non-linear optics by means of dynamical berry phase

Diapositiva 1

Non-linear optics
by means of
dynamical Berry phase

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

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

From Gauss's law:

Materials equations:

Electric Displacement

Electric Field

Polarization

In general:

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)

and even more ..

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

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

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)

The problem
of bulk polarization

How to define polarization as a bulk quantity?

Polarization for isolated systems is well defined

1)

2)

3)

Bulk polarization, the wrong way 1

1)


2)

Unfortunately Clausius-Mossotti
does not work for solids because
WF are delocalized


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!!


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

An exact formulation exists also for correlated wave-functions
R. Resta., Phys. Rev. Lett. 80, 1800 (1998)

From Polarization to the
Equations of Motion

It is an object difficult to calculate numerically
due to the gauge freedom of the Bloch functions

I. Souza, J. Iniguez and D. Vanderbilt, Phys. Rev. B 69, 085106 (2004)

Our computational setup

As expected we reproduce results obtained from linear response theory:

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

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

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

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
Phys. Rev. B 87, 161403(R) (2013)Observation of intense second harmonic generation from MoS2 atomic crystals
Phys. Rev. B 87, 201401(R) (2013)

Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation
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.

gap at the K point; a larger peak around 1.5 eV, which originates from two-photon resonances with transitions along the high symmetry axis between and K where the highest valence and lowest conduction bands are flat and there is a high density of states; a broad structure between 23.5 eV which originates from one-photon res-

Nonlinear optics in semiconductors
from first-principles real-time simulations

TDSE

What next?

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

SHG in liquid-liquid interfaces, nanostructures

Dissipation, coupling with phonons..... luminescence, light emission,strong fields...

Open questions?

Dissipative effects? How?

Coupling dynamical Berry phase with Green's functions?

Coupling dynamical Berry phase with density matrix hierarchy
equations, BBGKY?

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

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, PRB 84, 245110 (2011)

2) Nonlinear optics from ab-initio by means of the dynamical Berry-phase
C. Attaccalite and M. Grning. http://arxiv.org/abs/1309.4012 3) Second Harmonic Generation in h-BN and MoS2 monolayers: the role of electron-hole interaction
M. Grning and C. Attaccalite submitted to NanoLetters

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

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

non-linear optics by means of dynamical berry phase

Education