Step-by-step explanation:
To derive the average time between the emission of electron-hole pairs in a region of semiconductor, we can use the concept of carrier generation and recombination. In this scenario, carriers are generated by alternate emission of electrons and holes by centers, and we'll assume the emission cross-sections for electrons and holes are equal: σ_n = σ_p = σ.
The average time (τ) between such emission processes can be calculated using the following formula:
τ = 1 / (σ * v_th * n_i)
Where:
- σ is the emission cross-section (given as 2 x 10^(-16) cm^2).
- v_th is the thermal velocity of carriers, which can be calculated as sqrt(2 * k * T / m*), where k is the Boltzmann constant, T is the temperature in Kelvin (300 K in this case), and m* is the effective carrier mass.
- n_i is the intrinsic carrier concentration, which can be calculated as sqrt(N_c * N_v) * exp(-E_g / (2 * k * T)), where N_c and N_v are the effective density of states in the conduction and valence bands, E_g is the energy bandgap, k is the Boltzmann constant, and T is the temperature.
Let's calculate each of these components:
1. Calculate v_th:
m* (effective carrier mass) depends on the type of carriers (electrons or holes) and the semiconductor material. You'll need the specific values for your semiconductor material.
2. Calculate n_i:
N_c and N_v are the effective density of states, which also depend on the material. E_g is the energy bandgap of the semiconductor.
3. Finally, calculate τ using the formula provided.
Note that for an accurate calculation, you need specific material properties and the type of carriers (electrons or holes). The effective mass, bandgap, and density of states vary among different semiconductor materials.