2.1.2 · D5Band Theory & Carrier Physics
Question bank — Band gap and its meaning for conductivity
For every "true/false" item, the goal is not the verdict — it is the justification. A right guess with wrong reasoning still counts as a miss.
First, the pictures every trap refers to
Before the questions, pin down the three diagrams this page keeps pointing at. Every symbol below is earned here — refuse to guess at a trap until you can point to it on these figures.
True or false — justify
A completely full band cannot carry current.
True. A full band has every state occupied, so the adjacent -state a field would push an electron into is already taken — no electron can shift momentum, and the net current cancels exactly.
A material with no electrons in its conduction band cannot conduct.
False as stated. It cannot conduct via that band right now, but electrons in a partly-filled band (a metal) or thermally promoted carriers still conduct — "no electrons up there" is not the same as "no conduction anywhere."
The band gap is a range of energies that electrons pass through very quickly.
False. The gap is a forbidden range — no allowed electron states exist there at all, so an electron cannot occupy it even momentarily; it must land on the far side (from to ).
Insulators have no free charges because they contain no electrons.
False. Insulators are full of electrons; they just sit in a full valence band with a huge gap above, so almost none get promoted. The problem is empty states, not missing charges.
Raising temperature always raises conductivity.
False. It rises for a semiconductor (more carriers cross the gap) but falls for a metal, where extra lattice vibration scatters the already-abundant electrons more — see Temperature dependence of resistivity in metals.
A semiconductor at absolute zero behaves like an insulator.
True. At the thermal factor , so the conduction band is empty and the valence band full — no carriers, no current, exactly like an insulator.
Two materials with the same must have the same conductivity.
False. Since , conductivity also depends on the density of available states and on ==mobility == (drift velocity per unit field), so can differ even at equal — see Conductivity, mobility and drift current.
The Fermi level always lies inside a filled band.
False. In an intrinsic semiconductor sits inside the gap (mid-gap), where there are no states at all — it marks the 50% occupancy energy, not an occupied level. Revisit Fermi level and Fermi-Dirac distribution.
Doping changes the size of the band gap.
False (to first order). Doping adds shallow donor/acceptor levels and shifts the Fermi level, giving carriers cheaply, but the intrinsic gap stays essentially the same — see Doping and carrier concentration.
A larger band gap means a better conductor.
False. Larger gap → harder to promote electrons → fewer carriers → worse conduction. Metals conduct best precisely because their gap is effectively zero.
works for any semiconductor at any doping.
False. It only holds in the nondegenerate (Boltzmann) limit with mid-gap; heavy doping pushes into a band and this formula breaks — you then need the full Fermi–Dirac occupancy.
Spot the error
", using the full gap in the exponent."
The exponent should be . Because sits mid-gap, promoting a carrier to costs only , not the whole gap.
"Metals conduct because thermal energy lifts their electrons across a gap."
Wrong mechanism. Metals have a partly-filled band (or overlapping bands, ); electrons slide into the empty adjacent -state with no thermal jump required.
"Diamond doesn't conduct because it has too few electrons."
Diamond has plenty of electrons, all locked in a full valence band. Its eV gap makes vanishingly small, so almost none are ever promoted — it's the gap, not a shortage of electrons.
"Since eV and Si's gap is 1.12 eV, no electrons can ever cross — Si is an insulator."
The verdict is wrong. The Boltzmann tail is small but nonzero (), and that tiny fraction over huge atom counts gives usefully many carriers — that is why Si is a semiconductor, not an insulator.
"The gap exists because there simply aren't enough atomic levels to fill it."
Backwards. The gap arises from electron waves Bragg-reflecting at the zone boundary, forming two standing waves of different energy; the split between them is the gap — see Formation of energy bands from atomic orbitals.
"A hole is a real positive particle that we add to the crystal."
A hole is the absence of an electron in the valence band, which behaves like a positive carrier. No particle is added — an electron is removed, and the empty state moves like a positive charge.
"GaAs and Ge behave the same for LEDs because their gaps are similar in size."
Size is not the issue — it's where the band edges sit in . GaAs has a direct gap (edges at the same , efficient light emission); Ge and Si are indirect (edges at different , poor emitters).
Why questions
Why does the factor of 2 appear in ?
Because is mid-gap: you pay to lift the electron up to , and the hole it leaves sits below — the gap cost is split symmetrically.
Why does a full band contribute exactly zero current even though its electrons are moving?
For every electron moving one way there is an equal-and-opposite one at moving the other way; with all -states occupied these contributions cancel perfectly, leaving no net drift.
Why is a semiconductor's conductivity described as "exponentially tunable"?
Carrier count sits inside an exponential, , so small changes in (or added dopants, or light) produce huge multiplicative changes in .
Why do we compare to rather than to when classifying materials?
Because the actual climb from mid-gap to the conduction band is ; that half-gap is the real energy barrier a carrier faces.
Why does GaAs (1.42 eV) matter for optics while Ge (0.66 eV) is used differently?
Gap size sets carrier count and photon energy, but GaAs's direct gap (valence and conduction edges at the same ) lets electrons drop straight down and emit light efficiently, whereas Ge's indirect gap needs a phonon — see Direct vs indirect band gap.
Why is "metal, semiconductor, insulator" better seen as a spectrum than three walls?
They differ only by the continuous value of (0, small, large); nothing physically switches — the same Boltzmann factor smoothly interpolates the behaviour.
Why can two crystals with identical still have very different ?
Because : even with the same carrier count , a higher mobility (more drift velocity per volt) gives more current — the gap fixes , not .
Edge cases
What happens to carrier concentration as ?
, so : every intrinsic semiconductor becomes a perfect insulator with no thermal carriers.
What is the limiting case ?
The conduction and valence bands touch or overlap, carriers exist even at , and the material behaves as a metal with large, weakly temperature-dependent .
Is there a hard boundary in eV between a semiconductor and an insulator?
No. The cutoff (often quoted near 3–4 eV) is a convention about whether thermal/optical excitation gives useful carriers; the physics is continuous.
What if you shine light of photon energy exactly on a semiconductor?
An electron can absorb the photon and jump from valence to conduction band, creating an electron–hole pair — this is optical carrier generation, independent of temperature.
What if the applied field is extremely small — does a semiconductor still conduct?
Yes, to linear order: even a tiny field nudges the existing thermal carriers, giving a small drift current proportional to the field (Ohmic regime) — see Conductivity, mobility and drift current.
What happens to an intrinsic semiconductor's Fermi level as rises?
It stays close to mid-gap (drifting only slightly if the electron and hole effective masses differ), because electrons and holes are generated in equal numbers.
What breaks in the heavily-doped (degenerate) regime?
Doping so much pushes into the conduction (or valence) band, so is no longer — the Boltzmann tail and the mid-gap assumption both fail, and becomes almost metallic and only weakly -dependent. See Doping and carrier concentration.
Can an intrinsic semiconductor ever be degenerate just from heating?
Not realistically — thermal generation keeps near mid-gap. Degeneracy comes from doping (or extreme carrier injection), which is why the simple picture is safe for intrinsic material but not for heavily-doped devices.