# 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

σ = e [n |

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

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