Phonon-confinement theory of thermal conductivity in ultrathin silicon films

The thermal properties of solids under nanoscale confinement are currently not understood at the atomic level. Recent numerical studies have highlighted the presence of a minimum in the thermal conductivity as a function of thickness for ultrathin films at a thickness about 1-2 nm, which cannot be described by existing theories. We develop a theoretical description of thin films which predicts a new physical law for heat transfer at the nanoscale. In particular, due to the strong redistribution of phonon momentum states in reciprocal space (with a transition from a spherical Debye surface to a different homotopy group $\mathbb{Z}$ at strong confinement), the low-energy phonon density of states no longer follows Debye’s law but rather a cubic law with frequency, which then crosses over to Debye’s law at a crossover frequency proportional to the average speed of sound of the material and inversely proportional to the film thickness. Concomitantly, this implies that the phonon population becomes dominated by low-energy phonons as confinement increases, which then leads to a higher thermal conductivity under extreme confinement. The theory is able to reproduce the thermal conductivity minimum in recent molecular simulations data for ultrathin silicon and provides useful guidelines so as to tune the minimum position based on the mechanical properties of the material.

💡 Research Summary

This paper presents a novel theoretical framework to explain the anomalous thickness dependence of thermal conductivity in ultrathin silicon films, particularly the observed minimum at around 1-2 nm thickness, which cannot be accounted for by classical size-effect theories like the Fuchs-Sondheimer model.

The core of the theory lies in a fundamental geometric reconsideration of phonon momentum states in reciprocal space under strong spatial confinement. Confinement along the film thickness (L) direction restricts the maximum allowed phonon wavelength in that direction. This creates two symmetric spherical regions of “forbidden” states within the Debye sphere in k-space. When the film becomes so thin that the radius of these forbidden spheres (π/L) exceeds the Debye radius (k_D), the geometry of the occupied states in k-space is drastically altered. The Debye sphere becomes truncated, leading to a redistribution of phonon states from the surface towards the core of the sphere.

This redistribution fundamentally changes the low-frequency phonon density of states (DOS). Instead of following the conventional Debye law (g(ω) ~ ω²), the DOS in the confined film follows a cubic law (g(ω) ~ ω³) for frequencies below a crossover value ω× = 2πv/L, where v is the average speed of sound. Above ω×, it recovers the standard Debye ω² dependence. As the film thickness L decreases, ω× increases, extending the range of the cubic-law DOS to higher frequencies. Consequently, the relative population of low-frequency, long-wavelength phonons increases with increasing confinement.

Since thermal conductivity in dielectrics is dominated by these low-frequency acoustic phonons—which have long mean free paths and are less susceptible to scattering—the shift in the phonon population towards lower energies leads to an enhancement of thermal conductivity under extreme confinement. This effect competes with the boundary scattering effect described by Fuchs-Sondheimer theory, which reduces conductivity with decreasing thickness. The interplay between these two opposing mechanisms—confinement-induced DOS modification (increasing κ) and boundary scattering (decreasing κ)—naturally produces a minimum in thermal conductivity as a function of thickness.

The theory integrates this new DOS into the Debye-Peierls model for thermal conductivity, incorporating Fuchs-Sondheimer corrections for interface scattering. The resulting model provides a qualitative explanation for molecular dynamics simulation data on silicon films, successfully reproducing the minimum. Furthermore, it offers practical guidance: the position of the minimum depends on the material’s average sound speed (ω× ∝ v/L), suggesting that the thermal transport properties of nanoscale films can be tuned by material selection and engineering of mechanical properties.