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 upons 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 in order to develop a unified theory of
nanoelectromechanical systems out of equilibrium. For a slow mechanical mode
the current can be obtained from the Landauer–Büttiker 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 a bias
voltage. The principal results of the present paper are the 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
that 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 that 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