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When temperature is increased in case of a semiconductor the free electron gets more energy to cross the energy gap to the conduction band from the valence band.so now more electrons can go easily to the conduction band so resistance decreases with temperature.But due to temperature increase,in conductors the atoms of the conductor vibrate which gives a resistance to the free electrons.But why does this happen?In conductor temperature makes atoms vibrate but in semiconductors the electrons vibrate?

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There are two things that affect the resistance. The number of mobile charge carriers (typically electrons or holes) and their mobility (mainly limited due to scattering from atoms).

When you increase the temperature both the electrons and the atoms will obtain more energy. This holds for both a regular metallic conductor and for a semiconductor.

Both materials will see an increase in scattering of electrons from the increasingly vibrating atoms.

The big difference between the the two materials is the band gap of the semiconductor. Electrons in the semiconductor that do not have sufficient energy cannot contribute to the conductivity. Therefore increasing the energy of the electrons, increases the number of electrons or holes (charge carriers) that can contribute to the conductivity, thereby drastically lowering the resistance. This effect does not occur in metals.

Because the latter effect is typically much bigger than the first, even though there is an increase of scattering at higher temperatures, the resistance of semiconductors decreases with increasing temperature.

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As the temperature rises more electrons have sufficient energy to cross the conduction band. Electrons in the conduction band act as negative charge carriers under the action of the electric field. And the vacancies left by the electrons in the the valence band act as positive charge carriers

Increase in the temperature increases the lattice vibrations which hinders the movement/flow of electrons (resistance increases). But the effect due increase in lattice vibrations is overpowered by the increase in number of charge carriers hence the resistance decreases with the increase in temperature.

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