Heavy%20Fermion%20Superconductivity:%20Competition%20and%20Cooperation%20of%20Spin

1
Heavy Fermion Superconductivity Competition
and Cooperation of Spin Fluctuations and Valence
Fluctuations
K. Miyake KISOKO, Osaka University
KISOKO Graduate School of Engineering Science
2
Outline of the talk
Fundamental concepts of superconductivity of
heavy fermion metals (mainly Ce-based
compounds) Brief introduction of
experiments and theoretical wisdom
Two kinds of SC mechanism for heavy fermion
metals, spin fluctuations and valence
fluctuations Experiments and
theoretical attempts.
Effect of magnetic field on valence transition
Signatures of valence transition or crossover
in Fermi surface change of CeRhIn5
Outlook for the future New universality
class of QCP associated with critical
end point of valence transition in heavy fermions
Connection to high-Tc cuprates
3
Current collaborators S. Watanabe (Osaka
Univ. ? Kyushu
Institute of Technology), A. Tsuruta (Osaka
Univ. ), J. Flouquet (CEA / Grenoble, ESPCI)
Former collaborators Y. Onishi (NEC), O.
Narikiyo (Kyushu Univ.), H. Maebashi (ISSP,
Univ. Tokyo) , A. T. Holmes (Univ.
Birmingham), D. Jaccard (Univ. Geneva),
M. Imada (Univ. Tokyo) T. Sugibayashi (Ehime
Univ.),
4
Fundamental concepts of superconductivity of
heavy fermion metals (mainly Ce-based
compounds) experiments and theoretical
wisdom
5
1979 year of paradigm change of superconductivity
Report of superconductivity in CeCu2Si2 which is
barely magnetic material
Steglich et al PRL 43 (1979) 1892
zero resistance
C/T ? m(N/V)1/3
Meissner effect
?
heavy electron
103 times larger than usual metals
T(K)
T(K)
Long silence till1984 and Strum und Drang of
research developments after then
6
BCS superconductivity is very fragile against
magnetic impurities
La1-xGdx
Tc
T(K)
Ferromagnetic state
Gd ()
Matthais et al PRL (1958)
7
(Osaka Univ. 2002)
8
General clue for pairing interaction in heavy
fermions
Pairing interaction among quasi-particles
(weight of quasi-particle in electron)
Coupling constant for Cooper pairing l
In order that Tc is high enough to be observable,
Pairing and heaviness of quasi-particles should
be the same origin magnetic fluctuations,
quadruple fluctuations, etc.
9
Electronic (spin-fluctuation or
charge-fluctuation) mechanism
cf. Kohn-Luttinger PRL 15 (1965) 524
pairing interaction
finite Tc in principle even for purely
repulsive interaction
Pairing interaction in triplet channel
Theory for superfluid 3He
Nakajima Prog. Theor. Phys.
50 (1973) 1101. Anderson Brinkman Phys.
Rev. Lett. 30 (1973) 1108
Para-magnon ferromagnetic spin fluctuations
Spin triplet P-wave pairing
10
Ferromagnetic spin-fluctuation mechanism was so
successful for understanding the existence of
3He-A phase (ABM phase)
Anderson-Brinkman Phys. Rev. Lett. 30 (1973)
1108.
Kuroda Prog. Theor. Phys. 51 (1974) 1269 .
Analogy between heavy fermion SC and 3He was
stressed till mid 80, early stage of research
of heavy fermion SC
The same process is valid also for
antiferromagnetic spin fluctuations
KM, Schmitt-Rink, Varma PRB 34 (1986) 6554.
Pairing interaction in singlet channel
Scalapino, Loh, Hirsch PRB 34 (1986) 8190.
AF fluctuations promote d-wave
General expression of pairing interaction in RPA
spin triplet
spin singlet
11
Since mid 90, SCs appeared near pressure
induced AF-QCP
CePd2Si2
Grosche et al Physica B 223/224 (1995) 50
CeIn3
Non-Fermi liquid behavior
3d AF-QCP
moderate enhancement
Mathur et al Nature 394 (1998) 39
AF spin fluctuations should play an important
role Recent Dogma
12
Another SC mechanism for heavy fermions,
enhanced valence fluctuations
Experiments
13

Two-types of P-induced heavy fermion SC near AF
quantum critical point (QCP)
CeCu2Si2 Steglich et al, PRL 43 (1979)
1892
CePd2Si2 Grosche et al, Physica B
223/224 (1996) 50
CeIn3 Mathur et al, Nature 394 (1998) 39
Strong-coupling theory near magnetic QCP
Moriya et al JPSJ 52 (1900) 2905 Monthoux
Lonzarich PRB 63 (2001) 054529
CeCu2Si2 Bellarbi et al, PRB 30 (1984)1182
CeCu2Ge2 Jaccard et al, IRAPT97
Physica B 259/261 (1996) 1
CeRhIn5 Heeger et al, PRL 84 (2000) 4986
Suggesting new SC mechanism of repulsive origin
14
QCP of Valence transition in
Cu
Ge
Ce
D. Jaccard et al, Physica B 259-261 (1999) 1
Rapid decrease of
Kadowaki Woods SSC 58 (1986) 507
Rapid change of nf
(valence of Ce)
M. Rice K. Ueda, PRB 34 (1986) 6420
cf.
Gutzwiller arguments
K.M. et al, PhysicaB 259-261 (1999) 676
15
T-linear resistivity (TgtTc) in CeCu2Ge2 near PPv
Jaccard et al Physica B 230-232 (1997) 297
16
Pressure scale is shifted such that Pc0
CeCu2(Si1-xGex)2
x0 (Thomas et al 96)
x0 (Holmes et al 03)
x1 (Jaccard et al 97)
x0.1 (Yuan et al 03) Two distinct Tc domes !
Enhancement of r0 at P4GPa
r(T) - r0 ? Ta
r(T) - r0 ? T at P4GPa
H. Q. Yuan et al, Science 302 (2003) 2104
SCES02 Krakow
17

Signature of critical valence fluctuations
observed in CeCu2(Ge,Si)2 around the critical
pressure Pv
PV
1) Enhancement of Tc (SC) 2)
Enhancement of r0 3) T-linear reesistivity r(T)
T 4) Shoulder of gC/T (at TTc)
Holmes, Jaccard, KM PRB 69 (2004) 024508
Two separate domes of Tc and 2) 3)
H. Q. Yuan et al, Science 302 (2003) 2104
18
Fujiwara, Kobayashi et al JPSJ 77
(2008) 123711
(1/T1T) ? A ? ?2
NQR relaxation rates
Line nodes T3 law at all pressures
19
K. Fujiwara SCES2011
suggesting sharp valence crossover
6
20
Another SC mechanism for heavy fermions,
enhanced valence fluctuations
Theoretical attempts.
21

(Kondo regime)
22

23
Real-space picture of pairing potential G(0)(q)
Strong on-site repulsion
short-range attraction
d-wave pairing
24
Extended periodic Anderson model (PAM) with f-c
Coulomb repulsion Ufc
PAM
Superconductivity with d-wave symmetry in
the valence crossover region
pressure
Phase diagram at T0 K in U-ef
slave boson MF
slave-boson mean-field fluctuations
1st-order valence transition
Mixed Valence
QCP
1st-order valence transition
Kondo
crossover
paramagnetic metal
Onishi KM JPSJ 69 (2000) 3955
Watanabe et al. JPSJ 75 (2006) 043710
1d DMRG calculations
25
Intuitive picture for enhancement of residual
resistivity
Effect of impurity remains as short-ranged
r0 not enhanced
Effect of impurity extends to over long-range
region
xV diverging as P ? PV
r0 highly enhanced
26
Enhancement of r0 due to critical valence
fluctuations as many-body effect on impurity
potential
KM, Maebashi JPSJ 71 (2002) 1007
In the forward scattering limit (k 0),
Ward-Pitaevskii identity
cf. Betbeder-Matibet Nozieres Ann. Phys. 37
(1966) 17 for single component Fermi
liquid
Renormalized impurity potential
Residual resistivity
divergent at criticality ( )
higher order corrections do not change the
result. (cf. Rutherford scattering)
27

Self-energy due to critical valence fluctuations
1st version
A. T. Holmes, D. Jaccard, KM Phys. Rev B 69
(2004) 024508
CeCu2Si2
Umklapp scattering 0ltqlt3kF/2
Peak structure of effective mass at PPv
28
Variety of valence transition
Location of critical point (Pvc,Tvc) in P-T
plane depends on the details of materials
parameters
Assumption there exist compounds such that
Tvc0 or Tvc ltlt EF
Ce
critical point
Typical example of 1st order valence transition
Ce g- a transition
Z. Fisk et al J. Appl. Phys. 55 (1984) 1921
29
Effect of magnetic field on valence transition
and fluctuations causing a metamagnetic behavior

S. Watanabe, A. Tsuruta, KM, J. Flouquet
PRL 100 (2008) 236401
JPSJ 78 (2009) 104706.
30
Drastic effect of magnetic field on valence
transition
Expected Phase Diagram in P-T-B space
Drymiotis et al J. Phys. Condens. Matter
17 (2005) L77
How about in the case Tclt0 (i.e., in the
crossover regime) ?
Tc
B
B
31
Field-induced VQCP
3 dimension
Ufc
ef
S. Watanabe et al, PRL 100 (2008) 236401
32
CeIrIn5
1st order V-T
V-QCP
Ufc
Crossover of valence
ef (P)
In-NQR nQ starts to change at P 2.1GPa
1st-order transition at h42T
Yashima Kitaoka (2010)
S. Kawasaki, et al PRL 96(2006)147001
T. Takeuchi, et al JPSJ 70(2001) 877
E. C. Palm, et al Physica B329-333(2003) 587
33
Promising candidate for magnetic field induced
QCP-VT
Fine tuning possible by changing H and P
(P)
Raymond Jaccard J. Low Temp. Phys. 120 (2000)
107
Jaccard et al Physica B 259-261 (1999) 1
34
Magneto resistivity of CeCu6 under pressure
J // b, H // c
?
?
?
Signature of H and P induced QCP-VT
?
?
?
?
Measurements at higher H and P expected to
exhibit much sharper structure
Y. Hirose et al (Onuki group) J. Phys. Soc. Jpn.
Suppl. (2012)
accepted for publication
35
Signatures of valence transition
or crossover in Fermi surface
change of CeRhIn5
S. Watanabe K.M. JPSJ 79 (2010) 033707
36
NK Sato Group (Nagoya)
Knebel et al JPSJ 77 (2008) 144704
37
dHvA result in CeRhIn5
Pc
H. Shishido, R. Settai, H. Harima Y. Onuki, J.
Phys. Soc. Jpn. 74 (2005) 1103
Small Fermi surfaces similar to those of
LaRhIn5 for PltPc
drastic change at Pc2.35 GPa
Large Fermi surfaces similar to those of
CeCoIn5 for PgtPc
What is the nature of transition?
Knebel et al JPSJ 77 (2008) 144704
cf. Park et al Nature 440 (2006) 65
38
Transport anomalies in CeRhIn5
Pc
e
T. Muramatsu et al., JPSJ 70 (2001) 3362
e 1
G. Knebel et al., JPSJ 77 (2008) 114704
residual resistivity has a peak at PPc
Pc
P (GPa)
T. Park et al., Nature 456 (2008) 366
Signature of sharp valence crossover
T-linear resistivity emerges most prominently
near PPc
39
scaling under pressure
G. Knebel et al., JPSJ 77 (2008) 114704
at H15 T
Cyclotron mass of b2 branch by dHvA at H1217 T
Shishido et al., JPSJ 74 (2005) 1103
Pc
m scales with
Mass enhancement near PPc not from AF QCP
but from band effect
cf. K.M. et al., Solid State Commun. 71 (1989)
1149
40
Extended periodic Anderson model
S. Watanabe, A. Tsuruta, K. Miyake J.
Flouquet, JPSJ 78 (2009) 104706
2D-like b2 branch
square lattice
(cf. half filling n 1)
filling
E(k)
k
first-order transition
Mixed Valence
Kondo
41
Slave-boson mean field theory
G. Kotliar A. E. Ruckenstein, PRL 57 (1996) 1362
probability for empty, singly-,
doubly-occupied states
l , l , dl Lagrange multipliers
Q (p,p) AF-ordered vector
mas renormalization factor
7 equations are solved self-consistently
42
Ground state phase diagram
H0
n 0.9
At quantum critical end point (QCP) of
first-order valence transition, valence
fluctuation diverges
t 1, V 0.2, U 8
Ufc1.0
cv
Ufc
0.5
0.0
ef
ef
43
Drastic change of Fermi surface
S. Watanabe et al., JPSJ 79 (2010) 033707
h0.005
t 1, V 0.2, U 8, Ufc0.5,
n 0.9,
kF
kF for conduction band ek
at nc0.8
Small Fermi surface changes to large Fermi
surface at AF to paramagnetic transition
discontinously
44
Mass enhancement
S. Watanabe et al., JPSJ 79 (2010) 033707
h0.005
t 1, V 0.2, U 8, Ufc0.5,
n 0.9,
As ef increases toward , Zs increases
Gap between original lower hybridized band the
folded band increases
f-dominant flat part of the folded band
approaches Fermi level m
Mass enhancement by band effect
This explains scaling
G. Knebel et al., JPSJ 77 (2008) 114704
45
Comparison with dHvA measurement
Pc
Larger D(m) for ef gt
missing b2 branch for P gtPc
h makes D(m) small
CeCoIn5
For ef-0.4, D(m)0.84 is about 10 times larger
than Dc(m)0.092 at nc0.8
At P 0 g 50 mJ/molK2 in CeRhIn5 is about 10
times larger than g 5.7 mJ/molK2 in LaRhIn5
H. Shishido et al., JPSJ 74 (2005) 1103
46
Effect of hybridization strength on P-T phase
diagram
(C)
(B)
(A)
V
CeCu2(Si,Ge)2
ß-YbAlB4 etc.
CeRhIn5 under H
CeCoIn5, CeIrIn5
S. Watanabe KM J. Phys. Condens. Matter,
23 (2011) 094217.
47
New universality class of QCP Critical end point
of valence transition in heavy fermions
S. Watanabe, KM PRL 105 (2010) 186403
48
Unconventional criticality in b-YbAlB4
r T
c T -0.5
T(K)
S. Nakatsuji et al., Nature Phys. 4 (2008) 603
Y. Matsumoto et al., arXiv0908.1242
Uniform magnetic susceptibility is enhanced as c
T -0.5 even though the system in not close to
the FM phase
-logT
Enhanced Wilson ratio
49
Unconventional criticality
Self Consistent Renormalization (SCR) theory for
spin fluctuations
T. Moriya, Spin Fluctuations in Itinerant
Electron Magnetism (Springre-Verlag, Berlin,
1985) T. Moriya K. Ueda, Rep. Prog. Phys. 66
(2003) 1299 J. A. Hertz, PRB
14 (1976) 1165 A. J. Millis,
PRB 48 (1993) 7183
RG study
r C/T c0 cQ
1/T1T
Fermi liquid T 2 constant
3D AF T 3/2 c-T 1/2 c-T1/4 T
-3/2 C.W. T -3/4
3D F T 5/3 -lnT T -4/3 C.W.
T -4/3 2D AF T
-lnT c -lnT/T C.W. -lnT/T 2D
F T 4/3 T -1/3 -1/(T lnT ) C.W.
-1/(T lnT )3/2
YbRh2Si2 T -lnT T -0.6
T -0.5
b-YbAlB4 T 1.5 T -lnT T -0.5
exp. desired
50
Quantum criticality of VQCP
Periodic Anderson model
K. M, J. Phys.Condens. Matter 19 (2007)
125201 S. Watanabe K. M., Phys. Status Solidi B
247 (2010) 490
large U
small U
fc
fc
valence
T
T
T
crossover
critical
end point
1st order valence transition
VQCP
P
P
P
Ce
b-YbAlB4
Most of
Ce, Yb compounds
51
QCP due to Critical Valence Transition
Unconventional criticality is caused by quantum
valence criticality
Key Origin Strong locality of valence
fluctuation mode
arising from local correlations of f electrons
Uniform spin susceptibility diverges in
paramagnetic metal even without proximity to FM
phase
cv c(T) T -x
0.5 lt x lt 0.7
1/(T1T ) c(T) T -x


T-linear resistivity
C/T -lnT
Large Wilson ratio RWgtgt2
Unified understanding expected for unconventional
NFL in
b-YbAlB4 , YbRh2Si2 , YbAuCu4 (J. L. Sarrao et
al 1999) Ce0.9-xLaxTh0.1 (J. C. Lashley et al
2006) YbCu5-xAlx (E. Bauer et al 1997), etc.
52
Connection to high-Tc cuprates
53
True phase diagram in high-Tc cuprates free from
disorder
cf. Varma et al Solid State Commun.
62 (1987) 681 p-d charge
transfer mechanism for cuprates due to Udp
Mukuda et al JPSJ 77 (2008) 124706
54
5 layered high-Tc cuprate superconductor
S. Shimizu et al JPSJ 80 (2011) 043706
55
Summary
Fundamental concepts of superconductivity of
heavy fermion metals (mainly Ce-based
compounds)
Two kinds of mechanism for heavy fermion
metals, spin fluctuations and valence
fluctuations Experiments and
theoretical attempts.
Magnetic field is a good tuning parameter on
valence transition
Critical valence transition or crossover seems
to be crucial for understanding CeRhIn5,
and also for other Ce115 and Pu115
Outlook New universality class of QCP
associated with critical valence
transition in heavy fermions Connection
to high-Tc cuprates
56
Phonon mechanism seems to be irrelevant to heavy
fermion superconductivity
cf. Kondo volume collapse mechanism using
phonons (cf. Razafimandimby, Fulde,
Keller, Z. Phys. B 54 (1984) 111)
Static effect is very small according to
microscopic analysis based on periodic Anderson
model. Jich et al Phys. Rev B 35
(1987) 1692
57
Dynamical valence susceptibility
RPA
At critical end point as well as QCP,
58
YbCu5-xAlx
x 1.5
E. Bauer et al, PRB 56 (1997) 711
3.0
2.8
2.6
Yb valence
2.4
2.2
x
2.0
0
2
1.2
0.8
0.4
1.6
x
T (K)
x 1.5
T
Yb valence crossover occurs near x 1.5
300 K
X-ray LIII absorption edge measurements
10 K
cf. K. Yamamoto et al, JPSJ 76 (2007) 124705
H. Yamaoka et al, PRB 80 (2009) 035120
0 K
x
VQCP
59
CeIrIn5
convex shape
C. Capan et al PRB 70 (2004) 180502R
17T
15T
12T
FL region
Residual resistivity increases
T-linear resistivity
Valence crossover line in T-H phase diagram
Q-CEP of 1st-order valence transition
S. Watanabe et al, PRL 100 (2008) 236401
60
CeIrIn5
C. Capan et al., PRB 80 (2009) 094518
Fermi surface is always large i.e., c-f
hybridization is always finite (f electrons are
always itinerant)
Adiabatic continuation holds with Luttingers sum
rule satisfied
Hc28 T
MV
S. Watanabe et al., JPSJ 75 (2006) 043710
Kondo
r0 has a peak at Hc
Fermi surface volume does not change at HHc
Consistent with field-induced V-QCP
Watanabe, Tsuruta, K.M. Flouquet, PRL 100
(2008) 236401
JPSJ 78 (2009)
104706
61
Fermi surface in AF phase
h0.005
dHvA H15T
Contour plot of lower hybridized band

h0.005
t 1, V 0.2, U 8, Ufc0.5,
n 0.9,
Fermi surface of conduction band ek at
nc0.8

Small Fermi surface
i.e., V 0 in HPAM


1
0.8
Fermi surface in AF phase for V0.2 is nearly
the same as small Fermi surface for V0
For P ltPc Fermi surface in CeRhIn5 is very
similar to LaRhIn5
62
H
H0
AF
Knebel et al JPSJ 77 (2008) 144704
Watanabe (Next Talk)
cf. Park et al Nature 440 (2006) 65
63
Quantum valence criticality
valence crossover surface
T
1st-order valence transition surface
ef
0K
S. Watanabe et al, JPSJ 78 (2009) 104706
VQCP
Ufc
Mode-coupling theory for valence fluctuations
gives
cv c(T) T -x
0.5 lt x lt 0.7
1/(T1T ) T -x


T-linear resistivity
C/T -logT
Large Wilson ratio RWgtgt2
S. Watanabe K.M., PRL 105 (2010) 186403
r C/T c 1/(T1T )
S. Nakatsuji et al, Nature Phys. 4 (2008) 603
b-YbAlB4 T 1.5 T -logT T -0.5
exp. desired
P. Gegenwart et al, Nature 4 (2008) 186
YbRh2Si2 T -logT T -0.6
T -0.5
64
Quantum criticality in YbRh2Si2
c T -0.6
r T
O. Trovarelli, C. Geibel, S. Mederle, C.
Langhammer, F.M. Grosche, P. Gegenwart, M. Lang,
G. Sparn, F. Steglich, PRL 85 (2000) 626
Heavy electron metal YbRh2Si2 shows magnetic
transition at TN 65 mK
-logT
65
YbRh2Si2
1/(T1T ) T -1/2
K. Ishida et al., PRL 89 (2002) 107202
66
To construct mode-coupling theory for valence
fluctuations by Ufc after taking account of
local correlations by U, we employ slave-boson
large-N expansion framework
m 1, ..., N
HPAM
Lagrange multiplier li
bi slave-boson operator
Lagrangian
67
For with ,
we introduce identity
perturbed
unperturbed
Ufc
For , saddle point solution is
obtained by stationary condition dS00
lqldq , bqbdq
68
Gaussian fixed point
J. A. Hertz, PRB 14 (1976) 1165 A. J. Millis, PRB
48 (1993) 7183
Renormalization group analysis
qsq, wszw
w
d 3 , z 3
wc
wc/sz
q
qc
0
Higher order terms than Gaussian term are
irrelevant
qc/s
Gaussian fixed point
69
Mode coupling theory of valence fluctuations
Construct best Gaussian taking account of mode
couplings for up to j4 terms
Feynmans inequality
Variational principle
Self-consistent equation for h
li mean free path by impurity scattering
K.M., O. Narikiyo Y. Onishi, Physica B 259-261
(1999) 676
70
Divergence of uniform spin susceptibility
S. Watanabe K.M., arXiv0906.3986
Critical valence fluctuations are qualitatively
described by RPA framework with respect to Ufc
G
G
Dynamical f-spin susceptibility has common
structure to cv(q,iwl )
At V-QCP, renormalized valence fluctuation
cv(0,0) diverges
(0,0) diverges
DMRG
Uniform spin susceptibility diverges
result
Watanabe et al. JPSJ 75 (2006) 043710
71
Almost flat dispersion of valence fluctuation mode
D1, V0.5, U
D
ef-1.0
ef-0.5
-D
ef 0.0

Almost q-independent dispersion emerges in Kondo
regime also in valence-fluctuation regime
Local correlation effect U
A extremely small !!
72

73
Unconventional criticality by valence fluctuations
In clean system CqC/q in d3, for AqB2lth

A
A
when at V-QCP (y00)
shown below
cv(0,0) h-1
Now we consider low-T regime (TltltTF)
y11
T0 is extremely small due to small A
t T/T0 is enlarged even for low T
(V-QCP)
Least square fit of y(t) for 5lttlt100
cv c(t) t -x
0.5 lt x lt 0.7


1/(T1T ) c(t) t -x
74
T-linear resistivity
CqC/q
In y gt 1 ( t gt 5 ) regime, T-linear
resistivity appears


y11
y11
Ueda Moriya JPSJ 39 (1975) 605
If A is extremely small, dynamical exponent z
may be regarded as z in CqC/qz-2
When z , r(T ) T for T 0 limit
Locality of valence fluctuation is origin of
T-linear resistivity
cv
Specific heat
A. T. Holmes, D. Jaccard K.M., PRB 69 (2004)
024508
C/T -lnT
S. Watanabe K.M., arXiv0906.3986