Scaling Behavior of Individual Nanoparticle Plasmon Resonances

Subscriber access provided by SELCUK UNIV

Article

Scaling Behavior of Individual Nanoparticle Plasmon Resonances Reto Giannini, Christian Hafner, and Jörg F. Löffler J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp509245u • Publication Date (Web): 03 Feb 2015 Downloaded from http://pubs.acs.org on February 6, 2015

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Scaling Behavior of Individual Nanoparticle Plasmon Resonances Reto Giannini,†,* Christian V. Hafner,§ and Jörg F. Löffler† †

Laboratory of Metal Physics and Technology, Department of Materials, ETH Zurich, 8093

Zurich, Switzerland, and §Laboratory for Electromagnetic Fields and Microwave Electronics, Department of Information Technology and Electrical Engineering, ETH Zurich, 8092 Zurich, Switzerland. *

Corresponding author: [email protected]

Abstract: Via experiments and an intuitive model, this study reports on the dependence of different dipole resonances in gold nanoparticles (NPs) with rectangular, elliptical and diamondlike footprints on the length of the three principal axes and on particle geometry. The length of the two in-plane principal axes of the NPs studied is between 50 and 300 nm, while the particle height is between 10 and 50 nm. The scattering experiments reveal characteristic dependencies of the in-plane dipolar resonances on axis lengths and particle geometry. These experimental findings are discussed according to an intuitive resonance condition based on the electrostatic eigenmode method extended to include terms for retardation (dynamic depolarization) and particle curvature. Focus is laid on the dependence of the retardation term on the relevant

ACS Paragon Plus Environment

1


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 47

depolarization factor, the contributions of static and dynamic depolarization to the resonance wavelength, and to a generalization of the depolarization factors that describe the static depolarization in order to account for the differences in the dipolar surface modes of NPs with different geometries. The result of these studies is a detailed description and improved understanding of the scaling behavior of individual dipolar NP plasmon resonances.

Introduction The optical response of a metallic nanoparticle (NP) is dominated by the appearance of Localized Surface Plasmon Resonances (LSPRs) over a wide range of wavelengths. The LSPRs depend on the material of the NP,1 the surrounding material,2-5 particle shape and size,6-10 and particle excitation.11-13 In order to profit from the unique effects associated with LSPRs, such as strong near-fields and high scattering or absorption cross-sections, a general understanding of the involved interactive effects is needed. One of the most important parameters in the context of applications 14-21 is the position of the resonances in terms of frequency or wavelength. Without a knowledge of methods for tuning the resonance wavelength to the desired position and of the effects behind these methods, design construction is time-consuming and complex. Unfortunately, most bodies lead to equations that cannot be solved analytically which makes it difficult to establish a general understanding of the influence of the processes involved, such as particle depolarization and retardation, even for the dipolar modes of non-coupled particles of high symmetry.

In this paper, we follow the approach of defining an analytical resonance condition that is not restricted to the quasi-static case of ellipsoids followed by its liking to experimental data.22-25 To guarantee a formulation that is valid for arbitrary particle shapes, we start with the resonance


2

Page 3 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


condition which results from the electrostatic eigenmode method 26-28 and expand it by a generalized term for retardation (dynamic depolarization) and a parameter for particle curvature (curvature factor). The resulting shape-term, y, shows a characteristic relation to the dipolar resonance wavelength. In addition, the shape-term can be easily decomposed into its static and dynamic parts. Scattering experiments on NPs with rectangular footprints covering a wide range of axis configurations are then used to reveal some important basics of the scaling behavior of dipolar plasmon resonances of metallic NPs. It is shown that the retardation term depends strongly on the depolarization factor. The decomposition of the shape-term reveals that it is the static part that defines the scaling behavior in the cases studied, although the quasi-static approximation is clearly not valid for the axis lengths addressed. An estimation of the influence of changing particle cross-sections described by the curvature factor helps to bring differently shaped particle footprints into the discussion. It is shown that this influence can be explained by a changing relative polarization along the principal axis of the NP given by the depolarization factors and by a changing surface charge distribution described by their generalization. These findings in combination with the resonance condition introduced are enough to explain the experimentally observed scaling behavior of the dipolar resonances of gold NPs with elliptical, rectangular and diamond-like footprints (Figure 1) on an intuitive basis and in detail. Numerical simulations are not needed for these statements and are not considered throughout the paper.


3


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 47

Figure 1. Representative scanning electron microscopy images and schematic illustrations of the Au nanoparticles studied. The particle excitation performed along the in-plane principal axes is also indicated. The length of the a-axis is between 50 and 300 nm while the b-axis is kept constant at 123 ± 6 nm for all measured scattering spectra. The particle heights are between 10 and 50 nm.

Background and Model The electrostatic eigenmode method 26 (nonretarded boundary element method) leads to a simple expression for the NP permittivity at which an arbitrarily shaped particle shows a resonance in the electrostatic approximation under the assumption of a lossless body,

1 . 1

(1)


4

Page 5 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ε(λ) = ε1(λ) + iε2(λ) is the dielectric permittivity of the NP and εm is the dielectric constant of the surrounding medium. The parameter η represents the eigenvalues of the eigenvalue problem for the boundary integral equation in the quasi-static approximation 26,27

∮

∙

.

(2)

The particle is given by its boundary S, its outward unit normal at a point Q on S, nQ, and the vector between two points M and Q on S, rMQ. σ(Q) is the surface-charge distribution at Q for which the eigenfunctions (resonance modes) are determined over eq. (2). The NP permittivity at resonance can easily be transformed to a resonance condition in which eigenvalue and material properties appear in different terms, i.e. ! "#

#

0.

(3)

This formulation is formally equal to the resonance condition for dipolar resonances of ellipsoids and spheres in the quasi-static approximation.1,24,29 Therefore we set

%& '&

#

,

(4)

where Li is the depolarization factor for zero susceptibility along the principal axis under consideration and accordingly it is an entry in the diagonalized depolarization tensor L (see Supporting Information for details). In the case of bodies of arbitrary shapes, an appropriate definition of the depolarization tensor is needed in order to account for the appearance of nonuniform fields and to avoid a dependence of L on the position within the particle. A good approach to this issue can be found in the description of the magnetic properties of arbitrarily


5


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 47

shaped bodies, where the use of volume-averaged demagnetization tensors (or magnetometric demagnetization tensors) that only account for geometrical aspects is very popular.30,31,32 The latter is realized by considering only the case of zero susceptibility. The analogy between the macroscopic description of electrostatics and magnetostatics allows for the later use of analytically derived magnetometric demagnetization tensors 31-33 for particles with zero magnetic susceptibility in the study of the influence of different particle geometries on the plasmon resonance wavelengths. However, a volume-averaged depolarization dyadic 34 does not consider the established surface charge distribution and it neglects the interaction between surface charges. Therefore, a factor gi that accounts for these effects needs to be added in eq. (4). Berkovitch et al. 35,36 gave an intuitive picture of this by studying NPs with concave crosssections. Based on this picture, we call gi the curvature parameter in the following; gi also accounts for the (static) non-uniform polarization of the particle. The multiplication of gi breaks the relative character of the depolarization factor resulting from the unit trace condition for depolarization (demagnetization) tensors.30,34 In consequence, giLi can be seen as a generalized depolarization factor which allows us to directly compare the depolarization probabilities of NPs with different geometries. In the case of ellipsoidal bodies, there is no need to redefine or generalize the depolarization tensor and the formulas used in literature can be applied.29 These formulas equal those for the demagnetization tensor of ellipsoids.29,37 The validity of eq. (3) can be extended to the dynamic regime by considering the description of spheres. Based on this, retardation effects are taken into account by an additional term, Ri, which depends on the relevant axis length di.1,6,23,24,29,38 In order to maintain the general validity of the resonance condition, we do not further specify the dependence of Ri on di (or di/λ), but we


6

Page 7 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


consider an expected dependence on giLi. A significant dependence of Ri on ε(λ) is not expected for NPs with negligible ε2.6,23,24,38 This results in ! "#

(& 0 ,

(5)

with

(& %& '& )& %& '& , & .

(6)

In the following, we drop the index i indicating the principal axis under consideration. y can be interpreted as a shape-term that shows a characteristic relation to the resonance wavelength, λRes, via eq. (5). Figure 2 shows this relation for gold with the real part of the dielectric permittivity approximated by a Drude-Sommerfeld function, ε1(λ) = α – β2λ2, and negligible absorption, ε2 = 0. The influence of the surrounding media is also visualized by εm = 1, 1.5, 2 and 2.5. Fitting experimental gold bulk data 39 in the wavelength range between 600 and 1000 nm leads to α = 9.84 and β = 7.1 µm-1. Due to the fact that all NPs studied show λRes > 600 nm, this representation of ε1(λ) is sufficient for the discussion of their scaling behavior. For completeness, Figure 2 is extended to y-values in which ε1(λ) is no longer properly described by the parameters used (dashed lines). Here, a linear relation between ε1 and λ is used instead, i.e.

ε1(λ) = mλ + q, with m = -65.8 µm-1 and q = 30.787.


7


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 47

Figure 2. Dependence of the resonance wavelength, λRes, on the shape-term y for four different surrounding media, εm = 1, 1.5, 2 and 2.5. Figure 2 reveals an increasing red-shift of λRes for decreasing y-values. This red-shift is as expected at decreasing L-values (e.g. spheroids excited along the long axis with increasing aspect ratio) or at increasing retardation (e.g. increasing length of addressed axis at constant aspect ratio). The predicted red-shift for an increasing dielectric constant of the surrounding medium is also as expected. However, the increasing slope of λRes(y) with decreasing y-values is of importance for the later discussion of the experimental data.

Experimental Methods Sample Fabrication. In order to study the different parameters defining the resonance wavelengths we performed scattering experiments on NPs with rectangular, elliptical and diamond-like footprints of different heights. The NPs were produced by electron-beam lithography (Raith 150) and Physical Vapor Deposition (PVD). Their footprints were written in positive resist (PMMA 950K) on top of 8 nm ITO. Cleaned standard cover slips were used as substrates. After PMMA development, the NPs were grown within the pattern on top of 2 nm Ti using PVD. Finally, the remaining PMMA was removed, generating freestanding NPs.


8

Page 9 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The contribution of the substrate and the supporting layers to the scaling behavior of the NP plasmon resonances can be studied by analyzing the effect of different substrate configurations and their dependencies on particle size. We observed that a 5 nm ITO layer and a 2 nm Ti layer generate a red-shift of the dipolar in-plane resonance wavelength of nanocylinders by approximately 45 nm compared to the case of a pristine substrate. This shift is independent of the nanocylinder diameter in the range of 100 to 150 nm. Furthermore, a change of the cylinder height from 100 nm to 30 nm generates only a shift of the resonance wavelength on the order of 5 nm. Therefore, the influence of the substrate used can be considered as an offset that does not affect the experimentally observed scaling behavior presented in the following sections. The basic element of the sample layout used to study the scaling behavior of the NPs is an assembly containing 4×6 arrays of 5×5 NPs. The period within an array is 1.5 µm in both directions, which prevents the appearance of near-field coupling effects. The distance between two arrays is 4 µm. Within a basic element, the design parameters of the NPs are unchanged, i.e. for each NP footprint designed there are 24 arrays and 600 NPs of identical shape. The basic element is arranged in a systematic way over the sample. In y-direction, the aspect ratio of the two in-plane axes changes in (design-) steps of 0.25. In x-direction, one of the two inplane axes changes in steps of 10 nm. In other words, the basic elements are arranged as a grid on the sample. Within a row, one of the two in-plane axes is kept constant while the other inplane axis changes from one grid position to the next one according to the designed aspect ratio. However, the length of the axis that is kept constant within a row increases by 10 nm from one row to the next.


9


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 47

A systematic analysis of the influence of elliptical, rectangular and diamond-like footprints on the dipolar plasmon resonance positions was guaranteed by writing basic elements with the corresponding footprints but an identical set of design parameters next to each other at the same grid position. Markers were used to identify the designed length of the constant axis, the designed aspect ratio, and the footprint during the scattering experiments. The physical lengths of the principal axes and the overall shape of the nanoparticles were analyzed using a scanning electron microscope (SEM, Hitachi SU-70). The height of the NPs was set by a calibrated quartz crystal and verified with a profilometer (Dektak XT Advanced, Bruker) and the SEM using a stage tilted up to 45°. Nanoparticle Scattering Spectra. The scattering spectra of ensembles of 5×5 NPs were determined using an inverted microscope (Zeiss Axio Observer) equipped with a dark-field objective (EC Epiplan Neofluar 50×/0.8, Zeiss) coupled to a spectrometer (Spectra Pro 2500i, Princeton Instruments) and a liquid-nitrogen-cooled CCD detector (Spec-10, Princeton Instruments). Standard dark-field optical microscopy and spectroscopy (DFOMS) in reflection was used, i.e. NP excitation and collection of the scattered light was carried out by the same objective. For excitation, a Hg lamp was coupled to the rear port of the microscope via additional optics, including filters for intensity optimization and a polarizer. The measurement approach taken generates an electric field component parallel to the particle height. This component is of no importance for the NPs studied because the dipolar resonance associated with the particle height is blue-shifted out of the considered wavelength range of 550 – 950 nm for h ≤ 50 nm. All measured spectra were divided by the reflection spectra of an Ag mirror using exactly the same setup configuration but in bright-field mode. The exact resonance wavelengths were


10

Page 11 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


determined by fitting each scattering spectra with up to three Lorentzian curves, using resonance frequency, line width and amplitude as fitting parameters.

Nanocuboids Experimental Results. NPs with rectangular footprints (cuboids) are of special interest due to the availability of analytical formulas for appropriate depolarization factors (see Supporting Information for details).31,33 This allows us to determine the dependence of the retardation term R on the depolarization factor and to decompose the shape-term into its static and dynamic parts, gL and R.

Figure 3 summarizes the dipolar peak resonance wavelengths (Figure 3a,c) and scattering spectra (Figure 3b,d) observed experimentally in cuboids for a particle excitation along the two in-plane principal axes, a and b. One of these two axes was kept constant at b = 123 ± 6 nm, while the a-axis varied between 50 and 300 nm. Three different particle heights, h = 13, 20 and 50 nm, were studied. It is known that NPs of this shape show a variety of polarization modes of dipolar character.40,41 However, only two modes can be observed with the method chosen and wavelength range studied. One is observed best with a particle excitation along the constant baxis and the other with a particle excitation along the variable a-axis. The resonance observed along the constant b-axis in dependence on the variable a-axis, λb(a), shows at small a-values a clear and increasing deviation from the nearly constant value observed for large a-values (Figure 3a). Reduction of the particle height generates an additional red-shift. The absolute value of this offset increases with decreasing particle height (see also Figure 8a,b, discussed later). The resonance excited along the variable axis in dependence of its axis length, λa(a), shows the


11


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 47

expected red-shifting with increasing axis length (Figure 3c). A close to linear dependence of λa on a is observed. Again, a decreasing particle height leads to an offset and influences the slope of

λa(a). In addition, the measurements on the NPs with h = 50 nm show a deviation from the linear dependence at small a-values.

Figure 3. Dependence of the observed in-plane dipolar resonances of a cuboid defined by (a,b,h) on the variable a-axis length and the corresponding scattering spectra for the case of a particle height of 50 nm. The b-axis is kept constant, b = 123 ± 6 nm. (a) Peak resonance wavelength for particle excitation along the constant b-axis and the three different particle heights of h = 13, 20, 50 nm. (b) Corresponding experimental scattering spectra for h = 50 nm. (c) Peak resonance wavelength for particle excitation along the variable a-axis and the three different particle heights of h = 13, 20, 50 nm. (d) Corresponding experimental scattering spectra for h = 50 nm. See supporting information for further experimental scattering spectra (Figure S1).


12

Page 13 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Retardation Term and its Dependencies. A first step in studying the observed dependencies in detail is the calculation of the retardation term for each measurement point according to eqs. (5) and (6). Screening the values obtained for equal length of the relevant principal axis, d, reveals the dependence of the retardation term on L (Figure 4). Depolarization factors as given by Ref. [33] (see Supporting Information) and a constant curvature parameter of g = 0.94 were used (see below). The permittivity of the surrounding medium was set to εm = ½ (εsubstrate+1) = 1.625. The error bars indicate the effect of errors in g, λRes, L and εm of 0.12, 5 nm, 0.015 and 0.325, respectively. Figure 4 shows that the retardation term is always negative, generating a negative contribution to y (eq. (6)) and therefore the well-known red-shift of the resonance wavelength due to retardation. In addition, the retardation term observed for a constant length of the relevant axis scales strongly with L. A second-order polynomial, R(L) = pL2+ qL, leads to a good fit with d-dependent parameters p(d) and q(d). For d = 123 ± 6 nm the parameters are p = 1.2316 and q = -0.2542. A decreasing length of the relevant axis shifts the polynomial to lower absolute R-values, while a longer axis leads to a shift of R(L) to higher absolute values. However, L is always larger than the absolute value of R in the range of axis lengths considered. Concerning the slope of R(L), the case of d = 123 ± 6 nm shows that a change in L by ∆L causes a shift in R which is always smaller than ∆L as long as L is smaller than 0.30, i.e. ∆R < ∆L. For L > 0.30 we enter a region where ∆R (∆L) ≈ ∆L. Such a characteristic L-value also exists for the other axis lengths indicated.


13


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 47

Figure 4. Dependence of the retardation term R on the depolarization factor L and the length of the relevant principal axis d. The legend indicates the average measured value of d and the maximum deviation from this value for the corresponding set of data points. The dashed line represents R = -L. Figure 4 also gives indications regarding the interplay between the dependence of R on L and d. In the frequently discussed case of constant aspect ratios, a change in the relevant axis does not produce a changing L, and the pure d-dependence of R can be observed. In the case of particles with one principal axis kept constant, the L-dependence of R can be studied by a particle excitation along this constant axis as done in Figure 4. For the case discussed below of two constant principal axes and an excitation along the only variable axis, the influences of d and L on R are competitive, i.e. the change in R triggered by L is compensated by the change in R triggered directly by d. For example, an increase in the variable axis length generates a decreasing depolarization factor along this axis (i.e. a decreasing R) and a d-driven increase of R at the same time. Static and Dynamic Contributions. Based on eqs. (5) and (6) it is possible to decompose the shape-term y into its dynamic and static parts, R and gL, for all measurement points shown in


14

Page 15 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 3. This allows us to understand the influence of R and gL on y and therefore on the resonance wavelengths. Figure 5 shows this decomposition of the resonances measured along the constant in-plane axis of length b = 123 ± 6 nm (Figure 5a-c) and along the variable in-plane axis (Figure 5d-f), both in dependence on the variable axis length. The same values for εm and error estimations as mentioned above were used. First, we look at the resonance excited along the variable axis (Figure 5d-f). In all configurations addressed, y is dominated by the depolarization factor and its dependence on the axis length follows the trend given by gL. This means that it is the changing static particle depolarization that accounts for the length-dependent shift of λRes and not the retardation. In addition, Figure 5d-f shows that the absolute value of R does not increase with increasing axis length. Therefore, R is not dominated by the axis length but is strongly influenced (in the case of h = 50 nm even dominated) by the changing L. However, the axis length dependence of R compensates for these L driven changes at low L-values in the configurations studied.


15


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 47

Figure 5. Decomposition of the shape-term y in its static and dynamic parts, gL and R, for NPs with rectangular footprints. (a)-(c) Particle excitation along the constant in-plane axis, b = 123 ± 6 nm. (d)-(f) Particle excitation along the variable in-plane axis. The particle heights are h = 50 nm (a,d), h = 20 nm (b,e) and h = 13 nm (c,f). In the case of the resonance excited along the constant in-plane axis (Figure 5a-c), retardation depends on L only. Although R contributes to y in such a way that it reduces the trend given by L, it cannot completely compensate for changes in L. This is a direct consequence of the ∆R(∆L)


16

Page 17 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


< ∆L statement and again leads to a shape-term that follows the trend given by L. However, the higher the L-value (long variable axis, high particles) the more equal are ∆R(∆L) and ∆L. This leads to a flattened y with a nearly constant value over a wide range of the variable axis length.

Nanoparticles with Elliptical, Diamond-like and Rectangular Footprints Experimental Results. Figure 3 revealed characteristic dependencies of the two in-plane dipolar resonances of cuboids on the axis configuration consisting of a fixed height, and a variable and a constant in-plane principal axis. In order to study the influence of particle geometry on these dependencies, we repeated the scattering experiments on NPs with elliptical and diamond-like footprints. Figure 6 shows the resulting peak resonance wavelengths for particle heights of 50 (a,d), 20 (b,e) and 13 nm (c,f) in dependence on the variable axis. Representative experimental scattering spectra are given as Supporting Information (Figure S2). In Figure 6a-c, the NPs are excited along the constant in-plane axis, λb(a), and in Figure 6d-f they are excited along the variable in-plane axis, λa(a). The influence of the NP footprint on the overall scaling behavior of the resonances is very small. A red-shift of both λb(a) and λa(a) for rectangular footprints compared to elliptical and diamond-like footprints is observed. Surprisingly, there is no offset between elliptical and diamond-like footprints. In addition, there is a tendency towards an increased slope of λa(a) for rectangular compared to elliptical/diamondlike footprints. In the following, we first discuss the observed overall scaling behavior of the dipolar resonances before we turn to the influence of changing footprints. Table 1 summarizes all observations on NPs with widely varying axis configurations and different footprints.


17


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 47

Figure 6. Dependence of the dipolar resonance wavelengths of nanoparticles with rectangular (R), diamond-like (D) and elliptical (E) footprints on the variable in-plane axis. (a)-(c) Particle excitation along the constant in-plane axis, b = 123 ± 6 nm. (d)-(f) Particle excitation along the


18

Page 19 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


variable in-plane axis. (a), (b) Particle height h = 50 nm. (c), (d) Particle height h = 20 nm. (e), (f) Particle height h = 13 nm.


19


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 47

Table 1. Experimentally observed influence of changing length configurations and particle footprints on the in-plane dipolar resonances. The NPs studied have three principal axes. One inplane axis is kept constant at b = 123 ± 6 nm; the other in-plane axis, a, varies between 50 and 300 nm. The particle height, h, is between 10 and 50 nm. Indexes indicate the principal axis along which the resonance is excited.

Exclusive axis dependence

Constant value of λb over a wide range of a,h values. Linear dependence of λa(a) over a wide range of b,h values. Deviation from constant value for λb(a) for small a-values.

Non-exclusive axis

Deviation of linear a-dependence of λa(a) for small a-values.

dependence Deviation of linear a-dependence of λa(a) is more pronounced for higher NPs, but almost not observable for very low particles. Red-shift of λb(a) and λa(a) for decreasing h. Height influence

Red-shift more pronounced for small particle height. Increasing slope of λa(a) for decreasing h. Red-shift of λa(a) and λb(a) for rectangular (R) footprints compared

Geometry influence

to elliptical (E) and diamond-like (D) footprints. No offset of λb(a) and λa(a) for D compared to E footprints. Tendency towards an increased slope of λa(a) for R compared to E,D.


20

Page 21 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Discussion: The Shape Term. The various features of the overall scaling behavior of the dipolar resonances can be explained by further analyzing the static and dynamic parts, their contributions to the shape-term y, and the transformation of the resulting shape-term to the resonance wavelength. As pointed out, the depolarization factors are key to understand the shape-term corresponding to a given axis configuration due to the dependence on it of both static and dynamic parts. Figure 7 shows the characteristic dependence of the two in-plane depolarization factors on the variable in-plane axis for cuboids33 and ellipsoids31 (see also Supporting Information). The configurations chosen are similar to the configurations studied experimentally.

Figure 7. Calculated depolarization factors along the in-plane principal axes for cuboids 33 (open symbols) and ellipsoids 31 (filled symbols). One in-plane axis is kept constant, b = 125 nm.


21


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 47

Particle height h = 13, 20, 50 and 100 nm. (a) Depolarization factor along the constant principal axis. (b) Depolarization factor along the variable principal axis. Figure 8 visualizes the dependencies that are discussed in the following using the example of NPs with elliptical footprints. The shape-term of the resonance excited along the constant inplane axis (b-axis) in dependence on the variable a-axis shows a nearly constant value for a > 100 nm (Figure 8a). This can be explained by the dependence of the corresponding depolarization factor on a, Lb(a) (Figure 7a), and the discussed onset of compensation of changes in L by R (Figure 4). For low particle heights, Lb(a) itself is nearly constant while for an increasing particle height Lb increases in absolute value and in its dependence on a. The latter is reduced by an onset of compensation, leading to a nearly constant shape-term. This also explains the decreasing yb-value for a < 100 nm. Lb(a) decreases with decreasing a for all particle heights. This reduces the compensation effect and makes the a-dependence of Lb visible in yb, including its more pronounced decrease for higher particles. The decreasing influence of a change in height on the shape-term for increasing particle height can be explained accordingly. An increase in height increases Lb and therefore the compensation of the h-induced changes in yb by R(Lb). At a certain particle height, this compensation makes yb nearly independent of h.


22

Page 23 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 8. Shape-term y and resonance wavelengths from experimental data on cylinders with elliptical footprints. (a,b) In-plane resonance excited along the constant axis, b = 123 ± 6 nm. (c,d) In-plane resonance excited along the variable axis. Particle height h = 13, 20, 50 and 110 nm. The transformation of yb to the resonance wavelength, λRes, is straightforward. According to Figure 2, lower y-values result in higher resonance wavelengths. In addition, the slope of λRes(y) increases for decreasing y. This explains why the observed overall red-shift at lower a-values is on the same order for all three particle heights although the decrease in yb is less pronounced for lower particles. It also explains why data points for low particles tend to scatter more: the resonance wavelengths become more sensitive to changes in y, caused e.g. by production-based shape variations.


23


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 47

Only one argument needs to be added to discuss particle excitation along the variable axis (Figure 8 c,d): the additional dependence of R on a. As pointed out in the discussion of Figure 4, this dependence causes reduced compensation for the a-driven influence of La on ya. Therefore, R(La, a) does not completely compensate changes in La and ya follows the trend given by La(a) (Figure 7b) for all addressed a-values. In combination with the still-valid arguments for the decreasing height influence, it produces the characteristic curves for ya(a) shown in Figure 8c. The transformation of the shape-term values to the corresponding resonance wavelengths is again straightforward. Particles of lower height cover a range of lower ya-values, resulting in the reading-out of red-shifted wavelengths in the steep, nearly linear part of λRes(y) while higher particles address the rounding towards a constant value (Figure 2). The result is a seemingly reduced slope and a deviation from linear behavior for higher particles. Finally, we state that with the chosen axis configurations, there are no ya,b-values higher than 0.2 and no dipolar resonance wavelengths below 600 nm. This means that we do not leave the region of correct description of ε1(λ) via the chosen Drude-Sommerfeld function and that the assumption of negligible damping is correct. Discussion: Curvature Parameter. A discussion of the influence of a changing particle footprint based on the resonance condition (eqs. (5) and (6)) is only possible if the depolarization factors used are generalized by the curvature parameter g. In the following we estimate g for the footprints studied. Additional measurements addressing g for different particle shapes specifically are needed in order to determine more reliable values and to study the curvature factor in detail. However, the estimations made are sufficient to explain the observed influence of NP footprint on the dipolar plasmon wavelength.


24

Page 25 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The curvature parameter g along a principal axis of the geometry under question (index q in the following) can be estimated from experimental data and known depolarization factor if a reference geometry with known g and known corresponding depolarization factor is available (index r), i.e.,

%+ ≈ - Δ( Δ) % ' , .

(12)

where ∆y = yq – yr is the measured difference in the shape-terms and ∆R = Rq – Rr accounts for the difference in retardation. If particles of identical lengths of the principal axes, (a,b,h)q = (a,b,h)r, and the resonances excited along the same principal axis are compared, ∆R is on the order of the difference of the depolarization factors or smaller. For similar particle shapes this difference is expected to be small. However, the assumption ∆R ≈ 0 ignores the influence of gL on R and contributes to the approximate character of the g values given below. Ellipsoids are good reference geometries due to their known curvature parameter, gr = 1, and the availability of analytical formulas for calculating their depolarization factors (see Supporting Information).29,31,37 Thanks to the similarity between a prolate ellipsoid excited along its minor axis and a cylinder excited along its base, gr = 1 can also be used as approximation for the inplane curvature parameter of cylinders. This makes the measurements on cylinders in combination with the depolarization factors given in Ref. [33] good references (see also Supporting Information). For such reference geometries and under the assumptions made it follows that

%+ ≈ - Δ( ' . .

(13)


25


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 47

With this, the g-values for NPs with square footprints and appropriate lengths of the in-plane axes or diagonals can be derived from the measured resonance wavelengths. The data points obtained from measurements on NPs with rectangular footprints, but a ≈ b ≈ 123 nm, lead to a gvalue of 0.94 ± 0.12. The corresponding measurements on NPs with diamond-like footprints (diagonals on the order of 123 nm) lead to a g-value of 0.87 ± 0.10. Only a very weak dependence on particle height is observed (0.95 and 0.88 for h = 50 nm, 0.94 and 0.86 for h = 20 nm). The given errors are based on estimated errors in gr, λq,r, and εm of 0.035, 10 nm and 0.325, respectively. The error in R is set to 1/2(yq – yr). The difference in the g-values estimated for NPs with square footprints excited along the side and along the diagonal of the footprint is a manifestation of the anisotropy introduced by considering the surface charge distribution. In other words: whereas the entries of the depolarization tensor do not change under rotation around the h-axis for a particle with a square footprint, the entries of the curvature tensor do. However, direct experimental proof of this curvature tensor property via polarization-dependent scattering experiments on the very same NP with a square footprint is still pending. The small observed difference in the g-value for the various footprints in combination with the small influence of a changing particle height indicate that the surface charge distributions vary slowly with changing aspect ratios but constant particle geometry. In addition, the depolarization factor itself considers to some extent the relative aspects of changing aspect ratios. In consequence, the g-values obtained can be taken as approximations for particles with the same footprint but varying aspect ratios. Discussion: Influence of Particle Footprint. Figure 9 shows the effect of a changing footprint on the depolarization factor for the in-plane dipolar resonance of NPs with circular, square and diamond-like (square along diagonals) footprints. In sum, this effect, at constant


26

Page 27 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


length of the principal axes, is twofold: first, the polarizability along the axis addressed changes relative to the other principal axes. This is considered via the depolarization factor L (open symbols in Figure 9). Second, the charge distribution and interaction between the charges change. The influence of this change on the resonance wavelength is considered via the curvature parameter g and leads to the generalized depolarization factors gL (filled symbols in Figure 9).

Figure 9. Standard (open symbols, see also Supporting Information) and generalized (filled symbols) in-plane depolarization factors in dependence on particle height for various particle footprints. The length of the in-plane principal axes is a = b = 125 nm. (a) Absolute values. (b) Values relative to a circular footprint.


27


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 47

As an example, we will discuss NPs with rectangular (index R) and elliptical (index E) footprints. Figure 9 indicates that the depolarization of a cuboid along its in-plane axes relative to its out-of-plane depolarization is less pronounced than that of a cylinder with identical lengths of the principal axes, LR < LE. In addition, gR is smaller than gE. Due to the additive character of the two effects caused by a change from elliptical to rectangular footprints, a well-pronounced redshift of the resonances is observed for particles with rectangular footprints. The situation is different for particles with diamond-like footprints. For these footprints, the inplane depolarization is more pronounced compared to the out-of-plane depolarization than for particles with elliptical footprints, LD > LE. Again, gD is smaller than gE. In combination, a change in footprint from elliptical to diamond-like generates an increase in the resonance wavelength caused by the changing surface charge distribution (gD < gE), which is compensated by the higher relative depolarization of the in-plane resonances caused by this change (LD > LE). As a consequence the resonance wavelengths for NPs with elliptical and diamond-like footprints tend to coincide with each other, as is seen with the experimentally addressed configurations. To conclude the discussion of the items listed in Table 1, the explanation of the observed tendency towards an increased slope of λa(a) for R-shaped compared to E,D-shaped particles is straightforward. Red-shifted resonances indicate lower corresponding y values. Therefore, their slopes are increased according to eq (5) (Figure 2).

Summary We have discussed the dependence of dipolar plasmon resonances on the length of the principal axes of nanoparticles with rectangular, elliptical and diamond-like cross-sections (footprints) in detail. The discussion is based on an intuitive resonance condition derived from


28

Page 29 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the electrostatic eigenmode method, generalized volume-averaged depolarization factors and dynamic depolarization (retardation). The depolarization factors are generalized by introducing a curvature parameter that accounts for the effects resulting from the mode-dependent surface charge distribution. In consequence, a direct comparison of differently shaped NPs based on generalized depolarization factors was possible. We showed that it is the static particle depolarization given by the depolarization factors that dominates the scaling of the observed dipolar resonance wavelengths. Using the picture of a shape-dependent resonance wavelength in the quasi-static regime that red-shifts due to sizedependent retardation, this means that the shift in the quasi-static resonance position due to changing axis configurations dominates the resonance position even for axis lengths between 50 and 300 nm. The appearance of retardation could not compensate completely the scaling behavior given by the quasi-static response. In addition, it was shown that the retardation term itself is not dominated by its dependence on the length of the relevant principal axis, but strongly influenced by the changing depolarization factor. In consequence, considering only axis lengths is not sufficient to describe retardation effects. The dependence of the retardation term on the depolarization factor was isolated by choosing axis configurations with one in-plane axis held constant. A strong scaling that can be described by a second-order polynomial is observed. The absolute value of the retardation term is smaller than the value describing the static particle depolarization. However, the slope of R(L) is around -1 for frequently addressed, moderate axis configurations. With these findings, it was possible to understand the experimentally observed characteristic dependence of the dipolar resonance wavelengths of particles with different footprints and various configurations on the length of the principal axes. The scaling of the resonance


29


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 47

wavelengths can be explained by three arguments: the dependence of the relevant depolarization factor on the axis configuration, the contribution of the static and dynamic parts to the shapeterm (especially the onset of compensation of changes in the depolarization factor by retardation), and the characteristic dependence of the resonance wavelength on the shape-term defined by the resonance condition. The observed influence of a changing particle footprint was explained by estimating the curvature parameter for the footprints studied. Based on this, we showed that the effect of a changing footprint can be interpreted as the result of a change in the depolarization along the axis in question relative to the other principal axes and the change in the surface charge distribution. The cases studied show that these effects can be cumulative or destructive.

Acknowledgments The authors thank Y. Ekinci for fruitful discussion. This work was supported by the Swiss National Science Foundation (SNF Grant No. 200021-125149). The samples were produced at FIRST, Center for Micro- and Nanotechnology, ETH Zurich.

Associated Content Supporting Information Available. Experimental scattering spectra for NPs with different footprints and axis configurations. Formulas used to calculate the depolarization factors for NPs with rectangular and circular footprints as well as ellipsoids; discussion of their usability in the corresponding cases. This material is available free of charge via the Internet at http://pubs.acs.org.


30

Page 31 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


References 1. Kreibig, U.; Vollmer, M. Optical Properties of Metal Clusters; Springer Series in Materials Science 25: Berlin, 1995. 2. Noguez, C. Surface Plasmons on Metal Nanoparticles: the Influence of Shape and Physical Environment. J. Phys. Chem. C 2007, 111, 3806-3819. 3. Mock, J.J.; Smith, D.R.; Schultz, S. Local Refractive Index Dependence of Plasmon Resonance Spectra from Individual Nanoparticles. Nano Lett. 2003, 3, 485-491. 4. Miller, M.M.; Lazarides, A.A. Sensitivity of Metal Nanoparticle Surface Plasmon Resonance to the Dielectric Environment. J. Phys. Chem. B 2005, 109, 21556-21565. 5. Päivänranta, B.; Merbold, H; Giannini, R.; Büchi, L.; Gorelick, S.; David, C.; Löffler, J.F.; Feurer, T.; Ekinci, Y. High Aspect Ratio Plasmonic Nanostructures for Sensing Applications. ACS Nano 2011, 8, 6374-6382. 6. Kelly, K.L.; Coronado, E.; Zhao, L.L.; Schatz, G.C. The Optical Properties of Metal Nanoparticles: the Influence of Size, Shape, and Dielectric Environment. J. Phys. Chem. B 2003, 107, 668-677. 7. Myroshnychenko, V.; Rodriguez-Fernandez, J.; Pastoriza-Santos, I.; Funston, A.M.; Novo, C.; Mulvaney, P.; Liz-Marzan, L.M.; Garcia de Abajo, F.J. Modelling the Optical Response of Gold Nanoparticles. Chem. Soc. Rev. 2008, 37, 1792-1805.


31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 47

8. Kooij, E.; Poelsema, B. Shape and Size Effects in the Optical Properties of Metallic Nanorods. Phys. Chem. Chem. Phys. 2006, 8, 3349-3357. 9. Wiley, B.J.; Chen, Y.; McLellan, J.M.; Xiong, Y.; Li, Z.-Y.; Ginger, D.; Xia, Y. Synthesis and Optical Properties of Silver Nanobars and Nanorice. Nano Lett. 2007, 7, 1032-1036. 10. Ringe, E.; Zang, J.; Langille, M.R.; Mirkin, C.A.; Marks, L.D.; Van Duyne, R.P. Correlating the Structure and Localized Surface Plasmon Resonance of Single Silver Right Bipyramids. Nanotech. 2012, 23, 444005. 11. Muskens, O.L.; Bachelier, G.; Del Fatti, N.; Vallée, F.; Brioude, A.; Jiang, X.; Pileni, M.P. Quantitative Absorption Spectroscopy of a Single Gold Nanorod. J. Phys. Chem. C 2008, 112, 8917-8921. 12. Nehl, C.L.; Liao, H.; Hafner, J.H. Optical Properties of Star-Shaped Gold Nanooparticles. Nano Lett. 2006, 6, 683-688. 13. Huang, Y.; Kim, D-H. Dark-Field Microscopy Studies of Polarization-Dependent Plasmonic Resonance of Single Gold Nanorods: Rainbow Nanoparticles. Nanoscale 2011, 3, 3228-3232. 14. Liu, N.; Tang, M.L.; Hentschel, M.; Giessen, H.; Alivisatos, A.P. Nanoantenna-Enhanced Gas Sensing in a Single Tailored Nanofocus. Nature Materials 2011, 10, 631-636. 15. Nie, S.; Emroy, S.R. Probing Single Molecules and Single Nanoparticles by SurfaceEnhanced Raman Scattering. Science 1997, 275, 1102-1106.


32

Page 33 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


16. Anker, J.N.; Hall, W.P.; Lyandres, O.; Shah, N.C.; Zhao, J.; Van Duyne, R.P. Biosensing with Plasmonic Nanosensors. Nature Materials 2008, 7, 442-453. 17. Jha, S.K.; Ahmed, Z.; Agio, M.; Ekinci, Y.; Löffler, J.F. Deep-Ultraviolet SurfaceEnhanced Resonance Raman Scattering of Adenine on Aluminum Nanoparticle Arrays. J. Am. Chem. Soc. 2012, 134, 1966-1969. 18. O’Carroll, D.M.; Hofmann, C.E.; Atwater, H.A. Conjugated Polymer/Metal Nanowire Heterostructure Plasmonic Antennas. Adv. Mater. 2010, 22, 1223-1227. 19. Atwater, H.A.; Polman, A. Plasmonics for Improved Photovoltaic Devices. Nature Materials. 2010, 9, 205-213. 20. Van Maltzahn, G.; Park, J.-H.; Agrawal, A.; Bandaru, N.K.; Das, S.K.; Sailor, M.J.; Bhatia, S.N. Computationally Guided Photothermal Tumor Therapy using LongCirculating Gold Nanorod Antennas. Cancer Res. 2009, 69, 3892-3900. 21. Leung, J.P.; Wu, S.; Chao, K.C.; Signorell, R. Investigation of Sub-100 nm Gold Nanoparticles For Laser-Induced Thermotherapy for Cancer. Nanomaterials 2013, 3, 86106. 22. Encina, E.R.; Coronado, E.A. Resonance Conditions for Multipole Plasmon Excitations in Noble Metal Nanorods. J. Phys. Chem. C 2007, 111, 16796-16801. 23. Meier, M.; Wokaun, A. Enhanced Fields on Large Metal Particles: Dynamic Depolarization. Opt. Lett. 1983, 8, 581-583.


33


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 47

24. Kuwata, H.; Tamaru, H.; Esumi, K.; Miyano, K. Resonant Light Scattering from Metal Nanoparticles: Practical Analysis beyond Rayleigh Approximation. Appl. Phys. Lett. 2003, 83, 4625-4627. 25. Kooij, E.; Ahmed , W.; Zandvliet, H.W.; Poelsema, B. Localized Plasmons in Noble Metal Nanospheroids. J. Phys. Chem. C 2011, 115, 10321-10332. 26. Mayergoyz, I.D.; Fredkin, D.R.; Zhang, Z. Electrostatic (Plasmon) Resonances in Nanoparticles. Phys. Rev. B 2005, 72, 155412. 27. Mayergoyz, I.D.; Zhang, Z.; Miano, G. Analysis of Dynamics of Excitation and Dephasing of Plasmon Resonance Modes in Nanooparticles. Phys. Rev. Lett. 2007, 98, 147401. 28. Vernon, K.C.; Funston, A.M.; Novo, C.; Gomez, D.E.; Mulvaney, P.; Davis, T.J. Influence of Particle-Substrate Interaction on Localized Plasmon Resonances. Nano Lett. 2010, 10, 2080-2086. 29. Bohren, C.F.; Huffman, D.R. Absorption and Scattering of Light by Small Particles; Wiley: Weinheim, 2004. 30. Moskowitz, R.; Della Torre, E. Theoretical Aspects of Demagnetization Tensors. IEEE Trans. on Magn. 1966, 2, 739-744. 31. Chen, D.-X.; Pardo, E.; Sanchez, A. Demagnetizing Factors of Rectangular Prisms and Ellipsoids. IEEE Trans. on Magn. 2002, 38, 1742-1752.


34

Page 35 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


32. Chen, D.-X.; Brug, J.A.; Goldfarb, R.B. Demagnetizing Factors of Cylinders. IEEE Trans. on Magn. 1991, 27, 3601-3619. 33. Aharoni, A. Demagnetizing Factors for Rectangular Ferromagnetic Prisms. J. Appl. Phys. 1998, 83, 3432-3434. 34. Yaghjian, A.D. Electric Dyadic Green’s Functions in the Source Region. Proc. IEEE. 1980, 68, 248-263. 35. Berkovitch, N.; Ginzburg, P.; Orenstein, M. Concave Plasmonic Particles: Broad-Band Geometrical Tunability in the Near-Infrared. Nano Lett. 2010, 10, 1405-1408. 36. Berkovitch, N.; Ginzburg, P.; Orenstein, M. Nano-Plasmonic Antennas in The Near Infrared Regime. J. Phys.: Condens. Matter 2012, 24, 073202. 37. Osborn; J.A. Demagnetizing Factors of the General Ellipsoid. Phys.Rev. 1945, 67, 351357. 38. Schmucker, A.L.; Harris, N.; Banholzer, M.J.; Blaber, M.G.; Osberg, K.D.; Schatz, G.C.; Mirkin, C.A. Correlating Nanorod Structure with Experimentally Measured and Theoretically Predicted Surface Plasmon Resonance. ACS Nano. 2010, 4, 5453-5463. 39. Olson, C.G.; Lynch, D.W.; Weaver, J.H. In Handbook of Chemistry and Physics; 90th ed.; Lide, D.R., Ed.; CRC Press: Boca Raton, FL, 2009-2010; p. 12-127. 40. Hung, L.; Lee, S.Y.; McGovern, O.; Rabin, O.; Mayergoyz, I. Calculation and measurement of radiation corrections for plasmon resonances n nanoparticles. Phys. Rev. B 2013, 88, 075424.


35


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 47

41. Cortie, M.B..; Liu, F.; Arnold, M.D.; Niidome, Y.; Mayergoyz, I. Multimode Resonances in Silver Nanocuboids. Langmuir 2012, 28, 9103-9112.


36

Page 37 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


TOC Graphic


37


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 1. Representative scanning electron microscopy images and schematic illustrations of the Au nanoparticles studied. The particle excitation performed along the in-plane principal axes is also indicated. The length of the a-axis is between 50 and 300 nm while the b-axis is kept constant at 123 ± 6 nm for all measured scattering spectra. The particle heights are between 10 and 50 nm. 177x85mm (300 x 300 DPI)


Page 38 of 47

Page 39 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 2. Dependence of the resonance wavelength, λRes, on the shape-term y for four different surrounding media, εm = 1, 1.5, 2 and 2.5. 53x34mm (300 x 300 DPI)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 3. Dependence of the observed in-plane dipolar resonances of a cuboid defined by (a,b,h) on the variable a-axis length and corresponding scattering spectra for a particle height of 50 nm. The b-axis is kept constant, b = 123 ± 6 nm. Three different particle heights of h = 13, 20 and 50 nm are shown. (a) Peak resonance wavelength for particle excitation along the constant b-axis. (b) Corresponding experimental scattering spectra. (c) Peak resonance wavelength for particle excitation along the variable a-axis. (d) Corresponding experimental scattering spectra. See supporting information for further experimental scattering spectra (Figure S1). 105x62mm (300 x 300 DPI)


Page 40 of 47

Page 41 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 4. Dependence of the retardation term R on the depolarization factor L and the length of the relevant principal axis d. The legend indicates the average measured value of d and the maximum deviation from this value for the corresponding set of data points. The dashed line represents R = -L. 53x34mm (300 x 300 DPI)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 5. Decomposition of the shape-term y in its static and dynamic parts, gL and R, for NPs with rectangular footprints. (a)-(c) Particle excitation along the constant in-plane axis, b = 123 ± 6 nm. (d)-(f) Particle excitation along the variable in-plane axis. The particle heights are h = 50 nm (a,d), h = 20 nm (b,e) and h = 13 nm (c,f). 158x141mm (300 x 300 DPI)


Page 42 of 47

Page 43 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 6. Dependence of the dipolar resonance wavelengths of nanoparticles with rectangular (R), diamondlike (D) and elliptical (E) footprints on the variable in-plane axis. (a)-(c) Particle excitation along the constant in-plane axis, b = 123 ± 6 nm. (d)-(f) Particle excitation along the variable in-plane axis. (a), (b) Particle height h = 50 nm. (c), (d) Particle height h = 20 nm. (e), (f) Particle height h = 13 nm. 194x213mm (300 x 300 DPI)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 7. Calculated depolarization factors along the in-plane principal axes for cuboids 34 (open symbols) and ellipsoids 35 (filled symbols). One in-plane axis is kept constant, b = 125 nm. Particle height h = 13, 20, 50 and 100 nm. (a) Depolarization factor along the constant principal axis. (b) Depolarization factor along the variable principal axis. 105x134mm (300 x 300 DPI)


Page 44 of 47

Page 45 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 8. Shape-term y and resonance wavelengths from experimental data on cylinders with elliptical footprints. (a,b) In-plane resonance excited along the constant axis, b = 123 ± 6 nm. (c,d) In-plane resonance excited along the variable axis. Particle height h = 13, 20, 50 and 110 nm. 107x64mm (300 x 300 DPI)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 9. Standard (open symbols) and generalized (filled symbols) in-plane depolarization factors in dependence on particle height for various particle footprints. The length of the in-plane principal axes is a = b = 125 nm. (a) Absolute values. (b) Values relative to a circular footprint. 107x139mm (300 x 300 DPI)


Page 46 of 47

Page 47 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


TOC image 36x26mm (600 x 600 DPI)


Scaling Behavior of Individual Nanoparticle Plasmon Resonances

Recommend Documents