Explain how doping silicon with either phosphorus or gallium increases the electrical conductivity over that of pure silicon.

An intrinsic semiconductor is a pure semiconductor material without any impurities. In the case of silicon, its atoms are arranged in a crystalline structure. Each silicon atom has four valence electrons that form covalent bonds with neighboring silicon atoms, creating a full shell of electrons in the valence band. As a result, under normal conditions, there are very few free electrons or holes in the system, which makes the conductivity relatively low. When temperature increases, some of the bonds break, creating electron-hole pairs which enhance the conductivity.

Extrinsic semiconductors are obtained by introducing impurities or dopants to an intrinsic semiconductor, such as silicon, to increase its conductivity. The impurity atoms either donate extra electrons (N-type doping) to the system or accept electrons to create holes (P-type doping). This process controls the conductivity of the semiconductor much more effectively than relying on temperature.

Phosphorus has five valence electrons, one more than silicon. When phosphorus atoms are added to a silicon crystal, the extra electron from the phosphorus forms a weak bond with one of the neighboring silicon atoms. However, this bond is easily broken, releasing the free electron into the system. Since the extra electron is negatively charged, this type of extrinsic semiconductor is called N-type (negative charge carriers). The presence of these free electrons significantly increases the electrical conductivity of the doped silicon compared to pure silicon.

Gallium has three valence electrons, one less than silicon. When gallium atoms are introduced to a silicon crystal, they form covalent bonds with neighboring silicon atoms but leave one of the bonds incomplete due to the lack of one electron in the gallium atom. This creates a hole in the crystal structure which is positively charged. Electrons from neighboring atoms can fill this hole, but this process creates another hole in the system. Such movement of holes forms a positive current flow within the P-type semiconductor (positive charge carriers). This increased presence of holes in the doped semiconductor results in a higher electrical conductivity than that of pure silicon. In conclusion, doping silicon with either phosphorus or gallium significantly increases its electrical conductivity compared to that of pure silicon by introducing large numbers of free electrons (in the case of phosphorus doping) or holes (in the case of gallium doping) into the system.