Simulation of Film Growth Contour in a Narrow Deep Trench and Film Crystallinity in LPCVD Process
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Pd=
Ed)
Id exp(
(1)
k T, Ea
Pr
kT
v aexp(
(2)
Em
Pm
kT
vm exp(
(3)
The sticking coefficient, S, is then given by Eq.(4) as a function of the relative rates of the desorption and the immobilization. SoX So X
Pr,S Pd+Pr
So___________
1+ -d
exp(
Va
(4)
( EEd k Tý,
The rate that the adsorbed reactants are consumed by either the desorption or the surface immobilization per hopping is given by dM Pd +P, dN Pd+Pr+Pm
Here, the amount of the adsorbed reactants, M, is expressed by a fraction of the initial amount. N is th number of hopping. Integration of Eq. (5) with the initial condition, M - I at N = 0, gives Eq. (6). Pd+Pr
M =
exp(
-
Pd+Pr+Pm x N
)
(6)
N is related to the migration distance, 1, as follows. I = Ax
N
(7)
X is a distance per hopping, assumed to be the same as the lattice constant of the substrate crystal. For example, AL is 0.543 nm for silicon. Integration of M between -1 and 1 gives an accumulated amount of the adsorbed reactants that are consumed while migrating from their initial position to distance 1. From Eqs. (6) and (7), the maximum number of hopping, Nmax, and the maximum migration distance, lmax, defined as those required for consuming 99 % of the adsorbed reactants, are given by Eq. (8). Nmax
-
In ( 0.01 )x
Pd +Pr +Pro "Na Pd+PrP
8
/a
X Nmax
I
(8)
The simulation based on the above analysis is done as follows. The domain and the boundaries of the system considered for the simulation are shown in Figure 1. SOURCE PLANE
REFLECTING BOUNDARY
Figure 1. Domain and boundaries of the system for simulation.
SUBSTRATE
126
A reactant is generated at a random position on the top boundary, which travels in an arbitrary direction determined by a random number. The trajectory of the reactant is traced until it is either adsorbed on or scattered elastically from the substrate surface. The scattered reactant is assumed to be emitted from the surface by the specular pattern, with the re emission angle equal to the incidence one[3]. Unless adsorbed on the substrate at another position, the scattered reactant eventually disappears from the domain through the top or the side boundary. The migration distance of the adsorbed reactant is also determined by a random number. The probability that the reactant is either immobilized on or desorbed from the surface is given by Eq. (6). The adsorbed reactant is assumed to be desorbed (inelastic scattering) from the surface by the cosine pattemr[3], with the re-emission angle independent of the incidence angle. The sticking coefficient, S, and the maximum migration distance, lmax, are calculated using Eqs. (4) and (8). For the Monte Carlo computation, the domain is divided into a matrix so that the length of 70 elements in series corresponds to 1 gm. Each element is given an identification number, e. g., 1 for the substrate, 2 for the gas phase, and 3 for the film.
When the reactant is immobilized on the surface, the mesh element corresponding to the position is converted from 2 to 3. The film profile is obtained by grouping the
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