k·p perturbation theory

In solid-state physics, the k·p perturbation theory is an approximated semi-empirical approach for calculating the band structure (particularly effective mass) and optical properties of crystalline solids. It is pronounced 'k dot p', and is also called the 'k·p method'. This theory has been applied specifically in the framework of the Luttinger–Kohn model (after Joaquin Mazdak Luttinger and Walter Kohn), and of the Kane model (after Evan O. Kane). According to quantum mechanics (in the single-electron approximation), the quasi-free electrons in any solid are characterized by wavefunctions which are eigenstates of the following stationary Schrödinger equation: where p is the quantum-mechanical momentum operator, V is the potential, and m is the vacuum mass of the electron. (This equation neglects the spin–orbit effect; see below.) In a crystalline solid, V is a periodic function, with the same periodicity as the crystal lattice. Bloch's theorem proves that the solutions to this differential equation can be written as follows: where k is a vector (called the wavevector), n is a discrete index (called the band index), and un,k is a function with the same periodicity as the crystal lattice. For any given n, the associated states are called a band. In each band, there will be a relation between the wavevector k and the energy of the state En,k, called the band dispersion. Calculating this dispersion is one of the primary applications of k·p perturbation theory. The periodic function un,k satisfies the following Schrödinger-type equation (simply, a direct expansion of the Schrödinger equation with a Bloch-type wave function): where the Hamiltonian is