*2.2. Relationship between the Total Process Gas Flow and the Resulting Film Stress*

Knisely [14] introduced a model describing the power law relation between the RF substrate bias and the resulting AlN film stress. For a fixed Sc alloy this power law equation can be adapted to instead describe the total Al1-xScxN films stress Tave as a function of N<sup>2</sup> process gas flow

$$\mathbf{T\_{ave}} = \mathfrak{a}\left(\mathfrak{F}(\mathbf{t}\_{\mathbf{f}})^{\mathcal{Y}} + \mathbf{F\_{N\_2}}\right) \tag{2}$$

where α is determined from the slope of the stress versus N<sup>2</sup> gas flow, FN<sup>2</sup> is the N<sup>2</sup> gas flow in the chamber, and β and γ are empirical fits based on the deposition parameters and environment. Similar to Knisely [14] we implement a film that utilizes multiple layers deposited under different sputtering gas conditions where the average film stress of each layer is used to compensate for the through-thickness stress gradient. Where Knisely [14] utilizes a different RF substrate bias to control the stress of each AlN layer, we utilize a different N<sup>2</sup> flow to control the stress of each Al0.68Sc0.32N layer. The layer film stress, t<sup>f</sup> , based on the thickness and N<sup>2</sup> flow of the layer is derived from integrating the average film stress [14] and is given by

$$\mathbf{T\_{f}(t, F\_{N\_{2}})} = \alpha \left(\boldsymbol{\beta} (1 + \boldsymbol{\gamma}) (\mathbf{t\_{f}})^{\boldsymbol{\gamma}} - F\_{N\_{2}, \boldsymbol{\mu}}\right) \tag{3}$$

where FN<sup>2</sup> is the constant flow applied to the layer and t is the thickness of the layer. Equation (3) can be used to calculate the layer stresses needed to compensate for the through-thickness stress gradient within the film. The stress gradient is calculated from the average stress measurements taken from different wafers deposited using identical deposition process parameters and at varying AlScN thicknesses. Equation (3) is then used to interpolate between the experimental measurements of films with different thicknesses to find the through-thickness stress gradient.
