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In p-type semiconductors, when the acceptor gains an electron, it creates a hole in the valence band of the semiconductor. From what I understand, the holes create the current in this case. Does it mean that the current flows in the valence band as opposed to the conduction band?

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We get p-type semiconductors by doping intrinsic semiconductors with atoms that can easily accept electrons, and thus, leave empty spaces or "holes" in the atoms from which they accept electrons. This means, the current flow here is due to holes.

For example, lets consider silicon (Si). Each Si atom has 4 electrons in its valence shell and the atoms are bonded together to form the Si crystal like this : https://commons.wikimedia.org/wiki/File:Covalent_bonding_in_silicon.svg. Suppose we dope a Boron (B) atom (3 electrons in its valence shell) in it to make it p-type. B replaces one atom of Si in the semiconductor. Now the surrounding Si atoms will see an empty spot in Bo's valence shell. So one electron from a Si atom will jump into this spot, leaving behind a vacancy or "hole" in the Si atom from which it has come out. Similarly, another electron from an adjacent Si atom may occupy this vacancy or recombine with the hole, and the chain follows.
So it is apparent that the conduction started with that one hole in the Boron atom and is guided by the creation and recombination of subsequent holes in the Silicon atoms. We thus term it, the hole current. We do not have an excess of free electrons here, which would be the case for n-type semiconductors - so yes, it is as opposed to the conduction band. The hole current occurs just due to jumping of bound electrons into a nearby available hole; that's also why we have the acceptor levels just above the valence band.

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Holes that is quasi-particle in semiconductor. Real current is electron flow through almost full valence band, but it is much easier describe it as holes flow.

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