Donor - acceptor energy levels in the band gap: Foundations

Before we can talk about donor and acceptor levels, we must earn every piece of notation the parent note throws around: $E_{C}$ , $E_{V}$ , $E_{D}$ , $E_{A}$ , the "band gap", "meV", " $k T$ ", and the little $+ / -$ ion symbols. We build them one at a time, each on a picture.

1. Energy — the vertical axis of everything

Every diagram in this topic is a picture where up = more energy. Not position, not time — energy. An electron high on the page has lots of energy; an electron low on the page has little.

Figure — Donor - acceptor energy levels in the band gap

2. The energy unit — eV and meV

We need this unit because the numbers in this topic are tiny. A silicon band gap is about $1.1 eV$ ; a donor's ionization energy is about $45 meV = 0.045 eV$ . Writing " $45 meV$ " is far friendlier than " $0.045 eV$ ".

3. Bands — where electrons are allowed

In a single atom an electron may only sit on certain rungs. Bring $1 0^{23}$ atoms together into a crystal and those rungs smear into thick bands of allowed heights.

$E_{V}$ (say "E-vee") = the top edge of the valence band — the highest bound electron height.
$E_{C}$ (say "E-cee") = the bottom edge of the conduction band — the lowest free-electron height.

4. The band gap — the forbidden zone

Why does this appear as a subtraction? Because a gap is a distance, and a distance on a vertical axis is always (top height) − (bottom height). Reading $E_{C} - E_{V}$ literally means "how far up is the conduction floor above the valence ceiling."

5. Dopant levels — a single allowed step inside the gap

Doping inserts one lonely allowed height into the forbidden gap. There are two flavours.

The two "ionization energies" are — again — just vertical distances:

6. $k T$ — the size of a thermal "nudge"

Why does room temperature free these electrons? Because heat is random jiggling, and its typical energy has a name.

Why do we compare $k T$ against the ionization energy? Because a level is emptied when a thermal nudge is big enough to make the jump. Since $k T \approx 26 meV$ is comparable to the $\approx 45 meV$ donor jump, nudges succeed constantly — nearly every donor gives up its electron. That is complete ionization at room temperature.

7. The ion symbols $N_{D}^{+}$ and $N_{A}^{-}$

Once a dopant loses/gains its electron, the atom itself is left charged and fixed in the lattice — it cannot move.

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer before revealing.

What does "up" mean on every diagram in this topic?

More electron energy (the vertical axis is energy, not position).

How many meV are in one eV?

1 eV = 1000 meV

What is the valence band vs the conduction band?

Valence = allowed energies of bound (bonded) electrons; conduction = allowed energies of free, current-carrying electrons.

What do

E_{V}

and

E_{C}

label?

The top edge of the valence band and the bottom edge of the conduction band.

Write the band gap as a formula.

E_{g} = E_{C} - E_{V}

(about

1.1 eV

in silicon).

Where does a donor level

E_{D}

sit, and where does an acceptor level

E_{A}

sit?

E_{D}

just below

E_{C}

;

E_{A}

just above

E_{V}

What are the two ionization energies?

Donor:

E_{C} - E_{D}

; acceptor:

E_{A} - E_{V}

— both tens of meV.

Roughly how big is

k T

at room temperature, and why does it matter?

About

26 meV

; it is comparable to the ionization jump, so dopants ionize almost completely.

What are

N_{D}^{+}

and

N_{A}^{-}

, and do they move?

Fixed positive donor ions and fixed negative acceptor ions — they stay locked in the lattice; only the freed carriers move.

Recall Two-line self-summary

Doping plants a shallow allowed level near a band edge; because the jump to the band is only tens of meV and $k T \approx 26 meV$ , room-temperature heat empties it, freeing a carrier and leaving a fixed ion behind.

🇮🇳 See the Hinglish version: 2.2.03 Donor - acceptor energy levels in the band gap (Hinglish)