Equilibrium First-Order Melting and Second-Order Glass Transitions of the Vortex Matter
in Bi
2Sr
2CaCu
2O
8H. Beidenkopf,1,* N. Avraham,1Y. Myasoedov,1H. Shtrikman,1E. Zeldov,1B. Rosenstein,1,2 E. H. Brandt,3and T. Tamegai4
1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
2National Center for Theoretical Sciences and Electrophysics Department, National Chiao Tung University, Hsinchu 30050,
Taiwan, Republic of China
3Max-Planck-Institut fu¨r Metallforschung, Heisenbergstrasse 3, D-70506 Stuttgart, Germany 4Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
(Received 16 July 2005; published 16 December 2005)
The thermodynamic H-T phase diagram of Bi2Sr2CaCu2O8was mapped by measuring local
equilib-rium magnetization MH; T in the presence of vortex shaking. Two equally sharp first-order
magneti-zation steps are revealed in a single temperature sweep, manifesting a liquid-solid-liquid sequence. In addition, a second-order glass transition line is revealed by a sharp break in the equilibrium MT slope. The first- and second-order lines intersect at intermediate temperatures, suggesting the existence of four phases: Bragg glass and vortex crystal at low fields, glass and liquid at higher fields.
DOI:10.1103/PhysRevLett.95.257004 PACS numbers: 74.25.Qt, 64.70.Pf, 74.25.Dw, 74.72.Hs
The magnetic field vs temperature (H-T) phase diagram of the vortex matter in high-temperature superconductors, and in Bi2Sr2CaCu2O8 (BSCCO), in particular, has drawn extensive scientific attention [1]. The commonly cited thermodynamic phase diagram of BSCCO currently con-sists of a single unified first-order (FO) melting line. It separates the low-field quasi-long-range ordered Bragg glass (BrG) phase from the high-field liquid and glass phases [1– 6]. It is not clear, however, whether the two high-field disordered phases are thermodynamically dis-tinct, or rather reflect a gradual dynamic crossover from liquid into a frozen, pinned state upon cooling [7,8]. In this Letter we show that the equilibrium phase diagram of the vortex matter is indeed more diverse than the one usually considered.
Experimentally, one of the main obstacles in mapping the low-temperature thermodynamics of the vortex matter is its logarithmically slow relaxation rate. Consequently, the phase diagram has been studied in the past mostly through dynamic phenomena. Two prime examples are the irreversibility line itself, marking the onset of hystere-sis, and the second magnetization peak (SMP), observed along such hysteretic magnetization loops [9].
Recently, though, vortex shaking was shown to be ex-tremely effective in catalyzing relaxation at low tempera-tures [2,10,11]. Its application unveiled the inverse melting and the thermodynamic FO transition as the phenomenon underlying the nonequilibrium SMP [2]. The shaking method employs the segregated penetration of an in-plane field component into the highly anisotropic BSCCO samples in the form of Josephson vortices, which are confined in between the CuO2 planes. In the presence of an ac in-plane field, the Josephson vortices instantaneously bisect the pancake vortex (PV) stacks on their passage, interacting mainly with adjacent PVs, while most of the
PVs in the stack remain at rest [12]. These occasional interactions agitate pinned PVs, assisting them in assuming their equilibrium configuration.
Within the present study we performed local magneti-zation measurements by field and temperature sweeps, while utilizing the shaking method, to map the equilibrium phase diagram of the vortex matter. The cross mapping of the FO melting line along both sweeping directions shows an excellent agreement. Temperature sweeps provided par-ticularly sharp features, with which we demonstrate a liquid-solid-liquid sequence of phases. Ertas¸ and Nelson have predicted such a liquid-solid-liquid sequence to occur within a single temperature sweep [13], but it was never observed experimentally. We further find evidence of a novel second-order (SO) phase transition within the vortex solid phase, which bears important consequences regard-ing the nature of the BrG phase.
The reported results were obtained with a slightly over-doped BSCCO crystal with Tc 90 K grown by the trav-eling solvent floating zone method [14]. This specific sample was polished into a triangular prism of base 660 270 m2 and height 70 m [15] (other samples yielded similar results, to be presented elsewhere). The sample was attached onto an array of eleven 10 10 m2 GaAs=AlGaAs Hall sensors. In all measurements taken below 60 K the sample was subject to a 10 Hz in-plane acfield of amplitude 350 Oe, which was aligned parallel to the planes to an accuracy of a few millidegrees. Note that according to the anisotropic scaling theory [1] this in-plane field is effectively attenuated by a factor ’ 200— the anisotropy constant in BSCCO. We found that at higher temperatures shaking had no effect on the FO transition besides a small broadening (see below).
The field sweep mapping of the FO melting line is shown in Fig. 1. The collapse of the hysteretic magnetiza-PRL 95, 257004 (2005) P H Y S I C A L R E V I E W L E T T E R S 16 DECEMBER 2005week ending
tion into a reversible behavior upon shaking is demon-strated in Fig. 1(a) taken at 32 K. A reversible magnetiza-tion step appears instead of the SMP. To better resolve the step we plot in Fig. 1(b) the derivative of the measured induction with respect to applied field dB=dH. The FO transition thus appears reversibly as a -like peak on top of the dB=dH 1 background. Figure 1(c) shows a color scheme of the derivatives dB=dH measured by field sweeps within the temperature range 28–80 K. The indi-vidual melting peaks combine to give the locus of the FO transition line HmT. Its negative slope at elevated tem-peratures becomes positive below 38 K. This nonmono-tonic behavior marks the change in the character of the transition from thermally induced above the extremum to disorder driven in the inverse-melting region [5,6,13].
In addition to the field sweeps, our experimental setup also enables temperature sweeps in the presence of vortex shaking. It thus allows one to measure directly the tem-perature dependence of the equilibrium magnetization at a constant applied c-axis field. At fields slightly lower than 390 Oe [e.g., at 380 Oe along the dashed line in Fig. 2(a)] these sweeps should cross the melting line twice. Re-markably, the measured local induction in Fig. 2(b) indeed shows two very clear and opposite equilibrium magnetiza-tion steps on both descending and ascending sweeps. Note that for clarity we have subtracted from the data a linear slope T. It is contributed both by the slight temperature dependence of the Hall coefficients of the sensors and by the linear term of the magnitude of the diamagnetic equi-librium magnetization, which monotonically decreases with temperature. This is the first observation of two FO transitions obtained in a single temperature sweep. More-over, the two steps are equally sharp and with comparable
heights of about 0.15 G. This demonstrates that the ther-mally and disorder-driven processes, responsible for the melting and the inverse melting, respectively, are equiva-lent mechanisms leading to a FO destruction of the qua-siordered vortex solid.
The temperatures at which the magnetization steps ap-pear along the temperature sweep of Fig. 2(b) are in complete agreement with the melting behavior deduced from field sweeps (dotted lines). Therefore, the transition line in Fig. 2(a) is independent of the specific H-T path along which it is approached —a mandatory equilibrium property. The remaining small hysteresis of 0.1 to 0.2 G between downward and upward sweeps apparently results from surface barrier effects, while the vortices in the bulk are well equilibrated by the shaking. The finite widths (about 0.7 K) that the melting steps attain are mainly due to a spatial and a temporal averaging mechanisms. The first is introduced by the sensor’s finite active area that averages over the propagating melting front [16], which results from the spatially inhomogeneous equilibrium magnetization profile [17]. The temporal one is a by-product of the shaking technique. The in-plane field component is known to slightly reduce the melting temperature [12,18,19]. Consequently, our time averaged measurement in the pres-ence of the ac in-plane shaking field results in an addi-tional broadening due to the instantaneous periodic shift of the effective local melting temperature.
We thus turn to report the detection of a novel phase transition, whose signature is a distinct break in the slope of the magnetization MT. It is visible around 37 K in Fig. 2(b) at 380 Oe, and becomes much more pronounced
30 35 40 45
B −
T (G)
T (K)
Liquid Solid Liquid 380 Oe
b
0.1 T (K) H (Oe) 350 420 34.2 42.3a
H m(T) 380 αFIG. 2 (color online). (a) The FO melting line HmT mapped
via field sweeps (open circles). (b) Local induction BT in the presence of shaking, measured upon temperature sweep at 380 Oe along the dashed line in (a). A linear slope T was subtracted for clarity. The two equally narrow FO magnetization steps (black segments) show a liquid-solid-liquid sequence. The temperatures at which the phase transitions occur coincide with those, extracted from field sweeps (doted lines). The color code reflects different phases in Fig. 4.
1.05 dB/dH 30 40 50 60 70 80 0 100 200 300 400 500 T (K) H (Oe) H (Oe) B − H (G)
Solid
Liquid
T=32 K 0.98 350 H (Oe) 410 b a T=32 Kc
Hm(T) -20 -10 350 410FIG. 1 (color online). (a) The 32 K field sweeps with (solid curve) and without (dotted curve) an in-plane 350 Oe-10 Hz shaking field. Reversible steps in magnetization (solid arrows) appear instead of the hysteretic SMP (open arrows) upon shak-ing. (b) The derivative of the induction with respect to applied field dB=dH at 32 K as a color scheme. The first-order transition appears as a paramagnetic peak on top of the dB=dH 1 background. (c) Successive mapping of the first-order melting line HmT measured by field sweeps.
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at fields further away from the extremum of the FO melting line, as depicted by Fig. 3. The 420 Oe temperature sweep [Fig. 3(a)] does not intersect with the FO line, hence no steps appear in the local induction. Nevertheless, a sharp break in slope is clearly resolved along both descending and ascending temperature sweeps at Tg. A sharp revers-ible break in the induction slope appears also in the 350 Oe temperature sweep of Fig. 3(b) (dotted line) in between the two melting steps, hence within the solid phase. This non-analytic behavior is emphasized by the sharp step in the derivative dB=dT shown in the insets. These kinks were found also in other samples and at various Hall-sensor locations, and did not depend on the sweeping rate. We thus conclude that this break in slope of the equilibrium magnetization MT indicates a thermodynamic SO phase transition.
Mapping of both the first-order HmT and second-order HgT transition lines onto the equilibrium H-T phase diagram is given in Fig. 4. The SO line (solid dots) inter-sects the melting curve (open circles) and shows weak field dependence throughout the mapped region (and therefore cannot be readily observed by field sweeps). The resulting phase diagram consists of four distinct thermodynamic phases.
The high-field part of the novel SO line can be naturally identified with the long sought glass transition line. It asserts that the low-temperature glass phase is indeed thermodynamically distinct from the high-temperature liq-uid one. Several experimental studies have observed bulk irreversibility features in BSCCO, which appeared above HmT at about 35 K [20,21]. However, all these studies probed dynamic or nonequilibrium vortex properties. The glass line of Fig. 4 is the first experimental evidence of such a thermodynamic transition in BSCCO.
Yet, the most intriguing result in Fig. 4 is the detection of the SO line within the vortex solid region. This implies that two distinct thermodynamic phases are present in the low-field region below HmT, contrary to the common belief
that a single BrG phase prevails throughout this part of the phase diagram. A number of previous studies indicated a depinning line of similar topology within the BrG below HmT [20]. However, all these measurements probed only the nonequilibrium properties, which were consistent with the existing theoretical dynamic predictions and simula-tions of depinning [13,22]. In contrast, the present finding of a thermodynamic line requires a more fundamental reconsideration.
It is interesting to note that in YBa2Cu3O7 crystals a thermodynamic signature of a SO transition within the liquid phase has been reported [23]. There, however, the SO line emanates from the upper critical point of the FO line, directly extending it to higher fields. This topology is consistent with several dynamic measurements in YBa2Cu3O7 [24,25], although alternative topologies have been also suggested [26,27]. In contrast, our thermody-namic data of BSCCO show that the SO and the FO transitions are two independent lines that intersect each other nearly at a right angle.
Several theoretical studies have shown that under the elastic medium approximation quasi-long-range order of the vortex lattice is still retained in the presence of quenched disorder, giving rise to the BrG phase [28,29]. This phase was found to be stable at all temperatures (as long as topological excitations are excluded) in systems of dimensionality greater than two and lower than four. This is probably the reason why a nontopological thermody-namic phase transition of the BrG phase was hardly ever considered in 3D models. An exception is a Josephson-glass line that was suggested to exist within the BrG region [30]. Still, the general belief is that the BrG phase is robust until dislocations proliferate, which gives rise to the FO phase transition [6,8,28]. In 2D systems, however, the BrG models did find a possible finite-temperature depinning
25 30 35 40 45 50 55 60 B − T (G) T (K)
b
0.2 0.2 Tga
Tg 350 Oe 420 Oe 30 35 40 T (K) dB/dT (mG/K ) 40 Tg 35 36 37 38 T (K) dB/dT (mG/K ) 40 Tg αFIG. 3 (color online). Local induction BT, measured along temperature sweeps while shaking. A reversible sharp break in the slope (at the dotted lines) appears both above [(a) 420 Oe] and below [(b) 350 Oe] the melting line HmT. The insets show
a corresponding step in the derivative dB=dT, signifying a thermodynamic second-order phase transition.
H( O e)
Glass
Liquid
BrG
Crystal
H( O e) T (K) 40 80 0 400 200 60 Hm(T) Hg(T) Hg(T) Hm(T) T (K) 0 200 400 600 20 40 60 80FIG. 4 (color online). The thermodynamic phase diagram of BSCCO accommodates four distinct phases, separated by a first-order melting line HmT (open circles), which is intersected by
the second-order glass line HgT (solid dots). The inset plots an
equivalent phase diagram, calculated based on Ref. [31], con-sisting of a second-order replica symmetry breaking lines HgT
both above (dotted line) and below (dashed line) the first-order transition HmT (solid line).
PRL 95, 257004 (2005) P H Y S I C A L R E V I E W L E T T E R S 16 DECEMBER 2005week ending
transition, above which disorder is no longer relevant. Therefore, the observed SO transition could be accounted for within these models only in the extreme case of vanish-ing couplvanish-ing between the superconductvanish-ing layers.
In contrast, in a recent theoretical work [31] the free energy of the vortex matter in the presence of quenched disorder was explicitly calculated under the lowest-Landau-level approximation in a 3D model. It was found that two transitions are present: a FO melting, at which the quasi-long-range order is destroyed, and a SO glass tran-sition, below which the replica symmetry is spontaneously broken. The inset of Fig. 4 shows the phase diagram, calculated using a similar effective 2D model with parame-ters optimized for BSCCO [32]. The calculations repro-duce the measured features very well. Amongst them are the melting and inverse-melting behavior, the discontinuity of the magnetization slope dM=dT at the glass transition, and the HgT line itself, which resides at slightly higher temperatures as compared to experiment. In addition, the calculations show that the portions of the HgT line, lying above and below the FO transition HmT (dotted and dashed lines, respectively), are slightly shifted from each other. Therefore, they do not cross the melting line at a single point, but rather form two closely located tricritical points along it. This minute shift can hardly be seen in the inset of Fig. 4 and is below our current experimental resolution.
Within this model the high-field glass phase and the low-field BrG are strongly pinned and replica symmetry bro-ken, whereas the two high-temperature phases are replica symmetric and thus reversible. This conclusion of revers-ibility is consistent also with the existing dynamic mea-surements [20]. It is therefore tempting to speculate that the phase above HgT and below HmT should acquire a true crystalline order. However, since Rosenstein and Li did not calculate the structure factor and our measurements do not probe this quantity, the proposed vortex crystal phase certainly calls for further experimental and theoreti-cal investigations.
In summary, we present thermodynamic evidence of a possibly second-order glass transition line that splits the quasiordered vortex solid into two distinct phases. By comparing the results with existing dynamic measure-ments and a new theoretical study, we suggest that the two phases are BrG and a vortex crystal. The glass line crosses the first-order melting line near its extremum and extends to higher fields, giving rise to two thermodynami-cally distinct disordered phases —a glass and a liquid.
We thank D. Li, V. M. Vinokur, and B. Horovitz for stimulating discussions. This work was supported by the Israel Science Foundation Center of Excellence, by the German-Israeli Foundation G. I. F., by the Minerva Foundation, Germany, and by Grant-in-aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology, Japan. B. R. acknowledges the support of the Albert Einstein Minerva Center for
Theoretical Physics and E. Z. the U.S.-Israel Binational Science Foundation (BSF).
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