Upper critical magnetic field in NbRe and NbReN micrometric strips

Figure 1 displays the normalized resistive transitions in zero magnetic field of the NbRe and NbReN microstrips. The critical temperature, the low-temperature resistivity, and the residual resistivity ratio (RRR) are reported for both microstrips in Table 1. The RRR is defined as the ratio of the resistivity at room temperature and at 10 K, that is, RRR = ρ300K/ρ10K = R300K/R10K. The values of Tc are not significantly smaller than the values measured on unpatterned films of the same thickness [4,8]. The rounding present at the onset of both the curves is due to the paraconductivity phenomenon, whose nature has been analyzed in detail in the case of unstructured NbReN films of different thickness [4].

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Figure 1: Resistive transition in zero magnetic field of the NbRe (black squares) and NbReN (red circles) microstrips.

Table 1: Parameter values of NbRe and NbReN microstrips.

Parameters NbRe NbReN Film thickness (nm) 8 10 ρ (μΩ·cm) 248 220 RRR 0.92 0.94 D (× 10−4 m2/s) 0.49 0.48 Tc (K) 5.31 4.61 [Graphic 8] (T/K) 2.23 2.27 α = 3/(2kF[Graphic 9]) 0.51 0.51 μ0Hp(0) (T) 9.78 8.48 [Graphic 10] (T) 8.09 7.17 μ0Hc2(0) (T) 8.12 6.38

In Figure 2, the resistive curves of NbRe are shown for various values of H with the microstrip placed perpendicularly (Figure 2a) or parallely (Figure 2b) to the external field. The same quantities measured for NbReN are shown in Figure 2c and Figure 2d. In Figure 2e and Figure 2f, the field dependence of the width of the resistive transitions, ΔTc, is reported for NbRe and NbReN, respectively. We define [Graphic 11], where [Graphic 12]([Graphic 13]) is the critical temperature obtained with the 90% (10%) RN criterion. As it can be seen, when the samples are placed perpendicularly to the field the curves significantly broaden at high fields due to the entering of the vortices. However, the value of ΔTc is similar for both materials. In contrast, in a parallel field, ΔTc is almost constant in both cases.

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Figure 2: (a) Temperature dependence of the resistance of the NbRe microstrip in various magnetic fields in the perpendicular geometry. The curves have been measured for increasing magnetic fields (as indicated by the arrow) from μ0H = 0.003 to 4 T. (b) The same as panel (a) but in the parallel geometry. The field increases from μ0H = 0.003 to 8 T. (c) Temperature dependence of the resistance of the NbReN microstrip in various magnetic fields in the perpendicular geometry. The curves have been measured for increasing magnetic fields from μ0H = 0.001 to 3 T. (d) The same as panel (c) but in the parallel geometry. The field increases from μ0H = 0.007 to 5 T. (e, f) ΔTc field dependence of (e) the NbRe and (f) the NbReN microstrip for the perpendicular (black squares) and parallel (red circles) geometries.

The temperature dependence of Hc2 for NbRe and NbReN is displayed, respectively, in Figure 3a and Figure 3b, where both Hc2⟂(T) and Hc2∥(T) are reported. In the insets of Figure 3a and Figure 3b, the temperature dependence of the anisotropy coefficient γ(T) = Hc2∥(T)/Hc2⟂(T) is given for NbRe and NbReN, respectively. For both the materials, γ shows a nonmonotonic behavior with a fast increase followed by a smooth decrease when the temperature is lowered. However, for NbReN, γ is larger by a factor of almost two than for NbRe. We will comment on this point later on.

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Figure 3: (a, b) H–T phase diagram of (a) the NbRe and (b) the NbReN microstrip. Black squares and red circles indicate the temperature dependence of Hc2⟂ and Hc2∥, respectively. The insets show the behavior of the anisotropy coefficient γ as a function of the temperature.

In the dirty limit and assuming that spin–orbit scattering is negligible with respect to spin-independent scattering, the temperature dependence of Hc2 is given by the implicit equation [19]

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where t = T/Tc, [Graphic 14]. λso = ℏ/(3πkBTcτso) with τso being the mean free time for spin–orbit scattering. We have used Equation 1 with λso = 0 to describe the experimental data for both materials using the measured values of Tc and the slope of Hc2⟂ close to Tc, which can be accurately determined via the many transition curves measured at very low fields. Moreover, the values of α for both materials have been obtained from the normal-state properties. Therefore, the WHH curves obtained from Equation 1 do not contain any fitting parameter [19,40]. All these quantities together with other superconducting and normal-state properties of the two materials are summarized in Table 1. In Figure 4 the perpendicular H–T phase diagram is reported for NbRe (Figure 4a) and NbReN (Figure 4b) microstrips together with the prediction given by the WHH theory. As far as NbRe is concerned, the experimental data are not described by the WHH theory considering α = 0.51, the value obtained from the normal-state properties (see solid line in Figure 4a). In fact, kF[Graphic 15] = 3Dm/ℏ [47], where D = [Graphic 16][48] is the quasiparticle diffusion coefficient and m is the mass of the electron. From the value of D reported in Table 1, we get kF[Graphic 17] ≈ 1.3 and then α = 0.51. In contrast, data are very well reproduced by the WHH theory with α = 0, even though the points at the lowest temperature lay above the curve (see dashed line in Figure 4a). The value of the zero-temperature critical magnetic field [Hc2⟂(0) = 8.12 T] is below Hp(0) = 9.78 T, and since α = 0, the orbital limiting to Hc2⟂ is the only contribution that should be considered. Our result, which is in line with other studies on Hc2⟂(T) made on non-centrosymmetric materials in bulk forms [37,39,41], may suggest the presence of a triplet component of the order parameter. This result was even more evident in the case of polycrystalline NbRe samples, for which the experimental points in the H–T phase diagram stand well above the WHH line with α = 0 [14]. This result, interpreted in [14] as an indication of unconventional superconducting pairing, may be weakened in our case due to the poorer crystal quality of our disordered films. In the case of NbReN, the data are well described by the WHH theory with α = 0.51, as shown by the solid line in Figure 4b. In this case, the critical field is paramagnetically limited with the Pauli contribution that lowers the value of the critical field with respect to the pure orbital-limited case (α = 0). This suppression of the perpendicular critical field in conjunction with the steepest behavior of Hc2∥ in the studied temperature range could also be the cause of the larger value of γ measured on the NbReN microstrip. Again, we ascribe this effect to the crystallographic properties of the films. Indeed, it is reasonable to suppose that the presence of the N atoms in the atomic cell may break the non-centrosymmetricity of the system, thus depressing the spin-triplet component of the order parameter. For this reason, Hc2⟂ becomes paramagnetically limited [24,35]. In order to confirm these results, we are currently working on different experiments that may give more direct access to the order parameter in these systems. Regarding the aforementioned purpose, while in the case of NbRe it is now even more evident that films with larger crystallites are mandatory [7,8,12], detailed analyses of the NbReN crystal structure are still lacking and need to be performed.

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Figure 4: (a, b) Temperature dependence of the perpendicular upper critical field of (a) the NbRe and (b) the NbReN microstrip. The lines represent the WHH calculations. Details of the procedure are given in the text.

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