Low-Power Photothermal Self-Oscillation of Bimetallic Nanowires

Low-power photothermal self-oscillation of bimetallic nanowires

Roberto De Alba,1 T. S. Abhilash,1 Richard H. Rand,2, 3 Harold G. Craighead,4 and Jeevak M. Parpia1 1Department of Physics, Cornell University, Ithaca NY, USA 2Department of Mathematics, Cornell University, Ithaca NY, USA 3Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca NY, USA 4School of Applied and Engineering Physics, Cornell University, Ithaca NY, USA We investigate the nonlinear mechanics of a bimetallic, optically absorbing SiN-Nb nanowire in the presence of incident laser light and a reflecting Si mirror. Situated in a standing wave of optical intensity and subject to photothermal forces, the nanowire undergoes self-induced oscillations at low incident light thresholds of < 1 µW due to engineered strong temperature-position (T -z) coupling. Along with inducing self-oscillation, laser light causes large changes to the mechanical resonant frequency ω0 and equilibrium position z0 that cannot be neglected. We present experimental results and a theoretical model for the motion under laser illumination. In the model, we solve the governing nonlinear differential equations by perturbative means to show that self-oscillation amplitude is set by the competing effects of direct T -z coupling and 2ω0 parametric excitation due to T -ω0 coupling. We then study the linearized equations of motion to show that the optimal thermal time constant τ for photothermal feedback is τ → ∞ rather than the widely reported ω0τ = 1. Lastly, we demonstrate photothermal quality factor (Q) enhancement of driven motion as a means to counteract air damping. Understanding photothermal effects on micromechanical devices, as well as nonlinear aspects of optics-based motion detection, can enable new device applications as oscillators or other electronic elements with smaller device footprints and less stringent ambient vacuum requirements.

Micro- and nano-mechanical resonators are widely cal system is said to be in the “sideband-resolved regime,” studied for applications including electro-mechanical cir- and radiation-pressure effects are enhanced at laser fre- 1 2 18,19 cuit elements and sensing of ultra-weak forces, masses, quencies of Ωc ± ωm. Radiation-pressure-based feed- 3 and displacements. An integral part of these systems is back with red detuning (Ωc − ωm) is currently one of the the detection method employed to readout motion, which most promising experimental techniques for suppressing must itself be extremely sensitive and inevitably imparts thermal motion and thereby accessing quantum behavior its own force on the resonator, influencing the dynamics. in mechanical systems.20 Such low-κ optical systems can, The phase relation between mechanical motion and the however, be difficult to attain and miniaturize. resulting detector back-action determines whether this interaction will serve to dampen vibrations or amplify Photothermal feedback places less stringent require- them, potentially leading to self-oscillation if the detector ments on the optical system (as we show in this work), supplies enough energy per cycle to overcome mechanical and has been explored in a broad range of mechan- 9,10,12,21–24 damping. ical device geometries through experiment, simulation,25,26 and theoretical studies.10,21,27 While Feedback due to external amplifiers has been used these works provide many insights into the underlying to generate self-oscillation of micro-mechanical res- physics, some neglect the thermally-induced change in 4–8 onators; in such systems the oscillation amplitude R resonator equilibrium position z0, while others neglect is set either by nonlinearity of the amplifier or of the the change in resonant frequency ω0. In this work we resonator. Systems in which mechanical motion influ- have developed bimetallic nanowires that are designed ences the amount of laser light circulating in an opti- to be especially susceptible to the photothermal force 9–12 cal cavity or magnetic flux through a Superconduct- – devices in which optically-induced changes to z0 and 13,14 ing QUantum Interference Device (SQUID) have also ω0 cannot be neglected. Temperature-position coupling been shown to self-oscillate under the right experimen- dz/dT is provided by supporting bimetallic cantilevers tal conditions. In these systems R is set largely by the at either end of the nanowire (shown in Fig. 1 (a)), and periodicity of the detection scheme – either R ≈ λ/4 induces self-oscillation as well as changes in z0. At room

arXiv:1610.07591v4 [cond-mat.mes-hall] 28 Dec 2016 where λ is the laser wavelength or R ≈ Φ0/2 where Φ0 temperature these cantilevers apply an upward torque is the displacement needed to change the SQUID flux by on the nanowire and change its z position when its ten- one flux quantum. In the case of a mechanical resonator sion changes due to thermal expansion. Temperature- coupled to an optical cavity, back-action can arise either frequency coupling dω0/dT , also due to thermal expan- from radiation pressure or photothermal force – that is, sion, produces an overall shift in ω0 (Fig. 1 (c)) and mod- thermally-induced deflection caused by optical absorp- ifies motion through 2ω0 parametric excitation of the res- tion. The effects of these two forces are identical if the onant frequency. We adapt the perturbation theory first cavity resonance (with frequency Ωc and width κ) is suffi- discussed in Ref. 10, and present our results for a general ciently broad;11,12,15–17 however if κ is much smaller than optical intensity profile g(z). We then linearize the gov- the mechanical vibration frequency ωm the optomechani- erning coupled z, T equations to study nanowire behavior 2

high speed sample stage with piezo actuator (b) photo- Si chip (a) detector with nanowire

abs -

55nm 10µm beam focusing splitter objective vacuum chamber (d) (c) 30 = 11.2µW 5 95 = 13,400 = 8.4µW 2 4 96 20 = 5.6µW / (%) = 9,700 abs = 7,600 = 2.8µW

3 97 Voltaeg (AU) = 6,200 Photo-detector Amplitude ( µV/µW ) 10 1 2 98 ( ), In-plane Power/ 1 99 0 3.02 3.025 3.03 0 0 100 Frequency (MHz) -1/2 -1/4 0 1/4 1/2 (+)/ FIG. 1. The optomechanical system and experimental setup. (a) False-color scanning electron micrograph of our suspended device; blue: the SiN/Nb bilayer. Arrows indicate the competing tensile force and bimetallic “torque” that provide dz/dT coupling. Inset: magnified top-down image of the nanowire. (b) The experimental setup: nanowire absorption modulates the reflected laser power, which is recorded by a high-speed photo-detector. (c) Nanowire resonance at laser powers below the threshold for self-oscillation, driven inertially by a piezo actuator; solid lines are Lorentzian fits. Considerable frequency softening dω0/dT and Q-enhancement can be seen as P increases. (d) The optical intensity profile g(z) versus distance z + φ to the Si mirror. Because the nanowire is much narrower than the incident laser beam, only ≈ 3% of laser light interacts with the nanowire; of this 3%, the nanowire absorbs ≈ 70%. Self-oscillation occurs if the static nanowire is located in a shaded region and the power P is sufficiently high. A dashed line indicates the Taylor-series approximation for g(z) used in the perturbation theory. at laser powers below the threshold for self-oscillation. Measured in the time domain (Fig. 2 (a)), purely sinu- soidal motion with R ≈ λ/4 results in a detected signal Our optomechanical system is depicted in Figure 1 that saturates as z traverses the extremes of g(z). This (a,b). The nanowire has dimensions of ∼ (50 nm)2 × results in detected harmonics of the vibration frequency 40 µm and is suspended 8 µm above a Si back-plane. In- that can readily be measured in the frequency domain. cident laser light (beam diameter d ≈ 2.5 µm) is focused L Figure 2 (b) shows the nanowire motion as measured by a near the wire center, and reflects off of the underlying Si multi-channel lock-in amplifier whose reference frequency to form a standing wave of optical intensity; our one- is centered at the resonant frequency ω ≈ 2π × 3 MHz mirror optical system thus functions similarly to a very 0 with a 10 kHz bandwidth for three different laser powers; low-finesse two-mirror cavity. The total optical power (or the three harmonics shown (ω , 2ω , 3ω ) were measured ~ 2 0 0 0 more precisely, the electric field energy density |E(z)| ) simultaneously. The reference frequency was adjusted at in a plane parallel to the mirror at a distance z is given each power to follow the resonance. Nanowire motion is by P g(z), where g(z) is the dimensionless intensity pro- plotted as X and Y quadratures, or in-phase and out-of- file and P is the incident beam power; all P values given phase components relative to a fixed phase. The lower throughout this work signify this total beam power. Be- panel displays nanowire motion just below the critical cause the nanowire is extremely narrow, it covers only power (Pcrit = 22 µW), which is a combination of thermal ≈ 3% of the incident beam by area and is therefore as- motion and electrical noise about the origin; this has the sumed not to influence g(z). It does, however, absorb expected Gaussian distribution. As P is increased above a small portion of the local power Pabs, and nanowire Pcrit, all three harmonics demonstrate sharply-defined motion generates fluctuations in the reflected laser beam nonzero amplitudes. This optically-induced motion has a that can be measured using a high-speed photodetector. phase that randomly cycles through all possible angles as The detected signal is proportional to P −Pabs, as shown time progresses at nearly constant amplitude. All plots in Fig. 1 (d). This detection method has the benefit of show 1,000 data points except for the lower panels which utilizing the same light which induces self-oscillation, but each contain 2,000 points. Figure 2 (c) shows the ampli- is highly nonlinear for oscillation amplitudes R λ/8, √ & tude of these harmonics ( X2 + Y 2) for many values of where λ = 660 nm is the laser wavelength used. If the P . Solid lines are a best fit (with a total of 4 free param- optical field profile g(z) is known, this detector nonlinear- eters) based on the model described below. Deviation of ity can be used to deduce the absolute size of mechanical the fit at high powers could be due to aberrations of the motion. optical plane wave g(z) caused by the nanowire, as stud- Self-oscillation of the nanowire is shown in Figure 2. ied previously by Refs. 24 & 28. All measured signals are 3

ﬁeld to be:

0 (a) 2/ detected harmonics 50 0 2π(z + φ) π 3 2 0 g(z) = α + β sin − (3)

Voltage (mV) 0 Photo-detector 2 0 λ 4 -50 0 0.2 0.4 0.6 0.8 1 Time (µs) 0 0 0 Here α, β are determined by the reflection coefficient of 2 3 =58µW the Si back-plane, and φ is the P = 0 nanowire position (b) 200 300 0 (c) within the standing wave. The factor of −π/4 is added to 0 0 2 250 0 center the self-oscillation region (negative dg/dz region, -200 3 =31µW Fig. 1 (d)) about z + φ = 0. The total mirror-nanowire 200 200 distance is z + φ + (λ/2)(n − 1/4), where the integer n is 0 150 -200 irrelevant to our measurements. Amplitude ( µV/µW ) component ( µV/µW ) -200 0 200 -200 0 200 -200 0 200 100 In other device geometries, large mechanical resonators =21µW 10 can generate significant internal and external optical 50 0 reflections, producing a Fabry-Perot interference effect

-10 0 which results in g(z) having sharper peaks and wider 0 20 40 60 counts -10 0 10 -10 0 10 -10 0 10 (µW) valleys, or skewing its peaks left or right. For this reason component (µV/µW) FIG. 2. Photothermal self-oscillation. (a) Measured photo- we present our theoretical results for a general intensity detector signal during nanowire self-oscillation (circles), and profile g(z). In all cases, however, g(z) is periodic in λ/2. its decomposition into Fourier components (solid lines). Al- During self-oscillation, the resonator position is well though the nanowire motion is a near-pure sinusoid, the non- modeled by z(t) = z0 + R cos(ωt) where z0 is the linear optical readout results in detected harmonics at integer temperature-dependent equilibrium position. This value multiples of the oscillation frequency. (b) Phase portraits of can be estimated by solving Eqs. 1 & 2 for the case undriven nanowire motion as measured in the frequency do- of a static nanowire, which give the implicit equation main by a multi-channel lock-in amplifier centered about the resonant frequency. X and Y denote cosine and sine compo- z0 = τDP Ag(z0). Near P = 0 this formula has only one nents of motion. The critical power needed for self-oscillation solution for z0, but more solutions become available as is Pcrit = 22 µW. Data below this power (lowest row) is a P increases. For high enough P values, solutions near- combination of thermal motion and detector noise, while data est z = 0 can cease to be valid; this suggests that the above this power (upper rows) has a well-defined nonzero am- static wire exhibits discontinuous jumps in z0 as P is plitude. (c) Data points: amplitudes of the self-oscillation increased quasi-statically. The static solution to Eqs. 1 signals shown in (b) versus laser power P . Solid lines are a & 2 is studied further in the Supplementary Information. best fit based on the IPT model described in the text. While the static solution for z0 (and the corresponding temperature T0 = z0/D) is a useful starting point for an- alyzing the self-oscillating nanowire, in what follows we normalized by P , the laser power used. will show that typical oscillation amplitudes R produce The governing differential equations for the position sizable changes in T0 (and z0). and temperature of our photothermal system10 are: Although Eqs. 1-3 are nonlinear and cannot be solved exactly, perturbative methods can be applied. Here we employ the Poincaré-Lindstedtmethod, which requires 2 z¨ + γz˙ + ω0(1 + CT )(z − DT ) = 0 (1) scaling γ, C, and D in Eq. 1 by a small dimensionless 1 parameter ε 1. Eqs. 1 & 2 can then be solved for T˙ + T = P Ag(z) (2) τ z(t),T (t), and ω1 (the self-oscillation frequency) to any desired order in ε. The method also requires approxi- mating g(z) by the first few terms of its Taylor series. Here ω0, γ are the intrinsic resonant frequency and damping of the nanowire. T denotes the temperature above We expand g(z) about z + φ = 0 and keep enough terms ambient and C,D are the changes in resonator frequency such that the optical field is accurately modeled over an and position per unit temperature, respectively. The sec- entire period |z + φ| < λ/4: ond equation is Newton’s law of cooling, where τ denotes 3 the thermal diffusion time constant, and the right-hand- g(z) ≈ k0 + k1(z + φ) + k3(z + φ) 5 7 (4) side describes heat absorption from the incident laser. A +k5(z + φ) + k7(z + φ) includes the thermal mass and optical absorption of the nanowire, as well as its ≈ 3% area coverage of the in- 3 where k0 = (α + β/2), k1 = −2πβ, k3 = (16/3)π β, k5 = cident laser beam. Detailed calculations of the thermal 5 7 −(64/15)π β, and k7 = (512/315)π β. A comparison of parameters in Eqs. 1 & 2 based on the materials and this approximation with the exact g(z) is shown in Fig. 1 dimensions of our system are presented in the Supple- (d). The perturbation theory is presented in its entirety mentary Information. in the Supplementary Information, but the main results Because the nanowire does not interact appreciably are given below. with the incident laser, we can approximate the optical Using Eq. 4 and solving Eqs. 1 & 2 to order ε1 gives 4 the following equation for R: As mentioned above, FOPT predicts a change in the time-averaged temperature of the nanowire due to self- 2 4 6 0 = c0 + c1R + c2R + c3R (5) oscillation. This addition to T0 is where 3 2n (2n) X R gz0 δT = −T + τP A (7) ω2D γ 0 0 22n(n!)2 c = 1 g(1) + n=0 0 2 2 z0 2 1 + ω1τ τ PA (3) (2) The nanowire equilibrium position thus relocates to z = ω2D g ω2C g 0 1 z0 0 z0 D (T + δT ) during self-oscillation. Although one could c1 = 2 2 2 − 2 2 1 0 0 1 + ω1τ 2 1!2! 1 + 4ω1τ 2 0!2! proceed to order ε2 in perturbation theory to account for ω2D g(5) ω2C g(4) this equilibrium shift, the resulting algebraic expressions c = 1 z0 − 0 z0 2 1 + ω2τ 2 242!3! 1 + 4ω2τ 2 231!3! quickly become cumbersome. An approach that is easier 1 1 to implement and was used to fit the data in Fig. 2 (c) 2 (7) 2 (6) ω1D gz0 ω0C gz0 is to recursively perform FOPT while updating T0 and c3 = 2 2 6 − 2 2 5 1 + ω1τ 2 3!4! 1 + 4ω1τ 2 2!4! z0 with successive δT0 values. Starting with the static nanowire solution (z0 = τDP Ag(z0)), R and δT0 are it- 2 2 Here we have introduced ω1 = ω0(1 + CT0) as the new eratively calculated until R converges on a fixed value (n) th resonant frequency and gz0 as the n derivative of g(z) and δT0 converges on zero. This scheme is hereafter re- evaluated at z = z0. This result is hereafter referred to as ferred to as Iterated Perturbation Theory (IPT). We find the First Order Perturbation Theory (FOPT) solution. in practice that IPT converges most reliably if δT0 is mul- The number of terms in Eq. 5 increases if more terms tiplied by a small scaling factor (0.05 was used) before are kept in the Taylor expansion Eq. 4 (following the being added to T0; convergence typically occurs within clear pattern in c0 . . . c3), however the terms shown are 20-100 iterations. sufficient to accurately model our experimental data. A comparison of FOPT, IPT, and numerical integra- Eq. 5 indicates that R is influenced by both the tion of Eqs. 1–3 is shown in Figure 3 (a,b). The param- temperature-position coupling D and the temperature- eters used are derived from the IPT fit to experimental frequency coupling C. Interestingly, D influences self- data in Fig. 2 (c) – in this fit the only free parameters oscillation via temperature fluctuations at the oscillation were φ, τ, dL, and an overall vertical scaling factor. It frequency ω1, while C does so via temperature fluctua- should be noted that while IPT reproduces the results of tions at 2ω1; the effect of C is thus equivalent to paramet- numerical integration almost exactly, the former required ric 2ω1 excitation of the resonant frequency. Eq. 5 also only ∼1 second of computation time while the latter re- suggests that as z0 changes, C dominates near points of quired 4−5 hours. Fig. 3 (c) shows the nanowire position g (z0) with even symmetry (extrema) while D dominates as it moves through the optical field. The deviation of z0 near points with odd symmetry (inflection points). The away from its static value due to δT0 is clearly visible in threshold for self-oscillation occurs when R = 0 in Eq. 5; Fig. 3 (b). Interestingly, z0 trajectories from numerical this leads to c0 = 0 and gives a critical laser power of: integration and static theory intersect at z0 +φ = 0 – i.e. at the inflection point of g(z); here the odd symmetry of 2 2 γ 1 + ω1τ g(z) results in δT0 = 0 in Eq. 7. The inflection point is Pcrit = − (6) 2 2 (1) crossed by z at P ≈ 47 µW, while the maximum R value ω τ DAgz 0 1 0 occurs at the slightly higher power of P ≈ 56 µW. This expression reveals the source of low critical power As shown in the numerical integration results of Fig. 3 in our nanowire: a combination of low thermal mass A, (d), self-oscillation requires roughly 104 oscillation cy- long thermal time constant ω1τ ≈ 400, and large coupling cles to reach steady state at P = 60 µW. We note that D = 1.64 nm/◦C afforded by our cantilevers. Further, this “equilibration time” drastically increases for P val- 5 because γ, D, A are all positive, a negative optical gradi- ues approaching Pcrit = 22 µW; a maximum of 3 × 10 ent is needed for self-oscillation. While the sensitivity of cycles were required just above the transition. Also Pcrit on τ is rather weak for ω1τ > 1, it is noteworthy shown in Fig. 3 (d), the shift δz0 = 0.0251λ due to self- that short time constants τ → 0 inhibit self-oscillation. oscillation exactly matches the observed change in tem- ◦ ◦ We revisit this later in the paper where we discuss oper- perature δT0 = δz0/D = 10.1 C, where D = 1.64 nm/ C ation of the wires in the presence of N2 gas. For the case for this system. The numerical results in the Fig. 3 (d) in- D = 0,C 6= 0 Eq. 5 still supports limit cycle oscillations, set show that during self oscillation z(t) is a nearly pure but has no R = 0 solution. This suggests that z(t) = z0 tone at frequency ω1. A Fourier series fit to this data remains a stable equilibrium point even for P > Pcrit, (not shown) reveals that the next largest harmonic com- and only initial conditions of (z, z˙) sufficiently close to ponent is 2ω1, with 0.001% the amplitude of ω1 motion. z(t) = z0 + R cos(ωt) will lead to oscillation. One can It is the pureness of this tone that leads to the excel- therefore draw an attractor diagram to describe which lent agreement between numerical integration and IPT initial conditions lead to limit cycle behavior and which – after all the perturbation theory is predicated on the 12 approach the stable equilibrium. assumption z(t) = z0 + R cos(ω1t). Numerical results for 5

tive or positive. Therefore

2 (3) 2 (2) (a) (b) ω Dgz 2ω Cgz 0.2 1/16 1 0 0 0 2 2 < 2 2 (8) 1 + ω1τ 1 + 4ω1τ 0.15 + )/

/ onset of 0 0

( self-oscillation prediction for is the necessary condition for hysteresis. Because C < 0 0.1 static nanowire in this experiment, we would expect hysteretic behav- -1/16 numerical numerical ior when z0 is near a maximum of g(z). The width of 0.05 integration integration FOPT FOPT the hysteresis region (i.e. how low P can be while still IPT -1/8 IPT maintaining self-oscillation) is calculated in the Supple- 0 0 20 40 60 0 20 40 60 mentary Information. (µW) (µW) Lastly, we focus on the behavior of our nanowire for 0 (c) 1 approximate () (d) laser powers P < P . Since the vibration amplitude in exact () 60 crit

(°C) 68.62 0µW 1/4 T( ° C) ( + )/ this case is typically much smaller than λ/4, it suffices 0.8 22µW 40 1/8 68.6 0 68.58 to approximate g(z) by a linear expansion about z = z0 -1/8 68.56 20 4 0.6 0 1 2 03 (1) time-10 (2/ ) in Eq. 2: g(z) ≈ g(z0) + gz0 (z − z0). Furthermore, we 1/4 ( ) (normalized) 0.4 =60µW can neglect any time-dependent CT terms in Eq. 1. This 1/8 0 ( + )/ then leads to the linearized equations 0.2 60µW 0 x¨ + γx˙ + ω2(x − Du) = f eiωt (9) -1/8 1 d 0 4 -1/4 -1/8 0 1/8 1/4 0 0.25 0.5 0.75 1 × 10 1 0 u˙ + u = P Ag(1)x (10) (+)/ time (2/ ) z0 FIG. 3. Detailed behavior of the nanowire according to fits τ of the experimental data. (a,b) Comparison of the oscillation where we have introduced the new variables x = z − z , amplitude R and equilibrium position z calculated by pertur- 0 0 u = T −T and added the driving term f at frequency ω. bation theory and numerical integration, with φ/λ = −0.114. 0 d In this linearized system we can safely use the complex Note that z0 = 0 at P = 0. The shift in z0 due to self- iωt iωt oscillation is clearly visible in (b). (c) Nanowire position solutions x =xe ˜ and u =ue ˜ . Based on Eq. 10, these (1) within the optical field g(z) as P increases. Red points are related by u = x (τP Agz0 )/(1 + iωτ). Substituting (spaced every 1 µW) indicate the changing z0 value, while this into Eq. 9 and collecting real and imaginary terms, horizontal lines indicate the extent of R. (d) Numerical in- 2 one can recast the mechanical system asx ¨+γeff x˙+ωeff x = tegration results at P = 60 µW with the initial condition iωt fde where the effective resonant frequency ωeff and (z, z,˙ T ) = (0, 0, 0); only the upper and lower envelopes of os- damping γeff are: cillation are shown. In the lower panel, a solid line signifies the peak-peak moving average, which is an indication of z0. (1) ! 3 τDP Agz The shift in z0 after t = 5 × 10 closely follows the trend in ω2 = ω2 1 − 0 (11) T (t) shown in the upper panel. Inset: magnified image of eff 1 1 + ω2τ 2 these results near t = 104, showing the harmonic content of ω2τ 2DP Ag(1) z(t) and T (t). γ = γ + 1 z0 (12) eff 1 + ω2τ 2

Firstly, we note that the photothermal terms in ωeff con- 6 P > 60 µW reveal that higher harmonics of the ω1 mo- stitute a roughly 1 part in 10 correction for the experi- tion grow steadily as P increases (2ω1 reaching 0.004% at mental parameters used in this work; thus to very good 80 µW), possibly explaining the growing deviation from approximation ωeff = ω1. Next, we should expect self- IPT seen in Fig. 3 (a). The oscillation frequency in the oscillation to occur when γeff = 0. Substituting P = Pcrit Fig. 3 (d) inset√ is 0.93 ω0, in close agreement with the ex- from perturbation theory (Eq. 6) and ω = ω1 indeed gives pected ω1 = ω0 1 + CT0 ≈ 0.92 ω0, where C = −0.0022 γeff = 0, showing compatibility of these two models. In- ◦ and T0 = 68.58 C. The 1% increase in frequency is likely terestingly, the photothermal damping shift on resonance 1 τ 2 due to ω2, the -order correction to the oscillation fre- is ∆γ = |γeff − γ| ∝ 1+ω2τ 2 , which increases monotoni- quency, which is calculated in the Supplementary Infor- 1 cally as τ → ∞. Long time constants ω1τ 1 therefore mation. strengthen the photothermal effect. This can also be seen 1 Perturbation theory can also be used to predict by setting τ = 0 in Eq. 10, which results in u ∝ ix. In whether the onset of self-oscillation will be exhibit hys- this case, u is perfectly out of phase with x, meaning it teresis. Such behavior is referred to as a subcritical Hopf contributes entirely to damping in Eq. 9. bifurcation, and would manifest as a continuation of sta- These results appear to be counter to those of previous ble self-oscillation for some range of powers as P is de- theoretical studies which model the photothermal effect creased below Pcrit. The distinction between a hysteretic as a time-delayed back-action force F (x) that responds or non-hysteretic transition (subcritical or supercritical to changes in x after a time constant τ.12,21 Such a model τ dF bifurcation) depends upon whether c1 in Eq. 5 is nega- produces the result ∆γ ∝ 1+ω2τ 2 dx , which is maximized 6

(1) if z0 can extend to the next negative region of gz0 , near z + φ = λ/2, P would be large enough to support self- 2.5 1.005 (a) Pressure (Torr) (b) Pressure (Torr) oscillation. We note that for the four values of pres- 2.0 2.0 1.0 1 1.0 sure where self-oscillation is seen, the two lowest pres- 2 0.4 0.4 0.02 0 0.02 sures yield P ≈ 22 µW identical to the value with / crit 0.001 0.995 0.001 0,eff

1.5 /2 (kHz) no N2 gas added, and are consistent with the Q value eff 0.99 seen with no added gas. For the case of the two higher 1 0.985 pressures, 0.4 Torr and 1 Torr, the introduction of gas increases the damping (higher γeﬀ ) and shortens the τ, re- 0.5 0.98 quiring an additional power to overcome damping. Above

0 0.975 this pressure, self-oscillation cannot be reached in our 0 20 40 60 80 100 120 0 20 40 60 80 (µW) (µW) present setup. Even so, the results of Fig. 4 demonstrate

0 15 (c) 5000 (d) the capability of photothermal feedback to counteract air 1/ (kHz) 3000 10 damping at low pressures. Such optical Q-enhancement

1000 could lower the stringent vacuum requirements of typical 5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 micro-electro-mechanical device applications. Pressure (Torr) Pressure (Torr)

FIG. 4. Nanowire behavior for P < Pcrit under various N2 pressures. (a,b) Nanowire effective damping γeff and resonant We have presented an experimental and theoret- frequency ωeff . These values were obtained from Lorentzian ical study of photo-thermal feedback in mechanical fits to piezo-driven resonance peaks such as those shown in nanowires. While the device tested self-oscillates under Fig. 1 (c). Stars in (a) indicate the measured onset of self- the illumination of a 22 µW laser beam, only ∼ 3% of this oscillation. Solid lines are fits to Eqs. 11 & 12. (c) Q factors at beam is incident on the ultra-fine nanowire – suggesting P = 0 extrapolated from fits in (a,b). (d) Thermal diffusion that incident powers of < 1 µW are ultimately neces- rate 1/τ versus gas pressure. sary to induce motion. This is significantly lower than the 300 µW to few mW required in previously studied free-space photothermal structures,10,24,29 and lower still (in magnitude) when ωτ = 1 and vanishes as τ → ∞. than the ≈ 10 µW reported for an optical-cavity-coupled The discrepancy here lies in dF /dx. Adapting our Eqs. 9 photothermal structure,12 where the two-mirror cavity & 10 to such a model reveals that the thermal force mag- results in much higher optical field gradients dg/dz. The nitude (i.e. the asymptotic value after a change in x) is low power needed in our system is attributable to the (1) low thermal mass of the nanowire and large temperature- F (x) = kDu = kDτP Agz0 x, where k is the mechani- 2 position coupling D afforded by the supporting can- cal spring constant. This then leads to ∆γ ∝ τ , in 1+ω2τ 2 tilevers. A simple beam-theory calculation suggests that agreement with our earlier result. D scales with cantilever length L and width w as L3/w We have experimentally tested Eqs. 11 & 12 for several (see Supplementary Information), suggesting that even values of γ as shown in Figure 4. In these measurements stronger photothermal effects can readily be achieved. γ was varied by introducing pure N2 gas into our sample We have observed that the equilibrium position z0 of this test chamber; doing so added drag to the nanowire mo- system is strongly tunable with incident laser power and tion, resulting in higher intrinsic damping γ and lowered can drastically affect nanowire dynamics. Self-oscillation Q factors (Fig. 4 (c)). All preceding measurements were in this system is due in part to temperature oscillations −3 performed with pressure 10 Torr. The fits shown in at the vibration frequency ω1 and to parametric 2ω1 os- Fig. 4 (a,b) were constrained at the lowest two pressures cillations of the resonant frequency. The perturbation to maintain consistent thermal parameters with the fit theory used here can readily be adapted for systems in in Fig. 2 (c). At higher pressures τ was allowed to vary, which micro-mechanical resonators are coupled to mag- as nanowire interaction with ambient gas likely increases netic SQUID circuits, optical cavities, or other periodic its thermal dissipation rate. The laser waist diameter external systems. dL and initial optical field position φ were also allowed to differ from Fig. 2 (c) as each change in pressure re- It is well established that a self-oscillating system can quired manual refocusing, and the roughness of the Si become entrained if a sufficiently strong driving force is back-plane led to changes in φ based on exact laser po- applied – i.e. the system will oscillate at the driver fre- sitioning. Here φ/λ = 0.044 compared to the value of quency rather than its own natural frequency.9,30–32 Such −0.114 in Fig. 2 (c); dL = 2.5 µm for the two highest a system is promising for a number of electro-mechanical pressures and dL = 2.0 µm for all lower pressures. applications, including narrow bandpass filters and re- Curvature in the γeff and ωeff fits is due to the chang- lated electrical signal processing devices. Although we ing equilibrium position z0 as P increases, and the re- have observed such behavior in our nanowires (not shown (1) sulting change in gz0 . Because of this curvature, the here), further work is needed to extend the perturbation γeff trajectory for 2.0 Torr is not expected to enter self- theory to predict the entrainment bandwidth as a func- oscillation at higher P values. It is however possible that tion of driver strength and laser power. 7

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