GoKawiilTEGNet architecture
The TEGNet architecture is a streamlined, fully connected, multilayer perceptron, consisting of one input layer, three hidden layers and one output layer. The input layer contains six neurons representing the dimensions of a TE leg (width a, depth b and length c) and the boundary conditions (hot-side temperature T h , cold-side temperature T c and applied current I). The output layer has two neurons corresponding to the voltage V 0 and heat flow Q 0 at the cold side. Each hidden layer contains 128 neurons, with SiLU as the activation function. The loss function is defined as the mean squared error between the predicted values and the data generated from COMSOL Multiphysics (abbreviated as COMSOL in the main text and the following), allowing for backpropagation during training. All algorithms were implemented in Python (version 3.10) using the PyTorch module.
Data generation
The training data for TEGNet is generated using COMSOL owing to its high accuracy in solving coupled thermo-electrical PDEs. All simulations used a ‘normal’ mesh size and a steady-state solver. Each data point contains eight columns: the TE leg dimensions (a, b and c), boundary conditions (T h , T c and I) and the corresponding V 0 and Q 0 . The dimensions a, b and c range from 1 to 12 mm. Temperature ranges for T h and T c were selected according to the intrinsic operating limits of each material to avoid unreal TE properties arising from material degradation or phase changes. An extra 20 K margin was included to ensure full coverage of the TE property data and allow reasonable extrapolation. For example, NbFeSb operates up to about 1,203 K, so T h and T c were sampled in the range 283–1,223 K, with T h > T c . For MgAgSb, which operates up to about 573 K, the corresponding range was 283–593 K. The Sobol sequence is used to sample a, b, c, T c and T h , for which T h should exceed T c . The maximum current I is dynamically determined on the basis of a, b, c, T c , T h and the TE properties of the material to avoid non-convergent COMSOL cases and ensure operation within the power-generation regime. The current I is then sampled uniformly from 0 to the maximum value. These considerations effectively eliminate unphysical conditions and yield a high-quality dataset requiring minimal preprocessing. These combined input parameters (a, b, c, T c , T h , I) are fed into COMSOL and V 0 and Q 0 are extracted from the solved spatial distributions of voltage and temperature. Because n-type and p-type materials produce voltages of opposite sign under identical temperature gradients, all COMSOL-generated V 0 values were converted to positive. Sample sizes of 300, 600, 1,200, 1,800 and 2,400 were tested for MgAgSb and an optimal sample size of 1,200 was chosen on the basis of a balance between accuracy and computational cost for all TE materials.
TEGNet training
For TEGNet training, the input data are fed into the network and the weights and biases of each neuron are updated by means of backpropagation to minimize the loss function. The optimizer is Adam and a CosineAnnealingLR scheduler is used with an initial learning rate of 1 × 10−3. The total number of epochs is set to 6,000. The TEGNet parameters were initially evaluated using MgAgSb data, which were split into a training set (80%), a validation set (10%) and a test set (10%). The validation set serves as an indicator for overfitting or underfitting during training. The model performance was evaluated after confirming the suitability of the network parameters. The data were divided into a training set (80%) and a test set (20%) to fully use the available data. To reduce potential bias from data splitting, five independent random splits were performed for both parameter evaluation and model performance assessment and the results were averaged. The final model for each material was trained on the entire dataset to maximize data use. Through training, the temperature-dependent TE properties are effectively learned and embedded within the network weights for each material. All training was conducted on a personal computer with a 12-core CPU and 16 GB RAM.
Materials synthesis
The MgAgSb used in this work contains 0.625 wt% C 18 H 36 O 2 and Mg 3 Bi 1.4 Sb 0.6 has the composition Mg 3.2 In 0.02 Sb 0.595 Bi 1.4 Te 0.005 , whereas Bi 0.4 Sb 1.6 Te 3 is used as is. For simplicity, MgAgSb + 0.625 wt% C 18 H 36 O 2 and Mg 3.2 In 0.02 Sb 0.595 Bi 1.4 Te 0.005 are abbreviated as MgAgSb and Mg 3 Bi 1.4 Sb 0.6 , respectively, throughout the main text and the following. MgAgSb and Mg 3 Bi 1.4 Sb 0.6 were prepared by ball milling (SPEX 8000D) in an Ar atmosphere for 5 h, whereas Bi 0.4 Sb 1.6 Te 3 was prepared by melting at 1,273 K in a quartz tube for 12 h, followed by ball milling of the resulting ingot for 1 h. The raw materials used are high-purity Mg turnings (99.95%), Ag powders (99.99%), Sb shots (99.999%), Bi shots (99.999%), Te shots (99.999%) and In powders (99.99%).
TE generator fabrication
For the MgAgSb/Bi 0.4 Sb 1.6 Te 3 segmented TE generator, the MgAgSb part was fabricated with Sb powders as interface materials, whereas the Bi 0.4 Sb 1.6 Te 3 part was fabricated without interface materials. Both components were spark plasma sintered using SPS-322LX (Dr. Sinter) at 573 and 693 K for 5 and 10 min under 60 MPa, respectively. The Sb/MgAgSb/Sb piece was cut parallel to the pressing direction, whereas Bi 0.4 Sb 1.6 Te 3 was cut perpendicular to the pressing direction. Segmented TE generators with different dimensions were assembled by connecting Sb/MgAgSb/Sb and Bi 0.4 Sb 1.6 Te 3 using liquid Ga–In to form Sb/MgAgSb/Sb/Ga–In/Bi 0.4 Sb 1.6 Te 3 , which is abbreviated as MgAgSb/Bi 0.4 Sb 1.6 Te 3 in the main text for simplicity. The contact resistance of the Sb/MgAgSb/Sb/Ga–In/Bi 0.4 Sb 1.6 Te 3 interface was measured using a resistance distribution instrument (S1331, Mottainai Energy). Notably, even without interface materials, Bi 0.4 Sb 1.6 Te 3 in contact with Sb through liquid Ga–In exhibited negligible contact resistance. For the Mg 3 Bi 1.4 Sb 0.6 –MgAgSb n–p paired TE generator, the Mg 3 Bi 1.4 Sb 0.6 part was fabricated with stainless-steel powders as interface materials and sintered at 973 K under 60 MPa for 10 min (SPS-1080 System, SPS Syntex Inc.). The MgAgSb part was prepared in the same manner as for the segmented TE generator. Two-pair Mg 3 Bi 1.4 Sb 0.6 –MgAgSb n–p paired TE generators were assembled on an AlN ceramic plate with copper electrodes.
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