Question #43837

A semiconductor device has a temperature-dependent resistance given by $R_S(T) = R_0 \exp\left(\frac{E_g}{2k_B T}\right)$, where $E_g$ is the energy band gap, $k_B$ is the Boltzmann constant, and $T$ is the absolute temperature. The device is connected in series with a fixed load resistor $R_L$ and an ideal DC voltage source $V_0$. The device dissipates heat to the environment at a rate $P_{diss} = K(T - T_a)$, where $K$ is the thermal conductance and $T_a$ is the ambient temperature. The system is currently operating in thermal equilibrium at a temperature $T$ such that $R_S > R_L$. Thermal runaway occurs if a small positive temperature fluctuation results in the rate of electrical heat generation increasing faster than the rate of heat dissipation. What is the critical minimum value of the thermal conductance $K$ required to prevent thermal runaway at this operating point?