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Transport, magnetic and optical properties of Weyl materials

Abstract

Weyl fermions were originally proposed as massless Dirac fermions in relativistic quantum field theory. In solids, Weyl fermions often appear when either time-reversal symmetry or spatial-inversion symmetry is broken, which lifts the Kramer’s degeneracy of Bloch states at each crystal momentum k. Weyl fermions in solids are characterized by two novel features: spin–momentum locking and a diverging Berry curvature corresponding to a magnetic monopole forming in momentum space. These two effects bring about many novel transport phenomena, magnetic excitations and nonlinear optical effects. In this Review, we discuss some of these phenomena in various systems, in particular, the anomalous Hall effect and magnon excitation in 3D metallic ferromagnets, transition-metal-oxide semiconductors and semimetals; the quantum anomalous Hall effect, axion insulator state and magnetic skyrmions at the 2D magnetic surface of topological insulators; and nonlinear phenomena in noncentrosymmetric Weyl materials.

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Fig. 1: Properties of the ferromagnetic state of SrRuO3.
Fig. 2: Magnetotransport in the half-Heusler alloy GdPtBi.
Fig. 3: Weyl semimetal phases observed in pyrochlore iridates.
Fig. 4: QAH and axion insulator states and their topological responses.
Fig. 5: Edge channels and skyrmions on the surface of topological insulators.
Fig. 6: Nonlinear Hall effect in bilayer WTe2.
Fig. 7: Quantized circular photovoltaic effect in Weyl semimetals.
Fig. 8: Giant SHG response of the Weyl semimetal TaAs.

References

  1. Dirac, P. A. M. The quantum theory of the electron. Proc. R. Soc. A 117, 610–624 (1928).

    Google Scholar 

  2. Weyl, H. Elektron und gravitation. I. Z. Phys. 56, 330–352 (1929).

    Google Scholar 

  3. Vafek, O. & Vishwanath, A. Dirac fermions in solids: from high-Tc cuprates and graphene to topological insulators and Weyl semimetals. Annu. Rev. Condens. Matter Phys. 5, 83–112 (2014).

    CAS  Google Scholar 

  4. Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018).

    CAS  Google Scholar 

  5. Yan, B. & Felser, C. Topological materials: Weyl semimetals. Annu. Rev. Condens. Matter Phys. 8, 337–354 (2017).

    Google Scholar 

  6. Bernevig, B. A., Weng, H., Fang, Z. & Dai, X. Recent progress in the study of topological semimetals. J. Phys. Soc. Jpn. 87, 041001 (2018).

    Google Scholar 

  7. Yang, B. J. & Nagaosa, N. Classification of stable three-dimensional Dirac semimetals with nontrivial topology. Nat. Commun. 5, 4898 (2014).

    CAS  Google Scholar 

  8. Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).

    CAS  Google Scholar 

  9. Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall effect. Rev. Mod. Phys. 82, 1539–1592 (2010).

    Google Scholar 

  10. Fang, Z. et al. The anomalous Hall effect and magnetic monopoles in momentum space. Science 302, 92–95 (2003).

    CAS  Google Scholar 

  11. Murakami, S. Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase. New J. Phys. 9, 356 (2007).

    Google Scholar 

  12. Burkov, A. A. & Balents, L. Weyl semimetal in a topological insulator multilayer. Phys. Rev. Lett. 107, 127205 (2011).

    CAS  Google Scholar 

  13. Fang, C., Gilbert, M. J., Dai, X. & Bernevig, B. A. Multi-Weyl topological semimetals stabilized by point group symmetry. Phys. Rev. Lett. 108, 266802 (2012).

    Google Scholar 

  14. Lu, L., Fu, L., Joannopoulos, J. D. & Soljacic, M. Weyl points and line nodes in gyroid photonic crystals. Nat. Photon. 7, 294–299 (2013).

    CAS  Google Scholar 

  15. Bradlyn, B. et al. Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals. Science 353, aaf5037 (2016).

    Google Scholar 

  16. Chiu, C.-K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).

    Google Scholar 

  17. Berry, M. V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984).

    Google Scholar 

  18. Adams, E. N. & Blount, E. I. Energy bands in the presence of an external force field—II: Anomalous velocities. J. Phys. Chem. Solids 10, 286–303 (1959).

    CAS  Google Scholar 

  19. Sundaram, G. & Niu, Q. Wave-packet dynamics in slowly perturbed crystals: gradient corrections and Berry-phase effects. Phys. Rev. B 59, 14915–14925 (1999).

    CAS  Google Scholar 

  20. Shindou, R. & Nagaosa, N. Orbital ferromagnetism and anomalous Hall effect in antiferromagnets on the distorted fcc lattice. Phys. Rev. Lett. 87, 116801 (2001).

    CAS  Google Scholar 

  21. Chen, H., Niu, Q. & MacDonald, A. H. Anomalous Hall effect arising from noncollinear antiferromagnetism. Phys. Rev. Lett. 112, 017205 (2014).

    Google Scholar 

  22. Nakatsuji, S., Kiyohara, N. & Higo, T. Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature. Nature 527, 212–215 (2015).

    CAS  Google Scholar 

  23. Kuebler, J. & Felser, C. Non-collinear antiferromagnets and the anomalous Hall effect. EPL 108, 67001 (2014).

    Google Scholar 

  24. Kuroda, K. et al. Evidence for magnetic Weyl fermions in a correlated metal. Nat. Mater. 16, 1090–1096 (2017).

    CAS  Google Scholar 

  25. Takahashi, K. S. et al. Anomalous Hall effect derived from multiple Weyl nodes in high-mobility EuTiO3 films. Sci. Adv. 4, eaar7880 (2018).

    CAS  Google Scholar 

  26. Nielsen, H. B. & Ninomiya, M. The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal. Phys. Lett. B 130, 389–396 (1983).

    Google Scholar 

  27. Fujikawa, K. & Suzuki, H. Path Integrals and Quantum Anomalies (Oxford Univ. Press, 2004).

  28. Fukushima, K., Kharzeev, D. E. & Warringa, H. J. Chiral magnetic effect. Phys. Rev. D. 78, 074033 (2008).

    Google Scholar 

  29. Parameswaran, S. A., Grover, T., Abanin, D. A., Pesin, D. A. & Vishwanath, A. Probing the chiral anomaly with nonlocal transport in three-dimensional topological semimetals. Phys. Rev. X 4, 031035 (2014).

    Google Scholar 

  30. Son, D. T. & Spivak, B. Z. Chiral anomaly and classical negative magnetoresistance of Weyl metals. Phys. Rev. B 88, 104412 (2013).

    Google Scholar 

  31. Ishizuka, H. & Nagaosa, N. Robustness of anomaly-related magnetoresistance in doped Weyl semimetals. Phys. Rev. B 99, 115205 (2019).

    CAS  Google Scholar 

  32. Weng, H., Fang, C., Fang, Z., Bernevig, B. A. & Dai, X. Weyl semimetal phase in noncentrosymmetric transition-metal monophosphides. Phys. Rev. X 5, 011029 (2015).

    Google Scholar 

  33. Huang, X. et al. Observation of the chiral-anomaly-induced negative magnetoresistance in 3D Weyl semimetal TaAs. Phys. Rev. X 5, 031023 (2015).

    Google Scholar 

  34. Xiong, J. et al. Evidence for the chiral anomaly in the Dirac semimetal Na3Bi. Science 350, 413–416 (2015).

    CAS  Google Scholar 

  35. Hirschberger, M. et al. The chiral anomaly and thermopower of Weyl fermions in the half-Heusler GdPtBi. Nat. Mater. 15, 1161–1166 (2016).

    CAS  Google Scholar 

  36. Liu, Z. K. et al. A stable three-dimensional topological Dirac semimetal Cd3As2. Nat. Mater. 13, 677–681 (2014).

    CAS  Google Scholar 

  37. Uchida, M. et al. Quantum Hall states observed in thin films of Dirac semimetal Cd3As2. Nat. Commun. 8, 2274 (2017).

    Google Scholar 

  38. Moll, P. J. W. et al. Transport evidence for Fermi-arc-mediated chirality transfer in the Dirac semimetal Cd3As2. Nature 535, 266–670 (2016).

    CAS  Google Scholar 

  39. Zhang, C. et al. Quantum Hall effect based on Weyl orbits in Cd3As2. Nature 565, 331–336 (2019).

    CAS  Google Scholar 

  40. Nishihaya, S. et al. Quantized surface transport in topological Dirac semimetal films. Nat. Commun. 10, 2564 (2019).

    Google Scholar 

  41. Potter, A. C., Kimchi, I. & Vishwanath, A. Quantum oscillations from surface Fermi arcs in Weyl and Dirac semimetals. Nat. Commun. 5, 5161 (2014).

    CAS  Google Scholar 

  42. Soluyanov, A. A. et al. Type-II Weyl semimetals. Nature 527, 495–498 (2015).

    CAS  Google Scholar 

  43. Wang, Z. et al. MoTe2: a type-II Weyl topological metal. Phys. Rev. Lett. 117, 056805 (2016).

    Google Scholar 

  44. Jiang, J. et al. Signature of type-II Weyl semimetal phase in MoTe2. Nat. Commun. 8, 13973 (2017).

    CAS  Google Scholar 

  45. Ali, M. N. et al. Large, non-saturating magnetoresistance in WTe2. Nature 514, 205–208 (2014).

    CAS  Google Scholar 

  46. Shimano, R. et al. Terahertz Faraday rotation induced by an anomalous Hall effect in the itinerant ferromagnet SrRuO3. EPL 95, 17002 (2011).

    Google Scholar 

  47. Onoda, M., Mishchenko, A. S. & Nagaosa, N. Left-handed spin wave excitation in ferromagnet. J. Phys. Soc. Jpn. 77, 013702 (2008).

    Google Scholar 

  48. Itoh, S. et al. Weyl fermions and spin dynamics of metallic ferromagnet SrRuO3. Nat. Commun. 7, 11788 (2016).

    CAS  Google Scholar 

  49. Jenni, K. et al. Interplay of electronic and spin degrees in ferromagnetic SrRuO3: anomalous softening of the magnon gap and stiffness. Phys. Rev. Lett. 123, 017202 (2019).

    CAS  Google Scholar 

  50. Witczak-Krempa, W., Chen, G., Kim, Y. B. & Balents, L. Correlated quantum phenomena in the strong spin-orbit regime. Annu. Rev. Condens. Matter Phys. 5, 57–82 (2014).

    CAS  Google Scholar 

  51. Machida, Y., Nakatsuji, S., Onoda, S., Tayama, T. & Sakakibara, T. Time-reversal symmetry breaking and spontaneous Hall effect without magnetic dipole order. Nature 463, 210–213 (2010).

    CAS  Google Scholar 

  52. Moon, E.-G., Xu, C., Kim, Y.-B. & Balents, L. Non-Fermi-liquid and topological states with strong spin-orbit coupling. Phys. Rev. Lett. 111, 206401 (2013).

    Google Scholar 

  53. Kondo, T. et al. Quadratic Fermi node in a 3D strongly correlated semimetal. Nat. Commun. 6, 10042 (2015).

    Google Scholar 

  54. Matsuhira, K., Wakeshima, M., Hinatsu, Y. & Takagi, S. Metal-insulator transitions in pyrochlore oxides Ln2Ir2O7. J. Phys. Soc. Jpn. 80, 094701 (2011).

    Google Scholar 

  55. Ueda, K., Fujioka, J. & Tokura, Y. Variation of optical conductivity spectra in the course of bandwidth-controlled metal-insulator transitions in pyrochlore iridates. Phys. Rev. B 93, 245120 (2016).

    Google Scholar 

  56. Witczak-Krempa, W., Go, A. & Kim, Y. B. Pyrochlore electrons under pressure, heat, and field: Shedding light on the iridates. Phys. Rev. B 87, 155101 (2013).

    Google Scholar 

  57. Ueda, K. et al. Spontaneous Hall effect in the Weyl semimetal candidate of all-in all-out pyrochlore iridate. Nat. Commun. 9, 3032 (2018).

    Google Scholar 

  58. Ueda, K. et al. Magnetic-field induced multiple topological phases in pyrochlore iridates with Mott criticality. Nat. Commun. 8, 15515 (2017).

    CAS  Google Scholar 

  59. Ueda, K. et al. Magnetic field-induced insulator-semimetal transition in a pyrochlore Nd2Ir2O7. Phys. Rev. Lett. 115, 056402 (2015).

    CAS  Google Scholar 

  60. Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009).

    CAS  Google Scholar 

  61. Das Sarma, S., Adam, S., Hwang, E. H. & Rossi, E. Electronic transport in two-dimensional graphene. Rev. Mod. Phys. 83, 407–470 (2011).

    Google Scholar 

  62. Kotov, V. N., Uchoa, B., Pereira, V. M., Guinea, F. & Castro Neto, A. H. Electron-electron interactions in graphene: Current status and perspectives. Rev. Mod. Phys. 84, 1067–1125 (2012).

    CAS  Google Scholar 

  63. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    CAS  Google Scholar 

  64. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    CAS  Google Scholar 

  65. Bernevig, B. A. & Hughes T. L. Topological Insulators and Topological Superconductors (Princeton Univ. Press, 2013).

  66. Tokura, Y., Seki, S. & Nagaosa, N. Multiferroics of spin origin. Rep. Prog. Phys. 77, 076501 (2014).

    Google Scholar 

  67. Nomura, K. & Nagaosa, N. Electric charging of magnetic textures on the surface of a topological insulator. Phys. Rev. B 82, 161401 (2010).

    Google Scholar 

  68. Han, J. et al. Room-temperature spin-orbit torque switching induced by a topological insulator. Phys. Rev. Lett. 119, 077702 (2017).

    Google Scholar 

  69. Yasuda, K. et al. Current-nonlinear Hall effect and spin-orbit torque magnetization switching in a magnetic topological insulator. Phys. Rev. Lett. 119, 137204 (2017).

    CAS  Google Scholar 

  70. Slonczewski, J. C. Current-driven excitation of magnetic multilayers. J. Magn. Magn. Mater. 159, L1–L7 (1996).

    CAS  Google Scholar 

  71. Jackiw, R. Fractional charge and zero modes for planar systems in a magnetic field. Phys. Rev. D. 29, 2375–2377 (1984).

    CAS  Google Scholar 

  72. Tokura, Y., Yasuda, K. & Tsukazaki, A. Magnetic topological insulators. Nat. Rev. Phys. 1, 126–143 (2019).

    Google Scholar 

  73. Chang, C.-Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).

    CAS  Google Scholar 

  74. Mogi, M. et al. Magnetic modulation doping in topological insulators toward higher-temperature quantum anomalous Hall effect. Appl. Phys. Lett. 107, 182401 (2015).

    Google Scholar 

  75. Qi, X. L., Hughes, T. L. & Zhang, S. C. Topological field theory of time-reversal invariant insulators. Phys. Rev. B 78, 195424 (2008).

    Google Scholar 

  76. Morimoto, T., Furusaki, A. & Nagaosa, N. Topological magnetoelectric effects in thin films of topological insulators. Phys. Rev. B 92, 085113 (2015).

    Google Scholar 

  77. Mogi, M. et al. Tailoring tricolor structure of magnetic topological insulator for robust axion insulator. Sci. Adv. 3, eaao1669 (2017).

    Google Scholar 

  78. Mogi, M. et al. A magnetic heterostructure of topological insulators as a candidate for an axion insulator. Nat. Mater. 16, 516–521 (2017).

    CAS  Google Scholar 

  79. Chang, C.-Z. et al. High-precision realization of robust quantum anomalous Hall state in a hard ferromagnetic topological insulator. Nat. Mater. 14, 473–477 (2015).

    CAS  Google Scholar 

  80. Allen, M. et al. Visualization of an axion insulating state at the transition between 2 chiral quantum anomalous Hall states. Proc. Natl Acad. Sci. USA 116, 14511–14515 (2019).

    CAS  Google Scholar 

  81. Peccei, R. D. & Quinn, H. R. CP conservation in the presence of pseudoparticles. Phys. Rev. Lett. 38, 1440–1443 (1977).

    CAS  Google Scholar 

  82. Okada, K. N. et al. Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state. Nat. Commun. 7, 12245 (2016).

    CAS  Google Scholar 

  83. Tse, W.-K. & MacDonald, A. H. Giant magneto-optical Kerr effect and universal Faraday effect in thin-film topological insulators. Phys. Rev. Lett. 105, 057401 (2010).

    Google Scholar 

  84. Maciejko, J., Qi, X.-L., Drew, H. D. & Zhang, S.-C. Topological quantization in units of the fine structure constant. Phys. Rev. Lett. 105, 166803 (2010).

    Google Scholar 

  85. Wu, L. et al. Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator. Science 354, 1124–1127 (2016).

    CAS  Google Scholar 

  86. Dziom, V. et al. Observation of the universal magnetoelectric effect in a 3D topological insulator. Nat. Commun. 8, 15197 (2017).

    CAS  Google Scholar 

  87. Yasuda, K. et al. Quantized chiral edge conduction on domain walls of a magnetic topological insulator. Science 358, 1311–1314 (2017).

    CAS  Google Scholar 

  88. Rosen, I. T. et al. Chiral transport along magnetic domain walls in the quantum anomalous Hall effect. npj Quantum Mater. 2, 69 (2017).

    Google Scholar 

  89. Yasuda, K. et al. Geometric Hall effects in topological insulator heterostructures. Nat. Phys. 12, 555–559 (2016).

    CAS  Google Scholar 

  90. Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotechnol. 8, 899–911 (2013).

    CAS  Google Scholar 

  91. Tokura, Y. & Nagaosa, N. Nonreciprocal responses from non-centrosymmetric quantum materials. Nat. Commun. 9, 3740 (2018).

    Google Scholar 

  92. Krstic´, V., Roth, S., Burghard, M., Kern, K. & Rikken, G. L. J. A. Magneto-chiral anisotropy in charge transport through single-walled carbon nanotubes. J. Chem. Phys. 117, 11315–11319 (2002).

    Google Scholar 

  93. Rikken, G. L. J. A. & Wyder, P. Magnetoelectric anisotropy in diffusive transport. Phys. Rev. Lett. 94, 016601 (2005).

    CAS  Google Scholar 

  94. Pop, F., Auban-Senzier, P., Canadell, E., Rikken, G. L. J. A. & Avarvari, N. Electrical magnetochiral anisotropy in a bulk chiral molecular conductor. Nat. Commun. 5, 3757 (2014).

    Google Scholar 

  95. Morimoto, T. & Nagaosa, N. Chiral anomaly and giant magnetochiral anisotropy in noncentrosymmetric Weyl semimetals. Phys. Rev. Lett. 117, 146603 (2016).

    Google Scholar 

  96. Ma, Q. et al. Observation of the nonlinear Hall effect under time-reversal-symmetric conditions. Nature 565, 337–342 (2019).

    CAS  Google Scholar 

  97. Sodemann, I. & Fu, L. Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials. Phys. Rev. Lett. 115, 216806 (2015).

    Google Scholar 

  98. Zhang, Y., Brink, J. V. D., Felser, C. & Yan, B. Electrically tuneable nonlinear anomalous Hall effect in two-dimensional transition-metal dichalcogenides WTe2 and MoTe2. 2D Mater. 5, 044001 (2018).

    CAS  Google Scholar 

  99. Morimoto, T. & Nagaosa, N. Scaling laws for nonlinear electromagnetic responses of Dirac fermion. Phys. Rev. B 93, 125125 (2016).

  100. Glazov, M. M. & Ganichev, S. D. High frequency electric field induced nonlinear effects in graphene. Phys. Rep. 535, 101–138 (2014).

    CAS  Google Scholar 

  101. Hendry, E., Hale, P. J., Moger, J., Savchenko, A. K. & Mikhailov, S. A. Coherent nonlinear optical response of graphene. Phys. Rev. Lett. 105, 097401 (2010).

    CAS  Google Scholar 

  102. Yoshikawa, N. A., Tamaya, T. & Tanaka, K. High-harmonic generation in graphene enhanced by elliptically polarized light excitation. Science 356, 736–738 (2017).

    CAS  Google Scholar 

  103. Hafez, H. A. et al. Extremely efficient terahertz high-harmonic generation in graphene by hot Dirac fermions. Nature 561, 507–511 (2018).

    CAS  Google Scholar 

  104. de Juan, F., Grushin, A. G., Morimoto, T. & Moore, J. E. Quantized circular photogalvanic effect in Weyl semimetals. Nat. Commun. 8, 15995 (2017).

    Google Scholar 

  105. Morimoto, T., Zhong, S., Orenstein, J. & Moore, J. E. Semiclassical theory of nonlinear magneto-optical responses with applications to topological Dirac/Weyl semimetals. Phys. Rev. B 94, 245121 (2016).

    Google Scholar 

  106. Taguchi, K., Imaeda, T., Sato, M. & Tanaka, Y. Photovoltaic chiral magnetic effect in Weyl semimetals. Phys. Rev. B 93, 201202 (2016).

    Google Scholar 

  107. Ishizuka, H., Hayata, T., Ueda, M. & Nagaosa, N. Emergent electromagnetic induction and adiabatic charge pumping in noncentrosymmetric Weyl semimetals. Phys. Rev. Lett. 117, 216601 (2016).

    Google Scholar 

  108. Huang, S. et al. New type of Weyl semimetal with quadratic double Weyl fermions. Proc. Natl Acad. Sci. USA 113, 1180–1185 (2016).

    CAS  Google Scholar 

  109. Hirayama, M., Okugawa, R., Ishibashi, S., Murakami, S. & Miyake, T. Weyl node and spin texture in trigonal tellurium and selenium. Phys. Rev. Lett. 114, 206401 (2015).

    Google Scholar 

  110. Chan, C.-K., Lindner, N. H., Refael, G. & Lee, P. A. Photocurrents in Weyl semimetals. Phys. Rev. B 95, 041104 (2017).

    Google Scholar 

  111. König, E. J., Xie, H.-Y., Pesin, D. A. & Levchenko, A. Photogalvanic effect in Weyl semimetals. Phys. Rev. B 96, 075123 (2017).

    Google Scholar 

  112. Golub, L. E. & Ivchenko, E. L. Circular and magnetoinduced photocurrents in Weyl semimetals. Phys. Rev. B 98, 075305 (2018).

    CAS  Google Scholar 

  113. Ma, Q. et al. Direct optical detection of Weyl fermion chirality in a topological semimetal. Nat. Phys. 13, 842–847 (2017).

    CAS  Google Scholar 

  114. Flicker, F. et al. Chiral optical response of multifold fermions. Phys. Rev. B 98, 155145 (2018).

    CAS  Google Scholar 

  115. Chang, G. et al. Unconventional chiral fermions and large topological Fermi arcs in RhSi. Phys. Rev. Lett. 119, 206401 (2017).

    Google Scholar 

  116. Rees, D. et al. Quantized photocurrents in the chiral multifold fermion system RhSi. Preprint at arXiv 1902.03230 (2019).

  117. Xu, S.-Y. et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015).

    CAS  Google Scholar 

  118. Lv, B. Q. et al. Experimental discovery of Weyl semimetal TaAs. Phys. Rev. X. 5, 031013 (2015).

    Google Scholar 

  119. Yang, L. X. et al. Weyl semimetal phase in the non-centrosymmetric compound TaAs. Nat. Phys. 11, 728–732 (2015).

    CAS  Google Scholar 

  120. Wu, L. et al. Giant anisotropic nonlinear optical response in transition metal monopnictide Weyl semimetals. Nat. Phys. 13, 350–355 (2017).

    CAS  Google Scholar 

  121. Patankar, S. et al. Resonance-enhanced optical nonlinearity in the Weyl semimetal TaAs. Phys. Rev. B 98, 165113 (2018).

    CAS  Google Scholar 

  122. Morimoto, T. & Nagaosa, N. Topological nature of nonlinear optical effects in solids. Sci. Adv. 2, e1501524 (2016).

    Google Scholar 

  123. von Baltz, R. & Kraut, W. Theory of the bulk photovoltaic effect in pure crystals. Phys. Rev. B 23, 5590–5596 (1981).

    Google Scholar 

  124. Sipe, J. E. & Shkrebtii, A. I. Second-order optical response in semiconductors. Phys. Rev. B 61, 5337–5352 (2000).

    CAS  Google Scholar 

  125. Young, S. M. & Rappe, A. M. First principles calculation of the shift current photovoltaic effect in ferroelectrics. Phys. Rev. Lett. 109, 116601 (2012).

    Google Scholar 

  126. Young, S. M., Zheng, F. & Rappe, A. M. First-principles calculation of the bulk photovoltaic effect in bismuth ferrite. Phys. Rev. Lett. 109, 236601 (2012).

    Google Scholar 

  127. Cook, A. M. et al. Design principles for shift current photovoltaics. Nat. Commun. 8, 14176 (2017).

    CAS  Google Scholar 

  128. Nagaosa, N. & Morimoto, T. Concept of quantum geometry in optoelectronic processes in solids: application to solar cells. Adv. Mater. 29, 1603345 (2017).

    Google Scholar 

  129. Morimoto, T., Nakamura, M., Kawasaki, M. & Nagaosa, N. Current-voltage characteristic and shot noise of shift current photovoltaics. Phys. Rev. Lett. 121, 267401 (2018).

    CAS  Google Scholar 

  130. Osterhoudt, G. B. et al. Colossal mid-infrared bulk photovoltaic effect in a type-I Weyl semimetal. Nat. Mater. 18, 471–475 (2019).

    CAS  Google Scholar 

  131. Ma, J. et al. Nonlinear photoresponse of type-II Weyl semimetals. Nat. Mater. 18, 476–481 (2019).

    CAS  Google Scholar 

  132. Zhang, Y. et al. Photogalvanic effect in Weyl semimetals from first principles. Phys. Rev. B 97, 241118 (2018).

    CAS  Google Scholar 

  133. Yang, X., Burch, K. & Ran, Y. Divergent bulk photovoltaic effect in Weyl semimetals. Preprint at arXiv 1712.09363 (2018).

  134. Bergfeld, S. & Daum, W. Second-harmonic generation in GaAs: experiment versus theoretical predictions of \({\chi }_{xyz}^{2}\). Phys. Rev. Lett. 90, 036801 (2003).

    CAS  Google Scholar 

  135. Wagner, H. P., Kuhnelt, M., Langbein, W. & Hvam, W. Dispersion of the second-order nonlinear susceptibility in ZnTe, ZnSe, and ZnS. Phys. Rev. B 58, 10494 (1998).

    CAS  Google Scholar 

  136. Ju, S., Cai, T.-Y. & Guo, G.-Y. Electronic structure, linear, and nonlinear optical responses in magnetoelectric multiferroic material BiFeO3. J. Chem. Phys. 130, 214708 (2009).

    Google Scholar 

  137. Shoji, I., Kondo, T., Kitamoto, A., Shirane, M. & Ito, R. Absolute scale of second-order nonlinear-optical coefficients. J. Opt. Soc. Am. B 14, 2268–2294 (1997).

    CAS  Google Scholar 

Download references

Acknowledgements

We thank Hiroaki Ishizuka, Kentaro Ueda, Kenji Yasuda, Masahi Kawasaki, Joel Moore and Joe Orenstein for the useful discussion and support in preparing the manuscript. This work was supported by JST CREST (nos. JPMJCR16F1 and JPMJCR1874) and JSPS KAKENHI Grant number 18H03676.

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Nagaosa, N., Morimoto, T. & Tokura, Y. Transport, magnetic and optical properties of Weyl materials. Nat Rev Mater 5, 621–636 (2020). https://doi.org/10.1038/s41578-020-0208-y

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