SALMON TUTORIAL, TSUKUBA, 2017

How to Use SALMON-2:
Periodic Systems
Overview

Mitsuharu UEMOTO
Center for Computational Sciences, University of Tsukuba
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Introduction of RT-TDDFT for Solid
Real-Time TDDFT for “Nonlinear Optics” in Solid• First-principle TDDFT-based electron dynamics simulation• Light-matter interaction between “strong laser pulse” and “solid crystal”
Real-Time Calculation

Multiscale calculation• Linear response• Dielectric Function• Nonlinear response:• SHG, THG, Kerr effects• Light Propagation• Damaging / Ablation• Non-thermal laser processing

P = (1) E + (2) E2 +• Light matter energy transfer(3)E3 + · · ·S.A. Sato, K. Yabana et al, PRB 92, 1 (2015).

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2

Solid Crystal
Si

Silicon (Wikipedia.org)~a2~a1•

Solid Crystal• Periodic structure of atoms• Result is unmodified to the shift by the lattice vector:

R =n1 a1 + n2 a2 + n3 a3Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience
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Periodic Condition and Bloch Wavefunction[TR , H] = 0• Hamiltonian is invariant to the shift by R

V (r + R) = V (r) ! T̂RTranslation Operator

Hamiltonian and Translation operator has common eigenstate:

H =✏TR=

Bloch’s theorem
The electron’s wavefuntion in periodic system satisfy:b,k (r + R) = eikRb,k (r)k: Bloch wavevector
b: Band Index

The Bloch wavefunction can rewritten by using the periodic function ubk :b,k (r) =eikr

ub,k (r)Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience
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Brilliouin Zone and K points
The k values with arbitrary integer m corresponds to the same eigenstate✓◆2⇡m1 2⇡m2 2⇡m3ikR
ik0 R
0ub,k e$ ub,k0 e
k =k+,,
a1
a2
a3
k-vectors are restricted in the region of the reciprocal space✓⇡⇡ k1 a1a1◆ ✓,⇡⇡ k2 a2a2◆ ✓,⇡⇡ k3 a3a3◆→ Brilliouin Zone (BZ)

Discretize BZ to finite number of k-points:

ZZZub,k ! ub,kidkf (k) !

X!(ki )f (ki )kiUniformly choose the k-points in the Brilliouin Zone
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Governing Equation in Periodic System

Incident Field

E(t) =J(t)Ȧ(t)/c

Time-Dependent Kohn-Sham (TDKS) equation in Periodic System:⌘2@1 ⇣ei~ ub,k (r, t) =i~r + ~k + A(t) + vion + vH + vXC ub,k (r, t)@t2mc
KS orbitub,kAtomic potential

vion

Electron Interaction

vHvXC

Compute TDKS equation by using Finite Difference Method on Real Space Gridub,k (r) ! zu(NL,NK,NB)
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Example of RT-TDDFT Calculation

A(t) =A0 cos2⇡t
sin !1 t
TFrequency = 1.55 [eV]Intensity = 109 [W/cm2]
Length = 10 [fs]•••Si[001] (Å)

43Si2Si

1001Si2345

Si[110] (Å)

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PART 1
Ground State Calculation

Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience
http://salmon-tddft.jp/8Ground State Calculation
Solve Kohn-Sham Equation"b,k ub,k (r) =

12( ~r + ~k) + vion + vH + vXC ub,k (r)2m→ Initial State of the real-time calculation• Electron Single-particle Eigenenergy Spectra• Electron Density of States (DoS)"b,k⇢(E) =X(E"b,k )fkb,k• Charge Density Profile• Total Energyn(r) =X
b,kEtot|ub,k (r)|2 fk

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Example of Ground State Calculation
Top of View

Crystal Structure
S-layerS-layerS

Ti-layerTi
Density of StatesSALMON

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Ti-layer10

PART 2
Linear Response Calculation(Dielectric Function)

Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience
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11Linear Response Calculation
Solve Time-dependent Kohn-Sham Equation

⌘2@1 ⇣ei~ ub,k (r, t) =i~r + ~k + A(t) + vion + vH + vXC ub,k (r, t)@t
2mc(with weak incident field: A(t), E(t))A(t) = A ✓(t)

E(t) =• Impulse responseJ(t)• Dielectric Function✏ij (!)• Conductivityij (!)P(!) =E (t)1(✏(!)
4⇡I) E(!)

J(!) = (!)E(!)• …
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Linear Response (Dielectric Function)
Induced Current Density
J(t) =e X1m
⌦
b,kZ

⌦

Re u?b,k (r, t)

Incident Pulse⇣⌘2ei~r + ~k + A(t) ub,k (r, t) dr + JNL
cWeak incident field
A(t) = A ✓(t)

Induced Current

Response to weak incident

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Linear Response (Dielectric Function)
Fourier transformed spectra
Conductivity

Dielectric Const.

Re[✏(!)]

R
J(t)ei!t dt(!) = RE(t)ei!t dt

4⇡i✏(!) =(!)!

E0Z(t)e i!t dt

Im[✏(!)](Experimental value from D. E. Aspnes et al. Phys. Rev. B, 27, 2 (1983))
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PART 3
Pulse Response Calculation

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15

Pulse Response Calculation
Solve Time-dependent Kohn-Sham Equation

⌘2
@1 ⇣e
i~ ub,k (r, t) =i~r + ~k + A(t) + vion + vH + vXC ub,k (r, t)@t
2mc…with Strong Incident Pulse

A(t)• Induced Current, Polarization• Nonlinear optical constants• Excitation Energy

J(t), P(t) =(2),(3)

,···

Z

J(t) dt

Eex (t) = Etot (t)Etot (0)• Projection to Ground State and Number of Excited Electrons:
nex = Nelec

X (occ)X ⌦kuGSb,k ub0 ,k (t)b,b0Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience
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↵2fk16

Applied electric field

Incident pulse

Electric field (GV/m)

Nonlinear Responses from SiO2 Crystal• Pulse shape function

1.00.5
0.0-0.5-1.0Ez (t)0A(t) =A0 cos2

510Time (fs)15

20

10Time (fs)

1520

⇡t
sin !1 tT• Central frequency ω1 = 1.55 eV
Phase length Tp = 20 fs

Polarization (C/m2)

Calculated polarization
0.010
Pz (t)
0.005
0.000-0.005-0.010
05

Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience
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Nonlinear Responses from SiO2 Crystal• Numerically expanded the SiO2 responses to the p(1) and p(3) components

P(t) =p(1) (t)E + p(2) (t)E 2 + p(3) (t)E 3 + · · ·↑Linear↑Second order↑Third orderp(1) (t)pz(1)(t) (au)0.15

I 1, I 3
I 2, I 3
E z(t)

0-0.150510Time t (fs)

15Pulse intensity
I1 = 1 X 1010 W/cm2I2 = 5 X 1010 W/cm2I3 = 1 X 1011 W/cm2

20p(3)(t)pz(3)(t) (au)6

• χ(3) = 4.32 X 10-22 V2/m2 (LDA),•= 1.34 X 10-22 V2/m2 (mGGA)• Experiment χ(3) = 2.5 X 10-22 V2/m

0-60510Time t (fs)1520

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Excitation Energy of Silicon Crystal
Excitation energyEex (t) = Etot (t)Etot (0) orZ t

J(t0 )E(t0 ) dt0

S. A. Sato, K. Yabana et al. J. Chem. Phys. 143, (2015).
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PART 4Multiscale Maxwell+TDDFT
Calculation

Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience
http://salmon-tddft.jp/20Light Propagation in Solid State Material
Consider the light irradiation into the solid material,+++E(t) : Applied Electric Field
P(t) : Induced Polarization---Incident EM field

E(t)

P(t)

Weak intensity light• Dielectric response to applied field → Linear Response

P=(1)E = ("1)/4⇡ · E• Light propagation in the solid media → Maxwell’s wave equation.

r ⇥ r ⇥ A(r, t)" @2A(r, t) =0
c2 @t2Strong intensity light
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Light Propagation in Solid State Material
Induced current density in matter

J(t) =e X1m⌦b,kZ

⌦Re u?b,k (r, t)⇣⌘2e
i~r + ~k + A(t) ub,k (r, t) dr + JNLc

• “Constitution Relation” of Electromagnetics:(Current density can be described as the functional of the EM field)

J =J[A(t)]Light propagation in the media

Couple• Governing equation of EM field (Maxwell’s wave eq.)

r⇥r⇥A

1 @
A=2
2c @t

4⇡
Jc

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Multiscale Maxwell-TDDFT calculation
Macroscopic system (EM field)

Microscopic system (Electron dynamics)x ⇠ 0.05 nmx ⇠ 10 nm

Solve Maxwell’s Eq. in Macroscopic grid:1 @ 2 A 4⇡r⇥r⇥A+ 2 2 = Jc @tc
FDTD-based EM calculation:J(tti ) ! A(t)Solve TD-KS Eq. in Microscopic GridÊunk = Ĥ(k)unk

RT-TDDFT calculation:A(tti ) ! J(t)Exchange J and A at every time step
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Example of Multiscale Calculation
2 Dimensional problem• Cylindrical symmetric case
• Light propagation to thin film semiconductor• Application• Simulation of Laser Processing by Cylindrical Vector Beam
Computation modelMartin Kraus et al, OpEx, 18,. 21• Consider the laser pulse irradiation to thin-film (d~7μm) Silicon crystal
Vacuum region

Matter region

7um

Incident pulse
90°

135°

45°

20406080

180°

0°

225°

315°

270°

Si

Vacuum region• Incident pulse• Frequency: 1.55 eV• Pulse length: 16 fs• Azimuth polarized beam
• I = 1012 W/cm2Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience
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Example of Multiscale Calculation
EM Field
VacuumVacuumMatter Region

Penetration Depth (nm)Electronic Excitation Energy

Energy absorption by themulti-photon excitation process

Penetration Depth (nm)• 512 X 64 = 32768 macroscopic points
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Conclusion

Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience
Isolated System (Molecule)

Periodic System (Solid material)

Ground State Calculation:

Total energy, DoS, Charge density profile …

Linear Response Calculation:

Dielectric function, Oscillator strength …

Pulse Reponse Calculation:

Nonlinear excitation, Energy transfer ...
Maxwell+TDDFT Multiscale Calculation

Let’s enjoy your research life with SALMON !
Scalable Ab-initio Light-Matter simulator for Optics and Nanoscience
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