N-type doping with donor atoms (phosphorus, arsenic): Visual walkthrough

We are chasing one number: the donor ionisation energy $E_{D}$ — the energy you must pay to lift the 5th electron off its donor atom and into the "free to conduct" state.

Step 1 — What is holding an electron to an atom at all?

WHAT. Before we can free an electron, we need to know what pins it down. A negative electron is pulled toward a positive core by the Coulomb force — the plain electric attraction between opposite charges.

WHY this tool. We use Coulomb's law and not gravity or magnetism because inside an atom the electric attraction is the overwhelmingly dominant force between the electron and the nucleus. Everything about "how tightly bound" comes from it.

PICTURE. Look at the figure: a small blue electron circles a positive core (red +), the arrow showing the inward pull. Two knobs control how strong that pull is — the distance and the charge in between. Those two knobs are exactly what silicon will change later.

Figure — N-type doping with donor atoms (phosphorus, arsenic)

Step 2 — The hydrogen atom: the simplest "one electron bound to +1" system

WHAT. The cleanest system with exactly one electron bound to a $+ 1$ core is the hydrogen atom. Nature already solved it for us: the energy needed to rip its electron away is a famous, measured number — $13.6 eV$ .

WHY this tool. A donor core, after keeping its four bonding electrons, has a net charge of $+ 1$ with one lonely electron circling it. That is structurally a hydrogen atom. So instead of solving quantum mechanics from scratch, we borrow hydrogen's answer and adjust it. This is the whole trick.

PICTURE. On the left, the hydrogen atom in vacuum: one proton, one electron, a tight little orbit, binding energy $13.6 eV$ . Note the units box — an electron-volt (eV) is the energy an electron gains crossing a $1$ -volt battery; $13.6 eV$ is a lot on the atomic scale.

Step 3 — Change #1: silicon screens the pull (the $ε_{r}$ knob)

WHAT. The donor is not in vacuum — it is buried inside a silicon crystal. The other silicon atoms full of electrons sit between the core and the 5th electron and partly cancel the pull. This weakening is called screening.

WHY this matters. Weaker pull ⇒ easier to free the electron ⇒ smaller binding energy. Silicon's screening ability is measured by its relative permittivity $ε_{r} \approx 11.7$ : the medium is $11.7 \times$ better at damping electric forces than vacuum. Concretely, we swap $ε_{0} \to ε_{r} ε_{0}$ everywhere.

PICTURE. Compare the two panels: in vacuum the field lines run straight and strong; inside silicon a cloud of silicon electrons crowds the space, and the surviving field arrow is much shorter. The pull is diluted.

Step 4 — Change #2: the electron feels "lighter" inside the lattice ( $m^{*}$ )

WHAT. A free electron in vacuum has mass $m_{0}$ . But an electron moving through the repeating pattern of silicon atoms responds to forces as if it had a different mass, the effective mass $m^{*} \approx 0.26 m_{0}$ . It behaves lighter.

WHY this tool. Effective mass is a bookkeeping trick: it lets us keep using simple $F = ma$ -style physics while hiding all the complicated push-and-pull from the crystal inside one number. In $E_{H}$ the mass sits on top, so a smaller $m^{*}$ makes the binding energy smaller too. We swap $m_{0} \to m^{*}$ .

PICTURE. A ball rolling across a bumpy, periodic track (the lattice) versus a smooth vacuum floor. On the bumpy track the same push moves it more readily — as if it weighed only $0.26$ of its true weight. The label shows $m^{*} / m_{0} = 0.26$ .

Step 5 — Plug both changes in: the donor binding energy

WHAT. Take hydrogen's formula and apply the two swaps at once: $m_{0} \to m^{*}$ and $ε_{0} \to ε_{r} ε_{0}$ .

WHY. Each swap is a physical fact about being inside silicon (Steps 3 & 4). Doing both gives the real energy to free the donor electron, $E_{D}$ . Because both changes shrink the energy, we expect $E_{D} ≪ 13.6 eV$ .

PICTURE. A "scaling machine": $13.6 eV$ enters the left, gets multiplied by $m^{*} / m_{0} = 0.26$ (small), then divided by $ε_{r}^{2} \approx 137$ (large), and drips out the right as $\approx 0.026 eV$ . The bar shrinks from a tall column to a sliver.

Step 6 — Why "tiny" means "already free": compare to thermal energy

WHAT. A number alone means nothing until we compare it to the energy available in the room. That budget is the thermal energy $k_{B} T \approx 0.026 eV$ at room temperature ( $T \approx 300 K$ ).

WHY this comparison. Atoms jiggle with heat, and each jiggle can transfer roughly $k_{B} T$ of energy to an electron. If the price to free the electron ( $E_{D}$ ) is smaller than or equal to the thermal energy available, then ordinary room warmth pays the bill — the electron gets kicked loose all by itself.

PICTURE. Two bars side by side: the "cost" bar $E_{D} = 26 meV$ and the "budget" bar $k_{B} T = 26 meV$ . They are essentially the same height — the little electron easily hops over the tiny wall into the conduction band (green arrow up).

Step 7 — Reality check & the edge cases

WHAT. Our one-line hydrogen model gave $26 meV$ . Measured values are $E_{D} (P) \approx 45 meV$ , $E_{D} (As) \approx 54 meV$ . Different donors, different numbers — and two extreme cases deserve their own look.

WHY these cases. A model you trust must survive the corners: what if it's very cold, and what if the donor is not perfectly hydrogen-like?

Cold case ( $T \to 0$ ): as $T$ drops, $k_{B} T$ shrinks below $E_{D}$ . The thermal budget can no longer pay, electrons freeze back onto donors, and $n$ collapses — "carrier freeze-out". The material stops conducting like a metal. Our formula for $E_{D}$ is unchanged; only whether the room can afford it changes.
Different donors (P vs As): the hydrogen model assumes the core is a featureless $+ 1$ point. Real donor cores have a bit of internal structure (called central-cell correction), so heavier As binds slightly tighter than P. That's why measured values sit a little above our $26 meV$ estimate — same order of magnitude, model still a win.

PICTURE. Left: an energy-level ladder near $E_{C}$ showing our estimate ( $26$ ), P ( $45$ ), As ( $54$ ) meV — all clustered just under the conduction band. Right: an $n$ -versus- $T$ curve with three zones — freeze-out (cold, $n$ small), saturation (room temp, $n \approx N_{D}$ , flat), intrinsic (very hot, $n$ shoots up as silicon itself ionises).

The one-picture summary

Everything above, compressed: hydrogen's $13.6 eV$ enters at the top; screening ( $\div ε_{r}^{2}$ ) and lightness ( $\times m^{*} / m_{0}$ ) crush it to $\approx 26 meV$ ; that sliver sits just under $E_{C}$ , and because room-temperature $k_{B} T$ matches it, the electron is already free.

Recall Feynman: tell the whole walkthrough in plain words

Ask "how hard is it to steal the phosphorus atom's spare electron?" Pretend that electron plus its $+ 1$ core is just a hydrogen atom — and hydrogen we already know costs $13.6 eV$ to break. But this atom lives inside silicon, which changes two things. First, all the surrounding silicon electrons crowd between the core and the spare, muffling the pull — that "muffling number" is $11.7$ , and because it appears squared, it slashes the energy by about $137$ . Second, the electron slips through the crystal's regular pattern as if it weighed only a quarter as much, cutting the energy by another factor of $0.26$ . Multiply it out: $13.6 eV$ becomes about $26 meV$ — five hundred times smaller. Now the killer comparison: room-temperature heat itself carries about $26 meV$ of jiggle energy per kick. Same size! So warmth alone frees the electron; every donor gives up its spare, and $n \approx N_{D}$ . Only if you freeze the crystal near absolute zero does the heat budget run dry and the electrons snap back onto the donors.

Recall Quick self-test

Which two silicon properties shrink $13.6 eV$ ? → the dielectric constant $ε_{r}$ (screening, divides by $ε_{r}^{2}$ ) and the effective mass $m^{*}$ (lightness, multiplies by $m^{*} / m_{0}$ ).
Why does $ε_{r}$ enter squared? → it replaces $ε_{0}$ , which appears as $ε_{0}^{2}$ in $E_{H}$ .
Numerically, $E_{D} \approx$ ? → $13.6 \times 0.26/11. 7^{2} \approx 0.026 eV = 26 meV$ .
Why does that mean "free at room temperature"? → because $k_{B} T \approx 26 meV \approx E_{D}$ , so thermal energy pays the ionisation cost.
What happens as $T \to 0$ ? → carrier freeze-out: electrons return to donors, $n$ collapses.