.67) epitaxial single domain ilms, while a (111) oriented epitaxial ilm will have the maximum e 31, in BaTiO 3 system. (6) Piezoelectric orce microscopy can be used to characterize piezoelectric responses o erroelectric ilms. In the quantitative d measurement, the eective longitudinal piezoelectric coeicient o a erroelectric ilm can be indirectly measured by calibrating the raw signal o the piezoelectric response with a X-cut quartz plate (d 11 =2.3pm/V). [8],[82], [83] On the other hand, the contours o domain patterns on the surace can be detected by imaging the local piezoelectric response with both amplitude and phase inormation, with the aid o a lock-in ampliier. [86] 115

130 HAPTER 3, EXTRINSI PIEZOELETRI RESPONSE IN THIN EPITAXIAL FERROELETRI FILMS 3.1 Orientation dependence o 9 o domain pattern In tetragonal erroelectric ilms, 9 o domains orm as elastic domains to release the internal growth stresses. For (1) oriented ilm, the neighboring elastic domains have polar vectors aligned out-o-plane and in plane, alternatively, which is usually called a c/a/c/a 9 o domain pattern. Under an applied electric ield, the extrinsic contribution to d rom the 9 o domain wall movement can be comparable to or even larger than the intrinsic d, as has been calculated by Yu and Roytburd et al. [9] N. Pertsev et al. also calculated the extrinsic piezoelectric response rom domain wall motion or a tetragonal erroelectric thin ilm with a dense c/a domain structure. [91] In apparent piezoelectric coeicient d or a (1) oriented tetragonal lead titanate thin ilm, their calculation showed an increase o ~7pm/V, which is about the same as the intrinsic d. It can be expected that, as the composition o the tetragonal PbZr 1-x Ti x O 3 ilm approaches the morphotropic phase boundary (x~.5), the extrinsic contribution to d rom the 9 o domain wall movement will be enhanced, as both the intrinsic piezoelectric properties and the sotness o the material with respect to the change o polar state peak at the MPB. Furthermore, there is an orientation dependence o the 9 o domain patterns. For a (1) oriented tetragonal ilm, a 3-domain pattern will orm due to the symmetrical biaxial in-plane stress (Fig 3.1 a, b). [92] The two sets o a domains are orthogonal and thereore are pinned by each other. They are not as easy to move under the electric ield as a c/a 2-domain pattern which has been theoretically modeled aorementioned. For a (111) oriented tetragonal ilm, all three variants o the tetragonal phase are equivalent 116

131 under an applied electric ilm along the ilm normal ( Engineered Domain oniguration ), thereore there will be no extrinsic contribution rom domain wall movement toward the longitudinal piezoelectric response. For a (11) oriented tetragonal ilm, the in-plane stress is close to a uniaxial one due to the anisotropy in misit strains (Section 3.2), a dominant c/a 2-domain pattern (may still contain the third domain variants in minor ractions) with canted and in-plane polar vectors alternatively aligned in neighboring domains is expected (Fig 3.1 c, d). As the ilm composition approaches the MPB, the sotness o the PZT material in shear directions [27],[29] also eases the 9 o domain movement in a non-polar (11) orientation in addition to the domain pattern. Unlike the c/a two-domain pattern calculated by Perterv et al. in a (1) orientation [91], which is in a stressed state, the two-domain pattern in (11) is a ully relaxed structure. Thereore it is possible that the (11) orientation can have a larger extrinsic contribution to d rom its 9 o domain wall movement than the (1) orientation or MPB-PZT ilms. 117

132 [ 1 ] #=9 o [1 ] (a) (b) [1 1-1] [1 1 1] #~7 o (c) (d) Fig (a) topography, (b) out-o -plane piezoelectric orce microscopy (PFM) or illustration o the domain structure o a tetragonal (1) ilm: (PZT 2/8, 5 µm 5 µm, the bright matrix areas are c domains while the orthogonal gray strips are a domains which have zero longitudinal piezoelectric response); (c) topography, (d) in-plane PFM or illustration o the domain structure o a tetragonal (11) ilm (PZT 52/48, 1 µm 1 µm, the areas with grey contrast are c domains which have small piezoelectric shear deormation; while the strips with white and dark contrasts are a domains which have large shear piezoelectric deormation. The opposite contrasts in a domains indicate the opposite signs o the in-plane polarizations). 118

133 Anisotropic misit in a (11) oriented epitaxial ilm For a tetragonal phase with sel strain 1 =(a T - a c )/a c, 3 =(c T -a c )/ a c (a T and c T are the lattice constants o tetragonal c and a axis, respectively, and a c is the lattice constant o the cubic phase, thereore 1 < and 3 >,), 1 = 2 12 s P Q and 3 = s s P Q P Q =. [6] = ~ / = Q Q. For a (11) plane, the sel strain or a and c domains are: = = ^ s a P Q ^ c = ) ( 2 ) ( = P s Q For a c/a/c/a two domain structure with being the a domain raction, the average selstrain is: = + = ) ( ) ( ) ( ) ( ) ( ) (1 ) ( ^ ^ ^ c a = ) 1( 2 1 ) 1( ) (1 ) ( s Q P For the equilibrium a domain raction ~(1+) -1, [92] it can be seen that ) ( 2 = and it is uniaxial strain in the (11) ilm growth plane. In reality, the a domain ractions in a tetragonal ilm is directly related with the ratio o eective misit strain ( _ M ) and its tetragonality ( T ) by a =- _ M / T. [92] I we consider that the ilm is ully relaxed (or

134 thick ilms) on the substrate in its cubic paraelectric phase, then _ M ~ a, the compressive tetragonal strain, and T ~ a + c with c being the elongational tetragonal strain. An estimation or the a domain raction in thick tetragonal ilms can be made by substituting 2 2 a =Q 12 P s and c =Q 11 P s, which results a ~(1+) -1, where =-Q 11 /Q 12 is characteristic o the anisotropy in electrostrictive eects. This gives a ~3% or PZT 5/5 ilms in a ully relaxed state. For PZT ilms epitaxial grown on STO substrate, _ M is smaller than a due to the compressive state in the ilm rom thermal mismatch, [92] thereore a ~3% is the upper bound or the a domain raction. In our experiment, a ~2%is estimated rom X-ray rocking curve and PFM results (Fig 3.4) or the ilms in remnant state, which is also supported by the remnant polarization measurement results. [6] 3.3 Experiment discovery o highly mobile 9 o domain walls and ultra high extrinsic piezoelectric responses due to 9 o domain wall movement BAKGROUND Lead Zirconate Titanate solid ceramics (PbZr 1-x Ti x O 3 ) with compositions close to the morphotropic boundary are used in sensors, actuators, etc., because o their high piezoelectric coeicients. Due to the substrate clamping eect, the extrinsic contributions to the piezo-response are reduced and the eective piezoelectric coeicients o PbZr 1-x Ti x O 3 ilms are degraded relative to bulk materials (typically on the order o 1 pm/v). We recently discovered reversible strain as high as.35% (longitudinal) in the (11) oriented epitaxial PbZr.52 Ti.48 O 3 ilms under applied ac 12

135 electric ields o <3kV/cm, resulting in the eective piezoelectric coeicients on the order o 1 pm/v. Transmission electron microscopy, X-ray diraction analysis, and atomic orce microscopy revealed that the (11) ilm eatures a sel-assembled polydomain structure, consisting o two domain sets o a tetragonal phase. Thermodynamic analysis suggests the high strain level in this ilm may be understood on the basis o a ield-induced 9 o domain wall movement, which is greatly enhanced due to the special domain pattern and sotness o the ilm material. PbZr 1-x Ti x O 3 (abbreviated as PZT) solid solution system is o many practical applications, among which piezoelectric transducers (sensors, actuators) are major application ields especially or PZT materials with compositions near its morphotropic phase boundary (abbreviated as MPB-PZT, x~.5). One o the important piezoelectric constants is the converse longitudinal piezoelectric coeicient, namely, d, which is a measure o how much normal strain the piezoelectric material can achieve per unit o applied electric ield and hence a measure o the perormance o piezoelectric actuators. MPB-PZT bulk ceramic materials show large d coeicients (~3-8pm/V), which has long been known to have a dominant extrinsic contribution rom 9 o domain movement. [93]-[95] However, or erroelectric ilms which are widely used in integrated systems such as micron-electrical-mechanical-systems (MEMS), it was ound that clamping o substrate greatly reduces the d coeicient. [36], [96] Particularly, it was reported that the extrinsic contributions by 9 o domain wall movement to the piezoelectric response are strongly suppressed in most ilms under a micron in [97], [98] thickness. There are many reports on the d coeicient o erroelectric ilms by dierent measurement techniques and most o them showed similar results on the order 121

136 [38], [99]-[13] o 1 pm/v or ilms less than a micron in thickness. However, most o these measurements were carried out under relatively small signal excitation o the ilm with applied electric ield along the polar axis, or in ilms with polycrystalline quality. Thereore, it is interesting to ask whether or not larger piezoelectric responses can be generated or epitaxial ilms o dierent orientations under excitation ield o variable strengths. EXPERIMENT RESULTS In this study, epitaxial PbZr 1-x Ti x O 3 ilms were grown on SrTiO 3 substrates with 2 dierent crystallographic orientations, (1) and (11), using an on-axis radio-requency magnetron sputtering technique. The nominal Zr/Ti ratio o the sputtering target was 52/48. Prior to the PZT ilm deposition, 2nm thick epitaxial SrRuO 3 bottom electrodes were deposited on the substrate by a 9 o-axis sputtering technique. [83] During the PZT ilm deposition, the substrate temperature was maintained at 6 with an oxygen pressure o 4 mtorr, ollowed by cooling under an oxygen pressure o 3 Torr. The thicknesses o the PZT ilms were about 7~1 nm. The composition o the PZT ilms is the same as the target as conirmed by the Energy Dispersion Spectroscopy. For the purpose o electric and piezoelectric measurements, Pt electrodes were abricated via UV lithography and pulsed laser deposition. X-ray diraction analysis showed good epitaxy and tetragonally distorted structures or both (1) and (11) ilms. As an example, Fig 3.2 shows the dierent scans used to veriy the crystalline structure o the (11) ilm, the splitting in the {2} peak in Fig 3.2 (b) and the non-splitting (222) peak in Fig 3.2 (c) support a tetragonal symmetry against a rhombohedral one. The a domain ractions or both (1) and (11) ilms are estimated to be ~2% rom x-ray rocking 122

137 curves (not shown here). ross-section and plain view transmission electron microscopy (TEM) were perormed or the (11) ilms. In Fig 3.3 (a) (b), it clearly showed that two sets o striplike parallel domains were ormed in this ilm with a <111> type domain boundary trace lying on the (11) growth plane. The TEM results were interpreted in Fig. 3.3 (c). A tetragonal polydomain structure with alternating c and a domains has a {11} domain boundary. This domain boundary will have a <111> trace when the ilm is grown on a (11) crystalline plane. Imaging-mode piezoresponse microscopy [81] was perormed or the (11) ilm. A dominant 2-domain pattern with two sets o plane parallel large domains was discovered in this ilm rom the topography and in-plane piezoelectric images as shown in Fig. 3.1 (c) (d). It was also illustrated that the angle between the traces o the a domains is ~ 68 o, which is in a good agreement with the angle between two <111> traces (~7 o ). This observation supported the X-ray and TEM analysis results. Furthermore, in Fig 3.4 (a)-(c), it is illustrated that these a-domains are easy to move under a local writing ield provided by the PFM tip, which was not discovered or (1) oriented ilms with symmetrical 3-domain architectures. Thereore, it is interesting to know how the domain wall movement will contribute to a global measured strain under an uniorm ield applied through a top electrode pad. 123

138 (11) ilm (11) plane scan Log (Intensity) (1) phi scan Intensity (Arb. Units) (degrees) PZT 11 STO 11 SRO 11 Intensity (arbi. unit) (11) ilm (2) plane scan PZT 2 PZT 2 STO (11) ilm (222) plane scan (a) (b) Intensity (arbi. unit) PZT 222 STO theta (degree) (c) Fig (a)-(c) X-ray result or (11) PZT 52/48 ilm showing a tetragonal distorted structure. 124

139 #~7 o (a) (c) (b) Fig. 3.3 (a) ross-sectional TEM image (twin traces are // <111>); (b) Plain-view TEM image (dark-ield image near <11> zone axis.; the twin traces are // <111>) or (11) oriented PZT 52/48 ilm. (c) Schematics o 9 o domain pattern in a (11) ilm. 125

140 (a) Fig. 3.4 Piezoelectric Force Microscopy (PFM) results o (11) PZT 52/48 ilm (6µm 6µm), Top: topography, center: out-o plane PFM, bottom: in-plane PFM: (a) Original. 126

141 (b) Fig. 3.4 Piezoelectric Force Microscopy (PFM) results o (11) PZT 52/48 ilm (6µm 6µm), Top: topography, center: out-o plane PFM, bottom: in-plane PFM: (b) Ater writing 1V in a 3µm 3µm area. 127

142 (c) Fig. 3.4 Piezoelectric Force Microscopy (PFM) results o (11) PZT 52/48 ilm (6µm 6µm), Top: topography, center: out-o plane PFM, bottom: in-plane PFM: (c) Poling back using +1V in the same 3µm 3µm area. 128

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