Hot-electron effects in metals

When sufficient electrical power P is dissipated in a thin metal film at millikelvin temperatures, the electrons can be driven far out of thermal equilibrium with the phonons. For uniform power dissipation in a volume \ensuremath{\Omega} we show that the electrons attain a steady-state temperature ${\mathit{T}}_{\mathit{e}}$=(P/\ensuremath{\Sigma}\ensuremath{\Omega}+${\mathit{T}}_{\mathit{p}}^{5}$${)}^{1/5}$, where ${\mathit{T}}_{\mathit{p}}$ is the phonon temperature and \ensuremath{\Sigma} is a parameter involving the electron-phonon coupling. We have used a sensitive ammeter based on a dc superconducting quantum interference device (SQUID) to measure the Nyquist current noise in thin films of AuCu as a function of P, and thus inferred ${\mathit{T}}_{\mathit{e}}$. We fitted our data to the theory with the single parameter \ensuremath{\Sigma}, and found good agreement for \ensuremath{\Sigma}=(2.4\ifmmode\pm\else\textpm\fi{}0.6)\ifmmode\times\else\texttimes\fi{}${10}^{9}$ ${\mathrm{Wm}}^{\mathrm{\ensuremath{-}}3}$ ${\mathrm{K}}^{\mathrm{\ensuremath{-}}5}$. When we increased the volume of the resistor by attaching a thin-film cooling fin, there was a much smaller increase in ${\mathit{T}}_{\mathit{e}}$ for a given power dissipation in the resistor, in qualitative agreement with a simple model for nonuniform heating. We also measured the flux noise in dc SQUIDs at low temperatures, and found that the white noise was limited by heating of the electrons in the resistive shunts of the Josephson junctions. We were able to reduce these effects substantially by attaching cooling fins to the shunts.