http://salmon-tddft.jp

Practical Aspects of TDDFT Calculations
partly as an introduction to the afternoon tutorial session
using SALMON: Scalable Ab Initio Light-Matter simulator
for Optics and Nano-science

Kazuhiro Yabana
Center for Computational Sciences
University of Tsukuba

1

Classifications of TDDFT calculations for optical responses
that can be done in current SALMON
Isolated Systems Periodic Systems(Molecules,(Crystalline solids)
Nano-particles)
Weak fields(Linear response)

Polarizabilitya(w)Strong fields
Excitation energy(Nonlinear dynamics) Atomic motion

Dielectric functione(w)Excitation energy
Carrier density
Atomic motion

Light propagationin bulk materials(Maxwell + TDDFT)

1D light
propagation
E(x,t), J(x,t)

To be developed:
spin degrees (at present, only spin-saturated system: LSDA, spin-orbit,…)
1D, 2D systems
Electromagnetic field analyses with various options of electron dynamics
2Basic features of light-matter interactions- Interaction dominates between light electric field and electrons.
- For ordinary (weak) light, one may use perturbation theory in quantum mechanics.
- There are two spatial scales and single time scale ➔ Dipole approximation

Laser pulse

Electron dynamicsℏ" ∼ 1.55 eV) ~ 800 nm/012 ≈ 1~10 eV
Δ5~1 Å(Ti:Sapphire laser)(Typical dielectric)
3

Linear optical response of molecule is characterized by polarizability
in dipole approximation
Electric field E⌫ (t)PolarizationZpµ (t) =d~r ( erµ ) ⇢(~r, t)

Polarizability (time domain)ℏ" ∼ 1.55 eV) ~ 800 nm/012 ≈ 1~10 eV
Δ5~1 Å

pµ (t) =XZ

dt0 ↵µ⌫ (tt0 )E⌫ (t0 )⌫Relation to frequency-dependent polarizability(polarizability for fixed-frequency field)R
dtei!t pµ (t)↵µ⌫ (!) = R
dtei!t E⌫ (t)4Polarizability: a classical spring-mass modelm, - exk, gNewton equationmẍ + g ẋ + kx =Polarization andPolarizabilityp(t) =

ex(t) =eE(t)Z

dt0 ↵(tt0 )E(t0 )We can solve the equation for
Sinusoidal field

Impulsive field

E(t) = E0 e i!t

E(t) =<latexit sha1_base64="PJpi6ymYAIVDPEGe/vnKjuhLHzo=">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</latexit><latexit sha1_base64="0+vLkLRSY2IvnJtmF6jy6Y6WVLE=">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</latexit>

I
(t)
e

5

Sinusoidal external field (forced oscillator)eE0 e i!tmẍ + g ẋ + kx =x(t) =em!2m, - e1
E0 eı!ti ! + !02

k
!02 =
m

E(t)

x

k, g

g
=
m

Polarization and polarizabilityp(t) =

ex(t) = ↵(!)E(t)e2↵(!) =m!2

1
i ! + !02

Ima (w )

e2
k

w

w0

Re a (w )6

Impulsive external field (Damped oscillation)mẍ + g ẋ + kx = I (t)I
x(0) = 0 ẋ(0) =
m
p
2 t/4
!02
I
t/2 sinpx(t) = e
2 /4
m
!02

m, - eImpulse
!02 =

E(t)
k
m

Polarization and polarizability

x

k, g

g
=
m

↵(t)
<latexit sha1_base64="kFjXG+TfEGTyfQ8YkM3B9HxlqoM=">AAACbHichVG7SgNBFD1Z3/EVH4UgghgUbcJdERQr0cYyMcZEEpHdddTFfbE7CWjwB2wtLNRCQUT8DBt/wCKfIIKNgo2FN5sFUVHvMDNnztxz58yM7llmIIlqMaWpuaW1rb0j3tnV3dOb6OtfC9yyb4ic4VquX9C1QFimI3LSlJYoeL7QbN0SeX1vqb6frwg/MF1nVe57YsPWdhxz2zQ0ydR6SbO8XW1STm0mkpSiMEZ/AjUCSUSRdhPXKGELLgyUYUPAgWRsQUPArQgVBI+5DVSZ8xmZ4b7AIeKsLXOW4AyN2T0ed3hVjFiH1/WaQag2+BSLu8/KUYzTA93QC93TLT3S+6+1qmGNupd9nvWGVnibvUdD2bd/VTbPErufqj89S2xjLvRqsncvZOq3MBr6ysHJS3Z+Zbw6QZf0xP4vqEZ3fAOn8mpcZcTKKeL8Aer35/4J1qZTKqXUzExyYTH6inYMYwyT/N6zWMAy0sjxuTaOcYbz2LMyqAwrI41UJRZpBvAllIkPjbOM9w==</latexit>
sha1_base64="bs8QlLljyYde7WbNFuyzJJm+ZMk=">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</latexit>
sha1_base64="9R3XVOon34pyAPQKMY9Th93DIFc=">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</latexit>
sha1_base64="oeNDMxTYdYa0KSp4NRuVOFrmrN4=">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</latexit>p(t) =

ex(t) =e2↵(t) = ✓(t)emZ

dt0 ↵(tt0 )E(t0 )

p
!02
sin
t/2
p
!02

t

2 t/4
2 /4

7

A classical spring-mass model: Compare two descriptions
Polarizability as a function of time
= damped oscillation induced by an impulse2e↵(t) = ✓(t)emp!02
t/2 sinp!02

Polarizability as a function of frequency= forced oscillatione2↵(!) =
m

2 t/4
2 /4

!2

1
i ! + !02Z1↵(t) =d!↵(!)e i!t
2⇡ C
Im w

Time and frequency domain
Polarizabilities are related by
Fourier transformation.

Contour for integral and poles´

Two description involves
the same physical information
!=

i
±
2

r

!02

Re w

´

2

4

8

A classical spring-mass model: Photoabsorption cross sectionm, - eFext (t) =<latexit sha1_base64="PXdNDdD/TL8JkF7ULzZEj9rmvlA=">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</latexit>p(t) =<latexit sha1_base64="W9t1HFTsDOxGEnPcgzu1zQhJ/nA=">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</latexit>

eE(t)Z
ex(t) = dt0 ↵(t

E(t)

0

x

k, g

0

t )E(t )Work done by light electric field = Energy absorbed by the spring
W =<latexit sha1_base64="AhT/uV07f8sktJAhlYFOOhyLpHI=">AAADAHichVFLaxRBEK4ZX3F9ZKMXwUvjEt1clpoQMAQCIaLoLQ83G8gkQ09v76bJPJqZ3iWbYS4e/QMeclIQFQ/i1asX/4CH/ATxGMGLB2tmByQJag09VfVVfdVfd/s6UKlBPLLsc+cvXLw0cbl25eq165P1qRsbaTxIhGyLOIiTTZ+nMlCRbBtlArmpE8lDP5Adf+9BUe8MZZKqOHpqRlpuh7wfqZ4S3BDk1Q87bJG5KjKsa5jbjQ3bb5oZ9sjL3CRkct/kRXqqRRfYw6rQS7jInDxztcrLNg93MvI9M8pZ141D2eescuXQJ2Hu8kDv8uYYnWEuKTQ08ES+M+vVG9jC0tjZwKmCBlS2EtffgAtdiEHAAEKQEIGhOAAOKX1b4ACCJmwbMsISilRZl5BDjbgD6pLUwQndo3+fsq0KjSgvZqYlW9AuAa2EmAym8Su+w2P8gu/xG/7666ysnFFoGZH3x1ypvcnnt9Z//pcVkjew+4f1T80GejBfalWkXZdIcQox5g8PXhyvL6xNZ3fxFX4n/S/xCD/TCaLhD/F6Va4dQo0ewDl93WeDjdmWgy1nda6xtFw9xQTchjvQpPu+D0vwGFagDcKyrHsWWo79zH5rf7A/jlttq+LchBNmf/oNCD+/qg==</latexit>Z

dtẋ(t)Fext (t) =Z

1dtṗ(t)E(t) =⇡Z 1d!!Im↵(!)|E(!)|20

Photoabsorption cross section is given byimaginary part of the polarizability

(!) =

4⇡!
Im↵(!)c9

A general (accurate) formula of polarizability for molecules
Electronic description
for a molecule()

Hamiltonian

H=

X

~2 2
r
2m i

i

Schroedinger eq.

H

Dipole operator

D=

n = EnXXa

Z a e2~ a||~ri R

+

X
i<j

e2
|~ri

~rj |

n

zi

i

Polarizability

e2 X↵(!) =
fn0
m n
!2
2m

Oscillator strength fn0 = ~2 (EnE0 ) |h

1
i !+
n |D|

0 i|

En En 2~
2

Each excited state contributes as a classical oscillator.
e2
Polarizability of↵(!) =
m !2classical oscillator

1
i ! + !0210

Polarizability for molecules in TDDFTTDDFT provides time-dependent density for a given external potential

Vext (~r, t)⇢(~r, t)<latexit sha1_base64="XnIvEqj6ZvTbfkN50R/iCd76mzQ=">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</latexit><latexit sha1_base64="rvn5oD2N14WtLcj1yNYq+djXjTI=">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</latexit>

External potential of uniform electric field

Induced polarizationin moleculeZpµ (t) =Vext (~r, t) = eE(t)r⌫<latexit sha1_base64="TAmGeadLaX2iDBx0RtUWlTsKmGI=">AAACi3ichVFNSyNBEH2O7pqNuka9LHgZDEoWJNSooIiCIIJHvxIFI2Fm7OjgfNHTCcaQPyDePXjahWURb17d2172D+zBnyAeFbx4sDIZWFR0a5ju16/qVb/utkLXiRTRdYfW2fXhY3fqU7qnt+9zf2ZgsBgFVWmLgh24gdyyzEi4ji8KylGu2AqlMD3LFZvWwWIrv1kTMnICf0PVQ7HjmXu+U3FsUzFVzowVy42S9HRxqJq5Uk3YuhxXX/V5XehLOQaS0361Wc5kKU9x6K+BkYAsklgJMj9Rwi4C2KjCg4APxdiFiYi/bRgghMztoMGcZOTEeYEm0qytcpXgCpPZAx73eLWdsD6vWz2jWG3zLi7/kpU6RukvndMd/aELuqHHN3s14h4tL3WerbZWhOX+4y/rD/9VeTwr7P9TvetZoYKZ2KvD3sOYaZ3CbutrR6d367Nro40x+k637P8bXdNvPoFfu7d/rIq1M6T5AYyX1/0aFCfyBuWN1answnLyFCkMYwQ5vu9pLGAZKyjwvie4xBV+aX3apDarzbVLtY5EM4RnoS09ARv3ltQ=</latexit>

d~rrµ ⇢(~r, t)<latexit sha1_base64="Gk7Km84YLJx/dV9fD63GpsxYqlI=">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</latexit>pµ (t) =<latexit sha1_base64="Hyub+w9i2UMhe+bkGy70IZCbUqk=">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</latexit>

Z

Polarizability is given by
dt0 ↵µ⌫ (tRdtei!t pµ (t)↵µ⌫ (!) = R
dtei!t E⌫ (t)t0 )E(t0 )11Calculation of Polarizability in TDDFT
Time-domain method

@
i~
@t

r, t) =i (~(Time-Dependent Kohn-Sham equation

~2 2
r
2m

Z
X Z a e2e2
+ d~r0⇢(~r0 , t) + µxc [⇢(~r, t)] + Vext (~r, t)~ a||~r ~r0 ||~r R
a⇢(~r, t) =

X
i

)

r, t)
i (~

| i (~r, t)|2

Computational procedure
Prepare ground state
Apply impulsive external potentialhKS<latexit sha1_base64="4Bqyi9LK1JZoVzIiHLvGNyXcbYg=">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</latexit>r) = ✏i i (~r)
i (~Vext (~r, t) = I (t)z<latexit sha1_base64="+RqhRXkuXhhZ5ChdO6QmDKLdM+I=">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</latexit>

Newton mechanics

Orbitals immediately after the impulse<latexit sha1_base64="Lt1c/RlP/Oz+FU5KVxzaB5EO8PY=">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</latexit>

Calculate polarizationr, t = 0+ ) = ei (~p(t) =e<latexit sha1_base64="5J3xDRybGj3Se/TL4+OggINJXy0=">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</latexit>

Fourier transformation
Frequency-dependent polarizability

1↵(!) =
I<latexit sha1_base64="uWSKZlbQoBiihzpj5+LTYVxx/r8=">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</latexit>

Z

Z

iIz/~

r)
i (~

ẋ(t = 0+ ) =<latexit sha1_base64="iKwKVfw/vZMeRX8Pd4bQ8DVLWyg=">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</latexit>d~rz⇢(~r, t)
dtei!t p(t)

12

I
m

An example:
photoabsorption in ethylene moleculeElectron density
in the ground stateElectron density change
in ethylene molecule13

Real-time response: Optical response of Ethylene (C2H4) molecule
Dipole moment
Electron density after impulsive kick

p-p* oscillation

time

Oscillator strength
p-p* oscillations electron
emission

Excitation energy

14

Photoabsorption cross section of typical molecules
K. Yabana et.al, Chap.4, Charged Particle and Photon Interactions with Matter, CRC Press, 2010.

C6H6

N2

CO2

C60

15

Surface plamons (Mie plasmon): Collective excitation in metallic cluster
K. Yabana, G.F. Bertsch, Phys. Rev. B54, 4484 (1996).

Time-dependence of dipole moment

Li147+!
!
d (t ) µ ò dr zr (r , t )[eV-1]

Optical absorption spectrum
Exp:Li139+
TDDFT

Electron density change from the ground states (w ) µ1
dt eiwt d (t )òkAssume icosahedral geometry
16

Dynamical screening effect:
Time-dependence of the Kohn-Sham Hamiltonian@i~@tr, t) =
i (~

(

~2 2
r
2m

Z
X Z a e2e2
+ d~r0⇢(~r0 , t) + µxc [⇢(~r, t)] + Vext (~r, t)
0~|~r ~r ||~r Ra |a⇢(~r, t) =Xix

| i (~r, t)|2

Photoabsorption of rare gas atom
Zangwill, Soven 1980

+

r, t)
i (~

- - - --

Change of Hartree potential
by the motion of electrons

)

E

-

+ + + ++

Without dynamical screening(Time-independent Hamiltonian)Dynamical screening effect is importanteven for very small system (atom).

17

Calculation of large systems using massively parallel supercomputers
Surface plasmon resonance of Au clusters

4 nm

40 6

Oscillator strength (1 / eV)

Au1414Au560

35 4 146
30 2

1414
54

25 02.0 2.5 3.0
2.0

922

3.0

20
560

15

308

10
5
0
1.5

2.0

2.5

3.0

3.5

4.0

Energy (eV)15,554 electrons

0.08

ized strength (1 / eV)

Au54

0.07

0.020.2

0.06

0.010.12.1 JPCA,
K. Iida, M. Noda, K. Ishimura, K. Nobusada,118,
2.3
2.5 11317 (2014)

0.05
0.04
0.03

2.1

54
146
308
560

2.3

2.5

18

Thomas-Reiche-Kuhn (TRK) sum rule
Sum of oscillator strength over all excited states is equal to the number of electrons.

Z 1<latexit sha1_base64="i+b1IXMGl79syf9R7SvF5n5ikGY=">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</latexit>

4⇡d! (!) =
c

0

Z 1
0

2⇡ 2 e2
d!!Im↵(!) =Nemc
Number of electrons

TRK sum rule is related to the initial velocity after impulsive excitation
Z
Z
d↵(t)
1 1
dt i!t=d!!Im↵(!)↵(t) =
e↵(!)
dt t=0⇡ 02⇡<latexit sha1_base64="u+lnICGfTIIcekWurWkDp4MdqAg=">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</latexit><latexit sha1_base64="yxjZZO2pJV9b20kx2ePV8xVzNYY=">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</latexit>

Velocity immediately after the kickD(t)<latexit sha1_base64="IDT2GZ9n4WI2PcJrcU/Xv/7yhZU=">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</latexit>

dD(t)
I
= Ne
dt
m
t

19

Computational Schemes for linear response calculations
Orbital representation- Real space grid with pseudpotential- Plane wave with pseudopotential- Basis functions referring to atomic positions

SALMON adopts
Real-space grid +norm conserving pseudopotential

Solve linear response problem- Time domain method
solve TDKS equation in real time
- Frequency domain method (Sternheimer method)linear algebraic equation- Eigenvalue method
an option of quantum chemistry methods

Merit of real-space + real-time- Easy to parallelize by space division (real space)- Good for excitations involving huge number ofparticle-hole excitations (real time)- Only choice for nonlinear dynamics (real time)20

Classifications of TDDFT calculations for optical responses
that can be done in current SALMON
Isolated Systems Periodic Systems(Molecules,(Crystalline solids)
Nano-particles)
Weak fields(Linear response)

Polarizabilitya(w)Dielectric functione(w)Strong fields
Excitation energy(Nonlinear dynamics) Atomic motion

Excitation energy
Carrier density
Atomic motion

Light propagationin bulk materials(Maxwell + TDDFT)

1D light
propagation
E(x,t), J(x,t)

21Interaction of intense and ultrashort laser pulses with matters
What is strong? What is ultrashort?
Intensity

duration

To be compared: atomic unit
- Time: period of hydrogen atom = 2p

Bohr model for hydrogen atom

- Field strength: electric field felt by electron
in classical hydrogen atom

22

Development of Intense Laser Pulses

QED Vacuum breakdown
Nuclear reaction,
Acceleration
Relativistic
Classical

Nonlinearelectron
dynamics

Nonrelativistic
Quantum

G. Mourou and T. Tajima, SPIE News

23

Extremely nonlinear optics --- Intense laser pulse --1 atomic unit *+,-. : 3.51×1056W/cm7

Solar constant

Strongest laser pulse

0.1366 W/cm
Linearopticse (w )

2

10 - 10 W/cm
13

Nonlinear
Optics(Perturbative)

15

10 22 W/cm 22

Nonlinear(Non-perturbative)

Laser
acceleration

Vacuumbreakdownc (2 ) (w1 , w2 ), c (3) (w1 , w2 , w3 )High harmonic generation

Takahashi et.al,Appl. Phys. Lett. 93, 041111 (2008)

femtosecond
laser processing

B.N. Chichkov et.al,Appl. Phys. A63, 109 (1996)24

Extremely nonlinear optics --- Ultrashort laser pulse ---Picosecond

Femtosecond

Attosecond

10-12 s10-15 s

10-18 sPeriod of
Optical phonon(Si, 64 fs)

10-23 sPeriod of electron
orbital in Hydrogen,
150as
Period of
Ti-sapphirelaser pulse
1.55eVTime

Nucleon motion
in nucleus
P.B. Corkum, F. Krausz,
Nature Phys. 3, 380 (2007)

Shortest laser
pulse
67as

25

High harmonic generation spectrum from atom
Takahashi et.al,Appl. Phys. Lett. 93, 041111 (2008)

I = 1.5 x 1014 W/cm2
Red: 1400nm (0.88 eV)
Blue: 800nm (1.55 eV)Three step model (Rescattering)

P.B. Corkum, Phys. Rev. Lett. 71, 1994 (1993)

26

Figure taken from P.B. Corkum, F. Krausz, Nature Phys. 3, 380 (2007)

Ethylene molecule (C2H4 )Rescattering by TDDFT

I (w ) µ ò dteiwt d A (t )! !
dD (t ) d
= ò dr zr (r , t )
dt
dt

2








































27

Coulomb explosion of molecules
under intense and ultrashort laser pulse
N2 molecule
I=3.35x1015W/cm2, 27fs
Electron dynamics +Atomic motion(Ehrenfest dynamics)
Final charge states are
fractional, and not simple
to compare withmeasurements.

28Collision of highly charged ion with atom
Ar8+ - Ar at 18 keV v=2.5 au

Ar

Neutralatom

Ar8+Highlycharged
ion

29

Ar8+ - Ar Collision

Nagano, Yabana, Tazawa, Abe, Phys. Rev. A62(2000)062721

E=18.4keV

v=2.5 au

E=400keV

v=11.7 au

E=3200keV

v=33 au30

Electron transfer to highly excited orbitals:
Classical Overbarrier ModelzB
Neutral atom

z

R
-Iq
-Iq-q/R
Ion (+q charge)

1q- z R-zCondition for charge transfer- IP -q
1
q=- R
zB R - zB
31

TDDFT School and SALMON Hands-on Seminar
Center for Computational Sciences, University of Tsukuba
Nov. 11-12, 2018
http://salmon-tddft.jp

Practical Aspects of TDDFT Calculations
partly as an introduction to the afternoon tutorial session
using SALMON: Scalable Ab Initio Light-Matter simulator
for Optics and Nano-science

Kazuhiro Yabana
Center for Computational Sciences
University of Tsukuba

32

Classifications of TDDFT calculations for optical responses
that can be done in current SALMON
Isolated Systems Periodic Systems(Molecules,(Crystalline solids)
Nano-particles)
Weak fields(Linear response)

Polarizabilitya(w)Strong fields
Excitation energy(Nonlinear dynamics) Atomic motion

Dielectric functione(w)Excitation energy
Carrier density
Atomic motion

Light propagationin bulk materials(Maxwell + TDDFT)

1D light
propagation
E(x,t), J(x,t)

33

Schrödinger equation in a periodic potential: Bloch’s theorem⇢~2 d 2+ V (x)2m dx2i (x) = ✏i i (x)i (x) = eikx

V (x + a) = V (x)unk (x)unk (x + a) = unk (x)
(

1
2m

✓

d
i~
+ ~k
dx

◆2

)

+ V (x) unk (x) = ✏nk unk (x)

⇡
⇡
<k<
a
a
Brillouin zone (k-points)

Energy band of
crystalline silicon✏n~k<latexit sha1_base64="efvqLcucaVZbPfnCoE7IXNlMRT8=">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</latexit>

34

Using potential (“length gauge”), lattice periodicity is violated

V (x)<latexit sha1_base64="o+3qfr9s4GypcAtz6DC0NQSCUDE=">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</latexit>xeE(t)xa<latexit sha1_base64="4pFRrW5/0Kn6XfZ7st61CcT6LIc=">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</latexit>

V (x + a) 6= V (x)<latexit sha1_base64="kmVEyYeIE6mnVPs8ZYLWTzk78UU=">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</latexit>

We make use of gauge freedom to overcome the problem.

35

A general scheme of gauge transformation
⇢

⌘2
@
1 ⇣
~ + eA
~ +
i~
=
i~r
@t
2m
c

ie
0
(~r, t) = exp
f (~r, t) (~r, t)
~c

@
i~
@t

0

=

⇢

⌘2
1 ⇣
~ + eA
~0 +
i~r
2m
c

~ 0 (~r, t) = A(~~ r, t)A
0(~r, t) = (~r, t)0~ (~r, t)rfe @f (~r, t)c @t

Physical quantities are invariant under the transformation.
~ r, t), B(~~ r, t)⇢(~r, t), ~j(~r, t), E(~⇢(~r, t) = | (~r, t)|2
⇢
⌘
1 ⇣
e~
~
~j(~r, t) = Re(~r, t)i~r + A(~r, t) (~r, t)mc~ r, t)@A(~
1~ r, t) = r~ (~r, t)
E(~
c
@t
~
~
~
B(~r, t) = r ⇥ A(~r, t)
36

0Gauge transformation: from “length gauge” to “velocity gauge”

@
i~
@t

Length gauge

0

= exp

Velocity gauge



=

⇢

⌘2
1 ⇣
~
~
i~r
+ V (~r) + eE(t)
· ~r
2m

ie ~
A(t) · ~r
~c

@
i~
@t

0

=

~
E(t)
=

⇢

~
1 dA(t)
c dt

⌘2
1 ⇣
~ + e A(t)~i~r
+ V (~r)2mc0Expressing the spatially-uniform electric field by a vector potential,
Hamiltonian has the lattice periodicity.

h(~r + ~a) = h(~r )37Time-Dependent Kohn-Sham equation@i~@tr, t) =
i (~

(

⌘2
1 ⇣
~ + e A(t)
~
i~r
2m
c

Z
X Z a e2e2
+ d~r0⇢(~r0 , t) + µxc [⇢(~r, t)]~ a||~r ~r0 ||~r R
a⇢(~r, t) =

X
i

)

r, t)
i (~

| i (~r, t)|2

At each time, we may apply the Bloch’s theoremr, t) = ei (~@i~ un~k (~r, t) =
@t

(

i~
k~
r

⌘2
1 ⇣
~ + ~~k + e A(t)
~
i~r
2m
c

un~k (~r, t)

i ! (n~k)<latexit sha1_base64="IbXw2SRbmjWY6G3NthS0evuuMXk=">AAACfXichVHNSltBGD259iemP4l2E3BzaUixUMJ3pRBxFXDjMv7ECEbCvdcxDrl/zJ1EYtAH8AVcdNWiSCn2JbrxBbrwEcRlCt0U2i83F0ortd8wM2fOfOebMzNO5MlYE11njKkHDx89zk7nnjx99jxfmJndjMOeckXDDb1QbTl2LDwZiIaW2hNbkRK273ii6XSXx/vNvlCxDIMNPYjEjm93ArknXVsz1S4UpdlSsrOvbaXCA3M+MFt94Zrd1+1CiSqUhHkXWCkoIY16WLhAC7sI4aIHHwIBNGMPNmJu27BAiJjbwZA5xUgm+wJHyLG2x1mCM2xmuzx2eLWdsgGvxzXjRO3yKR53xUoTZfpKH2lEV/SJbujHP2sNkxpjLwOenYlWRO38SXH9+39VPs8a+79V93rW2MNi4lWy9yhhxrdwJ/r+4elofWmtPHxFH+iW/b+na/rCNwj639zzVbH2Djn+AOvv574LNhcqFlWs1bel2kr6FVnM4SXm+b2rqGEFdTT43GOc4RKfMz+NsvHGqExSjUyqeYE/wqj+Avdgku8=</latexit>)
Z
X Z a e2
e2
0
0
+ d~r
⇢(~r , t) + µxc [⇢(~r, t)] un~k (~r, t)~ a||~r ~r0 ||~r R
aun~k (~r + ~a, t) = un~k (~r, t)38

Electron dynamics in a unit cell of solid (bulk silicon)
Ground state band structure
static Density Functional Theory (DFT)
8&' %&' ( =,-.

/ + ℏ1 - + 56 ( + 573 ( %&' (Electron dynamics in a unit cellTime-dependent DFT (TDDFT)
!ℏ

#
#$

%&' (, * =

,
-.

2

/ + ℏ1 + 4 *-3+ 56 (, * + 573 (, *%&' (, *$4 * = −: ∫ < *′ dt’

Energy band of silicon
Change of electron density in crystalline silicon!ω = 1.55eV
I = 1.0 ×1014 W/cm 2
39

Further gauge transformation is possible: “length gauge” in k-space
Already discussed@i~
Length gauge@t@
Velocity gauge i~
@t

=
0

⇢

=

⌘2
1 ⇣
~
~
i~r
+ V (~r) + eE(t)
· ~r
2m

⇢

1 ⇣
e ~ ⌘2~
i~r + A(t) + V (~r)
2mc~1 dA(t)
c dt

~
E(t)
=
0

Time-dependent
Bloch equation⇢⇣@1e ~ ⌘2~~i~ un~k =i~r + ~k + A(t) + V (~r) un~k
@t
2m
c
New

Length gauge
in k-space

u0n~k (~r, t) = exp@
i~ u0n~k =
@t

⇢

h

e ~ ~ iA(t)rk un~k (~r, t)
~c

⌘2
1 ⇣
~ + ~~k + V (~r) + ieE(t)~ r~ k u0i~rn~k2mCoordinate operator in Bloch space ~r ! irk

~
eE(t)
· ~r
40

Modern theory of polarization using Berry’s phase
R.D. King-Smith, D. Vanderbilt, Phys. Rev. B47, 1651 (1993)

1
P~ =
V
P~ =

Dynamical
Berry theory

1 Xe(2⇡)3 n@i~ u0n~k =
@t

Z

⇢

Z

BZ

d~r ~r⇢(~r)~ k |u ~ id~khun~k |irnk~r ! irk

⌘2
1 ⇣
~
~
~ r
~ k u0i~r + ~k + V (~r) + ieE(t)n~k2mCoordinate operator in Bloch space ~r ! irk

~
eE(t)
· ~r

I. Souza, J. Iniguez, D. Vanderbilt, Phys. Rev. B69, 085106 (2004)
C. Attaccalite, M. Gruening, Phys. Rev. B88, 235113 (2013)41

Time-dependent Kohn-Sham equation for Bloch orbitals(implemented in SALMON)@
i~ un~k (~r, t) =
@t

(

⌘2
1 ⇣
~ + ~~k + e A(t)
~
i~r
2m
c

)
Z
X Z a e2
e2
0
0
+ d~r
⇢(~r , t) + µxc [⇢(~r, t)] un~k (~r, t))~ a||~r ~r0 ||~r R
aun~k (~r + ~a, t) = un~k (~r, t)

~
E(t)
=

~
1 dA(t)c dt- Calculation of dielectric function in real time
- Electron dynamics in crystalline solids
under intense and ultrashort light pulse

42Electric field, Electric current density, Polarization density,
Conductivity, Dielectric function
E(t)Current density fromZ Microscopic to Macroscopic1
J(t) =
d~rj(~r, t)
V V<latexit sha1_base64="YlEGjFv8aVvl1+eZdcutG8xk1VM=">AAACZ3ichVHLSsNAFD2Nr1ofrQpScKMWRTflRgTFVUGELlu1KqhIEkcNpklIpoVa/AEXbiu4UhARP8ONP+Cin1BcKrhx4U0aEBX1DjNz5sw9d87M6K5l+pKoGVM6Oru6e+K9ib7+gcFkamh4w3cqniFKhmM53pau+cIybVGSprTElusJraxbYlM/Xg72N6vC803HXpc1V+yWtUPbPDANTQbUyoyc3UtlKEthjP8EagQyiKLgpG6xg304MFBBGQI2JGMLGnxu21BBcJnbRZ05j5EZ7gucIsHaCmcJztCYPebxkFfbEWvzOqjph2qDT7G4e6wcxxQ90R290CPdU4vef61VD2sEXmo8622tcPeSZ+m1t39VZZ4ljj5Vf3qWOMBi6NVk727IBLcw2vrqSeNlbWl1qj5N1/TM/q+oSQ98A7v6atwUxeolEvwB6vfn/gk25rIqZdXifCaXj74ijjFMYobfewE55FFAic89wjkauIi1lKQyqqTbqUos0ozgSygTH43xiq4=</latexit><latexit sha1_base64="LrawCNN8XmuaTlK1H7N+2i8ih2c=">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</latexit>

Electric field induces
Z electric current in a unit cell of solidtJ(t) =

dt0 (tt0 )E(t0 )j(~r, t)conductivity<latexit sha1_base64="T6829aiN5WlGLm68c1PQh5QBZ9U=">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</latexit><latexit sha1_base64="iZ5VZWgkXj4Mtq03GIGx5k2MylU=">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</latexit>

Polarization is given
Z as time-integral of the currentt

P (t) =

dt0 J(t0 )<latexit sha1_base64="FjGbP9fU2zyhu0ilR5ePSPVk/vw=">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</latexit>

Dielectric function

D(t) = E(t) + 4⇡P (t) =<latexit sha1_base64="bs721mVIt+XpyIXLZ2BOBQBQv1U=">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</latexit>
sha1_base64="w8je/sSwRPK4BGR63IjQA5yQbvg=">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</latexit>
sha1_base64="b9ZdzfgfPly2xO8wMO4ZBqHhG+I=">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</latexit>Z t

dt0 ✏(tt0 )E(t0 )Dielectric function
43

Classical spring-mass model for dielectric functionm, - eE(t)Impulsive force
Sinusoidal fieldxk, gNewtonian dynamics

Newtonian dynamicsmẍ + kx = I (t)eE0 e i!tmẍ + kx =<latexit sha1_base64="ktqRkV+YD1Ua1r9usm88/VliQCw=">AAACgXichVHLLgRBFD3ae7wGGwmLjgkhksltkRAiETbsvAYJMunuKXSmX+mumWBiY+kHLKxIBLHiF2z8gIVPEEsSGwt3ejoRBLdSVadO3XPrVJXh21YoiR6rlOqa2rr6hsZEU3NLa1uyvWMl9AqBKTKmZ3vBmqGHwrZckZGWtMWaHwjdMWyxauRnyvurRRGElucuyz1fbDr6tmttWaYumcomexx1I5fzpLqrDqn5XXVSnWNC2FIfkIPZZIrSFIX6E2gxSCGOeS95gQ3k4MFEAQ4EXEjGNnSE3NahgeAzt4kScwEjK9oXOECCtQXOEpyhM5vncZtX6zHr8rpcM4zUJp9icw9YqaKPHuiKXuierumJ3n+tVYpqlL3s8WxUtMLPth11Lb39q3J4ltj5VP3pWWILY5FXi737EVO+hVnRF/ePX5bGF/tK/XRGz+z/lB7pjm/gFl/N8wWxeIIEf4D2/bl/gpXhtEZpbWEkNTUdf0UDutGLAX7vUUxhFvPI8LmHuMQNbpVqZVAhZbiSqlTFmk58CWXiA+b5kks=</latexit><latexit sha1_base64="wMsD3jni4f2J0DBJpw3OlIxWEK4=">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</latexit>
sha1_base64="ZmJZr5ZegrPDsSoDWqlPzkPFbA0=">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</latexit>
sha1_base64="/6jFeL9FFe/K0asQQnlILhzxz2Y=">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</latexit>

x(0) = 0ẋ(0) =

x(t) = ✓(t)<latexit sha1_base64="px5u8zPnNpPqtNm9zlmwGIPR5NQ=">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</latexit>Imx(t) =

I
sin !0 t
m!0<latexit sha1_base64="/xBbEPFM8QhwdKdYkSfyt43a2+g=">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</latexit>
sha1_base64="AJAutlrgtSU2cIdUGgstdUW11BU=">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</latexit>
sha1_base64="WkFMGxnhxbEMlSVCNK8/FFFtmGw=">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</latexit>!02 =<latexit sha1_base64="MgzJLmfZrPkxikWMen6hl8qd6Sw=">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</latexit>kmElectric current
J(t) =enẋ(t)<latexit sha1_base64="4r/7XlxKqmnN+7+sn+10xx70qpg=">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</latexit>e2 n(t) = ✓(t)cos !0 tm✏(t) = (t) + ✓(t)<latexit sha1_base64="nggnFPu3Eo52HacLZLCuCEIFHRg=">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</latexit>
sha1_base64="zHrL85MkIM0K9pOc7g8HV6po8LU=">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</latexit>
sha1_base64="5uO/3+6TBfT50H+gzUIuZz92drs=">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</latexit>enx(t) =<latexit sha1_base64="cFcPX7kkzWbaNwNHWxJA/YF4BrM=">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</latexit>
sha1_base64="TbRWnqWjO6xvAMTXbIItdLRcJMg=">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</latexit>
sha1_base64="q7dlhxREqqsKnMMaZH3MPgEhHaM=">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</latexit>

e2 n
m

1
E(t)
! 2 + !02<latexit sha1_base64="0G1vh50F+56kRWm57wdKg3dYgvw=">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</latexit><latexit
sha1_base64="gPajX1iPBL4O7aVYJ19ZuO4whoE=">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</latexit>
sha1_base64="n3Ret3+DEi2ll7STapg/5gL4/v4=">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</latexit>

Conductivity and dielectric function<latexit sha1_base64="8gD9AnxppeXYaNzyNI5o1Weanrs=">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</latexit>
sha1_base64="Yet+wds6stKyfiUGehinElSxOno=">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</latexit>
sha1_base64="NUYTMRKPf5K7wGOKxuSlgYFq4EA=">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</latexit>

1E0 e i!t2
2! + !0

Polarization
P (t) =

n oscillator
densityem4⇡e2 nsin !0 t
m!0Dielectric function
D = E + 4⇡P = ✏(!)E4⇡e2 n1✏(!) = 1 +2m! + !02<latexit sha1_base64="Jk4O/ulmqGnoPSyGzpg0u6LiHMc=">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</latexit>
sha1_base64="aHvRFU3Kc5wf/0tpdemJkGZU9TQ=">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</latexit>
sha1_base64="ENohsLFgO2yYQNBpB66FO58mIgY=">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</latexit>

44Calculation of Dielectric function in TDDFT
Time-domain method
Time-dependent Kohn-Sham equation for Bloch orbitals (implemented in SALMON)@
i~ un~k (~r, t) =
@t

(

⌘2
1 ⇣
~ + ~~k + e A(t)
~
i~r
2m
c

)
Z
2
X Z a e2e
+ d~r0⇢(~r0 , t) + µxc [⇢(~r, t)] un~k (~r, t))~ a||~r ~r0 ||~r R
aun~k (~r + ~a, t) = un~k (~r, t)

~
E(t)
=

~
1 dA(t)c dtComputational procedure
Prepare ground statehKS<latexit sha1_base64="4Bqyi9LK1JZoVzIiHLvGNyXcbYg=">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</latexit>

Apply impulsive external potential= shift of k-value
Calculate electric current

E(t) = k (t)A(t) =

ck✓(t)<latexit sha1_base64="FYuGj5RhngfB+TRv6D08AOW2n08=">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</latexit>
sha1_base64="koUSqQhc+1tabamWPRCsDfa6Ivw=">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</latexit>
sha1_base64="CTHkbInTb5YEA5NTmAdIz7FjHU4=">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</latexit>

1J(t) =
V<latexit sha1_base64="4yaAKzxbTsf5tPTayItPLuTwa9I=">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</latexit>
sha1_base64="pyHdAFvUvpljSsM7tVJlJ6RRDns=">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</latexit>
sha1_base64="IntBEw8Fn5uVnBVBiWQ66OCj47Q=">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</latexit>

Fourier transformation
Conductivity and dielectric functionr) = ✏i i (~r)
i (~Z

d~rj(~r, t) ! k (t)(!), ✏(!) = 1 +<latexit sha1_base64="X4sS17sLBVkunkRBgGewPF4SUQ4=">AAACvXichVFNaxRBEK2MxiTrRzbmInhpXCIRZakJgkGQBL14zNcmC5mw9LS1myY9M8307Eoc9g/4BzzklICI+B+8ePEPeMjdi0i8RPCSQ2pnJ0gMag3T/fpVverX3aE12mWIhyPepcujV8bGJypXr12/MVmdurnukm6qqKESk6TNUDoyOqZGpjNDTZuSjEJDG+HOs0F+o0ep00m8lu1a2opkJ9ZtrWTGVKvaDJzuRHI2SCLqyHsPRLDtrFSUz0VRvxKQddok8VlaPBG+uC+CdipV/jCwWmhxvkE/H4J+q1rDOhYhLgK/BDUoYympvoMAXkACCroQAUEMGWMDEhx/m+ADgmVuC3LmUka6yBP0ocLaLlcRV0hmd3js8GqzZGNeD3q6Qq14F8N/ykoBM/gF3+MxfsYP+A1P/torL3oMvOzyHA61ZFuTr2+t/vqvKuI5g+3fqn96zqAN84VXzd5twQxOoYb63qs3x6uPV2byu3iA39n/Ph7iJz5B3Pup3i7Tyh5U+AH8P6/7Ilifq/tY95extvi0fIpxuA13YJbv+xEswnNYggbv+xG+whH88BY88owXD0u9kVIzDefCe3kKYy6rCA==</latexit>
sha1_base64="Qme6RGv+QwuxMUS2MRbk6VcHZys=">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</latexit>
sha1_base64="TcaR6hdf658t9vl+rVKsSFE1c5o=">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</latexit>

4⇡i (!)!
45

Dielectric function for bulk Si TDDFT-ALDAE(t) = k (t)A(t) =

kc✓(t)

Results using LDA is rather poor.

46

Classifications of TDDFT calculations for optical responses
that can be done in current SALMON
Isolated Systems Periodic Systems(Molecules,(Crystalline solids)
Nano-particles)
Weak fields(Linear response)

Polarizabilitya(w)Strong fields
Excitation energy(Nonlinear dynamics) Atomic motion

Dielectric functione(w)Excitation energy
Carrier density
Atomic motion

Light propagationin bulk materials(Maxwell + TDDFT)

1D light
propagation
E(x,t), J(x,t)

47Electron dynamics in a-quartz (SiO2)
Electron density in the ground state
In a plane containing two O atoms

O
Si

48

Electron density change
Laser pulse: 1013 W/cm2, 800nm (1.55eV)

Multiphoton/Tunneling excitation
Conduction bandbandgap~10 eV

1.55eV*7

Valence band

O
Si

Red: increase, Blue: decrease

49

Electron density change
Laser pulse: 1014 W/cm2, 800nm (1.55eV)

Multiphoton/Tunneling excitation
Conduction bandbandgap~10 eV

1.55eV*7

Valence band

O
Si

Red: increase, Blue: decrease50Energy transfer from a pulsed electric field to electrons in solids
Multiphoton/Tunneling excitation

Pulse duration (FWHM) 5.5 fs

Conduction bandbandgap~10 eV

1.55eV*7

Valence band
A. Yamada, K. Yabana, arXiv:1807.02733

51

Possible applications in forefront optical sciences
Optical field induced current in SiO2

Experiment:
Schiffrin et.al,
Nature 493, 70 (2013).

1013W/cm2

High Harmonic Generation in solids

T.T. Luu et.al, Nature 521, 498 (2015)

52

Possible applications in forefront optical sciences
Ultrafast energy transferfrom laser pulse to electronsin SiO2
A. Sommer et.al, Nature 534, 86 (2016)

Creation of crator by strong laser pulse:
Initial stage of nonthermal laser processing
B. Chimier et al, Phys. Rev. B84, 094104 (2011)

53

Classifications of TDDFT calculations for optical responses
that can be done in current SALMON
Isolated Systems Periodic Systems(Molecules,(Crystalline solids)
Nano-particles)
Weak fields(Linear response)

Polarizabilitya(w)Strong fields
Excitation energy(Nonlinear dynamics) Atomic motion

Dielectric functione(w)Excitation energy
Carrier density
Atomic motion

Light propagationin bulk materials(Maxwell + TDDFT)

1D light
propagation
E(x,t), J(x,t)

54

“Calculations” of light-matter interaction
Macroscopic Electromagnetism (EM)Quantum Mechanics (QM)

Light propagation description by Maxwell
equations. Materials’ properties comes into
through constitutive relations (dielectric
constant).

First-principles calculations for dielectric
function.
Perturbation theory in quantum mechanics.

D = εE
Constitutive relation
connects two theories
55

Constitutive Relation bridges EM and QM! ", $ = & ", $ + 4)* ", $Ordinary assumption: Local + Linear* ", $ = 2 -$′. / $ − $ 0 & ", $ 0* ", 5 = . / 5 & ", 5General form (nonlocal, nonlinear)* ", $ = * & "′, $′= ∫ -$′-"′. / ", " 0 , $ − $ 0 & " 0 , $ 0+ 2 -$ 0 -$ 00 -" 0 -"′′ . 3 ", " 0 , " 00 , $ − $ 0 , $ − $′′ & " 0 , $ 0 & " 00 , $′′ + ⋯In recent optical sciences,
Nano-structure
Intense laser pulse

nonlocalnonlinear

Require unified approachof EM and QM.
56

Propagation of light electromagnetic fields
Maxwell eq. + ”modelling for electron dynamics”
Empirical,
computationallycheap

Drude or Lorentz model for dielectric function
electron dynamics = classical Newtonian motion
Empirical models
Rate equation for carriers, Two temperature model
Semiconductor Bloch Equation (SBE)Time evolution of density matrix, with (ab-initio) band structure,empirically introducing dephasing, exciton, e-phonon,…

Ab-initio,
computationallyexpensive

TDDFT in real time

57

Light propagation using dielectric constant
Incident pulse
λ = 800nm,!ω = 1.55eV(below direct gap)

A/c

Vacuum
e=1

Si
e=16

Wave equatione (z ) ¶ 2
c 2 ¶tA( z , t ) 2¶2
A( z , t ) = 0
¶z 2

!# 16
(Si)
ε ( z) = "$# 1 (vacuum)[mm]

When light pulse is very strong, we cannot use dielectric function.

58

Maxwell + TDDFT multiscale simulation
Weak pulsed light irradiating on a surface of bulk Si
I=1010W/cm2
A/c
VacuumSi

Frequency: 1.55eV
Direct bandgap: 2.4eV(LDA
200 macroscopic grid ponts= 200 electron dynamics in parallel[µm]

Z=0 [µm]

Z=0.8 [µm]Z=1.6 [µm]59