Download Lecture: The Free-Electronic Theory and more Lecture notes Solid State Physics in PDF only on Docsity!
Lecture -18-
The Free-electron theory
1. Introduction.
Metals are of great importance in our daily lives. Iron is used in automobiles, copper in electrical wiring, silver and gold as jewelry. Metals are characterized by common physical properties: great physical strength, high density, good electrical and thermal conductivities, and high optical reflectivity, which is responsible for their characteristic bright appearance.
2. Conduction electrons
What are the conduction electrons?
A valence electron really belongs to the whole crystal, since it can move readily from one ion to its neighbor, and then the neighbor's neighbor, and so on. This mobile electron, which is called a valence electron in a free atom, becomes a conduclion electron in a solid.
In Na the number of conduction electrons is the same as the number of atoms, and the same is true for K, and also for the noble metals Cu, Ag, Au, all of which are monovalent.In
divalent metals-such as Be, Mg, Zn, and Cd-the number of electrons is twice the number of atoms, and so on.
We can find the electron concentration by this equation.
= ( ) NA
Where:
is the number of electrons per unit volume, is the atomic valence, NA is Avogadro's number,
ρ is density of the substance.
- The Free-electron gas
In the free-electron model, the conduction electrons are assumed to be completely free, except for a potential at the surface.
According to this model, the conduction electrons move about inside the specimen without any collisions, except for an occasional reflection from the surface ( see figure below). Because of this, we speak of a free-electron gas.
- Free-electron gas is charged, while in ordinary gases the molecules are mostly neutral. So We may thus think of free-electron gas is thus actually similar to a plasma.
- The concentration of electrons in metals is large:(10^29 ) el./m^3. By contrast, the ordinary gas has about 10^25 molecules/m^3. 3. Static Electrical conductivity(Drude calculation
The law of electrical conduction in metals-Ohm's law- is :
I=V/R ……….. 1
Suppose that L and A are, respectively, the length and cross section of the wire; then
J=I/A …………. 2
……… 3
R= ………. 4
: is the electric field , and is the electrical resistivity.
…………5 , J = …………
which is the form of Ohm's law.
The velocity of the electron ( is:
is a constant called the collision time. This, then, is the steady-state velocity of the electron. It is opposite to because the charge on the electron is negative.
The current density J can be calculated from eq. 7
J= …….. 8
Comparing (8) with Ohm's law (6), one finds the following expression for the conductivity.
σ = ne μd ….. 10
where μd = = (^) d / ……..
μd: is drift mobility , it's unit m^2 V-^1 S-^1
