The Pairing Symmetry Of The Multi-Orbital Superconductors With Spin-Orbit Coupling

The early discovered superconductors are generally thought to be the isotropic s-wave type,which can be described by an electron-phonon coupling mechanism from the BCS theory and that's why they are named as the“BCS superconductors".However the heavy-fermion superconductors and Copper-based superconductors later found can not be explained by the BCS theory and their pairing symmetries are far beyond the s-wave pairing.On one hand the theoretical researchers have spent lots of effort on the pairing mechanism of these unconventional superconductors.On the other hand,what pairing symmetries of them are and how they affect the physical properties become very important and further attractive nowadays.Especially,recent studies on the topological superconductors have triggered plenty of people to find the non-s-wave pairing symmetries,like the p-wave,d-wave,f-wave pairing states and etc.,which are expected to have many novel topological properties.The two-dimensional superconductivity usually displays unique quantities which are absent in other dimensions,and the spin-orbit coupling always plays an essential role on the pairing symmetry.However the actual two-dimensional materials usually have multiple orbitals or layers,which makes the superconductivity more complex and interesting.So our researches are mainly focused on the multi-layer 2H-TaS₂and the multi-orbital quasi-two-dimensional Sr₂Ru0₄.The former is a kind of the so called Ising superconductor as there exists strong Ising spin-orbit couping in the layer,which may cause novel mixed superconducting sates.The later has been thought to be the most promising time-reversal symmetry-breaking+4)topological superconductor over twenty years,which can be helpful in realizing the topological quantum computation.Until now there's no definitive answers on their pairing symmetries,which requires further studies.We analyze the point-group symmetry of the crystal and find all the relevant basis gap functions,then the most stable superconducting order is a certain superposition of them and can be obtained by solving the linearized gap equations near the critical temperature.Our findings show that for the mono-and tri-layer 2H-TaS₂,the inversion-symmetry breaking will lead to the nodal s+f-wave Ising pairing state when the nearest-neighbor pairing interaction is dominant.This time reversal invariant nodal superconducting state causes topological flat zero-energy bands on the edges,giving rise to an enhancement of the local density of states there,which is consistent with the zero-bias conductance peak observed by the STM.The conclusion above is based on the freestanding conditions,or in the case there's a weak coupling between the thin-layer TaS₂and the substrate.In the superlattices with mirror symmetry along z axis,there will induce Rashba splitting on the top and bottom layers of the trilayer,which will lead to the s+f+p-wave pairing.Our result demonstrates that the nodal gap struc-ture could survive even under a rather strong Rashba coupling.As the trilayer is highly hole-doped or compressed,it can become a time-reversal symmetry-breaking d+p-wave Ising superconductor.These superconductors are full-gap in these two cases,and the corresponding Chern numbers are-6 and-18,thus they are distinct high-Chern-number topological superconductors.For the complex multi-orbital superconductor Sr₂RuO₄,we highlight the pairings between the quasi one-dimensional and two-dimensional orbitals.We find that when the inter-orbital pairing interaction is much larger than the others,a form of inter-orbital time-reversal invariant p-wave pairings could be the most stable state.Our calculations demonstrate that this form of pairings open near-nodal gaps on the Fermi surface,and the in-plane Knight shift under the critical temperature will reduce while the out-of-plane one would keep nearly flat,besides,the p-wave order won't split under unaxial stress.All these properties are consistent with the experimental facts.Furthermore,the p-wave order is composed of two degenerate states:one is spin singlet and the other triplet.This allows the system to form a kind of superconducting multi-domain structure breaking the time-reversal symmetry in the bulk,which is in line with theSR and other experiments.Under a compression,our results support the proposal that the superconductor will undergo a phase transition from the p-wave to an inter-orbital-dominant d-wave order,accompanied by a strong enhancement of the critical temperature.This conclusion is also in agreement with some strain experiments.