6. Differential Method

As outlined in the previous chapter a rigorous calculation
of the EM field becomes necessary to cope with the phenomena that determine
the performance of today's semiconductor photolithography.
Difficulties arise from the EM scattering caused by the inhomogeneous resist and
by the increasingly nonplanar topography as well as the oblique wave propagation
due to the usage of high numerical apertures.
We chose the *differential method* among the various rigorous techniques
described in Section 5.3.
The major reasons for this choice are listed below by summarizing
some important advantages of the differential method in comparison to the other
three available rigorous techniques (cf. Table 5.1):

**Spatial domain method.**The spatial domain method is based on the finite-element method. Hence the simulation domain has to be adequately meshed which is an extremely difficult task especially in three space dimensions. Complicating the situation, the mesh has to be adapted to the solution of the Maxwell equations because otherwise the curls of the EM field cannot be properly discretized and the numerical algorithm is unstable. A second problem arises from the radiation*boundary conditions (BCs)*as they establish tight relations between all boundary points. Thus the bandwidth of the system matrix, and to a large degree also the CPU time, is determined by the number of the boundary elements. The differential method neither suffers from the*meshing*nor from the*boundary problem*. Both are avoided by the usage of Fourier series since only an equally spaced ortho-product-tensor-grid is required in the lateral directions and the radiation BCs are optimally matched by the Fourier expansions.**Time-domain finite-difference method.**The time-domain finite-difference method uses an equally spaced ortho-tensor-product-grid and thus avoids the meshing problem. Problems arise from the extreme computational requirements that prohibit a workstation-based three-dimensional simulation. Because of its high numerical costs the algorithm is primarily restricted to*massively parallel computer architectures*. Here the full power of parallel computers can be exploited as the equations are predisposed for a parallel implementation. The average computation time is about 20 minutes on an appropriate supercomputer, the storage requirements are, however, extremely high around 16 GB of memory. Thus the algorithm is*memory-limited*rather than computation time-limited. The differential method realizes a compromise between storage and CPU costs--typical values are 500 MB of memory and 6 hours run-time on a modern engineering workstation--and thus enables a workstation-based simulation also for three-dimensional applications. A second problem stems from the fact that the stead-state solution is calculated iteratively since the number of required iterations is not known in advance. Due to the*accumulation of numerical errors*the solution potentially starts to diverge after the steady-state is reached unless the real part of the refractive index is smaller than its imaginary part [6,7]. This is a severe restriction since a lot of commonly used materials in IC fabrication are highly dispersive, i.e., the imaginary part of the refractive index is larger than the real part [6,7]. An automatic termination of the iteration cannot be implemented since it is difficult to decide whether the solution is still converging towards the steady-state or already diverging from it.**Waveguide method.**Beside the differential method the waveguide method is the second frequency-domain method. Hence waveguide and differential method are closely related. The differences are of sophisticated nature and will therefore be discussed in greater detail in Section 6.5.3. Anticipating this thorough discussion, two important advantages of the differential method shall be stated beforehand: The waveguide method divides the simulation domain into thin layers. Within one layer an eigenvalue problem is solved. By matching recursively adjacent layers a linear algebraic system is established. If the layers are too absorptive*stability problems*occur. The differential method transforms the Maxwell equations into a two-point*boundary value problem (BVP)*of ordinary*differential equations (ODEs)*. The*zero-order discretization*, i.e., the system matrix and thus the medium is assumed to be constant within a discrete interval, refers to the waveguide method since the solution of an ODE system with constant coefficients can be reduced to an eigenvalue problem. Hence the discretization order of the differential method is implicitly higher than that of the waveguide method. Furthermore it can be chosen arbitrarily, and numerical instabilities can thus simply be avoided by increasing it. Computationally efficient methods exist to solve two-point BVPs that assure convergence of the algorithm in even strongly absorptive media. This is another advantage of the differential method over the waveguide method.

As for the outline of this chapter: In Section 6.1 we first pose the problem formulation and then briefly sketch the operation principle of the differential method. In Section 6.2 the lateral discretization of the Maxwell equations is discussed in great detail. Especially the compact matrix notation introduced significantly simplifies all further investigations. Next, in Section 6.3 the BCs are formulated. As they are posed at two different spatial locations, i.e., at the two boundary points of the simulation interval, a two-point BVP of ODEs has to be solved. Important solution algorithms for BVPs are summarized in Section 6.4, whereby the ``shooting method'' is best suited for our application since it can be implemented in a very memory-efficient way. Its theoretical background and implementation is thus discussed at greater length. The final Section 6.5 is devoted to the performance and limitation of the differential method as well as to a comparison of differential and waveguide methods.

- 6.1 Fundamentals
- 6.2 Lateral Discretization of the Maxwell Equations
- 6.3 Formulation of the Boundary Conditions
- 6.4 Vertical Discretization of the Maxwell Equations
- 6.5 Discussion

1998-04-17