Abstract
Based on the spin-generalized Bogoliubov–de Gennes theory, we investigate the topological defect configurations in a mesoscopic superconducting square with spin–orbit (SO) interaction. The mixed even-parity d-wave and extended s-wave components can be obtained by suitable choice of the chemical potential in such a system. We find that several novel types of topological defect states can be generated in the presence of Rashba SO coupling when the external magnetic flux turns on. Unclosed domain-wall states carrying even or odd number of one-component vortices as well as double-quanta skyrmionic patterns can appear for different Rashba SO-coupling strengths. The next-nearest-neighbor hopping effect on the evolution of topological structures is further examined. A skyrmionic chain feature with one-component vortex–antivortex pairs can show up in the present mixed-pairing system. Our investigation may provide useful information for future experiments and shed new light on device designing.
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This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The results and data presented in this work can be replicated using the numerical procedures described in the text.]
References
B.J. Baelus, F.M. Peeters, Phys. Rev. B 65, 104515 (2002)
V.F. Becerra, M.V. Milosevic, Phys. Rev. B 94, 184517 (2016)
V.F. Becerra, E. Sardella, F.M. Peeters, M.V. Milosevic, Phys. Rev. B 93, 014518 (2016)
H.-B. Braun, Adv. Phys. 61, 1 (2012)
P.M.R. Brydon, A.P. Schnyder, C. Timm, Phys. Rev. B 84, 020501(R) (2011)
R.-F. Chai, G.-Q. Zha, Eur. Phys. J. B 94, 193 (2021)
L.F. Chibotaru, A. Ceulemans, V. Bruyndoncx, V.V. Moshchalkov, Nature 408, 833 (2000)
L. Da-Chuan, Y.-Y. Lv, J. Li, B.-Y. Zhu, Q.-H. Wang, H.-B. Wang, P.-H. Wu, NPJ Quantum Mater. 3, 12 (2018)
P.G. de Gennes, Superconductivity of Metals and Alloys (Addison-Wesley, New York, 1994)
G. Dresselhaus, A.F. Kip, C. Kittel, Phys. Rev. 95, 568 (1954). (G. Dresselhaus, Phys. Rev. 100, 580 (1955))
E.T. Filby, A.A. Zhukov, P.A.J. de Groot, M.A. Ghanem, P.N. Bartlett, V.V. Metlushko, Appl. Phys. Lett. 89, 092503 (2006)
J. Garaud, E. Babaev, Phys. Rev. B 86, 060514 (2012). (Scientific Reports 5, 17540 (2015))
J. Garaud, E. Babaev, Phys. Rev. Lett. 112, 017003 (2014)
J. Garaud, E. Babaev, Phys. Rev. B 91, 014510 (2015)
J. Garaud, J. Carlstrom, E. Babaev, Phys. Rev. Lett. 107, 197001 (2011)
J. Garaud, J. Carlstrom, E. Babaev, M. Speight, Phys. Rev. B 87, 014507 (2013)
A.K. Geim, I.V. Grigorieva, S.V. Dubonos, J.G.S. Lok, J.C. Maan, A.E. Filippov, F.M. Peeters, Nature 390, 259 (1997)
A.K. Geim, S.V. Dubonos, J.G.S. Lok, M. Henini, J.C. Maan, Nature 396, 144 (1998)
A.K. Geim, S.V. Dubonos, I.V. Grigorieva, K.S. Novoselov, F.M. Peeters, V.A. Schweigert, Nature 407, 55 (2000)
R. Geurts, M.V. Milosevic, F.M. Peeters, Phys. Rev. Lett. 97, 137002 (2006). (Phys. Rev. B 79, 174508 (2009))
L.P. Gorkov, E.I. Rashba, Phys. Rev. Lett. 87, 037004 (2001)
V. Grinenko, P. Materne, R. Sarkar, H. Luetkens, K. Kihou, C.H. Lee, S. Akhmadaliev, D.V. Efremov, S.-L. Drechsler, H.-H. Klauss, Phys. Rev. B 95, 214511 (2017)
S. Ikegaya, W.B. Rui, D. Manske, Andreas, P. Schnyder, Phys. Rev. Res. 3, 023007 (2021)
K. Jiang, X. Wu, J. Hu, Z. Wang, Phys. Rev. Lett. 121, 227002 (2018)
M. Kheirkhah, Z. Yan, Y. Nagai, F. Marsiglio, Phys. Rev. Lett. 125, 017001 (2020)
K. Kuboki, J. Phys. Soc. Jpn. 70, 2698 (2001)
W.-C. Lee, S.-C. Zhang, C. Wu, Phys. Rev. Lett. 102, 217002 (2009)
N.D. Mermin, Rev. Mod. Phys. 51, 591 (1979)
R. Micnas, J. Ranninger, S. Robaszkiewicz, Rev. Mod. Phys. 62, 113 (1990)
V.V. Moshchalkov, L. Gielen, C. Strunk, R. Jonckheere, X. Qiu, C. Van Haesendonck, Y. Bruynseraede, Nature 373, 319 (1995)
E.I. Rashba, Sov. Phys. Solid State 2, 1109 (1960). (Y. A. Bychkov and E. I. Rashba, JETP Lett. 39, 78 (1984))
M. Sato, S. Fujimoto, Phys. Rev. Lett. 105, 217001 (2010)
M. Sato, Y. Takahashi, S. Fujimoto, Phys. Rev. Lett. 103, 020401 (2009). (Phys. Rev. B 82, 134521 (2010))
J.D. Sau, R.M. Lutchyn, S. Tewari, S. Das Sarma, Phys. Rev. Lett. 104, 040502 (2010)
A.P. Schnyder, S. Ryu, Phys. Rev. B 84, 060504(R) (2011)
A.P. Schnyder, P.M.R. Brydon, C. Timm, Phys. Rev. B 85, 024522 (2012)
Y. Tanaka, Y. Mizuno, T. Yokoyama, K. Yada, M. Sato, Phys. Rev. Lett. 105, 097002 (2010)
Y. Tanaka, M. Sato, N. Nagaosa, J. Phys. Soc. Jpn. 81, 011013 (2012)
M. Veldhorst, C.G. Molenaar, X.L. Wang, H. Hilgenkamp, A. Brinkman, Appl. Phys. Lett. 100, 072602 (2012)
R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (Springer, Berlin, 2003)
C.L.M. Wong, J. Liu, K.T. Law, P.A. Lee, Phys. Rev. B 88, 060504(R) (2013)
K. Yada, M. Sato, Y. Tanaka, T. Yokoyama, Phys. Rev. B 83, 064505 (2011)
G.-Q. Zha, Phys. Rev. B 95, 014510 (2017). (Solid State Communications 302, 113730 (2019))
G.-Q. Zha, EPL 130, 67005 (2020)
G.-Q. Zha, S.-P. Zhou, B.-H. Zhu, Y.-M. Shi, H.-W. Zhao, Phys. Rev. B 74, 024527 (2006)
G.-Q. Zha, H.-W. Zhao, S.-P. Zhou, Phys. Rev. B 76, 132503 (2007)
G.-Q. Zha, L. Covaci, F.M. Peeters, S.-P. Zhou, Phys. Rev. B 92, 094516 (2015)
H.-M. Zhang, Z.-X. Li, J.-P. Peng, C.-L. Song, J.-Q. Guan, Z. Li, L. Wang, K. He, S.-H. Ji, X. Chen, H. Yao, X.-C. Ma, Q.-K. Xue, Phys. Rev. B 93, 020501(R) (2016)
L.-F. Zhang, V.F. Becerra, L. Covaci, M.V. Milosevic, Phys. Rev. B 94, 024520 (2016)
L.-F. Zhang, L. Covaci, M.V. Milosevic, Phys. Rev. B 96, 224512 (2017)
L.-F. Zhang, Y.-Y. Zhang, G.-Q. Zha, M.V. Milosevic, S.-P. Zhou, Phys. Rev. B 101, 064501 (2020)
Y. Zhong, Y. Wang, S. Han, Y.-F. Lv, W.-L. Wang, D. Zhang, H. Ding, Y.-M. Zhang, L. Wang, K. He, R. Zhong, J.A. Schneeloch, G.-D. Gu, C.-L. Song, X.-C. Ma, Q.-K. Xue, Sci. Bull. 61, 1239 (2016)
X. Zhu, Phys. Rev. Lett. 122, 236401 (2019)
A.A. Zyuzin, J. Garaud, E. Babaey, Phys. Rev. Lett. 119, 167001 (2017)
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This work was supported by National Natural Science Foundation of China under Grants No. 62171267 and No. 61771298.
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Chai, RF., Zha, GQ. Topological defect states and phase transitions in mesoscopic superconducting squares with Rashba spin–orbit interaction. Eur. Phys. J. B 95, 101 (2022). https://doi.org/10.1140/epjb/s10051-022-00369-y
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DOI: https://doi.org/10.1140/epjb/s10051-022-00369-y