Strong and tunable mode coupling in carbon nanotube resonators

The nonlinear interaction between two mechanical resonances of the same freely suspended carbon nanotube resonator is studied. We find that, in the Coulomb-blockade regime, the nonlinear modal interaction is dominated by single-electron-tunneling processes and that the mode-coupling parameter can be tuned with the gate voltage, allowing both mode-softening and mode-stiffening behaviors. This is in striking contrast to tension-induced mode coupling in strings where the coupling parameter is positive and gives rise to a stiffening of the mode. The strength of the mode coupling in carbon nanotubes in the Coulomb-blockade regime is observed to be 6 orders of magnitude larger than the mechanical-mode coupling in micromechanical resonators.

Figure 1

(a) Schematic of the carbon nanotube device freely suspended over a trench between drain and source electrodes. The Si substrate is employed as a back gate. (b) and (c) Resonance spectra measured by means of the rectification method at fixed gate voltage and low excitation power for two mechanical eigenmodes of the carbon nanotube. The quality factor is obtained by fitting the measured spectra to the response of a dampened harmonic oscillator (solid red lines). (d) dc current through the nanotube vs gate voltage showing single-electron tunneling and Coulomb-blockade electronic behavior. (e) and (f) Color map showing the absolute value of the rectified current through the nanotube as a function of the rf frequency and gate voltage ($V_{\mathrm{sd}}$ $=$ 200 μV). The tuned mechanical resonance shows up as the gray (black) curve with a dip at the Coulomb peak.

Figure 2

(color scale) dc rectified current through the nanotube as a function of the rf frequency of the two signal generators connected to the antenna ($V_{\mathrm{sd}}$ $=$ 200 μV). When the mechanical resonance of mode A is excited, the current shows a sudden change. The resonance of mode A is marked by a drop in the rectified current (shown in black on the color maps). (a) The resonance frequency of mode A shifts to a lower frequency when the rf signal of the second generator hits the resonance frequency of mode B ($V_{\mathrm{g}}$ $=$ −3.003 V). (b) and (c) The same as (a) but with a $V_{\mathrm{g}}$ value of −3.0045 and −3.0054, respectively, showing (b) mode stiffening and (c) negligible coupling. The green-dotted lines are a guide to the eye. The blue-line profiles inserted in (a)–(c) show the oscillation amplitude of mode B (calculated from the measured change in the dc current through the nanotube using an electrostatic model and assuming that the mode shape resembles that of the third bending mode of a doubly clamped beam) as a function of the (slowly incremented) frequency of the second generator.

Figure 3

(a) Maximum change in the resonance frequency of mode A as a function of the gate voltage. The resonance frequency of mode A increases with gate voltages when the nonlinear spring constant is positive (i.e., for the gate voltages at which Coulomb peaks occur). On the other hand, the resonance frequency of mode A decreases for the gate voltages when the nonlinear spring constant is negative. (b) Frequency dip of mode A when the gate voltage is swept across a Coulomb peak. The sign of the nonlinear spring constant α can be determined by the sign of the curvature of the frequency vs gate voltage trace.

Figure 4

Resonance frequency shift in mode A as a function of the oscillation amplitude of mode B for two different gate voltages, showing softening (blue squares) and stiffening (red circles) behaviors. The modal coupling strength can be determined from the quadratic dependence of the frequency shift with the oscillation amplitude (dashed lines).