Theory of giant diode effect ofd-wave superconductor junction on the surfaceof topological insulator
Yukio Tanaka
Nagoya University

NQS2022, Yukawa Institute: November 17, 2022

Y. Tanaka, Bo Lu and N. Nagaosa
arXiv:2205.13177

Contents of this talkBackground of this talk1 d-wave superconductor,
topological surface Andreev bound states
2 Josephson effect in d-wave superconductor
junctions without TI
3 Superconducting junctions on TI
4 Diode effect by d-wave superconductor
junctions on TI

Unconventional superconductor
In strongly correlated systems, pair potential changes signon the Fermi surface.

Isotropic pairing

Anisotropic pairing

Tunneling effect in unconventional superconductors
Unconventional
superconductors-wave
Normalmetal

CuprateImportant issue ofcuprate in the 90s.

？

Properties of unconventional
superconductors
Pair potential becomes zero on the Fermi surface.
We call it nodal points or nodes.
Quasiparticle （Bogoliubov particle） feels different sign(phase) of pair potential depending on their directionsof motions.
ｄ-wave caseー＋＋ー

Wave function in N/S junction (2)Spin-singlet s-wave superconductorFormula of differential conductance １
transparency（Tanaka and Kashiwaya PRL 74 3451)

Conductance formula & Surface Andreev
bound state (SABS)
Condition for SABS
dxy-wave

Flat zero energy bandsurface
C.R. Hu : Phys. Rev. Lett. 72 (1994) 1526.

px-wave

Flat zero energy bandsurface

J. Hara, K. Nagai : Prog.Theor. Phys. 74 (1986) 1237.

d-wave pairing in cuprate

１００ surface

１１０ surface

Conductance in d-wave superconductor junctions 1
Y. Tanaka & S. Kashiwaya: Phys. Rev. Lett. 74 (1995) 3451.

Angle between the crystal axis andnormal to the interface

Exceptional case without
zero bias conductance peaktransparency(d) LDOS in bulk
S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63 (2000) 1641.

Conductance in d-wave superconductor junctions 2
Y. Tanaka & S. Kashiwaya: Phys. Rev. Lett. 74 (1995) 3451.

Angle between the crystal axis andnormal to the interface(d) LDOS in bulk

S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63 (2000) 1641.

Conductance in d-wave superconductor junctions 3
Y. Tanaka & S. Kashiwaya: Phys. Rev. Lett. 74 (1995) 3451.

Angle between the crystal axis andnormal to the interface(d) LDOS in bulk

S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63 (2000) 1641.

STS measurement in high Tc cuprate

Strong evidence supportingd-wave pairing symmetry in cuprate
S. Kashiwaya & Y.Tanaka : Rep. Prog. Phys. 63 (2000) 1641.

Physical origin of zero bias conductance peak?
Why zero energy surface Andreev bound state appears?
It appears not accidentally.
NodeEnergy gap closes

Topological invariant defined in momentum space.
Bulk-edge correspondence
M. Sato, Y. Tanaka, K. Yada, T.Yokoyama、Phys. Rev. B 83, 224511 (2011)

Topological invariant
BdG Hamiltonian

matrixWinding numbersurface

superconductoryxSato, Tanaka, Yada , Yokoyama, PRB 83 224511 (2011)

Topological invariant

Topological invariant is defined for effective 1d Hamiltonian for fixed ky.
Single Fermi surface

Sign change of the pair potential
at the reflection
Sato, Tanaka, et al, PRB 83 224511 (2011)

Winding number
Winding number defined around nodes

d-wave superconductor−1+1+1−1Nodal structure and winding number

d-wave superconductor(110) oriented surface

dxy wave superconductor-1+1+1-1ZESABSWinding number & Index theorem
From the bulk-edge correspondence, there exists the gapless stateson the edge only when integer w1d is nonzero.

BdG Hamiltonian has a symmetry (chiral symmetry)
Zero energy ABS is an eigenstate of σy.
Number of ZES where the eigenvalue of σy is 1
Number of ZES where the eigenvalue of σy is −1

Index Theorem
ySato, Tanaka, Yada , Yokoyama, PRB 83 224511 (2011)

SuperconductorxNodal structure and winding number
d-wave superconductor Misorientation Angle(100) direction-1+1+1-1

ZESABS vanishes due to the topological cancellation

Contents of this talkBackground of this talk1 d-wave superconductor,
topological surface Andreev bound states
2 Josephson effect in d-wave superconductor
junctions without TI
3 Superconducting junctions on TI
4 Diode effect by d-wave superconductor
junctions on TI

Various types of Josephson junctions
Current phase relations

00-junction

0π-junction

0-junction

Various types of Josephson junctions
Free energy of the junctions

00-junction

0π-junction
L. N. Bulaevskii (1977) F/S
D. J. van Harlingen (1995) ｄ-wave
V. V. Ryazanov (2001) F/S0-junction
Y. Tanaka and S. Kashiwaya (1997)
d-waveJosephson current formula available forunconventional superconductors
Phys. Rev. B 53 (1996) 11957.

(1)SABS is taken into account.
(2)Coexistence of 0 and π junctions(3)Realization of φ-junction
Including spin-triplet case; Y. Asano PRB 2001

Insulator

Scattering Process

Electron-like
quasiparticle

Hole-likequasiparticle
Electron-like
quasiparticle

Electron-like
quasiparticle

Hole-likequasiparticle

Andreev bound states in d-wave junctions
Phase differencetransmissivity

injection angleθθ

4π periodicity
PRB 53 9371 (1997)

Enhanced d.c. Josephson current by ABSeIC RN∆0T-1T/Tc

D.C. Josephson current does not have 4π periodicity(conventional 2π periodicity)
Ic maximum Josephson current
Barash (1996)

Tanaka, Kashiwaya(1996)

A.C. Josephson current has 4π periodicity
Yakovenko et al (2004)

Josephson current in mirror type junctions
αβ

α=−β3(a) α=0(b) α=0.1π2(c) α=0.25π
1Non-monotonic
temperature dependence
001

Phys. Rev. B 53 11957(1996)
Phys. Rev. B 56 892(1997)

Barash, Burkhardt, Rainer, Phys. Rev. Lett.77 4070 (1996)

Josephson
incurrent
mirrorjunctionsDCcurrent
Josephsonin typed/I/d junctions
α

α=−ββTransition from 0 junction to π junction
Τ/Τd=0.05

0.5

Τ/Τd=0.3
Τ/Τd=0.6

00.5

–0.5
00–101

Physical Review B 56 892(1997)
Physical Review B 53 11957(1996)

1Mirror type junctionsααChange of the position ｏｆ the free energy minima
ϕ-junction
The position of the free energy
minima, ϕm can locate neither
0 nor π.
ground state isdoubly degenerate.

Y. Tanaka and S. Kashiwaya, Phys. Rev. B 53 11957(1996)

Yip J. Low Temp. Phys. 1997

Experiments of dc Josephson current
non-monotonic temperature dependence
Testa et. al, Phys. Rev. B 71, 134520 (2005)

Ilichev, PRL 86 5369 (2001)

Experiments of dc Josephson current
non-monotonic temperature dependence
Testa et. al,
Phys. Rev. B 71, 134520 (2005)

Experiments

TheoryIlichev, PRL 86 5369 (2001)

Contents of this talkBackground of this talk1 d-wave superconductor,
topological surface Andreev bound states
2 Josephson effect in d-wave superconductor
junctions without TI
3 Superconducting junctions on TI
4 Diode effect by d-wave superconductor
junctions on TI

Surface state of topological insulator
Massless Dirac cone (Weyl cone)Momentum spin locking
Fermi level: above Dirac point

Fermi level: below Dirac point
Conduction band

Valence bandHamiltonian of the surface state ofTopological insulator
S: s-wave pair potential

N/TI/S
S/TI/S
Tanaka, Yokoyama, Nagaosa, PRL 103, 107002 (2009)

SABS and differential conductance

SABS is obtained when denominator becomes zero forConductance formula obtained in unconventional superconductors

We assume my=0. The component mx does not influence the charge conductance.
Tanaka, Yokoyama, Nagaosa, PRL 103, 107002 (2009)Chiral Majorana mode (CMM) on topological insulator
ABSConductancezyxabcABS as a chiral Majorana mode (CMM)
Tanaka, Yokoyama, Nagaosa, PRL 103, 107002 (2009)x

S/FI/S junction on top of TI 1abcy

Anomalous current phase relation by mx

φAnomalous current phase relation can bedetected by interferometerϕ0 junctions

000-junction
0- junction

π- junction
L. N. Bulaevskii (1977) FS junction
D. J. van Harlingen (1995) ｄ-wave
V. V. Ryazanov (2001) FS junction

Y. Tanaka et al (2009)
R. Grein et al., (2009);
M. Eschrig et al., (2007);
Y. Asano et a(2007)
A. Buzdin, et al (2008)

Both time reversal andspatial inversion symmetry
should be broken!!

S/FI/S junction on top of TI 2

Tanaka, Yokoyama, Nagaosa, PRL 103, 107002 (2009)d-wave superconductor junctions on TI

Linde Tanaka et al, PRL(2010)

Diode effect in Josephson junctions
Background of this talk1 d-wave superconductor,
topological surface Andreev bound states
2 Josephson effect in d-wave superconductor
junctions without TI
3 Superconducting junctions on TI
4 Diode effect by d-wave superconductor
junctions on TI

arXiv:2205.1317
7

Diode effect in Josephson junctions
Positive maximum Josephson current
Negative maximum Josephson current

0If the value of Q is nonzero, we can expect a diode effect.

Superconducting diode effectObservation of diode effect
Recent experiments••••

F. Ando et al, Nature 584 373 (2020)

B. Pal, et al, Nature Physics 18, 1228 (2022).
H. Narita, et al, Nature Nanotechnology 17, 823 (2022).
K.-R. Jeon, et al, Nature Materials 21, 1008 (2022).
L. Bauriedl, et al, Nature Communications 13, 4266 (2022),……

Theoretical works•
•
•••
•••
•
•

K. Misaki and N. Nagaosa, Phys. Rev. B 103, 245302 (2021).
J. J. He, Y. Tanaka and N. Nagaosa, New J. Phys. 24, 053014 (2022).
A. Daido, Y. Ikeda, and Y. Yanase, Phys. Rev. Lett. 128, 037001 (2022).
N. F. Q. Yuan and L. Fu, Proc. Nat. Acad. of Sci. 119, e2119548119 (2022).
S. Ili´c and F. S. Bergeret, Phys. Rev. Lett. 128, 177001 (2022).
T. Karabassov, et al, arXiv:2203.15608 (2022).
R. S. Souto, et al, arXiv:2205.04469 (2022).
J. Jiang, et al, Rev. Applied 18, 034064 (2022).
A. Daido and Y. Yanase, arXiv:2209.03515.
T. Kokkeler, et al, arXiv:2209.13987

d-wave superconductor junctions on TI
We are aware ofanomalous current phase relation

Bo Lu, Yada, Golubov Tanaka (PRB 2015)

Merit of d-wave / ferromagnet
junctions on TI• High transition temperature compared to conventionalBCS superconductor.
• Wide variety of temperature dependence• Enhancement of Q factor at low temperature• Simultaneous breaking of both time reversal andspatial inversion symmetry

We can expect the same order of these three terms!!
arXiv:2205.13177

ModelFerromagnet Insulator

Topological insulator

Electron-like
quasiparticle

Hole-likequasiparticle
Electron-like
quasiparticle

Electron-like
quasiparticle

Hole-likequasiparticle

ModelHamiltonian on the surface state of TI
Magnetization
Pair potential

Model (d/FI/d junction on TI)

ｄ-wave pair potential

arXiv:2205.13177

ModelHamiltonian on the surface state of TI
Magnetization
Pair potentialOutline of calculation 1
Josephson current formula expressed by Andreev reflection coefficient
A. Furusaki and M. Tsukada, Solid State Commun. Vol. 78, 299 (1991).

(1)Retarded Green’s function is calculated by using thescattering state of the wave function.
(2)Matsubara Green’s function by analytical continuation.
(3) Josephson current by Matsubara Green’s function.
(4)Josephson current given by Andreev reflection
coefficient which is expressed by Matsubara frequency.
d-wave superconductor
Y. Tanaka and S. Kashiwaya, PRB 53 11957 (1996),
PRB 56 892 (1997)

Josephson junctions on TI

Lu Bo and T. Yukio, Phil. Trans. R. Soc. A. 376, 20150246 (2018).

Andreev reflection and Josephson current

Cooper pairelectron

Cooper pair

holeS

N(I)

S

Josephson current I(ϕ) by Andreev reflection coefficient

Andreev reflection coefficient by an electron-likequasiparticle injection
Andreev reflection coefficient by an hole-like
quasiparticle injection

Josephson current• The simultaneous existence of I1, I2 J1 at thesame time in the same order is the source ofthe giant diode effect.

Josephson current 1transmissivity at the interface

Josephson current 2

Josephson current 3

Condition for nonzero cos ϕ term

Condition for diode effectCurrent phase relation

Josephson current and sign inversion of mz

Josephsoncurrent
3
Josephson current 2

Current phase relation

Quality factor 1
Positive maximum Josephson current
Negative maximum Josephson current0

We plot magnitude of Q in the following calculation.

QQuality
quality factor
factor 12

arXiv:2205.13177

Q
温度依存性
Qualityfactor ３

Large magnitude of Q factor
Strong temperature dependenceTemperature dependence of maximum
Josephson current

Inverse the direction ofthe magnetization

Temperature dependence of Fourier component 1

The sign of J1 changes by the inversion of the sign of mz.

Temperature dependence of Fourier component 2

Andreev bound state at the interface 1

Andreev bound states at the interface becomes Majorana fermions(chiral Majorana mode)
We can know the energy level by the zero of the denominator of theAndreev reflection coefficients.

Andreev bound state at the interface 2

Andreev bound states condition

Andreev bound state at the interface 3

Limiting case where we can obtain analytical result

Andreev bound state at the interface 4

Limiting case where we can obtain analytical result

Andreev bound state at the interface 5

In general, it is impossible to solve Eb analytically.
We plot inverse of

Andreev bound states(θ dependence）

Q is nonzero (diode effect)アンドレーエフ束縛状態(角度依存性）
Andreev bound states(θ dependence）

No Diode effect case.

Andreev bound states (Phase dependence）

D:

Q is nonzero (diode effect)

Zero energy Andreev bound states andJosephson current

Zero energy Andreev bound state (Majorana mode) and Josephson current areclosely related to each other.

Summary and Conclusions(1)The large magnitude of quality factor Q is realizedby tuning the crystal axis of both left and right d-wave
superconductors.
(2)The magnitude of Q becomes almost 0.4 at low
temperatures and its sign is reversed by changing thedirection of the magnetization in the FI.
(3)The large Q stems from the simultaneous existence ofsin φ, cos φ and sin 2φ component.
(4)The strong temperature dependence of Q stems from theexistence of the low energy Andreev bound state appearingas Majorana bound states.
arXiv:2205.13177