Electrical probe for mechanical vibrations in suspended carbon nanotubes

The transport properties of a suspended carbon nanotube probed by means of a scanning tunnel microscope (STM) tip are investigated. A microscopic theory of the coupling between electrons and mechanical vibrations is developed. It predicts a position-dependent coupling constant, sizable only in the region where the vibron is located. This fact has profound consequences on the transport properties, which allow to extract information on the location and size of the vibrating portions of the nanotube.

Figure 1

Schematic setup of a CNT, suspended between the two substrates S1 and S2 at positions $x_1$ and $x_2$. A quantum dot with ends at $x=0$ and $x=L$ is embedded in the CNT.

Figure 2

Fermi velocity $v_{\mathrm{F}}$ of a semiconducting CNT of length $L=400$ nm with $N_0$ electrons and different CNT waist lengths $|{\mathbf{w}}_{n,m}|$: blue (solid) 1 nm is for (3,2), purple (dashed) 2 nm is for (5,4), yellow (dotted) 3 nm is for (8,6), green (dash-dotted) 4 nm is for (11,7).

Figure 3

Energy of the lowest eigenmode. (a) Plot of ${\Omega }_0/{\omega }_0$ as a function of $\alpha$. (b) Plot of ${\Omega }_1/{\omega }_1$ as a function of $\alpha$. In all figures, $\delta =1$, $x_{v0}=0$, $p=1$, $g=1$, and ${\lambda }_{\mathrm{m}}=2$.

Figure 4

Schematic setup of an STM transport experiment performed on a suspended CNT. The quantum dot is the nanotube itself biased with respect to the tip and two lateral contacts. Electrons can flow through the system via the STM tip at position ($x$) and at the contacts ($x_{1,2}$). A back gate allows to tune the effective charge on the dot.

Figure 5

Local electron-vibron coupling $\lambda \left(x\right)$ as a function of the tip position $x$ for (a) $\delta =1$, $x_{v0}=0$, and $p=1$, (b) as above but $p=2$, (c) $\delta =0.4$, $x_{v0}=0.15L$, and $p=1$, (d) as above but $p=2$. In all panels, red (solid) lines denote a metallic CNT with $\alpha =32$, blue (dashed) lines denote a semiconducting CNT with $\alpha =5$. Other parameters: ${\lambda }_{\mathrm{m}}=2$ and $g=1$. The shaded plots on top of the panels depict the amplitude of the strain field ${\widehat{u}}_p\left(x\right)$. The arrow in panel (c) denotes the tip position for the conductance shown in Fig. 8.

Figure 6

Local electron vibron coupling $\lambda \left(x\right)$ as a function of $x$ for $\delta =1$, $x_{v0}=0$ and different vibron modes: red (solid) $p=1$, green (dashed) $p=2$, blue (dotted) $p=3$. The thin lines are the maxima of $\lambda \left(x\right)$. Other parameters: $\alpha =5$, ${\lambda }_{\mathrm{m}}=2$, and $g=1$.

Figure 7

Ratio of the coupling strengths ${\lambda }_{\mathrm{L}}/{\lambda }_{\mathrm{R}}$ (see text) as a function of $\delta$ for a vibron with origin at $x_{v0}=0.1L$ and $p=1$. The shaded plots schematically depict the amplitude of the strain field of the vibronic mode ${\widehat{u}}_p\left(x\right)$. Other parameters: $\alpha =5$, ${\lambda }_{\mathrm{m}}=2$, and $g=1$.

Figure 8

Semiconducting CNT. Differential conductance $\mathcal{G}$ (units $e^2{\Gamma }_0^{\left(\mathrm{T}\right)}/{\Omega }_0$) as a function of $N_{\mathrm{g}}$ and $V$ (units ${\Omega }_0/e$) for a vibron originating at $x_{v0}=0.15L$ with $\delta =0.4$, $p=1$, and a tip at $x=0.35L$. Other parameters: ${\lambda }_{\mathrm{m}}=2$, $\alpha =5$, $g=1$, $k_{\mathrm{B}}T=0.15{\Omega }_0$, and $A={\Gamma }_0/{\Gamma }_0^{\left(\mathrm{T}\right)}=100$.

Figure 9

Semiconducting CNT: Plot of $\mathcal{G}$ (units $e^2{\Gamma }_0^{\left(\mathrm{T}\right)}/{\Omega }_0$) as a function of the tip position $x$ and bias voltage $V$ (units ${\Omega }_0/e$) at resonance $N_{\mathrm{g}}=(1/2)+(3{\omega }_1/8E_{{\rho }_{+}})$ and (a) $\delta =1$, $x_{v0}=0$, and $p=1$, (b) same as in (a) but for $p=2$, (c) $\delta =0.4$, $x_{v0}=0.15L$, and $p=1$, (d) same as in (c) but for $p=2$. Other parameters: ${\lambda }_{\mathrm{m}}=2$, $\alpha =5$, $g=1$, $k_{\mathrm{B}}T=0.15{\Omega }_0$, and $A=100$.

Figure 10

Metallic CNT: Plot of $\mathcal{G}$ (units $e^2{\Gamma }_0^{\left(\mathrm{T}\right)}/{\Omega }_0$) as a function of the tip position $x$ and bias voltage $V$ (units ${\Omega }_0/e$) at resonance $N_{\mathrm{g}}=(1/2)+(3{\omega }_1/8E_{{\rho }_{+}})$ and (a) $\delta =1$, $x_{v0}=0$, and $p=1$, (b) same as in (a) but for $p=2$, (c) $\delta =0.4$, $x_0=0.15L$, and $p=1$; (d) same as in (c) but for $p=2$. Other parameters: ${\lambda }_{\mathrm{m}}=2$, $\alpha =32$, $g=1$, $k_{\mathrm{B}}T=0.15{\Omega }_0$, and $A=100$.

Figure 11

Differential conductance $\mathcal{G}$ (units $e^2{\Gamma }_0^{\left(\mathrm{T}\right)}/{\Omega }_0$) as a function of the tip position $x$ for $N_{\mathrm{g}}=(1/2)+(3{\omega }_1/8E_{{\rho }_{+}})$ and $eV=2{\Omega }_0$ (red solid line) or $eV=4{\Omega }_0$ (blue dashed line) for a metallic CNT: (a) $\delta =1$, $x_{v0}=0,$ and $p=1$, (b) same as in (a) but for $p=2$, (c) $\delta =0.4$, $x_{v0}=0.15L$, and $p=1$, (d) same as in (c) but for $p=2$. Other parameters: ${\lambda }_{\mathrm{m}}=2$, $\alpha =32$, $g=1$, $k_{\mathrm{B}}T=0.15{\Omega }_0$, and $A=100$.

Figure 12

(a) Conductance $\mathcal{G}$ (units $e^2{\Gamma }_0^{\left(\mathrm{T}\right)}/{\Omega }_0$) as a function of the tip position $x$ for decreasing values of $g$ from $g=1$ (noninteracting, lightest gray) to $g=0.2$ (strongly interacting, black); (b) Plot of $\mathcal{G}$ for different asymmetries (see key) and $g=1$. Other parameters: $\delta =1$, $x_{v0}=0$, $p=1$, ${\lambda }_{\mathrm{m}}=2$, $\alpha =5$, $k_{\mathrm{B}}T=0.15{\Omega }_0$ and $eV=2{\Omega }_0$.