We have investigated the temperature dependence of the H || c flux line
lattice structural phase transition from square to hexagonal symmetry, in the
tetragonal superconductor LuNi_2B_2C (T_c = 16.6 K). At temperatures below 10 K
the transition onset field, H_2(T), is only weakly temperature dependent. Above
10 K, H_2(T) rises sharply, bending away from the upper critical field. This
contradicts theoretical predictions of H_2(T) merging with the upper critical
field, and suggests that just below the H_c2(T)-curve the flux line lattice
might be hexagonal.Comment: 4 pages, 3 figure

A. B. Abrahamsen

B. Rosenstein

D. K. Christen

D. McK. Paul

I. Affleck

J. Schelten

K. Mortensen

K. O. Cheon

M. Ichioka

M. R. Eskildsen

M. Yethiraj

N. H. Andersen

P. C. Canfield

P. L. Gammel

U. Klein

U. Yaron

V. G. Kogan

Y. DeWilde

Z. D. Hao

English

arXiv

Eskildsen, MR

Abrahamsen, AB

Kogan, VG

Gammel, PL

Mortensen, Kell

Andersen, NH

Canfield, PC

Copenhagen University Research Information System

Temperature dependence of the flux line lattice transition into square symmetry in superconducting LuNi2B2C

We have investigated the temperature dependence of the H parallel to c flux line lattice structural phase transition from square to hexagonal symmetry, in the tetragonal superconductor LuNi2B2C (T-c = 16.6 K). At temperatures below 10 K the transition onset field, H-2(T), is only weakly temperature dependent. Above 10 K, H-2(T) rises sharply, bending away from the upper critical field. This contradicts theoretical predictions of H-2(T) merging with the upper critical field and suggests that just below the H-c2(T) curve the flux line lattice might be hexagonal

Eskildsen, M.R.

Abrahamsen, A.B.

Kogan, V.G.

Gammel, P.L.

Mortensen, K.

Andersen, N.H.

Canfield, P.C.

Online Research Database In Technology

Temperature dependence of the flux line lattice transition into square symmetry in superconducting LuNi<sub>2</sub>B<sub>2</sub>C

Crossref

Physical Review Letters

Temperature Dependence of the Flux Line Lattice Transition into Square Symmetry in Superconducting<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>LuNi</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mi>C</mml:mi></mml:math>

We have investigated the temperature dependence of the H parallel to c flux line lattice structural phase transition from square to hexagonal symmetry, in the tetragonal superconductor LuNi2B2C (T-c = 16.6 K). At temperatures below 10 K the transition onset field, H-2(T), is only weakly temperature dependent. Above 10 K, H-2(T) rises sharply, bending away from the upper critical field. This contradicts theoretical predictions of H-2(T) merging with the upper critical field and suggests that just below the H-c2(T) curve the flux line lattice might be hexagonal.This article is published as Eskildsen, M. R., A. B. Abrahamsen, V. G. Kogan, P. L. Gammel, K. Mortensen, N. H. Andersen, and P. C. Canfield. "Temperature Dependence of the Flux Line Lattice Transition into Square Symmetry in Superconducting LuNi 2 B 2 C." Physical Review Letters 86, no. 22 (2001): 5148.
DOI: 10.1103/PhysRevLett.86.5148.
Copyright 2001 American Physical Society.
Posted with permission

Eskildsen, M. R.

Abrahamsen, A. B.

Kogan, V. G.

Gammel, P. L.

Andersen, N. H.

Canfield, Paul

Digital Repository @ Iowa State University (ISU)

VOLUME 86, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 28 MAY 20015148Temperature Dependence of the Flux Line Lattice Transition into Square Symmetryin Superconducting LuNi2B2CM. R. Eskildsen,1,* A. B. Abrahamsen,1 V. G. Kogan,2 P. L. Gammel,3K. Mortensen,1 N. H. Andersen,1 and P. C. Canfield21Risø National Laboratory, P.O. Box 49, DK-4000 Roskilde, Denmark2Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 500113Agere Systems, 700 Mountain Avenue, Murray Hill, New Jersey 07974(Received 11 December 2000)We have investigated the temperature dependence of the H k c flux line lattice structural phase tran-sition from square to hexagonal symmetry, in the tetragonal superconductor LuNi2B2C (Tc  16.6 K).At temperatures below 10 K the transition onset field, H2T, is only weakly temperature dependent.Above 10 K, H2T rises sharply, bending away from the upper critical field. This contradicts theoreticalpredictions of H2T merging with the upper critical field and suggests that just below the Hc2T curvethe flux line lattice might be hexagonal.DOI: 10.1103/PhysRevLett.86.5148 PACS numbers: 74.60.Ge, 74.70.DdStudies of the topology of the magnetic flux line lat-tice (FLL) in type-II superconductors have a long history.Early neutron scattering experiments on low-k supercon-ductors such as niobium revealed a multitude of differentFLL symmetries and orientations, mainly determined bythe symmetry of the atomic crystal structure in the planeperpendicular to the applied field [1]. This is not surprisingsince deviations from the hexagonal FLL characteristic ofan isotropic superconductor, and the locking to the crys-talline lattice are driven by the symmetry of the screeningcurrent plane and by nonlocal flux line interactions withina range determined by the coherence length j0, and theelectronic mean free path . Later, similar effects werealso observed in the strong type-II superconductor V3Siwith k  17 [2], demonstrating that nonlocal effects canbe equally important in high-k materials.Over the last couple of years, effects of nonlocality havebeen clearly observed in the borocarbide superconductors.The borocarbides are quaternary intermetallics withstochiometry RNi2B2C (R  Y, Gd–Lu) and a tetragonalunit cell (I4mmm) [3]. These materials have attractedattention due to the coexistence of superconductivity(R  Y, Dy–Tm, Lu) and antiferromagnetic ordering(R  Gd Tm). The borocarbides are strong type-IIsuperconductors with Ginzburg-Landau (GL) parameter,k  6 15. The discovery of a square FLL in most ofthe H k c phase diagram [4], which undergoes a smoothtransformation into hexagonal symmetry at fields below1 kOe [5], was the first observation of a purely field drivenFLL symmetry transition.Using nonlocal corrections to the London model and in-corporating the symmetry of the screening current planeobtained from band structure calculations, one is able tocalculate the FLL free energy and thereby to determinethe stable FLL configuration in different fields in the boro-carbides [6]. The model successfully describes the natureof the FLL square to hexagonal symmetry evolution in the0031-90070186(22)5148(4)$15.00borocarbides with H k c as the applied field is reduced.Qualitatively, this can be understood as driven by the four-fold basal plane anisotropy which makes the vortex currentpaths “squarish” close to the core. At high densities thisleads to a square FLL, whereas at low fields the system ap-pears isotropic resulting in a hexagonal FLL [7]. The onsetof the transition occurs as the field decreases, commencingat a critical field H2, determined by the range of the non-local interactions [6]. Later studies of the FLL in a set ofsamples doped with various amounts of impurities to sys-tematically change the nonlocality range showed a quanti-tative agreement between the predicted and measured onsetof the transition [8,9]. The model also predicted a reorien-tation transition of the rhombic FLL at a field H1 ø H2,which has been subsequently observed [10,11].Whereas there is a growing experimental and theoreti-cal understanding of the FLL geometry and transitions forfields well into the mixed state, there has been little at-tempt to understand what happens at fields just below theupper critical field. In this Letter we report on detailedmeasurements of the temperature dependence of the struc-tural transition, H2T , in LuNi2B2C with the applied fieldparallel to the crystalline c axis, over a broad range oftemperatures and applied fields. Below 10 K, H2T  is es-sentially constant. Above 10 K, H2T  rises sharply, even-tually bending away from the Hc2T  curve, suggesting theexistence of a hexagonal FLL in the region just below theupper critical field.The FLL in LuNi2B2C (Tc  16.6 K) was studied usingthe small angle neutron scattering spectrometer on the coldneutron beam line at the Risø National Laboratory DR3research reactor. The sample used was a 1 g single crystal,grown from a high temperature flux [3] using isotopicallyenriched 11B to enhance the neutron transmission. Incidentneutrons with wavelengths in the range ln  6 15.6 Åand a bandwidth Dlnln  24% were used, and the FLLdiffraction pattern was collected by a position sensitive© 2001 The American Physical SocietyVOLUME 86, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 28 MAY 2001detector at the end of a 6 m evacuated tank. Magneticfields in the range 1.5 to 10 kOe were applied parallel tothe crystalline c axis and to the incoming neutrons. Inorder to keep the resolution of the spectrometer unchanged,the neutron wavelength was varied in concert with theapplied field to keep the scattering angle, 2u  q10ln2p ,constant. Here q10 is the FLL scattering vector shown inFig. 1. The measurements were performed at temperaturesFIG. 1. FLL diffraction patterns from LuNi2B2C obtained attemperatures of 10 and 12 K and fields of 1.5 and 3 kOe. Theorientation relative to the crystallographic axes, and the FLL1, 0-peak azimuthal width is shown in the top panel. Eachdiffraction pattern is a sum of four measurements with thesample oriented to satisfy the Bragg condition for each of themain peaks/groups of peaks, and subtracted by background mea-surements taken at 20 K. The neutron wavelengths used were,respectively, ln  15.6 Å (1.5 kOe) and 11.0 Å (3 kOe). Thescattering close to q  0 is an artifact of imperfect backgroundsubtraction.between 2 and 20 K, following a constant field coolingprocedure. A background signal obtained above Tc wassubtracted from the data.In Fig. 1 we show three different FLL diffraction pat-terns obtained in LuNi2B2C. In quasi-2D structures suchas that of the FLL, the real space flux line configurationand the reciprocal space diffraction pattern are related by asimple 90± rotation and relabeling of the axes. The diffrac-tion pattern therefore directly shows the FLL symmetryand orientation, with the caveat that scattering from dif-ferently oriented domains is superposed. For the squareFLL there is no orientational degeneracy with respect tothe atomic lattice, due to the tetragonal crystal structureof the borocarbides, as reflected in the top panel of Fig. 1.Here both the FLL 1, 0 and 1, 1 reflections are clearlyvisible, and resolution limited both in the radial and theazimuthal direction. The square FLL is aligned with thescattering vector q10 along the 110 crystal direction.However, as soon as the FLL is distorted away from asquare symmetry there is a twofold degeneracy, since ori-entations with the long rhombic unit cell diagonal alongeither the a or the b axis have the same energy. Experimen-tally this will first manifest itself as an azimuthal broad-ening and, when the distortion increases as a splitting ofthe 1, 0 reflections, with the two peaks each belongingto a different domain orientation [5]. The magnitude ofthe splitting will eventually reach 30± if the FLL is trans-formed to a perfect hexagonal symmetry. The azimuthalbroadening is evident in the middle panel of Fig. 1, and inthe bottom panel the 1, 0 reflections are clearly split. Theeffect of the transition is also reflected in the square FLL1, 1 reflections. Ideally a radial splitting should be ob-served, with one peak moving towards shorter q and gain-ing intensity and the other moving to longer q and fading,to become, respectively, hexagonal 1, 0 and 1, 1 reflec-tions belonging to different domain orientations. However,due to the very rapid decrease in the scattered intensitywith increasing q, one observes only a radial broadeningand a relative increase in the intensity.In general, the azimuthal splitting was only directlyobservable when above 10±, comparable to experimen-tal resolution. To determine the transition onset fieldH2T , we analyzed the azimuthal intensity distributionin each diffraction pattern at a scattering vector q  q10,where q10  a 3 2pBf012, f0  hc2e is the fluxquantum, and a is a constant which depends on the FLL5149VOLUME 86, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 28 MAY 2001symmetry: With a FLL apex angle b as shown in Fig. 3,a  sinb212, and hence a  1 for the square case(b  90±) and 2p3 12  1.075 for the hexagonal FLL(b  60± or 120±). In the top panel of Fig. 2 we showthe temperature dependence of the azimuthal width forH  3 kOe, obtained by fitting to a single Gaussian. Aclear broadening is seen for T . 10 K. The low-T widthof 13.8± 6 0.2± is a direct measure of the azimuthal resolu-tion, which was kept constant throughout the experiment.To determine the azimuthal splitting, the intensity distri-bution at each temperature and field was then fitted to twoGaussians with a width fixed of 13.8±, as demonstrated inthe top panel inset. In the bottom panel of Fig. 2 the azi-muthal splitting is plotted versus temperature for a numberof fields between 1.5 and 6 kOe. At low temperatures thesplitting goes to 0± 6 2± as the field is increased. As T israised the splitting increases smoothly for H # 2 kOe andabruptly above a certain threshold for H $ 3 kOe. Thethreshold is determined by H2T . In addition, one notesFIG. 2. Top panel: The FLL 1, 0-reflection azimuthal widthobtained by fitting a single Gaussian to the azimuthal intensitydistribution, as a function of temperature at an applied field of3 kOe. Inset: FLL azimuthal intensity distribution folded intoone quadrant, at 3 kOe and 2 and 12 K. The 2 K data (offsetfor clarity) are well fitted by a single Gaussian of width 13.8±equal to the experimental resolution. The 12 K data are fittedby two equal weight Gaussians with the same width, yieldinga splitting of 10.5± 6 0.4±. Bottom panel: Azimuthal splittingversus temperature for applied fields between 1.5 and 6 kOe.The error bars include both the error on the direct fit of thesplitting and the uncertainty in the experimental resolution. Thedashed lines are guides to the eye.5150a crossing of the curves for 4.5 and 6 kOe, which meansthat H2T  is a multivalued function of T .Figure 3 is a contour plot of the azimuthal splitting in aH, T -phase diagram. Here we have used the 3± contour,well above the zero-splitting error of 2±, as the criterionfor the transition onset field. Below 10 K, H2T  is nearlyconstant, equal to 2.25 kOe. This is somewhat higher thanthe literature values [5], but as shown by Gammel et al. [9],H2 depends strongly on the sample purity as this affects therange of the nonlocal flux line interactions. Above 10 K,H2T  increases sharply, bending away from Hc2T  andthereby avoiding to cross the upper critical field. Fromthis it is clear that the square FLL is not stable at Hc2T .The strong temperature dependence contradicts predictionsbased on extended GL theory [12,13], where H2T  inter-cepts Hc2T  at a finite field. An extrapolation of our mea-surements suggests that in the vicinity of Hc2T , the FLLmay become hexagonal.Theoretically the nonlocal contribution to the vortex-vortex interaction weakens when the temperature rises,and H2T  should somewhat increase. This relativelyweak increase is essentially captured in the London ap-proach, where the order parameter jDj is taken as a con-stant in space [14]. Moreover, jDj is assumed equal tothe zero-field uniform value D0T  which determines thecondensation energy FN 2 FS of uniform superconduc-tors (~D20 at low temperatures and ~D40 near Tc). Beingconstant in space and field independent, the condensationenergy so defined is commonly disregarded in calculationsof the equilibrium FLL. Within this scheme, jDj affectsonly the value of the penetration depth, l ~ 1jDj. Themodel is good for flux line spacings large relative to theFIG. 3. H, T-phase diagram showing constant azimuthalsplitting contours. The dashed line corresponds to a splittingof 3± and is taken as the FLL symmetry transition onset fieldH2T . Approaching Hc2T the scattered intensity vanishesand the shaded area shows the range of our measurements. Aschematic illustration of the FLL symmetry and apex angleis shown for the square (center) and hexagonal case (low H,high T).VOLUME 86, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 28 MAY 2001core size jT , i.e., in small fields, and for temperatureswell below Tc.However, with increasing density of vortices at a fixedT , the spatial average jDj	 decreases thus causing an over-all increase of the system energy [15]. The jDj	 sup-pression depends on the ratio ajT , with a being theintervortex spacing. Therefore, at a given flux line den-sity, Bf0, the system energy can be reduced if the inter-vortex distance is maximized. This can be interpreted asan extra (to the London interaction) repulsion of vortices.This repulsion is isotropic for cubic materials or for te-tragonal crystals with H k c and will lead to a deviationfrom square symmetry and eventually to a hexagonal FLL(or triangular with the structure determined only by the ra-tio jajb for H k c). We speculate that this effect shouldbecome dominant on approaching Hc2T  where jDj isstrongly suppressed, in accord with the data presented inthis Letter.In addition, we have earlier reported that in TmNi2B2Cat low temperatures the FLL is hexagonal near Hc2T .With decreasing field, the FLL transforms first to a rhom-bic structure followed by the square [16]. As H is furtherreduced, the FLL should go through the same evolutionin inverse order, confirmed by low field decoration data[11]. This material is antiferromagnetic below 1.5 K, andit is possible that the magnetic ordering has a detrimentaleffect on the nonlocality range in fields near Hc2T . Ifso, both the antiferromagnetism and the order parametersuppression would have similar effects on the FLL, whichmay explain why the domain of “reverse evolution” is sobroad in TmNi2B2C. Finally, also in agreement with ourhypothesis are data on the FLL in V3Si that show howa distorted triangular lattice which is seen at T , 5 Kand H  10 kOe evolves toward hexagonal when T ap-proaches TcH [17].Within the microscopic theory applied to the mixedstate, there is no artificial separation of the electromagnetic(London) intervortex forces and those due to the varyingorder parameter. This problem, however, has never beenaddressed for the FLL [18]. The GL approach does pro-vide such a complete description, but it is valid only nearthe zero-field Tc0. The accuracy of results obtained withinGL for T , Tc0 is difficult to control. Moreover, the veryexistence of the GL expansion in powers of the order pa-rameter gradients away from Tc0 is questionable. The co-herence length jT  does not diverge as T ! TcH; as aconsequence the gradients of D are not necessarily small,and the convergence of the GL series is difficult (if notimpossible) to monitor. In this respect, calculations basedupon keeping certain quartic terms in the GL expansionwhile discarding others [12,13] can only be justified by amicroscopic theory, which is still to be done.In conclusion, we have in this Letter reported measure-ment of the temperature dependence of the FLL squareto hexagonal symmetry transition in LuNi2B2C. Our datashow that the transition field H2T  bends away from theupper critical field line Hc2T . This suggests that whileapproaching the Hc2T  curve, the FLL is driven towardshexagonal symmetry. We propose a possible qualitativeargument for why this might be the case. This poses newquestions and challenges concerning the phase diagram ofthe flux line solid.This project has received support from the Danish Tech-nical Research Council and the Danish Energy Agency.M. R. E. has received support from the Christian and AnnyWendelbo Foundation. A. B. A. is supported by the DanishResearch Academy. V. G. K. and P. C. C. are supported bythe Director of Energy Research, Office of Basic EnergyScience under Contract No. W-7405-Eng.-82.*Present address: DPMC, Université de Genève, 24 QuaiE.-Ansermet, CH-1211 Genève 4, Switzerland.Email address: morten.eskildsen@physics.unige.ch[1] J. Schelten, G. Lippmann, and H. Ullmaier, J. Low Temp.Phys. 14, 213 (1974); D. K. Christen et al., Phys. Rev. B21, 102 (1980).[2] D. K. Christen et al., Physica (Amsterdam) 135B, 369(1985).[3] For a review, see P. C. Canfield, P. L. Gammel, and D. J.Bishop, Phys. Today 51, No. 10, 40 (1998).[4] U. Yaron et al., Nature (London) 382, 236 (1996);M. Yethiraj et al., Phys. Rev. Lett. 78, 4849 (1997).[5] M. R. Eskildsen et al., Phys. Rev. Lett. 78, 1968 (1997);M. R. Eskildsen et al., Phys. Rev. Lett. 79, 487 (1997).[6] V. G. Kogan et al., Phys. Rev. B 55, R8693 (1997); V. G.Kogan et al., Phys. Rev. Lett. 79, 741 (1997).[7] I. Affleck, M. Franz, and M. Amin, Phys. Rev. B 55, R705(1997).[8] K. O. Cheon et al., Phys. Rev. B 58, 6463 (1998).[9] P. L. Gammel et al., Phys. Rev. Lett. 82, 4082 (1999).[10] D. McK. Paul et al., Phys. Rev. Lett. 80, 1517 (1998).[11] A. B. Abrahamsen et al., Bull. Am. Phys. Soc. 44, 1483(1999).[12] Y. DeWilde et al., Phys. Rev. Lett. 78, 4273 (1997).[13] B. Rosenstein and A. Knigavko, Phys. Rev. Lett. 83, 844(1999).[14] V. G. Kogan, P. Miranović, and D. McK. Paul, in TheSuperconducting State in a Magnetic Field, edited byC. A. R. Sà de Melo (World Scientific Publishing, Singa-pore, 1998).[15] Using a variational procedure, Z. D. Hao et al., Phys.Rev. B 43, 2844 (1991), have found that a certain sup-pression of jDj by the average field is required for a betteragreement with magnetization data.[16] M. R. Eskildsen et al., Nature (London) 393, 242 (1998).[17] M. Yethiraj et al., Phys. Rev. Lett. 82, 5112 (1999).[18] The microscopic description of the mixed state at arbitraryT and B is due to U. Klein [J. Low Temp. Phys. 69, 1(1987)] for isotropic materials with k  1. M. Ichioka,A. Hasegawa, and K. Machida, Phys. Rev. B 59, 8902(1999), used the same approach to compare the squareand triangular FLL in YBCO but did not try to find theminimum energy FLL structure.5151

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Temperature Dependence of the Flux Line Lattice Transition into Square
  Symmetry in Superconducting LuNi$_2$B$_2$C

Temperature Dependence of the Flux Line Lattice Transition into Square Symmetry in Superconducting LuNi $_2$ B $_2$ C

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Temperature Dependence of the Flux Line Lattice Transition into Square Symmetry in Superconducting LuNi2_22​B2_22​C

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Temperature Dependence of the Flux Line Lattice Transition into Square Symmetry in Superconducting LuNi $_2$ B $_2$ C