Direct Observation of Superheating and Supercooling of Vortex Matter using Neutron Diffraction

We report the first observation of a striking history dependence of the structure function of the vortex matter in the peak effect regime in a Nb single crystal by using small angle neutron scattering combined with in situ magnetic susceptibility measurements. Metastable phases of vortex matter, supercooled vortex liquid and superheated vortex solid, have been identified. We interpret our results as direct structural evidence for a first-order solid-liquid transition at the peak effect. PACS numbers: 74.60.Ge, 61.12.Ex Typeset using REVTEX 1 A current subject of wide interest concerns the existence of a vortex solidliquid transition in type-II superconductors [1]. In addition to providing a model system for the study of melting and freezing, the vortex matter system offers unprecedented opportunities in studying the effects of quenched disorder on phase transitions. Recently, the magnetization jump in high-Tc superconductors YBa2Cu3O7−δ (YBCO), a widely accepted thermodynamic signature of a vortex solid-liquid transition [2], and the peak effect, a wellknown vortex-lattice softening anomaly in many low-Tc [3,4] and high-Tc [5–8] superconductors in which the critical current exhibits a peak rather than decreasing monotonically with increasing temperature, are found to occur at the same temperature [7,8]. However, there is no direct structural evidence indicating whether the underlying phase transition is solid-to-solid, solid-to-liquid, or even liquid-to-liquid in origin. Since quenched disorder is known to have important consequences for phase transitions [9,10], whether a solid-liquid transition can occur when random pinning is effective has broad implications in condensed matter physics. Small angle neutron scattering (SANS) [11–15] is presently the most powerful technique for probing the vortex structure in bulk superconductors. For high-Tc superconductors, the magnetic penetration depth is typically quite long, yielding a very weak vortex signal in SANS experiments, which has so far prevented a direct structural study of the peak effect in high-Tc systems. Recent effort has therefore focused on low-Tc systems where one can detect the peak effect and measure the vortex structure using SANS in the same temperature and field regime [14,15], and for these purposes Nb is experimentally the most favorable system because of the short penetration depth and the availability of large, ultrahigh quality single crystals. However, it has been controversial as to whether there is a vortex solid-liquid transition in Nb and the two previous SANS studies of this issue were not conclusive 2 [14,15]. In the first study [14], a characteristic temperature at which the azimuthal width of the Bragg peaks starts to increase was interpreted as the vortex-lattice melting transition. In the second study [15], a nearly isotropic ring of scattering was observed in the peak-effect regime, but was interpreted as a disordered solid rather than a liquid. In both studies, only field-cooled vortex states were measured. We report here striking hysteresis in neutron diffraction patterns of the vortex state in the peak-effect regime of Nb. The metastability of supercooled and superheated vortex states is directly demonstrated by applying a weak perturbation and observing the evolution of the SANS patterns. These data provide direct evidence for a first-order vortex solid-liquid transition. Our experiments were performed using the 30-m SANS instrument NG-7 at the NIST Center for Neutron Research. The increased flux on the SANS instrument due to a new liquid-hydrogen cold source turns out to be important, but the key improvement provided by the present experiment is that the SANS and the ac magnetic response of the vortex array are measured simultaneously in situ. Thus one can correlate the ordering in the vortex state directly with the macroscopic vortex dynamics. The coil for the magnetic ac susceptibility measurements can also be used to apply a small ac field to dynamically perturb the vortex system, allowing an in situ determination of the metastability of the vortex states. The sample is a Nb single crystal of 99.998% purity, a cylinder with a slightly uneven diameter from 1.316 cm at one end to 1.169 cm at the other, and 2.48 cm in length. The incident neutron beam has a mean wavelength λ=6.0 Å and a bandwidth ∆λ/λ=0.11. The neutron beam traverses through the central region of the sample, defined by a cadmium mask (0.7 cm in diameter), along the cylindrical axis which coincides with the three-fold symmetric <111> crystallographic direction. The dc magnetic field is applied by 3 a horizontal superconducting magnet along the same direction. The absolute accuracy of the measured sample temperature is ± 0.20 K with a temperature stability better than ± 0.025 K. The peak-effect regime of the Nb sample is determined in situ by measuring the characteristic dip in the temperature dependence of the real-part of the ac magnetic susceptibility χ′(T ), as shown in Fig.1(a) for H=3.75 kOe. The pronounced diamagnetic dip in χ′(T ) of the ac susceptibility corresponds to a strong peak effect in the critical current (or nonlinear conductance) in the sample [5,8]. The onset, the peak, and the end of the peak effect are denoted by To(H), Tp(H), and Tc2(H), respectively. (In YBCO, the end of the peak effect is still far below Tc2(H) [5–8].) Fig.1(b) shows the window of our experiment. The upper critical field Hc2(4.20K) = 4.23 kOe of this sample is higher than that of Ref. [15] but similar to that of Ref. [14]. The lower critical field line, separating the Abrikosov and Meissner phases, is estimated from the first appearance of vortex scattering in the SANS. For each (T,H), we measure the vortex SANS patterns for different thermal paths. For low temperatures the vortex SANS images show sharp Bragg peaks with sixfold symmetry in agreement with previous studies [11,14,15], independent of the thermal history. An example is shown in the inset of Fig.1(b) for H=3.75 kOe and T=3.50 K. However, the vortex SANS pattern starts to show striking history dependence as the peak-effect regime is approached. For clarity, we define the field-cooled (FC) state as when the sample is cooled to the target temperature in a magnetic field, while the zero-field-cooled (ZFC) state is reached by cooling the sample in zero field to the target temperature and then increasing the magnetic field to the desired value. A field-cooledwarming (FC-W) state is when the system is cooled in field to a much lower temperature (∼ 2 K) then warmed back to the final temperature. For the FC path, the vortex SANS patterns show nearly isotropic rings for 4 Tp < T < Tc2 and broad Bragg spots for T < Tp. There is no clear sharpening in the Bragg peaks when To is crossed. Only at a lower T < Tp do the Bragg peaks become sharp. In contrast, for the ZFC and the FC-W paths, the sharp Bragg spots are observed for all temperatures up to Tc2. Shown in the top panel of Fig.2 are the ZFC and FC images at H=3.75 kOe and T=4.40 K, which is just below To(3.75kOe)=4.50 K. The images in the mid panel are for H=4.00 kOe and T=4.40 K, which is 0.10 K above Tp(4.0kOe)=4.30 K. The intensities at the radial maximum for the mid panel SANS data are plotted in the lower panel. The sharp Bragg spots for the ZFC state indicate a vortex lattice with long-range-order (LRO) [16], while the broad spots for the FC state suggest a disordered phase with short-range-order. The orientational order of the vortex assembly can be quantified by the azimuthal widths of the Bragg peaks at the radial position of the intensity maximum. The azimuthal widths ∆θ of the Bragg peaks can be obtained by fitting six Gaussian peaks to the data, for each (T,H) and path. Likewise, the translational order of the vortex assembly can be quantified by the radial widths ∆Q of the Bragg peaks. The Gaussian widths are plotted in Fig.3. Clearly, the ZFC states are more ordered than the FC states, translationally and orientationally, across the peak-effect regime. To determine the positional correlation of the vortex lines along the field direction, the rocking curves (Bragg peak intensity vs. the relative angle between the neutron beam and the applied magnetic field) of the ZFC and FC states at H=3.75 kOe and T=4.40 K are also measured. The rocking-curve width for the FC state is 30% larger than that of the ZFC state, suggesting that the vortex lines are entangled [17] in the FC states, but nearly straight in the ZFC (and FC-W) states. Since a simultaneous broadening in radial and azimuthal widths is characteristic of a liquid (or glass) [18], the hysteresis in both ∆θ and ∆Q suggests 5 a first-order vortex solid-liquid (or glass) transition. A controversial issue is the location of the underlying equilibrium phase transition in the vortex matter relative to the position of the peak effect [4,19–21]. One interpretation [4,19,20] places the conjectured vortex solid-liquid transition Tm at Tp, consistent with the recent experiments [7,8] in YBCO where the magnetization jump was found to coincide with the peak effect (the two are also related in other high-Tc systems [22]). Another widely held view, is based on the classical Lindemann criterion which would place Tm at Tc2(H) for Nb, provided the vortex-lattice elastic moduli are well-behaved [23]. In this scenario, the FC disordered phase seen here (as well as in [14,15]) is a supercooled liquid and the thermodynamic ground state is an ordered solid across the entire peak-effect regime. The third scenario places Tm at or below the onset of the peak effect [21]. We find that it is possible to experimentally determine the ground state, and consequently the approximate position of Tm, of the vortex system. For this purpose, we use the ac susceptibility coil to apply a small ac magnetic field to shake the vortex assembly and use the SANS to observe how the structure of the system evolves. Even though the true ground state of the vortex system may not have been reached in the time scale of the shaking experiment due to random pinning, the evolution of the diffraction patterns leaves little doubt regarding the nature of the ground state. As shown in Fig.4(a), after applying an ac field of 3.3 Oe at 100 Hz (1 kHz gives similar results) for 103 sec, the ordered ZFC vortex lattice becomes completely disordered. Preliminary time-dependent data show that the Bragg peaks start to disappear within the first 102 sec of the shaking experiment. In contrast, no measurable difference can be found for the disordered FC vortex states before and after the ac field is applied (for up to 104 sec). By using the same approach, we find that the FC disordered states for T < Tp are 6 metastable and the ordered ZFC state is the ground state, opposite to that for T > Tp. In the T < Tp regime, the metastability is obviously stronger since a much larger ac field is needed to change the metastable state. We find that an ac field of 50 Oe (at ≈ 0.1 Hz, using the superconducting magnet at non-persistent mode) can crystallize the disordered FC states at T < Tp. The shaking effects of an ac magnetic field were also observed in transport [24] and ac magnetization [25] in the peak-effect region in 2H-NbSe2, and were interpreted as a structural re-organization in the vortex matter, consistent with our direct SANS observations here. The metastable nature of the ZFC ordered state for T > Tp can also be observed in a field ramping experiment. For example, at a field ramp (increase) rate of less than 5 Oe/sec, the final vortex state (at T > Tp) is always ordered. In contrast, a ramp rate larger than 40 Oe/sec always results in a disordered state. However, the field ramping experiment alone cannot rule out a trivial possibility that the sample was heated to above Tc2(H) during the field ramping by the induced screening current. If this happens, after the field ramping stops, the sample cools back to the set temperature in field and the final state is actually a FC state. In the shaking experiment, the ac susceptibility of the sample is monitored and serves as an in situ thermometer to ensure that the sample temperature never fluctuates to above Tc2(H) during the entire SANS run (1 hour). Thus we conclude that, for T > Tp, the ordered ZFC vortex lattice is a superheated state and the ground state of the vortex system is a disordered vortex liquid, while for T < Tp, the ground state is a vortex crystal (or Bragg glass [16]), and the disordered FC state is a supercooled vortex liquid. A thermodynamic phase transition must have taken place in the region of the peak effect, Tm ≈ Tp. This is consistent with the observations in YBCO where the magnetization jump was found to coincide with the peak 7 effect [7,8]. In principle, one should also observe a discontinuous change of Qo when the vortex solid melts. Unfortunately, the expected shift in Qo due to the density change at the vortex solid-liquid transition is below our resolution limit, as we estimate here. From thermodynamic considerations [2], using the Clausius-Clapeyron relations, the vortex density change at Tm is of the order ∆B = − 4πcL C66 TmdHm/dT where cL is the Lindemann number and C66 ≈ B c2 4π b(1−b) 8κ is the vortex-lattice shear modulus, b = B/Bc2. For Nb at H=4.00 kOe, Tm ≈ Tp=4.30 K (Tc2=4.50 K), dHm/dT= -0.846 kOe/K, and C66≈ 0.6x10 4 erg/cm3 assuming a reasonable κ ≈1 and Lindemann number cL=0.1, the upper bound for the jump ∆B at melting is about 0.2 G. For a triangular lattice before melting Qo(B = H) = 2π √