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TEMPERATURE DEPENDENCE OF DISLOCATION
RESONANCE DAMPING
H. Schmidt, D. Lenz, K. Lücke
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque C5, suppZ4ment au nO1O, Tome 42, octobre 2981 page c5-351
TEMPERATURE DEPENDENCE OF D I S L O C A T I O N RESONANCE DAMPING
H. Schrnidt, D. Lenz and K . Liicke
I n s t i t u t
Fiir
AZZgemeine MetaZZkunde und MetaZZphysik,RWTH
Aachen, F.R. G.A b s t r a c t . - We have s i m u l t a n e o u s l y measured t h e f r e q u e n c y dependence (10 t o
- -
--
90 MHz) o f t h e u l t r a s o n i c a t t e n u a t i o n a and sound v e l o c i t y v on h i g h p u r i t y Cu as f u n c t i o n o f t e m p e r a t u r e (4.2
s
T<
300K). I r r a d i a t i o n p i n n i n g was used t o s e p a r a t e t h e d i s l o c a t i o n resonance (DR) c o n t r i b u t i o n s aD and AVD/v0. The e v a l u a t i o n o f d a t a i s c o n s i d e r a b l y improved b y t h e use o f t h e f r e q u e n c y depen- dence o f A v ~ / v ~ , e s p e c i a l l y a t T < 50K where aD becomes masked by t h e phonon- e l e c t r o n a t t e n u a t i o n . The observed D R - e f f e c t s have been analyzed i n terms o f t h e Granato-Lucke DR-theory. From t h e d e r i v e d t e m p e r a t u r e dependences o f A L ~ , B/A and t 2 6 ( A d i s l o c a t i o n d e n s i t y , L d i s l o c a t i o n mean f r e e l o o p l e n g t h be- tween p i n n i n g p o i n t s , B d i s l o c a t i o n d r a g c o e f f i c i e n t ) we f i n d ( i ) B(T) i n agreement w i t h c u r r e n t t h e o r i e s and B(273K (4+
2 ) . 1 0 - ~ ~ s / m ~ i n c l u d i n g t h e e l e c t r o n d r a g c o e f f i c i e n t Be = ( If
2)aIO-&;/m2 ; ( i i ) L t o be independent o f T b u t A t o decrease w i t h T down t o t h e l o w e s t T. T h i s A(T) b e h a v i o u r i s d i s - cussed.1. I n t r o d u c t i o n . - MHz-measurements o f d i s l o c a t i o n resonance (DR) damping a r e a v a l u - a b l e t o o l f o r t h e i n v e s t i g a t i o n o f t h e i n t e r a c t i o n o f d i s l o c a t i o n s w i t h phonons and e l e c t r o n s ( d r a g c o e f f i c i e n t B /1,2/). F o r t h e e v a l u a t i o n o f B(T) f r o m such measurements i t has been t a c i t l y assumed t h a t t h e " d i s l o c a t i o n resonance s t r e n q t h " ( b e i n g p r o p o r t i o -
2
n a l t o AL ( w i t h
A
d i s l o c a t i o n d e n s i t y and L d i s l o c a t i o n l o o p l enoth) i s independent o f T. However, t h e r e a r e MHz-results which i n d i c a t e t h a t such a n assumption m i g h t n o t be j u s t i f i e d : ( i ) w i t h t h e s t a n d a r d MHz-pulse-echo t e c h n i q u e T-dependent changes o f L e a s i l y r e s u l t f r o m d i s l o c a t i o n break-away caused by t h e d i f f e r e n t thermal expansion o f t r a n s d u c e r and sample ( Q S D - e f f e c t /3,4/) ; ( i i ) sound v e l o c i t y measurements by A l e r s2
and Zimmermann 15 y e a r s ago showed t h a t t h e d i s l o c a t i o n modulus d e f e c t F.1D
-
AL decrea- ses w i t h d e c r e a s i n g T down t o 0.3K. B o t h p o i n t s c a s t doubts on MHz-B(T)-measurements. F u r t h e r m o r e t h e MD r e s u l t s cannot be e x p l a i n e d i n t h e frame o f t h e c l a s s i c a l v i b r a t i n g s t r i n g model as worked o u t by K o e h l e r /6/ and Granato and Lucke /7/. W i t h t h e p r e s e n t work we i n t e n d t o c o n t r i b u t e t o c l a r i f i c a t i o n o f t h e M H Z - " A L ~ " problem by simultaneous measurements o f t h e f r e q u e n c y dependence o f d i s l o c a t i o n damping & d i s l o c a t i o n modulus d e f e c t betweenR'r
and 4.2K.2. Experimental.- Our experiments were performed on a < I l l > o r i e n t e d Cu s i n g l e c r y s t a l w i t h RRR 1100. T h i s RRR r e f e r s t o e l e c t r i c a l p u r i t y s i n c e i t i s measured a f t e r i n - t e r n a l o x i d a t i o n (170h, IIOOK, po2 = 0.1 Pa) o f a 100 ppm Fe doped c r y s t a l (PRR z 15). T h i s t r e a t m e n t r e s u l t s i n an i n c r e a s e d d i s l o c a t i o n l o o p l e n g t h L as compared w i t h h i g h c h e m i c a l l y p u r e c r y s t a l s and a l s o r e n d e r s t h e sample i n s e n s i t i v e t o quartz-sample-de-
C S - 3 5 2 JOURNAL DE PHYSIQUE
f o r m a t i o n (QSD) e f f e c t s /8,4/. [Both i s e x p l a i n e d by assuming t h a t t h e p r e c i p i t a t i o n o f Fe,OY d u r i n g i n t e r n a l o x i d a t i o n a t t h e same t i m e cleans t h e l a t t i c e f r o m Fe and some r e s i d u a l i m p u r i t i e s w h i c h o t h e r w i s e form C o t t r e l l c l o u d s d u r i n g h e a t t r e a t m e n t and thus make t h e h i g h p u r i t y Cu s e n s i t i v e t o QSD]. The sample p r e p a r a t i o n , y - i r r a d i - a t i o n and t h e t e c h n i q u e s o f a t t e n u a t i o n and v e l o c i t y measurements a r e d e s c r i b e d i n /g/. The c o o l i n g was performed by a c o n v e n t i o n a l ~ e 4 - e v a p o r a t i o n c r y o s t a t . F o r t h e temperature measurements a c a l i b r a t e d P t r e s i s t o r resp. Si-Diode was used. D u r i n g t h e a ( f ) , v ( f ) measurements t h e temperature was h e l d c o n s t a n t w i t h i n
+
0.1 K.3. Experimental R e s u l t s and E v a l u a t i o n . - Fig.1 ( a ,b) shows t h e a ( T ) ( f i g . l a ) and v ( T ) ( f i g . l b ) d a t a which c o u l d be measured o n l y up t o 90 MHz because o f h i g h a t t e n u a t i o n i n t h e p r e s e n t c r y s t a l (measurable l i m i t
2
: 4.5 dB/us). A t T > 100K a ( T ) i s d i f f e r e n t a t l o w and h i g h f: a t 10 MHz a c o n t i n u o u s a ( T ) decrease, a t 90 MHz a broad maximum around 150K i s observed. The s t e e p a ( T ) i n c r e a s e a t T < 50K i s due t o phonon-electron ab- s o r p t i o n (PEA, c . f . / I D / ) . The d i s l o c a t i o n c o n t r i b u t i o n a D ( T ) i s o b t a i n e d b y s u b t r a c - t i o n o f t h e background a t t e n u a t i o n aB(T) measured a f t e r s t r o n g y - i r r a d i a t i o n and i n - c l u d i n g t h e PEA c o n t r i b u t i o n apE ( c . f . dashedc u r v e f o r 50 MHz i n f i g . l a ) . I n c o n t r a s t t o t h e u n i r r a d i a t e d s t a t e s a f t e r i r r a d i a t i o n a Q S D - e f f e c t a i s o b s e r v e d ( f i g . l a ) w h i c h QSD i s due t o d i s l o c a t i o n break-away f r o m i r r a - d i a t i o n p i n n i n g p o i n t s (PP). A f t e r s t r o n g
-
- 3 0 i r r a d i a t i o n a << a(gy=O); we use t h e e x t r a - QS D p o l a t e d ag (dashed d o t t e d l i n e ) as n o n - d i s l o - c a t i o n background f o r 50 < T < 150K. We p o i n t 0 50 100 150 200 250 300o u t t h a t a t T < 50K aD 2 O.la and apE 0.9a Temperature T [K]
whereas a t T > 100K aD 2 0,9a, i . e . a t l o w T FIG l a
52801 -- .
aD becomes masked by PEA. Cv.long wave, q ~ ~ ( l ~ l ) . ~ = S D M H Z
I
F i g . l b shows v ( T ) (measured s i m u l - ~ ~ ~ ~ ~ ! t a n e o u s l y w i t h u ( T ) ~ ~ ~ ~ ~ , f i g . l a ) and t h e "background" v e l o c i t y v, (T) (measured a f t e r s t r o n g y i r r a d i a t i o n s i m u l t a n e o u s l y w i t h aB, f i g . l a ) . The v ( T ) - d a t a f o r t h e o t h e r frequen- c i e s b e i n g c l o s e t o v ( T ) have been o m i t t e d ~ ~ ~ ~ ~ 5,ml L -. .-
_
A -f o r sake o f c l a r i t y ; vo(T) i s f - i n d e p e n d e n t
o
50loo
150 200 250 360and g i v e n by v ~ , ~ ~ ~ > t h e v e l o c i t y o f t h e " d i s - Temperature T [K]
l o c a t i o n f r e e " < I l l > - o r i e n t e d c r y s t a l . We n o t e FIG I b
FIG 1 T-dependence o f a t t e n u a t i o n t h a t t h e s l i g h t QSD-effect (seen on aB around
-
(fig.1a) (aB = 50 MHz
-
non-
150K) does n o t i n f l u e n c e v,(T) t o any measura- d i s l o c a t i o n background a f t e rU
b l e extend. The i n s e r t i n f i g . l b b r i n g s o u t y - i r r a d i a t i o n ) and sound ve- l o c i t y ( f i g . l b ) ( v e = V L ~ ~ ~ ,
t h e remarkable LT behaviour: Whereas f o r T + 0 measured a f t e r bac ground
v(T) e x h i b i t s a f i n i t e and even i n c r e a s i n g slope w i t h T + 0. The same v(T)-behaviour
has been observed a t a l l frequencies. Q u i t e s i m i l a r v(T) e f f e c t s have been r e p o r t e d by A l e r s and Zimmermann /5/ a t even lower T (T 2 0.3K).
-p ...
F i g.2 shows t h e (T ,f )-dependence o f t h e
d i s l o c a t i o n modulus d e f e c t ,_p --__ I [ M H Z ]
a
MD : AM/Mo = (Mo-M)/Mo = 2 ( ~ ~ - v ) / v ~ (1 )
which i s more i n f o r m a t i v e than v(T) and e a s i e r \
t o compare w i t h aD(T) ( f i g . l a ) . With decrea- s i n g T t h e MD passes through a broad maximum
I: ard thendecreases almost l i n e a r i l y down t o
4.2K. As w i t h an(T) t h e MD-maximum shows a 0 50 100 150 200 2x300
c h a r a c t e r i s t i c 7 - s h i f t from 200K a t 10 MHz t o Temperature T !K] 50K a t 70 MHz. We p o i n t o u t t h a t a t T < 50K FIG 2 T-dependence o f d i s l o c a t i o n ND(T,f) i s n o t i n f l u e n c e d by phonon-electron
-
modulus d e f e c t . e f f e c t s w i t h i n t h e present Av/vo s e n s i t i v i t y /11/ and t h e r e f o r e much b e t t e r s u i t e d f o r t h e i n v e s t i g a t i o n o f d i s l o c a t i o n e f f e c t s than t h e PEA-rnas ked aD.4 . D i s c u s s i o n . - F i g . 3 s h o w s ( u p p e r p a r t : T > IOOK lower p a r t : 1 IOOK) t h e f-dependence of our MD- and aD-data (converted i n t o decre-
ment 6)
.
\'\ \ \ \ 6 = 0.115 (a-clg) [ d B / p ]/f [MPz]',
\ \ (2) 0The (6,MD)-data a r e compared w i t h t h e theore-
,,---_____
t i c a l f-dependence o f 6 resp. MD according t o t h e GL-theory f o r an exponential L - d i s t r i - b u t i o n /7/. Our reasons f o r t h e choice o f t h e GL-frequency p r o f i l e (GLP) have been given elsewhere /g/. Here we l i k e t o remind t h a t t h e GLP i s a i n t e r l i n k e d combination ( l a r g e
c i r c l e s i n f i g . 3 ) o f t h e 6-branch ( s o l i d cur- Frequency f [MHz]
v e ) and t h e MD-branch (dashed curve) which FIG 3 Comparison of present (&,MD)- a r e t o be simultaneously compared w i t h t h e
-
data w i t h t h e GC-theory. 6 ( f ) - r e s p . MD(f)-data. As seen i n f i g . 3 ane x c e l l e n t f i t o f t h e present data i s obtained
C5-354 JOURNAL DE PHYSIQUE
w i t h
R
o r i e n t a t i o n f a c t o r , G shear modulus, b Burgers v e c t o r , C l i n e t e n s i o n . F o r l a - t e r purposes we add t h e decrement f o r f << fblAX4 2
dLOW = 19.7 Q G ~ ~ B AL f / C (7
As f i g . 3 shows fMAX i n c r e a s e s w i t h d e c r e a s i n g T ( f r o m 13 MHz a t 273K t o 100 MHz a t 50K). Q u a l i t a t i v e l y s u i h a s h i f t i s p r e d i c t e d by t h e t h e o r e t i c a l /12/ B
-
T dependence a t T/QD>
0.2 (QD = 345K Debeye-T). However, i t i s t o be p o i n t e d o u t t h a t t h e " d i s l o c a t i o _ ? resonance s t r e n g t h " GMAX w h i c h i s independent o f B (equ. ( 3 ) ) decreases w i t h decrea- s i n g T. Such a GMAX(T)-behaviour p o i n t s t o a more complex T-dependence o f DR t h a n h i t h e r t o t r e a t e d by t h e t h e o r y and m o s t l y assumed f o r e v a l u a t i o n o f e x p e r i m e n t a l d a t a2
(GMAX
-
!L = c o n s t /13,14/, see however /3,15/). T h i s c r u c i a l p o i n t , t h e T-dependen- c e o f A L ~ , i s even b e t t e r demonstrated by comparing t h e measured MDLOW(T)- b e h a v i o u r w i t h t h e o r y (equ.(5).
MDLOW b e i n g independent o f B and any s t r o n g l y T-dependent2 2
q u a n t i t i e s : C(-Gb ) c a n c e l s t h e Gb -dependence i n equ. (5)): As f i g . 2 and 3 show MD W(T) decreases c o n t i n u o u s l y w i t h d e c r e a s i n g T down t o l o w e s t T. Adding A l e r s and -L 0
Zimmermann's /5/ r e s u l t s we c o n c l u d e t h a t t h e MD(T) decrease c o n t i n u o u s down t o even
0.3K. 18'
i
'1
Fig.4 shows t h e T-dependence o f t h e DR parameters w h i c h can be d e r i v e d f r o m t h e GL- t h e o r y w i t h o u t r e s o r t t o any a p r i o r i unknown
2
q u a n t i t y : AL ( d e r i v e d f r o m 6MAX and f r o m MD* equ. ( 3 ) and ( 5 ) ) , B/A ( d e r i v e d f r o m equ. (3,4)
2
resp. equ. (5,6)) and L B ( d e r i v e d f r o m equ.
( 4 ) resp. e q u . ( 6 ) ) w i t h R = 0.088, G = 10 4.08 X 10 N /m2, C = 0.55 X I O - ~ ~ N , b = 2.55 X 1 0 - ~ ~ m ( a l l v a l u e s f r o m / 1 3 / ) . Whereas t h e t e m p e r a t u r e i n f l u e n c e on G
2
and C n e a r l y c a n c e l each o t h e r f o r AL .we o 50 1 0 0 1 5 0 2 0 0 2 % 300 e v a l u a t e B L ~ w i t h C = c o n s t which r e s u l t s Temperature T {K,
i n a s m a l l e r r o r ( 5%) between 273 and FIG 4 T-dependence o f d i s l o c a t i o n
2 resonance parameters.
4.2K. As f i g . 4 shows AL decreases by 2 80%
between 273 and 4.2K. With t h e p l o t o f B / A
2
we e l i m i n a t e L ( T ) ( w h i c h c o u l d be t h e reason f o r t h e 11L b e h a v i o u r ) a t t h e c o s t o f i n t r o d u c i n g B(T). I f we would assume A = c o n s t t h e d e p i c t e d B/A-behaviour would d i - r e c t l y r e p r e s e n t t h e T-dependence o f B. The a l m o s t l i n e a r B/A-behaviour f o r T > 50K c o u l d t h e n be e x p l a i n e d by phonon s c a t t e r i n g (Bph
-
T ) b u t t h e i n c r e a s e i n B/A a t T < 30K t h e n i s i n c o n t r a d i c t i o n t o any c u r r e n t t h e o r y / l ,16/. Furthermore, B/ A de- creases by t h e f a c t o r o f 3 between 273 and 4.2K whereas f r o m t h e o r e t i c a l arguments a n o r d e r o f magnitude r e d u c t i o n o f B i s p r e d i c t e d/?/.
2
consider t h e T-dependence o f LB i n f i g . 4 , assume L = const and f i n d B
-
T down t o 50K, B = const a t T < 30K and B(273K)/B(4.2K) 2 10. Such a B(T)-behaviour i s i n com-p l e t e accordance w i t h t h e t h e o r e t i c a l p r e d i c t i o n s .
measurements. With counted l o c a l A - v a l u e s
i n t h e range (1 t o 4).10'crn-~ and B(273)/ A = 02 1 . 7 - 1 0 - ~ ~ ~ s ( f i g . 4 ) we g e t B(273K) =
2
T-dependence o f B: I n f i g . 5 we p l o t B* = ( L ' B ) ~ / ( L B)213K = B(T)/B(273K) f o r L inde- pendent o f T versus T/QD. For t h i s p l o t we have taken care o f C = C ( T ) . Since i n n o r - mal metals B i s t h e sum o f t h e phonon (Bph) and e l e c t r o n (Be) drag and s i n c e f o r T -t 0 Bph -t 0 we o b t a i n from our r e s u l t s B(T + O)/B(273K) = Be/B(273K) = 1/10 and
Be = 0.11 Bph(273K). S u b t r a c t i o n o f t h e T-independent B,-contribution (dashed d o t t e d l i n e i n f i g . 5 ) y i e l d s t h e phonon c o n t r i b u t i o n Temperature T [Hi
0 50 100 150 200 250 300 345
Bph(T)/Bph(273K) (open squares) i n good
( 4 2 2). 1 0 - ~ ~ s / r n ~ . T h i s value conf inns pre- , 4 . , , , . , . . .
0 01 02 03 04 05 06 07 08 09 10
vious B-values obtained by theory and o t h e r
TIB,
agreement w i t h theory /12/. As can be seen
from f i q . 5 f o r Cu t h e approximation B/B(273K)= 'bO
-
"
experiments ( c . f . c o m p i l a t i o n given by /15/).
FIG 5 T-dependence o f t h e normalized For t h e e l e c t r o n drag c o e f f i c i e n t we o b t a i n
-
d i s l o c a t i o n drag c o e f f i c i e n t s .-3- BIB(273K)
- - - B,/B(273)0
+ (B-Be)/Bph(273K)
- - - Bp,, theorte
l
(Bra~lsford)T-dependence o f A : The reasonable values f o r ~ ( 2 7 3 K ) and B, and t h e good agreement o f B(T) w i t h phonon s c a t t e r i n g t h e o r y a p o s t e r i o r i j u s t i f y our L = const assumption.
2
This L = const i s even s t r o n g e r supported by t h e behaviour o f L B a t T < 30K ( f i g . 3 ) : Since a t these temperatures Bph << Be and thus B = Be = const i t f o l l o w s t h a t L =
const. Therefore t h e l i n e a r T-dependence o f f,~' i n t h i s temperature range can o n l y be due t o t h e 1-dependence o f A w h i c h i s a d m i t t e d l y a strange conclusion. However, recent measurements of DR i n s l i g h t l y y - i r r a d i a t e d samples u s i n g t h e SSC-technique
(SSC = sandwich s i n g l e c r y s t a l , a novel b u f f e r technique designed f o r t h e protec- t i o n o f t h e pinned d i s l o c a t i o n s a g a i n s t QSD induced break-away 1171) show t h a t any L ( T ) - e f f e c t between 4.2 and 300K must be excluded and prove t h a t A = A (T) 1181.
2
Fig.4 (curve AL w i t h L = const) d i r e c t l y shows t h a t P decreases l i n e a r l y from 150 t o 4.2K (and probably even t o lower T 151). A l i n e a r e x t r a p o l a t i o n p o i n t s t o r e s i d u - a l DR damping a t T
-.
OK. However, more work i s needed a t very low T s i n c e A l e r s and Zimmermann /5/ measured MD e f f e c t s o n l y a t a s i n g l e f. Since o f course t h e d i s l o - T/273 ( d o t t e d l i n e ) f o r T > 50K and B/B(273K)=E
l.
0.1 (dashed d o t t e d l i n e ) f o r T < 30K can be
g
09 U
used. For numerical e v a l u a t i o n o f B we need
F
08t o know e i t h e r A o r L. Since L cannot be mea-
-J 0 7
sured independent1 y o n l y t h e A
-
measurement 0 6 (e.g. by t h e e t c h - p i t technique) o f f e r s/ /
C5-356 JOURNAL DE PHYSIQUE
c a t i o n d e n s i t y ( i n t h e sense o f t o t a l d i s l o c a t i o n length/cm3 ) i s T-independent we ha- v e t o c o n c l u d e t h a t t h e number o f l o o p s c o n t r i b u t i n g t o d i s l o c a t i o n resonance i s tem- p e r a t u r e dependent. Such a n e f f e c t p o i n t s towards some i n t i m a t e p r o p e r t y o f t h e d i s - l o c a t i o n s i n v o l v e d i n MHz-DR and p r o b a b l y has t o be d i s c u s s e d i n terms o f T-dependent k i n k - (and perhaps s t a c k i n g f a u l t - ) b e h a v i o u r . I t seems i m p o s s i b l e t o a t t r i b u t e t h e observed
A
( T ) t o t h e t e m p e r a t u r e dependent d e n s i t y o f t h e r m a l l y generatedk i n k
-
p a l r s , s i n c e t h i s cannot e x p l a i n e.g. A (T) a t <100K. Thus we a t t r i b u t e t h e A ( T ) be- h a v i o u r t o an u n e x p l a i n e d T - i n f l u e n c e on t h e m o t i o n o f g e o m e t r i c a l k i n k s i n Cu. Acknowledgements
.-
Our thanks a r e due t o o u r c o l 1 eagues I r e n e Z e f e r e r who c a r e f u l l y t y p e d o u r m a n u s c r i p t s , t o Fred Ramm who d i d t h e s k i l l f u l d r a w i n g and p h o t o g r a p h i c work, t o L o t h a r Kaul and K u r t D i e s i n g e r f o r g r o w i n g t h e s i n g l e c r y s t a l s and l a s t n o t l e a s t t o Ludwig C o s l e r who cooperated w i t h sample p r e p a r a t i o n and m e t a l l o g r a p h i c work. Work s u p p o r t e d by t h e DFG (SF0 125).References
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