The temperature dependence of resistances of Cu and undoped Si in the temperature range 300-400 K, i

Q: Why doesn't the number of free electrons in Copper increase with temperature?

Metals already have a massive sea of free electrons (about one per atom) even at room temperature. The thermal energy added between 300 K and 400 K is negligible compared to the Fermi energy, so it doesn't significantly increase the number of free electrons. It only increases their collision rate with the lattice.

Q: Does the collision rate (relaxation time) change in Silicon too?

Yes, the lattice vibrations increase in Silicon as well, which decreases the relaxation time $\tau$. However, the exponential increase in the number of charge carriers $n$ completely overpowers the linear decrease in $\tau$, leading to a net exponential decrease in resistance.

Q: Is the increase in Copper's resistance always perfectly linear?

No. At very low temperatures (near absolute zero), the resistance of a metal follows a $T^5$ dependence (Bloch-Grüneisen law). However, near room temperature and above, the linear approximation $R_T = R_0(1 + \alpha \Delta T)$ is highly accurate.

Q: What does 'undoped' Silicon mean?

Undoped Silicon is an intrinsic semiconductor. This means it has no intentionally added impurities. Its electrical properties are purely determined by the thermal excitation of electrons from the valence band to the conduction band.

Q: Why is the specific temperature range 300-400 K mentioned?

This range (roughly 27°C to 127°C) ensures we are in a regime where the linear approximation for metals holds true, and for the semiconductor, intrinsic carrier generation is the dominant mechanism affecting resistance.

Question

The temperature dependence of resistances of Cu and undoped Si in the temperature range 300-400 K, is best described by

Options:

(A) linear increase for Cu, linear increase for Si

(B) linear increase for Cu, exponential increase for Si

(C) linear increase for Cu, exponential decrease for Si

(D) linear decrease for Cu, linear decrease for Si

Sigma Percentile · Accepted Answer

### The Tale of Two Materials

When we heat up a material, its electrical properties can change drastically depending on its fundamental nature. In this problem, we are asked to compare the temperature dependence of resistance for two very different materials: **Copper (Cu)**, a classic metal, and **Silicon (Si)**, a standard semiconductor, specifically in the temperature range of $300$ K to $400$ K.

### The Conductor's Plight: Copper

Imagine a crowded hallway where you (an electron) are trying to run from one end to the other. The people standing in the hallway represent the positive lattice ions. At lower temperatures, these people are relatively still, making it somewhat easy to navigate through them.

However, as the temperature increases, thermal energy causes these lattice ions to vibrate more vigorously. The people in the hallway start dancing around wildly! Naturally, you will bump into them more often. In physics terms, the **relaxation time** $	au$ (the average time between collisions) decreases.

Since resistance $R$ is inversely proportional to $	au$, the resistance of the metal increases. For a relatively small temperature range like $300$ K to $400$ K, this increase is well approximated by a linear relationship:

$R_T = R_0(1 + \alpha \Delta T)$

Thus, Copper exhibits a **linear increase** in resistance.

### The Semiconductor's Awakening: Silicon

Now, let's look at Silicon. At absolute zero, an intrinsic (undoped) semiconductor behaves like a perfect insulator because all its electrons are tightly bound in the valence band. There is an energy gap, $E_g$, that they must cross to reach the conduction band and become free to move.

As we increase the temperature to $300$ K and beyond, thermal energy acts like a key, unlocking electrons and promoting them across the bandgap. The number density of these free charge carriers, $n$, follows an exponential Boltzmann distribution:

$n \propto e^{-\frac{E_g}{2k_B T}}$

Because the number of available charge carriers explodes exponentially with temperature, the material becomes vastly more conductive. Even though the lattice ions are vibrating more (just like in Copper), this effect is completely overwhelmed by the sheer exponential influx of new charge carriers.

Therefore, the resistivity $ho$, and consequently the resistance $R$, drops exponentially:

$R \propto e^{\frac{E_g}{2k_B T}}$

Thus, Silicon exhibits an **exponential decrease** in resistance.

### The Final Verdict

Comparing our findings, Copper's resistance increases linearly, while Silicon's resistance decreases exponentially. This perfectly aligns with the physical models of metallic conduction and intrinsic semiconductor excitation.

Animated Solution for Physics - Current Electricity: The temperature dependence of resistances of Cu and undoped Si in the temperature range 300-400 K, is best described by

Select Answer:

Visualized Solution

Analyzing the Materials

Temperature Dependence of Conductors

Carrier Concentration in Semiconductors

Temperature Dependence of Semiconductors

Conclusion

The Sigma Insight: Ohm's Law, Resistance and Electrical Power

The Tale of Two Materials

The Conductor's Plight

The Semiconductor's Awakening

The Final Verdict

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