  An important goal in nanoelectromechanics is to cool the vibrational motion,
ideally to its quantum ground state. Cooling by an applied charge current is a
particularly simple and hence attractive strategy to this effect. Here, we
explore this phenomenon in the context of the general theory of
thermoelectrics. In linear response, this theory describes thermoelectric
refrigerators in terms of their cooling efficiency and figure of merit ZT. We
show that both concepts carry over to phonon cooling in nanoelectromechanical
systems. As an important consequence, this allows us to discuss the efficiency
of phonon refrigerators in relation to the fundamental Carnot efficiency. We
illustrate these general concepts by thoroughly investigating a simple
double-quantum-dot model with the dual advantage of being quite realistic
experimentally and amenable to a largely analytical analysis theoretically.
Specifically, we obtain results for the efficiency, the figure of merit, and
the effective temperature of the vibrational motion in two regimes. In the
quantum regime in which the vibrational motion is fast compared to the
electronic degrees of freedom, we can describe the electronic and phononic
dynamics of the model in terms of master equations. In the complementary
classical regime of slow vibrational motion, the dynamics is described in terms
of an appropriate Langevin equation. Remarkably, we find that the efficiency
can approach the maximal Carnot value in the quantum regime, with large
associated figures of merit. In contrast, the efficiencies are typically far
from the Carnot limit in the classical regime. Our theoretical results should
provide guidance to implementing efficient vibrational cooling of
nanoelectromechanical systems in the laboratory.
