Thermal boundary resistance at Si/Ge interfaces determined by approach-to-equilibrium molecular dynamics simulations

The thermal boundary resistance of Si/Ge interfaces has been determined using approach-to-equilibrium molecular dynamics simulations. Assuming a reciprocal linear dependence of the thermal boundary resistance, a length-independent bulk thermal boundary resistance could be extracted from the calculation resulting in a value of $3.76\times {10}^{-9}$ ${\mathrm{m}}^2$ K/W for a sharp Si/Ge interface and thermal transport from Si to Ge. Introducing an interface with finite thickness of 0.5 nm consisting of a SiGe alloy, the bulk thermal resistance slightly decreases compared to the sharp Si/Ge interface. Further growth of the boundary leads to an increase in the bulk thermal boundary resistance. When the heat flow is inverted (Ge to Si), the thermal boundary resistance is found to be higher. From the differences in the thermal boundary resistance for different heat flow direction, a rectification factor of the Si/Ge interface can be determined and is found to significantly decrease when the sharp interface is moderated by introduction of a SiGe alloy in the boundary layer.

Figure 1

Schematic representation of a Ge/Si interface [in the (001) crystallographic plane] with boundary thickness $d_I$. The lattice spacing in in-plane direction (orthogonal to thermal transport) of Si has been adopted from the equilibrium lattice spacing of Ge $(a_{\mathrm{Ge},0}=5.6567$ Å). The out-of-plane lattice spacing of Si has been calculated from the elastic constants $(a_{\mathrm{Si}},\perp =5.1913$ Å).

Figure 2

Inverse of the thermal conductivity 1/$\kappa$ as a function of the inverse sample length 1/$L_z$ of crystalline Ge (c-Ge, diamonds), crystalline Si (c-Si, triangles up), and pseudomorphic Si (p-Si, triangles down). The bulk thermal conductivity ${\kappa }_{\infty }$ of these materials has been approximated based on a linear relationship between 1/$\kappa$ and 1/$L_z$. It resulted in 93.3, 233.4, and 204.3 W/mK for c-Ge, c-Si, and p-Si, respectively.

Figure 3

Total thermal conductivity ${\kappa }_{\mathrm{all}}$ as a function of the sample size $L_z$ of Si/Ge interfaces with a sharp interface. Thermal transport has been simulated from Si to Ge (diamonds) and vice versa (circles). The behavior of $\kappa$ to $L_z$ has been approximated with a $2\mathrm{nd}$ order polynomial function [Eq. (5)] (dashed line).

Figure 4

Thermal boundary resistance $R_I$ as a function of the sample size $L_z$. The behavior of $R_I$ to $L_z$ follows a linear reciprocal trend $R_I=R_{I,\infty }(1+\frac{{\lambda }_I}{L_z})$.

Figure 5

Bulk thermal boundary resistance $R_{I,\infty }$ as a function of the interfacial thickness $d_I$ for thermal transport from Si to Ge (diamonds) and from Ge to Si (circles).

Figure 6

Rectification $f_{\mathrm{Rect}}$ of the bulk thermal boundary resistance of Si/Ge interfaces as a function of the interfacial thickness $d_I$. It has been calculated according to Eq. (10).