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RECOVERY PROCESSES IN THE
HIGH-TEMPERATURE DEFORMATION OF GERMANIUM, SILICON AND INDIUM
ANTIMONIDE
H. Siethoff
To cite this version:
H. Siethoff. RECOVERY PROCESSES IN THE HIGH-TEMPERATURE DEFORMATION OF GERMANIUM, SILICON AND INDIUM ANTIMONIDE. Journal de Physique Colloques, 1983, 44 (C4), pp.C4-217-C4-225. �10.1051/jphyscol:1983426�. �jpa-00223045�
RECOVERY PROCESSES IN THE HIGH-TEMPERATURE DEFORMATION OF GERMANIUM, SILICON AND INDIUM ANTIMONIDE
H. Siethoff
Physikalisches Institut dev UniversitSt Wiirzbupg, Rontgenving 8, D-8700 Wurzbupg, F.R.G.
RESUME - Les courbes c o n t r a i n t e - d é f o r m a t i o n de Ge, Si e t InSb montrent, indépendam- ment du stade I I I r e s t a u r a t i o n bien connu, e t aux températures élevées, un autre
stade de r e s t a u r a t i o n (stade V ) . Tandis qu'un processus de d i f f u s i o n permet de rendre compte de l ' e x i s t e n c e du stade I I I , un mécanisme de "glissement dévié" a été proposé pour rendre compte de l ' e x i s t e n c e du stade V. Ces i n t e r p r é t a t i o n s sont déduites de l ' a n a l y s e de l a dérivée première des courbes c o n t r a i n t e - d é f o r m a t i o n , c ' e s t à d i r e du c o e f f i c i e n t d'écrouissage e t conduisent f i n a l e m e n t à l a conclusion que l e régime où la l o i - p u i s s a n c e n ' e s t plus s u i v i e e t qui e s t obtenue dans l e stade I I I pour l e s f o r t e s c o n t r a i n t e s , p o u r r a i t également ê t r e gouverné par un phénomène^
de "glissement d é v i é " . Le modèle de traînage des crans de B a r r e t t e t Nix appliqué à l a r e s t a u r a t i o n au stade I I I conduit à des c o e f f i c i e n t s d ' a u t o d i f f u s i o n des mpno- lacunes D = 41 exp ( - 3 , 0 eV/kT) cn^S"1 pour Ge e t D = 0,63 exp (3,7 eV/kTJcni^s"*
pour S i . Ces valeurs c o n f o r t e n t c e l l e s obtenues à p a r t i r d'expériences de d i f f u s i o n de t r a c e u r s .
ABSTRACT - The s t r e s s - s t r a i n curves of Ge, Si and InSb show, besides the well known recovery stage I I I , a t high temperatures a second recovery stage (stage V ) . While stage I I I has been explained i n terms of a d i f f u s i o n - c o n t r o l l e d recovery process, f o r stage V a c r o s s - s l i p mechanism has been proposed. These ideas are corroborated by an a n a l y s i s of the f i r s t d e r i v a t i v e of the s t r e s s - s t r a i n c u r v e s , i . e . the work- hardening c o e f f i c i e n t , which f i n a l l y leads t o the c o n c l u s i o n , t h a t the regime of power-law breakdown, which i s observed i n stage I I I a t high s t r e s s e s , may also be governed by c r o s s - s l i p . The j o g - d r a g g i n g model of B a r r e t t and N i x , when applied t o stage I I I r e c o v e r y , y i e l d s c o e f f i c i e n t s of monovacancy s e l f - d i f f u s i o n of 0=41 exp ( - 0 . 3 eV/kT) cm2 s "1 f o r Ge and D=0.63 exp(3.7 eV/kT) cm2 s "1 f o r S i . These para- meters f a v o r a b l y compare t o those evaluated from t r a c e r d i f f u s i o n experiments.
I - INTRODUCTION
At the Hu'nefeld conference the author / l / reported on f i r s t r e s u l t s concerning the recovery mechanism working i n stage I I I of the s t r e s s - s t r a i n curves of Ge and S i . Since t h a t time the understanding of the high-temperature deformation of these m a t e r i a l s has been improved. The development was promoted by the d e t e c t i o n HI of a
h i t h e r t o unknown second recovery stage i n the s t r e s s o r ) - s t r a i n ( a ) curves a t tempe- r a t u r e s above about 0.8 T (T absolute m e l t i n g p o i n t ) , which i s demonstrated f o r Si by curve (a) i n F i g . l : Beyond stage I I I the s t r e s s increases again forming a stage of l i n e a r hardening (stage I V ) , f o l l o w e d by the second recovery stage (stageV).
The curves (c) t o (e) i n F i g . 1 show the usual t h r e e - s t a g e hardening behaviour a t lower temperatures, w h i l e curve (b) i s of an intermediate t y p e .
The usual way t o i n v e s t i g a t e recovery phenomena i n the deformation curves i s t o de- f i n e / 2 , 3 / a s t r e s s a t the beginning of the actual recovery s t a g e s , i . e . T , , r and T ^ ( c f : , F i g . l ) , and t o examine i t s s t r a i n - r a t e and temperature dependences. The r e s u l t s of such experiments f o r Si,Ge,and InSb are presented i n the next two chapters
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983426
C4-218 JOURNAL DE PHYSIQUE
and a r e d i s c u s s e d i n terms of d i f f u s i o n - c o n t r o l l e d and c r o s s - s l i p r e c o v e r y mechanism f o r
TIII and tv, r e s p e c t i v e l y . An a I t e r n a t i v e methoa /4/ f o r a n a l y s i n g s t r e s s - s t r a i n curves i s t o i n v e s t i g a t e t h e f i r s t d e r i v a t i v e d r / d a a , t h e s o - c a l l e d h a r d e n i n g co- e f f i c i e n t . The a p p l i c a t i o n o f t h i s method t o t h e r e c o v e r y stages I 1 1 and V i n S i and Ge /5/ n o t o n l y c o r r o b o r a t e s t h e c o n c l u s i o n s drawn f r o m ~ 1 1 1 - and r y - a n a l y s i s , b u t a l s o leads t o t h e idea, t h a t t h e regime o f power-law breakdown, w h i c h i s observed i n s t a g e I 1 1 a t h i g h s t r e s s e s , may a l s o b e governed by c r o s s - s l i p . These phenomena w i l l be d e a l t w i t h i n c h a p t e r I V .
F i g . 1 - S t r e s s - s t r a i n curves o f S i a t d i f f e r e n t temperatures and ( i n i t i a l ) s t r a i n r a t e s ; t h e y i e l d p o i n t r e g i o n and s t a g e I a r e o m i t t e d ; t h e d i f f e r e n t d e f o r - m a t i o n stages and t h e s t r e s s e s 50
~III and TV a r e i n d i c a t e d f o r '7
c u r v e ( a ) . E
z E F .
40
Tl°C 1036 15.' l a ) 1300 051
_ I b ) 1200 L 8
I c ) 1000 048 I d ) 1000 1.2 IC) 1000 12
I 1 - ANALYSIS OF THE tIII-DATA
F i g . 2 shows t h e s t r e s s a t t h e o n s e t o f s t a g e 111, n o r m a l i z e d b y t h e shear modulus G, as a f u n c t i o n o f t h e temperature compensated s t r a i n r a t e f o r Ge /2,6,7/, S i /3,8/, and InSb /9,10/. The measurements p r e s e n t e d h e r e c o v e r a temperature range from 0.65 Tm t o t h e me1 t i n g p o i n t ( f o r S i t o 1300 OC) and a s t r a i n - r a t e r a n g e o f more t h a n two o r d e r s o f magnitude. T h i s p l o t i s based on a model /16/ w h i c h o r i g i n a l l y d e s c r i - bes s t e a d y - s t a t e c r e e p o f m a t e r i a l s a t h i g h temperatures by t h e d i f f u s i o n - c o n t r o l l e d c l i m b of edge d i s l o c a t i o n s as t h e u n d e r l y i n g r e c o v e r y mechanism:
DGb Y 3 T n ic = A - (-)
kT Gb . . . . . ( 1 )
Here t h e c r e e p r a t e ic depends on s t r e s s by a power law ( n i s t h e s t r e s s exponent), w h i l e i t s t e m p e r a t u r e dependence i s m a i n l y overned by t h e c o e f f i c i e n t of s e l f - d i f f u s i o n ~ = ~ , e x p ( - Q s D / t ~ ) , where Do and Q s ~ a r e t h e p r e - e x p o n e n t i a l f a c t o r and t h e
-
lo-4
Gc 5 1 JnSb dD/ev 2.8 3 5 I 5 1lrn3rn~' 75 58 38
F i g . 2 - S t r e s s a t t h e b e g i n n i n g o f s t a g e 111, c o r r e c t e d b y t h e s h e a r modulus G, as a f u n c t i o n o f t h e temperature-compensated s t r a i n r a t e iIII f o r Ge /2,6,7/, S i /3,8/, and InSb /9,10/ i n a d o u b l e - l o g a r i thmi c p l o t a c c o r d i n g t o e q u a t i o n (2);
d e v i a t i o n s f r o m t h e s t r a i g h t l i n e s a t h i g h s t r e s s e s a r e a s c r i b e d t o t h e regime o f power-law breakdown; G = { c ~ ~ ( c ~ ~ - c ~ ~ ) / ~ } ~ / ~ , w i t h t h e e l a s t i c m o d u l i t a k e n f r o m r e f s . 111-13/; s t a c k i n g f a u l t e n e r g i e s y from r e f s . /14,15/.
a c t i v a t i o n energy, r e s p e c t i v e l y , o f s e l f - d i f f u s i o n ( k i s B o l tzmann's c o n s t a n t ) . The c r e e q r a t e i s c o r r e c t e d by t h e s t a c k i n g - f a u l t energy y ( 5 8 m~m-2, 75 m~m-2, and 38 mJm- f o r S i /14/, Ge /14/, and InSb /15/, r e s p e c t i v e l y ) ; b i s t h e magnitude o f t h e Burgers v e c t o r , and A i s a c o n s t a n t .
F o r t h e i n t e r p r e t a t i o n o f s t a g e I11 r e c o v e r y e q u a t i o n ( 1 ) was used /2,3,6/ i n i n v e r - t e d form, r e p l a c i n g T by '111 and ic by iIII ( t h e s t r a i n r a t e a s s o c i a t e d w i t h ~ 1 1 ~ ) :
w i t h A* = ( ~ T / A D , G ~ ) ' / ~ ( G ~ / ~ ) ~ / ~ .
F i g . 2 demonstrates t h a t t h i s f o r m a l i s m s u i t a b l y d e s c r i b e s t h e T ~ ~ ~ - d a t a f o r n e a r l y t h e whole s t r e s s range. Only f o r t h e h i g h e s t s t r e s s e s t h e r e a r e d e v i a t i o n s f r o m t h e power law. T h i s regime o f t h e s o - c a l l e d power-law breakdown, w h i c h i s a l s o observed i n c r e e p experiments f o r most m a t e r i a l s , w i l l b e f u r t h e r d i s c u s s e d i n c h a p t e r I V . E q u a t i o n ( 2 ) has been f i t t e d t o t h e d a t a shown i n F i g . 2 f o r Ge, S i , and InSb.
The o p t i m a l v a l u e s o f QSD and n a r e g i v e n i n Table 1 i n t h e f i r s t 1 i n e (Do cannot b e determined w i t h o u t knowing t h e c o n s t a n t A ) . The magnitude o f t h e s t r e s s exponent n i s i n accordance w i t h t h e creep model used h e r e /16/. The ~ ~ ~ - v a l u e s , w h i c h have a l s o been used f o r p l o t t i n g Fig. 2, a r e i n a s a t i s f a c t o r y , though n o t p e r f e c t agreement w i t h t h e a c t i v a t i o n e n e r g i e s o f monovacancy s e l f - d i f f u s i o n f r o m t r a c e r experiments ( c f . T a b l e 1 ) . As t h e d i f f u s i o n o f t h e I n and Sb atoms i s a s e q u e n t i a l process f o r t h e c l i m b mechanism i n InSb, t h e d i f f u s i o n parameters g o t f r o m t h e ~ 1 1 1 - a n a l y s i s have t o b e a s c r i b e d t o t h e s l o w e s t moving s p e c i e s ( I n ) . F o r S i t h e s e l f - d i f f u s i o n f r o m t r a c e r experiments /17/ a t temperatures above a b o u t 1050 OC shows an a c t i v a t i o n energy n e a r 5 eV, which has been a t t r i b u t e d t o i n t e r s t i t i a l d i f f u s i o n . A change i n mechanism has been found, however, f o r t h e d i f f u s i o n o f Ge i n S i around 1050 OC /19/, w i t h a c t i v a t i o n e n e r g i e s o f 3.93 eV and 4.97 eV f o r t h e low- and t h e h i g h - t e m p e r a t u r e
regime, r e s p e c t i v e l y . The 1 ow-temperature value, w h i c h has been a s c r i b e d t o mono- vacancy d i f f u s i o n , has been adopted i n Table 1. The '111-measurements, which c o v e r t h e temperature range from 800 OC t o 1300 OC, do n o t show any change i n mechanism /8/. Therefore t h e a c t i v a t i o n energy measured h e r e i m p l i e s , t h a t s t a g e 111 r e c o v e r y i s governed by a vacancy mechanism f o r t h e whole t e m p e r a t u r e range.
C4-220 JOURNAL DE PHYSIQUE
Table 1 - Results o f the ~III- and r ~ ~ ~ ~ - a m l y s i s . A c t i v a t i o n energies f o r mono- vacancy s e l f - d i f f u s i o n from t r a c e r experiments a r e given f o r comparison; f o r InSb t h e value o f t h e ( s l o w e r ) I n - d i f f u s i o n i s used.
Method Ge S i InSb
1
QsD/eV n Ref.I
QsD/eV n Ref.I
I?"/ev n Ref* f i t t o the data from r e f s . /3,8/; ** f i t t o t h e data from r e f s . /9,10/.
r1 I 1- a n a l y s i s '111m- a n a l y s i s tracer diffusion
Measurements o f TI I, though i n l i m i t e d temperature and s t r a i n - r a t e ranges, have a l s o been performea by Alexander and Haasen /21/ (who f i r s t proposed t h e climb mecha- nism) and by Kojima and Sumino /22/ f o r Ge and by Mousset /23/ f o r S i . The accordance w i t h t h e parameters o f Table 1 i s r a t h e r good. The p r e f a c t o r , however, shows l a r g e r v a r i a t i o n s . The reason may be sought i n d i f f e r e n t experimental procedure.
I 1 1 - ANALYSIS OF THE TV*
2.8 3.7 /2/
2.9 3.4 / 5 /
3.0 - 8
The i n t e r p r e t a t i o n o f TV as a f u n c t i o n o f temperature and t h e associated s t r a i n r a t e
i v i s n o t as s t r a i g h t - f o r w a r d as t h a t o f f o r two reasons: The temperature range a c c e s s i b l e t o measurements i s s m a l l e r ( e s p e c ~ a l l y f o r S i ) and t h e s c a t t e r o f t h e data i s increased on account o f t h e specimen d i s t o r t i o n s a t the r e l a t i v e l y h i g h de- formation degrees a t r v . F.ig. 3 shows t h e data f o r Ge /2/ together w i t h creep measurements /24/, which w i l l be discussed l a t e r . I n t h i s double-logarithmic p l o t , the t v ( 6 ~ ) - r e l a t i o n seems t o be curved, i . e . n o t t o obey a power law. I n t h e i r f i r s t paper /2/, however, t h e authors approximated these measurements by s t r a i g h t p a r a l l e l l i n e s ( a procedure n o t u n r e a l i s t i c because o f t h e data s c a t t e r i n g ) and used t h e same formalism f o r t h e i n t e r p r e t a t i o n o f r V ( i V , ~ ) as f o r m e r l y f o r rIII(ciI~~,T). By t h i s way an a c t i v a t i o n energy o f 3.1 eV and a n-value o f 5 was obtained, which arereason- a b l e parameters i n t h e framework o f t h a t formalism ( c f . chapter 11).
F i g . 3 - Double-logarithmic p l o t o f s t r e s s vs. s t r a i n r a t e a t d i f f e r e n t tempera- tures f o r t h e r v ( i V , ~ ) - d a t a /2,25/ and f o r the measurements o f steady-state creep /24/ i n Ge.
3 . 5 3.5 *
3.7 3.4 /5/
3.9 - 1
1.5 3.8 **
- - -
1.8 - /20/
S i m i l a r i n v e s t i g a t i o n s on S i and InSb, however, q u e s t i o n e d t h i s view, and arguments became e v i d e n t , t h a t s t a g e V m i g h t b e c o n t r o l l e d by a c r o s s - s l i p mechanism /25/.
From t h e two c r o s s - s l i p t h e o r i e s /26,27/, which can b e compared t o macroscopic de- f o r m a t i o n experiments, t h a t o f W o l f /26/ c o u l d be r u l e d o u t f o r t h e i n t e r p r e t a t i o n o f t h e r V ( a v , T ) - d a t a , w h i l e t h e c r o s s - s i i p model o f E s c a i g /27/ seemed t o b e a s u i t - a b l e approach t o t h e s e experiments /25/.
I n E s c a i g ' s model t h e f u n c t i o n r V ( & , T ) may be w r i t t e n as f o l l o w s : Method
e q u a t i o n s ( 3 ) and ( 4 ) c r o s s - s l i p o f s i n g l e screw d i s l o c a t i o n s /28/
a n a l y s i s o f x v /2,25/
a n a l y s i s /25/
o f creep d a t a /4/
a n a l y s i s /25/ o f c r e e p d a t a /29,30/
a n a l y s i s o f TV
t o g e t h e r w i t h h i g h - s t r e s s d a t a
O f =111
w i t h t h e parameters l n ( i o x l s ) = 2 4 + 2 and -co=y/ab (a.3). The maximum c r o s s - s l i p energy E0 i s g i v e n by
Q
G~ b 4 Gb 1/2
E; = ( I n ... ( 4
I n e q u a t i o n ( 3 ) t h e s t r e s s ~v i s p r o p o r t i o n a l t o t h e temperature and t o t h e l o g a r i t h m o f t h e s t r a i n r a t e , i n accordance w i t h t h e c u r v a t u r e of t h e s t r e s s vs. s t r a i n - r a t e r e l a t i o n i n t h e d o u b l e - l o g a r i t h m i c p l o t o f F i g . 3. U s i n g t h e s t a c k i n g - f a u l t e n e r g i e s i n t r o d u c e d e a r l i e r /14/, ~8 and ro can b e c a l c u l a t e d f r o m t h e above e q u a t i o n s and a r e 1 i s t e d i n t h e f i r s t l i n e o f T a b l e 2. F o r comparison, E and ro d e r i v e d f r o m t h e c r o s s - s l i p b e h a v i o r o f s i n g l e screw d i s l o c a t i o n s /28/ a r e i v e n i n t h e second l i n e . E q u a t i o n ( 3 ) has been f i t t e d w i t h t h e t h r e e f r e e parameters EQ, ro and i0 t o t h e rv- d a t a o f Ge /2/ and S i /25/; t h e o p t i m a l values a r e p r e s e n t e d i n t h e t h i r d l i n e o f T a b l e 2. F o r Ge t h e r e i s a f a i r agreement w i t h E s c a i g ' s t h e o r y . F o r S i t h e agreement i s p o o r e r f o r reasons mentioned above; t h i s problem w i l l b e d e a l t w i t h a g a i n i n c h a p t e r I V .
Ge
E;/~V 1 n(;,xls) ro/Nmm -2
4.3 24 62.5
3.9 - 22.7
3.8 27.2 112
3.95 28.9 460
4.5 25.3 349
3.5 24.3 114
Measurements o f s t e a d y - s t a t e creep p u b l i s h e d i n t h e l i t e r a t u r e /24,29,30/ have a l s o been e x p l a i n e d b y c r o s s s l i p /25/. A f i t o f E s c a i g l s model t o these d a t a ( r e p l a c i n g
TV by t h e creep s t r e s s r and iV by t h e creep r a t e ac i n e q u a t i o n ( 3 ) ) y i e l d s t h e parameters a l s o l i s t e d i n T a b l e 2. The h i g h t o - v a l u e s have been e x p l a i n e d /25/ i n terms o f t h e h a r d {Ill) o r i e n t a t i o n o f t h e creep specimens and o f t h e o r i e n t a t i o n dependence h i d d e n i n E s c a i g ' s t h e o r y . F i g . 3 demonstrates t h e c o i n c i d e n c e between TV and one s e t o f t h e s e c r e e p d a t a .
S i
E;/~v l n ( i , x l s ) io/Nmm - 2
7.4 2 4 49.7
6.1 - -
4.4 21 - 9 123
4.9 - 296
5.6 25.3 338
4.9 26.1 152
JOURNAL DE PHYSIQUE
Fig. 4 - Work-hardening c o e f f i c i e n t a as a f u n c t i o n o f t h e shear s t r e s s f o r S i , s t a r t i n g a t the onset o f stage 111; the corresponding s t r e s s - s t r a i n curves are shown i n F i g . 1; the maximum s t r e s s f o r stage 111, TIII~, i s d e f i n e d by the l i n e a r e x t r a - polation @ - t o .
IV - ANALYSIS OF THE WORK-HARDENING COEFFICIENT
An a1 t e r n a t i v e method t o 3nalyse s t r e s s - s t r a i n curves, f i r s t a p p l i e d t o metals /4/, i s t o i n v e s t i g a t e t h e work-hardening c o e f f i c i e n t O = d d d a . I t w i l l be shown i n t h e following, t h a t t h i s method leads t o an improvement o f the QSD-values obtained from stage I 1 1 a n a l y s i s and o f t h e c r o s s - s l i p parameters f r a n t v - a n a l y s i s f o r S i .
Fig. 4 shows, s t a r t i n g a t t h e onset o f stage 111, e a s a f u n c t i o n o f t h e shear s t r e s s f o r the deformation curves o f F i g . 1. A t h i g h temperatures, i .e. i n the curves ( a ) and (b), one c l e a r l y d i s t i n g u i s h e s between the deformation stages 111, I V , and V.
W i t h i n stage 111 one always f i n d s a r e l a t i o n 8-(1-r/rII g ) f o r S i and Ge, and a maxi- mum s t r e s s f o r stage 111, r l ~ m can be d e f i n e d by the { I n e a r e x t r a p o l a t i o n 0+0. The s r r e r s ~ 1 1 ~ . has been anaiysed i i t h respect t o i t s temperature and s t r a i n - r a t e
( a l l I m ) dependence /5/. The r e s u l t was a s t r i k i n g s i m i l a r i t y w i t h t h e r111-data d i s - cussed i n chapter 11. Therefore, equation ( 2 ) has been compared t o the T~II~(~III~)- data, r e p l a c i n g irII by a I I I m and TIII by 'IIIm; the optimal values o f the f i t s a r e given i n Table 1 i n t h e second l i n e . I t i s s a t i s f a c t o r y t o see t h a t t h e QSD-values obtained f o r S i and Ge a r e now close t o t h e values found i n t r a c e r experiments. The '111~-data, which cover temperature ranges from 640 OC t o 920 OC f o r Ge and from 1000 OC t o 1300 OC f o r S i , a r e shown i n F i g . 5 i n a p l o t s i m i l a r t o F i g . 2. -
The i n v e s t i g a t i o n o f t h e maximum s t r e s s tIIIm thus corroborates the conclusion drawn i n chapter 11, t h a t i n S i and Ge t h e same recovery mechanism, i .e. t h e d i f f u s i o n - c o n t r o l l e d c l i m b o f edge d i s l o c a t i o n s , operates n o t o n l y a t t h e beginning b u t a l s o i n t h e whole stage 111.
R e f e r r i n g again t o F i g . 4, t h e r e i s no l i n e a r r e l a t i o n s h i p between t h e work-harden- i n g c o e f f i c i e n t and the shear s t r e s s i n stage V. I n t h e curves ( c ) and ( d ) stage V enlarges a t t h e expense o f stages I V and V. I n curve (e), f i n a l l y , the beginning of stage I11 seems t o be no longer governed by t h e climb mechanism working i n stage I 1 1 a t h i g h temperatures, b u t by t h e recovery mechanism r e s p o n s i b l e f o r stage V, i .e.
cross s l i p . On the o t h e r hand, TIII evaluated from curve ( e ) (and from o t h e r curves g o t a t h i g h s t r e s s l e v e l s ) belongs t o the regime o f power-law breakdown, which i s
p l o t according t o equation ( 2 ) ; G and y a s i n F i g . 2 .
Fig. 6 - S t r e s s e s T I I I and
TV as a function of t h e tem- p e r a t u r e a t a f i x e d s t r a i n r a t e f o r Si and Ge. A t low temperatures t h e TI 1 d a t a i n t h e regime of power-Lw break- down approximate t h e 1 i n e a r e x t r a p o l a t i o n of t h e TV-data.
C4-224 JOURNAL DE PHYSIQUE
shown i r i F i g . 2 a t h i g h s t r e s s e s ( c f . a l s o c h a p t e r I I ) . These f i n d i n g s i m m e d i a t e l y l n a d t o t h e conc!usion, t h a t t h e regime o f power-law breakdown s h o u l d b e governed b y c r o s s s l i p . T h i s i d e a i s c o n f i r m e d by F i g . 6. Here t h e s t r e s s e s TIII and a r e p i o t t e d l i n e a r l y as a f u n c t i o n o f t h e t e m p e r a t u r e a t a f i x e d s t r a i n r a t e . I n Ge t h e -cy(T)-curve i s s t r a i g h t as p o s t u l a t e d by E s c a i g ' s theory, w h i l e t h e TIII(T)-curve n e c e s s a r i ! ~ i s c u r v e d i n t h i s p l o t . A t temperatures below a b o u t 600 OC, where t h e power l a w breaks down, t h e ~ 1 1 1 - d a t a a p p a r e n t l y approximate t h e l i n e a r e x t r a p o l a t i o n t a l o w e r temperatures o f t h e Tv-data, thirs c o r r o b o r a t i n g t h e above i d e a ; f o r S i t h e s i t u a t i o n i s l e s s c l e a r (see, however, b e l o w ) .
Nhat remains t o do i s t o compare E s c a i g ' s t h e o r y , i . e . e q u a t i o n ( 3 ) , t o t h e r v - d a t a t o g e t h e r w i t h t h e ~ 1 1 1 - d a t a from t h e regime of power-law breakdown. I t i s s a t i s f a c - t o r y t o see t h a t such a f i t can b e s u c c e s s f u l l y performed, w i t h t h e parameters g i v e n i n t h e l a s t 1 in e of T a b l e 2. The parameters f o r Ge have n o t changed \jery much, i f compared t o those f r o m t h e T v - a n a l y s i s . The parameters f o r S i , however, have improved a p p r e c i a b l y , t h u s subsequently j u s t i f y i n g t h e e x t r a p o l a t i o n o f t h e r v - d a t a f o r S i i n F i g . 6.
V - THE RECOVERY MODEL OF BARRETT AND NIX
Besides t h e creep t h e o r y used i n c h a p t e r I 1 f o r t h e i n t e r p r e t a t i o n o f t h e ~ 1 1 1 - measurements, i n p r e v i o u s works /2,3,6/ a l s o t h e r e c o v e r y model of B a r r e t t and N i x /31/ has been discussed. T h i s model, which i s based on t h e d i f f u s i o n - c o n t r o l l e d d r a g g i n g o f j o g s on screw d i s l o c a t i o n s , can b e c h a r a c t e r i z e d by t h e f o l l o w i n g r e l a - ti on :
Here h i s t h e average j o g s p a c i n g and m i s a c o n s t a n t . I t has been shown /2,3,6/ t h a t t h i s model p r o v i d e s f o r a s u i t a b l e d e s c r i p t i o n of t h e T ~ I I - d a t a , i n c l u d i n g those f r a n t h e regime o f power-law breakdown. The c o n c l u s i o n s drawn i n t h e p r e c e d i n g chap- t e r , where power-law breakdown has been a s c r i b e d t o c r o s s s l i p , does n o t on p r i n c i p l e d i s c r e d i t t h e t h e o r y o f B a r r e t t and N i x : A f i t o f e q u a t i o n 5 ) t o thetIIIm-data
y i e l d s values f o r t h e a c t i v a t i o n energy of s e l f - d i f f u s i o n Q ~ D o f 3.7 eV and 3.0 eV f o r S i and Ge, r e s p e c t i v e l y , and values f o r t h e p r e - e x p o n e n t i a l f a c t o r Do o f 0.63 cm2s-1 f o r S i and o f 4 1 cmzs-1 f o r Ge. These parameters f a v o r a b l y compare t o those e v a l u a t e d f r o m t r a c e r experiments f o r monovacancy s e l f - d i f f u s i o n /18,19/.
From t h e f i t t h e parameter m emerges t o b e 2.2 f o r b o t h m a t e r i a l s . As t h e s t r e s s term t m e n t e r s e q u a t i o n ( 5 ) v i a t h e d i s l o c a t i o n d e n s i t y , t h i s v a l u e i n d i c a t e s a n e a r l y q u a d r a t i c a l s t r e s s dependence o f t h e d i s l o c a t i o n d e n s i t y , i n accordance w i t h d i r e c t measurements o f t h i s r e l a t i o n s h i p . F i n a l l y , f o r t h e j o g s p a c i n g t h e f i t y i e l d s h = l l b f o r Ge and h = 17b f o r S i .
V I - COMPARISON TO METALS
I n v e s t i g a t i o n s of t h e work-hardening c o e f f i c i e n t of f c c m e t a l s /4,32/ a t h i g h tempe- r a t u r e s r e s u l t e d i n B ( T ) - r e l a t i o n s h i p s , w h i c h were r a t h e r s i m i l a r t o t h o s e found f o r S i and Ge, b u t which were n o t r e a l l y understood u n t i l now. The s i m i l a r i t y o f o ( r ) now i n d i c a t e s t h e s t r e s s - s t r a i n curves o f f c c m e t a l s a l s o t o b e c h a r a c t e r i z e d by f i v e d e f o r m a t i o n stages i n t h e h i g h - t e m p e r a t u r e regime. An e v a l u a t i o n /5/ o f TI^^^ f r o m B(-c)-data f o r g o l d /4/ a c c o r d i n g t o e q u a t i o n ( 2 ) y i e l d e d parameters w h i c h were com- p a t i b l e w i t h a d i f f u s i o n - c o n t r o l l e d r e c o v e r y mechanism. A s i m i l a r c o n c l u s i o n has been reached e a r l i e r /33/ f o r A1 f r o m an e v a l u a t i o n o f TIII from p u b l i s h e d s t r e s s - s t r a i n c u r v e s /34/ a t h i g h temperatures.
V I I - CONCLUDING REMARK
A t temperatures c l o s e t o t h e me1 t i n g p o i n t t h e s t r e s s - s t r a i n curves o f S i and Ge show new f e a t u r e s , w h i c h may b e e x p l a i n e d by t h e occurence o f a t h i r d r e c o v e r y stage.
The narrow range o f temperature and s t r a i n r a t e where t h i s e f f e c t can b e observed, however, does n o t a l l o w any i n t e r p r e t a t i o n .
The a u t h o r i s i n d e b t e d t o P r o f . W. S c h r o t e r and Dr. H. G. B r i o n f o r t h e i r engaged and continuous cooperation, t o P r o f . P. Haasen f o r p r o v i d i n g t h e f a c i l i t i e s o f h i s i n s t i t u t e and t o Dr. W. Blum f o r h i s s u g g e s t i o n t o i n v e s t i g a t e t h e work-hardening c o e f f i c i e n t .
REFERENCES
/1/ S i e t h o f f , H., J. Physique 40 (1979) C6-177.
/2/ B r i o n , H.G., S i e t h o f f , H. , a n d S c h r o t e r , W., P h i l . Mag. A 43 (1981) 1505.
/3/ S i e t h o f f , H. and S c h r o t e r , W., P h i l . Mag. A 37 (1978) 711.-
/4/ Mecking, H., Work Hardening i n Tension and F z i g u e , A. W. Thompson, ed. (TMS- AIME) 1977, p.67.
/5/ S i e t h o f f , H. and S c h r o t e r , W., S c r i p t a Met., March 1983.
/6/ B r i o n , H.G., S c h r o t e r , W., and S i e t h o f f , H., I n s t . Phys. Conf. Ser. 5 (1979) 508.
/7/ Schafer, S., Diploma Thesis, G o t t i n g e n (1963).
/8/ Schroter, W., Srion, H.G., and S i e t h o f f , H., Phys. S t a t . S o l . ( a ) 52 (1979) K165.
/9/ Schafer, S., Alexander, H., and Haasen, P., Phys. S t a t . S o l . 5 ( 1 x 4 ) 247.
/ l o / S i e t h o f f , H. and B r i o n , H.G., unpublished r e s u l t s .
/11/ Burenkov, Yu.A., Nikanorov, S.P., and Stepanov, A.V., Sov. Phys. S o l i d S t . 12
(1971) 1940.
/12/ Burenkov, Yu.A., Nikanorov, S.P., Sov. Phys. S o l i d S t . 16 (1974) 963.
/13/ P o t t e r , R.F., Phys. Rev. 103 (1956) 47.
/14/ G o t t s c h a l k , H., J. P h y s i q u e s (1979) C6-127.
/15/ G o t t s c h a l k, H., Patzer, G., and Alexander, H., Phys. S t a t . S o l . ( a ) 45 (1978) 207.
/16/ Mohamed, F.A. and Langdon, T.G., J. Appl. Phys. 3 (1974) 1965.
/17/ Mayer, H. J., Mehrer, H., and Maier, K., I n s t . Phys. Conf. Ser. 3 1 (1977) 186.
/18/ Widmer, H. and Gunther-Mohr, G.R., H e l v . Phys. Acta 34 (1961) 6 3 .
/19/ H e t t i c h , G., Mehrer, H., and Maier, K., I n s t . Phys. E n f . Ser. 46 (1979) 500.
/20/ Eisen, F.H. and B i r c h e n a l l , C.E., A c t a Met. 5 (1957) 265.
/21/ Alexander, H., and Haasen, P., A c t a Met. 9 (T961) 1001.
/22/ Kojima, K. and Sumino, K., C r y s t a l L a t t i c e D e f e c t s 2 (1971) 147; and p r i v a t e connnunication.
/23/ Mousset, C., Thesis, P o i t i e r s (1979).
/24/ B e t e k h t i n , V . I . and Bakhtibaev, A.N., Sov. Phys. Dokl. - 14 (1970) 1007.
/25/ S i e t h o f f , H., P h i l . Mag. A (1983), i n t h e p r e s s . /26/ Wolf, H., Z. N a t u r f o r s c h g . 15a (1960) 180.
/27/ Escaig, B., D i s l o c a t i o n Dynamics, A.R.Rosenfield e t a l . , eds., (McGraw H i l l ) 1968. o. 655.
/28/ ~ o l l e r ; H.-J., Ewaldt, H., and Haasen, P., Phys. S t a t . S o l . ( a ) 55 (1979) 469.
/29/ Myshlyaev, M.M., N i k i t e n k o , V. I., and Nesterenko, V.I., Phys. S t a t . S o l . 2
(1969) 89.
/30/ ~ y s h l y a e v , M.M. and Khodos, I.I., Phys. S t a t . S o l . ( b ) 3 (1971) 83.
/31/ B a r r e t t , C.R. and Nix, W.D., Acta Met. 13 (1965) 1247.
/32/ N i c k l a s , B. and Mecking, H., Proc. ICSMA-5, P. Haasen e t a l . , eds. (Pergamon) 1979, p. 351.
/33/ S i e t h o f f , H., Phys. S t a t . S o l . ( a ) 62 (1980) K13.
/34/ Howe, S., Liebmann, B., and Lucke, iTT, Acta Met. - 9 (1961) 625.
