Identifying geometric third-order nonlinear transport in disordered materials

Abstract

In third-order nonlinear transport, a voltage can be measured in response to the cube of a driving current as a result of the quantum geometric effects, which has attracted tremendous attention. However, in realistic materials where disorder scattering also contributes to nonlinear transport, identifying the geometric mechanisms remains a challenge. We find a total of 20 mechanisms of third-order nonlinear transport by developing a comprehensive theory that treats the geometric effects and disorder scattering on an equal footing. More importantly, we find that 12 of these mechanisms can be unambiguously identified, by deriving a scaling law that expresses the third-order nonlinear Hall conductivity as a polynomial in the linear longitudinal conductivity. We apply this theory to identify the geometric mechanisms of third-order nonlinear transport in materials both with and without time-reversal symmetry, such as 2D materials, topological materials, and altermagnets. This theory further promotes nonlinear transport as a probe of geometric effects and phase transitions in quantum materials.

Figures (3)

• Figure 1: (a) Third-order nonlinear Hall effect is measured experimentally as a third-harmonic transverse voltage $V_{3\omega}$ driven by a low-frequency current $I_\omega$ ($\omega \in$10-1000 Hz), or calculated theoretically as a current density $J_{3\omega}$ driven by an electric field $E_\omega$. (b) Schematic illustration of the wavefunction ($\psi$) manifold in momentum ($\mathbf{k}$) space crane, the Berry curvature and quantum metric are the imaginary (Im) and real (Re) parts of the quantum geometric tensor, respectively.
• Figure 2: We use symbols to visualize the mechanisms of third-order nonlinear transport, which consist of three major elements: the quantum geometry (+ $i$), side jump (), which shifts electron coordinates sideways by $\delta\mathbf{r}$, and skew scattering (), which asymmetrically deflects electrons. The horizontal and vertical axes indicate how these elements contribute to the components of the velocity $\dot{\mathbf{r}}$ (and their dependence on the electric field $\mathbf{E}$) and to the nonequilibrium distribution function $\delta f$, respectively. Here $v^\mathrm{band}$ is the group velocity defined by the band dispersion, and $f^{in}$ is the distribution function in the relaxation time approximation Mahan1990. The mechanisms of the third-order nonlinear transport are from the combinations of $\dot{\mathbf{r}}$ and $\delta f$ up to third-order of $\mathbf{E}$. Besides the known Berry curvature quadrupole () KTLow23prb, quantum metric quadrupole () Yang22prb, and third-order intrinsic () mechanisms Fang24prlWangJian23prb, we discover a total of 20 mechanisms, including 17 more mechanisms related to the disorder scattering, such as the third-order skew scattering (). Therefore, how to distinguish the geometric mechanisms in realistic materials, where disorder scattering is ubiquitous, remains a challenge. Mechanisms marked with a red asterisk do not require broken time-reversal symmetry ($\mathcal{T}$-even). Mechanisms without a red asterisk require broken time-reversal symmetry ($\mathcal{T}$-odd).
• Figure 3: The results of fitting the experimentally measured third-order nonlinear Hall conductivity $\chi_{y;xxx}$ as a function of the longitudinal conductivity $\sigma_{xx}$ (solid curves with circles), using the scaling law in Eq. (\ref{['Eq:Scaling']}). [(a1)--(a2)] Results for MoTe$_2$ in the temperature range $T \in [10,100]~\mathrm{K}$ (adapted from Ref. Lai21nn). (a1) The green solid and dashed lines represent the quantum metric quadrupole (QMQ, ) mechanism and the mixed Berry-curvature-plus-side-jump (BSJ, ) mechanism, respectively. (a2) The corresponding scaling-law weights of the QMQ and BSJ contributions at $\sigma_{xx}=5.28~\mathrm{S}\cdot\mu\mathrm{m}^{-1}$. [(b1)--(b2)] Results for 8-layer WTe$_2$ in the temperature range $T \in [2,80]~\mathrm{K}$ (adapted from Ref. Ye22prb). (b1) The green solid and purple dashed lines represent the QMQ mechanism and the Drude mechanism, respectively. (b2) The scaling-law weights of the Drude and QMQ contributions at $\sigma_{xx}=13.3~\mathrm{S}\cdot\mu\mathrm{m}^{-1}$. [(c1)--(c2)] Same analysis as in (b1) and (b2), but for FeSn in the temperature range $T \in [90,330]~\mathrm{K}$ (adapted from Ref. Sankar24prx). [(d1)--(d2)] Same as in (a1) and (a2), but for Fe$_5$GeTe$_2$ in the temperature range $T \in [40,275]~\mathrm{K}$ (adapted from Ref. Yu25ncomms). The dominant mechanism is the third-order intrinsic mechanism (TOI, ) when $T < 100~\mathrm{K}$. For $T > 100~\mathrm{K}$, the response is jointly dominated by the second-order side-jump (2SJ, ) mechanism and the mixed Berry-curvature-plus-skew-scattering (BSK, ) mechanism, with their scaling-law weights at $\sigma_{xx}=1.36~\mathrm{S}\cdot\mu\mathrm{m}^{-1}$ shown in the inset. The gray arrows indicate the values of $\sigma_{xx}$ used for extracting the weight ratios. The fitted scaling parameters and their confidence intervals for each experimental dataset are summarized in Sec. SIII of the Supplemental Material Supp.