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What is the relation between electronic band structure that is calculated by using free electron approximation (FEA) and tight binding approximation (TBA)?

My understanding:

  1. In FEA, it seems we start with a parabolic dispersion relation (the relation between the energy of an electronic state and the crystal momentum of the state). Then we adjust the shape of the curve near the boundaries of Brillouin zones because of the perturbation of the periodic potential. At last, we just "project" the resulting dispersion relation to those paths in the first Brillouin zone.

  2. In TBA, we don't calculate the general band structure directly, instead, we specifically calculate the bands formed by some atomic orbital. For example, for graphene, we can specifically calculate the energy band of $2P_z$ orbital of $C$ atoms.

My confusion:

  1. In FEA, for each calculated energy band, how do we know it is formed by which atomic orbital? On the other hand, when using TBA, how do we know that we have exhausted every possible combination of atomic orbitals?

  2. Is there a relation between the energy bands calculated by the two different methods?

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The relationship of free electron to tight binding is understood via the Kronig-Penney model. In the Kronig-Penney model a series of quantum wells (particle-in-a-box) are separated by somewhat low walls which allow tunneling between the wells. In the free electron model we start by ignoring the walls and just "folding back" the parabolic energy vs. wave vector relationship and then using the potential of the walls as a perturbation. In the tight binding model we look at solutions to the states in the individual quantum wells and then see how they are modified by interacting with their neighbors. So, from a free electron view point, the lowest state has $k=0$ would model an $s$ orital. The next state up is just a single wave cycle, so it looks like a $p$ orbital, etc... Within either model you never exhaust the possibilities, you just go higher and higher in energy with more nodes. When the height of the walls is low relative to the energy, the states look like free electrons and when the walls are high they look like the solutions of stand-alone quantum wells (and the band structure is relatively "flat"). For intermediate cases the solutions seamlessly transition betweeen the two models.

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