Tuning mechanical modes and influence of charge screening in nanowire resonators

We probe electromechanical properties of InAs nanowire (diameter $\sim 100\text{nm}$) resonators where the suspended nanowire is also the active channel of a field-effect transistor. We observe and explain the nonmonotonic dispersion of the resonant frequency with dc gate voltage $\left(V_g^{\text{dc}}\right)$. The effect of electronic screening on the properties of the resonator can be seen in the amplitude. We observe the mixing of mechanical modes with $V_g^{\text{dc}}$. We also experimentally probe and quantitatively explain the hysteretic nonlinear properties, as a function of $V_g^{\text{dc}}$, of the resonator using the Duffing equation.

Figure 1

(Color online) (a) Tilted-angle SEM image with the circuit used for actuation and detection of resonance for an InAs NW resonator. The diameter of the wire is 100 nm and the length of NW is $2.9\mu \text{m}$. The scale bar indicates a length of 200 nm. (b) The mixing current $\left(\Delta \omega /2\pi =17\text{KHz}\right)$ as a function of frequency for two values of $V_g^{\text{dc}}$ (red circles for positive $V_g^{\text{dc}}$ and blue triangles for negative $V_g^{\text{dc}}$, mixing current shown for positive $V_g^{\text{dc}}$, is five times of its original value). (c) Variation in the conductance as a function of dc gate voltage. (d) The plot of sound velocity $V_s=\frac{f_0{l}^2}{1.78r}$ calculated using the measured frequency $\left(\omega /2\pi \right)$ of the fundamental mode of the InAs NW resonators and geometrical values. The dashed line indicates the speed of sound obtained for $V_s=\sqrt{E/\rho }$ using bulk values for $E$ and $\rho$ (Ref. 32).

Figure 2

(Color online) [(a) and (b)] Color logscale plots of mixing current as a function of $V_g^{\text{dc}}$ and $\omega /2\pi$ for two devices [for device-1, Fig. 2a, diameter $d=100\text{nm}$, length $l=2.9\mu \text{m}$ and for device-2, Fig. 2b, $d=120\text{nm}$, $l=3.1\mu \text{m}$]. (c) Calculated dispersion as a function of $V_g^{\text{dc}}$. (d) Line plot at 29 MHz for device-1 [dashed line in Fig. 2a]. (e) The plot of the $Q$ as a function of $V_g^{\text{dc}}$ for device-1. (f) The plot of the “amplitude” $z_{amp}^{reso}$ as a function of $V_g^{\text{dc}}$ for device-1. [The dark red (black) and light blue (light gray) traces in Figs. 2e, 2f, show the data for negative and positive gate voltages.]

Figure 3

(Color online) [(a) and (b)] Shows the result of a FEM based self-consistent solution of Poisson’s equation giving the potential around a doubly clamped suspended nanowire device 100 nm in diameter and $1.5\text{−}\mu \text{m}$ long. The separation of the NW and 300-nm-thick ${\text{SiO}}_2$ is 100 nm. The back plane of the nanowire is the gate, held at $-5\text{V}$ and the two terminals of the wire are grounded. The maximum of the color-scale bar (white) is 0 V and the minimum (black) is $-5\text{V}$. (c) The space charge distribution in the cross section of the nanowire for $V_g^{\text{dc}}=-25\text{V}$ shows the gradient. The log-color scale below varies from $10\text{C}/{\text{m}}^3$ (black) to ${10}^6\text{C}/{\text{m}}^3$ (white). (d) Plot of space charge density in the vertical direction at $V_g^{\text{dc}}=-25\text{V}$ through the middle of the wire. The asymmetry along the vertical direction due the gate below the wire is clearly seen. (e) The space charge distribution in the cross section of the nanowire for $V_g^{\text{dc}}=-5\text{V}$ shows the gradient. The log-color scale below varies from ${10}^3\text{C}/{\text{m}}^3$ (black) to ${10}^5\text{C}/{\text{m}}^3$ (white). (f) The space charge distribution in the cross section of the nanowire for $V_g^{\text{dc}}=5\text{V}$ is uniform as the electron density in the nanowire is increased when the nanowire field effect transistor is turned on. The log-color scale below varies from ${10}^3\text{C}/{\text{m}}^3$ (black) to ${10}^5\text{C}/{\text{m}}^3$ (white).

Figure 4

(Color online) [(a) and (b)] Shows mode mixing for device-3 and device-1, respectively (for device-3, $d=103\text{nm}$, and $l=3.1\mu \text{m}$). (c) Calculated dispersion for mode mixing using asymmetry in the wire (see Appendix ). (d) Zoomed-in view for device-1 [Fig. 4b] which shows the mixing of the modes as a function of $V_g^{\text{dc}}$. Minimum separation between the modes, indicated by the double-headed arrow, is $0.25\pm 0.02\text{MHz}$.

Figure 5

(Color online) [(a)–(d)] Color logscale plots of mixing current from device-1 as a function of $V_g^{\text{dc}}$ and $\omega ∕2\pi$ in nonlinear regime with increasing amplitude of driving force starting from 100 to 400 mV, respectively. [(e)–(h)] Color-scale plots of calculated amplitude of a resonator in the nonlinear regime using Duffing equation with increasing amplitude of driving force from Figs. 5e, 5f, 5g, 5h. (i) Shows the amplitude for the two directions of $V_g^{\text{dc}}$ (solid red and dashed blue lines) which indicates hysteresis at constant frequency 31 MHz (solid red and dashed blue arrows show corresponding gate sweep direction).

Figure 6

Shows fitting of measured data.

Figure 7

(Color online) Shows the calculated ratio of $z$ and $y$—amplitudes of two modes (red circles and blue triangles) for natural oscillations as $V_g^{\text{dc}}$ is swept.

Figure 8

(Color online) [(a) and (b)] Color-scale plots of calculated amplitude of a resonator in the nonlinear regime using Duffing equation for increasing and decreasing frequency sweeps, respectively. [(c) and (d)] Show the line plots of dashed line in 8a and 8b, respectively. Arrows are showing frequency sweep direction.