Osmium-Based Superconducting HEAs

• Osmium-based superconducting high entropy alloys are multicomponent solid solutions where superconductivity arises from intricate disorder, varied local chemistry, and distinct crystal lattices.
• They are synthesized by arc melting stoichiometric elemental mixtures to form defect-tolerant, single-phase structures, with superconducting transition temperatures between 2.90 K and 5.11 K.
• Tuning strategies focus on adjusting valence electron count, crystal symmetry, and disorder to optimize electron correlations and explore unconventional pairing mechanisms.

Osmium-based superconducting high-entropy alloys (HEAs) constitute a distinct materials platform where superconductivity emerges as a result of the unique interplay between multicomponent disorder, varied local chemical environments, and an underlying simple (often high-symmetry, but sometimes noncentrosymmetric) crystal lattice. In these systems, osmium (a heavy 5d transition metal) is incorporated alongside four or more other elements, resulting in solid solutions whose superconducting properties cannot be trivially predicted from the parent elements or traditional binary/ternary compounds. The development of osmium-based HEA superconductors has expanded the landscape of unconventional superconductivity and provided new approaches to investigating the combined effects of disorder, correlation, and strong spin-orbit coupling.

1. Synthesis, Structure, and Phase Stability

Osmium-based HEA superconductors are synthesized primarily by arc melting stoichiometric mixtures of high-purity elemental precursors in an inert argon atmosphere, with repeated remelting cycles to ensure chemical homogeneity and uniform random mixing. Exemplary systems include the Re₃.₅Os₃.₅Ta₀.₅Hf₀.₅Nb₃ and (Ru/Re)₀.₃₅Os₀.₃₅Mo₀.₁₀W₀.₁₀Zr₀.₁₀ compositions, as well as several closely related compounds within this quinary and septenary compositional landscape (Chen et al., 7 Aug 2025, Bhatt et al., 26 Sep 2025).

X-ray diffraction (XRD) and Rietveld refinement have verified that these alloys crystallize into well-defined, defect-tolerant crystal lattices. The most frequently observed structures include:

• Noncentrosymmetric α-Mn type (space group $I$–43$m$) for Re–Os-rich alloys (lattice parameter $a \sim 9.6–9.8$ Å).
• Hexagonal close-packed (hcp) for Ru–Os-rich variants.

Single-phase samples are attainable within narrow compositional ranges, with the random arrangement of atoms confirmed by the absence of superstructure reflections and minimal impurity phases. Notably, osmium does not disrupt phase stability, supporting its compatibility in HEA formation and providing access to magnetic and electronic phases not observed in simpler binary/ternary systems.

These osmium-based HEAs display exceptional chemical robustness: even after immersion for one month in 0.5 mol/L HCl, they retain both structural integrity and superconducting properties—evidenced by unchanged XRD patterns and invariance of $T_c$ and superconducting transition widths (Chen et al., 7 Aug 2025).

2. Superconducting Properties

Osmium-based HEA superconductors exhibit type-II superconductivity. Transport, magnetization, and specific heat measurements establish superconducting transition temperatures ($T_c$) between $2.90$ K and $5.11$ K, depending on composition, crystal structure, and valence electron count (VEC) (Chen et al., 7 Aug 2025, Bhatt et al., 26 Sep 2025).

Critical field parameters are extracted as follows:

• Upper critical field, $H_{c2}(T)$: Fitted using the Ginzburg–Landau equation:

$$H_{c2}(T) = H_{c2}(0)\frac{1-\left(\frac{T}{T_c}\right)^2} {1+\left(\frac{T}{T_c}\right)^2}$$

Reported zero-temperature upper critical fields range from $5.99$ T to $7.71$ T in the Re–Os–based materials, with some systems approaching the Pauli paramagnetic limit ($H_{\mathrm{Pauli}} = 1.86\,T_c$) (Chen et al., 7 Aug 2025).

• Lower critical field, $H_{c1}(T)$: Extracted via

$$H_{c1}(T) = H_{c1}(0)\left[1-\left(\frac{T}{T_c}\right)^2\right]$$

Specific heat measurements reveal $\Delta C/\gamma T_c$ in the $1.38$–$1.48$ range, consistent with Bardeen-Cooper-Schrieffer (BCS) weak-coupling superconductivity (1.43), indicating moderate electron–phonon coupling (Chen et al., 7 Aug 2025). Debye-model fits take the form $C_p/T = \gamma + \beta T^2 + \delta T^4$.

Critically, the Kadowaki–Woods ratio (KWR) is anomalously enhanced ($$\sim 10^{-4}\ \mu\Omega\,\mathrm{cm}\,(\mathrm{mol K}/J)^2$$), signifying pronounced electron–electron correlations akin to heavy fermion systems. Low-temperature resistivity data follow $\rho(T) = \rho_0 + AT^2$.

3. Electronic Structure, Disorder, and the Role of Valence Electron Count

The electronic properties in osmium-based HEA superconductors are intricately connected to disorder at both the compositional and structural levels. Core level photoemission experiments show significant elemental charge transfer, giving rise to cumulative core level shifts commensurate with differences in electronegativity and VEC (Bhatt et al., 26 Sep 2025). For instance, moving from Ru-based (VEC ≈ 7.2) to Re-based (VEC ≈ 6.85) alloys shifts the Fermi level due to a rigid band shift, verifying the sensitivity of the density of states (DOS) to subtle changes in electron count.

Density functional theory (DFT) calculations demonstrate that $d$-states (predominantly from Os, Ru/Re) dominate near the Fermi level, while $s$-states reside at deeper binding energies. The precise crystal symmetry—e.g., comparing hcp and α-Mn for the same alloy composition—gives rise to measurable shifts in the Fermi level, again underlining the nontrivial role of lattice symmetry (Bhatt et al., 26 Sep 2025).

Disorder fundamentally suppresses the spectral density of states (SDOS) at the Fermi level, as revealed by high-resolution photoemission. The SDOS exhibits a $|E - E_F|^{1/2}$ dependence with energy and a $T^{1/2}$ temperature dependence, in quantitative agreement with the Altshuler-Aronov theory, indicative of strong carrier localization induced by electron–electron interactions in a highly disordered background.

Feature	Ru–Os-based HEA	Re–Os-based HEA
Crystal structure	hcp	α-Mn (noncentrosymmetric)
VEC (approximate)	7.2	6.85
SDOS suppression	Present, Altshuler-Aronov	Present, Altshuler-Aronov

4. Pairing Mechanism and the Effects of Disorder

Results from both experiment and theory converge on a predominantly phonon-mediated mechanism for superconductivity in osmium-based HEAs (Bhatt et al., 26 Sep 2025). The electron–phonon coupling strength ($\lambda$) is calculated via DFPT (virtual crystal approximation):

$$\lambda = 2 \int_{0}^{\omega_{\rm max}} \frac{\alpha^2 F(\omega)}{\omega} d\omega$$

For Ru–HEA, $\lambda_{\mathrm{DFT}} \approx 0.60$; experiment yields $\lambda_{\exp} \approx 0.52$ with $T_c (\mathrm{DFT}) \approx 3.77$ K compared to $T_c (\exp) = 2.90$ K. In the Re–HEA, calculations assuming hcp symmetry yield much higher $\lambda$ and $T_c$ than observed experimentally for the α-Mn structure ($\lambda_{\exp} \approx 0.53$, $T_c \approx 4.9$ K), highlighting a suppression of pairing in the structurally disordered state.

The reduction in $T_c$ relative to nominally “clean” band structure-based calculations is attributable to disorder-induced localization, as evidenced by the suppressed SDOS and the strong Altshuler-Aronov effects (Bhatt et al., 26 Sep 2025). Disorder thus acts as a pair-breaking mechanism, lowering $T_c$ and altering low-energy excitation spectra.

5. Tuning Strategies: VEC, Structural Design, and Disorder Control

Empirical and theoretical results underscore the critical importance of valence electron count (VEC), crystal symmetry, and disorder management in optimizing superconducting properties:

• VEC dependence: In the Re–Os–based HEAs, $T_c$ increases monotonically with VEC, consistent with classic Matthias-type rules ($6.45 < {\rm VEC} < 6.81$ for studied systems), providing a straightforward axis for materials design (Chen et al., 7 Aug 2025).
• Structure selection: Controlled selection of lattice symmetry (hcp vs. α-Mn) allows for tunable electronic structure and $T_c$. The α-Mn phase stabilizes noncentrosymmetric superconductivity, with possible routes toward unconventional states such as singlet–triplet mixing.
• Disorder engineering: Compositional disorder can be varied by adjusting element selection and mixing ratios; structural disorder is managed through annealing and synthetic control. Modulating disorder tunes the extent of carrier localization and may optimize superconductivity by balancing band structure effects against localization.

6. Robustness, Correlation Phenomena, and Prospective Applications

Osmium-based HEAs remain chemically and structurally robust under aggressive conditions, retaining superconductivity and phase purity after prolonged exposure to strong acids (Chen et al., 7 Aug 2025). The large Kadowaki–Woods ratio and renormalized electronic specific heat coefficients point to strong electronic correlations, approaching those in heavy fermion systems – a property unusual for conventional phonon-mediated superconductors.

Potential application domains include:

• Devices and wiring in corrosive or mechanically stressful environments, leveraging the mechanical toughness and chemical durability of the HEA matrix.
• Magnet technology and superconducting electronics, where disorder tolerance and robust superconductivity under pressure or strain are advantageous.
• Platforms for investigating the interplay between disorder-induced localization, strong correlations, and unconventional superconducting states.

7. Current Challenges and Outlook

The continued development of osmium-based HEA superconductors faces several barriers:

• The synthesis of single-phase alloys with controlled VEC and symmetry is sensitive to composition; not all candidate alloys yield homogeneous phases.
• Theoretical modeling of disorder—especially beyond effective medium approximations—remains complex; standard DFT approaches do not fully capture the effects of disorder on pairing or localization.
• Thin film growth and high-quality single crystal synthesis have not yet been realized, limiting the scope of advanced spectroscopic and transport investigations.
• The interplay between strong spin-orbit coupling, noncentrosymmetric crystal field effects, and strong correlations remains an open area for detailed experimental and theoretical research.

Ongoing studies are poised to clarify these points, suggesting that targeted control of composition, structure, and disorder may yield new osmium-based HEAs with elevated $T_c$, enhanced upper critical fields, and potentially unconventional superconducting ground states (Sun et al., 2019, Chen et al., 7 Aug 2025, Bhatt et al., 26 Sep 2025).