Twenty-five years of heavy-fermion superconductivity

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Physica B 359–361 (2005) 326–332 www.elsevier.com/locate/physb

Twenty-five years of heavy-fermion superconductivity Frank Steglich Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany

Abstract We discuss first the physical properties of the prototypical heavy-fermion superconductor CeCu2Si2 and, second, the relationship of its superconductivity with both a spin-density-wave-type quantum phase transition and a weak valence transition of Ce that occurs at high pressure. We, then, briefly address the isostructural heavy-fermion compound YbRh2Si2, displaying a local type of quantum critical behavior which appears to be detrimental to superconductivity. r 2005 Elsevier B.V. All rights reserved. PACS: 71.27.+a; 74.70.Tx; 75.30.Mb Keywords: Kondo effect; Heavy fermions; Superconductivity; Quantum criticality

1. Heavy-Fermion superconductivity: a short history Until the late 1970s superconductivity and magnetism were considered antagonistic phenomena. Since the local magnetic moments break up the spin-singlet state of the Cooper pairs [1], already very low concentrations of paramagnetic impurities suppress superconductivity in a classical (BCS) superconductor. In view of this antagonistic nature of superconductivity and magnetism, the discovery of the stoichiometric tetragonal compound CeCu2Si2 adopting a bulk superconducting state below T c
0:6 K (Fig. 1) was a big surprise Tel.: +49 351 46463900; fax: +49 351 46 463902.

E-mail address: [email protected] (F. Steglich).

[2]. Since the Ce ion in this compound is trivalent (4f1), it possesses a local magnetic moment. While usually less than 1 at% of Ce3+ ions are sufficient to suppress superconductivity in a classical superconductor, in CeCu2Si2 100 at% of them are necessary to generate the superconducting state. The reference compound based on nonmagnetic La3+ (4f0) is not a superconductor, and already a few at% La3+ substituted for Ce3+ destroy the superconducting state in CeCu2Si2. In contrast to what is known for the classical superconductors, nonmagnetic impurities are, therefore, pair breaking in this material. The normal (n)-state properties of CeCu2Si2 are as anomalous as the superconducting ones. Despite the large concentration of magnetic moments, CeCu2Si2 seems to exhibit a nonmagnetic

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Fig. 1. Low-temperature properties of polycrystalline CeCu2Si2 indicating a bulk superconducting transition: (a) resistivity (main part) and low-field ac susceptibility (inset) as a function of temperature as well as (b) specific heat as C/T vs. T for two samples [2].

low-temperature (T) state, cf. the almost Tindependent (Pauli-like) susceptibility below T ¼ 1 K [inset of Fig. 1(a)]. It is the action of the local Kondo interaction which leads to a demagnetization of the Ce3+ ions well below the Kondo temperature, T K
15 K for CeCu2Si2: The local Kramers degeneracy of the S eff ¼ 12 crystal-field (CF) ground state doublet is lifted by a ‘‘dynamical’’ antiferromagnetic (AF) coupling between the 4f electrons and the conduction electrons (of s-, pand d-symmetry), which thus become dressed by the local spin entropy and gain large effective masses m : From Fig. 1(b) it may be inferred that the specific heat of n-state CeCu2Si2 in the asymptotic limit T ! 0; realized when applying [2] an overcritical magnetic field B
3 T; is both very large and proportional to temperature, C ¼ g0 T: g0
1 J=K2 mol is exceeding the Sommerfeld coefficient of Cu by more than three orders of magnitude. Furthermore, the jump height at Tc, DC=T c ; is comparable to the giant value gðT c Þ ¼ CðT c Þ=T c : From these observations it had been concluded [2] that C(T) is indeed of electronic origin, with g(T) having the meaning of a T-dependent Sommerfeld coefficient and, in addition, that Cooper pairs are formed by these ‘‘heavy fermions’’ (HF). HF quasiparticles carrying an elementary charge |e| and a spin 12 (as concluded from the empirical Kadowaki–Woods [3] and Sommerfeld–Wilson [4] ratios) may be considered ‘‘weakly delocalized felectrons’’ or ‘‘composite fermions’’, consisting of

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a dominating f-part and weak admixtures of itinerant conduction-electron contributions. Of course, the HF are responsible for the pronounced T-dependence of the electrical resistivity of CeCu2Si2 at TXT c ; too, cf. Fig. 1(a). The large m values are the result of a correspondingly small renormalized Fermi velocity vF of the charge carriers. Instead of being, say, a thousand times larger than the velocity of sound vS as for elements in the main groups of the periodic table and often even for transition-metal compounds, vF is at best of the same order of magnitude as vS in CeCu2Si2. Thus, the retardation of the electron–phonon interaction, an essential ingredient of the BCS theory by which the (renormalized) Coulomb repulsion between the quasiparticle partners of the Cooper pair is avoided, appears to be absent here. Consequently, phonon-mediated Cooper pairing a` la BCS is more than unlikely. Because of the seeming phenomenological similarity between HF superconductivity and superfluidity [5] in 3He (see, e.g. Ref. [6]), magnetic pairing mechanisms have been favored rather early (see, e.g. Ref. [7]). At present we know about 20 HF superconductors, mainly Ce- and U-based compounds. Four of the latter, i.e., UPt3 [8], URu2Si2 [9], UNi2Al3 [10], and UPd2Al3 [11], are behaving exceptionally: (i) they order antiferromagnetically below TN, ranging from 5 to 17 K, and (ii) they exhibit, well below TN and coexisting with AF order, a heavy Landau–Fermi-Liquid (LFL) state which becomes unstable against a HF-superconducting transition at Tc (ranging between 0.5 and 2 K). Recent work by Zwicknagl et al. [12], however, strongly suggests that the heavy quasiparticle masses in these U-based HF-metals are due to the interaction of the charge carriers with the CF rather than being a result of the local Kondo interaction.

2. Spin-density-wave quantum phase transition, valence change and heavy-fermion superconductivity in CeCu2Si2 Most of the HF superconductors, including the U-based compound UBe13 [13], show pronounced deviations from the properties of a LFL in their

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n-state. Such ‘‘non-Fermi-liquid’’ (NFL) phenomena are ascribed, in these clean stoichiometric compounds, to the existence of a ‘‘nearby’’ continuous AF quantum phase transition (QPT) [14]. In contrast to its classical counterpart at finite temperature, the QPT between an antiferromagnetically ordered state and a magnetically disordered one which comes about at T ¼ 0; say, by the application of pressure, is driven by quantum fluctuations rather than by thermal ones (see, e.g. Ref. [15]). The existence of such a QPT at p ¼ pc(T ¼ 0) causes anomalous (NFL) properties of the material in a wide range of its generic phase diagram, in particular, at surprisingly high temperatures. In case of the rare-earth-based HF metals, two competing fundamental interactions completely cancel each other at the QPT, the RKKY interaction on the one hand and the local Kondo interaction on the other. This gives way to the action of ‘‘residual interactions’’ between the quasiparticles. In fact, a number of Ce-based HF metals show superconductivity at very low temperatures near p ¼ pc where TN-0. CePd2Si2 is a prototypical example for such p-induced superconductors [16]. The latter material requires samples of high quality to become a superconductor, pointing towards the fact that already tiny amounts of ordinary potential scatterers suppress an apparently highly anisotropic, unconventional superconducting order parameter. It is widely accepted by now that extended AF spin fluctuations (‘‘paramagnons’’) cause the NFL phenomena in the n-state of Ce-based HF metals as well as their superconductivity. While the quantum critical fluctuations are low-lying in energy, highfrequency fluctuations are required to form Cooper pairs [17]. However, it should be stressed that the experimental verification of paramagnonmediated superconductivity is still lacking—as holds true for other classes of unconventional superconductors as well. Since CeCu2Si2, in contrast to CePd2Si2 [16], behaves as a ‘‘NFL superconductor’’ already at ambient pressure, its physical properties could be explored by a variety of techniques in the past years. Some of these results will be presented in the following. Early experiments on the former

compound had been severely plagued by ‘‘sample dependences’’ of its physical properties, cf. Fig. 1(b). These were resolved many years later by Geibel and collaborators who thoroughly studied the ternary chemical Ce–Cu–Si phase diagram [18]. They found that within the narrow homogeneity range of the primary 1:2:2 phase, which allows for an exchange between Cu and Si atoms by not more than 1%, several ground-state properties were realized. These occur in the physical phase diagram of Fig. 2 at different strengths of the 4f-conduction electron hybridization, measured by a general coupling parameter g [18]. Strong coupling induced by pressure and/or Cu excess is found to favor superconductivity. On the other hand, weak coupling due to Cudeficiency and/or Ge substitution for Si results in the formation of another ordered state, the socalled ‘‘A phase’’ [19]. From direction-dependent resistivity measurements on CeCu2Si2 single crystals, the A phase was concluded [20] to be a conventional SDW with very small ordered moment, i.e., some kind of AF ordering in the system of the itinerant heavy quasiparticles. Recently, this conclusion was confirmed [21] surprisingly well by neutron-diffraction studies [22]. The continuous

Fig. 2. Generic phase diagram of CeCu2Si2 combining data for high-quality polycrystals, undoped ones from the homogeneity range (hatched) and Ge-doped ones, CeCu2(Si1xGex)2. Since the AF transition temperature TA increases linearly with the Ge-concentration x, the coupling constant g was assumed to be linear in (1x), i.e., (1TA). Sectors I, II and III indicate samples of ‘‘A’’, ‘‘A/S’’ and ‘‘S’’ type, respectively, see text [18].

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Fig. 3. Normalized resistivity as r=r300 K vs. T2 (upper scale) and vs. T3/2 (lower scale) (a) as well as Ce-increment to the specific heat as g ¼ DC=T vs. T (b), at B ¼ 2 T and p ¼ 0 as well as two finite pressure values for an A-type CeCu2Si2 polycrystal. The low-T drop in rðTÞ at p ¼ 0 is due to the onset of a superconducting transition. Solid lines display T2 and T3/2 dependences of rðTÞ (a) and a fit of the theory by Moriya and Takimoto (Ref. [24]) to the gðTÞ data [20,23].

CeCu2(Si1-xGex)2 Tc

2

SC TN,Tc (K)

QPT established for CeCu2Si2 at g ¼ gc (Fig. 2), where the A phase disappears continuously as a function of pressure and/or Cu excess, is thus called an itinerant or SDW-type QPT. The anomalous NFL power laws observed [20,23] in the temperature dependences of both the electrical resistivity (DrT 1:5 ) and the Sommerfeld coefficient (g ¼ g0  bT 0:5 ), cf. Fig. 3, are typical for three-dimensional (3D) critical fluctuations [24]. A quantum critical state that is characterized by these power laws is often called a ‘‘nearly antiferromagnetic Fermi liquid’’ (NAFFL), which stresses that the quasiparticles are well defined at the QPT, i.e., possess a finite, though critically enhanced, effective mass m : Based upon present knowledge, it appears possible that all NFL superconductors of the HF variety do exhibit such a NAFFL n-state. One of several puzzles raised by CeCu2Si2 concerns the pressure dependence of its superconducting transition temperature Tc (Fig. 4). Compared to CePd2Si2 [16], superconductivity in undoped CeCu2Si2 is surprisingly robust against the application of pressure (Fig. 4). Yuan et al. [25] have recently addressed this interesting problem. They assumed a strongly anisotropic, unconventional superconducting order parameter to exist in CeCu2Si2 as well. Scattering of the charge carriers from a small number of normal (nonmagnetic) defects is then expected to efficiently break up the

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TN 1 x=0

Tc

AF

x=0.1 x=0.1

Tc

SC

x=0.25

SC 0

-2

0

2

4

6

8

∆p(x) (GPa)

Fig. 4. Experimental phase diagram of CeCu2(Si1xGex)2 showing transition temperatures into the antiferromagnetically ordered (TN, closed symbols) and the superconducting state (Tc, open symbols) vs. relative pressure Dp ¼ pFpc ðxÞ; which reflects the inverse unit cell volume. The pc(x) values are chosen so that the magnetic transition lines for x ¼ 0:1 (pc ¼ 1:5 GPa; circles) and x ¼ 0:25 (pc ¼ 2:4 GPa; squares) coincide. Pure CeCu2Si2 is assumed here to have pc ¼ 0:4 GPa (open triangles) [25].

Cooper pairs. In fact, CeCu2(Si1xGex)2 containing 25 at% Ge is not a superconductor, and the x ¼ 0.1 sample does superconduct only below a substantially reduced transition temperature. By Ge doping one introduces ‘‘negative pressure’’ due to an effective expansion of the lattice, which weakens the 4f-conduction electron hybridization. By this means the existence range of the A phase can be explored in much detail. The Ge-induced expansion of the average cell volume can, however, be experimentally compensated for by applying hydrostatic pressure, cf. Fig. 4. Here, the values of the applied pressure have been reduced by pc ðxÞ; the critical pressure necessary to achieve the QPT (T N ! 0) for an alloy with Ge concentration x. The phase diagram of CeCu2 (Si0.9Ge0.1)2 indeed shows striking similarity to the one of CePd2Si2 [16]: Superconductivity exists only below a ‘‘dome’’ with relatively low Tc,max and centered around the QPT. However, at higher p a second superconducting ‘‘dome’’ develops (Fig. 4), which seemingly coincides with a weak instability of the Ce valence, Ce3+2Ce(3+d)+ [26].

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3. Unconventional quantum criticality and lack of superconductivity in YbRh2Si2 The compound YbRh2Si2, like CeNi2Ge2 isostructural to CeCu2Si2, undergoes a weak AF phase transition at a Ne´el temperature as low as 70 mK, cf. inset of Fig. 5 (Ref. [34]). By applying a

0.8 YbRh2(Si0.95Ge0.05)2

~T

0.75

0.6

8

0.4

χ (10-6m3/mol)

To conclude, when reducing the mean free path of the charge carriers by doping CeCu2Si2 with 10 at% Ge, two superconducting regimes can be separated from each other which, as had been anticipated by Jaccard and collaborators [27], may be caused by different pairing mechanisms. Both of them are very likely of an unconventional nature, mediated by the exchange of virtual fluctuations of the spin density [17] (at low p) and of the charge density [28] (at high p), respectively. Needless to stress that this presumption, too, has yet to be verified by experiments. At this point it should be noted that unconventional, i.e., non-phonon-mediated Cooper pairing, could so far be demonstrated convincingly only for one single material, the LFL superconductor UPd2Al3: By combining quasiparticle tunneling [29] and INS [30] measurements, Sato et al. were able to show that in this strong coupling superconductor (T c
2 K well below T N ¼ 14:3 K) it is the acoustic magnon in the center of the AF Brillouin zone, which acts as the ‘‘exchange boson’’ and, thus, replaces the optical phonons in classical strongcoupling superconductors like Pb and Hg [30]. We now return to the Ce-based NFL superconductors. As mentioned before, it is highly probable that they all are located, in their generic phase diagram, close to a QPT that causes extended critical fluctuations. A new era in studying quantum critical phenomena began with detailed INS investigations on the quantum critical alloy CeCu5.9Au0.1 [31], for these experiments revealed local critical fluctuations being important at the QPT. Theoretically, these local fluctuations either coexist with [32] or replace [33] the (2D) correlated ones associated with the magnetic instability. In the subsequent chapter, we will introduce the tetragonal compound YbRh2Si2, an exciting new example for local quantum critical behavior.

χ -1(106mol/m3)

330

0.2

6 4

YbRh2(Si1-xGex)2

2

x=0 x = 0.05

0 0.01

0.0 -0.4

0.1

1

4

T (K)

0

1

2 T (K)

3

4

Fig. 5. Initial magnetic ac susceptibility as w vs. T (on a logarithmic scale) for YbRh2(Si1xGex)2 with x ¼ 0 and 0.05 (inset) and as w1 vs. T (on a linear scale) for the Ge-doped compound (main part). From the Curie–Weiss law found at To0:3 K; a Weiss temperature Y ¼ 0:3 K and a very large fluctuating moment, meff ¼ 1:4mB =Yb3þ ; are obtained [36].

small critical magnetic field Bc one can suppress the magnetic order and tune the material to a (‘‘field-induced’’) QPT [35]. Upon further increasing the field one enters a heavy LFL phase at sufficiently low temperatures. NFL phenomena are observed [36] in the close vicinity of Bc down to the lowest accessible temperature (E20 mK) and, in addition, relatively far away from Bc at surprisingly high temperatures (p10 K). Since the ionic radius of the nonmagnetic Yb2+(4f14) configuration exceeds that of the magnetic Yb3+(4f13) one, ‘‘demagnetization’’ of Yb, opposite to the case of Ce, requires the application of ‘‘negative pressure’’. Therefore, an YbRh2(Si1xGex)2 single crystal with nominal Ge concentration x ¼ 0:05 exhibits a reduced Ne´el temperature of only 20 mK [36]. Consequently, the critical field to drive the system towards the QPT is substantially lower than for the undoped compound. In Fig. 6(a) it is shown that for this system, the Sommerfeld coefficient gðTÞ ¼ C el ðTÞ=T of the electronic specific heat and, thus, the effective quasiparticle mass m diverge logarithmically within one and a half decades of temperature, i.e., for 0:3 KoTo10 K: In the same temperature window, the resistivity depends linearly on T, DrT (Fig. 6(b)). This is equivalent to an effective

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frequency ac magnetic susceptibility (main part of Fig. 5, Ref. [36]) and the electron paramagnetic resonance [37]. These results reveal the presence of antiferromagnetically correlated, but very large paramagnetic Yb3+ moments far below the Kondo temperature which amounts to T K
25 K for this compound [34]. The large paramagnetic moment contrasts with the extremely small ordered moment [38] below T N ¼ 70 mK: Further support for the QPT being of a local kind stems from the observation of a fractional exponent in the temperature dependence of a diverging Gru¨neisen ratio [39]. Fig. 6. Zero-field results of both the electronic specific heat as Cel/ T vs. T (on a logarithmic scale) for YbRh2(Si1xGex)2 with x ¼ 0 and the nominal Ge concentration x ¼ 0:05; respectively, below T ¼ 2 K (a) and the electrical resistivity as r vs. T (on a linear scale) for the Ge-doped single crystal below T ¼ 10 K (b) [36].

quasiparticle–quasiparticle scattering cross-section, proportional to A ¼ DrðTÞ=T 2 ; which diverges as 1/T. These unique temperature dependences of gðTÞ and DrðTÞ indicate that at the QPT, both m and the scattering cross-section are becoming singular on the whole Fermi surface, i.e., in a way predicted by the SDW scenario assuming strictly two-dimensional spin fluctuations [24]. At temperatures To0:3 K; gðTÞ shows a stronger than logarithmic divergence, T Z ; Z
0:4 [Fig. 6(a)], while Dr keeps following the linear Tdependence to the lowest accessible temperature of 10 mK [Fig. 6(b)]. This apparent disparity between the T-dependences of the thermodynamic quantity gðTÞ and the transport property DrðTÞ was taken as evidence of a break up of the ‘‘composite fermions’’ on the approach to the (B ¼ 0) QPT [36]. For B ¼ 0; the QPT is expected to occur at a slightly higher Ge concentration. Loosely speaking, it seems as if the dominating local 4f component of the heavy quasiparticles behaves more sensitively on the incipient AF order than their itinerant (conduction-electron) component. In fact, striking evidence for a break up of the local Kondo singlets, which are commonly considered constituent to the formation of the HF, was provided by measurements of both the low-

4. Outlook Most remarkably, YbRh2Si2 is not a superconductor, although high-purity single crystals are now available. One may, therefore, speculate that, while a conventional (SDW) QPT favors an unconventional superconducting state to form in Ce-based HF metals, an unconventional QPT is unfavorable for superconductivity. Understanding the intimate relationship between superconductivity and quantum criticality more profoundly will remain a challenge for condensed matter physicists.

Acknowledgments I am grateful for valuable discussions with J. Custers, P. Gegenwart, C. Geibel, F.M. Grosche, R. Ku¨chler, S. Paschen, G. Sparn, O. Stockert, S. Wirth, H.Q. Yuan, P. Coleman, C. Pe´pin, Q. Si and G. Zwicknagl. References [1] A.A. Abrikosov, L.P. Gor’kov, Zh. Exsp. Teor. Fiz. 39 (1960) 1781 [Sov. Phys. JETP 12 (1961) 1243]. [2] F. Steglich, J. Aarts, C.D. Bredl, W. Lieke, D. Meschede, W. Franz, H. Scha¨fer, Phys. Rev. Lett. 43 (1979) 1892. [3] K. Kadowaki, S.B. Woods, Solid State Commun. 58 (1986) 507. [4] W. Lieke, U. Rauchschwalbe, C.D. Bredl, F. Steglich, J. Aarts, F.R. de Boer, J. Appl. Phys. 53 (1982) 2111.

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