Tight binding model contradiction

Question

I have been studying recently the tight binding model and there is a point I cannot understand. First, it starts from the idea that the electrons belong to the atom more than to the crystal, so they are bound to their atoms but there is certain probability for the to hop to the adjacent atom , and the wave function takes the form of linear combination of atomic orbitals. However, after this the model states that the wave function must be Bloch like, which to me means that the electron must be spread along the crystal.

I cannot reconcile these two ideas. At the end, you get the bands as it is supposed to be, but the recipe to obtain the wave functions that later on you use to diagonalize the hamiltonian does not convince me. Any help?

Answer 1

In the tight binding model, the wavefunction of a given electron is a linear combination of "atomic wavefunctions" (which are localized at lattice sites). A single one of these wavefunction is indeed localized only at a particular site and does not extend to the whole crystal, but a linear combination of many such wavefunctions can satisfy Bloch theorem, for each atomic wavefunction is localized on a different lattice site, covering the whole crystal. See for instance this reference.

There is no contradiction.

This model implies that the electrons which are eigenstates of the Hamiltonian are spread entirely on the crystal (just like in the free and nearly free electron models), but that they are localized at lattice sites (unlike the free and nearly free electron models which have no such requirement).

Other references all pointing at the same explanation: Ashcroft and Mermin's expression 10.4 and its associated text, extending to the next page (179-180). Ziman's "Principle of the theory of solids" expression 3.24 page 91 and its associated text. Or again, Wikipedia's corresponding article.

Answer 2

Tight-binding model defines crystal states in terms of the orbitals of independent atoms. The confusion arises from the fact that, firstly, one talks about the states in the independent atoms, and then about the states in the crystal.

This confusion is more than just semantic, since the atomic states are modified when the atoms are brought together to form a crystal, whereas hopping integrals are quantities that are understandable intuitively, but impossible to define rigorously mathematically.

A good way to grasp it is to read about a hydrogen molecule, where two orbitals of equal energy $E_0$ are mixed via hopping of electrons with probability $\Delta$. The whole Hamiltonian is then $$ \hat{H}=\begin{bmatrix} E_0 & \Delta\\ \Delta &E_0 \end{bmatrix},$$ With the energies $E_0\pm\Delta$. Either of the corresponding states mixes two orbitals, which is the minimal equivalent of the Bloch functions.

Answer 3

The central idea of tight-binding method is to approximate Wannier states using atomic orbitals that are localized at atoms.