Quantum Transport in Crystals: Effective Mass Theorem and K·P Hamiltonians

Abstract

In this paper the effective mass approximation and k·p multi-band models, describing quantum evolution of electrons in a crystal lattice, are discussed. Electrons are assumed to move in both a periodic potential and a macroscopic one. The typical period ǫ of the periodic potential is assumed to be very small, while the macroscopic potential acts on a much bigger length scale. Such homogenization asymptotic is investigated by using the envelope-function decomposition of the electron wave function. If the external potential is smooth enough, the k·p and effective mass models, well known in solid-state physics, are proved to be close (in strong sense) to the exact dynamics. Moreover, the position density of the electrons is proved to converge weakly to its effective mass approximation.

A Hybrid Classical-Quantum Transport Model for the Simulation of Carbon Nanotube Transistors

“…It is however clear that the model and the numerical algorithm can be applied to general strongly confined nanostructures (such as silicon nanowires or different CNTs), with different gate geometries. We consider a (10,0) zig-zag single-walled CNT (see Fig.1 In [6], a novel quantum effective mass model has been derived by performing an asymptotic process which consists in using an envelope function decomposition to obtain a new effective mass approximation (see also [2] for a similar approach for 3D periodic crystals). We recall it briefly here.…”

An Effective Mass Model for the Simulation of Ultra-Scaled Confined Devices

“…It gives the following corollary. 6 for obtaining in a rigorous way the effective mass for 3D periodic crystals). As we shall see in the next section, they do not allow to completely diagonalize the periodic part of the Hamiltonian.…”

“…Consequently, with a slight abuse of notation, we follow the denomination of Ref. 6 and throughout the paper we shall refer to the 1-periodic functions χ n (y, z ), eigenvectors of (2.3), as Bloch functions whereas the quasi-periodic functions e iky χ n (y, z ) will be called Bloch waves. For each pair of Bloch functions (χ n (y, z ), χ n (y, z )), we define averaged quantities on the periodic direction as the functions g nn (z ) = 1/2 −1/2 χ n (y, z )χ n (y, z )dy.…”

Mathematical Analysis of a Nonparabolic Two-Band Schrödinger-Poisson Problem

“…The study of multiband models is a very active area of research (Ben Abdallah and Kefi, 2008;Barletti and Frosai, 2010;Barletti, Frosali, and Demeio, 2007;Barletto and Mahats, 2010;Mahats, 2005;Ben Abdallah, Jourdana, and Pietra, 2012;Pinaud, 2004;Barletti and Ben Abdallah, 2011;Morandi and Schuerrer, 2011;Morandi, Hervieux, and Manfredi, 2010;Morandi and Demeio, 2008). A considerable effort has been made in order to develop mathematical models that reproduce the steady states and the out-of-equilibrium dynamics in heterostructure devices (Ben Abdallah and Mehats 2004).…”