Thermal transport across metal silicide-silicon interfaces: First-principles calculations and Green's function transport simulations

Heat transfer across metal-semiconductor interfaces involves multiple fundamental transport mechanisms such as elastic and inelastic phonon scattering, and electron-phonon coupling within the metal and across the interface. The relative contributions of these different transport mechanisms to the interface conductance remains unclear in the current literature. In this work, we use a combination of first-principles calculations under the density functional theory framework and heat transport simulations using the atomistic Green's function (AGF) method to quantitatively predict the contribution of the different scattering mechanisms to the thermal interface conductance of epitaxial ${\mathrm{CoSi}}_2$-Si interfaces. An important development in the present work is the direct computation of interfacial bonding from density functional perturbation theory (DFPT) and hence the avoidance of commonly used “mixing rules” to obtain the cross-interface force constants from bulk material force constants. Another important algorithmic development is the integration of the recursive Green's function (RGF) method with Büttiker probe scattering that enables computationally efficient simulations of inelastic phonon scattering and its contribution to the thermal interface conductance. First-principles calculations of electron-phonon coupling reveal that cross-interface energy transfer between metal electrons and atomic vibrations in the semiconductor is mediated by delocalized acoustic phonon modes that extend on both sides of the interface, and phonon modes that are localized inside the semiconductor region of the interface exhibit negligible coupling with electrons in the metal. We also provide a direct comparison between simulation predictions and experimental measurements of thermal interface conductance of epitaxial ${\mathrm{CoSi}}_2$-Si interfaces using the time-domain thermoreflectance technique. Importantly, the experimental results, performed across a wide temperature range, only agree well with predictions that include all transport processes: elastic and inelastic phonon scattering, electron-phonon coupling in the metal, and electron-phonon coupling across the interface.

Figure 1

Schematic of various mechanisms involved in heat transfer between the dominant energy carriers, i.e., electrons in the metal and phonons in the semiconductor. Phonon-phonon energy transfer across the interface could involve elastic and inelastic interfacial scattering processes. Electron-phonon coupling could involve coupling between electrons in the metal with phonons in the metal and with phonons in the semiconductor.

Figure 2

Comparison of computational times for different device lengths obtained using direct inversion and the RGF algorithm.

Figure 3

Atomic structures of $\mathrm{Si}-{\mathrm{CoSi}}_2$ interface supercells used in DFPT calculations (two unit cells shown along the in-plane direction for clarity). The red dotted boxes indicate the region around the interface for which IFCs are extracted from the interface supercell calculation. (a) 8A configuration. (b) 8B configuration.

Figure 4

Results from ballistic phonon transport calculations for a $\mathrm{Si}-{\mathrm{CoSi}}_2$ interface. (a) Phonon transmission function at normal incidence computed using average and DFPT force constants for the 8A interface. The inset shows the same graph for small phonon frequencies or long wavelengths. (b) Thermal interface conductance for 8A and 8B $\mathrm{Si}-{\mathrm{CoSi}}_2$ interfaces.

Figure 5

(a)–(c) Device temperature profile for cases A, B, and C, where case B corresponds to nominal scattering rates in Si and ${\mathrm{CoSi}}_2$, while cases A and C correspond to artificially decreased and increased scattering rates, respectively. The magenta lines correspond to linear fits of the temperature profiles on either side of the interface. (d)–(f) Spectral variation of the energy flux from Si and ${\mathrm{CoSi}}_2$ contacts for cases A, B, and C, respectively. (g)–(i) Accumulation of energy flux in the Si and ${\mathrm{CoSi}}_2$ contacts with respect to phonon frequency for cases A, B, and C, respectively.

Figure 6

(a) Spatial variation of the electron-phonon coupling coefficient $G_{ep}$ across a $\mathrm{Si}-{\mathrm{CoSi}}_2$ interface for different supercell lengths. The coupling coefficient in bulk strained ${\mathrm{CoSi}}_2$ and bulk Si are also shown for comparison. (b) Spatial variation of the local electron DOS at the Fermi energy across the $\mathrm{Si}-{\mathrm{CoSi}}_2$ structure shown in Fig. 3.

Figure 7

(a) Partitioning of different regions in the $\mathrm{Si}-{\mathrm{CoSi}}_2$ interface supercell used for the classification of phonon modes. (b) and (c) Contribution of Si, ${\mathrm{CoSi}}_2$, interfacial, and delocalized phonon modes to the total DOS (b) and Eliashberg function (c) of the $\mathrm{Si}-{\mathrm{CoSi}}_2$ interface supercell.

Figure 8

(a) Electron and lattice temperature profile across $\mathrm{Si}-{\mathrm{CoSi}}_2$ interface with electron-phonon coupling inside the metal region only. (b) Electron and lattice temperature profile across $\mathrm{Si}-{\mathrm{CoSi}}_2$ interface with electron-phonon coupling inside the metal region and in two unit cells of Si closest to the interface. In both (a) and (b), the red line corresponds to a linear fit of the lattice temperature profile in Si and the green line corresponds to a linear fit of the electron temperature profile in ${\mathrm{CoSi}}_2$ away from the interface region. (c) Heat flux distribution in the Büttiker probes across the $\mathrm{Si}-{\mathrm{CoSi}}_2$ interface corresponding to the temperature profiles in (a) and (b). For the simulation with direct electron-phonon coupling, the first Büttiker probe in Si closest to the interface has a nonzero energy flux.

Figure 9

Comparison of simulation predictions with experimental data (blue squares with error bars). A (black solid curve): phonon-only simulation with elastic interface scattering. B (magenta circles): phonon-only simulation with anharmonic phonon scattering in both Si and ${\mathrm{CoSi}}_2$. C (red hexagrams): electrons and phonons considered in the simulation with electron-phonon energy transfer inside the metal region only. D (green diamonds): electrons and phonons considered in the simulation with electron-phonon energy transfer included in two (1.9 nm) unit cells of Si closest to the interface.