Hi everyone,
I've been trying to understand k-points recently, especially as it relates to DFT calculations, and I'm still confused about the Brillouin zone and an apparent paradox:
1. For periodic systems, per Bloch's theorem, we would expect physical quantities (e.g. charge density) to repeat with the same periodicity as the primitive cell, just as we know the potential for that system repeats with the periodicity of the unit cell ( V(r) = V(r+T), where T some linear combination of primitive vectors)
2. The periodicity of the unit cell can be conveniently described in frequencies via the Fourier transform. Since the unit cell has a very specific periodicity, there are only very specific allowable frequencies that can describe functions that repeat in the unit cell. These frequencies are given by the reciprocal lattice.
3. Here comes the confusing part: since any physical quantity should be a linear combination of points on the reciprocal lattice (since these are the only frequencies that match the periodicity of the unit cell), it seems natural that evaluation of physical quantities ought to be done by integrating over the reciprocal lattice, or "G-vectors." Instead, as everyone knows, it's the Brillouin zone, the first Wigner-Seitz cell in reciprocal space, that always gets integrated over. This is highly confusing to me.
I know there's something like any point in reciprocal space can be moved into the Brillouin zone by some combination of the primitive vectors in reciprocal space, but I'm not following why this is important.
Moving points of higher Brillouin zones into the first zone is the reduced zone scheme. This scheme works because physical experiments involving orbits of electrons in k space measure changes in k vectors versus the value of k. These changes can be translated into the first zone intact.
Integrals then can be performed over the reduced states in the zone as well as any original states in the first Brillouin zone. As a practical matter, the states near the Fermi surface are the most important in determining the characteristics of a semiconductor or insulator and the Fermi surface can be represented within the first Brillouin zone.
Moving points of higher Brillouin zones into the first zone is the reduced zone scheme. This scheme works because physical experiments involving orbits of electrons in k space measure changes in k vectors versus the value of k. These changes can be translated into the first zone intact.
Integrals then can be performed over the reduced states in the zone as well as any original states in the first Brillouin zone. As a practical matter, the states near the Fermi surface are the most important in determining the characteristics of a semiconductor or insulator and the Fermi surface can be represented within the first Brillouin zone.
In DFT calculations you define a limited direct-space cell that simulates your crystal under Periodic Boundary Conditions. To perform DFT you move in the K-Space. As well as the direct space, also the k-space keeps the PBC. It is then necessary to calculate only the point inside the 1 BZ, because all the other point are trivially obtained through the Bloch theorem (i.e. you would obtain a point in a cell next to the initial one, which by definition has to behave in the same way as the one you consider).
More simple, as you reduce the infinite crystal to a single unit cell, the same happens in the IBZ where all the reciprocal space is reproduced with PBC applied to 1BZ.
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