  We show that in mesoscopic four-terminal thermoelectric devices with two
electrodes (the source and the drain) and two heat baths, inelastic scattering
processes can lead to unconventional thermoelectric transport. The source (or
the drain) can be cooled by passing a thermal current between the two heat
baths, with no net heat exchange between the heat baths and the electrodes.
This effect, termed as cooling by heat current, is a mesoscopic heat drag
effect. In addition, there is a transverse thermoelectric effect where
electrical current and power can be generated by a transverse temperature bias
(i.e., the temperature bias between the two heat baths). This transverse
thermoelectric effect, originates from inelastic scattering processes, may have
advantages for improved figures of merit and power factor due to spatial
separation of charge and heat transport. We study the Onsager current-affinity
relations, the linear-response transport properties, and the transverse
thermoelectric figure of merit of the four-terminal thermoelectric devices for
various system parameters. In addition, we investigate the efficiency and power
of the cooling by transverse current effect in both linear and nonlinear
transport regimes. We also demonstrate that by exploiting the inelastic
transport in the quantum-dot four-terminal systems, a type of Maxwell's demon
can be realized using nonequilibrium heat baths.
