Predicting doped Fe-based superconductor critical temperature from structural and topological parameters using machine learning

1 Introduction

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, T_c, is of most importance as it determines the applicability in practical situations. T_c is influenced by several factors, including lattice disorders and electronic structures. Generally, high- T_c 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 T_c of 8 K [7]. 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 [8]. 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 T_c [9].

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.

2 Methodology

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, xi,yi;i=1,2,…,nwhere 2 xi∈Rdandyi∈R,from an unknown distribution. A trained GPR predicts values of the response variable y^new given an input matrix x^new. 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, T_c(K

Recall a linear regression model, y=xTβ+ε,where ε∼N0,σ2.A GPR aims at explaining y by introducing latent variables, l(x_i) 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 lxi′scaptures 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 kx,x′=Covl(x),lx′the covariance function, and consider now the GPR model, y=b(x)Tβ+l(x), wherel(x)∼GP0,kx,x′and b(x)∈Rp.kx,x′is often parameterized by the hyperparameter, θ, and thus might be written as kx,x′∣θ.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.

Fig. 1

Data visualization. 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, T_c(K).

Fig. 2

Model performance and training data sizes. When the training dataset size is between 10 and 28, we draw 2000 random sub-samples from the whole sample without replacements to train models. When the training dataset size is 29, 30, or 31, we draw 31C₂₉, ₃₁C₃₀, or ₃₁C₃₁ sub-samples from the whole sample without replacements based on exhaustive sampling to train models. Each trained model based on a certain sub-sample is utilized to score the whole sample and calculate the associated performance measurements. The GPR here uses the Rational Quadratic kernel and Constant basis function with standardized predictors. Provided a performance measure, box plots show the median, 25^th percentile, and 75^th percentile. The whiskers extend to the most extreme values (i. e. ± 2.7 standard deviation coverage) not considered as outliers, and the outliers are plotted using the ‶+" symbol.

Fig. 3

Superconducting transition temperature predictions based on different models. The BNPP (Back Propagation Neural Network) prediction is obtained from [44]. The GPR model of the current study is constructed with the whole sample with the Rational Quadratic kernel, Constant basis function, and standardized predictors. It has a β̂ of 37.0582, σ̂ of 0.1994, σ̂_l of 0.0935, σ̂_f of 17.3098, and α̂ of 0.5223. Detailed numerical predictions based on different models are reported in Table 1 (Columns 8 and 9).

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 σ_l is the characteristic length scale defining how far apart x's can be for y's to become uncorrelated, r_f is the signal standard deviation, r=xi−xjT×xi−xj,and α is a positive-valued scale-mixture parameter. Note that σ_l and σ_f should be positive. This could be enforced through θ such thatθ₁ = log σ_l and θ₂ = log σ_f .

Similarly, four basis functions are investigated here, namely Empty, Constant, Linear, and Pure Quadratic, whose specifications are listed in Eqs. (5) –(8), respectively, where

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 β^θ,σ2,maximizing the log likelihood function with respect to β given θ and σ². It then obtains the β-profiled likelihood, logPy∣X,β^θ,σ2,θ,σ2,

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).

3 Dataset

The experimental data in Table 1 (Columns 2 –7) are from [9]. 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, T_c(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.

4 Result

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 [44]. 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: β^=37.0582,σ^=0,1994, σ^l=0.0935,σ^f=17.3098,and σ^=0,5223.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.

Fig. 4

Bootstrap analysis of GPR prediction stability. We draw 9000 bootstrap samples from the whole sample with replacements. Each bootstrap sample is utilized to train the GPR based on the Rational Quadratic kernel, Constant basis function, and standardized predictors, and calculate the associated performance measurements. The histograms plot distributions of the CC, RMSE, and MAE across the 9000 bootstrap samples, whose averages are 99.99%, 0.0014, and 0.0011, respectively.

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.

5 Conclusion

We develop the Gaussian process regression (GPR) model to predict critical temperature, T_c, 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 T_c.