Exact Transmission Coefficients Research Articles

Reflection and transmission coefficients from the superposition of various potentials

The reflection and transmission coefficients describe the behavior of the matter wave incident on a potential barrier. They can be expressed in terms of the probability with which the matter wave can be reflected or transmitted. The central equation accounting for the behavior of the matter wave is the Schrödinger equation. The Schrödinger equation is the second order partial differential equation. However, in a stationary state, the Schrödinger equation is reduced to the time independent Schrödinger equation. This time independent Schrödinger equation is the second order linear ordinary differential equation. Since the time independent Schrödinger equation is linear, superposition of any of the two solutions to the time independent Schrödinger equation is also a solution. In this paper, we focus on the superposition of various potentials. The reflection and transmission coefficients from the superposition of various potentials are obtained. A comparison between the exact coefficients and those obtained by the 2 × 2 transfer matrix is made. The relationship between the transmission coefficient of the superposed potential and that of each individual potential is found. The results show that the transmission coefficient obtained from the 2 × 2 transfer matrix is of a lower bound on the exact transmission coefficient.

An exact transmission coefficient with one and two barriers in graphene

In this work, we find the exact transmission coefficient across one and two potential barriers in graphene. We obtain theoretical expressions for this coefficient in terms of the characteristics of the potential barriers, height, width and separation and the characteristics of the carriers, Fermi energy and incidence angle. This coefficient is essential in order to characterize the conductance in several electronic devices based on graphene. Since the calculated transmission coefficient is highly non-isotropic, we extend the Landauer formalism and obtain a closed expression for transistor conductance.

Analytical study of the propagation of acoustic waves in a 1D weakly disordered lattice

This paper presents an analytical approach of the propagation of an acoustic wave through a normally distributed disordered lattice made up of Helmholtz resonators connected to a cylindrical duct. This approach allows to determine analytically the exact transmission coefficient of a weakly disordered lattice. Analytical results are compared to a well-known numerical method based on a matrix product. Furthermore, this approach gives an analytical expression of the localization length apart from the Bragg stopband which depends only on the standard deviation of the normal distribution disorder. This expression permits to study on one hand the localization length as a function of both disorder strength and frequency, and on the other hand, the propagation characteristics on the edges of two sorts of stopbands (Bragg and Helmholtz stopbands). Lastly, the value of the localization length inside the Helmholtz stopband is compared to the localization length in the Bragg stopband.

이방성 매질의 편광투과특성 분석을 위한 확장된 존스 행렬식의 개선

확장된 존스 행렬법을 응용하여 등방성 매질과 이방성 매질의 경계면에서 투과계수를 근사적인 방법을 사용하지 않고 정확히 계산하였다. 광학 이방성이 작은 경우(<TEX>$|n_e-n_o|\;{\ll}\;n_o,\;n_e$</TEX>), 계산된 투과계수가 근사식에 의한 투과계수와 일치하는 것을 확인하였다. 정확한 투과계수 표현을 사용하여 임의의 편광상태로 입사한 빛이 단축이방성 a-판을 투과한 후의 편광상태를 임의의 입사각과 방위각에 대하여 계산하였다. 또한 광축이 수직으로 교차된 이상적인 두 o-타입 편광자에 편광되지 않은 빛이 투과한 경우 투과율을 임의의 입사각과 방위각에서 계산한 후, 기존의 근사식과 비교하였고, 완전하지 않은 두 편광자의 소광계수와 두께의 변화에 따른 투과율을 계산하여 실제의 편광자를 상정한 소광도를 평가하였다. 근사식이 적용되지 않는 이방성이 큰 이방성 매질을 통과한 후의 빛의 편광상태를 분석할 때 적용될 수 있도록 정확한 투과계수를 사용하는 방법을 제시함으로써 액정 디스플레이 분야 광소자의 편광 분석에 기여하고자 하였다. The exact transmission coefficients at the interface between a uniaxial anisotropic medium and an isotropic medium at? oblique incidence are derived by applying the extended Jones matrix method. When the birefringence of the uniaxial anisotropic medium is small (<TEX>$|n_e-n_o|\;{\ll}\;n_o,\;n_e$</TEX>), the exact transmission coefficients are compared with those by the conventional extended Jones matrix method by Yeh et al. They showed an excellent agreement with each other. In addition, using the exact transmission coefficients, we calculated the polarization characteristics of a light through a uniaxial medium to an incident light with arbitrary polarization state at? oblique incidence. We compared the transmittances of an unpolarized light through a pair of crossed o-type polarizers by two different methods and calculated the transmittance as the variation of the optical constants of the polarizers to evaluate of the extinction ratio. The polarization analysis method using the exact transmission coefficients can be applied to polarization characteristics of a light through a uniaxial medium with large birefringence as well as to liquid crystals and to optical anisotropic material.

Transmission coefficient for a point scatterer at specific energies is affected by the presence of the scatterer but independent of the scatterer’s characteristics

We calculate the exact transmission coefficient of a quantum wire in the presence of a single point defect at the wire's cut-off frequencies. We show that while the conductance pattern (i.e., the scattering) is strongly affected by the presence of the defect, the pattern is totally independent of the defect's characteristics (i.e., the defect that caused the scattering cannot be identified from that pattern).

Above-barrier resonances: analytical expressions for energy and width

We construct a smooth realistic barrier potential that can generate resonances above the top of the barrier provided a parameter λ controlling the flatness of the barrier is larger than the critical value λ = (2)1/2. The energies and widths of resonances are expressed analytically in terms of the characteristics of the barrier, namely height, range and flatness. In order to obtain a more versatile and asymmetric barrier, two symmetric barriers are merged together side by side and the exact transmission coefficient across such a barrier is derived.

Asymmetric parabolic effective barrier model for heavy ion fusion and its relation to coupled channel effects

Using the exact transmission coefficient across an asymmetric parabolic barrier derived by Zafar Ahmed and the ideas used earlier in our effective fusion barrier transmission model we develop a new asymmetric parabolic effective fusion barrier model for heavy ion fusion. A closed form expression for fusion cross section (F) depending only on the s-wave characteristics of the fusion barrier is derived. The expected change of shape of the barrier in the interior side due to channel coupling is represented by a variable curvature parameter 2 whereas the outer curvature 1 is kept unchanged. We apply this model to fit the fusion data, namely fusion cross section, spin distribution, average angular momenta and barrier distribution, in the cases of the following representative pairs of nuclei at energies around the Coulomb barrier: 16O+ 144Sm, 16O+ 152Sm, 40Ca+ 46,48,50Ti, 58Ni+ 64Ni and 64Ni+ 92Zr. These examples include cases of symmetric, asymmetric and nearly symmetric fusion. We obtain good fits to the experimental data. The asymmetry parameter = (2/1) 1/2 used in the present macroscopic formulation is correlated to the order of coupling in coupled channel calculations.

二重強磁性トンネル接合における電気伝導とＴＭＲ

The transport properties and tunnel magnetoresistance (TMR) of ferromagnetic tunnel junctions with double barriers are investigated in the free-electron approximation. Tunnel currents are calculated on the basis of a Tsu-Esaki-type equation using exact transmission coefficients for the double junctions. Enhancement and oscillation of the tunnnel magnetoresistance occur as a function of the bias voltage. It is shown that these can be attributed to spin-dependent resonant tunneling through resonant states formed in the quantum well.

Tunnelling through asymmetric parabolic potential barriers

An exact transmission coefficient for the asymmetric parabolic barrier potential, i.e. , where and , is rederived.

Analytic expressions for emission characteristics as a function of experimental parameters in sharp field emitter devices

An analytical treatment of elliptical field emitting structures which includes image charge effects off axis and the variation of field along the tip is given. The angular distribution of electron emission along the tip, the total emitted current, and the area factor are calculated as a function of emitter to anode distance, tip radius, tip height, and the anode-tip voltage difference. Errors arise if the planar Fowler–Nordheim (FN) equation is used to govern electron emission; we show how to correct this by a modification of the FN equation which includes changes in fields and image charge due to curvature. An analytic form of the area factor is given and compared to a calculation based on the exact transmission coefficient and various Wentzel–Kramers–Brillouin-based approximations to it. The analysis is extended beyond diode geometries with the inclusion of additional electrodes, allowing devices such as triodes to be modeled.

Generalized measures for physical properties of nonperiodic chains.

The physical properties associated with nonperiodic deterministic sequences (such as the Fibonacci sequence) are studied by means of an information-theoretic measure, which is a generalization of one previously defined to measure the disorder of such sequences. In application to electronic or optical transmission, the measure allows the definition of an effective transmission coefficient, which is computationally more efficient to calculate, while still exhibiting the same structure (e.g., self-similarity), as the exact transmission coefficient.

Tunneling through a one-dimensional potential barrier.

The exact transmission coefficient of a one-dimensional potential barrier, V(x)=${\mathit{V}}_{0}$(1-[{1-exp(x/a)}/{1+c exp(x/a)}${]}^{2}$), of which the Morse and the Eckart barriers are special cases, has been obtained. Comparing the exact and WKB transmission coefficients, the limitation of the semiclassical method has been quantified. Another branch of this potential (c0) has been shown useful in nucleus-nucleus fusion. Using the transmission coefficients, the exact energy eigenvalues of the inverted (${\mathit{V}}_{0}$\ensuremath{\rightarrow}-${\mathit{V}}_{0}$) potential (oscillator) have been derived.

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.

Investigation of a new numerical method for the exact calculation of one-dimensional transmission coefficients: Application to the study of limitations of the WKB approximation

A new technique for an exact, numerical calculation of one-dimensional transmission coefficients has been recently derived by Vigneron and Lambin. This has been extended to allow different right and left asymptotic values of the potential, V(+∞) ≠ V(−∞). The algorithm is assessed by using it to calculate the J- V characteristics for a variety of model tungsten (W-W) metal-vacuum-metal junctions with different interelectrode separations. Comparison of these results with the corresponding J- v characteristics previously calculated using the WKB approximation reveals some unexpected limitations on the accuracy of the WKB method. These are consistent with well known inadequacies of the WKB approximation. Namely, the WKB barrier transmission coefficient T WKB always exceeds the exact transmission coefficient T for energies close to the bottom of the barrier. Furthermore, at the top of the barrier, T WKB assumes the infinite energy value of unity. This suggests that either T WKB> T for all energies or there exist one or more energy intervals between the bottom and the top of the barrier in which T WKB ⩽ T. For all barriers investigated here and by Vigneron and Lambin, exactly one such interval was found. This result can also be proven analytically for the square barrier. It is also found that for a given barrier shape the location of the intersection of T WKB and T may depend on the barrier width. Furthermore, the energies for which T WKB = T determine whether J WKB approaches the exact current density J as the barrier width increases. For some barriers, with widths larger than some optimum value, J WKB diverges from J with increasing barrier thickness. It is concluded that the Vigneron-Lambin algorithm is generally faster and more accurate than the WKB approximation.

Investigation of a new numerical method for the exact calculation of one-dimensional transmission coefficients: Application to the study of limitations of the WKB approximation

Abstract A new technique for an exact, numerical calculation of one-dimensional transmission coefficients has been recently derived by Vigneron and Lambin. This has been extended to allow different right and left asymptotic values of the potential, V (+∞) ≠ V (−∞). The algorithm is assessed by using it to calculate the J - V characteristics for a variety of model tungsten (W-W) metal-vacuum-metal junctions with different interelectrode separations. Comparison of these results with the corresponding J - v characteristics previously calculated using the WKB approximation reveals some unexpected limitations on the accuracy of the WKB method. These are consistent with well known inadequacies of the WKB approximation. Namely, the WKB barrier transmission coefficient T WKB always exceeds the exact transmission coefficient T for energies close to the bottom of the barrier. Furthermore, at the top of the barrier, T WKB assumes the infinite energy value of unity. This suggests that either T WKB > T for all energies or there exist one or more energy intervals between the bottom and the top of the barrier in which T WKB ⩽ T . For all barriers investigated here and by Vigneron and Lambin, exactly one such interval was found. This result can also be proven analytically for the square barrier. It is also found that for a given barrier shape the location of the intersection of T WKB and T may depend on the barrier width. Furthermore, the energies for which T WKB = T determine whether J WKB approaches the exact current density J as the barrier width increases. For some barriers, with widths larger than some optimum value, J WKB diverges from J with increasing barrier thickness. It is concluded that the Vigneron-Lambin algorithm is generally faster and more accurate than the WKB approximation.

Tunneling in asymmetric double-well potentials

Methods previously developed for the case of a symmetric double-well potential are here extended to study tunneling in an asymmetric, piecewise harmonic, double-well potential. The approach involves the decomposition of the real stationary states into complex right- and left-moving states and the determination of exact transmission coefficients for the latter. Numerical solutions for a selection of representative parameter values are obtained for the time evolution of states which correspond initially to a Gaussian wave packet localized in the upper well. Both resonant and nonresonant cases are studied and the significance of the transmission coefficient is established for each.

Polarizabilities of shallow donors in silicon : J. Bethin, T. G. Castner and N. K. Lee. Solid St. Commun.14, 1321 (1974)

A new technique for an exact, numerical calculation of one-dimensional transmission coefficients has been recently derived by Vigneron and Lambin. This has been extended to allow different right and left asymptotic values of the potential, V(+∞) ≠ V(−∞). The algorithm is assessed by using it to calculate the J-V characteristics for a variety of model tungsten (W-W) metal-vacuum-metal junctions with different interelectrode separations. Comparison of these results with the corresponding J-v characteristics previously calculated using the WKB approximation reveals some unexpected limitations on the accuracy of the WKB method. These are consistent with well known inadequacies of the WKB approximation. Namely, the WKB barrier transmission coefficient TWKB always exceeds the exact transmission coefficient T for energies close to the bottom of the barrier. Furthermore, at the top of the barrier, TWKB assumes the infinite energy value of unity. This suggests that either TWKB>T for all energies or there exist one or more energy intervals between the bottom and the top of the barrier in which TWKB ⩽T. For all barriers investigated here and by Vigneron and Lambin, exactly one such interval was found. This result can also be proven analytically for the square barrier. It is also found that for a given barrier shape the location of the intersection of TWKB and T may depend on the barrier width. Furthermore, the energies for which TWKB = T determine whether JWKB approaches the exact current density J as the barrier width increases. For some barriers, with widths larger than some optimum value, JWKB diverges from J with increasing barrier thickness. It is concluded that the Vigneron-Lambin algorithm is generally faster and more accurate than the WKB approximation.