  Recent discoveries of materials with ultra-low thermal conductivity open a
pathway to significant developments in the field of thermoelectricity. Here, we
conduct a comparative study of three chemically similar antimonides to
establish the root causes of their extraordinarily low thermal conductivity
($0.4-0.6$ Wm$^{-1}$K$^{-1}$ at 525 K). The materials of interest are: the
unconventional type-XI clathrate K$_{58}$Zn$_{122}$Sb$_{207}$, the tunnel
compound K$_{6.9}$Zn$_{21}$Sb$_{16}$, and the type-I clathrate
K$_8$Zn$_{15.5}$Cu$_{2.5}$Sb$_{28}$ discovered herein. Calculations of the
phonon dispersions show that the type-XI compound exhibits localized (i.e.,
rattling) phonon modes with unusually low frequencies that span the entire
acoustic regime. In contrast, rattling in the type-I clathrate is observed only
at higher frequencies, and no rattling modes are present in the tunnel
structure. Modeling reveals that low-frequency rattling modes profoundly limit
the acoustic scattering time; the scattering time of the type-XI clathrate is
half that of the type-I clathrate and a quarter of the tunnel compound. For all
three materials, the thermal conductivities are additionally suppressed by soft
framework bonding that lowers the acoustic group velocities, and structural
complexity that leads to diffusonic character of the optical modes.
Understanding details of thermal transport in structurally complex materials
will be crucial for developing the next generation of thermoelectrics.
