Y.%20Tanaka

1
Odd frequency pairing in superconducting
heterostructures
Alexander Golubov Twente University, The
Netherlands
Y. Tanaka Nagoya University, Japan Y.
Asano Hokkaido University, Japan Y. Tanuma Akita
University, Japan
2
Contents

• (1)What is odd-frequency pairing
• (2)Normal metal / Superconductor junctions
• (3)Ferromagnet/Superconductor junctions

3
Conventional Classification of Symmetry of
Cooper pair
Spin-singlet Cooper pair
Even Parity
d-wave
Cuprate
s-wave
BCS
Spin-triplet Cooper pair
Odd Parity
p-wave
3He
Sr2RuO4
4
Odd-frequency pairing
even-frequency superconductivity
Fermi-Dirac statistics
D(w)
D(?), f(?)
Symmetry of pair wave functions
Momentum x Spin x Frequency
w
odd-frequency superconductivity
Berezinskii (1974)
D(?), f(?)
Spin-triplet s-wave BalatskyAbrahams (1992) Spi
n-singlet p-wave
w
5
Pair amplitude(pair correlation)
Exchange of two electrons
Fermi-Dirac statistics
6
Pair amplitude
Exchange of time
Even-frequency pairing (conventional pairing)
Odd-frequency pairing
7
Symmetry of the pair amplitude
symmetric, - anti-symmetric
Frequency (time)
Spin
Orbital
Total
BCS
- (singlet)
-
(even)
(even)
ESE
Cuprate
(even)
-
-(odd)
(triplet)
3He
ETO
Sr2RuO4
-(odd)
-
(even)
(triplet)
OTE
-(odd)
-
- (singlet)
-(odd)
OSO
ESE (Even-frequency spin-singlet even-parity)
ETO (Even-frequency spin-triplet odd-parity) OTE
(Odd-frequency spin-triplet even-parity)
Berezinskii OSO (Odd-frequency spin-singlet
odd-parity) Balatsky, Abrahams
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• Odd-frequency pairing state is possible in
  inhomogeneous superconductors even for
  conventional even-frequency paring in the bulk
• Broken spin rotation symmetry or spatial
  invariance symmetry can induce odd-frequency
  pairing state
• - ferromagnet/superconducor junctions
  Bergeret,VolkovEfetov, 2001
• - non-uniform systems
• Junctions TanakaGolubov, 2007
  EschrigLofwander, 2007
• Vortices Yokoyama et al., 2008 Tanuma et
  al., 2009)

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Contents

• (1)What is odd-frequency pairing
• (2) Ballistic normal metal junctions
• (3)Diffusive normal metal junctions
• (4)Ferromagnet/Superconductor junctions

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Ballistic junction
Ballistic Normal metal (semi-infinite)
Superconductor (semi-infinite)
Y. Tanaka, A. Golubov, S. Kashiwaya, and M. Ueda
Phys. Rev. Lett. 99 037005 (2007)
M. Eschrig, T. Lofwander, Th. Champel, J.C.
Cuevas and G. Schon J. Low Temp. Phys 147
457(2007)
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Eilenberger equation
(explicitly denote direction of motion)
Pair potential
Quasiparticle function
Form factor
Pair amplitudes
Bulk state
Only
S
N
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spin-triplet p-wave superconductor
Normal metal
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Symmetry of the bulk pair potential is ETO
(low-transparent)
Pair potential
(high-transparent)
px-wave component of ETO pair amplitude
s-wave component of OTE pair amplitude
ETO (Even-frequency spin-triplet odd-parity) OTE
(Odd-frequency spin-triplet even-parity)
Y. Tanaka, et al PRL 99 037005 (2007)
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Underlying physics

• Near the interface, even and odd-parity
  pairing states (pair amplitude) can mix due to
  the breakdown of the translational symmetry.

Fermi-Dirac statistics
The interface-induced state (pair amplitude)
should be odd in frequency where the bulk pair
potential has an even -frequency component since
there is no spin flip at the interface.
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Andreev bound states in inhomogeneous systems are
manifestations of odd-frequency pairing amplitude
Andreev bound states
Positive pair potential
Electron-like QP
Cooper pair
Hole-like QP
Negative pair potential
Surface Tanaka et al, 2007
Vortex Tanuma et al, 2009
Scattering direction of QP
Phase change due to a vortex
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Mid gap Andreev resonant (bound) state (MARS)
Local density of state has a zero energy peak.
(Sign change of the pair potential at the
interface)
Tanaka Kashiwaya PRL 74 3451 (1995), Rep. Prog.
Phys. 63 1641 (2000) Buchholz(1981) Hara
Nagai(1986) Hu(1994) Matsumoto Shiba(1995) Ohashi
Takada(1995) Hatsugai and Ryu (2002)

?
?

Interface (surface)
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Odd-frequency pairing state in N/S junctions (N
finite length)
Bounds state are formed in the normal metal
Y. Tanaka, Y. Tanuma and A.A.Golubov, Phys. Rev.
B 76, 054522 (2007)
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Ratio of the pair amplitude in the Nregion
(odd/even)
At some energy, odd-frequency component can
exceed over even frequency one.
Odd-frequency pairing
Even-frequency pairing
Hidden odd-frequency component in the s-wave
superconductor junctions
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Ratio of the pair amplitude at the N/S interface
and the bound state level
Bound states condition (Z0) (McMillan Thomas
Rowell)
Bound states are due to the generation of the
odd-frequency Cooper pair amplitude
Y. Tanaka, Y. Tanuma and A.A. Golubov, PRB 76
054522 (2007)
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Symmetry of the Cooper pair (No spin flip)
Sign change (MARS)
Interface-induced symmetry (subdominant component
)
Bulk state
ESE (s,dx2-y2 -wave)
(1)
No
ESE (OSO)
(2)
OSO (ESE)
ESE (dxy-wave)
Yes
(3)
ETO (px-wave)
OTE (ETO)
Yes
ETO (py-wave)
ETO (OTE)
(4)
No
(4)
(1)
(2)
(3)

• ESE (Even-frequency spin-singlet even-parity)
• ETO (Even-frequency spin-triplet odd-parity)
• OTE (Odd-frequency spin-triplet even-parity)
• OSO (Odd-frequency spin-singlet odd-parity)

Phys. Rev. Lett. 99 037005 (2007)
21
Contents

• (1)What is odd-frequency pairing
• (2) Ballistic normal metal junctions
• (3)Diffusive normal metal junctions
• (4)Ferromagnet/Superconductor junctions

22
Impurity scattering effect
Tanaka and Golubov, PRL. 98, 037003 (2007)
Ballistic Normal metal
Superconductor
Only s-wave pair amplitude exists in DN
Impurity scattering (isotropic)
Diffusive Normal metal (DN)
Superconductor
(1)ESE (2)OTE
ESE (Even-frequency spin-singlet even-parity) OSO
(Odd-frequency spin-singlet odd-parity)
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Even frequency spin singlet even parity (ESE)
pair potential
Even frequency spin singlet s-wave (ESE) pair is
induced in DN.
ESE pair /ESE pair potential
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Odd frequency spin triplet s-wave (OTE) pair is
induced in DN
Px-wave case
New type of proximity effect
Y.Tanaka, A.A.Golubov, Phys.Rev.Lett. 98, 037003
(2007)
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Density of states in DN
Conventional proximity effect with
Even-frequency Cooper pair in DN
Unconventional proximity effect with
Odd-frequency Cooper pair in DN
TanakaKashiwaya, 2004
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Summary of Proximity effect
(diffusive normal metal, s-wave pairing state
only)
Induced pair amplitude in DN
Symmetry of the pair potential
Even frequency spin singlet even parity (ESE)
ESE
(1)
Even frequency spin triplet odd parity (ETO)
(2)
OTE
Odd frequency spin triplet even parity (OTE)
OTE
(3)
Odd frequency spin singlet odd parity (OSO)
(4)
ESE
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How to detect odd-frequency paring amplitude
measuring electrical conductivity
Asano, Tanaka, Golubov,
Kashiwaya, PRL 99, 067005 (2007)
OTE (Odd-frequency spin-triplet even-parity)
ESE (Even-frequency spin-singlet even-parity)
Kashiwaya, Maeno 2007
OTE proximity
ESE proximity (conventional)
Zero energy peak
No Zero energy peak
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Andreev bound states in inhomogeneous systems are
manifestations of odd-frequency pairing amplitude
Andreev bound states
Positive pair potential
Electron-like QP
Cooper pair
Hole-like QP
Negative pair potential
Surface Tanaka et al, 2007
Vortex Tanuma et al, 2009
Scattering direction of QP
Phase change due to a vortex
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Symmetry of the Cooper pair in a vortex core
Center of the vortex core
l angular momentum
m vorticity
bulk
Even
ESE
Even
ESE (s-wave..)
Even
Odd
ESE (s-wave..)
OSO
Odd
Even
ETO (chiral p-wave)
ETO
Odd
Odd
ETO (chiral p-wave)
OTE
ESE (Even-frequency spin-singlet even-parity) ETO
(Even-frequency spin-triplet odd-parity) OTE
(Odd-frequency spin-triplet even-parity) OSO
(Odd-frequency spin-singlet odd-parity
Yokoyama et al., Physical Review B, Vol. 78,
012508, 2008
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Difference of the angular momentum of the
odd-frequency pair at the core center
Angular momentum at the center of core lm
l angular momentum
m vorticity
Tanuma, Hayashi, Tanaka Golubov Phys. Rev. Lett.
102, 117003 (2009).
Chirality and vorticity Y. Kato and N. Hayashi
(2000, 2001,2002)
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Contents

• (1)What is odd-frequency pairing
• (2)Ballistic normal metal junctions
• (3)Diffusive normal metal junctions
• (4)Ferromagnet/Superconductor junctions

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Odd-frequency pair amplitude (not pair
potential) is generated in ferromagnet junctions
Odd frequency spin-triplet s-wave pair
spin-singlet s-wave pair
_

Superconductor
Ferromagnet
Bergeret, Efetov, Volkov, (2001) Eschrig, Buzdin,
Kadigrobov,Fominov, Radovic
Generation of the odd-frequency pair amplitude
in ferromagnet
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Josephson current in S/HM/S
Half metal CrO2
Keizer et.al., Nature (2006)
Bergeret et. al., PRL(01), Kadigrobov et. al.,
Europhys Lett.(01)
Spin active interface
Theory in the clean limit
Eschrig et. al., PRL(03)
Lofwander and Eschrig, Nature Physics (2008)
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Recursive GF
Furusaki, Physica B(92), Asano, PRB(01)
Advantages
SNS, SFS, S/HM/S
Parameters
exchange spin-flip
Y.Asano, Y. Tanaka, A.A.Golubov, Phys.Rev.Lett.
98, 107002 (2007)
35
Spin active interface
No sign change
odd-frequency pairs
36
Odd-frequency pairs
37
Quasiparticle DOS in Half Metal
S/HM/S
SFS
only
even-odd mix
pure odd
Zero Energy Peak
can be detected by tunneling spectroscopy
38
Summary

• Ubiquitous presence of the odd-frequency pairs in
  inhomogeneous systems.
• Low energy Andreev bound states can be expressed
  in terms of odd-frequency pairing (proximity
  effect and vortices).
• The origin of the anomalous proximity effect in
  DN/spin-triplet p-wave junction is the generation
  
• of the odd-frequency pairing state.
• (4) Odd-frequency triplet pairs carry
  supercurrent in
• S/Half Metal/S Josephson junctions.