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Solid State Physics 3

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Solid State Physics3

Section 10-4,6

- Heat Capacity of Electron Gas
- Band Theory of Solids
- Conductors, Insulators and Semiconductors
- Summary

As can be seen from the

graph, the prediction

fails at very low

temperatures. This is

due, in part, to the

failure of the

equipartition theorem

at low temperatures.

Challenge: create a

better model!

Derive the temperature

dependence of R/R0 by

computing the average potential energy <E>

of a lattice ion

assuming that the energy

level of the nth

vibrational state is

rather than En = ne as

Einstein had assumed

Due:

before classes end

By definition, the heat capacity (at constant

volume) of the electron gas is given by

where U is the total energy of the gas. For a gas

of N electrons, each with average energy <E>,

the total energy is given by

Total energy

In general, this integral must be done

numerically. However, for T << TF, we can use

a reasonable approximation.

At T= 0, the total energy of the electron gas is

For 0 < T << TF, only a small fraction kT/EF

of the electrons can be excited to higher energy

states

Moreover, the energy of

each is increased by

roughly kT

Therefore, the total energy can be written as

where a = p2/4, as first shown by Sommerfeld

The heat capacity of the

electron gas is predicted to

be

Consider 1 mole of copper. In this case Nk = R

For copper, TF = 89,000 K. Therefore, even at

room temperature, T = 300 K, the contribution

of the electron gas to the

heat capacity of copper is

small:

CV = 0.018 R

So far we have neglected the lattice of positively charged ions

Moreover, we have ignored the Coulomb repulsion between the electrons and the attraction between the lattice and the electrons

The band theory of solids takes into account the interaction between the electrons and the lattice ions

Consider the potential energy of a

1-dimensional solid

which we approximate by the Kronig-Penney Model

The task is to compute the quantum states and

associated energy levels of this simplified model

by solving the Schrödinger equation

1

2

3

For periodic potentials, Felix Bloch showed that

the solution of the Schrödinger equation must

be of the form

and the wavefunction must

reflect the periodicity of

the lattice:

1

2

3

By requiring the wavefunction and its derivative

to be continuous everywhere, one finds energy

levels that are grouped into bands separated by

energy gaps. The gaps occur at

The energy gaps

are basically energy

levels that cannot

occur in the solid

1

2

3

Completely free

electron

electron in a

lattice

When, the wavefunctions become

standing waves. One wave peaks at the lattice

sites, and another peaks between them. Ψ2, has

lower energy

than Ψ1. Moreover, there is a jump in energy

between these states, hence the energy gap

The allowed ranges of the wave vector k are

called Brillouin zones.

zone 1: -p/a < k < p/a; zone 2: -2p/a < - p/a;

zone 3: p/a < k < 2p/a etc.

The theory can explain why some

substances are conductors, some

insulators and

others

semi

conductors

Sodium (Na) has one electron in the 3s state, so

the 3s energy level is half-filled. Consequently, the

3s band, the valence band, of solid

sodium is also half-filled. Moreover,

the 3p band, which for Na is

the conduction band, overlaps with

the 3s band.

So valence electrons can easily be

raised to higher energy states.

Therefore, sodium is a good

conductor

NaCl is an insulator, with a band gap of 2 eV,

which is much larger than the thermal energy at

T=300K

Therefore, only a tiny fraction of electrons are

in the conduction band

Silicon and germanium have band gaps of 1 eV and

0.7 eV, respectively. At room temperature, a small

fraction of the electrons are in the conduction

band. Si and Ge are intrinsic semiconductors

- The heat capacity of the electron gas is small compared with that of the ions
- Energy gaps arise in solids because they contain standing wave states
- The size of the energy gap between the valence and conduction bands determines whether a substance is a conductor, an insulator or a semiconductor