Chapter 7 Conclusion 57
7.2 Future studies
In the studies for the candidate materials for the spin-triplet superconductors such as Sr2RuO4, UPt3, and CuxBi2Se3, the contributions of the multi-band and the multi-gap superconductivities have been investigated since the mechanisms of the superconductivities are expected to be related to the multi-orbital electronic states. If the multi-band superconductivity realizes, physical quantities such as the NMR relaxation rate and the DOS are quantitatively modified, because they are given by the Fermi surface average of the multi-band, mixed up the band-dependent chirality and the nodes-structures of the pairing function. For example, in iron-based superconductors, the multi-gap contribution on the coherence effect in the NMR relaxation rate have been investigated to distinguish the pairing symmetries, i.e., s++ or s+− wave pairings [90]. In addition, in the multi-band superconductors, it has been suggested that the momentum parity derived from the internal-orbital exchange also reflects the NMR relaxation rates and the appearance of odd-frequency Cooper pairs [154, 155].
Therefore, we need to investigate the material-dependent multi-band contribution on the local NMR relaxation rate in the vortex state when we quantitatively analyze the experimental data to identify the pairing symmetry of candidate materials for spin-triplet multi-band superconductors by the site-selective NMR measurement. However, if the chirality in the multi-band superconductivity exists, the anomalous suppression of the local NMR relaxation rates by the negative coherence effects is expected to be observed, but it might be smeared by the multi-band superconductivity.
Consideration of the spin-orbit coupling effect is important to investigate the direction ofd-vector since the spin-orbit coupling plays a main role in the pinning interaction of the d-vector. If the
Chapter 7 Conclusion 60 spin-orbit coupling effect is weaker so that the electron spins can easily reorient toward the applied magnetic fields so thatd⊥H. In addition, in the surface state measurements such as the STM/STS measurements, the antisymmetric spin-orbit coupling, such as Rashba interaction, is allowed to be exist by the local inversion symmetry breaking, and it reflects the pairing symmetry. Therefore, we need to investigate the spin-orbit coupling effects on the physical quantities such as the local NMR relaxation rates and the spin-polarized LDOS, and the effect on the pairing symmetry of candidate materials for spin-triplet superconductors observedw by the site-selective NMR measurement and spin-polarized STM/STS measurements.
Acknowledgments
I would like to express my gratitude to my supervisor, Professor Masanori Ichioka, whose comments and suggestions were of inestimable value for my study. He has guided me to accomplishment of my research by proper advice and prepared environment in which I can concentrate on my investigation.
I would like to thank Professor Kazushige Machida who cared and discussed about the progress of the research. He has guided me efficiently at the beginning stage of my research.
I would like to thank Professor Seiichiro Onari for invaluable comments and teaching the band calculation and computational technic. His clear suggestions made my research move forward.
I would also like to thank Professor Hiroto Adachi for helpful discussions. In particular, I can understand quasiclassical Green’s function technic for the NMR relaxation rate and the thermal conductivity in the vortex state of superconductors by his instruction.
I am indebted to Professor Takeshi Mizushima for helpful discussions about the Bogoliubov-de Gennes theory and topological superconductivities, and Professor Guo-qing Zheng for helpful discussions about the experiments. I would also like to thank everyone in the Mathematical Physics Laboratory for advice about research and life.
In my personal life, I thank my parents and all of my friends for the great supports.
Finally, I would like to acknowledge the financial support of the Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
2018, March Kenta Tanaka
62
A
Riccati equation
In this appendix, the matrix Riccati equation is introduced to solve the quasiclssical Eilenberger equation in the matrix representation for the spin-space [87, 88]. By numerically calculating the matrix Riccati equation, we can obtain solutions of the quasiclassical Eilenberger equation in the spin-triplet pairing states. Moreover, in the calculation of the Riccati equation the solutions can be obtained for arbitrary initial values. This technique for the numerical calculations is called matrix Riccati method.
A.1 Matrix Riccati equation
The Eilenberger equation is iv·∇gˇ+ 1
2[ ˇM ,g] = ˇˇ 0, Mˇ≡
[iMˆg −Mˆf
Mˆf¯ −iMˆ¯g
]
=i(ωn+iv·A)ˇτ3+ ˇK·2 ˇS−∆ˇ −Σˇ (A.1) where ˇg is the quasiclassical Green’s function, ˇ∆ order-parameter, ˇΣ self-energy, ˇS spin-matrix defined as [ ˇS]11 = [ ˇS]22 = ˆσ/2 with the Pauli matrix ˆσ, and ˇK Zeeman effect and spin-orbit coupling term. These terms are defined as
ˇ
g(ωn,k,r) =−iπ
[ gˆ ifˆ
−fˆ† −gˆ¯ ]
(A.2)
∆(kˇ F,r) =
[ ˆ0 ∆ˆ
−∆ˆ† ˆ0 ]
, Σ(ωˇ n,r) =
[−iΣˆg Σˆf
−Σˆf¯ iΣˆ¯g
]
(A.3) Sˇ=
[Sˆ ˆ0 ˆ0 Sˆtr
] , Kˇ =
[Kˆ+ ˆ0 ˆ0 Kˆ−
]
=
[(µBH−αSLL(k))ˆ1 0
0 (µBH−αSLL(−k))ˆ1 ]
(A.4) Then, we introduce the projection matrix ˇP± related to the quasiclassical Green’s functions as.
Pˇ± = 1 2(ˇ1± ˇg
iπ). (A.5)
Moreover, ˇP± satisfy the condition that
Pˇ±Pˇ±= ˇP±, Pˇ+Pˇ−= ˇP−Pˇ+= 0, (A.6) Pˇ++ ˇP−= ˇ1, Pˇ−−Pˇ+ =iˇg. (A.7) We introduce the Riccati amplitudes ˆaand ˆbas
Pˇ+ = ( ˆ1
−iˆb
) (ˆ1 + ˆaˆb)−1(ˆ1 iˆa)
, Pˇ− = (−iˆa
ˆ1
) (ˆ1 + ˆbˆa)−1( iˆb ˆ1)
. (A.8)
A Riccati equation 63 Therefore, from the Eilenberger equation (A.1), the matrix Riccati equation is obtained as
v·∇ˆa+1
2( ˆMgaˆ+ ˆaMˆg¯+ ˆaMˆf¯ˆa−Mˆf) = 0, (A.9)
−v·∇ˆb+ 1
2( ˆMg¯ˆb+ ˆbMˆg+ ˆbMˆfˆb−Mˆf¯) = 0. (A.10) Using the relation
ˇ g=−iπ
((ˆ1 + ˆaˆb)−1 ˆ0 ˆ0 (ˆ1 + ˆbˆa)−1
)
·
(ˆ1−ˆaˆb 2iˆa
−2iˆb −(ˆ1−ˆbˆa) )
, (A.11)
we can obtain the quasiclassical Green’s function ˇgfrom the solutions ˆaand ˆbof the Riccati equation.
A.2 Symmetry of g, and ˇ a, ˆ ˆ b
Matrix elements of ˇM defined as Eq.(A.1) are given by
iMˆg =i(ωn+iv·A+ ˆΣg) + ˆK+·2 ˆS (A.12)
Mˆf = ˆ∆ + ˆΣf (A.13)
Mˆf¯= ˆ∆†+ ˆΣf¯ (A.14)
−iMˆ¯g =−i(ωn+iv·A+ ˆΣ¯g) + ˆK−·2 ˆStr. (A.15) And, the second term of Eilenberger equation (A.1) is
−1
2[ ˇM ,g] =ˇ i 2
[ fˆMˆf¯−Mˆffˆ¯+ ˆgMˆg−Mˆgˆg gˆMˆf + ˆMfgˆ¯−Mˆgfˆ−fˆMˆ¯g
−( ˆMf¯ˆg+ ˆg¯Mˆf¯−Mˆ¯gfˆ¯−fˆ¯Mˆg) −( ˆf¯Mˆf −Mˆf¯fˆ+ ˆ¯gMˆ¯g−Mˆg¯ˆ¯g)).
]
(A.16) Therefore, we obtain the four Eilenberger equations
v·∇gˆ= 1
2( ˆfMˆf¯−Mˆffˆ¯+ ˆgMˆg −Mˆgg)ˆ
= 1
2[ ˆf( ˆ∆†+ ˆΣf¯)−( ˆ∆ + ˆΣf) ˆf¯]− i
2[ˆg( ˆK+·2 ˆS)−( ˆK+·2 ˆS)ˆg], (A.17) v·∇gˆ¯= 1
2( ˆMf¯fˆ−fˆ¯Mˆf + ˆM¯gˆ¯g−ˆ¯gMˆg¯)
= 1
2[( ˆ∆†+ ˆΣf¯) ˆf −fˆ¯( ˆ∆ + ˆΣf)]− i
2[ˆ¯g( ˆK−·2 ˆStr)−( ˆK−·2 ˆStr)ˆ¯g], (A.18) v·∇fˆ= 1
2(ˆgMˆf + ˆMfˆ¯g−Mˆgfˆ−fˆMˆg¯)
= 1
2[ˆg( ˆ∆ + ˆΣf) + ( ˆ∆ + ˆΣf)ˆ¯g] + i
2[( ˆK+·2 ˆS) ˆf−f( ˆˆK−·2 ˆStr)]
−(ωn+iv·A) ˆf −1
2( ˆΣgfˆ+ ˆfΣˆg¯), (A.19)
−v·∇fˆ¯= 1
2( ˆMf¯gˆ+ ˆ¯gMˆf¯−Mˆg¯fˆ¯−fˆ¯Mˆg),
= 1
2[( ˆ∆†+ ˆΣf¯)ˆg+ ˆ¯g( ˆ∆†+ ˆΣf¯)] + i
2[ ˆf¯( ˆK+·2 ˆS)−( ˆK−·2 ˆStr) ˆf¯]
−(ωn+iv·A) ˆf −1
2( ˆΣg¯fˆ¯+ ˆf¯Σˆg). (A.20)
A Riccati equation 64 By performing the transformation ωn→ −ω∗n and the Hermite conjugate, and using the relation (K±)∗=K±, we obtain the transformed Eilenberger equations
v·∇gˆ†= 1
2[ ˆf†( ˆ∆ + ˆΣ†f¯)−( ˆ∆†+ ˆΣ†f) ˆf¯†] + i
2[ˆg†( ˆK+·2 ˆS)−( ˆK+·2 ˆS)ˆg†], (A.21) v·∇gˆ¯†= 1
2[( ˆ∆ + ˆΣ†f¯) ˆf†−fˆ¯†( ˆ∆†+ ˆΣ†f)] + i
2[ˆg¯†( ˆK−·2 ˆStr)−( ˆK−·2 ˆStr)ˆg¯†], (A.22) v·∇fˆ†= 1
2[ˆg†( ˆ∆†+ ˆΣ†f) + ( ˆ∆†+ ˆΣ†f)ˆg¯†]− i
2[( ˆK+·2 ˆS) ˆf†−fˆ†( ˆK−·2 ˆStr)]
−(−ωn∗−iv·A) ˆf†− 1
2( ˆΣ†gfˆ†+ ˆf†Σˆ†g¯), (A.23)
−v·∇fˆ¯†= 1
2[( ˆ∆ + ˆΣ†f¯)ˆg†+ ˆ¯g†( ˆ∆ + ˆΣ†f¯)]− i
2[ ˆf¯†( ˆK+·2 ˆS)−( ˆK−·2 ˆStr) ˆf¯†]
−(−ωn∗+iv·A) ˆf†− 1
2( ˆΣ†¯gfˆ¯†+ ˆf¯†Σˆ†g), (A.24) where the quasiclassical Green’s functions ˇgindicate ˇg(−ωn∗)→ˇg. Then, we compare the Eilenberger equations (A.17)-(A.20) and (A.21)-(A.24), and obtain the symmetry relation of the quasiclassical Green’s functions in the Matsubara representation
ˆ
g(ωn) =−gˆ†(−ω∗n), ˆ¯g(ωn) =−gˆ¯†(−ω∗n), (A.25) fˆ(ωn) = ˆf¯†(−ω∗n), f(ωˆ¯ n) = ˆf†(−ω∗n). (A.26) The symmetry of the self-energy Σ(ωn) considering the s-wave impurity potential is the same relation to Eqs.(A.25) and (A.27) from ΣOˆ= τ1
0⟨Oˆ⟩k (∀Oˆ ∈[ˇg]i,j).
On the other hand, by performing the transformation ωn→ −ωn∗ and the Hermite conjugate to Riccati equation, we obtain the symmetry relation
ˆ
a(ωn) =−ˆb†(−ω∗n), ˆb(ωn) =−ˆa†(−ω∗n). (A.27) In addition, from the relation v(−k) = −v(k),∆(ˆ −k,r) = eiα∆(k,ˆ r) where α is a constant comming from the chirality, ˆK±(−k) = ˆK∓(k), we transform the Riccati equation (k→−k, r→−r).
If the simple caseK+ =K− is satisfied, we obtain the symmetry relation for r→ −rand k→ −k as
ˆ
a(k,r) =∓e−iαˆa(−k,−r), ˆb(k,r) =∓e+iαˆb(−k,−r), (A.28) when ˆ∆(k,−r) =∓∆(k,ˆ r). From the definitionsK±=µBH−αSLL(±k), the conditionK+ =K− is satisfied when the spin-orbit coupling is absent so that αSL = 0.
On the other hand, in the real-energy representation, the symmetric relation between advanced and retarded quasiclassical Green’s functions
ˆ
g(iωn→ω+iη) =−[ˆg(iωn→ω−iη)]∗, fˆ†(iωn→ω+iη) = [ ˆf(iωn→ω−iη)]∗. (A.29) By using these symmetry relations Eqs.(A.25), (A.27), (A.28), and (A.29), the numerical calcula-tion for the Eilenberger equacalcula-tion can be performed efficiently.
65
B
NMR relaxation rates in the low temperature limit
In this appendix, I consider the NMR relaxation rate in the low temperature limit. In the low temperature limitT→0,
B =
∫ ∞
−∞
A(E)
4Tcosh2(E/2T)dE→A(E = 0). (B.1)
Therefore, from the Eqs. (3.84) and (3.97), in the lowT limit, (T1(r)T)−1
(T1(Tc)Tc)−1 =Wslgg(E= 0,r) +Wslf f(E= 0,r), (B.2) and
2 (T2zz(r)T)−1
(T2zz(Tc)Tc)−1 =W↑↑gg(E= 0,r) +W↑↑f f(E = 0,r) +W↓↓gg(E= 0,r) +W↓↓f f(E = 0,r)
−W↑↓gg(E= 0,r)−W↑↓f f(E = 0,r)
−W↓↑gg(E= 0,r)−W↓↑f f(E= 0,r). (B.3) Here, Wslgg, Wslf f, Wσσgg′ and Wσσf f′ are defined by aijσσ′ as Eqs. (3.86), (3.87), (3.99) and (3.100).
aijσσ′ are defined by Eqs. (3.80), (3.81), (3.82), and (3.83).
Between advanced and retarded Green’s functions, there are relationsgσσ′(E+iη) =−gσ∗′σ(E−iη), f¯σσ′(E + iη) =fσ∗′σ(E−iη). From the parity in the E-dependence of odd-frequency Cooper pair, f¯σσ′(E+iη) =−fσ∗′σ(−E+iη). In addition, in the spin-triplet superconducting state without Zeeman effect, g↑↑(E±iη) = g↓↓(E±iη), g↑↓(E±iη) = g↓↑(E±iη) = 0,f↑↓(E±iη) = f↓↑(E±iη). Using these
B NMR relaxation rates in the low temperature limit 66 relations, in the limitE →0 we obtain we obtain the aijσσ′ as
a11σσ′(E,k,r) = 1
2[gσσ′(E+ iη,k,r) +gσ′σ(E−iη,k,r)]
= 1
2[gσσ′(E+ iη,k,r) +gσ∗′σ(E+ iη,k,r)]
→Re[gσσ′(0 + iη,k,r)]δσσ′
a22σσ′(E,k,r) = 1
2[¯gσσ′(E+ iη,k,r) + ¯gσ′σ(E−iη,k,r)]
= 1
2[¯gσσ′(E+ iη,k,r) + ¯gσ∗′σ(E+ iη,k,r)]
→Re[¯gσσ′(0 + iη,k,r)]δσσ′
a12σσ′(E,k,r) = i
2[fσσ′(E+ iη,k,r)−fσ′σ(E−iη,k,r)]
= i
2[fσσ′(E+ iη,k,r)−f¯σ∗′σ(E+ iη,k,r)]
→i[fσσ′(0 + iη,k,r)], a21σσ′(E,k,r) = i
2[ ¯fσσ′(E+iη,k,r)−f¯σ′σ(E−iη,k,r)]
= i
2[ ¯fσσ′(E+iη,k,r)−fσ∗′σ(E+iη,k,r)]
→i[ ¯fσσ′(0 + iη,k,r)].
(B.4) where the Kronecker deltaδσσ′ = 1 when σ=σ′,δσσ′ = 0 when σ̸=σ′.
Therefore, in the low temperature limit, we have (T1gg(T)T)−1
(T1(Tc)Tc)−1=N↑(E = 0)2, (T1f f(T)T)−1
(T1(Tc)Tc)−1=−|Fs,↑↓(E = 0)|2, 2 (T2zzgg (T)T)−1
(T2zz(Tc)Tc)−1=N↑(E = 0)2+N↓(E = 0)2 2 (T2zzf f(T)T)−1
(T2zz(Tc)Tc)−1=−|Fs,↑↑(E = 0)|2− |Fs,↓↓(E = 0)|2 +|Fs,↑↓(E= 0)|2+|Fs,↓↑(E = 0)|2
(B.5) with the definitions of thes-wave Cooper pair amplitude (5.1) and the spin-resolved LDOS (6.3).
When (↑,↓), (↓,↑) spin-components of thes-wave Cooper pairs have a finite value, the coherence terms become
(T1f f(T)T)−1
(T1(Tc)Tc)−1 =−|Fs,↑↓(E = 0)|2<0, (B.6) 2 (T2zzf f(T)T)−1
(T2zz(Tc)Tc)−1 =|Fs,↑↓(E= 0)|2+|Fs,↓↑(E = 0)|2>0. (B.7) When (↑,↑), (↓,↓) spin-components of thes-wave Cooper pairs have a finite value, the coherence
B NMR relaxation rates in the low temperature limit 67 terms become
(T1f f(T)T)−1
(T1(Tc)Tc)−1 = 0, (B.8) 2 (T2zzf f(T)T)−1
(T2zz(Tc)Tc)−1 =−|Fs,↑↓(E = 0)|2− |Fs,↓↑(E = 0)|2<0. (B.9)
68
C
Kramer-Pesch approximation
Kramer and Pesch developed the analytic solution of the quasiclassical Eilenberger equations for the vortex core structure focusing on the low-energy state of clean superconductors [156, 157]. In this appendix, the quasiclassical Green’s functions and the local NMR relaxation rates around the vortex core are obtained by the Kramer-Pesch approximation for the Riccati equation. The Kramer-Pesch approximation is performed as the Matsubara frequency and impactparameter expansion. As shown in Fig. C.1,y is the impact parameter, v-axis is parallel to the quasiparticle trajectory, the origin o corresponds to the vortex center.
a b
v
o r
ar
bθ θ
ru r(x)
v
Fv
center Vortex Quasiparticle
: Trajectory
Fig. C.1 Coordinate system and schematic figure.
C.1 Analytic solution of the Riccati equation
The order-parameter in the single vortex state is defined as
∆(θ, x, y) =|∆(r)|ei(θr−θ)eiL′zθ (C.1) where the coordinate system is replaced; (x, y)→r = (ra, rb) in Fig. C.1, eiL′zθ is phase factor of chiralityLz and vorticityW (L′z≡Lz+W).
In addition, on the xaxis, the order-parameter is written as
∆(θ, x, y) =|∆(r(x))|sign(x)f(θ) (C.2)
C Kramer-Pesch approximation 69 where sign(x) = +1 (x >0),−1 (x <0) is signature function for sigh change of vortex’sπ-shift, and f(θ) =eiL′zθ,
On the other hand, we consider the solutions of Riccati equation ˆa(x, y),ˆb(x, y). From Appendix A, the relation between quasiclassical Green’s functions and the solutions ˆa(x, y),ˆb(x, y) is that
ˇ g=−iπ
((ˆ1 + ˆaˆb)−1 ˆ0 ˆ0 (ˆ1 + ˆbˆa)−1
)
·
(ˆ1−ˆaˆb 2iˆa
−2iˆb −(ˆ1−ˆbˆa) )
, (C.3)
where ˇgin this thesis is defined as ˇ g=−iπ
( gˆ ifˆ
−ifˆ† −ˆg )
(C.4) .
In this Appendix, we represent ˆX→X. Then, we introduce the analytic solution for a, b by the low-energy expansion about ωn, y, and take no account of the second and the higher orders.
a(x, y;iωn)∼ a0+a1 (C.5)
b(x, y;iωn)∼ b0+b1 (C.6)
∆(x, y)∼ ∆0+ ∆1 (C.7)
ωn ∼ 0 +ωn (C.8)
y ∼ 0 +y (C.9)
The Riccati equation is written as vF
∂
∂x(a0+a1) + 2ωn(a0+a1) + (∆∗0+ ∆∗1)(a0+a1)(a0+a1)−(∆0+ ∆1) = 0, (C.10) vF
∂
∂x(b0+b1)−2ωn(b0+b1)−(∆0+ ∆1)(b0+b1)(b0+b1) + (∆∗0+ ∆∗1) = 0. (C.11) In this approximation, ∆(x, y) is
∆(θ, x, y) =|∆(r)|eiϕf(θ) (ϕ=θr−θ) (C.12)
=|∆(r)|(cosϕ+isinϕ)f(θ) (C.13)
=|∆(r)| x+iy
√x2+y2f(θ) (C.14)
∼ |∆(r)|( x
|x|+i y
|x|)f(θ) (Taylor expansion) (C.15)
=|∆(r)|(sign(x) +isign(x)y
x)f(θ) (C.16)
= ∆(θ, x) +iy
x∆(θ, x), (C.17)
where the first term corresponds to ∆0, the second term corresponds to ∆1.
C.2 Zeroth-order Riccati equation
From Eq.(C.10), (C.11), we obtain the zeroth-order Riccati equation vF
∂
∂xa0(θ, x, y= 0)−∆0(θ, x){1−a20}= 0, (C.18) vF
∂
∂xb0(θ, x, y= 0) + ∆0(θ, x){1−b20}= 0. (C.19)
C Kramer-Pesch approximation 70 Then, we use ∆0= ∆∗0. The solution of the non-linear differential equation (C.18) is
a0(θ, x, y= 0) = tanh(u(θ, x) +Ca) (Ca : constant of integration) (C.20) u(θ, x) =vF−1
∫ x 0
dx′sign(x′)|∆(x′)|f(θ). (C.21)
Then, considering the boundary condition of vortex state (Bulk state at x=xa), a0(θ, x=xa, y= 0) = ∆(θ, xa)
0 +√
02+|∆0|2 (C.22)
= ∆(θ, xa)
|∆(θ, xa)| (C.23)
= sign(xa)sign(f(θ)) (C.24)
= sign(xaf(θ)). (C.25)
where the solution of bulk state isa0= ∆/(ωn+√
ωn2+|∆0|2).
The constant of integration Ca is
Ca = arctanh(sign(xaf(θ)))−u(xa). (C.26) (C.27) When sign(xaf(θ)) =±1,Ca=±∞.
Therefore, from Eq. (C.20),
a0(θ, x, y = 0) = sign(xaf(θ)). (C.28) In the same way, we obtain
b0(θ, x, y= 0) = sign(xbf(θ)) (C.29) from the zeroth-order Riccati equation (C.19).
C.3 First-order Riccati equation
From Riccati equations (C.10), (C.11), vF
∂
∂xa1(θ, x, y) + 2∆(θ, x)a0a1−iy
x∆(θ, x)(a20+ 1) + 2ωna0= 0, (C.30) vF
∂
∂xb1(θ, x, y)−2∆(θ, x)b0b1−iy
x∆(θ, x)(b20+ 1)−2ωnb0= 0. (C.31) These non-homogeneous differential equations is solved by using solutions of the homogeneous differential equations. From the solution of homogeneous differential equations
ahomo1 (θ, x, y) =Ce−2a0u(θ,x), (C.32) the solution of non-homogeneous differential equation (C.30) is
a1= 2 vF
∫ x xa
dx′[−ωna0+iy
x′∆(θ, x′)]e2a0u(θ,x′)e−2a0u(θ,x). (C.33) Then, we use the boundary condition of vortex state (Bulk state atx=xa).
In the same way, we obtain b1= 2 vF
∫ x xa
dx′[ωnb0+iy
x′∆(θ, x′)]e−2b0u(θ,x′)e2b0u(θ,x). (C.34)
C Kramer-Pesch approximation 71
C.4 Quasiclassical Green’s functions
Substitutinga0, a1, b0, b1 for Eq. (C.3), the first order quasiclassical Green’s functions are described as
g(θ, x, y;iωn) = 1−ab
1 +ab (C.35)
∼1−(a0b0+a0b1+a1b0)
1 + (a0b0+a0b1+a1b0) (a1b1∼0) (C.36)
= 2−(a0b1+a1b0) a0b1+a1b0
(xa(b)→ −(+)∞:a0b0=−1) (C.37)
= 2
a0b1+a1b0
(C.38)
= 2e−2sign(f(θ))u(θ,x)
2 vF{∫x
xa −∫x
xb}dx′{ωn+ixy′sign(f(θ))∆(θ, x′)}e−2sign(f(θ))u(θ,x′) (C.39)
= 2e−2sign(f(θ))u(θ,x)
2 vF
∫xb
xadx′{ωn+ixy′sign(f(θ))∆(θ, x′)}e−2sign(f(θ))u(θ,x′) (C.40)
= 2e−2sign(f(θ))u(θ,x)
4 vF
∫∞
0 dx′{ωn+ixy′sign(f(θ))∆(θ, x′)}e−2sign(f(θ))u(θ,x′) (C.41)
= vFe−2sign(f(θ))u(θ,x)
2C{ωn+iE(θ, y)}, (C.42)
whre
C≡
∫ ∞
0
dx′e−2sign(f(θ))u(θ,x), (C.43)
E(θ, y)≡y C
∫ ∞
0
dx′sign(f(θ))∆(θ, x′)
x′ e−2sign(f(θ))u(θ,x′). (C.44) In the same way,
fˆ(θ, x, y;iωn) = 2ˆa
1 + ˆaˆb (C.45)
=− 2 ˆds
ˆ
a0ˆb1+ ˆa1ˆb0
(C.46)
=−g(θ, x, y;ˆ iωn)·dˆs, (C.47)
fˆ†(θ, x, y;iωn) = 2ˆb
1 + ˆaˆb (C.48)
= 2 ˆds
ˆ
a0ˆb1+ ˆa1ˆb0
(C.49)
= ˆg(θ, x, y;iωn)·dˆs (C.50) where
dˆs =
((−dx+idy)/| −dx+idy+δ| dz/|dz+δ|
dz/|dz+δ| (dx+idy)/|dx+idy+δ| )
(C.51)
C Kramer-Pesch approximation 72 with small parameterδ∼0.
Therefore, we obtain the analytic solution of quasiclassical Eilenberger equation in the low-energy vortex core state. And we obtain the relation between quasiclassical Green’s functions
|g(Eˆ ±iη)·dˆs|=|fˆ(E±iη)|=|fˆ¯(E±iη)| (C.52)
C.5 NMR relaxation rate around the vortex center in the low T limit
From the Appendix B, the NMR relaxation rates in the low temperature limit is given as (T1gg(T)T)−1
(T1(Tc)Tc)−1 =⟨Re[g↑↑(0 + iη)]⟩2k
=N↑(E = 0)2, (T1f f(T)T)−1
(T1(Tc)Tc)−1 =−|⟨f↑↓(0 + iη)⟩2k|
=−|Fs,↑↓(E= 0)|2, 2 (T2zzgg (T)T)−1
(T2zz(Tc)Tc)−1 =⟨Re[g↑↑(0 + iη)]⟩2k+⟨Re[g↓↓(0 + iη)]⟩2k
−⟨Re[g↑↓(0 + iη)]⟩2k− ⟨Re[g↓↑(0 + iη)]⟩2k
=N↑(E= 0)2+N↓(E = 0)2, 2 (T2zzf f(T)T)−1
(T2zz(Tc)Tc)−1 =−|⟨f↑↑(0 + iη)⟩2k| − |⟨f↓↓(0 + iη)⟩2k| +|⟨f↑↓(0 + iη)⟩2k|+|⟨f↓↑(0 + iη)⟩2k|
=−|Fs,↑↑(E= 0)|2− |Fs,↓↓(E = 0)|2 +|Fs,↑↓(E = 0)|2+|Fs,↓↑(E = 0)|2.
(C.53) By using the relation |g(E)ˆ ·dˆs|= |fˆ(E)|=|f(E)ˆ¯ | derived from the Kramer-Pesh approximation (C.52) at lowT andB limit (low-energy limit), we obtain the relation in the chiralp-waved∥zstate
(T1f f(T)T)−1
(T1(Tc)Tc)−1 =−(T1gg(T)T)−1
(T1(Tc)Tc)−1, (C.54)
(T2zzf f(T)T)−1
(T2zz(Tc)Tc)−1 =−(T2zzgg (T)T)−1
(T1(Tc)Tc)−1. (C.55)
at the vortex core region, since|⟨f↑↓(0 + iη)⟩k|=|⟨f↓↑(0 + iη)⟩k|̸=0. In the chiralp-waved∥xstate, we obtain
(T2zzf f(T)T)−1
(T2zz(Tc)Tc)−1 =−(T2zzgg (T)T)−1
(T1(Tc)Tc)−1, (C.56)
since |⟨f↑↑(0 + iη)⟩k| = |⟨f↓↓(0 + iη)⟩k|̸=0. These relations for the chiral p-wave states in Eqs.
(C.54)-(C.56) consistent with the calculation results of the negative coherence terms in the vortex core region at the finite temperature and magnetic field in Table 5.1 and Fig. 5.4(a).
In addition, up-spin pairs and excitations in the helical p-wave state satisfy the relations
|⟨g↑↑(E±iη)⟩k|=|⟨f↑↑(E±iη)⟩k|=|⟨f¯↑↑(E±iη)⟩k|given by the Kramer-Pesch approximation, since
C Kramer-Pesch approximation 73 up-spin pairs’ order-parameter ∆↑↑ has the pairing function ϕp− with the chirality Lz = −1. On the other hand,|⟨f↓↓(0 + iη)⟩k|= 0 at the vortex center. Therefore, in the helicalp-wave state,
2 (T2zzf f(T)T)−1
(T2zz(Tc)Tc)−1 =−|⟨g↑↑(0 + iη)⟩k|2− |⟨f↓↓(0 + iη)⟩k|2
=−N↑(E= 0)2− |⟨f↓↓(0 + iη)⟩k|2
∼ −N↑(E = 0)2,
(C.57) since |⟨f↓↓(0 + iη)⟩k| shows a small amplitude in the vortex core region compared to the large amplitude|⟨f↑↑(0 + iη)⟩k|.
From the definitions of the local DOSNσ(E,r) =⟨Re[gσσ(E+ iη,r,k)]⟩k and the spin-dependent s-wave Cooper pair Fs,σσ′(E,r) = ⟨fσσ′(E + iη,r,k)⟩k, the relations between the |⟨gσσ′(E)⟩| and
|⟨fσσ′(E)⟩|obtained from Kramer-Pesch approximation are rewritten byNσ(E,r) and|Fs,σσ′(E,r)|. The relation|⟨g↑↑(E)⟩k|=|⟨f↑↑(E)⟩k|, i.e., |Fs,↑↑(E,r= 0)|=|N↑(E,r= 0)|for the helical p-wave state is confirmed at the low|E|region, not only E= 0, by our numerical calculations at a low field H/Hc2 = 0.023 and a finite temperature T /Tc0 = 0.5. The chiralp-wave cases are also confirmed.
Therefore, the relations between the DOS term and the coherence term for the chiralp-wave states in Eqs. (C.54)-(C.56) consistent with the our calculation results in Table 5.1 and Fig. 5.4(a).
74
Bibliography
[1] H. Kamerlingh Onnes, Comm. Phys. Lab. Univ. Leiden, Nos. 119, 120, 122 (1911).
[2] V. L. Ginzburg, and L. D. Landau, Zh. Exptl. Teor. Fiz.20, 1064 (1950).
[3] J. Bardeen, L. N. Cooper, and J .R. Schrieffer, Phys. Rev.108, 1175 (1957).
[4] F. Steglich, J. Aarts, C. D. Bredl, W. Lieke, D. Meschede, W. Franz, and H. Sch¨afer, Phys.
Rev. Lett.43, 1892 (1979).
[5] J. G. Bednorz and K. A. M¨uller, Z. Phys. B 64, 189 (1986).
[6] Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J. G. Bednorz, and F. Lichtenberg, Nature(London) 372, 532 (1994).
[7] A. P. Mackenzie and Y. Maeno, Rev. Mod. Phys.75, 657 (2003).
[8] Y. Maeno, S. Kittaka, T. Nomura, S. Yonezawa, and K. Ishida, J. Phys. Soc. Jpn. 81, 011009 (2012).
[9] G. R. Stewart, Z. Fisk, J. O. Willis, and J. L. Smith, Phys. Rev. Lett. 52, 679 (1984).
[10] K. Machida, T. Ohmi, and M. Ozaki, J. Phys. Soc. Jpn.64, 1067 (1995).
[11] R. Joynt and L. Taillefer, Rev. Mod. Phys.74, 235 (2002).
[12] Y. S. Hor, A. J. Williams, J. G. Checkelsky, P. Roushan, J. Seo, Q. Xu, H. W. Zandbergen, A.
Yazdani, N. P. Ong, and R. J. Cava, Phys. Rev. Lett.104, 057001 (2010).
[13] S. Sasaki, M. Kriener, K. Segawa, K. Yada, Y. Tanaka, M. Sato, and Y. Ando, Phys. Rev. Lett.
107, 217001 (2011).
[14] R. Balian and N. R. Werthamer, Phys. Rev. 131, 1553 (1963).
[15] P. W. Anderson and P. Morel, Phys. Rev. 123, 1911 (1961).
[16] P. W. Anderson and W. F. Brinkman, Phys. Rev. Lett.30, 1108 (1973).
[17] M. M. Salommaa and G. E. Volovik, Rev. Mod. Phys.59, 533 (1987); M. M. Salommaa and G.
E. Volovik, Rev. Mod. Phys. 60, 573 (1988).
[18] T. M. Rice and M. Sigrist, J. Phys.: Condens. Matter7, L643 (1995).
[19] K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z. Q. Mao, Y. Mori, and Y. Maeno, Nature 396, 658 (1998).
[20] H. Murakawa, K. Ishida, K. Kitagawa, Z. Q. Mao, and Y. Maeno, Phys. Rev. Lett.93, 167004 (2004); H. Murakawa, K. Ishida, K. Kitagawa, H. Ikeda, Z. Q. Mao, and Y. Maeno, J. Phys.
Soc. Jpn.76, 024716 (2007).
[21] H. Tou, Y. Kitaoka, K. Asayama, N. Kimura, Y. ¯Onuki, E. Yamamoto, and K. Maezawa, Phys.
Rev. Lett.77, 1374 (1996).
[22] H. Tou, Y. Kitaoka, K. Ishida, K. Asayama, N. Kimura, Y. Onuki, E. Yamamoto, Y. Haga, and K. Maezawa, Phys. Rev. Lett. 80, 3129 (1998).
[23] C. M. Aegerter, S. H. Lloyd, C. Ager, S. L. Lee, S. Romer, H. Keller, and E. M. Forgan, J.
Phys.: Condens. Matter10, 7445 (1998).
[24] G. M. Luke, A. Keren, L. P. Le, W. D. Wu, Y. J. Uemura, D. A. Bonn, L. Taillefer, and J. D.
Garrett, Phys. Rev. Lett. 71, 1466 (1993).
[25] J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, Phys. Rev. Lett.97, 167002 (2006).
[26] E. R. Schemm, W. J. Gannon, C. M. Wishne, W. P. Halperin, and A. Kapitulnik, Science345, 190 (2014).
[27] S. Takamatsu and Y. Yanase, J. Phys. Soc. Jpn. 82, 063706 (2013).
Bibliography 75 [28] T. Scaffidi, J. C. Romers, and S. H. Simon, Phys. Rev. B89, 220510(R) (2014).
[29] S. Kittaka, T. Nakamura, Y. Aono, S. Yonezawa, K. Ishida, and Y. Maeno, Phys. Rev. B 80, 174514 (2009).
[30] J. J. Neumeier, M. F. Hundley, M. G. Smith, J. D. Thompson, C. Allgeier, H. Xie, W. Yelon, and J. S. Kim, Phys. Rev. B50, 17910 (1994).
[31] C. Bergemann, S. R. Julian, A. P. Mackenzie, S. NishiZaki, and Y. Maeno, Phys. Rev. Lett.84, 2662 (2000).
[32] D. Forsythe, S. R. Julian, C. Bergemann, E. Pugh, M. J. Steiner, P. L. Alireza, G. J. McMullan, F. Nakamura, R. K. W. Haselwimmer, I. R. Walker, S. S. Saxena, G. G. Lonzarich, A. P.
Mackenzie, Z. Q. Mao, and Y. Maeno, Phys. Rev. Lett.89, 166402 (2002).
[33] A. Damascelli, D. H. Lu, K. M. Shen, N. P. Armitage, F. Ronning, D. L. Feng, C. Kim, Z.-X.
Shen, T. Kimura, Y. Tokura, Z. Q. Mao, and Y. Maeno, Phys. Rev. Lett. 85, 5194 (2000).
[34] A. Sekiyama, S. Kasai, M. Tsunekawa, Y. Ishida, M. Sing, A. Irizawa, A. Yamasaki, S. Imada, T. Muro, Y. Saitoh, Y. ¯Onuki, Y. Tokura, and S. Suga, Phys. Rev. B 70, 060506(R) (2004).
[35] N. Hiraoka, T. Buslaps, V. Honki¨aki, T. Nomura, M. Itou, Y. Sakurai, Z. Q. Mao, and Y.
Maeno, Phys. Rev. B74, 100501(R) (2006).
[36] T. Oguchi, Phys. Rev. B51, 1385(R) (1995).
[37] M. W. Haverkort, I. S. Elfimov, L. H. Tjeng, G. A. Sawatzky, and A. Damascelli, Phys. Rev.
Lett.101, 026406 (2008).
[38] E. Pavarini and I. I. Mazin, Phys. Rev. B74, 035115 (2006); E. Pavarini and I. I. Mazin, Phys.
Rev. B76, 079901 (2007).
[39] T. Nomura and K. Yamada, J. Phys. Soc. Jpn.69, 3678 (2000); T. Nomura and K. Yamada, J.
Phys. Soc. Jpn.71, 1993 (2002).
[40] Q. H. Wang, C. Platt, Y. Yang, C. Honerkamp, F. C. Zhang, W. Hanke, T. M. Rice, and R.
Thomale, Europhys. Lett.104, 17013 (2013).
[41] R. Arita, S. Onari, K. Kuroki, and H. Aoki, Phys. Rev. Lett.92, 247006 (2004).
[42] K. Hoshihara and K. Miyake, J. Phys. Soc. Jpn.74, 2679 (2005).
[43] W. Kohn and J. M. Luttinger, Phys. Rev. Lett.15, 524 (1965).
[44] M. Tsuchiizu, Y. Yamakawa, S. Onari, Y. Ohno, and H. Kontani, Phys. Rev. B 91, 155103 (2015).
[45] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, J. Luitz, An Augmented Plane Wave and Local Orbitals Program for Calculation Crystal Properties (Technical University of Wien, Vienna).
[46] S. NishiZaki, Y. Maeno, and Z. Mao, J. Low Temp. Phys.117, 1581 (1999).
[47] M. Thinkham Introduction to Superconductivity, second edition. (McGraw-Hill, Singapore) 1996.
[48] T. M. Riseman, P. G. Kealey, E. M. Forgan, A. P. Mackenzie, L. M. Galvin, A. W. Tylor, S. L.
Lee, C. Ager, D. Mck. Paul, C. M. Agerter, R. Cubitt, Z. Q. Mao, T. Akima, and Y. Maeno, Nature396, 242 (1998).
[49] P. G. Kealey, T. M. Riseman, E. M. Forgan, L. M. Galvin, A. P. Mackenzie, S. L. Lee, D. McK.
Paul, R. Cubitt, D. F. Agterberg, R. Heeb, Z. Q. Mao, and Y. Maeno, Phys. Rev. Lett. 84, 6094 (2000).
[50] G. M. Luke, Y. Fudamoto, K. M. Kojima, M. I. Larkin, B. Nachumi, Y. J. Uemura, J. E. Sonier, Y. Maeno, Y. Mori, D. F. Agterberg, Physica B 289, 373 (2000).
[51] A. A. Abrikosov, Zh. Eksp. Teor. Fiz.32, 1442 (1957). [Sov. Phys. JETP5, 1174 (1957)].
[52] S. L. Thiemann, Z. Radovi´c, and V. G. Kogan, Phys. Rev. B 39, 11406 (1989).
[53] P. G. Kealey, D. Charalambous, E. M. Forgan, S. L. Lee, S. T. Johnson, P. Schleger, R. Cubitt, D. McK. Paul, C. M. Aegerter, S. Tajima, and A. Rykov, Phys. Rev. B64, 174501 (2001).
[54] M. Ishihara, Y. Amano, M. Ichioka, and K. Machida, Phys. Rev. B87, 224509 (2013).
[55] Y. Amano, M. Ishihara, M. Ichioka, N. Nakai, and K. Machida, Phys. Rev. B90, 144514 (2014).
[56] S. J. Kuhn, W. Morgenlander, E. R. Louden, C. Rastovski, W. J. Gannon, H. Takatsu, D. C.
Bibliography 76 Peets, Y. Maeno, C. D. Dewhurst, J. Gavilano, and M. R. Eskildsen, Phys. Rev. B96, 174507 (2017).
[57] P. G. Kealey, D. Charalambous, E. M. Forgan, S. L. Lee, S. T. Johnson, P. Schleger, R. Cubitt, D. McK. Paul, C. M. Aegerter, S. Tajima, and A. Rykov, Phys. Rev. B64, 174501 (2001).
[58] G. M. Luke, Y. Fudamoto, K. M. Kojima, M. I. Larkin, J. Merrin, B. Nachumi, Y. J. Uemura, Y. Maeno, Z. Q. Mao, Y. Mori, H. Nakamura, and M. Sigrist, Nature394, 558 (1998).
[59] K. Yosida, Phys. Rev. 110, 769 (1958).
[60] H. L. Fine, M. Lipsicas, and M. Strongin, Phys. Lett. 29A, 366 (1969).
[61] J. A. Duffy, S. M. Hayden, Y. Maeno, Z. Mao, J. Kulda, and G. J. Mclntyre, Phys. Rev. Lett.
85, 5412 (2000).
[62] K. Deguchi, Z. Q. Mao, H. Yaguchi, and Y. Maeno, Phys. Rev. Lett. 92, 047002 (2004); K.
Deguchi, Z. Q. Mao, and Y. Maeno, J. Phys. Soc. Jpn.73, 1313 (2004).
[63] S. E. Barrett, D. J. Durand, C. H. Pennington, C. P. Slichter, T. A. Friedmann, J. P. Rice, and D. M. Ginsberg, Phys. Rev. B 41, 6283 (1989).
[64] M. Ichioka, K. M. Suzuki, Y. Tsutsumi, and K. Machida, in Superconductivity − T heory and Applications, edited by A. M. Luiz (InTech, Croatia, 2011), Chap. 10.
[65] M. Ichioka, H. Adachi, T. Mizushima, and K. Machida, Phys. Rev. B 76, 014503 (2007).
[66] K. M. Suzuki, Y. Tsutsumi, N. Nakai, M. Ichioka, and K. Machida, J. Phys. Soc. Jpn. 80, 123706 (2011).
[67] K. Kumagai, H. Shishido, T. Shibauchi, and Y. Maeno, Phys. Rev. Lett. 106, 137004 (2011).
[68] K. K. Tanaka, M. Ichioka, N. Nakai, and K. Machida, Phys. Rev. B89, 174504 (2014).
[69] H. Mukuda, K. Ishida, Y. Kitaoka, K. Asayama, R. Kanno, and M. Takano, Phys. Rev. Lett.
60, 12279 (1999).
[70] K.Ishida, H. Mukuda, Y. Kitaoka, Z. Q. Mao, Y. Mori, and Y. Maeno, Phys. Rev. Lett. 84, 5387 (2000).
[71] K.Ishida, Y. Kitaoka, K. Asayama, S. Ikeda, S. NishiZaki, Y. Maeno, K. Yoshida, and T. Fujita, Phys. Rev. B 56, R505(R) (1997).
[72] H. F. Hess, R. B. Robinson, R. C. Dynes, J. M. Valles, Jr., and J. V. Waszczak, Phys. Rev.
Lett.62, 214 (1989).
[73] Ø. Fischer, M. Kungler, I. Maggio-Aprile, and C. Berthod, Rev. Mod. Phys. 79, 353 (2007).
[74] J.-P. Xu, M.-X. Wang, Z. L. Liu, J.-F. Ge, X. Yang, C. Liu, Z. A. Xu, D. Guan, C. L. Gao, D.
Qian, Y. Liu, Q.-H Wang, F.-C. Zhang, Q.-K. Xue, and J.-F. Jia, Phys. Rev. Lett.114, 017001 (2015).
[75] Z.-Z. Li, F.-C. Zhang, and Q.-H. Wang, Sci. Rep. 4, 6363 (2014).
[76] T. Kawakami and X. Hu, Phys. Rev. Lett. 115, 177001 (2015).
[77] H.-H. Sun, K.-W. Zhang, L.-H. Hu, C. Li, G.-Y. Wang, H.-Y. Ma, Z.-A. Xu, C.-L. Gao, D.-D.
Guan, Y.-Y. Li, C. Liu, D Qian, Y. Zhou, L. Fu, S.-C. Li, F.-C. Zhang, and J.-F. Jia, Phys.
Rev. Lett.116, 257003 (2016).
[78] Y. Nagai, H. Nakamura, and M. Machida, J. Phys. Soc. Jpn. 83, 064703 (2014).
[79] T. Yokoyama, Y. Tanaka, and A. A. Golubov, Phys. Rev. B 78, 012508 (2008).
[80] Y. Tanuma, N. Hayashi, Y. Tanaka, and A. A. Golubov, Phys. Rev. Lett 102, 117003 (2009).
[81] Y. Tanaka, M. Sato, and N. Nagaosa J. Phys. Soc. Jpn.81, 011013 (2012).
[82] T. Daino, M. Ichioka, T. Mizushima, and Y. Tanaka, Phys. Rev. B86, 064512 (2012).
[83] Y. Nagai and H. Nakamura, J. Phys. Soc. Jpn.85, 074707 (2016).
[84] J. A. Sauls and M. Eschrig, New J. Phys.11, 075008 (2009).
[85] N. Schopohl ”Transformation of the Eilenberger equations of superconductivity to a scalar Riccati equation” arXiv:cond-mat/9804064
[86] N. Schopohl and K. Maki, Phys. Rev. B52, 490 (1995).
[87] M. Eschrig, Phys. Rev. B61, 9061 (2000).
[88] A. B. Vorontsov and J. A. Sauls, Phys. Rev. B68, 064508 (2003).
[89] N. Hayashi, and Y. Kato, Physica C 388, 513 (2003); J. Low Temp. Phys.131, 893 (2003).
Bibliography 77 [90] Y. Nagai, N. Hayashi, N. Nakai, H. Nakamura, M. Okumura, and M. Machida, New Journal of
Physics10 103026 (2008).
[91] K. K. Tanaka, M. Ichioka, and S. Onari, Phys. Rev. B93, 094507 (2016).
[92] F. Schwabl, Statistical Mechanics (springer) p.16.
[93] R. Kubo, J. Phys. Soc. Jpn.81, 063701 (2012).
[94] R. Kubo, J. Phys. Soc. Jpn.81, 063701 (2012).
[95] H. B. Callen, and T. A. Welton, Phys. Rev.83, 34 (1951).
[96] G. C. Wick, Phys. Rev.80, 268 (1950).
[97] T. Matsubara, Prog. Theor. Phys.14, 351 (1955).
[98] C. Bloch, and C. T. de Dominics, Nucl. Phys.7, 459 (1958).
[99] R. K. Wangsness and F. Bloch, Phys. Rev. 89, 728 (1953).
[100] M. Takigawa, M. Ichioka, K. Machida, and M. Sigrist, J. Phys.: Chem. Solids63, 1333 (2002).
[101] Y. Kato and N. Hayashi, Physica C388, 519 (2003).
[102] N. J. Curro, C. Milling, J. Haase, and C. P. Slichter, Phys. Rev. B 62, 3473 (2000).
[103] V. F. Mirtovi´c, E. E. Sigmund, M. Eschrig, H. N. Bachman, W. P. Halperin, A. P. Reyes, P.
Kuhns, and W. G. Moulton, Nature (London)413, 501 (2001).
[104] K. Kakuyanagi, K.-ichi Kumagai, and Y. Matsuda, Phys. Rev. B 65, 060503(R) (2002); K.
Kakuyanagi, K.-ichi Kumagai, Y. Matsuda, and M. Hasegawa, Phys. Rev. Lett. 90, 197003 (2003).
[105] Y. Nakai, Y. Hayashi, K. Kitagawa, K. Ishida, H. Sugawara, D. Kikuchi, and H. Sato, J. Phys.
Soc. Jpn. 77, 333 (2008); Y. Nakai, Y. Hayashi, K. Ishida, H. Sugawara, D. Kikuchi, and H.
Sato, Phys. Rev. B 403, 1109 (2008).
[106] T. Mizushima, Phys. Rev. B 90, 184506 (2014).
[107] S. B. Chung and S.C. Zhang, Phys. Rev. Lett.103, 235301 (2009).
[108] M. J. Graf, S. K. Yip, and J. A. Sauls, Phys. Rev. B62, 14393 (2000).
[109] K. K. Tanaka, M. Ichioka, and S. Onari, Physica C530, 62 (2016).
[110] P. W. Anderson, J. Phys. Chem. Solids11, 26 (1959).
[111] K. K. Tanaka, M. Ichioka, S. Onari, N. Nakai, and K. Machida, Phys. Rev. B 91, 014509 (2015).
[112] K. K. Tanaka, M. Ichioka, S. Onari, Phys. Rev. B 95, 134502 (2017).
[113] K. K. Tanaka, M. Ichioka, S. Onari, J. Phys.: Conf. Ser.,871, 012024 (2017).
[114] H. F. Hess, R. B. Robinson, R. C. Dynes, J. M. Valles, Jr., and J. V. Waszczak, Phys. Rev.
Lett 64, 2711 (1989).
[115] N. J. Curro, C. Milling, J. Haase, and C. P. Slichter, Phys. Rev. Lett62, 3473 (2000).
[116] V. F. Mitrovic, E. E. Sigmund, M. Eschrig, H. N. Bachman, W. P. Halperin, A. P. Reyes, P.
Kuhns, and W. G. Moulton Nature 413, 501 (2001).
[117] K. Kakuyanagi, K. Kumagai and Y. Matsuda, Phys. Rev. B65, 060503(R) (2002).
[118] Y. Nakai, Y. Hayashi, K. Kitagawa, K. Ishida, H. Sugawara, D. Kikuchi, and H. Sato, J. Phys.
Soc. Jpn.77, Suppl. A 333 (2008).
Y. Nakai, Y. Hayashi, K. Ishida, H. Sugawara, D. Kikuchi, and H. Sato, Physica B. 403, 1109 (2008).
[119] G. Eilenberger, Z. Phys.214, 195 (1968).
[120] U. Kein, J. Low. Temp. Phys.69, 1 (1987).
[121] M. Ichioka, N. Hayashi, and K. Machida, Phys. Rev. B 55, 6565 (1997).
[122] N. Hayashi, M. Ichioka, and K. Machida, Phys. Rev. Lett77, 4074 (1996).
[123] N. Nakai, P. Miranovic, M. Ichioka, H. F. Hess, K. Uchiyama, H. Nishimori, S. Kaneko, N.
Nishida, and K. Machida, Phys. Rev. Lett97, 147001 (2006).
[124] R. Heeb and D. F. Agterberg, Phys. Rev. B59, 7076 (1999).
[125] M. Ichioka and K. Machida, Phys. Rev. B65, 224517 (2002).
[126] M. Ichioka, Y. Matsunaga, and K. Machida, Phys. Rev. B71, 172510 (2005).
[127] W. Fite, II, and A. G. Redfield, Phys. Rev. Lett.17, 381 (1966).