Reevaluating the Hartman effect for field emission

An electron wave packet tunneling through a barrier has a transmission (or “group delay”) time τg that, for a rectangular barrier, is commonly held to become independent of the barrier width L as the width increases (the McColl-Hartman effect). In the present study, it is shown that first, the McColl-Hartman effect for a rectangular barrier is dependent upon L as the Gamow tunneling factor θ (k) vanishes, and τg is only independent of L when θ (k) is large; and second, for a triangular barrier to model field emission, although τg can be large for small field, it vanishes when the energy matches the barrier height.