Structural stability of the square flux line lattice in YNi 2 B 2 C and LuNi 2 B 2 C studied with small angle neutron scattering

We have studied the ﬂux line lattice in YNi (cid:50) B (cid:50) C and LuNi

The borocarbide intermetallic superconductors have shown a wide range of unusual and interesting properties.These compounds, with chemical formula ͑R͒Ni 2 B 2 C assume the filled ThCr 2 Si 2 structure, ubiquitous to ternary intermetallics [1].For R Y, Dy, Ho, Er, Tm, or Lu the materials are strongly type-II superconductors (k ϳ 5 15) with T c up to 16 K and H c2 which can exceed 10 T [2][3][4][5][6].Several members of this class show magnetic transitions which persist in the superconducting phase, reviving interest in the detailed coexistence of superconductivity and magnetism [7,8].The structure and dynamics of the flux line lattice (FLL) in these materials are also remarkable.At high fields, a square FLL is seen, as opposed to the hexagonal structure generic to high k materials [9,10].This undergoes a field induced square to hexagonal transition at low field, driven by a coupling to the anisotropic Fermi surface [11].This symmetry transition, and the associated changes in the FLL elastic moduli, were shown to influence the microwave transport data.Further, the FLL has been found to couple to magnetic order in ErNi 2 B 2 C (Er:1221), as evidenced by a rotation and disordering of the FLL in the metamagnetic superconducting phase below 2.3 K [9].
In this Letter we report on studies of YNi 2 BC2 (Y:1221) and LuNi 2 B 2 C (Lu:1221), the two nonmagnetic end members of this series, using both small angle neutron scattering (SANS) and dc transport measurements.Our data allow two important conclusions to be reached.First, both materials have a low field hexagonal FLL which transforms into a square FLL in the 1 kOe range, in agreement with recent calculations based on nonlocal corrections to the London model [12].Second, we find that the FLL has a maximum in its longitudinal correlation length near H͞H c2 ϳ 0.2.Below this field, the SANS data can be understood within the traditional collective pinning model using the measured critical currents and the FLL tilt modulus [13].Above this field, the longitudinal correlation lengths are dramatically reduced below this estimate.
We can fit our high field data for H k c and at an angle from the c axis using the field dependence of the shear modulus c 66 .This is in disagreement with the weak collective pinning model, in which the longitudinal correlations are determined by the tilt modulus.The shear modulus has a peak at H͞H c2 ϳ 0.3, after which it decreases, ultimately leading to the peak effect in the transport critical current when the FLL is completely amorphous at H͞H c2 ϳ 0.9.Our data then represent direct measurements of the gradual disordering of the FLL in high k materials.Previous studies in low k, marginally type-II niobium showed a disordering in the transverse FLL correlations, in agreement with the simple weak collective pinning model [14].Understanding the shear properties plays a central role in the dynamics of the FLL, particularly the phase transitions, such as FLL melting seen in both the high temperature and borocarbide [15] superconductors.Measurements of c 66 have mostly been inferred from its dynamical consequences [16], leaving static data as an important point of comparison.
The single crystal materials used in these experiments were flux grown platelets with the c axis perpendicular to the flat surface [5].Isotopically enriched B 11 was used to increase thermal neutron penetration.Y:1221 has a superconducting transition temperature T c 15.6 K, and Lu:1221 has T c 16.6 K [3].Sample dimensions were typically 8 3 8 3 1 mm 3 with masses of 200-400 mg.For transport studies, 3 3 1 3 0.3 mm 3 bars were cut from the large crystals, and etched in HF and aqua regia to reduce surface pinning.
The SANS experiments were performed in the cold neutron guide hall of the Risø National Laboratory DR3 reactor.Neutrons with wavelengths l n between 3.8 and 12.4 Å, energy spread Dl n ͞l n 0.18, and angular divergence of ϳ0.1 ± were used.The Bragg scattered neutrons were counted using an area detector located 6 m from the sample.Magnetic fields from 500 Oe to 35 kOe were applied parallel to the neutron beam.In most of the experiments, the c axis was parallel to the field, but we also performed experiments with the field rotated in the ac plane.In all cases a field cool to T 2.2 K was performed.
The reflectivity of a single FLL Bragg peak allows detailed measurements of the penetration depth l and coherence length j, the two microscopic length scales in the superconducting state.The reflectivity is given by R ͑2pg 2 l 2 n t͞16f 2 0 q͒H 2 1 , where g 1.91 is the gyromagnetic ratio of the neutron, t is the sample thickness, f 0 is the flux quantum, and q 2p p ͑B͞f 0 ͒ is the scattering vector of the first order square FLL Bragg reflection.First order Ginzburg-Landau corrections to the London model result in a magnetic form factor for the square FLL H 1 ͑f 0 ͞4p 2 l 2 ͒ exp͑22p 2 Bj 2 ͞f 0 ͒ [17].In Fig. 1, the field dependence of H 2 1 is plotted.In this semilog plot, the data lie on a straight line with a zero field intercept determined by l and a slope determined by j.We find l 1060 6 30 Å for both compounds and j 88 6 3 Å for Y:1221 and j 82 6 2 Å for Lu:1221, in good agreement with literature values [4], and transport measurements yielding H c2 65 kOe.The j 71 Å inferred from transport is about 10% below our estimate from SANS, as we have typically found with this model for the form factor.
SANS experiments can also be used to determine the FLL correlation lengths [18].In addition to the detailed data on the rocking curve widths described below, we measured the radial and azimuthal widths of the FLL Bragg reflections in the plane of the detector, which give the positional and orientational correlation lengths of the FLL, respectively.Above 2 kOe, these widths stayed constant within the experimental resolution for all FIG. 1. Shown is the field dependence of the FLL magnetic form factor for Y:1221 and Lu:1221 at T 2.2 K, from which j͑T͒ and l͑T ͒ are extracted.materials, fields, and angles studied.For fields H k c above 2 kOe, the FLL was square, rotated 45 ± with respect to the underlying crystal and field independent except for the rocking curve width and the trivial field dependence of the flux line density.For rotation of the field by 30 ± in the ac plane away from c in Y:1221, the FLL was found to be hexagonal and distorted by the effective mass anisotropy m c ͞m a 1.8 inferred from H c2 measurements, up to the highest measured field of 20 kOe.This rotation corresponds to having the field roughly in the (105) reciprocal lattice direction.
In low field measurements performed on Y:1221 the azimuthal width of the FLL peaks were seen to broaden towards a ring of scattering below 1 kOe, similar to results in the magnetic compound Er:1221.In Er:1221, this is the onset of the square to hexagonal symmetry transition [11].Our SANS data on Y:1221 show that both the high field square FLL and the square to hexagonal transition are generic to this class of materials independent of the presence of magnetism.The low field hexagonal lattice in Lu:1221 and Y:1221 was verified in decoration studies at 60 Oe and 4.2 K. Together, these data support recent calculations of FLL symmetry transitions based on nonlocal corrections to either the London model [12] or the Abrikosov model [10].Corrections to the London model, which describe the low field, low temperature regime of the square-hexagonal transition, result from the anisotropy of the Fermi surface and produce a fourfold anisotropy.This anisotropy causes the FLL to undergo a second order phase transition from a low field, hexagonal lattice to a high field square lattice in clean systems, and provides quantitative comparison to our results in Er:1221.Magnetism does not appear to enter into the stability of the FLL, except insofar as it modifies certain fourth order Fermi surface averages.Additionally, the square to hexagonal transition field should increase as the field is rotated in the ac plane away from the c axis.Since our data at 30 ± tilt show a hexagonal FLL, the angle driven symmetry transformation is in qualitative agreement with this model.
In the remainder of this paper, we focus on the stable, high field FLL and how it disorders to accommodate the pinning force introduced by random disorder.Shown in the inset to Fig. 2 as a function of field, are the widths s m (FWHM), obtained by Lorentzian fits to the rocking curves and corrected for instrumental resolution [18].The rocking curve width in this geometry measures how well correlated the flux lines are along their length, parallel to the applied field.In Fig. 2, the longitudinal correlation lengths are shown, determined from the rocking curves using j L ͞a 0 ͑ps m ͒ 21 .Here a 0 2p͞q 0 p ͑f 0 ͞B͒ is the lattice parameter for the square FLL.For both materials and orientations, j L ͞a 0 shows a similar behavior, increasing at low fields to a broad maximum near 10 kOe, and then gradually decreasing.It is important to emphasize that this maximum is not directly associated with either the square to hexagonal transition below 1 kOe or the pronounced "peak effect" at H͞H c2 ϳ 0.9 (H 60 kOe) shown in Fig. 3.The maximum at 10 kOe is a quite unexpected behavior.In all previously studied high k materials, and within the theoretical collective pinning framework, the longitudinal correlation length increases monotonically with field up to the peak effect.This general trend is understood as arising from vortex-vortex interactions which increasingly order the lattice with field.In particular, SANS studies of Nb show a longitudinal correlation length which increases monotonically, reaching a maximum at the peak effect, where the positional correlation length R c has dropped to ϳ1a 0 [19].In the case of the borocarbides, the low field estimate R c ͑a 0 ͞l͒j L [20] yields positional correlations ϳ50a 0 throughout the regime of the 10 kOe maximum, far removed from the amorphous limit.Decoration studies show positional correlation lengths .10a0 in Lu:1221 at 600 Oe, the maximum field studied.We speculate that the anomalous maximum seen here, at which point we also observe strong departure from the collective pinning model, arises from the shear properties of the FLL.The finite critical current throughout this regime further suggests that the high field disordering is a static effect, rather than a dynamic disordering such as melting.
In Fig. 2, we also show data for the field rotated by 30 ± from the c axis, where the FLL has distorted hexagonal symmetry.In this orientation we also find that j L is dominated by the shear strength, although the low field j L is considerably reduced in this case.Within the 3D collective pinning model, the longitudinal Larkin-Ovchinnikov length can be expressed as L c 2c 66 c 44 j 2 ͑1͞nf 2 ͒ where c 44 is the tilt modulus, c 66 is the shear modulus, and nf 2 is the elemental pinning force [13].As pointed out by Giamarchi and Le Doussal [21], L c in this theory differs from j L measured in a SANS experiment by constant times ͑a 0 ͞j͒ 2 .This leads to j L ϳ 2c 66 c 44 a 2 0 ͑1͞nf 2 ͒.In the same model, the elemental pinning force is related to the critical current via J c B͞c n 2 f 4 ͞2c 44 c 2 66 j 3 , leading to j L ͞a 0 A͑2c 44 ca 2 0 ͞j 3 J c B͒ 1͞2 , with A being a constant.Note that only the tilt modulus enters into this equation.Shown in Fig. 3  Clearly, the structural and transport data agree within the framework of the collective pinning model for fields below the 10 kOe, but sharply disagree for fields greater than this.We have similar qualitative results for Y:1221.Even the weak maximum in J c B at 10 kOe would produce a local minimum in j L , rather than the observed maximum.In previous studies of Nb and NbSe 2 this collective pinning analysis works well up to the peak effect in relating J c B and j L [18,19].We believe that the breakdown of this analysis and the observed high field disordering is related to the anomalous shear properties of the FLL in this system.The square to hexagonal transition at low fields further suggests that the shear modulus of the square lattice is unusually small.For an hexagonal FLL (we do not have expressions valid for the square FLL), the shear modulus can be approximated by c 66 ~͑H 2 c ͞16p͒b͑1 2 0.29b͒ ͑1 2 b͒ 2 where b B͞B c2 [22].This expression has a peak near b ϳ 0.3, and provides an excellent overall fit to our data, although the fitted values of H c2 tend to be about 15% low.
There are two routes which may lead to the shear modulus c 66 controlling the behavior of j L .First, since the square FLL is stabilized by nonlocal corrections to the London model, the statistical summations which lead to the cancellation of c 66 in the expressions for the longitudinal correlation length may need to be modified.Second, the assumption of isotropic continuum elasticity may be too gross an oversimplification.Clearly, the shear properties are quite anisotropic at the square-hexagonal FLL transition, at which the shear strength was found to vanish only in one particular direction in numerical studies [23].
In conclusion, we have used SANS and transport to study the FLL in Lu:1221 and Y:1221.In both cases, the low field, hexagonal FLL transforms into a square FLL for fields greater than ϳ1 kOe.We see a significant, static disordering in the square lattice phase at H͞H c2 ϳ 0.2, at which point the simple collective pinning model breaks down.We attribute this disordering to the anomalous shear properties of the square FLL, although there is clearly a need for more theoretical understanding of the correlation lengths in this case.It is interesting to compare these data to our studies in Er:1221 [9,11].In that case, the low field data also followed the collective pinning model.However, the data departed abruptly at 1.5 kOe, above which j L was field independent.In that case we believe the abrupt departure is related to strong pinning to magnetic domain walls in the antiferromagnetic superconducting phase, which has also not been considered in collective pinning models.

FIG. 2 .
FIG. 2. The longitudinal FLL correlation lengths at T 2.2 K are shown, extracted from the rocking curve widths via j L ͞a 0 ͑ps m ͒ 21 .For H k c, the Lu:1221 data are shown as the solid squares and the Y:1221 data are shown as the solid circles.Data on Y:1221 with the field rotated 30 ± in the ac plane are shown as the open circles.The longitudinal correlation length extracted from transport data using the collective pinning model for Lu:1221 and H k c are shown as the solid line, which agrees with the low field data.In the inset the raw rocking curve widths are shown.

FIG. 3 .
FIG. 3. Shown are the transport data for the pinning force density versus field at T 2.2 K for Lu:1221 (solid squares) with H k c.The relative longitudinal Larkin-Ovchinnikov length (open squares) extracted using simplest form of the collective pinning model is also shown.The line is a fit to B 3͞2 , the result for a constant pinning force.
are the results from critical current measurements in Lu:1221 with H k c.Voltage criteria between 0.1 and 0.35 mV͞cm were used to determine the onset of flux flow resistivity.Plotted is the pinning force density J c B as a function of field.The data show a large peak effect near 60 kOe [H c2 ͑T 2.2 K͒ 65 kOe] and a weak maximum near 10 kOe.Also shown in the figure is L c ͞a 0 ignoring dispersion in c 44 BH͞4p.The solid line is a simple expression valid when J c B is constant.The value of j L with the constant A 1͞70 is plotted in Fig. 2 as the solid line for comparison with the SANS data.
This work was supported by NATO under Grant No. CRG960550.M. R. E. is supported by the Danish Research Academy.P. C. C. is supported by the Director of Energy Research, Office of Basic Energy Science under