Bloch's theorem for finite systems ?

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Hi all

I have a question regarding Bloch's theorem (also known as Floquet's theorem) and its use. I have seen in many solid state textbooks the famous problem of N coupled oscillators where one finds the dispersion relation analytically by using Bloch's theorem. However many times, authors add the comment that N tends to infinity.
My question is:
Is it "right" to use Bloch's theorem for finite,periodic structures ?
By "right" i mean if it is physically correct - not if one can do that as an approximation.

To my understanding (so far) Bloch's theorem has also to do with the boundary conditions (right?). In a finite structure there would be a problem at the endpoints unless the boundary conditions are cyclic. Then the theorem works, but can one do the same in a more general case without cyclic boundary conditions ?



Thanks

 

[genneth]

The study of finite-sized systems is massive and complicated. Any textbook on basic condensed matter will always work in the limit where N is very large. In most cases, you can expect a 1/sqrt(N) correction for very large but finite N. (There are times when this breaks down, but they are active research directions.)

 

[kanato]

In the strictest sense of the definitions, finite and periodic are mutually exclusive, so I assume you mean a finite structure which is approximately periodic in some region of space. Mainly I'm being pedantic on this point because you are asking a pedantic question of whether it's "right" to use Bloch's theorem rather than is it just approximate, and below where I use the term periodic, I really mean periodic in the mathematical sense, ie. periodic over the entire space.

For a free particle in quantum mechanics, you find that the momentum is a good quantum number, and this follows from the infinitesimal translational symmetry in space. The cyclic boundary condition you refer to that makes a finite solid periodic is generally called the Born-von Karman boundary condition, and from it you can derive Bloch's theorem. There you find the pseudo-momentum vector k is a good quantum number, and this follows from the discrete translational symmetry of space in a periodic solid. If you don't have the periodicity, you break the translational symmetry and k is no longer a good quantum number (very strictly speaking, of course).

Realistically, in a solid, 100 unit cells, ~50 nm or so is usually more than enough for the electrons to feel as if they are in an infinite periodic crystal. So Bloch's theorem works to a very excellent approximation for bulk properties of a crystal that is anywhere near macroscopically sized, and there is a whole crapload of experiments which show this.

 

[Modey3]

Hi all

Is it "right" to use Bloch's theorem for finite,periodic structures ?
By "right" i mean if it is physically correct - not if one can do that as an approximation.

Of course. This is how a lot of ab-initio codes that handle infinite systems work. This is how one does a calculation on a defect in a crystal. So to answer your question: It is right because it gives the correct answer, but it isn't "real" in the sense that in a real material defects would be repeated in a orderly fashion throughout space.