  Dirac semimetals (DSM) and Weyl semimetal (WSM) fall under the generic class
of three-dimensional solids, which follow relativistic energy-momentum relation
$\epsilon_\mathbf{k}= \hbar v_F |\mathbf{k}|$ at low energies. Such a linear
dispersion when regularized on a lattice can lead to remarkable properties such
as the anomalous Hall effect, presence of Fermi surface arcs, positive
longitudinal magnetoconductance, and dynamic chiral magnetic effect. The last
two properties arise due to the manifestation of chiral anomaly in these
semimetals, which refers to the non-conservation of chiral charge in the
presence of electromagnetic gauge fields. Here, we propose the planar Nernst
effect, or transverse thermopower, as another consequence of chiral anomaly,
which should occur in both Dirac and Weyl semimetals. We analytically calculate
the planar Nernst coefficient for DSMs (type-I and type-II) and also WSMs
(type-I and type-II), using a quasi-classical Boltzmann formalism. The planar
Nernst effect manifests in a configuration when the applied temperature
gradient, magnetic field, and the measured voltage are all co-planar, and is of
distinct origin when compared to the anomalous and conventional Nernst effects.
Our findings, specifically a 3D map of the planar Nernst coefficient in type-I
Dirac semimetals (Na$_3$Bi, Cd$_3$As$_2$ etc) and type-II DSM (PdTe$_2$, VAI3
etc), can be verified experimentally by an in-situ 3D double-axis rotation
extracting the full $4\pi$ solid angular dependence of the Nernst coefficient.
