1. 1State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China 2. 2Key Laboratory of Micro-Nano Measurement-Manipulation and Physics (Ministry of Education), School of Physics, Beihang University, Beijing 100191, China 3. 3School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China 4. 4Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
Ferromagnetism and superconductivity are generally considered to be antagonistic phenomena in condensed matter physics. Here, we theoretically study the interplay between the ferromagnetic and superconducting orders in a recent discovered monolayered CoSb superconductor with an orthorhombic symmetry and net magnetization, and demonstrate the pairing symmetry of CoSb as a candidate of non-unitary superconductor with time-reversal symmetry breaking. By performing the group theory analysis and the first-principles calculations, the superconducting order parameter is suggested to be a triplet pairing with the irreducible representation of 3B2u, which displays intriguing nodal points and non-zero periodic modulation of Cooper pair spin polarization on the Fermi surface topologies. These findings not only provide a significant theoretical insight into the coexistence of superconductivity and ferromagnetism, but also reveal the exotic spin polarized Cooper pairing driven by ferromagnetic spin fluctuations in a triplet superconductor.
L. Fu and C. L. Kane, Superconducting proximity effect and Majorana fermions at the surface of a topological insulator, Phys. Rev. Lett. 100(9), 096407 (2008) https://doi.org/10.1103/PhysRevLett.100.096407
J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Generic new platform for topological quantum computation using semiconductor heterostructures, Phys. Rev. Lett. 104(4), 040502 (2010) https://doi.org/10.1103/PhysRevLett.104.040502
8
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(4), 047001 (2016) https://doi.org/10.1103/PhysRevLett.117.047001
9
N. Read and D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect, Phys. Rev. B 61(15), 10267 (2000) https://doi.org/10.1103/PhysRevB.61.10267
N. Read and G. Moore, Fractional quantum Hall effect and nonabelian statistics, Prog. Theor. Phys. Suppl. 107, 157 (1992) https://doi.org/10.1143/PTPS.107.157
C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80(3), 1083 (2008) https://doi.org/10.1103/RevModPhys.80.1083
S. R. Elliott and M. Franz, Majorana fermions in nuclear, particle, and solid-state physics, Rev. Mod. Phys. 87(1), 137 (2015) https://doi.org/10.1103/RevModPhys.87.137
17
R. Aguado, Majorana quasiparticles in condensed matter, Riv. Nuovo Cim. 40, 523 (2017)
18
V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Signatures of Majorana fermions in hybrid superconductor–semiconductor nanowire devices, Science 336(6084), 1003 (2012) https://doi.org/10.1126/science.1222360
19
M. Deng, C. Yu, G. Huang, M. Larsson, P. Caroff, and H. Xu, Anomalous zero-bias conductance peak in a Nb– InSb nanowire-Nb hybrid device, Nano Lett. 12(12), 6414 (2012) https://doi.org/10.1021/nl303758w
20
A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrikman, Zero-bias peaks and splitting in an Al–InAs nanowire topological superconductor as a signature of Majorana fermions, Nat. Phys. 8(12), 887 (2012) https://doi.org/10.1038/nphys2479
21
M. T. Deng, S. Vaitiekènas, E. B. Hansen, J. Danon, M. Leijnse, K. Flensberg, J. Nygård, P. Krogstrup, and C. M. Marcus, Majorana bound state in a coupled quantum-dot hybrid-nanowire system, Science 354(6319), 1557 (2016) https://doi.org/10.1126/science.aaf3961
22
H. Zhang, C. X. Liu, S. Gazibegovic, D. Xu, J. A. Logan, G. Wang, N. van Loo, J. D. S. Bommer, M. W. A. de Moor, D. Car, R. L. M. Op het Veld, P. J. van Veldhoven, S. Koelling, M. A. Verheijen, M. Pendharkar, D. J. Pennachio, B. Shojaei, J. Sue Lee, C. J. Palmstrøm, E. P. A. M. Bakkers, S. Das Sarma, and L. P. Kouwenhoven, Quantized Majorana conductance, Nature 556(7699), 74 (2018)
23
S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yazdani, Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor, Science 346(6209), 602 (2014) https://doi.org/10.1126/science.1259327
24
J. P. Xu, M. X. Wang, Z. L. Liu, J. F. Ge, X. Yang, C. Liu, Z. A. Xu, D. Guan, C. L. Gao, D. Qian, Y. Liu, Q. H. Wang, F. C. Zhang, Q. K. Xue, and J. F. Jia, Experimental detection of a Majorana mode in the core of a magnetic vortex inside a topological insulator-superconductor Bi2Te3/NbSe2 heterostructure, Phys. Rev. Lett. 114(1), 017001 (2015) https://doi.org/10.1103/PhysRevLett.114.017001
25
H. H. Sun, K. W. Zhang, L. H. Hu, C. Li, G. Y. Wang, H. Y. Ma, Z. A. Xu, C. L. Gao, D. D. Guan, Y. Y. Li, C. Liu, D. Qian, Y. Zhou, L. Fu, S. C. Li, F. C. Zhang, and J. F. Jia, Majorana zero mode detected with spin selective Andreev reflection in the vortex of a topological superconductor, Phys. Rev. Lett. 116(25), 257003 (2016) https://doi.org/10.1103/PhysRevLett.116.257003
26
A. Banerjee, C. A. Bridges, J. Q. Yan, A. A. Aczel, L. Li, M. B. Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu, J. Knolle, S. Bhattacharjee, D. L. Kovrizhin, R. Moessner, D. A. Tennant, D. G. Mandrus, and S. E. Nagler, Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet, Nat. Mater. 15(7), 733 (2016) https://doi.org/10.1038/nmat4604
27
J. X. Yin, Z. Wu, J. H. Wang, Z. Y. Ye, J. Gong, X. Y. Hou, L. Shan, A. Li, X. J. Liang, X. X. Wu, J. Li, C. S. Ting, Z. Q. Wang, J. P. Hu, P. H. Hor, H. Ding, and S. H. Pan, Observation of a robust zero-energy bound state in iron-based superconductor Fe(Te,Se), Nat. Phys. 11(7), 543 (2015) https://doi.org/10.1038/nphys3371
28
Q. Liu, C. Chen, T. Zhang, R. Peng, Y. J. Yan, C. H. P. Wen, X. Lou, Y. L. Huang, J. P. Tian, X. L. Dong, G. W. Wang, W. C. Bao, Q. H. Wang, Z. P. Yin, Z. X. Zhao, and D. L. Feng, Robust and clean Majorana zero mode in the vortex core of high-temperature superconductor (Li0.84Fe0.16)OHFeSe, Phys. Rev. X 8(4), 041056 (2018) https://doi.org/10.1103/PhysRevX.8.041056
29
M. Chen, X. Chen, H. Yang, Z. Du, and H. H. Wen, Superconductivity with twofold symmetry in Bi2Te3/FeTe0.55Se0.45 heterostructures, Sci. Adv. 4(6), eaat1084 (2018) https://doi.org/10.1126/sciadv.aat1084
30
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(6412), 333 (2018) https://doi.org/10.1126/science.aao1797
31
S. Zhu, L. Kong, L. Cao, H. Chen, S. Du, Y. Xing, W. Liu, D. Wang, C. Shen, F. Yang, J. Schneeloch, R. Zhong, G. Gu, L. Fu, Y. Y. Zhang, H. Ding, and H. J. Gao, Nearly quantized conductance plateau of vortex zero mode in an iron-based superconductor, Science 367, eaax0274 (2019) https://doi.org/10.1126/science.aax0274
32
C. Chen, K. Jiang, Y. Zhang, C. Liu, Y. Liu, Z. Wang, and J. Wang, Atomic line defects and zero-energy end states in monolayer Fe(Te,Se) high-temperature superconductors, arXiv: 2003.04539 (2020) https://doi.org/10.1038/s41567-020-0813-0
33
C. Liu, C. Chen, X. Liu, Z. Wang, Y. Liu, S. Ye, Z. Q. Wang, J. P. Hu, and J. Wang, Zero-energy bound states in the high-temperature superconductors at the twodimensional limit, Sci. Adv. 6(13), eaax7547 (2020) https://doi.org/10.1126/sciadv.aax7547
V. Kozii, J. W. F. Venderbos, and L. Fu, Threedimensional Majorana fermions in chiral superconductors, Sci. Adv. 2(12), e1601835 (2016) https://doi.org/10.1126/sciadv.1601835
H. Tou, Y. Kitaoka, K. Ishida, K. Asayama, N. Kimura, Y. Ōnuki, E. Yamamoto, Y. Haga, and K. Maezawa, Nonunitary spin-triplet superconductivity in UPt3: Evidence from 195Pt Knight shift study, Phys. Rev. Lett. 80(14), 3129 (1998) https://doi.org/10.1103/PhysRevLett.80.3129
A. D. Hillier, J. Quintanilla, and R. Cywinski, Evidence for time-reversal symmetry breaking in the noncentrosymmetric superconductor LaNiC2, Phys. Rev. Lett. 102(11), 117007 (2009) https://doi.org/10.1103/PhysRevLett.102.117007
44
J. Quintanilla, A. D. Hillier, J. F. Annett, and R. Cywinski, Relativistic analysis of the pairing symmetry of the noncentrosymmetric superconductor LaNiC2, Phys. Rev. B 82(17), 174511 (2010) https://doi.org/10.1103/PhysRevB.82.174511
45
A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett, and R. Cywinski, Nonunitary triplet pairing in the centrosymmetric superconductor LaNiGa2, Phys. Rev. Lett. 109(9), 097001 (2012) https://doi.org/10.1103/PhysRevLett.109.097001
46
S. K. Ghosh, G. Csire, P. Whittlesea, J. F. Annett, M. Gradhand, B. Újfalussy, and J. Quintanilla, Quantitative theory of triplet pairing in the unconventional superconductor LaNiGa2, Phys. Rev. B 101, 100506(R) (2020) https://doi.org/10.1103/PhysRevB.101.100506
47
C. Ding, G. Gong, Y. Liu, F. Zheng, Z. Zhang, H. Yang, Z. Li, Y. Xing, J. Ge, K. He, W. Li, P. Zhang, J. Wang, L. Wang, and Q. K. Xue, Signature of superconductivity in orthorhombic CoSb monolayer films on SrTiO3(001), ACS Nano 13(9), 10434 (2019) https://doi.org/10.1021/acsnano.9b04223
K. V. Samokhin and M. B. Walker, Order parameter symmetry in ferromagnetic superconductors, Phys. Rev. B 66(17), 174501 (2002) https://doi.org/10.1103/PhysRevB.66.174501
51
D. J. Singh and L. Nordstrom, Planewaves, Pseudopotentials, and the LAPW Method, 2nd Ed., Springer-Verlag, Berlin, 2006
52
P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz, in: WIEN2K, An Augmented PlaneWave+ Local Orbitals Program for Calculating Crystal Properties, edited by K. Schwarz, Technical University Wien, Austria, 2001
W. Li, X. Y. Wei, J. X. Zhu, C. S. Ting, and Y. Chen, Pressure-induced topological quantum phase transition in Sb2Se3, Phys. Rev. B 89(3), 035101 (2014) https://doi.org/10.1103/PhysRevB.89.035101
W. Ding, J. Zeng, W. Qin, P. Cui, and Z. Zhang, Exploringhigh transition temperature superconductivity in a freestanding or SrTiO3-supported CoSb monolayer, Phys. Rev. Lett. 124(2), 027002 (2020) https://doi.org/10.1103/PhysRevLett.124.027002
57
M. Y. Zou, J. N. Chu, H. Zhang, T. Z. Yuan, P. Cheng, W. T. Jin, D. Jiang, X. G. Xu, W. J. Yu, Z. H. An, X. Y. Wei, G. Mu, and W. Li, Evidence the ferromagnetic order on CoSb layer of LaCoSb2, Phys. Rev. B 101(15), 155138 (2020) https://doi.org/10.1103/PhysRevB.101.155138
58
W. Li, J. X. Zhu, Y. Chen, and C. S. Ting, First-principles calculations of the electronic structure of iron-pnictide EuFe2(As,P)2 superconductors: Evidence for antiferromagnetic spin order, Phys. Rev. B 86(15), 155119 (2012) https://doi.org/10.1103/PhysRevB.86.155119
59
X. G. Xu and W. Li, Electronic and magnetic structures of ternary iron telluride KFe2Te2, Front. Phys. 10(4), 107403 (2015) https://doi.org/10.1007/s11467-015-0495-8
60
D. J. Singh, Electronic structure and doping in BaFe2As2 and LiFeAs: Density functional calculations, Phys. Rev. B 78(9), 094511 (2008) https://doi.org/10.1103/PhysRevB.78.094511
61
W. C. Huang, W. Li, and X. Liu, Exotic ferromagnetism in the two-dimensional quantum materials C3N, Front. Phys. 13(2), 137104 (2018) https://doi.org/10.1007/s11467-017-0741-3
62
Z. Zhou, W. T. Jin, W. Li, S. Nandi, B. Ouladdiaf, Z. Yan, X. Wei, X. Xu, W. H. Jiao, N. Qureshi, Y. Xiao, Y. Su, G. H. Cao, and Th. Brückel, Universal critical behavior in the ferromagnetic superconductor Eu(Fe0.75Ru0.25)2As2, Phys. Rev. B 100, 060406(R) (2019) https://doi.org/10.1103/PhysRevB.100.060406
63
J. B. Goodenough, Theory of the role of covalence in the perovskite-type manganites [La,M(II)]MnO3, Phys. Rev. 100(2), 564 (1955) https://doi.org/10.1103/PhysRev.100.564
64
S. Maekawa, T. Tohyama, S. E. Barnes, S. Ishihara, W. Koshibae, and G. Khaliullin, Physics of Transition Metal Oxides, Berlin Heidelberg: Springer-Verlag, 2004 https://doi.org/10.1007/978-3-662-09298-9
65
W. Li, Z. Liu, Y. S. Wu, and Y. Chen, Exotic fractional topological states in a two-dimensional organometallic material, Phys. Rev. B 89(12), 125411 (2014) https://doi.org/10.1103/PhysRevB.89.125411
66
W. Li, J. Li, J. X. Zhu, Y. Chen, and C. S. Ting, Pairing symmetry in the iron-pnictide superconductor KFe2As2, EPL 99(5), 57006 (2012) https://doi.org/10.1209/0295-5075/99/57006
Y. Tada, S. Takayoshi, and S. Fujimoto, Magnetism and superconductivity in ferromagnetic heavy-fermion system UCoGe under in-plane magnetic fields, Phys. Rev. B 93(17), 174512 (2016) https://doi.org/10.1103/PhysRevB.93.174512
