Materials Science

Free electron theory of metals

 

1. Free electron theory of metals

Drude model [ Classical theory of free electrons]

1.1 Introduction

This classical theory of free electron in metals was introduced by Paul Drude in 1900.

1.2 Assumptions

This theory is based on the following assumptions.
1. Metal consists of positive immobile ions formed by the detachment of valence (free) electrons from neutral atoms.
2. Free electrons are non-interacting particles inside metals and they behave similar to gas molecules in a container.
3. They move in random manner and interact momentarily with positive ions through elastic collisions. The average distance between two successive collisions is known as mean free path and the average time between subsequent collisions is known as collision time.
4. Thermal equilibrium is attained through collisions by electrons with positive ions. According to equipartition theorem, the average energy possessed by an electron is 3/2 KBT. KB is the Boltzmann constant and T is the absolute temperature.

If τ is the collision time and λ is the mean free path, then the average velocity is given as λ / τ.

1.3 Derivation of electrical conductivity

Assume that a constant electric field, Ex is applied to a metallic substance along x direction. Electrons are accelerated in the direction opposite to the applied of electric field. Let vx be the average drift velocity of electrons due to the applied field.

If FE is the force acting on electrons due to the electric field, E, then

F=-eE_{x} —– (1)

According to Newton’s Law of motion,

F=m\frac{dv_{x}}{dt} —– (2)

From equations (1) and (2),

m\frac{dv_{x}}{dt}=-eE_{x} —– (3)

dv_{x}=-\frac{eE_{x}}{m}dt

By integrating the above equation, we obtain

v_{x}=-\frac{eE_{x}}{m}t —– (4)

This equation indicates that the electron velocity is increasing indefinitely with time. It is not true. Collisions of electrons with positive ions introduced an opposing force to the electron current. Hence, the equation (3) is modified by adding the retardation force as follows

m\frac{dv_{x}}{dt}=-eE_{x}-\frac{mv_{x}}{\tau }

\frac{dv_{x}}{dt}=-\frac{eE_{x}}{m}-\frac{v_{x}}{\tau }

\tau \frac{dv_{x}}{dt}=-\frac{eE_{x}\tau }{m}-v_{x}

\tau \frac{dv_{x}}{dt}=-\left( \frac{eE_{x}\tau }{m}+v_{x} \right )

\frac{dv_{x}}{\left( \frac{eE_{x}\tau }{m}+v_{x} \right )}=-\frac{dt}{\tau }

By Integrating the above equation, we obtain

ln\left( \frac{eE_{x}\tau }{m}+v_{x} \right )=-\frac{t}{\tau }+Constant

\frac{eE_{x}\tau }{m}+v_{x}=e^{-\frac{t}{\tau}+Constant}

\frac{eE_{x}\tau }{m}+v_{x}=Ae^{-\frac{t}{\tau}} —– (5)

A is the constant which is found by applying the initial condition [vx=0 at t=0].

A=\frac{eE_{x}\tau }{m}

Now equation (5) becomes

\frac{eE_{x}\tau }{m}+v_{x}=\frac{eE_{x}\tau }{m}e^{-\frac{t}{\tau}}

v_{x}=-\frac{eE_{x}\tau }{m}\left( 1-e^{-\frac{t}{\tau}} \right)

By applying steady state condition, we get

v_{x}=v_{d}=\frac{eE_{x}\tau }{m} —– (6)

vd is known as the drift velocity of electrons.

The current density under electric field is expressed as

J=\rho v_{d}=-nev_{d}

ρ is the charge density and n is the number of electrons per unit volume. Substituting equation (6) in the above expression, we obtain

J=\frac{ne^{2}\tau }{m}E_{x} —– (7)

The general expression of current density as a function of electric field is written as

J=\sigma E_{x} —– (8)

Comparing equations (7) and (8), we get

\boxed{\sigma=\frac{ne^{2}\tau }{m}} —– (9)