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?