How to Use SALMON-2 Overviewsalmon-tddft.jp/mediawiki/images/7/7d/HowtoUseSALMON-2_Overview.pdfScalable Ab-initio Light-Matter simulator for Optics and Nanosciences How to Use SALMON-2:

ScalableAb-initioLight-MattersimulatorforOpticsandNanoscienceshttp://salmon-tddft.jp/

How to Use SALMON-2:Periodic Systems

Overview

1

Mitsuharu UEMOTOCenter for Computational Sciences, University of Tsukuba

SALMONTUTORIAL,TSUKUBA,2017

http://salmon-tddft.jp/ScalableAb-initioLight-MattersimulatorforOpticsandNanoscience

IntroductionofRT-TDDFTforSolid


Real-TimeTDDFTfor“NonlinearOptics”inSolid

• First-principleTDDFT-basedelectrondynamicssimulation• Light-matterinteractionbetween“stronglaserpulse” and“solidcrystal”

Multiscalecalculation

• LightPropagation• Damaging/Ablation• Non-thermallaserprocessing

S.A.Sato,K.Yabana etal,PRB92,1(2015).

P =�(1)E+�(2)E2 + �(3)E3 + · · ·

Real-TimeCalculation

• Linearresponse• DielectricFunction

• Nonlinearresponse:• SHG,THG,Kerreffects

• Lightmatterenergytransfer


SolidCrystal


Silicon(Wikipedia.org)

~a1

~a2

Si

• SolidCrystal• Periodicstructureofatoms• Resultisunmodifiedtotheshiftbythelatticevector:

R =n1a1 + n2a2 + n3a3


PeriodicConditionandBlochWavefunction


b,k(r+R) = eikR b,k(r)

V (r+R) = V (r)

k:Blochwavevectorb:BandIndex

TheBlochwavefunction canrewrittenbyusingtheperiodicfunctionubk :

b,k(r) =eikrub,k(r)

! TR

H = ✏

TranslationOperator

[TR,H] = 0

TR = �

• HamiltonianisinvarianttotheshiftbyR

HamiltonianandTranslationoperatorhascommoneigenstate:

Theelectron’swavefuntion inperiodicsystemsatisfy:

Bloch’stheorem


Brilliouin ZoneandKpoints


k0 = k+

✓2⇡m1

a1,2⇡m2

a2,2⇡m3

a3

◆Thek valueswitharbitraryintegerm correspondstothesameeigenstate

✓� ⇡

a1 k1 ⇡

a1

◆,

✓� ⇡

a2 k2 ⇡

a2

◆,

✓� ⇡

a3 k3 ⇡

a3

◆k-vectorsarerestrictedintheregionofthereciprocalspace

→ Brilliouin Zone(BZ)

Uniformlychoosethek-pointsintheBrilliouin Zone

ZZZdkf(k) !

X

ki

!(ki)f(ki)

ub,k ! ub,ki

DiscretizeBZtofinitenumberofk-points:

ub,keikR $ ub,k0eik

0R


GoverningEquationinPeriodicSystem


i~ @

@tub,k(r, t) =

1

2m

⇣�i~r+ ~k+

e

cA(t)

⌘2

+ vion + vH + vXC

�ub,k(r, t)

vion

vH vXCub,k

E(t) =� A(t)/cIncidentField J(t)

KSorbit Atomicpotential ElectronInteraction

Time-DependentKohn-Sham(TDKS)equationinPeriodicSystem:

ub,k(r) ! zu(NL,NK,NB)Compute TDKS equation by using Finite Difference Method on Real Space Grid


ExampleofRT-TDDFTCalculation


0 1 2 3 4 5

Si[110] (Å)

0

1

2

3

4

Si[

001]

(Å)

Si Si

Si

A(t) =A0 cos2 ⇡t

Tsin!1t

• Frequency=1.55[eV]• Intensity=109 [W/cm2]• Length=10[fs]

http://salmon-tddft.jp/ScalableAb-initioLight-MattersimulatorforOpticsandNanoscience 8

PART 1

Ground State Calculation


GroundStateCalculation


SolveKohn-ShamEquation

• ElectronSingle-particleEigenenergy Spectra

• ElectronDensityofStates(DoS)

• ChargeDensityProfile

• TotalEnergy

"b,k

n(r) =X

b,k

|ub,k(r)|2fk

⇢(E) =X

b,k

�(E � "b,k)fk

"b,kub,k(r) =

1

2m(�~r+ ~k)2 + vion + vH + vXC

�ub,k(r)

→ InitialStateofthereal-timecalculation

Etot


ExampleofGroundStateCalculation


SALMON

Ti-layer

S-layer

S-layer

Ti-layer

Ti

S

TopofViewCrystalStructure

DensityofStates


PART 2

Linear Response Calculation(Dielectric Function)


LinearResponseCalculation


SolveTime-dependentKohn-ShamEquation

• Impulseresponse

• DielectricFunction

• Conductivity

• …

(withweakincidentfield:A(t),E(t))

i~ @

@tub,k(r, t) =

1

2m

⇣�i~r+ ~k+

e

cA(t)

⌘2

+ vion + vH + vXC

�ub,k(r, t)

J(t)

✏ij(!)

�ij(!) J(!) = �(!)E(!)

P(!) =1

4⇡(✏(!)� I)E(!)

A(t) = �A ✓(t)

E(t) = ��E �(t)


LinearResponse(DielectricFunction)


Incident Pulse Weakincidentfield

InducedCurrent

J(t) =� e

m

X

b,k

1

⌦

Z

⌦Re u?

b,k(r, t)⇣�i~r+ ~k+

e

cA(t)

⌘2ub,k(r, t) dr+ JNL

Responsetoweakincident

InducedCurrentDensity

A(t) = �A ✓(t)


LinearResponse(DielectricFunction)


�(!) =

RJ(t)ei!t dtRE(t)ei!t dt

✏(!) =4⇡i

!�(!)

Re[✏(!)] Im[✏(!)]

Conductivity

DielectricConst.

E0

Z�(t)e�i!t dt

(ExperimentalvaluefromD.E.Aspnes etal.Phys.Rev.B,27,2(1983))

Fouriertransformedspectra


PART 3

Pulse Response Calculation


PulseResponseCalculation


SolveTime-dependentKohn-ShamEquation

• InducedCurrent,Polarization

• Nonlinearopticalconstants

• ExcitationEnergy

• ProjectiontoGroundState andNumberofExcitedElectrons:

…withStrongIncidentPulse

i~ @

@tub,k(r, t) =

1

2m

⇣�i~r+ ~k+

e

cA(t)

⌘2

+ vion + vH + vXC

�ub,k(r, t)

A(t)

Eex

(t) = Etot

(t)� Etot

(0)

J(t),P(t) =

ZJ(t) dt

�(2),�(3), · · ·

nex

= Nelec

�X

k

(occ)X

b,b0

��⌦uGS

b,k

��ub0,k(t)↵��2 fk


NonlinearResponsesfromSiO2 Crystal


Incidentpulse

A(t) =A0 cos2 ⇡t

Tsin!1t

• Centralfrequencyω1 =1.55eVPhaselengthTp =20fs

1.00.50.0

-0.5-1.0

Elec

tric

field

(GV/

m)

0 5 10 15 20Time (fs)

Ez(t)

0.0100.0050.000

-0.005-0.010

Pola

rizat

ion

(C/m

2 )

0 5 10 15 20Time (fs)

Pz(t)

Appliedelectricfield

Calculatedpolarization

• Pulseshapefunction


NonlinearResponsesfromSiO2 Crystal


↑Linear ↑Second order ↑Third order

p(1)(t)

p(3)(t)

I 1, I 3I 2, I 3Ez(t)

0.15

0

-0.15

p z(1

) (t) (a

u)

0 5 10 15 20Time t (fs)

6

-6

0

p z(3

) (t) (a

u)

0 5 10 15 20Time t (fs)

• NumericallyexpandedtheSiO2 responsestothep(1) andp(3) components

P(t) =p(1)(t)E + p(2)(t)E2 + p(3)(t)E3 + · · ·

PulseintensityI1 =1X1010 W/cm2I2 =5X1010 W/cm2I3 =1X1011 W/cm2

• χ(3)=4.32X10-22 V2/m2 (LDA),• =1.34X10-22 V2/m2 (mGGA)• Experimentχ(3)=2.5X10-22 V2/m2


ExcitationEnergyofSiliconCrystal


Eex

(t) = Etot

(t)� Etot

(0)

Excitationenergy Z t

J(t0)E(t0) dt0or

S.A.Sato,K.Yabanaetal.J.Chem.Phys.143,(2015).


PART 4

Multiscale Maxwell+TDDFTCalculation


LightPropagationinSolidStateMaterial


Weakintensitylight

Strongintensitylight

• Lightpropagationinthesolidmedia→Maxwell’swaveequation.

E(t)IncidentEMfield

+++

- - -P(t)

Considerthelightirradiationintothesolidmaterial,E(t):AppliedElectricFieldP(t):InducedPolarization

• Dielectricresponsetoappliedfield→ LinearResponse

P = �(1)E = ("� 1)/4⇡ ·E

�r⇥r⇥A(r, t)� "

c2@2

@t2A(r, t) =0


LightPropagationinSolidStateMaterial


J(t) =� e

m

X

b,k

1

⌦

Z

⌦Re u?

b,k(r, t)⇣�i~r+ ~k+

e

cA(t)

⌘2ub,k(r, t) dr+ JNL

J =J[A(t)]

• “ConstitutionRelation”ofElectromagnetics:(CurrentdensitycanbedescribedasthefunctionaloftheEMfield)

Inducedcurrentdensityinmatter

�r⇥r⇥A� 1

c2@

@t2A =� 4⇡

cJ

• GoverningequationofEMfield(Maxwell’swaveeq.)Couple

Lightpropagationinthemedia


MultiscaleMaxwell-TDDFTcalculation


SolveTD-KSEq.inMicroscopicGrid

RT-TDDFTcalculation:

Macroscopicsystem(EMfield)

SolveMaxwell’sEq.inMacroscopicgrid:

Microscopicsystem(Electrondynamics)

FDTD-basedEMcalculation:

r⇥r⇥A+1

c2@2A

@t2=4⇡

cJ

A(t��ti) ! J(t)J(t��ti) ! A(t)

ExchangeJandAateverytimestep

Eunk = H(k)unk

�x ⇠ 10 nm

�x ⇠ 0.05 nm


ExampleofMultiscaleCalculation


Computationmodel

Incidentpulse

Vacuumregion VacuumregionMatter region

• Considerthelaserpulseirradiationtothin-film(d~7μm)Siliconcrystal

• Incidentpulse• Frequency:1.55eV• Pulselength:16fs• Azimuthpolarizedbeam• I=1012 W/cm2

7um

Si

0°

45°

90°

135°

180°

225°

270°

315°

2040

6080

MartinKrausetal,OpEx,18,.21

2Dimensionalproblem

• Cylindricalsymmetriccase• Lightpropagationtothinfilmsemiconductor

• Application• SimulationofLaserProcessingbyCylindricalVectorBeam


ExampleofMultiscaleCalculation


EMFieldVacuum MatterRegion Vacuum

PenetrationDepth(nm)

ElectronicExcitationEnergy

PenetrationDepth(nm)

Energyabsorptionbythemulti-photonexcitationprocess

• 512X64=32768macroscopicpoints


Conclusion


ScalableAb-initioLight-MattersimulatorforOpticsandNanoscience

IsolatedSystem(Molecule) PeriodicSystem(Solidmaterial)

GroundStateCalculation: Totalenergy,DoS,Chargedensityprofile…

LinearResponseCalculation: Dielectricfunction,Oscillatorstrength…

PulseReponseCalculation: Nonlinearexcitation,Energytransfer...

Maxwell+TDDFT MultiscaleCalculation

Let’s enjoy your research life with SALMON !

Documents

How to Use SALMON-2 Overviewsalmon-tddft.jp/mediawiki/images/7/7d/HowtoUseSALMON-2_Overview.pdfScalable Ab-initio Light-Matter simulator for Optics and Nanosciences How to Use SALMON-2: