A numerical simulation to verify the stress-free growth of silicon crystal ribbon

0045.7949187 13.00 + 0.00 Pcrgsmon Journals Ltd.

A NUMERICAL SIMULATION TO VERIFY STRESS-FREE GROWTH OF SILICON CRYSTAL

THE RIBBON-f

SUJIT K. RAY and SENOL UTKU Duke University, Durham, NC 27706, U.S.A. and BEN K. WADA

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, U.S.A. (Received 11 August 1986) Altstract-Thermal stresses developed during the growth of silicon crystal ribbon have been shown to be negligible, thus eliminating residual stresses and dislocations, if the temperature profile satisfies a second-order partial differential equation inside the ribbon. This has been numerically verified through a finite element model, an outline of which is presented here. This model shows that, for homogeneous isotropic material with temperature independent thermal expansion co-eflicicnts, thermal stresses will vanish if the temperature profile satisfies the Laplacian. A comparison of stresses due to uniform and non-uniform temperature gradients in the plane of the ribbon is also pnsented. The strategies employed to control the round-off error and to validate the computer model are discussed.

1. INTRODUCTION Silicon crystals are important material for the conversion of solar energy into electrical energy. Tra+tionally, they are grown in the ingot shape. They need to be cut in the form of a yafer and polished before they can be used. These costly and time-consuming processes can be eliminated however if the crystals are grown in the shape of ribbon. The Dendritic Web Growth process is one of those alternatives which produces a thin long ribbon of silicon crystal drawn between two dendrites. There is no shaping die, thus avoiding possible contamination. These ribbons were found to be very pure and shown electrically as good as Czochralski or float-zone silicon wafers [I, 21. It was also shown that by controlling the temperature of the melt, the impurities can be segregated, thereby permitting the use of a cheaper, solar grade silicon containing a higher concentration of contaminants [3,4]. The ribbon is essentially formed by the interaction of crystallographic and surface tension forces. This process of growing silicon crystal web requires that the latent heat of crystallization be removed by conduction through the crystal thereby requiring negative temperature gradient in the growth direction. The ribbon temperature is generally brought down from 1685 K at the melt-crystal interface to 300 K (i.e. room temperature) within 200-4OOmm, using some kind of exponential temperature profile [5,6]. Usually, there is a very steep thermal gradient near the melt-crystal interface resulting in the rapid drop of ribbon temperature. During this cooling

process, residual thermal stresses were observed to develop. They were also found to be sufficiently large, causing defective crystals, especially in ribbons with high growth rates and/or large width. It was also noted that the thermal stresses were the principal source of plastic deformation and buckled shape of the ribbon. In an earlier paper [7], a displacement formulation for the behavior of the visco-elasto-plastic variable thickness ribbon under thermal load and the conditions of zero thermal stress were presented. It was shown that the thermal stresses and therefore, all other chain effects (caused by thermal stress) will vanish if the temperature profile satisfies a secondorder partial differential equation inside the ribbon. For a homogeneous isotropic elastic material with temperature independent thermal expansion coefficients, any temperature profile, satisfying the Laplacian inside the ribbon, will produce negligible thermal stresses. This paper presents an outline of a numerical finite element model for the simulation of dendritic web growth process. The errors associated with this model are discussed and a scheme for controlling these errors is briefly explained. This model is validated by the proofs of zero thermal stresses for temperature profile varying only in growth direction as well as in growth and width directions simultaneously. Finally, the model is used to prove that the ribbon growth is indeed stress-free if the temperature profile satisfies the Laplacian. 2. OUTLINE OF A DISPLACEMENT-BASED FINITE ELEMENTFORMULATION

t This paper presents results of research performed for the FSA project, Jet Propulsion Laboratory, California Insti-

tute of Technology sponsored by the U.S. Department of Energy through an interagency agreement with NASA.

Figure 1 shows the physical dimensions of the ribbon and the reference co-ordinate system used in 725

SUJIT K.

726

RAY CI al h/2 h/2

I I

I I 1

I I

/Silicon

ribbon

I I I I I I

I I

Interface

Melt

I I I I I

L

II I:

I

\

I:

Fig. I. The ribbon geometry and the reference co-ordinate system.

the numerical model. Using finite element technique, the ribbon is divided into a suitable number of finite elements. Figure 2 shows the node and element numbering scheme for the mesh. In order to reduce

the bandwidth of the resulting system of equilibrium equations, the nodes are numbered first across the width of the ribbon. Since the thickness of the ribbon is very small as compared to its lateral dimensions

M, * No. of rrctongular

elements

M,

elements

l

No. of rectangular

in Y -dlrectlan in X-dlrectlon

Fig. 2. Node and element numbering schemes.

727

Stress-free growth of silicon crystal ribbon

and the temperature is uniform across the thickness, it has been assumed that the stresses do not vary in the z-direction. In addition, displacements are assumed to be small, resulting in a small displacement-plane stress model for the ribbon. In a displacement based finite element model, displacements of the nodes are the primary unknowns. The equations of equilibrium are therefore written in terms of unknown nodal displacements. Once the displacements are solved for, the elemental strains and stresses are computed by numerically differentiating the known displacements. In the following, the basic steps of the model are briefly outlined. 2.1. Stiffness malrix For in-plane deformations, it is being assumed that the deformed pattern of finite element mesh can be represented by the nodal displacements u and u which are positive when in the positive x- and y-directions respectively. Rectangular element: referring to Fig. 3(a) and using the following interpolation rules u = a, + a,.~ + azy + a,xy

(1)

v = 60+ 6,x + 6,y + b,xy,

(2)

the element stiffness matrix .P in its free-free state is obtained as shown in [8]. Dendrite: referring to Fig. 3(b) and assuming u = a, + a,x,

(3)

the element stiffness matrix X” in its free-free state is calculated as shown in [9]. The free-free element stiffness matrices are then assembled onto the stiffness matrix K of the ribbon using any suitable boundary conditions.

the elemental thermal load vector p’ is obtained as shown in 181.This load vector is then assembled onto the thermal load vector p using the same set of boundary conditions. 2.3. Nodal displacements

Using the Cholesky decomposition method [IO], the unknown nodal displacements w are computed from the following set of nodal equilibrium equations: Kw=p.

(6)

2.4. Elemental strains and stresses Using the interpolation rule of qns (1) and (2) and nodal displacements w of qn (6), the elemental strains 6”’ can be estimated from

3

1

au au au do ax ay ay ax

I

--,--,-+-

.

In general, L” is a function of local x- and ycoordinates and varies from point to point in any element. However for simplification, strains were calculated only at the centroid of the element, i.e. at the origin of the local co-ordinate system. From strains cm, the elemental stresses urn (at the centroid) can be computed by using

urn= D(cm - CT),

(9)

where D is a 3 x 3 material, (7 is the strain due to temperature increase and

2.2. Thermal load Using the interpolation

rule

3, INTERNAL NODAL FORCES

T = I,, + t,x + t,y + r3xy

(for rectangular

elements)

T = I, + t,x (for dendrites),

The internal vertex forces q” equilibrating elemental stresses urn may be estimated from

the

(4) (5)

9“=XX.

(11)

Y

Lx m--x t

(a 1

Roctangulor

rlrment

lb 1

Fig. 3. Local co-ordinate system of the elements.

Dandrltr

SUJIT K. RAY et al.

128

The vertex forces q” can then be assembled onto the internal nodal force vector, q. The nodal force vector q is finally compared with the thermal load vector p to determine the equilibrium.

: Round-off :,error

I 4. CONTROL OF ERRORS

The finite element formulation can be thought of as curve fitting-where an approximate function is used to fit the curve of primary unknown (i.e. displacements) between two points (i.e. nodes of finite element mesh). In reality, these unknown functions are smooth and possess continuous higher-order derivatives. The approximating functions, on the other hand, are usually low order polynomials and generally have discontinuous derivatives at the nodes of the mesh. The error introduced in the analysis by using an approximating function is known as truncation error and can be controlled by refining the mesh successively. For an infinitely refined mesh, the nodal displacements of the finite element mesh represent the neighboring points on the exact displacement curve and, therefore, the finite element solution converges to the true solution. But the quality of a numerical solution, to a great extent, depends on how the numbers are represented in the computer system. The stored numbers are never exact-they are always represented in a fixed number of computer memory locations. For numbers requiring larger memory locations, extra digits are either discarded, thereby truncating the number or the numbers are rounded off so as to fit’within the specified memory locations. This type of error is known as round-off error. Each arithmetic operation introduces round-off errors, the net effect of which could be devastating. Round-off errors though can be controlled by minimizing the number of arithmetic operations leading to a coarse mesh or by increasing the computer storage locations for each number, at the expense of huge storage requirements. The latter option has its own limitation-there is always an upper limit on the number of the computer storage locations that can be assigned to store a number. Clearly, the error-controlling techniques for truncation and round-off errors are contradictory to each other. Figure 4 schematically shows the variation of these errors with the mesh size. While the truncation error depends mostly on the choice of the approximating functions, round-off errors depend on many factors-omputer systems, precision of the stored number, sequence and number of arithmetic operations, choice of algorithms, etc. In any case, there exists an ‘optimum’ mesh size for which the total error of truncation and round-off is minimum (Fig. 4). This mesh size is most likely to produce an inexpensive but reasonably accurate numerical solution. To control the errors, all of the above strategies have been used in the present numerical model. First, the important arithmetic operations such as fac-

Truncation error

,

I /

‘I

: ,I , , \ ,

I

Optimum / mesh size’

Mesh size

Fig. 4. Variations of the errors with the mesh size. torization of the stiffness matrix, forward and backward passes of the Cholesky method and the computation of elemental strains were carried out in double precision. However, the results were stored as single precision numbers to reduce the computer storage requirements. In addition, the sums of positive and negative numbers were accumulated separately and the negative sum was subtracted from the positive sum at the end, thereby preventing the loss of significant digits. Finally a convergence study was undertaken to find the ‘optimum’ mesh size. Figure 5 shows the keysketch and the geometrical and material properties of the ribbon used for this comparison of mesh performance. The meshes were without dendrites and subjected to a uniform cooling of 1000 K. From elasticity, such a determinate structure undergoing uniform temperature loading should be stress-free and, therefore, the true response is given by

uxx=cTyy=uxy=o.

(12)

The positive directions of these stresses and the convention used in identifying a mesh are shown in Fig. 6, while Fig. 7 presents the results of this convergence study. For convenience, stresses along two parallel lines, y = 6.75 mm and y = 20.25 mm, are plotted. It can be seen that the results of 3 x 18, 9 x 18 and 27 x 18 are quite good while the results of 27 x 54 and 81 x 18 are discouraging. These, being the two most refined meshes, have maximum number of unknowns to be solved, requiring maximum arithmetic operations and resulting in somewhat wrong stresses. The erratic shapes of these curves can also be interpreted as a manifestation of round-off errors.

Stress-free growth of silicon crystal ribbon

I

729

162 mm

IYoung’s modulus =200,000

N/mm’

Poisson’s ratio = 0.3 Thermal expansion co-efficient

* 0.126 x IOe4/K

Thickness =0.1 mm Dend,rite diameter .O

Fig. 5. Key-sketch for the convergence study.

Of the three acceptable meshes-3 x 18, 9 x 18 and 27 x ll-the most refined one, i.e. 27 x 18, can be chosen as the ‘optimum’ mesh. 5. VALIDATION

OF THE NUMERICAL

MODEL

To validate the numerical model, the ribbon was further subjected to two different thermal loading: Loading I: A cooling of -7x-100 in degrees Kelvin. Loading II: A cooling of -5x-10~ - 50 in degrees Kelvin. Figure 8 shows the keysketch and the geometrical and material properties of the ribbon. For these loadings, only rigid body movements of the ribbon are prevented (Fig. 8). The meshes 27 x 54 and

81 x 18 are not considered because of their disastrous performances during the uniform cooling case. Stresses are again plotted along the same two lines, y = 6.75 mm and y = 20.25 mm, and are shown in Figs 9 and 10. Theoretically, they should be zero since the temperature changes satisfy the Laplacian. It is interesting to see how the shapes of the stress-plots change when the meshes are refined. For coarse meshes, the truncation errors prevail resulting in incorrect representation of the mesh behavior. This can be seen from the stresses which are way off from their true values. For the refined mesh, on the other hand, the approximating function converges more and more to the true solution and stress-plots rotate and/or get flattened out. For 27 x 18 mesh, the stresses everywhere are very close to zero, further reinforcing the choice of 27 x 18 as ‘optimum’ mesh.

I

6 x 2 mesh elements No. of elements In X-dir.

in Y-dir.

t

Directions of positive

Fig. 6. Conventions used in the simulation.

stresses

SUJITK. RAY e! al.

730

COMPARISON OF STRESSES SIGMA-XX

STRESSES

ALONG Y=6 75

-b

-lb

-17 ._/-16

_-0

20

i . . ..I ,,,, 40

60 DISTANCE

,,..I ,.., 60

.,.,! ,,., 100

FROM THE MELT

Fig. 7a. Typical result of the convergence study (uniform

,.,,! ,,,, 120

,,,,! ,,,, 140

(mm)

temp. drop

of lOOOK).

., 160

Stress-free growth of silicon cryslal ribbon

731

COMPARISON OF STRESSES SIGMA-W

STRESSES ALONG Y=20 25

J. -

‘---

15

&_W

. d\

----___

-0

I

I 3x18

:27x6-

r.f; _.__-----

I

I

.

,

27

?

--,--.

q_“.&

L ---

9

4

---

---

-

-.

-

A__

1.4

1 .a \

x18

I p1

-----~--

x 1%

\ -0

3 --

I I i

-0.q

-0 6

-0.7

-0 8

-0.9

-1.0

-1.1 0

20

40

60

80

100

DISTANCEFROMTHE MELT

120

140

160

(mm)

Fig. 7b. Typical result of the convergence study (uniform temp. drop of lOOOK).

732

SIJJITK.

Fb,u el

al.

COMPARISON OF STRESSES SIGMA-XY

STRESSES

I

ALONG Y=ZO 25

I,,

,

iii i ,

,

,

,

I t

-0 21

-0

24 0

20

‘IO

60

80

100

120

140

160

DISTANCE FROM THE MELT inun)

Fig. 7c. Typical result of the convergence study (uniform temp. drop of lOOOK).

Stress-free

6.75

growth

of silicon crystal

ribbon

mm

162 mm

I

Young’s modulus ~200,000 Poisson’s

rotlo 9 0.3

Thrmol Thickness

exponslon lO.l mm

Dendrite

133

diameter

Fig. 8. Key-sketch

*

N/mm2

co-efficient

9 0.126

x IO-‘/K

90 for the validation

tests.

734

SUJIT K.

by

er ai.

COMPARISON OF STRESSES SIGMA-XX STRESSES ALONG Y=6 75

N -1

/ m m l

:

-2

20

90

60

80

100

120

DISTANCE FROtl THE MELT (ml

Fig. 9a. Typical result of the validation tests (Loading 1)

140

lb0

Stress-free

growth

of silicon crystal

COMPARISON

OF

ribbon

735

STRESSES

SIGMA-W STRESSES ALONG

Y=20

25

35

30

\

25

\

20

\

15

\

10

\

I

: -5 : m m .

i

-lO-.

‘77

l

2

,/ I

‘. ‘. ” L

: i -15

\

-20

-25

\

-30

\ \

-35 0

20

10

60

80

100

120

DISTANCE FROM THE MELT (mm) Fig. 9b. Typical

result of the validation

tests (Loading

I).

190

160

SIJJIT K. RAY e/ al.

736

COMPARISON OF STRESSES ~=2025

s~cbtkx~ ~F~ESSES mm

/

7

\,

/ /

6

I

\

F -- 4-

0

20

40

60

80

100

120

DISTANCE FROM THE MELT (nun)

Fig. 9c. Typical result of the validation tests (Loading I)

190

160

731

Stress-free growth of silicon crystal ribbon

COMPARISON OF STRESSES SIGMA-XX STRESSES ALONG Y=6 75

20

40

60 DISTANCE

80

100

FROM THE MELT

120

(nun)

Fig. 10a. Typical result of the validation tests (Loading II).

C.A.S. 15I5-0

140

160

COMPARISON OF STRESSES SIGMA-YY STRESSES ALONG

Y=20.25

0 s ;

5

-5

: N -10 / m m l

0 2 -15

0

20

40

60

80

100

DISTANCE FROM THE MELT

120

(mm)

Fig. lob. Typical result of the validation tests (Loading II).

140

lb0

139

Stress-free growth of silicon crystal ribbon

COMPARISON OF STRESSES SICNA-XY

0

20

YO

STRESSES

GO

ALONG Y=20.25

80

100

120

DISTRNCEFROM THE MELT (mm) Fig. IOc. Typical result of the validation tests (Loading II).

140

160

740

SUJIT

6. VERIFICATION

OF STRESS-FREE

K. RAY el al.

GROWTH

T(x, c) = T(x, -c) = -0.065(x

+ 5.3(x - IO)*- 86.5(x - 10)

Reference [7] gives a theoretical formulation for stress-free growth of silicon crystal ribbons. It was shown that if the temperature loading T(s._r) satisfies

+ 700

d2T a’T ax2+2=0

the ribbon will be stress-free, provided it is isotropic, homogeneous and initially stress-free with temperature independent thermal expansion co-efficients. To show this is so, the numerical model was used to apply the following two different temperature loading on the ribbon [5, 61: Loading III:

ZY

for

-c
(14)

T(x, v) = T(x, -_y) = 120~ -” - 86.5~ +I565 T(x, y) = 7(x, -y)

for

O
(15)

= -0.065 (x - lO)3

+ 5.3(x - lO)2- 86.5(x - IO) +700 T(L,y)=351.94K

for for

lOGxG20

(16)

-cGy
(17)

-c Qy Gc

(18)

Loading IV: T(O,y)=

1685K

for

T(x, c) = T(x, -c) = 120em5” -86.5x+1565

for

O
(19)

for

7’&~~)=351.94K

(13)

T(O,y) = 1685 K

for and

-c <)‘<
(20) (21)

-c
O
(22)

Figure 11 presents the keysketch and the geometrical and material properties of the ribbon used for this demonstration. The resulting isostress plots of OX,, aYYand rrXvfor loading III are shown in Figs 12a-c while those for loading IV are shown in Figs 12d-f (all stresses are in N/mm*). The ‘optimum’ 27 x 18 mesh with dendrites was used to obtain the stresses. To prevent the singularity of the stiffness matrix and subsequent failure of the numerical model, only rigid body movements of the ribbon were prevented. Figures 12a-c show that for loading III, u,, stresses range from -18 N/mm* to lON/mm* while cvv ranges from - 31 N/mm2 to 13 N/mm2. Since the yield stress of silicon decreases with the increasing temperature, a high concentration of uYYstress near the melt-crystal interface is more likely to cause yielding, thereby introducing plastic deformation in the ribbon. Loading IV, on the other hand, produces stresses that are almost negligible (Figs 12d-f). These are not likely to cause any defect in the crystal. The only difference between loading III and loading IV is the thermal gradient in the width directionfor loading III, the gradient is zero, that is, the

162 mm

Young’s modulus.

IO < x < 20

for

27 x 18 mesh

I

- IO)’

200,000

N/mm2

Poisson’s rotlo 0.2 Thermal expanslon co-rfflcient Thickness no.1 mm l

Dendrite diometer * 0.6 mm (All stresses ore In N/mm’

8 0.042

x IO-‘/K

unit)

Fig. il. Key-sketch for the verification of stress-free growth.

Stress-free growth of silicon crystal ribbon

ISOSTRESS

LINES FOR

741

SIGMA-XX

(27 X 18 MESH)

38.25 33.75

. \,~--_____________--.-;/

\'. -/-.~~--.___.-----~-...,_,

-

_A/,,,/

‘. --_-____-----

29.25

/.I

--

"\ : S 1 A N c E

24.75

k 0 II

1125

“\ \

.--.--------.

“\

_.’

‘--.__--’

20.25

..’ /-

_--_

\

/’

1s 75

‘\

/’

‘\

/

2.25

4

-2.25

E’

I

I

T z

N

I

/ 6.75

I

-6. 15

\

R

L i

‘\

-15.75

\

I \

-\ --__* / 1’ 0

I -20.25 N

..---1

“l__

_A

m -29.75 m -29.25

I

\

-11 25

1..

,/

--.-~._~-- J’

\_

/

-33.75

/” ~‘_,_‘-----‘._____-.~

-38 25

/-/_-\ ‘15

_/.C.---------.__

d’

. .

_’

---

/

27

I

‘39

1. 51

163

,75

7

I.8’

I‘,‘, 99

111

123

.

‘.

-. 135 I.

1 147 I 15

DISTANCE FROH THE MELT (mm) LEGEND. SIGWlXX

..-.

-18 1

....... -13 --_ 5

__ ---

Fig. 12a. Isostress plot for loading III.

-8 10

____

-4

742

SUIT K. RAY et

ISOSTRESS

al.

LINES FOR SIGMA-YY (27 x 18

MESH)

38.25

29.25

! s

\

‘\

2q.'5

__I---‘,

I

I

20.25

I’

I

i 1s.75

f;

l1.25

!l

6.15

\

I

\

:

/

I

I i I

I

I

I

I

I

; E

2.25

C

-2.25

: :

i

I

I

-6.75 -11.25

I

I I

I

i

\ I

; \ \

\

\

“\

i

m -24.75 m

i

I

i

I -20.25 N

i

I

i

i -15.75

I

I

I i ; i

R L

I

/

\ -,

/

1’

/ I

=.__/’

I

-29.25 -33.75 -38.25 '3

I15

121

,39

151

'63

'75

'81

'99

1’11

123 1 . 135 ,

147 , 1:

DISTFINCE FRON THE t!ELT bun) LEGEND

SIGMYY

__-.

-31 -2

_______ -23 _-_ 6

-_ ---

Fig. 12b. Isostress plot for loading III.

-16 13

_-_-

-9

Stress-free growth of silicon crystal ribbon

143

ISOSTRESSLINESFOR SIGMA-XY (27 X 18

MESH)

38 25

I i I i i i I i i i

33.75 29.2s

Y

2q.7s

s

20.25

; c E

15.75

;

'lLS

z

6.75

A E

2.2s -2.25

Y E R

-6 .lS

I

I ,I I i

i

---_-_-_--

‘L-_-_-_-_-_-_-_,

,---_

s

L

I ‘\\ -\ I \ _\i/ i i

T

/’

I

‘c -----_---_-_-

\

‘,

-11 2s

tf4-15.75 E

--1 /

I -20 2s

1

‘\

N

i i i i

!

m -24.75 m -29.25

I$

-33.15

\

\, !

1

‘k__/

I

-38 25 ‘3

I15

139

I27

I51

I63

‘75

I

07

99

1

111 I

123 I

\ 135 I

147 I .-1

DISTANCE FROM THE MELT (mm) LEGEND, SIGMAXY

---.

-9.7 3.2

_______ -6.4 -6.4

Fig. 12~. Isostress plot for loading III.

----

-3.2 9.7

_-_-

-0.0

SUJIT

K. RAY et al.

ISOSTRESS LINES FOR SIGMA-XX (21X 1.3 MESH) 38 25 ,’

33.15

:

,’

_I’

I

_/I’

_______---1

29.25

_.--

‘\ ‘\ --___-

Cl I

2’

_.

,’

24.75

s. 20.25 ; c E

15.75

L

11 25

:

6.75

T

E”

2.25

:

-2.25

Y

-6.15

ii L I

-11.25

! -15.75 I -20.25 N

m -24.75 m

__=----___ I *.

#’

-29.25

*.__

,’

--__

‘\

----------_

! I

-33.15

--. \. -. -\

\

‘.

.\.

II

-38 25

‘15

81

6

I

99

111 I.

123 h

135 1.

DISTANCE FROM THE MELT (mm) LEGEND, SIGHRXX

111

-0.16 "0::;

-______ -0.05 __-. 0.29

Fig. 12d. Isostress plot for loading IV.

----

0.06 0.90

1$7 15

745

Stress-free growth of silicon crystal ribbon

ISOSTRESS

LINES FOR SIGMA-YY (27

X 18

MESH)

I I I

29.25

Y s T

I I I I I I I

24.75 20.25

A ;

15.75

k

l*.25

f:

6.75

- I

T

2.25

; C

-2.25

: :

-6.75

R L

-11

_I I

I I

25

-20.25

I I

In -24.75 m

\

: -15.75 E I

N

-29.25 -33.75 -38.25

5

15

4.

27

t

I

39

51

1

DISTRNCE LEGEND

SIGMYY

e-m.

-0.8 0.6

63

0

‘75

87

I.

99

8.

111 1.

123 0.

135 I.

147 1

159

FROM THE MELT tmm) _______ _-_

-0, q 0.9

----

Fig. 12e. Isostress plot for loading IV.

-0.1 1.3

----

0.3

746

K.

SUJIT

RAY er al.

ISOSTRESSLINESFOR SIGMA-XY (27 X 18

MESH)

38.25

I \ -‘\ \,I, \

33 75 29.25 D I S T A

i ‘\

‘1

‘\ ‘\ \

24.75 20.25

! E

15.75

F

1125

i M

I

I i

‘\ i \ \

I

I

1

6.75

‘\

H' E

2.25 /’

_--____-

‘\ --_

/

/

I

---_

----

/’

i

.L_ -___-__--

-_

--

-/

-l

:

-2.25

N :

-6.15

R L

\

-‘.

-11 25

: -15.75 E

\ \

I -20 25 N m -24 15 m -29.25 -33.15

_.i

\

-38 25

lL

3

,

15

! '27

6

'39

51

'63

‘15

‘8 1

‘99

1'1 I

Ii3

135

Ii7 15'

DISTANCE FROM THE MELT (mm) LEGEND

SIGMRXY

---.

-1.4

0.5

__---__

---

9 09

-0

-__-

-0.5

----

0 fJ

14

Fig. 12f. Isostress plot for loading IV.

temperature profile T is a function of x only. For loading IV, the profile T is a function of both x and y. As shown in [7j, this results in frowning of the temperature profile where the edge temperature (at y = + c) is less than the temperature at the middle (at y = 0). Figures 12a-f conclusively prove that a properly frowned temperature profile will not produce any thermal stress and consequently, the ribbon growth will be stress-free.

7. SUMMARY

OF RESULTS

This work verifies numerically that the dendritic web growth of silicon crystal ribbon will be stress-free if the temperature profile satisfies a second-order partial differential equation. Further, the round-off and truncation errors associated with a finite element model can be controlled by a careful choice of ‘optimum’ mesh size.

Stress-free growth of silicon crystal ribbon Acknowle~gemenrs-The model presented here is a part of the computer program being developed to simulate the thermo-visco-elasto-plastic behavior of silicon crystal ribbon during its growth process. The first author would like to extend his sincere appreciation to the staff of the Duke University Computation Center for their numerous valuable suggestions. REFERENCES

R. Ci. Seidensticker, Dendritic web silicon for solar cell application. J. CrJ~.rf.Growth 39, 17-22 (1977). R. H. Hopkins, R. G. Seidensticker and J. Schruben. Modeling thermal stress effects in silicon web growth. J: Crvsr. Growth 65. 307-313 (19831. R.-G. Seidenstickkr, A. M. Stewart and R. H. Hopkins, Solute partitioning during silicon dendritic web growth. J. Cryst. Growth 46, 51-54 (1979).

R. G. Seidensticker and R. H. Hopkins, Silicon ribbon growth by the dendritic web process. J. Cryst. Growrh SO, 221-235 (1980).

147

5. R. F. Sekarka, Lateral temperature modeling of web. Presentation in the Third JPLlFSA Stress-strain Workshop, Mobil Solar Energy Corporation, Waltham, MA, 23-24 Jan., 1985. 6. Westinghouse Corporation, Dendritic web ribbon and cell processing development program reports for JulyNov. 1984. Reoorts WAESD-TR-84-0037. WAESDTR-84-0037A, ‘WAESD:TR-84-0037B, WAESD-TR84-0037C, WAESD-TR-84-0037D (1984). 7. S. Utku, S. K. Ray and 8. K. Wada, On control of stresses in silicon web growth. Compur. Strucr. 23, 657664 (1986). 8. S. K. Ray, S. Utku and B. K. Wada. Generalization of rectangular element stiffness matrix and thermal load vector associated with u,, + u,x + 09 + u,xy type interpolation rule. Compur. Srruct. 24, 949-951 (1986). 9. Robert D. Cook, Concepts and Applications of Finite Elemenf Analysb. Wiley, New York (1981). 10. C. H. Norris, J. B. Wilbur and S. Utku, Elemenrary Srrucrurul Analysis. McGraw-Hill, New York (1976).