Study of flux pinning mechanism under hydrostatic pressure in optimally doped (Ba,K)Fe2As2 single crystals

Strong pinning depends on the pinning force strength and number density of effective defects. Using the hydrostatic pressure method, we demonstrate here that hydrostatic pressure of 1.2 GPa can significantly enhance flux pinning or the critical current density (Jc) of optimally doped Ba0.6K0.4Fe2As2 crystals by a factor of up to 5 in both low and high fields, which is generally rare with other Jc enhancement techniques. At 4.1 K, high pressure can significantly enhance Jc from 5 × 105 A/cm2 to nearly 106 A/cm2 at 2 T, and from 2 × 105 A/cm2 to nearly 5.5 × 105 A/cm2 at 12 T. Our systematic analysis of the flux pinning mechanism indicates that both the pinning centre number density and the pinning force are greatly increased by the pressure and enhance the pinning. This study also shows that superconducting performance in terms of flux pinning or Jc for optimally doped superconducting materials can be further improved by using pressure.

Flux pinning has been a topic of much interest in field of superconductivity because of its importance for applications and fundamental aspects.This interest stems from the significance of flux pinning for high critical current density (J c ) in superconductors, which is the defining property of a superconductor.Generally, various types of random imperfections, such as cold-workinduced dislocations [4], secondary-phase precipitates [3], defects induced by high energy ion irradiation [5], etc., can be used to enhance flux pinning.Unfortunately, it is difficult to discern the maximum potential of a superconductor from these techniques, and the outcomes hold up only to a certain level.Furthermore, the critical current is only enhanced, in most of cases, either in low or high fields, with degradation of the superconducting critical temperature (T c ) another drawback.For instance, proton irradiation can only enhance flux pinning in high fields by inducing point defects in Ba122:K.Similarly, light ion C 4+ irradiation of Ba122:Ni crystals can only enhance J c in low fields at high temperatures [1].High energy particle irradiation can also decrease T c by more than 5 K for cobalt and nickel doped Ba-122 [2,3].
As is well known, J c is mostly limited by weak links (in the case of polycrystalline bulks), and thermally activated flux creep (an intrinsic property) emerges from weak pinning [4][5][6][7][8][9][10].Strong pinning can be achieved either by inducing effective pinning centres with strong pinning force.
Our previous results show that J c is enhanced significantly under hydrostatic pressure at high fields (i.e., over one order of magnitude) in comparison to low fields, along with enhancement of the closely related T c by more than 5 K in Sr 4 V 2 O 6 Fe 2 As 2 polycrystalline bulks and NaFe 0.97 Co 0.03 As single crystals [11,12].Until now, however, it has been unclear that the observed J c enhancement under pressure is correlated with improved T c or flux pinning.The primary motivation for the present work is to use optimally doped single crystal samples (which has an unchanged T c under hydrostatic pressure), to elucidate the contributions of flux pinning to J c enhancement in Fe-based superconductors.The secondary motivation is to investigate further the contributions from both N p and F p to strong pinning.
The argument is as follows: Hydrostatic pressure can induce pinning centres, which, in turn, enhance the pinning force.The total pinning force and the pinning centres are correlated by F p = N p f p where N p is the number density of pinning centres and f p is the elementary pinning force, which is the maximum pinning strength of an individual pinning centre, with a value that depends on the interaction of the flux line with the defect.According to the flux pinning theory, strongly interacting defects can contribute to F p individually, provided that F p ∝ N p , and weakly interacting defects can contribute only collectively; the collective theory leads to F p ∝ (N p ) 2 for small defect numbers [13].
The Ba122:K compound is believed to be the most technologically suitable because of its isotropic nature and high T c , upper critical field (H c2 ), and J c values (J c > 10 6 A/cm 2 at 2 K and 0 T) [14][15][16][17][18].The depairing current density (J d ) is the maximum current density that superconducting electrons can support before de-pairing of Cooper pairs, and is given as The   value that is found is roughly 0.3 GA/cm 2 by using values of penetration depth, λ = 105 nm and coherence length,  = 2.7 nm [19,20].Our estimation indicates that there is a significant potential to further enhance flux pinning in the (Ba,K)Fe 2 As 2 .
In this paper, we investigate the flux pinning of optimally doped (Ba,K)Fe 2 As 2 under hydrostatic pressure.We demonstrate that hydrostatic pressure causes little change in T c , but leads to significant enhancement in flux pinning or J c by a factor of 10 in both low and high fields in optimally doped Ba 0.6 K 0.4 Fe 2 As 2 crystals.Our analysis shows that the both N p and F p are increased by the pressure and contribute to strong pinning.
High quality 122 crystals were grown by using a flux method.The pure elements Ba, K, Fe, As, and Sn were mixed in a mol ratio of Ba 1−x K x Fe 2 As 2 :Sn = 1:45-50 for the self-flux.A crucible with a lid was used to reduce the evaporation loss of K as well as that of As during growth.The crucible was sealed in a quartz ampoule filled with Ar and loaded into a box furnace.The temperature dependence of the magnetic moments and the M-H loops at different temperatures and pressures were measured on a Quantum Design Physical Property Measurement System (QD PPMS 14 T) by using the Vibrating Sample Magnetometer (VSM).We used an HMD high pressure cell and Daphne 7373 oil as a pressure transmitting medium to apply hydrostatic pressure on the samples.
Figure 1 shows the temperature dependence of the magnetic moments for zero-field-cooled (ZFC) and fieldcooled (FC) measurements at different pressures.T c remains almost unchanged at different pressures.T c ≈ 37.95 K was found at P = 0 GPa and P = 1 GPa.Similar results were also reported for Ba 0.6 K 0.4 Fe 2 As 2 thin film [21].Furthermore, a temperature independent magnetic moment at low temperatures was observed, along-with a small transition width, indicating the high quality of the crystals.The field dependence of J c at different temperatures (4.1, 16, and 24 K) and pressures (0 and 1.2 GPa), obtained from the M-H curves by using Bean's model, are shown in Fig. 2. Nearly five-fold J c enhancement can be seen at 16 K and 24 K in both low and high fields at P=1.2GPa.It is noteworthy that J c is enhanced for the Ba 0.6 K 0.4 Fe 2 As 2 crystal at 1.2 GPa in both low and high fields.This has not been found by other approaches for pinning enhancement reported so far.At 16 K and self-field, the J c is 2 × 10 5 and increases up to 6 × 10 5 A/cm 2 by pressure of 1.2GPa and retains as high as 3 × 10 5 A/cm 2 at 12 T.At 24 K, J c at zero field is 9×10 4 A/cm 2 and raises up to 2.5 × 10 5 A/cm 2 at P=1.2GPa and remains the same level at 12 T.At 4.1 K, the J c is nearly 1 × 10 6 A/cm 2 at 2 T and 5 × 10 5 A/cm 2 at 12 T under P=1.2GPa.The pinning force (  =  c × ) as a function of field at 8 K, 12 K, 24 K, and 28 K is plotted in Figure 3 [22].At high fields and pressures, the F p is found to be nearly 5 times higher at 8, 12, 24, and 28 K as compared to P = 0 GPa, which corresponds nicely to J c enhancement.Figure 4 shows a comparison of F p obtained in our Ba 0.6 K 0.4 Fe 2 As 2 under pressure with those of several other low and high temperature superconducting materials [23][24][25][26].The (Ba,K)Fe 2 As 2 shows better in-field performance under pressure.Pressure can significantly improve F p values to greater than 60 GN/m 3 at H > 10 T, which are even superior to those of Nb 3 Sn and NbTi.With respect to the N p , pressure can also increase the number of point pinning centres, which can suppress thermally activated flux creep, leading to J c enhancement [11].The N p can be calculated from the following relation: Where Σ  is the accumulated pinning force density,    is the maximum elementary pinning force � p �, which is the interaction between a flux line and a single defect, and  is an efficiency factor. = 1 corresponds to a plastic lattice, and the  value is otherwise    /, where B is the bulk modulus of the material.We assume to a second order of approximation that the interaction between a flux line and a single defect is nearly same under pressure.Therefore, we can use    ≈ 3 × 10 -13 N for a similar superconductor (i.e., Ba122:Co) to estimate N p [27].At 4.1 K, N p ≈ 7.3 × 10 24 m -3 at P = 0 Gpa, which increased to N p ≈ 1.2 × 10 25 m -3 for P = 1.2 GPa, while at 24 K, N p ≈ 6.6 × 10 23 m -3 at P = 0 Gpa, which increased to N p ≈ 3.8 × 10 24 /m 3 for P = 1.2 GPa.To examine whether the observed J c enhancement is likely affected by volume change of the samples under high pressure, we have performed the following analysis.According to the Wentzel-Kramers-Brillouin (WKB) approximation, high pressure can modify grain boundaries through reduction of the tunnelling barrier width and the tunnelling barrier height for polycrystalline bulks, correlated to following simple mathematical expression [28][29][30]: Where W is the barrier width, k = (2mL) 1/2 /ℏ is the decay constant, which is barrier height (L) dependent, ℏ is the reduced Planck constant, and J c0 is the critical current density at 0 K and 0 T. The relative pressure dependence of J c can be determined from Eq. ( 1) as [31]: Where the compressibility in the width and height of the grain boundary are defined by   = − ln / and   = − ln /, respectively.
For the (Ba,K)Fe 2 As 2 single crystals, we assume to a first approximation that κ GB and κ L are nearly comparable to the average linear compressibility values κ a = -dlna/dP (κ a ≈ 0.00318 GPa -1 ) and κ c = -dlnc/dP (κ c ≈ 0.00622 GPa -1 ), respectively, in the FeAs plane, where a and c are the inplane and out-of-plane lattice parameters [32] .Therefore, we can write Eq. ( 3) correspondingly as By using J c ≈ 10 5 A/cm 2 at 24 K and J c0 ≈ 10 6 A/cm 2 , we find that (   ln(    ⁄ ) ) ≈ 0.0073 GPa -1 and (1/2   ln(    ⁄ ) ) ≈ 0.0071 GPa -1 , so both only contributed less than 2% to the experimental value, i.e., dlnJ c /dP = 0.92 GPa -1 from the inset of Figure 5.This illustrates that the origin of the strong flux pinning under pressure does not arise from the change in volume.
The J c value as a function of reduced temperature (i.e.1-T/T c ) at 0 and 10 T under different pressures is shown in Fig. 6.The data points in different fields and pressures follow a power law description [i.e.J c ∝ (1-T/T c ) β ], where β is a critical exponent [33][34][35].Ginzburg-Landau theory predicts different vortex pinning mechanisms at specified fields, with different values of exponent β.It was found that β = 1 corresponds to non-interacting vortices and β > 1.5 refers to the core pinning mechanism.The exponent β (i.e., slope of the fitting line) is found to be 1.74 and 1.85 for zero field, and 1.20 and 1.43 at 10 T, at 0 and 1.2 GPa, respectively, which reveals a strong J c dependence on pressure.The low values of β at high pressure show that the J c decays rather slowly in comparison to its values at low pressure.Different values of exponent β have also been observed in MgB 2 and yttrium barium copper oxide (YBCO) [36,37] Figure 6: lnJ c versus reduced temperature at different fields and pressures.
The pinning mechanisms in Ba 0.6 K 0.4 Fe 2 As 2 have been analysed by using collective pinning theory.Generally, core pinning comprises 1)ℓ pinning, which comes from spatial variation in the charge carrier mean free path, ℓ, and 2)  c pinning due to randomly distributed spatial variation in T c .Referring to the theoretical approach proposed by Griessen et al.: in the case of ℓ pinning, while � (1 +  2 ) applies in the case of  c pinning, where  =   c � [38].Fig. 7 shows almost perfect overlapping of the experimentally obtained J c values and the theoretically expected variation for the ℓ pinning mechanism at 0.05 T. This is in agreement with the observation of little change in T c under high pressure.We also observed similar results in BaFe 1.9 Ni 0.1 As 2 and SiCl 4 doped MgB 2 [39,40].Furthermore, ℓ pinning has also been reported in FeTe 0.7 Se 0.3 crystals [41].
We have studied the flux pinning in optimally doped Ba 0.6 K 0.4 Fe 2 As 2 crystal under hydrostatic pressure, analysing the critical current density determined experimentally.We propose that strong flux pinning in a wide range of fields can be achieved by improving the pinning force under pressure.The pressure of 1.2 GPa improved the F p by nearly 5 times at 8, 12, 24, and 28 K, which can increase J c by nearly two-fold at 4.1 K and five-fold at 16 K and 24 K in both low and high fields, respectively.This study also demonstrates that such an optimally doped superconductor's performance in both low and high fields can also be further enhanced by pressure.

Figure 1 :
Figure 1: Magnetic moments versus temperature at P = 0 GPa and P = 1.2 GPa

Figure 2 :
Figure 2: J c as a function of field at P = 0 and 1.2 GPa at 4.1, 16 , and 24 K.

Figure 4 :
Figure 4: Comparison of F p for different superconductors.

Figure 5 :
Figure 5: J c -F p ratios at P = 1.2 GPa and P = 0 GPa.The relative change of J c with pressure as a function of T is given in the inset.In order to examine if the pinning force enhancement is a major factor responsible for the observed J c enhancement in our crystal under pressure, we have calculated the difference in ratio of   1.2

Figure 7 :
Figure 7: J c as a function of T/T c .Experimental data points are in agreement with  pinning.