  The phonon thermal transport properties of eight ternary intermetallic
semiconductors are investigated by accounting for higher-order four-phonon
scattering, phonon renormalization, and multi-channel thermal transport. The
commonly used lowest-order theory, which accounts only for three-phonon
scattering and without phonon renormalization, fails drastically for considered
materials and underpredicts the thermal conductivity by up to a factor of two.
The thermal conductivity decreases for three compounds and increases for five
compounds with the application of higher-order corrections owing to a
contrasting role of four-phonon scattering and phonon stiffening on the
predicted thermal conductivity. Using the higher-order theory, at a temperature
of 300 K, the lowest obtained thermal conductivity is 0.31 W/m-K for BiCsK2 and
three other compounds (SbCsK2, SbRbNa2, and SbRbK2) have thermal conductivities
lower than 0.5 W/m-K via the particle-like phonon transport channel. The
contribution from the wave-like coherent transport channel is lower than 0.05
W/m-K in all of these ultra-low thermal conductivity compounds. The
higher-order theory is a must for the correct description of thermal transport
physics, failing which the thermal transport is wrongly characterized as
multi-channel transport by the lowest-order theory.
