Umklapp scattering occurs when a phonon (or electron, magnon, etc...) is created
ID: 2072943 • Letter: U
Question
Umklapp scattering occurs when a phonon (or electron, magnon, etc...) is created outside of the first Brilliouin zone of a crystal. Thermal resistivity is driven by a three phonon collision process where two low k phonons combine to create a phonon outside of the first Brillouin zone (k_1 + k_2 = k_ 3), which then manifests as a phonon traveling the opposite direction. Using the Debye model for a 3D crystal, estimate the temperature (as a fraction of the Debye temperature) where Umklapp scattering begins to freeze out.Explanation / Answer
A) Debye model for 3 D crystal
The Debye model is a solid-state equivalent of Planck's law of black body radiation, where one treats electromagnetic radiation as a gas of photons in a box. The Debye model treats atomic vibrations as phonons in a box. Most of the calculation steps are identical.
q is now vector--instead of dq we must integrate over dq=4pi q^2dq, q2=qx2+qy2+qz2.
There are three waves for every q two transversal and one longitudinal
Instead of integrating by dq we integrate by domega Number of modes:
g(omega),domega= 3V 4pi q^2(dq)(2(pi)^3)
Limits: lower limit omega= upper limit should be Theta_E, but we used omega^2=c^2 q^2 instead of exact equation (7), so we correct this by taking Theta_Dinstead:
int_0^Theta_Dg omega,domega=3N
C=3NkBD(T/Theta_D)
Theta_D directly proportional omega_0 /k_B --- Debye temperature
High temperatures:
for u >>1 upper limit in the integral is small (1/u),
and C=3NkB
Low temperatures:
for u << 1 integral is constant, and C proportional T^3
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