Quantum Mechanical Transmission Research Articles

Tunneling near the peaks of potential barriers: Consequences of higher-order Wentzel-Kramers-Brillouin corrections.

We derive the quantum-mechanical tunneling transmission coefficient for energies near the peak of a general potential barrier, to fourth order in the Wentzel-Kramers-Brillouin (WKB) approximation, using a method based on an eighth-order Taylor expansion of the potential near the peak. The result agrees with contour-integral formulations of WKB tunneling. For the P\"oschl-Teller potential the WKB series converges rapidly to the known exact transmission coefficient. Higher-order WKB corrections may be important in attempts to determine the internuclear potential by inversion using low-energy fusion cross-section data.

Effect of Barrier Thickness on the Quantum Mechanical Transmission across Parabolic Barriers

physica status solidi (b)Volume 145, Issue 1 p. K27-K30 Short Note Effect of Barrier Thickness on the Quantum Mechanical Transmission across Parabolic Barriers F. Berz, F. Berz Philips Research Laboratories, Solid State Electronic Division, Surrey Search for more papers by this author F. Berz, F. Berz Philips Research Laboratories, Solid State Electronic Division, Surrey Search for more papers by this author First published: 1 January 1988 https://doi.org/10.1002/pssb.2221450141 Cross Oak Lane, Redhill, Surrey RH1 5HA, England. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volume145, Issue11 January 1988Pages K27-K30 RelatedInformation

Rectification Properties of Asymmetrical Superlattices

A rectifying superlattice which can be used for detecting THz waves is proposed. The device is created by using an effective-mass superlattice in which the superlattice cycle is gradually changed. The operating principle is based on the fact that the quantum mechanical transmission coefficient with asymmetrical geometry is asymmetrical with respect to the applied voltage. The forward current is several orders in magnitude greater than the reverse current at a relatively low applied voltage. This feature is potentially useful for detecting and mixing small signals at THz frequencies.

Calculation of transmission tunneling current across arbitrary potential barriers

This paper presents a simple method for accurately calculating quantum mechanical transmission probability and current across arbitrary potential barriers by using the multistep potential approximation. This method is applicable to various potential barriers and wells, including continuous variations of potential energy and electron effective mass. Various potential barrier structures and a hot-electron transistor are analyzed to show the feasibility of this method.

Negative resistance in a finite superlattice owing to quantum mechanical reflection

Sasaki [Jpn. Inst. Electron. Commun. Eng. Tech. Rep. ED-84, 27 (1984)] has proposed a new type of superlattice, a modulation mass superlattice where the effective mass of electrons is changed periodically and the conduction-band edge is aligned so as to eliminate potential discontinuity. We investigate the quantum mechanical transmission of electrons for the case where a voltage is applied perpendicularly to the layers. We find that the modulation mass superlattice exhibits a new type of negative resistance owing to quantum mechanical reflection of electrons. The threshold voltage of negative resistance is much lower (around 10 mV than that (above 0.1 V) of compositional superlattice. The current density of the modulation mass superlattice is two orders of magnitude greater than that of a compositional superlattice. Such properties are very desirable for achieving a low-power ultrafast logic device.

Exact solution of the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential barrier

A simple yet exact method of solving the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer-matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.

Electron Transport across a Semiconductor Heterojunction

An electron-transport theory is presented for an isotype semiconductor heterojunction diode with energy-band and effective-mass discontinuities. During an interface electron transfer, the electron total energy and transverse momentum are assumed to be conserved. After the quantum-mechanical transmission coefficient is determined with a tight-binding connection of the electron wave functions, the boundary conditions for the momentum distribution function are obtained, assuming a conservation of electron flux. These conditions are applied to quasi-Fermi energy formulation in an approximate way. Numerical examples in an AlxGa1-xAs/GaAs junction indicate that due considerations concerning interface electron-transfer effects are necessary for the design and simulation of longitudinal-type heterojunction devices.

Analytical calculation of the quantum-mechanical transmission coefficient for a triangular, planar-doped potential barrier

The thermionic emission limit for current transport in planar-doped triangular barrier devices is determined by the quantum-mechanical reflection of the charge carriers at the barrier. In this paper we present an analytical calculation for the quantum-mechanical transmission and reflection coefficients for such a barrier.

Interface resistance

Interface resistance is studied by using the Landauer formula which relates the resistance to the quantum mechanical transmission coefficient. A simple rederivation of the Landauer formula is given. Using a step-like potential barrier as a model for the metal-semiconductor contact an analytical expression for the effective Richardson constant is derived. As an other application the grain boundary resistance in polycrystalline semiconductors is studied. The short-range potential fluctuation associated with the grain boundary is described by a rectangular potential barrier. The results for the grain boundary limited mobility cover both the strong and weak scattering regimes.

Calculation of the transmission coefficient of low-energy (

A simple approximation method is developed for calculating the quantum-mechanical transmission coefficient T( E) of a slow electron (in the energy range E < 20 eV) for the case of a one-dimensional surface potential barrier model which simulates most closely the real shape of the potential acting on an electron at a metal-vacuum interface. Although this method is developed for this particular case, it should permit, in principle, the direct investigation of arbitrary potential-energy functions.

Quantum mechanical reflection at triangular ‘‘planar-doped’’ potential barriers for transistors

The quantum mechanical reflection of electrons attempting to overcome the quasitriangular collector potential barrier in a planar-doped barrier transistor (PDBT) is an undesirable phenomenon that needs to be minimized. We describe a numerical technique for calculating the quantum mechanical reflection and transmission coefficients at triangular potential profiles that approximate the collector barrier of the PDBT. The results of these calculations, and the conclusions drawn, are presented.

Tunnel effect on the kinetics of a sigmatropic shift

Non-linear Arrhenius plots and large kinetic isotope effects observed in sigmatropic hydrogen shifts of photochemically generated transient di- and hexa-hydrocarbazoles are interpreted as tunnel contributions. An Eckart barrier was used to calculate the quantum-mechanical transmission probabilities of the tunneling protons.

Impact ionization by electrons and holes in InP

Ionization coefficients of holes in InP exceed those of electrons by a factor of approximately four over the electric field range from 2.5 to 4.5 × 10 5 V/cm and have been measured in Pt on p-type 〈100〉 orientation InP Schottky diodes. The shape of the baseline for the hole induced multiplication has been fitted by using one adjustable parameter, namely: the mean free path for optical phonon scattering. The model includes effects of quantum mechanical transmission, barrier height lowering and phonon scattering before the barrier maximum. The shape of the baseline for electron multiplication was likewise fitted by one adjustable parameter: the minority carrier diffusion length. Both models produce an excellent fit to low field characteristics that have appreciable curvature. The average phonon scattering mean free path for holes is ∼ 35 A ̊ roughly consistent with that required to characterize the hole ionization coefficient. The electron diffusion length in p-type InP was found to be ∼ 18 μ for all diodes. Ionization coefficients were obtained for diodes in the doping range from 5.0 × 10 15 to 7.8 × 10 15 cm −3. The impact ionization curves were fitted by Baraff-Chwang curves with slightly field dependent average energies for pair production. The threshold energies have been calculated from a consideration of energy and momentum conservation in a theoretical band structure. The deduced mean free paths for phonon scattering and breakdown voltages calculated from the ionization coefficient measurements are in excellent agreement with other independent experiments.

Application of variational transition-state theory and the unified statistical model to atomic hydrogen + molecular chlorine .fwdarw. hydrochloric acid + atomic chlorine

Microcanonical variational theory with a semiclassical adiabatic ground-state transmission coefficient predicts the thermal rate constant for collinear H + Cl/sub 2/ within 6% for 200 to 1000 K and the kinetic isotope effect k/sub H/k/sub D/ within 6 or 9% (depending on the detailed assumptions in the transmission coefficient) over the same temperature range. Other versions of variational transition-state theory and even conventional transition-state theory with semiclassical or quantum mechanical adiabatic ground-state transmission coefficients are also tested and show similar accuracy. The variational calculations predict a greater breakdown of conventional transition-state theory at higher temperatures and for three-dimensional reactions. The unified statistical theory is less accurate than variational transition-state theory for the magnitudes of the collinear rate constants, but more accurate for collinear kinetic isotope effects. 5 tables.

Photoelectron injection at metal-semiconductor interfaces

A modelling of the photoinjection process is developed which permits fitting of the spectral photoresponse of Schottky barriers including the electric field dependence of barrier height and photoresponse by means of two adjustable parameters: the zero field barrier qφ BO and λ 0 the zero temperature mean free path for optical phonon scattering of high energy electrons. The model assumes an image force potential barrier with Thomas-Fermi screening in the metal. Effects of optical phonon scattering and quantum mechanical transmission are convoluted on the Fowler photoelectron supply function. The effects of phonon scattering are frequently large because the ranges in energies associated with the transverse momentum and normal momentum are approximately the amount by which the quantum energy hv exceeds the barrier energy qφ B. At high fields, quantum mechanical tunneling dominates the response when hv < qφ B. At low fields, phonon assisted transmission is appreciable for the same quantum energy range. The calculation of the collection probability includes effects of multiple scattering even for electrons that do not lie initially within the cone of acceptance at the barrier maximum. An approach that considers the probability of collection the same as that of reaching the potential maximum without scattering is found to be acceptable only at high fields. Experimental results are reported from oxide-passivated epitaxial Pt x Si-〈111〉 n-type Si Schottky barrier diodes with annular Schottky barrier guard rings measured at temperatures of 90 and 298 K for an electric field range from 5 × 10 3 to 9 × 10 4 V cm . The field, spectral and temperature dependences of the photoresponse data are in excellent agreement with theoretical predictions with λ 0 = 110 Å at both 90 and 298 K. The zero field barrier height obtained from fitting photoresponse curves at a number of electric fields is also in excellent agreement with I-V and C-V measurements.

Carrier transport across heterojunction interfaces

A general theory is presented to describe the carrier transport across heterojunction interfaces. In matching the boundary conditions at the interface, the conservation of total energy and perpendicular momentum is assumed and the difference of effective masses on two sides of the junction is taken into account. The quantum mechanical transmission coefficient is calculated by a combined numerical and WKB method. Application of the present model to an Al xGa 1−xAsGaAs N- n heterojunction is performed and it gives rise to rectifying characteristics together with non-saturated reverse current. Comparison with the classical thermionic emission model is made to show the significance of tunneling and effect of quantum mechanical reflection.

A model of the ion sputtering process

This paper proposes a new model of the ion sputtering process. According to this model: (1) Sputtering atoms or ions originate almost entirely from the surface layer. (2) Atoms initially leave the surface in the same charge state in which they exist in the crystal, and are largely neutralized in traversing a region near the surface. (3) Neutralization can be explained in terms of quantum mechanical electron transmission and reflection phenomena that occur when the atom is within about three lattice spacings from the surface. The model upon which the theory is based pictures a potential well at the metal surface caused by the electric field of the departing ion. This potential well completely cancels the effect of the potential barrier normally present at the surface of the undisturbed metal, and for certain distances of the ion from the surface, results in a very high probability of emission of an electron from the metal surface, which then immediately neutralizes the emitted ion. This high neutralization probability explains the very low (10 −2 to 10 −6) sputtered ion yields observed experimentally. The ion sputtering yields for a number of elements were computed and compared with values obtained experimentally. It is found that the theory gives results compatible with experiment, and also provides an, at least qualitative, explanation for some of the heretofore rather puzzling experimentally observed features of ion sputtering.

Internal photoemission and quantum-mechanical transmission coefficient

The effect of the quantum-mechanical transmission coefficient on the photoemissive quantum yield in sandwich structures with thin insulators is investigated for the trapezoidal barrier and for the barrier with a parabolic top. The transmission coefficient evaluated in an explicit form from the exact solution of the Schrodinger equation is compared with that obtained from the modified WKB approximation and its applicability to real barriers is discussed. The resulting Fowler plots (square root of the quantum yield plotted against photon energy) deviate from the straight line in the low-energy region; this deviation corresponds in its magnitude and shape to the so-called low-energy tail known from experiment. Moreover, the Fowler plots obtained using the transmission coefficient evaluated exactly from the Schrodinger equation show weak superimposed oscillations; this effect results from the oscillations in the transmission coefficient which have their origin in the interference of electron waves within the insulator. Es wird der Einflus des quantenmechanischen Transmissionskoeffizienten auf die Quantenausbeute der Photoemission in Sandwichstrukturen mit dunnen Isolatoren fur die trapezformige Barriere und fur die Barriere mit parabolischem Gipfel untersucht. Der in expliziter Form aus der exakten Losung der Schrodingergleichung berechnete Transmissionskoeffizient wird mit dem aus der modifizierten WKB-Naherung verglichen und seine Anwendung auf reale Barrieren diskutiert. Die resultierenden Fowler-Kurven (Quadratwurzel der Quantenausbeute uber der Photonenenergie) weichen im niederenergetischen Bereich von der Geraden ab; diese Abweichung entspricht in Grose und Form dem niederenergetischen Teil, wie er experimentell bekannt ist. Daruberhinaus zeigen die Fowler-Kurven, die mit dem aus der Schrodingergleichung exakt berechneten Transmissionskoeffizienten erhalten wurden, schwache uberlagerte Oszillationen; dieser Effekt ruhrt von Oszillationen im Transmissionskoeffizienten her, die ihren Ursprung in der Interferenz der Elektronenwellen innerhalb des Isolators haben.

Effects of thermal excitation and quantum-mechanical transmission on photothreshold determination of Schottky barrier height

Quantum-mechanical electron transmission effects on photothreshold determination in Schottky barriers with high electric fields are shown to produce a strong effect on apparent photothresholds without a concomitant large change in shape of the photoresponse curve. Quantum-mechanical electron transmission probabilities are determined by numerical integration of the Schroedinger equation for a Schottky barrier model which includes Thomas-Fermi shielding in the metal, image force lowering in the semiconductor, and conservation of transverse momentum at the metal-semiconductor interface. Results are presented for <111> n-type Si and n-type GaAs. In the case of Si, conservation of transverse momentum at the interface produces a discontinuity in the electron kinetic energy associated with motion normal to the interface but does not greatly affect the transmission probability. Using these results, the difference between the true barrier height, qΦ B , and the apparent barrier height, qΦ A , is calculated as a function of temperature, doping, and electric field at the metal-semiconductor interface. qΦ A is defined as the intercept on the hv axis of the slope of a plot of (photocurrent per absorbed photon) 1 2 versus quantum energy, hv. Effects of barrier curvature due to doping are negligible for impurity densities below about 10 18 cm −3 for Si and 10 17 cm −3 for GaAs. Values of q( Φ B - Φ A ) in excess of 50 meV occur in many cases. The corrections derived are used to reinterpret previously reported measurements of the dependence of Schottky barrier height on temperature and impurity concentration.

Reflections on a Wave Packet Approach to Quantum Mechanical Barrier Penetration

We consider the time development of a one-dimensional Gaussian wave packet. Noting that the quantum mechanical predictions concerning position and momentum measurements of a free particle described by a Gaussian packet are consistent with the predictions one would make on the basis of a classical “Naive Model,” we investigate whether this model can account for the scattering of a particle by a rectangular barrier. Obtaining numerical solutions of the time dependent Schrödinger equation for a series of packets with varying incident momenta, we find that the Naive Model is inadequate to explain the quantum mechanical transmission. We study the relative speeds of the transmitted and reflected packets, discussing a number of possible interpretations and indicating the quantum mechanical result. Finally, we note that for certain choices of the parameters describing the packet and the barrier, the reflected packet is composed of two distinct humps.