Pairing symmetry in monolayer of orthorhombic CoSb
Tian-Zhong Yuan, Mu-Yuan Zou, Wen-Tao Jin, Xin-Yuan Wei, Xu-Guang Xu, Wei Li
NNon-unitary superconductivity in the monolayer of orthorhombic CoSb
Tianzhong Yuan, Muyuan Zou, Wentao Jin, Xinyuan Wei, and Wei Li
1, 3, ∗ State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China Key Laboratory of Micro-Nano Measurement-Manipulation and Physics (Ministry of Education),School of Physics, Beihang University, Beijing 100191, China Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Jiangsu 210093, China (Dated: April 23, 2020)Ferromagnetism and superconductivity are generally considered to be antagonistic phenomena incondensed matter physics. Here, we theoretically study the interplay between the ferromagneticand superconducting orders in the monolayered CoSb with an orthorhombic symmetry, and suggestCoSb as a non-unitary superconductor with time-reversal symmetry breaking. By performing thegroup theory analysis and the first-principles calculations, the superconducting order parameter isfound to be a triplet pairing with the irreducible representation of B u , which displays intriguingnodal points and non-zero periodic modulation of Cooper pair spin polarization on the Fermi surfacetopologies. These findings not only provide a significant insight into the coexistence of superconduc-tivity and ferromagnetism, but also reveal the enhancement of exotic spin polarized Cooper pairingby ferromagnetic spin fluctuations in a triplet superconductor. Introduction.—
The search for exotic unconventionalsuperconductivity with time-reversal symmetry break-ing is one of the most challenging tasks in condensedmatter physics. Among of them, the prominent chiralsuperconductors originated from the contribution of or-bital angular momentum of Cooper paired electrons, suchas the chiral p -wave topological superconductors [1, 2],have received great attentions as they host the Majoranaquasiparticles at the boundaries [3–8], which is equivalentto the non-Abelian Moore-Read (Pfaffian) spin-tripletpaired state in the fractional quantum Hall effect withfilling factor of 5 / s -wave superconductor [24, 25], and the quantumspin liquids [26], as well as the iron-based superconduc-tors [27–33].Additionally, another class of superconductors, the in-triguing non-unitary superconductors with time-reversalsymmetry breaking [34], originated from the contribu-tion of spin angular momentum of Cooper paired elec-trons, are inspiring enormous research interests in thecondensed matter communities recently. The richnessof existing Majorana quasiparticles in three-dimensionalhigh-symmetry non-unitary superconductors has beentheoretically proposed [35]. So far, however, the onlyexperimentally established non-unitary pairing is in the A phase of superfluid He in an applied high magneticfield [36–38], although non-unitary paired states havebeen extensively reported in the heavy fermion super-conductor UPt related to the B phase at low temper-ature in an applied magnetic field [39–42], and in the noncentrosymmetric LaNiC [43, 44] and centrosymmet-ric LaNiGa superconductors [45, 46] with the absence ofan applied magnetic field.In this Letter, we theoretically propose the mono-layered orthorhombic CoSb as a non-unitary super-conductor, which has been successfully grown on theSrTiO (001) substrate by molecular beam epitaxy. Ex-perimentally, symmetric superconducting gap around theFermi level with coherence peaks at around ± in - situ scanning tunneling spectroscopy(STS), accompanied by a weak net ferromagnetic (FM)moment lying in the basal plane found by ex - situ mag-netization measurements [47]. Group symmetry analy-sis suggests the pairing symmetry of monolayered CoSbto be a non-unitary triplet gap function of B u or B u symmetry with nodes. Within the framework ofdensity-functional theory, the calculations demonstratethe ground state of monolayered CoSb to be a half metalwith the easy axis of FM magnetization along the ˆ y axislying in the basal plane, which is consistent with exper-imental observation [47] and supports the group theoryanalysis that only spin-down electrons are responsible forthe Cooper pairing in the non-unitary superconductingstate. In the strong-coupling approach, the supercon-ducting order parameter in the monolayered CoSb is fi-nally solidified to be a triplet pairing with the irreduciblerepresentation of B u , displaying intriguing nodal pointsand non-zero periodic modulation of Cooper pair spin po-larization on the Fermi surface topologies. These findingssuggest the coexistence of ferromagnetism and supercon-ductivity and the enhancement of exotic spin polarizedCooper pairing by FM spin fluctuations in the tripletsuperconductor CoSb. Group symmetry analysis.—
Considering the D h pointgroup of the superconducting orthorhombic CoSb mono-layer shown in Fig. 1(a) with the time-reversal symmetrybreaking [47], the superconducting gap function ∆( (cid:126)k ) can a r X i v : . [ c ond - m a t . s t r- e l ] A p r be factored into the basis functions with the irreduciblerepresentation of the group SO (3) × D h in the weakspin-orbit coupling limit [48], where × represents thedirect product and SO (3) represents all spin rotations.Similar to that in the centrosymmetric superconductorLaNiGa [45], this product group has a total of eight ir-reducible representations listed in Table I, including fourone-dimensional singlet representations and four three-dimensional triplet representations. The latter have agap function that transforms like a vector under spin ro-tations, resulting in the two possible ground states ina general Ginzburg-Landau theory [48]. This leads totwelve possible gap functions listed in Table I, of whicheight are unitary and four are non-unitary. Only thefour non-unitary gap functions are non-trivially complexthat can break the time-reversal symmetry. Furthermore,in accordance with the two-dimensionality of monolay-ered CoSb, we further eliminate the two states showingstrong k z dependence of the gap function. In contrast,there are only four possible gap functions and none ofthem could break time-reversal symmetry in the strongspin-orbit coupling limit, as listed in Table I. Therefore,from the viewpoint of symmetry, the superconductingorthorhombic CoSb monolayer has to be a non-unitarytriplet superconductor with a weak spin-orbit coupling,so that the possible gap functions with B u and B u symmetry are compatible with the experimental obser-vation of time-reversal symmetry breaking [47]. In them,only spin-down electrons participate in pairing, and thusthere is an ungapped Fermi surface coexisting with an-other one with nodes ( B u or B u ). TABLE I. The upper and lower tables show the gap func-tions of the homogeneous superconducting states allowed bysymmetry for a weak and a strong spin-orbit coupling, re-spectively. We have used the standard notation [34] ˆ∆( (cid:126)k ) =∆( (cid:126)k ) i ˆ σ y for singlet states and ˆ∆( (cid:126)k ) = i [ d ( (cid:126)k ) · ˆ σ ]ˆ σ y for triplets,where ˆ σ is the vector of Pauli matrices, and (cid:126)k is the momen-tum. SO (3) × D h unitary state non-unitary state A g ∆( (cid:126)k ) = 1 0 B g ∆( (cid:126)k ) = k x k y B g ∆( (cid:126)k ) = k x k z B g ∆( (cid:126)k ) = k y k z A u d ( (cid:126)k ) = (0 , , k x k y k z d ( (cid:126)k ) = (1 , − i, k x k y k z B u d ( (cid:126)k ) = (0 , , k z d ( (cid:126)k ) = (1 , − i, k z B u d ( (cid:126)k ) = (0 , , k y d ( (cid:126)k ) = (1 , − i, k y B u d ( (cid:126)k ) = (0 , , k x d ( (cid:126)k ) = (1 , − i, k x D h Gap functions with strong spin-orbit coupling A u d ( (cid:126)k ) = ( Ak x , Bk y , Ck z ) B u d ( (cid:126)k ) = ( Ak y , Bk x , Ck x k y k z ) B u d ( (cid:126)k ) = ( Ak z , Bk x k y k z , Ck x ) B u d ( (cid:126)k ) = ( Ak x k y k z , Bk z , Ck y ) The First-Principles Calculations.—
The calculationsare performed using the all-electron full potential linearaugmented plane wave method [49] as implemented in theWIEN2k code [50]. The exchange-correlation potential iscalculated using the generalized gradient approximationas proposed by Perdew, Burke, and Ernzerhof [51]. Al-though the conduction electrons mainly originated fromthe light atoms of cobalt have a weak spin-orbit cou-pling, consistent with the group analysis, the heavy me-diated anion of antimony has a strong spin-orbit cou-pling, whose strength is proportional to Z (where Z is the atomic number; Z = 51 for Sb) [52], leading toa significant changes of the overlapped wave functionsbetween the Co 3 d and Sb 5 p orbitals [53]. Therefore,the spin-orbit coupling is included with the second varia-tional method throughout the calculations. Furthermore,a 3000 (cid:126)k -point is chosen to ensure the calculation withan accuracy of 10 − eV, and all structural parameters(lattice constants, a = 5 .
92 ˚A and a = 3 .
24 ˚A, as wellas internal coordinates) are performed using the valuesof experimental crystal structure [47] shown in Fig. 1(a).To reduce the interaction between neighboring layers ofCoSb, a vacuum slab of 15 ˚A along the ˆ z axis is intro-duced.Figs. 1(b)-(d) show the non-magnetic (NM) electronicstructures of monolayered orthorhombic CoSb, where nospin polarization is allowed on the Co ions. Such a studycan provide a benchmark for inspecting whether the mag-netically ordered state is favorable. From the calculatedenergy band structure and the corresponding Fermi sur-face topologies shown in Figs. 1(b) and (c), there aremainly two bands crossing the Fermi level contributingto the electron conduction in orthorhombic CoSb, in con-trast to the four bands across the Fermi level in thetetragonal CoSb [54, 55]. Verifying the orbital charactersof the energy bands around the Fermi level (see detailsin Fig. S1 in supplementary material), we notice thatthe five Co 3 d orbitals participate the electron conduc-tions, implying the strong Hund’s coupling in the Co 3 d orbitals.The calculated density of states (DOS) and the pro-jected DOS (PDOS) on Co 3 d and Sb 5 p orbitals for theNM state of monolayered orthorhombic CoSb are shownin Fig. 1(d). It can be seen that the conduction elec-trons mainly come from the contribution of Co 3 d statespartially hybridized with mediated Sb 5 p states. Inspect-ing the value of DOS at the Fermi level, N ( E f ) = 3 . N ( E f ) × I >
1, is met, where I is the Stoner parameter,taking values of 0 . − . d series (note that the effective I can be reducedby hybridization) [57], implying the NM state is unstableagainst the magnetic states for monolayered CoSb.In order to capture the magnetic behavior of Co 3 d states in the monolayered orthorhombic CoSb, we con-sider a two-dimensional phenomenologically theoreticalHeisenberg model on the Co ion sites as follow [55, 56]:ˆ H = J x (cid:88) i (cid:126)S i (cid:126)S i +ˆ x + J y (cid:88) i (cid:126)S i (cid:126)S j +ˆ y + J (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) (cid:126)S i (cid:126)S j , (1)where (cid:126)S is the magnitude of Co spin. The (cid:104)(cid:104) i, j (cid:105)(cid:105) de-notes the summation over the next-nearest neighbor Coion sites. The parameters J x and J y describe thenearest neighboring exchange interactions along the ˆ x and ˆ y direction, respectively, and J denotes the next-nearest neighboring exchange interaction. From the cal-culated energies for various magnetic configurations [58],the magnetic exchange couplings J x = 1 .
05 meV, J y = − .
46 meV, and J = − .
98 meV are found for themonolayered orthorhombic CoSb. The strong FM su-perexchange coupling strength along the ˆ y axis could beunderstood through the Goodenough-Kanamori orthog-onal rule [53, 59] that the interacting cations of Co atomsconnected to the intervening anions of Sb form an angleof 74 . ◦ ( ∼ ◦ ), which promotes the mediated Sb 5 p or-bitals to be orthogonal to the two nearest neighboringCo 3 d orbitals. Considering the strong Hund’s couplingon Co 3 d orbitals, the Co ion with 3 d electronic con-figuration favors the unpaired spin on Sb 5 p orbitals tobe parallelly aligned to the spin of the Co 3 d orbitals,resulting in the FM exchange coupling. However, whenthe distance of 3 .
24 ˚A (= a ) between two nearest Co ionsites along ˆ y axis is changed to 2 .
96 ˚A (= a /
2) alongthe ˆ x axis and the bond angle of Co-Sb-Co is changedto 66 . ◦ , the orthogonality between the Sb 5 p and thetwo nearest neighboring Co 3 d orbitals is weakened sig-nificantly and thus the antiferromagnetic superexchangecoupling could be gradually enhanced along the ˆ x direc-tion. Due to the strong FM exchange couplings on theCoSb layer, it suggests the ground state of CoSb to be aFM order [60], which is consistent with the magnetizationmeasurements on the monolayered films of orthorhombicCoSb [47]. Furthermore, the strong anisotropic FM su-perexchange interaction along the ˆ y axis drives the easyaxis of magnetization of CoSb towards the ˆ y axis lyingin the basal plane, which is also confirmed by the totalenergy calculations. The magnetic momentum of 1.83 µ B on Co ion sites are found (see details in Table S1 insupplementary material).The calculated low-energy band structure, the corre-sponding Fermi surface topologies, and the PDOS on thespin-up and spin-down species of total, Co 3 d and Sb5 p orbitals for the FM ordered state with fixed magne-tization along the ˆ y axis in the monolayered orthorhom-bic CoSb are shown in Fig. 2. Compared with the NMstate shown in Fig. 1, we find that most of the bandsaround the Fermi level are gapped by the FM order. The corresponding electronic DOS at the Fermi levelis N ( E f ) = 0 .
12 and N ( E f ) = 1 .
48 states per eV perCo atom for spin-up and spin-down species, respectively,which is significantly less than that of the NM state (3.58states per eV per Co atom), demonstrating a half metalnature of the monolayered orthorhombic CoSb that thespin-up orbitals are fully occupied while the spin-downorbitals are partially occupied (see details in Figs. S2and S3 in supplementary material). This finding is con-sistent with the group theory analysis that only spin-down electrons are responsible for the Cooper pairs inthe non-unitary superconducting state.
Theoretical Model Calculations.—
A simplified theoret-ical model of low-energy excitations in the non-unitarysuperconducting state is provided for further understand-ing of the behaviors of superconducting electrons basedon the following Bogoliubov-de Gennes (BdG) Hamilto-nian: ˆ H sc = (cid:32) ˆ H ( (cid:126)k ) ˆ∆ ( (cid:126)k )ˆ∆ † ( (cid:126)k ) − ˆ H ( (cid:126)k ) (cid:33) , (2)where (cid:126)k is the momentum of the excitation, ˆ H ( (cid:126)k ) de-scribes an effective spin-dependent four-band normal-state free electron Hamiltonian obtained by projectingthe first-principles calculated bands shown in Fig. 2 ontothe lowest two spin-dependent bands around the Fermilevel [61, 62], and ˆ∆ ( (cid:126)k ) = ∆ ˆ∆( (cid:126)k ) ⊗ iτ y representsthe pairing potential with a pairing amplitude of ∆ .In the tensor products, the first sector represents thespin channels σ = ↑ , ↓ shown in the caption of Table Iwhile the second represents the two band channels [46].Following the group symmetry analysis, the d ( (cid:126)k ) vec-tor has two possible choices of d ( (cid:126)k ) = (1 , − i,
0) sin( k y a )and d ( (cid:126)k ) = (1 , − i,
0) sin( k x a ), as listed in the Table I,corresponding to the irreducible representations of B u and B u , respectively. Here we have assumed thatthe Cooper pairs carry the spin magnetization with thevalue of (cid:104) ˆ S (cid:126)k (cid:105) = i d × d ∗ [34] along the ˆ y axis in accor-dance with the FM magnetization obtained by the first-principles calculations. Since the pairing amplitude of ∆ is proportional to FM superexchange coupling strengthwithin strong-coupling approach, the triplet pairing statewith the irreducible representations of B u is energeti-cally unfavorable rather than that of B u , to avoid theshort-range repulsion caused by the antiferromagnetic ex-change coupling along the ˆ x axis [62, 63]. Therefore, thenon-unitary paired B u state induced by the FM spinfluctuations results in the formation of Cooper pairingin monolayered CoSb superconductor. The gap zeros of B u state ( k y = 0 and k y = π/a ) cross the Fermi sur-face topologies, shown in Fig. 3(a), leading to intriguingnodal behavior. Additionally, it is interesting to pointout that the amplitude of Cooper pair spin polarization (cid:104) ˆ S (cid:126)k (cid:105) on the counters of Fermi surface topologies displaysa periodic modulations and the Cooper pair spin polar-ization (cid:104) ˆ S (cid:126)k (cid:105) vanishes at the nodal points on the Fermisurface topologies, which are the typical characters ofnon-unitary superconductivity. The DOS of supercon-ducting state with the non-unitary pairing of B u sym-metry is also calculated and shown in Fig. 3(b). As isexpected, the V-shaped DOS is clearly visible, qualita-tively consistent with the experimentally observed STSspectra [47]. Conclusion.—
By performing the group theory anal-ysis and the first-principles calculations, we systemi-cally study the electronic and magnetic properties in themonolayered orthorhombic CoSb superconductor, andfind the normal state of CoSb to be a half metal withthe easy axis of FM magnetization along the ˆ y axis lyingin the basal plane, suggesting the orthorhombic CoSb asa non-unitary superconductor in which only spin-downelectrons are responsible for the Cooper pairing. In thestrong-coupling approach, we solidify the pairing symme-try of CoSb to be a triplet pairing with the irreduciblerepresentations of B u that displays intriguing nodalpoints and non-zero periodic modulation of Cooper pairspin polarization on the Fermi surface topologies. Thesefindings indicate the novel coexistence of FM and super-conducting orders in CoSb and the enhancement of exoticspin polarized Cooper pairing by FM spin fluctuationsdriving in a triplet superconductor.This work was supported by the National Natural Sci-ence Foundation of China (Grant No. 11927807) and theNatural Science Foundation of Shanghai of China (GrantNo. 19ZR1402600). W. L. also acknowledges the start-up funding from Fudan University. ∗ w [email protected][1] X.-L. Qi and S.-C. Zhang, Topological insulators and su-perconductors , Rev. Mod. Phys. , 1057 (2011).[2] M. Sato and Y. Ando, Topological superconductors: areview , Rep. Prog. Phys. , 076501 (2017).[3] A. Y. Kitaev, Unpaired Majorana fermions in quantumwires , Phys.-Usp. , 131 (2001).[4] D. A. Ivanov, Non-Abelian statistics of half-quantum vor-tices in p -wave superconductors , Phys. Rev. Lett. , 268(2001).[5] L. Fu and C. L. Kane, Superconducting proximity effectand Majorana fermions at the surface of a topologicalinsulator , Phys. Rev. Lett. , 096407 (2008).[6] K. T. Law, P. A. Lee, and T. K. Ng,
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FIG. 2. (Color online) (a) The electronic band structure and (b) the corresponding Fermi surface topologies for the FM stateof monolayered CoSb. (d) The PDOS on the spin-up and spin-down species of total, Co 3 d , and Sb 5 p orbitals for the FMstate of monolayered CoSb. The Fermi energies are set to zero. k x k y −6 −4 −2 Energy (meV) DO S (a) (b)0 � /a FIG. 3. (Color online) (a) A schematic plot for the gap nodal structure of non-unitary pairing on the Fermi surface topologies.The green dotted lines denote the Fermi surface topology of monolayered CoSb. The magenta dashed lines denote the zero gapvalue of the order parameter d ( (cid:126)k ) = (1 , − i,
0) sin( k y a ) with the irreducible representation of B u . The vector plot of Cooperpair spin polarization (cid:104) ˆ S (cid:126)k (cid:105) is also shown on the counters of Fermi surface topologies. (b) The DOS as a function of energy forthe non-unitary superconducting state. The parameter of pairing amplitude is set as ∆ = 5 meV. Supplemental Material “Non-unitarysuperconductivity in the monolayer oforthorhombic CoSb”
In this supplementary material, we firstly present thedetailed calculations of the orbital resolved energy bandsof the non-magnetic and ferromagnetic states of monolay-ered orthorhombic CoSb, as shown in Fig. S1- S3. Hereit is interesting to point out that a Dirac like pocket ap-peared at the high symmetric line of Y − G shown in Fig. 2in the main text mainly stems from the contributions ofspin-up components of Sb 5 p orbitals by inspecting theorbital resolved energy bands shown in Fig. S3. Secondly,we also perform the calculations on the energetic proper-ties of the various Co spin ordered orientations for mono- layered orthorhombic CoSb with ferromagnetic orderingstate, as listed in Table S1. The calculations demonstratethe ground state of monolayered CoSb is a ferromagneticorder with the magnetization along the ˆ y axis lying inthe basal plane and the magnetic momentum of 1.83 µ B on Co ion sites. TABLE S1. Energetic properties of the different Co spin con-figurations for monolayered orthorhombic CoSb. Results arethe total energy difference per Co atom for different Co spindirections in the ferromagnetic CoSb layer.CoSb (100) (010) (001)∆ E (meV/Co) 0.13 0.0 0.03 m Co ( µ B ) 1.83 1.83 1.84 G X S Y G -1-0.500.51 E n e r g y ( e V ) E F G X S Y G -1-0.500.51 E n e r g y ( e V ) E F (c) d xy (d) d xz G X S Y G -1-0.500.51 E n e r g y ( e V ) E F (a) d z G X S Y G -1-0.500.51 E n e r g y ( e V ) E F (b) d x y G X S Y G -1-0.500.51 E n e r g y ( e V ) E F (e) d yz FIG. S1. (Color online) The orbitals resolved energy bands projected onto the Co 3 d for the non-magnetic state of monolayeredorthorhombic CoSb. The Fermi energies are set to zero. G X S Y G -1-0.500.51 E n e r g y ( e V ) E F G X S Y G -1-0.500.51 E n e r g y ( e V ) E F G X S Y G -1-0.500.51 E n e r g y ( e V ) E F G X S Y G -1-0.500.51 E n e r g y ( e V ) E F G X S Y G -1-0.500.51 E n e r g y ( e V ) E F G X S Y G -1-0.500.51 E n e r g y ( e V ) E F G X S Y G -1-0.500.51 E n e r g y ( e V ) E F G X S Y G -1-0.500.51 E n e r g y ( e V ) E F G X S Y G -1-0.500.51 E n e r g y ( e V ) E F G X S Y G -1-0.500.51 E n e r g y ( e V ) E F (a) dz2 spin up(b) dx2-y2 spin up(c) dxy spin up(d) dxz spin up(e) dyz spin up (f) dz2 spin down(g) dx2-y2 spin down(h) dxy spin down(i) dxz spin down(j) dyz spin down FIG. S2. (Color online) The spin dependent orbitals resolved energy bands projected by the Co 3 d for the ferromagnetic stateof monolayered orthorhombic CoSb with the magnetization along the ˆ y axis lain in the CoSb plane. The Fermi energies are setto zero. G X S Y G -1-0.500.51 E n e r g y ( e V ) E F G X S Y G -1-0.500.51 E n e r g y ( e V ) E F G X S Y G -1-0.500.51 E n e r g y ( e V ) E F G X S Y G -1-0.500.51 E n e r g y ( e V ) E F G X S Y G -1-0.500.51 E n e r g y ( e V ) E F G X S Y G -1-0.500.51 E n e r g y ( e V ) E F (a) p x spin up(b) p y spin up(c) p z spin up (d) p x spin down(e) p y spin down(f) p z spin down FIG. S3. (Color online) The spin dependent orbitals resolved energy bands projected by the Sb 5 d for the ferromagnetic stateof monolayered orthorhombic CoSb with the magnetization along the ˆ yy