  We investigate non-equilibrium transport of charge and heat through an
interacting quantum dot coupled to a finite electron reservoir. Both the
quantum dot and the finite reservoir are coupled to conventional electric
contacts, i.e., infinite electron reservoirs, between which a bias voltage can
be applied. We develop a phenomenological description of the system, combining
a rate equation for transport through the quantum dot with standard linear
response expressions for transport between the finite and infinite reservoirs.
The finite reservoir is assumed to be in a quasi-equilibrium state with
time-dependent chemical potential and temperature which we solve for
self-consistently. We show that the finite reservoir can have a large impact on
the stationary state transport properties, including a shift and broadening of
the Coulomb diamond edges. We also demonstrate that there is a region around
the conductance lines where a heat current flows out of the finite reservoir.
Our results reveal the dependence of the temperature that can be reached by
this thermoelectric cooling on the system parameters, in particular the
coupling between the finite and infinite reservoirs and additional heat
currents induced by electron-phonon couplings, and can thus serve as a guide to
experiments on quantum dot-enabled thermoelectric cooling of finite electron
reservoirs. Finally, we study the full dynamics of the system, with a
particular focus on the timescales involved in the thermoelectric cooling.
