Bidirectional Learning of Relationships between Atomic Environments and Electronic Band Dispersion in Semiconductor Heterostructures

TL;DR

A bidirectional learning approach that links local atomic environments to electronic band dispersion using atomically resolved spectral functions as information-dense representations is introduced, providing a physics-informed route for interpreting spectroscopic data and for data-driven exploration of electronic properties in complex semiconductor heterostructures.

Abstract

Atomic-scale variations in semiconductor heterostructures, arising from strain, interfaces, and compositional modulation, strongly influence electronic band dispersion but remain difficult to probe and compare using first-principles methods alone. Here, we introduce a bidirectional learning approach that links local atomic environments to electronic band dispersion using atomically resolved spectral functions as information-dense representations. This formulation enables a forward model that predicts how atomic environments shape electronic bands, and a reverse model that infers atomic-environment descriptors directly from band dispersion images, including angle-resolved photoemission spectra. Applied to silicon/germanium superlattices and heterostructures, the approach reveals how inner and interfacial atomic environments give rise to distinct spectral signatures. The coupled forward-reverse framework enables self-consistent validation by reconstructing electronic band structures from inferred descriptors. By treating electronic bands as decomposable, learnable objects, this work provides a physics-informed route for interpreting spectroscopic data and for data-driven exploration of electronic properties in complex semiconductor heterostructures.

Figures (7)

• Figure 1: Outline of the forward learning approach. (a) Training data consist of strain-symmetrized and strained Si/Ge superlattices spanning a range of layer periods and compositions, including representative examples: (i) Si$_4$Ge$_4$, (ii) Si$_{14}$Ge$_{14}$, (iii) Si$_{13}$Ge$_{13}$Si$_{13}$Ge$_{13}$ and (iv) Si$_{26}$Ge$_{26}$. Atomic environments are described using element type and local structural descriptors. (b) Atomically resolved spectral functions (ASFs) for atoms highlighted in (a) (red circles), illustrating how distinct local atomic environments give rise to diverse electronic band dispersion features across superlattices. (c) Forward model predictions for a Si$_{28}$Ge$_{28}$ superlattice not included in the training set, demonstrating the model’s ability to generalize beyond the training data.
• Figure 2: Relationships between atomic-environment descriptors and spectral functions (SFs) in Si/Ge systems. (a) Atomic-environment descriptors and (b) SFs for relaxed bulk Si (row 1), strained bulk Si (row 2), relaxed bulk Ge (row 3) and strained bulk Ge (row 4). (c,d) Descriptors and ASFs for inner Si atoms and (e,f) interface Si atoms in Si$_{26}$Ge$_{26}$ (row 1), Si$_{12}$Ge$_{12}$ (row 2), Si$_6$Ge$_6$ (row 3) and Si$_4$Ge$_4$ (row 4) superlattices.
• Figure 3: Forward learning model predictions and validation. (Column 1) (a) Representative supercell configuration of a test heterostructure Si$_{8}$Ge$_{8}$Si$_{20}$Ge$_{20}$, with selected Si atoms from (b) the inner Si$_{8}$ layer, (c) the inner Si$_{20}$ layer, (d) the Si$_{8}$Ge$_{8}$ interface, and (e) the Si$_{20}$Ge$_{20}$ interface regions highlighted. (Column 2) Atomic-environment descriptors used as inputs to the forward models. (Column 3-4) ASFs predicted by the RF and NN models, respectively. (Column 5) DFT-computed ASFs for comparison. (Column 6) Normalized energy-resolved intensity profiles, $I_n(E)$. (Column 7) Mean absolute errors (MAEs) for RF (blue) and NN (red) predictions. (f) Total SFs obtained by summing ASFs over all atoms in the heterostructure.
• Figure 4: Outline of reverse learning approach. (a) Training images consisting of example ASFs for inner Si atoms in (i) Si$_{4}$Ge$_{4}$ and (ii) Si$_{14}$Ge$_{14}$ superlattices. (b) Atomic-environment descriptors associated with the training images, including elemental identity, effective bond lengths, and local order parameters. (c) A trained convolutional neural network (CNN) maps input ASF images to predicted atomic-environment descriptors for atoms in the heterostructure Si$_{8}$Ge$_{8}$Si$_{20}$Ge$_{20}$. Predicted descriptors are compared with those obtained directly from DFT.
• Figure 5: Reverse learning model predictions for a model heterostructure. (a) Supercell of the strain-symmetrized Si$_{8}$Ge$_{8}$Si$_{20}$Ge$_{20}$ heterostructure. Predicted (b) atomic species, (c) effective bond lengths and (d-f) spatially resolved local order parameters, $Q_i^{order}$, where $i=(x,z)$ and $order = 1,2,3$, for all atoms in the heterostructure. Predicted descriptor averages (symbols) are compared with DFT reference values (solid lines). Error bars indicate the standard deviation across input ASF images with different Fermi-level alignments. MAEs are reported in each panel.