Local and Global Reciprocity in Orbital-Charge-Coupled Transport

The coupled transport of charge and orbital angular momentum (OAM) lies at the core of orbitronics. Here, we examine the reciprocal relation in orbital-charge-coupled transport in thin films, treating bulk and surface contributions on equal footing. We argue that the conventional definition of orbital current is ill-defiled, as it violates reciprocity due to the nonconservation of OAM. This issue is resolved by adopting the so-called \emph{proper} orbital current, which is directly linked to orbital accumulation. We establish the reciprocal relation for the \emph{global} (spatially integrated) response between orbital and charge currents, while showing that their \emph{local} (spatially resolved) responses can differ significantly. In particular, we find large surface contributions that may lead to nonreciprocity when currents are measured locally. These findings are supported by first-principles calculations on W(110) and Pt(111) thin films. In W(110), orbital-charge interconversion is strongly nonreciprocal at the layer level, despite exact reciprocity in the integrated response. Interestingly, spin-charge interconversion in W(110) remains nearly reciprocal even locally. In contrast, Pt(111) exhibits local nonreciprocity for both orbital-charge and spin-charge conversions, which we attribute to strong spin-orbit coupling. We propose that such local distinctions can be exploited to experimentally differentiate spin and orbital currents.