Reevaluating the Hartman effect for field emission

Abstract

An electron wave packet tunneling through a barrier has a transmission (or ``group delay'') time ${\ensuremath{\tau}}_{g}$ 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 $\ensuremath{\theta}(k)$ vanishes, and ${\ensuremath{\tau}}_{g}$ is only independent of $L$ when $\ensuremath{\theta}(k)$ is large; and second, for a triangular barrier to model field emission, although ${\ensuremath{\tau}}_{g}$ can be large for small field, it vanishes when the energy matches the barrier height.