Majorana fermions in semiconducting nanowire and Fulde–Ferrell superconductor hybrid structures

Abstract

The novel idea that spin-orbit coupling (SOC) and an s-wave pairing system can lead to induced pwave pairing with a strong magnetic limit, has stimulated widespread interest in searching for Majorana fermions (MFs) in semiconductor-superconductor hybrid structures. However, despite major advances in the semiconductor nanotechnology, this system has several inherent limitations that prohibit the realization and identification of MFs. We overcome these limitations by replacing the s-wave superconductor with the type-II Fulde–Ferrell (FF) superconductor, in which the center-of-mass momentum of the Cooper pair renormalizes the in-plane Zeeman field and chemical potential. As a result, MFs can be realized in semiconductor nanowires with small values of the Landé g-factor and high carrier densities. The SOC strength directly influences the topological boundary; thus, the topological phase transition and associated MFs can be engineered by an external electric field. Theoretically, almost all semiconductor nanowires can be used to realize MFs by using the FF superconductor. However, we find that InP nanowire is more suitable for the realization of MFs compared to InAs and InSb nanowires. Thus, this new scheme can take full advantage of the semiconductor nanotechnology for the realization of MFs in semiconductor-superconductor hybrid structures.

Keywords

Majorana fermion / topological transition / Pfaffian / FF-superconductor / hybrid structure

References

[1] F. Wilczek, Majorana returns, Nat. Phys. 5(9), 614 (2009)

[2] E. Majorana, Teoria simmetrica dell’elettrone e del positrone, Nuovo Cim. 14(4), 171 (1937) (in Italian)

[3] M. Z. Hasan and C. L. Kane, Topological insulators, Rev. Mod. Phys. 82(4), 3045 (2010)

[4] X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83(4), 1057 (2011)

[5] 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)

[6] A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys. Uspekhi 44(10S), 131 (2001)

[7] S. Das Sarma, M. Freedman, and C. Nayak, Topologically protected qubits from a possible non-Abelian fractional quantum Hall state, Phys. Rev. Lett. 94(16), 166802 (2005)

[8] 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)

[9] 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)

[10] J. Alicea, Majorana fermions in a tunable semiconductor device, Phys. Rev. B 81(12), 125318 (2010)

[11] Y. Oreg, G. Refael, and F. von Oppen, Helical liquids and Majorana bound states in quantum wires, Phys. Rev. Lett. 105(17), 177002 (2010)

[12] R. Lutchyn, J. Sau, and S. Das Sarma, Majorana fermions and a topological phase transition in semiconductorsuperconductor heterostructures, Phys. Rev. Lett. 105(7), 077001 (2010)

[13] 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(22), 227003 (2010)

[14] J. Liu, Q. Han, L. B. Shao, and Z. D. Wang, Exact solutions for a type of electron pairing model with spinorbit interactions and Zeeman coupling, Phys. Rev. Lett. 107(2), 026405 (2011)

[15] T. H. Hsieh and L. Fu, Majorana fermions and exotic surface andreev bound states in topological superconductors: Application to CuxBi2Se3, Phys. Rev. Lett. 108(10), 107005 (2012)

[16] L. Mao, M. Gong, E. Dumitrescu, S. Tewari, and C. W. Zhang, Hole-doped semiconductor nanowire on top of an s-wave superconductor: A new and experimentally accessible system for Majorana fermions, Phys. Rev. Lett. 108(17), 177001 (2012)

[17] 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)

[18] E. J. H. Lee, X.-C. Jiang, M. Houzet, R. Aguado, C. M. Lieber, and S. De Franceschi, Spin-resolved Andreev levels and parity crossings in hybrid superconductor–semiconductor nanostructures, NaT. NanotechnoL. 9, 79 (2014)

[19] M. T. Deng, S. Vaitiekėnas, 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)

[20] C. W. Zhang, S. Tewari, R. M. Lutchyn, and S. Das Sarma, p_x+ i_pySuperfluid from s-wave interactions of fermionic cold atoms, Phys. Rev. Lett. 101(16), 160401 (2008)

[21] M. Sato and S. Fujimoto, Existence of Majorana fermions and topological order in nodal superconductors with spinorbit interactions in external magnetic fields, Phys. Rev. Lett. 105(21), 217001 (2010)

[22] M. Gong, S. Tewari, and C. W. Zhang, BCS-BEC crossover and topological phase transition in 3D spinorbit coupled degenerate Fermi gases, Phys. Rev. Lett. 107(19), 195303 (2011)

[23] M. Gong, G. Chen, S. T. Jia, and C. W. Zhang, Searching for Majorana fermions in 2D spin-orbit coupled fermi superfluids at finite temperature, Phys. Rev. Lett. 109(10), 105302 (2012)

[24] H. Hu and X. J. Liu, Fulde–Ferrell superfluidity in ultracold Fermi gases with Rashba spin–orbit coupling, New J. Phys. 15(9), 093037 (2013)

[25] X. J. Liu and H. Hu, Topological superfluid in onedimensional spin-orbit-coupled atomic Fermi gases, Phys. Rev. A 85(3), 033622 (2012)

[26] L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, Majorana fermions in equilibrium and in driven cold-atom quantum wires, Phys. Rev. Lett. 106(22), 220402 (2011)

[27] L. P. Gor’kov and E. I. Rashba, Superconducting 2D system with lifted spin degeneracy: Mixed singlet-triplet state, Phys. Rev. Lett. 87(3), 037004 (2001)

[28] I. Bonalde, W. Bramer-Escamilla, and E. Bauer, Evidence for line nodes in the superconducting energy gap of noncentrosymmetric CePt₃Si from magnetic penetration depth measurements, Phys. Rev. Lett. 94(20), 207002 (2005)

[29] N. Kimura, K. Ito, K. Saitoh, Y. Umeda, H. Aoki, and T. Terashima, Pressure-induced superconductivity in noncentrosymmetric heavy-fermion CeRhSi₃, Phys. Rev. Lett. 95(24), 247004 (2005)

[30] I. Sugitani, Y. Okuda, H. Shishido, T. Yamada, A. Thamizhavel, E. Yamamoto, T. D. Matsuda, Y. Haga, T. Takeuchi, R. Settai, and Y. Ōnuki, Pressureinduced heavy-fermion superconductivity in antiferromagnet CeIrSi₃ without inversion symmetry, J. Phys. Sov. Jpn. 75(4), 043703 (2006)

[31] H. Mukuda, T. Fujii, T. Ohara, A. Harada, M. Yashima, Y. Kitaoka, Y. Okuda, R. Settai, and Y. Onuki, Enhancement of superconducting transition temperature due to the strong antiferromagnetic spin fluctuations in the noncentrosymmetric heavy-fermion superconductor CeIrSi₃: A ²⁹Si NMR study under pressure, Phys. Rev. Lett. 100(10), 107003 (2008)

[32] M. Nishiyama, Y. Inada, and G. Q. Zheng, Spin triplet superconducting state due to broken inversion symmetry in Li₂Pt₃B, Phys. Rev. Lett. 98(4), 047002 (2007)

[33] S. K. Goh, Y. Mizukami, H. Shishido, D. Watanabe, S. Yasumoto, M. Shimozawa, M. Yamashita, T. Terashima, Y. Yanase, T. Shibauchi, A. I. Buzdin, and Y. Matsuda, Anomalous upper critical field in CeCoIn₅/YbCoIn₅ superlattices with a Rashba-type heavy fermion interface, Phys. Rev. Lett. 109(15), 157006 (2012)

[34] J. Alicea, New directions in the pursuit of Majorana fermions in solid state systems, Rep. Prog. Phys. 75(7), 076501 (2012)

[35] C. W. J. Beenakker, Search for Majorana fermions in superconductors, Annu. Rev. Con. Mat. Phys. 4(1), 113 (2013)

[36] 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)

[37] M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caroff, and H. Q. Xu, Anomalous zero-bias conductance peak in a Nb–InSb nanowire–Nb hybrid device, Nano Lett. 12(12), 6414 (2012)

[38] 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)

[39] L. P. Rokhinson, X. Y. 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(11), 795 (2012)

[40] T. D. Stanescu, S. Tewari, J. D. Sau, and S. Das Sarma, To close or not to close: The fate of the superconducting gap across the topological quantum phase transition in Majorana-carrying semiconductor nanowires, Phys. Rev. Lett. 109(26), 266402 (2012)

[41] C. H. Lin, J. D. Sau, and S. Das Sarma, Zero-bias conductance peak in Majorana wires made of semiconductor/ superconductor hybrid structures, Phys. Rev. B 86(22), 224511 (2012)

[42] J. Liu, A. C. Potter, K. T. Law, and P. A. Lee, Zero-bias peaks in the tunneling conductance of spin-orbit-coupled superconducting wires with and without Majorana endstates, Phys. Rev. Lett. 109(26), 267002 (2012)

[43] G. Ben-Shach, A. Haim, I. Appelbaum, Y. Oreg, A. Yacoby, and B. I. Halperin, Detecting Majorana modes in one-dimensional wires by charge sensing, Phys. Rev. B 91(4), 045403 (2015)

[44] Y. X. Zeng, C. Lei, G. Chaudhary, and A. H. MacDonald, Quantum anomalous Hall Majorana platform, Phys. Rev. B 97(8), 081102 (2018)

[45] D. K. Finnemore, D. E. Mapother, and R. W. Shaw, Critical field curve of superconducting mercury, Phys. Rev. 118(1), 127 (1960)

[46] J. W. Rohlf, Modern Physics from A to Z, Wiley, 1994

[47] C. E. Pryor and M. E. Flatte, Landé gfactors and orbital momentum quenching in semiconductor quantum dots, Phys. Rev. Lett. 96(2), 026804 (2006)

[48] M. Gong, L. Mao, S. Tewari, and C. W. Zhang, Majorana fermions under uniaxial stress in semiconductorsuperconductor heterostructures,Phys. Rev. B 87, 060502(R) (2013)

[49] P. Ghosh, J. D. Sau, S. Tewari, and S. Das Sarma, Non-Abelian topological order in noncentrosymmetric superconductors with broken time-reversal symmetry, Phys. Rev. B 82(18), 184525 (2010)

[50] P. Fulde and R. A. Ferrell, Superconductivity in a strong spin-exchange field, Phys. Rev. 135(3A), A550 (1964)

[51] G. Koutroulakis, H. Kühne, J. A. Schlueter, J. Wosnitza, and S. E. Brown, Microscopic study of the Fulde–Ferrell– Larkin–Ovchinnikov state in an all-organic superconductor, Phys. Rev. Lett. 116(6), 067003 (2016)

[52] H. Maya, S. Krämer, M. Horvatić, C. Berthier, K. Miyagawa, K. Kanoda, and V. F. Mitrović, Evidence of Andreev bound states as a hallmark of the FFLO phase in κ-(BEDT-TTF)₂Cu(NCS)₂, Nat. Phys. 10, 928 (2014)

[53] J. Wosnitza, FFLO states in layered organic superconductors, Ann. Phys. 530(2), 1700282 (2018)

[54] Y. W. Guo and Y. Chen, Topological Fulde–Ferrell and Larkin–Ovchinnikov states in spin-orbit-coupled lattice system, Front. Phys. 13, 137402 (2018)

[55] W. Chen, M. Gong, R. Shen, and D. Y. Xing, Detecting Fulde–Ferrell superconductors by an Andreev interferometer, New J. Phys. 16(8), 083024 (2014)

[56] C. F. Chan and M. Gong, Pairing symmetry, phase diagram, and edge modes in the topological Fulde–Ferrell– Larkin–Ovchinnikov phase, Phys. Rev. B 89(17), 174501 (2014)

[57] C. L. Qu, Z. Zheng, M. Gong, Y. Xu, L. Mao, X. B. Zou, G. C. Guo, and C. W. Zhang, Topological superfluids with finite-momentum pairing and Majorana fermions, Nat. Commun. 4(1), 2710 (2013)

[58] W. Zhang and W. Yi, Topological Fulde–Ferrell–Larkin–Ovchinnikov states in spin–orbit-coupled Fermi gases, Nat. Commun. 4(1), 2711 (2013)

[59] I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, Band parameters for III–V compound semiconductors and their alloys, J. Appl. Phys. 89(11), 5815 (2001)

[60] J. P. Heida, B. J. vanWees, J. J. Kuipers, T. M. Klapwijk, and G. Borghs, Spin-orbit interaction in a two-dimensional electron gas in a InAs/AlSb quantum well with gate-controlled electron density, Phys. Rev. B 57(19), 11911 (1998)

[61] V. A. Guzenko, A. Bringer, J. Knobbe, H. Hardtdegen, and TH. Schäpers, Rashba effect in Ga_xIn_1–xAs/InP quantum wire structures, Appl. Phys. A Mater. Sci. Process. 87(3), 577 (2007)

[62] W. A. Zein, N. A. Ibrahim, and A. H. Phillips, Spin polarized transport in an AC-driven quantum curved nanowire, Phys. Res. Int. 2011, 505091 (2011)

[63] TH. Schäpers, V. A. Guzenko, A. Bringer, M. Akabori, M. Hagedorn, and H. Hardtdegen, Spin–orbit coupling in Ga_xIn_1–xAs/InP two-dimensional electron gases and quantum wire structures, Semicond. Sci. Technol. 24(6), 064001 (2009)

[64] J. G. Powles, B. Holtz, and W. A. B. Evans, New method for determining the chemical potential for condensed matter at high density, J. Chem. Phys. 101(9), 7804 (1994)

[65] Z. Wilamowski, W. Jantsch, H. Malissa, and U. Rössler, Evidence and evaluation of the Bychkov–Rashba effect in SiGe/Si/SiGe quantum wells, Phys. Rev. B 66(19), 195315 (2002)

[66] J. Luo, H. Munekata, F. F. Fang, and P. J. Stiles, Observation of the zero-field spin splitting of the ground electron subband in GaSb-InAs-GaSb quantum wells, Phys. Rev. B 38, 10142(R) (1988)

[67] S. Lamari, Rashba effect in inversion layers on p-type InAs MOSFET’s, Phys. Rev. B 64(24), 245340 (2001)

[68] K. L. Litvinenko, L. Nikzad, C. R. Pidgeon, J. Allam, L. F. Cohen, T. Ashley, M. Emeny, W. Zawadzki, and B. N. Murdin, Temperature dependence of the electron Landé g factor in InSb and GaAs, Phys. Rev. B 77(3), 033204 (2008)

[69] M. Oestreich and W. Rühle, Temperature dependence of the electron Landé gfactor in GaAs, Phys. Rev. Lett. 74(12), 2315 (1995)

[70] R. Zielke, F. Maier, and D. Loss, Anisotropic gfactor in InAs self-assembled quantum dots, Phys. Rev. B 89(11), 115438 (2014)

[71] S. Åsbrink and A. Waśkowska, Pressure-induced critical behavior of KMnF3 close to Pc= 3.1 GPa: X-ray diffraction results, Phys. Rev. B 53, 12 (1996)

[72] C. Hermann and C. Weisbuch, k→·p→perturbation theory in III-V compounds and alloys: A reexamination, Phys. Rev. B 15(2), 823 (1977)

[73] Z. W. Zheng, B. Shen, Y. S. Gui, Z. J. Qiu, C. P. Jiang, N. Tang, J. Liu, D. J. Chen, H. M. Zhou, R. Zhang, Y. Shi, Y. D. Zheng, S. L. Guo, J. H. Chu, K. Hoshino, and Y. Arakawa, Enhancement and anisotropy of the Landau gfactor in modulation-doped Al_0.22Ga_0.78N/GaN heterostructures, J. Appl. Phys. 95(5), 2473 (2004)

[74] F. Maier, C. Klöffel, and D. Loss, Tunable g factor and phonon-mediated hole spin relaxation in Ge/Si nanowire quantum dots, Phys. Rev. B 87, 161305(R) (2013)

[75] H. Kosaka, A. Kiselev, F. Baron, K. W. Kim, and E. Yablonovitch, Electron g factor engineering in III-V semiconductors for quantum communications, Electron. Lett. 37(7), 464 (2001)

[76] 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(19), 195125 (2008)

[77] J. F. Wang, M. S. Gudiksen, X. F. Duan, Y. Cui, and C. M. Lieber, Highly polarized photoluminescence and photodetection from single indium phosphide nanowires, Science 293(5534), 1455 (2001)

[78] K. Storm, F. Halvardsson, M. Heurlin, D. Lindgren, A. Gustafsson, P. M. Wu, B. Monemar, and L. Samuelson, Spatially resolved Hall effect measurement in a single semiconductor nanowire, Nat. Nanotechnol. 7(11), 718 (2012)

[79] D. Liang and X. P. A. Gao, Strong tuning of spin orbit interaction in an InAs nanowire by surrounding gate, Nano Lett. 12, 6 (2012)

[80] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Gate control of spin-orbit interaction in an inverted In_0.53Ga_0.47As/In_0.52Al_0.4As heterostructure, Phys. Rev. Lett. 78(7), 1335 (1997)

[81] X. W. Zhang and J. B. Xia, Rashba spin-orbit coupling in InSb nanowires under transverse electric field, Phys. Rev. B 74(7), 075304 (2006)

[82] The topological gapless phase III is still topological protected because it is impossible to adiabatically tune this phase to a trivial phase without closes the energy gap at zero momentum, see ReF. [56].

[83] DetH_BdG(k_x) = A²+α²k²_x , where A=h¯x2+h¯y2−k4/(4m*2)−Δ2−μ¯2+k2(α2+μ/m*), so the energy gap can close only at the critical boundary; see also discussion in ReF. [22].

[84] R. M. Lutchyn, T. D. Stanescu, and S. Das Sarma, Search for Majorana fermions in multiband semiconducting nanowires, Phys. Rev. Lett. 106(12), 127001 (2011)

[85] S. K. Yip, Two-dimensional superconductivity with strong spin-orbit interaction, Phys. Rev. B 65(14), 144508 (2002)

[86] D. F. Agterberg, Novel magnetic field effects in unconventional superconductors, Physica C 387(1–2), 13 (2003)

[87] O. Dimitrova and M. V. Feigel’man, Theory of a twodimensional superconductor with broken inversion symmetry, Phys. Rev. B 76(1), 014522 (2007)

[88] D. F. Agterberg and R. P. Kaur, Magnetic-field-induced helical and stripe phases in Rashba superconductors, Phys. Rev. B 75(6), 064511 (2007)

[89] Z. Zheng, M. Gong, X. B. Zou, C. W. Zhang, and G. C. Guo, Route to observable Fulde-Ferrell-Larkin- Ovchinnikov phases in three-dimensional spin-orbitcoupled degenerate Fermi gases, Phys. Rev. A 87(3), 031602 (2013)

[90] Z. Zheng, M. Gong, Y. C. Zhang, X. B. Zou, C. W. Zhang, and G. C. Guo, FFLO superfluids in 2D spin-orbit coupled Fermi Gases, Sci. Rep. 4(1), 6535 (2015)

[91] A. I. Larkin and Y. N. Ovchinnikov, Nonuniform state of superconductors, Zh. Eksp. Teor. Fiz. 47, 1136 (1964)

RIGHTS & PERMISSIONS

2019 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature