# Effective-mass approximation

The basic optical properties of ISB transitions can be derived using the envelope function formalism [Bast 81, Whit 81, Bast 82]. This formalism is less general than *ab initio* methods, and its validity is limited to the vicinity of high-symmetry points of the Brillouin zone. However, due to the flexibility and the relative computational simplicity of this model it is widely used to describe the quantum confinement in semiconductor heterostructures. In the envelope- function approximation, the wavefunction is developed over the basis formed by the periodic components of the Bloch functions of the bulk material, and then only a small number of terms corresponding to the nearest energy bands is retained. In this way, the wavefunction component varying slowly on the crystal-cell scale-the envelope function-is separated from the rapidly varying component:

where wy_{k}=_{0}(r) is the periodic part of the Bloch function at the center of an energy band *i**,* and *фг**(г)* is the corresponding envelope function.

To describe the ISB transitions, the sum can be often limited to only one energy band—the so-called effective-mass approximation [BenD 66].

The envelope function satisfies the Schrodinger equation:

where *U _{H}*(г) is the heterostructure potential and m*(t) is the inverse effective mass tensor which can be diagonalized:

For the QW case, the translational invariance in the *(**x, y**)* layer plane allows factorization of the envelope function into plane waves in the *x* and *y* direction and z-dependent envelope function ф_{;} (г) = е^{гк}и ^{г}иф_{г} (z), where k||and r||are the wavevector and the coordinate in the layer plane, and *S* is the considered crystal surface. The function T(z) satisfies the one-dimensional Schrodinger equation:

with continuity of the T(z) and —1rjy^(z) at the QW interfaces [BenD 66].