Last time we have discussed the conductivity of semiconductor. Let us today derive the expression the expression for conductivity of semiconductor.

In a Semi-Conductor of length (ℓ) and area of Cross-Section. Let ne and nh be the no. of electrons and holes with drift velocities Ve and Vh respectively.

So, the total current in the semi-conductor will be the sum of current due to electrons as well as holes,

i.e.        I = Ie + Ih                                                             (i)

As we know that

I = ne A Vd

\Ie = ne e A Ve

And Ih = nh e A Vh

I = ne e A Ve + nh e A Vh

I = eA[ne  Ve + nh vh]

                   I/A = e[ne  Ve + nh vh]                  (2)

As we know that

E = V/l      (in magnitude)

Also, R = ρl/A

Or ρ = RA/l

Where ρ is resistivity and R is resistance

Or E/ρ = (V/l)/ ρ

Or E/ρ = (V/l)l/ RA

E/ρ = (V/RA)

E/ρ = I/A                         (3)

Put (3) in (2) we get,

E/ρ = e[ne  Ve + nh vh]

I/ò ρ = e / E[ne  Ve + nh vh]

σ = e [ne  Ve/E + nh vh/E]                  (σ = Conductivity)

Here, mobility (m) = Drift vel./ Electric field

σ = e [ne  me + nh mh]

 

This is the derivation and expression for the conductivity of a semiconductor.