Recently, Fe-based superconductors have shown promising properties of high critical temperature and high upper critical fields, which are prerequisites for applications in high-field magnets. Critical temperature, Tc, is an important characteristic correlated with crystallographic and electronic structures. By doping with foreign ions in the crystal structure, Tc can be modified, which however requires significant manpower and resources for materials synthesis and characterizations. In this study, we develop the Gaussian process regression model to predict Tc of doped Fe-based superconductors based on structural and topological parameters, including the lattice constants, volume, and bonding parameter topological index H31. The model is stable and accurate, contributing to fast Tc estimations.
High magnetic fields are strongly sought in many application areas. Superconducting wires are among the top candidates as they allow for large electric current densities to flow without resistance. However, low temperature superconductors, such as NbTi and Nb3Sn, can only generate magnetic fields up to 10.5 T and 20 T due to their upper critical fields being less than 25 T at 4.2 K. Thus, high-temperature superconductors with upper critical fields greater than 50 T, are promising candidates for magnet fabrication [1, 2, 3, 4, 5, 6].
Iron-based superconductors have high critical temperature next to cuprates, an upper critical field above 50 T, a relatively high irreversibility field, and a high crystallographic symmetry, which are appropriate for fabrication of superconducting wires, tapes, and coated conductors. Among materials characteristics, critical temperature, Tc, is of most importance as it determines the applicability in practical situations. Tc is influenced by several factors, including lattice disorders and electronic structures. Generally, high- Tc superconductivity can be induced and tuned by varying dopant types and levels, where antiferromagnetism is diminished by carrier doping, structural modifications under external pressure, or chemical pressure via isovalent substitutions. For exmaple, FeSe has the simplest structure among the known iron-based superconductors with a Tc of 8 K . Combined with different doping mechanisms, a large variety of Fe-based superconductors were synthesized, such as AFeSe (A = alkali), AeFe2As2 (Ae = alkali-earth), and LnOFeAs (Ln = lanthanide). Previous studies have indicated that changes in lattice constants reflect the expansion or contraction of the interlayer spacing of FeAs layers and local geometry of the FePn(Ch)4 (Pn = pnictide, Ch = chalcogenides) tetrahedron . Furthermore, inspired by the theoretical characterization of molecular branching that uses topological indices, the bonding parameter topological index H31 has been used to describe characteristics of the electronic structure of doped Fe-based superconductors. H31 is defined based on the relative atomic position, Pauling electronegativity, ionic radius, and valence state, which are strongly correlated with Tc .
Therefore, it is important to investigate the relationship between doping mechanisms and changes in critical temperature. However, this type of investigation is usually carried out by extensive experimental approaches, which involve resource-intensive and time-consuming synthesis and characterizations [10, 11, 12, 13, 14, 15, 16]. Recently, data-driven analysis has been proved to facilitate the understanding of the relationship between materials structures and performance by finding statistical correlations between physical attributes and property parameters [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]. This study sheds light on the relationship between superconducting transition temperature of iron-based superconductors and crystal cell structures via the Gaussian process regression (GPR) model. The model leads to accurate predictions of superconducting transition temperature and can be used to help understandings of superconducting transition temperature based on structural and topological parameters.
The introduction of the methodology follows, e. g., [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. GPRs are nonparametric kernel-based probabilistic models. Consider a training dataset, where 2 from an unknown distribution. A trained GPR predicts values of the response variable ynew given an input matrix xnew. In the current study, x'is (i =1, 2, 3, 4) are the lattice constants – a and c, crystal cell volume – V , and bonding parameter topological index – H31, y is the super-conducting transition temperature, Tc(K
Recall a linear regression model, where A GPR aims at explaining y by introducing latent variables, l(xi) where i = 1; 2; . . . ; n, from a Gaussian process such that the joint distribution of l(xi)'s is Gaussian, and explicit basis functions, b. The covariance function of captures the smoothness of y and basis functions project x into a feature space of dimension p.
A GP is defined by the mean and covariance. Let m(x) = E(l(x)) be the mean function and the covariance function, and consider now the GPR model, and is often parameterized by the hyperparameter, θ, and thus might be written as In general, different algorithms estimate β, σ2, and θ for model training and would allow specifications of b and k, as well as initial values for parameters.
|Index||Sample||a (nm–1)||c (nm–1)||V (nm3)||H31||Tc (K)|
|CC w. Tc (K)||–58.75 %||–57.58 %||–58.88 %||36.42 %||–||94.05 %||99.99 %|
Notes: Predictors include the lattice constants – a and c, crystal cell volume – V , and bonding parameter topological index – H31. The target variable is the superconducting transition temperature, Tc(K). The BPNN (Back Propagation Neural Network) prediction is obtained from . The GPR prediction is from the current study. All predictions are visualized in Fig. 3.
The current study explores four kernel functions, namely Exponential, Squared Exponential, Matern 5/2, and Rational Quadratic, whose specifications are listed in Eqs. (1) – (4), respectively, where σl is the characteristic length scale defining how far apart x's can be for y's to become uncorrelated, rf is the signal standard deviation, and α is a positive-valued scale-mixture parameter. Note that σl and σf should be positive. This could be enforced through θ such thatθ1 = log σl and θ2 = log σf .
To estimate β, σ2, and θ, the marginal log likelihood function in Eq. (9) is to be maximized, where K(X; X |θ) is the covariance function matrix given by
The algorithm first computes maximizing the log likelihood function with respect to β given θ and σ2. It then obtains the β-profiled likelihood,
which is to be maximized over θ and σ2 to compute their estimates.
Model performance is assessed with the CC (correlation coefficient), MAE (mean absolute error), and RMSE (root mean square error).
The experimental data in Table 1 (Columns 2 –7) are from . Predictors include the lattice constants – a and c, crystal cell volume – V , and bonding parameter topological index – H31. The target variable is the superconducting transition temperature, Tc(K). This dataset covers a wide range of doped Fe-based oxy-arsenides. Data are visualized in Fig. 1, which reveals nonlinear patterns modeled via the GPR.
We investigate the relationship between model performance and the size of training data in Fig. 2, which shows the benefit of training the GPR with all observations. The stability of the GPR approach is confirmed by bootstrap validation analysis.
The final GPR model is reported in Fig. 3, whose performance is compared with that based on the BNPP (Back Propagation Neural Network) model . The final GPR model is based on the Rational Quadratic kernel (Eq. (4)), Constant basis function (Eq. (6)), and standardized predictors, whose estimated parameters are: 1994, and It is found that the GPR model provides more accurate superconducting transition temperature predictions. Specifically, the CC, RMSE, and MAE based on the GPR model are 99.99%, 0.0027, and 0.0024, respectively, while these measurements based on the BNPP model are 94.05%, 6.1932, and 4.2906.
Given the small sample size (see Table 1) utilized, the prediction stability of the GPR is evaluated via bootstrap validation analysis in Fig. 4, which reveals that the modeling approach maintains high CCs, low RMSEs, and low MAEs across the bootstrap samples. This result suggests that the GPR might be generalized for superconducting transition temperature modeling based on larger samples for iron-based superconductors.
Table 2 shows that the Rational Quadratic kernel and Constant basis function are the optimal choice in terms of performance measurements considered. This choice balances the simplicity and accuracy of the model.
|Sample Mean||Sample Mean|
|Rational Quadratic||Pure Quadratic||74.69%||11.5790||33.2575%||8.8586||25.4440%|
Notes: The final GPR model is based on the Rational Quadratic kernel and Constant basis function, with predictors standardized.
We develop the Gaussian process regression (GPR) model to predict critical temperature, Tc, of doped Fe-based superconductors based on structural and topological parameters. The model is accurate and stable, which suggests the GPR’s usefulness in modeling and understanding the relationship between crystal cell structures and superconducting transition temperature. The modeling exercise might also contribute to effective doping design of Fe-based superconductors for enhanced Tc.
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