3D Simulations of Electromagnetic Fields in Nanostructures using the Time-Harmonic Finite-Element Method

📝 Abstract

Rigorous computer simulations of propagating electromagnetic fields have become an important tool for optical metrology and optics design of nanostructured components. As has been shown in previous benchmarks some of the presently used methods suffer from low convergence rates and/or low accuracy of the results and exhibit very long computation times which makes application to extended 2D layout patterns impractical. We address 3D simulation tasks by using a finite-element solver which has been shown to be superior to competing methods by several orders of magnitude in accuracy and computational time for typical microlithography simulations. We report on the current status of the solver, incorporating higher order edge elements, adaptive refinement methods, and fast solution algorithms. Further, we investigate the performance of the solver in the 3D simulation project of light diffraction off an alternating phase-shift contact-hole mask.

💡 Analysis

Rigorous computer simulations of propagating electromagnetic fields have become an important tool for optical metrology and optics design of nanostructured components. As has been shown in previous benchmarks some of the presently used methods suffer from low convergence rates and/or low accuracy of the results and exhibit very long computation times which makes application to extended 2D layout patterns impractical. We address 3D simulation tasks by using a finite-element solver which has been shown to be superior to competing methods by several orders of magnitude in accuracy and computational time for typical microlithography simulations. We report on the current status of the solver, incorporating higher order edge elements, adaptive refinement methods, and fast solution algorithms. Further, we investigate the performance of the solver in the 3D simulation project of light diffraction off an alternating phase-shift contact-hole mask.

📄 Content

With the advances of micro-and nanotechnology simulation tools for rigorous solutions of Maxwell’s equations have become an important tool in research and development. Designing, e.g., a nanooptical component or a metrology tool is usually assisted by computer simulations, and simulations are used in most scientific works on nanooptical research to support theoretical and experimental findings. It exists a variety of different methods for solving Maxwell’s equations, and generally also a variety of different numerical implementations of each method. Prominent examples of different methods are the finite-element method (FEM), the finite difference time domain method (FDTD), the boundary element method (BEM), and rigorously coupled wave analysis (RCWA).

We address 3D simulation tasks occuring in microlithography by using the frequency-domain finite-element solver JCMsuite. This solver has been successfully applied to a wide range of 3D electromagnetic field computations including microlithography, [1][2][3] left-handed metamaterials in the optical regime, 4, 5 photonic crystals, 6 and nearfield-microscopy. 7 The solver has also been used for pattern reconstruction in EUV scatterometry, 8,9 and it has been benchmarked to other methods in typical DUV lithography 1,2 and other 6,10 projects.

In this paper we report on the current status of the finite-element solver JCMsuite, and we present simulations of light transition through periodic arrays of alternating phase-shift contact-holes which have to be printed on wafers with the same reticle with high overlapping process window. 3D effects cause imbalances of intensities of the 0 • -and 180 • -holes, and of different behavior in defocus if the etch depth is not adjusted. The evaluation of the impact of these effects has been chosen as a test case for JCMsuite.

The finite-element package JCMsuite allows to simulate a variety of electromagnetic problems. It incorporates a scattering solver (JCMharmony), a propagating mode solver (JCMmode) and a resonance solver (JCMresonance). The scattering, eigenmode and resonance problems can be formulated on 1D, 2D and 3D computational domains. Admissible geometries can consist of periodic or isolated patterns, or a mixture of both. Further, solvers for problems posed on cylindrically symmetric geometries are implemented.

. Schematics of the 3D test structure: cross-section in a x-y-plane. 0-degree (180-degree) rectangular phase-shift holes are depicted in grey (light grey). The elementary vectors, a1, a2, of the periodic pattern are indicated by arrows.

In this paper we concentrate on light-scattering off a 3D pattern (photomask) which is periodic in the xand y-directions and is enclosed by homogeneous substrate (at z sub ) and superstrate (at z sup ) which are infinite in the -, resp. +z-direction (see Figures 1 and2). Light propagation in the investigated system is governed by Maxwell’s equations where vanishing densities of free charges and currents are assumed. 11 The dielectric coefficient ε( x) and the permeability µ( x) of the considered photomasks are periodic and complex, ε

Here a is any elementary vector of the periodic lattice. For given primitive lattice vectors a 1 and a 2 an elementary cell Ω ⊂ R 3 can be defined as

As can be seen from the computational domain in Figure 3 elementary cells Ω of the same volume can be defined also by more complicated polygons which are adapted to the actual geometry. A time-harmonic ansatz with frequency ω and magnetic field H( x, t) = e -iωt H( x) leads to the following equations for H( x):

• The wave equation and the divergence condition for the magnetic field:

• Transparent boundary conditions at the boundaries to the substrate (at z sub ) and superstrate (at z sup ), ∂Ω, where H in is the incident magnetic field (plane wave in this case), and n is the normal vector on ∂Ω:

The DtN operator (Dirichlet-to-Neumann) is realized with an adaptive PML method. 12,13 This is a generalized formulation of Sommerfeld’s radiation condition.

• Periodic boundary conditions for the transverse boundaries, ∂Ω, governed by Bloch’s theorem 14 :

where the Bloch wavevector k ∈ R 3 is defined by the incoming plane wave H in .

Similar equations are found for the electric field E( x, t) = e -iωt E( x); these are treated accordingly. The finite-element method solves Eqs. ( 1) -( 4) in their weak form, i.e., in an integral representation.

The finite-element methods consists of the following steps:

• The computational domain is discretized with simple geometrical patches, JCMsuite uses linear (1D), triangular (2D) and tetrahedral or prismatoidal (3D) patches. The use of prismatoidal patches is well suited for layered geometries, as in photomask simulations.

• The function spaces in the integral representation of Maxwell’s equations are discretized using Nedelec’s edge elements, which are vectorial functions of polynomial order defined on the simple geometrical patches. 15 In the current imp

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