Theory of extrinsic contributions to the full orbital current

Abstract: The orbital Hall effect (OHE) underpins the emerging field of orbitronics. A complete quantum-mechanical evaluation of the usual orbital-current operator must retain all matrix elements of the position operator, including its band-diagonal differential part; intrinsic calculations based on this full evaluation can differ by orders of magnitude from the conventional truncation. A relaxation-time estimate of disorder effects on the full current has been reported, but a microscopic treatment of extrinsic scattering has remained unavailable because the position operator requires wavevector-off-diagonal density-matrix elements. Here we develop such a theory by solving the quantum kinetic equation for the wavevector-off-diagonal density matrix, including band-structure, field-corrected side-jump, and skew-scattering terms within the non-crossing approximation. We apply the theory to a massive Dirac cone and to the same model with a particle-hole-symmetry-breaking quadratic term. Disorder-generated contributions of order $τ<sup>0$ are generically comparable to the intrinsic current in the metallic regime and cannot be separated from it by simple disorder-strength scaling. For the bare massive Dirac cone the surviving non-crossing extrinsic correction cancels the conventional intrinsic value and leaves a total current exactly twice the quantum correction, whereas particle-hole asymmetry activates conventional skew scattering for disorder with a non-zero third moment, which dominates in sufficiently clean samples. The wavevector-off-diagonal construction is independent of the two-band model and provides a general route to disorder corrections for position-dependent observables in multiband solids.