  A valley-contrasting Berry curvature in bilayer transition metal
dichalcogenides with spin-orbit coupling can generate valley magnetization when
the inversion symmetry is broken, for example, by an electric field, regardless
of time-reversal symmetry. A nontrivial Berry curvature can also lead to
anomalous transport responses, such as the anomalous Hall effect and the
anomalous Nernst effect. Applied to a bilayer WSe$_2$, an electric field can
tune the Berry curvature and orbital magnetic moment, which has important
consequences for the orbital magnetization and the anomalous Nernst responses.
The orbital magnetization and its two contributions, one due to the magnetic
moment and one due to the Berry curvature, are calculated and interpreted in
terms of opposite circulating currents of the bands in the two layers. The
valley anomalous Nernst coefficient and spin Nernst coefficient are also
calculated. We find that a finite electric field leads to peaks and dips in the
Nernst responses that have the signs of the Berry curvatures of the bands and
are proportional to their magnitudes; it also enhances the valley Nernst
responses. These experimentally verifiable findings may be promising for
caloritronic applications.
