  We study current-induced vibrational cooling, heating, and instability in a
donor-acceptor rectifying molecular junction using a full counting statistics
approach. In our model, electron-hole pair excitations are coupled to a given
molecular vibrational mode which is either harmonic or highly anharmonic. This
mode may be further coupled to a dissipative thermal environment. Adopting a
master equation approach, we confirm the charge and heat exchange fluctuation
theorem in the steady-state limit, for both harmonic and anharmonic models.
Using simple analytical expressions, we calculate the charge current and
several measures for the mode effective temperature. At low bias, we observe
the effect of bias-induced cooling of the vibrational mode. At higher bias, the
mode effective temperature is higher than the environmental temperature, yet
the junction is stable. Beyond that, once the vibrational mode (bias-induced)
excitation rate overcomes its relaxation rate, instability occurs. We identify
regimes of instability as a function of voltage bias and coupling to an
additional phononic thermal bath. Interestingly, we observe a reentrant
behavior where an unstable junction can properly behave at a high enough bias.
The mechanism for this behavior is discussed.
