Topics: Electrons in solids

Electrons in solids

Abstract

This page is an introduction to electron transport in solids. Additional derivation and information will be required to obtain a full understanding of the material. Please submit corrections when you feel necessary.

Electronic structure and quantization

In solids, electrons are confined to a number of energy bands and are not allowed in other regions (band gaps). This forbidden region, or band gap, is the region between the top of the valence band and the bottom of the conduction band. Electrons my not exist in this region, but they are able to cross or jump the band gap to move up in energy to the conduction band or drop to the valence band.

Quantization of energy was first suggested by Ludwig Boltzmann and was first explained by Planck who hypothesised that energy was radiated and absorbed in discrete energy elements that were proportional to the frequency of the waves radiated. This lead to the well know proportionality constant, h, known as Planck’s Constant. The derivation of energy and momentum quantization can most easily be performed for a free particle in a one-dimensional, infinite potential energy well. It begins with the one-dimensional, time-independent form of the Schrodinger Equation:

$-\frac{\hbar^2}{2 m} \frac{d \Psi (x)}{d x} + U(x) \Psi (x) = E \Psi (x)$

Where the general solutions are given by:

$Ψ(x) = A exp i k x + B exp - i k x$

where

$k = \frac{\sqrt{2 m (E-U)}}{\hbar}$

Due to the infinite potential barrier at $x = 0, L$ we apply the following boundary conditions:

$(1)Ψ(0) = 0$

$(2)Ψ(L) = 0$

$(3) \int_{-\infty}^{\infty} \Psi(x)^* \Psi(x) dx = 1$

We obtain the following solution:

$\Psi(x)_n = \sqrt{\frac{2}{L}} \sin{\frac{n \pi x}{L}}$ , 0 \leq x \leq L

$Ψ(x) n = 0$ , x < 0 or x > L

$E_n = \frac{\hbar^2 \pi^2}{2 m L^2} n^2$

$n = 1,2,3,...$

Due to the constraint that n must be an integer shows that there are quantized energy levels.

Statistics of electons

In this section, the density of states and the occupation of energy band states will be discussed. The density of states is the density of total allowable states for a given crystal. The simplest approach for obtaining the density of states for a crystal will be described here. The first step is to solve the time-independent Schrodinger Equation for the crystal of interest (here we will use a three-dimensional crystal). The three-dimensional, time-independent for of the Schrodinger Equation is given by:

$\nabla^2 \Psi (x,y,z) + k^2 \Psi (x,y,z) = 0$

where the overall wavefunction solution is given by:

$Ψ(x, y, z) = A sin k x x sin k y y sin k z z$

Next we need to calculated the number of solutions per unit volume of k-space,

$(\frac{\textrm{Solutions}}{\textrm{Unit volume of k-space}})=\frac{a b c}{\pi^2}$

We can then calculated the allowed energy states by dividing by the redundant energy states and multiplying by two to account for spin up and spin down,

$(\frac{\textrm{Allowed energy states}}{\textrm{Unit volume of k-space}})=\frac{a b c}{4 \pi^2}$

Now we need to determine the number of states for a given k value by adding the number of states between k and k+dk,

$(\textrm{Energy states with k between k and k + dk})=4 \pi k^2 dk \frac{a b c}{\pi^2}$

We then convert the discrete k-states to energy states using the relationship between k and E and dk and dE to obtain

$k^2 = \frac{2 m E}{\hbar^2}$

$dk = \frac{1}{\hbar} \sqrt{\frac{m}{2}} \frac{dE}{\sqrt{E}}$

Finally we divided this result by the system volume and the differential energy to obtain the three-dimensional electron density of states given by:

$g(E) = \frac{m \sqrt{2 m E}}{\pi^2 \hbar^3}$

Solutions of the density of states can be obtained for the conduction and valence bands. The approach is the same, the differences come from the location and shape of the bands near the band edges (top of valence and bottom of conduction bands).

The Fermi function is the commonly used probability of finding a state occupied in text books and is what will be discussed here. Other methods can be found in reference texts.

Band structure

In a lattice electrons are not in an isolated state which causes the formerly degenerate energy levels to split and cause bands, with some higher energy states in the band becoming unbound. Within the bands, free particle-like solutions are allowed while in the region between the bands, known as the band gaps, these solutions are not allowed.

Electron scattering

Electronic specific heat and thermal conductivity

Last modified on 24 Jun, 2009

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