Structure and superconductivity in the binary Re$_{1-x}$Mo$_x$ alloys

TL;DR

This study synthesizes and characterizes Re$_{1-x}$Mo$_x$ alloys, revealing their complex phase diagram, superconducting properties, and potential for exploring symmetry-breaking phenomena in superconductivity.

Contribution

It provides a comprehensive investigation of the crystal structures and superconducting behavior across the full Re$_{1-x}$Mo$_x$ phase diagram, highlighting new insights into electron-phonon coupling and symmetry considerations.

Findings

Alloys are superconducting with critical temperatures higher than pure Re and Mo.

The phase diagram includes four distinct crystal structures with varying symmetry.

Superconducting state is fully gapped with evidence of moderate electron-phonon coupling.

Abstract

The binary Re$_{1-x}$Mo$_x$ alloys, known to cover the full range of solid solutions, were successfully synthesized and their crystal structures and physical properties investigated via powder x-ray diffraction, electrical resistivity, magnetic susceptibility, and heat capacity. By varying the Re/Mo ratio we explore the full Re$_{1-x}$Mo$_x$ binary phase diagram, in all its four different solid phases: hcp-Mg ($P6_3/mmc$), $\alpha$-Mn ($I\overline{4}3m$), $\beta$-CrFe ($P4_2/mnm$), and bcc-W ($Im\overline{3}m$), of which the second is non-centrosymmetric with the rest being centrosymmetric. All Re$_{1-x}$Mo$_x$ alloys are superconductors, whose critical temperatures exhibit a peculiar phase diagram, characterized by three different superconducting regions. In most alloys the $T_c$ is almost an order of magnitude higher than in pure Re and Mo. Low-temperature electronic specific-heat data evidence a fully-gapped superconducting state, whose enhanced gap magnitude and specific-heat discontinuity suggest a moderately strong electron-phonon coupling across the series. Considering that several $\alpha$-Mn-type Re$T$ alloys ($T$ = transition metal) show time-reversal symmetry breaking (TRSB) in the superconducting state, while TRS is preserved in the isostructural Mg$_{10}$Ir$_{19}$B$_{16}$ or Nb$_{0.5}$Os$_{0.5}$, the Re$_{1-x}$Mo$_x$ alloys represent another suitable system for studying the interplay of space-inversion, gauge, and time-reversal symmetries in future experiments expected to probe TRSB in the Re$T$ family.

Tables

Table 1. Table 1.: Crystal structure, space group, and refined lattice parameters for four representative phases of the Re 1-x Mo x binary alloy, with Mo concentrations x = 0.12 𝑥 0.12 x=0.12 , 0.23, 0.4, and 0.6.

Sample	Re0.88Mo0.12	Re0.77Mo0.23	Re0.6Mo0.4	Re0.4Mo0.6
Structure	hexagonal hcp-Mg	cubic $\alpha$-Mn	tetragonal $\beta$-CrFe	cubic bcc-W
Space group	$P{6}_3/mmc$ (No. 194)	$I\overline{4}3m$ (No. 217)	$P{4}_2/mnm$ (No. 136)	$Im\overline{3}m$(No. 229)
$a$ (Å)	2.76798(2)	9.58476(3)	9.58514(4)	3.12627(10)
$b$ (Å)	2.76798(2)	9.58476(3)	9.58514(4)	3.12627(10)
$c$ (Å)	4.48728(5)	9.58476(3)	4.97891(2)	3.12627(10)
$V_{\mathrm{cell}}$ (Å3)	29.7743(5)	880.529(6)	457.437(4)	30.5549(17)

Table 2. Table 2.: Refined atomic coordinates and site occupancy factor (SOF) of four different crystal structures of Re 1-x Mo x alloys with x = 0.12 𝑥 0.12 x=0.12 , 0.23, 0.4, and 0.6.

$x$	$T_c^{Cp}$(K)	${\gamma }_n$(mJ/mol-K2)	${\Theta }_D$(K)	$\Delta C/\left({\gamma }_n{T}_c\right)$	$\Delta /\left(k_B{T}_c\right)$	${\lambda }_{\mathrm{ep}}$	$N({ϵ}_F$)/(eV-f.u.)	$N_{\mathrm{band}}({ϵ}_F$)/(eV-f.u.)	$m^{\ast }$/$m_e$
0.00111Data from Refs. Rorer et al., 1965; Heiniger et al., 1966; Smith and Keesom, 1970; Sundar et al., 2015b, a.	1.69	2.30	415.0	1.30	1.71	0.46	0.33	0.27	1.46
0.12	7.45	3.80	402.9	1.18	1.85	0.66	1.61	0.97	1.66
0.20	9.02	3.77	363.7	0.98	1.80	0.72	1.60	0.93	1.72
0.23222centrosymmetric	9.43	3.53	333.0	1.07	1.80	0.77	1.50	0.85	1.77
0.23333non-centrosymmetric	8.65	3.66	311.4	1.90	2.00	0.76	1.55	0.88	1.76
0.35	6.30	3.20	391.1	1.51	1.90	0.63	1.36	0.83	1.63
0.40	6.07	3.01	520.0	1.44	1.82	0.58	1.28	0.81	1.58
0.45	6.60	3.45	397.2	1.59	1.85	0.64	1.46	0.89	1.64
0.60	13.00	4.05	341.8	2.00	2.14	0.87	1.72	0.92	1.87
0.65	12.05	3.89	397.3	1.96	2.10	0.79	1.65	0.92	1.79
0.75	10.30	3.69	480.6	1.64	1.90	0.69	1.57	0.93	1.69
0.80111Data from Refs. Rorer et al., 1965; Heiniger et al., 1966; Smith and Keesom, 1970; Sundar et al., 2015b, a.	8.50	3.65	420.7	1.51	1.87	0.68	1.55	0.92	1.68
0.85111Data from Refs. Rorer et al., 1965; Heiniger et al., 1966; Smith and Keesom, 1970; Sundar et al., 2015b, a.	6.74	3.41	436.2	1.50	1.85	0.62	1.45	0.89	1.62
0.90111Data from Refs. Rorer et al., 1965; Heiniger et al., 1966; Smith and Keesom, 1970; Sundar et al., 2015b, a.	3.02	2.39	429.9	1.35	1.80	0.51	1.01	0.66	1.51
1.00111Data from Refs. Rorer et al., 1965; Heiniger et al., 1966; Smith and Keesom, 1970; Sundar et al., 2015b, a.	0.92	1.83	460.0	1.25	1.70	0.41	0.28	0.20	1.41

Table 3. Table 3.: Superconducting transition temperatures T c subscript 𝑇 𝑐 T_{c} , specific-heat fit parameters, estimated Debye temperatures, density of states, and electron-phonon coupling constants for the Re 1-x Mo x series.

$x$	$T_c^{Cp}$(K)	${\gamma }_n$(mJ/mol-K2)	${\Theta }_D$(K)	$\Delta C/\left({\gamma }_n{T}_c\right)$	$\Delta /\left(k_B{T}_c\right)$	${\lambda }_{\mathrm{ep}}$	$N({ϵ}_F$)/(eV-f.u.)	$N_{\mathrm{band}}({ϵ}_F$)/(eV-f.u.)	$m^{\ast }$/$m_e$
0.00111Data from Refs. Rorer et al., 1965; Heiniger et al., 1966; Smith and Keesom, 1970; Sundar et al., 2015b, a.	1.69	2.30	415.0	1.30	1.71	0.46	0.33	0.27	1.46
0.12	7.45	3.80	402.9	1.18	1.85	0.66	1.61	0.97	1.66
0.20	9.02	3.77	363.7	0.98	1.80	0.72	1.60	0.93	1.72
0.23222centrosymmetric	9.43	3.53	333.0	1.07	1.80	0.77	1.50	0.85	1.77
0.23333non-centrosymmetric	8.65	3.66	311.4	1.90	2.00	0.76	1.55	0.88	1.76
0.35	6.30	3.20	391.1	1.51	1.90	0.63	1.36	0.83	1.63
0.40	6.07	3.01	520.0	1.44	1.82	0.58	1.28	0.81	1.58
0.45	6.60	3.45	397.2	1.59	1.85	0.64	1.46	0.89	1.64
0.60	13.00	4.05	341.8	2.00	2.14	0.87	1.72	0.92	1.87
0.65	12.05	3.89	397.3	1.96	2.10	0.79	1.65	0.92	1.79
0.75	10.30	3.69	480.6	1.64	1.90	0.69	1.57	0.93	1.69
0.80111Data from Refs. Rorer et al., 1965; Heiniger et al., 1966; Smith and Keesom, 1970; Sundar et al., 2015b, a.	8.50	3.65	420.7	1.51	1.87	0.68	1.55	0.92	1.68
0.85111Data from Refs. Rorer et al., 1965; Heiniger et al., 1966; Smith and Keesom, 1970; Sundar et al., 2015b, a.	6.74	3.41	436.2	1.50	1.85	0.62	1.45	0.89	1.62
0.90111Data from Refs. Rorer et al., 1965; Heiniger et al., 1966; Smith and Keesom, 1970; Sundar et al., 2015b, a.	3.02	2.39	429.9	1.35	1.80	0.51	1.01	0.66	1.51
1.00111Data from Refs. Rorer et al., 1965; Heiniger et al., 1966; Smith and Keesom, 1970; Sundar et al., 2015b, a.	0.92	1.83	460.0	1.25	1.70	0.41	0.28	0.20	1.41

Equations

$$\to \text{bcc-W}+\beta \text{-CrFe}\text{and}$$

$$\to \beta \text{-CrFe}.$$

$$\beta \text{-CrFe + hcp-Mg}$$

$$\beta \text{-CrFe}$$

$${\lambda }_{\mathrm{ep}}=\frac{1.04+{\mu }^{\ast }\ln ({\Theta }_{\mathrm{D}}/1.45T_c)}{(1-0.62{\mu }^{\ast })\ln ({\Theta }_{\mathrm{D}}/1.45T_c)-1.04}.$$

$$S=\frac{6{\gamma }_{\mathrm{n}}}{{\pi }^2}\frac{\Delta (0)}{k_{\mathrm{B}}}{\int }_0^{\infty }[f\ln f+(1-f)\ln (1-f)]\mathrm{d}x,$$

Full text

††preprint: Preprint: , 18:44.

I Introduction

Time reversal and spatial inversion are two key symmetries which radically influence electron pairing in the superconducting state. Superconductors with a space-inversion center can host either even-parity spin-singlet (e.g., $s$- or $d$-wave) or odd-parity spin-triplet (e.g., $p$-wave) pairing states. These strict symmetry requirements, however, are relaxed in non-centrosymmetric superconductors (NCSCs), where the antisymmetric spin-orbit coupling (ASOC) allows, in principle, the occurrence of parity-mixed superconducting states, whose mixing degree is related to the strength of the ASOC and to other microscopic parameters. E. and Sigrist (2012); Smidman et al. (2017) Because of the mixed pairing, NCSCs frequently display interesting properties. For instance, some NCSCs, such as CePt3Si,Bonalde et al. (2005) CeIrSi3,Mukuda et al. (2008) Li2Pt3B,Yuan et al. (2006); Nishiyama et al. (2007) K2Cr3As3,Pang et al. (2015); Adroja et al. (2015) and YBiPt Kim et al. (2018) exhibit line nodes in the gap, while others, such as LaNiC2 Chen et al. (2013) and (La,Y)2C3,Kuroiwa et al. (2008) show multiple-gap superconductivity. The external pressure drives CeIrSi3 into a gapless superconductivity, Landaeta et al. (2018) while a nodal behavior has been observed in LaNiC2 and Y2C3. Bonalde et al. (2011); Landaeta et al. (2017); Chen et al. (2011) Recently, many NCSCs,Sato and Fujimoto (2009, 2009); Tanaka et al. (2010); Chadov et al. (2010); Meinert (2016); Sun et al. (2015); Smidman et al. (2017); Kim et al. (2018); Ali et al. (2014) in particular YPtBi,Kim et al. (2018) BiPd, Sun et al. (2015) and PbTaSe2,Ali et al. (2014) have been closely investigated as possible models of topological superconductors.

Interestingly, numerous muon-spin relaxation/rotation ($\mu$SR) studies have revealed that some NCSCs exhibit also time-reversal symmetry breaking (TRSB), concomitant with the onset of superconductivity. Examples include LaNiC2,Hillier et al. (2009) La7(Ir,Rh)3,Barker et al. (2015); Singh et al. (2018a) and several Re-based binary alloys Re$T$($T$= transition metal, e.g., Ti, Zr, Nb, Hf).Singh et al. (2014, 2017a, 2018b); Shang et al. (2018a, b) In general, the breaking of time-reversal symmetry below $T_{c}$ and a lack of space-inversion symmetry of the crystal structure are independent events, not required to occur together. For instance, TRS is broken in several $\alpha$-Mn-type Re$T$ compounds and in the pure elementary Re, Shang et al. (2018b) yet it is preserved in the isostructural Mg10Ir19B16 or Nb0.5Os0.5,Aczel et al. (2010); Singh et al. (2018c) clearly suggesting that TRS breaking is most likely related to the presence of Re atoms, rather than to a generic lack of space-inversion symmetry. Indeed, by converse, the centrosymmetric Sr2RuO4, PrOs4Ge12, and LaNiGa2 also exhibit a broken TRS in the superconducting state.Luke et al. (1998); Aoki et al. (2003); Hillier et al. (2012)

To further study the TRSB in Re$T$ materials, one should identify a system that exhibits both centro- and non-centrosymmetric structures, while still preserving its basic stoichiometry. For instance, depending on synthesis protocol, Re3W can be either a centro- (hcp-Mg-type) or a non-centrosymmetric ($\alpha$-Mn-type) superconductor,Biswas et al. (2011) yet neither is found to break TRS.Biswas et al. (2012) On the other hand, other superconducting Re$T$ compounds, with $T$ = Ti, Zr, Nb, Hf, indeed break TRS, yet mostly adopt the same ($\alpha$-Mn-type) structure. Similar to the Re3W case, the Re1-xMox binary alloys discussed here represent another candidate system. For different Re/Mo ratios, depending on synthesis protocol, they adopt either centro- or non-centrosymmetric structures.Massalski et al. (1996) Although the superconductivity of several Re1-xMox alloys was reported decades ago, only recently the Mo-rich side was studied by different techniques.Roberts (1976); Shum et al. (1986); Okada et al. (2013); Ignatyeva and Velikodny (2004) To date, a systematic study of the full range of Re1-xMox solid solution is missing. In particular, due to synthesis difficulties, its Re-rich side remains largely unexplored. Yet, in view of the non-centrosymmetric structures adopted, it is precisely this part of the phase diagram to be the most interesting one.

In this paper, based on systematic physical-property measurements, we explore the full superconducting phase diagram of the Re1-xMox system. To this aim, polycrystalline Re1-xMox samples, with $0.12\leq$ x $\leq 0.75$, were successfully synthesized. Although samples with different Re/Mo ratios exhibit different crystal structures, they all become superconductors (whose highest $T_{c}$ reaches 12.4 K). All the relevant superconducting parameters, including gap values and symmetries, were determined by magnetometry, transport, and specific-heat measurements, thus allowing us to present the complete Re1-xMox superconducting phase diagram.

After briefly describing the experimental methods in Sec. II, we present the key results in Sec. III, including those of EDX and XRD, electrical resistivity, magnetization, and specific heat. Finally, the overall superconducting phase diagram is presented and discussed in Sec. IV.

II Experimental details

Polycrystalline Re1-xMox (${0.12}\leq x\leq{0.75}$) alloys were prepared by arc melting Re and Mo metals with different stoichiometric ratios in high-purity argon atmosphere. To improve the homogeneity, samples were flipped and remelted several times and, for some of them, the as-cast ingots were annealed at 900∘C for two weeks. The $\beta$-CrFe phase (e.g, Re0.6Mo0.4) was obtained by interrupting the heating immediately after the melting of the precursors. Hence, all the measurements reported here for the $\beta$-CrFe phase refer to as-cast samples. The extra phases were obtained by further annealing the as-cast samples. The $\alpha$-Mn phase with a non-centrosymmetric crystal structure (Re0.77Mo0.23) was stabilized by annealing the sample over one week at 1400∘C in argon or hydrogen atmosphere. Unlike the rather malleable Mo-rich alloys, their Re-rich counterparts turned out to be extremely hard. In addition, the Re0.77Mo0.23 alloy resulted fragile after annealing at 1400∘C. Previously, the same arc-melting processes were adopted to cover the whole $x$ range,Farzadfar et al. (2009); Yang et al. (2010) with the samples being annealed at 1200∘C and then quenched in water. However, these early attempts failed to produce clean $\alpha$-Mn phase.

The x-ray powder diffraction (XRD) patterns were measured at room temperature by using a Bruker D8 diffractometer with Cu K$\alpha$ radiation. The atomic ratios of the Re1-xMox samples were measured by x-ray fluorescence spectroscopy (XRF) on an AMETEK Orbis Micro-XRF analyzer. The magnetic susceptibility, electrical resistivity, and specific-heat measurements were performed on a 7-T Quantum Design Magnetic Property Measurement System (MPMS-7) and a 9-T Physical Property Measurement System (PPMS-9).

III Key experimental results

III.1 Crystal structures

As shown in Fig. III.1(a), the Re1-xMox binary alloys exhibit a very rich phase diagram. While pure Mo and Re form body-centered-cubic (bcc) and hexagonal-close-packed (hcp) structures, respectively, at intermediate Re/Mo ratios two other phases appear, a tetragonal $\beta$-CrFe and a cubic $\alpha$-Mn phase. By suitably combining arc melting and annealing processes, we could successfully synthesize pure-phase alloys representative of all four crystal structures. Although the Re1-xMox phase diagram has been studied before, most of the obtained samples contained at least two different solid phases, thus preventing a systematic study of their physical properties.Farzadfar et al. (2009); Yang et al. (2010) Conforming to the binary phase diagram, the bcc-W and hcp-Mg solid phases were easily obtained via arc melting of Re and Mo and resulted to be stable at 1400∘C. Two reactions take place during the solidifying process:

[TABLE]

In order to synthesize the pure $\beta$-CrFe phase, the first reaction was blocked by interrupting the heating imediately after the melting of both the Re and Mo metals. As for the $\alpha$-Mn phase, this cannot be synthesized via arc melting, since no such phase appears during the liquid-mixture cooling. Yet, two other reactions include this phase:

[TABLE]

Since in both cases solid-state reactions are involved, the pure $\alpha$-Mn phase was obtained by annealing the melted Re and Mo metals at 1400∘C.

Subsequently, the Re/Mo atomic ratio was determined via EDX on polished samples. The estimated Mo (or Re) concentration vs. its nominal value is presented in Fig. III.1(b). For all the samples, the measured Re (Mo) concentration was slightly smaller (larger) than the nominal value, reflecting the preferential evaporation of Re during the arc melting process. For clarity, since typical deviations do not exceed ca. 12%, the nominal concentrations will be used hereafter.