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July 2007 Cargese
50 years of BCS School
Quasiparticles in field
Ilya Vekhter
Louisiana State University, USA
July 2007 Cargese
50 years of BCS School
Quasiparticles in field
Ilya Vekhter
Louisiana State University, USA
Field Quasiparticles(spherical cows)
50 years of BCS School
Quasiparticles in field
Ilya Vekhter
Louisiana State University, USA
July 2007 Cargese
Unconventional superconductors under rotated magnetic field:
searching for zeroes
Ilya Vekhter
Louisiana State University, USA
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References: theory AV & IVPRL 96, 237001 (2006)cond-mat/0606390 (M2S proc.)PRB 75, 224501 (2007)PRB 75, 224502 (2007)+ unpublished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
Impurities and universal conductivity
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
Impurities and universal conductivity
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
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
H=0, finite energy ε
BCS peaks at ε=|∆(φ)|
A. Vorontsov and I.Vekhter, ‘06
Low ε, low H: Nodal contribution, agreement
with semiclassical method
July 2007 Cargese
Angle-resolved density of states
H=0, finite energy ε
BCS peaks at ε=|∆(φ)|
A. Vorontsov and I.Vekhter, ‘06
Low ε, low H: Nodal contribution, agreement
with semiclassical method
Intermed ε, low H: BCS peaks at ε=∆ preserved for H|| node,
destroyed for H|| antinodeanisotropy inversion
July 2007 Cargese
Angle-resolved density of states
Intermed ε, low H:
H=0, finite energy ε
BCS peaks at ε=|∆(φ)|
Low ε, low H: Nodal contribution, agreement
with semiclassical method
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
A. Vorontsov and I.Vekhter, ‘06
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
A. Vorontsov and I.Vekhter, ‘06
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)
A. Vorontsov and I.Vekhter, ‘06
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
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