Temperature profiles for silicon ribbon are presented. Ribbon dislocations and elastic stress are discussed. Ribbon quality is important for the ribbon's function as a solar cell

Dillon, O. W., Jr.

English

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https://ntrs.nasa.gov/search.jsp?R=19850011215 2020-03-20T20:16:13+00:00Z
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UNIVERSITY OF KENTUCKY
February 1, 1985
Technical Report No. JPL 956571-1
For
Contract No. 956571 between the
University of Kentucky Research Foundation and
The California Institute Df Technology, Jet Pro ,^ulsion Laboratory
Stress-Strain Analysis of Silicon Ribbon
t
r
by
0. W. Dillon, Jr.
Principal Investigator
(NASI -CE-175459)	 STRESS-S'IFIIN ANALYSIS CF
SILICCN BIBBCH f9ertucky Uciv.) 	 11
8C AJ2/hF A0 1 	 CSCL 1CA
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N65-1S525
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G3/44 14314
INTRODUCTION
At the Mini-workshop on stresses and strains held at Mobil Solar on
January 23-24, 1985, we reported on our numerical results for the in plane
stresses and the dislocation fields in silicon ribbon. We stressed that we
had obtained a number of convergent solutions which could be relied upon to
provide insight into the physical phenomena involved. It was emphasized at
the meeting that the approach to the calculation for the dislocation density
was unique and is the FIRST such calculation ever done for any problem. In
our model the internal structure (here the dislocation density) of the
material itself changes due to the temperature field that is imposed and to
the related stress field. The dislocation density changes from point to
point and the result is a prediction of the dislocation density in the final
ribbon. This appears to us to be a valuable aid to the improvement of
quality of the ribbon and its ability to function as a solar cell. We have
had no major difficulty with convergence when the dislocation density is kept
fixed.
In the presentation it was honestly stated that we also had attempted a
number of cases which did not converge. We had previously assumed that this
lack of convergence modeled some physical phenomena and that the only open
question was precisely what phenomena was involved. In short it did not
bother us what phenomena it signaled. Fortunately other participants did
not see that our divergence was not purely numerical in nature. This concern
for the meaning of the divergence has encouraged us to perform that additional
computations described below. There is no question that the changing dis-
location density makes it more difficult to obtain convergence from a purely
numerical modeling point of view. When a solution converges it usually does
so beautifully. For example in most of the ribbon, the effective stress
changes by 1 Pa in 107 Pa from one iteration to the next after 40 iterat";ons.
Near the melt interface the changes are somewhat larger than 1 Pa but they are
still very small.
RESULTS
Upon returning to Lexington from the Mini-workshop, we ran calculations
on two 'Liermal profiles closely related to the (old) EFG one. These calcula-
tions were done for a 3 cm by 5 cm ribbon just because our program was al-
ready set at that geometry.
The ofd EFG thermal profile was
T4 = 437e
-1.36x 
cos,rx + 1157e_* 066x
- 317e ' 47 sin( 2 + 6)
The two new ones to be considered are
T 5 = 437e -1 '
36
 cos-rx + 1157e_* 066x
- 158.5e .47x
and
T6 = 437e-1.36x + 1157e-' 066x
- 158.5e-'47x
These new profiles are shown in Fig. 1 and Fig. 2. Clearly the profile
T6 is very smooth and T 5 has a "bump" that occurs in the hot part of the
ribbon.
The original (old) EFG profile has an additional annealing "bump" at
larger	 Shan the one shorn in Fig. 1.
{after 43 iterations absolutely beautifully conve.-gent solution for both
new profiles are obtained while the one for the (old) EFG profile (eg. T4)
does nct converge. The dislocation density at the melt interface in these
calculations was chosen as 0.001/cm` . The main results are shown in Figs.
3 and 4. The final dislocation density for profile T6 is only 7.1/cm2 while
it is 2375/cm2 for the thermal field T 5 . These results illustrate the role
of "wiggles" in the thermal profiles and their adverse effect on generating
dislocations. Subsequently we have tried additional calculations for profile
T 5
 and a melt interface dislocation density of 0.0015/cm2 . This case also
converges , eautifully and results in a final dislocation density (at y = .9 cm)
of 3560/cm 2 which is 1.5 times the value fot the case when No = .001/cm
2
 .
lie note in passing that there is divergence for profile T 5 and a melt
interface density No = 0.011cm2.
The "effective stress" is shorn in Fig. 5. It is clear that the mechanism
which keeps the dislocations from getting out of hand in these calculations is
that the stress falls below the "back stress" and hence the dislocation den-
sity does not further change with increasing x.
detailed look at the results also shows that the dislocaticil density
changes (almost) abruptly from No = 0.001/cm 2
 at x = 0 to N = 1.5/cm, 2 at
x = .25 cm from the melt interface. Perhaps this value at x = .25 cm is a
better physical parameter to relate to ones experiences.
For the case when the thermal profile is T 5 and the ribbon is 3 cm x
5 cm, the elastic stress 
6yy 
at the melt interface is ayy = -.9607 x 10 8 Pa
at y = 0.90 cm when the rielt interface dislocation density is No = O.CO1/cm2,
this same point has o,,y = -.8469 x 10 8
 Pa. When the melt interface
dislocation density is No = 0.0015/cm 2 , this same point has oyy = -.8462 x 108
Pa. The dislocation density at y = 0.9 cm and x = 2.375 cm is N = 3517/cm2
for the case where No = 0.0015/cm 2 . In all cases the Qyy stresses in the
plastic and elastic cases are already much closer to being the same by
x = 0.25 cm. However the ayy stresses at x = 5 cm are actually larger in the
plastic domain than when considered as being elastic. They are however
smaller than at the melt interface.
In view of the magnitude of the dislocations generated (as shown in Fig.
3) and the corresponding extremely small plastic strains they represent, we
do not see how much plastic defornation can be accomodated and still produce
good solar cells. We therefore consider the case of a small dislocation den-
sity at the melt interface is realms the one that is needed.
This investigator believes that his ability to return from the meeting
and at once select a di--ferent profile that converges so well, reinforces the
notion that he has a good understanding of what really causes the divergences
to occur.
Fig. 6 shows the structure ^-`iere a single crystal was used as the seed
in an EFG furnace in 1974. One can note a short region of new ribbon single
crystal and its subsequent break down into a polycrystal. This photo was
first used in a paper by Leipold, Stirn, Fouterdyk and DeAnaelis in the
Eleventh IEEE Photovoltoris Specialists conference - 1975. It ap pears there
but reprcduces poor. ,
 hence is included herein.
CONCLUSIMIS
In surunary we believe that results such as those shown in Figs. 3 and 4
are of importance to this program for producing elastically good solar calls.
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T2 = 437e -1.36x cosTrx + 1157e -0.066x 
_ 158.2e-0.47x
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Stress-strain analysis of silicon ribbon

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