Silicon Photonics: Fundamentals and Devices

3.5 Si-Based Quantum Structures

Sections 3.2 through 3.4 presented the basic ideas of quantum confinement; simplified expressions for subband energies in QWs, QWRs, and QDs; and expressions for DOS functions in the structures by considering isotropic effective masses for electrons and holes.

In the present section the modifications to be introduced for materials having anisotropic mass will be presented, taking Si as the example.

3.5.1 Electron Subband Structure

Since Si and Ge have ellipsoidal conduction band valleys, the effective mass is expressed as a tensor. The kinetic energy operator for the Schrödinger equation is

(3.31)

where 's are the elements of the reciprocal effective mass tensor for the particular conduction band minimum being considered. Since the potential energy V(z) is a function of z only, we may seek a solution of the Schrödinger equation as 3

(3.32)

where 1, ... 3 denote the principal axes of the constant energy ellipsoid. Substituting this in the effective mass Schrödinger equations in which the kinetic energy operator is given by Eq. (3.31), one obtains

(3.33)

where .

Making now the substitution

(3.34)

to eliminate the first derivative with respect to z, we ...