Ultrasonic attenuation in UPt3

386

Journal of Magnetism and Magnetic Materials 63 & 64 (1987) 386-388 North-Holland, Amsterdam

U L T R A S O N I C A T T E N U A T I O N IN UPt3 V MULLER,

D MAURER,

K -D SCHOTTE

and U SCHOTTE*

Fachberelch Physlk, Frele Unloersuat Berlin, Armmallee 14, D-IO00 Berhn 33, Germany * Hahn- Meaner- InsutuI fur Kernforschung, Ghemcker So" 100, D- 1000 Berlin 39, Germany The peak m the attenuation for longitudinal ultrasound as a function of temperature around 12 K is interpreted as being caused by a heating phenomenon, this is supported by anlsotropy of the effect The A-hke peak at T~ could also have its origin in this mechanism depending on large Grunetsen parameters

T h e i n t e r m e t a l h c c o m p o u n d UPt~ IS a s o - c a l l e d " h e a v y f e r m l o n " m e t a l , w h i c h m e a n s that the specific h e a t C o r the coefficient 3' = C~ T has the v e r y l a r g e v a l u e of 420 m J / K 2 T h i s reflects a high e l e c t r o n i c level d e n s i t y T h e o r i g i n of this high d e n s i t y Is the o p e n f-shell of the u r a m u m Ions T h e m a g n e t i c m o m e n t s , c o n n e c t e d with c o n f i g u r a tions of p r o b a b l y f2 or f3 c h a r a c t e r , a r e s u p p r e s s e d since the f e l e c t r o n s are i t i n e r a n t a n d f o r m v e r y n a r r o w e l e c t r o n i c b a n d s T h e i t i n e r a n t n a t u r e of th~s states ~s b e s t i l l u s t r a t e d b y the o c c u r r e n c e of s u p e r c o n d u c t i v i t y M u c h of the c u r r e n t i n t e r e s t in UPt~ ~s the p o s s l b l h t y of t r i p l e t i n s t e a d of slnglet p a i r i n g , so far triplet C o o p e r p a i r s h a v e , h o w e v e r . only b e e n identified in ~He A c t u a l l y the low t e m p e r a t u r e specific h e a t has a c e r t a i n similarity with that of ~He A l s o the u l t r a s o n i c a t t e n u a t i o n o b s e r v e d in o u r l a b o r a t o r y [ l ] has a p r o n o u n c e d p e a k as a f u n c t i o n of t e m p e r a t u r e just b e l o w the t r a n s i t i o n t e m p e r a t u r e T h i s IS v e r y similar to the findings for the a t t e n u a t i o n in n o r m a l a n d superfluld 3He However, one should not forget that metalhc systems c o n t a i n usually i m p u r i t i e s a n d o t h e r imp e r f e c t i o n s u n k n o w n in liquid h e l i u m S u p e r c o n d u c t w l t y , also of slnglet t y p e in h e a v y f e r m l o n materials should not necessarily be a "scaled" v e r s i o n of s u p e r c o n d u c t i v i t y in n o r m a l m e t a l s in the sense that o n e has just a h i g h e r d e n s i t y of states a n d a s h o r t e r c o r r e l a t i o n l e n g t h b e c a u s e of the h i g h e r m a s s of the c h a r g e c a r r i e r s T h i s p a p e r discusses a m a y b e r e l e v a n t d i f f e r e n c e n a m e l y the v e r y l a r g e e l e c t r o m c G r u n e l s e n p a r a m e t e r s of the o r d e r of 60 for UPt~ [2] T h i s can be d e r i v e d f r o m the unusual tern-

p e r a t u r e d e p e n d e n c e for l o n g i t u d i n a l u l t r a s o u n d at f r e q u e n c i e s of a few 100 M H z [3], at h i g h e r t e m p e r a t u r e s a b o v e the critical t e m p e r a t u r e T~ T h e G r u n e l s e n p a r a m e t e r g w e s the r e l a t i v e c h a n g e m t e m p e r a t u r e u n d e r the influence of a d e f o r m a t i o n u,j = (Ou,/Oxj + Ouj/Ox,)/2 at c o n s t a n t e n t r o p y , (Ul.U2,UO are the d i s p l a c e m e n t s at the p o s m o n (x~,xz,xO. the t h e r m o d y n a m i c d e f i n m o n of the G r u n e l s e n c o n s t a n t ~ b e i n g ~,, = - ( 0 In T/Ou.)~

(1)

A d e f o r m a t i o n of 10 s for u,j, w h i c h is c o m m o n m u l t r a s o n i c e x p e r i m e n t s , t o g e t h e r with a G r u n e i s e n c o n s t a n t of 60 for UPt3, will Imply a t e m p e r a t u r e c h a n g e of 6 m K at 10 K If o n e t a k e s into a c c o u n t the high specific h e a t it IS n o t difficult to find that a sizable a m o u n t of e n e r g y is s t o r e d as t h e r m a l e n e r g y m an elastic w a v e O f c o u r s e at low t e m p e r a t u r e s any effect. c o n n e c t e d to a t h e r m a l w a v e w h i c h is a c c o m p a n y i n g an elastic w a v e , s h o u l d d i s a p p e a r , since the h e a t c a p a c i t y v a n i s h e s T h e s a m e a r g u m e n t a p p l i e s at high t e m p e r a t u r e s , say 30 K. since the h e a t c a p a c i t y d u e to the e l e c t r o m c states d e c r e a s e s s t e e p l y T h i s e x p l a i n s the a b s o r p t i o n d a t a for l o n g i t u d i n a l s o u n d m two d i f f e r e n t cryst a l l o g r a p h i c d i r e c t i o n s d i s p l a y e d in the figure, w h i c h show a p e a k s t r u c t u r e Th~s is u n c o m m o n for n o r m a l m e t a l s w h e r e this t h e r m a l effect is completely negligible It is not difficult to a r g u e m o r e q u a n t i t a t i v e l y [4] A c c o r d i n g to L a n d a u a n d Llfschltz [5] the t h e r m a l c o n d u c t i v i t y c a n be an i m p o r t a n t loss m e c h a n i s m for s o u n d w a v e s (see also ref [6]) W e use this i d e a for the following d i s c u s s i o n In

0 3 0 4 - 8 8 5 3 / 8 7 / $ 0 3 50 © E l s e v i e r S c i e n c e P u b h s h e r s B V ( N o r t h - H o l l a n d Physics P u b l i s h i n g D i w s i o n )

V Muller et al / Ultrasomc attenuatmn m UPt~

ordinary metals such an assumption would not be reasonable since there the electrons with their m u c h higher fermi velocity would easily transfer the heat and no thermal d e s e q u I h b r l u m could e v o l v e H e a v y f e r m l o n materials b e h a v e m o r e like insulators since their fermi and their sound velocities are not so different Using the formula of L a n d a u and Llfschltz for the dissipated e n e r g y E , K is the thermal conductivity, E=--~

(gradT) 2dV,

(2)

one can calculate the attenuation coefficient In the figure this coefficient is actually defined as an inverse extinction length for the amplitude 2oq = - E / ( v, Eel) ,

(3)

v, IS the longitudinal sound velocity In the direction t, and Ee~ is the elastic e n e r g y of the sound w a v e Eel = p J" u 2 d V where p is the density In o r d e r to c h e c k w h e t h e r the adiabatic (or lsentropic) condition is really satisfied we define first f r o m the e q u a t i o n of heat a thermal diffusion time 1/'rd,tt = q 2 r / C ,

(4)

w h e r e C is the specific heat and q the w a v e v e c t o r of the ultrasonic w a v e If tO~'d,n>>1, to IS the f r e q u e n c y , the thermal c o n d u c t i v i t y would be too low to short-circuit the t e m p e r a t u r e differences in an ultrasonic w a v e For UPt~ at 10 K one finds experimentally C = 3 6 J/(mol K) [7], r = 14 x 10 -3 J / ( s c m K) [8], and v~ = 3 9 x 105 cm/s With the density of 19 4 g / c m 3 or a m o l a r v o l u m e of 42 c m 3 one gets finally the large value tO~'d,tr= 300 at 500 M H z T h e r e f o r e the s o u n d w a v e leads to nearly adiabatic c o m p r e s s i o n s R e p l a c i n g in (2) grad T by l ) , T g r a d u,,, in using (1), we obtain after a few manipulations from

(3) a,-

y~2TC 1 2pv3 Td,ff

(5)

T h a t a, IS p r o p o r t i o n a l to q2 or co2 has been verified experimentally [3] F u r t h e r the anlsotropy, as seen in the figure, seems to c o m e out correctly UPt3 is a hexagonal system, so the c o n v e n t i o n a l

387

definition of the Grunelsen p a r a m e t e r 1) = ilK~ C, where K 1s the elastic m o d u l and /3 the thermal expansion p a r a m e t e r , must be replaced by the tensorlel form [9] C f ~ 1 = (cii ÷ c12)/31--{- C13/3~,

Cfl3 = 2c1~/31 + c~3/3~

(6)

If one uses the experimental values for the elastic constants [9] cll + Cl2 = 45, c ~ = 17, C33 ~ 29 (In units of 10 ~ e r g / c m 3) and c o m p a l r s the specific heat, which is linear in T, with the coefficient To = 1 0 × 1 0 1 5 e r g / ( c m ~ K 2) with the thermal expansion parameters, which are also linear in T, that is /31/T---- 1 5 × l 0 6 K - 2 and / 3 2 / T = --1 0 × 10 - 6 K -2 one obtains for the Grunelsen p a r a m e t e r 1~1 = 51 and ~ = 22 So the ratio of the absorption should be, a c c o r d i n g to (5), ( l h / f i l ) 2 = 0 2 which IS not far off the experimental values in fig 1 With (4), (5) and (6) and the experimental data for the specific heat [7] and the thermal c o n d u c tivity [8] one can calculate the absorption as a function of t e m p e r a t u r e and obtains two curves for a~, and a3 with a p e a k close to 10 K and a width which c o r r e s p o n d s nearly to the experimental findings T h e absorption rate at the maxim u m is h o w e v e r only 0 6 d B / c m c o m p a r e d to 3 d B / c m o b s e r v e d experimentally for a~ T h e thermal c o n d u c t i v i t y has been m e a s u r e d for a different polycrystalhne sample T h e r e f o r e , G o t t w l c k [11] c h e c k e d the b-axis crystal He o b t a i n e d a c o n d u c t i v i t y which was five times lower at 10 K c o m p a r e d to the polycrystalhne reference E v e n allowing a factor of 2 for the large g e o m e t r i c a l uncertainty in the e x p e r i m e n t one has to assume that o t h e r loss m e c h a n i s m s are m o r e efficient than the thermal c o n d u c t i v i t y For e x a m ple a local transfer of heat between electrons and p h o n o n s could be responsible for the large absorption, since the p h o n o n Grunelsen p a r a m e t e r is only 2 to 3 For the s u p e r c o n d u c t i n g phase one can use (5) in describing the low t e m p e r a t u r e absorption peak [1] at Tc Below 1 K the ratio of K / C increases by a factor of about three c o m p a r e d to the value at 5 or 10 K T h e value of I ) 2 T C would, h o w e v e r , d e c r e a s e by a factor of 100 f r o m 10 to 1 K because of the T 2 d e p e n d e n c e of T C Since the effect itself

388

V Muller et al / Ultrasomc attenua#on in UPh

100

I

'

I

,

I

,

75

)

90

73

80

71

"o ¢m D

0 cQ. 0

70

! -

69

8

•

o~

.

B 520 MHz

•'

6O

67

50

, 0

) 20

,

I 40

,

~ 60

,

t

,

80

65 100

T)n K Fig 1 Longitudinal sound attenuatmn along the c-axis (C 467 MHz) and along the b-axis (B 520 MHz) Note the dflferent scales

is not much smaller than before, again one has to look for a more efficient relaxation mechanism than provided by the thermal conductivity, especially since the Grunelsen parameter for the superconductor d In Tc/d In V has experimentally been found to be of the same size as in the normal state [12] We think that the order parameter relaxation could be slow enough to be responsible for the absorption peak In conclusion we suggest that the unusual behavior of longltudmal attenuation of UPt~ is due to temperature vartatlons in the sound wave Whether this is a general phenomenon for heavy fermlon materials which have usually quite large electronic Grunelsen parameters remams to be shown Also the relaxation mechanism must stdl be identified Contrary to normal metals the elastic wave couples also to the density of states This is a very c o m m o n p h e n o m e n o n in case of antlferromagnets, where one also sees strong absorption of longitudinal ultrasound at the onset of ordering [10] So we want to make the point that sound absorption in thts unusual superconductor is of a rather usual type

References [1] V Muller, D Maurer, E W Scheldt, Ch Roth and K Luders, Solid State Commun 57 (1986) 319 [2] B Luthl, J Magn Magn Mat 52 (1985)70 [3] V Muller, D Maurer, K de Groot, E Bucher and H E Bommel, Phys Rev Lett 56 (1986) 248 [4] K D Schotte, D Forster and U Schotte, Z Phys B 64 (1986) 165 [5] L D Landau and E M Llfschltz, Elastlzltatstheorle § 35 (Akademle-Verlag, Berlin, 1975) [6] P Fulde, C Pethlk, D Pines and K J Quader, preprlnt M L Kuhc, K H Bennemann and V Muller, to appear [7] J J M Franse, A de Vnsser, A Menovsky and P H FrIngs, J Magn Magn Mat 52 (1985)61 [8] J J M Franse, A Menovsky, A de Vlsser, C D Bredl, U Gottwlck, W Lleke, H M Mayer, U Rauchschwalbe, G Sparn and F Steghch, Z Phys B 59 (1985) 15 [9] M Yoshlzawa, B Luthl and K D Schotte, Z Phys B 64 (1986) 169 [10] C W Garland, m Physical Acoustics, vol VII, Ultrasonic Investigation of Phase Transitions and Critical Points, eds W P Mason and R N Thurston (Academic Press, New York, 1970) [11] U Gottwlck, TH Darmstadt, private commumcatmon [12] J O Willis, J D Thompson, Z Fmk, A de Vlsser, J J M Franse and A Mevosky, Phys Rev B 31 (1985) 1654