Bandstructures and Density of States
Recap
The Brillouin zone
Band structure
DOS
Phonons
Bandstructures and Density of States
P.J. Hasnip
DFT Spectroscopy Workshop 2009
Recap of Bloch's Theorem
Recap
The Brillouin zone
Band structure
DOS
Phonons
Bloch's theorem: in a periodic potential, the density has the same periodicity. The possible wavefunctions are all `quasi-periodic':
k (r) = eik.ruk (r).
We write uk (r) in a plane-wave basis as:
uk (r) =
cGk eiG.r,
G
where G are the reciprocal lattice vectors, defined so that G.L = 2m.
First Brillouin Zone
Recap
The Brillouin zone
Band structure
DOS
Phonons
Adding or subtracting a reciprocal lattice vector G from k leaves the wavefunction unchanged ? in other words our system is periodic in reciprocal-space too.
We only need to study the behaviour in the reciprocal-space unit cell, to know how it behaves everywhere. It is conventional to consider the unit cell surrounding the smallest vector, G = 0 and this is called the first Brillouin zone.
First Brillouin Zone (2D)
Recap
The Brillouin zone
Band structure
DOS
Phonons
The region of reciprocal space nearer to the origin than any other allowed wavevector is called the 1st Brillouin zone.
First Brillouin Zone (2D)
Recap
The Brillouin zone
Band structure
DOS
Phonons
The region of reciprocal space nearer to the origin than any other allowed wavevector is called the 1st Brillouin zone.
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