Quick question about semiconductor alloys and band gaps?

[jeebs]

This seems like a straightforward thing but no textbook I've seen addresses it. Take the direct gap semiconductors GaAs (band gap 1.42 eV) and InAs (band gap 0.42 eV). If these two were alloyed together, would the band gap become some sort of average between the two values?

As an example, say we had Ga_0.4In_0.6As. First question, are the subscripts talking about relative mass densities or number densities of Ga and In in the alloy?

Secondly, do I get some sort of weighted average band gap for this alloy if I calculate

E_g = 0.4*1.42eV + 0.6*0.42eV = 0.82eV ?
According to this wikipedia graph

http://en.wikipedia.org/wiki/File:InGaAs_Energy_band_composition.PNG

it appears to be about 0.7eV for a Ga fraction of 0.4. Am I calculating this the wrong way or what?

 

[nickjer]

The percentages are concentration. That is why the sum of the concentrations is always 1.0. So in your case, 40% of the cations are Ga atoms and 60% of the cations are In.

A weighted average (like what you did) assumes a linear dependence of the band gap on concentration, which isn't necessarily true in the real world. It is good for a quick estimate of the band gap, but most semiconducting alloys follow a quadratic dependence. So experimentalists try to measure the quadratic coefficient called the bowing parameter. You usually can find the bowing parameter in tables.

Now, not all alloy band gaps follow a quadratic dependence on concentration. Sometimes it can be cubic or higher order. There is no exact formula, you just have to measure and find out.