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MICROLITHOGRAPHY

9.16 WAFER PROCESSING

called the extinction coefficient. This latter name comes from the relationship between the imaginary
part of the refractive index and the absorption coefficient of the material

a = 4 pk (9.10)
l

where a is the absorption coefficient and l is the vacuum wavelength.
Consider now the propagation of light through this simple film stack. We will begin with simple
illumination of the stack by a monochromatic plane wave normally incident on the resist. When this
plane wave strikes the resist surface, some of the light will be transmitted and some will be reflect-
ed. The amount of each is determined by the transmission and reflection coefficients. Defined as the
ratio of the transmitted to incident electric field, the transmission coefficient t for a normally inci-
ij
dent plane wave transmitting from layer i to j is given by
r = 2n i (9.11)
ij
n + n
i j
In general, the transmission coefficient will be complex, indicating that when light is transmitted
from one material to another, both the magnitude and the phase of the electric field will change.
Similarly, the light reflected off layer j back into layer i is given by the reflection coefficient r
ij
n − n
r = i j (9.12)
ij
n i + n j
If an electric field E is incident on the photoresist, the transmitted electric field will be given by
I
t E .
12 I
The transmitted plane wave will now travel down through the photoresist. As it travels, the wave
will change phase sinusoidally with distance and undergo absorption. Eventually, the wave will trav-
el through the resist thickness D and strike the substrate, where it will be partially reflected.
So far, our incident wave (E ) has been transmitted in the photoresist (E ) and then reflected off
I 0
the substrate (E ), as pictured in Fig. 9.10b. The total electric field in the photoresist (so far) will be
1
the sum of E and E . Before evaluating mathematically what this sum will be, consider the physi-
0 1
cal result. When two waves are added together, we say that the waves interfere with each other. If
the waves are traveling in opposite directions, the result is a classical standing wave, a wave whose
phase is fixed in space (as opposed to a traveling wave whose phase changes). Of course, we would
expect the mathematics to confirm this result.
Allowing the light to reflect off the top surface of the resist, it again propagates down. If all the
reflections and propagations inside the resist are accounted for, the resulting intensity in the thin pho-
toresist film becomes
−
−
≈
l
Iz) ( e −a z | + r | 2 e −a (2 D z) ) − 2 | r e | −a D cos( p n D z)/ ) (9.13)
(
4
(
23 23 2
This equation is graphed in Fig. 9.11 for a photoresist with typical properties on a silicon sub-
strate. By comparing the equation to the graph, many important aspects of the standing wave effect
become apparent. The most striking feature of the standing wave plot is its sinusoidal variation. The
cosine term in Eq. (9.13) shows that the period of the standing wave is given by
Period = λ/2n
2
The amplitude of the standing waves is given by the multiplier of the cosine in Eq. (9.13). It is
quite apparent that there are two ways to reduce the amplitude of the standing wave intensity. The
first is to reduce the reflectivity of the substrate (reduce r ). As such, the use of an antireflection
23
coating (ARC) is one of the most common methods of reducing standing waves. The second method
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