Ir 5$d$-band Derived Superconductivity in LaIr$_3$

TL;DR

This study investigates the superconducting properties of LaIr$_3$, revealing weak-coupling, fully gapped s-wave superconductivity with a transition at 2.5 K and detailed insights into its magnetic and electronic characteristics.

Contribution

First comprehensive analysis of LaIr$_3$'s superconductivity using muon-spin rotation and heat capacity, highlighting weak-coupling s-wave behavior and detailed magnetic properties.

Findings

Superconducting transition at 2.5 K

Type-II superconductivity with Hc2(0) = 3.84 T

Weak-coupling s-wave superconductivity with 2Δ(0)/kBTc = 3.31

Abstract

We have studied the superconducting properties of LaIr$_3$ with a rhombohedral structure using magnetization, heat capacity, and muon-spin rotation/relaxation ($\mu$SR) measurements. The zero-field cooled and field cooled susceptibility measurements exhibit a superconducting transition below $T_{\mathrm{C}}$ = 2.5 K. Magnetization measurements indicate bulk type-II superconductivity with upper critical field $\mu_0H_{\mathrm{c2}}(0)$ = 3.84 T. Two successive transitions are observed in heat capacity data, one at $T_{\mathrm{C}}$ = 2.5 K and a second at 1.2 K below $T_{\mathrm{C}}$ whose origin remain unclear. The heat capacity jump reveals $\Delta C$/$\gamma T_{\mathrm{C}} \sim$ 1.0 which is lower than 1.43 expected for BCS weak coupling limit. Transverse field-$\mu$SR measurements reveal a fully gapped $s-$wave superconductivity with 2$\Delta(0)/k_{\mathrm{B}}T_{\mathrm{C}}$ = 3.31, which is small compared to BCS value 3.56, suggesting weak coupling superconductivity. Moreover the study of the temperature dependence of the magnetic penetration depth estimated using the transverse field-$\mu$SR measurements gives a zero temperature value of the magnetic penetration depth $\lambda_{\mathrm{L}}(0)$ = 386(3) nm, superconducting carrier density $n_{\mathrm{s}}$ = 2.9(1) $\times$10$^{27}$ carriers $m^{-3}$ and the carriers' effective-mass enhancement $m^{*}$ = 1.53(1) $m_{\mathrm{e}}$. Our zero-field-$\mu$SR measurements do not reveal the spontaneous appearance of an internal magnetic field below the transition temperature, which indicates that time-reversal symmetry is preserved in the superconducting state of LaIr$_3$.

Figures

Equations

$$G_{z1}(t)=A_{1}\cos(\omega_{1}t+\phi)\exp\bigg{(}\frac{-\sigma^{2}t^{2}}{2}\bigg{)}+A_{2}\cos(\omega_{2}t+\phi)$$

$$\frac{{\sigma }_{\mathrm{sc}}(T)}{{\sigma }_{\mathrm{sc}}(0)}$$

$${\lambda }_{\mathrm{e}-\mathrm{ph}}=\frac{1.04+{\mu }^{\ast }\ln ({\Theta }_D/1.45T_{\mathrm{C}})}{(1-0.62{\mu }^{\ast })\ln ({\Theta }_D/1.45T_{\mathrm{C}})-1.04}$$

$$G_{z2}(t)=A_0{G}_{\mathrm{KT}}(t)e^{-\lambda t}+A_{\mathrm{bg}}$$

$$G_{\mathrm{KT}}(t)=\left[\frac{1}{3}+\frac{2}{3}(1-{\sigma }_{\mathrm{KT}}^2{t}^2)e^{\frac{-{\sigma }_{\mathrm{KT}}^2{t}^2}{2}}\right]$$

Full text

I Introduction

The quest for unconvetional superconductivity in materials based on 5$d$ transition metal elements is one of the most fascinating and significant topics in condensed matter physics B. J. Kim ; B. J. Kim1 ; S. J. Moon ; Hirai2010 . Due to the presence of strong spin-orbit (SO) coupling effects, the 5$d$ transition metal compounds have been comprehensively investigated to find a correlation between strong SO coupling and unconventional superconductivity J. H. Kim ; G. Jackeli . The close correspondence of comparatively weak electronic correlations (0.5-3 eV), strong crystal field effects (1-5 eV) and strong relativistic SO coupling effects (0.1-1 eV) offers an remarkably promising opportunity for the study of the physics resulting from competing spin, orbital, charge, and lattice degrees of freedom Y. Singh ; S. K. Choi . For example, pressure induced superconductivity in 1T-TaS2 B. Sipos , in noncentrosymmetric CePt3Si E. Bauer , and in the geometrically frustrated pyrochlore oxides Cd2Re2O7 M. Hanawa and KOs2O6 S. Yonezawa . In iriduim-containing compounds, superconductivity has been reported in materials such as IrSe2, Cu1-xZnxIr2S4, CeIrSi3, ScIrP, LaIrP and LaIrAs, etc. and in the ternary ThCr2Si2-type compounds BaIr2P2 and SrIr2As2 J. G. Guo ; H. Suzuki ; I. Sugitani ; Y. Okamoto ; Y. P. Qi ; N. Berry . In the case of rare-earth Ir based superconductors most of their electronic characteristics are due to the rare earth components rather than Ir. Nevertheless, there are a few cases, such as CaIr2 H. M. Tutuncu , IrGe D. Hirai and Mg10Ir19B16 T. Klimczuk , where the superconductivity arises from Ir 5$d$ states at the Fermi surface. Until the discovery of LaIr3, no simple example of a La-Ir superconductor has been found whose properties are controlled by the Ir 5$d$ bands at the Fermi level and with strong spin-orbit-coupling.

Recently Haldolaarachchige et al. N. Haldolaarachchige reported that LaIr3 shows superconductivity with $T_{\mathrm{C}}$ = 2.5 K, where the bands near the Fermi surface are governed by the Ir 5$d$ states that are heavily influenced by spin-orbit coupling and there is no strong contribution from the La-orbitals near $E_{\mathrm{F}}$. LaIr3 is therefore one of the few superconductors where 5$d$ electrons play the principal role in the appearance of superconductivity N. Haldolaarachchige . Furthermore, three-dimensional metallic character is seen from the band structure calculations; many bands with large dispersion cross $E_{\mathrm{F}}$ N. Haldolaarachchige . LaIr3 crystallizes in the PuNi3-type rhombohedral structure with the space group $R\bar{3}m$ (166, $D^{5}_{\mathrm{3d}}$ ) N. Haldolaarachchige . There are two crystallographically inequivalent La sites and three Ir sites N. Haldolaarachchige . To examine the role of lanthanide elements and the spin-orbit coupling effects arising from the Ir 5$d$ bands in determining the superconducting properties, the study of other rhombohedral structure type $R$Ir3 materials with $R$-lanthanides (with 4$f$) J. G. Huber ; Y. J. Sato will be important. In particular, materials with partially filled 4$f$ orbitals, such as Ce, Pr, etc., may exhibit magnetism effects and possibly superconductivity. Sato et al. Y. J. Sato reported that CeIr3 is a bulk type-II superconductor with a $T_{\mathrm{C}}$ = 3.4 K, which is the second highest $T_{\mathrm{C}}$ among the Ce-based intermetallic compounds. The low value of the electronic heat capacity coefficient, 23 mJ K*-2* mole*-1*, shows CeIr3 is a weakly correlated electron system Y. J. Sato .

In this paper, we have investigated the superconducting properties of LaIr3 by means of dc magnetization, heat capacity ($C_{\mathrm{P}}$), tranverse field muon spin rotation (TF$-\mu$SR), and zero-field muon spin relaxation (ZF$-\mu$SR) measurements. An analysis of the temperature dependence of the magnetic penetration depth measured using the TF$-\mu$SR measurements suggests an isotropic $s-$wave character for the superconducting gap. The ZF$-\mu$SR result does not show spontaneous appearance of an internal magnetic field in the superconducting state, suggesting time-reversal symmetry is preserved for LaIr3.

II Experimental Details

Polycrystalline samples of LaIr3 were prepared by arc melting of the constituent elements as reported by Haldolaarachchige et al. N. Haldolaarachchige . Powder X-ray diffraction measurement was carried out using a Panalytical X-Pert Pro diffractometer. Magnetic susceptibility measurements were made using a Magnetic Property Measurement System (MPMS) superconducting quantum interference device (SQUID) magnetometer (Quantum Design). Heat capacity measurements were performed by the relaxation method in a Quantum Design Physical Property Measurement System (PPMS). Temperatures down to 0.35 K were attained by a 3He insert in the PPMS. We used the $\mu$SR technique to examine the superconducting ground state of LaIr3. ZF$-\mu$SR measurements were performed on the MUSR spectrometer with the detectors in the longitudinal configuration at the ISIS Muon Facility located at the Rutherford Appleton Laboratory, United Kingdom. For the TF$-\mu$SR measurements the spectrometer were rotated by 90*∘*. The powdered sample of LaIr3 was mounted on a high purity (99.995%) silver plate using diluted GE varnish which was cooled down to 0.1 K using a dilution refrigerator. Using an active compensation system the stray magnetic fields at the sample position were canceled to a level of 1 mG. Spin-polarized muon pulses were implanted into the sample and the positrons from the resulting decay were collected in the detectors positions either forward or backward of the initial muon spin direction. For ZF$-\mu$SR measurements the asymmetry of the muon decay is calculated by, $G_{\mathrm{z}}(t)=[{N_{\mathrm{F}}(t)-\alpha N_{\mathrm{B}}(t)}]/[{N_{\mathrm{F}}(t)+\alpha N_{\mathrm{B}}(t)}]$, where $N_{\mathrm{B}}(t)$ and $N_{\mathrm{F}}(t)$ are the number of counts at the detectors in the forward and backward positions and $\alpha$ is a constant determined from calibration measurements made in the normal state with a small (20 Oe) applied transverse magnetic field. The data were analyzed using the software WiMDA FPW .

III Results and Discussion

III.1 Crystal structures and physical properties

Powder x-ray diffraction investigation confirmed that the sample is single phase with a rhombohedral crystal symmetry with the space group 166 ($R\bar{3}m$) N. Haldolaarachchige . Fig. 1(a) shows the crystal structure of LaIr3. Red symbols represent La1 atoms, yellow symbols represent La2 atoms and small grey symbols represent Ir atoms. A rhombohedral unit cell contains two distinct La atoms (La1 and La2) and three distinct Ir atoms (Ir1, Ir2, and Ir3). The bulk nature of superconductivity in LaIr3 was confirmed by the magnetic susceptibility $\chi(T)$, as shown in Fig. 1(b). The low field $\chi(T)$, measurements displays a strong diamagnetic signal due to a superconducting transition temperature $T_{\mathrm{C}}$ = 2.5 K. The magnetization $M(H)$ curve (not shown here) at low temperature suggest type-II superconductivity.

Fig. 1(c) shows the $C_{\mathrm{P}}(T)/T$ at different applied magnetic fields. At 2.5 K, a clear anomaly is seen showing the superconducting transition which matches well with $\chi(T)$ data. We have observed a second transition in the heat capacity at 1.2 K, which is reminiscent of the two peaks in the heat capacity of UPt3 UPt3 . Nevertheless, our TF-$\mu$SR studies do not show any clear changes at the lower transition and further investigations are needed in order to identify the origin of the second transition. As we do not see any change in ZF-$\mu$SR data at 0.09 K compare with 4 K, this also supports that second transition is not due to a magnetic impurity. In an applied magnetic field of 60 kOe, the heat capacity jumps fully suppressed. Since the normal-state specific heat was found to be invariant under external magnetic fields, the normal-state electronic specific heat coefficient $\gamma$ and the lattice specific heat coefficient $\beta$ were deduced from the data in a field of 60 kOe by a least-square fit of the $C_{\mathrm{P}}(T)/T$ data to $C_{\mathrm{P}}(T)/T=\gamma+\beta T^{2}$, which gives a Sommerfeld constant $\gamma$ = 15.32(3) mJ/(mol-K2), $\beta$ = 0.56(1) mJ/(mol-K4), and from this value of $\beta$ (= ${nN_{A}}\frac{12}{5}\pi^{4}R\Theta_{D}^{-3}$, where $R$ = 8.314 J/mol-K, $n$ is the number of atoms per formula unit, and $N_{A}$ is Avogadro’s number), we have estimated the Debye temperature $\Theta_{D}$ = 430(4) K. The value of $\gamma$ obtained is comparable to the cubic Laves phase superconductor CaIr2, and some other Ir-based heavy element superconductors, which further suggests the common effect of Ir-sub-lattice on many Ir based superconducting materials. In the case of LaIr3 the low value of $\gamma$ may be due to the absence of $f-$ orbitals near the Fermi level. Using the heat capacity jump we have calculated the dimensionless parameter $\Delta C_{\mathrm{P}}$/$\gamma T_{\mathrm{C}}\sim$1 which is lower than 1.43 expected for BCS weak coupling limit.

III.2 Superconducting gap structures

Fig. 2(a-b) shows the TF-$\mu$SR spectra above and below $T_{\mathrm{C}}$ and Fig. 2(b) shows the corresponding maximum entropy plots. It is clear that above $T_{\mathrm{C}}$ the $\mu$SR spectra show a very small relaxation mainly from the quasi-static nuclear moments, and the internal field distribution is very sharp and centered near the applied field. However at 0.1 K the $\mu$SR spectra show moderate damping and the internal field distribution has two components, one sharp near the applied field and one broad which is shifted lower than applied field. In order to shed light on the pairing mechanism and the superconducting gap structure of LaIr3 compound, we have examined the temperature dependence of the TF-$\mu$SR data. Below $T_{\mathrm{C}}$ the asymmetry decay with time due to inhomogeneous field distribution of the flux-line lattice. TF-$\mu$SR asymmetry spectra at all temperatures below and above $T_{\mathrm{C}}$ could be best fitted by using an oscillatory decaying muon spin depolarization function Bhattacharyya1 ; Bhattacharyya2 ; Adroja ; Bhattacharyyarev ,

[TABLE]

where $A_{1}$ and $A_{2}$ are the transverse field asymmetries results from the sample and background from the Ag sample holder respectively, and $\omega_{1}$ and $\omega_{2}$ are the frequencies of the muon precession in the sample and background respectively, $\phi$ is the initial phase and $\sigma$ is the total Gaussian muon depolarization rate. The superconducting contribution to the muon depolarization rate $\sigma_{\mathrm{sc}}$ is estimated employing [$\sigma_{\mathrm{sc}}=\sqrt{\sigma^{2}-\sigma_{\mathrm{n}}^{2}}$], where $\sigma_{\mathrm{n}}$ is the nuclear contribution which is assumed to be fixed over the whole temperature range and was obtained by quadratically subtracting the background nuclear part obtained from the value estimated above $T_{\mathrm{C}}$. In the fitting of TF$-\mu$SR data given in Fig. 2(c), we kept $A_{2}$ fixed at its low-temperature value and this value was estimated first by fitting the lowest temperature data. $A_{1}$ was allowed to vary and its value is nearly temperature independent. The value of the phase was first estimated from the lowest temperature and kept fixed for all other temperatures. We can model the temperature dependence of the penetration depth/superfluid density using the following equation ThCoC2 ; Bhattacharyya1 ; Adroja

[TABLE]

where $f=[1+\exp(-E/k_{B}T)]^{-1}$ is the Fermi function, $\phi$ is the angle along the Fermi surface, and $\Delta_{i}(T,0)=\Delta_{0,i}\delta(T/T_{c})g(\phi)$. The temperature variation of the superconducting gap is approximated by the relation $\delta(T/T_{\mathrm{C}})=\tanh[{1.82(T_{\mathrm{C}}/T-1)]^{0.51}}$ where g($\phi$) refers to the angular dependence of the superconducting gap function and $\phi$ is the polar angle for the anisotropy. g($\phi$) is substituted by (a) 1 for an $s$-wave gap, (b) $|\cos(2\phi)|$ for a $d$-wave gap with line nodes Annett ; Pang . The data can be well modeled using a single isotropic $s-$wave gap of 0.35(1) meV. This gives a gap ratio of 2$\Delta(0)$/kB$T_{\mathrm{C}}$ = 3.31(1), which lower than the value as expected from BCS theory 3.56. This suggest weak coupling superconductivity in the case of LaIr3 which is in agreement with heat capacity data. TF$-\mu$SR measurements of Mg10Ir19B16 reveal spin-singlet $s-$wave pairing, even though the lack of inversion symmetry and the large SO coupling of Mg10Ir19B16 should produce a mixed pairing state, likely with a large spin-triplet contribution A. A. Aczel . Furthermore, TF$-\mu$SR measurements on La7Ir3 suggest that the superconducting gap is isotropic and that the pairing symmetry of the superconducting electrons is predominantly $s-$wave with an enhanced binding strength J.A.T. Barker .

The observed large value of the muon spin depolarization rate below the superconducting transition temperature is related to the superfluid density or penetration depth. For a triangular Sonier ; Chia ; Amato lattice $\frac{\sigma_{\mathrm{sc}}^{2}}{\gamma_{\mu}^{2}}=\frac{0.00371\times\phi_{0}^{2}}{\lambda^{4}}$, where $\phi_{0}$ is the flux quantum number (2.07 $\times$10*-15Tm2*) and $\gamma_{\mu}$ is the muon gyromagnetic ratio $\gamma_{\mu}/2\pi$ = 135.5 MHz T*-1*. As with other phenomenological parameters characterizing the superconducting state, the superfluid density can also be related to quantities at the atomic level. Using London’s theory Sonier $\lambda_{\mathrm{L}}^{2}=\frac{m^{*}c^{2}}{4\pi n_{\mathrm{s}}e^{2}}$, where $m^{*}=(1+\lambda_{\mathrm{e-ph}})m_{\mathrm{e}}$ is the effective mass and $n_{\mathrm{s}}$ is the density of superconducting carriers. Within this simple picture, $\lambda_{L}$ is independent of magnetic field. $\lambda_{\mathrm{e-ph}}$ is the electron-photon coupling constant that can be estimated from the Debye temperature ($\Theta_{D}$) and $T_{\mathrm{C}}$ using McMillans relation McMillan

[TABLE]

Here $\mu^{*}$ is the repulsive screened Coulomb parameter with a typical value of $\mu^{*}$ = 0.15, give $\lambda_{\mathrm{e-ph}}$ = 0.53. As LaIr3 is a type II superconductor, the magnetic penetration depth $\lambda$, superconducting carrier density $n_{\mathrm{s}}$, the effective-mass enhancement $m^{*}$ have been estimated to be $\lambda_{\mathrm{L}}(0)$ = 386(3) nm, $n_{\mathrm{s}}$ = 2.9(1) $\times$1027 carriers $m^{-3}$, and $m^{*}$ = 1.53 $m_{\mathrm{e}}$ respectively, for LaIr3.

III.3 Zero-field muon spin relaxation

Fig. 3(a) shows the time evolution of the zero-field muon spin relaxation asymmetry spectra in LaIr3 at temperatures above and below $T_{\mathrm{C}}$. Below $T_{\mathrm{C}}$, we do not observe any change in the muon spin relaxation rate with decreasing temperature down to the lowest temperature (0.09 K) of measurements, which indicates the absence of any spontaneous internal field in the superconducting phase. The ZF$-\mu$SR spectra for LaIr3 can be well described by the damped Gaussian Kubo-Toyabe (K-T) function Adroja1 ; Adroja2 ; Adroja3 ; Bhattacharyya3 ,

[TABLE]

where

[TABLE]

is the Kubo-Toyabe (KT) functional form expected from an isotropic Gaussian distribution of randomly oriented static (or quasi-static) local fields at muon sites. $\lambda$ is the electronic relaxation rate, $A_{\mathrm{0}}$ is the initial asymmetry, $A_{\mathrm{bg}}$ is the background. The parameters $\sigma_{\mathrm{KT}}$, $A_{\mathrm{0}}$, and $A_{\mathrm{bg}}$ are found to be temperature independent. Our zero-field$-\mu$SR measurements indicates that the time-reversal symmetry is preserved in the superconducting state of LaIr3. In Mg10Ir19B16, ZF$-\mu$SR measurements also find no evidence for time-reversal symmetry-breaking fields A. A. Aczel . In the case of La7Ir3, ZF$-\mu$SR measurements reveal the presence of spontaneous static or quasistatic magnetic fields below the superconducting transition temperature suggest that in this case the superconducting state breaks time-reversal symmetry J.A.T. Barker .

IV Summary

In summary, we have examined the characteristics of the superconducting ground state in LaIr3 using TF and ZF$-$muon spin relaxation/rotation measurements. Bulk type-II superconductivity is observed in the susceptibility measurements with $T_{\mathrm{C}}$ = 2.5 K. Similar to UPt3 two consecutive transitions are seen in the heat capacity data and the origin of the lower temperature feature remains unclear at this time. Transverse field$-\mu$SR measurements reveal a fully gapped $s-$wave type superconductivity with the dimensionless ratio, 2$\Delta(0)/k_{\mathrm{B}}T_{\mathrm{C}}$ = 3.31(1), compared to 3.56 (BCS value) suggesting weak coupling superconductivity. Our zero-field$-\mu$SR measurements do not reveal the spontaneous appearance of an internal magnetic field below the transition temperature, which indicates that the time-reversal symmetry is preserved in the superconducting state. The results underline the need for further research into the properties of Ir-based intermetallic superconductors and especially those that have a noncentrosymmetric structure. Strong spin-orbit coupling in these systems would then be antisymmetric, which often leads to fascinating and unconventional superconductivity with mixed pairing.

V Acknowledgments

AB would like to acknowledge DST India, for Inspire Faculty Research Grant (DST/INSPIRE/04/2015/000169). DTA and ADH would like to thank CMPC-STFC, grant number CMPC-09108, and also DIST for financial support. DTA thanks to JSPS for invitation fellowship.