Question

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Question :- Why conductivity Of Semi - Conductor Increase on increase the Temperature....
Derive this With Formula ????
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Answer 1

As the temperature of the semi-conductor is increased, the electrons in the valence band gain sufficient energy to escape from the confines of their atoms. As a result, in highertemperatures, a semi-conductor's valence electrons are free = conduction results, resistivity decreases.

Answer 2

Formula:
conductivity(T) = conductivity_0 * exp{ - Eg /(2kT)}

Eg = Energy gap between conduction band and valance orbit.
Eg (T) = Eg (0°K) - A T^2 /(T + B).
A and B are constants for a substance.
k = Boltzmann constant.

Thus if T increases Eg decreases.

Also more thermal energy with electrons will increase intrinsic carriers. Increase in number of collisions is not significant (relatively) .

From the formula we see that conductivity increases.

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Conductivity = q [ n * μ_n + p * μ _p ]
q= charge of electron.
n & p : densities of electrons and holes.
μ_n , μ_p = mobilities of electrons & holes.

If it's a doped semiconductor, either electrons or holes will be majority carriers. One of above two terms is too small (in comparison) . For a pure semiconductor there is one term of intrinsic carriers.

1. Low temperatures:

At low temperatures (<200°K), mobility is reduced as carriers collide with lattices. They are attracted to lattices due to low speed. Lattice scattering mobility is proportional to T^{-3/2}. Impurity (doping) scattering mobility is proportional to T^{3/2}. So impurity scattering dominates.
Here carriers come from ionization.

2. Temperatures 200°K to 400°K.

Extrinsic conductivity dominates. Dopants get ionized more as T increases. But the increase is small.

3. At high temperatures (>400°K).

Carrier concentration n_i is mainly intrinsic. By using Fermi distribution and Boltzmann distribution we get

n_i (T) =2 [ 2 pi k T/h^2]^1.5 × [ μ_n * μ_p ]^0.75 * exp{ - Eg /(2kT)}.

Substituting for mobilities we get:
n_i (T) = n_0 * T^1.5 * exp{ - Eg/(2kT)}.

Finally conductivity is derived as in the equation 1.

See the picture for a graph.