process compensated micromechanical resonators - Semantic Scholar
PROCESS COMPENSATED MICROMECHANICAL RESONATORS Gavin K. Ho, John K. C. Perng, and Farrokh Ayazi School of Electrical and Computer Engineering, Georgia Institute of Technology [email protected] (404)944-6009, [email protected] (404)894-9496
ABSTRACT Manufacturability and yield are the major challenges prior to adoption of micromechanical resonators as frequency references. In this paper, a design for manufacturability (DFM) technique to achieve absolute frequency accuracy is presented. Non-idealities of a deep reactive ion etching process are examined and determined to be random. The variations in resonator geometry are assumed to be locally systematic and are represented as a process bias. The effect of process bias on resonator center frequency is modeled and the procedure for optimizing for zero sensitivity is explained. Process bias on a 10MHz optimized design was replicated with electron-beam lithography and supporting data demonstrating DFM is reported.
INTRODUCTION
123-4
Micromechanical resonators [1-7] are strong candidates to complement quartz technology in frequency references. Frequency accuracy, the focus of this work, is a key technological hurdle that must be addressed. Deviations in center frequency can be attributed to material properties and geometry. To address this issue, center frequency trimming by iterative laser ablation [5] and by selective deposition [ 6 ] have been proposed. Individuallyprogrammed synthesizers utilizing fractional-N PLLs have also been demonstrated [ 7 ]. However, a design for manufacturability (DFM) technique employing batch compensation addresses the issue at its roots. The material properties must be consistent and repeatable to enable DFM. Single-crystal silicon (SCS) is the choice material since it is the best controlled and most characterized. Its ideal crystalline nature also has potential for very high Q and minimal aging. In reference, quartz crystal units typically have absolute frequency tolerances up to ±20ppm. Hence, the applicability of micromechanical resonators is contingent on meeting similar performance metrics. In this article, we demonstrate process compensation (PC) for micromechanical resonators, such that its center frequency is robust to dimensional variations caused by lithography and micromachining. First, the non-idealities of silicon bulk micromachining (Figure 1) and the effects on resonator characteristics are analyzed. Next, we present a very simple lumped model of the I-shaped bulk acoustic resonator (IBAR) [1]. See Figure 2. PC techniques for IBARs are discussed and validated with finite element simulations. Finally, the design of experiments and verification of the process compensation scheme are presented.
1-4244-0951-9/07/$25.00 ©2007 IEEE.
183
Striations Scalloping ≈ 50nm Trench Profile Variation (Bowing)
Figure 1: SEM of a typical 10µm DRIE trench in SOI
SILICON MICROMACHINING AND PROCESS COMPENSATION High performance capacitive resonators must have large transduction area. This requirement calls for deep reactive ion etching (DRIE) of thick resonators whether trenches are used for transduction or used in defining sacrificial gaps. In trench etching, non-idealities such as scalloping, striations, bowing, and footing exist (Figure 1). The first three phenomena are generally random and are typically within 50nm. In optimized processes, footing can be avoided and random non-idealities are further reduced. Lithography and pattern transfer account for the majority of micromachining variations. They limit the dimensional accuracy which generally compromises center frequency accuracy. Although these variations are temporally random, they are spatially systematic. Electrode
Anchor
IBAR
Anchor
Figure 2: SEM of a DFM-optimized 10MHz IBAR
MEMS 2007, Kobe, Japan, 21-25 January 2007.
Process compensation of center frequency is conceptually straightforward. Since critical dimension (CD) variations lead to deviations from the modal stiffness k and mass m, the frequency k fn = 1 (1) m 2π may also deviate from the ideal design value. A tolerant design ensures that variations in k are proportional to variations in m, thus maintaining a constant fn.
RESONATOR DESIGN & MODELING
Flange wr Hybrid Mode
Lr Rod
Le Lf wf Lr wr t
uB=yB+yA
mB
uA=yA
cB
kB
wf Lf
uA
mA cA
kA
wr/2
Lr/2
Figure 4: Lumped 2DOF ¼-model of the IBAR
In this section, a simple lumped mechanical model of the IBAR is presented. Three categories of IBARs: the Qenhanced (QE), compliance-enhanced (CE), and semicompliant high-Q (CQ), are discussed and contrasted. A 10MHz optimized CQ IBAR design is shown and its frequency-sensitivity is modeled. A simple 2DOF system is presented to analyze the resonator (Figure 4). The model comprises of an extensional component and a flexural component. Damping in the model is assumed to be intrinsic (i.e. by material losses and not air damping or anchor loss) and proportional to velocity. The coupled 2DOF system consists of the lumped parameters mi, ki, and ci for i=A,B which are derived from the parameters of the constituent components. The local coordinates yi model the local displacements of each component and ui represent the global displacements. The three categories of IBARs can be described using Figure 4. The eigenvalue problem describing the 2DOF system yields eigenvectors (i.e. mode shapes) that are dependent on the lumped parameters. Clearly, the system can have two modes. The mode of interest is the lower frequency mode in which the mass elements are in phase. For the sake of brevity, the three classifications are described without mathematical detail. For the case in which kB is very large, the mode is predominantly in extension of the rod (i.e. yB
ABSTRACT Manufacturability and yield are the major challenges prior to adoption of micromechanical resonators as frequency references. In this paper, a design for manufacturability (DFM) technique to achieve absolute frequency accuracy is presented. Non-idealities of a deep reactive ion etching process are examined and determined to be random. The variations in resonator geometry are assumed to be locally systematic and are represented as a process bias. The effect of process bias on resonator center frequency is modeled and the procedure for optimizing for zero sensitivity is explained. Process bias on a 10MHz optimized design was replicated with electron-beam lithography and supporting data demonstrating DFM is reported.
INTRODUCTION
123-4
Micromechanical resonators [1-7] are strong candidates to complement quartz technology in frequency references. Frequency accuracy, the focus of this work, is a key technological hurdle that must be addressed. Deviations in center frequency can be attributed to material properties and geometry. To address this issue, center frequency trimming by iterative laser ablation [5] and by selective deposition [ 6 ] have been proposed. Individuallyprogrammed synthesizers utilizing fractional-N PLLs have also been demonstrated [ 7 ]. However, a design for manufacturability (DFM) technique employing batch compensation addresses the issue at its roots. The material properties must be consistent and repeatable to enable DFM. Single-crystal silicon (SCS) is the choice material since it is the best controlled and most characterized. Its ideal crystalline nature also has potential for very high Q and minimal aging. In reference, quartz crystal units typically have absolute frequency tolerances up to ±20ppm. Hence, the applicability of micromechanical resonators is contingent on meeting similar performance metrics. In this article, we demonstrate process compensation (PC) for micromechanical resonators, such that its center frequency is robust to dimensional variations caused by lithography and micromachining. First, the non-idealities of silicon bulk micromachining (Figure 1) and the effects on resonator characteristics are analyzed. Next, we present a very simple lumped model of the I-shaped bulk acoustic resonator (IBAR) [1]. See Figure 2. PC techniques for IBARs are discussed and validated with finite element simulations. Finally, the design of experiments and verification of the process compensation scheme are presented.
1-4244-0951-9/07/$25.00 ©2007 IEEE.
183
Striations Scalloping ≈ 50nm Trench Profile Variation (Bowing)
Figure 1: SEM of a typical 10µm DRIE trench in SOI
SILICON MICROMACHINING AND PROCESS COMPENSATION High performance capacitive resonators must have large transduction area. This requirement calls for deep reactive ion etching (DRIE) of thick resonators whether trenches are used for transduction or used in defining sacrificial gaps. In trench etching, non-idealities such as scalloping, striations, bowing, and footing exist (Figure 1). The first three phenomena are generally random and are typically within 50nm. In optimized processes, footing can be avoided and random non-idealities are further reduced. Lithography and pattern transfer account for the majority of micromachining variations. They limit the dimensional accuracy which generally compromises center frequency accuracy. Although these variations are temporally random, they are spatially systematic. Electrode
Anchor
IBAR
Anchor
Figure 2: SEM of a DFM-optimized 10MHz IBAR
MEMS 2007, Kobe, Japan, 21-25 January 2007.
Process compensation of center frequency is conceptually straightforward. Since critical dimension (CD) variations lead to deviations from the modal stiffness k and mass m, the frequency k fn = 1 (1) m 2π may also deviate from the ideal design value. A tolerant design ensures that variations in k are proportional to variations in m, thus maintaining a constant fn.
RESONATOR DESIGN & MODELING
Flange wr Hybrid Mode
Lr Rod
Le Lf wf Lr wr t
uB=yB+yA
mB
uA=yA
cB
kB
wf Lf
uA
mA cA
kA
wr/2
Lr/2
Figure 4: Lumped 2DOF ¼-model of the IBAR
In this section, a simple lumped mechanical model of the IBAR is presented. Three categories of IBARs: the Qenhanced (QE), compliance-enhanced (CE), and semicompliant high-Q (CQ), are discussed and contrasted. A 10MHz optimized CQ IBAR design is shown and its frequency-sensitivity is modeled. A simple 2DOF system is presented to analyze the resonator (Figure 4). The model comprises of an extensional component and a flexural component. Damping in the model is assumed to be intrinsic (i.e. by material losses and not air damping or anchor loss) and proportional to velocity. The coupled 2DOF system consists of the lumped parameters mi, ki, and ci for i=A,B which are derived from the parameters of the constituent components. The local coordinates yi model the local displacements of each component and ui represent the global displacements. The three categories of IBARs can be described using Figure 4. The eigenvalue problem describing the 2DOF system yields eigenvectors (i.e. mode shapes) that are dependent on the lumped parameters. Clearly, the system can have two modes. The mode of interest is the lower frequency mode in which the mass elements are in phase. For the sake of brevity, the three classifications are described without mathematical detail. For the case in which kB is very large, the mode is predominantly in extension of the rod (i.e. yB
