Self consistent ion trajectories in electron shading damage

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Self consistent ion trajectories in electron
shading damage

• T.G. Madziwa, F.F. Chen D. Arnush
• UCLA Electrical Engineering
• ltptl
• November 2002

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Abstract
In electron-shading damage, the photoresist is
charged negatively, preventing electrons from
entering the trench, while ions are accelerated
toward the bottom of the trench. We have
numerically calculated the effect of these fields
on the ion trajectories. The ions are injected
at acoustic speed from a sheath edge far from the
substrate, and the electrons have a
Maxwell-Boltzmann distribution. The photoresist
and trench walls are assumed to be insulators,
and the trench bottom a conductor at various
potentials relative to the sheath edge. The
potentials on all surfaces are given initial
values, and a Poisson solver is used to compute
the electric field everywhere. The ions
trajectories in this field are then computed.
Setting the flux of ions to each dielectric
surface equal to the Maxwellian electron flux
yields a new value of the surface charge. The
E-fields and trajectories are then recomputed,
and the process iterated until the values
converge. It is found that the E-field is
concentrated near the entrance to the trench, the
only place where the charges matter. The ions
receive a kick there and then coast the rest of
the way. Thus the trajectories are very
sensitive to the exact shape of the photoresist
and will change as the etch progresses.
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Motivation
1. Verification of the physical picture of
electron shading damage mechanism. New
result ion orbits are ballistic inside the
trench and are determined by fields at the
entrance There are no significant numbers of
ions that hit the sidewalls.
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Methodology 1

• Define space with dielectric and trench, and
  metal collector.
• Define sheath edge.
• Assume ions are emitted from sheath edge with
  directed velocity cs.
• Assume electrons are Maxwellian everywhere.
• Assume j 0 on dielectric surfaces. This
  causes the plane surface to charge to 15.5 V,
  according to the plane floating potential
  formula.
• On the trench walls, potential will self-adjust
  so that the ion and electron fluxes are equal.

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Methodology 2

• Ion flux depends on the E-field in trench, which
  depends on wall potential.
• We use a 2D Poisson solver to get E-fields for
  given boundary potentials and then calculate the
  ion orbits in this E-field.
• We then iterate until the potential distribution
  and ion orbits converge to a stable solution.
• Initially, the walls are assumed to be a
  potential that gradually changes from the one on
  the dielectric to the one on the collector.
• To avoid a singularity when no ions are
  collected in a cell, we approximate that to be
  0.1 ion received. This makes an empty bin much
  more negative (-40V) than bin with one ion.

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Setup II

• The purple region is the region of interest and
  it lies in the plasma sheath
• Ions (Nions) are emitted from the plasma-sheath
  boundary with the Bohm velocity
• We count the ions that fall in the vertical and
  horizontal bins

diagram is upside down
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Current densities
The ion current densities per bin
The electron current density
where Vx is the potential in bin x and is
the random velocity of the electrons.
The condition for surface voltage calculation is
that
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Scale Invariance I

• Our simulations were done in mm dimensions.
• We want to show that by going down to more
  realistic dimensions, the results remain valid.
• We look at the invariance of the Poisson
  solution, scaling of the time of flight of ions
  and the equation of motion.
• We look at the case where all lengths are scaled
  by a uniform factor

W
HS
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Scale Invariance II
Start with Poisons equation
Define
Then
Let s be the scale length of the gradient ?, and
define , so that
Now we have
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Sidewall ions AR7, -40V bias
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Ion trajectories AR7, -26V
bias
A few ions can hit the sidewall here
Inside the trench, the ions go in straight paths
ions bend at the entrance to the trench
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Sidewall ions AR5, -26V
bias
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Sidewall ions AR5, -26V
bias
Ion orbits appear to be straight inside the trench
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Sidewall ions AR5, -26V
bias
Aspect ratio 5 to 1 Bias -26 Volts
Scale 80 to 1
With increased magnification, ion orbits also
cross
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Trajectories AR3, -22V
The ions bend a lot more with AR3 and so more
ions fall on the sidewalls.
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Sidewall ions -26V bias
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Why more sidewall ions at lower AR?
AR7 -26V No sidewall ions
Answer lies in distribution of equipotential
lines.
AR3 -22V Sidewall ions
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Collector Ions AR 3

• All aspect ratios follow the same trend.As
  V increases, so does the ions on collector
  (and sidewall ions decrease)

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Limit cycles isolated periodic solutions

• A typical simulation will come to one of these
  types of solutions (and some more types not shown
  here)

a limit cycle when the trajectory repeats itself
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Sidewall ions at AR5 -18V bias
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Dependence on overall structure

• Not much change in the number of ions deposited
  on sidewalls.
• With the rounded edges, there are significantly
  more ions on the collector than with the sharp
  corners.

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What about a minor change in structure like a
kink on surface?
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Kinks on surface with a bias of 18V

• A tiny kink on surface gives rise to ion
  distribution changes
• A small kink on the surface changes the sidewall
  profiles much more than a complete change in
  structure. (compare with the case of an arc
  versus straight edges)
• A kink close to the opening of the trench
  increases sidewall ions from a max of 0.29 to
  1.8

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AR5, -60V bias, 2 bins in metal
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Effect of neighboring trenches

• More sidewall ions when there are adjacent
  trenches
• Ions collected are symmetrical (just like the
  geometry)
• More ions are collected on the inner walls

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Double trench, AR5, -22V bias
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Conclusions

• insignificant number of ions hit the wall with
  single trenches.
• Ions drift freely inside trench once their orbits
  are set by the fields at the entrance.
• Fewer ions hit the wall at high potentials
  because of the large acceleration.
• Fewer ions hit the sidewall at large aspect ratio
  because the field lines are straighter.
• Fewer sidewall ions with large collectors because
  the field lines are straighter.
• There are still too few ions hitting the sides to
  change the etching profile.