The intrinsic carrier concentration of silicon sample of 300 K is 1.5 ×1016/m3 . If after doping, the number of majority carriers is 5 × 1020/m3 , the minority carrier density is (a) 4.50 × 1011/m3 (b) 3.33 × 104 /m3 (c) 5.00 × 1020/m3 (d) 3.00 × 10-5/m3
The Correct Answer and Explanation is:
To solve this, we use the mass action law in semiconductors: ni2=n⋅pn_i^2 = n \cdot p
Where:
- nin_i is the intrinsic carrier concentration
- nn is the concentration of electrons (majority carriers in n-type)
- pp is the concentration of holes (minority carriers in n-type)
Given:
- ni=1.5×1016 m−3n_i = 1.5 \times 10^{16} \, \text{m}^{-3}
- n=5×1020 m−3n = 5 \times 10^{20} \, \text{m}^{-3} (majority carriers after doping)
We are to find pp, the minority carrier concentration.
Using the mass action law: p=ni2np = \frac{n_i^2}{n}
Substitute values: p=(1.5×1016)25×1020=2.25×10325×1020=0.45×1012=4.5×1011 m−3p = \frac{(1.5 \times 10^{16})^2}{5 \times 10^{20}} = \frac{2.25 \times 10^{32}}{5 \times 10^{20}} = 0.45 \times 10^{12} = 4.5 \times 10^{11} \, \text{m}^{-3}
✅ Correct Answer: (a) 4.50 × 10¹¹/m³
📘 Explanation:
In intrinsic (pure) silicon at 300 K, the concentration of electrons and holes is equal and given by the intrinsic carrier concentration nin_i. When a semiconductor is doped (with donor atoms to make it n-type), the majority carrier concentration (electrons) increases significantly.
However, the product of electron and hole concentrations must still satisfy the mass action law n⋅p=ni2n \cdot p = n_i^2. This means if one carrier type increases, the other must decrease to maintain this equilibrium.
Here, after doping, n=5×1020 m−3n = 5 \times 10^{20} \, \text{m}^{-3}, much larger than nin_i. So, the minority carrier concentration (holes in n-type) must decrease.
Using the mass action law, we find the hole concentration pp drops to 4.5×1011 m−34.5 \times 10^{11} \, \text{m}^{-3}, which is logical: heavily doped semiconductors have very few minority carriers.
This principle is critical in designing diodes and transistors, where control over minority and majority carrier populations determines electrical behavior.
