On the nature of superconductivity in the anisotropic dichalcogenide NbSe₂{CoCp₂}_x

The temperature dependent magnetic dc-susceptibility (χ_V(T)) of NbSe₂{CoCp₂}_0.26 is depicted in figure 3 for zero field cooled (ZFC) and field cooled (FC) cycles in a low magnetic field applied perpendicular and parallel to the ab-planes. For the ZFC sequence the sample was first cooled down to 2 K in a zero field, where the earth magnetic field was compensated within an error bar of 1 µT by the so-called TinyBee setup described in detail in [22]. In the subsequent heating run a small magnetic field of μ₀H = 0.5 mT was applied to record the susceptibility data up to 8 K. For the susceptibility measurements with H perpendicular to the ab-plane (H⊥ab) a demagnetization factor of N = 0.8 was taken into account. This factor is estimated experimentally by measuring a lead plate with a shape nearly identical to the investigated sample (≈4 mm × 2.5 mm; slight trapezoid). For the volume susceptibility χ_V we calculated the density of NbSe₂{CoCp₂}_0.26 from the lattice parameters and the stoichiometry of the sample known from the ICP-OES and elementary analysis, resulting in a value of ρ = 4.052 g cm⁻³.

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In figure 3 all χ_V(T) curves exhibit a sharp superconducting transition at T_c = 7.18 K for the ZFC and FC procedures. This critical temperature is virtually identical to the one of the host substance and is slightly smaller than $T^{{\rm onset}}_{{\rm c}} = 7.5\,{\rm K}$ estimated from our resistivity studies (section 4.3, figure 7(a)). This result is remarkable due to the fact that previous intercalations of 2H-NbSe₂ with electronically innocent guest molecules yielded a significant reduction of T_c and values between 0.6 K and 6.88 K [9, 19, 21, 23, 24].

Furthermore, the χ_V(T)-data of NbSe₂{CoCp₂}_0.26 show some deviation from an ideal bulk superconducting behavior. In the ideal case, the ZFC procedure in low magnetic fields results in a complete shielding state of the sample with χ_V = −1 at low temperatures. The FC measurement provides information about the strength of the flux expulsion due to the Meissner effect. In the case of hard type-II superconductors, such as the high temperature superconductor YBaCuO [25], a superconducting volume susceptibility in the range of −1 ⩽ χ_V ⩽ −0.5 is usually expected, in contrast to our findings for NbSe₂{CoCp₂}_0.26 with χ_V ≈ −0.05 (H ⊥ ab).

For NbSe₂{CoCp₂}_0.26 we observed in the case of H || ab that the ZFC procedure results in a partial screening (χ_V > −1), mainly due to the presence of a normal state region in the van der Waals gap between the (NbSe₂)₂ layers. On the other hand for H ⊥ ab a full screening is expected. The deviation of 7% from the ideal value may be due to the error in the determination of the demagnetization factor or the sample density. For the FC measurement with H || ab the sample is exposed to a maximal field of μ₀H = 0.5 mT due to the vanishing demagnetization factor. This leads to a flux expulsion in the superconducting planes but not in the inter-layer space. Therefore, the resulting intermediate phase gives rise to a small but not negligible Meissner-phase. On the contrary, for H ⊥ ab the demagnetization factor is close to one, due to the plate-like shape of the sample. This may lead to local magnetic fields higher than μ₀H_c1⊥, resulting in flux-penetration with vortex field strengths higher than μ₀H_c1⊥. As a result, the Meissner effect may be superimposed by this flux pinning process and therefore only a negligible Meissner effect is observed.

This deviation from an ideal superconducting behavior strongly points to a pronounced anisotropic superconductor with a layered structure of superconducting planes which are weakly coupled by van der Waals forces.

In order to calculate the microscopic superconducting parameters like the London penetration depth (λ), the GL coherence length (ξ), and the GL parameter (κ), the lower and upper critical fields perpendicular and parallel to the ab-planes are determined from magnetization curves and from temperature dependent specific heat and resistivity measurements in various magnetic fields.

The lower critical fields of 2H-NbSe₂ and NbSe₂ {CoCp₂}_0.26 are estimated from magnetization measurements μ(μ₀H) at very low magnetic fields, using the high resolution of μ₀ ΔH = 1 µT increments accomplished by the TinyBee setup [22]. In figure 4(a) the magnetic moment versus magnetic field for 2H-NbSe₂ in the case of H perpendicular to the ab-planes at 2 K is plotted in the range between 0 and 4 mT using minute increments of μ₀ ΔH = 0.025 mT. The linear fit (employing μ(H) data below 0.5 mT) shows the first deviation from the magnetization curve at approximately 1 mT. According to a common procedure in the literature (see, e.g. [26]) the latter value can therefore be employed to specify μ₀H_c1 at 2 K. This is in accordance with the kink in the derivative $\frac{{\rm d}\mu}{\mu_0 {\rm d} H}$, as shown in figure 4(b). This kink in the derivative $\frac{{\rm d}\mu}{\mu_0 {\rm d} H}$ is, however, a more stringent criterion for the determination of μ₀H_c1, because its location is less dependent on the choice and size of the fitting range and was therefore used to extract the H_c1 values. This assessment is a sensitive indicator to identify the magnetic field μ₀H_c1, which is defined as the field where the first flux in the form of a quantized vortex penetrates the sample and starts to increase continuously with an increasing magnetic field [1]. It should be mentioned that for NbSe₂, $\frac{{\rm d}\mu}{\mu_0 {\rm d} H}$ (μ₀H) shows a slight increase for small fields in both field orientations, which may be due to a small flux penetration caused by sample inhomogeneities, thereupon the magnetic flux penetrates the sample in the form of quantized vortexes visible by a pronounced increase in the derivative $\frac{{\rm d}\mu}{\mu_0 {\rm d} H}$, as shown in figures 4(b) and (c).

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In contrast to this, figures 4(d) and (e) show for the intercalated sample NbSe₂{CoCp₂}_0.26 a strong initial flux penetration. At a distinct magnetic field the slope of the derivative $\frac{{\rm d}\mu}{\mu_0 {\rm d} H}$ changes to smaller values indicating a reduced rate of flux penetration. This unusual behavior hints at unquantized flux penetration in the intercalated 2H-NbSe₂ already below H_c1. This observation is likely explained by the large spacial separation of the superconducting layers. In the case of magnetic fields above H_c1, the screening currents become sufficiently large to channel the flux into quantized vortices. Since the spacial extent of a vortex, given by the penetration depth λ, is far larger than the layer distance in 2H-NbSe₂, the vortex penetration rate above H_c1 remains essentially unaffected by the subtle and additional increase in the inter-layer distance due to the intercalation of the cobaltocene molecules.

The lower critical field μ₀H_c1 at zero temperature is determined by extrapolating the magnetic field data at 2 K and T_c, using the empiric parabolic relation [27]

$$\begin{equation} \mu_0 H_{{\rm c1}}(T)=\mu_0 H_{{\rm c1}}(0)\times[1-(T/T_{{\rm c}})^2] \end{equation} \tag{ 1 }$$

by taking into account a demagnetization factor of N = 0.8 for the applied field H perpendicular to the ab-planes. The resulting lower critical field values with H parallel and perpendicular to the ab-planes are depicted in figures 4(b)–(e) and are summarized in table 1 for NbSe₂ and NbSe₂{CoCp₂}_0.26, respectively.

The upper critical fields H_c2 perpendicular and parallel to the ab-planes are extracted from specific heat and resistivity measurements, respectively (for details see below). For NbSe₂{CoCp₂}_0.26, the electronic contribution to the specific heat divided by temperature ΔC/T is plotted for H perpendicular to the ab-planes versus T² in various magnetic fields up to 5 T in figure 5(a). The critical temperature is suppressed gradually by the magnetic field from T_c = 7.12 K to a temperature below 2 K for μ₀H > 5 T. The specific heat discontinuity at T_c(0) of δC/T ≃ 31.6 mJ mol⁻¹ K² divided by the Sommerfeld coefficient in the normal state γ ≃ 22.1 mJ mol⁻¹ K² yields the thermodynamic ratio δC/γT_c = 1.43, which surprisingly matches the expected value of 1.43 for a weakly coupled BCS superconductor [28]. In contrast to this, a somewhat higher δC/γT_c ratio is observed in the parent compound 2H-NbSe₂ with varying values between 1.96 and 2.12, which may be connected to a more pronounced electron-phonon coupling (see table 1 and [29, 30]). Such enhanced values are generally known for layered superconductors, e.g. FeSe (δC/γT_c = 1.65) [31].

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Furthermore, it should be mentioned that the nearly linear trend in the ΔC/T versus T² plot below T_c of NbSe₂{CoCp₂}_0.26 (figure 5(a)) is also observed in the parent species [13, 29, 30]. This linearity is unusual for single band s-wave superconductors. A similar superconducting behavior is also found in the intermetallic compound MgB₂, which is discussed with respect to a second superconducting band at the Fermi level [32]. In particular, for the host sample 2H-NbSe₂, Huang et al proposed that a two-gap scenario is more favorable than an anisotropic s-wave model [29]. We will outline in the following that the latter proposal is not supported by our own study. We give a more detailed discussion and comparison to the model calculations in section 5.

For NbSe₂{CoCp₂}_0.26 the temperature dependency of the in-plane resistivity in various magnetic fields is shown in figure 5(b) in the temperature range between 1.8 K and 8 K. Towards lower temperatures, after the initial steep drop of the resistivity due to the onset of superconductivity, the sample shows a finite resistivity, which again vanishes on further cooling. This observed reentrant behavior is also known from the host sample 2H-NbSe₂ [33]. The out-of-plane resistivity behavior could, however, not be determined, although a special set-up was used as described in [34].

In figure 6 the phase diagrams of the upper (a) and lower (b) critical magnetic fields parallel and perpendicular to the ab-planes are presented. For μ₀H_c2∥ the critical temperatures are obtained from the 50% values of the normal state resistivity (figure 5(b)) slightly below the onset of superconductivity. For H_c2⊥ the T_c data are determined at that temperature, where the specific heat discontinuity of the superconducting anomaly reaches half of the values δC/T (figure 5(a)). For both branches (H || ab and H ⊥ ab in figure 6(a)) we extrapolate the μ₀H_c2 values in the linear region between 2 and 6 K down to zero temperature taking into account the WHH-theory [35]. It should be mentioned that a positive curvature of μ₀H_c2∥(T) is observed just below T_c, which is not in line with the WHH-theory. However, this feature appears to be a characteristic H_c2∥(T) signature observed in many layered superconductors (see chapter 6 in [1]), and in particular also in the host substance 2H-NbSe₂ [8]. This positive, upward curvature may be due to the onset of a dimensional cross-over [36], or caused by an anisotropic Fermi-surface [20]. The phase boundaries of the lower critical fields down to zero temperatures are generated from the values of the lower critical fields at 2 K (see figures 4(d) and (e), 5 K and from the T_c values at H = 0, using the empiric relation for μ₀H_c1(T) (see equation (1)). The resulting lower and upper critical field values at T = 0 K are tabulated in table 1, in comparison with selected results of the host sample 2H-NbSe₂.

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The upper critical field ratio H_c2||/H_c2⊥ of NbSe₂ {CoCp₂}_0.26 defines the anisotropy factor Γ, which is about 4.4 at T = 0. This value is 34% larger than that of 2H-NbSe₂ and is similar to those values found in different single crystalline MgB₂ samples (for an overview see [37]). For both species, NbSe₂ and MgB₂, a multi-band scenario is discussed in the literature, which goes along with a temperature dependent anisotropy factor Γ(T) [8, 37]. Such behavior is also observed in NbSe₂{CoCp₂}_0.26 (insert of figure 6(a)), but the change of the anisotropy per Kelvin $\frac{{\rm d} \Delta \Gamma}{{\rm d}T}$ of approximately 0.075 K⁽⁻¹⁾ is two times smaller than in MgB₂—the benchmark system for multi-band superconduction. This may be due to different determination methods of Γ(T) [8, 37, 38]. Even though the change of Γ(T) is smaller in comparison to MgB₂, this criterion does not disqualify NbSe₂{CoCp₂}_0.26 as suitable candidate for a multi-band scenario at this stage of our analysis.

For a detailed comparison between 2H-NbSe₂ and NbSe₂{CoCp₂}_0.26 we also calculated the microscopic superconducting parameters ξ, λ, and κ directly from the critical magnetic fields according to the anisotropic GL theory [1]. Here the GL coherence length ξ_GL is extracted from the upper critical field using the following equations:

$$\begin{eqnarray} &&\fl \mu_0 H_{{\rm c2}\parallel} = \frac{\Phi_0}{2 \pi \xi_{\parallel} \xi_{\bot}} \qquad {\rm and} \qquad \mu_0 H_{{\rm c2}\bot} = \frac{\Phi_0}{2 \pi \xi_{\parallel}^2} \end{eqnarray} \tag{ 2 }$$

with the flux quantum Φ₀ = h/2e. The London penetration depth λ is extracted from the lower critical field:

$$\begin{eqnarray} &&\fl \mu_0 H_{{\rm c1}\parallel} = \frac{\Phi_0}{4 \pi \lambda_{\parallel} \lambda_{\bot}} \times \ln \kappa_{\parallel} \qquad {\rm and} \qquad \mu_0 H_{{\rm c1}\bot} = \frac{\Phi_0}{4 \pi \lambda_{\parallel}^2}\nonumber\\ &&\qquad\qquad \times \ln \kappa_{\bot} \end{eqnarray} \tag{ 3 }$$

with the GL-parameters

$$\begin{equation} \kappa_{\parallel} = \left(\frac{\lambda_{\parallel} \lambda_{\bot}}{\xi_{\parallel} \xi_{\bot}}\right)^{1/2} \qquad {\rm and} \qquad \kappa_{\bot} = \frac{\lambda_{\parallel}}{\xi_{\parallel}} \end{equation} \tag{ 4 }$$

which are extracted from microscopic superconducting parameters.

All the quantities are presented in table 1 together with those of selected studies in order to compare our results of the host species 2H-NbSe₂ with former findings in the literature (see, [8, 40–42]). First of all it should be mentioned that within the variation of different calculations and measurement procedures, the superconducting quantities like ξ, λ, and κ of the parent compound 2H-NbSe₂ are in good agreement with previous results. The direct comparison between the host species and NbSe₂{CoCp₂}_0.26 surprisingly reveals that the microscopic superconducting GL quantities, ξ_|| and λ_||, remain almost unchanged within the layers upon CoCp₂ intercalation. On the contrary, the coherence length perpendicular to the layers ξ_⊥ diminishes by 25% after CoCp₂ intercalation, and is of the order of the inter-layer distance of the intercalated species. In addition, the penetration depth parallel to the layers λ_|| is approximately of the same order of magnitude for both species, whereas for λ_⊥ a drastic increase by a factor of 4.5 is observed between the host and the intercalated sample, respectively.

This analysis emphasizes the outstanding character of the intercalated samples, as the superconducting properties within the layers remain almost unchanged but change significantly perpendicular to the layer along the c lattice parameter. This is a surprising result since cobaltocene (CoCp₂) is an electronic donor molecule, which is capable of transferring its unpaired electron into the conduction band of suitable host materials which might act as electron acceptors. Indeed, the electron transfer is indicated by an enhancement of the Sommerfeld coefficient γ in the intercalated sample (NbSe₂: γ = 18 mJ mol⁻¹·K⁻²; NbSe₂{CoCp₂}_0.26: γ = 22 mJ mol⁻¹ · K⁻²) and a vanishing local magnetic moment at the intercalated cobaltocene molecules in the normal state. However, the superconducting properties of the parent lattice do not change significantly within the layers upon the intercalation.

Within the framework of the anisotropic GL theory, it is expected for a single gap anisotropic superconductor that the ratio of the upper critical field H_c2||/H_c2⊥ equals the ratio of ξ_|| / ξ_⊥, λ_⊥ / λ_|| and κ_|| / κ_⊥ [1]. The anisotropy factor Γ ≈ 3.3 of 2H-NbSe₂ can be extracted from its H_c2- and ξ-ratio. In 2H-NbSe₂, the ξ-ratio is slightly larger than the corresponding λ- (1.6) and κ-ratio (2.2). For the intercalated sample NbSe₂{CoCp₂}_0.26 we observe a higher anisotropy factor of about 4.4, which is equal to the ξ ratio, but is significantly smaller than the λ- (7.8) and κ-ratio (5.9). These results therefore suggest that the anisotropic superconducting behavior in NbSe₂{CoCp₂}_0.26 cannot be described and modeled within the framework of the anisotropic GL theory involving an anisotropy in the effective masses [1]. This is especially true for a pronounced anisotropic system where the layer distance is of the same order of magnitude as the coherence length [10]. An alternative approach for the description of a layered superconductor based on Josephson-coupled 2D layers leads to a somewhat different expression for H_c1|| [39]. However, the numerical values for the λ- and κ-ratio for NbSe₂{CoCp₂}_0.26 are almost identical to those obtained above.

As mentioned in section 4.2 the temperature-dependent in-plane resistivity in various magnetic fields shows a reentrant superconducting behavior (see figure 5(b)). This behavior is directly associated with a peak effect which is frequently found in layered type-II superconductors, including the host sample 2H-NbSe₂ (see, for example, [43]). The observed peaks in the resistivity data ρ(T) in figure 5(b) might be a consequence of two competing mechanisms: superconductivity and flux motion controlled resistivity. Therefore, an analysis of the temperature-dependent resistivity could provide a measure of the average vortex velocity. For conventional low temperature superconductors this effect is primarily observed in the regime of the upper critical field μ₀H_c2 and indicates the presence of different superconducting phases arising from a change of flux motion [44]. For example, a highly dense vortex liquid phase is expected below μ₀H_c2, when the layer separation, d, is of the same order of magnitude as the coherence length ξ [10]. For NbSe₂{CoCp₂}_0.26 this criterion is nearly fulfilled when the magnetic field is applied perpendicular to the layers. In that case d becomes compatible in magnitude with ξ_⊥/2 (see table 1).

In order to deal more precisely with the temperature dependent resistivity in the superconducting phase of NbSe₂{CoCp₂}_0.26 we present in figures 7(a) and (b) enlarged plots of ρ(T) near T_c in two selected magnetic fields, 0 T and 10 T, respectively. In a zero magnetic field, NbSe₂{CoCp₂}_0.26 exhibits the highest superconducting critical temperature, $T^{{\rm onset}}_{{\rm c}} = 7.5\,{\rm K}$, observed so far in the intercalated or substituted 2H-NbSe₂ samples (see section 4.1). Towards lower temperatures and after the initial superconducting drop, the resistivity shows a pronounced shoulder around 7.4 K, followed again by a steep drop until ρ(T) vanishes on further cooling. This observation hints at two competing mechanisms which control the ρ(T) behavior of the intercalated NbSe₂ phases. In order to identify this additional contribution a single superconducting transition has been modeled by a Boltzmann-type sigmoidal function to fit the experimental data (disregarding the temperature region between 7.2 K and 7.4 K). In figure 7(a) the dashed red line depicts the superconducting transition according to the Boltzmann-type sigmoidal function:

$$\begin{equation} \rho_{{\rm fit}} = \rho_{{\rm high}} - \frac{\rho_{{\rm high}}}{1+{\rm e}^{(T-T_{{\rm c}})/\Delta T}}, \end{equation} \tag{ 5 }$$

with ρ_high being the normal state resistivity at T = 8 K and ΔT the slope factor. Here, T_c = 7.39 K is calculated from the ρ_fit-data (the dashed red line in figure 7(a)) and taken as the temperature, where the resistivity drops by 50% with respect to the value of the normal state resistivity at 8 K. Subtracting the received superconducting contribution from the experimental data results in a peak-like feature (the solid green line in figure 7(a)). From this differential curve two characteristic temperatures are obtained. T_p1(T) is defined by the intersection of the Boltzmann curve with the difference curve, and T_p2(T) marks the position of the maximum of the extracted peak. With increasing magnetic fields the shoulder shifts to lower temperatures, until it renders into a separate peak, whose height increases with increasing fields (figure 7(b)). In this case, T_p1 and T_p2 are marked as the local minimum and maximum of ρ(T), respectively. The results of this investigations are summarized in the H-T-phase diagram for H_|| in figure 9(a). It should be mentioned that the H-T-phase diagrams of the intercalated sample and the host substance in the highly dense vortex liquid phase are very similar, with exception of the upper critical field area [14].

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The presence of a peak effect is also supported by AC-susceptibility measurements. In figures 8(a) and (b) the imaginary and real part of the AC-susceptibility, $\chi_{V}^{\prime\prime}(T)$ and $\chi_{V}^{\prime}(T)$, are plotted for NbSe₂{CoCp₂}_0.26 in an applied magnetic DC-field of 2 T and 3T perpendicular to the layers. Similar to the resistivity measurements, a pronounced peak effect is determined from the AC-susceptibility data. For $\chi_{V}^{\prime}(T)$ a distinct peak develops, again characterized by the two temperatures T_p1 and T_p2, which mark the local minimum and the local maximum, respectively. However, in the cases of $\chi_{V}^{\prime\prime}(T)$ the situation is reversed (see figures 8(a) and (b)). The resulting temperature-dependent characteristic magnetic fields, μ₀H_p1⊥(T) and μ₀H_p2⊥(T), are summarized in the H-T-phase diagram for H_⊥ in figure 9(b). In the case of H_||(T) (figure 9(a)) $\chi_{V}^{\prime}(T)$ and $\chi_{V}^{\prime\prime}(T)$ could only be analyzed up to 0.6 T. Nevertheless, for this low field region the H_||(T) values of the characteristic temperatures, determined from the susceptibility data, nicely agree with the values from the resistivity measurements [14].

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The phase diagrams in both directions (figures 9(a) and (b)) exhibit three different phases in the mixed state which broaden with increasing magnetic fields. Such behavior is also found in 2H-NbSe₂ for H perpendicular to the layers [33, 45]. For the dilute vortex state in low magnetic fields, no phase separation up to 2 T for H_|| and 0.5 T for H_⊥ is observed. For the host sample a clear separation occurs in this region, which broadens with decreasing fields [46]. This different behavior in the dilute vortex region may be due to an unusual quantized flux penetration, as discussed in section 4.2.

In weakly pinned type-II superconductors, the vortex assemblage acts more or less as an elastic medium in a random pinning environment which can be activated thermally or driven by disorder. In both cases the shear forces between the single vortices get stronger than the pinning force. Intrinsic pinning centers are most likely in the large gaps between the superconducting layers; a scenario which is also discussed for HTC superconductors [47]. In the latter case, the mixed phase is not homogeneous but consists of different complex vortex phases [43]. In the simplest case the vortex lattice changes from a solid state, where the vortex array is fixed in an ordered Abrikosov- or in a slightly disordered Bragg phase, to an amorphous state (glass phase) and, finally, into a vortex liquid state, where the vortex flux motion is reversible. In comparison to the phase diagram of 2H-NbSe₂, the region between the μ₀H_c2 and μ₀H_p1 borderlines is related to the liquid state, followed by an amorphous state between the μ₀H_p1 and μ₀H_p2 lines. Due to the complex nature of vortex phases, a vast number of detailed studies have been performed in the case of high T_c as well as low T_c superconductors, see, e.g. [10, 33, 48, 49]. To analyze the complicated vortex phase structure in NbSe₂{CoCp₂}_0.26, further investigations are warranted which also consider the subtle pinning behavior of this type-II superconductor in greater detail, as well as the presence of large van der Waals gaps and intercalated guest molecules which might act as pinning centers.