GATE 2023 EC Q.17: solution and answer

Official answer: ASource: Official answer key, GATE 2023 EC (IIT Kanpur)

1. An intrinsic semiconductor has no dopant atoms, so any electrons in the conduction band or holes in the valence band arise purely from thermal excitation of an electron across the bandgap $E_g$.

2. The probability that a state at energy $E$ is occupied by an electron is given by the Fermi-Dirac distribution: $f(E)=\frac{1}{1+e^{(E-E_F)/kT}}$, where $E_F$ is the Fermi level (near mid-gap for an intrinsic semiconductor).

3. As $T\to 0\text{K}$, $kT\to 0$. For any conduction-band state ($E>E_F$), the exponent $(E-E_F)/kT\to +\infty$, so $f(E)\to 0$ — every conduction-band state becomes empty. For any valence-band state ($E<E_F$), the exponent $\to -\infty$, so $f(E)\to 1$ — every valence-band state becomes fully occupied.

4. This matches the intrinsic carrier concentration relation $n_i=\sqrt{N_C{N}_V}{e}^{-E_g/(2kT)}$, which gives $n_i\to 0$ as $T\to 0\text{K}$ — no free electrons or holes exist at absolute zero.

5. Therefore, at $T=0\text{K}$: the valence band is completely filled with electrons and the conduction band is completely empty of electrons — option (A).

6. Option (B) describes an unphysical population-inverted state that never occurs in an ordinary semiconductor. Options (C) and (D) are physically meaningless: a band cannot be described as simultaneously filled with holes (absence of electrons) while also being called filled, and both bands cannot be entirely electron-filled without contradiction, since a hole is by definition an unoccupied electron state.