Visualizing a dilute vortex liquid to solid phase transition in a Bi2Sr2CaCu2O8 single crystal

Using high-sensitivity magneto-optical imaging, we find evidence for a jump in local vortex density associated with a vortex liquid to vortex solid phase transition just above the lower critical field in a single crystal of Bi2Sr2CaCu2O8. We find that the regions of the sample where the jump in vortex density occurs are associated with low screening currents. In the field–temperature vortex phase diagram, we identify phase boundaries demarcating a dilute vortex liquid phase and the vortex solid phase. The phase diagram also identifies a coexistence regime of the dilute vortex liquid and solid phases and shows the effect of pinning on the vortex liquid to vortex solid phase transition line. We find that the phase boundary lines can be fitted to the theoretically predicted expression for the low-field portion of the phase boundary delineating a dilute vortex solid from a vortex liquid phase. We show that the same theoretical fit can be used to describe the pinning dependence of the low-field phase boundary lines provided that the dependence of the Lindemann number on pinning strength is considered.


Introduction
The conventional [1] field (H)-temperature (T) phase diagram for an ideal type-II superconductor consists of a mixed state with a hexagonally ordered vortex lattice present between the lower critical field (H c1 (T )) and the upper critical field (H c2 (T )). Numerous studies have established that long-range spatial order in the ideal hexagonal Abrikosov lattice is destroyed due to pinning in a non-ideal superconductor. The vortex state in a superconductor could either be a reasonably well-ordered glassy phase devoid of topological defects such as dislocations, namely a Bragg glass, or be a disordered phase with topological defects, namely the vortex glass phase [2,3]. At high H when the inter-vortex spacing, a 0 ∝ √ φ 0 /B (where φ 0 is the quantum of magnetic flux and is equal to 2.07 × 10 -7 G cm 2 , and B ∼ µ 0 H ) is less than the superconducting penetration length λ (λ is the range of vortex-vortex interaction), non-local effects lead to a softening of the elastic moduli of the vortex lattice [4,5], thereby making the vortex lattice susceptible to thermal fluctuation and pinning effects [2,5]. In high-T c superconductors, high superconducting transition temperatures (T c (0)), high anisotropy and small condensation energy result in the vortices being highly susceptible to thermal fluctuations. Across the melting phase boundary at high fields, large thermal fluctuations cause the soft vortex solid (VS) to melt into a vortex liquid (VL) phase, which is characterized by zero shear elastic moduli [2,6,7]. Akin to ice-to-water transformation, the VS to VL melting transition produces an enhancement in the density of vortices. The melting at higher fields has been well studied both theoretically and experimentally. Experiments on melting at high H have established the presence of a jump in the equilibrium magnetization [8] and the presence of latent heat [9] associated with the first-order VS to VL melting phase transition. However, the nature of the vortex phase at low fields is less well investigated. At low fields (near H c1 ), where a 0 > λ, the weak inter-vortex interaction leads to exponentially small elastic moduli [2,4] of the lattice, making the vortex state susceptible to thermal fluctuations and pinning effects. In the H-T 3 phase diagram at intermediate fields, one encounters the VS phase. It has been theoretically proposed [2,6,10] that at fixed T, by increasing H or decreasing H, the VS undergoes a phase transformation into the VL phase across a phase boundary close to H c2 (T ) and another phase boundary close to the H c1 (T ). The VS therefore can melt at both high and low fields and in the H-T vortex phase diagram the VS phase is bounded by two lines, across which the VS melts into a VL phase. In the H-T vortex phase diagram, at high fields, the line across which the VS melts follows a monotonic behavior with negative slope tracking in the behavior of H c2 (T ), albeit with a different temperature dependence. In the low-field regime near H c1 (T ), the line across which the dilute VS transforms into a VL is associated with an exponentially small shear modulus at low H (for a schematic of the melting phase boundary see [11]). Apart from thermal fluctuations at low magnetic fields due to the weakened shear modulus, the elastic vortex lattice is highly susceptible to the disordering influence of pinning. Experiments on low-T c superconductors with weak pinning have suggested a disordered vortex phase [12][13][14][15] present at low H which is distinct from the elastic, ordered vortex phase present at higher fields. Recently, scanning hall probe imaging of the vortex state in disordered thin films of Bi 2 Sr 2 CaCu 2 O 8 (BSCCO) [16] at low fields (above 2.4 Oe) have captured images with fluctuating contrast. The authors ascribe the fluctuating contrast to the presence of a VL phase, wherein the rapidly fluctuating vortices in the liquid phase are intermittently trapped and pinned on the strong pinning centers, leading to enhancement in contrast and its subsequent decay as the vortex escapes from the pin. Using the sensitive magneto-optical imaging (MOI) technique, we have identified a change in the equilibrium vortex density associated with a VL to VS phase transition at low fields near H c1 . We construct an H-T vortex phase diagram identifying a line across which the VL transforms into a VS phase via an intermediate regime of phase coexistence comprising VL and VS phases. The low-field phase transformation boundary we have identified fits to the theoretically proposed expression [10] for the low-field line across which a VL transforms into a VS phase. Using the MOI technique, we have also spatially resolved the magnetization relaxation in different regions of the sample to enable the construction of a coarse map of the pinning landscape in the sample. We investigate the effect of pinning on the low-field VL to VS transformation line by showing a significant increase in the Lindemann number with pinning strength.

Experimental details
We chose a high-quality single crystal of BSCCO [17] with dimensions of (0.8 × 0.5 × 0.03 mm 3 ) and T c = 90 K. To introduce vortices in the sample, a magnetic field was applied parallel to the c-axis of the single crystal (H c) using a copper coil solenoid magnet. Highsensitivity and high-field homogeneity across the sample is maintained within the solenoid in the field range 0-200 Oe. We employ the conventional MOI [18] as well as the differential magnetooptical (DMO) imaging technique [19,20] to image the changes in local vortex density. A schematic diagram and details of our MOI setup have been presented elsewhere [20]. The DMO technique is sensitive to small changes (δ B z (x, y)) in the local field distribution B z (x, y) (where (x, y) defines the sample plane perpendicular to H, and z is along H). We obtain differential (δ(x, y)) images by increasing H by an amount δ H = 1 Oe and capturing the Faraday rotated images I i (H ) and I i (H + δ H ) i number of times at H and H + δ H ,  where usually k = 20. Appropriate calibration converts δ(x, y) to provide a measure of δ B z (x, y) [19,20]. Unlike conventional DMO [19], due to irreversible magnetization response of BSCCO at low fields, we usually do not average δ(x, y) by repeated modulation of the external field by δ H about H. In the δ(x, y) images (see, e.g., figure 1), the bright and dark contrasts correspond to high and low δ B z .  figure 1(e) has been artificially offset by 3 G for the sake of clarity. The bright contrast observed along the sample edges (cf figure 1(a)) is due to strong edge screening currents (discussed later in the context of figures 2(a) and 4(b)). The gray contrast outside the sample and away from the sample edges in the differential images corresponds to the change in the Faraday rotated magneto-optical intensity due to the increase in H, namely δ H , by 1 Oe. Note that the δ B z (r ) ∼ 1 G far away from the sample edges (close to the 0 µm tick in the graph shown in figure 1(e)). At 30 Oe (figure 1(a)), due to strong diamagnetic screening of flux inside the superconductor, most of the regions inside the sample possess a dark contrast, where δ B z = 1 G. This corresponds to the vortex-free diamagnetic Meissner state. Flux enters the sample interior through a cigar-shaped arm, cf the gray location near the * marked in figure 1(a) at 30 Oe. When H is increased by 1 Oe, δ H = 1 Oe, the local δ B z near * increases by ∼1 G, as seen in figure 1(e) (red curve). When H is increased to 36 Oe, δ B z abruptly increases to ∼2 G (see the blue curve in figure 1(e) and the corresponding bright region around the * in figure 1(b)). When H is increased further to 42 Oe (green curve in figure 1(e)), δ B z near the * region (figure 1(c)) decreases back to a value of ∼1 G, while the region with enhanced mean δ B z ∼ 2 G spreads to other locations of the sample where flux has already penetrated (cf the double-headed arrow marked on the green curve in figure 1(e)). At 57 Oe the enhanced δ B z reduces down to 1 G = δ H = 1 Oe, which corresponds to the onset of a weakly pinned vortex phase (the VS phase), uniformly over the entire sample (cf figure 1(d) and the corresponding pink curve of figure 1(e)). Figure 1(f) compares the δ B z signal at 36 and 57 Oe in a 50 µm wide region in the sample, namely a region between 550 and 600 µm in the line scans of figure 1(e), where the anomalous jump in δ B z occurs at 36 Oe. It is clear from figure 1(f) that δ B z is anomalously enhanced at 36 Oe compared to that at 57 Oe. As will be shown later in figure 2(b), at 50 K, upon increasing the field from 36 to 57 Oe, a VL phase transforms into a VS phase. At this juncture, it is worth mentioning that the observation of a brightening in 6 a DMO image corresponding to an enhanced δ B z over and above the background intensity is used to signify [19,22] a vortex phase transformation associated with a change in the density of vortices. The (H, T) where δ B z signal locally becomes maximum (and well above δ H = 1 Oe) is situated deep inside the VL phase and signals the transformation into the VS phase. This behavior occurs reproducibly and repeatedly at the same location in the sample over repetition of the above measurements.

The nature of vortex distribution in the sample and determination of the width of the jump in B z
Along with differential δ B z (x, y) images, we have also measured the B z (x, y) distribution across the sample using conventional MOI (images not shown). Figure 2(a) shows the variation of B z (r ) with increasing H across the red line shown in figure 1(a), with the distance (r) being measured along the red line. In a sample with weak pinning and large edge currents, vortices are pushed toward the center of the sample, resulting in the dome-shaped field profile [23].
where δ H = 1.5 Oe and | 1,2 represents points of intersection along line 1 or 2 in figure 2(a).
Oe is comparable with the δ B z ∼ 2 G noted earlier in the DMO imaging procedure (figure 1). In figure 2(b), beyond H c1 ∼ 32 Oe, δ B z | 1,2 begins deviating systematically from 0 G. As H > H c1 , for the curve labeled 1 in figure 2(b) (black squares), δ B z | 1,2 increases and crosses a value of 1.5 G before reaching a maximum value of ∼3 G at H M = 39 Oe. Beyond 39 Oe, δ B z | 1,2 begins to decrease from ∼3 G and settles down to a constant value of ∼1.5 G = δ H . In figure 2(b), the black dashed curve through the data points is a guide to the eye representing the above behavior for the δ B z found across line 1 in figure 2(a), namely δ B z | 1 . At higher H (45 Oe and above) the anomalous enhancement in the peak height of B z (r ) ceases, cf figure 2(a), whereas above 45 Oe the consecutive B z (r ) profiles (taken in intervals of 1.5 Oe) are almost equi-spaced along the B z -axis. At fields above 50 Oe in figure 2(b) δ B z | 1,2 = 1.5 Oe = δ H ; therefore in figure 1(d) at 57 Oe, one observes a uniform grayish contrast acquired over the entire sample. In comparison with line 1, across line 2 (see figure 2(a)), the peak in δ B z | 1,2 occurs later, namely at H M = 40.5 Oe (see red circles with the dotted red curve in figure 2(b)). The δ B z (r ) behavior across line 2 shows that a behavior similar to line 1 exists at different regions of the sample, and the peak in δ B z (r ) which corresponds to a brightening in the DMO image occurs at slightly different fields at different regions of the sample. The above discussion suggests that the jump in δ B z progressively spreads to neighboring regions of the sample with increasing H.

7
The horizontal dashed line in figure 2(b) represents δ B z = δ H , namely the line across which the amount of change in local flux density is equal to the change in external field. Note, for the sake of comparison, we have shown with a dotted blue curve in figure 2(b) a representative sketch of a monotonic variation in δ B z with increasing H for depicting a hypothetical situation where no jump in B z exists, namely the VS phase is gradually attained as the field is increased from H c1 . Note that both δ B z | 1,2 (H ) curves in figure 2(b) exceed the dotted blue curve. We would like to mention that by comparing figure 1 with the B z (r ) profile in figure 2, it is clear that the jump δ B z ∼ 2 G noted in figure 1 occurs in a region of the sample where vortices have already penetrated and the dome-shaped B z (r ) profile is established at a lower field. For example, near the location * in the sample where the jump (bright contrast) in δ B z has been noted at 36 Oe (figures 1(a) and (b)), vortices had already penetrated into this region prior to 36 Oe as indicated by the presence of the dome-shaped profile from 32 Oe onwards (cf figure 2(a)). At low fields just above H c1 , we propose that the change δ B z ∼ 2 G observed in the DMO images is associated with the appearance of a first-order-like vortex phase transition from a VL to a VS phase. The can sustain greater changes in vortex density compared to a VS phase; namely we identify the peak δ B z (H ) signal corresponding to the limit of a low-field VL phase above H c1 (we identify this limit as H M in figure 2(b)). Beyond the maximum in δ B z | 1,2 the signal gradually decreases with increasing H (> H M ), which corresponds to the regime of coexistence of a VL and a VS phase, before saturating to a value = 1.5 G = δ H . In figure 2(b) the field regime of VL and VS coexistence for line 1 is identified with a cross-shaded region, while the coexistence field regime for line 2 overlaps with that of line 1 and is identified with a transparent gray-shaded region overlapping the cross-shaded region. Beyond the coexistence region, we identify the VS phase in figure 2(b). We would like to emphasize that above 55 Oe (inside the VS phase) no further brightening in the DMO image is found in response to δ H modulation, suggesting that the brightening observed in figure 1 is due to an additional enhancement in δ B z over and above the conventional effect of B z (r ) dome height modulation when the external H is modulated.

Location of jump in B z at low H with respect to the vortex solid phase
where A is the area over which averaging is performed and the local field B z (x, y) is determined from conventional MOI (without DMO) [18,21]. M(H) for the whole sample corresponds to A (A is the entire sample area), while the local M(H) for * corresponds to averaging over a region around the location marked with * in figure 1 with area A = 25 µm 2 . From the initial linear portion of the M(H) curves (blue and red curves for bulk and local responses, respectively), it is evident that the vortex penetration field is higher for the bulk. The higher bulk penetration field is associated with surface and geometric barriers [24][25][26], whose evidence was found in  enhancement in the local vortex density (ρ = B z φ 0 ). The jump in δ B z suggests that in regions of the sample where vortices have penetrated, the local vortex density is anomalously enhanced as the external magnetic field is increased just above H c1 , indicating the presence of the VL phase. On further increasing H beyond the sharp drop, the local (red curve) diamagnetic response begins to increase and tends to approach the bulk diamagnetic magnetization value (namely the red curve shows a tendency to increase toward the blue curve in figure 3(a)) with a bump-like feature. Over this region in figure 2 [27] due to weakening of the diamagnetic screening currents with increasing H. Recall that in the field regime of around 60 Oe and beyond, we have already noted the presence of a VS phase (identified with δ B z = δ H = 1.5 Oe) in figure 2(b). In DMO, the observation of δ B z = δ H which is characteristic of the onset of a reversible, weak pinning VS phase [20] appears in the field regime of around 60 Oe. The B z (r ) line scan in figure 2(a) shows the presence of a dome-shaped field profile above 57 Oe at 50 K, which is characteristic of uniform vortex distribution associated with a weakly pinned VS phase in the interior of the sample. Furthermore, in figures 4(a) and (b) we also show the is reminiscent of the second magnetization peak anomaly observed in HTSCs [28]. It may be recalled that earlier suggestions [15] for the presence of a low-field disordered vortex state in low-T c superconductors had detected the presence of a similar second magnetization peak like anomaly close to H c1 . Here, the bump-like feature in the local M(H) curve demarcates the approximate H range over which the VL and VS phases coexist (identified with the crossshaded region in figure 3(a); a similar shaded area in a different location of the sample has been shown in figure 2(b)). The location of H M (cf figure 2(b)) is indicated by an arrow in figure 3(a).  figure 3(c), which corresponds to a regime just above H c1 (T ) (cf figure 3(a)), is reproduced almost identically at H = 45 Oe (50 K) in figure 3(d) where the measurement is made in the fifth quadrant (namely measurements made while increasing the H from 0 G to the desired +H value, where prior to increasing H from 0 G, the field was originally cycled across four quadrants, namely from 0 to +200 G (I), from +200 to 0 G (II), from 0 to −200 G (III) and from −200 to +0 G (IV)). The same is confirmed by the observation of a similar anomalous brightening in measurements made at different temperatures, as seen in the DMO images in figures 3(e) and (f) which are obtained at 60 K, 33 Oe in the first and fifth quadrants, respectively. Therefore, the brightening associated with the enhancement in δ B z at H M (T ) is not related to H c1 (T ). Note that in the second quadrant, due to a strong irreversible magnetization response which leads to a positive magnetization background, it becomes difficult to discern the jump in magnetization when reducing the magnetic field from the VS phase in the low H regime.

Estimating the equilibrium height of jump in B z
While the jump in B z provides evidence for the existence of a VL phase prior to the onset of a VS phase, it is possible that the jumps measured in δ B z in figures 1 and 2 need not be the equilibrium value associated with this transition due to the metastable effects associated with pinning. To estimate the size of equilibrium jump in magnetization, corresponding to a dense VL phase transforming into the VS phase (cf figure 1(f)), in figure 3(b) we investigate the effect on δ B z of repeated modulating by δ H = 1 Oe about H (N times), namely, we obtain the average differential image, δ(x, y) N at a given H and T. From δ(x, y) N we define δ B z bright = δ B z in a region in the δ(x, y) N image with bright contrast (like the * region in figure 1(b)) −1 G (where 1 G is the δ B z produced in response to δ H = 1 Oe in a reversible H, T regime). In figure 3(b) for H = 36 Oe and T = 50 K, we plot δ B z bright versus N in the * region where one observes a bright contrast (cf figure 1(b)). From figure 3(b) we see that beyond N = 10, the δ B z bright approaches an asymptotic non-zero constant value of about 0.1 G, which implies that over and above the 1 G change in δ B z due to δ H = 1 Oe, there exists an additional equilibrium enhancement in δ B z of ∼0.1 G. In a region of the sample where a uniform VS phase has formed, for example at 57 Oe, at 50 K in figure 1(d), over most of the sample, δ B z bright = 0. The quantity δ B z bright is a measure of the jump in equilibrium vortex density due to a denser VL phase, which is different from the VS phase that sets in at higher fields. Inset (b) of figure 5 shows a DMO image from an isofield measurement, where we keep H fixed and vary T. At each T we modulate the temperature between T and T + δT , where δT = 0.4 K, and obtain differential MO images. We observe bright regions in the sample at 50 K in this varying temperature measurement in a constant dc field of 42 Oe, which coincides with similar bright regions observed at 42 Oe in the varying magnetic field experiment at 50 K in figure 1(c) (also reproduced as figure 5(c)), indicating that the brightening feature observed is not path dependent in the H-T space, but is a signature of a thermodynamic phase transformation.

Distribution of screening currents in the low-field regime
Using an inversion technique [29,30] we determine the distribution of the absolute value of the superconducting screening current density, j (x, y) = |J(x, y)| =| j x (x, y) 2 + j y (x, y) 2 |, across the sample from B z (x, y) obtained from the conventional MO images, where J = j x (x, y)x + j y (x, y)ŷ is the net current density at (x, y) and j x and j y are the x and y components of the current density. The image in figure 4(a) shows the j(x, y) distribution across the sample at 57 Oe and 50 K (cf figure 1(d)).
The bright regions in the image represent significant j(x, y), while dark regions correspond to negligible j(x, y). Note from the line scan in figure 4(b) that the j(x, y) across major portions of the sample is almost zero while a large j (x, y) of ∼10 4 A cm −2 is confined only along the sample edges and along the linear (diagonal) defects inside the sample which are regions with enhanced pinning or large screening currents. Such linear defects in single crystals of BSCCO are microscopic regions with compositional inhomogeneities incorporated into the crystal during its growth and have been reported earlier [22,31]. These regions possess enhanced vortex density compared to their neighboring regions [31] and as discussed later they may also act as nucleation centers for initiating the low-field melting of the vortex state. From the earlier discussion of figures 2(b) and 3(a), we believe that H around 60 Oe and beyond (like 57 Oe in figure 4(a)) corresponds to field values at which interactions between the vortices become sufficiently strong to promote the formation of a VS phase. The onset of an equilibrium vortex distribution produces negligible gradients in the vortex distribution which is seen as low j(x, y), as in figure 4(a), and present over a majority of the sample interior where the VS exists. The inset of figure 4(b) shows an image of the j(x, y) distribution at 36 Oe, at 50 K, in which we observe large diamagnetic screening currents distributed across the Meissner-like regions of the sample. However, it is noteworthy that the bright regions, where a jump in δ B z was observed in figure 1(b), have low j(x, y) values (cf within the region encircled by the red dashed curve in figure 4(b), inset). We reaffirm our original inference that the brightening observed in figure 1, upper panels, and figures 3(c)-(f) is a feature which is not associated with penetrating vortices. Had the brightening in δ B z been only related to penetrating vortices, then we should have observed large gradients and hence large screening currents j(x, y) in regions where the jump occurs, which is absent as observed in figure 4. The above is true for all the bright regions we have observed and for those shown in figures 1 and 3.

Mapping the pinning landscape and the location of regions with jumps in B z
By capturing images in the interval of 200 ms, using conventional MOI technique we capture the time (t) evolution of B z (x, y, t) and hence of M(x, y, t) of the remnant magnetized state of the superconductor (created from 150 Oe). Figure 4(c) shows the typical relaxation of the MOI intensity count versus t at 50 K, observed in the small region of area 25 µm 2 located in a region marked * in figure 1(a). The intensity is calibrated to obtain the remnant magnetization M rem (x, y, t). Note that the magneto-optical intensity I(x, y, t) is related to M rem (x, y, t) as M rem (x, y, t) ∝ [I (x, y, t)] 0.5 . By fitting the M rem (x, y, t) data to the expression [32] figure 4(c)), we determine the local pinning potential U c . Using the above procedure, from the relaxation of M rem determined at different positions in the sample, we obtain the coarse pinning potential landscape U c (x, y) in the sample. The coarse graining is over a 100 × 100 µm 2 area. The U c values are normalized to the maximum value of ∼85k B found in the sample and color coded as shown in figure 4(d). Figure 4(d) reveals that the enhanced δ B z bright = 0.1 G is primarily nucleated first along the linear defect regions with large U c (red). From here, it first spreads to the nearby regions with lower U c (cooler (blue) colored regions), and then finally to other regions with larger U c (the warmer (red) colored regions in figure 4(d)). By comparing images of figure 1 with the map of the pinning landscape in figure 4(d), it appears that the vortices penetrate into the superconductor preferably through the regions with linear defects. Subsequently, the jump in δ B z is also nucleated in the same neighborhood of linear defects after which the jump propagates through the weaker pinning regions of the sample before reaching the stronger pinning regions of the sample at higher fields.
The jump in B z , namely δ B z bright , which is the contrast difference encountered in the VL measured w.r.t. to the VS phase (with δ B z = 1 G), is associated with the abrupt enhancement in local vortex density, which we propose is due to melting into a low-field VL phase from a higher field VS phase. Recall here that such jumps in local field have been observed for melting into the VL phase at high fields [8,19]. Furthermore, as mentioned earlier, j(x, y) distribution shows that regions with δ B z bright = 0.1 G are devoid of screening currents, which is characteristic of a VL phase associated with almost zero critical current density due to the predominance of thermal fluctuations overcoming pinning in this phase. Figure 1 suggests that after the vortices enter the superconductor above H c1 , one encounters a thermal fluctuationdominated VL phase which at higher fields (namely at higher vortex densities) transforms into a VS phase via a first-order-like transition. Referring to figure 3(b), above H c1 the density of the VL phase is higher than the VS phase at higher fields; namely, it is ρ + δρ, where this increase in vortex density, δρ, associated with the equilibrium δ B z bright ∼ 0.1 G jump, is due to the presence of a denser VL phase compared to the VS phase. With further increase in H, the excess δρ decreases as the VS phase crystallizes out of the VL phase.

H-T vortex phase diagram identifying the low-field vortex liquid, VS phases and the coexistence regime
In different regions of the sample, the upper limit of the VL phase (H M (T )) is determined from the peak in the δ B z signal measured as a function of H (cf figure 2(b)) at different T. In inset (c) of figure 5, four regions labeled 1, 2, 3 and 4 in the sample are identified. By comparing with the pinning landscape in figure 4(d), two are located within weak pinning regions (labeled 1 and 2) and the other two are located within strong pinning (labeled 3 and 4) regions of the sample. Figure 5(a) shows the location of the field H M (T ) in region 4 (the strongest pinning region). Using the criterion that at H, just greater than H c1 , the δ B z begins to systematically deviate from 0 G as in figure 2(b), we estimate the local H c1 (T, r ), where r denotes the location of the region where H c1 (T ) is being determined. The crosses indicate the values of H c1 (T, r ) in region 4 of the sample. The pink line is a guide to the eye. As shown in figure 5(a), the VL phase exists between this H c1 (T ) line and up to the H M (T ) curve for region 4 (solid red line through squares). Here the cross stripped region identifies the width of the VL phase regime for the strongest pinned regions of the sample. From data recorded at different H and T, we determine the (H, T) at which δ B z decreases and saturates to a value of δ H (cf at ∼50 G in figure 2(b)) and corresponds to the onset of a VS phase. In figure 5(a) (3,4) the threshold H or T for the transformation from a VL to VS phase is higher compared to the relatively weaker pinning regions (1,2). The H M (T ) lines for the four regions of the sample indicate that the phase boundary across which the VL phase transforms into the VS phase progressively shifts upwards with increasing pinning. It is noteworthy that at high T close to T c , one finds that the different H M (T ) curves merge together, corresponding to the uniform appearance of the VL phase across the entire sample (see DMO images at 85 K in figure 6). Figures 6(a)-(c) show DMO images of VL phase spreading across the sample at T = 85 K with H = 9, 12 and 15 Oe and δ H = 1 Oe, respectively. Note that at 12 Oe, brightening corresponding to a peak in δ B z appears across the entire sample and the bright contrast disappears altogether by 15 Oe. This suggests that at 85 K, the low-field first-order-like VL to VS transition is sharper (with a width of <3 Oe) as compared to the broader width of the transition at 50 K, namely a width of ∼5-10 Oe. At T close to T c the VL phase appears almost uniformly across the entire sample, which is due to thermal fluctuations smearing out the variations in the pinning landscape, thereby making the phase transition set in more uniformly across the entire sample. The H M (T ) of 12 Oe at T close to T c (85 K) coincides with the point on the high-field melting line in BSCCO [20,30,33]; the changes in B z observed in this (H, T) regime are also comparable. In our sample there are significant effects of pinning which lead to a broadening of the transformation from the VL to VS phase at low fields. It has been shown earlier [34] that at low T the jump in magnetization associated with the high-field VS to VL transition is smeared out due to strong pinning-induced irreversible magnetization response. This high-field jump in magnetization in this situation is discernable only after suppressing the underlying irreversible magnetization response with an in-plane oscillating magnetic field. We have made no such attempts to suppress the underlying irreversibility in magnetization to identify the high-field jumps at low T. Therefore within the limited low-field range of our setup, at low T we are unable to identify the high-field melting line in our sample, which is presumably broadened out due to pinning effects. In our sample within the temperature (20-90 K) and field range (0-200 Oe) of our investigation, we found evidence for only the low-field VL to VS phase transformation. Due to the large depinning current density at low H, transport measurements in this regime suffer from significant joule heating effects. Therefore, it is difficult to identify the low-field VL to VS phase transition line using standard bulk transport measurement techniques as have been used for identifying the high-field melting line in the past [35]. As indicated by the dotted (blue) arrow at the edge of the cross-shaded region in figure 3(a), the VL to VS transformation may appear only as a slight bump on the bulk magnetization response, making it difficult to discern the low-field VL to VS transition line from bulk magnetization measurements.

Fitting the theoretical expression for the low-field melting line
Theoretically, it is predicted [10] that at low fields, the boundary in the H-T vortex phase diagram across which the VL to VS phase transformation occurs obeys Here, λ is the penetration depth and ε 0 = (φ 0 /4πλ) 2 is the vortex line energy. Usually, thermal melting of the vortex state always occurs in the presence of quenched random pinning centers. The above expression is based on considering a softening of the elastic moduli of the VS due to weak inter-vortex interaction at low fields, which lead to enhanced wandering of flux lines from their mean position. Enhanced rms wandering of the flux lines has been considered to arise not only from thermal fluctuations but also from quenched random disorder [36], leading to a net rms fluctuation of a vortex line about the mean position ( u 2 1/2 th ), which is determined phenomenologically using the Lindemann criterion, namely c L = u 2 figure 5(e)). In figure 5(f) we plot the c L value we found from fitting the different H M (T ) lines as a function of the normalized U c . We observe that the value of c L changes from close to 0.25 for low pinning regions to a value close to 0.4 for regions with strong pinning. An increase in the value of c L with increasing pinning strength indicates that in the dilute vortex regime an increase in the pinning strength results in an enhanced stability of the vortex state against thermal fluctuations. The upward shift of the H M (T ) line in the field temperature phase diagram with an increase in pinning strength is distinct from the behavior known for the high-field melting line; namely an increased point disorder results in shifting of the high-field melting line to lower fields [37].
The changes in the configuration of vortices triggered in response to an external magnetic field modulation help us in determining the presence of a phase transformation in the vortex state. From our experiments at low fields, we find a significant change in the density of vortices generated in response to a small modulation in external field which suggests a change in entropy. For the low-field VL to VS phase transition, using the Clausius-Clapeyron relation for vortex lattice melting, namely we estimate the change in entropy ( s) per vortex per copper oxide plane [8] in BSCCO as ∼0.05k B , where we use the size of the jump in B in the low-field VL phase, namely Oe at 50 K, CuO interlayer spacing d = 15 Å and a mean slope (cf figure 5(a)) of the low-field melting line dH M /dT ∼ 0.8 Oe K −1 . The latent heat for the low-field melting, L ∼ 0.05k B T M , appears to be comparable with that observed for highfield melting (∼0.1k B T M ) in BSCCO [8,9] at similar temperatures of 50 K. It appears that the pinning seems to have the opposite effect on melting of the dilute vortex state as compared to that on melting at higher fields. Point-pin-induced wandering of the flux lines about their mean position, especially in a soft VS, promotes easier melting of the VS, thereby causing the high-field melting boundary to shift down in the H-T vortex phase diagram with enhanced pinning [37]. Contrary to known behavior for melting at high fields [37], we observe that with enhanced pinning the low-field melting boundary moves up in the H-T vortex phase diagram. Therefore, it appears that the VS phase is shrinking from both the high-field and the low-field ends in the H-T phase diagram. The distinct behavior of low-field melting line near H c1 (T ) from the high-field melting line suggests that the behavior of the two melting lines may have to be treated separately theoretically.
From figures 1 and 6 it appears that the initial nucleation and growth of the low-field VL phase occurs along the linear defects in the sample [22] where vortex chain states [22,38] are generated by the pancake vortices in BSCCO decorating a plane of Josephson vortices which prefer to align along the linear sample inhomogeneities [31,39]. We believe that enhanced longitudinal fluctuations along the defects [40] help in thermally destabilizing the stack of pancake vortices penetrating along the Josephson vortex planes. Spread of the VL phase to sample regions with higher pinning is delayed as pinning appears to strengthen the vortex state against thermal fluctuations, which is indicated by the increase in c L with pinning strength in figure 5.
Before concluding, we would like to clarify that the anomalous brightening feature associated with a jump in B z (cf figure 1) is not related to an inhomogeneous vortex state. Note that regions of the sample where the anomalous jump in B z occurs (namely regions with enhanced bright contrast in figures 1(b) and (c)) possess a dome-shaped B z (r ) field distribution (e.g. see navy blue curve corresponding to 37.5 Oe, as well as the ones above it in figure 2(a)). We recall here that a similar dome-shaped magnetic field distribution at the onset of the high-field vortex melting phenomenon in BSCCO single crystals has been reported in the past (cf [19]). A dome-shaped B z (r ) profile is characteristic of a sample with negligible bulk screening currents, with the dome-shaped field distribution being associated with the screening current circulating around the sample edges (for example, for details of the effect of geometrical barriers in the absence of bulk shield currents in high-T c superconductors leading to the domeshaped field profile, see [23]). We argue that if an inhomogeneous vortex state were to be associated with the regions of the sample with anomalous enhancement of magneto-optical intensity, then these regions should have exhibited Bean-like profiles in the local magnetic field distribution, B z (r ). Consequently, in these regions, significant bulk screening currents (related to the gradient in Bean-like B z (r ) profile) should have been present along with screening currents already circulating at the sample edges. Such a current distribution would be inconsistent with observing a dome-shaped field distribution. Therefore, the dome-shaped field distribution we observe in figure 2(a) suggests the absence of an inhomogeneous vortex state in the regions with the anomaly. The above assertion is also supported by our observation in figure 4(b), inset, which shows the absence of screening currents within regions of the sample where the anomalous enhancement in magneto-optical intensity is found (cf the dark region within the encircled region in figure 4(b), inset). The above suggests that the observed anomalous enhancement in the magneto-optical intensity is not associated with inhomogeneity of the vortex state; rather it is a signature of a phase transformation at low vortex densities.

Conclusion
In conclusion, using MOI in a BSCCO single crystal, we have identified signatures of a low-field VL to VS phase transition via a coexistence regime and have identified their location on the H-T vortex phase diagram. The first-order-like transition into the low-field VL from the VS phase is marked by enhanced equilibrium vortex densities. The location of the low-field phases identified is found to be sensitive to the pinning strength. The locus of the low-field phase boundary in the field-temperature phase diagram fits to the theoretical curve predicted for low-field VL to VS melting. We find that the effect of enhanced pinning strength suggests that at low fields the vortex lines are stiffened against thermal wandering leading to a higher Lindemann number. We hope that the present work will stimulate future theoretical and experimental work study the peculiarities of the dilute vortex phase.