Vegard predicted to calculate theoretically the optical energy gap of crystalline systems for the III-V systems. Does this law valid to calculate the optical energy gap of glassy chalcogenides (II-VI) alloys?
In many binary semiconducting systems, the band gap in semiconductors is approximately a linear function of the lattice parameter.
Therefore, if the lattice parameter of a semiconducting system follows Vegard's law, one can also write a linear relationship between the band gap and composition.
For example using InPxAs(1-x), the band gap energy can be written as:
Eg = xEg(In-P) + (1-x)Eg(In-As)
Sometimes, the linear interpolation between the band gap energies is not accurate enough, and a second term to account for the curvature of the band gap energies as a function of composition is added.
This curvature correction is characterized by the bowing parameter, b:
Eg = xEg(In-P) + (1-x)Eg(In-As) - bx(1-x)
You need be careful with this. Vegard's law is just an observation that only works in some III-V and II-VI semiconductor alloys where linear behavior has been observed, but it is not at all valid generally. The GaN - GaAs case is a good example. The effects of alloying on the band gap are strongly dependent on the interactions of the host lattice and the alloying elements. Some details about these things are found in the linked paper.
Article Origin of band gap bowing in dilute GaAs1−xNx and GaP1−xNx a...
As already mentioned above, Vegards rule is actually formulated for lattice parameter variation. But as far as I know it is only a rule but no law! This already explains that the deviation from Vegards rule is more the rule than the exception :-). You can only use this in mathematics called linear inter- or extrapolation if this has been proven somewhere for exactly this system you are interested in. You cannot use it if you do not have any arguments. It is as with the mentioned interpolation: if you have a number of points you can use a line for their approximation, but commonly the line as description is far from beeing good. More complicated functions are typically better which you already mentioned in your own answer :-). Conclusion: Vegards rule is no mystic. He only analyzed the lattice parameters for mixed crystals and found an acceptable linear correlation as long as the structure type keeps unchanged and bonding relationships do not change remarkably.
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