A study of the dependence of the heat capacity Cp(alpha) on field angle in
LuNi2B2C reveals an anomalous disorder effect. For pure samples, Cp(alpha)
exhibits a fourfold variation as the field H < Hc2 is rotated in the [001]
plane, with minima along  (alpha = 0). A slightly disordered sample,
however, develops anomalous secondary minima along  for H > 1 T, leading
to an 8-fold pattern. The anomalous pattern is discussed in terms of coexisting
superconducting gap anisotropy and non-local effects.Comment: 5 pages, 4 figure

B. H. Brandow

E. D. Bauer

Elbert E. M. Chia

Eun Mi Choi

G. E. Volovik

Heon Jung Kim

I. Vekhter

J. D. Thompson

J. F. Annett

M. B. Salamon

M. Nohara

P. C. Canfield

Sung-Ik Lee

Tuson Park

English

arXiv

A study of the dependence of the heat capacity C-p(alpha) on the field angle in LuNi2B2C reveals an anomalous disorder effect. For pure samples, C-p(alpha) exhibits a fourfold variation as the field H (alpha=0). A slightly disordered sample, however, develops anomalous secondary minima along  for mu(0)H>1 T, leading to an eightfold pattern at 2 K and 1.5 T. The anomalous pattern is discussed in terms of coexisting superconducting gap anisotropy and nonlocal effects.This article is published as Park, Tuson, Elbert EM Chia, M. B. Salamon, E. D. Bauer, I. Vekhter, J. D. Thompson, Eun Mi Choi, Heon Jung Kim, Sung-Ik Lee, and P. C. Canfield. "Evidence for the Coexistence of an Anisotropic Superconducting Gap and Nonlocal Effects in the Nonmagnetic Superconductor LuNi 2 B 2 C." Physical Review Letters 92, no. 23 (2004): 237002.
DOI: 10.1103/PhysRevLett.92.237002.
Copyright 2004 American Physical Society.
Posted with permission

Park, Tuson

Chia, Elbert E. M.

Salamon, M. B.

Bauer, E. D.

Vekhter, I.

Thompson, J. D.

Choi, Eun Mi

Kim, Heon Jung

Lee, Sung-Ik

Canfield, Paul

Digital Repository @ Iowa State University (ISU)

P H Y S I C A L R E V I E W L E T T E R S week ending11 JUNE 2004VOLUME 92, NUMBER 23Evidence for the Coexistence of an Anisotropic Superconducting Gap and Nonlocal Effectsin the Nonmagnetic Superconductor LuNi2B2CTuson Park,1 Elbert E. M. Chia,2 M. B. Salamon,2 E. D. Bauer,1 I. Vekhter,1,* J. D. Thompson,1 Eun Mi Choi,3Heon Jung Kim,3 Sung-Ik Lee,3 and P. C. Canfield41Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA2Department of Physics and Material Research Laboratory, University of Illinois at Urbana-Champaign, Illinois 61801, USA3National Creative Research Initiative Center for Superconductivity and Department of Physics,Pohang University of Science and Technology, Pohang 790-784, Republic of Korea4Ames Laboratory, Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA(Received 17 October 2003; published 10 June 2004)237002-1A study of the dependence of the heat capacity Cp on the field angle in LuNi2B2C reveals ananomalous disorder effect. For pure samples, Cp exhibits a fourfold variation as the field H <Hc2 isrotated in the 001 plane, with minima along h100i (  0). A slightly disordered sample, however,develops anomalous secondary minima along h110i for 0H > 1 T, leading to an eightfold pattern at2 K and 1.5 T. The anomalous pattern is discussed in terms of coexisting superconducting gapanisotropy and nonlocal effects.DOI: 10.1103/PhysRevLett.92.237002 PACS numbers: 74.25.Bt, 74.20.Rp, 74.25.Op, 74.62.Dhthe upper critical field Hc2 [13] and magnetization angle. Three samples of LuNi2B2C with different Tc’s,Exotic superconductors, defined as those that followthe Uemura relation [1] between the superconductingtransition temperature Tc and the magnetic penetrationdepth , Tc / 	2, have physical properties that differfrom conventional (BCS) superconductors. High-Tc cup-rates, bismuthates, Chevrel phases, organic, and heavy-fermion superconductors were suggested to constitutethis class. For most of the members, the superconductinggap function is highly anisotropic or has gap zeros on theFermi surface [2]. The rare-earth nickel borocarbidesRNi2B2C (R  Y, Lu, Tm, Er, Ho, and Dy) remain achallenge because they share many features in commonwith the exotic superconductors but, similar to elementalBCS superconductors, fall below the Uemura trend [3].Unlike BCS superconductors, which exhibit expo-nential temperature dependence in density-of-states(DOS)-dependent quantities below Tc, the borocarbideshave power-law temperature dependences in their low-temperature specific heat [4], NMR 1=T1 [5], and thermalconductivity [6], indicating that electronic excita-tions persist even well below Tc. Recently, compellingevidence for the presence of nodes along h100i directionshas been reported both from field-angle thermal conduc-tivity [7] and field-angle heat capacity measurements [8]of YNi2B2C.Another interesting feature of the borocarbides is thatthe flux line lattice (FLL) undergoes a field-driven tran-sition from hexagonal to square with increasing magneticfield H k 001 [9–11]. Kogan et al. successfully incorpo-rated nonlocal corrections to the London model andFermi surface anisotropy in these materials [12] to de-scribe the transition within a BCS scheme, i.e., an iso-tropic s-wave gap with electron-phonon coupling.Nonlocal effects were also used to explain, without re-sorting to any exotic order parameter, the modulation of0031-9007=04=92(23)=237002(4)$22.50M [14] with the angle  between the magnetic fieldand the crystal axes. The two seemingly irreconcilableviewpoints have only increased the confusion about thenature of the order parameter of the borocarbides.Recently, Nakai et al. considered the coexistence of ananisotropic gap and nonlocal effects in the borocarbides[15] and thereby explained the reentrant FLL transforma-tion [16] in terms of the interplay between the two effects.In this Letter, we present further evidence for the coex-istence of the gap anisotropy and nonlocal effects. Inclean systems, the gap anisotropy effects dominate whileboth effects are comparable in less clean samples.Single crystals of LuNi2B2C were grown in a Ni2B fluxas described elsewhere [17]. Samples A and C were notannealed while sample N was annealed at T  1000 Cfor 100 h under high vacuum. Typical sample dimensionswere 1 1 0:1 mm3. The crystal axes of the samplewere determined by two independent methods, x-ray andupper critical field measurements as a function of mag-netic field angle [13], which were consistent with eachother. The Tc’s, determined by the point where the steep-est drop occurs in the resistive superconducting transi-tion, are 15.5, 15.9, and 16.1 K for samples A, C, and N,respectively (not shown). The resistivity at Tc is 2.34 and1:44 cm for samples A and sample N, correspondingto mean-free paths of 144.5 and 234 A, respectively.Assuming that 16.1 K is the transition temperature for apure sample, sample A with Tc of 15.5 K is equivalent to0:8% of Co doping on the Ni site; i.e., LuNi1	xCox2B2Cwith x  0:008 [18]. The disorder in sample A may beassociated with defects which can be removed by judi-cious postgrowth annealing [19].Using an ac technique [8], we have measured the low-temperature specific heat of nonmagnetic LuNi2B2C as afunction of magnetic field intensity and magnetic field 2004 The American Physical Society 237002-1P H Y S I C A L R E V I E W L E T T E R S week ending11 JUNE 2004VOLUME 92, NUMBER 23labeled A, C, andN, were studied and revealed an anoma-lous disorder effect. The heat capacity of the samples withhigher Tc’s shows a fourfold pattern as a function ofmagnetic field angle, confirming the result from the othernonmagnetic borocarbide YNi2B2C that the supercon-ducting gap is highly anisotropic with nodes alongh100i [8]. In contrast, the heat capacity of the disorderedsample with the lowest Tc (sample A) shows a dramaticchange in the field-angle heat capacity above 0.8 T at 2 K:The maxima along h110i split, giving rise to minimaseparated by =4 (eightfold pattern) and, further, Cpdeviates from a square-root field dependence at thesame field, labeled Hs1. The eightfold pattern and thedeviation from H1=2 dependence in sample A are consis-tent with the coexistence of an anisotropic gap and non-local effects.Figure 1(a) shows the magnetic field dependence ofthe heat capacity of sample C at 2.5 K with increas-ing (circles) and decreasing (crosses) magnetic field.The dashed line represents the least-square fit of C0bH	H0, where C0 is zero-field heat capacity and0H0  0:1 T is a fitting parameter that takes accountof the Meissner effect. The best fit is obtained when  0:46, namely, theVolovik effect for nodal superconductors[20,21], and is consistent with previous reports on thenonmagnetic borocarbides [4]. Figures 1(b)–1(d) showthe heat capacity of sample A at 2.5, 4, and 8 K, respec-tively. The dashed line is the square-root field dependenceand the arrows indicate where the data deviate from theµFIG. 1 (color online). (a) Heat capacity of sample C at 2.5 Kfor H k 110. (b)–(d) Heat capacity of sample A at 2.5, 4, and8 K, respectively, for H k 110. Dashed lines are H1=2 fit andarrows indicate the deviation point from H1=2.237002-2fit. The deviation field shows a systematic increase withtemperature, i.e., 0.8 T at 2.5 K, 1.8 T at 4 K, and no cleardeviation at 8 K. Above the deviation field, the data fallbelow the H1=2 line.Field-angle heat capacity directly measures the changein the DOS with magnetic field direction. The DOS of ad-wave superconductor has been shown to exhibit a four-fold oscillation with field-angle against crystal axes [22].At T  0 K, the DOS has a simple form:NE;H;=N0 ’ D41 j sin2j; (1)where D4 is a Doppler-shift-induced coefficient and describes the oscillation amplitude. The field-angle sensi-tive Doppler effect, arising from the supercurrent flowscirculating around the vortices, leads to maxima in theDOS when the field is along gap maxima and DOSminima when the field is along nodes. A 3D supercon-ductor has a much reduced oscillation amplitude 6%compared to that of a 2D system 40% due to contri-butions from the out-of-plane component. We note that asimilar effect is predicted for an s g-wave supercon-ductor [23].Figures 2(a) and 2(b) show the low-temperature field-angle heat capacity of sample C at 2.5 K and sample A at2 K, respectively. The samples were field cooled to 2 K (or2.5 K) and were rotated within the ab plane by a computercontrolled stepping motor at increments of 3. The heatcapacity with increasing and decreasing field angleshowed reversible behavior for all measured fields, indi-cating that the flux pinning effect is negligible in ourmeasurement. Background contributions (Cbkg) from lat-tice vibrations and thermometry were subtracted in theusual manner [8], and the remaining field-induced heatcapacity C  Ctotal 	 Cbkg was analyzed in terms ofC  c1 j sin2j for pure samples. At lowfields, there is a clear fourfold oscillation with minimaalong h100i for both samples, indicating that the zeros ofthe gap are located along those directions, consistent with∆∆αFIG. 2 (color online). (a) Normalized heat capacity of sampleC at 2.5 K and a fourfold oscillation (solid line), C=c 1 j sin2j. (b) Field-induced heat capacity of sample Aat 2 K, C  Ctotal 	 Cbkg, and a 4 4 pattern (solid line)(see text).237002-2Γγµ0ααΓFIG. 3 (color online). Oscillation amplitude  of sample C(circles) and sample N (squares). The dashed line is from the3D nodal quasiparticle theory with vF  1:5 107 cm=s [8].The bottom panel contrasts the oscillation amplitude due to thenonlocal effect (circles) and gap anisotropy (squares).P H Y S I C A L R E V I E W L E T T E R S week ending11 JUNE 2004VOLUME 92, NUMBER 23those of YNi2B2C [7,8]. The oscillation amplitude  isabout 4% for all three samples (sample N is not shown,but is essentially the same as sample C); this is consistentwith the 3D superconductivity in the borocarbides [24].At 1 T, surprisingly, the heat capacity of sample Adevelops minima along h110i, producing two sets of four-fold patterns or eightfold, an effect not observed in eithersample C or sample N with higher Tc’s. The crossoverfield from the fourfold to the 4 4 pattern of sample Alies between 0.6 and 1 T, which is also the point where theheat capacity of sample A deviates from the square-rootfield dependence (see Fig. 1). With increasing field, thesplitting in sample A gradually disappears and the field-angle heat capacity recovers its fourfold pattern above 4 T.We also measured the field-angle heat capacity of sampleA at 4 K to check if the anomalous peak splitting persistsat higher temperatures (not shown). The fourfold patternnow persists to 1 T, evolving into two sets of fourfoldpatterns above 2 T. The 4 T data at 4 K have a shapesimilar to the 2 T data at 2 K. The crossover field Hs1increases with increasing temperature.We focus on the fact that the anomalous eightfoldpattern occurs only at sample A which has half theelectronic mean-free path of sample N while the Tc isslightly decreased. According to the nonlocal theory byKogan et al. [12], the hexagonal-to-square FLL transitiondepends on the electronic mean-free path l and the super-conducting coherence length  of the sample. Gammelet al. found that a mere 9% of Co doping onto the Ni sitein Lu1221 can make the FLL transition field at least 20times higher than that of a pure matrix forH k 001 [25].Because the FLL transition field for the pure sample isexpected to be small [12]—possibly below our measure-ment range —the nonlocal effects would not influence thefield-angle heat capacity of sample N (or C). In contrast,the disorder in sample A is expected to increase thetransition to a higher field, i.e., to a field relevant in thefield-angle heat capacity measurement.When the magnetic field is rotated within the ab plane,the transition field may differ with different field direc-tions because of the different nonlocal range, i.e., =l.The two different transition fields can be characterized byHs1 and Hs2. As a magnetic field rotates within the abplane for Hs1  H  Hs2, the FLL will experience astructural change (or distortion), i.e., hexagonal for H k100 and square for H k 110. Since the borocarbideshave nodes on the Fermi surface, the DOS will differdepending on the FLL structure [21]. The negative devia-tion from the H1=2 above Hs1 in the heat capacity ofsample A [see Fig. 1(b)] also indicates that the DOS ofthe hexagonal FLL is larger than that of the square FLL.The additional FLL anisotropy in the DOS will modulatethe gap-anisotropy oscillation and leads to a 4 4-foldpattern.We hypothesize that the two effects are independent ofeach other and have a form of two cusped fourfoldoscillations in the DOS:237002-3C  p1 p21 j sin2j1 j sin2	 45j;(2)where p1 and p2 are fitting parameters. The value represents the oscillation due to gap anisotropy in puresamples (see Fig. 3). The nonlocal effects give rise to a45-shifted fourfold pattern and are accounted for by .The solid line in Fig. 2(b) is the least-square fit of Eq. (2)and represents the data very well. The oscillations due tothe nonlocal effects () and the gap anisotropy () at 2 Kare compared as a function of magnetic field in thebottom panel of Fig. 3. The FLL effect  increases sharplyabove 0.6 T and decreases gradually to zero at 4 T, in-dicating that the low field corresponds toHs1 and the highfield to Hs2.Figure 4 summarizes the H 	 T phase diagram of thedisordered sample A. Unlike pure samples, it has addi-tional phase lines in the superconducting state where thefield-angle heat capacity shows a crossover from the four-fold to the 4 4 pattern (Hs1) or vice versa (Hs2). Theincrease in the crossover fields with increasing tempera-ture is consistent with the nonlocal effects [12], addingstrength to our viewpoint that the anomalous eightfoldpattern is due to the coexistence of nonlocal effects andgap anisotropy. We note, however, that the differencebetween the two transition fields Hs1 and Hs2 is largerthan that expected within the nonlocal theory [12]. Fordiscussion, we assume that the anisotropy in the FLLtransition field between h001i and h100i is similar tothat between h110i and h100i because the Hc2 anisotropy237002-3µ 0Η FIG. 4 (color online). H 	 T phase diagram of sample A withlowest Tc. Hs1 is the FLL transition for H k 110 and Hs1 forH k 100. The dotted lines are a guide to the eye. Triangles andsquares were sketched to show the corresponding FLL.P H Y S I C A L R E V I E W L E T T E R S week ending11 JUNE 2004VOLUME 92, NUMBER 23is almost the same for the two configurations. The ob-served ratio Hs2=Hs1  4 is larger than the predictedratio 2 [12], but is smaller than the reported ratio of10 in YNi2B2C [26]. The difference between experi-ments and theory may attest that we need to account forboth nonlocality and the anisotropic gap nature of theborocarbides.In summary, we have studied the nonmagnetic super-conductor LuNi2B2C via the dependence of heat capacityon a magnetic field angle and a magnetic field. Unlikepure samples, a slightly disordered sample A shows adeviation from H1=2 and a 4 4-fold pattern in Cpabove 1 T. The anomalous properties were explained interms of the coexistence of gap anisotropy and nonlocaleffects. These experiments resolve the apparently irrec-oncilably different views on the nature of the order pa-rameter in the nonmagnetic borocarbides.This project was supported in part by NSF GrantNo. DMR 99-72087 and at Pohang by the Ministry ofScience and Technology of Korea through the CreativeResearch Initiative Program. This manuscript has beenauthored by Iowa State University of Science andTechnology under Contract No. W-7405-ENG-82 withthe U.S. Department of Energy. X-ray measurementswere carried out in the Center for Microanalysis ofMaterials, University of Illinois, which is partially sup-ported by the U.S. Department of Energy under GrantNo. DEFG02-91-ER45439.We thank S. Baily, N. Moreno-Salazar, and M. Hundley for their help in perform-ing experiments. T. P. and M. S. acknowledge benefitsfrom the discussion with S. Balatsky, T. Leggett, andK. Machida.2370*Current address: Department of Physics and Astronomy,Louisiana State University, Baton Rouge, LA 70803,USA.02-4[1] Y. J. Uemura et al., Phys. Rev. Lett. 66, 2665 (1991).[2] J. F. Annett, Physica (Amsterdam) 317C, 1 (1999).[3] B. H. Brandow, Philos. Mag. 83, 2487 (2003), and refer-ences therein.[4] M. Nohara, M. Isshiki, H. Takagi, and R. J. Cava, J. Phys.Soc. Jpn. 66, 1888 (1997).[5] G.-Q. Zheng, Y. Wada, K. Hashimoto, Y. Kitaoka,K. Asayama, H. Takeya, and K. Kadowaki, J. Phys.Chem. Solids 59, 2169 (1998).[6] E. Boaknin, R.W. Hill, C. Proust, C. Lupien, L. Taillefer,and P. C. Canfield, Phys. Rev. Lett. 87, 237001 (2001).[7] K. Izawa, K. Kamata, Y. Nakajima, Y. Matsuda,T. Watanabe, M. Nohara, H. Takagi, P. Thalmeier, andK. Maki, Phys. Rev. Lett. 89, 137006 (2002).[8] T. Park, M. B. Salamon, E. M. Choi, H. J. Kim, and S.-I.Lee, Phys. Rev. Lett. 90, 177001 (2003).[9] Y. DeWilde, M. Iavarone, U. Welp, V. Metlushko, A. E.Koshelev, I. Aranson, G.W. Crabtree, and P. C. Canfield,Phys. Rev. Lett. 78, 4273 (1997).[10] M. R. Eskildsen, P. L. Gammel, B. P. Barber, A. P.Ramirez, D. J. Bishop, N. H. Andersen, K. Mortensen,C. A. Bolle, C. M. Lieber, and P. C. Canfield, Phys. Rev.Lett. 79, 487 (1997).[11] M. Yethiraj, D. M. Paul, C.V. Tomy, and E. M. Forgan,Phys. Rev. Lett. 78, 4849 (1997).[12] V. G. Kogan, M. Bullock, B. Harmon, P. Miranovic,L. Dobrosavljevic-Grujic, P. L. Gammel, and D. J.Bishop, Phys. Rev. B 55, R8693 (1997).[13] V. Metlushko, U. Welp, A. Koshelev, I. Aranson, G.W.Crabtree, and P. C. Canfield, Phys. Rev. Lett. 79, 1738(1997).[14] L. Civale, A.V. Silhanek, J. R. Thompson, K. J. Song, C.V.Tomy, and D. M. Paul, Phys. Rev. Lett. 83, 3920 (1999).[15] N. Nakai, P. Miranovic, M. Ichioka, and K. Machida,Phys. Rev. Lett. 89, 237004 (2002).[16] M. R. Eskildsen, A. B. Abrahamsen, V. G. Kogan, P. L.Gammel, K. Mortensen, N. H. Andersen, and P. C.Canfield, Phys. Rev. Lett. 86, 5148 (2001).[17] B. K. Cho, P. C. Canfield, L. L. Miller, D. C. Johnston,W. P. Beyermann, and A. Yatskar, Phys. Rev. B 52, 3684(1995).[18] K. O. Cheon, I. R. Fisher, V. G. Kogan, P. C. Canfield,P. Miranovic, and P. L. Gammel, Phys. Rev. B 58, 6463(1998).[19] X.Y. Miao, S. L. Bud’ko, and P. C. Canfield, J AlloysCompd. 338, 13 (2002).[20] G. E. Volovik, JETP Lett. 58, 469 (1993).[21] M. Ichioka, A. Hasegawa, and K. Machida, Phys. Rev. B59, 184 (1999).[22] I. Vekhter, P. J. Hirschfeld, J. P. Carbotte, and E. J. Nicol,Phys. Rev. B 59, R9023 (1999).[23] K. Maki, H. Won, P. Thalmeier, Q. Yuan, K. Izawa, andY. Matsuda, Europhys. Lett. 64, 496 (2003).[24] L. F. Mattheiss, Phys. Rev. B 49, 13 279 (1994).[25] P. L. Gammel, D. J. Bishop, M. R. Eskildsen, K.Mortensen, N. H. Andsersen, I. R. Fisher, K. O. Cheon,P. C. Canfield, and V. G. Kogan, Phys. Rev. Lett. 82, 4082(1999).[26] H. Sakata, M. Oosawa, K. Matsuba, N. Nishida,H. Takeya, and K. Hirata, Phys. Rev. Lett. 84, 1583(2000).237002-4

Evidence for the coexistence of an anisotropic superconducting gap and nonlocal effects in the nonmagnetic superconductor LuNi2B2C

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Evidence for the Coexistence of an Anisotropic Superconducting Gap and Nonlocal Effects in the Nonmagnetic Superconductor
                    
                      
                        
                          LuNi

The nonmagnetic superconductor LuNi2B2C via the dependence of heat capacity on a magnetic field angle and a magnetic field was studied. The heat capacity exhibits a fourfold variation as the field H[001] plane with minimum (100). It was found that a disorderd sample A shows a deviation from H1/2 and a (4+4)- fold pattern in heat capacity. The anomaloous pattern in terms of coexixting superconducting gap anisotropy and nonlocal effects was also discussed

Chia, Elbert E.M.

Lee, Sung Ik

Canfield, P. C.

Louisiana State University

Evidence for the coexistence of an anisotropic superconducting gap and nonlocal effects in the nonmagnetic superconductor LuNi2B 2C

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