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DYNAMIC MODELING CONTROL AND MANIPULATION AT THE NANOSCALE

# Modeling Piezoresponse Force Microscopy for Low-Dimensional Material Characterization: Theory and Experiment

[+] Author and Article Information
Amin Salehi-Khojin, Saeid Bashash

Department of Mechanical Engineering, Smart Structures and Nanoelectromechanical Systems Laboratory, Clemson University, Clemson, SC 29634

Department of Mechanical Engineering, Smart Structures and Nanoelectromechanical Systems Laboratory, Clemson University, Clemson, SC 29634jalili@clemson.edu

Gary Lee Thompson, Alexey Vertegel

Department of Bioengineering, Clemson University, Clemson, SC 29634

During the experiments, the displacement of cantilever tip, i.e., $x=L$, is monitored as system output.

Note that the middle area of the microcantilever is cut out; hence, it maintains a constant cross-sectional area for most parts of its length, representing the properties of a rectangular beam, even though at the first look it seems to be triangular.

Due to the presence of nonhomogeneities and difference in the fabrication process, these values could be different from one sample to another.

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Corresponding author.

J. Dyn. Sys., Meas., Control 131(6), 061107 (Nov 06, 2009) (7 pages) doi:10.1115/1.4000161 History: Received June 13, 2008; Revised July 20, 2009; Published November 06, 2009; Online November 06, 2009

## Abstract

Piezoresponse force microscopy (PFM) is an atomic force microscopy-based approach utilized for measuring local properties of piezoelectric materials. The objective of this study is to propose a practical framework for simultaneous estimation of the local stiffness and piezoelectric properties of materials. For this, the governing equation of motion of a vertical PFM is derived at a given point on the sample. Using the expansion theorem, the governing ordinary differential equations of the system and their state-space representation are derived under applied external voltage. For the proof of the concept, the results obtained from both frequency and step responses of a PFM experiment are utilized to simultaneously identify the microcantilever parameters along with local spring constant and piezoelectric coefficient of a periodically poled lithium niobate sample. In this regard, a new parameter estimation strategy is developed for modal identification of system parameters under general frequency response. Results indicate good agreements between the identified model and the experimental data using the proposed modeling and identification framework. This method can be particularly applied for accurate characterization of mechanical and piezoelectric properties of biological species and cells.

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## Figures

Figure 1

(a) A schematic of a tip-sample junction in the PFM system and (b) schematic model of vertical PFM system and sample

Figure 2

(a) The Asylum Research MFP-3D, (b) the triangular microcantilever (TR400PB, Olympus), and (c) the PPLN chip on the MFP-3D stage

Figure 3

(a) Height, (b) PFM amplitude, and (c) PFM phase images of a PPLN

Figure 9

Frequency response of the identified model compared with the experimental resonances

Figure 10

Response of a PPLN to the unit step input voltage at test point depicted in Fig. 4

Figure 4

Experimental frequency response plot of PFM on a PPLN sample

Figure 5

Experimental setup for a microcantilever under a microsystem analyzer (MSA-400)

Figure 6

Examples for 3D demonstration of the microcantilever vibrations at (a) pure bending and (b) mixed bending/torsion motions

Figure 7

Modal system identification flowchart subjected to uncertainties

Figure 8

(a) Average error percentage trajectory in random optimization and (b) evolution of selected frequencies associated with the transversal motion

## Errata

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