Simple view of the ${\text{Mg}}_2{\text{Si}}_{1-x}{\text{Sn}}_x$ phonon spectrum: Sn resonances and mean field

We have studied the phonon spectrum of ${\text{Mg}}_2{\text{Si}}_{1-x}{\text{Sn}}_x$ alloys using both density-functional theory calculations and neutron measurements. In the low tin concentration limit a resonance is obtained in the lower part of the spectrum. At higher concentrations comparison between theory and experiments shows that a simple averaging of the force constants of the perfect crystals ${\text{Mg}}_2\text{Si}$ and ${\text{Mg}}_2\text{Sn}$ can explain the main features of the density of states. However we suggest that the tin resonance persists up to large tin concentrations and may therefore be important for transport properties.

Figure 1

The density of states of ${\text{MgSi}}_{1-x}{\text{Sn}}_x$ for $x=0,1/32,31/32$, and 1 shown in (a), (b), (c), and (d), respectively. The green dashed line in (b) is the result for Sn using a $3\times 3\times 3$ supercell corresponding to the concentration $x=1/108$.

Figure 2

Theoretical (black) and experimental (blue) density of states for the alloys ${\text{Mg}}_2{\text{Si}}_{1-x}{\text{Sn}}_x$. The different curves are described in the text.