Diffusive Phonons in Nongray Nanostructures

Abstract

Nanostructured semiconducting materials are promising candidates for thermoelectrics due to their potential to suppress phonon transport while preserving electrical properties. Modeling phonon-boundary scattering in complex geometries is crucial for predicting materials with high conversion efficiency. However, the simultaneous presence of ballistic and diffusive phonons challenges the development of models that are both accurate and computationally tractable. Using the recently developed first-principles Boltzmann transport equation (BTE) approach, we investigate diffusive phonons in nanomaterials with wide mean-free-path (MFP) distributions. First, we derive the short MFP limit of the suppression function, showing that it does not necessarily recover the value predicted by standard diffusive transport, challenging previous assumptions. Second, we identify a Robin type boundary condition describing diffuse surfaces within Fourier's law, extending the validity of diffusive heat transport in terms of Knudsen numbers. Finally, we use this result to develop a hybrid Fourier/BTE approach to model realistic materials, obtaining excellent agreement with experiments. These results provide insight on thermal transport in materials that are within experimental reach and open opportunities for large-scale screening of nanostructured thermoelectric materials.Due to their ability to convert heat directly into electricity, thermoelectric (TE) materials have a wide range of applications, including waste heat recovery [1], wearable devices [2], and deep-space missions [3]. Widespread of TE materials is limited, however, by the simultaneous requirement for low thermal conductivity and high electrical conductivity, a condition that is rarely met in natural materials [4]. Nanostructured materials overcome this limitation in that heat-carrying phonons have mean free paths MFPs (Λ) larger than the limiting dimension, L c , resulting in strong thermal transport suppression [5]. On the other side, electrons have MFPs that are typically as small as a few nanometers thus their size effects are mostly negligible [6]. Notable nanostructures, including thin films [7], nanowires [8,9], and porous materials [10][11][12][13][14][15], show a significant suppression in thermal conductivity with respect to the bulk, holding promises for high-efficiency thermal energy conversion.In the case of materials with wide bulk MFP distribution, K(Λ), the effective thermal conductivity (κ eff ) can be conveniently computed by κ eff /κ bulk = B 0 (Λ)S(Kn)dΛ, where κ bulk is the bulk thermal conductivity, Kn = Λ/L c is the Knudsen number, S(Kn) is the material-independent, suppression function [16], andis a bulk material property, which can be computed from first-principles [17]. This approach, which we refer to as the "non-interacting model," treats phonons with different MFPs separately, with the diffusive regime (i.e., for short Kns) as described by standard Fourier's law; on the other hand, the suppression function associated with ballistic phonons (i.e....

Parameter-free model to estimate thermal conductivity in nanostructured materials

“…In fact, as shown in Fig. 3(a), for L = 10 nm S(Λ) strongly depends on the material while for L = 100 nm the suppression functions approaches a universal value, called diffusive material limit [26]. Such a limit assumes that all phonon MFPs are smaller than L c and is practically achieved with L= 1 µm.…”

“…according to [26], the gradient of < T (Λ) > reaches a plateau for small Kns. Therefore, for macroscopic materials, i.e.…”

“…As mentioned above, this is the case of a diffusive material. It is possible to show that, in this regime, S(0) only includes geometric effects thus it is equal to r [26]. For this reason, the performance of S MG (Λ) in estimating S(Λ) increases with the size of the unit cell.…”

“…Periodic boundary conditions are applied to the heat flux on the boundaries of the unit cell. Along the walls of the pore we apply diffuse scattering boundary conditions, i.e., incoming phonons are scattered back isotropically; the temperature of the phonons leaving the pore's surface is set so that zero normal thermal flux is guaranteed along the boundary [26]. Heat flux is ensured by applying a temperature difference ∆T = 1 K between the hot and cold contacts.…”

“…For clarity, we will drop all the space dependencies from the notation. Along the walls of the pore we apply diffuse scattering boundary conditions, i.e., incoming phonons are scattered back isotropically; the temperature of the phonons leaving the pore's surface, T B , is set so that zero normal thermal flux is guaranteed along the boundary and is given by [26] T…”