# Why does the conductivity $\sigma$ decrease with the temperature $T$ in a semi-conductor?

We performed an undergrad experiment where we looked at the resistance $\rho$ and Hall constant $R_\text H$ of a doped InAs semiconductor with the van der Pauw method. Then we cooled it down to around 40 K and did temperature-dependent measurements up to around 270 K. We were asked to create the following three plots from our measurements and interpret them.

This is conductivity $\sigma = 1 / \rho$ versus the inverse temperature $T^{-1}$. I see that increasing the temperature (to the left) decreases the conductivity. I do understand that higher temperatures do that since the electrons (or holes) have more resistance due to phonon scattering. However, since higher temperatures mean a higher amount of free electrons, I would think that $\sigma$ should go up, not down.

The density of holes $p = 1/(e R_\text H)$ does increase with the temperature, that is what I would expect:

And the electron mobility $\mu = \sigma R_\text H$ decreases with the temperature as well:

Now, I am little surprised that even though $p$ goes up with $T$, $\mu$ and $\sigma$ go down with $T$. Are the effects of phonon scattering and other things that increase the resistance that strong?

• Maybe the "more free electron" part is somewhere near 20K? Maybe it is a single-crystal so the direction of flow is important? Maybe 4-point-method or whatever method you use for I/V characteristic measurement has some flaws such that contact points generate reverse-effect(s)? Dec 7, 2013 at 20:17
• I assume that we used the 4-point-method, it was called van der Pauw in the manual. I added it in the text. The semi-conductor was InAs. We did 8 measurements with various directions of current and voltage for the resistance and 10 for the Hall constant, with changed direction of the magnetic field and with no field at all. Dec 7, 2013 at 20:36
• Oh, and if the “more free electron” part is around 20 K, why does $p$ go up when $T$ goes over 100 K? Dec 7, 2013 at 20:37
• Yes, Im wrong about 20K part as it cannot be that low.Im sorry. What about irregularities between metal and semiconductor caused by doping? Dec 7, 2013 at 20:57
• I would understand that $p$ does not rise above a given value. However, it does not fall below a certain value. What do you mean with those irregularities? Dec 7, 2013 at 22:33

Phonon scattering goes up a lot as temperature increases -- faster than electron numbers increase in the conduction band.

Keep in mind that phonons obey the Bose-Einstein distribution, so their numbers scale like

$$N_{BE}=\frac{1}{e^{\frac{\hbar\omega}{k_b T}}-1}$$

In the large $T$ limit, this becomes

$$\frac{k_b T}{\hbar\omega}$$

So their numbers roughly scale linearly with temperature at "high temperature". For phonons, "high temperature" means above the Debye temperature, but that's only ~650K for silicon; you're a good chunk of the way there at room temperature.

However, electrons follow a Fermi-Dirac distribution, so you'd expect their numbers to scale like.

$$N_{FD}=\frac{1}{e^{\frac{\epsilon}{k_b T}}+1}$$

In the large T limit, this goes to $\frac{1}{2}$.

There's also a chemical potential for the electrons that limits their numbers. Phonons have no such restriction; given the energy, you can have as many phonons as you want.

Even if you're not talking about high temperatures, note that the $N_{BE}>N_{FD}$ is always true.