ULTRASONIC ABSORPTION AND DISPERSION IN AMORPHOUS SUPERCONDUCTORS

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ULTRASONIC ABSORPTION AND DISPERSION IN

AMORPHOUS SUPERCONDUCTORS

W. Arnold, A. Billmann, P. Doussineau, C. Frenois, A. Levelut

To cite this version:

JOURNAL DE PHYSIQUE

CoZZoque C6, supptdment au nO1 2, Tome 42, de'cembre 1981 page C6-37

U L T R A S O N I C ABSORPTION AND D I S P E R S I O N I N AMORPHOUS SUPERCONDUCTORS

W. Arnold,

A.

~illrnann*, P. ~oussineau*, C.

ren no is*

and

A.

~evelut*

E'raunhofer-Institute, 6600 Saarbrticken, F . R. G.

" ~ n i v e r s i t d Paris V I , Laboratoire drUZtrasons, 75230 Paris, France

L i k e disordered i n s u l a t o r s , amorphous metal s c o n t a i n tunnel1 i n g centers (TLS) which dominate t h e i r low-temperature p r o p e r t i e s . This has been shown m a i n l y by u l t r a s o n i c experiments /I/. I n c o n t r a s t t o amorphous i n s u l a t o r s , t u n n e l l i n g centers i n a-metals r e l a x very r a p i d l y due t o t h e i r coup1 i n g t o e l e c t r o n s /2/. It i s obvious, t h a t t h i s c o u p l i n g could be v e r i f i e d i n amorphous superconductors.

The e f f e c t o f s u p e r c o n d u c t i v i t y on the acoustic p r o p e r t i e s o f a-metals has been c a l - c u l a t e d r e c e n t l y /3/. For instance, the absorption o f ultrasound i s p r e d i c t e d t o drop r a p i d l y f o r T

<

Tc r e l a t i v e t o the value i n t h e normal state. The o r i g i n of t h i s e f f e c t i s explained i n the f o l l o w i n g way: the acoustic wave p e r t u r b s the popu- l a t i o n d i f f e r e n c e o f t h e TLS and t h e r e f o r e gives r i s e t o a r e l a x a t i o n a l process.

In

t h e normal s t a t e the r e l a x a t i o n time T1 o f the TLS i s extremely s h o r t because o f the s t r o n g c o u p l i n g between TLS and electrons. Due t o t h e p a i r condensation f o r e l e c t r o n s t h i s r e l a x a t i o n channel disappears i n the superconducting s t a t e : TI becomes l o n g e r and the r e s u l t i n g a t t e n u a t i o n i s m o d i f i e d g r e a t l y .

I n order t o v e r i f y these p r e d i c t i o n s , an experiment has been c a r r i e d o u t on a-PdZr a1 l o y (Tc = 2.62 K), using l o n g i t u d i n a l waves o f frequencies w / 2 n = .74 GHz and 1.3 GHz /4/. As expected, a t very low temperatures (T = .4 K) t h e r e l a x a t i o n attenua- t i o n becomes very small and m a g n e t i c - f i e l d dependent up t o f i e l d s t r e n g t h s o f t h e order o f Hc2. I n c r e a s i n g the temperature, the a t t e n u a t i o n r i s e s r a p i d l y and reaches the normal s t a t e value a t a temperature Tn which, s u r p r i s i n g l y , turned o u t t o be much lower than Tc ( F i g .

1).

Here, we present an e x p l a n a t i o n o f t h i s unexpected behavior. We show t h a t i t i s necessary to take i n t o account n o t o n l y the r e l a x a t i o n o f the TLS by t h e r m a l l y e x c i t e d q u a s i - p a r t i c l e s s t i l l present a t T

<

T,, b u t a1 so t h e r e l a x a t i o n o f the TLS by thermal phonons. Moreover, i t i s necessary t o i n v o l v e a stronger TLS e l e c t r o n c o u p l i n g than p r e v i o u s l y assumed. Furthermore, we corroborated these ideas by c a r r y i n g o u t an experiment i n a-CuZr w i t h a low Tc ( = .4 K), such t h a t the r o l e o f phonons becomes almost n e g l i g i b l e .

Our explanation o f the experiments i n PdZr i s based on the f o l l o w i n g hypotheses: a) When the r e l a x a t i o n o f the TLS caused by the e l e c t r o n s i s dominant (H

>

Hc2), the

r e l a x a t i o n r a t e becomes so strong t h a t even a t the lowest temperatures (

.4

K)

C6-38 JOURNAL DE PHYSIQUE

the c o n d i t i o n w ~ :

<<

1 holds and the 20

a t t e n u a t i o n i s then weakly temperature-

-

la

dependent. This i s a consequence o f

1

o 1

x

5

the d i s t r i b u t i o n f u n c t i o n o f r e l a x a t i o n

3

s

w

times P ( T ) i n h e r e n t i n the tunnel1 i n g w E lf

s

model, and T: i s the f a s t e s t r e l a x a t i o n

E

Y Lo~ltudlnal waves

time o f t h i s d i s t r i b u t i o n . VI

F;

J -Theory a1

When the r e l a x a t i o n o f the TLS by elec-

0.m

t r o n s i s suppressed

(H

=

0,

T

<

Tc),

the c o n t r i b u t i o n o f the thermal phonons 0.1 a 2 0.5 1 2 s 10

lWF€RANRf l K l t o the re1 a x a t i o n r a t e becomes n o t i c e a b l e

.

A t the lowest temperatures i t manifests Fig. 1: U l t r a s o n i c absorption i n PdZr i t s e l f by a Ts-dependence o f the attenua-

t i o n , i n d i c a t i n g t h a t as i n i n s u l a t o r s t h e one-phonon process dominates /5/.

A t T

>

1.5 K the r e l a x a t i o n r a t e even w i t h o u t any a p p l i e d f i e l d i s so s t r o n g t h a t WT;

<<

1 holds and a plateau i s obtained f o r the a t t e n u a t i o n .

We have c a r r i e d o u t numerical c a l c u l a t i o n s which can be found i n d e t a i l i n /6/. For the c a l c u l a t i o n o f the r e l a x a t i o n a t t e n u a t i o n we use the f o l l o w i n g equation / 3 / :

rm

Here N i s the number o f TLS per u n i t volume and energy,

P

i s the d e n s i t y o f the mate- r i a l , v i s the sound v e l o c i t y ,

P =

( k ~ ) - l ,

Y

i s the deformation p o t e n t i a l between TLS and phonons. 2E i s the l e v e l s p l i t t i n g o f the TLS. S i m i l a r l y , the r e l a t i v e change o f the sound v e l o c i t y i s given by:

_,.

,TP

d L ? L

(2)

avlv

=-de

d((3E)sech2(flE)

-

2 ~ V 2

1

+

w

2~

2(E,T]

t,min

I n Eqs (1,2) the re1 a x a t i o n r a t e

ax is

now determined by two mechanisms:

i) by e l e c t r o n s

(*TI

and f o r T<Tc by q u a s i - p a r t i c l e s ( e s s e n t i a l l y +exp(-Tc/T) and i i ) by phonons ( 4 T 3 )

/5/.

Assuming Yc0.8 eV and a r a t h e r s t r o n g c o u p l i n g o f

PV'=

0.8 between e l e c t r o n and TLS, we c a l c u l a t e the s o l i d l i n e s i n F i g . 1 which agree v e r y w e l l w i t h the experimental curves. Having a c l o s e r l o o k t o t h e behavior o f t h e ab- s o r p t i o n i n the v i c i n i t y o f Tc, we can say t h a t i t i s i n p a r t i c u l a r the s t r o n g coup- l i n g of the TLS t o e l e c t r o n s which dominates the r e l a x a t i o n mechanism a t Tc, a1 though the phonons s t i l l c o n t r i b u t e . Because

~ T Y

already (c 1, t h e r e l a x a t i o n a b s o r p t i o n e x h i b i t s a plateau, whatever the re1 a x a t i o n process might be /5,6/.

( E q . ( l ) ) . We added the c o n t r i b u t i o n due t o resonant c a r - ' - ' q J i n t e r a c t i o n o f the TLS w i t h phonons, which v a r i e s

I-

f a m i l i a r l o g a r i t h m i c c o n t r i b u t i o n (

-<

C I n (T/To) due as C I n (T/To). To i s a reference temperature.

The r e s u l t i s t h a t a t Tc the v e l o c i t y should show a

i

change o f slope i n c o n t r a s t t o behavior o f the ab- sorption. This can e a s i l y be explained. Provided

a

T: a 1, the i n n e r i n t e g r a l i n Eq. ( 2 ) r e s u l t s i n

i.

I ~ ( U T ~ ( E , T ) ) /6/. Therefore the r e l a x a t i o n r a t e i s

5

s t i l l e x p l i c i t l y present i n the v e l o c i t y through

_P

t

<,

which o f course contains the r a p i d change o f T i 1 8 - caused by the e l e c t r o n s a t Tc, and the change o f

slope depends on the r e l a t i v e weight o f the phonon and e l e c t r o n processes. I n order t o v e r i f y t h i s , we

t o the resonant process plays a r o l e below 200 ;K. With i n c r e a s i n g T, however, the

0 eo - _{0 oo0} 0

IT

-

cu Zr T,: .4 K Shear WOW U S 745 MHz

-

f(K)

c o n t r i b u t i o n o f the re1 a x a t i on process becomes n o t i c e a b l e

.

Because Tl changes r a p i d -

1 1

have c a r r i e d o u t an experiment i n t h e amorphous super- T*mparatua conductor C U ~w i t h a Tc ~ Z ~= ~0.4 ~K ( t r a n s v e r s e waves F i g . 2: A v / v i n CuZr,

w / ~ T T = 740

MHz).

As can be seen i n Fig. 2, o n l y the sol i d 1 in e : theory

l y w i t h T, and a l s o because the r e l a x a t i o n process decreases the sound v e l o c i t y w i t h i n c r e a s i n g T, the t o t a l change o f v e l o c i t y passes through a maximum and then de- creases w i t h i n c r e a s i n g T. For T > T c , however, TI o n l y c o n t a i n s t h e weak temperature dependence due t o normal e l e c t r o n s . Then, a t temperatures f o r which wTm c 1 holds,

A V / V i s given by

-

C/2 I n (T/To). Adding the c o n t r i b u t i o n due t o resonant i n t e r a c - t i o n which i s s t i l l present, t h i s r e s u l t s i n a p o s i t i v e slope o f e x a c t l y one h a l f o f i t s magnitude a t low temperatures, which we indeed observe experimentally.

I n sumnary we have shown experimentally and t h e o r e t i c a l l y t h a t t u n n e l l i n g c e n t e r s i n an amorphous superconductor do couple t o e l e c t r o n s as o r i g i n a l l y proposed /3/.

References

/1/ J .L.Black, i n Metal1 i c Glasses, Ed. H.J. Guntherodt (Springer, N.Y.) 1980 /2/ B.Golding, J.E.Graebner, A.B.Kane, and J.L.Black, Phys.Rev.Lett.

9,

1478 (1978) /3/ J.L.Black and P.Fulde, Phys.Rev.Lett.

2,

453 (1979)

/4/ G.Weiss, W.Arnold, and H.J .Guntherodt, J .Phys.Colloq.

41,

C8-742 (1980)

/5/ J.Jackle, Z.Phys.

3,

212 (1972)