The notion of a particle in nonrelativistic quantum mechanics is very general: anything that can have a wavefunction, a probability amplitude for being at different locations, is a particle. In a metal, electrons and their associated elastic lattice deformation clouds travel as a particle. These effective electron-like negative carriers are electron quasiparticles, and these quasiparticles have a negative charge, which can be seen by measuring the Hall conductivity. Their velocity gives rise to a potential difference transverse to a wire in an external magnetic field which reveals the sign of the carriers.
But in a semiconductor, the objects which carry the charge can be positively charged, which is physically accurate--- a current in such a material will give an opposite sign Hall effect voltage.
To understand this, you must understand that the electron eigenstates in a periodic lattice potential are defined by bands, and these bands have gaps. When you have an insulating material, the band is fully filled, so that there is an energy gap for getting electrons to move. The energy gap generically means that an electron with wavenumber k will have energy:
$$ E= A + B k^2 $$
Where A is the band gap, and B is the (reciprocal of twice the) effective mass. This form is generic, because electrons just above the gap have a minimum energy, and the energy goes up quadratically from a minimum. This quadratic energy dependence is the same as for a free nonrelativistic particle, and so the motion of the quasiparticles is described by the same Schrodinger equation as a free nonrelativistic particle, even though they are complicated tunneling excitations of electrons bound to many atoms.
Now if you dope the material, you add a few extra electrons, which fill up these states. These electrons fill up k up to a certain amount, just like a free electron Fermi-gas and electrons with the maximum energy can be easily made to carry charge, just by jumping to a slightly higher k, and this is again just like a normal electron Fermi gas, except with a different mass, the effective mass. This is a semiconductor with a negative current carrier.
But the energy of the electrons in the previous band has a maximum, so that their energy is generically
$$ E = -Bk^2$$
Since the zero of energy is defined by the location of the band, and as you vary k, the energy goes down. These electrons have a negative nonrelativistic effective mass, and their motion is crazy--- if you apply a force to these electrons, they move in the opposite direction! But this is silly--- these electron states are fully occupied, so the electrons don't move at all in response to an external force, because all the states are filled, they have nowhere to move to.
So in order to get these electrons to move, you need to remove some of them, to allow electrons to fill these gaps. When you do, you produce a sea of holes up to some wavenumber k. The important point is that these holes, unlike the electrons, have a positive mass, and obey the usual Schroedinger equation for fermions. So you get effective positively charged positive effective mass carrier. These are the holes.
The whole situation is caused by the generic shape of the energy as a function of k in the viscinity of a maximum/minimum, as produced by a band-gap.
Bohr model holes
You can see a kind of electron hole already in the Bohr model when you consider Moseley's law, but these holes are not the physical holes of a semiconductor. If you knock out an electron from a K-shell of an atom, the object you have has a missing electron in the 1s state. This missing electron continues to orbit the nucleus, and it is pretty stable, in that the decay takes several orbits to happen.
The many-electron system with one missing electron can be thought of as a single-particle hole orbiting the nucleus. This single particle hole has a positive charge, so it is repelled by the nucleus, but it has a negative mass, because we are not near a band-gap, it's energy as a function of k is the negative of a free electron's energy.
This negative-mass hole can be thought of as orbiting the nucleus, held in place by its repulsion to the nucleus (remember that the negative mass means that the force is in the opposite direction as the acceleration). This crazy system decays as the hole moves down in energy by moving out from the nucleus to higher Bohr orbits.
This type of hole-description does not appear in the literature for Moseley's law, but it is a very simple approximation which is useful, because it gives a single particle model for the effect. The approximation is obviously wrong for small atoms, but it should be exact in the limit of large atoms. There are unexplained regularities in Moseley's law that might be explained by the single-hole picture, although again, this "hole" is a negative mass hole, unlike the holes in a positive doped semiconductor.
