The article information
 Zhe Zheng, Bo Da, Kejun Zhang, Zejun Ding
 郑哲, 达博, 章可俊, 丁泽军
 Simulation Study of Electron Beam Induced Surface Plasmon Excitation at Nanoparticles
 电子束诱导的纳米颗粒表面等离激元激发模拟
 Chinese Journal of Chemical Physics, 2018, 31(5): 655660
 化学物理学报, 2018, 31(5): 655660
 http://dx.doi.org/10.1063/16740068/31/cjcp1805096

Article history
 Received on: May 6, 2018
 Accepted on: June 21, 2018
b. CAS Key Laboratory of Geospace Environment, Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China;
c. Center for Materials research by Information Integration (CMI2), Research and Services Division of Materials Data and Integrated System (MaDIS), National Institute for Materials Science, 121 Sengen, Tsukuba, Ibaraki 3050047, Japan
With the rapid development of nanotechnology in recent years the synthesis and characterization of nanostructured materials which can be modulated both in size and shape have attracted much attention for their unique electromagnetic characteristics [1]. One of the most important physical properties of nanomaterials is the local surface plasmon excitation occurring on the surface of particles or in the gap between them. Different from the usual surface plasmon which propagates along the surface plane, the local surface plasmon is well localized at the curved surface and, hence, the dispersion is closed related to the morphology of nanostructure. It is because of such locality the electromagnetic field can be greatly enhanced nearby the particular structure and lead to related physical phenomenon.
It was found several hundred years ago that metallic nanoparticles have selective optical absorption to visible light, this property is used to produce color window glass in church in Europe. The earliest theory about the unique optical property of metallic nanoparticles is developed by Mie who derived the absorption and scattering of light by the isotropic nanoparticles via solving Maxwell equation [2]. Due to its exact analytical form, the Mie's theory is still important nowadays for the validation of other theories. However, the Mie's theory is limited to the spherical or elliptical particle shape cases and for other complex particle shapes one can only numerically calculate their optical properties. For this, it has been developed many numerical simulation methods to deal with the interaction of light with the structures in arbitrary shape; among them the widely used ones are Tmatrix method [3], finite difference time domain (FDTD) method [4], boundary element method (BEM) [5], multipole methods [6], discrete dipole approximation (DDA) method [7] etc. Among them DDA method has attracted much interest and became one of the most important methods for study of nanoparticles in arbitrary shape. It was firstly proposed by Purcell and Pennypacker for detail analysis of dust matter [8]; then many ones have made significant improvements to make it a powerful tool for calculating optical properties, i.e. the absorption, extinction and scattering of light by nanoparticles [912]. However, at present this method can only deal with the interaction of external light field with nanostructured materials. Therefore, for application to the investigation of surface plasmon excitation problem by scanning transmission electron microscopy combined with electron energy loss spectroscopy (STEMEELS) [1315], it is highly expected to extend the traditional DDA method to the case of surface plasmon excitation induced by an external electron beam. In this work we aimed to improve the DDA method for incorporation of incident electrons in addition to light field as excitation source. The surface plasmon excitation at Ag particles is studied via the calculation of EELS spectra and results have demonstrated the effectiveness of the extended DDA method.
Ⅱ. THEORY A. DDA method for light fieldBy the DDA method a nanostructured material is spatially divided into
$ \begin{align} {{\bf{P}}_j} = {α _j}{\bf{E}}_j^{{\rm{loc}}} = {α _j}\left( {{\bf{E}}_j^{{\rm{app}}}  \sum\limits_{k ≠ j} {{{\bf{A}}_{jk}}{{\bf{P}}_k}} } \right) \end{align} $  (1) 
where
$ \begin{align} {\bf{E}}_j^{{\rm{app}}} = {{\bf{E}}_0}\exp \left( {i{\bf{k}} \cdot {{\bf{r}}_j}  i\omega t} \right) \end{align} $  (2) 
where
By using the lattice dispersion relation (LDR), the dipole polarizability
$ \begin{eqnarray} \alpha _{j, ll}^{{\rm{LDR}}} = \frac{\alpha _{j, ll}^{\rm{CM}}}{1 + \left( \alpha _{j, ll}^{\rm{CM}}/ d^3\right)\left[b_1(k_0d )^2( 2/3)i(k_0d)^3 \right]} \end{eqnarray} $  (3) 
where
$ \begin{eqnarray} \alpha _{j, ll}^{\rm{CM}}= \frac{3d^2}{4\pi}\frac{\varepsilon _{ll}  1}{\varepsilon _{ll} + 2} \end{eqnarray} $  (4) 
In DDA the matrix describing the interaction of light field with a nanostructured material is written as,
$ \begin{align} {\bf{A}}_{jk}{\bf{P}}_k& = \frac{\exp \left( ikr_{jk}\right)}{r_{jk}^3}\Bigg\{k^2{\bf{r}}_{jk}\times \left( {\bf{r}}_{jk}\times {\bf{P}}_k \right) + \\ & \frac{1  ikr_{jk}}{r_{jk}^2} \times\left[r_{jk}^2{\bf{P}}_k3{\bf{r}}_{jk}\left( {\bf{r}}_{jk}\cdot {\bf{P}}_k\right) \right]\Bigg\}\quad\\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (j\neq k) \end{align} $  (5) 
where
$ \begin{eqnarray} \sum\limits_{k = 1}^N {{{\bf{A}}_{jk}}{{\bf{P}}_k} = {\bf{E}}_j^{{\rm{app}}}} \;\;\left( {j = 1, ..., N} \right) \end{eqnarray} $  (6) 
For a system composed of
$ \begin{eqnarray} {\bf{\tilde A\tilde P}} = {{\bf{\tilde E}}^{{\rm{app}}}} \end{eqnarray} $  (7) 
By solving this equation of
$ \begin{eqnarray} {C_{{\rm{ext}}}} = \frac{{4\pi k}}{{{{\left {{{\bf{E}}^{{\rm{app}}}}} \right}^2}}}\sum\limits_{j = 1}^N {{\rm{Im}}\left( {{\bf{E}}{{_j^{{\rm{app}}}}^{, *}} \cdot {{\bf{P}}_j}} \right)} \end{eqnarray} $  (8) 
where
$ \begin{eqnarray} {Q_{{\rm{ext}}}} = \frac{{4\pi k}}{{\pi r_{{\rm{eff}}}^2{{\left {{\bf{E}}_j^{{\rm{app}}}} \right}^2}}}\sum\limits_{j = 1}^N {{\rm{Im}}\left\{ {{\bf{E}}_j^{{\rm{app}}, *} \cdot {{\bf{P}}_j}} \right\}} \end{eqnarray} $  (9) 
where
The plane wave electromagnetic field is used for the study of local surface plasmon excitation by external light field; while in the case of incident electron beam we have to consider the action of a specific external electric field on the nanostructure. In fact, in study of surface excitation at semiinfinite medium by external charges one usually considers that the excitations induced by external light and external charges have no essential difference except that the form of electric fields employed is different [16].
We consider now a charge
$ \begin{align} {\bf{E}}_{j, x}^{{\rm{app}}} = \frac{{q\omega }}{{2\pi {\varepsilon _0}{v^2}}}\exp \left( {i\omega \frac{{{z_j}}}{v}} \right)\frac{{{d_{j, x}}}}{{{d_j}}}{K_1}\left( {\frac{{\omega {d_j}}}{v}} \right) \end{align} $  (10) 
$ \begin{align} {\bf{E}}_{j, y}^{{\rm{app}}} = \frac{{q\omega }}{{2\pi {\varepsilon _0}{v^2}}}\exp \left( {i\omega \frac{{{z_j}}}{v}} \right)\frac{{{d_{j, y}}}}{{{d_j}}}{K_1}\left( {\frac{{\omega {d_j}}}{v}} \right) \end{align} $  (11) 
$ \begin{align} {\bf{E}}_{j, z}^{{\rm{app}}} =  \frac{{q\omega }}{{2\pi {\varepsilon _0}{v^2}}}\exp \left( {i\omega \frac{{{z_j}}}{v}} \right)i{K_0}\left( {\frac{{\omega {d_j}}}{v}} \right) \end{align} $  (12) 
where
$ \begin{eqnarray} \Delta E = \int_0^\infty {\hbar \omega \Gamma \left( \omega \right){\rm{d}}\left( {\hbar \omega } \right)} \end{eqnarray} $  (13) 
where
$ \begin{eqnarray} \Gamma \left( \omega \right) = \frac{1}{{\pi {\hbar ^2}}}\sum\limits_{j = 1}^N {{\rm{Im}}\left\{ {{\bf{E}}_j^{{\rm{app}}} \cdot {{\bf{P}}_j}} \right\}} \end{eqnarray} $  (14) 
For a single dipole (
$ \begin{eqnarray} \Gamma \left( \omega \right) = \frac{1}{{\pi {\hbar ^2}}}{\rm{Im}}\left\{ {{\alpha _1}} \right\}{\left {{\bf{E}}_1^{{\rm{app}}}} \right^2} \end{eqnarray} $  (15) 
which is the same as the result obtained for dielectric sphere [17].
C. Simulation of local surface plasmon excitationIn order to verify the effectiveness of the present DDA method we have applied it to study the STEMEELS spectrum of metallic nanoparticles. In recent years many experimental studies have been done on surface plasmon with the use of STEMEELS instrument. Koh et al. have measured EELS spectra for an electron beam incident on different positions of the surface of a single Ag particle [19]. In FIG. 1(a), the experimental spectra were obtained at interval of 2 nm for an electron beam of 10 keV incident on an Ag spherical particle of diameter of 24 nm. It is clear to see that the EELS spectrum varies significantly with beam location. In the spectra the 3.33.4 eV peak corresponds to surface plasmon of particle and the 3.8 eV peak is due to bulk plasmon, which agrees reasonably with other experimental observations [2022]. The relative intensity of the surface plasmon peak to the bulk plasmon peak changes significantly with the beam location from particle edge to the center. Qualitative explanation of such competitive behavior between surface plasmon and bulk plasmon is easy because the possibility of surface plasmon excitation strongly is dependent on the distance between the moving electron and the boundary of particle, however, it is difficult to make a quantitative evaluation. In fact, even the quantitative explanation is also difficult for a bulk Ag sample because of its unique character. The energies of surface plasmon and bulk plasmon peak are quite close. In the reflection electron energy loss spectroscopy (REELS) spectra measured for bulk Ag shown in FIG. 1(b), only one broad peak around 3.8 eV is seen, which may be contributed from both surface and bulk plasmon excitations and is almost independent of the incident electron energy. Precisely because of the lack of quantitative experimental evidence from other techniques like REELS, the explanation about those features observed in EELS spectra for Ag particle is far away from wellestablished. In this experiment, the observed local surface plasmon energy of 3.3 eV is slightly smaller than the minimum plasmon energy of silver spherical dipole medium. Koh et al. [19] attributed it to a hypothetical thin dielectric layer of unknown matter adhered to the silver sphere surface. But here we consider another possibility; that is, the response of nanoparticle to the incoming electron beam differs from that to the external light field. We will use the DDA method to clarify the issue.
FIG. 2 shows a single Ag nanoparticle in diameter of 24 nm constructed with different number of dipole. For the simulation we will firstly evaluate the minimum required number of dipole from the calculations until the result becomes stable by increasing the number, and in the simulation a larger value than the minimum number is used. Then we use DDA method to study the response of Ag nanoparticles and Ag semiinfinite bulk material to the external light field. FIG. 3 shows the optical response characteristics, the extinction efficiency, when a plane wave in incident on a single Ag particle. For comparison, it is also shown the normalized Ag bulk energy loss function,
Then we used formula, Eqs. (10)(15), to simulate the EELS spectrum for an electron beam passing by the edge of an Ag nanoparticle. Here we used different numbers of dipoles to construct a nanoparticle: we divided the side length (24 nm) of an Ag nanocube by integers of 20, 30, and 40 to form an Ag nanosphere in which the number of dipoles is 4224, 14328 and 33552, respectively. With the increasing of the division, the structure approaches to sphere. FIG. 4 shows the simulated EELS spectrum for different numbers of dipoles. When the number is greater than 14328 the spectrum becomes stable and we will use 33552 in further discussion. The EELS spectrum has two peaks, one is at 3.3 eV and the other at 3.7 eV. According to previous discussion, the 3.7 eV peak corresponds to bulk plasmon excitation while 3.3 eV peak is attributed to the local surface plasmon excitation of Ag nanoparticle. Where the surface plasmon is excited by electrons flight in the tangent line of the sphere the peak intensity of the surface mode is much stronger than that of the bulk mode.
By comparing the light field excitation and electron beam excitation, we can know that the local surface plasmon peak is at 3.45 eV in case of plane wave electric field of light and 3.3 eV in case of electric field of electrons while the EELS experimental measurement result [19] is also at 3.3 eV. Therefore, our present DDA calculation agrees with experiment very well. The calculation clearly indicates that 3.3 eV peak position is attributed to the nanoparticle character of surface plasmon, which differs from the semiinfinite surface plasmon, but not to the hypothetical thin dielectric layer on the sphere. In fact, similar behavior of surface plasmon peaks for a semiinfinite Ag sample was observed in Ref.[25] in which the excitations of surface plasmon by parallel electron beam were calculated by a traditional analytical theory.
Furthermore, we have simulated EELS spectra for an electron beam incident on different locations of Ag nanoparticle. FIG. 5 shows the normalized EELS spectra varying by beam location. This result is rather close to the experimental observation in FIG. 1 [19]. When the electron beam passing through the nanosphere center, only the bulk plasmon peak at 3.7 eV can be clearly seen while the surface plasmon excitation presents a shoulder. With changing of the position to be closer to the edge, the bulk plasmon peak intensity decreases while the surface plasmon peak intensity increases, which confirms further that the 3.3 eV peak is indeed due to surface plasmon but not to the thin dielectric layer.
Ⅲ. CONCLUSIONWe have extended the DDA method, which has been widely used for the calculation of interaction of light filed with nanostructured materials in arbitrary morphology, to the simulation of EELS spectrum for investigating the local surface plasmon excitation induced by an electron beam. The method has been verified to be effective, through a study for a silver particle, for an arbitrary nanostructure. Since there is no limitation on the modeling of structure, the method should play an important role in the nanomaterial characterization by EELS.
Ⅳ. ACKNOWLEDGEMENTSWe thank Professor K. Goto K. Goto from Advanced Industrial Science and Technology, Nagoya, Japan for helpful comments and the measurement of REELS spectrum for bulk Ag sample. This work was supported by the National Natural Science Foundation of China (No.11574289), Special Program for Applied Research on Super Computation of the NSFCGuangdong Joint Fund (2nd phase) (No.U1501501), "111" Project by Education Ministry of China and "Materials research by Information Integration" Initiative (MI2I) Project of the Support Program for Starting Up Innovation Hub from Japan Science and Technology Agency (JST). We thank Dr. H. M. Li and the supercomputing center of USTC for their support in performing parallel computations.
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b. 中国科学技术大学近代物理学院, 近地空间环境重点实验室, 合肥 230026;
c. 日本国家物质材料研究机构, 筑波 3050047