Plasmonic devices such as nano antennas are a large interest for researchers due to possible large field enhancement in the nearfield of the device. This strong field enhancement might be used for single molecule detection and spectroscopy (SERS). In the paper "Comparison of electromagnetic field solvers for the 3D analysis of plasmonic nano antennas" by Johannes Hoffmann, Christian Hafner, Patrick Leidenberger, Jan Hesselbarth, Sven Burger, Proc. SPIE Vol. 7390, pp. 73900J-73900J-11, a simple plasmonic nano antenna consisting of two 80 nm diameter Gold spheres with a 1 nm gap in between the sphere is analyzed. The main quantities of interest are the field distribution and field enhancement inside the small gap. This rather simple geometry might still offer some challenges for electromagnetic field solvers because of the large aspect ratio of the small gap compared to the wavelengths of interest and the strong dispersive behavior of gold at optical frequencies. The paper will demonstrate how both general purpose solvers of CST MICROWAVE STUDIO® can be used to simulate such a device. The simulation results agree closely with the reference solution....
Simulation with the frequency-domain solver
Frequency-domain solvers are very well suited for the simulation of dispersive devices since the simulation is performed in terms of a frequency sweep. For each simulated frequency point, the correct materials properties at this frequency can be used. The main challenge for the FD Solver is the geometry discretization. Since the FD Solver uses a tetrahedral mesh, any round geometry is segmented and will create a local geometry error. Virtually all frequency-domain electromagnetic field solvers rely on an automatic mesh adaptation scheme to refine the simulation mesh for an accurate field solution. Usually, the applied mesh adaption is based on the initial discretization mesh and might not refine the geometry. Any “built in” geometry error is not corrected. Starting with CST MWS 2009, a true geometry adaptation has been added to the tetrahedral frequency domain solver to take care of this problem. Any mesh refinement in critical regions will not only create a better field solution but will refine the geometry as well by literally snapping new mesh points onto the original surface and thus reduce the local geometry error.
Figure 1 shows the very coarse starting mesh. The segmentation of the sphere is clearly visible. Based on this mesh, the frequency domain solver of CST MWS 2010 is used to solve this device using the “conventional” and “snapped” refinement. The nano antenna is excited by a plane wave propagating in the Z direction and polarized in x direction.
Figure 2 shows the resulting mesh after adaptation. Zooming in around the gap clearly reveals the difference and demonstrates the advantages of the true geometry adaptation.
Figure 3 shows the calculated local field strength in between the spheres as a function of the mesh adaptation passes. The calculation is performed close to the resonance frequency of the setup at 473 THz. The results for the “snapped” mesh converges quickly to the results for the semi analytical solution of about 730 V/m as published in the paper. The solution for the unsnapped mesh converges as well but to the wrong value. The difference of several 100 V/m is entirely due to the coarse geometry representation of the initial mesh.
Please note: even with an conventionally refined mesh (CST MWS 2008 and before) accurate results can be achieved, if the initial mesh/geometry is refined in critical geometry regions before the mesh adaptation. A local refinement in the gap region can be achieved within minutes using global and local mesh settings.
Figure 4 now shows the field enhancement in between the spheres as a function of the wavelength compared to the highly accurate semi analytical reference solution. The broadband solution is extracted by using a local field probe and running a frequency sweep with 5 THz step width. The results are virtually identical.
Simulation with the time domain solver
>In addition to the geometry discretization with the high aspect ratio, the TD solver faces the additional challenge to model this highly dispersive material in a single simulation run. With CST MWS 2010 higher order material dispersion has been added and the user has now the possibility to fit the dispersion parameters and order of the fitting scheme automatically to a user defined table. Figure 5 shows a plot of the automatically fitted dispersion curve and the user supplied material data. The curves show very good agreement and the maximum error is less the 1%.
Figure 6 depicts again the local field enhancement in between the spheres as calculated by the TD solver using a manual pre-refined calculation mesh. The mesh inside the gap is defined with a step width of 0.08 nm whereas for the rest spheres a mesh step of 0.3 nm is used. Again, a very close agreement with the reference solution has been reached. The resonance peak shows a shift of less than 1 % and can be well explained by the slightly different material definition.
Please note, because of the high aspect ratio and the small gap width, the simulation model requires a very small time step, which makes the simulation time considerably longer then the FD solver time. From this point of view, the FD solver shows the better performance. Nevertheless, the TD results are still valuable as a result cross check. Furthermore, the simulation time for the TD solver can be reduced by using GPU and cluster computing options.
Comments on published results
"Comparison of electromagnetic field solvers for the 3D analysis of plasmonic nano antennas by "Johannes Hoffmann, Christian Hafner, Patrick Leidenberger, Jan Hesselbarth, Sven Burger", Laboratory for Electromagnetic Fields and Microwave Electronics, ETH Zurich, Gloriastrasse 35, 8092 Zurich, Switzerland Proc. SPIE Vol. 7390, pp. 73900J-73900J-11 (2009).
The authors have performed simulations of the model shown above using CST MWS 2008. They concluded in the cited paper, that CST MWS 2008 generates highly inaccurate results which are partially several orders of magnitude away from the reference solution. It was however revealed that wrong material settings were used by the authors, that eventually lead to erroneous results. If the correct settings are applied, CST MWS 2008 generates results close to the reference solution.
These results have been published without consulting CST to check if there might be an error in the usage of the program and if there is any possibility to improve the results.
For detailed information please contact CST