Analytical expressions for tunneling time through single and double barrier structures

In the past, the quantum mechanical tunneling time through simple rectangular barrier has been obtained by various theoretical approaches including the dwell time, the phase delay time, the Larmor clock time and also using the numerical analysis of wave packets. The agreement among these approaches over a range of incident electron energy is far from satisfactory. In this manuscript, analytical expressions for the tunneling time are derived based on the group velocity approach (referred hereafter as the average particle time, τ_APT) for single and double rectangular potential barriers under zero bias. The results of the single barrier case, including the limiting value of the tunneling time for various energy limits, are compared with these previous tunneling time calculations. The τ_APT results provide physically meaningful tunneling times for zero and infinite incident energy limits of the electron. The τ_APT for the double barrier structure is computed from the analytical solution as a function of the incident energy of the electron for two experimentally studied resonant tunneling structures. For both the single and double barrier cases, the effect of the structure parameters such as barrier width, height, and well width on the τ_APT are obtained and reported