1.IntroductionRecent developments in microelectromechanical systems (MEMS), non-volatile ferroelectric memories (FeRAMs) [1], ferroelectric field effect transistors [2], and ferroelectric heterostructures [3, 4], as well as in micron- and nanoscale electromechanical sensors and actuators have attracted significant interest to the ferroelectric behaviour on the nanoscale [5, 6]. Of particular interest are details of electromechanical coupling and dielectric response related to the intrinsic ferroelectric size effect, surface and interface phenomena, and domain wall dynamics. Progress in these fields requires an understanding of local ferroelectric properties at the nanometre level.Rapid developments in scanning probe microscopy (SPM) techniques have given rise to a range of local probe techniques capable of accessing polarization-dependent electromechanical, elastic, and electric properties on the nanoscale [5–7]. Among these techniques, the most popular one is piezoresponse force microscopy (PFM), due to ease of implementation, high resolution, and its relative insensitivity to topography. Application of periodic bias to the conductive tip in contact with ferroelectric surface results in periodic surface displacement due to inverse piezoelectric effect. Mapping of the amplitude and phase of the displacement allows imaging of ferroelectric domain structures with about 3–10?nm resolution [5, 6]. Therefore, PFM imaging provides direct insight into the nanoelectromechanics of ferro- and piezoelectric materials on the length scales defined by radius of the tip-surface contact and the radius of curvature of the tip. The tip bias and the indentation force can be varied in a broad range, thus providing an experimental approach to study field- and stress-induced phenomena in ferroelectrics. A number of other SPM techniques based on mechanical and electromechanical phenomena such as scanning near field acoustic microscopy (SNAM) [8], atomic force acoustic microscopy (AFAM) [9], and heterodyne ultrasonic-electrostatic force microscopy (HUEFM) [10] were also developed and applied for local characterization of ferroelectric and piezoelectric materials. It can be expected that PFM and other electromechanical SPMs can provide a wealth of quantitative information on the elastic, electromechanical, and dielectric properties of ferroelectrics on micro- and nanoscales. However, quantitative interpretation of these techniques requires knowledge of the contact mechanics of piezoelectric materials, in order to relate indentation depth, tip bias, and indentation force with indenter parameters and relevant material properties. While Hertzian contact mechanics is widely used in the interpretation of acoustic microscopy on non-piezoelectric materials, analysis of PFM, SNAM, AFAM, and HUEFM on ferroelectric and piezoelectric materials requires Hertzian contact mechanics to be extended to materials with strong electromechanical coupling.Another broad set of SPM applications is based on tip-induced changes in material properties. Application of high voltage or stress to the PFM tip can induce local 180° or 90° polarisation switching, providing an approach to engineer and control domain structures at the nanoscale. This approach can potentially be used for high-density ferroelectric storage [11–13]. Minimal switched domain size was experimentally demonstrated by Tybell et al.[12] to be as small as about 40?nm, corresponding to recording densities of the order of 400?Gb/in 2 . Theoretical predictions suggest that the minimal stable domain size is of the order of several unit cells; combined with demonstrated ferroelectric properties in the films of two unit cells' thickness, this opens way to the nearly-atomic level data storage [14]. Polarisation-dependent reactivity of the surface in the acid etching [15] or metal photodeposition processes [16, 17] can be used to engineer nanoscale structures (ferroelectric lithography). The practical viability of these SPM applications is critically dependent on the minimal stable domain size that can be formed during polarisation switching induced by a voltage bias or stress applied to the probing tip [18]. Addressing this issue requires quantitative knowledge of tip-induced electroelastic fields inside the material to predict and control polarisation-switching processes.In the present work, we analyse problems of piezoelectric indentation involving indenters of several different shapes. Exact solutions for the electroelastic fields are obtained in elementary functions. In addition to being of interest for continuum mechanics applications, these results are used for modelling and quantitative interpretation of the electromechanical SPMs on ferroelectric surfaces.2.Background: the elastic–piezoelectric correspondence principle (CP)Coupled electromechanical problems of piezoelectric indentation present a difficult mathematical challenge. The following earlier works should be mentioned in this connection.Chen [19] and Chen and Ding [20] derived electroelastic fields for the circular flat and spherical punch problem, respectively. However, their results are given in a form that does not explicitly identify the combinations of electroelastic constants in whose terms the fields are expressed. These combinations are identified in the analysis to follow.In the work of Giannakopoulos and Suresh [21] and the follow-up work of Giannakopoulos [22], three punch geometries were considered: spherical, conical, and flat circular. In these works, electroelastic fields were given in the closed form in the plane z?=?0. Inside the material, the fields are given in the form of integral representations, which makes it more difficult (as compared to solutions in elementary functions) to distinguish the contributions of the bias- and the stress effects. In addition, any subsequent use of these solutions (e.g. for the analysis of the polarisation switching phenomena), requires non-trivial numerical procedures that obscure the physical origins of the studied process. We also remark that their boundary conditions contain the statement that szz?=?0 at the edge of the contact zone (??=?a): being correct for the spherical and conical shapes, it is incorrect for the flat punch; moreover, ??=?a is actually a singularity point in this case, as seen from the table 3 in Giannakopoulos and Suresh [21]. Note, also, that the stiffness relations derived in the mentioned works contain constants that are numerically different from the ones derived here; it appears that our relations are correct since they have been verified (by rather lengthy calculations) to be in agreement with independently obtained results of Chen and Ding [20].A universal approach to solving the piezoelectric indentation problems for various indenter geometries is given by the recently established correspondence principle (CP) between the elastic and the piezoelectric problems for transversely isotropic materials [23]. This principle yields the coupled electroelastic fields provided that the purely elastic solution for the corresponding elasticity problem is known, and that this solution is written in a certain form emanating from theory developed by Fabrikant [24, 25]. Most of the known elastic solutions for the transversely isotropic material actually do have this form; those few that do not, can be reduced to it.The correspondence principle of Karapetian et al.[23] is based on the following key observations: the general solution of the field equations for (1) the purely elastic medium in terms of three harmonic potential functions, Fj = F (x,?y,?zj) (where zj?=?z / ? j and ?j are certain combinations of anisotropic elastic constants, j?=?1,?2,?3) and (2) for the piezoelectric medium in terms of four harmonic potential functions Fj = F (x, ?y, ?zj), (where and are certain combinations of elastic, piezoelectric, and dielectric constants, j?=?1,?...?,?4) have a remarkably similar form. For the purely elastic medium, such representations in terms of potential functions were established by Fabrikant [24]. The CP is based on extending these representations to the piezoelectric medium and the solution for Green's function for the piezoelectric space and half-space in the form obtained by Karapetian et al.[26]; we note that the mentioned Green's functions were also given by Dunn and Wienecke [27] in a different form that is not immediately suited for establishing the CP.More specifically, the CP establishes the following correspondences: it defines which elastic and piezoelectric boundary value problems are considered to be analogous;it identifies correspondence between “clusters” of the piezoelectric constants and “clusters” of the elastic constants;it provides tables of correspondence that yield explicit solutions of the piezoelectric boundary value problems on the basis of the ones for the corresponding purely elastic problems. There are two such tables: one for the problems that involve an infinite space that may contain distributed “sources” (for example, an infinite space containing an inhomogeneity) and another one for the half-space related problems (crack and punch problems, for example); certain problems that involve elements of both types (such as the interaction problems) may require utilization of both tables.The implementation of CP was illustrated by Karapetian et al.[23] on several electromechanical problems including crack and punch problems as well as crack–point force interaction. In the text to follow, we utilize this principle to obtain exact solutions to problems of indentation of the piezoelectric half-space. Full electroelastic fields in the entire half-space are derived in elementary functions, for the spherical, conical, and flat indenters. These geometries correspond to the limiting cases of SPM tip geometry. Solutions are given for the uniform indenter potential, corresponding to the uniformly biased conductive tips used in the SPM techniques.3.Indentation problem for piezoelectric half-spaceWe consider indentation of the piezoelectric half-space by punches of three geometries: circular flat, spherical, and conical. The half-space is transversely isotropic, with the isotropy plane parallel to the boundary. In section 3.1, constitutive equations and relevant combinations of material constants are introduced. Sections 3.2, 3.3, and 3.4 present solutions for the flat, spherical, and conical punches. In all three cases, we identify the proper boundary conditions and derive full coupled electroelastic fields. The latter fields are represented as superpositions of solutions of two sub-problems, with zero potential and with zero displacement under the punch (“purely mechanical” and “purely electrical” boundary conditions, respectively). Although this separation is not needed for obtaining the solution (the correspondence principle provides the solution in the general case), it provides physically useful insights: (1) solution of the purely electrical sub-problem is found to be identical for all three geometries and (2) such representation allows bias- and force-induced phenomena in electromechanical SPMs to be separated. We then examine the electromechanics of piezoelectric indentation, by deriving stiffness relations that relate the indentation force, indenter bias, indentation depth, and indenter charge. This extends the corresponding results of Hertzian mechanics to piezoelectric materials. We also derive the intensity factors for the stresses and for the electric displacements.An important observation is that solutions of the sub-problems with purely electrical boundary conditions are identical for all three punch geometries, provided that the radii of the contact areas are the same, as becomes clear from the text to follow. For the spherical and conical geometries, these radii are expressed in terms of the imposed indentation displacement, while for the flat geometry, the contact area is constant. Therefore, the solution to the sub-problem with purely electrical boundary conditions is given only once for the flat circular punch.3.1.Notations and relevant combinations of constantsWe retain notations used by Karapetian et al.[23]. We consider the case when the material is transversely isotropic with respect to all three groups of properties (elastic constants, piezoelectric coupling and dielectric permeabilities), with z axis being the axis of symmetry for each of them. Then the linear constitutive equations have the form For displacements, ui, stresses, sij, electric potential, ?, and electric displacement components, Di the following complex notations will be used: The transversely isotropic elastic and piezoelectric constants entering equations (3.1) are defined as follows: cij are elastic stiffnesses, eij are piezoelectric constants, ?ij are dielectric permeabilities, and Constants are defined by the following relations (j?=?1,?2,?3): where are roots of the cubic equation with coefficients The cubic equation (3.4) follows from the algebraic equations [26]: The following combinations of the piezoelectric constants will be used: and The following geometric parameters are used (j?=?1,?2,?3) In derivation of the fields in limiting cases the following relations are utilized: 3.2.Solution for the circular flat indenter3.2.1.Boundary conditionsA rigid flat punch with circular base of radius a indents a transversely isotropic piezoelectric half-space, z?>?0, as shown in figure 1a. Results are presented as a superposition of the fields in two sub-problems, with (A) purely mechanical boundary conditions and (B) purely electrical boundary conditions. We note that solution of sub-problem (B) is identical for all three punch geometries, provided the circular contact areas have the same radius. These two sub-problems are characterized by the following boundary conditions on plane z?=?0:Figure 1. Geometrical parameters for flat (a), spherical (b), and conical (c) indentation.(A) Purely mechanical boundary conditions (zero electric conditions, ??=?0 for 0 = < 8): (B) Purely electrical boundary conditions (zero mechanical conditions, uz = 0 for 0 = ?< 8): where prescribed vertical displacement, w, and electric potential, ?0, are constant and the cylindrical coordinates (?, ?, z) are used. We seek the electroelastic fields in the half-space.3.2.2.Electroelastic fieldsAs mentioned above, utilisation of CP requires that solution of the corresponding purely elastic problem is available and is written in a certain form. The available purely elastic solution [24] needs to be modified: being given in terms of prescribed displacement, w, it has to be re-expressed in terms of total contact force, P, as required by Correspondence Tables (see Karapetian et al.[23]). It is done by using the “stiffness relation” between w and P. With this adjustment, the mentioned Correspondence Table 2 gives piezoelectric fields in terms of force P (required to maintain w) and charge Q (required to maintain potential ?0). Focusing on values of uz and ? on the boundary, as given by this solution, re-expressing P, Q in terms of w, ?0 (from two algebraic equations) and inserting these expressions into the solution yields the following results.For sub-problem A: For sub-problem B: The structure of the obtained electroelastic field is illustrated in figures 2 and 3 for the values of materials parameters of BaTiO3 and LiNbO3 and the following indentation parameters: radius of the contact area a?=?3?nm, indentation depth w?=?2?nm and tip bias ?0?=?1?V. Relevant material parameters are listed in table 1. Note that, at large distances from the contact area, the displacement field is determined primarily by the anisotropy of the elastic constants tensor cij which for ferroelectric perovskites depends only weakly on the chemical composition of material. The potential in sub-problem A is prescribed to be zero under the tip; it reaches maximum at a certain depth. The maximum value of the potential in this case is determined by the strength of the electromechanical coupling in the material.Figure 2. 2D spatial distribution of the normal displacement (a, b), electrostatic potential (c, d), normal stress (e, f), and electric displacement (g, h) in the mechanical (a, c, e, g) and electrical (b, d, f, h) sub-problems for flat indenter for BaTiO3. Images are shown in the logarithmic scale. Contact radius a?=?3?nm and indentation depth w?=?2?nm. In electrical problem, tip potential ?0?=?1V. These conditions correspond to indentation force P?=?1.54?µN.Figure 3. 2D spatial distribution of the normal displacement (a, b), electrostatic potential (c, d), normal stress (e, f), and electric displacement (g, h) in the mechanical (a, c, e, g) and electrical (b, d, f, h) sub-problems for flat indenter for LiNbO3. Images are shown in the logarithmic scale. Contact radius a?=?3?nm and indentation depth w?=?2?nm. In electrical problem, tip potential ?0?=?1V. These conditions correspond to indentation force P?=?2.47?µN.1Table 1. Coupling constants for different materials.Material, 10 11 N/m 2 ?=?-?, N/Vm, 10 -9 C/mVBaTiO34.0315.4048.54LiNbO36.477.523.11LiTaO37.808.802.81PZT6B3.6025.6023.63In sub-problem B, the potential field is determined primarily by the anisotropy of the dielectric constants tensor ?ij. Note the difference in potential distributions for BaTiO3 and LiNbO3 in figure 2d and figure 3d caused by significantly higher dielectric anisotropy (?11/?33?=?17.4 for BaTiO3 and 2.8 for LiNbO3) of the former. Normal displacement in this sub-problem is prescribed to be zero at the surface, and attains maximal values at a certain distance from the surface. It can be shown that this situation corresponds to negative force acting on the surface. Note that a qualitative similarity exists between the shapes of strain distribution in sub-problem B and of potential distribution in sub-problem A; both have local maximums at certain distances from the surface.The normal stress szz under the indenter, in both sub-problems, has a well-known square root singularity at the perimeter of the contact area. A similar singularity exists in the normal component of the electric displacement Dz in both problems.A prominent feature of the field distributions in figures 2 and 3 is that nontrivial behaviour persists on a length scale comparable to the contact radius. For the distances substantially larger than contact radius, the field approaches the asymptotic power law behaviour expected for the point-charge/point-force model, which are the limiting case of the indentation geometry (see results of Kalinin et al.[28] for spherical indentation). This implies that for separations from the indentation zone exceeding the contact radius, the indenter can be modelled, with very good accuracy, as a point charge or point force, considerably simplifying the description of the bias- and stress-induced phenomena at large distances from the contact. Relevant parameters such as force and charge magnitudes (including elastic, dielectric, and electromechanical coupling effects) can be determined from stiffness relations.We now specify the electroelastic fields on the surface (z?=?0) and at ??=?0. The former are relevant for analyses of the tip-induced surface dynamics, the latter for analyses of the bias- and stress-induced phenomena directly below the tip.For fields in plane (z = 0) and sub-problem A:For ? a (outside of contact zone): For sub-problem B: For ??a (outside of contact zone): For fields at ??=?0, in the limit of ??0, we have l1j ?0, and some of the terms contain indeterminate ratios of the 0 / 0 type that have to be evaluated using L’Hospital's rule. After some calculations, the solution simplifies as follows:For sub-problem A: For sub-problem B: 3.2.3.Stress- and electric displacement intensity factors (SIF and EDIF)Both szz and Dz components of the solution (formulae 3.15 and 3.17) have the square-root singularity at the edge of the punch. The values of (Mode I) SIF and EDIF can be obtained from the fields above (at ??a (outside of contact zone): For sub-problem B the solution is identical to the one for the flat circular indenter, see formulae (3.17)–(3.18).Fields at ??=?0, for sub-problem A: For sub-problem B the solution is identical to the one for the flat circular indenter, see formulae (3.20).3.3.3.Stress- and electric displacement intensity factors (SIF and EDIF)We observe that in the case of the spherical indenter, szz and Dz in sub-problem A [purely mechanical boundary conditions, formulae (3.28)] do not have the square-root singularity at the punch edge, whereas they do have the singularity in sub-problem B (purely electrical boundary conditions, formulae (3.17)). The values of (Mode I) SIF and EDIF are obtained by substituting szz and Dz of formulae (3.17) into formulae (3.21): 3.3.4.Piezoelectric stiffness relationsThe above derived solution implies the following stiffness relations that interrelate applied force P and concentrated charge Q (required to maintain prescribed w0 and ?0) to w0 and ?0; they are obtained by integrating szz, Dz at z?=?0 over the contact region.Integration of the stress components over the contact area yields: Similar integration of the electric displacement components over the contact area yields: 3.4.Solution for the conical indenterBoundary conditions for the conical indenter are formulated as follows: where w0?+?? is the vertical displacement of the cone and ? is the “depth of penetration” as shown in figure 1c. Thus, boundary condition w(?,??) is prescribed as a function containing three parameters: a (radius of the contact zone), w0 and ? that are to be found in the process of solution. Since e?=?a cot (a) where a is the cone angle and w0?+?? is the prescribed vertical displacement of the cone, there remains only one independent parameter, taken to be ? in the text to follow. Electric potential ?0 is prescribed to be constant in the contact zone, equal to the tip potential.The sub-problems A and B are defined by the following boundary conditions:(A) Purely mechanical boundary conditions (zero electric conditions, ??=?0):(B) Purely electrical boundary conditions (zero mechanical conditions, uz?=?0): 3.4.1.Electroelastic fieldsApplying the same logic as in the flat punch problem and utilizing solution of the elastic problem for the transversely isotropic half-space [30], the correspondence principle yields the following electroelastic fields.For sub-problem A:For sub-problem B the solution is identical to the one for the flat circular indenter, see formulae (3.14).Electroelastic fields in the sub-problem A under the conical indenter with a?=?30° for BaTiO3 and LiNbO3 are illustrated in figure 5. The behaviour at large distances from the contact area is qualitatively similar to that for flat and spherical indenters, figures 2, 3, and 4. The normal displacement directly below the tip is no longer constant and decreases to the periphery of the contact area. Unlike the flat indenter, there is no singularity in the normal stress szz and electric displacement Dz fields at the boundary of contact area, ??=?a. However, both szz and Dz diverge at the centre of contact zone, ?,?z?=?0, as expected for the conical indenter. Note that conical tip shape is extremely unstable with respect to wear during imaging and can be expected to be a good approximation for tip shape only for imaging and spectroscopic measurements on soft materials such as ferroelectric polymers.Figure 5. 2D spatial distribution of the normal displacement (a, b), electrostatic potential (c, d), normal stress (e, f), and electric displacement (g, h) in BaTiO3 (a, c, e, g) and LiNbO3 (b, d, f, h) for the mechanical problem for conical indenter geometry. Images are shown in the logarithmic scale. Indentation parameters are a?=?30° and indentation depth w?=?2?nm, corresponding to indentation force P = 2.44?µN for BaTiO3 and 3.96?µN for LiNbO3.Fields in plane (z?=?0), sub-problem A:For ??a (outside of contact zone): For sub-problem B the solution is identical to the one for the flat circular indenter, see formulae (3.17)–(3.18).The formula for uz above yields the following relation between the “penetration depth” and the displacement of the cone: For fields at ??=?0: For sub-problem B the solution is identical to the one for the flat circular indenter, see formulae (3.20).3.4.2.Stress- and electric displacement intensity factors (SIF and EDIF)Similar to the case of the spherical indenter, szz and Dz in sub-problem A (purely mechanical boundary conditions, formulae (3.40)) do not have the square-root singularity at the punch edge, whereas they do have the singularity in sub-problem B (purely electrical boundary conditions, formulae (3.17)). Values of and are given by formulae (3.31).3.4.3.Piezoelectric stiffness relationsThe above solution implies the following stiffness relations that interrelate displacement of the cone w0?+??, electric potential ?0, applied force P and indenter charge Q. They are obtained by integrating and at z?=?0 (sub-problem A) over the contact area that yields and using e?=?(2 / p) (w0 + e) where second terms are obtained from the solution of sub-problem (B), as a result of integration of and over the contact area.Other useful relations for sub-problem A that follow from the results above and from the geometrical relation for the radius of the contact zone a?=?(2 / p) (w0 + e)tan a are:We note that the limiting cases of the purely elastic problems and of the purely electrostatic problem are recovered from the results obtained above by setting the coupling constants eij?=?0. This implies the following replacement of the constants involved: 4.Nanoelectromechanics of SPMsThe theoretical results of section 3 yield full electroelastic fields in the half-space expressed in elementary functions and stiffness relations for the indenters of various geometries. Here, we discuss these results in the context of quantitative description of the imaging mechanisms of SPM of ferroelectric and piezoelectric materials. We will utilize the fact that , as follows from somewhat tedious algebraic derivations.4.1.Stiffness relations for an arbitrary indenter shapeThe stiffness relations derived in section 3 interrelate applied force, P, and concentrated charge, Q, with indenter displacement, w0, indenter potential, ?0, indenter geometry and materials properties. For the indenter geometries studied in section 3, these stiffness relations have the following phenomenological structure: where h is total indenter displacement (h?=?w0 for flat and spherical indenters and h?=???+?w0 for conical indenter), ? is a geometric factor (??=?a for a flat indenter, ??=?(2 / 3)R 1 / 2 for spherical indenters and ??=?(1 / p) tan a for a conical indenter) and n?=?0 for flat, n?=? 1 / 2 for the spherical and n?=1 for the conical indenters, respectively.These stiffness relations provide an extension of the corresponding results of Hertzian mechanics and continuum electrostatics to the transversely isotropic piezoelectric medium. By comparing equation (4.1) with stiffness relations for isotropic elastic solid for three indenter geometries studied, the indentation elastic stiffness for the piezoelectric indentation problem is analogous to the effective Young's modulus for isotropic material, E*?=?E / (1?-?? 2 ), where E is Young's modulus of the material below the indenter and ? is Poisson's ratio, . Similarly, by comparing the indenter charge (4.2) with the capacitance of the conductive disc on dielectric half-plane, indentation dielectric constant is found to be analogous to the dielectric constant for the uniform material, . Electromechanical coupling is determined by indentation piezocoefficient. In the contact problem, ratio describes coupling between the force a [ABSTRACT FROM AUTHOR]