Coefficient of thermal expansion and elastic constants of metastable <i>β</i> -phase tungsten thin films as a function of nitrogen content

The metastable A15 \ensuremath{\beta} phase of tungsten exhibits the largest spin Hall angle among all transition metals and is of interest for applications in spintronic devices. However, \ensuremath{\beta}-W is known to support very high stresses in such applications, which may affect properties and reliability. To understand these stresses, the thermomechanical properties must be known. The coefficient of thermal expansion (CTE) and elastic constants of \ensuremath{\beta}-W thin films containing 0--11.1 at. % nitrogen were determined within the ${24}^{\ensuremath{\circ}}\ensuremath{-}100{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}\mathrm{C}$ temperature range using three methods: substrate curvature measurements of thermoelastic curves for films deposited on $\mathrm{Si}\phantom{\rule{0.16em}{0ex}}\ensuremath{\langle}100\ensuremath{\rangle}$ and fused silica substrates, ab initio molecular dynamics (AIMD) simulations, and classical molecular dynamics (MD) simulations using a Spectral Neighbor Analysis Potential fitted from AIMD results using machine learning. Single crystal elastic constants obtained from the simulations were used along with the measured textures of the films to calculate biaxial elastic moduli. Results from all three approaches were consistent. The results of the same methods applied to the well-known BCC \ensuremath{\alpha} phase of W were within 3% of reference values, and those for \ensuremath{\beta}-W are expected to have a similar accuracy. It was found that the CTE of \ensuremath{\beta}-W increases while the biaxial modulus decreases with increasing nitrogen content. The average CTE and biaxial modulus of pure \ensuremath{\beta}-W within ${24}^{\ensuremath{\circ}}\ensuremath{-}100{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}\mathrm{C}$ were determined to be 4.931 $\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}{/}^{\ensuremath{\circ}}\mathrm{C}$ and 503 GPa, respectively, which are 10% higher and 11% lower than those of \ensuremath{\alpha}-W.