Current-induced forces in mesoscopic systems: a scattering matrix approach

Abstract

Nanoelectromechanical systems are characterized by an intimate connection between electronic and mechanical degrees of freedom. Due to the nanoscopic scale, current flowing through the system noticeably impacts the vibrational dynamics of the device, complementing the effect of the vibrational modes on the electronic dynamics. We employ the scattering matrix approach to quantum transport to develop a unified theory of nanoelectromechanical systems out of equilibrium. For a slow mechanical mode, the current can be obtained from the Landauer-B\"uttiker formula in the strictly adiabatic limit. The leading correction to the adiabatic limit reduces to Brouwer's formula for the current of a quantum pump in the absence of the bias voltage. The principal result of the present paper are scattering matrix expressions for the current-induced forces acting on the mechanical degrees of freedom. These forces control the Langevin dynamics of the mechanical modes. Specifically, we derive expressions for the (typically nonconservative) mean force, for the (possibly negative) damping force, an effective "Lorentz" force which exists even for time reversal invariant systems, and the fluctuating Langevin force originating from Nyquist and shot noise of the current flow. We apply our general formalism to several simple models which illustrate the peculiar nature of the current-induced forces. Specifically, we find that in out of equilibrium situations the current induced forces can destabilize the mechanical vibrations and cause limit-cycle dynamics.