  The discovery of Berry curvature (BC) has spurred a tremendous surge of
research into various quantum phenomena such as the anomalous transport of
electrons and the topological phases of matter. In two-dimensional crystalline
systems, the conventional definition of the BC lacks the in-plane components
and thus it cannot explain the transverse transport along the plane-normal
direction. Here, we modify the BC to newly provide in-plane components in two
dimensions, giving rise to the vertical Hall effects that describe out-of-plane
transports in response to in-plane perturbations and their Onsager reciprocity.
Our first-principles calculations show that a large in-plane BC can appear even
in an atomic-thick GdAg2 monolayer, and a hexagonal BiAg2 monolayer can host a
large BC dipole known to vanish in the conventional BC. The quantum transports
driven by the hitherto-hidden BC will become more significant in recently
emerging two-dimensional platforms, including van der Waals heterostructures.

Mesoscale and Nanoscale Physics;Materials Science