Multiple magnon modes in the Co$_3$Sn$_2$S$_2$ Weyl semimetal candidate

TL;DR

This study reveals multiple magnon modes in the Weyl semimetal candidate Co$_3$Sn$_2$S$_2$, showing spin-transfer effects at low currents, which could be promising for spintronics applications.

Contribution

First experimental observation of multiple magnon modes in a magnetic Weyl semimetal, demonstrating spin-transfer effects at low current densities in a single crystal.

Findings

Multiple $dV/dI$ spikes indicating magnon modes

Spin-transfer effects observed at low current densities

Potential for spintronics applications

Abstract

We experimentally investigate electron transport in kagome-lattice ferromagnet Co$_3$Sn$_2$S$_2$, which is regarded as a time-reversal symmetry broken Weyl semimetal candidate. We demonstrate $dV/dI(I)$ curves with pronounced asymmetric $dV/dI$ spikes, similar to those attributed to current-induced spin-wave excitations in ferromagnetic multilayers. In contrast to multilayers, we observe several $dV/dI$ spikes' sequences at low, $\approx$10$^4$ A/cm$^2$, current densities for a thick single-crystal Co$_3$Sn$_2$S$_2$ flake in the regime of fully spin-polarized bulk. The spikes at low current densities can be attributed to novel magnon branches in magnetic Weyl semimetals, which are predicted due to the coupling between two magnetic moments mediated by Weyl fermions. Presence of spin-transfer effects at low current densities in Co$_3$Sn$_2$S$_2$ makes the material attractive for applications in spintronics.

Equations

$$I_{sw}(H)=\alpha \gamma e\sigma [H+H_{an}-H_{ex}+2\pi M]/g(0),$$

Full text

I Introduction

A strong area of interest in condensed matter physics is topological semimetals armitage . Dirac semimetals host the 4-fold degenerate topologically protected Dirac points, which are the special points of Brillouin zone with three dimensional linear dispersion. In Weyl semimetals (WSM), by breaking time reversal or inversion symmetries, the 4-fold degeneration of Dirac points declines to the 2-fold one, so every Dirac point splits into two Weyl nodes with opposite chiralities. Similarly to topological insulators and quantum Hall insulators, Weyl semimetals have topologically protected surface states, which are Fermi arcs connecting projections of Weyl nodes on the surface Brillouin zone armitage . First experimentally investigated WSMs were non-centrosymmetric crystals with broken inversion symmetry. Spin- and angle- resolved photoemission spectroscopy data indeed demonstrate spin-polarized surface Fermi arcs das16 ; feng2016 .

Also, ferromagnetic and antiferromagnetic WSMs with broken time-reversal symmetry can be discussed armitage . There are only a few candidates of magnetically ordered materials for the realization of WSMs mag1 ; mag2 ; mag3 ; mag4 . Recently, giant anomalous Hall effect was reported kagome ; kagome1 for the kagome-lattice ferromagnet Co3Sn2S2, as an indication for the existence of a magnetic Weyl phase. Fermi arcs were also visualized kagome_arcs for Co3Sn2S2 by scanning tunneling spectroscopy.

Another specifics of time reversal symmetry broken WSMs is a large anomalous Hall conductivity kagome ; kagome1 . The anomalous Hall effect (AHE) manifests itself as non-zero Hall conductance in zero magnetic field. AHE is known in a large class of magnetically ordered materials due to the two qualitatively different microscopic mechanisms: extrinsic processes due to scattering effects, and an intrinsic mechanism connected to the Berry curvature ahe1 ; ahe2 ; ahe3 . The latter variant is realized for WSMs, where the Berry curvature is enhanced at Weyl nodes armitage .

It is well known, that the magnetically ordered materials allows spin-wave excitations. For example, current-induced excitation of spin waves, or magnons, was demonstrated in ferromagnetic multilayers at large electrical current densities myers ; tsoi1 ; tsoi2 ; katine ; single ; balkashin ; balashov . Spin-wave excitations are observed only for one current polarity and is presented as a sharp peak in differential resistance myers ; tsoi1 ; tsoi2 ; katine ; single ; balkashin ; balashov . Electric field assisted magnetization dynamics was also predicted weyl_magnon ; dirac_magnon for WSMs, which requires experimental investigations. In particular, novel magnon branches are predicted weyl_magnon in magnetic Weyl semimetals, which can be understood as a result of the coupling between two magnetic moments mediated by Weyl fermions.

Here, we experimentally investigate electron transport in kagome-lattice ferromagnet Co3Sn2S2, which is regarded as a time-reversal symmetry broken Weyl semimetal candidate. We demonstrate $dV/dI(I)$ curves with pronounced asymmetric $dV/dI$ spikes, similar to those attributed to current-induced spin-wave excitations in ferromagnetic multilayers. In contrast to multilayers, we observe several $dV/dI$ spikes’ sequences at low, $\approx$104 A/cm2, current densities for a thick single-crystal Co3Sn2S2 flake in the regime of fully spin-polarized bulk. The spikes at low current densities can be attributed to novel magnon branches in magnetic Weyl semimetals, which are predicted due to the coupling between two magnetic moments mediated by Weyl fermions. Presence of spin-transfer effects at low current densities in Co3Sn2S2 makes the material attractive for applications in spintronics.

II Samples and technique

Co3Sn2S2 single crystals were grown by the gradient freezing method. Initial load of high-purity elements taken in stoichiometric ratio was slowly heated up to 920∘ C in the horizontally positioned evacuated silica ampoule, held for 20 h and then cooled with the furnace to the ambient temperature at the rate of 20 deg/h. The obtained ingot was cleaved in the middle part. The Laue patterns confirm the hexagonal structure with $(0001)$ as cleavage plane. Electron probe microanalysis of cleaved surfaces and X-ray diffractometry of powdered samples confirmed stoichiometric composition of the crystal.

Despite it is possible to form contacts directly on the cleaved Co3Sn2S2 crystal plane, large samples are not suitable for transport experiments: it is impossible to create high current density in a thick sample, which is crucial for the current-induced magnetization experiments.

Instead, the leads pattern is formed on the insulating SiO2 substrate by lift-off technique after thermal evaporation of 100 nm Au. The gold leads are separated by 5 $\mu$m intervals, see Fig. 1 (a), and (b). Since the kagome-lattice ferromagnet Co3Sn2S2 can be easily cleaved along (0001) crystal plane, small (about 100 $\mu$m size and 1 $\mu$m thick) Co3Sn2S2 flakes are obtained by a mechanical cleaving method. Then we select the most plane-parallel flakes with clean surface, where no surface defects could be resolved with optical microscope. They are transferred to the Au leads pattern and pressed slightly with another oxidized silicon substrate. A special metallic frame allows us to keep the substrates parallel and apply a weak pressure to the sample. No external pressure is needed for a Co3Sn2S2 flake to hold on to a substrate with Au leads afterward. This procedure provides reliable Ohmic contacts, stable in different cooling cycles, which has been also demonstrated before cdas ; nbwte ; inwte .

We check by standard magnetoresistance measurements that our Co3Sn2S2 flakes demonstrate giant anomalous Hall effect ahe1 ; ahe2 ; ahe3 in normal to the flake’s plane magnetic field. Fig. 2 (a) shows hysteresis behavior and sharp switchings in Hall resistance $R_{xy}$, the switchings’ positions $\approx 0.5$ T even quantitatively coincide with the previously reported ones kagome ; kagome1 . Fig. 2 (b) shows positive, non-saturating longitudinal magnetoresistance, which is also consistent with the reported behavior kagome for Co3Sn2S2 semimetal. Thus, magnetoresistance measurements confirms high quality of our Co3Sn2S2 samples.

We study electron transport along the Co3Sn2S2 surface by a standard four-point technique. The principal circuit diagram is depicted in Fig. 1 (b). To obtain $dV/dI(I)$ characteristics, a large (up to 3 mA) dc current $I$ is additionally modulated by a low ($\approx$5 $\mu$A) ac component. We measure both dc ($V$) and ac ($\sim dV/dI$) components of the voltage drop in Fig. 1 (b) with a dc voltmeter and a lock-in, respectively, after a broad-band preamplifier. The lock-in signal is checked to be independent of the modulation frequency. The obtained $dV/dI(I)$ curves are qualitatively independent on the particular choice of current and voltage probes. We check by comparison of four- and three-point $dV/dI(I)$ curves, that contact resistance is neglegible, see also Refs cdas ; nbwte ; inwte for details.

The measurements are performed in a usual He4 cryostat equipped with a superconducting solenoid. Co3Sn2S2 magnetic properties arise from the kagome-lattice cobalt planes, whose magnetic moments order ferromagnetically kagome out of plane below 175 K. We do not see noticeable temperature dependence in the interval 1.4-4.2 K, so all the results below are obtained at 4.2 K.

III Experimental results

Examples of $dV/dI(I)$ characteristics are shown in Fig. 3 (a) and (b). The curves demonstrate clear non-Ohmic behavior: $dV/dI$ is rising with both bias current polarities without any saturation at high biases. This behavior is inconsistent with trivial reasons like tunneling through different sample defects, which generally leads to decreasing $dV/dI(I)$. Moreover, the experimental $dV/dI(I)$ curves are well reproducible for different samples with very different zero-bias resistance, cp. Fig. 3 (a) and (b). On the other hand, an overall symmetric increase in $dV/dI$ is usually attributed to electron scattering in the magnetically ordered materials concave_shape ; myers .

For the curves in Fig. 3 we observe well-developed $dV/dI$ spikes for certain (positive) current polarity. These $dV/dI$ features are well reproducible in different cooling cycles. The spike position is sensitive to the magnetic field, as depicted in Fig. 3, it moves to higher currents with increasing the magnetic field. In the same time, there is no noticeable hysteresis with the current sweep direction , see the inset to Fig. 3 (a). This behavior is very similar to one reported for ferromagnetic multilayers myers ; tsoi1 ; tsoi2 ; katine ; single ; balkashin ; balashov , where the $dV/dI$ spikes have been attributed to spin-wave excitation modes.

In addition, small spikes are sometimes present at the counter (negative) current polarity, as it is depicted in Fig. 3 (a) for $B$ = 2.5 T and for $B$ = 2.7 T in the inset. These minor spikes have qualitatively the same magnetic field behavior, so we regard only the major ones at positive currents below.

The colormap in Fig. 4 (a) demonstrates evolution of $dV/dI$ spikes with magnetic field, which is applied normally to the Co3Sn2S2 flake. Since the general $dV/dI(I)$ shape is practically independent of the magnetic field, see Fig. 3, it is subtracted from the data. Thus, the colors correspond only to the spikes’ amplitudes in Fig. 4 (a).

It is clear, that the $dV/dI$ spikes’ positions depend linearly on the magnetic field, as it has been also demonstrated myers ; tsoi1 ; tsoi2 ; katine ; single ; balkashin ; balashov for magnons katine in multilayers. In contrast to multilayers, we observe several excitation branches, which are highlighted by white dashed lines on the colormap. The line slopes coincide within experimental accuracy, they also independent of the field direction. The spikes are better visible at highest currents, where they can be demonstrated even in zero magnetic field.

On the other hand, the relative amplitudes of the spikes are different for different branches, the spikes even vanish and reappear in some regions of the map. Also, we observe splitting of a particular branch in high magnetic fields, see also Fig. 4 (b): while increasing the magnetic field, the spike’s amplitude is diminishing, an additional spike appears at different current position; in higher fields, two spikes coexist, see Fig. 4 (b). Such interplay generally reflects the ground state reconstruction eisen ; caf ; smet .

IV Discussion

$dV/dI(I)$ curves with an overall symmetric increase in $dV/dI$ and asymmetric $dV/dI$ spikes have been reported for spin valves myers ; tsoi1 ; tsoi2 ; katine ; single ; balkashin ; balashov . The spin valves are the sandwich structures, where spin-dependent scattering affects the magnetic moments of two spin-polarized layers, while their mutual orientation defines the differential resistance. We observe qualitatively similar effects in bulk properties of single-crystal Co3Sn2S2 flakes. Since Co3Sn2S2 is a quasi-two-dimensional ferromagnet, it can be regarded as a layered structure consisting of multiple Co monolayers. Thus, possible spin-transfer effects in a single Co layer should be discussed firstly.

For spin valves, the hysteresis in $dV/dI(I)$ curves in low magnetic fields indicates current-induced switching in the orientation of layers’ magnetic moments between parallel and antiparallel orientations. In the high-field regime, the high-bias $dV/dI$ spikes correspond to a precessing spin-wave state with increasing deviations from the parallel alignment, which leads to increases in $dV/dI$ resistance. In experiments on multilayers myers ; tsoi1 ; tsoi2 ; katine ; single ; balkashin ; balashov , spin-waves are excited by injection of extremely high ( 109 A/cm2) current density for thin, 2-4 nm, Co layers.

We estimate the maximum current density as $\approx 10^{4}$ A/cm2 for $I=1$ mA through $S\sim 1\times 10\mu$m2 contact area, since the current density is maximal near the contact. Below, we rely on the model of Slonczewski slonczewski , which was successfully adapted for quantitative analysis of spin-transfer effects in Co/Cu/Co nanopillars katine . All the model parameters are well known for Co katine .

Within the model slonczewski ; katine , the $dV/dI$ spikes positions $I_{sw}$ are described by

[TABLE]

where $\alpha$ is the damping parameter, $\gamma$ is the gyromagnetic ratio, $\sigma$ is the total spin of the free layer, $M$ is the magnetization, $H_{ex}$ and $H_{an}$ are the exchange and the anisotropy fields, respectively. The total spin is $\sigma=MV/\gamma\hbar$, where $V=Sd$ is the free layer’s volume. $g(0)\leq$ 0.25 is the scalar function, which depends on the mutual orientation of the free and static layers’ magnetization slonczewski . The $dI_{sw}/dH$ slope is defined by $\alpha\gamma e\sigma/g(0)$, so all the branches are parallel in Fig. 4 (a).

A single Co monolayer plays the role of the free layer and the rest of the sample is the static layer. Then, one may estimate the free layer thickness as $d\sim(dI_{sw}/dH)\cdot\hbar g(0)/\alpha eMS$. For known Co parameters katine , $\alpha$ = 0.005, $g(0)$ = 0.14, and $M$ can be estimated as $\approx 100$ emu/cm3 for Co3Sn2S2. It gives inappropriately small $d\approx 3\cdot 10^{-2}$ $\AA$. The damping parameter $\alpha$ can only be larger in a ferromagnet tsoi1 ; tsoi2 ; single , which leads to even smaller $d$. To obtain realistic $d\sim 4$ $\AA$, which is approximately equal to the distance between the Co layers in Co3Sn2S2, one should use very small $S\approx$ 100$\times$100 nm2, which is impossible even for inhomogeneous large 1$\times$10 $\mu$m2 planar contacts. We wish to note, that $dV/dI$ spikes are observed for different samples and are independent of the particular contacts, see Fig. 3 (a) and (b).

Thus, possible spin-transfer effects between Co layers can not explain the experimental results. The bulk of the sample is fully spin-polarized kagome above $\approx 0.5$ T, see Fig. 2, so we can not attribute $dV/dI$ spikes to any bulk spin structures like domains. The domains are always accompanied by hysteresis in $dV/dI(I)$ curves myers ; katine , which can not be seen in the inset to Fig. 3 (a). We have to search for Weyl specifics of the Co3Sn2S2 flake. In principle, there are two possibilities, which refer to surface or bulk WSM properties, respectively:

(i) Broken time-reversal symmetry WSMs are characterized by topologically protected Fermi-arc surface states, which survive in high magnetic fields armitage . Low current density $\approx 10^{4}$ A/cm2 can reflect the small total spin $\sigma$ in the surface state, which is unreachable in traditional ferromagnets.

(ii) An additional bulk magnon branch is predicted weyl_magnon in magnetic WSMs. Physically, this magnon excitation can be understood as a direct result of the coupling between two magnetic moments mediated by Weyl fermions. The branch is also described by the modified Landau-Lifshitz equation weyl_magnon , so $I_{sw}(H)$ is still linear slonczewski ; katine . However, the gapless nature of Weyl fermions leads to long-range correlation of this magnon excitation weyl_magnon , which results in the dramatic decrease dirac_magnon in the damping parameter $\alpha$. Thus, observation of low current density $\approx 10^{4}$ A/cm2 well correspond to the predicted weyl_magnon novel magnon mode in magnetic WSMs.

In both cases, spin-unpolarized current from Au leads is injected into the magnetically-ordered WSM, which creates spin precession, or, in other words, spin-wave excitation. The process is somewhat similar to the tip experiments tsoi1 ; tsoi2 ; single , while the current density is much smaller in our case. In multilayers, the asymmetry of $dV/dI$ spikes reflects the intrinsic asymmetry of the spin diode. The real Co3Sn2S2 flakes are also not symmetric, see Fig. 1 (a), which creates higher current density for one region of the sample and is responsible for the spikes asymmetry in Fig. 3. The latter variant (ii) is based on the solid theoretical background weyl_magnon ; dirac_magnon ; magnon-first and seems to be more realistic. In the terms of Eq. 1, different modes in Fig. 4 (a) are characterized by different fields $H_{ex}$ and $H_{an}$, due to the complicated band structure of the real Co3Sn2S2 WSM kagome_arcs , which is reflected in different onset current at zero magnetic field. Moreover, band bifurcation has been predicted in magnetic field weyl_magnon , which can be an origin of the $dV/dI$ spikes’ interplay in Fig. 4 (b).

V Conclusion

As a conclusion, we experimentally investigate electron transport in kagome-lattice ferromagnet Co3Sn2S2, which is regarded as a time-reversal symmetry broken Weyl semimetal candidate. We demonstrate $dV/dI(I)$ curves with pronounced asymmetric $dV/dI$ spikes, similar to those attributed to current-induced spin-wave excitations in ferromagnetic multilayers. In contrast to multilayers, there are several $dV/dI$ spikes’ sequences at low, $\approx$104 A/cm2, current densities for a thick single-crystal Co3Sn2S2 flake in the regime of fully spin-polarized bulk. We attribute $dV/dI$ spikes at low current densities to novel magnon branches in magnetic Weyl semimetals, which can be understood as a direct result of the coupling between two magnetic moments mediated by Weyl fermions. In this case, the observed splitting of the magnon branches in magnetic field may reflect the predicted band bifurcation for magnetic Weyl semimetal. Presence of spin-transfer effects at low current densities in Co3Sn2S2 makes the material attractive for applications in spintronics.

Acknowledgements.

We wish to thank V.T. Dolgopolov and Yu.S. Barash for fruitful discussions, and S.V Simonov for X-ray sample characterization. We gratefully acknowledge financial support partially by the RFBR (project No. 19-02-00203), RAS, and RF State task.