Temperature Dependence of Resistivity

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The resistivity of a metallic conductor nearly always increases with increasing temperature. As temperature increases, the ions of the conductor vibrate with greater amplitude, making it more likely that a moving electron will collide with an ion. This impedes the drift of electrons and hence the current. Over a small temperature range, the resistivity of a metal can be represented by a linear relation

where r_o is the resistivity at a reference temperature T_o and r (T) is the resistivity at temperature T. a is called the temperature coefficient of resistivity and has dimensions of (^oC)^-1.

However, the temperature dependence of r at low temperatures is non-linear as shown in figure given below.

Fig (a) - Resistivity r_T of copper as a function of temperature T

In metallic alloys, the resistivity is very large, but has a weak temperature dependence, as seen in below figure.

Fig(b) - Resistivity r_T

of nichrome as a function of absolute temperature T

Alloys have a residual resistivity even at absolute zero, but a pure metal has a vanishingly small resistivity. This can be used to check the purity of metals.

The resistivity of a semiconductor decreases rapidly with increasing temperature as shown in fig (c).

Fig (c) - Temperature dependence of resistivity for a typical semiconductor

This means that a is negative. The resistivity of an insulator too decreases exponentially with increase in temperature.

These observations may be understood qualitatively using the equation for r.

Since m and e are constants,

In metals, the number of free electrons, n does not change with temperature. But, as temperature increases, the atoms/ions vibrate with increasing amplitude. Therefore, the collisions of electrons with them become more frequent, resulting in a decrease in t. This means an increase in r with increase in temperature.

In both insulators and semiconducotors, t remains almost constant, but the number of free charge carriers increases with temperature. At any temperature T, the number of carriers is given by

n(T) = n₀ exp (-E_g/k_B T)

where E_g is the energy gap between the conduction and valence bonds. From this, we can get the temperature dependence of r to be

r(t) = r₀ exp (E_g/k_B T)

In semiconductors, E_g ~ 1 eV, \ r is not very high.

In insulators, E_g >> 1 eV; \ r is very high.

Also, this last equation shows that for semiconductors and insulators, resistivity increases with decreasing temperature.