Intuition The ONE core ideaA semiconductor's usefulness comes down to counting two things: how many mobile electrons sit in the upper band and how many empty spots ("holes") sit in the lower band. Every symbol on the parent page exists to answer just one question — how many seats are there, and what fraction are filled?
Before you can read the carrier-concentration equations, you must own every letter and squiggle in them. This page introduces each one from absolute zero : what it means in plain words, the picture it stands for, and why the topic can't live without it. Nothing here assumes you've seen band diagrams before.
We build the vocabulary in the order the physics needs it — energy first, then bands, then "seats", then "occupancy", then the counting itself. Come back here whenever a symbol on the parent note feels unearned.
E E how much energy one electron has , measured in electron-volts (eV). Think of it as height on a vertical ladder : the higher up an electron sits, the more energy it carries.
Every diagram in this topic uses energy going up the page . That single choice is why we can talk about electrons "falling" and holes "rising" later — it's a real height axis, not a metaphor.
Worked example Figure 1 — the energy ladderWhat it shows: a vertical arrow labelled Energy E , with a horizontal-rung "ladder" of allowed energy levels. The orange dot is one electron sitting on a low rung; the plum dot is that same electron promoted to a higher rung. Read it as: the only thing that changes between the two dots is height = energy, not left–right position. This is why every later diagram plots energy upward.
Intuition Why energy and not position?In everyday physics we plot where things are. Here we plot how energetic they are, because whether an electron can move and conduct depends entirely on its energy level — not on where it is in the crystal. So the whole subject is drawn on an energy axis .
Electrons in a solid are not allowed to have any energy — only energies inside certain bands (allowed ranges), separated by a forbidden zone. For a semiconductor there are two bands that matter.
Definition Band edges and gap
E v valence band edge — the top of the lower, normally-full band (the "main floor").E c conduction band edge — the bottom of the upper, normally-empty band (the "balcony").E g = E c − E v band gap — the height of the forbidden step between them.
The subscripts are memory hooks: c for c onduction, v for v alence.
Worked example Figure 2 — bands and the forbidden gapWhat it shows: the same energy axis. The teal filled block at the bottom is the valence band (full of electrons), ending at the teal line E v faint orange block on top is the conduction band (empty), starting at the orange line E c plum double-arrow between them measures the gap E g = E c − E v Read it as: nothing is allowed inside the plum span — that's the forbidden zone an electron must jump over to conduct.
Intuition Why the gap mattersE g E g ≈ 1.12 n i Band gap Eg .
Common mistake "The gap is where electrons live."Feels right because we draw the gap as a big empty region. Truth: the gap is forbidden — no allowed states exist there. Electrons live in the bands around it. The only special energy inside the gap is a bookkeeping marker, E F
E F E F 50% chance of being filled. Picture it as the water line on the energy ladder: states well below are "underwater" (filled), states well above are "in the air" (empty), and right at E F
Worked example Figure 3 — the Fermi level as a water lineWhat it shows: the horizontal axis is probability filled f ( E ) plum S-curve is the occupancy: near 1 at the bottom (filled seats "underwater"), near 0 at the top (empty seats "in the air"). The dashed teal line marks E F f = 0.5 Read it as: E F
Intuition Why we need a water lineWe can't list all 1 0 22 single number , E F E F E F Fermi level position and Doping and charge neutrality .
Notice E F difference : E c − E F E F − E v
Definition The thermal energyk T k Boltzmann's constant (a fixed conversion number, k ≈ 8.62 × 1 0 − 5 eV/K T absolute temperature in kelvin. Their product k T typical jiggle-energy that heat gives to one particle. At room temperature (300 K), k T ≈ 0.0259
k T k T k T E c − E F pure number answering "how many thermal kicks tall is this step?" A step of 3 k T
k T No — k T energy (eV), because k
Definition Density of statesg ( E ) g ( E ) how many available electron-seats exist per unit energy and per unit volume at height E states ⋅ − 1 ⋅ − 3 (or per m3 g g
Worked example Figure 4 — the square-root shape ofg c ( E ) What it shows: energy on the vertical axis, density of states g c ( E ) E c zero conduction seats (nothing is drawn). Above E c orange curve grows like E − E c Read it as: the higher you climb above the band edge, the more available seats appear, but right at the edge there are essentially none.
The subscript picks the band: g c ( E ) g v ( E ) g c ( E ) ∝ E − E c Density of states .
Intuition Why "density" — and why per volume tooThere are infinitely many possible energies, so you can't count seats at an exact energy — you count seats within a thin slice of energy. That's what "per unit energy" means: g ( E ) d E E E + d E g per unit volume . That is exactly why the final carrier counts n , p 3
f ( E ) f ( E ) = 1 + e ( E − E F ) / k T 1 f ( E ) probability that a seat at energy E — a number between 0 and 1.
Read it against the water-line picture:
Far below E F e negative → 0 f → 1
At E F = 0 f = 1 + 1 1 = 2 1 why E F Far above E F f → 0
Intuition Why this exact shape and not a hard cutoffIf electrons ignored each other you'd expect a sharp "filled below, empty above" step. But two electrons can't share a seat (Pauli exclusion), and heat blurs the edge. f ( E ) T k T + 1 Fermi-Dirac distribution .
1 − f ( E ) hole is a missing electron, so the probability a valence seat is empty is 1 − f ( E ) 1 − f ( E ) f ( E )
+ 1 E ≥ E c E F well below the band edge — specifically E c − E F ≫ k T ∼ 3 k T E − E F ≫ k T e ( E − E F ) / k T huge number. Next to a huge number the "+ 1 f ( E ) = 1 + e ( E − E F ) / k T 1 ≈ e − ( E − E F ) / k T . Boltzmann tail . Calling this the non-degenerate case (water line safely inside the gap) is exactly what lets the messy Fermi–Dirac integral collapse into the clean formulas in Section 9. If instead E F + 1 Intrinsic vs extrinsic semiconductors .
Definition Effective massm n ∗ m p ∗ apparent masses the carriers behave with inside the crystal , which differ from the free-electron mass because the atomic lattice pushes back.
Intuition Why we fake the massInstead of tracking every atom's tug on an electron, we bundle all those forces into one number: pretend the electron is a free particle but heavier or lighter . That trick lets us reuse simple free-particle formulas — and it's exactly what sets the size of N c , N v Effective mass .
The star ∗
Two more ingredients hide inside N c N v
Definition Planck's constanth h h ≈ 6.626 × 1 0 − 34 J ⋅ s h 2 how tightly seats are packed in the band. (The parent note's derivation uses ℏ = h /2 π
Intuition Why a factor of 2 sits in front ofN c , N v two electrons, because an electron carries a spin that can point "up" or "down" — Pauli's rule forbids identical electrons, but opposite spins count as different. So each seat is really a double seat . That extra bookkeeping is the leading 2 N c = 2 ( … ) N v = 2 ( … ) spin degeneracy ; forget it and you undercount every carrier by half.
N c N v N c effective number of conduction-band seats per unit volume , as if the whole spread-out band were squeezed into one single shelf sitting right at E c N v E v states per unit volume (cm− 3 .
N c = 2 ( h 2 2 π m n ∗ k T ) 3/2 , N v = 2 ( h 2 2 π m p ∗ k T ) 3/2 2 h m ∗ k T
Intuition Why bother collapsing the bandThe real integral (seats × occupancy over all heights) is messy. In the non-degenerate case (Section 6) it turns out to give the same answer as a much simpler picture: one shelf of N c N c T 3/2
Definition Electron and hole concentrationsn , p
n electron concentration = the number of mobile electrons in the conduction band per unit volume (units cm− 3 p hole concentration = the number of empty spots in the valence band per unit volume (units cm− 3
Now every letter in the master results is earned:
Read the mnemonic on the parent page — "electrons fall from the top, holes rise from the bottom" — and every symbol should now feel like a labelled part, not a mystery.
Band gap Eg = Ec minus Ev
Fermi level EF the water line
Effective states Nc and Nv
Planck constant h and spin factor 2
Non-degenerate Boltzmann tail
Intrinsic ni and np equals ni squared
This map is the whole topic in one glance: energy feeds the bands and the water line; density of states plus occupancy plus effective mass feed the carrier count; the gap and effective states feed n i Intrinsic vs extrinsic semiconductors .
Test yourself — cover the right side and answer out loud.
What does the vertical axis in every band diagram represent? Electron energy E
What are E c E v The conduction band edge (bottom of upper band) and valence band edge (top of lower band).
Define the band gap E g E g = E c − E v
What is special about the Fermi level E F It's the 50% occupancy line — states below are mostly filled, above mostly empty.
What are the units and value of k T Energy (eV), with k ≈ 8.62 × 1 0 − 5 k T ≈ 0.0259
In plain words, what is g ( E ) Available seats per unit energy and per unit volume; units states⋅ − 1 ⋅ − 3
What does f ( E ) E = E F Probability a seat at E f ( E F ) = 2 1
Why does the hole integral use 1 − f ( E ) A hole is an empty seat, so its probability is one-minus-occupancy.
What condition lets f ( E ) Non-degenerate case E c − E F ≫ k T + 1
What is h N c h 6.626 × 1 0 − 34 ⋅
What does N c Effective conduction seats per unit volume (cm− 3 E c
What do n p Electron and hole concentrations, both in cm− 3
Why does n i e − E g /2 k T Because
n i = n p and
n p ∝ e − E g / k T , so the square root halves the exponent.
