3.1 Link between the iso-focal point and the DOF 3.1.1 Definition of the iso-focal point
The best contrast of the image is located at best focus. Around this point, the image degrades as a function of defocus. One of the characteristics of the intensity profile is that each of them cross at a same point called iso-focal point (figure 3-1). It is an invariant point in CD as function of defocus. At the best focus (f=0µ m), the image contrast is the highest and this contrast decreases as function of defocus down to zero for the largest defocus. In this case, no image is formed.
3.1.2. Influence of the iso-focal position on the DOF
Practically, the depth of focus corresponds to the focus range where the image is considered to be acceptable. In terms of CD, it corresponds to a focus range where the CD is between +/- 10% of the target CD. Generally, for a given mask dimension, the target CD corresponds to the CD at the best focus. But the measured CD at each focus increases or decreases more or less rapidly depending on the position of the iso-focal point compared to the position of the target CD. This limits the depth of focus. DOF is maximum when the
CD is invariant as function of defocus. This is only possible when the target CD is close to the iso-focal point.
In practice, the depth of focus is determined from the Bossungs curves, variation of the CD versus dose and focus. At first order, it is possible to fit this CD variation as a function of defocus, intensity threshold and position of the iso-focal point with this simple expression [36-37]:
CD = c
1
(f-f0
)2
(t-t0
) (3.1) Wherec
1
: constant function of the processf et f
0
: the position of the focus and the position of the best focus.t et t
0
: the intensity threshold of the measured CD and the iso-focal intensity threshold.From the expression (3.1), the depth of focus can be approximated by:
0
DOF k
t t η
= +
− (3.2) Where
k et η : constants depending on the tolerance (+/-10% of the target CD) and the process
t et t
0
: the intensity threshold of the measured CD and the iso-focal intensity threshold.This expression illustrates the fact that the DOF is closely linked to the position of the iso-focal point compared to the position of the target CD. When these two points are at the same level the depth of focus is maximum. Figure 3-2 shows the evolution of the iso-focal intensity threshold and the evolution of the depth of focus of a 150nm line as a function of the pitch with a 193nm wavelength and for an annular illumination. For a 1:1 feature (equal line and space) the iso-focal intensity threshold is at the same level as the intensity threshold of the target CD. Then the iso-focal intensity threshold increases to a constant value for isolated lines. The evolution of the DOF is function of the evolution of the iso-focal intensity threshold. It is maximum for the pitch 1:1 then it decreases when the iso-focal intensity threshold moves further from the threshold of the 1:1 pitch. DOF reaches a constant level when the iso-focal intensity threshold reaches its constant value for the isolated lines.
Generally for a given pitch and illumination conditions, the intensity threshold of the target CD is not at the iso-focal level. The aerial image being function of the mask dimension, illumination conditions (NA, σ) and resist parameters (resist thickness, contrast…), the intensity threshold of the target CD is also function of these same parameters. It is then possible to correct the position of the intensity threshold of the target CD and to have it overlapping with the iso-focal point.
3.2 Evolution of the iso-focal point
3-2.1 Theoretical background
Fig 3-3a shows a schematic sketch of a lithographic system It consists of four parts: illumination optics, mask, projection optics and wafer. The light that falls onto the mask can be considered as a sum of plane waves with different angles of incidence.
These plane waves are diffracted by the mask into the several directions Here we will limit our study to the zero and first orders of diffraction (0, 1 and −1) because in the case of low k
1
imaging most of the contribution comes from these orders. We will consider only one-dimensional patterns.A feature on a mask is reproduced on the wafer by a light interference process from the diffracted orders. Considering the coherence properties of lithographic light sources, we consider that two beams interfere if they come from the same source point.
In the case where the feature pitch is small enough, as we can see in Fig 3-4, from the source S1, P
−1
interacts with P0
. The first-order contribution P1
is not captured by the pupil. We have here only an interaction between two beams (−1 and 0). We have the same with another source point S2 where only P0
and P1
interact together because the pupil does not capture P−1
.The iso-focal intensity threshold corresponds to the continuous component (background intensity) of the aerial image. The intensity expression of the aerial image can be obtained from the Fourier decomposition. We can then easily deduce the background intensity (zero order of the Fourier serie of the aerial image) and thus also the variation of the iso-focal point. In the following, we will study the theoretical variation of the iso-focal intensity threshold for three types of masks: binary mask, alternated mask and attenuated phase shift mask.
3.2.2 Binary and attenuated masks [38]
Image formation using a binary mask or an attenuated PSM is shown in Fig.
3-5(a).
For a pattern with a pitch of 2L and aperture width of 2w (Fig. 3-5(b)), the amplitude at the
mask can be expressed as T(x)=1 for∣ ∣< w, and T(x)=t for xx ∣ ∣> w. The value of t represents the transmittance of dark region at the mask. The case of t=0
corresponds to a binary mask, and t < 0 indicates an attenuated PSM. For an opening ratio β=w/L and fundamental frequency f
0
=1/2L, the one-dimensional Fourierspectrum F(f) of the binary mask or attenuated PSM is given as
F(f)={β+(1-β)t}δ(f-f’)+{(1-t)sin(βt)/π}{δ(f+f
0
-f’)+ δ(f-f0
-f’)} (3.3)and C
0
=β+(1-β)t; C1
=(1-t)sin(βt)/ πWhen the 0th, -1st, and +1st components are transmitted for the set of S
3
(Fig. 3-5(c)), the image at the wafer is formed by three-beam interference and its intensity isexpressed as
I
1
(X)={C0
+2C1
cos(2πX)}2
(3.4) Similarly, elimination of either the -1st or the +1st component for the case of S2
gives I2
(X)=C0 2
+2C1 2
+2C1
C0
cos(2πX) (3.5) and the elimination of both the -1st and +1st components givesI
3
(X)=C0 2
(3.6) Here, let A3
be the source element ratio for three-beam interference, A2
that fortwo-beam interference, and A
1
that for one beam transmission. The equation for defining the image width is then expressed with a certain threshold level Ith
as follows. pattern f0
/fC and the illumination conditions. For conventional circular illumination,
five cases (Fig. 3-6) can be considered to find the A3
, A2
, and A1
values. They are,From equation 3.7, we can determine the intensity threshold of the iso-focal point.
It corresponds to the continuous components and is written as : For a binary mask: The threshold of the iso-focal point is mainly a function of the mask aperture ratio (β)