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Indian 1 Phys. 78(8), 809-814 (2004)

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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,

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(

. 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 . tanh and the plot of ^ v.s / for fixed values of z = 0, vti = 0, Aa = 0.130362. o . i £ | r f s , o 2 'Ik ^2t This choice of coupling constants corresponds to the (10) systems with > Tc- The two individual parameters ^(0 and g(t) in absence of each other exhibit usual BCS like where = A^(0) and /ii = N(0)U arc the dimensionless mean-field temperature dependence. The SC transition SC and SDW coupling constants respectively with A^(0) temperature tc z 0.017 and the SC gap z{t = 0) n 0.03 being the density of states at the Fermi level. Further which corresponds to lA iO W c z 3.52. Similarly the *norc, the cut off energy Es of the SDW state is defined SDW system gives a N6el temperature tN z 0.011 and as the energy above the Fermi energy which destorys the g(t = 0) - 0.019. Since the long range orders are present nesting of the Fermi surface completely. Similarly (Op Is 812 K C Bishoyi, G C Rout and S N Behera

simultaneously i.e. for the case of co-cxistence of SC and SDW, the eqs. (9) and (10) are solved self- cosnsistently for the same values of coupling constant as shown in Figure 2.

Figure 3. The self-coiisistcnt plot of c j? vs t for fixed value of = 0.130363 and for different values of A, = 0.238605, 0 237355,0.236105.

Figure 2. The self-consislenl plot of i j? r for fixed values of = 0.238605, /l2 = 0 130362.

In the co-existent phase, both the SC and SDW states are suppressed considerably. However, even though the coupling constants are same, the SDW transition temperature In is greatly enhanced as if the roles of the SC and the SDW are reversed with > t, as shown in Figure 2. This situation depicts a similar situation > f<. observed in boracarbide system DyNi2B2C with Tn = 10 K and Tc = 6 K. The SC order parameter shows a mean field behaviour with the value l^hgT,- -2.66 which is Hgure 4. The self-consistent plot of z,gvsi for fixed value of At = 0.23800^ slightly less than the universal BCS constant. The SDW and for different values of = 0.133862,0.132862,0.131862, 0.130.362 order parameter at the outset increa.ses as the temperature is lowered below attains a maximum around tfJ2 and The SDW coupling Aj enhances the SC gap parameter then starts decreasing at the onset of SC i.e. in the through out the temperature range. In the process, SC transition temperature t^ is enhanced. It shows as if the temperature range t < r,.. SDW coupling acts as the SC coupling. On the other The effect of SC coupling constant on SC order hand, the SDW coupling constant enhances the SDW parameter z(t) and SDW order parameter g(t) is shown in gap parameter g(t) in the temperature range where the Figure 3. The increase in SC coupling constant At co-existent pha.se exists. However, the SDW gap remains decreases the SC order parameter and pushes the transition unaltered for the temperature t > tc, keeping ts constant. temperature tc to lower temperatures. However, the (B) Case Tn > Tc : parameter z{t = 0) remains unaltered with change in Another set of coupling constants At = 0.188605 and A: value of This contradicts the BCS type behaviour. It At. = 0.130362 are selected to give the transition temperatures is obtained that the SDW gap g(f) increases throughout Ts > Tc in their individual phases [see Figure 5]. the temperature range with increase of At. But the N6el When the two phases co-exist and interplay each temperature (tu) is enhanced in the co-existent phase. other, the self-consistent solutions of the eqs. (9) and The effect of SDW coupling constant Ai on SC gap (10) for the above set of parameters give temperature parameter z{t) and the SDW gap parameter z(r) in co­ dependences of the SC gap z(t) and SDW gap g(0 existent phase is shown in Figure 4. shown in Figure 6. The spin-density-wave (SOW) in quaternary borocarbide superconductors 813

The effect of SDW coupling A\ on the SC gap zU) and SDW gap g{t) is depicted in Figure 8. 0 01

Figure 5. I'he plot of g vs t for fixed values of z = 0, = 0, A; = 0 130'^62 ,uui the pl

Figure 1, flic sclf-consistcnl plot of g, z vs t for fixed values of Aj =: 0.130362 wul for different values of A, =0.191105, 0.188605, 0.187355. The SDW coupling suppresses the SDW gap through out the temperature range and also suppresses the N6cl temperature. The increase of SDW coupling Xi enhances the SC gap z(t) through out the temperature range and pushes the SC transition temperature to higher temperatures. It is clear from the Figures 7 and 8 that

Figure 6. llic self-consistent plot of .v. z va / for fixed values of Aj = 0.188605 and/t. rrO L10362. This corresponds to a system with This situation is realized in the TmNi2B2C system with 7]/!]^ -'7.33 which is observed experimentally. Both the gap parameters exhibit mean-field lemparature dependence. Ihc SC order parameter at the outset increases as the temperature is lowered below attains a maximum at around tJ2 and there after starts decreasing towords the low temperatures. Figure 8. The self-consistent plot of g, z vs / for fixed values of .i| = 0.188605 and for different values of = 0.130362, 0.130062,0.129762. The effect of SC coupling constant A\ on the temperature dependence of both the gap parameters is the roles of the SC coupling Ai and SDW coupling A-z shown in Figure 7. are reversed. In other words, the SC coupling Ai acts as the SDW coupling while the SDW coupling Az acts as An increase in the SC coupling constant A\ enhances the SC coupling. both the SDW gap and N6el temperature where as it enhances the SC gap parameter z(t) in the temperature 5. Conclusion range r < in the co-existent phase. However the gap *^^mains unaltered for the temperature t > keeping We have considered a model Hamiltonian consisting of constant. BCS type mean field term for superconductivity and the 814 K C Bishoyi, G C Rout and S N Behera mean field spin density wave term for the conduction [5] M Inui, S Doniach, P J Hirschfcid and A E Ruckenstein Phys. Re\ electrons of the borocarbide system. The Hamiltonian is B3 2320 (1988) solved for the SC gap parameter and the SDW gap [6] 1 K Yanson Symmetry and Pairing in Superconductors (eds) M parameter by using Zubarev type Green’s function Ausloos and S Kruchinin (Dordecht; Kluwer) 271 (1999) technique. These two gap parameters are solved self- [7] H Schm idt and H F Braun Quaternary Borocarbides, SupeKonductors and Hg Based High-Tc superconductors (cd) A V consistently by adjusting the two coupling constants i.e. NarlikaifNew York: Nova) ch3 p47 (1998). SC coupling y^l and SDW coupling A: for the situation (8| G Hilscher and H Michor Micro Structural Studies of High 7, T( > Tn and T,. < appropriate for the general Superconductors and more on Quaternary Borocarbides (ed) A V borocarbide systems RNi;BjC (R= Lu, Ho, Tm, Dy.... ). Narlikar (New York: Nova) 241 (1999) The numerical .solutions show that both the SC and [9] P C Canfield, S L Bud’ko, B K Cho, W P Beyermann and A Yatskar SDW gap parameters exhibit mean field temperature J. Alloys Compounds 250 596 (1997) dependence. However one gap parameter shows [ I0| J W Lynn. S Skanthakumar, Q Huang. S K Sinha, Z. Hossain, L C suppression at low temperatures where the other phase Gupta. R Nagarajan and C Godart Phys. Rev, B55 6584 (1997) co-exists with it and the vice-versa. Moreover, the SC [11] C Stassis, M Bullock. J Zaresiky. P Canfield, Z Honda, G Shiranc coupling constant acts as the SDW coupling constant and and S M Shapiro Phys. Rev. BS5 8678 (1997) the vice-versa. Similar observations are indicated in the [12] M Bullock, J Zarestky, C Stassis, A Goldman, P Canfield, Z Honda. communication [23] consisting of a model Hamiltonian G Shirane and S M Shapiro Phys. Rev. B57 7916 (1998) containing superconductivity and anti-ferromagnetism 113] SBDugdic,M A Alam, 1 Wilkinson,RJ Hughes, 1RFisher,PC arising due to the same conduction electrons and a weak Canfield, T Jarlborg and G Shanti Phys. Rev. Lett. 83 4824 (1999) hybridization between the conduction electrons of the [14] P Dervenagas, J Zarestky, C Stassis. A 1 Goldman, P C Canfield and nickel atom and the /-electrons of the rare-earth atom. B K Cho Phys. Rev B53 8506 (1996) Acknowledgments 115] J Zarestky, C Stassis. A I Goldman, P C Canfield, P Dervenaga.s, 0 K Cho and D C Johnston Phy.s. Rev. B5I 678 (1995) One of the authors (GCR) would like to acknowledge [16] S K Sinha, J W Lynn, T E Gregercit, Z Hossain, L C Gupta, R the University Grants Commission for the financial Nagarajan and C Godart Phys. Res'. BSl 681 (1995) assistance. The authors (KCB and GCR) gratefully [17] T Vogt, A Goldman, B Sternlieb and C Stassis Phys. Rev. Lett 75 acknowledge the research facilities offered by the Institute 2628(1995) of Physics, Bhubaneswar, Orissa, India during their short [18] C Dctlefs, A 1 Goldman, C Stassis, P Canfield, B K Cho, J P Hill slay. and D Gibbs Phys. Rev. 53 6355 (1996)

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