On the local aspect of valley magnetic moments

Valley magnetic moments play a crucial role in valleytronics in 2D hexagonal materials. Traditionally, insights drawn from the study of quantum states in homogeneous bulks have led to a widespread belief that only materials with broken structural inversion symmetry can exhibit nonvanishing valley magnetic moments. This belief, however, limits the scope of relevant applications, especially for materials with inversion symmetry, such as gapless monolayer graphene, despite its advantage in routine growth and production. This work revisits valley-derived magnetic moments in a broad context covering inhomogeneous structures as well. It generalizes the notion of a valley magnetic moment for a state from an integrated quantity to the local field called the “local valley magnetic moment” with space-varying distribution. It explores the local magnetic moment analytically both within the Dirac model and through a symmetry argument. Numerical investigations are conducted within the tight-binding model. Overall, we demonstrate that the breaking of inversion symmetry in the electron probability distribution leads to nonvanishing local magnetic moments. This probability-based breaking can occur in both structural inversion symmetric and symmetry-broken structures. In suitable inversion-symmetric structures with inhomogeneity, e.g., zigzag nanoribbons of gapless monolayer graphene, it is shown that the local moment of a state can be nonvanishing while the corresponding integrated moment is subject to the broken symmetry constraint. Moreover, it is demonstrated that the local moment can interact with space-dependent magnetic fields, resulting in field effects such as valley Zeeman splitting. Such effects can be exploited for local valley control as a conduit for the implementation of valleytronics.