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Optimizing proximitized magnetic topological insulator nanoribbons for Majorana bound states

Eduárd Zsurka1,2,3,*, Daniele Di Miceli3,4, Julian Legendre3, Llorenç Serra4,5, Detlev Grützmacher1,2, Thomas L. Schmidt3, and Kristof Moors1,2,†,‡

  • *Contact author: eduard.zsurka@uni.lu
  • †Contact author: kristof.moors@imec.be
  • ‡Present Address: Imec, Kapeldreef 75, 3001 Leuven, Belgium.
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Phys. Rev. B 112, 094504 – Published 4 September, 2025

DOI: https://doi.org/10.1103/x737-p9mq

Abstract

Heterostructures comprised of a magnetic topological insulator (MTI) placed in the proximity of an s-wave superconductor have emerged as a platform for the practical realization of Majorana bound states (MBSs). More specifically, it has been theoretically predicted that MBS can appear in proximitized MTI nanoribbons (PNRs) in the quantum anomalous Hall regime. As with all MBS platforms, disorder and device imperfections can be detrimental to the formation of robust and well-separated MBSs that are suitable for fusion and braiding experiments. Here, we identify the optimal conditions for obtaining a topological superconducting gap that is robust against disorder, with spatially separated stable MBSs in PNRs, and introduce a figure of merit that encompasses these conditions. Particular attention is given to the thin-film limit of magnetic topological insulators (MTIs), where the hybridization of the surface states cannot be neglected, and to the role of electron-hole asymmetry in the low-energy physics of the system. Based on our numerical results, we find that (1) MTI thin films that are normal (rather than quantum spin Hall) insulators for zero magnetization are favorable, (2) strong electron-hole asymmetry causes the stability and robustness of MBS to be very different for chemical potentials above or below the Dirac point, and (3) the magnetization strength should preferably be comparable to the hybridization or confinement energy of the surface states, whichever is largest.

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References (72)

  1. R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, and Z. Fang, Quantized anomalous Hall effect in magnetic topological insulators, Science 329, 61 (2010).
  2. C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang et al., Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator, Science 340, 167 (2013).
  3. J. Wang, B. Lian, H. Zhang, Y. Xu, and S.-C. Zhang, Quantum anomalous Hall effect with higher plateaus, Phys. Rev. Lett. 111, 136801 (2013).
  4. X. Kou, M. Lang, Y. Fan, Y. Jiang, T. Nie, J. Zhang, W. Jiang, Y. Wang, Y. Yao, L. He, and K. L. Wang, Interplay between different magnetisms in Cr-doped topological insulators, ACS Nano 7, 9205 (2013).
  5. J. Wang, B. Lian, and S.-C. Zhang, Quantum anomalous Hall effect in magnetic topological insulators, Phys. Scr. T164, 014003 (2015).
  6. K. Yasuda, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, M. Kawasaki, F. Kagawa, and Y. Tokura, Quantized chiral edge conduction on domain walls of a magnetic topological insulator, Science 358, 1311 (2017).
  7. Y. Tokura, K. Yasuda, and A. Tsukazaki, Magnetic topological insulators, Nat. Rev. Phys. 1, 126 (2019).
  8. H. Sun, B. Xia, Z. Chen, Y. Zhang, P. Liu, Q. Yao, H. Tang, Y. Zhao, H. Xu, and Q. Liu, Rational design principles of the quantum anomalous Hall effect in superlatticelike magnetic topological insulators, Phys. Rev. Lett. 123, 096401 (2019).
  9. Y. Deng, Y. Yu, M. Z. Shi, Z. Guo, Z. Xu, J. Wang, X. H. Chen, and Y. Zhang, Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi2Te4, Science 367, 895 (2020).
  10. K. Yasuda, T. Morimoto, R. Yoshimi, M. Mogi, A. Tsukazaki, M. Kawamura, K. Takahashi, M. Kawasaki, N. Nagaosa, and Y. Tokura, Large non-reciprocal charge transport mediated by quantum anomalous Hall edge states, Nat. Nanotechnol. 15, 831 (2020).
  11. P. Wang, J. Ge, J. Li, Y. Liu, Y. Xu, and J. Wang, Intrinsic magnetic topological insulators, Innovation 2, 100098 (2021).
  12. G. Qiu, P. Zhang, P. Deng, S. K. Chong, L. Tai, C. Eckberg, and K. L. Wang, Mesoscopic transport of quantum anomalous Hall effect in the submicron size regime, Phys. Rev. Lett. 128, 217704 (2022).
  13. J. Wang, Q. Zhou, B. Lian, and S.-C. Zhang, Chiral topological superconductor and half-integer conductance plateau from quantum anomalous Hall plateau transition, Phys. Rev. B 92, 064520 (2015).
  14. Y. Zeng, C. Lei, G. Chaudhary, and A. H. MacDonald, Quantum anomalous Hall Majorana platform, Phys. Rev. B 97, 081102(R) (2018).
  15. C.-Z. Chen, Y.-M. Xie, J. Liu, P. A. Lee, and K. T. Law, Quasi-one-dimensional quantum anomalous Hall systems as new platforms for scalable topological quantum computation, Phys. Rev. B 97, 104504 (2018).
  16. O. Atanov, W. T. Tai, Y.-M. Xie, Y. H. Ng, M. A. Hammond, T. S. Manfred Ho, T. H. Koo, H. Li, S. L. Ho, J. Lyu, S. Chong, P. Zhang, L. Tai, J. Wang, K. T. Law, K. L. Wang, and R. Lortz, Proximity-induced quasi-one-dimensional superconducting quantum anomalous Hall state, Cell Rep. Phys. Sci. 5, 101762 (2024).
  17. P. Schüffelgen, D. Rosenbach, C. Li, T. W. Schmitt, M. Schleenvoigt, A. R. Jalil, S. Schmitt, J. Kölzer, M. Wang, B. Bennemann, U. Parlak, L. Kibkalo, S. Trellenkamp, T. Grap, D. Meertens, M. Luysberg, G. Mussler, E. Berenschot, N. Tas, A. A. Golubov et al., Selective area growth and stencil lithography for in situ fabricated quantum devices, Nat. Nanotechnol. 14, 825 (2019).
  18. A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78, 195125 (2008).
  19. S. Tewari and J. D. Sau, Topological invariants for spin-orbit coupled superconductor nanowires, Phys. Rev. Lett. 109, 150408 (2012).
  20. A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys.-Usp. 44, 131 (2001).
  21. M. Freedman, A. Kitaev, M. Larsen, and Z. Wang, Topological quantum computation, Bull. Amer. Math. Soc. 40, 31 (2003).
  22. J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, Non-Abelian statistics and topological quantum information processing in 1D wire networks, Nat. Phys. 7, 412 (2011).
  23. P. Mandal, N. Taufertshöfer, L. Lunczer, M. P. Stehno, C. Gould, and L. W. Molenkamp, Finite field transport response of a dilute magnetic topological insulator-based Josephson junction, Nano Lett. 22, 3557 (2022).
  24. A. Uday, G. Lippertz, K. Moors, H. F. Legg, R. Joris, A. Bliesener, L. M. C. Pereira, A. A. Taskin, and Y. Ando, Induced superconducting correlations in a quantum anomalous Hall insulator, Nat. Phys. 20, 1589 (2024).
  25. H. Yi, Y.-F. Zhao, Y.-T. Chan, J. Cai, R. Mei, X. Wu, Z.-J. Yan, L.-J. Zhou, R. Zhang, Z. Wang, S. Paolini, R. Xiao, K. Wang, A. R. Richardella, J. Singleton, L. E. Winter, T. Prokscha, Z. Salman, A. Suter, P. P. Balakrishnan et al., Interface-induced superconductivity in magnetic topological insulators, Science 383, 634 (2024).
  26. P. Mandal, S. Mondal, M. Stehno, S. Ilić, F. Bergeret, T. Klapwijk, C. Gould, and L. Molenkamp, Magnetically tunable supercurrent in dilute magnetic topological insulator-based Josephson junctions, Nat. Phys. 20, 984 (2024).
  27. Y. Sato, S. Nagahama, I. Belopolski, R. Yoshimi, M. Kawamura, A. Tsukazaki, N. Kanazawa, K. S. Takahashi, M. Kawasaki, and Y. Tokura, Molecular beam epitaxy of superconducting FeSexTe1−x thin films interfaced with magnetic topological insulators, Phys. Rev. Mater. 8, L041801 (2024).
  28. W. Yuan, Z.-J. Yan, H. Yi, Z. Wang, S. Paolini, Y.-F. Zhao, L. Zhou, A. G. Wang, K. Wang, T. Prokscha, Z. Salman, A. Suter, P. P. Balakrishnan, A. J. Grutter, L. E. Winter, J. Singleton, M. H. W. Chan, and C.-Z. Chang, Coexistence of superconductivity and antiferromagnetism in topological magnet MnBi2Te4 films, Nano Lett. 24, 7962 (2024).
  29. Y. Huang, Y. Fu, P. Zhang, K. L. Wang, and Q. L. He, Inducing superconductivity in quantum anomalous Hall regime, J. Phys.: Condens. Matter 36, 37LT01 (2024).
  30. Y.-H. Wan and Q.-F. Sun, Quarter-quantized thermal Hall effect with parity anomaly, Phys. Rev. B 109, 195408 (2024).
  31. L.-J. Zhou, R. Mei, Y.-F. Zhao, R. Zhang, D. Zhuo, Z.-J. Yan, W. Yuan, M. Kayyalha, M. H. W. Chan, C.-X. Liu, and C.-Z. Chang, Confinement-induced chiral edge channel interaction in quantum anomalous Hall insulators, Phys. Rev. Lett. 130, 086201 (2023).
  32. E. D. L. Rienks, S. Wimmer, J. Sánchez-Barriga, O. Caha, P. S. Mandal, J. Růžička, A. Ney, H. Steiner, V. V. Volobuev, H. Groiss, M. Albu, G. Kothleitner, J. Michalička, S. A. Khan, J. Minár, H. Ebert, G. Bauer, F. Freyse, A. Varykhalov, O. Rader et al., Large magnetic gap at the Dirac point in Bi2Te3/MnBi2Te4 heterostructures, Nature 576, 423 (2019).
  33. D. Burke, D. Heffels, K. Moors, P. Schüffelgen, D. Grützmacher, and M. R. Connolly, Robust Majorana bound states in magnetic topological insulator nanoribbons with fragile chiral edge channels, Phys. Rev. B 109, 045138 (2024).
  34. T. Förster, P. Krüger, and M. Rohlfing, Two-dimensional topological phases and electronic spectrum of Bi2Se3 thin films from GW calculations, Phys. Rev. B 92, 201404(R) (2015).
  35. T. Förster, P. Krüger, and M. Rohlfing, GW calculations for Bi2Te3 and Sb2Te3 thin films: Electronic and topological properties, Phys. Rev. B 93, 205442 (2016).
  36. E. Zsurka, C. Wang, J. Legendre, D. Di Miceli, L. Serra, D. Grützmacher, T. L. Schmidt, P. Rüßmann, and K. Moors, Low-energy modeling of three-dimensional topological insulator nanostructures, Phys. Rev. Mater. 8, 084204 (2024).
  37. H.-Z. Lu, W.-Y. Shan, W. Yao, Q. Niu, and S.-Q. Shen, Massive Dirac fermions and spin physics in an ultrathin film of topological insulator, Phys. Rev. B 81, 115407 (2010).
  38. W.-Y. Shan, H.-Z. Lu, and S.-Q. Shen, Effective continuous model for surface states and thin films of three-dimensional topological insulators, New J. Phys. 12, 043048 (2010).
  39. S.-Q. Shen, Topological Insulators: Dirac Equation in Condensed Matters, Springer Series in Solid-State Sciences (Springer, Heidelberg, 2012).
  40. H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface, Nat. Phys. 5, 438 (2009).
  41. C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, Model Hamiltonian for topological insulators, Phys. Rev. B 82, 045122 (2010).
  42. I. A. Nechaev and E. E. Krasovskii, Relativistic k·p Hamiltonians for centrosymmetric topological insulators from ab initio wave functions, Phys. Rev. B 94, 201410 (2016).
  43. L. Fu and C. L. Kane, Superconducting proximity effect and Majorana fermions at the surface of a topological insulator, Phys. Rev. Lett. 100, 096407 (2008).
  44. 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).
  45. P. Sitthison and T. D. Stanescu, Robustness of topological superconductivity in proximity-coupled topological insulator nanoribbons, Phys. Rev. B 90, 035313 (2014).
  46. D. Heffels, D. Burke, M. R. Connolly, P. Schüffelgen, D. Grützmacher, and K. Moors, Robust and fragile Majorana bound states in proximitized topological insulator nanoribbons, Nanomaterials 13, 723 (2023).
  47. C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, Kwant: a software package for quantum transport, New J. Phys. 16, 063065 (2014).
  48. B. Nijholt, J. Weston, J. Hoofwijk, and A. Akhmerov, Adaptive parallel active learning of mathematical functions (Zenodo, 2019), doi:10.5281/zenodo.1182437.
  49. M. Wimmer, Efficient numerical computation of the Pfaffian for dense and banded skew-symmetric matrices, ACM Trans. Math. Softw. 38, 1 (2012).
  50. R.-X. Zhang, H.-C. Hsu, and C.-X. Liu, Electrically tunable spin polarization of chiral edge modes in a quantum anomalous Hall insulator, Phys. Rev. B 93, 235315 (2016).
  51. B. Zhou, H.-Z. Lu, R.-L. Chu, S.-Q. Shen, and Q. Niu, Finite size effects on helical edge states in a quantum spin-Hall system, Phys. Rev. Lett. 101, 246807 (2008).
  52. J. H. Bardarson, P. W. Brouwer, and J. E. Moore, Aharonov-Bohm oscillations in disordered topological insulator nanowires, Phys. Rev. Lett. 105, 156803 (2010).
  53. Y. Zhang, K. He, C.-Z. Chang, C.-L. Song, L.-L. Wang, X. Chen, J.-F. Jia, Z. Fang, X. Dai, W.-Y. Shan, S.-Q. Shen, Q. Niu, X.-L. Qi, S.-C. Zhang, X.-C. Ma, and Q.-K. Xue, Crossover of the three-dimensional topological insulator Bi2Se3 to the two-dimensional limit, Nat. Phys. 6, 584 (2010).
  54. J. Legendre, E. Zsurka, D. Di Miceli, L. Serra, K. Moors, and T. L. Schmidt, Topological properties of finite-size heterostructures of magnetic topological insulators and superconductors, Phys. Rev. B 110, 075426 (2024).
  55. A. C. Potter and P. A. Lee, Multichannel generalization of Kitaev's Majorana end states and a practical route to realize them in thin films, Phys. Rev. Lett. 105, 227003 (2010).
  56. T. D. Stanescu, R. M. Lutchyn, and S. Das Sarma, Majorana fermions in semiconductor nanowires, Phys. Rev. B 84, 144522 (2011).
  57. D. Rainis and D. Loss, Majorana qubit decoherence by quasiparticle poisoning, Phys. Rev. B 85, 174533 (2012).
  58. A. B. Odobesko, S. Haldar, S. Wilfert, J. Hagen, J. Jung, N. Schmidt, P. Sessi, M. Vogt, S. Heinze, and M. Bode, Preparation and electronic properties of clean superconducting Nb(110) surfaces, Phys. Rev. B 99, 115437 (2019).
  59. I. Lee, C. K. Kim, J. Lee, S. J. L. Billinge, R. Zhong, J. A. Schneeloch, T. Liu, T. Valla, J. M. Tranquada, G. Gu, and J. C. S. Davis, Imaging Dirac-mass disorder from magnetic dopant atoms in the ferromagnetic topological insulator Crx(Bi0.1Sb0.9)2−xTe3, Proc. Natl. Acad. Sci. 112, 1316 (2015).
  60. Y. X. Chong, X. Liu, R. Sharma, A. Kostin, G. Gu, K. Fujita, J. C. S. Davis, and P. O. Sprau, Severe Dirac mass gap suppression in Sb2Te3-based quantum anomalous Hall materials, Nano Lett. 20, 8001 (2020).
  61. A. Park, A. Llanos, C.-I. Lu, Y. Chen, S. N. Abadi, C.-C. Chen, M. L. Teague, L. Tai, P. Zhang, K. L. Wang, and N.-C. Yeh, Phonon and defect mediated quantum anomalous Hall insulator to metal transition in magnetically doped topological insulators, Phys. Rev. B 109, 075125 (2024).
  62. H. Beidenkopf, P. Roushan, J. Seo, L. Gorman, I. Drozdov, Y. S. Hor, R. J. Cava, and A. Yazdani, Spatial fluctuations of helical Dirac fermions on the surface of topological insulators, Nat. Phys. 7, 939 (2011).
  63. H. Nam, Y. Xu, I. Miotkowski, J. Tian, Y. P. Chen, C. Liu, M. Z. Hasan, W. Zhu, G. A. Fiete, and C.-K. Shih, Microscopic investigation of Bi2−xSbxTe3−ySey systems: On the origin of a robust intrinsic topological insulator, J. Phys. Chem. Solids 128, 251 (2019).
  64. V. N. Men'shov, V. V. Tugushev, and E. V. Chulkov, Anomalous Hall and spin Hall conductivities in three-dimensional ferromagnetic topological insulator/normal insulator heterostructures, Europhys. Lett. 114, 37003 (2016).
  65. V. N. Men'shov, V. V. Tugushev, and E. V. Chulkov, Quantum anomalous Hall effect in magnetically modulated topological insulator/normal insulator heterostructures, JETP Lett. 104, 453 (2016).
  66. J. Wang, B. Lian, and S.-C. Zhang, Electrically tunable magnetism in magnetic topological insulators, Phys. Rev. Lett. 115, 036805 (2015).
  67. S. K. Chong, P. Zhang, J. Li, Y. Zhou, J. Wang, H. Zhang, A. V. Davydov, C. Eckberg, P. Deng, L. Tai, J. Xia, R. Wu, and K. L. Wang, Electrical manipulation of topological phases in a quantum anomalous Hall insulator, Adv. Mater. 35, 2207622 (2023).
  68. L. Chen, X.-H. Pan, Z. Cao, D. E. Liu, and X. Liu, Electrostatic environment and Majorana bound states in full-shell topological insulator nanowires, Phys. Rev. B 109, 075408 (2024).
  69. P. Rüßmann and S. Blügel, Proximity induced superconductivity in a topological insulator, arXiv:2208.14289.
  70. M. Lanius, J. Kampmeier, C. Weyrich, S. Kölling, M. Schall, P. Schüffelgen, E. Neumann, M. Luysberg, G. Mussler, P. M. Koenraad, T. Schäpers, and D. Grützmacher, P–N junctions in ultrathin topological insulator Sb2Te2/Bi2Te2 heterostructures grown by molecular beam epitaxy, Cryst. Growth & Des. 16, 2057 (2016).
  71. https://github.com/ezsurka/mti-nanoribbon-fom.
  72. D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed. (Cambridge University Press, Cambridge, 2018).

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Phys. Rev. B 112, 094504– Published 4 September, 2025
  • Received 22 May 2025
  • Revised 6 August 2025
  • Accepted 12 August 2025
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