Derivation of expression for the conductivity of a Semi-Conductor

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 n_e and n_h be the no. of electrons and holes with drift velocities V_e and V_h 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 = I_e + I_h                                                             (i)

As we know that

I = n_e A V_d

\I_e = n_e e A V_e

And I_h = n_h e A V_h

I = n_e e A V_e + n_h e A V_h

I = eA[n_e  V_e + n_h v_h]

                   I/A = e[n_e  V_e + n_h v_h]                  (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[n_e  V_e + n_h v_h]

I/ò ρ = e / E[n_e  V_e + n_h v_h]

σ = e [n_e  V_e/E + n_h v_h/E]                  (σ = Conductivity)

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

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