What is the cause of formation of indirect band gap in semiconductors?
Background: I am an electronics (device level) student who suddenly wished to understand semiconductor stuff from basics (at solid state level, without any background in that).
What I understood: From this video, <PERSON> said that in a crystal there is some periodicity of arrangement of atoms, hence a periodicity in potential. Thus, he formulated some equation that helped solve <PERSON>'s equation and an E-K (Energy-Wave vector) diagram was formed. And then suddenly direct and indirect band were introduced. Nevertheless, I got the difference between direct and indirect band gap. In indirect, the lowest state of conduction band is shifted to a permissible value of K vector. And to shift an electron from valence band to conduction band we would need momentum along with energy.
What I am having trouble in understanding: Why in some elements like Silicon, the conduction band is shifted? What makes it different from a direct band gap material? Equation wise the lowest energy level of conduction band (E) should be now some (K-a) instead of only K (where a is a constant, the shift). But what causes this?
What I have searched: The first answer in this link mentions "Indirect band gaps only happen when your perturbing coupling are strong enough that avoided level crossings at different points get mixed up". 346
I didn't get the statement because it talks about crossings that weren't there in the E-K band diagram in above linked video. What I think is that the answer means to say an effect like crystal field does something. But how it does so and why not in another material?
From this answer, again the presumption is that minimum is shifted due to some potential. But isn't that potential not existing in a direct band gap material. Why does it exist in Silicon? Is it something to do with nuclear charge?
What I am expecting: Some stuff happens in Silicon (or any indirect band gap material) that doesn't happen/or isn't strong enough in a direct band gap material. That reason causes a shift.
Edit: This link states that lesser is the lattice constant (interatomic distance) stronger is the binding between valence electron and nuclei, meaning more energy gap (harder to make an electron jump). However, for Silicon (indirect band gap), lattice constant is 5.4 angstrom while for GaAs (direct band gap), lattice constant is 5.65 angstrom. The difference is very less, but is is it enough to create 2 separate structures? 969