Thermal expansion coefficient of single-crystal silicon from 7 K to 293 K

We measured the absolute lengths of three single-crystal silicon samples by means of an imaging Twyman-Green interferometer in the temperature range from 7 K to 293 K with uncertainties of about 1 nm. From these measurements we extracted the coefficient of thermal expansion with uncertainties on the order of $1\times {10}^{-9}/\mathrm{K}$. To access the functional dependence of the length on the temperature, usually polynomials are fitted to the data. Instead we used a physically motivated model equation with seven fit parameters for the whole temperature range. The coefficient of thermal expansion is obtained from the derivative of the best fit. The measurements conducted in 2012 and 2014 demonstrate a high reproducibility, and the agreement of two independently produced samples supports single-crystal silicon as a reference material for thermal expansion. Although the results for all three samples agree with each other and with measurements performed at other institutes, they significantly differ from the currently recommended values for the thermal expansion of crystalline silicon.

Figure 1

Schematic of the beam path in PTB's Ultraprecision Interferometer. Prismatic samples are placed in the extended probe beam on the left. Its temperature can be varied between 7 K and 293 K by a pulse tube cryocooler.

Figure 2

(a) Measured length versus temperature for sample 2, fit (red line) according to Eq. (5), and, below, residuals from the fit and overall uncertainty of the length versus temperature data (Appendix pp1). (b) Deduced CTE according to Eq. (4) and overall uncertainty (Appendix pp2).

Figure 3

Comparison of all three measurements performed at PTB: In 2012 on sample 1 (light blue), and in 2014 on samples 2 (dark blue) and 3 (purple). Plotted is a band with a half-width of 1 $\sigma$ around the respective CTE minus the average CTE of all three measurements.

Figure 4

Comparison of high-accuracy CTE measurements performed on crystalline silicon. Plotted are bands with a half-width of 1 $\sigma$ around the respective CTE minus recommended reference data [23]. Compared are the results from this work on sample 2 (blue), the data from the reference certificate acquired from NMIJ [18] (red), measurements from JPL [6] (green), and the $1-\sigma$ uncertainty of the reference data (gray).

Figure 5

Comparison of thermal strain results. Plotted are bands with a half-width of 1 $\sigma$ around the respective strain minus the strain measured for sample 2. Compared are the results from this work on sample 1 (light blue), on sample 2 (dark blue), and on sample 3 (purple) with the recommended reference data [14] (black line), which are provided without uncertainty.