nonlinear and nonreciprocal responses of

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Nonlinear and Nonreciprocal Responses of Noncentrosymmetric Quantum Matters Naoto Nagaosa RIKEN Center for Emergent Matter Science (CEMS) and Dept. Applied Phys. Univ. Tokyo Nov.9, 2017 @NQS2017 Kyoto

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Page 1: Nonlinear and Nonreciprocal Responses of

Nonlinear and Nonreciprocal Responses

of Noncentrosymmetric Quantum Matters

Naoto Nagaosa

RIKEN Center for Emergent Matter Science (CEMS) and

Dept. Applied Phys. Univ. Tokyo

Nov.9, 2017 @NQS2017 Kyoto

Page 2: Nonlinear and Nonreciprocal Responses of

Collaborators U.Tokyo: Ryohei Wakatsuki, Keita Hamamoto, Kou Misaki, Hiro Ishizuka, Motohiko Ezawa, S. Miyashita RIKEN CEMS: Shintaro Hoshino, Masahito Ueda U.C. Berkeley: Takahiro Morimoto Delware: B. Nikolic Chuo Univ.: Tomya Hayata Exp: Y. Tokura, Y. Iwasa, M. Kawasaki, S. Koshikawa, S. Shimizu, Y. Kaneko, Y. Saito, T. Ideue

Page 4: Nonlinear and Nonreciprocal Responses of

V(x)

x

ikxe ikxteikxreβˆ’

V(x)

x

ikxeβˆ’ikxet βˆ’'ikxer '

22 |||'| tt = 22 |||'| rr =

Right and Left directions of flow

Incident from left

Incident from right

tt ='Unitary nature of S matrix: conservation of prob.

Time-reversal symmetry T: S=S^Transpose

Page 5: Nonlinear and Nonreciprocal Responses of

Asymmetry between t and -t

1. Microscopic time-reversal symmetry breaking external magnetic field B magnetic ordering M 2. Macroscopic irreversibility dissipation of energy diffusion

Page 6: Nonlinear and Nonreciprocal Responses of

Non-reciprocal transport in non-centrosymmetric system

R (+I) β‰  R (-I) +I

-I

R = R0 (1 + Ξ²B2 + Ξ³BI)

Broken inversion symmetry ⇔ Ξ³ β‰  0

”dichroism” of the electric current magnetochiral anisotropy

),,(),,( BkBk βˆ’βˆ’= ωσωσ ν¡¡νOnsager’s reciprocal relation for linear response

Magnetochiral optical effect directional dichroism

BkBk β‹…+= αΡωΡ ¡¡ 0),,(

Ik β†’ G. L. J. A. Rikken (2001)

Nonlinear response 2BEEJ Ξ±Οƒ +=

e.g. electromagnon in multiferroics A. Loidl, Y.Tokura

Time-reversal symmetry of microscopic dynamics vs irreversibility

Page 7: Nonlinear and Nonreciprocal Responses of

K. Ishizaka et al. Nat. Mater. 10, 521 (2011).

Band structure in noncentrosymmetric crystal

)()( kk βˆ’= ↓↑ ΡΡTime-reversal symmetry

Page 9: Nonlinear and Nonreciprocal Responses of

Nonreciprocal Response

Linear Response

Nonlinear Response

Time-reversal Unbroken

Time-reversal Broken

Forbidden

Shift current

Nonlinear Hall effect

pn junction

Optical ME effect

Magnetochiral effect

Nonreciprocal magnon

Nonreciprocal nonlinear optical effect

Magnetochiral anisotropy

Electric magnetochiral effect

Inverse Edelstein effect

Table 1

Page 10: Nonlinear and Nonreciprocal Responses of

Intra- and Inter-band matrix elements of current

k

E

wave packet nkevnkJnk βˆ’>=< ||

>< mkJnk ||

>βˆ‚βˆ‚

<βˆ’βˆ’>=βˆ‚

βˆ‚<βˆ’>=<

βˆ‚βˆ‚

βˆ’>=βˆ‚

βˆ‚<βˆ’>=<

mkk

nkemkkkHnkemkJnk

kkenk

kkHnkenkJnk

mknk

n

||)(|)(|||

)(|)(|||

ΡΡ

Ξ΅

Wavefunction matters !!

Of-diagonal matrix elements - Distort the band structure by E - Additional current

Even a filled band can support current e.g., polarization current quantum Hall current

)()( kbkb βˆ’βˆ’=

0)( =kb

)()( kbkb βˆ’=Time-reversal

Inversion

Page 11: Nonlinear and Nonreciprocal Responses of

Berry Phase Curvature in k-space Bloch wavefucntion )()( ruer nk

ikrnk =ψ

>βˆ‡<βˆ’= nkknkn uuika ||)( Berry phase connection in k-space

)()( kaikarx nknii i+βˆ‚=+= covariant derivative

)())()((],[ kibkakaiyx nznxknyk yx=βˆ‚βˆ’βˆ‚= Curvature in k-space

yVkb

mk

yVyxi

mkHxi

dttdx

nzxx

βˆ‚βˆ‚

+=βˆ‚βˆ‚

βˆ’=βˆ’= )(],[],[)(

xk yk

zk

>β€’ βˆ†+ knku|>β€’ nku|kβˆ†

Anomalous Velocity and Anomalous Hall Effect

)(| kiaknknk

neuu >=< βˆ†+ )(kan

Intracell coordinates

Wannier coordinates

Page 12: Nonlinear and Nonreciprocal Responses of

DC nonreciprocal responses

eAp +=Ο€ Expansion of Keldysh Green’s function in constant E and B. Multiband effect fully taken into account.

Non-interacting case No E-dependence of Ξ£

Page 13: Nonlinear and Nonreciprocal Responses of

AC nonreciprocal response – Shift current

Page 14: Nonlinear and Nonreciprocal Responses of
Page 15: Nonlinear and Nonreciprocal Responses of

electron flow

Page 16: Nonlinear and Nonreciprocal Responses of

Steve M. Young and Andrew M. Rappe, Phys. Rev. Lett. 109, 116601 (2012)

BaTiO3

'nnφ

nχphase of the transition matrix element

Berry connection (vector potential)

First-principles calculation of shift current

R. von Baltz & W. Kraut, PRB1981 J.E. Sipe & A.I. Shkrebtii, PRB2000

Page 17: Nonlinear and Nonreciprocal Responses of

Shift-current Even order response to E

Conduction and valence bands are coupled to particle bath Independently

kdb βˆ‚βˆ‚= / Current

Topological nature of Floquet bands in Keldysh formalism T. Morimoto, NN Science Adv. 2016

See also A.M. Cook et al., Nature Commun. 2017

Page 18: Nonlinear and Nonreciprocal Responses of

Exciton formation

Bethe-Salpeter equation for exciton formation

Hanai-Littilewood-Ohashi

T. Morimoto, NN PRB 2016

Page 19: Nonlinear and Nonreciprocal Responses of

Position dependence Nakamura et al.

Nonlocal nature of shift current Y. Nakamura et al.

Page 20: Nonlinear and Nonreciprocal Responses of

Localization and shift current

Lead Lead 0 1 2 3 4 5 6 7 8 N

π‘₯π‘₯𝐴𝐴

𝑙𝑙𝐴𝐴 L

N

Disorder strength

L=N-1

0 <|

J|>

L<<N-1

Positon of local excitation

Pure system Excite whole system

Localization Length <<N

H. Ishizuka, NN NJP 2017 P50 preliminary

Page 21: Nonlinear and Nonreciprocal Responses of

Shift current on the surface of TI K.W.Kim, T.Morimoto, NN arXiv:1607.03888, PRB 2017

Page 22: Nonlinear and Nonreciprocal Responses of

Emergent electric field of 3d Weyl fermion in momentum space H. Ishizuka et al., PRL 2016

c.f. Moore-Orenstein PRL 2010 Sodeman-Fu PRL 2015

Page 23: Nonlinear and Nonreciprocal Responses of

Theorems

Nonreciprocal dc transport with time-reversal symmetry

in noncentrosymmetric systems:

dc E-field: requires both irreversibility and electron correlation

virtual excitation by interband matrix element of J

ac E-field: the quantum geometry of the multiband structure even

in noninteracting systems

robust against the exciton formation and localization

real excitation by interband matrix element of J

Page 24: Nonlinear and Nonreciprocal Responses of

Nonreciprocal Response

Linear Response

Nonlinear Response

Time-reversal Unbroken

Time-reversal Broken

Forbidden

Shift current

Nonlinear Hall effect

pn junction

Optical ME effect

Magnetochiral effect

Nonreciprocal magnon

Nonreciprocal nonlinear optical effect

Magnetochiral anisotropy

Electric magnetochiral effect

Inverse Edelstein effect

Table 1

Page 25: Nonlinear and Nonreciprocal Responses of

Ξ³ (Tβˆ’1Aβˆ’1) 10-3 10-2 10-1

G. L. J. A. Rikken et al. PRL 87, 236602

(2001).

Bi helices Organics Si FET

F. Pop et al. Nat. Commun. 5, 3757

(2014).

G. L. J. A. Rikken et al. PRL 94, 016601

(2005).

I

E B

Magnetochiral anisotropy – tiny effect

FSOIB B ΡΡ¡ << ,R = R0 (1 + βB2 + γBI)

11410~ βˆ’βˆ’ ATΞ³

Page 26: Nonlinear and Nonreciprocal Responses of

Enhanced magnetochiral anisotropy in BiTeBr Y.Iwasa G, Y.Tokura G, NN G Nature Phys. 2017

Page 27: Nonlinear and Nonreciprocal Responses of

𝐻𝐻 =π‘˜π‘˜π‘§π‘§2

2π‘šπ‘šβˆ₯+π‘˜π‘˜π‘₯π‘₯2 + π‘˜π‘˜π‘¦π‘¦2

2π‘šπ‘šβŠ₯+ Ξ» π‘˜π‘˜π‘₯π‘₯πœŽπœŽπ‘¦π‘¦ βˆ’ π‘˜π‘˜π‘¦π‘¦πœŽπœŽπ‘₯π‘₯ βˆ’ π΅π΅π‘¦π‘¦πœŽπœŽπ‘¦π‘¦

Magnetochiral anisotropy in Rashba model

βˆ’π‘’π‘’π‘¬π‘¬ β‹…πœ•πœ•πœ•πœ•πœ•πœ•π’Œπ’Œ

= βˆ’1πœπœπœ•πœ• βˆ’ πœ•πœ•0 πœ•πœ• = πœ•πœ•0 + πœ•πœ•1 + πœ•πœ•2 + β‹― ,

Boltzmann equation Expand in E

𝐽𝐽π‘₯π‘₯ = 𝐽𝐽π‘₯π‘₯(1) + 𝐽𝐽π‘₯π‘₯

(2) = 𝜎𝜎1𝐸𝐸π‘₯π‘₯ + 𝜎𝜎2𝐸𝐸π‘₯π‘₯2

increasing Fermi energy

B

Ξ³ diverges as n 0 Ξ³ is zero when two FSs coexist

Ξ³ is independent of Ο„ in the single relaxation time approx. like Hall coefficient

K. Hamamoto

Page 28: Nonlinear and Nonreciprocal Responses of

AΞ³Ξ³ =' A cross section of the sample

No fitting parameters !

111~ βˆ’βˆ’ ATΞ³

Page 29: Nonlinear and Nonreciprocal Responses of

Weyl semimetals X. Huang et al., PRX 2015

TaAs

negative magnetoresistance due to chiral anomaly

Page 30: Nonlinear and Nonreciprocal Responses of

Chiral anomaly in Weyl semimetals

chemical potential difference in non-equilibrium steady state

BEBJ )( β‹…βˆβˆ† Ο„

5Β΅

negative magnetoresistance due to chiral anomaly

X. Huang et al., PRX 2015

TaAs

K. Fukushima, D. Kharzeev

Page 31: Nonlinear and Nonreciprocal Responses of

Weyl fermions in noncentrosymemtric semimetals

Page 32: Nonlinear and Nonreciprocal Responses of

Magnetochial anisotropy in noncentrosymemtric Weyl semimetals

TaAs sec/104~ 5 mv Γ— meV10|~| Β±Ξ΅

T. Morimoto, NN PRL2016

1110~ βˆ’βˆ’ ATΞ³

Page 33: Nonlinear and Nonreciprocal Responses of

Giant enhancement of non-reciprocal response in superconductor

FSOIBSC B ΡΡ¡ <<βˆ† ,~

Wakatsuki, Saito et al. Science Adv. 2017

Page 34: Nonlinear and Nonreciprocal Responses of

Band structure and spin splitting in MoS2

Page 35: Nonlinear and Nonreciprocal Responses of

π»π»π‘˜π‘˜π‘˜π‘˜π‘˜π‘˜ =Δ§2π‘˜π‘˜2

2π‘šπ‘š+ πœπœπ‘§π‘§πœ†πœ†π‘˜π‘˜π‘₯π‘₯ π‘˜π‘˜π‘₯π‘₯2 βˆ’ 3π‘˜π‘˜π‘¦π‘¦2 βˆ’ π›₯π›₯π‘π‘πœŽπœŽπ‘§π‘§ βˆ’ π›₯π›₯π‘†π‘†π‘†π‘†πœŽπœŽπ‘§π‘§πœπœπ‘§π‘§,

𝐹𝐹 = οΏ½d π’“π’“π›Ήπ›Ήβˆ— π‘Žπ‘Ž +𝒑𝒑2

4π‘šπ‘š+𝛬𝛬𝐡𝐡ℏ3

𝑝𝑝π‘₯π‘₯3 βˆ’ 3𝑝𝑝π‘₯π‘₯𝑝𝑝𝑦𝑦2 𝛹𝛹 +𝑏𝑏2�𝑑𝑑 𝒓𝒓|𝛹𝛹 𝒓𝒓 |4,

𝛬𝛬 =93𝜁𝜁 528𝜁𝜁 3

π‘”π‘”πœ‡πœ‡Bπ›₯π›₯SOπœ†πœ†πœ‹πœ‹π‘˜π‘˜B𝑇𝑇c 2

𝒋𝒋 =𝑒𝑒2

16Δ§πœ–πœ–βˆ’1𝑬𝑬 βˆ’

πœ‹πœ‹π‘’π‘’3π‘šπ‘šπ›¬π›¬π΅π΅64ℏ3π‘˜π‘˜B𝑇𝑇c

πœ–πœ–βˆ’2𝑭𝑭 𝑬𝑬

πœ–πœ– =𝑇𝑇 βˆ’ 𝑇𝑇𝑐𝑐𝑇𝑇𝑐𝑐

𝑭𝑭 𝑬𝑬 = 𝐸𝐸π‘₯π‘₯2 βˆ’ 𝐸𝐸𝑦𝑦2,βˆ’2𝐸𝐸π‘₯π‘₯𝐸𝐸𝑦𝑦

𝛾𝛾𝑆𝑆𝛾𝛾𝑁𝑁

βˆΌπœ€πœ€πΉπΉπ‘˜π‘˜π΅π΅π‘‡π‘‡π‘π‘

3

Paraconductivity due to SC fluctuation in noncentrosymmetric MoS2

Page 36: Nonlinear and Nonreciprocal Responses of

400

Tc

Page 37: Nonlinear and Nonreciprocal Responses of

..))(()( 2 chkdiikH kykpkykskDk

kk +β‹…βˆ†+βˆ†+β‹…+= +βˆ’

++βˆ’

++βˆ‘ ΟˆΟƒΟƒΟˆΟˆΟƒΟˆΟˆΟƒΞ»ΞΎΟˆ

)(2

1 ),(2

1βˆ’+β†“βˆ’

βˆ’+↑ βˆ’=+= kk

ikk

ikk ccecec kk θθ ψψChiral base

..)()(|)|( chcceccecckH kki

pskki

pskkk

kkk +βˆ†+βˆ†+βˆ†+βˆ†βˆ’+Β±= +

βˆ’βˆ’+βˆ’

++βˆ’

++

βˆ’Β±

+Β±βˆ‘ θθλξ

Rashba superconductor

Frigeri et al. 2004

+ and – bands are p+ip superconductor

xk

yk

k

kΞΈ

Fu-Kane, 2008 Proximity effect of 3D topological

insulator and s-wave SC

|||| ps βˆ†>βˆ†

|||| ps βˆ†<βˆ†

Z2 classification of DIII in 2D

Non-topological

Topological helical Majorana

Y.Yanaka-Yokoyama-Balatsky-N.N. Fujimoto-Sato

Page 38: Nonlinear and Nonreciprocal Responses of

- Non-linear and non-reciprocal responses in nontrosymmetric systems contain rich physics - Time-reversal symmetry plays an important role - Nonreciprocal response without T-breaking is related to β€œinterband current” and β€œquantum geometry” - Electron correlation gives the dc nonreciprocal response - Shift current in photovoltaic effect from Berry phase - Enhancement of Magnetochiral anisotropy - Polar Rashba semiconductor - Weyl semimetals due to chiral anomaly - Bosonic transport due to Cooper pairing

Summary

Symmetry Quantum Geometry Electron Correlation Irreversibility