Sublinear temperature dependence of thermal conductivity in the incommensurate phase of TlInTe2

Thermal conductivity $(\ensuremath{\kappa})$ of semiconducting and insulating solids generally show inverse linear temperature dependence above the Debye temperature $({T}_{\mathrm{D}})$ owing to dominant phonon-phonon scattering, i.e, $\ensuremath{\kappa}\ensuremath{\propto}{T}^{\ensuremath{-}1}$. Recently, in ultralow $\ensuremath{\kappa}$ materials, $\ensuremath{\kappa}$ is found to decrease sublinearly ($\ensuremath{\kappa}\ensuremath{\propto}{T}^{\ensuremath{-}\ensuremath{\alpha}}$ where $0<\ensuremath{\alpha}<1$) above ${T}_{\mathrm{D}}$ as interbranch wavelike tunneling contribution becomes significant. Here we show that the deviation from linearity can be unprecedently large in incommensurate (IC) phases as exemplified by archetypal Zintl-like semiconductor ${\mathrm{TlInTe}}₂$. This happens because two mutually incompatible translational symmetries allow spatially and temporally varying phase shifts, thus giving rise to two new heat-carrying modes: phasons and amplitudons. Using comprehensive transport and spectroscopy measurements combined with first-principles simulations of multichannel thermal transport, i.e., phonon-phonon scattering and tunneling contributions, we find that new modes contribute nearly 10-30% of the total $\ensuremath{\kappa}$ near the IC transition. The origin of this IC transition is rooted in the Tl lone pair and large polarizability of the electronic cloud, whose fingerprints are visible in the phonon linewidths. Our study paves the way for understanding ultralow $\ensuremath{\kappa}$ in IC phases, particularly for charge-density-wave materials where IC modulation is ubiquitous.