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New directions of superconducting nanostructures 2009 (NDSN2009)
4-5 September, Nagoya Univ.
How to determine pairing symmetries of
chiral p-wave superconductors
from vortex-core tunneling spectroscopy
via odd-frequency pairings
Yasunari Tanuma (Akita University)
Collaborators:
N. Hayashi (Osaka Prefecture University)
Y. Tanaka (Nagoya University)
A. A. Golubov (University of Twente)
Contents
1. Introduction
- Chiral p-wave superconductors
- Andreev bound states & odd-frequency pairing states
2. Theoretical model
- Quasiclassical theory in the presence of impurity effect
- Vortex-core tunneling spectroscopy
3. Our results
- Parallel & antiparallel vortex states
- odd-frequency s-wave & d-wave pair amplitude
4. Summary
Sr2 RuO4
- Unconventional Superconductors
Y. Maeno et al., Nature (London) 372 (1994) 532.
Critical temperature Tc = 1.5[K]
- Crystal structure -
- Electronic states Quasi-2D Fermi surface
Ru : 4d -orbitals
Layered perovskite
eg
d3z2 −r2
dx2 −y2
β γ α
dxy
c
t2g
b
a
dyz
dzx
A.P. Machenzie et al., Phys. Rev. Lett. 76 (1996) 3786.
RuO2 plane
dyz
α band （ orbital
）
three bands
dzx
β band （ orbital
）
γ band （ orbital
dxy
）
Possible candidate for spin-triplet superconductors
Sr2 RuO4 - pairing symmetry in bulk states
Sr2 RuO4
orbital part：
Tc = 1.5[K]
μSR
Spontaneous Magnetization
（time-reversal symmetry breaking states）
G.M. Luke et al., Nature (London) 394 (1998) 558.
spin part：
NMR
Spin-polarizability for
H ! ab
is independent of temperature.
K. Ishida et al., Nature (London) 396 (1998) 658.
（frequency part）：
For ω → −ω pair wave function is even.
even frequency
d -vector notation
chiral p-wave pairing symmetry
d = z(sin kx ± i sin ky )
Non-uniform superconducting systems
- Andreev bound states (ABS) positive pair potential
∆+
electron-like quasiparticle
Cooper pair
hole-like quasiparticle
∆−
∆+ ∆− < 0
negative pair potential
Surface ABS
∆−
Vortex ABS
∆+
-
+
+
-
∆+
Odd-frequency
pairings
∆−
(110) surface
quasiparticle s moving direction
phase shift by applying magnetic field
The proposal of this study：
Vortex-core tunneling spectroscopy of
chiral p-wave superconductors
via odd-frequency pairing states.
・spin-triplet pairing symmetry
・chiral-domain structure
- Chirality and vorticity
Antiparallel vortex
（APV）
-
+
Y. Kato: J. Phys. Soc. Jpn. 69 (2000) 3378.
Y. Kato & N. Hayashi: J. Phys. Soc. Jpn. 70 (2001) 3368.
Y. Kato & N. Hayashi: J. Phys. Soc. Jpn. 71 (2002) 1721.
Parallel vortex
-
+
（PV）
- Nonmagnetic impurity scattering effect
N. Hayashi, Y. Kato & M. Sigrist: J. Low Temp. Phys. 79 (2005) 893.
The quasiclassical Green s function theory
- impurity scattering effect
・The Eilenberger equation
!
"
ˆ − Σ,
ˆ gˆ
−ivF · ∇ˆ
g = iωn τˆz − ∆
!
"
g
if
gˆ = −iπ
quasiclassical Green s function
−if¯ −g
Matsubara frequency
- self-energy
Born limit
ωn = (2n + 1)πT
ˆ n , r) → Σ(E,
ˆ
Σ(iω
r)
¯
ˆ
k)
- pair potential ∆(r,
}
SCF calculation
Y. Tanaka, Y. Tanuma & S. Kashiwaya:
Phys. Rev. B 64 (2001) 054510.
・LDOS（Local Density of states）
!
"
N (r, E) = NF Re g R iω → E + iδ
n
Self-consistent calculation
・Pair potential（PP）
∆(r, θ) = ∆+ (r)e
+ilθ
∆± (r) = πT V
+ ∆− (r)e
−ilθ
l = 0, 1, 2, · · ·
s-wave, p-wave, d-wave,
! "
|ωn |<ωc
Matsubara frequency
・Pair amplitude （PA）
F
(l)
(iωn , r, θ) =
(l)
F+ (iωn , r)e+ilθ
(l)
+
(l)
F− (iωn , r)e−ilθ
!
F± (iωn , r) = e∓ilθ
!
"
f (iωn , r, θ" )
- even frequency
f (iωn , r, θ! ) = f (−iωn , r, θ! )
- odd frequency
f (iωn , r, θ! ) = −f (−iωn , r, θ! )
e∓ilθ
!
#
f (iωn , r, θ" )
ωn = (2n + 1)πT
PP & odd-ω PA near vortex core
Angular momentum at the center of core： l + m
Impurity scattering：weak
m = +1
m = −1
p−wave
pair potential
(antiparallel)
m = +1
(parallel)
m = −1
(antiparallel)
(parallel)
1
px−ipy −wave
px+ipy −wave
0.5
px−ipy −wave
px+ipy −wave
0
−0.5
2
odd− ω
pair amplitude
Impurity scattering：strong
1
0
1
0.5
0
1.5
1
0.5
0
0
Γ = 0.1Δ 0 T = 0.1Tc
Γ = 0.3Δ 0 T = 0.1Tc
s−wave
s−wave
dx2−y2+idxy−wave
dx2−y2+idxy−wave
APV
PV
dx2−y2−idxy−wave
r / ξ0
0
For antiparallel vortex (APV) l + m = 0
For parallel vortex (PV)
l + m = −2
Γ：impurity
scattering rate
dx2−y2−idxy−wave
1
r / ξ0
2
3
‘odd-ω s-wave’
‘odd-ω d-wave’
Vortex-core tunneling spectroscopy
in order to identify pairing symmetry
Antiparallel vortex
vorticity
+iφ
STM tip
e
impurity
Parallel vortex
vorticity
−iφ
STM tip
e
-
+
−iθ
e
chirality
impurity
+
−iθ
e
chirality
odd-ω s-wave
odd-ω d-wave
s-wave pair amplitude is
robust against the impurity.
d-wave pair amplitude is
sensitive to the impurity.
LDOS near the vortex-core
Born impurity scattering
(a) Antiparallel chiral p-wave vortex
(b) Parallel chiral p-wave vortex
−2
4
−1
3
−1
3
2
1
m=+1
Γ = 0.1Δ0
2
−2
E / Δ0
−1
1
2
0
1
1
0
4
2
−2
3
−1
2
0
m=+1
Γ = 0.3Δ0
1
2
3
N(0,E)
4 0
0.5
1
1.5
r / ξ0
odd-frequency s-wave
2
2
0
1
m=−1
Γ = 0.1Δ0
0
4
3
E / Δ0
0
E / Δ0
4
E / Δ0
−2
2
0
1
1
0
2
0
1
m=−1
Γ = 0.3Δ0
1
2
3
N(0,E)
4 0
0.5
1
1.5
r / ξ0
odd-frequency d-wave
The measurements of ZEP under the influence of impurities.
The observations of the symmetry of those odd-ω pair amplitude.
2
0
The ZEP as a function of Γ
AbrikosovGorkov plot
3
N(r=0,E=0) / N F
10
antiparallel
parallel
2
Pair amplitude
1
0.5
10
odd-ω s-wave
0
0
0.2
! / "0
0.4
odd-ω d-wave
1
10
10
0
0
# = 0.005 "0
0.2
! / "0
Chirality & vorticity：
0.4
Y. Kato: J. Phys. Soc. Jpn. 69 (2000) 3378.
Y. Kato & N. Hayashi: J. Phys. Soc. Jpn. 70 (2001) 3368.
Y. Kato & N. Hayashi: J. Phys. Soc. Jpn. 71 (2002) 1721.
Summary
Vortex-core tunneling spectroscopy
of chiral p-wave superconductors
in the presence of impurity effect
- It enables us to detect the existence of
chirality and odd-frequency pairings.
Y. Tanuma N. Hayashi, Y. Tanaka & A.A. Golubov:
Phys. Rev. Lett. 102 (2009) 117003.