  The thermoelectric properties of conductors with low electron density can be
altered significantly by an applied magnetic field. For example, recent work
has shown that Dirac/Weyl semimetals with a single pocket of carriers can
exhibit a large enhancement of thermopower when subjected to a sufficiently
large field that the system reaches the extreme quantum limit, in which only a
single Landau level is occupied. Here we study the magnetothermoelectric
properties of compensated semimetals, for which pockets of electron- and
hole-type carriers coexist at the Fermi level. We show that, when the
compensation is nearly complete, such systems exhibit a huge enhancement of
thermopower starting at a much smaller magnetic field, such that $\omega_c\tau
> 1$, and the stringent conditions associated with the extreme quantum limit
are not necessary. We discuss our results in light of recent measurements on
the compensated Weyl semimetal tantalum phosphide, in which an enormous
magnetothermoelectric effect was observed. We also calculate the Nernst
coefficient of compensated semimetals, and show that it exhibits a maximum
value with increasing magnetic field that is much larger than in the single
band case. In the dissipationless limit, where the Hall angle is large, the
thermoelectric response can be described in terms of quantum Hall edge states,
and we use this description to generalize previous results to the multi-band
case.
