View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by IACS Institutional Repository Indian 1 Phys. 78(8), 809-814 (2004) %■ .5 U P ? The spin-density-wave (SDW) in quaternary borocarbide superconductors K r Bishoyi'^, G C Roiil- and S N Behera^ ‘P G Department of Physics, I’ M College (Autonomous), BaIasore-756 (K)l, Orissa, India ^Condensed Matter Physics Group, (jovernmc’iit Scijpncc College, Chatrapur-761 020, Ori.ssu, India institute of Physics, Sachivalaya Marg, ||lliubaneswar"751 005. Orissa, India h-mail' bishoyi^iopb.rcs.in Abstract : A large family of qualemaiy borocarbidcs RNi.B^C (R » Lu, Y, Ho, Dy) was discovered which have separate phase of anti- lcrr(»magnetism (AFM), superconductivity (SC) and co existent anii-lerromagnctic superconductivity. We propose a spin*dcnsity-wave (SDW) state as a possible mechanism of the anomalours anli-ferromagnciism in the borocarbide superconductors. In this SDW slate, the electron-hole pair amplitude changes its sign in the momentum space The BCS type mean field superconductivity and a mean field SDW in Hubbard model arc considered. The superconducting gap and SDW gap are calculated by Zubarev's technique of equation of motion method and are solved self- coiisi.siently for different model parameters. The co-exislence oi'superconductivity and anli-ferromagnciism for both of the cases ie. 1\ > and T, < /v are explained. Keyword.s : Quaternary borocarbidcs, spin density waves. PACS Nos. : 74.70.Dd, 75.30.Fv 1. Introduction Recently, the large family of quaternary borocarbidcs was discovered, which have separate phases of anti- ll IS well known that the spin density wave (SDW) and ferromagnetism (AFM), SC and coexisting AFM superconductivity (SC) co-exist in a variety ol systems, supcrconductiviy [6-10]. It is possible to study the e.ii, the co-cxistence is well documented in the itinerant interplay of AFM and SC for both the ca.ses, 7; > unii-fciTomagnetic metal Cr and its alloys [1], organic an d ; < Tn. Incommensurate magnetic structures (SDWs) layered superconductors, the Bcchgaard salts (TMTSF)X, 7 with the wave vector ( z 0.55, 0, 0), originating from the with X = PFfi, AsFft and CIO4 [2] and heavy fermion Fermi surface nesting are found for LuNi2B 2C [11-13], systems URu2S i2 and Ui„jcThjcBei3 [3]. It is observed that Y N i B C [12], TbNi B C [14], ErNi B C [10,15,16], URu Si , U N i A l an d U P d A i compounds are SDW 2 2 2 2 2 2 2 2 2 3 2 3 H o N i B C [10,171, GdNi B C [18] and with wave vector super-conductors but with different degrees of magnetic 2 2 2 2 (~ 0.093, 0.093, ) fo r T m N i B C [10]. It is natural to moment localization. All these systems have a low 0 2 2 make an extrapolation that other members of the family dimensional structure and exhibit Fermi surface instabilities may possess the same property. Especially interesting is which drive the SDW transition. One can tentatively add the situation in H0N 12B2C w ith Tn 8.5 K and ^ 8 .0 to these classes of materials, the high-r,. perovskite oxide K [10]. Here, superconductivity tends to be almost re­ superconductors, La -jtSrjfCu a n d YB C\X) ^^s w h ich 2 04 2 0 entrant near 5 K, which is revealed by the critical field have two dimensional structures and inhibit anli- H c2 measurement. l^errornagnetic order [4J, as the probable candidates lor the co-existence of SDW and SC. In these correlated In view of the simultaneous presence of the SDW and SC states in ^these wide class of metals, it is but superconductors, the porbability of co-existence of SDW SC has been demostrated [5] at the mean field level. natural to enquire as to how the properties of each of the Corresponding Author 2 0 0 4 lA C S 810 K C Bishoyi, G C Rout and S N Behera individual phases become modified due to the presence «- S'. -f cj; ) of the other, In Section 2 of this paper, we formulate the it,cr k mean field theory of the coexistent, SDW-SC phase. In Section 3, the single particle Green’s functions and the (2) corresponding gap equations of the SDW and SC order k parameters are obtained by equations of motion method of Zubarev |I9J. The interplay of SDW and SC parameters where are the annihilation (creation) operators further establish the fact that the two states tend to inhibit each other. The SDW and SC coupled gap for the electrons of the nickel atom and €i is the kinetic equations are solved self-consistcntly and the results are energy of the electron with momentum k and spin a The discussed in Section 4. In the concluding Section 5, G and A are the SDW and SC order parameters given by attempts are made to correlate the results of this calculation to those observed by experiments in (3) borocarbide systems. 2. Formalism (4) In case of a correlated system, the SDW state arises due to the repulsive intratomic (Hubbard) Coulomb repulsion with U and V being respectively the repulsive Coulomb (t/) between the electrons, driven by a Fermi surface and attractive interaction strengths. instability. This arises predominantly due to the existence of nested pieces of Fermi surface with electron-hole 3. Calculation of gap equation symmetry given by In order to determine the gap equations, the following four Green's functions are defined in terms of electron = S k , (1) operators as where Q is the SDW wave vector equal to twice the Fermi wave vector in a particular direction and is the G)(A:,£U) = ^ electron enei;gy. This nesting property is a manifestation of the low dimensionality of the system. If nesting is not perfect then the Fermi surface is only partially destroyed G2ik,0))-{ due to the appearance of the SDW gap (2G). This allows for the possibility of occurrence of a superconducting instability provided there is an effective attractive G2(k,0»^{A- interaction between the electrons. A mean field Hamiltonian can be constructed for the SDW state arising G^(k,Q)) = { from the Hubbard Hamiltonian [20]. Similarly we assume A). (5) that the SC state is described by the BCS reduced The coupled Green's functions are evaluated explicitly by Hamiltonian, where the effective interactive interaction using the Hamiltonian given in eq. (2) using Zubarev's between electrons is produced by the suitable exchange equation of motion method [19]. of an excitation which gives rise to the high transition temperature as shown by the authors [21] Behera et al have studied the phonon response of the high temperature (o)-6 * )G i(k,<u)= -^ + GG2{*,<u)+ i4G3(*,a)), 27T superconductors in the co-existence of SC and SDW phase [22]. Recently Rout et al [23] reported a model {q)+ e I, )02{k,a))= GG, {k,0))- AG4 {k,(u), study of borocarbide system describing the co-existence of the SC and AFM in presence of hybridization. The {(i}+Sic)G2{k,ct))=G^{k,(t))-^-AGi{k,(o), mean field Hamiltonian describing the coexistent SDW- SC state is given by (o)+e*)G4(ifc,a))= ~GG2ik,(o)-AG2{k,(o). The spin-density-wave (SDW) in quaternary borocarbide superconductors 811 In order to determine the gap eqs. (3) and (4) for the the highest frequency of the boson whose exchange SdW and SC states respectively, it is necessary to evaluate brings about the attractive interaction between the the Green's functions. The above coupled equations are electrons. All the quantities entering eqs. (9) and (10) are solved to get these Green’s functions which are given made dimensionless after dividing those by cop. Thus the below. dimensionless order parameters are redefined as AlcOp = Z and G/(Op = g, the variables as EiJcop = a\ kuTIcop = t, WIcop^W, Giik,(o)=‘ 2;r|D(ft>)| 4. Results and discussion there is a family of borocarbide superconductors where 2n\D(0>f the N^el temperature 7’yv is greater than the super­ conducting transition temperature T,. There is also another A((o‘' ~ g u ) ^mily of this system where To discuss our Ch{k,(0} = ~2^Di(0)\ tfecoretical model for these two families of borocarbide system, we have selected two sets of coupling constants, one for the SC {Le, /li) and another for magnetism {i.e. — 2GA{o)-^Ej^ ) C nik,(0) = (7) ^q). The interplay between the SC and the SDW states is 2;rlD(w)| discussed below on the basis of our model. where (A) Case < Tc : The temperature dependences of the SC order parameter z{t) for the SC coupling constant = 0.238605 and the E{L ’ ^2k - +^2 • A,2 SDW order parameter g(t) for the SDW coupling constant /I2 = 0.130362 are shown in Figure 1. <’U = 7^4 +A^ • ^2k - “ A'A ’ 0.03 (8) 0.025 The expressions for the order parameters G and A defined by eqs. (3) and (4), are calculated from Green's functions 0.02 G2((o) and Gj(<y) respectively. After simplification, those are written as 0.015 . A, f+"o , . PE2k 0.01 4 = — 1 de,,A -^ tan h f + -^ ta n h 4 * L^u l, 2 J E2k 1 0.005 (9) and 0 0.005 0.01 0.015 0.02 Figure 1. The plot of z v'S / for fixed values of ^ = 0. * 0.238605, = 0 -tanh ■ A .