Two scenarios for superconductivity in CeRh_2As_2
TTwo scenarios for superconductivity in CeRh As David M¨ockli and Aline Ramires Instituto de F´ısica, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, RS, Brazil Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland (Dated: February 19, 2021)CeRh As , a non-symmorphic heavy fermion material, was recently reported to host a remarkablephase diagram with two superconducting phases. In this material, the two inequivalent Ce sites perunit cell, related by inversion symmetry, introduce a sublattice structure corresponding to an extrainternal degree of freedom. Here we propose a classification of the possible superconducting states inCeRh As from the two Ce-sites perspective. Based on the superconducting fitness analysis and thequasiclassical Eilenberger equations, we discuss two limits: Rashba spin-orbit coupling and inter-layer hopping dominated normal state. In both limits, we are able find two scenarios that generatephase diagrams in qualitative agreement with experiments: i) intra-sublattice pairing with an even-odd transition under magnetic field, and ii) inter-sublattice pairing with an odd-odd transition undermagnetic field. The heavy fermion CeRh As was recently discov-ered to host remarkable properties in the superconduct-ing state [1]. Under c-axis magnetic field, a transi-tion between a low-field and a high-field superconductingphase is observed by measurements of magnetization andmagnetostriction. The presence of two superconductingphases is unusual, and has only been reported in othertwo stoichiometric heavy fermion materials: UPt [2–4]and UTe [5]. The high effective electronic mass inferredfrom the low temperature value of the specific heat coef-ficient indicates that CeRh As is in the heavy fermionregime, but power-law temperature dependence of thespecific heat below 4K suggests the proximity to a quan-tum critical point, whose fluctuations can be playing animportant role for pairing [6–9]. Furthermore, the high-field phase has an upper critical field of 14T, exceptionalfor a material with a superconducting critical tempera-ture ( T c ) of 0.26K, suggesting that CeRh As hosts anunconventional triplet superconducting state.Theory developed in the context of layered super-conductors with local inversion symmetry breaking ac-counts for the qualitative features of the temperature ver-sus magnetic field phase diagram of CeRh As [10; 11].More recently, these studies were supplemented by de-tailed investigations including also orbital depairing ef-fects [12; 13]. Intra-layer singlet pairing (referred hereas BCS) is assumed to be the stable superconductingstate within each layer. Once a magnetic field is appliedperpendicular to the layers, a pair-density wave (PDW)state, with a sign change of the order parameter betweenlayers, is favoured under the requirement that Rashbaspin orbit coupling (SOC) is comparable to inter-layerhopping (ILH) amplitudes.Nevertheless, the phenomenology of CeRh As indi-cates that other scenarios might be possible. In partic-ular, the small anisotropy of the effective mass, inferredfrom the slope of the upper critical field around T c , in-dicates that the system is rather three-dimensional [1].This is in agreement with recent first-principles calcula- Ce(1)Ce(2)As/RhRh/AsAs/RhRh/AsCe(1) Ce(1)Ce(2) Intra-SLSingletIntra-SLSingletInter-SLTriplet a) b) t u t d t p t p t p FIG. 1. Schematic depiction of CeRh As . The spherescorrespond to the two inequivalent Ce atoms and the rectan-gular structures to the two types of Rh/As layers. The bigarrows on the left correspond to the effective electrical fieldgenerating a staggered Rashba-SOC. The dotted lines corre-spond to in-plane intra-SL hopping with amplitude t p (samefor all layers), and the dashed lines correspond to inter-SLhopping, with amplitude t u or t d , depending if the hopping isto a neighbor in the layer above or below. tions [14], and with other 122-materials in this family. Asan example, CeCu Si crystalizes in the ThCr Si -typestructure (the centrosymmetric analog of the CaBa Ge -type structure of CeRh As ), displays a 3-dimensionalspin density wave state supported by Fermi surface nest-ing, and hosts superconductivity around the pressure in-duced quantum critical point [15; 16].Motivated by these facts, here we analyse the possi-ble unconventional superconducting states that can behosted by CeRh As . We start with focus on the mainingredient, the Ce ions, and propose a microscopic modelfor the normal state Hamiltonian based on the spin and a r X i v : . [ c ond - m a t . s up r- c on ] F e b sublattice (SL) degrees of freedom (DOF). This modelallows us to discuss two limits: i) two-dimensional, dom-inated by Rashba-SOC, and ii) three-dimensional, dom-inated by ILH. We then classify all possible types ofCooper pairs that can be formed in this material. Weperform the superconducting fitness analysis and discussthe relative stability of different superconducting statesin both limits. Finally, we obtain temperature versusmagnetic field phase diagrams from the quasiclassicalEilenberger equations, based on which we can proposetwo scenarios for superconductivity in CeRh As .CeRh As has a CaBa Ge -type structure with thenon-symmorphic centrosymmetric space group P4/nmm (No. 129). Given the heavy fermion nature of the elec-tronic structure around the Fermi energy inferred by re-cent experiments, we start modelling the electronic DOFfrom the Ce sites perspective. The Ce atoms are locatedin between Rh-As layers which appear intercalated in twoflavors: with Rh atoms tetrahedrally coordinated by As,or vice-versa, as schematically shown in Fig. 1. The in-tercalation of two types of Rh-As layers generates twoinequivalent Ce sites with C v point group symmetry.Importantly, the Ce sites are not centers of inversion.The point group C v can be generated by C z , a rota-tion along the z-axis by π/
2, and σ xz , a mirror reflectionalong the xz-plane. The complete space group includesinversion, which can be made a symmetry at the Ce sitesif composed with half-integer translation vectors. Wedefine i / as inversion composed with a translation by( a/ , a/ , c/ a and c are the unit cell dimensions in the planeand along the z-axis, respectively). These three opera-tions generate the complete space group. Here we notethat this space group is isomorphic to D h up to integerlattice translations [17]. Given these generators, TableI summarizes the properties of the irreducible represen-tations. For simplicity, here we focus on the symmetryanalysis around the Γ point.We start with the most general non-interacting nor-mal state Hamiltonian considering Wannier functions lo-calised at the Ce atoms accounting for a two SL structure: H = (cid:88) k ˆΨ † k ˆ H ( k ) ˆΨ k , (1)where ˆΨ † = ( c † ↑ , c † ↓ , c † ↑ , c † ↓ ) encodes the two SLs (1 , ↑ , ↓ ) DOF. The 4 × H ( k ) can beparametrized as:ˆ H ( k ) = (cid:88) a,b h ab ( k )ˆ τ a ⊗ ˆ σ b , (2)where ˆ τ a and ˆ σ b are Pauli matrices, { a, b } = { , , } , orthe two-dimensional identity matrix, { a, b } = { } , corre-sponding to the SL and spin DOF, respectively. In pres-ence of inversion (implemented as τ ⊗ σ accompaniedby k → − k ) and time-reversal symmetry (implemented Irrep C z σ xz i / Basis A g +1 +1 +1 x + y , z A g +1 -1 +1 xy ( x − y ) B g -1 +1 +1 x − y B g -1 -1 +1 xyE g { xz, yz } A u +1 +1 -1 zA u +1 -1 -1 xyz ( x − y ) B u -1 +1 -1 z ( x − y ) B u -1 -1 -1 xyzE u { x, y } TABLE I. Irreducible representations (irrep) at the Γ-pointassociated with the three symmetry operations at the Ce sites.The last column shows examples of polynomials in each irrep.( a, b ) Irrep k Process Parameter(0 , A g Even Intra-SL hopping t p (1 , A g Even Inter-SL hopping t u + t d (2 , A u Odd Inter-SL hopping t u − t d (3 , E u Odd Rashba-SOC α (3 , , A u Odd Ising-SOC λ TABLE II. Symmetry allowed ( a, b ) terms in the normal stateHamiltonian, as given in Eq. 2. The table highlights the irrep,the even or odd k dependence, the associated physical processand dominant parameter for each term. as iτ ⊗ σ , accompanied by complex conjugation and k → − k ), only the ( a, b ) pairs summarized in Table IIare symmetry allowed.We now associate each term with specific physi-cal processes. (0 ,
0) concerns intra-SL hopping, with h ( k ) = 2 t p [cos( k x a ) + cos( k y a )], dominated by intra-layer hopping to nearest neighbours with amplitude t p . (1 ,
0) and (2 ,
0) stem from inter-SL hoppingwhich in this case is necessarily out-of-plane, with h ( k ) = 4( t u + t d ) cos( k x a/
2) cos( k y a/
2) cos( k z c/
2) and h ( k ) = − t u − t d ) cos( k x a/
2) cos( k y a/
2) sin( k z c/ t u and t d , the hoppingamplitudes to the nearest neighbors in the layer aboveand below, which are inevitably distinct due to inver-sion symmetry breaking. (3 ,
1) and (3 ,
2) are concernedwith intra-SL staggered Rashba-type SOC, h ( k ) = − α sin( k y a ) and h ( k ) = α sin( k x a ), parametrized by α . Finally, (3 ,
3) is an Ising-type SOC, h ( k ) = λ sin( k x a ) sin( k y a ) sin( k z c )[cos( k x a ) − cos( k y a )], associ-ated with hopping to neighbors of the same SL in thenext nearest layer, parametrized by λ . This Hamiltonianis the same as the one proposed in Ref. [1].The superconducting order parameter can be writtenin a similar fashion:ˆ∆( k ) = (cid:88) a,b d ab ( k )ˆ τ a ⊗ ˆ σ b ( iσ ) . (3) Irrep [ a, b ] Spin SL k Parity A g [0 ,
0] S Intra E E[1 ,
0] S Inter E E A g [0 ,
3] T Intra O O[1 ,
3] T Inter O O A u [3 ,
0] S Intra E O[2 ,
0] S Inter O E A u [2 ,
3] T Inter E O[3 ,
3] T Intra O E E g { [0 , , [0 , } T Intra O O { [1 , , [1 , } T Inter O O E u { [2 , , [2 . } T Inter E O { [3 , , [3 , } T Intra O ETABLE III. Symmetry classification of the [ a, b ] matrices as-sociated with all order parameters, defined in Eq. 3, organizedby irreducible representations (irreps) around the Γ point.Here E/O stands for even/odd and S/T for singlet/triplet.The irrep associated with the complete order parameter is ob-tained by taking the product with the irrep of d ab ( k ), whichis always nontrivial for the odd- k order parameters. In presence of inversion symmetry, we can distinguishbetween even or odd in k and even or odd parity gapfunctions. Table III lists all the order parameters, theirnature in terms of the microscopic DOFs, and their sym-metry properties. The different brackets distinguish thenormal state Hamiltonian parameters, ( a, b ), from thesuperconducting order parameters, [ a, b ].Within the standard weak-coupling formalism, the su-perconducting fitness analysis allows us to discuss the rel-ative stability of superconducting states based on proper-ties of the normal electronic state [18; 19]. In particular,it was shown that the larger the average over the FSsof Tr [ ˆ F A ( k ) † ˆ F A ( k )], the more robust the superconduct-ing instability and the higher critical temperature. Here,ˆ F A ( k ) = ˜ H ( k ) ˜∆( k )+ ˜∆( k ) ˜ H ∗ ( − k ) is the superconduct-ing fitness matrix, written in terms of the normalizednormal state Hamiltonian ˜ H ( k ) = [ ˆ H ( k ) − h ( k ) τ ⊗ σ ] / | h ( k ) | , where | h ( k ) | = (cid:113)(cid:80) ( a,b ) (cid:54) =(0 , | h ab ( k ) | , andthe normalized gap matrix ˜∆( k ) with (cid:104) ˜∆( k ) (cid:105) F S = 1.The superconducting fitness analysis for CeRh As issummarized in Table IV. We start discussing the scenariowith dominant ILH, such that we assume | t p | > | t u + t d | > | t u − t d | > | α | > | λ | . From Table IV, we find that Cooperpairs with a = 0 are the most stable since they have thecontribution from the two largest terms in the normalstate Hamiltonian, (1 ,
0) and (2 , a = 1, supported only by (1 ,
0) hopping,controlled by ( t u + t d ) . Pairs with a = 2 are less sta-ble since these are supported only by (2 , t u − t d ) , while pairs with a = 3 are not stabilized by anyhopping term in the normal state Hamiltonian. Amongthe intra-SL pairs ( a = 0 , (1,0) (2,0) (3,1) (3,2) (3,3) ILH SOC[0 ,
0] 1 1 1 1 1 2( t u + t d ) 2 α + λ [0 ,
1] 1 1 1 0 0 α [0 ,
2] 1 1 0 1 0[0 ,
3] 1 1 0 0 1 λ [1 ,
0] 1 0 0 0 0 ( t u + t d ) ,
1] 1 0 0 1 1 α + λ [1 ,
2] 1 0 1 0 1[1 ,
3] 1 0 1 1 0 2 α [2 ,
0] 0 1 0 0 0 ( t u − t d ) ,
1] 0 1 0 1 1 α + λ [2 ,
2] 0 1 1 0 1[2 ,
3] 0 1 1 1 0 2 α [3 ,
0] 0 0 1 1 1 0 2 α + λ [3 ,
1] 0 0 1 0 0 α [3 ,
2] 0 0 0 1 0[3 ,
3] 0 0 0 0 1 λ TABLE IV. Superconducting fitness analysis for all or-der parameters. Each line corresponds to an order pa-rameter labelled by [ a, b ], as in Eq. 3. The nu-merical entries correspond to Tr [ ˆ F A ( k , s ) † ˆ F A ( k , s )] =4 (cid:80) cd (table entry) | h cd ( k , s ) | / | h ( k , s ) | , for each term ( c, d )in the normal-state Hamiltonian. The last two columns sum-marize the effects of ILH and SOC dropping the accompany-ing momentum dependence of the respective functions. Among the inter-SL pairs ( a = 1 , ,
0] and [1 ,
3] as the two mostrobust superconducting states for the ILH dominated sce-nario. Moving now to the SOC dominated scenario, weassume | t p | > | α | > | t u + t d | > | t u − t d | > | λ | . Forthe intra-SL Cooper pairs ( a = 0 , b = 0), while for inter-SLpairs SOC stabilizes triplet states with d-vector along thez-axis ( b = 3). Among the intra-SL pairs, a finite ILHstabilizes [0 , , , t u = t d and λ = 0, the [1 ,
3] order parameter candi-date, an odd parity SL-symmetric spin triplet (SLS-ST),would have the same transition temperature. These aregood candidates for the low-field superconducting phaseof CeRh As .In presence of a magnetic field along the z-axis, theproposed PDW state, an odd parity intra-SL spin singlet,here captured by [3 , ,
0) and (2 ,
0) do not contribute to α / | t u + t d | = α / | t u + t d | = / FIG. 2. Identical phase diagrams for the even-odd intra-SL[0 , → [3 ,
0] and odd-odd inter-SL [1 , → [2 ,
3] scenarioswith | t u + t d | = T c . The blue curves correspond to the SOC-dominated regime, while the green curves correspond to theILH-dominated regime. The dotted (dashed) lines are theextensions of the low (high) field solutions. The inset dis-plays schematic two-dimensional cuts of the FSs. Left: SOC-dominated regime. The lighter (darker) color indicates FSsstemming from Ce(1) [Ce(2)] layers, and the arrows indicatethe helicity. Turning on ILH connects FSs with same helicityand complementary Ce-content (dotted lines). Right: ILH-dominated regime. The lighter (darker) color indicates spinup (down). Turning on SOC connects FSs with opposite spinand bonding state (dotted lines). its stabilisation. Interestingly, from the analysis above,within the assumption of t u = t d and λ = 0, the orderparameter [2 , | t u − t d | > λ , the SLA-ST is more robustthan the PDW state. Order parameters with a similarnature, antisymmetric in an internal DOF, were recentlyproposed in multiple contexts [20–23].With these facts at hand, we now examine the be-haviour under magnetic field of the four order param-eters identified above. The hierarchy of energy scalesmotivates the writing of a quasiclassical theory assuming | t p | (cid:29) {| t u + t d | , | t u − t d | , α, λ } . For simplicity, here wetake t u = t d and λ = 0 and study the interplay of themagnetic field B , Rashba SOC α and ILH | t u + t d | . Weextend the linearized Eilenberger equations of Ref. [24] to include | t u + t d | , which allows us to obtain the transitionlines of the best low and high field phase candidates. Foreach pairing irrep-channel, we associate a superconduct-ing critical temperature T irrep that is defined in the ab-sence of magnetic field and SOC. For simplicity, we con-sider the same critical temperatures for all fitness favoredchannels by setting T A g = T A g = T A u = T A u = T c .We find two promising scenarios that display the samephase diagram: (i) inter-SL scenario, with a transitionfrom a low field odd SLS-ST state [1 ,
3] to a high fieldodd SLA-ST state [2 , ,
0] to a high fieldodd PDW state [3 , As are likely to be inthe dirty limit [1], in which case a k -independent orderparameter in the microscopic basis would be robust [24–27]. This suggests the intra-SL scenario for the low fieldphase, since in the inter-SL d ( k ) is odd in momentum.If cleaner samples become available and display a sig-nificantly enhanced T c , then the inter-SL scenario wouldremain a good candidate. Furthermore, the presence andlocation of nodes in the superconducting gap can give usimportant information since the inter-SL involves odd-parity states which necessarily display line nodes in thesuperconducting gap at k z = 0.In this work, we do not discuss the pairing mechanism.For a realistic discussion, details of the FS in the heavyfermion regime are needed in order to investigate possi-ble spin-fluctuation mechanisms. Also, the clarificationof the nature of the hidden order observed at 0.4K andits association with multipolar order brings an interestingpossibility for exotic pairing from multipolar interactions[28–30]. Moreover, the presence of quantum critical be-havior evidenced by the temperature dependence of thespecific heat at low temperatures suggests a scenario sim-ilar to β -YbAlB , for which a careful description of thecrystal electric field states was key to understand its phe-nomenology [31–34].In summary, we have analyzed all possible supercon-ducting order parameters for CeRh As within the Ce-sites perspective. We find temperature versus magneticfield phase diagrams in qualitative agreement with exper-iments for both SOC and ILH dominated normal states.We have identified two possible scenarios for the two su-perconducting phases observed in CeRh As : i) even-oddintra-SL, associated with the previously proposed PDWscenario, and ii) new odd-odd inter-SL scenario. 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