Electron direct tunneling time in heterostructures with nanometer-thick trapezoidal barriers

An analytical expression of direct tunneling time of an electron through a nanometer-thick trapezoidal barrier has been derived. It is found that the direct tunneling time is independent of the thickness of the SiO2 barrier for the SiO2 layers thicker than 1 nm, which are used in advanced MOS (metal-oxide-semiconductor) devices. By applying a voltage to the barrier layer, the direct tunneling time becomes shorter than that obtained without the applied voltage. The nonparabolic energy-momentum dispersion of the barrier layer increases the direct tunneling time as compared to the parabolic one. However, the nonparabolic effect is negligible for high electron energy. It is also shown that the tunneling time obtained by the phase time is shorter than that calculated by the semiclassical approach for high electron energy. However, both of the calculated tunneling times have not been able to explain the reason why the calculated tunneling time are orders of magnitude shorter than the highest time resolution achievable in the silicon-based MOS devices.