SUPPLEMENTARY INFORMATION

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1 SUPPLEMENTARY INFORMATION DOI: /NNANO Graphene mechanical oscillators with tunable frequency Changyao Chen, Sunwoo Lee, Vikram V. Deshpande, Gwan Hyoung Lee, Michael Lekas, Kenneth Shepard, James Hone 1 Measurement circuit of graphene mechanical oscillators The oscillator circuit is built on resonant channel transistor (RCT) circuit 1 with a positive feedback. It is important to obtain open-loop mechanical resonances with large signal to background ratio (SBR), in order to observe stable closed-loop oscillations. If the quality of as-fabricated samples fails to provide an adequate SBR, we feed an additional out-of-phase signal to a nearby electrode, which actively reduce the out-of-resonance background 2. The complete circuit setup for open-loop characterizations is shown in Fig. S1. In the open-loop configuration, transmission coefficient S 21 is measured by vector network analyzer (VNA, Agilent E5071C). During the open-loop measurement, we implement all the electronic elements necessary for positive feedback, including a fixed-gain amplifier (MITEQ AU-1447-BNC, 60 db gain), a variable amplifier (Mini Circuits ZFL-1000G, -35 to 27 db gain), phase shifters (Lorch Microwave), bandpass filters (Mini Circuits) and a directional coupler (Mini Circuits ZFDC-20-4L). Once the open-loop condition is optimized to satisfy the Barkhausen criteria, we connect node 1 and 2, and monitor the self-sustained oscillation from the coupler port (node 3). NATURE NANOTECHNOLOGY 1

2 10 MΩ β Figure S1 A complete circuit diagram for open loop and closed loop characterizations. 2 Mechanical oscillations with different feedback gain First we have to confirm the origin of the oscillation is mechanical, not spurious electrical signal. Since the open loop gain is proportional to the applied bias 1, when V d is below certain value, the open loop gain will drop below unity, failing the Barkhausen criterion. Only if the oscillation is purely mechanical, the change in bias V d will switch on and off the oscillation. Fig. S2 shows a typical response of mechanical oscillation due to the change of V d (sample 1), and it is clear that 2

3 with V d smaller than certain threshold, the oscillation goes away. a b Powder (dbm) V d = -0.5V V d = -0.4V V d (V) Figure S2 a, Contour-plot of oscillator output power as a function of frequency and V d. osc_vd_mega b, The oscillator is on with V d = -0.5V, and off at V d = -0.4V. Data acquired from sample 1 at room temperature with V g =10V. Next, we test how the feedback parameters, namely, feedback gain and feedback phase, will affect the mechanical oscillation. We want to study how robust graphene mechanical oscillators are, e.g., what is the range of feedback gain and phase for self-sustained oscillations. From the Barkhausen criterion, the loop gain needs to be exactly unity for sustained oscillation. The oscillation starts with perturbation (for example, thermal vibration), and with positive feedback (loop gain greater than 1), the electrical signal is amplified and grow gradually, until some negative feedback of the system starts to dominate. Such negative feedback prevents the oscillation from blowing-up, and keeps the loop gain equals to 1. In LC oscillators, such nega- 3

4 tive feedback usually arises from the nonlinearity of the amplifiers, where in our graphene mechanical oscillators, all the amplifiers are operated in linear regime and the negative feedback is from the nonlinearity of the mechanical resonators. z G m P P Figure S3 a. Nonlinearity of mechanical resonators, the vibrational amplitude z will no longer be linear with applied power P at large P. The red dot denotes the onset of nonlinearity. b. Mechanical gain G m as function of applied power P. This concept is illustrated in Fig. S3. The total loop gain G loop is consisted of electrical gain G e and mechanical gain G m, and at steady states, G loop = G e G m = 1. The electrical gain G e is the gain of all amplifiers, and the mechanical osc_stable gain G m can be considered as dz/dp, where z is the vibrational amplitude of the mechanical resonators, and P is the applied power. In linear regime, a small vibration z will generate an oscillating current linearly. If we only consider the simplest Duffing nonlinearity, the P -dependence of z is shown in fig S3a, and the mechanical gain G m is shown in Fig. S3b. It is clear that, in the linear regime (z grows linearly with P ), mechanical gain G m is constant. Once the feedback power increases, the resonator will become nonlinear, therefore, G m will start dropping 4

5 (blue dots). Therefore, the minimum G e needed is to accommodate the largest G m (red dots, corresponding to the onset of mechanical nonlinearity), but G e can be larger with smaller G m (blue dots) by pushing the mechanical resonator deeper into the nonlinear regime. a Feedback gain (db) b Feedback gain (db) P (dbm) V Gain (V) V Gain (V) Width (khz) Power (nw) Figure S4 a,color-plot of oscillatorosc_gain_mega power as function of frequency and feedback electrical gain. b. Oscillator power and spectrum linewidth with different feedback electrical gain G e. Data acquired from sample 1 at room temperature, V d = -0.5V, V g =10V. The result of tuning electrical feedback gain G e is shown in Fig. S4. The gain is tuned by voltage controlled variable amplifier (Mini Circuit ZFL-1000G, gain range -35 db to 27 db). As we can see, the self-oscillations only happen above certain electrical feedback gain G e (65 db in Fig. S4), corresponding the condition where total loop gain G loop equals to 0 db. With increasing of G e, we have found three observations. First, the oscillation frequency starts to drift; second, the oscillation power first increase, then decrease; third, the linewidth of oscillation 5

6 spectrum first decrease, then increase as well. The drift of oscillation frequencies can be explained within the frame of mechanical nonlinearity and Barkhausen criterion: when the open-loop resonators are driven deeper into the nonlinear regime, both the amplitude-frequency and phase-frequency response will be bifurcated, with two possible stable states 3. Therefore, the Barkhausen criterion is satisfied at different frequencies. The details depend on the type of nonlinearities present in the system. In graphene mechanical resonators, there are various types of nonlinearities, including nonlinear damping 4, therefore, it is difficult to analyze this frequency drift quantitatively. The second and third observations are related to nonlinearity. When the electrical gain G e is increased, the inherit resonators are being driven with larger power, resulting larger vibrational amplitude z. This also increases the carrier signal power of the oscillation, as shown in Fig. S4b. The linewidth of the spectrum is related to phase noise, which will decrease with larger carrier power P C. This is in good agreement with our observation: the narrowest linewidth is accompanied with largest oscillation power. 6

7 103.4 Feedback gain (db) P (dbm) V Gain (V) Figure S5 Oscillator power spectrum with excessive different feedback gain G e. At large G e, the oscillation disappears. Data acquired from sample 5. osc_gain_mega_2 However, when G e is too large, pushing the mechanical resonators deeper in nonlinear regime, the resonance and phase will deviated even more from openloop characterization (performed in linear regime). Therefore, when at large G e (and large P ), it is expected that the oscillation power starts to drop, and even the self oscillation will seize, as shown in Fig. S5. 3 Mechanical oscillations with different feedback phase The result with different feedback phase is shown in Fig. S6. The feedback gain is set to be 2 db higher than unity under open-loop condition. We observed similar behaviors with tuning feedback gain: oscillation frequency drift, oscillation power 7

8 and spectrum linewidth optimized at certain values of feedback phase (gain). a 53.0 Feedback phase (degrees) b Feedback phase (degrees) P (dbm) V Phase (V) Width (khz) V Phase (V) 1 Power (nw) Figure S6 a. Oscillator power spectrum at different feedback phases. b. Oscillator power and spectrum linewidth with different feedback phases. Data acquired from sample 1 at room temperature, V d = -0.5V, V g =10V. osc_phase_mega 8

9 4 Open-loop data for Fig. 2 and Fig. 4 in main text 0-2 S 21 (db) Figure S7 Open-loop response for sample 2 in Fig. 2 in main text, with Q res 15. The mechanical resonance is around 25 MHz, indicated by the red arrow. Data acquired at room temperature. The open-loop transmission for sample tested in Fig. 2 in main text (sample 2) is shown in Fig. S7. In Fig. 2a, the oscillation peak amplitude is about 0.4 mv, and the total amplification of the feedback loop is 60 db, minus 20 db from the coupler, therefore, the voltage fed in to the amplifier is 40 nv. Assuming a 50 Ohm input resistance, the oscillation creates current with peak value of 0.8 na. The transconductance of the very sample is of 1 us/v, then the vibration amplitude is 0.14 nm. The open-loop and corresponding closed-loop oscillation power spectrum for 9

10 sample tested in Fig. 4 (sample 4) are shown in Fig. S8. In Fig. S8a, between V g of 2.5V and 5V, the open-loop frequency tunability is about 8 MHz/V. When the feedback loop condition are set to satisfy Barkhausen criterion at V g = 4.2V (shown as yellow arrow in Fig. S8b), it is clear that, due to phase delay in the long transmission lines, only within vicinity of V g = 4.2V there are pronounced self-oscillation. a S 21 (a.u.) b Power (dbm) V g (V) 0 5 V g (V) Figure S8 Open-loop transmission (S 21 in linear scale) a, and corresponding closedloop power spectrum b of sample 3 described in main text. The yellow arrow in b indicates where the Barkhausen criterion is set. Data are acquired at room temperature, with V d = -0.9V, and drive power of -40 dbm for open-loop characterization. 10

11 5 Modulation bandwidth of graphene VCO The modulation bandwidth quantifies how rapidly the VCO can respond to changes in tuning voltage. Due to the small oscillation power and phase delay in our prototype graphene VCO, it is very difficult to perform standard wideband modulation bandwidth measurements 5. Therefore, we instead characterize the modulation bandwidth of the open-loop resonator, as shown in Fig. S9a (sample 4, 2 µm diameter drum, 200 nm vacuum gap). The dashed black line shows the open-loop transmission S 21 as a function of RF drive frequency at V g = 4.2V, with a single resonance around 107 MHz. We next modulate V g with a 0.4 Vpp square wave to shift the frequency by ± 3 MHz. At low modulation frequency, we observe two disctinct resonances at 104 MHz and 110 MHz. As the modulation frequency is increased, these peaks disappear, and a single central peak emerges at around 107 MHz. We fit the peaks with Lorentzian functions and plot their heights as a function of modulation frequency (Fig. S9b); their amplitudes are equal at the Chen, et al. Figure 3! modulation bandwidth of 15 khz. a! b! Transmission (a.u.) 1! 0.8! 0.6! 0.4! No modulation! at V g = 4.2V! Peak height (a.u.) 2! 1! 0! 2! 1! 0! Peak height (a.u.) 100! 105! 110! 115! 1k! 10k! Frequency (Hz) Figure S9 Modulation bandwidth of graphene VCO. a, Open-loop transmission of graphene 11

12 mechanical resonator with different rate of frequency modulation. At low modulation rate, we can clearly resolve two side bands, and at modulation rate higher than 15 khz, the resonator response can no longer follow the modulation of controlled voltage (V g ). b, Heights of side band and central peak as function of modulation frequency, showing the modulation bandwidth of 15 khz. Both data are acquired from sample 4 at room temperature, with V d = - 0.9V, V g = 4.2V, and driving power of -40 dbm. In modulation bandwidth test, we add a square-wave with 0.4 V peak-to-peak value for modulation (Stanford Research System DS345). The applied modulation frequency is from of 1 Hz to 100 khz. The DC voltage and low frequency modulation signal are combined with summing amplifier (Stanford Research System SIM 980) then applied to the DC port of the bias-tee while the RF excitation is applied to the RF port of the bias-tee. However, the DC port the bias-tee (Mini Circuits ZFBT-6GW+) only has bandwidth of less than 20 khz, therefore, the observed modulation bandwidth is limited by this component in the test circuit. 6 White noise in graphene mechanical oscillators From the data shown in Fig. 2b, we have phase noise of -43 dbc/hz at offset frequency of 100 Hz. This corresponds to a virtual damping rate about 1 to 4 khz: an open-loop resonance fluctuation rate of 1 to 4 khz may cause the lowfrequency plateau in phase noise, which deteriorates oscillators performance 6. There are multiple sources of white noise in our measurement setup. One of them is the noise from cascade ampilifers. We have used up to three amplifiers in the feedback loop: Miteq AU-1447-BNC (60 db gain, noise figure 1.4 db, noise 12

13 temperature 110 K), HD (40 db, gain, noise figure 1 db, noise temperature 75 K), and Mini Circuits ZFL-1000G (-35 to 27 db gain, noise figure 8.94 db, noise temperature 1982 K). Apparently, the variable gain amplifier will introduce largest amount of noise, with equivalent voltage noise of 2.3 nv/ Hz for a 50 Ω system. At 100 MHz, the amplitude of the voltage noise is 23 µv. Another noise source is voltage source (Keithley 2400) we used for supplying DC gate voltage. With setting of 20 V range, the rated peak to peak voltage noise is 500 µv, assume a tunability of 1 MHz/V, this voltage noise will create frequency fluctuation of 0.5 khz. 7 Flicker noise in graphene mechanical oscillators Due the nonlinearity inherently built-in with the electrostatic actuation of graphene mechanical oscillators, closed-to-dc 1/f (flicker) noises are up-converted to the vicinity of carrier frequency f 0. There are three major contributions of the flicker noise aliasing, originating from different types of nonlinearity. The noise current i n can be defined as i n = 2Γu ac u n, (1) where Γ is called current aliasing factor, u ac is actuation voltage, and u n is voltage noise. The first noise aliasing is from electrical mixing (electrical nonlinearity): all the close-to-dc noise will mix with the mechanical resonant motion, act like electrical mixer, resulting in sideband noise around f 0. The current aliasing factor Γ C 13

14 due to electrical mixing is 7 : Γ C = πqf 0η 2 kv d, (2) where Q is the quality factor of the embedded mechanical resonator, η is electromechanical coupling coefficient, k is spring constant and V d is applied bias voltage. The second noise aliasing is from force mixing (electrostatic force nonlinearity): all the close-to-dc (voltage) noise will act like force noise, which is squared, and the multiplication will result in sideband noise around f 0. The current aliasing factor Γ F due to force mixing is 7 : Γ F = πqf 0η 2 kv d (1 + QηV d ), (3) kd where d is the distance between the graphene and the local gate. Third noise aliasing is from spring constant mixing (spring constant nonlinearity): all the close-to-dc (voltage) noise will change the tension of the graphene sheet, and produce sideband noise. In mechanical oscillators built based on microelectromehanical (MEMS) resonators, such nonlinearity results in capacitive spring constant softening, and can be ignored compared to previous flicker noise aliasing sources 7. However, in case of graphene oscillators, such nonlinearity results in spring constant hardening, and the magnitude is comparable with other nonlinear mixing effects. The current aliasing factor Γ k for the fully clamped drum resonator is: Γ k = 2πQ2 f 0 η 3 k 3 16πEtx c (1 ν 2 )R 2, (4) where Et = 340 N/m is the 2D Young s modulus of graphene, x c is the critical 14

15 vibrational amplitude, ν = is Poission s ratio for graphene and R is radius of the drum. Therefore, the total current aliasing factor Γ = Γ C + Γ F + Γ k, and the phase noise (including thermal noise) at offset frequency f is: S(f) = 10 log{[ 4k B(T R m + T N R i ) ][1 + ( f 0 u 2 ac 2Qf )2 (1 + 2Γ 2 Rmu 2 2 f 1/f ac )]}, (5) f where k B is Planck s constant, T is temperature, R m is motional resistance, T N and R i is the noise temperature and input resistance of the amplifier, respectively. f 1/f is the corner frequency of the flicker noise. For sample 2 shown in the main text, Fig. S10 shows the effect of different values of f 1/f, Figure S10 Phase noise of sample 2 with different values of corner frequency f 1/f. 15

16 8 Phase noise data sample with higher Q In Fig. S11, we show additional phase noise data taken from sample 6 (3 µm diameter drum, 200 nm vacuum gap), at room temperature. The open-loop response (Fig. S11a) shows quality factor Q res 84, and oscillator spectrum shown in Fig. S11b inset shows effective quality factor Q osc eft 1, 800. The phase noise spectrum in Fig. S11b reveals similar low frequency plateau below 2 khz, and roll off with slope between 1/f 2 and 1/f 3. The compression ratio is about 21. S 21 (db) a 0 Q res ~ Phase noise (dbc/hz) b Power (dbm) Q osc ~1, Offset frequency (Hz) Figure S11 Additional phase noise data. a Open-loop response and b corresponding oscillation spectrum (inset) and phase noise. Data acquired from sample 6 (3 µm diameter drum, 200 nm vacuum gap) at room temperature, with V g = -4.8V and V d = -0.4V. The possible explanation is that, with higher open-loop Q, the (vibrational amplitude) onset of nonlinearity is decreased. Such reduced amplitude results in smaller oscillation power, makes graphene oscillator more vulnerable to ambient noise. Although a higher Q ensures less energy dissipation, the reduced power (energy per cycle) offsets such the benefit. 16

17 Compared with other NEMS oscillators 8, 9, the prototype graphene oscillators show poorer phase noise performance. This is however, expected: since the oscillation frequency of graphene VCO is tuned by external force, it makes them vulnerable to external perturbation, such as ambient white noise, and 1/f noise from electronics. Further engineering improvements, such as low-noise amplifiers and on-chip circuitry, should improve the phase noise performance of graphene oscillators. 9 Additional data with voltage controlled oscillation a V g (V) b Power (dbm) c Power (dbm) V g = -6.3V V g = -7.6V Figure S12 Graphene VCO operation with manual feedback parameter adjustments. a, osc_vco_3 Open-loop transmission S 21. There is also an additional resonance visible with higher frequency. b, Oscillation spectrum at V g = -6.3V. c, Oscillation spectrum at V g = -7.6V. Data 17

18 acquired from sample 7 (2 µm diameter drum, 200 nm vacuum gap) at room temperature, with V d = -0.5V. During the VCO operation, the large phase delay greatly comprise the efficient frequency tuning range. However, we can manually adjust the feedback phase at different operating point (gate voltage V g ) to maintain self-oscillation. In Fig. S12, we show graphene self-oscillations at 42.7 MHz with V g = -6.3 V and 48.5 MHz with V g = -7.6V. References S1 Xu, Y. et al. Radio frequency electrical transduction of graphene mechanical resonators. Appl. Phys. Lett. 97, (2010). S2 Ekinci, K. L., Yang, Y. T., Huang, X. M. H. & Roukes, M. L. Balanced electronic detection of displacement in nanoelectromechanical systems. Applied Physics Letters 81, (2002). S3 Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. In Reviews of Nonlinear Dynamics and Complexity, 1 52 (Wiley-VCH Verlag GmbH & Co. KGaA, 2009). S4 Eichler, A. et al. Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene. Nature Nanotechnol. 6, (2011). S5 Mini-Circuit Application Note AN S6 Ham, D. & Hajimiri, A. Virtual damping and einstein relation in oscillators. IEEE Journal of Solid-State Circuits 38, (2003). 18

19 S7 Kaajakari, V., Koskinen, J. K. & Mattila, T. Phase noise in capacitively coupled micromechanical oscillators. Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions on 52, (2005). S8 Feng, X. L., White, C. J., Hajimiri, A. & Roukes, M. L. A self-sustaining ultrahigh-frequency nanoelectromechanical oscillator. Nature Nanotechnology 3, (2008). S9 Villanueva, L. G. et al. A nanoscale parametric feedback oscillator. Nano Lett. 11, (2011). 19