Low-temperature small-angle electron-electron scattering rate in Fermi metals

Abstract

Abstract Low-temperature elementary conductivity formula in Fermi metals is reviewed starting from Ohm’s law. This provides the background needed to understand why and how the equations exploited are complicated due to effective mass effect and complex scattering rate even in the presence of small-angle electron-electron scattering at low temperatures. Using the mathematical conditions and physical arguments exploited to derive the Drude conductivity formula, we arrive at our main result—the analytic scattering rate formula at low temperatures that gives rise to the famous T 2 dependence without any ad hoc constants. Our derivation formally proves that the formula, 1 / τ = ( A / ℏ ) ( k B T ) 2 / E F $1/\tau =\left(A/\hslash \right){\left({k}_{\text{B}}T\right)}^{2}/{E}_{\text{F}}$ first guessed by Ashcroft and Mermin to be correct where A = N impurity/4π 2 and N impurity is the number of impurities (or scattering centers) present in a given sample. We also highlight the discovery of a new fundamental physical constant, λ Arulsamy = 3 ℏ 2 ( 4 π ϵ 0 ) 2 / m el e 4 ${\lambda }_{\text{Arulsamy}}=\left[3{\hslash }^{2}{\left(4\pi {{\epsilon}}_{0}\right)}^{2}\right]/\left[{m}_{\text{el}}{e}^{4}\right]$ that associates quantum mechanical energy with that of thermal energy, and is also related to Rydberg constant.