A silicon crystal doped with indium has an electron concentration \( n_0 \) at room temperature, its hole concentration being much larger than \( n_i \). A second silicon crystal at the same temperature contains twice as many indium atoms per unit volume. If each indium atom gives one hole, the electron concentration in the second crystal is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( n_0/2 \)
In a semiconductor in thermal equilibrium the electron and hole concentrations obey the relation \( n_en_h=n_i^{2} \). At a fixed temperature \( n_i \) does not change, so the minority carrier concentration varies inversely as the majority carrier concentration.
At room temperature almost every indium atom is ionised and leaves a hole. So the hole concentration depends on the indium concentration. Twice the indium gives twice the hole concentration, \( 2n_h \).
For the first crystal:
\( \displaystyle n_0=\frac{n_i^{2}}{n_h} \)
For the second crystal:
\( \begin{aligned} n_e &= \frac{n_i^{2}}{2n_h} \\ &= \frac{n_0}{2} \end{aligned} \)
Therefore, the electron concentration in the second crystal is half that in the first.
NCERT (2026-27):
Extrinsic Semiconductor – “… At room temperature, most of the acceptor atoms get ionised leaving holes in the valence band. Thus at room temperature the density of holes in the valence band is predominantly due to impurity in the extrinsic semiconductor. The electron and hole concentration in a semiconductor in thermal equilibrium is given by”
\( n_en_h=n_i^{2} \)\( \qquad (14.5) \)