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T-square resistivity without Umklapp scattering in dilute metallic Bi$_2$O$_2$Se

The electrical resistivity of Fermi liquids (FLs) displays a quadratic temperature ($T$) dependence because of electron-electron (e-e) scattering. For such collisions to decay the charge current, there are two known mechanisms: inter-band scattering (identified by Baber) and Umklapp events. However, dilute metallic strontium titanate (STO) was found to display $T^2$ resistivity in absence of either of these two mechanisms. The presence of soft phonons and their possible role as scattering centers raised the suspicion that $T$-square resistivity in STO is not due to e-e scattering. Here, we present the case of Bi$_2$O$_2$Se, a layered semiconductor with hard phonons, which becomes a dilute metal with a small single-component Fermi surface upon doping. It displays $T$-square resistivity well below the degeneracy temperature where neither Umklapp nor interband scattering is conceivable. We observe a universal scaling between the prefactor of $T^2$ resistivity and the Fermi energy, which is an extension of the Kadowaki-Woods plot to dilute metals. Our results imply the absence of a satisfactory theoretical basis for the ubiquity of e-e driven $T$-square resistivity in Fermi liquids.