Nodeless superconductivity and preserved time-reversal symmetry in the noncentrosymmetric ${\mathrm{Mo}}_3\mathrm{P}$ superconductor

We report a comprehensive study of the noncentrosymmetric superconductor ${\mathrm{Mo}}_3\mathrm{P}$. Its bulk superconductivity, with $T_c=5.5\mathrm{K}$, was characterized via electrical-resistivity, magnetization, and heat-capacity measurements, while its microscopic electronic properties were investigated by means of muon-spin rotation/relaxation ($\mu \mathrm{SR}$) and nuclear magnetic resonance (NMR) techniques. In the normal state, NMR relaxation data indicate an almost ideal metallic behavior, confirmed by band-structure calculations, which suggest a relatively high electron density of states, dominated by the Mo $4d$ orbitals. The low-temperature superfluid density, determined via transverse-field $\mu \mathrm{SR}$ and electronic specific heat, suggest a fully gapped superconducting state in ${\mathrm{Mo}}_3\mathrm{P}$, with zero-temperature gap ${\mathrm{\Delta }}_0=0.83\mathrm{meV}$, the same as the BCS gap value in the weak-coupling case, and a zero-temperature magnetic penetration depth ${\lambda }_0=126\mathrm{nm}$. The absence of spontaneous magnetic fields below the onset of superconductivity, as determined from zero-field $\mu \mathrm{SR}$ measurements, indicates a preserved time-reversal symmetry in the superconducting state of ${\mathrm{Mo}}_3\mathrm{P}$ and, hence, spin-singlet pairing.

Figure 1

Room-temperature x-ray powder diffraction pattern and Rietveld refinement for ${\mathrm{Mo}}_3\mathrm{P}$. The open red circles and the solid black line represent the experimental pattern and the Rietveld-refinement profile, respectively. The blue line at the bottom shows the residuals, i.e., the difference between calculated and experimental data. The vertical bars mark the calculated Bragg-peak positions for ${\mathrm{Mo}}_3\mathrm{P}$ (green), ${\mathrm{MoO}}_2$ (blue), and ${\mathrm{Mo}}_3{\mathrm{P}}_4$ (purple). The crystal structure (unit cell) is shown in the inset.

Figure 2

Temperature dependence of the ${\mathrm{Mo}}_3\mathrm{P}$ electrical resistivity. The solid gray line through the data is a fit to Eq. (1). The inset shows the enlarged low-temperature data region, highlighting the superconducting transition.

Figure 3

Temperature dependence of the ${\mathrm{Mo}}_3\mathrm{P}$ magnetic susceptibility ${\chi }_{\mathrm{v}}$ ($\equiv \chi$), measured using ZFC and FC protocols. Data were collected in a 1 mT applied field in a 1.8–12 K temperature range.

Figure 4

(a) Field-dependent magnetization $M\left(H\right)$ recorded at various temperatures up to $T_c$. For each temperature, the lower critical field ${\mu }_0{H}_{c1}$ was determined as the value at which $M\left(H\right)$ deviates from linearity (dashed line). (b) Estimated lower critical field ${\mu }_0{H}_{c1}$ as a function of temperature, with the solid line representing a fit to ${\mu }_0{H}_{c1}\left(T\right)={\mu }_0{H}_{c1}\left(0\right)[1-{(T/T_c)}^2]$.

Figure 5

Temperature-dependent electrical resistivity $\rho (T,H)$ (a) and specific heat $C/T(T,H)$ (b) for different applied magnetic fields, up to 1.4 T. In the first case, $T_c$ was defined as the onset of zero resistance. (c) Field-dependent magnetization $M(H,T)$ measured at various temperatures and in applied magnetic fields up to 1.0 T. From $\rho (T,H), C(T,H)/T$, and $M(H,T)$ one can derive the upper critical field ${\mu }_0{H}_{c2}$, as shown in (d). The solid lines represent fits to an effective GL model, whereas the dash-dotted lines are fits based on a WHH model without spin-orbit scattering.

Figure 6

Normalized electronic specific heat $C_{\mathrm{e}}/{\gamma }_{n}T$ for ${\mathrm{Mo}}_3\mathrm{P}$ versus $T/T_c$. Inset: The measured specific heat $C/T$ as a function of $T^2$. The dashed line in the inset is a fit to $C/T=\gamma +\beta T^2$, while the solid line in the main panel is the electronic specific heat calculated by considering a fully gapped $s$-wave model.

Figure 7

(a) Time-domain $\mathrm{TF}-\mu \mathrm{SR}$ spectra of ${\mathrm{Mo}}_3\mathrm{P}$ measured in its superconducting state (at $T=1.5\mathrm{K}$) at 50 and 300 mT. (b) Field-dependent Gaussian relaxation rate ${\sigma }_{\mathrm{FLL}}\left(H\right)$. The arrows indicate the field values (50 and 500 mT) used in the temperature-dependent $\mathrm{TF}-\mu \mathrm{SR}$ studies. The solid line is a fit to Eq. (8).

Figure 8

$\mathrm{TF}-\mu \mathrm{SR}$ time-dependent spectra, collected at 1.5 K (a) and 7.5 K (b) in an applied field of 50 mT. Fourier transforms of the above time spectra at 1.5 K (c) and 7.5 K (d). The solid lines are fits to Eq. (4) using two oscillations, while the dash-dotted line in (c) represents a fit with a single oscillation. The vertical dashed line indicates the applied magnetic field. Note the clear diamagnetic shift below $T_c$ in (c).

Figure 9

Superfluid density vs temperature, as determined from $\mathrm{TF}-\mu \mathrm{SR}$ measurements in an applied magnetic field of 50 mT (a) and 500 mT (b). The insets show the temperature dependence of the muon-spin relaxation rate $\sigma \left(T\right)$. Two components are required to describe the 50 mT experimental data [see details in Fig. 8]. The different lines represent fits to various models, including $s$-, $p$-, and $d$-wave pairing (see text for details).

Figure 10

(a) Representative $\mathrm{ZF}-\mu \mathrm{SR}$ spectra for ${\mathrm{Mo}}_3\mathrm{P}$ in the superconducting (1.5 and 4 K) and the normal state (8 K). The solid lines are fits to Eq. (10), as described in the text. Temperature dependence of the Lorentzian relaxation rate ${\mathrm{\Lambda }}_{\mathrm{ZF}}$ (b) and Gaussian relaxation rate ${\sigma }_{\mathrm{ZF}}$ (c). The dashed line, which marks the bulk $T_c$ at 5.5 K, is a guide to the eyes. None of the reported fit parameters show distinct anomalies across $T_c$.

Figure 11

Normalized ${}^{31}\mathrm{P}$ NMR Knight shifts $({\delta }_{\mathrm{iso}}-{\nu }_0)/{\nu }_0$ (left) and linewidths (right) vs temperature, as measured at 0.5 and 7 T. While there is a clear drop of frequency and increase of width below $T_c$, the data in the normal phase are practically temperature and field independent, indicating an almost ideal metallic behavior.

Figure 12

(a) $1/T_{1}T$ vs temperature measured at 0.5 and 7 T. Above $T_c$ the product is almost constant (0.054 ${\mathrm{s}}^{-1}{\mathrm{K}}^{-1}$). Inset: $1/T_1$ vs temperature measured at the same fields. The field-independent spin-lattice relaxation rate is linear in temperature and changes slope at $T^{★}\sim 103\mathrm{K}$. (b) The Korringa ratio vs temperature is almost constant down to $T_c$. An average ${\alpha }^{-1}$ value much lower than 1 (here shown by a dashed line) suggests ferromagnetic electronic correlations in ${\mathrm{Mo}}_3\mathrm{P}$.

Figure 13

Calculated electronic band structure for ${\mathrm{Mo}}_3\mathrm{P}$. Total and partial (Mo and P) density of states (a). Orbital-resolved density of states for the Mo (b) and P atoms (c). Band structure of ${\mathrm{Mo}}_3\mathrm{P}$ without (d) and with SOC interactions (e), calculated within $\pm 1$ eV from the Fermi energy level.

Figure 14

Uemura plot for different types of superconductors. With $1/100<T_c/T_{\mathrm{F}}<1/10$, the gray region indicates the band of unconventional superconductors, including heavy fermions, organics, fullerenes, pnictides, and high-$T_c$ cuprates. Selected noncentrosymmetric and elementary superconductors are shown as symbols (adopted from Refs. [16, 65, 66]). The dotted line corresponds to $T_c=T_{\mathrm{F}}$ (here $T_{\mathrm{F}}$ is the temperature associated with the Fermi energy, $E_{\mathrm{F}}=k_{\mathrm{B}}{T}_{\mathrm{F}}$), while the dash-dotted line indicates $T_c/T_{\mathrm{F}}=3.09\times {10}^{-4}$ for ${\mathrm{Mo}}_3\mathrm{P}$.

Figure 15

Selected ${\mathrm{Mo}}_3\mathrm{P}{}^{31}\mathrm{P}$ NMR line shapes measured at (a) 0.5 and (b) 7 T. While at low field a temperature-dependent line broadening and frequency shift are observable (especially below $T_c$), at high field the line shapes are practically temperature independent. The vertical lines indicate the ${}^{31}\mathrm{P}$ NMR reference frequencies at the respective fields. The black line on top of the 295 K data in panel (b) represents a fit using a Knight-shift anisotropy model (see text for details).

Figure 16

Clogston-Jaccarino $K-\chi$ plot for ${\mathrm{Mo}}_3\mathrm{P}$ as measured at 7.067 T. The low-temperature part was fitted by using a linear relation, as given in Eq. (A1). Inset: ${}^{31}\mathrm{P}$ NMR shift vs temperature measured at the same field. The dash-dotted line highlights the shift deviations in the 50–200 K range.