Coherent resonant tunneling time and velocity in finite periodic systems

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The velocity ${v}_{\mathrm{res}}$ of resonant tunneling electrons in finite periodic structures is analytically calculated in two ways. The first method is based on the fact that a transmission of unity leads to a coincidence of all still-competing tunneling time definitions. Thus, having an indisputable resonant tunneling time ${\ensuremath{\tau}}_{\mathrm{res}}$, we apply the natural definition ${v}_{\mathrm{res}}=L∕{\ensuremath{\tau}}_{\mathrm{res}}$ to calculate the velocity. For the second method, we combine Bloch's theorem with the transfer matrix approach to decompose the wave function into two Bloch waves. The expectation value of the velocity is then calculated. Both approaches lead to the same result, showing their physical equivalence. The obtained resonant tunneling velocity ${v}_{\mathrm{res}}$ is smaller than or equal to the group velocity times the magnitude of the complex transmission amplitude of the unit cell. Only at energies at which the unit cell of the periodic structure has a transmission of unity does ${v}_{\mathrm{res}}$ equal the group velocity. Numerical calculations for a $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ superlattice are performed. For typical parameters, the resonant velocity is below one-third of the group velocity.