Dr. Jie Zou PHY 1371 1

Chapter 43

Molecules and Solids (Cont.)

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Outline

Free electron theory of metals Band theory of solids

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Free-electron theory of metals Classical free-electron theory of

electrical conduction in metals: Model: Treats a metal as an electron gas and

uses the kinetic theory of gases. Predicts Ohm’s law Difficulties: Does not predict the correct

values of electrical and thermal conductivities. Quantum-based free-electron theory

of metals: Model: The outer-shell electrons are free to

move through the metal but are trapped within a three-dimensional box formed by the metal surfaces.

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Fermi-Dirac distribution function

Quantum statistics: Required by the Pauli exclusion principle that each state of the system can be occupied by only two electrons (one with spin-up and the other with spin down).

The probability that a particular state having energy E is occupied by one of the electrons in a solid is given by

Function f (E) is called the Fermi-Dirac distribution function. EF is called the Fermi energy.

Quick quiz: Physical meaning of the two plots on the left.

1

1)( /)( TkEE BFe

Ef

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Electron in a three-dimensional box

For a particle in a one-dimensional box of length L, the allowed value of energy is

For one electron in a solid cube of sides L and volume L3, the energy for such an electron is (see Problem 30)

For example, the ground state, nx = ny = nz =1 and E = 3ħ22/2meL2, can be occupied by two electrons corresponding to spin-up and spin-down (ms = +1/2 and ms = -1/2).

,...,,nnmL

En 321 2

22

22

1integers are ),,( )(2

2222

22

zyxzyxe

n nnnnnnLm

E

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Density-of-states function The number of allowed

states per unit volume that have energies between E and E + dE is

Function g(E) is called the density-of-states function.

dEEh

mdEEg e 2/1

3

2/328)(

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Quick quiz

The Fermi energy for silver is 5.48 eV. Near which of these energies are the energy levels closer together? (a) 2 eV. (b) 6 eV. (c) The spacing of

energy levels is the same near both energies.

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Electron distribution function versus energy

Set N(E)dE = The number of electrons per unit volume that have energy between E and E + dE, where

N(E) is called the electron distribution function.

1

28

)()()(

/)(

2/1

3

2/3

TkEEe

BFe

dEE

h

m

dEEgEfdEEN

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Problem 35 (a) Consider a system of electrons

confined to a three-dimensional box. Find the ratio of the number of allowed energy levels at 8.50 eV to the number at 7.00 eV.

(b) Copper has a Fermi energy of 7.0 eV at 300 K. Calculate the ratio of the number of occupied levels at an energy of 8.50 eV to the number at the Fermi energy.

(c) Compare answers for (a) and (b).

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Electron concentration Set ne = The total number of electrons

per unit volume. ne is called the electron concentration.

Find Fermi energy at T = 0 K from

Solve for See Table 43.4 for the values of ne for

different metals.

0 /)(

2/1

3

2/3

0 1

28)( TkEE

ee BFe

dEE

h

mdEENn

2/3

0 3

2/32/1

3

2/3 28

3

228F

Eee

e Eh

mdEE

h

mn

F

3/22

8

3

2)0(

e

eF

n

m

hE

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Band theory of solids

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Energy bands of a sodium crystal

Blue represents energy bands occupied by the sodium electrons.

Gold represents energy bands that are empty.

Energy gaps or forbidden energies (white regions) between the allowed bands; electrons cannot occupy states that lie in these gaps.

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Homework

Chapter 43, P. 1436, Problems: #33, 35.