Enhanced $T_c$ and multiband superconductivity in the fully-gapped ReBe$_{22}$ superconductor

In search of the origin of superconductivity in diluted rhenium superconductors and their significantly enhanced $T_c$ compared to pure Be (0.026 K), we investigated the intermetallic ReBe$_{22}$ compound, mostly by means of muon-spin rotation/relaxation ($\mu$SR). At a macroscopic level, its bulk superconductivity (with $T_c=9.4$ K) was studied via electrical resistivity, magnetization, and heat-capacity measurements. The superfluid density, as determined from transverse-field $\mu$SR and electronic specific-heat measurements, suggest that ReBe$_{22}$ is a fully-gapped superconductor with some multigap features. The larger gap value, $\Delta_0^l=1.78$ k$_\mathrm{B}T_c$, with a weight of almost 90\%, is slightly higher than that expected from the BCS theory in the weak-coupling case. The multigap feature, rather unusal for an almost elemental superconductor, is further supported by the field-dependent specific-heat coefficient, the temperature dependence of the upper critical field, as well as by electronic band-structure calculations. The absence of spontaneous magnetic fields below $T_c$, as determined from zero-field $\mu$SR measurements, indicates a preserved time-reversal symmetry in the superconducting state of ReBe$_{22}$. In general, we find that a dramatic increase in the density of states at the Fermi level and an increase in the electron-phonon coupling strength, both contribute to the highly enhanced $T_c$ value of ReBe$_{22}$.


Introduction
As one of the lightest elements, beryllium exhibits high-frequency lattice vibrations, a condition for achieving superconductivity (SC) with a sizeable critical temperature. Yet, paradoxically, its T c =0.026 K is so low [1], that its SC is often overlooked. Clearly, T c is affected also by the electron-phonon coupling strength (typically large in elements with covalent-bonding tendencies) and the density of states (DOS) at the Fermi level N(ò F ) (rather low in pure Be). The latter depends on the details of crystal structure and on atomic volume, both effects being nicely illustrated by metal-hydride SCs under pressure (see, e.g. [2]). In this regard, recently researchers could demonstrate a purely phonon-mediated SC with T c up to 250 K in actinium hydrides at 200 GPa [3]. The key insight of this work was the discovery of a link between chemical composition and SC. Namely, that SC is more likely to occur in materials containing metal atoms that are close to populating a new electronic subshell, such as the d 1 -(Sc, Y, La, and Ac) or p 0 (Be, Mg, and Ca) elements. In these cases, the electronic structure becomes highly sensitive to the positions of the neighboring atoms [4], resulting in stronger electron-phonon interactions and a higher N(ò F ). Based on this intuition, Be-rich alloys may achieve a T c much higher than elementary beryllium, a prediction which turns out to be true for ReBe 22 [5], whose T c ∼9.6 K is almost 400(!) times higher than that of Be. This is a remarkable result, deserving more attention and a detailed investigation of the ReBe 22 electronic properties.
ReBe 22 represents also a very interesting case in an entirely different aspect. Recently, a number of studies have shown that Re-based superconductors exhibit unconventional superconducting behavior. For example, in non-centrosymmetric α-Mn-type ReT alloys (T=transition metal), the time-reversal symmetry (TRS) is broken, and the upper critical field is close to the Pauli limit [6][7][8]. Surprisingly, our previous results show that, below T c , even pure Re breaks TRS, thus behaving as an unconventional superconductor [8]. While binary Rebased superconductors have been investigated in both the full-(pure Re) and the intermediate Re limit (ReT), it is not clear if the unconventional behavior, in particular the TRS breaking, persists also in the dilute Re limit. With only 4% of Re content, ReBe 22 is a good test case to verify such scenario.
In this paper, we report on an extensive study of the physical properties in the normal and superconducting state of ReBe 22 , by means of electrical resistivity, magnetization, thermodynamic, and muon-spin relaxation (μSR) methods. In addition, we also present numerical density-functional-theory (DFT) band-structure calculations. ReBe 22 exhibits a fully-gapped, spin-singlet superconducting state with preserved TRS. Despite the very small amount of Re, the ReBe 22 alloy shows a remarkable increase in T c compared to its elementary constituents, which we mostly attribute to the significant increase of DOS at the Fermi level.

Methods
Polycrystalline samples of ReBe 22 were prepared by arc melting of elementary Be (Heraeus, 99.9%) and Re (Chempur, 99.97%) in an argon-filled glove box (MBraun, p(H 2 O/O 2 ) < 0.1 ppm), dedicated to the handling of Be-containing samples [9]. To compensate for the evaporation losses and to avoid the formation of spurious Re-Be binary phases, a small excess of beryllium was used. Powder x-ray diffraction (XRD) measurements were performed on a Huber G670 image-plate Guinier camera (Ge-monochromator, Cu Kα 1 radiation). The lattice parameter of cubic ReBe 22 was determined from a least-squares fit to the experimental peak positions. The sample purity was then checked by electron microscopy and energy-dispersive x-ray spectroscopy (EDX) on a JEOL JSM-6610 scanning electron microscope equipped with secondary electron-, electron backscatter-, and UltraDry EDS detectors (see figure A1 in the appendix). Besides traces of elemental Be, no chemical impurities or secondary phases could be detected.
The magnetic susceptibility, electrical resistivity, and specific-heat measurements were performed on a 7-T Quantum Design Magnetic Property Measurement System (MPMS-7) and on a 14-T Physical Property Measurement System (PPMS-14) equipped with a 3 He option. The μSR measurements were carried out at the GPS spectrometer of the Swiss muon source at Paul Scherrer Institut, Villigen, Switzerland [10]. The μSR data were analyzed by means of the musrfit software package [11].
The band structure of ReBe 22 was calculated by means of density-functional theory. Here we used the fullpotential nonorthogonal local orbital code (FPLO) [12]. To calculate the nonmagnetic band structure we employed the local-density approximation parametrized by the exchange-correlation potential of Perdew and Wang [13]. The strong spin-orbit coupling of Re atoms was taken into account by performing full-relativistic calculations by solving the Dirac Hamiltonian with a generic potential.

Results and discussion
3.1. Crystal structure As shown in figure 1, the complex intermetallic compound ReBe 22 adopts a cubic ZrZn 22 -type structure with space group Fd m 3 (No. 227) and Z=8 formula units per cell. The lattice parameter a=11.5574(4) Å, determined from the XRD pattern (see figure 1(d)), is consistent with the previously reported value [14]. No obvious impurity phases could be detected, indicating the high quality of the synthesized samples. The crystal structure can be visualized by means of two structural motifs. In the ReBe 16 motif (see figure 1(a)), Re is coordinated by 12 Be atoms, lying 2.53 Å apart at the vertices of a truncated tetrahedron, also known as a Friauf polyhedron. Four additional Be atoms lie atop the hexagonal faces of the truncated tetrahedron at a distance of 2.50 Å from the center. A similar motif is also found in the NbBe 2 superconductor [15,16]. As for the rest of Be atoms, these form distorted Be-centered Be 13 icosahedra, with the short interatomic distances ranging from 2.05 to 2.29 Å ( figure 1(b)). Such Be-icosahedra represent the structural building blocks of the complex MBe 13 (M = rare earths and actinides) phases with a NaZn 13 -type structure [17].
As shown in figure 1(c), the ReBe 16 and Be 13 units are connected by sharing the polyhedra vertices. The arrangement of the two types of polyhedra within a unit cell can be described as hierarchically derived from the MgCu 2 -type structure, where the Mg positions are occupied by ReBe 16 units and those of Cu by Be 13 icosahedra [18]. Both motifs, the truncated tetrahedron and the icosahedron, are found in the close-packed Laves phase structures. As a consequence of the high Be content, ReBe 22 features structural motifs typically found in Be-rich intermetallic compounds, dominated by close-packing structures similar to that of hcp-Be [19,20]. Since the ratio of metallic radii [r Re /r Be =1.223] is close to the ideal value of 1.225, this facilitates the close-packing of unequal spheres in ReBe 22 and the accommodation of Re in the structure [21,22]. The high packing fraction in this deltahedral structure is an important factor for the stabilization of this unusual stoichiometry, also found in the isostructural MoBe 22 and WBe 22 compounds, both featuring similar ratios of radii [23].

Electrical resistivity
The temperature dependence of the electrical resistivity ρ(T) of ReBe 22 was measured in zero magnetic field from 300 down to 2 K. As shown in figure 2, the resistivity exhibits metallic features down to base temperature, dropping to zero at the superconducting transition at = T 9.42 K c zero (see inset). Between T c and 300 K the electrical resistivity can be modeled by the Bloch-Grüneisen (BG) formula [24,25]: Here, the first term ρ 0 is the residual resistivity due to the scattering of conduction electrons on the static defects of the crystal lattice, while the second term describes the electron-phonon scattering, with Q D R being the characteristic (Debye) temperature and A a coupling constant. The fit (black-line) in figure 2 results in ρ 0 =2.72(5) μΩ cm, A=95(3) μΩ cm, and Q = ( ) 590 5 K D R . Such a large Q D R value is consistent with the heatcapacity results (see below) and reflects the high frequency of phonons in ReBe 22 . This is compatible with the high Debye temperature of elemental Be (∼1031 K) [26], in turn reflecting the small mass of beryllium atoms. A relatively large residual resistivity ratio [RRR=ρ(300 K)/ρ 0 ∼16] and a sharp superconducting transition (ΔT=0.23 K) both indicate a good sample quality.

Magnetization measurements
The bulk SC of ReBe 22 can be probed by magnetization measurements. The temperature evolution of the magnetic susceptibility χ(T), measured at 1 mT using both field-cooled (FC) and zero-field-cooled (ZFC) protocols, is shown in figure 3(a). The splitting of the FC-and ZFC-susceptibilities is typical of granular superconductors, where the magnetic-field flux is trapped (in open holes) upon cooling the material in an

Specific heat
The temperature dependence of the heat capacity C(T) of ReBe 22 was also measured in zero-field conditions from 300 K down to 2 K. Although a single Debye-or Einstein model cannot describe the data, as shown in figure 4, the normal-state C(T) can be fitted by a combined Debye and Einstein model, with relative weights x and -( ) x 1 [29]: The number of atoms per ReBe 22 formula-unit (n=23) is considered in the above equation. The first term represents the electronic specific heat, which can be determined from the low-T heat-capacity data (see below). The second and the third terms represent the acoustic-and optical phonon-mode contributions, described by the Debye and Einstein model, respectively [29]: Here Q D C and Q E C are the Debye and Einstein temperatures, while R=8.314 J mol −1 K −1 is the molar gas constant. The best fit curve (solid line in figure 4) is obtained for Q = ( ) 545 5 K D C and Q = ( ) 792 5 K E C , with x=0.35 (2). The resulting Debye temperature is comparable to that derived from electrical resistivity data (see figure 2).
The low-T specific-heat data were further analyzed, since they can offer valuable insight into the superconducting properties of ReBe 22 through the evaluation of the quasiparticle DOS at the Fermi level. As shown in figure 5, the sharp specific-heat jump at T c again indicates a bulk superconducting transition and a good sample quality. The electronic specific heat C e /T was obtained by subtracting the phonon contribution from the experimental data. The DOS at the Fermi level N(ò F ) can be evaluated from the expression states/eV-f.u. (accounting for spin degeneracy) [30], where k B is the Boltzmann constant and γ n =15.3(2)mJ mol −1 K −2 is the electronic specific-heat coefficient. The electronphonon coupling constant l ep , a measure of the attractive interaction between electrons due to phonons, was estimated from the Q D C and T c values by means of the semi-empirical McMillan formula [31]: The Coulomb pseudo-potential  m was fixed to 0.13, a typical value for metallic samples. From equation (5) we obtain λ ep =0.64(1) for ReBe 22 , almost three times larger than the reported value for elemental Be (0.21) [32]. By using this value, finally, the band-structure DOS N band (ò F ) can be estimated from the relation states/eV-f.u. After subtracting the phonon contribution from the specific-heat data, the electronic specific heat divided by the electronic specific-heat coefficient, i.e. C e /γ n T, is obtained (main panel in figure 5). The temperaturedependent superconducting-phase contribution to the entropy was calculated by means of the BCS expression [33]: is the excitation energy of quasiparticles, with ò the electron energies measured relative to the chemical potential (Fermi energy) [33,37]. [38], with Δ 0 the gap value at zero temperature. The temperature-dependent electronic specific heat in the superconducting state can be calculated from The dashed-dotted line in figure 5 represents a fit with an s-wave model with a single gap Δ 0 =1.40(1) meV. While this reproduces well the experimental data in the 0.4<T/T c <0.8 range, out of it the single-gap model clearly deviates from the data (see lower inset). On the contrary, the two-gap model exhibits a better agreement, both at low temperatures as well as near T c . The solid line in figure 5 is a fit to the two-gap s-wave model, known also as α model [39]: .  9). In addition, the larger gap is comparable to the weak-coupling BCS value (1.4 meV), indicating weakly-coupled superconducting pairs in ReBe 22 . The specific-heat discontinuity at T c , i.e. ΔC/γ n T c =1.24, is smaller than the BCS value of 1.43. There are two possibilities for such a reduced specific-heat discontinuity, despite a good sample quality and full superconducting volume fraction: (i) gap anisotropy, including a nodal gap, as observed in some heavy-fermion superconductors or in Sr 2 RuO 4 [40,41], or (ii) multiband SC, as e.g. in MgB 2 or LaNiGa 2 [36,42]. Due to a highly-symmetric crystal structure and to a lack of gap nodes (see below), only the second scenario is applicable to the ReBe 22 case.
The multiband SC of ReBe 22 can be inferred also from the field dependence of the electronic specific heat coefficient γ H . As shown in figure 6(a), at a given applied field, γ H is obtained as the linear extrapolation of C/T versus T 2 (in the superconducting phase) to zero temperature. The dependence of the normalized γ H /γ n versus the reduced magnetic field H/H c2 (0) is shown in figure 6(b) (here γ n is the zero-field normal-phase value). Note that, the field dependence of γ H /γ n at 2 K exhibits similar features to that evaluated at zero temperature. Due to the multiband effects, it is difficult to describe the field dependence of γ H in ReBe 22 with a simple formula. As can be seen in figure 6(b), γ H (H) clearly deviates from the linear field dependence (dashed-dotted line) expected for single-gap BCS superconductors [43], or from the square-root dependence H (solid line) expected for nodal superconductors [44,45]. In fact, ReBe 22 exhibits similar features to other multiband superconductors, as e.g. LaNiC 2 [34], FeSe [35], and MgB 2 [36] (the latter being a prototypical two-gap superconductor), although the slopes of γ H (H) close to H=0 are different.

Upper critical field
The upper critical field μ 0 H c2 of ReBe 22 was determined via temperature-dependent resistivity ρ(T, H) and specific heat C(T, H)/T measurements at various applied magnetic fields, as well as from the field-dependent     was analyzed by means of three different models, i.e. a Ginzburg-Landau (GL) [46], a Werthamer-Helfand-Hohenberg (WHH) [47], and a two-band (TB) [48] model. As can be seen in figure 7(a), at low fields, both GL and WHH models reproduce very well the experimental data. However, at higher magnetic fields, both models deviate significantly from the experimental data, providing underestimated near T c is considered a typical feature of multiband superconductors, as e.g. MgB 2 [49,50]. It reflects the gradual suppression of the small superconducting gap with increasing magnetic field, as evidenced also by the specific-heat data shown in , the resulting magnetic penetration depth λ GL =109(1) nm, is comparable to 87(1) nm (40 mT) and 104(1) nm (120 mT), the experimental values evaluated from TF-μSR data (see section 3.6). A GL parameter κ∼4.7(3), much larger than the threshold value of 1 2 , clearly indicates that ReBe 22 is a type-II superconductor.
3.6. Transverse-field μSR μSR measurements in an applied transverse field (TF) were carried out to investigate the superconducting properties of ReBe 22 at a microscopic level. Preliminary field-dependent μSR depolarization-rate measurements at 1.5 K were carried out to determine the optimal field value for the temperature-dependent study (see figure A2 in Appendix). To track the additional field-distribution broadening due to the flux-line-lattice (FLL) in the mixed superconducting state, the magnetic field was applied in the normal state, prior to cooling the sample below T c . After the field-cooling protocol, which ensures an almost ideal FLL even in case of pinning effects, the TF-μSR measurements were performed at various temperatures upon warming. Figures 8(a) and (b) show two representative TF-μSR time spectra collected in the superconducting (1.5 K) and the normal state (10 K) in an applied field of 40 mT at the GPS spectrometer. The enhanced depolarization rate below T c reflects the inhomogeneous field distribution due to the FLL, causing an additional distribution broadening in the mixed state (see figure 8(c)). The μSR spectra can be modeled by the following expression: Here A i and A bg represent the initial muon-spin asymmetries for muons implanted in the sample and sample holder, respectively, with the latter not undergoing any depolarization. B i and B bg are the local fields sensed by implanted muons in the sample and sample holder, γ μ =2π×135.53 MHz T −1 is the muon gyromagnetic ratio, f is a shared initial phase, and σ i is a Gaussian relaxation rate of the ith component. The number of required components is material dependent, generally in the 1n5 range. For superconductors with a large 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), as e.g. in ReT [7,8] or Mo 3 Rh 2 N [51]. In case of a small κ (1 2), as e.g. in ReBe 22 , the magnetic penetration depth is comparable to the coherence length. The rather small λ implies fast-decaying fluxon field profiles and a broad field distribution, in turn requiring multiple oscillations to describe A(t) [52].
The fast-Fourier-transform (FFT) spectra of the TF-μSR datasets at 1.5 and 10 K are shown in figures 8(c) and (d).
The solid lines represent fits to equation (8) using three oscillations (i.e. n=3) in the superconducting state and one oscillation in the normal state. The TF-μSR spectra collected at 120 mT require only two oscillations (i.e. n=2), indicating a narrower field distribution compared to the 40 mT case. The derived Gaussian relaxation rates as a function of temperature are summarized in the insets of figure 9. Above T c , the relaxation rate is small and temperature-independent, but below T c it starts to increase due to the onset of the FLL and the increase in superfluid density. In case of multi-component oscillations, the firstterm in equation   , where σ n is the nuclear relaxation rate. The superconducting gap value and its symmetry can be investigated by measuring the temperaturedependent σ FLL (T), which is directly related to the magnetic penetration depth and thus the superfluid density (σ FLL ∝ 1/λ 2 ).
Since the upper critical field of ReBe 22 is relatively small (600 mT) compared to the applied fields used in the TF-μSR study (40 and 120 mT), the effects of the overlapping vortex cores with increasing field ought to be considered when extracting the magnetic penetration depth λ from s FLL . For ReBe 22 , λ was calculated by means of [28,53]: where h=H appl /H c2 , with H appl the applied magnetic field. The above expression is valid for type-II superconductors with  k 5 in the 0.25/κ 1.3  h1 field range. With k~4.7 and h=0.067 (TF-50 mT) and 0.2 (TF-120 mT), ReBe 22 fulfills the above condition. Note that, in the above expression, only the absolute value of the penetration depth, but not its temperature dependence is related to the h value. By using equation (12), we calculate the inverse-square of the magnetic penetration depth, which is proportional to the superfluid density, i.e. l r µ -( ) ( ) T T

sc
. As can be seen in figure 9, below T c /3, ρ sc (T) is practically independent of temperature, in agreement with the specific-heat results shown in figure 6, once more indicating a nodeless SC in ReBe 22 . r ( ) T sc was further analyzed by means of a two-gap s-wave model, previously applied to the well-established two-gap superconductor MgB 2 [38,54]. In general, the superfluid density can be described by: As in the specific-heat case, r D sc s and r D sc l are the superfluid densities related to the small (Δ s ) and large (Δ l ) gaps, and w is a relative weight. For each gap, ρ sc (T) is given by: where f and Δ are the Fermi-and the gap function, respectively, as in section 3.4. Here, the gap value at zero temperature Δ 0 is the only adjustable parameter. As can be seen in figure 9(a), for TF-40 mT, the temperatureindependent behavior of l -( ) , with a weight w=0.1. The latter are consistent with the gap values obtained from specific-heat data. For the single-gap model, the estimated gap value is Δ 0 =1.33 meV, with the same λ 0 as in the two-gap case. In the TF-μSR with m = H 120 mT 0 appl (see figure 9(b)), the applied field suppresses the smaller gap (see details in figures 6(b) and 7). Hence the λ −2 (T) dependence is consistent with a single-gap s-wave model, leading to l = ( ) 104 1 0 nm and Δ 0 =1.10(1) meV.

Zero-field μSR
To search for a possible weak magnetism or TRS breaking in the superconducting state of ReBe 22 , ZF-μSR measurements were performed in the 1.5-20 K temperature range. Normally, in the absence of external fields, there is no change in the ZF muon-spin relaxation rate across T c . However, in case of a broken TRS, the onset of tiny spontaneous currents gives rise to associated (weak) magnetic fields, causing an increase in the muon-spin relaxation rate in the superconducting state. Representative ZF-μSR spectra for ReBe 22 collected above (15 K) and below (1.5 K) T c are shown in figure 10. No oscillations could be observed, implying a lack of magnetic order in ReBe 22 . In such case, in absence of applied fields, the relaxation is mainly determined by the randomly oriented nuclear moments. Consequently, the ZF-μSR spectra of ReBe 22 can be modeled by means of a combined Lorentzian and Gaussian Kubo-Toyabe relaxation function [55,56]: Here A s and A bg are the same as in the TF-μSR case in equation (8). In polycrystalline samples, the 1/3nonrelaxing and 2/3-relaxing components of the asymmetry correspond to the powder average of the local internal fields with respect to the initial muon-spin orientation. The resulting fit parameters versus temperature, including the Lorentzian-Λ ZF and Gaussian relaxation rates σ ZF , are shown in figures 10(b) and (c). Here A s was fixed to its average value of 0.205, however, the same features are also found in fits with released A s . The large relaxation rates reflect the significant nuclear magnetic moments present in ReBe 22 . A similarly fast Gaussian relaxation was also found in other Re-based alloys [7,8]. This is in contrast to superconductors containing nuclei with small magnetic moments, as e.g. Mo 3 Rh 2 N [51], which exhibit a negligibly small relaxation. Despite the clear difference in the ZF-μSR spectra recorded in the normal and superconducting states ( figure 10(a)), neither Λ ZF (T) nor σ ZF (T) show distinct changes across T c . The enhanced s ZF below 6 K in figure 10(c) might be caused by tiny amounts of magnetic impurities, below the XRD and EDX detection threshold. This is also indicated by the Curie-Weiss-like behavior of σ ZF (T) in figure 10(c), i.e. s = --T 0.242 0.047 ZF 1 , whose positive curvature is opposite to the negative one, common in case of TRS breaking [7,8]. To further distinguish the intrinsic versus extrinsic effects in σ ZF (T), samples synthesized using even higher purity chemicals are desirable. We also performed auxiliary longitudinal field (LF)μSR measurements at 1.5 K. As shown in figure 10(a), a field of 10 mT is already sufficient to lock the muon spins and to completely decouple them from the weak magnetic fields, confirming the sparse presence of magnetic impurities. In conclusion, the ZF-μSR results indicate a preserved TRS in the superconducting state of ReBe 22 .

Electronic band structure
To shed more light on the underlying electronic properties of ReBe 22 , we performed electronic band-structure calculations based on DFT, including spin-orbit coupling. Figure 11 shows the total-, atomic-, and orbitalprojected DOS, disclosing the metallic nature of the system through its nonzero DOS at the Fermi level. The main contributions to the latter arise from the Re-d and Be-p orbitals. While the Be-Be bonding is comprised primarily of 2s orbitals, the Re-Be hybridization consists of Re-5d and Be-2p states. Notwithstanding a 4% Re-to-Be ratio in a ReBe 22 formula unit, Re atoms are over-represented with an almost 3 times larger weight of 12% in the DOS at the Fermi level. Our calculations estimate a total DOS at the Fermi level of N(ò F )=4 states/ eV-f.u., comparable to the 3.25 states/eV-f.u. extracted from specific-heat data. Both values are significantly larger (∼50 times) than that estimated for elemental Be [57] and, consequently, may justify the huge increase in T c with respect to Be (from 0.026 to 9.4 K). Interestingly, a similar T c value has been observed also when Be is deposited as a quenched condensed film [58]. Also in this case, the surge in T c was shown to originate from the increase of DOS at E F in the structurally disordered condensate [57].
The ReBe 22 band structure shown in figure 12 reveals multiple dispersive bands crossing the Fermi energy. In particular, the electron pockets centered around the Γ point are much larger than the hole pockets centered around the L point. This circumstance is typical of multigap/multiband superconductors, as clearly reflected also in our experimental results. Finally, the band splitting due to the spin-orbit coupling of Re is barely visible here due to the low Re/Be ratio.

Discussion
The different families of superconductors can be classified according to the ratio of the superconducting transition temperature T c to the effective Fermi temperature T F , in a so-called Uemura plot [63]. As can be seen in figure 13, several types of unconventional superconductors, including heavy-fermion, organic, high-T c iron pnictides, and cuprates, all lie in a 1/100<T c /T F <1/10 range, here indicated by the shadowed region. Conversely, conventional BCS superconductors exhibit < T T 1 1000 c F , here exemplified by the elemental Sn, Al, and Zn. Three typical examples of multiband superconductors, LaNiC 2 , ThFeAsN, and MgB 2 , are also shown in figure 13. According to the superconducting parameters obtained from our measurements (here summarized in table 1), the calculated T c /T F value for ReBe 22 is~´-9.5 20700 4.6 10 4 (diamond in figure 13). Although it cannot be classified as an unconventional superconductor, ReBe 22 is far away also from the region of conventional superconductors and shows practically the same ratio as the multiband superconductor LaNiC 2 (both lying in the same dashed-dotted line). Compared to pure Be ( =´=´-T T 0.026 1.64 10 1.58 10 c F 5 7 ) [1,30], the T c /T F value of ReBe 22 is enhanced due to the presence of diluted Re, the latter being characterized by a lower Fermi temperature and, hence, by a larger T T c F ratio (see Re in figure 13). Such conclusion is further supported by our electronic band-structure calculations, which show that, although Re contributes only 4% to the atomic ratio, with its 12% weight, it is over-represented in the DOS at the Fermi level.

Conclusion
To summarize, we investigated the physical properties of the ReBe 22 superconductor by means of electrical resistivity, magnetization, heat capacity, and μSR measurements, as well as by electronic band-structure calculations. We find that ReBe 22 is a type-II superconductor (κ∼4.7), with a bulk T c ∼9. . The temperature dependence of the zero-field electronic specific heat and superfluid density reveal a nodeless SC, well described by an isotropic s-wave model, which is more consistent with a multigap-rather than a single-gap SC. The multigap features are further supported by the  22 . The data of the reference samples were adopted from [8,[59][60][61][62][63]. Table 1. Normal-state and superconducting properties of ReBe 22 , as determined from electrical resistivity, magnetic susceptibility, specific-heat, and μSR measurements. The London penetration depth λ L , the effective mass  m , bare band-structure effective mass  m band , carrier density n s , BCS coherence length ξ 0 , electronic mean-free path l e , Fermi velocity v F , and effective Fermi temperature T F were estimated following the equations (40)- (50) in [62].

Property
Value Property Value Appendix. EDX and field-dependent muon-spin relaxation A typical energy-dispersive x-ray spectrum of ReBe 22 and the respective electron micrograph are shown in figure A1. The high chemical purity of the sample is reflected in the lack of unknown peaks. Figure A2(a) shows the time-domain TF-μSR spectra of ReBe 22 , collected in two applied fields, 40 and 400 mT. The solid lines represent fits using the same model as that described in equation (8). The resulting superconducting Gaussian relaxation rate σ FLL (H) is summarized in figure A2(b). Above the lower critical field m H 0 c1 (28.1 mT), the Gaussian relaxation rate decreases continuously. By considering the decrease of the inter vortex distance with the field and the vortex-core effects, a field of 40 mT was chosen for the temperaturedependent TF-μSR studies. For a comparison, the TF-μSR relaxation was also measured in a field of 120 mT, here expected to suppress the small superconducting gap. Figure A1. Energy-dispersive x-ray spectrum recorded on a ReBe 22 sample. The x-ray emission lines attributed to rhenium, carbon, and oxygen are indicated by maroon, red, and blue lines, respectively. Residual carbon and oxygen signals originate from the atmosphere in the microscope chamber. The absence of unidentified peaks reflects the high chemical purity of the sample. The beryllium K-line is too low in energy (at 0.11 keV) to be detected. Note the logarithmic scale. Inset: electron micrograph (electron backscatter detector) of the investigated sample area. The dark spots represent residual elemental beryllium inside the ReBe 22 grains.