Quantum waveguide theory: A direct solution to the time-dependent Schrödinger equation

Abstract

In this paper, we present a highly accurate and effective theoretical model to study electron transport and interference in quantum cavities with arbitrarily complex boundaries. Based on this model, a variety of quantum effects can be studied and quantified. In particular, this model provides information on the transient state of the system under study, which is important for analyzing nanometer-scale electronic devices such as high-speed quantum transistors and quantum switches. ͓S0163-1829͑99͒02739-3͔

pyCTQW: A continuous-time quantum walk simulator on distributed memory computers

“…An alternative algorithm that has recently found traction in the field of computational physics and quantum chemistry is the so-called Chebyshev series approximation, which takes its name from the Chebyshev polynomials that occur in the series expansion [34][35][36][37][38][39]. A huge part of what makes the Chebyshev expansion so attractive is the use of Bessel J zero functions as series coefficients, leading to exceptionally fast convergence without sacrificing a high level of accuracy.…”

Guided-wave electron optics for the integration of nanophotonic devices with nanoelectronic devices

“…In spinor-electron optics we are interested on monoenergetic waves with relativistic energy E, that is, the fourcomponent spinor-electron wave, Ψ i (r, t) = ψ i (r)exp{−iEt/ }, with i = (1,2,3,3) and r = (ξ, γ, η), will be governed by the monoenergetic Dirac equation, which in its standard representation can be written as follows:…”

“…Next, by inserting the spinor given by equations (2,3) into equation (1) it is obtained the monoenergetic Dirac equation for the 2D electron waveguide:…”

“…1,2 The electron waveguided optics is necessary since in low-dimensional structures, such as 1D and 2D Al C Ga 1−C As nanostructures (where C is a concentration) implementing quantum wells, quantum wires and so on, a pronounced wave nature of the electrons is obtained and therefore both interference and spinor effects must be taken into account; accordingly, the electron optics of these devices would must be formulated by a rigorous quantum spinor wave theory, that is, by solving the Dirac equation. Theoretical and applied studies about electron waveguides based on the Schršdinger equation have been reported in the last years, [1][2][3][4][5] however, to our knowledge, the fully relativistic spinor nature of the guided electron-waves has not been taken into account; that is, a spinor-electron waveguided optics fully based on the Dirac equation has not been developed for 1D and 2D electron wave devices.…”

“…1,2 The electron waveguided optics is necessary since in low-dimensional structures, such as 1D and 2D Al C Ga 1−C As nanostructures (where C is a concentration) implementing quantum wells, quantum wires and so on, a pronounced wave nature of the electrons is obtained and therefore both interference and spinor effects must be taken into account; accordingly, the electron optics of these devices would must be formulated by a rigorous quantum spinor wave theory, that is, by solving the Dirac equation. Theoretical and applied studies about electron waveguides based on the Schršdinger equation have been reported in the last years, [1][2][3][4][5] however, to our knowledge, the fully relativistic spinor nature of the guided electron-waves has not been taken into account; that is, a spinor-electron waveguided optics fully based on the Dirac equation has not been developed for 1D and 2D electron wave devices. It must be noted that some works, in the recent past, have developed an electron optics based on approximate solutions of the Dirac equation but very far from these kind of devices, that is, they have been oriented, for instance, to the study of imaging processes with bulk electron lenses; 6,7 only recently a preliminary study 8 has put the basis to find Dirac modes and moreover a particular slab electron guide was solved: a symmetric 2D electron guide, in such a way that original results related to spinor dispersion equations, spinor mode cuttof conditions, and so on, were obtained; these results can not be derived by using or Schrödinger equation either Pauli-Schrödinger equation (that is, the Schrödinger equation corrected with the relativistic term of Pauli); in short, the primary aim of this work is to generalize the above results by solving the spinor-electron waves propagation in asymmetric 2D and 1D electron waveguides and in particular in a channel electron waveguide, which is, to a certain extent, analogous to the vector light waves propagation in dielectric channel waveguides (the building blocks in integrated photonics and optoelectronics) which confine the light in a bidimensional region with a width of the order of the light wavelength.…”