Defect-induced weak collective pinning in superconducting YB6 crystals

In a previous study (2017 Phys. Rev. B 96 144501), a strong variation in the superconducting transition temperature T c of YB6 differing by a factor of two has been explained by a change in the density of yttrium and boron vacancies tuning the electron–phonon interaction. Here, by using an array of miniature Hall probes, we address the penetration of the magnetic field, pinning, and critical current density on a series of YB6 single crystals with T c variation between 4.25 and 7.35 K. The analysis of the superconducting and normal-state specific heat characteristics allowed us to determine T c and the stoichiometry of our samples. We observed almost no pinning in the most stoichiometric YB6 crystal with the lowest T c. Upon increasing the number of vacancies weak pinning appears, and the critical current density is enhanced following the increased transition temperature in a linear variation. We argue that such an increase is, within weak collective pinning theory, consistent with the increasing number of vacancies that serve as pinning centers.


Introduction
After superconducting MgB 2 with a critical temperature of T c = 39 K, yttrium hexaboride YB 6 exhibits the second highest transition temperature T c = 8.4 K among borides [1][2][3][4][5]. Recently, the properties of YB 6 have been investigated rather intensively, above all its electronic band structure [6][7][8][9], the optical and Raman spectra [10][11][12], as well as its low temperature and superconducting properties. Studies on the superconducting properties of this compound can be found e.g. in [3,[13][14][15][16][17] where the relevant superconducting parameters as the coherence length ξ, the penetration depth λ, the Ginzburg-Landau parameters κ, and the superconducting energy gap ∆ were determined. They show that YB 6 compound exhibits type II superconductivity in dirty limit with medium electron-phonon interaction and s-wave pairing of charge carriers with 2∆/k B T c ≈ 4, where k B is the Boltzmann constant. Several experimental works concluded that the soft phonon modes around 7.5 meV, related to the rattling motion of Y atom in the boron cage, make the biggest contribution to the electron-phonon coupling [3,7,[18][19][20]. Contrary, the theoretical paper [21] suggested that it is the boron sub-lattice (B 6 octahedra as a whole) that is mainly responsible for superconductivity in YB 6 .
One of the open questions in YB 6 is what affects a large variation of the superconducting transition temperature of different YB 6 samples, which is in the range 4-8 K. Lortz et al [3] noticed that the transition temperature was controlled by the B/Y ratio and the highest T c was obtained for a non-stoichiometric composition with B/Y < 6. In [15] the relation between T c and stoichiometry was addressed systematically and it was argued that the T c enhancement in YB 6 single crystals is determined by the increase of the number of vacancies, both at boron and yttrium sites, leading to such a nonstoichiometric composition, which is accompanied by enhancement of the electron-phonon interaction.
So far, very little is known about the penetration of the magnetic field, the pinning strength, and the critical current density j c in YB 6 . Here, by using an array of miniature Hall probes we have investigated these phenomena in four single crystalline YB 6 samples with transition temperatures T c of 7.35 K, 7.05 K, 5.7 K, and 4.25 K. We obtained the chemical composition and various superconducting parameters by analysis of our measurements of superconducting and normal-state specific heat. These results confirmed that the samples with the higher T c are more non-stoichiometric which allows enhanced electron-phonon interaction. They also exhibit stronger pinning, whereas the sample with the lowest T c shows almost no pinning. Surprisingly, the critical current densities obtained from magnetization hysteresis loops reveal almost linear dependence on the superconducting critical temperature. Following the weak collective pinning theory we show that the increase of j c can be explained by an increasing number of vacancies that serve as the pinning centers. We argue that increasing amount of vacancies is thus not merely responsible for higher T c but also serves for enhanced collective pinning augmenting the critical current.

Experiment
The samples studied in this work were prepared from four crystals grown in two centers-National Institute for Materials Science, Tsukuba, Japan (A and C samples) and Institute for Problems of Materials Science, Kyiv, Ukraine (B and D samples). In both cases, the same method was used: the traveling-solvent floating zone technique, due to both the structural features of yttrium hexaboride and its phase diagram. For details of the synthesis see supplementary information. The dimensions of the crystals are sketched in figure 1 and are given in table 1.
The heat capacity C(T)/T at low temperatures was measured in the 3 He refrigerator using an accalorimetry technique with the light emitting diode as a heat source, the chromel-constantan thermocouple as a sample holder and a thermometer at the same time. This method provides only relative values of the heat capacity of a sample but with very high precision [22]. Corrections of both the main thermometer (Cernox) and the thermocouple in the magnetic field were carefully inspected and implemented in the data treatment. Heat capacity measurements at higher temperatures from 4 K up to room temperature were obtained in a standard Quantum Design Physical Property Measurement System using a relaxation technique.
The local magnetometry measurements were performed using an array of miniature Hall probes based on semiconductor heterostructure GaAs/AlGaAs with a two-dimensional electron gas in the active layer. The Hall crosses are arranged evenly in a line, with dimensions of individual probes being 10 × 10 µm 2 and with distances between the centers of two adjacent probes being 25 or 35 µm. The samples are placed on top of the array and installed in a superconducting horizontal magnet. Probes with the sample were oriented perpendicular to the applied magnetic field. Before each measurement, the sample was cooled in a zero magnetic field compensating for the remanent field of the superconducting coil. The array was powered by constant current and voltage across the probes was measured. During measurement, the applied magnetic field was increased and then decreased gradually, and the voltage of the probes was recorded in order to measure the local magnetic induction of the sample. For more details about the method see e.g. [23] and [24].
The Hall-probes arrangement sketch, together with the photo of the sample placed on top of the Hall-probes array is depicted in figure 1. This figure also contains the sketch of the sample and its dimensions (a, b, and c) in respect to the Hall-probe array orientation. Figure 2 shows the plot of the field-dependent part of electronic heat capacity (empty symbols), calculated as a difference between zero-and half-Tesla magnetic field ∆C/T = C(0 T)/T−C(0.5 T)/T for the samples A, B, C, and D in blue circles, red squares, green up-triangles, and gray down-triangles, respectively (this color code is maintained all through this report). Calculating ∆C/T, the lattice contribution from the total heat capacity was subtracted. The curves in figure 2 corresponding to the individual samples are adjusted in the y-axis for clarity. Note that not all of the data points from our measurements are shown in the figure, only one point out of 20 is displayed for readability. The solid lines in matching colors are the theoretical curves   according to the α-model [25] corresponding to the specific heat of a single s-wave gap superconductor with the coupling ratio 4.1, 4.3, 4.1, and 4 for the samples A, B, C, and D, respectively. The theoretical lines follow the experimental data in very good agreement for all crystals, proving a medium superconducting coupling in YB 6 independently on T c but also good quality and homogeneity of the samples. The superconducting transition temperature T c of the samples was determined at the mid-point of the specific-heat anomaly, in correspondence with the theoretical curve.  [15]. We also performed a thorough analysis of the normal-state heat capacity of all investigated samples to determine their stoichiometry following the model developed in [15]. For details of the procedure, see [15]  Profiles of magnetic field penetration into sample A for increasing magnetic field (upper panel) and decreasing magnetic field (lower panel), measured at 1.9 K. Vertical dashed lines correspond to the position of sample edges, arrows depict the evolution of the magnetic field (pointing up/down for increasing/decreasing field, resp.). Points depict magnetic induction of the sample at different applied magnetic fields (different colors) and at designated Hall probe position in respect to the sample placement. Lines that connect the points are guides to the eyes. and supplementary information (SI). From this analysis, we obtained the chemical composition of the crystals reported in table 1 revealing amounts of Y and B vacancies. Consistently with the previous report [15], we observe that the sample whose stoichiometry is closest to YB 6 has the lowest T c and the highest T c is found on the sample with the largest amount of yttrium and boron vacancies.

Results and discussion
In the following, we focus on the magnetization measurements. By the parallel recording of the Hall voltage of all probes across the sample with a gradually increasing/decreasing magnetic field, we were able to construct the magnetic profiles of all inspected crystals. Figure 3 displays such a profile for sample A, the one with the highest T c . The crystal was significantly wider than the span of the Hall-probes array, therefore it was necessary to measure the profile in parts by progressively shifting the sample. The resulting profile was constructed from six overlapping sample positions on the array. Probes located between the two dashed lines in figure 3 are covered by the sample and thus reflect a distribution of the magnetic field inside the crystal as a response to the applied field. In low magnetic fields these probes are shielded by the sample being in Meissner state (see the lowest curve in the upper panel of the figure). When the applied magnetic field exceeds a critical value, the so-called penetration field H p that is closely related to the lower critical magnetic field H c1 through sample geometry, the vortices start to enter the sample. If pinning centers are present in the sample, vortices bearing the magnetic field are trapped right after they penetrate inside. As a result, they accumulate near the sample edges. The magnetic profile of the sample is then V-shaped [26]. When the magnetic field is decreased again, the V-profile gets inverted since at first the vortices close to the edges are leaving the sample. Clearly, the observed profile of sample A forms a V shape in an increasing magnetic field. The lowest point of the 'valley' is slightly shifted toward the right side of the sample and does not match the center of the sample. This is related to the non-uniform thickness of the sample. Similar V-shape profiles were observed also for samples B and C (not shown here), however, it was not the case for sample D, the sample with the lowest T c . On the contrary, the profile forms a dome (see figure 4, upper panel), indicating vortices accumulating close to the sample center soon after the first penetration. Such a scenario corresponds to the absence of the vortex pinning, when the spatially extended Meissner current results in the effective trapping of vortices in the center of the sample [27].
Once the profile of the magnetic field penetrating the sample is clear, it is possible to determine the lower critical magnetic field H c1 . It is derived from the penetration field H p which is a field at which the first complete vortex penetrates the sample. In samples A, B, and C with V-shaped magnetic profiles the first vortex settles near the edge, so the Hall probe near the sample edge was selected to measure H p (for more details about the procedure see e.g. [24]). In sample D the profile was dome-like, so the probe at the sample center was selected for H p measurement.
The lower critical magnetic field can be calculated from the penetration field using an expression , where b and a are the thickness and width of the sample (see table 1), and the coefficient α is 0.67 for a disk-like crystal and 0.36 for an infinite stripe. For our samples, we considered α to be an average of the two extreme cases. Such an approach was justified e.g. in [28], where a collection of nine single crystalline MgB 2 samples with various aspect ratios was studied in detail. The resulting values of µ 0 H c1 for  [15] except sample A which showed H c1 lower by 35% compared to sample No. 1 from [15].
The pinning of the vortex matter prevents dissipative processes in superconductors and its strength directly determines the size of the critical current. In order to compare the pinning strength of different samples, we calculated the critical current density of the crystals. In a superconductor with pinning centers, when the applied field drops to zero, some vortices persist pinned, and the magnetic induction of the sample is thus not equal to zero. Its value acquires a maximum value B r (remanent magnetic induction) at the peak of the inverted V-shape, while it decreases toward the sample edge (see figure 3, lower panel, the lowest curve). The pinning strength can be expressed by the critical current density j c , given by the formula j c ≈ B r /w, where w is the sample half-width (in our case the distance between the peak of inverted V and sample edge). Thus, the gradient of magnetic induction inside the sample mirrors the pinning strength. Figure 5 displays the hysteresis loops for all four crystals measured at the position of the Hall probe corresponding to the V profile peak for crystals A, B, and C, and at the sample center for crystal D. The loops are distinguished in color code as indicated above. For sample A the loop was measured at T = 1.9 K, and for other samples at T = 1.5 K. The arrows illustrate the sequence of the applied magnetic field-starting from zero, increasing to a maximum positive value, decreasing to a minimum negative value, and again increasing. The value of remanent magnetic induction B r (T) is taken at zero applied field, H = 0, in the decreasing branch of the loop. Figure 6 represents the main result of this study. It depicts the critical current density calculated at T ∼ T c /4 from the remanent magnetic induction in the sample and the sample geometry (j c = B r /w), as described above, for each crystal. Note, that we did not measure the complete loop for sample B at 1.8 K at the position of the V profile peak, instead the measurement was performed at 1.5 K. In order to compare with other samples, for sample B we estimated B r (1.8 K) by scaling with the evolution of the hysteresis loops with temperature on a probe slightly away from the profile peak. In the plot, the estimated value is presented. This introduced uncertainty, however, does not affect the conclusions of this paper.
Remarkably, the obtained critical current densities shown in figure 6 are extremely small in all our samples. Low-temperature values of critical current density in superconducting single crystals are typically higher by orders [29]. Even in MgB 2 , where a dome-shaped profile of magnetic field penetration was observed, the value of the critical current density was ∼5 × 10 4 A cm −2 [28]. On the other hand, j c values of our YB 6 crystals are similar to values measured in 2H-NbSe 2 single crystals showing intrinsic low pinning values and critical currents by many orders smaller than the depairing current. For example, specially selected single crystals of NbSe 2 with very weak pinning revealed a critical current density of about 1.6 × 10 4 A cm −2 obtained using the width of the magnetization curve [30], but also 2H-NbSe 2 crystals with 500 A cm −2 were prepared [31]. Similarly, such a low critical current density (∼500 A cm −2 ) was observed also in the crystals of Cu x TiSe 2 [32].
The essential message of this report shown in figure 6 is that the pinning strength clearly follows the evolution of the transition temperature-it decreases with decreasing T c .
The previous study [15] has shown the role of yttrium and boron vacancies for the lattice stability, sample homogeneity, Cooper-pair breaking, and the resulting superconducting transition temperature of yttrium hexaboride. The strong variation of the superconducting transition temperature of YB 6 differing by a factor of 2 has been explained, with T c most suppressed near stoichiometric composition due to bcc lattice instability, and contrary, T c enhanced at high density of yttrium and boron vacancies allowing for increased electron-phonon interaction. Due to the small size of metallic (Y 3+ ) ion, YB 6 is close to instability and susceptible to decomposition into neighboring binary phases which can be healed by the introduction of vacancies [11,12]. Then, a near stoichiometric YB 6 reveals the highest resistivity, the lowest T c , and the smallest mean free path. The increase in vacancy concentration serves as a stabilizing factor of the YB 6 lattice leading to an enhancement of the electron-phonon interaction and a T c increase. The analysis of new samples approved these conclusions (see supplementary information). The question now arises of how all this can influence the character of pinning and the resulting critical current density in our samples.
In general, there are two limiting regimes of vortex pinning: 'strong' and 'weak collective' pinning [33,34]. Strong pinning originates from large defects, such as nm-size heterogeneities, while weak collective pinning results from the high number of atomic-size defects.
The weak collective pinning theory has been successfully applied for the analysis of pinning in several low T c type-II superconductors e.g. [35][36][37] with randomly distributed weak pinning centers. Critical current density j c (see [38] and references therein) reads where j 0 = 0.54 Hc λL is depairing current, H c is the thermodynamic critical field, λ L London penetration depth, n d is the density of pinning sites, D v is effective ion radius, ε λ is anisotropy of the penetration depth (= 1 in our case), ξ is Ginzburg-Landau coherence length and ξ 0 is Bardeen-Cooper-Schrieffer coherence length. At distances L > L c , where the longitudinal correlation length is L c = ξ 0 √ j 0 j c , the vortex can readjust itself elastically to the optimal local configuration. The vortex then 'breaks up' into segments of length L c , each of which is pinned independently [34].
In the following, we will analyze our data in the frame of weak collective pinning. For further calculations it is important to note that our samples are in the dirty limit, therefore from measurements of H c2 and H c1 we can derive only dirty limit values ξ and λ using relations µ 0 H c2 (0) = Φ0 2πξ 2 and µ 0 H c1 (0) = Φ0 4πλ 2 (ln κ + α), where κ = λ/ξ is Ginzburg-Landau parameter and α = .5 + 1+ln 2 2κ− √ 2+2 . ξ and λ are related to clean-limit values ξ 0 and λ L through expressions involving electronic mean free path l (λ = λ L √ 1 + ξ 0 /l, ξ = 0.85 √ ξ 0 l). λ L can be computed knowing the carrier density n from Hall effect measurements from [15] using expression λ L = √ m µ 0 ne 2 , where m and e are free electron mass and charge, respectively. Taking the resulting λ L , λ, and ξ, we can calculate l and ξ 0 . Having all these parameters, we can calculate depairing current j 0 ∼ 1.3 − 4 × 10 7 A cm −2 for our samples. These values largely exceed the measured critical current density j c which is on the order of 10 3 A cm −2 in our case (see table 1). Thus we can conclude that the condition for weak collective pinning: j c /j 0 ≪ 1 is fulfilled for our samples. The longitudinal correlation length L c is much smaller than the sample thickness for samples A, B, and C (L c ∼ 12-20 µm), so we can consider pinning to be 3D collective pinning [34]. Moreover, the shape of the profiles indicates that the pinning is in a single vortex regime since the slope is not changing with the magnetic field. It is also interesting to note, that for all our samples A, B, C, and D, λ L (45.4-50.3 nm) is much smaller than ξ 0 (120-301.5 nm) leading to κ 0 = λ L /ξ 0 ≪ 2 −1/2 , while it is opposite for λ and ξ. The dirty limit values of penetration depth and coherence length lead to κ = λ/ξ ≫ 2 −1/2 . It means, that if the crystals were clean, having a mean free path much longer than the coherence length, they would be type-I superconductors. Type-II superconductivity is in YB 6 thus only induced by the sample being in the dirty limit.
Next, from equation (1) it is possible to extract the term related to the pinning centers n d D v 4 . Let us now consider the number of boron vacancies to be also the number of pinning sites. For sample A, with the chemical composition Y 0.9 B 5.929 , we have 0.071 missing boron atoms per unit cell. Taking the lattice constant 4.1 Å [15], the unit cell volume is 68.9 Å 3 , thus we arrive to the density of pinning sites n d = 10.3 × 10 26 per m 3 . Similarly, it is 9 × 10 26 per m 3 for sample B with composition Y 0.96 B 5.938 and 4.8 × 10 26 per m 3 for sample C with composition Y 0.97 B 5.967 . Inserting these values into equation (1) we arrive to the size of ion radius D v = 0.67 ± 0.06 × 10 −10 m for samples A, B, and C. This value is very reasonable compared to boron radius (0.85 × 10 −10 m) and is consistent with the concept of missing atoms serving as the pinning centers. Increasing critical current density, in line with increasing critical temperature, thus may be interpreted as a result of an increased number of vacancies, pointing to their common origin.
A similar effect was observed in iron-based superconductors, e.g. PrFeAsO 1 − y , in which it was found that the more oxygen vacancies are added, the higher the T c and also the stronger the pinning [38]. Also in Cr 0.0005 NbSe 2 , it was shown [36] that hydrostatic pressure increases both, T c and j c due to point-like defects.
The only sample that sticks out of the picture is sample D (similar to sample No. 4 in [15]). Despite a non-zero number of vacancies, this sample shows almost no pinning (note the nearly reversible magnetization loop in figure 5 together with the magnetic profile depicted in figure 4). The reason for this behavior remains an open question.

Conclusions
In conclusion, we have performed local magnetization measurements on four crystals of YB 6 with critical temperatures varying between 4.25 and 7.35 K. We have observed that critical current density j c increases with increasing critical temperature. Applying weak collective pinning theory we have shown that the increase of j c can be consistently explained by an increasing number of vacancies that serve as the pinning centers. Thus the increasing amount of vacancies is not merely responsible for higher T c in YB 6 but also serves for enhanced collective pinning augmenting the critical current density. It follows that by adjusting the number of vacancies in YB 6 crystals it is possible to tune the pinning in order to test the single vortex collective pinning theory.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.