The monoclinic 2M phase of WS2_2, synthesized by deintercalation of K0.7_{0.7}WS2_2 and built from stacked 1T′'-type monolayers, is an intrinsic superconductor with Tc=8.8T_c = 8.8 K — the highest among untuned transition metal dichalcogenides — and first-principles calculations give it a strong topological invariant Z2=(1;000)Z_2=(1;000) with a single Dirac cone of topological surface states on the (100) surface, making it a candidate intrinsic topological superconductor. The discovery paper left four linked issues open: (i) the microscopic origin of TcT_c, since first-principles electron–phonon coupling combined with the Allen–Dynes modified McMillan formula predicts only ≈2\approx 2 K; (ii) the origin of a pronounced Shubnikov–de Haas frequency near 80 T that matches no calculated bulk Fermi-surface extremal orbit and was proposed to come from the surface states; (iii) whether the topological surface states actually become superconducting (through the natural bulk–surface proximity effect or intrinsic pairing) and whether the resulting surface state is topological, supporting Majorana bound states; and (iv) the multiband, anisotropic gap structure of the superconducting state. Later work has substantially constrained the gap structure (nodeless anisotropic multiband s++s^{++}, reproduced by ab initio Migdal–Eliashberg theory) and produced scanning-tunnelling evidence for probable Majorana bound states in vortices, but a definitive account of the pairing mechanism and its reconciliation with unconventional two-dimensional-limit behaviour, the assignment of the 80 T frequency, and definitive confirmation of topological superconductivity on the surface states remain open.

Background

Transition metal dichalcogenides (TMDs) are layered compounds MX2MX_2 (M=M= Mo, W; X=X= S, Se, Te) whose electronic properties depend strongly on how the MX2MX_2 monolayers are stacked. The metastable 1T′' distortion of these monolayers features zigzag metal chains and, in monolayer form, a band inversion that produces quantum spin Hall behaviour. In 2019, Fang and co-workers synthesized a new bulk polymorph of WS2_2, called 2M-WS2_2 (two 1T′'-type monolayers per monoclinic unit cell, space group C2/mC_{2/m}, obtained by chemical deintercalation of K0.7WS2K_{0.7}WS_2), and showed that it is an intrinsic (ambient-pressure, ungated) type-II superconductor with a transition temperature Tc=8.8T_c = 8.8 K — the highest reported for any TMD without external tuning — with an in-plane coherence length near 8.8 nm and a large upper-critical-field anisotropy.

Several standard probes frame the open questions. Shubnikov–de Haas (SdH) quantum oscillations — periodic magnetoresistance oscillations whose frequency FF is proportional to the extremal cross-sectional area of a Fermi-surface sheet — were used together with density-functional theory (DFT) to map the Fermi surface, which is semimetallic, with hole pockets near the Y point and an electron pocket near Γ\Gamma. The electron–phonon coupling constant λ\lambda (the dimensionless integral of the Eliashberg spectral function α2F(ω)\alpha^2F(\omega), which quantifies how strongly phonons scatter electrons at the Fermi level) was computed from first principles, and TcT_c was estimated with the Allen–Dynes modified McMillan formula, a semi-empirical interpolation formula that is known to underestimate TcT_c when the coupling is anisotropic or the phonon spectrum is complex. The topological character was established by the Fu–Kane parity method: for an inversion-symmetric crystal, the product of parity eigenvalues of the occupied bands at the eight time-reversal-invariant momenta yields the strong Z2Z_2 invariant, which for 2M-WS2_2 takes the value (1;000)(1;000) due to a band inversion between S-pp and W-dd orbitals at Γ\Gamma; a surface Green's-function calculation based on maximally localized Wannier functions (localized, tight-binding-like orbitals constructed from the first-principles wavefunctions) predicts topological surface states — helical, spin–momentum-locked states protected by time-reversal symmetry — including a single Dirac cone (a linear band crossing) on the (100) cleavage surface.

The coexistence of bulk superconductivity and topological surface states in one chemically simple, stable crystal immediately suggested a route to topological superconductivity: Fu and Kane (Physical Review Letters 100, 096407 (2008)) showed that conventional ss-wave pairing proximitized onto spin–momentum-locked surface states of a topological insulator produces a topological superconducting phase whose vortices host Majorana bound states — zero-energy excitations that are their own antiparticles and are pursued as building blocks for fault-tolerant topological quantum computation, distinguishable in principle from the ordinary (Caroli–de Gennes–Matricon) vortex core states of a conventional superconductor. The discovery paper explicitly left the microscopic and topological content of this route unverified, in the four linked respects stated in the problem statement below.

Problem Statement

Determine the superconducting and topological character of 2M-WS2_2 — the intrinsic Tc=8.8T_c = 8.8 K superconducting, semimetallic polymorph of WS2_2 with calculated strong Z2=(1;000)Z_2=(1;000) topology and topological surface states — resolving the four linked issues that its discovery paper (Fang et al., Advanced Materials 2019) left open:

  1. Microscopic origin of the transition temperature. First-principles electron–phonon calculations give a coupling constant λ=0.49\lambda = 0.49, and the Allen–Dynes modified McMillan formula with Coulomb pseudopotential μ∗=0.1\mu^* = 0.1 then yields Tc≈2T_c \approx 2 K, about one quarter of the measured 8.8 K. Establish whether the bulk state is a conventional anisotropic electron–phonon superconductor whose TcT_c is merely underestimated by the McMillan-type formula (as ab initio Migdal–Eliashberg theory — the standard many-body framework for phonon-mediated superconductivity that treats the full anisotropic gap equation rather than an interpolation formula — suggests), or whether additional pairing channels or correlation effects are operative, and reconcile this account with the unconventional superconducting signatures observed in thin (sub-20 nm) flakes and with the strange-metal and stripe-charge-order phenomenology.

  2. Origin of the unassigned quantum-oscillation frequency. Identify the origin of the pronounced SdH frequency near F≈80F \approx 80 T, which matches no extremal cross-section of the calculated bulk Fermi surface (the ≈150\approx 150 T and ≈167\approx 167 T branches are assigned to the hole pockets and the ≈807\approx 807 T branch to the electron pocket), and confirm or refute the proposal that it arises from the topological surface states.

  3. Superconductivity of the topological surface states. Determine whether the calculated topological surface states (single Dirac cone with helical spin texture on the (100) surface) become superconducting — through the natural bulk–surface proximity effect or intrinsic surface pairing — and establish the symmetry and topology of the resulting surface superconducting state; in particular, whether 2M-WS2_2 realizes a topological superconducting phase whose vortices host genuine Majorana bound states, as distinguished from trivial Caroli–de Gennes–Matricon vortex core states.

  4. Reconciliation of parameters with the established gap structure. The bulk gap form is taken as established context (nodeless anisotropic multiband s++s^{++}, reproduced by ab initio Migdal–Eliashberg theory); the open part of this item is to reconcile the calculated and measured superconducting parameters (TcT_c, gap, upper critical fields, penetration depth) across the bulk and few-layer limits.

These four items are independent sub-questions, each of them mandatory, that share one scientific objective: a complete, self-consistent microscopic and topological account of the superconducting state of this candidate intrinsic topological superconductor, such that the measured transition temperature, gap structure, quantum-oscillation spectrum, and surface-state phenomenology are jointly explained. The task is resolved only when all four items are jointly explained; resolving a proper subset of them does not resolve it.

References

  1. Yuqiang Fang, Jie Pan, Dongqin Zhang, Dong Wang, Hishiro T. Hirose, Taichi Terashima, Shinya Uji, Yonghao Yuan, Wei Li, Zhen Tian, Jiamin Xue, Yonghui Ma, Wei Zhao, Qikun Xue, Gang Mu, Haijun Zhang, Fuqiang Huang, 'Discovery of Superconductivity in 2M WS2_2 with Possible Topological Surface States', Advanced Materials 31 (30), 1901942 (2019). DOI: 10.1002/adma.201901942, https://doi.org/10.1002/adma.201901942
  2. Yuqiang Fang, Jie Pan, Dongqin Zhang, Dong Wang, Hishiro T. Hirose, Taichi Terashima, Shinya Uji, Yonghao Yuan, Wei Li, Zhen Tian, Jiamin Xue, Yonghui Ma, Wei Zhao, Qikun Xue, Gang Mu, Haijun Zhang, Fuqiang Huang, 'Coexistence of Intrinsic Superconductivity and Topological Insulator State in Monoclinic Phase WS2_2', arXiv:1808.05324 (2018; preprint of the preceding entry). DOI: 10.48550/arXiv.1808.05324, https://arxiv.org/abs/1808.05324
  3. Yonghao Yuan, Jie Pan, Xintong Wang, Yuqiang Fang, Canli Song, Lili Wang, Ke He, Xucun Ma, Haijun Zhang, Fuqiang Huang, Wei Li, Qi-Kun Xue, 'Evidence of anisotropic Majorana bound states in 2M-WS2_2', Nature Physics 15, 1046–1051 (2019). DOI: 10.1038/s41567-019-0576-7, https://doi.org/10.1038/s41567-019-0576-7
  4. Zurab Guguchia, Dariusz J. Gawryluk, Marta Brzezinska, Stepan S. Tsirkin, Rustem Khasanov, Ekaterina Pomjakushina, Fabian O. von Rohr, Joel A. T. Verezhak, M. Zahid Hasan, Titus Neupert, Hubertus Luetkens, Alex Amato, 'Nodeless superconductivity and its evolution with pressure in the layered Dirac semimetal 2M-WS2_2', npj Quantum Materials 4, 50 (2019). DOI: 10.1038/s41535-019-0189-5, https://doi.org/10.1038/s41535-019-0189-5
  5. Chao-Sheng Lian, Chen Si, Wenhui Duan, 'Anisotropic Full-Gap Superconductivity in 2M-WS2_2 Topological Metal with Intrinsic Proximity Effect', Nano Letters 21, 709–715 (2021; published online December 2020). DOI: 10.1021/acs.nanolett.0c04357, https://doi.org/10.1021/acs.nanolett.0c04357
  6. Soohyun Cho, Soonsang Huh, Yuqiang Fang, Chenqiang Hua, Hua Bai, Zhicheng Jiang, Zhengtai Liu, Jishan Liu, Zhenhua Chen, Yuto Fukushima, Ayumi Harasawa, Kaishu Kawaguchi, Shik Shin, Takeshi Kondo, Yunhao Lu, Gang Mu, Fuqiang Huang, Dawei Shen, 'Direct Observation of the Topological Surface State in the Topological Superconductor 2M-WS2_2', Nano Letters 22, 8827–8834 (2022). DOI: 10.1021/acs.nanolett.2c02372, https://doi.org/10.1021/acs.nanolett.2c02372 (preprint arXiv:2212.06703, https://arxiv.org/abs/2212.06703)
  7. Piumi Samarawickrama, Rabindra Dulal, Zhuangen Fu, Uppalaiah Erugu, Wenyong Wang, John Ackerman, Brian Leonard, Jinke Tang, TeYu Chien, Jifa Tian, 'Two-Dimensional 2M-WS2_2 Nanolayers for Superconductivity', ACS Omega 6, 2966–2972 (2021). DOI: 10.1021/acsomega.0c05327, https://doi.org/10.1021/acsomega.0c05327
  8. Sunil Ghimire, Kamal R. Joshi, Marcin Kończykowski, Romain Grasset, Amlan Datta, Makariy A. Tanatar, Damien Bérubé, Su-Yang Xu, Yuqiang Fang, Fuqiang Huang, Peter P. Orth, Mathias S. Scheurer, Ruslan Prozorov, 'Anisotropic multiband superconductivity in 2M-WS2_2 probed by controlled disorder', Physical Review Research 6, 013124 (2024). DOI: 10.1103/PhysRevResearch.6.013124, https://doi.org/10.1103/PhysRevResearch.6.013124 (preprint arXiv:2307.14891, https://arxiv.org/abs/2307.14891)
  9. Piumi Samarawickrama, Joseph McBride, Sabin Gautam, Zhuangen Fu, Kenji Watanabe, Takashi Taniguchi, Wenyong Wang, Jinke Tang, John Ackerman, Brian Leonard, Jifa Tian, 'Crossover from Conventional to Unconventional Superconductivity in 2M-WS2_2', Nano Letters 24, 16184–16190 (2024). DOI: 10.1021/acs.nanolett.4c05257, https://doi.org/10.1021/acs.nanolett.4c05257
  10. Xuemin Fan, Xiao-Qi Sun, Penghao Zhu, Yuqiang Fang, Yongkang Ju, Yonghao Yuan, Jingming Yan, Fuqiang Huang, Taylor L. Hughes, Peizhe Tang, Qi-Kun Xue, Wei Li, 'Stripe charge order and its interaction with Majorana bound states in 2M-WS2_2 topological superconductors', National Science Review 12, nwae312 (2025; published online September 2024). DOI: 10.1093/nsr/nwae312, https://doi.org/10.1093/nsr/nwae312
  11. Kebin Xiao, Yunkai Guo, Daran Fu, Yuqiang Fang, Yating Hu, Jingming Yan, Yucong Peng, Yuyang Wang, Yongkang Ju, Peizhe Tang, Xiangang Wan, Fuqiang Huang, Qi-Kun Xue, Wei Li, 'Lattice-decoupled rotatable stripe-like charge order within the strange metal phase of 2M-WS2_2', Proceedings of the National Academy of Sciences 122 (48), e2513493122 (2025). DOI: 10.1073/pnas.2513493122, https://doi.org/10.1073/pnas.2513493122
  12. Lixuan Xu, Yuqiang Fang, Fuqiang Huang, Shihao Zhang, Shiheng Liang, Yulin Chen, Lexian Yang, Zhongkai Liu, Nan Xu, Yiwei Li, 'Dual topology and versatile Rashba-split surface state configurations in 2M-WS2_2 and 2M-WSe2_2', Physical Review B 111, 115151 (2025). DOI: 10.1103/PhysRevB.111.115151, https://doi.org/10.1103/PhysRevB.111.115151
  13. Liang Fu, C. L. Kane, 'Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator', Physical Review Letters 100, 096407 (2008). DOI: 10.1103/PhysRevLett.100.096407, https://doi.org/10.1103/PhysRevLett.100.096407