Non-unitary superconductivity in the monolayer of orthorhombic CoSb

Ferromagnetism and superconductivity are generally considered to be antagonistic phenomena in condensed matter physics. Here, we theoretically study the interplay between the ferromagnetic and superconducting orders in the monolayered CoSb with an orthorhombic symmetry, and suggest CoSb as a non-unitary superconductor with time-reversal symmetry breaking. By performing the group theory analysis and the first-principles calculations, the superconducting order parameter is found to be a triplet pairing with the irreducible representation of $^3B_{2u}$, which displays intriguing nodal points and non-zero periodic modulation of Cooper pair spin polarization on the Fermi surface topologies. These findings not only provide a significant insight into the coexistence of superconductivity and ferromagnetism, but also reveal the enhancement of exotic spin polarized Cooper pairing by ferromagnetic spin fluctuations in a triplet superconductor.

Additionally, another class of superconductors, the intriguing non-unitary superconductors with time-reversal symmetry breaking [34], originated from the contribution of spin angular momentum of Cooper paired electrons, are inspiring enormous research interests in the condensed matter communities recently. The richness of existing Majorana quasiparticles in three-dimensional high-symmetry non-unitary superconductors has been theoretically proposed [35]. So far, however, the only experimentally established non-unitary pairing is in the A 1 phase of superfluid 3 He in an applied high magnetic field [36][37][38], although non-unitary paired states have been extensively reported in the heavy fermion superconductor UPt 3 related to the B phase at low temperature in an applied magnetic field [39][40][41][42], and in the noncentrosymmetric LaNiC 2 [43,44] and centrosymmetric LaNiGa 2 superconductors [45,46] with the absence of an applied magnetic field.
In this Letter, we theoretically propose the monolayered orthorhombic CoSb as a non-unitary superconductor, which has been successfully grown on the SrTiO 3 (001) substrate by molecular beam epitaxy. Experimentally, symmetric superconducting gap around the Fermi level with coherence peaks at around ± 6 meV was observed by in-situ scanning tunneling spectroscopy (STS), accompanied by a weak net ferromagnetic (FM) moment lying in the basal plane found by ex-situ magnetization measurements [47]. Group symmetry analysis suggests the pairing symmetry of monolayered CoSb to be a non-unitary triplet gap function of 3 B 2u or 3 B 3u symmetry with nodes. Within the framework of density-functional theory, the calculations demonstrate the ground state of monolayered CoSb to be a half metal with the easy axis of FM magnetization along theŷ axis lying in the basal plane, which is consistent with experimental observation [47] and supports the group theory analysis that only spin-down electrons are responsible for the Cooper pairing in the non-unitary superconducting state. In the strong-coupling approach, the superconducting order parameter in the monolayered CoSb is finally solidified to be a triplet pairing with the irreducible representation of 3 B 2u , displaying intriguing nodal points and non-zero periodic modulation of Cooper pair spin polarization on the Fermi surface topologies. These findings suggest the coexistence of ferromagnetism and superconductivity and the enhancement of exotic spin polarized Cooper pairing by FM spin fluctuations in the triplet superconductor CoSb.
Group symmetry analysis.-Considering the D 2h point group of the superconducting orthorhombic CoSb monolayer shown in Fig. 1(a) with the time-reversal symmetry breaking [47], the superconducting gap function ∆( k) can be factored into the basis functions with the irreducible representation of the group SO(3) × D 2h in the weak spin-orbit coupling limit [48], where × represents the direct product and SO(3) represents all spin rotations. Similar to that in the centrosymmetric superconductor LaNiGa 2 [45], this product group has a total of eight irreducible representations listed in Table I, including four one-dimensional singlet representations and four threedimensional triplet representations. The latter have a gap function that transforms like a vector under spin rotations, resulting in the two possible ground states in a general Ginzburg-Landau theory [48]. This leads to twelve possible gap functions listed in Table I, of which eight are unitary and four are non-unitary. Only the four non-unitary gap functions are non-trivially complex that can break the time-reversal symmetry. Furthermore, in accordance with the two-dimensionality of monolayered CoSb, we further eliminate the two states showing strong k z dependence of the gap function. In contrast, there are only four possible gap functions and none of them could break time-reversal symmetry in the strong spin-orbit coupling limit, as listed in Table I. Therefore, from the viewpoint of symmetry, the superconducting orthorhombic CoSb monolayer has to be a non-unitary triplet superconductor with a weak spin-orbit coupling, so that the possible gap functions with 3 B 2u and 3 B 3u symmetry are compatible with the experimental observation of time-reversal symmetry breaking [47]. In them, only spin-down electrons participate in pairing, and thus there is an ungapped Fermi surface coexisting with another one with nodes ( 3 B 2u or 3 B 3u ). TABLE I. The upper and lower tables show the gap functions of the homogeneous superconducting states allowed by symmetry for a weak and a strong spin-orbit coupling, respectively. We have used the standard notation [34]∆( k) = ∆( k)iσy for singlet states and∆( k) = i[d( k)·σ]σy for triplets, whereσ is the vector of Pauli matrices, and k is the momentum.
Gap functions with strong spin-orbit coupling A1u The First-Principles Calculations.-The calculations are performed using the all-electron full potential linear augmented plane wave method [49] as implemented in the WIEN2k code [50]. The exchange-correlation potential is calculated using the generalized gradient approximation as proposed by Perdew, Burke, and Ernzerhof [51]. Although the conduction electrons mainly originated from the light atoms of cobalt have a weak spin-orbit coupling, consistent with the group analysis, the heavy mediated anion of antimony has a strong spin-orbit coupling, whose strength is proportional to Z 4 (where Z is the atomic number; Z = 51 for Sb) [52], leading to a significant changes of the overlapped wave functions between the Co 3d and Sb 5p orbitals [53]. Therefore, the spin-orbit coupling is included with the second variational method throughout the calculations. Furthermore, a 3000 k-point is chosen to ensure the calculation with an accuracy of 10 −5 eV, and all structural parameters (lattice constants, a 1 = 5.92Å and a 2 = 3.24Å, as well as internal coordinates) are performed using the values of experimental crystal structure [47] shown in Fig. 1(a). To reduce the interaction between neighboring layers of CoSb, a vacuum slab of 15Å along theẑ axis is introduced.
show the non-magnetic (NM) electronic structures of monolayered orthorhombic CoSb, where no spin polarization is allowed on the Co ions. Such a study can provide a benchmark for inspecting whether the magnetically ordered state is favorable. From the calculated energy band structure and the corresponding Fermi surface topologies shown in Figs. 1(b) and (c), there are mainly two bands crossing the Fermi level contributing to the electron conduction in orthorhombic CoSb, in contrast to the four bands across the Fermi level in the tetragonal CoSb [54,55]. Verifying the orbital characters of the energy bands around the Fermi level (see details in Fig. S1 in supplementary material), we notice that the five Co 3d orbitals participate the electron conductions, implying the strong Hund's coupling in the Co 3d orbitals.
The calculated density of states (DOS) and the projected DOS (PDOS) on Co 3d and Sb 5p orbitals for the NM state of monolayered orthorhombic CoSb are shown in Fig. 1(d). It can be seen that the conduction electrons mainly come from the contribution of Co 3d states partially hybridized with mediated Sb 5p states. Inspecting the value of DOS at the Fermi level, N (E f ) = 3.58 states per eV per Co atom, we notice that this value is much larger than that in the tetragonal CoSb [54,55] and the iron-based superconductors [56]. While magnetism may occur with lower values of the DOS, it must occur within a band picture if the Stoner criterion [56,57], N (E f ) × I > 1, is met, where I is the Stoner parameter, taking values of 0.7 − 0.9 eV for ions near the middle of the 3d series (note that the effective I can be reduced by hybridization) [57], implying the NM state is unstable against the magnetic states for monolayered CoSb.
In order to capture the magnetic behavior of Co 3d states in the monolayered orthorhombic CoSb, we consider a two-dimensional phenomenologically theoretical Heisenberg model on the Co ion sites as follow [55,56]: where S is the magnitude of Co spin. The i, j denotes the summation over the next-nearest neighbor Co ion sites. The parameters J 1x and J 1y describe the nearest neighboring exchange interactions along thex andŷ direction, respectively, and J 2 denotes the nextnearest neighboring exchange interaction. From the calculated energies for various magnetic configurations [58], the magnetic exchange couplings J 1x = 1.05 meV, J 1y = −46.46 meV, and J 2 = −2.98 meV are found for the monolayered orthorhombic CoSb. The strong FM superexchange coupling strength along theŷ axis could be understood through the Goodenough-Kanamori orthogonal rule [53,59] that the interacting cations of Co atoms connected to the intervening anions of Sb form an angle of 74.1 • (∼ 90 • ), which promotes the mediated Sb 5p orbitals to be orthogonal to the two nearest neighboring Co 3d orbitals. Considering the strong Hund's coupling on Co 3d orbitals, the Co 3+ ion with 3d 6 electronic configuration favors the unpaired spin on Sb 5p orbitals to be parallelly aligned to the spin of the Co 3d orbitals, resulting in the FM exchange coupling. However, when the distance of 3.24Å (= a 2 ) between two nearest Co ion sites alongŷ axis is changed to 2.96Å (= a 1 /2) along thex axis and the bond angle of Co-Sb-Co is changed to 66.8 • , the orthogonality between the Sb 5p and the two nearest neighboring Co 3d orbitals is weakened significantly and thus the antiferromagnetic superexchange coupling could be gradually enhanced along thex direction. Due to the strong FM exchange couplings on the CoSb layer, it suggests the ground state of CoSb to be a FM order [60], which is consistent with the magnetization measurements on the monolayered films of orthorhombic CoSb [47]. Furthermore, the strong anisotropic FM superexchange interaction along theŷ axis drives the easy axis of magnetization of CoSb towards theŷ axis lying in the basal plane, which is also confirmed by the total energy calculations. The magnetic momentum of 1.83 µ B on Co ion sites are found (see details in Table S1 in supplementary material).
The calculated low-energy band structure, the corresponding Fermi surface topologies, and the PDOS on the spin-up and spin-down species of total, Co 3d and Sb 5p orbitals for the FM ordered state with fixed magnetization along theŷ axis in the monolayered orthorhombic CoSb are shown in Fig. 2. Compared with the NM state shown in Fig. 1, we find that most of the bands around the Fermi level are gapped by the FM order.
The corresponding electronic DOS at the Fermi level is N (E f ) = 0.12 and N (E f ) = 1.48 states per eV per Co atom for spin-up and spin-down species, respectively, which is significantly less than that of the NM state (3.58 states per eV per Co atom), demonstrating a half metal nature of the monolayered orthorhombic CoSb that the spin-up orbitals are fully occupied while the spin-down orbitals are partially occupied (see details in Figs. S2 and S3 in supplementary material). This finding is consistent with the group theory analysis that only spindown electrons are responsible for the Cooper pairs in the non-unitary superconducting state.
Theoretical Model Calculations.-A simplified theoretical model of low-energy excitations in the non-unitary superconducting state is provided for further understanding of the behaviors of superconducting electrons based on the following Bogoliubov-de Gennes (BdG) Hamiltonian:Ĥ where k is the momentum of the excitation,Ĥ 0 ( k) describes an effective spin-dependent four-band normalstate free electron Hamiltonian obtained by projecting the first-principles calculated bands shown in Fig. 2 onto the lowest two spin-dependent bands around the Fermi level [61,62], and∆ 0 ( k) = ∆ 0∆ ( k) ⊗ iτ y represents the pairing potential with a pairing amplitude of ∆ 0 . In the tensor products, the first sector represents the spin channels σ =↑, ↓ shown in the caption of Table I while the second represents the two band channels [46]. Following the group symmetry analysis, the d( k) vector has two possible choices of d( k) = (1, −i, 0) sin(k y a 2 ) and d( k) = (1, −i, 0) sin(k x a 1 ), as listed in the Table I, corresponding to the irreducible representations of 3 B 2u and 3 B 3u , respectively. Here we have assumed that the Cooper pairs carry the spin magnetization with the value of Ŝ k = id × d * [34] along theŷ axis in accordance with the FM magnetization obtained by the firstprinciples calculations. Since the pairing amplitude of ∆ 0 is proportional to FM superexchange coupling strength within strong-coupling approach, the triplet pairing state with the irreducible representations of 3 B 3u is energetically unfavorable rather than that of 3 B 2u , to avoid the short-range repulsion caused by the antiferromagnetic exchange coupling along thex axis [62,63]. Therefore, the non-unitary paired 3 B 2u state induced by the FM spin fluctuations results in the formation of Cooper pairing in monolayered CoSb superconductor. The gap zeros of 3 B 2u state (k y = 0 and k y = π/a 2 ) cross the Fermi surface topologies, shown in Fig. 3(a), leading to intriguing nodal behavior. Additionally, it is interesting to point out that the amplitude of Cooper pair spin polarization Ŝ k on the counters of Fermi surface topologies displays a periodic modulations and the Cooper pair spin polar-ization Ŝ k vanishes at the nodal points on the Fermi surface topologies, which are the typical characters of non-unitary superconductivity. The DOS of superconducting state with the non-unitary pairing of 3 B 2u symmetry is also calculated and shown in Fig. 3(b). As is expected, the V-shaped DOS is clearly visible, qualitatively consistent with the experimentally observed STS spectra [47].
Conclusion.-By performing the group theory analysis and the first-principles calculations, we systemically study the electronic and magnetic properties in the monolayered orthorhombic CoSb superconductor, and find the normal state of CoSb to be a half metal with the easy axis of FM magnetization along theŷ axis lying in the basal plane, suggesting the orthorhombic CoSb as  Supplemental Material "Non-unitary superconductivity in the monolayer of orthorhombic CoSb" In this supplementary material, we firstly present the detailed calculations of the orbital resolved energy bands of the non-magnetic and ferromagnetic states of monolayered orthorhombic CoSb, as shown in Fig. S1-S3. Here it is interesting to point out that a Dirac like pocket appeared at the high symmetric line of Y −G shown in Fig. 2 in the main text mainly stems from the contributions of spin-up components of Sb 5p orbitals by inspecting the orbital resolved energy bands shown in Fig. S3. Secondly, we also perform the calculations on the energetic properties of the various Co spin ordered orientations for mono-layered orthorhombic CoSb with ferromagnetic ordering state, as listed in Table S1. The calculations demonstrate the ground state of monolayered CoSb is a ferromagnetic order with the magnetization along theŷ axis lying in the basal plane and the magnetic momentum of 1.83 µ B on Co ion sites.