the Creative Commons Attribution 4.0 License.

Special issue: Sensors and Measurement Science International SMSI 2020

**Regular research article**
03 May 2021

**Regular research article** | 03 May 2021

# Determination of optimal crystallographic orientations for LiNbO_{3} and LiTaO_{3} bimorph actuators

_{3}and LiTaO

_{3}bimorph actuators

Oleh Buryy Ihor I. Syvorotka Yuriy Suhak Uliana Yakhnevych Dmytro Sugak Sergii Ubizskii and Holger Fritze

^{1},

^{2},

^{3},

^{1},

^{1,2},

^{1},

^{3}

**Oleh Buryy et al.**Oleh Buryy Ihor I. Syvorotka Yuriy Suhak Uliana Yakhnevych Dmytro Sugak Sergii Ubizskii and Holger Fritze

^{1},

^{2},

^{3},

^{1},

^{1,2},

^{1},

^{3}

^{1}Semiconductors Electronics Department, Lviv Polytechnic National University, Lviv, 79013, Ukraine^{2}Scientific Research Company “Electron-Carat”, Lviv, 79031, Ukraine^{3}Institute for Energy Research and Physical Technologies, Clausthal University of Technology, 38640, Goslar, Germany

^{1}Semiconductors Electronics Department, Lviv Polytechnic National University, Lviv, 79013, Ukraine^{2}Scientific Research Company “Electron-Carat”, Lviv, 79031, Ukraine^{3}Institute for Energy Research and Physical Technologies, Clausthal University of Technology, 38640, Goslar, Germany

**Correspondence**: Yuriy Suhak (yuriy.suhak@tu-clausthal.de)

**Correspondence**: Yuriy Suhak (yuriy.suhak@tu-clausthal.de)

Received: 12 Nov 2020 – Revised: 15 Mar 2021 – Accepted: 23 Mar 2021 – Published: 03 May 2021

The actuators for precise positioning based on bimorph
structures of piezoelectric LiNbO_{3} and LiTaO_{3} crystals are
considered. The optimal orientations of the actuator plates ensuring the
highest possible displacements are determined by the extreme surfaces
technique and the finite-element method. The simulated displacements for optimal orientations of LiNbO_{3} and LiTaO_{3} plates are compared with
those obtained experimentally for manufactured LiNbO_{3} and LiTaO_{3} actuators, whose orientations are not optimal. As is shown, the optimal configuration of the actuator allows us to significantly increase its displacement
for both LiNbO_{3} and LiTaO_{3} specimens.

_{3}and LiTaO

_{3}bimorph actuators, J. Sens. Sens. Syst., 10, 121–126, https://doi.org/10.5194/jsss-10-121-2021, 2021.

The systems for precise positioning are widely used to ensure high accuracy,
linearity and reproducibility of the small movements of probes in scanning
probe microscopes, micro-electromechanical systems, micro-dispensers,
micro-motors for surgery, laser gyroscopes, mechanisms for laser resonator
adjusting, piezo-drives for control systems of car suspensions and lamps,
etc. (see, e.g., Segel, 2012; Uchino, 2017; Vijaya, 2017). To date, the most frequently used material for electromechanical actuators is the lead
zirconate titanate-based piezo-ceramics (PbZr_{x}Ti_{1−x}O_{3}, PZT). However, this material reveals two main fundamental properties which can
limit its use. Firstly, the presence of lead in its composition prevents
application of PZT in medicine. It is therefore necessary to note that in
accordance with the regulations of the EU, lead-containing compounds are not allowed to be used in technical devices in the near future (Panda, 2009).
Secondly, the low values of Curie temperature in PZT do not allow
application of ceramic actuators at the temperatures higher than 300^{∘}. Therefore, searching for new lead-free materials operating at high
temperatures is continuously performed. In this regard, piezoelectric
ferroelectric crystals, particularly lithium niobate (LiNbO_{3}) and
lithium tantalate (LiTaO_{3}), can be considered an alternative to PZT under certain operating conditions. More than 30 years of investigations
have proven the use of LiNbO_{3}-based actuators (Nakamura et al., 1989, 1995; Ueda et al., 1990; Wakatsuki et al., 1998; Randles et al., 2006; Kawamata et al., 2007; Antipov et al., 2008; Matsunami et al.,
2008; Bykov et al., 2014; Shur et al., 2015; Kubasov et al., 2016; Turutin
et al., 2018; Buryy et al., 2019; Jiang et al., 2020), in particular
in medicine (Randles et al., 2006). The active elements of LiNbO_{3}-based actuators could be manufactured in the form of bimorph crystalline plates
(Nakamura et al., 1989, 1995; Ueda et al., 1990; Kawamata
et al., 2007; Antipov et al., 2008; Bykov et al., 2014; Shur et al., 2015;
Kubasov et al., 2016; Turutin et al., 2018; Buryy et al., 2019), multi-layer
structures (Matsunami et al., 2008) and thin films (Jiang et al., 2020).
These crystals possess many advantages compared to PZT, namely, higher Curie temperature, the virtual absence of hysteresis and creep, and weak
dependences of the piezoelectric constants on temperature (Antipov et al., 2008; Shur et al., 2015). The main drawback of LiNbO_{3} and LiTaO_{3} compared to PZT is the lower value of piezoelectric constants by approximately an order of magnitude.

To increase the extent of deformation under the influence of the electric
field, bimorph structures can be used. Such structures are formed by two
bonded plates of piezoelectric crystals in such a way that the vectors of polarization (or its components) of both parts are anti-parallel and
perpendicular to the bonding interface between them. In other words, such a configuration is a bidomain structure, which functions in accordance
with the bimorph principle: applying a voltage leads simultaneously to the
expansion of one layer and compression of the other layer; as the result,
the element bends (Shur et al., 2015). Lithium niobate bimorph plates can be
manufactured by a few techniques, i.e., by the thermo-chemical formation of the so-called inverse layer on a surface of a single-domain LiNbO_{3} plate (Nakamura et al., 1989), by the electro-thermal method, particularly during infrared heating (Antipov et al., 2008; Bykov et al., 2014), and by direct
bonding of high-quality polished and cleaned plates (Shur et al., 2015). We
have manufactured such a bimorph structure using two lithium niobate plates
bonded by means of diffusion, using copper from films deposited on the
surfaces of the plates as a diffusant (Buryy et al., 2019). In this paper
the orientations of connected crystalline plates were chosen in such a way that the perpendicular to the surfaces of the plates coincided with the *z* or (*y*+128^{∘}) directions. The chosen orientations were not optimal;
however, it is known that in anisotropic crystals, including LiNbO_{3},
the phenomena of interaction of internal and external physical fields could
be described by complex surfaces (see, e.g., Buryy et al., 2013). The developed methods of construction of such surfaces enable determination of optimal crystal cuts corresponding to the maxima of displacement.

It should also be noted that lithium tantalate crystals are isostructural and close in properties to LiNbO_{3} and, therefore, could
also potentially be used for production of actuators. However, to the best
of our knowledge, there are no experimental or theoretical works where
application of LiTaO_{3} for actuator manufacturing was considered.

This work focuses on the determination of optimal spatial orientations of
crystalline LiNbO_{3} and LiTaO_{3} plates for actuating applications to ensure the highest possible displacement of such actuators.

The considered design of the actuator is shown in Fig. 1. The unit vector
** m** is parallel to the long edges of the bimorph (and perpendicular
to the plane where the bimorph is fixed). The electric field

**is applied along the short edges of the bimorph: $\mathit{E}=E\mathit{n}\perp \mathit{m}$, where**

*E**E*=|E|,

**is the unit vector that determines the direction of the electric field.**

*n*Let us consider one plate of the bimorph separately. Under the influence of the electric field, its deformation can be described by tensor

where $\stackrel{\mathrm{\u0303}}{d}$ is the tensor of piezoelectric coefficients. The
relative expansion of the plate along the vector ** m** is equal to

and, as is seen from Eq. (2), depends on the directions of ** m** and

**; here $\mathrm{\Delta}u={u}_{\mathrm{0}}\mathit{m}\stackrel{\mathrm{\u0303}}{\mathit{\epsilon}}\mathit{m}$ is the absolute displacement of the plate, Δ**

*n**u*

*>*0 in case of plate expansion and Δ

*u*

*<*0 in case of plate compression, and

*u*

_{0}is the actuator length. Therefore the directions of

**and**

*m***, which maximize the value of relative expansion**

*n**δ*

*l*, must be determined in order to optimize the actuator. Despite lithium niobate and lithium tantalate being crystals of sufficiently high symmetry (trigonal), here we intentionally solve the optimization problem in the most general formulation, while the same approach could also be used in future for low-symmetry crystalline materials, too.

The optimization was performed using the approach developed for the analysis of induced and nonlinear optical effects in crystals (see, e.g., Buryy et al., 2013). This approach is based on the construction and analysis of the special-type (extreme) surfaces. Such surfaces comprise all possible maxima of the investigated effect which are achieved by
determination of the optimal orientations of the determinant factor (electric field) ** n** for all possible directions of crystal orientation
determined by vector

**. If the direction of**

*m***is defined by the angles**

*n**θ*

_{n},

*ϕ*

_{n}of the spherical coordinate system and the direction of

**by the angles**

*m**θ*

_{m},

*ϕ*

_{m}, then

*δ*

*l*(the objective function of the optimization) will depend on four variables

*θ*

_{n},

*ϕ*

_{n},

*θ*

_{m}, and

*ϕ*

_{m}. Now for each pair of angles

*θ*

_{m},

*ϕ*

_{m}we can determine such angles

*θ*

_{n max},

*ϕ*

_{nmax}which maximize the value of

*δ*

*l*. Obviously, these angles depend on

*θ*

_{m},

*ϕ*

_{m},

*θ*

_{n max}≡

*θ*

_{n max}(

*θ*

_{m},

*ϕ*

_{m}), and

*ϕ*

_{nmax}=

*ϕ*

_{n max}(

*θ*

_{m},

*ϕ*

_{m}), and

*δ*

*l*can be considered a function of two variables

*θ*

_{m}and

*ϕ*

_{m}. The dependence

*δ*

*l*

_{max}(

*θ*

_{m},

*ϕ*

_{m}) at

*θ*

_{n}=0 …

*π*,

*ϕ*

_{n}=0 … 2

*π*can be represented as a surface; the designation “

*δ*

*l*

_{max}” used here emphasizes that the dependence of

*δ*

*l*

_{max}is obtained after maximizing on

*θ*

_{n},

*ϕ*

_{n}(parameters of the optimization). In accordance with the rule of determination of

*δ*

*l*

_{max}, we use the term “extreme” for such a surface. Thus the value of

*δ*

*l*

_{max}determines the length of the radius vector of this surface for the given direction of

**. The angles**

*m**θ*

_{n},

*ϕ*

_{n}are varied during the optimization process in such a manner that the condition

**⊥**

*n***is always fulfilled; i.e., the vector**

*m***rotates in the plane perpendicular to**

*n***. Subsequently, the piezoelectric extreme surface is given by**

*m*
where *n*_{max} is the vector from a set of vectors
** n**⊥

**which maximize the value of**

*m**δ*

*l*for the fixed direction of

**. The piezoelectric coefficients used in these calculations are indicated in Table 1. The extreme surfaces for both LiNbO**

*m*_{3}and LiTaO

_{3}crystals are shown in Fig. 2. As is mentioned above, each point of the surface represents the maximal value of the relative expansion

*δ*

*l*

_{max}for a given

*θ*and

*ϕ*, but the highest possible expansions correspond to the global maxima of the relative expansion, which, in their turn, correspond to the points which are the outermost from the origin of coordinates. The optimal directions of

**and**

*m***as well as the corresponding values of**

*n**δ*

*l*

_{max}are indicated in Table 1. It should be noted that alternatively the optimization can also be performed when the angles

*θ*,

*ϕ*coincide with

*θ*

_{n},

*ϕ*

_{n}and the angles

*θ*

_{m},

*ϕ*

_{m}are considered the variables. The obtained results for both optimization cases are the same; therefore, the latter case is not shown.

Only one set of the angles is given for each crystal. Other sets can be
obtained using the symmetry elements of the $\stackrel{\mathrm{\u203e}}{\mathrm{3}}m$ point group (point group of crystals (3*m*) + center of inversion).

Since the maximal displacement of the actuator occurs when one plate
maximally expands and the other one maximally compresses, the plates of the actuator have to be rotated relative to each other by 180^{∘} in the YOZ plane. It is equivalent to application of the electric field of the opposite
polarity to the plates. In this case the absolute values of the displacement
will be exactly the same but with different signs. These optimal configurations of actuators are shown in Fig. 3. As is seen from the figure, they are almost similar for both LiNbO_{3} and LiTaO_{3}: only
the angles of rotation around the *x* axis show an insignificant difference (about 3^{∘}).

To determine the maximum extent of displacements of actuators, their
simulation by the finite-element method was performed. In this simulation the geometry of actuators is similar to that used in our previous work (Buryy et al., 2019): width (*x*), length (*y*), and height (*z*) of each plate are 15, 65 and 0.75 mm, respectively; see Fig. 4 (dependence of actuator displacements on its geometry is analyzed in Buryy et al., 2019). The simulated excitation voltages are in the range from −300 to 300 V. Three
different cases were considered: (i) the “initial” orientations of the
plates – crystallographic *z* axes are perpendicular to the plates; (ii) the
optimal orientations indicated in Table 1; (iii) *y*+127^{∘} and
*y*+36^{∘} cuts for LiNbO_{3} and LiTaO_{3} correspondingly.

The cases for comparison were chosen for the following reasons. Case (i) corresponds to the orientation of the already manufactured LiNbO_{3}-based actuator (Buryy et al., 2019). Moreover, in this paper the dependencies of
actuator displacements are theoretically analyzed with variations of
actuator length (from 10 to 80 mm) and thickness (from 0.4 to 2.4 mm). Since the main peculiarities of these dependencies were revealed in Buryy et al. (2019), we do not repeat the same analysis here. Case (ii)
corresponds to the optimal orientations determined in the current study (see Table 1). Case (iii) corresponds to the commercially available cuts of LiNbO_{3} and LiTaO_{3} characterized by the highest values of
transversal piezoelectric coefficients (Matsunami et al., 2008; Kubasov et al., 2016).

The results of the simulation, i.e., the absolute values of displacements of bimorph plates, are shown in Figs. 5–6 and indicated in Table 2. As well
as in Buryy et al. (2019), the dependencies of actuator displacement on excitation voltage are linear in all the studied voltage ranges, i.e., up to a field strength of 200 V/mm. As is shown, the optimal configuration of the actuator allows us to sufficiently increase the displacement value (with other
parameters fixed): the displacement is 26 times higher for lithium niobate
and 4.6 times higher for lithium tantalate compared to the actuator, studied in Buryy et al. (2019) (case (i)). Compared to the commercially
available plates (case (iii)), the displacement increases for about 11 %
for LiNbO_{3} and 67 % for LiTaO_{3}.

The displacement of the LiNbO_{3} actuators with optimized orientation is
about 3 times higher compared to that of LiTaO_{3}; see Fig. 6. However, the displacement of the LiTaO_{3} actuator is 2 times higher when case (ii) is considered. So, in case the use of z-cut plates is desired from the
technological point of view, the application LiTaO_{3} is preferable.

The obtained simulation results will be used for preparation of bimorph actuators with optimal orientations in the forthcoming research.

The actuators for precise positioning based on bimorph structure formed by
joining lithium niobate and lithium tantalate plates of different
crystallographic orientations are considered. To ensure the highest possible
displacements of the actuator, the optimal orientations of their plates were
determined by the extreme surfaces technique. The possible absolute values of displacements are determined by simulation using the finite-element method. It is shown that the optimal orientations of the long edges of the actuator
plates are defined by the angles *θ*_{m}=40^{∘}, *ϕ*_{m}=90^{∘} for LiNbO_{3} crystal and by *θ*_{m}=43^{∘}, *ϕ*_{m}=90^{∘} for LiTaO_{3}. The electric field is applied
in the directions determined by the angles *θ*_{n}=130^{∘},
*ϕ*_{n}=90^{∘} and *θ*_{n}=133^{∘}, *ϕ*_{n}=90^{∘} for LiNbO_{3} and LiTaO_{3}, respectively. The optimal
configuration of the actuator allows us to sufficiently increase its displacement (with other parameters fixed): in 26 times for LiNbO_{3} and in 4.6 times for LiTaO_{3} compared to the previously manufactured actuator with the *z* axes of the plates perpendicular to the surface of bonding. In comparison with the commercially available LiNbO_{3} *y*+127^{∘} cut and LiTaO_{3} *y*+36^{∘} cut, the displacement increases for about 11 % for LiNbO_{3} and 67 % for LiTaO_{3}. For the optimized design, the
displacement of the LiNbO_{3} actuator is about 3 times higher than that of LiTaO_{3}. However, for the case when the *z* axes of the plates are perpendicular to the bonding surface, the displacement is twice as high for
the LiTaO_{3} actuator compared to LiNbO_{3}.

All relevant data presented in the article are stored according to institutional requirements and as such are not available online. However, all data used in this paper can be made available upon request to the authors.

OB developed the methodology and software and determined the optimal orientations of the actuator plates. IIS and DS developed the design of the actuator. IIS and UY produced the experimental sample of the actuator. YS and HF developed the equipment and measured the actuator displacements. UY simulated the bimorph actuator and calculated the absolute values of its displacements. DS, SU and HF formulated the concept of the work. OB, YS and DS prepared the manuscript with contributions from all the co-authors.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Sensors and Measurement Science International SMSI 2020”. It is a result of Sensor and Measurement Science International, Nuremberg, Germany, 22–-25 June 2020.

The authors from Clausthal University of Technology were supported by the Energie-Forschungszentrum Niedersachsen, Goslar, Germany.

The work was carried out in the framework of the joint German–Ukrainian project “Nanocrystalline piezoelectric compounds LiNb_{1−x}Ta_{x}O_{3} for high-temperature applications” (M/48-2020) and project DB/MODUS (no. 0121U107736) of the Ukrainian Ministry of Education and Science. Further, research grants from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, FR1301/42-1, SU1261/1-1) supported this work.

This open-access publication was funded by Clausthal University of Technology.

This paper was edited by Stefan J. Rupitsch and reviewed by one anonymous referee.

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_{3}or LiTaO

_{3}bimorph structures are considered. They consist of two joined plates: one of them lengthens and the other one shortens under the influence of an electric field and, as a result, the actuator bends. Such a device ensures small movements of probes in scanning probe microscopes, micro-electromechanical systems, micro-motors, etc. We determine the optimal orientations of the plates, ensuring the highest possible actuator displacements.

_{3}or LiTaO

_{3}bimorph structures are...