Kaleidoscope of Heavy Electrons: Super-conductivity, Magnetism, Quantum Critical Points

F. SteglichMPI for Chemical Physics of Solids, Dresden, Germany

In collaboration with:J. Custers, P. Gegenwart, C. Geibel, T. Cichorek, M. Grosche§, R. Küchler, M. Lang#,T. Lühmann, T. Nakanishi, M. Nicklas, S. Paschen, N. Sato+, R. Shiina§§, J. Sichelschmidt, G. Sparn, O. Stockert, T. Tayama##, K. Tenya++, P. Thalmeier, Y. Tokiwa, H. Wilhelm, H.Q. Yuan, S. Wirth

at present: § Royal Holloway College, Univ. of London; # Univ. of Frankfurt, + Nagoya Univ.,§§ Tokyo Metropolitan Univ.; ## ISSP, Univ. of Tokyo; ++ Hokkaido Univ.

M.M. Abd-Elmeguid, Univ. of Cologne K. Neumaier, WMI GarchingK. Ishida, Kyoto Univ. Y. Kitaoka, Osaka Univ.D.E. MacLaughlin, UC Riverside T. Komatsubara, Tohoku Univ.P. Coleman, Rutgers Univ. P. Fulde, MPI-PKS DresdenK. Miyake, Osaka Univ. C. Pépin, CEA SaclayQ. Si, Rice Univ. G. Varelogiannis, MPI-PKS DresdenG. Zwicknagl, TU Braunschweig

Crystal structures and phenomena

ferroelectricity

charge density wave

quantum Hall effect

conductivity in polymers

superfluidity

superconductivity (SC)

high-Tc SC in cuprates

colossal magnetoresistance

heavy fermions

quantum phase transitions

C60

1985, 98

2000 (C)

1996, 2003

1913, 72, 73, 20031987

YBa2Cu3O7UPd2Al3

Outline

• Heavy-fermion (HF) metals

• Superconductivity

• Quantum criticality

• Outlook

Superconductivity in CeCu2Si2

100 at% Ce3+ ions necessaryfor superconductivity

(LaCu2Si2 is not a superconductor)

Superconductivity of Heavy Fermions

(J/K2mole) ~ m*

TK ≈ 15 K

• Superconducting phase transition(B = 0)∆C/Tc ≈ (Tc)Cooper pairs formed by "HeavyFermions"Anisotropic order parameter

• Normal state(Bc2 ≈ 3 Tesla)

→ 1 J/K2mole (Cu: 0.7 · 10-3 J/K2mole)Heavy Electrons("Heavy Fermions")

Heavy-Fermion metals

T >> TK (10 ... 100 K)

T << TK: Heavy electrons ("compositefermions")

→ m* ≈ 1000 mel: "strongly correlated electron system"

vF ≈ 106sm

vF* ≈ 103sm

Groundstate properties - Heavy Landau Fermi liquid (LFL) : CeCu6- Non-Fermi liquid (NFL) : YbRh2Si2- Magnetic order : NpBe13- Superconductivity : UPd2Al3

Heavy-Fermion Superconductors

U-basedTc(K)

UBe13 0.9 ('83 Z/LANL)

Ce-basedTc(K)

CeCu2Si2 0.7 ('79 DA/K)[p = 2.9 GPa: 2.3 ('84 GE/GR)]CeNi2Ge2 0.2 ('97 DA,

'98 CA/GR)CeIrIn5 0.4 ('00 LANL)CeCoIn5 2.3 ('00 LANL)Ce2CoIn8 0.4 ('02 NA)CePt3Si 0.7 ('03 VI)

p > 0CeCu2Ge2 0.6 ('92 GE)CeRh2Si2 0.4 ('95 LANL)CePd2Si2 0.4 ('94 CA)CeIn3 0.2 ('98 CA)CeRhIn5 2.1 ('00 LANL)Ce2RhIn8 1.1 ('03 LANL)

UPt3 0.5 ('84 LANL)URu2Si2 1.4 ('84 K)UNi2Al3 1.2 ('91 DA)UPd2Al3 2.0 ('91 DA)

Pr-basedPrOs4Sb12 1.85 ('01 UCSD)

Pu-basedPuCoGa5 18.5 ('02 LANL)PuRhGa5 8.7 ('03 KA)

Non-Fermi-Liquid Superconductor: CePd2Si2[N.D. Mathur et al., Nature 394, 39 (1998)]

p →

↑T CePd2Si2

• AF QCP at pc = 28 kbar• Tc = 0.4 K at p=pc

• NFL normal state• SC mediated by strong spin-fluctuations ?

Magnetic Cooper pairing in UPd2Al3

C. Geibel et. al., 1991

Tmagn ≈ 14 Kµs ≈ 0.85 µB

γ ≈ 140 mJ/K2molTc ≈ 2 K2∆0/kBTc ≈ 6 (Kyougaku

et al., 1993)

•Two more localized ("core") electrons: magnetism• one less localized ("heavy" itinerant) electron:

heavy LFL state (Tc< T < TN)heavy-fermion SC (T < Tc)

U3+ (5f3)coexisting with local AF

Quasiparticle tunneling[M. Jourdan et. al., Nature 398, 47 (1999)]

tunnel diode UPd2Al3 - AlOx - Pb (Pb normal conducting : B = 0.3 T)

dI/dV

(T = 0.15 K)

Inelastic neutron scattering[N.K. Sato et al., Nature 410, 340 (2001)]

acoustic magnon("magnetic exciton")

= = (0, 0, 1/2)Tc = 1.8 K

Cooper pairs formed by heavyelectrons ("itinerant" 5f electrons)

superconducting glue provided bymagnetic excitons in the system of

"localized" 5f electrons

Q 0Q

Itinerant (SDW) scenario

g

T

TN

gc

TK

NFL

LFLSDW

Moriya, Hertz,Millis,Continentino,Lonzarich, …

• local moments screened• heavy QPs formed ("itinerant f-electrons")• weak coupling: SDW phase• QPs scattered off critical spin fluctuations

• in wide regions of phase diagram (T, lg-gcl)• thermal excitation of quantum critical modes• coupling of statics and dynamics: τc ~ ξz (AF: z = 2); deff = d + z

T < TK

NFL

0

CeCu2Si2: "A-S" transition at p < 0.7 GPa[G. Sparn et al., Rev. High Press. Sci. Technol. 7, 431 (1998)]

p > pc:

∆ρ = αT1.5

γ = γ0- βT0.5

(3D-SDW scenario)

SC

Tc

0.1

-2 0 2 4 6 80.0

0.5

1.0

1.5

2.0

2.5CeCu2(Si1-xGex)2

Tc

Tem

pera

ture

[K]

∆p=p-pc1(x) [GPa]

Tc TN x 0

Tc

AFM

S C

TN

0.1 0.25

Observation of two distinct superconductingphases in CeCu2Si2

[H. Q. Yuan et al., Science 302, 2104 (2003)]

Locally critical (local moment) scenario

T TK

NFL

LFL

• Two QCPs coinciding:

• New quantum criticality dominating: local spin fluctuations• At QCP: Kondo resonance not fully developed• NFL: fluctuations between small and large Fermi surface

Schröder, Coleman, Si,Pépin,Ingersent,Ramazashvili

TN

T*

ggc0

AFM - PMLFL - NFL

experimental techniques: T-dependence (at g = gc)g (p or B) -dependence (at T = 0)

YbRh2(Si0.95Ge0.05)2[P. Gegenwart et al., Acta Pol. Phys. B 34, 323 (2003)]

χ = C/(T−θ)θ = − 0.3 KC → µpara = 1.4 µB/Yb3+

χ−1 ∼ Tα with α ≈ 0.75

c

a

Yb

Rh

Si

-0.4 0 1 2 3 40.0

0.2

0.4

0.6

0.8

~ T0.75χ-1

(106 m

ol/m

3 )

T (K)

0.01 0.1 1 40

2

4

6

8

YbRh2(Si1-xGex)2x=0x = 0.05χ

(10-6

m3 /m

ol)

T (K)

0.12 0.18 0.24 0.30

B (T)

T=12K

T=5K

YbRh2Si2

dP/d

B (a

rb. u

nits

)

B ⊥ c (T)

0.3 0.6 0.9

-90

0

90

Θ (d

eg.)

Yb3+ ESR in YbRh2Si2[J.Sichelschmidt et al., PRL 91, 156401 (2003)]

B

a a

cΘ

g⊥ = 3.561 ± 0.006g|| = 0.17 ± 0.07

linewidth ∆Bexp = 0.02T << ∆BK = kBTK/µB = 37 T (TK = 25 K)

intensity I ~ χ (ω = 0, q = 0)

anisotropy

large local Yb3+ moments exist well below TK = 25 K

Kondo temperature TK from thermodynamics

0.1 1 10 100 3000.0

0.5

1.0

B = 0

YbRh2Si2

∆C /

T (J

/ m

ol K

2 )

T (K)

0 100 2000

1

2

3

4

RLn8

∆S

(Rln

2)

T (K)

TK = 20-30 K

~ log(T0/T)

T0 = 24 K

Rln8

Kondo temperature TK from transport

0 50 100 150 200 250 300

-100

-80

-60

-40

-20

0

S (µ

V/K

)

T (K)0 100 200 300

0

20

40

60

80

YbRh2Si2

j // c

j // a

ρ (µ

Ω c

m)

T (K)

0.02 0.1 10

1

2

3

a

x =0.05

x = 0 YbRh2(Si1−xGex)2

C el/T (J

mol−1

K− 2)

T (K)0 2 4 6 8 10

0

5

10

15

20

25

x=0.05

b

ρ (µΩ

cm)

T (K)

0.0 0.1 0.20.8

1.2

j // c

x=0 j ⊥c

Disparity between Cel/T and ρ in YbRh2Si2[J. Custers et. al., Nature 524, 424 (2003)]

Characteristic energy scale T0(b) ~ b vanishes at QCP

Scaling behavior in Cel(T,b)/T; b = B - Bc

0.02 0.1 1 20

1

2

B (T)

0.8

B ⊥ c

0.40.2

0.1

0.05

YbRh2 (Si0.95 Ge0.05 )2

Cel /

T (J

mol

-1 K

-2 )

T (K) 0.1 1 10 1000.2

1

B (T)0.050.10.20.40.8

φ(T

0.33

J K

-2 m

ol-1

T / (B - Bc ) (KT-1)

T0(b)

⇒T/T0 scaling for Φ(T,b) with Φ(T,b) = b1/3⋅C(T,b)/T)

⇒ no residual entropy at QCP⇒ S(T,b)=b2/3 S(T/T0) →0 at QCP

b 0:

0.02 0.1 1 100

1

2

a

γ 0 (J

mol

-1 K -2

)

(B - Bc ) (T)

~b−1/3

b =

γ0 YbRh2(Si.95Ge.05)2

Field dependence of γ0(b), b⊥ c (Bc = 0.027 T)

0.02 0.1 1 20

1

2

3

TN

B ⊥ c

B (T) 0 0.025

YbRh2 (Si0.95 Ge0.05 )2

Cel /

T (J

mol

-1 K

-2 )

T (K)

~ T−1/3

T 0:

Similar: ~12

−=∆

ATρ

T (b → 0)b (T → 0)

⇒ b-dependence at T=0 similar to

T-dependence at b=0

γ ~ (-log b) for b ≥ 0.3 Tb−1/3 (b → 0)

⇒ contradiction to 2D-SDW scenario

Scaling behavior in ρ(T,b), b = B - Bc

0 5 100

5

10

15

~ 1/b B ⊥ c 1/10 B || c

A(µΩcmK-2)

B (T) 0.1 1 10 1000.1

1

5B ⊥ cj ⊥ c

B (T) 0.1 0.2 0.4 0.5 0.7 1

dρ/d

T (µ

Ωcm

/K)

T / (B - Bc ) (K T -1 )

b>0: ∆ρ=A(b)T2

Grüneisen ratio[R. Küchler et al., PRL 91, 066405 (2003)]

Specific heat: C/T ∼ ∂S/ ∂TThermal expansion: β ~ (– ∂S/ ∂p)( β = 1/V dV/dT) LFLAF

p

T

0 2 4

100

150

Γ

T (K)

CeNi2Ge2

Γcr ~ 1/T

L. Zhu, M. Garst, A. Rosch, Q. Si, PRL 91, 066404 (2003):Grüneisen ratio Γcr = βcr/Ccr diverges at any QCP, Γcr ~ T-x

Critical exponent x nature of QCP

x = 1:

3D-SDW scenario

Outlook

Importance of local f-degrees of freedom, e.g.,

UPd2Al3 - Cooper pairing via (propagating) local 5f-excitation

break-up of heavy ("composite") quasiparticles

Is local QCP unfavorable for superconductivity?

CeCu2Si2 - compatible with (3D) SDW scenario, superconductors

CeNi2Ge2...

- no superconductor

YbRh2(Si, Ge)2 - suggestive of local QCP

non-Kondo-screened fluctuating Yb3+ moments

fractional exponent for Grüneisen ratio

However:

Superconductivity vs. magnetism

IA VIIIA1

H1.008 IIA IIIA IVA VA VIA VIIA

2

He4.003

3

Li6.941

4

Be9.012

5

B10.81

6

C12.01

7

N14.01

8

O16.00

9

F19.00

10

Ne20.18

11

Na22.99

12

Mg24.31 IIIB IVB VB VIB VIIB VIIIB IB IIB

13

Al26.98

14

Si28.09

15

P30.97

16

S32.07

17

Cl35.45

18

Ar39.95

19

K39.10

20

Ca40.08

21

Sc44.96

22

Ti47.87

23

V50.94

24

Cr52.00

25

Mn54.94

26

Fe55.85

27

Co58.93

28

Ni58.69

29

Cu63.55

30

Zn65.39

31

Ga69.72

32

Ge72.61

33

As74.92

34

Se78.96

35

Br79.90

36

Kr83.80

37

Rb85.47

38

Sr87.62

39

Y88.91

40

Zr91.22

41

Nb92.91

42

Mo95.94

43

Tc(98)

44

Ru101.1

45

Rh102.9

46

Pd106.4

47

Ag107.9

48

Cd112.4

49

In114.8

50

Sn118.7

51

Sb121.8

52

Te127.6

53

I126.9

54

Xe131.3

55

Cs132.9

56

Ba137.3

57

La138.9

72

Hf178.5

73

Ta180.9

74

W183.8

75

Re186.2

76

Os190.2

77

Ir192.2

78

Pt195.1

79

Au197.0

80

Hg200.6

81

Tl204.4

82

Pb207.2

83

Bi209.0

84

Po(209)

85

At(210)

86

Rn(222)

87

Fr(223)

88

Ra(226)

89

Ac(227)

104

Rf(261)

105

Db(262)

106

Sg(266)

107

Bh(264)

108

Hs(269)

109

Mt(268)

58

Ce140.1

59

Pr140.9

60

Nd144.2

61

Pm(145)

62

Sm150.4

63

Eu152.0

64

Gd157.3

65

Tb158.9

66

Dy162.5

67

Ho164.9

68

Er167.3

69

Tm168.9

70

Yb173.0

71

Lu175.0

90

Th232.0

91

Pa(231)

92

U238.0

93

Np(237)

94

Pu(244)

95

Am(243)

96

Cm(247)

97

Bk(247)

98

Cf(251)

99

Es(252)

100

Fm(257)

101

Md(258)

102

No(259)

103

Lr(262)

• Pairbreaking by magneticimpurities (xc < 1 at%)

LaAl2 TCo = 3.3 KGd-doping: xc = 0.6 at%Ce-doping: xc = 0.9 at%

TK = 0.5 K MHZ

(Riblet, Winzer 1971)

CeCu2Si2: Phase diagrams

chemical (C. Geibel et al., 1995)

physical (P. Gegenwart et al., 1998)

29Si NMR[K. Ishida et al., PRL 89, 107202 (2002)]

(1/T1T)4f ∝ T-½

0.1 1 10 100

0.1

1

10

~ T − 1/2

T (K)

NMR 0.15 T 0.50 T 2.42 T

(1/T

1T)4f

NM

R (s

-1K-1

)

0.1

1

10 ESR

0.19 T 0.68 T

(Im χ

/ω) 4f

ES

R

(arb

. uni

ts)

B →Bc+

B > Bc : LFL state below T*(B)

T → 0