Machine learning mirrors finite element physics to predict photoionization in quantum dot rings

Researchers have trained machine learning models to predict the photoionization cross section of a donor impurity in a GaAs/AlGaAs quantum dot–ring nanostructure, achieving agreement with finite element method calculations so close that the coefficient of determination exceeded 0.9999 for the two best algorithms. The study, published in Discover Nano, combined numerical solutions of the effective-mass Schrödinger equation with Gradient Boosting Regression, K-Nearest Neighbors, and LightGBM models, offering a computationally cheap surrogate for simulations that are otherwise expensive to run across large parameter spaces.

The physical quantity at the center of the work is the photoionization cross section, or PICS, which measures the probability that an electron bound to a donor impurity absorbs an incident photon and transitions to a higher level, leading to ionization. The team computed this quantity using the electric dipole approximation, with the electronic states and their wavefunctions determined by solving the effective-mass Schrödinger equation through the Finite Element Method while explicitly accounting for the presence of the impurity. Because the cross section is expressed as an effective interaction area, it reflects the strength of the coupling between the confined electron and electromagnetic radiation at a given photon energy, and it is strongly influenced by quantum confinement, the size and shape of the dot, and the material composition.

The computational setup was deliberately detailed. The GaAs/AlGaAs heterostructure geometry was discretized in COMSOL Multiphysics with a non-uniform mesh refined near the GaAs/AlGaAs interfaces and around the donor impurity region. Homogeneous Dirichlet boundary conditions were imposed at outer boundaries chosen large enough to eliminate boundary-induced effects on the confined states. A systematic mesh-convergence analysis refined the mesh until the relative variation of the ground-state energy between consecutive refinements fell below 10⁻⁵, with the final mesh comprising roughly 25,000 to 35,000 second-order triangular finite elements. The confinement potential arose from the conduction band offset between GaAs and AlGaAs, modeled as a piecewise constant potential, while the effective mass was treated as position-dependent, taking different values in the GaAs and AlGaAs regions.

From these converged FEM solutions the researchers derived two observables: the donor impurity binding energy and the photoionization cross section. The dataset used for machine learning consisted of 8 FEM-generated samples, with 1400 PICS data points each, totaling 11,200 data points. Photon energies were sampled systematically across the 10 to 35 meV range for each radial impurity position, creating sufficient data to compare resonance features and observe how the radial impurity position affects the positions and intensities of the PICS peaks. Input variables for the models were the photon energy and the radial impurity position r0, while all geometrical and material parameters, including quantum dot height and width, ring height, ring diameter, ring width, wetting layer thickness, and barrier extents, were held constant at reference values.

Data preprocessing used Min–Max normalization to the range [0,1], and the dataset was randomly split into 80 percent training and 20 percent testing subsets. The test set was used exclusively for performance evaluation and was not involved in training or parameter selection, preventing data leakage, and all candidate models were trained and tested on the exact same split for consistent comparison. Performance was assessed with Mean Squared Error, Mean Absolute Error, Root Mean Squared Error, Mean Absolute Percentage Error, and the coefficient of determination. Because percentage errors become undefined for zero-valued targets, data points with zero or negligibly small PICS values were excluded from the MAPE calculation using a small numerical threshold. MAE and RMSE carry the physical units of the PICS, MSE is reported in squared units, and R² is dimensionless.

The headline result concerns how sharply the PICS responds to the impurity’s location. As the radial distance r0 increases, the resonance peaks shift significantly in both height and position. The authors attribute this to impurity-induced changes in electronic-state localization: moving the donor away from the center modifies the Coulomb potential experienced by the confined electron, redistributing the electronic probability density and altering the localization of the bound-state wavefunction. This changes the energy separation between the initial bound state and the final photoionized state, shifting the resonance energy. The displacement also changes the spatial overlap between the initial and final wavefunctions, modifying the electric-dipole transition matrix element, and since the PICS is proportional to the square of that matrix element, the peak intensities change as well.

Among the tested algorithms, K-Nearest Neighbors delivered the best performance, with MAE of approximately 1.58×10⁻⁴, RMSE of about 7.39×10⁻⁴, and R² of roughly 0.99992, matching both the peak locations and peak heights of the FEM curves even at the largest impurity positions where the system is most sensitive. The Gradient Boosting Regressor followed closely, with MAE of about 3.06×10⁻⁴, RMSE near 1.16×10⁻³, and R² around 0.9998 over the entire photon energy range. Linear regression plots of predicted versus FEM values showed nearly perfect diagonal clustering for both models, with R² above 0.9997 for GBR and above 0.9999 for KNN, and little scatter with no indication of systematic deviation. In comparative heat maps, KNN’s MAE ranged from 1.58×10⁻⁴ to 2.39×10⁻⁴ and RMSE from 7×10⁻⁴ to 1.07×10⁻³, while GBR’s MAE spanned 2.87×10⁻⁴ to 5.59×10⁻⁴ with RMSE up to 9.3×10⁻³ and R² near 0.9997. LightGBM was the least accurate of the three but the most computationally efficient.

The learning curves reinforced the picture of stable, well-behaved models. For GBR, the MAE decreased monotonically as the proportion of training data increased and stabilized around 3×10⁻⁴, indicating convergence without further improvement as more data was added, while RMSE stayed below 2.4×10⁻³ across all training-set configurations and R² approached 0.9999. For KNN, the MAE fell to a minimum of 1.6×10⁻⁴ with increasing training data, the MSE reached values on the order of 10⁻⁷, one order of magnitude below GBR, and RMSE remained below 1.1×10⁻³ in all configurations. MAPE analysis showed the lowest errors for impurity positions near the center of the dot, at r0 = 0 and 1 nm, where the PICS varies more smoothly with photon energy, while errors increased for r0 ≥ 2 nm as the impurity-induced Coulomb potential produced a more nonlinear PICS response. KNN’s non-parametric interpolation preserved the local characteristics of the FEM data best under these conditions.

The authors also explain why such exceptionally high accuracy is plausible rather than suspicious. Unlike experimental datasets, which contain measurement uncertainties and random noise, the training data were generated from numerical solutions of the effective-mass Schrödinger equation under well-defined physical conditions, making the dataset deterministic. Because the PICS varies smoothly with photon energy and impurity position, neighboring spectral points are strongly correlated, so the models approximate a continuous nonlinear physical relationship rather than fitting noisy observations. The high R² values and low errors are therefore consistent with the underlying data characteristics and do not necessarily indicate overfitting. KNN’s advantage is likewise physical: its local interpolation among neighboring training samples suits smooth nonlinear functions, whereas the tree-based ensembles in GBR and LightGBM approximate the target with piecewise constant or piecewise linear partitions that can introduce small approximation errors on highly smooth functions.

The hyperparameters were selected through preliminary optimization on a validation set: GBR used 300 estimators, a learning rate of 0.05, a maximum tree depth of 3, and a subsampling ratio of 1.0; LightGBM used 400 boosting estimators, a learning rate of 0.05, and a maximum depth of 3; KNN used 5 nearest neighbors with distance-based weighting and the Euclidean metric; and a stacking ensemble combining KNN, GBR, and Linear Regression with a 5-fold cross-validation meta-estimator performed comparably to the best individual models without substantial improvement. All models were implemented in Python 3.9.12 with scikit-learn 1.2.2 and LightGBM 3.3.5 on a workstation with an Intel Core i7 processor and 32 GB RAM. The authors caution that the trained models apply only to the investigated nanostructure and should not be expected to generalize directly to structures with different geometrical or material characteristics; extending the framework to include structural parameters as inputs would require a substantially larger FEM-generated dataset. Within those bounds, the work demonstrates that machine learning can serve as an efficient surrogate for computationally intensive FEM simulations, enabling rapid prediction of the photoionization cross section across photon energies and donor impurity positions in a fixed GaAs/AlGaAs quantum dot–ring nanostructure.

Subject of Research: Machine learning prediction of the photoionization cross section of donor impurities in GaAs/AlGaAs quantum dot–ring nanostructures

Article Title: Machine learning prediction of photoionization cross section in GaAs AlGaAs quantum dot ring nanostructures

Article References: Jaouane, M., Zeiri, N., Arraoui, R., Sali, A., Murshed, M. N., & Hayrapetyan, D. (2026). Machine learning prediction of photoionization cross section in GaAs AlGaAs quantum dot ring nanostructures. Discover Nano, 21(1), Article 561. https://doi.org/10.1186/s11671-026-05086-4

Image Credits: AI Generated

DOI: 10.1186/s11671-026-05086-4

Keywords: photoionization cross section, GaAs/AlGaAs, quantum dot–ring nanostructure, donor impurity, finite element method, machine learning, K-nearest neighbors, gradient boosting regression, LightGBM, effective-mass Schrödinger equation, COMSOL Multiphysics, surrogate modeling