Competition between phonon superconductivity and Kondo screening in mixed valence and heavy fermion compounds. 6 0 Victor Barzykin 0 2 Department of Physics and Astronomy, n a University of Tennessee, Knoxville, TN 37996-1200 J 8 1 L. P. Gor’kov∗ ] n National High Magnetic Field Laboratory, Florida State University, o c 1800 E. Paul Dirac Dr., Tallahassee, Florida 32310 - r p u s . t a m - d n o c [ 1 v 2 1 4 1 0 6 0 / t a m - d n o c : v i X r a 1 Abstract We consider competition of Kondo effect and s-wave superconductivity in heavy fermion and mixed valence superconductors, using the phenomenological approach for the periodic Anderson model. Similar to the well known results for single-impurity Kondo effect in superconductors, we have found principal possibility of a re-entrant regime of the superconducting transition tempera- ture, T , in heavy fermion superconductors in a narrow range of model parameters and concentra- c tionof f-electrons. SuppressionofT inmixedvalencesuperconductorsismuchweaker. Ourtheory c has most validity in the low-temperature Fermi liquid regime, without re-entrant behavior of T . c To check its applicability, we performed the fit for the x-dependence of T in Ce La Ru Si and c 1−x x 3 2 obtained an excellent agreement with the experimental data, although no re-entrance was found in this case. Other experimental data are discussed in the light of our theoretical analysis. In particular, we compare temperatures of the superconducting transition for some known homologs, i.e., the analog periodic lattice compounds with and without f-elements. For a few pairs of ho- mologs superconductivity exists only in the heavy fermion materials, thus confirminguniquenessof superconductivity mechanisms for the latter. We suggest that for some other compoundsthe value of T may remain of the same order in the two homologs, if superconductivity originates mainly c on some light Fermi surface, but induces sizable superconducting gap on another Fermi surface, for which hybridization or other heavy fermion effects are more significant. By passing, we cite the old results that show that the jump in the specific heat at the transition reflects heaviness of carriers on this Fermi surface independently of mechanisms responsible for superconductivity. PACS numbers: 74.70.Tx,74.62.-c, 74.20.-z,74.20.Fg ∗ Also at L.D. Landau Institute for Theoretical Physics, Chernogolovka,142432,Russia 2 I. INTRODUCTION Since the discovery of the first heavy fermion (HF) system, CeAl , in 1975 [1], these com- 3 pounds have attracted enormous theoretical and experimental interest due to their fascinat- ing properties. The most striking feature of these systems, their extremely large effective mass m∗ (100 1000)m of charge carriers, is qualitatively understood in terms of unifica- e ∼ − tion of magnetic degrees of freedom with the ones of itinerant electrons. The magnetic and superconducting properties of these compounds are both rich and puzzling (For a review, see [2, 3]). Unconventional superconducting properties may be independent of mechanism, and related to the unusual symmetry of the order parameter (for reviews, see Refs. [4, 5]). However, unlike common superconductors, the microscopic mechanisms of superconductiv- ity (SC) in these materials for the most part remain unknown. At least a partial answer to it could be in the question whether conventional, i.e., phonon-mediated superconductivity[6] is excluded for them. In what follows, we address this issue. The theoretical framework for studying heavy fermion metals is the periodic Anderson model,consideredbelowinsectionII.SectionIIIsumsuptheresultsofthephenomenological approach. In sections IV and V the main theoretical formulas for s-wave superconductivity are derived in the same frameworks. In section VI we discuss some scarce experimental data to show that, although there are cases where new mechanisms seem to be necessary, in many other cases phonon-mediated exchange may remain as the cause of superconductivity. In section VII we draw our conclusions. The single site Anderson impurity problem in a normal metal has been solved[2, 7]. However, even for the low impurity concentration, the competition of superconductivity and Kondo effect or mixed valency remains a problem, for which an exact answer was not obtained theoretically. It is well known that the pair-breaking action of scattering on magnetic impurities in s-wave superconductors leads to a drastic suppression of the transition temperature with increasing impurity concentration. Taking the Kondo screening into account, however, this answer is less obvious, since at low temperatures the Kondo singlet state acts as a non-magnetic impurity[2], and thus a finite concentration of such impurities may not significantly change T . Competition of superconductivity and Kondo c effect in alloys has been studied using various approximate methods, starting from the pioneering work of Mu¨ller-Hartmann and Zittartz[8], and we summarize the results below 3 briefly. We stress, however, that our main interest lies in the study of superconductivity of dense systems, especially of stoichiometric heavy fermion compounds, where at low temperatures theFermiliquidregimebecomesrestored,andtheoreticalmethodsbasedonimpurityscatter- ing lose ground. We shall tryto make use of the fact that many Ce- andU-based compounds have their homologs, i. e., stoichiometric compounds with non-magnetic elements like Y, La, Lu, substituting the rare-earth or actinide elements. For a number of them continuous alloy composition range is available to trace whether the superconducting transition temperature varies drastically from a phonon-like T for non-magnetic compounds to a new mechanism c in HF- or mixed valence (MV) compounds. II. LIMITING REGIMES IN CONCENTRATION AND MODEL PARAMETERS. Mu¨ller-Hartmann and Zittartz[8] used Nagaoka decoupling scheme for the Green’s func- tions. The Nagaoka approximation fails at temperatures below the Kondo temperature, yielding non-analytic features in all physical properties. Thus, this scheme is expected to fail in the Fermi liquid regime, when the superconducting transition temperature is much less than the Kondo temperature. Nevertheless, this theory[8] has been successfully applied to many Kondo alloys, such as (La, Ce) Al , (La,Th)Ce, and other Ce compounds. 2 The main result[8] is that, instead of the usual paramagnetic pair breaking curve[9], the dependenceofT onconcentrationacquires, duetotheKondoscreening, acharacteristic“S”- c shape with a re-entrant behavior. However, T never goes to zero at low temperatures. Such c re-entrant behavior of T has been observed in the heavy fermion alloys (La,Ce)Al [10, 11], c 2 (La,Th)Ce[12], and (La,Y)Ce[13, 14]. In particular, three transition temperatures were clearly seen in La Ce Y [13]. As the temperature was lowered, the first transition .7915 0.0085 .20 T (from normal to superconducting state) was observed at 0.55K, the second transition c1 T (from superconducting to normal state) at 0.27K, and the third transition (back to c2 superconducting state) at 0.05K. Significant deviations from the Abrikosov-Gor’kov theory were also observed in Kondo superconductors LaCe and LaGd[15], and PbCe and InCe films [16]. 4 In its simplest form, the single-impurity Anderson Hamiltonian has the following form: Hsingle = ǫkc†kσckσ + E0fσ†fσ (1) kσ σ X X +V (c† f +H.c.)+Un n , kσ σ ↑ ↓ kσ X where E is the energy of a localized orbital, U is the on-site Coulomb repulsion energy, 0 and V is the hybridization integral between localized states and conduction states ǫ . The k broadening of the local level due to hybridization is given by the golden rule: Γ = πρV2, (2) where ρisthesingle-spindensity ofstatesattheFermienergy. Thepossible f-configurations for Ce and U are f0, f1, and f2. For Yb (f12, f13, f14), one can treat Eq.(1) as the Hamiltonian for holes. The renormalization group analysis of the model Eq.(1) has been performed by Haldane[17], who has shown that there are two fixed points: (1) The Kondo regime, ǫ Γ, (3) f − ≫ in which the renormalized impurity level ǫ stops well below the chemical potential and at f very large U > 0 is indistinguishable from a local spin. Charge fluctuations on the site are negligible, while the local spin interacts antiferromagnetically with spins of conduction electrons via exchange coupling J = V2/ǫ < 0. At high temperatures local spins behave as f pair-breaking paramagnetic centers[9, 18]. Below a characteristic temperature, T , the local K moments become screened, and the Fermi liquid regime sets in [2]. For Ce, it corresponds to the f1 configuration. (2) The mixed valence regime, ǫ Γ. (4) f | | ∼ Inthisregimethetwoconfigurations, say, f0 andf1 forCe, areapproximatelydegenerate. The system is characterized by a time scale for spin fluctuations (charge fluctuations are strong as well). If the f-level is taken above the chemical potential, and the hybridization, V, is weak: E ǫ Γ, (5) 0 f ≃ ≫ 5 then, theimpurityismostly“empty”, sothatthescatteringismostlynonmagneticincharac- ter. Nevertheless, finite hybridization of electrons with correlated impurity levels introduces an effective repulsion between conduction electrons with opposite spin, which grows with increased concentration of impurities, and causes pair weakening. In the Hartree-Fock ap- proximation such pair weakening caused by resonant impurity scattering has been studied by Kaiser[19], Shiba[20], and Schlottmann[21]. This results in a modified exponential decay of T with increased impurity concentration[19]. The Hartree-Fock approach is only valid c for small enough Coulomb repulsion[21], U/Γ 1. Nevertheless, it has produced a good ≪ description of ThCe[22] and some lanthanide alloys with large Kondo or spin fluctuation scales. The presence of Kondo effect significantly complicates the treatment of T suppression. c The approach[8] that uses the Nagaoka decoupling only works well for small values of the Kondo scale T < T . In the Fermi liquid regime, the theory was developed by Mat- K c0 suura, Ichinose, and Nagaoka[23], and by Sakurai[24]. For the Fermi liquid fixed point, pair weakening occurs through virtual polarization of the Kondo ground state[23, 24]. This theory is valid when T T . The behavior of magnetic impurities in strongly coupled K c0 ≫ superconductors has been studied numerically[25] and analytically[26], with the result that strong coupling weakens the effect of Kondo impurities by a factor 1 + λ, where λ is due to electron-phonon interaction. The low-temperature regime has also been studied in the slave boson 1/N formalism[27]. A unified treatment of superconductivity in presence of Anderson impurities in the NCA approximation (T T ) has been done by Bickers and c K ≫ Zwicknagl[28]. Analysis of superconducting properties of alloys in this regime has been done pertur- batively by Schlottmann[29]: T decreases at first linearly with concentration, then expo- c nentially. In addition, this regime was studied at zero temperatures[30] using the large- degeneracy expansion of Gunnarsson and Sch¨onhammer[31]. For the periodic lattice, su- perconductivity in a mixed valence compound was analyzed using the Green function approach[32]. Experimentally, superconductivity in mixed valence regime has been stud- ied in detail for CeRu Si [33], CeRu , and CeIr [34]. 3 2 2 3 Asitwasmentionedinintroduction, belowwestudythecompetitionofsuperconductivity and Kondo effect in concentrated alloys and Anderson lattices at low temperatures. The influence of impurities on the superconducting state in heavy fermions has also attracted 6 some experimental interest. For example, superconductivity in presence of non-magnetic impurities has been studied inheavy fermion superconductor CeCu Si [35, 36]. The peculiar 2 2 properties of U Th Be are also well known[37, 38]. Numerous experiments have been 1−x x 13 done in other alloys, such as Th Ce , Th U , Al Mn , La Ce In. We refer the 1−x x 1−x x 1−x x 3−x x reader to Ref.[39] for a detailed review of relevant experiments. However, the theoretical model developed below leaves scattering effects aside. We apply our results only to systems where the latter does not play important role, because the f-electrons go into bands (i.e., where no sharp decrease of T at low concentrations was observed). For the heavy fermion c systems the common assumption since Ref.[40] is that SC forms on a heavy Fermi surface due to electron-electron interactions, and then induces SC on other parts (see, e.g., in Ref. [41]). To the contrary, we begin with the phonon mechanism. III. HEAVY FERMION LIQUID AND RENORMALIZED BANDS. As usual, we start the consideration of a heavy fermion liquid by writing the periodic Anderson model, H = H +H +H , (6) 0 V ef where H0 = ǫkc†kσckσ + E0fi†σfiσ + Ufi†↑fi†↓fi↓fi↑. (7) kσ iσ i X X X Here the creation and annihilation operators for the f-electrons, f† and f , carry the site iσ iσ index i, and there is a Coulomb interaction at each site for the f-electrons. The operators c†kσ and ckσ correspond to delocalized Bloch states. The hybridization term in the model Hamiltonian, H , accounts for the s d hybridization between the f-electrons and the Bloch V − states: HV = Vkeik·Rifi†σckσ +Vk∗e−ik·Ric†kσfiσ . (8) iX,k,σ(cid:16) (cid:17) Finally, the third term in the Hamiltonian Eq.(6) corresponds to the attraction, caused by the electron-phonon interaction, which we consider in the weak coupling limit here: λ Hef = 2 Ψ†σ(r)Ψ†σ′(r)Ψσ′(r)Ψσ(r)dr, (9) Z 7 where the Ψ (r) and Ψ†(r) are operators, which correspond to the itinerant band: σ σ 1 Ψσ(r) = (2π)3 eik·rckσdk, (10) Z 1 Ψ†(r) = e−ik·rc† dk. (11) σ (2π)3 kσ Z The on-site Coulomb repulsion U is usually very large for f-electron materials, and it will be taken infinite below. To account for this, the creation/annihilation operators for f- electrons in H would have to be taken with projection operators, which project out doubly V occupied f-electron states[42]. A convenient way to rewrite this Hamiltonian, invented by Coleman[43], Read and Newns[44], and Barnes[45], is to introduce a new slave boson field b+, which creates a hole on site i, and to rewrite the Anderson Hamiltonian in a way, which i allows a 1/N expansion in the number of orbitals. However, in what follows, we resort to a more phenomenological approach of Edwards[47] and Fulde[48] (see also Ref.([2])). Introducing the set of the fermion Green’s functions (for imaginary time τ), Gmcc(k,τ) ≡ −hTτckm(τ)c†km(0)i, (12) Gmfc(k,τ) ≡ −hTτfkm(τ)c†km(0)i, (13) Gmff(k,τ) ≡ −hTτfkm(τ)fk†m(0)i, (14) and transforming to Matsubara frequencies, we find for them the diagrammatic expansion in powers of U and Vk for the conduction and f-electron Green’s functions: iωn −ǫf +µ−Σσ(ωn,k) −Vk Gfkf,σ(ωn) Gckf,σ(ωn) = Iˆ. (15)  −Vk iωn −ǫk +µ Gfkc,σ(ωn) Gckc,σ(ωn)  Assumingthat Σ (ω ,k) can be expanded near theFermi surface, k = k, and retaining σ n F | | the first order terms: ∂Σ(ω ,k ) Σ(ω ,k) Σ(0,k )+(k k ) ∇Σ(0,k) +ω n F , (16) n ≃ F − F · k=kF n ∂ω (cid:18) n (cid:19)ωn=0 theGreen’sfunctioncannowbewrittenintheformanalogoustothenon-interacting(U = 0) problem: 1 Gcc (ω ) = , (17) k,σ n iωn −ǫk+µ− iω|nV˜−k|ǫ˜2fk where ǫ˜fk = Z(ǫf µ+ΣR(0,kF)+(k kF) Σ(0,kF)), (18) − − ·∇ V˜k 2 = Z Vk 2. | | | | 8 Since ǫ˜fk is only weakly k-dependent, we can replace it with a constant, ǫefff. Furthermore, in what follows we also assume that Vk does not depend on the direction of k, so that we can replace V˜k by a constant, V˜ = V˜kF. As usual, the quasiparticle residue Z is given by 1 Z = . (19) 1 ∂Σ(ωn,kF) − ∂ωn ωn=0 h i After making use of Eqs (15) and (19), the Green’s functions acquire the form: iω ǫeff Gm(k,ω ) = n − f , cc n (iωn −ǫefff)(iωn −ξk)−V˜2 Gm(k,ω ) = iωn −ξk , (20) ff n (iωn −ǫefff)(iωn −ξk)−V˜2 V˜ Gm(k,ω ) = . fc n (iωn −ǫefff)(iωn −ξk)−V˜2 From the poles of the Green’s functions Eq.(20), the renormalized energy spectrum has the following form: ǫeff +ξ 1 ξ˜ = f k (ǫeff ξ )2 +4 V˜ 2, (21) k1,2 2 ± 2 f − k | | q where ξk ǫk µ. In the limit U the effective f-band ǫ˜fk must lie above the Fermi ≡ − → ∞ surface, so that the total occupation of the f-level, n , is such that 0 < n < 1. For the f f ˜ effective mass of quasiparticles, after expanding ξ in the vicinity of the Fermi surface, one k2 obtains: m∗ V˜ 2 = | | +1. (22) m (ǫeff)2 f IntheKondolimit,ǫeff = T canbethoughtofastheKondotemperature. Theconservation f K of the total number of quasiparticles (Luttinger theorem) leads to the shift of the chemical potential, given by V˜ 2 µ˜ = µ+ | | . (23) ǫeff f Note, however, that the parameters V˜ and ǫeff are, in general, temperature-dependent. f Then, their temperature dependence can be studied within a specific model, such as slave boson 1/N approach. Due to the restriction of the slave boson approach to low tempera- tures, our results for phonon-mediated superconductivity are applicable to the case when both V˜ and ǫeff are much greater than T , so that these parameters could be regarded as f c temperature-independent. Below we merely consider V˜ and ǫeff > 0 as two free parameters f of our theory. 9 IV. GENERAL FORMULA FOR T . c We can now evaluate the superconducting transition temperature, using the phonon attraction Hamiltonian Eq.(9), and our new energy spectrum Eq.(21). In what follows, we assume that the phonon cutoff ω in Eq.(9) is much greater than the Kondo temperature D ǫeff, and the effective hybridization V˜ , f | | ω max V˜ ,ǫeff . (24) D ≫ {| | f } Following Ref.[49], T is obtained by evaluating the Cooper diagram, c dk 1 T Gcc(ω )Gcc ( ω ) = . (25) c (2π)3 k n −k − n λ n Z | | X As usual, to get rid of the cutoff dependence arising from the frequency summation, we have to introduce a new energy scale T , for the superconducting temperature in absence of the c0 f-electrons: 2ω γ T = D e−2π2/|λ|mkF, (26) c0 π and rewrite Eq.(25) as dk dk T Gcc(ω )Gcc ( ω ) = T G0(ω )G0 ( ω ), (27) c (2π)3 k n −k − n c0 (2π)3 k n −k − n |ωnX|<ωDZ |ωnX|<ωDZ where 1 G0(ω ) . (28) k n ≡ iω ξ n k − (Note that we introduced the cutoff ω in the sum over n, while the integral over ξ goes D from to + .) After some simple but tedious transformations, we find that the Cooper −∞ ∞ bubble on the left-hand side of Eqs(25),(27) can be written as: ω2 +(ǫeff)2 Π(ω ,k) Gcc(ω )Gcc ( ω ) = n f , (29) n ≡ k n −k − n (ω2 +ξ˜2 )(ω2 +ξ˜2 ) n k1 n k2 ˜ where ξ are given by Eq.(21). Integrating Eq.(27) by ξ , we get: k1,2 k π(ω2 +(ǫeff)2 π n f T = T , (30) c|ωnX|<ωD |ωn|(ωn2 +[ǫefff]2 +V˜2) c0|ωnX|<ωD |ωn| which can be rewritten using the definition of the digamma function, ∞ 1 1 Ψ(z) γ , (31) ≡ − − z +n 1 − n n=1(cid:18) − (cid:19) X 10