Intuition The one core ideaA semiconductor is a material sitting on a knife-edge between "conductor" and "insulator": its electrons need a specific packet of energy to become free to move. Everything in this topic is just bookkeeping of how many free charges exist (n p n i how easily they're made (E g T
This page assumes you have seen none of the notation. We build each symbol on the previous one, anchored to a picture. Parent topic: Intrinsic vs extrinsic semiconductors .
Before symbols, the picture. A semiconductor like silicon is a huge, orderly 3‑D grid of atoms, each atom sharing electrons with its neighbours through covalent bonds (a shared pair of electrons acting like a handshake that glues two atoms together).
A shared pair of electrons between two atoms. In the picture it's the line joining two silicon cores. When both electrons of a bond stay put, nothing moves → no current. See Silicon crystal structure and covalent bonding for the full 3‑D geometry.
Why the topic needs this: conduction = charges that can move. In a perfect, cold crystal every electron is trapped in a bond, so we start from zero movers and ask "how do we free some?"
Definition Electron and the charge symbolq == q == size of its charge, a fixed positive number q = 1.602 × 1 0 − 19 − q + q
A deliberate naming choice: many books call this e reserve the letter e for the mathematical constant e = 2.71828 … q e always the number 2.71828 …
The picture: a small blue dot in the bond. Why we need it: current is charge in motion , and every carrier we count carries exactly ± q q
To free an electron we must pay energy . So we need a unit of energy sized for single atoms.
Definition Electron-volt, eVThe energy one electron gains falling through one volt . It's a tiny unit: 1 eV = 1.602 × 1 0 − 19
Picture: think of energy as the height an electron must be lifted. A joule would lift it kilometres; an eV lifts it just off the ground — the right ruler for atoms.
Here is the single most important picture in the whole topic. Instead of drawing every atom, we draw allowed energy heights an electron may occupy.
Definition Valence band, conduction band, andE g
Valence band (E v stuck in bonds .Conduction band (E c free to roam and carry current.Band gap E g = E c − E v forbidden height in between — no electron may sit there. To conduct, an electron must be lifted the full height E g
Why the topic needs it: E g Band gap and energy bands .
When thermal shaking pays the fee E g
Definition Free electron and hole
A free electron is a mover in the conduction band, charge − q
A hole is the empty seat left in the valence band, charge + q drifts like a positive particle .
Intuition Why holes are worth inventingTracking every shuffling electron in a nearly-full band is hopeless. Tracking the one empty seat in a full theatre row is easy. The empty seat is the hole. One "broken bond" always creates exactly one electron and one hole — a matched pair.
Why the topic needs it: because pairs are created together, in a pure crystal the counts of the two carriers are forced to be equal. That fact becomes the definition of n i
Now we can name the numbers we spend the rest of the topic computing.
Definition The three carrier-count symbols
== n == free electrons per cubic centimetre (a concentration , not a total).== p == holes per cubic centimetre .== n i == intrinsic carrier concentration — the value of n p pure crystal, where pairs force n = p
Picture: imagine a fixed 1 cm³ box; n p
n = p = n i pure crystal every carrier came from a pair, so electrons and holes must be equal. We give that shared value its own name n i n above n i
What supplies the entrance fee E g Heat. So we need a symbol for "how much energy heat provides."
T k B k B T
== T == kelvin (0 K = the coldest possible, no thermal shaking).== k B == 8.617 × 1 0 − 5 k k B k k k B == k B T == typical thermal energy available to jiggle one particle. At room temperature (T = 300 k B T ≈ 0.0259
Intuition The tug-of-war that runs the whole topicCompare two numbers: the fee E g k B T lucky, rare electron gathers enough — which is why pure silicon conducts only weakly. Raise T k B T why semiconductor resistance drops when heated .
The master formula is built around e − E g / k B T
Definition What the lettere == e == mathematical constant e = 2.71828 … e x exp ( x ) e x not the electric charge — that is q
Intuition Why an exponential, not a ratioStatistical physics says: the chance a particle has grabbed energy E falls off exponentially — the Boltzmann factor e − E / k B T combinatorially harder to collect, and exponentials are exactly the functions that shrink by a fixed multiple for each fixed step of energy. So doubling the gap doesn't halve the carriers — it squares the smallness. That violent sensitivity is why E g
e − E g / k B T
The minus sign → bigger gap E g
E g / k B T pure number (fee ÷ pocket money): energies cancel, leaving "how many times harder than typical."
We now have the Boltzmann factor — the chance part. But a chance alone isn't a carrier count; we still need to know how many seats the carriers can land in. Those seats are N c N v n i
To turn "chance of a jump" into "number of carriers" we need two more ideas: where the electron sea level sits, and how many landing seats each band offers.
E F == E F == Fermi level , is the energy height at which a state has a 50% chance of holding an electron — a "sea level" for electrons. Doping moves this sea level up (n-type) or down (p-type). Full treatment: Fermi level and Fermi-Dirac statistics .
Definition Effective massesm e ∗ m h ∗ heavier or lighter . The effective mass is the mass it behaves as if it has when accelerated by a force. m e ∗ m h ∗ m 0 = 9.11 × 1 0 − 31 m e ∗ ≈ 1.08 m 0 m h ∗ ≈ 0.81 m 0
Why they enter the seat-count: counting quantum states packs more states into a band the heavier the carrier is (heavier carriers move slower, so more of them fit into a given range of momentum). That is why m ∗ 3/2 3/2
Definition Effective densities of statesN c , N v == N c == == N v == all the available seats near each band edge into one equivalent number sitting right at the edge:
N c = 2 ( h 2 2 π m e ∗ k B T ) 3/2 , N v = 2 ( h 2 2 π m h ∗ k B T ) 3/2 h
Temperature dependence: both grow like T 3/2 Silicon at 300 K: N c ≈ 2.8 × 1 0 19 cm − 3 N v ≈ 1.0 × 1 0 19 cm − 3 1 0 10
Intuition Fermi–Dirac, and why it simplifies to BoltzmannElectrons obey Fermi–Dirac statistics : the exact chance a state at energy E f ( E ) = 1 + e ( E − E F ) / k B T 1 . E c far above the sea level E F k B T E c − E F k B T ≈ 0.026 E − E F ≫ k B T 1 + f ( E ) ≈ e − ( E − E F ) / k B T . Boltzmann approximation : for the rare, high-energy carriers we care about, the fancy Fermi–Dirac curve collapses into the simple Boltzmann factor of §7.
means physically
The full fee E g whole pair (one electron and one hole). When we ask for the concentration of one carrier type, we're effectively splitting that cost — each carrier of the pair "carries" half the exponent, E g /2
Intuition Why the cancellation ofE F n p E F shifts with doping . But their product does not — it depends only on knowns (N c , N v , E g , T n p = n i 2 any doping, letting us find the tiny minority carrier from the big majority one in the parent note.
Finally, the deliberate impurities.
N D N A
== N D == donor concentration (per cm³): group‑V atoms (P, As, Sb) each donate one loosely-bound electron → n-type , so n ≈ N D == N A == acceptor concentration (per cm³): group‑III atoms (B, Al, Ga) each accept an electron, leaving a hole → p-type , so p ≈ N A
Picture: replace one silicon atom in the grid with an intruder that has one electron too many (donor) or one too few (acceptor). Why the topic needs it: these are the dials engineers turn to design n p PN junction diode and Carrier mobility and drift current .
N D count , not the actual n n ≈ N D N D ≫ n i and every donor is actually ionised. When they're comparable, or when donors are not fully ionised, you must be more careful (see the edge cases below).
Common mistake Low temperature: incomplete ionisation ("carrier freeze-out")The assumption: "each donor gives exactly one free electron, so n = N D When it breaks: at low temperature the donor electron (bound by only ≈0.05 eV) may not have enough thermal energy k B T un-ionised , holding their electron, so n < N D T n → 0 n ≈ N D extrinsic region of temperature, where k B T ≪ E g
Common mistake Both dopants present: compensationThe assumption: "a crystal is either n-type or p-type."
When it breaks: real crystals can contain both donors (N D N A cancel each other. Only the net doping survives:
n ≈ N D − N A ( if N D > N A ) , p ≈ N A − N D ( if N A > N D ) . compensation . If N D = N A charge neutrality : n + N A − = p + N D +
covalent bond shared pair
conduction band bottom edge Ec
band gap Eg equals Ec minus Ev
carrier counts n and p and ni
Boltzmann factor exp minus Eg over kB T
densities of states Nc and Nv
effective masses me star and mh star
law of mass action np equals ni squared
donors ND and acceptors NA
intrinsic vs extrinsic topic
Each node names a symbol and its meaning together (for example "band gap Eg equals Ec minus Ev") so a symbol never appears without its definition attached.
Cover the right side and test yourself. If any answer surprises you, re-read that section above.
What does the symbol q The magnitude of electron charge, 1.602 × 1 0 − 19 − q + q
On this page, what does the letter e The mathematical constant e = 2.71828 … q
What is an electron-volt (eV)? The energy an electron gains crossing one volt, 1.602 × 1 0 − 19
Define the band gap E g The forbidden energy height between valence and conduction bands, E g = E c − E v
What two carriers appear when one bond breaks? One free electron (charge − q + q
What do n p n i n p n i
What is k B T Typical thermal energy per particle; at 300 K, k B T ≈ 0.0259
Why is the Boltzmann factor an exponential e − E g / k B T The chance of gathering energy E g
Why can we use the Boltzmann factor instead of full Fermi–Dirac? Because E c − E F ≫ k B T 1 + f ( E ) = 1/ ( 1 + e ( E − E F ) / k B T ) e − ( E − E F ) / k B T
Where does the 2 1 e − E g /2 k B T From taking the square root of
n p = N c N v e − E g / k B T when
n = p = n i ;
e x = e x /2 halves the exponent.
Why does multiplying n × p E F The two chance factors carry E F + E F − E F N c , N v , E g , T
What are N c , N v Effective densities of available seats at the band edges; both scale as T 3/2
What is the effective mass m ∗ N c , N v The mass a carrier behaves as if it has inside the lattice; heavier carriers pack more states into a band, so m ∗ 3/2
What are N D N A N D N A
When does the shortcut n ≈ N D When N D n i
What is compensation, and what survives it? Donors and acceptors cancel; only the net doping survives, n ≈ N D − N A p ≈ N A − N D
