| The effective-mass approximation is shown to be invalid for calculation of subband splitting in semiconductor under an external field when the bulk band-structure has two or more minima in the direction of external field. It has to be replaced by the original effective hamiltonian method in which the characteristics of the whole band structure is retained. The effective hamiltonian formalism in the Bloch's representation is established. While the effective hamiltonian with a triangular external potential is approximated by the multi-effective-mass approximation, the saddle point approximation is applied to its boundary condition to find the quantization condition. The WKB formalism is generalized to effective hamiltonian with nonquadratic band structure, based on a newly developed method called spatio-expansion. It is then shown that the saddle point approximation is equivalent to the WKB approximation, their semi-classical characters are therefore the same. The WKB wave functions and the Bohr-Sommerfeld quantization rules for effective hamiltonians with two symmetric band valleys and three band valleys are obtained. Finally, the semi-classical form of space-time propagator for hamiltonian with non-quadratic kinetic energy and time-dependent potential energy, which is useful for the three-dimensional band structure and external potential with nonseparable variables, is obtained by using the saddle point approximation. |
