Scaling law and extrinsic mechanisms for time-reversal-odd second-order nonlinear transport

Time-reversal-odd ($\mathcal{T}$-odd) second-order nonlinear current response has been theoretically proposed and experimentally confirmed recently. However, the role of disorder scattering in the response, especially whether it contributes to the ${\ensuremath{\sigma}}_{xx}$-independent term, has not been clarified. Here, we derive a general scaling law for this effect, which accounts for multiple scattering sources. We show that the nonlinear conductivity is generally a quartic function in ${\ensuremath{\sigma}}_{xx}$. The microscopic mechanisms underlying each scaling term are revealed, which include multiple previously unknown extrinsic processes unique to $\mathcal{T}$-odd nonlinear transport. Besides intrinsic contribution, we find that extrinsic contributions also enter the zeroth-order term, and their values can be comparable to or even larger than the intrinsic one. This offers a solution to the conflicting interpretations of recent experiments. In addition, cubic and quartic terms must involve skew scattering, and they signal competition between at least two scattering sources. Our finding reveals the significant role of disorder scattering in $\mathcal{T}$-odd nonlinear transport, and it establishes a foundation for analyzing experimental results.