I. INTRODUCTION

Recently, Light Detection and Ranging (LiDAR) sensors are being used for autonomous driving in cars and logistics transport robots equipped with artificial intelligence ^[1,2]. Initially, mechanical type LiDAR driven by a motor was used, and research is being conducted on MEMS type LiDAR, which is smaller, cheaper, and consumes less power ^[3,4].

MEMS type LiDAR eliminates the need for a motor, resulting in a more compact and vibration-resistant design. In addition, the micromirrors, core component of MEMS LiDAR system, can be fabricated using semiconductor processes, potentially reducing production costs.

Micromirrors can be driven by piezoelectric, electromagnetic, or electrostatic methods. In general, the piezoelectric method has a small displacement and requires the formation of an efficient piezoelectric thin film, while the electromagnetic method is known to require a coil and consume a lot of current. On the other hand, in the case of the electrostatic method, a conductive structure is sufficient, so it is simple to manufacture and has the advantage of low current consumption.

Vertical comb drive actuators are the most common type of electrostatic actuator for micromirror ^[5,6]. However, vertical comb-type electrostatic actuators require high voltages for operation. For example, R. Farrugia et al. presented a resonant micromirror using the SOIMUMP process and applied a driving voltage of 200 Vpk to drive the micromirror ^[5]. V. Milanovic et al. developed and utilized a micromirror including up/down comb and rotary transformer, and a driving voltage of up to 80 Vpk was required to drive the micromirror ^[6,7]. S. Nabavi et al. proposed a vertical actuator using the electrostatic force of the lateral comb on a plane, but using a structure that rotates well in the vertical direction ^[8]. Only very limited applications are possible because a long spring with a cross-sectional area of 10 ${\mu}$m ${\times}$ 10 ${\mu}$m and a length of 2 mm and a special structure must be used.

In this paper, a tilted plate electrostatic actuator (TPEA) that can drive large electrostatic forces with a lower driving voltage than previous works is proposed. Section II explains the working principle of the proposed TPEA. And based on theoretical analysis, the performance of TPEA is compared with previous electrostatic force actuators. Section III compares the performance of the proposed TPEA and the previous electrostatic force actuator based on simulation using the finite element method and Section IV concludes.

II. WORKING PRINCIPLE AND THEORETICAL ANALYSIS

Fig. 1(a) shows a 3-dimensional view of the TPEA proposed in this paper. Fig. 1(b) shows a cross section cut along the FF line in Fig. 1(a). TPEA consists of a bottom electrode and a tilted top plate. There is no spring restraining the z-axis direction on the A side of the top plate, and a y-axis torsion spring can be connected to the B side of the top plate. The object to which z-axis force is transmitted and driven is connected to D. When a DC voltage difference is applied between the bottom electrode and the top plate, side A of the top plate touches point C of the substrate along the S1 path and is positioned at the initial position. The angle formed by the initial position with the substrate is ${\theta}$$_{0}$. When AC voltage is added to the DC voltage between the bottom electrode and the top plate, the top plate rotates around point C. At this time, a vertical force in the z-axis is generated at point B of the tilted plate, and the vertical force is transmitted to point D. As the applied AC voltage decreases, the electrostatic force decreases and the tilted plate begins to return to its initial position. Depending on the frequency of the applied AC voltage, the tilted plate is repeatedly driven along path S2.

Fig. 1(c) shows an example of applying TPEA to a seesaw mirror to help understand the operation of TPEA. The seesaw mirror consists of an optical mirror plate (P$_{\mathrm{mirror}}$), torsion springs (k$_{\mathrm{m}}$) fixed to R$_{1}$ and R$_{2}$, and two TPEAs. Points R$_{1}$ and R$_{2}$ have a fixed boundary condition. The TPEA is connected to the mirror plate via a torsion spring (k$_{\mathrm{a}}$). When TPEA1 is driven, the mirror rotates to the -y axis, and when TPEA2 is driven, the mirror rotates to the +y axis. As the TPEA moves down toward the bottom electrode, the actuator torsion spring (k$_{\mathrm{a}}$) twists as shown in the inset of Fig. 1(c) and transmits driving force to the mirror.

In the following subsections, the electrostatic forces generated by the proposed structure, the previous parallel plate structure, and the previous comb structure are theoretically derived and compared.

Fig. 1. Tilted plate electrostatic actuator (TPEA): (a) 3-dimnsional view; (b) cross-sectional view at FF; (c) application of TPEA to seesaw mirror.

A. Electrostatic Force of TPEA

Fig. 2 shows the shape and main geometric parameters for calculating the electrostatic force of the tilted plate structure. The symbols L$_{t}$ and W$_{t}$ represent the length and width of the top plate, respectively. The symbol L$_{e}$ represents the length of the lower electrode. The symbols g$_{t}$ and z$_{t}$ represent the z-axis distance between the top plate and bottom electrode at points M and N, respectively. The symbol l represents the x-axis distance from C to N, which is the non-overlap section between the top plate and bottom electrode. The symbol ${\theta}$ refers to the inclination angle that the top plate forms with the substrate.

Fig. 2. Simple model of the tilted plate electrostatic actuator.

Basically, electrostatic force can be obtained by differentiating the energy stored in a capacitor made of two electrodes in the direction of movement. The capacitance and stored energy formed between the tilted top plate and the bottom electrode are C$_{t}$ and E$_{t}$, respectively, and can be expressed as Eqs. (1) and (2).

(1)

$ C_{t}=\varepsilon \frac{W_{t}\cdot (l+L_{e})}{g_{t}}\ln \left(1+\frac{L_{e}}{l}\right) \\ $

(2)

$ E_{t}=\frac{1}{2}\varepsilon \frac{W_{t}\cdot (l+L_{e})}{g_{t}}\ln \left(1+\frac{L_{e}}{l}\right)V^{2} $

In Eq. (2), the symbol V represents the voltage difference applied between the inclined top plate and the bottom electrode, and the symbol ${\varepsilon}$ represents the dielectric constant. The electrostatic force F$_{t}$ can be expressed as Eq. (3).

(3)

$ F_{t}=F_{t0}\cdot \alpha \\ $

(4)

$ F_{t0}=\frac{1}{2}\varepsilon \frac{W_{t}L_{e}}{g_{t}^{2}}V^{2} \\ $

(5)

$ \alpha =1+\frac{L_{e}}{l} $

In Eq. (2), the electrostatic force F$_{t0}$ can be expressed as Eq. (3), and means the electrostatic force generated when the gap between the upper plate and the bottom electrode is uniform with g$_{t}$. The symbol ${\alpha}$ in Eq. (2) is defined as in Eq. (5) and is an amplification coefficient that indicates how much the electrostatic force increases when the top plate is tilted compared to when it is parallel. It can be seen that the amplification coefficient is at least greater than 1, and is proportional to the length ratio of the length L$_{e}$ of the bottom electrode to the length l between point C and point N. If the length ratio is 1, the amplification coefficient becomes 2, and if the length ratio is 9, the amplification coefficient becomes 10.

B. Electrostatic force of conventional parallel plate actuator

Fig. 3(a) shows the shape and main geometric parameters for calculating the electrostatic force of the conventional parallel plate structure. The symbols L$_{p}$ and W$_{p}$ represent the length and width of the overlap area between top plate and bottom electrode, respectively. The symbol g$_{p}$ represents the z-axis distance between the top plate and bottom electrode.

Fig. 3. Simple model of conventional electrostatic actuator: (a) Parallel plate type; (b) Vertical comb type.

The capacitance and stored energy formed between the top plate and the bottom electrode are C$_{p}$ and E$_{p}$, respectively, and can be expressed as Eqs. (6) and (7).

(6)

$ C_{p}=\varepsilon \frac{W_{p}\cdot L_{p}}{g_{p}} \\ $

(7)

$ E_{p}=\frac{V^{2}}{2}\varepsilon \frac{W_{p}\cdot L_{p}}{g_{p}} $

In Eq. (5), the symbol V represents the voltage difference applied between the inclined top plate and the bottom electrode, and the symbol ${\varepsilon}$ represents the dielectric constant. The electrostatic force F$_{p}$ in the z-axis direction can be obtained by differentiating the energy E$_{p}$ with respect to g$_{p}$ and can be summarized as Eq. (8).

(8)

$ F_{p}=\frac{1}{2}\varepsilon \frac{W_{p}\cdot L_{p}}{g_{P}^{2}}V^{2} $

III. SIMULATION RESULTS

Two-dimensional (2D) finite element analysis (FEA) simulations were performed using the dimensions provided in Table 1. Compared to three-dimensional (3D) FEA simulations, 2D FEA simulations offer the benefits of reduced computational time and simplified interpretation. In this study, 2D FEA simulation was employed to evaluate the electrostatic force performance of TPEA, parallel plate drive actuator, and comb drive actuator. In the simulation, the lower edge of the bottom electrode or lower electrode of the model was set as a fixed constraint. A boundary condition was given to the upper plate or upper electrode so that it could move in the y direction from the initial position. The initial position of the tilted plate actuator was made so that one side of the top plate was moved by -4 ${\mu}$m in the y direction and touched the floor. Ground was set on the top plate and DC voltage was applied to the bottom electrode. A boundary probe was assigned to the bottom electrode, and the electrostatic force was measured by specifying the expressions maxwell stress, outer nominal traction, and y component.

Fig. 4 shows the 2D model and electric field during simulation to extract electrostatic force from the parallel plate actuator, comb actuator, and TPEA.

Fig. 5 shows the simulated electrostatic force values according to the applied voltage. Table 2 summarizes the force per unit area generated by the three actuators when 50 V is applied. The comb actuator, parallel plate actuator, and TPEA generated electrostatic forces of 195N, 731N, and 4.7 kN per unit area, respectively. These electrostatic force values are 6.4 times that of the parallel plate actuator and 24.1 times that of the TPEA compared to the comb actuator, which is consistent with the theoretically calculated values of 6.5 times and 22.8 times.

Fig. 6 shows a comparison of the results when electrostatic force simulation was performed on a 2D or 3D basis. Even if the electrostatic force is simulated based on 2D, the error is smaller than the 3D-based result, so when considering the resources and time of simulation, performing 2D-based simulation is sufficient to identify the characteristics.

Table 3 presents a performance comparison of TPEA with previously reported actuators. Comb sizes required for calculating electrostatic force were estimated using photographic data if not specified in the reference literature. If the electrode gap was not clearly posted in the reference paper, it was assumed to be 4 ${\mu}$m, which is the same value used to calculate electrostatic force in this paper.

Fig. 4. Result of electric field analysis simulation: (a) Parallel plate type; (b) Vertical comb type; (c) Tilted plate type.

Fig. 5. Result of electrostatic force analysis simulation.

Fig. 6. Electrostatic force simulation results: (a) 3D model of the proposed TPEA; (b) Electrostatic force comparison in 2D and 3D models.

IV. CONCLUSIONS

A new tilted plate electrostatic actuator (TPEA) is proposed. The proposed TPEA has a tilted top plate with one side touching the floor and the spacing between electrodes is uneven. As a result of applying reasonable values to the theoretical analysis results, TPEA generated a force approximately 6.5 times greater than that of the conventional parallel plate electrostatic actuator and approximately 22.8 times greater than that of the conventional vertical comb actuator. Through finite element analysis, the results were confirmed to be consistent with the theoretical analysis results. For the same applied voltage, TPEA generates a force 22.8 times greater than that of the vertical comb actuator, so to generate the same force, the applied voltage of the TPEA requires only 21% of that of the vertical comb actuator. TPEA, which can generate large electrostatic forces with such a small driving voltage, is expected to be applied to several MEMS applications, including micromirrors.