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2.1.1 · D5Band Theory & Carrier Physics

Question bank — Energy bands - valence band and conduction band

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Before you start, keep these five plain-word anchors in mind (built fully in the parent):

  • Valence band (VB) — the highest band that is completely full at absolute zero temperature.
  • Conduction band (CB) — the lowest band that is empty at absolute zero.
  • Band gap — the forbidden energy stretch between VB top () and CB bottom (), where no allowed states exist.
  • Carrier — a mobile charge: an electron in the CB, or a hole (missing electron) in the VB.
  • Fermi level — the energy where the occupation probability is one-half (see Fermi level and Fermi-Dirac distribution).

True or false — justify

A completely filled band can still carry a large current if you apply a strong enough field.
False. For every electron pushed one way there is one moving the opposite way, and with no empty states to move into the net current is exactly zero regardless of field strength.
At K a pure (intrinsic) semiconductor behaves as a perfect insulator.
True. With zero thermal energy no electron can cross , so the CB stays empty and the full VB cannot conduct — identical band picture to an insulator, just with a smaller gap.
A material with more electrons in its valence band is automatically a better conductor.
False. A fuller VB conducts less, not more; conduction needs empty adjacent states, so a brim-full band is inert while a half-empty band conducts.
Metals have zero band gap.
Misleading. The precise statement is that a metal's highest occupied band is partially filled (or VB and CB overlap), so empty states sit right above filled ones — the Fermi level lies inside a band, not in a gap.
Doping a semiconductor shrinks its band gap so that carriers can cross more easily.
False. Doping adds shallow donor/acceptor levels inside the gap and supplies carriers directly; itself barely changes. See Intrinsic and extrinsic semiconductors.
The band gap is the energy an electron has while it sits between the two bands.
False. The gap is a forbidden range — no allowed states exist there, so an electron never possesses an energy inside the gap (defect/trap levels are separate localized states, not part of the gap).
Raising temperature always increases the intrinsic carrier concentration.
True. In the exponent grows toward zero as rises, so the exponential — and hence — increases.
Two isolated atoms already form a "band."
False. Two atoms split one level into just two levels (bonding + antibonding). A band is the near-continuous smear you get only when levels pack into a few eV.
Glass (an insulator) is transparent to visible light because its band gap is large.
True. A visible photon (~1.8–3.1 eV) carries less energy than glass's large gap, so it cannot lift a VB electron across — no interband absorption, hence transparency.

Spot the error

"Silicon at room temperature has almost no carriers because is essentially zero."
The tiny exponential is multiplied by an enormous prefactor , giving a usable . "Small exponential" is not "no carriers."
"A photon with energy just below is partly absorbed and partly transmitted across the gap."
Interband absorption is a threshold effect: if there is no final state to reach, so it is not absorbed at all across the gap — the material is transparent, not "partly" transparent.
"Because the gap in the diagram is drawn as a wide empty space, electrons drift slowly through it from VB to CB."
Electrons do not pass through the gap; there are no states to occupy en route. A crossing is an instantaneous jump from a VB state to a CB state, driven by absorbing energy .
"The factor of 2 in is just a Boltzmann convention."
It is physical: exciting one carrier creates both a CB electron and a VB hole, so the gap energy is shared between two species (see Holes as charge carriers), which halves the effective activation energy per carrier.
"Ge conducts better than Si at room temperature because Ge atoms are bigger."
Atom size is not the reason. Ge's smaller band gap (0.66 vs 1.12 eV) sits in the exponent, giving thousands× more intrinsic carriers — a gap effect, not a size effect.
"Adding heat lets electrons ignore the Pauli exclusion principle and pile into the valence band."
Heat never suspends Pauli exclusion. It supplies energy for electrons to jump into empty CB states; the VB has no room to accept more electrons.
"A direct-gap material and an indirect-gap material with the same absorb a threshold photon equally easily."
An indirect gap also needs a momentum change (a phonon), so absorption near threshold is far weaker even at equal . See Direct vs indirect band gap.

Why questions

Why does bringing atoms together split each sharp atomic level rather than leaving it untouched?
Overlapping identical orbitals would force electrons into the same quantum state; Pauli exclusion forbids that, so each level splits into slightly different levels to keep every state distinct.
Why does a partly filled band conduct while a completely filled band does not?
A partly filled band has empty states immediately above filled ones, so a field can nudge electrons into them and create net drift; a full band offers nowhere to move, so all shifts cancel.
Why does the intrinsic carrier density depend exponentially on instead of, say, linearly?
Boltzmann statistics make the probability of a thermal fluctuation of size scale as ; climbing the gap is such a fluctuation, so the dependence is exponential.
Why can a small band gap material be an insulator at yet a useful semiconductor at 300 K?
At no thermal energy exists to cross even a small gap, so it is an insulator; at 300 K enough electrons are thermally excited across the modest gap to give controllable conduction.
Why do the effective densities of states and enter at all?
They count how many CB and VB states are actually available near the band edges for carriers to occupy; more available states means more carriers can be created. They come from the Density of states.
Why is a metal's Fermi level inside a band while a semiconductor's lies in the gap?
In a metal the highest band is partly filled, so the half-occupation energy falls within that band; in a semiconductor the VB is full and CB empty, so the half-occupation point sits in the forbidden gap between them.

Edge cases

What happens to a completely full band if you cool it to exactly — does it become an even better insulator?
It was already non-conducting; cooling removes any thermally excited CB electrons, making the material a purer insulator, but a full band already carried zero current, so behaviour at is the limiting inert case.
If (VB and CB just touch), what kind of material do you get?
A zero-gap or semimetal case — empty states sit infinitesimally above filled ones, so it conducts like a (poor) metal with carriers available at any .
If is enormous (say 6 eV, like diamond), can any temperature make it a good conductor before it melts?
No practical one; the exponential stays vanishingly small until approaches an eV, temperatures far above where the crystal survives — so it stays an insulator throughout.
For a photon with energy exactly equal to , is it absorbed?
It is right at the threshold: a VB electron can just reach the CB bottom, so absorption becomes possible (edge case), though the available final states there are few, making absorption weak just at threshold.
At , is the conduction band strictly empty for every real semiconductor?
For a pure intrinsic crystal, yes — but real doped crystals have donor electrons that can sit in the CB even at if levels are shallow enough; the "empty CB at " rule is the intrinsic idealization.
Recall One-line self-test

If you can answer "why does a full band carry no current?" and "why is in the exponent, not out front?" in your own words, you have the two ideas most D5 traps are built on.

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