Figure of merit for piezoelectric MEMS speakers

Abstract

Piezoelectric MEMS speakers, an emerging technology with great promise, face significant challenges in performance evaluation and rational design. Their broadband nature means that responses at every frequency point across the whole operating bandwidth contribute to performance, yet there is no widely recognized weighting approach for fair evaluation. This absence of quantitative criteria makes objective comparisons of different designs difficult, slowing the adoption of new design concepts; and it leads to ambiguous design goals without response balance across frequency bands. Additionally, the current design methods rely on labor-intensive simulations, further prolonging the development process. To address these challenges, two figures of merit (FOMs) obtained via theoretical deduction are proposed in this study. These FOMs facilitate the evaluation of key metrics, such as sound pressure level and energy efficiency over a wide frequency range, enabling quantitative comparisons among various speaker designs. On the basis of FOMs, the design process can be simplified into a single-objective optimization problem, significantly streamlining the speaker design. Using this method, piezoelectric MEMS speakers with ultra-high FOMs and superior performance are demonstrated. The normalized SPLs at 1 and 10 kHz reach an impressive 76.6 and 86.6 dB/mm²/V_rms, respectively, with normalized sensitivities of 91.2 and 91.5 dB/mm²/mW. This achievement validates our FOM theory, representing a notable advancement in the field.

Introduction

In recent years, piezoelectric MEMS speakers have garnered significant interest as audio solutions across the consumer, medical, and industrial fields. They offer unique advantages over conventional micro speakers, such as compact size, excellent high-frequency response, low power consumption, robustness against dust and water, ease of mass production, and integration compatibility with electronic components^1,2,3,4. Despite these benefits, MEMS speakers have yet to challenge the market dominance of traditional electrodynamic and balanced armature speakers. This is primarily due to their limitations, such as low output sound pressure level (SPL), high driving voltage, and narrow bandwidth^1,5,6.

In response to these challenges, researchers have conducted many valuable explorations. In 2018, Stoppel et al. proposed a diaphragm structure composed of four triangular cantilever beams^7,8, which significantly increased deflection and provided a basis for mechanical decoupled diaphragms as the mainstream in piezoelectric MEMS speaker design. They subsequently modified their design from a triangular cantilever beam to a rectangular shape to increase the effective radiating area, achieving a 10 dB increase in the normalized SPL at 30 Hz⁹. Cheng et al. proposed a series of speakers that use spring actuators to drive a central diaphragm, thereby enhancing the low-frequency response^10,11,12. Many subsequent works^13,14,15 focused on developments of this design philosophy, such as improving the size and shape of the actuators and central diaphragm and enhancing connection methods. However, despite these improvements in the quasi-static response, these new designs also cause a forward shift in the resonant frequency, which subsequently reduces the high-frequency response. This is because the increase in quasi-static sound pressure is largely due to reduced structural stiffness, which in turn lowers the resonant frequency. Additionally, there is another type of speaker that does not use a piezoelectric actuator as the acoustic surface but rather as a driving source for a passive diaphragm^16,17. The displacement is transferred through a heavy coupling block. Despite this maximization of the equivalent radiating area, this approach introduces an additional mass into the system, which also causes a decrease in the high-frequency response.

The aforementioned works reflect two major issues in MEMS speaker unit research. First, the structure design is tricky, as it requires balancing various metrics. Enhancing the SPL in specific frequency bands often involves other trade-offs, such as a reduced response in other frequency ranges. Achieving high performance requires a careful balance. However, the current design of MEMS speakers relies heavily on time-consuming FEM simulations, making it difficult to achieve this balance. Second, it is challenging to evaluate and compare different designs due to the wideband nature of speakers and the unclear effects of performance variations. Notably, an improvement in performance within one frequency band is often accompanied by changes in many other bands, making it difficult to determine whether improvements arise from a truly better design or are simply the result of compensation. Additionally, as speakers are wideband devices, each frequency point within the operating frequency range contributes to the overall evaluation, further complicating the fair comparison of different designs. These two issues make it difficult to integrate advanced design concepts, leading to a time-consuming and labor-intensive design process, which ultimately hampers fast development in the field.

Given the aforementioned issues, the development of speaker units has reached a stalemate. Recent research has focused predominantly on topics other than speaker units, such as bandwidth enhancement based on multi-resonant monolithic integration^15,18,19,20 and polymer coatings^21,22,23,24. However, this does not imply that the optimization of a single unit is unimportant and that considerable research has not been focused on this issue; rather, research has reached a stage where further progress cannot be made without the aid of theoretical guidance. When a system has several competing metrics, introducing a figure of merit (FOM) to facilitate fair comparison is common. For MEMS speakers, establishing an appropriate FOM could help address the major challenges in design evaluation. A theoretical approach capable of quantitatively revealing the intrinsic correlations among performance parameters, on the basis of which an FOM can be derived, is crucial. Only through such an approach can the full potential of MEMS speakers be realized, enabling balanced audio designs. While there have been several theoretical studies of piezoelectric MEMS speakers, most have concentrated on building equivalent circuit models to enable rapid performance predictions^25,26,27 rather than investigating the relationships among design variables. These models are complex, making it difficult to derive generalizable and abstract conclusions. Additionally, other studies have been limited by their focus on a single performance metric²⁸, restricting their broad applicability. To date, no integrated metric such as an FOM has been proposed for evaluating MEMS speakers, nor has a method been introduced to shift design approaches away from repetitive trial-and-error simulations.

In this paper, two FOMs are proposed for evaluating the output frequency response and energy efficiency of MEMS speakers: broadband normalized voltage sensitivity (FOM_vol) and broadband normalized power sensitivity (FOM_sen). Additionally, precise expressions for these two FOMs are derived for cantilever-type diaphragms of arbitrary shape. Through analysis, the independence of these FOMs from planar dimensions is confirmed, and this conclusion extends to many other structures in addition to cantilevers. The rationality of using these two FOMs as broadband performance indicators for MEMS speakers is demonstrated. By optimizing the FOM as part of the design philosophy, we demonstrate a piezoelectric MEMS speaker that exhibits better normalized output SPL and energy efficiency than any current MEMS speakers, thus validating the effectiveness of the FOMs and the related theory.

Results and discussion

FOM theory

The cantilever piezoelectric speaker is one of the most representative types of piezoelectric MEMS speakers, and many other structures can be seen as variants of it. Therefore, the theoretical analysis in this paper is based on a cantilever beam piezoelectric MEMS speaker. The theoretical analysis presented in this work is based on several assumptions:

• 1. A multilayered speaker integrates layers made of homogeneous and isotropic materials;

• 2. The nonlinearity and viscosity of the layers can be ignored, and the speaker works in a given linear region;

• 3. The residual stress in the speaker is neglected;

• 4. The maximum section width is not greater than five times the root width, so the displacement at each point in the same section can be considered the same.

The primary application scenario for MEMS speakers is in-ear applications, where the speaker’s sound pressure level (SPL) frequency response often resembles that shown in Fig. 1a. The sound pressure $\Delta p$ can be expressed as follows^29,30:

$$\Delta p=\frac{1.4\Delta V{p}_{0}}{{V}_{0}}\cdot Gain$$

(1)

where V₀ is the initial volume of the ear cavity, p₀ is the ambient pressure, Gain is the gain of the human ear for sounds, which is a function of frequency, and ΔV is the air volume change associated with the vibration of the speaker, which can be represented as¹⁰:

$$\Delta V=SU{\bar{\delta }}_{dy}$$

(2)

$${\bar{\delta }}_{dy}={\bar{\delta }}_{st}\left/\sqrt{{\left(1-{\left(\frac{{f}_{in}}{{f}_{r}}\right)}^{2}\right)}^{2}+{\left(2\xi \frac{{f}_{in}}{{f}_{r}}\right)}^{2}}\right.$$

(3)

where S represents the active area of the speaker, U is the driving voltage, ${\bar{\delta }}_{{st}}$ and ${\bar{\delta }}_{{dy}}$ are the average displacement of the diaphragm per unit voltage for static and dynamic conditions, respectively, ξ is the damping ratio, f_in is the driving frequency, and f_r is the first-order resonant frequency of the beam. By substituting (2) and (3) into (1) and excluding constants unrelated to device characteristics, the determinants of ${\overline{{\Delta p}_{k}}}$ and ${\overline{{\Delta p}_{m}}}$, which represent the output sound pressure per unit area per unit voltage for the stiffness control region (${f}_{{in}}\ll {f}_{r}$) and the mass control region (${f}_{{in}}\gg {f}_{r}$), respectively, are as follows:

$$\overline{\Delta {p}_{k}}\sim {\overline{\delta }}_{st}$$

(4)

$$\overline{\Delta {p}_{m}}\sim {\overline{\delta }}_{st}\cdot {f}_{r}^{2}$$

(5)

Fig. 1: Typical piezoelectric MEMS speaker.

a A typical SPL frequency response curve of a piezoelectric MEMS speaker. b Three types of optimization for the frequency response in a pure cavity. i: costly quasi-static enhancements; ii: costly high-frequency enhancements; iii: perfect optimization. c Structural schematic of a MEMS speaker with a random planar shape

From (4) and (5), ${\bar{\delta }}_{{st}}$ is the only determinant of the quasi-static response, whereas this is not the case for the response in the mass control region, where f_r must also be considered. This can be interpreted as follows: the speaker undergoes an amplitude decay with a slope of 12 dB/octave after the first-order resonance (ignoring the higher-order modes). For a given ${f}_{{in}}\gg {f}_{r}$, the larger f_r is, the lower the amplitude decay of the output.

Therefore, both ${\bar{\delta }}_{{st}}$ and f_r must be considered. As shown in Fig. 1b, optimizing only ${\bar{\delta }}_{{st}}$ may result in (i), and optimizing only f_r may result in (ii). It is not conclusive that (i) or (ii) are better than the result indicated by the original curve, as they sacrifice high- and low-frequency performance, respectively. What is needed is (iii), which corresponds to larger ${\bar{\delta }}_{{st}}$ and f_r. However, the two are, to some extent, mutually constrained: increasing the displacement usually reduces the resonant frequency. Therefore, the geometric mean of the determinants of ${\overline{{\Delta p}_{k}}}$ and ${\overline{{\Delta p}_{m}}}$ is used to determine a weighting scheme for ${\bar{\delta }}_{{st}}$ and f_r in the optimization process, which we define as the speaker FOM_vol. f_in is not reflected in FOM_vol because of its irrelevance given the device characteristics.

$$\sqrt{\overline{\Delta {p}_{k}}\cdot \overline{\Delta {p}_{m}}}\sim {\overline{\delta }}_{st}\cdot {f}_{r}$$

(6)

$$FO{M}_{vol}={\bar{\delta }}_{st}\cdot {f}_{r}$$

(7)

As shown in (7), FOM_vol can also be regarded as the product of the quasi-static response and the resonant frequency, through which the entire frequency response curve can essentially be reconstructed. Therefore, FOM_vol can represent the ability of a speaker to output sound pressure over its entire operating bandwidth per unit area and per unit voltage.

In addition, the energy efficiency ratio is another important metric that can be characterized in terms of normalized power sensitivity. A piezoelectric MEMS speaker can be regarded electrically as a capacitor, and its power consumption per unit area can be expressed as²⁸:

$$\overline{P}=\frac{{U}^{2}\omega C}{S}=\frac{{U}^{2}\omega {\varepsilon }_{r}{\varepsilon }_{0}\lambda \gamma }{{t}_{p}}\begin{array}{cc}; & \end{array}\begin{array}{c}{\lambda }_{unimorph}=1\\ {\lambda }_{bimorph}=2\end{array}$$

(8)

where γ is the ratio of the area covered by the piezoelectric material to the total diaphragm area. The normalized sensitivity, which represents the sound pressure level produced per milliwatt in a unit area, can be expressed as:

$$\overline{Sensitivity}=20lg\left(\frac{\overline{\Delta p}\cdot U}{{p}_{ref}}\sqrt{\frac{1}{\overline{P}}}\right)=20lg\left(\frac{\overline{\Delta p}\sqrt{{t}_{p}}}{{p}_{ref}\sqrt{\omega {\varepsilon }_{r}{\varepsilon }_{0}\lambda \gamma }}\right)$$

(9)

Eliminating the irrelevant terms in (9) and considering (6) together, we define a parameter FOM_sen that can directly represent the energy efficiency ratio of a MEMS speaker:

$$FO{M}_{sen}={\bar{\delta }}_{st}\cdot {f}_{r}\cdot \sqrt{\frac{{t}_{p}}{{\varepsilon }_{r}\lambda \gamma }}$$

(10)

The derivation of both FOMs as functions of design parameters requires first determining the expressions of ${\bar{\delta }}_{{st}}$ and f_r.

Mean displacement

For a cantilever beam piezoelectric MEMS speaker, the beam radius of curvature R (Fig. 1c) can be obtained by applying the force boundary condition (11) to solve the piezoelectric Eq. (12).

$$\left\{\begin{array}{c}M=\mathop{\sum }\limits_{i=1}^{n}{\int }_{{z}_{i}}^{{z}_{i}+{t}_{i}}{T}_{i}w(x)ydy=0\\ F=\mathop{\sum }\limits_{i=1}^{n}{\int }_{{z}_{i}}^{{z}_{i}+{t}_{i}}{T}_{i}w(x)dy=0\end{array}\right.$$

(11)

$${T}_{i}=\left\{\begin{array}{l}{Y}_{i}(\frac{y}{R})\pm {Y}_{i}{d}_{31}{E}_{3}\quad{\rm{for}}\,{\rm{piezoelectric}}\,{\rm{layers}}\\ {Y}_{i}(\frac{y}{R})\qquad\qquad\quad\,\,\,{\rm{for}}\,{\rm{other}}\,{\rm{layers}}\end{array}\right.$$

(12)

where F and M are the resultant force and moment of the cantilever beam actuator, respectively; y is the distance to the neutral-plane; z_i and t_i are the bottom position coordinate and thickness of each layer, respectively; T_i is the mechanical stress in the length direction of each layer; Y_i is the Young’s modulus of each layer; and w(x) is the section width at location x. Since the curvature can be regarded as the second-order derivative of the deflection curve, the deflection curve function $\delta (x)$ can be obtained by integrating the curvature²⁸.

$$\frac{1}{R}=\frac{6{d}_{31}U}{{t}_{p}^{2}}\cdot \frac{1}{{\phi }_{1}}$$

(13)

$$\delta (x)=\frac{3{d}_{31}U{x}^{2}}{{t}_{p}^{2}}\cdot \frac{1}{{\phi }_{1}}$$

(14)

For the bimorph:

$${\phi }_{1}=\frac{13.5{A}_{e}{{B}_{e}}^{3}+24{A}_{e}{{B}_{e}}^{2}+12{A}_{e}{B}_{e}+3{{B}_{e}}^{2}+6{B}_{e}+4}{{B}_{e}+1}$$

(15)

For the unimorph:

$$\begin{array}{l}{\phi }_{1}=\frac{1}{(A{B}^{2}\,+\,2A{B}_{e}B\,+\,AB)}\cdot \left(16{A}_{e}^{2}{B}_{e}^{4}+24{A}_{e}^{2}{B}_{e}^{3}+12{A}_{e}^{2}{B}_{e}^{2}\right.\\\qquad\,\,+\,32{A}_{e}A{B}_{e}^{3}B+24{A}_{e}A{B}_{e}^{2}{B}^{2}+36{A}_{e}A{B}_{e}^{2}B+8{A}_{e}A{B}_{e}{B}^{3}\\ \qquad\,\,+\,12{A}_{e}A{B}_{e}{B}^{2}+12{A}_{e}A{B}_{e}B+8{A}_{e}{B}_{e}^{3}+12{A}_{e}{B}_{e}^{2}\\\qquad\,\,+\,8{A}_{e}{B}_{e}+{A}^{2}{B}^{4}+12A{B}_{e}^{2}B+12A{B}_{e}{B}^{2}+12A{B}_{e}B\\\left. \qquad\,\,+\,4A{B}^{3}+6A{B}^{2}+4AB+1\right)\end{array}$$

(16)

where A = Y_m/Y_p, A_e = Y_e/Y_p, B=t_m/t_p, B_e = t_e/t_p and t_p, t_m and t_e denote the thicknesses of the piezoelectric layer, the support layer and the electrode layer, respectively. Notably, to obtain a concise representation, the thicknesses of all electrode layers are assumed to be the same. The static air volume change ΔV_st caused by the vibration of the diaphragm can be obtained via integration. Normalizing this factor based on the area and voltage yields ${\bar{\delta }}_{{st}}$:

$$\Delta {V}_{st}=N{\int_{0}^{L}}w\delta (x)f(x)dx$$

(17)

$${\bar{\delta }}_{st}=\frac{\Delta {V}_{st}}{SU}=\frac{{\int }_{0}^{L}w\delta (x)f(x)dx}{U{\int }_{0}^{L}wf(x)dx}=\delta (L)\cdot \frac{{\int }_{0}^{L}{x}^{2}{f}(x)dx}{U{L}^{2}{\int }_{0}^{L}f(x)dx}$$

(18)

where N is the number of array elements and f(x) is the ratio of the cross-sectional width at position x to the anchor width.

Resonant frequency

For out-of-plane bending of beams with width-line density m(x) and modal function g(x), the maximum potential energy U_m and kinetic energy K_m are determined as follows³¹:

$${U}_{m}=\frac{1}{2}{\int_{0}^{L}}{E}_{p}I(x){\left(\delta (L)\cdot \frac{{\partial }^{2}g(x)}{\partial {x}^{2}}\right)}^{2}dx$$

(19)

$${K}_{m}=\frac{{(2\pi f)}^{2}}{2}{\int_{0}^{L}}m(x){(\delta (L)\cdot g(x))}^{2}dx$$

(20)

By setting K_m = U_m, f_r can be expressed as:

$${f}_{r}=\frac{1}{2\pi }{\left(\frac{{\int_{0}^{L}}{E}_{p}I(x){(\frac{{\partial }^{2}g(x)}{\partial {x}^{2}})}^{2}dx}{{\int_{0}^{L}}m(x)g{(x)}^{2}dx}\right)}^{\frac{1}{2}}$$

(21)

The width-line density m(x) and the equivalent moment of inertia for the piezoelectric layer I(x) are given by:

$$\begin{array}{cc}m(x)=\mathop{\sum }\limits_{i=1}^{n}{\rho }_{i}{t}_{i}wf(x)={\rho }_{p}{t}_{p}w{\phi }_{2}f(x); & {\phi }_{2}=\frac{\mathop{\sum }\nolimits_{i=1}^{n}{\rho }_{i}{t}_{i}}{{\rho }_{p}{t}_{p}}\end{array}$$

(22)

$$I(x)=f(x)\cdot \mathop{\sum }\limits_{i=1}^{n}\left(\frac{w{t}_{i}^{3}}{12}+w{t}_{i}({y}_{c}-{y}_{i})\right)=\frac{1}{6}w{{t}_{p}}^{3}f(x){\phi }_{1}{\phi }_{3}$$

(23)

where y_c is the neutral axis position and ${\phi }_{2}$ and ${\phi }_{3}$ are dimensionless functions:

$${y}_{c}=\mathop{\sum }\nolimits_{i=1}^{n}{E}_{i}{z}_{i}{t}_{i}w\left/\mathop{\sum }\nolimits_{i=1}^{n}{E}_{i}{t}_{i}w\right.$$

(24)

For the bimorph:

$${\phi }_{2}=3BC+2$$

(25)

$${\phi }_{3}=B+1$$

(26)

For the unimorph:

$${\phi }_{2}=2{B}_{e}{C}_{e}+BC+1$$

(27)

$${\phi }_{3}=\frac{AB(2{B}_{e}+B+1)}{2(AB+2{A}_{e}{B}_{e}+1)}$$

(28)

g(x) is the ratio of the displacement at each point relative to that at the tip of the cantilever beam. The static deflection mode function is not g(x), as there are differences between the resonant and quasi-static modes. However, the exact analytical expression of g(x) for an arbitrarily planar beam is difficult to derive and complex in form. Therefore, the Ritz method was used to obtain an approximate solution of g(x) with a simple form. Notably, the accuracy of this approach is sufficient for engineering applications. In this method, arbitrary second-order derivable functions ${\varphi }_{i}$(x) need to be chosen, and they need to satisfy only the displacement boundary conditions. Then, the stiffness matrix K and the mass matrix M can be derived³²:

$$\begin{array}{cc}{\rm K}=\left(\begin{array}{ccc}{k}_{11} & \cdots & {k}_{1n}\\ \vdots & \ddots & \vdots \\ {k}_{n1} & \cdots & {k}_{nn}\end{array}\right) & M=\left(\begin{array}{ccc}{m}_{11} & \cdots & {m}_{1n}\\ \vdots & \ddots & \vdots \\ {m}_{n1} & \cdots & {m}_{nn}\end{array}\right)\end{array}$$

(29)

$$\left\{\begin{array}{c}{k}_{ij}={\int }_{0}^{L}{f(x)}{{\varphi}^{\prime\prime} }_{i}{{\varphi}^{\prime\prime} }_{j}dx\\ {m}_{ij}={\int }_{0}^{L}{f(x)}{\varphi }_{i}{\varphi }_{j}dx\end{array}\begin{array}{cc} & i,j=1,2,\cdots ,n\end{array}\right.$$

(30)

The eigenvalues ${\bar{\omega }}_{i}^{2}$ and their corresponding eigenvectors α_i can be obtained by solving the following matrix eigenvalue problem:

$$(K-{\bar{\omega }}^{2}M)\alpha =0$$

(31)

$$\alpha ={[\begin{array}{ccc}{a}_{1} & \cdots & {a}_{n}\end{array}]}^{T}$$

(32)

The number of eigenvalues agrees with the number of ${\varphi }_{i}$(x). By substituting the eigenvector ${\alpha }_{1}$ corresponding to the smallest eigenvalue ${\bar{\omega }}_{1}^{2}$ into (33), the Ritz approximate solution of g(x) for the first-order modes can be obtained:

$$g(x)\approx \mathop{\sum }\limits_{i=1}^{n}{a}_{i}{\varphi }_{i}(x)$$

(33)

Notably, the above method can be used to find the resonant frequency and mode at higher orders up to order n.

For subsequent derivations, g(x) and f(x) are normalized with respect to the beam length L to obtain $\hat{g}$(t) and $\hat{f}$(t), which are independent of any factor except shape. Below, t is equal to x/L.

$$\begin{array}{c}\hat{g}(t)=g(L\cdot t)\\ \hat{f}(t)=f(L\cdot t)\end{array}$$

(34)

$$\begin{array}{l}{f}_{r}=\frac{\sqrt{6}}{12\pi }\cdot \sqrt{\frac{{E}_{p}}{{\rho }_{p}}}\cdot \frac{{t}_{p}\sqrt{{\phi }_{3}}}{\sqrt{{\phi }_{1}{\phi }_{2}}}\cdot \sqrt{\frac{{\int }_{0}^{L}{f}(x){(\frac{{\partial }^{2}g(x)}{\partial {x}^{2}})}^{2}dx}{{\int }_{0}^{L}{f}(x){g}^{2}(x)dx}}\mathop{\rightleftharpoons }\limits^{x\,=\,L\cdot t}\\\quad\quad\;\frac{\sqrt{6}}{12\pi }\cdot \sqrt{\frac{{E}_{p}}{{\rho }_{p}}}\cdot \frac{{t}_{p}\sqrt{{\phi }_{3}}}{{L}^{2}\sqrt{{\phi }_{1}{\phi }_{2}}}\cdot \sqrt{\frac{{\int }_{0}^{1}{\hat{f}}(t){(\frac{{\partial }^{2}\hat{g}(t)}{\partial {t}^{2}})}^{2}dt}{{\int }_{0}^{1}{\hat{f}}(t){\hat{g}}^{2}(t)dt}}\end{array}$$

(35)

Analysis of the FOMs

Substituting (35) and (18) into (7) and (10) yields the following expressions for FOM_vol and FOM_sen:

$$FO{M}_{vol}=\underbrace{\frac{\sqrt{6}}{4\pi }}_{{\rm{constant}}}\cdot \underbrace{{d}_{31}\sqrt{\frac{{E}_{p}}{{\rho }_{p}}}}_{{\rm{piezoelectric}}\,{\rm{material}}}\cdot \underbrace{\frac{\sqrt{{\phi }_{3}}}{{t}_{p}\sqrt{{\phi }_{1}{\phi }_{2}}}}_{{\rm{laminated}}\,{\rm{structure}}}\cdot \underbrace{\varOmega }_{{\rm{planar}}\,{\rm{shape}}}$$

(36)

$$FO{M}_{sen}=\underbrace{\frac{\sqrt{6}}{4\pi }}_{{\rm{constant}}}\cdot \underbrace{{d}_{31}\sqrt{\frac{{E}_{p}}{{\rho }_{p}{\varepsilon }_{r}}}\cdot }_{{\rm{piezoelectric}}\,{\rm{material}}}\,\,\underbrace{\sqrt{\frac{{\phi }_{3}}{{t}_{p}\lambda {\phi }_{1}{\phi }_{2}}}}_{{\rm{laminated}}\,{\rm{structure}}}\cdot \underbrace{\varOmega }_{{\rm{planar}}\,{\rm{shape}}}$$

(37)

where Ω is a formula related only to the planar shape and not to size.

$$\varOmega ={\int_{0}^{1}}{t}^{2}{\hat{f}}(t)dt\cdot \sqrt{{\int_{0}^{1}}{\hat{f}}(t){\left(\frac{{\partial }^{2}{\hat{g}}(t)}{\partial {t}^{2}}\right)}^{2}dt}\left/{\int_{0}^{1}}{\hat{f}}(t)dt\cdot \sqrt{{\int_{0}^{1}}{\hat{f}}(t){\hat{g}}^{2}(t)dt}\right.$$

(38)

The calculated Ω values for seven different shapes are given in Fig. 2a, and the corresponding $\hat{g}(t)$ and $\hat{f}(t)$ values for these shapes are given in Table 1. The mean value of Ω is 1.1812 with a standard deviation of 0.0242, indicating that the variation in Ω is negligible. Therefore, Ω can be treated as a constant in subsequent analyses. Additionally, it is clear that L and w do not appear in (36) and (37). This suggests that planar factors have a negligible effect on the FOMs.

Fig. 2: Effect of planar variables on the FOM of cantilever-type speakers.

a Calculated Ω values for seven different planar structures. b The irrelevance of the FOMs to beam length

Table 1 Normalized shape functions and normalized mode functions for different planar structures

The value of the dimensionless function $\sqrt{\phi_3/\phi_1\phi_2}$ is determined by the laminated structure, which involves the selection of materials for electrodes and structural layers given the ratio between the thickness of each of these layers and the piezoelectric layer. Unlike Ω, this term cannot be treated as a constant. Thus, the FOMs are clearly influenced mainly by the layer thickness and material selected and are almost fully independent of the planar shape and size. An essential and interesting conclusion is that the correspondence between the FOMs and the planar structure is one-to-many. To illustrate this point, Fig. 2b provides a concrete example with the beam length L as the variable. When the design in the thickness direction is fixed, altering L merely involves a trade-off between the quasi-static output and the resonant frequency, with their product remaining unchanged. Similarly, replacing L in Fig. 2b with w or the planar shape $\hat{f}$(t) does not significantly influence the FOMs.

Although the derivation above was specifically focused on the cantilever beam, it is likely that similar principles apply to other structures. However, conducting a rigorous theoretical derivation for each structure is challenging. Therefore, three typical designs were selected to extend the theory through FEM simulations. They share the same thickness configurations: a unimorph structure comprising a 5 μm silicon support layer, 0.1 μm molybdenum electrodes, and a 0.5 μm AlN piezoelectric layer. Detailed planar geometries for these designs can be found in the literature^4,10,33.

Herein, FOM_vol was calculated by first obtaining simulated resonant frequencies f_r and mean displacement ${\bar{\delta }}_{{st}}$, then incorporating them into (7). Figure 3 illustrates the trend of the simulated FOM_vol as a function of the planar dimensions for these structures. Figure 3a shows the trend for a cantilever-type structure. As depicted in Fig. 3b–d, similar to the situation with cantilever beams, the FOM is almost unaffected by the planar dimensions. This indicates that the independence of the FOM from planar dimensions may be a general characteristic applicable to a wide range of structures. In contrast, the independence of the FOM from planar shape does not hold as a universal rule. This is evident from the differing FOM values observed in Fig. 3 for the four structures, despite their identical thickness structure. Based on current evidence, only the cantilever beam type has demonstrated planar shape independence of the FOMs.

Fig. 3: Simulated relationship between the FOMs and planar dimension for different structures.

a Cantilever-type⁷. b–d Typical structures^4,10,33

The independence of FOMs from planar dimensions enables them to capture the intrinsic characteristics of a structure, remaining unaffected by variations in size. This means that the broadband characteristics of a design can be easily derived from a single example via the relevant FOMs. In other words, once the frequency response of one speaker is established, the responses of other speakers with the same design but varying sizes can be predicted. Moreover, since FOMs are the product of the quasi-static response and natural frequency, a structure with high FOMs will exhibit excellent performance at a given natural frequency. This suggests that when comparing two speakers with different resonant frequencies, adjusting their planar dimensions to match the resonant frequencies for normalized output comparison is unnecessary, and simply comparing their FOMs will suffice. The independence of FOMs from planar dimensions thus makes them convenient and practical criteria for evaluating the wideband performance of speakers.

In addition to serving as practical evaluation criteria, FOMs can aid in simplifying the design process. Design variables can be categorized into two types on the basis of their impact on the FOM(s): relevant variables and irrelevant variables. During the design process, these two types of variables can be determined independently without mutual influence. The determination of the relevant variables can be prioritized, and this process can be transformed into a single-objective, multivariable optimization problem. The corresponding mathematical model is presented as follows:

$$\begin{array}{cc}\mathop{\min }\limits_{x} & -FOM({\boldsymbol{x}})\\ s.t. & ub\ge {\boldsymbol{x}}\ge lb.\end{array}$$

(39)

where x represents the vector of design variables and ub and lb represent the vectors of the upper and lower bounds, which are related to the fabrication process. After the relevant variables are identified, the irrelevant variables can be determined either through solving the associated equations or via simulation on the basis of the desired resonant frequency. The effective diaphragm area of the device can then be adjusted by varying the number of elements N. These two steps can be carried out without mutual interference.

FOM-based design

To demonstrate the proposed design method and validate its effectiveness, we designed, fabricated and characterized a set of MEMS speakers. The objective was set as developing a compact, low-voltage, low-power MEMS tweeter. To achieve this, AlN was selected as the piezoelectric material. Notably, AlN has a well-established manufacturing process and a low dielectric constant. Additionally, the resonant frequency was targeted at approximately 5 kHz to ensure a balanced response across the target frequency range of 2–16 kHz. The first step was to determine the relevant variables. The corresponding mathematical model is shown in Eq. (39). Given the relatively low dimensionality of the vector x in this problem, the optimal solution for the FOM was determined by analyzing the graphs of the FOMs concerning the relevant variables. As shown in Fig. 4, for the bimorph structure, the FOM_vol increases monotonically as both t_p and t_e decrease, with the steepest gradient of increase at small thicknesses. FOM_sen exhibits a similar trend; however, the key difference is that for a fixed t_e, there is an optimal point for t_p, which decreases as t_e decreases. For the unimorph structure, minimizing t_p within the process constraints yields better results. Within the same fabrication boundaries, the bimorph structure has a higher upper limit for both efficiency and output capability than the unimorph structure does. On the basis of practical considerations, the minimum thicknesses for Mo, AlN, and Si are set to 50 nm, 100 nm, and 1 µm, respectively. Within the fabrication boundaries, the maximum value of FOM_vol corresponds to the combination of a bimorph structure and the minimum permissible thickness. In this configuration, FOM_sen also reaches a highly desirable level. Therefore, the designed thicknesses of the piezoelectric layers and the electrodes are set to 100 nm and 50 nm, respectively.

Fig. 4

FOM trends with respect to relevant variables

Since there are no theoretical FOM differences in planar shapes for cantilever speakers, we selected a triangular design scheme for the planar design. The dimensional parameters were specifically designed to ensure that the resonant frequency falls within the 4–6 kHz range, thereby achieving a balanced response in the mid-to-high frequency spectrum. The lengths L of the three different variants were set to 425 µm, 400 µm, and 375 µm, respectively. The details are illustrated in Fig. 5a. To evaluate the acoustic performance of the speakers, a simulation model was established using COMSOL Multiphysics 6.1, as illustrated in Fig. 5b. The speaker was connected on one side to the IEC 60318-4 ear simulator from the COMSOL library, while the opposite side was placed in a free sound field environment bounded by perfectly matched layers to absorb outgoing sound pressure¹³. Notably, the characteristic mesh element size throughout the model was constrained to be smaller than 1/7 of the wavelength of 20 kHz (maximum frequency) to ensure numerical accuracy. The FEM simulation results for the acoustic response, presented in Fig. 5c, reveal that all design variants exhibit a consistent peak above 10 kHz. This artifact originates from the half-wavelength standing wave resonance inherent to the IEC 60318-4 ear simulator^2,13.

Fig. 5: Design and simulation of high-FOM speakers.

a Schematic diagram of the proposed design. b FEM modeling and setup. c Simulated SPL frequency response with a driving voltage of 2.5 V_rms

Performance results and comparison

Figure 6a, b display a photograph of the fabricated device and its surface profile obtained by the confocal microscope, respectively. Figure 6c shows the cross-sectional SEM image. To evaluate the crystalline quality of the 100-nm-thick AlN film, the X-ray rocking curve of the as-deposited AlN piezoelectric layer is shown in Fig. 6d. The fabricated piezoelectric MEMS speaker was acoustically characterized by using an IEC 60318-4 ear simulator. Under a 2.5 V_rms excitation, the SPL at 1 kHz for the three variants reached 95.4, 94.4, and 92.6 dB, respectively (Fig. 7a). Among these, the variant with a length L of 425 µm maintained an SPL above 100 dB across the core high-frequency range of 2–13 kHz, with a maximum SPL of 115 dB. The resonant frequencies were 4.5, 5, and 5.8 kHz, respectively. A noticeable low-frequency degradation was observed in the frequency band below 400 Hz due to cantilever warping induced by residual stress, which resulted in an increased slit width. Wider slits lead to more severe acoustic short-circuiting, a phenomenon where out-of-phase acoustic waves from opposite sides of the diaphragm cancel each other out. It becomes particularly pronounced in the low-frequency range^{1,10,21,23,24}. The warping-induced slit width enlargement was not accounted for in the numerical model, resulting in discrepancies between simulated and experimentally measured responses within this frequency band. Overall, the results align well with those of the finite element simulation presented in Fig. 5c. Discrepancies may be attributed to factors not accounted for in the simulation, such as residual stresses, manufacturing tolerances in the back cavity boundaries, and viscous effects in air domains outside the slit region.

Fig. 6: Fabricated high-FOM speaker.

a Photograph of the device fabricated on a coin. b Surface profile (L = 375 μm) obtained by the confocal microscope. c Cross-sectional SEM image. d X-ray rocking curve of the as-deposited AlN piezoelectric layer

Fig. 7: Performance of the high-FOM speaker.

a SPL test results with a driving voltage of 2.5 V_rms. b–e Comparison of the proposed speaker with other designs^9,14,28,34 in terms of b normalized SPL, c FOM_vol, d normalized sensitivity, and e FOM_sen

Figure 7b, c show the normalized SPL and FOM_vol. Figure 7d, e show the normalized sensitivity and FOM_sen. As predicted, the FOMs for the three variants are arguably the same, demonstrating good agreement with theoretical values; designs with longer beams exhibit a superior frequency band response before the resonant frequency is reached, whereas designs with shorter beams display a better response above the resonant frequency. The variant with a beam length of 400 µm is used as an example; its normalized SPL at 1 and 10 kHz reaches an impressive 76.6 and 86.6 dB/mm²/V_rms, respectively, with normalized sensitivities of 91.2 and 91.5 dB/mm²/mW. The normalized performance and FOMs of four state-of-the-art MEMS speakers are also shown in Fig. 7b–e^9,14,28,34. 1 kHz and 10 kHz were selected for performance comparisons, as they serve as common metrics for assessing mid- and high-frequency response characteristics. The proposed bimorph piezoelectric speaker yields the highest FOMs, normalized voltage sensitivity and normalized power sensitivity to date, indicating that the speaker is well suited for low-voltage drive conditions. The two FOMs are relatively well balanced in the proposed speaker design, with both metrics reaching high levels. This balance is attributed to the precise theoretical expressions that aid in finding optimal parameter combinations.

Discussion

Validation of FOM theory

Despite the use of AlN, a piezoelectric material not typically known for its actuation ability, the proposed speaker with optimized FOMs demonstrates remarkable normalized output performance, exceeding the performance of a wide range of PZT-driven speakers. This finding indicates that regardless of the materials or structures used, designs with high FOMs consistently exhibit superior performance. It is hypothesized that the use of materials with a high d₃₁, such as PZT, through the optimization of FOMs could further enhance speaker performance. Furthermore, this result strongly supports the effectiveness of the design method, in which the FOMs are optimized considering the potential of the speaker driving materials, potentially providing a basis for speaker optimization.

On the other hand, the effective range of FOM theory has been validated. The measured performance of the speaker and the theoretical predictions align well. When the diaphragm is sufficiently thin, various issues can arise, such as residual stress and deterioration of crystal quality, which make simplified theoretical analysis challenging. Therefore, experimental validation is necessary. The test results for the proposed structure indicate that the FOM theory remains valuable for reference at a total diaphragm thickness as thin as 0.4 μm, fully covering the thickness range of all existing MEMS speaker designs.

Significance and limits of FOM theory

In this work, FOMs represent the output capability across a wide frequency range and are easy to calculate, making them suitable for comparing the performance of different designs. Furthermore, FOMs can be used to guide design, as their independence with respect to planar dimensions allows the speaker design to be simplified from a multi-objective, multivariable optimization problem into a systematic problem of single-objective, multivariable optimization combined with solving the relevant equations. This strategy transforms the design of MEMS speakers from a simulation-based trial-and-error approach into a more efficient mathematical approach.

Theoretical and experimental results show that the influence of planar dimensions on the FOMs is negligible, especially for cantilever-type speakers, for which even the shape has a negligible effect on FOM values. However, this does not imply that planar shapes are not important. In fact, the FOMs do not reflect all aspects of speaker performance and only incorporate basic parameters such as output sensitivity and efficiency, which is the greatest limitation of the proposed approach. Planar factors significantly influence other aspects of performance, such as total harmonic distortion (THD), high-order modes, and the fill factor. For example, shapes with a large equivalent radiating area require less displacement to achieve the same SPL. Reduced displacement leads to weaker geometric nonlinearity, which generally results in lower THD^13,14. Additionally, the planar design can be adjusted to change the positions of high-order modes, even when the first-order resonant frequency is fixed, such as to fulfill the requirements of ultrasonic applications^35,36. In other words, FOM theory provides a simple way to control fundamental performance parameters such as SPL and efficiency, allowing designers to focus on other capabilities of MEMS speakers.

Although theoretical calculations were validated through FEM simulations and experiments, the designed resonant frequency tends to be slightly lower than that predicted. This is attributed to the fact that the radiation impedance, also referred to as the “co-vibrating air mass” in the audio field, is not considered in the model. This issue becomes increasingly pronounced as the diaphragm thickness decreases. In future work, we will incorporate this factor into the model and further optimize the corresponding method.

Conclusion

In this work, we derive expressions for the average displacement, resonant frequency and sensitivity of cantilever beams of arbitrary shapes. We define the products of the quasi-static normalized voltage sensitivity and power sensitivity and the natural frequency as FOM_vol and FOM_sen, respectively. The analysis reveals that planar parameters are independent of the FOMs and that FOMs can serve as a reliable metric for evaluating speaker designs for certain structural classes. In addition, FOMs can be used to enhance speaker design. Under the guidance of FOM theory, the developed MEMS speaker demonstrates outstanding performance, with normalized sound pressure and sensitivity values ranking higher than those of other speakers. The success of this speaker confirms the reliability of FOM theory, even at the microscale, with a total speaker thickness of only 0.4 μm.

FOMs offer a comprehensive basis for evaluating speaker performance across a wide bandwidth and provide a systematic and quantifiable framework for assessing MEMS speakers. The introduction of FOMs transforms the design process, shifting it from time-consuming trial-and-error methods based on FEM or LEM simulations to a streamlined, single-objective, multivariable problem-solving approach, significantly improving both efficiency and focus.