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Rapid switching in high-Q mechanical resonators

Hajime Okamoto, Imran Mahboob, Koji Onomitsu, and Hiroshi Yamaguchi

Citation: Applied Physics Letters 105, 083114 (2014); doi: 10.1063/1.4894417View online: http://dx.doi.org/10.1063/1.4894417

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Rapid switching in high-Q mechanical resonators

Hajime Okamoto,a) Imran Mahboob, Koji Onomitsu, and Hiroshi YamaguchiNTT Basic Research Laboratories, Nippon Telegraph and Telephone Corporation, 3-1 Morinosato-Wakamiya,Atsugi 243-0198, Japan

(Received 14 July 2014; accepted 20 August 2014; published online 27 August 2014)

Sharp resonance spectra of high-Q micromechanical resonators are advantageous in their

applications, such as highly precise sensors and narrow band-pass filters. However, the high-Qcharacteristics hinder quick repetitive operations of mechanical resonators because of their long

ring-down time due to their slow energy relaxation. Here, we demonstrate a scheme to solve this

trade-off problem in paired GaAs micromechanical resonators by using parametrically induced

intermode coupling. The strong intermode coupling induced by the piezoelectric modulation of ten-

sion allows on-demand energy transfer between closely spaced mechanical modes of the resonator

via coherent control of the coupling. This enables rapid switching of the vibration amplitude within

the ring-down time, leading to quick repetitive operations in high-Q mechanical resonators. VC 2014

AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4894417]

Semiconductor micro/nano-mechanical resonators offer

a wide range of applications, such as sensors,15 filters,6

amplifiers,79 memories,10,11 and logics.1215 The character-

istically high quality factor (Qxm/c 103106, where xmis the angular eigenfrequency and c the damping factor)

results in sharp resonance spectra, leading to the excellent

device performance, e.g., highly sensitive detectors1 and

ultralow power consumption logic gates.14 On the other

hand, the high-Qcharacteristics pose an obstacle to the oper-

ation speed of mechanical resonators. This is because the

mechanical vibration lasts for a finite period of time once the

resonators are harmonically triggered, where energy-

relaxation time t1, at which the vibration energy is reduced

by 1/e, is proportional to Q as t1Q/xm. The slow energyrelaxation requires a long waiting time for initializing the

mechanical device, i.e., a trade-off exists between Q andoperation speed. For example, the operation speed of me-

chanical logic gates using a sub-megahertz parametric reso-

nator with Q 105 is as slow as a few seconds,14 but such ahighQ is necessary for the ultralow-power logic operation.

A straightforward approach to solving this trade-off prob-

lem could be Q-control. Both active and passive feedback

Q-control schemes have been demonstrated by optical and

electrical means.1622 However, these schemes require either a

feedback circuit or a cavity, which need to be integrated into

the mechanical resonator without degrading its performance,

resulting in a complicated device.20,21 Therefore, a simpler

and more compact scheme is desired to solve the trade-off

problem betweenQand operation speed.A promising candidate for this is to utilize highly control-

lable coupling between two mechanical modes (orresonators)

induced by the parametric mode mixing technique.2326 In this

approach, energy is transferred from one mode to the other

within the intrinsic ring-down time, leading torapid amplitude

switching in a high-Q mechanical resonator.27 Here, we dem-

onstrate the experimental implementation of the rapid switch-

ing using piezoelectrically induced parametric mode mixing

in paired GaAs micromechanical resonators.

Figure 1 shows an optical micrograph of the sample.

The 100-lm-long and 34-lm-wide two doubly clamped

beams (beam L and beam R), which consist of 400-nm-thick

i-GaAs, 100-nm-thickn-GaAs, 300-nm-thick Al0.25Ga0.75As,

and 60-nm-thick Au gates, are geometrically interconnected

via the coupling overhang. The fabrication details are

described elsewhere.24 In this sample, harmonic driving, ten-

sion modulation, and detection of the mechanical motion are

available by using the piezoelectric effect in Al0.25Ga0.75As

through the gates when the conductive n-GaAs layer is

grounded via the ohmic contacts (Fig. 1).24 The measure-

ments were performed using the setup shown in Fig. 1with

the sample cooled to 1.5 K in a vacuum (5 105Pa) with a4He cryostat.

Figure2(a)shows the frequency response of the ampli-

tude of beam L measured while beam L is driven withVd 0.4 mVrms. The resonance at 291.51 2p kHz (x1)with the larger amplitude corresponds to mode 1, which is

dominated by the vibration of beam L. In contrast, the reso-

nance at 291.95 2pkHz (x2) with the smaller amplitudecorresponds to mode 2, which is dominated by the vibration

FIG. 1. Optical micrograph of the sample and schematic drawing of the

measurement setup. The coupling overhang is indicated by hatching.a)Electronic mail: [email protected]

0003-6951/2014/105(8)/083114/4/$30.00 VC 2014 AIP Publishing LLC105, 083114-1

APPLIED PHYSICS LETTERS105, 083114 (2014)


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of beam R. This indicates that the two beams are elastically

coupled through the coupling overhang (Fig.1), but the cou-

pling is weak because the eigenfrequency of beam L, xL,

does not matchwith that of beam R, xR, or ratherx1

xLand x2 xR.24 Figure2(b)shows the time response of theamplitude of beam L at x1measured after the harmonic driv-

ing is switched off. This reveals the long ring-down time,

where the extracted t1 at which the square amplitude (or the

vibration energy) is reduced by 1/e is 14ms and thus

Q 25 000. The switch-off time toff, defined as when thevibration energy becomes three-orders of magnitude smaller

than the initial value, is 0.1 s. This means one has to wait at

least 0.1 s to initialize this mechanical system.

If the two mechanical modes are strongly coupled or

mixed by parametric pumping, the vibration energy in mode

1 (beam L) can be quickly transferred to mode 2 (beam R),

leading to the significant reduction oftoff.

27

Such intermodecoupling can be induced by the parametric pump whi chperi-

odically modulates the tension with frequency xp.24 The

parametric pump results in sidebands developing around

both modes at x16xp and x26xp, and their subsequent

overlap when xpx2 x1 results in the intermode cou-pling. This interaction can be described by the following

equations of motion for the two modes:24

x1 c _x1 x21x1 K cosxptx2 F1cosxdt /; (1a)x2 c _x2 x22x2 K cosxptx1 F2cosxdt /; (1b)

where K is the intermode coupling coefficient that is propor-

tional to the pump amplitude,Fi is the drive force for the ith

mode, andxdis the drive frequency. Note that the intramode

coupling, which plays a role only for higher order coupling,24

is neglected in Eqs.1(a)and1(b). These equations indicatex1(x2) is influenced by x2(x1) when xpx2 x1, i.e., the twomodes are parametrically coupled.

The time response of the two beams under parametric

pumping is simultaneously measured via gates L and R,

while the sinusoidal pump voltage is applied to gate P (Fig.

1) just after the harmonic driving is switched off. Figure 3

shows the change in the amplitude of beam L at x1 and

beam R at x2, measured forxpx2x1 0.44 2p kHzand pump voltage Vp 0.55 Vrms. The result clearly showsthat the amplitude of the two beams alternately oscillates,indicating coherent energy exchange between beam L (mode

1) and beam R (mode 2) via the intermode coupling. This

indicates that coupling rateg can exceed damping rate c, i.e.,

g K=2 ffiffiffiffiffiffiffiffiffiffiffix1x2p c, thus confirming the availability of

FIG. 2. (a) Frequency response of the amplitude of beam L, measured while

beam L is driven withVd 0.4 mVrmswith the setup schematically shown inthe inset. The corresponding mode shapes obtained by finite element method

calculations are also shown. (b) Time response of the amplitude of beam L

atx1, measured with an oscilloscope via a lock-in amplifier after the driving

is switched off. The inset is the schematic drawing of the pulse sequence for

the measurement.

FIG. 3. Time response of the amplitude of beam L atx1 and the amplitude

of beam R at x2, measured after switching on the parametric pumping with

xpx2 x1 and Vp 0.55 Vrms while the harmonic driving is switchedoff. Top: The pulse sequence and measurement setup.

FIG. 4. Time response of the amplitude of beam L atx1underp-pulse pump-

ing with voltageVp 0.55 Vrmsand durationtp tp 9 ms. A comparison ofthe time response with and withoutp-pulse pumping is shown in the inset on

a logarithmic scale. Top: The pulse sequence and measurement setup.

083114-2 Okamotoet al. Appl. Phys. Lett.105, 083114 (2014)


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coherent control to this system. Such strong intermode cou-

pling can be obtained in the present two-beam system,

whereas theintermode coupling in a single-beam system is

weaker.2426 The realization of the strong coupling provides

the necessary tool for implementing rapid switching of the

vibration amplitude in high-Qmechanical resonators.27

A strong advantage of this coupling is that the time-

domain on-off control is available by adjusting the pumping

duration. The result in Fig. 3indicates that if pumping dura-

tion tp is set to a half period of the coherent oscillation, i.e.,

the p-pulse length (tp 9 ms), all of the vibration energy in

beam L (mode 1) is transferred to beam R (mode 2), leading to

quick damping of beam L. Figure4shows the demonstration of

thisp-pulse operation, in which the suppression of the vibration

of beam L is confirmed aftertp

. Thisp-pulse operation reduces

the switch-off time by an order of magnitude, enabling rapidinitialization of this mechanical system (see the inset in Fig.4).

Next, we compare the response of beam L (at x1) under

repetitive operations with and without p-pulse pumping.

Figure5(a) shows the response with no pumping while the

drive voltage is periodically switched on and off every 0.1 s

(toff). The result clearly shows the delayed response, notonly for the damping butalso for the rising of the amplitude,

reflecting the Q value.27 Here, the delay for raising the am-

plitude canbe reduced by applying the two-step driving tech-

nique.27,28 If the beam is first driven by the stronger short

pulse with voltageVd1 0.9 mVrmsand durationtd1 14 ms,and subsequently driven by the weaker long pulse with volt-

ageVd2 0.4 mVrmsand duration td2 86 ms, then the delayis drastically reduced as shown in Fig.5(b). This enables the

quick start-up of the high-Qmechanical resonator. The quick

shut-down operation is also achieved by applying the afore-

mentioned p-pulse pumping. A combination of p-pulse

pumping with the two-step driving results in the square-

wave-like response [Fig. 5(c)], realizing rapid switching of

the vibration amplitude in the high-Qmechanical resonator.

In conclusion, to solve a trade-off problem between Q and

operation speed in a mechanical resonator we have demon-

strated a scheme using the parametrically induced intermode

coupling. The piezoelectric p-pulse control of the intermode

coupling in paired GaAs micromechanical resonators enables

quick vibration damping within the intrinsic long ring-down

time. When the scheme is used in combination with the two-

step driving technique, rapid amplitude switching is realized

in the high-Q mechanical resonators. Because the coherent

control enables the transfer of the vibrational states between

the adjacent mechanical resonators, this technique also opens

up the possibility of rapid signal processing and information

transfer using micro/nano-mechanical resonator arrays.10,15,25

The authors thank Chia-Yuan Chang, Edward Yi Chang,

Adrien Gourgout, and Ahmet Taspinar, for their early

contributions to this work. This work was partly supported

by JSPS KAKENHI (23241046).

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FIG. 5. (a) Time response of the ampli-

tude of beam L at x1 under the repeti-

tive driving with voltage Vd 0.4mVrms and duration td 0.1s (with nopumping). (b) Time response of the am-

plitude of beam L atx1under repetitive

two-step driving, where the voltage and

duration for the first step are Vd1 0.9mVrms and td1

14 ms, respectively,

and those for the second step areVd2 0.4 mVrms and td2 86 ms (withno pumping). (c) Time response of the

amplitude of beam L atx1under repeti-

tive two-step driving in combination

with subsequent p-pulse pumping

with pump voltage Vp 0.55 Vrmsand duration tp tp 9 ms. Right:Corresponding pulse sequences.



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