In Situ Wafer Curvature Relaxation Measurements to Determine Surface Exchange Coefficients and Thermo-chemically Induced Stresses

Abstract

The curvature relaxation technique is a new, electrode-free, in situ technique for simultaneously measuring the chemical oxygen surface exchange coefficient (k) and stress state $\left( \lambda \right)$ of a thin or thick, dense or porous mechano-chemically active film atop a dense inert substrate. This chapter presents (1) an overview of the technique, (2) a detailed derivation of the fitting equations used to extract k from curvature relaxation experiments, (3) a discussion of the technique’s benefits and limitations, and (4) sample MATLAB code to predict the stress distributions found within dense thin or thick film bilayers as a function of temperature.

Appendices

Appendix 1: Sample Film Stress and Bilayer Curvature MATLAB Code

Appendix 2: Relating Normalized Oxygen Concentration and Film Strain Changes

Based on the relationship between the instantaneous oxygen vacancy concentration (c), the number of oxygen atoms per formula unit of non-defective material (N, which is 3 for ABO₃ perovskite materials), the oxygen nonstoichiometry ($\delta$, which is defined as the number of oxygen atoms missing per formula unit of material) and the volume occupied by one formula unit of material ($V^{*}$, which is equal to the unit cell volume for materials such as ABO₃ perovskites which contain 1 formula unit per unit cell), i.e.,

$$c = \frac{N - \delta }{{V^{*} }}$$

the difference between an instantaneous and original oxygen vacancy concentration that exists at a particular point within an oxygen exchange material that is equilibrating to a new ${p\text{O}}_{ 2}$:

$$c - c_{\text{o}} = \frac{N - \delta }{{V^{*} }} - \frac{{N - \delta_{\text{o}} }}{{V_{\text{o}}^{*} }}$$

where the naught subscript denotes the original state of the material. Applying the fact that:

$$\delta = \delta_{\text{o}} +\Delta \delta$$

results in:

$$c - c_{\text{o}} = \frac{{N - \delta_{\text{o}} }}{{V^{*} }} - \frac{{\Delta \delta }}{{V^{*} }} - \frac{{N - \delta_{\text{o}} }}{{V_{\text{o}}^{*} }}$$

Rearranging and applying the linear relationship between δ, the volumetric chemical strain $( \epsilon_{\text{c}} )$, and the volumetric chemical expansion coefficient $(\alpha_{\text{c}} )$ i.e.,

$$\epsilon_{\text{c}} = \alpha_{\text{c}}\Delta \delta$$

results in:

$$c - c_{\text{o}} = \frac{{N - \delta_{\text{o}} }}{{V^{*} }} - \frac{{N - \delta_{\text{o}} }}{{V_{\text{o}}^{*} }} - \frac{{\epsilon_{\text{c}} }}{{\alpha_{\text{c}} V^{*} }}$$

Combining the first two right-hand terms and factoring out a $- \left( {N - \delta_{\text{o}} } \right)$ results in:

$$c - c_{\text{o}} = \frac{{ - \left( {N - \delta_{\text{o}} } \right)\left( {V^{*} - V_{\text{o}}^{*} } \right)}}{{V^{*} V_{\text{o}}^{*} }} - \frac{{\epsilon_{\text{c}} }}{{\alpha_{\text{c}} V^{*} }}$$

Applying the definition of volumetric chemical strain , i.e.,

$$\epsilon_{\text{c}} = \frac{{V^{*} - V_{\text{o}}^{*} }}{{V_{\text{o}}^{*} }}$$

results in:

$$c - c_{\text{o}} = \frac{{ - \epsilon_{\text{c}} \left( {N - \delta_{\text{o}} } \right)}}{{V^{*} }} - \frac{{\epsilon_{\text{c}} }}{{\alpha_{\text{c}} V^{*} }}$$

Factoring out a $- \frac{{\epsilon_{\text{c}} }}{{\alpha_{\text{c}} V^{*} }}$ results in:

$$c - c_{\text{o}} = - \frac{{\epsilon_{\text{c}} }}{{\alpha_{\text{c}} V^{*} }}\left( {1 + \alpha_{\text{c}} \left( {N - \delta_{\text{o}} } \right)} \right)$$

Performing the same analysis on $c_{\infty } - c_{\text{o}}$ with the assumption that the final and instantaneous values of the film strain and cell volume per formula unit can be different, but that the chemical expansion coefficient is constant yields:

$$c_{\infty } - c_{\text{o}} = - \frac{{\epsilon_{{\text{c},\infty }} }}{{\alpha_{\text{c}} V_{\infty }^{*} }}\left( {1 + \alpha_{\text{c}} \left( {N - \delta_{\text{o}} } \right)} \right)$$

where $\epsilon_{{\text{c},\infty }}$ and $V_{\infty }^{*}$ are the final amount of strain, and the final unit cell per formula unit, in the material. Therefore:

$$\frac{{c - c_{\text{o}} }}{{{c}_{\infty } - \text{c}_{\text{o}} }} = \frac{{\epsilon_{\text{c}} V_{\infty }^{*} }}{{\epsilon_{{\text{c},\infty }} V^{*} }}$$

Substituting rearranged versions of the definition of volumetric chemical strain , i.e.,

$$\begin{aligned} V^{*} & = V_{\text{o}}^{*} (1 + \epsilon_{\text{c}} ) \\ V_{\infty }^{*} & = V_{\text{o}}^{*} (1 + \epsilon_{{\text{c},\infty }} ) \\ \end{aligned}$$

yields:

$$\frac{{c - c_{\text{o}} }}{{c_{\infty } - c_{\text{o}} }} = \frac{{\epsilon_{\text{c}} \left( {1 + \epsilon_{{\text{c},\infty }} } \right)}}{{\epsilon_{{\text{c},\infty }} \left( {1 + \epsilon_{\text{c}} } \right)}}$$

Distributing the strain terms through the bracketed terms yields:

$$\frac{{c - c_{\text{o}} }}{{c_{\infty } - c_{\text{o}} }} = \frac{{\epsilon_{\text{c}} + \epsilon_{\text{c}} \epsilon_{{\text{c},\infty }} }}{{\epsilon_{{\text{c},\infty }} + \epsilon_{\text{c}} \epsilon_{{\text{c},\infty }} }}$$

Since the strains encountered during a small ${p\text{O}}_{ 2}$ change are expected to be less than a few percent, the $\epsilon_{\text{c}} \epsilon_{{\text{c},\infty }}$ terms are much smaller in magnitude than the $\epsilon_{\text{c}}$ or $\epsilon_{{\text{c},\infty }}$ terms with the result of that in a $\kappa R$ experiment:

$$\frac{{c - c_{\text{o}} }}{{c_{\infty } - c_{\text{o}} }} = \frac{{\epsilon_{\text{c}} }}{{\epsilon_{{\text{c},\infty }} }}$$