Lattice Thermal Conductivity of the Binary and Ternary Group-IV Alloys Si-Sn, Ge-Sn, and Si-Ge-Sn

Efficient thermoelectric (TE) energy conversion requires materials with low thermal conductivity and good electronic properties. Si-Ge alloys, and their nanostructures such as thin films and nanowires, have been extensively studied for TE applications; other group-IV alloys, including those containing Sn, have not been given as much attention as TEs, despite their increasing applications in other areas including optoelectronics. We study the lattice thermal conductivity of binary (Si-Sn and Ge-Sn) and ternary (Si-Ge-Sn) alloys and their thin films in the Boltzmann transport formalisms, including a full phonon dispersion and momentum-dependent boundary-roughness scattering. We show that Si-Sn alloys have the lowest conductivity ($3\mathrm{W}/\mathrm{mK}$) of all the bulk alloys, more than 2 times lower than Si-Ge, attributed to the larger difference in mass between the two constituents. In addition, we demonstrate that thin films offer an additional reduction in thermal conductivity, reaching around $1\mathrm{W}/\mathrm{mK}$ in 20-nm-thick Si-Sn, Ge-Sn, and ternary Si-Ge-Sn films, which is near the conductivity of amorphous ${\mathrm{SiO}}_2$. We conclude that group-IV alloys containing Sn have the potential for high-efficiency TE energy conversion.

Figure 1

Phonon dispersion curves [the vibrational frequencies (THz)] for $\alpha \text{−}\mathrm{Sn}$, ${\mathrm{Ge}}_{0.5}{\mathrm{Sn}}_{0.5}$, and ${\mathrm{Si}}_{0.5}{\mathrm{Sn}}_{0.5}$ showing vs the phonon wave vector. The symbols represent the experimental measurement of $\alpha \text{−}\mathrm{Sn}$ from Ref. [56], and solid lines are the numerical simulation of $\alpha \text{−}\mathrm{Sn}$. Dotted lines represent the dispersion of ${\mathrm{Ge}}_{0.5}{\mathrm{Sn}}_{0.5}$, and dashed lines depict the dispersion of ${\mathrm{Si}}_{0.5}{\mathrm{Sn}}_{0.5}$.

Figure 2

(Top) A 3D surface plot of thermal conductivity vs Sn and Ge composition for Si-Sn, Ge-Sn, and Si-Ge of bulk material. Si-Sn has the lowest thermal conductivity in the bulk form. (Bottom) A ternary plot of thermal conductivity vs alloy composition.

Figure 3

(Top) A 3D surface plot of the lowest achievable thermal conductivity vs alloy composition for Si-Ge-Sn. Binary alloys correspond to the edges of the triangle of data points, showing that Sn has the lowest thermal conductivity in the amorphous (disordered) limit.

Figure 4

(Top) Lattice thermal conductivity of binary alloy Si-Ge vs Ge composition for bulk, 500-, 100-, and 20-nm thickness at room temperature. The sample thickness is $1\mu \mathrm{m}$ for nanostructures with 0.45-nm roughness. The bottom line shows the lowest achievable thermal conductivity amorphous limit of Si-Ge vs Ge composition. (Bottom) Cumulative thermal conductivity of pure Si (dashed lines) and Si-Ge alloy (solid lines) for acoustic branches.

Figure 5

(Top) Lattice thermal conductivity of binary alloy Si-Sn vs Sn composition for bulk, 500-, 100-, and 20-nm thickness. The simulation is done at room temperature with a roughness of 0.45 nm and a sample thickness of $1\mu \mathrm{m}$ for nanostructures. The bottom line depicts the amorphous-limit thermal conductivity of the alloy against Sn composition, while the top line shows the thermal conductivity of bulk Si-Sn. (Bottom) Cumulative thermal conductivity vs mean free path of pure Si (dashed lines) and Si-Sn alloy (solid lines).

Figure 6

(Top) Lattice thermal conductivity of binary Ge-Sn vs Sn composition for bulk, 500-, 100-, and 20-nm thickness. The simulation is done at room temperature with a roughness of 0.45 nm and $1\text{−}\mu \mathrm{m}$ in-plane sample length. The bottom line depicts the amorphous-limit thermal conductivity of the alloy against Sn composition, while the top line shows the thermal conductivity of bulk Si-Sn. (Bottom) Cumulative thermal conductivity vs mean free path of pure Si (dashed lines) and Si-Sn alloy (solid lines).

Figure 7

(Top) Lattice thermal conductivity of ternary alloy Si-Ge-Sn vs Ge or Sn composition for bulk, 500-, 100-, and 20-nm thickness at room temperature with 0.45-nm roughness and $1\text{−}\mu \mathrm{m}$ in-plane film length. The bottom black line shows the lowest achievable thermal conductivity of the alloy against Ge or Sn composition. (Bottom) Cumulative thermal conductivity vs mean free path of pure Sn (dashed lines) and ${\mathrm{Si}}_{0.3}{\mathrm{Ge}}_{0.3}{\mathrm{Sn}}_{0.3}$ alloy (solid lines).