Thermal effects on conductivity: Visual walkthrough

Step 1 — What "current" even is: a single drifting speck

WHAT. Put a material in front of you and imagine one free electron inside it. With no push, it wanders randomly and goes nowhere on average.

WHY. Before we can talk about how much current flows, we must agree on what flows. The whole formula is just "how many carriers × how much charge each × how fast they slide."

PICTURE. In the figure below the red dot bounces around, but its average position (the red arrow) barely moves — random wandering carries no net current.

Figure — Thermal effects on conductivity

Step 2 — Turn on a field: the speck gains a drift

WHAT. Apply a field $E$ pointing left-to-right. Now every random bounce is nudged slightly downfield. The average of all those nudges is a slow, steady slide — the drift.

WHY. Current is not the fast random motion; it's this gentle net slide. We use Newton's law $force = mass \times acceleration$ because a field exerts a force $q E$ on a charge $q$ , and force is exactly what Newton's law converts into motion.

$a = force q E divided by mass m \frac{q E}{m}$

$q$ — the charge on one carrier (how strongly the field grabs it).
$E$ — the applied push.
$m$ — the carrier's mass (how sluggish it is to accelerate).
$a$ — the resulting acceleration downfield.

PICTURE. The random cloud from Step 1 now leans. The red arrow (the average) has grown and points along the field.

Step 3 — The collision clock $τ$ : why drift doesn't run away

WHAT. Multiply the acceleration by the time it acts. Over one free run of length $τ$ :

$v_{d} = a τ = acceleration \times time = velocity gained \frac{q E τ}{m}$

$τ$ — how long the field gets to push before a reset.
Bigger $τ$ ⇒ more time to speed up ⇒ bigger drift.

WHY. This is the entire heart of the temperature story. Everything about heat will act through $τ$ (for metals) — so we isolate it now.

PICTURE. A sawtooth: the drift climbs during each free run, then drops to zero at a collision (the red spike), then climbs again. The average height is $v_{d}$ .

Step 4 — Package it as mobility, then as conductivity

WHAT. Divide the drift by the field:

$μ = \frac{v _{d}}{E} = \frac{q τ}{m}$

Notice $E$ has cancelled — good, because $μ$ should describe the material, not how hard you push. Now count carriers. If there are $n$ carriers per unit volume, each of charge $q$ , drifting at $v_{d}$ , the current density (charge crossing per area per second) gives:

$σ = n q μ = \frac{n q ^{2} τ}{m}$

$n$ — carriers per unit volume (how crowded the hallway is).
$q^{2}$ — one $q$ from the force, one from the charge each carrier delivers.
$τ$ — the free-run time (mobility's home).
$m$ — carrier mass.

WHY. This is the fork in the road. Only two symbols, $n$ and $τ$ , care about temperature. Everything downstream is just asking which one moves and how.

PICTURE. The formula as a machine with two temperature dials, $n$ and $τ$ ; the others ( $q$ , $m$ ) are bolted shut.

Step 5 — Metal branch: $n$ frozen, $τ$ shrinks with heat

WHAT. With $n$ fixed, only $τ$ moves. Hotter lattice = atoms vibrate with bigger swings = bigger "targets" for the electron to hit = collisions come sooner = $τ$ falls. The scattering rate $1/ τ$ grows in step with the thermal energy, which is $\propto T$ :

$\frac{1}{τ} \propto T ⟹ τ \propto \frac{1}{T} .$

Feed that into resistivity $ρ = 1/ σ = m / (n q^{2} τ)$ :

$ρ = \frac{m}{n q ^{2} τ} \propto \frac{1}{τ} \propto T .$

$ρ$ up because $τ$ down.
$n$ , $q$ , $m$ all constant — so $ρ$ tracks $T$ linearly.

Near room temperature we write this straight line as:

$R (T) = R_{0} [1 + α (T - T_{0})]$

$R_{0}$ — resistance at the reference temperature $T_{0}$ .
$α$ — the temperature coefficient, the slope of the line (positive for metals).

WHY. Because the only moving part scales like $1/ T$ , the resistivity scales like $T$ — a line, not a curve, over normal temperatures.

PICTURE. Cold lattice = tiny wiggles, long clear runs (long $τ$ ). Hot lattice = violent wiggles, the red electron path is chopped into short pieces (short $τ$ ).

Step 6 — Semiconductor branch: the $n$ dial explodes

WHAT. Creating a carrier by lifting one electron over the gap also leaves a hole behind — two carriers born together, and the pair statistics put half the gap in the exponent:

$n \propto e^{- E_{g} / (2 k_{B} T)} .$

$E_{g}$ — the wall height (bigger wall ⇒ far fewer carriers).
$k_{B} T$ — the thermal energy budget (bigger $T$ ⇒ easier jump).
the factor $2$ — because one jump makes two carriers (electron + hole).

Put this into $σ = n q μ$ . Mobility $μ$ still sags slightly with heat, but it is a mild power-law drop, while $n$ explodes:

$σ \propto e^{- E_{g} / (2 k_{B} T)}$

As $T$ rises the exponent's magnitude shrinks toward zero, so $σ$ shoots up and resistance falls. The exponential crushes the small $μ$ decrease.

WHY. An exponential always eventually beats any power law. So even though both $n$ and $μ$ change, the fate of the semiconductor is decided entirely by the $n$ explosion.

PICTURE. A wall of height $E_{g}$ ; at low $T$ a trickle of red dots clear it, at high $T$ a flood. The count on the far side climbs like a rocket.

Step 7 — Edge & degenerate cases (never leave a gap)

$T \to 0$ in a metal. $τ$ would grow huge as vibrations die, so $ρ \to$ a small leftover from impurities and defects (which don't melt away). This flat floor is the residual resistivity — the line $R_{0} [1 + α (T - T_{0})]$ bends into a plateau, it does not reach zero (unless the metal becomes a superconductor, where $ρ$ snaps to exactly zero below a critical temperature — a different mechanism entirely).
$T \to 0$ in a semiconductor. The exponent $- E_{g} / (2 k_{B} T) \to - \infty$ , so $n \to 0$ : a cold intrinsic semiconductor is essentially an insulator. No thermal carriers, no current.
$E_{g} \to 0$ . The gap vanishes; carriers are always available regardless of temperature — the material behaves metal-like (a "zero-gap" conductor). The exponential flattens to $1$ .
Doped semiconductor ( $E_{g}$ effectively tiny for donors). With doping, carriers are supplied by impurity atoms at low temperature, so $n$ is already large and roughly constant — over a wide "extrinsic" range the material can even act metal-like ( $ρ$ rising with $T$ ) before the intrinsic exponential takes over at high $T$ . See Intrinsic vs Extrinsic Semiconductors.

PICTURE. Two curves on one temperature axis: the metal line flattening to a residual floor, the semiconductor curve diving from insulator toward high conductivity — with the crossover labelled.

The one-picture summary

Everything above is one equation, $σ = n q^{2} τ / m$ , with two temperature dials. Turn only $τ$ (metal): conductivity slowly falls. Turn $n$ exponentially (semiconductor): conductivity rockets. This final figure holds both stories side by side.

Recall Feynman: the whole walkthrough in plain words

Picture a single runner in a room (Step 1). With no wind he wanders and goes nowhere. Switch on a fan (the field) and he leans forward — he drifts (Step 2). But the room is full of shaking poles; every time he hits one he forgets which way he was leaning and starts over. How far he gets depends on the average time between smacks, $τ$ (Step 3). Count how crowded the room is ( $n$ ), how hard the fan pushes, and you get the whole flow: $σ = n q^{2} τ / m$ (Step 4). Now the punchline — heat. In a metal the crowd size $n$ is fixed, but heat makes the poles shake harder, so smacks come sooner, $τ$ shrinks, and flow drops: resistance rises with heat (Step 5). In a semiconductor most runners are locked behind a door of height $E_{g}$ ; heat unlocks exponentially more of them, the room floods, and flow soars even though the poles still shake: resistance falls with heat (Step 6). Check the corners: freeze a metal and you hit a floor from permanent scratches in the walls; freeze a semiconductor and the doors stay shut so nothing moves; dope it and the doors are pre-opened so it acts metal-like for a while (Step 7). One equation, two dials, two opposite fates.

Active Recall

Connections

Thermal effects on conductivity — the parent result this page derives.
Free Electron Model of Metals — where the fixed electron sea $n$ comes from.
Mobility and Drift Velocity — Steps 2–4 in full.
Band Theory & Energy Gaps — the wall $E_{g}$ of Step 6.
Intrinsic vs Extrinsic Semiconductors and Doping and Carrier Concentration — the edge cases of Step 7.
Superconductivity — the $ρ \to 0$ degenerate limit.