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May31-09, 02:54 AM
P: 34
Quote Quote by saaskis View Post
Bloch wave function in a perfect metal is extended throughout the structure, so I would call the states delocalized.
No - i disagree. That's misconception that is why so many people think its triviality ask these questions. Bloch wavefunction is (orthonormal) basis function in which electronic state can be represented - in momentum space and yes indeed - its delocalized. However that doesn't mean that a particular electronic state (which can easily be in superposition of these eigen-states) is also delocalized... thats the whole point of this thread. this has been talked about near the beginnings and mentioned by several people.

Quote Quote by saaskis View Post
Remember that it is all about length scales. Your theory should be able to tackle the whole complex dependence on the relative sizes of Fermi wavelength, elastic mean free path, dephasing length, energy relaxation length and the size of your structure.
That is true, i agree! and i have been trying to talk about length scales and energy scales in the problem in several of my posts. nobody has ever commented on the content of those posts...

Quote Quote by saaskis View Post
Yes, but the lattice periodicity is quite awkward to take into account in a structure that is not periodic. Take graphene, for example. If you terminate the lattice on zigzag edge, you always have zero energy states at the Dirac point, no matter how large your structure. But in a perfectly periodic structure, the density of states at the Dirac point should be zero.
Sorry, not that its not interesting to talk about hexagonal 2D lattices, why is there a need to bring up some specifics again?

Let me define a problem:

we have a perfectly-periodic (no impurity) 1D lattice of scale 'a' and bounding potential of scale 'b'. we have non-interacting electrons (so ignoring elastic scattering here) and electron-phonon scattering (inelastic scattering). we also have a temperature T that describes both electron and phonon distributions (assuming equilibrium). This is a toy model of a solid - true. But adopting such model can we now answer the question: are electrons in localized or delocalized states? And even more interestingly, what aspects of condensed-matter physics such model recovers (we agree that it omits plenty, like nanotubes for instance).

So, as a starting point, can we, within the constraints stated above, come to some agreements, for example:

1) electrons are definitely delocalized because they are described by Bloch states (i'm saying thats wrong, but i'm open for discussion)

2) electrons are definitely localized (in a sense of classical particles, there are no other localizations -- we have perfect lattice without external fields).

3) neither of the above: the relevant energy/length scale is .....

4) the constraints are not sufficient to talk answer the posed question.

Can we 'solve' this problem (which is in essence how i took the original post and therefore found it interesting to participate in this thread) first?