Donor - acceptor energy levels in the band gap: Question bank

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Figure — Donor - acceptor energy levels in the band gap

Look at the figure above: the two grey blocks are the valence and conduction bands, the white strip between them is the forbidden gap, the lavender dashes near the top are donor levels $E_{D}$ (a short hop $E_{C} - E_{D}$ from freedom), and the coral dashes near the bottom are acceptor levels $E_{A}$ (a short hop $E_{A} - E_{V}$ above the valence band). Keep this picture in mind for every question.

True or false — justify

Doping fills electron states inside the band gap that were empty in the pure crystal.

False — doping doesn't fill the gap; each dopant atom introduces one new localized state inside the gap, near a band edge. The gap of the host crystal is still "forbidden" everywhere else.

A donor level

E_{D}

sits below the conduction-band edge

E_{C}

True —

E_{D}

lies just below

E_{C}

so that the small energy

E_{C} - E_{D}

(tens of meV) lets the extra electron jump up into the conduction band.

An acceptor level

E_{A}

sits below the valence-band edge

E_{V}

False —

E_{A}

sits just above

E_{V}

; an electron from the valence band jumps up into

E_{A}

, and the ionization energy is

E_{A} - E_{V}

A shallow level means a small ionization energy.

True — "shallow" means close to the nearest band edge, so the gap the carrier must cross (

E_{C} - E_{D}

E_{A} - E_{V}

) is small and thermal energy easily bridges it.

When a donor is ionized, it releases a hole into the valence band.

False — a donor releases an electron into the conduction band; it is the acceptor that produces a hole (in the valence band).

An ionized donor becomes a fixed positive ion, and an ionized acceptor becomes a fixed negative ion.

True — the donor lost an electron so its core charge is

+ 1

(one of the

N_{D}^{+}

ions); the acceptor gained an electron so it carries

- 1

(one of the

N_{A}^{-}

ions). Both are locked into the lattice and cannot move.

Deeper levels (further from the band edge) ionize more easily than shallow ones.

False — deeper means a larger ionization energy, so it's harder to promote the carrier; deep levels stay mostly un-ionized at room temperature and act as traps, not good dopants.

The dopant's free carrier and the fixed ion it leaves behind add up to zero net charge.

True — the crystal stays overall neutral: the electron donated to the conduction band is balanced by the

+ 1

ionized donor core (and similarly for acceptors).

Spot the error

"Phosphorus in silicon donates a hole because it has one extra electron."

Error: an extra electron becomes a free electron, not a hole. Extra electron → donor → n-type. Holes come from a missing electron (acceptor, group III).

"Boron sits below

E_{V}

and gives its extra electron to the conduction band."

Error: boron is group III — it has one electron too few, so it accepts an electron. Its level

E_{A}

is just above

E_{V}

, and it creates a hole, not a conduction electron.

"The fifth electron of phosphorus is strongly bound, so its level is deep."

Error: the fifth electron is only weakly bound (it feels a screened

+ 1

core, like a giant hydrogen atom), so the level is shallow — close to

E_{C}

"At room temperature the ionization energy

\sim 45 meV

is much larger than

k T \sim 26 meV

, so almost no donors ionize."

Error: even though

45 > 26

, the ionization energy is only comparable to

k T

, so a large fraction ionizes; near-complete ionization at room temperature is exactly why doping works.

"An ionized donor can drift toward a contact and carry current."

Error: the ionized donor ion is a fixed lattice site — it cannot move. Only the electron it released moves and carries current.

"Because the level is inside the forbidden gap, an electron can permanently rest in the conduction band the instant we add phosphorus, no energy needed."

Error: the electron starts bound at

E_{D}

; it still needs the small ionization energy

E_{C} - E_{D}

(supplied by thermal energy) to reach the conduction band.

Why questions

Why does the dopant state land inside the gap rather than inside a band?

Because the extra weakly-bound electron is a localized state around one impurity atom — its binding energy pulls it just below

E_{C}

(or the hole state just above

E_{V}

), landing it in the otherwise-forbidden gap.

Why is the donor level so shallow (only tens of meV)?

The extra electron orbits a screened

+ 1

core inside the high-permittivity silicon — a hydrogen-like atom with a huge Bohr radius and tiny binding energy, so its level lies very near the band edge.

Why does a shallow level mean "many free carriers at room temperature"?

Small ionization energy

≲

thermal nudge

k T

means essentially every dopant gives up its carrier, so the free-carrier count

\approx

the doping concentration.

Why does adding acceptors create holes rather than electrons?

The group-III atom is missing a bonding electron; an electron from the valence band hops up to fill

E_{A}

, and the vacancy it leaves behind in the valence band is a mobile hole.

Why does the crystal stay electrically neutral after doping and ionization?

Every free carrier is exactly balanced by the fixed ionized dopant core of opposite sign (

N_{D}^{+}

per electron,

N_{A}^{-}

per hole), so charges cancel.

Edge cases

What happens to donor carriers as temperature drops toward

0 K

The thermal energy

k T

vanishes, so electrons can no longer be promoted from

E_{D}

E_{C}

; they "freeze out" back onto the donors and the material loses its free carriers.

If a dopant level were placed at mid-gap instead of near a band edge, what would it be good for?

A mid-gap ("deep") level is a poor carrier source (huge ionization energy) but an efficient recombination centre — it grabs and combines electrons and holes rather than releasing free carriers.

Can a single crystal region contain both donors and acceptors at once?

Yes — they compensate: donated electrons fill acceptor levels first, so only the excess of whichever dopant is in majority (

N_{D}

N_{A}

) contributes net free carriers (net type = larger concentration).

What does the ionization energy tend to as the level sits exactly at the band edge?

It tends to zero — the carrier is essentially already in the band, needing no thermal nudge; this is the idealized "always fully ionized" limit.

Is a hole a real particle sitting at

E_{A}

No — the hole is the absence of an electron in the valence band;

E_{A}

is where the missing electron went (up from the valence band), and the hole is the vacancy left behind, which behaves like a positive mobile carrier.

If we keep raising temperature far past room temperature, do carriers keep rising with doping?

Eventually intrinsic generation across the full band gap dominates over dopant ionization, so the carrier count is set by the host gap, not the dopants — the doped behaviour is swamped.