# Modeling of MEMS Mirrors Actuated by Phase-Change Mechanism

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}), which goes through a phase transition that can be induced by a gradient of temperature. VO

_{2}has a reversible solid-to-solid phase transition that comes with drastic changes in the mechanical [11], electrical [12], and optical properties [13] of the material. When induced thermally, the transition of VO

_{2}occurs at ≈$68$ °C, but this transition temperature can be reduced by doping [14] or adding extrinsic stress to the material [15,16]. The integration of VO

_{2}with the MEMS mirror technology decreases the temperature required in the conventional TE mechanism, from $300$ °C to $90$ °C for full actuation, which lowers the total power consumed by the device. Another advantage of using VO

_{2}as the actuation mechanism is the large strain energy density generated during the transition, with values higher than conventional actuation mechanisms such as thermal expansion, electrostatic, electromagnetic, and piezoelectric [17]. Furthermore, the intrinsic hysteretic behavior of VO

_{2}properties (including the mechanical stress that generates deflection in VO

_{2}-based MEMS [18,19]) across the phase transition has been exploited to design programmable MEMS actuators [20] and resonators [21], and can be used as well to program tilting angles in MEMS mirrors. However, all of these advantages come at the cost of added nonlinear effects that make the modeling and control more complicated than other actuation mechanisms.

_{2}-based MEMS devices. Nonlinear mathematical models such as the Prandtl–Ishlinskii model [25] and the Preisach model [26] have been adopted to capture and estimate the hysteresis behaviors. Unlike the identification of the Prandtl–Ishlinskii model, which requires solving a nonlinear optimization problem, the Preisach model identification problem can be reformulated as a linear least-squares problem and solved efficiently [26]. The Preisach model is thus adopted in this work. In order to control the systems with hysteresis, feedforward control can be realized by inverting the hysteresis nonlinearity [26], and feedback control can also be implemented, where the feedback signal can be obtained based on external sensors or with self-sensing methods [27]. In self-sensing, the correlation between the electrical and mechanical properties across the transition is utilized [28].

_{2}-based MEMS mirror. The modeling is focused on one of the four actuators of the device. First, the mechanical model of the system is derived, where the nonlinear behavior of the VO

_{2}is incorporated in the model as an external force applied to the system. A Preisach model is used to capture the hysteresis behavior of the VO

_{2}. The parameters for the whole model are identified using simulation and experimental results. Finally, the hysteresis model is validated with a different set of experimental results including quasi-static and dynamic responses. The proposed model can be translated to other actuators of the MEMS mirror, and this work facilitates the control of the device.

## 2. Experimental Procedures

_{2}-based MEMS mirror used in this paper is shown in Figure 1 and the design has been reported in [7,29,30]. The device consists of four mechanical actuators (legs) coupled with a reflective platform (mirror). Tilting of the mirror platform is achieved by individual actuation of the legs, which is independently controlled, or by actuation of all the legs using the same input signal simultaneously, which generates a piston-like movement. There are two actuation mechanisms: stress due to the thermal expansion difference of the materials forming the bimorph regions of the device, and stress generated during the phase-change transition of the VO

_{2}. Outside the phase transition region of VO

_{2}, the only active mechanism is thermal expansion, while, during the phase transition, both mechanisms exist, but the phase transition of VO

_{2}dominates [26]. The generated stress is capable of bending a thin bimorph rectangular structure composed of continuous SiO

_{2}(≈$1.4\mathsf{\mu}\mathrm{m}$) and VO

_{2}($250\mathrm{n}\mathrm{m}$) layers and a thin patterned metal ($130\mathrm{n}\mathrm{m}$) layer. To increase the vertical displacement of the leg, a rigid structured (frame), composed of a thick ($50\mathsf{\mu}\mathrm{m}$) layer of Si, connects the bimorphs. The transition of the VO

_{2}film is induced using Joule heating, where an input current is applied through the monolithically integrated resistive heater of the leg. The metal traces are designed to have a smaller width in the bimorph parts of the leg. This is done to create a higher resistance in these regions of the heater, which increases the dissipated power and localizes the generated heat in the bimorph regions.

#### 2.1. Design and Fabrication of VO_{2}-Based MEMS Mirrors

_{2}-based MEMS mirror presented here follows the same fabrication process as in [10], but different metal layers are used to increased the yield per wafer—this is discussed in more detail in Section 2.2. Device fabrication starts with a two–inch double–sided p-type <111> polished Si wafer as the substrate with a thickness of $300\mathsf{\mu}\mathrm{m}$. A thin SiO

_{2}layer ($1\mathsf{\mu}\mathrm{m}$) is deposited on both sides of the wafer by plasma enhance chemical vapor deposition (PECVD) at a temperature of $300$ °C. One of the SiO

_{2}layers is used as a mask for the Si back-side etch, while the other (top side) forms the first layer of the bimorphs over which the VO

_{2}films are deposited. A thin film from any material that can survive most chemicals used in standard MEMS processing (i.e., Si${}_{3}$N${}_{4}$) would have been an acceptable choice for the backside. However, the selection of SiO

_{2}material for the top side is based on the larger mechanical actuation across the VO

_{2}phase transition, which is due to the higher orientation of the VO

_{2}grains with monoclinic (011)${}_{M}$ planes parallel to the substrate [20]. Although higher VO

_{2}orientations are expected from crystalline substrates (i.e., quartz, sapphire), their processing represents major fabrication processing hurdles. The VO

_{2}is deposited by pulse laser deposition (PLD) and patterned with dry etching using reactive ion etch (RIE), following the procedure shown in [10].

_{2}, the remaining processes are performed at temperatures lower than $250$ °C to avoid any degradation in the VO

_{2}due to over-oxidation of the film. A $200\mathrm{n}\mathrm{m}$ SiO

_{2}layer is grown by PECVD using a temperature of $250$ °C on top of the VO

_{2}, for electrical isolation from the metal traces that will be deposited next. The electrical connections and resistive heaters are fabricated by depositing and patterning via lift-off layers of Cr ($20\mathrm{n}\mathrm{m}$)/Au ($110\mathrm{n}\mathrm{m}$), where the Cr is used as an adhesion layer between the SiO

_{2}and the Au. Another $200\mathrm{n}\mathrm{m}$ of SiO

_{2}is deposited to insulate the metal traces from the ambient (air), reducing the thermal losses. This is followed by a sequence of SiO

_{2}dry etch steps by RIE in order to expose the metal contacts, pattern the legs and platform of the device, and expose the Si substrate. The same SiO

_{2}etching is repeated on the back-side SiO

_{2}layer to expose the Si substrate. During the processing of the back-side, the top side was protected by spinning PMGA (polymethylglutarimide) resist. After processing the backside, the PMGA is removed by submerging the sample in photoresist stripper (Microposit Remover 1165). Using the SiO

_{2}as a hard mask, the exposed Si layer on the backside was etched with deep reactive ion etch (DRIE). The DRIE etching is timed to remove $250\mathsf{\mu}\mathrm{m}$ of the Si layer, reducing the Si substrate from $300\mathsf{\mu}\mathrm{m}$ to $50\mathsf{\mu}\mathrm{m}$. The mirror structure is released by etching the remaining $50\mathsf{\mu}\mathrm{m}$ of the Si substrate from the top by DRIE. Finally, to remove the Si from certain parts of the legs and create the bimorph sections, a Si isotropic etch is performed using XeF

_{2}gas. This process is timed to only etch the desired parts avoiding any undesired over etch that would affect the frame regions of the legs.

#### 2.2. Increasing Yield by Reducing Intrinsic Stress

_{2}after the metallization. However, the use of Ti/Pt created a low yield (≈12.5%) in the final devices, due to peeling of the metal. We thought this was due to the intrinsic stress of the evaporated Ti/Pt metal layer on SiO

_{2}, which could be as high as 340 MPa (compressive stress) [31,32]. In order to address the issue, in the present work, we have substituted the Ti/Pt layer with evaporated Cr/Au, which has lower intrinsic stress (250 MPa tensile stress) [33,34]. This has increased the yield to ≈75%.

#### 2.3. Experimental Setup

## 3. Modeling

_{2}-based MEMS mirror to have a 2D movement upon actuation. Therefore, the description and modeling of the system will involve movements along two perpendicular axes: pitch and roll. Figure 3 shows a schematic of the platform with the two axes used to describe the tilting movement of the platform. The force ($\overrightarrow{F}$) represents the actuation generated by the legs. A set of two equations (one per degree of freedom: pitch and roll) is used to model the movement of the mirror. The inclusion of the VO

_{2}in the device adds a nonlinear term to the equations due to the hysteretic behavior of the material. A non-monotonic Preisach model is developed to capture the hysteresis term. The effect of the VO

_{2}is included in the external force that generates actuation. The parameters for the linear part of the equation that describe the system’s mechanical response are obtained from a combination of experimental measurements and finite element method (FEM) simulations (details in Section 3.1). The coefficients of the nonlinear part of the equation are calculated from a set of experiments (details in Section 3.2).

#### 3.1. Linear Model

_{2}is represented in $\overrightarrow{T}$ via

_{2}can be expressed as:

#### 3.2. Nonlinear Model

_{2}. Similar to [26], a non-monotonic hysteresis model is developed:

#### 3.2.1. Phase Transition-Induced Force

#### 3.2.2. Differential Thermal Expansion-Induced Force

_{2}and SiO

_{2}layers. This component was modeled as a linear term and a quadratic term in previous studies [26,27]. The following linear model is adopted in this work:

_{2}layer and SiO

_{2}layer, and the negative term is introduced due to the fact that the thermal expansion-induced force has an opposite direction as the phase transition-induced force.

## 4. Results and Discussion

#### 4.1. Simulation Results

#### 4.2. MEMS Mirror Mechanical Model

_{2}when actuating only one leg. Before each experiment, a pre-heating stage is performed to improve the stability and repeatability of the measurements, caused by the use of gold as the metal trace [40,41]. A similar process was performed in [7], where a sine wave was applied as the input voltage to anneal the metal layers. For the VO

_{2}-based MEMS mirrors in this work, the pre-heating stage consisted of applying a 12 mA to all of the actuators for a total of 10 min. An input sequence of increasing voltage steps is used to measure the thermal time response of the actuated leg. The input is applied to the base of the transistor and had increasing amplitude steps of 0.5 V, which corresponds to ≈0.7 mA (once the transistor is on). Each step is held for 1 s before the next step started. The thermal time response (${\tau}_{th}$) within steps is calculated from the rise time using the following equation:

_{2}, the values for the pitch and roll movements are 14 ms and 14.79 ms. During the transition of the VO

_{2}, the system showed a pseudo-creep effect where each step took longer to reach steady state compared to outside the transition. This effect can be caused by the added stress from the legs that are not actuated. The added stress can move the transition temperature of the VO${}_{2}$, which has been observed previously in VO

_{2}thin films [42,43]. Even more relevant to the present case, this effect was also observed in VO

_{2}-based MEMS mirrors [10], where it was found that individual leg actuation and piston-like actuation required different actuation voltages—note that, during individual actuation, the remaining mirror legs add a stress that is not present during piston-like movement. The pseudo-creep is not included in the modeling of the device, in order to focus on the fundamental thermal and mechanical dynamics in the general case, and, as verified in later experiments, the presented model (ignoring the creep effect) shows adequate capability in predicting the mirror dynamics.

_{2}, it does capture the mechanical response of the system. The values for the resonance frequency (${\omega}_{n}$) and the damping ratio ($\zeta $) for each degree of freedom are found by a curve fit, and fitting parameters are shown in Figure 6. It is worth noted that the presented curve fitting method works well at low frequencies and produces larger errors at higher frequencies above 150 Hz. This is likely due to the fact that the fitting uses linear approximation and that the mechanical couplings between each leg and between the legs and the mirror are not fully captured. The highest frequency considered in the mechanical response of the system is 10 Hz, and the model follows the experimental results fairly well in this frequency range. Analysis and modeling at higher frequencies are potential extensions to this study:

#### 4.3. Identification and Verification

#### 4.3.1. Identification

#### 4.3.2. Quasi-Static Verification

#### 4.3.3. Frequency Verification

#### 4.3.4. Multi-Frequency Verification

## 5. Conclusions

_{2}, the work can be extended to simpler electrothermal designs based on typical TEC difference or phase-change materials. Therefore, the present work presents a platform that can be adapted for the design of a broad scope of MEMS mirrors. Future work will focus on generalizing the presented model to other actuation modes and observing the effect of the other legs in the actuated leg and studies at higher frequencies,introducing a closed-loop control design, based on the present model, to accurately manipulate the different tilting angles of the mirror. Furthermore, future work will focus on incorporating the control system on each actuator of the VO

_{2}-based MEMS mirror for different purposes such as creating a 2D image or laser tracking. Additionally, a comprehensive model will be studied to incorporate the pseudo-creep behavior of the device, which is relevant for quasi-static positioning applications.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Xie, H.; Pan, Y.; Fedder, G.K. A CMOS-MEMS mirror with curled-hinge comb drives. J. Microelectromech. Syst.
**2003**, 12, 450–457. [Google Scholar] - Koh, K.H.; Kobayashi, T.; Lee, C. A 2-D MEMS scanning mirror based on dynamic mixed mode excitation of a piezoelectric PZT thin film S-shaped actuator. Opt. Express
**2011**, 19, 13812–13824. [Google Scholar] [CrossRef] [PubMed] - Hung, A.C.L.; Lai, H.Y.H.; Lin, T.W.; Fu, S.G.; Lu, M.S.C. An electrostatically driven 2D micro-scanning mirror with capacitive sensing for projection display. Sens. Actuators A Phys.
**2015**, 222, 122–129. [Google Scholar] [CrossRef] - Naono, T.; Fujii, T.; Esashi, M.; Tanaka, S. Non-resonant 2-D piezoelectric MEMS optical scanner actuated by Nb doped PZT thin film. Sens. Actuators A Phys.
**2015**, 233, 147–157. [Google Scholar] [CrossRef] - Yalcinkaya, A.; Urey, H.; Brown, D.; Montague, T.; Sprague, R. Two-axis electromagnetic microscanner for high resolution displays. J. Microelectromech. Syst.
**2006**, 15, 786–794. [Google Scholar] [CrossRef] - Cho, A.R.; Han, A.; Ju, S.; Jeong, H.; Park, J.H.; Kim, I.; Bu, J.U.; Ji, C.H. Electromagnetic biaxial microscanner with mechanical amplification at resonance. Opt. Express
**2015**, 23, 16792–16802. [Google Scholar] [CrossRef] [PubMed] - Wu, L.; Dooley, S.; Watson, E.; McManamon, P.F.; Xie, H. A Tip-Tilt-Piston Micromirror Array for Optical Phased Array Applications. J. Microelectromech. Syst.
**2010**, 19, 1450–1461. [Google Scholar] [CrossRef] - Jain, A.; Qu, H.; Todd, S.; Xie, H. A thermal bimorph micromirror with large bi-directional and vertical actuation. Sens. Actuators A Phys.
**2005**, 122, 9–15. [Google Scholar] [CrossRef] - Samuelson, S.R.; Xie, H. A Large Piston Displacement MEMS Mirror With Electrothermal Ladder Actuator Arrays for Ultra-Low Tilt Applications. J. Microelectromech. Syst.
**2014**, 23, 39–49. [Google Scholar] [CrossRef] - Torres, D.; Wang, T.; Zhang, J.; Zhang, X.; Dooley, S.; Tan, X.; Xie, H.; Sepúlveda, N. VO
_{2}-Based MEMS Mirrors. J. Microelectromech. Syst.**2016**, 25, 780–787. [Google Scholar] [CrossRef] - Sepulveda, N.; Rua, A.; Cabrera, R.; Fernández, F. Young’s modulus of VO
_{2}thin films as a function of temperature including insulator-to-metal transition regime. Appl. Phys. Lett.**2008**, 92, 1913. [Google Scholar] [CrossRef] - Zylbersztejn, A.; Mott, N. Metal-insulator transition in vanadium dioxide. Phys. Rev. B
**1975**, 11, 4383. [Google Scholar] [CrossRef] - Barker, A., Jr.; Verleur, H.; Guggenheim, H. Infrared optical properties of vanadium dioxide above and below the transition temperature. Phys. Rev. Lett.
**1966**, 17, 1286. [Google Scholar] [CrossRef] - Mlyuka, N.R.; Niklasson, G.A.; Granqvist, C.G. Mg doping of thermochromic VO
_{2}films enhances the optical transmittance and decreases the metal-insulator transition temperature. Appl. Phys. Lett.**2009**, 95, 171909. [Google Scholar] [CrossRef] - Cao, J.; Gu, Y.; Fan, W.; Chen, L.; Ogletree, D.; Chen, K.; Tamura, N.; Kunz, M.; Barrett, C.; Seidel, J.; et al. Extended mapping and exploration of the vanadium dioxide stress-temperature phase diagram. Nano Lett.
**2010**, 10, 2667–2673. [Google Scholar] [CrossRef] [PubMed] - Breckenfeld, E.; Kim, H.; Burgess, K.; Charipar, N.; Cheng, S.F.; Stroud, R.; Piqué, A. Strain Effects in Epitaxial VO
_{2}Thin Films on Columnar Buffer-Layer TiO_{2}/Al_{2}O_{3}Virtual Substrates. ACS Appl. Mater. Interfaces**2017**, 9, 1577–1584. [Google Scholar] [CrossRef] [PubMed] - Merced, E.; Tan, X.; Sepúlveda, N. Strain energy density of VO
_{2}-based microactuators. Sens. Actuators A Phys.**2013**, 196, 30–37. [Google Scholar] [CrossRef] - Rúa, A.; Fernández, F.l.E.; Sepúlveda, N. Bending in VO
_{2}-coated microcantilevers suitable for thermally activated actuators. J. Appl. Phys.**2010**, 107, 074506. [Google Scholar] [CrossRef] - Cabrera, R.; Merced, E.; Sepúlveda, N. Performance of Electro-Thermally Driven VO
_{2}-Based MEMS Actuators. J. Microelectromech. Syst.**2014**, 23, 243–251. [Google Scholar] [CrossRef] - Cabrera, R.; Merced, E.; Sepúlveda, N. A micro-electro-mechanical memory based on the structural phase transition of VO
_{2}. Phys. Status Solidi**2013**, 210, 1704–1711. [Google Scholar] [CrossRef] - Merced, E.; Cabrera, R.; Dávila, N.; Fernández, F.E.; Sepúlveda, N. A micro-mechanical resonator with programmable frequency capability. Smart Mater. Struct.
**2012**, 21, 035007. [Google Scholar] [CrossRef] - Bai, Y.; Yeow, J.T.W.; Wilson, B.C. A Characteristic Study of Micromirror with Sidewall Electrodes. Int. J. Optomech.
**2007**, 1, 231–258. [Google Scholar] [CrossRef] - Isikman, S.O.; Urey, H. Dynamic Modeling of Soft Magnetic Film Actuated Scanners. IEEE Trans. Magn.
**2009**, 45, 2912–2919. [Google Scholar] [CrossRef] - Han, F.; Wang, W.; Zhang, X.; Xie, H. Modeling and Control of a Large-Stroke Electrothermal MEMS Mirror for Fourier Transform Microspectrometers. J. Microelectromech. Syst.
**2016**, 25, 750–760. [Google Scholar] [CrossRef] - Zhang, J.; Merced, E.; Sepúlveda, N.; Tan, X. Optimal compression of generalized Prandtl–Ishlinskii hysteresis models. Automatica
**2015**, 57, 170–179. [Google Scholar] [CrossRef] - Zhang, J.; Merced, E.; Sepúlveda, N.; Tan, X. Modeling and Inverse Compensation of Nonmonotonic Hysteresis in VO
_{2}-Coated Microactuators. IEEE/ASME Trans. Mech.**2014**, 19, 579–588. [Google Scholar] [CrossRef] - Zhang, J.; Torres, D.; Ebel, J.L.; Sepúlveda, N.; Tan, X. A Composite Hysteresis Model in Self-Sensing Feedback Control of Fully Integrated VO
_{2}Microactuators. IEEE/ASME Trans. Mech.**2016**, 21, 2405–2417. [Google Scholar] [CrossRef] - Merced, E.; Torres, D.; Tan, X.; Sepúlveda, N. An Electrothermally Actuated VO
_{2}-Based MEMS Using Self-Sensing Feedback Control. J. Microelectromech. Syst.**2015**, 24, 100–107. [Google Scholar] [CrossRef] - Xie, H. Vertical Displacement Device. US Patent 6,940,630, 6 September 2005. [Google Scholar]
- Wu, L.; Xie, H. A large vertical displacement electrothermal bimorph microactuator with very small lateral shift. Sens. Actuators A Phys.
**2008**, 145, 371–379. [Google Scholar] [CrossRef] - Zhang, L.; Tsaur, J.; Maeda, R. Residual Stress Study of SiO
_{2}/Pt/Pb(Zr,Ti)O_{3}/Pt Multilayer Structure for Micro Electro Mechanical System Applications. Jpn. J. Appl. Phys.**2003**, 42, 1386. [Google Scholar] [CrossRef] - Matsui, Y.; Hiratani, M.; Kumagai, Y.; Miura, H.; Fujisaki, Y. Thermal Stability of Pt Bottom Electrodes for Ferroelectric Capacitors. Jpn. J. Appl. Phys.
**1998**, 37, L465. [Google Scholar] [CrossRef] - Kinbara, A.; Haraki, H. Internal Stress of Evaporated Thin Gold Films. Jpn. J. Appl. Phys.
**1965**, 4, 243. [Google Scholar] [CrossRef] - Kebabi, B.; Malek, C.; Ladan, F. Stress and microstructure relationships in gold thin films. Vacuum
**1990**, 41, 1353–1355. [Google Scholar] [CrossRef] - Leo, D.J. Engineering Analysis of Smart Material Systems; John Wiley & Sons: Hoboken, NJ, USA, 2007. [Google Scholar]
- Mayergoyz, I. Mathematical Models of Hysteresis and Their Applications; Springer: New York, NY, USA, 1991. [Google Scholar]
- Tan, X.; Baras, J. Modeling and control of hysteresis in magnetostrictive actuators. Automatica
**2004**, 40, 1469–1480. [Google Scholar] [CrossRef] - Nye, J. Physical Properties of Crystals: Their Representation by Tensors and Matrices; Oxford University Press: Oxford, UK, 1985. [Google Scholar]
- Wortman, J.J.; Evans, R.A. Young’s Modulus, Shear Modulus, and Poisson’s Ratio in Silicon and Germanium. J. Appl. Phys.
**1965**, 36, 153–156. [Google Scholar] [CrossRef] - Gall, K.; Dunn, M.L.; Zhang, Y.; Corff, B.A. Thermal cycling response of layered gold/polysilicon MEMS structures. Mech. Mater.
**2004**, 36, 45–55. [Google Scholar] [CrossRef] - Gall, K.; West, N.; Spark, K.; Dunn, M.L.; Finch, D.S. Creep of thin film Au on bimaterial Au/Si microcantilevers. Acta Mater.
**2004**, 52, 2133–2146. [Google Scholar] [CrossRef] - Tsai, K.Y.; Chin, T.S.; Shieh, H.P.D.; Ma, C.H. Effect of as-deposited residual stress on transition temperatures of VO
_{2}thin films. J. Mater. Res.**2004**, 19, 2306–2314. [Google Scholar] [CrossRef] - Case, F.C. Modifications in the phase transition properties of predeposited VO
_{2}films. J. Vac. Sci. Technol. A Vac. Surf. Films**1984**, 2, 1509–1512. [Google Scholar] [CrossRef]

**Figure 1.**SEM image of a VO

_{2}-based micro-electro-mechanical system (MEMS) mirror (top view), where the different parts of an actuator leg are labeled: frame, bimorph and the connector between the mirror platform and the actuator leg.

**Figure 3.**Schematic of the mirror platform showing the force ($\overrightarrow{F}$) applied by the actuated leg, and the axes of rotation for pitch and roll angles, where ${a}_{r}$ ($115\mathrm{m}\mathrm{m}$) and ${a}_{p}$ ($300\mathrm{m}\mathrm{m}$) are the distance between the force and each axis. In this case, the current i is applied to the bottom-left leg.

**Figure 4.**Finite element method (FEM) model schematic of the VO

_{2}-based MEMS mirror used to find the rotational spring constant by applying a sequence of increasing force as a point load. The force is applied at different locations (1, 2, 3 & 4) for each simulation.

**Figure 5.**Time response measurements from actuating one leg for both variables: pitch (

**left**) and roll (

**right**) angles.

**Figure 6.**Frequency response for the actuation of one leg. A fitted curve is used to find the damping ratio ($\zeta $) and the gain ${A}_{T}$. Both pitch (

**left**) and roll (

**right**) angles have the same resonant frequency with the value of 739 Hz.

**Figure 7.**Identification plots of the pitch (

**left**) and roll (

**right**) angles, used to find the coefficients of the hysteresis model.

**Figure 8.**Parameters values (weights) used in the Preisach model for the (

**a**) pitch and (

**b**) roll; (

**c**) the modeling performance; and (

**d**) modeling error for the hysteresis between pitch angle and the current input; (

**e**) the modeling performance; and (

**f**) modeling error for the hysteresis between roll angle and the current input.

**Figure 9.**(

**a**) A current step input for model verification; the measured and estimated steady-state (

**b**) pitch angle; and (

**c**) roll angle.

**Figure 10.**The pitch and roll angle verification performances for current inputs with different frequencies.

**Figure 11.**(

**a**) A multi-frequency current input for model verification; the measured and estimated (

**b**) pitch angle; and (

**c**) roll angle.

**Table 1.**Parameters of the materials used in finite element method (FEM) simulations, where the Si, SiO

_{2}, and Au are obtained from the COMSOL library, while the VO

_{2}properties are reported in [19].

Properties | Materials | |||
---|---|---|---|---|

Si | SiO_{2} | Au | VO_{2} | |

Density [Kg/m${}^{3}$] | 2320 | 2200 | 19300 | 4670 |

Young’s Modulus [GPa] | 187 [38,39] | 70 | 70 | 140 |

Poisson Ratio | 0.22 | 0.17 | 0.44 | 0.33 |

Point Load Location | Rotational Spring Constant | |
---|---|---|

Pitch ($\times {10}^{-9}\frac{\mathbf{N}\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}\mathbf{m}}{\mathbf{deg}}$) | Roll ($\times {10}^{-9}\frac{\mathbf{N}\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}\mathbf{m}}{\mathbf{deg}}$) | |

1 | 2.29 | 1.219 |

2 | 2.29 | 1.217 |

3 | 2.29 | 1.217 |

4 | 2.29 | 1.217 |

Constant | Name and Units | Pitch (${\mathit{\theta}}_{\mathit{p}}$) | Roll (${\mathit{\theta}}_{\mathit{r}}$) |
---|---|---|---|

${A}_{T}$ | Gain [deg/A${}^{2}$] | 79,743 | 33,871 |

${\tau}_{th}$ | Time response [s] | 0.0014 | 0.001479 |

${\omega}_{n}$ | Resonant Frequency [rad/s] | 4643 | 4643 |

$\zeta $ | Damping ratio | 0.00363 | 0.00447 |

J | Moment of Inertia [Kg$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$m${}^{2}$] | $6.10\times {10}^{-15}$ | $3.23\times {10}^{-15}$ |

G | Rotational Damping coefficient [N$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$m$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$s/rad] | $205.6\times {10}^{-15}$ | $134\times {10}^{-15}$ |

k | Rotational Spring coefficient [N$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$m/rad] | $132\times {10}^{-9}$ | $69.7\times {10}^{-9}$ |

a | Position of the force with respect to the axis [ $\mathrm{m}\mathrm{m}$] | 600 | 115 |

${c}_{0}$ | Constant bias of Preisach model [ $\mathrm{deg}/\mathrm{m}\mathrm{m}$] | 0.99 | 0.38 |

${k}_{0}$ | Thermal expansion-induced force term [N/°C] | $1.4\times {10}^{4}$ | $3.8\times {10}^{3}$ |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Torres, D.; Zhang, J.; Dooley, S.; Tan, X.; Sepúlveda, N. Modeling of MEMS Mirrors Actuated by Phase-Change Mechanism. *Micromachines* **2017**, *8*, 138.
https://doi.org/10.3390/mi8050138

**AMA Style**

Torres D, Zhang J, Dooley S, Tan X, Sepúlveda N. Modeling of MEMS Mirrors Actuated by Phase-Change Mechanism. *Micromachines*. 2017; 8(5):138.
https://doi.org/10.3390/mi8050138

**Chicago/Turabian Style**

Torres, David, Jun Zhang, Sarah Dooley, Xiaobo Tan, and Nelson Sepúlveda. 2017. "Modeling of MEMS Mirrors Actuated by Phase-Change Mechanism" *Micromachines* 8, no. 5: 138.
https://doi.org/10.3390/mi8050138