Intuition The ONE core ideaIn a single atom, electrons can only live on a few sharp "shelves" of allowed energy; jam 1 0 23 band of nearly-touching shelves. Everything in this topic — metals, insulators, semiconductors, doping, light absorption — is just the story of which bands are full, which are empty, and how big the forbidden gap between them is.
This page assumes nothing . Every letter, arrow, and word used in the parent note Energy bands (2.1.1) is built here from scratch, in an order where each idea leans only on the ones before it.
Before any band, we need the vertical axis every diagram in this topic uses.
E Energy E ::= a number saying how much "oomph" a particle has. For an electron bound near an atom, we measure energy so that more tightly bound = lower (more negative) energy , and freer = higher energy .
Plain words: how hard it would be to yank the electron away. A deeply-bound electron sits low; a nearly-free one sits high.The picture: a vertical number line. Up = more energy = freer. We will stack every allowed state on this line.Why the topic needs it: every band diagram is a plot with energy on the vertical axis . Without agreeing "up = higher energy" none of the pictures mean anything.
The unit we use for these energies is the electron-volt .
Definition Electron-volt, eVeV ::= the energy one electron gains crossing a voltage of one volt. It is tiny: 1 eV = 1.602 × 1 0 − 19
Why this unit and not joules? Atomic energy gaps are around 1 1.8 × 1 0 − 19 1.12 only reason physicists prefer it here.
Intuition Why a lone atom has onlycertain allowed energies
An electron trapped around a nucleus behaves like a wave that must "fit" around the atom. Only special wave-shapes fit — like only certain notes fit on a guitar string. Each fitting shape has one exact energy. Energies between them are simply not allowed.
Energy level ::= one of the discrete (separated, countable) allowed energies of an electron in a single isolated atom.
The picture: a few horizontal lines drawn at separate heights on the energy axis — sharp shelves with clear empty space between them (left side of figure below).Why the topic needs it: the whole story starts from "one atom has separated levels." Bands are what happens to these shelves when atoms crowd together.
Two symbols we will pin to specific shelves later:
E v E c
E v top of the valence band (the highest filled level, once bands form).E c bottom of the conduction band (the lowest empty level).The picture: two special heights on the energy axis; everything between them is forbidden.We cannot define these yet — we first need the word "band." Hold them.
Quantum state ::= a complete "address" for an electron: its energy shelf plus its spin (a tiny built-in up/down property). Two electrons in the same address = same quantum state.
Intuition Overlap + Pauli = splittingPush two identical atoms close. Their identical shelves start to overlap. If both electrons kept the exact same energy they would share one quantum state — Pauli forbids it. Nature dodges this by nudging the one shelf into two slightly different shelves (one a touch lower, one a touch higher). Add more atoms → more nudged copies → a dense cluster of shelves.
Energy band ::= the near-continuous cluster of allowed energies you get when a single atomic level splits into N N
The picture: on the far right of the figure, the sharp line has become a solid-looking vertical ribbon — so many shelves so close they look like a filled strip.Why "near-continuous"? For N ≈ 1 0 23 1 0 23 1 0 − 23 looks solid.
Now we can finally name the two bands the topic cares about:
Definition Valence band (VB) and Conduction band (CB)
Valence band (VB) ::= the highest band that is completely full of electrons at temperature zero. These are the bonding electrons holding the crystal together. Its top edge is ==E v Conduction band (CB) ::= the lowest band that is empty (or only partly full) at temperature zero. Electrons up here can roam and carry current. Its bottom edge is ==E c
E g Band gap ::= the energy range between the top of the VB and the bottom of the CB where no allowed states exist at all .
E g = E c − E v
Plain words: the height of the empty forbidden strip separating "full" from "empty."The picture: a blank horizontal band on the energy axis — a floor with no shelves and no ramp. An electron is either below it (VB) or above it (CB); it can never rest inside.Why the topic needs it: the size of this gap decides everything — metal vs insulator vs semiconductor, whether light gets absorbed, how many carriers appear when heated.
Common mistake "Electrons sit in the gap."Why it feels right: we draw the gap as open space, so it looks like somewhere to stand.
Fix: the gap is forbidden — zero allowed states. It is a height an electron passes through in one jump, never a place it lingers.
The parent note reaches for a few mathematical tools. Here is each one, and why that tool and no other .
Definition The exponentiale − x this function
e − x x e ≈ 2.718 x = 0 1 x = 10 0.00005
Why an exponential and not, say, 1/ x Random thermal kicks combine multiplicatively : needing twice the energy is not twice as unlikely, it is the unlikeliness squared . That "unlikeliness multiplies" behaviour is exactly what e − x Fermi level and Fermi-Dirac distribution .The picture: a curve that plunges toward zero. A tall gap E g − E g / ( 2 k B T ) astronomically rare.
n i N c N v
n i N c , N v Density of states .They combine as n i = N c N v e − E g / ( 2 k B T ) Holes as charge carriers and Intrinsic and extrinsic semiconductors .
Definition Photon energy andh c = 1240
A photon is one particle of light; its energy is E = h c / λ λ
h c = 1240 E ( eV ) = 1240/ λ ( nm ) Why keep it handy? It converts a colour (wavelength) straight into an energy in eV, which we can compare to E g E ≥ E g
Energy E on a vertical axis
Levels split when atoms overlap
Band gap Eg = Ec minus Ev
Thermal crossing exp factor
Light absorption if E photon at least Eg
Metal insulator semiconductor
Test yourself — cover the right side and answer before revealing.
What does "up" mean on the energy axis? Higher energy = a more loosely bound / freer electron.
What is 1 eV in everyday terms? The energy an electron gains crossing 1 volt; 1.6 × 1 0 − 19
Why does a single atom have only discrete energy levels? The electron is a standing wave that must "fit" the atom; only certain shapes fit, each with one exact energy.
State the Pauli exclusion principle in one line. No two electrons can occupy the same quantum state (same shelf and spin).
Why does one atomic level become a band in a solid? Overlapping identical levels of N N
Define E v E c E v E c
Write the band gap formula. E g = E c − E v
Can an electron rest inside the gap? No — the gap is forbidden, zero allowed states; electrons only cross it in a jump.
What is k B T E g 0.02585 E g
Why is the thermal-crossing factor an exponential e − E g /2 k B T Rare thermal events multiply in unlikeliness; the exponential is the natural law for such Boltzmann probabilities.
When is a photon of wavelength λ When E = 1240/ λ eV ≥ E g
