9.2 Neutral transport
9.2.0.3 Analytical fitting function
An analytical function f = f(AR,Psw,Pbt)was fitted to the neutral flow simulation data:
f(AR,Psw,Pbt) =arctan
1
(AR+ARn)·(m·Pbtp+n·Pswq)
· 1
π/2, (9.3) where the parameters are listed in table 9.1.
The fitting function is also depicted in figures 9.7, 9.8, 9.9, 9.10.
The neutral flow to the surface of the profile in the etching profile simulation is then calculated from this fitting function, in the following way: First, an aspect ratio value is assigned to each node of the string representing the surface, where the mask thickness is considered. If the node is within the mask opening, then the local flow is calculated using the fitting function equ. (9.3) (fig. 9.11 (a)).
If the node is without the mask opening, i.e. its x-value is lower than the x-value of the left mask edge or larger than the x-value of the right mask opening, the fitting function is multiplied with a ”diffusion”-term exp(∆λx)
f(AR,Psw,Pbt) =exp(∆x
λ )·arctan
1
(AR+ARn)·(m·Pbtp+n·Pswq)
· 1
π/2, (9.4)
9.2. Neutral transport 103
0 5 10 15 20 25 30
0.0 0.2 0.4 0.6 0.8 1.0
normalized particle density [1]
aspect ratio AR [1]
0.01 0.025 0.05 0.10 0.50
Figure 9.12: Normalized neutral flow to the bottom for smooth (dashed line) and rippled sidewalls (solid line). The absorption probability was equal for sidewalls and bottom, and varied from 0.01 to 0.50. Only for low absorption probabilities (≤0.025) there is a significant difference between smooth and rippled sidewalls.
where∆x=|xNode−xmask edge|, andλis a ”diffusion length” (fig. 9.11 (b)).
It should be emphasized that this diffusion term is only a mathematical construct used to estimate the neutral flux and does not represent a physical reality, since surface diffu-sion of adsorbed fluorine is not probable [52].
The analytical fitting function was derived from simulation results for smooth side-walls. A comparison of neutral flow for smooth and rippled sidewalls (fig. 9.12) revealed that there is no significant difference between smooth and rippled sidewalls, as long as the absorption probabilities are above 0.025. Only for very low absorption probabilities, a difference between rippled and smooth sidewalls exists. The rippled profiles used for theses simulations are same as for the ion flow simulations inside rippled trenches, and are shown in fig. 9.3.
The size of the ripples used in these simulation was comparable to the trench open-ing (feature size). Therefore, the simulations for the rippled sidewall profiles represent simulations for an upper limit of the sidewall roughness.
From these results, it can be concluded, that in terms of neutral transport due to dif-fuse reflection, the roughness of the sidewalls is not an important parameter. Hence, the simulations of smooth sidewalls provides reliable data for the etching profile simulations.
The analytical fitting function can reproduce the numerical simulations of neutral transport with good agreement. Therefore, the analytical fitting function is used in the etching profile simulation to calculate the neutral flux.
Chapter 10
Aspect ratio dependent plasma polymer deposition
The term ”aspect ratio dependent plasma polymer deposition” is used here to summarize the dependence of polymer film thickness distribution or film composition on the aspect ratio of the structure on which is was deposited. Usually, studies on plasma polymeriza-tion have been carried out on unstructured, flat samples, hence aspect ratio effects could not be observed. The knowledge of the dependence of the deposited film thickness on the aspect ratio is an important factor for the simulation of gas chopping etching profiles.
Since no data on the aspect ratio dependent plasma polymer deposition was available, experiments to clarify this problem had to be carried out1.
10.1 Theory and experimental setup
The local polymer film thickness Tpoly(~x)at position~x can be calculated from the deposi-tion time∆t by
Tpoly(~x) =DR(~x)∆t, (10.1)
where DR(~x)is the local deposition rate given by equ. (8.76). Normalizing the local film thickness with respect to the film thickness at the top of the trench, yields
Tpoly,norm(~x) = DR(~x)∆t
DRAR=0∆t. (10.2)
Substituting the local deposition rates, yields
Tpoly,norm(~x) = VmonoPdepoΦmono∆t
VmonoPdepoΦmono,AR=0∆t, (10.3)
which simplifies to
1A large portion of this chapter has been submitted to the Journal of Vacuum Science and Technology.
(a) (b)
Figure 10.1: Polymer film deposited on long trenches with rectangular cross section (a).
Film deposited at the bottom of a trench (b).
Tpoly,norm(~x) = Φmono
Φmono,AR=0
, (10.4)
which is the definition of the normalized monomer flowΦmono,norm, where normaliza-tion is made with respect to aspect ratio AR=0:
Tpoly,norm(~x) =Φmono,norm. (10.5)
Equ. (10.5) is only true if in steady state, only deposition or desorption are allowed (refer to section 8.2.1.3: Polymer deposition model). In this case, the absorption proba-bility is equal to the deposition probaproba-bility.
If furthermore the thickness of the deposited polymer film is thin compared to the feature dimension, the polymer deposition does not considerable vary the profile geome-try. In other words, the geometry alteration due to the deposition of polymer film can be neglected.
In this case, the normalized polymer film thickness is exactly the normalized particle distribution as calculated by the Monte-Carlo flow simulator (refer to section 8.2.1.3:
Polymer deposition model).
Deposition experiments have been carried out on samples with trenches of nearly rectangular cross sections with different aspect ratios (fig. 10.1). By comparing the ex-perimental film thickness distributions with the Monte-Carlo simulations, the effective polymerization or absorption probability can be determined (fig 10.2).
The process conditions of the deposition processes have been chosen identical to the deposition recipe employed in the gas chopping etching processes, in order to gather relevant data. The only exception was the deposition time. Since the film thickness was measured using scanning electron microscopy, the film thickness should be at least about 100 nm. Therefore, the deposition was extended to 25-30 minutes.
In order to investigate the temperature dependence of the deposition probability and film thickness distribution, deposition experiments have been carried out at 3 different temperatures, 0◦C, 60◦C, and 100◦C.