Cross-correlated measurements of thermal noise are performed to determine the electron temperature in nanopatterned channels of a GaAs/AlGaAs heterostructure at 4.2 K. Two-dimensional (2D) electron reservoirs are connected via an extended one-dimensional (1D) electron waveguide network. Hot electrons are produced using a current $I_{ \text{ h}}$ in a source 2D reservoir, are transmitted through the ballistic 1D waveguide and relax in a drain 2D reservoir. We find that the electron temperature increase ${ \Delta}T_{ \text{ e}}$ in the drain is proportional to the square of the heating current $I_{ \text{ h}}$, as expected from Joule's law. No temperature increase is observed in the drain when the 1D waveguide does not transmit electrons. Therefore, we conclude that electron-phonon interaction is negligible for heat transport between 2D reservoirs at temperatures below 4.2 K. Furthermore, mode control of the 1D electron waveguide by application of a top-gate voltage reveals that ${ \Delta}T_{ \text{ e}}$ is not proportional to the number of populated subbands $N$, as previously observed in single 1D conductors. This can be explained with the splitting of the heat flow in the 1D waveguide network.
Categories: cond-mat.mes-hall