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Theory of creep of germanium crystals
Van Bueren, H . G . 1959
Physica 25 775-791
THEORY OF CREEP OF GERMANIUM CRYSTALS b y H. G. V A N B U E R E N Philips Research Laboratories, N.V. Philips Gloeilampenfabrieken, Eindhoven, Nederland
Synopsis To explain the shape of the creep curves of germanium single crystals loaded in tension and in bending, a simple kinetic model is proposed, in which the dislocations are generated by (surface) sources and move with a uniform velocity over their glide planes. In this model a quantitative interpretation of the parameters of the creep curve in terms of the velocity of the dislocations, the incubation time of the sources and the density of the sources is possible. From the observations at "high" stress levels the velocity of dislocations in the germanium lattice can be determined; from those at "low" stress levels the rate of generation of the dislocations from the sources. The observed stress and temperature dependence of the creep process leads to similar dependences of incubation time and velocity. These dependences are used to form a quantitative theory of dislocation production and motion in the germanium lattice. This theory is shown to reflect semi-quantitatively various observed peculiarities of the creep phenomenon. The influence of other dislocations and of oxygen as an impurity on the elementary creep process can now also be qualitatively understood.
1. Creep o/ diamond structure materials. Crystals of g e r m a n i u m , silicon, i n d i u m antimonide, a n d p r e s u m a b l y o t h e r d i a m o n d s t r u c t u r e c o m p o u n d s as well, show a peculiar form of creep when loaded in tension, compression or b e n d i n g at a t e m p e r a t u r e which is a b o u t half of the (absolute) melting t e m p e r a t u r e . Several publications h a v e a p p e a r e d in the last few years dealing with this p h e n o m e n o n 1-5), b u t these were all based on relatively few e x p e r i m e n t a l data. I n view of the wide scatter t h a t i n v a r i a b l y occurs between different specimens tested u n d e r c o m p a r a b l e conditions (such a scatter is logically connected with the characteristics of the d e f o r m a t i o n process, see section 3) a reliable c o m p a r i s o n of these d a t a with a n y theoretical predictions could n o t . b e made. P e n n i n g a n d d e W i n d 6) h a v e r e c e n t l y o b t a i n e d a v e r y extensive set of experimental d a t a on bent g e r m a n i u m crystals, however, a n d this enables us to acquire a more q u a n t i t a t i v e insight in t h e n a t u r e of the d e f o r m a t i o n process. (Their p a p e r wit1 be indicated b y the a b b r e v i a t i o n P W in the following). I n the present p a p e r a detailed theoretical description of the e x p e r i m e n t a l - - 775 - -
776
~. G. VAN B U E R E N
data obtained both b y P W and b y v a n B u e r e n 4) is attempted. Section 9. deals with the form of the creep curve; the simplest possible dislocation model which can explain the observed time dependence of the creep phenomenon is introduced. With the aid of this model observed parameters can now be translated into properties of the individual dislocations in the germanium lattice. The variation of these properties with the temperature and especially with the loading stress is investigated in section 3. This enables us in section 4 to speculate on the generation process of the dislocations and the kinetics of these imperfections moving through the diamond lattice. Finally, in section 5, the effect of other dislocations and of chemical impurities like oxygen on the motion and generation of dislocations will be discussed.
z~ 2
Bending: __
O~ 7_=/+~0oC;, O.=13kg/rnm 2 qm=2turn ~J dislocation free ltz =lOS~rain
150
0.1°/o-~-~
55
0
I0
,/p
N
~ 't (m/n)
Fig. 1. Creep c u r v e of g e r m a n i u m single b e n t at 440°C. The axis of t h e crystal bar (with a square cross-section of 4 m m 2) was along t h e <111 > direction, t h e stress in t h e o u t e r layer was 13 kg/mm~. Deflection is plotted against t i m e ; 1/z deflection corresponds to a strain in t h e o u t e r layer of 0.001%. D o t s : e x p e r i m e n t a l points o b t a i n e d b y P W a), d r a w n c u r v e : theoretical curve.
2. Kinetics o/the creep process. In bending as well as in tensile tests the general form of the creep curve is the same: a slow initial period is followed b y an interval in which the creep rate is remarkably constant. This interval is finally terminated by the onset of work-hardening, a process which will not be discussed here. Figs. 1 and 2 illustrate two typical observed curves. P W described their curves b y three parameters, namely tp, a measure for
THEORY OF CREEP OF GERMANIUM CRYSTALS
777
the length of the initial period, z~, the deflection at t~ and a, the ultimate stationary creep rate. The definition of these parameters follows immediately from the figures. We shall use this characterization, also for the extension curves, throughout this paper.
900 2
P
Extension: / T=~?O°C; (T=6.0kg/mm2 =2ram dislocation density _ 7OO _ _ --3xlOZ~cm~_ 2 -~,, ltt= 3x I0- strain 8OO
2%
6O5
505 /.,00
2O5
,/
lOG
~p o
lo,
ts tp
20
30 = t(min)
Fig. 2. Creep curve of g e r m a n i u m single crystal stressed in tension at 470°C along a <110> direction. Tensile stress 6 k g / m m s, round cross-section 3 m m 2, 1/, e x t e n s i o n = 0.003% strain. Dots are e x p e r i m e n t a l points o b t a i n e d b y V a n B u e r e n 4), t h e d r a w n c u r v e is the theoretical curve.
To explain the general form as observed, the following model is proposed. A number of dislocation sources is present in the material. To the nature of these sources we shall revert, in section 5, presumably they are in some way connected with the surface. In this section it is sufficient to call their density per unit of volume N cm -3. On application of a stress (at an elevated temperature) the sources emit dislocations in the form of loops that expand with a vetocity v. When a loop has moved a sufficient distance b away from the dislocation source a new loop can be emitted. The distance ~ is determined by the condition that the backstress excerted by the loop on the source has dropped sufficiently below the critical shear stress ~er of the source to enable the latter to work again. This critical shear stress can be estimated from the necessary value of the applied stress (see below) as a few kg/mm 2. As ~ m Gb/~er, one thus finds, inserting for the shear modulus G and Burgers vector b the values pertaining to germanium, ~ m (1-2) × l0 -4 cm. We shall take simply b = 1 Ft in the following. Even when the preceding Physica 25
778
H.G. VAN
BUEREN
loop is 1 # away, and the shear stress has again reached about the critical value, the generation of a new loop m a y still require in principle an activation period (incubation time) O. Thus loops will be produced b y the source in regular intervals of time to = ~/v + O,
(1)
and they will all expand with the velocity v. It follows easily that the total length of dislocation A in the material increases with time as A = (N/to) ~rvt2.
(2)
At each moment the rate of strain, ~, contributed to b y these expanding dislocation loops, is: = Abv;
hence ~_
~rNbv 2
t 2.
to
which after integration yields a cubic time law for the creep process. As is discussed b y PW, such a cubic time law describes the initial part of the creep in a satisfactorily manner. However, the expansion of the loops does not continue indefinitely, nor does their contribution to the strain rate increase for ever. In the tensile specimens, the dislocations can only travel on their glide plane over a distance not greater than the cross dimension d of the crystal. In the most favourable case the dislocations are generated at one side surface and travel right through the crystal to the other side. Once a distance of the order of this maximum distance has been covered, thus after a time ts = d / v ,
(3)
the efficiency of the loop in contributing to the strain rate decreases. Formula (2) thus holds only for t < t,. For t > ts a situation gradually builds up in which the dislocation density becomes stationary: the regular periodic emission of new loops b y the source being compensated exactly b y the loss of dislocations at the surface. Therefore (after a transition period), the creep rate now becomes constant. In the bending experiments a similar saturation takes place, only there d is the distance the dislocations must move in order to reach the neutral zone (thus about half the cross dimension of the sample). Once having arrived there, they do not expand any further (because no stress acts upon them any longer), but just expand sidewise on their glide planes until again somewhere the surface is reached. Then again the dislocation density becomes stationary.
THEORY OF CREEP OF GERMANIUM CRYSTALS
779
The s t a t i o n a r y density is easily estimated, b y inserting (3) into (2) as, As =
(4)
~Nd2/vto,
and it gives rise to a s t a t i o n a r y creep rate (5)
~8 = z~Nbd2/to,
in which the velocity of the dislocations does not appear explicitly, but only through to. At t = ts the transition occurs from the initial cubic creep law to the s t a t i o n a r y state. Integrating (3) and (5) and adjusting the integration constant to join the curves at t ----- ts leads to the following analytical expressions for the whole creep curve: ~Nbv 2 8--
-
-
t3
3to e--
~Nbd2 ( 2d) to t - - - - -3
(t < ts),
(t > ts),
(6)
joined b y a transition region. The parameters used b y P W : a, t~ a n d z~, can easily be expressed in the theoretical quantities v, to and N. After d u l y converting deflections or extensions into strains, and minutes into seconds respectively, one finds: a ~
~8 = ~Nbd2/to sec -1,
(7)
tp
2 = §d/v sec, -gt8
(8) (9)
~
z2o -+ t(tp) = 8nNbd3/81 vto
The full-drawn curves in Figs. 1 and 2 represent the theoretical relations (6). The agreement is as good as one might expect, in view of the accuracy of the measurements (about 1 # in the bending tests and 3 # in the tensile tests respectively). We refer to P W for a more detailed discussion of the shape of the creep curves. The slight hump of the measured points over the theoretical curve in Fig. 2 might well be due to mutual influencing between the sources, in that they trigger each other into action. A similar hump has also been observed in a number of other tests both by PW and Van Bueren. It is not an essential feature of the creep curve, however, and we shall not pay attention to it here. P W found the ratio (at~/z~) to be r e m a r k a b l y constant in nearly all their experiments, n a m e l y on the average equal to 6.7, with a mean deviation of only 1.5. This is now easily explained in our t h e o r y b y noting from (7), (8) and (9) t h a t this ratio should always be 27/4 = 6.75, independent of a n y parameter. The quantities a and tp are very sensitive to stress and temper-
780
H.G.
VAN BUEREN
ature variations, but z~ is not. The product at~ itself is therefore practically constant, which was already noticed in 4). P W also tried to fit their data to an exponential curve. It should be noted that then the theoretical value of the ratio at~/z~ is only 2.7 (e), much too small. The two curves illustrated in Figs. I and 2 lend themselves to a numerical evaluation of the three important theoretical quantities N, to and v. It should be remarked that in the formulae (7) to (9) invariably the combination N/to occurs, and only two of the three equations are mutually independent. However, a third equation exists, namely equation (1), which enables the separation of N and to. As will be shown in the next section, under the circumstances of both tests quoted the time 0 was negligible compared to d/v, and (1) reduced to
v t o = 6 ~ 10 -4cm,
(la)
which constitutes the required third condition. From Vhe observed parameters of the curve and 6 we can now derive in a unique way the three basic quantities of the theory. This has been done in Table I. TABLE
I
N u m e r i c a l e v a l u a t i o n of the creep curves of F i g s .
[ Test conditions
Parameters observed
Derived quantities
Bending test T
=
(Fig.
1)
440°C
[
I and
2.
Extension T
=
test (Fig.
a
=
6 kg/mm
d
=
0.2 cm
a
=
53 p/rain
tp
=
10.6
zp = 36 p
zv
= 84/2
to = 0.5 sec N'= 3 x 10 a c m - a
t0 = 0 . 5 sec N
=
2.5
×
103cm
v
v
=
2.0
x
1 0 -'1
o
=
13 k g / m m
d
=
0.1 cm
a
=
43
t~ =
=
A~s =
2
p/min
6.4 rain
1.8 ×
10 - 4 c m / s e c
I 0 o cm -2
2)
470°C z
rain
-3
cm/sec
A s = = 3 × 10acm-2
That both sets of numerical values are about equal is to be regarded as accidental, in so far as two examples have been chosen out of the available experimental material that would show this property, for purposes to become obvious below. In the next section we shall show how it follows from the experiments that the quantities to and v as dislocation properties are extremely sensitive to the test conditions, but that N and As as material properties are not. This fourth quantity As, the saturated dislocation density defined by (4), has been included in the Table, n o t because it is an independent parameter, but because it lends itself also to experimental verification. It is well-known that the dislocation density in a slightly
THEORY OF CREEP OF GERMANIUM CRYSTALS
781
deformed germanium crystal, as derived from etch pit countings, tends to a saturation value 3) of the order of the value qubted for As in Table I. Source density N and saturation dislocation density As are to be regarded as properties inherent to the material studied, viz. germanium (or other diamonds structure materials). The fact that they always come out comparably in the tests is therefore satisfactory. The density of sources is found to be extremely low. As it is not connected with the dislocation density in the undeformed material (see section 5), and moreover surface treatments like rapid etching strongly influence the whole creep process (A. G. T w e e t , P. P e n n i n g, private comminications) it seems plausible that the sources are in some way connected with the surface. Which way that is, is unknown at present. Taking them to be surface sources, their surface density comes out to be equal in both tests, namely about 150 cm -z. Assuming further only one source to be active per glide plane, one derives a mean distance between active glide planes of about 0. I mm, in fair agreement with the slipline spacings observed under the microscope. This comparison bears considerably less weight than that carried out in Table I, however, a number of rather uncertain suppositions are underlying it. T h e c o m p a r i s o n of a b e n d i n g w i t h a n e x t e n s i o n t e s t is n o t w i t h o u t danger, t h e stress d i s t r i b u t i o n b e i n g m u c h less h o m o g e n e o u s in t h e first case t h a n in t h e s e c o n d (where i t is n e i t h e r v e r y h o m o g e n e o u s , t h o u g h , as follows a l r e a d y f r o m t h e low source d e n s i t y ) . D u e t o t h e i n v a r i a n c e of t h e a p p l i e d b e n d i n g m o m e n t , t h e stress d i s t r i b u t i o n t h r o u g h a b e n t s a m p l e c h a n g e s w h e n t h e d e f o r m a t i o n zone progresses f r o m t h e surfaces i n t o t h e interior. I t b e c o m e s u l t i m a t e l y v e r y simple, n a m e l y a u n i f o r m c o m p r e s s i v e stress a t o n e side a n d a tensile stress a t t h e o t h e r side, s e p a r a t e d b y a v e r y t h i n n e u t r a l zone m i d w a y t h e s p e c i m e n (cf 7)). Therefore, e v e r y w h e r e in de/ormed zones a n d a t all t i m e s t h e stress is fairly c o n s t a n t , a n d in t h e s t a t i o n a r y s t a t e we m a y w i t h some c o n f i d e n c e e v e n c o n s i d e r t h e stress as f a i r l y h o m o g e n e o u s t h r o u g h o u t t h e whole s p e c i m e n ( a p a r t f r o m t h e difference in sign). T h i s m a k e s t h e c o m p a r i s o n b e t w e e n b e n d i n g , a n d e x t e n s i o n e x p e r i m e n t s m o r e v a l u a b l e t h a n m i g h t a t first b e believed, a n d e x p l a i n s t h e s i m i l a r i t y in s h a p e b e t w e e n t h e creep c u r v e s in b o t h cases. W h e r e a s t h e n u m e r i c a l d e t e r m i n a t i o n of t h e s h e a r stress a c t i n g o n t h e d i s l o c a t i o n s is v e r y e a s y in a n e x t e n s i o n t e s t - t h e r e it 'is t h e a p p l i e d stress t i m e s t h e o r i e n t a t i o n f a c t o r (sin 2 cos ¢) of t h e o p e r a t i v e slip planes, viz. 0.41 in ( 1 1 0 ) c r y s t a l s - , t h i s is m o r e difficult in a b e n d i n g test. A t t h e b e g i n n i n g of s u c h a test, one m i g h t t h i n k t h a t t h e s h e a r stress s h o u l d be c a l c u l a t e d f r o m t h e elastic tensile or c o m p r e s s i v e stress in t h e o u t e r l a y e r of t h e sample, w h i c h in t u r n c a n easily b e c a l c u l a t e d f r o m t h e a p p l i e d m o m e n t a n d t h e d i m e n s i o n s of t h e s a m p l e a n d h a s also b e e n e n t e r e d in T a b l e I. H o w e v e r , t h i s m o m e n t was n o t i n v a r i a n t a l o n g t h e t e s t piece u n d e r t h e c o n d i t i o n of t h e t e s t b y P W . F u r t h e r , as soon as t h e d e f o r m a t i o n s t a r t s , t h e stress drops, t o r e a c h u l t i m a t e l y 2/3 of its o r i g i n a l v a l u e a t t h e surface, b u t n o w e v e r y w h e r e t h r o u g h o u t t h e m a t e r i a l . Since m o r e o v e r , o w i n g to t h e difference in c r y s t a l l o g r a p h i c o r i e n t a t i o n i n t h e e x p e r i m e n t s , t h e o r i e n t a t i o n f a c t o r for t h e (111) [110] glide s y s t e m s in b e n d i n g was a b o u t 2/3 of t h a t in t e n s i o n (viz. 0.27), a n y n u m e r i c a l c o m p a r i s o n s h o u l d s t a r t b y r e d u c i n g t h e c a l c u l a t e d elastic bending stress b y a b o u t t w o f a c t o r s 2/3 in o r d e r t o c o m p a r e it t o t h e t e n s i l e stress. I n t h e n e x t section a n a r g u m e n t will b e a d v a n c e d t o s h o w t h a t t h e a c t u a l c o n v e r s i o n f a c t o r is 0.34, w h i c h is e v e n less.
782
H. G. VAN BUEREN
It should finally be remarked in this section already that the theory predicts a considerable size e//ect in the creep characteristics. Assuming surface sources to be present ordy, the stationary creep rate and the length of the initial period should vary approximately proportionally to d (in a bending test the rate of deflection is thus independent of d), and the third parameter, z~, should even be porportional to d~ (or to d in a bending test). It is difficult to detect these variations owing to the large spread in the individual results (small stress and temperature differences have much more effect), b u t that the size of the sample influences the creep process to a large extent appears already from a superficial inspection of the data published in litterature. No detailed investigations on this aspect of the theory are available to us, however, all our own samples and those of P W having been ground to the same thickness within a few percents, whereas the spread in the values of z~, the quantity most sensitive to d (and least to the other variables) between individual tests is many tens of percents.
3. Stress and temperature dependence o/ velocity and incubation time. From the preceding section it will be clear that the stud), of the creep curve can give information on the generation and the motion o/ individual dislocations through the germanium lattice. It has already been attempted earlier 4), from extension measurements alone, to determine the temperature and stress dependence of the creep process. The result was that the two parameters here named ~8 and t~, varied according to one single formula of the type
I/t~
= const, exp
,
(10)
where a is the applied tensile stress and w = 1.1 × 10-21 cm 3. The insertion of a factor kT in the coefficient of a is more or less arbitrary; it has been done for reasons to become clear below. The more numerous and accurate bending data of P W verify the exponential form of the stress dependence to a large extent s). Only, some complications are present, and in order to bring out the essential features and enable us to apply the preceding simple theoretical considerations, we have combined and redrawn in a more schematical manner and on a different scale, a number of diagrams already presented b y the latter authors in their paper. The results are depicted as logarithmic plots in Figs. 3 and 4. In this section we are interested predominantly in the /ull-drawn curves, which refer to the most common initial conditions. The stationary strain rate ~s is again seen to be representable b y an exponential expression of the type (I0), only the parameter w in this expression changes discontinuously from a smaller to a larger value when the bending stress drops below a certain transition stress at. At the temperature
T H E O R Y OF C R E E P OF GERMANIUM CRYSTALS
783
of m e a s u r e m e n t (440°C), one has
{
~ s = 1.2 × 10-7 e x p { 3 . 2 × 10-9~a/kT) sec -1: a > a t ; ~s = 5.2 × 10-15 exp {2.7 × 10-21a/kT} sec -1: a < as;
(11)
as = 7 kg/mm2; f
e*" ,I
../
J
[
,'~ ~ I /
" #1 ,/
~
/ . •t ' ~ "
I
I
11 /
/s
,f
/
/' 5
6
7
/
/
B
9
10
n
12
13
1l~
Cr (kg/mm 2)
15
Fig. 3. Stationary creep rate ~8as a function of the loading stress a in bending experiments. This diagram represents the average of about 150 experiments, in which the original dislocation density D as well as the oxygen content of the samples was varied as indicated (from PW 8)). a again denoting the a p p l i e d stress (elastic stress in o u t e r layer). The description of the initial time t~ b y an exponential expression is s o m e w h a t less satisfactory (Fig. 4). To m a k e m a t t e r s not too complicated we shall nevertheless do so and write l~ = 1.7 X 104 exp { - - 3 . 2 X
lO-2~a/kT} sec.
(12)
which covers most of the e x p e r i m e n t a l points sufficiently a c c u r a t e l y (thin d r a w n straight line in Fig. 4), and takes care of the observed small v a r i a t i o n of the p r o d u c t at~ for a > at. A sudden transition at a = at like t h a t in ~s is not present in t~. At stresses of the order of at or lower, however, the initial time begins gradually to increase s o m e w h a t more r a p i d l y t h a n according to (12). The deviation n e v e r a m o u n t s to more t h a n a f a c t o r 2, e x c e p t for o x y g e n doped crystals, and we shall neglect it in this section.
784
H.G.
VAN B U E R E N
B y comparing (10) and (11) we note t h a t the exponents can be b r o u g h t into m u t u a l numerical accord b y suitably defining the scales o~ st~e'ss in extension and bending experiments respectively. This can be done b y comparing the coefficients (called w) of the applied stress in b o t h cases. As we have already seen, the active shear stress in the bending test is considerably lower t h a n the applied s.tress. F o r t h r i g h t agreement between the results of the two types of experiment can be reached b y assuming 1) t h a t the extension experiments also a p p l y to the "high-stress region" as defined b y (11) - which we shall call region B, to distinguish it from the "low-stress t0 *
\ o,
7---~*0°C
•p(~) , \ ".y l x ~,[, °° -.
-"~*,o~.~.
..
\
"~,
5
6
7
8
9
.10
~, q" (kg/mm2)
Fig. 4. As Fig. 3, now for the length of the initial period, tp. Thin drawn curve represents formula (12), other curves derived from PW 6). Open circles denote measurements of G i l m a n and J o h n s t o n 8) of the dislocation velocity in LiF, plotted on an arbitrary scale of ordinates. region" A -, and 2) t h a t the conversion factor used to convert into each other the scales of applied stress in extension and bending tests is 0.34. The transition stress in tension t h e n comes out as 0.65 k g / m m 2, m u c h lower t h a n the applied stress in tension, hence we are indeed in region B. As the orientation factor in tension was 0.41, in the bending experiments the active shear stress c o m p o n e n t , must be t a k e n as 0.41 × 0.34 = 0.14 times the applied stress. We shall use this q u a n t i t y T as the independent stress variable in the following considerations. B y the aid of (7) and (8) and numerous d a t a such as those presented in Table I we now deduce from the bending ob.servations
THEORY OF CREEP OF GERMANIUM CRYSTALS
785
(made at 440°C) as given by (11) and (12) ; v = 4.0 × 10-6 exp{2.3 × lO-21"r/kT} cm/sec to = 25 exp{-- 2.3 x lO-~'l'r/kT}
sec in region B" ~- > ~'t
t 0 = 5 X l0 s e x p { - 1.9 X lO-2°'r/kT} Tt = 1.0 kg/mm 2.
secinregionA: T<,t
(13)
Here we have made use of the fact that only the generation time to and the dislocation velocity v are stress dependent. This assumption is based on the observation, already mentioned repeatedly before, of the relative constancy of the parameter z~ (cf. formula (9)), in remarkable contrast to the several powers of 10 that a and t~ each can vary. The product at~ should itself also become fully independent of stress in the r e g i o n , > *t, because there, as we shall see below, formula (la) should apply, which infers this invariance (According to (7) and (8) at~ = ~-zcNbdS/vto; thus vto constant implies atx constant when N is constant). Mutual agreement between all this evidence can indeed only be obtained by taking the theoretical quantity N to be independent of the stress. According to (4) also As should be invariant, and the remark to this effect made in the previous section finds its foundation here. The exponential stress dependence of the dislocation velocity has already been encountered earlier 4), and will be discussed in more detail in section 4. The peculiar properties of the core structure of a dislocation in the diamond lattice lead to a formula of the type (10) or (13): v = v0 exp{-- (Q1 - colr)/kT}
(14)
where Q1 denotes the activation energy to "rearrange" the atomic or electronic structure of a dislocation. From the available experimental d a t a it is now evident that we must take wl = 2.3 X 10-21 cm 3.
(15)
As to the temperature dependence of the creep rate, the observations of the various authors are not in very good accord. From the extension experiments alone was derived Q1 ~ 2 eV 4); other authors have quoted lower values, between 1.5 and 1.8 eV, and from a comparison between bending measurements at 440°C (PW) and our own extension measurements at 470°C we find Q1 somewhere between 1.1 and 1.6 eV. This large scatter is due to the accumulation of a number of errors, which are not very important individually. We shall adopt Q1 = 1.5 eV as the best possible average, but realize that this may be in error by a few tenths of an eV to both sides. The assignment of a numerical value to v0 in (14) is then even more dubious. With the value of Q1 adopted we find from (13): v0 = 105 cm/sec,
(16)
786
H . G . VAN BUEREN
which seems not unreasonable, after all, as the limiting velocity of the dislocations. We now consider the stress dependence of the generation time to. As we have seen above, this is the inverse of that of v in region B (large stress). In accordance herewith, we regard the generation of dislocations to be /ully controlled by the velocity o/ the expanding loops in region B. In other words, in equation (1) :
oto
*t)'.
(17a)
The numerical factors in (I 3) are in agreement with this assumption. As the stress is lowered, the velocity of the dislocations decreases according to (13) or (14), b u t the incubation time 0 increases faster than b/v, until at * = m both have become equal:
0to = = ~/v 0 + b/v t, at , = Tt,
(17b)
again in agreement with (13). Below m, the generation of dislocations becomes rate-controlled by the incubation time o/the sources rather than b y the velocity of the dislocations: 0 >> 5/v } in region A (r < rt) to ~ 0
(17c)
This is due to the much stronger stress dependence of 0, which is reflected b y the stress dependence of to in region A. Use of formula (13) leads to 0 = 00 exp{--1.9 X lO-2°r/kT};
(18)
00 = 5 x 108 sec.
(19)
where About the temperature dependence of to insufficient observations exist. Equations (14) to (19) form the starting point of the theoretical description of the generation and motion of dislocations in germanium, to be given in section 4. An outline of such a theory has already been given, but it was based on relatively scanty data, and inclusion of the wealth of new of new observational material gathered b y P W has made some alterations necessary. There exists a group of observations of the motion of dislocations which are intimately connected with our results, and should be considered together with them. G i l m a n and J o h n s t o n 8) have published extensive data on the stress and temperature dependence of the velocity of dislocations in lithium fluoride crystals. This velocity was observed visually (by aid of an etching technique), and an exponential stress dependence, very much resembling that of the stationary creep rate and of the initial time for creep
T H E O R Y OF C R E E P OF GERMANIUM CRYSTALS
787
of germanium crystals, was found. Some of the results have been reproduced in Fig. 4 (since it is t~ which reflects the velocity of dislocations) ; the ordinate scale of this reproduction is of course arbitrary. B y the technique of G il m a n and J o h n s t o n (discontinuous observation of the location of dislocations after application of a stress) only the velocity of the dislocations is measured and delay time effects are not noted,but the theory expanded above m a y well apply to the observations on LiF also. However, G i l m a n and J o h n s t o n postulate further the existence of a dislocation multiplication mechanism, which leads theoretically to an exponentialshape of the initial part of the flow curve, in contrast to the 3rd power law derived here. Macroscopic observations to decide on this point are not sufficiently accurate. Another group of interesting observations concerns the creep of ceramic oxide materials, especially of single crystals of sapphire (A1203) and rutile (Ti02). W a c h t m a n and M a x w e l l 9) have shown such crystals to deform in creep above 900 and 600°C respectively; the form of the creep curves obtained resembles very much that of the curve of germanium crystals. It appears as if creep as described in this paper might be a general property of nearly perfect crystals, containing less than 105 dislocations/cm 2.
4. Phenomenological description o/the generation and motion o/ dislocations in germanium crystals. V a n B u e r e n a) considered the motion of a dislocation in the diamond lattice to be primarily characterized b y a stressdependen t energy of activation, which yields a formula of the type (14) (the occurrence of another term with Q1 + wl* can be neglected here). The activation energy is connected with the existance of local dragging points along a dislocation line, that must be carried with it but can move only with the aid of. thermal activation. The stress dependence comes in because the force excerted on the dragging points depends on the measure to which the length of free dislocation between two neighbouring dragging points bows out under the applied stress. We refer to the paper cited for a further discussion. The quantity Q1 (called Qo in 4)) is the energy that must be supplied to bring a dragging point from its stable into its rnobile configuration; it ~hould be about 1.5 eV according to the new experimental evidence. The quantity Vo as given b y (16) should be of the order bvo, where v0 is the atomic vibration frequency. As v0 ~ 1014 sec -1, this condition is indeed obeyed ; however, not too much weight should be attached to this agreement. The "activation volume" wl should, according to this theory, be equal to
Wl = lb 2,
(20)
where l is the mean distance along the dislocation line between the dragging points. From (13) we then derive l = 1.4 X 10-6cm.
(21)
788
H.G. VAN
BUEREN
T h e difference b e t w e e n the values q u o t e d here a n d in 4) depends largely on tile r e a d j u s t m e n t of the stress scale. There is a n o t h e r difference, however, which reaches far deeper, between the t h e o r y proposed earlier a n d t h a t considered now. I n the previous p a p e r the initial period a n d the s t a t i o n a r y strain r a t e were considered as d e t e r m i n e d b y di//erent processes, n a m e l y b y the generation a n d the m o t i o n of the dislocations respectively. I n the light of the new evidence this conclusion can not be maintained. It, follows f r o m t h e kinetical considerations of section 2 a n d 3 t h a t in region B (T > Tt) the whole creep curve is d e t e r m i n e d b y the velocity o/the dislocations alone. The coincidence b e t w e e n the numerical values of the two a c t i v a t i o n energies g o v e r n i n g the t e m p e r a t u r e dependence of the length of the initial period a n d of the s t a t i o n a r y strain rate, respectively, which was r e g a r d e d in 4) as more or less accidental, becomes now a n a t u r a l consequence of the theory. Nevertheless, the generation process of dislocations does p l a y a certain role in d e t e r m i n i n g the creep characteristics, n a m e l y in region A (T < Tt), where 0, the i n c u b a t i o n time, r a t h e r t h a n v determhaes the stationary creep rate (but not the l e n g t h of the initial period, as this remains at every stress level d e t e r m i n e d b y the velocity alone, see (8)). As the n a t u r e of the dislocation sources is u n k n o w n , only speculations can exist a b o u t the theoretical b a c k g r o u n d of the observed s t r o n g stress dependence of 0 ((18) a n d (19)). Assuming that the sources are of the generalized Frank-Read type, that is, that they consist of segments of dislocations of length L, and that the stress dependence of their activity arises again from the occurence of a stress-dependent activation energy - the simplest way in which such an exponential stress dependence can be described - we can write 0 = (L/b) vo exp{(Q2 - ~o2-r)/kT}. (22) Taking L to be 10-4 (to comply with the observed values of the critical shear stress) in order to obtain numerical agreement with experiment we must have: Q2 = 2.5 eV, w2 = 1.9 X 10-2°cm a.
(23)
A formula of the type (22) could be understood in principle, as explained in 4), by assuming that the sources are blocked by impurities of some kind, and that thermal activation is needed to pull the dislocation segments free. However, H a as e n 10) has recently pointed out that C o t t r e l l ' s mathematical analysis leading to a linear stress dependence of Q2 should be extended and that instead for 0 a stress dependence of the form 0 ~ exp {Q2" (~'/7o)-~/kT} (24) should apply, where ~-o is some constant stress. Now, this conclusion is based on a detailed consideration of the purely elastic interaction between impurity atom and dislocation, which may be indeed important for the case considered by the author cited (C in c~-Fe), but is certainly not a priori to be expected to play a dominating part in the relatively spacious, covalent diamond lattice. There the preference of an impurity for a dislocation site is in first instance determined by the possibility of better accomodation of the valence bonds of the impurity atom near the dislocation
T H E O R Y OF C R E E P OF G E R M A N I U M C R Y S T A L S
789
axis. This better accomodation can for instance be associated with the occurrence of u n s a t u r a t e d v a l e n c e b o n d s a l o n g t h e d i s l o c a t i o n line, atad is e s s e n t i a l l y r e s p o n s i b l e f o r short range i n t e r a c t i o n o n l y . I n t h i s c a s e a l i n e a r s t r e s s d e p e n d e n c e o f t h e a c t i v a t i o n e n e r g y f o r d i s l o c a t i o n release c a n a g a i n b e e x p e c t e d , a s it is p r e d o m i n a n t l y t h e l o n g r a n g e e l a s t i c i n t e r a c t i o n w h i c h c a u s e s t h e ~--~ v a r i a t i o n .
From (14), (17b), and (22) together with v0 = bvo, we derive the following theoretical expression for the transition stress Tt: ¢t =
Q2 - Q1 wz -- wl
AQ k T log(L/5) m - ; Aw
(25)
the second term can be neglected since L and 6 are about equal (to 10-4 cm; this is simply equivalent to saying that the successive loops of a Frank-Read source follow each other in a distance about equal to the length of the source). This means that, if wl and w2 are constants and the description in terms of a stress-dependent activation energy is correct, the transition stress should be independent (or only slightly dependent) of the temperature. If, however, the stress dependence arises from another cause, Wl,2 should be expected to be at least proportional to T, and it would follow t h a t Tt would decrease with increasing temperature. 5. Influence o/ other dislocations and o/ impurities (oxygen) on the creep process. From the observations (PW) it follows t h a t the creep process is essentially independent of the original dislocation density D of the material as long as this is less than about l0 s cm -2 *). Even in so-called dislocation /ree crystals the values of the parameters a and t~ and their stress dependence do not deviate markedly from those in pulled crystals with D ~ 104 -- l05 cm -2 (Figs. 3 and 4).' This observation indicates clearly that the dislocation sources discussed in the previous sections are not connected with the dislocations already present before the deformation started (e.g. in the form of a network) That the surface of the crystal has probably a large influence instead has already been mentioned in section 2. When the original dislocation density becomes larger than Do ~ 105 c m -2, some influence of it on the creep process can be detected, but only in stage B (, > *t). There, the stationary creep rate then begins to increase roughly proportional to D, and the initial period tp decreases in more or less the same ratio. In region A (~ < *t) any dependence of the creep rate a on D seems to be absent (cf P W Fig. 3), but observations are scarce; about the influence of D on tp there are even less observations in this region, but they also indicate a much smaller effect then in region B (see lower curve in Fig. 4 of this paper). *) All dislocation densities quoted here have been determined by counting etch pits on suitably etched crystal surfaces. There may, of course, have been dislocation like structures in the material that did not show up in this way.
~90
H~ G. V A N B U E R E N
This evidence, however meagre, seems at first sight to point to the conclusion that the effect of the dislocation density should be understood as an effect on v, the dislocation velocity, because only this quantity determines a and t~ in an inverse manner in region B, and is no longer of significance to a in region A. - B y this argument t~ should be expected to decrease with D also in region A, though -. However, it seems improbable that v should be affected in the w a y cited, because the presence of other dislocations could only retard dislocation motion, not accelerate it. It appears more likely that the increase of the creep rate should actually be interpreted as an increase of the density of dislocations that can participate in the deformation process (thus not of sources). The strain rate is no longer given b y Abv at each moment, but b y (A + D)bv, and the stationary creep rate is increased accordingly. This becomes notable when D is comparable to A in the early creep stage; that is, one should expect Do to be of the order of a few times 2~N-d, which is indeed the case. To explain that the initial period is also shortened, it is sufficient to assume that the "network" dislocations can only become active when they have been first "activated" b y a source dislocation. Apparently the total dislocation density then increases faster than when no network dislocations were present (multiplication ?). In this respect it is of interest that the t3-form of the initial creep curve is best realized in dislocation free material, but approaches the exponential shape in high-density crystals, where at~/z~ decreases to about 3 6). One remark m a y yet be made on the subject of workhardening. After a few percents of creep invariably the creep rate decreases again because the interaction between dislocations begins to become of importance. Not all new dislocations move out of the material but some remain and the dislocation density thus gradually increases above As. This is borne out b y etch pit countings. As soon as A ~ 107 cm -2, work-hardening begins. A possible reason for this has been given in 4). The introduction of about 0.001% of oxygen into a germanium crystal decreases the stationary creep rate and increases the length of the inital period b y the same slight amount in region B, but it diminishes the creep rate in region A b y a factor of about 50. Another way of expressing this is b y noting that the transition stress Tt has increased b y about 0.3 kg/mm 2
(30%). We can understand this effect, at least phenomenologically, more easily than the influence of dislocations. Oxygen clearly inhibits the activity o/the sources. The incubation time ~ becomes longer, apparently b y a factor of about 50 at the temperature of the tests. Although some variation of the stress dependence can also be observed (see below), according to (25) this increase could be due to a change of Q2 alone b y an amount of approximately 0.26 eV. The form of the stress dependence of the stationary strain rate has become less steep, presumably because the interaction between oxygen
T H E O R Y OF C R E E P OF G E R M A N I U M CRYSTALS
791
atoms and dislocations differs from already existing blocking actions in that it extends over longer distances.
6. Conclusions. The study of the creep curve of germanium single crystals and its stress and temperature dependence is especially suited to obtain a quantitative knowledge of the kinematic properties o/ dislocations in this material. Presumably this same conclusion holds for other nearly perfectly crystalline materials such as for instance lithium iluoride. The generation of dislocations, at least its time dependence, can be studied in this way too, yielding quantitative evidence in the rate of production of dislocations by sources. About the physical nature of the sources this type of investigation gives little insight; for that purpose it should be combined with optical invest!gations. Received 14-5-59
I.ITTERATURE I) 2) 3) 41 5) 6) "I) 8) 9) I0)
G a l l a g h e r , C. F., Phys. Rev. 8 8 (1952) 721; 92 (1953) 846. P e a r s o n , G. L., R e a d , W. T. and F e l d m a n n , W. L., Acta Met. 5, (1957) 181. P a t e l , J. R., and A l e x a n d e r , B. H., Acta Met. 4 (1956) 385. V a n B u e r e n , H. G., Physica 24 (1958) 831. A l l e n , J. W., Phil. Mag. 3 (1958) 1297. P e n n i n g , P. and De W i n d , C., Physica 25 (1959) 765. H o r n s t r a , J. and P e n n i n g , P., Philips Research Reports 14, (1959). G i l m a n , J. J. and J o h n s t o n , W. P., J. applied Phys. 2!} (1959) 131. W a c h t m a n , J. B. and M a x w e l l , L. H., J. Am. Ceramic Soc. 37 (1954) 291. H a a s e n , P., AIME Conference, Cleveland 1958, to be published. I am indebted to Prof. H a a s e n for showing me his manuscript in advance of publication.
Physica 25 775-791
THEORY OF CREEP OF GERMANIUM CRYSTALS b y H. G. V A N B U E R E N Philips Research Laboratories, N.V. Philips Gloeilampenfabrieken, Eindhoven, Nederland
Synopsis To explain the shape of the creep curves of germanium single crystals loaded in tension and in bending, a simple kinetic model is proposed, in which the dislocations are generated by (surface) sources and move with a uniform velocity over their glide planes. In this model a quantitative interpretation of the parameters of the creep curve in terms of the velocity of the dislocations, the incubation time of the sources and the density of the sources is possible. From the observations at "high" stress levels the velocity of dislocations in the germanium lattice can be determined; from those at "low" stress levels the rate of generation of the dislocations from the sources. The observed stress and temperature dependence of the creep process leads to similar dependences of incubation time and velocity. These dependences are used to form a quantitative theory of dislocation production and motion in the germanium lattice. This theory is shown to reflect semi-quantitatively various observed peculiarities of the creep phenomenon. The influence of other dislocations and of oxygen as an impurity on the elementary creep process can now also be qualitatively understood.
1. Creep o/ diamond structure materials. Crystals of g e r m a n i u m , silicon, i n d i u m antimonide, a n d p r e s u m a b l y o t h e r d i a m o n d s t r u c t u r e c o m p o u n d s as well, show a peculiar form of creep when loaded in tension, compression or b e n d i n g at a t e m p e r a t u r e which is a b o u t half of the (absolute) melting t e m p e r a t u r e . Several publications h a v e a p p e a r e d in the last few years dealing with this p h e n o m e n o n 1-5), b u t these were all based on relatively few e x p e r i m e n t a l data. I n view of the wide scatter t h a t i n v a r i a b l y occurs between different specimens tested u n d e r c o m p a r a b l e conditions (such a scatter is logically connected with the characteristics of the d e f o r m a t i o n process, see section 3) a reliable c o m p a r i s o n of these d a t a with a n y theoretical predictions could n o t . b e made. P e n n i n g a n d d e W i n d 6) h a v e r e c e n t l y o b t a i n e d a v e r y extensive set of experimental d a t a on bent g e r m a n i u m crystals, however, a n d this enables us to acquire a more q u a n t i t a t i v e insight in t h e n a t u r e of the d e f o r m a t i o n process. (Their p a p e r wit1 be indicated b y the a b b r e v i a t i o n P W in the following). I n the present p a p e r a detailed theoretical description of the e x p e r i m e n t a l - - 775 - -
776
~. G. VAN B U E R E N
data obtained both b y P W and b y v a n B u e r e n 4) is attempted. Section 9. deals with the form of the creep curve; the simplest possible dislocation model which can explain the observed time dependence of the creep phenomenon is introduced. With the aid of this model observed parameters can now be translated into properties of the individual dislocations in the germanium lattice. The variation of these properties with the temperature and especially with the loading stress is investigated in section 3. This enables us in section 4 to speculate on the generation process of the dislocations and the kinetics of these imperfections moving through the diamond lattice. Finally, in section 5, the effect of other dislocations and of chemical impurities like oxygen on the motion and generation of dislocations will be discussed.
z~ 2
Bending: __
O~ 7_=/+~0oC;, O.=13kg/rnm 2 qm=2turn ~J dislocation free ltz =lOS~rain
150
0.1°/o-~-~
55
0
I0
,/p
N
~ 't (m/n)
Fig. 1. Creep c u r v e of g e r m a n i u m single b e n t at 440°C. The axis of t h e crystal bar (with a square cross-section of 4 m m 2) was along t h e <111 > direction, t h e stress in t h e o u t e r layer was 13 kg/mm~. Deflection is plotted against t i m e ; 1/z deflection corresponds to a strain in t h e o u t e r layer of 0.001%. D o t s : e x p e r i m e n t a l points o b t a i n e d b y P W a), d r a w n c u r v e : theoretical curve.
2. Kinetics o/the creep process. In bending as well as in tensile tests the general form of the creep curve is the same: a slow initial period is followed b y an interval in which the creep rate is remarkably constant. This interval is finally terminated by the onset of work-hardening, a process which will not be discussed here. Figs. 1 and 2 illustrate two typical observed curves. P W described their curves b y three parameters, namely tp, a measure for
THEORY OF CREEP OF GERMANIUM CRYSTALS
777
the length of the initial period, z~, the deflection at t~ and a, the ultimate stationary creep rate. The definition of these parameters follows immediately from the figures. We shall use this characterization, also for the extension curves, throughout this paper.
900 2
P
Extension: / T=~?O°C; (T=6.0kg/mm2 =2ram dislocation density _ 7OO _ _ --3xlOZ~cm~_ 2 -~,, ltt= 3x I0- strain 8OO
2%
6O5
505 /.,00
2O5
,/
lOG
~p o
lo,
ts tp
20
30 = t(min)
Fig. 2. Creep curve of g e r m a n i u m single crystal stressed in tension at 470°C along a <110> direction. Tensile stress 6 k g / m m s, round cross-section 3 m m 2, 1/, e x t e n s i o n = 0.003% strain. Dots are e x p e r i m e n t a l points o b t a i n e d b y V a n B u e r e n 4), t h e d r a w n c u r v e is the theoretical curve.
To explain the general form as observed, the following model is proposed. A number of dislocation sources is present in the material. To the nature of these sources we shall revert, in section 5, presumably they are in some way connected with the surface. In this section it is sufficient to call their density per unit of volume N cm -3. On application of a stress (at an elevated temperature) the sources emit dislocations in the form of loops that expand with a vetocity v. When a loop has moved a sufficient distance b away from the dislocation source a new loop can be emitted. The distance ~ is determined by the condition that the backstress excerted by the loop on the source has dropped sufficiently below the critical shear stress ~er of the source to enable the latter to work again. This critical shear stress can be estimated from the necessary value of the applied stress (see below) as a few kg/mm 2. As ~ m Gb/~er, one thus finds, inserting for the shear modulus G and Burgers vector b the values pertaining to germanium, ~ m (1-2) × l0 -4 cm. We shall take simply b = 1 Ft in the following. Even when the preceding Physica 25
778
H.G. VAN
BUEREN
loop is 1 # away, and the shear stress has again reached about the critical value, the generation of a new loop m a y still require in principle an activation period (incubation time) O. Thus loops will be produced b y the source in regular intervals of time to = ~/v + O,
(1)
and they will all expand with the velocity v. It follows easily that the total length of dislocation A in the material increases with time as A = (N/to) ~rvt2.
(2)
At each moment the rate of strain, ~, contributed to b y these expanding dislocation loops, is: = Abv;
hence ~_
~rNbv 2
t 2.
to
which after integration yields a cubic time law for the creep process. As is discussed b y PW, such a cubic time law describes the initial part of the creep in a satisfactorily manner. However, the expansion of the loops does not continue indefinitely, nor does their contribution to the strain rate increase for ever. In the tensile specimens, the dislocations can only travel on their glide plane over a distance not greater than the cross dimension d of the crystal. In the most favourable case the dislocations are generated at one side surface and travel right through the crystal to the other side. Once a distance of the order of this maximum distance has been covered, thus after a time ts = d / v ,
(3)
the efficiency of the loop in contributing to the strain rate decreases. Formula (2) thus holds only for t < t,. For t > ts a situation gradually builds up in which the dislocation density becomes stationary: the regular periodic emission of new loops b y the source being compensated exactly b y the loss of dislocations at the surface. Therefore (after a transition period), the creep rate now becomes constant. In the bending experiments a similar saturation takes place, only there d is the distance the dislocations must move in order to reach the neutral zone (thus about half the cross dimension of the sample). Once having arrived there, they do not expand any further (because no stress acts upon them any longer), but just expand sidewise on their glide planes until again somewhere the surface is reached. Then again the dislocation density becomes stationary.
THEORY OF CREEP OF GERMANIUM CRYSTALS
779
The s t a t i o n a r y density is easily estimated, b y inserting (3) into (2) as, As =
(4)
~Nd2/vto,
and it gives rise to a s t a t i o n a r y creep rate (5)
~8 = z~Nbd2/to,
in which the velocity of the dislocations does not appear explicitly, but only through to. At t = ts the transition occurs from the initial cubic creep law to the s t a t i o n a r y state. Integrating (3) and (5) and adjusting the integration constant to join the curves at t ----- ts leads to the following analytical expressions for the whole creep curve: ~Nbv 2 8--
-
-
t3
3to e--
~Nbd2 ( 2d) to t - - - - -3
(t < ts),
(t > ts),
(6)
joined b y a transition region. The parameters used b y P W : a, t~ a n d z~, can easily be expressed in the theoretical quantities v, to and N. After d u l y converting deflections or extensions into strains, and minutes into seconds respectively, one finds: a ~
~8 = ~Nbd2/to sec -1,
(7)
tp
2 = §d/v sec, -gt8
(8) (9)
~
z2o -+ t(tp) = 8nNbd3/81 vto
The full-drawn curves in Figs. 1 and 2 represent the theoretical relations (6). The agreement is as good as one might expect, in view of the accuracy of the measurements (about 1 # in the bending tests and 3 # in the tensile tests respectively). We refer to P W for a more detailed discussion of the shape of the creep curves. The slight hump of the measured points over the theoretical curve in Fig. 2 might well be due to mutual influencing between the sources, in that they trigger each other into action. A similar hump has also been observed in a number of other tests both by PW and Van Bueren. It is not an essential feature of the creep curve, however, and we shall not pay attention to it here. P W found the ratio (at~/z~) to be r e m a r k a b l y constant in nearly all their experiments, n a m e l y on the average equal to 6.7, with a mean deviation of only 1.5. This is now easily explained in our t h e o r y b y noting from (7), (8) and (9) t h a t this ratio should always be 27/4 = 6.75, independent of a n y parameter. The quantities a and tp are very sensitive to stress and temper-
780
H.G.
VAN BUEREN
ature variations, but z~ is not. The product at~ itself is therefore practically constant, which was already noticed in 4). P W also tried to fit their data to an exponential curve. It should be noted that then the theoretical value of the ratio at~/z~ is only 2.7 (e), much too small. The two curves illustrated in Figs. I and 2 lend themselves to a numerical evaluation of the three important theoretical quantities N, to and v. It should be remarked that in the formulae (7) to (9) invariably the combination N/to occurs, and only two of the three equations are mutually independent. However, a third equation exists, namely equation (1), which enables the separation of N and to. As will be shown in the next section, under the circumstances of both tests quoted the time 0 was negligible compared to d/v, and (1) reduced to
v t o = 6 ~ 10 -4cm,
(la)
which constitutes the required third condition. From Vhe observed parameters of the curve and 6 we can now derive in a unique way the three basic quantities of the theory. This has been done in Table I. TABLE
I
N u m e r i c a l e v a l u a t i o n of the creep curves of F i g s .
[ Test conditions
Parameters observed
Derived quantities
Bending test T
=
(Fig.
1)
440°C
[
I and
2.
Extension T
=
test (Fig.
a
=
6 kg/mm
d
=
0.2 cm
a
=
53 p/rain
tp
=
10.6
zp = 36 p
zv
= 84/2
to = 0.5 sec N'= 3 x 10 a c m - a
t0 = 0 . 5 sec N
=
2.5
×
103cm
v
v
=
2.0
x
1 0 -'1
o
=
13 k g / m m
d
=
0.1 cm
a
=
43
t~ =
=
A~s =
2
p/min
6.4 rain
1.8 ×
10 - 4 c m / s e c
I 0 o cm -2
2)
470°C z
rain
-3
cm/sec
A s = = 3 × 10acm-2
That both sets of numerical values are about equal is to be regarded as accidental, in so far as two examples have been chosen out of the available experimental material that would show this property, for purposes to become obvious below. In the next section we shall show how it follows from the experiments that the quantities to and v as dislocation properties are extremely sensitive to the test conditions, but that N and As as material properties are not. This fourth quantity As, the saturated dislocation density defined by (4), has been included in the Table, n o t because it is an independent parameter, but because it lends itself also to experimental verification. It is well-known that the dislocation density in a slightly
THEORY OF CREEP OF GERMANIUM CRYSTALS
781
deformed germanium crystal, as derived from etch pit countings, tends to a saturation value 3) of the order of the value qubted for As in Table I. Source density N and saturation dislocation density As are to be regarded as properties inherent to the material studied, viz. germanium (or other diamonds structure materials). The fact that they always come out comparably in the tests is therefore satisfactory. The density of sources is found to be extremely low. As it is not connected with the dislocation density in the undeformed material (see section 5), and moreover surface treatments like rapid etching strongly influence the whole creep process (A. G. T w e e t , P. P e n n i n g, private comminications) it seems plausible that the sources are in some way connected with the surface. Which way that is, is unknown at present. Taking them to be surface sources, their surface density comes out to be equal in both tests, namely about 150 cm -z. Assuming further only one source to be active per glide plane, one derives a mean distance between active glide planes of about 0. I mm, in fair agreement with the slipline spacings observed under the microscope. This comparison bears considerably less weight than that carried out in Table I, however, a number of rather uncertain suppositions are underlying it. T h e c o m p a r i s o n of a b e n d i n g w i t h a n e x t e n s i o n t e s t is n o t w i t h o u t danger, t h e stress d i s t r i b u t i o n b e i n g m u c h less h o m o g e n e o u s in t h e first case t h a n in t h e s e c o n d (where i t is n e i t h e r v e r y h o m o g e n e o u s , t h o u g h , as follows a l r e a d y f r o m t h e low source d e n s i t y ) . D u e t o t h e i n v a r i a n c e of t h e a p p l i e d b e n d i n g m o m e n t , t h e stress d i s t r i b u t i o n t h r o u g h a b e n t s a m p l e c h a n g e s w h e n t h e d e f o r m a t i o n zone progresses f r o m t h e surfaces i n t o t h e interior. I t b e c o m e s u l t i m a t e l y v e r y simple, n a m e l y a u n i f o r m c o m p r e s s i v e stress a t o n e side a n d a tensile stress a t t h e o t h e r side, s e p a r a t e d b y a v e r y t h i n n e u t r a l zone m i d w a y t h e s p e c i m e n (cf 7)). Therefore, e v e r y w h e r e in de/ormed zones a n d a t all t i m e s t h e stress is fairly c o n s t a n t , a n d in t h e s t a t i o n a r y s t a t e we m a y w i t h some c o n f i d e n c e e v e n c o n s i d e r t h e stress as f a i r l y h o m o g e n e o u s t h r o u g h o u t t h e whole s p e c i m e n ( a p a r t f r o m t h e difference in sign). T h i s m a k e s t h e c o m p a r i s o n b e t w e e n b e n d i n g , a n d e x t e n s i o n e x p e r i m e n t s m o r e v a l u a b l e t h a n m i g h t a t first b e believed, a n d e x p l a i n s t h e s i m i l a r i t y in s h a p e b e t w e e n t h e creep c u r v e s in b o t h cases. W h e r e a s t h e n u m e r i c a l d e t e r m i n a t i o n of t h e s h e a r stress a c t i n g o n t h e d i s l o c a t i o n s is v e r y e a s y in a n e x t e n s i o n t e s t - t h e r e it 'is t h e a p p l i e d stress t i m e s t h e o r i e n t a t i o n f a c t o r (sin 2 cos ¢) of t h e o p e r a t i v e slip planes, viz. 0.41 in ( 1 1 0 ) c r y s t a l s - , t h i s is m o r e difficult in a b e n d i n g test. A t t h e b e g i n n i n g of s u c h a test, one m i g h t t h i n k t h a t t h e s h e a r stress s h o u l d be c a l c u l a t e d f r o m t h e elastic tensile or c o m p r e s s i v e stress in t h e o u t e r l a y e r of t h e sample, w h i c h in t u r n c a n easily b e c a l c u l a t e d f r o m t h e a p p l i e d m o m e n t a n d t h e d i m e n s i o n s of t h e s a m p l e a n d h a s also b e e n e n t e r e d in T a b l e I. H o w e v e r , t h i s m o m e n t was n o t i n v a r i a n t a l o n g t h e t e s t piece u n d e r t h e c o n d i t i o n of t h e t e s t b y P W . F u r t h e r , as soon as t h e d e f o r m a t i o n s t a r t s , t h e stress drops, t o r e a c h u l t i m a t e l y 2/3 of its o r i g i n a l v a l u e a t t h e surface, b u t n o w e v e r y w h e r e t h r o u g h o u t t h e m a t e r i a l . Since m o r e o v e r , o w i n g to t h e difference in c r y s t a l l o g r a p h i c o r i e n t a t i o n i n t h e e x p e r i m e n t s , t h e o r i e n t a t i o n f a c t o r for t h e (111) [110] glide s y s t e m s in b e n d i n g was a b o u t 2/3 of t h a t in t e n s i o n (viz. 0.27), a n y n u m e r i c a l c o m p a r i s o n s h o u l d s t a r t b y r e d u c i n g t h e c a l c u l a t e d elastic bending stress b y a b o u t t w o f a c t o r s 2/3 in o r d e r t o c o m p a r e it t o t h e t e n s i l e stress. I n t h e n e x t section a n a r g u m e n t will b e a d v a n c e d t o s h o w t h a t t h e a c t u a l c o n v e r s i o n f a c t o r is 0.34, w h i c h is e v e n less.
782
H. G. VAN BUEREN
It should finally be remarked in this section already that the theory predicts a considerable size e//ect in the creep characteristics. Assuming surface sources to be present ordy, the stationary creep rate and the length of the initial period should vary approximately proportionally to d (in a bending test the rate of deflection is thus independent of d), and the third parameter, z~, should even be porportional to d~ (or to d in a bending test). It is difficult to detect these variations owing to the large spread in the individual results (small stress and temperature differences have much more effect), b u t that the size of the sample influences the creep process to a large extent appears already from a superficial inspection of the data published in litterature. No detailed investigations on this aspect of the theory are available to us, however, all our own samples and those of P W having been ground to the same thickness within a few percents, whereas the spread in the values of z~, the quantity most sensitive to d (and least to the other variables) between individual tests is many tens of percents.
3. Stress and temperature dependence o/ velocity and incubation time. From the preceding section it will be clear that the stud), of the creep curve can give information on the generation and the motion o/ individual dislocations through the germanium lattice. It has already been attempted earlier 4), from extension measurements alone, to determine the temperature and stress dependence of the creep process. The result was that the two parameters here named ~8 and t~, varied according to one single formula of the type
I/t~
= const, exp
,
(10)
where a is the applied tensile stress and w = 1.1 × 10-21 cm 3. The insertion of a factor kT in the coefficient of a is more or less arbitrary; it has been done for reasons to become clear below. The more numerous and accurate bending data of P W verify the exponential form of the stress dependence to a large extent s). Only, some complications are present, and in order to bring out the essential features and enable us to apply the preceding simple theoretical considerations, we have combined and redrawn in a more schematical manner and on a different scale, a number of diagrams already presented b y the latter authors in their paper. The results are depicted as logarithmic plots in Figs. 3 and 4. In this section we are interested predominantly in the /ull-drawn curves, which refer to the most common initial conditions. The stationary strain rate ~s is again seen to be representable b y an exponential expression of the type (I0), only the parameter w in this expression changes discontinuously from a smaller to a larger value when the bending stress drops below a certain transition stress at. At the temperature
T H E O R Y OF C R E E P OF GERMANIUM CRYSTALS
783
of m e a s u r e m e n t (440°C), one has
{
~ s = 1.2 × 10-7 e x p { 3 . 2 × 10-9~a/kT) sec -1: a > a t ; ~s = 5.2 × 10-15 exp {2.7 × 10-21a/kT} sec -1: a < as;
(11)
as = 7 kg/mm2; f
e*" ,I
../
J
[
,'~ ~ I /
" #1 ,/
~
/ . •t ' ~ "
I
I
11 /
/s
,f
/
/' 5
6
7
/
/
B
9
10
n
12
13
1l~
Cr (kg/mm 2)
15
Fig. 3. Stationary creep rate ~8as a function of the loading stress a in bending experiments. This diagram represents the average of about 150 experiments, in which the original dislocation density D as well as the oxygen content of the samples was varied as indicated (from PW 8)). a again denoting the a p p l i e d stress (elastic stress in o u t e r layer). The description of the initial time t~ b y an exponential expression is s o m e w h a t less satisfactory (Fig. 4). To m a k e m a t t e r s not too complicated we shall nevertheless do so and write l~ = 1.7 X 104 exp { - - 3 . 2 X
lO-2~a/kT} sec.
(12)
which covers most of the e x p e r i m e n t a l points sufficiently a c c u r a t e l y (thin d r a w n straight line in Fig. 4), and takes care of the observed small v a r i a t i o n of the p r o d u c t at~ for a > at. A sudden transition at a = at like t h a t in ~s is not present in t~. At stresses of the order of at or lower, however, the initial time begins gradually to increase s o m e w h a t more r a p i d l y t h a n according to (12). The deviation n e v e r a m o u n t s to more t h a n a f a c t o r 2, e x c e p t for o x y g e n doped crystals, and we shall neglect it in this section.
784
H.G.
VAN B U E R E N
B y comparing (10) and (11) we note t h a t the exponents can be b r o u g h t into m u t u a l numerical accord b y suitably defining the scales o~ st~e'ss in extension and bending experiments respectively. This can be done b y comparing the coefficients (called w) of the applied stress in b o t h cases. As we have already seen, the active shear stress in the bending test is considerably lower t h a n the applied s.tress. F o r t h r i g h t agreement between the results of the two types of experiment can be reached b y assuming 1) t h a t the extension experiments also a p p l y to the "high-stress region" as defined b y (11) - which we shall call region B, to distinguish it from the "low-stress t0 *
\ o,
7---~*0°C
•p(~) , \ ".y l x ~,[, °° -.
-"~*,o~.~.
..
\
"~,
5
6
7
8
9
.10
~, q" (kg/mm2)
Fig. 4. As Fig. 3, now for the length of the initial period, tp. Thin drawn curve represents formula (12), other curves derived from PW 6). Open circles denote measurements of G i l m a n and J o h n s t o n 8) of the dislocation velocity in LiF, plotted on an arbitrary scale of ordinates. region" A -, and 2) t h a t the conversion factor used to convert into each other the scales of applied stress in extension and bending tests is 0.34. The transition stress in tension t h e n comes out as 0.65 k g / m m 2, m u c h lower t h a n the applied stress in tension, hence we are indeed in region B. As the orientation factor in tension was 0.41, in the bending experiments the active shear stress c o m p o n e n t , must be t a k e n as 0.41 × 0.34 = 0.14 times the applied stress. We shall use this q u a n t i t y T as the independent stress variable in the following considerations. B y the aid of (7) and (8) and numerous d a t a such as those presented in Table I we now deduce from the bending ob.servations
THEORY OF CREEP OF GERMANIUM CRYSTALS
785
(made at 440°C) as given by (11) and (12) ; v = 4.0 × 10-6 exp{2.3 × lO-21"r/kT} cm/sec to = 25 exp{-- 2.3 x lO-~'l'r/kT}
sec in region B" ~- > ~'t
t 0 = 5 X l0 s e x p { - 1.9 X lO-2°'r/kT} Tt = 1.0 kg/mm 2.
secinregionA: T<,t
(13)
Here we have made use of the fact that only the generation time to and the dislocation velocity v are stress dependent. This assumption is based on the observation, already mentioned repeatedly before, of the relative constancy of the parameter z~ (cf. formula (9)), in remarkable contrast to the several powers of 10 that a and t~ each can vary. The product at~ should itself also become fully independent of stress in the r e g i o n , > *t, because there, as we shall see below, formula (la) should apply, which infers this invariance (According to (7) and (8) at~ = ~-zcNbdS/vto; thus vto constant implies atx constant when N is constant). Mutual agreement between all this evidence can indeed only be obtained by taking the theoretical quantity N to be independent of the stress. According to (4) also As should be invariant, and the remark to this effect made in the previous section finds its foundation here. The exponential stress dependence of the dislocation velocity has already been encountered earlier 4), and will be discussed in more detail in section 4. The peculiar properties of the core structure of a dislocation in the diamond lattice lead to a formula of the type (10) or (13): v = v0 exp{-- (Q1 - colr)/kT}
(14)
where Q1 denotes the activation energy to "rearrange" the atomic or electronic structure of a dislocation. From the available experimental d a t a it is now evident that we must take wl = 2.3 X 10-21 cm 3.
(15)
As to the temperature dependence of the creep rate, the observations of the various authors are not in very good accord. From the extension experiments alone was derived Q1 ~ 2 eV 4); other authors have quoted lower values, between 1.5 and 1.8 eV, and from a comparison between bending measurements at 440°C (PW) and our own extension measurements at 470°C we find Q1 somewhere between 1.1 and 1.6 eV. This large scatter is due to the accumulation of a number of errors, which are not very important individually. We shall adopt Q1 = 1.5 eV as the best possible average, but realize that this may be in error by a few tenths of an eV to both sides. The assignment of a numerical value to v0 in (14) is then even more dubious. With the value of Q1 adopted we find from (13): v0 = 105 cm/sec,
(16)
786
H . G . VAN BUEREN
which seems not unreasonable, after all, as the limiting velocity of the dislocations. We now consider the stress dependence of the generation time to. As we have seen above, this is the inverse of that of v in region B (large stress). In accordance herewith, we regard the generation of dislocations to be /ully controlled by the velocity o/ the expanding loops in region B. In other words, in equation (1) :
oto
*t)'.
(17a)
The numerical factors in (I 3) are in agreement with this assumption. As the stress is lowered, the velocity of the dislocations decreases according to (13) or (14), b u t the incubation time 0 increases faster than b/v, until at * = m both have become equal:
0to = = ~/v 0 + b/v t, at , = Tt,
(17b)
again in agreement with (13). Below m, the generation of dislocations becomes rate-controlled by the incubation time o/the sources rather than b y the velocity of the dislocations: 0 >> 5/v } in region A (r < rt) to ~ 0
(17c)
This is due to the much stronger stress dependence of 0, which is reflected b y the stress dependence of to in region A. Use of formula (13) leads to 0 = 00 exp{--1.9 X lO-2°r/kT};
(18)
00 = 5 x 108 sec.
(19)
where About the temperature dependence of to insufficient observations exist. Equations (14) to (19) form the starting point of the theoretical description of the generation and motion of dislocations in germanium, to be given in section 4. An outline of such a theory has already been given, but it was based on relatively scanty data, and inclusion of the wealth of new of new observational material gathered b y P W has made some alterations necessary. There exists a group of observations of the motion of dislocations which are intimately connected with our results, and should be considered together with them. G i l m a n and J o h n s t o n 8) have published extensive data on the stress and temperature dependence of the velocity of dislocations in lithium fluoride crystals. This velocity was observed visually (by aid of an etching technique), and an exponential stress dependence, very much resembling that of the stationary creep rate and of the initial time for creep
T H E O R Y OF C R E E P OF GERMANIUM CRYSTALS
787
of germanium crystals, was found. Some of the results have been reproduced in Fig. 4 (since it is t~ which reflects the velocity of dislocations) ; the ordinate scale of this reproduction is of course arbitrary. B y the technique of G il m a n and J o h n s t o n (discontinuous observation of the location of dislocations after application of a stress) only the velocity of the dislocations is measured and delay time effects are not noted,but the theory expanded above m a y well apply to the observations on LiF also. However, G i l m a n and J o h n s t o n postulate further the existence of a dislocation multiplication mechanism, which leads theoretically to an exponentialshape of the initial part of the flow curve, in contrast to the 3rd power law derived here. Macroscopic observations to decide on this point are not sufficiently accurate. Another group of interesting observations concerns the creep of ceramic oxide materials, especially of single crystals of sapphire (A1203) and rutile (Ti02). W a c h t m a n and M a x w e l l 9) have shown such crystals to deform in creep above 900 and 600°C respectively; the form of the creep curves obtained resembles very much that of the curve of germanium crystals. It appears as if creep as described in this paper might be a general property of nearly perfect crystals, containing less than 105 dislocations/cm 2.
4. Phenomenological description o/the generation and motion o/ dislocations in germanium crystals. V a n B u e r e n a) considered the motion of a dislocation in the diamond lattice to be primarily characterized b y a stressdependen t energy of activation, which yields a formula of the type (14) (the occurrence of another term with Q1 + wl* can be neglected here). The activation energy is connected with the existance of local dragging points along a dislocation line, that must be carried with it but can move only with the aid of. thermal activation. The stress dependence comes in because the force excerted on the dragging points depends on the measure to which the length of free dislocation between two neighbouring dragging points bows out under the applied stress. We refer to the paper cited for a further discussion. The quantity Q1 (called Qo in 4)) is the energy that must be supplied to bring a dragging point from its stable into its rnobile configuration; it ~hould be about 1.5 eV according to the new experimental evidence. The quantity Vo as given b y (16) should be of the order bvo, where v0 is the atomic vibration frequency. As v0 ~ 1014 sec -1, this condition is indeed obeyed ; however, not too much weight should be attached to this agreement. The "activation volume" wl should, according to this theory, be equal to
Wl = lb 2,
(20)
where l is the mean distance along the dislocation line between the dragging points. From (13) we then derive l = 1.4 X 10-6cm.
(21)
788
H.G. VAN
BUEREN
T h e difference b e t w e e n the values q u o t e d here a n d in 4) depends largely on tile r e a d j u s t m e n t of the stress scale. There is a n o t h e r difference, however, which reaches far deeper, between the t h e o r y proposed earlier a n d t h a t considered now. I n the previous p a p e r the initial period a n d the s t a t i o n a r y strain r a t e were considered as d e t e r m i n e d b y di//erent processes, n a m e l y b y the generation a n d the m o t i o n of the dislocations respectively. I n the light of the new evidence this conclusion can not be maintained. It, follows f r o m t h e kinetical considerations of section 2 a n d 3 t h a t in region B (T > Tt) the whole creep curve is d e t e r m i n e d b y the velocity o/the dislocations alone. The coincidence b e t w e e n the numerical values of the two a c t i v a t i o n energies g o v e r n i n g the t e m p e r a t u r e dependence of the length of the initial period a n d of the s t a t i o n a r y strain rate, respectively, which was r e g a r d e d in 4) as more or less accidental, becomes now a n a t u r a l consequence of the theory. Nevertheless, the generation process of dislocations does p l a y a certain role in d e t e r m i n i n g the creep characteristics, n a m e l y in region A (T < Tt), where 0, the i n c u b a t i o n time, r a t h e r t h a n v determhaes the stationary creep rate (but not the l e n g t h of the initial period, as this remains at every stress level d e t e r m i n e d b y the velocity alone, see (8)). As the n a t u r e of the dislocation sources is u n k n o w n , only speculations can exist a b o u t the theoretical b a c k g r o u n d of the observed s t r o n g stress dependence of 0 ((18) a n d (19)). Assuming that the sources are of the generalized Frank-Read type, that is, that they consist of segments of dislocations of length L, and that the stress dependence of their activity arises again from the occurence of a stress-dependent activation energy - the simplest way in which such an exponential stress dependence can be described - we can write 0 = (L/b) vo exp{(Q2 - ~o2-r)/kT}. (22) Taking L to be 10-4 (to comply with the observed values of the critical shear stress) in order to obtain numerical agreement with experiment we must have: Q2 = 2.5 eV, w2 = 1.9 X 10-2°cm a.
(23)
A formula of the type (22) could be understood in principle, as explained in 4), by assuming that the sources are blocked by impurities of some kind, and that thermal activation is needed to pull the dislocation segments free. However, H a as e n 10) has recently pointed out that C o t t r e l l ' s mathematical analysis leading to a linear stress dependence of Q2 should be extended and that instead for 0 a stress dependence of the form 0 ~ exp {Q2" (~'/7o)-~/kT} (24) should apply, where ~-o is some constant stress. Now, this conclusion is based on a detailed consideration of the purely elastic interaction between impurity atom and dislocation, which may be indeed important for the case considered by the author cited (C in c~-Fe), but is certainly not a priori to be expected to play a dominating part in the relatively spacious, covalent diamond lattice. There the preference of an impurity for a dislocation site is in first instance determined by the possibility of better accomodation of the valence bonds of the impurity atom near the dislocation
T H E O R Y OF C R E E P OF G E R M A N I U M C R Y S T A L S
789
axis. This better accomodation can for instance be associated with the occurrence of u n s a t u r a t e d v a l e n c e b o n d s a l o n g t h e d i s l o c a t i o n line, atad is e s s e n t i a l l y r e s p o n s i b l e f o r short range i n t e r a c t i o n o n l y . I n t h i s c a s e a l i n e a r s t r e s s d e p e n d e n c e o f t h e a c t i v a t i o n e n e r g y f o r d i s l o c a t i o n release c a n a g a i n b e e x p e c t e d , a s it is p r e d o m i n a n t l y t h e l o n g r a n g e e l a s t i c i n t e r a c t i o n w h i c h c a u s e s t h e ~--~ v a r i a t i o n .
From (14), (17b), and (22) together with v0 = bvo, we derive the following theoretical expression for the transition stress Tt: ¢t =
Q2 - Q1 wz -- wl
AQ k T log(L/5) m - ; Aw
(25)
the second term can be neglected since L and 6 are about equal (to 10-4 cm; this is simply equivalent to saying that the successive loops of a Frank-Read source follow each other in a distance about equal to the length of the source). This means that, if wl and w2 are constants and the description in terms of a stress-dependent activation energy is correct, the transition stress should be independent (or only slightly dependent) of the temperature. If, however, the stress dependence arises from another cause, Wl,2 should be expected to be at least proportional to T, and it would follow t h a t Tt would decrease with increasing temperature. 5. Influence o/ other dislocations and o/ impurities (oxygen) on the creep process. From the observations (PW) it follows t h a t the creep process is essentially independent of the original dislocation density D of the material as long as this is less than about l0 s cm -2 *). Even in so-called dislocation /ree crystals the values of the parameters a and t~ and their stress dependence do not deviate markedly from those in pulled crystals with D ~ 104 -- l05 cm -2 (Figs. 3 and 4).' This observation indicates clearly that the dislocation sources discussed in the previous sections are not connected with the dislocations already present before the deformation started (e.g. in the form of a network) That the surface of the crystal has probably a large influence instead has already been mentioned in section 2. When the original dislocation density becomes larger than Do ~ 105 c m -2, some influence of it on the creep process can be detected, but only in stage B (, > *t). There, the stationary creep rate then begins to increase roughly proportional to D, and the initial period tp decreases in more or less the same ratio. In region A (~ < *t) any dependence of the creep rate a on D seems to be absent (cf P W Fig. 3), but observations are scarce; about the influence of D on tp there are even less observations in this region, but they also indicate a much smaller effect then in region B (see lower curve in Fig. 4 of this paper). *) All dislocation densities quoted here have been determined by counting etch pits on suitably etched crystal surfaces. There may, of course, have been dislocation like structures in the material that did not show up in this way.
~90
H~ G. V A N B U E R E N
This evidence, however meagre, seems at first sight to point to the conclusion that the effect of the dislocation density should be understood as an effect on v, the dislocation velocity, because only this quantity determines a and t~ in an inverse manner in region B, and is no longer of significance to a in region A. - B y this argument t~ should be expected to decrease with D also in region A, though -. However, it seems improbable that v should be affected in the w a y cited, because the presence of other dislocations could only retard dislocation motion, not accelerate it. It appears more likely that the increase of the creep rate should actually be interpreted as an increase of the density of dislocations that can participate in the deformation process (thus not of sources). The strain rate is no longer given b y Abv at each moment, but b y (A + D)bv, and the stationary creep rate is increased accordingly. This becomes notable when D is comparable to A in the early creep stage; that is, one should expect Do to be of the order of a few times 2~N-d, which is indeed the case. To explain that the initial period is also shortened, it is sufficient to assume that the "network" dislocations can only become active when they have been first "activated" b y a source dislocation. Apparently the total dislocation density then increases faster than when no network dislocations were present (multiplication ?). In this respect it is of interest that the t3-form of the initial creep curve is best realized in dislocation free material, but approaches the exponential shape in high-density crystals, where at~/z~ decreases to about 3 6). One remark m a y yet be made on the subject of workhardening. After a few percents of creep invariably the creep rate decreases again because the interaction between dislocations begins to become of importance. Not all new dislocations move out of the material but some remain and the dislocation density thus gradually increases above As. This is borne out b y etch pit countings. As soon as A ~ 107 cm -2, work-hardening begins. A possible reason for this has been given in 4). The introduction of about 0.001% of oxygen into a germanium crystal decreases the stationary creep rate and increases the length of the inital period b y the same slight amount in region B, but it diminishes the creep rate in region A b y a factor of about 50. Another way of expressing this is b y noting that the transition stress Tt has increased b y about 0.3 kg/mm 2
(30%). We can understand this effect, at least phenomenologically, more easily than the influence of dislocations. Oxygen clearly inhibits the activity o/the sources. The incubation time ~ becomes longer, apparently b y a factor of about 50 at the temperature of the tests. Although some variation of the stress dependence can also be observed (see below), according to (25) this increase could be due to a change of Q2 alone b y an amount of approximately 0.26 eV. The form of the stress dependence of the stationary strain rate has become less steep, presumably because the interaction between oxygen
T H E O R Y OF C R E E P OF G E R M A N I U M CRYSTALS
791
atoms and dislocations differs from already existing blocking actions in that it extends over longer distances.
6. Conclusions. The study of the creep curve of germanium single crystals and its stress and temperature dependence is especially suited to obtain a quantitative knowledge of the kinematic properties o/ dislocations in this material. Presumably this same conclusion holds for other nearly perfectly crystalline materials such as for instance lithium iluoride. The generation of dislocations, at least its time dependence, can be studied in this way too, yielding quantitative evidence in the rate of production of dislocations by sources. About the physical nature of the sources this type of investigation gives little insight; for that purpose it should be combined with optical invest!gations. Received 14-5-59
I.ITTERATURE I) 2) 3) 41 5) 6) "I) 8) 9) I0)
G a l l a g h e r , C. F., Phys. Rev. 8 8 (1952) 721; 92 (1953) 846. P e a r s o n , G. L., R e a d , W. T. and F e l d m a n n , W. L., Acta Met. 5, (1957) 181. P a t e l , J. R., and A l e x a n d e r , B. H., Acta Met. 4 (1956) 385. V a n B u e r e n , H. G., Physica 24 (1958) 831. A l l e n , J. W., Phil. Mag. 3 (1958) 1297. P e n n i n g , P. and De W i n d , C., Physica 25 (1959) 765. H o r n s t r a , J. and P e n n i n g , P., Philips Research Reports 14, (1959). G i l m a n , J. J. and J o h n s t o n , W. P., J. applied Phys. 2!} (1959) 131. W a c h t m a n , J. B. and M a x w e l l , L. H., J. Am. Ceramic Soc. 37 (1954) 291. H a a s e n , P., AIME Conference, Cleveland 1958, to be published. I am indebted to Prof. H a a s e n for showing me his manuscript in advance of publication.