A Multi-Scale Model for Optical Responses of Nano-Structures · A Multi-Scale Model for Optical Responses of Nano-Structures Songting Luo Department of Mathematics Iowa State University

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Introduction Ehrenfest+TDCDFT Linear Response Theory Examples Conclusion

A Multi-Scale Model for Optical Responses of

Nano-Structures

Songting Luo

Department of Mathematics

Iowa State University

Gang Bao (Zhejiang Univ., China) and Di Liu (Michigan State Univ.)

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Outline

1 Light-Matter Interactions

2 Semiclassical Models

3 Linear Response Formulation

4 Numerical Examples

5 Conclusion

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Light-Matter Interactions

Light-Matter interactions at nano or atomic scale: simultaneous

determination of{Motion of matter.

Evolution of Electromagnetic (EM) field.

• Semiclassical Theories:

Quantum mechanical description of matter.

But for EM field: use classical treatments.

Compared to QED, lower computational burden.

(Cohen-Tannoudji et al 1989, Cho 2003, Keller 1996, Stahl 1987,

etc. )

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Light: Evolution of the EM Field

Maxwell’s equations:

∇× E = −1

c

∂B

∂t, ∇×H =

1

c

∂D

∂t+

4π

cJ,

∇ ·D = 4πρ, ∇ · B = 0,

Continuity equation:

∇ · J = −∂ρ∂t

Constitutive relations:

D = εE, B = µH, J = j + σE,

• c: speed of light in vacuum.

• ε, µ, σ: permittivity, permeability, and conductivity. resp..

• ρ, j: charge density and impressed current density, resp..

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Evolution of the EM Field cont’ed

In Vector and Scalar Potentials (under Coulomb gauge: ∇ ·A = 0):

E = −∇φ− 1

c

∂A

∂t,

B = ∇× A.

Therefore, given input {j, ρ}, the EM field given as (i.e., integral

with scalar/dyadic Green’s functions),

(φ,A) =M[j, ρ]. (1)

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Matter: Motion of the Matter

Time-dependent Schrodinger eqn. in a given transverse EM field:

ι∂ψ(r1, r2, . . . , rN , t)

∂t= HM (t)ψ(r1, r2, . . . , rN , t).

where the (nonrelativistic) Hamiltonian (Cho 2003):

HM ≡∑l

1

2ml

[pl −

elc

A(rl)]2

+ U ≡ H0 +Hint,

• ml, el, rl,pl: mass, charge, coordinate and conjugate

momentum of the coordinate, resp. of l-th particle.

• Mutual Coulomb interaction: U = 12

∑l 6=k

elek|rl−rk| .

• Matter Hamiltonian: H0 =∑

l1

2mlpl

2 + U ;

• Light-Matter Interaction:

Hint =∑

l1ml

(−elc Al · pl +

e2l2c2

A2l

).

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Motion of the Matter cont’ed

Ground state ψ0: H0ψ0 = Eψ0. The matter state will experience

time evolution after switching on radiation-matter interaction.

Then, charge and current densities (ρ(r, t), j(r, t)) are given as

expectation values,

ρ(r, t) = 〈ψ|ρ|ψ〉, j(r, t) = 〈ψ|j|ψ〉,

with,

ρ =∑

l elδ(r− rl),

j =∑

lel2ml

[plδ(r− rl) + δ(r− rl)pl]−∑

le2lmlc

A(rl, t).

Therefore, given input {A, φ}, the current and charge densities are

given as

(j, ρ) = H[φ,A]. (2)

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Self-Consistent Determination of the Interactions

The current and charge densities (j, ρ) and the EM field (φ,A)

must be determined self-consistently, through the coupled system,

Maxwell + Schrodinger

{By Maxwell’s eqn.: (φ,A) =M[j, ρ].

By Schrodinger eqn.: (j, ρ) = H[φ,A].

The system can be solved iteratively until self-consistent

convergence to determine the motion of the matter and the

evolution of the EM field.

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Self-Consistent Determination: High Complexity

Difficulty: Need N -body wavefunction to compute current and

charge densities, i.e.,

ψ(r1, ..., rN , t)⇒ (j(r, t), ρ(r, t)).

N -body system ⇒ 3N spatial variables.

Solving multi-component N -body Schrodinger equations ⇒ too

expensive!

Note: Only (j(r, t), ρ(r, t)) are needed in Maxwell’s equations.

They have only 3 spatial variables!

Therefore: focus on computing (j(r, t), ρ(r, t)).

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Ehrenfest Molecular Dynamics

Denote r = (r1, . . . , rNe), R = (R1, . . . ,RNn) as coordinates of

electrons and nuclei, resp..

Separation of electrons and nuclei (with Born-Oppenheimer

approximation): the electronic wavefunction ψ for electrons,

ι~∂ψ(r, t)

∂t=(

Ne∑l=1

1

2me

(prl −

elc

A(rl))2

+ 〈Wee +Wnn +Wen〉R

)ψ(r, t),

and the nuclear wavefunction E for nuclei,

ι~∂E(R, t)∂t

=

(Nn∑k=1

1

2Mk

(pRk −

Zkc

A(Rk)

)2

+ 〈He〉r

)E(R, t),

(Marx and Hutter 2009, Ullrich 2012, etc.)

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Ehrenfest Molecular Dynamics cont’ed

Where

He =

Ne∑l=1

1

2me

(prl −

elc

A(rl))2

+Wee +Wnn +Wen,

Wee =1

2

Ne∑j 6=k

ejek|rj − rk|

,

Wnn =1

2

Nn∑j 6=k

ZjZk|Rj − Rk|

,

Wen =

Ne∑j=1

Nn∑k=1

ejZk|rj − Rk|

,

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Electrons: Motion of Electrons

The electronic wavefunction ψ(r, t)

ι~∂ψ

∂t=

(Ne∑l=1

1

2me

(prl −

elc

A(rl))2

+ 〈Wee +Wnn +Wen〉R

)ψ,

The electronic charge and current densities:

ρe(r, t) = 〈ψ|ρ|ψ〉, je(r, t) = 〈ψ|j|ψ〉,

The difficulty is still the Ne-body wavefunction (Schrodinger

equation) involved in the computation of charge and current

densities.

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Time-Dependent Current Density Functional Theory

Hohenberg-Kohn (HK) theorems and time-dependent analogue:

The density and/or current density determine uniquely the external

potentials and therefore all physical observables.

Density/Current Density ⇔ External Potential ⇔ Wavefunction

Kohn-Sham (KS) system:

The density and/or current density of the interacting system can

be calculated as the density and/or current density of an auxiliary

non-interacting system.

(Hohenberg and Kohn 1964, Kohn and Sham 1965, Runge and

Gross 1984, Ghosh and Dhara 1988, etc.)

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Density Functional Theory (DFT)

Time-independent Schrodinger equation:

H0ψ0(r1, ..., rNe) = Eψ0(r1, ..., rNe).

Time-independent Kohn-Sham system (single-particle):

HKS0 ψl(r) = εlψl(r), l = 1, 2, ..., Ne.

HKS0 =

1

2p2 + v(r) + vH(r) + vxc(r).

Density:

ρe(r) =∑l

fl|ψl(r)|2,

with fl the occupation number.

v: external potential; vH : Hartree potential;

vxc: scalar exchange-correlation potential.

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Time-Dependent Current Density Functional Theory

(TDCDFT)

Time-dependent Schrodinger equation:

ι∂ψ(r1, ..., rNe , t)

∂t= HMψ(r1, ..., rNe , t).

Time-dependent Kohn-Sham system (single-particle):

ι∂ψl(r, t)

∂t= HKS

M (t)ψl(r, t), l = 1, ..., Ne.

HKSM (t) =

1

2

[p +

1

cA(r, t) + Axc(r, t)

]2+ v(r, t) + vH(r, t) + vxc(r, t).

v: external potential; vH : Hartree potential;

vxc: scalar exchange-correlation potential.

Axc: vector exchange-correlation potential.

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TDCDFT cont’ed

Density and Current density are computed as:

ρe(r, t) =∑l

fl|ψl(r, t)|2;

je(r, t) =−ι2

∑l

fl [ψ∗l∇ψl − ψl∇ψ∗l ] +

ρ

c(A + Axc).

Complexity: 3Ne spatial variables ⇒ 3 spatial variables.

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Exchange-Correlation Potentials

Local Density Approximation (LDA):

Exchange-correlation energy functional:

ELDAxc [ρ] =

∫ρ(r)εhom.xc (ρ(r))dr,

εhom.xc (ρ(r)) = εx(ρ) + εc(ρ).

εx(ρ) = −3

4(3ρ

π)1/3

; εc(ρ) is fitted into analytical functions.

Then

vLDAxc (r) =δELDAxc [ρ]

δρ.

Adiabatic Local Density Approximation (ALDA): extension of LDA.

vALDAxc (r, t) =δELDAxc [ρ]

δρ|ρ=ρ(t).

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Exchange-Correlation Potentials

The Vignale-Kohn (VK) functional is derived and has the following

expression in frequency domain,

ιωAxc,l(r, ω) = ∂lvALDAxc − 1

ρ0(r)

∑j

∂jσxc,lj(r, ω), for l = 1, 2, 3.

σxc is of the form of a viscoelastic stress tensor.

(Dirac 1930, Vosko et. al 1980, Vignale and Kohn 1996, Vignale

et. al 1997)

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Nuclei: Motion of Nuclei

The nuclear wavefunction E(R, t),

ι~∂E∂t

=

(Nn∑k=1

1

2Mk

(pRk −

Zkc

A(Rk)

)2

+ 〈He〉r

)E.

Assume semiclassical-limit (mean-field) approximation:

E(R, t) ≈ a(R, t)eiS(R,t)/~, as ~→ 0,

where a is the amplitude and S is the phase. Substitution implies

Hamilton-Jacobi equation for the phase S(R, t),

∂S

∂t+

Nn∑k=1

1

2Mk|∇RkS|

2 +

Nn∑k=1

1

2Mk

(−2Zk

cA(Rk) · ∇RkS +

Z2k

c2A2(Rk)

)+ 〈He〉r = 0.

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Motion of Nuclei cont’ed

Denoting Pk = ∇RkS, we have the Hamilton’s ODEsdRk

dt=

Pk

Mk− ZkMkc

A,

dPk

dt=

ZkMkc∇RkA(Rk) · Pk −

Z2k

Mkc2A · ∇RkA−∇Rk〈He〉r.

By taking the time derivative of first equation and using the

second equation, we have the Hamilton’s equations,

Mkd2Rk

dt2=−Zkc

(∂A

∂t

)−(∇RkWnn +∇Rk

∫drρe(r, t)Wen

),

for k = 1, . . . , Nn, with ρe the electron density.

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Motion of Nuclei cont’ed

Then the nuclear charge and current densities are given as,

ρn(R) =

Nn∑k=1

Zkδ(R− Rk), jn(R) =

Nn∑k=1

Zkvkδ(R− Rk),

with

vk =dRk

dt.

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A Semiclassical Model

A simpler semiclassical model is derived,

Maxwell + Schrodinger

⇓

Maxwell + Ehrenfest Molecular Dynamics

⇓

Maxwell + TDCDFT for Electrons + Hamilton’s Eqn’s for

Nuclei

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A Semiclassical Model: A Coupled System

The semiclassical model ⇒ the coupled system,

• (φ,A) =M[j, ρ]. By Maxwell’s equations of EM field.

• (je, ρe) = H[φ,A]. By TDCDFT for electrons.

• (jn, ρn) = N [φ,A, je, ρe]. By Hamilton’s Eqn’s for nuclei.

with

j = je + jn, ρ = ρe + ρn.

This coupled system can be solved iteratively until self-consistent

convergence. Note: the system is 3-D.

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Linear Response Theory

Perturbed quantities (in frequency domain ω):

δA(r, ω) = −1

c

∫G(r− r′)δj(r′)dr′,

δφ(r, ω) = −∫G(r− r′)δρ(r′)dr′,

δje(r, ω) =

∫ (χjj(r, r

′, ω)− χjj(r, r′, 0))· δAKS(r

′, ω)dr′

+

∫χjρ(r, r

′, ω)δvKS(r′, ω)dr′,

δρe(r, ω) =

∫χρj(r, r

′, ω) · δAKS(r′, ω)dr′

+

∫χρρ(r, r

′, ω)δvKS(r′, ω)dr′,

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Linear Response Theory cont’ed

δRk =

Zk

(− ιωc

A0(R0k))−∇RkWnn(R

0)−∫dr(ρ0e∇RkWen(R

0))

+ Zk

(− ιωcδA(R0

k))−∫drδρe∇RkWen(R

0)

Zkιωc∇Rk

A0(R0k)−ω2Mk

.

δjn(r, ω) =Nn∑k=1

ZkιωδRkδ(R− R0k),

δρn(r, ω) = −Nn∑k=1

Zk∇Rδ(R− R0k) · δRk.

(Casida 1995, van Faassen et al 2003, Ullrich 2012, Bao et al

2014, 2016, etc. )

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G (G): appropriate scalar (tensorial) Green’s function;

χ: linear response functions:

χαβ(r, r′, ω) =

∑i,a

fi

(ψi(r)αψa(r)ψa(r′)βψi(r′)

(εi − εa) + ω

)

+ fi

(ψi(r)αψa(r)ψa(r′)βψi(r′)

(εi − εa)− ω

)∗;

(ψi, εi): occupied ground-state KS orbitals;

(ψa, εa): unoccupied ground-state KS orbitals.

α, β: density operator ρ = 1 and/or paramagnetic current density

operator jp = −ι(p− p†)/2.

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Kohn-Sham (KS) potentials:

δvKS(r, ω) = δvH(r, ω) + δvxc(r, ω);

δAKS(r, ω) = 1/c(A0(r, ω) + δA(r, ω)) + δAxc(r, ω).

Hartree potentials:

δvH(r, ω) =

∫δρe(r′, ω)

|r− r′|dr′.

Exchange-correlation potentials:

δvxc(r, ω) =

∫fxc(r, r

′, ω)δρe(r′, ω)dr′;

δAxc(r, ω) =

∫fxc(r, r

′, ω)δje(r′, ω)dr′,

with fxc and fxc being the scalar and tensor xc-kernels respectively.

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P-matrix Formulation

Define P-matrix:

Plm =

−ωεm−εl

∫drψl(r)jpψm(r)δAKS +

∫drψl(r)ψm(r)δvKS

εm − εl + ω,

and Plm = Pml − Plm, the above perturbed quantities can be

rewritten as,

δje(r, ω) =∑ia

fi−ω

εi − εaψi(r)jpψa(r)Pai(ω),

δρe(r, ω) =∑ia

fiψi(r)ψa(r)Pai(ω),

(Casida 1995, van Faassen et al 2003, Ullrich 2012, , Bao et al

2014, 2016, etc. )

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P-matrix Formulation cont’ed

δA(r, ω) =∑ia

fiω

c(εi − εa)

∫G(r− r′)ψi(r

′)jpψa(r′)dr′Pai(ω)

− 1

c

∫G(r, r′)δjndr′,

δφ(r, ω) =∑ia

fi

∫G(r− r′)ψi(r

′)ψa(r′)dr′Pai(ω) +

∫G(r, r′)δρndr′.

From the classical MD equation for nuclei, assume we can solve to

obtain:

δRk =∑ia

CRkia Pia + DRk ,

with appropriate coefficients CRk and DRk .

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Then, we have

δρn = −∑ia

Nn∑k=1

Zk∇Rδ(R− R0k) · C

Rkia Pia −

Nn∑k=1

Zk∇Rδ(R− R0k) ·DRk ,

δjn =∑ia

Nn∑k=1

Zkιωδ(R− R0k)C

Rkia Pia +

Nn∑k=1

Zkιωδ(R− R0k)D

Rk .

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Therefore, all the terms can be written as spectral expansions

w.r.t. the P-matrix (w.l.o.g.):

δA =∑ia

CAiaPia + DA, δφ =

∑ia

CφiaPia +Dφ,

δje =∑ia

CjeiaPia, δρe =

∑ia

CρeiaPia,

δjn =∑ia

CjniaPia + Djn , δρn =

∑ia

Cρnia Pia +Dρn ,

δWen =∑ia

CWenia Pia +DWen ,

where the coefficients C’s and D’s can be derived from previous

equations.

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Then we can derive,

(εa − εi + ω)Pai =−ω

εa − εi

∫ψi(r)jpψa(r)

A0

cdr

+−ω

εa − εi

∫ψi(r)jpψa(r)D

Adr +

∫ψi(r)ψa(r)D

Wendr

+∑jb

−ωεa − εi

∫ψi(r)jpψa(r)C

AjbdrPjb

+∑jb

−ωεa − εi

∫ ∫ψi(r)jpψa(r)fxc(r, r

′, ω)Cjejbdr′drPjb

+∑jb

∫ψi(r)ψa(r)C

Wenjb drPjb

+∑jb

∫ ∫ψi(r)ψa(r)fHxc(r, r

′, ω)Cρejb dr′drPjb.

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and,

(εi − εa + ω)Pia =−ω

εi − εa

∫ψa(r)jpψi(r)

A0

cdr

+−ω

εi − εa

∫ψa(r)jpψi(r)D

Adr +

∫ψa(r)ψi(r)D

Wendr

+∑jb

−ωεi − εa

∫ψa(r)jpψi(r)C

AjbdrPjb

+∑jb

−ωεi − εa

∫ ∫ψa(r)jpψi(r)fxc(r, r

′, ω)Cjejbdr′drPjb

+∑jb

∫ψa(r)ψi(r)C

Wenjb drPjb

+∑jb

∫ ∫ψa(r)ψi(r)fHxc(r, r

′, ω)Cρejb dr′drPjb.

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A Linear System on P-matrix

Let us denote:

Fia ≡−ω

εa − εi∫ψi(r)jpψa(r)

A0c dr + −ω

εa−εi

∫ψi(r)jpψa(r)D

Adr +∫ψi(r)ψa(r)DWendr′.

Mia,jb ≡−ω

εa − εi∫ψi(r)jpψa(r)C

Ajbdr.

Kia,jb ≡−ω

εa − εi∫ ∫

ψi(r)jpψa(r)fxc(r, r′, ω)C

jejbdr′dr +∫

ψi(r)ψa(r)CWenjb dr +

∫ ∫ψi(r)ψa(r)fHxc(r, r

′, ω)Cρejb dr′dr.

⇒

(εa − εi + ω)Pai = Fia +

∑jb

(Mia,jb +Kia,jb)Pjb,

(εi − εa + ω)Pia = Fai +∑jb

(Mai,jb +Kai,jb)Pjb.

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A Linear System on P

By simple subtraction and addition, we have a linear system on Pia,

(S− ω2I)P = F , (3)

with

Sia,jb = (εa − εi)2δijδab − 2(εa − εi)(Mia,jb +Kia,jb),

and Fia = 2(εa − εi)Fia.

From the system, we can study

1. Given incident light A0 with frequency ω, solve the system ⇒self-consistent determination of the interactions.

2. Determination of the resonant eigenmodes ω ⇔det(S− ω2I) = 0.

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A Linear System on P cont’ed

About the linear system:

• The system is multiscale in nature: a multiscale scheme needs

to be designed and applied.

• S is of size (occ · unocc)× (occ · unocc). occ, unocc: # of

occupied and unoccupied KS orbitals.

• S is nonlinear in ω. Explicit form is difficult to use.

Fortunately, matrix-vector product can be evaluated, which is

good enough for Krylov subspace methods.

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A Multiscale Scheme

A multi-scale scheme has the following steps: at iteration m,

1 Micro solver: Linear response of TDCDFT with (δφm, δAm)

and δRm((δjmn , δρ

mn )) to obtain (δjm+1

e , δρm+1e ); and Classical

MD with (δφm, δAm) and (δjme , δρme ) to obtain

δRm+1

((δjm+1n , δρm+1

n )). Microscale on a smaller domain.

2 Macro solver: Maxwell’s equations with inputs

(δjm+1, δρm+1) to update (δφm+1, δAm+1); Macroscale on a

larger domain.

3 Repeat until self-consistent convergence.

Linear interpolation is used for the communication between the

numerical solutions of the micro and macro solvers.

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Linear Solver and Eigensolver

Linear solver: BiCGSTAB algorithm (Van der Vorst 1992, Bao et

al 2014, 2016).

Eigensolver: an iterative Jacobi-Davidson algorithm (Sleijpen and

Van der Vorst 1996, Bao et al 2014, 2016):

1. Initialization: an initial guess of ω0.

2. With ωm, update S(ωm).

3. With S(ωm), update ωm+1.

4. Repeat until convergence.

The solvers are incorporated into the multiscale scheme.

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Numerical Implementation

Within the BiCGSTAB and Jacobi-Davidson algorithms:

Macro solver: hybrid edge-nodal finite element method for

Maxwell’s equations.

Micro solver: TDCDFT and Hamilton’s Eqn’s.

Meshes for macro and micro solvers:

Figure: Left: domain and mesh for Maxwell’s eqn; Right: a slice at z = 0.

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Example 1

Computed lowest eigenfrequency: 0.3452 (a.u.) with

4 occ, 8 unocc.

Computed linear response spectra:

0.33 0.34 0.35 0.36

0.005

0.01

0.015

0.02

Frequency

Ab

sorp

tio

n C

ross

Sec

tio

n

Figure: For methane (CH4): Response spectra (frequency in a.u.).

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Example 2


4 occ, 8 unocc.


0.29 0.3 0.31 0.32

2

4

6

8

10

x 10−3

Frequency

Abs

orpt

ion

Cro

ss S

ectio

n

Figure: For silane (SiH4): Response spectra (frequency in a.u.).

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Example 3


34 occ, 34 unocc.


0.13 0.135 0.14 0.145 0.15

0.5

1

1.5

2

2.5

Frequency

Ab

so

rpti

on

Cro

ss S

ecti

on

Figure: For trans-Azobenzene (C12H10N2): Response spectra (frequency

in a.u.).

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Example 3 cont’ed


34 occ, 34 unocc.


0.13 0.14 0.15

2000

4000

6000

8000

10000

Frequency

|Dip

ole

Mo

men

t|

0.13 0.135 0.14 0.145 0.15

0.2

0.4

0.6

0.8

1

1.2

Frequency

|δR|

Figure: Left: dipole moment; Right: |δR|.

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Example 3 cont’ed

With surface plasmon enhancement:

Au Au

trans-‐Azobenzene x

z

Figure: Gold nano particles and molecule

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Example 3 cont’ed


2.03

3.47

4.91

5.870e-01

6.352e+00|E|

1.12

2.24

3.37

4.49

5.303e-01

5.020e+00|E|

1.9

2.85

3.79

5.491e-01

4.343e+00|E|

Figure: Plots of |E| with ω = 0.15 (a.u.). Radius of the nano particle is

10nm, and inter-distance of the nano particles is 2, 4, 6nm. The

maximum magnitude is 6.352e0, 5.050e0, 4.434e0 corresponding to the

inter-distances.

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Example 3 cont’ed

With surface plasomon enhancement:

2.45

4.3

6.16

5.917e-01

8.017e+00|E|

3.11

5.57

8.02

6.538e-01

1.048e+01|E|

4.2

7.66

11.1

7.370e-01

1.459e+01|E|

Figure: Plots of |E| with ω = 0.15(a.u.). Radius of the nano particle is

15, 20, 25nm, and inter-distance of the nano particles is 2nm. The

maximum magnitude is 8.017e0, 1.048e1, 1.459e1 corresponding to the

inter-distances.

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Example 3 cont’ed


Figure: Dipole moment |µ|. Size of nano particles = 10nm, inter-distance

= 10nm; Angle between the polarization direction and the x-axis is

0, π/6, π/3, π/2. Red: sphere; Black: no Gold particles

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Example 3 cont’ed

With surface plasomon enhancement:

Figure: Dipole moment |µ|. Size of nano particles = 20nm, inter-distance

= 10nm; Angle between the polarization direction and the x-axis is

0, π/6, π/3, π/2. Red: sphere; Black: no Gold particles

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Conclusion

• Multiphysics modeling and multiscale computation of optical

response of nanostructure with semiclassical theory.

• Semiclassical model: Maxwell + TDCDFT + Hamilton’s Eqn.

Linear response formulation is derived.

• Multiscale Scheme: micro solver and macro solver.

Self-consistent determination of the interactions, and

computation of resonant eigenfrequencies.

• Future plans: more real problems; time-domain

implementations.

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Thank You

Supported by NSF DMS.

Documents

A Multi-Scale Model for Optical Responses of Nano-Structures · A Multi-Scale Model for Optical Responses of Nano-Structures Songting Luo Department of Mathematics Iowa State University