Umklapp Scattering: Causes, Importance, and Phonon Momentum

[Altruist]

what causes this? why is it important? I don't understnad why the phonon momentum is different but the k-vectors outside the first Brillouin zone are physically equivalent to vectors inside it. I can see that physically the vector is shifted by pi back into the Brillouin zone by G. Is the k vector representing momentum of the phonon? if not, what is?

n process on the left, u process on the right

 

[bpsbps]

First draw a grid of points (2 or 3 points is fine) in 1D separated by the distance a. The grid points are the atoms

Next draw the curve Cos[k*x] for the below k values. Place the atoms on the curve. The oscillation is the displacement of atoms in a transverse lattice vibration.

k=0
k=0.5*pi/a
k=pi/a
k=2*pi/a
k=4*pi/a

Note that:

for k=0 all the atoms are uniformly shifted.

for k=0.5* pi/a it takes two periods for the atoms to be completely out of phase (one all the way up and the other all the way down)

for k=pi/a nearest neighbor atoms are completely out of phase

for k=2*pi/a again all the atoms are uniformly shifted. k=0 and k=2*pi/a are identical

for k=4*pi/a again the same... and so on for each additional n*2*pi/a (n=0,1,2,3,...)

So k values that are differ by 2*pi/a are identical.

You can test this by plotting Cos[(0.5*pi/a) * x] and Cos[(0.5*pi/a + 2*pi/a) * x].

 

[charng]

Umklapp scattering is a type of phonon scattering process in which the momentum of a phonon changes by a reciprocal lattice vector, causing it to scatter back into the first Brillouin zone. This phenomenon occurs due to the periodicity of the crystal lattice and is caused by the interaction between phonons and the periodic potential of the lattice.

This type of scattering is important because it can lead to a decrease in the thermal conductivity of a material, as the scattering of phonons reduces their ability to transport heat. This is particularly relevant in materials used for thermal insulation or in thermoelectric devices.

The difference in phonon momentum outside and inside the first Brillouin zone is due to the fact that the Brillouin zone is a mathematical construct and does not necessarily represent physical momentum. The k-vector in this case represents the wavevector of the phonon, which is related to its momentum but not equivalent to it. The shift by pi back into the Brillouin zone is due to the periodicity of the lattice and the interaction with the reciprocal lattice vector, which causes the phonon to scatter.

It is important to note that while the k-vector represents the wavevector of the phonon, it is not the same as the momentum of the individual atoms in the lattice. The momentum of the atoms can be affected by other factors such as the crystal structure and defects in the lattice.

In summary, umklapp scattering is an important phenomenon in solid state physics that can significantly affect the thermal properties of materials. It is caused by the periodicity of the crystal lattice and can lead to a change in the momentum of phonons, ultimately impacting their ability to transport heat. The k-vector represents the wavevector of the phonon, but not necessarily its momentum, which is influenced by other factors.