Sensitivity of mixing-current technique to detect nanomechanical motion

Detection of nanomechanical displacement by transport techniques has reached a high level of sensitivity and versatility. In order to detect the amplitude of oscillation of a nanomechanical oscillator, a widely used technique consists of coupling this motion capacitively to a single-electron transistor and to detect the high-frequency modulation of the current through the nonlinear mixing with an electric signal at a slightly detuned frequency. The method known as mixing-current technique is employed in particular for the detection of suspended carbon nanotubes. In this paper we study theoretically the limiting conditions on the sensitivity of this method. The sensitivity is increased by increasing the response function to the signal, but also by reducing the noise. For these reasons we study systematically the response function, the effect of current and displacement fluctuations, and finally the case where the tunneling rate of the electrons are of the same order or larger of the resonating frequency. We find thus upper bounds to the sensitivity of the detection technique.

Figure 1

Schematic of the typical experimental setup used to measure the displacement of a mechanical oscillator by detection of the mixing current (adapted from Ref. [31]).

Figure 2

$S_{xx}^{\mathrm{add}}$ as a function of $\delta ={\epsilon }_P/eV$ for $\nu \left(\delta \right)$ minimizing the function. Inset: the value of $\nu$ that minimizes the function for given $\delta \left[{\nu }_m\left(\delta \right)\right]$.

Figure 3

Electric scheme of a single-electron transistor.