Silicon Photonics: Fundamentals and Devices

4.7 Absorption and Gain

In Section 5, the transition rate from a valence band state to a conduction band state conserving momentum has been calculated and from the rate the expression for the absorption coefficient has been obtained. The same transition rate can be used to obtain the expression for Einstein's B₁₂ coefficient (see Eq. (4.27)), and from it the coefficients A and B₂₁.

Under usual circumstances, a light beam falling on a semiconductor sample is absorbed. However, if a condition for population inversion is created in the semiconductor, then instead of absorption, the electromagnetic radiation is amplified. In this section we shall establish the condition for population inversion in a semiconductor and then obtain the expression for the gain coefficient in terms of the B coefficient. It is straightforward to relate the gain coefficient with the absorption coefficient.

Calculation of the emission and absorption rates in a semiconductor must take into consideration the band picture as well as the Fermi occupational probabilities in the conduction and valence bands. Let us consider an energy level E₂ in the conduction band and an energy level E₁ in the valence band of a direct-gap semiconductor. Spontaneous emission from E₂ to E₁ can occur only when the upper state is occupied and the lower state is empty (i.e., occupied by a hole). The occupational probabilities for the electrons in the conduction and valence bands are expressed by Fermi–Dirac statistics and are given ...