Question

(a) please write down the resistivity as a function of temperature for a metal. please separate it into four regions such as resistivity is proportional to T, proportional to T^5,proportional to T^2 and proportional to T^0. Please also explain why is that ?
(b) what will happened if the doped impurities in the non-magnetic metal is magnetic ions?
(c) what is the resistivity as a function of temperature for an instrinsic semiconductor ?
(d) how about the resistivity change as a function of temperatures in an insulator?

Answer 1

(a) Write down the resistivity as a function of temperature for a metal.

_$\rho$ = _$\rho$0 + A (T / _$\Theta$R)ⁿ_{$\int_{0}^{\Theta R}$}{xⁿ / [(e^x - 1) (1 - e^-x)]} dx

where, _$\rho$0 = residual resistivity due to defect scattering

A = constant term

_$\Theta$R = debye temperature

n = integer value

(c) What is the resistivity as a function of temperature for an instrinsic semiconductor ?

$\Rightarrow$ For an intrinsic semiconductor, the resistivity generally decreases with increasing temperature. The electrons are bumped to the conduction energy band by thermal energy, where they flow freely and in doing so leave behind holes in the valence band which also flow freely. The electric resistance of a typical intrinsic semiconductor decreases exponentially with temperature :

_$\rho$ = _$\rho$0 e^-$\alpha$T

where, $\alpha$ = temperature coefficient of resistivity

_$\rho$0 = resistivity at a standard temperature

An approximation of the temperature dependence of resistivity of a semiconductor is given by the Steinhart-Hart equation :

1 / T = A + B ln (_$\rho$) + C [ln (_$\rho$)]³

where A, B and C are also known as "Steinhart-Hart coefficients".

Silicon is an example of semiconductor. The conductivity increases means the "resistivity decreases". When the temperature is increased in a semiconductor, then the density of charge carriers which also increases and the resistivity decreases.

Thus, temperature coefficient of resistivity of semiconductor is negative.

(d) How about the resistivity change as a function of temperatures in an insulator?

$\Rightarrow$ When the temperature is increased, then the atoms of a material vibrate and it makes the valence electrons present in the valence band to shift to the conduction band. When the conductivity of a material increases, it means that the resistivity decreases and so the current flow increases. For an insulator, they have a negative temperature coefficient.

Diamond is an example of insulator.