question on Bloch wave

[fermions]

hello,

I would like to know briefly how the Bloch wave is simliar to a wave function of an electron and what's the difference between them

also how did the Bloch wave affect the energy of the electronic band

I hope you can understand what I ask as I was just reading some elementary books about it

thanks

[timo]

If I remember correctly the Bloch wave is the part of the wavefunction that is periodic with the lattice (we are talking about wave functions in a crystal lattice, are we?). Or in other words: psi_k(x) = exp(i kx) B_k (x) with psi being the wavefunction and B_k being the lattice-periodic Bloch function.

Different k -their number is limited to the number of elemental crystal cells but tends towards infinity for macroscopic crystals- lead to different energies which is what you plot in the E(k) diagrams. Dunno about how the energy bands appear in this picture atm, but I'll look it up later.

I'm at work atm so I can't really give any more detailed answers but I'll give a more detailed answer when I'm at home later (isn't THAT long since I had my final exams in solid state physics so some knowledge should still be there ...).

[timo]

Addendum on how the different energy bands appear:

 

Writing the wavefunction as above has the disadvantage that this notation is not unique. Any factor exp(iKx) with K being a vector of the reciprocal lattice can be pulled out of the exp(i kx) term and applied to the Bloch wave, since exp(i Kx) is also lattice periodic.

 

You can get a unique representation by restricting k to the 1st Brillouin zone (that´s the set of vectors in the reciprocal space for which the origin is the closest reciprocal lattice point, in case you don´t know the Brillouin zones). So for any k in the exp(i kx) not being in the 1st Brillouin zone you substract a suited reciprocal lattice point K so that the resulting k' is in the first Brillouin zone: exp( i kx) = exp(i (k' + K)x) = exp(i k'x) * exp(i Kx). The exp(i Kx) is then considered part of the new Bloch wave. This new Bloch wave can result in a different energy than the old one.

Hence, for any given restricted k you have multiple solutions (because of having multiple Bloch waves) and multiple energies. That´s the energy bands (in 1D, each band would correspond to a certain K being substracted).

[fermions]

thanks... I understand a little bit now...

does it mean that k varies with the number of atoms in the lattice? and is the probability of finding an electron near an atom is greater than that of between the atoms? If so, is it because the atoms (or should I say the positive ions) attract the electrons so it changed the wave function of it?

[timo]

>> does it mean that k varies with the number of atoms in the lattice?

Not really. The k-value is a quantum number characterizing the quantum state. For phonons, there´s as many possible k-values as there are primitive crystal cells (but still an infinite number of vibrational states since the number of phonons in the crystal is arbitrary). For electrons, it´s almost the same, although I wouldn´t know the reason for it atm (it pobably comes from periodic boundary conditions).

 

>> and is the probability of finding an electron near an atom is greater than that of

>> between the atoms?

That entirely depends on the state the electron is in. But as a rule of thumb I´d guess that states with the electron density closer to the ions have lower energies than states with electron density farther away from the ions. Reason: The potential energy is lower if the electrons are closer to the positively charged ions.

 

>> If so, is it because the atoms (or should I say the positive ions) attract the

>> electrons so it changed the wave function of it?

There is something similar to that idea in solid state phyiscs, indeed. As you might or might not know, there are so called band gaps at the end of the brillouin zone. That is: The energy of the lowest band and the 2nd lowest band do not have the same value at the end of the Brillouin zone which they had if the electrons weren´t influenced by the ions. Due to reasons I can´t remember atm (the experimentalists' reasoning as being shown in Kittel seemed a bit weird to me - I once had a better one but I forgot it) you can assume the Bloch waves for k near the Brillouin zone to be standing sine- or cosine-waves. Both those waves have different electron densities near the ion. This results in a difference of the potential energy which results in the band gap.