Porous silicon heat transport splits by doping: molecular dynamics for columnar films, physics-informed networks for spongy ones
Synopsis
Effective thermal conductivity of porous silicon has long scattered because pore morphology and substrate doping differ, and no single model covered both. Gas-microphone photoacoustic measurements, atomistic molecular dynamics and physics-informed neural networks were compared on the same 48% to 75% porosity layers: molecular dynamics matched heavily doped columnar structures, while the networks matched the stochastic spongy networks of lightly and moderately doped silicon. Symbolic regression then combined porosity and temperature into one analytical expression, with mean absolute percentage error no more than 6.5% on the training set over 5% to 85% porosity and 300 to 900 K.
Interpretation
Predicting porous silicon thermal conductivity now runs along two computational routes chosen by substrate doping, rather than one model for every pore structure. Effective medium approximations and percolation models gave analytical porosity dependence but often simplified real pore morphology and neglected nanoscale phonon transport such as boundary scattering. On electrochemically etched p+-type layers about 20 μm thick with 48% to 75% porosity, photoacoustic, molecular dynamics and network results were compared with literature data for p-type, n-type and part of the p+-type sets; molecular dynamics gave slightly lower conductivity for porosity below 0.5.
Each route owns one morphology: the atomistic simulation captures phonon confinement, the continuum network captures macroscopic percolation. Molecular dynamics computed conductivity from the heat current autocorrelation function via the Green-Kubo formalism, while the physics-informed network embedded Fourier's law and the steady heat equation in its loss and predicted temperature and heat flux fields from a conductivity map. Molecular dynamics removed one spherical pore from a 12×12×12 unit-cell supercell of 13824 atoms and averaged ten independent runs per temperature; the network trained on 128×128 matrices with 16384 samples per temperature and averaged 1024 structures.
Porosity and temperature were folded into one analytical expression that can be evaluated directly. Most earlier studies examined the temperature dependence of porous silicon conductivity up to 20 K, 300 K or at most 350 K, with only one reporting to 800 K. Symbolic regression trained on a grid of 5% to 85% porosity in 5% steps and 300 K to 900 K in 100 K steps, tested on 70 randomly chosen porosity values, reaching mean absolute percentage error no more than 6.5% on the training set.
Perspective
The demonstrated scope is electrochemically etched p+-type porous silicon at 48% to 75% porosity and about 20 μm thickness, paired with a single-spherical-pore molecular dynamics model and a stochastic-pore network model; the analytical expression is given for 5% to 85% porosity and 300 to 900 K, limited by the authors to randomly distributed pores where phonon confinement is not yet dominant.
The authors note considerable scatter among p+ samples, indicating isotropic structures similar to lightly doped systems can also form under heavy doping; the expression's error is reported as mean absolute percentage error on the training set, and its use is limited to porosity regimes where phonon confinement is not yet dominant.