interactions between superconductivity and quantum ... · pdf file(300k,h=0) 8 t 7.5t 7 t 6.7t...

Introduction CeCoIn5 Ferromagnetic superconductors URhGe & UCoGe Conclusion

Interactions between Superconductivity and Quantum Criticality inCeCoIn5, URhGe and UCoGe

Ludovic Howald

IMAPEC/SPSMS/INAC/DSM/CEA17 Rue des Martyrs38054 Grenoble

France

11 February 2011

Panel:H. SuderowC. MeingastC. Berthier

Thesis supervisor: J.P. Brison

1 / 33

Interactions between Superconductivity and Quantum Criticality in CeCoIn5, URhGe and UCoGe


My PhDwork

CeCoIn5

Transport: Resistivity under magnetic field.

Field induced QCP

Analysis of the upper critical field.

Effect of magnetic fluctuations on SC

2 / 33



My PhDwork

CeCoIn5

Transport: Resistivity under magnetic field.

Field induced QCP

Analysis of the upper critical field.

Effect of magnetic fluctuations on SC

Ferromagnetic superconductors URhGe & UCoGe

First Thermal conductivity measurements

Bulk superconducting transitionOther low T contributions than e−, magnetic fluctuations?Two band superconductivity?Large and anisotropic thermoelectric power

2 / 33



Experimental setups

Low temperature (8mK) high field (8.5T) resistivity on CeCoIn5

First low T thermal conductivity measurement on URhGe and UCoGe

Design of 2 new setups with:

rotating stage

sample holder in Ag to allow high fieldmeasurements

low temperature transformer

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Heavy Fermions

S. Nakatsuji et al.: Phys. Rev. Lett., 89, 106402 (2002) G. Knebel et al.: J. Phys. Soc. Jpn., 77, 114704 (2008)

Large effective mass,

Proximity to magnetic phase transition.

4 / 33



Quantum Critical Points (QCP)

second order phase transition at T = 0

Characterized by critical exponents, effective dimension d+ z, z ∈ [1, 3]

Non Fermi Liquid (Fermi liquid region vanishes at QCP)Experimentally:

ρ(T) = AT2 + ρ0 (T < TFL), ρ(T) ∝ T(T >> TFL)TFL → 0 at QCPA diverges at QCP

G. Knebel et al.: J. Phys. Soc. Jpn., 77, 114704 (2008) G. Knebel et al.: Phys. Rev. B, 65, 024425 (2001)

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Superconductivity

TSC = Ω exp(

−1.04(1+λ)λ−µ⋆(1+0.62λ)

)

λ = N(EF)V

V(~r, t) =Charges interactions

ee′g2eχe(~r, t)+

Spins interactions

~s · ~s′g2sχs(~r, t)

At a magnetic QCP soft modes re-enforced λ

What is the pairing mechanism? → Experimental probe of λ?

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Upper critical field Hc2

Hc2 → λ

TSC under field is limited by two mechanisms:

Kinetic energy, (given by 12m

(p− e~A)2): HOrbital ∝(

TSCvF

)2

Zeeman splitting: HPauli∼= ∆

gµB

J. P. Brison: Habilitation a Diriger des Recherches (1997)

Parameters:

effective mass: vF ∝ 1/m⋆

gyromagnetic ratio: g

characteristic energy scale: Ω

coupling constant: λ

TSC

m⋆ = mb(1 + λ)HPauli

∼= ∆

gµB⇒ HPauli/TSC ր if λ ր

7 / 33



Phase diagram of CeRhIn5

In CeRhIn5 one critical pressure Pc = 2.5 GPa.

Hc2 can be fitted with:

λ maximum at Pc,Ω constant,g smoothly evolves with pand vF only depend on λ.

G. Knebel et al.: J. Phys. Soc. Jpn., 77, 114704 (2008)

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Phase diagram of CeRhIn5 & CeCoIn5


CeCoIn5 p = 0 ⇔CeRhIn5 p ∼= 2GPa

G. Knebel et al.: Phys. Status Solidi B, 247, 557 (2010)

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CeCoIn5 p = 0 ⇔CeRhIn5 p ∼= 2GPa

No sign of QCP under p

G. Knebel et al.: Phys. Status Solidi B, 247, 557 (2010)

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No AFM phase detected but close toAFM (FFLO/Q-phase, Cd doping, ...)

Proximity to a field induced QCPH ‖~c 0.0 0.5 1.0 1.5 2.0 2.5

0

1

2

3

4

5

6

P = 0 GPa P = 0.45 GPa P = 1.34 GPa

H (T

esla

)

T (K)

CeCoIn5

H//c

G. Knebel et al.: Phys. Status Solidi B, 247, 557 (2010)C. F. Miclea et al.: Phys. Rev. Lett., 96, 117001 (2006)

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Previous experiments; Field induced QCP

J. Paglione et al.: Phys. Rev. Lett., 91, 246405 (2003) A. Bianchi et al.: Phys. Rev. Lett., 91, 257001 (2003)

QCP obtained from limit of the Fermi-liquid domain. (ρ(T) = AT2 + ρ0)

H(QCP)=Hc2?

magneto-resistance problems at low temperatures (ωcτ > 1),

specific heat data only available down to∼ 80mK. At 100mK 70% signal fromhyperfine contribution.

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This experiment

3 samples

A CeCoIn5~j ‖ a-axis

B CeCoIn5~j ‖ c-axis

C Ce0.99La0.01CoIn5~j ‖ c-axis

2 fields orientations:

H‖ c-axisH 45 c-axis

~j ‖ c-axis more sensitive to NFL[9]

ωcτ < 1 sample B, C and for 3samples when H 45 c-axis

Low noise high resolution

ρ(T) = AT2 + ρ0 (T < TFL)

TFL determined from χ2

A

M. A. Tanatar et al.: Science, 316, 1320 (2007)

0,0 0,1 0,2 0,3 0,4 0,52,0

2,5

3,0

3,5

4,0

4,5

5,0

5,5

B (x=0) C (x=0.01)

T (K)

(T) (

cm) j

//[10

0]

(T

) (cm

) j//[

001]

0,6

0,8

1,0

1,2

1,4

j//[100] A (x=0)

j//[001]

H=7T//[001]CexLa1-xCoIn5

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Phase diagrams H‖ c-axis

0 2 4 6 8 10 12 140

500

1000

1500

2000

2500

Paglione et al. Our data sample A

CeCoIn5 j//a

H (T)

Fermi-Liquid

T (m

K)

Sup

erco

nduc

tivity

J. Paglione et al.: Phys. Rev. Lett., 91, 246405 (2003)

3 4 5 6 7 8 90.0

0.1

0.2

0.3

0.4

0.5

0.61% La

cross

over

T (K

)

SC

Quantum

Critical

FL

a)

H (T)

Previous results reproduced with unfavourable geometry (~j ‖~c-axis),

No true coincidence between Hc2(0) and HQCP.

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All curves

0,0 0,1 0,2 0,3 0,4

0,050

0,055

0,060

0,065

0,015

0,020

0,025

0,030

0,035

0,0 0,1 0,2 0,3 0,4

0,015

0,020

0,025

0,03

0,04

0,05

0,06

0,07

0,0 0,1 0,2 0,3 0,4

0,06

0,07

0,045

0,050

H//[011]

H//[001]

7 T8 T8.5T

Sample A

8T7T6T

(T)/

(300

K,H=0

)

8 T7.5T7 T6.7T6.5T

Sample C

Sample B

8.5T7 T6 T5.7T

5.5T5.3T

T (K)

8.5T8 T7.5T7 T

5.7T6.7T7 T5.3T8.5T

5 sets of data can be used tofit

A ∝ |H − HQCP|−α

TFL ∝ |H − HQCP|z/2

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Dynamical exponent

3 4 5 6 7 8 90

10

20

30

40

50

60

HQCP

A (

cmK-2

)j//[0

01]

H (T)

(H-4.81)-1.08

Hc2

SC

b)0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7T

(K)

SC

cross

over

Quantum

Critical

FL

a) CeCoIn5

A ∝ |H −HQCP|−α. Fits found

α = 1.09± 0.37

TFL ∝ |H −HQCP|z/2

Hertz-Millis theory for AFM z = 2

for coincidence of divergence of Acoefficient and TFL = 0 we needz = 1.16± 0.14

Single energy scale:

ρ(T) = a(T/T0)2 + ρ0 → A = a/T2

0

A ∝ |H − HQCP|−α

→ T0 ∝ |H − HQCP|α/2

TFL ∝ |H − HQCP|z/2

TFL ∝ T0 → α = z

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QCP points scenarios

Divergence of m⋆ along hot spot directions

Mostly developed theory(Hertz-Millis-Moriya) Predicts z = 2,

ρ(T) ∝ T3/2(3d),...

Other models: Disorder (Rosch et al.), KondoNecklace model (Reyes et al.), ...

Complete reconstruction of the Fermi surfaceat QCP: divergence of m⋆ in all directions.

Few theoretical predictions.

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CeCoIn5 Phase diagram suggested by Zaum et al.

S. Zaum et al.: arXiv:1010.3175v1 (2010)F. Ronning et al.: Phys. Rev. B, 73, 064519 (2006)

Conclusion

Proximity between QCP and Hc2 at p = 0is a coincidence

divergence of A under p (Ronning et al.)

Hall effect anomaly (Singh et al.)

S. Singh et al.: Phys. Rev. Lett., 98, 057001 (2007)

How to explain Hc2?

G. Knebel et al.: Phys. StatusSolidi B, 247, 557 (2010)

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Decoupling between maximum of TSC and maximum of λ:Magnetic Pair breaking mechanisms

In CeCoIn5 at p = 0,∆C/C ∼= 4.5,BCS value 1.43,

Kos et al. and Bang et al. explain thisjump with magnetic pair breakingeffect. [13] T⋆ ∼ 6K → TSC = 2.3K(coupling between SC andmagnetization needed),

Monthoux et al. show for SC withstrong coupling & AFM pairing

⇒ pair breaking associated to the QCP⇒ the maximum of TSC is not at the

QCP

S. Kos et al.: Phys. Rev. B, 68, 052507 (2003)Y. Bang and A. V. Balatsky: Phys. Rev. B, 69, 212504 (2004)P. Monthoux and G. Lonzarich: Phys. Rev. B, 63, 054529 (2001)

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Data of Hc2

First measurements fromMiclea et al.to pmax and recent measurement ofKnebel et al. up to more than 2 · pmax

0,0 0,5 1,0 1,5 2,0 2,50

2

4

6

8

10

12

14

P = 0 GPa P = 0.45 GPa P = 1.34 GPa

H//[100]

H (T

esla

)

T (K)

H//[001]

C. F. Miclea et al.: Phys. Rev. Lett., 96, 117001 (2006)

0,0 0,5 1,0 1,5 2,0 2,50

1

2

3

4

5

6 P = 0 GPa P = 0.35 GPa P = 1.3 GPa P = 1.5 GPa P = 2.6 GPa P = 4 GPa

H (T

esla

)

T (K)

H//[001]

0,0 0,5 1,0 1,5 2,0 2,50

2

4

6

8

10

12

14 P = 0 GPa

P = 1 GPa P = 1.5 GPa P = 2.6 GPa

H (T

esla

)

T (K)

H//[100]

G. Knebel et al.: J. Phys.: Condens. Matter, 16, 8905 (2004)

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Parameters of the model

TSC(p,H) fitted with an Eliashberg model

We include magnetic pair breaking in the calculation:

TSC(H = 0)/Ω = F(λ, µ⋆, TM)And defined T⋆ = ΩF(λ, µ⋆, TM = 0)

+ Orbital and paramagnetic limit for field dependence.

Ω const.µ⋆ const. ∼= 0.1λ vary with pTM vary with p, TM = 0 at p = 4GPavF vary with p as: vF = vF0(1+ λ(p = 0))/(1+ λ(p))g vary with p and field orientation

λ(p) given by vF(p) ∝ TSC/dHc2dT

|T=TSC

Ω, TM(0) and λ0 are related through the condition T⋆(p = 0) = 6K.(∆C/C)

TM(p) given by TSC(p)

L. N. Bulaevskii et al.: Phys. Rev. B, 38, 11290 (1988)

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Fits of Hc2

0,0 0,5 1,0 1,5 2,0 2,50

2

4

6

8

10

12

14

P = 0 GPa P = 0.45 GPa P = 1.34 GPa

H//[100]

H (T

esla

)

T (K)

H//[001]

fixed parameters

vF0, Ω, µ⋆

Pressure dependent parameters

λ, TM, ga, gc

C. F. Miclea et al.: Phys. Rev. Lett., 96, 117001 (2006)

0,0 0,5 1,0 1,5 2,0 2,50

1

2

3

4

5

6 P = 0 GPa P = 0.35 GPa P = 1.3 GPa P = 1.5 GPa P = 2.6 GPa P = 4 GPa

H (T

esla

)

T (K)

H//[001]

0,0 0,5 1,0 1,5 2,0 2,50

2

4

6

8

10

12

14 P = 0 GPa

P = 1 GPa P = 1.5 GPa P = 2.6 GPa

H (T

esla

)

T (K)

H//[100]

G. Knebel et al.: J. Phys.: Condens. Matter, 16, 8905 (2004)

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Resulting parameters

0

1

2

3

4

TM

a

0

2

4

g

ga

P (GPa)

gc

0 1 2 3 40

5

10

gc

0

2

4

6

T (K

) b

Tc

T*

0

10

20

30

40

T(K

)

Maximum of ga, gc, T⋆, λ and TM

around 0.4GPa in agreement withQCP at this pressure,

M. Yashima et al.: J. Phys. Soc. Jpn., 73, 2073 (2004)M. Nicklas et al.: J. Phys.: Condens. Matter, 13, L905 (2001)

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Resulting parameters

0

1

2

3

4

TM

a

0

2

4

g

ga

P (GPa)

gc

0 1 2 3 40

5

10

gc

0

2

4

6

T (K

) b

Tc

T*

0

10

20

30

40

T(K

)

Maximum of ga, gc, T⋆, λ and TM

around 0.4GPa in agreement withQCP at this pressure,

Relatively large value of gc ∼= 8 (couldbe reduced to 6 with a lower value ofλ0)

Difference between λz → m⋆ &λ∆ → TSC,Contribution of localized momentmay leads to large g.

T. Tayama et al.: Journal of the Physical Society of Japan,74, 1115 (2005)

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Proposed phase diagram

02

46

0 1 2 3 4

0

1

2

3

57

H(T)

P(GPa)

T(K)

10 · TFL

T⋆

Conclusion

Features of CeCoIn5

(∆C/C, pressuredependence of: TSC,∆0/TSC, paramagneticlimit, ...) in this scenario.

Phase diagram of CeCoIn5

is a paradigm of an(almost 2D) stronglycoupledanti-ferromagneticallymediated superconductor.

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Ferromagnetic superconductors

Upper critical field of Ferromagnetic superconductors?

D. Aoki et al.: J. Phys. Soc. Jpn., 78, 113709 (2009)

23 / 33



Introduction

0 2 4 6 80

20

40

60

80

100

120

140

160

PM

SC

FM

(cm

)

T (K)

H=0T T2 fits

UCoGe j//c

Co-existence SC+Ferro →Triplet superconductivity,

Unusual Hc2: Re-entrance,positive curvature, strongangular dependence.

W. A. Fertig et al.: Phys. Rev. Lett., 38, 987(1977)D. Aoki et al.: J. Phys. Soc. Jpn., 78, 113709(2009)

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Measured Thermal Conductivity

0.1 1 10500

1000

1500

2000

2500

TSC 0T 0.2T 1T

/T(Wcm

-1K

-2)

T (K)

TCurie

UCoGe H//c j//c

Large residual term,

sharp superconducting phase transition,

Sample Quality?

25 / 33



Superconducting Phase diagram H‖~c-axis

0.0 0.2 0.4 0.6 0.80.0

0.2

0.4

0.6

(T) (T)

H (T

)

T (K)

UCoGe H//c j//c

Unusual shape of Hc2

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Superconducting Phase diagram H‖~c-axis

0.0 0.2 0.4 0.6 0.80.0

0.2

0.4

0.6

(T) (T)

H (T

)

T (K)

UCoGe H//c j//c

0 1 2 30

2

4

A(cm

K-2)

H (T)

UCoGe J//cH//c

Unusual shape of Hc2 requires either:

increase of λdecrease of orbital limitation

Possible explanations:

Increasem⋆ meta-magnetic transition→ increase λ (Miyake et al.),Field⊥moment superconductingpair→ increase λ (Mineev)

A. Miyake et al.: J. Phys. Soc. Jpn., 77, 094709 (2008)V. P. Mineev: arXiv:1011.3753v1 (2010)

F. Hardy et al.: to be publiehed in: J. Phys. Soc. Jpn. (2011)

26 / 33



Superconducting Phase diagram H‖ ~b-axis

0

2

4

6

8

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

(T) (T) (T) H 5° b-axis (T) H 5° b-axis

T (K)

H (T

)

UCoGe H//b j//c

0.1 1

500

1000

1500

0T 1T 2T 4T 6T 8.5T

/T(Wcm

-1K

-2)

T (K)

UCoGe H//b j//c

We confirm by bulk measurements:

strong angular dependence

re-entrance and positive curvature forHc2

importance of rotation mechanism


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Model of Lifshitz phase transition

Developed from an idea of(and with) Vincent Michaland V. Mineev

Lifshitz phase transition =topological anomaly on FS

cyclotronic vF → 0 for someH orientations

28 / 33



Fits with a divergence of the effective mass

0.0 0.2 0.4 0.6 0.80

5

10

15

20

H(T)

T(K)D. Aoki et al.: J. Phys. Soc. Jpn., 78, 113709 (2009)

0 10 200

1

2

m*(

arbi

trary

uni

ts)

H(T)

m*

m⋆ = m0 · log(1+ α| Hcrit.H−Hcrit.

|)

29 / 33



Fits with a divergence of the effective mass

0.0 0.2 0.4 0.6 0.80

5

10

15

20

H(T)

T(K)D. Aoki et al.: J. Phys. Soc. Jpn., 78, 113709 (2009)

0 10 200

1

2

m*

m*(

arbi

trary

uni

ts)

H(T)

m*

m⋆ = m0 · log(1+ α| Hcrit.H−Hcrit.

|)

α size of the “S” (α = 0.2)

29 / 33



Other experimental support for a Lifshitz scenario

Strong increase of thermolectricpower (TEP) at HR unrelated to FMphase (measurements done with L.Malone at LNCMI)

Strong anisotropy on TEP~j ‖~c-axis ∼= −30,~j ‖ ~a-axis ∼= −3(measurements done with L. Maloneat LNCMI)

0 5 10 15 20-7

-6

-5

-4

-3

-2

-1

0

1

2,9K 0,48K

S/T

(V

K-2)

H(T)

UCoGe H//bj//a

30 / 33



Other experimental support for a Lifshitz scenario

Strong increase of thermolectricpower (TEP) at HR unrelated to FMphase (measurements done with L.Malone at LNCMI)

Strong anisotropy on TEP~j ‖~c-axis ∼= −30,~j ‖ ~a-axis ∼= −3(measurements done with L. Maloneat LNCMI)

2D character of the compoundobserved from slope dHc2/dT

Small specific heat γ ∼= 50 and smallFermi velocity (dHc2/dT large) →small number of quasi-particles withlarge effective masses. → Small FSpockets of heavy carriers


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Fits of upper critical field

0

5

10

15

0 0.2 0.4 0.6 0.8 T (K)

H (Tesla)bc

bc

bc

bc

bc

bc

bc

bc

bc

bc

bc

bc

bc

bc

bc

bc

bc

bcbc rrrr rrrrrr

rrrrrrrrrr

rrrr

r

r

rrrrrrrrr

r

r

r

r

r

H ‖ ~a H ‖ ~b

u

u

u

u

u

u

u

u

u

u

u

u

uu

u

Shape of Hc2 can be reproduced for:

H‖ ~b-axis Hcrit. = 12TH‖~a-axis Hcrit. = 30Tχb

∼= 2χa (Huy et al.)

Does not explain Hc2 for H ‖~c-axis

Pair breaking like in CeCoIn5?

However suppression of orbitallimitation must happen in case ofLifshitz phase transition and gives anexplanation for the shape of Hc2.

N. T. Huy et al.: Phys. Rev. Lett., 100, 077002 (2008)

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Conclusion and Prospectives

CeCoIn5

no true QCP at Hc2(0)

Field dependence of TFL suggestsz = 1 QCP type?

Inclusion of pair breaking due tomagnetic fluctuations explain the SCpressure phase diagram.

L. Howald et al.: Journal of the Physical Society of Japan,80, 024710 (2011)

Prospectives

confirm position QCP at 0.4GPa withmore measurements of Hc2,

dHvA measurements to get moreinformation on the type of QCP,

Study of the pressure field phasediagram of other compounds withlarge strong coupling constant:NpPd5Al2?

Ferromagnetic superconductors

First thermal conductivity and firstbulk measurements of SC!

Confirmation of the unusualcurvature of Hc2 by bulkmeasurements.

We propose a new scenario to explainthe “re-entrance” of SC.

Prospectives

Lots to do... Test different qualitiessamples / other geometry, ...

32 / 33



Thank you for your attention

33 / 33


interactions between superconductivity and quantum ... · pdf file(300k,h=0) 8 t 7.5t 7 t 6.7t...

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