Scaling parameters in anomalous and nonlinear Hall effects depend on temperature

In the study of the anomalous Hall effect, the scaling relations between the anomalous Hall and longitudinal resistivities play the central role. The scaling parameters by definition are fixed as the scaling variable (longitudinal resistivity) changes. Contrary to this paradigm, we unveil that the electron-phonon scattering can result in apparent temperature dependence of scaling parameters when the longitudinal resistivity is tuned through temperature. An experimental approach is proposed to observe this hitherto unexpected temperature dependence. We further show that this phenomenon also exists in the nonlinear Hall effect in nonmagnetic inversion-breaking materials and may help identify experimentally the presence of the side-jump contribution besides the Berry-curvature dipole.

Figure 1

Temperature dependence of $\beta$'s of Eq. (3b) in the 2D massive Dirac model (5) in the presence of both zero-range scalar impurities and acoustic phonons. $T_{BG}$ is the Bloch-Gruneisen temperature [42] which plays the basic role instead of the Debye temperature in 2D metallic systems and marks qualitatively the lower boundary of the high-$T$ classical equipartition regime $\left(T>T_{BG}\right)$ where the resistivity is linear in $T$. When $T$ drops below the equipartition regime the $\beta$'s become $T$ dependent.

Figure 2

Temperature dependence of (a) ${\sigma }_{\text{AH}}^{\text{SJ,ei}}$ and ${\sigma }_{\text{AH}}^{\text{SJ,ep}}$, and of (b) ${\sigma }_{\text{AH}}^{\text{SJ}}+{\sigma }_{\text{AH}}^{\text{sk}}={\sigma }_{\text{AH}}-c$ for ${\epsilon }_F/{\mathrm{\Delta }}_0=2$ in model (5) in the presence of zero-range scalar impurities and acoustic phonons. Larger values of the parameter $\eta$ correspond to smaller impurity density. The dashed curves in (b) are obtained by assuming the scaling relation (17) in the presence of both scattering sources. In the weak-scattering regime $D_F\left|V_i\right|\ll 1$ and ${\mathrm{\Delta }}_0{\tau }_0^{\text{ei}}/\hslash \gg 1$. Thus in the calculation of ${\sigma }_{\text{AH}}^{\text{sk}}$ we take $D_F{V}_i={10}^{-3},{\mathrm{\Delta }}_0{\tau }_0^{\text{ei}}/\hslash ={10}^{j+2}$ for $\eta =3\times {10}^j\left(j=0,1,2\right)$, and ${\mathrm{\Delta }}_0{\tau }_0^{\text{ei}}/\hslash =5\times {10}^4$ for $\eta =1500(n_i$ is tuned).