# Design and Modeling of Polysilicon Electrothermal Actuators for a MEMS Mirror with Low Power Consumption

^{1}

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## Abstract

**:**

^{®}) process, obtaining a small footprint size (1028 μm × 1028 µm) for actuators of 550 µm length. The actuators have out-of-plane displacements caused by low dc voltages and without use material layers with distinct thermal expansion coefficients. The temperature behavior along the actuators is calculated through analytical models that include terms of heat energy generation, heat conduction and heat energy loss. The force method is used to predict the maximum out-of-plane displacements in the actuator tip as function of supplied voltage. Both analytical models, under steady-state conditions, employ the polysilicon resistivity as function of the temperature. The electrothermal-and structural behavior of the actuators is studied considering different beams dimensions (length and width) and dc bias voltages from 0.5 to 2.5 V. For 2.5 V, the actuator of 550 µm length reaches a maximum temperature, displacement and electrical power of 115 °C, 10.3 µm and 6.3 mW, respectively. The designed actuation mechanism can be useful for MEMS mirrors of different sizes with potential application in endoscopic OCT systems that require low power consumption.

## 1. Introduction

^{®}) process from Sandia National Laboratories. This electrothermal actuation mechanism has a simple structural configuration composed by an array of four polysilicon actuators, which can achieve out-of-plane displacements with low dc voltages. These actuators do not require materials with different thermal expansion coefficients due to that employ polysilicon layers with distinct wide, which are separated by 2 μm gap. This device has a small footprint size (1028 μm × 1028 μm), compact structure and simple performance with reduced temperatures. The proposed design includes the modeling of temperature behavior and maximum displacements of the actuators under steady-state conditions. Our actuation mechanism can be used for the rotation of MEMS mirrors of different sizes. The rotation orientation of the mirror can be adjusted through the selective biasing of the four actuators. Thus, the proposed design could be considered for potential applications in endoscopic OCT systems.

## 2. Design and Modeling

#### 2.1. Structural Configuration

_{i}) and width (ω

_{h}and ω

_{c}) of the upper (hot) and bottom (cold) beams are studied. The first structural layer is formed by a polysilicon beam (ω

_{c}) and the second layer is composed by three polysilicon beams of width ω

_{h}each one, in which ω

_{h}<< ω

_{c}. Figure 3a,b depicts views of the hot and cold beams in two electrothermal actuators with deflections in opposite directions. In addition, the mirror and springs are designed using the poly4 layer of SUMMiT V process. In this fabrication process, on the mirror surface can be deposited an aluminum layer (96 μm × 96 μm × 0.7 μm). The springs have a connection with low stiffness between the actuators and mirror, which lets higher mirror tilting.

_{h}= t

_{c}= 2.25 μm) and variable width (i.e., w

_{h}of 2 μm to 5 μm and w

_{c}of 20 μm to 30 μm).

#### 2.2. Electrical Model of Electrothermal Actuators

_{1}, R

_{2}and R

_{3}are the electrical resistance values obtained for each hot beam (ω

_{h}), cold beam (ω

_{c}) and connection between both beams, respectively. These resistances are calculated including the dimensions of the beams and the resistivity of the polysilicon layers. For instance, Table 1 shows the values of the electrical resistances for an electrothermal actuator with the following dimensions: L

_{h}= L

_{c}= 450 μm, ω

_{h}= 5 μm, ω

_{c}= 30 μm and t

_{h}= t

_{c}= 2.25 μm.

#### 2.3. Analytical Modeling of the Electrothermal and Structural Behavior

_{h}) is equal to the length of the bottom beam (L

_{c}): L

_{h}= L

_{c}= L. Figure 5 shows a differential element for the thermal analysis of the actuator.

_{p}is the thermal conductivity and ρ is the resistivity of the polysilicon, T is the operation temperature, T

_{0}is the substrate temperature, Q is the shape factor that includes the impact of the element shape on heat conduction to the substrate and R

_{t}is the thermal resistance generated by the substrate and actuator that are considered wide enough [31]:

_{a}is the distance between both the bottom beam of the actuator and Si

_{3}N

_{4}surface, t

_{n}is the thickness of the Si

_{3}N

_{4}film, t

_{s}is the thickness of the SiO

_{2}film and k

_{a}, k

_{n}and k

_{s}are the thermal conductivity of air, Si

_{3}N

_{4}and SiO

_{2}films, respectively.

_{h}= t

_{c}and w = w

_{c}.

_{h}(s) and T

_{c}(s) are the temperature distribution along the upper (hot) and bottom (cold) beams, respectively, and J

_{h}and J

_{c}are the current density through the upper and bottom beams, respectively.

_{i}, we assume a temperature on the anchor pads equal to the substrate temperature (i.e., T

_{h}(0) = T

_{0}and T

_{c}(2L + g) = T

_{0}), a continuity of both temperature (i.e., ${T}_{h}\left(L\right)={T}_{c}\left(L\right)$) and rate of heat conduction (i.e., 3w

_{h}dT

_{h}(L)/ds = w

_{c}dT

_{c}(L)/ds) across the join point of the upper and bottom beams. By assuming these boundary conditions, the following matrix equation is determined as:

_{i}of Equation (12) are determined using operations on matrices. Next, these coefficients are employed into Equations (8) and (9) to calculate the temperature increase along the upper and bottom beams due to bias voltages. These coefficients are calculated as:

_{h}) and bottom (ΔL

_{c}) beams can be determined as:

_{1}, X

_{2}and X

_{3}) is studied using the force method [35]. These unknowns are internal forces (horizontal force X

_{1}, vertical force X

_{2}and bending moment X

_{3}). The force method will be used to find the redundant unknowns followed by the virtual work method to obtain the deflection at the tip of the frame.

_{1}, X

_{2}and X

_{3}) are calculated through the canonical equations of the force method, which satisfy the compatibility conditions of the deformations [36]. For this case, the canonical equations are given by the following matrix form:

_{i}caused by action of unit unknown X

_{j}. ${\delta}_{ij}$ can be determined by the diagram product of the bending moments related with the unit unknowns X

_{i}and X

_{j}. These coefficients are obtained as:

_{h}and I

_{c}are the moment of inertia of the hot and cold beams, respectively.

_{F}is the bending moment due to the virtual unit force and M the bending moment related with the thermal expansion. The physical and mechanical properties of the polysilicon used in the above analysis are listed in Table 2.

_{1}, X

_{2}and X

_{3}are the follows:

_{max}) of the electrothermal actuator:

## 3. Results and Discussions

_{h}of 2 μm to 5 μm and w

_{c}of 20 μm to 30 μm) for the upper and bottom beams and a constant thickness (i.e., t

_{h}= t

_{c}= 2.25 μm). We compared the results of our models with respect to analytical models of temperature and displacements of electrothermal actuators reported by reference [31]. For this, we use Equations (7), (8) and (23) of reference [31] and assume negligible the flexure beam length (i.e., L

_{f}= 0). However, these models are applied for electrothermal actuators of variable cross-section area with in-plane deflections. In order to employ these models to our actuators with out-of-plane deflections, we considered that the variables of width and thickness of their hot and cold beams are equals to the thickness and width of our hot and cold beams. Figure 7a,b shows the results of the temperature along of the surface of the upper (hot) and bottom (cold) beams, which are generated by a bias voltage of 2.5 V. This distribution considers different lengths (350 and 550 μm) and two values of width for each upper beam (2 and 5 μm). For all the cases, the maximum temperature is achieved close to the half of the length of the upper beam. The shorter beams present higher temperatures than the larger beams due to their less electrical resistance, which produce higher currents for a bias voltage. For the upper beams of 5 μm width, the temperature decays more slowly along of the electrothermal actuator, as shown in Figure 7b. In the actuator tip, we observed a significant variation in the behavior of the temperature distribution along of the hot and cold beams. The results of our analytical models have a similar behavior respect to those of reference [31]; although, our results register the highest temperature values in all the cases. Next, we calculate the temperature distribution regarding two actuators of different lengths (450 and 550 μm), which are supplied by different dc bias voltages, as shown in Figure 8a,b. The maximum voltage of 2.5 V generates the higher temperature magnitudes (147.3 °C and 114.9 °C) for both actuators, considering our models. For the actuator of 450 μm length, the bias voltages of 1.0 V, 1.5 V and 2.0 V increase the temperature up 38.2 °C, 62.1 °C and 97.6 °C, respectively. For the same voltages, the actuator of 550 μm length has an increment of temperature of 34.0 °C, 52.0 °C and 78.5 °C, respectively. Also, the temperature distribution along the actuator of 450 μm length was determined varying the width of the upper and bottom beams, as shown in Figure 9a,b. For upper beams of 2 μm width and bias voltage of 2.5 V, the temperature has a low increment of 16.6 °C when the width of the bottom beam increases from 20 to 30 μm. Instead, the temperature distribution decays more slowly for upper beams of 5 μm width, keeping 30 μm width for the bottom beam. For these cases, the results of our models have good agreement respect to those of reference [31].

_{h}= 2 μm and w

_{c}= 30 μm, respectively. These displacements have direction down due to the higher temperature of the upper beams. However, if the position of the beams is inverted then the motion of the actuator will be upward. If the length of the actuator is 450 μm and the bias voltage is 2.5 V then the maximum displacements are 8.9 μm and 5.6 μm, respectively. The response of our models has good agreement respect to results of reference [31]. Although the displacements obtained with our analytical models have higher values than those of the reference [31]. Figure 11a,b shows the maximum out-of-plane displacements of the actuator (450 μm length and 2.5 V voltage) considering different dimensions in the width of its upper and bottom beams. For these cases, the larger displacement (8.9 μm) is obtained with 2.5 V voltage for beams with w

_{h}= 2 μm and w

_{c}= 30 μm, respectively. In addition, the displacement of the actuator tip decreases when the width of the upper beams increases. Moreover, if the width of the bottom beam increases then the actuator tip will have larger displacements. The electrical power of each actuator is determined using the equivalent electrical circuit of Figure 5. For an actuator with w

_{h}= 2 μm, w

_{c}= 30 μm and three different lengths L

_{h}: 350 μm, 450 μm and 550, we obtain the following electrical power: 9.9 mW, 7.7 mW and 6.3 mW. Finally, the displacements of the actuator tip can be increased with bias voltages higher than 2.5 V, which also will increment the electrical power. For instance, if the actuator of L

_{h}= 550 μm and w

_{h}= 2 μm is biased with 5 V then its maximum displacement, temperature and power are increased up 59.2 μm, 570.3 °C and 25.2 mW, respectively. Furthermore, the mirror surface area can be scalable to achieve larger values than 10000 μm

^{2}. On the other hand, the surface of the silicon substrate below of the actuators array and mirror must be etched using DRIE process to allow the free motion of the actuators and mirror under different bias voltages. Nevertheless, the maximum displacement of the actuators must generate stress less than the rupture stress of the polysilicon.

^{®}software (version 15.0, ANSYS, Berkeley, CA, USA) to predict the out-of-plane displacements of the proposed actuation mechanism. For this, the pads were negligible and the initial end of each actuator was considered as fixed support. For these supports were applied a bias voltage of 2.5 V and initial temperature of 20 °C. The FEM models regard polysilicon actuators with the following dimensions: L

_{h}= L

_{c}= 550 μm, ω

_{h}= 2 μm, ω

_{c}= 30 μm, t

_{i}= 2.25 μm and g = 2 μm. Our FEM models include elements solid226 type with a hexahedral mesh. First, we use a FEM model of a single electrothermal actuator under 2.5 V bias voltage. Figure 12 depicts the out-of-plane displacements of this actuator, achieving a maximum downward deflection of 10.3 μm that well agree with the results (10.3 μm and 8.8 μm) of both our analytical model and that of the reference [31], as shown in Figure 10a. Next, we used a FEM model composed by four polysilicon electrothermal actuators, four springs (508 μm length, 5 μm width and 2.25 μm thickness) and a mirror. Each one of these actuators has the same dimension respect to the previous actuator. The initial ends of the four actuators have boundary conditions of clamped support and temperature of 20 °C. For this FEM model, we studied four different cases modifying the bias voltage values of the four actuators. For the first case, one actuator was only supplied with a voltage of 2.5 V, keeping the other three actuators without bias voltage (see Figure 13). Thus, the actuator and mirror have maximum out-of-plane deflections of 7.4 μm and 4.8 μm, respectively. For this case, the displacement of the actuator decreases (3.9 μm) respect the response of a single actuator without connection with springs and mirror. This displacement reduction is due to an increment of the model stiffness when the four actuators are joined to the mirror. In the second case two actuators are biased with 2.5 V, obtaining out-of-plane displacements with opposite directions (downward and upward) that allow the mirror rotation with respect to two of its vertices, as shown in Figure 14. The absolute value of the maximum displacement of the two biased actuators is 6.7 μm, which is 3.5 μm less than that obtained with a single actuator. Two mirror vertices reach maximum displacements of 3.7 μm and −3.7 μm, respectively. For the third case, a 2.5 V bias voltage is applied for three actuators, achieving maximum displacements of 9.2 μm, 7.7 μm and −4.5 μm (see Figure 15). Indeed, two mirror vertices have displacements of 6.2 μm and −1.4 μm that enable the mirror tilting. In the last case all the actuators are biased with 2.5 V, obtaining the downward and upward deflection of two actuator pairs as well as the mirror rotation along the x-axis (see Figure 16). The larger displacements of the actuators and mirror are 7.1 μm, −7.1 μm, 3.9 and −3.9 μm, respectively. In order to reach larger deflection and tilting of the actuators and mirror, the bias voltage can be increased. Moreover, the rotation orientation of the mirror can be regulated through the selective biasing of the four actuators. Also, the proposed actuation mechanism can be employed for MEMS mirrors of larger surface area and their rotation angles can be controlled using different bias voltages.

^{2}, which can be suitable for potential applications in endoscopic OCT systems.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Design of an electrothermal actuation mechanism for the rotation of a microelectromechanical systems (MEMS) mirror.

**Figure 3.**View of out-of-plane displacements, y

_{i}, with directions (

**a**) downward and (

**b**) upward of two electrothermal actuators with inverted structural layers due to Joule effect; (

**c**) geometrical parameters of the hot and cold beams of an electrothermal actuator.

**Figure 5.**(

**a**) Schematic of the one-dimensional model for an electrothermal actuator; (

**b**) its differential element; and (

**c**) cross-section of the different layers for the thermal analysis.

**Figure 6.**Rigid structure simplified for the electrothermal actuator regarding three redundant forces and moments (X

_{1}, X

_{2}and X

_{3}).

**Figure 7.**Distribution of the temperature along of the upper (hot) and bottom (cold) beams of an electrothermal actuator, which considers different lengths (350 μm to 550 μm) and two width values for the upper beams: (

**a**) 2 μm; and (

**b**) 5 μm.

**Figure 8.**Distribution of the temperature along of the upper (hot) and bottom (cold) beams of two electrothermal actuators with lengths of (

**a**) 550 μm and (

**b**) 450 μm. This temperature is due to different bias voltages, whose values change from 0.5 to 2.5 V.

**Figure 9.**Distribution of the temperature along of the upper (hot) and bottom (cold) beams of an electrothermal actuator, modifying the width of the (

**a**) upper and (

**b**) bottom beams. For both cases, the length of the actuator is 450 μm and bias voltage is 2.5 V, respectively.

**Figure 10.**Maximum out-of-plane displacements of the electrothermal actuator tip as a function of bias voltage, regarding different lengths and two width values for the upper beams: (

**a**) 2 μm and (

**b**) 5 μm.

**Figure 11.**Maximum out-of-plane displacements of the electrothermal actuator tip as a function of bias voltage, varying the width of the (

**a**) upper and (

**b**) bottom beams. For both cases, the length of the actuator is 450 μm and bias voltage is 2.5 V, respectively.

**Figure 12.**Out-of-plane displacements of one polysilicon electrothermal actuator (L

_{h}= L

_{c}= 550 μm) caused by a 2.5 V bias voltage.

**Figure 13.**Out-of-plane deflections of the MEMS mirror when one polysilicon electrothermal actuator (L

_{h}= L

_{c}= 550 μm) is biased with 2.5 V.

**Figure 14.**Out-of-plane displacements of the MEMS mirror when two polysilicon electrothermal actuators (L

_{h}= L

_{c}= 550 μm) are biased with 2.5 V.

**Figure 15.**Out-of-plane displacements of the MEMS mirror when three polysilicon electrothermal actuators (L

_{h}= L

_{c}= 550 μm) are biased with 2.5 V.

**Figure 16.**Out-of-plane displacements of the MEMS mirror when four polysilicon electrothermal actuators (L

_{h}= L

_{c}= 550 μm) are biased with 2.5 V.

**Table 1.**Resistance values of the equivalent electrical circuit of an electrothermal actuator considering the following dimensions: ω

_{h}= 2 μm, ω

_{c}= 30 μm and t

_{h}= t

_{c}= 2.25 μm.

Parameter | Electrical Resistance (Ω) | ||
---|---|---|---|

L_{h} = 350 μm | L_{h} = 450 μm | L_{h} = 550 μm | |

R_{1} | 1576.6 | 2027 | 2077.4 |

R_{2} | 105.1 | 135.1 | 165.2 |

R_{3} | 1.7 | 1.7 | 1.7 |

Property | Value |
---|---|

Young’s Modulus, $E$ | 169 GPa |

Thermal expansion, $\alpha $ | 2.5 × 10^{−6} K^{−1} |

Thermal conductivity, ${k}_{p}$ | 125 W·m^{−1}·K^{−1} |

Substrate Temperature, ${T}_{0}$ | 300 K |

Linear temperature coefficient, ξ | 1.25 × 10^{−3} K^{−1} |

Resistivity at ${T}_{0}$, ${\rho}_{0}$ | 20.27 × 10^{−6} Ω·m |

Density | 2330 kg·m^{−3} |

Poisson ratio | 0.23 |

Authors | Mirror Size | Device Footprint (μm × μm) | Maximum Displacement (μm) | Bias Voltage (V) |
---|---|---|---|---|

Zhang et al. [18] | 900 μm × 900 μm | 2500 × 2500 | 312 | 3 |

Kawai et al. [37] | 3000 μm diameter | 5000 × 5000 | *- | 20 |

Zhang et al. [38] | 1000 μm × 1000 μm | 1500 × 1500 | 70 | 2 |

Li et al. [39] | 1000 μm diameter | 2000 × 2000 | 227 | 0.8 |

Espinosa et al. [40] | 1000 μm × 1000 μm | 1500 × 1500 | 174 | 3.5 |

Koh et al. [41] | 1500 μm × 1000 μm | 6000 × 6000 | *- | 5 |

Our work | 100 μm × 100 μm | 1028 × 1028 | 59.2 | 5 |

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Lara-Castro, M.; Herrera-Amaya, A.; Escarola-Rosas, M.A.; Vázquez-Toledo, M.; López-Huerta, F.; Aguilera-Cortés, L.A.; Herrera-May, A.L. Design and Modeling of Polysilicon Electrothermal Actuators for a MEMS Mirror with Low Power Consumption. *Micromachines* **2017**, *8*, 203.
https://doi.org/10.3390/mi8070203

**AMA Style**

Lara-Castro M, Herrera-Amaya A, Escarola-Rosas MA, Vázquez-Toledo M, López-Huerta F, Aguilera-Cortés LA, Herrera-May AL. Design and Modeling of Polysilicon Electrothermal Actuators for a MEMS Mirror with Low Power Consumption. *Micromachines*. 2017; 8(7):203.
https://doi.org/10.3390/mi8070203

**Chicago/Turabian Style**

Lara-Castro, Miguel, Adrian Herrera-Amaya, Marco A. Escarola-Rosas, Moisés Vázquez-Toledo, Francisco López-Huerta, Luz A. Aguilera-Cortés, and Agustín L. Herrera-May. 2017. "Design and Modeling of Polysilicon Electrothermal Actuators for a MEMS Mirror with Low Power Consumption" *Micromachines* 8, no. 7: 203.
https://doi.org/10.3390/mi8070203