Switchability and controllability of a thermal transistor

A thermal transistor consists of three segments, the drain, the source, and the gate. We study the influence of the thermal resistance R_{G} in the gate segment systematically and reveal its key importance. In contrast to the negative differential thermal resistance between the drain and the source that has been studied extensively due to its indispensable function, the role of the gate segment has been neglected for long. It is found that the thermal transistor is switchable, i.e., it can be set exactly to on and off states by setting the control temperature T_{G} to T_{on} and T_{off}, only when R_{G} is lower than a critical value. The transistor is controllable, i.e., it can be set to any stable steady state, only when R_{G} is lower than an even lower critical value. In this case, we are able to build a useful device, a thermal sensor with arbitrarily large temperature sensitivity, by setting a proper value of R_{G}. However, when these conditions are not satisfied, multiple stable steady states may coexist then the heat currents cannot be exactly determined by the control temperature.