In this topic, we describe a simulation technique for studying scattering from isolated small particles on a surface. The incident beam is a plane wave at a grazing angle of incidence injected from TFSF source.
We will test this technique by calculating the far field scattering intensity profile from a PolyStyrene Latex (PSL) sphere on a Si substrate, as shown in the figure below. Scattering from PSL spheres can be used to calibrate and test defect detection systems. The simulation results can be compared to those published by S. Stokowski.
In the above screenshot, the Si substrate is shown in red and the PSL sphere in cyan. The TFSF source is represented by a box in light-grey, with its injection plane at the top plane of the box. The pink arrow and the blue arrows represent the light propagation direction and the polarization, respectively. This type of simulation can be challenging due to the large difference in the refractive indices of the particle and the substrate.
A PSL sphere (index 1.6) is located in air above a Si substrate.
We are interested in the far field intensity in this example. The farffield is calculated from the near field recorded by a power monitor. The required size of the monitor strongly depends on the scattering property of the structures and should be large enough to collect as much light as possible from the near field. However, this will inevitably result in a large memory requirement and a longer simulation time. Some test runs might be necessary to find out an appropriate monitor size and the simulation spans.
In this example, a large simulation span of 7 um is used and, due to the large incidence angle of 70º , the monitor is positioned very close to the source in order to collect the scattered light propagating almost parallel to the substrate. Current settings allow the monitor to pick up field components propagating at angles of up to 86.5º with respect to the surface normal. Since most of scattered near field lies on the right side of the particle, the simulation region is shifted to the right. Below is the near field of the sphere with a radius 0.11 um sphere with P polarization of 70 degree incidence:
Near field intensity of the sphere with a radius 0.11um, P polarization at 70 deg incidence.
In general, conformal technique should be used to reduce the error of material interface. However, for the very coarse mesh of the smallest sphere with a radius of 0.03um, its performance is degraded since it ha only 5 meshes in each direction. In addition, for such small sphere, its scattering is relatively uniform and is slightly stronger in the backward direction. Thus, the shifted monitor may not collect all the near field information. To increase the accuracy, one may further increase the monitor size.
The simulation region uses PML absorbing boundary conditions on all boundaries. Note that for P- and S- polarizations the incident light is angled only in XZ plane. So, symmetric/anti-symmetric boundary condition in y min can be used to speed up the simulation. This is included in the script. For circular polarization at normal incidence, neither symmetric nor anti-symmetric boundary condition can be applied, since there are two different polarizations on the symmetric line of the object, please refer to the symmetric/anti-symmetric boundary condition section for more details.
The source is initially P polarized with a wavelength of 488 nm. The angle of incidence is 70 degrees. The source polarization can be changed to S by editing the script file settings, which sets the polarization to 90 degrees. A normally incident circularly polarized beam can also be created by editing the script file. First, it sets the source angle theta to 0. Then, it creates a second source with identical settings, except with a polarization angle of 90 degrees and a phase of 90 degrees. Each source will have a different name.
The 2D power monitor located above the source will be used to calculate far field scattering. Only light that passes through this monitor will be included in the far field projection. So it is important to position it as close as possible to the sphere. Since we are only interested in scattered field it is placed above the source. In order to calculate the far field projection, an analysis group is used, in which the script outputs the far field projection parameters.
- Run a test without particle: To have an initial test, you should run a simulation without the particle. The noise intensity should be less than 1e-12.
- Simulation Mesh: Even though the angle of incidence is large, light inside the silicon substrate travels very close to Z axis, due to the large refractive index of silicon at this wavelength. To accelerate the simulation, we use a relatively coarse mesh in XY for Si by using an override mesh. In addition, we use finer mesh for the TFSF region including the particles.
- TFSF region: The TFSF region can be chosen small to enclose only the particle. But it should be in a uniform mesh, particularly in the directions parallel to the injection plane. As like other sources, no monitor can penetrate or pass through the grey lines.
- Angle dependence of broadband source: Similar to plane wave, if a broadband source is injected, the real incidence angle changes with frequency, please refer the description for broadband injection. Therefore, angled TFSF is usually used for single wavelength illumination.
- Auto shutoff min: You may notice that the simulation ends at an auto shutoff level larger than the its default value (1e-5) but the results are still accurate. This is because in this case, the equivalent intensity inside the simulation region is calculated relative to the TFSF injection power which is small. Increasing the simulation time to reach the default auto shutoff min does not improve the results.
- Initial simulation vs. final results: It is always a good practice to have relatively low accuracy to speed up the simulation initially. To get the final results, we strongly recommend to use larger simulation region, and do convergence tests.
We are interested in the scattering profile from the PSL sphere. Open psl.fsp and run the script psl_analysis.lsf. It will run a series of simulations, calculating the scattering for various polarizations. Results for the other polarizations can be obtained by modifying psl_analysis.lsf and re-running the script. The following figures show the normalized scattering profile as a function of defect diameter and source settings. Please note that for comparison purpose, the far field intensity is normalized to its maximum.
|Diameter 60nm, 70 deg incidence, P polarization||Diameter 60nm, 70 deg incidence, S polarization||Diameter 60nm, 0 deg incidence, Circular polarization|
|Diameter 100nm, 70 deg incidence, P polarization||Diameter 100nm, 70 deg incidence, S polarization||Diameter 100nm, 0 deg incidence, Circular polarization|
|Diameter 140nm, 70 deg incidence, P polarization||Diameter 140nm, 70 deg incidence, S polarization||Diameter 140nm, 0 deg incidence, Circular polarization|
|Diameter 180nm, 70 deg incidence, P polarization||Diameter 180nm, 70 deg incidence, S polarization||Diameter 180nm, 0 deg incidence, Circular polarization|
|Diameter 220nm, 70 deg incidence, P polarization||Diameter 220nm, 70 deg incidence, S polarization||Diameter 220nm, 0 deg incidence, Circular polarization|
The scattering profile is clearly a function of particle diameter when using a 70o angle of incidence. This is not the case when using a normally incident beam. When using a 70o angle of incidence, it is possible to estimate the defect size by knowing the scattering profile. These results are very similar to Figure 3 from S. Stokowski.
Next, we calculated the scattered power in the far field within an angular cone between 25 and 72 degrees. The following plot is generated by changing the plot_scattering flag in the script file. These results are similar to Figure 5 from S. Stokowski.
P-polarized light at grazing angles of incidence is best for detecting small defects because the scattering profile is a strong function of defect size and has a relatively large scattering intensity.
S. Stokowski and Vaez-Iravani, „Wafer Inspection Technology Challenges for ULSI Manufacturing,” KLA-Tencor.