Abstract Traversal time in tunneling has become a somewhat controversial subject, and this note presents a personal perspective on some of these discussions. There is a widely invoked linear relationship yielding dwell time as a weighted average of traversal time and reflection time. I assert that it is totally without basis. Many authors follow wave packets through barriers. The results of such calculations are hard to correlate with experiment. Stevens has provided a wave packet approach which avoids some of the problems typically faced by such analyses; nevertheless its analytical details are questioned. The significance of complex traversal times, favored by some investigators, is questioned. The relative merit of various possible “clocks” in the analysis of traversal time is discussed, but without definitive conclusions.
Abstract Traversal time in tunneling has become a somewhat controversial subject, and this note presents a personal perspective on some of these discussions. There is a widely invoked linear relationship yielding dwell time as a weighted average of traversal time and reflection time. I assert that it is totally without basis. Many authors follow wave packets through barriers. The results of such calculations are hard to correlate with experiment. Stevens has provided a wave packet approach which avoids some of the problems typically faced by such analyses; nevertheless its analytical details are questioned. The significance of complex traversal times, favored by some investigators, is questioned. The relative merit of various possible “clocks” in the analysis of traversal time is discussed, but without definitive conclusions.
Over sixty years ago, it was suggested that there is a time associated with the passage of a particle under a tunneling barrier. The existence of such a time is now well accepted; in fact the time has been measured experimentally. There is no clear consensus, however, about the existence of a simple expression for this time, and the exact nature of that expression. The proposed expressions fall into three main classes. The authors argue that expressions based on following a feature of a wave packet through the barrier have little physical significance. A second class tries to identify a set of classical paths associated with the quantum-mechanical motion and then tries to average over these. This class is too diverse to permit assessment as a single entity. The third class invokes a physical clock involving degrees of freedom in addition to that involved in tunneling. This not only is a prescription for the derivation of expressions for the traversal time but also leads to a direct relationship to experiment.
The desire to minimize the response time of tunneling devices has led to renewed interest in the unresolved fundamental question How much time, on average, does an incident electron that is ultimately transmitted spend inside a one-dimensional potential barrier? In addition to this transmission time ⊤t ·,there are two other times of interest: the reflection time ⊤r for electrons that are ultimately reflected and the dwell time TD, which is the average time spent inside the barrier regardless of the ultimate fate of the incident particle. An exact expression for TD and the time-modulated barrier, Feynman path, Larmor clock, and wave-packet evolution approaches for calculating ⊤t and ⊤r arc discussed. It is argued that none of these approaches is completely satisfactory because they all lead to transmission and reflection times that either are complex or, if real, do not satisfy: all of a few reasonable (but not universally accepted) criteria, such as consistency with the exact expression for the dwell time.
The old question of "How long does it take to tunnel through a barrier?" has acquired new urgency with the advent of techniques for the fabrication of semiconductor structures in the nanometer range. For the restricted problem of tunneling in a scattering configuration, a coherent picture is now emerging. The dwell time ${\ensuremath{\tau}}_{D}$ has the status of an exact statement of the time spent in a region of space, averaged over all incoming particles. The phase times ${\ensuremath{\tau}}_{T}^{\ensuremath{\phi}}$ and ${\ensuremath{\tau}}_{R}^{\ensuremath{\phi}}$ are defined separately for transmitted and reflected particles. They are asymptotic statements on completed scattering events and include self-interference delays as well as the time spent in the barrier. Consequently, neither the dwell time nor the phase times can answer the question of how much time a transmitted (alternatively, reflected) particle spent in the barrier region. Our discussion of this question relies on a few simple criteria: (1) The average duration of a physical process must be real. (2) Since transmission and reflection are mutually exclusive events, the times ${\ensuremath{\tau}}_{T}$ and ${\ensuremath{\tau}}_{R}$ spent in the barrier region are, if they exist, conditional averages. Consequently, they must obey the identity ${\ensuremath{\tau}}_{D}=T{\ensuremath{\tau}}_{T}+R{\ensuremath{\tau}}_{R}$, where T and R are the transmission and reflection probabilities, respectively. The existence of this identity distinguishes tunneling in a scattering configuration from tunneling out of a metastable state. (3) Any proposed ${\ensuremath{\tau}}_{T}$ and ${\ensuremath{\tau}}_{R}$ must meet every requirement that can be constructed from ${\ensuremath{\tau}}_{D}$. On the basis of (2), the naively extrapolated phase times, as well as the B\"uttiker-Landauer time, must be rejected. The local Larmor times, as introduced by Baz', satisfy (2), but not every criterion of type (3). The local Larmor clock is therefore unreliable. The asymptotic Larmor clock shows the phase times, as it should. Finally, the inverse characteristic frequency of an oscillating barrier cannot always be defined. It is shown not to represent the duration of the tunneling process. This leaves the dwell time and the phase times as the only well-established times in this context. It also leaves open the question of the length of time a transmitted particle spends in the barrier region. It is not clear that a generally valid answer to this question exists.
The proper basis for the calculation of transmission and reflection times for wave packets scattered off arbitrary tunneling structures in one dimension is considered. With packets narrow in wave-number space, we demonstrate that the classic phase times are indeed correct to lowest order. Explicit, general expressions for the leading correction terms for finite wave packets are given. The physics associated with these corrections is discussed. We also consider the dwell time, as it is currently defined, and derive a general relation between this dwell time and the phase times. This relation shows when the dwell time can and cannot be used. Finally, we discuss wave packets transmitted from narrow resonances, and derive an explicit, exact formula for the tunneling time with resonance transmission from a symmetric double barrier. Comparison with earlier approximate results is made.
The tunneling time of a particle through a given barrier is commonly defined in terms of “internal clocks” that effectively measure the interaction time with internal degrees of freedom of the barrier. It is known that this definition of the time scale for tunneling is not unique in the sense that it depends on the clock used to define it. For the case of resonance tunneling, a particular choice that in the limit of a high/broad square barrier yields the original result of Büttiker and Landauer ( Phys. Rev. Lett. 1982, 49, 1739) is correlated to the lifetime of the resonance state. This is illustrated for analytically solvable one-dimensional double barrier models and for a realistic model of electron tunneling through a static water barrier. The latter calculation constitutes a novel application of this concept to a 3-dimensional model, and the observed structure in the energy dependence of the computed traversal time reflects the existence of transient tunneling resonances associated with instantaneous water structures. These models, characterized by the existence of shape resonances in the barrier, make it possible to examine different internal clocks that were proposed for measuring tunneling times in situations where a “clock independent” intrinsic time scale (the resonance lifetime) for the tunneling time exists. It is argued that this time may be used in order to estimate the relative importance of dynamical barrier processes that affect the tunneling probability.
The authors analyse tunnelling through time-modulated barriers. The numerical results for the transmitted sideband intensities, obtained with wave-packet simulations, are in quantitative agreement with the analytical results of Buttiker and Landauer (1982, 1985). For opaque barriers the asymmetry of the sideband intensities can be characterized with a single frequency whose inverse Buttiker and Landauer interpret as the traversal time. This simple behaviour does not hold in general, and the authors discuss some of the difficulties one meets in an attempt to generalise the one-parameter description to arbitrary barriers. Furthermore, they examine analytically the transmitted wavefunction in the adiabatic limit, and show that the related density and current involve both the dwell time and the Buttiker time, defined in the context of a Larmor clock. This suggests the possibility that experiments probing different aspects of the transmitted wavefunction may yield different, but complementary, information about the tunnelling process.