# Why does the Gallum-Arsenite bandgap narrow for higher temperatures and is this because of unavoidable impurities?

I found this sentence:

GaAs at 300K contains 10^15 acceptor atoms per cubic centimeter.

The bandgaps for Gallium Arsenite are:

0 Kelvin : 1.518 eV 300 Kelvin: 1.424 eV

Now I am wondering:

1. Why does Gallium Arsenite have this impurity?

2. Is the bandgap at 300 Kelvin is stated for a pure GaAs semiconductor or does it already incorporate the influence of the impurity?

3. If it doesn't what else does then cause the narrowing of the bandgap?

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2. Yes, impurities do have an effect on the narrowing of the bandgap. But there magnitude is significant only for very high concentrations of impurities. Now, from Eq. (15) of http://jap.aip.org/resource/1/japiau/v47/i2/p631_s1 you can see that the expression for the bandgap as a function of doping for p-GaAs (at 300 K) is given by $$E_g (eV) = 1.424 - 1.6\times10^{-8}p^{1/3}\equiv E_{g0} - \Delta E_g(p)$$ where $p$ is in $cm^{-3}$. If you use $p = 10^{15} cm^{-3}$ then you get $\Delta E_g=1.6 meV$. Yes, $\Delta E_g$ may in principle have a temperature dependence. But considering the order of magnitude corrections that impurities make to the bandgap, impurities alone are not able to explain a variation of close to a $100meV$ from 0 K to 300 K. Then a natural question to ask is: what are the most important things that change in a solid as you change temperature? The atoms start to vibrate more vigorously (increase in phonon energy) and the solid expands (increase in lattice constant).
3. In Eq. (45) of http://jap.aip.org/resource/1/japiau/v53/i10/pR123_s1 you can see the empirical expression: $$\epsilon_i(T) = 1.519 - \frac{5.405\times 10^{-4}T^2}{T+204}$$ where $T$ in obviously in Kelvins. It can be noted that $\epsilon_i(297K) = 1.424eV$. The above expression holds for high-purity materials. Or a better way to say this is that the impurity levels that are so low that they influence the energies by less than $1meV$ which are below our tolerance. By the way, don't consider the above formula is magically perfect. There are a couple of articles which follow the one cited above which come up with better and better microscopic models to explain the temperature dependence of the bandgap. But these are just quantitative differences in fitting. The qualitative behavior of the temperature dependence is still the same.