5.2.1
SETTIG UP THE COMSOL MODEL
The initial simulation structure is developed in the COMSOL graphic user interface; this enables the background equation systems, geometries and program features to be created semi-automatically. Although a similar system could have been created through the MATLAB interface the additional support of the COMSOL interface ensures a sound basis for the simulation and enables quick experiments and adaptations to be made. This is partially reliant on COMSOL’s accuracy, however COMSOL benefits from frequent updates and wide use and validation has been performed on the models used. For sim- plicity the same geometry and has been used for both wavelengths considered, similar boundary conditions have been used except when variation is required due to different incident waveforms.
GEOMETRY
The generic model geometry built in this thesis (Fig 5.2) has a rectangular domain which varies in height and width but is generally in the region of hundreds of nanometers; although a dimensionless form could have been used this is the scale where lithography operates and material parameters vary widely with dimensions and hence it was not thought useful. The geometry is viewed as a vertical cross section of a resist stack on a wafer, with light incident from the top of the geometry and a substrate being the final layer. Depending on the simulation the rectangular domain is divided into two or more different regions, representing different materials, where the base layer is always a PML to model a thick, absorbing, substrate.
BOUDARY SETTIGS
The boundaries of the simulation are defined as periodic on the left and right sides, envisioning an infinite layer in the − plane, which represents a reasonable assump- tion as many current chip features, such as gratings, extend over large dimensions and this method reduces the domain size greatly; however edge effects will alter results for finite gratings. Depending on the incident waveforms the periodic conditions may
change between periodicity and anti-periodicity in order to allow a correct match (e.g. a plane wave would have periodic conditions, whereas a one period of a sinusoidal intensi- ty waveform would have anti-periodicity – consider the underlying sinusoid). In this work only normal incidence is considered, if oblique incidence was of interest then Floquet periodicity would likely be used, with the phase change between boundaries dependent on the refractive index of the layer.
As explained in the geometry section a PML is used to model the substrate, and a scattering boundary condition with no incident wave is used at the external boundary of this to further reduce any reflections. Similar to the model of the dielectric stack the incident field is specified as either a TE or TM or combined incident field specified on a scattering boundary condition. The scattering boundary condition models the incidence of a coherent plane wave, from say a laser source, as an intensity level is set and then a standing wave from the reflection is able to form without affecting the intensity level. In this formulation of the model there is complete control over the incident angle and
Figure 5.2 Schematic diagram showing the geometry and boundary conditions of the generic simulation. The substrate and PML were present in all simulations, however a variety of material layers were used, including an AML and photoresist and possibly a plasmonic layer or air spacer layer.
5.2 AMOL MODEL IN MATLAB
power for both wavelengths. Example code for setting this incident wave is included in Appendix D. The incident light can be specified as a sum of plane waves at the top surface with free choice of amplitude and direction. This allows the incident wave to be easily specified as that from a Lloyd’s mirror arrangement involving two waves interfer- ing from opposite angles.
5.2.2
EXPORTIG TO MATLAB
After the AMOL geometry and system has been set out in COMSOL it is then exported into MATLAB to allow greater control over the simulation and for automating opera- tion. The FEM is represented in MATLAB as a structure which lists the relevant equations, geometries, variables and parameters; in MATLAB the FEM structures for both wavelengths are combined in a cell matrix and then this file is saved to allow it to be easily loaded in the future. When the file is loaded later care must be taken that all required variables are set to appropriate values for the simulation, otherwise they remain at the initial levels.
5.2.3
PHOTOCHROMIC LAYER I MATLAB
The kinetics of the photochromic reaction require knowing the intensity of both wave- lengths involved in the simulation at each point in the photochromic layer, furthermore the absorbance pattern of this layer will change with the change of intensity caused by the change in absorbance. To handle this situation an assumption is made that a stable photochromic state formed in a small time period; the simulation may then be repeated multiple times updating the absorbance in the AMOL layer based upon the incident intensities until the change between resulting photochromic states is less than a given tolerance.
The two simulation models described above are used to enable this process in MATLAB, for the first iteration of the simulations an opaque absorbance matrix is assumed for both wavelengths. The absorbance is set by creating an array of the refrac-
tive index in the photochromic layer for each simulation (this array is manually set to opaque for the first iteration), which is set as a global variable so that it may be refe- renced throughout the simulation. The absorbance values of the photochromic layer are then set as a function of the co-ordinates in the & plane, with an additional specifier to differentiate in the case of different absorbances for each wavelength. This function interpolates from the refractive index array to allow an approximation of the absorbance to be made for any point in the photochromic layer.
The calculation of the refractive index at each point in the photochromic layer requires knowledge of the light intensities, thus after the first iteration these must be calculated. This is achieved through the construction of the refractive index array as a rectangular grid with regular spacing (taken at 1 nm generally-this is limited by an increase in time required to interpolate a larger array and the limit in accuracy is the mesh of the simula- tion), the co-ordinates of this grid may then be used to interpolate the light intensity (normE_rfweh^2) for both wavelengths in the photochromic layer. Once the light intensity of both wavelengths has been calculated it must be converted into a corres- ponding refractive index, as described in Chapter 3. This calculation is held in a separate function to allow simple modification of the parameters used and any assumption, such as a constant refractive index.
5.2.4
CHAGIG PARAMETERS I MATLAB
Throughout the exploration of the AMOL system various parameters of the system must be altered so that appropriate simulations may be performed. Primarily adjustments are made to the geometry being used, the incident waveforms, or the material parameters of the simulation. The location of the setting in the FEM structure depends on the parame- ter being changed; the geometry has a separate part of the FEM structure to itself, the primary parameters are located in a variables structure and the material parameters and definitions are defined inside the FEM equation variables.
5.2 AMOL MODEL IN MATLAB
First considering the geometry; the COMSOL scripts must be used to generate a geome- try structure that is consistent with the COMSOL requirements; in this work this consists of a set of rectangular regions, where each of these regions then requires an appropriate set of equations. The straightforward way of ensuring that the model remains consistent is to maintain a number of regions the same as the initial simulation, reducing the complexity in making the changes. With this practice the geometry width and height dimensions can be changed by creating a new geometry and substituting it. The variables structures inside the FEM structure contain large variable lists; when a variable is changed the specific COMSOL variable name must be used and then code allows the variable list to be searched and updated appropriately (it should be noted that many variables are specified as tensors and hence may require multiple updates in the struc- ture).
5.2.5
SOLVIG AMOL I MATLAB
The solution of a complete AMOL system model may now be found using the COMSOL FEM model with the created MATLAB functions and environment. Assum- ing an initial geometry has already been created a generic simulation would begin by reloading this basic structure for both wavelengths and setting initial properties such as the wavelength values, refractive indexes of all layers (including an initial setting for the refractive index of the photochromic layer), model dimensions and incident waveforms and photochromic reaction kinetics; the simulation parameters such as the tolerance and maximum iterations must also be set. Once all the parameters are set the iterations may begin, repeatedly meshing and solving the model for both wavelengths until the refrac- tive index in the photochromic layer changes by less than the pre-defined tolerance level and hence the solution is returned. Figure 5.3 demonstrates the formation of an aperture for a representative case, with a funnel shape forming within a few apertures before slowly widening as increased propagates through it. The rapid decrease in difference between consecutive iterations is shown in the log plot in Fig 5.3d.
5.3
PLAE WAVE ICIDECE I AMOL
Warner and Blaikie [Warner 2009] introduced an analytical model for an AMOL type system which was described in Chapter 3 (Section 3.4.3); here the results of this analyti- cal system are matched to the results produced to the presented FEM AMOL model for normally incident plane waves (i.e. no grating) where the scalar approach is valid. Taking the closed form of the intensity fraction at equilibrium (Eq 3.34) a figure may be drawn showing the decay of light intensity (ℐ, ℐ) with normalized depth (*), repro- duced in Fig 5.4. The figure shows the weaker beam asymptote to zero whereas the
Figure 5.3 Absorbance pattern in the AML at iteration number (a) 1, (b) 3 and (c) 10, with the measured change in absorbance plotted in (d).