We study the dynamics of transverse oscillations of a suspended carbon nanotube into which electron current is injected from the tip of a scanning tunneling microscope (STM). In this case the correlations between the displacement of the nanotube and its charge state, determined by the position-dependent electron tunneling rate, can lead to a “shuttle-like” instability for the transverse vibrational modes. We find that selective excitation of a specific mode can be achieved by an accurate positioning of the STM tip. This result suggests a feasible way to control the dynamics of this nano-electromechanical system (NEMS) based on the “shuttle instability.”

There are several reasons for the considerable current interest in nano-electromechanical systems (NEMSs), both for technological applications and fundamental research. The peculiar combination of several features such as high vibrational frequencies and small masses which characterize most NEMS makes these systems very suitable for the realization of new measurement tools with extremely high sensitivity in mass sensing and force microscopy applications [

The physical basis for many of the interesting functionalities of NEMS is the strong interplay between mechanical and electronic degrees of freedom [

The typical setup for many NEMS includes a spatially extended movable element such as a suspended carbon nanotube, whose dynamics has been demonstrated to be characterized by a number of different vibrational modes [

The selective promotion of the electromechanical instability for different vibrational modes provides an interesting perspective for probing the dynamics of NEMS. Here we show that such selective excitation can be achieved by means of local injection of electric charge. The main idea is to optimize the electromechanical coupling for the mode(s) which we want to make unstable. The local character of the electric charge injection makes the selective excitation of the nanotube transverse modes possible by varying the position of the STM tip.

We will consider here the same device analyzed by Jonsson et al. since it provides a convenient set-up to control the electromechanical coupling of different vibrational modes. The system is sketched in Figure

Sketch of the model system considered. A metallic carbon nanotube is suspended over the trench between two metallic leads which are mantained at the same electrochemical potential, while an STM is put at distance

We take the

In order to describe the motion of the nanotube we model it as a classical elastic beam of length

The motion of the nanotube in the

In (

The precise spatial distribution of

The displacement field

The expansion of

In (

We introduced in (

An approximate expression for the force coefficients

The size of the correction in (

For what concerns the transport of charge the system is equivalent to a double tunnel junction, having one potential barrier localized between the STM tip and the nanotube and the another one between the nanotube and the leads. In our analysis we will consider the case of electrons for which the decoherence rate is much greater than the tunneling rates, so that the description of tunneling as a stochastic (rather than coherent) process is sufficient.

Furthermore, we consider the system in the Coulomb blockade regime and limit to one the number of extra electrons which can charge the nanotube, that is,

Following the approach presented in [

The coupling between the mechanical and electronic degrees of freedom arises because the tunneling rate between the STM tip and the nanotube,

We remark that all the tunneling rates are generally functions of the

The time evolution of the probability densities

From (

The set of equations (

The static solution of the linearized equations of motion,

This procedure leads to an algebraic equation which in general cannot be solved analytically. However, if the dimensionless parameters

For realistic values of the parameters which are consistent with the conditions of validity of our model (

The sign of the real part of

For fixed values of

The set of values of

In order to map the instability regions we have to specify an analytic expression for the damping rates

In order to include this effect in our model we follow the approach introduced by Zener and formally replace the Young modulus with a frequency dependent complex function which results in the following expression for the damping rates:

A large class of dissipative phenomena in solids (e.g., thermoelasticity, dislocations, and defects dynamics) can be parametrized though the dimensionless coefficient

We first consider the limit in which the characteristic inverse time of the retarded mechanical response is much smaller than the frequencies of the nanotube eigenmodes:

In Figure

(Color online) Regions of instability in the parameter plane (

The physical picture presented in Figure

(Color online) Regions of instability in the parameter plane (

The dynamical behaviour of the nanotube in the regime of single-mode instability is qualitatively the same of the ordinary “shuttle” system [

In the phase space of the system the dynamical state in this situation is described by a limit cycle, that is, an isolated closed trajectory characterized by finite amplitude oscillations [

In conclusion in the present work we studied the dynamics of the flexural vibrations of a suspended carbon nanotube in which extra electrons are injected at a position-dependent rate. We showed that a localized constant electrostatic field can excite many transverse vibrational modes of the nanotube into a “shuttle-like” regime of charge transport. For a fixed bias voltage and in presence of dissipative processes with inverse characteristic times much smaller than the frequencies of the nanotube vibrational modes, we found that it is possible to induce a selective instability through an accurate positioning of an STM. It thus seems possible to extend the approach followed here to other systems characterized by a nontrivial coupling between charge transport and mechanical degrees of freedom.

The author wants to thank L. Y. Gorelik, R. I. Shekhter, and M. Jonson for fruitful discussions and support. Partial financial support from the Swedish VR and from the Faculty of Science at the University of Gothenburg through its “Nanoparticle” Research Platform is gratefully acknowledged.