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Using the non-equilibrium Keldysh formalism, we investigate the spatialrelation between the electro-chemical potential measured in scanning tunnelingspectroscopy, and local current patterns over the entire range from the quantumto the classical transport regime. These quantities show similar spatialpatterns near the quantum limit, but are related by Ohm's law only in theclassical regime. We demonstrate that defects induce a Landauer residualresistivity dipole in the electro-chemical potential with the concomitantspatial current pattern representing the field lines of the dipole.

Scanning Tunneling Potentiometry, Charge Transport and Landauer's Resistivity Dipole from the Quantum to the Classical Transport Regime