Animated Solution for Physics - Current Electricity: Read the following statements carefully $Y$ : The resistivity of semiconductor decreases with increase of temperature. $Z$ : In a conducting solid, the rate of collisions between free electrons and ions increase with increase of temperature. Select the correct statement (s) from the following

To truly understand the electrical behavior of materials, we must shrink down to the atomic level. The fundamental equation governing resistivity $\rho$ for any material is given by:

Here, $m$ is the mass of an electron, $e$ is the elementary charge, $n$ is the number density of free charge carriers, and $\tau$ is the relaxation time (the average time between collisions). The battle for conductivity is fought between $n$ and $\tau$.

In a semiconductor like silicon at absolute zero, all valence electrons are locked tightly in covalent bonds. The number density of free carriers $n$ is practically zero, making it an insulator.

However, as the temperature $T$ increases, thermal energy begins to agitate the lattice. This energy is sufficient to break some of the covalent bonds, freeing electrons and leaving behind positively charged 'holes'. The crucial point here is that the number of charge carriers $n$ increases exponentially with temperature.

While the increased thermal agitation also causes more collisions (decreasing $\tau$), the massive, exponential explosion in the number of carriers $n$ completely dominates the equation. Since $n$ is in the denominator, the overall resistivity $\rho$ drops dramatically.

Now, let's look at a metallic conductor like copper. Metals are characterized by a "sea" of free electrons. Even at room temperature, the number density $n$ is enormous (around $10^{29}{\text{ m}}^{-3}$) and remains practically constant regardless of normal temperature changes.

So, what happens when we heat a conductor? The thermal energy is absorbed by the massive positive ions forming the crystal lattice. These ions begin to vibrate with greater amplitude around their mean positions.

Imagine running down a hallway. If the people in the hallway are standing still, you can navigate easily. But if they start dancing wildly, you are going to bump into them much more often. Similarly, the vibrating ions create a larger effective cross-section for collisions. The free electrons crash into the ions more frequently, meaning the relaxation time $\tau$ decreases.

Looking back at our master equation, since $n$ is constant and $\tau$ decreases (and is in the denominator), the resistivity $\rho$ must increase.

We have established that both statements are factually correct. However, Statement Z describes the mechanism for why conductors increase in resistivity when heated. It has absolutely nothing to do with the bond-breaking mechanism that causes semiconductors to decrease in resistivity (Statement Y).

Because they describe two entirely different physical phenomena in two different classes of materials, Z is not the reason for Y.