Homework Set 12
-1-
6.781
Submicrometer and Nanometer Technology
Homework Set #12
12-1
An optical projection system with a numerical aperture of 0.62, a
demagnification factor of 4X, a field diameter of 20 mm, and an operating
wavelength of 193 nm (ArF excimer laser) is characterized by the following
dependence of contrast, K, on spatial frequency, 1/p, for 3 values of the aperture
filling, σ.
λλ == 193
nmnm
248.4
NA ==0.62
0.4
NA
σ=0
1.0
σ = 0.7
K
0.5
σ = 1.0
1/p
s1
0.5
1.0
1.0
1.0
2.0
13.0
.5
spatial frequency
2.0
4.0
(X10
-3
25.0
.5
36.0
.0
nm -1 )
(over)
3.5
7.0
Homework Set 12
-2-
A grating reticle with a transmission as sketched below is illuminated with σ = 0.7.
transmission
1.0
0.5
s2
5
2
10
4
( µm )
(µm)
15
6
(a)
What is the contrast, K, in the image of this reticle?
(b)
What fraction of the zero-order diffraction cone falls within the
entrance aperture?
(c)
What fraction of the 1st order diffraction cone falls within the
entrance aperture?
(d)
What fraction of the 2nd order diffraction cone falls within the
entrance aperture?
(e)
What is the ratio Imax/Imin in the image?
(f)
Plot the irradiance (intensity) distribution of the image.
(g)
Calculate the approximate depth of focus.
Assume that the substrate is coated with a special resist, type MIT-1, whose
development rate, R, is linearly proportional to the energy absorbed per unit
volume, E.
R(nm/sec) = a E(erg/c m3)
X
Homework Set 12
-3-
The energy absorbed per unit volume, E, is, of course, proportional to the product
of the intensity (ergs/cm2 ) and the exposure time, t.
E ∝ It
Assume that you expose a l-µm-thick film for a sufficiently long time that the
development rate at the point of maximum intensity is R = 100 nm/sec. Also
assume that the energy absorbed is uniform with depth in the resist (an unrealistic
assumption). Thus, development rate will vary as a function of x only.
R = R(x)
h)
What development time should be used to achieve equal widths for the
spaces and the lines.
i)
Sketch the resist profile.