Energy balance and heat exchange in mesoscopic systems

The spatial distribution of the power transferred from the electrons to the phonon bath is calculated for several mesoscopic structures with use of the nonequilibrium Green’s-function formalism. We introduce the concept of a ‘‘current potential’’ ${\mathrm{\mu }}_{\mathit{J}}$ which has the property that the local power transfer is proportional to its spatial derivative. At low bias, the local power transfer between the electrons and the phonon bath can be separated into a component that is linear in the applied bias and a component that goes quadratically with bias. The linear component represents the thermoelectric heat, which can be locally positive or negative, while the quadratic component represents the Joule heating, which is positive everywhere. The numerical calculations presented in this paper are based on a local phonon model. However, the basic formulation is quite general. Energy-balance equations are derived in the absence of magnetic fields for systems with local or nonlocal electron-phonon interactions.