View
225
Download
3
Embed Size (px)
Citation preview
Dr. Jie Zou PHY 1371 3
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.
Dr. Jie Zou PHY 1371 4
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
Dr. Jie Zou PHY 1371 5
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
Dr. Jie Zou PHY 1371 6
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)(
Dr. Jie Zou PHY 1371 7
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.
Dr. Jie Zou PHY 1371 8
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
Dr. Jie Zou PHY 1371 9
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).
Dr. Jie Zou PHY 1371 10
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
Dr. Jie Zou PHY 1371 12
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.