We have talked about bands but what do the electron states look like? The central idea around crystal theory is the the world looks the same from any lattice point, therefore so must all the ovservable electron properties.
6.2 Bloch's Theorem
Bloch (1920) showed that the electron wavefunction must have the form
Y (r
)=e
k.r
Un,k(r)
where Un,k is the cell periodic, this means it repeats with the lattice's periodicity. Note that
|Y (r)|2=Y |r+R|2
where R is any lattice vector. The subscript n in the electrons wavefunction is called the band index. Each band originates in an atomic or, if theres more than one atom per lattice point, molecular level.
k is the momentum quantum number. Electron states in crystals are plane-wave like. Remember though that Y (r) is a stationary state with perfectly defined k leads to a perfectly undefined position.
6.3 Consequences of Bloch's Theorem
In a perfect crystal Quantum Mechanic solutions are plane waves. This means that the electrons move through the crystal forever and without friction. The problem of solving Quantum Mechanics for 1018 particles has now been reduced to finding U(r) for a given k . Almost all of the crystal properties are contained in dispersion curves otherwise known as E(k) Curves. This implies that the graph of energies of Quantum Mechanic states against their Bloch wave vector looks like
Figure 6.1 - Bloch Diagram for Pure Silicon
Note that when electrons scatter from one state to another k is conserved (give or take a recipical lattice vector G ). We identify k as the Crystal Momentum belonging to the electrons and the crystal simultaneously, implying that it is not just for the electron.
6.4 Limitations
It applies only to infinite perfect crystals, not to regious close to crystal surfaces or defects/impurities. The latter of these flaws produce localised states which don't fall on E(k) curves. At finite temperatures real crystals are shaking and so the atoms are not in their proper positions. This leads to the electron scattering between k states to form electrical resistance. Electrons don't quite move without friction however they very nearly do so.
