Question

For an indirect gap semiconductor such as silicon, the difference
in wave vector between the VBM and the CBM is on the order of ?k = ?∕a, where a is the lattice constant of the material. (a) Explain why this
is so. (b) Estimate ?k for Si, where the lattice constant is 0.543 nm. (c) For
an optical transition to take place, the photon needs to have at least the gap
energy (1.11 eV for Si) and it has to have a wave vector corresponding to ?k.
What is the actual size of the wave vector of 1.11 eV photons? For which type
of electromagnetic radiation does the modulus of the wave vector become
comparable to the reciprocal lattice distances in a solid?

Answer 1

a) We know that for an indirect semiconductor like Si, the photon can not be emitted directly as the transition between the Conduction band and Valence band can not take place directly but passes through an intermediate state by transferring the momentum through the crystal lattice,, having lattice spacing a. The process needs to be take place in such a manner that the energy and momentum conservation must hold . The study of Brillouin zone revealed that the wavevector of the electron in a crystal lattice is ranging from -pi/a to +pi/a. So when an electron recombines with a hole in the valence band, the difference in momentum transfers to the lattice leading to change in momentum delta k as

It met the condition of interference.

(b) Putting the value of a in above we get

(b) The energy E can be given as

Solving for k and treating ko as reference level.

The actual wave vector will be 5.42nm.

(c) The corresponding wavelength lambda becomes

It represents the X-ray module system