Navigation:  The optical model > Layer stacks > Layer types > Concentration gradient

This type of layer has been introduced for the description of gradually changing composition in a part of the layer stack. A 'concentration gradient layer' must use an effective dielectric function for which the first fit parameter is the volume fraction. It is recommended that you use one of those effective medium theories that have as their only parameter the volume fraction, such as the Maxwell Garnett, Bruggeman or Looyenga formulas.

Like simple layers concentration gradient layers have a thickness and a dielectric function (once again: this must be an effective dielectric function). However, the layer will be divided into a set of sublayers (whose number can be specified by the user). All sublayers have the same thickness (the total layer thickness divided by the number of sublayers). Each sublayer can have its own value of the volume fraction which is defined by a user-defined formula.

 

The graph of the user-defined formula is displayed in a subwindow of the object. It can be displayed in the main view as well: Just drag&drop the concentration gradient layer from the treeview into the view element list.

 

The number of sublayers of a concentration gradient appears as fit parameter. This makes it easy to find out how many sublayers are sufficient for reasonable computations.

 

Simple example

The application of concentration gradient layers is demonstrated using the following example. We consider a reflectance spectrum in the visible spectra range. The layer stack is a concentration gradient layer on top of a silver halfspace. The light enters from vacuum. The concentration gradient uses a Bruggeman effective medium with Ag particles embedded in vacuum. The dielectric function of silver is taken from the dielectric function database (Ag(JC)).

The layer stack is this:

 

 

Note the number three in the 7th column of the concentration gradient layer: This is the number of sublayers to be used for dividing the total thickness which is 10 nm in this case. With a right mouse click or the menu command Edit you can open a window which shows the volume fraction profile:

 

 

In the example a linear profile is defined with the simple formula 'X'. At the top of the layer (the left side of the graph) the silver volume fraction is 0 and at the bottom solid silver (volume fraction 1.0) is reached.

The reflectance spectrum (normal incidence of light, blue curve) is not very different from that of a sharp vacuum-air interface (red curve):

Note that you should sample the total thickness into sublayers whose optical thickness is much less than the wavelength of light. Otherwise you will see features in the spectra caused by the steps in the profile. If we increase the total thickness to 100 nm 3 sublayers are not sufficient. The following picture compares the computed spectrum with 3 sublayers (red curve) to that with 10 (blue):

The difference between 10 (in red) and 20 sublayers (blue) is not very large:

Hence 20 sublayers seem to be sufficient for this case. Going from 20 (red) to 30 (blue) doesn't change the spectrum any more:

Of course, you can define more advanced profiles with the flexible choices of user-defined formulas. A smoother profile than the linear one is defined by the formula SQR(0.5-0.5*COS(PI*X)), for example:

This leads to the following spectrum (red: linear profile, blue: smooth profile):

This example is closed with an interesting spectrum of the smooth profile for a thickness of 500 nm (blue curve) which leads to a very low reflectivity (for comparison the 100 nm spectrum is repeated here in red):

 

Noble metal/air-composites with low volume fractions are called 'blacks' for obvious reasons!

 

Advanced example

Now we show some more features of concentration gradient layers using an example of silicon oxide layers of various oxygen concentration on a 1 mm thick glass plate. The example makes use of two database entries, namely those of SiO and SiO2. A concentration gradient layer on the glass substrate is used with a mixture of SiO particles embedded in an SiO2 matrix. The SiO volume fraction is modeled by the formula C2-C2*COS(2*PI*C1*X) which produces profiles like the following (for C1=10 and C2=0.25):

 

The periodical repetition leads to constructive superposition of partially reflected waves producing pronounced interference maxima:

You can use the six constants C1, ..., C6 in the formula. Together with the total layer thickness these constants appear as fit parameters. You can set these constants and their names in the dialog that you open with the Parameters command in the Gradient definition window:

 

 

Hence you can select each of the constants as fit parameters and use sliders to learn how profile changes influence your optical spectra. If you have a fast PC and a large screen SCOUT can be a very powerful tool for intuitive thin film design!