- Letter
- Open Access
Multiband superconductors under magnetic fields: Fractional vortices and Ising superconductivity
Phys. Rev. Research 7, L012010 – Published 15 January, 2025
DOI: https://doi.org/10.1103/PhysRevResearch.7.L012010
Abstract
Inspired by the recent experiments in monolayer iron-based superconductors, we theoretically investigate the properties of a two-dimensional multiband superconductor under magnetic fields, focusing on two aspects. First, for vortex bound states under out-of-plane magnetic fields, the spatial anisotropy and positions of electron density peaks are associated with interband couplings. Second, under in-plane magnetic fields, even with inversion symmetry, a Ising-type spin-orbit coupling is allowed, leading to an enhanced in-plane upper critical field. Applications to other two-dimensional multiband superconductors are also discussed.
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References (65)
- H. J. Choi, D. Roundy, H. Sun, M. L. Cohen, and S. G. Louie, The origin of the anomalous superconducting properties of , Nature (London) 418, 758 (2002).
- S. Souma, T. Sato, T. Takahashi, H. Matsui, S.-C. Wang, H. Ding, A. Kaminski, J. C. Campuzano, S. Sasaki, and K. Kadowaki, The origin of multiple superconducting gaps in , Nature (London) 423, 65 (2003).
- G. Blumberg, A. Mialitsin, B. S. Dennis, M. V. Klein, N. D. Zhigadlo, and J. Karpinski, Observation of Leggett's collective mode in a multiband superconductor, Phys. Rev. Lett. 99, 227002 (2007).
- K. H. Jin, H. Huang, J. W. Mei, Z. Liu, L. K. Lim, and F. Liu, Topological superconducting phase in high- superconductor with Dirac–nodal-line fermions, npj Comput. Mater. 5, 57 (2019).
- X. Zhou, K. N. Gordon, K.-H. Jin, H. Li, D. Narayan, H. Zhao, H. Zheng, H. Huang, G. Cao et al., Observation of topological surface states in the high-temperature superconductor , Phys. Rev. B 100, 184511 (2019).
- X. Xi, Z. Wang, W. Zhao, J.-H. Park, K. T. Law, H. Berger, L. Forró, J. Shan, and K. F. Mak, Ising pairing in superconducting atomic layers, Nat. Phys. 12, 139 (2016).
- W. Y. He, B. T. Zhou, J. J. He, N. F. Q. Yuan, T. Zhang, and K. T. Law, Magnetic field driven nodal topological superconductivity in monolayer transition metal dichalcogenides, Commun. Phys. 1, 40 (2018).
- D. Shaffer, J. Kang, F. J. Burnell, and R. M. Fernandes, Crystalline nodal topological superconductivity and Bogolyubov Fermi surfaces in monolayer , Phys. Rev. B 101, 224503 (2020).
- S. Kezilebieke, M. N. Huda, V. Vaňo, M. Aapro, S. C. Ganguli, O. J. Silveira, S. Głodzik, A. S. Foster, T. Ojanen, and P. Liljeroth, Topological superconductivity in a van der Waals heterostructure, Nature (London) 588, 424 (2020).
- A. Hamill et al., Two-fold symmetric superconductivity in few-layer , Nat. Phys. 17, 949 (2021).
- H. Ding, P. Richard, K. Nakayama, K. Sugawara, T. Arakane, Y. Sekiba, A. Takayama, S. Souma, T. Sato, T. Takahashi et al., Observation of Fermi-surface–dependent nodeless superconducting gaps in , Europhys. Lett. 83, 47001 (2008).
- T. Qian, X.-P. Wang, W.-C. Jin, P. Zhang, P. Richard, G. Xu, X. Dai, Z. Fang, J.-G. Guo et al., Absence of a holelike Fermi surface for the iron-based superconductor revealed by angle-resolved photoemission spectroscopy, Phys. Rev. Lett. 106, 187001 (2011).
- P. Zhang, K. Yaji, T. Hashimoto, Y. Ota, T. Kondo, K. Okazaki, Z. Wang, J. Wen, G. D. Gu, H. Ding, and S. Shin, Observation of topological superconductivity on the surface of an iron-based superconductor, Science 360, 182 (2018).
- V. Grinenko, R. Sarkar, K. Kihou, C. H. Lee, I. Morozov, S. Aswartham, B. Büchner, P. Chekhonin, W. Skrotzki, K. Nenkov, R. Hühne, K. Nielsch, S.-L. Drechsler, V. L. Vadimov, M. A. Silaev, P. A. Volkov, I. Eremin, H. Luetkens, and H.-H. Klauss, Superconductivity with broken time-reversal symmetry inside a superconducting -wave state, Nat. Phys. 16, 789 (2020).
- V. Grinenko, D. Weston, F. Caglieris, C. Wuttke, C. Hess, T. Gottschall, I. Maccari, D. Gorbunov, S. Zherlitsyn, J. Wosnitza et al., State with spontaneously broken time-reversal symmetry above the superconducting phase transition, Nat. Phys. 17, 1254 (2021).
- I. Shipulin, N. Stegani, I. Maccari, K. Kihou, C.-H. Lee, Q. Hu, Y. Zheng, F. Yang, Y. Li, C.-M. Yim, R. Hühne et al., Calorimetric evidence for two phase transitions in with fermion pairing and quadrupling states, Nat. Commun. 14, 6734 (2023).
- S. Deng, L. Viola, and G. Ortiz, Majorana modes in time-reversal invariant -wave topological superconductors, Phys. Rev. Lett. 108, 036803 (2012).
- G. Xu, B. Lian, P. Tang, X.-L. Qi, and S.-C. Zhang, Topological superconductivity on the surface of Fe-based superconductors, Phys. Rev. Lett. 117, 047001 (2016).
- X. Wu, S. Qin, Y. Liang, H. Fan, and J. Hu, Topological characters in Fe() thin films, Phys. Rev. B 93, 115129 (2016).
- E. Babaev, Vortices with fractional flux in two-gap superconductors and in extended Faddeev model, Phys. Rev. Lett. 89, 067001 (2002).
- F. Wilczek, Magnetic flux, angular momentum, and statistics, Phys. Rev. Lett. 48, 1144 (1982).
- D. Wang, L. Kong, P. Fan, H. Chen, S. Zhu, W. Liu, L. Cao, Y. Sun, S. Du, J. Schneeloch, R. Zhong, G. Gu, L. Fu, H. Ding, and H.-J. Gao, Evidence for Majorana bound states in an iron-based superconductor, Science 362, 333 (2018).
- Y. Iguchi, R. Shi, K. Kihou, C.-H. Lee, V. Grinenko, E. Babaev, and K. A. Moler, Superconducting vortices carrying a temperature-dependent fraction of the flux quantum, Science 380, 1244 (2023).
- Y. Zheng, Q. Hu, H. Ji, I. Timoshuk, H. Xu, Y. Li, Y. Gao, X. Yu, R. Wu, X. Lu, V. Grinenko, E. Babaev, N. F. Q. Yuan, B. Lv, C.-M. Yim, and H. Ding, Direct observation of quantum vortex fractionalization in multiband superconductors, arXiv:2407.18610.
- M. Sigrist and K. Ueda, Phenomenological theory of unconventional superconductivity, Rev. Mod. Phys. 63, 239 (1991).
- H. Suhl, B. T. Matthias, and L. R. Walker, Bardeen-Cooper-Schrieffer theory of superconductivity in the case of overlapping bands, Phys. Rev. Lett. 3, 552 (1959).
- L. Fu and E. Berg, Odd-parity topological superconductors: Theory and application to , Phys. Rev. Lett. 105, 097001 (2010).
- N. V. Orlova, A. A. Shanenko, M. V. Milošević, F. M. Peeters, A. V. Vagov, and V. M. Axt, Ginzburg-Landau theory for multiband superconductors: Microscopic derivation, Phys. Rev. B 87, 134510 (2013).
- N. F. Q. Yuan, K. F. Mak, and K. T. Law, Possible topological superconducting phases of , Phys. Rev. Lett. 113, 097001 (2014).
- J. Garaud, J. Carlström, and E. Babaev, Topological solitons in three-band superconductors with broken time reversal symmetry, Phys. Rev. Lett. 107, 197001 (2011).
- L. Salasnich, A. A. Shanenko, A. Vagov, J. A. Aguiar, and A. Perali, Screening of pair fluctuations in superconductors with coupled shallow and deep bands: A route to higher-temperature superconductivity, Phys. Rev. B 100, 064510 (2019).
- A. A. Shanenko, T. T. Saraiva, A. Vagov, A. S. Vasenko, and A. Perali, Suppression of fluctuations in a two-band superconductor with a quasi-one-dimensional band, Phys. Rev. B 105, 214527 (2022).
- H. Tajima, A. Perali, and P. Pieri, BCS-BEC crossover and pairing fluctuations in a two band superfluid/superconductor: A matrix approach, Condens. Matter 5, 10 (2020).
- A. Perali, C. Castellani, C. Di Castro, M. Grilli, E. Piegari, and A. A. Varlamov, Two-gap model for underdoped cuprate superconductors, Phys. Rev. B 62, R9295(R) (2000).
- Y. Wang, P. J. Hirschfeld, and I. Vekhter, Theory of quasiparticle vortex bound states in iron-based superconductors: Application to scanning tunneling spectroscopy of LiFeAs, Phys. Rev. B 85, 020506(R) (2012).
- M. Marciani, L. Fanfarillo, C. Castellani, and L. Benfatto, Leggett modes in iron-based superconductors as a probe of time-reversal symmetry breaking, Phys. Rev. B 88, 214508 (2013).
- J. Böker, P. A. Volkov, K. B. Efetov, and I. Eremin, superconductivity with incipient bands: Doping dependence and STM signatures, Phys. Rev. B 96, 014517 (2017).
- A. A. Vargas-Paredes, A. A. Shanenko, A. Vagov, M. V. Milošević, and A. Perali, Crossband versus intraband pairing in superconductors: Signatures and consequences of the interplay, Phys. Rev. B 101, 094516 (2020).
- A. Benfenati, M. Barkman, and E. Babaev, Demonstration of skyrmions in three-band superconductors by self-consistent solutions of a Bogoliubov–de Gennes model, Phys. Rev. B 107, 094503 (2023).
- I. Timoshuk and E. Babaev, Microscopic properties of fractional vortices and domain walls in three-band superconductors, arXiv:2407.20132.
- S. Graser, A. F. Kemper, T. A. Maier, H. P. Cheng, P. J. Hirschfeld, and D. J. Scalapino, Spin fluctuations and superconductivity in a three-dimensional tight-binding model for , Phys. Rev. B 81, 214503 (2010).
- V. B. Zabolotnyy, D. S. Inosov, D. V. Evtushinsky, A. Koitzsch, A. A. Kordyuk, G. L. Sun, J. T. Park, D. Haug, V. Hinkov, A. V. Boris, C. T. Lin, M. Knupfer, A. N. Yaresko, B. Büchner, A. Varykhalov, R. Follath, and S. V. Borisenko, electronic order in iron arsenide superconductors, Nature (London) 457, 569 (2009).
- P. Zhang, T. Qian, P. Richard, X. P. Wang, H. Miao, B. Q. Lv, B. B. Fu, T. Wolf, C. Meingast et al., Observation of two distinct band splittings in FeSe, Phys. Rev. B 91, 214503 (2015).
- R. M. Fernandes and A. V. Chubukov, Low-energy microscopic models for iron-based superconductors: A review, Rep. Prog. Phys. 80, 014503 (2017).
- K. Nakayama, T. Sato, P. Richard, Y. M. Xu, T. Kawahara, K. Umezawa, T. Qian, M. Neupane, G. F. Chen, H. Ding, and T. Takahashi, Universality of superconducting gaps in overdoped observed by angle-resolved photoemission spectroscopy, Phys. Rev. B 83, 020501(R) (2011).
- S. V. Borisenko, D. V. Evtushinsky, Z.-H. Liu, I. Morozov, R. Kappenberger, S. Wurmehl, B. Büchner, A. N. Yaresko, T. K. Kim, M. Hoesch, T. Wolf, and N. D. Zhigadlo, Direct observation of spin–orbit coupling in iron-based superconductors, Nat. Phys. 12, 311 (2016).
- Y. S. Kushnirenko, D. V. Evtushinsky, T. K. Kim, I. Morozov, L. Harnagea, S. Wurmehl, S. Aswartham, B. Büchner, A. V. Chubukov et al., Nematic superconductivity in LiFeAs, Phys. Rev. B 102, 184502 (2020).
- S. Qin, C. Fang, F. C. Zhang, and J. Hu, Topological superconductivity in an extended -wave superconductor and its implication to iron-based superconductors, Phys. Rev. X 12, 011030 (2022).
- T. Sato, K. Nakayama, Y. Sekiba, P. Richard, Y.-M. Xu, S. Souma, T. Takahashi, G. F. Chen, J. L. Luo et al., Band structure and Fermi surface of an extremely overdoped iron-based superconductor , Phys. Rev. Lett. 103, 047002 (2009).
- Y. Zhang, L. X. Yang, M. Xu, Z. R. Ye, F. Chen, C. He, H. C. Xu, J. Jiang, B. P. Xie, J. J. Ying, X. F. Wang, X. H. Chen, J. P. Hu, M. Matsunami, S. Kimura, and D. L. Feng, Nodeless superconducting gap in (, Cs) revealed by angle-resolved photoemission spectroscopy, Nat. Mater. 10, 273 (2011).
- N. Xu, P. Richard, X. Shi, A. van Roekeghem, T. Qian, E. Razzoli, E. Rienks, G. F. Chen, E. Ieki, K. Nakayama, T. Sato, T. Takahashi, M. Shi, and H. Ding, Possible nodal superconducting gap and Lifshitz transition in heavily hole-doped , Phys. Rev. B 88, 220508(R) (2013).
- Y. Ota, K. Okazaki, Y. Kotani, T. Shimojima, W. Malaeb, S. Watanabe, C. T. Chen, K. Kihou, C. H. Lee, A. Iyo, H. Eisaki, T. Saito, H. Fukazawa, Y. Kohori, and S. Shin, Evidence for excluding the possibility of -wave superconducting-gap symmetry in Ba-doped , Phys. Rev. B 89, 081103(R) (2014).
- D. Wu, J. Jia, J. Yang, W. Hong, Y. Shu, T. Miao, H. Yan, H. Rong, P. Ai, X. Zhang, C. Yin, J. Liu, H. Chen, Y. Yang, C. Peng, C. Li, S. Zhang et al., Nodal pairing symmetry in an iron-based superconductor with only hole pockets, Nat. Phys. 20, 571 (2024).
- B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 314, 1757 (2006).
- E. I. Rashba and V. I. Sheka, Symmetry of energy bands in crystals of wurtzite type: II. Symmetry of bands including spin-orbit interaction, Fiz. Tverd. Tela 1, 162 (1959).
- Y. A. Bychkov and E. I. Rashba, Oscillatory effects and the magnetic susceptibility of carriers in inversion layers, J. Phys. C: Solid State Phys. 17, 6039 (1984).
- J. M. Lu, O. Zeliuk, I. Leermakers, N. F. Q. Yuan, U. Zeitler, K. T. Law, and J. T. Ye, Evidence for two-dimensional Ising superconductivity in gated , Science 350, 1353 (2015).
- C. Wang, B. Lian, X. Guo, J. Mao, Z. Zhang, D. Zhang, B.-L. Gu, Y. Xu, and W. Duan, Type-II Ising superconductivity in two-dimensional materials with spin-orbit coupling, Phys. Rev. Lett. 123, 126402 (2019).
- J. Falson et al., Type-II Ising pairing in few-layer stanene, Science 367, 1454 (2020).
- H. Liu, H. Liu, D. Zhang, and X. C. Xie, Microscopic theory of in-plane critical field in two-dimensional Ising superconducting systems, Phys. Rev. B 102, 174510 (2020).
- P. Fulde and R. A. Ferrell, Superconductivity in a strong spin-exchange field, Phys. Rev. 135, A550 (1964).
- A. I. Larkin and Yu. N. Ovchinnikov, Nonuniform state of superconductors, Zh. Eksp. Teor. Fiz. 47, 1136 (1964), Sov. Phys. JETP 20, 762 (1965).
- N. F. Q. Yuan and L. Fu, Supercurrent diode effect and finite-momentum superconductors, Proc. Natl. Acad. Sci. USA 119, e2119548119 (2022).
- A. Daido, Y. Ikeda, and Y. Yanase, Intrinsic superconducting diode effect, Phys. Rev. Lett. 128, 037001 (2022).
- J. J. He, Y. Tanaka, and N. Nagaosa, A phenomenological theory of superconductor diodes, New J. Phys. 24, 053014 (2022).