NEMS Oscillator Frequency Pulling and Nonlinear Behavior

Self-Sustaining Oscillator Based upon Vibrating Nanomechanical Resonator

3.6 NEMS Oscillator Frequency Pulling and Nonlinear Behavior

We have also demonstrated that the NEMS oscillator frequency can be pulled by the loop gain tuning and the magnetic field change. The NEMS oscillator frequency pulling mechanisms are based on the fact that both the loop gain and magnetic field change induce change of the driving force upon the doubly-clamped resonator, which is readily described by the forced Duffing equation (see Chapter 2 for general discussion),

(

t y

dt y dy dt

y

d²₂ +2µ +ω₀² +α ³ =ΓcosΩ

)

, (3-10)

where y is the beam displacement (in-plane), 2µ≡ω0/Q with µ the damping coefficient, Γ≡F/m with m the device mass; F the driving force, and Ω the driving frequency. The cubic nonlinearity coefficient is α =

(

2π L

) (

⁴E 18ρ

)

, determined by the geometry and elastic properties of the beam. This coefficient α can also be related to the critical displacement aC (defined at the onset of nonlinearity), α =

(

8 3ω0²

) (

9Qa_C²

)

. Based on the Duffing equation description of the doubly-clamped beam resonator, both the amplitude and the resonance peak frequency are dependent on the driving force. The relationship between the NEMS resonator beam amplitude a and the frequency pulling σ (σ≡ωpeak−ω0, with ωpeak the resonance peak frequency) is readily described by the frequency-response equation,

2 0 2 2 0 2

4 8

3

ω ω

σ α

µ ⎥ = ^Γ

⎦

⎢ ⎤

⎣

⎡ ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ −

+ a , (3-11)

in which

( )

^=∞

aC

da

dσ sets the onset of nonlinearity and the critical amplitude aC. As the drive strength Γ is increased, the relationship between the resonance peak frequency ωpeak and the amplitude at this peak frequency, i.e., the backbone curve, is

2 0 0

3 ⎟⎟

ak ⎞

, (3-12)

⎜⎜ ⎠

⎝

= ⎛

−

≡

C pe peak

peak a

a Q ω ω

ω σ

where a_peak =ΓQ ω₀² , and ^a_C =

( )

3 2^a_peak,_C =

( )

3 2Γ_C^Q ω0², with apeak,C the peak amplitude at the critical driving ΓC. With these relations, eq. (3-12) can be rewritten as

2 0 0

3 3

4 ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ Γ

= Γ

−

≡

C peak

peak Q

ω ω ω

σ , (3-13)

where the drive force Γ is readily determined by the transduction scheme and RF driving power sent to the device in a real measurement, and thus the backbone behavior in eqs.

(3-12) and (3-13) can be calibrated by sweeping the driving force in measurements.

Therefore, both the backbone curve and the frequency-response curves can be experimentally determined.

-4.0M -2.0M 0.0 2.0M 4.0M 6.0M 0.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5

0 1x10¹⁴ 2x10¹⁴ 428.45

428.50 428.55 428.60 428.65 428.70

Resonance Peak Freq. (MHz)

Drive Level, (Force/Mass)², (m²/s⁴)

45 23

Displacement, RMS (nm)

Frequency Pulling (Hz)

1

RF Drive 1. -38dBm 2. -35dBm 3. -32dBm 4. -29dBm 5. -26dBm

Fig. 3.14 Duffing behavior of the NEMS resonator calibrated in measurements: the frequency-response curves and the backbone curve (resonator peak amplitude versus resonance peak frequency). The family of frequency-response curves shows the NEMS beam resonator displacement in nanometers versus frequency (as described in eq. (3-11)). Inset: the linear fit of measured frequency pulling versus driving force squared (Γ²=(F/m)²), according to eq. (3-13).

As shown in the inset of Fig. 3.14, the measured frequency pulling versus squared driving force is fit linearly according to eq. (3-13), and then the behavior described in eqs.

(3-11), (3-12) and (3-13) can be quantitatively determined. By combining the extracted data from measurements and the theory of Duffing nonlinearity, the frequency-response curves are reproduced and plotted in Fig. 3.14, which show the onset of nonlinearity is attained when the RF power sent to the NEMS device is about -29dBm, exactly the same as observed in network analysis measurements. This verifies the validity of the above analyses, and also demonstrates reliable prediction of device absolute displacement (in nanometers) as the driving force is increased. It is noted that because the cubic nonlinearity coefficient α>0, the frequency pulling is a spring stiffening effect.

424 426 428 430 432

-60 -50 -40 -30 -20 -10 0 10

0 100k 200k 300k 400k 500k -30

-25 -20 -15 -10 -5 0 5 10

NEMS Oscillator Output (dBm)

Frequency Pulling (Hz)

95.2 dB [-32 dBm]

94 dB [-33 dBm]

93 dB [-34.1 dBm]

92.3 dB [-34.9 dBm]

92 dB [-35.2 dBm]

91 dB [-36.3 dBm]

Os c ill a tor O utp u t Power (dBm )

F requency (MHz)

Amplification Gain (dB) [Power to Device (dBm)]

Fig. 3.15 Measured NEMS oscillator output frequency pulling with calibrated sustaining amplification gain change (i.e., the change of RF power sent to the NEMS device). Inset: NEMS oscillator output power versus frequency pulling. Note when the sustaining amplification gain is smaller than some certain value (here ~90dB) there is no measurable oscillation.

The above analyzed and calibrated effect of NEMS resonator directly determines the behavior and performance of the NEMS oscillator. As shown in Fig. 3.15, when the

feedback amplification gain is changed and thus the RF power driving the device is changed accordingly, the NEMS oscillator output spectrum changes both its peak power and peak frequency (in each case, the loop phase change is tuned to optimize the oscillation to happen at the resonance peak frequency, as addressed in the previous section). The oscillation frequency is pulled upward, as a result of the resonator’s frequency stiffening effect, when the loop gain is increased. Shown in the inset of Fig.

3.15 is the relation of oscillator output versus oscillation frequency, which follows the tendency as depicted by the backbone curve of the NEMS resonator in Fig. 3.14.

The nature of the frequency pulling effect of the NEMS oscillator is based on the transition from linear to nonlinear regime of a doubly-clamped beam Duffing-type resonator, as the driving force is increased. Spontaneously, it becomes very intriguing to drive the NEMS resonator into the nonlinear regime, and to realize and then characterize the nonlinear oscillation with NEMS. Moreover, once self-oscillation is realized with a nonlinear NEMS resonator, the noise-induced switching of self-oscillations between the bistable states of the same frequency-determining resonator could be very interesting and important for both fundamental research and technological applications [29].