Optimizing the Electrode Geometry of an In-Plane Unimorph Piezoelectric Microactuator for Maximum Deflection

Abstract

Piezoelectric microactuators have been widely used for actuation, sensing, and energy harvesting. While out-of-plane piezoelectric configurations are well established, both in-plane deflection and asymmetric electrode placement have been underexplored in terms of actuation efficiency. This study explores the impact of asymmetric electrode geometry on the performance of slender unimorph actuators that deflect in-plane, where actuator length is much larger than width or thickness. After validating the finite element modeling method against experimental data, the geometric parameters of the proposed unimorph model are manipulated to explore the effect of different electrode geometries and layer thicknesses on actuation efficiency. Four key findings were that (1) the fringing field within the piezoelectric material plays a measurable role in performance, (2) symmetry in electrode placement is generally nonoptimal, (3) optimal electrode geometry is independent of scale, and (4) the smaller the ratio of width to thickness, the larger the deflection. The findings contribute to the development of efficient design strategies that optimize the performance of planar actuators for potential implications for microelectromechanical systems (MEMS). To aid designers of piezoelectric unimorph actuators in determining the optimal electrode geometry, three types of parameterized figures and two types of simulation apps are provided.

1. Introduction

Piezoelectric transducers produce voltage when subjected to mechanical stress or, conversely, undergo mechanical deformation when subjected to an applied electric field. Energy-efficient piezoelectric transducers convert electrical energy into mechanical movements or vice versa, which establishes them as useful components in actuation, fine resolution sensing, or energy harvesting implementations for applications such as biomedical, microrobotics, and precision instrumentation [1,2,3,4,5,6]. As sensors, piezoelectric transducers can detect minute mechanical changes, converting them into measurable electrical signals, enabling real-time monitoring and feedback in various systems. Their capability in energy harvesting facilitates sustainable energy solutions, where ambient mechanical disturbances are transformed into usable electrical energy. As actuators, piezoelectric transducers can deliver large forces that can be leveraged or small displacements that can be magnified. Practical applications benefit from transduction efficiency. Efficiency improves performance and contributes to energy conservation [7,8]. Understanding the role of design parameters for efficient actuation, such as electrode geometry, width, and layer thickness, can measurably improve performance.

While conventional out-of-plane piezoelectric bimorph and unimorph configurations are common and have been well explored in the literature, e.g., [9,10,11,12,13,14,15,16], uncommon in-plane configurations have been sparsely explored [17,18,19,20,21]. Compared to out-of-plane transducers, in-plane piezoelectric transducers often achieve a faster response, a larger generative force, and improved electromechanical coupling [17].

The following publications were found from a large search on planar piezoelectric actuation: [17,18,19,20,21]; their planar lengths and widths are on the order of millimeters, and their bimorph structures avoid the fringing electric field issue by using three electrodes that cover all or most of the entire upper and lower surfaces. Compared to out-of-plane piezoelectric transducers that bend in the direction of thickness, in-plane configurations that bend in the direction of width are capable of faster responses, greater deflection, greater resonance frequency, and greater blocking force [17,20]. In [17], the wide, thin, rectangular planar actuator utilizes alternating layers of vertically stacked piezoelectric material and metal electrodes. Planar deflection is achieved by applying a unidirectional electric field through a dually poled piezoelectric material, where half of the cantilever is poled into the plane, and half is poled out of the plane. Experiments resulted in a planar deflection of 1.3 µm for a cantilever length of 9.2 mm, a width of 3.5 mm, a total thickness of 0.945 mm, and an applied voltage of 12 V. The computer model for this article is validated using the experimental results of [17]. In [18], a single-layer version of [19] is utilized, where dually poled top surface electrodes act upon a uniformly poled piezoelectric material. A deflection of 300 µm was measured for a cantilever of length 40 mm, a width 5 mm, a thickness of 0.5 mm per layer, and an applied voltage of 100 V. In [19], a trapezoidal version of [17] is utilized to achieve a larger deflection of 1.54 µm, with a length 10.14 mm, a base width of 3.08 mm, a tip width of 1.54 mm, 13 layers with a 35 µm thickness, and an applied voltage of 10V. In [20,21], two applications of planar piezoelectric actuation are given with the designs of “gripper” and “extender” actuators. Both actuator designs utilized two electrodes on the upper faces with opposing polarities similar to the electrodes used in [17,18,19].

Conversely, the configuration presented herein is a slender, two-electrode unimorph that is subject to a significant fringing field. While three electrodes would increase deflection performance, doing so would increase the number of fabrication steps, wiring complexity, and cost, where planar arrays can be used to regain any loss in deflection [22]. The proposed in-plane unimorph actuator in this article consists of one electrode partially covering the piezoelectric top surface and a second electrode covering the entire bottom surface. This electrode structure facilitates the connection of one planar transducer to a multitude of others in large arrays to increase overall deflection or force. The consequence of fabrication simplicity and electrode connectivity simplicity is the interplay of fringing electric fields. Fringing electric fields are usually a topic of interest for the designers of electrostatic actuators [23], where most out-of-plane and in-plane piezoelectric transducers are sandwiched between parallel conductive electrodes that fully cover parallel outer surfaces. However, the fringing electric fields play a measurable role in performance for electrode geometries where the electrodes do not fully and equally cover opposing surfaces. Due to the complex interplay between the fringing electric fields, the width and thickness of the piezoelectric material, and the asymmetric widths of the surface electrodes, finite element modeling is used to parameterize and understand the effects on performance. By sweeping the geometric parameters, optimal parameter combinations are established.

Section 2 discusses the computational method and model validation steps. Section 3 provides parameterized simulation results and discusses them. Section 4 concludes the findings.

2. Materials and Methods

The basic design used in this study was motivated by the simple two-electrode, three-layer piezoelectric fabrication process, where piezoelectric material is sandwiched between thin conductive electrodes. The fabrication process can be carried out using sol-gel [17,18,19,24], or be purchased prefabricated and poled in rolled piezoelectric sheets, e.g., lead zirconate titanate (PZT) [25,26] or polyvinylidene fluoride (PVDF) [27]. The basic geometry of the transducer is illustrated in Figure 1.

The slender structure has a length $L$ that is much larger than its width $w$ or layer thickness $h$. The thickness of the electrodes $h_m$ is much thinner than the thickness of the piezoelectric layer, resulting in a negligible effect on bending performance. The width of the electrode is $w_{elec}$, the width of the passive side of the cantilever is $w_{pass}$, the total width is $w=w_{elec}+w_{pass}$, and the layer thickness is $h$. Ratios of these are the geometric parameters that are swept in Section 3 to explore their effect on performance.

Modeling. The piezoelectric planar unimorph is modeled with finite element analysis (FEA) in COMSOL Multiphysics software [28], where the strain due to the applied electric field E is modeled as

$${\left[\begin{array}{cccccc}{\epsilon }_{xx}& {\epsilon }_{yy}& {\epsilon }_{zz}& {\epsilon }_{yz}& {\epsilon }_{xz}& {\epsilon }_{xy}\end{array}\right]}^T=d^T{\left[E_x,E_y,E_z\right]}^T$$

(1)

where the piezoelectric coupling matrix is

$$d=\left[\begin{array}{cccccc}0& 0& 0& 0& d_{15}& 0\\ 0& 0& 0& d_{15}& 0& 0\\ d_{31}& d_{31}& d_{33}& 0& 0& 0\end{array}\right]$$

(2)

The piezoelectric strain coefficients for PZT-5H are experimentally matched to those reported in [29], where $d_{31}$ = $-274$ pC/N, $d_{33}$ = 593 pC/N, and $d_{15}$ = 741 pC/N. The types of finite elements chosen are hexahedral and tetrahedral, and the element resolution is determined by increasing the number of degrees of freedom (DOF) necessary to achieve an acceptable absolute relative error beyond the finite element convergence knee (Figure 2).

Validation. The computer model is validated against the experimental results of the in-plane piezoelectric actuator from [17]. By substituting the geometry, material properties, and applied voltages from [17] into the simulation tool (Figure 3), planar deflection is produced that fairly agrees with experimental results (Figure 4) when the applied voltage is swept from –12 to 12 volts. Our computer model ignored hysteresis, used meshed 12,769 free tetrahedral mesh elements, used a fixed boundary constraint at the base, applied a domain voltage upon the metal of the electrodes, used a charge conservation upon the piezoelectric domain, and used a PARDSO linear solver with a relative tolerance of 0.001 as the voltage was swept. The substituted parameters from [17] were length = 9 mm, total width = 3.5 mm, piezoelectric thickness = 0.945 mm, electrode thickness = 3 μm, electrode material = Pd, piezoelectric material PZT with $d_{31}=175\times {10}^{-12}$ C/N, and Young’s modulus $E=6.5\times {10}^{10}$ N/m².

Fringing electric field. Due to the asymmetric electrode configuration of the proposed planar transducer shown in Figure 1, a nonuniform strain field within the material is induced that is spatially proportional to the fringing electric field. A cross-section of voltage and electric fields are simulated in Figure 5. The overall width is 4 μm, and the thickness is 2 μm. Only the electric field lines within the piezoelectric material contribute to in-plane deflection. Although the electric field passes through all of the piezoelectric material, the material on the left side of the cross-section induces much greater strain due to the field lines’ concentration and the field lines’ dominant vertical orientation.

Deflection. The largest possible deflection depends on the strength of the coercive field. The coercive field is the maximum electric field that can be applied within a polarized piezoelectric material, reducing the polarization and bringing the average polarization to zero [30]. The coercive field for any type of piezoelectric material will vary due to variations in the fabrication process [31,32,33,34,35,36]. Figure 6 shows the deflection of the transducer with a geometry of length $L$= 100 µm, width $w$= 4 µm, electrode width $w_{elec}$= 1.6 µm, thickness $h$= 2 µm, metal thickness $h_m$= 0.01 µm, and an applied voltage per unit thickness that is equal to or less than the coercive field. The fixed boundary is the vertical side wall of the wide anchor that is furthest from the tip.

While in-plane tip deflection along the y-axis can be on the order of 1% of the transducer length, the maximum deflection will depend on the values of the abovementioned parameters. Compared to the desired in-plane actuation deflection along the y-axis, the out-of-plane deflection along the z-axis is two orders smaller, and the out-of-plane deflection solely due to 1 G gravity is four orders smaller.

3. Results and Discussion

Based on the unimorph configuration model shown in Figure 6, simulations were performed for the dependence of planar deflection on four design parameters: electrode width $w_{elec}$, passive width $w_{pass}$, total width $w$, and layer thickness $h$, as defined in Figure 1. The parameters held constant are the unimorph’s length $L$ = 100 μm, the electrode thickness $h_m$ = 0.01 μm, and the applied electric field. To make the results applicable to other length scales, the parameters are plotted in the form of nondimensional ratios. This is because the cross-sectional shapes of the fringing electric fields are independent of the length scale, and the unimorph cross-sectional field determines the optimal deflection and not the unimorph length. The following simulation results suggest that the fringing electric field is the cause of nonmonotonic tip deflection $y$ when the parameters are linearly swept. These results are unlike what is conventionally obtained from the cross-sections of non-fringing, parallel electric field configurations [17,18,19,20,21] that would result in monotonic tip deflections if the design parameters were swept.

Since the fringing electric field (Figure 5a) is independent of the geometric length scale, we normalize the following results so that our results can be applied to the geometries of different scales. Tip deflection is expressed as $y/y_{\max }\le 1$, where $y_{\max }$ is the largest deflection found by sweeping the design parameters. The nondimensional ratios of the swept design parameters are $w_{elec}/w$, $w/w_0$, ${w_{elec}/w}_0$, and ${w_{pass}/w}_0$, where $w=w_{elec}+w_{pass}$ and $w_0$ is a normalizing constant. Figure 7 explores the dependence of the tip deflection $y/y_{\max }$ on the width of the electrode $w_{elec}$ for a constant aspect ratio of the width and thickness of $h=w/2$. The y-axis is swept as $0.1\le w_{elec}/w\le 0.9$, and the x-axis is swept as $0.1\le w/w_0\le 0.9$ while the constraint of $h=w/2$ is maintained. For example, if $w_0=4$ μm, then the maximum widths and thickness of the sweep are $w^{\max }=0.9\times 4$ μm, $w_{elec}^{\max }=0.9\times w_{\max }$, and $h^{\max }=0.9\times 4 \mathrm{\mu }\mathrm{m}/2$. Figure 8 explores the dependence of the tip deflection $y/y_{\max }$ on independent values of electrode width $w_{elec}$ and passive width $w_{pass}$ for a constant thickness. The x-axis is swept as $0.1\le w_{elec}/w_0\le 0.9$, and the y-axis is swept as $0.1\le w_{pass}/w_0\le 0.9$ for a constant thickness of $h=w_0/2$. For example, if $w_0=4$ μm, then the thickness and maximum width would be $h=4 \mathrm{\mu }\mathrm{m}/2$ and $w=w_{elec}^{\max }+w_{pass}^{\max }=2\times 0.9\times 4$ μm. Figure 9 explores the dependence of the tip deflection $y/y_{\max }$ on independent values of electrode width $w_{elec}$ and layer thickness $h$. The x-axis is swept as $0.1\le w_{elec}/w\le 0.9$, and the y-axis is swept as $0.1\le h/w\le 0.9$, where the passive width is determined by $w_{pass}/w=1-w_{elec}/w$ for a given value of $w_{elec}/w$. For example, if $w_{elec}=1$ μm, $w_{pass}=3$ μm, and $h=2$ μm, then $w=4$ μm, $w_{elec}/w=0.25$, and $h/w=0.5$.

A key finding from Figure 7 is the scale-independent agreement of an optimal electrode width ratio for a given layer thickness ratio. This can be seen by observing that the location of the maximum of each curve in Figure 7a is at $w_{elec}/w\approx 0.325$ for a thickness-to-width constraint of $h=w/2$. This is because the shape of the voltage contours and electric field lines (as shown in Figure 5) is independent of scale. That is, changing the width and thickness by the same scale factor does not affect the optimal electrode width ratio. Figure 7b shows a surface plot version of Figure 7a, where a superimposed red curve intersecting the optimal parameters is shown to be a straight line when the red curve is projected onto the independent parameter axes, i.e., the contour plane.

Figure 8 loosens the constraint on thickness-to-width that was applied in Figure 7 by allowing the active and passive region width to be independently chosen by parameters $w_{elec}/w_0$ and $w_{pass}/w_0$, where $w_0$ is a constant chosen using the value of layer thickness, $w_0=2h$. Given a thickness $h,$ the range of values explored in Figure 8 are $0.1\le w_{elec}/\left(2h\right)\le 0.9$ and $0.1\le w_{pass}/\left(2h\right)\le 0.9$, which implies an overall range of widths of $0.4h\le w\le 3.6h$, where $w=w_{elec}+w_{pass}$. From Figure 8, thinner widths yield larger deflections for optimized electrode region width pairs. A family of curves is shown in Figure 8a, and each curve represents an electrode width ratio of ${w_{elec}/w}_0$. As both ratios ${w_{elec}/w}_0$ and ${w_{pass}/w}_0$ are reduced, achieving greater deflections for an optimal electrode width is possible. The corresponding contour and surface versions of Figure 8a are shown in Figure 8b,c. The points along the superimposed red curve identify optimal sets of parameters. This finding is significant because it identifies the nonmonotonic behavior caused by the width of the singular upper electrode. This is a novel finding because of the symmetrical nature of electrodes in previous planar designs that do not have this issue. Due to this nonmonotonic behavior, the design of planar piezoelectric actuators that only have a single upper electrode must include the optimization of the electrode structure to maximize efficiency.

Figure 9 explores the parameter ratios $h/w$ and $w_{elec}/w$. Here, the ranges of explored ratios are $0.1\le h/w\le 0.9$ and $0.1\le w_{elec}/w\le 0.9$. Note that unlike $w_0$, the value of $w$ is not held constant, as the ratios $h/w$ and $w_{elec}/w$ are varied. It can be seen with the family of curves in Figure 9a that the curve representing $h/w=0.5$ peaks at an electrode width ratio of $w_{elec}/w\approx 0.325$, which agrees with Figure 7a. However, correlation with Figure 8 is not as straightforward because the parameterized ratios are not equivalent between the figures; that is, $w_{elec}/w$, where $w$ varies as $w_{elec}+w_{pass}$, is not equal to $w_{elec}/w_0,$ where $w_0$ is constant. Figure 9b shows the contour and surface plot version of Figure 9a, where the points along the superimposed red curve indicate the optimal sets of parameters.

Figure 10 and Figure 11 show the dependencies of tip twist angle and tip out-of-plane deflection on electrode width $w_{elec}$ and passive width $w_{pass}$. For the microscale parameters chosen, due to the asymmetry of the electrodes and fringing fields, there will be a small amount of twist of a fraction of a degree and a small amount of out-of-plane deflection on the order of nanometers. Although the FEA mesh used in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 is the same, the nanometer-scale deflections shown in Figure 11 reveal sub-nanometer noise, which could likely be reduced by further mesh refinement (recall Figure 2).

In addition to the use of parameter ratios, three freely available unimorph design tools in the form of a MATLAB function and two standalone COMSOL applications have been created and are available for download from the nanoHUB [37]. The uncompiled MATLAB function requires either MATLAB [38] or Octave [39].

The MATLAB function returns the optimal value of electrode width $w_{elec}$ given inputs of width $w$ and layer thickness $h$. The command line function is

$$w_{elec}=\mathrm{unimorph}\_\mathrm{optimal}\_\mathrm{welec} (w, h)$$

where $w_{pass}=w-w_{elec}$.

Standalone COMSOL applications have been designed and made publicly available [37] to allow designers to create their own versions of the proposed piezoelectric unimorph design. The COMSOL application, shown in Figure 12, allows one to simulate the deflection of the unimorph based on the design parameters of length, total width, electrode width, layer thickness, metal thickness, and electric field.

The two COMSOL applications provide simulation options for structures including or not including metal electrodes. Ideally, the thickness of the electrode is minimized to have a negligible impact on performance. After entering the desired parameters, the geometry is displayed by pressing the geometry button; deflection is computed and displayed by pressing the displacement button; and surface voltage is computed and displayed by pressing the voltage button. Color bars show the amounts of deflection or voltage. A selection of several piezoelectric materials is provided, while the default PZT-5H produces the best performance of the provided choices.

4. Conclusions

Optimal geometric design parameters were determined for the planar unimorph of the type for which piezoelectric material is interposed between the thin top and bottom electrodes, with the top electrode only covering a portion of the upper piezoelectric surface. The simulation method was validated against published experimental results of a wide planar piezoelectric actuator with full electrode coverage. The dependence of planar deflection on four design parameters, including electrode width $w_{elec}$, passive width $w_{pass}$, total width $w$, and layer thickness $h$, was observed by sweeping the parameters and analyzing their relationships. It was observed that variations in the top electrode’s width and the piezoelectric layer’s thickness resulted in nonmonotonic deflection characteristics. The results suggest that the deflection response of the unimorph is not linear or easily predictable based on incremental changes in these parameters. The results suggest that the cause of the behavior of this nonlinear design is due to the fringing electric field within the piezoelectric. A key finding of this study is that symmetry in electrode placement is often not optimal. For instance, if the thickness of the piezoelectric layer is half of the unimorph’s total width, the optimal electrode width is approximately 32.5% of the overall width. It was also found that the optimal electrode geometry ratio is independent of scale, and larger deflections resulted from minimizing the width-to-thickness ratios. To help assist designers in determining the optimal design parameters, three types of figure functionalities and a MATLAB function that are independent of scale are provided, but only for the PZT-5H material, and two downloadable COMSOL applications for modeling a variety of other piezoelectric materials are made publicly available. Although this study focused on the most popular piezoelectric material, PZT, a similar nonmonotonic behavior for other piezoelectric materials was found; however, the optimal geometric ratios of one piezoelectric material are not necessarily the optimal design ratios for another type of piezoelectric material.