Presentation given at the "Theory days" workshop in Toulouse 2015

1. Nonlinear response of solids
with Green's functions and TD­D(P)FT
Claudio Attaccalite
CINaM, Aix­Marseille Universite (FR)
Myrta Grüning, Queen's University, Belfast(UK)
Theory Days - Toulouse - 26-11-2015Theory Days - Toulouse - 26-11-2015

2. What is it non­linear optics?
P(r ,t)=P0+χ
(1)
E+χ
(2)
E
2
+O(E
3
)
First experiments on linear­optics
by P. Franken 1961
Ref: Nonlinear Optics and
Spectroscopy
The Nobel Prize in Physics 1981
Nicolaas Bloembergen

3. Ref: supermaket
Why non­linear optics?
..applications..

4. Ref: supermaket
Why non­linear optics?
..applications..

5. “Self­focusing limit in terms of peak power is of the order
of 4 MW in the 1­μm wavelength region. No method is known
for increasing the self­focusing limit of optical fibers
beyond that value.” RP Photonics
Why non­linear optics?
..applications.. (the darkside )

6. Why non­linear optics?
..research..
Linear
Science, 2014, vol. 344, no 6183, p. 488­490
Non­linear(COLOR)Non­linear(BW)

7. To see “invisible” excitations
The Optical Resonances in
Carbon
Nanotubes Arise
from Excitons
Feng Wang, et al.
Science 308, 838 (2005);
..research..

8. ...and more...
photon entanglement in vivo imaging
Ref: PNAS 107, 14535 (2007)

9. How to calculate non­linear
response?
1) Direct evaluation of X(1)
,X(2)
....
2) Sternheimer equation
R. M. Sternheimer, Phys. Rev. 96, 951(1954)
3) Real­time propagation
P(t)=P0+χ
(1)
E+χ
(2)
E
2
+O(E
3
)
(HKS
0
−ϵn
0
)ψn
1
=(H KS
1
−ϵn
1
)ψn
0
χ=χ
0
+χ
0
(v+f xc)χ
−i∂t ψ=Hks ψ

10. 1) Equations become more and more complex
with the response order
2) Difficult to include more than one
external field (FWM, etc..)
Ref: PRB 80, 165318(2009), PRA 83, 062122(2001),
JCP 126, 184106(2007)
Direct evaluation of X(1)
,X(2)
....
Disadvantages:
Sternheimer equation

11. Advantages
Ref: JCP, 127, 154114(2007)
PRB, 89, 081102(2014)
Real­time propagation
1) The same equation for all response
functions
2) Can deal with complex spectroscopic
tecniques (SFG, FWM, etc...)
Disadvantages
1) Results are more difficult to analize

12. What to propagate?
(Current) Density Density Matrix
Green's function Wave function

13. The best chioce for solids is.....
Wave function
Why?
1)We know how to express polarization in terms of
the wave­function (the King­Smith Vanderbilt formula).
R. D. King­Smith and D. Vanderbilt, PRB 47, 1651 (1993)
2)Polarization in terms of Green's function or
density matrix is very complicated
K.­T. Chen and P. A. Lee, PRB 84, 205137 (2011)
3)Polarization in terms of the density is impossible!
R. M. Martin and G. Ortiz, PRB 56, 1124(1997)

14. From Polarization to the
Equations of Motion
I. Souza, J. Iniguez and D. Vanderbilt, PRB 69, 085106 (2004)
Pα=
2ie
(2π)
3 ∫BZ
d k∑n=1
nb
〈un k∣
∂
∂ kα
∣unk 〉
King­Smith and Vanderbilt formula
Phys. Rev. B 47, 1651 (1993)
L=
i ℏ
N
∑n=1
M
∑k
〈 vkn∣˙vkn〉−〈H
0
〉−v Ε⋅P
Once we know how to write the polarization we
can define a Lagrangian
i ℏ ∂
∂t
|vk n⟩=Hk
0
|vk n⟩+i e Ε⋅|∂k vk n⟩
From the Lagrangian we derive the Hamiltonian
and the Eqs. of motion

15. Correlation
Ref: PRB 84, 245110(2011),PRB 25, 2867 (1982)
L. P. Kadanoff & G. A. Baym,
Quantum statistical mechanics, Benjamin(1962).
G. Strinati, La Rivista del Nuovo Cimento, 11,1­86 (1998)
Correlation effects are derived
from non­equilibrium Green's
theory within the GW approximation

16. GW band structure
Aryasetiawan, F., & Gunnarsson, O. The GW method.
Reports on Progress in Physics, 61(3), 237.(1998).
W (ω)=
V ee
ϵ(ω)
GW is the first
order correction in
terms of screened
interaction
H=H0+Δi, jk
GW
Δi, jk
GW
=⟨ΣGW ⟩i, jk
Mapping GW in an effective Hamiltonian

17. Bethe­Salpeter Equation
+ -=
G. Strinati, Rivista del Nuovo Cimento, 11, 1 (1988)
C. Attaccalite, M Gruning and A. Marini PRB, 84, 245110 (2011)
L=L0+L0[v+ δ Σ
δG
] L Exact Bethe-Salpeter (useless)
L=L0+ L0[v−W ]LApprox. Bethe-Salpeter (usefull)
H=H0+Σ
COHSEX
[ ^ρ(t=0)−^ρ(t)]
Mapping BSE in an effective Hamiltonian
∂ Σ
COHSEX
∂V ext

18. Our computational setup
Correlation:
Green's function
theory (GW+BSE)
Coupling:
Modern­Theory of
Polarization
Ref: PRB 88, 235113(2013)

19. Linear response
Technical details:
● Wave­function in plane­waves plus norm­conserving pseudo
● Propagation time about 50 fs, with a time­step 0.01 fs
● Smearing included as non­Hermitian operator ­iγ (Weisskopf­Wigner)
or in post­processing as in Octopus
● Laser shape: delta function for LR/ sinus for Non­LR

20. X(2)
results: semiconductors
CdTe
Ref:
E. Luppi el al.,PRB 82, 235201 (2010)
E. Ghahramani et al., PRB 43, 9700 (1991)
I. Shoji,et al. J.Opt.Soc.Am. B 14, 2268 (1997)
J.I.Jang,et al. J.Opt.Soc. Am.B 30, 2292 (2013)
M. Grüning at al. PRB 88, 235113(2013)
AlAs
Local field effects are
more important than in
the linear reposonse

21. Local­fields and excitonic effects
in h­BN monolayer
IPA
IPA + GW + TDSHF
independent particles
+quasi-particle corrections
+time-dependent Hartree (RPA)
+screend Hartree-Fock (excitonic effects)

22. MoS2
single­layer
10) Second harmonic microscopy of monolayer MoS2
Phys. Rev. B 87, 161403(R) (2013)
100) Probing Symmetry Properties of Few­Layer MoS2
and h­
BN by Optical Second­Harmonic Generation
NanoLetters, 13, 3329 (2013)
1000) Observation of intense second harmonic generation
from MoS2
atomic crystals
Phys. Rev. B 87, 201401(R) (2013)

23. X(3)
in silicon
Ref: PRB 41, 1542 (1990)
Optical letters, 14,57 (1989)
PRB 88, 235113(2013)

24. X(3)
in nanostructures
Interference between excitons
is not trivial the case of ANGR7
Ref: Nature Comm. 5, 4253 (2014)
PRL 94, 047404 (2005)

25. TD­D(P)FT … why and why not?
TDDFT is much faster than Green's functions
Runge­Gross theorem does not hold in periodic
system (PRB 68, 045109)
Density is not enough to describe the response of
periodic systems (PRB 56, 1124)
q⋅P(q)=in(q)
Current­density­FT works :­)
If there are not magnetic fields and
transverse reponse is not important it
reduces to
Density­Polarization­FT Exc(P ,n)

26. Searching for a functional
● We want a functional of the density and Polarization
● No functionals of the density only
● We do not want the exchange
● No divergences 1/q^2 etc...
Checklist:

27. Electron gas with gap (JGM)
P. Trevisanutto et al.
PRB 87, 205143(2013)
Dielectric constant
Corresponding exchange­corr functional
(with a Micro and Macroscopic contribution)
We map the Macroscopic part in a
functional of the polarization and the
microscopic part in a functional of the
density
N. T. Maitra, I. Souza, and K. Burke PRB 68, 045109

28. Some preliminary results
of the JGM functional

29. What we miss?
Exc (t )=f xc (n)+α P (t )+β P
2
(t )+ γ P
3
(t)+....
For linear optics For χ2, χ3... For χ3,χ4,....
It can be shown that second term is as large as
local field effects(PRB 54, 8540)
First problem
Second problem
α(t)
In principle all the coefficients depend on time.
This is important to describe complex excitons, etc...
β(t) γ(t)

30. Prospectives
Applications:
Developement:
● An exchange­correlation function beyond linear
response (that works)
● Spin and spin­orbit coupling
● Therahertz spectroscopy
● THG and Kerr parameters
● SHG and spin
● Electro­absorption
● Interfaces

31. Conclusions
Develop an approach that
'imports' successful GW+BSE
recipe into versatile real­time
approach:
correlation in nonlinear­optics
e­laser interaction treated
within modern polarization theory
e­h interaction key to
understand
SHG spectra of 2D materials
(such as hBN, MoS2)
Density­polarization
functional theory
for the linear­response
JGM and beyond

32. Acknowledgement:
Myrta Grüning,
Queen's University Belfast
Related work:
Implementation of dynamical Berry phase:
C. Attaccalite, M. Grüning, PRB 88, 235113 (2013)
Application to SHG of 2D materials:
M. Grüning. C. Attaccalite, PRB 89(R), 081102 (2014)
C. Attaccalite et al., PCPC 17, 9533 (2015)
Time­dependent BSE theory:
C. Attaccalite ,M. Grüning. A. Marini PRB 84, 245110 (2011)
Time­dependent density polarization theory:
M. Grüning, D. Sangalli and C. Attaccalite, in preparation(2015)
The codes:
A. Marini et al. Comp. Phys. Comm. 180, 1392 (2009)
P. Giannozzi et al. J. of Phys.: Cond. Matter, 21, 395502 (2009)

33. Does the velocity gauge make
the life easier?
It is true if you have a local Hamiltonian
Length gauge:
H =
p2
2 m
+r E+V (r)
Ψ(r ,t )
Velocity gauge:
H =
1
2 m
( p−e A)2
+V (r)
e
−i r⋅A(t)
Ψ(r ,t)
Analitic demostration:
K. Rzazewski and R. W. Boyd,
J. of Mod. optics 51, 1137 (2004)
W. E. Lamb, et al.
Phys. Rev. A 36, 2763 (1987)
Well done velocity gauge:
M. Springborg, and B. Kirtman
Phys. Rev. B 77, 045102 (2008)
V. N. Genkin and P. M. Mednis
Sov. Phys. JETP 27, 609 (1968)
... but when you have non­local pseudo, exchange,
damping term etc..
they do not commute with the phase factor

34. The King­Smith Vanderbilt
polarization
King­Smith 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

35. Post­processing real­time data
P(t) is a periodic function of period TL
=2p/wL
pn
is proportional to χn
by the n­th order of the external field
Performing a discrete­time signal
sampling we reduce the problem to
the solution of a systems of linear equations
Ref: PRB 88, 235113(2013),
JCP 138, 064104(2013)