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Research Papers

Effects of Thickness on the Responses of Piezoresponse Force Microscopy for Piezoelectric Film/Substrate Systems

[+] Author and Article Information
J. H. Wang

Department of Engineering Mechanics,
School of Mechanics,
Civil Engineering and Architecture,
Northwestern Polytechnical University,
Xi’an, Shaanxi 710129, China
e-mail: wangjh@nwpu.edu.cn

C. Q. Chen

Department of Engineering Mechanics,
AML & CNMM,
Tsinghua University,
Beijing 100084, China
e-mail: chencq@tsinghua.edu.cn

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 3, 2017; final manuscript received September 26, 2017; published online October 16, 2017. Assoc. Editor: Harold S. Park.

J. Appl. Mech 84(12), 121004 (Oct 16, 2017) (11 pages) Paper No: JAM-17-1483; doi: 10.1115/1.4038064 History: Received September 03, 2017; Revised September 26, 2017

Piezoresponse force microscopy (PFM) extends the conventional nano-indentation technique and has become one of the most widely used methods to determine the properties of small scale piezoelectric materials. Its accuracy depends largely on whether a reliable analytical model for the corresponding properties is available. Based on the coupled theory and the image charge model, a rigorous analysis of the film thickness effects on the electromechanical behaviors of PFM for piezoelectric films is presented. When the film is very thick, analytical solutions for the surface displacement, electric potential, image charge, image charge distance, and effective piezoelectric coefficient are obtained. For the infinitely thin (IT) film case, the corresponding closed-form solutions are derived. When the film is of finite thickness, a single parameter semi-empirical formula agreeing well with the numerical results is proposed for the effective piezoelectric coefficient. It is found that if the film thickness effect is not taken into account, PFM can significantly underestimate the effective piezoelectric coefficient compared to the half space result. The effects of the ambient dielectric property on PFM responses are also explored. Humidity reduces the surface displacement, broadens the radial distribution peak, and greatly enlarges the image charge, resulting in reduced effective piezoelectric coefficient. The proposed semi-empirical formula is also suitable to describe the thickness effects on the effective piezoelectric coefficient of thin films in humid environment. The obtained results can be used to quantitatively interpret the PFM signals and enable the determination of intrinsic piezoelectric coefficient through PFM measurement for thin films.

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Figures

Grahic Jump Location
Fig. 1

Schematic of PFM for a piezoelectric film/substrate system

Grahic Jump Location
Fig. 2

Distributions of numerically calculated −uz(r,0)/ϕ0 along the radial coordinate for different film thicknesses of the PZT4 film/substrate system. The closed-form solutions of IT film and half space are calculated from the corresponding expressions given in Table 1.

Grahic Jump Location
Fig. 3

Distributions of numerically calculated ϕ(r,0)/ϕ0 along the radial coordinate for different film thicknesses of the PZT4 film/substrate system. The closed-form solutions of IT film and half space are calculated from the expressions given in Table 1.

Grahic Jump Location
Fig. 4

Effects of film thickness t/d on the normalized image charge Q/(2πϕ0R0ε0) for the PZT4 film/substrate system

Grahic Jump Location
Fig. 5

Effects of film thickness to tip radius ratio t/R0 on the normalized thickness t/d for the PZT4 film/substrate system

Grahic Jump Location
Fig. 6

Effects of film thickness t/d on the effective piezoelectric coefficient d33eff for the PZT4 film/substrate system. The inset is the corresponding linear plot.

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Fig. 7

Effects of film thickness t/R0 on the effective piezoelectric coefficient d33eff for PZT4, PZT5H, PZT6B, PZT7A, and PZT8 film/substrate systems with the material constants listed in Table 2. The dotted lines are curve-fitting results.

Grahic Jump Location
Fig. 8

The normalized surface displacement radial distributions for the half space and the IT film cases in air, that is, εe=1 and in a humid environment, that is, εe=81

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Fig. 9

The normalized surface displacement radial distributions for a thin film with the normalized thickness t/R0=0.001, in air, that is,εe=1 and in a humid environment, that is,εe=81

Grahic Jump Location
Fig. 10

The normalized electric image charge Q/(2πϕ0R0εeε0) versus t/R0 from 10−5 to 10, in air, that is, εe=1 and in a humid environment, that is, εe=81

Grahic Jump Location
Fig. 11

The effective piezoelectric coefficient d33eff versus t/R0 from 10−5 to 10, in air, that is, εe=1 and in a humid environment, that is, εe=81, and other ambient dielectric constants, for example, εe=30 and εe=150. The dashed lines are calculated from Eq. (48).

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