The answer delves into solid state physics.
Basically in a large piece of material, the individual atomic orbitals mean nothing. You get larger, generalized states called the conducting band and valence bands. In the valence bands, electrons are confined to their respective atoms, but in energy states corresponding to the conduction band, they are free to move about.

Between these two "levels" is an energy barrier called the bandgap. In metals, the bandgap is extremely small and the electrons readily move about in the material, allowing conduction. In semiconductors, the bandgap is of a medium size, so current can flow, but not too well. In insulators, the bandgap is so large as to make the conduction band inaccessible.

When a dopant is added, you generally choose one that has either one more or one less electron than the parent material. Since the dopant is a tiny portion of the material, it is worked into the general band structure. However, the orbitals won't match up quite the same.

In an n-type, it's highest energy electron will fall into the conduction band. Effectively, we've changed the conducting properties by adding conducting electrons to the system.

In a p-type, we add something with one less electron than the main semiconductor. This introduces a "hole" in the valence layer. Electrons in the valence layer will jump from atom to atom to attempt to fill the hole, allowing conduction to occur in the valence layer.

The purpose of transition metals as dopants is very different. It's often to take advantage of their catalytic properties or d-orbital electron configuration for colors.

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Rinkoo Gupta
AskIITians Faculty