Thermal expansivity of ${\text{Ge}}_{1-y}{\text{Sn}}_y$ alloys

The temperature dependence of the lattice parameter of ${\text{Ge}}_{1-y}{\text{Sn}}_y$ alloys deposited on Si substrates has been determined from an analysis of their x-ray reciprocal-space maps. It is found that over the range $0<y<0.03$ the alloy thermal expansivity increases by up to 20% as a function of $y$. This implies a strong deviation from a linear interpolation between the end compounds since the thermal expansivities of pure Ge and $\alpha \text{-Sn}$ are nearly the same. Alternative interpolation formulas based on a Debye model and a mixed Debye-Einstein model of the phonon structure are tested and it is found that they also fail to explain the observed increase in thermal expansivity.

Figure 1

(Color online) High-temperature XRD plot recorded at $30°\text{C}$ showing the (224) reciprocal-space maps of a fully relaxed ${\text{Ge}}_{0.98}{\text{Sn}}_{0.02}$ film and the corresponding Si substrate.

Figure 2

Relaxed lattice parameter of ${\text{Ge}}_{1-y}{\text{Sn}}_y$ alloys as a function of temperature. The data is normalized to the relaxed lattice parameter at $30°\text{C}$. The slope of a linear fit to the data gives the average thermal expansivity within the temperature range of the fit. The compositions are obtained from fits of RBS spectra.

Figure 3

Compositional dependence of the linear expansivity in ${\text{Ge}}_{1-y}{\text{Sn}}_y$ alloys. Circles represent experimental values extracted from linear fits of the data in Fig. 2. The error bars correspond to the errors in those fits. The solid line represents a linear fit of the expansivity as a function of composition. The dashed line is a theoretical prediction based on the two models discussed in Sec. 3, which are undistinguishable over this narrow compositional range.

Figure 4

Circles represent the experimental linear thermal expansivity for Si, Ge, and $\alpha \text{-Sn}$. The data were taken from Ref. 13 (Si), Refs. 14, 15, 29 (Ge), and Ref. 16 (Sn). The dashed lines represent fits with Eq. (6). The adjusted values of ${\gamma }_D$ for each material are indicated in the three panels. The solid line is a fit with Eq. (8). The corresponding adjustable parameters, ${\gamma }_{\text{TA}}$ and ${\gamma }_{\text{opt}}$, are also indicated in the three panels.

Figure 5

Compositional dependence of the linear thermal expansivity at 300 K. Squares represent experimental data from Ref. 20. The dashed lines represents a prediction based on the Debye expansivity model in Eq. (6). The solid line is the corresponding calculation based on the mixed Debye-Einstein model of the expansivity in Eq. (8).