Magnetoresistance and anomalous Hall effect in micro-ribbons of the magnetic Weyl semimetal Co$_3$Sn$_2$S$_2$

TL;DR

This study demonstrates that micro-ribbons of the magnetic Weyl semimetal Co$_3$Sn$_2$S$_2$ retain their intrinsic anomalous Hall effect after focused ion beam fabrication, enabling detailed transport studies at micro-scales.

Contribution

It introduces a method to fabricate microstructures of Co$_3$Sn$_2$S$_2$ using focused ion beam, preserving its key transport properties for future research.

Findings

Micro-ribbons exhibit large intrinsic anomalous Hall effect.

Focused ion beam can effectively pattern single crystals without degrading properties.

Transport properties are consistent with bulk measurements.

Abstract

Magnetic Weyl semimetals exhibit intriguing transport phenomena due to their non-trivial band structure. Recent experiments in bulk crystals of the shandite-type Co$_3$Sn$_2$S$_2$ have shown that this material system is a magnetic Weyl semimetal. To access the length scales relevant for chiral transport, it is mandatory to fabricate microstructures of this fascinating compound. We therefore have cut micro-ribbons (typical size $0.3~\times~3~\times~50$\mu$m^3$) from Co$_3$Sn$_2$S$_2$ single crystals using a focused beam of Ga$^{2+}$-ions and investigated the impact of the sample dimensions and possible surface doping on the magnetotransport properties. The large intrinsic anomalous Hall effect observed in the micro ribbons is quantitatively consistent with the one in bulk samples. Our results show that focused ion beam cutting can be used for nano-patterning single crystalline Co$_3$Sn$_2$S$_2$, enabling future transport experiments in complex microstructures of this Weyl semimetal.

Figures

Full text

magnetic Weyl semimetal, anomalous Hall effect, magnetotransport, single crystal, microstructure, Kagome lattice, Weyl fermions

I Introduction

Weyl semimetals (WSM) are an interesting class of materials due to their non-trivial topological properties. They posses topologically robust or symmetry protected bulk band crossings in momentum space referred to as Weyl nodes.Burkov and Balents (2011) For WSM those nodes always appear in pairs of opposite chirality at different positions in momentum space, connected by Fermi arcs.Weng et al. (2015) The Fermi arcs and Weyl nodes have been probed experimentally in many different materials including \ceTaAs, \ceTaP, \ceNbAs, \ceNbP, \ceWTe2 and \ceMoTe2 via photoemission spectroscopy and magnetotransport measurements.Xu et al. (2015); Lv et al. (2015); Huang et al. (2015); Wu et al. (2016); Crepaldi et al. (2017) A prominent example for the transport signature of Weyl fermions presents the chiral magnetic effect (CME) which gives rise to a positive correction to the magnetoconductance for parallel magnetic and electric field components.Zhang et al. (2016) Weyl nodes can have further implications for transport experiments by acting as sources for Berry curvature.Wan et al. (2011) In other words, the crossing points in momentum space can contribute significantly to the transport in the system if they are located near the Fermi level.Burkov (2015) In turn, the enhanced Berry curvature can give rise to a large intrinsic anomalous Hall effect (AHE), which can be calculated by integrating the Berry curvature of the occupied electronic states.Fang et al. (2003); Xiao, Chang, and Niu (2010) One therefore expects a large AHE for magnetic WSM with Weyl nodes close to the Fermi level. Recently shandite type \ceCo3Sn2S2 was identified as a magnetic WSM. More specifically, \ceCo3Sn2S2 exhibits a large AHE as well as CME in bulk single crystals.Liu et al. (2018); Wang et al. (2018) To explore the transport properties arising from Weyl physics in more detail, it is important to realize microstructures of this fascinating compound. Here we show that \ceCo3Sn2S2 micro ribbons (typical dimensions of 0.3~{}\times~{}3~{}\times~{}$$50\text{\,}\mathrm{\SIUnitSymbolMicro m}) with transport properties quantitatively comparable to those of bulk crystals can be fabricated via focused ion beam cutting in combination with conventional lithography techniques.

II Materials and Methods

\ce

Co3Sn2S2 crystallizes in the shandite crystal structure featuring metallic layers of \ceCo-Sn stacked along the c-axis in an ABC fashion where the Co atoms build a corner sharing Kagome lattice (cf. Fig. 1a).Kassem et al. (2015) Magnetometry measurements on that compound reveal a strong uniaxial anisotropy with an easy axis along the $c$-direction.Schnelle et al. (2013); Kassem et al. (2016, 2017) The Kagome lattice with out of plane easy axis makes this material an ideal candidate to study the quantum anomalous Hall effect.Zyuzin, Wu, and Burkov (2012)

The single crystals used in this work are grown by a self-flux method with Sn as flux. They grow in thin (hundreds of microns thick) hexagonal shaped flakes with the $c$-direction as the surface normal. We refer to Liu et. al (ref. 13) for more details on the growth parameters and bulk properties. We used a focused beam of \ceGa^2+-ions to cut the micro-ribbons from the bulk crystals. Focused ion beam (FIB) cutting is a viable tool to fabricate nano- and microstructures based on single crystals.Moll et al. (2010, 2017) In our fabrication procedure, a bulk crystal is aligned for the FIB preparation to obtain ribbons with a $c$-direction out of plane. Fig. 1b illustrates the orientation of the [001], [110] and [$\bar{\text{1}}\text{10}$] within the ribbon. The acceleration voltage for the \ceGa^2+-ions is kept constant at $30\text{\,}\mathrm{kV}$ during all stages. For FIB preparation the beam aperture size is denoted as current to obtain an energy product when combined with the acceleration voltage. The aperture size is subsequently reduced from $21\text{\,}\mathrm{nA}$ for the large scale material removal to $80\text{\,}\mathrm{pA}$ for the final shaping in order to reduce the surface deterioration and \ceGa^2+ implantation. After FIB preparation the ribbons are lifted out of the single crystal and transferred to a glas substrate for further device preparation. Contacts to the micro-ribbons are fabricated with a standard optical lithography- and lift-off process. To ensure ohmic contacts, the contact areas are cleaned in-situ with \ceAr^+-ion etching before the sputter deposition of $10\text{\,}\mathrm{nm}$ Cr and $200\text{\,}\mathrm{nm}$ Au. The samples are subsequently mounted onto chip carriers and inserted in a superconducting magnet cryostat for the transport experiments. In the following, we concentrate on one particular ribbon with dimensions of 0.35 x 4.2 x $46.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$3 which we very systematically studied, but note that the results from this ribbon are consistent with data obtained in further samples. Fig. 1c depicts the device layout used for the magnetotransport measurements. The longitudinal resistance $R_{\mathrm{xx}}$ is obtained by applying a current of $100\text{\,}\mathrm{\SIUnitSymbolMicro A}$ between contact 1 and 4 while detecting the voltage drop between contact 6 and 7. The transverse (Hall) voltage drop $V_{\mathrm{y}}$ is simultaneously detected between contacts 2 and 6 while sweeping the magnetic field ($\mathrm{\mu}_{0}H_{\mathrm{z}}$) along the $z$-direction. For calculating the resistivities, the ribbon thickness is determined to $350\text{\,}\mathrm{nm}$ by scanning electron microscopy.

III Results and Discussion

\ce

Co3Sn2S2 features a positive temperature coefficient of the resistivity for the entire temperature range from $300\text{\,}\mathrm{K}$ to $10\text{\,}\mathrm{K}$ studied here. Indeed, as evident from Fig. 2, the resistivity decreases from $3.2\text{\,}\mathrm{\SIUnitSymbolMicro\SIUnitSymbolOhm}\text{\,}\mathrm{m}$ to $0.5\text{\,}\mathrm{\SIUnitSymbolMicro\SIUnitSymbolOhm}\text{\,}\mathrm{m}$. The resistivity is proportional to the temperature for $T$>$T_{\mathrm{c}}$ suggesting that the resistivity is governed by electron-phonon interactions in that region.Kasuya (1956) A pronounced change of slope is evident at $T$=$174\text{\,}\mathrm{K}$ which is consistent with the Curie temperature of this compound. Schnelle et al. (2013); Kassem et al. (2016); Kubodera et al. (2006); Corps et al. (2015) The steeper $\mathrm{d}\rho/\mathrm{d}T$ for $T$<$T_{\mathrm{c}}$ is related to the spin-disorder contribution to the resistivity which is frozen out below $T_{\mathrm{c}}$ as the moments align. Kasuya (1956) For bulk samples a residual resistance ratio (RRR) of 6.8 was found, which is closely comparable to the RRR of 6.5 which we obtain in our micro-ribbons.Liu et al. (2018)

We now turn to the Hall response. The bulk crystals feature a giant AHE of up to $0.44\text{\,}\mathrm{\SIUnitSymbolMicro\SIUnitSymbolOhm}\text{\,}\mathrm{m}$ being identified as contribution of the Weyl nodes to the Berry curvature.Liu et al. (2018) It has been pointed out that the intrinsic AHE is sensitive to the exact position of the Weyl nodes with respect to the Fermi level because only Weyl nodes close to the Fermi level strongly contribute to the transport properties. The dependence of the anomalous Hall conductance (AHC) on the Fermi level position was calculated by Liu and coworkers.Liu et al. (2018) A sharp peak of up to $1.1\text{\,}\mathrm{S}\text{\,}{\mathrm{mm}}^{-1}$ is expected for the pure \ceCo3Sn2S2 compound. From these calculations one would expect a strong decrease of the AHC for an upwards shift of the Fermi energy whereas it is rather robust against lowering the Fermi level. In previous investigations of magnetotransport in FIB fabricated micro-ribbons it has been found that the implantation of \ceGa^2+-ions leads to a noticeable surface doping effect.Niemann et al. (2017) Thus, to gauge possible effects of the sample preparation on the location of the Weyl points, we compare the AHC of the micro-ribbon with the values reported for bulk crystals. Exemplary Hall measurements for temperatures in the vicinity of $T_{\mathrm{c}}$ are depicted in Fig. 3a. The anomalous Hall resistivity is deduced by linear fitting the high field part (\mu_{0}H_{\mathrm{z}}>$$5\text{\,}\mathrm{T}) and determining the intercept ($\rho_{\mathrm{H}}^{\mathrm{A}}$) of the linear fit with the ordinate. This corresponds to the transverse voltage response arising from the saturation magnetization without any external magnetic field. Nagaosa et al. (2010) To correct possible offsets, arising e.g. due to non-ideal contact placement, $\rho_{\mathrm{H}}^{\mathrm{A}}$ is antisymmetrized following ($\rho_{\mathrm{H}}^{\mathrm{A}}=\mathrm{(}\rho_{\mathrm{H}}^{\mathrm{A}}(B)-\rho_{\mathrm{H}}^{\mathrm{A}}(-B)\mathrm{)/}2$). Above $220\text{\,}\mathrm{K}$, the transverse resistivity ($\rho_{\mathrm{xy}}=-V_{\mathrm{y}}/I_{\mathrm{x}}\cdot t$) depends linearly on the field as expected for the ordinary Hall effect not shown here.Nagaosa et al. (2010) For lower temperatures an nonlinear Hall contribution emerges, resulting in s-shaped Hall traces and thus a finite $\rho_{\mathrm{H}}^{\mathrm{A}}$. The extracted $\rho_{\mathrm{H}}^{\mathrm{A}}$ is plotted in Fig. 3b as a function of temperature where it displays a steep increase below $T_{\mathrm{c}}$ with a maximum of $0.4\text{\,}\mathrm{\SIUnitSymbolMicro\SIUnitSymbolOhm}\text{\,}\mathrm{m}$ between $140\text{\,}\mathrm{K}150\text{\,}\mathrm{K}$ as it was already shown for bulk crystals (cf. open squares in Fig. 3). Interestingly, \ceCo3Sn2S2 shows a finite AHE above $T_{\mathrm{c}}$ up to about $220\text{\,}\mathrm{K}$. To evaluate the microscopic mechanisms involved in the AHE, it has proven helpful to study the evolution of $\sigma_{\mathrm{H}}^{\mathrm{A}}$ with $\sigma_{\mathrm{xx}}$ (cf. Fig. 3c). Since the mechanisms exhibit a distinct scaling behavior with $\sigma_{\mathrm{xx}}$ such a study allows to probe the involved effects.Nagaosa et al. (2010) \ceCo3Sn2S2 shows $\sigma_{\mathrm{H}}^{\mathrm{A}}\propto\sigma^{0}$ for a conductance between $0.7\text{\,}{\mathrm{m\SIUnitSymbolOhm}}^{-1}\text{\,}{\mathrm{m}}^{-1}1.9\text{\,}{\mathrm{m\SIUnitSymbolOhm}}^{-1}\text{\,}{\mathrm{m}}^{-1}$.

Such a plateau is characteristic for the intrinsic AHE contribution or the side-jump mechanism. According to the calculations in ref. 13, the intrinsic contribution should be in the order of $1.1\text{\,}\mathrm{S}\text{\,}{\mathrm{mm}}^{-1}$ in good quantitative agreement with both the bulk and micro-ribbon results. It is further characteristic for the intrinsic AHE contribution that its magnitude is independent of temperature (for $T$<$T_{\mathrm{c}}$) but only depends on the magnetization. This behaviour is observed for \ceCo3Sn2S2 and becomes evident by comparing Fig. 3d to $M$(H) curves reported previously.Sakai, Kamihara, and Matoba (2013); Corps et al. (2015) Taken together, Fig. 3b-d shows that the micro-ribbon and the bulk crystals show quantitatively similar transport properties.

We now turn to evolution of the magnetoresistance (MR) with magnetic field and temperature. The MR is obtained from the longitudinal resistance $\rho_{\mathrm{xx}}$ during sweeps of the magnetic field along the $z$-direction (out of plane). Figure 4a depicts exemplary MR measurements (open squares), where the MR is derived as ($\rho_{\mathrm{xx}}(H)~{}-~{}\rho_{\mathrm{xx}}(0))/\rho_{\mathrm{xx}}(0)$. Extracting the MR magnitude for different field strength as a function of temperature (cf. Fig. 4b) gives further insight into the different contributions to the MR. The open squares in Fig. 4b show the data for the bulk crystal extracted from ref. 13 in the same manner for $9\text{\,}\mathrm{T}$. \ceCo3Sn2S2 exhibits a small negative MR (<0.5%) in the paramagnetic state which strongly increases in vicinity of $T_{\mathrm{c}}$ to a magnitude of 3.3% at $9\text{\,}\mathrm{T}$. The compound displays a constant negative MR of 2.2% at $9\text{\,}\mathrm{T}$ between $110\text{\,}\mathrm{K}150\text{\,}\mathrm{K}$. Below $110\text{\,}\mathrm{K}$ a crossover to positive non-saturating MR is observed reaching a maximum of 22.8% at $9\text{\,}\mathrm{T}$. This behavior can be explained by two different contributions, first the suppression of spin-fluctuations and second the Lorentz deflection of charge carries due to an external magnetic field. In the ferromagnetic state a linear negative MR is expected from the suppression of spin fluctuations. Ueda (1976) A maximum of the negative MR at $T_{\mathrm{c}}$ is also observed in other ferromagnetic compounds due to the significant suppression of critical fluctuations near the ordering temperature.Masuda, Hioki, and Oota (1977) This behavior is only apparent in the microstructures presented here, whereas no sharp maximum of the negative MR around $T_{\mathrm{c}}$ can be observed in the bulk crystals (cf. open squares in Fig. 4b). The second contribution from the Lorentz deflection is proportional to the charge carrier mobility. Semimetals typically exhibit high mobilities at low temperatures. Ali et al. (2014); Liang et al. (2015); Kumar et al. (2017) Therefore, when decreasing the temperature the Lorentz deflection becomes the governing MR mechanism and this deflection can be captured with Kohler’s law where the MR is proportional to $H^{2}$. A simple model with a negative term proportional to H and a positive term scaling with $H^{2}$ is consistent with the measured MR in \ceCo3Sn2S2 (cf. solid lines in Fig 4).

IV Conclusions

It is not straight forward for microstructures of exotic materials, such as the recently discovered magnetic WSM \ceCo3Sn2S2, to resemble the fascinating properties observed in bulk samples. We have therefore, cut several micro-ribbons from \ceCo3Sn2S2 single crystals using FIB and investigated their transport properties by complementary magnetoresistance- and Hall effect measurements. The magnetoresistance (for field perpendicular to the current flow direction) can be consistently explained by contributions from the suppression of spin-fluctuations and Lorentz deflection of high mobility charge carriers in semimetals. More importantly, the quantitative agreement between the AHE observed in micro-ribbons and the one reported for bulk samples makes clear that the intriguing Weyl properties are not affected by the FIB microstructuring process. In other words, our data consistently shows that reducing the sample dimensions down to hundreds of nanometers does not change the salient features observed in the transport properties. In particular, the implantation of \ceGa^2+-ions (which inevitably occurs during the FIB process) does not significantly alter the properties of \ceCo3Sn2S2. We thus have shown that FIB cutting can be used for nano-patterning single crystalline \ceCo3Sn2S2, enabling future experiments in complex nano- and microstructures. This will allow further study of the implications of Weyl node contributions to the charge transport.

V Acknowledegments

The authors gratefully acknowledge the FIB preparation by Almut Pöhl and Tina Sturm. Financial support via DFG SFB 1143 (project B05, C05 and C08) is gratefully acknowledged.