Specifically, the algorithm consists of the solution of an ensemble of single-particle SSEs that are coupled, one to each other, through effective Coulombic potentials. The resulting algorithm for quantum transport simulations reformulates the traditional “curse of dimensionality” that plagues all state-of-the-art techniques for solving the time-dependent Schrödinger equation (TDSE). Īs an example of the practical utility of the SSE, a Monte Carlo simulation scheme to describe quantum electron transport in open systems that is valid both for Markovian or non-Markovian regimes guaranteeing a dynamical map that preserves complete positivity has been recently proposed. Instead of directly solving equations of motion for the reduced density matrix, the SSE approach exploits the state vector nature of the so-called conditional states to alleviate some computational burden. A preferred technique has been the stochastic Schrödinger equation (SSE) approach. As such, one can then borrow any state-of-the-art mathematical tool developed to study open quantum systems. Ī formally exact approach to electron transport beyond the quasi-stationary regime relies in the modeling of the active region of electron devices as an open quantum system. At these frequencies, the discrete nature of electrons in the active region is expected to generate unavoidable fluctuations of the current that could interfere with the correct operation of such devices both for analog and digital applications. This limitation poses a serious problem in the near future as electron devices are foreseen to operate in the terahertz (THz) regime. The amount of information that these simulators can provide, however, is mainly restricted to the stationary regime, and therefore, their predicting capabilities are still far from those of the traditional Monte Carlo solution of the semi-classical Boltzmann transport equation. A number of quantum electron transport simulators are available to the scientific community. In the design of these nanostructures, simulation tools constitute a valuable alternative to the expensive and time-consuming test-and-error experimental procedure. In this respect, a number of nanodevices based on nanojunctions like single electron transistors, field effect transistors, and heterostructure nanowires have been recently reported, which promise great performance in terms of miniaturization and power consumption. Today, advances in fabrication techniques like direct growth of branched nanostructures, electron beam irradiation, thermal and electrical welding, or atomic force microscope have allowed controlling the size and composition of nanojunctions for creating devices with desired functionalities. Studies of such systems have been inspired by the pioneering investigations of Sharvin in the mid-1960s. Nanoscale constrictions (sometimes referred to as point contacts or nanojunctions) are unique objects for the generation and investigation of ballistic electron transport in solids.
1D SCHRODINGER EQUATION FULL
This technique achieves quantitative accuracy using an order less computational resources than the full dimensional simulation for a typical two-dimensional geometrical constriction and upto three orders for three-dimensional constriction. The resulting scheme consists of an eigenstate problem for the confinement degrees of freedom (in the transverse direction) whose solution constitutes the input for the propagation of a set of coupled one-dimensional equations of motion for the transport degree of freedom (in the longitudinal direction).
Here, we consider the use of a Born–Huang-like expansion of the three-dimensional time-dependent Schrödinger equation to separate transport and confinement degrees of freedom in electron transport problems that involve geometrical constrictions. Warning( 'Step size is wrong.The so-called Born–Huang ansatz is a fundamental tool in the context of ab-initio molecular dynamics, viz., it allows effectively separating fast and slow degrees of freedom and thus treating electrons and nuclei with different mathematical footings. Xl=input( 'Enter length of the well(in nm): ')
% = % Program to solve 1D infinite well problem % ***************************** By Mahesha MG ***************************** % This is to solve 1D Schrodinger wave equation (Time independent system) % by finite difference method - boundary value problem % Date: % =
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