Current-induced forces in nanosystems: A hierarchical equations of motion approach

An approach to calculating current-induced forces in charge transport through nanosystems is introduced. Starting from the fully quantum mechanical hierarchical equations of motion formalism, a timescale separation between electronic and vibrational degrees of freedom is used to derive a classical Langevin equation of motion for the vibrational dynamics as influenced by current-induced forces, such as the electronic friction. The resulting form of the friction is shown to be equivalent to previously derived expressions. The numerical exactness of the hierarchical equations of motion approach, however, allows the investigation of transport scenarios with strong intrasystem and system-environment interactions. As a demonstration, the electronic friction of three example systems is calculated and analyzed: a single electronic level coupled to one classical vibrational mode, an Anderson impurity model coupled to one classical vibrational mode, and a single electronic level coupled to both a classical and quantum vibrational mode.