Role of excitonic effects in nonlinear optical properties of 2D materials

1. Role of excitonic effects in nonlinear optical
properties of 2D materials
Myrta Grüning, Queen's University Belfast - Northern Ireland
!?!
2 4 6 8 10
with e-h
without e-h
Claudio Attaccalite - CNRS Marseille - France
OSI 12 - 27 June 2017
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+ ...
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2. Interest of nonlinear optics in 2D materials:
Second Harmonic Generation from
Artificially Stacked Transition Metal
Dichalcogenide Twisted Bilayers
ACS Nano 8, 2951 (2014)
Ultra-strong nonlinear optical processes
and trigonal warping in MoS2 layers
arXiv:1608.04101 (2016)
see as well 2D Mater. 4 (2017) 011006
Imagining:
Relatively strong (technological):
Scientific Reports 4: 5530 (2014)
Extraordinary Second Harmonic Generation
in Tungsten Disulfide Monolayers Giant two-photon absorption
in monolayer MoS2
Laser & Photonics Reviews (2015)

3. Several experiments argued the importance of
excitonic effects in nonlinear optics of 2D materials:
Excitonic effects important role in
SHG edge detection in MoS2 flakes
[Science 344, (2014) 488]

4. Key questions explored in this talk:
2 4 6 8 10
with e-h
without e-h
Calculated at the GW+
Bethe-Salpeter equation
How to get a feasible approach
equivalent to GW+BSE
Are excitonic effects important
for nonlinear properties?

5. Dynamical polarization obtained from real-time evolution
of natural orbitals
I. Souza et al, PRB 69, 085106 (2004)
light-matter
interaction
Energy functional
corresponding to
zero-field Hamitonian
Natural orbitals:
diagonalize the 1-e
Green's function at each t

6. Dynamical polarization as a Berry-phase (consistent with PBC):
I. Souza et al, PRB 69, 085106 (2004)
Euler-Lagrange EOMs:
Position operator consistent with PBC:
expressed in terms of

7. How do we approximate our effective Hamiltonian:
Constraint: Total wave-function can be written as a single Slater potential
Hamiltonian of the unperturbed system
(Kohn-Sham)
Independent particle level of approximation (IPA)

8. How do we approximate our effective Hamiltonian:
Constraint: Total wave-function can be written as a single Slater potential
Scissor operator:
QP renormalization effects
on the unperturbed system
Quasiparticle approximation (QPA or IPA+GW)

9. How do we approximate our effective Hamiltonian:
Constraint: Total wave-function can be written as a single Slater potential
Crystal local effects:
sensible to inhomogeneity
of the system
+ -
+
+
++
-
-
--
Hartree-only approximation(TDH)

10. How do we approximate our effective Hamiltonian:
Constraint: Total wave-function can be written as a single Slater potential
Static screened HF self-energy:
TD renormalization effects
on QP energies
on optical excitations (excitonic effects)
Screened Hartree-Fock (TDSHF)

11. By changing input/postprocessing in our computational set-up
we can obtain different (non)linear optical properties:
in
out
post-processing:
Obtain by
Fourier transform
in
out
Solve Euler-Lagrange equations:
Kohn-Sham:
C. Attaccalite, M. G. Phys. Rev. B 88, 235113 (2013)
Absorption

12. In the linear response limit the approach reduces
to GW+BSE results
0
10
20
30
40
50
60
Absorption
Polarization
4 5 6 7 8 9 10
Energy (eV)
0
10
20
30
40
50
60
Absorption
0 5 10 15
Time (fs)
Polarization
post-processing
TD-Hartree:
TD-BSE:
Exp
BSE
TD-BSE
Exp
RPA
TD-Hartree
C. Attaccalite, M.G, A Marini PRB 84, 245110 (2011)

13. By changing input/postprocessing in our computational set-up
we can obtain different (non)linear optical properties:
in
out
post-processing:
Obtain by
Fourier matrix inversion:
in
out
Solve Euler-Lagrange equations:
Kohn-Sham:
For do:
e.g.
C. Attaccalite, M. G. Phys. Rev. B 88, 235113 (2013)

14. In h-BN monolayer dramatic effect of e-h interaction on SHG
Intensities doubled at excitonic resonances
@810nm exp [Y. Li et al NL(2013)]: ~20 pm/V
ours : ~40 pm/V
0.2
0.4
0.6
0.8
1.0
(a): IPA
Arb.units
(b)
0.2
0.4
0.6
0.8
1.0
(c): IPA+GW
Arb.units
(d)
0.0
0.4
0.8
1.2 (e): TDSHF
2 4 6 8 10
Energy (eV)
Arb.units
(f)
M. Grüning and C. Attaccalite, Phys. Rev. B 89(R), 081102 (2014)
E: Phys. Rev. B 90, 199901 (2014) .
IPA results validated against Guo&Lin (PRB 72,075416)
EFFECTIVE
THICKNESS
0.33 nm

15. In MoS2 we found SHG@810nm of ~1 nm/V
(~3 orders of magnitude > nonlinear crystals)
S
Mo
EFFECTIVE
THICKNESS
0.615 nm
M. Grüning and C. Attaccalite, Phys. Rev. B 89(R), 081102 (2014).
E: Phys. Rev. B 90, 199901 (2014); agreement with Pedersen et al PRB 2014
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Energy (eV)
0.5
1.0
1.5
2.0
2.5
3.0
3.5 TDSHF
IPA
EXPERIMENT x8
MoS2
Comparison with experimental estimates:
+ > exp: ~0.1 nm/V (Malard PRB 2013; Li NL 2013);
+ << ~100 nm/V (Kumar PRB 2013)
+ ~ Janisch (SciRep 2014) for WS2
Again e-h effects 2x enhancement (important to include local fields as well)

16. Similar results found for SHG intensity in SiC, GaN and ZnO:
C. Attaccalite, A. Nguerc, E.Cannuccia, M.G PCCP (2015)
* Enhancement at excitonic resonances
* Transparent in the 'interesting' frequency region
* Intensities of 0.1 -1 nm/V (smaller than MoS2,
but still larger than conventional NL crystals)

17. In THG of 1D nanostructures many-body effects also key:
C. Attaccalite, E. Cannuccia M. G. PRB 95.125403
# reduction of factor
3-4@ excitonic resonances
# intensity redistribution 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Laser frequency (eV)
0
2
4
6
8
10
12
KS particles
QP particles
QP particles + e-h interaction
ThirdHarmonicintensity(esux10-8
)

18. By changing input/postprocessing in our computational set-up
we can obtain different (non)linear optical properties:
in
out
post-processing:
Obtain by
Fourier matrix inversion:
in
out
Solve Euler-Lagrange equations:
Kohn-Sham:
For do:
e.g.
C. Attaccalite, M. G. work in progress
+ extrapolation from
different intensities
For do:

19. Preliminary results for 2 photon absorption in 2D h-BN:
C. Attaccalite, M. G. in progress
2.5 3.0 3.5 4.0 4.5 5.0
Energy (eV)
0.0
0.2
0.4
0.6
0.8
1.0
χ(3)
(−ω;ω,−ω,ω)[esu]
1e 10
GW+BSE (0.025 eV)
(0.1 eV)QPA

20. Role of excitonic effects in nonlinear optical
properties of 2D materials
in
out
in
out
e.g.
e-
h+
with e-h interaction
e-
h+
independent e-h pairs
Real-time approach to calculate
non linear optics which includes
excitonic effects
(and consistent with PBC)
Role of excitonic effects for SHG
in 2D: enhancement at excitonic
resonances
(and THG in 1D ...)
nonlinear module of yambo-code
https://github.com/attacc/lumen
OSI 12 - 27 June 2017