Dynamics and Dissipation Induced by Single-Electron Tunneling in Carbon Nanotube Nanoelectromechanical Systems

We demonstrate the effect of single-electron tunneling (SET) through a carbon nanotube quantum dot on its nanomechanical motion. We find that the frequency response and the dissipation of the nanoelectromechanical system to SET strongly depends on the electronic environment of the quantum dot, in particular, on the total dot capacitance and the tunnel coupling to the metal contacts. Our findings suggest that one could achieve quality factors of ${10}^6$ or higher by choosing appropriate gate dielectrics and/or by improving the tunnel coupling to the leads.

Figure 1

False color SEM image of (a) a low capacitance device based on a ${\mathrm{SiO}}_2$ covered Si backgate (grey) and (b) a high capacitance device based on an ${\mathrm{Al}}_2{\mathrm{O}}_3$ covered local metallic gate (blue). The rf actuation signal is injected into the local metallic gate and Si backgate, respectively, through a home-built bias-$T$. As the induced mechanical motion changes the charge flow through the CNT quantum dot and vice versa, we can detect the CNT resonance through a change in zero bias conductance. (c) Mechanical resonance of a typical CNT NEMS at low driving power $P_{\mathrm{rf}}=-100\mathrm{dBm}$. The resonance width $\delta f=2.5\mathrm{kHz}$ leads to a quality factor of $Q\approx 140000$.

Figure 2

Modulation of the resonance frequency due to SET: (a) Frequency shift $\Delta f$ (scaled with $f{V}_g^2$) for different dot capacitance $C_{\mathrm{dot}}$, with comparable tunneling rates $\Gamma$. Inset shows a typical Coulomb peak centered at $\Delta V_g=0$. (b) Maximum scaled frequency shift at maximum tunnel current $G_{\max }(\Delta V_g=0)$ vs the dot capacitance for all measured devices. The solid red line corresponds to the fit using Eq. (1) with $C_g^{\prime 2}/k=8\times {10}^{-22}{\mathrm{F}}^2/\mathrm{N}\mathrm{m}$ and ${\Gamma }_d\approx 10\mathrm{GHz}$. The data are in fair agreement with the model. (c) Scaled frequency shift for different Coulomb peaks, i.e., tunneling rates $\Gamma$, with a fixed dot capacitance $C_{\mathrm{dot}}=260\mathrm{aF}$ (device 1). The tunneling rate $\Gamma$ is estimated from the line shape of the respective Coulomb peaks (see 13). (d) Maximum frequency shift $\Delta f_{\max }$ at maximum tunnel current $G_{\max }(\Delta V_g=0)$ vs temperature for a device yielding $C_{\mathrm{dot}}=160\mathrm{aF}$ (device 2). The inset shows the evolution of $G_{\max }$ as a function of temperature.

Figure 3

Modulation of the $Q$ factor (scaled with $f{V}_g^2$) due to SET: (a) For different dot capacitances, with comparable tunneling rates $\Gamma$. The solid lines are to guide the eye and the inset shows a representative Coulomb peak centered at $\Delta V_g=0$. (b) Maximum quality factor $Q_{\max }$ at minimal tunnel current $G(|\Delta V_g|\gg 0)$ vs the dot’s capacitance for all measured devices. The solid red line represents the fit with Eq. (2) with $C_g^{\prime 2}/k=1\times {10}^{-21}{\mathrm{F}}^2/\mathrm{N}\mathrm{m}$ and $\Gamma \approx 10\mathrm{GHz}$. Our findings are in good agreement with the model. (c) Modulation of the $Q$ factor (scaled with $f{V}_g^2$) for different tunneling rates $\Gamma$, with a fixed dot capacitance $C_{\mathrm{dot}}=160\mathrm{aF}$ (device 2). The solid lines are to guide the eye and the tunnel coupling $\Gamma$ is estimated from the line shape of the respective Coulomb peaks (see 13). (d) Maximum quality factor $Q_{\max }$ at minimal tunnel current $G(|\Delta V_g|\gg 0)$ vs temperature for a device yielding $C_{\mathrm{dot}}=160\mathrm{aF}$ (device 2). The inset shows the evolution of $G_{\min }$ as a function of temperature.