Q.The conductivity of a semiconductor increases with increase in temperature because

The key idea is that in a semiconductor, thermal energy excites electrons from the valence band to the conduction band, creating electron-hole pairs. This increases the number density of free charge carriers.

Reasoning:

1. As temperature rises, more covalent bonds break, generating more free electrons and holes. The number density of carriers ($n$) increases exponentially.
2. Relaxation time ($\tau$) — the average time between collisions — actually decreases with temperature because lattice vibrations (phonons) scatter carriers more strongly.
3. The increase in $n$ dominates over the decrease in $\tau$, so conductivity $\sigma = n e \mu$ (where $\mu \propto \tau$) still rises overall.

Why This Question Tests a Key Insight

Many students memorise that "conductivity increases with temperature for semiconductors" but miss the why. The trap is thinking that both factors — number of carriers and relaxation time — move in the same direction. They don't. Let's unpack the physics.

In an N-type semiconductor at room temperature, most conduction comes from donor electrons (from pentavalent impurities like phosphorus). But as temperature rises, something more dramatic happens: thermal energy becomes large enough to break covalent bonds in the silicon or germanium lattice itself. Each broken bond creates an electron-hole pair. This is called intrinsic carrier generation, and it floods the material with both electrons and holes.

So the number density of free carriers ($n$) rises sharply with temperature. That alone would increase conductivity.

But what about relaxation time ($\tau$)? That's the average time an electron travels between collisions. As temperature increases, lattice atoms vibrate more vigorously (phonon scattering increases). Electrons bump into these vibrations more often, so $\tau$ decreases.

The conductivity formula ties both together:

Here $\mu$ is mobility, $e$ is electron charge, $m^*$ is effective mass. So $\sigma \propto n \tau$. If $n$ increases and $\tau$ decreases, which one wins?

Step-by-Step Reasoning

1. At low to moderate temperatures, the donor electrons are already ionised, so $n$ is roughly constant (equal to donor concentration $N_D$). But $\tau$ decreases with temperature because lattice scattering intensifies. So conductivity decreases slightly with temperature in this range — a fact many miss.

2. At higher temperatures (typically above ~400 K for silicon), intrinsic carrier generation kicks in. The number density $n$ grows exponentially with temperature, roughly as $n \propto T^{3/2} e^{-E_g/(2kT)}$, where $E_g$ is the band gap energy. This exponential rise is enormous.

3. Relaxation time behaves as $\tau \propto T^{-3/2}$ for lattice scattering (the dominant mechanism at high temperatures). So $\tau$ decreases as a power law, not exponentially.

4. Compare the rates: The exponential increase in $n$ completely overwhelms the power-law decrease in $\tau$. For example, raising temperature from 300 K to 400 K might increase $n$ by a factor of 1000, while $\tau$ drops by only a factor of about 1.5. The product $n\tau$ — and hence $\sigma$ — increases strongly.

5. Option analysis:
   • (A) "Number density of free current carriers increases" — true, but incomplete. It ignores the relaxation time change.
   • (B) "Relaxation time increases" — false. Relaxation time decreases.
   • (C) "Both increase" — false for the same reason.
   • (D) "Number density increases, relaxation time decreases but effect of decrease in relaxation time is much less than increase in number density" — this is exactly what happens.

Method: Weighing Two Opposing Temperature Effects

Conductivity depends on two separate things: how MANY free carriers there are, and how EASILY they move between collisions. Temperature affects both, but in opposite directions -- so the question is really "which effect wins?"

Step 1 -- What happens to the number of carriers as T rises.

Thermal energy breaks covalent bonds in the lattice, freeing more electron-hole pairs. This effect is strong: raising the temperature noticeably generates a large number of new carriers, so the carrier density $n$ rises sharply.

Step 2 -- What happens to how easily carriers move as T rises.

As the lattice heats up, atoms vibrate more, so a moving carrier collides with the lattice more often. The average time between collisions (relaxation time) therefore falls as T rises -- carriers are scattered more frequently, not less.

Step 3 -- Compare which effect dominates.

The rise in carrier density with temperature is a much steeper effect than the fall in relaxation time. So even though carriers are individually scattered a bit more often, there are so many more of them that the net conductivity still goes up.

Step 4 -- Match to the options.

• (A) only mentions the carrier-density rise, missing that relaxation time also changes -- incomplete.
• (B) and (C) both claim relaxation time increases -- this is backwards; it decreases.
• (D) states both effects correctly and says the density rise dominates -- this matches the physics.

Final answer: Option (D) -- carrier density increases, relaxation time decreases, but the density increase dominates, so conductivity rises with temperature.