Owning the ability of converting between mechanical and electric energies, piezoelectric materials such as Lead Zirconate Titanate (PZT) are ubiquitous in many microelectromechanical systems (MEMS) including sensors and actuators. On the one hand, advanced micro-fabrication technique has provided the synthesis and fabrication of PZT layer, preferably deposited on a silicon/silicon oxide substrate, which is the key active component in MEMS. On the other hand, characterizing material properties of PZT layer plays an essential role in the design process, yet is a challenging task due to the small-scale and electromechanical coupling nature.
Nanoindentation is one of the most widely used technique in the determination of material properties of thin film. The technique involves the compression of a hard indenter into the surface of the film rest on a substrate, meanwhile the load magnitude and indentation depth are recorded to interpretate material properties such as hardness or Young’s modulus . Such depth-sensing approach not only has been successfully investigated purely mechanical properties of several materials but also employed in the studies of piezoelectric materials, including analytical models [2,3], numerical modelling [4,5] and experimental measurements [6,7]. However, there exist several unsolved challenges in nanoindentation of piezoelectric materials. The first key issue with most of the theoretical models (analytical and numerical), while the simplified axis-symmetric electromechanical governing equations in combination with frictionless assumption of the indenter has resulted in a reliable compression force-indentation depth curve, the unloading curve is overlooked. The slope of the unloading curve contains information of the material properties in linear regime. The second issue with the compression force-indentation depth curve is the assumption of material parameters such as elastic moduli, piezoelectric and dielectric coefficients. In practice, these parameters are rather the unknowns. The third problem is the interpretation of measured depth-sensing curve to estimate the material properties, where in most of reports, Oliver-Pharr formular was used for general anisotropic material but the formular is only valid for isotropic materials , leading to an overestimation of true Young’s modulus. Hence, there are still significant opportunities to develop a more robust nanoindentation model for piezoelectric materials which will effectively determine material coefficients and offers a reliable “virtual-indentation-test” for MEMS applications.
Besides, when PZT material is subjected to large electric field, its nonlinear behavior known as hysteresis effects emerges rendering linear relationship between polarization and applied electric field invalid. Hence, characterizing hysteresis behavior of PZT, in this case can be considered as ferroelectric material, is also an important task in MEMS design. Polarization hysteresis measurement method can be categorized as charge-based method such as Sawyer-Tower or current-based method such as virtual ground and shunt. Among these methods, a model of ferroelectric capacitor has been proposed and incorporated with Sawyer-Tower as a virtual-hysteresis-test . The model has been successfully demonstrating polarization hysteresis of PZT thin film, including the effects originated from the existence of passive layer, charge defects [10,11]. Nevertheless, Sawyer-Tower method in practice can be susceptible to parasitic effects from measuring components . Therefore, it is desirable to develop a more robust ferroelectric capacitor model based on virtual ground or shunt circuit so that more reliable polarization hysteresis of PZT thin film can be obtained.
 Oliver, W. C., & Pharr, G. M. (1992). An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. Journal of materials research, 7(6), 1564-1583.
 Giannakopoulos, A. E., & Suresh, S. (1999). Theory of indentation of piezoelectric materials. Acta materialia, 47(7), 2153-2164.
 Wang, J. H., Chen, C. Q., & Lu, T. J. (2008). Indentation responses of piezoelectric films. Journal of the Mechanics and Physics of Solids, 56(12), 3331-3351.
 Liu, M. (2012). Finite element analysis of the contact deformation of piezoelectric materials. University of Kentucky.
 Cheng, G., & Venkatesh, T. A. (2012). Nanoindentation response of anisotropic piezoelectric materials. Philosophical magazine letters, 92(6), 278-287.
 Fang, T. H., Jian, S. R., & Chuu, D. S. (2003). Nanomechanical properties of lead zirconate titanate thin films by nanoindentation. Journal of Physics: Condensed Matter, 15(30), 5253.
 Delobelle, P., Guillon, O., Fribourg-Blanc, E., Soyer, C., Cattan, E., & Remiens, D. (2004). True Young modulus of Pb (Zr, Ti) O 3 films measured by nanoindentation. Applied Physics Letters, 85(22), 5185-5187.
 Lamuta, C. (2019). Elastic constants determination of anisotropic materials by depth-sensing indentation. SN Applied Sciences, 1(10), 1-13.
 Hammerquist, C. C., & Nairn, J. A. (2018). Modeling nanoindentation using the material point method. Journal of Materials Research, 33(10), 1369-1381.
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The first objective of this work is to develop an advanced nanoindentation model for piezoelectric material, in which the frictionless assumption is removed and replaced by frictional model such that both loading and unloading force will be obtained. To achieve this goal, a formulation involving piezoelectric material constitutive laws and contact mechanics will be derived. Subsequently, the weak-form of the boundary value problem can be solved with appropriate methods, for instance Finite element method (by in-house code or commercial software) or discrete method such as Material Point Method (MPM) which has been proved to be efficient in modelling nanoindentation in elasticity . The numerical results will be validated with those of analytical models and experimental measurements for the loading curve. The second aim of this work is to “extract” properties of piezoelectric materials, including elastic moduli, piezoelectric and dielectric coefficients via inverse analysis. The loading/unloading-indentation depth will be served as the target of the inverse analysis, which can be based on heuristic approach such as Genetic Algorithm or gradient-based approach. It will be proved that the combination of loading and unloading phase can yield better prediction than the use of only loading curve as the loading-history is considered. To this end, the inversed predictions of anisotropic piezoelectric materials will be compared with available references. It should be noted that, there exists only analytical calculation of elastic constants from nanoindentation but not for coupling coefficients. Hence, suitable adjustment should be made to compare with the analytical model and further validate the results.
Besides, modelling of ferroelectric capacitor with virtual ground method can be considered as a contingency plan. In this innovative model, the ferroelectric capacitor response, which can be described through the rate of change of charge on its surface to the rate of change of input voltage, is incorporated with virtual ground electrical circuit. Kirchhoff’s law will be utilized to derive the current equilibrium such that ferroelectric charge can be related to output voltage. The resulting ODE will be solved numerically, for instance by forward Euler method. To this end, input parameters including ferroelectric coefficients such as saturation and remanent polarization, coercive electric field, dielectric permittivity will be calibrated with measured hysteresis curve, which will be obtained experimentally. Furthermore, the developed model will be modified for inverse analysis, in which the true polarization can be extracted from output voltage profile.