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Produced by the NASA Center for Aerospace Information (CASI)
MIrd - 9 9^

j&A/Tpc - 9St4S 71 -/

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

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

H

N65-1S525

1i DcIaS

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 dislocation 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 calculations were done for a 3 cm by 5 cm ribbon just because our program was already set at that geometry.

The ofd EFG thermal profile was

T 4 = 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

T 6 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 density 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 density 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 p ears 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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