Bandgap Energy and refractive index

[Martel_lu]

The statement that the higher the bandgap energy Eg, the lower the refractive index of material is confusing me.

If I start from two equations.
Eq1: Eg=hc/lamda_g
Eq2: lamda=lamda_0/n

Higher Eg means lower lamda_g. It means in order to have the light in semiconductor material with wavelength of lamda_g, we have to increase n.
That comes out the conclusion that higher Eg, Higher n.

I don't know where is the logic wrong.

 

[espen-4]

Neither equation relates the band gap to the refractive index. So, the logic ypu are seeking cannot be found from these equations alone.
A quick google search resulted in finding this paper that shows how complicated this relationship is: https://web.njit.edu/~sirenko/PapersNJIT/Ravi_IPT_2007.pdf.