ELECTROTHERMAL bimorph actuators are common in

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1 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO. 1, FEBRUARY An Electrothermomechanical Lumped Element Model of an Electrothermal Bimorph Actuator Shane T. Todd, Student Member, IEEE, and Huikai Xie, Senior Member, IEEE Abstract This paper reports a simple electrothermomechanical lumped element model (ETM-LEM) that describes the behavior of an electrothermal bimorph actuator. The ETM-LEM is developed by integrating an electrothermal LEM of a heater with a thermomechanical LEM of a bimorph actuator. This new LEM uses only one power source in both the electrical and thermal domains. The LEM provides a simple and accurate way of relating the output mechanical response of a bimorph actuator to the electrical inputs. The model shows that the tip angular rotation of the bimorph actuator is linearly proportional to its average temperature change. The LEM predicts a linear relationship between both the average temperature change and bimorph tip angular rotation versus voltage when operated above a certain voltage. The LEM is used to predict the rotation angle of a fabricated electrothermal bimorph micromirror in response to the electrical inputs and produces results that agree with finite element model simulations and experimental data within 15% for all measured parameters. [ ] Index Terms Bimorph, electrothermal modeling, electrothermomechanical modeling, lumped element modeling, micromirror, thermal actuator, thermomechanical modeling. I. INTRODUCTION ELECTROTHERMAL bimorph actuators are common in MEMS devices, and these actuators have been used in applications, including micromirrors [1] [5], RF switches [6], nanoprobes [7], IR detectors [8], and read write cantilevers for data storage [9]. Compared with other types of actuators such as electrostatic actuators, electrothermal bimorphs can achieve large mechanical displacements because of the large strain difference that is created when using materials with different coefficients of thermal expansion (CTEs). Electrothermal bimorphs can also be fabricated using standard IC processing methods and materials and can easily be integrated into CMOS compatible devices. To understand the actuation behavior of an electrothermal bimorph actuator, one must consider the actuator s electrothermal response to an electrical input and thermomechanical response to a rise in temperature. Thermomechanical models Manuscript received May 23, 2006; revised April 26, This work was supported by the National Science Foundation under award BES Subject Editor N. Aluru. S. T. Todd was with the Department of Electrical Engineering, University of Florida, Gainesville, FL USA. He is now with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA USA ( stodd@ece.ucsb.edu). H. Xie is with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL USA ( hkx@ufl.edu). Digital Object Identifier /JMEMS of bimorph actuators are well established [10] [12], and they have been used for MEMS bimorph actuators [1] [3]. Meanwhile, electrothermal models have been developed for various MEMS heaters, including microbridge heaters [13] [15], and free-end heaters [2], [7], [16] [19]. Furthermore, circuitequivalent lumped element models (LEMs) have been developed for structures with resistive heating elements [14], [20], [21]. For example, a simple electrothermal LEM (ET-LEM) with a single thermal power source was developed for a heater, assuming a uniform temperature distribution [20]. However, most heaters exhibit a nonuniform temperature distribution along with a nonzero thermal coefficient of electrical resistivity (TCR), which results in a complex relationship between the electrical inputs and the thermal outputs of a device. In this case, the heater may be divided into an electrical and thermal coupled network containing a finite number of elements. Each element consists of temperature and voltage nodes, a thermal power source, and thermal and electrical resistors. This approach has been demonstrated by Mastrangelo [14] and Manginell et al. [21]. Generally, these models are not solvable analytically and must be simulated in a circuit simulator such as SPICE. In this paper, we introduce an ET-LEM of a heater with a nonzero TCR and a nonuniform temperature distribution using a single equivalent thermal power source. Using this new ET-LEM, a simple electrothermomechanical LEM (ETM- LEM) has been developed to describe both the electrothermal and thermomechanical behaviors of a bimorph actuator with an embedded heater. Compared to previously reported models, the major advantages of this model include the use of only one power source in both the electrical and thermal domains of the LEM, the reduced complexity of the derived equations, and the ability to predict the bimorph mechanical response to electrical inputs (including current, voltage, and power). The modeling strategy employed here is to first develop a thermomechanical LEM (TM-LEM) of the bimorph actuator and an ET-LEM of the heater separately and then integrate them into a single ETM-LEM. The development of the LEMs follows the methods described by Senturia [20]. In the next section, the electrothermal bimorph used in the model will be briefly introduced. In Section III, the TM-LEM is developed using the bimorph thermomechanical equations. In Section IV, the ET-LEM is derived using a previously reported electrothermal transducer model [16]. In Section V the ET-LEM and TM-LEM are combined to form an integrated ETM-LEM that is used to derive equations for the bimorph tip angular rotation in response to power, current, and voltage. Section VI compares the model results to finite element model (FEM) simulations /$ IEEE

2 214 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2008 III. BIMORPH ACTUATOR TM-LEM The thermomechanical equations which describe a bimorph deformation in response to a temperature change are well known, and they were established a long time ago [10], [11]. However, to the best of the authors knowledge, an analytical model for the mechanical response of a bimorph to a temperature distribution has not been accurately established. In this section, it will be shown that the tip angular rotation of a bimorph is linearly proportional to its average temperature change. A TM-LEM will be developed where this relationship will be implemented. Fig. 1. Schematic of a cantilevered bimorph. A. Development of the TM-LEM Consider a bimorph actuator at an ambient temperature of T 0 that is uniform about the beam. An applied moment due to an internal strain in the bimorph can result from residual stress present in the material layers at T 0 or if the temperature changes. In this analysis, the residual stress is assumed to be uniform about the material layers. The curvature of a bimorph beam can be represented in terms of both the applied moment caused by the internal strain and the composite stiffness of the bimorph shown as [11], [22] and experimental measurements of an electrothermal bimorph micromirror. 1 = M 0 + M T ρ A EI (1) II. ELECTROTHERMAL BIMORPH ACTUATOR A thermal bimorph is composed of two material layers with different CTEs, as shown in Fig. 1. Typically, the layers consist of one material with a high CTE, such as a metal like Al, and another material with a low CTE, such as a dielectric like SiO 2. When the bimorph temperature decreases or increases, the high-cte material will contract or expand, respectively, more than the low-cte material, yielding a bending moment to change the bimorph curvature. This curvature change results in an angular rotation and a vertical displacement at the end of the bimorph, which can be used for actuation in many types of devices. In electrothermal bimorph actuators, a resistive heater is used to generate a temperature change. The heater can be externally attached to the bimorph [7] or embedded in the bimorph [4]. Since the embedded heater layer is typically much thinner than the two layers of the bimorph, its mechanical contributions are ignored in the following analyses. Fig. 1 shows a schematic of a cantilevered bimorph actuator with a top layer material of thickness t 1 and width w 1 and a bottom layer material of thickness t 2 and width w 2. All parameters of the top and bottom layers will be denoted by subscripts 1 and 2, respectively. For simplicity, the widths of the material layers are assumed to be equal (i.e., w A = w 1 = w 2 ). The length and the total thickness of the bimorph actuator are, respectively, L A and t A = t 1 + t 2. The radius of curvature of the bimorph ρ A is defined as positive when the bimorph curls in the positive z-direction, as shown in Fig. 1. In the next section, we will develop the TM-LEM of the bimorph actuator. where ρ A is the radius of curvature, M 0 and M T are the moments caused by the residual stress and temperature change, respectively, and EI is the composite stiffness of the bimorph. The composite stiffness of the bimorph can be found using the transformed-section method [11], [12], [22] and is expressed as EI = w ( ) A t 4 1E t 4 2E t 1 t 2 E 1E 2 4t t 1 t 2 +4t t 1 E 1 + t 2E 2 (2) where E is the biaxial elastic modulus of either layer, and all other parameters are defined in Fig. 1. The biaxial elastic modulus is given by E = E 1 ν where E is the elastic modulus of either layer, and ν is the Poisson ratio of either layer. The applied moment caused by the residual stress in the bimorph can be derived by integrating the stress across the thickness of the bimorph [22] and is given by (3) ( σ1 M 0 = m A E 1 σ ) 2 E 2 = m A ε 0 (4) where σ is the residual stress in either layer, m A is a parameter that we call the moment coefficient, and ε 0 is the difference in strain in the material layers caused by the residual stress. In this convention, tensile stress is positive, and compressive stress is

3 TODD AND XIE: ELECTROTHERMOMECHANICAL LUMPED ELEMENT MODEL OF ELECTROTHERMAL BIMORPH ACTUATOR 215 negative. The moment coefficient is defined to simplify notation in the model and is given by m A = w At A t 1 t 2 E 1E 2 2(t 1 E 1 + t 2E 2 (5) ). For a bimorph with an embedded heater, the temperature change resulting from actuation will be nonuniform about the bimorph length and is given by the temperature distribution T (s). This means that the strain due to the temperature distribution is nonuniform about the bimorph and will depend on the position along the length. The bimorph moment caused by the temperature distribution will also be distributed along the length and is given by or M T (s) =m A ε T (s) (6) M T (s) = m A (α 1 α 2 )(T (s) T 0 )= m A α T T (s) (7) where ε T (s) is the position-dependent strain difference caused by the temperature distribution, s is used to represent the position along the length of the bimorph, where s follows the contour of the bimorph deformation (as shown in Fig. 1), and α is the CTE of either layer. Since the total bimorph moment is distributed about the length, the curvature will also be position-dependent. This is why it is difficult to analytically derive the tip displacement of the bimorph. However, it is possible to analyze the tangential angle at the tip of the bimorph (tip angle) by integrating the moment distribution about the length. The tip angle of the bimorph can be found by integrating the arc angles of differential sections across the length of the bimorph, which yields [23] θ = L A EI = L Am A EI 1 L A L A 0 [( σ1 E 1 (M 0 + M T (s)) ds σ ) ] 2 E 2 α T T = θ 0 θ T (8) where T is the average temperature change of the bimorph, θ 0 is the initial tip angle at T 0 due to the residual stress, and θ T is the tip angular rotation due to the temperature distribution. The average moment due to the temperature distribution can be represented in terms of the average temperature change, i.e., M T = m A α T T. Although the temperature distribution along the bimorph beam is not uniform, it is the average temperature that determines the tip angular rotation. This is a convenient result because only a single parameter, which is the average temperature change of the bimorph, is needed from thermal analysis to predict the tip angular rotation. It should be noted that (8) is valid for large deflections because it is based on the moment curvature relationship that is valid for any curvature that causes stresses within the linear-elastic range of the beam materials [22]. The integration over ds in (8) follows the contour of the deformed bimorph, and it does not involve an approximation of the curvature. Fig. 2. TM-LEM of the bimorph actuator. The resultant TM-LEM is shown in Fig. 2, where the bimorph moment is the effort variable and the tip angular rotation rate θ is the flow variable. Thus, the tip angle is simply the integral of the flow variable. Therefore, the bimorph can be modeled as an equivalent capacitor, and the charge of the capacitor represents the output tip angle. In the TM-LEM, the effort and flow variables are conjugate power variables. The applied moment is divided into two voltage sources: a dc source, which represents the initial moment applied by the residual stress, and a voltage-controlled voltage source, which represents the additional moment applied by a change in temperature. The compliance of the bimorph is represented by a capacitor with a capacitance that is equal to the bimorph length-to-stiffness ratio. The analysis of the TM-LEM is trivial because there is only one passive element in the circuit. It is possible to include the transient behavior in the TM-LEM by inserting inductors to represent the equivalent masses of the structures and the dissipative behavior by inserting resistors to represent damping. Representing the curvature and tip angle in terms of the total moment and the stiffness is clumsy because these terms involve a complicated combination of variables. It is more convenient to rearrange the curvature in (1) and tip angle in (8) in terms of the total bimorph thickness and a parameter called the curvature coefficient. The curvature coefficient is given by [11], [24] (1 + m) 2 β ρ =6 1/mn + m 3 n +4m 2 +6m +4 where m = t 1 /t 2 is the thickness ratio, and n = E 1/E 2 is the biaxial elastic modulus ratio. The curvature coefficient is a unitless parameter that varies from 0 to 1.5. The ratio between the moment coefficient and the stiffness yields the curvature coefficient to the total thickness ratio (m A /EI = β ρ /t A ). Thus, the position-dependent curvature and the tip angle can be, respectively, expressed as 1 ρ A (s) = β [( ρ σ1 t A θ = β ρ L A t A E 1 σ 2 E 2 [( σ1 E 1 σ 2 E 2 ) ] α T T (s) (9) (10) ) ] α T T = θ 0 θ T. (11) These are much more convenient forms because they represent the curvature and tip angle in terms of the bimorph dimensions and a unitless variable that can only be equal to a value

4 216 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2008 Fig. 3. Schematic of the electrothermal bimorph with embedded heater used in the electrothermal model. within the range of Now that we have established that the average temperature change is the variable which controls the tip angular rotation of the bimorph, we must determine the average temperature change due to the actuation of the heater. IV. HEATER ET-LEM In this section, we will develop an ET-LEM that can be used to predict the average temperature change of an electrothermal heater. The ET-LEM can then be integrated with the TM-LEM to form the ETM-LEM. We are applying the ET-LEM to a bimorph actuator with an embedded heater, although the ET-LEM could be applied to many different types of electrothermal transducers. The ET-LEM uses temperature change and power as the effort and flow variables in the thermal domain, and voltage and current as the effort and flow variables in the electrical domain. Resistors are used to represent the thermal resistance in the thermal domain and the electrical resistance in the electrical domain. We will first develop the thermal domain LEM and then later add the electrical domain to form the ET-LEM. A. Development of the ET-LEM A schematic showing the structure of a bimorph actuator with an embedded heater used to generate the electrothermal model is shown in Fig. 3. The heater is actuated by applying a voltage or current to the embedded electrical resistor. Before the heater is actuated, the actuator temperature is equal to the substrate and ambient temperature T 0. Upon actuation, the temperature distribution is assumed to vary only along the length of the actuator and is given by T (s). The change in temperature at a position along the length of the actuator is T (s) =T (s) T 0. By ignoring convection and radiation on the actuator surface, the temperature distribution of the actuator is given by [16] T (s) =P [R TA ( s2 2L 2 + f s ) ] + fr TL (12) A L A where P is the total power dissipated by the electrical resistor, R TA is the conduction thermal resistance of the actuator, f is a parameter called the balancing factor, and R TL is the equivalent external thermal resistance that the actuator sees at its leftside boundary. The actuator conduction thermal resistance is given by R TA = L A (13) κ A w A t A where κ A is the composite combination of the thermal conductivities of the layers in the actuator. The balancing factor is a very important unitless parameter that varies from zero to one and is expressed as [16] f = R TA/2+R TR R TL + R TA + R TR (14) where R TR is the equivalent external thermal resistance that the actuator sees at its right-side boundary. The balancing factor measures the relative importance of the actuator conduction thermal resistance and external thermal resistances.

5 TODD AND XIE: ELECTROTHERMOMECHANICAL LUMPED ELEMENT MODEL OF ELECTROTHERMAL BIMORPH ACTUATOR 217 The thermal LEM will be developed by using the temperature distribution equation shown in (12) in solving for the maximum, average, and boundary temperature changes. Although the model ignores the surface convection on the actuator, it can include the surface convection that is present in the regions adjacent to the actuator, represented by the equivalent thermal resistances at the boundaries (R TL and R TR ). The equivalent thermal resistance between two temperature nodes is found by dividing the temperature difference between the nodes by the total power dissipated between the nodes. The maximum temperature is found by determining the position of the maximum temperature ŝ and then evaluating ˆT = T (s =ŝ). The position of the maximum temperature can easily be found by taking the derivative of (12) and setting it to zero, which yields ŝ = fl A. Thus, the maximum temperature change is given by ˆT = P (f 2 R TA /2+fR TL ). (15) At the position of maximum temperature, the slope of the temperature distribution will be zero, and the temperature will decrease directly on each side. This means that all of the power dissipated by the heater to the left of the maximum temperature node ˆT will flow to the left toward R TL. Similarly, all of the power dissipated by the heater to the right of ˆT will flow to the right toward R TR. The total power dissipated to the left of ˆT is equal to the left-side actuator volume times the power density which yields fp, and the total power dissipated to the right of ˆT is equal to the right-side actuator volume times the power density which yields (1 f)p. The current source that represents the total power dissipated is placed at ˆT.The temperatures at the left and right boundaries of the actuator can be found by, respectively, evaluating T L = T (s =0)and T R = T (s = L A ) using (12), which yield T L = fpr TL (16) T R =(1 f)pr TR. (17) Now, the question becomes how to split the actuator conduction thermal resistance on the left and right sides of ˆT.The actuator conduction thermal resistances on each side of ˆT are denoted as R AL and R AR. It was shown earlier that the total power flow at the left and right boundaries of the actuator are fp and (1 f)p, respectively. Thus, we have R AL =( ˆT T L )/(fp). (18) Substituting (15) and (16) into (18) yields Similarly, we have R AL = fr TA /2. (19) R AR =( ˆT T R )/ [(1 f)p ]=(1 f)r TA /2. (20) Note that (14) is needed to obtain (20). We now have enough information to construct a basic thermal LEM using a single power source at the maximum temperature node and equivalent thermal resistances between the maximum temperature and boundary nodes. The basic thermal LEM is shown in Fig. 4, Fig. 4. Basic thermal LEM. where the balancing factor plays a crucial role in the thermal behavior of the heater. The balancing factor f is given its name because it determines where the maximum temperature is located and then balances the power flow and actuator conduction thermal resistance on each side of the maximum temperature node. Notice that half of the total actuator conduction thermal resistance (R TA /2) is split on each side of the maximum temperature node by f. The quadratic temperature distribution across the actuator causes the actuator conduction thermal resistance to be divided by two. The thermal LEM in Fig. 4 does not include a node for the average temperature. As shown in (8) and (11), an average temperature node is needed to determine the tip angle of the bimorph actuator. The average temperature change is found by evaluating T =(1/L A ) L A 0 T (s)ds of (12), which yields T = P [(f 1/3)R TA /2+fR TL ]. (21) The location of the average temperature node T in the LEM depends on the relative values of the actuator conduction thermal resistance and external thermal resistances, as represented by the balancing factor. If f 2/3, T will exist to the left of ˆT.Iff 1/3, T will exist to the right of ˆT.If 1/3 <f<2/3, T can exist on either side of ˆT. In the LEM considered for a cantilevered bimorph actuator, it is assumed that the left-side external thermal resistance R TL is less than the right-side external thermal resistance R TR because R TL connects directly to the substrate, whereas R TR depends on convection to dissipate heat (see Fig. 3). This places T to the left of ˆT. To insert T into the thermal LEM, we must find the equivalent thermal resistance between ˆT and T as well as the equivalent thermal resistance between T and T L.The equivalent thermal resistance between ˆT and T is given by ( ˆT T )/(fp)=(f 1+1/3f)R TA /2 (22) which is obtained by using (15) and (21). Similarly, by using (16) and (21), the equivalent thermal resistance between T and T L is obtained as ( T T L )/(fp)=(1 1/3f)R TA /2. (23)

6 218 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2008 Fig. 5. ET-LEM, including the average temperature node. Notice that the sum of these equivalent thermal resistances is equal to the total equivalent thermal resistance that exists between ˆT and T L which is fr TA /2. The ET-LEM, which includes the average temperature node and the power supplied to the thermal domain by the electrical domain, is shown in Fig. 5. The power source in the thermal domain is replaced by a voltage- or currentcontrolled current source that represents the heat supplied to the thermal domain from the power dissipated by the electrical resistor. Notice that a temperature-dependent electrical resistor exists in the electrical domain. The total electrical resistance depends on the average temperature change and is given by R E = ρ E0L A w E t E (1 + ξ T )=R E0 (1 + ξ T ) (24) where w E is the width of the electrical resistor, t E is the thickness of the electrical resistor, ρ E0 and R E0 are, respectively, the initial electrical resistivity and resistance at T 0, and ξ is the TCR. This temperature-dependent resistor represents the thermal feedback to the electrical resistance that exists when the heater temperature rises. It is possible to expand the ET-LEM to include transient behavior by inserting capacitors which represent the capacitance in the electrical domain and heat capacity in the thermal domain. B. Electrothermal Response Now that we have a complete ET-LEM, the relationships between temperature changes and electrical inputs can be obtained using circuit analysis. We will limit this analysis to solve for the average temperature change since it is the parameter that determines the bimorph tip angular rotation, as shown in Section III. Solving for the average temperature change in terms of power yields T = P R T1 (25) where R T1 is defined as the equivalent average thermal resistance when convection is ignored on the surface of the actuator and is given by R T1 =(f 1/3)R TA /2+fR TL. (26) Solving for T in terms of current and voltage, respectively, yields T (I) = I2 R E0 R T1 (27) 1 I 2 R E0 ξr T1 T (V )= 1 4ξR T1 V 2ξ R E0 2V 2 R T1 /R E0 = (4ξRT1 ). (28) 1+ /R E0 V 2 +1 The positive TCR of the electrical resistor causes the electrical resistance to increase when the temperature increases. The temperature-dependent electrical resistance causes the current to provide positive feedback and the voltage to provide negative feedback to the power and temperature as shown in (27) and (28). At small current and voltage, the power and temperature are quadratically related to the current and voltage as shown in the numerators of (27) and (28). At larger current and voltage, the temperature increases enough to appreciably change the electrical resistance, causing the power and temperature to be nonquadratically related to the current and voltage as shown by the feedback terms in the denominators of (27) and (28). The positive feedback from the TCR to the current causes the average temperature change to approach infinity when the current approaches the critical current, which is given by I C = 1/R E0 ξr T1. The negative feedback from the TCR to the applied voltage causes the average temperature change to be approximately linear with the voltage when half of the critical voltage is passed, where the critical voltage is given by V C = R E0 /ξr T1. From the way we have defined the critical current and critical voltage, the initial electrical resistance is equal

7 TODD AND XIE: ELECTROTHERMOMECHANICAL LUMPED ELEMENT MODEL OF ELECTROTHERMAL BIMORPH ACTUATOR 219 Fig. 6. ETM-LEM of the electrothermal bimorph actuator. to the ratio between them, which is given by R E0 = V C /I C. The linear relationship between the average temperature change and applied voltage is expressed as T (V V C /2) R T1 /ξr E0 V. (29) The linear actuation range of the heater can be very beneficial to many types of devices where linear actuation is important. By equating (27) and (28), we obtain the I V characteristic of the heater, which is given by 2V/R E0 I = (4ξRT1 ). (30) 1+ /R E0 V 2 +1 This equation is a much simpler form of the heater I V characteristic compared with other models reported in the literature [13] [15], [18]. The I V characteristic demonstrates that voltage provides negative feedback to current. At small voltage, the current is linearly related to voltage by Ohm s Law as shown in the numerator of (30). At larger voltage, the increased electrical resistance causes the current to become nonlinear with voltage as shown by the feedback term in the denominator of (30). C. Electrothermal Model With Actuator Convection Although the ET-LEM ignored the convection on the surface of the actuator, the aforementioned equations can represent actuator convection by using a more general expression for the equivalent average thermal resistance. The actuator convection thermal resistance is defined as R CA = 1 2hL A (w A + t A ) (31) where h is the average convection coefficient on the surface of the actuator. The factor of two exists because convection dissipates heat on all four sides of the actuator. In general, we assume that the convection coefficient is constant and uniform on the actuator and on the regions adjacent to the actuator. The average convection coefficient could represent nonuniform convective heat transfer on a cross section of the actuator by including a shape factor, as discussed by Lin and Chiao [13] and Huang and Lee [18]. The convection thermal resistance could include a different-shaped cross section by modifying the area dimensions used in (31). It could also include conductive heat transfer normal to the actuator surface by modifying (31) to include a conduction term. If convection is considered on the surface of the actuator, the equivalent average thermal resistance is given by [16] R T2 [ ] 2[1 cosh(a)] a(r L +r R )sinh(a) =R CA a 2 (r L +r R )cosh(a)+a (1+a 2 r L r R ) sinh(a) +1 (32) where a = R TA /R CA, r L = R TL /R TA, and r R = R TR /R TA. The relationships derived in (25) (30) can include actuator convection by replacing R T1 with R T2. Equation (32) was derived from an electrothermal transducer model that considered convection on the surface of the actuator [16]. V. I NTEGRATED ETM-LEM The TM-LEM and ET-LEM given in Sections III and IV can be integrated to form the complete ETM-LEM shown in Fig. 6. The ETM-LEM demonstrates that, in the electrical domain, an input voltage or current causes the electrical resistor to dissipate power and raise the actuator temperature in the thermal domain, delivering an applied moment and angular rotation to the actuator in the mechanical domain. The link between the electrical and thermal domains is a voltage- or current-controlled current source, which represents the dissipated power delivered to the thermal domain from the electrical domain. The feedback from the thermal domain to the electrical domain is represented by the temperature-dependent electrical resistor. The link between the thermal and mechanical domains is a voltagecontrolled voltage source that represents the applied moment due to the expansion of the materials in response to a rise in

8 220 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2008 Fig. 7. (a) Top-view schematic of the micromirror device. (b) Side view of a two-beam section of the micromirror device. temperature. By performing a standard circuit analysis on the LEM, one obtains the tip angular rotation in terms of power, current, and voltage, which are given, respectively, by θ T (P )=β ρ L A t A α T R T1 P (33) L A I 2 R E0 R T1 θ T (I) =β ρ α T (34) t A 1 I 2 R E0 ξr T1 L A α T θ T (V )=β ρ 4ξR T1 V 2t A ξ R (35) E0 These equations can also include convection on the surface of the actuator by replacing R T1 with R T2, as discussed in Section IV. It should also be noted that we only chose to derive the tip angular rotation due to the temperature change and not the total tip angle, which includes the initial angle due to the residual stress. This was done because we will only report experimental results which measure the angular rotation in the next section. However, as shown in (11) and in the LEM in Fig. 6, the model can measure the total angle due to both the residual stress and the temperature change represented by θ = θ 0 θ T. Fig. 8. SEM photograph of the micromirror device. VI. ELECTROTHERMAL BIMORPH MICROMIRROR The ETM-LEM described earlier is used to model the behavior of a previously reported electrothermal bimorph micromirror [4]. The micromirror design consists of four regions a bimorph-actuator region, a mirror plate region, a substrate thermal isolation region, and a mirror thermal isolation region, as shown in Fig. 7. An SEM photograph of the micromirror

9 TODD AND XIE: ELECTROTHERMOMECHANICAL LUMPED ELEMENT MODEL OF ELECTROTHERMAL BIMORPH ACTUATOR 221 TABLE I GEOMETRIC AND MATERIAL PARAMETERS OF ONE TWO-BEAM SECTION OF THE MICROMIRROR DEVICE device is shown in Fig. 8. The geometry and parameters of the design are listed in Table I. A. Device Design The bimorph-actuator region is composed of an array of 72 bimorph beams that attach the mirror plate to the substrate. Each bimorph beam comprises an SiO 2 bottom layer, an Al top layer, and a polysilicon layer embedded in the SiO 2. The embedded polysilicon layer serves as an electrical resistor for heat dissipation. The mirror plate is composed of an Al reflective surface and a thick single-crystal-silicon bottom layer to ensure mirror flatness. The thermal isolation regions are composite structures of Al and SiO 2. The mirror plate tilts at an angle equal to the bimorph tip angle. Thus, the ETM-LEM can be used to predict the mirror rotation angle in response to an applied electrical input. Since the device is symmetric about the bimorph-actuator array, one two-beam section [shown in Fig. 7(a)] can be used in the models. A cross-sectional side view of a two-beam section is shown in Fig. 7(b). The material properties used in the analytical models were found from published data. The geometric and material properties of one two-beam section of the micromirror device are given in Table I. The regions adjacent to the bimorph-actuator array dissipate heat through conduction and convection. The external thermal resistance to the left of the bimorph R TL is due to the conduction and convection in the substrate thermal isolation region and is approximately equal to R TL 2R TisR Cis (36) R Tis +2R Cis where R Tis is the conduction thermal resistance of the substrate thermal isolation region, and R Cis is the convection thermal resistance of the substrate thermal isolation region. The external thermal resistance to the right of the bimorph R TR is due to the conduction and convection in the mirror thermal isolation and is approximately equal to R TR R Tim + R Tm /2+R Cm (37) where R Tim and R Tm are, respectively, the conduction thermal resistances of the mirror thermal isolation and mirror plate regions, and R Cm is the convection thermal resistance of the mirror plate region. The more accurate forms of R TL and R TR were given previously in [16]. The expressions for the bimorph actuator and external thermal resistances complete the models of the electrothermal micromirror.

10 222 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2008 TABLE II TEMPERATURE-DEPENDENT MATERIAL PROPERTIES USED IN FEM SIMULATIONS B. Models Compared to FEM Simulations and Experimental Results DC characterization of the micromirror was conducted in an open-air environment by measuring the I V characteristic and mirror rotation angle over a range of applied voltages. The mirror was positioned on a rotation stage. The electrical resistance was determined at each actuation point by taking the ratio of the applied voltage to the measured current. Although the average temperature change could not be measured directly, it was calculated from the measured electrical resistance by using the polysilicon TCR with (24). The polysilicon TCR was found to be K 1 in a previous experiment [23]. The micromirror tilts at an angle of approximately 32 at room temperature due to the residual stress present in the material layers of the bimorph. We did not make precise measurements of the residual stress present in the material layers; therefore, we did not model the total angle that includes both the initial tip angle due to the residual stress and the tip angular rotation due to a temperature change. Thus, the following analysis focuses only on the tip angular rotation resulting from actuation. In the following analyses, the experimental data are compared to the models and FEM simulations. Both the LEM and the model which includes actuator convection (using R T1 and R T2, respectively) are considered. The model that considers the actuator convection is labeled as actuator convection model in the following figures. FEM simulations of the electrothermomechanical behavior of the device were conducted in CoventorWare. In the models, we assumed that most material properties were independent of temperature. This assumption is not accurate in reality because many of the thermal and mechanical properties of Al, SiO 2, and polysilicon are temperature-dependent. To simulate the effect of temperaturedependent material properties on the behavior of the device, we conducted one set of FEM simulations with mostly temperature-independent material properties (all material properties being temperature-independent except for polysilicon electrical resistivity) and another set which included first-order temperature dependence of selected material properties. In the following figures, we refer to the temperature-independent FEM simulations as FEM simulation 1 and the temperature- Fig. 9. Plots of the mirror rotation angle (i.e., bimorph tip angular rotation) versus the average temperature change. dependent simulations as FEM simulation 2. The first-order temperature-dependent terms used in the FEM simulations are listed in Table II. Fig. 9 shows the thermomechanical data where the mirror rotation angle (equivalently, the bimorph tip angular rotation) is plotted against the average temperature change for model, FEM simulation, and experimental results. The LEM and temperature-independent FEM simulation curves match very closely, showing that the analytical model is very accurate when the material properties are temperature-independent. The temperature-dependent FEM simulation curve lies slightly above the other curves. This is largely due to the increase of the Al CTE with temperature. The experimental curve lies between the temperature-dependent FEM simulation curve and the LEM curve, indicating that the temperature dependence of the Al CTE may have been overestimated in the FEM simulation. Fig. 10 shows the electrothermal data where the average temperature change is plotted versus the following: 1) power; 2) voltage; and 3) current for model, FEM simulation, and experimental results. Before the experiments were conducted, the convection coefficient on the surface of the device was

11 TODD AND XIE: ELECTROTHERMOMECHANICAL LUMPED ELEMENT MODEL OF ELECTROTHERMAL BIMORPH ACTUATOR 223 Fig. 10. Plots of the average temperature change versus (a) power, (b) current, and (c) voltage. unknown. In the models, we assumed that the convection coefficient was the same in all regions of the device. To estimate the convection coefficient, the experimental data of the average temperature change versus the total power were fitted using a quadratic equation, and the convection coefficient used in the equivalent average thermal resistance R T2 in (32) was adjusted until R T2 equaled the linear term of the quadratic experimental fit. This approach yielded a convection coefficient of approximately 133 W/(K m 2 ) on the surface of the device. The LEM curve in Fig. 10(a) produces a slightly higher average temperature change compared to the actuator convection model curve. This is due to the fact that the LEM ignores the heat dissipation caused by convection on the surface of the actuator. Notice that there is only a slight difference between these curves in Fig. 10(a). This result suggests that the ET-LEM can accurately predict the electrothermal behavior of a bimorph actuator despite the fact that it ignores the convection on the surface of the actuator. Remember that, although the LEM ignores the convection on the actuator, it includes the convection in the regions adjacent to the actuator. Therefore, as long as much more heat is dissipated in the actuator by conduction compared to convection, the LEM can provide a reasonable approximation of the electrothermal behavior of the device. In other words, the LEM is a good approximation as long as the actuator convection thermal resistance is much greater than the actuator conduction thermal resistance (i.e., R CA R TA ). The experimental data in Fig. 10(a) show that the average temperature change was nonlinear versus the power. At higher power, the slope of the average temperature change versus the power decreased. The temperature-dependent FEM simulation data also show this trend, but not to the same extent as the experimental data. This could mean that the thermal conductivities in the real device were more temperature-dependent than the thermal conductivities used in the simulation. It could also mean that the tendency of the slope to decrease at higher power was caused by a larger heat dissipation resulting from the following: 1) a convection coefficient that increased with temperature; 2) radiation; and/or 3) the Thompson effect [14]. The plots of the average temperature change versus the current and voltage shown in Fig. 10(b) and (c), respectively, demonstrate the effect of the polysilicon TCR on the electrothermal behavior of the device. Fig. 10(b) shows that the average temperature change increases dramatically as the current approaches the critical current. This behavior demonstrates the positive feedback of the increasing electrical resistance to the applied current. Fig. 10(c) shows that, for voltages greater than half the critical voltage, the average temperature change becomes approximately linear with the voltage. This behavior demonstrates the negative feedback of the increasing electrical resistance to the applied voltage. Model, FEM simulation, and experimental results of the I V characteristic of the device are shown in Fig. 11. The slope of the initial electrical conductance is also included in Fig. 11. As was mentioned previously, the initial electrical resistance is equal to the ratio of the critical voltage to the critical current, which is given by R E0 = V C /I C. This relationship is shown in Fig. 11. Notice that, for small voltages, the slope of the I V characteristic is approximately linear and equal to the initial electrical conductance. This occurs because, at small voltages, the temperature change is too small to cause an appreciable increase in the electrical resistance. At larger voltages, the

12 224 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2008 to rotate about a fixed axis [34]. The experimental data match the model and FEM simulation results within 15% for all of the reported measurements on the micromirror device. Fig. 11. Plots of the I V characteristic of the electrothermal bimorph actuator. VII. CONCLUSION ET-LEM and TM-LEM of an electrothermal bimorph actuator have been developed. The models were integrated to form an ETM-LEM which predicts the bimorph tip angular rotation in response to an applied power, current, and voltage. The thermal LEM demonstrated the importance of the balancing factor in predicting the location of the maximum temperature and the division of heat flow and conduction thermal resistance on each side of the maximum temperature node. The ET-LEM demonstrates how the TCR provides feedback from the thermal domain to the electrical domain. The ET-LEM was applied to a bimorph actuator and could also be applied to a wide variety of electrothermal transducers. It was analytically and experimentally verified that the bimorph tip angular rotation is linearly proportional to the average temperature change. It was also verified that the average temperature change and tip angular rotation of an electrothermal bimorph actuator are linear with respect to the voltage when actuated past half the critical voltage. All of the experimental measurements agree with the model and FEM simulation results within 15%. ACKNOWLEDGMENT The authors would like to thank A. Jain for helping in collecting the dc characterization data. Fig. 12. Plots of the mirror rotation angle (i.e., bimorph tip angular rotation) versus voltage. slope of the I V curve decreases. The decrease in the slope becomes more significant as the voltage becomes larger because the temperature change becomes large enough to sufficiently increase the electrical resistance. As the voltage becomes very large, the slope of the I V characteristic approaches zero, demonstrating that the current of the device is always less than the critical current. The critical current is labeled in Fig. 11 to demonstrate this phenomenon. Fig. 12 shows the electrothermomechanical data where the mirror rotation angle is plotted against the voltage for model, FEM simulation, and experimental results. The experimental data in Fig. 12 demonstrate that the mirror rotation angle is approximately linear with the voltage for voltages greater than half the critical voltage. This experimental result verifies that an electrothermal bimorph actuator can be linearly actuated with the voltage when operated in a certain range. This property was used in a multidegree-of-freedom micromirror design, where the application of equal and opposite voltages to bimorph actuators on either side of the mirror plate caused the mirror REFERENCES [1] J. Buhler, J. Funk, O. Paul, F. P. Steiner, and H. Baltes, Thermally actuated CMOS micromirrors, Sens. Actuators A, Phys.,vol.47,no.1 3, pp , Mar./Apr [2] G. Lammel, S. Schweizer, and P. Renaud, Optical Microscanners and Microspectrometers Using Thermal Bimorph Actuators. Boston, MA: Kluwer, [3] L. A. Liew, A. Tuantranont, and V. M. Bright, Modeling of thermal actuation in a bulk-micromachined CMOS micromirror, Microelectron. J., vol. 31, no. 9/10, pp , Oct [4] H. Xie, A. Jain, T. Xie, Y. Pan, and G. K. Fedder, A single-crystal siliconbased micromirror with large scanning angle for biomedical applications, in Proc. Tech. Dig. CLEO, Baltimore, MD, 2003, pp [5] H. Xie, Y. T. Pan, and G. K. Fedder, Endoscopic optical coherence tomographic imaging with a CMOS-MEMS micromirror, Sens. Actuators A, Phys., vol. 103, no. 1/2, pp , Jan [6] P. Robert, D. Saias, C. Billard, S. Boret, N. Sillon, C. Maeder-Pachurka, P. L. Charvet, G. Bouche, P. Ancey, and P. Berruyer, Integrated RF-MEMS switch based on a combination of thermal and electrostatic actuation, in Proc. 12th Int. Conf. Solid-State Sens., Actuators, Microsyst. Dig. Tech. Papers, 2003, vol. 2, pp [7] X. F. Wang, D. A. Bullen, J. Zou, C. Liu, and C. A. Mirkin, Thermally actuated probe array for parallel dip-pen nanolithography, J. Vac. Sci. Technol. B, Microelectron. Process. Phenom., vol. 22, no. 6, pp , Nov [8] S.-H. Lim, J. Choi, R. Horowitz, and A. Majumdar, Design and fabrication of a novel bimorph microoptomechanical sensor, J. Microelectromech. Syst., vol. 14, no. 4, pp , Aug [9] P. Vettiger, G. Cross, M. Despont, U. Drechsler, U. Durig, B. Gotsmann, W. Haberle, M. A. Lantz, H. E. Rothuizen, R. Stutz, and G. K. Binnig, The millipede Nanotechnology entering data storage, IEEE Trans. Nanotechnol., vol. 1, no. 1, pp , Mar [10] A.-J. Villarceau, Recherches sur le mouvement et la compensation des chronometers, Annales de l Observatoire Imperial de Paris, [11] S. Timoshenko, Analysis of bi-metal thermostats, J. Opt. Soc. Amer., vol. 11, no. 3, pp , 1925.

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He is currently working toward the Ph.D. degree in electrical and computer engineering at the University of California, Santa Barbara. His graduate study is supported by a National Defense Science and Engineering Graduate Fellowship. He has coauthored three journal papers and five conference papers and has one pending patent. His current research interests include design, modeling, fabrication, and characterization of RF and optical MEMS devices. Huikai Xie (S 00 M 02 SM 07) received the B.S. and M.S. degrees in electronics engineering from Beijing Institute of Technology, Beijing, China, the M.S. degree in electrical engineering from Tufts University, Medford, MA, in 1998, and the Ph.D. degree in electrical and computer engineering from Carnegie Mellon University, Pittsburgh, PA, in From 1992 to 1996, he was a Research Faculty Member and Lecturer with the Tsinghua University, Beijing, working on various silicon-based chemical and mechanical sensors. He spent summer 2001 at Robert Bosch Corporation, Broadview, IL, designing 6-DOF inertial measurement units. He is currently an Associate Professor with the Department of Electrical and Computer Engineering, University of Florida, Gainesville. He has published over 90 technical papers and is the holder of four U.S. patents, with eight pending patents. His present research interests include micro/nanofabrication, integrated inertial sensors, microactuators, integrated power converters, optical MEMS, optical imaging and fiber-optic sensors. Dr. Xie received the 1996 Tsinghua University Motorola Education Award and the 1996 Best Paper Award from Chinese Journal of Semiconductors. He was named the 2006 Small Times magazine Best of Small Tech Researcher of the Year Finalist.