- Letter
- Open Access
Moiré dispersion of edge states in spin chains on superconductors
Phys. Rev. Research 4, L032010 – Published 22 July, 2022
DOI: https://doi.org/10.1103/PhysRevResearch.4.L032010
Abstract
Our calculations of ferromagnetic spin chains on -wave superconductors show that the energy oscillations of edge states with the chain's length are due to a moiré pattern emerging from the electronic interactions in the superconductor and the discreteness of the spin-chain lattice. By modifying the spin lattice, the moiré dispersion of edge states can be controlled. In particular, we can engineer nondispersive edge states that remain at fixed energy regardless of the size distribution of the spin chains. This is an important step in the study of edge states of spin chains that can be fabricated with a certain size dispersion.
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References (46)
- C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008).
- B. Field and T. Simula, Introduction to topological quantum computation with non-Abelian anyons, Quantum Sci. Technol. 3, 045004 (2018).
- A. Yu Kitaev, Unpaired Majorana fermions in quantum wires, Phys. Usp. 44, 131 (2001).
- M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
- X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
- M. Sato and Y. Ando, Topological superconductors: A review, Rep. Prog. Phys. 80, 076501 (2017).
- R. Aguado, Majorana quasiparticles in condensed matter, Riv. del Nuovo Cim. 40, 523 (2017).
- D.-J. Choi, N. Lorente, J. Wiebe, K. von Bergmann, A. F. Otte, and A. J. Heinrich, Colloquium: Atomic spin chains on surfaces, Rev. Mod. Phys. 91, 041001 (2019).
- 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, 602 (2014).
- S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and A. Yazdani, Majorana fermions in chains of magnetic atoms on a superconductor, Phys. Rev. B 88, 020407(R) (2013).
- M. Ruby, F. Pientka, Y. Peng, F. von Oppen, B. W. Heinrich, and K. J. Franke, Tunneling Processes into Localized Subgap States in Superconductors, Phys. Rev. Lett. 115, 087001 (2015).
- R. Pawlak, M. Kisiel, J. Klinovaja, T. Meier, S. Kawai, T. Glatzel, D. Loss, and E. Meyer, Probing atomic structure and Majorana wavefunctions in mono-atomic Fe chains on superconducting Pb surface, npj Quantum Inf. 2, 16035 (2016).
- S. Jeon, Y. Xie, J. Li, Z. Wang, B. A. Bernevig, and A. Yazdani, Distinguishing a Majorana zero mode using spin-resolved measurements, Science 358, 772 (2017).
- H. Kim, A. Palacio-Morales, T. Posske, L. Rózsa, K. Palotás, L. Szunyogh, M. Thorwart, and R. Wiesendanger, Toward tailoring Majorana bound states in artificially constructed magnetic atom chains on elemental superconductors, Sci. Adv. 4, eaar5251 (2018).
- E. Liebhaber, L. M. Rütten, G. Reecht, J. F. Steiner, S. Rohlf, K. Rossnagel, F. von Oppen, and K. J. Franke, Quantum spins and hybridization in artificially-constructed chains of magnetic adatoms on a superconductor, Nat. Commun. 13, 2160 (2022).
- L. Schneider, P. Beck, T. Posske, D. Crawford, E. Mascot, S. Rachel, R. Wiesendanger, and J. Wiebe, Topological Shiba bands in artificial spin chains on superconductors, Nat. Phys. 17, 943 (2021).
- L. Schneider, P. Beck, J. Neuhaus-Steinmetz, L. Rzsa, T. Posske, J. Wiebe, and R. Wiesendanger, Precursors of Majorana modes and their length-dependent energy oscillations probed at both ends of atomic Shiba chains, Nat. Nanotechnol. 17, 384 (2022).
- 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, 887 (2012).
- L. P. Rokhinson, X. Liu, and J. K. Furdyna, The fractional a.c. Josephson effect in a semiconductor–superconductor nanowire as a signature of Majorana particles, Nat. Phys. 8, 795 (2012).
- 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, 1003 (2012).
- H. O. H. Churchill, V. Fatemi, K. Grove-Rasmussen, M. T. Deng, P. Caroff, H. Q. Xu, and C. M. Marcus, Superconductor-nanowire devices from tunneling to the multichannel regime: Zero-bias oscillations and magnetoconductance crossover, Phys. Rev. B 87, 241401(R) (2013).
- M. Cheng, R. M. Lutchyn, V. Galitski, and S. Das Sarma, Splitting of Majorana-Fermion Modes due to Intervortex Tunneling in a Superconductor, Phys. Rev. Lett. 103, 107001 (2009).
- S. Das Sarma, J. D. Sau, and T. D. Stanescu, Splitting of the zero-bias conductance peak as smoking gun evidence for the existence of the Majorana mode in a superconductor-semiconductor nanowire, Phys. Rev. B 86, 220506(R) (2012).
- J. Klinovaja and D. Loss, Composite Majorana fermion wave functions in nanowires, Phys. Rev. B 86, 085408 (2012).
- A. Theiler, K. Björnson, and A. M. Black-Schaffer, Majorana bound state localization and energy oscillations for magnetic impurity chains on conventional superconductors, Phys. Rev. B 100, 214504 (2019).
- C. Mier, D.-J. Choi, and N. Lorente, Calculations of in-gap states of ferromagnetic spin chains on -wave wide-band superconductors, Phys. Rev. B 104, 245415 (2021).
- E. Prada, P. San-Jose, and R. Aguado, Transport spectroscopy of NS nanowire junctions with Majorana fermions, Phys. Rev. B 86, 180503(R) (2012).
- J. Cayao, P. San-Jose, A. M. Black-Schaffer, R. Aguado, and E. Prada, Majorana splitting from critical currents in Josephson junctions, Phys. Rev. B 96, 205425 (2017).
- N. Leumer, M. Marganska, B. Muralidharan, and M. Grifoni, Exact eigenvectors and eigenvalues of the finite Kitaev chain and its topological properties, J. Phys.: Condens. Matter 32, 445502 (2020).
- R. Bistritzer and A. H. MacDonald, Moiré bands in twisted double-layer graphene, Proc. Natl. Acad. Sci. USA 108, 12233 (2011).
- S. Lisi, X. Lu, T. Benschop, T. A. de Jong, P. Stepanov, J. R. Duran, F. Margot, I. Cucchi, E. Cappelli, A. Hunter, A. Tamai, V. Kandyba, A. Giampietri, A. Barinov, J. Jobst, V. Stalman, M. Leeuwenhoek, K. Watanabe, T. Taniguchi, L. Rademaker et al., Observation of flat bands in twisted bilayer graphene, Nat. Phys. 17, 189 (2021).
- J.-X. Zhu, Bogoliubov–de Gennes Method and its Applications (Springer, Heidelberg, 2016).
- C. Mier, J. Hwang, J. Kim, Y. Bae, F. Nabeshima, Y. Imai, A. Maeda, N. Lorente, A. Heinrich, and D.-J. Choi, Atomic manipulation of in-gap states in the superconductor, Phys. Rev. B 104, 045406 (2021).
- L. Yu, Bound state in superconductors with paramagnetic impurities, Acta Phys. Sin. 21, 75 (1965).
- H. Shiba, Classical spins in superconductors, Prog. Theor. Phys. 40, 435 (1968).
- A. I. Rusinov, Superconductivity near a paramagnetic impurity, JETP Lett. 9, 85 (1969).
- F. Pientka, L. I. Glazman, and F. von Oppen, Topological superconducting phase in helical Shiba chains, Phys. Rev. B 88, 155420 (2013).
- K. Pöyhönen, A. Westström, J. Röntynen, and T. Ojanen, Majorana states in helical Shiba chains and ladders, Phys. Rev. B 89, 115109 (2014).
- G. M. Andolina and P. Simon, Topological properties of chains of magnetic impurities on a superconducting substrate: Interplay between the Shiba band and ferromagnetic wire limits, Phys. Rev. B 96, 235411 (2017).
- S. Tewari and J. D. Sau, Topological Invariants for Spin-Orbit Coupled Superconductor Nanowires, Phys. Rev. Lett. 109, 150408 (2012).
- A. Heimes, D. Mendler, and P. Kotetes, Interplay of topological phases in magnetic adatom-chains on top of a Rashba superconducting surface, New J. Phys. 17, 023051 (2015).
- J. Li, S. Jeon, Yonglong Xie, A. Yazdani, and B. A. Bernevig, Majorana spin in magnetic atomic chain systems, Phys. Rev. B 97125119 (2018).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.4.L032010 for extra information on the decay length of MBS, profiles of MBS and low-energy modes, and the moiré behavior of wave functions, which includes Ref. [46].
- C. Bena, Friedel oscillations: Decoding the hidden physics, C. R. Phys. 17, 302 (2016).
- G. C. Ménard, S. Guissart, C. Brun, S. Pons, V. S. Stolyarov, F. Debontridder, M. V. Leclerc, E. Janod, L. Cario, D. Roditchev, P. Simon, and T. Cren, Coherent long-range magnetic bound states in a superconductor, Nat. Phys. 11, 1013 (2015).
- Y. Peng, F. Pientka, L. I. Glazman, and F. von Oppen, Strong Localization of Majorana End States in Chains of Magnetic Adatoms, Phys. Rev. Lett. 114, 106801 (2015).