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Non­linear response of two dimensional 
crystals and layered materials
Claudio Attaccalite 
CNRS/CINaM, Aix­Marseille Univ...
What is it non­linear optics?
P(r ,t)=P0+χ
(1)
E+χ
(2)
E
2
+O(E
3
)
First experiments on linear­optics 
by P. Franken 1961...
Why non­linear optics?
..research..
Linear
Science, 2014, vol. 344, no 6183, p. 488­490
Non­linear(COLOR)Non­linear(BW)
Probing number of layers
Probing stacking order
Stacking Order Dependent Second Harmonic Generation and Topological Defect...
To see “invisible” excitations
The Optical Resonances in 
Carbon 
Nanotubes Arise
from Excitons
Feng Wang, et al.
Science ...
Non­linear response 
from real­time simulation
Nonlinear optics from an ab-initio approach by means of the dynamical Berry...
An example for 
linear response:
C. Attaccalite, M. Gruning, A. Marini, Phys. Rev. B 84, 245110 (2011)
Non­linear response  
from real­time simulations
Real-time dynamics
Polarization
SHG
Second Harmonic Generation
 in h­BN monolayer
IPA
IPA
IPA + GW + TDSHF
independent particles
+quasi-particle corrections
+time-dependent Hartree (RPA)
+screend Hartree-Fock...
IPA
IPA + GW + TDSHF
independent particles
+quasi-particle corrections
+time-dependent Hartree (RPA)
+screend Hartree-Fock...
MoS2 
single­layer
10) Second harmonic microscopy of monolayer MoS2
Phys. Rev. B 87, 161403(R) (2013)
100) Probing Symmetr...
Nature of excitons in 
single­layer h­BN
Tight-binding amplitudes for the two
degenerate states, symmetric and
antisymmetr...
Nature of excitons in bulk h­BN
Excitons in van der Waals materials: From monolayer to bulk hexagonal
boron nitride
J. Kos...
Two­photon absorption
Monolayer h­BN 
Two-photon absorption in two-dimensional materials: The case of hexagonal boron nitr...
Two­photon absorption
Two-photon absorption in two-dimensional materials: The case of hexagonal boron nitride
C. Attaccali...
Tight­binding modeling 1/2 
Monolayer h­BN  1 - Photon
The excitonic states can then
be classified according to the
repres...
Tight­binding modeling 2/2 
2 – Photon
In the presence of a symmetry
centre odd (even) states are
one(two)-photon allowed....
Experimental results
Experimental results 1/2
Giant Enhancement of the Optical Second-Harmonic Emission of WSe2 Monolayers
by Laser Excitation ...
Experimental results 2/2
Hexagonal boron nitride is an indirect band­gap 
semiconductor
G. Cassabois et al.,  Nature Photo...
●
Real-time simulations are a powerful tool to
study non-linear response of solids
●
Second Harmonic Generation is not zer...
Acknowledgments 
François Ducastelle Hakim AmaraMyrta Grüning
References 
Two-photon absorption in two-dimensional materia...
Let's add some correlation in 4 steps
We start from the DFT
(Kohn-Sham) Hamiltonian:
hk
universal, parameter free
approach...
TPA coefficients 
from real­time simulations
Richardson
extrapolation
Real-time dynamicsReal-time dynamics
Polarization
… but continue to be rediscovered... 
“Optical selection rule of excitons in gapped chiral
fermion systems,”
Phys. Rev. Le...
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Non-linear response of two dimensional crystals and layered materials

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Non-linear response of two-dimensional crystals and layered materials

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Non-linear response of two dimensional crystals and layered materials

  1. 1. Non­linear response of two dimensional  crystals and layered materials Claudio Attaccalite  CNRS/CINaM, Aix­Marseille Universite (FR)  Montpellier
  2. 2. What is it non­linear optics? P(r ,t)=P0+χ (1) E+χ (2) E 2 +O(E 3 ) First experiments on linear­optics  by P. Franken 1961 Ref: Nonlinear Optics and  Spectroscopy  The Nobel Prize in Physics 1981 Nicolaas Bloembergen
  3. 3. Why non­linear optics? ..research.. Linear Science, 2014, vol. 344, no 6183, p. 488­490 Non­linear(COLOR)Non­linear(BW)
  4. 4. Probing number of layers Probing stacking order Stacking Order Dependent Second Harmonic Generation and Topological Defects in h‐BN Bilayers Nano Lett. 13, 5660 (2013) Probing stacking order Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation Nano Lett. 13, 3329 (2013)
  5. 5. To see “invisible” excitations The Optical Resonances in  Carbon  Nanotubes Arise from Excitons Feng Wang, et al. Science 308, 838 (2005); ..research.. Right interpretation of the experiment “Selection rules for one-and two-photon absorption by excitons in carbon nanotubes,” E. B. Barros et al. PRB 73, 241406 (2006).
  6. 6. Non­linear response  from real­time simulation Nonlinear optics from an ab-initio approach by means of the dynamical Berry phase C. Attaccalite, M. Grüning, PRB, 88(23), 235113. (2013)
  7. 7. An example for  linear response: C. Attaccalite, M. Gruning, A. Marini, Phys. Rev. B 84, 245110 (2011)
  8. 8. Non­linear response   from real­time simulations Real-time dynamics Polarization SHG
  9. 9. Second Harmonic Generation  in h­BN monolayer IPA
  10. 10. IPA IPA + GW + TDSHF independent particles +quasi-particle corrections +time-dependent Hartree (RPA) +screend Hartree-Fock (excitonic effects) Second Harmonic Generation  in h­BN monolayer
  11. 11. IPA IPA + GW + TDSHF independent particles +quasi-particle corrections +time-dependent Hartree (RPA) +screend Hartree-Fock (excitonic effects) zero in the hydrogen model Second Harmonic Generation  in h­BN monolayer
  12. 12. MoS2  single­layer 10) Second harmonic microscopy of monolayer MoS2 Phys. Rev. B 87, 161403(R) (2013) 100) Probing Symmetry Properties of Few­Layer MoS2  and h­ BN by Optical Second­Harmonic Generation NanoLetters, 13, 3329 (2013) 1000) Observation of intense second harmonic generation  from MoS2  atomic crystals Phys. Rev. B 87, 201401(R) (2013)
  13. 13. Nature of excitons in  single­layer h­BN Tight-binding amplitudes for the two degenerate states, symmetric and antisymmetric with respect to the y- axis. Excitons in boron nitride single layer T. Galvani et al., Phys. Rev. B 94, 125303 (2016) Schematic splitting scheme of the 2p levels. (Lowest states are degenerate, one bright and one dark)
  14. 14. Nature of excitons in bulk h­BN Excitons in van der Waals materials: From monolayer to bulk hexagonal boron nitride J. Koskelo, et al, Phys. Rev. B 95, 035125 (2017) Combinations with respect to the exchange of the e-h pair between two inequivalent layers The two lowest excitons Third and fourth excitons Splitting due to the interlayer hopping
  15. 15. Two­photon absorption Monolayer h­BN  Two-photon absorption in two-dimensional materials: The case of hexagonal boron nitride C. Attaccalite, M. Grüning, H. Amara, S. Latil, and F. Ducastelle Phys. Rev. B 98, 165126 (2018)
  16. 16. Two­photon absorption Two-photon absorption in two-dimensional materials: The case of hexagonal boron nitride C. Attaccalite, M. Grüning, H. Amara, S. Latil, and F. Ducastelle Phys. Rev. B 98, 165126 (2018) Monolayer h­BN  Bulk h­BN 
  17. 17. Tight­binding modeling 1/2  Monolayer h­BN  1 - Photon The excitonic states can then be classified according to the representations of the C3v point group. Among the three representations A1, A2 and E, only the two-dimensional representation E is optically active. 2 - Photon In the discrete which indicates also that all excitons are in principle bright. We have seen in particular that the oscillator strength for the ground state 1s exciton is very strong.
  18. 18. Tight­binding modeling 2/2  2 – Photon In the presence of a symmetry centre odd (even) states are one(two)-photon allowed. In the case of the AA’ stacking combining both processes can be used to discriminate between the components of the Davydov doublets. Bulk h­BN 
  19. 19. Experimental results
  20. 20. Experimental results 1/2 Giant Enhancement of the Optical Second-Harmonic Emission of WSe2 Monolayers by Laser Excitation at Exciton Resonances Phys. Rev. Lett. 114, 097403 (2015) Probing the 1s state in WS2
  21. 21. Experimental results 2/2 Hexagonal boron nitride is an indirect band­gap  semiconductor G. Cassabois et al.,  Nature Photonics, 10, 262 (2016)
  22. 22. ● Real-time simulations are a powerful tool to study non-linear response of solids ● Second Harmonic Generation is not zero for the 1s exciton in two-dimensional hexagonal crystals ● Two-photon absorption can probe dark excitons in h-BN Conclusions {at zero momentum}
  23. 23. Acknowledgments  François Ducastelle Hakim AmaraMyrta Grüning References  Two-photon absorption in two-dimensional materials: The case of h-BN C. Attaccalite, M. Grüning, H. Amara, S. Latil, and F. Ducastelle Phys. Rev. B 98, 165126 (2018) Second harmonic generation in h-BN and MoS2 monolayers: Role of electron-hole interaction M. Grüning, C. Attaccalite, PRB 89, 081102 (2013) Sylvain Latil
  24. 24. Let's add some correlation in 4 steps We start from the DFT (Kohn-Sham) Hamiltonian: hk universal, parameter free approach 1) 2) 4) Renormalization of the band structure due to correlation (GW) Electron-hole interactionCharge fluctuations (time-dependent Hartree) Δρ→ΔV H 3)
  25. 25. TPA coefficients  from real­time simulations Richardson extrapolation Real-time dynamicsReal-time dynamics Polarization
  26. 26. … but continue to be rediscovered...  “Optical selection rule of excitons in gapped chiral fermion systems,” Phys. Rev. Lett. 120, 077401 (2018). “Unifying optical selection rules for excitons in two- dimensions: Band topology and winding numbers,” Phys. Rev. Lett. 120, 087402 (2018) Part of the selection rules were  already published in the literature “Optical selection rule of excitons in gapped chiral fermion systems,” PRB 91 075310 (2015) “Nonlinear optical selection rule based on valley- exciton locking in monolayer ws2,” Light: Science &Amp; Appli-cations 4, e366 (2015). “Optical selection rules for excitonic rydberg series in the massive dirac cones of hexagonal two- dimensional materials,” Phys. Rev. B 95, 125420 (2017). “Intrinsic exciton-state mixing and non-linear optical properties in transition metal dichalcogenide monolayers,” Phys. Rev. B 95, 035311 (2017).

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