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Two pair-breaking mechanisms for a conventional superconductor with spin-singlet pairing: the Lorentz force increases the kinetic energy of the superconducting ground state, while the spin alignment due to Zeeman coupling causes energy penalty for forming spin-singlet Cooper pairs.
In a conventional superconductor the non-dissipative supercurrents are carried by Cooper pairs in spin-singlet configuration. An applied magnetic field is generally detrimental to such a superconducting state: on one hand, the coupling of charged superfluid to gauge fields generates orbital motion which leads to an increase in the kinetic energy of the superconductor; on the other, the Zeeman coupling tends to align the electron spins to the same direction, which makes it energetically costly to form spin-singlets with opposite-spin configuration. In low-dimensional systems where the orbital motion is suppressed, the limiting field Bp at which the Zeeman energy saved by polarizing electron spins wins against forming Cooper pair condensates is called the Pauli paramagnetic limit where the superconductor undergoes a phase transition to become a normal (i.e., non-superconducting) metal.
Upper left: Crystalline structure of a monolayer transition-metal dichalcogenide (TMD) with a prismatic unit cell that breaks the spatial inversion. Lower left: Electrons in K and K' valleys experience opposite effective out-of-plane Zeeman fields due to the broken inversion symmetry. Right: Ising superconductivity with enhanced in-plane upper critical fields in superconducting MoS2 thin films, which exceeds the conventional Pauli limit by 5-6 times.
Since 2015, Ising superconductors with strongly enhanced in-plane upper critical field Bc2 have been found in a large family of two-dimensional transition-metal dichalcogenides, such as MoS2, NbSe2 and TaS2. The enhancement in in-plane Bc2 was explained to result from (i) suppressed orbital motion in 2D limit, and (ii) the Ising spin-orbit coupling (SOC) in TMD materials due to broken inversion symmetry, which protects the spins of electrons forming Cooper pairs from being aligned by external magnetic fields.
Left panel: Topological band crossing with nontrivial gap induced by spin-orbit-parity coupling (SOPC). Right panel: Magnetic field - temperature superconducting phase diagram under SOPC with Pauli limit violation.
Figure adapted from Phys. Rev. Lett. 125, 107001 (2020).
Spin-orbit-parity-coupled superconductivity
Within the framework established for Ising superconductivity, the spin-orbit band splitting caused by inversion symmetry breaking plays an absolutely critical role for the robust superconducting state against strong magnetic fields - the equal-spin Cooper pairs generated by the spin-orbit splitting endow the non-centrosymmetric superconductor with the ability to withstand magnetic fields beyond the conventional pair-breaking Pauli paramagnetic limit. In two experiments in 2018 [Science 362, 926 (2018), V. Fatemi et al.; Science 362, 922 (2018), E. Sajadi et al.], however, superconductivity with Pauli limit violation was observed in monolayer 1T'-WTe2, a topological insulator where inversion symmetry is well respected - this observation thus cannot be explained by the theory for Ising superconductors, as spin-orbit splitting vanishes otherwise in the material with global inversion symmetry.
Within the framework established for Ising superconductivity, the spin-orbit band splitting caused by inversion symmetry breaking plays an absolutely critical role for the robust superconducting state against strong magnetic fields - the equal-spin Cooper pairs generated by the spin-orbit splitting endow the non-centrosymmetric superconductor with the ability to withstand magnetic fields beyond the conventional pair-breaking Pauli paramagnetic limit. In two experiments in 2018 [Science 362, 926 (2018), V. Fatemi et al.; Science 362, 922 (2018), E. Sajadi et al.], however, superconductivity with Pauli limit violation was observed in monolayer 1T'-WTe2, a topological insulator where inversion symmetry is well respected - this observation thus cannot be explained by the theory for Ising superconductors, as spin-orbit splitting vanishes otherwise in the material with global inversion symmetry.
In this Letter, we propose that spin and momentum can indeed couple in a centrosymmetric topological insulator (e.g., a monolayer WTe2 in 1T'-phase), through the involvement of an extra band parity degree of freedom --- we refer to this special effect as spin-orbit-parity coupling (SOPC). The SOPC effect not only opens up the nontrivial gap in the inverted bands of the topological insulator (see figure on the left), but results in a rich variety of distinctive superconducting properties: (i) a first-order superconductor-metal transition at critical fields violating the Pauli limit, thus explaining the experimental observation of enhanced critical fields; (ii) in-plane anisotropy in spin-susceptibility and critical fields; and (iii) strong gating dependence in the enhanced critical fields. The SOPC also stabilizes a possible nontrivial inter-orbit pairing phase with exotic p±ip' pairing symmetry.
In this Letter, we propose that spin and momentum can indeed couple in a centrosymmetric topological insulator (e.g., a monolayer WTe2 in 1T'-phase), through the involvement of an extra band parity degree of freedom --- we refer to this special effect as spin-orbit-parity coupling (SOPC). The SOPC effect not only opens up the nontrivial gap in the inverted bands of the topological insulator (see figure on the left), but results in a rich variety of distinctive superconducting properties: (i) a first-order superconductor-metal transition at critical fields violating the Pauli limit, thus explaining the experimental observation of enhanced critical fields; (ii) in-plane anisotropy in spin-susceptibility and critical fields; and (iii) strong gating dependence in the enhanced critical fields. The SOPC also stabilizes a possible nontrivial inter-orbit pairing phase with exotic p±ip' pairing symmetry.
Our prediction of SOPC superconductivity has been verified in a recent experiment in 2M-WS2 [Nature Physics 19, 106-113 (2023)] by Prof. Faxian Xiu's group at Fudan University. See also News and Views by Joseph Falson in Nature Physics - "A highly anisotropic polymorph".
Our prediction of SOPC superconductivity has been verified in a recent experiment in 2M-WS2 [Nature Physics 19, 106-113 (2023)] by Prof. Faxian Xiu's group at Fudan University. See also News and Views by Joseph Falson in Nature Physics - "A highly anisotropic polymorph".
Left panel: Superconducting phase diagram of the Ising superconductor NbSe2. In the large area where the external field is higher than the Pauli limit the system is a nodal topological superconductor. Right panel: Local spectral density on the arm-chair edge along which Majorana flat bands emerge and connect nodes with opposite topological charges in the bulk.
Figure adapted from Communications Physics 1, 40 (2018)
volume
1, Article number: 40 (2018) .
Nodal topological superconductivity in monolayer TMDs
In this article featured as Editor's highlights in Communications Physics, we propose that a monolayer NbSe2, which was demonstrated to be an Ising superconductor with strongly enhanced in-plane critical fields by Kin Fai Mak's group, is in fact a nodal topological superconductor in a large portion of its superconducting phase diagram where the in-plane field is higher than the Pauli limit (blue area in the left panel of the figure). The pairing nodes characterized by opposite topological charges in the bulk are connected by a large number of non-dispersive Majorana fermions localized on one of the system edges - a feature analogous to Fermi arc states connecting Weyl nodes in a Weyl or Dirac semimetal. This proposal provides one of the first realistic candidate material for nodal topological superconductivity.
Upper panel: Thanks to the equal-spin Cooper pairs in Ising superconductors, electrons with in-plane spin component can tunnel into the Ising superconductors via equal-spin Andreev reflection. Bottom: A half-metal wire with in-plane spin polarization h placed on top of an Ising superconductor becomes a spinless Kitaev chain and supports Majorana fermions on its ends. Right panel shows the spatial profile of the Majorana wave function.
Figure adapted from Phys. Rev. B 93, 180501(R) (2016)
Ising superconductivity & Majorana fermions in TMDs
In this Rapid Communication featured as Editor's suggestion in Physical Review B, we made the first theoretical proposal that the Ising spin-orbit coupling (SOC) in Ising superconductors such as MoS2, NbSe2 and TaS2 generates equal-spin Cooper pairs with their spins pointing in in-plane directions. These unconventional spin-triplet Cooper pairs are compatible with in-plane spin magnetism, thus allowing electrons from a spin-polarized half-metal to tunnel into the Ising superconductor via equal-spin Andreev reflections (see upper panel in the figure). We further propose that these spin-triplet Cooper pairs induce a topologically nontrivial proximity gap in a half-metal wire placed on top of the Ising superconductor, which results in an effective spinless Kitaev chain supporting Majorana fermions on the ends of the wire.
The presence of spin-triplet Cooer pairs established in this work also provides the basis for our understanding of the microscopic mechanism behind Ising superconductivity: the equal-spin Cooper pairs endow Ising superconductors with non-vanishing spin susceptibility, which allows the superconductor to lower its energy by aligning the spins of its Cooper pairs with the magnetic field. A detailed explanation is presented in a later work [Phys. Rev. Research 2, 013026 (2020)].
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