Superconductivity in LiGa2Ir Heusler type compound with VEC = 16

Polycrystalline LiGa2Ir has been prepared by a solid state reaction method. A Rietveld refinement of powder x-ray diffraction data confirms a previously reported Heusler-type crystal structure (space group Fm-3m, No. 225) with lattice parameter a = 6.0322(1) Å. The normal and superconducting state properties were studied by magnetic susceptibility, heat capacity, and electrical resistivity techniques. A bulk superconductivity with Tc = 2.94 K was confirmed by detailed heat capacity studies. The measurements indicate that LiGa2Ir is a weak-coupling type-II superconductor (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\uplambda }$$\end{document}λe–p = 0.57, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta }$$\end{document}ΔC/\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\upgamma }$$\end{document}γTc = 1.4). Electronic structure, lattice dynamics, and the electron–phonon interaction are studied from first principles calculations. Ir and two Ga atoms equally contribute to the Fermi surface with a minor contribution from Li. The phonon spectrum contains separated high frequency Li modes, which are seen clearly as an Einstein-like contribution in the specific heat. The calculated electron–phonon coupling constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\uplambda }$$\end{document}λe–p = 0.68 confirms the electron–phonon mechanism for the superconductivity. LiGa2Ir and recently reported isoelectronic LiGa2Rh are the only two known representatives of the Heusler superconductors with the valence electron count VEC = 16.


Results and discussion
Experimental studies. Figure 1 presents the pXRD pattern and results of the Rietveld analysis for the synthesized LiGa 2 Ir. The pXRD analysis indicates an excellent quality of the examined sample and the refinement confirms that the compound crystallizes in the cubic L2 1 crystal structure (space group Fm-3m, No. 225). A difference plot (between experimental and fitted data) and the Bragg positions are also shown in Fig. 1. The refined lattice parameter a = 6.0322(1) Å is in a good agreement with the previously reported for LiGa 2 Ir 36,37 and slightly larger than refined for LiGa 2 Rh (a = 5.9997(8) Å) 26 .
In the analysis, the atomic positions were fixed by symmetry. The refinement of isotropic thermal displacement parameters B iso yielded: 1.7(6) Å 2 , 3.18(7) Å 2 and 3.16(6) Å 2 for Li (at the 4b site), Ir (4a), and Ga (8c), respectively. The conventional, background-corrected Rietveld reliability factors for the refinement are R p = 10.9%, R wp = 13.2%, R exp = 7.27%, and χ 2 = 3.32. A LeBail refinement leads to only slightly lower χ 2 value (~ 3.3), indicating that the discrepancy between model and observed intensities is dominated by modelling of peak shapes and background and not by structure parameters.
The superconducting transition of LiGa 2 Ir, was first investigated by the temperature-dependent volume magnetic susceptibility, defined as χ = M/H (M -magnetization, H-applied magnetic field), with zero-field-cooled (ZFC) and field-cooled (FC) measurement modes under H = 10 Oe. As depicted in the main panel of Fig. 2(a), the measured magnetization was multiplied by 4π and corrected for the demagnetization effect − 4πχ V = 1/(1 − N), where the demagnetizing factor N is taken to be 0.58 (estimated from the M V (H) fit discussed later). A Meissner transition, corresponding to the onset of superconductivity, appears at T c = 2.95 K, where the superconducting transition temperature (T c ) was estimated as the point at which the line set by the steepest slope of the magnetization in the ZFC data set intersects with the extrapolation of the normal-state magnetic susceptibility 38 . The shielding volume fraction at 1.95 K is ~ 100%, confirming that the sample is a bulk superconductor. Compared with the ZFC data, the observed FC diamagnetic signal is much weaker, which is likely caused by the porous www.nature.com/scientificreports/ nature of the polycrystalline sample. In the inset of Fig. 2(a), the low-field regions of the isothermal dc magnetization curves measured at various temperatures ranging from 1.9 K to 2.8 K are presented. For each temperature, the experimental data obtained in low magnetic fields were fitted using a linear relation M fit = − aH, which is suitable for a perfect shielding effect. Assuming that the initial response to the magnetic field is completely diamagnetic, the demagnetization factor N = 0.58 was found. The N value is reasonably consistent with the expected (calculated) N z value derived for a circular cylinder sample with the height to radius ratio of approx. 0.5 39 . At each temperature, the value of the lower critical field H * c1 is defined as the point of deviation of the data curve from the pure Meissner response. At each T, this point was precisely calculated following the methodology described elsewhere 40 . The estimated H * c1 values are shown with the corresponding temperatures in the main panel of Fig. 2(b). At T = 1.9 K the H * c1 is 68 Oe and decreases monotonically with an increase in temperature, to 10 Oe at T = 2.8 K. The data points were analyzed with the equation: where H * c1 (0) is the critical field at 0 K and T c is the superconducting critical temperature. Our experimental data is well described with the above formula and the fit (red solid line) yields H * c1 (0) = 113(3) Oe and T c = 3.03(3) K. Taking into account the demagnetization factor (N = 0.58) derived above, the lower critical field value H c1 = H * c1 /(1 − N) = 268 Oe. The obtained value is slightly higher than these reported for the other full-Heuslers compounds 24,26 . The inset in Fig. 2(b) presents the full magnetization loop versus applied magnetic field measured in the superconducting state at T = 1.9 K. It is evident that LiGa 2 Ir exhibits conventional type-II superconductivity.
The measurement of the heat capacity is reliable evidence of the presence of bulk superconductivity. Figure 3(a) shows in more detail the superconducting transition for LiGa 2 Ir plotted as C p /T versus T under µ 0 H = 0 T. The sharp anomaly visible in the specific heat data confirms bulk superconductivity and suggests a good quality of the sample. From the graphical equal-area construction, represented by green solid lines, the T c is estimated to be 2.94 K, which is consistent, with the value determined by magnetic measurements. The specific heat jump at T c is found to be about C/T c = 7.7 mJ mol −1 K −2 . Figure 3(b) shows the heat capacity data plotted as C p /T versus T 2 , under µ 0 H = 0.15 T. In the normal state, the raw data can be fitted using the expression C p /T = γ + βT 2 + δT 4 , where the first term is the electronic specific heat coefficient and the second and third terms are attributed to the lattice contributions to the heat capacity (the δT 5 term in the heat capacity was added after analyzing the computed phonon spectrum, discussed below). The extrapolation, expressed as the red solid line, gives γ = 5.5(1) mJ mol −1 K −2 , β = 0.366(1) mJ mol −1 K −4 and δ = 0.0052(3) mJ mol −1 K −6 . In a Debye model for the phonon contribution, the β coefficient is related to the Debye temperature Θ D through � D = 12π 4 5β nR 1/3 , where R = 8.31 J mol −1 K −1 and n = 4 for LiGa 2 Ir. The resulting value of Θ D is 277(1) K, which is significantly lower than the Debye temperature for Rh analog LiGa 2 Rh (Θ D = 320 K 26 ). Using the previously obtained specific heat jump at T c and the Sommerfeld coefficient (γ = 5.5(1) mJ mol −1 K −2 ), the ratio ΔC/γT c = 1.40 can be calculated. The estimated value is close to the BCS value of 1.43, suggesting that LiGa 2 Ir is a weakly coupled superconductor and is close to that of LiGa 2 Rh (ΔC/γT c = 1.48 26 ).
(1) Figure 1. Powder X-ray diffraction pattern of LiGa 2 Ir (red points) together with the Rietveld refinement profile (black solid line where μ * is the Coulomb pseudopotential factor, usually assumed to be μ * = 0.13 for conventional intermetallic superconductors 24,42 . Taking T c = 2.94 K and Θ D = 277 K, the calculated λ e−p = 0.57, implying that LiGa 2 Ir can be classified as a weakly coupled BCS superconductor. The temperature-dependent electrical resistivity for LiGa 2 Ir, ρ(T), is depicted in the main panel of Fig. 4(a), in the range of temperature 1.8 − 300 K, without the application of an external magnetic field. In the normal state, the resistivity reveals a metallic behavior (dρ/dT > 0) with rather small residual resistivity ratio (RRR = ρ(300 K)/ρ(5 K) = 2.1). That characteristic can be attributed to the polycrystalline nature of the sample investigated that probably contained many macroscopic defects. The value obtained is comparable to those reported for full-Heusler compounds 24 . The resistivity undergoes a sudden drop at 2.96 K, that perfectly agrees with the T c obtained from magnetic and heat capacity measurements. The inset of Fig. 4(a) emphasizes the low-temperature resistivity under various magnetic fields from 0 to 1400 Oe. As expected, the superconducting transition becomes slightly broader and the T c shifts to a lower temperature as the applied magnetic field is increased. Using the criterion that the point with 50% normal state resistivity (ρ 0 ) is the transition temperature, we determined the upper critical field μ 0 H c2 (T) for LiGa 2 Ir at various temperatures ( Fig. 4(b)). The solid line is a fit to the Ginzburg-Landau equation 43 : www.nature.com/scientificreports/ where t = T/T c and T c is the transition temperature at zero magnetic field. Equation (3) describes the experimental data well, and yields µ 0 H c2 (0) = 0.31(1) T. The Pauli limiting field within the BCS theory for a weak electron-phonon coupling 44,45 gives H p c2 (0) = 1.85 T c = 5.4 T, which is eighteen times larger than estimated upper critical field value for LiGa 2 Ir. An identical value of µ 0 H c2 (0) has been reported for isostructural and isoelectronic LiGa 2 Rh superconductor 26 . Consequently, assuming that the upper critical field is purely orbital, using the GL formula where Ф 0 = hc/2e is the flux quantum, the superconducting coherence length is calculated to be ξ GL = 322 Å. Similarly, from the relation H c1 = � 0 4π 2 GL ln GL ξ GL , a superconducting penetration depth λ GL (0) = 443 Å is found for LiGa 2 Ir. The GL parameter κ GL = λ GL /ξ GL can then be estimated as κ GL = 1.38 > 1/ √ 2 , confirming the type-II nature of the superconductivity. Finally, the thermodynamic critical field can be obtained from κ GL , H c1 and H c2 using the formula H c1 H c2 = H 2 c ln κ GL . The resulting value of H c is 1633 Oe (µ 0 H c = 0.16 T). With the Sommerfeld coefficient γ and the electron-phonon coupling parameter λ e−p known, the non-interacting density of states at the Fermi level N(E F ) can be calculated using the formula: where k B is the Boltzmann constant. For LiGa 2 Ir, N(E F ) is estimated to be 1.49 states eV −1 per formula unit (f.u.). Superconducting and normal parameters are gathered in Table 1.
Pressure dependence of the T c for LiGa 2 Ir is shown in Fig. 5. LiGa 2 Ir shows an exceptionally low pressure coefficient compared to other Heusler compounds for which high-pressure studies were reported 24,46,47 . It is however worth noting that all of the compounds reported to date are Pd-based systems with a valence electron count of    47 and HfPd 2 Al 24 the suppression of T c by high pressure likely stems from the stiffening of the lattice, yet in the case of LiGa 2 Ir the effect is much weaker.
Theoretical calculations. First, the unit cell was relaxed with the Broyden-Fletcher-Goldfarb-Shanno algorithm starting from the experimental lattice constant. Atomic positions were fixed by the symmetry constraints of the full-Heusler structure. The relaxation was repeated including spin-orbit coupling (SOC) because the effect might be important due to the presence of a heavy Ir atom. For calculations with SOC, the scalarrelativistic pseudopotential of Ga and Ir were replaced with the full-relativistic ones. The obtained relaxed lattice constants are in a very good agreement with the experimental one, and SOC was found to have a negligible effect on the lattice constant (Table 2). Figure 6 shows the electronic dispersion relations and total DOS. Three bands cross the Fermi level forming three Fermi surface sheets visualized using XCrysDen 48 in Fig. 7. SOC has a small effect on the electronic bands and DOS near the Fermi energy (E F ), however much stronger SOC effects are seen for electronic states with energies below E F , like an anticrossing of bands in the Ŵ -K direction around −1.5 eV. The total and partial DOS for each atom in LiGa 2 Ir are shown in Fig. 8. States near E F are built mainly from Ga-4p and Ir-5d orbitals, whereas the contribution to the DOS(E F ) from Li is negligible. Interestingly, the Fermi level is located in the local minimum of the DOS(E), formed from a superposition of a decreasing DOS of Ir and increasing DOS of Ga. From the calculated DOS(E F ) values, slightly increased in the relativistic case (see Table 3) , the bandstructure value of the Sommerfeld electronic specific heat coefficient is calculated, γ band = π 2 3 k 2 B DOS(E F ) , and used to estimate the electron-phonon coupling parameter as γ expt = γ band 1 + γ . This results in γ = 0.48 , a slight underestimate when comparing to e−p = 0.57 obtained from T c using McMillan's formula.
Phonon dispersion relations ω(q) and phonon density of states F(ω ) of LiGa 2 Ir with atomic contributions are shown in Fig. 9. Modes associated with different atoms are well separated because of large differences in mass (M Li = 6.94u, M Ga = 69.72u, M Ir = 192.22u). Three optic modes of Li form an Einstein-like peak in F(ω ) around 11 THz, a much higher frequency than the Ga, and Ir-dominated parts of the phonon spectrum. The acoustic part is mostly contributed by the heaviest Ir vibrations. Although SOC had a small effect on the electronic DOS(E F ), it affected phonons, slightly pushing the Ir branches towards higher frequencies and visibly lowering the Ga and Li frequencies. This is shown in Table 4, where the average phonon frequencies are collected. The global effect of SOC is a small decrease in the average frequency, from 5.77 THz to 5.75 THz.  www.nature.com/scientificreports/ Looking again at the phonon dispersion relations, we notice the presence of a small dip in the first acoustic branch in the Γ-K direction. Such an anomaly is frequent in Heusler compounds, and was already reported in HfPd 2 Al 49 , LiGa 2 Rh 50 or LiPd 2 X (X = Si, Ge, Sn) 51 , where in the last case it evolved into a soft mode with an imaginary frequency for X = Ge and Sn. As our ongoing investigation of the LiPd 2 Ge case showed 51 , this may be related to the anharmonic features of the crystal potential.
The theoretical phonon spectrum allows us to analyze the lattice specific heat in more detail. The constant volume C v is computed directly from the phonon density of states F(ω ) as: The computed curve (red line) is compared to the measured constant pressure C p data (open circles) in Fig. 10, where we notice a good overall agreement.
If one wishes to describe the heat capacity using an approximate model, a combination of Debye and Einstein terms (C Debye + C Einstein ) should be considered: Figure 6. Electronic dispersion relation and total DOS of LiGa 2 Ir calculated with and without SOC. www.nature.com/scientificreports/ where n D and n E are the number of phonon modes treated as Debye and Einstein type, respectively. Looking at the phonon DOS in Fig. 9 we see that the three high-frequency Li modes may be described as an Einstein term with the average frequency corresponding to about 540 K, whereas the remaining part, containing nine Ir and Ga modes, could be roughly approximated by the Debye spectrum. Hence, we assume n D = 9 and n E = 3 and we use only two fitting parameters: the Debye and Einstein temperatures. The fit in the temperature range 1.85 K -200 K gave the values D = 242(1) K and E = 550(10) K. Contributions from each of these terms are shown in Fig. 10 and the combined heat capacity describes the experimental data reasonably well. Deviations are seen in the lower temperature range due to the non-Debye-like phonon spectrum, captured accurately in the direct calculation using Eq. (5). Note that the low-temperature fit, described before, yielded a larger value of Θ D = 277 K as the whole heat capacity was ascribed to the Debye-like phonon spectrum (n = 4 in eq. � D = 12π 4 5β nR 1/3 corresponds to 12 phonon modes, 3 per each of the atom). Since the Einstein term in our case gives no contribution to the specific heat at low temperatures, adopting to the combined model (i.e. changing to n = 3) we get Θ D = 252 K, very close to the value obtained from the fit for the broad temperature range.
Moving on to the electron-phonon interactions, the magnitude of the electron-phonon interaction for a given phonon branch is represented in Fig. 11 by the phonon linewidth γ qν which is computed from the electron-phonon interaction matrix elements g qν as 52-54 :  www.nature.com/scientificreports/ where In the formulas above, ω qν is the phonon frequency at the wavevector q for the mode ν, M s is mass of atom s, ψ i,k is an electron wavefunction for a given band i and wavevector k, ê ν is a phonon polarization vector and dV scf dû ν,s is a change of the electronic potential due to a displacement of the atom s in the direction u.
The strongest electron-phonon interactions, seen as the largest phonon linewidths, are associated with the optic modes of Ga near 5 THz and the Einstein-like Li branch around 11 THz. Next, the electron-phonon interaction function α 2 F(ω) (Eliashberg function) is calculated by summing the contributions from each of the phonon branches, weighted by their inverse frequency: Fig. 11 and has three peaks associated with enhanced electron-phonon interactions: at 1.78 THz, 5.22 THz, and 11.49 THz . The first peak is associated with acoustic Ir vibrations, having moderate phonon linewidths but low frequencies, effectively increasing the Eliashberg function. The second and third (8)   www.nature.com/scientificreports/ maxima are associated with the above-mentioned Ga and Li branches. Comparing α 2 F(ω) with the phonon DOS function F(ω) (see also Fig. 12) we see, that the Eliashberg function is enhanced over the phonon DOS at lower frequencies, and as a consequence the height of all three α 2 F(ω) maxima become comparable. The overall electron-phonon coupling parameter is calculated from the Eliashberg function: and reaches = 0.68, slightly lowered by the spin-orbit coupling from the scalar-relativistic value of 0.70 (see Table 4). The frequency distribution of  www.nature.com/scientificreports/ is plotted in Fig. 12 and the mode contribution from all 12 phonon branches is displayed in Table 5. The acoustic phonons contribute in approximately 72% to the electron-phonon coupling constant, therefore, Ir vibrations are the most important factor in the superconductivity of LiGa 2 Ir. The small and decreasing effect of SOC on the electron-phonon coupling parameter is related to the small SOC effect on the Fermi surface of the compound and slight increase of the Ir phonon frequencies.
The superconducting critical temperature T c is calculated using the Allen-Dynes formula 55 : where Taking the standard value of the Coulomb screening parameter μ * = 0.13, the calculated value T c = 2.85 K is in a very good agreement with the experimental value of 2.95 K. The experimental T c is exactly reproduced using an only slightly smaller μ * = 0.121. The spin-orbit coupling has a small effect on the calculated critical temperature, slightly decreasing its value to T c = 2.69 K. The agreement between calculations and experiment clearly confirms that superconductivity in LiGa 2 Ir is mediated by phonons.
To investigate exceptionally low decrease of T c with pressure we have calculated electronic structure and lattice dynamics under the pressure of 1 GPa. All further calculations were done including SOC. The lattice constant relaxed under 1 GPa is a = 6.0021 Å. Electronic structure was almost unchanged, with only slight decrease of DOS(E F ) by 0.016 eV −1 (see Table 3). Moreover, only small changes were induced by the pressure in the phonon dispersion relations and in the electron-phonon coupling. Small effect of the lattice stiffening under the external pressure is observed in phonon dispersion relations shown in Fig. 13, where Ga and Li modes moved towards higher frequencies, but modes of heaviest Ir changed only slightly. The small dip in the acoustic mode at Γ-K was not affected either. Average phonon frequencies, electron-phonon coupling constant and critical temperature are collected in Table 4. Assuming the same value of µ*, which reproduces the experimental T c under ambient pressure, the obtained change of T c is equal −0.048 K/GPa, which is in excellent agreement with the experiment.
To understand why the effect of pressure on T c is so weak in LiGa 2 Ir, comparing to other Heusler compounds, we have to compare the values of characteristic parameters which determine the response of superconductor to external pressure. In our analysis we will compare LiGa 2 Ir to MgPd 2 Sb, for which we have recently found much stronger response to pressure: −0.23 K/GPa 46 . The most important parameter which determines the change of T c with respect to pressure is the variation of the electron-phonon coupling constant , which in the case of (13)  (see Eqs. 6-9) is composed of two factors, the frequency-independent electronic contribution expressed by phonon linewidths γ qν and the phonon frequency ω qν , we may analyze what is the origin of such differences. To do so, we calculate the first moment of Eliashberg function 41 : This quantity is frequency-independent because: At 0 and 1 GPa the first moment of Eliashberg function is equal respectively: 4.144 THz 2 , 4.217 THz 2 in LiGa 2 Ir and 1.869 THz 2 , 1.890 THz 2 in MgPd 2 Sb. Thus the changes in I are + 1.8% in LiGa 2 Ir and + 1.12% in MgPd 2 Sb. As far as the electronic part of λ is concerned, stronger increase in the electronic contribution in the case of LiGa 2 Ir is found.
Much larger difference is found in the change of the "average square phonon frequency" defined as With such definition = 2I/ ω 2 and in the case of a weak frequency dependent electron-phonon interaction ω 2 is close to similar quantity determined from the pure phonon DOS 41 . For LiGa 2 Ir we get 12.273 THz 2 (0 GPa) and 12.574 THz 2 (1 GPa), i.e. 2.4% increase. On the other hand, for MgPd 2 Sb we have 6.116 THz 2 (0 GPa) and 6.491 THz 2 (1 GPa), which is 6.13% increase. When analogical quantities are computed from the pure phonon DOS, the increases are 2.6% for LiGa 2 Ir and 5.5% for MgPd 2 Sb, confirming the trend. Thus, the effect of lattice stiffening takes over the increase in electronic contribution to electron-phonon coupling constant in both materials, explaining the decrease of λ with pressure. The much weaker effect on λ in LiGa 2 Ir is explained by stronger increase in the electronic contribution (parameter I) accompanied by the smaller increase in the average square phonon frequency, while compared to MgPd 2 Sb.
Two other global parameters are important in determining the pressure evolution of T c . In the McMillan or Allen-Dynes formulas for T c we have T c ∝ ω c , with ω c being the characteristic phonon frequency ( ω α 2 F log in the Allen-Dynes formula and D in McMillan formula). The evolution of the exponential part of T c equation is governed by the evolution of λ, discussed above, thus we should also take a look on how the multiplicator in T c formula is affected by pressure. The more intuitive picture is provided by the McMillan formula, thus calculating the pressure derivative of Debye temperature we get:  and B is the bulk modulus defined by the approximate pressure-volume relation V = V 0 exp(-p/B) which holds in the small pressure ranges where the variation in bulk modulus with pressure can be neglected. In our case we fitted this relation in the pressure range 0-5 GPa obtaining B = 151(2) GPa (see Fig. 14(b)) for LiGa 2 Ir and 118(2) GPa in MgPd 2 Sb, whereas the average Grüneisen parameter is 1.9 in LiGa 2 Ir compared to 2.7 in MgPd 2 Sb. The larger B, along with the smaller γ G additionally lower the pressure dependence of T c in LiGa 2 Ir, compared to MgPd 2 Sb and other Heusler compounds.
To complete the analysis of bulk modulus for LiGa 2 Ir we have fitted p(V) relation using the Birch-Murnaghan equation of state 56 : where Pressure volume relation was obtained by relaxing cells up to 20 GPa with a 1 GPa step, whereby the starting lattice constant was the one calculated at p = 0 GPa. Fitted equation of state and fit parameters with standard deviations are shown in Fig. 14(a). Bulk modulus B 0 equal 139 GPa is quite high and confirms LiGa 2 Ir resistance to pressure. Similarly calculated value for MgPd 2 Sb is 106 GPa, confirming the difference between the two compounds.

Summary and conclusions
We have made high-quality polycrystalline LiGa 2 Ir using a solid state reaction method. LiGa 2 Ir forms in a full-Heusler crystal structure type with a refined lattice parameter a = 6.0322(1) Å, in agreement with that reported by Czybulka, et al. in ref. 36,37 . The heat capacity, electrical resistivity, and magnetic susceptibility confirm the bulk superconductivity with T c = 2.94 K. Analysis of our data shows that LiGa 2 Ir is a weak-coupling type-II superconductor ( e-p = 0.57 from the McMillan formula, C/γT c = 1.4). Theoretical calculations show that 5d states of Ir and 4p states of Ga equally contribute to the Fermi surface, which consists of three sheets. Although the compound contains heavy Ir, the spin-orbit coupling does not modify the electronic structure near the Fermi level, with the influence only visible for the deeper-lying electronic states. In the phonon spectrum we may distinguish three groups of modes, lined up according to the atomic mass: high-frequency Einstein-like optic Li vibrations, a medium-frequency group of mostly Ga optic modes and a low-frequency acoustic Ir part. The calculations of the Eliashberg function gave e-p = 0.68 with the dominating contribution from the heaviest iridium. SOC slightly lowers the coupling constant, as the scalar-relativistic value is e-p = 0.70. The computed superconducting critical temperature agrees very well with the measurements, confirming the phonon mechanism of superconductivity. www.nature.com/scientificreports/ Comparing the superconducting properties of LiGa 2 Ir and LiGa 2 Rh (see Table 1) we see that LiGa 2 Ir is another Ir-based superconductor with a T c higher than that observed for the isostructural and isoelectronic compound containing Rh. The other examples are SrRh 2 vs. SrIr 2 , CaRh 2 vs. CaIr 2 , IrGe vs. RhGe 57 . This is caused by the larger electron-phonon coupling constants e-p originating from the larger mass of Ir versus Rh. The heavier atom oscillates with lower frequency, thus the inverse proportionality of enhances λ e-p if in both compounds the frequency-independent phonon linewidths are similar.
The observed weak pressure dependence of T c in LiGa 2 Ir originates from the large bulk modulus, relatively small Grüneisen parameter and compensating increase of the electronic contribution to the electron-phonon coupling constant.
More than 60 years ago, Berndt Matthias proposed that the critical temperature changes with the valence electrons and there are three peaks at VEC ~ 3, 5 and 7 el./at. All known Heusler-type superconductors belong to the third maximum, with two exceptions: LiGa 2 Ir and LiGa 2 Rh 26 for which VEC = 4 el./at. Superconductivity reported for LiGa 2 Rh 26 and LiGa 2 Ir (this work) will shed light on the validity of the Matthias T c vs. VEC diagram and can be a stimulus for the future studies and experimental effort to find other Heusler-type superconductors with VEC ~ 20 (5 el./at) for which superconducting transition temperature should be higher.