A twist in the way an electron’s wave changes as the electron moves through a crystal. It acts like a hidden magnetic field, pushing moving electrons sideways even when no real magnetic field is applied.
Going deeper
In a honeycomb with inequivalent sites, Berry curvature concentrates at the zone corners with opposite signs at K and K′. Under an electric field it deflects carriers sideways, K and K′ to opposite edges – the valley Hall effect, which gives a Hall voltage without a magnetic field once one valley is filled preferentially.
A magnetic field in momentum space
As an electron’s crystal momentum changes, the shape of its wavefunction within the changes too. Carried around a closed loop in momentum space, the wavefunction picks up a geometric phase, the Berry phase, in addition to the ordinary dynamical one. The Berry curvature is that phase per unit area of momentum space; mathematically it behaves like a magnetic field, except that it lives in momentum rather than real space.
Its physical consequence is an anomalous velocity. An electron pushed by an electric field moves not only along the field but also sideways, in proportion to the Berry curvature at its momentum. Summed over a completely filled band, the curvature gives an integer, the , which fixes the quantised Hall conductance of a or Chern insulator.
Where it is large
Berry curvature concentrates where bands come close together, so small produce large, sharply peaked curvature. Symmetry controls its sign. makes it opposite at opposite momenta; makes it equal there; with both, it vanishes everywhere for -degenerate bands.
A gapped breaks inversion but keeps time reversal, so K and K′ carry equal and opposite curvature and the total is zero – yet each alone has a strong, well-defined sign. This is the case in hBN, in and in bilayer graphene with a gap opened by a perpendicular field. When time reversal is also broken, by magnetism or by spontaneous valley polarisation in , the curvature need not cancel, giving anomalous and even quantised effects without an external field.
Seeing its effects
The valley Hall effect makes Berry curvature visible. In monolayer MoS2, circularly polarised light of one excites carriers mainly in one valley; under an in-plane field they drift sideways, producing a Hall voltage with no magnetic field, and the sign reverses with the light’s handedness. In graphene aligned with hBN and in gapped bilayer graphene, large nonlocal resistances have been attributed to valley currents, although other can contribute and interpretations have been debated.
Other signatures include the anomalous Hall effect of magnetic metals, the nonlinear Hall effect from a Berry curvature dipole in few-layer WTe2, and the valley-selective optical selection rules of TMDC monolayers, which share their origin with the orbital magnetic moment tied to the curvature.
For specialists
The imaginary part of the quantum geometric tensor – an effective magnetic field in momentum space responsible for anomalous and valley Hall effects.