An investigation of the blowing up of UK numerical results for the inplane stress calculations for silicon ribbon is presented. The inplane stress that exists in a thin plate is governed by two general equations of equilibrium and compatibility

Dillon, O. W., Jr.

English

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https://ntrs.nasa.gov/search.jsp?R=19850018948 2020-03-20T19:02:43+00:00Z
IOct- 11PL-950571
8 15 _0Z
ci cl so - to 65
UNIVERSITY OF KENTUCKY
(NASA-CR - 175775) IN FLAME STBtiSS ANALYSIS 	 685-27259
(Kentucky Univ.)	 16 p 8C A, 2/ 14F A0 1
CSCL 200
Unclas
G3/39 21248
May 1, 1985
Technical Report No. JPL 956571-2
^ QQ	 rrC"I.
For
Contract No. 956571 between the
University of Kentucky Research Foundation and
The California Institute of Technology, Jet Propulsion Laboratery
Stress-Strain Analysis of Silicon Ribbon
by
0. W. Dillon, Jr.
Principal Investigator
This report was prepared for the Jet Propulsion Laboratory,
California Institute of Technology, sponsored by the
National Aeronautics and Space Administration.
C
1
.^ f
T
e .
IN PLANE STRESS ANALYSIS
Summary
In light or the discussions concerning the "blowing up" of the UK
numerical results for the inplane stress calculations for silicon ribbon,
I hava decided to pre,are this report that ties (I hope) the loose ends
together.
The results obtained since January 1985 show that the "blowing up"
of computer solutions at least for smooth thermal orofiles, is indeed
a real physical phenomena. It is our view that they are precisely the
"Luder's band" described in the Mobil Solar Quarterly report to JPL
for January through March, 1984.
ANALYSIS
The inplane stresses that exist in a thin plate are governed by two
general equations (equilibrium and compatability). Equilibrium is used in
the form
a2oXX — 
a2 r
ax2 	ay 
which contains the normal stresses, axx and oyy . The coordinate system used
is shown in Fig. 1. Compatability is assumed in the form
a
2
E
2#
 zazE
xx { 
__ X
L- 2
	 ax2	 axay
y
•	 r
which contains the (total) strain rates e
ii
, Physically compatability is
i
f^
(1)
,(2)
0i
h = Nm be- Q/ kT B (,/J 2  - D 3Nm)/ 3J2 Tom	 (5)
when Vi  > DAM , and the parameter J 2
 is the second invariant of the deviatoric
stress tensor. If D3Nm > 3J 2 the value of h is zero.
The really novel and extremely important aspect of the Sumino--Haasen
a .
model is that the dislocation density chances where shown to vary according
to the relationship,
Nm = K1 Nm (T - DM M)X+me-Q/V	 (6)
We now postulate that this same law applies to the situation that occurs
in ribbon growth. It will be shown below that having a changing dislocation
s
	 density is a major effect in modeling real material behavior for the silicon
used in solar cell applications.
	 in
r
The major assumption that we are making here is that silicon is assumed
to be isotropic in both its elastic and plastic constitutive relations.
Eq. (6) happens to fit into the general framework of internal variable
theories of viscoplasticity of Dillon and Kratochvil [3]. We sometimes change
the parameter K 1
 in Eq. (6) to K 1 /10 in order to approximate a "shape factor"
for the ribbon. We now assume that the total strain rate s.. is the sum of
the elastic and plastic parts. Thus our constitutive equation is
I
	
Eij	 l E o ij	 E oKK 6 i + a T 6 
i + EijPL	 (1)
r
F 
t
r^
in
Steady State
We henceforth assume that the growth process is in a steady state. Thus
one can replace derivatives in time by those in the spatial coordinate x.
By combining Eqs. (2) and (7), in the steady state we obtain
2	 2	
1 fx 
a2 ExxPL	a2E PL	 a2ex PL
V (Q 
+ayy)
	
aE0 T +
	 (	 - + —y	- 2	 ) t du	 (8)
xx	
0	 ay
2 	 x
	
ax	 y
where v is the pull speed, In deriving Eq. (8) we neglected spatial deriva-
tives of a and E. In the solution of Eqs. (8), we do allow a and E to vary
with temperature and hence w i th x. In earlier numerical work where da/dx
and dE/dx were retained this assumption was shown to be more than adequate.
When '.ire material is elastic, the solution of Eqs. (1) and (8) is readily
accomplished on the digital computer for a specified thermal profile. We have
also shown in previous reports that the more analytical procedures given in
Boley and Weiner are not adequate for the thermal profiles of interest in the
Mobil Solar and Westingnouse processes for growing silicon ribbon. In effect
these profiles change too rapidly in space for the Boley-Weiner 14] method to
yield good results. On the other hand if V 
2 
T is constant, the Boley-Weiner
method gives excellent agreement with the stresses obtained using finite
Jifferences and the digital computer. The ribbon has no external loads applied
to it so that stresses are due entirely to the thermal profile.
Dislocation Density
To th?s writer's knowledge this project is the first ever to use equa-
tior.s such as Eq. (6) for any manufacturing process. That is we explicitly
consider the changing internal structure and its effect on stresses and
strains tnat develop during the growth of ribbon.
k
y^i
Most materials researchers normally use materials with large values of
the dislocation density. In this case changes in N do not drastically affect
the materiai response, Some solar cell researchers however believe that
electronic performan,:e is reduced if the cells have a large number of
active dislocations. Silicon people use words sL,ch as "dislocation free"
silicon for many applications. Certainly it appears that we should consider
"low" dislocation densities and one is then forced to consider that they also
change when silicon is subjected to stresses. In the growth processes, it is
clear that a constitutive model with a changing dislocation density is an
important improvement over a model that assumes Nm remains constant.
Westinghouse claims average values of N are 10 2 to 10 3 /cm 2 while Mobil
Solar material p robably contains 10 F, to 10 6/cm2 . These numbers apply to the
respective final prodLIcts in both cases.
As will be discussed below, to study how dislocations grow one needs to
assume the initial value of N at the melt interface, i.e. the value of N in
the solid state at a temperature equal to the solidifying temperature. It
turns out that this is a crucial assumption. We decided to try various valves
for N0 at the melt interface and to evaluate the validity by comparing the
predicted final values of N in the real product. Sometimes this is perfectly
satisfactory approach, However we encountered "instabilities" in the numerical
scheme and thus could not obtain valid solutions for all of the intended values
of No at the melt interface. This raised the question about the validity of our
entire numerical code.
Oxygen
While we do not explicitly show how in Eqs. (4) and (6,, our computer
programs also allow for using different (but uniform) values for the oxygen
content. Oxygen content is known to be different for float zone and CZ
I
IN
iIN PLANE STRESS ANALYSIS
Summary
In light cf the discussions concerning the "blowing up" of the UK
numerical results for the inplane stress calculations for silicon ribbon,
1 have decided to {prepare this report that ties (I hope) the loose ends
together.
The results obtained since January 1985 show that the "blowing up"
of computer solutions at least for smooth thermal profiles, is indeed
a real physical phenomena. It is our view that they are precisely the
"Luder's band" described in the Mobil Solar Quarterly report to JPL
for January through March, 1984.
,MALfSIS
The inplane stresses that exist in a thin plate are governed by two
general equations (equilibrium and Compatability). Equilibrium is used in
the form
2	 2dQ2 
a 
ax	 ay
which contains the normal stresses, axx and ayy . The coordinate system used
is shown in Fig. 1. Compatability l5 ac.sumed in the form
a e	 aZ e
	
2a E:
aye + 
a
^ _ 
a
^
r
which contains the (total) strain rates E
ij
. 
, Physically Compatability is
t
(1)
(2)
A
^^') it
silicon and to be a major actor in their having differing properties. It is
probably different yet for the case of ribbon,
The old EFG thermal profile used is
T4
 = 437c-1 .36x cosTrx + 1157e - • 066x
- 317e
-.47
 sin(2 + 6)
Tho modified EFG profile that is used is
T 5 = 437e
-1.36 
cos-,7x + 1157e .066x
1
I	 - 158.5e-.47x
We continue to use
	
± f1
n !
T3 = 1600e-' 0827x + 85e
-5x cosnx
+ 75 sin(7rx/4) - 273
r
as the Westinghouse profile.
RESULTS
We assume thermal profiles which only vary with the distance x from the
melt interface. We write Eqs. (1) and (8) in their finite difference approx-
imations and treat the plastic incompatability integral in Eq. (8) as a pseudo
	 ,.
thermal field. We then solve Eqs. (1) and (8) by an iterative procedure. The
KOF
a
first iteration is obtained by assuming that the material is elastic, and
hence the integral in Eq. (8) is zero. The stresses so obtained are then
used in Eqs. (5) and (6) to calculate the plastic strain rates and the dis-
location density Nm(x,y) as functions of spatial position. These new plastic
strain rates are then used in the integral in Eq. (8), and the whole process
repeated. In particular the Qyy stresses at x = 0 are determined in the
standard way. They are very large and do not relax very much from their
elastic values.
Usually 20 iterations suffices for excellent results. Typically values
of 3J 2 are of order 108 Pa, while changes in succeeding iterations are
usually only 10 2 Pa after ?0 iterations.
Typical results for the stresses and the dislocation density are shown
in Figs. 2-4. The dislocation density changes by six (or more) orders of
R
magnitude in Fig. (3). As shown in Fig. (2), the stresses eventually decrease
in magnitude and N increases so much that of the ribbon is elastic and Nm
#	 becomes zero.
The data in Fig. (3) show that the value of the melt interface dislocation
density that was used is (almost) ridicously small. On the other hand, the
final value appears to be representative of ribbon products inferring at least
that No is perhaps not too bad.
The plastic strain rates 
ExxPL 
are shown in Fig. 4. Due to the low values
of No , the plastic strain rates are almost always small compared to the elastic
strain rate. Moreover, one finds that the strain rates in the "cusp" region 	 •
shown in Fig. 4 are very sensitive to the value of N o assumed at the melt
interface, as can be seen by comparing with the second curve given in Fi g . 4.
2It is relevant that the numerical scheme does not converge for N o
 = 0.6/cm.
r
A review of the numLri , .I problem reveals that it is the plastic com-
patabilit;, integral in Eq. (8) which causes the solutions not to converge.
This can be understood in view of the "cusp" region in Fig. 4 and noting that
compatability requires spatial differentiation of the strain rates. When one
considers the basic meaning of compatability, namely that the velocities are
required to be single valued functions of space, one decides to look for
experimental data for multivalued displacements or velocities.
Such data exists in the Fig. 15 in the Mobil Solar report for January
through March of 1984 15] and has further been seen by us in EFG ribbo;:. In
some cases one can actually see large "jumps" in the deformation field indica-
ting multivalued displacements really do develop. According to personal
communication with Dr. J. Kalejs of Mobil Solar, these regions of "Luder's
bands" or some other forms of shear banding are developed at 1000°F at stresses
of about 25 MPa. Our calculations, show the initial strain rate cusp develops
at 1100°F or below and at stresses of 19.7 MPa or above.
EFG Thermal Profile
At this writing, we have obtained convergent solutions for the "old EFG'
thermal profile for the case where No = 1,25 x 10^ 13/cm2 . However we obtain
nicely convergent solutions for a modified EFG profile for larger values of
No . The dislocatiGo density at the melt interface for the modified EFG pro-
file to result in convergent solutions is much below the value of N o used in
the Westinghouse thermal profile. Numerical sulutions for the EFG profile
where No
 is close to the value which results in "unstable" solutions, shows
two regions of "cusps" developing in plastic strain rate field in this thermal
profile. The cusp near x = 0 is very large due to the very large values of
oyy 
at x = 0 in these profiles. Considerably more care is involved in getting
stable solutions with an EFG thermal profile as compared to the Westinghouse
one.
1
Ij
Said differently, some computer solutions for EFG profile that blo•.0-up
are clue to the spacing used. However this is not normally the case with
the Westinghouse profile, Very very small spacings are needed near x = 0
to avoid blowing up for EFG. The root cause of this problem is the large
stresses there.
Inhomogeneoas plastic strain rates
Cusps in the plastic strain (rates) such as those shown in Fig. 4 occur
also in the case of localized necking of ductile materials. Our localization
is (probably) more accurately associated with shear b , nding than necking.
However the analysis below, which parallels that used in necking, is believed
to be useful in understanding the comp l exity of the phenoinend shown in Fig. 4.
One solves Eq. (3) for the stress and obtains
Q/kT•PL
The spatial derivative of the stress is
aT _ aT aN	 aT aT	 aT	
aePL
ax TN ax + ^ ax + at L ax	 (1G)
Since the process is a steady state one, we can write M/=x = I/v and
3T/M, aT/aT and aT/ae PL can be easily obtained from Eq. ( g ). Therefore Ne
have
DT _ aT N _ aT aT
De_	 ax aN v aT ax
ax	 aT	 (1')
aePL
If one were to consider necking for example, in a constant temperature test
in a material where the dislocation density is uniform and constant force,
then the R.H.S. of Eq. (11) is zero and the plastic strain rate is uniform
in space as well.
In the ribbon growing process however none of the terms on the R.H.S.
of the analogue of Eq. (11) vanish and the strain rate in nonuniform is
space. As shown in Fig. 4 there are regions where the strain rate is
several orders of magnitude larger than it is elsewhere in the ribbon. There
is no real simple interpretation of what causes the cusps to develop. Each
f
term in the analogue of Eq. (11) makes a contribution that adds together
to cause the "cusps". In the Westinghouse profile with a low melt face
dislocation density, N is also small near x = 0 and the first and last terms
on the R.H.S. of Eq. (11) tend to cancel one another. This keeps both
epL and 3c /ax small. As the dislocation density grows A does as well and
at x = 2.5 cm tha cusp has developed as shown in Fig. 4.	 4
The stresses in the (old) EFG profile are very large and nonuniform and
this in turn requires very small values of N in order to keep 
ePL from
getting lac_! and therefore inhomogeneous. Small values of N exist if very
small values of N o a re assumed at the melt interface.
The stresses ayy at x = 0
Hutchinson and Lambropoulos [6] used a finite element method to calculate
the stresses in ribbon. They assumed a constitutive equation of the power law
type and thereby :,ssume (in effect) that the dislocation density did not
change. They also assumed a y = 0 at the melt interface, x = 0. Subsequent
work by Hutchinson justified this assumption.
k
Al
Our analysis shows that the ayy stresses at x = 0, are in fact very large
and do not relax very much. An examination of the reason for tnis reveals
tnat it is because of the low values of N o
 assumed. Both analyses are in fact.
"correct" and the question is then which is applicable to the growth proccss.
W! belie-,e that our analysis -is mere applicable. We believe that the dis-
locatior density is indeed small and therefore one cannot assume it remains
constant.
Adaptive grids
In recent years, it has become good computing practice to let the sG?ution
of a problem help distribute the grid points in order to concentrate more
points in regions where rapid changes occur. We have incorporated that
practice in many of our calculations. Caution is required though lest, the
aspect ratio of the grid becomes unsatisfactory. Precautions such as using
adaptive grids change the exact values where solutions blow up, but not enough 	
I
to warrant discussion, at least for a Westinghouse profile.
i
References	
I
[ 11 Yonenaga, I. and Sumino, K., Pis. Stat. Sol.(a), Vol. 50, (1978), p. 685.
Suezawa, M., Sumino, K. and Yonenaga, I., Phys. Stat. Sol.(a), Vol. 51,
(1979), p. 217.
[21 Haasin, P., "Dislocations in the Diamond Structure", Dislocation Dynamics,
Edited by Rosenfield, Hahn, Bennett, Joffee, McGraw Hill (1968).
Sciroter, W., Brion, H. G. and Sietoff, H., J. Appl. Phys.. Vol. 54, (1783),
p. 1815.
[31 Dillon, 0, W. and Kratochvil, J. Appl. Phys., Vol. 40, (1969), p. 3207.
[41 Boley, B. and Weiner, J., Lh.eorL or Thermal Stresses, Wiley, New York ( 1960).
/^b^
;Dill
 
n .
	 A
[5] Kaleis, J., Stress Studies in EFG Quarterly Progress Report - Subcontract
No. 956312 for January 1, 1984 to March 31, 1984. Mobil Solar Corporation,
Walthorn, Mass.
[6] Hutchinson, J. W. and Lambropoulos, J. C., Proceedings of the Flat-Plate
Solar Array Project Research Forum on the Hijn-Speed Growth and Characteri-
zation of Crystals for Solar Cells, Publication 84•-23, Jet Propulsiur,
Laboratory, Pasadena, California.
11
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o =0
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(NO BUCKLING)
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In plane stress analysis

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