Why minibands are formed in superlattices?

Question

In a single, finite quantum well, there are energy levels defined by the eigenstates - the solutions of the Schroedinger's Equation. The corresponding wavefunctions leak to the barrier because of its finite height. If we bring two identical QWs together, the wavefunctions of corresponding states will mix and, as far as I understand, the energy levels will have to split in order to satisfy the exlusion principle. If we then add another QW, the levels will split again, and so on. A structure containing many adjacent QWs with thin barriers is called a superlattice and instead of discrete energy levels it has "minibands", where many levels are very close to each other.

What is the mechanism behind the creation of minibands?

I think I understand a similar situation in a homonuclear molecules: if two atoms of Hydrogen are brought together, their atomic orbitals mix in a process that is called LCAO-MO (Linear Combination of Atomic Orbitals into Molecular Orbitals), leaving a bonding and an antibonding $\sigma$ molecular orbital.

When we bring together many atoms and create a crystal, their valence orbitals also mix, giving rise to valence and conduction bands (altough here I find it somewhat more confusing than in a H-H case).

Should I therefore think of miniband creation as the "next step of mixing"? I.e. the already mixed orbitals (valence and conduction band) mix once again and create just another configuration? I will be grateful for an explanation.

Answer 1

This is just Bloch's theorem--- there is a parameter which is the mixing phase between adjacent wells, which is the lattice analog of momentum, and for a large translation invariant lattice of wells, this momentum is a conserved quantity, so you have an eigenstate of H for each value of this phase.

Define the translation operator T to be the operator that moves an infinite line of wells one step to the left. This is a symmetry, and it is a unitary operator, so the eigenvalue of T is a pure phase $e^{i\phi}$. As $\phi$ varies continuously, you parametrize the band. The theory is the same as the band states in a regular conductor.