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)
Bulk polarization, the wrong way 2
2)
Unfortunately Clausius-Mossotti
does not work for solids because
WF are delocalized
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!!
King-Smith and Vanderbilt formula
.. 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
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
Top Related