Designing Nanostructures for Phonon Transport via Bayesian Optimization

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 the $\mathrm{Si}/\mathrm{Ge}$ 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 nonintuitive 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 the crossover of the two effects on phonon transport: as the layer thickness in the superlattice increases, the impact of Fabry-Pérot interference increases, and the rate of reflection at the layer interfaces decreases. An aperiodic superlattice with spatial variation in the layer thickness has a degree of freedom to realize optimal balance between the above two competing mechanisms. Furthermore, the spatial variation enables weakening the impact of constructive phonon interference relative to that of destructive interference. The present work shows the effectiveness and advantage of material informatics in designing nanostructures to control heat conduction, which can be extended to other nanostructures and properties.

Popular Summary

Successful nanotechnologies rely on the efficient management, storage, and conversion of heat, which depend on the development of materials that enhance thermal insulation or heat conduction. At the heart of these efforts is the design of nanostructures for the transport of phonons, which are packets of vibrational motion in a lattice of atoms (similar to how a photon is a packet of light). These designs are challenging because of the sheer variety of nanostructures, as well as interference between phonons at device interfaces. We have developed a new and universal framework for designing nanostructures that combines phonon transport calculations with insights from informatics, the science of information systems.

By alternating between transport calculations and an informatics-based optimization technique, our method enables an efficient search for optimal structures that exhibit the preferred thermal properties. As case studies, we used our method to design silicon-germanium (Si-Ge) composite nanostructures that minimize or maximize the thermal conductance across the Si-Si and Si-Ge interfaces, which are important in thermoelectrics. The optimal structures were obtained by calculating only a few percent of the more than 60,000 candidate structures, considerably accelerating the design process and saving computational resources. The structures are also nonintuitive; the structure with minimum thermal conductance, for example, was found to be an aperiodic superlattice with a significant reduction in conductance compared to the best traditional periodic superlattice.

Our results show the effectiveness and advantage of material informatics in designing nanostructures to control heat conduction, and they can be applied to nanostructure design of a variety of materials with any desired property.

Figure 1

Schematics of the materials informatics method combing the atomistic Green function (AGF) and Bayesian optimization. In the actual AGF calculation, the buffer region is installed between the interfacial region and the lead, which is omitted in the sketch for simplicity.

Figure 2

Interfacial $\mathrm{Si}/\mathrm{Ge}$ alloy structure optimization. (a)–(d) Optimal structures with the maximum and minimum interfacial thermal conductance (ITC) for Si-Si and Si-Ge interface with different sizes of supercell cross section ($1\mathrm{UC}\times 1\mathrm{UC}$ and $2\mathrm{UC}\times 2\mathrm{UC}$). The thickness of the interfacial region is 1.1 nm. (e),(f) The 10 optimization runs with different initial choices of candidates, where the insets show the probability distributions of ITC obtained from calculations of all the candidates. (g),(h) Phonon transmission functions of the optimized interfacial structure (maximum and minimum ITC) and bare Si-Si (i.e., no interfacial structure) and Si-Ge interfaces.

Figure 3

Sketch of the 8-UL superlattice structure with equal numbers of Si and Ge atoms at Si-Si interface to illustrate the descriptor. Eight binary flags are used to indicate the specie of each UL (“1” for Ge and “0” for Si), and thus, the descriptor of this structure is (10110010).

Figure 4

(a) Comparison of ITC of interfacial structures at Si-Si and Si-Ge interfaces with variable ULs. (b) The reduction ratio $|({\sigma }_{\text{aperiodic}}-{\sigma }_{\text{periodic}})|/{\sigma }_{\text{periodic}}$, where ${\sigma }_{\text{aperiodic}}$ is the ITC of the optimized aperiodic superlattice, and ${\sigma }_{\text{periodic}}$ is the ITC of the corresponding periodic superlattice with period thickness optimized for each number of ULs (i.e., total length). The comparison of (c) the phonon dispersion and DOS, and (d) the phonon transmission for three superlattice structures optimal aperiodic superlattice (1101010001), and periodic superlattice with the largest (1111100000) and smallest (1010101010) periodic thickness.

Figure 5

(a),(b) ITC and phonon transmission versus the layer thickness. (c),(d) ITC and phonon transmission versus the number of interfaces with layer thickness of 1 UL, where 1 UL is 5.43 Å thick. (e),(f) ITC versus number of interfaces for cases of 14-UL superlattice with equal $\mathrm{Si}/\mathrm{Ge}$ layer number and 10-UL superlattice with variable $\mathrm{Si}/\mathrm{Ge}$ fraction.

Figure 6

Comparision between the phonon transmission obtained from the cascade model and full AGF calculation. (a) Periodic superlattices with different number of periods, where one period consists of a Si UL and a Ge UL denoted as “10.” (b) The optimal aperiodic superlattice structure with total thickness of 10 UL. Here, the thickness of a unit layer is 5.43 Å.