self consistent ion trajectories in electron shading damage
DESCRIPTION
Self consistent ion trajectories in electron shading damage. T.G. Madziwa, F.F. Chen & D. Arnush UCLA Electrical Engineering ltptl November 2002. Abstract. - PowerPoint PPT PresentationTRANSCRIPT
Self consistent ion trajectories in electron
shading damage
T.G. Madziwa, F.F. Chen & D. Arnush UCLA Electrical Engineering
ltptlNovember 2002
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.
Motivation
Hashimoto et al. JJAP 1994
sheath
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.
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.
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.
diagram is upside down
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
L
S
HS
W
x1
y1
x3 x4
collector
Dielectric material
Vacuum region x1
y2
y3
y4
Current densities
y
eJ Jy iy Wi
exp( ( ))e
J en v V Ve e r x sk TB e
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.
r
The condition for surface voltage calculation is that:
J Je i
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
HS
W
Scale Invariance II2
0
( )e i
eV n n
Start with Poison’s equation:
Then:
2 2 20/ , / , /e D e e s eeV KT KT n e c KT M Define:
2 2 20
21 ( / )D
ei e
e
KTn n
n e
Let s be the scale length of the gradient , and define , so that
/ sr r
2 2 2s
2
2
21 i
eD
ns
n
Now we have:
For, s << D we can solve Laplace’s equation:2 0
( ) atb b b r r r
subject to the boundary conditions:
Sidewall ions AR=7, -40V bias
0 5 10 4 0.001 0.0015 0.0020
0.002
0.004
0.006
0.008
0.01
0.012
0.01415par
0
Y final
2.5par0 X final
zoom
Ion trajectories: AR=7, -26V bias
0 1 10 4 2 10 4 3 10 4 4 10 4 5 10 4 6 10 40
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
0.011
0.012
0.013
0.0
ynb
1.2W0 xnb
ions bend at the entrance to the trench
A few ions can hit the sidewall here
Inside the trench, the ions go in straight paths
zoom
Sidewall ions AR=5, -26V bias
0.001 0.0020.005
0.006
0.007
0.008
0.009
0.01
0.011
trace 1
0.012
5par
Y final
2.5par0.25W X final
Sidewall ions AR=5, -26V bias
Ion orbits appear to be straight inside the trench
Sidewall ions AR=5, -26V bias
Scale: 80 to 1
With increased magnification, ion orbits also cross
Aspect ratio: 5 to 1Bias: -26 Volts
2.510 4 3.12510 4 3.7510 4 4.37510 4 5 10 40.006
0.0065
0.007
0.0075
0.008
0.0085
0.009HS
6par
ynb
1W0.5W xnb
The ions bend a lot more with AR=3 and so more ions fall on the sidewalls.
Trajectories AR=3, -22V
Sidewall ions -26V bias
0 5 10 4 0.001 0.0015 0.0020
0.002
0.004
0.006
0.008
0.01
0.012
0.01415par
0
Y final
2.5par0 X final0 5 10 4 0.001 0.0015 0.002
0
0.002
0.004
0.006
0.008
0.01
12par
0
Y final
2.5par0 X final
0 5 10 4 0.001 0.0015 0.0020.005
0.006
0.007
0.008
0.009
10par
5par
Y final
2.5par4.292 10
4
X final
AR=7:1 AR=3:1AR=5:1
Why more sidewall ions at lower AR?
zoom
zoom
Answer lies in distribution of equipotential lines.
AR=7 -26V
No sidewall ions
AR=3 -22V
Sidewall ions
Collector Ions: AR = 3 All aspect ratios follow the same trend.As |V| increases, so
does the ions on collector… (and sidewall ions decrease)
760
800
840
880
-70 -60 -50 -40 -30 -20 -10 0
Bias voltage/V
colle
ctor
ions
Limit cycles: isolated periodic solutions
A typical simulation will come to one of these types of solutions (and some more types not shown here):
Monotonic increase/decrease
Damping oscillations 3 limit cycle
2 limit cycle
Chaos
a limit cycle when the trajectory repeats itself
0 5 10 4 0.001 0.0015 0.002
0
0.002
0.004
0.006
0.008
0.01
12par
0
Y final
2.5par0 X final0 5 10 4 0.001 0.0015 0.002
0
0.002
0.004
0.006
0.008
0.01
12par
0
Y final
2.5par0 X final0 5 10 4 0.001 0.0015 0.002
0
0.002
0.004
0.006
0.008
0.01
12par
0
Y final
2.5par0 X final
0
2
4
6
8
10
12
14
16
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Bin #
Ions c
ollec
ted
Sidewall ions at AR=5 -18V bias:
0
2
4
6
8
10
12
14
16
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Bin #Ion
s coll
ected
0
2
4
6
8
10
12
14
16
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Bin #
Ions c
ollec
ted3 limit cycle: 62 ions, 56 ions, 34 ions
0
10
20
30
40
50
60
70
80
0 5 10 15 20 25
Iteration #
Side
wal
l Ion
s
Dependence on overall structure
Case 1: straight edges Case 2: arc at entrance
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.
What about a minor change in structure like a kink on surface?
KnL02, case c: kink close to trench opening
Case a: no kink
KnL01, case b: kink not too close to trench opening
Kinks on surface with a bias of –18V
13.5%15.3%17.5%
10%9.6%11%
1.8%0.15%0.29%
Case
13.5%15.3%17.5%arc
10%9.6%11%Collector
1.8%0.15%0.29%Sidewalls
cbaCase
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%
AR=5, -60V bias, 2 bins in metal
Sheath region
Inside the trench
Inside the metal trench,
0 5 10 5 1 10 4 1.510 4 2 10 4 2.510 4 3 10 4 3.510 4 4 10 4 4.510 40
9.6153810 4
0.00192
0.00288
0.00385
0.00481
0.00577
0.00673
0.00769
0.00865
0.00962
0.01058
0.01154
0.0125d
0.000
ynb
1W0 xnb
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
Double trench, AR=5, -22V bias
0.006 0.004 0.002 0 0.002 0.004 0.0060.004
0.0051
0.0063
0.0074
0.0086
0.0097
0.0109
0.01212par
4par
Yfinal
0.0060.006 Xfinal
Ion shadowing effect
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.