Foundations — Band gap and its meaning for conductivity

This page is the toolbox. Before you touch the parent note the band-gap topic, you must be able to read every symbol it throws at you. We build each one from nothing: plain words → a picture → why the topic can't live without it.

0. The starting picture: what is "energy" for an electron?

Forget circuits for a second. Picture a single electron near an atom. It is like a marble resting on a shelf. A low shelf = low energy (tightly bound, close to the nucleus). A high shelf = high energy (loosely held, far out). To lift the marble to a higher shelf you must give it energy; when it falls back it releases energy.

Why the topic needs this: the whole idea of a "forbidden gap" is just the space between shelves, scaled up from one atom to a whole crystal. If you don't accept that electrons can only sit on allowed shelves, nothing else makes sense.

1. From single shelves to bands — the symbol $N$

Now bring $N$ atoms together into a solid. ($N$ just means "how many atoms" — for a real crystal it is astronomically large, around $10^{22}$ per cubic centimetre.)

When two atoms get close, their identical shelves cannot stay identical — nature refuses to let two electrons share the exact same state, so each shelf splits into two slightly-different shelves. Bring $N$ atoms together and each original shelf splits into $N$ shelves packed so tightly they look like a continuous smear. That smear is a band.

Why the topic needs this: the parent note's central object, $E_g$, is exactly the height of this leftover empty stripe. See Formation of energy bands from atomic orbitals for the full splitting story.

2. The two star bands: $E_v$, $E_c$, and the gap $E_g$

Of all the bands, only two matter for conduction: the highest band that electrons actually fill, and the empty one just above it.

3. The unit of the whole topic: the electron-volt (eV)

Energies of electrons are tiny in everyday units (joules), so we use a natural electron-sized unit.

Picture: roll our marble down a hill whose "steepness" is 1 volt; the energy it picks up is 1 eV. It's just a convenient ruler sized to electrons instead of to trucks.

Why the topic needs this: every gap in the parent note is quoted in eV — Si is $1.12$ eV, diamond $\approx 5.5$ eV. Without this ruler you cannot compare a gap to the thermal energy available (next section), and comparison is the entire game.

4. Temperature as an energy: $k_B$, $T$, and the combo $k_{B}T$

Heat is random jiggling. To ask "can heat kick an electron across the gap?" we must convert a temperature into an energy — otherwise we'd be comparing a gap (in eV) to a temperature (in kelvin), which is meaningless.

Why the topic needs this: the make-or-break comparison in the parent note is "is $E_g$ big or small compared to $k_{B}T$?" You literally cannot ask that until $k_{B}T$ exists.

5. The exponential $e^{-x}$ — the shape of "rarely, but not never"

The parent note's headline formula is $n\propto e^{-E_g/(2k_{B}T)}$. To read it you need to know what $e^{-x}$ does.

Why the topic needs this: conductivity's dependence on the gap is entirely this curve. Big gap → far out on the tail → almost no carriers → insulator. Small gap → near the top of the curve → many carriers → semiconductor. The Fermi level and Fermi-Dirac distribution page justifies where this exponential comes from.

6. Charge carriers: $n$, electrons, and holes

Picture: count the electrons that made it up to the empty conduction band — those are free to roam and carry current. $n$ is that headcount, per box of material.

Why the topic needs this: the parent's "factor of 2" (the $E_g/2$ in the exponent) comes precisely from the electron climbing up half a gap and the hole sitting half a gap below — the cost is shared. See Intrinsic vs extrinsic semiconductors and Doping and carrier concentration for how $n$ is engineered.

7. The referee: the Fermi level $E_F$

Picture: imagine filling the shelves with water up to a line. Below the line, shelves are (mostly) full; above, (mostly) empty. That line is $E_F$. Mid-gap means an electron at the top of the valence band must climb only half the gap to reach $E_F$, and then another half to reach $E_c$.

Why the topic needs this: the whole $E_g/2$ story rests on $E_F$ being mid-gap. Full detail in Fermi level and Fermi-Dirac distribution.

8. Conductivity $\sigma$ — the thing we're ultimately predicting

Why the topic needs this: $\sigma$ is the payoff — the observable property that the invisible gap $E_g$ secretly controls. How $\sigma$ also depends on how fast carriers move (mobility) is covered in Conductivity, mobility and drift current.

How these foundations feed the topic

Equipment checklist

Connections

• Parent: Band gap and conductivity
• Formation of energy bands from atomic orbitals
• Fermi level and Fermi-Dirac distribution
• Intrinsic vs extrinsic semiconductors
• Doping and carrier concentration
• Conductivity, mobility and drift current
• Temperature dependence of resistivity in metals