In the study of stress-strain relationships in silicon ribbon, numerous solutions were calculated for stresses, strain rates, and dislocation densities through the use of the Sumino model. It was concluded that many cases of failure of computer solutions to converge are analytical manifestations of shear bands (Luder's band) observed in experiments

Dillon, O.

English

Results are presented for work on stress-strain relationships in silicon ribbon. Calculations of stress fields, dislocation desities, and buckling were made; uniaxial tensile tests were made on silicon at 1150 C; and dislocation motion studies were performed

NASA Technical Reports Server

, ADVANCEDS,L,CONSHEET N8 7 " 1 6 4 0 8
" _" ANALYSIS OF SILICON STRESS/STRAIN
• ; RELATIONSHIPS -,
:i UNIVERSITY OF KENTUCKY
O. Dillon
i
'1 _ ":' Dislocation Density Contour Plot
• 6=,L_ J
1_% T-1 412-1 10. "/4"X,3.5"X""2
, UNIT OF X RND T-CM. Z=l PER OH,,,,.2
, 7.8 '
7.2
6.6
,,- I.O
5.4
4.8
: 4.2
3,6
3.0
2.4
! .4
,_ 1.9
• .
1.2
0.8
O,O i
O.O0 1.75 3.50
¥
LEGEND= Z ....... 0.$ ..... 9.6
•------ 18, ----..- 27.
-.--,--- 36. ----- 46.
: .... 55. ------- 64.
- -- - 73. - ..... 02.
91. ------- tO0.
/ .... 109. ----- 118.
' ------ 127.
j.
"_, Dislocation density contour plot /'or the
- parabolic therm_l profile of Eq.Ca-8), 8=7
"_! cm ribbon and an ini_tnl dislocotion denstt},
: O.S ca"2.
-;j
_:] 255
®'
1987006957-255
https://ntrs.nasa.gov/search.jsp?R=19870006975 2020-03-20T12:13:32+00:00Z
• _i_,l_,_l,m_p.-tt__,, .w...,_.... • • • . -ll,,-.2.. • -
!
f
ADVANCED SILICON SHEET
Dislocation Density Along Y = 0 (Centerline)
T=1412-I I0."/4_X+3.5,X_.2
_. LINE 1 FOR WIDTH=8 CM LINE 2 FOR WIDTH-?. CI_I
,: LINE 3 FOR WIDTH=6 CM LINE 4 FOR WIDTH=4 CM
/ Dislocation den#ity along the ¢enterline of the
, ribbon for the paribollc thermml profile of %q,
;_ (4-8), initial dlslocatlon denmlty : 0.5 cm_"2,
,_ and width • 8, 7, B, 4 CI.
256
1987006957-256
__,_;,,._ ..2:_"-. ,_
ADVANCED SILICON SHEET
Final Dislocation Density Along the Ribbon
Width for Westinghouse Profile
LENGTH = 12 CM, HIDTH = 3.5 CM
STAR FJR NO = 13 ICMww2
DIAMOND FOR MO = 5 /CMww2
SQUARE FOR NO = i /CMww2
2500-
° II
_ooo- '¢° i _/'''"
L "
! 0C
! A 1500-
T
' fro .r[=.,
N lO00-
i
:! O
_- E
_ N 500- ,.,.,,//_,,, IS Jb(.., 3 ,•" I " ""
,. T
_- Y 0-
......... I ...... " ' ' ! _ "' ....... '1 ' "" .... q,, 1
i
• 0 -1.5 -I.0 -0,5 0.0
' CO'"7"-c'ID_- T (CM} *
,.b
• il
! 2s7
1987006957-257
ADVANCED SILICON SHEET
Dislocation Density Contour Plot
(
i HESTINOHOUSE PROFILE, N0=0. 25/CM"M2, R/T=0 4,
__ UNIT OF X RN0 T'CR, Z=I PER CM..2
x 6,/z H. /I
"i 7.8-'. ' '
' I I
.! 7.5- I 'i 7.2- , I
.m 6.9-." I '
6.6- I I
I
. 6.3- I I6.0-
5.4-
5.1- ' ' !
4.6 I ,I _.
' I4.5- I
I
4.2- I I3.9- I !
3.6 j , / / /
_- I , i
. /
3,_ -'._./ , ,," / ./ / ,) -3-- -......"/ ./ / / , '
_:_ -:--:--.--:-.;.-......] .:._;-:--.
.O.O 1.5 3.0
1
LEGEND: Z ....... 10 ..... . 201
39-" .... 583
774 .... 965
- -,- - - 1156 I347
., 1538 1729
" 1920 .... 2 1 I,l
2302 ---- 2493
-" 2684
i
:!
258
_o- ,. , -.... ®ii V ,.
1987006957-258
=o-
Be
_., 259
o"
1987006957-259
., • q ' °
_: ADVANCED SILICON SHEET
,kI
Effective Plastic Strain Rate
NESTINGHOU5E PROFILE. NO=0. 25/CMwM2
' WIDTH = 6 CM. LENGTH =8 CM, FI/T=0.
UNIT OF X AND T=CM, Z=IOww-5 PER SEC
S0.57
0.29
0.00
8
2
X 2 T i
• I
i
: O. OC_
i:
"( b
,/
I
I
. 260
Jr/it-- , ql - ,
1987006957-260
ADVANCED SILICON SHEET
Effective Plastic Strain Rate
WESTINGHOUSE PROFILE, NO_O. 3?5/CMww2
HIOTH = 6 CM, LENGTH =8 CH, R/T=I.6867 MR
UNIT OF X RND T=CM Z=jOww-5 PER 5EC
Z
0.41 _'_
, 0.28
0.14
0.00
8.
/ 5,
2 IP
X 2. T
!
0.00C)/
261
®'i
1987006957-261
" _, °'"L
°s
• • #
_' ° _ 6•, _ ....... • ._
YL _-
|,
ADVANCED SILICON SHEET
-,*o Residual Stress XX Along Ribbon
' Width for Westinghouse Profile
LENGTH - 12 CM. NIDTH - 3. 5 CM
• STRRFOR NO = 13 /C-H-.2
DIAMOND FOR NO = 5 /CH..2
. t,,4_ 50UARE FOR NO = l ICHor2
R
L -3-
-
.Z" P
R -12
-I. 75 -1.25 -0.75 -0.25
T (CM)
,." ,°
oo
®262 i
1987006957-262
I ADVANCEDSILICONSHEET ORIGINALPAGEI_
,. k Deflection Shape Of POORQUAIJTY
* HRLP "ld ] I_¶H (C}"I. , LI[NGTH"S. 0
_" OIRMf.TE[R Off OENORITE$ IS 0.0 INCHES
CRITICRL THICKNESS = 0.00528 INCHE3
r
r/x/_ f
_._ ii
=. !
• II
!! 263
I# -i,
1987006957-263
r,
" e L
i
!
ADVANCED SILICON SHEET
.- Deflection Shape
d
/
HRLF-NIOTH (C)_l,o LENGTHeg. 0
DIRMETER OF DENDRITES I5 0.2 INCHES
' CRITICRL THICKNESS - 0.05"/120 |NCHE$
:'8
Z
O.044
0.030
$. O0
: 0.015
i' " ADVANCED SILICON SHEET
Topics
IDislocationMotion
k
R) ]5"oble_._orMttla.tion
' B) Cal©ulationof Forces
• C) Trackingthe .orionor a singleDislocation
,_.c,;_ 2 DislocationMultiplication&Densit9
• R) Three_tbods ot calculations - basedon resolvedsbe.r stresses on
eachslip sgs_e.
• B) Dislocationdensit9 b9averaging the I shear st_sses I in 0,5 c,,
widthsof ribbon,"(starting at x : 0,2 cM,)
I
/
jo, ,,-e
.:..tiY 265
v
?
1987006957-265
- -_..L,_..2" _ • -|
'. l ADVANCED SILICON SHEET
Possible Dislocations in the Silicon Crystal7
b_gets voctor tangontvoctor slip plane typeof dislocation
<-I0-1> <-I0-I> (-I-1 l),left,screw
' = <Ol I> = ,_fL60'(-120_I"
= <1-I O> ---,eft,60.'(+120')
_: = <1 -2 -I> = ,Left, edge
t_! <0 1 l> <0 1 l> = ,e_ screw= <1- O> ',eft,60'(+120')
• = <-I..0-1> = ,[eft,60'(-120'),"
:; = <-2t-I> = ,[e_ edge
<=I1 O> <-I1 O> = ,left, screw
= <O-l-l> = ,left,O'(-i20')v"
= = ,left,9)(+120')v"
- = _l-1 2> = ,left, dge
< 1 0 -l> < l O-l> (-I 1 -1),ri it, screw , i
• = <-l-1 0> = ,rigl 60 (-120) v
= <0 1 l> = ,rig_ 60' (¥120')_
' = <-l-2-I> = .n.g edge
<0 -1-I> <0-1-l>
_ = ,,_,,_ It, screw
= < 1 1 0> ,,,._!it, 60' (-120'1 /
\ = <-1 0 1> ,n.g tt, 60' (+120)
= <-2 -I I> ,ng it, cage
 <1 1 O> = ,ngat, screw
= <-l 0 I> ,right, 60', (+120,')__ i
_ = <O-1-l> ,right, 60 (-120) v"
= <-l 1-2> = ,right, edge
<0-1 l> <0-1 1> (_ 1-ly, transv_.,_screw ..
- = <1 1 0> ,transv., 60 (+120).,
- = . <-I 0 -l> ,transv., 6.0' (-120') ,/
(. = <2 1 I> ,transv., edge
 = ,tmsv., screw
= ,> . 60'
- <-10-1> ,transv., 60' (-!20')/
h
= < 1 -1 2> ,transv., cage
 = ,transv.,screw
- <-I-1 0> = ,tmnsv., 60' (-l:_0') "v"
ffi <0 1-1> ,= ,, ansv., 60' (+120')
= <-I -2 I> ,- ,transv., cage
)_m maMmm_emmmaml
Growth direction to the n_lt i,, ,"2 -1:1 >.
:, For the motion of the dislocations, 60' dislocations that have
•.120 .degreewith the burger'svector will be chos.cnbec.aus, these._.
_., 60' dmlocattons may .multiplymem_lves m.om than +lZO"type Off
dislocations as many investigators o_ervecL
'o*) ,,..#
,.P .
"_:. ORIGINAL PAGE
. Of POOR QUALITY,
266
1987006957-266
1i i ADVANCEDS;LICONSHEET
Assumptions
_ (l) The density of dislocation at the liquid solid inte_ace is uniform
_I _) (2) .'The,pulling rate of the ribbon is 3crn/min.
,,_ (3) Dislocations can move only in active slip systems that have their
rosoNed, shear stress higher than 95% of the most active slip system that
t
,' hasmaximum Schmid Factor.
(4) Average velocity of the dislocations in the preser(ce -f other
"dg_,locationsi almost same as the velocity of isolated dislocation.
Equivalent to low dislocation density.
(5) The velocity equation proposed by K.Sumino
V = Vo _ exp(-E/kT)
where E is 2.2 eV for 60' and 2.35 eV for screw,
Vo is 0.035 for screw and 0.01 m3 / MN.sec for 60'. _
4
is stLllvalid at high temFeraturelike around melting temperature.
/
_°_
267
1987006957-267
i|
• ! ADVANCED SILICON SHEET ORDINAL p.._-_,
ResolvedShear Stress OF Pi3G_ QUALITY
_ X
i S.et
?
"= ¢e2'.... . _. c+_e=__g
.......... LeSt
/'_* e
3.5+ ,,,
;.
o 4P
'; s.o "
\ ' /
\ o t
C2.5, ',,1%,
2. 0, "+,
I ,, \%
I "%_%
1.5. "%,,
• %% %
O.5, ._ . . '_ ".. ,'s,"s,_ .___.(...__4...!__ .""../_
' (ia_-]l =O.II_" • "="=
-. I.EGIE,OoZl --- -I047111S -- -7022'I140 • -- ,-4'IgT'PIS . i
-- -_17Z750 ---- -117701 -- |0?7)40
...... 31o25e$
" II
268
=,. . ®:
1987006957-268
WJi I
- ADVANCED S:LICON SHEET
Motion of Dislocation
I ol/ll-i 1 II.T-ll I l).il,,,I-I -I 1)
,. 2. 251
,--]
i' e 1.75-
1.25.
1.00-
00 75"
0.25
." O.00, _'9° ° "° '"f '"
/ 0.0 0.2 8.4 0,6 0.8 1.4
UNIT OF AXES ARE CM
#°, ,...p
Ip
269
¢
._..a ql" ,_,
1987006957-269
, ADVANCED SILICON SHEET
Motion of the Dislocations Emerging to the Surface
, possible60fdislocationsemergingtothesurface ,,
,t t
• _.
burger'svector tangentve._to: plane motion
J
3"
a/2(-10-1) (0 1 I) (-1-i I) leftstrong
a/2(-1 I O) (0-1-1) (-1-1 1) *
,( _/2(-1 I O) (1 0 1) (-1-1 1) split
,_ -=/2(0 1 1) (-1 0 -1) (-1 -1 1) right weak
a/2( I 0-1) ( 0 1 1) (-1 1 -_) right strong
: a/2( I 0-I) (-1-1 O) (-1 1-I) *
,.. a/2(l I O) (0-1-1) (-1 1-1) *
/
:'" a/2( 0-1 -1) ( 1 1 "0) (-1 1 -1) right weak
a/2( 0-1 1) (-1 0 -1) ( 1 -1 -I) right strong
i
a/2(0 -I I) (-I-I O) ( I-I-I)leftstrong
a/2(I 0 1) (-I-I O) ( 1-I-I)leftweak
1
_2( I I O) (-I0 -I) ( 1-I-I)leftweak
/
J
,,r
ll_@"o
* ; These arc __rc_d into the liquidi
"=_ == - - I II
@,,.
ivll '
27O
1987006957-270
ADVANCED SILICON SHEET
, Calculation of the Density of Dislocations
_- From the K, Sumino's equation of dislocation multiplication
•,-_ dNm = K Ko Nml (Ta-G b SQRT(N_/13 ) r,m+ X ) exp(-Q/kT) dt _A_4 Imams,
where K,Ko,b, _ ,m, X ,k,Q are constm',ts given by K.Sumino
'w
'" Nm's ; dislocation"donsity6t:'
Nml .,xsource density
. Nm2 is the density controlling the back stress
Ta ; applied stresses
T ; temperature
G ; shear modulus
t ; rime
Tht_ possible ways of application of the ,'.quation (A)
(1) Nml and Nm2 are total density of dislocations
_. (2) Nml, is the partial density of dislocations on each slip system and
Nm2 is the total density of dislocaitons.
(3) BothNm's arepartialdensitiesof disloations oneach slipsystem
i'
,/
,.7
",_r
t
"_ I@
2,1
1987006957-271
ADVANCEDSILICONSHEET
• _ Method Calculating Dislocation Density
\ .
:_ Nml Nm2ir ' 4
+,,
,, } (I) J.->RSSi Ill total density =Nml = Nm2dNm -- dNml - dNm2
It
(21 ].->RSSi Ilk total density = _ (Nml)i --Nm2
,dNm2 = (dNm)i
I
(3) I--_.RSSi ._L total density = _ (Nml)i = _ (Nm2,_i
(dNml)i = (dNm2)i
, 272
19B7006957-272
tADVANCED SILICON SHEET
Total Density of Dislocation Using YI, YTOT and DGEAR
INITIRL l"01"qL DENSITT IS 90/I,I,,,,2 LAMOR I5 |.0
2.0
t.8
1.6
1.4
1.2
i.0
0.8
0. S
',1 :,
0,4
0.2
o.o I . -- o'. o' o' o'. " ' '--_-"'o.oo o. ls _o o_._s .6o .:s 9o l.os 1.2o l._Sl._O 4
T
L[GENO; OEN_ _ g. Sg|SE*l? _ .:_.4366E.18
--" _ $.O14_E,L8 _ 8.5915[*t8
/ ....... 1,. l189E.lg ....... |. 3746E-|9
...... I. S324Eo19
.°_ o
_r
w
273
®'f
1987006957-27:3
t ADVANCEDSILICONSHEET
Total Density of Dislocation Using YI, YI and DGEAR
|NITIRL TOTqL DEN$|TT [5 90/_,.m2 .LRHOR I5 1.0
Y ' 4
2.0
• I
1.8
1.6_
1° -_ I
t
1.2
e
1.0
0.8
0.6
0.4
' 84"# Z2 t 4_ft .'t 4",."._; ;;.'.t'.-.'.-_-#_###a# B#W_ t 8........ - - ".'.. • o o
o.o _ , o' o'. ' o' o' l' .o.oo o.15 ._o _s o.so ._5 .9o o_ i _o z._s_._o
T
LEGENO: 01_N58 _ ?. 7220E-11 _ 3. 0888l[- 12
-" _ 5. 4054F-12 - ._ 7. 7220(-!2
• ****.. ! rift It ollr. t ql ....... 1 _ lll_ I_IF ,i, | II
/
"j
]987006957-274
vl ADVANCED SILICON SHEET
-_ Averaging the Ira l in Calculating Density
.1
"Ji
.-_ 5.0 __
: X
.
cm
/
. _" ,
}
I
I
I 2 3
i , _ srr=.,,.,._,.._ O. Z G
0.0 - --i--....... ----->liq-sol,interface
-p
0.0 0.5 1.0 1.5 --> Y (cm)
I )
!
i l!
center edge
cohzrrmI;*,column2 ;,) column3 •12
' 1
./
%
)
__ 275
t
!
ADVANCED SILICON SHEET
Nine Slip Systems .'
.0 |
4
b_rge.r's vector slip plane type of plane
s/2(1 0 1) (-1-1 I) left• o
a/2(O I I) (-1-1 I) left
a/2(-11 0 ) (-1-1 1) left '
a/2 ( 1 0 -1 ) ( -1 1 -1 ) right
a/2(1 1 O) (-1 1-I) right
a/2(O 1 1) (-I 1-1) right
=/2(0 -1 1 ) ( I-I-I) trmrssvcrsc
a/2(I 0 1) ( 1-I-I) transverse
a_2(1 1 0 ) ( 1 -I-I) transverse
,/
0_
276
p=,
1987006957-276
_: ADVANCED SILICON SHEET
Density of DislocationsWhen NO is 1 at x = 0.2 cm
-- . T"r.IP.IE {He DIiJLOCRT'ION, B-.IV'_(,1 ! O-),T'm{l ! O'),N--(-J. ! -J.)
: T-O. 225-S-T'RA, Y'_O. ?$'-OIRHONO . t",i,. 2"_-3QURI_ USING _.¥_. _I'IRF.._S
X
3. 25-' ,l_
=" iC •C •
• 3. 00. ,c .
c o .*
,. 2.75. c •
e:' °" "C • • iC • •
C • *
2. SO-_ •
°- (III) (nr) "2.2S.a •
• p,
2.o.._ •° =b °:C • * ,
u o ,0 •
1.7S0 •
_o :4'
g. :4)
I.SO -°D 4)e ,e
= 4) g
1.2_.r ." :_-"- ,_
4' i
4) •
4) q,
I. 00- 4) •
4) *
4, •
4) •
4, *O..-'/S- 4) ,
4, *
4) •
ql, *
IO •O.SO' _•
II
0.00 ,........ ,....... ,...... , , . .... , ---,--
0 1'00 _JO 3"0"0 400 SO10 109 ?09
Z'I i
UNIT OF X IS CM,.Z IS DENSITY IN 1/M*'2 i
i;
_-.°
- !
._- 277
g! ,
1987006957-277
'1 i
i
ADVANCED SILICON SHEET "-)
Density of Dislocations When NO is 1 at x = 0.2 crn
I
" TI'IPAE_1r 0J'SLOCR.TI.0N,, llaR/2(l 0 1).T,,(1 0 l),lk,,( I -4,-,II
¥-0. 2.2'S-S.TAR. 1'-0.7S-0J;liI_ND , ¥-1.. 2"J'S-S_UNM£ A,
X
_" $.2S-
/
• • O
• 6 O
l 4) a
3.00- - • o
• • O
• • O
• o O
,. i O O
2.'/5- i) O 13_dl,ltl • o °
II 41, Q
41 ti D
i 41 0 1
2. S0. • I li
• Q
; * • Q
o
•
• • D
• ti D i
" " " _'_dl i a.. 2. 00- • 4. o
• 4) o
• e. O
II 4) O
1.75- * * o
• • o
• • o
• • o
• • o
1. SO- , ,i o
i, ,0, O
• O 0
• O. G
• • im
I. 25- • a
"" ° _ _Ig) T• i O if
,.00. ,,.,o"° (III. N .) /I'DO ,_ll, O. iIQ "
0.75- llO .." ' ,l_lm.
OO
•o-" *\/i
;. 0.50- *oeO
,in
•° ._" bO. 25- _ _,
O. 00'
l 41 !i e/ l0 111 14 16 ;8 20 2:_ ill
UNIT OF X IS CM, Z IS DENSITY IN 1/M*'2
i
iI
; 275
, (-:_l
1987006957-278
ADVANCED SILICON SHEET
Density of Dislocations When NO is 1 at x = 0.2 cm
,,0
,-t. . T_PIE OF Oi_u..0cR:rl;0N. e,_l/200 I I),T-tl ! tl.lei-I I -11
, _ TO..2_..-S'=AR. I'-0. ?S.-01_I140_) . T-I. 27S--MIJIIO, VelIIG IW,EItG. S)':l_i ,s
:_ 3.25-
t • i •
_- • • Q
I t • Q3.00- ,, • •
• 4 O"
• • •
• • •
a 4. 0
'*,: 2.75- * * 0 i
• • O
• • 00 I
• 02.50- • , o
• 4 O
el • 0
• • O
• • rl
• 2.25- • • o
m • o
• • 0 ;
• • 0
• • 0
2.00- * * o
• 4) O
• 4* 0
"" R• • 01.?5- • • o .• O 0
4 4, 0
• 4, 0
1.50- * * 0
. 4_ O
• 4 O "
e* "o i
• • 01.25- . * o
I • 0 __t
• O 0
eO 0
me 0
1.00- mr, o
° (nr) *
el O
Ill 0
"° (III)0.'/3- * o• O
'4, Q
4, D
• 0
O. SO . a
• O "'
e.o /
e.O . '1
eOO.2"S, ,o
0.0"0 ,.,......... I ................ I ......... I ......... I ......... _ ......
O S SO IS 20 2S )0
ZI
UNIT OF X IS CM, Z IS DENSITY IN f/M**2
I 279
1987006957-279
fADVANCED SILICON SHEET
Dislocation Multiplication by Stress Averaging Over 0.5 cm
di.
Dislocation Multiplicatior
Factor
[101] [_J,l] L 750 50 1[m] [_l] _ 2,_ 3.5 _ ,
[110] [ill] L 23 I 2
[.I.10._[.h_] ]_ ?50 50 1
rlo:j ELlzJ e lz i _0 ;;m] [m] _ i _-.5 3_ :
Elot: tIiiJT : 3 24
[011] Lill] T 1.5 66 37
_icuhtionssta'te¢[=tx:-O,2c ,
!
s (-TcN _, , ,_.
I I
l I
280
1987006957-280
•_- ". ... . ..'." •
• . _ • •
• "iiV..i;.:..,":i.,.
DENDRITE ' "-" •-'-" ".'. :". • : " '
_\/- ,..... ., . .... , .. ,,
1 _.: , _:" .. -. : ..' "%.', .:.
! ,; _,,'._ .. .;... . " ",%.."':.'i,1Iw,_ :, • • :..., .. ,
TOPVI[_ OF L__J
ETCHEDWEBRIBBON 100microns :
(a) ',
/
t
1
oe '
28!
_ .
- ¢"_I
1987006957-281
5.'N_ "
:t
7, ADVANCED SILICON SHEET
Cz Temperature Dependence
-- • dl_
IE-03s,-"
/
.-, 2G- _O->StL-I • IE-04S_"
o • IE-055¢c"
•-" 21-
i.n
i,i
u_
,,-: 11- zd_ l
IO $e¢. _ •
800 gO0 1000 _IO0 1200
TEMPERATURE(de9.C)
-. Cz Strain Rate Dependence
"1
--1200C
I IIOOC i
• IO00C
0
CL
z 1 o 900C ,,
_J IE1 4
r'_ j ...... _-_ ......../-
,; . j2_oC
.w_ ! , ..... I. I ....... I ....... l
"-t.., ]E-8 IE-5 IE-4 IE-3
' STRAINRATE (Isec)
1
282
1987006957-282
t
" I .i
ADVANCED SILICON SHEET
Cz Temperature Dependence
-, IE-03
;' • IE-04
o • IE-05
O_
M _
IE0 i i J i
, 55 .7 .75 .8 .85
IOOOlT(/DEC. K)
t
Web Ribbon: Strain Rate Dependence
o IO00C
] o IIOOC
5 t ,, 115oc
" I i/ A 1200C
t,D 0
_- IEI
•"_ J No:2_10_G',_'_"
r_ J
m . " I_,00C
;_" IE0 ...... ""n ................ _'"l ....................... n'-_
IE-6 lE-5 IE-4 lE-3
"' STRAINRATE (Isle)
"_¢' " I
283
t
1987006957-283
|
• )
', i '* ADVANCEDSILICONSHEET
• , Web Ribbon: Temperature bependence
.Q
o
0
o_ I
-' z [] STRAINRATE= lE-04
c_. o STRAINRATE- IE-OS
.
° _
.o t
0
•_ t- 4." _,_i_ _- z,,o_"_ '
• .10 "I _= ].Z'_ I0 $8E(_ .i'" _J
tJ3 0 I"
•° o
_ o
IEO , , - , .....'" I '_ '(
•65 . 7 •75 . B ,.8_
IO001T(/de9. K)
4
/ .t
e
' 284
rr
1987006957-284

Analysis of silicon stress/strain relationships

":_ ADVANCED SILICON SHEET N 8 6 " 2 9 _ 9 5_
::_ ANALYSIS OF SILICON STRESS/STRAIN RELATIONSHIPS
I
_. UNIVERSITY OF KENTUCKY
e
_. O. E ;lion ,6
h
= _ V;KK6ij
" _ij E "Jij E + ¢ijPL (1)
where the inelastic strain rate tensor _ij PL is the plastic strain rate and
is
PL f S.. (2)
_ij : 13
.L
where Sij are the deviatoric stress components and where
f = bB _ D_m) e-QlkT _ i
• _T Nm(/J2 _ (3) ' :'i
0
!
I
t '
__. 543 ",-_i_
,_.
t
I q_ _ _
1986019875-529
https://ntrs.nasa.gov/search.jsp?R=19860019923 2020-03-23T07:35:00+00:00Z
ADVANCEDSILICONSHEET
-_,
r
r, 544
®i
198S01 qR7_-_n
ADVANCED SILICON SHEET .t
T_
,'• (@2 _2_xyPL
Y2(exx+Oyy) = - _E_/2T + _ + - 2 • ) E du (8)
_ 0 @y2 @x2 @x_y
t
wherewc is the creep(viscoplastic)portionof the transversedisplacement.
The momentintensityis relatedto the st,_ssby the basicdefinition.
hl2L Mxx= " Oxx z dz
-h12
, The "elasticmoment"H is then
XX f
xx 12_L_ By2
_. whilethe corresponding"inelastic"mment componentis _
2,c h3
: Mxx -_?I[_'Wc_x2 + ½ _y__2_]w (6)
,,/
Sincethe momentsare the same,the displacementsere clearlyrelatedby
D94e o ;)2_e o @2_e o @2_____ee
" Nxx _?- + 2Nxy_ + Nyy _y2
" o _2we o _2we
,.,, + f E h3 o ;t2,,e + 2Nxy +
. C" x-E2- "yy -ry] (g>
I
w
1986019875-531
k
,, ADVANCED SILICON SHEET
2.
we(x,y,t),_g(t)W(x,y) (10)
8
and obtain
"g-xZg= 0 (11)
y;
for the time partand
0
D V4W = N B (I+ f F ) B2W Cl2)12(l_v2)42 Bx¢@xB
t
Henceone can see thatthe inelasticmaterialbehaviorresultsin bucklin_
very much like the elastic case but with the pseudo in plane forces given by
• o _
N B (I+ f E l!_)
.. 12(i+v2)42 (13)
, The separationparameter42 in Eq (13)reflectshow "fast"the lateraldeflec-
2 tionsgrow from someinitialvalue. Clearlythe presenceof f(x,y)iJlthe
numeratorof Eq _13)makes sim,Ir interpretationimpossible_or_2 exceptas
given in Eq (11).
TO obtainValuesfor 42, we use a Galerkinmetl_don Eq (12)and find
that
42 2 h_rF.[fE2 V4_ da (14)
= 3(h-'('_'_-h_r)fyEV4WWda
1986019875-532
'_ ADVANCED SILICON SHEET
' Normal Strain Rate XX Along Y 0 (Centerline)
for T = 1440*Exp (-O.O8X) Width = 6.0 CM "
t
LINE I IS TOTAL. LINE 2 15 ELASTIC, LINE 3
15 PLASTIC AND LINE 4 15 THERMAL STRAIN RATE
i
T 0 _, "-.,- --"
R "/_ *,
_} N -5- : "
R
A
" T -IO-
E
t
° :1111 -
i -20"
i '-2s-
) , I
-_0-
0 I 2 3 4 5 6 ? 8
LENGTH IN CM
,I
547
1986019875-533
ADVANCED SILICONSHEET
The Dislocation Density Contour Plot
for T = 1440- Exp (- O.08X) .
Unit ofXandY = CM, Unit of:,' = 103 PerCM :/-
X
' ,; ,tJ''. l, i I
7.e i , I I i!I I ' I '0 I I
, I I
1 1
!8.1 Ji 1
! I
i ! I
'-° ', i
l t
" i
S.t : e
I iI
° _ i
I 1 I '
I
4.2 ' I
I !
,., j :ll I "-
! . :Ii _ "/ I '_l !
"°,/ il Ii!'..
" //i ///ili i I
...,,/////,,./////
... / /
,., / II//;//, / / /X r I t •l, I I . "_ ./
.1" J .i , ,, ., I.,/ ,
." /_ s / / .. /* ./ l
o.c - * " " ..... '''- :2i..... 1O,O0 f', ?S 1. So ;_ S'.O0 S."_S
¥
t.l[Oi[NOo| .... o. 0,*_ .... O.OTS _--O. lSO _-- 0.225
--O.qO0 _ O,$_S ..... O.4SO _ _- O.S2S
- _- O.800 --_ O.r'S _ O.?SO --_ O.1125
"'' i O, _00 il O. 9T5
J
'i
_48
e
w_.. g / .
]986019875-534
i
.i
---'_;_ ADVANCED SILICON SHEET
l
4
The resultsare
;J T
2
,_!,/cm Nf max OyymaX OxxmaX t_cr t2cr
/cm2 t4Pa MPa Ein m
. 3 2497 -22.64 15.23 .1936(c) .1375(t)
C
.15 I092 -22.65 15.99 .1965(c) .1388(t)
.Of 266 -23.69 17.1 .1982Cc) .1394(t)
D_lOD Largerbackstress A
.3 1811 -22.64 15.41 .1948(c) .1381(t) '
,/
2
Table I
°,_
1986019875-535
t_ ADVANCED SILICON SHEET
if
; The 20th Elastic Buckling Mode
for Westinghouse Temperature Profile
Critical Thickness = 0, 198 ! 7 MM
Unit of X and Y = CM
0.$1
-0. 58
-1, O0 ',
-1
m,,,
G
_,_ 55O
°
1986019875-536
ADVANCED SILICON SHEET
_2hcr
Mode (mm) (sec-I)
[ 1 "i 1 .1936 +.06668 critical.... . case
2 .1375 +.01124
3 .I132 +.00742
4 .lOS1 _.00452
5 .0893 +.00365
6 .08503 +.00]53
7 .0756C +.00698
8 .06755 +.00117 ;
9 .06644 +.006555
lO .0498_ -.0000416
i 11 .04086 +.0000832
T = Tw :
No : .3/cm2 i
6cm x 6cm
i
551 I|
1986019875-537
ADVANCED SILICON SHEET
),2hcr
Mode (mm) (sec -1 )
1 critical
J 1 .1965 +.02937 I case
, 2 .1388 _,._04919
3 .1099 +.003094
4 .1074 +.0018414
5 .091_.6 +.OO16733
6 .08562 +.000734
7 .076924 +.OO1504
8 .06980S +.000597
9 .06800 +.U00367
l0 .062011 +.006800
T :T w
NO = .15/cm 2
'i
6cm x 6cm
m.
-j
-: 552
/" jl
1986019875-538
|ADVANCED SILICONSHEET
The First Positive Buckling Mode
for Westinghouse Temperature Profile
Critical Thickness = 0.761 54 MM
Unit of X _nd Y = CM
' ,87
tADVANCED SILICON SHEET i
hcr
Mode (mm) xZsec .I
I .4204 -1.706x 10-3
2 .3872 -7.164x 10-3
3 3346 -5.623x 10-4
4 .3080 +2.8909x 10-3
[ [critical5 .2789 +4,9641x 10-3 case
6 .2663 -7.8054x 10-4
7 2337 +2.3418x 10-4*
8 .2281 -1.1527x 10-3
I
9 .2044 -6.0790x 10-4
lO .1974 +1.2900x 10-4
II 1801 -2.0870x 10"4 i
12 .1724 +1.0930x 10-4
T = ModifiedEFG
No = 2 x lO'7/cm2
6cm x 6 cm
Run V-0545
8 June
i
r ,, 554
1" !
1986019875-540
T .:
., :_, ADVANCED SILICON SHEET jc
i
The Effectiveness Stress Contour Plot
for Modified EFG Profile
• Unit of X and Y = CM, Z = MPA
: NO = 2Xl0-7/cm 2
×
(
• ,q, o
%
OJ _ | • !
t l
t,
%%
3.9 "''-
wU_m _m _ _ _m_ _ _
17,? "',,.,
..... .__.. ---..,:..._.-._ \
, 0.0 1.5 3,0
T
" LEGEND: Z 1.5 ...... 17.9
------ 34. _ 50. ?
67.1 .... 83.5
99,9 ----- 116.3 ,,
---- 132. '7 149.1
• : 165.5 .... 181.9
.... 198.3 _ 214, ?
,[
1986019875-54
q: ADVANCEDSILICONSHEET
The Normal Stress YY Contour Plot
for Modified EFG Profile
Unit of X and Y = CM, Z = MPA ,
.J[
NO = 2XlO-7/cm 2
X
! , , i
5.8-
I
.... _.__..__..___
...... ____.-_.3 )
0.0 ..... -_
O.O 1.5 3.0
T
LEGEND: Z 215.4 191.8
.... 168.2 144.6
121. O -97.4
..... 73.6 .... 50.2
-26.6 ....... 3.
-------- 20.6 44.2
...... 67.8 ------- 91.4
,:°
1986019875-542
ADVANCED SIL.ICONSHEET
The Normal Stress XX Contour Plot
for Modified EFG Profile
Unit of X and Y = CM, Z = MPA
J
NO = 2XlO-7/cm 2
X
5,8--
3.9
1.9 _ ,r..
. , \
j
O.O .......... , • , , • , ,.... • •
O.O 1.5 3.0
Y
LEGEND: Z ........ 150.0 ...... 137.3
.... 124.6 ..... III.9
-99.2 -86.5
-73. B -B 1. 1
-48.4 , -35.7
-23.0 ..... 10.3
2.4 15.1
|
557 i'_ ;;i
]9860]9875-543
r-t_
ADVANCED SILICON SHEET
The 13:h Buckling Mode
for Modified EFG Profile
Critical Thickness = O, 027877 MM
e
Unit of X and Y = CM
0.64
0.29
-0.07
558
1986019875-544
ADVANCED SILICON SHEET
EDGE CENTER EDGE
Fig. I. Surface profile traee_ i11uscrating typical edge
buckling for ribbon no. 18-102-2 grown at a speed
of 3.0 cmlmin. Traces are taken along the growth
direction, with respect to the width dimension as
marked.
559 _
1986019875-545
ADVANCED SIliCON SHEET
for the case of a 6 cmx 6 cm ribbon pulled at vo = .0005 m/sec. The results
are sho_nin Table III.
H _;o Nf Ntel/cm o E_.,/cm cm/2 /cm2 2 YYmaxHPa _
1.75 ,5 Diverge 4.65 x 108 -151.1.* • _(_ _
1.0 .3 Diverge 3.175 x 108 -101.7"
.25 .3 Diverge .5941 x 104 - 17.4"
.240625 .3 1.06 x 104 .3137 x 104 - 17.8
.2375 .3 1984 .2527 x 104 - 16.8 • ;-/3(3
.225 .3 963 1049 - 15.0
.200 .3 173 174 - 12.1 • ._7_ u
Table Ill
;" The * in the last column indicates these are the elastic stresses,
becauseplastic ones are not obtained.
TNEFG= 1200 -125____x+ 485e"1"75x3
This led ;o divergent solutions under conditions whenthe Westinghouseprofile
did not. Wtth this situation tn mind we considered the famtly of thermal
profiles, defined by
m,,,
..J
" T(x) = 1200 - 125x + 485 e"Hx
,, 3
,,J
J
•_ 560
r
1986019875-546
561
_1_ .
W il .,,.,m_.,,_ i,l,i'llIIIlllll _- _ "-- -....... i
1986019875-547
• _ ,_,._,_P._ -.,,L_" "" _-
ADVANCED SILICON SHEET
, OF _-L .,,'
• rl
t
--- ,,,
' '--- '"' .... ' " ii  sii.-!;s
' ti: .... _ _' _l . l i . ''" .I,,,:',;;:':
• _i
• . . , ' . , ::1 i:::I::
' --_-_----,_,_I_ ...._-- ......._ ......--
-.i_.. ' !, lil, .'1 , . . ....... _!.._.
,1t :._, .-'.?'_ '!' , i , .... ,_ , "_':ii, ....:_' I........ , .... :, _i t , , ' ...... ', ' _ .......
_'___!l _._',_ ' ....... ,.:' ,.,, ......
:: ;'. -. . ._, _.-_..F_ _:,;, - , ,
.._Ii_*_,_,::_.,__i!iI.____..;__i.l,::i,', i,..: .._:.. :'_._:'.. ..,_',::.li::l
' "_ ' ,,',-' - I* ' ....... ' i'.:i J,..._i
I
I+ _" ':1_. il_ *' _ "" '*' _' _; ..... _:" : .... '"
. '__.L-._L.___ . , i ; ,'_ _.... _:. ,:l ......
'" ,_;:: ',.I " -': ........ _....... : ......... _ .... .............
,_ , _,,'_,_,' , ,',._,,, .. . _. i!i_,,!,, I;';iI: •
' ' ....... C ."--i --=.z i;,.=.iL, ' .......... , ...... _1_?,. T? ---:-_
........... IIJ, I ., .];! i',_
562 (_
1986019875-548
ADVANCED SILICON SHEET 'j
Really New Science
Dislocations as part of the stress analysis.
That is N =_ constant! "
l
; New
Creep buckling (lowest mode does not dominate!)
Practical
Elastic very useful
Plastic - residual stress
• cr - _cr (elastic)
Keep N small !,
Very sensitive :'
(ala melting)
4
I
t
i
5,3 I
1986019875-549


Analysis of silicon stress/strain relationships

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