Exposure tool control for advanced semiconductor lithography

Pupilgram modulation model

Though the model was originally proposed for pupilgram error analysis, it can be used for freeform pupilgram adjustment. In the model, pupilgram modulation can be expressed by linear combinations of Zernike intensity modulation functions and Zernike distortion modulation functions. These functions are orthogonal and can be expressed by a combination of Zernike polynomials. Some of these polynomials are graphically described in Figure 16. These polynomials are suitable for a Zernike linear combination analysis method to predict the OPE response to changes in the pupilgram. By using this model, we can optimize the intensity distribution of the pupil to minimize the OPE error relatively gently comparing with grid-based optimizations and therefore retaining the pupilgram’s original SMO solution characteristics.

We define the modulated pupilgram I_modulated(x,y) using the equation below.

(2)Imodulated(x, y)=T(x, y)[Ioriginal(x+Dx(x, y), y+Dy(x, y))⊗PSF]+C (2)

where (x, y) is the pupil coordinate, I_original(x+D_x(x, y), y+D_y(x, y)) is the original pupilgram intensity distribution, T(x, y) is the intensity modulation term, D_x(x, y) is the distortion function in x, D_y(x, y) is the distortion function in y, PSF is a Gaussian point-spread function in pupil that generates blur and C is a constant to express a background intensity offset.

In our definition of distortion, the pure shape modulations are expressed without any intensity modulations. It only describes a ‘remapping’ of the coordinate positions in the pupil. This equation also defines the hierarchy of the modulation components. The equation is also used for pupilgram error component analysis, which will be discussed in the last section of this paper.

The total intensity filtering effect can be expressed by a sum of component filtering effects T_i(x, y) as

(3)T(x, y)=T1(x, y)×T2(x, y)×⋯=∏iTi(x, y), (3)

where i is a positive integer.

For intensity modulation, we use Fringe Zernike functions, which are very familiar in the microlithography industry as a means to express wavefront aberrations. However, direct use of Fringe Zernike functions would not be so useful because Fringe Zernike functions take on negative values, which are not realistic for expressing the intensity filtering distribution. In order to make the intensity filtering description physically meaningful, we would need to use a combination of Fringe Zernike functions as basis functions, but such combinations are no longer mutually orthogonal.

Therefore, we propose to use Fringe Zernike functions applied in exponential as

(4)Tm(x, y)≡exp[cmZm(x, y)], (4)

where m is a positive integer.

In this case, negative values of the Zernike functions are physically meaningful and, when zero, give no filtering effect over the entire pupil.

Now the intensity filtering (modulation) effect can be expressed as a linear combination of Fringe Zernike functions as basis functions, as shown below.

(5)T(x, y)=exp[c1Z1(x, y)]×exp[c2Z2(x, y)]×⋯=exp[∑mcmZm(x, y)] (5)

One example of the effect of the Zernike intensity modulation is shown in Figure 3.

As for distortion modulation, we cannot directly use the Fringe Zernike functions since they are not sufficient to describe the two-dimensional distortion functions. We therefore propose to use the orthogonal distortion functions shown in Table 3. These are expressed by simple linear combinations of Fringe Zernike functions. The functions form an orthogonal series. We call these Zernike distortion modulation functions.

(6)(Dx(x,y),Dy(x,y))≡∑dk(Dkx(x,y),Dky(x,y)) (6)

Figure 16:

Graphical example of Zernike distortion modulation functions and Zernike intensity modulation functions.

(A) Zernike (pupil) distortion modulation functions.

(B) Zernike (pupil) intensity modulation functions.