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

1. Non­linear response of two dimensional
crystals and layered materials
Claudio Attaccalite
CNRS/CINaM, Aix­Marseille Universite (FR)
Montpellier

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. Why non­linear optics?
..research..
Linear
Science, 2014, vol. 344, no 6183, p. 488­490
Non­linear(COLOR)Non­linear(BW)

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. 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. 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. An example for
linear response:
C. Attaccalite, M. Gruning, A. Marini, Phys. Rev. B 84, 245110 (2011)

8. Non­linear response
from real­time simulations
Real-time dynamics
Polarization
SHG

9. Second Harmonic Generation
in h­BN monolayer
IPA

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. 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. 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. 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. 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. 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. 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. 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. 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. Experimental results

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. Experimental results 2/2
Hexagonal boron nitride is an indirect band­gap
semiconductor
G. Cassabois et al., Nature Photonics, 10, 262 (2016)

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. 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. 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. TPA coefficients
from real­time simulations
Richardson
extrapolation
Real-time dynamicsReal-time dynamics
Polarization

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).