NQS2022, Yukawa Institute, Kyoto Uniersity, Oct. 31-Dec. 2 (Nov. 24)

Nonreciprocal responses in superconductors:
Finite-q pairing, parity mixing, and quantum geometry

Akito Daido(Kyoto)

Hikaru Watanabe(Univ. Tokyo)

Yuhei Ikeda(Kyoto)

Hiroto Tanaka(Kyoto)

Kyoto University
Youichi YanaseWhy nonreciprocal responses?
Nonreciprocal response: left-moverright-mover

Diode: Building block of modern technology
LED = light-emitting diodePhotocurrent
Solar cellLight sensor

Second harmonic generation
Green laser

Why nonreciprocal responses?
Sensitive to Inversion symmetry breaking, since it is needed.

Probing the quantum materials
Visualization of ferro toroidal domains by SHG
N. A. Spaldin, M. Fiebig, and M. Mostovoy,
J. Phys. Condens. Matter 20, 434203 (2008).

Application to many parity-breaking magnet

Question: probing parity-breaking superconducting phases possible?
Order parameter, symmetry, topology, criticality ……
This talk: finite-q pairing, parity mixing in Cooper pairs, quantum geometry
Studied for a long time, but clarification has been awaited.

Contents
1. Superconducting diode effect (SDE)
Finite-q pairing in the SC state

Zeeman field(FFLO, helical, anapole)

2. Nonreciprocal optics
Quantum geometry of Bloch electrons
Parity mixing in Cooper pairs
Bulk Photocurrent

Contents
1. Superconducting diode effect (SDE)
Finite-q pairing in the SC state(FFLO, helical, anapole)
Akito Daido(Kyoto)

2. Nonreciprocal optics
Quantum geometry of Bloch electrons
Parity mixing in Cooper pairs

Yuhei Ikeda(Kyoto)

Josephson diodeY. Zhang et al. Phys. Rev. X 12, 041013 (2022)•••••••••••

J. Hu, C. Wu, and X. Dai, Phys. Rev. Lett. 99, 067004 (2007).
A. A. Reynoso, G. Usaj, C. A. Balseiro, D. Feinberg, and M. Avignon, Phys. Rev. Lett. 101, 107001 (2008).
A. Zazunov, R. Egger, T. Jonckheere, and T. Martin, Phys. Rev. Lett. 103, 147004 (2009).
I. Margaris, V. Paltoglou, and N. Flytzanis, J. Phys. Condens. Matter 22, 445701 (2010).
T. Yokoyama, M. Eto, and Y. V. Nazarov, Phys. Rev. B 89, 195407 (2014).
M. A. Silaev, A. Y. Aladyshkin, M. V. Silaeva, and A. S. Aladyshkina, J. Phys. Condens. Matter 26, 095702 (2014).
G. Campagnano, P. Lucignano, D. Giuliano, and A. Tagliacozzo, J. Phys. Condens. Matter 27, 205301 (2015).
F. Dolcini, M. Houzet, and J. S. Meyer, Phys. Rev. B 92, 035428 (2015).
C.-Z. Chen, J. J. He, M. N. Ali, G.-H. Lee, K. C. Fong, and K. T. Law, Phys. Rev. B 98, 075430 (2018).
M. Minutillo, D. Giuliano, P. Lucignano, A. Tagliacozzo, and G. Campagnano, Phys. Rev. B 98, 144510 (2018).
S. Pal and C. Benjamin, EPL 126, 57002 (2019).

Superconducting diode effect
F. Ando, Y. Miyasaka, T. Li, J. Ishizuka, T. Arakawa, Y. Shiota, T. Moriyama, Y. Yanase, and T. Ono Nature 584, 373 (2020): flippedforfor 100% rectification
 Ideal diodeSuperconducting diode effect= nonreciprocity in critical current.

Rikken et al., PRL (2001), (2005);

Magnetochiral anisotropy

cf.

Experiments
Rashba system + in-plane field
F. Ando et al., Nature 584, 373 (2020)

Rashba system/ferromagnet hybrid
H. Narita et al., Nature Nanotech. 17, 823 (2022)

Ising (or Zeeman)-type SOC + out-of-plane field

Zero field (triggered by SSB)

NbSe2 + mag. field L. Bauriedl et al.,Twisted trilayer graphene/WSe2

Nat. Commun. 13, 4266 (2022)

NbSe2/AFM + mag. field

J. Shin et al., arXiv:2111.05627+ Remarkable development in Josephson diode effectJ.-X. Lin et al., Nature Phys. 12, 1221 (2022)

H. Wu et al., Nature 604, 653 (2022)

Extrinsic SC diode effect
How can SC diode effect occur?
Some experiments might be understood by extrinsic mechanisms
Asymmetric
Meissner screening currentY. Hou et al., arXiv:2205.09276
M. K. Hope, et al.,
PRB 104, 184512 (2021).

Vortex formationin asymmetric multilayer SC
A. Sundaresh et al., arXiv:2207.03633

SC diode effect as an intrinsic material property?

Our work
The mechanism of SC diode effect is not clarified.

We propose an intrinsic mechanism.
Target: depairing critical current
𝑒𝑒𝑒

Condensation energy
vs. Kinetic energy𝑒𝑒𝑒Depairing critical current in FeSe
Y. Sun et al., Phys. Rev. B 101, 134516 (2020).

Methods
How to evaluate?

No diode effect in the simplest GL theory
Smidman et al., Rep. Prog. Phys. 80, 036501 (2017).

GL with higher-order spatial derivatives

Microscopic analysis: Relation to finite-q helical SC[ -wave/d-wave Rashba-Zeeman model, Mean-field theory]

: Temperature dependence• Temperature scaling

GL:GLGL𝛼 ∝𝑇 −𝑇cf.) N. Yuan and L. Fu, PNAS 119, e2119548119 (2022).
J. J. He, Y. Tanaka, and N. Nagaosa, New J. Phys. 24, 053014 (2022).

consistent with microscopic calc.
• Enhancement at low temperatureWhy?

𝜖 𝑘 ≃𝜖 𝑘 +𝑞⋅𝑣

• Helical superconductivity• Tiny diode effect by the asymmetry of•achieved in the gapless region(i.e)

Order parameter
Excitation gap•
•𝒄𝒄achieved at Landau critical momenta

Phase diagram forS-wave SC
D-wave SC

previous Δ𝑗 (𝑇)• Large 𝒄 at low and sign reversals by• Typical behavior of noncentrosymmetric SCs

Why?

Nature of helical SC
Key: Fermi-surface shift

Shcherbakov et al., Sci. Adv. (2021)

Free-energy minimum atIdeue et al., Nat. Phys. (2017)“crossover” line
Both FS are
happy with.

Nature of helical SC
Key: Fermi-surface shift

Shcherbakov et al., Sci. Adv. (2021)

Free-energy minimum at

Ideue et al., Nat. Phys. (2017)“crossover” line
Both FS are
almost happywith

Nature of helical SC
Key: Fermi-surface shift

Shcherbakov et al., Sci. Adv. (2021)

Free-energy minimum at

Ideue et al., Nat. Phys. (2017)“crossover” line
Both FS areunhappy with
Pairing on one FS
is better:

Current density
Equilibrium solution at“crossover” lineCurrent density
Equilibrium solution at“crossover” line

Ideue et al., Nat. Phys. (2017)

Current densityEquilibrium solution at“crossover” linePhase diagram: origin of sign-reversals
A. Daido, Y. Ikeda, and YY, PRL 128, 037001 (2022)• Helical superconductivity is realized• Sign reversal of SDE Change in the nature of helical SC• SC diode effect: promising probe of finite-q helical SC!

Experiment: Polarity oscillation of SDE in H
R. Kawarazaki et al. Applied Physics Express 15, 113001 (2022)• Sign reversal of SDE• Magnetic field is smaller
than theoretical prediction

Future issues

Theory:• Layered structure• Vortex stateExperiment:• More 2D• Exclusion of extrinsic mechanism (small device)General theory of intrinsic SDE
General GL model

A. Daido and YY, arXiv:2209.03515

Effective SOC of Cooper pairs

Intrinsic SDE in GL region• All the 21 noncentrosymmetricpoint group allows the SDE.
(even non-gyrotropic)• Behaviors of SDE can be predictedby the point group.

Nonreciprocal phase transition lines
A. Daido and YY, arXiv:2209.03515
Solid lines = phase transition lines under supercurrent

Large current: field-enhanced SC

Moderate current:
nonreciprocal reentrant SCContents
1. Superconducting diode effect (SDE)
Finite-q pairing in the SC state(FFLO, helical, anapole)

2. Nonreciprocal optics
Quantum geometry of Bloch electrons
Parity mixing in Cooper pairs

Hikaru Watanabe(Univ. Tokyo)

Hiroto Tanaka(Kyoto)

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Formulation for nonlinear conductivity (normal state)
 Response formula for nonlinear conductivity(2nd-) nonlinear current• Dipole-approximation (no photo-magnetic field, not spatially-dispersive)• Non-interacting electrons + phenomenological scattering
J. E. Sipe & A. I. Shkrebtii, Phys. Rev. B 61, 5337 (2000); Perturbation from photo-electric field••G. B. Ventura et al., Phys. Rev. B 96, 035431 (2017)von Neumann equation
Density mat.
in eq

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Formulation for nonlinear conductivity (normal state)
 Second order term in electric field

Monochromatic light induced photocurrent

 Polarization analysis of photocurrent conductivity

Linearly polarized light

B. I. Sturman & V. M. Fridkin, (CRC Press,1992).
Photovoltaic and Photo-refractive Effects inNoncentrosymmetric Materials

Circularly polarized light

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Classification of photocurrent [Watanabe and YY, Phys. Rev. X 11, 011001 (2021)]
P × T 〇 PT ×

P × T × PT 〇T-symmetricPT-symmetric

Berry curvature dipole
J. E. Moore & J. Orenstein, Phys. Rev. Lett. 105, 026805 (2010).

Metal

Intrinsic Fermi surface term
F. de Juan et al., Phys. Rev. Research 2, 012017 (2020).

Metal
andinsulator

Electric injection current
J. E. Sipe & A. I. Shkrebtii, Phys. Rev. B 61, 5337 (2000).

Shift current
R. von Baltz & W. Kraut, Phys. Rev. B 23, 5590 (1981);
J. E. Sipe & A. I. Shkrebtii, Phys. Rev. B 61, 5337 (2000).

Drude photocurrent
T. Holder et al., Phys. Rev. Research 2, 033100 (2020) .

Intrinsic Fermi surface termThis work

Magnetic injection current
Y. Zhang et al., Nat. Commun. 10, 3783 (2019).

Gyration current
This work

Electric/Magnetic multipole ⇔ T/PT symmetry ⇔ Linearly/Circularly polarized light

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Classification of photocurrent based on quantum geometry

Director

Transition amp.

Group velocity difference

Shift vector(band-resolved) Berry curvature(band-resolved) quantum metric

Diverging photocurrent in AFM Dirac/Weyl semimetalsWatanabe and YY, Phys. Rev. X 11, 011001 (2021), J. Ahn et al., Phys. Rev. X 10, 041041 (2020).

Superconductors
H. Watanabe, A. Daido, and YY, Phys. Rev. B 105, 024308 (2022)
H. Watanabe, A. Daido, and YY, Phys. Rev. B 105, L100504 (2022)
H. Tanaka, H. Watanabe, and YY, arXiv:2205.14445

Optical responses unique to superconductors
Variational parameter :(= vector potential A)

Opposite charges of electrons and holes

λ-derivative and k -derivative are not equivalent.

Divergent response at THz or sub THz regimeNonreciprocal
superfluid weight
Conductivity
derivative

𝟏𝟏𝟐𝟏𝟏

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Photocurrent in superconductors [Watanabe, Daido, YY, PRB 105, 024308 (2022)]
T-symmetric

Metal

Metal
andinsulator

Superconductor

PT-symmetric

Berry curvature dipole

Drude photocurrent

Intrinsic Fermi surface term

Intrinsic Fermi surface term

Electric injection current

Magnetic injection current

Shift current

Gyration current

Nonreciprocal superfluid density
Berry curvature derivative
Drude derivative

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Giant nonreciprocal optical responses in superconductors

Nonreciprocal superfluid weightBerry curvature derivative

Superconductor

Nonreciprocal superfluid density
Berry curvature derivative
Drude derivative

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Microscopic conditions [Tanaka, Watanabe, and YY, arXiv:2205.14445]
2D noncentrosymmetric superconductors
ASOC
Parity mixing

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Microscopic conditions [Tanaka, Watanabe, YY, arXiv:2205.14445]•

Type B transition causes the nonlinear optical responses unique to superconductors•Intraband Cooper pairs and interband Cooper pairs are necessary

1. Finite intraband pairing
2. Finite interband pairing

Superconducting fitness[Ramires, 2018]

Conclusion:
Spin-triplet Cooper pairs are necessary for SC nonlinear optical responses.

Summary
Intrinsic SC diode effect•Nonreciprocal depairing critical current

A. Daido, Y. Ikeda, and YY

•

Large

Phys. Rev. Lett. 128, 037001 (2022)•

Sign reversals as

A. Daido and YY, arXiv:2209.03515

𝒄 at low& moderate

Promising probe of finite-q helical SC

SC nonreciprocal optics•

Nonlinear optical responses unique to SC

H. Watanabe, A. Daido, and YY,•Diverging photocurrent and SHG at sub THz•Sensitive to quantum geometry

Phys. Rev. B 105, 024308 (2022)
Phys. Rev. B 105, 024308 (2022)
Tanaka, Watanabe, and YY, arXiv:2205.14445

Promising probe of parity-mixed Cooper pairs