Piezoresistive heat engine and refrigerator

P.G. Steeneken,∗ K. Le Phan, M.J. Goossens, G.E.J. Koops, G.J.A.M. Brom, and J.T.M. van BeekNXP-TSMC Research Center, NXP Semiconductors, HTC 4, 5656 AE Eindhoven, the Netherlands.

Heat engines provide most of our mechanical power and are essential for long-range transportation.However, whereas significant progress has been made in the miniaturization of motors driven byelectrostatic forces1,2, it has proven difficult to reduce the size of conventional liquid or gas drivenheat engines3–5 below 107µm3.

Here we demonstrate an all-silicon reciprocating heat engine with a volume of less than 0.5 µm3.The device draws heat from a DC current using the piezoresistive effect and converts it into mechan-ical energy by expanding and contracting at different temperatures. It is shown that the engine caneven increase the mechanical energy of a resonator when its motion is governed by random thermalfluctuations.

When the thermodynamic cycle of the heat engine is reversed, it operates as a refrigerator or heatpump that can reduce motional noise in mechanical systems6–13. In contrast to the Peltier effect,the direction of the thermal current does not depend on the direction of the electrical current.

When an alternating mechanical displacement x =x0e

iωt with frequency ω is applied to a piezoresistivebeam with spring constant k, this will not just result inan elastic force, but will also cause a resistance variationrac via the piezoresistive effect. If a constant DC currentIdc is flowing through the spring, the resistive heatingpower therefore changes by ppr = I2

dcrac and causes atemperature change, which in turn will generate a ther-mal expansion force Fte that adds to the other forces Fextwhich act on the spring such that kx = Fte + Fext. Thisthermodynamic feedback mechanism is shown schemati-cally in figure 1A.

Due to thermal delay, the thermal expansion feedbackforce Fte has a phase-lag with respect to the mechanicaldisplacement and can be written as Fte = βI2

dckx. Asis shown in appendix A the complex feedback coefficientβ can be calculated from the material constants and ge-ometry of the spring. In the linear approximation theresponse of the piezoresistive spring can be described bythe complex spring constant k∗eff = Fext/x:

k∗eff = k(1− βI2dc) (1)

If the imaginary part of the feedback coefficient is neg-ative (Im β < 0), the direction of the imaginary part ofthe effective spring force Fs = −k∗effx is opposite to thespeed v = iωx of the spring and thus increases the damp-ing. However, for Im β > 0, the damping is decreasedand the spring will amplify its own motion by the feed-back from the thermal expansion force. The sign of Im βdepends on the sign of the material’s piezoresistive coeffi-cient, but can also be tuned using an external impedanceas will be shown below.

As a consequence of the combination of the cyclic re-ciprocating motion of the engine beam with the displace-ment dependent piezoresistive heating, the beam expandsat a higher beam temperature Tb than at which it con-tracts if Im β > 0. The piezoresistive spring thus gener-ates mechanical power from the resistive heat of the DCcurrent source using a thermodynamic cycle and is there-fore a heat engine. Its thermodynamic cycle is illustratedin figures 1B and 1C. When this thermodynamic cycle is

reversed (Im β < 0), the device operates as a heat pumpor refrigerator.

This piezoresistive heat engine with effective springconstant k∗eff will be investigated by connecting it to a me-chanical resonator with mass m, intrinsic Q-factor Qint

and resonance frequency ω0 =√k/m. The spectral dis-

placement noise density <x2ω> near the resonance fre-

quency of this resonator with feedback at a temperatureT can be determined6,9,10 from equation (1) and the fluc-tuation dissipation theorem:

<x2ω> =

2ω30kBT/(kQint)

(ω2 − ω20)2 + (ωω0/Qeff)2

(2)

1

Qeff=

1

Qint− I2

dcIm β (3)

Where kB is Boltzmann’s constant and it is assumed thatQeff 1 and |βI2

dc| 1. Equation (3) demonstratesthat the effective Q-factor Qeff can be controlled by DCcurrent. For I2

dc≥(QintImβ)−1>0 the displacement am-plitude tends to infinity and a sustained oscillation of theresonator driven by the heat engine occurs14.

Even at lower current levels the thermal fluctuationsin the resonance mode are amplified by the heat engine.This increases the stored energy in the resonance, suchthat it equals the energy which would be stored in themode if the resonator would be at an effective tempera-ture Teff which is given by 1

2kBTeff = 12k∫∞−∞<x

2ω>

dω2π .

Note that the temperature Teff represents the mechanicalenergy stored in the resonance mode6,9,10 correspondingto the degree of freedom x and is not representative forthe temperature of the other degrees of freedom of theresonator.

Experiments are performed on a homogeneous crys-talline silicon structure, which is shown in figure 1D. Thestructure consists of a mass measuring 12.5×60.0×1.5µm3, which is suspended by a 3 µm wide spring beam anda 280 nm narrow engine beam. Both beams have a lengthof 800 nm. The structure is made out of a 1.5 µm thicksilicon layer on a silicon-on-insulator wafer, with a phos-phor doping concentration Nd=4.5×1018 cm−3 giving a

arX

iv:1

001.

3170

v1 [

cond

-mat

.mes

-hal

l] 1

9 Ja

n 20

10

2

A

resistance rac

elasticity forceF

displacementx

piezo-resistive

effectthermal

expansion

resistive2heating (µI )dc

thermodynamicfeedback

heatingpower

T

ppr

temperature

heatcapacitance & conductance

+Fte

+FextB

V,e

P,-s

2

3

4

1

p µ-epr

W

p <0pr

T =Tb avg

2

p =0pr

T <Tb avg

3p >0pr

T =Tb avg

4

C

p =0pr

T >Tb avg

1

T >Tb avg

TbTb

TbTb

T2 T3

T4

Idc

T2

Springbeam

Engine beam

10 mm2 mm

[100]

[010]

T1

D

Resonator mass

T1

Idc

Idc

Zlo

ad

R +rdc ac

R0

V +vdc ac

iac

Spectrumanalyzer

Voltageamplifier

1:1

T1

T2

mx

54 nF

C1

E

r =K R xac pr tot

R =R +R =794 Wtot dc 070 pF

12 kWC0

Ztot

k*eff

FIG. 1: A Besides the elastic coupling between displacement and force, a thermally delayed feedback mechanism via theinternal electrical and thermal variables occurs in a piezoresistive spring in the presence of a DC current flow. B Schematicthermodynamic cycle in the piezoresistive spring, when the device is operating at a single frequency as a heat engine (Im β > 0and therefore ppr ∝ −ε). The axes indicate the longitudinal strain ε and stress σ, which are proportional to the volume changeV and pressure P respectively. The enclosed area W represents the work generated by the engine per cycle. C Illustration ofthe phases 1-4 of the thermodynamic cycle shown in 1B, which are identified by the AC resistive heating power ppr and beamtemperature Tb compared to the time averaged beam temperature Tavg. To illustrate the position dependent heating powera fictitious external heat sink (blue) and source (red) are drawn. In reality the internal heating power in the beam dependson its position as a result of the piezoresistive effect. D Scanning electron microscope (SEM) image of the heat engine. Theinset shows a magnification of the wide spring beam and narrow engine beam by which the resonator mass m is suspended. EDrawing of the mechanical resonator (not to scale) and the passive electrical measurement schematic. The total AC impedanceparallel to the resonator Ztot = (iωC0)−1 + (iωC1 +Z−1

load)−1 can be controlled by adjusting Zload. The x-displacement inducesa piezoresistive voltage vac = [Z−1

tot + R−1tot]

−1IdcKprx, which is measured by the spectrum analyzer. The piezoresistive factorKpr was measured14 to be Kpr = −0.66 × 106 m−1. The DC resistance Rtot = Rdc + R0 = 794Ω, bias C0 = 54 nF and cableC1 = 70 pF capacitances were measured using an impedance analyzer at 1.26 MHz.

specific resistivity of ρdc=10−4Ωm. The thin crystallinesilicon layer is structured by a deep reactive ion etch andthe buried SiO2 layer below the mass and beams is re-moved in a hydrogen fluoride vapour etch. The in-planemechanical bending resonance mode that determines thefundamental resonance frequency f0=1.255 MHz has aspring constant k=256 N/m as determined from finiteelement method (FEM) simulations14. As a result ofthe geometry, the strain and current density are con-centrated in the narrow engine beam, which thereforegenerates most of the mechanical power. Measurementsare performed at 315 K in a vacuum chamber at a pres-sure below 10−3 mbar. The DC current Idc is driventhrough the beams via terminals T1 and T2 as shownin figure 1E. Terminals T3 and T4 are grounded. Thetotal impedance Ztot parallel to the resonator to groundcan be adjusted via Zload. The AC voltage spectrum vacis detected by a 1 MΩ input impedance voltage buffer

amplifier and a spectrum analyzer. The detected voltagevac is proportional to the displacement x as a result ofthe piezoresistive effect.

The voltage spectral density <v2ω> of the resonator

is measured at several values of Idc, in the open state(Zload = ∞) and with a load impedance Zload = 40µHas shown in figures 2A and B. Although the spectra aresimilar at low currents (Qeff = 9 × 103), they stronglydiverge at larger currents. For Zload = ∞, the spectralshape narrows and Qeff increases to 1.5×106 at Idc=1.045mA. At currents above 1.05 mA the resonator is spon-taneously brought into sustained oscillation14. On theother hand for Zload = 40µH, the spectra broaden andQeff decreases to 1.9×103 at 3.0 mA.

The voltage spectral density in figures 2A and B is fitby dashed curves which are given by equation (2) with<v2

ω>= I2dcK

2pr<x

2ω>/[Z

−1tot +R−1

tot]2 as follows from figure

1E and which include a parameter to fit the white John-

3

1.2540 1.2560f (MHz)

10-16

10-14

10-12

10-10

2<v ù

2 > (V

2/H

z)

3.0 mA2.4 mA1.6 mA1.0 mA0.5 mA0.2 mA

1.2550 1.2552 1.2554 1.2556f (MHz)

10-16

10-14

10-12

10-10

2<v ù

2 > (V

2/H

z)

1.045 mA0.8 mA0.6 mA0.4 mA0.2 mA

A B

C

FIG. 2: Voltage spectral density<v2ω> measured by a spec-

trum analyzer as a function of frequency f = ω2π

. A Inthe open state (Zload = ∞) the thermal motion is ampli-fied when the heat engine is driven by a current Idc. B ForZload = 40µH, the mechanical energy decreases with increas-ing current and the device operates as a refrigerator. Note thefactor 4 difference in frequency scale. C 1/Teff as determinedfrom the data in A and B by evaluating 1

2kBTeff = 1

2k <x2>,

where <x2> is determined by integrating equation (2). Thedifference between the y-axis intersection of the data and theambient temperature is within experimental uncertainty lim-its.

son noise background. By integrating the fitted <x2ω>

curves, Teff is determined and is plotted in figure 2C.Depending on the load impedance Zload, the effectivetemperature Teff of the resonator either increases until itstarts to oscillate or is cooled down to 70 K at 3 mA. DCresistive heating will also lead to a temperature increaseof the beam which opposes the cooling mechanism. It isestimated from the shift in the resonance frequency andfrom FEM simulations14 that the maximum temperatureof the resonator is 355 K at 3 mA. This temperature in-crease of 13% by DC resistive heating is relatively smallcompared to the factor 5 decrease in Teff as a result of thereduction of Qeff . Although a current controlled modi-fication of the spring constant according to equation (1)is expected to occur at all frequencies, it is most clearlyobserved near a resonance mode of a resonator to whichthe spring is coupled, since it has a large effect on theQ-factor Qeff of the resonator.

The difference between the spectra in figures 2A andB is caused by the fact that the coefficient β is pro-portional to the ratio between the piezoresistive heat-ing power and the AC resistance γZ ≡ ppr/(I

2dcrac) as

can be seen from figure 1A. Therefore the coefficient βin equation (1) is the product of two complex numbers:β = γZβ0, where γZ can be controlled externally viaZload and β0 depends solely on the device geometry andmaterial parameters. Figure 1E shows that a resistance

B

0 2 4 6 8I2

dc (mA

2)

0

10

20

30

40

x10-5

Zload

=40 mH

22 mH

10 mH

5.6 nF

51pF 1.0 nF

A

¥

0 2 4 6 80

2

4

6

8

10

12

14

16

x10-5

1/Q

eff

Zload

=9 ?

91 ?

240 ?

554 ??

I2

dc (mA

2)

FIG. 3: Inverse effective Q-factor 1/Qeff versus DC currentI2dc for different values of the load impedances Zload indi-

cated in the graph. Dashed lines are fits using equation (3).Symbols on the x-axis correspond to the minimum currentlevel at which sustained oscillation was observed. A Resis-tive load impedances Zload. B Capacitive and inductive loadimpedances Zload.

change rac of the beam will induce both an AC currentiac = −vac/Ztot and AC voltage vac = Rtotiac + Idcrac.The total heating power Pdc + ppr in the engine beamis given by (Idc + iac)

2(Rdc + rac) and for rac Rdc itfollows that γZ = 1− 2Rdc/(Ztot +Rtot).

Voltage spectra like those in figure 2 are recorded forseveral real and imaginary values of Zload, using resistors,capacitors and inductors. The fitted values of Qeff areplotted in figure 3. For small values of Zload a moreaccurate determination of Qeff was made by generatinga random electrostatic noise force on the resonator usinga white voltage noise source connected to terminal T3.The observed linear dependence of 1/Qeff on I2

dc in figure3 corresponds well with equation (3). From the slopeof these linear fits, Im β is determined and is plottedagainst γZ as symbols in figure 4. As shown by solid anddashed lines in figure 4, an excellent fit of the Im β datais obtained using a multiple linear regression which yieldsthe fit parameters Re β0 = −123.7 A−2, Im β0 = 64.6A−2 and Rdc = 439.3Ω.

The mechanism proposed in figure 1A is in good quan-titative agreement with these measurements. In ap-pendix A an analytical model is derived for the com-plex Young’s modulus and spring constant k∗eff and itis used to derive an estimate for β0, which yields β0 =−132+157i A−2. A finite element simulation14 of the fullgeometry including the anisotropy of the silicon crystalresults in β0 = −111 + 58i A−2.

When Im β is positive, the device operates as a heatengine and its thermodynamic cycle is illustrated in fig-ures 1B and C. As a result of thermal delay, the temper-

4

1

Z =Lload

Z =CloadZ =Rload

Z =Lload

Z =Rload

Z =Cload

Cooling

Amplification

Cooling

Amplification

-Im (gb)Z 0

-0.2 0.0 0.2 0.4 0.6 0.8Re g

Z

-100

-80

-60

-40

-20

0

20

40

-0.4 -0.2 0.0 0.2 0.4 0.6Im g

Z

-100

-80

-60

-40

-20

0

20

40-2

-Im

b (

A)

FIG. 4: Measured slopes −Im β of the curves in figure 3 plot-ted along the real (left) and the imaginary (right) γZ-axis forresistive (red circle), capacitive (blue square) and inductive(green diamond) values of Zload. An excellent fit of the data isobtained by plotting −Im (γZβ0) with β0=−123.7+64.6iA−2,for resistive (dashed lines), capacitive and inductive (solidlines) values of Zload. If Im β is positive the engine beam actsas a heat engine that amplifies the effective temperature andif Im β is negative the engine beam acts as a refrigerator thatcools the effective temperature of the resonator.

ature T and stress σ are phase shifted with respect to thestrain ε and piezoresistive heating power ppr. Thereforethe engine beam expands at a higher temperature thanat which it is compressed. It thus generates an amount ofmechanical work W from piezoresistive heat during eachcycle.

When Im β is negative and ppr ∝ ε, the direction ofthe heat cycle in figure 1B is reversed and the deviceoperates as a heat driven refrigerator. Like Maxwell’sdemon it operates as a heat ratchet15–19 that generatesa unidirectional flow of heat out of the resonator, simi-lar to theoretically proposed devices17,18. This methodof refrigeration represents an interesting alternative tooptical6–11 or RF13 cooling methods, since it does not re-quire optical or RF sources and their alignment. Becausethe sign of Im β can be made frequency-dependent usingelectrical filters at the position of Zload, the piezoresistivedevice might also be operated simultaneously as a heatengine and refrigerator, amplifying the displacement atone resonance frequency of the resonator while cooling itat another resonance frequency.

In contrast to the Peltier effect the direction of theheat flow is independent of the direction of the electricalcurrent. Therefore the device does not necessarily have tobe powered by a battery or solar cell, but can be poweredby any electrical DC or AC waveform whose resistiveheating power is modulated by the piezoresistive effect.In theory the heat engine could even be driven by randomthermal Johnson-Nyquist noise, and might thus generatea unidirectional heat flow when it is only connected to ahot resistor17.

As shown in figure 2A, the piezoresistive heat engine

can amplify the random thermal, or Brownian motion ofthe resonator. We speculate that this amplified randommotion might still be converted to directional work if itis rectified. This might be done for example by usingthe resonator as a pawl in Feynman’s ratchet-and-pawlBrownian motor20, or by using it to increase the effectivetemperature of a resonator with a triangular mass21 suchthat it is a Brownian motor which can propel itself di-rectionally through a fluid of homogeneous temperature.

We have demonstrated a solid-state piezoresistive heatengine and refrigerator that can be reduced to micro-scopic dimensions. The device might drive microme-chanical oscillators14, motors22 and sensors. However,even though man-made heat engines outperform biologi-cal engines on the macroscopic scale and are essential forlong-rang transportation, it remains to be seen whetherthey can ever compete with biological23 or artificial16,24

molecular motors on the microscale. Anyhow, their man-ufacturability and the possibility to operate them over awider range of environmental conditions are significantadvantages.

Acknowledgments

We thank J.J.M. Ruigrok, C.S. Vaucher, K. Reimann,C. v.d. Avoort, R. Woltjer and E.P.A.M. Bakkers fordiscussions and suggestions and thank J. v. Wingerdenfor his assistance with the SEM measurements.

5

Appendix A: Analytic model for β0

In this appendix the thermodynamic coupling betweenstress and strain in piezoresistive materials carrying acurrent is discussed in more detail. An analytical expres-sion is derived for the effective complex Young’s modulusY ∗eff . This expression is used to provide an analytical es-timate for the feedback coefficient β, which expresses theeffect of a DC current on the effective spring constantk∗eff of a piezoresistive spring. The analytical result iscompared to the measurements.

To evaluate the effect of a DC electric current on themechanical properties of a piezoresistive solid, considera solid through which a current density Jdc is flowingin the y-direction. When a small uniaxial vibrationalstress σace

iωt with frequency ω is present along the y-direction the piezoresistive effect will induce an AC re-sistivity change ρac:

ρaceiωt = ρdcπlσace

iωt (A1)

Where ρdc is the unstressed resistivity and πl is the lon-gitudinal piezoresistive coefficient25. It is assumed thatρac ρdc. This resistivity change modifies the AC resis-tive heating power density ppr:

ppreiωt = γZJ

2dcρace

iωt (A2)

As discussed in the main manuscript, γZ is a correctionfactor which is needed if a finite impedance to ground ispresent parallel to the piezoresistive resonator, such thatAC currents iac can also contribute to ppr. From the heatequation p = −kh∇2T + cpρd∂T/∂t it follows that theheating power causes sinusoidal temperature fluctuationswith amplitude Tac:

Taceiωt =

ppreiωt

γh + iωcpρd(A3)

The specific heat capacity is given by cp and ρd is themass density. The effect of thermal heat conductivity khis captured by the factor γh:

γh = −kh(∇2Tac)/Tac (A4)

The factor γh depends on the resonator’s geometry as willbe discussed below. The temperature increase generatesa thermal expansion stress:

σac,te = αteY Tac (A5)

Y is Young’s modulus and αte is the thermal expansioncoefficient. This thermal expansion stress adds to the ex-ternally applied stress σac,ext, such that the total stressis given by σac = σac,ext + σac,te, and the correspondingstrain is given by εac = σac/Y . The stress-strain rela-tion of the piezoresistive solid is therefore the same asthat of a solid with an effective complex Young’s mod-ulus Y ∗eff = σac,ext/εac. The variables, coupling mech-anisms and multiplicative factors in equations (A1-A5)

T2

resistivity temperature

elasticity

stressstrain

piezo-resistive

effectthermal

expansionthermodynamic

feedback

resistiveheating

1

Tac

óacåac

Y

ð Ylñdc

heatingpowerdensity

ãZJ2dc

ppr

ã h +iùcpñd

1h +iùc d

Yáte

/Y +

heat conductivity and capacity

ñac

+ óac,ext

ó =âI ódc acac,te2

FIG. A1: Schematic of the thermodynamic coupling mecha-nism via the piezoresistive, resistive heating and thermal ex-pansion effects. Relevant variables in the mechanical, electri-cal and thermal domain are shown. Coupling mechanisms arerepresented by arrows.

have been schematically shown in figure A1 and usingthese equations the effective modulus can be expressedas:

Y ∗eff(ω) ≡ σac − σac,teεac

≈ Y(

1− γZρdcπlJ

2dcαteY

γh + iωρdcp

)(A6)

This relation is only valid for small stress and strain,such that the linear approximation is valid and materialparameters are constant. When all constants in equation(A6) are real and positive, equation (A6) is identical tothe Young’s modulus of the standard anelastic solid26,however if some of the constants are negative or have anon-zero imaginary part, it can lead to properties likemechanical self-amplification and negative creep. Thismodification of the effective Young’s modulus can occurin any piezoresistive solid in the presence of electricalcurrent. Besides its effect on the dynamics of mechanicalstructures it might thus also affect the propagation ofacoustic waves in solids.

As follows from appendix A of Nowick and Berry26,the effective spring constant of a resonance mode witha complex position dependent Young’s modulus is givenby:

k∗eff = k

(∫VY ∗eff(z)ε2

ac(z)dz∫VY ε2

ac(z)dz

)= k(1− βI2

dc) (A7)

Where the integrals run over the volume V of the res-onator. Combining this with equation (A6), it is foundthat the feedback coefficient β can be expressed as:

β = γkγZρdcπlαteY

A2(γh + iωρdcp)= γZβ0 (A8)

Where A is the cross-sectional area of the piezoresistiveconductor, and γk is a factor which accounts for the non-uniformity of Y ∗eff . If Y ∗eff is constant over the whole vol-ume, γk = 1.

For the resonator under consideration, the thermody-namic feedback effect is large inside the narrow enginebeam where the strain and current density are concen-trated. It is therefore a good approximation to assume

6

that Y ∗eff is given by equation (A6) inside the volumeVengine of the engine beam and is equal to Y everywhereelse in the resonator. In this case the factor γk is givenby:

γk =

∫Vengine

ε2ac(z)dz∫

Vε2ac(z)dz

(A9)

For the lowest frequency in-plane bending resonancemode this fraction is found to be γk = 0.11 using a finiteelement simulation.

To evaluate γh the AC heat equation is to be solved:

kh∇2Tac + ppr = icpρdωTac (A10)

Since the dimensions of the engine beam are much smallerthan the thermal wavelength λh =

√8π2kh/(cpρdω0) ≈

26 µm, the heat equation can be simplified by assumingthat the AC resistive heating density ppr is zero out-side the engine beam and that ppr and Tac are inde-pendent of position inside the engine beam. The beamcan be treated as a point source at y = 0 with tem-perature Tac,0e

iωt. The corresponding solution of the 1-dimensional heat equation outside the beam is:

T (x, t) = Tac,0e−(1+i)2π|y|/λh+iωt (A11)

Since the heat conduction from the 2 ends of the beamneeds to be equal to the difference between the generated

and stored energy in the beam, it follows from equation(A3) that:

− 2kh∇Tac|y=0 = L(ppr − icpρdωTac) = LγhTac (A12)

Where L=800 nm is the beam length. Substitution ofequation (A11) yields:

γh = 4π(1 + i)kh/(Lλh) (A13)

The material constants along the [100] direction of n-type silicon are Y = 130 GPa, πl = −102×10−11 Pa,αte = 2.6 × 10−6 m−1, ρd =2329 kg/m3, cp = 702J/kg·K and kh =113 W/(K·m). The cross-sectional areais A=280×1500 nm2. Substituting these values in equa-tions (A13) and (A8) and using γk=0.11 from equation(A9) yields β0 = −132 + 157i A−2. This value is higherthan the experimental value, which is mainly attributedto the fact that the actual heat flux outside the beamis not 1-dimensional but radial, which results in a largervalue of γh. A finite element simulation of the full ge-ometry, including the anisotropy of the silicon crystalresults14 in β0 = −111 + 58i A−2, in good agreementwith β0 = −123.7 + 64.6i A−2 as determined from thefits of the measurements in figure 4.

∗ Electronic address: peter.steeneken@nxp.com1 A. M. Fennimore, T. Yuzvinsky, W.-Q. Han, M. S. Fuhrer,

J. Cumings, and A. Zettl, Nature 424, 408 (2003).2 D. L. Fan, F. Q. Zhu, R. C. Cammarata, and C. L. Chien,

Phys. Rev. Lett. 94, 247208 (2005).3 A. H. Epstein, J. Eng. Gas Turbines Power 126, 205

(2004).4 S. A. Jacobson and A. H. Epstein, Int. Symp. Micro-Mech.

Eng. (ISMME) pK18 (2003).5 C. M. Spadaccini and I. A. Waitz, Comprehensive Mi-crosystems (Elsevier, 2008), chap. 3.15 Micro-Combustion.

6 P. Cohadon, A. Heidmann, and M. Pinard, Phys. Rev.Lett. 83, 3174 (1999).

7 C. H. Metzger and K. Karrai, Nature 432, 1002 (2004).8 O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and

A. Heidmann, Nature 444, 71 (2006).9 D. Kleckner and D. Bouwmeester, Nature 444, 75 (2006).

10 C. Metzger, I. Favero, A. Ortlieb, and K. Karrai, Phys.Rev. B 78, 035309 (2008).

11 T. J. Kippenberg and K. J. Vahala, Science 321, 1172(2008).

12 J. Teufel, T. Donner, M. A. Castellanos-Beltran, J. Harlow,and W. Lehnert, Nature Nanotech. 4, 820 (2009).

13 K. R. Brown, J. Britton, R. J. Epstein, J. Chiaverini,D. Leibfried, and D. J. Wineland, Phys. Rev. Lett. 99,137205 (2007).

14 K. L. Phan, P. G. Steeneken, M. J. Goossens, G. E. J.Koops, G. J. A. M. Verheijden, and J. T. M. van Beek

(2009), to be published, http://arxiv.org/abs/0904.3748.15 I. Derenyi, M. Bier, and R. D. Astumian, Phys. Rev. Lett.

83, 903 (1999).16 V. Serreli, C.-F. Lee, E. R. Kay, and D. A. Leigh, Nature

445, 523 (2007).17 J. P. Pekola and F. W. J. Hekking, Phys. Rev. Lett. 98,

210604 (2007).18 N. Li, P. Hanggi, and B. Li, Europhys. Lett. 84, 40009

(2008).19 P. Hanggi and F. Marchesoni, Rev. Mod. Phys. 81, 387

(2009).20 R. P. Feynman, R. B. Leighton, and M. Sands, The Feyn-

man Lectures on Physics, vol. I, ch. 46 (Addison-Wesley,1963).

21 C. Van den Broeck and R. Kawai, Phys. Rev. Lett. 96,210601 (2006).

22 T. Ebefors and G. Stemme, The MEMS Handbook: MEMSApplications (Taylor & Francis Group, 2006), chap. Micro-robotics.

23 M. Schliwa and G. Woehlke, Nature 422, 759 (2003).24 R. A. van Delden, M. K. J. ter Wiel, M. M. Pollard, J. Vi-

cario, N. Koumura, and B. L. Feringa, Nature 437, 1337(2005).

25 C. S. Smith, Phys. Rev. 94, 42 (1954).26 A. S. Nowick and B. S. Berry, Anelastic relaxation in crys-

talline solids (Academic Press, 1972).