Heavy fermion superconductivity

82

Physica 126B (1984) 82-91 North-Ilolland, Amsterdam

HEAVY FERMION SUPERCONDUCTIVITY F. STEGLICH, C.D. BREDL, W. LIEKE, U. RAUCHSCHWALBE and G. SPARN Institut

f u r F e s t k ~ r p e r p h y s i k , Technische Hochschule Darmstadt, D--6100 Darmstadt, Fed. Rep. Germany

We discuss the phenomenon of heavy-fermion s u p e r c o n d u c t i v i t y by i n v e s t i g a t i n g the exemplary system CeCu2Si 2 in i t s normal and superconducting s t a t e s . Like the non-superconducting compound CeAl3, CeCu2Si 2 behaves as a Kondo l a t t i c e . We f i n d no evidence f o r t r i p l e t s u p e r c o n d u c t i v i t y in CeCu2Si 2. i . HEAVY FERMION EFFECTS IN METALS I t is well known t h a t the l o w - t e m p e r a t u r e prop e r t i e s of o r d i n a r y , i . e . non-superconducting and non-magnetic, metals can be described f a i r l y well by the phenomenological theory of Fermi l i quids ( I ) . In t h i s way, the observed simple power laws in temperature, such as a T-independent spin ( " P a u l i " ) s u s c e p t i b i l i t y , xo, a l i n e a r e l e c t r o n i c s p e c i f i c heat, Ce = yT, or a q u a d r a t i c e l e c t r o n - e l e c t r o n s c a t t e r i n g c o n t r i b u t i o n to the electrical resistivity, Pe-e = AT2, are a s c r i b e d to l o w - l y i n g e x c i t a t i o n s of the non-magnetic met a l l i c ground s t a t e . For d-band m e t a l s , the spec i f i c - h e a t c R e f f i c i e n t - y c a n reach values up to about IOmJ/KL mole. Since y as w e l l as xo and /A are p r o p o r t i o n a l to the e f f e c t i v e ( q u a s i p a r t i c l e ) mass me, in d-band metals me is found to exceed the f r e e - e l e c t r o n mass mo by up to a f a c t o r o f about 10. Much h i g h e r y values are sometimes observed f o r metals c o n t a i n i n g paramagnetic i o n s , namely f o r d i l u t e Kondo a l l o y s l i k e CuFe (2) as well as c e r t a i n i n t e r m e t a l l i c rare-ea-r-th and a c t i n i d e compounds. For example, the C r - d e r i v e d s p e c i f i c heat in CuCr v a r i e s l i n e a r l y w i t h temperature as T + O, i ~ l y i n g the g i g a n t i c y value of 16J/K 2 mole-Cr ( 3 ) . In terms o f a l o c a l F e r m i - l i q u i d theory f o r the s i n g l e Kondo i m p u r i t y ( 4 ) , the corresponding huge q u a s i p a r t i c l e d e n s i t y of s t a t e s r e s u l t s from a resonant phase s h i f t a t the Fermi l e v e l EF due to those conduction e l e c trons t h a t "screen" the l o c a l i z e d magnetic moment well below the c h a r a c t e r i s t i c "Kondo temp e r a t u r e " TK. The e x i s t e n c e of such heavy-mass q u a s i p a r t i c l e s ("heavy f e r m i o n s " ) below TK is supported by recent numerical r e s u l t s obtained from the Bethe-Ansatz s o l u t i o n of the Kondo-imp u r i t y problem ( 5 ) . Enhanced F e r m i - l i q u i d e f f e c t s are also found in q u i t e a l a r g e number o f i n t e r m e t a l l i c s of - Ce, Yb, U o~ Np where y is ranging from R 2.1 to ~ 2 j / K mole-R. A v a r i e t y of grounds t a t e p r o p e r t i e s is r e a l i z e d in these m a t e r i a l s : While some o f them, l i k e CeAl 3 ( 6 ) , remain in the heavy F e r m i - l i q u i d s t a t e down to u l t r a l o w temperatures, in c e r t a i n systems the "heavy f e r mion" e f f e c t s a p p a r e n t l y c o e x i s t w i t h o t h e r c o l l e c t i v e phenomena, namely spin f l u c t u a t i o n s l i k e ~

0378-4363/84/$03.00

© Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

in UAI 2 (7) and magnetic o r d e r l i k e in NpSn3 (8) or CeAl 2 ( 9 ) . Renewed i n t e r e s t in t h i s kind of i n t e r m e t a l l i c compounds has emerged from the discovery of "heavy fermion" s u p e r c o n d u c t i v i t y in the t e t r a gonal system CeCu2Si 2 ( 1 0 ) , and l a t e r in two Ubased i n t e r m e t a l l i c s , UBe13 (11) and UPt 3 (12). Despite several i n t e r e s t i n g d i f f e r e n c e s in t h e i r physical b e h a v i o r , these three systems share the most e s s e n t i a l p r o p e r t y , namely the superconducting phase t r a n s i t i o n t a k i n g place in a system o f heavy fermions whose e f f e c t i v e mass is comparable to the mass of the myon. While the U compounds are the s u b j e c t of several c o n t r i butions to t h i s conference, in the present papec we wish to review the c u r r e n t s t a t u s of CeCu2Si 2 A f t e r surveying in Sect. I I f o r t h i s exemplary system the c h a r a c t e r i s t i c f e a t u r e s of a heavyfermion superconductor (HFS), we w i l l address in I l l the dependence of various physical prop e r t i e s on sample q u a l i t y , a s u b j e c t t h a t has played a c r u c i a l r o l e since the discovery of HFS. In IV, recent r e s u l t s f o r both n o r m a l - s t a t e CeCu2Si 2 and CeAl 3 are discussed, which s t r o n g l y support the " K o n d o - l a t t i c e " nature of these compounds. In V, we present new r e s u l t s t h a t we bel i e v e to be r e l e v a n t to the question of whether CeCu2Si 2 is a s i n g l e t superconductor l i k e conv e n t i o n a l superconductors or whether i t is the f i r s t t r i p l e t superconductor, in analogy to the s u p e r f l u i d s t a t e of the ( n e u t r a l ) heavy Fermi l i q u i d He3 (13). The paper w i l l be summarized and a b r i e f o u t l o o k w i l l be given in VI. 2. CHARACTERISTIC FEATURES OF A HEAVY FERMION SUPERCONDUCTOR L o c a l i z e d magnetic moments a t high temperat u r e s . The temperature dependence o f the magnetic s u s c e p t i b i l i t y is shown in Fig. I f o r two p o l y c r y s t a l l i n e CeCu2Si 2 samples, one prepared s t o i c h i o m e t r i c a l l y , the o t h e r one w i t h 5at1# Cu d e f i c i t . Here, the r e s u l t s of only the s t o i c h i o m e t r i c sample w i l l be discussed. The temperature dependence o f x becomes very weak ( " P a u l i l i k e " ) below T = 101<, w h i l e i t seems to approach a Cur i e - l i k e behavior above the temperature l i m i t of these measurements. This t r a n s i t i o n from a nonmagnetic, low-T phase to a local-moment, high-T

F. Steglich et al. / Heavy ferrnion superconductivity

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FIGURE 1 S u s c e p t i b i l i t y data f o r CeCu2Si2 (sample No. 10, • ) and CeCul.9Si2 ( Q ) polycrystals. Main part shows ×.T vs. T, while inset shows ×-1 vs. T to demonstrate the Curie-Weiss behavior of x(T). Hypothetical curves f o r free Ce3+-ions are shown as dashed l i n e s . phase follows roughly a Curie-Weiss dependence (inset) and agrees with the results for quite a large number of CeCu2Si2 samples (14-16). We note that, even at a temperature as high as 550K, the effective magnetic moment is found to be considerably reduced compared to that of the J = 5/2 state of Cej+. The existence of magnetic moments_associated with the localized 4f electron of Ce3+ is supported by the results of inelastic neutron-scattering experiments, which reveal a s p l i t t i n g of the J = 5/2 state by the tetragonal crystal f i e l d (CF) into three doublets with excitation energies 140K and 364K relative to the groundstate (17). Pronounced high-T anomalies in both the electrical r e s i s t i v i t y and thermoelectric power have also been ascribed to (incoherent) scattering from CF-split Ce3+ ions (18). Normal heavy F e r m i - l i q u i d state at low temperatures. The s u s c e p t i b i l i t y results of Fig. i show that CeCu2Si2 does not order magnetically as one might have expected from the considerable concentration of magnetic moments that e x i s t for T > IOK. The average value of the Tindependent s u s c e o t i b i l i t y at low temperature is Xo = (8±I) "lO-Bm3/m°le (16,19), which exceeds the Pauli s u s c e p t i b i l i t y of simple metals by more than two orders of magnitude. In the normal-state r e s i s t i v i t y a quadratic term ~s found (Fig. 2) whose c o e f f i c i e n t A lO~cm/K ~ is about 107 times larger than the corresponding value determined f o r the ordinary metal tungsten (20). Further, one sees in Fig. 3 that the normal-state s p e c i f i c heat as measured in magnetic f i e l d s B > 2T is dominated by a l i n e a r ~

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FIGURE 2 Normal-state r e s i s t i v i t y of CeCu2Si2 as a function of T2. term, with y being of the order of 1J/K2 mole. Taken together, these results prove the existence of heavy fermions in the low-T normal state of CeCu2Si2. Their effective mass can be estimated from a combined analysis of normal-state and superconducting properties and is found to be about 220 times larger than the free-electron mass (21). This implies a correspondingly high density of quasiparticle states at EF. As was already mentioned, very similar low-temperature properties have been found for CeAl3 (6). In contrast to this compound, however, CeCu2Si2 becomes superconducting below Tc ~ O.6K (10). Superconductin9 groundstate. The discovery of bulk superconductivity (10,15) in a system like CeCu2Si2 was a big surprise in view of the well-known trends among ordinary metals: there, localized magnetic moments as well as very high densities of state (namely due to the presence of spinfluctuations) s t r ~ d i s f a v o r the formation of s u p e r c o n d u c ~ ~ h e most i n t r i guing features associated with the superconducting phase transition in CeCu2Si2 are gigantic absolute values of both the specific heat jump AC (see Fig. 3) and the slope B~2 of the upper c r i t i c a l magnetic f i e l d , Bc2(T), at Tc(21). AC as well as B~2 scale with the specific-heat coe f f i c i e n t y which proves (10) that in CeCu2Si2 (as well as in other HFS) the superconducting Cooper pairs are formed by those heavy fermions which are responsible for the unique low-temperature properties in the normal state. Since neither the ordinary d-band metal LaCu2Si2 (with ~ 4mJ/K2 mole) nor any known MCu2Si2 homolog ~as been found superconducting, the formation of a heavy Fermi-liquid phase (in connection with the quenching of the localized magnetic moments) appears to be a necessary condition for

t£ Steglich et ai. / Heavy fermion supercm~¢htctivil.~

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FIGURE 3 Specific heat of a "high-Tc" CeCu2Si2 sample, plotted as C/T vs. T, at several magnetic f i e l d s . For B >--~T, sample is in the normal state (cf. Ref. 23). superconductivity in CeCu2Si2. This constitutes an important difference between HFS and the socalled "magnetic superconductors" l i k e HoMo6S8 or ErRh4B4: Whereas heavy-fermion superconductiv i t y cannot e x i s t in the absence of f electrons, in t h ~ e r compounds the superconductivity, here carried by d electrons, is possible despite the presence of l o c a l i z e d 4f electrons because of an extremely weak d - f i n t e r a c t i o n (22). 3. VARIATION OF PHYSICAL PROPERTIES IN CeCu2Si2 A sensitive dependence of both superconducting and normal-state properties of CeCu2Si2 on preparation and annealing conditions was already reported in Refs. I0 and 15. Later, a wide scatter of Tc's (between 0.11 and 0.65K) was found f o r s t o i c h i o m e t r i c a l l y prepared polycrystals (23). Moreover, CeCu2Si2 single crystals grown from stoichiometric melts were found to remain in the normal state down to mK temperatures when measured at ambient pressure (23-25). These single crystals e x h i b i t very high residual r e s i s t i v i t i e s , t y p i c a l l y lOOp~cm. S u r p r i s i n g l y enough, they become superconducting, with Tc ~ O.5K, i f subjected to an external pressure of only a few kbar (24). Subsequent e f f o r t s were directed toward ( i ) improvement of sample q u a l i t y (26,27), especially: growth of superconducting single crystals (27,28); and ( i i ) evaluation of possible correl a t i o n s between superconducting and normal-state properties (23,19,29). Unfortunately, the char a c t e r i z a t i o n of the samples is s t i l l not yet s a t i s f a c t o r y (28). One important r e s u l t can, however, be stated: Structure refinement performed with high r e l i a b i l i t y on single crystals at room temperature reveals an abnormally large

Debye-Waller f a c t o r f o r the Cu s u b - l a t t i c e (30), displaying a reduced electron concentration at the Cu sites. Though these results would also be compatible with a considerable concentration of Si atoms on Cu s i t e s , i t is more l i k e l y that they indicate the existence of about 5-10 at% Cu vacancies at room temperature (27,28).Recent hightemperature s p e c i f i c - h e a t results suggest that the majority of these vacancies may be "thermally activated" (31). I t might, however, be a n t i cipated that a s l i g h t Cu d e f i c i t already exists in the low-temperature phase of s t o i c h i o m e t r i c a l l y prepared CeCu2Si2. This led us to expect that (part of) these Cu vacancies w i l l be f i l l e d i f an appropriate excess of Cu is used f or sample preparation. In f a c t , single crystals e x h i b i t ing bulk superconductivity and low residual r e s i s t i v i t y at ambient pressure could be subsequently grown from melts with 20 - 30 at% Cu excess by W. Assmus and M. Herrmann using the Bridgman technique (28). In the case of p o l y c r y s t a l l i n e samples, an even lower Cu excess suffices to s t a b i l i z e superconductivity (26,27): the highest Tc of 0.69K along with a rather sharp t r a n s i tion was obtained f o r 10 at% Cu excess. On the other hand, a Cu d e f i c i t of not more than 3 at~ was found to suppress superconductivity completely . I t is i n t e r e s t i n g to note that such polycrys t a l l i n e samples prepared with Cu d e f i c i t show a remarkably d i f f e r e n t behavior from s t o i c h i o metric and Cu-excess material: They e x h i b i t ( i ) higher residual r e s i s t i v i t i e s (~ lO0~]cm) i n d i cating a less perfect l a t t i c e , ( i i ) larger concentrations of magnetic impurities (presumably "non-transformed" Ce3+ ions, r e t a i n i n g t h e i r f u l l magnetic moments at low temperatures) that give r i s e to spin-glass type effects at low T (19), and ( i i i ) greater i n t r i n s i c s u s c e p t i b i l i ties at elevated temperatures (see Fig. i ) . We ascribe t h i s l a t t e r observation (33) to a reduced Kondo-interaction between the excited CFstates of Ce3+ and the conduction electrons. A s i m i l a r trend concerning the CF qround states is also evident from recent low-temperature results of the s p e c i f i c heat (29) and, though less pronounced, of the magnetic s u s c e p t i b i l i t y (19). I t is tempting to c o r r e l a t e this "weakening of the Kondo e f f e c t " to changes in the underlying bandstructure (32), presumably induced by the changes in Cu concentration (33). Such a c o r r e l a t i o n appears quite plausible in view of results of resonant photoemission experiments on CeCu2Si2 which suggest considerable h y b r i d i z a t i o n between the Ce-derived 4f and the Cu-derived 3d wavefunctions (34). We now return to the p o l y c r y s t a l l i n e CeCu2Si2 samples that were prepared with stoichiometric composition. As mentioned before, they show a large scatter of t r a n s i t i o n temperatures. I f we e s p e c i a l l y consider those with "low Tc" (Tc
F. Steglich et al. / Heavy fermion superconductivity as well as the inverse of the s p e c i f i c - h e a t coe f f i c i e n t (taken at Tc), ¥-1 (Tc) ' on the other, Since both To (35) and T -z (36) can be considered as being roughly proportional to the " s i n g l e - i o n " Kondo temperature TK, one can conclude from these i n v e s t i g a t i o n s that already a TK reduction by not more than 30-40% is s u f f i c i e n t to suppress superconductivity in CeCu2Si2 (23). Furthermore, these "low-Tc" samples though being bulk superconductors according to the DC Meissner e f f e c t do not show much signature of superconductivity in the s p e c i f i c heat: When measured at zero magnetic f i e l d , the low-T spec i f i c heat is dominated by a l i n e a r term nearly as large as in the normal state, and no jump can be seen at Tc. We have concluded that these samples are "gapless superconductors" with a density of states near EF that is almost i d e n t i c a l to the one in the normal state (23). These l a t t e r samples are e s p e c i a l l y i n t e r e s t i n g since they allow f o r determining the "normal-state" specif i c heat at zero magnetic f i e l d , which is otherwise not possible f o r those samples e x h i b i t i n g a pronounced jump AC in the B=O s p e c i f i c heat at Tc. I t i s , perhaps, not too s u r p r i s i n g to see that " s t o i c h i o m e t r i c " CeCu2Si2 polycrystals seem to provide a (continuous) t r a n s i t i o n between the Cu-excess samples with large AC ( c f . Fig. 7a) and the Cu-deficient samples that do not superconduct at a l l . 4. CeCu2Si2 - A KONDO LATTICE There is growing evidence that a large number of (nearly) t r i v a l e n t Ce compounds behave as "Kondo l a t t i c e s " (37). The high-temperature properties of these systems can be q u a l i t a t i v e l y understood in terms of independent C F - s p l i t Kondo ions. This is demonstrated in Fig. 4 for both CeAI 3 and CeCu2.2Si 2 by the temperature dependence of the e l e c t r o n i c s p e c i f i c heat associated with the 4f electron of Ce3+. I t can be i n t e r preted using a superposition of a Schottky anomaly, which is due to the c r y s t a l - f i e l d s p l i t ting of the J = 5/2 state into three doublets and i s , therefore, connected with an entropy change AS=RZn3, and a broad anomaly at lower temperature. The l a t t e r represents the numerical res u l t of the Bethe-Ansatz calculations f o r the single spin - 1/2 Kondo impurity (5). I t implies an entropy change AS=R~n2 i n d i c a t i n g a complete demagnetization of the c r y s t a l - f i e l d ground-state doublets by the Kondo e f f e c t . The " s i n g l e - i o n Kondo temperature" TK can be obtained through TK : 2.2Tp where the peak temperature Tp was o b t a i n e d b y adjusting the experimental entropy AS(T) :offACd~nT') to the theoretical value AS(TD) : O.45RZn2 (5). This y i e l d s TK = 5.5K f o r CeAI~ and = 13.5K f o r CeCu2.2Si 2. A Bethe-Ansatz c a l c u l a t i o n f o r c r y s t a l - f i e l d s p l i t Ce3+ ions would be h i g h l y desirable to allow f o r a quantitat i v e f i t , notably of the CeAI3 data. For more det a i l s the interested reader is referred to the o r i g i n a l l i t e r a t u r e (38,29). When such Ce i n t e r m e t a l l i c s are cooled below

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t~ Steglich et al. / Heavy j~,rmion superconductirity

86

A recent i n v e s t i g a t i o n of the low-temperature s p e c i f i c heat of CeAl3 and of "gapless superconducting" CeCu2Si2 revealed (43) that e a r l i e r discovered (44,23) maxima in the temperature dependence of the s p e c i f i c - h e a t coeff i c i e n t y(T) = C(T)/T at To ~ O.4K are, in f a c t , consistent with the assumption of a r e l a t i v e minimum of the density of states in the v i c i n i ty of EF. In passing we note t h a t , since this feature is rather i n s e n s i t i v e to magnetic f i e l d s as high as 8T, i t is also v i s i b l e in the normalstate data (B ~ 2T) of Fig. 3. As was f u r t h e r demonstrated in Ref. 43, the y(T) maximum expectedly disappears when the p e r i o d i c i t y of the Ce ions becomes destroyed by a l l o y i n g , e . g . , with Y. However, since only the symmetric part of the density of states enters the s p e c i f i c heat, no information can be obtained from these C(T) experiments concerning the r e l a t i v e position of EF with respect to that minimum. To this purpose, a study of the thermoelectric power (TEP), S(T), of Kondo l a t t i c e systems was i n i t i a t e d in our laboratory (45). The B = 0 results f o r CeAI3 and CeCu2.o2Si1.98 are shown in Fig. 5a. In order to suppress superconductivity in the l a t t e r system, the experiments were extended to B > 2T. These results are also shown in Fig. 5a (46). I t is well known (47) that the TEP of Kondo l a t t i c e s e x h i b i t s a negative peak near TK, in agreement with the t h e o r e t i c a l prediction f o r independent Kondo ions (41,47). In the l a t t e r case, S(T) ought to be linear at s u f f i c i e n t l y low T, where the Kondo-resonance density of states is independent of temperature (41,47). In a l a t t i c e , however, a non-linear T-dependence of the TEP is expected when kBT becomes smaller than the width of the "pseudo gap". Obviously, this expectation is v e r i f i e d f o r both CeAI3 and normal-state CeCu2.o2Si I 98" Even more s a t i s f y i n g is the f i n d i n g of a change in the

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FIGURE 5 Thermoelectric power (TEP) of CeCu2.o2Sil.98 and CeAl3 (a) and of Ceo.9Yo.lSi2 (b). Dashed s t r a i g h t lines in (a) indicate e x t r a p o l a t i o n from data at elevated temperatures to T = O.

sign of S(T) at almost exactly the same temperature TO at which the s p e c i f i c heat c o e f f i c i e n t ¥(T) has a maximum (47). The p o s i t i v e TEP sign as T ÷ 0 unequivocally proves that EF i n t e r sects a hole part in the density of states. This is, again, in s u r p r i s i n g l y good agreement with the t h e o r e t i c a l prediction that EF is located s l i g h t l y below the d e n s i t y - o f - s t a t e s minimum (41,47). For completeness, we show in Fig. 5b that S(T) does not change sign in Ceo.9Yo.iCu2Si 2, i . e . when the Kondo l a t t i c e is disturbed by replacing 10 at% Ce by Y. This forms a convincing case that the peak in ¥(T) and the change of sign in S(T) are i n t i m a t e l y correlated with each other. They demonstrate, we believe, the development of a coherence-derived "pseudo gap" at EF in the huge q u a s i p a r t i c l e density of states of non-magnetic (and normal-state) Kondo l a t t i c e s , such as the prototypical systems CeAl 3 and CeCu2Si2. 5. CeCu2Si2 - A TRIPLET SUPERCONDUCTOR In view of the seemingly unfavorable conditions f or a conventional type of superconductiv i t y in a system l i k e CeCu2Si2 and because of the s i m i l a r i t y (15) between the superconducting phase t r a n s i t i o n in a HFS and the superfluid phase t r a n s i t i o n in the heavy Fermi l i q u i d He3 (13) i t is only natural to suspect that CeCu2Si2 and i t s U-based counterparts are the f i r s t superconductors with s p i n - t r i p l e t , i . e . anisotropic, pairing (48,49). This might be due to an a t t r a c t i v e i n t e r a c t i o n between the heavy f e r mions mediated by spin f l u c t u a t i o n s rather than by phonons. On the other hand, i t was recently shown (50,51) that superconductivity in CeCu2Si2 can be understood in the frame of the BCS theory f o r conventional s p i n - s i n g l e t p a i r i n g , i f the coherent action of the Kondo e f f e c t along with an electron-phonon coupling typical of Kondo l a t tices is taken into account. I t has, furthermore, been argued that in a Kondo l a t t i c e , even ani~ s i n g l e t pairing might occur under circumstances (52). In order to see, i f evidence exists for possible t r i p l e t supercond u c t i v i t y in CeCu2Si2 we shall discuss below ( i ) the "Sommerfeld r a t i o " R between the low-T spin s u s c e p t i b i l i t y ×o and the s p e c i f i c - h e a t c o e f f i c i e n t ~, ( i i ) the temperature dependence of both the specific heat Cs(T) and the thermal conductivity cs(T ) in the superconducting state and ( i i i ) the upper c r i t i c a l magnetic f i e l d curve, Bc2(T). Our results w i l l be compared with t heor et ic a l predictions f o r p-wave p a ir in g in i s o t r o p i c systems ( l i k e He3). We wish to note, however, that due to the symmetry introduced by the l a t t i c e , a modified c l a s s i f i c a t i o n of superconducting phases may be more appropriate (53~. For the "nearly l o c a l i z e d " Fermi l i q u i d He~ × is enhanced over y by a f a c t o r of about 4 (~4), whereas f o r a single spin - 1/2 Kondo imp u r i t y , R = 2 (the "Wilson r a t i o " ) is expected (4). In the case of a Ce3+ ion in i t s CF-ground-

87

F. Steglich et al. / Heavy fermion superconductivity

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state doublet t h i s r a t i o is given by (15) R = (p~Ixo/Y)[~2kB2/p~ff(O)] where ~o is the i n duction constant, kB Boltzmann's constant and ueff(O) the low-T e f f e c t i v e moment of Ce3+ as calculated from the CF groundstate wavefunctions. One finds ueff(O) = 1.65uR for CeCu2Si2 and ueff(O) = 1.29p B f o r CeAI~(15). Using recent res u l t s of Xo and y as determined f o r i d e n t i c a l samples, R of normal-state CeCu2Si2 (19) and of CeAI3 (38) can be deduced (Fig. 6a and b). The pronounced temperature dependence of R ref l e c t s the v a r i a t i o n of y-Z(T) as discussed in the preceeding section, while, s u r p r i s i n g l y enough, the normal-state s u s c e p t i b i l i t y of both compounds is almost temperature independent below T = IK (19,38). Though, because of spin-orb i t coupling, the s u s c e p t i b i l i t y as measured is l i k e l y to d i f f e r from the spin s u s c e p t i b i l i t y , i t is i n t e r e s t i n g to f i n d R(OT-~'-=2.9 in the case of CeAI 3 which is close to the "Wilson r a t i o " . Un the other hand, R is much smaller f o r the HFS CeCu2Si2 . We f i n d R(O) : 0.8 as is found f o r conventional superconducting materials, where the electron-phonon i n t e r a c t i o n causes an enhancement of the s p e c i f i c heat r e l a t i v e to the spin susceptibility. Anisotropic superconductors may have a vanishing gap in c e r t a i n momentum d i r e c t i o n s which leads to simple power laws in T f o r various quant i t i e s . Thus, f o r instance, a T2-dependence of the u l t r a s o n i c attenuation was recently observed (55) f o r UPt3 well below Tc, in agreement with the expectation f o r the "polar state",which has a vanishing energy gap along a l i n e on the Fermi surface. For t h i s same state, both the s p e c i f i c heat Cs(T ) and the e l e c t r o n i c thermal conductiv i t y Ks(T ) ought to be proportional to T2 f o r T << Tc (56). On the other hand, however, a cubic temperature dependence of Cs(T ) was reported

FIGURE 7 a: Specific heat of two "high-Tc" CeCu2Si2-samples in log-log-representation. Straight l i n e s indicate simple power laws for comparison. b: Thermal c o n d u c t i v i t y of "high-Tc" CeCu2 o2Sii 98 in the superconducting state, Ks(T)~normaiized to the normal state values
for UBel3 and was interpreted as being due to the "axial state" of a t r i p l e t superconductor (57). I t s energy gap vanishes on two points on the Fermi surface. The corresponding r e s u l t s f o r a CeCu2.2Si 2 and a CeCu2Si2 sample, both with pronounced s p e c i f i c heat jumps at Tc + 0.65K, are displayed in the log-log representation of Fig. 7a. Though d i f f e r i n g s l i g h t l y in magnitude, Cs(T) shows a rather s i m i l a r temperature dependence f o r these samples, which e x h i b i t quite d i f f e r e n t values of the residual r e s i s t i v i t y I ~ ~ lO~cm f o r CeCu2.2Si 2 and ~ 40~cm CeCu2Si2). I t is evident from the f i g u r e that the low-T Cs(T ) results cannot be described by a T~ law (with T-independent exponent) over any substantial temperature range (58). Of course, these data also cannot be f i t t e d by an exponent i a l as predicted by the BCS theory (cf. Ref. 23), i n d i c a t i n g that no temperature-independent energy gap exists in CeCu2Si2. A s i m i l a r deviation from the BCS p r e d i c t i o n (59) is found for the r a t i o
88

F~ Steglich et al. / Heavy ]'errnion superconductivity

mation about the e l e c t r o n i c c o n t r i b u t i o n could be obtained from the measured ~s(T) data. We now turn to the l a s t topic of this sect i o n , the temperature dependence of the upper c r i t i c a l f i e l d 8c2(T ). For an i s o t r o p i c p-wave superconductor in the "pure l i m i t " (large mean free path ~), i t was shown (61) that in high magnetic f i e l d s the polar state is energet i c a l l y the most favorable one, thus determining Bc2(T ). Since the polar (as well as the a x i a l ) state has at l e a s t one d i r e c t i o n of non-reduced spin s u s c e p t i b i l i t y , there should be no paramagnetic pair breaking e f f e c t by+the external Bf i e l d ("Pauli l i m i t i n g " ) , i f B is oriented along this axis. This, however, holds in a f u l l y isot r o p i c system, where the order parameter w i l l o r i e n t i t s e l f with an axis of non-reduced susc e p t i b i l i t y along the f i e l d (see Ref. 61). Fig. 8a shows the Bc2(T) results as obtained resistiv e T y f o r a single c r y s t a l , prepared with a Cu excess of 30 at% (28). In these experiments the f i e l d , which was always p a r a l l e l to the current, was applied p a r a l l e l to e i t h e r the tetragonal axis (c) or the basal (Ce) plane (a). An unannealed crystal was chosen in order to approach the " d i r t y - l i m i t " case (small mean free path). In this way, we intended to reduce the dominating e f f e c t of diamagnetic pair breaking by the external f i e l d and, thus, to be able to check f o r the (non-)existence of Pauli l i m i t i n g (61). To compare our results with the predict i o n f o r p-wave superconductors, the e x i s t i n g theory was extended in a straight-forward way to a r b i t r a r y values of the mean free path (62). The only free parameter l e f t in the theory is the r a t i o ~o/~, where ~o = hVF/2~kBTc is a coherence length (v F being the Fermi v e l o c i t y ) , This r a t i o can be obtained by f i t t i n g the theory to the absolute value of the slope, B ~ , of 8c2(T) at Tc. For convenience, normalize~:quant i t l e s are used in Fig. 8a, where Btyp = (e/~kB) • (Tc Ye)/(~o/~) is a " t y p i cal f i e l d " . Here e_is the elementary charge, while y = 0.73 J/K Z mole (as determined f o r another superconducting single crystal prepared with a Cu excess of 26 at%, Ref. ~8) and p = 55u~cm (corresponding to A~IOA) are the respective values of the normal-state s p e c i f i c heat c o e f f i c i e n t and r e s i s t i v i t y , both taken at Tc, In contrast to what one expects f o r a pwave suBerconductor, the exoerimental Bc2(T ) curve becomes very f l a t below T/T c ~ 0.8 (T ~ O.5K). The f a c t o r - o f - t h r e e d~screpancy found as T ÷ 0 can only be understood by a considerable Pauli l i m i t i n g . In a d d i t i o n , unphysic a l l y large values f o r both the t r a n s i t i o n temperature Ten (~ 16K) and the reduced s p e c i f i c heat jump ~ i g h t ACo/yTco (~ 35) in the "pure l i m i t " are obtained within this analysis (62). On the other hand, a reasonable f i t of the Bc2(T ) data of Fig. 8a can be achieved in the framework of standard theory f o r " d i r t y - l i m i t " BCS superconductors by including both the Pauli

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l i m i t i n g and pair-breaking effects (63); the existence of p a i r breaking being inferred from the observatlon that Bc2 = 23 T/K as measured is smaller than the " d i r t y - l i m i t " value, Bc2 ( d i r t y - ~ = (12e/~3kB)(py) = 35 T/K (62). For the t h e o r e t i c a l r e s u l t displayed in Fig. 8a we used a pair-breaking parameter ~ = 0.11 as deduced from the measured value of AC/yTc (1.27, Ref.28), under the assumption that this r a t i o should equal the BCS value (1.43) in the absence of pair breaking. The r e l a t i v e importance of Pauli l i m i ting is determined by both the value of the Sommerfeld r a t i o at Tc, R ~ 0.65 (28), and a spino r b i t scattering parameter which was chosen to be ~so = 1.04 to obtain the s o l i d l i n e in Fig.8a. A f i n a l comment is in order concerning the slope 8#2, which is found to be e s s e n t i a l l y independent of the f i e l d d i r e c t i o n r e l a t i v e to the Ce planes (see Fig. 8a). As was stressed in Ref. 28, this indicates a s u r p r i s i n g l y i s o t r o pic Fermi surface in the low-T F e r m i - l i q u i d phase of CeCu2Si2. On the i other hand, a considerable anisotropy of Bc2 was recently observed f o r UPt3 single crystals (64) and subsequently interpreted as another signature of t r i p l e t superconductivity in this material (48). D i f f e r e n t from the set-up used in our experiment, in Ref. 64 the d i r e c t i o n of the current I was+fixed ( p a r a l l e l to the hexagonal a x i s ) , and the Bf i e l d w~s applied both p a r a l l e l and perpendicul a r to I . To rule out the p o s s i b i l i t y that in our s i n g l e - c r y s t a l experiment some "configurat i o n a l anisotropy" was missed, the upper c r i t i cal f i e l d of a p o l y c r y s t a l l i n e CeCu2Si2 sample was determined f o r the two d i f f e r e n t configurations (~ e i t h e r p a r a l l e l or perpendicular to 7). •

i

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F. Steglich et al. / Heavy fermion superconductivity

As is seen in Fig. 8b, the resulting Bc2(T ) curve does no# depend on the r e l a t i v e orientation between B and I . 6. SUMMARYAND OUTLOOK CeCu2Si2 samples with reproducible properties, including high-quality single c r y s t a l s , can be prepared by making appropriate choices of the stoichiometry. In this way a v a r i e t y of situations can be realized, ranging from non-superconducting samples to "gapless-superconducting" samples of low Tc and further to "high-Tc" samples with pronounced superconducting properties. Normal-state CeCu2Si2 shows typical signatures of a (non-magnetic) Kondo-lattice, p a r t i c u l a r l y in i t s low-temperature, heavy F e r m i - l i quid phase, which is best documented by a giant l i n e a r specific-heat c o e f f i c i e n t y. In this respect i t is phenomenologically closely related to CeAI3. However, while CeAI3 remains a non-superconducting heavy Fermi l i q u i d to very low temperatures, CeCu2Si2 becomes a heavy-fermion superconductor below 0.65K. These e s s e n t i a l l y d i f f e r e n t ground-state properties seem to correl a t e with quite d i f f e r e n t values of t h e i r temperature-independent low-T s u s c e p t i b i l i t i e s Xo: whereas in CeAI3 ×o exceeds the specific-heat c o e f f i c i e n t y by more than a factor of two, for CeCu2Si2 y is found to be enhanced over xo the l a t t e r being well known f o r conventional superconductors. In contrast to the normal Fermi-liquid state which is characterized by simple power laws in temperature, no simple power laws are found in the superconducting state of CeCu2Si2, although s t r i k i n g deviations from the BCS r e s u l t e x i s t for the temperature dependences of both the spec i f i c heat and thermal conductivity. A substant i a l Pauli l i m i t i n g is inferred from the measured temperature dependence of the upper c r i t i c a l magnetic f i e l d . The results discussed in this paper cannot, in the present state of the a r t , contribute to the excitement about possible t r i p l e t supercond u c t i v i t y in heavy-fermion superconductors. However, more d i r e c t experimental.probes, such as the Josephson e f f e c t or the proximity e f f e c t , as well as more r e a l i s t i c calculations, which should take into account the respective c r y s t a l anisotropies (53), are c e r t a i n l y necessary to d e f i o i t i v e l y solve the " t r i p l e t vs. s i n g l e t issue~ Only when these additional e f f o r t s ar~ successful, can the more essential question of the microscopic mechanisms underlying heavyfermion superconductivity be answered. I t w i l l be p a r t i c u l a r l y interesting to see whether the same mechanism operates in the d i f f e r e n t systems or whether d i f f e r e n t microscopic interactions can r e s u l t in rather s i m i l a r phenomena. Turning back to CeCu2Si2, we wish to conclude by stating that the present results along with recent evidence f o r e l a s t i c anomalies in this compound (24,65,66) are highly consistent with those theories (50-52) that invoke a phonon-

89

mediated s i n g l e t pairing between the heavy f e r mions in a Kondo l a t t i c e . ACKNOWLEDGMENTS We g r a t e f u l l y acknowledge the f r u i t f u l coopera. tion with J. Aarts, W. Assmus, G. Cordier, F.R. de Boer, S. Horn, J. Klaasse, S. Riegel, H. Sch~f e r , H. S p i l l e and G. Weber. One of us (F.S.) enjoyed the warm h o s p i t a l i t y of the Physics Group at the IBM - Th.J. Watson Research Center, Yorktown Heights, where the s u s c e p t i b i l i t y results of Fig. I were taken in collaboration with H. L i l i e n t h a l , T. Penney and S. von Molnar. We have benefitted from numerous enlightening discussions with P. Fulde, N. Grewe and D. Rainer, who also kindly supplied the computer program for the c r i t i c a l - f i e l d analysis. Special thanks are due to D.M. Ginsberg, D. Rainer, H. Rietschel and G. Weber for t h e i r careful reading and R. Lamatsch for her expert typing of the manuscript . This work was supported by the Sonderforschungsbereich 65 Frankfurt / Darmstadt.

REFERENCES (I) (2) (3) (4) 5) 6) (7) (8) (9) (10) (Ii) (12) (13) (14) (15) (16)

L.D. Landau, Zh. Eksperim. Teor. Fiz. 32 (1957) 59 (Sov. Phys. JETP 5 (1957) 101). see, e.g.: W.M. Star, PhD Thesis, Univers i t y of Leiden (1971), (unpublished). B.B. T r i p l e t t and N.E. P h i l l i p s , Phys. Rev. Lett. 27 (1971) i001. P. Nozi~res, J. Low Temp. Phys. 17 (1974) 31; J. Phys. (Paris) 39 (1978) 1117. For a recent review, see: N. Andrei, K. Furuya, and J.H. Lowenstein, Rev. Mod. Phys. 55 (1983) 331. K. Andres, J.E. Graebner, and H.R. Ott, Phys. Rev. Lett. 35 (1975) 1779. R.J. Trainor, M.B. Brodsky, and H.V. Culbert, Phys. Rev. Lett. 34 (1975) 1019. R.J. Trainor, M.B. Brodsky, B.D. Dunlap, and G.K. Shenoy, Phys. Rev. Lett. 37 (1976) 1511. F. Steglich, C.D. Bredl, M. Loewenhaupt, and K.D. Schotte, J. Phys. (Paris) 40 (1979) C5-301. F. Steglich, J. Aarts, C.D. Bredl, W. Lieke, D. Meschede, W. Franz, and H. Sch~fer, Phys. Rev. Lett. 43 (1979) 1892. H.R. Ott, H. Rudigier, Z. Fisk, and J.L. Smith, Phys. Rev. Lett. 50 (1983) 1595. G.R. Stewart, Z. Fisk, J.O. W i l l i s , and J.L. Smith, Phys. Rev. Lett. 52 (1984) 679. For a review, see: A.J. Leggett, Rev. Mod. Phys. 47 (1975) 331. B.C. Sales and R. Viswanathan, J. Low Temp. Phys. 23 (1976) 449. W. Lieke, U. Rauchschwalbe, C.D. Bredl, F. Steglich, J. Aarts, and F.R. de Boer, J. Appl. Phys. 53 (1982) 2111. J. Aarts, PhD Thesis, University of Amsterdam (1984), (unpublished).

90

t( Steglich et al. / Heav.r ,[erpni¢)n Sul~er~'oHductivity

(17) S. Horn, E. Holland-Moritz, M. Loewenhaupt, F. S t e g l i c h , H. Scheuer, A. Benoit, and J. Flouquet, Phys. Rev. B23 (1981) 3171. 18) W. Franz, A. Griessel, F. S t e g l i c h , and D. Wohlleben, Z. Phys. B31 (1978) 7. 19) U. Rauchschwalbe, W. Baus, S. Horn, H. S p i l l e , F. Steglich, F.R. de Boer, J. Aarts, W. Assmus, and M. Herrmann, I n t . Conf. on Valence Fluctuations, Cologne (1984); to appear in J. Magn. Magn. Mat. 20 D.K. Wagner, J.C. Garland, and R. Bowes, Phys. Rev. B3 (1971) 3141. 21) U. Rauchschwalbe, W. Lieke, C.D. Bredl, F. S t e g l i c h , J. Aarts, K.M. M a r t i n i , and A.C. Mota, Phys. Rev. Lett. 49 (1982) 1448. (22) See, e.g.: M.B. Maple, J. Magn. Magn. Mat. 31-34 (1983) 479. (23) C.D. Bredl, H. S p i l l e , U. Rauchschwalbe, W. Lieke, F. S t e g l i c h , G. Cordier, W. Assmus, M. Herrmann, and J. Aarts, J. Magn, Magn. Mat. 31-34 (1983) 373. (24) F.G. A ] i e v , N.B. Brandt, V.V. Moshchalkov, and S.M. Chudinov, Solid State Commun. 45 (1983) 215. (25) G.R. Stewart, Z. Fisk, and J.O. W i l l i s , Phys. Rev. B28 (1983) 172. (26) M. Ishikawa, H.F. Braun, and J.L. Jorda, Phys. Rev. B27 (1983) 3092. (27) H. S p i l l e , U. Rauchschwalbe, and F. Stegl i c h , Helv. Phys. Acta 56 (1983) 165. (28) W. Assmus, M. Herrmann, U. Rauchschwalbe, S. Riegel, W. Lieke, S. Horn, G. Weber, F. S t e g l i c h , and G. Cordier, Phys. Rev. L e t t . 52 (1984) 469. (29) C.D. Bredl, W. Lieke, R. Schefzyk, M. Lang, U. Rauchschwalbe, F. S t e g l i c h , S. Riegel, R. Felten, G. Weber, J. Klaasse, J. Aarts, and F.R. de Boer, see Ref. 19. (30) G. Cordier, private communication. (31) R. Felten, U. Umhofer, U. Rauchschwalbe, and G. Weber, see Ref. 19. (32) T. Jarlborg, H.F. Braun, and M. Peter, Z. Phys. B52 (1983) 295. (33) Similar observations were recently made on CeCu2Si2 single c r y s t a l s grown from d i f f e r e n t m e t a l l i c solvents (B. Batlogg, J.P. Remeika, A.S. Cooper, G.R. Stewart, Z. Fisk, and J.O. W i l l i s , see Ref. 19). (34) R.D. Parks, B. Reihl, N. M~rtensson, and F. Steglich, Phys. Rev. B27 (1983) 6052. (35) M. Lavagna, C. Lacroix, and M. Cyrot, J. Phys. F 12 (1982) 845. (36) R.M. Martin, Phys. Rev. L e t t . 48 (1982) 362. (37) F. S t e g l i c h , J. Magn. Magn. Mat. (to appear). (38) F.R. de Boer, J. Klaasse, J. Aarts, C.D. Bredl, W. Lieke, U. Rauchschwalbe, F. Stegl i c h , R. Felten, U. Umhofer, and G. Weber, see Ref. 19.

(39) A.P. Murani, K. Knorr, K.H.J. Buschow, and J. Flouquet, Solid State Commun. 36 (1980) 523. (40) G. GrUner and A. Zawadowski, Rep. Progr. Phys. 37 (1974) 1497. (41) N. Grewe, Solid State Commun. 50 (1984) 19, and references cited therein. (42) A.S. Edelstein, C.J. Tranchita, O.D. Mc Masters, and K.A. Gschneidner, J r . , Solid State Commun. 15 (1974) 81. (43) C.D. Bredl, S. Horn, F. Steglich, B. LUthi, and R.M. Martin, Phys. Rev. Lett. 52 (1984) 1982. (44) J. Flouquet, J.C. Lasjaunias, J. Peyrard, and M. Ribault, J. Appl. Phys. 53 (1982) 2127. (45) W. Lieke, G. Sparn, U. Gottwick, F. Stegl i c h , and N. Grewe, see Ref. 19. (46) Similar results were recently also found by D. Jaccard and J. Flouquet, see Ref. 19. (47) C.D. Bredl, N. Grewe, F. Steglich, and E. Umlauf, c o n t r i b u t i o n to t h i s conference. (48) C.M. Varma, p r e p r i n t (1984). 49) P.W. Anderson, p r e p r i n t (1984). 50) H. Razafimandimby, P. Fulde, and J. K e l l e r , Z. Phys. B54 (1984) 111. 51) N. Grewe, Z. Phys. B,(to appear). 52) T. Matsuura, K. Miyake, H. Jichu, and Y. Kuroda, p r e p r i n t (1984). F.J. Ohkawa and H. Fukuyama, p r e p r i n t (1984) 53) G.E. Volovik and L.P. Gor'kov, p r e p r i n t (1984). P.W. Anderson, p r e p r i n t (1984). 54) For a recent review, see: D. V o l l h a r d t , Rev. Mod. Phys. 56 (1984) 99. 55) D.J. Bishop, C.M. Varma, B. Batlogg, E. Bucher, Z. Fisk, and J.L. Smith, p r e p r i n t (1984). 56) C.M. Varma, private communication (1984). 57) H.R. Ott, H. Rudigier, T.M. Rice, K. Ueda, Z. Fisk, and J.L. Smith, Phys. Rev. Lett. 52 (1984) 1915. 58) Note, however, that one of the s t o i c h i o m e t r i c a l l y prepared polycrystals (No. 7, Ref. 23, with D = 3.5u~cm) behaves somewhat d i f f e r e n t : ° I t shows a very small bump in C(T) at T = O.3.T c below which C(T) is almost ~ T3 when measured at 8 = O. At the same temperature, an additional diamagnetic signal is v i s i b l e in the a c - s u s c e p t i b i l i t y measurement. This feature is shifted downwards by a magnetic f i e l d and completely suppressed by Bc2(O) : 1.5T. (59) J. Bardeen, G. Rickayzen, and L. Tewordt, Phys. Rev. 113 (1959) 982. (60) G. Sparn, Diploma Thesis, TH Darmstadt (1984), (unpublished). (61) K. Scharnberg and R.A. Klemm, Phys. Rev. B22 (1980) 5233. (62) D. Rainer et a l . (to be published). (63) P. Fulde and K. Maki, Phys. Rev. 141 (1966) 275. K. Maki, Phys. Rev. 148 (1966) 362.

F. Steglich et aL / Heavy fermion superconductivity

(64) J.W. Chen, S.E. Lambert, M.B. Maple, Z. Fisk, J.L. Smith, G.R. Stewart, and J.O. Willis, preprint (1984). (65) B. Bellarbi, A. Benoit, D. Jaccard, J.M. Mignot, and H.F. Braun, Phys. Rev. B,(to appear). (66) R. Mock and G. GUntherodt, J. Phys. C: Solid State Physics,(forthcoming).

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