Читать книгу VCSEL Industry - Babu Dayal Padullaparthi - Страница 23

The relationship between the density of states and energy in semiconductors is shown in Figure 1.6. This is determined by the product of the parabolic density of states in the conduction band and the valence band and the Fermi‐Dirac distribution according to Pauli exclusion principle. In the case of bulk semiconductors, it has the shape shown in the Figure 1.6. E_c and E_v are Fermi levels in thermal equilibrium in the conduction band and valence band, respectively, and E_g is the bandgap energy. If the electrons and holes are excessive due to photoexcitation or current injection, the distribution changes. The electron and hole levels in each band are called quasi‐ or degenerate‐Fermi level and are indicated by E_fc and E_fv. The condition for the population inversion from the thermal equilibrium state (also called quasi equilibrium) is expressed by:

 Efc − Efv > Ec − Ev: negative temperature or amplifying (condition for gain).

When the light with angular frequency ω comes in the semiconductor, the following characteristics appear, where ħ is the reduced Planck’s constant: ħ = h/2π (h: Planck’s constant). Here, ħω is a quantized photon energy, which appears in Chapter 8.

Figure 1.6 Population distribution of electrons and holes in semiconductor well vs. energy.

Source:[18]. (After Masahiro Asada and Yasuharu Suematsu)

 ħω < Ec − Ev: transparent

 ħω > Ec − Ev: absorptive

As shown in Figure 1.6, in the excited state, the carrier distribution representing the gain is distributed in a mountain shape with respect to energy. The result of the calculation is also shown in Figure 1.7.

Figure 1.7 Optical gain of semiconductor vs. energy.

Source:[19]. (After Masahiro Asada and Yasuharu Suematsu)