#
Omnipresence of Weak Antilocalization (WAL) in Bi_{2}Se_{3} Thin Films: A Review on Its Origin

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## Abstract

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## 1. Introduction

## 2. Weak-Antilocalization (WAL) Effect

#### 2.1. Electronic Motion in the Quantum Diffusive Regime

#### 2.2. WAL in Relevant Materials

## 3. WAL in Bi_{2}Se_{3} Thin Films

#### 3.1. Growth Methods

#### 3.2. Magnetotransport Properties and WAL Effect in Bi_{2}Se_{3} Thin Films

## 4. Remarks and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Von Klitzing, K. The quantized Hall effect. Rev. Mod. Phys.
**1986**, 58, 519–531. [Google Scholar] [CrossRef] - Moore, J. Topological insulators: The next generation. Nat. Phys.
**2009**, 5, 378–380. [Google Scholar] [CrossRef] - Kane, C.L.; Mele, E.J. Quantum Spin hall effect in graphene. Phys. Rev. Lett.
**2005**, 95, 226801. [Google Scholar] [CrossRef][Green Version] - Hasan, M.Z.; Kane, C.L. Colloquium: Topological insulators. Rev. Mod. Phys.
**2010**, 82, 3045–3067. [Google Scholar] [CrossRef][Green Version] - Fu, L.; Kane, C.L. Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys. Rev. Lett.
**2008**, 100, 096407. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pesin, D.; MacDonald, A.H. Spintronics and pseudospintronics in graphene and topological insulators. Nat. Mater.
**2012**, 11, 409–416. [Google Scholar] [CrossRef] [PubMed] - Analytis, J.G.; Chu, J.H.; Chen, Y.; Corredor, F.; McDonald, R.D.; Shen, Z.X.; Fisher, I.R. Bulk Fermi surface coexistence with Dirac surface state in Bi
_{2}Se_{3}: A comparison of photoemission and Shubnikov-de Haas measurements. Phys. Rev. B Condens. Matter Mater. Phys.**2010**, 81, 205407. [Google Scholar] [CrossRef][Green Version] - König, M.; Molenkamp, L.W.; Qi, X.; Zhang, S. Quantum Spin Hall Insulator State in HgTe Quantum Wells. Science
**2007**, 766, 766–771. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hsieh, D.; Qian, D.; Wray, L.; Xia, Y.; Hor, Y.S.; Cava, R.J.; Hasan, M.Z. A topological Dirac insulator in a quantum spin Hall phase. Nature
**2008**, 452, 970–974. [Google Scholar] [CrossRef] [PubMed][Green Version] - Xia, Y.; Qian, D.; Hsieh, D.; Wray, L.; Pal, A.; Lin, H.; Bansil, A.; Grauer, D.; Hor, Y.S.; Cava, R.J.; et al. Observation of a large-gap topological-insulator class with a single Dirac cone on the surface. Nat. Phys.
**2009**, 5, 398–402. [Google Scholar] [CrossRef][Green Version] - Hsieh, D.; Xia, Y.; Qian, D.; Wray, L.; Meier, F.; Dil, J.H.; Osterwalder, J.; Patthey, L.; Fedorov, A.V.; Lin, H.; et al. Observation of time-reversal-protected single-dirac-cone topological-insulator states in Bi
_{2}Se_{3}and Sb_{2}Te_{3}. Phys. Rev. Lett.**2009**, 103, 146401. [Google Scholar] [CrossRef][Green Version] - Chen, J.; Qin, H.J.; Yang, F.; Liu, J.; Guan, T.; Qu, F.M.; Zhang, G.H.; Shi, J.R.; Xie, X.C.; Yang, C.L.; et al. Gate-voltage control of chemical potential and weak antilocalization in Bi
_{2}Se_{3}. Phys. Rev. Lett.**2010**, 105, 176602. [Google Scholar] [CrossRef][Green Version] - 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][Green Version] - Steinberg, H.; Laloë, J.B.; Fatemi, V.; Moodera, J.S.; Jarillo-Herrero, P. Electrically tunable surface-to-bulk coherent coupling in topological insulator thin films. Phys. Rev. B Condens. Matter Mater. Phys.
**2011**, 84, 233101. [Google Scholar] [CrossRef][Green Version] - Xue, L.; Zhou, P.; Zhang, C.X.; He, C.Y.; Hao, G.L.; Sun, L.Z.; Zhong, J.X. First-principles study of native point defects in Bi
_{2}Se_{3}. AIP Adv.**2013**, 3, 052105. [Google Scholar] [CrossRef][Green Version] - Scanlon, D.O.; King, P.D.C.; Singh, R.P.; De La Torre, A.; Walker, S.M.K.; Balakrishnan, G.; Baumberger, F.; Catlow, C.R.A. Controlling bulk conductivity in topological insulators: Key role of anti-site defects. Adv. Mater.
**2012**, 24, 2154–2158. [Google Scholar] [CrossRef] [PubMed][Green Version] - Zhang, G.; Qin, H.; Chen, J.; He, X.; Lu, L.; Li, Y.; Wu, K. Growth of topological insulator Bi
_{2}Se_{3}thin films on SrTiO_{3}with a large tunability in chemical potential. Adv. Funct. Mater.**2011**, 21, 2351–2355. [Google Scholar] [CrossRef] - Hor, Y.S.; Richardella, A.; Roushan, P.; Xia, Y.; Checkelsky, J.G.; Yazdani, A.; Hasan, M.Z.; Ong, N.P.; Cava, R.J. P -type Bi
_{2}Se_{3}for topological insulator and low-temperature thermoelectric applications. Phys. Rev. B Condens. Matter Mater. Phys.**2009**, 79, 2–6. [Google Scholar] [CrossRef][Green Version] - Fu, L. Topological crystalline insulators. Phys. Rev. Lett.
**2011**, 106, 106802. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hsieh, T.H.; Lin, H.; Liu, J.; Duan, W.; Bansil, A.; Fu, L. Topological crystalline insulators in the SnTe material class. Nat. Commun.
**2012**, 3, 1–7. [Google Scholar] [CrossRef][Green Version] - Tanaka, Y.; Ren, Z.; Sato, T.; Nakayama, K.; Souma, S.; Takahashi, T.; Segawa, K.; Ando, Y. Experimental realization of a topological crystalline insulator in SnTe. Nat. Phys.
**2012**, 8, 800–803. [Google Scholar] [CrossRef][Green Version] - Xu, S.Y.; Liu, C.; Alidoust, N.; Neupane, M.; Qian, D.; Belopolski, I.; Denlinger, J.D.; Wang, Y.J.; Lin, H.; Wray, L.A.; et al. Observation of a topological crystalline insulator phase and topological phase transition in Pb
_{1-x}Sn_{x}Te. Nat. Commun.**2012**, 3, 1–11. [Google Scholar] [CrossRef] [PubMed][Green Version] - Dziawa, P.; Kowalski, B.J.; Dybko, K.; Buczko, R.; Szczerbakow, A.; Szot, M.; Łusakowska, E.; Balasubramanian, T.; Wojek, B.M.; Berntsen, M.H.; et al. Topological crystalline insulator states in Pb
_{1-x}Sn_{x}Se. Nat. Mater.**2012**, 11, 1023–1027. [Google Scholar] [CrossRef][Green Version] - Dzero, M.; Sun, K.; Galitski, V.; Coleman, P. Topological Kondo insulators. Phys. Rev. Lett.
**2010**, 104, 106408. [Google Scholar] [CrossRef] [PubMed][Green Version] - Xu, N.; Biswas, P.K.; Dil, J.H.; Dhaka, R.S.; Landolt, G.; Muff, S.; Matt, C.E.; Shi, X.; Plumb, N.C.; Radović, M.; et al. Direct observation of the spin texture in SmB
_{6}as evidence of the topological Kondo insulator. Nat. Commun.**2014**, 5, 1–5. [Google Scholar] [CrossRef] - 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, Na
_{3}Bi. Science**2014**, 343, 864–867. [Google Scholar] [CrossRef] [PubMed][Green Version] - 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][Green Version] - Das, A.; Ronen, Y.; Most, Y.; Oreg, Y.; Heiblum, M.; Shtrikman, H. Zero-bias peaks and splitting in an Al-InAs nanowire topological superconductor as a signature of Majorana fermions. Nat. Phys.
**2012**, 8, 887–895. [Google Scholar] [CrossRef][Green Version] - Sasaki, S.; Kriener, M.; Segawa, K.; Yada, K.; Tanaka, Y.; Sato, M.; Ando, Y.; Cu, T. Topological Superconductivity in Cu
_{x}Bi_{2}Se_{3}. Phys. Rev. Lett.**2011**, 107, 217001. [Google Scholar] [CrossRef][Green Version] - Taskin, A.A.; Sasaki, S.; Segawa, K.; Ando, Y. Achieving surface quantum oscillations in topological insulator thin films of Bi
_{2}Se_{3}. Adv. Mater.**2012**, 24, 5581–5585. [Google Scholar] [CrossRef] - Kim, Y.S.; Brahlek, M.; Bansal, N.; Edrey, E.; Kapilevich, G.A.; Iida, K.; Tanimura, M.; Horibe, Y.; Cheong, S.W.; Oh, S. Thickness-dependent bulk properties and weak antilocalization effect in topological insulator Bi
_{2}Se_{3}. Phys. Rev. B Condens. Matter Mater. Phys.**2011**, 84, 073109. [Google Scholar] [CrossRef][Green Version] - Chen, J.; He, X.Y.; Wu, K.H.; Ji, Z.Q.; Lu, L.; Shi, J.R.; Smet, J.H.; Li, Y.Q. Tunable surface conductivity in Bi
_{2}Se_{3}revealed in diffusive electron transport. Phys. Rev. B Condens. Matter Mater. Phys.**2011**, 83, 1–5. [Google Scholar] [CrossRef][Green Version] - Bigi, C.; Orgiani, P.; Nardi, A.; Troglia, A.; Fujii, J.; Panaccione, G.; Vobornik, I.; Rossi, G. Robustness of topological states in Bi
_{2}Se_{3}thin film grown by Pulsed Laser Deposition on (001)-oriented SrTiO_{3}perovskite. Appl. Surf. Sci.**2019**, 473, 190–193. [Google Scholar] [CrossRef] - Lee, Y.F.; Punugupati, S.; Wu, F.; Jin, Z.; Narayan, J.; Schwartz, J. Evidence for topological surface states in epitaxial Bi
_{2}Se_{3}thin film grown by pulsed laser deposition through magneto-transport measurements. Curr. Opin. Solid State Mater. Sci.**2014**, 18, 279–285. [Google Scholar] [CrossRef] - Berry, M. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. London. A. Math. Phys. Sci.
**1984**, 392, 45–47. [Google Scholar] [CrossRef] - Bergmann, G. Weak localization in thin films: A time-of-flight experiment with conduction electrons. Phys. Rep.
**1986**, 107, 1–58. [Google Scholar] [CrossRef] - Sangiao, S.; Marcano, N.; Fan, J.; Morellón, L.; Ibarra, M.R.; De Teresa, J.M. Quantitative analysis of the weak anti-localization effect in ultrathin Bi films. Epl
**2011**, 95, 37002. [Google Scholar] [CrossRef] - Hasan, M. Berry’s phase and quantization in topological insulators. Physics
**2010**, 3, 62. [Google Scholar] [CrossRef][Green Version] - Hikami, S.; Larkin, A.I.; Nagaoka, Y. Spin-Orbit Interaction and Magnetoresistance in the Two Dimensional Random System. Prog. Theor. Phys.
**1980**, 63, 707–710. [Google Scholar] [CrossRef] - Brahlek, M.; Koirala, N.; Salehi, M.; Bansal, N.; Oh, S. Emergence of decoupled surface transport channels in bulk insulating Bi
_{2}Se_{3}thin films. Phys. Rev. Lett.**2014**, 113, 026801. [Google Scholar] [CrossRef] [PubMed][Green Version] - Altshuler, B.L.; Aronov, A.G.; Khmelnitsky, D.E. Effects of electron-electron collisions with small energy transfers on quantum localization. J. Phys. C Solid State Phys.
**1982**, 15, 7367–7386. [Google Scholar] [CrossRef] - Jing, Y.; Huang, S.; Zhang, K.; Wu, J.; Guo, Y.; Peng, H.; Liu, Z.; Xu, H.Q. Weak antilocalization and electron–electron interaction in coupled multiple-channel transport in a Bi
_{2}Se_{3}thin film. Nanoscale**2016**, 18, 1879–1885. [Google Scholar] [CrossRef] [PubMed][Green Version] - Zhang, Y.; He, K.; Chang, C.Z.; Song, C.L.; Wang, L.L.; Chen, X.; Jia, J.F.; 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, 584–588. [Google Scholar] [CrossRef][Green Version] - Liu, Y.H.; Chong, C.W.; Jheng, J.L.; Huang, S.Y.; Huang, J.C.A.; Li, Z.; Qiu, H.; Huang, S.M.; Marchenkov, V.V. Gate-tunable coherent transport in Se-capped Bi
_{2}Se_{3}grown on amorphous SiO_{2}/Si. Appl. Phys. Lett.**2015**, 107, 012106. [Google Scholar] [CrossRef] - Assaf, B.A.; Katmis, F.; Wei, P.; Satpati, B.; Zhang, Z.; Bennett, S.P.; Harris, V.G.; Moodera, J.S.; Heiman, D. Quantum coherent transport in SnTe topological crystalline insulator thin films. Appl. Phys. Lett.
**2014**, 105, 102108. [Google Scholar] [CrossRef][Green Version] - Lu, H.Z.; Shen, S.Q. Weak antilocalization and localization in disordered and interacting Weyl semimetals. Phys. Rev. B Condens. Matter Mater. Phys.
**2015**, 92, 1–13. [Google Scholar] [CrossRef][Green Version] - Lu, H.Z.; Shen, S.Q. Quantum transport in topological semimetals under magnetic fields. Front. Phys.
**2017**, 12, 1–18. [Google Scholar] [CrossRef][Green Version] - 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, 1–7. [Google Scholar] [CrossRef][Green Version] - 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][Green Version] - 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][Green Version] - 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 Condens. Matter Mater. Phys.**2011**, 83, 165440. [Google Scholar] [CrossRef][Green Version] - Li, H.D.; Wang, Z.Y.; Kan, X.; Guo, X.; He, H.T.; Wang, Z.; Wang, J.N.; Wong, T.L.; Wang, N.; Xie, M.H. The van der Waals epitaxy of Bi
_{2}Se_{3}on the vicinal Si(111) surface: An approach for preparing high-quality thin films of a topological insulator. New J. Phys.**2010**, 12, 103038. [Google Scholar] [CrossRef] - Le, P.H.; Wu, K.H.; Luo, C.W.; Leu, J. Growth and characterization of topological insulator Bi
_{2}Se_{3}thin films on SrTiO_{3}using pulsed laser deposition. Thin Solid Films**2013**, 534, 659–665. [Google Scholar] [CrossRef] - Yang, L.; Wang, Z.; Li, M.; Gao, X.P.A.; Zhang, Z. The dimensional crossover of quantum transport properties in few-layered Bi
_{2}Se_{3}thin films. Nanoscale Adv.**2019**, 1, 2303–2310. [Google Scholar] [CrossRef][Green Version] - Brom, J.E.; Ke, Y.; Du, R.; Won, D.; Weng, X.; Andre, K.; Gagnon, J.C.; Mohney, S.E.; Li, Q.; Chen, K.; et al. Structural and electrical properties of epitaxial Bi
_{2}Se_{3}thin films grown by hybrid physical-chemical vapor deposition. Appl. Phys. Lett.**2012**, 100, 162110. [Google Scholar] [CrossRef] - Lin, Y.C.; Chen, Y.S.; Lee, C.C.; Wu, J.K.; Lee, H.Y.; Te Liang, C.; Chang, Y.H. A study on the epitaxial Bi
_{2}Se_{3}thin film grown by vapor phase epitaxy. AIP Adv.**2016**, 6, 065218. [Google Scholar] [CrossRef][Green Version] - Wang, W.J.; Gao, K.H.; Li, Z.Q. Thickness-dependent transport channels in topological insulator Bi
_{2}Se_{3}thin films grown by magnetron sputtering. Sci. Rep.**2016**, 6, 1–9. [Google Scholar] [CrossRef][Green Version] - Richardella, A.; Zhang, D.M.; Lee, J.S.; Koser, A.; Rench, D.W.; Yeats, A.L.; Buckley, B.B.; Awschalom, D.D.; Samarth, N. Coherent heteroepitaxy of Bi
_{2}Se_{3}on GaAs (111)B. Appl. Phys. Lett.**2010**, 97, 262104. [Google Scholar] [CrossRef] - Bansal, N.; Kim, Y.S.; Edrey, E.; Brahlek, M.; Horibe, Y.; Iida, K.; Tanimura, M.; Li, G.H.; Feng, T.; Lee, H.D.; et al. Epitaxial growth of topological insulator Bi
_{2}Se_{3}film on Si(111) with atomically sharp interface. Thin Solid Films**2011**, 520, 224–229. [Google Scholar] [CrossRef][Green Version] - Kim, N.; Lee, P.; Kim, Y.; Kim, J.S.; Kim, Y.; Noh, D.Y.; Yu, S.U.; Chung, J.; Kim, K.S. Persistent topological surface state at the interface of Bi
_{2}Se_{3}film grown on patterned graphene. ACS Nano**2014**, 8, 1154–1160. [Google Scholar] [CrossRef][Green Version] - Kou, X.F.; He, L.; Xiu, F.X.; Lang, M.R.; Liao, Z.M.; Wang, Y.; Fedorov, A.V.; Yu, X.X.; Tang, J.S.; Huang, G.; et al. Epitaxial growth of high mobility Bi
_{2}Se_{3}thin films on CdS. Appl. Phys. Lett.**2011**, 98, 2011–2014. [Google Scholar] [CrossRef][Green Version] - He, L.; Xiu, F.; Yu, X.; Teague, M.; Fan, Y.; Kou, X. Surface-Dominated Conduction in a 6 nm thick Bi
_{2}Se_{3}Thin Film. Nano Lett.**2012**, 12, 1486–1490. [Google Scholar] [CrossRef] [PubMed] - Guo, X.; Xu, Z.J.; Liu, H.C.; Zhao, B.; Dai, X.Q.; He, H.T.; Wang, J.N.; Liu, H.J.; Ho, W.K.; Xie, M.H. Single domain Bi
_{2}Se_{3}films grown on InP(111)A by molecular-beam epitaxy. Appl. Phys. Lett.**2013**, 102, 151604. [Google Scholar] [CrossRef][Green Version] - Bansal, N.; Kim, Y.S.; Brahlek, M.; Edrey, E.; Oh, S. Thickness-independent transport channels in topological insulator Bi
_{2}Se_{3}thin films. Phys. Rev. Lett.**2012**, 109, 116804. [Google Scholar] [CrossRef][Green Version] - Brahlek, M.; Koirala, N.; Bansal, N.; Oh, S. Transport properties of topological insulators: Band bending, bulk metal-to-insulator transition, and weak anti-localization. Solid State Commun.
**2015**, 215, 54–62. [Google Scholar] [CrossRef][Green Version] - Orgiani, P.; Bigi, C.; Kumar Das, P.; Fujii, J.; Ciancio, R.; Gobaut, B.; Galdi, A.; Sacco, C.; Maritato, L.; Torelli, P.; et al. Structural and electronic properties of Bi
_{2}Se_{3}topological insulator thin films grown by pulsed laser deposition. Appl. Phys. Lett.**2017**, 110, 171601. [Google Scholar] [CrossRef][Green Version] - Zhang, M.; Wei, Z.; Jin, R.; Ji, Y.; Yan, Y.; Pu, X.; Yang, X.; Zhao, Y. Electrical transport properties and morphology of topological insulator Bi
_{2}Se_{3}thin films with different thickness prepared by magnetron sputtering. Thin Solid Films**2016**, 603, 289–293. [Google Scholar] [CrossRef] - Oveshnikov, L.N.; Prudkoglyad, V.A.; Nekhaeva, E.I.; Kuntsevich, A.Y.; Selivanov, Y.G.; Chizhevskii, E.G.; Aronzon, B.A. Magnetotransport in thin epitaxial Bi
_{2}Se_{3}films. JETP Lett.**2016**, 104, 629–634. [Google Scholar] [CrossRef] - Dey, R.; Roy, A.; Pramanik, T.; Guchhait, S.; Sonde, S.; Rai, A.; Register, L.F.; Banerjee, S.K. Localization and interaction effects of epitaxial Bi
_{2}Se_{3}bulk states in two-dimensional limit. J. Appl. Phys.**2016**, 120, 164301. [Google Scholar] [CrossRef] - You, A.; Be, M.A.Y.; In, I. Magnetotransport in Bi
_{2}Se_{3}thin films epitaxially grown on Ge (111). AIP Adv.**2018**, 8, 115125. [Google Scholar] [CrossRef][Green Version] - Han, W.; Otani, Y.C.; Maekawa, S. Quantum materials for spin and charge conversion. npj Quantum Mater.
**2018**, 3, 1–16. [Google Scholar] [CrossRef][Green Version] - Wang, H.; Kally, J.; Lee, J.S.; Liu, T.; Chang, H.; Hickey, D.R.; Mkhoyan, K.A.; Wu, M.; Richardella, A.; Samarth, N. Surface-State-Dominated Spin-Charge Current Conversion in Topological-Insulator-Ferromagnetic-Insulator Heterostructures. Phys. Rev. Lett.
**2016**, 117, 076601. [Google Scholar] [CrossRef] [PubMed] - Wang, M.; Liu, C.; Xu, J.; Yang, F.; Miao, L.; Yao, M.; Gao, C.L.; Shen, C.; Ma, X.; Chen, X.; et al. The coexistence of superconductivity and topological order in the Bi
_{2}Se_{3}thin films. Science**2012**, 336, 52–55. [Google Scholar] [CrossRef] - Peng, H.; Lai, K.; Kong, D.; Meister, S.; Chen, Y.; Qi, X.L.; Zhang, S.C.; Shen, Z.X.; Cui, Y. Aharonov-Bohm interference in topological insulator nanoribbons. Nat. Mater.
**2010**, 9, 225–229. [Google Scholar] [CrossRef] [PubMed][Green Version] - Yang, F.; Qu, F.; Shen, J.; Ding, Y.; Chen, J.; Ji, Z.; Liu, G.; Fan, J.; Yang, C.; Fu, L.; et al. Proximity-effect-induced superconducting phase in the topological insulator Bi
_{2}Se_{3}. Phys. Rev. B Condens. Matter Mater. Phys.**2012**, 86, 134504. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Scheme of a Dirac cone at the surface of a Topological Insulator (TI) showing the spin-momentum locking (spin orientations are indicated by red arrows). (

**b**) Angle-Resolved Photoemission Spectroscopy (ARPES) image of the ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ band structure showing the Dirac cone at the center of the Brillouin zone. Reprinted with permission from Reference [7]. Copyright 2010 American Physical Society.

**Figure 2.**Schemes showing the band structure and the different contributions to transport in TIs: (

**a**) n-type conduction with parallel contributions of the surface and the bulk. (

**b**) Topological regime with pure surface transport, also indicating the possibility of bulk presence due to thermal activation. (

**c**) p-type conduction with surface and bulk contributing. Blue and yellow regions in the bands indicate electron and hole populations, respectively.

**Figure 3.**Schematic representation of the unit cell of rhombohedral ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}\text{}$(blue: Bi, violet: Se1, orange: Se2) showing the Quintuple Layer (QL) arrangement and the c-axis direction.

**Figure 4.**Sketch representing the movement of electrons through scattering centers: (

**a**) two possible paths (1 and 2) for an electron going from A to B. (

**b**) A loop formed by time reversal partners.

**Figure 5.**Simulations within the HLN model of the magnetoconductance $\u2206{\mathrm{G}}_{\mathrm{xx}}\left(\mathrm{B}\right)$ for different values of ${\mathrm{l}}_{\mathsf{\phi}}$ at fixed $\mathsf{\alpha}$: (

**a**) a single coherent channel $\mathsf{\alpha}=-1/2$. (

**b**) Two independent channels $\mathsf{\alpha}=-1$.

**Figure 6.**(

**a**) Hall resistance against magnetic field for ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ films with different thicknesses (indicated in QLs on the right part of the plot) showing the non-linear dependence. (

**b**) Carrier densities obtained from the SdH oscillations against thickness. (

**c**) Mobilities obtained from the two-band analysis against thickness. In (

**b**) and (

**c**), the blue circles represent the bulk data $\left({\mathrm{n}}_{\mathrm{b}},{\text{}\mathsf{\mu}}_{\mathrm{b}}\right)$ and the red squares represent the surface data $\left({\mathrm{n}}_{\mathrm{s}},{\text{}\mathsf{\mu}}_{\mathrm{s}}\right)$, where empty and filled squares represent the two different surfaces. (

**d**) ${\mathrm{G}}_{\mathrm{s}}/{\mathrm{G}}_{\mathrm{tot}}$ against film thickness. (

**e**) 2D Magnetoconductance for different film thicknesses. (

**f**) Value of $\mathsf{\alpha}$ against film thickness. Inset in (

**e**) shows schematic energy bands above and below the critical thickness. Data were taken at $\mathrm{T}=1.6\text{}\mathrm{K}$. Reprinted with permission from Reference [13]. Copyright 2012 American Physical Society.

**Figure 7.**Transport data of ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ thin films against thickness: (

**a**) conductance, (

**b**) carrier density, (

**c**) mobility. The solid straight lines are guides for the eyes. The subscript ${\mathrm{SC}}_{1}$ corresponds to the topological surface states whereas the subscript ${\mathrm{SC}}_{2}$ corresponds to the 2DEG. (

**d**) Magnetoconductance for different thicknesses. (

**e**) ${\mathrm{l}}_{\mathsf{\phi}}$ against thickness. (

**f**) $\mathsf{\alpha}$ against thickness. Insets in (

**a**), (

**b**), and (

**c**) show the data for thinner films. Reprinted with permission from Reference [64]. Copyright 2012 American Physical Society.

**Figure 8.**Modulation of $\mathsf{\alpha}$: (

**a**) effect of negative gating voltage ${\mathrm{V}}_{\mathrm{G}}\text{}\mathrm{on}\text{}\mathsf{\alpha}$. The insets on the left and right are band diagrams showing the Fermi level modulation relative to the bands. (

**b**) Effect of Cu doping ($\mathrm{x}$ represent the Cu content) on $\mathsf{\alpha}.$ The numbers i and ii correspond to the coupled and decoupled case, respectively. Reprinted with permission from references [32,40]. Copyright 2011 and 2014 American Physical Society.

**Figure 9.**(

**a**) Mobility against thickness following the conventional dependence of bulk transport in films, which is indicated by the solid line. (

**b**) Magnetoresistance for lower thicknesses. (

**c**) Magnetoresistance for high thicknesses showing the ${\mathrm{B}}^{2}\text{}$ dependence characteristic of bulk dominance. Inset displays the low field region. (

**d**) Parameter $\mathsf{\alpha}$ against thicknesses. (

**e**) ${\mathrm{l}}_{\mathsf{\phi}}$ against thickness. Reprinted with permission from Reference [31]. Copyright 2011 American Physical Society.

**Figure 10.**(

**a**) Temperature dependence of resistivity for three ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ films with different thicknesses showing the metallic behaviour ($\mathrm{d}\mathsf{\rho}/\mathrm{dT}>0)$. (

**b**) Hall resistivity at 2 K against magnetic field for different thicknesses. (

**c**) Mobility at 2 K against film thickness. (

**d**) Normalized magnetoresistance at 2 K for different thicknesses. (

**e**) Normalized magnetoresistance for a 15-nm-thick film at different temperatures. (

**f**) Coherent transport parameters ${\mathrm{l}}_{\mathsf{\phi}}$ and $\mathsf{\alpha}$ against thickness at 2 K.

**Figure 11.**Representation of the phase coherence length ${\mathrm{l}}_{\mathsf{\phi}}$ versus mobility $\mathsf{\mu}$ values for different films. The dashed black lines are guides for the eyes. Red empty squares correspond to data obtained in our samples. The rest have been taken from literature: Red circles, [13]; blue square, [30]; light green upwards triangle, [31]; cyan left-pointing triangle, [42]; downwards magenta triangles, [54]; right pointing orange triangle, [56]; dark green hexagon, [64]; violet pentagon, [69]; maroon star, [70].

**Table 1.**Transport parameters of ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ thin films reported in the literature.

Reference | t (nm) | n_{2D}${\left({10}^{13}\mathbf{c}{\mathbf{m}}^{-2}\right)}^{\text{}}$ | n $\left({10}^{19}\mathbf{c}{\mathbf{m}}^{-3}\right)$ | μ $\left(\mathbf{c}{\mathbf{m}}^{2}/\left(\mathbf{V}\mathbf{s}\right)\right)$ | T $\left(\mathbf{K}\right)$ | ${\mathbf{l}}_{\mathsf{\phi}}$ $\left(\mathbf{n}\mathbf{m}\right)$ | $\mathsf{\alpha}$ |
---|---|---|---|---|---|---|---|

[13] | 2–50 | - | 4 | 100–1200 | 1.6 | 100–1000 | −0.5 |

[14] | 20 | - | - | - | 0.3–10 | 80–300 | −1~−0.5 |

[30] | ≈38 | 1 | 390–880 | 1.8 | 306–968 | −0.52 | |

[31] | 1–100 | - | 0.1–6 | 70–1150 | 1.5 | 150–1000 | −0.6~0.5 |

[32] | 5–20 | 0.8–8.6 | - | 20–1000 | 1.2 | 143–∞ | −0.5 |

[42] | 10 | - | 6 | 472 | 0.4–10 | 150–870 | −0.6 |

[44] | 7 | 1.5 | - | - | 2.5 | 55–90 | −0.6~−0.2 |

[51] | 1–6 | 3.5 | - | 31–350 | 1.5 | 75–200 | −0.6~−0.3 |

[54] | 6–22 | - | 3.5–6.5 | 80–530 | 2 | 200–900 | −0.55~−0.35 |

[56] | 30 | - | 1.1 | 954 | 2 | 640 | −0.56 |

[57] | 9–54 | ~100 | - | - | 2–9 | 10–159 | −1.08~0.16 |

[64] | 10–245 | 3 | - | 500 | 1.5 | 750 | −0.6 |

[67] | 30–300 | 0.81–3.25 | - | - | 10 | 318–879 | −0.72~−0.34 |

[68] | 9.8–23 | 0.3 | - | - | 0.3–8 | 300–800 | −0.4 |

[69] | 5 | - | 3 | 27 | 2–20 | 8–20 | −0.5 |

[70] | 12 | - | 4.6 | 37 | 1.6–6 | 65–110 | −0.7~−0.6 |

Our data | 15–60 | 7–50 | 7–50 | 50–150 | 2 | 120–325 | ~−0.5 |

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**MDPI and ACS Style**

Gracia-Abad, R.; Sangiao, S.; Bigi, C.; Kumar Chaluvadi, S.; Orgiani, P.; De Teresa, J.M. Omnipresence of Weak Antilocalization (WAL) in Bi_{2}Se_{3} Thin Films: A Review on Its Origin. *Nanomaterials* **2021**, *11*, 1077.
https://doi.org/10.3390/nano11051077

**AMA Style**

Gracia-Abad R, Sangiao S, Bigi C, Kumar Chaluvadi S, Orgiani P, De Teresa JM. Omnipresence of Weak Antilocalization (WAL) in Bi_{2}Se_{3} Thin Films: A Review on Its Origin. *Nanomaterials*. 2021; 11(5):1077.
https://doi.org/10.3390/nano11051077

**Chicago/Turabian Style**

Gracia-Abad, Rubén, Soraya Sangiao, Chiara Bigi, Sandeep Kumar Chaluvadi, Pasquale Orgiani, and José María De Teresa. 2021. "Omnipresence of Weak Antilocalization (WAL) in Bi_{2}Se_{3} Thin Films: A Review on Its Origin" *Nanomaterials* 11, no. 5: 1077.
https://doi.org/10.3390/nano11051077