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Applied Physics Letters

Time-evolved numerical simulation of a two-dimensional electron wave packet through a quantum point contact

Akira Endoh, Shigehiko Sasa, Shunichi Muto

Published Jul 6, 1992• 8 citations
Publisher

Abstract

We performed time-evolved numerical simulation of a two-dimensional electron wave packet through a quantum point contact using the finite difference method to simulate the diffraction phenomenon of a single electron. We found that the transmitted wave packet has diffracted subpeaks which show the following properties: (1) The traces of diffracted subpeaks start from the center of the slit entrance. (2) Lateral wave number of diffracted subpeaks is quantized. (3) The diffraction angles are determined by slit width and the initial electron wave packet energy. (4) The number of allowed diffracted subpeaks is determined by potential barrier thickness. (5) There is a delay in the motion of diffracted subpeaks due to quantization inside the slit.

DOI
10.1063/1.107666

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We performed time-evolved numerical simulation of a two-dimensional electron wave packet through a quantum point contact using the finite difference method to simulate the diffraction phenomenon of a single electron. We found that the transmitted wave packet has diffracted subpeaks which show the following properties: (1) The traces of diffracted subpeaks start from the center of the slit entrance. (2) Lateral wave number of diffracted subpeaks is quantized. (3) The diffraction angles are determined by slit width and the initial electron wave packet energy. (4) The number of allowed diffracted subpeaks is determined by potential barrier thickness. (5) There is a delay in the motion of diffracted subpeaks due to quantization inside the slit.
We performed time-evolved numerical simulation of a two-dimensional electron wave packet through a double slit by solving the time-dependent Schrödinger equation using the finite difference method to simulate the electron interference phenomenon of a single electron. The propagating directions of interference peaks show that the starting point of electron interference is the center between the two slit exits. The number of interference peaks and their angles are found to be independent of potential barrier thickness. These results are compared with the results of our previous calculation of electron diffraction through a single slit.
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Abstract The goal of this paper is to study double-slit diffraction at the nanometric scale. We treat the time evolution of a Gaussian wave packet in quantum mechanics traveling until it reaches a double-slit wall. A detailed investigation and analysis of the diffraction phenomenon and its progress is developed by numerically solving the Schrödinger equation. This is written using a potential intended to model the wall. It is solved by using finite differences in the time domain method, so that the probability amplitude, ψ , is obtained as a function of the spatial coordinates at discretized times. Then, the probability density, ρ , is found in a straightforward way and used to plot graphics or make videos. Furthermore, the time evolution of the wave packet is registered at several fixed points to acquire local information. Fourier transforms are calculated to complement the knowledge obtained. Students interested in didactic approaches, or in applications for nanodevices, could find interest in this approach.
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In a "thought experiment," now a classic in physics pedagogy, Feynman visualizes Young's double-slit interference experiment with electrons in magnetic field. He shows that the addition of an Aharonov-Bohm phase is equivalent to shifting the zero-field wave interference pattern by an angle expected from the Lorentz force calculation for classical particles. We have performed this experiment with one slit, instead of two, where ballistic electrons within two-dimensional electron gas diffract through a small orifice formed by a quantum point contact (QPC). As the QPC width is comparable to the electron wavelength, the observed intensity profile is further modulated by the transverse waveguide modes present at the injector QPC. Our experiments open the way to realizing diffraction-based ideas in mesoscopic physics.
A numerical solution to the time evolution equation of the Wigner distribution function (WDF) with an accuracy necessary to simulate the passage of a wave packet past a barrier is developed, where quantum effects require high accuracy and fine discretization. A wave packet incident on a barrier, a portion of which tunnels through, demonstrates behavior that can define various characteristic transmission and reflection delay (TARD) times useful in the simulation of electron emission. A model for the TARD times is proposed that relies only on the asymptotic maxima of the position $\ensuremath{\rho}(x,t)$ and wave number $\ensuremath{\rho}(k,t)$ densities given by the WDF and applied to a ballistic trajectory model for the (faster) transmitted and (slower) reflected parts. The dependence of the TARD times on barrier width, symmetry, and abruptness is analyzed. For symmetrical barriers with characteristics similar to field emission barrier heights and widths, TARD times are on the order of a fraction of a femtosecond. The TARD times for when tunneling predominates are contrasted to tunneling times in the literature. Use of the TARD times in simulations of field emission in nanogap devices or to model ultrashort pulses generated under rapidly changing conditions for electron sources are proposed.