Ab initio modeling of superconducting alloys

TL;DR

The paper tackles the challenge of ab initio thermodynamic modeling for superconducting alloys by introducing the Extended Generalized Quasi-Chemical Approximation (EGQCA), which minimizes the Gibbs mixing free energy $\ abla G$ over a compact set of non-equivalent clusters while explicitly incorporating vibrational contributions $\ abla A$. By treating the alloy as an ensemble of $J$ clusters with computed energies, degeneracies, and cluster-specific properties, EGQCA enables prediction of composition- and growth-condition–dependent superconducting properties such as the electron-phonon coupling parameter $\lambda$ and the critical temperature $T_c$, via $\,\alpha^2F(\omega)$ and the Allen–Dynes formula. The authors validate EGQCA on Mg$_{1-x}$Al$_x$B$_2$, Nb–Ti, Nb–V, and predict high-$T_c$ behavior in Y$_{1-x}$Ca$_x$H$_6$ at high pressure, demonstrating accurate lattice parameters, ordering tendencies, miscibility gaps, and $T_c$ trends, often with only a small set of adjustable parameters (e.g., interpolated $\mu^*$). This work highlights the potential of EGQCA for in silico high-throughput screening of superconducting alloys and superhydrides, bridging synthesis, thermodynamics, and electronic-phononic properties under realistic thermodynamic and vibrational conditions.

Abstract

Designing new, technologically relevant superconductors has long been at the forefront of solid-state physics and chemistry research. However, developing efficient approaches for modeling the thermodynamics of superconducting alloys while accurately evaluating their physical properties has proven to be a very challenging task. To fill this gap, we propose an ab initio thermodynamic statistical method, the Extended Generalized Quasichemical Approximation (EGQCA), to describe off-stoichiometric superconductors. Within EGQCA, one can predict any computationally accessible property of the alloy, such as the critical temperature in superconductors and the electron-phonon coupling parameter, as a function of composition and crystal growth conditions by computing the cluster occurrence probabilities that minimize the overall mixing Gibbs free energy. Importantly, EGQCA incorporates directly chemical ordering, lattice distortions, and vibrational contributions. As a proof of concept, we applied EGQCA to the well-known Al-doped MgB$_2$ and to niobium alloyed with titanium and vanadium, showing a remarkable agreement with the experimental data. Additionally, we model the near-room temperature sodalite-like Y$_{1-x}$Ca$_x$H$_6$ superconducting solid solution, demonstrating that EGQCA particularly possesses a promising potential for designing in silico high-$T_{\text{c}}$ superhydride alloys. Our approach notably enables the high-throughput screening of complex superconducting solid solutions, intrinsically providing valuable insights into the interplay between synthesis, thermodynamics, and physical properties.

Figures (7)

• Figure 1: The EGQCA workflow encompasses the following steps: (1) Construction of non-equivalent clusters using symmetry space group analysis; (2) Structural optimization and (3) electronic structure calculations employing the DFT formalism; (4) Calculation of the phonon frequencies and electron-phonon coupling through the first-order linear response theory within DFPT; (5) Application of the EGQCA minimization to calculate thermodynamic quantities and (6) the determination of any computationally accessible physical property as a function of temperature and composition utilizing the cluster occurrence probability that minimizes the global Gibbs mixing free energy. This approach allows for comprehensive and accurate modeling of alloyed superconductors, enabling the prediction of their physical properties and behavior.
• Figure 2: Thermodynamic properties for Mg$_{1-x}$Al$_{x}$B$_2$ solid solution. (a) Excess enthalpy, $\Delta_{j}$, as a function of composition. (b) Kullback-Leibler divergence, $\Delta_{\text{KL}}$, as a function of temperature for selected aluminum compositions below 400 K. (c) Gibbs mixing free energy, $\Delta G$, (d) Mixing enthalpy, $\Delta H$, (e) configurational entropy, $\Delta S$, and (f) mixing vibrational entropy, $\Delta A$, as a function of composition for crystal growth conditions between 300 K and 1400 K.
• Figure 3: Cluster fraction probabilities for the 22 non-equivalent Mg$_{1-x}$Al$_{x}$B$_2$ clusters as a function of temperature at representative compositions $x = 0.125, 0.25, 0.375, 0.50, 0.625, 0.75$.
• Figure 4: Structural and superconducting properties of Mg$_{1-x}$Al$_x$B$_2$ solid solution within EGQCA: (a) lattice parameters, $a$ and $c$; (b) superconducting critical temperature, $T_\text{c}$; (c-d) Eliashberg spectral function, $\alpha^2 F$, as a function of composition and frequency; (e) phonon density of states, $D(\omega)$, as a function of composition and frequency. The solid lines correspond to the configurationally averaged EGQCA values and the shaded areas to the EGQCA standard deviation. Experimental data were extracted from Refs. slusky2001postorino2001renker2002xiang2002li2002gonnelli2006putti2003karpinski2005mudgel2007.
• Figure 5: Superconducting critical temperature, $T_\text{c}$, of Nb$_{1-x}$Ti$_x$ solid solution as a function of composition within GQCA. The lines correspond to the configurationally averaged GQCA values and the shaded area to the GQCA standard deviation. The dashed red line represents $T_\text{c}$ obtained with $\mu^{*}$ = 0.24 for all clusters, whereas the solid blue line employs linear interpolation from 0.24 ($x = 0.00$) to 0.37 ($x = 0.75$). Experimental data were extracted from Ref. hulm1961.