5.3.1 Lithography Modeling

The crucial point throughout lithography simulation is the solution of the Maxwell equations. Several methods for the numerical solution of the Maxwell equations have been proposed, ranging from simple vertical scalar models to rigorous approaches based on FEM discretizations of the Maxwell equations. We have used a three-dimensional extension of the differential method [32], which will be recapitulated in the following.

As can be seen from Fig. 5.5, the absorbed light intensity
inside the optically nonlinear resist has to be determined.
The exposure state of the photo-resist is described by the concentration of the
photo-active compound
. Dill's
``ABC''-model [9] is used for the correlation between the exposure
intensity
and the bleaching of the resist, which determines
the change in the resist's refractive index

where is the wavelength used for the exposure. This relation requires the knowledge about the intensity distribution which has to be calculated from the solution of the Maxwell equations. Assuming a time-harmonic field distribution within a time-step , the EM field obeys the Maxwell equations in the form

Due to the spatially periodic nature of the incident light and the assumption
of a laterally periodic simulation domain the EM field inside the simulation
domain can be expressed by a Fourier expansion

The permittivity itself is related to the refractive index by Maxwell's formula

Additionally, the inhomogeneous permittivity
and its reciprocal
can be expanded in Fourier series

Insertion of (5.6) and (5.8) into (5.5) transforms the

For the numerical solution of the equation system, the application of appropriate boundary conditions is necessary. Above the simulation domain we have to consider incident and reflected waves, whereas below only outgoing waves occur. The incident light is known from the aerial image simulation, whereas the unknown reflected and outgoing fields are eliminated by applying radiation boundary conditions. The resulting boundary value problem is solved by a numerically efficient implementation of the shooting method [31] which supplies the EM field coefficients.

The EM field coefficients are transformed back to the spatial
domain and the solution of the EM field intensity

Because the bleaching rate
is almost negligible when
compared with the frequency of the EM field, the refractive index
varies only slowly with respect to the field propagation and
thus a quasi-static approximation

** Prev:** 5.3 Coupling with Lithography
**Up:** 5.3 Coupling with Lithography
** Next:** 5.3.2 Contact Hole Printing

W. Pyka: Feature Scale Modeling for Etching and Deposition Processes in Semiconductor Manufacturing