E2 Charge Carriers: Electrons and Holes

Topic

The effective density of states of the conduction band in this semiconductor material, together with the effective density of states of its valence band and the bandgap established between them by another model in this set, determine the intrinsic charge-carrier concentration of this undoped semiconductor material: thermal energy available at the material's temperature excites a number of electrons from the valence band across the bandgap into the conduction band, and it is this number of excited electrons—distributed between the two available effective densities of states—that sets the intrinsic charge-carrier concentration. A wider bandgap means that, at a given temperature, fewer electrons are excited across it, resulting in a lower intrinsic charge-carrier concentration.

This intrinsic charge-carrier concentration determines the behavior of the undoped semiconductor material: each electron excited by thermal energy from the valence band to the conduction band leaves behind a hole in the valence band; consequently, the undoped semiconductor material always contains an equal number of free electrons in the conduction band and holes in the valence band, with both quantities equal to the intrinsic charge-carrier concentration. A higher temperature leads to a higher intrinsic charge-carrier concentration—due to more electrons being excited across the bandgap—causing both the free electrons and the holes (which remain equal in number) to increase in the same proportion. Consequently, the intrinsic carrier concentration—and the equal numbers of free electrons and holes with which this model began—ensures that this same intrinsic concentration appears in any subsequent law of mass action describing other models in this set: the product of the free-electron concentration and the hole concentration in this semiconductor material remains fixed—at thermal equilibrium and a constant temperature—at the square of that intrinsic carrier concentration. This holds true even when the semiconductor is no longer undoped and a doping process—described in greater detail by another model in the set—shifts the concentration of one of the two carrier types far above the intrinsic level. Because this doping shifts the concentration of either free electrons or holes well above the intrinsic carrier concentration, the law of mass action dictates that the concentration of the other carrier type must fall correspondingly far below that level, thereby keeping the product fixed at the square of the intrinsic concentration established at the model's inception.

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