Fig. 3

Comparison of spin and phonon Kondo effect. a Schematic dispersion relations for electrons in heavy electron systems (left) and phonons in an analogously defined “heavy” phonon system (right). The blue and red curves represent the non-interacting dispersive and localized entities, respectively, the violet curves the hybridized interacting states. As function of temperature, the system evolves from non-interacting well above the Kondo temperature to interacting well below it. For simplicity, phonon branches above ω_E are neglected. This is justified in real clathrates by the presence of multiple Einstein modes, resulting in multiple anticrossings^{13, 14}. The Debye model (blue line, right) assumes ω = v_sq where v_s is the sound velocity. The new dispersion relation (violet, right) is characterized by the group velocity v_g = ∂ω/∂q. It equals the sound velocity only at low wave vectors and frequencies. b Temperature-dependent phonon thermal conductivity, normalized to its maximum, calculated using a modified Callaway model (Supplementary Note 1) for various Einstein temperatures Θ_E (left) and corresponding data for the Ge-based type-I clathrate BCGG1.0 (Supplementary Table 1) and elemental Ge, electron irradiated and annealed at 77 K to similar defect densities⁶⁴ (right). c Specific heat and thermal expansion phonon Kondo anomaly, obtained by subtracting the experimental from the theoretical specific heat and thermal expansion curves of Ba₈Ga₁₆Ge₃₀ (left axis, see Supplementary Figs. 1b and 2b and Methods) and the corresponding entropy (right axis). d The inverse difference of calculated and experimental⁴¹ phonon thermal conductivities of Ba₈Ga₁₆Ge₃₀ (Supplementary Fig. 3b), showing the −ln T hallmark (full red line) of incoherent Kondo scattering in the spin Kondo effect above the Kondo temperature (dashed vertical line). The error bars represent the error of ±3% specified for the thermal conductivity data⁴¹