In the evaporation process, the pressure within the chamber is typically maintained sufficiently low to ensure that molecules in the stream of evaporant travel in straight lines until they collide with a surface. To calculate the thickness distribution within a machine, it is generally assumed that every molecule of evaporant sticks where it lands.
Although this assumption is not always strictly correct, it allows for uniformity calculations that are accurate enough for most purposes. The distribution of thickness can then be calculated in the same manner as the intensity of illumination in an optical calculation. To estimate the thickness, all that is required is knowledge of the distribution of evaporant from the source.
Holland and Steckelmacher provided an early and detailed account of techniques for predicting layer thickness and uniformity, establishing a theory that remains the foundation of uniformity predictions today. Behrndt later extended their expressions. Holland and Steckelmacher categorized sources into two main types:
- Point Sources: Emit material evenly in all directions and are analogous to point light sources.
- Directed Surface Sources: Emit material in a manner similar to that from a flat surface, where the intensity decreases as the cosine of the angle between the direction of interest and the surface normal.
The expressions for the material distribution emitted from these two types of sources are as follows:
1. For the Point Source:
\[
dM = \frac{m}{4\pi} d\omega
\]
2. For the Directed Surface Source:
\[
dM = \frac{m}{\pi} (\cos\phi) d\omega
\]
where \( m \) is the total mass of material emitted from the source in all directions, \( dM \) is the amount passing through solid angle \( d\omega \), and \( \phi \) is the angle between the direction of interest and the normal to the surface in the case of the directed source.
If the material is being deposited on surface element \( dS \) of the substrate, which has its normal at angle \( \vartheta \) to the direction of the source from the element, then the amount condensing on the surface can be expressed as:
1. For the Point Source:
\[
dM = \frac{m}{4\pi r^2} (\cos\vartheta) dS
\]
2. For the Directed Surface Source:
\[
dM = \frac{m}{\pi r^2} (\cos\phi)(\cos\vartheta) dS
\]
where \( r \) is the distance from the source to the surface element.
To determine the thickness \( t \) of the deposited material, we also need to know the density \( \mu \) of the film. The thickness is then given by:
1. For the Point Source:
\[
t = \frac{m}{4\pi \mu r^2} (\cos\vartheta)
\]
2. For the Directed Surface Source:
\[
t = \frac{m}{\pi \mu r^2} (\cos\phi)(\cos\vartheta)
\]
These equations form the basis of the methods used by Holland and Steckelmacher for estimating thickness in uniformity calculations.
1. Flat Plate
The simplest case is a flat plate held directly above and parallel to the source. In this arrangement, the angle \( \phi \) is equal to the angle \( \vartheta \), and the thickness \( t \) can be calculated as follows:
For the point source:
\[
t = \frac{m}{4\pi\mu r^2} = \frac{mh}{4\pi\mu(h^2 + \rho^2)^{3/2}} \cos \vartheta
\]
For the directed surface source:
\[
t = \frac{m}{\pi\mu r^2} = \frac{mh}{\pi\mu(h^2 + \rho^2)^2} \cos \vartheta
\]
Here, the notation corresponds to Figure 13.1. These expressions simplify further:
– For the point source:
\[
t = t_0 \left(\frac{h^2}{h^2 + \rho^2}\right)^{3/2}
\]
– For the directed surface source:
\[
t = t_0 \frac{h^2}{h^2 + \rho^2}
\]
where \( t_0 \) is the thickness directly above the source (\( \rho = 0 \)). These relationships are plotted in Figure 13.2.
In both cases, the uniformity of the thickness is not particularly good. This geometry is only suitable for very accurate work if the substrate is extremely small and placed at the center of the machine.
2. Spherical Surface
A slightly better arrangement for achieving uniform deposition is a spherical geometry, where the substrates are placed on the surface of a sphere. A point source positioned at the center of the sphere will provide a uniform thickness of deposit on the inner surface of the sphere. Similarly, a directed surface source will produce a uniform distribution when it is part of the spherical surface itself.
In fact, the uniformity of coating within a spherical geometry led Knudsen to propose the cosine law for thin-film deposition. This method is frequently used in machines for basic applications, such as blooming optical components like lenses, where the uniformity requirement may be within 10% of the layer thickness at the center of the component.
However, for precision work, this level of uniformity is insufficient. To achieve higher uniformity, the rotation of the substrate carrier is often employed, which will be discussed in the next section.
3. Rotating Substrates
In this section, we examine the uniformity achieved when substrates are rotated during the coating process. The scenario is as if the surface for coating, depicted in Figure 13.1, is rotated about a normal axis at a distance \( R \) from the source.
As the surface rotates, the thickness deposited at any point is the average of the thickness that would have been deposited on a stationary substrate around a ring centered on the axis of rotation, provided that the number of revolutions during the deposition is sufficiently high to ensure that the incomplete revolutions contribute insignificantly to the total thickness.
By optimizing the distance between the source and the axis of rotation, uniformity can be significantly improved compared to stationary substrates.
Directed Surface Source
Figure 13.3 illustrates the setup for a directed surface source. Here, the mean thickness is calculated by integrating the thickness deposited around the circular path. For any point \( P \) on the circle, defined by angle \( \psi \), the thickness at \( P \) is given by:
\[
t = \frac{m}{\pi \mu} \cdot \frac{h}{(h^2 + R^2 + \rho^2 – 2\rho R \cos\psi)^{3/2}}
\]
To compute the average thickness for the rotating substrate:
\[
t = \frac{m h}{\pi \mu} \int_0^{2\pi} \frac{d\psi}{(h^2 + R^2 + \rho^2 – 2\rho R \cos\psi)^{3/2}}
\]
Using contour integration, the integral simplifies, and the final expression for the thickness becomes:
\[
t = t_0 \cdot \frac{1 + \frac{\rho^2}{h^2} + \frac{R^2}{h^2} – \frac{2\rho R}{h^2}}{\left[1 + \frac{\rho^2}{h^2} + \frac{R^2}{h^2} + \frac{2\rho R}{h^2}\right]^{3/2}}
\]
where \( t_0 \) is the central thickness above the substrate holder. Figure 13.4 shows this function plotted for various dimensions of typical medium-sized coating machines. The rotating configuration provides far superior uniformity compared to stationary substrates. For example, when \( R = 7 \), the uniformity is excellent over the central radius of 3.75.
This arrangement is commonly used in narrowband filter production, where high uniformity is essential. For broader filters or antireflection coatings, the sources can be moved outward, increasing the coated area at the cost of slight uniformity reduction.
Point Source
For a point source, the thickness at \( P \), assuming no rotation, is given by:
\[
t = \frac{m}{4 \pi \mu} \cdot \frac{1}{(h^2 + R^2 + \rho^2 – 2\rho R \cos\psi)^{3/2}}
\]
For a rotating substrate, the thickness is averaged over the circle’s circumference:
\[
t = \frac{m h}{4 \pi \mu} \int_0^{2\pi} \frac{d\psi}{(h^2 + R^2 + \rho^2 – 2\rho R \cos\psi)^{3/2}}
\]
This integral simplifies to involve elliptic integrals of the second kind \( E(k, \alpha) \), leading to:
\[
t = \frac{m}{4 \pi \mu} \cdot \frac{E(k, \alpha)}{\sqrt{h^2 + (R + \rho)^2}}
\]
where \( k \) is a function of \( R, h, \) and \( \rho \). Curves for this expression, similar in shape to those for directed surface sources, are provided by Holland and Steckelmacher.
Source Behavior and Adjustments
Sources like electron beam and howitzer types show variations in distribution. For instance, howitzer sources loaded with zinc sulfide may exhibit behavior between a point source and a directed surface source due to scattering in the high-pressure evaporant stream near the heater. Electron beam sources, studied by Graper [6], tend to be more directional and often follow a \( \cos^x \theta \) distribution, where \( x \) depends on power and material.
During machine setup, sources are positioned based on theoretical calculations. Minor adjustments can then be made by trial and error for optimal performance. Ensuring precise angular alignment is critical; even small tilts from the correct direction can result in uniformity errors.
Advanced Configurations
For better uniformity over a larger area, combinations like spherical surfaces and rotating plates are used. A domed holder, or calotte, rotated about its center, with sources adjusted to the sphere’s surface, provides excellent uniformity over an extended area. Figure 13.5 illustrates this setup.
When even greater uniformity is required, planetary geometries are employed. Here, substrates are mounted in small carriers that rotate individually while also revolving around the machine’s center. This arrangement further enhances the averaging effect, producing uniform coatings suitable for demanding applications.
4. Use of Masks
It is possible to improve thickness uniformity during the coating process by carefully utilizing masks. These masks are typically stationary and positioned just in front of the substrates, which rotate on a single carrier about a single axis. The masks are designed to adjust the radial distribution of thickness by blocking or modulating specific regions of the evaporant stream.
Stationary Masks
Theoretical calculations are used to determine the approximate dimensions of these masks, which are then fine-tuned experimentally to achieve the desired thickness distribution. The central monitor glass is usually left unmasked for several reasons:
1. Practicality: It is challenging to adjust the mask for the central part of the chamber, where the mask width approaches zero.
2. Monitoring Needs: The central monitor is typically stationary, and in some monitoring systems, having extra material on the monitor compared to the batch is advantageous.
Rotating Masks
A significant improvement in uniformity was introduced by Ramsay et al., who developed a system using a rotating mask. This method is particularly beneficial for large, flat substrates nearing the size of the coating machine, where conventional methods such as simple rotation or stationary masks are insufficient.
- Design: The mask rotates about a vertical axis at a speed significantly higher than that of the substrate carrier. This rotation effectively corrects the angular distribution of the evaporant from the source.
- Source Placement: The source is positioned at the center of the machine, with the mask rotation axis located near the source. A line drawn from the source through the mask’s center typically intersects the perimeter of the substrate carrier.
- Trimming and Stability: Slight adjustments to the rotation axis can be made for fine-tuning, making this setup highly stable. Uniformity as precise as 0.1% has been achieved over areas with a diameter of around 200 mm using this method.
The rotating mask setup is a robust solution for achieving high uniformity, particularly in situations where stationary masks are inadequate due to their sensitivity to source characteristics. This method offers exceptional stability and consistency in coating performance.
