Effect of asymmetric morphology on coupling surface plasmon modes and generalized plasmon ruler

Ultramicroscopy 185 (2018) 55–64

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Effect of asymmetric morphology on coupling surface plasmon modes and generalized plasmon ruler K.J. Zhang a, B. Da b, Z.J. Ding a,∗ a

Key Laboratory of Strongly-Coupled Quantum Matter Physics, Chinese Academy of Sciences; Hefei National Laboratory for Physical Sciences at Microscale and Department of Physics, University of Science and Technology of China, Hefei 230026, Anhui, PR China b Center for Materials research by Information Integration (CMI2), Research and Services Division of Materials Data and Integrated System (MaDIS), National Institute for Materials Science (NIMS), 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan

a r t i c l e

i n f o

Article history: Received 14 September 2017 Revised 8 November 2017 Accepted 19 November 2017 Available online 21 November 2017 Keywords: Electron energy loss spectrum (EELS) Boundary element boundary (BEM) Asymmetric Ag–Ag heterodimers Coupling surface plasmon modes

a b s t r a c t Plasmon coupling in aggregated noble metal systems can provide a path to manipulate the optical response purposefully and possesses a wide range of application. Previously, most studies focused on the coupling behavior of Ag–Ag dimers with the same shape. However, plasmon coupling between nanoparticles at different morphologies can provide a new way to modulate optical properties due to broken of symmetry. In this work, we investigate systematically the coupling modes of asymmetric Ag–Ag heterodimers consisting of different morphologies by the boundary element method (BEM). Herein nanoparticles with different surface curvatures (modified by rounding parameter e) are constructed and combined as asymmetric Ag–Ag heterodimers. Simulated electron energy loss spectroscopy (EELS) spectra and eigenmodes are combined to analyze the evolution of coupling modes. The mode energy degeneracy and degeneracy breaking phenomena are found, while the charge states are always not degenerate, for the first time by modulating symmetry of the morphology. It is also found that coupling gap mode G2 can only be excited for Ag–Ag heterodimers with quite small separation distance, and will be greatly influenced by nanogap morphology. The rounded effect can also cause distinct blue shift of bounding dipolar modes. These results provide the possibility to modulate optical response by using different asymmetric dimers effectively. In contrast, optical response of high-order coupling modes is less sensitive to topographic effect than that of low-order coupling modes. Moreover, plasmon ruler for asymmetric Ag–Ag heterodimers is investigated and we demonstrate that a generalized plasmon ruler is applicable to predict the relative shift of coupling dipolar due to change of separation distance. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Electron energy loss spectroscopy (EELS) technique is often used to characterize surface plasmon (SP) mode of nanomaterials [1–3], which is a significant phenomenon arising from interaction between the free electrons in metal and applied electromagnetic field of incident electron beam. The external field will result in a strong induced field at surface and cause SP excitation. For semi-infinite metallic films, SPs are recognized as propagating waves of surface charges since the pioneering work of Ritchie [4]. Actually, SP depends strongly on the boundary condition and can be modified dramatically by changing the morphology and size of nanomaterials. For example, two different SP modes (tangential mode and normal mode) occur by reducing the thickness of a film [5]. For nanoparticles (NPs) with finite size, the nature of ∗

Corresponding author. E-mail address: [email protected] (Z.J. Ding).

https://doi.org/10.1016/j.ultramic.2017.11.011 0304-3991/© 2017 Elsevier B.V. All rights reserved.

SPs is standing waves of surface charges, known as localized surface plasmons (LSPs), with the local enhancement of electromagnetic field. In recent years, lots of applications have been developed based on the properties of LSPs, such as surface-enhanced spectroscopies [6], chemical and biomedical applications [7], as well as nano-photonics [8]. Except various LSP modes due to different shapes of single NP, coupling LSPs of adjacent NPs also enriches the study and applications of plasmonic NPs. As two or more NPs are brought in close proximity, strong near-field interaction with each other will induce hybrid resonance modes, analogy to quantum chemical molecular orbital theory. It has been reported that these hybrid resonance modes are sensitive to the number of NPs and spacing distance between them [9,10], both experimentally and theoretically. In addition, quantum effects in plasmonic structures, tunneling charge transfer plasmon (tCTP) modes, will play an important role when the spacing is reduced to below 1 nm [11–13]. In this case, quantum effect must be taken into account in a theoretical analysis. Various approaches have been applied

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to study LSP modes, such as far-field extinction and absorption spectra [14–16], near-field optical spectroscopy [17,18], cathodoluminescence (CL) spectroscopy [19–21] and EELS [22–24]. It has been shown that EELS is an important tool for investigating various LSP modes, owing to not only its outstanding spatial resolution (<1 nm) and energy resolution (<100 meV) [10,25,26], but also its ability to capture more information than optical excitation spectra (i.e. dark mode) [27]. Nelayah et al. have first recorded EELS maps (two-dimensional) of plasmons by using STEM-EELS [22]. Recently, Hohenster et al. have developed three-dimensional reconstruction of plasmonic eigenmodes tomography from EELS spectra [2,28]. In the past decades, many works have focused on the coupling behavior of LSPs for isometric and symmetrical NP dimers, e.g. sphere-sphere dimer [27,29–31], cube-cube dimer [9,32] and prism-prism dimer [33,34]. Hooshmand et al. also reported the coupling behavior of heteroid materials of NP dimers, Ag–Au dimer [35]. They pointed out that asymmetric distribution of induced Efield is caused by gold interband transition. However, for heteroid and asymmetric shapes of NP dimers with the same material, there is very little understanding of their coupling behavior. Experimentally, these asymmetric NP dimers can be assembled via asymmetric functionalization or electrostatic interactions [36,37]. Actually, such asymmetric NP dimers are also unavoidable during the synthesis process in an experiment. In addition, nanomanipulation of NPs into plasmonic structures has been developed successfully [38] and makes it possible to structure various NP dimers with different separation distances. Compared to symmetrical dimers, the eigenmodes of each adjacent NP for asymmetric NP dimers are different. Therefore, a theoretical investigation on the coupling behavior arising from these asymmetric NP dimers is very necessary. In this paper, we focus on the optical properties of heteroid shapes of Ag–Ag dimers. Firstly the evolution of LSP modes of single Ag NP, by morphing its structure from cube into sphere, was investigated. Theoretical EELS spectra were calculated by using boundary element method (BEM). In addition, we also analyzed the coupling behavior of LSP modes for asymmetric heterodimers consisting of Ag cube (or sphere) NP and Ag superellipsoid (SE) NP. Combined with the eigenmodes analysis of these asymmetric heterodimers, we show that the rounding parameter has paramount influence on the coupling behavior of LSP modes for different asymmetric heterodimers. Also, influence of spacing between asymmetric heterodimers on bounding dipolar mode is discussed. 2. Theoretical method Several numerical methods have been developed for study the optical properties of NPs, such as, discrete dipole approximation (DDA) [39–42], finite difference time domain method (FDTD) [43,44], boundary element method (BEM) [45,46], generalized multipole technique [47] and so on. Each numerical method has its own advantages and disadvantages. BEM was adopted in our research, and all of simulations were performed by using MNPBEM toolbox. The details of this software can be found in Refs. [39,40]. In the frame of BEM, the boundaries are discretized by finite triangular or quadrilateral surface elements. Besides, surface charge density and current density are introduced for solving the full Maxwell equations (retarded effect). For NPs small enough (the effective size is much smaller than the wavelength), the retardation effect can be ignored, and Maxwell equations can be regarded as electrostatic problem (so-called quasistatic approximation). Within the quasistatic approximation, the Poisson equation is converted to the boundary integral equation [45,48]:

(ω )σ (s ) +

 ∂V

  ∂ G s, s     ∂ φ (s ) σ s da = − ext ∂n ∂n

(1)

(ω ) = 2π

ε2 ( ω ) + ε1 ( ω ) , ε2 ( ω ) − ε1 ( ω )

(2)

where the Green function G(s, s ) connects surface elements, s and s , of the particle boundary, and external potential φ ext is due to applied planewave or swift electrons in STEM-EELS. The dielectric functions, ɛ1 (ω) and ɛ2 (ω), correspond respectively to the NP and surrounding environment. If there is no external potential φ ext (r), the eigenmode equation can then be obtained from Eq. (1) as:

 ∂V

  ∂ G s, s     σi s ds = 2π λi σi (s ). ∂n

(3)

The eigenenergy λi can be obtained from [(ωi ) + λi ] = 0 for ith mode. Note that the eigenmode equation is independent of the material property, and only relies on geometric shape of the particle. Therefore, we can analyze the effect of shape of LSP modes within the help of eigenmode equation. In the electron-driven case, an incident electron beam passes along zˆ-direction with a constant velocity vector v. The applied electric field generated by fast electron can be written as [49]:

Einc (r, ω ) =

2 eω

v2 γε ε



e iω z / v

i

γε



K0

    ωR ωR ˆ zˆ − K1 R , vγε vγε

(4)

where r = (R, z ) with R denotes radial distance between point r and electron trajectory, γε = 1/ 1 − ε v2 /c2 is the Lorentz contraction factor, Km is the modified mth order Bessel function. In the quasistatic approximation, the probability of energy loss of the incident electrons, due to interaction with induced field from the surface of NPs, is given by [48,50]:

 SP (R0 , ω ) =

4

[−gi (ω )] π v2 i    2  R0 − s⊥   ω   × dsσ i (s )e−iω/vs K0 , v  

(5)

with the weight factor,

gi ( ω ) =

2

ε1 (ω )(1 + λi ) + ε2 (ω )(1 − λi )

,

(6)

where R0 is the impact parameter of the incident electrons, and s (s⊥ ) is the component of s parallel (perpendicular) to the trajectory of an electron. By inspection of Eq. (5), the total loss probability  SP can be decomposed into summation of contributions from each eigenstate σ i . This clearly reveals that EELS spectra can reflect the information of plasmon eigenmodes. Meanwhile, for some eigenstates, their weight factor distributions can be very close. Therefore, the excitation energies of these modes will be almost the same from the perspective of the EELS spectra. In the retarded case, although there is no such accessible decomposition expression for  SP , it is reasonable to analyze EELS spectra combined with eigenmode decomposition. Generally, the difference of EELS spectra performed under quasistatic limit and retarded case is mainly displayed in the peak positions, especially for those peaks with lower excitation energies. We focus on the coupling behavior of Ag NP dimers from now on. Following previous works [51], we morph the rounded nanocube to superellipsoid gradually by changing the rounding parameter e (nanosphere for e = 1). The boundary elements can be structured by:

x(u, v ) = a · s(u, e ) · c (v, e )

(7a)

y(u, v ) = a · s(u, e ) · s(v, e )

(7b)

z (u, v ) = a · c (u, e ),

(7c)

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Fig. 1. Modulating symmetry of NPs by changing the rounding parameter e: rounded cube, superellipsoid, sphere and rounded octahedron correspond to e = 0.25, 0.5, 1, 1.3 respectively. The initial edge length a is set as 30 nm.

where s(ζ , e ) = sign(sin ζ )|sin ζ |e and c (ζ , e ) = sign(cos ζ )|cos ζ |e , u ∈ [0, π ), v ∈ [−π , π ), and side length, a, of original nanocube is 30 nm. Fig. 1 shows single NP morphology with different rounding parameter values (e = 0.25, 0.5, 1 and 1.4). When e > 1, the NP morphology becomes rounded octahedron. An asymmetrical structure Ag dimer is composed of two NPs with different rounding parameters, i.e. e1 =e2 . The coupling effects of these asymmetrical dimers were performed by BEM-EELS simulation. In our study, we have not taken the substrate effect into account, and set the surrounding dielectric constant ε2 = 1 (i.e. vacuum environment). Throughout this paper, the energy of incident electron was chosen as 100 keV. The dielectric function of Ag was adopted from Johnson and Christy’s data [52]. 3. Results and discussion 3.1. LSP modes of single Ag NPs Many previous studies [27,29–32] have reported the primary LSP modes of Ag nanocube and nanosphere separately. It is well known that the LSP mode varies strongly with the geometrical morphology of nanoparticles. However, few studies have paid attention to the effect of continuous morphological changes on LSP mode. Here, we morph nanocube into nanosphere continuously, and aim to study the effect of morphology on LSP modes. Actually, many eigenmodes of nanosphere are highly degenerate due to the spherical symmetry, and many higher-order modes cannot be not excited in small size nanoparticles. For nanocube with C4v symmetry, some degenerate eigenmodes can be distinguished because of the lower symmetry. Fig. 2(a) shows BEM-simulated EELS spectra of four Ag NPs (the rounding parameter e = 0.25, 0.5, 0.9 and 1), performed at two different electron beam incident positions. For non-spherical symmetry system, the mode excitation depends on the position of electron beam strongly. Actually, energy loss of incident electron beam (incident direction along z-axis) arises from the work done by the induced electric field of the excited modes on the incident electrons. Thus, electron beam at different incident positions can excite specific modes which have strong z-axis electric component near the beam position. In rounded nanocube case (e = 0.25), there are four primary LSP modes. Here, we denote them as modes C1 (3.27 eV), C2 (3.31 eV), E (3.54 eV) and F (3.69 eV), respectively. Up-most panel in Fig. 2(b) shows the corresponding 2D EELS maps from left to right. EELS maps reflect the z-axis electromagnetic local density of states (zEMLODS) under quasistatic approximation [50], and can be used to characterize different LSP modes. Furthermore, Kociak et al. [53] pointed out that, in retarded case, the link between EELS map and the zEMLDOS still holds qualitatively. At first glance, corner localized modes (C1 and C2 ) in Fig. 2(b) are almost unable to be distinguished, indicating that the electrical field distributions of mode C1 and C2 are very similar. Recently, similar experimental and simulated result are reported for gold concave nanocube [54]. For modes E and

F with higher excited energy, the surface charges are localized on the edges and faces, respectively. With increasing the e value (approaching to spherical symmetry), we trace the LSP modes when nanocube is changed to nanosphere. Modes C1 , C2 and E all are blue shifted, and modes C2 , E become degenerate gradually (indicated by the colored dash lines in Fig. 2(a)). Finally, mode C1 is evolved into the dipolar mode (mode D, 3.47 eV) of nanosphere, and modes C2 and E are evolved into quadrupole mode (Q) of nanosphere. Different from the first three modes, mode F is red-shifted and evolved into higher-order mode of nanosphere. We also notice that surface charges are localized on the sharp vertexes or edges for modes C1 , C2 and E, and charges are localized on flat surface for mode F. That is, the sharp corner effect and flat surface will cause opposite shift of LSP modes. Similarly, Schmidt et al. [26] reported this opposite shift when morphing a nanodisk into nanotriangle. They introduced a perturbation approach that uses the nanodisk eigenmodes as basis vectors, and treats morphology difference as a perturbation to describe the different geometries modes. This allows a good interpretation of peak shift trends. In addition, they further pointed out that hybridizations from different eigenmodes, such as dipole, quadrupole and face mode (higher-order), can reduce futher energy separation (red-shift and blue-shift). EELS maps of above modes of gradually rounded nanoparticles are shown in Fig. 2(b). For the third and fourth rows of Fig. 2(b), EELS maps of different modes tend to be monistic and spherical symmetry. Especially for nanosphere (e = 1), EELS maps of multipolar modes are almost identical since all the points around the sphere are equivalent. In other words, broken symmetry or spatial anisotropy is favorable for EELS mapping to capture the spatial distributions of LSP modes. Therefore, for the NP systems with lower-symmetry (broken symmetry), EELS map is more helpful to study LSP modes. 3.2. Coupling modes of Ag cube-superellipsoid (CSE) dimer As mentioned above, for Ag NP dimers, most previous works [27,29–32] focus on symmetric configurations, e.g. sphere-sphere dimer and cube-cube dimer. However, asymmetric configurations under broken symmetry can provide an interesting insight about the coupling behavior between different LSP modes, which has been discussed barely before. Herein, we construct cubesuperellipsoid (CSE) dimer (face-to-face arrangement) by fixing one as cube (e1 = 0.25), and morphing the other one from cube to sphere and to rounded octahedron (see insets in Fig. 3(a) and (b)). The coaxial line of CSE dimer is perpendicular to the adjacent surface of the cube. We have also considered CSE dimers in diagonal alignment (corner-to-corner) to study the effect of orientation of the coupled particles (details see Fig. S1-S2 in the supplementary material). The shortest distance d between the two NPs is set as 2 nm. Actually, the spacing distance plays an important role in coupling behavior of LSP modes. In the frame of classical theory, the relative variation of excited wavelength ( λ/λ0 ) for bonding

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Fig. 2. (a) BEM-simulated EELS spectra of single Ag NPs with four rounding parameters (e = 0.25, 0.5, 0.9 and 1) by solving the full Maxwell equations. Two different electron beam impact positions are shown as rightmost schemes. (b) Simulated EELS maps (each row) of above four Ag super-ellipsoid NPs. From left to right: the characteristic loss energies correspond to the EELS peaks as indicated by the red, blue, green and purple color respectively in (a). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

dipolar (BD) mode varies exponentially with the relative separation distance d/a, which is called plasmon ruler [55] (the effect of spacing distance will be discussed in the last section). For sufficiently closer dimers (d < 0.5 nm), quantum mechanical effects become important [13]. In this case, tunneling electrons across the gap lead to the emergence of new charge transfer plasmons (CTPs). In our work, it is adequate to using classical electrodynamics without consideration of the tunneling effect. Figs. 3(a) and 3(b) show the simulated EELS spectra of CSE dimers (with different e2 values) taken at the two electron beam positions, a-case and b-case, respectively. Firstly, we consider the coupling behavior of symmetric Ag cube-cube dimer, corresponding to the bottom EELS spectra in Fig. 3(a) and (b). Compared with the EELS spectra of single Ag nanocube (top panel in Fig. 2(a)), there are four new peaks arising in Ag cube-cube dimer case. We assign these coupling modes as G1 (2.55 eV, dipole orientation↑↓), bonding dipolar (BD, dipole orientation → → , 2.94 eV), G2 (3.13 eV), and bonding corners (BC, dipole orientation↑↑, 3.31 eV). Their intrinsic charge distributions are shown in Fig. 4(b(i)). Obviously, the charge distributions of modes G1 and G2 are mainly located on the surfaces facing the gap, and they can only be excited for small spacing distance case. According to ref. [49] these two resonances are denoted as gap modes. Longitudinal antibonding (↑↓) of mode C1 produces mode G1 , and strong electrical mutual attraction induces to gap localized distribution of charges. Mode G2 arises from the coupling of highorder modes. Mode BD arises from transverse bonding ( → → ) of the initial mode C1 between adjacent Ag nanocubes. Strong enhancement of electric field occurs around the gap center (hot spot). It can be excited prominently in the a-case of electron incident position. For mode BC, the characteristic excitation energy (3.31 eV) is almost the same as mode C2 of single Ag nanocube. If we increase the spacing distance, the coupling effect will decrease. However, the energy of mode BC will not change (always be 3.31 eV). Al-

though the characteristic energy is consistent, the charge distributions of mode BC differ from original mode C2 . Interestingly, longitudinal bonding (↑↑) of initial mode C2 will decrease the charge distribution around the gap (see Fig. 4(b)(i-BC)). Thus, electric field around the gap closes to zero. For the coupling of higher-order modes (E and F), we find that coupling effect is very weak. The energy of coupling mode E (F) is consistent with that of mode E (F) for single nanocube. Therefore, for the different asymmetric CSE dimers, we mainly focus on the evolution of above four coupling modes. As shown in Fig. 3(c), with increasing e2 to break symmetry, the modes G1 and G2 blue shift more strongly than the modes BD and BC. It is accessible that these two gap modes are more sensitive to the morphology of adjacent gaps. When e2 approaching unit modes G1 and BD becomes degenerate; but the degeneracy is breaking by further changing morphology; at e2 = 1, the energy of mode G1 is higher than that of mode BD, while the energies of modes G2 and BC are degenerate. By further increasing e2 value above unit, the degeneracy of modes G2 and BC keeps while the separation between modes G1 and BD becomes more evident (see Fig. 3(c)). Therefore, the mode energy degeneracy and crossover phenomena are found for the CSE dimers. However, even the energy degeneracy exists we will show later that the charge states are not degenerate. EELS maps of above four modes for four representatives Ag CSE dimers are shown in Fig. 4(a). As mentioned above, EELS map only reflects one component of EMLDOS. With the help of eigenvalue equation (Eq. (3)), the surface charge distributions of eigenmodes are simulated in the quasistatic approximation. Four eigenmodes (corresponding to modes G1 , BD, G2 and BC) for the four representative CSE dimers are shown in Fig. 4(b). Obviously, for Ag cubecube dimer case, the maximum EELS intensity of mode G1 appears around the gap in Fig. 4(a(i)). As e2 increases, the symmetry of system is further reduced, the EELS maps of mode G1 mainly highlight around the cube right side, and are no longer symmetric as seen in

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59

Fig. 3. Simulated EELS spectra taken at two beam positions for different Ag NP dimers (indicated by red and blue dots in the insets): (a) electron beam is incident at the left side of long axis; (b) by the side of the cube in the CSE dimer. Fixing the left NP as nanocube (e1 = 0.25), the right one is morphed from a nanocube (bottom, e2 = 0.25) into a nanosphere (e2 = 1) and to a rounded octahedron (top, e2 = 1.4). The labeled peaks below 3.4 eV correspond to four coupling modes: G1 (blue triangle), BD (orange circle), G2 (red diamond) and BC (purple square). (c) The variation of excited energies of the four coupling modes of CSE dimer with rounding parameter e2 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

first column of Fig. 4(a). From the inspection of corresponding surface charge distributions (the first column in Fig. 4(b)), the opposite charges concentrate on the adjacent surfaces. Since gap mode arises from strong Coulomb interaction between the adjacent surfaces, the morphology of gap can affect the strength of Coulomb interaction. Therefore, the excitation energy of mode G1 strongly depends on the morphology of gap. As for the other gap mode G2 , the EELS maps and corresponding eigenmodes are shown in the third column in Figs. 4(a) and 4(b). In symmetric case, EELS map of mode G2 shows strong loss probability around the endmost corners and the gap region. Interestingly, for CSE dimers with increasing e2 (asymmetric case), EELS maps of mode G2 mainly highlight the endmost corners of the left cube side, and become dim at the right sphere side gradually. This indicates the asymmetry of surface charge distribution due to the symmetry broken. The evolution of surface charge for this eigenmode also reflects this change (see Fig. 4(b) (G2 )). Compared to the eigenmode G1 , the opposite charges are located on the corners and centroid of the adjacent surfaces for eigenmode G2 . When e2 approaches to 1, the charge distribution of the right NP shows significant deviation from that of left one due to the disappearance of the corners. In a word, modes G1 and G2 are strongly dependent on the morphology of the gap. Of course, distance d is also an important parameter that affects gap modes. There will be more gap modes occurred for narrower gap case (d ≤ 1 nm) [56]. For mode BD, the EELS map of each Ag CSE dimer shows the same characteristic with EELS intensity reaching a maximum by

the two ends of dimer. Similar results can also be found in other dimers, i.e. rod dimer [31,57] and prism dimer [44,58]. The intrinsic surface charge distributions reveal strongly bounding coupling, and opposite charges always highly fasten on each center of two adjacent faces with increasing e2 (see the second column in Fig. 4(b)). This will lead to strong enhancement of E-field around the gap center (hot spot). Therefore, the energy of mode BD shows weaker dependence on the value of e2 (Fig. 3(c)). For mode BC, EELS map is only evident at the end-most corners for Ag cube-cube dimer. With increasing e2 , EELS signal is still obvious at the left end-most cube corners but vanished at the right sphere side gradually. Eventually, EELS map of mode BC is the same as that of mode G2 due to the energy degeneracy (modes BC and G2 in Fig. 4(a)). However, it is important to note that the charge distributions of these two modes are completely different, indicating that the charge states are always not degenerate. The eigenmode BC shows that charges are highly located on the end-most corners and rarely distributed around the gap region, as shown in Fig. 4(b) (BC). Two dimensional EELS map is difficult to distinguish these energy degenerate modes. Interestingly, in asymmetric dimer cases, only the cube part (endmost corners) is excited and the right part seems to be suppressive. Meanwhile, the excitation energy of mode BC is almost unchanged for different Ag CSE dimers and close to the energy of mode C2 of single Ag nanocube (∼3.31 eV). This can be confirmed by the nearly same eigenvalues and imaginary part of weight factor of mode BC for different Ag CSE dimers (see Fig. 5).

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Fig. 4. (a) BEM-simulated EELS maps of four coupling modes (G1 , BD, G2 , and BC) for four representative Ag CSE dimers (separation distance d = 2 nm) by solving full Maxwell equations; (b) BEM-simulated distributions of surface charges corresponding these four eigenmodes in the quasistatic approximation. The adjacent NPs are rotated separately for a clear view.

3.3. Coupling modes of Ag superellipsoid-sphere (SES) dimer So far, we have discussed the evolution of coupling modes of Ag CSE dimers, which are composed of one fixed cube (e1 = 0.25) and the other alterable superellipsoid (0.25 ≤ e2 ≤ 1.4). Furthermore, we can also model the asymmetric Ag superellipsoid-sphere (SES) dimers by morphing one from cube into sphere (0.25 ≤ e1 ≤ 1) and fixing the other one as sphere (e2 = 1). Fig. 6 shows the evolutions in simulated EELS spectra for various Ag SES dimers with in-

creasing e2 from 0.25 to 1 (bottom to top). Two incident positions, a-case and b-case, are performed in our simulation, as shown in the insets of Fig. 6(a) and (b), respectively. In the a-case, there are two noticeable modes excited. The lower energy one is labeled as bonding dipolar mode (BD), and the other one is labeled as coupling collinear high-order mode (CH). Meanwhile, for different Ag SES dimers, the energies of modes BD and CH change slightly. In the b-case, we can also see the signal of mode BD. Besides, longitudinal antibonding mode (A, dipole orientation↑↓) and longi-

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18 16 14

-Im(gi)

12

e2=0.25 e2=0.4 e2=0.7 e2=1

10 8 6 4 2 0

3

4 Energy (eV)

Fig. 5. The imaginary part of weight factor of mode BC for four different Ag CSE dimers: e2 = 0.25, 0.4, 0.7 and 1 (with corresponding eigenvalues of λi =−2.943, −2.959, −2.960 and −2.946, respectively).

tudinal bonding mode (P, dipole orientation↑↑) are blue-shifted with increasing e1 (from bottom to top in Fig. 6(b)). Modes A and P are common longitudinal coupling modes for sphere-sphere dimer case. Actually, comparing Fig. 3(b) and Fig. 6(b), modes A and P correspond to modes G1 and BD, respectively. By morphing the morphology, these longitudinal coupling modes can be adjusted in large energy range. For higher-order coupling case, the coupling

61

effect is very weak as mentioned before. In Fig. 6(b), signals of modes E and F are distinct and arise from left nanocube. They will fade away and become coupling higher-order modes (HO) of sphere-sphere dimer gradually. Fig. 7(a) shows EELS maps of four coupling modes (BD, A, P and CH) for four representative Ag SES dimers (e1 = 0.3, 0.5, 0.7 and 1). For mode BD, just as in Ag CSE dimer case, all EELS maps present the same feature with EELS intensity reaching a maximum at the end of dimer. The corresponding surface charge of eigenmode confirms the collinear dipolar bonding and highlighted charges distribute around the gap area due to strong Coulomb attraction (see Fig. 7(b) (BD)). For mode A, EELS map shows obvious asymmetry and mainly highlights at the right rounded corners of superellipsoid (see Fig. 7(a) (A)). This is attributed to asymmetric distribution of charge density between the superellipsoid (left one) and sphere (right one), as shown in Fig. 7(b) (i-iii, A). With increasing e1 to 1, the difference of charge distributions between two NPs vanishes (see Fig. 7(b) (iv, A)). Meanwhile, for Ag sphere-sphere case (e1 = 1), the longitudinal antibonding mode (mode A) has zero net dipole moment and is invisible by optical excitation. In addition, the longitudinal bonding mode (mode P), for Ag spheresphere case, is shown in Fig. 7(b) (iv, P). Mode P can be excited by planewave because of the non-zero net dipole moment. The evolution of EELS map (charge distribution) of mode P is shown in Fig. 7(a) and (b), from Ag SES dimer to Ag sphere-sphere dimer. When e1 = 0.3, both EELS map and charge distribution of mode P indicate that massive charges locate on the left end corners of rounded cube and fewer charges locate on the right sphere. Thus, the enhanced E-field distribution only occurs on the left corners of cube side. Following the plasmon hybridization model [35], the coupling mode P can be expressed as P = aSE + bShpere (a > b).

Fig. 6. (a-b) Simulated EELS spectra for different Ag SSE dimers taken at two beam positions, indicated by the red and blue dots in the insets of (a) and (b), respectively. We fix the right NP as nanosphere (e2 = 1) and morph the left one from nanocube (bottom, e1 = 0.25) into nanosphere (top, e1 = 1). The labeled peaks correspond to coupling modes: BD (blue triangle), A (orange circle), P (red diamond), CH (purple square) and HQ. (c) The excited energies of the four coupling modes of CSE dimers with various rounding parameters e1 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 7. (a) BEM-simulated EELS maps of four coupling modes (BD, A, P and CH) for four representative Ag SES dimers (separation distance d = 2 nm) by solving full Maxwell equations; (b) BEM-simulated distributions of surface charges corresponding these four eigenmodes in the quasistatic approximation. The adjacent NPs are also rotated separately for a clear view.

The contributions from superellipsiod NP will be greater than the spherical one. With increasing e1 , the difference of relative intensity of charge distribution between left and right NPs decreases gradually. It is worth mentioning that there are also charges existed around the gap area owing to the narrow gap effect. Moreover, EELS maps (charge distribution) of mode CH for different Ag SES dimers are shown in Fig. 7(a) (CH) (Fig. 7(b) (CH)). Obviously, mode CH arises from collinear multipolar coupling between two NPs.

3.4. Influence of spacing distance In above sections, we have discussed the relationship of coupling modes of different asymmetric Ag–Ag dimers for a fixed spacing distance (d = 2 nm). On the other hand, spacing distance between two NPs also have significant effect on the coupling LSP modes. Previous researches have confirmed that plasmon ruler is appropriate for symmetric Ag–Ag dimers [9]. However, for asymmetric case, the effect of plasmon ruler has not been studied yet.

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Fig. 8. Influence of spacing distance for mode BD of two heteroid Ag-Ag dimers (e1 = 0.25): (a) e2 = 0.5, (b) e2 = 1; the incident electron positions are labeled by the red dots. The relative variation of wavelength varies with d/a in (c) and (d) corresponding to (a) and (b), respectively. The lines represent exponential curve fitting. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

In this section, we discussed the influence of spacing distance on mode BD for asymmetric Ag–Ag dimers. As we know, plasmon ruler of symmetric dimers can be expressed as:

λ/λ0 = c1 exp (c2 d/a )

(8)

where d is the spacing distance and a is the characteristic length of NP. λ0 is the dipolar peek of individual NP, and λ denotes the variation of excited wavelength of mode BD compared to λ0 . For symmetric dimers case, the dipolar peeks of two identical NPs are consistent. However, for asymmetric dimers case, the dipolar peeks of two NPs are different, so it is not obvious to choose the reference λ0 . Fig. 8(a) and (b) show the evolution of mode BD for two asymmetric Ag–Ag dimers: e2 = 0.5 and 1, respectively (both e1 = 0.25). The wavelengths of dipolar mode for above three individual Ag NPs (e = 0.25, 0.5 and 1) are 379.2 nm (λ10 , 3.27 eV), 366.75 nm (λ20 , 3.381 eV) and 357.04 nm (λ30 , 3.473 eV). In a-case, the plasmon ruler is still suitable when choosing parameter λ0 in Eq. (8) as λ10 . The fitted exponential function is shown in Fig. 8(c). When choosing parameter λ0 as λ20 , we cannot have a suitable fitting. Similarly, in b-case, plasmon ruler is also only applicable when choosing λ0 as λ10 (see Fig. 8(d)). Therefore, for asymmetric dimers, if we choose the maximal wavelength of dipolar modes (λmax ) between two NPs as initial reference λ0 , we can still use plasmon ruler to predict optical property of asymmetric dimers. The generalized plasmon ruler can then be expressed as

λ/λmax = c1 exp(c2 d/a ). In addition, for asymmetric cube-sphere dimers (diagonal alignment), the generalized plasmon ruler is also suitable (see Fig. S3).

obtained under quasistatic approximation. First, by morphing nanocube into nanosphere gradually, the evolutions of LSP modes for single Ag NP are obtained. Low-order and high-order LSP modes will blue shift and red shift respectively, with increasing the rounding parameter e. EELS maps can reflect effectively the spatial information of corner, edge and face modes for cube case, but is restricted for sphere case due to the spherical symmetry. Furthermore, the evolution of coupling modes between different Ag NPs is studied in detail. For Ag CSE dimers, our analyzes point that the low-energy coupling modes (especially for mode G1 , relative change of energy E ∼ 0.55eV) are very sensitive to the effect of morphology. The modes can be degenerate in energy but none for the corresponding charge states. For Ag SES dimers, the responses of longitudinal antibonding and bonding modes strongly depend on morphology changing. Besides, the mode with asymmetrical charge distribution will induce asymmetrical electric field enhancement. This opens a new way to fabricate functional heteroplasmonic devices. Finally, the influence of separation distance for asymmetric Ag–Ag dimers on coupled dipolar mode is investigated. Compared to symmetrical dimers, a generalized plasmon ruler is suggested, for asymmetric dimers case, to predict the energy shift of bounding dipolar mode with the change of separation distance. Acknowlegments This work was supported by the National Natural Science Foundation of China (No. 11574289) and Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (2nd phase). We also thank supercomputing center of USTC for the support of parallel computing.

4. Conclusion In summary, theoretical studies of surface plasmon coupling behavior for Ag–Ag heterodimers have been performed by MNPBEM. EELS spectra and maps are simulated by solving full Maxwell equations (retarded case), and assistant eigenmode analyzes are

Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ultramic.2017.11.011.

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