Superconductivity in a breathing kagome metals ROs2 (R = Sc, Y, Lu)

We have successfully synthesized three osmium-based hexagonal Laves compounds ROs2 (R = Sc, Y, Lu), and discussed their physical properties. LeBail refinement of pXRD data confirms that all compounds crystallize in the hexagonal centrosymmetric MgZn2-type structure (P63/mmc, No. 194). The refined lattice parameters are a = b = 5.1791(1) Å and c = 8.4841(2) Å for ScOs2, a = b = 5.2571(3) Å and c = 8.6613(2) Å for LuOs2 and a = b = 5.3067(6) Å and c = 8.7904(1) Å for YOs2. ROs2 Laves phases can be viewed as a stacking of kagome nets interleaved with triangular layers. Temperature-dependent magnetic susceptibility, resistivity and heat capacity measurements confirm bulk superconductivity at critical temperatures, Tc, of 5.36, 4.55, and 3.47 K for ScOs2, YOs2, and LuOs2, respectively. We have shown that all investigated Laves compounds are weakly-coupled type-II superconductors. DFT calculations revealed that the band structure of ROs2 is intricate due to multiple interacting d orbitals of Os and R. Nonetheless, the kagome-derived bands maintain their overall shape, and the Fermi level crosses a number of bands that originate from the kagome flat bands, broadened by interlayer interaction. As a result, ROs2 can be classified as (breathing) kagome metal superconductors.

DOS that stems from the kagome flat bands that are broadened by interaction between the layers.ROs 2 can be thus considered as the kagome metal superconductors.

Experimental section
Polycrystalline compounds ScOs 2 , YOs 2 and LuOs 2 were prepared by using the standard arc-melting method.High-purity elements, i.e., (> 99.9 wt%, Onyxmet, Poland) were weighed in nominal stochiometric ratios and arcmelted together in an inert argon atmosphere on a water-cooled cooper hearth.A piece of zirconium was used as a getter material during the melting process.To improve chemical homogeneity, the ingots were remelted three times, flipping them over after each melting.Weight losses upon melting were negligible (< 0.5%).All samples were hard, silver in color and stable against air and moisture.Parts of each sample were wrapped in tantalum foil, sealed under vacuum in a quartz tube and annealed at 800 • C for 7 days.The annealing temperature (800 • C ) for all samples was chosen based on a known Y-Os phase diagram 14 .No melting was observed during the heating process.
Powder X-ray diffraction (pXRD) measurements of as-cast and annealed samples were carried out at room temperature using Bruker D2 Phaser diffractometer [Cu Kα radiation (λ = 1.5406Å)] equipped with a LynxEye-XE detector.Full LeBail analysis of the obtained XRD patterns was performed using the Bruker DIFFRAC.SUITE TOPAS software.The magnetization measurements were performed using a Quantum Design Evercool II Physical Property Measurement System (PPMS) with a Vibrating Sample Magnetometer (VSM) function.Both zero-field-cooled (ZFC) and field-cooled (FC) data were collected from 1.7 to 7 K under an applied field of 10 Oe.The magnetization was also measured at various temperatures in the superconducting state (T < T c ) as a function of the applied field.Magnetic measurements were performed on samples of arbitrary shape with a mass of about 15 mg.All thermodynamic and transport measurements were also performed in a PPMS system.Specific-heat measurements were carried out in zero field and field up to 3 T, using the two-τ time-relaxation method.Each sample was cut to a suitable size and mounted with the Apiezon N grease onto the α-Al 2 O 3 measurement platform to ensure good thermal contact.Temperature-and magnetic-field-dependent electrical transport measurements were tested using a standard four-probe technique, in which Pt wires ( ∅ = 50 µm ) were attached to the surface of polished samples by spot welding.
Electronic band structure and density of states calculations were performed by means of the density functional theory with the Perdew-Burke-Ernzerhof generalized gradient approximation (PBE GGA) 15 of the exchange-correlation potential utilizing the ELK 5.2.14 all-electron full-potential linearized augmented plane wave + local orbitals (FP-LAPW + lo) code 16 .Crystal structures were taken from the Materials Project database 17 (MP id: mp-567612, mp-570670, mp-567590 for Sc-, Y-, and Lu-bearing compound, respectively) and were used without further relaxation.Calculations were conducted in the full-(with spin-orbit coupling) and scalarrelativistic (neglecting the SOC) on an 8 × 8 × 6 Monkhorst-Pack k-point mesh.
Tight binding models of kagome networks were created and solved using the Pybinding package 18 .

Results
The room-temperature pXRD patterns of ScOs 2 , YOs 2 and LuOs 2 materials are presented in Fig. S1 in Supplementary Materials (SM).All compounds are reported to crystallize in a hexagonal centrosymmetric MgZn 2 -type structure (P6 3 /mmc, No. 194) 13 .The pXRD confirmed a good quality of all samples with a small amount of impurity phases (Os; P6 3 /mmc, No. 194).Annealing at 800 • C does not effect on the XRD patterns, indicating that all compounds melt congruently.In a more detailed analysis of the data, the P6 3 /mmc phase was refined with the LeBail method.The LeBail fit to the powder diffraction pattern is represented by a black solid line (Fig. S1).All values are in very good agreement with the data reported previously 13 .Figure S2 in SM presents a schematic view of the hexagonal structure of ROs 2 .The 4f (1/3, 2/3, z) site is occupied by R atoms and the 2a (0,0,0) and 6h (x, 2x, 1/4) sites are occupied by Os atoms.
Figure 2 shows the unit cell volume (V) versus the atomic radius ratio of the rare earth metal to osmium metal (r R /r Os ).The value of radii ratio is an important parameter governing the formation of the Laves phase structure, which ideally is 1.225 19,20 .The ratio of the known Laves phases often deviates from this ideal value (ranges from 1.05 to 1.70).The r R /r Os ratio was calculated based on atomic radii values given by S.M. McLennan 21 .The unit cell volumes for the hexagonal structure of ROs 2 (R = lanthanides from La to Yb) were taken from the ICSD database.As expected, the unit cell volume increases with an increase in the radius of the rare earth metal i.e., the smallest value is noted for ScOs 2 and the largest for LaOs 2 .It is worthwhile to mention that for ROs 2 compounds one can observe polymorphic transition point (PTP), where the crystal structure changes 22,23 .Typically, by conventional arc melting techniques, the C15 (cubic) phase is formed with the light R's (La-Pr) and the C14 (hexagonal) phase with the heavier R's (Pr-Lu).However, it has been observed that for LaOs 2 , CeOs 2 , and PrOs 2 annealing under high pressure leads to the transformation of the cubic (MgCu 2 -type) to the hexagonal (MgZn 2 -type) crystal structure 24,25 .Moreover, M.S. Torikachvili reported 26 that the transformation of CeOs 2 can be accomplished at ambient pressure by annealing at temperatures between 500 and 950 °C.
The superconducting properties of all compounds were first characterized by the measurement of temperature-dependent magnetization under zero-field-cooled and field-cooled conditions.Panels (a-c) of Fig. 3 present the volume magnetic susceptibility (defined as χ = M/H where M is the magnetization and H is the applied magnetic field) measured under an applied field of 10 Oe.The bifurcation of the ZFC and FC magnetic susceptibilities indicates the transition into the superconducting state.It can be seen that for the ZFC and FC signals, the transition is slightly broadened for all samples and reaches saturation at lower temperature.When corrected for the demagnetization effect, N = 0.33 for ScOs 2 and LuOs 2 , and N = 0.73 for YOs 2 (estimated from the M(H) data, discussed in SM), χ ZFC approaches a value of − 1 at the lowest temperatures, indicating volume superconductivity.It should be noted that since the measured samples were in the form of individual chunks whose shape was not well defined, it is difficult to estimate the theoretically expected values of the N-factor.However, for the YOs 2 sample, the rather large value of N and the absence of χ ZFC (T) saturation at the lowest temperatures may indicate that the superconducting Meissner fraction is not 100%.The diamagnetic signal of the FC measurement is weaker, likely caused by the flux line pinning, typically seen for polycrystalline samples of superconductors.The divergence of ZFC and FC signals is more pronounced for ScOs 2 probably due to smaller grains and greater number of grain boundaries.The critical temperature was estimated as an intersection point between the extrapolated lines corresponding to the normal and superconducting state magnetic susceptibilities 27 .The T c value is 5.54 K for ScOs 2 , 4.31 K for YOs 2 and 3.45 K for LuOs 2 .It is worthwhile to mention that the values of critical temperature for compounds with Y and Lu agree well with the previous report 13 , while for ScOs 2 the superconducting Figure 2. The unit cell volume vs the atomic radius ratio of the rare earth metal to osmium metal, r R /r Os .LaOs 2 , CeOs 2 , and PrOs 2 form in the hexagonal phase under high pressure.
The field-dependent magnetization at different temperatures (T < T c ) was measured to determine both the demagnetization factor, N, and estimate the value of the lower critical field, H c1 (0).M(H) measured at selected temperatures for all compounds are depicted in Fig. S3 in SM.For all investigated samples the magnetization exhibits behavior observed for the conventional type-II superconductors [31][32][33] .The demagnetization factor was found assuming that the initial linear response to the field for an isotherm taken at T = 1.7 K is ideally diamagnetic.For an analysis of the lower critical field ( H * c1 ) the point corresponding to the first deviation from a linear response was estimated at each temperature.To precisely calculate this point, we followed the methodology described elsewhere [34][35][36][37] .In panels (d-f) of Fig. 3 the values of H * c1 are plotted as a function of temperature for all compounds.An additional point for H = 0 Oe is a zero-field transition temperature taken from the electrical resistivity measurement.The experimental H * c1 (T) data points were analyzed using the equation 38 : where T c is the superconducting critical temperature and H * c1 (0) is the lower critical field at 0 K.The solid red line through the data points shows a good agreement of the Ginzburg-Landau (GL) theory.Considering the demagnetization factor, the lower critical field, H c1 (0) = H c1 * (0)/(1 − N), at 0 K is calculated to be 187 Oe for ScOs 2 , 83 Oe for YOs 2 , and 48 Oe for LuOs 2 .It should be noted that since the demagnetization factor N for YOs 2 is likely overestimated, the value of H c1 (0) for this compound is likely smaller.
The results of low-temperature heat capacity (C p ) measurements are summarized in Fig. 4. Panels (a-c) present the zero-field data plotted as C p /T versus temperature.The bulk nature of the superconductivity for all samples is confirmed by the pronounced heat-capacity jump on cooling through T c .To determine the critical temperature, we employed idealized equal entropy construction, which reflects the expected entropy balance between the normal state and the superconducting state at the superconducting phase transition.The T c 's equal 5.36 K, 4.55 K and 3.47 K for ScOs 2 , YOs 2 and LuOs 2 , respectively, and agree with the magnetization data presented above.
The highest T c is noticed for ScOs 2 and may be related to the strengthening of the electron-phonon interaction.The highest and the lowest value of Θ D is observed for ScOs 2 and LuOs 2 , respectively, which can be qualitatively explained as the effect of the larger atomic weight of Lu comparing to Sc.The Sommerfeld coefficient, related to the density of states at the Fermi level, is slightly different for all compounds, with the largest value for LuOs 2 .
In addition, having the Sommerfeld coefficient and the electron-phonon coupling constant, the density of states at the Fermi energy N(E F ) can be estimated: where k B is the Boltzmann constant.N(E F ) = 3.30 (ScOs 2 ), 2.90 (YOs 2 ) and 3.86 (LuOs 2 ) states eV −1 per formula unit (f.u.).
The results of electrical resistivity measurements for all samples are summarized in Fig. 6a-c.Resistivity shows metallic behavior for all studied compounds with a residual resistivity ratio (RRR = ρ(300)/ρ(7)) of 5.7 for ScOs 2 , 7.1 for YOs 2 , and 6.4 for LuOs 2 , which is either attributable to the sample's polycrystalline nature and grain boundaries or intrinsic.One can observe that the resistivity drops abruptly to zero, confirming that a superconducting transition occurs in all investigated compounds, which was also confirmed by a large diamagnetic signal and a significant specific heat jump at T c .To obtain the upper critical field (see insets of Fig. 6a-c), H c2 (T), we measured the resistivity at various magnetic fields (μ 0 H = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.75, 1, 1.25, 1.5, 1.75, and 2 T for ScOs 2 ; μ 0 H = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.75, 1, 1.25, 1.5, and 1.75 T for YOs 2 , and μ 0 H = 0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.4, 0.6, 0.8, and 1 T for LuOs 2 ).As expected, with increasing magnetic field, the superconducting transition shifts to lower temperature.For ScOs 2 , a two-step transition is seen when the magnetic field is applied.The origin of this behavior is unknown, although, it might originate from the surface or filamentary superconductivity, with a higher critical field.For all investigated compounds, the upper critical field (μ 0 H c2 (0)) is determined by the temperature when the resistivity drops to 50% of the normal-state value and is plotted as a function of temperature in the Fig. 6d.According to the Ginzburg-Landau (GL) theory, the μ 0 H c2 value at 0 K can be estimated using the expression [42][43][44] : where t = T/T c and T c is a fitting parameter (transition temperature at zero magnetic field).The fitting line from the GL relation fairly well describes the experimental data for all compounds and one can obtain the values of μ 0 H c2 (0): 2.58(1) T for ScOs 2 , 2.23(2) T for YOs 2 , and 1.64(5) T for LuOs 2 .The paramagnetic limiting field (μ 0 H P ) is given by μ 0 H P = � 0 / √ 2µ B ( 0 is the zero-temperature superconducting gap, and µ B is the Bohr magneton), which can be expressed as μ 0 H P = 1.86 T c , yielding μ 0 H P ~ 9.9, 8.5, and 6.5 T for ScOs 2 , YOs 2 and LuOs 2 , respectively.In all cases the experimental values of µ 0 H c2 (0) are much smaller than the Pauli limiting field, suggesting that all compounds are the conventional type-II superconductors.Table S1 (SM) gathers μ 0 H c2 (0) values obtained from GL and WHH models.
To further understand the electronic structure of ScOs 2 , YOs 2 and LuOs 2 we performed electronic DOS and band-structure calculations (see Fig. 7).DFT calculations show that the DOS in the vicinity of the Fermi level in all three cases is dominated by Os 5d states, with electropositive elements (Sc, Y, Lu) acting mostly as electron donors.Inclusion of the spin-orbit coupling does not significantly affect the DOS(E F ). ( 4) The broad peak-like feature of DOS within 1 eV around the Fermi level stems from several weakly dispersive Os d-dominated bands, followed by a number of highly dispersive bands between ca.0 and − 6 eV (Fig. 8a,b).This is highly reminiscent of a generic tight-binding kagome band structure, as shown in Fig. 8c,d   Obviously, the band structure is much more complicated than e.g. in AV 3 Sb 5 kagome superconductors 46 or in CoSn 10 .This is due to: (1) the presence of two Os layers (kagome + trigonal) that are within the interacting range, resulting in significant hybridization of kagome bands, (2) fairly large unit cell consisting of 2 individual kagome planes, (3) trigonal distortion (breathing kagome), lifting the degeneracy at the K point and resulting in a gap between the Dirac bands.
Nevertheless, the set of weakly dispersive bands forming the DOS around the Fermi level can be traced back to the kagome flat band that is "bent" by interactions (Fig. 8c).

Summary
In summary, a detailed investigation of superconducting and normal state properties of ROs 2 (R = Sc, Y, Lu) hexagonal Laves compounds is presented.Magnetic susceptibility, electrical resistivity, and specific heat capacity measurements showed that ROs 2 (R = Sc, Y, Lu) are type-II superconductors with transition temperatures of T c = 5.36, 4.55, and 3.47 K, respectively.For ScOs 2 the superconducting transition is observed at a higher temperature than reported previously, but the lack of details in the previous reports does not allow us to speculate about why the T c is different from ours.The normalized specific heat jumps, ΔC/γT c , is calculated to be 1.51, 1.22, and 1.41 for ScOs 2 , YOs 2 , and LuOs 2 , evidencing the bulk nature of the superconductivity in these materials.Our band structure calculations showed that the dominating contribution to DOS(E F ) came from 5d states of Os atoms.The overall domination of Osmium 5d states suggests that ROs 2 compounds are Os 5d-band metals and that 5d electrons play the dominant role in superconductivity.
In addition to the superconducting properties, the studied compounds also exhibit a unique lattice structure.The Os atoms in these compounds form a trigonal breathing kagome lattice, a distorted variant of the hexagonal kagome.The band structure of the ROs 2 , although complicated, can be traced back to a generic kagome band model, modified by the breathing distortion and interlayer interactions.Os1 dominates the kagome-like bands between 0 and − 6 eV, while Os2 contributed mostly to a number of weakly dispersive bands between − 2 and − 3 eV.Panel (d) shows the tight binding band structure of a kagome system within 3 approximations: in the simplest case (thick gray lines) only nearest-neighbor interactions are considered and all the nearest neighbor tight binding hopping integrals are set to be equal (t 1, t 2 = − 1), resulting in a perfect p6m kagome.When the hexagonal symmetry is broken in breathing kagome (brown line; t 1 = − 1, t 2 = 0.9), the Dirac point at K is gapped, but the flat band remains intact.Inclusion of next-nearest neighbor interaction (orange line, t 1 = − 1, t 2 = 0.9, t 3 = 0.1) results in the flat band attaining some dispersion.The three tight-binding models are schematically drawn in panel (c).

Figure 1 .
Figure 1.(a) 3-orbital tight-binding band structure of a kagome network with nearest-neighbor interactions only, showing a pair of Dirac bands crossing (DP) at the K point of the Brillouin zone.Locations of BZ points are shown schematically in (b).Crystal structure of ROs 2 (c,d) shown as a stacking of Os1 breathing kagome layers separated by triangular planes of R and Os2.Note that purple and gray triangles highlighted in panel (c) are not equal in size, thus the symmetry of the 2D kagome network is reduced from p6m to p3m1.

Figure 5 .
Figure 5.The critical temperature, the Debye temperature, and the Sommerfeld coefficient versus atomic mass of the rare earth atom in ROs 2 compounds (R = Sc, Y and Lu).
. Kagome-like bands bear a strong contribution of Os1 d states, while the Os2 d mostly contributes to a set of weakly dispersive bands around − 2 to − 3 eV below the E F .

Figure 6 .
Figure 6.The electrical resistivity versus temperature measured in zero applied magnetic field for ScOs 2 (a), YOs 2 (b), and LuOs 2 (c).Insets show the superconducting transition under various magnetic fields.(d) The temperature dependence of the upper critical field of all compounds, determined from electrical resistivity measurements.

Figure 7 .
Figure 7. Band structure and electronic density of states for ScOs 2 (a,b), YOs 2 (c,d), and LuOs 2 (e,f).In all three compounds the DOS(E F ) is dominated by the contribution of Os 5d states.Besides the splitting of the completely occupied 4f band in LuOs 2 (peak ca.− to − 6 eV below the E F ), the difference between fully-(FR; blue line in panels b,d,f) and scalar-relativistic (SR; gray line) is rather small.

Figure 8 .
Figure 8. Band structure of ScOs 2 with Os1 d (a) and Os2 d (b) contribution highlighted (proportional to the color intensity).Os1 dominates the kagome-like bands between 0 and − 6 eV, while Os2 contributed mostly to a number of weakly dispersive bands between − 2 and − 3 eV.Panel (d) shows the tight binding band structure of a kagome system within 3 approximations: in the simplest case (thick gray lines) only nearest-neighbor interactions are considered and all the nearest neighbor tight binding hopping integrals are set to be equal (t 1, t 2 = − 1), resulting in a perfect p6m kagome.When the hexagonal symmetry is broken in breathing kagome (brown line; t 1 = − 1, t 2 = 0.9), the Dirac point at K is gapped, but the flat band remains intact.Inclusion of next-nearest neighbor interaction (orange line, t 1 = − 1, t 2 = 0.9, t 3 = 0.1) results in the flat band attaining some dispersion.The three tight-binding models are schematically drawn in panel (c).