Semiclassical dynamics and nonlinear charge current

Abstract

Electron conductivity is an important material property that can provide a wealth of information about the underlying system. Especially, the response of the conductivity with respect to electromagnetic fields corresponds to various nonlinear charge currents, which have distinct symmetry requirements and hence can be used as efficient probes of different systems. To help the band-structure engineering of such nonlinear currents, a universal treatment of electron dynamics up to second order expressed in the basis of the unperturbed states are highly useful. In this work, we review the general semiclassical framework of the nonlinear charge currents.

Keywords

nonlinear charge current / nonlinear electron dynamics / nonlinear anomalous Hall effect / linear magnetoresistance / negative longitudinal magnetoresistance / Berry curvature / quantum metric / positional shift

References

[1] N. W. Ashcroft and M. Mermin, Solid State Physics, Harcourt, Orlando, New York, 1976

[2] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous Hall effect, Rev. Mod. Phys.82(2), 1539 (2010)

[3] D. Xiao, M. C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys.82(3), 1959 (2010)

[4] X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys.83(4), 1057 (2011)

[5] T. Morimoto and N. Nagaosa, Topological nature of nonlinear optical effects in solids, Sci. Adv.2(5), e1501524 (2016)

[6] S. Zhong, J. E. Moore, and I. Souza, Gyrotropic magnetic effect and the magnetic moment on the Fermi surface, Phys. Rev. Lett.116(7), 077201 (2016)

[7] J. Ma and D. A. Pesin, Chiral magnetic effect and natural optical activity in metals with or without Weyl points, Phys. Rev. B92(23), 235205 (2015)

[8] D. T. Son and B. Z. Spivak, Chiral anomaly and classical negative magnetoresistance of Weyl metals, Phys. Rev. B88(10), 104412 (2013)

[9] N. Armitage, E. Mele, and A. Vishwanath, Macroscopic polarization in crystalline dielectrics: The geometric phase approach, Rev. Mod. Phys.66, 899 (2018)

[10] The highest order of τ is the same as the highest order of the field, which is the property of the asymptotic solution to the Boltzmann equation, as implied in Eq. (64).

[11] J. Zak ,Magnetic translation group, Phys. Rev.134(6A), A1602 (1964)

[12] J. Zak ,Magnetic translation group (II): Irreducible representations, Phys. Rev.134(6A), A1607 (1964)

[13] D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B14(6), 2239 (1976)

[14] M. C. Chang and Q. Niu, Berry phase, hyperorbits, and the Hofstadter spectrum, Phys. Rev. Lett.75(7), 1348 (1995)

[15] G. Sundaram and Q. Niu, Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects, Phys. Rev. B59(23), 14915 (1999)

[16] D. Culcer, Y. Yao, and Q. Niu, Coherent wave-packet evolution in coupled bands, Phys. Rev. B72(8), 085110 (2005)

[17] R. Shindou and K. I. Imura, Noncommutative geometry and non-Abelian Berry phase in the wave-packet dynamics of Bloch electrons, Nucl. Phys. B720(3), 399 (2005)

[18] M.-C. Chang and Q. Niu, Berry curvature, orbital moment, and effective quantum theory of electrons in electromagnetic fields, J. Phys.: Condens. Matter20, 193202 (2008)

[19] M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A Math. Phys. Sci.392, 45 (1984)

[20] D. Xiao, J. Shi, and Q. Niu, Berry phase correction to electron density of states in solids, Phys. Rev. Lett.95(13), 137204 (2005)

[21] J. E. Moore and J. Orenstein, Confinement-Induced Berry phase and helicity-dependent photocurrents, Phys. Rev. Lett.105(2), 026805 (2010)

[22] S. Zhong, J. Orenstein, and J. E. Moore, Optical gyrotropy from axion electrodynamics in momentum space, Phys. Rev. Lett.115(11), 117403 (2015)

[23] T. Morimoto, S. Zhong, J. Orenstein, and J. E. Moore, Semiclassical theory of nonlinear magneto-optical responses with applications to topological Dirac/Weyl semimetals, Phys. Rev. B94(24), 245121 (2016)

[24] E. Deyo, L. E. Golub, E. L. Ivchenko, and B. Spivak, Semiclassical theory of the photogalvanic effect in noncentrosymmetric systems, arXiv: 0904.1917 (2019)

[25] R. Resta, Manifestations of Berry’s phase in molecules and condensed matter, J. Phys.: Condens. Matter12(9), R107 (2000)

[26] R. D. King-Smith and D. Vanderbilt, Theory of polarization of crystalline solids, Phys. Rev. B47, 1651(R) (1993)

[27] R. Resta, Macroscopic polarization in crystalline dielectrics: The geometric phase approach, Rev. Mod. Phys.66(3), 899 (1994)

[28] M. C. Chang, and Q. Niu, Berry phase, hyperorbits, and the Hofstadter spectrum: Semiclassical dynamics in magnetic Bloch bands, Phys. Rev. B53(11), 7010 (1996)

[29] I. Souza and D. Vanderbilt, Dichroic f-sum rule and the orbital magnetization of crystals, Phys. Rev. B77(5), 054438 (2008)

[30] W. Yao, D. Xiao, and Q. Niu, Valley-dependent optoelectronics from inversion symmetry breaking, Phys. Rev. B77(23), 235406 (2008)

[31] L. L. Foldy and S. A. Wouthuysen, On the Dirac theory of spin 1/2 particles and its non-relativistic limit, Phys. Rev.78(1), 29 (1950)

[32] E. Blount, Extension of the Foldy–Wouthuysen transformation, Phys. Rev.128(5), 2454 (1962)

[33] D. Xiao, Y. Yao, Z. Fang, and Q. Niu, Berry-phase effect in anomalous thermoelectric transport, Phys. Rev. Lett.97(2), 026603 (2006)

[34] T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Orbital magnetization in periodic insulators, Phys. Rev. Lett.95(13), 137205 (2005)

[35] D. Ceresoli, T. Thonhauser, D. Vanderbilt, and R. Resta, Orbital magnetization in crystalline solids: Multi-band insulators, Chern insulators, and metals, Phys. Rev. B74(2), 024408 (2006)

[36] O. Gat and J. E. Avron, Magnetic fingerprints of fractal spectra and the duality of Hofstadter models, New J. Phys.5, 44 (2003)

[37] O. Gat, and J. E. Avron, Semiclassical analysis and the magnetization of the Hofstadter model, Phys. Rev. Lett.91(18), 186801 (2003)

[38] J. Shi, G. Vignale, D. Xiao, and Q. Niu, Quantum theory of orbital magnetization and its generalization to interacting systems, Phys. Rev. Lett.99(19), 197202 (2007)

[39] T. Qin, Q. Niu, and J. Shi, Energy magnetization and the thermal Hall effect, Phys. Rev. Lett.107(23), 236601 (2011)

[40] Y. Gao, S. A. Yang, and Q. Niu, Field Induced Positional Shift of Bloch electrons and its dynamical implications, Phys. Rev. Lett.112(16), 166601 (2014)

[41] Y. Gao, S. A. Yang, and Q. Niu, Geometrical effects in orbital magnetic susceptibility, Phys. Rev. B91(21), 214405 (2015)

[42] E. Blount, Bloch electrons in a magnetic field, Phys. Rev.126(5), 1636 (1962)

[43] J. P. Provost, and G. Vallee, Riemannian structure on manifolds of quantum states, Commun. Math. Phys.76(3), 289 (1980)

[44] T. Neupert, C. Chamon, and C. Mudry, Measuring the quantum geometry of Bloch bands with current noise, Phys. Rev. B87(24), 245103 (2013)

[45] J. Anandan and Y. Aharonov, Geometry of quantum evolution, Phys. Rev. Lett.65(14), 1697 (1990)

[46] R. Resta, The insulating state of matter: A geometrical theory, Eur. Phys. J. B79(2), 121 (2011)

[47] M. V. Berry, Quantum phase corrections from adiabatic iteration, Proc. R. Soc. Lond. A414(1846), 31 (1987)

[48] A. M. Essin, A. M. Turner, J. E. Moore, and D. Vanderbilt, Orbital magnetoelectric coupling in band insulators, Phys. Rev. B81(20), 205104 (2010)

[49] L. Onsager, Interpretation of the de Haas–van Alphen effect, Philos. Mag.43(344), 1006 (1952)

[50] K. Reijnders, T. Tudorovskiy, and M. Katsnelson, Semiclassical theory of potential scattering for massless Dirac fermions, Ann. Phys.333, 155 (2013)

[51] M. C. Gutzwiller, Periodic orbits and classical quantization conditions, J. Math. Phys.12(3), 343 (1971)

[52] M. Wilkinson, An example of phase holonomy in WKB theory, J. Phys. A17(18), 3459 (1984)

[53] R. Rammal, Landau level spectrum of Bloch electrons in a honeycomb lattice, J. Phys. France46(8), 1345 (1985)

[54] G. P. Mikitik and Y. Sharlai, Manifestation of Berry’s phase in metal physics, Phys. Rev. Lett.82(10), 2147 (1999)

[55] P. Carmier and D. Ullmo, Berry phase in graphene: Semiclassical perspective, Phys. Rev. B77(24), 245413 (2008)

[56] J. N. Fuchs, F. Piéchon, M. O. Goerbig, and G. Montambaux, Topological Berry phase and semiclassical quantization of cyclotron orbits for two dimensional electrons in coupled band models., Eur. Phys. J. B77(3), 351 (2010)

[57] Y. Gao and Q. Niu, Zero-field magnetic response functions in Landau levels, Proc. Natl. Acad. Sci. USA114(28), 7295 (2017)

[58] A. A. Taskin, K. Segawa, and Y. Ando, Oscillatory angular dependence of the magnetoresistance in a topological insulator Bi_1-xSb_x, Phys. Rev. B82(12), 121302 (2010)

[59] J. G. Analytis, R. D. McDonald, S. C. Riggs, J. H. Chu, G. S. Boebinger, and I. R. Fisher, Two-dimensional surface state in the quantum limit of a topological insulator, Nat. Phys.6(12), 960 (2010)

[60] Z. Ren, A. A. Taskin, S. Sasaki, K. Segawa, and Y. Ando, Large bulk resistivity and surface quantum oscillations in the topological insulator Bi₂Te₂Se, Phys. Rev. B82(24), 241306 (2010)

[61] B. Sacépé, J. B. Oostinga, J. Li, A. Ubaldini, N. J. Couto, E. Giannini, and A. F. Morpurgo, Gate-tuned normal and superconducting transport at the surface of a topological insulator, Nat. Commun.2, 575 (2011)

[62] C. Brüne, C. X. Liu, E. G. Novik, E. M. Hankiewicz, H. Buhmann, Y. L. Chen, X. L. Qi, Z. X. Shen, S. C. Zhang, and L. W. Molenkamp, Quantum Hall effect from the topological surface states of strained bulk HgTe, Phys. Rev. Lett.106(12), 126803 (2011)

[63] F. Xiu, L. He, Y. Wang, L. Cheng, L. T. Chang, M. Lang, G. Huang, X. Kou, Y. Zhou, X. Jiang, Z. Chen, J. Zou, A. Shailos, and K. L. Wang, Manipulating surface states in topological insulator nanoribbons, Nat. Nanotechnol.6(4), 216 (2011)

[64] F. X. Xiang, X. L. Wang, M. Veldhorst, S. X. Dou, and M. S. Fuhrer, Observation of topological transition of Fermi surface from a spindle torus to a torus in bulk Rashba spin-split BiTeCl, Phys. Rev. B92(3), 035123 (2015)

[65] G. Gómez-Santos, and T. Stauber, Measurable lattice effects on the charge and magnetic response in graphene, Phys. Rev. Lett.106(4), 045504 (2011)

[66] A. Raoux, M. Morigi, J. N. Fuchs, F. Piéchon, and G. Montambaux, From dia- to paramagnetic orbital susceptibility of massless fermions, Phys. Rev. Lett.112(2), 026402 (2014)

[67] M. Ogata and H. Fukuyama, Orbital magnetism of Bloch electrons (I): General formula, J. Phys. Soc. Jpn.84(12), 124708 (2015)

[68] In Ref. [67], it is found that except the energy polarization contribution, all the other terms are consistent. For the energy polarization contribution, Ref. [41] contains a typo. When inserting the second order energy in Eq. (42), the energy polarization in Ref. [41] has an additional 1/4 factor by mistake. After removing such factor as given in Eq. (43), the energy polarization has the same expression with Eq. (2.31) in Ref. [67].

[69] N. Marzari and D. Vanderbilt, Maximally localized generalized Wannier functions for composite energy bands, Phys. Rev. B56(20), 12847 (1997)

[70] I. Souza, N. Marzari, and D. Vanderbilt, Maximally localized Wannier functions for entangled energy bands, Phys. Rev. B65(3), 035109 (2001)

[71] J. R. Yates, X. Wang, D. Vanderbilt, and I. Souza, Spectral and Fermi surface properties from Wannier interpolation, Phys. Rev. B75(19), 195121 (2007)

[72] N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally localized Wannier functions: Theory and applications, Rev. Mod. Phys.84(4), 1419 (2012)

[73] M. M. Vazifeh and M. Franz, Electromagnetic response of Weyl semimetals, Phys. Rev. Lett.111(2), 027201 (2013)

[74] L. Fu, and E. Berg, Odd-parity topological superconductors: Theory and application to Cu_xBi₂Se₃, Phys. Rev. Lett.105(9), 097001 (2010)

[75] H. K. Pal and D. L. Maslov, Necessary and sufficient condition for longitudinal magnetoresistance, Phys. Rev. B81(21), 214438 (2010)

[76] K. Ohgushi, S. Murakami, and N. Nagaosa, Spin anisotropy and quantum Hall effect in the kagomé lattice: Chiral spin state based on a ferromagnet, Phys. Rev. B62(10), R6065 (2000)

[77] R. Shindou and N. Nagaosa, Orbital ferromagnetism and anomalous Hall effect in antiferromagnets on the distorted fcc Lattice, Phys. Rev. Lett.87(11), 116801 (2001)

[78] M. Taillefumier, B. Canals, C. Lacroix, V. K. Dugaev, and P. Bruno, Anomalous Hall effect due to spin chirality in the Kagomé lattice, Phys. Rev. B74(8), 085105 (2006)

[79] A. Kalitsov, B. Canals, and C. Lacroix, Anomalous Hall effect due to magnetic chirality in the pyrochlore lattice, J. Phys. Conf. Ser.145, 012020 (2009)

[80] H. Takatsu, S. Yonezawa, S. Fujimoto, and Y. Maeno, Unconventional anomalous Hall effect in the metallic triangular-lattice magnet PdCrO₂, Phys. Rev. Lett.105(13), 137201 (2010)

[81] M. Udagawa and R. Moessner, Anomalous Hall effect from frustration-tuned scalar chirality distribution in Pr₂Ir₂O₇, Phys. Rev. Lett.111(3), 036602 (2013)

[82] H. Chen, Q. Niu, and A. MacDonald, Anomalous Hall effect arising from noncollinear antiferromagnetism, Phys. Rev. Lett.112(1), 017205 (2014)

[83] M. T. Suzuki, T. Koretsune, M. Ochi, and R. Arita, Cluster multipole theory for anomalous Hall effect in antiferromagnets, Phys. Rev. B95(9), 094406 (2017)

[84] G. Y. Guo and T. C. Wang, Large anomalous Nernst and spin Nernst effects in the noncollinear antiferromagnets Mn₃X (X= Sn, Ge, Ga), Phys. Rev. B96(22), 224415 (2017)

[85] L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media, New York: Pergamon Press, 1984

[86] I. Sodemann and L. Fu, Quantum nonlinear Hall effect induced by berry curvature dipole in time-reversal invariant materials, Phys. Rev. Lett.115(21), 216806 (2015)

[87] S. Y. Xu, Q. Ma, H. Shen, V. Fatemi, S. Wu, T. R. Chang, G. Chang, A. M. M. Valdivia, C. K. Chan, Q. D. Gibson, J. Zhou, Z. Liu, K. Watanabe, T. Taniguchi, H. Lin, R. J. Cava, L. Fu, N. Gedik, and P. Jarillo-Herrero, Electrically switchable Berry curvature dipole in the monolayer topological insulator WTe₂, Nat. Phys.14(9), 900 (2018)

[88] K. Kang, T. Li, E. Sohn, J. Shan, and K. F. Mak, Observation of the nonlinear anomalous Hall effect in 2D WTe₂, arXiv: 1809.08744 (2018)

[89] Q. Ma, S.Y. Xu, H. Shen, D. Macneill, V. Fatemi, T.-R. Chang, A. M. M. Valdivia, S. Wu, Z. Du, C.-H. Hsu, S. Fang, Q. D. Gibson, K. Watanabe, T. Taniguchi, R. J. Cava, E. Kaxiras, H.-Z. Lu, H. Lin, L. Fu, N. Gedik, and P. Jarillo-Herrero, Observation of the nonlinear Hall effect under time reversal symmetric conditions, Nature565, 337 (2019)

[90] A. Malashevich and I. Souza, Band theory of spatial dispersion in magnetoelectrics, Phys. Rev. B82(24), 245118 (2010)

[91] J. Orenstein and J. E. Moore, Berry phase mechanism for optical gyro ropy in stripe-ordered cuprates, Phys. Rev. B87(16), 165110 (2013)

[92] S. Nandy and I. Sodemann, Symmetry and quantum kinetics of the non-linear Hall effect, arXiv: 1901.04467 (2019)

[93] J. R. Reitz, and A. W. Overhauser, Magnetoresistance of potassium, Phys. Rev.171(3), 749 (1968)

[94] P. A. Penz and R. Bowers, Strain-dependent magnetoresistance of potassium, Phys. Rev.172(3), 991 (1968)

[95] B. K. Jones, Strain-dependent magnetoresistance of sodium and potassium, Phys. Rev.179(3), 637 (1969)

[96] A. L. Friedman, J. L. Tedesco, P. M. Campbell, J. C. Culbertson, E. Aifer, F. K. Perkins, R. L. Myers-Ward, J. K. Hite, C. R. Jr Eddy, G. G. Jernigan, and D. K. Gaskill, Quantum linear magnetoresistance in multilayer epitaxial graphene, Nano Lett.10(10), 3962 (2010)

[97] D. X. Qu, Y. S. Hor, J. Xiong, R. J. Cava, and N. P. Ong, Quantum oscillations and Hall anomaly of surface states in the topological insulator Bi₂Te₃, Science329(5993), 821 (2010)

[98] H. Tang, D. Liang, R. L. J. Qiu, and X. P. A. Gao, Twodimensional transport-induced linear magneto-resistance in topological insulator Bi₂Se₃ nanoribbons, ACS Nano5(9), 7510 (2011)

[99] X. Wang, Y. Du, S. Dou, and C. Zhang, Room temperature giant and linear magnetoresistance in topological insulator Bi₂Te₃ nanosheets, Phys. Rev. Lett.108(26), 266806 (2012)

[100] J. Tian, C. Chang, H. Cao, K. He, X. Ma, Q. Xue, and Y. P. Chen, Quantum and classical magnetoresistance in ambipolar topological insulator transistors with gatetunable bulk and surface conduction, Sci. Rep.4(1), 4859 (2015)

[101] L. He, X. Hong, J. Dong, J. Pan, Z. Zhang, J. Zhang, and S. Li, Quantum transport evidence for the threedimensional Dirac semimetal phase in Cd₃As₂, Phys. Rev. Lett.113(24), 246402 (2014)

[102] T. Liang, Q. Gibson, M. N. Ali, M. Liu, R. J. Cava, and N. P. Ong, Ultrahigh mobility and giant magnetoresistance in the Dirac semimetal Cd₃As₂, Nat. Mater.14(3), 280 (2015)

[103] J. Feng, Y. Pang, D. Wu, Z. Wang, H. Weng, J. Li, X. Dai, Z. Fang, Y. Shi, and L. Lu, Large linear magnetoresistance in Dirac semimetal Cd₃As₂with Fermi surfaces close to the Dirac points, Phys. Rev. B92(8), 081306 (2015)

[104] M. Novak, S. Sasaki, K. Segawa, and Y. Ando, Large linear magnetoresistance in the Dirac semimetal TlBiSSe, Phys. Rev. B91(4), 041203 (2015)

[105] A. Narayanan, M. Watson, S. Blake, N. Bruyant, L. Drigo, Y. Chen, D. Prabhakaran, B. Yan, C. Felser, T. Kong, P. Canfield, and A. Coldea, Linear magnetoresistance caused by mobility fluctuations in n-doped Cd₃As₂, Phys. Rev. Lett.114(11), 117201 (2015)

[106] X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen, Observation of the chiral-anomaly-induced negative magnetoresistance in 3D Weyl semimetal TaAs, Phys. Rev. X5(3), 031023 (2015)

[107] A. A. Abrikosov, Quantum magnetoresistance, Phys. Rev. B58(5), 2788 (1998)

[108] A. A. Abrikosov, Quantum linear magnetoresistance, Europhys. Lett.49(6), 789 (2000)

[109] C. M. Wang and X. L. Lei, Linear magnetoresistance on the topological surface, Phys. Rev. B86(3), 035442 (2012)

[110] C. Herring, Effect of random inhomogeneities on electrical and galvanomagnetic measurements, J. Appl. Phys.31(11), 1939 (1960)

[111] M. M. Parish and P. B. Littlewood, Non-saturating magnetoresistance in heavily disordered semiconductors, Nature426(6963), 162 (2003)

[112] N. A. Porter and C. H. Marrows, Linear magnetoresistance in n-type silicon due to doping density fluctuations, Sci. Rep.2(1), 565 (2012)

[113] N. Kozlova, N. Mori, O. Makarovsky, L. Eaves, Q. Zhuang, A. Krier, and A. Patanè, Linear magnetoresistance due to multiple-electron scattering by low-mobility islands in an inhomogeneous conductor, Nat. Commun.3(1), 1097 (2012)

[114] H. Chen, Y. Gao, D. Xiao, A. H. MacDonald, and Q. Niu, Semiclassical theory of linear magnetoresistance in crystalline conductors with broken time-reversal symmetry, arXiv: 1511.02557 (2015)

[115] A. B. Pippard, Magnetoresistance in Metals, Cambridge University Press, Cambridge, England, New York, 1989

[116] H. B. Nielsen and M. Ninomiya, The Adler–Bell–Jackiw anomaly and Weyl fermions in a crystal, Phys. Lett. B130(6), 389 (1983)

[117] V. Aji, Adler–Bell–Jackiw anomaly in Weyl semimetals: Application to pyrochlore iridates, Phys. Rev. B85(24), 241101 (2012)

[118] A. A. Burkov, Chiral anomaly and transport in Weyl metals, J. Phys.: Condens. Matter27(11), 113201 (2015)

[119] M. A. Stephanov, and Y. Yin, Chiral kinetic theory, Phys. Rev. Lett.109(16), 162001 (2012)

[120] D. T. Son and N. Yamamoto, Kinetic theory with Berry curvature from quantum field theories, Phys. Rev. D87(8), 085016 (2013)

[121] B. Z. Spivak and A. V. Andreev, Magnetotransport phenomena related to the chiral anomaly in Weyl semimetals, Phys. Rev. B93(8), 085107 (2016)

[122] Y. Hidaka, S. Pu, and D. L. Yang, Relativistic chiral kinetic theory from quantum field theories, Phys. Rev. D95(9), 091901 (2017)

[123] A. Sekine, D. Culcer, and A. H. MacDonald, Quantum kinetic theory of the chiral anomaly, Phys. Rev. B96(23), 235134 (2017)

[124] A. A. Burkov, Chiral anomaly and diffusive magnetotransport in Weyl metals, Phys. Rev. Lett.113(24), 247203 (2014)

[125] A. A. Burkov, Negative longitudinal magnetoresistance in Dirac and Weyl metals, Phys. Rev. B91(24), 245157 (2015)

[126] A. Andreev and B. Spivak, Longitudinal negative magnetoresistance and magnetotransport phenomena in conventional and topological conductors, Phys. Rev. Lett.120(2), 026601 (2018)

[127] C. Shekhar, A. K. Nayak, Y. Sun, M. Schmidt, M. Nicklas, I. Leermakers, U. Zeitler, Y. Skourski, J. Wosnitza, Z. Liu, Y. Chen, W. Schnelle, H. Borrmann, Y. Grin, C. Felser, and B. Yan, Extremely large magnetoresistance and ultrahigh mobility in the topological Weyl semimetal candidate NbP, Nat. Phys.11(8), 645 (2015)

[128] X. Yang, Y. Li, Z. Wang, Y. Zheng, and Z. A. Xu, Chiral anomaly induced negative magnetoresistance in topological Weyl semimetal NbAs, arXiv: 1506.03190 (2015)

[129] Z. Wang, Y. Zheng, Z. Shen, Y. Lu, H. Fang, F. Sheng, Y. Zhou, X. Yang, Y. Li, C. Feng, and Z. A. Xu, Helicityprotected ultrahigh mobility Weyl fermions in NbP, Phys. Rev. B93(12), 121112 (2016)

[130] C. L. Zhang, S. Y. Xu, I. Belopolski, Z. Yuan, Z. Lin, B. Tong, G. Bian, N. Alidoust, C. C. Lee, S. M. Huang, T. R. Chang, G. Chang, C. H. Hsu, H. T. Jeng, M. Neupane, D. S. Sanchez, H. Zheng, J. Wang, H. Lin, C. Zhang, H. Z. Lu, S. Q. Shen, T. Neupert, M. Z. Hasan, and S. Jia, Signatures of the Adler–Bell–Jackiw chiral anomaly in a Weyl fermion semimetal, Nat. Commun.7(1), 10735 (2016)

[131] H. J. Kim, K. S. Kim, J. F. Wang, M. Sasaki, N. Satoh, A. Ohnishi, M. Kitaura, M. Yang, and L. Li, Dirac versus Weyl fermions in topological insulators: Adler-Bell-Jackiw anomaly in transport phenomena, Phys. Rev. Lett.111(24), 246603 (2013)

[132] J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan, M. Hirschberger, W. Wang, R. Cava, and N. Ong, Evidence for the chiral anomaly in the Dirac semimetal Na₃Bi, Science350(6259), 413 (2015)

[133] J. Feng, Y. Pang, D. Wu, Z. Wang, H. Weng, J. Li, X. Dai, Z. Fang, Y. Shi, and L. Lu, Large linear magnetoresistance in Dirac semimetal Cd₃As₂ with Fermi surfaces close to the Dirac points, Phys. Rev. B92(8), 081306 (2015)

[134] C. Z. Li, L. X. Wang, H. Liu, J. Wang, Z. M. Liao, and D. P. Yu, Giant negative magnetoresistance induced by the chiral anomaly in individual Cd₃As₂ nanowires, Nat. Commun.6(1), 10137 (2015)

[135] H. Li, H. He, H. Z. Lu, H. Zhang, H. Liu, R. Ma, Z. Fan, S. Q. Shen, and J. Wang, Negative magnetoresistance in Dirac semimetal Cd₃As₂, Nat. Commun.7(1), 10301 (2016)

[136] C. Zhang, E. Zhang, W. Wang, Y. Liu, Z. G. Chen, S. Lu, S. Liang, J. Cao, X. Yuan, L. Tang, Q. Li, C. Zhou, T. Gu, Y. Wu, J. Zou, and F. Xiu, Room-temperature chiral charge pumping in Dirac semimetals, Nat. Commun.8, 13741 (2017)

[137] Y. Gao, S. A. Yang, and Q. Niu, Intrinsic relative magnetoconductivity of nonmagnetic metals, Phys. Rev. B95(16), 165135 (2017)

[138] X. Dai, Z. Du, and H. Z. Lu, Negative magnetoresistance without chiral anomaly in topological insulators, Phys. Rev. Lett.119(16), 166601 (2017)

[139] H. W. Wang, B. Fu, and S. Q. Shen, Intrinsic magnetoresistance in three-dimensional Dirac materials with low carrier density, Phys. Rev. B98(8), 081202 (2018)

[140] K. Yoshida, Transport of spatially inhomogeneous current in a compensated metal under magnetic fields (III): A case of bismuth in longitudinal and transverse magnetic fields, J. Appl. Phys.51(8), 4226 (1980)

[141] R. D. Reis, M. O. Ajeesh, N. Kumar, F. Arnold, C. Shekhar, M. Naumann, M. Schmidt, M. Nicklas, and E. Hassinger, On the search for the chiral anomaly in Weyl semimetals: The negative longitudinal magnetoresistance., New J. Phys.18(8), 085006 (2016)

[142] Z. Yuan, H. Lu, Y. Liu, J. Wang, and S. Jia, Large magnetoresistance in compensated semimetals TaAs₂ and NbAs₂, Phys. Rev. B93(18), 184405 (2016)

[143] R. Shindou and L. Balents, Artificial electric field in Fermi liquids, Phys. Rev. Lett.97(21), 216601 (2006)

[144] R. Shindou and L. Balents, Gradient expansion approach to multiple-band Fermi liquids, Phys. Rev. B77(3), 035110 (2008)

[145] D. Culcer, A. Sekine, and A. H. MacDonald, Interband coherence response to electric fields in crystals: Berryphase contributions and disorder effects, Phys. Rev. B96(3), 035106 (2017)

[146] Y. Tian, L. Ye, and X. Jin, Proper scaling of the anomalous Hall effect, Phys. Rev. Lett.103(8), 087206 (2009)

[147] S. H. Chun, Y. S. Kim, H. K. Choi, I. T. Jeong, W. O. Lee, K. S. Suh, Y. S. Oh, K. H. Kim, Z. G. Khim, J. C. Woo, and Y. D. Park, Interplay between carrier and impurity concentrations in annealed Ga_1-xMn_xAs: Intrinsic anomalous Hall effect, Phys. Rev. Lett.98(2), 026601 (2007)

[148] W. L. Lee, S. Watauchi, V. L. Miller, R. J. Cava, and N. P. Ong, Dissipationless Anomalous Hall Current in the Ferromagnetic Spinel CuCr₂Se_4-xBr_x, Science303(5664), 1647 (2004)

[149] R. Mathieu, A. Asamitsu, H. Yamada, K. S. Takahashi, M. Kawasaki, Z. Fang, N. Nagaosa, and Y. Tokura, Scaling of the anomalous Hall effect in Sr_1-xCa_xRuO₃, Phys. Rev. Lett.93(1), 016602 (2004)

[150] B. C. Sales, R. Jin, D. Mandrus, and P. Khalifah, Anomalous Hall effect in three ferromagnetic compounds: EuFe₄Sb₁₂, Yb₁₄MnSb₁₁, and Eu₈Ga₁₆Ge₃₀, Phys. Rev. B73(22), 224435 (2006)

[151] C. Zeng, Y. Yao, Q. Niu, and H. H. Weitering, Linear magnetization dependence of the intrinsic anomalous Hall effect, Phys. Rev. Lett.96(3), 037204 (2006)

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature