Measuring Properties of the Piezoelectric Crystal Using A Quadrature Michelson Interferometer

Alexis Valencia and Putheawin Samphea

Abstract

In this experiment, we aim to study the converse piezoelectric effect of an unknown crystal by determining the piezoelectric strain coefficient and the material of the crystal using a Quadrature Michelson Interferometer. Initially, an electric field is applied to induce mechanical strain on the crystal producing interference in the interferometer, which allows us to detect the intensity and analyze the phase change of the light wave. The results are then compared to established piezoelectric coefficient values identified for various piezoelectric materials.

Introduction

Piezoelectricity is an electric charge that accumulates in certain materials when mechanical stress is applied and was discovered in 1880 by the French physicists, Jacques and Pierre Curie [1]. Piezoelectric crystals have asymmetric molecular structure but are electrically neutral; thus, any mechanical stress on the crystal would throw the charges out of balance creating a positive or negative net charge [2]. The effect is a result of linear electro-mechanical interaction between mechanical and electrical states that showcase no inversion symmetry and is similar to the occurrence of electric dipole moments in solid matter.

The Piezoelectric Effect has various practical uses including the production and detection of sound, generation of high voltages, and operation as an ultrasound [1]. It's the ability to create a potential difference and act as a power source is best seen in the electric cigarette lighter where pressing a button activates a spring-loaded hammer to hit a piezoelectric crystal. This causes a high voltage electric current to flow across a spark gap, thereby igniting the gas.

This experiment seeks to build upon previous studies by inspecting the relationship between the contraction of a crystal and the converse piezoelectric effect is applied. Namely, when a voltage is applied onto its opposite sides, the crystal shrinks, expands, or experiences shear strain depending on the polarity of the voltage.

Such deformation is on a scale of a few Angstroms requiring sensitive equipment such as an interferometer that is capable of measuring minuscule displacements. On top of that, a Quadrature Michelson Interferometer uses a combination of circularly and linearly polarized light waves providing the direction of the crystal's displacement.

Thus, this experiment focuses on studying the correlation between the applied voltage and the deformation of the crystal, by utilizing the sensitivity of a Quadrature Michelson Interferometer.

Theory

As a voltage is applied to the crystal, it creates electrical stress resulting in a dimensional strain. This effect can be expressed in the converse piezoelectric equation ε = Ed where ε is strain, E is the electric field, and d is the coefficient. Next, if we treat the unknown crystal similar to a capacitor, the applied voltage (V) can be displayed as V = EL with E being the electric field and L as the length of the crystal. The voltage equation and the general strain equation, ε = ΔL/L , are then substituted back into the piezoelectric equation to obtain ΔL = Vd where we can solve for d, the coefficient. More specifically, we apply a range of voltages on the crystal and measure the deformation along the z-axis, so the experiment focuses on the coefficient in that direction. A Quadrature Michelson Interferometer utilizes linearly and circularly polarized light waves to obtain both the path length and its direction. It allows collections of data with accuracy in the order of picometer.

Experimental Setup and Apparatus

The experiment began with using a quarter-wave plate to circularly polarized lights from a He-Ne laser. The circularly polarized beam passes through a non-polarized beamsplitter separating the light into two paths (see Figure below). One path takes the beam to the reference arm while the other path directs the beam to the measuring arm where a 2-mm-thick crystal is attached to. Lights from these two directions combine together in the middle before they are sent to a polarizing beamsplitter. Here, vector components of the combined lights are sent off to two photo-detectors where the light intensity is measured as an electrical signal. This signal (V_x, V_y) are pre-amplified, then passed to an oscilloscope.

Data Analysis and Results

The xy-graph of the measured intensities (V_x,V_y) follows the curvature of a circle, as shown in the figure (left) below. The blue plot represents the data when the measuring arm was moved by one wavelength. The red plot represents data when the crystal is under piezoelectric effects. V_x and V_y were used to calculate the phase difference φ using the following relationship

The phase change can then be used to find the change in distance that the interferometer detected and that represents the expansion and contraction of the crystal.

We used phase unwrapping function in MatLab to find the net displacement represented by the phase change. Distance vs. Time graph (figure on the right) shows the crystal deformation as a function of time.

The graph of the crystal deformation (Δx) with respect to the applied voltage (V) shows an almost-perfect linear relationship (figure on the left below). The gap in this graph indicates that Δx and V are not synchronized. In other words, there is a time delay in the data. Thus, we used a cross-correlation analysis to determine the lags and adjust the two data, as seen in the figures on the right. Note that the graph shows a step side of approximately 2 nm with respect to distance; therefore, the uncertainty of Δx (dependent variable) is estimated to be 1 nm.

After cross-correlation analysis, we observed that part of the graph with positive voltages has a smaller slope than that at negative voltages. By applying a least squares fit algorithm to the two segments of the line, their slopes were determined. However, these results have reduced chi-squared larger than one due to the minuscule uncertainty of the data and the imperfect linearity in the relationship between crystalline deformation and applied voltage in our data.

We collected four data sets and obtained results, as shown in the table below.

Conclusion

Overall, the purpose of this project was to determine the piezoelectric strain coefficient a Quadrature Michelson Interferometer. With the data we calculated, we found the range of values for the coefficient and ascertained two possible material candidates that match this range to be lead-free ceramics KNN (498 pm/V)^[3] and PZT-5B (480 pm/V)^[4].

References

[1] Johnson Electric Company. “The Piezoelectric Effect - Piezoelectric Motors and Motion Systems.” Nanomotion, Nanomotion, 28 Aug. 2018, www.nanomotion.com/piezo-ceramic-motor-technology/piezoelectric-effect.

[2] Woodford, Chris. "Piezoelectricity". explainthatstuff.com/piezoelectricity (Aug.21, 2018).

[3] M. Sutapun, R. Muanghlua, and N. Vittayakorn, Ferroelectrics 490, 1, 2016.

[4] Omega Piezo Technologies, “Ceramic Components”