  Solid-state control of the thermal conductivity of materials is of
exceptional interest for novel devices such as thermal diodes and switches.
Here, we demonstrate the ability to continuously tune the thermal conductivity
of nanoscale films of $\rm La_{0.5}Sr_{0.5}CoO_{3-\delta}$ (LSCO) by a factor
of over 5, via a room-temperature electrolyte-gate-induced non-volatile
topotactic phase transformation from perovskite (with $\delta \approx 0.1$) to
an oxygen-vacancy-ordered brownmillerite phase (with $\delta=0.5$), accompanied
by a metal-insulator transition. Combining time-domain thermoreflectance and
electronic transport measurements, model analyses based on molecular dynamics
and Boltzmann transport, and structural characterization by X-ray diffraction,
we uncover and deconvolve the effects of these transitions on heat carriers,
including electrons and lattice vibrations. The wide-range continuous
tunability of LSCO thermal conductivity enabled by low-voltage (below 4 V)
room-temperature electrolyte gating opens the door to non-volatile dynamic
control of thermal transport in perovskite-based functional materials, for
thermal regulation and management in device applications.
