Observation of superconductivity in the intermetallic compound β-IrSn₄

Among various subjects of condensed matter physics, superconductivity has been attracting much attention. Interest at present mainly concerns the search for superconductors with high critical temperatures or establishing the proper mechanisms of Cooper pairing in so-called unconventional superconductors. The theme of the type-I BCS superconductivity is generally considered to be less interesting. This is owing to the fact that the type-I superconductivity occurs mostly in simple elemental metals or metalloids and the mechanism underlying the superconductivity is basically the well-known electron–phonon interaction. Curiously, type-I superconductivity has recently been found in several binary and ternary compounds, for instance, YbSb₂ [1, 2], TaSi₂ [3], (Sc,Lu)Ga₃ [4], LaPd₂Ge₂ [5], LaRh₂Si₂ [6], (Lu,Y,La)Pd₂Si₂ [6] and LaRhSi₃ [7]. These discoveries have not only broken the exclusive association between type-I superconductivity and elemental metals, but have encouraged further investigation of type-I superconductivity, since a field-dependent anisotropic order parameter in the superconducting state has been considered for LaRhSi₃ [7].

Intermetallic IrSn₄ binary crystallizes in three modifications; the low-temperature form, so-called α-IrSn₄, adopts the trigonal IrGe₄-type structure (space group P3₁21) [8], the high-temperature form, β-IrSn₄  has the tetragonal MoSn₄-type structure (space group I4₁/acd) [9], and the high-pressure form, HP-IrSn₄, possesses the orthorhombic PtSn₄-type structure (space group Ccca) [10, 11]. The structures of the two latter modifications stand in close relation to each other. To the best of our knowledge, only α-IrSn₄ was reported to be a Pauli paramagnet and non-superconducting down to 2 K [8], whereas none of physical properties of β-IrSn₄ and HP-IrSn₄ have been reported yet.

In this work, we present magnetization, electrical resistivity and specific heat measurements on single crystals of intermetallic compound β-IrSn₄, which becomes superconducting with a transition temperature T_c of 0.9 ± 0.05 K. Although the zero-field data characterize β-IrSn₄ well as a s-wave weakly coupled type-I superconductor in a dirty limit, from an estimation of some thermodynamic parameters we arrive at a conclusion that superconductivity in β-IrSn₄ can be ascribed to a type-II superconductor in a clean limit with an assumption of a non-spherical Fermi surface.

Single crystals of β-IrSn₄ were grown by the self-flux method. A mixture of high-quality elemental metals Ir (purity 99.99%) and Sn (at. 99.999%) with the atomic ratio Ir:Sn = 1:80 was placed in an alumina crucible, sealed in an evacuated quartz ampule, and heated up to 900 °C. After cooling down to 600 °C at a rate of 5 °C h⁻¹, the furnace was turned off and the quartz ampule was slowly cooled down to 300 °C within 3 h. Then, liquid Sn flux was decanted and the excess of Sn was removed by etching in HCl. Single crystals were obtained in the form of regular geometric shapes (see the left upper panel of figure 1).

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X-ray powder diffraction of powdered small crystals and single crystal diffraction of a crystal were performed at room temperature using a X'Pert PRO x-ray diffractometer with monochromatized Cu Kα radiation. Since powdered small crystals possess texture, lattice parameters were calculated from the positions of the Bragg peaks only. Two alternative models of the crystal structure were considered, i.e., the tetragonal MoSn₄-type structure and the orthorhombic PtSn₄-type structure. Our analysis indicated that all the observed Bragg lines on the x-ray diffraction patterns could be indexed on the basis of the tetragonal MoSn₄-type structure with the lattice parameters a = 0.6308(2) nm and c = 2.2739(4) nm (see the bottom panel of figure 1). The lattice parameters of our sample are in good agreement with those previously reported (a = 0.6309 nm and c = 2.2770 nm) [9]. The single crystal diffraction pattern (solid line) shows only 00l reflections, indicating that the crystallographic c-axis is perpendicular to the widest surface of the crystals.

Taking into account the atomic positions of Ir(8b) and Sn(32g) [9], we show the crystal structure of β-IrSn₄4 in the right upper panel of figure 1. In this tetragonal MoSn₄-type structure, the Ir atoms are coordinated by eight Sn atoms. The eight Sn atoms form a square antiprism, which together with other antiprisms establish infinite layers. Further, the structure can be described as a stacking of four layers of IrSn₈ antiprisms. The atoms are arranged in the sequence, 4Sn–Ir–4Sn...4Sn–Ir–4Sn...4Sn–Ir–4Sn...4Sn–Ir–4Sn along the c-direction. The minimum interatomic distances within layers are d_Sn–Sn = 0.309 and d_Ir–Ir = 0.446 nm. Because of a large d_Ir–Ir, the direct Ir–Ir interactions are almost excluded. Thus, the closer Sn–Sn contacts are favorable for an in-plane charge-carrier flow within the Sn layers. However, we recognize that interatomic distances d_Ir–Sn = 0.275 nm and d_Sn–Sn = 0.296 nm between layers are even shorter than these within the plane. This means that dominant charge carriers would take a path ...Ir–Sn–Sn–Ir–Sn–Sn–Ir... along the shortest interatomic distances, i.e., the chain along the c-axis. In considering the origins of the physical behavior discussed below, the fact that the Ir and Sn atoms in the unit cell are nearest neighbors may suggest overlap of the Sn sp-orbitals with Ir d-orbitals as mostly responsible for the electron transport properties in β-IrSn₄.

Dc-magnetization measurement was carried out using a commercial SQUID magnetometer (Quantum Design MPMS-XL) with an iQuantum ³He outfit. For the measurement, several single crystals were glued together to get a specimen of 18 mg. In figure 2(a) we present the temperature dependence of the magnetization M(T) collected at several magnetic fields. The data at 1.5 mT reveal a large diamagnetic signal below T_c(1.5 mT) = 0.95 K, indicating a superconducting transition. It is interesting to note that the onset of the superconductivity in 2 mT occurs at the same T_c as in 1.5 mT. With further increasing fields, T_c shifts down to lower temperatures, the diamagnetic signal at 0.45 K becomes stronger and simultaneously the transition becomes sharper. The latter finding may suggest that superconductivity becomes more homogeneous with magnetic fields. Figure 2(b) shows the isothermal magnetization measured at a temperature around 0.5 K. The hysteresis of the magnetization in the superconducting state resembles the behavior of both type-I and type-II superconductors with comparable coherence lengths and penetration depths. If β-IrSn₄ is a type-I superconductor then the departure from the discontinuous jump transition at the critical field can be ascribed to a strong pinning of domain walls in the intermediate state or to an unaccounted demagnetization factor. In this case, the thermodynamic critical field μ₀H_c can be estimated to be approximately 4.7 mT. If β-IrSn₄ is considered as a type-II superconductor then one can estimate the lower critical field μ₀H_c1 and the upper critical field μ₀H_c2. μ₀H_c1 = 3 mT, if is defined as the field where the deviation of the M(H) curve from linearity in the Meissner state sets in, while μ₀H_c2 = 6.5 mT at 0.5 K, determined as the intersection between linear interpolations of the magnetization in the superconducting and normal states. The positions of the critical fields are denoted by solid arrows in figure 2(b). Above μ₀H_c2 = 6.5 mT or in the normal state the studied compound behaves like a Pauli paramagnet.

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Electrical resistivity was measured using a conventional four-probe ac-technique with an in-plane current of 0.5 mA and frequency of 37 Hz. The gold wires for electrical contracts were attached to samples using silver paste. The two measured samples have dimensions of 0.36 × 1.68 × 0.18 mm³, 1.2 × 0.74 × 0.1 mm³, respectively. The measurements were carried out using a Quantum Design physical property measurement system (PPMS) with a ³He cooler. Several selected magnetic fields up to 10 mT were applied parallel to the c-axis. The experimental data obtained on the two samples were reproducible. The typical value of the remnant field is less than 0.5 mT when decreasing the field from 10 mT in the oscillating mode. For measurements in fields below 2 mT, the bias field was further removed to get a lower remnant field, below 0.2 mT, before cooling the sample from room temperature. The strength of the remnant fields was controlled by an Applied Physics Systems Fluxgate Model 150–6325 magnetometer.

Low-temperature data shown in the upper inset of figure 3(a) indicate that the resistivity starts to drop abruptly at 1 K and arrives at zero value at 0.84 K. The critical temperature, T_c, defined as a middle point of the superconducting transition, and the transition width, ΔT_c = T_c(90%) − T_c(10%), are determined to be 0.93 K and 0.15 K, respectively. Upon increasing the magnetic fields, the superconducting transition shifts to lower temperatures and even a small field of 10 mT is sufficient to reduce the T_c to less than 0.5 K. From the field dependence of the resistivity we have determined the upper critical field. At 0.45 K μ₀H_c2 = 6.8 mT (see figure 3(b)), which is in good agreement with the value obtained from the magnetization measurements.

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In figure 4 we show the electrical resistivity of β-IrSn₄ in the temperature range 0.35–300 K. The temperature dependence of ρ(T) indicates that this intermetallic is a good metal. At room temperature the resistivity amounts to 80 μΩ cm and decreases to approximately 3 μΩ cm at 1 K. A large residual resistivity ratio RRR = 27, sharp superconducting transition and small residual resistivity value manifest the good sample quality and negligible scattering by defects and impurities. The resistivity just above T_c follows a quadratic temperature dependence ρ(T) = ρ₀ + AT², as shown in the inset of figure 4 by the solid line. By fitting this formula to the data between 1 and 15 K we obtained the residual resistivity of ρ₀ = 3.2 μΩ cm and the coefficient A = 0.9 × 10⁻³ μΩ cm K⁻². It is apparent that the A-coefficient is greatly enhanced, as compared to those observed for transition metals with A of the order of magnitude of 10⁻⁵ μΩ cm K⁻² [12]. The observation of the ρ ∝ T² dependence and a relatively large A coefficient are symptomatic of strong electron–electron interactions in the Fermi-liquid state of β-IrSn₄. At higher temperatures, a curvature of the resistivity suggests to analyze it in terms of the Bloch–Grüneisen relation [13]. We fitted the experimental data to equation (1) and found that the residual resistivity ρ₀ = 3.17 μΩ cm, the coefficient k = 112 μΩ cm, the Debye temperature Θ_D = 216 K and the exponent n = 3. The observed value n = 3 and a good agreement between theory and the experimental data up to room temperature implies a significant s–d band scattering.

$$\begin{eqnarray} &&\rho (T)={\rho }_{0}+k\left (\frac{T}{{\Theta }_{\mathrm{D}}}\right )^{n}\int \nolimits _{0}^{{\Theta }_{\mathrm{D}}/T}\frac{{x}^{n}\mathrm{d}x}{({\mathrm{e}}^{x}-1)(1-{\mathrm{e}}^{-x})}. \end{eqnarray} \tag{ 1 }$$

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The Hall effect was measured on one of the samples used in the resistivity measurements. We utilized a Quantum Design Horizontal Rotator in a Quantum Design PPMS platform to rotate the sample from 0° to 180°. The Hall resistivity data, determined as ${\rho }_{\mathrm{H}}=\frac{1}{2}({\rho }_{\mathrm{H},0^{\circ}}-{\rho }_{\mathrm{H},1 8 0^{\circ}})$, were collected at 2 K and in fields up to 9 T using a conventional four-probe ac-technique with an in-plane current of 0.5 mA and frequency of 37 Hz. Figure 5 depicts the experimental Hall resistivity. The data show that ρ_H is linearly proportional to μ₀H up to 2 T. At high fields, ρ_H deviates from the initial linear dependence. The reason for the nonlinear ρ_H(H) behavior is not ascertained as yet. One suspects that it may arise from a new contribution associated with spin–orbit coupling, which is negligible in a low-field regime but becomes perceptible in high fields. Alternatively, one can consider the observed field dependence of the Hall resistivity as due to competing contributions of two carrier types with different densities and mobilities. In the low-field limit, the Hall coefficient is dominated by the ordinary term R₀ and can be estimated by R₀ = ρ_H/(μ₀H) = 8.95 × 10⁻¹⁰ m³ C⁻¹. In the simple one-band approximation, the ratio 1/(R₀e), where e is the elementary charge, may be interpreted as an effective charge-carrier concentration. We estimated the number of carriers, n = 1/(R₀e) = 6.98 × 10²⁷ m⁻³. Because the Hall coefficient is positive at all fields measured, it implies that the charge carriers are predominantly hole-like. The Hall ordinary term R₀ can be combined with the resistivity data to obtain the Hall mobility μ_H and Hall mean free path l_H. From the equations μ_H = R₀/ρ and ${l}_{\mathrm{H}}={\hbar }^{2}(3{\pi }^{3}/{e}^{4})^{1/3}{R}_{0}^{2/3}/\rho$ we estimated μ_H = 280 cm² V⁻¹ s⁻¹ and l_H = 109 nm, respectively.

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The specific heat was measured by the thermal relaxation method on a sample of ∼1 mg. The specific heat measurements were done using the same setup and experimental conditions as the electrical resistivity. We carried out careful calibration of the calorimeter and measurement of addenda before the proper measurements were done. The zero-field specific heat of β-IrSn₄ divided by temperature, C_p/T, versus T² is depicted in figure 6(a). The normal state specific heat follows the standard law for metals C_p = γ_nT + βT³, where γ_n and β are the normal state electronic heat and phonon specific heat coefficients, respectively. A linear fit of the data to C_p/T = γ_n + βT² yields γ_n = 5.2 ± 0.2 mJ mol⁻¹ K⁻² and β = 0.91 ± 0.15 mJ mol⁻¹ K⁻⁴. Knowing the β value, we can determine the Debye temperature: Θ_D = (12π⁴rR/5β)^1/3 = 221 ± 8 K, in a good agreement with the value Θ_D deduced from the resistivity data (where r = 5 is the number of atoms per formula unit and R is the gas constant). In order to get some information on the strength of the electron–phonon coupling we estimate the average electron–phonon coupling constant ${\bar {\lambda }}_{\mathrm{e}\text{{\ndash}}\mathrm{ph}}$ using the McMillan equation [14]:

$$\begin{eqnarray} &&{\bar {\lambda }}_{\mathrm{e}\text{{\ndash}}\mathrm{ph}}=\frac{1.0 4+{\mu }^{\ast }\ln ({\Theta }_{\mathrm{D}}/1.4 5{T}_{\mathrm{c}})}{(1-0.6 2{\mu }^{\ast })\ln ({\Theta }_{\mathrm{D}}/1.4 5{T}_{\mathrm{c}})-1.0 4} \end{eqnarray} \tag{ 2 }$$

where μ* is the Coulomb pseudopotential, usually assumed to be between 0.1 and 0.15 [14, 15]. For μ* = 0.15,Θ_D = 221 K and T_c ≈ 0.9 K, equation (2) yields ${\bar {\lambda }}_{\mathrm{e}\text{{\ndash}}\mathrm{ph}}=0.5$, suggestive of a weak electron–phonon coupling in β-IrSn₄.

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The bulk nature of superconductivity in β-IrSn₄ is evidenced by a clear peak in the C_p(T)/T dependence. The critical temperature T_c = 0.85 K, taken as the middle point of the ΔC_p(T)/T jump, agrees fairly with that deduced from the resistivity. We determined the ratio ΔC_p/γ_nT_c = 1.29, which is noticeably smaller than 1.43 given by the BCS theory for weakly coupled superconductors. In figure 6(b) we show the ln(C_el(T)) versus 1/T dependence, where C_el(T) = ΔC_p(T) = C_p(T) − βT³ is the electronic specific heat. It is clear that in the superconducting state C_el(T) follows a single-gapped BCS-like function C_el(T) = Aγ_nT_cexp(−Δ₀/k_BT). For the temperature range 0.4–0.7 K we obtained an energy gap Δ₀/k_B = 1.51 K, corresponding to 2Δ₀/k_BT_c = 3.55 and parameter A = 12.8. The two latter values are comparable to those of weak coupling BCS superconductors: 3.52 and 8.5, respectively. This and the value of the ratio ΔC_p/γ_nT_c consistently imply that β-IrSn₄ is a weak coupled superconductor.

The plot of C_p(T)/T measured in various magnetic fields up to 7 mT is shown in figure 7. The main points to note from this figure are the following: first, the specific heat jump ΔC_p/T_c in small magnetic fields, namely in 1, 1.5 and 2 mT, is larger than that in zero-field. The observed field dependence of ΔC_p/T_c means that the transition order changes from the second order in zero-field to the first order in applied magnetic fields. A similar behavior has been observed in the type-I superconductors (Sc,Lu)Ga₃ [4] and LaRhSi₃ [7], thus strongly supporting a type-I superconductivity in β-IrSn₄. Second, perhaps most surprisingly, the critical temperature increases a little, up to 0.94 K, on the application of a field of 1.5 mT. The same T_c value was obtained from the C_p-data at 2 mT. Because the used field strength is much larger than the remnant field (<0.5 mT), the measurement artifact can be excluded. Moreover, the values of the critical temperature T_c are reproduced in independent measurements of specific heat and magnetization on different samples. Therefore, it is most likely that the slight enhancement of T_c upon application of the magnetic fields is an intrinsic property of the studied samples.

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In figure 8(a) we show the effect of applied magnetic fields on the Sommerfeld ratio C_p(H)/T at 0.4 K. Apparently, C_p(H)/T is not linear in field. Similarly to the case of LaRhSi₃ [7] one describes the field dependence of the C_p(H)/T data by the exponential function C_p(H)/T = γ(0.4 K) + A × exp(−H*/H), where γ(0.4 K),A and μ₀H* are fitting parameters. The best fit for data collected in magnetic fields below 7 mT gives γ(0.4 K) = 3.9 mJ mol⁻¹ K⁻²,A = 15.8 mJ mol⁻¹ K⁻² and μ₀H* = 9.5 mT. The exponential field dependence of C_p(H)/T in the tetragonal superconducting LaRhSi₃ has been attributed to a change in the superconducting gap, which is isotropic in zero-field but becomes anisotropic in applied magnetic fields [7]. Since a similar nonlinearity of C_p(H)/T is observed in LaRhSi₃ and β-IrSn₄, one may think that the effect reported for β-IrSn₄ can be explained in the same frame as has been used for LaRhSi₃. However, because our specific heat data were collected on single crystals, i.e. for μ₀H ∥ c-axis, our data should reflect the field dependence of the superconducting gap along one direction only, thus contradict the anisotropic model of the superconducting gap. To unravel the strong field dependence of C_p(H)/T in β-IrSn₄, we propose to consider C_p(H)/T in two field ranges: below 2.5 mT, where C_p(H)/T is independent of magnetic fields, and in the field range 2.5–6 mT, where C_p(H)/T follows magnetic fields linearly. These two field regimes correspond approximately to those below μ₀H_c1, and between μ₀H_c1 and μ₀H_c2, respectively. We note that for a standard BCS superconductor with a single isotropic energy gap, C_p(H)/T should increase linearly in magnetic fields, until the flux lines start to overlap. Such a linear field dependence of C_p(H)/T is found for applied fields larger than the lower critical field, when the non-superconducting density of states develops with an increasing number of flux lines. Obviously, in fields smaller than μ₀H_c1 ∼ 3 mT,C_p(H)/T should be practically independent of fields since the magnetic flux penetrating into the material is only in a very thin layer of the surface and the contribution of the non-superconducting density of states to the total density at the Fermi level is negligible.

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From our measurements of magnetization, electrical resistivity and specific heat we deduced the critical fields μ₀H_c2(T). The resulting H–T phase diagram is shown in figure 8(b). In this figure we compare also the temperature dependences of the thermodynamic critical fields μ₀H_c(T) calculated from the C_p-data at 0 and 1.5 mT. The theoretical μ₀H_c(T) values have been obtained by integrating the difference of specific heats between the superconducting (s) and normal (n) state and integrating the difference of the specific heats divided by temperature according to the formulas:

$$\begin{eqnarray} -\frac{1}{2}{\mu }_{0}{V}_{\mathrm{M}}{H}_{\mathrm{c}}^{2}(T)&=&\Delta F(T)=\Delta U(T)-T \Delta S(T) \end{eqnarray} \tag{ 3a }$$

$$\begin{eqnarray} \Delta U(T)&=&\int \nolimits _{T}^{{T}_{\mathrm{c}}}[{C}_{s}({T}^{\prime})-{C}_{n}({T}^{\prime})]\mathrm{d}({T}^{\prime}) \end{eqnarray} \tag{ 3b }$$

$$\begin{eqnarray} \Delta S(T)&=&\int \nolimits _{T}^{{T}_{\mathrm{c}}}\frac{[{C}_{s}({T}^{\prime})-{C}_{n}({T}^{\prime})]}{{T}^{\prime}}\mathrm{d}({T}^{\prime}), \end{eqnarray} \tag{ 3c }$$

where ΔU(T) is the internal energy difference and ΔS(T) is the entropy difference. Using the relation ${\mu }_{0}V{H}_{\mathrm{c}}^{2}(0)=(3{\gamma }_{n}/2{\pi }^{2}{k}_{\mathrm{B}}^{2})({\Delta }_{0}/{k}_{\mathrm{B}})^{2}$, the values γ_n = 5.2 × 10⁻³ J mol⁻¹ K⁻²,V_M = 68.24 × 10⁻⁶ m³ mol⁻¹ and Δ₀ = 1.51 K we obtained μ₀H_c(0) = 5.76 mT at 0 K. This value is denoted by a solid diamond in figure 8(b). Clearly, there is an essential deviation of the experimental μ₀H_c2(T) data from the thermodynamic field deduced from the entropy balance. This behavior indicates that β-IrSn₄ should be a type-II superconductor in magnetic fields. A comparison of the calculated thermodynamic critical fields at 0 and 1.5 mT indicates that the temperature dependence of μ₀H_c(T) is sensitive to applied magnetic fields.

A remarkable finding of our measurements is that T_c(H_c2) = 0.85 K at zero field attains its maximum value of 0.94 K in fields 1.5 and 2 mT. This behavior resembles the situation found for Eu_xSn_1−xMo₆S₈ [16], λ-(BETS)₂FeCl₄ [17], URhGe [18] and Eu(Fe_0.81Co_0.18)₂ As₂ [19], namely superconductivity can be induced or the critical temperature can be increased by the application of suitable magnetic fields. The analogy between β-IrSn₄ and the mentioned superconductors is not obvious because the observed effect in β-IrSn₄ occurs just below μ₀H_c1 and is rather obscure. Therefore, further investigations will be required in order to establish the proper mechanism of the unusual T_c(H) dependence observed in β-IrSn₄.

Another aspect worth considering is the almost linear behavior of H_c2(T_c) above 2 mT. In fact, a fit of the conventional relation H_c2(T) = H_c2(0)[1 − (T/T_c(0))ⁿ] to the experimental data gives the critical field H_c2(0) = 9.55 mT, critical temperature T_c(0) = 1.13 K and exponent n = 1.25. The latter value is significantly reduced in respect to n = 2 expected from the BCS theory.

We calculated the upper critical field using the model developed by Werthamer et al [20] (WHH model) for a type-II superconductor. Taking the slope of μ₀H_c2(T) in the temperature range 0.63–0.94 K, dμ₀H_c2(T)/dT =− 9.83 mT K⁻¹, and the orbital-limiting field relation ${\mu }_{0}{H}_{\mathrm{c}2}^{\mathrm{orb}}=0.7 3{T}_{\mathrm{c}}(0)[-\mathrm{d}{\mu }_{0}{H}_{\mathrm{c}2}(T)/\mathrm{d}T]_{{T}_{\mathrm{c}}}$ for a clean limit superconductor, we obtain ${\mu }_{0}{H}_{\mathrm{c}2}^{\mathrm{orb}}=8.1~\mathrm{mT}$ for T_c(0) = 1.13 K. It appears that all observed values of the critical fields are small and certainly indicate the absence of the Pauli limiting field μ₀H_p = 1.86T_c, which is of the order of ∼2 T. In order to support the dominant orbital-pair breaking effect we determined the value of the Maki parameter [20, 21], ${\alpha }_{\mathrm{M}}=(3{e}^{2}\hbar {\gamma }_{V}{\rho }_{0})/(2{\pi }^{2}{k}_{\mathrm{B}}^{2}{m}_{e})=5.4\times 1{0}^{-3}$. In the formula, γ_V = 76.2 J m⁻³ K⁻² is the electronic specific heat coefficient per volume and m_e is the electron mass. On the other hand, from the relation α_M = 0.528 × [−dH_c2(T)/dT]_Tc [20, 21], with dH_c2(T)/dT in the units of T K⁻¹, the value of the Maki parameter is ≈5.2 × 10⁻³, i.e., very close to the above value.

It is of considerable interest to estimate some thermodynamic parameters characterizing the superconducting state. First, we calculated the Fermi wavenumber k_F = (3π²n_s)^1/3 = 4.65 × 10⁹ m⁻¹, where the superconducting carrier density n_s = 3.39 × 10²⁷ m⁻³ is a half of the total electron density n and e is the elementary charge. Then, we determined the effective mass ${m}^{\ast }=3{\hbar }^{2}{\gamma }_{V}/({k}_{\mathrm{B}}^{2}{k}_{\mathrm{F}})\sim 3.1~{m}_{e}$. The London penetration depth λ_L(0) = 162 nm can be obtained from the formula: ${\lambda }_{\mathrm{L}}(0)=\sqrt{\frac{{m}^{\ast }{c}^{2}}{4 \pi {n}_{s}{e}^{2}}}$, where c is the speed of light. On the other hand, using the BCS relation for the coherence length ξ₀ = 0.18ħυ_F/k_BT_c, we obtained ξ₀ = 261 nm, where υ_F = ħk_F/m* = 1.71 × 10⁵ m s⁻¹ is the Fermi velocity. The obtained ratio $\kappa (0)={\lambda }_{\mathrm{L}}(0)/{\xi }_{0}\approx 0.6 2{\lt }1/\sqrt{2}$ suggests ascribing the superconductivity in β-IrSn₄ to either a type-I or to a type-II/1 in the classification of Auer and Ullmaier [22]. Further, using the formulas given by Orlando et al [23], we can estimate the BCS coherence length ${\xi }_{0}=7.9 5\times 1{0}^{-1 7}{n}_{s}^{2/3}S/{S}_{\mathrm{F}}/[{\gamma }_{V}{T}_{\mathrm{c}}]$ and London penetration depth ${\lambda }_{\mathrm{L}}(0)=1.3 3\times {1 0}^{8}{\gamma }_{V}^{1/2}/[{n}_{s}^{2/3}S/{S}_{\mathrm{F}}]$. In this estimation, uncertainties may appear due to the unknown ratio of the Fermi surface of the superconducting electrons, S, to the surface of the free electrons S_F,S/S_F. Therefore, we will consider two extreme cases, namely assuming a spherical Fermi surface (S/S_F = 1) or non-spherical Fermi surface (e.g. S/S_F = 0.35). The first assumption yields ξ₀ = 262 nm,λ_L = 163 nm, whereas with the latter one we obtain ξ₀ = 92 nm,λ_L = 465 nm. Correspondingly, the electronic mean free path ${l}_{e}=1.2 7\times {1 0}^{4}/[{\rho }_{0}{n}_{s}^{2/3}(S/{S}_{\mathrm{F}})]$ amounts to 521 or 1489 nm, respectively. Because l_e is always larger than ξ₀, the superconductivity in β-IrSn₄ is certainly in the clean limit. The estimation leads to an interesting conclusion that if the Fermi surface is spherical then κ = 0.62, so supporting type-I and type-II/1 superconductivity in β-IrSn₄. Conversely, if the Fermi surface is non-spherical, κ takes a value of 5, implying that the superconductivity is of type-II.

In the clean limit, the Ginzburg–Landau coherence length ξ_GL and Ginzburg–Landau parameter κ_GL can be estimated using the respective relations ${\xi }_{\mathrm{GL}}=5.8 7\times 1{0}^{-1 7}/[{n}_{s}^{2/3}(S/{S}_{\mathrm{F}})]$ and ${\kappa }_{\mathrm{GL}}=1.6\times {1 0}^{2 4}{T}_{\mathrm{c}}{\gamma }_{V}/[{n}_{s}^{2/3}(S/{S}_{\mathrm{F}})]^{2}$ [23]. We obtained ξ_GL and κ_GL values of 68 nm and 4.9, respectively, and the resulting Ginzburg–Landau penetration depth is λ_GL = κ_GLξ_GL = 327 nm. Also, a method of estimation of ξ_GL is based on the knowledge of the upper critical field. Assuming ${\mu }_{0}{H}_{\mathrm{c}2}^{H\parallel c}={\mu }_{0}{H}_{\mathrm{c}2}^{\mathrm{orb}}$, an anisotropic value of the Ginzburg–Landau coherence length ξ_ab, which is related to ${\mu }_{0}{H}_{\mathrm{c}2}^{H\parallel c}$ by the relation ${\mu }_{0}{H}_{\mathrm{c}2}^{H\parallel c}={\Phi }_{0}/(2 \pi {\xi }_{a b}^{2})$, where Φ₀ is the flux quantum. For ${\mu }_{0}{H}_{\mathrm{c}}^{c}=8.1~\mathrm{mT}$ this results as ξ_ab = 206 nm.

The first picture which emerges from our measurements on single crystals of β-IrSn₄ is that this binary intermetallic compound becomes superconducting below 0.9 ± 0.05 K. The zero-field data listed in table 1 are in accordance with the conventional s-wave singlet pairing mechanism and would also provide a hint of either the type-I or type-II/1 superconductivity with a weak electron–phonon coupling.

Table 1.  Thermodynamic parameters of the superconductor β-IrSn₄: critical temperature T_c, critical field μ₀H_c0(0), electron–phonon coupling constant ${\bar {\lambda }}_{\mathrm{e}\text{{\ndash}}\mathrm{ph}}$, Maki parameter α_M, electronic heat coefficient γ_n, lattice heat coefficient β, specific heat jump ΔC_p/γ_nT_c, superconducting energy gap ratio 2Δ₀/k_BT_c, Fermi wavevector k_F, effective masses of superconducting carriers m*.

Tc (K)	0.9 ± 0.05
μ0Hc0(0) (mT)	5.8 ± 0.1
αM	5.3 × 10−3
${\bar {\lambda }}_{\mathrm{e}\text{{\ndash}}\mathrm{ph}}$	0.5 ± 0.05
γn (mJ mol−1 K−2)	5.2 ± 0.2
β (mJ mol−1 K−4)	0.91 ± 0.15
ΔCp/γnTc	1.3 ± 0.1
2Δ0/kBTc	3.55 ± 0.05
kF (×109 m−1)	4.7 ± 0.5
m* (me)	3.1 ± 0.3

In order to get insight into the physics underlying the superconductivity in β-IrSn₄ we can discuss several factors which would be involved in the Cooper pairing mechanism: (i) anisotropy of the Fermi surface and (ii) electron correlations. First of all, on account of the fact that β-IrSn₄ crystallizes in the tetragonal MoSn₄-type structure, an anisotropy of the Fermi surface is expected. Moreover, the assumption of a non-spherical Fermi surface leads to a deduction of the type-II superconductivity, which coincides with the observation of the upper critical field μ₀H_c2. To examine the electron correlation factor, a determination of the Kadowaki–Wood ratio, R_KW, [24, 25] is useful. Using the electronic heat coefficient γ_n = 5.2 mJ mol⁻¹ K⁻² and the T²-resistivity coefficient A = 0.9 × 10⁻³ μΩ cm, the R_KW is estimated to 3.3 × 10⁻⁵ μΩ cm/(mJ mol⁻¹ K)⁻². This value is essentially larger than the value of 1 × 10⁻⁵ μ Ω cm/(mJ mol⁻¹K)⁻² commonly found for many correlated electron systems. Thus, we are dealing with the fact that both R_KW and the effective mass of ∼3 m_e are enhanced in β-IrSn₄. This implies that the electron correlation effect, presumably due to an overlap of Sn sp-orbitals with Ir d-orbitals, would have an impact on the Cooper pairing mechanism.

Finally, an interesting outcome is obtained from comparison of the studied compound with the intermetallic compound AuSn₄, which was previously reported to undergo a superconducting transition at T_c = 2.4 K [26, 27]. The latter compound has been shown to have an orthorhombic PdSn₄ structure with the noncentrosymmetric space group Aba2 [28]. Aside from the different crystallographic structures, the intermetallic β-IrSn₄ and AuSn₄ compounds differ in the number of d-electrons. So, the microscopic mechanisms of the electron pairing in these superconductors may be quite different.

To summarize, we have grown single crystals of intermetallic compound β-IrSn₄, crystallizing in the tetragonal MoSn₄-type structure. We have measured low-temperature magnetic, electrical resistivity and specific heat properties of the compound. We found a superconducting transition T_c(0T) = 0.9 ± 0.05 K and an upper critical magnetic field μ₀H_c2(0) ∼ 8 mT. We have determined basic thermodynamic parameters characterizing the superconducting state, such as the specific heat jump at T_c,ΔC_p(T_c)/γ_nT_c, electron–phonon coupling constant, λ_e–ph, coherence length ξ, Fermi wavevector k_F, effective mass m*, magnetic penetration depth λ, and Ginzburg–Landau parameter κ. The determined parameters are gathered in tables 1 and 2. The analysis of the experimental data allows us to characterize β-IrSn₄ as either a type-I as well as type-II/1 or a type-II clean limit superconductor, depending on assumption of a spherical or non-spherical Fermi surface, respectively. We favor the assumption of a non-spherical Fermi surface, since β-IrSn₄ crystallizes in the tetragonal MoSn₄-type structure. Moreover, the behavior of the upper critical field data indicates that a type-II superconductivity is realized in β-IrSn₄  and this interpretation does not contradict the unusual phenomena observed, including the nonlinear field dependence of the Sommerfeld coefficient and a jump in the critical magnetic field around the critical temperature, since all these anomalies may occur in the vicinity of the lower critical field μ₀H_c1. In the light of the observation of superconductivity on the border between the type-II and type-I in β-IrSn₄, the physical properties of this intermetallic compound are interesting and demand further investigation.

The authors acknowledge for the financial support from the project 2011/01/B/ST3/04553 of the National Science Centre of Poland.