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 $omega_ctau > 1$, 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.
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