NQS 2022 tutorial lecture, YITP (Kyoto), Nov. 1, 2022

Geometrical nonlinear optical effects incorrelated electron systems
Takahiro Morimoto
University of Tokyo, Department of Applied Physics

Topology
Geometry

Correlatedelectrons

Superconductors

Multiferroic magnets

Nonlinear
responses
Photovolatic effects

High harmonic
generation

Mott insulators

Nonlineartransport

Plan of this talk• Shift current as a geometric nonlinear response– Floquet theory approach

Morimoto, Nagaosa, Sci. Adv. (2016)• Nonlinear optical effects of correlated electrons– Exciton shift current

Morimoto, Nagaosa, PRB (2016)– Shift current of magnons in multiferroic magnets

Morimoto, Nagaosa, PRB (2019)
Morimoto, Kitamura, Okumura, PRB (2021)– Shift current in phonons

Okamura et al. PNAS (2022)– Photocurrent of exciton polaritons
Morimoto, Nagaosa, PRB (2020)

Second order nonlinear optical effects
J(ω+ω’) = σ(ω+ω’; ω,ω’) E(ω) E(ω’)

Amplitude

𝐸𝐸 𝑡𝑡 = 𝐸𝐸𝑒𝑒𝑒𝑒𝑒𝑒 (𝑡𝑡)𝑒𝑒 −𝑖𝑖𝜔𝜔0 𝑡𝑡

DC/THz0SHGω02ω0Frequency

Bulk photovoltaic effect

Shift current

Circular photogalvanic effect (CPGE)Hodes,Science (2013)

Nature 2013

Mechanism of photovoltaic effect• p-n junction– Potential gradient byartificial structure• Perovskite solar cell
– Bulk crystal– NoncentrosymmetricShift current
Sipe, Shkrebtii, PRB (2000)• dc current proportional to E2• Nonvanishing in noncentrosymmetric crystals

BaTiO3Young and Rappe, PRL (2015)

Rappe group, J. Phys. Chem. Lett., (2015)• Concise descriptions of second
order nonlinear optical effects• Not a complicated perturbation
theory• Connection to k-space geometry
 New formulation from Floquet theory

Amplitude

Motivation of this study
DC/THz0SHG

ω02ω0ωFloquet theory describes nonlinear optical effects
Floquet formalism: an analog of Bloch theorem in the time direction
Conduction band
1 photon dressedvalence band
Valence band・Floquet two band model

with・Nonequilibrium steady state = Anticrossing of Floquet bands
・Photocurrent = sum of velocities of occupied Floquet states
 Keldysh Green’s function

Derivation of shift current formula= 𝒅𝒅 ⋅ 𝝈𝝈𝑖𝑖Γ
𝑅𝑅/𝐴𝐴𝐺𝐺= 𝜔𝜔 − 𝐻𝐻𝐹𝐹 ±2

𝐽𝐽 = ∫ 𝑑𝑑𝒌𝒌−1𝑣𝑣 = 𝜕𝜕𝑘𝑘 𝐻𝐻𝐹𝐹 = 𝒃𝒃 ⋅ 𝝈𝝈

𝐺𝐺 < = 𝐺𝐺 𝑅𝑅 Σ < 𝐺𝐺 𝐴𝐴= ∫ 𝑑𝑑𝒌𝒌

Compact expression of shift current
Joint DOSTransition rate

Saturation effect

Shift vector: RR�Berry connection
Morimoto, Nagaosa, Sci. Adv. (2016)[cf. perturbation theory：Sipe, Shkrebtii, PRB (2000)]

electron flow

A semiclassical picture of shift current
electron flow

Lightirradiation

Optical excitation of electron hole pairs with electric polarization
 Dc current flow according to J=dP/dt
Momentum space

Real spaceRAdvantage of Floquet formalism for shift current
Advantages
1. Requires information of only 2 bands- c.f. perturbation theory including all bands

Von Baltz, Kraut, PRB (1981); Sipe, Shkrebtii, PRB (2000)

2. Floquet  Saturation effect (nonperturbative in E )3. Keldysh  Interaction effect, relaxation effectApplications
1.Saturation effect of shift
current measured in SbSI

Sotome, TM, Ogawa, et al., PNAS (2019)

2. I-V characteristics and shot noise

Morimoto, Nakamura, Kawasaki,Nagaosa, PRL (2018)

Geometry matters in nonlinear responses!
EOptical transition
between the manifoldsk

Plan of this talk• Shift current as a geometric nonlinear response– Floquet theory approach

Morimoto, Nagaosa, Sci. Adv. (2016)• Nonlinear optical effects of correlated electrons– Exciton shift current

Morimoto, Nagaosa, PRB (2016)– Shift current of magnons in multiferroic magnets

Morimoto, Nagaosa, PRB (2019)
Morimoto, Kitamura, Okumura, PRB (2021)– Shift current in phonons

Okamura et al. PNAS (2022)– Photocurrent of exciton polaritons
Morimoto, Nagaosa, PRB (2020)

How does electron correlation
modify/enhance shift current response?
Weakly correlated systems:
Photoexcitation of bound states

Shift current of excitons• Attractive interaction between electrons and holes Exciton formation−++−

No free electron hole pair

Eg: band gap, Vex: exciton binding energy, v12: current matrix element,
R2 : shift vector:

Charge neutral excitons can induce dc current!

Morimoto, Nagaosa, PRB (2016)

Observation of exciton shift current
THz spectroscopy ofshift current in CdS

Photovoltaic effect of WSe2/BP
heterostructure

Exciton resonanceof WSe2Sotome, Ogawa et al. PRB(L) (2021)

Akamatsu, Ideue, Morimoto, Iwasa et al.
Science (2021)

Plan of this talk• Shift current as a geometric nonlinear response– Floquet theory approach

Morimoto, Nagaosa, Sci. Adv. (2016)• Nonlinear optical effects of correlated electrons– Exciton shift current

Morimoto, Nagaosa, PRB (2016)– Shift current of magnons in multiferroic magnets

Morimoto, Nagaosa, PRB (2019)
Morimoto, Kitamura, Okumura, PRB (2021)– Shift current in phonons

Okamura et al. PNAS (2022)– Photocurrent of exciton polaritons
Morimoto, Nagaosa, PRB (2020)

How does electron correlation
modify/enhance shift current response?
Strongly correlated systems:
Photoexcitation of collective mode
 Photocurrent (hopefully giant)

How does electron correlation
modify/enhance shift current response?
Strongly correlated systems:
Photoexcitation of collective mode
 Photocurrent (hopefully giant)

Magnets: Spin systems

Multiferroics

Multiferroics and electromagnon

Magnetochiraldichroism

https://cems.riken.jp/jp/laboratory/estru
S. Kibayashi, et al., Nat. Commun. (2014).

Shift current from electromagnon?

YES!

Electric polarization in magnets

Tokura, Seki, Nagaosa, Rep. Prog. Phys. (2014)

Electronic picture• Superexchange in a Hubbard model Heisenberg coupling in a spin modelElectronic picture• Electric polarization in a Hubbard model withinversion breaking (staggered potential)
 Polarization in a spin model: Exchange striction

P𝑃𝑃 ∝ 𝑆𝑆𝑖𝑖 ⋅ 𝑆𝑆𝑖𝑖+1

Exchange striction in
Rice-Mele-Hubbard modelt−δt
Um-mt+δt

Exchange striction in
Rice-Mele-Hubbard modelt−δt
Um-mEt+δt

Exchange striction in
Rice-Mele-Hubbard modelt−δt
Um-mEt+δtCoupling to electric field in magnets
𝐻𝐻𝐸𝐸𝐸𝐸 = −𝐸𝐸 ⋅ 𝑃𝑃 = 𝐸𝐸Σ𝑖𝑖 Π𝑖𝑖 𝑆𝑆𝑖𝑖 ⋅ 𝑆𝑆𝑖𝑖+1

External electric field can induce magnon excitation through Pfrom exchange striction

Shift current

Π

q=0,
ω+δω

Π

q=0, −δωΠ

q=0, −ωMorimoto, Kitamura, Okumura, PRB (2021)

Shift current of magnons
Morimoto, Kitamura, Okumura, PRB (2021)

Dipole matrix
element

Shift vector ofmagnons

Optical excitation of magnons with electric polarization
 Dc current flow according to J=dP/dt

Light irradiationelectron flow

J1-J2 model• Cycloidal spin structure

J1-J2 model

Magnon shift current can be comparable ofelectronic shift current!

Spin shift current of magnons incollinear antiferromagnets

Magnon shift vector
Polarization operator(exchange striction)

Candidate materisl:
Multiferroic AF magnet
Mn2Mo3O8

Fujiwara, Kitamura, Morimoto, arXiv:2211.*****

Plan of this talk• Shift current as a geometric nonlinear response– Floquet theory approach

Morimoto, Nagaosa, Sci. Adv. (2016)• Nonlinear optical effects of correlated electrons– Exciton shift current

Morimoto, Nagaosa, PRB (2016)– Shift current of magnons in multiferroic magnets

Morimoto, Nagaosa, PRB (2019)
Morimoto, Kitamura, Okumura, PRB (2021)– Shift current in phonons

Okamura et al. PNAS (2022)– Photocurrent of exciton polaritons
Morimoto, Nagaosa, PRB (2020)

Phonon shift current
Soft phonons intypical ferroelectrics: BaTiO3Okamura et al., PNAS (2022)

Phonon shift current
Soft phonons intypical ferroelectrics: BaTiO3

Phonon excitation

Electronic excitation

Electric current flows even with photoabsorption much below
the electronic band gap!!
Okamura et al., PNAS (2022)

Phonon shift current
1†
𝐻𝐻RM = � � {𝑡𝑡 + (−1)𝑖𝑖 𝛿𝛿𝛿𝛿 }�𝑐𝑐𝑖𝑖+1𝑐𝑐𝑖𝑖 + ℎ. 𝑐𝑐. � + (−1)𝑖𝑖 𝑚𝑚𝑐𝑐𝑖𝑖† 𝑐𝑐𝑖𝑖 � ,2𝑖𝑖

𝐻𝐻el −ph =𝑔𝑔

� sin𝑘𝑘𝑘𝑘𝑐𝑐𝑘𝑘† 𝜎𝜎𝑦𝑦 𝑐𝑐𝑘𝑘 (𝑏𝑏 + 𝑏𝑏 † ) .

Ab initio calculation for phonon shift current√𝑉𝑉 𝑘𝑘Photoexcited phonons can also carry heat Heat current analog of phonon shift current:
Onishi, Morimoto, Nagaosa, arXiv:2201.02431

Plan of this talk• Shift current as a geometric nonlinear response– Floquet theory approach

Morimoto, Nagaosa, Sci. Adv. (2016)• Nonlinear optical effects of correlated electrons– Exciton shift current

Morimoto, Nagaosa, PRB (2016)– Shift current of magnons in multiferroic magnets

Morimoto, Nagaosa, PRB (2019)
Morimoto, Kitamura, Okumura, PRB (2021)– Shift current in phonons

Okamura et al. PNAS (2022)– Photocurrent of exciton polaritons
Morimoto, Nagaosa, PRB (2020)

Nonlinearresponses

Topology
Geometry

Nonequilibirum
systems
Polaritons

Photovolatic effects

Driven systems

Nonlineartransport

High harmonicgeneration

How does electron correlation andnonequilibrium states modify/enhance shift
current response?

Exciton polaritons

Do polaritons support photocurrent like excitons?
Byrnes et al. Nat. Phys. (2014)

Photocurrent from polariton excitations
Polariton excitations:ExcitonCavity photon−+

Morimoto & Nagaosa, PRB (2020)

Photocurrent from polariton excitations
Polariton excitations:Photocurrent:

Photocurrent is proportional to polariton density
difference between two branches.
Morimoto & Nagaosa, PRB (2020)

Photocurrent from polariton excitations

Microscopic derivation from Feynman diagram Similar mechanism to shift current
Expressions for coupling constants

J=Photocurrent from polariton excitations

Time reversal symmetry for g1*g2 is pure imaginary Polariton photocurrent requires T symmetry breaking in addition to inversion symmetry breaking

cf. J ∝polariton density

Possible observation in transition metal
dichalcogenides
Effective two band model:

Liu et al., PRB (2013)
Contour plot of energy dispersion

Two ways to break TRS:(i) Application of magnetic field to induce
valley asymmetry(ii) Cavity photons of circular polarization
Order estimate:forJ ~ 30nA
for 10um polariton condensatePhotocurrent with time reversal symmetry?
Role of dissipation?

Dissipative polariton condensates
Gross-Pitaevskii eq. with non-Hermitian Hamiltonian

Hanai, Edelman, Ohashi, Littlewood , PRL (2019)

Photocurrent of polariton condensates
Eigenstates of non-Hermitian Hamiltonian

Photocurrent

Photocurrent of polariton condensates

Time reversal symmetry for g1*g2 is pure imaginary

Photon decay induces polariton photocurrent even with TRS!

Photocurrent of polariton condensates
ExctionPolariton

Cavity photon

Polarization−−++

Photon decay induces polariton photocurrent even with TRS!

Summary• Shift current is a geometric nonlinear response– Exciton shift current
Morimoto, Nagaosa, Sci. Adv. (2016)
Morimoto, Nagaosa, PRB (2016)• Shift current of correlated and open electron systems
Magnons

Morimoto, Nagaosa, PRB (2019)
Morimoto, Kitamura, Okumura, PRB (2021)

Phonons

Okamura et al. PNAS (2022)

Polaritons

Morimoto, Nagaosa, PRB (2020)
Morimoto, Nagaosa, in prep.

Outlook
Geometric nonlinear optical effects
in strongly correlated and nonequilibrium systems

Geim et al., Nature (2013)

Superconductors:

Mott insulators:

Heterostructures: