Measuring the Spin Polarization with a Superconducting Point Contact and Machine Learning

Abstract

Measuring spin polarization (P) of materials is essential for understanding their fundamental properties and for their application in spintronics. Point contact Andreev reflection (PCAR) spectroscopy is a straightforward yet powerful technique for measuring P. However, conventional analysis methods depend on iterative fitting procedures that are time-consuming, subjective, and often lead to non-unique solutions. This complexity arises from the interplay of multiple physical parameters with pressure, including temperature, superconducting gap, and interfacial barrier strength. Here, we present a machine learning (ML) approach that utilizes convolutional neural networks (CNNs) to facilitate the rapid and automated extraction of P from PCAR spectra. We validate the ML model by analyzing experimental PCAR spectra from various materials reported in the literature. The predicted parameters by the CNN model show excellent agreement with the literature values, demonstrating its robust performance across a wide range of materials and parameter sets. This approach significantly reduces analysis time while maintaining accuracy, providing a practical tool for material characterization, thus accelerating materials discovery for spintronics.

1. Introduction

Point-contact Andreev reflection (PCAR) spectroscopy is a versatile and powerful technique to investigate the fundamental properties of superconductors [1,2,3,4,5]. In addition to its well-established role in determining superconducting characteristics, PCAR spectroscopy has also been used to measure the spin polarization (P) of ferromagnetic materials, a pivotal factor in spintronic applications [6,7,8,9,10,11,12,13]. When a spin-polarized current from a ferromagnet enters a superconductor, the Andreev reflection process becomes spin-dependent, leading to changes in the differential conductance spectra, which directly correlate with the degree of P (Figure 1).

The theoretical framework for spin-polarized PCAR, extending the Blonder-Tinkham-Klapwijk (BTK) theory [14], incorporates the P along with other parameters such as temperature (T), superconducting gap (Δ), and interfacial barrier strength (Z) [15,16]. The shape of the resulting spectra is highly sensitive to these parameters (Figure 2). In particular, P suppresses Andreev (hole) reflection, thereby reducing the zero-bias conductance—at full polarization (P = 1), Andreev reflection is completely suppressed, leading to dI/dV = 0 (Figure 1c). This rich parameter space, while providing comprehensive information about the ferromagnet–superconductor interface, presents significant challenges for spectral analysis.

Typical approaches to extracting P from PCAR spectra rely on iterative fitting procedures that are both time-consuming and subjective. The presence of experimental artifacts—including thermal noise, shot noise, and conductance anomalies arising from non-ballistic transport regimes—further complicates the fitting process [17,18]. Moreover, the high dimensionality of the parameter space often results in non-unique solutions, requiring extensive expertise and careful consideration of physical constraints to obtain reliable results. Such challenges become particularly pronounced when analyzing multiple samples for a systematic study.

Machine learning (ML) has been widely applied across various scientific and engineering disciplines [19,20,21]. Recent advances in ML, particularly deep learning architectures, have demonstrated remarkable success in analyzing complex spectroscopic data across various domains of materials science [22,23,24]. Convolutional neural networks (CNNs), due to their ability to automatically extract relevant features from data, have been suggested as particularly well-suited for spectral analysis tasks [25,26,27,28,29].

In this study, we demonstrate a CNN-based approach for rapid and automated analysis of spin-polarized PCAR spectra, enabling simultaneous extraction of P and other fitting parameters. We train the models using theoretical spectra generated through modified BTK theory for a superconductor–ferromagnet junction with various combinations of the parameters (i.e., T, Δ, Z, and P). The model is then applied to experimental spectra reported in the literature to extract the parameters. The model validation is demonstrated by comparing the simulated spectra using the extracted parameters to the experimental spectra. This study thus serves as another compelling demonstration of the applicability of ML to complex spectral analyses where multiple physical parameters are intricately intertwined.

2. Methods

Figure 3 illustrates the workflow of this study. We trained a CNN model using theoretical data to encompass a sufficient range of parameters. The theoretical data was augmented by adding noise to improve the robustness of the model against experimental artifacts [30]. In PCAR measurements, temperature is a control parameter; we thus set T as input along with spectra for the model, and the other parameters (i.e., Δ, P, and Z) were designated as output data for model training. The trained model was validated by analyzing experimental PCAR spectra obtained from the literature.

2.1. Theoretical Spectra Generation

PCAR spectra for ferromagnet/superconductor junction were simulated using the modified BTK theory, which incorporates P effects [15]. When the current flows from the ferromagnet to the superconductor, the incident and reflected waves are described by:

$${\psi }_{\mathrm{inci}}=\left(\begin{array}{c}1\\ 0\end{array}\right)e^{ i{q}^{+}x}$$

(1)

$${\psi }_{\mathrm{refl}}=a\left(\begin{array}{c}0\\ 1\end{array}\right)e^{ (\alpha +i)q^{-}x}+b\left(\begin{array}{c}1\\ 0\end{array}\right)e^{-i{q}^{+}x},$$

(2)

where $\alpha$ is a dimensionless real number, and P is expressed as a function of $\alpha$:

$$P = {\displaystyle \frac{{\alpha }^2}{{\alpha }^2+4}}.$$

(3)

The probabilities of Andreev reflection ($A$) and ordinary reflection ($B$) are given by $a{a}^{*}$ and $b{b}^{*}$, respectively. The resulting conductance is proportional to $1 + A\left(E\right)-B\left(E\right)$. Including thermal effects via the Fermi-Dirac distribution, the differential conductance is given by:

$$G \left(V\right)={\displaystyle \frac{Z^2+1}{k_{\mathrm{B}}T}}{\int }_{-\infty }^{\infty }{\displaystyle \frac{{e }^{E - \mathit{eV} - \mu / k_{\mathrm{B}}T}}{{(1+e^{ E - \mathit{eV} - \mu / k_{\mathrm{B}}T})}^2}}\times \left[1 + A\left(E\right)-B\left(E\right)\right]\mathrm{d}E,$$

(4)

where $k_{\mathrm{B}}$ and $\mu$ are Boltzmann constant and chemical potential, respectively. Z, T, e, and V are interfacial barrier strength, temperature, elementary charge, and bias voltage, respectively. The differential conductance as a function of bias voltage was calculated by considering the spin-dependent transmission and reflection probabilities at the interface. The P modifies the Andreev reflection probability, resulting in suppressed conductance within the Δ.

We generated theoretical PCAR spectra under various combinations of parameters. The parameters are summarized in

Table 1. Parameters for simulating PCAR spectra. The simulation parameters and their ranges were determined based on the experimental spectra used for model evaluation.

Parameters	Range (Step)
Bias (mV)	0–10 (per 1)
T (K)	1.2–6.6 (per 0.6)
Δ (meV)	0.6–1.6 (per 0.2)
Z	0–1.1 (per 0.1)
P	0.3–0.9 (per 0.05)
# of data	9360

. For each parameter combination, the normalized differential conductance was computed with a bias voltage step size of 0.1 mV over a range of 10 mV, ensuring adequate resolution for distinguishing spectral features associated with different P values. The simulation parameters and their ranges were determined based on typical experimental conditions for ferromagnetic materials interfaced with conventional superconductors.

2.2. Data Preprocessing

To enhance model robustness against experimental artifacts, we augmented the theoretical PCAR spectra by adding random values (using “random.randint” function) within the amplitude range of ±0.02 [30]. For each original spectrum, we generated three spectra with randomly added noise. The final augmented dataset consisted of approximately 40,000 spectra, providing sufficient diversity for robust training.

2.3. Machine Learning Model Training

We implemented a one-dimensional convolutional neural network (1D-CNN) using the PyTorch 2.3.1 library in Python 3.12.3 to predict the fitting parameters (Δ, Z, and P) from input data (spectrum and T). Each parameter was individually Min-Max normalized to 0–1 to unify their different scales and prevent any single parameter from dominating the learning process, ensuring stable and efficient model training. The CNN architecture consisted of two convolutional layers with the same padding and rectified linear unit (ReLU) activation functions, followed by max pooling layers with a pool size of 2. The model architecture and hyperparameters were optimized through a systematic manual search guided by the validation loss. We adopted a constructive approach, starting from a basic neural network and incrementally adding layers and neurons until convergence in prediction accuracy was achieved. Specifically, we evaluated various kernel sizes and dense layer configurations. The final architecture, consisting of two convolutional layers and nine dense layers, was chosen as it minimized the validation loss while maintaining high prediction accuracy.

The extracted feature maps were flattened and concatenated with the temperature input (reshaped as a 1 × 1 matrix) before being passed to fully connected layers. The fully connected network comprised nine dense layers with ReLU activation functions. The output layer consisted of three nodes with linear activation functions corresponding to the predicted parameters (Δ, Z, and P). The model was trained using the adaptive moment estimation (ADAM) optimizer with mean squared error as the loss function and a learning rate of 0.001. The batch size was set to 50, and the dataset was divided into training and test sets in an 8:2 ratio.

2.4. Evaluation

Model performance was preliminarily evaluated using simulated test data. For the test dataset, we computed the coefficient of determination (R²) between the actual and predicted parameter values. The trained model was validated by applying experimental PCAR spectra from ferromagnet/superconductor junctions to find the best fitting parameters. The experimental PCAR spectra on ferromagnets were obtained from the literature—Fe [31], MnAs [32], Ni [33], CrO₂ [33], EuB₆ [34], and Zn_0.95Fe_0.05Al_0.01O [35].

3. Results and Discussion

To determine the optimal number of epochs for model training, we analyze the loss function (or loss) with respect to the number of epochs. An epoch represents one complete pass through the entire training dataset, and the loss measures the difference between the model’s predictions and the actual values. During typical training, both training and validation losses usually decrease. However, if the number of epochs becomes excessively high, the training loss may continue to decrease while the validation loss starts to increase. This phenomenon is commonly seen as a sign of overfitting. Therefore, we select the epoch at which the validation loss reaches its minimum. Figure 4 shows the learning curves (i.e., epochs vs. loss) over 3000 epochs, with the loss displayed on a logarithmic scale for a clearer observation of the changes relative to the number of epochs. Both training and validation losses decrease rapidly up to 300 epochs, after which they become saturated, showing no clear indication of overfitting. Consequently, we chose 514 epochs for model training, where the validation loss reaches its minimum value of 6.64 × 10⁻⁵. The optimal model gives training and test R² values of 99.98% and 99.93%, respectively, indicating nearly perfect prediction performance. Although other deep-learning architectures, such as recurrent neural networks or transformers, could in principle be applied to spectral analysis, the high R² score suggests that our CNN model effectively captures the relevant local spectral features and their nonlinear correlations with the physical parameters, enabling accurate PCAR analysis. This demonstrates that the ML model effectively learns the hidden correlation between spectral shapes and the parameters used to create each spectrum with remarkable accuracy.

To assess the practical applicability of the ML model, we apply it to analyze the parameters for experimental spectra. We gathered experimental spectra for various materials from the literature (Figure 5). This includes typical 3d transition metal ferromagnets Fe [31] and Ni [33], as well as spectra for MnAs [32], CrO₂ [33], EuB₆ [34], and Fe- and Al-co-doped ZnO (Zn_0.95Fe_0.05Al_0.01O) [35].

Figure 5. Analysis of experimental PCAR spectra from various ferromagnet–superconductor (F/S) junctions using the ML model. The materials of each junction are indicated at the top of each panel: Fe/Nb [31], MnAs/Pb [32], Ni/Nb [33], CrO₂/Pb [33], EuB₆/Pb [34], and Zn_0.95Fe_0.05Al_0.01O/Pb [35]. The experimental data (open circles) are compared with the simulated spectra (red solid lines) obtained from the modified BTK theory using the parameters predicted by the ML model. The predicted parameters (Δ (unit: meV), Z, and P) with T (unit: K) are shown in each graph, and the fitting parameters from both the literature and the predicted in this study are summarized in Table 2 for comparison.

Figure 5. Analysis of experimental PCAR spectra from various ferromagnet–superconductor (F/S) junctions using the ML model. The materials of each junction are indicated at the top of each panel: Fe/Nb [31], MnAs/Pb [32], Ni/Nb [33], CrO₂/Pb [33], EuB₆/Pb [34], and Zn_0.95Fe_0.05Al_0.01O/Pb [35]. The experimental data (open circles) are compared with the simulated spectra (red solid lines) obtained from the modified BTK theory using the parameters predicted by the ML model. The predicted parameters (Δ (unit: meV), Z, and P) with T (unit: K) are shown in each graph, and the fitting parameters from both the literature and the predicted in this study are summarized in Table 2 for comparison.

Table 2. Fitting parameters for simulating experimental spectra (Figure 5) extracted from the ML model (this study) and the literature.

Materials	Predictions	Literature	References
Fe/Nb	T = 4.2 K Δ = 1.33 meV Z = 0.00 P = 0.41	T = 4.2 K Δ = 1.35 meV Z = 0 P = 0.42	G. Strijkers et al. [31]
MnAs/Pb	T = 4.2 K Δ = 1.11 meV Z = 0.07 P = 0.51	T = 4.2 K Z = 0.15 P = 0.52	R. Panguluri et al. [32]
Ni/Nb	T = 4.2 K Δ = 1.39 meV Z = 0.07 P = 0.32	T = 4.2 K Δ = 1.35 meV Z = 0.19 P = 0.33	Y. Ji et al. [33]
Ni/Nb	T = 4.2 K Δ = 1.42 meV Z = 0.01 P = 0.37	T = 4.2 K Δ = 1.32 meV Z = 0 P = 0.37	Y. Ji et al. [33]
CrO2/Pb	T = 1.85 K Δ = 1.60 meV Z = 0.65 P = 0.77	T = 1.85 K Δ = 1.51 meV Z = 0.76 P = 0.77	Y. Ji et al. [33]
CrO2/Pb	T = 1.85 K Δ = 1.12 meV Z = 0.00 P = 0.96	T = 1.85 K Δ = 1.14 meV Z = 0 P = 0.96	Y. Ji et al. [33]
EuB6/Pb	T = 1.4 K Δ = 1.33 meV Z = 0.35 P = 0.45	T = 4.2 K Δ = 1.32 meV Z = 0.52 P = 0.47	X. Zhang et al. [34]
Zn0.95Fe0.05Al0.01O/Pb	T = 2 K Δ = 1.02 meV Z = 0.1 P = 0.72	T = 2 K Δ = 1.26 meV Z = 0.23 P = 0.65 Γ = 0.39	T. Xu et al. [35]

Table 2. Fitting parameters for simulating experimental spectra (Figure 5) extracted from the ML model (this study) and the literature.

Materials	Predictions	Literature	References
Fe/Nb	T = 4.2 K Δ = 1.33 meV Z = 0.00 P = 0.41	T = 4.2 K Δ = 1.35 meV Z = 0 P = 0.42	G. Strijkers et al. [31]
MnAs/Pb	T = 4.2 K Δ = 1.11 meV Z = 0.07 P = 0.51	T = 4.2 K Z = 0.15 P = 0.52	R. Panguluri et al. [32]
Ni/Nb	T = 4.2 K Δ = 1.39 meV Z = 0.07 P = 0.32	T = 4.2 K Δ = 1.35 meV Z = 0.19 P = 0.33	Y. Ji et al. [33]
Ni/Nb	T = 4.2 K Δ = 1.42 meV Z = 0.01 P = 0.37	T = 4.2 K Δ = 1.32 meV Z = 0 P = 0.37	Y. Ji et al. [33]
CrO2/Pb	T = 1.85 K Δ = 1.60 meV Z = 0.65 P = 0.77	T = 1.85 K Δ = 1.51 meV Z = 0.76 P = 0.77	Y. Ji et al. [33]
CrO2/Pb	T = 1.85 K Δ = 1.12 meV Z = 0.00 P = 0.96	T = 1.85 K Δ = 1.14 meV Z = 0 P = 0.96	Y. Ji et al. [33]
EuB6/Pb	T = 1.4 K Δ = 1.33 meV Z = 0.35 P = 0.45	T = 4.2 K Δ = 1.32 meV Z = 0.52 P = 0.47	X. Zhang et al. [34]
Zn0.95Fe0.05Al0.01O/Pb	T = 2 K Δ = 1.02 meV Z = 0.1 P = 0.72	T = 2 K Δ = 1.26 meV Z = 0.23 P = 0.65 Γ = 0.39	T. Xu et al. [35]

The P of MnAs has garnered attention, especially since significant tunnel magnetoresistance (TMR) has been observed in MnAs/AlAs/(Ga,Mn)As junctions. Understanding the P is crucial for explaining the observed high TMR and for utilizing MnAs in spin injection into semiconductors. Measuring the P of this material has been a key focus due to its potential near-half-metallic state and high Curie temperature.

CrO₂ has been theoretically predicted to be a half-metal and a strong candidate for low-field MR devices. PCAR measurements on this material have shown exceptional P, with P ≈ 96%, which supports the notion of its half-metallicity.

EuB₆, a ferromagnetic semimetal among divalent hexaborides, has been extensively studied due to its unusual transport properties and low-temperature magnetic behaviors. In addition to the possibility of half-metallicity, the electronic band structure at the X point has been debated—whether it is semi-metallic or semiconducting with a large gap. PCAR measurements on this material have suggested that EuB₆ is not a half-metal, revealing P = 56 ± 9%. Along with Hall and magnetoresistance measurements, the spin-dependent band structure has also been discussed.

In ZnO, Al doping provides donor electrons that enhance conductivity, which is considered essential for stabilizing ferromagnetism and spin-dependent band splitting induced by magnetic Fe doping. Magneto-transport behaviors such as magnetoresistance and the anomalous Hall effect do not provide conclusive evidence of carrier spin polarization. Therefore, direct measurement of spin polarization near the Fermi level has become a critical issue for these materials. PCAR measurements on Zn_1-xFe_xAl_0.01O/Pb junction provide P = 65 ± 8% for x = 0.05 and P = 40 ± 5% for x = 0.1. Combined with magnetoresistance and anomalous Hall effect, these results confirm spin-polarized carrier transport while revealing that non-uniform magnetization and interfacial spin-flip scattering reduce the measured polarization.

Note that Nb and Pb have been generally used for PCAR measurements owing to their relatively high superconducting critical temperature (T_c) and well-known Δ (Nb: 1.5 meV; Pb: 1.3 meV), which can be used as a reference for analyzing PCAR spectra.

By applying the ML model to the experimental spectra of various materials, we can evaluate its practical capability across a wide range of parameters, including P. Although P is an intrinsic property of a material, it has been suggested that the measured P value can vary due to surface oxidation, which leads to spin-mixing effects and reduces the intrinsic P near the surface. Since the barrier parameter Z reflects the degree of surface oxidation, which depends on the contact condition, we selected two different spectra obtained from Ni and CrO₂, measured under different contacts.

For practical validation, we input the experimental spectra and the measurement temperature into the model, which then outputs the parameters Δ, Z, and P. To verify the reliability of these extracted parameters, we simulate theoretical spectra using them and overlay the results (represented by red lines) with the corresponding experimental spectra (shown as circle symbols) (Figure 5). As mentioned previously, the analyzed parameters cover a wide range, with Z varying from 0 to over 0.5 and P ranging from approximately 0.3 to 1. The model demonstrates excellent predictive performance across these parameter values. To quantitatively compare the predicted parameters (Δ, Z, and P) with those reported in the literature, we compile them in Table 2 and visualize the results in parity plots (Figure 6). The Δ, Z, and P data points lie very close to the ideal prediction lines (dashed lines), directly demonstrating strong agreement between our predictions and the reported results. It can be seen that one data point in each parameter slightly deviates from the line. The single outlier is the data point for Zn_0.95Fe_0.05Al_0.01O (marked by the red dashed circle in Figure 6). In the literature for Zn_0.95Fe_0.05Al_0.01O, an additional broadening term (Γ) was taken into account in BTK fitting for spectral analysis. The Γ term accounts for quasiparticle lifetime, inelastic scattering, and impurity effects, which influence the predicted Z and Δ, and consequently P. If the predicted Δ significantly deviates from the expected values, the analysis should be revised using a modified theoretical model. The same principle applies to ML-based analysis: the ML model can also be further improved by training with spectra generated using different theoretical models.

The Z values predicted by the ML model are overall slightly smaller than those reported in the literature. This discrepancy can be attributed to the use of different theoretical models (i.e., equations). Several studies have employed a modified BTK theory, where P is treated as a coefficient in the linear combination of fully polarized current and unpolarized current [11,16,31]:

$$G\left(V\right) = \left(1-P_l\right)G_{\mathit{unpolarized}}\left(V, Z, \Delta , T\right) + P_l{G}_{\mathit{polarized}}\left(V, Z, \Delta , T\right)$$

(5)

The P values predicted by this theoretical model were found to be highly dependent on Z. To address this issue, an alternative theoretical model was proposed, in which P is defined as a mathematically independent parameter rather than as a weighting factor in the linear combination [15]. It has been demonstrated that the alternative model yields p values that are less dependent on Z and smaller Z values than those obtained from the previous model. The marginal differences between the ML-predicted values and those reported in the literature stem from the theoretical models adopted for the analysis. If necessary, the ML model can be retrained using a different dataset. We also note that P values appear less sensitive to the selection of theoretical models.

Regardless of the theoretical framework used, the present study indeed demonstrates the practical applicability of PCAR measurements combined with ML-assisted analysis for measuring P in materials, thereby accelerating material discovery for spintronic applications. Moreover, this work highlights the potential of ML as a robust tool for analyzing spectra involving competing or correlated physical parameters, enabling rapid and reliable quantitative analysis.

To further demonstrate the practical applicability of ML-assisted spectrum analysis compared with conventional fitting methods, we compare our ML approach with two traditional fitting methods: least square fitting (using the SciPy library) and dual annealing (a combination of simulated annealing and L-BFGS-B) (Figure 7). All methods were applied to the same synthetic spectrum, which was intentionally generated using the parameters T = 2.7 K, Δ = 1.5 meV, Z = 0.7, and P = 0.53, not included in the training dataset, and with noise of ±0.01 and ±0.05. For the spectrum with the noise level of ±0.01, both least-squares fitting and the ML model accurately reproduce the ground-truth parameters (i.e., the same values used to generate the spectrum) to two decimal places, while dual annealing fails to obtain accurate fits. A similar trend can be seen for the spectrum with the noise level ±0.05. Notably, computation differed significantly; the CNN model required only 0.1 s, whereas least-squares fitting took approximately 700 s and dual annealing around 6500 s, respectively. We have tested our model with varying noise level, and the CNN model provides reasonable predictions for spectra with noise levels up to approximately 10% of a given spectral dI/dV range. Along with fitting accuracy, this result attests to the advantage of ML-assisted spectral analysis in terms of robustness against noise and computational efficiency. It can also be extended to new superconducting systems through transfer learning.

4. Conclusions

ML enables rapid and objective analysis of spectra, overcoming the ambiguity and computational cost of conventional fitting methods. Ferromagnet–superconductor PCAR spectra represent a typical example of complex experimental spectra, where multiple parameters are strongly correlated and often competing. The present study demonstrates that ML remains effective even under such conditions, providing reliable parameter extraction and robust interpretation. This approach paves the way for high-throughput material screening and discovery for spintronics. The trained model can be extended to batch processing through simple loop-based execution and integrated into automated workflows for sample preparation, measurement, and analysis aimed at closed-loop optimization.