The document discusses light-induced real-time electron dynamics in solids. It begins by introducing an effective Hamiltonian approach that accounts for band structure renormalization, charge fluctuations, and electron-hole interactions. It then discusses linear response simulations using plane waves and norm-conserving pseudopotentials. Challenges with different gauges in the presence of non-local operators are also covered. The document concludes by mentioning work on non-linear response, exciton relaxation mechanisms, and alternatives to the Berry phase for calculating polarization in insulators.
In this second lecture, I will discuss how to calculate polarization in terms of Berry phase, how to include GW correction in the real-time dynamics and electron-hole interaction.
Neutral Electronic Excitations: a Many-body approach to the optical absorptio...Claudio Attaccalite
Neutral Electronic Excitations: a Many-body approach to the optical absorption spectra.
Introduction to Bethe-Salpeter equation and linear response theory.
We present an ab-initio real-time based computational approach to nonlinear optical properties in Condensed Matter systems. The equation of mot ions, and in particular the coupling of the electrons with the external electric field, are derived from the Berry phase formulation of the dynamical polarization. The zero-field Hamiltonian includes crystal local field effects, the renormalization of the independent particle energy levels by correlation and excitonic effects within the screened Hartree- Fock self-energy operator. The approach is validated by calculating the second-harmonic generation of SiC and AlAs bulk semiconductors : an excellent agreement is obtained with existing ab-initio calculations from response theory in frequency domain . We finally show applications to the second-harmonic generation of CdTe the third-harmonic generation of Si.
Reference :
Real-time approach to the optical properties of solids and nanostructures : Time-dependent Bethe-alpeter equation Phys. Rev. B 84, 245110 (2011)
Nonlinear optics from ab-initio by means of the dynamical Berry-phase
C. Attaccalite and M. Gruning Phys. Rev. B 88 (23), 235113 (2013)
In this second lecture, I will discuss how to calculate polarization in terms of Berry phase, how to include GW correction in the real-time dynamics and electron-hole interaction.
Neutral Electronic Excitations: a Many-body approach to the optical absorptio...Claudio Attaccalite
Neutral Electronic Excitations: a Many-body approach to the optical absorption spectra.
Introduction to Bethe-Salpeter equation and linear response theory.
We present an ab-initio real-time based computational approach to nonlinear optical properties in Condensed Matter systems. The equation of mot ions, and in particular the coupling of the electrons with the external electric field, are derived from the Berry phase formulation of the dynamical polarization. The zero-field Hamiltonian includes crystal local field effects, the renormalization of the independent particle energy levels by correlation and excitonic effects within the screened Hartree- Fock self-energy operator. The approach is validated by calculating the second-harmonic generation of SiC and AlAs bulk semiconductors : an excellent agreement is obtained with existing ab-initio calculations from response theory in frequency domain . We finally show applications to the second-harmonic generation of CdTe the third-harmonic generation of Si.
Reference :
Real-time approach to the optical properties of solids and nanostructures : Time-dependent Bethe-alpeter equation Phys. Rev. B 84, 245110 (2011)
Nonlinear optics from ab-initio by means of the dynamical Berry-phase
C. Attaccalite and M. Gruning Phys. Rev. B 88 (23), 235113 (2013)
In this talk I will present real-time spectroscopy and different code to perform this kind of calculations.
This presentation can be download here:
http://www.attaccalite.com/wp-content/uploads/2022/03/RealTime_Lausanne_2022.odp
Non-interacting and interacting Graphene in a strong uniform magnetic fieldAnkurDas60
We study monolayer graphene in a uniform magnetic field in the absence and presence of interactions. In the non-interacting limit for p/q flux quanta per unit cell, the central two bands have 2q Dirac points in the Brillouin zone in the nearest-neighbor model. These touchings and their locations are guaranteed by chiral symmetry and the lattice symmetries of the honeycomb structure. If we add a staggered potential and a next nearest neighbor hopping we find their competition leads to a topological phase transition. We also study the stability of the Dirac touchings to one-body perturbations that explicitly lowers the symmetry.
In the interacting case, we study the phases in the strong magnetic field limit. We consider on-site Hubbard and nearest-neighbor Heisenberg interactions. In the continuum limit, the theory has been studied before [1]. It has been found that there are four competing phases namely, ferromagnetic, antiferromagnetic, charge density wave, and Kekulé distorted phases. We find phase diagrams for q=3,4,5,6,9,12 where some of the phases found in the continuum limit are co-existent in the lattice limit with some phases not present in the continuum limit.
[1] M. Kharitonov PRB 85, 155439 (2012)
*NSF DMR-1306897
NSF DMR-1611161
US-Israel BSF 2016130
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
First-order cosmological perturbations produced by point-like masses: all sca...Maxim Eingorn
This presentation based on the paper http://arxiv.org/abs/1509.03835 was made at Institute of Cosmology, Tufts University, on November 12, 2015. The abstract follows:
In the framework of the concordance cosmological model the first-order scalar and vector perturbations of the homogeneous background are derived without any supplementary approximations in addition to the weak gravitational field limit. The sources of these perturbations (inhomogeneities) are presented in the discrete form of a system of separate point-like gravitating masses. The obtained expressions for the metric corrections are valid at all (sub-horizon and super-horizon) scales and converge in all points except the locations of the sources, and their average values are zero (thus, first-order backreaction effects are absent). Both the Minkowski background limit and the Newtonian cosmological approximation are reached under certain well-defined conditions. An important feature of the velocity-independent part of the scalar perturbation is revealed: up to an additive constant it represents a sum of Yukawa potentials produced by inhomogeneities with the same finite time-dependent Yukawa interaction range. The suggesting itself connection between this range and the homogeneity scale is briefly discussed along with other possible physical implications.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
Physics, Astrophysics & Simulation of Gravitational Wave Source (Lecture 1)Christian Ott
Lecture on the physics, astrophysics, and simulation of gravitational wave sources delivered in March 2015 at the International School on Gravitational Wave Physics, Yukawa Institute for Theoretical Physics, Kyoto University
Introduction to computation material science.
The presentation source can be downloaded here:
http://www.attaccalite.com/wp-content/uploads/2022/11/CompMatScience.odp
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In this talk I will present real-time spectroscopy and different code to perform this kind of calculations.
This presentation can be download here:
http://www.attaccalite.com/wp-content/uploads/2022/03/RealTime_Lausanne_2022.odp
Non-interacting and interacting Graphene in a strong uniform magnetic fieldAnkurDas60
We study monolayer graphene in a uniform magnetic field in the absence and presence of interactions. In the non-interacting limit for p/q flux quanta per unit cell, the central two bands have 2q Dirac points in the Brillouin zone in the nearest-neighbor model. These touchings and their locations are guaranteed by chiral symmetry and the lattice symmetries of the honeycomb structure. If we add a staggered potential and a next nearest neighbor hopping we find their competition leads to a topological phase transition. We also study the stability of the Dirac touchings to one-body perturbations that explicitly lowers the symmetry.
In the interacting case, we study the phases in the strong magnetic field limit. We consider on-site Hubbard and nearest-neighbor Heisenberg interactions. In the continuum limit, the theory has been studied before [1]. It has been found that there are four competing phases namely, ferromagnetic, antiferromagnetic, charge density wave, and Kekulé distorted phases. We find phase diagrams for q=3,4,5,6,9,12 where some of the phases found in the continuum limit are co-existent in the lattice limit with some phases not present in the continuum limit.
[1] M. Kharitonov PRB 85, 155439 (2012)
*NSF DMR-1306897
NSF DMR-1611161
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UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
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This presentation based on the paper http://arxiv.org/abs/1509.03835 was made at Institute of Cosmology, Tufts University, on November 12, 2015. The abstract follows:
In the framework of the concordance cosmological model the first-order scalar and vector perturbations of the homogeneous background are derived without any supplementary approximations in addition to the weak gravitational field limit. The sources of these perturbations (inhomogeneities) are presented in the discrete form of a system of separate point-like gravitating masses. The obtained expressions for the metric corrections are valid at all (sub-horizon and super-horizon) scales and converge in all points except the locations of the sources, and their average values are zero (thus, first-order backreaction effects are absent). Both the Minkowski background limit and the Newtonian cosmological approximation are reached under certain well-defined conditions. An important feature of the velocity-independent part of the scalar perturbation is revealed: up to an additive constant it represents a sum of Yukawa potentials produced by inhomogeneities with the same finite time-dependent Yukawa interaction range. The suggesting itself connection between this range and the homogeneity scale is briefly discussed along with other possible physical implications.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
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____________________________________________________________
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Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
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6. We introduce an effective Hamiltonian
that contains...
We start from the DFT
(Kohn-Sham) Hamiltonian:
hk
universal, parameter free
approach
1)
7. We introduce an effective Hamiltonian
that contains...
We start from the DFT
(Kohn-Sham) Hamiltonian:
hk
universal, parameter free
approach
1) 2)Renormalization of the band
structure due to correlation (GW)
hk+Δhk
8. We introduce an effective Hamiltonian
that contains...
We start from the DFT
(Kohn-Sham) Hamiltonian:
hk
universal, parameter free
approach
1) 2)Renormalization of the band
structure due to correlation (GW)
Charge fluctuations
(time-dependent Hartree)
3)
hk+Δhk
hk+Δhk+V H [Δρ]
9. We introduce an effective Hamiltonian
that contains...
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 interaction
Charge fluctuations
(time-dependent Hartree)
3)
hk+Δhk
hk+Δhk+V H [Δρ] hk+Δhk+V H [Δρ]+Σsex [Δ γ]
10. Linear response
● Wave-function in plane-waves plus norm-conserving pseudo
● Propagation time about 50 fs, with a time-step 0.01 fs
● Smearing included as non-Hermitian operator -ig (Weisskopf-
Wigner)or in post-processing as in Octopus
● Laser shape: delta function for LR/ sinus for Non-LR
11. Linear response
● Wave-function in plane-waves plus norm-conserving pseudo
● Propagation time about 50 fs, with a time-step 0.01 fs
● Smearing included as non-Hermitian operator -ig (Weisskopf-
Wigner)or in post-processing as in Octopus
● Laser shape: delta function for LR/ sinus for Non-LR
12. The Gauge problem
H =
p2
2 m
+r E+V (r) Length gauge:
H =
1
2 m
( p−e A)2
+V (r) Velocity gauge:
13. The Gauge problem
H =
p2
2 m
+r E+V (r) Length gauge:
H =
1
2 m
( p−e A)2
+V (r) Velocity gauge:
Quantum Mechanics is gauge invariant,
both gauges must give the same results
14. The Gauge problem
H =
p2
2 m
+r E+V (r) Length gauge:
H =
1
2 m
( p−e A)2
+V (r) Velocity gauge:
Quantum Mechanics is gauge invariant,
both gauges must give the same results
… but in real calculations the each gauge choice
has its advantages and disadvantages
15. The Gauge problem
H =
p2
2m
+r E+V (r)+V nl (r ,r ' )
H =
1
2m
( p−e A)
2
+V (r)+V nl (r ,r ' )
In presence of a non-local
operator
these Hamiltonians
Are not equivalent anymore
≠
W. E. Lamb, et al. Phys. Rev. A 36, 2763 (1987)
M. D. Tokman, Phys. Rev. A 79, 053415 (2009)
16. The Gauge problem
H =
p2
2m
+r E+V (r)+V nl (r ,r ' )
H =
1
2m
( p−e A)
2
+V (r)+V nl (r ,r ' )
≠
moral of the story:
non-local potential should be introduce
in length gauge and then transformed as
W. E. Lamb, et al. Phys. Rev. A 36, 2763 (1987)
M. D. Tokman, Phys. Rev. A 79, 053415 (2009)
H =
1
2m
( p−e A)
2
+V (r)+e
i Ar
V nl (r ,r ' )e
−i Ar '
In presence of a non-local
operator
these Hamiltonians
Are not equivalent anymore
17. The Gauge problem
H =
p2
2 m
+r E+V (r)
Length gauge:
H =
1
2 m
( p−e A)2
+V (r)
Velocity gauge:
● Non-local operators can be easily introduced
● The dipole operator <r> is ill-defined in solids
you need a formulation in term of Berry-phase
● Non-local operators acquires a dependence
from the external field
● The momentum operator <p> is well defined
also in solids
In recent years different wrong papers using velocity gauge
have been published (that I will not cite here)
18. In solids the polarization is written in terms
of wave-function phase
King-Smith and Vanderbilt formula
Phys. Rev. B 47, 1651 (1993)
Pα=
2ie
(2π)
3 ∫BZ
d k∑n=1
nb
〈un k∣
∂
∂ kα
∣unk 〉
Berry's connection !!
1) it is a bulk quantity
2) time derivative gives the current
3) reproduces the polarizabilities at all orders
4) is not an Hermitian operator
23. Intra-exciton spectroscopy (exp)
C. Poellmann, et al .Nature Materials 14, 889 (2015)
S. Cha, et al. Nature Communications 7, 10768 (2016)
P. Steinleitner et al. Nano Letters 18, 1402 (2018)
P. Merkl,et al. Nature Materials 18, 691 (2019)
24. Intra-exciton spectroscopy (exp)
C. Poellmann, et al .Nature Materials 14, 889 (2015)
S. Cha, et al. Nature Communications 7, 10768 (2016)
P. Steinleitner et al. Nano Letters 18, 1402 (2018)
P. Merkl,et al. Nature Materials 18, 691 (2019)
26. Δ P(t)=Ppp (t )−Pp(t)
Δ P(ω)=∫tpp
∞
Δ P(t)eiωt−τ(t−tpp)
Induced polarization from the pump
χ(ω)=
Δ P(ω)
Eprobe (ω)
Pump and probe (theory real-time)
27. Pump and probe (theory real-time)
Δ P(t)=Ppp (t )−Pp(t)
Δ P(ω)=∫tpp
∞
Δ P(t)eiωt−τ(t−tpp)
Induced polarization from the pump
χ(ω)=
Δ P(ω)
Eprobe (ω)
28. Pump and probe (theory real-time)
Δ P(t)=Ppp (t )−Pp(t)
Δ P(ω)=∫tpp
∞
Δ P(t)eiωt−τ(t−tpp)
Induced polarization from the pump
χ(ω)=
Δ P(ω)
Eprobe (ω)
29. Pump and probe (linear response)
+ -
=
|λ⟩=∑c v k
Ac v k
λ
|c v k ⟩
μλ ,λ '
α
=⟨λ'|μα
|λ⟩=∑c v k ∑c ' v' k'
Ac v k
λ ∗
Ac ' v ' k'
λ '
⟨c v k|μα
|c ' v' k' ⟩
χα ,β
λ
(ω)=
2
V
N λ ∑λ '
μλ ' λ
α
μλ λ '
β
Eλ '−Eλ−ω+i η
We can get excited states from the solution
of the Bethe-Salpeter Equation (a Casida like equation)
excited states are in the form:
Intra-exciton dipoles
Linear response from an excited state
30. χλ
α ,β
(ω)=
2
V
N λ ∑λ '
μλ ' λ
α
μλ λ '
β
Eλ '−Eλ−ω+i η
Linear response vs real-time
Pump and probe (linear response)
For m we used
velocity gauge
in order to avoid ill-defined
and complicated
intra-bands dipole matrix
elements
D. Sangalli, M. D’Alessandro, C. Attaccalite
https://arxiv.org/abs/2211.12241 (2022)
36. Exciton relaxation
M. Bernardi et al.
Phys. Rev. Lett. 125, 107401 (2020)
E. Malic
Nano Lett. 20, 4, 2849(2020)
Exciton-lifetime
C
C
37. Simple relaxation term in the
Hamiltonian C
Finite temperature
absorption GaN
~
H=H +i Γ
non-Hermitian term
Γi∝ℑΣii
ph
(ω=ϵi)
Derived from the
electron-phonon self-energy
~
ϵi(T)=ϵi+i η(T)
38. Simone Sanna group
University Giessen
Acknowledgments
Davide Sangalli
CNR -Milan
Marco D’Alessandro
CNR - Rome
Thank you for your attention
●
Non-linear response from real-time simulations including
excitonic effect
●
Intra-exciton transition from first-principle
●
How to address relaxation in an efficient way?
Pierre Lechifflart
Aix-Marseille Univ.
39. The Gauge problem
The guage transformation connect the different guages
A2=A1+∇ f
ϕ2=ϕ1−
1
c
∂ f
∂ t
Non-local operators do not commute
with gauge transformation
vector potential
scalar potential
wave-function phase
f (r ,t) Arbitrary scalar
function
Vnl ,Σxc ,ΔGW
,etc .…
ψ2=ψ1 e
−ie f (r, t)/ℏ c
40.
41.
42. Design materials for non-linear optics
with ab-initio codes
Simone Sanna group
Justus Liebig University
Giessen
Design Synthesis Properties
New molecular crystals with non-linear response comparable to
LiNbO3
J. Phys. Chem. C, 126, 7, 3713–3726(2022)
Tetraphenyl Tetrel
Molecular Crystals
KNbO3
/ LiNbO3
/ LiTaO3
Our prediction:
Tunable c2
in
LiNbx
Ta1-x
O3
without affecting c3
Nonlinear optical response of ferroelectric oxides
Phys. Rev. Mat. 6 (6), 065202 (2022)
48. Alternatives to the Berry-phase
Unified formalism for calculating
polarization, magnetization, and
more in a periodic insulator
Kuang-Ting Chen and Patrick A. Lee
Phys. Rev. B 84, 205137(2011)
J. Phys. Chem. Lett. , 9, 24, 7045–7051 2018