User:Sebseb22/sandbox

To add in after the “measuring stress and strain” paragraph:

A common method for determining the stress evolution of a film is to measure the wafer curvature during its deposition. Stoney [4] relates a film’s average stress to its curvature through the following expression:

$$\kappa=\frac{6\langle \sigma \rangle h_f}{M_sh^2_s}$$,

where $$M_s = \frac{\Epsilon}{1-\upsilon}$$, where $$\Epsilon$$ is the bulk elastic modulus of the material comprising the film, and $$\upsilon$$ is the Poisson’s ratio of the material comprising the film, $$h_s$$ is the thickness of the substrate, $$h_f$$ is the height of the film, and $$\langle \sigma \rangle$$ is the average stress in the film. The average stress thickness of a given film can be determined by integrating the stress over a given film thickness:

$$\langle \sigma \rangle = \frac{1}{h_f} \int_{0}^{h_f} \sigma(z) dz$$

where $$z$$ is the direction normal to the substrate and $$\sigma(z)$$ represents the in-place stress at a particular height of the film. The stress thickness (or force per unit width) is represented by $$\langle \sigma \rangle h_f$$ in equation 1.1 and is an important quantity as it is directionally proportional to the curvature by $$\frac{6}{M_s h_s^2}$$. Because of this proportionality, measuring the curvature of a film at a given film thickness allows us to directly determine the stress in the film at that thickness. The curvature of a wafer is determined by the average stress of in the film. However, if stress is not uniformly distributed in a film (as it would be for epitaxially grown film layers that have not relaxed so that the intrinsic stress is due to the lattice mismatch of the substrate and the film), it is impossible to determine the stress at a specific film height without continuous curvature measurements. If continuous curvature measurements are taken, the time derivative of the curvature data [5]: $$\frac{d\kappa}{dt} \propto \sigma(h_f) \frac{\partial h_f}{\partial t} + \int_{0}^{h_f} \frac{\partial \sigma(z,t)}{\partial t}dz$$

can show how the intrinsic stress is changing at any given point. Assuming that stress in the underlying layers of a deposited film remains constant during further deposition, we can represent the incremental stress $$\sigma(h_f)$$ as [5]:

$$\sigma (h_f) \propto \frac{\frac{\partial \kappa}{\partial t}}{\frac{\partial h_f}{\partial t}} = \frac{d \kappa}{dh}$$

This thickness vs. stress-thickness plot provides a visual representation of incremental and average stress for a deposition of Ag on SiO2. Here, the average stress $$\bar{\sigma}$$ is found by dividing the stress-thickness by a specific thickness, and the incremental stress $$\sigma(h_f)$$ is obtained by taking the slope of the stress-thickness vs. thickness curve at a given thickness. The incremental stress represents the additional stress added to the film from the most recently deposited layer, whereas the average stress represents the average in-plane stress over the entire thickness of the film. For polycrystalline films, the stress is generally not uniform over the entire film’s thickness due to the effects that grain growth and size has on residual film stress.

To include after Volmer weber:

There are three distinct stages of stress evolution that arise during Volmer-Weber film deposition [6]. The first stage consists of the nucleation of individual atomic islands. During this first stage, the overall observed stress is very low. The second stage commences as these individual islands coalesce and begin to impinge on each other, resulting in an increase in the overall tensile stress in the film [2]. This increase in overall tensile stress can be attributed to the formation of grain boundaries upon island coalescence that results in interatomic forces acting over the newly formed grain boundaries. The magnitude of this generated tensile stress depends on the density of the formed grain boundaries, as well as their grain-boundary energies [7]. During this stage, the thickness of the film is not uniform because of the random nature of the island coalescence but is measured as the average thickness. The third and final stage of the Volmer-Weber film growth begins when the morphology of the film’s surface is unchanging with film thickness. During this stage, the overall stress in the film can remain tensile, or become compressive.

On a stress-thickness vs. thickness plot, an overall compressive stress is represented by a negative slope, and an overall tensile stress is represented by a positive slope. The overall shape of the stress-thickness vs. thickness curve depends on various processing conditions (such as temperature, growth rate, and material). Koch [8] states that there are three different modes of Volmer-Weber growth. Zone I behavior is characterized by low grain growth in subsequent film layers and is associated with low atomic mobility. Koch suggests that Zone I behavior can be observed at lower temperatures. The zone I mode typically has small columnar grains in the final film. The second mode of Volmer-Weber growth is classified as Zone T, where the grain size at the surface of the film deposition increases with film thickness, but the grain size in the deposited layers below the surface does not change. Zone T-type films are associated with higher atomic mobilities, higher deposition temperatures, and V-shaped final grains. The final mode of proposed Volmer-Weber growth is Zone II type growth, where the grain boundaries in the bulk of the film at the surface are mobile, resulting in large yet columnar grains. This growth mode is associated with the highest atomic mobility and deposition temperature. There is also a possibility of developing a mixed Zone T/Zone II type structure, where the grains are mostly wide and columnar, but do experience slight growth as their thickness approaches the surface of the film. Although Koch focuses mostly on temperature to suggest a potential zone mode, factors such as deposition rate can also influence the final film microstructure [2].