We investigate the dynamical control of the heat flux exchanged in near-field regime between a membrane made with a phase-change material and a substrate when the temperature of the membrane is tuned around its critical value. We show that in interaction with an external source of thermal radiation, this system is multistable and behaves as a thermal transistor, being able to dynamically modulate and even amplify super-Planckian heat fluxes. This behavior could be used to dynamically control heat fluxes exchanged at the nanoscale in systems out of thermal equilibrium and to process thermal information employing suspended membranes.
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