Design specific variation in pattern transfer by via/contact etch ...

Mass-balance
Differential equation for across-die distribution of
the radicals
Due to complex character of dependency of the introduced
parameters on the plasma recipe as well as the difference in
generation rates of different radicals it is reasonable to introduce a
dimensionless form for radical concentration:
This transformation leads to the dimensionless form of the mass
balance equation:
2
γ
⎛ γλ ⎞ cn0
n0
= andθ
0 = n0
/ =
Here:
k
⎜
V
D ⎟
⎝ ⎠ 3γλ
Solution of these mass balance equations generates across-die
distributions of concentrations of all radicals participating in etch
reactions. Gas-kinetic properties of radicals (D, λ, c) are calculated
based on Chapman-Enskog kinetic theory.
Approximate solution of the mass-balance equation
clearly demonstrates that an across-die distribution
of radicals is characterized by a complex PD
dependency, more complicated than the
“traditional” reverse PD: 1/PD.
n
θ =
2 ⎛ γλ ⎞
⎜
⎟
⎝ D ⎠
r ⎡ 1 3 λ r ⎤ r
1−θ
( r)
⎢ + F ⎥ r
⎣θ0
4 L ⎦
2 ( r ) + λ ∆θ
( ) = 0
Approximate solution
2 ⎡ 1 3 λ r ⎤
1−
λ ∆⎢
+ F(
r ) ⎥⎦
r
≈
⎣θ0
4 L
θ ( r )
⎣θ0
4 L
θ ( r ) ≈
⎦
⎡ 1 3 λ r ⎤
⎢ + F(
r ) ⎥⎦
⎣θ0
4 L
F = ∫
r r r r
PD – GDSII
2r
d r′
r 2 ⎛ r′
⎞
⎜1+
⎟ 2
⎝ h ⎠
( r,
φ)
ˆ χ(
r′
+ r ) ρ(
r′
+ r ) 2
Sticking
coefficients
Visibility factor
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