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Ab-initio real-time spectroscopy: application to non-linear optics

C. Attaccalite, Institut Nel Grenoble
M. Grning, Queen's University, Belfast

Etant donne que celui-ici est un seminaire theorique je vais commencer avec quelques formules

What is it non-linear optics?

References:

Nonlinear Optics and Spectroscopy
The Nobel Prize in Physics 1981
Nicolaas Bloembergen

First experiments on linear-optics by P. Franken 1961

Materials equations:

Electric Field

Electric Displacement

In general:

Polarization

Imaginez prendre un solide et le plonger dans un champ lectrique

Le champ lectrique modifie le diples a lintrieur du matriel et le diples gnrent, un autre champ qui se oppose a le champ extrieur, la polarisation

Si vous clairez un objet avec de la lumire rouge et vous voyiez en transmission ou rflexion un couleur diffrent, a c'est de l'optique non linaire

C'est pour ca que la premiere experience d'optique non lineaire remont a le 1961 par Frenkel

The first motivation to
study non-linear optics is
in your (my) hands

This is a red laser

This is not a green laser!!

How it works
a green laser pointer

To see invisible excitations

The Optical Resonances in Carbon
Nanotubes Arisefrom ExcitonsFeng Wang, et al.Science 308, 838 (2005);

Probing symmetries and crystal structures

Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation Nano Lett. 13, 3329 (2013)

Second harmonic microscopy of MoS2 PRB 87, 161403 (2013)

and even more ..

Second harmonic generating
(SHG) nanoprobes for
in vivo imaging
PNAS 107, 14535 (2007)

Photon entanglement

Spectroscopy:
experimental point of view

Spectroscopy:
theoretical point of view

Linear response theory:
+ fast approach
+ analysis of the results

- difficult for non-linear response
- limited to equilibrium phenomena

Par contre avec cet approche est difficile a calculer la rponse au-de la de la linaire et aussi traiter de phnomne hors dquilibre

Real-time approach

Time-dependent Schrodinger equations

External perturbation
a uniform electric field

A partir de la polarisation en temps rel je peux extraire les diffrentes polarisabilites

Non-linear optics

Non-linear optics can calculated in the same way of TD-DFT as it is done in OCTOPUS or RT-TDDFT/SIESTA codes.

Quasi-monocromatich-field

p-nitroaniline

Y.Takimoto, Phd thesis (2008)

Cette approche tais dj utiliser avec succs sur des molcules.

Mais aprs le 2008 personne a jamais essayer a faire la mme chose pour de solides

La raison est trs simple.

The problem of bulk polarization

How to define polarization as a bulk quantity?

Polarization for isolated systems is well defined

Calculer la moyenne de l'operatore dipole sur tout le solide

Bulk polarization, the wrong way

The bulk polarization!!
King-Smith-Vanderbilt formula

King-Smith and Vanderbilt formula
Phys. Rev. B 47, 1651 (1993)

Berry's phase !!

it is a bulk quantity

time derivative gives the current

reproduces the polarizabilities at all orders

is not an Hermitian operator

Our computational setup

Let's add some correlation in 4 steps

We start from the DFT
(Kohn-Sham) Hamiltonian:

universal, parameter free approach

1)

2)

4)

Renormalization of the band
structure due to correlation (GW)

Electron-hole interaction

Charge fluctuations
(time-dependent Hartree)

3)

fluctuation de densit gnre une champ lectrique a travers lquation de Poisson

We reproduce results obtained from linear response theory:

C. Attaccalite, M. Gruning, A. Marini, Phys. Rev. B 84, 245110 (2011)

SHG in bulk semiconductors: SiC, AlAs, CdTe

AlAs

SiC

CdTe

E. Ghahramani, D. J. Moss, and J. E. Sipe,
Phys. Rev. B 43, 9700 (1991)

I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito,
J. Opt. Soc. Am. B 14, 2268 (1997) J. I. Jang, et al.
J. Opt. Soc. Am. B 30, 2292 (2013)

E. Luppi, H. Hbener, and V. Vniard Phys. Rev. B 82, 235201 (2010)

The main objective of this section is to validate the computational approach described in Secs. II and III against results in the literature for SHG obtained by the response theory based approach in frequency domain.

he minor discrepancies between the curves are due to the different choice for the k-grid used for integration in momentum space: we used a -centered uniform grid (for which we can implement the numerical derivative) whereas Ref. 6 used a shifted grid.

In order to interpret those spectra, note that SHG resonances occur when eitherw or 2w equals the difference between two single-particle energies. Then one can distinguish two energy region: below the single-particle minimum direct gap where only resonances at 2 can occur, and above where both resonance can occur.Local-field reduce from 15% to 30%

Cadmium telluride

THG in silicon

D. J. Moss, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 41, 1542 (1990)

D. Moss, H. M. van Driel, and J. E. Sipe, Optics letters 14, 57 (1989)

For ener-gies below 1 eV, our QPA spectra is in good agreementwith results obtained from semi-ab-initio tight-bindingand with the experimental measurement.

For higher energies our spectra are less structured with respect both the semi-ab-initio tight-binding and the experiment, in particular missing the peak at 1-1.1 eV. The intensities of the spectra however are more consistent with the ex-periment than the previous theoretical results

Local-fields and excitonic effects
in h-BN monolayer

IPA

IPA + GW + TDSHF

independent particles
+quasi-particle corrections
+time-dependent Hartree (RPA)
+screend Hartree-Fock (excitonic effects)

In fact, the IPA+GW shows two peaks: the first at about 4 eV is the shifted two-photon resonances peak which is attenuated by 40% with respectto IPA [Fig. 2 (a)]; the second very pronounced peak at about 8 eV comes from the interference of one-photon resonances and two-photon resonances.

MoS2 single-layer

Second harmonic microscopy of monolayer MoS2
N. Kumar et al.
Phys. Rev. B 87, 161403(R) (2013)Observation of intense second harmonic generation from MoS2 atomic crystals
L. Malard et al.
Phys. Rev. B 87, 201401(R) (2013)

Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation
Y. Li et al.
NanoLetters, 13, 3329 (2013)

MoS2 differs from h-BN in several aspects. First, while the h-BN has an indirect minimum band gap as its bulk counterpart, in MoS2 an indirect-to-direct bandgap transition occurs passing from the bulk to the monolayer due to the vanishing interlayer interaction. Second, spin-orbit coupling plays an important role in this material, splitting the top valence bands, as visible from the absorption spectrum, presenting a double peak at the onset.7 Third, Mo and S atoms in the MoS2 monolayer are on different planes resulting in a larger inhomogeneity than for theH-BN.Second harmonic microscopy of monolayer MoS2
N. Kumar et al. [USA 2-3 order larger]
Phys. Rev. B 87, 161403(R) (2013)Observation of intense second harmonic generation from MoS2 atomic crystals
L. Malard et al. [Brasil 21x smaller]
Phys. Rev. B 87, 201401(R) (2013)

Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation
Y. Li et al. [ratio BN/MoS2 correct]
NanoLetters, 13, 3329 (2013)

What next?

SFG, DFG, optical rectification, four-wave mixing,
electron-optical effect, Fourier spectroscopy, etc....

SHG in liquid-liquid interfaces, nanostructures

Pump and probe experiments

Dissipation, coupling with phonons.....
time resolved luminescence....

Acknowledgement

Myrta Grning,
Queen's University Belfast

Reference:

1) Real-time approach to the optical properties of solids and nanostructures:
Time-dependent Bethe-Salpeter equation
C. Attaccalite, M. Gruning and A. Marini PRB 84, 245110 (2011)

2) Nonlinear optics from ab-initio by means of the dynamical Berry-phase
C. Attaccalite and M. Grning, Phys. Rev. B 88, 235113 (2013) 3) Second Harmonic Generation in h-BN and MoS2 monolayers: the role of electron-hole interaction
M. Grning and C. Attaccalite, Phys. Rev. B 89, 081102(R) (2014)

The King-Smith and Vanderbilt formula

We introduce the Wannier functions
Blount, 1962

We express the density in terms of Wannier functions

Polarization in terms
of Wannier functions [Blount 62]

How to perform k-derivatives?

Solutions: In mathematics the problem has been solved by using
second, third,... etc derivatives
SIAM, J. on Matrix. Anal. and Appl. 20, 78 (1998)

Global-gauge transformation
Phys. Rev. B 76, 035213 (2007)

Phase optimization
Phys. Rev. B 77, 045102 (2008)

Covariant derivative
Phys. Rev. B 69, 085106 (2004)

Wrong ideas on velocity gauge

In recent years different wrong papers using velocity gauge
have been published (that I will not cite here) on:
1) real-time TD-DFT
2) Kadanoff-Baym equations + GW self-energy
3) Kadanoff-Baym equations + DMFT self-energy

Length gauge:

Velocity gauge:

Analitic demostration:K. Rzazewski and R. W. Boyd,
Journal of modern optics 51, 1137 (2004)
W. E. Lamb, et al.
Phys. Rev. A 36, 2763 (1987)

Well done velocity gauge:M. Springborg, and B. KirtmanPhys. Rev. B 77, 045102 (2008)
V. N. Genkin and P. M. Mednis
Sov. Phys. JETP 27, 609 (1968)

Post-processing real-time data

P(t) is a periodic function of period TL=2p/wL

pn is proportional to cn by the n-th order of the external field

Performing a discrete-time signal
sampling we reduce the problem to
the solution of a systems of linear equations

SHG in frequency domain

King-Smith and Vanderbilt formula
Phys. Rev. B 47, 1651 (1993)

The idea of Chen, Lee, Resta.....

Berry's phase and Green's functions

Z. Wang et al. PRL 105, 256803 (2010)
Chen, K. T., & Lee, P. A. Phys. Rev. B, 84, 205137 (2011)
R. Resta, www-dft.ts.infn.it/~resta/sissa/draft.pdf

A bit of theory

Which is the link between
Berry's phase and SHG?

The Berry phase

IgNobel Prize (2000) together
with A.K. Geim
for flying frogs

A generic quantum Hamiltonian with a parametric dependence

phase difference between two ground eigenstates at two different x

cannot have
any physical meaning

Berry, M. V. . Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 392(1802), 45-57 (1984).

...connecting the dots...

the phase difference of a closed-path is gauge-invariant
therefore is a potential physical observable

g is an exotic observable which cannot be expressed in terms
of any Hermitian operator

Berry's geometric phase

Berry's Phase and Geometric Quantum Distance:
Macroscopic Polarization and Electron Localization
R. Resta, http://www.freescience.info/go.php?pagename=books&id=1437

Berry's connection

Berry's phase exists because the system is not isolated
x is a kind of coupling with the rest of the Universe

In a truly isolated system, there can be no manifestation of
a Berry's phase

Examples of Berry's phases

Molecular AB effect

Aharonov-Bohm effect

Correction to the Wannier-Stark ladder spectra of semiclassical electrons

Ph. Dugourd et al.
Chem. Phys. Lett. 225, 28 (1994)

R.G. Sadygov and D.R. Yarkony
J. Chem. Phys. 110, 3639 (1999)

J. Zak, Phys. Rev. Lett. 20, 1477 (1968)

J. Zak, Phys. Rev. Lett. 62, 2747 (1989)

Let's add some correlation in 4 steps

1) We start from the Kohn-Sham Hamiltonian:

universal, parameter free approach

2) Single-particle levels are renormalized within the G0W0 approx.

3) Local-field effects are included in the response function

Time-Dependent Hartree

4) Excitonic effects included by means of the Screened-Exchange

Bulk polarization, the wrong way 3

3)

intra-bands terms undefined

diverges close to the bands crossing

ill-defined for degenerates states

Electrons in a periodic system

Born-von-Karman
boundary conditions

Bloch orbitals solution of
a mean-field Schrdinger eq.

Bloch functions
u obeys to periodic boundary conditions

We map the problem in k-dependent Hamiltonian
and k-independent boundary conditions

k plays the role of
an external parameter

What is the Berry's phase related to k?

King-Smith and Vanderbilt formula
Phys. Rev. B 47, 1651 (1993)

Berry's connection
again!!