I am confused about what you want. Do you want suggestions for literature articles on the topic? Do you want suggestions for how to measure the effect in the lab? Do you want explanations of why the effect happens from first principles?

Please be a bit clearer in your question.

Dear Jeffrey

with the focus on the measure the effect in the lab, Exactly I want to know if the temperature increases or decreases 1 degree, how the extension of piezoelectric for a constant applied voltage change? I guess this effect depends on the different parameters such as applied voltage value.

Also I like to have the literature articles :)

That is the linear thermal expansion coefficient modified to also be at constant voltage

$\left.\frac{1}{L_o}\frac{dL}{dT}\right|_V$

(to see this LaTeX code as an equation, use this link ... http://www.codecogs.com/latex/eqneditor.php)

A literature search on "thermal expansion coefficient AND PZT" would be a reasonable starting point.

Do you mean the effect of temperature on the PZT is only thermal expansion behavior like the other material?

Yep. This is after all what the derivative means translated in to physical sense... how the relative length of a material changes with temperature while holding something (voltage) constant. The behavior is non-isotropic I would certainly imagine.

Hope that makes a useful connection for you the lessons in your undergraduate or graduate physical chemistry or materials properties courses.

because of my little knowledge about piezoelectric behavior of materials, I asked this question. however, in AFM the sawtooth voltage was applied to piezoelectric to provide the scan and finally achieve the image. Is the story same compared with apply a constant voltage? (This phenomena would affect the calibration of AFM)

Interesting follow-up question. You have two components that affect the uncertainty. One is the thermal component. The other is the voltage component. For each component, you have an offset (bias) term and a random (fluctuation) term. The offset term is what you calibrate. The random term is why you measure more than one time.

The best you can hope is to calibrate each term separately while the other term remains constant. So, you calibrate piezo voltage at constant temperature(s) and calibrate for thermal offset at constant voltage(s).

In practice, for most measurement systems in operation, once you have calibrated at or around the nominal conditions of typical use, your random uncertainty term tends to dominate all else. In other words, once you have done enough measurements, your standard uncertainty will be larger than your calibration uncertainty. Put another way, on a properly design and carefully calibrated instrument, you would have to do a very large number of measurements to reduce your random uncertainty to a range that is below any issues with calibration uncertainties.

You are on the right track to calibrate your AFM system in every way that you can. Gather the statistical information properly, and you can gain great insight in to how precisely you can really report a measurement value from it.

Thank you Mr. Weimer

One remaining note here. The thermal term would be an issue when you have one of two cases. Either, your work is done at a temperature that is significantly different enough from the calibration temperature or your work is done in an environment where the temperature fluctuates dramatically enough over the course of a measurement. In either case, the relative uncertainty in z position of the piezo will be on the order of $\alpha_T\ \Delta T$.