The demand for smaller, faster and lower power semiconductor devices continues to drive improvements in optical lithography. Currently very high numerical aperture (NA) exposure tools combined with resolution enhancement techniques (RET) are used to produce state of the art devices with critical dimensions (CD) less than 100 nm. For example, at the 45 nm node, some of the features to be imaged are less than a quarter of the wavelength of the 193 nm light source used, requiring the use of alternating phase shift masks (APSM). The associated pitches are sub-wavelength (~130 nm), which leads to severe proximity effects requiring optical proximity correction (OPC). These effects need to be understood using lithography simulation so that they may be taken into account during reticle design in order to achieve a predictable and reliable process. Lithography simulation can assist with improving device yields and reducing the number of reticle iterations, allowing a fabrication house to ramp products faster and save substantially in production costs.

As optical lithography techniques have continued to improve lithography simulation techniques do too. FDTD simulations uses the finite difference time domain technique to rigorously solve for the object fields at the mask. All diffraction, refraction, interference, absorption and polarization effects are calculated in the near field of the mask without approximation. FDTD also incorporates an automatic graded meshing, which greatly reduces memory requirements and time per simulation. By post-processing the FDTD simulation data, the aerial image at the wafer may be calculated.

### Magnification and coherence

In DUV lithography, the light source typically has a wavelength of 248 nm (KrF laser) or 193 nm (ArF excimer laser). At these wavelengths, the photomask pattern is normally etched into a chrome layer on a glass substrate. The numerical aperture of the imaging optics (NAo) can range from 0.5 to greater than 1 for immersion lithography techniques. A demagnification factor M = 4 or 5 is used, such that the aerial image is reduced by a factor of M relative to the pattern on the photomask.

The numerical aperture of the illumination (NAi) optics can be used to adjust the degree of coherence of the illuminating beam. The degree of partial coherence σ is given by σ = NAi/(M*NAo). Partial coherence values ranging from 0.3 to 0.7 are commonly used to optimize a particular lithographic process.

### Resolutions and depth of focus

Two figures of merit are often discussed for such a projection lithography system: the resolution and the depth of focus. The resolution, W, is the minimum printable feature size and is determined by W=k1*λ/NA. The depth of focus, DOF, is the range over which the image is adequately sharp, and is given by DOF=k2* λ/NA². The two constants k1 and k2 are characteristics of the given lithography process, both ranging from 0.4 to 1.0.

One can see that these figures of merit can lead to different strategies for printing progressively smaller and smaller features. However, it is clear that printing subwavelength features on a wafer requires wavelength (and possibly sub-wavelength) scale features on the photomask. And as mask features get smaller, approximate techniques for simulating the lithographic process will break down. Thus, accurate prediction of the aerial image on the wafer requires rigorous simulation of the interaction of the source light with the photomask. Such simulations are possible using FDTD.

### Rigorous solution of light propagating through the photomask

FDTD allows for the rigorous simulation of the interaction between illumination light and a mask, from which the aerial image on the wafer can be calculated. We will consider a few illustrative examples that show some of the issues of projection lithography such as the resolution limits, corner rounding and line shortening. We will then briefly discuss two reticle enhancement techniques (RET) that improve resolution and reduce corner rounding and line-shortening. These are alternating phase shift masks and optical proximity correction using assist features.

### Source and Illumination optics

To use FDTD to rigorously calculate the aerial image printed on the wafer, we assume Kohler illumination for the source and illumination optics. With Kohler illumination, points on an extended source are assumed to radiate incoherently. The optics are arranged so that any point on the extended source arrives at the photomask as a plane wave, each with slightly different angle of incidence. Thus, for a circular illumination pupil, one simulation must be performed for each orthogonal polarization state, and the results are added in intensity. Other more complex illumination schemes (e.g. annular) may also be treated rigorously by performing a larger set of simulations.

### See also

Simplified Microscopic Imaging

Photolithographic projection to photoresist