# STUDY ON THE EFFECT OF HEAT LOSS IN POST EXPOSURE BAKE PROCESS by ghkgkyyt

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```									     STUDY ON THE EFFECT OF HEAT LOSS IN POST
EXPOSURE BAKE PROCESS

Sarah KIM 1 , Do Wan KIM 1

1) Department of Applied Mathematics, Hanyang University, Ansan 426-791, KOREA

Corresponding Author : Sarah KIM, sarahkim@hanyang.ac.kr

ABSTRACT
In post exposure bake (PEB) process of semiconductor devices production, a shape of critical
dimension (CD) could be changed into an inappropriate form by superﬂuous heating tempera-
ture and time of a hot-plate. This is because redundant diffusion of acid can occur from them.
In the past, there was little concern about such deformation of CD because the line width of CD
that we wanted was relatively thick. However, now we need to control with prudence heating
temperature and time of a hot-plate for a thinner line width (ex. sub-32nm) of CD that is used
lately. These reasons let us have a desire to start this work.
Here we present a mathematical heat conduction model during PEB process for getting suitable
heating temperature and time and compute it by using numerical method.
Computation of the heat conduction is a very delicate matter, especially because of the mul-
tiscale conductivity and thickness of stacked sublayers of the wafer. However, through our
method we could markedly reduce the unknowns. Thus, our method brings more cost efﬁcient
beneﬁts because we can compute heat transfer only with the averaging interface temperatures
which are the reduces unknowns. Also, a system of Volterra-type integral equations for the same
number of unknowns could be drived. In other words, we used integral scheme that has several
kernels. It is more stable and robust than a difference method. These points could be merits of
our work.
There are previous works, [1] and [2], on this topic. The previous papers stated that the PEB
wafer stack was insulated from the surrounding air temperature. That is, there was no consider-
However, in the PEB process, when heat transfer occurs from one layer to another, there must
be temperature leakage into the surrounding air. We propose a new numerical algorithm in
which effect of heat loss is added. Finally, we obtain the accurate and efﬁcient results including
realistic wafer tests. Reliable results are shown for heat conduction to photoresist on top of
wafer during PEB process.

PROBLEM SETTING

Let us deﬁne u(x, t) as an averaging temperature over each cross section i.e,
1
u(x, t) =           T (x, y, z, t)dydz                           (1)
|Sx |
where T (x, t), (x = (x, y, z)), is a three-dimensional temperature function in the stack. More-
over, Sx is the cross section of the stack at x position on x-axis and the area |Sx | has a constant
value. Besides, assume that the stack has thin n-materials which each material has a thermal
conductivity κi , a thickness di and a negative constant ci is the ratio of heat loss by the differ-
ence of temperature for i = 1, · · · , n.

PROBLEM MODELING

With an initial condition, u(x, 0) = 0, and two boundary conditions, u(xi−1 , t) = uL (t) and
u(xi , t) = uR (t), the averaging temperature u(x, t) in our problem could be governed by the
following heat equation.
ut = κi uxx + ci (u − u∗ )                             (2)
where the constant u∗ is the temperature of surrounding air. To make more easy computation,
we let the the two boundaries of each material having reference region [0, 1] and not x but x
denotes the new region. And then, after solving the PDE, we applied the result into the below
interface (or transmission) condition.
κi ∂ui        κi+1 ∂ui+1
|x=1 =            |x=0                                 (3)
di ∂x         di+1 ∂x

Figure 1. Wafer stack and one-dimensional model

REFERENCES

1.   KIM, D.W., LEE, J.-E. and OH, H.-K., “Heat Conduction to Photoresist on Top of Wafer
during Post Exposure Bake Process: I. Numerical Approach”, Japanese Journal of Applied
Physics, Vol. 47, No. 11, pp 8338-8348, 2008.
2.   OH, H.-K., KIM, D.W. and LEE, J.-E., “Heat Conduction to Photoresist on Top of Wafer
during Post Exposure Bake Process: II. Application”, Japanese Journal of Applied Physics,
Vol. 47, No. 11, pp.8349-8353, 2008.

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