~pure Si at 300 K has equal electron (n_e) and hole (n_h ) concentration of 2x 10¹⁶ per m³ .doping by indium increases n_h to 4 x 10²² per m³ .what will be n_e (per m³) in the doped silicon ?

To determine the electron concentration (n e) in the doped silicon after introducing indium, we can utilize the principle of charge neutrality, which states that the total positive charge must equal the total negative charge in a semiconductor. In pure silicon at 300 K, the electron concentration (n e) and hole concentration (n h) are equal, both being 2 x 10^16 per m³. When indium, a p-type dopant, is introduced, it increases the hole concentration to 4 x 10^22 per m³. Let's break down the calculations step by step.

Understanding Charge Neutrality

In a semiconductor, the relationship between electron and hole concentrations can be described by the equation:

• n e * n h = n i²

Here, n i is the intrinsic carrier concentration of silicon, which at 300 K is approximately 1.5 x 10^10 per m³. This relationship holds true because the product of the electron and hole concentrations equals the square of the intrinsic carrier concentration.

Calculating the Initial Product of n e and n h

Initially, we have:

• n e = n h = 2 x 10^16 per m³

Thus, the product is:

• n e * n h = (2 x 10^16) * (2 x 10^16) = 4 x 10^32 per m^6

Applying the Doping Effect

After doping with indium, the hole concentration increases to:

• n h = 4 x 10^22 per m³

Using the charge neutrality condition, we can express the new electron concentration (n e) as follows:

• n e * n h = n i²

Substituting the known values:

• n e * (4 x 10^22) = (1.5 x 10^10)²

Calculating n i² gives:

• (1.5 x 10^10)² = 2.25 x 10^20 per m^6

Finding the New Electron Concentration

Now, we can rearrange the equation to solve for n e:

• n e = (2.25 x 10^20) / (4 x 10^22)

Calculating this gives:

• n e = 5.625 x 10^-3 per m³

Final Result

Therefore, the electron concentration in the doped silicon after introducing indium is approximately:

• n e ≈ 5.625 x 10^-3 per m³

This result illustrates how doping significantly alters the charge carrier concentrations in semiconductors, shifting the balance towards holes in the case of p-type doping. If you have any further questions about this topic or related concepts, feel free to ask!