Q.Temperature dependence of resistivity $\rho(T)$ of semiconductors, insulators and metals is significantly based on the following factors:

Concept: Temperature Dependence of Resistance/Resistivity

Resistivity is $\rho = \dfrac{1}{ne\mu} = \dfrac{m}{ne^2\tau}$, where $n$ = number density of charge carriers and $\tau$ = mean time between collisions (relaxation time). Temperature affects $\rho$ through exactly these two quantities:

• (a) True. In semiconductors/insulators, $n$ grows sharply with $T$ as more electrons are thermally excited across the band gap ($n \propto e^{-E_g/2k_BT}$) — the dominant T-dependence there.
• (b) True. In metals, $n$ is essentially fixed, but $\tau$ (time between successive collisions with vibrating lattice ions) shortens as $T$ rises because increased lattice vibration means more frequent scattering — the dominant T-dependence there. …

Starting point: what $\rho$ actually depends on

From the microscopic (Drude) model of conduction,

where $n$ is the number density of charge carriers, $e$ the carrier charge, $m$ the carrier mass, and $\tau$ the average time between two successive collisions (relaxation time). Since $e$ and $m$ are fixed constants of the carrier, any temperature dependence of $\rho$ must come through $n$ and/or $\tau$.

(a) Number of charge carriers, $n$ — genuinely $T$-dependent

In a metal, essentially all the free electrons are already in the conduction band, so $n$ barely changes with $T$. But in a semiconductor or insulator, the valence band is full and the conduction band is empty at $T=0$; carriers appear only when thermal energy excites an electron across the band gap $E_g$:

As $T$ rises, $n$ increases sharply (exponentially), which is exactly why semiconductor/insulator resistivity falls so fast with temperature. This makes (a) a real, significant temperature-dependence mechanism.

(b) Time between successive collisions, $\tau$ — genuinely $T$-dependent

In a metal, $n$ is fixed, but the lattice ions vibrate more vigorously as $T$ increases, scattering the drifting electrons more often. This shortens the mean free time $\tau$ between collisions (roughly $\tau \propto 1/T$ at ordinary temperatures), which is exactly why metallic resistivity rises with temperature. This makes (b) the dominant mechanism for metals.

(c) Length of material — not a real mechanism …

Method: Testing "What Does X Depend On?" Statement Questions

This method applies to any question that lists several candidate factors and asks which ones genuinely affect a physical quantity (like $\rho$, $R$, or any derived quantity).

Steps

Step 1: Write down the defining formula for the quantity in question.

Before judging any listed factor, write out the microscopic (or macroscopic) formula that DEFINES the quantity. For resistivity, that's the Drude relation:

Only variables that appear in this formula can possibly cause the quantity to change.

Step 2: Classify every symbol in the formula as "fixed constant" or "genuinely variable."

$e$ (carrier charge) and $m$ (carrier mass) are fundamental constants — they never change with $T$, sample size, or any external condition. $n$ (carrier density) and $\tau$ (relaxation time) are the only two quantities that can physically change with temperature.

Step 3: Match each candidate statement in the question against this classification. …