Larmor-clock transmission times for resonant double barriers

Recent generalizations of Büttiker’s analysis of the Larmor clock to any region $z_1$≤z≤$z_2$ within an arbitrary one-dimensional barrier lead to two local transmission ‘‘times’’ for an incident electron of energy E=${ħ}^2$ $k^2$/2m. These are conveniently regarded as the real and (minus) the imaginary parts of a complex time ${\tau }_T^V$(k;$z_1$,$z_2$). In this paper the properties of ${\tau }_T^V$(k;$z_1$,$z_2$) are investigated for the double-rectangular-barrier potential V(z)=$V_1$Θ(z)Θ(a-z) +$V_2$Θ(z-b)Θ(d-z). Results are presented for the dependence of the real and imaginary parts of the Larmor-clock transmission time on incident energy and barrier and well widths. The local transmission time ${\tau }_T^V$(k;$z_1$,$z_2$) is real for symmetric double barriers when the energy of the incident electron is exactly on resonance. Remarkably, the behavior of the real quantity ${\tau }_T^V$(k;0, z≪a) for this special case provides further evidence for the importance of the imaginary part of the local Larmor-clock transmission time for the corresponding (isolated) single barrier V(z)=$V_1$Θ(z)Θ(a-z). At resonance, for symmetric double barriers, (real) local transmission speeds $v_T$(z) can be very much in excess of the speed of light in the well region for z very close to a quasinode of the Schrödinger stationary-state wave function. It is proven that $v_T$(z)≤c for a symmetric double barrier at resonance when the Dirac equation is used in place of the Schrödinger equation. On the other hand, it is shown that application of the Dirac equation to a single opaque rectangular barrier does not alter the well-known difficulty that a/Re[${\tau }_T^V$(k;0,a)] exceeds c for sufficiently large barrier width a.