Time Domain Control of Ultrahigh Frequency Nanomechanical Systems

Transcription

1 Supplementary Information Time Domain Control of Ultraig Frequency Nanomecanical Systems N. Liu, F. Giesen, M. Belov, J. Losby, J. Moroz, A.E. Fraser, G. McKinnon, T.J. Clement, V. Sauer, W.K. Hiebert, M.R. Freeman 1 Time-domain actuation and detection: timing diagrams and electrical pulse caracterization Figure S1 contains timing diagrams to elp illustrate te measurement tecnique. Panel S1a sows a set of pulses corresponding to te initiation and observation of a single event. Te two laser pulses, indicated in red, are derived from a single optical pulse using te first beamsplitter (BS) after te laser in Fig. 1. Te red "trigger" pulse commences te countdown of a pre-set (but computer controllable, Stanford Researc Systems DG535) electronic delay, after wic an avalance transistor-based electronic pulser is fired (Picosecond Pulse Labs 000D, wit "Turbo" option and no external carge line). Tis applies an electrostatic impulse excitation to te sample. Motion of te cantilever ensues, wit te displacement during tis specific event being measured or sampled just once, by a "probe" pulse arriving at a particular instant pre-determined by te setting of te electronics delay (tis 100 fs-wide pulse is effectively instantaneous on te time scale of te motion). Tere is insufficient signal from te reflection of a single probe pulse (typically < 50 pj probe pulse energy in te present setup), and signal is accumulated over many repeats of te event wit te same electronic delay setting. Most of te measurements reported ere were performed at an 800 khz repetition rate and averaged wit ~ 1s time 1

2 constants. Te repetitive nature of te measurement is illustrated in Fig. S1b. Te repetition rate is set low enoug for te cantilever to return to static equilibrium following eac pulse. Finally, in order to furter enance te sensitivity of te measurement, te application of te electrostatic impulses to te sample is modulated on and off on a longer time scale, allowing a pase-sensitive measurement of te probe response by a lock-in amplifier. In tis way, te interferometric signature of te cantilever motion becomes a small alternating (square wave) modulation of te reflected probe intensity, and is extracted background-free by te lock-in. Te modulation principle is illustrated in Fig S1c. It is implemented using a semiconductor RF switc (Mini Circuits ZYSWA-- 50DR), driven by a square TTL waveform from te frequency clock in Fig. 1, to alternate te electronic trigger pulse (derived from te optical trigger, recall panel a) between te trigger input of te avalance transistor pulser and te "trigger dump", a reflection-less 50 Om termination (see Fig. 1). Te freq. clock simultaneously feeds a reference input to te lock-in. As a result of tese steps, a displacement sensitivity of ~1 pm / Hz is acieved (see te following section, on te calibration of te cantilever displacement, and Fig. 1c), for reflections from 10 nm wide cantilevers, using a probe beam average power of < 50 µw. Comparable sensitivity as been obtained by oters for comparable [1] and even narrower (nanowire) [] resonators, in frequency-domain interferometric measurements. Altoug te amount of ligt reflected from a structure decreases rapidly wit decreasing size in te sub-diffraction limit regime, te key to te peraps non-intuitive robustness of te interferometric approac lies in te fact tat it arises from te superposition of

3 reflected amplitudes rater tan intensities. (Te flip-side of tis being tat it can in fact be difficult to get rid of interference in situations were it is undesired.) Te sape of te output pulse from te avalance transistor may be dispersed by stray capacitance along te excitation patway. Te electric field versus time pulse sape was measured non-invasively on te mounting circuit board close to te NEMS cip, using a small piece of litium niobate wafer (TorLabs LNW-0.5-ZC, 0.5mm tick, z- cut) placed in te fringe electric field between transmission line traces on te circuit board. A time-resolved measurement of te polarization cange of te probe beam upon transmission troug te litium niobate yields a result approximately proportional to E(t).[3] Te trace is sown in Figure S. Delay "time zero" is defined ere by coincidence of te onset of te electrical pulse wit arrival of te optical probe pulse at te cip. Te 50 ps 10% - 90% rise-time of te pulse is as specified for te PSPL 000D Turbo pulser. Te full-widt-alf-max agrees witin error to te 0.96 ns specification, despite a small electrical reflection causing a soulder on te falling side of te pulse, at delay time ~ 0.9 ns. An additional reflection contributes a small satellite peak at ~ 1.8 ns delay. Note tat te corresponding electrostatic force varies as E. Tis pulse sape may be sligtly dispersed before it reaces te cantilevers by te stray capacitance on te cip itself, an effect wic becomes more important at iger speeds/frequencies. Future designs will incorporate electro-optic sampling locations rigt on te cips, near te NEMS supports. 3

4 Calibration of te cantilever displacement To convert te measured lock-in signal into te cantilever displacement in real pysical units, calibration must be made. Te displacement sensitivity of te measurement can be calibrated by tuning te wavelengt of te Ti:sappire oscillator to measuring te interference-induced cange in reflectance for fixed cantilever-substrate spacing, and relating tis to te separation dependence at fixed wavelengt. According to te Fabry- Perot interference condition: U = AI 0[1 + γ ( γ 1)(1 cos( 4π )] (1) λ U is te voltage readout from potodiode; I 0 is te intensity of te incident laser; γ is te ratio of reflected beam from te cantilever vs from te substrate, and is te distance between two reflection surfaces. Te cange of U can be induced by canging eiter wavelengt λ (equation ()) or (equation (3)): di0 ΔU = A [ 1+ γ ( γ 1)(1 cos( 4π )] Δλ 8AI0γ ( γ 1) sin( 4π ) π Δλ λ dλ λ λ () ΔU π = 8AI 0γ ( γ 1) sin(4π ) Δ λ λ (3) Let, ten () and (3) can be rewritten as: By measuring te voltage cange of a laser beam reflected back from a single surface as a function of te cange of wavelengt, we can find di 0 /I 0 dλ. Rewriting equation (4) as α = 8AI ( 1)sin(4π 0γ γ ) λ di ΔU = A[ 1+ γ ( γ 1)(1 cos(4π )] Δλ απ Δ λ dλ λ απ ΔU = Δ λ (5) 0 λ (4) ΔU = U di I 0 Δλ απ Δλ 0dλ λ (6) 4

5 te parameter α can ten be extracted by recording te corresponding voltage cange, reflected back from a stationary cantilever, caused by canging te wavelengt of te incident laser beam. Plugging α into equation (5), te voltage cange, ΔU, arising from te displacement of te cantilever can be calculated (tis ΔU is twice te signal from te lock-in readout, wit on-off modulation). However, equation (5) only applies to te ideal situation were te laser beam is considered as a single point. Assuming a Gaussian transverse profile for te laser beam intensity, equation (5) becomes: απ ΔU = Δ λ (7) r0 e Δ( x) dx were Δ( x ) =, 0 ( x x0 ) e ( x x0 ) r0 θ ( x) dx 0 θ ( x) = 1 x < 0, x > L 0 x L Neverteless, te laser beam profile does not ave effect on equation (6). As one example, we calibrated te displacement of 400 nm cantilever. Around λ=770 nm, di 0 /I 0 dλ 1.7% nm -1, απ/λ 1.1 mv/nm, and laser beam diameter 880 nm. During te measurement, we positioned te center of te beam at te free end of te cantilever, wic gives te averaged Δ being 0.48Δ. Te maximum amplitude we obtained from te lock-in amplifier is ~100 μv, corresponding to a displacement of 14 pm. Te calibrated displacement of te 400 nm long cantilever is sown in Fig. 1c. 5

6 3 Simple model to predict te cantilever displacement by electrostatic actuation Let te tickness of te Si cantilever be 1, and te distance between te bottom of te cantilever and te substrate be. If we assume te energy stored between te cantilever and te substrate can be approximated by a parallel capacitor, te total energy is ten: E = ε 0 εav (ε ε ) (8) ε is te dielectric constant of Si, A is te surface area of te cantilever. Te force between te cantilever and te substrate is: F = de d = ε 0 ε AV (ε ε ) (9) If we assume te force is evenly distributed along te cantilever, te total torque against te clamping end is ½ Fl. As a first order approximation, we assume tis torque is equivalently exerted at te free end of te cantilever, and te bending of te cantilever at te free end becomes: 3Fl y = Yw = Y 4 3ε ε l V ( ε ε ) (10) Plugging in numbers for te 400 nm cantilever, te predicted y is 116 pm. Instead of being perfectly isolated, tere was a leakage resistance between te capacitor electrodes of approximately 0 Om. Te small sorts on te cip cause voltage division and lead to a substantial reduction of te actual displacement. If we assume only resistive impedance matcing, tat is, tat te capacitive reactance is small compared to 50 Om, tis 0 Om leakage by itself reduces te voltage between te cantilever and te substrate 6

7 from 50 V to V. Since te amplitude scales as te square of te voltage, tis reduces te expected amplitude from 116 pm to 3 pm. Wit te calibration procedure detailed in, a maximum displacement of 14 pm was measured for te 400 nm long cantilever. Additional voltage division from stray capacitance could account for some of te remaining discrepancy. 4 Simulation of te spatio-temporal image Te spatio-temporal simulation Fig. d is calculated from te continuum mecanics for te fundamental and first modes, as well as taking into account te upward curling of 80 nm (due to compressive stress on te Si from te Al layer). Te latter is accounted for in te detection sensitivity by te multiplicative term sin(4π( x )/λ), were 0 is te initial (design) distance between te two reflection surfaces at te clamped end of te cantilever, 1 is te total static bending displacement at te free end, and x is te position along te cantilever as a fraction of te total lengt. A dramatic example of suc curling is sown in te SEM micrograp of Fig. S3. A single fundamental mode is normally observed for sorter cantilevers. Te simulated images usually yield good agreement wit te direct measurement. Te recorded and calculated spatio-temporal images of a 0.9 µm long cantilever are sown in Fig. S4. Te simulated spatio-temporal image is also Gaussian-blurred to te diffractionlimited focus of te probe beam. Te comparison in line cross-section is sown in Panel S4c. For te longer cantilevers as in Fig., te calculation yields reasonable agreement wit te measurement, owever te modeled signal is larger near te clamped end. Tis discrepancy could be diminised by using a modified model, to better describe te 7

8 reflectivity cange close to te clamped end of te cantilever and te bending from compressive stress. Te reflectivity near te clamped end cannot be described by te same Fabry-Perot interference pattern as olds along te lengt of te cantilever, owing to te canging geometry. Te effect is visible in te reflected intensity map sown as te inset of Fig. b. 5 Determination of te Young s modulus of Si beams Te formula to determine te effective EI of a composite beam is [4]: EI = w( E Si 4 Si + E Al 4 Al + E Si 1( E E Si Al Si Si Al + E (4 Al Al Si ) + 6 Si Al + 4 Al )) (11) If te effect of te Al is only to mass load te resonator, te composite EI reduces to te values for Si (were I = widt tickness 3 /1 for a rectangular beam) yielding E Si = 1 GPa. Taking full account for te Al as an elastic layer (assuming E Al = 71 GPa) gives E Si = 77 GPa. Since some granularity is visible in Fig. 1b (likely reducing te Al modulus compared to bulk Al), te actual Young's modulus for our Si cantilevers is estimated to lie in te range between 10 and 77 GPa. Tis value for E Si is lower tan tat expected for a <110> symmetry axis of 169 GPa and cannot be explained by a damping sift or electrostatic tuning. Frequency softening due to electrostatic dc voltage does not apply in tis case since te excitations are impulses (tat is, te electric field is already off during te fitted ringdown). Even te 50 V pulse is a small fraction of te pull-in voltage as tere is negligible cange in te instantaneous frequencies in Fig. 4a for pulse-on compared to pulse-off states. In ig 8

9 damping regime, te frequency can also soften. However, te effect for Q = 5 is only about 0.5% and most measured ring-downs ad Q >> 5. Two separate surface effects, (i) nanoscale surface material relaxation, and (ii) surface stress caused by te Al layer owever, could influence te resonance frequencies of te devices and cause te apparent reduction in Young's modulus for Si. Te first effect is in relation to speculated relaxation of te few atomic layers near te surface tat sould alter te effective Young's modulus for ultratin layers. Ono et al. found a significant apparent reduction of Young's modulus for top-down patterned Si resonators tinner tan 40 nm.[5] Te trend-line troug teir data would indicate tat a 150 nm tick Si layer sould ave E Si<110> of about 10 GPa, wic is in te range for wat we find for te E Si<110> of our 147 nm device. However, frequency canges induced by termal treatment and gas adsorption were more modest in additional exploration of te issue from te same group.[6] Furter, bottom-up <111> Si nanowires as small as 40 nm do not appear to ave a noticeable reduction in Young's modulus.[] Te second effect is surface stress caused by te Al film. Our Al films ave tensile stress, wic puts a compressive stress on te Si-Al interface, as sown by te curling (bending) up of te cantilever out of plane (Fig. S3). For a μm long cantilever wit static bending of 80 nm at te free end, te estimate of differential surface stress between te top and bottom layers is about -100 N/m.[7] Precise calculation of te effect of tis surface stress on te cantilever stiffness remains an active area of researc [8-10] wit effects broken down into strain-dependent and strain-independent surface stress. Using formulae from tese works, results for our case range from miniscule [8] to a few percent [9] to a large effect in one case.[10] Finally, it sould be noted tat te plasma 9

10 processing of our devices could also lead to a substantial reduction in E Si, particularly since teir widt:tickness aspect ratio is sligtly less tan one. 6 Comparison of te experimental Q factor wit teoretical predictions Figure S5e sows te measured quality factor, Q, as a function of lengt for a series of silicon cantilevers from te same 147 nm tick primary set. Panels S5a-S5c sow caracteristic time traces for tese specimens. Te experimental Q factors are determined by fitting te time domain data to decaying sinusoids, as te total duration of te time record is insufficient to resolve te intrinsic peak widt by Fourier transformation. Panel S5e also sows tree calculations of Q wic take into account clamping loss at te support and momentum-excange damping from te air (measurements performed under ambient conditions) [11-13] 1 Q = Q clamp Q air (1) were Q air = 93 L E ρ P. At 10 nm, te resonator widts are small enoug (about twice te mean free pat) to be in te molecular flow vacuum regime at atmosperic pressure and room temperature. Tus, air damping can be calculated from momentumexcange rater tan viscous damping formulae. Tree different clamping loss cases are considered: a beam attaced to a semi-infinite support [13] (black curve) wit Q clamp = L 4 L. 4 w 3, a beam wit infinite widt attaced to a semi-infinite support [1] (ligt grey curve) wit Q clamp =.17 L 3 3, and a beam attaced to a support wit te same tickness as te beam [13] (dased line curve) wit Q clamp = 1.05 L. Tese are plotted for comparison. 10

11 For te first two cases, a maximum Q is expected for cantilevers around and below 1 µm in lengt, wit Q falling off towards sorter lengts due to clamping loss and towards longer lengts due to air damping. Rater tan following a single calculated curve, a great deal of device-to-device variation is found in te measured damping rate. Te black curve sould ave given te closest estimation for te Q values of te devices if teir beaviour were ideal. Te small undercut (~100 nm) due to te release procedure could greatly decrease te Q factor of te final devices, but sould not reduce it all te way down to te dased line (wic corresponds to infinite undercut). Oter sources of damping can also be present tat are not accounted for in te simple model (te most likely of tese could stem from te morpology or adesion of te metal film). Canges in te damping, and to a lesser extent te resonant frequency, of some structures as been observed following imaging at ig probe power, and correlated to canges in te metal layer observed in later electron micrograps. Panel S5d is included for interest as it demonstrates tat te time-domain tecnique remains applicable for overdamped structures wic display a relaxation (in contrast to ringing) response. Suc structures are sometimes found in fluidic environments, and are very difficult to caracterize in te frequency domain. Te sensitivity of frequencydomain measurements improves wit increasing quality factor, wereas time-domain data are broadband and ave signal strengt independent of Q. Te cantilever of S5d ad been operated in pentane and was contaminated by residue after drying but remained functional and overdamped. 11

12 References: 1. Kou, T., Karabacak, D., Kim, D.H., & Ekinci, K.L. Diffraction effects in optical interferometric displacement detection in nanoelectromecanical systems. Appl. Pys. Lett. 86, (005).. Belov, M. et al. Mecanical resonance of clamped silicon nanowires measured by optical interferometry. J. Appl. Pys 103, (008). 3. Valdmanis, J. A., Mourou, G. & Gabel, C. W. Picosecond electro-optic sampling system. Appl. Pys. Lett. 41, 11 (198). 4. Bergaud, C., Nicu, L. & Martinez, A. Multi-mode air damping analysis of composite cantilever beams. Jpn. J. Appl. Pys. 38, 651 (1999). 5. Li, X., Ono, T., Wang, Y. & Esasi, M. Ultratin single-crystalline-silicon cantilever resonators: Fabrication tecnology and significant specimen size effect on Young's modulus. Appl. Pys. Lett. 83, (003). 6. Wang, D. F., Ono, T. & Esasi, M. Termal treatments and gas adsorption influences on nanomecanics of ultra-tin silicon resonators for ultimate sensing. Nanotecnology 15, (004). 7. Godin, M., Tabard-Cossa, V. & Grutter, P. Quantitative surface stress measurements using a microcantilever. Appl. Pys. Lett. 79, (001). 8. Lacut, M. J. & Sader, J. E. Effect of Surface Stress on te Stiffness of Cantilever Plates. Pys. Rev. Lett. 99, 0610 (007). 9. Lu, P., Lee, H. P., Lu, C. & O'Sea, S. J. Surface stress effects on te resonance properties of cantilever sensors. Pys. Rev. B 7, (005). 10. Wang, G-F & Feng, X-Q Effects of surface elasticity and residual surface tension on te natural frequency of microbeams. Appl. Pys. Lett. 90, (007). 11. Newell, W. E. Miniaturization of Tuning Forks. Science 161, 130 (1968). 1. Jimbo, Y. & Itao, K. Energy loss of cantilever vibrator. J. Horological Inst. Jpn. 47, 1 (1968). 13. Potiadis, D. M. & Judge, J. A. Attacment losses of ig Q oscillators. Appl. Pys. Lett. 85, 48 (004). 1

13 Figure S1: Timing Diagrams. a Timing diagram to illustrate te relative timing sequence of te pump optical pulse (red), pulser output (blue), NEMS mecanical response (green), and te probe optical pulse (red). b Zoom out in time scale of panel a, te repetition rate is 800 KHz, set by a laser pulse picker. c Furter zoom out in time scale, sowing te effect of te on-off modulation set by a frequency clock (~ 3 KHz). 13

14 Figure S: Non-invasive measurement of pulse sape. Electro-optic sampling of te electrical excitation pulse sape on te cip-mounting circuit board, wit no external carge line on te PSPL 000D pulser (minimum pulse widt). Figure S3: Upward-curling of long cantilevers. Electron micrograp sowing te curvature of a cantilever from te tensile stress in te aluminum metallization, compressing te top of te silicon. Tis is te 3μm long cantilever wic yielded a measured resonant frequency muc lower tan expected; also apparent is an extra mass attaced near te center of te beam. 14

15 Figure S4: Reflected intensity and spatio-temporal images (measured and simulated). a, Optical reflectance image of a 0.9 um long cantilever. b, Spatio-temporal cantilever response, displacement (color map) sown versus position (along te dased wite line in a), and delay time. c, Te measured displacement versus position at t = 7 ns (solid blue circles), and comparison to te continuum mecanics result (solid line). Te latter is Gaussian-blurred to account for te finite focus of te probe beam. d, Simulated spatio-temporal cantilever response, using parameters extracted from te fitting in c. 15

16 Figure S5: Measured ring-downs and Q-factor summary for a series of cantilevers. 147 nm tick silicon, 10 nm wide in a to c, te measurements extend into te regime dominated by clamping loss. 340 nm tick silicon, 300 nm wide in d, tis structure was immersed in pentane and overdamped by residue. e Black, ligt grey and dased curves: expected quality factor versus lengt, for te series of 147 nm tick, 10 nm wide Si cantilevers, incorporating clamping loss for tree different cases and momentumexcange damping only. Symbols: quality factors determined by fitting te time-domain ringdown data to decaying sinusoids. 16