# Weak Localization and Antilocalization in Topological Materials with Impurity Spin-Orbit Interactions

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## Abstract

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## 1. Introduction

## 2. General Model

## 3. Topological Insulators

## 4. Weyl Semimetal Thin Films

## 5. Other Topological Materials

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Hasan, M.Z.; Kane, C.L. Topological Insulators. Rev. Mod. Phys.
**2010**, 82, 3045. [Google Scholar] [CrossRef] - Xiao, D.; Liu, G.B.; Feng, W.; Xu, X.; Yao, W. Coupled Spin and Valley Physics in Monolayers of MoS
_{2}and Other Group-VI Dichalcogenides. Phys. Rev. Lett.**2012**, 108, 196802. [Google Scholar] [CrossRef] [PubMed] - Burkov, A.A.; Balents, L. Weyl Semimetal in a Topological Insulator Multilayer. Phys. Rev. Lett.
**2011**, 107, 127205. [Google Scholar] [CrossRef] [PubMed] - Edmonds, M.T.; Hellerstedt, J.; O’Donnell, K.M.; Tadich, A.; Fuhrer, M.S. Molecular Doping the Topological Dirac Semimetal Na
_{3}Bi across the Charge Neutrality Point with F4-TCNQ. ACS Appl. Mater. Interfaces**2016**, 8, 16412. [Google Scholar] [CrossRef] [PubMed] - Hellerstedt, J.; Edmonds, M.T.; Ramakrishnan, N.; Liu, C.; Weber, B.; Tadich, A.; O’Donnell, K.M.; Adam, S.; Fuhrer, M.S. Electronic Properties of High-Quality Epitaxial Topological Dirac Semimetal Thin Films. Nano Lett.
**2016**, 16, 3210. [Google Scholar] [CrossRef] [PubMed] - Nayak, C.; Simon, S.H.; Stern, A.; Freedman, M.; Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys.
**2008**, 80, 1083. [Google Scholar] [CrossRef] - Wang, G.; Zhu, X.G.; Sun, Y.Y.; Li, Y.Y.; Zhang, T.; Wen, J.; Chen, X.; He, K.; Wang, L.L.; Ma, X.C.; et al. Topological Insulator Thin Films of Bi
_{2}Te_{3}with Controlled Electronic Structure. Adv. Mater.**2011**, 23, 2929. [Google Scholar] [CrossRef] [PubMed] - Sacépé, B.; Oostinga, J.B.; Li, J.; Ubaldini, A.; Couto, N.J.G.; Giannini, E.; Morpurgo, A.F. Gate-tuned normal and superconducting transport at the surface of a topological insulator. Nat. Commun.
**2011**, 2, 575. [Google Scholar] [CrossRef] [PubMed] - He, H.T.; Wang, G.; Zhang, T.; Sou, I.K.; Wong, G.K.L.; Wang, J.N.; Lu, H.Z.; Shen, S.Q.; Zhang, F.C. Impurity Effect on Weak Antilocalization in the Topological Insulator Bi
_{2}Te_{3}. Phys. Rev. Lett.**2011**, 106, 166805. [Google Scholar] [CrossRef] [PubMed] - Ren, Z.; Taskin, A.A.; Sasaki, S.; Segawa, K.; Ando, Y. Fermi level tuning and a large activation gap achieved in the topological insulator Bi
_{2}Te_{2}Se by Sn doping. Phys. Rev. B**2012**, 85, 155301. [Google Scholar] [CrossRef] - Kim, D.; Cho, S.; Butch, N.P.; Syers, P.; Kirshenbaum, K.; Adam, S.; Paglione, J.; Fuhrer, M.S. Surface conduction of topological Dirac electrons in bulk insulating Bi
_{2}Se_{3}. Nat. Phys.**2012**, 8, 459. [Google Scholar] [CrossRef] - Zhang, J.; Chang, C.Z.; Tang, P.; Zhang, Z.; Feng, X.; Li, K.; Wang, L.L.; Chen, X.; Liu, C.; Duan, W.; et al. Topology-Driven Magnetic Quantum Phase Transition in Topological Insulators. Science
**2013**, 339, 1582. [Google Scholar] [CrossRef] [PubMed] - Barreto, L.; Khnemund, L.; Edler, F.; Tegenkamp, C.; Mi, J.; Bremholm, M.; Iversen, B.B.; Frydendahl, C.; Bianchi, M.; Hofmann, P. Surface-Dominated Transport on a Bulk Topological Insulator. Nano Lett.
**2014**, 14, 3755. [Google Scholar] [CrossRef] [PubMed] - Tang, J.; Chang, L.T.; Kou, X.; Murata, K.; Choi, E.S.; Lang, M.; Fan, Y.; Jiang, Y.; Montazeri, M.; Jiang, W.; et al. Electrical Detection of Spin-Polarized Surface States Conduction in (Bi
_{0.53}Sb_{0.47})_{2}Te_{3}Topological Insulator. Nano Lett.**2014**, 14, 5423. [Google Scholar] [CrossRef] [PubMed] - Kozlov, D.A.; Kvon, Z.D.; Olshanetsky, E.B.; Mikhailov, N.N.; Dvoretsky, S.A.; Weiss, D. Transport Properties of a 3D Topological Insulator based on a Strained High-Mobility HgTe Film. Phys. Rev. Lett.
**2014**, 112, 196801. [Google Scholar] [CrossRef] [PubMed] - Kim, D.; Syers, P.; Butch, N.P.; Paglione, J.; Fuhrer, M.S. Coherent topological transport on the surface of Bi
_{2}Se_{3}. Nat. Commun.**2013**, 4, 2040. [Google Scholar] [CrossRef] [PubMed] - Cacho, C.; Crepaldi, A.; Battiato, M.; Braun, J.; Cilento, F.; Zacchigna, M.; Richter, M.; Heckmann, O.; Springate, E.; Liu, Y.; et al. Momentum-Resolved Spin Dynamics of Bulk and Surface Excited States in the Topological Insulator Bi
_{2}Se_{3}. Phys. Rev. Lett.**2015**, 114, 097401. [Google Scholar] [CrossRef] [PubMed] - Zhang, J.; Feng, X.; Xu, Y.; Guo, M.; Zhang, Z.; Ou, Y.; Feng, Y.; Li, K.; Zhang, H.; Wang, L.; et al. Disentangling the magnetoelectric and thermoelectric transport in topological insulator thin films. Phys. Rev. B
**2015**, 91, 075431. [Google Scholar] [CrossRef] - Hellerstedt, J.; Edmonds, M.T.; Chen, J.H.; Cullen, W.G.; Zheng, C.X.; Fuhrer, M.S. Thickness and growth-condition dependence of in-situ mobility and carrier density of epitaxial thin-film Bi
_{2}Se_{3}. Appl. Phys. Lett.**2014**, 105, 173506. [Google Scholar] [CrossRef] - Kastl, C.; Karnetzky, C.; Karl, H.; Holleitner, A.W. Ultrafast helicity control of surface currents in topological insulators with near-unity fidelity. Nat. Commun.
**2015**, 6, 6617. [Google Scholar] [CrossRef] [PubMed] - Fan, Y.; Kou, X.; Upadhyaya, P.; Shao, Q.; Pan, L.; Lang, M.; Che, X.; Tang, J.; Montazeri, M.; Murata, K.; et al. Electric-field control of spin-orbit torque in a magnetically doped topological insulator. Nat. Nano.
**2016**, 11, 352. [Google Scholar] [CrossRef] [PubMed] - Yoshimi, R.; Tsukazaki, A.; Kozuka, Y.; Falson, J.; Takahashi, K.S.; Checkelsky, J.G.; Nagaosa, N.; Kawasaki, M.; Tokura, Y. Quantum Hall effect on top and bottom surface states of topological insulator (Bi
_{1−x}Sb_{x})_{2}Te_{3}films. Nat. Commun.**2015**, 6, 6627. [Google Scholar] [CrossRef] [PubMed] - Hwang, E.H.; Das Sarma, S. Dielectric function, screening, and plasmons in two-dimensional graphene. Phys. Rev. B
**2007**, 75, 205418. [Google Scholar] [CrossRef] - Jung, J.; MacDonald, A.H. Enhancement of nonlocal exchange near isolated band crossings in graphene. Phys. Rev. B
**2011**, 84, 085446. [Google Scholar] [CrossRef] - Durst, A.C. Low-temperature thermal transport at the interface of a topological insulator and a d-wave superconductor. Phys. Rev. B
**2015**, 91, 094519. [Google Scholar] [CrossRef] - Lu, H.Z.; Shen, S.Q. Weak localization and weak anti-localization in topological insulators. Proc. SPIE
**2014**, 9167, 91672E. [Google Scholar] - Zhang, W.; Yu, R.; Zhang, H.J.; Dai, X.; Fang, Z. First Principles Studies on 3-DimenSional Strong Topological Insulators: Bi
_{2}Te_{3}, Bi_{2}Se_{3}and Sb_{2}Te_{3}. New J. Phys.**2010**, 12, 065013. [Google Scholar] [CrossRef] - Liu, S.; Kim, Y.; Tan, L.Z.; Rappe, A.M. Strain-Induced Ferroelectric Topological Insulator. Nano Lett.
**2016**, 16, 1663. [Google Scholar] [CrossRef] [PubMed] - Culcer, D.; Das Sarma, S. Anomalous Hall response of topological insulators. Phys. Rev. B
**2011**, 83, 245441. [Google Scholar] [CrossRef] - Adam, S.; Hwang, E.H.; Das Sarma, S. Two-dimensional transport and screening in topological insulator surface states. Phys. Rev. B
**2012**, 85, 235413. [Google Scholar] [CrossRef] - Yoshida, T.; Fujimoto, S.; Kawakami, N. Correlation effects on a topological insulator at finite temperatures. Phys. Rev. B
**2012**, 85, 125113. [Google Scholar] [CrossRef] - Das Sarma, S.; Li, Q. Many-body effects and possible superconductivity in the two-dimensional metallic surface states of three-dimensional topological insulators. Phys. Rev. B
**2013**, 88, 081404. [Google Scholar] [CrossRef] - Liu, W.E.; Liu, H.; Culcer, D. Screening, Friedel oscillations, and low-temperature conductivity in topological insulator thin films. Phys. Rev. B
**2014**, 89, 195417. [Google Scholar] [CrossRef] - Lu, H.Z.; Shen, S.Q. Finite-Temperature Conductivity and Magnetoconductivity of Topological Insulators. Phys. Rev. Lett.
**2014**, 112, 146601. [Google Scholar] [CrossRef] [PubMed] - Das Sarma, S.; Hwang, E.H.; Min, H. Carrier screening, transport, and relaxation in three-dimensional Dirac semimetals. Phys. Rev. B
**2015**, 91, 035201. [Google Scholar] [CrossRef] - Kane, C.L.; Mele, E.J. Quantum Spin Hall Effect in Graphene. Phys. Rev. Lett.
**2005**, 95, 226801. [Google Scholar] [CrossRef] [PubMed] - Qi, X.L.; Zhang, S.C. Topological insulators and superconductors. Rev. Mod. Phys.
**2011**, 83, 1057. [Google Scholar] [CrossRef] - Moore, J.E.; Balents, L. Topological invariants of time-reversal-invariant band structures. Phys. Rev. B
**2007**, 75, 121306. [Google Scholar] [CrossRef] - Ando, Y. Topological insulator materials. J. Phys. Soc. Jpn.
**2013**, 82, 102001. [Google Scholar] [CrossRef] - Culcer, D. Linear response theory of interacting topological insulators. Phys. Rev. B
**2011**, 84, 235411. [Google Scholar] [CrossRef] - Culcer, D. Transport in three-dimensional topological insulators: theory and experiment. Physica E
**2012**, 44, 860. [Google Scholar] [CrossRef] - Tkachov, G.; Hankiewicz, E.M. Spin-helical transport in normal and superconducting topological insulators. Phys. Status Solidi B
**2013**, 250, 215. [Google Scholar] [CrossRef] - Li, Y.Q.; Wu, K.H.; Shi, J.R.; Xie, X.C. Electron transport properties of three-dimensional topological insulators. Front. Phys.
**2012**, 7, 165. [Google Scholar] [CrossRef] - Moore, J.E. The birth of topological insulators. Nature
**2010**, 464, 194. [Google Scholar] [CrossRef] [PubMed] - Fu, L.; Kane, C.L.; Mele, E.J. Topological Insulators in Three Dimensions. Phys. Rev. Lett.
**2007**, 98, 106803. [Google Scholar] [CrossRef] [PubMed] - Chen, Y.L.; Analytis, J.G.; Chu, J.H.; Liu, Z.K.; Mo, S.K.; Qi, X.L.; Zhang, H.J.; Lu, D.H.; Dai, X.; Fang, Z.; et al. Experimental Realization of a Three-Dimensional Topological Insulator, Bi
_{2}Te_{3}. Science**2009**, 325, 178. [Google Scholar] [CrossRef] [PubMed] - Hor, Y.S.; Roushan, P.; Beidenkopf, H.; Seo, J.; Qu, D.; Checkelsky, J.G.; Wray, L.A.; Xia, Y.; Xu, S.Y.; Qian, D.; et al. The development of ferromagnetism in the doped topological insulator Bi
_{2−x}Mn_{x}Te_{3}. Phys. Rev. B**2010**, 81, 195203. [Google Scholar] [CrossRef] - Collins-McIntyre, L.J.; Harrison, S.E.; Schönherr, P.; Steinke, N.J.; Kinane, C.J.; Charlton, T.R.; Alba-Veneroa, D.; Pushp, A.; Kellock, A.J.; Parkin, S.S.P.; et al. Magnetic ordering in Cr-doped Bi
_{2}Se_{3}thin films. EPL Europhys. Lett.**2014**, 107, 57009. [Google Scholar] [CrossRef] - Wang, J.; Lian, B.; Zhang, S.C. Electrically Tunable Magnetism in Magnetic Topological Insulators. Phys. Rev. Lett.
**2015**, 115, 036805. [Google Scholar] [CrossRef] [PubMed] - Yoshimi, R.; Yasuda, K.; Tsukazaki, A.; Takahashi, K.S.; Nagaosa, N.; Kawasaki, M.; Tokura, Y. Quantum Hall states stabilized in semi-magnetic bilayers of topological insulators. Nat. Commun.
**2015**, 6, 8530. [Google Scholar] [CrossRef] [PubMed] - Li, M.; Chang, C.Z.; Wu, L.; Tao, J.; Zhao, W.; Chan, M.H.W.; Moodera, J.S.; Li, J.; Zhu, Y. Experimental Verification of the Van Vleck Nature of Long-Range Ferromagnetic Order in the Vanadium-Doped Three-Dimensional Topological Insulator Sb
_{2}Te_{3}. Phys. Rev. Lett.**2015**, 114, 146802. [Google Scholar] [CrossRef] [PubMed] - Kou, X.; Pan, L.; Wang, J.; Fan, Y.; Choi, E.S.; Lee, W.L.; Nie, T.; Murata, K.; Shao, Q.; Zhang, S.C.; et al. Metal-to-insulator switching in quantum anomalous Hall states. Nat. Commun.
**2015**, 6, 8474. [Google Scholar] [CrossRef] [PubMed] - Katmis, F.; Lauter, V.; Nogueira, F.S.; Assaf, B.A.; Jamer, M.E.; Wei, P.; Satpati, B.; Freeland, J.W.; Eremin, I.; Heiman, D.; et al. A high-temperature ferromagnetic topological insulating phase by proximity coupling. Nature
**2016**, 533, 513–516. [Google Scholar] [CrossRef] [PubMed] - Jiang, Z.; Chang, C.Z.; Tang, C.; Zheng, J.G.; Moodera, J.S.; Shi, J. Structural and proximity-induced ferromagnetic properties of topological insulator-magnetic insulator heterostructures. AIP Adv.
**2016**, 6, 055809. [Google Scholar] [CrossRef] - Fan, Y.; Kou, X.; Upadhyaya, P.; Shao, Q.; Pan, L.; Lang, M.; Che, X.; Tang, J.; Montazeri, M.; Murata, K.; et al. Electric-field control of spin-orbit torque in a magnetically doped topological insulator. Nat. Nanotechnol.
**2016**, 11, 352–359. [Google Scholar] [CrossRef] [PubMed] - He, Q.L.; Kou, X.; Grutter, A.J.; Yin, G.; Pan, L.; Che, X.; Liu, Y.; Nie, T.; Zhang, B.; Disseler, S.M.; et al. Tailoring exchange couplings in magnetic topological-insulator/antiferromagnet heterostructures. Nat. Mater.
**2017**, 16, 94–100. [Google Scholar] [CrossRef] [PubMed] - Lu, H.Z.; Zhao, A.; Shen, S.Q. Quantum Transport in Magnetic Topological Insulator Thin Films. Phys. Rev. Lett.
**2013**, 111, 146802. [Google Scholar] [CrossRef] [PubMed] - Xiao, D.; Yao, W.; Niu, Q. Valley-Contrasting Physics in Graphene: Magnetic Moment and Topological Transport. Phys. Rev. Lett.
**2007**, 99, 236809. [Google Scholar] [CrossRef] [PubMed] - Yao, W.; Xiao, D.; Niu, Q. Valley-dependent optoelectronics from inversion symmetry breaking. Phys. Rev. B
**2008**, 77, 235406. [Google Scholar] [CrossRef] - Ulstrup, S.; Johannsen, J.C.; Cilento, F.; Miwa, J.A.; Crepaldi, A.; Zacchigna, M.; Cacho, C.; Chapman, R.; Springate, E.; Mammadov, S.; et al. Ultrafast Dynamics of Massive Dirac Fermions in Bilayer Graphene. Phys. Rev. Lett.
**2014**, 112, 257401. [Google Scholar] [CrossRef] [PubMed] - Yu, R.; Zhang, W.; Zhang, H.J.; Zhang, S.C.; Dai, X.; Fang, Z. Quantized Anomalous Hall Effect in Magnetic Topological Insulators. Science
**2010**, 329, 61. [Google Scholar] [CrossRef] [PubMed] - Jiang, H.; Qiao, Z.; Liu, H.; Niu, Q. Quantum anomalous Hall effect with tunable Chern number in magnetic topological insulator film. Phys. Rev. B
**2012**, 85, 045445. [Google Scholar] [CrossRef] - Chang, C.Z.; Zhang, J.; Liu, M.; Zhang, Z.; Feng, X.; Li, K.; Wang, L.L.; Chen, X.; Dai, X.; Fang, Z.; et al. Thin Films of Magnetically Doped Topological Insulator with Carrier-Independent Long-Range Ferromagnetic Order. Adv. Mater.
**2013**, 25, 1065. [Google Scholar] [CrossRef] [PubMed] - Chang, C.Z.; Zhang, J.; Feng, X.; Shen, J.; Zhang, Z.; Guo, M.; Li, K.; Ou, Y.; Wei, P.; Wang, L.L.; et al. Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator. Science
**2013**, 340, 167. [Google Scholar] [CrossRef] [PubMed] - Weng, H.; Yu, R.; Hu, X.; Dai, X.; Fang, Z. Quantum anomalous Hall effect and related topological electronic states. Adv. Phys.
**2015**, 64, 227. [Google Scholar] [CrossRef] - Liu, C.X.; Zhang, S.C.; Qi, X.L. The Quantum Anomalous Hall Effect: Theory and Experiment. Annu. Rev. Condens. Matter Phys.
**2016**, 7, 301. [Google Scholar] [CrossRef] - Chang, C.Z.; Zhao, W.; Kim, D.Y.; Zhang, H.; Assaf, B.A.; Heiman, D.; Zhang, S.C.; Liu, C.; Chan, M.H.W.; Moodera, J.S. High-precision realization of robust quantum anomalous Hall state in a hard ferromagnetic topological insulator. Nat. Mater.
**2015**, 14, 473477. [Google Scholar] [CrossRef] [PubMed] - Mahoney, A.C.; Colless, J.I.; Pauka, S.J.; Hornibrook, J.M.; Watson, J.D.; Gardner, G.C.; Manfra, M.J.; Doherty, A.C.; Reilly, D.J. On-Chip Microwave Quantum Hall Circulator. Phys. Rev. X
**2017**, 7, 011007. [Google Scholar] [CrossRef] - Kandala, A.; Richardella, A.; Kempinger, S.; Liu, C.X.; Samarth, N. Giant anisotropic magnetoresistance in a quantum anomalous Hall insulator. Nat. Commun.
**2015**, 6, 7434. [Google Scholar] [CrossRef] [PubMed] - Chang, C.Z.; Zhao, W.; Kim, D.Y.; Wei, P.; Jain, J.K.; Liu, C.; Chan, M.H.W.; Moodera, J.S. Zero-Field Dissipationless Chiral Edge Transport and the Nature of Dissipation in the Quantum Anomalous Hall State. Phys. Rev. Lett.
**2015**, 115, 057206. [Google Scholar] [CrossRef] [PubMed] - Chang, C.Z.; Li, M. Quantum anomalous Hall effect in time-reversal-symmetry breaking topological insulators. J. Phys. Condens. Matter
**2016**, 28, 123002. [Google Scholar] [CrossRef] [PubMed] - Sochnikov, I.; Bestwick, A.J.; Williams, J.R.; Lippman, T.M.; Fisher, I.R.; Goldhaber-Gordon, D.; Kirtley, J.R.; Moler, K.A. Direct Measurement of Current-Phase Relations in Superconductor/Topological Insulator/Superconductor Junctions. Nano Lett.
**2013**, 13, 3086. [Google Scholar] [CrossRef] [PubMed] - Wang, M.X.; Li, P.; Xu, J.P.; Liu, Z.L.; Ge, J.F.; Wang, G.Y.; Yang, X.; Xu, Z.A.; Ji, S.H.; Gao, C.L.; et al. Interface structure of a topological insulator/superconductor heterostructure. New J. Phys.
**2014**, 16, 123043. [Google Scholar] [CrossRef] - Li, Z.Z.; Zhang, F.C.; Wang, Q.H. Majorana modes in a topological insulator/s-wave superconductor heterostructure. Sci. Rep.
**2014**, 4, 6363. [Google Scholar] [CrossRef] [PubMed] - Potter, A.C.; Kimchi, I.; Vishwanath, A. Quantum oscillations from surface Fermi arcs in Weyl and Dirac semimetals. Nat. Commun.
**2014**, 5, 5161. [Google Scholar] [CrossRef] [PubMed] - Burkov, A.A. Topological semimetals. Nat. Mater.
**2016**, 15, 1145. [Google Scholar] [CrossRef] [PubMed] - Pixley, J.H.; Huse, D.A.; Das Sarma, S. Rare-Region-Induced Avoided Quantum Criticality in Disordered Three-Dimensional Dirac and Weyl Semimetals. Phys. Rev. X
**2016**, 6, 021042. [Google Scholar] [CrossRef] - Yang, S.A. Dirac and Weyl Materials: Fundamental Aspects and Some Spintronics Applications. SPIN
**2016**, 6, 1640003. [Google Scholar] [CrossRef] - Zhang, Z.; Feng, X.; Guo, M.; Li, K.; Zhang, J.; Ou, Y.; Feng, Y.; Wang, L.; Chen, X.; He, K.; et al. Electrically tuned magnetic order and magnetoresistance in a topological insulator. Nat. Commun.
**2014**, 5, 4915. [Google Scholar] [CrossRef] [PubMed] - Peng, X.; Yang, Y.; Singh, R.R.P.; Savrasov, S.Y.; Yu, D. Spin generation via bulk spin current in three-dimensional topological insulators. Nat. Commun.
**2016**, 7, 10878. [Google Scholar] [CrossRef] [PubMed] - Li, X.; Zhang, G.; Wu, G.; Chen, H.; Culcer, D.; Zhang, Z. Proximity effects in topological insulator heterostructures. Chin. Phys. B
**2013**, 22, 097306. [Google Scholar] [CrossRef] - Culcer, D.; Winkler, R. External gates and transport in biased bilayer graphene. Phys. Rev. B
**2009**, 79, 165422. [Google Scholar] [CrossRef] - Liu, H.; Liu, W.E.; Culcer, D. Coulomb drag in topological insulator films. Physica E
**2016**, 79, 72. [Google Scholar] [CrossRef] - Liu, H.; Liu, W.E.; Culcer, D. Anomalous Hall Coulomb drag of massive Dirac fermions. Phys. Rev. B
**2017**, 95, 205435. [Google Scholar] [CrossRef] - Culcer, D.; Hwang, E.H.; Stanescu, T.D.; Das Sarma, S. Two-dimensional surface charge transport in topological insulators. Phys. Rev. B
**2010**, 82, 155457. [Google Scholar] [CrossRef] - Oh, S. The Complete Quantum Hall Trio. Science
**2013**, 340, 153–154. [Google Scholar] [CrossRef] [PubMed] - Xu, S.Y.; Liu, C.; Kushwaha, S.K.; Sankar, R.; Krizan, J.W.; Belopolski, I.; Neupane, M.; Bian, G.; Alidoust, N.; Chang, T.R.; et al. Observation of Fermi arc surface states in a topological metal. Science
**2015**, 347, 294–298. [Google Scholar] [CrossRef] [PubMed] - Zhang, S.B.; Lu, H.Z.; Shen, S.Q. Linear magnetoconductivity in an intrinsic topological Weyl semimetal. New J. Phys.
**2016**, 18, 053039. [Google Scholar] [CrossRef] - Lau, A.; van den Brink, J.; Ortix, C. Generic coexistence of Fermi arcs and Dirac cones on the surface of time-reversal invariant Weyl semimetals. arXiv, 2017; arXiv:arXiv:1701.01660. [Google Scholar]
- Xiao, D.; Chang, M.C.; Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys.
**2010**, 82, 1959–2007. [Google Scholar] [CrossRef] - Wang, Z.; Sun, Y.; Chen, X.Q.; Franchini, C.; Xu, G.; Weng, H.; Dai, X.; Fang, Z. Dirac semimetal and topological phase transitions in A
_{3}Bi (A = Na, K, Rb). Phys. Rev. B**2012**, 85, 195320. [Google Scholar] [CrossRef] - Wang, Z.; Weng, H.; Wu, Q.; Dai, X.; Fang, Z. Three-dimensional Dirac semimetal and quantum transport in Cd
_{3}As_{2}. Phys. Rev. B**2013**, 88, 125427. [Google Scholar] [CrossRef] - Young, S.M.; Zaheer, S.; Teo, J.C.Y.; Kane, C.L.; Mele, E.J.; Rappe, A.M. Dirac Semimetal in Three Dimensions. Phys. Rev. Lett.
**2012**, 108, 140405. [Google Scholar] [CrossRef] [PubMed] - Liu, Z.K.; Zhou, B.; Zhang, Y.; Wang, Z.J.; Weng, H.M.; Prabhakaran, D.; Mo, S.K.; Shen, Z.X.; Fang, Z.; Dai, X.; et al. Discovery of a Three-Dimensional Topological Dirac Semimetal, Na3Bi. Science
**2014**, 343, 864–867. [Google Scholar] [CrossRef] [PubMed] - Xiong, J.; Kushwaha, S.K.; Liang, T.; Krizan, J.W.; Hirschberger, M.; Wang, W.; Cava, R.J.; Ong, N.P. Evidence for the chiral anomaly in the Dirac semimetal Na
_{3}Bi. Science**2015**, 350, 413–416. [Google Scholar] [CrossRef] [PubMed] - Liu, Z.K.; Jiang, J.; Zhou, B.; Wang, Z.J.; Zhang, Y.; Weng, H.M.; Prabhakaran, D.; Mo, S.K.; Peng, H.; Dudin, P.; et al. A stable three-dimensional topological Dirac semimetal Cd
_{3}As_{2}. Nat. Mater.**2014**, 13, 677–681. [Google Scholar] [CrossRef] [PubMed] - Neupane, M.; Xu, S.Y.; Sankar, R.; Alidoust, N.; Bian, G.; Liu, C.; Belopolski, I.; Chang, T.R.; Jeng, H.T.; Lin, H.; et al. Observation of a three-dimensional topological Dirac semimetal phase in high-mobility Cd
_{3}As_{2}. Nat. Commun.**2014**, 5, 3786. [Google Scholar] [CrossRef] [PubMed] - Yi, H.; Wang, Z.; Chen, C.; Shi, Y.; Feng, Y.; Liang, A.; Xie, Z.; He, S.; He, J.; Peng, Y.; et al. Evidence of Topological Surface State in Three-Dimensional Dirac Semimetal Cd
_{3}As_{2}. Sci. Rep.**2014**, 4, 6106. [Google Scholar] [CrossRef] [PubMed] - Borisenko, S.; Gibson, Q.; Evtushinsky, D.; Zabolotnyy, V.; Büchner, B.; Cava, R.J. Experimental Realization of a Three-Dimensional Dirac Semimetal. Phys. Rev. Lett.
**2014**, 113, 027603. [Google Scholar] [CrossRef] [PubMed] - Burkov, A.A.; Kim, Y.B. $\mathbb{Z}$
_{2}and Chiral Anomalies in Topological Dirac Semimetals. Phys. Rev. Lett.**2016**, 117, 136602. [Google Scholar] [CrossRef] [PubMed] - Kushwaha, S.K.; Krizan, J.W.; Feldman, B.E.; Gyenis, A.; Randeria, M.T.; Xiong, J.; Xu, S.Y.; Alidoust, N.; Belopolski, I.; Liang, T.; et al. Bulk crystal growth and electronic characterization of the 3D Dirac semimetal Na
_{3}Bi. APL Mater.**2015**, 3, 041504. [Google Scholar] [CrossRef] - Xiong, J.; Kushwaha, S.; Krizan, J.; Liang, T.; Cava, R.J.; Ong, N.P. Anomalous conductivity tensor in the Dirac semimetal Na
_{3}Bi. EPL**2016**, 114, 27002. [Google Scholar] [CrossRef] - Li, H.; He, H.; Lu, H.Z.; Zhang, H.; Liu, H.; Ma, R.; Fan, Z.; Shen, S.Q.; Wang, J. Negative magnetoresistance in Dirac semimetal Cd
_{3}As_{2}. Nat. Commun.**2016**, 7, 10301. [Google Scholar] [CrossRef] [PubMed] - Pixley, J.H.; Goswami, P.; Das Sarma, S. Anderson Localization and the Quantum Phase Diagram of Three Dimensional Disordered Dirac Semimetals. Phys. Rev. Lett.
**2015**, 115, 076601. [Google Scholar] [CrossRef] [PubMed] - Xiao, X.; Yang, S.A.; Liu, Z.; Li, H.; Zhou, G. Anisotropic Quantum Confinement Effect and Electric Control of Surface States in Dirac Semimetal Nanostructures. Sci. Rep.
**2015**, 5, 7898. [Google Scholar] [CrossRef] [PubMed] - Zhao, B.; Cheng, P.; Pan, H.; Zhang, S.; Wang, B.; Wang, G.; Xiu, F.; Song, F. Weak antilocalization in Cd
_{(3)}As_{(2)}thin films. Sci. Rep.**2016**, 6, 22377. [Google Scholar] [CrossRef] [PubMed] - Weng, H.; Fang, C.; Fang, Z.; Bernevig, B.A.; Dai, X. Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides. Phys. Rev. X
**2015**, 5, 011029. [Google Scholar] [CrossRef] - Huang, S.M.; Xu, S.Y.; Belopolski, I.; Lee, C.C.; Chang, G.; Wang, B.; Alidoust, N.; Bian, G.; Neupane, M.; Zhang, C.; et al. A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class. Nat. Commun.
**2015**, 6, 7373. [Google Scholar] [CrossRef] [PubMed] - Xu, S.Y.; Belopolski, I.; Alidoust, N.; Neupane, M.; Bian, G.; Zhang, C.; Sankar, R.; Chang, G.; Yuan, Z.; Lee, C.C.; et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science
**2015**, 349, 613–617. [Google Scholar] [CrossRef] [PubMed] - Lv, B.Q.; Weng, H.M.; Fu, B.B.; Wang, X.P.; Miao, H.; Ma, J.; Richard, P.; Huang, X.C.; Zhao, L.X.; Chen, G.F.; et al. Experimental Discovery of Weyl Semimetal TaAs. Phys. Rev. X
**2015**, 5, 031013. [Google Scholar] [CrossRef] - Wu, D.; Liao, J.; Yi, W.; Wang, X.; Li, P.; Weng, H.; Shi, Y.; Li, Y.; Luo, J.; Dai, X.; Fang, Z. Giant semiclassical magnetoresistance in high mobility TaAs2 semimetal. Appl. Phys. Lett.
**2016**, 108, 042105. [Google Scholar] [CrossRef] - Soluyanov, A.A.; Gresch, D.; Wang, Z.; Wu, Q.; Troyer, M.; Dai, X.; Bernevig, B.A. Type-II Weyl semimetals. Nature
**2015**, 527, 495–498. [Google Scholar] [CrossRef] [PubMed] - Borisenko, S.; Evtushinsky, D.; Gibson, Q.; Yaresko, A.; Kim, T.; Ali, M.N.; Buechner, B.; Hoesch, M.; Cava, R.J. Time-Reversal Symmetry Breaking Type-II Weyl State in YbMnBi
_{2}. arXiv, 2015; arXiv:arXiv:1507.04847. [Google Scholar] - Wang, Y.; Liu, E.; Liu, H.; Pan, Y.; Zhang, L.; Zeng, J.; Fu, Y.; Wang, M.; Xu, K.; Huang, Z.; et al. Gate-tunable negative longitudinal magnetoresistance in the predicted type-II Weyl semimetal WTe
_{2}. Nat. Commun.**2016**, 7, 13142. [Google Scholar] [CrossRef] [PubMed] - Wang, Y.; Wang, K.; Reutt-Robey, J.; Paglione, J.; Fuhrer, M.S. Breakdown of compensation and persistence of nonsaturating magnetoresistance in gated WTe
_{2}thin flakes. Phys. Rev. B**2016**, 93, 121108. [Google Scholar] [CrossRef] - Shan, W.Y.; Lu, H.Z.; Shen, S.Q. Effective continuous model for surface states and thin films of three-dimensional topological insulators. New J. Phys.
**2010**, 12, 043048. [Google Scholar] - Abanin, D.A.; Pesin, D.A. Ordering of Magnetic Impurities and Tunable Electronic Properties of Topological Insulators. Phys. Rev. Lett.
**2011**, 106, 136802. [Google Scholar] [CrossRef] [PubMed] - Burkov, A.A. Chiral Anomaly and Diffusive Magnetotransport in Weyl Metals. Phys. Rev. Lett.
**2014**, 113, 247203. [Google Scholar] [CrossRef] [PubMed] - Lu, H.Z.; Zhang, S.B.; Shen, S.Q. High-field magnetoconductivity of topological semimetals with short-range potential. Phys. Rev. B
**2015**, 92, 045203. [Google Scholar] [CrossRef] - Huang, X.; Zhao, L.; Long, Y.; Wang, P.; Chen, D.; Yang, Z.; Liang, H.; Xue, M.; Weng, H.; Fang, Z.; et al. Observation of the Chiral-Anomaly-Induced Negative Magnetoresistance in 3D Weyl Semimetal TaAs. Phys. Rev. X
**2015**, 5, 031023. [Google Scholar] [CrossRef] - Liu, M.; Chang, C.Z.; Zhang, Z.; Zhang, Y.; Ruan, W.; He, K.; Wang, L.l.; Chen, X.; Jia, J.F.; Zhang, S.C.; et al. Electron interaction-driven insulating ground state in Bi
_{2}Se_{3}topological insulators in the two-dimensional limit. Phys. Rev. B**2011**, 83, 165440. [Google Scholar] [CrossRef] - Yang, Q.I.; Dolev, M.; Zhang, L.; Zhao, J.; Fried, A.D.; Schemm, E.; Liu, M.; Palevski, A.; Marshall, A.F.; Risbud, S.H.; et al. Emerging weak localization effects on a topological insulator-insulating ferromagnet (Bi
_{2}Se_{3}-EuS) interface. Phys. Rev. B**2013**, 88, 081407. [Google Scholar] [CrossRef] - Burkov, A.A. Negative longitudinal magnetoresistance in Dirac and Weyl metals. Phys. Rev. B
**2015**, 91, 245157. [Google Scholar] [CrossRef] - Goswami, P.; Pixley, J.H.; Das Sarma, S. Axial anomaly and longitudinal magnetoresistance of a generic three-dimensional metal. Phys. Rev. B
**2015**, 92, 075205. [Google Scholar] [CrossRef] - Lu, H.Z.; Shen, S.Q. Weak antilocalization and interaction-induced localization of Dirac and Weyl Fermions in topological insulators and semimetals. Chin. Phys. B
**2016**, 25, 117202. [Google Scholar] [CrossRef] - Li, Y.; Wang, Z.; Li, P.; Yang, X.; Shen, Z.; Sheng, F.; Li, X.; Lu, Y.; Zheng, Y.; Xu, Z.A. Negative magnetoresistance in Weyl semimetals NbAs and NbP: Intrinsic chiral anomaly and extrinsic effects. Front. Phys.
**2017**, 12, 127205. [Google Scholar] [CrossRef] - Niemann, A.C.; Gooth, J.; Wu, S.C.; Bäbler, S.; Sergelius, P.; Hühne, R.; Rellinghaus, B.; Shekhar, C.; Süb, V.; Schmidt, M.; et al. Chiral magnetoresistance in the Weyl semimetal NbP. arXiv, 2016; arXiv:1610.01413. [Google Scholar]
- Evers, F.; Mirlin, A.D. Anderson transitions. Rev. Mod. Phys.
**2008**, 80, 1355–1417. [Google Scholar] [CrossRef] - Lee, P.A.; Ramakrishnan, T.V. Disordered electronic systems. Rev. Mod. Phys.
**1985**, 57, 287–337. [Google Scholar] [CrossRef] - Mühlbauer, M.; Budewitz, A.; Büttner, B.; Tkachov, G.; Hankiewicz, E.M.; Brüne, C.; Buhmann, H.; Molenkamp, L.W. One-Dimensional Weak Antilocalization Due to the Berry Phase in HgTe Wires. Phys. Rev. Lett.
**2014**, 112, 146803. [Google Scholar] [CrossRef] [PubMed] - Lu, H.Z.; Shi, J.; Shen, S.Q. Competition between Weak Localization and Antilocalization in Topological Surface States. Phys. Rev. Lett.
**2011**, 107, 076801. [Google Scholar] [CrossRef] [PubMed] - Imura, K.I.; Kuramoto, Y.; Nomura, K. Weak localization properties of the doped Z
_{2}topological insulator. Phys. Rev. B**2009**, 80, 085119. [Google Scholar] [CrossRef] - Tkachov, G.; Hankiewicz, E.M. Weak antilocalization in HgTe quantum wells and topological surface states: Massive versus massless Dirac fermions. Phys. Rev. B
**2011**, 84, 035444. [Google Scholar] [CrossRef] - Kammermeier, M.; Wenk, P.; Schliemann, J.; Heedt, S.; Schäpers, T. Weak (anti)localization in tubular semiconductor nanowires with spin-orbit coupling. Phys. Rev. B
**2016**, 93, 205306. [Google Scholar] [CrossRef] - Koga, T.; Nitta, J.; Akazaki, T.; Takayanagi, H. Rashba spin-orbit Coupling Probed by the Weak Antilocalization Analysis in InAlAs/InGaAs/InAlAs Quantum Wells as a Function of Quantum Well Asymmetry. Phys. Rev. Lett.
**2002**, 89, 046801. [Google Scholar] [CrossRef] [PubMed] - Shan, W.Y.; Lu, H.Z.; Shen, S.Q. spin-orbit scattering in quantum diffusion of massive Dirac fermions. Phys. Rev. B
**2012**, 86, 125303. [Google Scholar] [CrossRef] - Hikami, S.; Larkin, A.I.; Nagaoka, Y. spin-orbit Interaction and Magnetoresistance in the Two Dimensional Random System. Progr. Theor. Phys.
**1980**, 63, 707–710. [Google Scholar] [CrossRef] - McCann, E.; Kechedzhi, K.; Fal’ko, V.I.; Suzuura, H.; Ando, T.; Altshuler, B.L. Weak-Localization Magnetoresistance and Valley Symmetry in Graphene. Phys. Rev. Lett.
**2006**, 97, 146805. [Google Scholar] [CrossRef] [PubMed] - Adroguer, P.; Carpentier, D.; Cayssol, J.; Orignac, E. Diffusion at the surface of topological insulators. New J. Phys.
**2012**, 14, 103027. [Google Scholar] [CrossRef] - Garate, I.; Glazman, L. Weak localization and antilocalization in topological insulator thin films with coherent bulk-surface coupling. Phys. Rev. B
**2012**, 86, 035422. [Google Scholar] [CrossRef] - König, E.J.; Ostrovsky, P.M.; Protopopov, I.V.; Gornyi, I.V.; Burmistrov, I.S.; Mirlin, A.D. Interaction and disorder effects in three-dimensional topological insulator thin films. Phys. Rev. B
**2013**, 88, 035106. [Google Scholar] [CrossRef] - Checkelsky, J.G.; Hor, Y.S.; Cava, R.J.; Ong, N.P. Bulk Band Gap and Surface State Conduction Observed in Voltage-Tuned Crystals of the Topological Insulator Bi
_{2}Se_{3}. Phys. Rev. Lett.**2011**, 106, 196801. [Google Scholar] [CrossRef] [PubMed] - Zhang, L.; Dolev, M.; Yang, Q.I.; Hammond, R.H.; Zhou, B.; Palevski, A.; Chen, Y.; Kapitulnik, A. Weak localization effects as evidence for bulk quantization in Bi
_{2}Se_{3}thin films. Phys. Rev. B**2013**, 88, 121103. [Google Scholar] [CrossRef] - Lang, M.; He, L.; Kou, X.; Upadhyaya, P.; Fan, Y.; Chu, H.; Jiang, Y.; Bardarson, J.H.; Jiang, W.; Choi, E.S.; et al. Competing Weak Localization and Weak Antilocalization in Ultrathin Topological Insulators. Nano Lett.
**2013**, 13, 48–53. [Google Scholar] [CrossRef] [PubMed] - Bao, L.; Wang, W.; Meyer, N.; Liu, Y.; Zhang, C.; Wang, K.; Ai, P.; Xiu, F. Quantum Corrections Crossover and Ferromagnetism in Magnetic Topological Insulators. Sci. Rep.
**2013**, 3, 2391. [Google Scholar] [CrossRef] [PubMed] - Bardarson, J.H.; Moore, J.E. Quantum interference and Aharonov–Bohm oscillations in topological insulators. Rep. Prog. Phys
**2013**, 76, 056501. [Google Scholar] [CrossRef] [PubMed] - Akiyama, R.; Fujisawa, K.; Sakurai, R.; Kuroda, S. Weak antilocalization in (111) thin films of a topologial crystalline insulator SnTe. J. Phys. Conf. Ser.
**2014**, 568, 052001. [Google Scholar] [CrossRef] - Shekhar, C.; Nayak, A.K.; Sun, Y.; Schmidt, M.; Nicklas, M.; Leermakers, I.; Zeitler, U.; Skourski, Y.; Wosnitza, J.; Liu, Z.; et al. Extremely large magnetoresistance and ultrahigh mobility in the topological Weyl semimetal candidate NbP. Nat. Phys.
**2015**, 11, 645–649. [Google Scholar] [CrossRef] - Louvet, T.; Carpentier, D.; Fedorenko, A.A. New quantum transition in Weyl semimetals with correlated disorder. Phys. Rev. B
**2017**, 95, 014204. [Google Scholar] [CrossRef] - Keldysh, L.V. Diagram Technique for Nonequilibrium Processes. Sov. Phys. JETP
**1965**, 20, 1018. [Google Scholar] - Vasko, F.T.; Raichev, O.E. Quantum Kinetic Theory and Applications; Springer Science+Business Media, Inc.: New York, NY, USA, 2005. [Google Scholar]
- Rammer, J.; Smith, H. Quantum field-theoretical methods in transport theory of metals. Rev. Mod. Phys.
**1986**, 58, 323. [Google Scholar] [CrossRef] - Schwab, P.; Raimondi, R.; Gorini, C. Spin-Charge Locking and Tunneling into a Helical Metal. Eur. Phys. Lett.
**2011**, 93, 67004. [Google Scholar] [CrossRef] - Adroguer, P.; Liu, W.E.; Culcer, D.; Hankiewicz, E.M. Conductivity corrections for topological insulators with spin-orbit impurities: Hikami-Larkin-Nagaoka formula revisited. Phys. Rev. B
**2015**, 92, 241402. [Google Scholar] [CrossRef] - Lu, H.Z.; Shen, S.Q. Quantum transport in topological semimetals under magnetic fields. Front. Phys.
**2016**, 12, 127201. [Google Scholar] [CrossRef] - Ngabonziza, P.; Stehno, M.P.; Myoren, H.; Neumann, V.A.; Koster, G.; Brinkman, A. Gate-tunable transport properties of in-situ capped Bi
_{2}Te_{3}topological insulator thin films. Adv. Electron. Mater.**2016**. [Google Scholar] [CrossRef] - Linder, J.; Yokoyama, T.; Sudbø, A. Anomalous finite size effects on surface states in the topological insulator Bi
_{2}Se_{3}. Phys. Rev. B**2009**, 80, 205401. [Google Scholar] [CrossRef] - Liu, C.X.; Zhang, H.; Yan, B.; Qi, X.L.; Frauenheim, T.; Dai, X.; Fang, Z.; Zhang, S.C. Oscillatory crossover from two-dimensional to three-dimensional topological insulators. Phys. Rev. B
**2010**, 81, 041307. [Google Scholar] [CrossRef] - Lu, H.Z.; Shan, W.Y.; Yao, W.; Niu, Q.; Shen, S.Q. Massive Dirac fermions and spin physics in an ultrathin film of topological insulator. Phys. Rev. B
**2010**, 81, 115407. [Google Scholar] [CrossRef] - Park, K.; Heremans, J.J.; Scarola, V.W.; Minic, D. Robustness of Topologically Protected Surface States in Layering of Bi
_{2}Te_{3}Thin Films. Phys. Rev. Lett.**2010**, 105, 186801. [Google Scholar] [CrossRef] [PubMed] - Zhang, Y.; He, K.; Chang, C.Z.; Song, C.L.; Wang, L.; Chen, X.; Jia, J.; Fang, Z.; Dai, X.; Shan, W.Y.; et al. Crossover of the three-dimensional topological insulator Bi
_{2}Se_{3}to the two-dimensional limit. Nat. Phys.**2010**, 6, 584. [Google Scholar] [CrossRef] - Hirahara, T.; Sakamoto, Y.; Saisyu, Y.; Miyazaki, H.; Kimura, S.; Okuda, T.; Matsuda, I.; Murakami, S.; Hasegawa, S. Topological metal at the surface of an ultrathin Bi
_{1−x}Sb_{x}alloy film. Phys. Rev. B**2010**, 81, 165422. [Google Scholar] [CrossRef] - Sakamoto, Y.; Hirahara, T.; Miyazaki, H.; Kimura, S.i.; Hasegawa, S. Spectroscopic evidence of a topological quantum phase transition in ultrathin Bi
_{2}Se_{3}films. Phys. Rev. B**2010**, 81, 165432. [Google Scholar] [CrossRef] - Taskin, A.A.; Sasaki, S.; Segawa, K.; Ando, Y. Manifestation of Topological Protection in Transport Properties of Epitaxial Bi
_{2}Se_{3}Thin Films. Phys. Rev. Lett.**2012**, 109, 066803. [Google Scholar] [CrossRef] [PubMed] - Seradjeh, B.; Moore, J.E.; Franz, M. Exciton Condensation and Charge Fractionalization in a Topological Insulator Film. Phys. Rev. Lett.
**2009**, 103, 066402. [Google Scholar] [CrossRef] [PubMed] - Seradjeh, B. Topological exciton condensate of imbalanced electrons and holes. Phys. Rev. B
**2012**, 85, 235146. [Google Scholar] [CrossRef] - Efimkin, D.K.; Lozovik, Y.E.; Sokolik, A.A. Electron-hole pairing in a topological insulator thin film. Phys. Rev. B
**2012**, 86, 115436. [Google Scholar] [CrossRef] - Kim, Y.; Hankiewicz, E.M.; Gilbert, M.J. Topological excitonic superfluids in three dimensions. Phys. Rev. B
**2012**, 86, 184504. [Google Scholar] [CrossRef] - Tilahun, D.; Lee, B.; Hankiewicz, E.M.; MacDonald, A.H. Quantum Hall Superfluids in Topological Insulator Thin Films. Phys. Rev. Lett.
**2011**, 107, 246401. [Google Scholar] [CrossRef] [PubMed] - Ando, T.; Fowler, A.B.; Stern, F. Electronic properties of two-dimensional systems. Rev. Mod. Phys.
**1982**, 54, 437–672. [Google Scholar] [CrossRef] - Chiu, S.P.; Lin, J.J. Weak antilocalization in topological insulator Bi
_{2}Te_{3}microflakes. Phys. Rev. B**2013**, 87, 035122. [Google Scholar] [CrossRef] - Zhao, Y.; Chang, C.Z.; Jiang, Y.; DaSilva, A.; Sun, Y.; Wang, H.; Xing, Y.; Wang, Y.; He, K.; Ma, X.; et al. Demonstration of surface transport in a hybrid Bi
_{2}Se_{3}/Bi_{2}Te_{3}heterostructure. Sci. Rep.**2013**, 3, 3060. [Google Scholar] [CrossRef] [PubMed] - Culcer, D.; Yao, Y.; MacDonald, A.H.; Niu, Q. Electrical generation of spin in crystals with reduced symmetry. Phys. Rev. B
**2005**, 72, 045215. [Google Scholar] [CrossRef] - Rossi, E.; Bardarson, J.H.; Fuhrer, M.S.; Das Sarma, S. Universal Conductance Fluctuations in Dirac Materials in the Presence of Long-range Disorder. Phys. Rev. Lett.
**2012**, 109, 096801. [Google Scholar] [CrossRef] [PubMed] - Culcer, D.; Winkler, R. Weak momentum scattering and the conductivity of graphene. Phys. Rev. B
**2008**, 78, 235417. [Google Scholar] [CrossRef] - Kechedzhi, K.; McCann, E.; Fal’ko, V.I.; Suzuura, H.; Ando, T.; Altshuler, B.L. Weak localization in monolayer and bilayer graphene. Eur. Phys. J. Spec. Top.
**2007**, 148, 39–54. [Google Scholar] [CrossRef]

**Figure 1.**Phase diagram representing the evolution of the quantum correction to the conductivity ${\sigma}^{\mathrm{qi}}(0)$ at zero magnetic field for massless and massive Dirac fermions. The parameters a, b and $\lambda $ are defined in Section 2. The extrinsic spin-orbit scattering strength is quantified by $\lambda $, while the ratio $a/b$ represents the ratio of the spin-orbit energy to the quasiparticle mass, evaluated at the Fermi energy. The conductivity is expressed in units of ${e}^{2}/h$ with the color bar on the right. The blue dashed line separates the WAL and WL regimes. The numerical parameters here are the same as those used in Figure 6 below.

**Figure 2.**The diagrams for the weak (anti-)localization conductivity of Dirac fermions. (

**a**) definition of the Green’s functions as arrowed solid lines in which Greek letters are spin indices; (

**b**) definition of dashed lines: impurities lines expressed in both retarded and advanced cases; (

**c**) the retarded self-energy in the first-order Born approximation, where ${G}_{0}^{\mathrm{R}}$ is the bare retarded Green’s function. This contribution to the self energy represents the classical picture of electrons scattering off randomly distributed impurities; (

**d**,

**e**) are Keldysh self-energies $({\Sigma}_{\mathit{k},\gamma \delta}^{\mathrm{K},\mathrm{b}}$ and ${\Sigma}_{\mathit{k},\alpha \beta}^{\mathrm{K},\mathrm{R}})$ of maximally crossed diagrams in the bare and the retarded dressed cases, respectively, where $\Gamma $ is the Cooperon structure factor. Both of these represent contributions due to quantum interference in scattering processes: an electron travelling through a disordered region can backscatter and return to its starting point. The loop can be performed clockwise or anticlockwise, and, in quantum mechanics, the two trajectories interfere. Note that (

**e**) vanishes in the absence of spin-orbit coupling; (

**f**) the Bethe-Salpeter equation for the twisted Cooperon structure factor $\tilde{\Gamma}.$

**Figure 3.**Plot of the zero-field quantum-interference conductivity correction in terms of the spin-orbit scattering strength $\lambda $ for massless Dirac fermions (solid line), HLN formula for 2DEG (dashed line) and 3DEG (dotted line). $\delta {\sigma}_{0}=({e}^{2}/\pi h)ln({\tau}_{\varphi}/\tau )$. In order to emphasize that the WAL conductivity of massless Dirac fermions is also dependent upon the spin-orbit impurity strength, the inset shows WAL in units independent of $\lambda $, i.e., $({e}^{2}/\pi h)ln({\tau}_{\varphi}/{\tau}_{0}),$ where ${\tau}_{0}$ is the scattering time in the absence of spin-orbit impurities and the ratio ${\tau}_{\varphi}/{\tau}_{0}$ varies from 5 (purple), 10 (blue), 50 (green), 100 (orange) to 500 (red). Adapted from Figure 3 in [154].

**Figure 4.**The magnetic field dependence of the quantum correction to the conductivity for massless Dirac fermions (solid lines) and the 3D HLN formula (dashed lines) at different values of the spin-orbit impurities concentrations. Purple plots are for purely scalar disorder ($\lambda =0$), while blue (green) plots for $\lambda =0.2$ ($\lambda =1.25$). The magnetic field is given in the unit of ${\tilde{B}}_{0}=e/(2\pi \hslash {v}_{\mathrm{F}}^{2}{\tau}_{0}^{2})$, where ${\tau}_{0}$ is the elastic scattering time in the absence of spin-orbit impurities and ${v}_{\mathrm{F}}$ the Fermi velocity. Adapted from Figure 1 in [154], where $\delta \sigma (B)\equiv \Delta \sigma (B)$ as defined in Equation (16).

**Figure 5.**(

**a**) a minimal sketch of the energy dispersion of a Weyl semimetal. We have defined ${k}_{\parallel}^{2}={k}_{x}^{2}+{k}_{y}^{2}$ while $({k}_{x},{k}_{y},{k}_{z})$ represents the 3D wavevector. ${k}_{z}$ points along the preferred direction. The conductance and valence bands cross linearly at the Weyl nodes, i.e., the left and right cones, and the nodes appear in pairs with opposite chirality number. A Dirac node appears when two oppositely-chiral Weyl nodes emerge together; (

**b**) a schematic picture of the band structure in WSM thin films, where $\Delta $ is the band gap and ${\epsilon}_{\mathrm{F}}$ is the Fermi energy.

**Figure 6.**The magnetoconductivity $\Delta \sigma (B)$ plots at $\lambda =0.5$ for different masses M. The parameters used are $A=300$ meV·nm, ${n}_{e}=0.01\phantom{\rule{4.pt}{0ex}}{\mathrm{nm}}^{-2},$ ${n}_{\mathrm{i}}=0.0001\phantom{\rule{4.pt}{0ex}}{\mathrm{nm}}^{-2}$ and ${\ell}_{\varphi}=500$ nm, according to [131].

**Figure 7.**(

**a**–

**c**) the carrier density dependence of the quantum-interference conductivity ${\sigma}^{\mathrm{qi}}$ in the massless $(M=0$ meV), small mass $(M=10\phantom{\rule{4.pt}{0ex}}\mathrm{meV}),$ and large mass $(M=100\phantom{\rule{4.pt}{0ex}}\mathrm{meV})$ cases. The parameters are the same as in Figure 6 and $L={\ell}_{\varphi}$. Note that $\lambda ={\lambda}_{c}{n}_{e}$ and $A{k}_{\mathrm{F}}=106$ meV for ${n}_{e}=0.01\phantom{\rule{0.166667em}{0ex}}{\mathrm{nm}}^{-2}$.

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Liu, W.E.; Hankiewicz, E.M.; Culcer, D.
Weak Localization and Antilocalization in Topological Materials with Impurity Spin-Orbit Interactions. *Materials* **2017**, *10*, 807.
https://doi.org/10.3390/ma10070807

**AMA Style**

Liu WE, Hankiewicz EM, Culcer D.
Weak Localization and Antilocalization in Topological Materials with Impurity Spin-Orbit Interactions. *Materials*. 2017; 10(7):807.
https://doi.org/10.3390/ma10070807

**Chicago/Turabian Style**

Liu, Weizhe Edward, Ewelina M. Hankiewicz, and Dimitrie Culcer.
2017. "Weak Localization and Antilocalization in Topological Materials with Impurity Spin-Orbit Interactions" *Materials* 10, no. 7: 807.
https://doi.org/10.3390/ma10070807