50 years of BCS School Quasiparticles in field · 50 years of BCS School Quasiparticles in field...
Transcript of 50 years of BCS School Quasiparticles in field · 50 years of BCS School Quasiparticles in field...
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July 2007 Cargese
50 years of BCS School
Quasiparticles in field
Ilya Vekhter
Louisiana State University, USA
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July 2007 Cargese
50 years of BCS School
Quasiparticles in field
Ilya Vekhter
Louisiana State University, USA
Field Quasiparticles(spherical cows)
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50 years of BCS School
Quasiparticles in field
Ilya Vekhter
Louisiana State University, USA
July 2007 Cargese
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Unconventional superconductors under rotated magnetic field:
searching for zeroes
Ilya Vekhter
Louisiana State University, USA
July 2007 Cargese
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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References: theory AV & IVPRL 96, 237001 (2006)cond-mat/0606390 (M2S proc.)PRB 75, 224501 (2007)PRB 75, 224502 (2007)+ unpublished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)
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What we learned 0
Week I:BCS solved the problem of superconductivity
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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
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What we learned ISuperconductor: phase coherence + gap for excitations
Anisotropic gap (d-wave)Isotropic gap (s-wave)
φφ 2cos)( 0∆=∆0)( ∆=∆ φ
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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
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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)
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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.
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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
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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
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Impurities and universal conductivity
0/~ ∆γ
0/~ ∆γ
line nodes
Festkörperphysik ist eine Schmutzphysik. W. Pauli
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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
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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
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Power laws + universal transport →Existence but not position of nodes:
need to break symmetry
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Magnetic field as a probe
j=2ensvsB
core: ∆=0
Type-II superconductors: vortex state 21 cc HHH ≤≤
vs~h/2mr
∆
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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
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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
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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
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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
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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
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Wanteddead or
alive
I. Vekhter , P. Hirschfeld, J. Carbotte and E. Nicol, 1999
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SuccessesYNi2B2C
205.0 cHH ≥
T. Park et al, 2003
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SuccessesYNi2B2C
T. Park et al, 2003
205.0 cHH ≥
CeCoIn5
H. Aoki et al 2004
22.0 cHH ≥
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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
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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
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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
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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
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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
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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
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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
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Angle-resolved density of statesA. Vorontsov and I.Vekhter, ‘06
H=0, finite energy ε
BCS peaks at ε=|∆(φ)|
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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July 2007 Cargese
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July 2007 Cargese
Thanks to everyone!
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July 2007 Cargese
I am ready for questions
Photo credit: A. Vorontsov