Numerical Simulation of Nanoscale Materials by Discrete Dipole Approximation

- 15 - http://www.ivypub.org/mc

Mathematical Calculation Dec 2012, Volume 1, Issue 1, PP.15‐19

Numerical Simulation of Nanoscale Materials by Discrete Dipole Approximation Wu Wang #

Supercomputing Center, Computer Network Information Center, Chinese Academy of Sciences, Beijing 100190, PR. China

#Email: [email protected]

Abstract

In this paper, the discrete dipole approximation (DDA) is applied to calculate the scattering property of optical materials in

nanoscale. First of all, the scattering electric field of golden nanosphere is calculated to validate the DDA. Then the DDA is used

to calculate the reflectance spectrum of a photonic crystal, an artificial material with periodic nanostructure. At last the DDA is

used for the electromagnetic cloaking structure of infinite 3D cylinder-shaped anisotropic metamaterial. The results are in

accordance with the MIE theory, FDTD, FEM method and predicted physical phenomenon.

Keywords: Nanoscale Materials; Discrete Dipole Approximation; Nanoparticle; Photonic Crystal; Electromagnetic Cloaking

I. INTRODUCTION

Nanoparticle is a widely researched concept since it has potential applications in optical, electronic and biomedical fields, such as quantum confinement in semiconductor particles, surface plasmon resonance in some metal particles [1], it’s an effective bridge between bulk materials and atomic. Metamaterials are artificial materials with periodic nano-structures, and have some physical property which not exist in nature. For example, negative index metamaterial can transmit light to a opposite direction compared to normal materials; electromagnetic cloaking metamaterial with anisotropic distribution enables light to bend in a path around an object [2]. Photonic crystals with periodic nano-structures can control light at certain frequency range due to its bandgap and guided resonance [3].

Different numerical methods have been used to solve Maxwell’s equations for wave or optical scattering, such as FDTD, FEM, method of moment (MoM). Fast multipole method can be used to accelerate MoM in O(NlogN) by integral kernel expansion and hierarchical partition [4]. The discrete dipole approximation (DDA) discretizes targets by dipole sites [5], and solves the polarization of each dipole by FFT-based iteration [6], so the complexity of DDA is also O(NlogN). DDA is effective for optical scattering problem in nanoscale. There are some implementations of DDA, such as DDSCAT [7] and OpenDDA [8]. In this paper DDA is used to simulation three types of nanoscale materials: nanosphere, metamaterial, photonic crystal, the analysis of calculation results is also provided.

II. FORMULATION OF DDA

The DDA method solves the volume integral equation discretized from Maxwell’s equations. When the scattering target is discretized with Nd dipole sites, the polarizations {Pj} of the dipoles satisfy following linear equations

2

10

, 1, ,dN

incij j i d

j

kP E i NA

(1)

where k=2/ is the wave number, 0 is the dielectric constant in vacuum, Einc is the incident electric field, then non-diagonal elements of coefficient matrix can be described as (r r denotes the tensor product)

23 2 2

exp( ) 1(3 1) ( 1) , ,

4ij ij

ij ij ij ijij ij

kr krr i j

r k rA

r r I (2)

for the diagonal elements, we can use Clausius-Mossotti polarizability [8, 10] of each dipole 3

13

13, ,

1 / 6 4 2

CMCMi i

ii i i iCMi i

d

kA

(3)


where d is the dipole size, it satisfies the criterion |nkd|<1, n is the refractive index of the material. Equations (1) can be solved by Krylov subspace iteration method (such as CG, BiCGStab, etc). The complexity of

the iteration depends on the matrix-vector product, so the runtime and storage is O(Nd2). Since dipoles are distributed

at the lattice sites, rij = ri - rj, the coefficient matrix A can be changed to be a three-level Toplitz matrix after extending polarization vector with 0 at the vacuum sites. . If Ne denotes the number of extended lattice sites, FFT can be used to calculate the matrix-vector product with a complexity of O(NelogNe). So DDA is a FFT- based fast algorithm. When the polarizations of all the dipoles are solved, the scattering electric field can be calculated by

2

1

exp( )exp( )( ) ,

dN

sca j jj

krE k k P

r r

rr r r r I r (4)

The cross section of scattering, absorption and extinction can be also evaluated.

III. NUMERICAL SIMULATION

The following results are calculated on SGI Altix4700 (installed in Supercomputing Center, Computer Network Information Center, CAS), the CPU is Intel Itanium2 (1.66 GHz)the number of OpenMP threads is 8.

A. Scattering of Nanosphere

The radius of golden nanosphere (Figure 1, left) is a=398 nm, the wavelength of incident light is =500 nm. The propagation direction k is along x axis, Einc is along y axis, the number of dipoles for DDA model is Nd= 515 811. The complex refractive index of Au at 0.5um is (0.96, 1.01). The iteration solver is BiCGStab [9] with point Jacobi preconditioner, and the tolerance of normalized residual |AP-Einc|/|Einc| is 10-6. Figure 1 (right) shows the normalized electric field intensity |E|/|Einc| distribution on xOy plane passing through the center of the sphere.

FIG. 1 DIPOLE DISCRETISATION OF NANOSPHERE (LEFT) AND ITS NEAR FIELD (RIGHT)

TABLE 1 ABSOPTION AND SCATTERING EFFICIENCY FACTOR

ka DDA MIE Theory Relative Error

Qabs Qsca Qabs Qsca Qabs Qsca

1 1.7189 0. 5993 1.7294 0.5955 0.607% 0.638%

2 1.5093 1.1141 1.5091 1.1192 0.0133% 0.456%

3 1.3474 1.2875 1.3446 1.2972 0.208% 0.748%

4 1.2390 1.3549 1.2335 1.3678 0.459% 0.943%

5 1.1630 1.3844 1.1564 1.3983 0.571% 0.994%

6 1.1074 1.3981 1.1011 1.4123 0.572% 1.005%

7 1.0649 1.4044 1.0582 1.4184 0.633% 0.987%

8 1.0313 1.4067 1.0242 1.4207 0.693% 0.985%


Keep and Nd unchanged with varying radius a, the absorption (Qabs) and scattering (Qsca) efficiency factor for ka = 1,...,8 compared with the results of MIE theory are listed in Table1. The average number of iterations is 13 and the runtime is 7.2 second per iteration. From this table we can see that DDA has an acceptable accuracy (less than 1%).

B. Photonic Crystal of Slab with Lattice Holes

Then DDA simulation of the photonic crystal slab (PCs) with lattice holes is shown in Figure 2 (left), and one unit cell of this triangular periodic structure is shown in Figure 2 (right). The material is GaAs (a widely used semiconductor), lattice size (hole distance or periodic length) is a=300 nm. The dielectric constant of this slab, the radius of air holes and the thickness of the slab are chosen to be 12, 0.2a and 0.5a, respectively. The incident wave is propagating along z direction, its wavelength is from 0.6 um to 2 um where no diffraction occurs. Set the dipole size d=10nm for DDA, since m=3.464, it satisfies the criterion of DDA: kd|m|<1.

FIG. 2 PHOTONIC CRYSTAL OF SLAB WITH LATTICE HOLES (LEFT) AND UNIT CELL (RIGHT)

The reflectance spetrum (S11 of Mueller matrix) calculated by DDA is shown in Figure 3. From the figure we can see a small shift in the resonant frequency can lead to a drastic change in the reflection (Lorentzian reflectivity line

shape) at 732 nm and 779 nm. Compared the result of FDTD for the same structure in [13], the difference is small.

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normali Frequency (in 2c/a)

Re

flect

an

ce

FIG. 3 THE REFLECTANCE SPECTRUM OF PCs WITH LATTICE HOLES

C. Cloaking Metamaterial

For a cylindrical cloaking metamaterial, the ideal parameters are given in cylindrical coordinate [10] as 2

, ,r r z z

r a r b r a

r r a b a r

(5)

in which a and b are the radius of inner and outer cylinder (shown in Figure 4, left), respectively. These parameters come out from transformations optics. Set a=400nm and b=800nm, the incident wavelength =600 nm. A DDA method of 2D version is used in [11], but here the nearfield distribution is calculated by DDA using a real 3D-cylinder with a smaller cylindrical hole, the hight is limited to be 50a in order to restrict the scale of unknowns. The nearfield through the cylinder is demonstrated in Figure 4 (right).

From the figure we can see that a cylindrical metamaterial described above can bend light around an object, the electric field intensity is nearly zero inside the inner cylinder, so cloaking (invisible) phenomenon works for this material with anisotropic parameters. The field intensity of cloaking metamaterial can be validated by FEM [12].


FIG. 4 CYLINDRICAL CLOAK (LEFT) AND ITS NEARFIELD DISTRIBUTION (RIGHT)

IV. CONCLUSIONS

The DDA method is effective for calculation of electromagnetic scattering of materials with nanostructures. Time-domain FEM and FDTD are full-wave solver but need thousands of timestep iterations, while DDA solve linear equations by FFT without timestep iteration. The results of a golden nanosphere, a lattice photonic crystal and an anisotropic cloaking metamaterial calculated by DDA are consistent with the MIE theory, FDTD and predicted physical phenomenon. Parallel implementation of DDA by MPI on cluster with multi-processors is our future work.

ACKNOWLEDGMENT

This work is supported by the National High-tech R&D Program (2012AA01A309) and National Natural Science Foundation of China (61202054). The author would like to thank Prof. N. Nishimura of Kyoto University for the knowledge and numerical simulation method of metamaterials. The author would also like to thank Prof. Y. J. Song of Beihang University for the fruitful discussion on the DDA.

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AUTHORS

Wu WANG was born in Hubei, China, in

1982. He received the B.S. degree in

mathematics from Peking University,

Beijing, China, in 2004 and the Ph.D.

degree in engineering from Chinese

Academy of Sciences, Beijing, China, in

2010.

From 2011 to 2012, he was a Postdoctoral Researcher at the

Graduate School of Informatics, Kyoto University at Kyoto,

Japan. He is currently an Assistant Researcher of

Supercomputing Center, Computer Network Information Center,

Chinese Academy of Sciences. His research interests are in

computational electromagnetics and parallel computing.

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Numerical Simulation of Nanoscale Materials by Discrete Dipole Approximation