Mixtures of classical and quantum-mechanical particles and the ionic response in electromigration

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Abstract
The authors discuss the result of taking the classical limit on the ions of a metal while retaining a quantum-mechanical description of the electrons. One finds that when this limit is possible the ionic motion is determined by a classical Hamiltonian, but in general this gives rise to velocity-dependent forces. These forces are non-adiabatic in origin and the motion of the ions can be fully classical only if the non-adiabatic terms in the Hamiltonian are small enough for second-order terms to be negligible, although a quasiclassical description may still be possible if this condition is not fulfilled. The existence of the Hamiltonian is of fundamental importance to the theory of electromigration as it clarifies the origin of the driving forces and enables the authors to discuss the response of the ions to the driving forces in terms of the usual formulation. The formal solution of the linearised Liouville equation is obtained to first order and is discussed for the case where the driving forces are obtained to second order in the electron-ion scattering.

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