#
Characterization of Pure Face-Shear Strain in Piezoelectric α-Tellurium Dioxide (α-TeO_{2})

^{*}

## Abstract

**:**

_{123}= d

_{14}: such unique behavior has the potential to enable novel gyroscopic sensors and high-precision torsional microelectromechanical systems (MEMS) actuators, as pure face-shear can be used to induce pure torsion. Although α-TeO

_{2}is one of the few known materials belonging to this symmetry class, considerable uncertainty in its single piezoelectric coefficient exists, with the few reported literature values ranging from 6.13 to 14.58 pC/N; this large uncertainty results from the difficulty in using conventional piezoelectric characterization techniques on paratellurite, limiting measurements to indirect methods. The novel applications that would be enabled by the adoption of this extraordinary material are frustrated by this lack of confidence in the literature. We therefore leverage, for the first time, a first-principles analytical physical model with electrochemical impedance spectroscopy (EIS) to determine, directly, the lone piezoelectric coefficient d

_{123}= d

_{14}= 7.92 pC/N.

## 1. Introduction

_{15}), thickness-extension (d

_{13}), and length or width extension deformation modes are used (d

_{33}). Face-shear (d

_{14}) is a more complex type of deformation, and is of great potential interest for torsional applications, like novel gyroscopic sensors or high-precision torsional MEMS actuators. However, although there exist 20 piezoelectric crystal classes, these various modes occur simultaneously and are typically interdependent and electromechanically coupled; very few materials offer a pure face-shear mode that is unpolluted by other, often undesirable excitation modes. One of the few alternative materials includes β-quartz, which allows pure-shear mode in certain limited orientations [1], is thermodynamically unstable except at high temperatures. For room temperature applications, α-phase tellurium dioxide (paratellurite, TeO

_{2}) is particularly interesting: as a rare 422 symmetry class crystal [2,3], its only non-zero piezoelectric coefficient is d

_{123}= d

_{14}. Thermodynamics and crystal symmetry arguments impose an elastic mechanical stiffness tensor c

_{ijkl}= c

_{IJ}, a dielectric permittivity tensor ε

_{ij}, and a piezoelectric strain tensor d

_{ijk}= d

_{iJ}with few independent coefficients [4,5,6]:

_{123}= d

_{14}exists in the literature, due to the difficulty in characterizing this unique mode, with the few available studies estimating values ranging from 6.13 to 14.58 pC/N [16,17,18,19]. This large uncertainty is a direct result of the challenges faced by employing traditional piezoelectric characterization techniques to a pure face-shear piezoelectric deformation that lacks piezoelectric deformation components in other directions.

## 2. Materials and Methods

_{2}for the first time.

_{1}, x

_{2}, x

_{3}) space at a time t is expressed as a weighted sum of the contribution of the three bulk waves f(x

_{i}) that propagate in that given direction (x

_{1}, x

_{2}, or x

_{3}):

_{kl}, the subscripts k and l represent the propagation direction and the displacement polarity of the wave component, respectively, and $\omega $ is the angular frequency. Fortunately, as paratellurite has a high level of symmetry, all nine bulk waves f(x

_{i}) exhibit a pure displacement, simplifying the computational effort in obtaining an analytical solution for the displacement vector.

_{k}, one has to satisfy the Christoffel equation coming from the combination of plane wave propagation and the equations of motion [8]:

_{kl}. The solution gives the expression of the unknowns A

_{kl}as a function of voltage V(ω), dielectric charge Q, the frequency ω, and the material parameters ($g,{\epsilon}_{S},{c}^{E}$).

_{1}× 2a

_{2}× 2a

_{3}:

_{2}grown by the Czochralski method was studied (5 mm thick by 10 mm wide by 10 mm long). Silver paste electrodes were used on the square faces, respectively; as the crystal is unconstrained and large compared to the electrodes, electrical and mechanical effects of the electrodes can be neglected. By imposing the electric field E in the x

_{1}-direction (1-direction), only the S

_{23}and S

_{32}deformations are induced by the sole d

_{14}= d

_{123}piezoelectric coefficient, as shown in Figure 1 below.

## 3. Results and Discussion

_{44}and dielectric coefficient ε

_{11}are generally consistent with the literature for α-tellurium dioxide. Our method is, however, the only direct analytical measurement reported in the literature for d

_{123}= d

_{14}, the sole piezoelectric coefficient. An additional benefit of using this analytical model is that because the entire electrochemical impedance spectrum is described using these material properties, it is possible to fit the complete experimental spectrum (over a wide frequency range) at once, thereby reducing uncertainty and experimental error.

_{14}≤ 6.88 pC/N. However, our previous FEA simulation was unable to provide as satisfactory a fit to the experimental EIS data (after extensive simulation/computation time) compared to the present analytical model that is easily computed with a personal computer in a few seconds. In particular, our previous numerical approach did not reliably converge, as the lack of multiple resonant frequencies in the impedance spectrum of paratellurite provides too many possible solutions for the material parameters. In addition, use of Doppler interferometry required invoking the quasi-static assumption while applying an alternating electric field at 40 kHz—an obvious contradiction. We therefore find that our previous results using numerical FEA and Doppler interferometry are likely an underestimate of the piezoelectric coefficient d

_{14}, while not providing for the extraction of other material properties.

_{123}= d

_{14}, this suggests that the other higher values reported for d

_{123}= d

_{14}also obtained by indirect measurements using the mechanical resonant ultrasound method, including Ogi and Fukunage [17] and Ledbetter et al. [18], may be significant overestimates.

## 4. Conclusions

_{2}as ${d}_{14}\equiv {d}_{123}$= 7.92 pC/N. We found that our measurements are partly consistent with the literature, in particular with the measurements of Ohmachi and Uchida [16]; given that our measurements are the first direct measurements reported on this material using analytical EIS, we believe that other authors that used more indirect methods may have overestimated this piezoelectric coefficient. Although our findings confirm the exciting potential of paratellurite, further work exploring the dielectric loss mechanism will be necessary before this interesting material can be applied in next-generation devices.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Piezoelectric shear strains S

_{23}and S

_{32}induced on the x

_{2}and x

_{3}faces of a paratellurite crystal by applying electric field E between the x

_{1}faces using silver (Ag) electrodes.

**Figure 2.**Electrical impedance modulus spectrum in monocrystalline paratellurite about the fundamental face-shear piezoelectric excitation.

**Figure 3.**Electrical impedance phase spectrum in monocrystalline paratellurite about the fundamental face-shear piezoelectric excitation.

**Table 1.**Material properties of α-tellurium dioxide (paratellurite, α-TeO

_{2}) experimentally determined in this study by analytical Electrochemical Impedance Spectroscopy (EIS) compared to values reported in the literature.

Measurement | ${\mathit{d}}_{14}\equiv {\mathit{d}}_{123}$ (pC/N) | ${\mathit{c}}_{44}^{\mathit{E}}$ (GPa) | ${\mathbf{\epsilon}}_{11}^{\mathit{T}}$ (1) |
---|---|---|---|

Current Study (Analytical EIS) | 7.92 | 25.4 | 20.2 |

Boivin et al. [19] (Finite Element Analysis & Doppler Interferometry) | 6.13–6.88 | n/a | n/a |

Ohmachi and Uchida [16] (Resonant Ultrasound Method) | 8.13 | 26.5 | 22.9 |

Ogi and Fukunage [17] (Resonant Ultrasound Method) | 12.41 | 26.6 | 22.7 |

Ledbetter et al. [18] (Resonant Ultrasound Method) | 14.58 * | 26.9 | n/a |

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**MDPI and ACS Style**

Boivin, G.; Bélanger, P.; Zednik, R.J.
Characterization of Pure Face-Shear Strain in Piezoelectric α-Tellurium Dioxide (α-TeO_{2}). *Crystals* **2020**, *10*, 939.
https://doi.org/10.3390/cryst10100939

**AMA Style**

Boivin G, Bélanger P, Zednik RJ.
Characterization of Pure Face-Shear Strain in Piezoelectric α-Tellurium Dioxide (α-TeO_{2}). *Crystals*. 2020; 10(10):939.
https://doi.org/10.3390/cryst10100939

**Chicago/Turabian Style**

Boivin, Guillaume, Pierre Bélanger, and Ricardo J. Zednik.
2020. "Characterization of Pure Face-Shear Strain in Piezoelectric α-Tellurium Dioxide (α-TeO_{2})" *Crystals* 10, no. 10: 939.
https://doi.org/10.3390/cryst10100939