  Clathrates are a materials class with an extremely low phonon thermal
conductivity, which is a key ingredient for a high thermoelectric conversion
efficiency. Here, we present a study on the type-I clathrate
La$_{1.2}$Ba$_{6.8}$Au$_{5.8}$Si$_{38.8}\square_{1.4}$ directed at lowering the
phonon thermal conductivity even further by forming mesoscopic wires out of it.
Our hypothesis is that the interaction of the low-energy rattling modes of the
guest atoms (La and Ba) with the acoustic modes, which originate mainly from
the type-I clathrate framework (formed by Au and Si atoms, with some vacancies
$\square$), cuts off their dispersion and thereby tilts the balance of phonons
relevant for thermal transport to long-wavelength ones. Thus, size effects are
expected to set in at relatively long length scales. The structuring was
carried out using a top-down approach, where the wires, ranging from 1260 nm to
630 nm in diameter, were cut from a piece of single crystal using a focused ion
beam technique. Measurements of the thermal conductivity were performed with a
self-heating $3\omega$ technique down to 80 K. Indeed, they reveal a reduction
of the room-temperature phonon thermal conductivity by a sizable fraction of
$\sim$40 % for our thinnest wire, thereby confirming our hypothesis.
