Download - Nonlinear and Nonreciprocal Responses of
![Page 1: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/1.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/2.jpg)
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 3: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/3.jpg)
Chirality of structure
From Wikipedia
No Inversion I No Mirror M
![Page 4: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/4.jpg)
V(x)
x
ikxe ikxteikxre−
V(x)
x
ikxe−ikxet −'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](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/5.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/6.jpg)
Non-reciprocal transport in non-centrosymmetric system
R (+I) ≠ R (-I) +I
-I
R = R0 (1 + βB2 + γBI)
Broken inversion symmetry ⇔ γ ≠ 0
”dichroism” of the electric current magnetochiral anisotropy
),,(),,( BkBk −−= ωσωσ νµµνOnsager’s reciprocal relation for linear response
Magnetochiral optical effect directional dichroism
BkBk ⋅+= αεωε µµ 0),,(
Ik → G. L. J. A. Rikken (2001)
Nonlinear response 2BEEJ ασ +=
e.g. electromagnon in multiferroics A. Loidl, Y.Tokura
Time-reversal symmetry of microscopic dynamics vs irreversibility
![Page 7: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/7.jpg)
K. Ishizaka et al. Nat. Mater. 10, 521 (2011).
Band structure in noncentrosymmetric crystal
)()( kk −= ↓↑ εεTime-reversal symmetry
![Page 8: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/8.jpg)
http://quantumwise.com/publications/ tutorials/item/828-silicon-p-n-junction
http://www.optique-ingenieur.org/en/courses/ OPI_ang_M05_C02/co/Contenu_09.html
pn junction
![Page 9: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/9.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/10.jpg)
Intra- and Inter-band matrix elements of current
k
E
wave packet nkevnkJnk −>=< ||
>< mkJnk ||
>∂∂
<−−>=∂
∂<−>=<
∂∂
−>=∂
∂<−>=<
mkk
nkemkkkHnkemkJnk
kkenk
kkHnkenkJnk
mknk
n
||)(|)(|||
)(|)(|||
εε
ε
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 −−=
0)( =kb
)()( kbkb −=Time-reversal
Inversion
![Page 11: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/11.jpg)
Berry Phase Curvature in k-space Bloch wavefucntion )()( ruer nk
ikrnk =ψ
>∇<−= nkknkn uuika ||)( Berry phase connection in k-space
)()( kaikarx nknii i+∂=+= covariant derivative
)())()((],[ kibkakaiyx nznxknyk yx=∂−∂= Curvature in k-space
yVkb
mk
yVyxi
mkHxi
dttdx
nzxx
∂∂
+=∂∂
−=−= )(],[],[)(
xk yk
zk
>• ∆+ knku|>• nku|k∆
Anomalous Velocity and Anomalous Hall Effect
)(| kiaknknk
neuu >=< ∆+ )(kan
Intracell coordinates
Wannier coordinates
![Page 12: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/12.jpg)
DC nonreciprocal responses
eAp +=π Expansion of Keldysh Green’s function in constant E and B. Multiband effect fully taken into account.
Non-interacting case No E-dependence of Σ
![Page 13: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/13.jpg)
AC nonreciprocal response – Shift current
![Page 14: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/14.jpg)
![Page 15: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/15.jpg)
electron flow
![Page 16: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/16.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/17.jpg)
Shift-current Even order response to E
Conduction and valence bands are coupled to particle bath Independently
kdb ∂∂= / 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](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/18.jpg)
Exciton formation
Bethe-Salpeter equation for exciton formation
Hanai-Littilewood-Ohashi
T. Morimoto, NN PRB 2016
![Page 19: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/19.jpg)
Position dependence Nakamura et al.
Nonlocal nature of shift current Y. Nakamura et al.
![Page 20: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/20.jpg)
Localization and shift current
Lead Lead 0 1 2 3 4 5 6 7 8 N
𝑥𝑥𝐴𝐴
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](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/21.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/22.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/23.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/24.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/25.jpg)
γ (T−1A−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 εεµ << ,R = R0 (1 + βB2 + γBI)
11410~ −− ATγ
![Page 26: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/26.jpg)
Enhanced magnetochiral anisotropy in BiTeBr Y.Iwasa G, Y.Tokura G, NN G Nature Phys. 2017
![Page 27: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/27.jpg)
𝐻𝐻 =𝑘𝑘𝑧𝑧2
2𝑚𝑚∥+𝑘𝑘𝑥𝑥2 + 𝑘𝑘𝑦𝑦2
2𝑚𝑚⊥+ λ 𝑘𝑘𝑥𝑥𝜎𝜎𝑦𝑦 − 𝑘𝑘𝑦𝑦𝜎𝜎𝑥𝑥 − 𝐵𝐵𝑦𝑦𝜎𝜎𝑦𝑦
Magnetochiral anisotropy in Rashba model
−𝑒𝑒𝑬𝑬 ⋅𝜕𝜕𝜕𝜕𝜕𝜕𝒌𝒌
= −1𝜏𝜏𝜕𝜕 − 𝜕𝜕0 𝜕𝜕 = 𝜕𝜕0 + 𝜕𝜕1 + 𝜕𝜕2 + ⋯ ,
Boltzmann equation Expand in E
𝐽𝐽𝑥𝑥 = 𝐽𝐽𝑥𝑥(1) + 𝐽𝐽𝑥𝑥
(2) = 𝜎𝜎1𝐸𝐸𝑥𝑥 + 𝜎𝜎2𝐸𝐸𝑥𝑥2
increasing Fermi energy
B
γ diverges as n 0 γ is zero when two FSs coexist
γ is independent of τ in the single relaxation time approx. like Hall coefficient
K. Hamamoto
![Page 28: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/28.jpg)
Aγγ =' A cross section of the sample
No fitting parameters !
111~ −− ATγ
![Page 29: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/29.jpg)
Weyl semimetals X. Huang et al., PRX 2015
TaAs
negative magnetoresistance due to chiral anomaly
![Page 30: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/30.jpg)
Chiral anomaly in Weyl semimetals
chemical potential difference in non-equilibrium steady state
BEBJ )( ⋅∝∆ τ
5µ
negative magnetoresistance due to chiral anomaly
X. Huang et al., PRX 2015
TaAs
K. Fukushima, D. Kharzeev
![Page 31: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/31.jpg)
Weyl fermions in noncentrosymemtric semimetals
![Page 32: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/32.jpg)
Magnetochial anisotropy in noncentrosymemtric Weyl semimetals
TaAs sec/104~ 5 mv × meV10|~| ±ε
T. Morimoto, NN PRL2016
1110~ −− ATγ
![Page 33: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/33.jpg)
Giant enhancement of non-reciprocal response in superconductor
FSOIBSC B εεµ <<∆ ,~
Wakatsuki, Saito et al. Science Adv. 2017
![Page 34: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/34.jpg)
Band structure and spin splitting in MoS2
![Page 35: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/35.jpg)
𝐻𝐻𝑘𝑘𝑘𝑘𝑘𝑘 =ħ2𝑘𝑘2
2𝑚𝑚+ 𝜏𝜏𝑧𝑧𝜆𝜆𝑘𝑘𝑥𝑥 𝑘𝑘𝑥𝑥2 − 3𝑘𝑘𝑦𝑦2 − 𝛥𝛥𝑍𝑍𝜎𝜎𝑧𝑧 − 𝛥𝛥𝑆𝑆𝑆𝑆𝜎𝜎𝑧𝑧𝜏𝜏𝑧𝑧,
𝐹𝐹 = �d 𝒓𝒓𝛹𝛹∗ 𝑎𝑎 +𝒑𝒑2
4𝑚𝑚+𝛬𝛬𝐵𝐵ℏ3
𝑝𝑝𝑥𝑥3 − 3𝑝𝑝𝑥𝑥𝑝𝑝𝑦𝑦2 𝛹𝛹 +𝑏𝑏2�𝑑𝑑 𝒓𝒓|𝛹𝛹 𝒓𝒓 |4,
𝛬𝛬 =93𝜁𝜁 528𝜁𝜁 3
𝑔𝑔𝜇𝜇B𝛥𝛥SO𝜆𝜆𝜋𝜋𝑘𝑘B𝑇𝑇c 2
𝒋𝒋 =𝑒𝑒2
16ħ𝜖𝜖−1𝑬𝑬 −
𝜋𝜋𝑒𝑒3𝑚𝑚𝛬𝛬𝐵𝐵64ℏ3𝑘𝑘B𝑇𝑇c
𝜖𝜖−2𝑭𝑭 𝑬𝑬
𝜖𝜖 =𝑇𝑇 − 𝑇𝑇𝑐𝑐𝑇𝑇𝑐𝑐
𝑭𝑭 𝑬𝑬 = 𝐸𝐸𝑥𝑥2 − 𝐸𝐸𝑦𝑦2,−2𝐸𝐸𝑥𝑥𝐸𝐸𝑦𝑦
𝛾𝛾𝑆𝑆𝛾𝛾𝑁𝑁
∼𝜀𝜀𝐹𝐹𝑘𝑘𝐵𝐵𝑇𝑇𝑐𝑐
3
Paraconductivity due to SC fluctuation in noncentrosymmetric MoS2
![Page 36: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/36.jpg)
400
Tc
![Page 37: Nonlinear and Nonreciprocal Responses of](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/37.jpg)
..))(()( 2 chkdiikH kykpkykskDk
kk +⋅∆+∆+⋅+= +−
++−
++∑ ψσσψψσψψσλξψ
)(2
1 ),(2
1−+↓−
−+↑ −=+= kk
ikk
ikk ccecec kk θθ ψψChiral base
..)()(|)|( chcceccecckH kki
pskki
pskkk
kkk +∆+∆+∆+∆−+±= +
−−+−
++−
++
−±
+±∑ θθλξ
Rashba superconductor
Frigeri et al. 2004
+ and – bands are p+ip superconductor
xk
yk
k
kθ
Fu-Kane, 2008 Proximity effect of 3D topological
insulator and s-wave SC
|||| ps ∆>∆
|||| ps ∆<∆
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](https://reader036.vdocuments.us/reader036/viewer/2022070300/615723c65e27e87b7340050e/html5/thumbnails/38.jpg)
- 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 “interband current” and “quantum 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