# Does anyone know the difference and relation between $k\cdot p$ method and tight binding (TB) method?

Among the methods of calculating energy bands for crystals, first-principles method is the most accurate. Besides first principles, two commonly used modeling methods are the $k\cdot p$ method and tight binding (TB) method. They can both give a Hamiltonina matrix of wave vector $k$, i.e. $H(k)$.

I want to know the detaild difference and relation between $k\cdot p$ and TB method, especiall their relations. Does anyone know? Are there any books or literatures to cover it?

I know TB can be used to calculate the energy bands in the full Brillouin zone (BZ), while $k\cdot p$ generally used for neighbourhood of band edges. However, I know an article which uses $k\cdot p$ to calculate the bands in full BZ [Phys. Rev. 142, 530 (1966)]. Is $k\cdot p$ fully equivalent to TB method?

$k \cdot p$-method is based on the matrix version of the perturbation theory derived by Lowdin. It states that the energy spectrum at some point of the Brillouin zone is know, and the method allows to find the band structure in the vicinity of this point, where the additional term proportional to kp (product of the wave vector and momentum matrix element) appears. This additional term is treated as a small perturbation and the momentum matrix element are hidden into effective mass which is treated as a fitting parameter.
One may force the $k \cdot p$-method to work pretty far from the special points of the Brillouin zone, taking more bands to interact (to per tube each others) and, thus, building larger $k \cdot p$ matrices. In this case we have more matrix elements to fit with experimental data. Many of them may be equal due to symmetry constrains - the group theory may help to figure out which of them are such.