Chapter 11 Metals, Semiconductors and Insulators

11.1 Introduction

In the last lecture we showed that the electrons fill the available levels up to the Fermi Energy level E_F . This lecture adds the effects of band structure to the model.

11.2 The Facts

Metals	Non-Metals
shiny and opaque	rarely reflects, sometimes transparent
all conduct electricity	resistivity varies up to a factor of 10⁹ from sample to sample
conductivity falls as temperature rises	conductivity rises as temperature rises
obeys Ohm's Law	Sometimes obeys Ohm's Law
ductile	brittle

11.3 EF Position

Figure 11.1 - A Metal

Figure 11.2 - A Semiconductor

Figure 11.3 - An Insulator

11.4 Full Bands Carry No Current

Figure 11.4 - Full Bands Do Not Carry Current

In a full band equal numbers of electrons with positive and negative K exist. This creates a net charge motion of zero, so it cannot conduct electricity.

Figure 11.5 - Applying an Electric Field to a Partially Filled Band

If we apply an electric field to the bands we get figure 11.5. The carriers of current scatter into states that are further to the right in K space, the unpairer carriers carry the net current.

11.5 Mobile Carrier Concentrations in Pure Non-Metals

Figure 11.6 - A Pure Non-Metal

the density of mobile carriers in the conduction band is

n_iµ e

E_q

2kT

	Ge	Si	GaA	Diamond
E_g (eV)	0.66	1.12	1.42	~ 4
n_i (cm^-3 )	2e13	1.4e10	1.7e6	almost zero

11.6 When Does EF Fall in the Gap

Each band spans 2p/d in K space, the allowed states are determined by the crystals size which is spaced by 2p/L . In the one dimensional case there are L/d K states/bands. However in three dimensions the band contains (L/d)³ K states. This will hold N=2(L/d)³ electrons, so in terms of density

L³

d³

ie. when n=2 it represents the density of the unit cells.

A band, in a pure material, will be exactly full when the unit cell contains an even number of electrons.