Non linear electron dynamics in solids

Non-linear electron
dynamics in solids
Claudio Attaccalite
CNRS/CINaM, Aix-Marseille Universite (FR)
“Atto2Nano: modeling ultrafast dynamics
across time-scales in condensed matter”
2023, Lausanne

Light induced real-time
electron dynamics

Non-linear response
P(ω)=P0+χ
(1)
E+χ
(2)
E
2
+O(E
3
)
First experiments on linear-optics
by P. Franken 1961
Ref: Nonlinear Optics and
Spectroscopy
The Nobel Prize in Physics 1981
Nicolaas Bloembergen

Real-time approach 1/2
ℱ𝒯
• Different response functions can be calculated with the same Eqs.
• Results analysis is more difficult

Real-time approach 2/2
ℱ𝒯
•Many-body effects can be included in an effective Hamiltonian
•Different response functions can be calculated with the same eqs.
•Results analysis is more difficult
Heff
=hk
0
+Δhk+VH [Δρ]+Σsex [Δ γ]

We introduce an effective Hamiltonian
that contains...
We start from the DFT
(Kohn-Sham) Hamiltonian:
hk
universal, parameter free
approach
1)

that contains...
hk
approach
1) 2)Renormalization of the band
structure due to correlation (GW)
hk+Δhk

that contains...
hk
approach
1) 2)Renormalization of the band
Charge fluctuations
(time-dependent Hartree)
3)
hk+Δhk
hk+Δhk+V H [Δρ]

that contains...
hk
approach
1) 2)
4)
Renormalization of the band
Electron-hole interaction
Charge fluctuations
(time-dependent Hartree)
3)
hk+Δhk
hk+Δhk+V H [Δρ] hk+Δhk+V H [Δρ]+Σsex [Δ γ]

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

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

Effective Schrodinger equation

Design materials for non-linear optics
with ab-initio codes
Simone Sanna group
Justus Liebig University
Giessen
Lithium niobate

Simone Sanna group
Giessen
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)
Lithium niobate

Simone Sanna group
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
HOMO/LUMO of X(C6
H5
)4
Tetraphenyl derivatives are thermostable,
colorless solids that are not air sensitive

Non-linear response of excitons
P(ω)=P0+χ
(1)
E+χ
(2)
E
2
+χ
(3)
E
3
Two-photon absorption
Phys. Rev. B 98, 165126(2019)

Non-linear response of excitons
P(ω)=P0+χ
(1)
E+χ
(2)
E
2
+χ
(3)
E
3
Two-photon absorption
Phys. Rev. B 98, 165126(2019)
Hexagonal boron nitride is an indirect band-gap
semiconductor
G. Cassabois et al., Nature Photonics, 10, 262 (2016)

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)

S. Cha, et al. Nature Communications 7, 10768 (2016)
MoS2

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)

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)

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

For degenerate excitons we prepare the initial state
according to the laser polarization by diagonalizing

For degenerate excitons we prepare the initial state
according to the laser polarization by diagonalizing
jλ ,λ '
α
=⟨λ '|jα
|λ⟩=∑c v k ∑c' v' k '
Ac v k
λ ∗
Ac ' v ' k'
λ '
⟨c v k|jα
|c ' v ' k' ⟩
Intra-exciton dipoles in velocity gauge
Approximated velocity
operator

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
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
Phys. Rev. B 107, 205203 (2023)

Pump and probe (group theory)
E(2x)
Failure of hydrogen model for
localized excitons!!!

We need relaxation (for electron/excitons)

Exciton-phonon from first principle

Phonon-assisted absorption/emission
"First-principles study of luminescence in hexagonal boron nitride single layer: Exciton-phonon coupling and the role of
substrate." Lechifflart, Pierre, et al. Physical Review Materials 7, 2 (2023): 024006.

Exciton relaxation
M. Bernardi et al.
Phys. Rev. Lett. 125, 107401 (2020)
Exciton-lifetime
C
C
E. Malic
Nano Lett. 20, 4, 2849(2020)
Exciton-dynamics

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.
D. Sangalli, M. D’Alessandro, C. Attaccalite
Phys. Rev. B 107, 205203 (2023)

Phys. Rev. B 102, 201402(R) (2020)

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

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:

The Gauge problem
H =
p2
2 m
H =
1
2 m
( p−e A)2
Quantum Mechanics is gauge invariant,
both gauges must give the same results

The Gauge problem
H =
p2
2 m
H =
1
2 m
( p−e A)2
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

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)

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

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)

Row 1 Row 2 Row 3 Row 4
0
2
4
6
8
10
12
Column 1
Column 2
Column 3

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)

Real-time Green’s functions
Kadanoff-Baym
equations
From Green’s functions to density matrix
Generalized Kadanoff-Baym Ansatz (GKBA)
Correspond to the equilibrium
self-energy

Electronic relaxation
3 – Measurement
process
1 – Photo-excitation
process
2 – relaxation towards
quasi-equilibrium
Sangalli, D., & Marini, A.
EPL (Europhysics Letters), 110(4), 47004.(2015)

Local-fields and excitonic effects
in h-BN monolayer
IPA
independent particles
+quasi-particle corrections
+time-dependent Hartree (RPA)
+screend Hartree-Fock (excitonic effects)

in h-BN monolayer
IPA IPA + GW

in h-BN monolayer
IPA IPA + GW
IPA + GW + TDSHF

Non linear electron dynamics in solids

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