Nonlinear and Nonreciprocal Transport Effects in Untwinned Thin Films of Ferromagnetic Weyl Metal ${\mathrm{SrRuO}}_3$

The identification of distinct charge transport features, deriving from nontrivial bulk band and surface states, has been a challenging subject in the field of topological systems. In topological Dirac and Weyl semimetals, nontrivial conical bands with Fermi-arc surface states give rise to negative longitudinal magnetoresistance due to chiral anomaly effect and unusual thickness dependent quantum oscillation from Weyl-orbit effect, which were demonstrated recently in experiments. In this work, we report the experimental observations of large nonlinear and nonreciprocal transport effects for both longitudinal and transverse channels in an untwinned Weyl metal of ${\mathrm{SrRuO}}_3$ thin film grown on a ${\mathrm{SrTiO}}_3$ substrate. From rigorous measurements with bias current applied along various directions with respect to the crystalline principal axes, the magnitude of nonlinear Hall signals from the transverse channel exhibits a simple $\sin \alpha$ dependence at low temperatures, where $\alpha$ is the angle between bias current direction and orthorhombic $[001{]}_{\mathrm{o}}$, reaching a maximum when current is along orthorhombic $[1\overline{1}0{]}_{\mathrm{o}}$. On the contrary, the magnitude of nonlinear and nonreciprocal signals in the longitudinal channel attains a maximum for bias current along $[001{]}_{\mathrm{o}}$, and it vanishes for bias current along $[1\overline{1}0{]}_{\mathrm{o}}$. The observed $\alpha$-dependent nonlinear and nonreciprocal signals in longitudinal and transverse channels reveal a magnetic Weyl phase with an effective Berry curvature dipole along $[1\overline{1}0{]}_{\mathrm{o}}$ from surface states, accompanied by 1D chiral edge modes along $[001{]}_{\mathrm{o}}$.

Popular Summary

Topological materials—ones with certain properties that are robust to perturbations—exhibit unusual electronic states on their surfaces. One challenge in the study of such materials is uncovering the charge transport signatures associated with these states. Here, we report on experimental observations of some surprising charge transport signatures in one such material—thin films of ${\mathrm{SrRuO}}_3$—and how similar signatures might be used to probe for topological surface and edge states more broadly.

${\mathrm{SrRuO}}_3$ belongs to a class of topological materials known as a Weyl semimetal, a solid-state crystal whose conduction and valence bands cross in single points known as Weyl nodes. When grown in thin films just 10 nm thick, ${\mathrm{SrRuO}}_3$ exhibits surface-dominant charge transport at very low temperatures.

In our experiments, we pattern a thin film of ${\mathrm{SrRuO}}_3$ into a sunbeam-shaped device for measuring charge transport anisotropy. Surprisingly, we observe both longitudinal and transverse Hall current rectification at temperatures below 10 K. The amplitude of the Hall rectification follows a simple sinusoidal dependence on the orientation of an applied bias current relative to one of the crystal axes. This dependence is orthogonal to that of the longitudinal current rectification. Using electronic band structure calculations, we show that these behaviors can be attributed to topological surface states and edge states with broken inversion symmetry.

Our observed current rectification effects in an approximately 10-nm ${\mathrm{SrRuO}}_3$ thin film appear to stem from the inversion-asymmetric surface and edge states in a topological Weyl system. These results not only highlight the significance of current rectification effects as a charge transport probe for topological surface and edge states but also feature potential applications in nonreciprocal electronics and nonlinear optics using topological materials.

Figure 1

The resistivity anisotropy in an untwinned SRO thin film. (a) An optical image of a patterned sunbeam SRO device with a film thickness of $t\approx 13.7\mathrm{nm}$. The white scale bar is $500\mathrm{\mu }\mathrm{m}$. Green arrows indicate the SRO orthorhombic crystalline directions. The right-hand panel illustrates a Hall bar device with the definitions for the measured longitudinal and transverse resistivity as ${\rho }_{\mathrm{L}}$ and ${\rho }_{\mathrm{T}}$, respectively. $\alpha$ angle is defined as the angle between the bias current direction (red arrow) and $[001{]}_{\mathrm{o}}$. (b) The $\alpha$-dependent ${\rho }_{\mathrm{L}}$ and ${\rho }_{\mathrm{T}}$ for three different field values of 0, $-1$, and $+1\mathrm{T}$. The experimental data agree well with the simulated curves (red curves) based on a resistivity anisotropy model. The upper, middle, and lower panels of (c) show the $T$-dependent ${\rho }_{\mathrm{L}}$, ${\rho }_{\mathrm{T}}$, and the ${\rho }_{\mathrm{T}}/{\rho }_{\mathrm{L}}$ ratio, respectively, for different Hall bar devices with $\alpha$ values ranging from 0° to 180°. See the text for more detailed descriptions.

Figure 2

Intrinsic AHE and enhanced diamagnetism in SRO thin films. (a) The hysteresis loops of Hall resistivity ${\rho }_{\mathrm{xy}}$ at different temperatures ranging from 2 to 180 K for a SRO thin film with $t\approx 13.7\mathrm{nm}$. (b) The magnitude of the Hall conductivity at zero field $|{\sigma }_{\mathrm{xy}}|$ is plotted as a function of the corresponding zero-field conductivity ${\sigma }_{\mathrm{xx}}$ for SRO thin films with different thicknesses ranging from 3.9 to 37.1 nm. $|{\sigma }_{\mathrm{xy}}|$ at low temperatures approach a constant value of about $2.0\times {10}^4{\mathrm{\Omega }}^{-1}{\mathrm{m}}^{-1}$, which is relatively close to the intrinsic anomalous Hall conductivity of $e^2/h{c}_0\approx 5.0\times {10}^4{\mathrm{\Omega }}^{-1}{\mathrm{m}}^{-1}$. (c) The field dependence of background subtracted magnetization $M^{\prime }$ for a thicker SRO film with $t\approx 37.1\mathrm{nm}$, exhibiting an increased diamagnetic response for $H\ge 2\mathrm{T}$ as $T$ drops. (d) The $T$ dependence of averaged slope of $dM/dH$ (red circles) in high field regime from 2 to 7 T shows a progressive decrease with decreasing $T$, indicating an enhanced diamagnetic response in SRO film at low $T$, which is in big contrast to the nearly $T$-independent slope of $dM/dH$ (black square) for a bare STO substrate.

Figure 3

The angular dependent nonlinear and nonreciprocal signals in SRO sunbeam device with $t\approx 13.7\mathrm{nm}$. The upper and lower panels of (a) show the $R_{\mathrm{L}}^{2\omega }$ and $R_{\mathrm{T}}^{2\omega }$, respectively, at eight different temperatures ranging from 1.4 to 10 K for $\alpha =90°$ Hall bar device. For $T\ge 2\mathrm{K}$, the curves of $R_{\mathrm{L}}^{2\omega }\text{−}{\mu }_{0}H$ and $R_{\mathrm{T}}^{2\omega }\text{−}{\mu }_{0}H$ are systematically shifted upward by multiples of 100 and $50\mathrm{\mu }\mathrm{\Omega }$, respectively, for clarity. Similarly, (b) shows the results for $\alpha =180°$ Hall bar device, sharing the same color codes and also axis labels. The $\mathrm{\Delta }{R}_{\mathrm{T}}^{2\omega }$ and $\mathrm{\Delta }{R}_{\mathrm{L}}^{2\omega }$ at zero magnetic field are defined in the lower and upper panel of (a) and (b), respectively. A summary of the $\alpha$-dependent $\mathrm{\Delta }{R}_{\mathrm{L}}^{\omega }$ and $\mathrm{\Delta }{R}_{\mathrm{L}}^{2\omega }$ ($\mathrm{\Delta }{R}_{\mathrm{T}}^{\omega }$ and $\mathrm{\Delta }{R}_{\mathrm{T}}^{2\omega }$) are shown in the upper (lower) panel of (c). The resulting $\mathrm{\Delta }{R}_{\mathrm{T}}^{2\omega }$ data fall relatively close to a simple $\sin \alpha$ curve shown as the red dashed line.

Figure 4

The $I$ and $T$ dependences of second harmonic signals for SRO sunbeam device with $t\approx 13.7\mathrm{nm}$. Upper panel of (a) shows the $R_{\mathrm{L}}^{2\omega }Y\text{−}{\mu }_{0}H$ curves for $\alpha =180°$ with four different bias currents ranging from 0.3 to 0.9 mA. Similarly, the lower panel of (a) shows the $R_{\mathrm{T}}^{2\omega }Y\text{−}{\mu }_{0}H$ curves for $\alpha =90°$ with four different bias currents. The two panels share the same color codes, and the curves are shifted upward for clarity. The corresponding $I$-dependent second harmonic signals of $\mathrm{\Delta }{R}_{\mathrm{L}(\mathrm{T})}^{2\omega }X$ and $\mathrm{\Delta }{R}_{\mathrm{L}(\mathrm{T})}^{2\omega }Y$ for $\alpha =180°$ and 90° are shown in upper panel and lower panel, respectively, of (b). For $\alpha =180°$, only $\mathrm{\Delta }{R}_{\mathrm{L}}^{2\omega }Y$ is sizable with practical $I$-linear dependence. On the contrary, for $\alpha =90°$ with bias current along $[1\overline{1}0{]}_{\mathrm{o}}$, $\mathrm{\Delta }{R}_{\mathrm{L}}^{2\omega }Y$ becomes nearly zero with no apparent $I$ dependence, but instead the transverse channel of $\mathrm{\Delta }{R}_{\mathrm{T}}^{2\omega }Y$ bears a clear $I$ dependence, inferring the presence of an effective $\overrightarrow{D}$ along $[1\overline{1}0{]}_{\mathrm{o}}$. Panel (c) shows the progressive increases in the magnitude with decreasing $T$ for both $\mathrm{\Delta }{R}_{\mathrm{L}}^{2\omega }Y$ of $\alpha =180°$ and $\mathrm{\Delta }{R}_{\mathrm{T}}^{2\omega }Y$ of $\alpha =90°$. Panels (b) and (c) share the same symbols and color codes.

Figure 5

The transverse magnetoresistance crossover and the emergence of quantum oscillations below 10 K. (a) A crossover of transverse MR from negative to positive in the weak field regime as $T$ decreases from 10 to 1.4 K. (b) The $T$ dependence of pure oscillating component of ${\rho }_{\mathrm{L}}$ ($\delta {\rho }_{\mathrm{L}}$) as a function of $1/{\mu }_{0}H$ for $\alpha =90°$, and the corresponding FFT spectra are shown in (c) which shares the same color code as (b). It reveals a dominant contribution from a 2D-like quantum oscillation with a frequency of about 28 T, where its oscillating amplitude grows rapidly below 10 K. The $T$ dependence of FFT amplitudes, $\mathrm{\Delta }{R}_{\mathrm{T}}^{2\omega }$, and $\mathrm{\Delta }{R}_{\mathrm{L}}^{2\omega }$ for $\alpha =90°$ and 180° are shown in (d). It indicates a close connection of the observed $\mathrm{\Delta }{R}_{\mathrm{L}(\mathrm{T})}^{2\omega }$ to the transverse MR crossover and 2D-like quantum oscillations below 10 K, where the surface dominant charge transport plays an important role.

Figure 6

A proposed scenario for the observed NRTE in Weyl metal thin film of SRO/STO. (a) The temperature dependence of the scattering plane-averaged SHG intensity from a SRO/STO film with $t\approx 35\mathrm{nm}$ measured in $S_{\mathrm{in}}–S_{\mathrm{out}}$ polarization geometry. The dashed gray line marks the Curie temperature $T_{\mathrm{c}}$. A constant background based on the SHG intensity above $T_{\mathrm{c}}$ [60] was subtracted off. The orange curve is a guide to the eye. The error bar shows the standard deviation of the intensities over 5 independent measurements. (b) shows the proposed real space scenario of the SRO/STO system with tilted Weyl nodes, giving rise to an effective BCD $\overrightarrow{D}$ from surface states along $[1\overline{1}0{]}_{\mathrm{o}}$ and 1D chiral edge modes along $[001{]}_{\mathrm{o}}$. (c) An illustration of a minimum model of Weyl system with one Weyl-node pair of chiral charges $+1$ and $-1$. For 2D slices between the Weyl-node pair, it gives a Chern number 1 with 1D chiral edge modes on the boundary of the system as demonstrated in the upper panel of (c). (d) Calculated Weyl nodes locations projected on $k_{\parallel }$ $\text{−}{k}_{\mathrm{z}}$ plane, and only Weyl nodes with energies of $|ϵ-{ϵ}_{\mathrm{F}}|\le 30\mathrm{meV}$ are included. Open and closed symbols correspond to different magnetization orientation conditions, and the red (blue) color indicates the corresponding chiral charge of $+1$ ($-1$). The shaded yellow region indicates the nonzero total Chern number, supporting the existence of 1D chiral edge modes along $k_{\mathrm{z}}$. (e) The band dispersions along $k_{\parallel }$ and $k_{\mathrm{z}}$ for Weyl node of ${\mathrm{W}}_{\mathrm{III}}^1$ show a tilted Weyl node.