Polarons in Suspended Carbon Nanotubes

We prove theoretically the possibility of electric-field controlled polaron formation involving flexural (bending) modes in suspended carbon nanotubes. Upon increasing the field, the ground state of the system with a single extra electron undergoes a first-order phase transition between an extended state and a localized polaron state. For a common experimental setup, the threshold electric field is only of the order of $\simeq 5\times {10}^{-2}\mathrm{V}/\mu \mathrm{m}$.

Figure 1

Setup: A semiconducting single wall carbon nanotube cantilever of length $L$. The supported part of the tube rests on an insulating substrate. The voltage-biased metallic electrode A is connected to the supported part of the tube by means of a tunnel barrier. An electron that enters the suspended part of the tube experiences a force $\mathit{F}=-e\mathit{E}$ perpendicular to the tube. As a result, the tube is deformed so that each point $x$ on the tube undergoes a displacement $y(x)$ perpendicular to the $x$ axis. The electron wave function $\psi (x)$ is also indicated.

Figure 2

Results of the numerical calculation: In the top panels and in the bottom left panel, solid curves indicate ground state properties. Dashed curves indicate properties of the metastable polaron state. A thin vertical line indicates the value of $\alpha /{\alpha }_c$ below which no polaron solutions exist. The critical value of $\alpha$ is ${\alpha }_c=312.03$. Top left: The minimal values $h_0$ of the (dimensionless) energy $h[\varphi ,f]$ [cf. Eq. (2)] vs $\alpha /{\alpha }_c$. Top right: The dimensionless inverse localization length $\kappa$ vs $\alpha /{\alpha }_c$. Bottom left: The ratio $n=U/\hslash {\omega }_0$ between the bending energy and the lowest phonon energy vs $\alpha /{\alpha }_c$. Bottom right: The electric field $E$ as a function of $\alpha /{\alpha }_c$, for $L=1\mu \mathrm{m}$, and three different tube radii $r$. Solid curve, $r=0.5\mathrm{nm}$; dashed curve, $r=1\mathrm{nm}$; and dotted curve, $r=1.5\mathrm{nm}$.

Figure 3

First-order phase transition: The dimensionless energy $h_{\mathrm{var}}$ minimized with respect to $a_m$ and $b_m$, versus $\kappa$, for several values of the coupling constant $\alpha$. The bottommost curve corresponds to $\alpha =355.2$. In each subsequent curve, the value of alpha is decremented by 4.797. The curves in the white region of the graph ($\alpha >{\alpha }_c=312.03$) have minima at $\kappa >0$ where the energy is negative (black dots). These minima correspond to a stable polaron ground state. Curves in the light gray region (${\alpha }_c>\alpha >{\alpha }_{\min }=258.96$) have local minima at $\kappa >0$ (open circles). These minima correspond to a metastable polaron state. For $\alpha <{\alpha }_{\min }$ (dark gray region), curves have no minimum at $\kappa >0$ and no polaron state exists.

Figure 4

Phase diagram: The three different shaded regions correspond to different tube radii. Dark gray, $r=0.5\mathrm{nm}$; gray, $r=1\mathrm{nm}$; and light gray, $r=1.5\mathrm{nm}$. In each case the shaded region indicates where the ground state is a polaron. At large $L$, the upper planes have been cut away to reveal the planes below. The lower boundaries (black curves) indicate the first-order transition where $\alpha ={\alpha }_c$. The upper boundaries correspond to $y(L)=L$.