On the Traversal Time of Barriers

Fifty years ago Hartman studied the barrier transmission time of wave packets (J Appl Phys 33:3427–3433, 1962). He was inspired by the tunneling experiments across thin insulating layers at that time. For opaque barriers he calculated faster than light propagation and a transmission time independent of barrier length, which is called the Hartman effect. A faster than light (FTL or superluminal) wave packet velocity was deduced in analog tunneling experiments with microwaves and with infrared light thirty years later. Recently, the conjectured zero time of electron tunneling was claimed to have been observed in ionizing helium inside the barrier. The calculated and measured short tunneling time arises at the barrier front. This tunneling time was found to be universal for elastic fields as well as for electromagnetic fields. Remarkable is that the delay time is the same for the reflected and the transmitted waves in the case of symmetric barriers. Several theoretical physicists predicted this strange nature of the tunneling process. However, even with this background many members of the physics community do not accept a FTL signal velocity interpretation of the experimental tunneling results. Instead a luminal front velocity was calculated to explain the FTL experimental results frequently. However, Brillouin stated in his book on wave propagation and group velocity that the front velocity is given by the group velocity of wave packets in the case of physical signals, which have only finite frequency bandwidths. Some studies assumed barriers to be cavities and the observed tunneling time does represent the cavity lifetime. We are going to discus these continuing misleading interpretations, which are found in journals and in textbooks till today.     

Tunneling time and stochastic-mechanical trajectories for the double-barrier potential

Quantum motion of particles tunneling a double barrier potential is considered by using stochastic mechanics. Stochastic-mechanical trajectories give us information about complex motion of tunneling particles that is not obtained within the framework of ordinary quantum mechanics. Using such information, we calculate the tunneling times within each of the barriers which depend on the distance between them. It is found that the stochastic-mechanical tunneling time shows better asymptotic behavior than the quantum-mechanical dwell time and presence time.     

Mechanism of ionization of organic compounds on tip emitters

The current-voltage characteristics of organic compounds at a tip platinum emitter have been studied in 0.2-1.0 V/Å fields. It is shown that the parameter (V-?) does not affect the formation of the desorption ion barrier in the field ionization of atoms and molecules under a high potential gradient providing electron tunneling to the Fermi level of the emitter material and, therefore, the probability of the ion desorption cannot explicitly depend on this parameter.     

Quasi-distribution of tunneling time

Using real time Feynman histories, a quasi-distribution of tunneling time Q(τ) is introduced. For the tunneling time of resident time type, an explicit expression for Q is shown for square barriers. Q becomes oscillatory as the barrier becomes opaque. Some well-known tunneling times fall within the range of τ where Q takes non-negligible values. The formal “average” and the “variance” of the tunneling time are found to be related to known tunneling times. It is thus demonstrated that the quasi-distribution extracts the temporal information about tunneling from real time Feynman histories.     

Electron transport through a molecular conductor with center-of-mass motion

The linear conductance of a molecular conductor oscillating between two metallic leads is investigated numerically both for Hubbard interacting and noninteracting electrons. The molecule-leads tunneling barriers depend on the molecule displacement from its equilibrium position. The results present an interesting interference which leads to a conductance dip at the electron-hole symmetry point that could be experimentally observable. It is shown that this dip is caused by the destructive interference between the purely electronic and phonon-assisted tunneling channels, which are found to carry opposite phases. When an internal vibrational mode is also active, the electron-hole symmetry is broken but a Fano-like interference is still observed.     

Confronting the Hartman effect with data from frustrated total internal reflection (FTIR)

Over 40 years ago, Hartman noted that the tunneling time τ of a particle through a barrier becomes independent of width for thick barriers. Lately, the Hartman effect has been seen as a support for superluminal tunneling time. By interpreting the reflection and transmission amplitudes in terms of multiple reflection series, we show that τ is linear in barrier width for thin barriers and may be associated with actual traversal time; for thick barriers, τ saturates to the Hartman value because of the suppression of all but the first term of the series due to the smallness of the tunneling factor. For large widths, τ cannot be identified with the propagation time but may be associated with a time to penetrate to a characteristic depth into the barrier, which is independent of width. We discuss data from frustrated internal reflection experiments, which support this view.     

Stationary phase method and delay times for relativistic and non-relativistic tunneling particles

The stationary phase method is frequently adopted for calculating tunneling phase times of analytically-continuous Gaussian or infinite-bandwidth step pulses which collide with a potential barrier. This report deals with the basic concepts on deducing transit times for quantum scattering: the stationary phase method and its relation with delay times for relativistic and non-relativistic tunneling particles. After reexamining the above-barrier diffusion problem, we notice that the applicability of this method is constrained by several subtleties in deriving the phase time that describes the localization of scattered wave packets. Using a recently developed procedure - multiple wave packet decomposition - for some specifical colliding configurations, we demonstrate that the analytical difficulties arising when the stationary phase method is applied for obtaining phase (traversal) times are all overcome. In this case, we also investigate the general relation between phase times and dwell times for quantum tunneling/scattering. Considering a symmetrical collision of two identical wave packets with an one-dimensional barrier, we demonstrate that these two distinct transit time definitions are explicitly connected. The traversal times are obtained for a symmetrized (two identical bosons) and an antisymmetrized (two identical fermions) quantum colliding configuration. Multiple wave packet decomposition shows us that the phase time (group delay) describes the exact position of the scattered particles and, in addition to the exact relation with the dwell time, leads to correct conceptual understanding of both transit time definitions. At last, we extend the non-relativistic formalism to the solutions for the tunneling zone of a one-dimensional electrostatic potential in the relativistic (Dirac to Klein-Gordon) wave equation where the incoming wave packet exhibits the possibility of being almost totally transmitted through the potential barrier. The conditions for the occurrence of accelerated and, eventually, superluminal tunneling transmission probabilities are all quantified and the problematic superluminal interpretation based on the non-relativistic tunneling dynamics is revisited. Lessons concerning the dynamics of relativistic tunneling and the mathematical structure of its solutions suggest revealing insights into mathematically analogous condensed-matter experiments using electrostatic barriers in single- and bi-layer graphene, for which the accelerated tunneling effect deserves a more careful investigation.     

The Büttiker-Landauer model generalized

Büttiker and Landauer studied scattering off an oscillating rectangular barrier in order to shed light on the time aspects of tunneling. The expression for the traversal time resulting from this study is controversial. In addition, doubts have recently been expressed on technical aspects of their work. In an attempt to clarify these issues, we investigate a generalization of their model to arbitrary oscillating barriers,V(x, t)=V ₀(x)+V ₁(x)cos t. In the process, we confirm that Büttiker and Landauer's work is technically sound. However, we show, by several examples, that no direct general relation exists between the characteristic frequency of an oscillating barrier and the duration of the tunneling process. For a wide range of realistic parameters this characteristic frequency does not even exist.This paper is dedicated to E. G. D. Cohen.     

The Atomic Tunneling Current in Two—Species Bose—Einstein Condensates

It is shown that the atomic tunneling current and the Shapiro-like steps strongly depend on the initial number of atoms in each condensate and the initial phase difference between the two condensates which are initially in even(odd) coherent states.The nonlinearity of interatomic interactions in the two condensates may lead to the atomic tunneling current and Shapiro-like step between the two condensates.It is found that the interatomic nonlinear interactions can induce the atomic tunneling current and Shapiro-like step between two condensates even though there does not exist the interspecies Josephson-like tunneling coupling.The static atomic tunneling current flows in positive or negative direction,which depends on the phase difference of the two-species condensates.     

Quantum nature of proton transferring across one-dimensional potential fields

Proton transfer plays a key role in the applications of advanced energy materials as well as in the functionalities of biological systems.In this work,based on the transfer matrix method,we study the quantum effects of proton transfer in a series of one-dimensional(1 D) model potentials and numerically calculate the quantum probability of transferring across single and double barriers(wells).In the case of single barriers,when the incident energies of protons are above the barrier height,the quantum oscillations in the transmission coefficients depend on the geometric shape of the barriers.It is found that atomic resonant tunneling(ART) not only presents in the rectangular single well and rectangular double barriers as expected,but also exists in the other types of potential wells and double barriers.For hetero-structured double barriers,there is no resonant tunneling in the classical forbidden zone,i.e.,in the case when the incident energy(E_i) is lower than the barrier height(E_b).Furthermore,we have provided generalized analysis on the characteristics of transmission coefficients of hetero-structured rectangular double barriers.     

Resonant tunneling in double superlattice barrier heterostructures

We have investigated resonant tunneling in double barrier heterostructures in which the tunnel barriers have been replaced by short period superlattices, and have shown for the first time quantum well confinement in a single quantum well bounded by superlattices. These results also demonstrate the first utilization of short period binary superlattices as effective tunnel barriers to replace the conventional Al_xGa_1−xAs barriers. The superlattice structure does not exhibit the asymmetry around zero bias in the electrical characteristics normally observed in the conventional Al_xGa_1−xAs barrier structures, suggestive of reduced roughness at the inverted interface by superlattice smoothing. The superlattice barrier also exhibits an anomalously low barrier height. The performance of this symmetric superlattice structure is compared with an intentionally constructed asymmetric double barrier superlattice structure, which exhibits pronounced asymmetry in the electrical characteristics. The observed behavior supports the view that resonant enhancement occurs in the quantum well.     

Tunneling of electrons in quantum wells with indirect gap semiconductor barriers

One of the features peculiar to GaAs-Ga_1−xAl_x As quantum wells with x ⩾0.43 are barriers formed by an indirect gap semiconductor. We make use of a simple one-dimensional tight-binding model to study the tunneling properties of such a system. Wave-functions and probabilities associated with an electron in each spatial region as a function of time are computed and compared with the results of a simple square barrier model. It is shown that the states related to the indirect conduction band minima of the barrier act as a new channel and increase the tunneling current between the wells. We suggest that these states are the origin of an unexplained structure observed in photoemission from a double quantum well. The effect of an external electric field is analyzed as well.     

Comparative research on activation technique for GaAs photocathodes

The properties of GaAs photocathodes mainly depend on the material design and activation technique. In early researches, high-low temperature two-step activation has been proved to get more quantum efficiency than high-temperature single-step activation. But the variations of surface barriers for two activation techniques have not been well studied, thus the best activation temperature, best Cs–O ratio and best activation time for two-step activation technique have not been well found. Because the surface photovoltage spectroscopy (SPS) before activation is only in connection with the body parameters for GaAs photocathode such as electron diffusion length and the spectral response current (SRC) after activation is in connection with not only body parameters but also surface barriers, thus the surface escape probability (SEP) can be well fitted through the comparative research between SPS before activation and SEP after activation. Through deduction for the tunneling process of surface barriers by Schrödinger equation, the width and height for surface barrier I and II can be well fitted through the curves of SEP. The fitting results were well proved and analyzed by quantitative analysis of angle-dependent X-ray photoelectron spectroscopy (ADXPS) which can also study the surface chemical compositions, atomic concentration percentage and layer thickness for GaAs photocathodes. This comparative research method for fitting parameters of surface barriers through SPS before activation and SRC after activation shows a better real-time in system method for the researches of activation techniques.     

The phase delay and its complex time: From stationary states up to wave packets

Complex time is often invoked about tunneling effect where the classical phase delay is completed with a crucial filter effect. Usually the complex times are obtained by considering the flux–flux correlation function, but this can be obtained by a very simple approach using the search of the maximum of the generalized complex phase function, including the amplitude of the wave function. Various aspects of the phase delay are presented in the case of wave packets impinging on simple or resonant quantum barriers.     

Resonant tunneling through double-barrier structures on graphene

Quantum resonant tunneling behaviors of double-barrier structures on graphene are investigated under the tightbinding approximation. The Klein tunneling and resonant tunneling are demonstrated for the quasiparticles with energy close to the Dirac points. The Klein tunneling vanishes by increasing the height of the potential barriers to more than 300 meV. The Dirac transport properties continuously change to the Schro¨dinger ones. It is found that the peaks of resonant tunneling approximate to the eigen-levels of graphene nanoribbons under appropriate boundary conditions. A comparison between the zigzag- and armchair-edge barriers is given.     

Quantum ratchets and quantum heat pumps

Quantum ratchets are Brownian motors in which the quantum dynamics of particles induces qualitatively new behavior. We review a series of experiments in which asymmetric semiconductor devices of sub-micron dimensions are used to study quantum ratchets for electrons. In rocked quantum-dot ratchets electron-wave interference is used to create a non-linear voltage response, leading to a ratchet effect. The direction of the net ratchet current in this type of device can be sensitively controlled by changing one of the following experimental variables: a small external magnetic field, the amplitude of the rocking force, or the Fermi energy. We also describe a tunneling ratchet in which the current direction depends on temperature. In our discussion of the tunneling ratchet we distinguish between three contributions to the non-linear current–voltage characteristics that lead to the ratchet effect: thermal excitation over energy barriers, tunneling through barriers, and wave reflection from barriers. Finally, we discuss the operation of adiabatically rocked tunneling ratchets as heat pumps. Received: 8 February 2002 / Accepted: 11 February 2002 / Published online: 22 April 2002     

Spin-Polarized Electron Tunneling TD:\pdf\.PDFhrough a Quantum Dot

Nonequilibrium Green's function is uscd to study spin-polarized electron tunneling through a quantum dot connected to two ferromagnetic electrodes with different orientations via two insulating barriers (FM/I/QD/I/FA.f). Intra-level Coulomb interaction in the dot is considered. General formula of tunneling current which can be used for arbitrary angle between the two electrodes' magnetizations is derived for both the weak and strong intra-dot interactions.We find that the transport current can be divided into two parts: the current with the spin-flip and the current without the spin-flip, which critically depend on the linewidth function near the Fermi level of the ferromagnetic electrodes. If a magnetic field is applied in the quantum dot, different behaviors will be found for weak and strong interactions.     

Electronic transmission and dwell time on a double barrier system with an accelerating quantum well

Resonant tunneling quantum structures consist of asymmetric wells and barriers have been investigated to find their optimized geometrical parameters and potential profile by the numerical calculations. The results show that the widths and the depths of the asymmetric wells have a significant effect on the transmission coefficient and the dwell time. The properties exhibited in this work may establish guidance to the device applications.     

Electrochemical annealing and its relevance in metal electroplating: an atomistic view

Using electrochemical scanning tunneling microscopy we have studied the decay of monolayer high islands on Au(001) electrodes as a function of electrode potential and as a function of the specifically adsorbed ion on the surface. We find that island decay is diffusion-limited and transport rates depend strongly on electrode potential and on the specifically adsorbed ion, an effect qualitatively known for long now. In this study we quantitatively investigate the transport rates and find values for the relevant transport energy barriers in the different electrolytes. PACS 68.37.Ef; 68.37.-d; 68.43.Jk; 68.35.Fx; 68.35.Md;68.35.-p; 68.08.-p