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Tunable coupling of mechanical vibration in GaAs micro-resonators
Physica E 42 (2010) 2849–2852
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Tunable coupling of mechanical vibration in GaAs micro-resonators Hajime Okamoto a,, Takehito Kamada a,b, Koji Onomitsu a, Imran Mahboob a, Hiroshi Yamaguchi a,b a b
NTT Basic Research Laboratories, NTT Corporation, Atsugi, Kanagawa 243-0198, Japan Department of Physics, Tohoku University, Sendai 980-8578, Japan
a r t i c l e in fo
abstract
Article history: Received 31 August 2009 Accepted 22 December 2009 Available online 28 December 2009
Control of the vibrational coupling in two mechanically coupled GaAs micro-resonators is demonstrated. The coupling efficiency depends on the eigenfrequency difference between the resonators. While the fabrication results in the eigenfrequency difference of 1% due to the patterning error, frequency modulation by the laser irradiation via the photothermal effect can compensate for the initial difference and realize perfect tuning. The two coupled vibrational modes exhibit the avoided crossing with the adjustment of the laser power, indicating the tunable coupling in the micromechanical system. & 2009 Elsevier B.V. All rights reserved.
Keywords: Vibration Coupling Tuning GaAs Piezoelectric effect Photothermal effect
1. Introduction Coupled micromechanical resonators have recently attracted much interest because they not only allow the study of interesting physical phenomena, such as synchronization and mode localization [1–4], but also enable new applications in sensors using the dynamics of the coupled system [5–7]. Vibrational coupling between the resonators depends on the difference in their eigenfrequency. Therefore, eigenfrequency modulation is important and desired for controlling the coupled mechanical vibration. Here, we demonstrate frequency modulation of coupled GaAs micromechanical resonators by the piezoelectric effect and photothermal effect. By using the optically induced thermal stress, vibrational coupling can be effectively controlled.
2. Material and methods The coupled micromechanical system has two doubly clamped beams of 40-mm length, 10-mm width, and 0:8-mm thickness [Figs. 1(a) and (b)]. The beams consist of i-GaAs, n-GaAs, and GaAs/AlGaAs superlattice (SL) layers grown on an Al0:65 Ga0:35 As sacrificial layer on a GaAs ð0 0 1Þ substrate. Au electrodes were deposited on the top of the beams. The two beams (beams 1 and 2) are oriented to the [110] direction and mechanically coupled with each other through an etching overhang, which was formed Corresponding author. Tel.: + 81 46 240 2522; fax: + 81 46 240 4317.
E-mail address: [email protected] (H. Okamoto). 1386-9477/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2009.12.038
by isotropic etching of the sacrificial layer [Figs. 1(a) and (b)], while they are electrically isolated by mesa etching [Fig. 1(a)]. The two beams can be individually actuated by applying an ac voltage to the top electrode with the n-GaAs layer grounded via ohmic contacts [Fig. 1(b)]. Positive voltage results in the lattice expansion of the SL layer in the longitudinal ([1 1 0]) direction due to the piezoelectric effect, while negative voltage causes lattice contraction [Fig. 1(c)]. Because the piezoelectric effect is not induced in the i-GaAs layer, the beam bends as shown in Fig. 1(c). Therefore, application of the AC voltage enables the excitation of the mechanical vibration at the resonance frequency [8]. The vibration was detected with a cw He:Ne laser via Doppler interferometry. The mechanical resonance characteristics were measured by sweeping the actuation frequency and monitoring the frequency response of the output signal of the interferometer with a network analyzer. All the measurements were performed in a vacuum at room temperature. The coupled micromechanical resonators have two coupled vibrational modes for the fundamental vibration, i.e., the lower frequency mode (mode L) and the higher frequency mode (mode H), where the mode frequency is 783.9 and 791.5 kHz, respectively.
3. Results and discussion The fabrication process results in the eigenfrequency difference of 1% between the two resonators. Therefore, frequency modulation is needed to control the coupling efficiency and to realize perfect tuning. The eigenfrequency of a doubly clamped beam can be modulated by applying stress to the beam. If dc voltage is applied to the top electrodes, piezoelectrically
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H. Okamoto et al. / Physica E 42 (2010) 2849–2852
Fig. 2. (a) Change in the eigenfrequency by the piezoelectric effect. Df is proportional to the voltage. (b) Change in the eigenfrequency by the photothermal effect. Df is proportional to the laser power.
Fig. 1. (a) Illustration of the coupled GaAs micromechanical resonators. (b) Schematic of the setup for the beam actuation. (c) Illustration of the piezoelectric effect. Mechanical vibration can be excited by this piezoelectric effect. (d) Illustration of the photothermal effect by laser irradiation. The eigenfrequency of the beam is modulated by the optically induced lattice expansion.
generated stress changes the effective spring constant of the beam and therefore the eigenfrequency is shifted [8,9]. Fig. 2(a) shows a change in the eigenfrequency as a function of the voltage applied to the two electrodes on each beam. Since negative voltage results in the lattice contraction [Fig. 1(c)], beam tension increases and the frequency shifts upward [Fig. 2(a)]. This shift is in proportion to the voltage with the rate of Df ¼ 0:4 kHz=V. This p behavior can be described by the theoretical expression ffiffiffiffiffiffiffiffiffiffiffi Df ¼ 3E=r Cd13 V=ð2pt2 Þ, where E is the elastic modulus, r the density, C the structural parameter, d13 the piezoelectric constant, V the voltage, and t the thickness [9]. The frequency shift by the piezoelectric effect is however not strong enough to compensate for the initial eigenfrequency difference between the two beams ð 6:5 kHzÞ, because the applicable voltage is limited to the range between þ 0:5 and 5:5 V due to the Schottky characteristics of the material. More effective frequency modulation is given by the photothermal stress, which is induced by the irradiation of the cw He:Ne laser [Fig. 1(d)]. The optically induced thermal stress
results in lattice expansion along the beam. Therefore, beam tension is reduced. Increasing the laser power decreases the eigenfrequency, where the frequency shift is proportional to the laser power with the rate of Df ¼ 97 kHz=mW. This behavior can be described by the theoretical expression Df ¼ al2 f0 DT=6:8t 2 , where a is the linear thermal expansion coefficient, l the length, and DT the average temperature change in the beam [10]. Taking into account the one-dimensional heat conduction and the optical absorption coefficient (a), DT is given by DT ¼ lPð1eat Þ=4wtk, where w is the width, and k the thermal conductivity [10]. Thus, Df is proportional to P as Df ¼ al3 ð1eat ÞP=27:2t3 wk. Here, l is the effective length, which would be close to the free-standing length given by the sum of the beam length and overhang depth. The curve fitting provides l ¼ 52 mm, which is in good agreement with the free-standing length ð54256 mmÞ observed by optical microscopy. In our coupled resonators, the eigenfrequency of beam 2 is higher than that of beam 1. Therefore, the laser irradiation to beam 2 reduces the eigenfrequency difference between the two resonators with increasing the vibrational coupling. Fig. 3(a) shows a change in the frequency of modes L and H caused by the optical eigenfrequency modulation in beam 2. A change in the amplitude of the two modes in beam 2 measured with the actuation of beam 2 is shown in Fig. 4(a), and the resonance characteristics at P ¼ 7, 64, and 173 mW are shown in Figs. 3(b)– (d), respectively. While the amplitude of mode H is larger than that of mode L for P o 65 mW and vice versa for P 465 mW [Fig. 4(a)], the amplitudes of the two modes coincide with each other at P C 65 mW because the two beams are perfectly tuned by
H. Okamoto et al. / Physica E 42 (2010) 2849–2852
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Fig. 3. (a) Change in the frequency of modes L and H by the laser irradiation to beam 2. Resonance characteristics of beam 2 measured with the actuation of beam 2 at the laser power of P ¼ 7, 64, and 173 mW are shown in (b)–(d), respectively.
symmetric coupled vibration and purely anti-symmetric vibration is realized for mode L and mode H, respectively [Fig. 3(a)]. The coupling efficiency can be monitored by measuring the amplitude of beam 2 with the actuation of beam 1 [Fig. 4(b)]. In this case, the amplitudes of the two modes are always the same, but the amplitudes become small if the coupling efficiency is too low to propagate the mechanical vibration from beam 1 to beam 2. Fig. 4(b) indicates that the amplitude of the un-actuated beam becomes maximum at P C65 mW and the value coincides with that of the actuated beam [Fig. 4(a)], i.e., the coupling efficiency is maximized at the perfect-tuning condition. By adjusting the laser power, one can effectively control the vibrational coupling between the micromechanical resonators.
4. Conclusions Tunable coupling of mechanical vibration in GaAs microresonators has been demonstrated. Coupling efficiency between the two micromechanical resonators can be effectively controlled by optical tuning via the photothermal effect. The two coupled vibrational modes exhibit the avoided crossing with the adjustment of the laser power, indicating the tunable coupling in the micromechanical system. The control of the vibrational coupling is important for studying the dynamics of the coupled system as well as for expanding the applications of micromechanical resonators in sensors.
Acknowledgements
Fig. 4. (a) Relation between the amplitude of the two coupled modes in beam 2 and the laser power measured with the actuation of beam 2. (b) Relation between the amplitude of the two coupled modes in beam 2 and the laser power measured with the actuation of beam 1.
The authors thank Norihiro Kitajima for helpful discussions. This work was partly supported by Grant-in-Aid For Scientific Research from the Japan Society for the Promotion of Science (No. 20246064). References
the laser irradiation [Figs. 3(c) and 4(a)]. The avoided crossing of the two modes shown in Fig. 3(a) also indicates the tunable coupling in the micromechanical system. At P C 65 mW, purely
[1] M.G. Rosenbulm, A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 76 (1996) 1804. [2] M.C. Cross, A. Zumdieck, R. Litshitz, J.L. Rogers, Phys. Rev. Lett. 93 (2004) 224101.
2852
[3] [4] [5] [6]
H. Okamoto et al. / Physica E 42 (2010) 2849–2852
S.B. Shim, M. Imboden, P. Mohanty, Science 316 (2007) 95. M. Sato, et al., Phys. Rev. Lett. 90 (2003) 044102. M. Spletzer, et al., Appl. Phys. Lett. 88 (2006) 254102. M. Spletzer, A. Raman, H. Sumali, J.P. Sullivan, Appl. Phys. Lett. 92 (2008) 114102.
[7] [8] [9] [10]
D. Galayko, et al., Sens. Actuators A 126 (2006) 227. I. Mahboob, H. Yamaguchi, Nat. Nanotech. 3 (2008) 275. S.C. Masmanidis, et al., Science 317 (2007) 780. S.C. Jun, et al., Nanotechnology 17 (2006) 1506.
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Tunable coupling of mechanical vibration in GaAs micro-resonators Hajime Okamoto a,, Takehito Kamada a,b, Koji Onomitsu a, Imran Mahboob a, Hiroshi Yamaguchi a,b a b
NTT Basic Research Laboratories, NTT Corporation, Atsugi, Kanagawa 243-0198, Japan Department of Physics, Tohoku University, Sendai 980-8578, Japan
a r t i c l e in fo
abstract
Article history: Received 31 August 2009 Accepted 22 December 2009 Available online 28 December 2009
Control of the vibrational coupling in two mechanically coupled GaAs micro-resonators is demonstrated. The coupling efficiency depends on the eigenfrequency difference between the resonators. While the fabrication results in the eigenfrequency difference of 1% due to the patterning error, frequency modulation by the laser irradiation via the photothermal effect can compensate for the initial difference and realize perfect tuning. The two coupled vibrational modes exhibit the avoided crossing with the adjustment of the laser power, indicating the tunable coupling in the micromechanical system. & 2009 Elsevier B.V. All rights reserved.
Keywords: Vibration Coupling Tuning GaAs Piezoelectric effect Photothermal effect
1. Introduction Coupled micromechanical resonators have recently attracted much interest because they not only allow the study of interesting physical phenomena, such as synchronization and mode localization [1–4], but also enable new applications in sensors using the dynamics of the coupled system [5–7]. Vibrational coupling between the resonators depends on the difference in their eigenfrequency. Therefore, eigenfrequency modulation is important and desired for controlling the coupled mechanical vibration. Here, we demonstrate frequency modulation of coupled GaAs micromechanical resonators by the piezoelectric effect and photothermal effect. By using the optically induced thermal stress, vibrational coupling can be effectively controlled.
2. Material and methods The coupled micromechanical system has two doubly clamped beams of 40-mm length, 10-mm width, and 0:8-mm thickness [Figs. 1(a) and (b)]. The beams consist of i-GaAs, n-GaAs, and GaAs/AlGaAs superlattice (SL) layers grown on an Al0:65 Ga0:35 As sacrificial layer on a GaAs ð0 0 1Þ substrate. Au electrodes were deposited on the top of the beams. The two beams (beams 1 and 2) are oriented to the [110] direction and mechanically coupled with each other through an etching overhang, which was formed Corresponding author. Tel.: + 81 46 240 2522; fax: + 81 46 240 4317.
E-mail address: [email protected] (H. Okamoto). 1386-9477/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2009.12.038
by isotropic etching of the sacrificial layer [Figs. 1(a) and (b)], while they are electrically isolated by mesa etching [Fig. 1(a)]. The two beams can be individually actuated by applying an ac voltage to the top electrode with the n-GaAs layer grounded via ohmic contacts [Fig. 1(b)]. Positive voltage results in the lattice expansion of the SL layer in the longitudinal ([1 1 0]) direction due to the piezoelectric effect, while negative voltage causes lattice contraction [Fig. 1(c)]. Because the piezoelectric effect is not induced in the i-GaAs layer, the beam bends as shown in Fig. 1(c). Therefore, application of the AC voltage enables the excitation of the mechanical vibration at the resonance frequency [8]. The vibration was detected with a cw He:Ne laser via Doppler interferometry. The mechanical resonance characteristics were measured by sweeping the actuation frequency and monitoring the frequency response of the output signal of the interferometer with a network analyzer. All the measurements were performed in a vacuum at room temperature. The coupled micromechanical resonators have two coupled vibrational modes for the fundamental vibration, i.e., the lower frequency mode (mode L) and the higher frequency mode (mode H), where the mode frequency is 783.9 and 791.5 kHz, respectively.
3. Results and discussion The fabrication process results in the eigenfrequency difference of 1% between the two resonators. Therefore, frequency modulation is needed to control the coupling efficiency and to realize perfect tuning. The eigenfrequency of a doubly clamped beam can be modulated by applying stress to the beam. If dc voltage is applied to the top electrodes, piezoelectrically
2850
H. Okamoto et al. / Physica E 42 (2010) 2849–2852
Fig. 2. (a) Change in the eigenfrequency by the piezoelectric effect. Df is proportional to the voltage. (b) Change in the eigenfrequency by the photothermal effect. Df is proportional to the laser power.
Fig. 1. (a) Illustration of the coupled GaAs micromechanical resonators. (b) Schematic of the setup for the beam actuation. (c) Illustration of the piezoelectric effect. Mechanical vibration can be excited by this piezoelectric effect. (d) Illustration of the photothermal effect by laser irradiation. The eigenfrequency of the beam is modulated by the optically induced lattice expansion.
generated stress changes the effective spring constant of the beam and therefore the eigenfrequency is shifted [8,9]. Fig. 2(a) shows a change in the eigenfrequency as a function of the voltage applied to the two electrodes on each beam. Since negative voltage results in the lattice contraction [Fig. 1(c)], beam tension increases and the frequency shifts upward [Fig. 2(a)]. This shift is in proportion to the voltage with the rate of Df ¼ 0:4 kHz=V. This p behavior can be described by the theoretical expression ffiffiffiffiffiffiffiffiffiffiffi Df ¼ 3E=r Cd13 V=ð2pt2 Þ, where E is the elastic modulus, r the density, C the structural parameter, d13 the piezoelectric constant, V the voltage, and t the thickness [9]. The frequency shift by the piezoelectric effect is however not strong enough to compensate for the initial eigenfrequency difference between the two beams ð 6:5 kHzÞ, because the applicable voltage is limited to the range between þ 0:5 and 5:5 V due to the Schottky characteristics of the material. More effective frequency modulation is given by the photothermal stress, which is induced by the irradiation of the cw He:Ne laser [Fig. 1(d)]. The optically induced thermal stress
results in lattice expansion along the beam. Therefore, beam tension is reduced. Increasing the laser power decreases the eigenfrequency, where the frequency shift is proportional to the laser power with the rate of Df ¼ 97 kHz=mW. This behavior can be described by the theoretical expression Df ¼ al2 f0 DT=6:8t 2 , where a is the linear thermal expansion coefficient, l the length, and DT the average temperature change in the beam [10]. Taking into account the one-dimensional heat conduction and the optical absorption coefficient (a), DT is given by DT ¼ lPð1eat Þ=4wtk, where w is the width, and k the thermal conductivity [10]. Thus, Df is proportional to P as Df ¼ al3 ð1eat ÞP=27:2t3 wk. Here, l is the effective length, which would be close to the free-standing length given by the sum of the beam length and overhang depth. The curve fitting provides l ¼ 52 mm, which is in good agreement with the free-standing length ð54256 mmÞ observed by optical microscopy. In our coupled resonators, the eigenfrequency of beam 2 is higher than that of beam 1. Therefore, the laser irradiation to beam 2 reduces the eigenfrequency difference between the two resonators with increasing the vibrational coupling. Fig. 3(a) shows a change in the frequency of modes L and H caused by the optical eigenfrequency modulation in beam 2. A change in the amplitude of the two modes in beam 2 measured with the actuation of beam 2 is shown in Fig. 4(a), and the resonance characteristics at P ¼ 7, 64, and 173 mW are shown in Figs. 3(b)– (d), respectively. While the amplitude of mode H is larger than that of mode L for P o 65 mW and vice versa for P 465 mW [Fig. 4(a)], the amplitudes of the two modes coincide with each other at P C 65 mW because the two beams are perfectly tuned by
H. Okamoto et al. / Physica E 42 (2010) 2849–2852
2851
Fig. 3. (a) Change in the frequency of modes L and H by the laser irradiation to beam 2. Resonance characteristics of beam 2 measured with the actuation of beam 2 at the laser power of P ¼ 7, 64, and 173 mW are shown in (b)–(d), respectively.
symmetric coupled vibration and purely anti-symmetric vibration is realized for mode L and mode H, respectively [Fig. 3(a)]. The coupling efficiency can be monitored by measuring the amplitude of beam 2 with the actuation of beam 1 [Fig. 4(b)]. In this case, the amplitudes of the two modes are always the same, but the amplitudes become small if the coupling efficiency is too low to propagate the mechanical vibration from beam 1 to beam 2. Fig. 4(b) indicates that the amplitude of the un-actuated beam becomes maximum at P C65 mW and the value coincides with that of the actuated beam [Fig. 4(a)], i.e., the coupling efficiency is maximized at the perfect-tuning condition. By adjusting the laser power, one can effectively control the vibrational coupling between the micromechanical resonators.
4. Conclusions Tunable coupling of mechanical vibration in GaAs microresonators has been demonstrated. Coupling efficiency between the two micromechanical resonators can be effectively controlled by optical tuning via the photothermal effect. The two coupled vibrational modes exhibit the avoided crossing with the adjustment of the laser power, indicating the tunable coupling in the micromechanical system. The control of the vibrational coupling is important for studying the dynamics of the coupled system as well as for expanding the applications of micromechanical resonators in sensors.
Acknowledgements
Fig. 4. (a) Relation between the amplitude of the two coupled modes in beam 2 and the laser power measured with the actuation of beam 2. (b) Relation between the amplitude of the two coupled modes in beam 2 and the laser power measured with the actuation of beam 1.
The authors thank Norihiro Kitajima for helpful discussions. This work was partly supported by Grant-in-Aid For Scientific Research from the Japan Society for the Promotion of Science (No. 20246064). References
the laser irradiation [Figs. 3(c) and 4(a)]. The avoided crossing of the two modes shown in Fig. 3(a) also indicates the tunable coupling in the micromechanical system. At P C 65 mW, purely
[1] M.G. Rosenbulm, A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 76 (1996) 1804. [2] M.C. Cross, A. Zumdieck, R. Litshitz, J.L. Rogers, Phys. Rev. Lett. 93 (2004) 224101.
2852
[3] [4] [5] [6]
H. Okamoto et al. / Physica E 42 (2010) 2849–2852
S.B. Shim, M. Imboden, P. Mohanty, Science 316 (2007) 95. M. Sato, et al., Phys. Rev. Lett. 90 (2003) 044102. M. Spletzer, et al., Appl. Phys. Lett. 88 (2006) 254102. M. Spletzer, A. Raman, H. Sumali, J.P. Sullivan, Appl. Phys. Lett. 92 (2008) 114102.
[7] [8] [9] [10]
D. Galayko, et al., Sens. Actuators A 126 (2006) 227. I. Mahboob, H. Yamaguchi, Nat. Nanotech. 3 (2008) 275. S.C. Masmanidis, et al., Science 317 (2007) 780. S.C. Jun, et al., Nanotechnology 17 (2006) 1506.