Spatial nonuniformity in heat transport across hybrid material interfaces

Successful thermal management in nanostructured devices relies on control of interfacial thermal transport. Recent measurements have revealed poor thermal transport across interfaces between two dissimilar materials, e.g., organic semiconductors and metals. In such systems, the interfacial thermal conductance $G_b$ is dominated by the strength of interfacial bonding, but existing analytical models still fail to accurately predict $G_b$ especially for organic-metal interfaces. Growing interest in this research area calls for comprehensive understanding of interfacial thermal transport across hybrid material interfaces. Here we demonstrate that spatial nonuniformity has to be assessed in the calculation of $G_b$ for interfaces with partial coverage or incommensurate growth that is characteristic of these interfaces. The interface between copper phthalocyanine and fcc metals (Ag, Al, and Au) exhibits a sixfold difference between the metal's (∼4-Å) and the organic molecule's (∼25-Å) lattice constant. Molecular dynamics simulations reveal the spatial variation in $G_b$, and a model is developed that considers the spatial variations in phonon transmission, successfully predicting $G_b$ for many organic-metal interfaces.

Figure 1

Thermal boundary conductance for various interfaces. The measured values of thermal boundary conductance $G_b$ for various interfaces from multiple references. The interfaces are categorized into three groups: The left (blue) panel presents $G_b$ of metal-inorganic semiconductor interfaces; the middle (gray) panel contains $G_b$ data points of metal-organic interfaces; the right (red) panel is for organic-organic interfaces. Data points with the same marker type and color correspond to interfaces that share one material: This material is shown in a bold font in the same color. $G_b$ values are obtained from Refs. [27, 28, 29, 30, 31, 32, 33, 34, 35]. CNT: Carbon Nanotubes; SAM: Self-Assembled Monolayer.

Figure 2

CuPc-fcc metal (Ag, Al, and Au) interface. Depiction of CuPc-fcc metal interfaces: The atom coordinates are obtained from the relaxed structure of molecular dynamics simulation. (a) Side view of the interface: The metal is in a crystalline state, and CuPc molecules are disordered from the second layer upwards, whereas the first-layer molecules lay flat. (b) Top view of the interface alongside the structure of an enlarged CuPc molecule.

Figure 3

Effective spring constants of atoms in the first-layer CuPc. Effective spring constants $K_{sp}$ (cross plane) are sampled using molecular dynamics simulations. The distributions of $K_{sp}$ of each type of atom (H, C, N, and Cu) are presented as histograms: Cu and N atoms at the center of the molecule own larger spring constants compared to C and H atoms that are further from the center. The inset is a side view of one particular CuPc molecule vertically spanned according to $K_{sp}$.

Figure 4

Pack and void environments of the first layer of metal atoms. The organic molecules have larger intermolecular spacings (>10 Å) than that of the fcc metal (∼4 Å). The metal surface does not have full coverage by the organic molecules. If the metal atoms are topped by the CuPc molecules (within a 6.58-Å radius, centered at the copper atoms), those are grouped in the pack marked as blue; otherwise, the metal atoms belong to the void region.

Figure 5

Effective spring constants and phonon density of states of interfacial Ag atoms in pack, void, and bulk. (Left) Histograms of effective spring constants (cross plane). Peak value of $K_{sp}$ in void and pack regions are separated by ∼5 N/m, whereas both are weaker in bonding compared to the bulk silver—sampled a few layers underneath the interface. (Right) Phonon density of states (DOS) also show differences among the three groups: 5 THz (LA) peak cannot be observed at the interface; the phonon DOS of the void redshifted further compared to that of the pack.

Figure 6

Cross-plane interfacial thermal conductivity. Thermal conductivity in the cross-plane direction ($z$ axis in Fig. 4) calculated using the Green-Kubo formalism. Size effect extrapolation is needed due to the narrow length sampled in the cross-plane direction. $k$(0) is fitted using Eq. (3) for both pack and void starting from 2 THz (the first peak value in Fig. 5) onwards with thermal conductivity of the pack 25% larger than that of the void.

Figure 7

Effective ratio $K_b/K_0$ versus the coverage percentage of the interface $\left(x_c\%\right)$. The ratio is a single parameter function of $x_c\%$. The spring constant value that correctly calculates the boundary conductanceis $K_b=1/\langle K^{-1}\rangle$, which differs from $K_0=\langle K\rangle$ by the factor of this effective ratio. An example distribution of {$K_{sp}$} is presented in the inset. The ratio is 0.787 for the CuPc-fcc metal (Al, Ag, and Au) interfaces.

Figure 8

Thermal boundary conductance of the CuPc-fcc metal interfaces versus $K_{sp}$. (Blue-dotted line) Calculated by the model proposed here [Eq. (10)] considering diffuse and inelastic phonon scatterings with partial interfacial coverage. (Black-dashed line with black round markers) Molecular dynamics simulated $G_b$ with upper and lower bounds drawn separately in green lines.