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 behaviour known as hysteresis effects
emerges rendering linear relationship between polarization and applied electric
field invalid. Hence, characterizing hysteresis behaviour 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),
Giannakopoulos, A. E., & Suresh, S. (1999). Theory of indentation of
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 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),
 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),
 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.
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),
 Lamuta, C.
(2019). Elastic constants determination of anisotropic materials by
depth-sensing indentation. SN Applied Sciences, 1(10),
C. C., & Nairn, J. A. (2018). Modeling nanoindentation using the material
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L., Nasby, R. D., Schwank, J. R., Rodgers, M. S., & Dressendorfer, P. V.
(1990). Device modeling of ferroelectric capacitors. Journal of applied physics, 68(12), 6463-6471.
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L., Schwank, J. R., Nasby, R. D., & Rodgers, M. S. (1991). Modeling
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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.