Phonon confinement and transport in ultrathin films

Thermal transport by phonons in films with thicknesses of less than 10 nm is investigated in a soft system (Lennard-Jones argon) and a stiff system (Tersoff silicon) using two-dimensional lattice dynamics calculations and the Boltzmann transport equation. This approach uses a unit cell that spans the film thickness, which removes approximations related to the finite cross-plane dimension required in typical three-dimensional-based approaches. Molecular dynamics simulations, which make no assumptions about the nature of the thermal transport, are performed to obtain finite-temperature structures for the lattice dynamics calculations and to predict thermal conductivity benchmarks. Thermal conductivity decreases with decreasing film thickness for both the two-dimensional lattice dynamics calculations and the MD simulations, until the thickness reaches four unit cells (2.1 nm) for argon and three unit cells (1.6 nm) for silicon. With a further decrease in film thickness, thermal conductivity plateaus in argon while it increases in silicon. This unexpected behavior, which we identify as a signature of phonon confinement, is a result of an increased contribution from low-frequency phonons, whose density of states increases as the film thickness decreases. Phonon mode-level analysis suggests that confinement effects emerge below thicknesses of ten unit cells (5.3 nm) for argon and six unit cells (3.2 nm) for silicon. These transition points both correspond to approximately twenty atomic layers. Thermal conductivity predictions based on the bulk (i.e., three-dimensional) phonon properties combined with a boundary scattering model do not capture the low thickness behavior. To match the two-dimensional lattice dynamics and molecular dynamics predictions for larger thicknesses, the three-dimensional lattice dynamics calculations require a finite specularity parameter that in some cases approaches unity. These findings point to the challenges associated with interpreting experimental thermal conductivity measurements of ultrathin silicon films, where surface roughness and a native oxide layer impact phonon transport.

Figure 1

Silicon film in-plane thermal conductivity data from experimental measurements (markers) and predictions (lines) at a temperature of 300 K. The embedded films (cross/plus markers) were measured by $3-\omega$ [10, 11, 12, 13, 14] and thermoreflectance techniques [15, 16]. The suspended films (open markers) were measured using suspended island [17, 18], Raman thermometry [19, 20], and transient thermal grating (TTG) techniques [21]. The predictions are from an analytical model [22], the Matthiessen rule (MR) [23, 24], free path sampling (FPS) [23], and the discrete ordinate method (DOM) [25]. Predictions using the Fuchs-Sondheimer (FS) model [21] were made with bulk phonon mean free paths predicted by Esfarjani et al. [26] (FS $+$ Esfarjani et al.) or Holland [27] (FS $+$ Holland). The experimental bulk thermal conductivity [28] is represented by the dashed horizontal line.

Figure 2

Bulk and film unit cells in a three unit cell thick unrelaxed LJ argon film. The conventional cell with four atoms is used as the basis in bulk and to build the film unit cell.

Figure 3

(a) In-plane lattice constant versus film thickness for argon films at temperatures of 5 (blue), 10 (red), and 20 (black) K. The solid horizontal lines represent the bulk lattice constants. (b) Separations between atomic layers in the cross-plane direction versus the layer number, which represents the location in the film. For example, for the two unit cell film, there are four layers and three layer separations. “0” corresponds to the separation between the two central layers while “$-1$” and “1” correspond to the separations involving the surface layers. The solid horizontal line represents the bulk layer separation. The numbers in the legend correspond to the film thickness in conventional unit cells.

Figure 4

RMS displacement in the cross-plane and in-plane directions versus atomic layer number for argon at a temperature of 5 K. The solid line represents the bulk RMS displacement along one Cartesian direction. The dashed and dotted lines represent the values that the surface atoms approach in the cross-plane and in-plane directions. The numbers in the legend correspond to the film thickness in conventional unit cells.

Figure 5

Thermal conductivity versus thickness for argon films at temperatures of (a) 5 K and (b) 20 K. The results from the MD-GK, 2D-BTE, and 3D-BTE (with bulk or depleted phonons combined with the Fuchs-Sondheimer boundary scattering model) methods are presented. The Fuchs-Sondheimer boundary scattering model is applied with specularity parameters $p$ of 0 (both temperatures) and 0.62 (5 K). The horizontal solid lines indicate the bulk thermal conductivities from 3D-BTE and MD-GK.

Figure 6

Phonon DOS for the bulk, two unit cell film, and fifteen unit cell film argon systems at a temperature of 5 K. The vertical dashed line represents the end of the quadratic portion of the bulk DOS. Phonon modes with frequencies below this point are labeled as low frequency. The percentages provided above the dashed horizontal lines indicate the contribution of the low-frequency phonons modes for that system. The DOS are normalized to have the same integrated area.

Figure 7

(a) Argon frequency-dependent thermal conductivity accumulation functions for bulk and films predicted from 3D-BTE and 2D-BTE at a temperature of 5 K. The vertical dashed line at 6 Trad/s indicates the cutoff for low-frequency phonon modes. The numbers near the curves represent the film thickness in unit cells. (b) Accumulated thermal conductivity for phonon modes with frequencies below 6 Trad/s plotted versus film thickness. The horizontal solid line represents the bulk value. The vertical dashed line indicates the point where the accumulated thermal conductivity starts to increase with decreasing film thickness, which is independent of temperature.

Figure 8

(a) In-plane lattice constant versus film thickness for pristine silicon films at a temperature of 200 K. The solid horizontal line represents the bulk value. (b) Separations between atomic layers in the cross-plane direction versus layer number. For the two unit cell film, there are eight layers and seven layer separations. “0” corresponds to the separation between the two central layers while “$-3$” and “3” correspond to the separations involving the surface layers. The solid horizontal line represents the bulk value. The numbers in the legend correspond to the film thickness in conventional unit cells.

Figure 9

Cross-plane RMS displacement versus atomic layer number for pristine silicon films at a temperature of 200 K. The solid line represents the bulk RMS displacement along one Cartesian direction. The dashed line represents the value that the surface atoms approach in the cross-plane direction. The numbers in the legend correspond to the film thickness in conventional unit cells.

Figure 10

Thermal conductivity versus thickness of pristine [(a), (b)] and reconstructed [(a)] silicon films at a temperature of 200 K. The results from the MD-GK [(a), (b)], 2D-BTE [(a), (b)] and 3D-BTE [(b), with bulk phonons combined with the Fuchs-Sondheimer boundary scattering model] methods are presented. The film thermal conductivities from the 2D-BTE and 3D-BTE methods are normalized by the bulk value from 3D-BTE (464 W/m-K), while that from MD-GK is normalized by the bulk MD-GK value (414 W/m-K). The Fuchs-Sondheimer boundary scattering model is applied with constant ($p=0$ and 0.98) and wavelength-dependent [67] specularity parameters.

Figure 11

Pristine silicon DOS for bulk, the two unit cell film, and the six unit cell film using the atomic structures at a temperature of 200 K. The vertical dashed line indicates the end of the quadratic portion of the bulk DOS. Phonon modes with frequencies below this point are labeled as low frequency. The percentages provided above the horizontal dashed lines indicate the contribution of low-frequency phonon modes to the DOS for that system.

Figure 12

(a) Pristine silicon frequency-dependent thermal conductivity accumulation functions for bulk and films predicted from 3D-BTE and 2D-BTE at a temperature of 200 K. The vertical dashed line at 31 Trad/s indicates the cutoff for low-frequency phonon modes. (b) Accumulated thermal conductivity for phonon modes with frequencies below 31 Trad/s plotted versus film thickness for both pristine and reconstructed films. The horizontal solid line represents the bulk value. The vertical dashed indicates the point where the accumulated thermal conductivity starts to increase with decreasing film thickness.