Umklapp scattering is not necessarily resistive

Since Peierls's pioneering work, it is generally accepted that phonon-phonon scattering processes consist of momentum-conserving normal scatterings and momentum-destroying Umklapp scatterings, and that the latter always induce thermal resistance. We show in this work that Umklapp scatterings are not necessarily resistive—no thermal resistance is induced if the projected momentum is conserved in the direction of heat flow. This distinction is especially important in anisotropic materials such as graphite and black phosphorous. By introducing a direction-dependent definition of normal and Umklapp scattering, we can model thermal transport in anisotropic materials using the Callaway model accurately. This accuracy is physically rooted in the improved description of mode-specific phonon dynamics. With the new definition, we predict that second sound might persist over much longer distances than previously expected.

Figure 1

Schematic illustration of normal scattering (N-scattering) and Umklapp scattering (U-scattering) processes, as conventionally defined. (a) N scattering does not induce thermal resistance in either the $x$ or $y$ direction, while (b) U scattering induces resistance in the $x$ direction but not in the $y$ direction.

Figure 2

Thermal conductivity of (a) BP in the zigzag direction and (b) graphite in the basal direction. Insets show atomic structures of BP and graphite, as well as the reciprocal space for graphtie.

Figure 3

Effective lifetime at 300 K under a tempearature gradient in the $x$ direction obtained for BP (a)–(d) and graphite (e)–(h) based on (a)–(e) exact iterative solution; (b), (f) the Callaway model with new definition of N- and R-scattering rates; (c), (g) the Callaway model with orignal definition; and (d), (h) the RTA model. For phonons enclosed by the white semicircle at the zone boundary in the $y$ direction, conventional U-scattering events can readily take place. However, these scatterings are considered N-scatterings based on the new definition.

Figure 4

Distribution of U-scattering percentage in the reciprocal space for BP with (a) new definition and (b) original definition of N- and U-scattering at 300 K, under a temperature gradient in the $x$ direction.

Figure 5

Propagation length of second sound as a function of temperature computed using the Callaway model with different definitions of N-scattering and U-scattering in (a) BP and (b) graphite.