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.

Tunneling times: a critical review

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.

Barrier interaction time in tunneling

The authors review the physics of ultrafast dynamics in semiconductors and their heterostructures, including both the observed experimental phenomena and the theoretical description of the processes. These are probed by ultrafast optical excitation, generating nonequilibrium states that can be monitored by time-resolved spectroscopy. Light pulses create coherent superpositions of states, and the dynamics of the associated phase relationships can be directly investigated by means of many-pulse experiments. The commonly used experimental techniques are briefly reviewed. A variety of different phenomena can be described within a common theoretical framework based on the density-matrix formalism. The important interactions of the carriers included in the theoretical description are the phonon interactions, the interactions with classical and quantum light fields, and the Coulomb interaction among the carriers themselves. These interactions give rise to a strong interplay between phase coherence and relaxation, which strongly affects the non equilibrium dynamics. Based on the general theory, the authors review the physical phenomena in various semiconductor structures including superlattices, quantum wells, quantum wires, and bulk media. Particular results which have played a central role in understanding the microscopic origins of the relaxation processes are discussed in detail.

Theory of ultrafast phenomena in photoexcited semiconductors

Transport of nanoscale energy in the form of excitons is at the core of photosynthesis and the operation of a wide range of nanostructured optoelectronic devices such as solar cells, light-emitting diodes and excitonic transistors. Of particular importance is the relationship between exciton transport and nanoscale disorder, the defining characteristic of molecular and nanostructured materials. Here we report a spatial, temporal and spectral visualization of exciton transport in molecular crystals and disordered thin films. Using tetracene as an archetype molecular crystal, the imaging reveals that exciton transport occurs by random walk diffusion, with a transition to subdiffusion as excitons become trapped. By controlling the morphology of the thin film, we show that this transition to subdiffusive transport occurs at earlier times as disorder is increased. Our findings demonstrate that the mechanism of exciton transport depends strongly on the nanoscale morphology, which has wide implications for the design of excitonic materials and devices.

/pdf/visualization-of-exciton-transport-in-ordered-and-disordered-4rm8wlktwv.pdf

Visualization of exciton transport in ordered and disordered molecular solids

We directly monitor exciton propagation in freestanding and SiO_{2}-supported WS_{2} monolayers through spatially and time-resolved microphotoluminescence under ambient conditions. We find a highly nonlinear behavior with characteristic, qualitative changes in the spatial profiles of the exciton emission and an effective diffusion coefficient increasing from 0.3 to more than 30  cm^{2}/s, depending on the injected exciton density. Solving the diffusion equation while accounting for Auger recombination allows us to identify and quantitatively understand the main origin of the increase in the observed diffusion coefficient. At elevated excitation densities, the initial Gaussian distribution of the excitons evolves into long-lived halo shapes with μm-scale diameter, indicating additional memory effects in the exciton dynamics.

/pdf/exciton-diffusion-and-halo-effects-in-monolayer-3wmgl7t8r9.pdf

Exciton Diffusion and Halo Effects in Monolayer Semiconductors.

Tunneling in low‐barrier GaAs/AlxGa1−xAs/GaAs heterostructures has been systematically investigated over a wide range of barrier heights and thicknesses. Measured conductance data have been compared with tunnel conductances calculated with and without the image potential correction. Experimental evidence is found for the validity of the static image correction in the limit of large tunneling times, and for the occurrence of dynamic effects in the limit of short tunneling times.

Experimental observation of the dynamical image potential in extremely low GaAs/AlxGa1−xAs/GaAs tunnel barriers

The material interface of a molecular beam epitaxy grown Al0.5Ga0.5As‐GaAs heterostructure is investigated on a cross‐sectional (110) cleavage plane using tunnel spectroscopy. The depleted n‐type region and the electron confinement layer adjacent to the interface are identified with local current‐voltage spectroscopy. The spatial width of these layers is close to 15 nm. The spectroscopy can be interpreted with the valence‐band offset in the interface, and a value of 0.35 eV is found for this quantity.

Tunnel spectroscopy of the AlGaAs-GaAs heterostructure interface

Par simulation numerique on determine les diffusivites ambipolaires ainsi que les vitesses de recombinaison de surface et d'interface dans ces heterostructures. La diffusion due au desordre limite les mobilites ambipolaires entre 40 et 120 K dans les barrieres Al 0,5 Ga 0,5 As. L'analyse s'etend aussi au piegeage des porteurs des couches barriere dans le puits quantique, en fonction de la temperature

Optical studies of vertical ambipolar transport and interface recombination velocities in GaAs/Al0.5Ga0.5As double-quantum-well heterostructures.

Time resolved spectroscopy on etched GaAs/GaAlAs-quantum-microstructures

Scanning Tunneling Microscopy and Potentiometry at Gaas Algaas Heterojunctions

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