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Thermo-acoustic engineering of silicon microresonators via evanescent waves R. Tabrizian and F. Ayazi Citation: Applied Physics Letters 106, 263504 (2015); doi: 10.1063/1.4923056 View online: http://dx.doi.org/10.1063/1.4923056 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/106/26?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dispersion engineering of high-Q silicon microresonators via thermal oxidation Appl. Phys. Lett. 105, 031112 (2014); 10.1063/1.4890986 Investigation on the optimized design of alternate-hole-defect for 2D phononic crystal based silicon microresonators J. Appl. Phys. 112, 024910 (2012); 10.1063/1.4740085 Acceleration insensitive encapsulated silicon microresonator Appl. Phys. Lett. 93, 234103 (2008); 10.1063/1.3036536 Amorphous silicon electrostatic microresonators with high quality factors Appl. Phys. Lett. 84, 622 (2004); 10.1063/1.1644319 External acoustic load on the performance of a travelling-wave thermoacoustic engine AIP Conf. Proc. 613, 807 (2002); 10.1063/1.1472098
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APPLIED PHYSICS LETTERS 106, 263504 (2015)
Thermo-acoustic engineering of silicon microresonators via evanescent waves R. Tabrizian1,a) and F. Ayazi2 1
Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor, Michigan 48109, USA 2 School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30308, USA
(Received 14 November 2014; accepted 16 June 2015; published online 29 June 2015) A temperature-compensated silicon micromechanical resonator with a quadratic temperature characteristic is realized by acoustic engineering. Energy-trapped resonance modes are synthesized by acoustic coupling of propagating and evanescent extensional waves in waveguides with rectangular cross section. Highly different temperature sensitivity of propagating and evanescent waves is used to engineer the linear temperature coefficient of frequency. The resulted quadratic temperature characteristic has a well-defined turn-over temperature that can be tailored by relative energy distribution between propagating and evanescent acoustic fields. A 76 MHz prototype is implemented in single crystal silicon. Two high quality factor and closely spaced resonance modes, created from efficient energy trapping of extensional waves, are excited through thin aluminum nitride film. Having different evanescent wave constituents and energy distribution across the device, these modes show different turn over points of 67 C and 87 C for their quadratic temperature characterC 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4923056] istic. V
After successful demonstration of oscillators with sub part per million instabilities,1,2 microelectromechanical systems (MEMS) are now aiming at the realization of integrated frequency references with sub part per billion instability. Such a significant improvement in stability would require continuous ovenization of the resonator at a constant temperature to effectively blind the device from ambient temperature variations that induce frequency fluctuations.3–5 For this purpose, it is desirable for the resonator to have a turn-over, i.e., a local extremum in the temperature characteristic of frequency, at a point slightly above the operating temperature range of the device (e.g., 55 C to 85 C for consumer applications). This paper presents a technique to create a turn-over in the temperature characteristic of silicon microresonators at a well-defined temperature point. To compensate the temperature coefficient of frequency (TCF) of MEMS resonators, a large variety of devicelevel6–9 and system-level10 compensation techniques have been proposed. However, these techniques usually impose excessive power consumption or manufacturing complexities. We demonstrate a single crystal silicon acoustic resonator implemented based on energy trapping of in-plane extensional wave in waveguides with rectangular cross section. In a similar technique to thickness-mode quartz and thin-film bulk acoustic wave resonators,11,12 acoustic energy trapping in this device is done through coupling propagating waves in the central section of the structure into evanescent waves at flanks. This is unlike conventional in-plane silicon bulk acoustic resonators (SiBAR) where the cavity is defined entirely through trenches that bound the geometry.13 Dispersive characteristic of laterally propagating waves in waveguides with rectangular cross section is used for a)
Electronic mail: [email protected].
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acoustic engineering of the device. Fig. 1 shows the dispersion behavior and mode shape of several eigen-waves in a rectangular waveguide aligned to h100i crystallographic direction of a (100) silicon substrate, for both evanescent and propagating waves. The dispersion relation between the frequency (f0) and wavenumber (KX) of laterally propagating waves is defined by Christoffel equation14 considering the stress-free peripheral faces of the waveguide: 2
3 @ 0 0 7 6 6 @x 7 6 7 @ 6 7 0 7 6 0 6 7 @y 6 72 3 ið KX xxtÞ 6 @ 7 6 0 7 Ux ðy; zÞe 0 6 7 6 @z 7 6 76 Uy ðy; zÞeiðKX xxtÞ 7 4 5 6 7 6 0 @ @ 7 ið KX xxtÞ 6 7 U y; z e Þ zð @z @y 7 6 6 7 6 @ @ 7 6 7 6 @z 0 @x 7 6 7 4 @ @ 5 0 @z @x 2 3 Ux ðy; zÞeiðKX xxtÞ 6 7 iðKX xxtÞ 7 ¼ qx2 6 4 Uy ðy; zÞe 5; ð Þ Uz ðy; zÞei KX xxt rmn ¼ 0 at waveguide periphery;
m; n 2 f x; y; zg:
(1)
Here, U ¼ (Ux, Uy, Uz) is the polarization vector. KX, x, and q are wave number, angular frequency, and waveguide mass density, respectively, and r ¼ [rmn] is the stress tensor. Such a relation results in both propagating and evanescent wave solutions with real and imaginary wavenumbers,
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C 2015 AIP Publishing LLC V
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263504-2
R. Tabrizian and F. Ayazi
Appl. Phys. Lett. 106, 263504 (2015)
FIG. 1. (Left) Dispersion behavior of eigen-waves in a single crystal silicon rectangular waveguide with a width of 50 lm and thickness of 20 lm. Inset shows the waveguide with characteristic cross-sectional dimensions of W and H, which is aligned to h100i direction of a (100) silicon plate. (Right) Deformation mode shape of different eigen-waves for arbitrary propagating and evanescent solutions, as well as their cross-sectional polarization. Several eigen-waves with symmetric (Si) and anti-symmetric (Ai) polarizations across the central axis of the waveguide are demonstrated.
respectively. While for the propagating solutions, the acoustic energy is uniform across the length of the waveguide, the energy of evanescent solutions decays exponentially along the waveguide axis (X-axis in Fig. 1). The dispersion behavior of a rectangular waveguide can be tailored by changing it cross-sectional dimensions (i.e., W and H). This is shown in Fig. 1 for the first width-extensional branch (S1) where slight variations of the width (W 6 DW) results in frequency shift of the branch. Such a geometry dependence can be used to couple propagating and evanescent waves with different wavenumbers, at a specific frequency. This can be done by cascading waveguides with different dimensions, along their axis. In such a structure, a coupled solution may exist if displacement and strain continuity hold at transition interface between consecutive waveguides.15 The thermo-acoustic properties of such a coupled solution can then be defined as a superposition of the properties of constituent waveguides, weighted by their respective contribution in trapping acoustic energy. Each (f0, KX) solution on eigen-wave dispersion branches is a result of a unique interaction between transverse and longitudinal waves launched into the waveguide in a specific incident angle, while reflected and converted at stress-free boundaries. Therefore, considering anisotropic temperature characteristic of longitudinal and transverse waves in single crystal silicon, each solution has a specific temperature characteristic. Fig. 2 shows the simulated TCF1,2 and turn-over temperature (TTurn-Over) of the S1 branch for N-doped silicon substrates with different resistivities. When properly oriented with respect to crystallographic axes of doped single crystal silicon, evanescent waves can provide a highly different TCF compared to propagating waves. Furthermore, as opposed to propagating waves, evanescent waves show a large variation in temperature characteristics for different KX. This is due to effective contribution of both shear and extensional wave components in the formation of evanescent solutions and highly different thermoacoustic properties of these constituents.17 In this work, evanescent waves with highly positive TCF1 are used to design a temperature-compensated acoustic resonator with well-defined turn-over temperature. Several waveguides with different width, and hence different
dispersion behavior, are cascaded in series. In such a structure, a solution—called a synthesized resonance mode hereafter—may exist arising from acoustic coupling of propagating and evanescent waves in constituent waveguides. When such a cascaded group of waveguides is flanked with sufficiently long evanescent-supporting sections, the energy will be confined mainly in the central waveguide leaving negligible concentration at flank terminations. Therefore, the anchoring of the cascaded group will not perturb vibration dynamics of the structure or diminish the generality of the waveguide assumption of the constituents— even though their lengths are finite.18 Fig. 3 compares the mode shape and simulated TTurn-Over of an ideal (i.e., infinitely long) SiBAR with two different coupled waveguides, all aligned to h100i axes of 0.0011 X . cm single crystal silicon. The displacement mode shape of the coupled waveguides can be defined in different sections as 8 L1 L1 > > <x< > < sinð K1;1 xÞ 2 2 Waveguide 1 : > L L L 1 > > 6sin K1;1 1 eiK2;1 ðx7 21 Þ < jxj; : 2 2 and 8 L2 L2 > > <x< > < cosðK1;2 xÞ 2 2 Waveguide 2 : > L2 iK2;2 ðx7L2 Þ L2 > > 2 e < jxj: : cos K1;2 2 2 (2) In order to calculate the contribution of each eigen-wave in temperature characteristic of the vibration mode, acoustically coupled system can be simplified to mechanically coupled harmonic oscillators with lumped equivalent mass and spring defined at the interface (i.e., X ¼ L1,2).19 Therefore, TCFi (i ¼ 1, 2) of the coupled vibration mode (TCFi, C) can be defined as TCFi;C ¼
meq;1 TCFi;1 þ meq;2 TCFi;2 meq;1 þ meq;2
(3)
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263504-3
R. Tabrizian and F. Ayazi
Appl. Phys. Lett. 106, 263504 (2015)
FIG. 2. (Left) The simulated first order TCF (TCF1) for S1, S2, and A2 branches and (Middle) second order TCF (TCF2) and (Right) turn-over temperature (TTurn-Over) of propagating and evanescent waves for S1 eigenwave branch of the rectangular waveguide aligned to direction of N-doped (100) silicon substrates with different resistivities. The material properties used in the simulations are taken from Ref. 16.
coupled waveguides 1 and 2 (i.e., TTurn-Over,i, i ¼ 1, 2) can be approximated from the superposition of that of the constituents, weighted by their effective contribution in vibration mode
where TCFi,j and meq,j (j ¼ 1, 2) are the TCFi and equivalent mass of the constituent eigen-waves at waveguide transitional face and can be calculated using the displacement mode shape.20,21 Having TCFi,C in (3), the TTurn-Over of the
TTurnOver;1
and
TTurnOver;2
L1 L1 1 cos K1;1 sin K1;1 2 2 TTurnOver ðK1;1 Þ þ TTurnOver ðK2;1 Þ K1;1 K2;1 ; L1 L1 1 cos K1;1 sin K1;1 2 2 þ K1;1 K2;1 L2 L2 sin K1;2 cos K1;2 2 2 TTurnOver ðK1;2 Þ þ TTurnOver ðK2;2 Þ K1;2 K2;2 : L2 L2 sin K1;2 cos K1;2 2 2 þ K1;2 K2;2
Fig. 4 shows the SEM image of the silicon resonator designed to serve as a test vehicle to reproduce coupled waveguides coupled modes. The device is composed of a central section with a width of W0 ¼ 50 lm that is surrounded by flanks with gradually changing widths. Flanks are designed to facilitate excitation of two synthesized modes with closely similar mode shapes to the trapped modes in coupled waveguides 1 and 2 in Fig. 3. The structure is finally clamped to the substrate through a tether with a width chosen to support no solution, either propagating or evanescent, at resonance frequency. This is done to reduce extension of acoustic field into the substrate, therefore increasing the quality factor (Q). A thin aluminum nitride (AlN) film is used to provide electromechanical transduction to the silicon microstructure. Wide tethers facilitate integration of several electrically isolated electrodes on the top surface of AlN, which in turn facilitates independent excitation of different synthesized modes.18
(4)
Fig. 5 shows the mode shape of the synthesized modes in designed resonator in comparison with a SiBAR of the same length, anchored through narrow tethers at nodal points of the mode shape. The electrode configuration of AlN transducer facilitate efficient excitation of both synthesized modes. Fig. 6 shows the frequency response of the device providing two modes with high Q. The larger Q of the mode 2 is a result of its smaller energy concentration in the flanks, and hence lower energy leakage, which can be observed in the mode shapes of Fig. 5. Fig. 7 shows the COMSOL finite element simulated as well as measured temperature characteristics of the two modes of the device, in comparison with a regular SiBAR implemented on the same substrate. The devices are all implemented on a silicon substrate with a resistivity of 0.001(6%10) X . cm. Temperature characterizations are
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263504-4
R. Tabrizian and F. Ayazi
Appl. Phys. Lett. 106, 263504 (2015)
FIG. 3. The mode shape (top view) and simulated TTurn-Over of two coupled waveguides in comparison with an ideal SiBAR. Proper boundary conditions are considered to emulate infinite length of waveguides in flanks. Also a transduction stack that is composed of 500 nm AlN sandwiched between 50 nm molybdenum electrodes and is separated from the silicon substrate through a 200 nm silicon dioxide layer is considered in thermoacoustic simulations to guarantee the validity of comparison with experimental results.
FIG. 4. The SEM image of the device with AlN transducer. The inset shows a h100i-aligned SiBAR implemented on the same substrate to provide a comparison reference for temperature characteristics of resonance frequency.
done in an environmental chamber with a temperature accuracy of 60.5 C and under a 1 mTorr vacuum. The slight discrepancy between simulation and experimental results is due to the difference between doping concentration of silicon substrate compared to what assumed for material properties in finite element simulations as well as the variations in substrate thickness (20 lm 6 1 lm, as indicated by the substrate-providing vendor). Highly different temperature characteristic of the synthesized modes is a result of their different constituent propagating and evanescent waves. Specifically, in the synthesized mode 2, effective contribution of the evanescent wave with large wavenumber—and hence
highly positive TCF1—has resulted in a TTurn-Over exceeding 85 C, which facilitates implementation of an oven-controlled stable frequency reference.22 The larger turn-over temperatures of the synthesized modes in implemented device compared to designed values is due to slightly lower resistivity of the actual silicon substrate compared to the simulation assumptions. Furthermore, the TTurn-Over can be tailored to a well-defined value by proper acoustic engineering, which includes design for evanescent wavenumber and relative energy distribution between propagating and evanescent fields. Fig. 8 shows the temperature characteristic of the Q for the modes/devices of Fig. 7, over the temperature range of 40 C to 80 C. The temperature induced degradation in the Q of the vibration modes in acoustically engineered cavity and SiBAR is a result of an increase in dissipative thermo-acoustic interactions at higher temperatures.23 Slight irregularities in the Q characteristic trend for the modes in acoustically engineered cavity can be attributed to the change of evanescent wavenumber in the coupled solutions with temperature variations. Such a change results in different energy distribution profile in the device flanks and induces unexpected variations in acoustic energy leakage into the substrate. In conclusion a thermo-acoustic engineering technique, based on acoustic coupling of propagating and evanescent waves, is introduced. The temperature characteristics of propagating and evanescent waves in the width-extensional eigenwave branch is extracted through numerical simulations. The application of such characteristics in the turn-over temperature of coupled waveguides is presented through analytical derivation and experimentally verified through device implementation and characterization. TTurn-Overs of 67 C and 87 C are measured for two synthesized modes in a single device. Such a
FIG. 5. Displacement mode shapes (top view) of synthesized modes and a h100i-aligned SiBAR that is anchored through narrow tethers.
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263504-5
R. Tabrizian and F. Ayazi
Appl. Phys. Lett. 106, 263504 (2015)
device can serve as the frequency reference for implementation of highly stable ovenized oscillators. This work was supported by DARPA TIMU program through SSC Pacific Contract No. N66001-11-C-4176. 1
FIG. 6. Measured frequency response of the acoustically engineered device having two synthesized width-extensional modes with small frequency split.
FIG. 7. Simulated (top) and measured (bottom) temperature characteristic of the synthesized modes in the acoustically engineered waveguide in comparison with a h100i-aligned SiBAR implemented on the same substrate.
FIG. 8. Temperature characteristic of the Q for the two modes of the acoustically engineered cavity as well as the h100i-aligned SiBAR.
M. H. Perrott, J. C. Salvia, F. S. Lee, A. Partridge, S. Mukherjee, C. Arft, and F. Assaderaghi, “A temperature-to-digital converter for a MEMSbased programmable oscillator with frequency stability and integrated jitter,” IEEE J. Solid-State Circuits 48(1), 276–291 (2013). 2 R. Tabrizian, M. Pardo, and F. Ayazi, “A 27 MHz temperature compensated MEMS oscillator with sub-ppm instability,” in IEEE 25th International Conference on Micro Electro Mechanical Systems (MEMS) (2012), pp. 23–26. 3 J. C. Salvia, R. Melamud, S. A. Chandorkar, S. F. Lord, and T. W. Kenny, “Real-time temperature compensation of MEMS oscillators using an integrated micro-oven and a phase-locked loop,” J. Microelectromech. Syst. 19(1), 192–201 (2010). 4 Z. Wu, A. Peczalski, and M. Rais-Zadeh, “Low-power ovenization of fused silica resonators for temperature-stable oscillators,” in IEEE International Frequency Control Symposium (FCS) (2014), pp. 1–5. 5 A. Tazzoli, G. Piazza, and M. Rinaldi, “Ultra-high-frequency temperaturecompensated oscillators based on ovenized AlN contour-mode MEMS resonators,” in IEEE International Frequency Control Symposium (FCS) (2012), pp. 1–5. 6 R. Melamud, B. Kim, S. A. Chandorkar, M. A. Hopcroft, M. Agarwal, C. M. Jha, and T. W. Kenny, “Temperature-compensated high-stability silicon resonators,” Appl. Phys. Lett. 90(24), 244107 (2007). 7 R. Tabrizian, G. Casinovi, and F. Ayazi, “Temperature-stable silicon oxide (SilOx) micromechanical resonators,” IEEE Trans. Electron Devices 60(8), 2656–2663 (2013). 8 A. K. Samarao and F. Ayazi, “Temperature compensation of silicon resonators via degenerate doping,” IEEE Trans. Electron Devices 59(1), 87–93 (2012). 9 E. J. Ng, Y. Yang, Y. Chen, and T. W. Kenny, “An etch hole-free process for temperature-compensated, high Q, encapsulated resonators,” in SolidState Sensors, Actuators, and Microsystems Workshop, Hilton Head (2014), pp. 99–100. 10 A. Tazzoli, M. Rinaldi, and G. Piazza, “Ovenized high frequency oscillators based on aluminum nitride contour-mode MEMS resonators,” Tech. Dig. -Int. Electron Devices Meet. 2011, 20–22. 11 S. Galliou, M. Goryachev, R. Bourquin, P. Abbe, J. P. Aubry, and M. E. Tobar, “Extremely low loss phonon-trapping cryogenic acoustic cavities for future physical experiments,” Sci. Rep. 3(2132), 1–6 (2013). 12 K. Kokkonen, J. Meltaus, T. Pensala, and M. Kaivola, “Characterization of energy trapping in a bulk acoustic wave resonator,” Appl. Phys. Lett. 97(23), 233507 (2010). 13 S. Pourkamali, G. K. Ho, and F. Ayazi, “Vertical capacitive SiBARs,” in 18th IEEE International Conference on Micro Electro Mechanical Systems (MEMS) (2005), pp. 211–214. 14 J. F. Rosenbaum, Bulk Acoustic Wave Theory and Devices (Artech House, Boston, 1988), Vol. 147. 15 J. Kaitila, M. Ylilammi, J. Ella, and R. Aigner, “Spurious resonance free bulk acoustic wave resonators,” IEEE Ultrason. Symp. 2003, 84–87. 16 E. J. Ng, V. A. Hong, Y. Yang, C. H. Ahn, C. L. Everhart, and T. W. Kenny, “Temperature dependence of the elastic constants of doped silicon,” IEEE J. Microelectromech. Syst. 24, 730–741 (2014). 17 T. Pensala, A. Jaakkola, M. Prunnila, and J. Dekker, “Temperature compensation of silicon MEMS resonators by heavy doping,” IEEE Ultrason. Symp. 2011, 1952–1955. 18 R. Tabrizian and F. Ayazi, “Acoustically-engineered multi-port AlN-onsilicon resonators for accurate temperature sensing,” Tech. Dig. -Int. Electron Devices Meet. 2013, 1811–1814. 19 M. U. Demirci and C. C. Nguyen, “Mechanically corner-coupled square microresonator array for reduced series motional resistance,” J. Microelectromech. Syst. 15(6), 1419–1436 (2006). 20 C. C. Nguyen, “Frequency-selective MEMS for miniaturized low-power communication devices,” IEEE Trans. Microwave Theory Tech. 47(8), 1486–1503 (1999). 21 R. A. Johnson, Mechanical Filters in Electronics (Wiley, New York, 1983). 22 R. Tabrizian, A. Norouz-pour Shirazi, and F. Ayazi, “Temperature-compensated tetherless bulk acoustic phonon trap for self-ovenized oscillators,” in Solid-State Sensors, Actuators, and Microsystems Workshop (2014), pp. 109–110. 23 F. Ayazi, L. Sorenson, and R. Tabrizian, Proc. SPIE 8031, 803119 (2011).
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This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 143.215.27.39 On: Fri, 03 Jul 2015 19:38:21
APPLIED PHYSICS LETTERS 106, 263504 (2015)
Thermo-acoustic engineering of silicon microresonators via evanescent waves R. Tabrizian1,a) and F. Ayazi2 1
Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor, Michigan 48109, USA 2 School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30308, USA
(Received 14 November 2014; accepted 16 June 2015; published online 29 June 2015) A temperature-compensated silicon micromechanical resonator with a quadratic temperature characteristic is realized by acoustic engineering. Energy-trapped resonance modes are synthesized by acoustic coupling of propagating and evanescent extensional waves in waveguides with rectangular cross section. Highly different temperature sensitivity of propagating and evanescent waves is used to engineer the linear temperature coefficient of frequency. The resulted quadratic temperature characteristic has a well-defined turn-over temperature that can be tailored by relative energy distribution between propagating and evanescent acoustic fields. A 76 MHz prototype is implemented in single crystal silicon. Two high quality factor and closely spaced resonance modes, created from efficient energy trapping of extensional waves, are excited through thin aluminum nitride film. Having different evanescent wave constituents and energy distribution across the device, these modes show different turn over points of 67 C and 87 C for their quadratic temperature characterC 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4923056] istic. V
After successful demonstration of oscillators with sub part per million instabilities,1,2 microelectromechanical systems (MEMS) are now aiming at the realization of integrated frequency references with sub part per billion instability. Such a significant improvement in stability would require continuous ovenization of the resonator at a constant temperature to effectively blind the device from ambient temperature variations that induce frequency fluctuations.3–5 For this purpose, it is desirable for the resonator to have a turn-over, i.e., a local extremum in the temperature characteristic of frequency, at a point slightly above the operating temperature range of the device (e.g., 55 C to 85 C for consumer applications). This paper presents a technique to create a turn-over in the temperature characteristic of silicon microresonators at a well-defined temperature point. To compensate the temperature coefficient of frequency (TCF) of MEMS resonators, a large variety of devicelevel6–9 and system-level10 compensation techniques have been proposed. However, these techniques usually impose excessive power consumption or manufacturing complexities. We demonstrate a single crystal silicon acoustic resonator implemented based on energy trapping of in-plane extensional wave in waveguides with rectangular cross section. In a similar technique to thickness-mode quartz and thin-film bulk acoustic wave resonators,11,12 acoustic energy trapping in this device is done through coupling propagating waves in the central section of the structure into evanescent waves at flanks. This is unlike conventional in-plane silicon bulk acoustic resonators (SiBAR) where the cavity is defined entirely through trenches that bound the geometry.13 Dispersive characteristic of laterally propagating waves in waveguides with rectangular cross section is used for a)
Electronic mail: [email protected].
0003-6951/2015/106(26)/263504/5/$30.00
acoustic engineering of the device. Fig. 1 shows the dispersion behavior and mode shape of several eigen-waves in a rectangular waveguide aligned to h100i crystallographic direction of a (100) silicon substrate, for both evanescent and propagating waves. The dispersion relation between the frequency (f0) and wavenumber (KX) of laterally propagating waves is defined by Christoffel equation14 considering the stress-free peripheral faces of the waveguide: 2
3 @ 0 0 7 6 6 @x 7 6 7 @ 6 7 0 7 6 0 6 7 @y 6 72 3 ið KX xxtÞ 6 @ 7 6 0 7 Ux ðy; zÞe 0 6 7 6 @z 7 6 76 Uy ðy; zÞeiðKX xxtÞ 7 4 5 6 7 6 0 @ @ 7 ið KX xxtÞ 6 7 U y; z e Þ zð @z @y 7 6 6 7 6 @ @ 7 6 7 6 @z 0 @x 7 6 7 4 @ @ 5 0 @z @x 2 3 Ux ðy; zÞeiðKX xxtÞ 6 7 iðKX xxtÞ 7 ¼ qx2 6 4 Uy ðy; zÞe 5; ð Þ Uz ðy; zÞei KX xxt rmn ¼ 0 at waveguide periphery;
m; n 2 f x; y; zg:
(1)
Here, U ¼ (Ux, Uy, Uz) is the polarization vector. KX, x, and q are wave number, angular frequency, and waveguide mass density, respectively, and r ¼ [rmn] is the stress tensor. Such a relation results in both propagating and evanescent wave solutions with real and imaginary wavenumbers,
106, 263504-1
C 2015 AIP Publishing LLC V
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263504-2
R. Tabrizian and F. Ayazi
Appl. Phys. Lett. 106, 263504 (2015)
FIG. 1. (Left) Dispersion behavior of eigen-waves in a single crystal silicon rectangular waveguide with a width of 50 lm and thickness of 20 lm. Inset shows the waveguide with characteristic cross-sectional dimensions of W and H, which is aligned to h100i direction of a (100) silicon plate. (Right) Deformation mode shape of different eigen-waves for arbitrary propagating and evanescent solutions, as well as their cross-sectional polarization. Several eigen-waves with symmetric (Si) and anti-symmetric (Ai) polarizations across the central axis of the waveguide are demonstrated.
respectively. While for the propagating solutions, the acoustic energy is uniform across the length of the waveguide, the energy of evanescent solutions decays exponentially along the waveguide axis (X-axis in Fig. 1). The dispersion behavior of a rectangular waveguide can be tailored by changing it cross-sectional dimensions (i.e., W and H). This is shown in Fig. 1 for the first width-extensional branch (S1) where slight variations of the width (W 6 DW) results in frequency shift of the branch. Such a geometry dependence can be used to couple propagating and evanescent waves with different wavenumbers, at a specific frequency. This can be done by cascading waveguides with different dimensions, along their axis. In such a structure, a coupled solution may exist if displacement and strain continuity hold at transition interface between consecutive waveguides.15 The thermo-acoustic properties of such a coupled solution can then be defined as a superposition of the properties of constituent waveguides, weighted by their respective contribution in trapping acoustic energy. Each (f0, KX) solution on eigen-wave dispersion branches is a result of a unique interaction between transverse and longitudinal waves launched into the waveguide in a specific incident angle, while reflected and converted at stress-free boundaries. Therefore, considering anisotropic temperature characteristic of longitudinal and transverse waves in single crystal silicon, each solution has a specific temperature characteristic. Fig. 2 shows the simulated TCF1,2 and turn-over temperature (TTurn-Over) of the S1 branch for N-doped silicon substrates with different resistivities. When properly oriented with respect to crystallographic axes of doped single crystal silicon, evanescent waves can provide a highly different TCF compared to propagating waves. Furthermore, as opposed to propagating waves, evanescent waves show a large variation in temperature characteristics for different KX. This is due to effective contribution of both shear and extensional wave components in the formation of evanescent solutions and highly different thermoacoustic properties of these constituents.17 In this work, evanescent waves with highly positive TCF1 are used to design a temperature-compensated acoustic resonator with well-defined turn-over temperature. Several waveguides with different width, and hence different
dispersion behavior, are cascaded in series. In such a structure, a solution—called a synthesized resonance mode hereafter—may exist arising from acoustic coupling of propagating and evanescent waves in constituent waveguides. When such a cascaded group of waveguides is flanked with sufficiently long evanescent-supporting sections, the energy will be confined mainly in the central waveguide leaving negligible concentration at flank terminations. Therefore, the anchoring of the cascaded group will not perturb vibration dynamics of the structure or diminish the generality of the waveguide assumption of the constituents— even though their lengths are finite.18 Fig. 3 compares the mode shape and simulated TTurn-Over of an ideal (i.e., infinitely long) SiBAR with two different coupled waveguides, all aligned to h100i axes of 0.0011 X . cm single crystal silicon. The displacement mode shape of the coupled waveguides can be defined in different sections as 8 L1 L1 > > <x< > < sinð K1;1 xÞ 2 2 Waveguide 1 : > L L L 1 > > 6sin K1;1 1 eiK2;1 ðx7 21 Þ < jxj; : 2 2 and 8 L2 L2 > > <x< > < cosðK1;2 xÞ 2 2 Waveguide 2 : > L2 iK2;2 ðx7L2 Þ L2 > > 2 e < jxj: : cos K1;2 2 2 (2) In order to calculate the contribution of each eigen-wave in temperature characteristic of the vibration mode, acoustically coupled system can be simplified to mechanically coupled harmonic oscillators with lumped equivalent mass and spring defined at the interface (i.e., X ¼ L1,2).19 Therefore, TCFi (i ¼ 1, 2) of the coupled vibration mode (TCFi, C) can be defined as TCFi;C ¼
meq;1 TCFi;1 þ meq;2 TCFi;2 meq;1 þ meq;2
(3)
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FIG. 2. (Left) The simulated first order TCF (TCF1) for S1, S2, and A2 branches and (Middle) second order TCF (TCF2) and (Right) turn-over temperature (TTurn-Over) of propagating and evanescent waves for S1 eigenwave branch of the rectangular waveguide aligned to direction of N-doped (100) silicon substrates with different resistivities. The material properties used in the simulations are taken from Ref. 16.
coupled waveguides 1 and 2 (i.e., TTurn-Over,i, i ¼ 1, 2) can be approximated from the superposition of that of the constituents, weighted by their effective contribution in vibration mode
where TCFi,j and meq,j (j ¼ 1, 2) are the TCFi and equivalent mass of the constituent eigen-waves at waveguide transitional face and can be calculated using the displacement mode shape.20,21 Having TCFi,C in (3), the TTurn-Over of the
TTurnOver;1
and
TTurnOver;2
L1 L1 1 cos K1;1 sin K1;1 2 2 TTurnOver ðK1;1 Þ þ TTurnOver ðK2;1 Þ K1;1 K2;1 ; L1 L1 1 cos K1;1 sin K1;1 2 2 þ K1;1 K2;1 L2 L2 sin K1;2 cos K1;2 2 2 TTurnOver ðK1;2 Þ þ TTurnOver ðK2;2 Þ K1;2 K2;2 : L2 L2 sin K1;2 cos K1;2 2 2 þ K1;2 K2;2
Fig. 4 shows the SEM image of the silicon resonator designed to serve as a test vehicle to reproduce coupled waveguides coupled modes. The device is composed of a central section with a width of W0 ¼ 50 lm that is surrounded by flanks with gradually changing widths. Flanks are designed to facilitate excitation of two synthesized modes with closely similar mode shapes to the trapped modes in coupled waveguides 1 and 2 in Fig. 3. The structure is finally clamped to the substrate through a tether with a width chosen to support no solution, either propagating or evanescent, at resonance frequency. This is done to reduce extension of acoustic field into the substrate, therefore increasing the quality factor (Q). A thin aluminum nitride (AlN) film is used to provide electromechanical transduction to the silicon microstructure. Wide tethers facilitate integration of several electrically isolated electrodes on the top surface of AlN, which in turn facilitates independent excitation of different synthesized modes.18
(4)
Fig. 5 shows the mode shape of the synthesized modes in designed resonator in comparison with a SiBAR of the same length, anchored through narrow tethers at nodal points of the mode shape. The electrode configuration of AlN transducer facilitate efficient excitation of both synthesized modes. Fig. 6 shows the frequency response of the device providing two modes with high Q. The larger Q of the mode 2 is a result of its smaller energy concentration in the flanks, and hence lower energy leakage, which can be observed in the mode shapes of Fig. 5. Fig. 7 shows the COMSOL finite element simulated as well as measured temperature characteristics of the two modes of the device, in comparison with a regular SiBAR implemented on the same substrate. The devices are all implemented on a silicon substrate with a resistivity of 0.001(6%10) X . cm. Temperature characterizations are
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R. Tabrizian and F. Ayazi
Appl. Phys. Lett. 106, 263504 (2015)
FIG. 3. The mode shape (top view) and simulated TTurn-Over of two coupled waveguides in comparison with an ideal SiBAR. Proper boundary conditions are considered to emulate infinite length of waveguides in flanks. Also a transduction stack that is composed of 500 nm AlN sandwiched between 50 nm molybdenum electrodes and is separated from the silicon substrate through a 200 nm silicon dioxide layer is considered in thermoacoustic simulations to guarantee the validity of comparison with experimental results.
FIG. 4. The SEM image of the device with AlN transducer. The inset shows a h100i-aligned SiBAR implemented on the same substrate to provide a comparison reference for temperature characteristics of resonance frequency.
done in an environmental chamber with a temperature accuracy of 60.5 C and under a 1 mTorr vacuum. The slight discrepancy between simulation and experimental results is due to the difference between doping concentration of silicon substrate compared to what assumed for material properties in finite element simulations as well as the variations in substrate thickness (20 lm 6 1 lm, as indicated by the substrate-providing vendor). Highly different temperature characteristic of the synthesized modes is a result of their different constituent propagating and evanescent waves. Specifically, in the synthesized mode 2, effective contribution of the evanescent wave with large wavenumber—and hence
highly positive TCF1—has resulted in a TTurn-Over exceeding 85 C, which facilitates implementation of an oven-controlled stable frequency reference.22 The larger turn-over temperatures of the synthesized modes in implemented device compared to designed values is due to slightly lower resistivity of the actual silicon substrate compared to the simulation assumptions. Furthermore, the TTurn-Over can be tailored to a well-defined value by proper acoustic engineering, which includes design for evanescent wavenumber and relative energy distribution between propagating and evanescent fields. Fig. 8 shows the temperature characteristic of the Q for the modes/devices of Fig. 7, over the temperature range of 40 C to 80 C. The temperature induced degradation in the Q of the vibration modes in acoustically engineered cavity and SiBAR is a result of an increase in dissipative thermo-acoustic interactions at higher temperatures.23 Slight irregularities in the Q characteristic trend for the modes in acoustically engineered cavity can be attributed to the change of evanescent wavenumber in the coupled solutions with temperature variations. Such a change results in different energy distribution profile in the device flanks and induces unexpected variations in acoustic energy leakage into the substrate. In conclusion a thermo-acoustic engineering technique, based on acoustic coupling of propagating and evanescent waves, is introduced. The temperature characteristics of propagating and evanescent waves in the width-extensional eigenwave branch is extracted through numerical simulations. The application of such characteristics in the turn-over temperature of coupled waveguides is presented through analytical derivation and experimentally verified through device implementation and characterization. TTurn-Overs of 67 C and 87 C are measured for two synthesized modes in a single device. Such a
FIG. 5. Displacement mode shapes (top view) of synthesized modes and a h100i-aligned SiBAR that is anchored through narrow tethers.
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Appl. Phys. Lett. 106, 263504 (2015)
device can serve as the frequency reference for implementation of highly stable ovenized oscillators. This work was supported by DARPA TIMU program through SSC Pacific Contract No. N66001-11-C-4176. 1
FIG. 6. Measured frequency response of the acoustically engineered device having two synthesized width-extensional modes with small frequency split.
FIG. 7. Simulated (top) and measured (bottom) temperature characteristic of the synthesized modes in the acoustically engineered waveguide in comparison with a h100i-aligned SiBAR implemented on the same substrate.
FIG. 8. Temperature characteristic of the Q for the two modes of the acoustically engineered cavity as well as the h100i-aligned SiBAR.
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