Unexpected thermal conductivity enhancement in aperiodic superlattices discovered using active machine learning

The demand for efficient energy systems and high-performance electronic devices has created the need to rapidly identify new materials and design nanostructures with extreme transport properties. As intuition-driven trial-and-error approaches face limitations, machine learning (ML) and data informatics have emerged as powerful tools for design and optimization. In thermal transport, ML has succeeded in predicting material properties and accelerating the design of nanostructures targeting specific thermal conductivities. However, most ML applications to date focus on optimizing known effects (e.g., maximizing disorder-induced Anderson localization of phonons) rather than discovering exceptional, unexpected behaviors. Traditional ML models are largely interpolative, performing well within the domain spanned by known data (and physics) but failing for excursions outside the training set. Therefore, adaptations are required for ML to identify materials or nanostructures exhibiting exceptional physical properties. In this work, we demonstrate an adaptive ML approach to identify unexpected thermal transport behavior in aperiodic superlattices. Binary superlattices (SLs) of alternating layers typically have lower lattice thermal conductivity (κ) than their constituents, attractive for thermoelectric applications. Randomizing layer thickness in periodic SLs can further reduce κ below random alloy limits via destructive interference and Anderson localization of coherent phonons. ML methods such as Bayesian optimization and genetic algorithms (GA) have efficiently identified random multilayer (RML) structures with ultralow κ. Yet, it remains unclear whether certain random thickness distributions can yield higher κ than periodic SLs. Prior GA-driven work on graphene nanomeshes showed disordered pores can enhance κ compared to uniform spacing, motivating exploration of analogous exceptions in 1-D SLs/RMLs. However, GA searches can be computationally expensive for rare solutions. Thus, a systematic, ML-enabled, extrapolative approach is needed to efficiently identify exceptional solutions. Here, we identify RML structures with unexpectedly higher κ than corresponding periodic SLs with the same total length and average period. A CNN-based predictor accelerates κ estimation across a large design space. An iterative data-generation approach is employed to assemble a representative training dataset that enables the CNN to accurately predict high-κ target RML structures by dynamically learning spatial features associated with locally enhanced phonon transmission.

- Binary superlattices (SLs) often show reduced lattice thermal conductivity compared to constituent bulk materials, enabling thermoelectric applications. - Random multilayers (RMLs) with aperiodic layer thickness can further reduce κ below random alloy limits due to destructive interference and Anderson localization of coherent phonons. - ML methods (e.g., Bayesian optimization, genetic algorithms) have efficiently searched vast design spaces to identify nanostructures with ultralow κ, but typically optimize known phenomena rather than discovering unexpected transport behaviors. - Wei et al. found that disorder in graphene nanomesh pore spacing can enhance κ over uniformly spaced pores, challenging the conventional belief that randomness always lowers κ, and motivating analogous searches in 1-D multilayers. - Prior works decomposed coherent and incoherent phonon conduction in SLs and RMLs, highlighting that interface characteristics and aperiodicity can significantly alter coherent transport and localization, influencing κ.

Overall approach: An adaptive, iterative ML workflow combines high-fidelity non-equilibrium molecular dynamics (NEMD) simulations for ground-truth κ labels with a convolutional neural network (CNN) to rapidly predict κ over a large space of aperiodic Si/Ge superlattice (RML) structures. The training set is iteratively augmented with RMLs exhibiting structural features that locally enhance coherent phonon transport, improving extrapolative prediction of high-κ, low-probability structures. The trained CNN enables exhaustive scanning of the design space at negligible cost relative to NEMD. NEMD simulations: - Software: LAMMPS. - Interatomic potential: Three-body Tersoff potential for Si/Ge; Ge lattice constant set equal to Si within the potential parameters to remove cross-sectional strain that otherwise causes large interfacial oscillations. - Geometry: Cross-section of 6 × 6 unit cells (UCs), sufficient for κ convergence; SL/RML length composed of 20 or 40 UCs along transport. - Temperature: 300 K; time step 0.5 fs (resolves highest phonon frequencies). - Reservoirs: Two bulk regions (20 UCs of Si and Ge) attached to either side of SL/RML act as thermal baths. - Equilibration: 500 ps at 300 K under NPT with periodic boundaries in all directions, then 250 ps under NVE. - Driving non-equilibrium: Langevin thermostats set Si and Ge bulk regions to 330 K and 270 K, respectively; two UCs at each end fixed to impose fixed boundary conditions along the transport direction. - Steady state: Reached over ~500 ps; temperatures sampled in 1D bins along the cross-plane direction. - Data collection: Temperature and heat flux averaged over 4 ns. - κ calculation: κ = q″ L / ΔT, where q″ is steady-state heat flux, L is SL/RML length along transport, and ΔT is temperature difference across the structure. - Interfacial thermal boundary resistance: R = ΔT_interface / q″ from temperature drops at interfaces. CNN-based κ prediction: - Input encoding: Binary array of length N (N = 20 or 40 UCs). Each element is 1 (Si) or 2 (Ge), representing the material of each UC along the length. - Architecture: - Convolutional layers: 1D convolutions with 44–50 filters, filter lengths 5–9, stride 1, no padding; a max-pooling layer after every two convolutional layers for down-sampling and translational invariance. - For 20-UC systems: 2 convolutional layers + 1 max-pooling layer + flatten + 1 dense layer (100 nodes). - For 40-UC systems: 4 convolutional layers with 1 max-pooling layer after every two conv layers + flatten + 1 dense layer (100 nodes). - Activation: ReLU throughout. - Training: - Loss: Mean Absolute Percentage Error (MAPE); performance also monitored via RMSE. - Optimizer: Adamax. - Epochs: 300–500 with early stopping to prevent overfitting when test loss plateaus or rises. - Iterative dataset curation: The CNN is iteratively retrained by adding RML structures identified (via NEMD and/or prediction-feedback) to have locally enhanced coherent transport (e.g., closely spaced interfaces), enabling accurate prediction in regions outside the initial training distribution. - Search strategy: After training, the CNN rapidly evaluates the entire discrete design space of candidate RML sequences for given N, enabling exhaustive scanning to flag high-κ candidates for NEMD verification. Analysis of transport mechanisms: - Comparative evaluation of temperature profiles and apparent interfacial thermal resistances across RMLs and periodic SLs (e.g., 5–5 SL as reference) to attribute κ differences to coherent phonon contributions, interface spacing, and localization effects.

- Discovery of aperiodic (random multilayer, RML) Si/Ge structures with lattice thermal conductivity κ higher than corresponding periodic superlattices (SLs) of the same total length and average period. - Specific identification of RMLs that outperform a reference periodic SL with average period ~5.4 nm (e.g., 5–5 SL), demonstrating unexpected κ enhancement due to disorder. - Mechanism: Enhanced coherent phonon transport enabled by closely spaced interfaces in certain aperiodic configurations reduces apparent interfacial thermal resistance and total thermal resistance relative to periodic SLs. - Dual role of randomness: - Short-period SLs: Randomness can induce Anderson localization of coherent phonons and decrease κ (conventional understanding). - Larger-period SLs: Certain forms of aperiodicity avoid localization while strengthening coherent transport, thereby increasing κ. - The adaptive CNN-accelerated workflow efficiently finds low-probability high-κ RMLs in a large design space with high-fidelity NEMD validation, highlighting ML’s ability to uncover exceptions beyond interpolative regimes.

The study addresses whether aperiodic layer-thickness distributions can ever enhance κ relative to periodic superlattices—a counterintuitive outcome given the widely held view that disorder reduces κ via phonon localization. By coupling NEMD with an iteratively trained CNN that learns spatial features responsible for locally enhanced coherent transport, the workflow extrapolates beyond the initial training distribution and efficiently flags rare RML candidates with higher κ. Comparative analyses of temperature profiles and apparent interfacial resistances show that closely spaced interfaces in these RMLs enhance coherent phonon transmission and lower interfacial thermal resistance, explaining the κ enhancement relative to periodic SLs of the same average period. These findings refine the conventional understanding of disorder in phononic multilayers: while disorder can suppress κ through localization in short-period systems, appropriately engineered aperiodicity in larger-period systems can amplify coherent transport and increase κ. This has direct implications for thermal management in multilayer-based devices (e.g., quantum cascade lasers), where higher cross-plane κ is beneficial. Methodologically, the work demonstrates that adaptive ML can move beyond interpolation to discover low-probability, physically exceptional solutions more efficiently than heuristic search methods alone.

An iterative ML-driven search combining CNN prediction with NEMD validation discovers aperiodic superlattices (RMLs) exhibiting higher thermal conductivity than corresponding periodic SLs of the same length and average period (~5.4 nm). The κ enhancement arises from increased coherent phonon contributions due to closely spaced interfaces that reduce apparent interfacial thermal resistance. The results show that engineered aperiodicity can either suppress or enhance κ depending on layer-period regimes, challenging the traditional assumption that randomness invariably lowers κ. Future directions include: extending the approach to higher-dimensional or more complex multilayer design spaces where exhaustive scanning becomes impractical; integrating more sophisticated optimization strategies (e.g., genetic algorithms) with the CNN predictor to efficiently navigate exponentially large design spaces; and exploring broader material systems and interface engineering to generalize the discovered coherent transport enhancements.

- Computational bottleneck: The dominant cost lies in NEMD evaluations and CNN training; although exhaustive scanning added little cost here, this approach may become infeasible as the number of design variables grows and the design space expands exponentially. - Scalability: Exhaustive search over all candidates is impractical for larger systems; more sophisticated optimizers (e.g., GA) may be required to maintain efficiency. - Model dependence: CNN predictive accuracy relies on iterative inclusion of representative high-κ aperiodic structures; insufficient coverage of rare features could degrade extrapolative performance. - Simulation assumptions: The Ge lattice constant was matched to Si within the Tersoff potential to eliminate interfacial strain, which may affect quantitative transferability to real systems with lattice mismatch.