The concentration of intrinsic charge carriers of a semiconductor(A)…

The correct answer is Option (4) → (B) and (C) only

The intrinsic carrier concentration \( n_i \) is the number of electron–hole pairs created by thermal energy in the pure material. It depends only on the band gap and the temperature:

    \( \displaystyle n_i \propto T^{3/2}\,e^{-E_g/2kT} \)

(A) depends on doping. — Incorrect. \( n_i \) is a property of the pure material. Doping changes the electron and hole concentrations n and p, but their product stays \( n_i^{2} \).

(B) increases with the decrease in band gap of the semiconductor — Correct. A smaller \( E_g \) makes the exponential factor larger, so more bonds break; germanium has far more intrinsic carriers than silicon.

(C) increases with the increase in temperature — Correct. Both \( T^{3/2} \) and the exponential factor grow with temperature.

(D) decreases non-linearly with increase in temperature — Incorrect. It changes non-linearly (almost exponentially), but it increases with temperature.

NCERT (2026-27):
Classification of Metals, Conductors and Semiconductors – “… Here a finite but small band gap (E_g < 3 eV) exists. Because of the small band gap, at room temperature some electrons from valence band can acquire enough energy to cross the energy gap and enter the conduction band.”

Intrinsic Semiconductor – “… As the temperature increases, more thermal energy becomes available to these electrons and some of these electrons may break–away (becoming free electrons contributing to conduction).”

Intrinsic Semiconductor – “… In intrinsic semiconductors, the number of free electrons, n_e is equal to the number of holes, n_h. That is

    \( \displaystyle n_e = n_h = n_i \qquad (14.1) \)

where n_i is called intrinsic carrier concentration.”