Strong Coupling between Mechanical Modes in a Nanotube Resonator

We report on the nonlinear coupling between the mechanical modes of a nanotube resonator. The coupling is revealed in a pump-probe experiment where a mode driven by a pump force is shown to modify the motion of a second mode measured with a probe force. In a second series of experiments, we actuate the resonator with only one oscillating force. Mechanical resonances feature exotic line shapes with reproducible dips, peaks, and jumps when the measured mode is commensurate with another mode with a frequency ratio of either 2 or 3. Conventional line shapes are recovered by detuning the frequency ratio using the voltage on a nearby gate electrode. The exotic line shapes are attributed to strong coupling between the mechanical modes. The possibility to control the strength of the coupling with the gate voltage holds promise for various experiments, such as quantum manipulation, mechanical signal processing, and the study of the quantum-to-classical transition.

Figure 1

Device characterization. (a) Colored scanning electron microscopy image of the device measured in this work with source ($S$), drain ($D$), and gate ($G$) electrodes. The nanotube position is represented by a dashed line, and the clamping points are indicated by arrows. The suspended length of the nanotube is $1.77\mu \mathrm{m}$, and the depth of the trench is 370 nm. Scale bar: 600 nm. The image is recorded after the measurements of the resonator. (b) Schematic side view of the device. (c)–(f) Mechanical resonances for small and large driving forces obtained by measuring the mixing current ($I_{\mathrm{mix}}$) as a function of the driving frequency ($f$) with the two-source technique. The driving force is electrostatic and is proportional to the oscillating voltage $V^{ac}$ applied to the gate electrode. $V^{ac}=0.2\mathrm{mV}$ and $V_g=1.5\mathrm{V}$ in (c); $V^{ac}=0.4\mathrm{mV}$ and $V_g=1.5\mathrm{V}$ in (d); $V^{ac}=0.2\mathrm{mV}$ and $V_g=4\mathrm{V}$ in (e); $V^{ac}=0.4\mathrm{mV}$ and $V_g=4\mathrm{V}$ in (f). For the detection, we apply an oscillating voltage ($V_s^{ac}=0.056\mathrm{mV}$) to the source electrode. Black (red) curves correspond to upward (downward) sweeps.

Figure 2

Tuning resonance frequencies. (a) Map of resonance frequencies as a function of $V_g$ (by measuring $I_{\mathrm{mix}}$ as a function of $f$ and $V_g$ for $V^{ac}=2\mathrm{mV}$ with the FM technique). We clearly discern three modes, while a fourth one is weaker and is indicated by arrows. Color scale: 0 (black) to 0.1 nA (red). (b) Schematic of the map of resonance frequencies as a function of $V_g$. The four modes are represented by plain lines and labeled $M$, $N$, $O$, and $P$. Dashed lines correspond to the resonance frequencies of these modes multiplied by 2, 3, $1/2$, or $1/3$ (the values are indicated in the labels). Black arrows point to regions where line shapes are exotic and two modes are commensurate. Gray arrows point to exotic line shape regions for which we cannot assign the coupled mode. (c) Finite element simulation of the map of resonance frequencies as a function of $V_g$ obtained with ANSYS. The dashed line corresponds to a mode that we have not detected. The simulations show that this mode has one node (while the others have either zero or two nodes) and thus cannot be detected due to symmetry reasons. Inset: Schematic of the static shape of the nanotube when $V_g=0\mathrm{V}$. The deformation in the transverse direction is exaggerated with respect to the nanotube length. The largest deformation is $\sim 40\mathrm{nm}$ (Supplemental Material, Sec. VII [23]). (d) Static displacement of the center of the resonator ($z_s$) calculated with ANSYS using the static shape of the nanotube (when $V_g=0\mathrm{V}$) depicted in the inset of (c). (e) $z_s$ calculated from the Euler-Bernoulli equation for a straight nanotube (plain line). The dashed line corresponds to the result calculated with ANSYS for the same straight nanotube.

Figure 3

Mechanical coupling measured in a pump-probe experiment. (a) Representation of the two drive frequencies used in the pump-probe experiment. The pump force is swept in frequency, whereas $f_{\mathrm{probe}}$ is set to match the resonance frequency of the lowest mode ($M$). (b) Normalized mixing current of the probed mode as a function of $f_{\mathrm{pump}}$ at $V_g=3.6\mathrm{V}$ (measured with the FM technique). Before the scan, we set $f_{\mathrm{probe}}$ so that the current is maximal ($I_{\mathrm{mix}}^0$). We plot the measured current divided by $I_{\mathrm{mix}}^0$. The oscillating voltage of the pump is 5.6 mV, and the FM oscillating voltage of the probe is 2 mV. (c) Normalized mixing current of the probed mode as a function of $f_{\mathrm{pump}}$ and $V_g$ using the same parameters as in (b). The line graph in (b) is marked with a dashed line. Color scale: $I_{\mathrm{mix}}=0$ (dark red) to $I_{\mathrm{mix}}=1$ (white).

Figure 4

Resonances when the measured mode is commensurate or nearly commensurate with another mode. (a)–(d) Resonance line shapes of one mode for different $V_g$ measured with the two-source technique. The driving voltage applied to the gate electrode is $V^{ac}=1.1\mathrm{mV}$, and the voltage applied for the detection on the source electrode is $V_s^{ac}=0.56\mathrm{mV}$. Black (red) curves correspond to upward (downward) sweeps. $V_g=2.4\mathrm{V}$ in (a); $V_g=2.6\mathrm{V}$ in (b); $V_g=2.8\mathrm{V}$ in (c); $V_g=3\mathrm{V}$ in (d). (e) Map of the resonance frequency as a function of $V_g$ (obtained by measuring $I_{\mathrm{mix}}$ as a function of $f$ and $V_g$ for $V^{ac}=1.1\mathrm{mV}$ and $V_s^{ac}=0.56\mathrm{mV}$ using the two-source technique by increasing $f$). Color scale: 0 (black) to 1 nA (red). (f) Map of the resonance frequency as a function of $V_g$ (obtained for $V^{ac}=1.1\mathrm{mV}$ and $V_s^{ac}=0.28\mathrm{mV}$ using the two-source technique by decreasing $f$). Color scale: 0 (black) to 0.7 nA (red). (g)–(j) Resonance line shapes for different modes and different values of $V_g$ measured with the two-source technique. $V^{ac}=1.1\mathrm{mV}$, $V_s^{ac}=0.56\mathrm{mV}$, and $V_g=1.98\mathrm{V}$ in (g). $V^{ac}=17\mathrm{mV}$, $V_s^{ac}=1.1\mathrm{mV}$, and $V_g=3.41\mathrm{V}$ in (h). $V^{ac}=1.7\mathrm{mV}$, $V_s^{ac}=0.28\mathrm{mV}$, and $V_g=1.86\mathrm{V}$ in (i). $V^{ac}=5.6\mathrm{mV}$, $V_s^{ac}=0.56\mathrm{mV}$, and $V_g=3.6\mathrm{V}$ in (g).