Density matrix approach to photon-assisted tunneling in the transfer Hamiltonian formalism

The transfer Hamiltonian tunneling current is derived in a time-dependent density matrix formulation and is used to examine photon-assisted tunneling. Bardeen's tunneling expression arises as the result of first order perturbation theory in a mean-field expansion of the density matrix. Photon-assisted tunneling from confined electromagnetic fields in the forbidden barrier region occurs due to time-varying polarization and wavefunction overlap in the gap, which leads to a non-zero tunneling current in asymmetric device structures, even in an unbiased state. The photon energy is seen to act as an effective temperature dependent bias in a uniform barrier asymmetric tunneling example problem. Higher order terms in the density matrix expansion give rise to multi-photon enhanced tunneling currents that can be considered an extension of non-linear optics where the non-linear conductance plays a similar role as the non-linear susceptibilities in the continuity equations.