Electrical conductivity of the materials according to quantum mechanical consideration

We know that conduction current density,  J = NVe = σE

Where, N= number of electron

            V= electron velocity

           σ = conductivity

According to quantum mechanical consideration, when an electric field E is applied, only the number of electrons (N^/) participate in conduction those have approximately fermi velocity (V_F) 

So, we can write  J = N^/ V_F e --------(i)

Now number of displaced electrons can be found from the relation of population density which reflects the following figure-

Figure:. Population density N(E) versus energy for free electrons and displacement ∆E by an electric field. N^/ is the number of displaced electrons per unit volume in the energy interval ∆E. N(E) is defined per unit energy and in the present case, also per unit volume.

We get-      N^/  =  N(E_F) ∆E

Where, N(E_F) = population density with Fermi energy

     And  ∆E = (dE/dK) ∆K

Therefore, N^/  =  N(E_F) (dE/dK) ∆K

The factor dE/dk is calculated by using the E versus |k| relationship known for free electrons as

E = ( ђ² k )/ 2m

Or, dE/dK  = ( ђ² K) / m

                   = ( ђ²/ m)(P/ђ)      [ since, K = P/ђ  ]

                   = ( ђ²/ m)(mV_F/ђ)    [ since, P = mV_F  ]

                   = ђV_F

Where, P = mass velocity     

            V_F = electron velocity at fermi level

            Ђ = plank constant

Now we also know (Newton’s Law)  that  

         F = m (dV/dt)

Or, eE = m (dV/dt)  [ applied force]

Or, d(mV)/dt    = eE

Or, dP / dt  =  eE

Or, ђ ( dK/dt ) = eE

Or, dK = (eE / ђ )  dt

Or, ∆K = (eE / ђ )  ∆t

So,  ∆K = (eE / ђ )  τ

Where, τ = relaxation time = ∆t

Putting all the values in (i) we get-

J = V_F² e² N(E_F) E τ    

One more consideration needs to be made because all the displaced electrons mayn’t move towards the applied electric field. If the electric field vector points in the negative V_KX direction, then only the components of those velocities that are parallel to the positive V_KX direction contribute to the electric current. The V_KY components cancel each other pairwise.

In other words only the projections of the velocities V_F  on the positive V_KX  axis ( V_FX = V_F cosϴ ) contribute to the current.    

Therefore,

This is a two dimensional view consideration. But real world is three dimensional. A similar calculation for spherical Fermi surface yields as

J = (1/3) e² N(E_F) E τ V_F²

Then the conductivity, σ = J/E

                                    = (1/3) e² N(E_F)  τ V_F²

Therefore according to quantum mechanical consideration, conductivity depends on population density ( other parameters may be fixed ) . The higher the population densities greater the conductivity.

Monovalent metals have partially filled valance bands. Their electron population densities near the Fermi energy are high which results in a large conductivity. Bivalent metals electron population densities near Fermi energy is small which leads to a comparatively low conductivity.