Chinese Journal of Chemical Physics  2018, Vol. 31 Issue (5): 655-660

The article information

Zhe Zheng, Bo Da, Ke-jun Zhang, Ze-jun 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): 655-660
化学物理学报, 2018, 31(5): 655-660
http://dx.doi.org/10.1063/1674-0068/31/cjcp1805096

Article history

Received on: May 6, 2018
Accepted on: June 21, 2018
Simulation Study of Electron Beam Induced Surface Plasmon Excitation at Nanoparticles
Zhe Zhenga,b, Bo Dac, Ke-jun Zhanga, Ze-jun Dinga     
Dated: Received on May 6, 2018; Accepted on June 21, 2018
a. CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics, Hefei National Laboratory for Physical Sciences at the Microscale and Department of Physics, University of Science and Technology of China, Hefei 230026, China;
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, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan
*Author to whom correspondence should be addressed. Ze-jun Ding, E-mail:zjding@ustc.edu.cn; Bo Da, E-mail:DA.Bo@nims.go.jp
Abstract: Phenomenon of localized surface plasmon excitation at nanostructured materials has attracted much attention in recent decades for their wide applications in single molecule detection, surface-enhanced Raman spectroscopy and nano-plasmonics. In addition to the excitation by external light field, an electron beam can also induce the local surface plasmon excitation. Nowadays, electron energy loss spectroscopy (EELS) technique has been increasingly employed in experiment to investigate the surface excitation characteristics of metallic nanoparticles. However, a present theoretical analysis tool for electromagnetic analysis based on the discrete dipole approximation (DDA) method can only treat the case of excitation by light field. In this work we extend the DDA method for the calculation of EELS spectrum for arbitary nanostructured materials. We have simulated EELS spectra for different incident locations of an electron beam on a single silver nanoparticle, the simulated results agree with an experimental measurement very well. The present method then provides a computation tool for study of the local surface plasmon excitation of metallic nanoparticles induced by an electron beam.
Key words: Surface plasmon excitation    Nanostructured materials    Nanoparticles    Electron energy loss spectroscopy    
Ⅰ. INTRODUCTION

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 T-matrix 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 [9-12]. 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 (STEM-EELS) [13-15], 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 field

By the DDA method a nanostructured material is spatially divided into $N$ unit cubes, where each unit cube is treated as a dipole whose dipole moment is calculated through the response to the local electric field. The polarization vector of the $j$th dipole at the position ${\bf r}_j$ under the local field is,

$ \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 $\alpha_j$ is the dipole polarizability, and the local electric field ${\bf E}^{\rm{loc}}_j$ includes the external field ${\bf E}^{\rm{app}}_j$ at the $j$th dipole and the local dipole electric field, $-\displaystyle\sum\limits_{k \ne j} {{{\bf{A}}_{jk}}{{\bf{P}}_k}}$, which is resulted from the action of rest $N$-1 dipoles on the $j$th dipole. Usually in the DDA method the external electric field is considered as the plane wave form,

$ \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 $\bf k$ is the wavevector of an electromagnetic wave. All the dipoles for the simulation of material characteristics are located in a cubic lattice whose lattice constant is determined as, $d$=$(V/N)^{1/3}$, where $V$ is the volume of material and $N$ is the number of dipoles. Hence, the product of wavevector $k_0$ ($|k_0|$=$\omega/c$) of electromagnetic wave in vacuum and lattice constant, $k_0d$, is the best dimensionless parameter characterizing this dipole lattice, which describes the variation of wave phase relative to the lattice spacing.

By using the lattice dispersion relation (LDR), the dipole polarizability $\alpha(\omega)$ is expressed as a series of expansion of $k_0d$:

$ \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 $b_1$=$-(4\pi/3)^{1/2}$, and $\alpha^{\rm{CM}}_{j, ll}$ is the tensor form of Clausius-Mossotti polarizability [12],

$ \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}}_k-3{\bf{r}}_{jk}\left( {\bf{r}}_{jk}\cdot {\bf{P}}_k\right) \right]\Bigg\}\quad\\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (j\neq k) \end{align} $ (5)

where ${\bf r}_{jk}$$\equiv$${\bf r}_{j}$-${\bf r}_{k}$, and ${\bf A}_{jj}$=$\alpha_j^{-1}$ if $j$=$k$. Multiplying Eq.(4) by ${\mathit{\boldsymbol{\alpha}}}_j^{-1}$ in two sides, one obtains:

$ \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 $N$ dipoles, one may define two vectors of $3N$ dimension, i.e. ${\bf{\tilde P}}$=$\left({{{\bf{P}}_1}, {{\bf{P}}_2}, ..., {{\bf{P}}_N}} \right)$ and ${{\bf{\tilde E}}^{{\rm{app}}}}$=$\left({{\bf{E}}_1^{{\rm{app}}}, {\bf{E}}_2^{{\rm{app}}}, ..., {\bf{E}}_N^{{\rm{app}}}} \right)$, and another matrix ${\bf{\tilde A}}$ of $3N$$\times$$3N$ dimension, where ${{\bf{\tilde A}}_{3j - l, 3k - m}}$=${\left({{{\bf{A}}_{jk}}} \right)_{3 - l, 3 - m}}$. Then Eq.(6) can be rewritten as,

$ \begin{eqnarray} {\bf{\tilde A\tilde P}} = {{\bf{\tilde E}}^{{\rm{app}}}} \end{eqnarray} $ (7)

By solving this equation of $3N$ dimension one can derive polarization vector ${\bf{\tilde P}}$, and hence the total extinction cross-section ${C_{{\rm{ext}}}}$ with the relation,

$ \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 ${\bf{E}}_j^{\rm{app,*}} $ is the complex conjugate of ${\bf{E}}_j^{{\rm{app}}}$. Similarly, the excitation coefficient can also be expressed as,

$ \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 ${r_{{\rm{eff}}}}$ is the radius of effective sphere of the material in volume.

B. DDA method for electron beam

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 semi-infinite 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 $q$ moving in vacuum in a velocity $\bf v$, which is parallel to the $z$-axis (normal of sample plane) and in an impact parameter $r_q$ about the $xy$-plane. The dipole at position ${{\bf{r}}_j}$=$\left({{x_j}, {y_j}, {z_j}} \right)$ will experience an electric field brought by the charge, whose field components are given by [17]:

$ \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 ${{\bf{d}}_j}$=${{\bf{r}}_j}$-${{\bf{r}}_k}$ is located on the $xy$-plane, $K_m$ is the $m$th order Bessel function. It can be seen that because of the $d_j$-dependence of ${\bf{E}}_j^{{\rm{app}}}$, the excitation probability changes with position of moving electron. On substituting Eq.(10)-Eq.(12) into the formula of DDA method in replace of the plane wave form of electromagnetic wave, one can then describe the surface plasmon excitation induced by an electron beam. In an electron energy loss spectroscopic measurement, the energy of the incident electron beam ranges from several tens keV to several hundred keV while the energy loss of electrons due to plasmon excitation is usually limited to several tens eV, which is comparatively very small. Therefore, the electron trajectory can be regarded as a straight line without any deflection. The energy exchange between incident electrons and nanomaterial is defined as,

$ \begin{eqnarray} \Delta E = \int_0^\infty {\hbar \omega \Gamma \left( \omega \right){\rm{d}}\left( {\hbar \omega } \right)} \end{eqnarray} $ (13)

where ${\Gamma \left(\omega \right)}$ is the differential energy loss probability [18]:

$ \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 ($N$=1), we have ${{\bf{P}}_1}$=${\alpha _1}{\bf{E}}_1^{{\rm{app}}}$. Hence,

$ \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 excitation

In order to verify the effectiveness of the present DDA method we have applied it to study the STEM-EELS spectrum of metallic nanoparticles. In recent years many experimental studies have been done on surface plasmon with the use of STEM-EELS 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 3.3-3.4 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 [20-22]. 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 well-established. 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. 1 (a) Electron energy loss spectra measured on a single Ag nanoparticle at different beam locations. Reprinted with permission from [19], copyright©American Chemical Society (2018). (b) Reflection electron energy loss spectra measured on a bulk Ag sample with 100 eV incident electron energy.

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 semi-infinite 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, ${\rm Im}\{- 1/\varepsilon(\omega)\}$, is derived from optical constants measured by optical methods [23, 24], where $\varepsilon(\omega)$ is dielectric function. It is seen that, the bulk energy loss function gives the value of bulk plasmon energy as 3.8 eV, and the extinction efficiency yields the surface plasmon energy as 3.72 eV for semi-infinite material and 3.45 eV for Ag particle. The boundary condition of the material clearly alters the plasmon excitation modes. The local surface plasmon mode of 3.45 eV derived by the DDA method agrees well with the theoretical value of 3.5 eV [17]. Note that the particle size is very small here, in diameter of 24 nm, then the plasmon excitation is dominantly due to surface plasmon mode while the bulk plasmon has negligible contribution. For larger sizes the plasmon energy will approach to the semi-infinite case. Therefore, for very small particles, the local surface plasmon describes well the whole response of the particle to light.

FIG. 2 Modeling of Ag nanoparticle with discrete dipole approximation method with different numbers of dipoles, $N$=280, 912, 2176, and 17256.
FIG. 3 Extinction efficiency spectra of Ag nanoparticle and Ag semi-infinite solid excited by a plane wave. 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.

FIG. 4 The influence of the number of discrete dipole on the electron-beam induced localized surface plasmon excitation of Ag nanoparticle.

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 semi-infinite surface plasmon, but not to the hypothetical thin dielectric layer on the sphere. In fact, similar behavior of surface plasmon peaks for a semi-infinite 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.

FIG. 5 Simulated electron energy loss spectra of Ag nanoparticle for different landing positions of electron beam.
Ⅲ. CONCLUSION

We 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. ACKNOWLEDGEMENTS

We 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 NSFC-Guangdong 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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电子束诱导的纳米颗粒表面等离激元激发模拟
郑哲a,b, 达博c, 章可俊a, 丁泽军a     
a. 中国科学技术大学物理系, 合肥微尺度物质科学国家研究中心, 合肥 230026;
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 305-0047
摘要: 本文基于离散偶极子近似方法,发展出了一套计算任意纳米结构材料的电子能量损失谱的方法和程序.模拟了在单个银纳米颗粒附近不同入射位置下的电子能量损失谱,其计算结果与实验能谱非常吻合.虽然离散偶极子近似法早在很多年前就已经被应用到处理外加光场激发的情形,但这套方法才真正提供了一个研究电子束诱导金属纳米颗粒局域表面等离激元激发的计算工具.
关键词: 表面等离激元    纳米材料    纳米颗粒    电子能量损失谱