Intuition The ONE core ideaA semiconductor is a material with a wall of energy (the bandgap) that electrons must climb to become free and carry current. Silicon's wall is taller than germanium's, so heat leaks fewer electrons over it, and — luckily — silicon also grows a perfect glassy insulator when you bake it in air; those two facts decide the whole story.
Before you can understand why silicon beats germanium , you must be able to read every symbol in the parent note without flinching. This page builds each one from nothing — plain words first, then the picture, then why the topic needs it. Nothing here assumes you have seen a physics class.
Definition Atom, valence electron, bondAn atom is a tiny lump (the nucleus) with electrons orbiting it. The electrons in the outermost ring are the valence electrons — the only ones that ever do anything chemically.
Silicon and germanium each have exactly 4 valence electrons.When four such atoms each share electrons with four neighbours, they lock into a repeating 3-D lattice called a diamond-cubic crystal .
Look at the figure. Each blue atom holds hands (shares an electron pair) with 4 neighbours. Every valence electron is busy holding a bond — none are free to wander. A crystal where every electron is bonded conducts nothing . That is the starting picture; everything below is about how an electron escapes a bond .
Intuition Why we even need this pictureCurrent = moving charge. If every electron is glued into a bond, nothing moves. So the whole subject is one question: what frees an electron from its bond, and how do we count how many get free? That count is what leaks, heats, and switches.
Definition Energy and the electron-voltEnergy = how much "effort" it takes to do something — here, to rip an electron out of its bond. We measure it in electron-volts , written eV
1 eV tiny amount, perfectly sized for single electrons.
Intuition Why eV and not joules?A joule is a human-scale unit (lifting an apple). One electron's escape energy in joules is a number like 0.0000000000000000001 eV "about 1" . That is the only reason we pick this tool: it fits the scale of the thing we're counting.
So when the parent says E g ≈ 1.12 eV "a bit more than one electron-volt of effort frees a bonded electron in silicon."
Instead of drawing millions of bonds, physicists draw an energy ladder . Every electron sits at some height on this ladder; height = energy.
Definition Valence band, conduction band, bandgap
Valence band (lower shaded block): the energies of electrons that are stuck in bonds . Bonded = low, comfortable.Conduction band (upper block): the energies of electrons that have broken free and can drift → carry current.Bandgap E g (the empty stripe between them): a forbidden zone — no electron is allowed to sit inside it. To go from bonded to free, an electron must jump the whole gap in one leap . Its height is E g eV
E g E E nergy, subscript g g ap. It is the toll to leave the bonded band and reach the free band. Silicon's toll (1.12 0.66
Common mistake "Bigger gap = worse material?"It feels like a bigger wall (fewer free electrons, higher resistance) is bad. But a semiconductor's job is to be quiet when OFF and conduct only when we tell it to. A clean, tall gap gives a clean OFF state. Bigger E g
When an electron jumps the gap, it leaves an empty bond behind. That empty spot acts like a positive particle called a hole . Neighbouring electrons can shuffle into it, so the empty spot appears to move — a hole carries current too.
Key fact: breaking one bond always makes one electron AND one hole together — a pair . Never one alone.
Intuition Why the "pair" fact matters laterThe parent note has a mysterious factor of 2 in e − E g /2 k B T entirely from this pairing idea. Hold onto it — Section 6 pays it off.
T k B k B T
T absolute temperature in kelvin (K). Room temperature ≈ 300 K k B Boltzmann constant — a fixed conversion number that turns temperature into energy.Their product k B T typical jiggle energy each particle has just from being warm. At 300 K k B T ≈ 0.02585 eV
k B T label of hotness. To compare it against the gap E g eV eV k B why this tool and not another : we need an apples-to-apples energy so we can ask "is a warm jiggle big enough to pay the toll E g
Now compare: jiggle energy 0.026 eV 1.12 eV ~43× too small on average. Only rare, lucky, extra-big jiggles pay the toll — which is exactly why so few electrons get free. To count those rare winners, we need the next tool.
Definition The exponential decaye − x e ≈ 2.718 e − x 1 x = 0 fast as x
Intuition WHY an exponential and not a simple fraction?Nature distributes jiggle energies so that the fraction of particles with at least energy E e − E / k B T Boltzmann factor , the topic of Fermi-Dirac distribution and thermal excitation ). Each extra chunk of required energy multiplies your chances down by the same factor — that "multiply-down repeatedly" behaviour is precisely what an exponential is. A straight-line or simple-fraction model would badly mispredict how steeply the free-electron count collapses as the gap grows. So the exponential is the right tool because the physics is multiplicative, not additive.
The bigger E g / k B T E g / k B T E g
n i n i how many free-electron/hole pairs exist per cubic centimetre in a pure (undoped) crystal, purely from heat. Subscript i i ntrinsic (built-in, before we add anything). More on this in Bandgap and intrinsic carrier concentration .
Worked example The Si-vs-Ge ratio, symbol by symbolDivide germanium's n i n i ( Si ) n i ( Ge ) ≈ exp ( 2 k B T E g S i − E g G e ) = exp ( 2 ( 0.02585 ) 1.12 − 0.66 ) = exp ( 8.9 ) ≈ 7.3 × 1 0 3 . thousands of times more heat-made carriers → thousands of times more OFF leakage. Now every symbol in that headline formula is earned.
Definition Quick glossary the parent leans on
N D d onor/dopant concentration: how many deliberately added impurity atoms per cm³ supply carriers on purpose (see Doping n-type and p-type ). A device fails when n i N D Mobility μ — how fast a carrier drifts for a given push; higher = faster switching (Carrier mobility and drift velocity ). Ge's is higher — its one real advantage.SiO₂ / GeO₂ — the oxides that grow on each material in air. SiO₂ is a hard, insulating glass; GeO₂ dissolves in water. This is the process story, expanded in SiO2 and the planar process and used in MOSFET operation and the gate oxide .MOSFET — the switch built on top of that oxide; needs a good insulating gate, which only Si provides cleanly.Intrinsic vs extrinsic — pure crystal (carriers from heat, n i N D Intrinsic vs extrinsic semiconductors .
Atom + 4 valence electrons
Covalent bonds in a crystal
Valence + conduction bands
Exponential Boltzmann factor
Why silicon beats germanium
Self-test: can you answer each without peeking? Read the left, then reveal.
What does the subscript in E g "g" for gap ; measured in electron-volts (eV
What is the valence band vs the conduction band? Valence = energies of electrons stuck in bonds; conduction = energies of electrons free to carry current.
Why can no electron sit inside the bandgap? It is a forbidden energy zone; electrons must leap the whole gap E g
What always accompanies a freed electron, and why? A hole — the empty bond it left behind; bonds break in electron-hole pairs .
What is k B T The typical thermal jiggle energy per particle; ≈ 0.02585 eV
Why convert temperature to energy with k B So we can compare the warm jiggle directly against the toll E g
Why an exponential e − E g / k B T The fraction of particles clearing an energy barrier falls off multiplicatively per extra chunk of energy — that is what an exponential encodes.
What does n i i Intrinsic carrier concentration — free pairs per cm³ in a pure crystal from heat alone; i
Where does the factor of 2 in e − E g /2 k B T n i = n p and
n p ∝ e − E g / k B T ; the square root halves the exponent.
What does ∝ "Proportional to" — same growth pattern, ignoring a constant multiplier.
Roughly how many more intrinsic carriers does Ge have than Si at 300 K? About 7 × 1 0 3
