Two step disordering of the vortex lattice across the peak effect in a weakly pinned Type II superconductor, Co0.0075NbSe2

The vortex lattice in a Type II superconductor provides a versatile model system to investigate the order-disorder transition in a periodic medium in the presence of random pinning. Here, using scanning tunnelling spectroscopy in a weakly pinned Co0.0075NbSe2 single crystal, we show that at low temperatures, the vortex lattice in a 3-dimensional superconductor disorders in two steps across the peak effect. At the onset of the peak effect, the equilibrium Bragg glass transforms into an orientational glass through the proliferation of dislocations. At a higher field, the dislocations dissociate into isolated disclination giving rise to an amorphous vortex glass. We also show the existence of a variety of additional non-equilibrium metastable states, which can be accessed through different thermomagnetic cycling.

Understanding the evolution of the structure of the vortex lattice (VL) in a weakly pinned type II superconductor is of paramount importance since it determines superconducting properties that are directly relevant for applications, i.e. critical current and the onset of electrical resistance. Over the past two decades, there have been intense efforts to understand the nature of the order-disorder transition of the VL with temperature or magnetic field 1,2,3 . It is generally accepted that in a clean system the hexagonal VL realised at low temperature and magnetic field, can transform to vortex liquid above a characteristic temperature (T) and magnetic field (H). Random pinning, arising from crystalline imperfection in the superconductor significantly complicates this scenario. It has been argued that since the system can no longer sustain true long-range order, both the ordered and the disordered state can become of glassy nature 4,5 , characterised by different degree of positional and orientational order. In addition, the VL can exist in a variety of nonequilibrium metastable states 6,7 , depending on the thermomagnetic history of the sample.
In contrast to the VL in superconducting thin films, where the order-disorder transition can be understood within the framework of Berezinski-Kosterlitz-Thouless (BKT) theory of 2-dimensional (2-D) melting 8,9,10 , the VL in a 3-dimensional (3-D) superconductor presents a more challenging problem. In this case the vortex line is rigid only up to a length scale much shorter than sample dimensions. Thus in a weakly pinned single crystal, the vortex line can bend considerably along the length of the vortex. It is generally accepted that in the presence of weak pinning the Abrikosov VL can transform into a quasi long range ordered state such as Bragg glass 11 (BG), which retains long-range orientational order of a perfect hexagonal lattice but where the positional order decays algebraically with distance.
Theoretically, both the possibility of a direct first order transition from a BG to a vortex glass (VG) state 12,13 (with short range positional and orientational order) as well as transitions through an intermediate state, such as multi-domain glass or a hexatic glass 14, 15 , have been discussed in the literature. While many experiments find evidence of a first-order orderdisorder transition 16,17,18,19 , additional continuous transitions and crossovers have been reported in other regions 20,21,22 of the H-T parameter space, both in low-Tc conventional superconductors and in layered high-Tc cuprates.
Experimentally, the order-disorder transition in 3-D superconductors has been extensively studied through bulk measurements, such as critical current 23 , ac susceptibility 24,25 and dc magnetisation 26,27 . These studies rely on the fact that in the presence of random pinning centres, the VL gets more strongly pinned to the crystal lattice as the perfect hexagonal order of the VL is relaxed 28 . The order-disorder transition thus manifests as sudden non-monotonic enhancement of bulk pinning 29 , and consequently of the critical current and the diamagnetic response in ac susceptibility measurements. Known as the "peak effect", this has been a central theme of many studies on the static and dynamic properties of VLs. These measurements, although valuable in establishing the phase diagram of type II superconductors, do not reveal the evolution of the microscopic structure of the VL across the order-disorder transition. A more direct, though less used method, is through direct imaging of the VL using scanning tunnelling spectroscopy 30,31,32,33,34 (STS). The main challenge in this technique is to get large area images that are representative of the VL in the bulk crystal.
Here, we track the evolution of the equilibrium state of the VL across the magnetic field driven peak effect at low temperature using direct imaging of the VL using STS in an NbSe2 single crystal, intercalated with 0.75% of Co. The intercalated Co atoms act as random pinning centres, making the peak effect more pronounced compared to pure NbSe2 single crystal 35 . We analyse VL images consisting of several hundred to a thousand vortices at 350 mK, taken across the magnetic field driven the peak effect. For low fields the stable state of the VL has nearly perfect hexagonal structure, with long range orientational order and a slowly decaying positional order. At the onset of the peak effect, dislocations proliferate in the VL, transforming the VL to an orientational glass (OG) with slowly decaying orientational order. Above the peak of the peak effect, dislocations dissociate into isolated disclinations driving the VL into an amorphous vortex glass (VG) which connects smoothly to the liquid state close to the upper critical field, Hc2. Our results points towards the possibility that the vortex lattice disorders in two steps, through successive destruction of positional and orientational order.

Results
Bulk pinning properties. Fig.1(a) shows the bulk pinning response of the VL at 350 mK, measured from the real part of the linear ac susceptibility (χ') (see Supplementary   Information) when the sample is cycled through different thermomagnetic histories. The χ'-H for the zero field cooled (ZFC) state (red line) is obtained while ramping up the magnetic field after cooling the sample to 350 mK in zero magnetic field. The "peak effect" manifests as a sudden increase in the diamagnetic response between 16 kOe ( on p H ) to 25 kOe (Hp) after which χ' monotonically increases up to Hc2 ~ 38 kOe. When the magnetic field is ramped down after reaching a value H > Hc2 (black line, henceforth referred as the ramp down branch), we observe a hysteresis starting below Hp and extending well below on p H . A much more disordered state of the VL with stronger diamagnetic response is obtained when the sample is field cooled (FC), by applying a field at 7 K and cooling the sample to 350 mK in the presence of the field (solid squares). This is however a non-equilibrium state: When the magnetic field is ramped up or ramped down from the pristine FC state, χ' merges with the ZFC branch or the ramp down branch respectively. In contrast, χ' for the ZFC state is reversible with magnetic field cycling up to on p H , suggesting that it is the more stable state of the system. Fig. 1 Real space imaging of the VL. The VL is imaged using STS over a 1 µm × 1 µm area close to the center of the cleaved crystal surface. We first focus on the VL along the ZFC branch. Figure 2 shows the representative conductance maps over the full scan area at 15 kOe and 24 kOe where vortices manifest as a local minima in the conductance. Figure 3  The orientational correlation function is defined as, is the angle between the bonds located at i r and the bond located , r ∆ defines a small window of the size of the pixel around r and the sums run over all the bonds. We define the position of each bond as the coordinate of the mid-point of the bond. Similarly, the spatial correlation function, , where K is the reciprocal lattice vector obtained from the Fourier transform, Ri is the position of the i-th vortex,  lattice constants, indicating that the presence of long-range orientational order. The envelope of ( ) r G K decays slowly but almost linearly with r. Since the linear decay cannot continue for large r, this reflects our inability to capture the asymptotic behaviour at large r at low fields due to limited field of view. While we cannot ascertain whether ( ) r G K decays as a power-law for large r as predicted for a BG, the slow decay of ( ) The possibility of a state with hexatic correlations between the onset and the peak of the "peak effect" has earlier been suggested from the field variation of the positional correlation length of the VL parallel (ξ || ) and perpendicular (ξ^) to the reciprocal lattice vector K, from neutron scattering studies in Nb single crystal 37 . In that measurement, ξ || and ξ^ was inferred from the radial width and azimuthal width of the six first order scattering in the Ewald sphere, projected on the plane of the detector 38 . While the relation between these correlation length and the ones obtained from the decay of or ( ) r G 6 is not straightforward, it is nevertheless instructive to compare the corresponding correlation lengths obtained from our data. We obtain the corresponding lengths in our experiment from the average radial (∆k||) and azimuthal ( 6 decays faster for the ramp down branch. However, analysis of the data shows that in both cases ( ) r G 6 decays as a power-law ( Fig. 5(f) We can now follow the magnetic field evolution of the FC state (Fig. 6). The FC state show an OG at 10 kOe (not shown) and 15 kOe (free dislocations), and a VG above 20 kOe (free disclinations). The FC OG state is however extremely unstable. This is readily seen by applying a small magnetic pulse (by ramping up the field by a small amount and ramping back), which annihilates the dislocations in the FC OG ( Fig. 7) eventually causing a dynamic transition to the QLRPO state. It is interesting to note that metastability of the VL persists above Hp where the ZFC state is a VG. The FC state is more disordered with a faster decay in ( ) r G 6 ( Fig. 6, lower panels), and consequently is more strongly pinned than the ZFC state.

Discussion
The two step disordering observed in our experiment is reminiscent of the two-step melting observed in 2-D systems 39

Material and Methods
Sample preparation. The Co0.0075NbSe2 single crystal was grown by iodine vapour transport method starting with stoichiometric amounts of pure Nb, Se and Co, together with iodine as the transport agent. Stoichiometric amounts of pure Nb, Se and Co, together with iodine as the transport agent were mixed and placed in one end of a quartz tube, which was then evacuated and sealed. The sealed quartz tube was heated up in a two zone furnace for 5 days, with the charge-zone and growth-zone temperatures kept at, 800 0 C and 720 0 C respectively.           The Co0.0075NbSe2 single crystal was characterised using 4-probe resistivity measurements from 286 K to 4 K (Fig. S1). The superconducting transition temperature, defined as the temperature where the resistance goes below our measurable limit is 5.3 K. The transition width, defined as the difference between temperatures where the resistance is 90% and 10% of the normal state resistance respectively is ~ 200 mK. The same crystal was used for both a.c. susceptibility and STS measurements.

II. a.c. susceptibility response as a function of excitation field
The a.c. susceptibility reported in Fig. 1 was performed with an a.c. excitation amplitude of 10 mOe at frequency 31 kHz. Since at large amplitudes, the a.c. drive can significantly modify the susceptibility response of the VL through large scale rearrangement of vortices, we performed several measurements with different a.c. excitation amplitudes to determine the range of a.c. field over which the χ' is independent of excitation field. We observe (Fig. S2) that below 3K, χ' shows significant dependence on the magnitude of the a.c. excitation only above 14 mOe.

III. Calculation of the correlation lengths ξ || and ξ^ from VL images
For an infinite lattice the correlation lengths along and perpendicular to the reciprocal lattice vectors K can be obtained from the width of the first order Bragg peaks (BP) of the reciprocal lattice using the relations, ξ || = 1/∆k|| and ξ^ = 1/∆k^, where ∆k|| and ∆k^ are the width of the first order Bragg peaks parallel and perpendicular to K. This method has been used to determine the correlation lengths from the Bragg spots in neutron diffraction measurements. However, when using the Fourier transform of VL images obtained from STS measurements, additional precaution has to be taken to account for contributions arising from the finite size of images and any inaccuracy in position resulting from the finite pixel resolution.
To obtain ξ || and ξ^ from our data, at every field a binary lattice is first constructed using the position of each vortex (Fig. S3(a)) obtained the VL images as explained in the Methods section. Similarly, the profile along the radial direction is determined by taking a line cut along the direction of the reciprocal lattice vector. We tried to fit peak profile averaged over the six symmetric BPs with both Lorentzian and Gaussian functions. The goodness of fit can be estimated from the residual sum of squares (χ-square) and the coefficient of determination 1 (adjusted R 2 ) which should be 1 when the fit is perfect. We observe that the Lorentzian function gives a much better fit with smaller χ-square and R 2 value close to unity. Therefore we calculate the corresponding peak width from the full width at half maxima of the best fit Lorentzian function (Fig. S3(c)-(d)).
determined using the same procedure as before (see Fig. S3(e)-(h)). In principle the experimental peak width is a convolution of intrinsic and extrinsic factors, and to correct for the contributions arising from extrinsic factors one needs to follow an elaborate deconvolution procedure. However, when the peak can be fitted with a pure Lorentzian function, the situation is simpler and the peak widths arising from different contributions are additive. Since in our case we can fit the Bragg peaks with a pure Lorentzian function, we subtract the peak width of the ideal lattice from the peak width obtained from the actual image, to obtain ∆k^ and ∆k|| arising from the lattice disorder alone.

IV. Filtering the STS conductance maps
For better visual depiction, the conductance maps obtained from STS are digitally filtered to remove the noise and scan lines which arise from the raster motion of the tip. The filtering procedure is depicted in Fig. S5. Fig. S5(a) shows the raw conductance map obtained at 24 kOe.
To filter the image we first obtain the 2D Fourier transform (FT) of the image Fig. S5(b). In addition to six bright spots corresponding to the Bragg peaks we observe a diffuse intensity at small k corresponding to the random noise and a horizontal line corresponding to the scan lines.
We first remove the noise and scan line contribution from the FT by suppressing the intensity along the horizontal line and the diffuse intensity within a circle at small k (Fig. S5(c)). The filtered image shown in Fig. S5(d) is obtained by taking a reverse FT Fig. 5(c).