Umklapp scattering

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In physics in general, and in particular in condensed matter and particle physics, the systems of interest are the so-called interacting systems. Thus, particles or excitations, such as electrons or phonons may interact through various processes depending on the nature of their interaction. For example in quantum electrodynamics one finds processes called Moeller scattering, Compton scattering, etc.

Umklapp scattering (also U-process or Umklapp process) is an anharmonic phonon-phonon (or electron-phonon) scattering process creating a third phonon with a momentum k-vector outside the first Brillouin zone. Umklapp scattering is one process limiting the thermal conductivity in crystalline materials, the others being phonon scattering on crystal defects and at the surface of the sample.

Figure 1.: Normal process (N-process) and Umklapp process (U-process). While the N-process conserves total phonon momentum, the U-process changes phonon momentum.

Figure 2.: k-vectors exceeding the first Brillouin zone (red) do not carry more information than their counterparts (black) in the first Brillouin zone.

Figure 1 schematically shows the possible scattering processes of two incoming phonons with wave-vectors (k-vectors) k₁ and k₂ (red) creating one outgoing phonon with a wave vector k₃ (blue). As long as the sum of k₁ and k₂ stay inside the first Brillouin zone (gray squares) k₃ is the sum of the former two conserving phonon momentum. This process is called normal scattering (N-process).

With increasing phonon momentum and thus wave vector of k₁ and k₂ their sum might point outside the Brillouin zone (k'₃). As shown in Figure 2, k-vectors outside the first Brillouin zone are physically equivalent to vectors inside it and can be mathematically transformed into each other by the addition of a reciprocal lattice vector G. These processes are called Umklapp scattering and change the total phonon momentum.

Umklapp scattering is the dominant process for thermal resistivity at low temperatures for low defect crystals.