Silicon Photonics: Fundamentals and Devices

Appendix A: k.p Method

A.1 Band Theory: Bloch Functions

A solid consists of many atoms and electrons. The total energy of the system is therefore the sum of the kinetic energies of all the nuclei and electrons, the potential energy due to nuclear forces, the potential energies of electrons in the field of nuclei, the potential energy due to electron–electron interactions, and the magnetic energy associated with the spin and the orbit. The total Hamiltonian of the system may be constructed accordingly. The formidable problem of solving the resultant Schrödinger equation is bypassed by introducing several approximations. Since the motion of nuclei is sluggish, the electrons instantaneously adjust their motion to that of the ions. The total wavefunction is then written as a wavefunction for ions and that for all electrons instantaneously dependent on all ionic positions R. An approximation, known as the adiabatic approximation, is introduced to decouple the Schrödinger equation into a purely ionic and a purely electronic equation, which are expressed, respectively, as

(A.1)

and

(A.2)

where r denotes the electronic coordinates [1, 2].

The electron potential energy is due to electron–electron ...