Heat generation by electric current in mesoscopic devices
  We study the heat generation in a nano-device with an electric current passing through the device. For the first time, a general formula for the heat generation is derived by using the nonequilibrium Keldysh Green functions. This formula can be applied in both the linear and nonlinear transport regions, for time-dependent systems, and with multi-terminal devices. The formula is also valid when the nano-device contains various interactions. As an application of the formula, the heat generation of a lead-quantum dot-lead system is investigated. The dc and ac biases are studied in detail. We find several interesting behaviors that are unique to nanostructures, revealing significant difference from heat generation in macroscopic systems. 
Qing-feng Sun and X. C. Xie