Foundations — Compare band gaps - conductor - semiconductor - insulator

This page assumes you have seen none of the symbols in the parent note. We build each one from a picture, in an order where every new idea rests on the previous. By the end you will read $n_i\propto e^{-E_g/2kT}$ and know exactly what every letter means.

0 · What is "energy" for one electron here?

Before bands, before gaps, we need the most basic axis: energy.

We will draw every later idea against this one vertical energy axis. Look at the figure: it is nothing but that ruler, with a low mark and a high mark.

Why the topic needs it: the whole comparison of conductor / semiconductor / insulator is a statement about heights on this energy ruler. Without an energy axis there is nothing to compare.

1 · Allowed energies clump into bands

An electron in a single atom can only sit at specific energy levels — never in between. In a solid, trillions of atoms sit close together, and those sharp levels smear into thick bands of allowed energies, with forbidden zones in between.

Why the topic needs it: "conductor vs insulator" is decided by which bands are filled and how far apart they are. Bands are the stage on which everything happens. Deeper study of why levels smear into bands lives in Band Theory & Carrier Physics.

2 · The two bands that matter: valence and conduction

Of all the bands, only two decide conduction: the highest full one and the lowest empty one.

Why the topic needs it: $E_v$ and $E_c$ are the two heights whose difference is the band gap. No VB/CB, no gap.

3 · The band gap $E_g$ — the "jump height"

Now we can name the star of the topic.

Why the topic needs it: $E_g$ is the single number that separates conductor ($E_g\approx 0$), semiconductor (small $E_g$), and insulator (large $E_g$). Everything else is machinery for computing how many electrons clear a gap of height $E_g$.

4 · Temperature and the packet $kT$

Electrons don't climb the gap for free — they need energy, and heat supplies it. We need a way to say "how much energy does warmth hand out?"

Why the topic needs it: the question is always is the gap $E_g$ big or small compared to the thermal packet $kT$? Compare the tall gap stripe to the short $kT$ stick in the figures — if the gap dwarfs $kT$, almost no electron makes the jump.

5 · The exponential $e^x$ — why a small gap change means a huge carrier change

The parent note's headline is $n_i\propto e^{-E_g/2kT}$. That $e$ is doing violent work; we must earn it.

Why the topic needs it: without the exponential you'd expect a big gap to reduce conduction "a lot." With it, a big gap reduces conduction unimaginably, which is exactly what turns a material from semiconductor into insulator.

6 · The symbols you'll meet later (previewed, not yet derived)

These appear in the parent's derivation. You only need to recognize them now; each has its own deep-dive later.

You do not need to derive these here — this page just guarantees no symbol ambushes you. How these feed conductivity is Conductivity and Mobility; why metals behave oppositely with heat is Temperature Dependence of Resistance in Metals.

How these foundations feed the topic

Equipment checklist

Answer each before moving to the parent note.

Connections

• Band Theory & Carrier Physics
• Fermi–Dirac Distribution
• Density of States
• Doping and the Fermi Level
• Intrinsic vs Extrinsic Semiconductors
• Conductivity and Mobility
• Temperature Dependence of Resistance in Metals