Introduction

Superconductivity in the transition metal dichalcogenides (TMDs) and their intercalated layered structure compounds have special features associated with extreme anisotropy of the superconducting materials1. Superconductivity and density waves are competing orders that derive from instabilities2 due to internal and external perturbations such as chemical pressure, external pressure and magnetic field. Spontaneous formation of periodic lattice distortions and Charge Density Waves (CDW) could be thermodynamically favorable under certain conditions in low dimensional metals where the wave vector is generally known to depend on the nesting properties of the Fermi surface3. The symmetry breaking suggest that phonon−electron coupling usually occurs at certain transition temperature (TCDW) and so called Peierls phase transition2. Recently, the TMDs are natural layered materials provided with a new platform to study superconductivity due to the tunable nature of the superconducting properties and coexistent with other collective electronic excitations as well as strong intrinsic spin-orbit coupling. The bulk crystals of TMDs are formed of monolayers and bound to each other by van der Waals attraction, which makes it feasible to investigate experimentally. Niobium diselenide (NbSe2) is one of the most studied layered TMDs which has van der Waals attraction between the layers and it has generated much attention due to interplay of superconducting transition temperature (Tc)4,5,6,7 and TCDW2,8,9,10 is around 7 K and 33 K respectively. Anisotropy in TMDs, particularly in NbSe2 compounds, having the highest Tc among TMDs, could be significantly enhanced by introducing foreign atoms or molecules in the interlayer space (intercalation process)6,9,11,12. The intercalation ability of these compounds is related to the expected implementation of high temperature superconductivity in the sandwich type structures described by the excitonic mechanism6,9,13. Further remarkably, Zeeman-protected Ising superconductivity is expected in NbSe2 due to non-centrosymmetric structure with in-plane inversion symmetry breaking and strong spin-orbit coupling and the anomalous large in-plane critical magnetic field has become one important direction in crystalline 2D superconductors11,12,14,15. The vortex movement (i.e., vortex entry into or exit from a single-crystalline superconductor) is possibly pinned at the edges when applied very low magnetic field which can be generally attributed to the translational symmetry breaking at the edges and it is dependent on both shape and dimension of the superconductors13,16. The applied magnetic field would significantly modify the pinning and subsequently influence the magnetic behavior of the samples.

The TMDs superconductors have revealed wonderful superconducting properties including high values of Tc, critical current density (Jc), upper critical field (Hc2) and irreversible field (Hirr). In the presence of strong pinning, the vortex state of type-II superconductors is usually characterized by Jc that decreases monotonically with an increasing field (H) or temperature (T). In the weakly pinned superconductors, interplay between intervortex interface and flux pinning produces an unusual peak in Jc as a function of both field and temperature which are just below the normal-state boundary and it is usually designated as secondary peak effect17,18. Both strong pinning and high Jc depend on variation in the grain size and the coherence length (ξ)19. Vortex pinning arises from the interplay of several competing energies, namely the self-energy of the flux lines, vortex-vortex interactions, vortex inhomogeneity interactions and thermal excitations20. A magnetic field generates an array of vortices in type-II superconductors and the vortices strongly interact with each other forming highly correlated configurations such as the vortex lattice. In high-Tc cuprates at relatively high temperatures, vortices move and vibrate due to thermal fluctuations to the extent that the lattice can melt becoming a vortex liquid21,22,23. Growth of the structure of the vortex lattice in a weakly pinned high-Tc superconductor is of paramount importance, since it determines superconducting properties that are directly suitable for applications24,25. As the temperature is raised, the vortex lattice undergoes a first-order transition to a stable disordered state24,26,27,28. A thermal fluctuation permits pinned vortices to fluctuate around the potential energy and reduces the effective pinning energy due to thermal smearing. The vortices also escape totally from the pinning centers through a variety of de-pinning excitations. Vortex motion still occurs for currents lower than Jc at a much slower rate. This flux creep mechanism implies a residual dissipation and it is responsible for the time relaxation of persistent currents flowing in a superconducting closed loop.

Hydrostatic pressure effects on the Tc enhancement shows more advantages that are relevant to the flux pinning compared to other perturbations. The application of high pressure leads to changes in the electronic bands leading to original properties, which may be associated with a structural phase transition. It always reduces the lattice parameters and causes the shrinkage of unit cells, giving rise to the reduction of anisotropy. Grain connectivity improvement should also be expected, as pressure can compress both grains and grain boundaries. The formation of point defects can be more favorable under high pressure, since it is well known that the formation energy of point defects decreases with an increasing pressure. High pressure can cause low-angle grain boundaries to migrate in polycrystalline bulk samples, resulting in the emergence of giant grains, sacrificing surface pinning thereafter. Hence, a ratio of point pinning centers to surface pinning centers is expected to be higher due to an increase of formation energy under high pressure. Several examples can be recalled here. Pressure increases the superconducting transition from 3.5 K to 6.5 K and also the semiconducting to metallic transition in LaO0.5F0.5BiS2 single crystals29. Whereas a large enhancement of Tc from 26 K (0 GPa) to 43 K (4 GPa) in LaO0.95F0.05FeAs30 with application of external hydrostatic pressure up to 3 GPa using piston cylinder pressure device, Tc was reduced with the application of pressure above 3 GPa to 30 GPa, using a diamond anvil pressure device. Further, pressure induced superconductivity has been observed in the pnictides such as LaFeAsO1−xFx, LaFePO and SrFe2As231. In the ladder compounds, pressure induces the metal-insulator transition generating hole carriers and eventually superconductivity occurs in Ca14−xSrxCu24O1132. Pressure is also an effective approach to improve the Jc significantly in FeSe33, Sr4V2O6Fe2As234 YBa2Cu3O7−x35 superconductors, as the pressure induces more point pinning centers and subsequently affects the pinning mechanism33,34. The investigation of the pressure dependence of thermodynamic magnitudes proved to be a useful method of studying the properties of anisotropic compounds. Such investigations were performed on the layered compound NbSe236,37,38,39. Pressure not only enhanced Tc with a rate of dTc/dP is 0.86 K/GPa (NbSe2) and 1.47 K/GPa (Fe0.0011NbSe2) but also increased Jc and Hc238. In the case of the intercalated NbSe2 pressure increased Tc and simultaneously suppressed TCDW40. Suderow et al., described the pressure dependence of anisotropy, of the electron-phonon coupling and Fermi velocities, which influence the peculiar interplay between CDW, Fermi surface complexity and superconductivity in NbSe241. Pressure induces a transition from spatial variation in the δTc pinning to δℓ pinning mechanism in the undoped NbSe2 superconductors38.

From above, it is clear that high pressure is exclusively unique in fine-tuning superconducting states with the benefit of without introducing disorder effects in comparison with chemical doping. High pressure is, thus, an important tool to study intrinsic properties of the materials to understand the enhancement of Tc, Jc and other physical properties23,33,34,36,38,40,42,43,44. These facts motivated our present study on the pressure effects on the superconducting properties of single crystalline CrxNbSe2. We anticipated that hydrostatic pressure would increase the superconducting volume, Hirr, and Hc2 due to enhancement of Tc, increase the point defect and reduction in the anisotropy of single crystalline Cr intercalated NbSe2 samples. Indeed, we observed such interesting properties when pressure was applied to Cr intercalated NbSe2 is investigated. These findings are reported below.

Results and Discussions

The temperature dependence of resistivity (ρ(T)) in the temperature range from 2 to 300 K under various hydrostatic pressures from 0 to 8 GPa [Fig. 1(a)], the expanded region near Tc [Fig. 1(b)] and zero-field-cooled (ZFC) and field-cooled (FC) dc magnetization (M(T)) at the constant magnetic field of 20 Oe [Fig. 1(c)] in the vicinity of the Tc under various hydrostatic pressures range from 0 to 1 GPa of the crystal flakes of Cr0.0005NbSe2 are shown respectively in Fig. 1(a,b and c). The Tc is defined as an onset, mid and offset of superconductivity with obtained from dR/dT in ρ(T) measurements and the onset of diamagnetic signals corresponds to superconductivity as observed from M(T) measurements. As shown in Fig. 1(a), the ρ decreases (0.44 to 0.04 mΩ-cm at 0 GPa and 0.20 to 0.018 mΩ-cm at 8 GPa) in the temperature region from 300 to 10 K. It is also clear from Fig. 1(c), that M(T) starts to deviate from the normal state behavior around 6.1 K due to the presence of a diamagnetic signal, which is slightly above the onset of Tc. The sharp superconducting transition along with the large residual resistivity ratio (~11) and superconducting width \(({\rm{\Delta }}{T}_{c}={T}_{c}^{onset}-{T}_{c}^{offset})\) indicate good quality of single crystals used in the present study. The observation of sharp superconducting transition and zero resistivity at each applied pressure ensure indicate the hydrostatic nature in our experiments. As shown in Fig. 1(a), normal state resistivity gradually decreases with the application of high pressure and it almost shows the metallic nature for all pressures up to ~ 8 GPa. Similar ρ(T) behavior under pressure has been reported for various superconducting materials33,34,36,38,40. This is associated with the fact that high pressure brings the layers in the unit cell closer together, and facilitates overlap of wave functions of the conduction electrons in the adjacent layers. The Tc (6.64 K) of Cr0.0005NbSe2 is found to be less than undoped NbSe2 single crystal, and similar trend has been reported for various intercalated compounds FexNbSe26,38, NbSe236,40, GaxNbSe211, PdxNbSe245, CuxNbSe212, (LaSe)1.14(NbSe2)46 pnictides34,44 at both ambient and pressure.

Figure 1
figure 1

(a) Temperature dependent of resistivity at various applied hydrostatic pressures from 0 to 8 GPa, (b) pressure evolution of onset (Tconset) and offset (Tcoffset) superconducting transition temperature from resistivity measurements c) temperature dependent of zero field cooled (ZFC) [closed symbols] and field cooled (FC) [open symbols] dc magnetization at various hydrostatic pressures from 0 to 1.02 GPa.

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Figure 1(c) shows M(T) in the ZFC and FC regimes with an applied magnetic field of 20 Oe under various hydrostatic pressures up to ~ 1 GPa in the vicinity of Tc up to 10 K; We clearly notice the diamagnetic transition at 6.14 K and 7.07 K at ambient and 1.02 GPa respectively. The diamagnetic transition shifts towards high temperature with the application of pressure as shown in Fig. 2(a). The hysteresis between ZFC and FC regimes indicates that Cr0.0005NbSe2 exhibits weak flux pinning centers. Without correcting the demagnetization factor, we estimate the superconducting volume fraction (ZFC) to be ~ 80% indicative of bulk superconductivity. At ambient pressure the sample shows a Tc of 6.14 K from M(T) at constant magnetic field of 20 Oe and it is ∼0.5 K less than that observed from ρ(T) measurements, as shown in Fig. 2(a). Evidently, \({T}_{c}^{onset}\) and \({T}_{c}^{offset}\) observed from ρ(T) is higher than M(T) measurements as shown in Fig. 2(a). With the application of high pressure, Tc onset steadily increases in entire pressure region due to the reduction of interlayer distances of Cr0.0005NbSe2 as shown in Fig. 2(a).

Figure 2
figure 2

(a) Pressure dependent of superconducting transition temperature (Tc) from resistivity and magnetization measurements, (b) pressure dependent of resistivity of Tconset (ρTc) and room temperature (ρ300K), (c) excellent fitting to the resistivity data from 7 to 50 K using Fermi-liquid model (ρ = ρ0 + AT 2) for various pressures, and (d) pressure dependent of residual resistivity (ρ0) and scattering factor (A) from FLM fitting for Cr0.0005NbSe2.

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With an application of pressure, the room temperature resistivity (ρ300K) decreases monotonically up to 8 GPa and favours enhancement of metallic nature [Fig. 1(a)]. However, ρ300K decreases at a faster rate (0.13 mΩ-cm/GPa) up to 1.5 GPa and then moderate decrease (0.01 mΩ-cm/GPa) is observed above 1.5 GPa to 8 GPa. Further, \({\rho }^{{{\rm{T}}}_{{\rm{c}}}({\rm{o}}{\rm{n}}{\rm{s}}{\rm{e}}{\rm{t}})}\) decreases with slower rates are 0.01 mΩ-cm/GPa and 0.001 mΩ-cm/GPa at below and above 1.5 GPa respectively. Hence, we found that both ρ300K and ρTc(onset) are sensitive to the low pressure as shown in Fig. 2(b). ρ(T) is found to be almost linear in the high temperature region for various applied pressures. The change in ρ(T) shows metallic behavior at all pressures of up to 8 GPa. The upward curvature in ρ(T) weakens progressively with increasing pressure. Figure 2(a) shows the rapid enhancement of Tc in the low pressure region and moderate increase in high pressure region as we observed from both ρ(T) and M(T) for Cr0.0005NbSe2. One can see that the both Tconset and Tcoffset shift towards higher temperature with the application of high pressure and the dTc/dP becomes 0.91 K/GPa in the 0 < P < 1 GPa from M(T) measurements. The rate of change of Tc with pressure dTc/dP = 0.75 K/GPa for P less than 1.5 GPa and 0.09 K/GPa in the interval 1.5 < P < 8 GPa as observed from ρ(T) measurements. These results suggest that the relative change in the density of states at the Fermi level is profound in low pressure region up to 1.5 GPa and the moderate increase in the high pressure region (1.5 < P < 8 GPa) which leads to corresponding variation of Tc. This is associated with the fact that the P brings the layers closer together, and it facilitates the overlap of the wave functions of the conduction electrons in the adjacent layers. Consequently, an increase in Tc is mainly determined by changes in the density of states at the Fermi level and the change in the phonon spectrum plays a minor role. The region above 100 K in Cr0.0005NbSe2 can be described reasonably well by straight line fits with different slopes for various pressures in ρ(T) suggesting that phonon scattering mechanism is dominant.

The relationship between interlayer spacing and Tc is a matter of fundamental importance in understanding of superconductivity in layer compounds. The effect of pressure upon the physical properties is due to the reduction in interlayer spacing. The band structure calculations of the Fermi level for NbSe2 at the middle of a narrow d sub-band situated between the unoccupied and primarily d bands of Nb and the fully occupied p bands of Se. Although differing in the details of the sub-band overlap with the p bands, both calculations highlight the importance of interband (hybridization) and interlayer interactions. These take the form of a nonzero empty site potential at the unoccupied interstitial positions between the Se layers due to wave function overlap in the modified muffin tin potential approach of Kasowski47. From earlier report, we inferred that neglecting a weak empty site potential broadens the sub-band and increases the overlap with the p bands. If the charge transfer occurs due to intercalation, shifts in the Fermi level towards the sloped region of the density of states curve and this can lead to a relatively a big change in density of states (N(εF)) under pressure, and consequently, in Tc. More detailed information about the electronic properties and superconductivity of the pure and intercalated NbSe2 solid solution was obtained from specific heat12 measurements.

The phonon mediated superconductors, the electron-phonon coupling constant have been estimated from McMilllan formula48 using Debye temperature49 and Tc,

$${T}_{c}=\frac{{\theta }_{D}}{{\rm{1.45}}}\exp (-\frac{{\rm{1.04}}({\rm{1}}+\lambda )}{\lambda -\mu ({\rm{1}}+{\rm{0.62}}\lambda )}),$$

where θD is the Debye temperature, µ* is the screened pseudo-potential (characteristic for the electron repulsion) and assumed to be 0.15 suggested by McMilllan48 for transition metallic superconductors and λ is the electron-phonon coupling constant and the parameters θD, µ* and λ are all pressure dependent. The values of λ is 0.84 at ambient pressure suggest that strong coupling in superconductivity. With the Sommerfeld parameter (γ) and the electron-phonon coupling constant (λ), the electron density of states at the Fermi level (N(εF)) can be obtained from \(N({\varepsilon }_{F})={\rm{3}}\gamma /{\pi }^{{\rm{2}}}{k}_{{B}^{{\rm{2}}}}({\rm{1}}+\lambda )\). The density of electronic states at the Fermi energy therefore clearly decreases when more metal ions intercalates into NbSe2. All the values of θD and dθD/dP for pure and intercalated samples are identical. The pressure dependence of Debye temperature θD(P) can be obtained from the Gruneisen’s formula48, \(\frac{{\rm{1}}}{{\theta }_{D}}=\frac{\partial {\theta }_{D}}{\partial P}=\frac{\alpha V}{{C}_{V}}\), where α is the thermal expansion coefficient, Cv is the heat capacity and the value of dθD/dP is one order of magnitude lower than the corresponding changes is Tc with pressure. The large relative growth of Tc can be qualitatively associated with the particularities of the NbSe2 band structure, and corresponding changes of λ(P)40. Earlier it was shown that the intercalated Fe and Cu is located in the interlayer space, when the superconducting properties are affected by the interlaminar intercalant12. A redistribution of the interlayer under pressure can serve as an additional reason for the changes in Tc39. The last idea is supported by an increase of the width of the superconducting transition under pressure and more pronounced manifestation of its stepped form. This type of evolution in the superconducting state under high pressure38,44,50 has been observed in layered structure superconductors with a small anisotropy parameter23,33,34.

It is known that Tc is found to increase for NbSe212,39, FexNbSe238 and Cr0.0005NbSe2 compounds by applying high pressure. Figure 2(c) examines the normal-state resistivity of Cr0.0005NbSe2 under hydrostatic pressure and its implication of electronic correlation. Low temperature region (7 ≤ T ≤ 50 K) of ρ(T) can be fitted using the Fermi liquid model, ρ = ρ0 + AT 2, where ρ0 is residual resistivity and A is a scattering factor. Figure 2(c) shows good fitting for both data at ambient and high pressure and it supports the Fermi liquid model. The value of ρ0 is 0.042 mΩ-cm: 0 GPa and 0.017 mΩ-cm (8 GPa) and the A [16.45 × 10−6 mΩ-cm/K2 (0 GPa) and 7.1 × 10−6 mΩ-cm/K2: 8 GPa] value shows clear indication of electron-electron interaction exhibits in both ambient and high pressure in this sample. The nature ρ0 and A under various hydrostatic pressure has been shown in Fig. 2(d) which suggests that Cr0.0005NbSe2 is a weakly correlated system.

Figure 3(a–c) [Fig. S2] shows the temperature dependent of dc magnetization with various applied magnetic fields (H) with the constant P of 0, 0.5 and 1.02 GPa respectively. It is found that both Tc and diamagnetic signal show decreasing trend and allows us to determine the upper critical field (Hc2) of this material at various pressures. Taking the onset of transition in M(T) at various magnetic fields with constant pressure as the upper critical field point Hc2(Tc) and infer that almost all Cooper pairs are broken at this temperature and H. Further, the Meissner signal is suppressed with the application of various H at 0 GPa. However, the Meissner signal enhances at various H with constant pressure of 0.5 and 1 GPa, and it confirms that the pinning centers increase due to the application of pressure. Figure 3(d–f) shows the field dependent isothermal magnetization (MHL) scan in a low field region under various temperatures with applied pressure of 0, 0.5 and 1 GPa respectively. These results reveal that magnetic moment increases with the application of constant pressure, and confirms that when the occurrence of pinning increases as pressure increases in the sample.

Figure 3
figure 3

(a), (b) and (c) Temperature dependent of dc magnetization at various magnetic fields for 0, 0.5 and 1.02 GPa respectively on Cr0.0005NbSe2, (d), (e) and (f) Field dependent of magnetization scan in a low field range (dH/dt ~ 20 Oe/s) at various temperatures for 0 GPa, 0.50 GPa & 1.02 GPa respectively on Cr0.0005NbSe2.

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According to the Ginzburg–Landau (GL) theory, the absolute zero temperature upper critical field Hc2(0) can be estimated by using formula, Hc2(T/Tc) = Hc2(0)(1 − [T/Tc]a)b with a = 1.39 and b = 1 associated with the large gaps that open in the Nb bands51 which gives values less than that observed from WHH approach and the values Hc2(0) [Fig. S1] are shown in Table 1. The orbital limited upper critical fields [\({H}_{c{\rm{2}}}^{orb}({\rm{0}})\)] are estimated using the Werthamer–Helfand–Hohenberg (WHH)52 empirical formula, \({H}_{c{\rm{2}}}^{orb}({\rm{0}})=-\,{\rm{0.693}}{T}_{c}{(d{H}_{c2}/dT)}_{T={T}_{c}}\) and these values are shown in Table 1. For the Superconducting materials exhibiting weak coupling case, the Pauli limited upper critical field is calculated from, Hp(0) = 1.84Tc, where Tc is taken from M(T) measurements and the values are shown in Table 1. Hp(0) indicating that both orbital effect and Pauli spin paramagnetic effect (PSP) which gives an influence on the pair-breaking mechanism through the entire pressure region. However, we haven’t excluded the effect of spin instability at low temperatures, which may very well play an important role in the occurrence of superconductivity. The absolute orbital upper critical field \({H}_{c{\rm{2}}}^{orb}({\rm{0}})\) [Fig. S1] destroys superconductivity characteristics by enhancing the pair breaking phenomena in the presence of magnetic field and the values of \({H}_{c{\rm{2}}}^{orb}({\rm{0}})\) are found to be higher at higher pressures than at ambient pressure. Further, the enhancement of critical field under pressure implies that the strong flux pinning exhibits in this sample. Since, Pauli limit superconductivity mechanism [\({H}_{c{\rm{2}}}({\rm{0}}) < {H}_{c{\rm{2}}}^{orb}({\rm{0}}) < {H}_{p}({\rm{0}})\)] is exhibited by this sample, the calculated Maki parameter \(\alpha =\sqrt{{\rm{2}}}{H}_{c{\rm{2}}}^{orb}({\rm{0}})/{H}_{p}({\rm{0}})\), is found to be nearly 1 at both ambient and at high pressures. Similar pressure dependences of Hc2(P) were reported in YBCO42, FeSe43 FexNbSe238 indicating clearly that pressure influences the upper critical field. The superconducting coherence length [ξGL(0)] can be estimated from the relation, ξGL(0) = [Φ0/2πHc2(0)]0.5 where Φo = 2.07X10−7G−cm2 and the values of ξGL(0) are calculated from ambient and high pressure and it is shown in Table 1.

Table 1 Summary of the upper and lower critical field for the Cr0.0005NbSe2 single crystal under pressure.
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The irreversible field (Hirr) is calculated from MHL using the criteria of the zero field current density [Jc(H = 0)] and it occurs due to depinning of the magnetic fluxes in this sample. Figure 4(c) shows temperature dependent of Hirr fitted with the equation, Hirr(T) = Hirr(0)(1 − (T/Tc)2)3/2, pressure up to ~ 1.02 GPa. It reveals that enhancement of Hirr with application of pressure and also provides evidence of the 3D nature of flux creep in the sample. This is an indication that Hirr is mainly controlled by the flux pinning. Hc1 is measured from MHL under various temperatures [Fig. 3(a–c)] and plotted as a function of temperature with the parabolic function fitting as shown in Fig. 4(d). The precise determination of Hc1(0) from MHL possibly suffers from demagnetization effect. Further, Hc1(0) can also be deduced from the first penetration field \({H}_{c{\rm{1}}}^{\text{'}}({\rm{0}})\), assuming that the magnetization M = −Hc1, when the first vortex enters into the sample. Thus, magnetic field has rescaled to Heff = H−NM and \({H}_{c{\rm{1}}}({\rm{0}})={H}_{c{\rm{1}}}^{\text{'}}({\rm{0}})/({\rm{1}}-N)\), where N is a demagnetization factor and H is a magnetic field. It has been shown by Brandt53 that a bar sample with a rectangle cross-section, the effective demagnetization factor \(N={\rm{1}}-\,\tanh \,\sqrt{{\rm{0.36}}t/b}\), where b and t width and thickness of the sample. Using this criterion, Hc1(0) was estimated and fitted to the parabolic temperature dependence Hc1(T) = Hc1(0)(1 − (T/Tc)2) and the experimental Hc1(0) is estimated at various pressures. The fit of our data to this expression (solid red line) suggests that vortex penetration in this material can be well described by BCS theory. Although the demagnetization effect is small and corrected lower critical field \({H}_{c{\rm{1}}}^{\text{'}}({\rm{0}})\) is calculated from Hc1(0) with demagnetization correction using Brandt’s formula,\({H}_{c{\rm{1}}}^{\text{'}}({\rm{0}})={H}_{c{\rm{1}}}({\rm{0}})/\,\tanh \,\sqrt{{\rm{0.36}}t/b}\) for approximate slab geometry53 and these values of \({H}_{c{\rm{1}}}^{\text{'}}({\rm{0}})\) listed in Table 1. Using the Hc1(0), it is possible to estimate the penetration depth from the relation, \(\lambda ({\rm{0}})={[(\Phi {}_{{\rm{0}}}/{\rm{2}}\pi {H}_{c{\rm{1}}}^{\text{'}}({\rm{0}}))(\mathrm{ln}\kappa +{\rm{0.5}})]}^{{\rm{0.5}}}\), where λ(0) is the penetration depth at 0 K. The ratio between the λ(0) and coherence length (ξGL(0) gives GL parameter (κ) through the expression κ = λ(0)/ξGL(0). The value of \(\kappa =\sqrt{{\rm{1}}/{\rm{2}}}\) has been conventionally used to classify superconductors as type I or type II based on whether κ value is higher or lower than \(\sqrt{{\rm{1}}/{\rm{2}}}\). Our analysis predicts that κ value is very much higher than critical value and indicates Cr0.0005NbSe2 as a type II superconductor. Similarly CuxNbSe212 and Zr0.96V0.04B254 compounds were shown to be type-II superconductor, since κ value has been reported larger than the critical value.

Figure 4
figure 4

(a) Temperature dependence of irreversible field (Hirr) for Cr0.0005NbSe2 sample under pressure. Solid lines represent how fit the relation Hirr(T) = Hirr(0)(1 − (T/Tc)2)3/2, (b) shows the lower critical field (Hc1) for Cr0.0005NbSe2 sample under pressure. Solid lines show the parabolic fitting.

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Figure 5(a–d) shows the field dependence of superconducting critical current density (Jc(H)) on Cr0.0005NbSe2 for various temperatures at constant hydrostatic pressures of 0, 0.03, 0.5, 0.90 and 1.02 GPa. The Jc(H) is estimated from MHL using Bean’s model55,56 Jc = 20ΔM/(b(3l − b)/3l), where width of the magnetization [ΔM = M + (H) − M−(H)], b is a width and l is a length of the samples (b < l) were in mm. In general, low-magnetic field region, the gap ΔM in magnetization loop is mainly caused by the intergranular current. However, in the high-field region, ΔM results largely due to the intragranular current. Similar results have been reported from M(H) measurements on cuprates57 and pnictides58,59.

Figure 5
figure 5

(a), (b), (c) and (d) Magnetic field dependent critical current density (Jc(H) on Cr0.0005NbSe2 for various hydrostatic pressures at various temperatures.

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High pressure enhances the Jc in Cr0.0005NbSe2 sample and subsequently enhances flux pinning with an increase of point pinning centers. The Jc(0) values determined from MHL curves at 2 K are 391884, 1029067, 1190414, 1169989, 1258268 A/cm2 for 0, 0.03, 0.5, 0.9 and 1.02 GPa respectively and the Jc(H = 0) increased by a factor of three at higher pressures. Figure 6(a–c) shows Jc(H) [Fig. S3] in the logarithmic scale for various temperatures and reveals that Jc exponentially decreases as the H increases. This result indicates that occurrence of weak pinning singularities, since magnetic impurities (Cr) act as point pinning centers. Further Fig. 6(d) shows exponential decay of Jc with an increase in applied magnetic field and these results compared theoretically, Jc(H) = Jc(0)exp{−(H/H0)3/2} and found excellent fitting for various pressure and temperatures.

Figure 6
figure 6

(a), (b), (c) and (d) Logarithmic scale of magnetic field dependent superconducting critical current density Jc(H) on Cr0.0005NbSe2 for various temperatures and a few fixed hydrostatic pressures. Solid lines are shown in the collective pinning model fitting at the log scale at a temperature of 2 K.

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Figure 7(a) shows Jc as a function of pressure in logarithmic scale at 2 K under various fields and the solid lines show linear fits to the data, which gives the slopes [d(logJc)/dP] of 0.40, 0.53, 0.71, 1.14 and 1.29 GPa−1 at 0, 0.02, 0.06, 0.1 and 0.2 T respectively. These results indicate that application of pressure leads to an enhancement in Jc and it is more significant at higher fields. The normalized Jc as a function of temperature at 0 and 0.1 T for various pressures is shown in Fig. 7(b,c) and fitted with the scaling relation Jc ∝ (1 − T/Tc)n where n is the critical exponent at each pressure. It is known that GL theory predicts distinct vortex pinning mechanisms in superconducting materials, with different values of exponent (n) at specific fields. The value of n = 1 and n >1.5 corresponds to non-interacting vortices and strong vortex core pinning mechanism respectively. The critical exponent is estimated from fitting the scaling relation in Fig. 7(b,c) for various magnetic fields at constant hydrostatic pressure. It reveals that the values of n are found to be 1.65 ≤ n ≤ 2.12(0 T) and 2.82 ≤ n ≤ 3.75 (0.1 T) under various hydrostatic pressure and shows higher value of n under pressure than at ambient pressure. Similar behaviour has been reported on Jc(P), Jc(T) and vortex dynamics properties in FexNbSe238 and in pnictides34.

Figure 7
figure 7

(a) Pressure dependence of Jc at 0, 0.02, 0.06, 0.1 and 0.2 T for 2 K. Solid lines are shown in the Linear fitting, (b) and (c) Temperature dependent normalized Jc at various pressures for constant magnetic field of 0.05 and 0.1 T. Solid lines are shown in the Jc ∝ (1 − T/Tc)n fitting.

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The core pinning mechanism is examined in the framework of collective pinning theory in this sample. Generally, core pinning comprises that pinning mechanism related to the spatial variation in the charge carrier mean free path (l), called δl pinning and pinning due to randomly distributed spatial variation in Tc, called δTc pinning, which is mostly due to crystal defects60. We analyzed the pinning mechanism as reported by Griessen et al.60 using the relation Jc/J0 ∝ (1 − t2)5/2/(1 + t2)1/2 for δl pinning mechanism, while Jc/J0 ∝ (1 − t2)7/6/(1 + t2)5/6 applies to δTc pinning, where t = T/Tc. Figure 8(a–f) shows almost perfect matching of the experimentally obtained Jc with the theoretically calculated values and favour δl pinning mechanism in the entire pressure region. Our analysis directly supports δl pinning mechanism whatever be the pressure, and it is responsible for increasing in Tc Our results suggest that there is a single vortex pinning due to spatial variations in the charge-carrier mean free path. The coherence length is proportional to the mean free path of the carriers, and therefore, the application of pressure leads to enhancement of δl pinning mechanism. One can note that similar results have been reported in case of FexNbSe26,38, FeTe0.7Se0.361 and pnictides34 superconducting compounds.

Figure 8
figure 8

(a–f) Normalized Jc (Jc/J0) as a function of reduced temperature (T/Tc) for various hydrostatic pressures on Cr0.0005NbSe2. The Solid line (red) is shown in δl pinning fitting and the solid line (blue) is shown in δTc pinning fitting.

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We have calculated the pinning force (Fp = JcH) as a function of magnetic field under various hydrostatic pressures and have investigated the magnetic field dependence of the pinning force density (Fp) in order to understand vortex pinning mechanisms in this sample. Field dependence of Fp [Fig. S4] for various temperatures in superconductors may be scaled into a unique curve if they are plotted as a function of reduced field, h = H/Hirr. The scaling of normalized pinning force from the Dew-Hughes formula62, fp = hp(1 − h)q, where p and q are the parameters describing the pinning mechanism. Further, Jc0.5H0.25 is plotted as a function of magnetic field is known as Kramer plot63 and it is used to determine Hirr, where Hirr is determined as the extrapolated to zero field from various isotherms of Jc and the temperature dependent of Hirr shows a linear behavior.

Figure 9(a–f) shows normalized pinning force (fp = Fp/Fp(max)) as a function of reduced critical field (h = H/Hirr) for various temperatures and constant P (0,0.5 and 1.02 GPa) [Fig. S5]. For the scaling of both point and surface pinning, we use the relation, fp = Ahp(1 − h)q, where p and q describes the nature of specific pinning mechanism62. It is known from Dew-Hughes model, when p = 1/2 and q = 2 and p = 1 and q = 2 describes the surface pinning and point pinning respectively as predicted by Kramer63. The best fit of the curves is obtained with fp(h) dependence given by, h0.89(1 − h)4.38 + Bh0.72(1 − h)1.38 at 0 GPa and h0.91(1 − h)5.70 + Bh0.79(1 − h)2.48 at 1.02 GPa. The application of pressure causes an increase of q parameter from 4.38 to 5.70 which is an indication of the increase of both pinning center and surface pinning mechanism. It reveals that the normal core point pinning is more dominant than surface pinning mechanism under high pressure [Fig. S5]. The broadening of the pinning force with the application of pressure indicates that the pinning centers are enhanced with the application of pressure for this sample. Cr intercalation of into Se-Nb-Se layers of NbSe2 single crystals leads to an increase of more pinning centers compared with the undoped NbSe2.

Figure 9
figure 9

(a–f) Normalized pinning force density (fp = Fp/Fp(max)) as a function of reduced magnetic field (h = H/Hirr) for various hydrostatic pressures on Cr0.0005NbSe2. Solid lines are shown in the collective pinning model fitting at the log scale at a temperature of 2 K.

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Conclusion

In summary, we have shown that hydrostatic pressure is a very effective means to significantly enhance Tc, Jc, Hirr, and flux pinning in the Cr0.0005NbSe2 superconductor. We demonstrate that hydrostatic pressure significantly increases Tc from M(T) (dTc/dP = 0.91 K/GPa; 0 < P < 1 GPa) and ρ(T) (0.75 K/GPa; P < 1.5 GPa & 0.09 K/GPa; 1.5 < P < 8 GPa) measurements. The pressure introduces more point defects in sample and it is responsible for enhancement Jc. We found that the hydrostatic pressure stabilizes a strong δl pinning mechanism. In addition, we found that the point pinning is more dominant than surface pinning under high pressure. The pressure enhanced the Hc1, Hc2, Hirr and reduces both coherence length and penetration depth which are responsible for the pinning mechanism.

Experimental Techniques

Cr0.0005NbSe2 single crystal has been synthesized using chemical vapour transport method. The essential commercially available high purity elemental metallic powders such as Nb (99.95%), Se (99.99%) and Cr (99.99%) procured from Alfa Aeser, which are mixed with suitable stoichiometric ratio and made of 6 mm (φ) pellet. The pellet kept in a sealed quartz tube with tiny amount of iodine in the presence of liquid nitrogen atmosphere. The sealed quartz tube is placed in a two-zone furnace with programmable temperature controller and the temperatures of charge and growth zone at 800 °C and 720 °C respectively for a period of seven days. The grown crystals are characterized structural phase and the elemental composition analyses are confirmed using x-ray diffraction and Energy dispersive X-ray Spectroscopy techniques respectively and further details about synthesis and characterization of these materials were recently reported by Rukshana Pervin et al.64. A cubic anvil pressure device, consisting of six tungsten carbide (WC) anvils, which have been used to produce homogeneous hydrostatic pressure up to 8 GPa for resistivity measurements65. The applied pressure is calibrated from the resistance changes of Bi with its phase transitions at room temperature such as Bi I-II (2.55 GPa), Bi II-III (2.77 GPa) Bi III’-V (7.68 GPa) and Daphne #7373 used as a pressure transmitting medium66. The temperature of sample inside the Teflon cell is monitored by measuring the resistivity of three calibrated Pt(Co) resistance thermometers attached to the neck of each anvil. The electrical contacts were made using gold wire of 20 µm φ and Ag paste is used to make contacts on the surface of the sample with the typical sample size is 0.8 × 0.5 × 0.5 mm3. The standard four probe method is used for doing resistivity measurements under ambient and high pressure upto 8 GPa and the anvils are through the gold foil. The magnetic properties (ZFC & FC) of Cr0.0005NbSe2 single crystals were investigated under various magnetic fields at ambient and high pressures. The dc magnetization measurements under pressure were carried out using Physical Property Measurements System – Vibrating Sample Magnetometer (PPMS-VSM, Quantum Design, USA). The external pressure was generated up to ~1 GPa by a clamp type miniature hydrostatic pressure cell which is made of specially heat treated nonmagnetic Cu-Be alloy. The fluorinert FC #70 and FC #77 (1:1) mixture was used as a pressure transmitting medium and the in-situ pressure was estimated from the superconducting transition temperature of pure Sn which was loaded along with the sample in the capsule.