What are the two main approaches used to determine the energy levels of electrons in a crystal?

The tight-binding method is based on the idea that the electronic wave functions in a crystal can be represented as a linear combination of atomic orbitals. This method begins by assuming that the electrons are tightly bound to the atoms and then considers the interactions between those orbitals when atoms are brought together to form a crystal. The tight-binding Hamiltonian can be represented as: \[H_{mn} = \int \phi_m^*(\vec{r})H \phi_n(\vec{r}) d^3r\] where \(H\) is the total single-electron Hamiltonian, and \(\phi_m(\vec{r})\) and \(\phi_n(\vec{r})\) are the atomic orbitals. Calculate the matrix elements of the Hamiltonian, and then find the energy eigenvalues and eigenvectors to determine the energy bands of the crystal.

The k·p perturbation method is an effective-mass approach for calculating electronic energy levels in a crystal. This method considers energy bands near the band edge, making use of the fact that electrons with wave vectors near the band edge have a long wavelength, and therefore, the crystal potential varies slowly on the scale of the electron wavelength. The k·p Hamiltonian can be expressed as: \[H_{n}(\vec{k}) = E_n(\vec{k}) + \sum_{m \neq n} \frac{\hbar^2}{2m_0} \langle u_{m,\vec{k}}|\nabla u_{n,\vec{k}}\rangle \cdot (\vec{k} - \vec{k}_n)\] where \(E_n(\vec{k})\) is the energy of the electron as a function of the wave vector \(\vec{k}\), \(u_{m,\vec{k}}\) and \(u_{n,\vec{k}}\) are the Bloch wave functions, and \(\vec{k}_n\) is the wave vector corresponding to the band edge. Calculate the matrix elements of the k·p Hamiltonian, then solve for the energy eigenvalues and eigenvectors to determine the energy bands of the crystal.