Q.Why are elemental dopants for Silicon or Germanium usually chosen from group XIII or group XV?

Silicon and germanium are group XIV (tetravalent) — each host atom forms four covalent bonds. A useful dopant should add or remove exactly one valence electron so it contributes one controllable carrier without disturbing the lattice.

• A group XV (pentavalent) atom has one extra electron. Four of its electrons bond to neighbours; the fifth is loosely bound and easily freed, giving an n-type donor.
• A group XIII (trivalent) atom has one electron too few, leaving one bond incomplete — a hole — so it acts as a p-type acceptor. …

The core idea — match the valence

Each silicon or germanium atom has four valence electrons and forms four covalent bonds with its neighbours. To make the crystal conduct in a controlled way, we substitute a few host atoms with impurity atoms that differ by just one valence electron. Group XIII and group XV are the immediate neighbours of group XIV in the periodic table, so their atomic sizes are similar and they fit into the lattice without distortion.

Group XV dopants (donors)

A pentavalent atom (phosphorus, arsenic, antimony) has five valence electrons. When it replaces a silicon atom, four electrons form the normal bonds and the fifth is left over, bound only weakly (about $0.045\ \text{eV}$ in silicon). This energy is far smaller than the $1.1\ \text{eV}$ band gap and comparable to thermal energy at room temperature, so the electron is easily released to the conduction band. The dopant becomes a fixed positive ion and donates one free electron — an n-type semiconductor.

Group XIII dopants (acceptors)

A trivalent atom (boron, aluminium, gallium) has only three valence electrons, so one of the four bonds is left incomplete — a hole. An electron from a neighbouring bond easily jumps in to fill it, moving the hole through the crystal like a positive carrier. The dopant becomes a fixed negative ion and provides one hole — a p-type semiconductor.

Why not other groups?

• Group I or II atoms have two or more electrons too few; they leave several unsatisfied bonds and create deep levels in the gap that trap carriers instead of releasing them. …

Method: Matching Dopant Valence to the Host Lattice's Bonding Requirement

The question is really asking "what property must a good dopant have?" -- and the answer follows from how many bonds each host atom forms and what a mismatch in electron count would do to the lattice.

Step 1 -- Note the host's bonding requirement.

Silicon and germanium are group XIV (tetravalent): each atom forms exactly four covalent bonds with its neighbours in the crystal.

Step 2 -- State the design goal for a dopant.

A useful dopant should differ from the host by exactly ONE valence electron, so that substituting it in creates precisely one loosely-bound extra carrier (electron or hole) without leaving multiple bonds unsatisfied.

Step 3 -- Check what a group XV (one more electron) atom does.

Four of its five valence electrons complete the normal four bonds; the fifth is left over, only weakly bound, and is easily freed at room temperature -- contributing one conduction electron. This is a clean, controllable n-type donor.

Step 4 -- Check what a group XIII (one fewer electron) atom does.

It has only three valence electrons, so one of the four bonds is left incomplete -- a hole. A neighbouring bond's electron can hop into this gap, letting the hole migrate like a positive carrier. This is a clean, controllable p-type acceptor.

Step 5 -- Check why atoms differing by two or more electrons are unsuitable. …