Resistance and Temperature in conductors and semiconductors

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Homework Statement

Revered Members,
When we increase the temperature, resistance increases in conductors. But resistance decreases in semiconductors. Why?

Help in this regard will be highly appreciated.

Homework Equations

The Attempt at a Solution

Due to increase in temperature, covalent bonds gets broken and free electrons are generated in semiconductors.
In the case of conductors, when we increase the temperature free electrons collide with the lattice and for every collision they lose energy and resistance increases.
In semiconductors too, free electrons collide with the lattice, but their resistance, in contrary, decreases? Why?
Also, R = ml/ne²A\tau
where m is the mass of electron, l is the length of the conductor, n is the number of free electrons per unit volume and e is 1.602*10^-19 Coulomb, A is cross sectional area of conductor and \tau is relaxation time.
In conductors, \tau, decreases when the temperature increases, so Resistance increases, I presume. But why \tau decreases, when temperature increases.

 

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Due to increase in temperature, covalent bonds gets broken and free electrons are generated in semiconductors.
In the case of conductors, when we increase the temperature free electrons collide with the lattice and for every collision they lose energy and resistance increases.
In semiconductors too, free electrons collide with the lattice, but their resistance, in contrary, decreases? Why?

You certainly learned that metals have free conduction electrons. All atoms share its outer electrons with the other atoms.

A semiconductor atom is covalently bound to its neighbours and there are no conduction electrons at zero K in the crystal. But the bond is not strong, the bonding energy is comparable with K_BT . If the temperature increases, more and more electrons get loose from their parent atom and behave like the electrons of a metal. The number of free electrons grows about exponentially with the temperature.

Also, R = ml/ne²A\tau
where m is the mass of electron, l is the length of the conductor, n is the number of free electrons per unit volume and e is 1.602*10^-19 Coulomb, A is cross sectional area of conductor and \tau is relaxation time.
In conductors, \tau, decreases when the temperature increases, so Resistance increases, I presume. But why \tau decreases, when temperature increases.

Tau is related to the mean time between two subsequent collision of the electrons. During this collision, the electron loses its "drift" velocity, the velocity gained from the electric field. So the collisions act against the increase of velocity, like a viscous force.
When the temperature increases the electrons move with higher speeds, the ions vibrate with greater amplitude, so the probability for the electron to hit an ion gets higher. The collisions become more frequent, tau decreases.
τ decreases with temperature both in metals and semiconductors, but the number of free electrons in unit volume is independent of temperature in the metals and highly depends on the temperature in semiconductors. The resistance decreases with increasing n and this overcomes the increase of resistance because of decreasing tau.

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