The discovery [ 1 ] of a first-order solid-liquid transition in the vortex matter of a classic type-II superconductor niobium was widely regarded as an important result since it resolved a long-standing issue of whether a genuine order-disorder transition can take place in the vortex system in type-II superconductors, and whether the anomalous peak effect is indeed caused by a structural phase transition in the vortex matter. However, in a follow-up SANS experiment [ 2 ] on a Nb crystal which has a lower upper critical field, suggesting even less bulk disorder, neither the first-order transition nor the peak effect was found. These two seemingly contradictory results suggest either a trivial technical error in one of the two experiments, or something more profound and interesting in the physics of vortex matter, namely the existence of multicritical behavior in the Bragg-glass transition. We have now carried out a systematic study of the Nb crystal used in our original experiment, and have found such a multicritical point [ 3 ].
Our experiment was carried out on a Nb single crystal in which both the peak effect and the first-order Bragg-glass melting transition were observed at the same temperatures [ 1 ]. The sample has a zero-field Tc = 9.16 K, and an estimated Ginzburg-Landau parameter ( 0 ) = 2.0. The experimental SANS configuration is shown in the inset of Figure 1 ( a ). The dc magnetic field was applied in the direction of the incoming neutron beam using a horizontal superconducting magnet. A coil was wound on the sample to allow in situ ac magnetic susceptibility measurements.
Figure 1 ( a ) shows the SANS data at H = 0.3 T. The Gaussian widths are obtained from fitting the Bragg peaks ( in intensity vs. azimuthal angle ) to six Gaussian peaks evenly spaced 60° apart. It is clear that the azimuthal widths — a measure of orientational disorder in the vortex array — are strongly history dependent. Supercooling and superheating effects are observed for field-cooling ( FC ) and field-cooled-warming ( FCW ) paths, respectively. As reported previously [ 1 ], the disordered phase at T > Tp and the ordered phase at T < Tp are their respective thermodynamic ground states. The abrupt change in the structure factor S ( q ) at the peak effect Tp depicts a symmetry-breaking phase transition from a vortex liquid with short-range order to a Bragg glass with quasi-long range order [ 1 ]. The phase transition is first order as evidenced by the strong thermal hysteresis in S ( q ). Compared to that at higher fields, the metastability region for H = 0.3 T is smaller but still pronounced.
It was found that the thermal hysteresis of S ( q ) observed in SANS is strongly field dependent, and the metastability region disappears completely at low fields. Figure 1 ( b ) shows the azimuthal width data for H = 0.2 T. For comparison, the real part ( T ) of the ac magnetic susceptibility is also shown in Figure 1 ( b ). The dip in ( T ) is the well-established signature of the peak effect [ 1 ]. The history dependence of the Bragg-peak width is only detectable within 100 mK of the peak-effect temperature Tp. A similar trend is observable in the history dependence of the radial widths of the Bragg peaks. At 0.3 T, there is a pronounced thermal hysteresis in the radial widths. At 0.2 T, however, the hysteresis is barely discernable. At an even lower field of 0.1 T ( data not shown ), the thermal hysteresis in S ( q ) is undetectable.
At H = 0.1 T, a very sharp peak effect ( the onset-toend width = 40 mK ) is still present. Thus we believe the phase transition at 0.1 T is still first-order but the metastability region is too narrow to be resolved in SANS ( the temperature resolution was 50 mK ). Nevertheless, the diminishing hysteresis in the low-field regime suggests that the phase transition is becoming continuous and meanfield- like, namely there is a multicritical point on the phase boundary bordering the Bragg glass on the H-T phase diagram. We show that this multicritical behavior is directly related to the appearance and the disappearance of the peak effect.Figure 2 shows a three-dimensional plot of the ( T ) as a function of temperature and magnetic field. At high fields, there is a pronounced peak effect, a characteristic dip in ( T ). At higher temperatures above the peak-effect temperature Tp ( H ) ( or Hp ( T ), used interchangeably ), there is a smooth step in ( T ). This step, Tc3 ( H ) ( or Hc3 ( T ) ), defined in Figure 1 ( b ), is the onset of surface superconductivity. In the mean-field theory of Saint-James and de Gennes, Tc3 ( H ) is a continuous phase transition. The separation between Tp ( H ) and Tc3 ( H ) grows larger with increasing magnetic field. Upon cooling, below Tc3 ( H ) and toward Tp ( H ), the screening effect in ( T ) increases gradually but there is no sharp feature to define another temperature scale. With decreasing field, the peak effect becomes narrower and smaller. For H < 0.08 T, there is only a single kink in ( T ) corresponding to the mean-field transition Tc2 ( H ) or Hc2 ( T ). There is no reentrant peak effect at low fields – the peak effect simply vanishes here. New theoretical studies are needed to elucidate the possible physical mechanisms of the multicritical point in a Bragg glass.
 X. S. Ling, S. R. Park, B. A. McClain, S. M. Choi, D. C. Dender, and J. W. Lynn, Phys. Rev. Lett. 86, 712 (2001).
 E. M. Forgan, S. J. Levett, P. G. Kealey, R. Cubitt, C. D. Dewhurst, and D. Fort, Phys. Rev. Lett. 88, 167003 (2002).
 S. R. Park, S.-M. Choi, D. C. Dender, J. W. Lynn, and X. S. Ling, submitted to Phys. Rev. Lett. (cond-mat-0305569).
S. R. Park and X. S. Ling
Providence, RI 02912
Korea Advanced Institute of Science and Technology
D. C. Dender and J. W. Lynn
NIST Center for Neutron Research
National Institute of Standards and Technology
Gaithersburg, MD 20899
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