Umklapp scattering

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

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 schematically shows the possible scattering processes of two incoming phonons with wave-vectors ("k"-vectors) "k"₁ and "k"₂ (red) creating one outgoing phonon with a wave vector "k"₃ (blue). As long as the sum of "k"₁ and "k"₂ stay inside the first Brillouin zone (gray squares) "k"₃ 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 "k"₁ and "k"₂ 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 high temperatures for low defect crystals. The thermal conductivity for an insulating crystal where the U-processes are dominant has 1/T dependance.

The name derives from the German word "umklappen" (to turn over). Rudolf Peierls in his autobiography "Bird of Passage" (ISBN 0-691-08390-8) states he was the originator of this phrase and coined it during his 1929 crystal lattice studies under the tutelage of Wolfgang Pauli. Peierls wrote "...I used the German term " Umklapp" (flip-over) and this rather ugly word has remained in use..." (page 43).