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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.