Superconductivity in bismuth pyrochlore lattice compounds RbBi 2 and CsBi 2 : the role of relativistic effects

: Superconducting properties of two bismuthide intermetallic compounds, RbBi 2 and CsBi 2 , were studied by means of experimental measurements and ab initio calculations. We show that in both compounds the superconductivity emerges from the pyrochlore Bi lattice and its formation is heavily influenced by relativistic effects. Based on our analysis of the effect of spin-orbit coupling on the electron-phonon coupling we suggest a possible criterion for finding new superconducting materials by looking for structures featuring relativistically-stabilized hypervalent networks of heavy p -block elements.


INTRODUCTION
Bismuth is the heaviest stable element of the periodic table (the half-life of 209 Bi τ1/2 = 2.0 × 10 19 yr is nine orders of magnitude longer than the age of the universe, t = 1.4 × 10 10 yr).This property makes it and its compounds interesting for studying effects of strong spin orbit coupling (SOC) on physical properties of materials.Relativistic effects often result in Bi compounds behaving differently than isoelectronic species.For example, the most common cationic oxidation state of Bi is +3, with +5 compounds being generally unstable towards reduction.Bi 3+ cation is very often lone-pair active, leading to complicated coordination environments.Less known are the effects of relativistic phenomena on structural chemistry of Bi intermetallic compounds.Bi is known to form several stable hypervalent polyanions, such as Bi -square network found in LiBi and NaBi but not observed in the isoelectronic NaSb [1][2][3][4] .Similarly, CaBi2 and the recently reported SrBi2 both host a hypervalent Bi -square lattice 5,6 , which is missing in the isoelectronic CaSb2 7 .
Here we present the results of our experimental and computational studies on RbBi2 and CsBi2, concentrated on the effects of strong SOC on superconducting properties.We show that in both compounds superconductivity emerges from the Bi4 2-pyrochlore lattice and that the relativistic effects play a complex role in stability and properties of the two phases.We found that strong SOC is crucial for the stability of charged Bi pyrochlore network and discuss the implications of this fact for the search for new heavy element-bearing superconducting materials.Figure 1.Crystal structure of the cubic Laves phases ABi2 (A = K, Rb, Cs).Bi atoms (drawn in purple) form a vertex-sharing tetrahedral (pyrochlore) lattice with voids filled with alkali atoms (green).

MATERIALS AND METHODS
Single crystals of RbBi2 and CsBi2 were grown using a Bi selfflux method.Pieces of bismuth (Alfa Aesar, 99.99%) and rubidium (Alfa Aesar, 99.9%) were put in an alumina crucible inside an Ar-filled glovebox.Cesium was first heated above its melting point by warming it in hand and then poured dropwise into the crucible and covered with Bi pieces.Crucibles were then put in fused silica tubes along with plugs of quartz wool, and the tubes were subsequently evacuated, back-filled with Ar, and sealed without exposing alkaline metals to air.They were then heated to 550°C, kept at that temperature for 8 hours, and then slowly cooled (5°C per hour) to 310°C at which temperature the excess Bi was spun off with the aid of a centrifuge.Clusters of octahedrally and tetrahedrally-shaped crystals were obtained, with sizes ranging up to ca. 1 mm.The phase identity of the crystals was examined using powder x-ray diffraction (PXRD) measurements employing a Bruker D8 FOCUS diffractometer with a Cu Kα radiation source and a LynxEye detector.A few single crystals were crushed for the PXRD measurements inside an Ar-filled glovebox and mixed with an Apiezon-N grease or petroleum jelly.Since the powdered samples quickly oxidize, reacting with oxygen and moisture from air, the scanning time had to be reduced to 8 min.PXRD patterns were then analyzed by means of the LeBail method 23 using the FullProf software package 24 .PXRD patterns taken after a prolonged exposition to air show that both compounds decomposed completely yielding weakly crystalline elemental Bi and an amorphous product (most probably a hydrated alkali metal hydroxide), similarly to what was observed in CaBi2 9 .Heat capacity measurements were performed using Quantum Design PPMS system by the standard relaxation technique.Magnetic susceptibility measurements were carried out using the ACMS option of the PPMS.Multiple randomly-oriented single crystals were put in the standard straw sample holders.Sample preparation was carried out inside a glovebox to prevent oxidation.Ab initio calculations were performed using the Quantum Espresso (QE) package 25 employing ultrasoft Perdew-Burke-Ernzerhof (PBE) GGA 26 pseudopotentials 27 with valence configuration 4s 2 5s 1 4p 6 for Rubidium, 5s 2 6s 1 5p 6 for Cesium and 6s 2 6p 3 5d 10 for Bismuth, whereas the wavefunction and charge density energy cut-off's were set to values 50 Ry and 500 Ry, respectively.For the consideration of the influence of the relativistic effects, two types of calculations were performed: scalar-relativistic (neglecting SOC) and fully-relativistic (with SOC effect included for Bi).As expected and tested, inclusion of SOC for Rb and Cs does not change the results of calculations, thus the SOC was include only for Bi in final calculations.In the first step lattice constant was optimized using the Broyden-Fletcher-Shanno-Goldfarb (BFGS) method.The electronic structure was calculated at the grid of (24) 3 k-points.Phonon structure and properties of electron-phonon coupling (EPC) were calculated at grid of (6) 3 q-points.Chemical bonding was investigated by means of Crystal Orbital Hamilton Population analysis 28,29 using the Stuttgart TB-LMTO-ASA code 30,31 .Calculations were performed using the experimental cell parameter and (32) 3 k-point mesh and employed the Perdew-Wang GGA 32 exchange-correlation (xc) potential.Electronic structure of an isolated tetrahedral Bi4 cluster was calculated using the ReSpect 5.1.0relativistic DFT code 33 .Scalar-relativistic calculations were performed by means of 1-component Kohn-Sham method with one-electron relativistic corrections due to Douglas-Kroll-Hess of second order (1c KS-DKH2).Fully relativistic electronic structure was calculated using the four-component Dirac-Kohn-Sham Hamiltonian (4c DKS).The Dyall core-valence double-zeta basis set 34 and the PBE xc potential were used in both cases.Molecular orbitals obtained from the 1c KS-DKH2 calculations were visualized using the IboView program 35,36 .

RESULTS AND DISCUSSION
Room temperature PXRD patterns of crushed crystals are presented in Fig. S1 of the Supplementary Material.All the Bragg lines in the PXRD pattern can be indexed to cubic (space group Fd-3m, #227) unit cell of ABi2 and elemental Bi (both from left-over flux droplets and decomposition of material).Lattice parameters resulting from LeBail fits are in agreement with previous reports 37,38 .Details of crystal structures are gathered in Table 1.The Rb-Rb and Cs-Cs distances are 4.20 Å and 4.26 Å, significantly shorter than in their elemental form (4.89 Å and 5.29 Å, respectively 39,40 ), while Bi-Bi spacing is larger than in elemental Bi (3.47 Å vs. 3.07 Å 41 ) and in the Bi4 cluster (2.98 Å as obtained by ab initio structural relaxation).This highlights the electron transfer from the electropositive alkali metal to the Bi network.To characterize superconducting transitions of ABi2, dc magnetization was measured in the temperature interval T = 1.9 K -5 K.The plot of low-field (Hdc = 20 Oe) volume susceptibility (  ) vs. temperature is shown in Fig. 2(a,b).Sharp transition to a full Meissner state (  reaching the value of -4π after correcting for the demagnetization effect) was observed in both materials.Very small difference between zero-field cooled (ZFC) and field-cooled (FC) magnetization (not shown) is indicative of weak (or nonexistent) vortex pinning.Field-dependent magnetization below the critical temperature is shown in Fig. 2(c,d).In case of RbBi2 the shape of magnetization curve is consistent with a type-I superconducting transition with an influence of the demagnetization effect due to randomly-oriented crystalline grains 9,42 .In CsBi2 the shape of the M vs. H curve has a tail-like feature at higher fields, which is more consistent with type-II SC.Assuming a perfect diamagnetic linear response to the magnetic field, the Mv vs H data at the lowest temperature (T = 1.showing an anomaly at the superconducting critical temperature (enlarged in insets).The normalized heat capacity jump ΔCp/γTc is calculated using γ values estimated from a fit presented in Fig. S2 of the Supplementary Material.5][46][47] ).
The heat capacity measured through the superconducting transition in ABi2 is shown in Fig. 3. Bulk nature of the superconductivity is confirmed by large, sharp anomalies at Tc = 4.24 and 4.65 K for RbBi2 and CsBi2, respectively.It is worth noting that under applied field the jump at Tc becomes larger than at zero field in both compounds.Similar effect was observed eg. in KBi2 16 , ScGa3, LuGa3 44 , YbSb2 45 , ReAl6 46 , and LiPd2Ge 47 type-I superconductors.The Debye temperature (ΘD ) is calculated from the β specific heat coefficients using the relation: This yields ΘD = 127(1) for RbBi2 and 123(1) K for CsBi2, both values being slightly higher than for elemental Bi (ΘD = 112 K).
Taking ΘD and Tc values and assuming the value of the Coulomb pseudopotential parameter μ ⋆ = 0.13, the electron-phonon coupling parameter (λel-ph) can be calculated using the modified McMillan formula: For RbBi2 this gives λel-ph = 0.82 and for CsBi2 λel-ph = 0.87, both values being indicative of moderate coupling strength Table 2 summarizes the normal and superconducting state parameters derived from experimental measurements.Charge transfer from the Rb/Cs to Bi is evident from the electronic density of RbBi2 and CsBi2 (Fig. 4) and the Bader charge analysis yields Rb +0.71 (Bi2) -0.71 and Cs +0.64 (Bi2) -0.64 .This is consistent with electronegativity difference (0.82/0.79 vs. 2.02 for Rb/Cs and Bi in the Mulliken scale, respectively).The density distribution along the Bi-Bi bonds has a strongly delocalized, metallic character.Since charge density is concentrated on the Bi framework, both compounds can be viewed as pyrochlore metals.

Table 2 Normal and superconducting state parameters for RbBi2 and CsBi2
RbBi2 CsBi2 γ (mJ mol -1 K -2 ) 5.0(4) 4.9( 2) The electronic band structure and electronic density of states (DOS) for both compounds are shown in Fig. 5.The band structure around the Fermi level is similar in both cases.The Fermi level lies within a peak of DOS, thus even a small difference of band structure detailed below, lead to higher DOS at Fermi level in case of CsBi2 (DOS(EF) = 1.51 eV -1 /f.u) than RbBi2 (DOS(EF) = 1.31 eV -1 /f.u), which is correlated with a higher transition temperature Tc, as discussed below.The plot of DOS shows that the p-states of Bi contribute the majority of the total DOS around Fermi level, being only slightly hybridized with s states of Rb/Cs.Bi 6s states are positioned well below the EF due to (scalar) relativistic effects 48 .Thus, both the metallic state and superconductivity in ABi2 arises from interacting 6p states of Bi atoms.In the energy range from -2.5 to 3 eV (with respect to EF) the shape of DOS is similar to the one of elemental bismuth.Particularly the minimum at -0.8 eV is similar to the one at Fermi level of pure Bi.Around the EF the DOS is similar to the case of elemental Bi under pressure p = 2.7 GPa, at which a phase transition to a monoclinic structure is observed and Bi becomes superconducting below Tc = 3.9 K 49 .In this sense, the presence of Rb/Cs atoms impose a chemical pressure on the lattice of Bi atoms, leading to stronger 6p orbital overlap and metallic bonding of the Bi framework.
Fig. 6.Electronic band structure (a, c) and DOS (b, d) of RbBi2 and CsBi2 respectively, calculated in a scalar-(without SOC; red line) and fully relativistic approach (with SOC; black).Panels (e) and (f) show the band structure around the Brillouin zone center (Γ) for RbBi2.At the X point the degeneracy is symmetryprotected, so the inclusion of SOC results only in shifting the band energies (points marked with green rectangles), while at L-point they are additionally split (purple rectangles).
In Figure 6 the scalar and fully-relativistic band structure is shown in a narrow energy range around EF.In the case of RbBi2, when SOC not included, the Fermi level is positioned in the flat part of DOS, while with SOC the local maximum appears and Fermi level is situated on its slope.The parabolic bands around Γ point with a large (3 eV) energy gap around the Fermi level are not affected by SOC.At many points of k-space (for example between K and U or X and Γ points), mainly below tge Fermi level, the band crossings are removed by SOC.
Figure 7 shows the Fermi surface (FS) of RbBi2 and CsBi2 calculated with and without SOC.Since the differences in FS between the two compounds are negligible, we will discuss the RbBi2 case only.The only difference between the two compounds lies in the Fermi velocity which is slightly smaller in the case of CsBi2 in the [1 0 0] direction.This is consistent with a less dispersed character of bands around EF and a higher DOS at the Fermi level of this compound.In the scalar relativistic case there are four bands per spin crossing the EF, and thus four FS branches.First one has a dodecahedral shape with almost completely flat faces.It is similar to the Fermi surface of Polonium which was concluded to lead to strong nesting, as one q-vector (connecting two flat faces) is able to connect many points of these faces of the Fermi surface, leading to the Kohn anomaly in phonon spectrum [50][51][52] .The hybridization causes the presence of cone-like corners, which at E = 0.35 eV below the EF are visible as Dirac-like bands at the L point.Additionally, there are small pockets around K and U points.This part of Fermi surface is associated with the band which is approximately parabolic around the Γ point and is contributing to Dirac-like cones at other points.
The second branch contains many cylinders and is associated with a flat band at the K-U line.The third and fourth branch of Fermi surface consist of pockets centered at X and L points.
At the X point all four bands are degenerate to two points, both ca.0.5 eV below the Fermi level.The SOC shifts them leaving one below and one above the EF.This reduces the Fermi surface to just two branches: the dodecahedral and the cylindrical one, both centered at Γ.
The degeneracy of bands at X-point is symmetry-enforced and leads RbBi2 and CsBi2 to be classified topologically as ESFD (enforced semimetals with Fermi degeneracy), i.e. it is a material, which could be driven to a topological insulator state by symmetry breaking 53 .In contrast to the behavior at the X point, at L the bands are not only shifted (one of them is shifted above the Fermi level), but also split.where rs(t) is position of atom s in time t, with crystallographic position R, νs is its polarization vector associated with phonon mode ν with frequency ων and all is considered in the q-point of the space of wave vectors.With one of the part (real or imaginary) of polarization vector equal to zero, atoms vibrate along the line with frequency ω and direction given by the polarization vector.When both parts are non-zero, the atomic trajectory is an ellipsoid.If additionally both parts are equal, the ellipsoid becomes a circle.Such a case has been found in many q points of phonon dispersion of RbBi2.Two examples are shown in Fig. 8(f-g).At the K point, Rb atoms are moving circularly, while Bi atoms move along a line.At the W point, all atoms are moving along a circle/ellipse.These modes might be candidates for chiral phonons which arise due to symmetry of the crystal structure and have been recently found in ABi compounds, including RbBi and CsBi 58 (the occurrence of chiral phonons in ABi2 compoinds is discussed in more detail in ref. 59 ).
The phonon spectrum of CsBi2 is qualitatively different from RbBi2.When SOC is neglected, the modes are not only softened as it is in the case of RbBi2, but the spectrum is rich in imaginary frequencies, suggesting structural instability.Inclusion of the SOC removes the imaginary frequencies, thus the relativistic effect of SOC is crucial for the stability of the cubic Laves structure of CsBi2.
As visualized in Fig. 9(e), the soft mode is associated with a movement of Bi atoms toward the center of faces of tetrahedron.
As is highlighted in the electronic charge density plot (Fig. 4(b)), this point is characterized by large electronic charge density associated with metallic bonds of close-packed Bi tetrahedron.Thus this mode may be seen as movement of Bi ions, which are attracted to electronic charge, leading to lowering of frequency.
When SOC is included, the electronic structure changes and Bi atoms move in different direction and the imaginary frequencies disappear.As we checked, similar change of displacement vectors of atoms is associated with the disappeareance of softened modes of RbBi2.This behavior may suggest that the softening arises from a Kohn anomaly caused by an electronic effect of nesting of the Fermi surface, which would be an interesting matter of further studies.
The influence of electronic surrounding on phonons is highlighted by large phonon linewidth of discussed modes, shown in Fig. 8(a) and Fig. 9 where Ek,n and fk,n are energy and occupancy of the n-th band at k-th point, N is a number of unit cells.The phonon frequencies are the eigenvalues of the dynamical matrix normalized by atomic masses.In consequence, the higher value of the EPC matrix leads to lower frequency.As a consequence, the phonon softening and phonon linewidth are associated with each other, as both depend on the EPC matrix and geometry of FS.It is visible in Fig. 8(a) and 9(a), where the softened modes are characterized by large phonon linewidth.
It is worth noting that not all softened modes disappeared with SOC.Many of softened optical modes survived and their large phonon linewidth demonstrates their importance for EPC and thus for superconductivity.The influence of SOC on ABi2 family is similar to the case of other family of Bi-rich superconductors, ABi3, which crystallize in a simple cubic AuCu3-type structure, within which the all A-Bi and Bi-Bi distances are equal.In case of lighter SrBi3 the SOC only removes softened modes with large phonon linewidth, while in case of heavier BaBi3 it also stabilizes the structure.In addition, both Ba-and Bi-derived modes of BaBi3 are lower than in the case of SrBi3, leading to nearly 30% larger value of EPC constant, which is equal to 1.43 and 1.11 respectively 12 .In the ABi2 family the substitution of lighter Rb with heavier Cs leads to only about 10% larger EPC constant.Modes associated with Cs atoms have lower frequency than those associated with Rb (average frequency of Cs is 9% lower than that of Rb, see Table 2).This is expected since Cs is over 55% heavier than Rb.The average frequency of Bi is of 5% lower in CsBi2 than in RbBi2.Surprisingly, not all modes of Bi are lowered leading to a completely different shape of phonon DOS, where the peak around 1.3 THz is spread out.
In order to better describe the difference between the phonon structures of CsBi2 and RbBi2, dispersion relations around the Г point are plotted and the modes associated with similar movement of atoms are marked with the same color in Fig. S3 of the Supplementary Material.Additionally, modes associated with the movement of Bi atoms are visualized in Fig. S3(c-f) of the Supplementary Material and force constants between atoms are shown in Fig. S3(g).The modes associated with a movement of both Bi (along Bi-Bi bonding) and Cs/Rb (Fig. S3(d)) are lowered from 1.31 in case of RbBi2 to 1.03 THz in CsBi2, while the frequency of modes associated with a movement of Bi towards the empty space around Cs/Rb (Fig. S3(c,e)) are increased from 1.22 THz and 1.32 THz in RbBi2 to 1.36 THz in CsBi2.In both compounds the phonon linewidths are large only in case of Bi modes.Since the FS is also dominated by 6p-states of Bi, the superconductivity in ABi2 is driven by the Bi pyrochlore lattice, as the electron-phonon coupling in both electronic and phonon ways is almost exclusively contributed by Bi.
This may be caused by a difference in the unit cell size, which is larger in the case of CsBi2, thus the Bi-Bi metallic bonding is slightly weaker leading to smaller force constants of Bi-Bi pairs (see Fig. S4(h) of the Supplementary Material) and lower frequencies of modes associated with a movement of Bi along Bi-Bi bonds.For the same reason, the highest optical mode associated with a movement of Bi toward the center of tetrahedron (Fig. S4(f)) is lowered from 2.37 THz in case of RbBi2 to 1.78 THz in CsBi2.It is similar to the case of soft mode shown in Fig. 10(e) and explains why the effect of softening is stronger in case of CsBi2.On the other hand, the force constants of Bi-Cs pairs are larger than of Bi-Rb pairs, leading to a higher frequencies of modes associated with a movements of Bi toward the space around Cs.These changes show that the substitution of Rb by Cs not only lowers the frequency due to a larger mass, but also influences the size of the unit cell and modifies the bonding of Bi network.
On the basis of the phonon linewidth the Eliashberg function is calculated: The Eliashberg function is effectively the phonon DOS weighted by a phonon linewidth divided by phonon energy and density of electronic states at Fermi level.
Eliashberg function of RbBi2, calculated without SOC, is the highest at low frequencies (up to 1.5 THz) due to the presence of softened acoustic modes and low-frequency optical modes.When SOC is included, the Eliashberg function becomes similar to phonon DOS, which means that electron-phonon coupling only weakly depends on phonon frequency.The largest contribution to Eliashberg function comes from highly-degenerate and partially flat Bi modes in the narrow frequency range.
Table 2.The summary of calculated properties of RbBi2 and CsBi2 compared to experimental results (marked with label expt.).The density of states is given in units of eV -1 /f.u., both mean frequency  and logarithmic frequency   in units of THz and critical temperatures in K.In a monoatomic case this integral corresponds to Hopfield parameter.The calculated value of the integral is 0.90 and 1.26 for RbBi2 and CsBi2, respectively.This suggests that, in addition to the lowering of frequency, the slight changes of the electronic structure resulting in lowering the Fermi velocity and increased DOS(EF) strengthen the EPC.
On the basis of calculated EPC constant the critical temperature can be calculated with help of Allen-Dynes equation where ) is a logarithmic average of frequency.
In the case of RbBi2  = 0.81 and   = 1.74 THz (equivalent to 83.5 K) leading to Tc = 3.23 K, reasonably close to the measured value 4.24 K.In case of CsBi2 EPC constant in larger ( = 0.92) leading to higher Tc = 3.85 K despite of lower logarithmic frequency.The estimated critical temperature is reasonably close to the experimental value Tc = 4.65 K.Both experimental and calculated values lead to a conclusion, that CsBi2 has higher critical temperature than RbBi2 due to slightly stronger electron-phonon coupling.
In summary, the structural arrangement of Bi atoms into a pyrochlore network of Bi tetrahedra intercalated with alkali metal atoms, results in a relatively high electron-phonon coupling constant.This emerges as a combination of both aforementioned factors: the large electronic density within the closepacked Bi atoms in Bi4 tetrahedra allows for the strong interaction of phonon modes with electrons (large phonon linewidths), whereas the tetrahedra themselves are relatively loosely packed within the unit cell with a large amount of "free space" filled only by Rb/Cs atoms.This, in combination with high atomic mass of Bi, results in relatively low Bi phonon frequencies, enhancing the EPC.Recently we discussed a similar situation in isostructural Laves phase superconductors: SrRh2 and SrIr2, where the pyrochlore lattice or Rh/Ir was found to be responsible for the emergence of strong-coupling superconductivity.
On the other hand RbBi2 and CsBi2 are in contrast with CaBi2, which crystallizes in an orthorhombic structure that can be viewed in a distorted Laves phase structure, in which tetrahedrons are distorted to Bi-Bi and Ca-Bi planes separated from each other.Here the phonon linewidths are smaller and average Bi frequencies are higher, leading to the EPC constant λep = 0.54, much lower than λep = 0.82 and λep = 0.87 for RbBi2 and CsBi2, respectively, despite the value of I = 1.17 lying between RbBi2 and CsBi2 cases.This results in a lower critical temperature (Tc = 2.0 K for CaBi2) 9,10 .It is worth mentioning that the average Bi frequency in elemental Bi, the phonon structure of which is presented in Fig. S8 of the Supplementary Material and summarized in Table 2, is also higher than in the studied Laves phases.Since Bi is a semimetal, the DOS(EF) and the value of I are small, leading to a low value of the EPC parameter λep = 0.23 and very low predicted critical temperature Tc = 0.7 mK in a fair agreement with the experimental Tc = 5 mK 61 .
Electron population analysis on RbBi2 shows that the DOS(EF) is contributed mostly by electrons occupying nearest-neighbor Bi-Bi antibonding states (see Fig. S4 of the Supplementary Material).A hypothetical neutral Bi pyrochlore lattice would have its EF lying close to a pseudogap.The two electrons donated by Rb increase the antibonding population, destabilizing the lattice.Unfortunately, with the currently available computational tools (either direct in case of local basis set or indirect -via projection of plane-wave wavefunctions to a local orbitals set [62][63][64][65] ) the bonding population analysis can only be performed in a scalar-relativistic case.While for bonding of lighter elements SR approximation is reasonably applicable, the chemical bonding in 6 th and 7 th period elements is heavily influenced by SOC effects 66,67 .A possible solution for reliable population analysis of compounds of the heaviest elements is via the localized Wannier spinors 68 .
The calculated electronic structure of the Bi4 cluster (shown in Fig. 11) are in general consistent with the previous results by Lohr and Pyykkö obtained by means of relativistically-parametrized extended Hückel method 69 , although the ordering of virtual orbitals is slightly different.The stabilizing effect of SOC on ABi2 can be qualitatively understood by considering the electronic structure of a Td Bi4 cluster.Firstly, in a heavy element, such as Bi scalar-relativistic effects result in a complete energetic separation of 6s and 6p orbitals 48 .Thus, only the p orbitals participate in the bonding.Secondly, the SOC splits the 6p atomic level into p1/2 and p3/2 separated by ca. 2 eV in a free atom.In the scalar-relativistic approximation the neutral Bi4 cluster is a closed-shell system with four electrons occupying the doubly-degenerate e HOMO level.With 2 additional electrons donated by electropositive species, the Bi4 2-is first-order Jahn-Teller (JT) active due to two electrons occupying a triply degenerate t1 level.SOC splits the 6p-derived levels (6s-derived lower t2 level is also weakly split due to small admixture of the p character), lifting the triple degeneracy of t1 and t2.The two additional electrons now occupy a e' (Γ6) level, quasi-degenerate (ΔE ≈ 0.02 eV) with the next e'' (Γ7) leading to structural stabilization.However, the splitting of the e' and the two u' (Γ8) levels is relatively small (ΔE ≈ 0.9 and 1.2 eV, respectively), resulting in a possibility of a second-order JT instability, as pointed out by Lohr and Pyykkö 69 .While the electronic structure of the crystalline ABi2 compound is much more complex than of isolated cluster, the stabilizing effects of SOC similarly stem from the lifted p orbital degeneracy while the weakened but persistent mode softening can be considered an effect of a second-order JT (SOJT) effect.
Fig. 11 Molecular orbitals of a tetrahedral Bi4 cluster.Note that for clarity of the picture the energy scale (right) is divided into two parts, as the separation between lower t2 and higher a1 levels is on the order of 8 eV, while for comparison the t2-e splitting is ca.0.5 eV.
This observation suggest a criterion for finding new superconductors: one may look for compounds stabilized by strong SOC, which should show relatively strong electron-phonon coupling while avoiding crystal structure distortion.
Examples of such SOC stabilized superconductors can be found among Sb-, Bi-and Pb-rich intermetallic phases featuring hypervalent polyanions, listed in Table 3.
KBi2, RbBi2, CsBi2 belong to a large family of cubic Laves (C-15) superconductors.The SuperCon database lists 192 C-15 superconductors, belonging to 16 different chemical systems, with critical temperatures ranging from 0.7 to 10.7 K.However, the three Bi-based compounds distinguish themselves from the whole family in a number of ways.Firstly, besides LaAl2 (Tc ≈ 3 K 75 ) and LaMg2 (Tc ≈ 1 K ), they are the only main-group element-based Laves superconductors.Secondly, we have shown that in RbBi2 and CsBi2 systems the superconductivity clearly emerges from the electron-rich Bi pyrochlore lattice, with Rb/Cs acting predominantly as electron donors to the Bi lattice.Such clear picture is not possible in case of transition metal based systems, where the bonding between atoms occupying the two crystal sites is expected to be much more metallic.Lastly, Bi is the heaviest stable element (Z = 83) and the relativistic effects on electronic properties in KBi2, RbBi2, and CsBi2 are the strongest of all of the Laves superconductors, even when compared with Ir-and Au-based systems, such as (La/Lu)Ir2 76 and (Pb,Bi)Au2 77 , since the SOC effect on 5d valnce electrons is weaker than on 6p shell.

CONCLUSIONS
We have analyzed the electronic and vibrational properties of two Bi-based cubic Laves phases: RbBi2 and CsBi2.Our results show that the occurrence of superconductivity in these two compounds and an isostructural KBi2 stems almost exclusively from the heavy element pyrochlore lattice.Both the stability of the phases and their electronic and vibrational properties are heavily affected by relativistic effects.The spin-orbit coupling is crucial for the dynamic stability of CsBi2, and to a lesser extent RbBi2.The stability enhancement can be explained by considering an isolated Bi4 cluster and the effects of SOC on its electronic structure.We postulate that crystal structures based on relativistically-stabilized hypervalent networks of heavy elements are interesting candidates for finding new superconductors.

ASSOCIATED CONTENT
Supporting Information.PXRD patterns for RbBi2 and CsBi2 samples.Crystallographic unit cell parameters for RbBi2 and CsBi2.Low temperature heat capacity.Details of phonon dispersion relations and visualization of Bi-associated modes.Crystal orbital Hamilton population (COHP) plot for RbBi2.This material is available free of charge via the Internet at http://pubs.acs.org.

Fig. 2 (
Fig. 2 (a,b) Temperature-dependent magnetic susceptibility of RbBi2 and CsBi2.A sharp transition to Meissner state is seen at the Tc.Susceptibility data was corrected for the demagnetization factor (see the text).(c,d) Field-dependent magnetization curves showing a typical type-I character of superconductivity for RbBi2 and a probable type-II character for CsBi2.

Fig. 3
Fig. 3 Low temperature heat capacity of (a) RbBi2 and (b) CsBi2,showing an anomaly at the superconducting critical temperature (enlarged in insets).The normalized heat capacity jump ΔCp/γTc is calculated using γ values estimated from a fit presented in Fig.S2of the Supplementary Material.Note that in both cases the jump under an applied magnetic field is found to be larger than at zero field, which is a typical behavior for type-I superconductors (see eg. refs.16,[44][45][46][47] ).

Fig. 4 .
Fig. 4. Charge density of supercell of RbBi2 plotted in (110) plane (a) and (111) plane (b); the charge density in (111) plane is similar to 2D Kagome lattice marked, however there are many of such planes in unit cell (c) resulting in 3D character of electronic bonds.Fig.S3of the Supplementary Material presents the low-temperature Cp/T vs. T 2 plots for both materials.Electronic (γ) and vibrational (β) heat capacity contributions were extracted by fitting the data with relation Cp/T = γ + βT 2 .The fit yields very similar γ values (5.0 and 4.9 mJ mol -1 K -2 ) for both materials.The normalized heat capacity jumps at Tc estimated by using the equal entropy construction method (see insets of Fig5(a,b)), are ΔCp/γTc = 2.43 for RbBi2 and ΔCp/γTc = 2.82 for CsBi2, both well above the BCS limit for a weakly-coupled superconductor (1.43).The Debye temperature (ΘD ) is calculated from the β specific heat coefficients using the relation:

Fig. 5 .
Fig. 5. Fully relativistic electronic band structure (a, c) and DOS (b, d) of RbBi2 and CsBi2 respectively.Atomic contributions to DOS are marked with green (Bi) and red (Rb/Cs) lines.The plot of DOS shows that the p-states of Bi contribute the majority of the total DOS around Fermi level, being only slightly hybridized with s states of Rb/Cs.Bi 6s states are positioned well below the EF due to (scalar) relativistic effects48 .Thus, both the metallic state and superconductivity in ABi2 arises from interacting 6p states of Bi atoms.In the energy range from -2.5 to 3 eV (with respect to EF) the shape of DOS is similar to the one of elemental bismuth.Particularly the minimum at -0.8 eV is similar to the one at Fermi level of pure Bi.Around

Fig. 7 .
Fig. 7. Fermi surface of RbBi2 (a, b) and CsBi2 (d, e) calculated without SOC (a, d) and with SOC (b, e), colored according to Fermi velocity.The Brillouin zone with high symmetry points is shown in panel (c).In the scalar relativistic case there are four bands per spin crossing the EF, and thus four FS branches.First one has a dodecahedral shape with almost completely flat faces.It is similar to the Fermi surface of Polonium which was concluded to lead to strong nesting, as one q-vector (connecting two flat faces) is able to connect many points of these faces of the Fermi surface, leading to the Kohn anomaly in phonon spectrum[50][51][52] .The hybridization causes the presence of cone-like corners, which at E = 0.35 eV below the EF are visible as Dirac-like bands at the L point.Additionally, there are small pockets around K and U points.This part of Fermi surface is associated with the band which is approximately parabolic around the Γ point and is contributing to Dirac-like cones at other points.The second branch contains many cylinders and is associated with a flat band at the K-U line.The third and fourth branch of Fermi surface consist of pockets centered at X and L points.At the X point all four bands are degenerate to two points, both ca.0.5 eV below the Fermi level.The SOC shifts them leaving one below and one above the EF.This reduces the Fermi surface to just two branches: the dodecahedral and the cylindrical one, both centered at Γ.

Fig. 8 .
Fig. 8. Panels (a, c): Phonon dispersion relation of RbBi2 with phonon linewidth proportional to the thickness of the purple line.Panels (b,d) show the phonon DOS (dashed black line) with atomic contribution marked in green (Bi) and red (Rb) and the Eliashberg function (purple line) normalized to 3N (N being a number of atoms per unit cell), calculated without SOC (a, b) and with SOC (c, d).Despite the vanishing FS pockets, the DOS(EF) or RbBi2 is almost unchanged by SOC (a negligible decrease from 1.35 to 1.31 eV -1 /f.u. is seen).The redction of the number of FS branches is compensate by the reduction of the Fermi velocity (see Fig.9a-b), as lower velocity leads to a higher density of states.Phonon structures of RbBi2 and CsBi2 are shown in Fig.8 and 9.In contrast to the electronic structure, the phonon dispersion curves (Fig.8 (a,c) and Fig.9 (a,c)) and density of states (Fig.8 (b,d) and Fig.9 (b,d)) of the two compounds differ significantly.The SOC has a major influence on phonons.When it is not taken into account, the vibrational structure of RbBi2 is rich in softened modes, suggesting the instability of the structure.When SOC is included, the softening of modes disappears, indicating a stabilization by relativistic effects.The phonon spectrum contains 18 modes, as the primitive cell of RbBi2 contains 6 atoms (2 formula units).Phonon modes coming from Bi and Rb atomic vibrations are not strongly separated from each other as it is eg. in the case of LiBi 4 or CaBi210 .Three acoustic modes and low-frequency optic modes (up to ω = 1.7 THz) are mainly associated with Bi atoms vibrations, while high-frequency modes (in the range ω = 1.7-2.5 THz) come from Rb. Transverse acoustic (TA) modes are degenerate along the X-Γ-L path and split in the xy plane along W-Γ-K.The optical modes coming from Bi lie within a narrow frequency range ω = 1.1 to 1.4 THz and a part of them is almost

Fig. 9 .
Fig. 9. Phonon dispersion relation of CsBi2 together with phonon linewidth (a, c) and its phonon DOS together with atomic contribution to phonon DOS and Eliashberg function, calculated w/o SOC (a, b) and with SOC (c, d).The color scheme is the same as in Fig.8.The soft mode, which appears along Г -W in the scalar relativistic case, is marked in real space with arrows (yellow, red, green, and purple spheres are Bi, light and dark blue are Cs).In both compounds the phonon linewidths are large only in case of Bi modes.Since the FS is also dominated by 6p-states of Bi, the superconductivity in ABi2 is driven by the Bi pyrochlore lattice, as the electron-phonon coupling in both electronic and phonon ways is almost exclusively contributed by Bi.
is a frequency of -th mode with a wave vector  (equal to a difference of wave vectors of two electronic states  ,+ ,   ) with a polarization vector ̂  , associated with a displacement  ̂ of -th atom of a mass   .Dirac delta functions ensure that only the electronic states near the FS are included in calculation.
× (  −   )( +, −   ) It is essentially proportional to phonon linewidth divided by frequency squared, thus the low frequency modes contribute more than higher ones.Obtained values are shown in Table2.When SOC is not included, the EPC constant of RbBi2 λep = 2.06, is more than twice larger than the experimental value.The cumulative EPC constant shown in Fig.10clearly shows that such a high value is caused by softened modes, which are characterized by low frequency and large phonon linewidth.When SOC is included, the softening disappears and EPC constant drops to λep = 0.81 (in excellent agreement with experimental value λep = 0.82), and is mostly delivered by acoustic and optical Bi modes, as the phonon linewidth is.The calculated EPC constant of CsBi2 λep = 0.92 is also in a good agreement with the experimental value λep = 0.87.It is slightly larger than in case of RbBi2 due to a shift of part of modes to lower frequencies and slight changes in band structure.The impact of the electronic structure on EPC can be measured using the integral: