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2019 | OriginalPaper | Chapter

3. Electronic Structure

Author : Hassan Raza

Published in: Nanoelectronics Fundamentals

Publisher: Springer International Publishing

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Abstract

Electronic structure describes how energy levels or spectra are distributed as a function of either real space \((\varvec{{ r}})\) or reciprocal space \((\varvec{{ k}})\). In this chapter, we focus on the Band Diagram or Energy Diagram \(E(\varvec{{ r}})\), and the Band Structure \(E(\varvec{{ k}})\). The method of choice for calculating the electronic structure of nanomaterials is the Quantum mechanics (also called wave mechanics), where the non-relativistic Schrödinger equation is the norm and not an exception. Before going into the details of this new kind of mechanics of the nanostructures and nanomaterials, let us first review some history.

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previous chapter Atomic Structure
next chapter Quantum Transport
Footnotes
1
1 eV = 1.6\(\times 10^{-19} \) J. The reason behind the wide spread use of the eV unit of energy is its convenience. For example, a voltage applied of a certain value, say 1 V, results in −1 eV shift in the potential energy for an electron, since potential energy of an electron is given as, \(-qV\), where V is the potential and \(q=1.6\times 10^{-19}\) C is the magnitude of the electronic charge.
 
2
Hamiltonian operator and Hamiltonian matrix (discussed later) should be Hermitian, i.e. \(H = H^\dag \), in order to obtain real energy values for equilibrium conditions. Conceptually, an imaginary energy value or frequency gives rise to wavefunction decay in time domain, and hence the probability of finding a particle becomes time dependent that is not consistent with the equilibrium quantum mechanics.
 
3
eig and inv are Matlab functions for calculating Eigen values and inverse of a matrix, respectively.
 
4
One should note that the phase velocity is simply given as, \(v_{px} = \omega /k_x \).
 
5
Euler’s identity gives, \(e^{i\theta } = \cos \theta +i\sin \theta \).
 
6
A standing wave is a superposition of two waves traveling in opposite directions.
 
7
\(\cos (k_xx) = (e^{ik_xx}+e^{-ik_xx})/2\), and \(\sin (k_xx) = (e^{ik_xx}-e^{-ik_xx})/j2\).
 
8
These wavefunctions correspond to a product state.
 
9
It is instructive to note the limits of these variables are given as, \(r = [0, \infty ]\), \(\theta = [0, \pi )\), and \(\phi = [0, 2\pi ).\)
 
10
In the absence of a periodic potential, and hence Bragg diffraction at the edge of the Brillouin zone.
 
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Metadata
Title
Electronic Structure
Author
Hassan Raza
Copyright Year
2019
Book
Nanoelectronics Fundamentals

Print ISBN: 978-3-030-32571-8

Electronic ISBN: 978-3-030-32573-2

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-32573-2_3

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