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So I was doing some problems and this came up : -

Two resistors $A$ and $B$ having resistances $R_1$ and $R_2$ respectively at $0^{\circ}C$ and temperature coefficients $α_1$ and $α_2$ are joined in series. If this combination gives the same resistance at all temperatures, then one possible correct choice is

(a) $A$ is a metal, $B$ is a semiconductor

(b) $A$ and $B$ both are metals

(C) $A$ and $B$ both are semiconductors.

Can anyone explain me the difference between electrical resistances of semiconductors and metals and their dependance on temperature.

Get as detailed as you can. And also any hint on the problem would be awesome.

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Key facts:

(1) Semiconductors have so called a band gap $E_{\Delta}$ with the fermi level sitting somewhere between the valence and conducting band. A semiconductor conducts via hole/electron excitations that jump to the conducting band. The number of free electrons/holes in the conducting band is suppressed by the boltzmann factor $e^{-\Delta E \over kT}$ where $\Delta E = E_{conducting}- \epsilon_{F}$, the difference between conducting and fermi energies.

(2) Metals have their fermi level sitting right in the middle of a conducting band. Any electron sitting on that fermi surface can conduct current. A metal would be in effect have no resistivity unless there's scattering events impeding the flow. A simple semiclassical model is for example shown in Griffiths Electrodynamics, where resistivity is derived for the electron mean free scattering time (in practice, there's a whole story about electron scattering with phonons also, and the distribution of excited phonons as a function of temperature). At higher temperature, there are more phonons to scatter due to equipartition, and therefore the resistivity increases.

From those 2 key facts we can infer that semiconductors conduct better at higher temperature, while metals conduct worse at higher temperature. And the rest is fairly trivial once you know these facts.

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Since, "in general", in semiconductors the resistance decreases as the temperature increases(due to gap becoming smaller). And in the conductors, the resistance increases as the temperature increases. So, as can be seen that both the materials have almost opposite nature with temperature dependence. So, I should say that one should be metal and other would be semiconductor so that the net resistance retains a value.

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  • $\begingroup$ Changes to the gap with temperature are negligible compared with changes in electron/hole concentration changes with temperature. $\endgroup$ – Jon Custer Jun 15 '17 at 12:30
  • $\begingroup$ i can't understand what you say, will you clarify it! $\endgroup$ – Vaibhav Dixit Jun 18 '17 at 3:14

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