Crystal growth, characterization, and point-contact Andreev-reflection spectroscopy of the noncentrosymmetric superconductor ${\mathrm{Mo}}_3{\mathrm{Al}}_2\mathrm{C}$

We report on the first successful growth of single crystals of the noncentrosymmetric superconductor ${\mathrm{Mo}}_3{\mathrm{Al}}_2\mathrm{C}$ obtained by means of a cubic-anvil, high-pressure, and high-temperature technique. Composition, structure, and normal-state transport properties of the crystals were studied by means of x-ray diffraction, energy-dispersive x-ray spectroscopy, magnetic susceptibility, and resistivity measurements as a function of temperature. Variations in critical temperature ($T_c$) between 8.6 and 9.3 K were observed, probably due to the slightly different carbon stoichiometry of the samples. Single-crystal x-ray refinement confirmed the high structural perfection of the grown crystals. Remarkably, the refined Flack parameter values for all the measured crystals using a $P{4}_{1}32$ space-group model were consistently close to either 0 or 1, hence indicating that the considered crystals belong to two enantiomorphic space groups, $P{4}_{1}32$ and $P{4}_{3}32$. An anomaly in the resistivity is observed at $\simeq 130$ K, most likely associated with the onset of a charge-density-wave phase. The superconducting properties (and in particular the symmetry, the amplitude, and the temperature dependence of the superconducting gap) were studied by using point contact Andreev-reflection spectroscopy. The results confirm that ${\mathrm{Mo}}_3{\mathrm{Al}}_2\mathrm{C}$ is a moderately strongly-coupled superconductor with $2\mathrm{\Delta }/k_{\mathrm{B}}{T}_c\simeq 4$ and unambiguously prove that the order parameter has an $s$-wave symmetry despite the asymmetric spin-orbit coupling arising from the lack of inversion symmetry.

Figure 1

The furnace heat-treatment protocols used for the high-pressure and high-temperature synthesis of ${\mathrm{Mo}}_3{\mathrm{Al}}_2\mathrm{C}$ single crystals. The range of critical temperatures ($T_c$) observed for each growth batch is reported in the bottom part of the figure.

Figure 2

Optical microscope images of ${\mathrm{Mo}}_3{\mathrm{Al}}_2\mathrm{C}$ crystals. (a) Top view of as-grown solidified lump after radial crushing. The lump was limited by the crucible and the circular cross section is due to the crucible walls. (b) Mechanically extracted ${\mathrm{Mo}}_3{\mathrm{Al}}_2\mathrm{C}$ crystals.

Figure 3

Zero-field-cooled magnetization curves for ${\mathrm{Mo}}_3{\mathrm{Al}}_2\mathrm{C}$ crystals extracted from various growth batches. The superconducting transition is rather narrow and the critical temperature varies between 8.6 and 9.3 K.

Figure 4

Reconstructed $h0l$ layer of the reciprocal space. Red circles mark some of the reflections that satisfy the reflection condition $h00:h=4n$.

Figure 5

Crystal structure of ${\mathrm{Mo}}_3{\mathrm{Al}}_2\mathrm{C}$ (space group: $P{4}_{1}32$) as viewed along the $c$ axis (left) and coordination polyhedra of the constituent atoms (right).

Figure 6

(a) Resistivity of the crystal B/2 (blue circles, left-hand vertical axis) and resistance of the crystal F/3 (red circles, right vertical axis) as a function of temperature up to 200 K. Inset: detail of the superconducting transitions. (b) Magnification of the normal-state resistivity of B/2 showing a clear minimum around 130 K.

Figure 7

(a) $I\text{−}V$ characteristics of the B/2 crystal, as a function of temperature. (b) Red solid circles: values of the pseudocritical current $j_m$ (minimum normal-zone existence current density) that marks the onset of an electric field of $1\mu \mathrm{V}$/cm (left scale). The solid line is a fit of the data with a function of the form $j_m=C\sqrt{1-(T/T_c)}$ that gives $T_c=8.48\pm 0.05$ K. Black open circles: resistivity of the same crystal as a function of temperature (right scale). Note that $T_c^{\mathrm{on}}=8.48\pm 0.02$ K.

Figure 8

(a) Normalized conductance curves of a Au/${\mathrm{Mo}}_3{\mathrm{Al}}_2\mathrm{C}$ contact (on the crystal B/3) with $R_N=250\mathrm{\Omega }$, recorded at different temperatures (symbols) with the relevant fit with a 2D BTK model [25] for a $s$-wave order parameter. (b) Temperature dependence of the energy gap extracted from the fit (red circles) and of the other parameters, $Z$ (blue squares) and $\mathrm{\Gamma }$ (black open circles). Note that neither $\mathrm{\Gamma }$ or $Z$ depend on temperature, as expected for a perfectly ballistic contact.

Figure 9

Differential conductance curves of a ${\mathrm{Mo}}_3{\mathrm{Al}}_2\mathrm{C}$/Ag-paste contact for different values of the contact resistance $R_N$, that was progressively reduced by applying short voltage pulses, as explained in Ref. [26]. Note the logarithmic scale on the vertical axis. Arrows indicate the position of the minima in the conductance (i.e., the dips).

Figure 10

Two examples of temperature dependencies of PCARS spectra. (a) Au/${\mathrm{Mo}}_3{\mathrm{Al}}_2\mathrm{C}$ contact on crystal B/2, having $R_N=24.6\mathrm{\Omega }$ and a large Maxwell contribution so that the Andreev features are almost completely eroded by the dips, and the zero-bias feature is unnaturally enhanced with respect to the high-bias tails. (b) Ag-paste ${\mathrm{Mo}}_3{\mathrm{Al}}_2\mathrm{C}$ contact on crystal F/3, having $R_N=76.5\mathrm{\Omega }$, and a smaller Maxwell contribution to the resistance. The dips here fall sufficiently far from the edges of the Andreev-reflection feature, at least at low temperature. In both (a) and (b) all the curves apart from the top one are vertically offset for clarity. The temperature corresponding to the achievement of the normal state is indicated in bold.

Figure 11

(a) The PCARS spectra obtained by normalizing the differential conductance curves of Fig. 10 (symbols) together with the relevant fit within the 2D BTK model. The best fit was obtained by minimizing the SSR in range $\pm 2$ mV. (b) Temperature dependence of the best-fitting parameters $\mathrm{\Delta }$ (solid circles), $\mathrm{\Gamma }$ (open circles), and $Z$ (solid squares).

Figure 12

(a) Amplitude of the energy gap $\mathrm{\Delta }$ extracted from the fit of low-temperature conductance curves, as a function of the contact resistance $R_N$. Different symbols refer to different crystals, as indicated in the label. Note that, despite the difference in $T_c$ among the different crystals, the gap amplitudes approximately follow a common trend. (b) Values of the gap ratio $2\mathrm{\Delta }/k_B{T}_c$ as a function of the contact resistance. As in (a), different symbols refer to different crystals. Both in (a) and (b), lines are only guides to the eye.