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Physical Review A

Reevaluating the Hartman effect for field emission

Kevin L. Jensen, Andrew Shabaev, Jeanne M. Riga, Donald A. Shiffler, Joel L. Lebowitz, Rebecca Seviour

Published Dec 9, 2021• 9 citations
Publisher

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.

DOI
10.1103/physreva.104.062203

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An electron wave packet tunneling through a barrier has a transmission (or ``group delay'') time \ensuremathτg{\ensuremath{\tau}}_{g}\ensuremathτg​ that, for a rectangular barrier, is commonly held to become independent of the barrier width LLL 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 LLL as the Gamow tunneling factor \ensuremathθ(k)\ensuremath{\theta}(k)\ensuremathθ(k) vanishes, and \ensuremathτg{\ensuremath{\tau}}_{g}\ensuremathτg​ is only independent of LLL when \ensuremathθ(k)\ensuremath{\theta}(k)\ensuremathθ(k) is large; and second, for a triangular barrier to model field emission, although \ensuremathτg{\ensuremath{\tau}}_{g}\ensuremathτg​ can be large for small field, it vanishes when the energy matches the barrier height.
Electron emission into a nanogap is not instantaneous, which presents a difficulty in simulating ultra-fast behavior using particle models. A method of approximating the transmission and reflection delay (TARD) times of a wave packet interacting with barriers described by a delta function, a metal–insulator–metal (MIM, rectangular) barrier, and a Fowler Nordheim (FN, triangular) barrier is given and has application to simulation. It is based on the superposition of a finite number of exact basis states obtained from Schrödinger’s equation, analogous to how quantum carpets are simulated. As a result, it can exactly and uniquely follow exponentially small tunneling currents. A Bohm-like trajectory is obtained from the time evolution of the density: it shows delay in both the transmitted and reflected packets that can be simply evaluated. The relations to prior studies of the analytic [Formula: see text]-function barrier and the Wigner distribution function (WDF) methods are described. A comparison of the TARD times is contrasted to alternate times in the Büttiker–Landauer (BL) and McColl–Hartman (MH) times; the MH approach is further reformulated explicitly in terms of Gamow factors to consider how the McColl–Hartman effect is to be related, particularly in the case of the FN barrier of field emission.
I show that the MacColl-Hartman effect, namely, the saturation of the group delay time of sub-barrier quantum tunneling as a function of the barrier width, comes from the saturating behavior of a more fundamental concept - the phase of the stationary wave function. The explanation of saturation is given based on the decomposition of the stationary wave function into the spectrum of wave numbers and formulation of the initial condition for the direction of propagation of the incident matter wave. It is also shown that the saturation plateau of MacColl and Hartman actually doesn't continue indefinitely, but has a finite length. After the plateau, the sub-barrier tunneling time monotonically increases with increasing width of the potential, and this applies both to the maximum's of the wave packet and to the average tunneling time.
A numerical solution to the time evolution equation of the Wigner distribution function (WDF) with an accuracy necessary to simulate the passage of a wave packet past a barrier is developed, where quantum effects require high accuracy and fine discretization. A wave packet incident on a barrier, a portion of which tunnels through, demonstrates behavior that can define various characteristic transmission and reflection delay (TARD) times useful in the simulation of electron emission. A model for the TARD times is proposed that relies only on the asymptotic maxima of the position $\ensuremath{\rho}(x,t)$ and wave number $\ensuremath{\rho}(k,t)$ densities given by the WDF and applied to a ballistic trajectory model for the (faster) transmitted and (slower) reflected parts. The dependence of the TARD times on barrier width, symmetry, and abruptness is analyzed. For symmetrical barriers with characteristics similar to field emission barrier heights and widths, TARD times are on the order of a fraction of a femtosecond. The TARD times for when tunneling predominates are contrasted to tunneling times in the literature. Use of the TARD times in simulations of field emission in nanogap devices or to model ultrashort pulses generated under rapidly changing conditions for electron sources are proposed.
We study the phase, Larmor and dwell times of a particle scattered off triangular barriers (TBs). It is interesting that the dependences of dwell, reflective phase and Larmor times on the wave number, barrier width and height for a pair of mirror-symmetric (MS) exact triangular barriers (ETBs) are quite different, as the two ETBs have quite distinct scattering surfaces. In comparison, the dependence of the transmitted phase or Larmor times is exactly the same, since the transmitted amplitudes are the same for a pair of MS TBs. We further study the Hartman effect by defining the phase and Larmor velocities associated with the phase and Larmor times. We find no barrier width saturation effect for the transmitted and reflected times. This is indicated by the fact that all the velocities approach finite constants that are much smaller than the speed of light in vacuum for TBs with positive-slope impact faces. As for ETBs with vertical left edges, the naive velocities seem to also indicate the absence of the Hartman effect. These are quite distinct from rectangular barriers and may shed new light on the clarification of the tunneling time issues.
Recent papers suggest that tunneling wave packets traverse the region of allowed propagation between two potential barriers with superluminal group velocity and in a time independent of barrier separation. This phenomenon has been termed the "generalized Hartman effect" and extended to multiple barriers. Here we show that this delay time is not a transit time but a cavity lifetime. It does not imply superluminal velocity. Reported experimental verifications of this effect are reinterpreted.
The time of passage of the transmitted wave packet in a tunneling collision of a quantum particle with a square potential barrier becomes independent of the barrier width in a range of barrier thickness. This is the Hartman effect, which has been frequently associated with ``superluminality.'' A fundamental limitation on the effect is set by nonrelativistic ``causality conditions.'' We demonstrate first that the causality conditions impose more restrictive bounds on the negative time delays (time advancements) when no bound states are present. These restrictive bounds are in agreement with a naive, and generally false, causality argument based on the positivity of the ``extrapolated phase time,'' one of the quantities proposed to characterize the duration of the barrier's traversal. Nevertheless, square wells may in fact lead to much larger advancements than square barriers. We point out that close to the thresholds of new bound states, the time advancement increases considerably, while, at the same time, the transmission probability is large, which facilitates the possible observation of the enhanced time advancement.