ULTRASONIC STUDY OF DISLOCATION TLS IN ALUMINUM SINGLE CRYSTALS

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ULTRASONIC STUDY OF DISLOCATION TLS IN ALUMINUM SINGLE CRYSTALS

A. Hikata, C. Elbaum

To cite this version:

A. Hikata, C. Elbaum. ULTRASONIC STUDY OF DISLOCATION TLS IN ALUMINUM SINGLE CRYSTALS. Journal de Physique Colloques, 1985, 46 (C10), pp.C10-293-C10-296.

�10.1051/jphyscol:19851065�. �jpa-00225449�

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JOURNAL D E PHYSIQUE

Colloque C10, supplement au n 0 1 2 , Tome 46, dBcembre 1 9 8 5 page C10-293

ULTRASONIC STUDY OF DISLOCATION TLS IN ALUMINUM SINGLE CRYSTALS

A. HIKATA AND C. ELBAUM

Metals Research Laboratory, Brown University, Providence, R.I. 02912, U.S.A.

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Nous avons observe' un comportement inattendu de la vitesse et de l'attenuation des ultrasons en fonction, respectivement, de la tempgrature et de l'amplitude, dans un monocristal d'aluminium dans l'btat supraconduc- teur, soumis 2 une contrainte de bias. Ce comportement ressemble 2 celui des mgtaux amorphes aux basses tempe'ratures (T < 1K) 02 il a 6te' interpret6 par l'effet tunnel dans des 6tats 5 deux niveaux. Nous interprgtons nos rgsultats en termes de l'effet tunnel pour des dislocations (crochets), entre deux niveaux (des puits de potentiel adjacents) produits par la contrainte de bias. Dans l'e'tat normal, la dissipation due aux e'lectrons, agissant sur l'effet tunnel, supprime ce processus.

Abstract

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We report unusual behavior of the temperature dependence of sound velocity and amplitude dependence of attenuation in an aluminum single crys- tal when the sample is in the superconducting state and is subjected to a bias stress. The observed behavior is similar to that found in metallic glasses at low temperatures (T < lK), where it has been explained in terms of two level tunneling systems (TLS). We interpret our results on aluminum in terms of tunneling of dislocations (kinks) between the two levels (of adjacent potential wells) produced by the bias stress. In the normal state, the effect of damping on the observed tunneling due to electrons, is to sup- press the process.

"Anomalous" temperature dependence, at low temperature (T < lK), of ultrasonic velocity, Av/v, has been observed in amorphous materials by many workers (1,2).

This dependence consists of a linear increase of Av/v with InT at low temperatures, followed by a maximum (usually around 1K) and a rapid decrease as the temperature increases further. Such £eatures, together with saturable ultrasonic attenuation, linear T dependence of specific heat, quadratic T dependence of thermal conductivi- ty, etc., have been explained by postulating the existence of low energy (< 1K) excitations referred to as two-level tunneling systems (TLS) ( 3 , 4 ) . The TLS model assumes that the low energy excitations arise from the quantum mechanical tunneling of some entity (an atom or group of atoms) between the two minima of an asymmetric double-well potential. In addition, the model requires a broad and smooth distribution in the energy of the tunneling states. It is this requirement that distin- guishes glasses from crystalline solids. However, many instances are reported that

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19851065

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JOURNAL DE PHYSIQUE

show g l a s s - l i k e c l i a r a c t e r i s t i c s i n d i s o r d e r e d c r y s t a l s . Ackerman e t a 1 (5) gave a n e x t e n s i v e review on t h i s s u b j e c t . The anomalous t e m p e r a t u r e dependence of Av/v d e s c r i b e d above i s c o n s i d e r e d t o be a c l e a r s i g n a t u r e of a TLS t y p e t u n n e l i n g mechanism. Here we r e p o r t r e s u l t s on t e m p e r a t u r e dependence of u l t r a s o n i c v e l o c i t y change a s w e l l a s t h e s a t u r a b l e a t t e n u a t i o n i n a n aluminum s i n g l e c r y s t a l which show c h a r a c t e r i s t i c s a n a l o g o u s t o m e t a l l i c g l a s s e s . Cle i n t e r p r e t t h e s e i n terms of TLS a r i s i n g from t h e t u n n e l i n g motion of k i n k s on d i s l o c a t i o n s . Samples used a r e aluminum s i n g l e c r y s t a l s grown from t h e m e l t . The p u r i t y o f t h e samples is c h a r a c - t e r i z e d by a r e s i s t L v i t y 2 a t i o o f % 1000 between 300K and 4.2K. Two u l t r a s o n i c waves a r e p r o p a g a t e d s i m u l t a n e o u s l y , one f o r measuring t h e v e l o c i t y and a t t e n u a t i o n changes and t h e o t h e r f o r t h e dynamic b i a s stress. The d e t a i l o f t h e dynamic b i a s s t r e s s method i s d e s c r i b e d i n Ref. 6. Three t r a n s d u c e r s a r e a t t a c h e d t o t h e sam- p l e , a p a i r of 20 o r 30 MHz s h e a r t r a n s d u c e r s o n (110) p l a n e s f a c i n g each o t h e r f o r s e n d i n g and r e c e i v i n g s i g n a l s r e s p e c t i v e l y , and t h e t h i r d (5 o r 1 0 MHz l o n g i t u d i - n a l ) t r a n s d u c e r o n a (100) f a c e f o r t h e b i a s stress. The p h a s e d e t e c t i o n method

(7) i s u s e d f o r t h e v e l o c i t y change measurements. The e r r o r o f t h e measurements i s

f 2 x S i m u l t a n e o u s l y w i t h t h e v e l o c i t y measurements, t h e c o r r e s p o n d i n g a t - t e n u a t i o n changes a r e monitored u s i n g a n a m p l i t u d e d e t e c t e d echo. F i g . 1 shows t h e t e m p e r a t u r e dependence of t h e v e l o c i t y change Av/v p l o t t e d i n t h e l o g a r i t h m i c s c a l e f o r f o u r d i f f e r e n t c a s e s : c u r v e A f o r no b i a s s t r e s s and no magnetic f i e l d ; c u r v e B f o r f u l l b i a s s t r e s s b u t no magnetic f i e l d ; c u r v e C f o r no b i a s s t r e s s b u t f u l l magnetic f i e l d (130 g a u s s ) ; c u r v e D f o r f u l l b i a s s t r e s s and f u l l magnetic f i e l d .

I n c u r v e A , t h e r e i s a s h a r p v e l o c i t y minimum a t a t e m p e r a t u r e j u s t below t h e su- p e r c o n d u c t i n g t r a n s i t i o n t e m p e r a t u r e Tc (1.18K), which i s c o n s i s t e n t w i t h t h e the- o r i e s (8) and t h e e x p e r i m e n t a l r e s u l t s o f o t h e r i n v e s t i g a t o r s ( 9 ) . A f t e r p a s s i n g through t h e minimum, t h e v e l o c i t y i n c r e a s e s w i t h d e c r e a s i n g t e m p e r a t u r e and r e a c h e s a v a l u e i n d e p e n d e n t of t e m p e r a t u r e . I n t h i s r e p o r t , we a r e concerned mainly w i t h t h e v e l o c i t y change t a k i n g p l a c e i n t h e t e m p e r a t u r e range below O.SK, where t h e e f - f e c t of t h e s u p e r c o n d u c t i n g t r a n s i t i o n i s s m a l l . When t h e sample i s s u b j e c t e d t o a b i a s s t r e s s ( c u r v e B), t h e v e l o c i t y d e c r e a s e s c o n s i d e r a b l y ; f o r example, a t 1.6K t h e v e l o c i t y d e c r e a s e d by a f a c t o r o f 0.82 x (from p o i n t a t o p o i n t b i n F i g . 1 ) . As t h e t e m p e r a t u r e i s d e c r e a s e d from t h e p o i n t b , t h e v e l o c i t y g o e s t h r 0 u g h . a minimum j u s t below Tc a s i n t h e c a s e o f c u r v e A. With f u r t h e r d e c r e a s e s of t h e t e m p e r a t u r e , however, t h e v e l o c i t y p a s s e s through a maximum a t % 0.9K and t h e n de- c r e a s e s l i n e a r l y w i t h 1nT. T h i s i s a marked d i f f e r e n c e from t h e c a s e of c u r v e A, and prompted us t o c o n s i d e r TLS a s a p o s s i b l e c a n d i d a t e f o r t h e mechanism. C%en a magnetic f i e l d (130 g a u s s ) l a r g e r t h a n t h e c r i t i c a l f i e l d i s a p p l i e d , t h e r e s u l t i n g v e l o c i t y changes a r e shown a s c u r v e s C and D. The InT dependence observed i n t h e c a s e of c u r v e B no l o n g e r e x i s t s when t h e magnetic f i e l d i s a p p l i e d ( c u r v e D).

F i g . 2 shows t h e a m p l i t u d e dependence of t h e measuring wave a t t e n u a t i o n w i t h and w i t h o u t b i a s s t r e s s . Without b i a s s t r e s s , t h e t y p i c a l a m p l i t u d e dependence was ob- s e r v e d , i . e . , up t o a c e r t a i n a m p l i t u d e , t h e r e i s no a p p r e c i a b l e a m p l i t u d e dependence, and w i t h f u r t h e r i n c r e a s e of a m p l i t u d e , t h e a t t e n u a t i o n grows v e r y r a p i d l y . Upon a p p l i c a t i o n o f a b i a s s t r e s s , t h e a t t e n u a t i o n of t h e measuring wave of s m a l l a m p l i t u d e (-60 dB) jumps t o a h i g h e r v a l u e . With t h e b i a s s t r e s s s t i l l on, t h e am- p l i t u d e of t h e measuring wave i s t h e n i n c r e a s e d . A s can b e s e e n , t h e a t t e n u a t i o n s t a r t s d e c r e a s i n g w e l l b e f o r e t h e a m p l i t u d e f o r t h e o n s e t o f t h e a m p l i t u d e dependence of t h e z e r o b i a s s t r e s s c a s e , goes through a minimum and a s y m p t o t i c a l l y ap- p r o a c h e s t o t h e v a l u e s o b t a i n e d i n t h e c a s e of t h e z e r o b i a s s t r e s s . The InT dependence of Avlv o b s e r v e d i n t h i s s t u d y i s a t t r i b u t e d t o p r o c e s s e s i n v o l v i n g d i s - l o c a t i o n s ( s p e c i f i c a l l y k i n k s on d i s l o c a t i o n s ) f o r t h e f o l l o w i n g two r e a s o n s ; i) i t i s produced by a s m a l l b i a s s t r e s s (".

l o 6

dynes/cm2) which i s known t o a f f e c t t h e r e s p o n s e of d i s l o c a t i o n s ( 6 ) , b u t does n o t produce measurable changes i n o t h e r p r o p e r t i e s of s o l i d s . i i ) The e f f e c t i s n o t p r e s e n t a f t e r c a r e f u l a n n e a l i n g of t h e sample a t 550C f o r 24 h r s . , which r e d u c e s t h e d i s l o c a t i o n d e n s i t y , and i t r e a p p e a r s a f t e r s l i g h t (0.15%) p l a s t i c d e f o r m a t i o n , which i n c r e a s e s t h e d i s l o c a t i o n d e n s i t y . I n view of t h e above, we o f f e r t h e f o l l o w i n g q u a l i t a t i v e d i s c u s s i o n s . Because of t h e s m a l l n e s s (19) of t h e k i n k - P e i e r l s p o t e n t i a l i n A l , i t i s e x p e c t e d t h a t even a t t e m p e r a t u r e s below 1 K k i n k s on d i s l o c a t i o n s w i l l b e s u b s t a n t i a l l y d e l o c a l i z e d . I n t h e p r e s e n c e of a b i a s s t r e s s , however, a l l t h e g e o m e t r i c a l k i n k s (11) on a g i v e n d i s l o c a t i o n a r e bunched up by b e i n g f o r c e d t o s h i f t toward a p i n n i n g p o i n t o r o t h e r o b s t a c l e , and t h u s t o i n t e r a c t more s t r o n g l y w i t h e a c h o t h e r . S i n c e t h e kink- P e i e r l s p o t e n t i a l i a i n v e r s e l y p r o p o r t i o n a l t o t h e s q u a r e of t h e k i n k w i d t h ( l o ) ,

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the potential barrier for kink motion will rise to the extent that the rate of over the barrier transmission of kinks becomes much less than that of subbarrier (tunneling) transmission below 2. 1K. Then the potential barriers adjacent to a kink form a double-well potential for each kink, and the dominant transmission between the two minima is by tunneling. This double-well potential is, however, not sym- metric because it has a superimposed asymmetric potential caused by the kink-kink interaction. Furthermore, the energy splitting due to these asymmetric double-well potentials has a broad distribution because; i) the distance between kinks on which the kink-kink interaction depends (11) varies from one end to the other on a given dislocation segment; ii) the angle between a dislocation line and the directions of the Peierls potential usually covers a wide range; iii) the dislocation loop length (between strong pinning points) extends over a wide range. Thus, the geometrical kinks subjected to a bias stress produce TLS similar to that believed to occur in amorphous materials. Conditions are thereby created for a temperature dependence of ultrasonic velocity characteristic of the tunneling TLS mechanism. For insu- lating glasses, the TLS model predicts that the logarithmic temperature dependence of Av/v (below the temperatures of the velocity maximum) arises from resonant interactions of TLS with the ultrasonic waves and has the form:

where n is the density of states in energy of the TLS, M is a coupling parameter, p is the mass density of the material, and TO is an arbitrary reference temperature.

Thus, in the presence of resonant interactions, the slope of Av/v vs. 1nT yields a value of nM2. In the case of Fig. 1, for example, the slope is 2 x lo7 erg/cm3

(we note that the raw data for the slope require a correction of ^Q^, 30% for the neg- ative slope observed in the absence of the bias stress; the corrected slope is shown by solid circles in Fig. 1). This value happens to be comparable to those found in amorphous solids. The TLS model also predicts the corresponding saturable attenuation a given by

where

2 3 2

aO = ( m M /pv ) ( 6 ~ /2kT) for .hw << 2kT

,

J is the acoustic intensity and Jc is the critical intensity. The solid curve drawn in Fig. 2 represents the best fit of the form (1

+

.J/J,)-~/~ to the data obtained by the amplitudes below the attenuation minimum. The arrow indicates the position of the Jc. The values of a thus obtained, however, show a linear dependence on frequency, as seen in Fig.

9.

This does not agree with the TLS model which predicts a quadratic frequency dependence. This discrepancy is exactly the same as observed between stages I and I1 of amplitude dependent attenuation in metallic glasses we reported previously (12). Recently, several authors (13) pro- posed mechanisms for the stage I1 of attenuation in metallic glasses but none seems to be completely satisfactory. The present results not only further the knowledge of dislocation dynamics but may also provide a clue to understanding the mechanisms in stage IS in metallic glasses. This research was supported by the National Sci- ence Foundation through the Materials Research Laboratory of Brown University.

1. See for example, S. Hunklinger and W. Arnold, in Physical Acoustics, ed. by W.P. Mason and R.U. Thurston, (Academic Press, New York and London) vol. 12, p. 155 (1976).

2. Glassy Metals I, IS, ed. by H. Beck and H.J. Gcntherodt, (Springer) vol. 46 (1931) and vol. 53 (1983).

3. P.11. Anderson, B.I. Halperin and C.M. Varma, Phil. Hag.

5,

1 (1972).

4. W.A. Phillips, J. Low Temp. Phys. _Z, 351 (1972).

5. D.A. Ackerman, D. Ploy, R.C. Potter, A.C. Anderson and T.7.N. Lawless, Phys. Rev.

B Z , 3886 (1981).

6. A. Hikata, R.A. Johnson and C. Elbaum, Phys. Rev.

BZ,

4856 (1970).

7. R.J. Blume, Rev. Sci. Inst.,

3,

1400 (1963).

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C10-296 J O U R N A L DE PHYSIQUE

8 . H. Ozaki and N. Mikoshiba, Phys. L e t t e r s

23,

52 (1966): I . E . Bulyzhenkov and V . I . I v l e v , Sov. Phys. JETP

9,

613 (1976): G. Kluge and V. Rudat, Sov. Phys.

Acoust.

2,

42 (1978).

9. V.D. F i l , V . I . Denisenko, P.A. Bezuglyi and A.S. P i r o g o v e , Sov. J. Low Temp.

Phys. 2 , 733 (1976): B.G.W. Tee and B.C. Deaton, Phys. Rev. L e t t e r s

3,

¹⁴³⁸

(1969):

10. G . S c h o t t k y , Phys. S t a t . S o l .

2,

697 (1964).

11. A. Seeger and P. S c h i l l e r , i n " P h y s i c a l A c o u s t i c s " , e d . by W.P. Mason, (Aca- demic, New York and London, 1 9 6 6 ) , v o l . I I I A , p. 361,

1 2 . A. H i k a t a , G. C i b u z a r and C . Elbaum, J. Low Temp. Phys.

49,

341 (1982).

1 3 . J . L . Black, i n r e f . 2, I , p . 167: M.A. C o n t i n e n t i n o , S o l i d S t a t e Corn.,

40,

781 (1981): W. Arnold, P. Doussineau and A. L e v e l u t , J. Physique

c,

^C9-553

(1983): Yu. M. ~ a l ' p e r i n , V.L. Gurevich and D.A. P a r s h i n , J . Physique L e t t e r s 45, L-747 (1984).

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0.2 0.3 0,5 1,0 2.0 3.0

TEMPERATURE (K)

160 MHz Measuring Wave 10 MHz Bias Wave

-60 . -40 -20 0

RELATIVE AMPLITUDE OF MEASURING WAVE (dB)

F i g . 1 Temperature dependence of v e l o c i t y change.

A: B i a s s t r e s s (0) i s z e r o , m a g n e t i c f i e l d

(H) i s z e r o ; A1 at 0.3 K

B: 0 on B = 0 ; C: Cf = 0 , H on; D: 0 on H on.

-

a F i g . 2 Amplitude dependence of

measuring wave w i t h and w i t h o u t b i a s s t r e s s . The s o l i d c u r v e r e p r e - s e n t s t h e b e s t f i t of t h e form ( 1

+

J / J c ) -112.

10 50 100 500 1000

Frequency (MHz) F i g . 3 Frequency dependence of

s a t u r a b l e a t t e n u a t i o n a