The problem asks for the analysis of carrier concentrations in an n-type silicon (n-Si) semiconductor when electron-hole pairs are generated due to light with photon energy greater than the energy bandgap (hν > Eg). The donor concentration is given as $10^{16} \, \text{cm}^{-3}$, and electron-hole pairs are generated at a density of $10^{12} \, \text{cm}^{-3}$.

Let’s break down each sub-part of the question:

b1) Plot the variation of majority carrier concentration with 'x' from 0 to L

• Majority carriers in n-type silicon are electrons. The donor concentration $N_D$ is $10^{16} \, \text{cm}^{-3}$. Before illumination, the majority carrier concentration (electrons) is approximately $n_0 \approx N_D$.

• When electron-hole pairs are generated ($10^{12} \, \text{cm}^{-3}$), this number is added to the initial electron concentration near the illuminated region. The concentration of electrons is highest near $x = 0$ and decreases towards $x = L$ because recombination reduces the excess carriers further into the material.

  The plot would start at $n(x = 0) = N_D + 10^{12} \, \text{cm}^{-3}$ and decrease towards $n(x = L) \approx N_D$.

b2) Plot the variation of minority carrier concentration with 'x' from 0 to L

• Minority carriers in n-type silicon are holes. Before illumination, the hole concentration is much smaller, $p_0 = \frac{n_i^2}{N_D}$, where $n_i$ is the intrinsic carrier concentration of silicon (around $1.5 \times 10^{10} \, \text{cm}^{-3}$).

• After the generation of electron-hole pairs, the hole concentration at $x = 0$ increases by $10^{12} \, \text{cm}^{-3}$, while further into the material, the hole concentration decays because of recombination.

  The plot would show a high hole concentration near $x = 0$ (starting at $p(x = 0) = p_0 + 10^{12} \, \text{cm}^{-3}$) and a decay towards $p(x = L) \approx p_0$.

b3) Which type of carrier will diffuse? Show the diffusion length, $L_p$, with corresponding value of the carrier concentration at $L_p$

• Holes (minority carriers) will diffuse because their concentration is significantly increased near $x = 0$ and they try to move towards regions with lower concentration (towards $x = L$).

• The diffusion length $L_p$ is a characteristic length over which the minority carriers (holes) will diffuse before they recombine. The diffusion length $L_p$ is given by:

  $L_p = \sqrt{D_p \tau_p}$

  where $D_p$ is the diffusion coefficient for holes and $\tau_p$ is the minority carrier (hole) lifetime.

• At $L_p$, the concentration of holes is reduced to $p(L_p) = p_0 + \Delta p \exp(-1)$, where $\Delta p = 10^{12} \, \text{cm}^{-3}$.

Let me know if you'd like any further details on the equations or plots!

Relative questions:

1. What factors influence the diffusion length of minority carriers in semiconductors?
2. How does the recombination rate affect the minority carrier concentration over distance?
3. How is the generation rate of electron-hole pairs related to the intensity of incident light?
4. How does temperature affect the intrinsic carrier concentration $n_i$ in semiconductors?
5. What are the differences between majority and minority carrier diffusion?

Tip: Understanding carrier dynamics in semiconductors is crucial for designing efficient devices like solar cells, transistors, and LEDs.