50 years of BCS School Quasiparticles in field · 50 years of BCS School Quasiparticles in field...

July 2007 Cargese

50 years of BCS School

Quasiparticles in field

Ilya Vekhter

Louisiana State University, USA

July 2007 Cargese



Ilya Vekhter


Field Quasiparticles(spherical cows)



Ilya Vekhter


July 2007 Cargese

Unconventional superconductors under rotated magnetic field:

searching for zeroes

Ilya Vekhter


July 2007 Cargese

July 2007 Cargese

Work with:Anton Vorontsov

References: theory AV & IVPRL 96, 237001 (2006)cond-mat/0606390 (M2S proc.)PRB 75, 224501 (2007)PRB 75, 224502 (2007)+ unpublished

References: expt. reviews

Specific heat: T. Park & M. SalamonMod. Phys. Lett. B 18, 1205 (2004)

Thermal cond.: Y. Matsuda, K. Izawa, IV J. Phys. Cond. Mat. 18, R705 (2006)

July 2007 Cargese

What we learned 0

Week I:BCS solved the problem of superconductivity

July 2007 Cargese

What we learned 0

Week I:BCS solved the problem of superconductivity

Week II“Science … never solves a problem without creating ten more”

G. B. Shaw

July 2007 Cargese

What we learned ISuperconductor: phase coherence + gap for excitations

Anisotropic gap (d-wave)Isotropic gap (s-wave)

φφ 2cos)( 0∆=∆0)( ∆=∆ φ

July 2007 Cargese

22 |)(|)()( kkk ∆+= ζE

0)ˆ( =∆ nk

0)ˆ( =∆FS

k

What we learned II

• Pairing at momentum Q may lead to anisotropic gap

Q

+

-• Phase change:

• zeroes (nodes)

• low-energy qp

unconventional superconductivity

+

-

node and low energy excitations

July 2007 Cargese

What this talk is about

How to determine position of nodes on the FS?

Are there nodes?

If yes, where?

What are the experimental options?

a) couple to phase (Josephson);b) couple to low energy qp (specific heat, thermal conductivity)

July 2007 Cargese

Testing the anisotropy I: phase• Josephson effect:

• Test for change of sign

• Phase-sensitive, but also surface sensitive

• Cuprates, but probably no other systems…

2,1|| 2,12,1φie∆=∆ 21 φφ −∝sj

s

L. Greene et al.D.Van Harlingen et al. J. R. Kirtley et al.

July 2007 Cargese

Testing the anisotropy II: nodes

T∆

No excitations at low TActivated behavior e-∆/T

T∆

node Density of qp ∝TSpecific heat C(T)∝ T2

NMR T1-1∝ T3

T0/)( ∆∝ωωN Power laws

July 2007 Cargese

Power Laws at low T

TTTTTTC

∝∆∝∆∝∝ −

λρ ;;;)( 31

12

A. Carrington et al. 1999

Measure properties of unpaired excitations: NMR, specific heat, thermal conductivity, superfluid density

YBa2Cu3O6.95

U1-xThxBe13

D. MacLaughlin1984

Power laws in low T properties

July 2007 Cargese

Impurities and universal conductivity

0/~ ∆γ

0/~ ∆γ

line nodes

Festkörperphysik ist eine Schmutzphysik. W. Pauli

July 2007 Cargese


L. Taillefer et al. 1997

0/~ ∆γ

0/~ ∆γ

line nodes

02

02 /vv)0(

/∆≈∝⎟⎟

⎠

⎞⎜⎜⎝

⎛FF NN

Tτ

κσ“Universal” transport

as T→0

E. Fradkin 1986, P. Lee 1993, M. Graf 1996, A. Durst and P. Lee 2000

July 2007 Cargese


0/~ ∆γ

0/~ ∆γline nodes “Universal” transport

as T→00

20

2

0/vv)0(/lim ∆≈∝

→ FFTNNT τκ

“Universal” thermal conductivity

July 2007 Cargese

Power laws + universal transport →Existence but not position of nodes:

need to break symmetry

July 2007 Cargese

Magnetic field as a probe

j=2ensvsB

core: ∆=0

Type-II superconductors: vortex state 21 cc HHH ≤≤

vs~h/2mr

∆

July 2007 Cargese

Low energy field-induced excitationsLocalized states in the vortex cores:

FEmE

2

20

2

2∆

≈≈∆ξh Caroli, DeGennes,Matricon

Unconventional SC: leak along the nodes, short ξ0

Extended near-nodal states in the bulk

vs·k∆

k(r)v(k)r)(k, ⋅−=′ sEEsemiclassical

At H<< Hc2 “Doppler shifted” quasiparticles contribute the most to measured properties

G. Volovik, 1993

July 2007 Cargese

Semiclassical methodIgnore core states, and use Doppler shift on extended states

Theory: G. Volovik, 1993 , C. Kübert et al. 1998

Example: DOS

0/)();0( ∆⋅≈= nsN krvrω

0/||)( ∆∝ ωωN with no field

Doppler shifted DOS

Average supercurrent 1v −Λ∝sMagnetic length/ intervortex distance H/0Φ=Λ

rr dNAHN ∫ =≈= − );0();0( 1 ωω

20

)0(

cHH

NN

∝Residual DOS

20

)0(

cHH

NN

∝Compare: s-wave

July 2007 Cargese

Semiclassical methodIgnore core states, and use Doppler shift on extended states

Theory: G. Volovik, 1993 , C. Kübert et al. 1998

Expt: K. Moler, 1994 , B. Revaz et al. 1998, Y. Wang et al. 2001,

YBCO

Example: DOS

0/)();0( ∆⋅≈= nsN krvrω

0/||)( ∆∝ ωωN with no field

Doppler shifted DOS

Average supercurrent 1v −Λ∝sMagnetic length/ intervortex distance H/0Φ=Λ

rr dNAHN ∫ =≈= − );0();0( 1 ωω

20

)0(

cHH

NN

∝Residual DOS

20

)0(

cHH

NN

∝Compare: s-wave

July 2007 Cargese

Magnetic field as a probe

B

Nodal quasiparticles are Doppler shifted

The shift depends on the angle between vs and nodal k

H α

• Quasiparticles moving || H are not Doppler shifted

• Quasiparticles moving ┴ H are Doppler shifted

• Directional probe

July 2007 Cargese

Anisotropic Density of States

I. Vekhter et al. 1999

H

active active

active active

H

active

active passive

passive

α

α

C/T

nodes

|)cos||,sinmax(|)( ααHN ∝H

Anisotropy under rotated H Minima in DOS, specific heat for H || nodes

July 2007 Cargese

Wanteddead or

alive

I. Vekhter , P. Hirschfeld, J. Carbotte and E. Nicol, 1999

July 2007 Cargese

SuccessesYNi2B2C

205.0 cHH ≥

T. Park et al, 2003

July 2007 Cargese

SuccessesYNi2B2C

T. Park et al, 2003

205.0 cHH ≥

CeCoIn5

H. Aoki et al 2004

22.0 cHH ≥

July 2007 Cargese

SuccessesYNi2B2C

T. Park et al, 2003

205.0 cHH ≥

Sr2RuO4

CeCoIn5

H. Aoki et al 2004

22.0 cHH ≥

22.0 cHH ≥

K. Deguchi et al. 2004

July 2007 Cargese

SuccessesYNi2B2C

T. Park et al, 2003

205.0 cHH ≥

Sr2RuO4

CeCoIn5

H. Aoki et al 2004

22.0 cHH ≥

22.0 cHH ≥

not too low

K. Deguchi et al. 2004

July 2007 Cargese

Thermal transport

• Semiclassical

– transport kernel?

– is there a local κ(r)?

– if so, what is the measured κ?

• Scattering on vortices?

– not included, but needed

– scattering vs DOS

• Twofold vs fourfold

• Maxima or minima at nodes?K. Izawa et al 2001

• many expts Yu. Matsuda et al 2001-06

τκ 2v ffNT∝

[ ])(),( rjrjT hh ′∝κ

0=⋅∇ hj

July 2007 Cargese

Experimental contradictions

Jq

Hα

CeCoIn5

C

H. Aoki et al 2004 K. Izawa et al 2001

22 yxd

−xyd vs

κlayered 3D structure

line nodes

need better theory with vortex scattering at moderate H/Hc2, T/Tc

July 2007 Cargese

ModelQuasi-2D Fermi surface (r=s=0.5):

3D vortex lattice, n=0,2,4; FS anisotropy S

Line nodes,

φφ 2sinor2cos)ˆ( =pY)ˆ()ˆ()ˆ,ˆ( 0 pppp ′=′ YYVV

φ azimuthal angleField in the plane at angle φ0 to the x-axis

Jq

H

φ0

xydJq

H

22 yxd

−

)/2cos( 222222FzFyxF prspprppp −+=

⎟⎟⎠

⎞⎜⎜⎝

⎛

Λ

Λ−Φ

Λ∆=∆ ∑ S

kSxS

eC yn

ySikn

kkn

n

y

y

y

2

4)(

,)(R

Heat flow along x

July 2007 Cargese

Method of solution),(~2)(2~2 φε RRAvF ∆=⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −∇+− igf

ciei R

Input: vortex lattice

-- self-consistency in T,H, impurities

-- DOS, specific heat, thermal conductivity

2/1

||

~2)ˆ(||

21 -i ),ˆ( 22

0

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ Λ′⎟⎟⎠

⎞⎜⎜⎝

⎛ Λ∆−= ⊥⊥

FFWYig

vp

vp εππε

H

vF

vF

Brandt-Pesch-Tewordt approximation: g → spatial average

Nearly exact near Hc2, good down to low fields

Closed form expression for the Green’s function

-- angle-dependent scattering on the vorticesA. Houghton and I. Vekhter ‘98, H. Kusunose ‘04, A. Vorontsov and I.Vekhter, ‘06

July 2007 Cargese

Angle-resolved density of statesA. Vorontsov and I.Vekhter, ‘06

H=0, finite energy ε

BCS peaks at ε=|∆(φ)|

July 2007 Cargese

Angle-resolved density of states



A. Vorontsov and I.Vekhter, ‘06

Low ε, low H: Nodal contribution, agreement

with semiclassical method

July 2007 Cargese







Intermed ε, low H: BCS peaks at ε=∆ preserved for H|| node,

destroyed for H|| antinodeanisotropy inversion

July 2007 Cargese


Intermed ε, low H:





BCS peaks at ε=∆ preserved for H|| node,

destroyed for H|| antinodeanisotropy inversion

Low ε, high H:pairbreaking by vortex scattering:

anisotropy inversion cf. M. Udagawa et al


July 2007 Cargese

Anisotropy reversal due to scatteringD

ensi

ty o

f sta

tes

H

Spec

ific

heat

moderate H, T

anisotropy inversion

“semiclassical” regime

TTNd

TC

2cosh)( 2

2 ωωωω −∫ ⎟⎠⎞

⎜⎝⎛=

In a region of a T-H phase diagram maxima, rather than minima of the specific heat correspond to nodal directions

July 2007 Cargese

Specific heat anisotropy

H

•Shaded area: C/T minimum for H||node

•Unshaded: C/T maximum for H||node

•weakly dependent on corrugation of FS

H/H

c2

July 2007 Cargese

Specific heat anisotropy

H

•Shaded area: C/T minimum for H||node

•Unshaded: C/T maximum for H||node

•weakly dependent on corrugation of FS

H/H

c2

Experiments on CeCoIn5:

suggestive of dx2-y2 pairing

not dxy

H. Aoki et al 2004

July 2007 Cargese

25.0=cT

T

22 yxd−

xyd

cn TT

//

κκ

H

JqJq

H

Thermal conductivitySingle particle scattering rate ≠transport scattering rate


July 2007 Cargese

25.0=cT

T

22 yxd−

agrees with experiment

cn TT

//

κκ

H

Jq

H

Thermal conductivity

K. Izawa et al 2001

both fourfold (nodes) and twofold (vortex)


July 2007 Cargese

Fermi surface effects

Important:relative orientation of near nodal vFand H determines both energy shift and scattering

2/1

||

~2)ˆ(||

21 -i ),ˆ( 22

0

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ Λ′⎟⎟⎠

⎞⎜⎜⎝

⎛ Λ∆−= ⊥⊥

FFWYig

vp

vp εππε DOS, specific

heat

Anisotropy of the specific heat across the T-H phase diagram is sensitive to the curvature of the Fermi surface in the vicinity of the nodal directions.

July 2007 Cargese

Summary

• Microscopic theory of the anisotropy of thermal/ transport properties of nodal sc under rotated field

•Finite energy, not just zero-energy DOS: vortex scattering and inversion of the DOS anisotropy

•Both vortex and nodal physics in thermal transport (2-4-fold)

•Resolved the controversy (likely): CeCoIn5 – dx2-y2

•Anisotropy depends on the curvature of the Fermi surface

•Details are important: not a straightforward probe

July 2007 Cargese

July 2007 Cargese

Thanks to everyone!

July 2007 Cargese

I am ready for questions

Photo credit: A. Vorontsov

50 years of BCS School Quasiparticles in field · 50 years of BCS School Quasiparticles in field...

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