What is the cause of formation of indirect band gap in semiconductors?

Question

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, F. Bloch 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 Schrodinger'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". 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?

Answer 1

I hesitate to recommend anyone to abandon a line of inquiry motivated essentially by curiosity. But in this case, well, it might be for the best. You see, there is no generally applicable intuition for this kind of question. The reason is that condensed matter systems are really complicated.

Take Si. Its solid has only one species of atom, but the crystal has 2 atoms per unit cell, and each atom has 14 electrons. That alone accounts for 28 bands, purely based on degrees of freedom. Now, there will be plenty of degeneracy, but we haven’t yet added in coupling. So the question of where the bands actually line up and why they are what they are usually can’t be answered specifically without a numerical calculation. Intuitive questions are generally reduced to symmetry arguments in real (non-simplified) systems. But for Si, the conduction band minimum does not even fall at one of the high-symmetry points. In a particular case such as this, you might find a nice underlying reason for an indirect gap, but it just won’t be broadly applicable.

Another illustrative point may be the comparison between Si and Ge. They are in the same column of the periodic table, they both have the diamond crystal structure, and both are indirect-gap. Yet Ge has a conduction band (local) minimum at the central $\Gamma$ point, while Si does not. The fact that Ge is indirect hinges on the minimum at the $L$ point being slightly lower (by ~0.14 eV) than that at the $\Gamma$ point. And why is $L$ lower than $\Gamma$? I don’t think there’s an intuitive answer; it just works out that way.

Bottom line is, materials are complicated, indirect happens. My advice: Just accept it and move on.

Answer 2

I don't think we should be so quick to say that there's no way to know without doing a calculation. In some cases, there are rules of thumb: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.98.245203

The real question is whether it's worth trying to develop intuition. For example, in German, there are roughly 10 different ways to pluralize nouns. I learned German as a foreign language, and when I did, they didn't teach any rules about how to pluralize nouns. All of my teachers said to just memorize the plurals. My Dad also studied German as a foreign language --- many years before I was born. At some point, I looked in my Dad's (very old) German textbook, and I was very surprised to learn that they taught rules of thumb for how to pluralize nouns. If rules of thumb existed, why didn't they teach them to me? I think the problem is that the rules were complicated, and there were enough exceptions to the rules, that many teachers decided it wasn't worth the trouble to teach the rules and then memorize the exceptions; students were better off just memorizing everything.

I'm guessing a similar situation exists for direct vs indirect band structure. I wouldn't be surprised that if you take into account things like the lattice structure, lattice constant, number of electrons, etc. you could figure out some rules of thumb. However, it would be complicated and exception riddled, so no one bothers. (Except for computers; machine learning is a hot new thing in materials discovery, and it is basically a way to develop complicated rules of thumb.)