  Effects of strong phonon anharmonicity of a type-I clathrate Ba$_{\rm
8}$Ga$_{\rm 16}$Sn$_{\rm 30}$ induced by quadruple-well potential of guest
atoms were investigated. Phonon transport including coherent interbranch
component was analyzed using a first-principles-based self-consistent phonon
(SCP) theory that gives temperature-dependent harmonic interatomic force
constants and by solving off-diagonal components of group velocity operator.
Experimentally observed thermal conductivities have been reasonably reproduced
by considering both lattice and electron contributions. Through the analysis
with the SCP theory, we found that hardening of guest modes leads to an
increase in lattice thermal conductivity at frequencies below those of
framework-dominant flat modes (< 40 cm$^{\rm -1}$), which finally results in
the slow decay and slight increase in the total lattice thermal conductivity
with increasing temperature. Detailed analyses revealed that the increase in
lattice thermal conductivity at low frequency is attributed to (a) the increase
in group velocities of phonon modes located at frequencies below that of the
flat guest modes and (b) abnormal increase in lifetimes of phonon modes located
between frequencies of the flat guest and framework modes with increasing
temperature. From an engineering point of view, this effect may lead to an
intriguing phenomenon, a larger decrease in thermal conductivity due to
nanostructuring at higher temperatures.
