Two-Dimensional Models of Thermoelastic Damping for Out-of-Plane Vibration of Microrings With Circular Cross-Section

Thermoelastic damping is an important dissipation mechanism in microresonators. This article presents an analytical model of thermoelastic damping for out-of-plane vibration of microrings with circular cross-section by considering two-dimensional heat conduction. The temperature field of the circular cross-section is calculated by using the Bessel function and free boundary conditions. The coupled motion of bending and torsion in out-of-plane mode has been considered to calculate the mechanical energy. The derivation shows that the analytical expression of thermoelastic damping can be considered as a product of Zener model and the energy ratio of pure bending energy stored to total elastic energy stored. The present model is verified by comparing with the finite-element method. The convergence of the analytical expression has been examined and the characteristics of the expression have been studied by using a normalized form. The effect of geometry on thermoelastic damping has been studied. The results show that thermoelastic damping in microrings of circular cross-section under out-of-plane mode depends on geometry, scales, and vibration frequencies.

SECTION I. Introduction

Microring resonator is a typical device of MEMS (microelectromechanical system), which is widely used in MEMS sensors and actuators [1]–​[4]. At present, two vibration modes are usually used in the design of ring resonator, i.e., in-plane mode and out-of-plane mode. Using in-plane mode, the ring resonator is suitable to design rate sensors that can detect the angular rate of only one direction [5]–​[7]. Using the coupling of in-plane and out-of-plane modes, the ring resonator can be applied to multi-axis rate sensors that can detect the angular rates more than one direction simultaneously [1], [2]. This kind of sensor has attracted the attention of researchers due to higher integration and advanced technology. In fact, microresonators have excellent mechanical properties. However, some dissipation mechanisms, which reduce the performance of the microresonators, become of significance at microscale due to the scale effect. The main dissipation mechanisms include air damping, support loss, and thermoelastic damping. Among them, thermoelastic damping is the intrinsic damping that is produced due to the thermoelastic effect of material and cannot be eliminated by proper design and manufacturing. Therefore, it is very important to investigate the mechanism of thermoelastic damping in microresonators.

Zener [8], [9] first investigated thermoelastic damping in beam resonator of transverse vibration and proposed an analytical model by using the thermal mode superposition method. Zener’s study has proved that the analytical model is available not only for a beam with rectangular cross-section but also for that with circular cross-section. Zener model for beam resonator with circular cross-section considers heat conduction in two directions of the section, given by [9] $$\begin{equation*} Q_{\textrm {Zener,circular}}^{-1} =\Delta _{E} \sum \limits _{q=1}^\infty {f_{q} \frac {\omega \tau _{q}}{1+\omega ^{2}\tau _{q}^{2}}}\tag{1}\end{equation*}$$ where $\Delta _{E} = E\alpha ^{2}T_{0}/C_{v}$ , $E$ is the Young’s modulus, $T_{0}$ is the ambient temperature, $\alpha$ is the coefficient of thermal expansion, $C_{v}$ is the heat capacity per unit volume, $f_{q}$ is the weight coefficient associated with the $q$ th thermal mode, $\tau _{q}$ is the relaxation time and $\omega$ is the angular frequency. With the development of MEMS, Zener model has attracted more and more attention, and its validity has been confirmed in resonators of microscale [10]. Lifshitz and Roukes (LR) [11] improved upon Zener model by using a complex temperature field and presented a closed-form expression of thermoelastic damping for a beam resonator with rectangular cross-section. However, LR model is not available for resonators with circular cross-sections.

Based on Zener and LR theory, various models of thermoelastic damping are derived for different structural shapes and vibration modes of microresonators. At present, the thermoelastic damping model has been studied in most common microresonators, such as microbeam [12], microring (in-plane vibration [13]–​[16], out-of-plane vibration [17]), and microplate [18], [19]. In some complex microresonators, analytical models of thermoelastic damping are also widely studied and obtained, such as composite laminated structure [20], [21], hemispherical structure [22], etc. However, to our knowledge, most works are focus on the resonator with rectangular cross-section, and only a few works have discussed the models of thermoelastic damping for the case of circular cross-section. This is because the normal methods of micromachining are planar technologies which can be used to fabricate microstructures with rectangular cross-section, and it is difficult to fabricate the structure with circular cross-section using the methods. Under such condition, this work focuses on the theoretical research and has no direct correlation with the current micro-scale technology. On the one hand, there are some works that have already studied the nano/micro-structures with non-rectangular cross-section. With the development of micromachining, the present model can be used for future devices with microring resonators with circular cross-sections. On the other hand, the present model is applicable to but not limited to microstructure, namely, it is also applicable to macrostructure. Therefore, the theory in this article is also effective in other fields and scales.

To date, thermoelastic damping in beam and ring resonators with circular cross-section can be calculated by Zener model and Li’s model [14], respectively. However, Li’s model is only suitable for the in-plane mode of a ring and cannot predict thermoelastic damping for the case of out-of-plane mode. Although in the previous work [17], we have studied the out-of-plane vibration of a circular ring with rectangular cross-section, the thermoelastic damping of a circular ring with circular cross-section cannot be calculated by the theory of rectangular cross-section due to the fact that the shape of the cross-section has a great influence on the heat conduction. In this article, we derive an analytical model for thermoelastic damping in microring resonators with circular cross-section under out-of-plane vibration. We first solve the two-dimensional heat conduction equation of the circular cross-section and obtain the temperature field of the ring resonator by using the thermal mode superposition method and the properties of the Bessel function. Then, we utilize the definition of the quality factor to derive an analytical expression of thermoelastic damping for out-of-plane vibration of a ring resonator with circular cross-section. The expression can be regarded as a product of Zener model and the energy ratio of pure bending energy stored to total elastic energy stored. The present model is validated by comparing its results with the finite-element method (FEM) solutions.

SECTION II. Problem Formulation

We consider a microring resonator of uniform circular cross-section under out-of-plane vibration with free boundary conditions as shown in Fig. 1. Three coordinate systems are defined, a global cylindrical coordinate system ($R$ , $\varphi$ , $Z$ ), a local Cartesian coordinate system ($x$ , $y$ , $z$ ), and a polar coordinate system ($r$ , $\beta$ ) attached to the circular cross-section where $x = r$ sin$\beta$ and $z = r$ cos$\beta$ . $R_{0}$ and $r_{0}$ are the radius of the undeformed centroidal line and the radius of circular cross-section, respectively. $u$ , v, and $w$ are the displacements along the radial, circumferential and axial directions, respectively, and $\phi$ is the torsional displacement about the $y$ -direction.

FIGURE 1.

Schematic diagram of the circular cross-section of a ring with coordinate systems. (a) The global cylindrical coordinate system ($R$ , $\varphi$ , $Z$ ) and the local Cartesian coordinate system ($x$ , $y$ , $z$ ). (b) The polar coordinate system ($r$ , $\beta$ ).

A. Coupled Motion of Out-of-Plane Vibration

For the out-of-plane vibration of a ring, the motion is coupling with bending and torsion. The coupling displacements consists of an axial displacement $w$ and a torsional displacement $\phi$ . Assume that the radius of the cross-section $r_{0}$ is small in comparison with the radius of the ring $R_{0}$ , and hence, warping is neglected. Under such condition, the coupled motion of out-of-plane vibration can be expressed as [2] $$\begin{align*} {\begin{cases} w\left ({{\varphi,t} }\right)=We^{\textrm {i}\omega _{n} t}=W_{0} \cos \left ({{n\varphi } }\right)e^{\textrm {i}\omega _{n} t} \\ \phi \left ({{\varphi,t} }\right)=-n^{2}\zeta We^{\textrm {i}\omega _{n} t}=-n^{2}\zeta W_{0} \cos \left ({{n\varphi } }\right)e^{\textrm {i}\omega _{n} t} \\ \end{cases}}\tag{2}\end{align*}$$ where $W$ and $W_{0}$ represent the amplitude and the maximum amplitude in the axial direction, $\omega _{n}$ is the natural frequency of out-of-plane vibration, $n$ is the mode number ($n = 2, 3, 4\ldots$ ), and $\zeta$ is a parameter, given by $$\begin{equation*} \zeta =\frac {1}{R_{0}}\left [{ {\frac {1+\mu }{1+n^{2}\mu }} }\right]\tag{3}\end{equation*}$$ where $$\begin{equation*} \mu =\frac {GJ}{EI_{x} }=\frac {1}{1+\upsilon }\tag{4}\end{equation*}$$ where $G$ is the shear modulus, $J$ is the torsion constant, $I_{x}$ is the second moment of area and $\upsilon$ is the Poisson’s ratio. For circular cross-section with radius $r_{0}$ , $J$ and $I_{x}$ can be expressed as $$\begin{equation*} J=2I_{x} =\frac {\pi r_{0}^{4}}{2}\tag{5}\end{equation*}$$

The out-of-plane displacement field of the ring in cylindrical coordinates can be expressed as [23], [24] $$\begin{align*} \begin{cases} \displaystyle u\left ({{R,\varphi,Z,t} }\right)=z\phi \\ \displaystyle v\left ({{R,\varphi,Z,t} }\right)=-z\frac {\partial w}{R\partial \varphi }\\ \displaystyle w\left ({{R,\varphi,Z,t} }\right)=w\left ({{\varphi,t} }\right)\\ \displaystyle \end{cases}\tag{6}\end{align*}$$

B. Strain and Stress Fields

For the thin ring ($r_{0} \ll R_{0}$ ), the differences of strains along the radius direction on the cross-section can be neglected. Therefore, using (2), (6) and the relevance between $w$ and $\phi$ , the strain field of the ring is obtained and can be expressed in polar coordinate [17], [25] $$\begin{align*} \begin{cases} \displaystyle \varepsilon _{\varphi } =\frac {u}{R_{0}}+\frac {1}{R_{0}}\frac {\partial v}{\partial \varphi }=-r\cos \beta \frac {1-R_{0} \zeta }{R_{0}^{2} }\frac {\partial ^{2}w}{\partial \varphi ^{2}} \\ \displaystyle \varepsilon _{R} =\varepsilon _{Z} =-\upsilon \varepsilon _{\varphi } +\left ({{1+\upsilon } }\right)\varepsilon _{\textrm {thermal}} \\ \displaystyle \gamma _{R\varphi } =\frac {1}{R_{0}}\frac {\partial u}{\partial \varphi }-\frac {v}{R_{0}}=r\cos \beta \frac {1-n^{2}R_{0} \zeta }{R_{0}^{2} }\frac {\partial w}{\partial \varphi } \\ \displaystyle \gamma _{\varphi Z} =-\frac {x}{R_{0}}\left ({{\frac {\partial \phi }{\partial \varphi }+\frac {\partial w}{R_{0} \partial \varphi }} }\right)=-r\sin \beta \frac {1-n^{2}R_{0} \zeta }{R_{0}^{2}}\frac {\partial w}{\partial \varphi } \\ \displaystyle \gamma _{RZ} =0 \\ \displaystyle \end{cases}\!\!\!\!\! \\{}\tag{7}\end{align*}$$ where $\varepsilon _{\mathrm {thermal}} = \alpha \theta$ represents the thermal strain caused by thermoelastic effect and $\theta$ is the change in temperature.

Typically, the temperature change caused by the thermoelastic effect is relatively small. Therefore, compared with the mechanical stresses, the thermal stress produced by $\theta$ can be ignored [26]. Then, the stress filed of the ring can be expressed as $$\begin{align*} \begin{cases} \displaystyle \sigma _{\varphi } =E\varepsilon _{\varphi } =-Er\cos \beta \frac {1-R_{0} \zeta }{R_{0}^{2}}\frac {\partial ^{2}w}{\partial \varphi ^{2}} \\ \displaystyle \tau _{r\varphi } =G\gamma _{r\varphi } =\frac {Er\cos \beta }{2\left ({{1+\upsilon } }\right)}\frac {1-n^{2}R_{0} \zeta }{R_{0}^{2}}\frac {\partial w}{\partial \varphi } \\ \displaystyle \tau _{\varphi Z} =G\gamma _{\varphi Z} =-\frac {Er\sin \beta }{2\left ({{1+\upsilon } }\right)}\frac {1-n^{2}R_{0} \zeta }{R_{0}^{2}}\frac {\partial w}{\partial \varphi } \\ \displaystyle \end{cases}\tag{8}\end{align*}$$

C. Heat Conduction Equation

Assume that the ring is subjected to time-harmonic force with the natural frequency $\omega _{n}$ and then operates in out-of-plane vibration. The steady-state responses of the translation displacement and the relative temperature filed in the ring have the form $$\begin{align*} \begin{cases} \displaystyle w\left ({{\varphi,t} }\right)=W\left ({\varphi }\right)e^{\textrm {i}\omega _{n} t}=W_{0} \cos \left ({{n\varphi } }\right)e^{\textrm {i}\omega _{n} t} \\ \displaystyle \theta \left ({{x,\varphi,z,t} }\right)=\theta _{0} \left ({{x,\varphi,z} }\right)\textrm {e}^{\textrm {i}\omega _{n} t} \\ \displaystyle \end{cases}\tag{9}\end{align*}$$

According to the Fourier Law, the thermoelastic temperature field is governed by the heat conduction equation, given by [27] $$\begin{equation*} \frac {\partial \theta }{\partial t}=\chi \nabla ^{2}\theta -\frac {E\alpha T_{0}}{\left ({{1-2\upsilon } }\right)C_{v}}\frac {\partial }{\partial t}\sum \limits _{j} {\varepsilon _{jj}}\tag{10}\end{equation*}$$ where $\nabla ^{2}$ (•) represents the Laplacian operator and $\varepsilon _{jj}$ is the normal strain. In fact, the bending component of out-of-plane vibration is the main cause of thermoelastic damping whereas the torsion component causes no local volume change and hence, suffers no damping [11]. For bending vibration, most heat flow transports along transverse directions ($x$ - and $z$ -axes). Accordingly, we assume that the heat flow along circumferential direction $\varphi$ is neglected. For such case, $\nabla ^{2}$ can be expressed as $$\begin{equation*} \nabla ^{2}=\frac {\partial ^{2}}{\partial x^{2}}+\frac {\partial ^{2}}{\partial z^{2}}=\frac {\partial ^{2}}{\partial r^{2}}+\frac {1}{r}\frac {\partial }{\partial r}+\frac {1}{r^{2}}\frac {\partial ^{2}}{\partial \beta ^{2}}\tag{11}\end{equation*}$$ Therefore, we obtain a two-dimensional heat conduction equation, given by $$\begin{align*}&\hspace {-0.5pc}\frac {\partial \theta }{\partial t}=\chi \left ({{\frac {\partial ^{2}\theta }{\partial r^{2}}+\frac {1}{r}\frac {\partial \theta }{\partial r}+\frac {1}{r^{2}}\frac {\partial ^{2}\theta }{\partial \beta ^{2}}} }\right) \\&\qquad\qquad\qquad\qquad \displaystyle {+r\cos \beta \frac {\Delta _{E}}{\alpha }\frac {\partial }{\partial t}\left ({{\frac {1-R\xi }{R^{2}}\frac {\partial ^{2}w}{\partial \varphi ^{2}}} }\right)} \tag{12}\end{align*}$$ The last term on the right side of the above equation is the internal heat source excitation. When the excitation is 0, we obtain a two-dimensional free heat conduction equation on a circular surface, given by $$\begin{equation*} \frac {\partial \theta }{\partial t}=\chi \left ({{\frac {\partial ^{2}\theta }{\partial r^{2}}+\frac {1}{r}\frac {\partial \theta }{\partial r}+\frac {1}{r^{2}}\frac {\partial ^{2}\theta }{\partial \beta ^{2}}} }\right)\tag{13}\end{equation*}$$ Equation (13) is a typical heat conduction equation of a circular plane. It is very mature to solve the eigenvalue and thermal mode of the equation by using the separation of variables method, and the form of the solution can be expressed as a sum of products of a thermal mode equation $\theta _{1}$ , a spatial frequency equation $\theta _{2}$ and a time-frequency equation $\theta _{3}$ , given by $$\begin{equation*} \theta \left ({{r,\beta,\varphi,t} }\right)=\theta _{1} \left ({{r,\varphi } }\right)\theta _{2} \left ({\beta }\right)\theta _{3} \left ({t }\right)\tag{14}\end{equation*}$$ where $\theta _{1}$ is a Bessel function of the first kind.

SECTION III. Solution of the Heat Conduction Equation

A. Temperature Field

The thermoelastic effect during vibration of the resonator results in the internal heat source, that is, the excitation of heat conduction. For (12), the last term on the right is the heat source term of the system, which excites the system with a time-frequency $\omega _{n}$ from exp($\text{i}\omega _{n}t$ ) and a spatial frequency $\omega _{k} = {1}$ from cos$\beta$ . Therefore, based on (14), the steady-state response of the temperature field can be expressed using the mode superposition method, given by $$\begin{align*} \theta \left ({{r,\beta,\varphi,t} }\right)=&\theta _{0} \left ({{r,\beta,\varphi } }\right)e^{i\omega _{n} t} \\=&\sum \limits _{q=1}^\infty {c_{q} J_{1} \left ({{\gamma _{q} r} }\right)} \cos \beta e^{\textrm {i}\omega _{n} t}\tag{15}\end{align*}$$ where $\gamma _{p}$ is a positive number and $c_{q}$ is the weight coefficient for different thermal modes, which can be determined by boundary conditions of heat conduction.

Assume that no heat transfer from the ring to the environment. The adiabatic boundary conditions are employed, i.e., $\partial \theta _{0}/\partial r = {0}$ at $r = \pm \,\,r_{0}$ . Substituting (15) into the boundary conditions, yields $$\begin{equation*} \left.{ {\frac {\partial \theta _{0}}{\partial r}} }\right |_{r=r_{0} } =\sum \limits _{q=1}^\infty {c_{q} \frac {d}{dr}J_{1} \left ({{\gamma _{q} r_{0}} }\right)} \cos \beta =0\tag{16}\end{equation*}$$ Using the properties of the Bessel function, we obtain $$\begin{equation*} J_{0} \left ({{a_{q}} }\right)-J_{2} \left ({{a_{q}} }\right)\textrm {=0} ~or~J_{1} \left ({{a_{q}} }\right)=a_{q} J_{0} \left ({{a_{q}} }\right)\tag{17}\end{equation*}$$ where $a_{q} = \gamma _{q}r_{0}$ is the root of (17).

Substituting (15) into (12), we obtain the distribution equation of temperature field (see APPENDIX A for details) $$\begin{equation*} \sum \limits _{q=1}^\infty {c_{q} \left ({{\gamma _{q}^{2}\! +\!\frac {i\omega _{n} }{\chi }} }\right)J_{1} \left ({{\gamma _{q} r} }\right)} =r\frac {i\omega _{n}}{\chi }\frac {\Delta _{E}}{\alpha }\frac {1\!-\!R_{0} \xi }{R_{0}^{2}}\frac {\partial ^{2}W}{\partial \varphi ^{2}}\tag{18}\end{equation*}$$ To solve the coefficient $c_{q}$ , the weighted orthogonality of the Bessel function is used, given by $$\begin{equation*} \int _{0}^{r_{0}} {rJ_{1} \left ({{\gamma _{p} r} }\right)J_{1} \left ({{\gamma _{q} r} }\right)dr} \textrm {=0},\quad \text {when}~p\ne q\tag{19}\end{equation*}$$ Let both sides of (18) be multiplied by $rJ_{1}(\gamma _{p}\beta)$ and integrated from $r = {0}$ to $r_{0}$ , and hence, we obtain $$\begin{align*}&\hspace {-0.5pc} c_{q} \left ({{\gamma _{q}^{2} +\frac {i\omega _{n}}{\chi }} }\right)\int _{0}^{r_{0}} {rJ_{1}^{2}\left ({{\gamma _{q} r} }\right)dr} \\&\qquad\quad \displaystyle {=\frac {i\omega _{n}}{\chi }\frac {\Delta _{E}}{\alpha }\frac {1-R_{0} \xi }{R_{0}^{2}}\frac {\partial ^{2}W}{\partial \varphi ^{2}}\int _{0}^{r_{0}} {r^{2}J_{1} \left ({{\gamma _{q} r} }\right)dr}} \tag{20}\end{align*}$$ Utilizing the boundary conditions (17), we can solve the above equation directly and then the coefficient $c_{q}$ is obtained as (see APPENDIX A for details) $$\begin{equation*} c_{q} =\frac {2\Delta _{E} r_{0} \left ({{\omega _{n}^{2}\! +\!i\omega _{n} \chi \gamma _{q}^{2}} }\right)}{\left ({{\chi ^{2}\gamma _{q}^{4}\! +\!\omega _{n}^{2}} }\right)\left ({{a_{q}^{2} -1} }\right)\alpha J_{1} \left ({{a_{q}} }\right)}\frac {1\!-\!R_{0} \xi }{R_{0}^{2}}\frac {\partial ^{2}W}{\partial \varphi ^{2}}\tag{21}\end{equation*}$$ Note that $c_{q}$ is a complex number that represents the out of phase relationship between temperature field and elastic vibration caused by thermoelastic damping.

B. Thermoelastic Damping

According to the definition of quality factor $Q$ , thermoelastic damping can be expressed as $$\begin{equation*} Q^{-1}=\frac {1}{2\pi }\frac {\Delta W}{W_{\textrm {stored}}}\tag{22}\end{equation*}$$ where $\Delta W$ is the energy loss per cycle due to thermoelastic damping and $W_{\mathrm {stored}}$ is the maximum energy stored per cycle of vibration. For vibrating resonators, $\Delta W$ is related to the coupling between stress field and thermal strain field that results in the conversion of mechanical energy into heat energy, given by $$\begin{equation*} \Delta W=-\pi \int \!\!\!\int \!\!\!\int _{V} {\hat {\sigma }_{\varphi } Im\left ({{\hat {\varepsilon }_{\textrm {thermal}}} }\right)dV}\tag{23}\end{equation*}$$ where the hat (&)hat; denotes the amplitude and the imaginary part of thermal strain can be expressed as $$\begin{align*} \text {Im}\left ({{\hat {\varepsilon }_{\textrm {thermal}}} }\right)=&\alpha \text {Im}\left ({{\theta _{0}} }\right) \\=&\alpha \cos \beta \sum \limits _{q=1}^\infty {\textrm {Im}\left ({{c_{q}} }\right)J_{1} \left ({{\gamma _{q} r} }\right)}\tag{24}\end{align*}$$ For out-of-plane vibration, $W_{\mathrm {stored}}$ is caused by the coupled motion of bending and torsion, given by $$\begin{equation*} W_{\textrm {stored}} =\frac {1}{2}\int \!\!\!\int \!\!\!\int _{V} {\left ({{\hat {\sigma }_{\varphi } \hat {\varepsilon }_{\varphi } +\hat {\tau }_{r\varphi } \hat {\gamma }_{r\varphi } +\hat {\tau }_{\varphi Z} \hat {\gamma }_{\varphi Z}} }\right)dV}\tag{25}\end{equation*}$$ Substituting (7) and (8) into (23) and (25), the energy loss produced per cycle and the maximum energy stored per cycle can be expressed as (see APPENDIX B for details) $$\begin{align*} \hspace {-0.5pc} \Delta W=&2\pi ^{3}E\Delta _{E} \omega _{n} \chi n^{4}W_{0}^{2} \\&\times \cdot \frac {r_{0}^{2}}{R_{0}^{3}}\frac {\left ({{n^{2}-1} }\right)^{2}\mu ^{2}}{\left ({{1\!+\!n^{2}\mu } }\right)^{2}}\sum \limits _{q=1}^\infty {\frac {1}{\left ({{\chi ^{2}\gamma _{q}^{4} \!+\!\omega _{n}^{2}} }\right)\left ({{a_{q}^{2} -1} }\right)}} \\{}\tag{26}\\ W_{\textrm {stored}}=&\frac {E\pi ^{2}r_{0}^{4}}{8R_{0}^{3} }n^{4}W_{0}^{2} \frac {\left ({{n^{2}-1} }\right)^{2}\mu ^{2}}{\left ({{1+n^{2}\mu } }\right)^{2}}\left [{ {1+\frac {1}{\left ({{1+\upsilon } }\right)n^{2}\mu ^{2}}} }\right] \\{}\tag{27}\end{align*}$$ Substituting (26) and (27) into (22), and using (4), the thermoelastic damping model for out-of-plane vibration of a ring with circular cross-section can be expressed as $$\begin{equation*} Q^{-1}=\frac {1}{1+{\left ({{1+\upsilon } }\right)} \mathord {\left /{ {\vphantom {{\left ({{1+\upsilon } }\right)} {n^{2}}}} }\right. } {n^{2}}}\Delta _{E} \sum \limits _{q=1}^\infty {f_{q} \frac {\omega _{n} \tau _{q}}{1+\omega _{n}^{2} \tau _{q}^{2}}}\tag{28}\end{equation*}$$ where $f_{q}$ is the weight coefficient and $\tau _{q}$ is the relaxation time, given by $$\begin{equation*} f_{q} =\frac {8}{a_{q}^{2} \left ({{a_{q}^{2} -1} }\right)}\tau _{q} =\frac {r_{0}^{2}}{a_{q}^{2} \chi }\tag{29}\end{equation*}$$

According to (1), (28) can also be expressed as $$\begin{equation*} Q^{-1}=\frac {1}{1+{\left ({{1+\upsilon } }\right)} \mathord {\left /{ {\vphantom {{\left ({{1+\upsilon } }\right)} {n^{2}}}} }\right. } {n^{2}}}Q_{\textrm {Zener,circular}}^{-1}\tag{30}\end{equation*}$$ Additionally, utilizing the energy expression, we can obtain the energy ratio $$\begin{equation*} \frac {W_{\textrm {bending}}}{W_{\textrm {stored}} }=\frac {1}{1+{\left ({{1+\upsilon } }\right)} \mathord {\left /{ {\vphantom {{\left ({{1+\upsilon } }\right)} {n^{2}}}} }\right. } {n^{2}}}\tag{31}\end{equation*}$$ where $W_{\mathrm {bending}}$ is the pure bending energy stored related to the bending component of the coupled motion. Thus, $Q^{-1}$ can be regarded as the product of the Zener model and the energy ratio of pure bending energy stored to total elastic energy stored.

C. Simplified Model

Table 1 lists the values of the weight coefficient $f_{q}$ for different thermal modes. Note that high order modes ($q = 2, 3, 4\ldots$ ) can be neglected by introducing a very little error. Hence, we remain only the first term ($q =1$ ) in the summation of (28), and then the simplified model of thermoelastic damping can be expressed as $$\begin{equation*} Q^{-1}=\frac {1}{1+{\left ({{1+\upsilon } }\right)} \mathord {\left /{ {\vphantom {{\left ({{1+\upsilon } }\right)} {n^{2}}}} }\right. } {n^{2}}}\Delta _{E} \frac {\omega _{n} \tau }{1+\omega _{n}^{2} \tau ^{2}}\tag{32}\end{equation*}$$ where $\tau = \tau _{1} = 0.295{r_{0}^{2}} \mathord {\left /{ {\vphantom {{r_{0}^{2}} \chi }} }\right. } \chi$ . The relaxation time of circular cross-section $\tau$ is different from that of square cross-section $\tau _{\mathrm {square}}$ , given by [9], [11] $$\begin{equation*} \tau _{\textrm {sqrare}} =\frac {b^{2}}{\pi ^{2}\chi }\tag{33}\end{equation*}$$ where $b$ is the beam thickness. When $b = 0.5431\pi r_{0}$ , we obtain $\tau _{\mathrm {square}} = \tau$ . Fig. 2 shows a comparison between $\tau _{\mathrm {square}}$ and $\tau$ for three different cases. From the figure, it shows that $\tau _{\mathrm {square}} > \tau$ , when the diameter of the circular section is equal to $b$ ($b = 2r_{0}$ ), or the two kinds of sections are of the same area ($b = \pi ^{0.5}r_{0}$ ). Also shown in the figure, $\tau _{\mathrm {square}} < \tau$ , when $r_{0}$ is the circumscribed circle of the square ($b = 2^{0.5}r_{0}$ ).

TABLE 1 The Values of $a_{q}$ and $f_{q}$ for the First Six Thermal Modes

FIGURE 2.

Variation of normalized relaxation time with $r_{0}$ .

Note that the thermoelastic damping model (30) is similar to that for rectangular sections in the reference [17]. However, the two equations are only similar in form and cannot be substituted for each other. Through comparison, we find that different cross-section shapes lead to different heat conduction paths besides the mechanical parameters such as the torsional constant $J$ and the second moment of area $I_{x}$ . This is also the main reason why the model for rectangular cross-section in the reference [17] cannot be used for circular cross-section. In this article, a two-dimensional heat conduction equation is adopted for circular cross-section, which considers the heat flow in the radial and axial directions (x- and z-axes), while the equation of rectangular cross-section [17] only considers the heat flow in the axial direction. Theoretically, the two-dimensional heat conduction model has higher accuracy than the one-dimensional one, but it is more difficult to obtain the analytical solution. However, for circular cross-section, the two-dimensional heat conduction equation is easier to solve than the one-dimensional heat conduction equation. Therefore, for the circular cross-section, we use the two-dimensional heat conduction equation with higher accuracy to describe the internal heat transfer phenomenon. On the contrary, for rectangular cross-section, it is easier to obtain the analytical solution of one-dimensional heat conduction equation with sufficient accuracy, because most of the heat conduction occurs in one direction.

SECTION IV. Results

In this section, we first verify the validity of the present model by comparing with FEM results. Then, the convergence of the present model as well as its characteristics of normalized expression are examined carefully. Next, the effect of ring geometry on thermoelastic damping is studied. Last, the differences of thermoelastic damping in the rings with circular cross-section and square cross-section are discussed. The material properties used for the theoretical calculation of this section are list in Table 2.

TABLE 2 Material Properties of Polysilicon at 300 K [28]

A. Verification

In this section, the present model of thermoelastic damping is verified by comparing with FEM. The present model is a two-dimensional model considering the heat flow along axial and radial directions. Theoretically, FEM is a very high accurate method to predict thermoelastic damping because three-dimensional heat conduction is considered in FEM. The FEM simulation results are obtained by using free boundary conditions and harmonic exciting force of axial direction applied to a small area of the ring.

Fig. 3 shows the imaginary part of the temperature of the ring vibrating in out-of-plane mode $n =2$ . The specifications of the ring are $r_{0} = 10\,\,\mu \text{m}$ and $R_{0} = 200\,\,\mu \text{m}$ . The imaginary part of the temperature represents the part of the material temperature that is out of phase with the mechanical vibration. As shown in the figure, the temperature gradient of the cross-section is mainly along the axial direction, but the role of radial direction cannot be ignored.

FIGURE 3.

The imaginary part of the temperature field of a ring resonator in out-of-plane mode ($n =2$ ). The inset is the temperature distribution of circular cross-section.

Fig. 4 shows the comparison of thermoelastic damping obtained by FEM and the present model for different structure dimensions and natural frequencies. The critical parameters are $r_{0} = 10\,\,\mu \text{m}$ and $R_{0} = 200\,\,\mu \text{m}$ , $600~\mu \text{m}$ , and $1000~\mu \text{m}$ . As shown in Fig. 4, the results of the present model are in good agreement with those of FEM for all cases. Note that the present model with $q =10$ is more accurate than the case of $q =1$ . However, the difference between the two cases of $q = {1}$ and $q =10$ is negligibly small. Fig. 5 shows the relative errors between the results of FEM and the present model. As shown in the figure, the analytical results obtained using $q =10$ are closer to FEM than those using $q =1$ . For all cases, the maximum error of the present model is within 15% and 4% for $q = {1}$ and $q =10$ , respectively. Also shown in the figure, for the case of thin ring (i.e., a large ratio of $R_{0}/r_{0})$ , the present model exhibits very high accuracy. However, for the case of $R_{0}/r_{0} =20$ , the relative error is remarkable at high order mode.

FIGURE 4.

Comparison between the results obtained by FEM and the present model with constant $r_{0} = 10\,\,\mu \text{m}$ .

FIGURE 5.

The relative error of thermoelastic damping between FEM and the present model for different modes. (a) $q =1$ . (b) $q =10$ .

B. Convergence of the Thermoelastic Damping Equation

The convergence of the thermoelastic damping equation in this article is checked carefully. Fig. 6 shows thermoelastic damping calculated by the analytical model (28) for the cases of $q =1$ , 2, 5, 10, and 20. The resonator dimensions are $r_{0} = 10\,\,\mu \text{m}$ and $R_{0} = 100\,\,\mu \text{m}$ . As shown in the figure, the curve of $q = {1}$ almost coincides with that of $q =20$ at low order modes. The curve of $q = {2}$ is sufficient precision at mode $n =20$ but not in good agreement with that of $q =20$ at mode $n =30$ . Note that the curve of $q = {5}$ is in excellent agreement with that of $q =20$ even at very high order modes. Thus, the convergence can be achieved by using $q = {1}$ or $q = {5}$ for low or high order modes, respectively.

FIGURE 6.

Thermoelastic damping calculated by the present model for different $q$ .

C. Normalized Equation of Thermoelastic Damping

To study the characteristics of the model of thermoelastic damping derived in this article, we normalize (32) and transform it into a Lorentzian with normalized frequency $\eta = \omega _{n}\tau$ , given by $$\begin{equation*} \frac {Q^{-1}}{\Delta _{E}}=\frac {1}{1+{\left ({{1+\upsilon } }\right)} \mathord {\left /{ {\vphantom {{\left ({{1+\upsilon } }\right)} {n^{2}}}} }\right. } {n^{2}}}L\left ({\eta }\right)\tag{34}\end{equation*}$$ where $L$ ($\cdot$ ) is the Lorentzian, given by $$\begin{equation*} L\left ({\eta }\right)=\frac {\eta }{1+\eta ^{2}}\tag{35}\end{equation*}$$ For any value of $\eta$ , the boundaries of $Q^{-1}/\Delta _{E}$ can be expressed as $$\begin{equation*} \frac {1}{1+{\left ({{1+\upsilon } }\right)} \mathord {\left /{ {\vphantom {{\left ({{1+\upsilon } }\right)} 4}} }\right. } 4}L\left ({\eta }\right)\le \frac {Q^{-1}}{\Delta _{E}}\le L\left ({\eta }\right)\tag{36}\end{equation*}$$

Fig. 7 shows the normalized thermoelastic damping varying with $\eta$ for the cases of $n =2$ , 5, 10, and $\infty$ . The curves of $n = {2}$ and $\infty$ are the boundaries as expected in (36). As shown in Fig. 7, Debye peak exists when $\eta$ equals to 1. The inset shows the Debye peaks for different mode numbers $n$ . It illustrates that increasing mode number $n$ increases the value of Debye peak. Moreover, at any frequency, the larger the mode number $n$ , the larger the thermoelastic damping value. Note that the curve of $n =10$ is very close to that of $n = \infty$ . Therefore, for the high order mode of out-of-plane vibration, the thermoelastic damping value of a ring is approaching to that of a beam with circular cross-section in transverse vibration.

FIGURE 7.

Variation of normalized thermoelastic damping with the dimensionless variable $\eta = \omega _{n}\tau$ .

D. Effect of Geometry

In this section, the effect of structure dimensions of ring resonators on thermoelastic damping is studied for different modes and ratios $R_{0}/r_{0}$ . Fig. 8 shows the dependence of thermoelastic damping on mode number $n$ with constant $r_{0} = 10\,\,\mu \text{m}$ for different $R_{0}$ . As shown in Fig. 8, increasing the value of $R_{0}$ increases the mode number $n$ which is associated with the Debye peak.

FIGURE 8.

Dependence of thermoelastic damping on mode number $n$ with constant $r_{0} = 10\,\,\mu \text{m}$ for different $R_{0}$ .

For ring resonators with circular cross-section, $R_{0}$ and $r_{0}$ are two key geometric dimensions. Fig. 9 shows the dependence of thermoelastic damping on varying ratios $R_{0}/r_{0}$ with constant $R_{0} = 1000\,\,\mu \text{m}$ for the cases of $n = {2}$ to 6. As shown in the figure, for any case, a peak value exists when changing the ratio $R_{0}/r_{0}$ . On both sides of the peak, the curve changes monotonically with the increase of $R_{0}/r_{0}$ . Also shown in the figure, the ratio $R_{0}/r_{0}$ , corresponding to the peak value, increases with increasing mode number $n$ .

FIGURE 9.

Dependence of thermoelastic damping on varying ratios $R_{0}/r_{0}$ with constant $R_{0} = 1000\,\,\mu \text{m}$ for different $n$ .

Fig. 10 shows thermoelastic damping varying with the radius $r_{0}$ for constant $R_{0}/r_{0} =40$ . As shown in this figure, there is a peak value of thermoelastic damping which corresponds to a certain $r_{0}$ for each curve of mode number $n$ . As the increase of $n$ , the radius $r_{0}$ , corresponding to the peak value, is getting smaller.

FIGURE 10.

Dependence of thermoelastic damping on varying $r$ with constant $R/r =40$ .

E. Comparison Between Circular and Square Cross-Sections

The microring resonators with rectangular cross-section have already been widely used in the MEMS field. In this section, we compare the results of thermoelastic damping between circular and square cross-sections under the same area of cross-sections ($b = \pi ^{0.5}r_{0}$ ).

Fig. 11 shows the ratio of thermoelastic damping of square section ring to circular section ring. The value of $R_{0}/r_{0}$ is fixed to 40. As shown in this figure, the ratio varies with mode numbers and scales, and a minimum value of the ratio exists ($>0.9$ ) for all cases of $r_{0}$ . In the regime of high order mode, the ratio increases monotonically with the increasing of mode number $n$ . Also shown in the figure, the differences of thermoelastic damping between circular and square cross-sections are not larger than 10% under such conditions. As can be seen from the figure, thermoelastic damping in circular cross-section is different from those in square cross-section and it is difficult to find a regular relationship between the two kinds of cross-sections. This is due to the different cross-section shapes that affect the modal frequency and heat conduction. Therefore, it is necessary to establish a new thermoelastic damping model for microresonators when the shape of the cross-section and the vibration mode are changed.

FIGURE 11.

The ratio of thermoelastic damping of square section ring to circular section ring for the case of equal cross-sectional area.

SECTION V. Conclusion

We presented an analytical model for thermoelastic damping in microring resonators with circular cross-section under out-of-plane vibration. We applied the mode superposition method to solve the heat conduction equation for two-dimensional heat flow across the circular cross-section and obtained the temperature field of the ring. The validity of the present model is verified by comparing with the FEM simulation. It is found that the results of the present model are in good agreement with those of FEM. The convergence of the present model is carefully checked and the characteristics of the present model are studied by using the normalized equation. The geometry effects on thermoelastic damping had been investigated for different ratios $R_{0}/r_{0}$ and scales. By comparison, it is found that the differences of thermoelastic damping between circular and square cross-sections under the same area of cross-sections depend on the mode number and size scale.

Appendix A Solution of Temperature Field

Substituting (15) into (12), we obtain $$\begin{align*}&\hspace {-0.5pc}\sum \limits _{q=1}^\infty {c_{q} \left [{ {J_{1}^{\prime \prime }\left ({{\gamma _{q} r} }\right)+\frac {1}{r}J_{1} ^{\prime }\left ({{\gamma _{q} r} }\right)-\frac {1}{r^{2}}J_{1} \left ({{\gamma _{q} r} }\right)-\frac {i\omega _{n}}{\chi }J_{1} \left ({{\gamma _{q} r} }\right)} }\right]} \\&\qquad\qquad\qquad\qquad\quad \displaystyle {=-r\frac {i\omega _{n}}{\chi }\frac {\Delta _{E}}{\alpha }\frac {1-R_{0} \xi }{R_{0}^{2}}\frac {\partial ^{2}W}{\partial \varphi ^{2}} } \tag{A.1}\end{align*}$$ According to the characteristics of the Bessel function, we obtain $$\begin{align*}&\hspace {-0.5pc} J_{1}^{\prime \prime }\left ({{\gamma _{q} r} }\right)+\frac {1}{r}J_{1}^{\prime }\left ({{\gamma _{q} r} }\right)-\frac {1}{r^{2}}J_{1} \left ({{\gamma _{q} r} }\right)-\frac {i\omega _{n}}{\chi }J_{1} \left ({{\gamma _{q} r} }\right) \\&\qquad\qquad\qquad\qquad\qquad \displaystyle { =-\left ({{\gamma _{q}^{2} +\frac {i\omega _{n}}{\chi }} }\right)J_{1} \left ({{\gamma _{q} r} }\right)} \tag{A.2}\end{align*}$$ Then, (A.1) can be expressed as $$\begin{equation*} \sum \limits _{q=1}^\infty {c_{q} \left ({{\gamma _{q}^{2}\! \!+\!\!\frac {i\omega _{n} }{\chi }} }\right)J_{1} \left ({{\gamma _{q} r} }\right)} \!=\!r\frac {i\omega _{n}}{\chi }\frac {\Delta _{E}}{\alpha }\frac {1\!-\!R_{0} \xi }{R_{0}^{2}}\frac {\partial ^{2}W}{\partial \varphi ^{2}}\tag{A.3}\end{equation*}$$ Using (19), we transform the above equation into the form as (20). The integrations in (20) can be solved, given by $$\begin{align*} \int _{0}^{r_{0}} {rJ_{1}^{2}\left ({{\gamma _{q} r} }\right)dr}=&\left \|{ {J_{1} \left ({{\gamma _{q} r} }\right)} }\right \|^{2} \\=&\frac {r_{0}^{2}}{2}\left ({\! {\frac {d}{dr}J_{1} \left ({{a_{q}} }\right)} \!}\right)^{2}\!\!+\!\!\frac {1}{2}\left ({\! {r_{0}^{2}\! \!-\!\!\frac {1}{\gamma _{q}^{2} }} \!}\right)J_{1}^{2} \left ({{a_{q}} }\right) \\{}\tag{A.4}\\ \int _{0}^{r_{0}} {r^{2}J_{1} \left ({{\gamma _{p} r} }\right)dr}=&\frac {r_{0}^{2}}{\gamma _{q} }J_{2} \left ({{a_{q}} }\right)\tag{A.5}\end{align*}$$ Based on the boundary conditions (17), (A.4) and (A.5) can be expressed as $$\begin{align*} \int _{0}^{r_{0}} {rJ_{1}^{2}\left ({{\gamma _{q} r} }\right)dr}=&\frac {1}{2}\left ({{r_{0}^{2} -\frac {1}{\gamma _{q}^{2}}} }\right)J_{1}^{2} \left ({{a_{q}} }\right)\tag{A.6}\\ \int _{0}^{r_{0}} {r^{2}J_{1} \left ({{\gamma _{p} r} }\right)dr}=&\frac {r_{0}^{2}}{a_{q} \gamma _{q} }J_{1} \left ({{a_{q}} }\right)\tag{A.7}\end{align*}$$ Substituting (A.6) and (A.7) into (20), yields $$\begin{align*}&\hspace {-0.5pc}c_{q} \left ({{\gamma _{q}^{2}\! \!+\!\!\frac {i\omega _{n}}{\chi }} }\right)\frac {1}{2}\left ({{r_{0}^{2}\! \!-\!\!\frac {1}{\gamma _{q}^{2}}} }\right)J_{1} \left ({{a_{q}} }\right) \\&\qquad\qquad\qquad\qquad\quad \displaystyle {=\frac {i\omega _{n}}{\chi }\frac {\Delta _{E}}{\alpha }\frac {1\!\!-\!\!R_{0} \xi }{R_{0}^{2}}\frac {\partial ^{2}W}{\partial \varphi ^{2}}\frac {r_{0}^{2}}{a_{q} \gamma _{q}}} \tag{A.8}\end{align*}$$ By solving the above equation, $c_{q}$ can be obtained as (21).

Appendix B Energy Calculation

Using (8), (23) and thermal strain expression $\varepsilon _{\mathrm {thermal}} = \alpha \theta$ , the energy loss per cycle over the entire structure is $$\begin{align*} \Delta W=&-\pi \!\int \!\!\int \! \int _{V} {\hat {\sigma }_{\varphi } \alpha \cos \beta \sum \limits _{q=1}^\infty {\textrm {Im}\left ({{c_{q}} }\right)J_{1} \left ({{\gamma _{q} r} }\right)} dV} \\=&2\pi E\Delta _{E} \omega \chi r_{0} \frac {\left ({{1-R_{0} \xi } }\right)^{2}}{R_{0}^{4}} \\&\times \sum \limits _{q=1}^\infty {\frac {\gamma _{q}^{2}}{\left ({{\chi ^{2}\gamma _{q}^{4} +\omega _{n}^{2}} }\right)\left ({{a_{q}^{2} -1} }\right)J_{1} \left ({{a_{q}} }\right)}} \\&\hspace {-2pc}\cdot \int \!\!\int \!\!\int _{V} {r^{2}J_{1} \left ({{\gamma _{q} r} }\right)\cos ^{2}\beta \left ({{\frac {\partial ^{2}W}{\partial \varphi ^{2}}} }\right)^{2}\left ({{R_{0}\!\! +\!\!r\sin \beta } }\right)drd\beta d\varphi } \\{}\tag{B.1}\end{align*}$$ The integration in (B.1) can be solved, given by $$\begin{align*}&\hspace {-1.1pc} \!\!\int \!\!\int _{V} {r^{2}J_{1} \left ({{\gamma _{q} r} }\right)\cos ^{2}\beta \left ({{\frac {\partial ^{2}W}{\partial \varphi ^{2}}} }\right)^{2}\left ({{R_{0} +r\sin \beta } }\right)drd\beta d\varphi } \\=&\int _{0}^{2\pi } {\left ({{\frac {\partial ^{2}W}{\partial \varphi ^{2}}} }\right)^{2}d\varphi } \left [{\! {R_{0} \int _{0}^{r_{0}} {r^{2}J_{1} \left ({{\gamma _{q} r} }\right)dr} \!\cdot \! \int _{0}^{2\pi } {\!\!\!\cos ^{2}\beta d\beta }} }\right. \\&\left.{ {+\int _{0}^{r_{0}} {r^{3}J_{1} \left ({{\gamma _{q} r} }\right)dr} \int _{0}^{2\pi } {\cos ^{2}\beta \sin \beta d\beta }} \!}\right] \\=&R_{0} \int _{0}^{r_{0}} {r^{2}J_{1} \left ({{\gamma _{q} r} }\right)dr} \!\cdot \!\int _{0}^{2\pi } {\!\!\cos ^{2}\beta d\beta } \cdot \int _{0}^{2\pi } {\left ({\! {\frac {\partial ^{2}W}{\partial \varphi ^{2}}} }\right)^{2}d\varphi } \\=&\frac {r_{0} R_{0} \pi ^{2}W_{0}^{2} n^{4}}{\gamma _{q}^{2} }J_{1} \left ({{a_{q}} }\right)\tag{B.2}\end{align*}$$ Thus, using (3) and (B.2), $\Delta W$ can be written as (26).

Substituting (7) and (8) into (25), the maximum energy stored per cycle is given by $$\begin{align*} W_{\textrm {stored}}=&\frac {E}{2}\frac {\left ({{1\!-\!R_{0} \zeta } }\right)^{2}}{R_{0}^{4}} \\&\cdot \int \!\!\!\int {r^{3}\cos ^{2}\beta \left ({{R_{0} \!+\!r\sin \beta } }\right)drd\beta }\! \cdot \! \!\int \!\! {\left ({{\frac {\partial ^{2}W}{\!\!\partial \varphi ^{2}}} }\right)^{2}d\varphi } \\&+\!\frac {E}{4\left ({{1\!\!+\!\!\upsilon } }\right)}\frac {\left ({{1\!-\!n^{2}R_{0} \zeta } }\right)^{2}}{R_{0}^{4}} \\&\cdot \int \!\!\!\int {r^{3}\cos ^{2}\beta \left ({{R_{0} \!+\!r\sin \beta } }\right)drd\beta } \cdot \!\int \!\! {\left ({{\frac {\partial W}{\partial \varphi }} }\right)^{2}d\varphi } \\&+\!\frac {E}{4\left ({{1\!+\!\upsilon } }\right)}\frac {\left ({{1\!-\!n^{2}R_{0} \zeta } }\right)^{2}}{R_{0}^{4}} \\&\cdot \int \!\!\!\int {r^{3}\sin ^{2}\beta \left ({{R_{0} +r\sin \beta } }\right)drd\beta } \cdot \!\int \!\! {\left ({{\frac {\partial W}{\partial \varphi }} }\right)^{2}d\varphi } \\=&\frac {E\pi ^{2}r_{0}^{4}}{8R_{0}^{3} }n^{4}W_{0}^{2} \left [{ {\left ({{1-R\zeta } }\right)^{2}+\frac {\left ({{1-n^{2}R\zeta } }\right)^{2}}{\left ({{1+\upsilon } }\right)n^{2}}} }\right]\tag{B.3}\end{align*}$$ Finally, using (3), $W_{\mathrm {stored}}$ can be expressed as (27).