Nanometric Rulers Based on Plasmon Coupling in Pairs of Gold

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Nanometric Rulers Based on Plasmon Coupling in Pairs of Gold Nanoparticles Anatolii I. Dolinnyi J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp5116614 • Publication Date (Web): 12 Feb 2015 Downloaded from http://pubs.acs.org on February 18, 2015

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The Journal of Physical Chemistry 1

Nanometric Rulers Based on Plasmon Coupling in Pairs of Gold Nanoparticles Anatolii I. Dolinnyi Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, Leninskii Prospect, 31, 119071 Moscow, Russia ABSTRACT: Using a generalized multiparticle Mie theory, we calculated the optical properties of gold nanoparticle (Au NP) pairs of 8-80 nm in diameter (D) and 0.1-120 nm in interparticle gap (s) under typical experimental conditions: an unpolarized incident light and random orientation of the pairs in space. By analyzing the extinction spectra of coupled spheres, three ranges of interparticle separations (long, middle and short) with different plasmon coupling regimes were distinguished. For long interparticle distances, a single plasmon peak in the spectrum at wavelength λp red-shifts exponentially relative to that of an isolated particle at wavelength λ0 as a function of x = s/D:

∆λ λp − λ0  = = aexp − λ0 λ0 

x  , with a decay constant (t = 0.19) being nearly independent of nanoparticle diameters at t

D < 50 nm. The stronger shifts (0.04 < a < 0.08) are observed for 30-60 nm Au NPs. In the middle distance range (0.02 < s/D < (s/D)split), the extinction spectra of dimers have two plasmon peaks: transverse and longitudinal. The shift of long-wavelength peak can be reasonably approximated by the equation

 x = a0 + a1 exp −  , where the λ0  t1 

∆λ

parameters a1 (= 0.352) and t1 (= 0.032) do not depend on the nanoparticle sizes, and a0 increases with particle size. The boundary between the long and middle interparticle distance ranges, (s/D)split, strongly varies with the Au NP diameter. At s/D < 0.02, the birth and evolution of third plasmon peak which locates between transverse and longitudinal peaks has strong effect upon the spectral properties of closely-coupled NPs. Now the fractional shift of the longitudinal peak obeys the equation

 x  x = a0 + a1 exp −  + a2 exp −  , where t2 = 0.004 and a2 = 0.643. The constancy of λ0  t1   t2 

∆λ

coefficients ai and ti for Au NPs of different sizes means that the fractional shifts of plasmon resonances of coupled pairs corrected by parameter a0 have to fall on a common curve. The obtained results point clearly that the Au NPs pairs can be used as the highly sensitive instruments to measure both absolute distances and their changes in the nanometric range of lengths.

INTRODUCTION The unique optical properties of noble metal nanoparticles, due to localized surface plasmons excited by the incident electromagnetic field of specific energy, and the changes of resonant excitation conditions with varying the size, shape and structure of the particles, the type of metal and

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surrounding medium have attracted great interest in the last decades because of an enormous potential into several practical applications such as ultrasensitive biological and chemical sensing112

. The localized surface plasmon resonance (LSPR) can be understood as a collective resonant

oscillation of free electrons within a metal nanoparticle in response to an incident electromagnetic radiation. When the plasmon-resonant nanoparticles are assembled, their plasmons are coupled. As a result, the LSPR-conditions for ensembles are dramatically modified compared with the isolated nanoparticles2,3,12-23. Now the plasmon excitation energy depends not only on the properties of individual metal particles but also on the number of particles in the ensemble, their relative location and interparticle distances, in other words, by varying the structural parameters only, the scattering and absorption of light of the nanoparticle clusters can be customized including the position and intensity of the plasmon bands and even their number. In this article, we will analyze the effect of the plasmon-resonant particle size and separation between nanoparticles on the extinction spectra of their coupled pairs which are the simplest element of ensembles. This topic is actively developed by many researchers2,3,12,19-36. One of the reasons, which provoke great interest in this topic, is to create a reliable tool that would allow high-precision measurement of distances between objects in the nanometric range. By using the diverse experimental and theoretical approaches, important information was obtained which concerns with the influence of incident light polarization, the spacing between particles, the type of metal, and other factors on extinction and scattering spectra of dimers that consist of the spherical, disk-shaped, elongated, etc. nanoparticles. Published results highlight that the plasmon resonance of coupled particle pairs radically depends on the interparticle distance19-23,27,28,32-34,37-58. Moreover, in the cases when the incident light polarization coincides with the direction of the axis passing through the λp   centers of particles, the fractional shift  ∆λ λ = λ - 1 of the LSPR-wavelength for a pair, λp, 0 0  


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relative to that for an isolated particle, λ0, can be represented as a damped exponential function of the distance between surfaces of particles, s, which is measured for spheres in units of their diameters, D23,33,39-44,46,49,53,54,58:

∆λ

λ0

(

= aexp − x

t

)

(1)

where x = s/D, a and t are coefficients. The empirically established relationship (1) which is known as a universal plasmon ruler equation makes it possible to measure the distance between two nanoparticles and to monitor the dynamics of its change in a variety of processes by quantifying the plasmon resonance shift for dimers. Jain et al.39 were among first who proposed Eq. (1) and evaluated the magnitudes of coefficients a and t (a = 0.18, t = 0.23 for protein environment). Using the Reinhard et al. experimental data37 on the plasmon shifts for gold nanoparticles (Au NPs) assembled through double-stranded DNA molecules with varying number of base pairs, they calculated the interparticle distances and established good agreement with experiment. In the table 1 we bring together parameters of Eq. (1) which were received by various authors following the best-fitting of both experimental and calculated shifts of LSPR-peaks for the absorption and scattering spectra of dimers. Analysis of these data allows us to make some conclusions about the properties of plasmon rulers constructed on the basis of Eq. (1) and, furthermore, to accentuate problems that have to be solved. (a) For coupled pairs of spherical, disk-shaped, spheroidal, and cylindrical nanoparticles, the values of the decay parameter t are close enough. For this reason, t is assumed to be equal approximately 0.2 and to be independent on the size and shape of particles, the optical properties of environment, and the type of metal, i.e., plasmon coupling decays over a length which is roughly 0.2D. However, recent studies44 show a significant influence of the nanoparticle shape (nanoprism, nanocube) and their orientation to each other on the magnitude of t (see table 1). According to the latest data45, the ACS Paragon Plus Environment


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nanoparticle size also affects the decay factor, but this effect has not been fully clarified.

Table 1. The parameters in Eq. (1) for coupled pairs which are produced from nanoparticles of different size, shape, type of the metal, and immersed in different environments. Shape of NPs

Size* and separation, nm Environment a

t

Method**

Reference

D = 42

nm = 1.6

0.13

0.24

Experiment, dark scattering microscopy

37

D = 10, s = 0.67-33.33

nm = 1.33 nm = 1.6

0.08±0.01 0.18

0.21±0.02 0.23

DDA calculations

39

D = 40, s < 3D; D = 5-50, s < 3D

εm = 1.00

0.045±0.004 0.22±0.03 0.23±0.03

DDA calculations

44

D = 34, s > 0.05D

neff = 1.35

0.12

0.22

Experiment, EELS 58

D = 41, s > 0.05D

nm = 1.474

0.16±0.03

Experiment, scattering spectroscopy

D = 26, s > 0.05D

neff = 1.25

0.23

0.20

Experiment, EELS 58

D = 50, 80, 95, h = 25, s = 10-(500-D)

neff = 1.25

0.37

0.22

Experiment, light scattering

33

D = 54, 68, 86, h ≈ 0.3D

εm = 1.0

0.14±0.01

0.23±0.03

DDA calculations

39

D = 86.5, h = 25.5, s = 2-208

εm = 1.0 εm = 1.38 εm = 2.25

0.16±0.02 0.23±0.02 0.29±0.02

0.23±0.04 0.24±0.03 0.27±0.03

DDA calculations

39

0.10±0.01

0.18±0.02

Experiment, microabsorption spectroscopy

39

Nanospheres, Au

Nanospheres, Ag

Nanodisks, Ag

D = 88, h = 25, s = 2-212

46

Nanodisks, Au Initial parameters: D = 50, h = 20, s = 30

nm = 1.52

0.33±0.01

0.25±0.01

Experiment, backscattering

40

Initial parameters: D = 50, h = 20, s = 30

nm = 1.52

0.16±0.01

0.29±0.01

MMP calculations

40

D = 30, h = 6; D = 10-100, h = D/5; D = 100, h = 10-30

εm = 1.00

0.10±0.01

0.22±0.03 0.22±0.03 0.30±0.02

DDA calculations

44

Hemispherically L = 72, ar = 4, capped s < 2.5D; head-to-tail nanorods, Au L = 60, ar = 2, s > 5; end-to-end

nm = 1.33

0.15±0.01

0.27±0.02

DDA calculations

41

nm = 1.56

0.12±0.01

0.26±0.04

DDA calculations

49

Nanospheroids, Au

L = 30, ar = 3, s < 2.5D; L = 20, ar = 2, s < 2.5D; head-to-tail

nm = 1.33 nm = 1.33

0.11±0.11 0.09±0.01

0.20±0.02 0.20±0.03

DDA calculations

41

Nanocylinders, Au

D = 17, L = 52, s < 2.5D; nm = 1.33 D = 17, L = 70, s < 2.5D; nm = 1.33 head-to-tail

0.20±0.02 0.21±0.01

0.24±0.03 0.23±0.02

DDA calculations

41


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The Journal of Physical Chemistry 5 bi = 38.1, b = 44.4, h = 13, s/bi < 4.5; tip-to-tip Nanoprisms, Au bi = 87.0, b = 79.7, si = 89.2, h = 20, s = 14.4-202.1; tip-to-tip Nanocubes, Au Infinitely long cylindrical wires, Au

L = 40; L = 10-50; side-to-side

εm = 1.00

0.09±0.01

0.33±0.02

DDA calculations

44

εm = 1.00

0.13±0.01

0.34±0.02

Experiment, microabsorption spectroscopy

44

εm = 1.00

0.07±0.01

0.41±0.04 0.37±0.03

DDA calculations

44

D = 20-50, s =(0.02-3)D;

0.112±0.010 0.19±0.04

D = 30-50, s =(0.02-3)D; edge-to-edge

0.098±0.010 0.28±0.07

FDTD calculations, 54 local FDTD calculations, nonlocal

*L is a long axis length of nanoparticles that is used to scale the gap distances; h is a height of disks or prizms, ar is an aspect ratio; for nanoprisms, bi is a bisector, b is a base, si is a side; nm and εm are the refraction index and dielectric function of medium, respectively. **FDTD is the finite difference time domain method, DDA is the discrete dipole approximation method, EELS is the electron energy-loss spectroscopy, MMP is the multiple multipole method.

(b) The amplitude of shift (pre-exponential factor a) depends on the size and shape of NPs, dielectric properties of the metal and environment. In particular, the magnitude of a varies from 0.08 to 0.23 for spherical particles and from 0.10 to 0.37 for disk-like particles. For Ag NPs, the magnitudes of a are larger than for Au NPs. (c) An experimental verification of Eq. (1) is mainly performed by means of dimers produced by electron-beam lithography (EBL). Benefits of advanced electron-beam technology are that it enables to fabricate the pairs of nanoparticles with high homogeneity in size, shape, and interparticle distance. For some time, the EBL-method was extensively employed as the primary technique to fabricate dimers and to experimentally calibrate the plasmon rulers. However, this technique has serious limitations associated with achievement of the gaps between nanoparticles less than ~5 nm. In addition, there are difficulties with the correct accounting of the substrate contribution in optical properties of nanoparticle dimers formed on its surface. (d) The exponential decay function (1) can, as it turned out, correctly describe the plasmon shifts for some (not complete) range of the interparticle distances. Significant deviations from Eq. (1) are



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observed at short distances between particles. According to various estimates, a single damped exponential underestimates the plasmon peak shift for s/D < 0.09-0.212,49. Kadkhodazadeh et al.58 have found that exponential fitting for pairs of coupled Au and Ag NPs is not good enough in the region 0.05 < s/D < 0.5, as evidenced by correlation coefficients (R2 = 0.88 and R2 = 0.94, respectively). The questions, where the lower boundary of this range is located and how its position depends on dimer properties, remain open. It should be noted that at present attempts are made to describe the distance dependence of the fractional plasmon resonance shift by using functions, such as powers of the inverse distance22,28,47,58. However, for closely spaced particles, neither exponential nor inverse power function is able to correctly predict the strong red-shifts of plasmon peaks when a gap decreases. It suggests that the more complex functional relationship between the plasmon shift and separation distance has to exist. In order to create a plasmon ruler, which is based on registration of the LSPR-wavelength shift, it is essential (i) to primarily obtain the exact relationship (or relationships) between the plasmon peak position and gap width, and (ii) to define the range of interparticle distances where this relationship is valid under typical experimental conditions. The precise measurements of the distances separating the nanoparticles in pairs and spectral properties of the latter should be made in the physical experiment to find such a relationship. If the spacing between particles is less than 2 nm, there are serious difficulties in their exact measurement. On the other hand, the theoretical calculations of the optical properties for pairs of coupled NPs with different separations (from extremely small, representing the small fractions of the particle size, till large, commensurate with their size) may help to determine the nature of plasmonic coupling between particles and to find laws of its change with varying the characteristics of both nanoparticles and environment. The main purposes of our study emerge from results mentioned above. Firstly, using a ACS Paragon Plus Environment

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rigorous theory, to calculate the extinction spectra of randomly oriented pairs of coupled Au NPs over wide ranges of sizes and interparticle distances, covering both weak and strong (with a significant shift of the plasmon resonance of dimers relative to the plasmon resonance of an isolated particle) interactions between particles. The use of the exact electrodynamic theory, which takes the electromagnetic retardation as well as the effect of dipole, quadrupole, and higher order multipoles into account, makes it possible to thoroughly explore the strong coupling of plasmonic NPs at very short distances that have not been systematically studied so far. Secondly, to find out conditions at which we can receive the greatest sensitivity of metrological measurements, based on plasmon coupling in pairs of gold nanoparticles. Thirdly, to analyze the possibility of presenting the data on plasmon resonance shifts for coupled nanosphere pairs with different sizes and different interparticle gap widths by using a common curve.

METHOD OF CALCULATION Various theoretical approaches have been adopted for modeling the optical properties of clusters of plasmonic NPs2,3,59-67. Some of them are mentioned in the table 1. We have employed the generalized multiparticle Mie theory (GMM) in this study. Based on the classical Mie theory for homogeneous spherical particles and the principle of superposition, this method allows one to calculate the spectral properties of many-particle clusters. The basic ideas of this approach as applied to the problem of a coupled pair of identical homogeneous spherical NPs, which is illuminated with light of a wavelength λ in the medium surrounding the pair, have been discussed in detail in the literature. A circumstantial description of calculation procedure can be found elsewhere (see, e.g., articles62-64,66 and references therein). Here we restrict ourselves to a very short summary. A bisphere is illuminated by a plate wave. The incident field and the scattered fields from two spherical particles are expanded as infinite series in terms of vector spherical harmonics in respective translated reference systems. The exact phase relations between expansion coefficients in coordinate systems ACS Paragon Plus Environment


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of particles are used to realize the transmission of fields. The expansions for mentioned fields are terminated at harmonics of the order Ns if the error from the truncation becomes negligible. As a result, we get a linear system set up by the standard electromagnetic boundary conditions. The solution of this system gives the coefficients of the expansions, which enable one to compute the extinction and scattering cross-sections. We have slightly modified the computer code gmm01TrA.f, which has originally been developed by Xu and which is freely available on the web, and used it to evaluate the spectral properties of coupled pairs. When choosing a computer code, we have been guided by comparative testing of several programs executed by Xu and Khlebtsov66. The extinction and scattering spectra for randomly oriented pairs of homogeneous spherical gold particles of the same size in water have been computed for the wavelength range of the incident unpolarized light 350-1100 nm with a step of 1 nm. The radius of the particle, R, is varied from 4 to 40 nm, the surface-to-surface distance from 0.1 nm to 3R. To make the model more realistic, the experimental dielectric data for gold, ε(ω) (where

ω = 2πc/λ is the circular frequency of the incident light, c is the speed of light), have been taken from the Johnson and Christy paper68 (for λ ~ 520-1400 nm) and the Irani et al. paper69 (for λ ~ 320505 nm). The nonlocality of dielectric response is not taken into account. The experimental data for

ε (ω) are approximated by cubic splines and the magnitudes of dielectric function are calculated in the wavelength region with step specified above. We have introduced the adjustment on the particle size, due to a limitation of the free path length of electrons (~42 nm for gold) at the transition from a bulk system to nanoparticles2,12: ε(ω, R ) = ε b (ω) + ∆ε(ω, R ) ,

∆ε(ω, R ) =


ω 2p

−

(

ω 2p

ω(ω + iγ b ) ω ω + iγ p

),

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ν γp = γb + A F , R

(2)

where γb, γp are the frequencies of collisions of free electrons in the bulk sample and particle, respectively (hγb/2π = 0.109 eV), ωp is the plasmon frequency of free electrons (hωp/2π = 8.55 eV), vF is the Fermi velocity (vF = 1.398×108 cm/s). The dimensionless parameter A, which is determined by the nature of the material particles and the details of the electron scattering at their surfaces, is fixed: A = 1. Dielectric properties of the solvent (water) are described by the equation given in the Khlebtsov et al. article15:

(ε (λ ))1/ 2= 1.32334 + 3.479 ×103 λ − 2 − 5.111×107 λ − 4 . The critical point registrated by all the researchers lies in the convergence of the extinction and scattering cross-sections for clusters of NPs. In the case of isolated spherical particles, there are several criteria in order to assess what number of spherical harmonics Ns in expansions for the incident and scattered fields is sufficient for the results to converge to a certain level. The Wiscombe empirical formulas70,4, obtained from the condition that the errors of the calculation of optical crosssections do not exceed 0.01%, have been widely exploited as a criterion: N s = 2 + α + 4α1/3 for α < 8

(α = 2πR λ ). Similar formulas are missing for dimers. Therefore, the spherical harmonics of higher order used to represent the incident and scattered waves are included in the calculations, and the results obtained in ascending order of harmonics are carefully compared to each other. According to the predictions of Quinten3, if metal nanospheres almost touch each other, Ns can grow up to 1001000. In the present work, the maximum number of harmonics Ns,max is limited by 50, and, for the investigated ranges of particle sizes and interparticle distances, the results are analyzed if the peak positions in the extinction spectra vary by no more than 1 nm when increasing the number of harmonics on 5. Simultaneously, the extinction and scattering effectiveness, Qext and Qsca, which are



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represented as ratios of cross-sections Cext and Csca to the cross-sectional area of the dimer: Qext( sca ) =

Cext( sca ) 2πR 2

, are changed

no more than 0.1%. These restrictions resemble the convergence

criteria used by Harris et al.22.

RESULTS AND DISCUSSION Spectra of extinction. The representative extinction spectra are shown in Figure 1 for randomly oriented pairs of 40 nm Au NPs, located at different distances from each other (from infinitely large to 0.2 nm). Nanoparticles are irradiated with unpolarized light. The extinction is submitted by the extinction efficiency, Qext.

Qext(λ)

a b c

e d

f

2

g 1

0 400

600

800 λ,

nm 1000

Figure 1.The extinction spectra, calculated in scope of the generalized multiparticle Mie theory for pairs of coupled 40 nm gold spheres with different distances between their surfaces: ∞ (a), 7 (b), 4 (c), 2 (d), 0.7 (e), 0.4 (f), and 0.2 nm (g). Ns = 30.

Before discussing the spectra of the dimers and analyzing their main characteristics, we consider in detail how at fixed interparticle distance the position and intensity of long-wavelength peaks for the spectra pictured in Figure 1 depend on the number of harmonics Ns, which are


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incorporated in the calculation program. A similar analysis has been performed for all the pairs of nanoparticles. As can be seen from Figure 2, the influence of a number of harmonics is very strong. If Ns is small (Ns = 5), there exist minor differences in positions of λp2 (Figure 2a) and in intensities of Qext(λp2) and Qsca(λp2) (Figure 2b) at s = 0.1-0.4 nm. However, when Ns increases, first we observe the strong deviations of λp2(s), Qext(λp2) and Qsca(λp2) at different interparticle distances from those for an isolated particle, and then they tend gradually to the certain levels. The limit values of λp2, Qext and Qsca in Figure 2 are achieved when Ns > 50 for s = 0.1 nm, Ns ~ 45 for s = 0.2 nm, and Ns < 30 for s = 0.4 nm, i.e., the closer the particles are to each other, the slower the results of the calculations converge to the corresponding levels and the more the number of harmonics must be considered to ensure the convergence. Many authors conclude12,71 that a high inhomogeneity of the local electromagnetic field in the gap between closely spaced particles is the main reason for such a strong dependence of the peak position on Ns. To correctly describe this field, it is necessary to use the higher order harmonics in the corresponding expansions. As a result, the process of convergence becomes very slow. By restricting the maximum number of harmonics in the computer program by Ns,max = 50, it is possible to ensure the convergence of the extinction spectra of coupled 40 nm Au NPs pairs, when s > 0.1 nm. For larger (smaller) nanoparticles, the convergence is achieved when the gap is larger (smaller) compared with the cases which are shown in Figure 2. By using the Tmatrix method, Khlebtsov et al.20 have found that, for the randomly oriented 15 nm Au bispheres in water, red-shifted resonance bands can be reproduced correctly if Ns = 30 (40) for s = 0.15 nm (0.075 nm). If incident light is polarized along dimer principal axis, Harris et al.22 have observed that, when a gap between 15 nm Au NPs is 0.5 nm, the number of Ns is 17 for convergence, while Pellegriini et al.21 have included up to 21 harmonics to calculate the scattering properties of 10 nm Au NPs pairs (s = 0.2 nm), embedded in silica. In contrast, in the region of the first plasmon peak, the number of harmonics, that are retained to achieve convergence, is much lower even near the contact of Au NPs ACS Paragon Plus Environment


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(not shown), i.e., here the effect of Ns on the spectral properties of pairs is very different.

λp2

(a)

s = 0.1 nm

900

s = 0.2 nm 800

s = 0.3 nm 700

600 0

20

40

Ns

60

(b)

3

Qext(λ p2)

s = 0.4 nm

2

s = 0.2 nm

1

s = 0.1 nm

Qsca(λ p2)

s = 0.4 nm

0

s = 0.1 nm 0

20

40

Ns

60

Figure 2. The long-wavelength peak position, λp2, (a) as well as the efficiencies of extinction, Qext(λp2), and scattering, Qsca(λp2), (b) for pairs of coupled 40 nm gold spheres versus the number of harmonics Ns. The surface-to-surface distance is 0.1, 0.2, and 0.4 nm.

An analysis of the changes in the dimer spectra (Figure 1) when decreasing the distance between the surfaces of particles allows us to select three distance ranges that exhibit distinctly different spectral behaviors except for very widely separated NPs where they behave independently.


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We denote them as ranges of long, middle, and short interparticle distances. When the gap between the particles is large enough (long gap, s/D > 0.1), the extinction spectrum has a single plasmon peak. If the particles do not feel each other, we discover the spectrum of isolated particles (Figure 1, spectrum a). When approaching particles, a plasmon band becomes broader (Figure 1, spectrum b, s = 7 nm) and red-shifts with respect to that of the individual particle (Figures 1, 3a). The maximum distance smax, where a particle begins to feel the presence of another particle, has been determined from conditions that the plasmon wavelength of dimers, λp, differs from that of an isolated particle by 1 nm. We have found smax ~ 40 nm (s/D ~ 1) for 40 nm Au NPs. (According to various estimates, the deviation of the dimer scattering spectrum from the monomer spectrum takes place at s/D ~ 2.5314,28,41,49.) Under conditions of unpolarized light and randomly oriented pairs of coupled Au NPs, the shifts ∆λ = λp - λ0 are small for the long distance range, but grow with increasing the particle size and reach the maximum value ~ 20 nm for 40 nm Au NPs (Figure 3a). The plasmon peak extinction decreases when approaching particles (Figure 3b): as s = 5 nm the extinction cross-section for a pair, Cext(λp), is approximately 1.851 times cross-section for an isolated particle, whereas the value 2 expected is achieved at longer distances between the particles. The decrease of Cext(λp) indicates that, compared to isolated particles, the dimer takes a smaller portion from the initial incident beam for elastic scattering and absorption (with subsequent transformation into heat).



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900

(a)

λpi, nm

2 700

3 1 500 0,01

0,1

Qext(λpi)

s/D

1

(b) 2

2,4

1

1,8

3 1,2 0,01

0,1

s/D

1

Figure 3. The position (a) and extinction efficiency (b) for the first (1), second (2), and third (3) peaks in the extinction spectra of pairs of coupled 40 nm gold spheres versus the surface-to-surface distance normalized by the particle diameter.

The transition into the middle distance range is accompanied by the splitting of a single plasmon band into two modes. Spectral peaks (λp1 ≈ 530 nm and λp2 ≈ 559 nm) can be observed when s = 4 nm (Figures 1, 3a). In comparison with the case considered above, now two coupled nanoparticles are capable of supporting more complex LSPR-modes (thereby demonstrating that


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oscillations of the conduction electrons take place within a pair as a whole). With decreasing s, the plasmon bands behave in different ways. The position of the short-wavelength peak, λp1, is shifted to the blue region of the spectrum, rapidly approaches to the position of the plasmon peak for an isolated particle (λp1 = 523 nm at s = 2.5 nm), and then remains almost unchanged (Figure 3a, the lower branch). The intensity of this peak decreases monotonically to the quantity Cext(λp1) = 1.657Cext,0 at s = 0.5 nm (where Cext,0 is the extinction cross-section for single NPs at the maximum of plasmon band). At the same time, the wavelength λp2, which corresponds to the second band maximum, increases when two particles go to each other (Figure 3a, the upper branch; Figure 1, spectra c-e): λp2 ≈ 595 nm at s = 2 nm and λp2 ≈ 665 nm at s = 0,7 nm. Simultaneously, Qext(λp2) first decreases, reaches a local minimum at s ≈ 2 nm (where Cext(λp2) = 1.679Cext,0), and then ascends sharply; the local maximum Qext(λp2) occurs at s ≈ 0.7 nm (where Cext(λp2) = 1.847Cext,0) (Figure 3b). The splitting of a single plasmon band is frequently noted in scientific literature to occur due to coupling of plasmon resonances of individual NPs. And the plasmon coupling (electromagnetic interactions between particles) increases, when the gap is reduced. The changes observed in the spectra of pairs of coupled spherical Au NPs, with their approaching to each other, remind us the changes of plasmon resonance spectra in a situation where the transition from spherical to rod-like NPs takes place when increasing the aspect ratio6. For randomly oriented nanorods, the single plasmon band also split into two modes: a strong longitudinal mode, corresponding to the oscillation of electrons along the long axis of the nanorods, and weak transverse one, corresponding to the oscillations perpendicular to the long axis. For gold nanorods, the latter is located in the Vis-region at wavelengths similar to those for spherical particles6. Finally, we address to the third distance range. One of the specific spectral properties for short distances between plasmonic NPs consists in a birth of the third band in their extinction ACS Paragon Plus Environment


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spectra. This band is covered first by the transverse one and observed as a weak shoulder on right side of the latter at λ ~ 580 nm for s = 0.4 nm (Figure 1, spectrum f). Further, when shifting in the region of larger wavelengths, it acquires a peak shape (for s < 0.3 nm) and takes up a position between two bands considered above: λp3 ≈ 617 nm at s = 0.2 nm (Figure 1, spectrum g; Figure 3a, curve 3). The existence of this band indicates clearly that a new mode of collective excitation of electrons inside dimers becomes important. In a very narrow interval 0.1 < s < 0.3 nm, the more pronounced red-shift of λp3 and change of Qext(λp3) can be observed (Figure 3, curves 3). The distance dependence for position of third and longitudinal bands is assumed to be similar34,50. Along with the qualitative changes of the extinction spectra of dimers, a rapid growth of λp2 and a sharp drop in Qext(λp2) occur for closely-coupled NPs (s < 0.7 nm) when s is reduced. Figure 3b reveals that Qext(λp2) < Qext(λp1) if s < 0.4 nm and Qext(λp2) < Qext(λp3) if s < 0.15 nm. The decrease of Qext(λp2), which gives a clear indication of redistribution of oscillation energy during the birth and evolution of the third band, is also noted by Pellegrini et al.21, Romero et al.34 and Marhaba et al.50. According to references18,19,34,67, the explanation for this behavior is that, when approaching particles, the induced surface charges are accumulated in the gap between particles, resulting in reorganization of the near fields, which acquire a complex spatial structure. For small gaps, the complex structure of field is retained due to the excitation of the higher order modes. When particles tend to each other, these modes first grow and then fall. This happens consistently for each longwavelength mode34. It is worth noting several points arising from the analysis of the extinction spectra of coupled Au NPs pairs. (i) The convergence in the region of the third band becomes more difficult as the particle size and the spacing between NPs are decreased. (ii) Despite the qualitative changes in the spectra which occur when reducing the surface-to-surface distance, the intensity and position of the first peak remain almost unchanged at s < 2 nm (Figures 1, 3). (iii) The sensitivity of the longitudinal ACS Paragon Plus Environment

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peak position relative to interparticle distance increases significantly with closer coupling. For comparison, as s belongs to the long distance range, the separation should be changed up to tens of nm if we want to displace the plasmon peak position by only 1 nm, whereas, for the short distance range, the 0.1 nm changes of s give the shifts of λp2 to tens of nm. In particular, reducing the distance between 40 nm Au NPs from 0.7 to 0.2 nm, one can observe a shift of longitudinal peak from 665 to 806 nm (~30 nm/1 Å). (iv) At very short interparticle distances (s < 0.15-0.2 nm), the intensity of the first band exceeds significantly those for the second and third bands, and the intensity of the third band, in turn, is higher than the second (Figure 3b). One feature of the extinction spectra transformation deserves consideration. Where is a boundary, (s/D)split, which separates the long distance and middle distance ranges? According to our results, a normalized interparticle distance where the plasmon spectrum of a coupled pair is split into two bands is not constant but varies strongly with particle size (Figure 4): (s/D)split grows quickly from ~0.027 at D = 8 nm up to ~ 0.10 at D = 40 nm, reaches its maximum in the size-interval of D = 40-60 nm, and then decreases to ~0.08 for D = 80 nm. The boundary positions correspond to s ~ 6.5 and ~ 4 nm for Au NPs with diameters 60-80 nm and 40 nm, respectively, and move to short distances for small particles (~ 0.2 nm for D = 8 nm).



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0,12

(s/D)split 0,08

0,04

0,00 0

40

D, nm

80

Figure 4. The position of the boundary between the long and middle distance ranges, (s/D)split, as a function of particle size.

Shifts of plasmon peaks as a function of interparticle spacing. Now we exploit the data about the plasmon bands in the extinction spectra of coupled Au NPs (in water) to discuss in detail the suitability of the exponential trend represented by Eq. (1). Two fitting parameters have been determined in order to describe the shifts of peaks by means of Eq. (1). The amplitude a and decay length t characterize, respectively, the maximum value of the fractional shift in the assumption that this dependence is correct up to touching particles, and the distance at which the coupled field of NPs in dimers fades away, when the particles move away from each other. The range of long interparticle distances (s/D > (s/D)split, weak coupling of plasmons) was defined above like the range in which the extinction spectrum of dimers has a single plasmon band for the investigated wavelength interval of the incident unpolarized light. Equation (1) makes it possible to approximate the data on fractional shifts of the plasmon peaks as a function of separation. The calculated values of t are shown in Figure 5a for Au NPs of different sizes. Its average value is 0.19 ± 0.03 at D ~ 8-50 nm (a correlation coefficient R2 > 0.97, for the 14 nm


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particles R2 = 0.90) that is quite well consistent with the literature results (see table 1). Figure 5b illustrates that the magnitude of a grows with the increase of diameter from D ~ 10 nm, reaches the maximum value ~ 0.08 for D ~ 40 nm, and further decreases. Such a change of the pre-exponential factor is caused by the fact that, for D > 50 nm, the shift of the plasmon band for Au NPs pairs slows down sharply when the ratio s/D is approaching to (s/D)split. Recently, using a single-nanoparticle photochemical imaging technique, Deeb et al. have performed measurements of the plasmonic near-field around individual Au NPs with diameters ranging from 12 to 86 nm [72]. They have established that the near-field enhancement initially increases with increasing the nanoparticle size, reaches a maximum at a size around 50 nm, beyond which it decreases. When two metal nanoparticles are brought in proximity to each other, the nearfield on one nanoparticle interacts with that on the other particle. Therefore, the local fields in the gaps between nanoparticles have to be modified in comparison with the fields near individual particles. Such a modification is small if plasmon coupling is weak (long distance range). And the trend observed in Figure 5b is closely reminiscent of the size dependence of the near-field enhancement for plasmonic NPs. As a gap is reduced, highly nonsymmetrical surface charge distributions have to develop in each of particles in response to the incident excitation and to their mutual electrostatic interaction. For small gaps, the strong coupling localizes the particle charge near the gap and results in a large field enhancement at the interparticle junction. In this case, the size dependence for the field intensity near isolated particles cannot be reproduced completely for dimers. Because of this, there appears the difference between size dependences for (s/D)split (Figure 4) and for parameter a (Figure 5b). According to Deeb et al. [72], the size dependence of near-field enhancement results from a simple interplay of three finite-size effects: nonradiative intrinsic and surface damping, dynamic depolarization, and radiative scattering. For dimers, it is necessary to take into account plasmon coupling of multipoles, whose role is increased for larger particles73. ACS Paragon Plus Environment


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t

(a)

0,4

0,2

0,0 0

0,08

20

40

D, nm 60

(b)

a

0,04

0,00 0

20

40

D, nm

60

Figure 5. The decay length (a) and amplitude (b) after approximating the dependence ∆λ/λ0(s/D) by using Eq. (1) in the range s/D > (s/D)split.

Thus, in the range of long interparticle distances, the plasmon peak position in the extinction spectrum of coupled pairs responds most strongly to the change of the surface-to-surface distance (0.04 < a < 0.08) when the particle diameter is 30-60 nm. For sizes outside this interval, the sensitivity of plasmon peak relative to variations in the interparticle distance is much less (Figure 5b). The intersection of the interval of the greatest sensitivity with the interval of weak sizeACS Paragon Plus Environment

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dependence of the parameter t leads to region 30 < D < 50 nm. The spheres from this region are probably most suitable for the design of plasmon rulers in the long interparticle distance range. In good agreement with the conclusions made previously by various researchers, the exponential function (1) provides a good fitting of the data (correlation coefficients R2 > 0.97) on the fractional shift of the plasmon resonance when the scaled distance between the surfaces of 30-50 nm particles exceed (s/D)split, i.e., when there are no abnormally large shifts and no new plasmon bands. The decay coefficient remains almost independent (universal) of NPs sizes. In the literature we can find the following estimates for the boundary of normalized interparticle distance where the data of experiments and calculations are very nearly fitted to an exponential decay (1): s/D ~ 0.0546,58, 0.0949, 0.123,45,67, and 0.212. Much effort was focused on the search for accurate and reliable relationship between the fractional shift of the plasmon resonance peaks and the interparticle distance for pairs of coupled NPs. The diverse theoretical methods are used to investigate the particles of various shapes, which are differently oriented relative to each other, embedded in a variety of nanostructures, and immersed in different dielectric environments. The first studies have revealed (see table 1) that the magnitude of t is universal and does not depend on the nanoparticle size, type of metal, and the dielectric properties of medium. Recent microabsorption experiments44 with dimers, fabricated by the EBL techniques, and DDA-calculations imply that the decay parameter in the plasmon ruler equation is very sensitive to the shape of particles (see table 1). Tabor et al.44 have studied the spectral properties of dimers, constructed from NPs of different forms and sizes, and concluded that nanospheres with a diameter of less than 50 nm were most suitable to create plasmon rulers because their decay coefficients are nearly independent of the nanoparticle diameters. This is consistent with our results. The dependence of t on the diameter in pairs of Ag NPs has been specified also by Encina and Coronado45. By using the FDTD-method and semi-analytical theory, based on the coupled dipole ACS Paragon Plus Environment


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approximation, in which are incorporated nonlocal effects for closely spaced Au NPs of cylindrical shape, Ben and Park54 have found that particles are most suitable for the design of plasmon rulers when their dimensions correspond to 20 < D < 70 nm. The plasmon ruler equation obtained by analysis of spectral properties of coupled pairs, as it turns out, works also in the case of the more complex structures such as linear chains and arrays of NPs. Ben and Park53 have performed DDA calculations for LSPR in infinite periodic 2D-arrays of 20-100 nm spherical Au NPs, separated by distances from 0.5D to 6D, and determined the fitting parameters for Eq. (1): t = 0.23 ± 0.02, a = 0.21 ± 0.02. The universality of t is observed when arrays consist of nanoparticles smaller ~70 nm; for larger particles, the reduction of extinction and decay coefficient are established, when the gap is reduced. According to the DDA spectral data for linear trimer of 10 nm gold spheres, Jain and El-Sayed41 have calculated the fitting parameters (t = 0.21

± 0.02 and a = 0.15 ± 0.01) and confirmed that t remains unchanged when the transition from dimers to more complex nanostructures takes place. In the range of middle distances between NPs, an incident light generates two types of excitations in coupled pairs, which are attributed to the excitations of transverse and longitudinal oscillations of conduction electrons. One can see in Figure 6 that the dependence of the fractional shift of the longitudinal resonant peak wavelength on the normalized separation distance is functionally similar for Au NPs of different sizes. In contrast to the case of long interparticle distances, for s/D < (s/D)split, each of the curves can be approximated with a high accuracy (correlation coefficient R2 > 0.99) by means of a sum of two exponentials:

∆λ

 x  x = a0 + a1 exp  −  + a 2 exp  −  . λ0  t1   t2 

(3)

Here, the decay coefficients t1 = 0.0320 ± 0.0038, t2 = 0.0040 ± 0.0007 for 14-80 nm Au NPs. By fixing the average values of t1 and t2, we have calculated that the pre-exponential factors a1 and a2 ACS Paragon Plus Environment

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do not depend on particle size in the D-interval studied: a1 = 0.352 ± 0.018 and a2 = 0.643 ± 0.046. At the same time, the free term in Eq. (3) takes non-zero values, which vary with a particle size. This effect is illustrated in Figure 7. A dependence a0(D) is reasonably approximated by a straight line: a0 = (0.0298 ± 0.0024) + (0.0017 ± 0.0001) R . It is obvious that a0 owes its origins to the shift of the plasmon peaks, emerging from the range of long interparticle distances. However, the parameters a and a0 differ, with the difference between them increasing significantly for larger Au NPs (D > 50 nm). Thus, when s/D < (s/D)split, the nature of the distance dependence of the plasmon band shift is more complicated due to a substantial contribution of the phase retardation and multipolar interactions in plasmon coupling at short and middle interparticle distances. For these ranges, it is the relation of interparticle distance to sphere diameter that determines the spectral properties of dimers rather than the absolute value of the first. Under conditions of a strong plasmon resonance shift (s/D < (s/D)split), the family of curves

∆λ  s    for Au NPs of different sizes (Figure 6) can be λ0  D 

reduced to a common curve by a simple displacement of each from them (Figure 8). The magnitude of the displacement equals to a0(D). (For the dimers formed from 8 nm Au NPs, (s/D)split ~ 0.027 (Figure 4), we have estimated the value of a0(8) via a straight line depicted in Figure 7.)



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0,8

∆λ/λ0

-

0,6

0,4

1 2 3 4 5 6 7 8

0,2

0,0 0,00

0,05

s/D

0,10

Figure 6. The fractional shifts of long-wavelength peak in extinction spectra of Au NPs pairs versus the normalized interparticle distance. Particle diameter is 8 (1), 14 (2), 20 (3), 30 (4), 40 (5), 46 (6), 60 (7), and 80 nm (8).

0,10

a0 0,06

0,02 0

20

40

60

80

D, nm

Figure 7. The free term in Eq. (3) as a function of the nanoparticle size in dimers.


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0,8

∆λ/λ0- a0

1

0,6

0,4

0,2

2 3

0,0 0,00

0,02

0,04

0,06

0,08

0,10

s/D

Figure 8. The universal curve (1) of the difference (∆λ/λ0 - a0) against s/D based on calculations of the extinction spectra for pairs of coupled 8-80 nm gold particles in water. Continuous curves in the figure correspond to the second (2) and third (3) summands in Eq. (3).

Comparing the data shown in Figures 3b and 8, one can notice that a sharp growth of λp2 begins when a dimer enters the range of short interparticle distances and the intensity of the longitudinal peak, Qext(λp2), starts to fall sharply. It happens if the ratio s/D becomes less than ~ 0.02 and suggests that the multipoles of higher order strongly affect the position of the longitudinal peak, pushing it in the near-infrared region. The common curve (Figure 8), which describes the relative shift as a function of distance between the particles, allows us to choose nanoparticles to test a particular range of distances and to ensure high sensitivity in the measurement of interparticle distances. Here, the distance sensitivity of this plasmon ruler constructed from a pair of coupled Au NPs is not below than that for constructions like Au film-Au NP74. Analyzing the results obtained, it is important to emphasize the following points. First, unlike the case of long distances between the particles, the pre-exponential coefficients in Eq. (3) are strongly independent of the Au NPs size, and their sum is close to 1, i.e., the shifts of the plasmon resonance for pairs relative to the plasmon resonance of the isolated particles are very large (~ λ0) ACS Paragon Plus Environment


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and, moreover, somewhat increase with particle size due to the growth of a0 (Figure 7). Second, as in the case of long interparticle distances, at 0.02 < s/D < (s/D)split, the dependence of the fractional LSPR-shift on the scaled distance can be approximated by a single damped exponential function (Figure 8, curve 2), with the decay parameter and amplitude being largely independent of the Au NPs size: t1 = 0.032, a1 = 0.352. The magnitude of decay coefficient t1 is considerably lower than t (t/t1 ~ 6), indicating that the interactions weaken much faster within the range of middle interparticle distances than within the range of large distances. Third, the hypersensitivity of the long wavelength peak relative to the mutual arrangement of the particles is observed at extremely short distances: s/D

 ∆λ  s   < 0.02, where the dependence    − a0  can be represented as a sum of two exponentials  λ0  D   (Figure 8) whose parameters are not practically changed with particle size, and t1/t2 ~ 8. One may point out that, for particles of small size (D = 8 nm), the middle distance ranges are very narrow and we can hardly find experimentally a distance range where a single damped exponential with parameters t1 and a1 performs approximation for data on spectral shifts. The complicated dependence of the LSPR-shift on the interparticle distance is noted by many scholars12,22,28,39,47,49,58. Here are some examples. Funston et al.49 have drawn attention to the fact that for a pair of Au NPs located at distances less than 2 nm (s/D < 0.09), the universal plasmon ruler equation (1) does not hold, whereas the exponential and the inverse cubic function ((s/D)-3) can be successfully used to fit data on the plasmon resonance shifts at s/D > 0.09. By using the T-matrix method, Harris et al.22 have studied the chains composed from 15 nm spherical NPs (s = 0.5-30 nm) and observed that the LSPR-shift for a pair is varied inversely proportional relative to the interparticle distance rather than exponential (exponential fitting is not particularly good, R2 = 0.88). Kadkhodazadeh et al.58 have received similar results: almost the same distance-dependence


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∆λ  s   s   ~  λ0  D   D 

−n

(n~0.9) is established for pairs of Au and Ag NPs at distances 0.05D < s < 0.5D.

It means that at s < 0.5D the interactions between two NPs significantly deviate from the interactions inherent in two classical dipoles. By using scattering spectroscopy, Yang et al.46 have measured the plasmon resonance energy, Eres(s/D), for coupled pairs, formed from 41 nm Ag NPs, at s = 1-25 nm and found the different regimes of plasmonic coupling. At large interparticle distances, the energy red-shifts continuously with decreasing distance up to s ~ 0.05D, and the single exponential with parameter t = 0.16 ± 0.03 provides good fitting for shifts. The distance dependence for the resonance energy is changed at s/D < 0.05: the red shift is absent and the resonance energy grows that are attributed to the change of the mechanism of interaction between NPs at short distances. At shorter distances, the energy does not further red-shift; instead the slope Eres(s/D) levels off. The results obtained in present work are in good agreement with the overall paradigm, according to which, when approaching plasmonic NPs, several mechanisms of plasmon coupling are accomplished. The model of dipole coupling (which leads to the inverse cubic dependence of fractional shift on a distance; according to Jain et al.39 the function (s/D)-3 can be approximated by Eq. (1) for a certain interval of its argument) is well suited to describe small red shifts of LSPR when the particle size is much less than the wavelength of the incident radiation and when the distance between particles is much greater than their size. The shift reflects the magnitude of interaction energy between two NPs like between two classical dipoles. In spite of the mentioned limitations, we observe that the exponential dependence is valid up to very small s (~0.2 nm) for 8 nm Au NPs. When the particles come closer together, a dipole mode of each particle begins to interact with modes of higher orders (quadrupole, octupole, etc.) on its pair. As a result, the LSPRwavelength becomes much more sensitive to variation of s. The multipolar nature of coupling has already been analyzed by Jain and El-Sayed23,39. Prodan et al.29,30 have proposed a concept of



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plasmon hybridization which provides a simple way to understand the coupling between the plasmons as the particles approach each other. According to these data, the higher order multipoles dominate at close interparticle distances that leads to the stronger shift of the LSPR compared to that expected from the pure dipole interactions. As soon as the modes of higher orders are actively involved in interactions (the indicator of this situation is the appearance of a third plasmon peak at

λp3 ~ 600 nm), a shift of the long wavelength peak into the near-infrared is significantly accelerated with decreasing interparticle distance. At the same time, the dipole oscillations in pairs of coupled NPs are inhibited and their response is attenuated.

CONCLUSION By a complete treatment of the electromagnetic retardation effect and higher-order multipolar interactions in scope of the generalized multiparticle Mie theory, we have performed the systematic studies of the plasmonic coupling for pairs of similar Au NPs of 8-80 nm in diameter, located from each other at distances 0.1-120 nm. (Note that the nonlocal and quantum effects which become important when the interparticle distances are below 1 nm are not captured in our model. Therefore, the obtained results may not be quantitative at very short spacing. The detailed analysis of these effects for strongly coupled plasmonic nanoparticles can be found in papers12,19,35,54,75 and references therein.) Calculations of extinction spectra relate to the following experimental conditions: unpolarized light, the random orientation of the dimers in space, and water as a dielectric medium. For convenience of analysis, we have specified three ranges of interparticle distances: long, middle, and short. 1. For the range of long interparticle distances (s/D > (s/D)split), there is a single plasmon peak in the extinction spectrum of a coupled pair (Figure 1), which is red-shifted relative to the peak of an isolated particle as the gap between spheres is reduced. Equation (1) describes the magnitude of the fractional shift, with the decay coefficient (t = 0.19 ± 0.03) being practically independent of the ACS Paragon Plus Environment

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nanoparticle size when D < 50 nm. The shift varies significantly with the particle diameter. The most significant shifts are observed for 30-60 nm Au NPs (Figure 5b) that makes these nanoparticles most suitable for the design of plasmon rulers, to reliably measure distances at s/D > (s/D)split. The sensitivity of such nanometric rulers is not so great: the amplitude of the plasmon peak shift reaches a maximum of ~20 nm for D ~ 40 nm. The mechanism of dipole-dipole interactions is an appropriate model to explain the influence of interparticle distances on a fractional shift of the LSPR-peak of dimers. 2. The middle range of interparticle distances (0.02 < s/D < (s/D)split). The value of (s/D)split, that marks the boundary between the long and middle distance ranges, significantly depends on the nanoparticle size (Figure 4). For the smallest NPs investigated (D = 8 nm), the width of this range is very small, therefore, it is very difficult to test the properties of such a coupled pair here. The extinction spectra of nanoparticle pairs are characterized by two plasmon peaks. The shift of the long-wavelength peak can be described by means of the truncated Eq. (3) with the parameters a1 (= 0.352 ± 0.018) and t1 (= 0.0320 ± 0.0038) which are independent of the sphere diameter (Figure 8). The sensitivity of the plasmon rulers relative to interparticle distances significantly increases in comparison with that for the range s/D > (s/D)split. The free term a0 increases with particle size (Figure 7): the larger the diameter, the higher the sensitivity of the nanometric plasmon rulers. Multipolar interactions impose an appreciable imprint on the surface-to-surface dependence of the long-wavelength peak position. 3. In the range of short interparticle distances (s/D < 0.02), the birth and evolution of the third plasmon peak make a significant contribution to the spectral properties of dimers. The fractional peak wavelength shift can be described using Eq. (3) with constant parameters ai and ti (i = 1, 2): a2 = 0.643 ± 0.046, t2 = 0.0040 ± 0.0007. The sum of the coefficients ai is about 1. The sensitivity of the longitudinal peak position relative to the interparticle distances increases sharply, reaching ~ 30 ACS Paragon Plus Environment


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nm/1 Å. The effect likely reflects the presence of highly inhomogeneous fields in close proximity to the surface of particles. The obtained results clearly highlight that the pairs of coupled Au NPs can be used (for instance, in biology, materials technology, nanoelectronics, and so on) as the accurate, highly sensitive metrological tool for measuring absolute distances and their changes in the nanometer range of lengths through registration of a plasmon peak position.

AUTHOR INFORMATION E-mail: [email protected]

Notes The authors declare no competing financial interest.

ACKNOWLEDGMENT This work was supported by the State grant for Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences (“Colloidal dispersions of plasmon-resonant nanoparticles and nanocontainers and ensembles based on them”, No. 01201353198).

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Nanometric Rulers Based on Plasmon Coupling in Pairs of Gold

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