logo

SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 64 , Issue 1 : 217062(2021) https://doi.org/10.1007/s11433-020-1597-6

Spin excitations and spin wave gap in the ferromagnetic Weyl semimetal Co$_3$Sn$_2$S$_2$

More info
  • ReceivedMay 31, 2020
  • AcceptedJun 24, 2020
  • PublishedAug 24, 2020
PACS numbers

Abstract

We report a comprehensive neutron scattering study on the spin excitations in the magnetic Weyl semimetal Co$_3$Sn$_2$S$_2$ with a quasi-two-dimensional structure. Both in-plane and out-of-plane dispersions of the spin waves were revealed in the ferromagnetic state. Similarly, dispersive but damped spin excitations were found in the paramagnetic state. The effective exchange interactions were estimated using a semi-classical Heisenberg model to consistently reproduce the experimental $T_{\rm~C}$ and spin stiffness. However, a full spin wave gap below $E_g=2.3$ meV was observed at $T=4$ K. This value was considerably larger than the estimated magnetic anisotropy energy ($\sim$0.6 meV), and its temperature dependence indicated a significant contribution from the Weyl fermions. These results suggest that Co$_3$Sn$_2$S$_2$ is a three-dimensional correlated system with a large spin stiffness, and the low-energy spin dynamics can interplay with the topological electron states.


Acknowledgment

This work was supported by the National Key Research and Development Program of China (Grant Nos. 2017YFA0303100, 2017YFA0302900, 2016YFA0300500, 2017YFA0206300, and 2019YFA0704900), the National Natural Science Foundation of China (Grant Nos. 11974392, 11974394, 11822411, 51722106, 11674372, 11774399, 11961160699, and 12061130200), the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (CAS) (Grant Nos. XDB07020300, XDB25000000, and XDB33000000), and the Beijing Natural Science Foundation (Grant Nos. JQ19002, Z180008, and Z190009). EnKe Liu and HuiQian Luo are grateful for the support from the Youth Innovation Promotion Association of CAS (Grant Nos. 2013002, and 2016004). HongMing Weng thanks the support from the K. C. Wong Education Foundation (GJTD-2018-01).


References

[1] Yan B., Felser C.. Annu. Rev. Condens. Matter Phys., 2017, 8: 337-354 CrossRef ADS arXiv Google Scholar

[2] Tokura Y., Yasuda K., Tsukazaki A.. Nat. Rev. Phys., 2019, 1: 126-143 CrossRef ADS Google Scholar

[3] Zou J., He Z., Xu G.. npj Comput. Mater., 2019, 5: 96 CrossRef ADS arXiv Google Scholar

[4] Weng H., Yu R., Hu X., Dai X., Fang Z.. Adv. Phys., 2015, 64: 227-282 CrossRef ADS arXiv Google Scholar

[5] Shekhar C., Nayak A. K., Sun Y., Schmidt M., Nicklas M., Leermakers I., Zeitler U., Skourski Y., Wosnitza J., Liu Z., Chen Y., Schnelle W., Borrmann H., Grin Y., Felser C., Yan B.. Nat. Phys., 2015, 11: 645-649 CrossRef ADS arXiv Google Scholar

[6] He Q. L., Pan L., Stern A. L., Burks E. C., Che X., Yin G., Wang J., Lian B., Zhou Q., Choi E. S., Murata K., Kou X., Chen Z., Nie T., Shao Q., Fan Y., Zhang S. C., Liu K., Xia J., Wang K. L.. Science, 2017, 357: 294-299 CrossRef ADS arXiv Google Scholar

[7] Chen H., Niu Q., MacDonald A. H.. Phys. Rev. Lett., 2014, 112: 017205 CrossRef ADS arXiv Google Scholar

[8] Tang P., Zhou Q., Xu G., Zhang S. C.. Nat. Phys., 2016, 12: 1100-1104 CrossRef ADS arXiv Google Scholar

[9] Wang Z., Vergniory M. G., Kushwaha S., Hirschberger M., Chulkov E. V., Ernst A., Ong N. P., Cava R. J., Bernevig B. A.. Phys. Rev. Lett., 2016, 117: 236401 CrossRef ADS arXiv Google Scholar

[10] Chang G., Xu S. Y., Zheng H., Singh B., Hsu C. H., Bian G., Alidoust N., Belopolski I., Sanchez D. S., Zhang S., Lin H., Hasan M. Z.. Sci. Rep., 2016, 6: 38839 CrossRef ADS arXiv Google Scholar

[11] Xu Q., Liu E., Shi W., Muechler L., Gayles J., Felser C., Sun Y.. Phys. Rev. B, 2018, 97: 235416 CrossRef ADS arXiv Google Scholar

[12] Hua G., Nie S., Song Z., Yu R., Xu G., Yao K.. Phys. Rev. B, 2018, 98: 201116(R) CrossRef ADS arXiv Google Scholar

[13] Shi Y., Kahn J., Niu B., Fei Z., Sun B., Cai X., Francisco B. A., Wu D., Shen Z. X., Xu X., Cobden D. H., Cui Y. T.. Sci. Adv., 2019, 5: eaat8799 CrossRef ADS arXiv Google Scholar

[14] Zhang D., Shi M., Zhu T., Xing D., Zhang H., Wang J.. Phys. Rev. Lett., 2019, 122: 206401 CrossRef ADS arXiv Google Scholar

[15] Liu J. Y., Hu J., Zhang Q., Graf D., Cao H. B., Radmanesh S. M. A., Adams D. J., Zhu Y. L., Cheng G. F., Liu X., Phelan W. A., Wei J., Jaime M., Balakirev F., Tennant D. A., DiTusa J. F., Chiorescu I., Spinu L., Mao Z. Q.. Nat. Mater, 2017, 16: 905-910 CrossRef ADS arXiv Google Scholar

[16] Ye L., Kang M., Liu J., von Cube F., Wicker C. R., Suzuki T., Jozwiak C., Bostwick A., Rotenberg E., Bell D. C., Fu L., Comin R., Checkelsky J. G.. Nature, 2018, 555: 638-642 CrossRef ADS arXiv Google Scholar

[17] Liu E., Sun Y., Kumar N., Muechler L., Sun A., Jiao L., Yang S. Y., Liu D., Liang A., Xu Q., Kroder J., Süβ V., Borrmann H., Shekhar C., Wang Z., Xi C., Wang W., Schnelle W., Wirth S., Chen Y., Goennenwein S. T. B., Felser C.. Nat. Phys., 2018, 14: 1125-1131 CrossRef ADS arXiv Google Scholar

[18] Wang Q., Xu Y., Lou R., Liu Z., Li M., Huang Y., Shen D., Weng H., Wang S., Lei H.. Nat. Commun., 2018, 9: 3681 CrossRef ADS arXiv Google Scholar

[19] Liu D. F., Liang A. J., Liu E. K., Xu Q. N., Li Y. W., Chen C., Pei D., Shi W. J., Mo S. K., Dudin P., Kim T., Cacho C., Li G., Sun Y., Yang L. X., Liu Z. K., Parkin S. S. P., Felser C., Chen Y. L.. Science, 2019, 365: 1282-1285 CrossRef ADS arXiv Google Scholar

[20] Morali N., Batabyal R., Nag P. K., Liu E., Xu Q., Sun Y., Yan B., Felser C., Avraham N., Beidenkopf H.. Science, 2019, 365: 1286-1291 CrossRef ADS arXiv Google Scholar

[21] Weng H. M.. Sci. China-Phys. Mech. Astron., 2019, 62: 127031 CrossRef ADS Google Scholar

[22] Belopolski I., Manna K., Sanchez D. S., Chang G., Ernst B., Yin J., Zhang S. S., Cochran T., Shumiya N., Zheng H., Singh B., Bian G., Multer D., Litskevich M., Zhou X., Huang S. M., Wang B., Chang T. R., Xu S. Y., Bansil A., Felser C., Lin H., Hasan M. Z.. Science, 2019, 365: 1278-1281 CrossRef ADS arXiv Google Scholar

[23] Otrokov M. M., Klimovskikh I. I., Bentmann H., Estyunin D., Zeugner A., Aliev Z. S., Gaβ S., Wolter A. U. B., Koroleva A. V., Shikin A. M., Blanco-Rey M., Hoffmann M., Rusinov I. P., Vyazovskaya A. Y., Eremeev S. V., Koroteev Y. M., Kuznetsov V. M., Freyse F., Sánchez-Barriga J., Amiraslanov I. R., Babanly M. B., Mamedov N. T., Abdullayev N. A., Zverev V. N., Alfonsov A., Kataev V., Büchner B., Schwier E. F., Kumar S., Kimura A., Petaccia L., Di Santo G., Vidal R. C., Schatz S., Kiβner K., ünzelmann M., Min C. H., Moser S., Peixoto T. R. F., Reinert F., Ernst A., Echenique P. M., Isaeva A., Chulkov E. V.. Nature, 2019, 576: 416-422 CrossRef ADS arXiv Google Scholar

[24] Deng Y., Yu Y., Shi M. Z., Guo Z., Xu Z., Wang J., Chen X. H., Zhang Y.. Science, 2020, 367: 895-900 CrossRef ADS arXiv Google Scholar

[25] Park P., Oh J., Uhlírová K., Jackson J., Deák A., Szunyogh L., Lee K. H., Cho H., Kim H. L., Walker H. C., Adroja D., Sechovsky V., Park J. G.. npj Quant. Mater., 2018, 3: , CrossRef ADS arXiv Google Scholar

[26] Itoh S., Endoh Y., Yokoo T., Kawana D., Kaneko Y., Tokura Y., Fujita M.. J. Phys. Soc. Jpn., 2013, 82: 043001 CrossRef ADS arXiv Google Scholar

[27] Haldane F. D. M.. Phys. Rev. Lett., 2004, 93: 206602 CrossRef ADS arXiv Google Scholar

[28] Xiao D., Chang M. C., Niu Q.. Rev. Mod. Phys., 2010, 82: 1959-2007 CrossRef ADS arXiv Google Scholar

[29] Burkov A. A.. Phys. Rev. Lett., 2014, 113: 187202 CrossRef ADS arXiv Google Scholar

[30] Onoda M., S. Mishchenko A., Nagaosa N.. J. Phys. Soc. Jpn., 2008, 77: 013702 CrossRef ADS Google Scholar

[31] Itoh S., Endoh Y., Yokoo T., Ibuka S., Park J. G., Kaneko Y., Takahashi K. S., Tokura Y., Nagaosa N.. Nat. Commun., 2016, 7: 11788 CrossRef ADS Google Scholar

[32] Jenni K., Kunkemoller S., Brüning D., Lorenz T., Sidis Y., Schneidewind A., Nugroho A. A., Rosch A., Khomskii D. I., Braden M.. Phys. Rev. Lett., 2019, 123: 017202 CrossRef ADS arXiv Google Scholar

[33] Weihrich R., Anusca I., Zabel M.. Z. Anorg. Allg. Chem., 2005, 631: 1463-1470 CrossRef Google Scholar

[34] Weihrich R., Anusca I.. Z. anorg. allg. Chem., 2006, 632: 1531-1537 CrossRef Google Scholar

[35] Vaqueiro P., Sobany G. G.. Solid State Sci., 2009, 11: 513-518 CrossRef ADS Google Scholar

[36] Schnelle W., Leithe-Jasper A., Rosner H., Schappacher F. M., Pottgen R., Pielnhofer F., Weihrich R.. Phys. Rev. B, 2013, 88: 144404 CrossRef ADS Google Scholar

[37] Kassem M. A., Tabata Y., Waki T., Nakamura H.. J. Phys. Soc. Jpn., 2016, 85: 064706 CrossRef ADS Google Scholar

[38] Yang R., Zhang T., Zhou L., Dai Y., Liao Z., Weng H., Qiu X.. Phys. Rev. Lett., 2020, 124: 077403 CrossRef ADS arXiv Google Scholar

[39] Kassem M. A., Tabata Y., Waki T., Nakamura H.. Phys. Rev. B, 2017, 96: 014429 CrossRef ADS arXiv Google Scholar

[40] Guguchia Z., Verezhak J. A. T., Gawryluk D. J., Tsirkin S. S., Yin J. X., Belopolski I., Zhou H., Simutis G., Zhang S. S., Cochran T. A., Chang G., Pomjakushina E., Keller L., Skrzeczkowska Z., Wang Q., Lei H. C., Khasanov R., Amato A., Jia S., Neupert T., Luetkens H., Hasan M. Z.. Nat. Commun., 2020, 11: 559 CrossRef ADS Google Scholar

[41] Danilkin S. A., Yethiraj M.. Neutron. News., 2009, 20: 37-39 CrossRef Google Scholar

[42] Wu C. M., Deng G., Gardner J. S., Vorderwisch P., Li W. H., Yano S., Peng J. C., Imamovic E.. J. Inst., 2016, 11: P10009 CrossRef ADS Google Scholar

[43] G. Wang, T. Lu, RHEED Transmission Mode and Pole Figures (Springer, New York, 2014). pp.7-22. Google Scholar

[44] E. Farhi, Y. Debab and P. Willendrup, J. Neut. Res. 17, 5 (2013). Google Scholar

[45] Gong D., Xie T., Zhang R., Birk J., Niedermayer C., Han F., Lapidus S. H., Dai P., Li S., Luo H.. Phys. Rev. B, 2018, 98: 014512 CrossRef ADS arXiv Google Scholar

[46] Van Kranendonk J., Van Vleck J. H.. Rev. Mod. Phys., 1958, 30: 1-23 CrossRef ADS Google Scholar

[47] D. Price and F. Fernandez-Alonso, Neutron Scattering-Magnetic and Quantum Phenomena (Elsevier, London, 2015). Google Scholar

[48] Korotin D. M., Mazurenko V. V., Anisimov V. I., Streltsov S. V.. Phys. Rev. B, 2015, 91: 224405 CrossRef ADS arXiv Google Scholar

[49] Liechtenstein A. I., Katsnelson M. I., Antropov V. P., Gubanov V. A.. J. Magn. Magn. Mater., 1987, 67: 65-74 CrossRef Google Scholar

[50] Shen J., Zeng Q., Zhang S., Tong W., Ling L., Xi C., Wang Z., Liu E., Wang W., Wu G., Shen B.. Appl. Phys. Lett., 2019, 115: 212403 CrossRef ADS arXiv Google Scholar

  • Figure 1

    (Color online) (a) Crystal and magnetic structure of Co$_3$Sn$_2$S$_2$. (b) The location of Weyl points in the Brillouin zone along the $z$ axis. (c) Dispersion of the Weyl Hamiltonian on the $k_x=0$ plane. (d), (e) Temperature dependence of the magnetization and resistivity. (f), (g) Temperature dependence of the peak intensity at $Q=~(1,~0,~1)$, and the deduced ordered moment $m$ compared with the anomalous Hall conductivity $\sigma^{\rm~A}_{\rm~H}$. (h) Spin excitations at 6 meV along $\mathbf{Q}=[0,~0,~L]$ direction. The red lines represent two Gaussian-fittings based on $L=3$ data, which are normalized by the magnetic form factor for $L=6$ and 9. The horizontal bars indicate the calculated instrument resolution.

  • Figure 2

    (Color online) (a)-(d) In-plane and out-of-plane spin waves at $T=~8$ K and paramagnetic excitations at $T=~200$ K around $Q=$ (0, 0, 3). The high energy data at low $Q$ side in panel (c) and (d) are missing due to scattering restrictions, and strong contaminations from quasi-elastic scattering prevent measurements of data below 2 meV. (e), (f) Dispersion of the spin excitations obtained from two Gaussian-fittings of the raw data. The solids lines are fitting results with $q^2$-dependence.

  • Figure 3

    (Color online) (a) Effective exchange interactions between Co atoms in our first principle calculations. (b) Estimation of the effective correlation length of the exchange interactions (main panel) and the total energy of the moment rotation (inset) for both in-plane and out-of-plane cases.

  • Figure 4

    (Color online) (a) Energy dependence of the spin excitations at $T=$ 8, 200, and 300 K measured at Taipan. (b) High resolution energy dependence of the spin excitations at $T=$ 4, 150, and 175 K measured at Sika. (c), (d) Constant-energy scans for $E=2$ and 2.5 meV at $T=4$ K, and $E=$ 0.3 meV at $T=175$ K. The solid lines are Gaussian fittings, and the horizontal bars indicate the instrument resolution. (e) Temperature dependence of the spin dynamic susceptibility $\chi^{\prime\prime}(Q,\omega)$ at low energy. (f) Temperature dependence of the spin wave gap and fitting results by eq. (2) with different parameters. Inset shows the integral intensity $\chi^{\prime\prime}(\omega)$ up to 4 meV.

  • Table 1  

    Table 1Experimental and calculated results of $D$ and $T_{\rm~C}$

    Experiment$D_H$(8 K) $D_L$(8 K) $D_H$(200 K) $D_L$(200 K)
    (meVÅ$^2$) (meVÅ$^2$) (meVÅ$^2$) (meVÅ$^2$)
    Value $803\pm46$ $237\pm13$ $360\pm30$ $169\pm10$
    Calculation$D_{xx}$ $D_{yy}$ $D_{zz}$ $T_{\rm~C}$
    (meVÅ$^2$) (meVÅ$^2$) (meVÅ$^2$) (K)
    Value $945$ $833$ $656$ 167

Copyright 2020  CHINA SCIENCE PUBLISHING & MEDIA LTD.  中国科技出版传媒股份有限公司  版权所有

京ICP备14028887号-23       京公网安备11010102003388号