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Designing nanostructures for interfacial phonon transport via Bayesian optimization

Shenghong Ju, Takuma Shiga, Lei Feng, Zhufeng Hou, Koji Tsuda, Junichiro Shiomi

arXiv:1609.04972v1cond-mat.mtrl-scicond-mat.mes-hallphysics.comp-ph

TL;DR

The paper asks how Si/Ge composite interfaces can be arranged to minimize or maximize interfacial thermal conductance. It combines atomistic Green's-function calculations with Bayesian optimization and finds a minimum-ITC aperiodic superlattice with 50% reduction relative to the best periodic superlattice.

  • Problem

    The paper asks how Si and Ge atoms should be arranged to obtain the largest or smallest interfacial thermal conductance.

    Source: §II · p. 4

  • Method

    The authors combine atomistic Green's-function calculations with Bayesian optimization to search Si/Ge interfacial structures for minimum and maximum ITC.

    Source: §IV · p. 14

  • Results

    The minimum-ITC aperiodic superlattice achieves a 50% reduction relative to the best periodic superlattice.

    Source: §IV · p. 14

  • Takeaways & Limitations

    The results demonstrate material informatics as a means of designing nanostructures to control heat conduction.

    Source: §IV · p. 14

Abstract

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We demonstrate optimization of thermal conductance across nanostructures by developing a method combining atomistic Green's function and Bayesian optimization. With an aim to minimize and maximize the interfacial thermal conductance (ITC) across Si-Si and Si-Ge interfaces by means of Si/Ge composite interfacial structure, the method identifies the optimal structures from calculations of only a few percent of the entire candidates (over 60,000 structures). The obtained optimal interfacial structures are non-intuitive and impacting: the minimum-ITC structure is an aperiodic superlattice that realizes 50% reduction from the best periodic superlattice. The physical mechanism of the minimum ITC can be understood in terms of crossover of the two effects on phonon transport: as the layer thickness in superlattice increases, the impact of Fabry-Pérot interference increases, and the rate of reflection at the layer-interfaces decreases. Aperiodic superlattice with spatial variation in the layer thickness has a degree of freedom to realize optimal balance between the above two competing mechanism. Furthermore, aperiodicity breaks the constructive phonon interference between the interfaces inhibiting the coherent phonon transport. The present work shows the effectiveness and advantage of material informatics in designing nanostructures to control heat conduction, which can be extended to other interfacial structures.

I. INTRODUCTION

The introduction motivates nanostructure design for controlling interfacial thermal conductance (ITC) and presents an atomistic Green’s function and Bayesian optimization framework to identify structures that maximize or minimize ITC.

  • At nanoscale, phonon transport becomes more ballistic, making ITC a determinant of whole-material heat conduction that can be controlled through interface structure.Source: §I · pp. 2–3
  • Coupled, configuration-sensitive tuning factors make total ITC controllability difficult, while interference and resonance complicate optimization of structures with multiple interfaces.Source: §I · p. 3
  • Although materials informatics has been applied to low-thermal-conductivity thermoelectrics and nanoparticle-size distributions [33], nanostructure optimization for thermal transport remains in its infancy.Source: §I · p. 3
  • The study combines atomistic Green’s function (AGF) with Bayesian optimization to identify non-trivial interfacial structures that maximize or minimize ITC.Source: §I · pp. 3–4

II. METHODOLOGY

The method searches Si/Ge atomic arrangements at fixed 50% composition to maximize or minimize interfacial thermal conductance, using atomistic Green’s function evaluations and Bayesian optimization to select candidates.

II. METHODOLOGY

  • The search arranges 16 Si/Ge atoms at 50% composition across 12,870 candidates, encoding Ge as 1 and Si as 0 and using ITC as the objective.The setup constrains the Si and Ge fractions to be equal.Source: §II · p. 4
  • The atomistic Green’s function method evaluates each configuration’s ITC from phonon transmission and relates heat current to the leads’ phonon distributions.The method derives ITC in the limit of small temperature differences.Source: §II · pp. 4–5
  • Bayesian optimization learns from evaluated descriptor–ITC pairs, estimates predictive ITC distributions for remaining candidates, and selects the next candidate by expected improvement.The study uses the COMBO library for this optimization process.Source: §II · pp. 5–6
  • The selected candidate’s ITC is calculated and added to the training examples, and the cycle repeats to schedule evaluations and find the best candidate.The passage states that this scheduling allows the best candidate to be found quickly.Source: §II · p. 6
  • The prediction model combines Bayesian linear regression with a random feature map; its hyperparameters are updated by maximizing type-II likelihood every 20 added training examples.COMBO initializes the hyperparameters using a heuristic procedure.Source: §II · p. 6

III. RESULS AND DISCUSSIONS

The results examine how superlattice structure, including layer thickness, interface spacing, and aperiodicity, affects interfacial thermal conductance through changes in phonon transport. They also describe Bayesian optimization’s identification of ITC extrema and how aperiodicity can lower ITC by disrupting constructive phonon interference.

III. RESULS AND DISCUSSIONS

  • All optimizations converged within 438 structure calculations, just 3.4% of 12,870 candidates; maximum-to-minimum ITC ratios were 2.4 for Si-Si and 2.2 for Si-Ge interfaces, respectively.Source: §III · p. 7
  • For phonon frequencies below 3 THz, transmissions of different structures are almost the same, whereas at higher frequencies transmission strongly depends on structure; maximum-ITC Si-Si provides a continuum path of Si.Source: §III · p. 7
  • Both minimum-ITC structures are aperiodic superlattices; in the 10-UL comparison, dispersion and DOS scarcely distinguish structures although the optimum has the lowest transmission.This shows that dispersion and DOS alone do not explain the minimum ITC.Source: §III · p. 8, §5.43 · p. 10

5.43 Å, and total thickness of interfacial structure ranges from 8 to 16 ULs (from 4.35

  • As total thickness increases, minimum ITC decreases; variable Si/Ge fractions lower it for Si-Si, while the difference between variable- and fixed-fraction cases is very small for Si-Ge.The optimized superlattices begin or end with a layer different from the lead material, enhancing phonon scattering.Source: §5.43 · p. 9
  • Aperiodic Si-Si superlattices with fixed Si/Ge fraction reduced ITC by 20–50% versus the best-period periodic superlattices; variable fractions reduced it further.The periodic comparison optimized period thickness separately for each total length.Source: §5.43 · p. 9
  • Increasing layer thickness or interface count lowers ITC toward a constant; Fabry–Pérot resonances proliferate with interface spacing, producing faster conductance loss at small spacing.As spacing increases, resonant frequencies eventually cover the full frequency range.Source: §5.43 · pp. 10–11
  • Aperiodic superlattices can outperform periodic counterparts for minimizing ITC by spatially distributing layers of different thicknesses and interface counts.The layer thickness and number of interfaces are competing parameters, and their balance determines the minimum-ITC structure.Source: §5.43 · p. 11
  • The ITC versus interface-count profile has a minimum in both tested superlattices, while structures with the same interface count can have significantly different ITC.The differences are attributed to the thicknesses of the constituent layers.Source: §5.43 · p. 12
  • As periodic-superlattice periods increase, full AGF transmission converges while cascade-model transmission keeps decreasing, indicating constructive interference and scope to reduce ITC by breaking it.The cascade model captures incoherent transport, whereas full AGF includes coherent transport across multiple layers.Source: §5.43 · p. 13
  • For an optimized aperiodic superlattice, the cascade model reproduces the general trend and magnitude of full-AGF transmission, consistent with suppressed constructive interference and transmission approaching the incoherent limit.The two transmission calculations differ in detail.Source: §5.43 · p. 13

IV. CONCLUSIONS

The study uses atomistic Green’s function and Bayesian optimization to identify Si/Ge interfacial structures that minimize or maximize ITC, finding that aperiodic superlattices can outperform optimal periodic ones at reducing ITC. Their low conductance arises from balancing competing transport effects and suppressing constructive phonon interference.

IV. CONCLUSIONS

  • The framework combines atomistic Green’s function and Bayesian optimization to identify structures minimizing or maximizing ITC across Si-Si and Si-Ge interfaces using only a few percent of candidates.This considerably saves computational resources.Source: §IV · p. 14
  • The method identifies thin interfacial structures with optimal Si/Ge configurations among all possible candidates.This demonstrates the method’s validity and capability.Source: §IV · p. 14
  • Aperiodic superlattices in searches extending to 8.69 nm have ITCs significantly smaller than those of optimal periodic superlattices.The minimum-ITC structures are non-intuitive and aperiodic.Source: §IV · p. 14
  • Aperiodic layer-thickness variation balances Fabry–Pérot interference, which strengthens with layer thickness, against interfacial scattering, which weakens, restraining constructive phonon interference toward incoherent transport.This balance accounts for the low ITC of the optimal aperiodic structure.Source: §IV · p. 14
  • The work demonstrates the effectiveness and advantage of material informatics for designing nanostructures to control heat conduction.Source: §IV · p. 14

FIGURES, TABLES AND CAPTIONS

The figures and table document the optimized interfacial structures and compare their ITC and phonon-transport behavior across structural variations. Supplementary figures examine transmission changes with scattering-region thickness and interface spacing.

Optimization results

  • Figure 1 presents maximum- and minimum-ITC structures for Si–Si and Si–Ge interfaces, optimization runs, ITC distributions, and phonon-transmission comparisons.The transmission comparisons include optimized structures and bare interfaces.Source: Figure 1 · p. 19

Superlattice comparisons

  • Figure 3 compares ITC across variable superlattice unit-layer counts, reduction ratios against optimized periodic structures, and phonon dispersion, density of states, and transmission for selected structures.The selected structures are the optimal aperiodic superlattice and periodic superlattices with the largest and smallest periodic thicknesses.Source: Figure 3 · p. 21

Structural parameter and model comparisons

  • ITC and phonon transmission are examined against layer thickness and interface count, including 14-UL structures with equal Si/Ge layer numbers and 10-UL structures with variable Si/Ge fractions.The caption also compares cascade-model transmission with full AGF calculations for periodic ‘10’ superlattices of different period counts and the 10-UL optimal structure.Source: FIGURES, TABLES AND CAPTIONS · pp. 22–23, Figure 5 · p. 23

Optimization results

  • Table 1 lists Bayesian-optimization superlattice structures for different setups and constraints.The caption notes that bracketed numbers indicate counts, but the supplied passage does not specify their values.Source: Table 1 · p. 24

Supplementary transmission analysis

  • For both scattering-layer thickness and interface spacing, normalized integrated phonon transmission converges to almost the same value despite substantially different transmission spectra.The supplementary figures show spectra for multiple thicknesses and interface distances, alongside their normalized integrated transmissions.Source: FIGURES, TABLES AND CAPTIONS · p. 25, Figure S1 · p. 26
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