Gordon-Volkov Wavefunctions for Field-Assisted Resonant Tunneling

2 Gordon-Volkov Wavefunctions for Field-Assisted Resonant Tunneling 

The potentials which will be considered here are stepwise constant. In each region of these potentials the time-dependent wavefunction is a linear combination of solutions of the wave equation for a particle interacting with an electromagnetic field of vector potential with A(t) [10]  

The equation is written in velocity gauge. Atomic units are used. The particle has charge unity and mass m in units of the free electron mass. V is the constant potential energy appropriate for the interval under consideration. The vector potential is supposed to be spatially constant at the length scale of the structure. With such a vector potential, the A2 term contributes an irrelevant phase factor which can be omitted. For a one-mode field A(t) is written as Ao cos(ut). The associated electric field is 0 sin(ut), with 0 = uAq. px is the linear momentum i ld/dx. For such a time-periodic Hamiltonian, a scattering approach can be developped, with a well-defined initial energy, and time-independent transition probabilities for reflection and transmission. 

We note that going to the length gauge provides a coupling of the form So x cos(uit). This is to be contrasted with the form Vo cos(ut) which is often assumed to investigate photon assisted tunneling [13, 14,15]. However this is not of major consequence on the characterization of the different regimes which can be delineated in the phenomenon [9], 

For a constant potential the solutions of the wave equation are Gordon-Volkov (G.V.) waves  

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Gordon-Volkov wavefunctions for field-assisted resonant tunneling... 

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