$s$-wave superconductivity in the noncentrosymmetric W₃Al₂C superconductor: An NMR study

In the normal phase, we observe an almost ideal metallic behaviour for all the applied magnetic fields (see figures 1(a) and 2(b)). For instance, the relaxation-rate data in figure 1(a) are well described by a power-law function with a scaling exponent of 1.0(1), i.e., the relaxation rate is perfectly proportional to temperature. We can also compute the Korringa product $1/T_{1}T$ (here, only 0.017 s^-1K^-1), which is proportional to the electronic density of states at the Fermi level, $N(E_{\mathrm{F}})$ (see figure 1(b)). As expected for an ideal metal, in the normal phase of W₃Al₂C, $N(E_{\mathrm{F}})$ is practically constant with temperature and independent of the applied magnetic field, but approximately thirty times smaller than the Korringa product in metallic Al (0.54 s^-1K^-1) [26, 27]). These widely differing relaxation rates are partly reflected also in the different Knight shifts: 450 ppm in W₃Al₂C vs 1636 ppm in metallic Al [28], resulting in a $T_{1}TK^{2}$ of $1.19\times 10^{-5}$ sK and $4.98\times 10^{-6}$ sK in W₃Al₂C and in metallic Al, respectively. The different $T_{1}TK^{2}$ values suggest that, although W₃Al₂C is an ordinary metal in its normal phase, it could still exhibit weak electronic correlations. The strong reduction in shift in W₃Al₂C suggests a very small contribution of $s$-electrons (responsible for the contact hyperfine interaction) at the Fermi energy. This is consistent with previous NMR studies of Al-based alloys [29] and with the electronic structure calculations of W₃Al₂C [11]. Indeed, the latter indicate that the $s$-electron band is shifted to higher energies, whereas the valence $d$-electron band of the W transition metal is prominent at the Fermi level.

In the superconducting phase, the electronic properties of the system can be described by the standard BCS theory, which predicts a gap ratio $2\Delta_{0}/k_{\mathrm{B}}T_{c}=3.53$ (see figure 2(a)) and a reduction of the NMR shift at low temperatures (see figure 2(b)). Here, the reduced shift reflects an electron pairing forming spin-compensated (typically $s$-wave) Cooper pairs. This result is consistent with a recent muon-spin rotation ($\mu$SR) study of W₃Al₂C [12], where $s$-wave superconductivity with the same gap ratio was reported. The superconducting gap value we find in W₃Al₂C is similar to the average SC gap reported in Mo₃Al₂C, where an NMR study claimed a $2\Delta_{0}/k_{\mathrm{B}}T_{c}$ ratio of $2.8(2)$ [14], while specific-heat measurements implied a ratio of 4.03 [13].

According to the Korringa relation [30], $T_{1}$ in metals depends strongly on $N(E_{\mathrm{F}})$. Hence, upon entering the superconducting state, we expect an exponential decay of $T_{1}$. Yet, preceding this decay, just below $T_{c}$ the spin-lattice relaxation rate first increases above its normal-state Korringa value, showing a so-called coherence (or Hebel-Slichter — HS) peak [31, 32]. Such increase in relaxation rate results partly from the enhanced density of electronic states, which pile up near the SC gap edges [33, 34]. Considering its key role in the experimental confirmation of the BCS theory, the observation of a Hebel-Slichter peak is recognized as a clear indication of BCS-type $s$-wave superconductivity.

However, its absence does not necessarily rule out the possibility of a standard $s$-wave pairing. This seems to be also the case for W₃Al₂C where, as illustrated in figure 1(a), we do not observe the expected peak in the ²⁷Al NMR relaxation rates below $T_{c}$. Several hypotheses can be put forward to explain this experimental result.

One possibility, relevant to type-II superconductors, was suggested by Goldberg and Weger [35]. Here, the total nuclear spin-relaxation rate consists of the sum of two terms, the first of which describes the relaxation in the normal-state vortex cores, while the other captures the relaxation in the remaining superconducting volume. This theory predicts that the contribution from the normal-state cores is proportional to the external magnetic field $H$. To test it, we measured the relaxation rates at different magnetic fields, in the range from 4 to 7 T. As can be seen in figure 2(a), the (scaled) relaxation times generally fall on the same exponential curve, indicating that no field-dependent term arises from vortex cores. Consequently, this hypothesis cannot explain the absence of an HS peak.

Another possibility is that the HS peak is suppressed by disorder, caused by intrinsic defects and/or magnetic impurities. To investigate the role of disorder, we studied the linewidths and the stretching coefficient $\beta$. As shown in figure 3, $\beta$ decreases continuously as the temperature is lowered, while remaining always smaller than 1. This indicates a distribution of relaxation rates, reflecting the inequivalence of NMR sites, in turn due to the intrinsic disorder (see, for instance, refs. [36, 37]). However, we find that both parameters depend only weakly on temperature and show no discontinuities or an unusual behaviour near $T_{c}$. Thus, we can rule out the possibility that disorder, although present, is affecting the HS peak.

Yet another possibility is that, for quadrupolar nuclei such as the spin-$\nicefrac{{5}}{{2}}$ ²⁷Al, also the quadrupole interaction contributes to the spin-lattice relaxation, tending to smear out the HS peak [38]. To test this possibility, we measured the ¹³C NMR relaxation rate in W₃Al₂C. Since ¹³C is a spin-$\nicefrac{{1}}{{2}}$ nucleus, it represents a purely dipolar probe, not undergoing any quadrupole interaction. As can be seen in figure 4(a), we indeed observe a small feature just below $T_{c}$. Here, the exact $T_{c}$ value at 5 T was determined by means of a standard resonance-detuning experiment (data not shown).

To model the feature near $T_{c}$, we begin with the usual expression for the HS peak [33]:

Here, $R_{s}$ and $R_{n}$ are the relaxation rates in the superconducting and the normal state, respectively, $M_{\mathrm{s}}(E)=N_{0}\Delta/\sqrt{E^{2}-\Delta^{2}}$ is the anomalous density of states (DOS) due to the coherence factor [33], and $N_{\mathrm{s}}(E)=N_{0}E/\sqrt{E^{2}-\Delta^{2}}$ is the DOS in the superconducting state.

In our case, two modifications were made. Firstly, we chose to convolute $M_{s}(E)$ and $N_{s}(E)$ with a triangular broadening function $B(E)$, characterized by a width $2\delta$ and a height $1/\delta$, with $\delta=0.2\Delta$. Secondly, we model the degree of coherence by the parameter $c$, by making the substitution $\Delta\rightarrow c\Delta$. The conventional BCS theory [39] predicts that, depending on the details of the scattering operator, the scattering matrix element can adopt a positive or a negative sign. The modelling of the HS peak typically requires a positive sign, but in other situations, e.g., in ultrasound absorption, the negative sign has to be used. Here, by setting $c$ as an adjustable parameter, one can model the case where the scattering matrix consists of both positive and negative components, which may cause a (partly) suppression of the HS peak. After these two modifications, $N_{s}(E)$ and $M_{s}(E)$ can be written as [40]:

The fit results are shown in figure 4(b). We find that the fit obtained by fixing $c=1$ drastically overestimates the relaxation rates below $T_{c}$. On the other hand, the fit obtained by fixing $c=0$ reproduces adequately the feature near $T_{c}$, although with a slightly overestimated relaxation below $T_{c}$.

From the ¹³C NMR relaxation rates in the superconducting state and the above analysis we infer with some confidence that W₃Al₂C indeed exhibits $s$-wave superconductivity, as evidenced by the observation of an (albeit reduced) HS peak. Yet, it appears that there are some decoherence effects, both at and below $T_{c}$, most likely due to a strong electron-phonon coupling in this material (see ref. [41] and the section below). In general, the exact mechanism of such decoherence effects is yet to be understood.