The thermal transport in partially trenched silicon nitride membranes has been studied in the temperature range from 0.3 to 0.6 K, with the transition edge sensor (TES), the sole source of membrane heating. The test configuration consisted of Mo/Au TESs lithographically defined on silicon nitride membranes 1 micron thick and 6 mm^2 in size. Trenches with variable depth were incorporated between the TES and the silicon frame in order to manage the thermal transport. It was shown that sharp features in the membrane surface, such as trenches, significantly impede the modes of phonon transport. A nonlinear dependence of thermal resistance on trench depth was observed. Partial perforation of silicon nitride membranes to control thermal transport could be useful in fabricating mechanically robust detector devices.
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