- Classical free electron theory:
The free electron theory of metals using classical laws was developed by Dude and Lorentz in the beginning of last century. That time the valance electrons in metals were regarded as the non-interacting particles of an ideal gas. The dude model is the application of Kinetic theory to electrons in a solid. It assumes that the material contains immobile positive ions and an electron gas of classical, non-interacting electrons of density n. Motion of these positive ions and electrons is damped by a fractional force due to their collisions with each other. This motion is characterized by a relaxation time 'τ'.
To obtain some useful results for conduction electrons in metals, let us start with some classical ideas.
(1) In the absence of an applied electric field, the electrons move in random directions. They collide with random impurities and/or lattice imperfections in the crystal. These imperfections arise from thermal motion of ions about their equilibrium position.
(2) The frequency of electron-lattice imperfections collisions can be described by a means free path λ. Mean free path (λ) can be defined as the average distance that an electron travels between collisions.
(3) When an electric field is applied. The electron drifts in the direction opposite to that of the field. The speed with which electrons drift is called drift speed.
(4) The drift speed is much less than the effective instantaneous speed v of the random motion. Let us now sound system with an example electric field. A given copper Rod of uniform cross-section (say 1sq-m) is subjected to a field. The possessive nature of electrons is now suppressed. The random motion is discouraged the charged electrons prefer to have a unidirectional motion. The direction of this motion is opposite to the direction of the applied electric field as sketched. Now the free electrons will bump into a caution of the lattice from time to time. Let the average time between such collisions be τ sec. immediately after a collision we suppose that the velocity of the electrons averages to zero. It means that the electron has no money of the momentum acquired from the field and that is thermal velocity averages to zero it means that its electron has no money of the momentum that its thermal velocity averages to zero. In a time τ sec the electron will at in a velocity given by:
V_{d} = aτ
_{ }Where V_{d} is called drift velocity, and the magnitude of a=e E/m
Thus V_{d}= -eEτ/m
Usually the term eτ/m is replaced by μ. Μ is called the mobility of charge carriers. It can be defined as the drift velocity in unit field. Thus
V_{d}= -μE
If n is the density of electron and -e is the charge of the electron, then charge flowing through unit area in one second is given by
J_{x} = δq/aδt
Here δq= net quantity of charge flowing an area A in time δT and
Δq = -enAδx
Jx is also called the current density.
σ =neμ
in a metal, when temperature increases, n remains constant. But μ decreases as lattice scattering increases and therefore conductivity decreases.