Superconductivity in Heavy-Fermion Materials

J. D. ThompsonLos Alamos National Laboratory

Outline:• the surprise and its generalization

• directions from new discoveries

• recent progress

• outlook

BCS@50, 2007

C = γTγ ≈1J/molK2

γ ⇒ m*≈1000me

the beginning: CeCu2Si2

⎨

a precedent for large m* but not superconductivity

♦ strong Cooper-pair breaking by Ce (Kondo) impurities in the conventional superconductor LaAl2 (M. B. Maple et al., SSC 11, 829 (1972))

♦ additional complication: materials issues with CeCu2Si2 -- some samples not superconducting, some even magnetic

♦ CeAl3: very large Sommerfeldcoefficient, reflecting a resonance in the electronic density of states produced by a collection of non-interacting Ce Kondo impurities in which γ ∝1/TK; in the T=0 limit, lattice periodicity of Ce ions ⇒ a collective band state formed by phase coherence among (Kondo) virtual bound states (K. Andres, J. E. Graebner, H. R. Ott, PRL 35, 1779 (1975))

T c/T

c0

n (a/o)

LaCeAl2

CeAl3

UPt3

UBe13

γ = 1.1J/mol K2

heavy-fermion superconductivity in UBe13 and UPt3

UBe13

♦ validation of CeCu2Si2: reproducible, bulk superconductivity of heavy quasiparticles in UBe13 (H. R. Ott et al., PRL 50, 1595 (1983) and soon thereafter in UPt3 (G. R. Stewart et al., PRL 52, 679 (1984))

♦ first suggestions that superconductivity might be unconventional

♦ C ∝T3lnT above Tc in UPt3 and enhanced Wilson ratio, analogous to 3He ⇒nearly ferromagnetic and possibly spin-triplet pairing

♦ C ∝ T3 below Tc in UBe13; not expected for s-wave pairing; comparison to Anderson-Brinkman-Morel ⇒ p-wave (H. R. Ott et al., PRL 52, 1915 (1984))

♦ but, straightforward analogy to 3He invalid due to inherent symmetry of crystal lattice and strong spin-orbit coupling

Tc~ 0.9K

Tc ~ 0.5K

evidence for unconventional superconductivity

UPt3

UPt3 H ⊥ c

UPt3 H // c

ab

c

♦UPt3 – two, bulk superconducting transitions in zero field and three phases when H>0 (R. A. Fisher et al. PRL 62, 1411 (1989); G. Bruls et al., PRL 65, 2294 (1990); S. Andenwalla et al., PRL 65, 2298 (1990)) plus power laws in specific heat, relaxation rate, thermal conductivity, etc. below Tcand weak magnetic correlations in phase ‘b’⇒ unconventional order parameter, possibly two-component♦Th-doped UBe13 – two transitions for 2<x<4% Th (H. R. Ott et al., PRB 31, 1651 (1985)), similar to UPt3; all phases superconducting, with weak (0.001μB) magnetism in phase ‘C’; inconsistent with expectations of conventional superconductivity

A

B

C

subsequent discoveries

CePd2Si2

D. Jaccard et al., Phys. Lett. A 163, 475 (1992)

F. M. Grosche et al., Physica B 224, 50 (1996)

R. Movshovich et al., PRB 53, 8241 (1996)

♦ antiferromagnetic to superconducting transition with applied pressure in a family of materials with the CeCu2Si2 structure type♦ near order of magnitude variation in TN’s but Tc’s comparable, ~ 0.5±0.1K, and essentially identical to Tc of CeCu2Si2♦ critical pressure Pc, where TN → 0, from 9 to 77 kbar (0.9-7.7 GPa); at material-dependent Pc, a material-independent critical cell volume of 168±4Å3, close to that (167Å3) of CeCu2Si2 at atmospheric pressure♦ a guiding principle for where to look: near the T=0 magnetic-nonmagnetic boundary; soft magnetic excitations competing with or promoting superconductivity?

CeRh2Si2TN

10Tc

an intuitive, simple perspective

♦ dilute Ce or U moments in a metallic host ⇒ Kondo impurity effect ⇒resonance in electronic density of states or equivalently large Sommerfeldcoefficient/Kondo ion and magnetic singlet state at T << TK

♦ periodic array of moments, e.g. CeCu2Si2, interacting through the indirect RKKY interaction ⇒ long range magnetic order

♦ competition between Kondo and RKKY, both of which depend on magnetic exchange J ⇒ non-monotonic variation of TN as a function of some parameter, e.g. pressure, that tunes magnitude of J♦ T-P phase diagrams of Ce122 compounds as expected from this simple model, with superconductivity appearing near the T=0 magnetic-nonmagnetic boundary, i.e., a magnetic quantum-critical point where long range, long lived quantum fluctuations of the magnetic order parameter induce a non-Landau Fermi liquid phase – found above Tc in most heavy-fermion superconductors♦ most obvious in Ce-based compounds but less so in U-based heavy-fermion superconductors, even though they appear in proximity to magnetism

AFMAFM FLFL

NFLNFL

Jc JN(EF)

TTKK~ e~ e--1/|J|1/|J|

TTRKKY RKKY ~ J~ J22

SCSC

S. Doniach, Physica 91 B, 231 (1977)

T m

CeIn3

ten-fold increase in Tc

layering

N. D. Mathur et al., Nature 394, 39 (1998)H. Hegger et al., PRL 84, 4986 (2000)

sequential stacking of CeIn3 and

RhIn2

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

♦ as with CeM2X2, maximum Tc in cubic CeIn3 near its T=0 antiferromagnetic-nonmagnetic boundary; non-Landau Fermi liquid resistivity, ρ ∝ T1.6 over an extended range above Tc♦ structurally layering CeIn3 to give the antiferromagnet CeRhIn5 ⇒ maximum Tcincreases from ~0.25 to 2.3 K♦ quasi-2D Fermi surface topology confirmed by dHvA (D. Hall et al., PRB 64, 064506 (2001); H. Shishido et al., JPSJ 71, 162 (2002))

0.0

0.5

1.0

1.5

SC

AFM+SC

T (K

)

P (GPa)

P1

CeRhIn5

AFM

TN

Tc

TN

10Tc

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

0.0

0.5

1.0

1.5

SC

AFM+SC

T (K

)

P (GPa)

P1

CeRhIn5

AFM

TN

Tc

M0

?

magnetism and superconductivity in CeRhIn5

S. Kawasaki et al., PRL 91, 137001 (2003)

♦ large-moment incommensurate antiferromagnetic order for 0 < P < 1.6 GPa(A. Llobet et al., PRB 69, 024403 (2004))

♦ microscopic coexistence of antiferromagnetism and nodal superconductivity:• at P=1.6 GPa, clear signature of

antiferromagnetism, followed at lower temperature by superconductivity; below Tc, 1/T1∝T3, as expected for a gap with nodes; T-linear 1/T1 at the lowest temperatures ⇒ presence of low-energy excitations• at P=2.1 GPa, where Tc > TN, no

evidence for magnetic order above 150 mK, even through extrapolated TN(P) gives TN ≈ 1K; 1/T1∝ T3 below Tc

120100 80 60 40 20 0 1.52.0

2.5

AFM

3.0

2.0

H (kOe)

T (K

)P (GPa)

P1

P2

MO

SC

NM

H = 0 kOe plane

T = 0.5 K plane

1.0

~P2 ~P2

field-induced criticality in CeRhIn5

♦ line of field-induced, second-order magnetic transitions connecting P1 and P2 inside the SC state; line separates a phase of coexisting magnetic order (MO) and superconductivity (SC) from a purely unconventional superconducting stateT. Park et al., Nature 440, 65 (2006); G. Knebel et al., PRB 74, 020501 (2006)

♦ diverging normal-state cyclotron mass and specific heat at P2♦ small-to-large Fermi surface and magnetic-nonmagnetic boundary at P2 (H. Shishido et al., JPSJ 74, 1103 (2005))♦ not expected at a conventional QCP

♦ layering CeIn3with other isovalenttransition elements ⇒ ambient-pressure superconductors CeCoIn5 and CeIrIn5 (C. Petrovic et al., J. Phys. Cond. Mat. 13, L337 (2001); EPL 53, 354 (2001))

similarity of CeCoIn5 to CeRhIn5 at P2

0 5 10 15 200

5

10

15

CeCoIn5

(ρ-ρ(0))=AT1.06

ρ (μ

Ω c

m)

temperature (K)

0 1 2 3 40.0

0.5

1.0

1.5

2.0 CeCoIn5, P=0 CeRhIn5, P~P2

C/T

(J/m

ol K

2 )

temperature (K)

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5

non-Fermi-liquid metal

superconductivity

T (K)

H /H

c2(0

)

heavy-fermionmetal

CeCoIn5

♦ smaller cell volume of CeCoIn5 ⇒ similar to CeRhIn5near P2 where dHvA frequencies are comparable♦ near-identical Tc and C/T, with ΔC/γTc ≈ 4 in both;entropy balance ⇒ C/T increases below Tc

♦ at Hc2(0), C/T ∝ -lnT/T* in CeCoIn5 and diverges in CeRhIn5 near P2

♦ CeCoIn5 at P=0,H=0, ρ∝ T from Tc to ~15K ≈TF*, similar to CeRhIn5near P2

A.Bianchi et al., PRL 91, 257001(2003);T. Park et al., Nature 440, 65 (2006)

♦ superconductivity of CeCoIn5 enclosed by a non-Fermi-liquid normal state that crosses over to Landau Fermi liquid behavior at H>Hc2(0) and with applied pressure (V. Sidorov et al., PRL 89, 157004 (2002); J. Paglione et al., PRL 91, 246405 (2003)); like CeCu2Si2, UBe13 and others

0.1 1

0.4

0.8

1.2

1.6

0.1 1

0.4

0.8

1.2

1.6

(C-C

Sch

-Cla

ttice

)/T (J

/mol

K2 )

temperature (K)

CeCoIn5 (H=Hc2(0))ΔC/T= -lnT/T*

C/T

(J/m

ol K

2 )

CeRhIn5 near P2(H=Hc2(0))

evidence for unconventional superconductivity in CeCoIn5

♦ power laws in C/T, κ, λ, and 1/T1-- all consistent with line nodes (R. Movshovich et al., PRL 86, 5152 (2001); S. Ozcan et al. EPL 62, 412 (2003); Y. Kohori et al., PRB 64, 134526 (2001))

♦ 4-fold modulation of in-plane thermal conductivity ⇒gap with dx2-y2 symmetry (K. Izawa et al., PRL 87, 057002 (2001))

♦ sign change of order parameter from point-contact spectroscopy ⇒ dx2-y2 gap and 2Δ/kTc≈ 6 (W. K. Park et al., PRL, submitted)

♦ Knight shift + strongly Pauli-limited Hc2 ⇒ spin-singlet pairing

antinodal nodal

magnetic order induced by hole doping CeCoIn5

0 2 4 60

500

1000

1500

2000

0 1 2 3 4 5 6 70.0

0.1

0.2

0.3

0.4

0.5

x% 0

0.25 0.50 0.75 1.0

1.25 1.5 2.5

[C-C

latt]

/T (m

J/m

ole-

K2 )

T (K)

CeCo(In1-xCdx)5

x% 0

2.5

temperature (K)

Sm

ag(R

Ln2)

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5 CeCo(In1-xCdx)5

TC

TN

T(K

)

x% Cd

AFM

SC

P

a

TcTN

♦ Cd substitution for In: adds 1 hole/substituted Cdand induces AFM that coexists with bulk superconductivity (L.Pham et al., PRL 97, 056404 (2006)); entropy below ~6K independent of ground state ⇒same electrons involved in both orders♦ microscopic coexistence of large-moment antiferromagnetism with Q = (½,½,½) for x=1% (from NQR: R. Urbano et al., PRL 99, 146402 (2007); from neutrons: M. Nicklas et al., PRB 76, 052401(2007)); same conclusion from neutron diffraction on x=0.75% ♦ development of magnetic intensity arrested abruptly at Tc and finite below; coexisting long-range magnetic and unconventional superconductivity

♦ intensity in elastic channel transferred to spin resonance below Tc?

another ten-fold increase in Tc

6 8 10 12 14 16 18 20 220

100

200

300

400

500

C/T

(m

J/m

ol K

2 )

T (K)

PuCoGa5

♦ replace Ce (4f1) with Pu (5f5), keeping CeCoIn5structure ⇒ increase Tc from 2.3 to 18.5 K, with Sommerfeld coefficient decreasing to ~ 100 mJ/mol K2; also not a singularity of Nature: PuRhGa5 with Tc = 9K (F. Wastin et al., JPCM 15, S1911(2003)) and closely related structure type NpPd5Al2 with Tc=4.9K and γ > 200 mJ/mol K2 (D. Aoki et al., JPSJ 76, 063701 (2007))

N. J. Curro et al., Nature 434, 622 (2005)

J. L. Sarrao et al., Nature 420, 297 (2002)

♦ 1/T1 ∝ T3 for T≤ Tc in both Pu115s and unconventional normal states in all three♦ same T-dependence of normalized relaxation rate in unconventional superconductors PuCoGa5, isostructural CeCoIn5 and high-Tc YBa2Cu3O7below and above Tc and qualitatively different than that of conventional superconductors Al and MgB2♦ new quasi-2D Ce, Pu and Np superconductors possibly a bridge between low Tc heavy-fermion and high Tc cuprates?

outlook

♦ basic problem posed originally by CeCu2Si2: Cooper pairs formed by heavy quasiparticles with Tc < TF* < ΘD• not BCS hierarchy of energy scales, but otherwise BCS-like: superconducting gap,

Meissner effect, lower and upper critical fields, etc. in all of ~ 20 examples• two superconducting transitions in UPt3 (and perhaps Th-doped UBe13) – not possible

for a conventional BCS order parameter • power laws below Tc in all heavy fermion superconductors – often is assumed to be and

probably is reflecting an unconventional order parameter; perhaps strongest case in UPt3and CeCoIn5

♦ superconductivity in proximity to f-electron magnetism and absence of superconductivity in non-magnetic (La/Th) analogues – suggest magnetic fluctuations play a (dominant) role in producing superconductivity, but evidence not definitive

♦ emerging similarities to cuprates with lessons to be learned from studying both; exceptionally high quality crystals, eg. Ce115s, and easily tuned ground states an advantage

♦ like cuprates, important to understand strongly correlated normal state out of which superconductivity develops; a long-standing problem with perhaps different answers for Ce(4f) vs U,Np,Pu (5f) materials; recent two-fluid phenomenology for Ce115s may hold promise