Electrical properties — conductors, semiconductors, insulators; doping (n-type, p-type): Visual walkthrough

This is the visual companion to the parent topic. If you have not met the idea of a band, read Band Theory of Solids first — but I will re-build the piece we need here.

Step 1 — What "energy of an electron" even means (the ladder picture)

WHAT. Picture a single electron sitting inside a solid. It is not allowed to have any energy it likes. It can only sit on certain allowed rungs of a ladder — these rungs are the energy levels.

WHY start here. Every symbol that follows ( $E_{C B}$ , $E_{V B}$ , $E_{g}$ ) is just "how high up this ladder." If you do not see the ladder, none of the letters mean anything.

PICTURE. Look at the ladder on the left of the figure. The vertical axis is energy (up = more energy). Each short horizontal mark is an allowed rung. Down low, the rungs are packed so tightly they merge into a solid grey block — that block is a band.

Figure — Electrical properties — conductors, semiconductors, insulators; doping (n-type, p-type)

Step 2 — The two bands that matter, and the empty gap between them

WHAT. Not the whole ladder matters — only two bands and the space between them:

the valence band (VB) — the top band that is full of electrons at absolute zero,
the conduction band (CB) — the next band up, empty at absolute zero,
the band gap — a stretch of ladder with no rungs at all. An electron may not sit there.

WHY these two. An electron can only move (carry current) if it has an empty rung right next to it to slide into. A completely full band offers no empty neighbour — so the full VB carries nothing. The empty CB is where the action is.

PICTURE. The green block is the full VB. The blue block above it is the empty CB. The white stripe between them is the forbidden gap — notice there are literally no rung-marks drawn inside it.

$E_{g}$ is measured in electron-volts (eV): one eV is the energy an electron gains crossing a 1-volt battery. For silicon $E_{g} \approx 1.1$ eV.

Step 3 — Heat kicks a few electrons across the gap

WHAT. At any temperature above absolute zero, the atoms jiggle. That jiggling energy occasionally slams into a valence electron hard enough to fling it up and over the forbidden gap into the empty conduction band.

WHY it matters. Now TWO good things happen at once:

There is an electron loose in the (mostly empty) CB — it has empty neighbours, so it can move.
It left behind an empty rung in the VB — a hole. Neighbouring VB electrons can shuffle into that hole, so the hole moves too, like a missing chair passed down a row.

PICTURE. The orange arrow shows one electron jumping the gap. Below it, the red circle marks the hole it left behind. Every jump makes a pair: one electron, one hole.

Step 4 — Why the tool here is the Boltzmann exponential (and not a straight line)

WHAT. We need to count how many electrons manage the jump at temperature $T$ . That number is $n$ .

WHY this tool. The jump costs energy $E_{g}$ . Nature does not give every electron enough — it hands out thermal energy at random, and the fraction of electrons lucky enough to have at least energy $E$ is governed by the Boltzmann factor $e^{- E / k_{B} T}$ . We reach for the exponential because the question is "what fraction of a random-energy crowd clears a fixed energy bar," and that is exactly what the Boltzmann factor answers. A straight line would wrongly say "twice the temperature, twice the jumpers" — but the truth is far more dramatic, because clearing a high bar is exponentially rare.

PICTURE. The curve shows the fraction of electrons with energy above the bar $E_{g}$ . Raise the temperature (orange curve vs blue) and the tail fattens — many more clear the bar. Notice the sharp bend: doubling $T$ can multiply the jumpers thousands-fold.

Step 5 — Turning "number of carriers" into "conductivity"

WHAT. More carriers should mean better conduction — but by how much? We need a bridge from count ( $n$ ) to conductivity ( $σ$ ).

WHY. Conductivity depends on three things: how many carriers there are, how much charge each carries, and how easily each drifts under a push. Multiply those three and you get $σ$ .

PICTURE. Three dials — count $n$ , charge $e$ , mobility $μ$ — feed one output dial $σ$ . Turn any input up, $σ$ goes up in proportion.

Step 6 — Substitute, and read the master equation

WHAT. Put the Step 4 count into the Step 5 formula. The constants ( $e$ , $μ$ , and the proportionality) all fold into one lump we name $σ_{0}$ .

WHY. We do not usually care about $e$ , $μ$ separately — we care how $σ$ changes with temperature. Bundling the temperature-independent junk into $σ_{0}$ leaves the one honest player, the exponential, standing alone.

PICTURE. The algebra collapses in one line; the constants slide into the box labelled $σ_{0}$ .

Step 7 — Making it a straight line (why we take the logarithm)

WHAT. Take the natural log of both sides.

WHY this tool. An exponential is hard to eyeball. The logarithm is the exact question "what power was the base raised to?" — it undoes the exponential and flattens the curve into a straight line. A straight line has a slope we can measure, and that slope hands us $E_{g}$ directly. We use $ln$ because it converts a multiplicative jump-law into an additive line we can fit with a ruler.

PICTURE. Left panel: the swooping $σ$ -vs- $T$ curve. Right panel: plot $ln σ$ against $1/ T$ and it becomes a clean downhill straight line.

Step 8 — The edge case: why metals do the OPPOSITE

WHAT. A metal has no gap — its VB and CB overlap ( $E_{g} = 0$ ). So the exponential factor is $e^{0} = 1$ : temperature does nothing to the carrier count.

WHY it flips. With carrier count fixed, the only free player is mobility $μ$ . Heating a metal makes the lattice jiggle harder, and those vibrations scatter the already-abundant electrons, so $μ$ falls. Fewer slippery rides ⇒ conductivity drops as $T$ rises.

PICTURE. Two curves on one temperature axis: the semiconductor (blue) rockets up with $T$ ; the metal (orange) sags gently down. Same axis, opposite slopes — the whole mystery from the top of the page, resolved.

Recall Cover the table and rebuild it

Metal band gap and carrier trend with heat? ::: $E_{g} = 0$ ; carriers fixed, mobility falls, so $σ$ decreases. Semiconductor band gap and carrier trend with heat? ::: $E_{g} \approx 0.1$ – $3$ eV; carriers rise exponentially, so $σ$ increases. Which factor dominates in each? ::: Metal → mobility $μ$ ; semiconductor → carrier count $n$ .

The one-picture summary

Everything above, on a single canvas: the ladder → the gap → the thermal jump → the Boltzmann count → $σ = n e μ$ → the master exponential → the straight-line log plot → the metal-vs-semiconductor flip.

Recall The whole walkthrough, told like Feynman

Electrons in a solid can only sit on certain energy rungs, and those rungs bunch into two blocks: a full lower one (valence) and an empty upper one (conduction), with a forbidden no-rung stripe between them of height $E_{g}$ . A full block can't carry current — no empty seat to slide into. Heat makes the atoms rattle, and once in a while a rattle is violent enough to fling an electron across the stripe. Now you've got a loose electron up top and an empty seat (a hole) down below — two things that can move. How many make the jump? Not a fixed few — a Boltzmann-exponential few, because clearing a fixed energy bar with random thermal energy is exponentially rare, so raising the temperature fattens the lucky tail dramatically. Each carrier's contribution to conduction is charge × how-slippery-it-is, so total conductivity is count × charge × mobility, $σ = n e μ$ . Bundle the boring constants into $σ_{0}$ and you're left with $σ = σ_{0} e^{- E_{g} /2 k_{B} T}$ — the 2 there because each jump makes a pair. Take a logarithm and this swoosh becomes a straight line whose slope literally is the band gap, so you can measure $E_{g}$ with a ruler. And a metal? Its blocks overlap, $E_{g} = 0$ , so heat can't add carriers; instead heat just shakes the lattice and trips up the electrons already there, so a metal conducts worse when hot. Same physics engine, opposite dial turned — mystery solved.

See also: p-n Junction Diodes · Thermistors (a device built directly on this exponential) · Hall Effect (how you tell electrons from holes) · Photovoltaic Cells · Transistors and Integrated Circuits · Crystal Defects · Covalent Network Solids · Metalic Bonding.