Time-reversal symmetry breaking in Re-based superconductors: Recent developments

In the recent search for unconventional- and topological superconductivity, noncentrosymmetric superconductors (NCSCs) rank among the most promising candidate materials. Surprisingly, some of them -- especially those containing rhenium -- seem to exhibit also time-reversal symmetry (TRS) breaking in their superconducting state, while TRS is preserved in many other isostructural NCSCs. To date, a satisfactory explanation for such discrepant behavior, albeit crucial for understanding the unconventional superconductivity of these materials, is still missing. Here we review the most recent developments regarding the Re-based class, where the muon-spin relaxation ($\mu$SR) technique plays a key role due to its high sensitivity to the weak internal fields associated with the TRS breaking phenomenon. We discuss different cases of Re-containing superconductors, comprising both centrosymmetric- and noncentrosymmetric crystal structures and ranging from pure rhenium, to Re$T$ ($T$ = 3$d$-5$d$ early transition metals), to the dilute-Re case of ReBe$_{22}$. $\mu$SR results suggest that the rhenium presence and its amount are two key factors for the appearance and the extent of TRS breaking in Re-based superconductors. Besides summarizing the existing findings, we also put forward future research ideas regarding the exciting field of materials showing TRS breaking.


I. INTRODUCTION
The combination of intriguing fundamental physics with far-reaching potential applications has made unconventional superconductors one of the most studied classes of materials. Standing out among them are the noncentrosymmetric superconductors (NCSCs) [1], whose crystal structures lack an inversion symmetry. As a consequence, in NCSCs, the strict symmetry-imposed requirements are relaxed, allowing mixtures of spin-singlet and spin-triplet Copper pairing channels, thus setting the scene for a variety of exotic properties, as e.g., upper critical fields beyond the Pauli limit, nodes in the superconducting gaps, etc. (see Refs. [1][2][3] for an overview). The degree of mixing in such combined pairings is related to the strength of the antisymmetric spin-orbit coupling (ASOC) and to other microscopic parameters, still under investigation. Currently, NCSCs rank among the foremost categories of superconducting materials in which to look for topological superconductivity (SC) or to realize the Majorana fermions, pairs of the latter potentially acting as noise-resilient qubits in quantum computing [4][5][6][7][8][9][10][11].
In general, the various types of NCSCs can be classified into two classes. One consists of strongly correlated materials, as e.g., CePt 3 Si [12], or Ce(Rh,Ir)Si 3 [13], which belong to the heavy-fermion compounds. Owing to the strong correlation and the interplay between dand f -electrons, these materials often exhibit rich magnetic and superconducting properties. Since their superconductivity is most likely mediated by spin fluctuations, this implies an unconventional (i.e., non phonon-related) pairing mechanism. Conversely, the other class consists mainly of weakly correlated materials, which are free of "magnetic" f -electrons, as e.g., LaNiC 2 , La 7 Ir 3 , CaPtAs, or ReT (T = 3d-5d early transition metals) [14][15][16][17][18][19][20]. Obviously, their superconductivity is not mediated by electrons' spin fluctuations. Hence, they lend themselves as prototype parent systems where one can study the intrinsic pairing mechanisms in NCSCs.
Recently, superconductivity with broken time-reversal symmetry (TRS) has become a hot topic in NCSCs. The main reason for this is the discovery of TRS breaking in some weakly-correlated NCSCs using muon-spin relaxation (µSR). Surprisingly, the superconducting properties of the latter largely resemble those of conventional superconductors, i.e., they are clearly distinct from those of the above mentioned strongly-correlated NCSCs. To date, only a handful of NCSC families have been shown to exhibit TRS breaking in the superconducting state, including LaNiC 2 [14], La 7 (Rh,Ir) 3 [15,21], Zr 3 Ir [22], CaPtAs [16], and ReT [14][15][16][17][18][19][20]. Except for the recently studied CaPtAs, where coexisting TRS breaking and superconducting gap nodes were observed below T c , in most of the above cases the superconducting properties evidence a conventional s-wave pairing, characterized by a fully opened superconducting gap. This leads to an interesting fundamental question: does the observed TRS breaking in NCSCs originate from an unconventional superconducting mechanism (i.e., from a pairing other than that mediated by phonons), or it can occur also in presence of conventional pairing (via some not yet understood mechanism) [2,3]? Why, among the many different NCSCs families, only a few exhibit a broken TRS in the superconducting state, also remains an intriguing open question.
In general, the causes behind the TRS breaking at the onset of superconductivity are mostly unknown. In particular, the α-Mn-type noncentrosymmetric ReT (T = Ti, Nb, Zr, and Hf) superconductors have been widely studied and demonstrated to show a superconducting state with broken TRS [17][18][19][20]. Yet, TRS seems to be preserved in the isostructural (but Re-free) Mg 10 Ir 19 B 16 and Nb 0.5 Os 0.5 [23,24]. Further, depending on synthesis protocol, Re 3 W is either a centro-(hcp-Mg-type) or a noncentrosymmetric (α-Mntype) superconductor, yet neither is found to break TRS [25].
In case of binary Re-Mo alloys, depending on the Re/Mo ratio, the compounds can exhibit up to four different crystal structures, including both centrosymmetric and noncentrosymmetric cases. Most importantly, all these alloys become superconductors at low temperatures [26]. A comparative µSR study of Re-Mo alloys, covering all the different crystal structures, reveals that the spontaneous magnetic fields occurring below T c (an indication of TRS breaking) were only observed in elementary rhenium and in Re 0.88 Mo 0.12 . By contrast, TRS was preserved in the Re-Mo alloys with a lower Re-content (below ∼ 88%), independent of their centro-or noncentrosymmetric crystal structures [27]. Since both pure rhenium and Re 0.88 Mo 0.12 have a simple (hcp-Mgtype) centrosymmetric structure, this strongly suggests that a noncentrosymmetric structure and the accompanying ASOC effects are not essential in realizing the TRS breaking in ReT superconductors. The µSR results regarding the Re-Mo family, as well as other Re-free α-Mn-type superconductors, clearly imply that not only the Re presence, but also its amount are crucial for the appearance and the extent of TRS breaking in the ReT superconductors. How these results can be understood within a more general framework requires further experimental and theoretical investigations.
This short review article focuses mostly on the experimental study of Re-based binary superconductors. In Sec. II, we discuss the basic principles of our probe of choice, the µSR, here used to detect TRS breaking and to characterize the superconducting properties. Section III describes the possible crystal structures and superconducting transition temperatures of ReT binary alloys. In Sec. IV, we focus on the upper critical fields and the order parameter in ReT superconductors. Section V discusses the TRS breaking in ReT superconductors and its possible origins. Finally, in the last section, we outline some possible future research directions.

II. MUON-SPIN RELAXATION AND ROTATION
Initially considered as an "exotic" technique, over the years muon-spin rotation, relaxation, and resonance (known as µSR), has become one of the most powerful methods to study the magnetic and superconducting properties of matter. This follows from a series of fortunate circumstances, related to the muon properties as a fundamental particle. Most notably, these include the 100% initial muon-spin polarization, following the two-body decay from pions, and the subsequent preservation of such information through the weak decay into positrons. In the search for unconventional superconductivity, as well as for TRS breaking effects, the very high sensitivity of the µSR technique to tiny magnetic fields is especially important [28]. Below we briefly outline the basics of the µSR technique and direct the reader to other references for more detailed information [29][30][31].

A. Principles of the µSR technique
Central to the µSR method is the availability of polarized positive muon (µ + ) beams, obtained by collecting the muons produced in the two-body decay of positive pions, π + → µ + + ν µ (with ν µ the muon neutrino), decaying at rest in the laboratory frame. Since pions have no intrinsic angular momentum and neutrinos have a fixed helicity (relative orientation of spin and linear momentum), the resulting muon beam is 100% spin polarized, with the muon spins directed antiparallel to the linear momentum (see Figure 1A). Having an energy of ∼4.12 MeV, muons can penetrate a sample between 0.1 mm and 1 mm, depending on the sample density. Once implantated, the monoenergetic muons decelerate within 100 ps through ionization processes (which do not interact with the muon spin) and finally come to rest at an interstitial site, practically without loss of their initial spin polarization. From this moment, if subject to magnetic interactions, the muon-spin polarization P(t) evolves with time (the muon spin precesses around the local magnetic field), thus providing important information on the sample's magnetism. The detection of the P(t) evolution is made possible by the parity-violating weak-decay interaction µ + → e + + ν e +ν µ (e + , ν e , andν µ are the positron, electron neutrino, and muon antineutrino, respectively), which implies a preferential emission of positrons along the muon-spin direction at the time of decay (see Figure 1A, which depicts also the anisotropic positron-emission pattern). Thus, by detecting the spatial distribution of positrons as a function of time, one can determine the time evolution of the muon-spin polarization P(t).
A schematic diagram of a time-differential µSR experiment is shown in Figure 1A. The incoming muon triggers a clock that defines the starting time t 0 . Once implanted, the muon spin precesses about the local magnetic field B(r) with a Larmor frequency ω µ = γ µ B(r), where γ µ /2π = 135.53 MHz/T is the muon gyromagnetic ratio. The clock stops when, after a mean lifetime of 2.197 µs, the muon decays into a positron e + , registered as an event by one of the positron detectors. The measured time intervals for ca. 10-50 millions of such events are stored in a histogram, given by (see Figure 1B): Here, the exponential factor accounts for the radioactive muon decay, N 0 is the initial count rate at time t 0 , while C is a time-independent background (due to uncorrelated start and stop counts). As shown in the inset of Figure 1B, unlike the inessential exponential decay, the physical information in a µSR experiment is contained in the A(t) = A 0 P(t) term (often known as the µSR spectrum). Here, A 0 is the so-called initial asymmetry (typically 0.3, depending on the detector's solid angle and efficiency), while P(t) is the muon-spin depolarization function, here given by the projection of P(t) on the unit vector describing the detector. Since P(t) represents the autocorrelation function of the muon spin S, i.e., P(t) = 〈S(t)S(0)〉 /S(0) 2 , it depends on the average value, the distribution, and the time evolution of the internal magnetic fields, thus reflecting the physics of the magnetic interactions in the sample under study To access the µSR signal we need to remove the extrinsic decay factor by combining the positron counts from pairs of opposite-lying detectors, for instance, N F and N B (for forward and backward), and obtain the asymmetry behaves as a normalized detector "contrast", proportional to A 0 . The parameter α is introduced to take into account the different efficiencies of the positron-detectors and needs to be determined by calibration.

B. Transverse-field µSR
Depending on the reciprocal orientation of the external magnetic field B with respect to the initial muon-spin direction S(0), in a µSR experiment, two different configurations are possible. (i) In transverse-field (TF) µSR the externally applied field B is perpendicular to S(0) and the muon spin An incoming polarized muon (with spin S µ antiparallel to momentum p µ ) is first detected by a (thin) muon detector, which starts the electronic clock. In the sample, the muon spin precesses in the internal/external field until the muon decays into a positron. This is emitted preferentially along the muon-spin direction and hits one of the positron detectors [here forward (F) or backward (B)], whose signal stops the clock. The gray curve depicts the anisotropic positron-emission pattern at the moment of muon implantation. The pattern rotates rigidly with the muon spin (initially pointing towards the B detector) at an angular frequency ω µ = γ µ B ext . (B) Detected positron counts in the F and B detectors as a function of time after ca. 10 7 events. Inset: The asymmetry signal, obtained as the normalized difference between the F and B counts, eliminates the inessential exponential decay and highlights the signal decay (here assumed to be Gaussian) reflecting the nature of the sample.
precesses around B (see Figure 1A). (ii) In a longitudinal field (LF) configuration the applied field is parallel to S(0), generally implying only a relaxing µSR signal.
Although, in principle, the TF scheme shown in Figure 1A works fine, strong transverse fields perpendicular to the muon momentum (p µ ) would deviate the muon beam too much from its original path. The Lorentz force can be zeroed by applying the field along the muon momentum. At the same time, to maintain the transverse geometry, the initial muon spin is rotated by 90 • (in the x or y direction) by using a so-called Wien filter, consisting of crossed electric and magnetic fields. Such a configuration is also known as transverse muon-spin mode, while Figure 1A plots the longitudinal muon-spin mode (i.e., p µ S µ ).
Since muons are uniformly implanted in the sample, they can detect the coexistence of different domains, characterized by distinct P i (t) functions, whose amplitudes A i represent a measure of the respective volume fractions. In case of superconductors, one can thus extract the SC volume fraction. More importantly, in a TF-µSR experiment one can directly probe the SC flux-line lattice (FLL). In this case, at the onset of superconductivity, the muon-spin precession in a TF field loses coherence, reflecting the magnetic field modulation (i.e., broadening) due to the FLL. The shape of the field distribution arising from the FLL can be analyzed and eventually used to extract the magnetic penetration depth λ and the coherence length ξ [32]. In many type-II superconductors, the simple relation, σ 2 sc /γ 2 µ = 0.00371Φ 2 0 /λ 4 eff , connects the muon-spin depolarization rate in the superconducting phase, σ sc , with the effective magnetic penetration depth, λ eff (here Φ 0 is the magnetic flux quantum) [33,34]. In case of superconductors with relatively low upper critical fields, the effects of the overlapping vortex cores with increasing field ought to be considered when extracting the magnetic penetration depth λ eff from σ sc . Since λ(T ) is sensitive to the low-energy excitations, its evolution with temperature is intimately related to the structure of the superconducting gap. Hence, µSR allows us to directly study the symmetry and value of the superconducting gap.
More in detail, in a TF-µSR experiment, the time evolution of the asymmetry can be modeled by: (2) Here A i , A bg and B i , B bg are the asymmetries and local fields sensed by implanted muons in the sample and the sample holder, γ µ is the muon gyromagnetic ratio, φ is a shared initial phase, and σ i is the Gaussian relaxation rate of the ith component. The number of required components is material dependent, typically in the 1 ≤ n ≤ 5 range. In general, for superconductors with a large Ginzburg-Landau parameter κ ( 1), the magnetic penetration depth is much larger than the coherence length. Hence, the field profiles of each fluxon overlap strongly, implying a narrow field distribution. Consequently, a single-oscillating component is sufficient to describe A(t). In case of a small κ (≥ 1/ 2), the magnetic penetration depth is comparable with the coherence length. The rather small penetration depth implies fast-decaying fluxon field profiles and a broad field distribution, in turn requiring multiple oscillations to describe A(t). The choice of n can be determined from the fast-Fourier-transform (FFT) spectra of the TF-µSR, which is normally used to evaluate the goodness of the fits. In case of multi-component oscillations, the first term in Equation (2) describes the field distribution as the sum of n Gaussian relaxations [35]: The first-and second moments of the field distribution in the sample can be calculated by: The total Gaussian relaxation rate σ eff in Equation (4) includes contributions from both a temperature-independent relaxation, due to nuclear moments (σ n ), and a temperature-dependent relaxation, related to the FLL in the superconducting state (σ sc ). The σ sc values are then extracted by subtracting the nuclear contribution following σ sc = σ 2 eff − σ 2 n . To get further insights into the superconducting gap value and its symmetry, the temperature-dependent superfluid density ρ sc (T ) [proportional to λ −2 eff (T )] is often analyzed by using a general model: Here, λ 0 is the effective magnetic penetration depth in the 0-K limit, f = (1 + e E/k B T ) −1 is the Fermi function and 〈〉 FS represents an average over the Fermi surface [36]. ∆ k (T ) = ∆(T )A k is an angle-dependent gap function, where ∆ is the maximum gap value and A k is the angular dependence of the gap, equal to 1, cos 2φ, and sin θ for an s-, d-, and pwave model, respectively, with φ and θ being the azimuthal angles. The temperature dependence of the gap is assumed to follow the relation 36,37], where ∆ 0 is the 0-K gap value.

C. Zero-field µSR
A particular case of LF, is that of zero-field (ZF) µSR, characterized by the absence of an external magnetic field. In this configuration the frequency of the µSR signal is exclusively proportional to the internal magnetic field, making it possible to determined the size of the ordered moments and, hence, the magnetic order parameter. Unlike various techniques, which require an external field to polarize the probe, µSR is unique in its capability of studying materials unperturbed by externally applied fields and in accessing their spontaneous magnetic fields. Due to the large muon magnetic moment (µ µ = 8.89 µ N ), ZF-µSR can sense even very small internal fields (∼ 10 −2 mT), hence probing local magnetic fields of either nuclear or electronic nature. In addition, since the muon is an elementary spin-1/2 particle, it acts as a purely magnetic probe, i.e., free of quadrupole interactions. All these features, make ZF-µSR an ideal technique for detecting TRS breaking in the superconducting state. This corresponds to the appearance (at the onset of SC) of spontaneous magnetic moments, whose magnitude can be very small, often lacking a proper magnetic order. As we show further on, in case of TRS breaking, we expect the appearance of an additional enhancement of relaxation below T c , reflecting the occurrence of these weak spontaneous fields. During the ZF-µSR measurements, to exclude the possibility of stray magnetic fields (typically larger than the weak internal fields), the magnets are quenched before starting the measurements, and an active field-nulling facility is used to compensate for stray fields down to 1 µT.
If the amplitudes of the local fields reflect a Gaussian distribution with zero average (a rather common circumstance), the µSR signal consists of overlapping oscillations with different frequencies. While at short times the spin dephasing is limited, at long times it becomes relevant and gives rise to a so-called Kubo-Toyabe (KT) relaxation function [31,38]. Two different models are frequently used to analyze the ZF-µSR data:

or
Equation (6) is also known as a combined Gaussian-and Lorentzian Kubo-Toyabe function, with the additional exponential relaxation describing the electronic contributions present in many real materials. In polycrystalline samples, the 1/3-non-relaxing and the 2/3-relaxing components of the asymmetry correspond to the powder average of the internal fields with respect to the initial muon-spin direction (statistically, with a 1/3 probability, the directions of the muon spin and of the local field coincide). Clearly, in case of single crystals, such prefactors might be different. The σ ZF and Λ ZF represent the zero-field Gaussian and Lorentzian relaxation rates, respectively. Typically, Λ ZF shows an almost temperature-independent behavior. Hence, an increase of σ ZF across T c can be attributed to the spontaneous magnetic fields which break the TRS, as e.g., in ReT [17,18,20].
In case of diluted nuclear moments, σ ZF is practically zero, hence, the TRS breaking is reflected in an increase of Λ ZF below T c , as e.g., in Zr 3 Ir and CaPtAs [16,22].

III. Re-BASED SUPERCONDUCTORS
In this section, we review the different phases of the binary ReT alloys. These are obtained when rhenium reacts with various early transition metals (see Figure 2A Among these the cubic α-Mn-type structure is noncentrosymmetric, while the rest are centrosymmetric [39]. Besides the above cases, a few other crystal structures have also been reported, including the cubic CsCl-(P m-3m, No. 221), cubic Cr 3 Si-(P m-3n, No. 223), and trigonal Mn 21 Zn 25 -type (R-3c, No. 167) [39]. As for the pure elements listed in Figure 2A, both Re and Os have an hcp-Mg-type structure, and show superconductivity below 2.7 and 0.7 K, respectively [20,40]; while V, Nb, Mo, Ta, and W all adopt a bcc-W-type structure, and become superconductors at ∼ 5.4, 9.0, 1.0, 4.5, and 0.015 K, respectively [40]. Unlike the above cases, Ti, Zr, and Hf can form either hightemperature bcc-W-type or low-temperature hcp-Mg-type structures, with T c ∼ 0.4, 0.6, and 0.13 K, respectively [40].
For T = Ti 3d metal, the known binary compounds are Re 24 Ti 5 , Re 6 Ti, and ReTi [42,43]. The former two adopt a noncentroysmmetric α-Mn-type structure and become superconductors below T c = 6 K [19,40,44], while the latter one crystallizes in a cubic CsCl-type structure. To date, no detailed physical properties have been reported for ReTi. For T = V, superconductivity has been reported in hcp-Mg-type Re 0.9 V 0.1 (T c = 9.4 K), β-CrFe-type Re 0.76 V 0.24 (T c = 4.5 K), and bcc-W-type Re 0.6 V 0.4 (T c = 2.2 K) [40,45]. Also for them, to date a microscopic study of their SC is still missing. The cubic Cr 3 Si-type Re 0.71 V 0.29 has also been synthesized, but its physical properties were never characterized [46]. For T = Zr 4d metal, the α-Mn-type Re 24 Zr 5 (T c = 5 K) and Re 6 Zr (T c = 6.7 K) have been investigated via both nuclear quadrupole resonance and µSR techniques [17,47]. Except for the α-Mn-type Re-Zr alloys, the MgZn 2 -type Re 2 Zr (similar to hcp-Mg-type) and Mn 21 Zn 25 -type Re 25 Zr 21 have been synthesized [39]. Re 2 Zr exhibits a T c value of ∼ 6-7 K [40,48], while Re 25 Zr 21 has not been studied. For T = Nb, depending on Re/Nb concentration, four different solid phases including hcp-Mg-, α-Mn-, β-CrFe-, and bcc-W-type have been reported. On the Re-rich side, the hcp-Mg-type Re-Nb alloys are limited to less than 3% Nb concentration [39], whereas many α-Mn-type Re-Nb binary alloys have been grown and widely studied by various techniques [20,[49][50][51][52], with the highest T c reaching 8.8 K in Re 24 Nb 5 (denoted as Re 0.82 Nb 0.18 in the original paper [20]). At intermediate Re/Nb values, for example, in β-CrFe-type Re 0.55 Nb 0.45 , T c s in the range of 2 to 4 K [40] have been reported, but no microscopic studies yet. As for the Nb-rich side (Nb concentration larger than 60%), here the Re-Nb alloys exhibit the same structure as that of pure Nb, but much lower T c values than Nb [39,40]. For T = Mo, the binary Re-Mo phase diagram (see Figure 2B) covers also four different solid phases [39]. The binary Re-Mo alloys have been characterized by different techniques and all of them shown to become superconductors at low temperature [26,27]. The T c varies nonmonotonically upon changing the Mo concentration, giving rise to three distinct superconducting regions. On the Re-rich side, the first SC region shows the highest T c ∼ 9.4 K in the hcp-Mg-type Re 0.77 Mo 0.23 . The same material but with an α-Mn-type structure can also be grown, with a T c value about 1 K lower than the hcp-Mg-type. In the second superconducting region, where the alloys adopt a β-CrFe-type structure, the superconducting transition tem-perature T c ∼ 6.3 K is almost independent of Mo content. Finally, on the Mo-rich side, all Re-Mo alloys display a cubic bcc-W-type structure and form a third superconducting region with the highest T c reaching 12.4 K in Re 0.4 Mo 0.6 .
For T = Hf 5d metal, the Re-Hf alloys show a similar phase diagram to Re-Zr. With only ∼3% Hf substitution, T c increases from < 3 K to 7.3 K in the hcp-Mgtype Re-Hf alloys [40]. Both the α-Mn-type Re 6 Hf and the MgZn 2 -type Re 2 Hf become superconductors below T c ∼ 6 K [18,40,48,53,54], whereas the physical properties of Mn 21 Zn 25 -type Re 25 Hf 21 remain largely unknown. On the Hf-rich side, the bcc-W-type alloys exhibit relatively low T c s, e.g., T c = 1.7 K for Hf 0.875 Re 0.125 [40]. For T = Ta, although the four different structures shown in Figure 2C-F can be synthesized, only the α-Mn-type Re-Ta alloys have been well studied. For example, Re 3 Ta and Re 5.5 Ta show T c values of 4.7 K and 8 K, respectively [55,56]. On the Ta-rich side, the bcc-W-type Re-Ta alloys become superconducting at T c < 3.5 K, lower than the T c of pure Ta [57]. We note that in case of the β-CrFe-type Re-Ta alloys, no superconducting transition has been observed down to 1.8 K in either Re 0.5 Ta 0.5 or Re 0.6 Ta 0.4 . For T = W, the Re-W alloys show a very similar phase diagram to Re-Mo in Figure 2B. As the W concentration increases, the highest T c values reach ∼ 8, 9, 6, and 5 K in the hcp-Mg-, α-Mn-, β-CrFe-, and the bcc-W-type alloys, respectively [39,40]. Among them, only the hcp-Mg-and the α-Mn-type Re 3 W have been investigated [25,58]. Finally, in case of T = Os, the Re-Os alloys show a rather monotonous phase diagram, since only hcp-Mg-type compounds with T c values below 2 K can be synthesized [39,40].

IV. UPPER CRITICAL FIELD AND NODELESS SUPERCONDUCTIVITY
As mentioned in the introduction, due to the mixture of singlet-and triplet paring, some NCSCs may exhibit relatively high upper critical fields, often very close to or even exceeding the Pauli limit, as e.g., CePt 3 Si [12], Ce(Rh,Ir)Si 3 [59,60], and recently (Ta,Nb)Rh 2 B 2 [61]. Therefore, the upper critical field can provide valuable clues about the nature of superconductivity. To investigate the temperature evolution of the upper critical field H c2 (T ), in general, the temperature-(or field-) dependent electrical resistivity ρ, magnetic susceptibility χ, and specific heat C/T at various magnetic fields (or at various temperatures) are measured [19,20,27]. As an example, Figure 3A shows the H c2 (T ) for Re 24 Nb 5 (α-Mn-type) and Re 0.4 Mo 0.6 (bbc-W-type) versus the normalized temperature T /T c (0). To obtain the upper critical field in the zero-temperature limit, H c2 (0), the Werthamer-Helfand-Hohenberg (WHH) or the Ginzburg-Landau (GL) models are usually applied when analyzing the H c2 (T ) data of ReT superconductors. Both models can adequately describe single-gap superconductors.
Here, in case of Re 24 Nb 5 and Re 0.4 Mo 0.6 , the WHH model (solid line in Figure 3A) reproduces the data very well and gives µ 0 H c2 (0) = 15.6 T, and 3.08 T, respectively. Figure 3B summarizes the µ 0 H c2 (0) values of the ReT and α-Mn-type NbOs 2 superconductors. As discussed in Section III, most of the previous studies focused exclusively on α-Mn-type ReT superconductors, the physical properties of the other ReT superconductors being practically neglected and requiring further studies. Unlike other ReT , all Re-Mo alloys belonging to four different structures have been studied via macro-and microscopic techniques [26,27]. The µ 0 H c2 (0) of centrosymmetric Re-Mo alloys, including hcp-Mg-, β-CrFe-, and bcc-W-type, are far away from the Pauli limit µ 0 H P = 1.86 T c (indicated by a dashed line in Figure 3B). Conversely, the α-Mn-type ReT and NbOs 2 both exhibit large upper critical fields, very close to or even exceeding the Pauli limit, despite their different T c values. For example, µ 0 H c2 (0) = 15.6 and 16.5 T for Re 24 Nb 5 and Re 5.5 Ta, while their µ 0 H P (0) are 16.4 and 14.9 T, respectively. The hcp-Mg-type Re 3 W also exhibits a relatively high H c2 , as determined from electrical resistivity data. However, its H c2 value might be overestimated since, e.g., at 9 T, no zero resistivity could be observed down to 2 K. Therefore, other bulk techniques, including magnetization-or heat capacity measurements are required to determine the intrinsic H c2 . In general, it would be interesting to know the H c2 values of other centrosymmetric ReT superconductors. Overall, the upper critical fields in Figure 3B indicate the possibility of singlet-triplet mixing in the noncentrosymmetric α-Mn-type superconductors.
Transverse-field µSR represents one of the most powerful techniques to investigate the superconductivity at a microscopic level. To illustrate this, in the inset of Figure 3C we show two typical TF-µSR spectra for bcc-W-type Re 0.4 Mo 0.6 in the normal and the superconducting states. Below T c , the fast decay induced by FLL (encoded into σ sc ) is clearly visible, while the slow decay in the normal state is attributed to the randomly oriented nuclear magnetic moments. By comparing the two spectra, one can also determine the superconducting volume fraction of a superconductor. As an example, the main panel of Figure 3C shows the normalized superfluid density calculated from σ sc (T ), which is proportional to [λ(T )/λ(0)] −2 (see details in Section II B), as a function of the reduced temperature T /T c (0) for Re 24 Nb 5 and Re 0.4 Mo 0.6 [20,27]. The low-T superfluid density is practically independent of temperature, clearly suggesting a lack of low-energy excitations and a fully-gapped superconductivity. Contrarily, such excitations exist in case of nodes in the superconducting gap, implying a temperaturedependent superfluid density below ∼ T c /3. As shown by solid lines in Figure 3C, the ρ sc (T ) of ReT superconductors is described very well by a fully-gapped s-wave model (see Equation 5). The other α-Mn-type ReT , TaOs, and NbOs 2 exhibit similar temperature-invariant superfluid densities [17-19, 25, 44, 56, 62, 63]. Although ReT alloys adopt different crystal structures (i.e., centrosymmetric or noncentrosymmetric, see Figure 2C-F) and have different T c values, they regularly exhibit low-T superfluid densities which are independent of temperature [27]. Except for T = Mo (and for some α-Mn structures), a systematic microscopic study of superconductivity in other ReT superconductors is still missing. Clearly, it would be interesting to know if their SC behavior is similar to that of Re-Mo alloys. The nodeless SC scenario in ReT alloys is also supported by other techniques, as the electronic specific heat, the magnetic penetration depth measured via the tunnel-diode-oscillator-based technique, or the point-contact Andreev reflection [19,20,27,53,54,58,[64][65][66]. In addition, some studies have found evidence of twogap superconductivity in Re 0.82 Nb 0.18 and Re 6 Zr [49,66]. Figure 3D summarizes the zero-temperature superconducting energy gap value for ReT and α-Mn-type NbOs 2 and TaOs superconductors as a function of their critical temperatures. Most of them exhibit a ∆ 0 /k B T c ratio larger than 1.76 (see dashed line in Figure 3D), the value expected for a weakly coupled BCS superconductor, which indicates a moderately strong coupling in these superconductors. In addition, the specific-heat discontinuity at T c (i.e., ∆C/γT c ) is larger than the conventional BCS value of 1.43, again indicating an enhanced electron-phonon coupling [19,20,27,53,54,58,64]. As mentioned above, it is worth noting that the superconducting parameters of all the other ReT materials (except for α-Mn-type and T = Mo) are missing, prompting further research efforts in this direction.
As discussed in the introduction, the lack of inversion symmetry in the NCSCs often induces an ASOC. This splits the Fermi surface by lifting the degeneracy of the conduction electrons, thus allowing admixtures of spin-singlet and spin-triplet pairing. In general, the strength of ASOC determines the degree of such an admixture and thus the superconducting properties of NCSCs [1,2]. A fully-gapped superconductor (i.e., dominated by spin-singlet pairing) can be tuned into a nodal superconductor (dominated by spintriplet pairing) by increasing the strength of ASOC. Such mechanism has been successfully demonstrated, e.g., in weakly-correlated Li 2 Pt 3 B (E SOC /k B T c ∼ 831) [67,68], CaP-tAs (E SOC /k B T c ∼ 800) [16,69], and in strongly-correlated CePt 3 Si (E SOC /k B T c ∼ 3095) superconductors [12,70], all exhibiting a relatively large band splitting E SOC compared to their superconducting energy scale k B T c . In the α-Mn-type ReT alloys, the density of states (DOS) near the Fermi level is dominated by the 5d orbitals of rhenium atoms, while contributions from the d orbitals of T atoms are negligible [71][72][73]. Therefore, a possible enhancement of SOC due to 3d-(e.g., Ti, V) up to 5d-electrons (e.g., Hf, Ta, W, Os) will, in principle, neither increase the band splitting E SOC nor affect the pairing admixture and thus the superconducting properties of α-Mn-type ReT superconductors. According to band-structure calculations, in Re 6 Zr, the SOC-induced band splitting is about 30 meV [73], implying a very small ratio E SOC /k B T c ∼ 25, comparable to that of fully-gapped Li 2 Pd 3 B, Mo 3 P, and Zr 3 Ir superconductors [22,68,74]. Therefore, despite the relatively large SOC of rhenium atoms, its effects are too weak to significantly influence the bands near the Fermi level. This might explain why all the α-Mn-type ReT superconductors exhibit nodeless superconductivity, more consistent with a spin-singlet dominated pairing [17-20, 25, 55]. However, we recall that often, due to the similar magnitude and same-sign of the order parameter on the spin-split Fermi surfaces, a possible mixed-pairing superconductor may be challenging to detect or to distinguish from a single-gap s-wave superconductor [75]. The almost spherical symmetry of the Fermi surface in these materials may also explain their BCS-like superconducting states [72]. As for the other centrosymmetric ReT alloys, in most of them the Re and T atoms occupy the same atomic positions in the unit cell. In this case, as the T -content increases, the contribution of T d orbitals to the DOS is progressively enhanced, at the expense of the Re 5d orbitals. Therefore, the chemical substitution of Re by another 3d, 4d, or 5d T metal (see Figure 2), should significantly tune the SOC and, hence, the band splitting, an interesting hypothesis waiting for theoretical confirmation. However, even for T = Hf, Ta, W, and Os, the maximum E SOC should still be comparable to that of α-Mn-type ReT alloys. Finally, irrespective of the strength of SOC, due to their centrosymmetric crystal structures, these compounds may exhibit either singlet-or triplet-pairing, but not an admixture of both. According to the TF-µSR results (see Figure 3C), despite a change in SOC and of the different crystal structures (see Figure 2C-F), all ReT superconductors exhibit fully-gapped superconducting states. This finding strongly suggests that, in the ReT superconductors, spin-singlet pairing is dominant.

V. TIME-REVERSAL SYMMETRY BREAKING
Owing to its very high sensitivity (see details in Section II C), ZF-µSR has been successfully used to search for spontaneous magnetic fields, reflecting the breaking of TRS in the superconducting states of different types of superconductors, as e.g., Sr 2 RuO 4 , UPt 3 , PrOs 4 Sb 12 , LaNiGa 2 , LaNiC 2 , La 7 (Rh,Ir) 3 , and α-Mn-ReT [14,15,[17][18][19][20][21][76][77][78][79][80][81]. The latter three are typical examples of weakly-correlated NSCSs, to be contrasted with strongly-correlated NCSCs, where either the TRS is broken by a coexisting long-range magnetic order, or the tiny TRS-breaking signal is very difficult to detect due to the presence of strong magnetic fluctuations [28]. In the former case, the broken TRS is unrelated to the superconductivity, while in the later case, a genuine TRS breaking effect is masked by the much faster muon-spin relaxation caused by magnetic fluctuations. Therefore, in general, a TRS breaking effect is more easily (and reliably) detected in weakly-correlated-or non-magnetic superconductors using µSR techniques. Normally, in the absence of external fields, the onset of superconductivity does not imply changes in the ZF-µSR relaxation rate. However, in presence of a broken TRS, the onset of a tiny spontaneous polarization or of currents gives rise to associated (weak) magnetic fields, readily detected by ZF-µSR as an increase Data were taken from Refs. [17,19,20,27].
in the relaxation rate. Given the tiny size of such effects, the ZF-µSR measurements are usually performed in both the normal-and the superconducting state with a relatively high statistics, at least twice that of the TF-µSR spectra. As an example, Figure 4A plots the ZF-µSR spectra of α-Mntype Re 25 Nb 5 , with the other ReT superconductors showing a similar behavior. The ZF-µSR spectra collected belowand above T c (at 1.5 and 12 K) exhibit small yet measurable differences. The lack of any oscillations in the spectra, implies the non-magnetic nature of ReT superconductors. Further, longitudinal-field µSR measurements under a relatively small applied field (typically a few tens of mT) in the superconducting state are usually performed to check if the applied field can fully decouple the muon spins from the weak spontaneous magnetic fields, and thus exclude extrinsic effects. In non-magnetic materials in the absence of external magnetic fields, the muon-spin relaxation is mostly determined by the interaction between the muon spins and the randomly oriented nuclear magnetic moments. Therefore, the spontaneous magnetic fields due to the TRS breaking will be reflected in an additional increase of muon-spin relaxation. The ZF-µSR asymmetry can be described by means of a Gaussian-or a Lorentzian Kubo-Toyabe relaxation, or a combination thereof (see Equations 6 and 7).  [17,20,27]. The other α-Mntype superconductors, e.g., Re 6 Ti, and Re 6 Hf [18,44], show similar σ ZF (T ) to Re 24 Nb 5 and Re 6 Zr, and thus the breaking of TRS in the superconducting state. At the same time, in the isostructural Re 3 Ta, Re 5.5 Ta, and Re 3 W cases, there is no clear increase in σ ZF (T ) upon crossing T c , implying a preserved TRS [25,55,56].
Recently, the breaking of TRS and the presence of nodes in the SC gap, attributed to an admixture of singlet-and triplet paring, has been reported in the noncentrosymmetric CaPtAs superconductor [16]. In general, however, the breaking of TRS in the superconducting state and a lack of space-inversion symmetry in the crystal structure are independent events, not necessarily occurring together. For instance, the unconventional spin-triplet pairing is expected to break TRS below T c , as has been shown, e.g., in Sr 2 RuO 4 , UPt 3 , and UTe 2 triplet superconductors [76-78, 80, 82-86]. An s + id spin-singlet state was proposed to account for the TRS breaking in some iron-based high-T c superconductors [87], where a nodal gap is also expected. The frequent occurrence of TRS breaking in the fully-gapped (i.e., dominated by spin-singlet pairing) ReT superconductors (see Section IV) is, therefore, rather puzzling. A similarly surprising result was the report that elementary rhenium also exhibits signatures of TRS breaking in its superconducting state (see Figure 4B), with ∆σ ZF (T ) being comparable to that of Re 6 Zr [20,27]. Since elementary rhenium adopts a centrosymmetric hcp-Mg crystal structure (see Figure 2C), this indicates that a lack of inversion symmetry and the accompanying ASOC effects are not crucial factors for the occurrence of TRS breaking in ReT superconductors. Further on, a comparison of ZF-µSR measurements on Re-Mo alloys with different Re/Mo contents, covering almost all the crystal structures reported in Figure 2, shows that only Re and Re 0.88 Mo 0.12 exhibit a broken TRS in the superconducting state, while those with a higher Mo-content (∼23%-60%), including both the centrosymmetric-and noncentrosymmetric Re 0.77 Mo 0.23 , preserve the TRS. Considering the preserved TRS in Mg 10 Ir 19 B 16 , and Nb 0.5 Os 0.5 [23,24], all of which share the same α-Mn-type structure, this implies that TRS breaking in ReT superconductors is clearly not related to the noncentrosymmetric crystal structure or to a possible mixed pairing but, most likely, is due to the presence of rhenium and to its amount. Such conclusion is further reinforced by the preserved TRS in many Re-based superconductors, whose Re-content is below a certain threshold. Such cases include, e.g., Re 3 W, Re 3 Ta, Re-Mo (with Mo-content higher than 12%) [25,27,55], the recently reported Re-B superconductors [88], and the diluted ReBe 22 superconductor [89]. Moreover, by comparing the ZF-µSR relaxation across various ReT superconductors, a clear positive correlation between ∆σ ZF (i.e., spontaneous fields) and the size of the nuclear magnetic moments µ n was identified [20]. For instance, among the ReT superconductors, Re 24 Nb 5 shows the largest spontaneous fields below T c (see Figure 4B), a fact compatible with the large nuclear magnetic moment of niobium, practically twice that of rhenium (6.17 vs. 3.2 µ N ). However, the correlation between µ n and ∆σ ZF alone cannot explain TRS breaking, considering that elementary Nb itself, despite having the highest µ n , does not break TRS. Clearly, the origin of such correlation is not yet understood and it requires further experimental and theoretical studies.
If SOC can be ignored, an alternative mechanism, which can account for the TRS breaking in ReT superconductors in presence of a fully-opened superconducting gap, is the internally-antisymmetric nonunitary triplet (INT) pairing. The INT pairing was originally proposed to explain the TRS breaking and nodeless SC in centrosymmetric LaNiGa 2 [3,81,90] and noncentrosymmetric LaNiC 2 [14,91], both exhibiting a relatively weak SOC. In case of INT pairing, the superconducting pairing function is antisymmetric with respect to the orbital degree of freedom, while remaining symmetric in the spin-and crystal-momentum channels [14,81,90,91]. Since in ReT superconductors, too, the SOC interaction is relatively weak (∼ 30 meV, see Section IV) [73] and since neither TRS breaking nor the nodeless SC are related to the symmetry of ReT crystal structures, the effect of SOC to the observed TRS breaking is insignificant. This could, therefore, explain why a lack of inversion symmetry (essential to SOC) is not a precondition for TRS breaking in ReT superconductors. Moreover, the occurrence of an INT state relies on the availability of a local-pairing mechanism driven by Hund's rules, e.g., by Ni 3d-electrons in LaNiC 2 and LaNiGa 2 [3,14,81,90,91]. Such local-pairing mechanism may also occur in ReT superconductors, since rhenium too can be magnetic [92,93]. This consideration is also in good agreement with the observation that TRS breaking depends on Re content, but not on a noncentrosymmetric crystal structure.

VI. CONCLUSION
In this short review we focused on recent experimental studies of ReT superconductors, where time-reversal symmetry breaking effects are often present and whose superconductivity can, therefore, be considered as unconventional. Due to its high sensitivity to the weak internal fields associated with TRS breaking, µSR represents one of the key techniques in the search for TRS-breaking effects in the superconducting state. Nonetheless, in certain cases, the amplitude of the spontaneous magnetic fields (the fingerprint of TRS breaking) may still be below the resolution of the µSR technique (∼ 10 −2 mT). Hence, the future use of other techniques, e.g., based on the optical Kerr effect [11], another very sensitive probe of spontaneous fields in unconventional superconductors, remains crucial. Due to their rich crystal structures, covering both centro-and noncentrosymmetric cases, and the pervasive presence of superconductivity at low temperatures, the nonmagnetic Re-based materials are the ideal choice for investigating the origin of TRS breaking. Here, we reviewed different cases of Re-containing superconductors, ranging from elementary rhenium, to ReT (T = 3d-5d early transition metals), to the dilute-Re case of ReBe 22 , all of which were investigated through both macroscopic and microscopic techniques. By a comparative study of ReT with different T metals mostly using µSR technique, we could demonstrate the secondary role played by SOC and why the spin-singlet pairing is dominant in ReT superconductors. This, however, brings up the question of reconciling the occurrence of TRS breaking with a fully-gapped SC state (spin-singlet pairing). A possible solution to this is offered by the so-called INT model, which requires an antisymmetric pairing function involving the orbital degree of freedom, making it insensitive to the presence (or lack) of inversion symmetry and SOC. Overall, the reported results suggest that the rhenium presence and its amount are two key factors for the appearance and the extent of TRS breaking in the Re-based superconductors. These key observations, albeit important, demand new experimental and theoretical investigations to further generalize them.
To date, as nearly all current studies have focused exclusively on α-Mn-type ReT superconductors (except for the Re-Mo series considered here), the superconducting properties of most other ReT alloys remain basically unexplored. Hence, the synthesis and characterization of non-α-Mn-type ReT alloys, including the study of their electrical, magnetic, and thermodynamic properties, is of clear interest. Similarly, systematic µSR measurements, crucial for detecting the presence of TRS breaking in Re-based superconductors, are in high demand. For instance, although both Re-Zr and Re-Nb alloys exhibit rich crystal structures and superconducting phase diagrams, only their α-Mn-type phase has been explored. In addition, most of the original measurements were performed only on polycrystalline samples. Hence, the synthesis of single crystals will be essential in the comprehensive search for possible superconducting nodes and, thus, for mixed singlet-triplet pairing. Finally, it would be of interest to extend the µSR studies on elementary rhenium from the bulk-to its thin-film form, where inversion symmetry is artificially broken. By checking if the TRS breaking is maintained or not, will help us to further clarify the rhenium conundrum.