Two-gap time reversal symmetry breaking superconductivity in noncentrosymmetric ${\mathrm{LaNiC}}_2$

We report a $\mu \mathrm{SR}$ investigation of a noncentrosymmetric superconductor (${\mathrm{LaNiC}}_2$) in single-crystal form. Compared to previous $\mu \mathrm{SR}$ studies of noncentrosymmetric superconducting polycrystalline and powder samples, the unambiguous orientation of single crystals enables a simultaneous determination of the absolute value of the magnetic penetration depth and the vortex core size from measurements that probe the magnetic field distribution in the vortex state. The magnetic field dependence of these quantities unambiguously demonstrates the presence of two nodeless superconducting energy gaps. In addition, we detect weak internal magnetic fields in the superconducting phase, confirming earlier $\mu \mathrm{SR}$ evidence for a time-reversal symmetry-breaking superconducting state. Our results suggest that Cooper pairing in ${\mathrm{LaNiC}}_2$ is characterized by an interorbital equal-spin pairing model introduced to unify the pairing states of ${\mathrm{LaNiC}}_2$ and the centrosymmetric superconductor ${\mathrm{LaNiGa}}_2$.

Figure 1

(a) TF-$\mu \mathrm{SR}$ asymmetry spectra recorded above and below $T_c$ for a magnetic field of $H=718$ Oe displayed in a 9.03-MHz rotating reference frame. Note, the precession frequency is related to the local field by $\nu =({\gamma }_{\mu }/2\pi )B$, where ${\gamma }_{\mu }/2\pi =13.5539$ MHz/kG is the muon gyromagnetic ratio. Also shown are fits that use Eq. (1) below $T_c$ (see Appendix pp4). (b) Fourier transform of the TF-$\mu \mathrm{SR}$ signal for $T=0.05$ K. The large peak at 9.73 MHz is due to muons stopping outside the sample. Left inset: Photograph of the ${\mathrm{LaNiC}}_2$ single crystals and GaAs wafers attached to an Ag backing plate on an Ag sample holder. Right inset: Fourier transform of TF-$\mu \mathrm{SR}$ signal for $T=5$ K.

Figure 2

(a) Temperature dependence of ${\lambda }_{bc}$ in ${\mathrm{LaNiC}}_2$ below $T\sim 0.38T_c$ and for $H=150$ Oe. The dashed curve is a fit of the form ${\lambda }_{bc}\left(0\right)+a{T}^4$ expected for point nodes along the $a$ axis. The dotted and solid curves are fits to the single-gap $s$-wave BCS expression [34] with zero-temperature energy-gap values ${\mathrm{\Delta }}_{bc}\left(0\right)=1.76k_B{T}_c$ and ${\mathrm{\Delta }}_{bc}\left(0\right)=1.16\left(3\right)k_B{T}_c$, respectively. Inset: Low-temperature (0.41 K $\le T\le 1$ K) electronic contribution to the heat capacity plotted vs $T^3$. (b) Magnetic field dependence of ${\lambda }_{bc}$ in ${\mathrm{LaNiC}}_2$ for $T=0.05$ K. The straight lines are linear fits for $H\le 0.4$ kOe and $H\ge 0.5$ kOe. The fit for $H\le 0.4$ kOe yields the ZF value ${\lambda }_{bc}\left(0\right)=1548\pm 24$ Å. Inset: Magnetic field dependence of ${\lambda }_{ab}$ in ${\mathrm{NbSe}}_2$ for $T=0.02$ K [35].

Figure 3

Magnetic field dependence of $r_0$ and ${\xi }_{bc}$ in ${\mathrm{LaNiC}}_2$ for $T=0.05$ K. Inset: Magnetic field dependence of ${\xi }_{ab}$ in ${\mathrm{NbSe}}_2$ for $T=0.02$ K [35].

Figure 4

Temperature dependence of the normalized superfluid density, ${\lambda }_{bc}^2\left(0\right)/{\lambda }_{bc}^2\left(T\right)$, in ${\mathrm{LaNiC}}_2$ for $H=150$ Oe. Circles denote $\mu \mathrm{SR}$ data points and error bars give the standard error at each temperature. The dashed curve is the superfluid density from single-band BCS theory. The upper solid curve is the total superfluid density in the two-band model, with contributions from the individual bands shown below it. Shaded areas denote the 1 $\sigma$ uncertainty regions associated with the model fit. Fit parameters are given in Table 1 of Appendix pp6.

Figure 5

Representative ZF-$\mu \mathrm{SR}$ asymmetry spectra. The solid curves are fits to Eq. (2). Inset: Temperature dependence of the exponential ZF relaxation rate $\mathrm{\Lambda }$. The open and solid squares correspond to two independent measurements of the ${\mathrm{LaNiC}}_2$ single crystals. The open diamonds are the exponential relaxation rate measured previously in polycrystalline ${\mathrm{LaNiC}}_2$ [3]. The dashed horizontal line denotes the average value of $\mathrm{\Lambda }$ above $T_c, {\langle \mathrm{\Lambda }\rangle }_{\mathrm{N}}$, for each data set. The data set from Ref. [3] and that denoted by the solid squares have been shifted vertically upward so that ${\langle \mathrm{\Lambda }\rangle }_{\mathrm{N}}$ for all three data sets coincide.

Figure 6

Powder x-ray-diffraction (XRD) spectrum of ${\mathrm{LaNiC}}_2$. The blue vertical lines indicate the calculated Bragg peak positions for ${\mathrm{LaNiC}}_2$, which crystallizes in the orthorhombic $Amm2$ space group. The three low intensity XRD Bragg peaks labeled with the pound sign are associated with an $\approx 5$% secondary phase of ${\mathrm{La}}_2{\mathrm{Ni}}_5{\mathrm{C}}_3$, which crystallizes in the tetragonal $P4/mbm$ space group.

Figure 7

Powder x-ray-diffraction (XRD) spectrum of ${\mathrm{La}}_2{\mathrm{Ni}}_5{\mathrm{C}}_3$. The blue vertical lines indicate the calculated Bragg peak positions for ${\mathrm{La}}_2{\mathrm{Ni}}_5{\mathrm{C}}_3$, which crystallizes in the tetragonal $P4/mbm$ space group. The lattice parameters for ${\mathrm{La}}_2{\mathrm{Ni}}_5{\mathrm{C}}_3$ determined from the XRD spectrum are $a=b=8.3266$ Å and $c=4.0244$ Å.

Figure 8

Temperature dependence of the resistivity of polycrystalline ${\mathrm{La}}_2{\mathrm{Ni}}_5{\mathrm{C}}_3$ down to 0.11 K.

Figure 9

(a) Temperature dependence of the heat capacity for different values of magnetic field applied parallel to the $a$ axis. (b) The dotted lines duplicate a procedure described in Ref. [48] to estimate $T_c$ and the heat-capacity jump $\mathrm{\Delta }C/T_c$ at $T_c$. Inset: Polynomial fit to the normal-state heat-capacity data for zero field over the temperature range 3.1–10.2 K, as described in the main text. (c) Temperature dependence of the upper critical field $H_{c2}^{\parallel a}\left(T\right)$ estimated from the heat-capacity data in (a). The solid red curve is a fit to the data, which is described in the main text.

Figure 10

(a) Temperature dependence of the dc magnetization down to 1.8 K measured for a magnetic field of 20 Oe applied perpendicular to the $a$ axis under zero-field cooled (ZFC) and field cooled (FC) conditions. (b) Magnetic hysteresis ($M$ vs $H$) loop at 1.8 K.

Figure 11

Schematic of the geometry for the TF-$\mu \mathrm{SR}$ experiments. The time evolution of the muon spin polarization $P_x\left(t\right)$ is monitored via detection of the muon decay positrons in a pair of counters positioned on opposite sides of the sample.

Figure 12

Fourier transforms of the TF-$\mu \mathrm{SR}$ asymmetry spectrum in ${\mathrm{LaNiC}}_2$ single crystals for a magnetic field of $H=150$ Oe applied parallel to the $a$ axis. The horizontal axis is the difference between the local field sensed by the muon and the external field. The peak at $B-B_{\mathrm{ext}}=0$ is a background signal originating from muons stopping outside the sample.

Figure 13

Fourier transforms of the TF-$\mu \mathrm{SR}$ asymmetry spectrum in ${\mathrm{LaNiC}}_2$ single crystals for $T=0.05$ K and different values of the magnetic field applied parallel to the $a$ axis.

Figure 14

Temperature dependence of the energy gaps, ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$, in the two-band model. Shaded areas denote the 1 sigma uncertainty regions associated with the model fit. The zero-temperature gap ratios, ${\mathrm{\Delta }}_i\left(0\right)/k_B{T}_c$, are 1.82 and 0.77, respectively.

Figure 15

Fits to the measured superfluid density for the two-phase superconductor model. $T_{c1}$ is fixed at 2.7 K and $T_{c2}$ is varied, taking on representative values of 1.5, 1.6, and 1.8 K in the three plots.