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J. Phys. Chem. C 2009, 113, 21604–21610
Localized Resonances of Composite Particles Alexander Moroz* WaVe-scattering.com ReceiVed: August 26, 2009; ReVised Manuscript ReceiVed: October 31, 2009
Composite particles (CPs) formed by a homogeneous distribution of much smaller constituent metal nanoparticles (MNPs) in a host dielectric can be designed to exhibit two physically different types of resonances: (1) remarkably sharp localized surface plasmon resonances (LSPRs) and (2) equally narrow and strong localized dielectric resonances (LDRs). Preliminary results for spherical CPs show that the line width (fwhm) of both types of resonances can be as narrow as 100 meV. The values of fwhm are smaller than for spherical MNPs of the same size and comparable to the smallest reported values in isolated MNPs of arbitrary shape. LDRs can be tuned over a broad range of almost 1 µm, for instance, from below 650 nm up to telecommunication bands above 1450 nm, by mere adjustment of the volume filling fraction of the constituent MNPs. The tunability of the LSPRs, which can also be designed to lie within the “biological window” of high optical transmission in blood and tissue between 700 and 1100 nm, is an order of magnitude smaller and amounts to ∼100 nm. The sensing potential of the LSPRs of spherical CPs is also promising and is quantified by a figure of merit of 1.3. In the case of LSPRs, the dielectric constant of the host dielectric is required to be preferably at least 4, whereas pronounced LDRs can be observed already when the dielectric constant of the host dielectric is as small as 2. The peculiar properties of the localized resonances of CPs have a potential for a variety of applications. 1. Introduction Over the past several years, there has been a large amount of interest in plasmonic applications of small metal nanoparticles (MNPs) in biology, medicine, sensing, energy conversion, and many other fields.1-23 These applications rely heavily on the fact that MNPs support localized surface plasmon resonances (LSPRs), which are excited when incident electromagnetic radiation creates coherent oscillations of the conduction electrons. LSPRs manifest themselves by selective photon absorption, which allows the optical properties of individual MNPs to be monitored with UV-vis spectroscopy.6,13-15,18,19 The selective absorption might eventually pave the way for multicolor optical molecular imaging, potentially enabling noninvasive high-resolution characterization and diagnosis of living tissue.6,17 LSPRs are responsible for the enhancement of the electromagnetic fields surrounding MNPs, which can be employed for LSPR sensing13-15,17-19,22,23 and for all surfaceenhanced spectroscopies.12,24 The performance of individual MNPs in the above applications is crucially influenced by the LSPR homogeneous line width (fwhm) Γ or, alternatively, the LSPR dephasing time T2 ) 2p/Γ.3 So far, two main paths have been pursued in order to improve the performance of individual spherical MNPs: (1) various core-shell MNP morphologies2,9,15,16 and (2) nonspherical MNP external shapes such as rods,4,5,10,21 cubes,11,18 and prisms.19 In this work, a third alternative path is proposed that consists of dielectric function engineering. It can be pursued independently or in combination with the above two approaches. Herein, we consider composite nano- and microparticles (CPs) formed by a homogeneous distribution of much smaller constituent spherical MNPs in a dielectric characterized by the dielectric constant εh. Although metal-dielectric composites, such as * E-mail:
[email protected]. URL: http://www.wave-scattering. com.
metal-doped glasses, have been used for centuries and their properties are well understood,25 the properties of such composites when carved as micro- and nanoscatterers remain to be entirely unexplored. Given a CP of radius rc, a CP will contain more than 103 constituent MNPs provided that the radius rs of the constituent MNPs does not exceed rs,max ) rcf 1/3/10 (e.g., rs ≈ 1.86 nm for rc ) 40 nm and f g 0.1), where f is the volume filling fraction of the constituent MNPs. Therefore, an effective medium description of the dielectric constant of such CPs for their applications in the visible and near-infrared ranges is justified. Assuming that the constituent MNPs characterized by the dielectric constant εs do not form aggregates inside a CP, the effective dielectric constant εG of a CP is well described by the Garnett theory25
εG ) εh
1 + 2fR , 1 - fR
R)
εs - εh εs + 2εh
(1)
The usual assumptions of the validity of the Garnett type of composite geometry are that (1) the inclusions are assumed to be spheres or ellipsoids of a size much smaller than the optical wavelength, (2) the distance between them is much larger than their characteristic size, and (3) the distance between them is also much smaller than the optical wavelength.26 The present work applies the Garnett theory to the case where hypothesis 2 has been relaxed. That this is justifiable and that a slightly amended Garnett formula holds reasonably well even up to the close packing [f ) fcp ) π/(32) ≈ 0.74] of the constituent particles,27-29 where two (i.e., hypotheses 1 and 2) of the usual three assumptions on the validity of the Garnett type of composite geometry are no longer valid, has been demonstrated by direct calculations for periodic and random arrangements of dielectric and metallic spheres.27-29 Using εG as the dielectric function of a CP is also in agreement with the basic assumptions underlying the dipole-dipole approximation (DDA). In the DDA, an arbitrary particle is modeled as an array of N
10.1021/jp9082568 2009 American Chemical Society Published on Web 12/08/2009
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polarizable subunits, each of which gives rise to only electric dipole radiation. A fundamental requirement of the DDA is that the interdipole separation is small compared to the wavelength of the incident radiation, exactly as in our case. Given the dielectric constant εG of the particle, one can use the Garnett formula to assign the polarizability of the elementary dipole subunits.30 We do here nothing else than apply the Garnett formula in reverse: given the polarizability of the elementary dipole subunits (i.e., of the constituent MNPs), the Garnett formula is used to determine the dielectric constant εG of the particle. In the discussion that follows, the homogeneous distribution of constituent MNPs should be understood as one that justifies the description of the effective dielectric constant of a CP by the Garnett formula (eq 1). The homogeneous distribution can be realized as either a random or a regular distribution of constituent MNPs.31 In the latter case, the proposed CPs are similar in structure to the so-called colloidal superparticles that comprise metal oxide nanoparticles and that experience growth controlled through solvophobic interactions.31 The required CPs may probably be realized by a modification of the method by Zhuang et al.31 starting with metal core-dielectric shell constituent NPs. The modification should allow the interstices between the nanoparticles to be filled with the dielectric material during or after nanoparticle aggregation. The initial encapsulation of MNPs can additionally overcome a weak photostability of some metals, such as silver, due to a photo-oxidation. This article is organized as follows: Using εG as the dielectric function of a CP, localized resonances of spherical CPs are characterized in section 2. The effect of size-corrections to the dielectric function of constituent MNPs on the resonance line width Γ is examined in section 3. The sensing potential of resonances of spherical CPs is then studied in section 4. Some applications are discussed in section 5. We conclude with section 6.
-10.56 j εs /εh j -0.209
(3)
which is obtained by setting f ) fcp in eq 2. In order that a spherical CP satisfies the LSPR condition in an ambient medium characterized by a dielectric constant εb, i.e., εG ) -2εb, the real part εs′ of εs has to satisfy
Figure 1. Extinction efficiency of spherical CPs of radius rc ) 80 nm formed by a constituent dielectric sphere with a dielectric constant of εh ) 4 for different volume filling fractions f of the constituent AgNPs. The CPs are supposed to be immersed in an aqueous solution with εb ) 1.7689 (refractive index nb ) 1.33). Starting with f ≈ 0.3, one observes for each f three LDRs, with each subsequent resonance further to the red having higher height.
2. Localized Resonances of Spherical CPs In what follows, εG ) εG′ + iεG′′, where εG′ and εG′′ are the real and imaginary parts, respectively, of εG, represents the dielectric function of a CP. Figure 1a shows model calculations of the extinction efficiency of CPs formed by a constituent dielectric sphere with a dielectric constant of εh ) 4 (e.g., ZnS in the visible and near-infrared regions) and radius rc ) 80 nm for different volume filling fractions f of the constituent AgNPs. Here and below, Palik data32 were employed for εs in eq 1 to determine the dielectric constant εG of a CP at a given wavelength. The results reported in this work are based on Mie theory and were obtained using the freely available code SPHERE.33 For each filling fraction, one observes at least two pronounced localized resonances. The positions of the outermost red and outermost blue resonances are summarized in Table 1. Obviously, the resonances in the blue part of the spectrum correspond to LSPRs of the respective CPs. The presence of the LSPRs follows from the fact that, in a narrow frequency window, ε′G takes on negative values (see Figure 1b). The latter can be qualitatively understood by assuming temporarily that εs has a zero imaginary part. Then, it follows from eq 1 that εG < 0 in a narrow region of the (f, εs/εh) plane limited by (see Figure 2)
-
εs(ω) 2+f 2(1 - f) < g+(f) ) ) g-(f) < 1-f εh 1 + 2f
(2)
A necessary condition for the inequalities to be satisfied is
Figure 2. Garnett’s εG given by eq 1 is positive almost everywhere in the (f, εs/εh) plane except for a small wedgelike region limited by the functions g-(f) and g+(f). On the solid (mangenta) line, εG ) 0, whereas on the dashed (blue) line, εG is both singular and discontinuous, approaching -∞ from within the wedgelike region and +∞ from outside the wedgelike region. εG < 0 can be attained only in a narrow wedgelike region for rather moderate negative values of εs/εh not smaller than approximately -10.5.
TABLE 1: Positions of LSPRs and LDRs and the Respective Values of εG at the Peak Positions for Spherical CPs with rc ) 80 nm Immersed in Aqueous Solution, with the CPs Consisting of Constituent AgNPs Dispersed in a Dielectric Matrix with εh ) 4 f
LSPR (nm)
0.1 0.2 0.3 0.4 0.5 0.6 0.7405
485.3 477.3 472.1 467.3 463.2 459.8 455.5
LDR (nm)
εG (-4.05, (-5.02, (-5.37, (-5.52, (-5.64, (-5.74, (-5.86,
4.32) 2.4) 1.75) 1.41) 1.19) 1.05) 0.91)
565.1 642 723.5 814.7 924.3 1068.1 1469.1
εG (11.46, (14.83, (18.79, (23.88, (31.05, (41.87, (80.48,
1.93) 1.58) 1.49) 1.4) 1.83) 2.55) 9.18)
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εs′(ω) 1 - fRm ) -2 , εh 1 + 2fRm
Rm )
εh - εb εh + 2εb
Moroz
(4)
Figure 3 shows the plots of the quasistatic solutions as a function of εh for different values of f. For εh g 2, the LSPR positions are blue-shifted for each increasing f, which is in agreement with and provides a qualitative understanding of the LSPR peak positions listed in Table 1. According to Table 1, the resonances in the red part of the spectrum in Figure 1a correspond to LDRs. In contrast to LSPRs, the appearance of LDRs is due to the fact that εG takes on values with a large positive εG′ . 1 and with εG′′ that is approximately at most 10% of ε′G. To explain the origin of LDRs, define the usual (external) size parameter x ) 2πrcεh1/2/λ and the intrinsic size parameter xi ) 2πrcεG1/2/λ. The condition for the occurrence of LDRs is then derived from the exact dipolar Mie coefficient by letting x f 0 while preserving the spherical Bessel function dependence of the Bessel function j1 with the argument xi.34 Because |xi| . x, the assumption x , 1 no longer implies |xi| , 1. Consequently, the usual Rayleigh limit cannot be taken. It turns out that the LDR condition
εˆ G ) -2εb
Fm(x) )
In the present case, the large values of εG′ are, as suggested by the plots of εG′ in Figure 1b, a consequence of εG exhibiting a polaritonic-like dependence. The polaritonic behavior can be qualitatively understood by substituting a Drude-like dielectric function
(5)
is formally identical to the LSPR condition, provided that one uses the renormalized dielectric function
εˆ G ≈ εGFm(xi),
Figure 4. Lewin’s function Fm for real arguments with first three poles at xp ≈ 2.7437, 6.1168, and 9.3166.
2(sin x - x cos x) x sin x - (sin x - x cos x)
εs(ω) ) 1 - ωp2 /[ω(ω + iγ)]
into eq 1, where ωp and γ are the plasma frequency and damping constant, respectively. Then, one obtains
2
εG ) ε∞
(6) The function Fm, which results if the j1’s with argument xi in the limiting form of the dipolar Mie coefficient are expressed in sines and cosines, has been introduced by Lewin.35 Figure 4 shows that Fm(x) monotonically increases with increasing x for purely real values of x up to the first pole of Fm(x) at xp ≈ 2.7437. In a narrow region just behind the pole, Fm(x) < 0, and hence, renormalized εˆ G < 0. Thus, the LDR condition (eq 5) can indeed be satisfied with purely positive εG′ . 1. The above features of Mie scattering are well-known in microwave down to infrared applications, where LDRs have been employed to provide metamaterials with a magnetic response and a negative refractive index.29 However, they have not been encountered in the visible region because of a lack of natural materials with sufficiently high values of εG′ .
(7)
ω(ω + iγ) - ωL2
(8)
ω(ω + iγ) - ωT2
which is the sought textbook polaritonic behavior of a singleresonance Drude-Lorentz model satisfying the Kramers-Kronig relations. In natural polaritonic materials, ε∞ would be the asymptotic value of the dielectric permittivity at high frequencies, whereas the respective variables ωT and ωL, ωT < ωL, would be the transverse and longitudinal optical phonon frequencies related by the Sachs-Teller relation ωL ) ωT[εG(0)/ε∞]1/2. In the present case 2
ωT
ωp2 ) , 1 - εhg-(f) ε∞ ) -εh
2
ωL
ωp2 ) 1 - εhg+(f)
2 1 - εhg+(f) g+(f) 1 - εhg-(f)
(9) (10)
and one can set ωT and ωL, to a large extent, at will by simply tuning f. The maximum value of εG′
[
′ εG,max ) ε∞ 1 +
Figure 3. Combinations of εs and εh as given by eq 4 that are necessary for a CP to satisfy the quasistatic LSPR condition in aqueous solution. The larger the filling fraction, the smaller the initial value of εs for εh ) 1 and the smaller the slope of the nearly linear decrease of εs with increasing εh.
ωL2 - ωT2 γ(2ωT - γ)
]
(11)
is attained at the frequency ωm ) (ωT2 - ωTγ)1/2. Given the functional dependence of g((f) shown in Figure 2, ωL increases, whereas ωT decreases with increasing f, which allows one to ′ over 2 orders of magnitude by varying tune the value of εG,max f. These features are clearly observed in the plots of ε′G in Figure 1b. In the examples shown in Figure 1a (see also Figure 7 below), up to three LDRs can be observed for each value of f J 0.3. Because of the proximity of those xi for which eq 5 is satisfied to the poles of Fm(xi), the three LDRs correspond roughly to the first three poles of Fm(xi) at xi ≈ 2.7437, 6.1168, and 9.3166. The latter sequence explains an almost equidistant separation of the LDRs. The extreme sharpness of LDRs is a direct consequence of the LSPR-like LDR condition of eq 5 combined with enormous variations of Fm(x) with x in the Fm(x) < 0
Localized Resonances of Composite Particles
Figure 5. Dependence of the extinction efficiency of spherical CPs with rc ) 40 nm comprising constituent AuNPs with f ) 0.3 on the dielectric constant εh of the constituent dielectric matrix. The CPs are supposed to be immersed in aqueous solution.
regions just behind the poles. A necessary condition for the LDR condition (eq 5) to be satisfied is that the imaginary part of xi is sufficiently small. Indeed, Fm(iκ) monotonically decreases with increasing magnitude of κ for purely complex values of x ) iκ from Fm(0) ) 1 to Fm(0) ) 0. For the CP parameters of Figure 1, the peak of the outermost red LDR for a given f = 0.3 is the highest, with the peak height of each subsequent LDR toward the blue part of the spectrum decreasing. This is because the loss tangent εG′′/εG′ increases as ωT is approached from the red part of the spectrum. A detailed calculation shows that the positions of LDRs are strongly size-dependent. For instance, in the example of a CP as in Figure 1a and Table 1, the position of the most pronounced LDR for f ) 0.5 can be varied from ∼780 to ∼1520 nm by varying rc from 50 to 160 nm. Surprisingly enough, the position of the LDR would hardly change if gold were to replace silver as the material of the constituent NPs. The position of the LDR for f ) 0.5 would then vary from ∼800 nm for rc ) 50 nm to ∼1490 nm for rc ) 160 nm. For both metals, the LDRs disappear for rc < 50 nm but become well pronounced and sharp for rc J 70 nm, with their height saturating for rc J 100 nm. The suppression of LDRs for small rc is illustrated by Figure 6, which shows the extinction efficiency of a CP with rc ) 40 nm consisting of constituent AuNPs dispersed in a Si matrix characterized by εh ) 12.25 (refractive index nh ) 3.5). See also the small LDRs in the red part of the spectrum in Figure 5 for CPs with constituent AuNPs as a function of εh and f ) 0.3. Turning back to the LSPRs, according to eq 9 one can decrease both ωT and ωL, and thereby move the range of εG′ < 0 to lower frequencies, by increasing εh, which can be clearly observed in Figure 5. Figure 6 and Table 2 focus on the case of εh ) 12.25, where the LSPRs can be placed in the near proximity of and within the biological window. Provided that silver constituent NPs instead of gold ones are used, the LSPR peaks will be slightly higher and blue-shifted by ca. 70 nm (e.g., that for f ) 0.3 would be located at ∼621 nm). The reason is the smaller damping constant γ and higher plasma frequency ωp of silver. 3. Effect of Size Corrections to the Dielectric Function of Constituent MNPs All listed LSPRs in Table 2 are extremely sharp with Γ e 80 meV and with the exceptionally high quality factor Q ) ωres/Γ J 22. However, the bulk dielectric functions of gold and silver have been employed in the plots of the extinction
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Figure 6. Extinction efficiency of spherical CPs of radius rc ) 40 nm formed by a constituent dielectric sphere with a dielectric constant of εh ) 12.25 (refractive index nh ) 3.5) of Si for different volume filling fraction f of the constituent AuNPs. The CPs are supposed to be immersed in aqueous solution. For rc < 50 nm, the LDRs are largely suppressed, and only the LSPRs remain.
TABLE 2: Positions, fwhm Values, and Quality Factors Q ) ωres/Γ of LSPRs for Spherical CPs of Figure 6 with the Respective Values of εG at the Peak Position f
LSPR (nm)
LSPR (eV)
0.1 0.2 0.3 0.4 0.5
744.84 715.95 692.74 675.58 661.23
1.665 1.732 1.790 1.835 1.875
εG (-3.72, (-4.40, (-4.50, (-4.67, (-4.75,
Γ (meV)
Q
73 76 75 74 80
22.8 22.8 23.9 24.8 23.4
3.80) 2.22) 1.62) 1.32) 1.16)
TABLE 3: Positions, fwhm Values, and Quality Factors of the LSPRs for Spherical CPs of Figure 6 and Table 2 after Inclusion of a Size Correction (Eqs 12 and 13) of the Au Dielectric Function for rs ) rs,maxa f
LSPR* (eV)
rs,max (nm)
Γ (meV)
Γ* (meV)
Q*
0.1 0.2 0.3 0.4 0.5
1.664 1.731 1.79 1.834 1.874
1.857 2.339 2.679 2.947 3.175
73 76 75 74 80
182 145 130 130 123
9 12 14 15 15
a
Corrected quantities indicated by an asterisk.
efficiency so far. It is generally accepted that, for radii of constituent MNPs smaller than the mean free path of conduction electrons (42 nm for Au and 52 nm for Ag), the dielectric function of free electron metals should be corrected to a reduced mean free path of the electrons because of the scattering of the electrons on the MNP surface1,36 D εs(ω) ) εb(ω) - εDb (ω) + εsd (ω)
(12)
Here, εb is the bulk metal dielectric function, and εDb is the Drude dielectric function (eq 7) describing the bulk metal conduction electrons with the bulk damping constant γ. The term εDsd differs from εbD in that γ is replaced by the size-corrected value1,36,37
γsd ) γ + AυF /rs
(13)
where υF is the Fermi velocity and A is a fitting parameter. Following experimental results for individual gold NPs down to rs ) 2.5 nm by Berciaud et al.,38 a value of A ) 0.25 was adopted in the simulations, together with the following parameters for gold: υF ) 1.4 nm fs-1, ωp ) 72800 cm-1, and γ ) 215 cm-1.39,40 Table 3 summarizes the line width and quality factors obtained with the size-corrected dielectric function of gold. Recall here that the radius rs,max ) rcf 1/3/10 of the
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TABLE 4: Positions, fwhm Values, and Quality Factors of LDRs for Spherical CPs of Figure 1a and Table 1 after Inclusion of a Size Correction (Eqs 12 and 13) of the Ag Dielectric Function for rs ) rs,maxa f
LDR* (eV)
rs,max (nm)
Γ (meV)
Γ* (meV)
Q
Q*
0.4 0.5 0.6
1.5243 1.3432 1.1621
5.894 6.35 6.747
103 96 64
129 105 90
15 14 18
12 13 13
a
Corrected quantities indicated by an asterisk.
constituent MNPs is determined by the condition that a CP contains at least 103 constituent MNPs. The values of Γ summarized in Table 3 show that, even with the size-corrected dielectric function of the constituent MNPs, the CPs are still a viable alternative for engineering NPs with desired properties. For comparison, for homogeneous gold spheres, Klar et al.3 reported Γ values between 120 and 250 meV, whereas So¨nnichsen et al.10 observed Γ values above 200 meV. Reported values of Γ for other NPs have been as follows: 180 meV for a gold nanoshell,15 146 meV for single silver nanocubes,18 and g166 meV for single silver nanoprisms.19 So far, sharper LSPRs have been reported only for a Fano resonance (Γ ) 32 meV) and for a dipolar bonding resonance (Γ ) 19 meV) of a complex concentric ring/disk cavity made from a silver nanostructure.22 The quality factor Q ) ωres/Γ determines the local-field enhancement and is a figure of merit for all nonlinear applications of LSPRs such as surface-enhanced Raman scattering (SERS), which is believed to be proportional to Q4.10 So far, quality factors as high as 23 have been reported only for gold nanorods.10 This fact was the reason why gold nanorods were claimed to be vastly superior to gold nanospheres in optical applications where large local-field enhancements are required.10 Table 3 shows that all listed LSPRs have a bare quality factor Q > 22, which matches that of gold nanorods with aspect ratios as high as 4:1.10 With the size-corrected dielectric function of gold the quality factor Q can still be as high as 15, which is 50% above the maximum reported values for homogeneous gold spheres and which compares to that of gold nanorods with an aspect ratio of 2:1.10 The effect of the size-corrected dielectric function on the LDRs in the example shown in Figure 1a is summarized in Table 4. Again, the value of A ) 0.25 was employed in the simulations, together with the following parameters for silver: υF ) 1.39 nm fs-1, ωp ) 72700 cm-1, and γ ) 145 cm-1.39,40 Obviously, the larger the CP is, the larger the value of rs ) rs,max is, and the smaller the effect of the size-corrected dielectric function can be. This is already obvious from Table 3. Given that recent measurements on individual gold particles have shown that the pure bulk dielectric function describes the scattering properties of such particles satisfactorily down to radii as small as ∼8 nm,3,10 the results obtained with the bulk dielectric function of constituent MNPs might not be affected by any size-dependent correction provided that rc J 100 nm and f g 0.5(100/rc)3. Indeed, the latter conditions ensure the possibility of designing a CP comprising at least 103 constituent MNPs, each with radius rs J 8 nm. Even if the size correction were included, an example presented in Figure 7 shows that, for rs ) 7.94 nm, the highest LDR would be broadened from ∼85 meV by ≈14% up to ∼97 meV, which is still very promising. 4. Sensing In some MNP sensing schemes, analytes are detected through their refractive-index-induced shift of the energy of a plasmon
Figure 7. Dependence of the extinction efficiency on the radius rs of the constituent AuNPs for spherical CPs with rc ) 100 nm. The CPs are formed by a dielectric matrix with εh ) 4; the volume filling fraction f of the constituent AuNPs is fixed to f ) 0.5. The radius rs is varied up to rs,max ) rcf1/3/10, so that each CP comprises at least 103 constituent MNPs. The CPs are supposed to be immersed in aqueous solution.
resonance. To directly compare the overall performance of single NPs as chemical sensors, a “figure of merit”
FOM )
m (eV RIU-1) fwhm (eV)
(14)
where m is the linear regression slope for the refractive index dependence, has been introduced18 and functions as a standard measure for assessing an NP sensing potential. Regarding sensing applications, the very same mechanism that gives rise to the extreme sharpness of LDRs, that is, large variations of Fm(xi) with xi, gives rise to the extreme stability of LDR positions with respect to changes in the dielectric constant εb of the ambient medium. Indeed, to maintain the LDR condition (eq 5), any change in εb can be compensated by a tiny variation in xi. For instance, the linear regression slope of the strongest LDR of a CP with f ) 0.5 shown in Figure 1a and Tables 1 and 4 is merely m ≈ 0.017, yielding a rather small FOM* of ≈ 0.16. The sensing potential of the LSPRs turns out to be one magnitude higher. Indeed, for the LSPR of a spherical CPs for f ) 0.4 shown in Figure 6 and Tables 2 and 3, one finds m ≈ 0.166, yielding FOM* ≈ 1.3 (FOM ≈ 2.24). This is more than 60% higher than the reported value of FOM ) 0.8 for a 20-nm-radius homogeneous gold sphere15 and only 38% short of the reported value of FOM ) 1.8 for a Au2S/Au nanoshell.15 5. Discussion and Outlook The peculiar properties of the localized resonances of CPs have potential for numerous applications. The LDRs can be employed to provide a ruler based on the formation of CP dimers. The working principle of the ruler is that the localized resonance wavelength of a CP is affected by other CPs that are in its immediate environment. When two CPs are brought into proximity, their localized resonances couple, which shifts the resonance wavelength depending on the particle separation. Because LDRs have been shown to be an order of magnitude less sensitive to refractive index changes than the dipolar LSPRs, the performance of a ruler employing LDRs could be an order of magnitude less sensitive to the variations in the refractive index of the background medium than that of a ruler based on the LSPRs studied by So¨nnichsen et al.41 Consequently, the distance changes could be determined more precisely. The tunability combined with the narrow line width makes LDRs a promising candidate for multicolor labeling. Because CPs can
Localized Resonances of Composite Particles be designed to have a plurality of LDRs, the CPs can serve as multicolor barcode labels. In another application, CPs can be employed as acceptors for fluorescence quenching.7 The use of CPs makes it possible to tune a localized resonance position to the emission wavelength of a given fluorophore without compromising the polarization- and direction-insensitive mode of operation. A superparticle employing a CP core and dielectric shell could provide a single platform for a set of multicolor spasers (surface plasmon analogues of lasers).42 In the latter case, at least one of the CP core26 and the dielectric shell are arranged to be an amplifying medium. By tuning a selected localized resonance wavelength of a CP and by selecting corresponding luminescent species, a plurality of spasers could be fabricated, with each spaser having a different lasing wavelength. Recall that the quality factor Q ) ωres/Γ determines the localfield enhancement and is the figure of merit for all nonlinear applications such as SERS.10 Therefore, it would be much easier to achieve optical bistability and other nonlinear phenomena, such as second- and third-order harmonic generation, with CPs than with homogeneous metallic particles of the same shape, size, and metal material. The use of CPs with a narrow line width, and hence large Q, to form nanogap regions in dimers, trimers, or other closely spaced clusters of CPs will be beneficial in that (i) higher field intensities can be achieved and (ii) the operating wavelength can be tuned over a broad range comprising telecommunication S-band (or short wavelength band) for wavelengths between 1450 and 1530 nm, C-band (or central wavelength band) for wavelengths between 1530 and 1565 nm, and L-band (or long wavelength band) for wavelengths between 1565 and 1610 nm. The CPs can be arranged on a waveguiding substrate in a regular two-dimensional (2D) lattice.43 In the latter case, an interaction between continuous spectrum of waveguide modes and discrete localized resonances of CPs would give rise to a sharp Fano resonance. Typically, the line width of the Fano resonance of a combined system would be much narrower than that of the individual localized resonance of a CP.43 Such narrow Fano resonances could lead to improved sensors. Whereas for spectroscopic and sensing applications one aims at possibly the smallest line widths, for other applications, broad line widths would be more preferable. CPs with broad line widths can be achieved by employing constituent MNPs with the smallest possible sizes. Such CPs could be then ordered into threedimensional (3D) arrays to form a highly efficient heat absorber for thermophotovoltaic applications. 6. Conclusions In summary, composite nanoparticles (CPs) consisting of a dielectric matrix with homogeneously dispersed MNPs have been shown to exhibit unique properties that can be controlled through the filling fraction f of the constituent MNPs, the dielectric constant εh of the host matrix, and the CP radius rc. CPs can be designed to exhibit (1) remarkably sharp localized surface plasmon resonances (LSPRs) and (2) a number of equally narrow and strong localized dielectric resonances (LDRs). Preliminary results for spherical CPs have shown that the line widths (fwhm) of both types of resonances can be as low as 100 meV. The fwhm values are smaller than for spherical MNPs of the same size and comparable to the smallest reported values in isolated MNPs of arbitrary shape. The LDRs can be tuned over a broad range of almost 1 µm, for instance, from below 650 nm up to telecommunication bands above 1450 nm, by mere adjustment of the volume filling fraction of the constituent nanoparticles. The tunability of the LSPRs, which
J. Phys. Chem. C, Vol. 113, No. 52, 2009 21609 can also be designed to lie within the biological window of high optical transmission in blood and tissue between 700 and 1100 nm, is an order of magnitude smaller and amounts to ∼100 nm. The basic principle behind the tunability of the localized resonances of CPs is a polaritonic-like behavior of the effective dielectric function of CPs that can be tuned over several orders of magnitude by varying simply the filling fraction f of the constituent MNPs and the dielectric constant εh of the host matrix. The sensing potential of the LSPRs was quantified by a FOM of 1.3, which is more than 60% higher than the reported FOM for a similar homogeneous gold sphere. In the case of the LSPR, the dielectric constant of the host dielectric matrix is required to be preferably at least 4, whereas pronounced LDRs can be observed (provided that the CP radius is large enough, rc J 70 nm) already when the dielectric constant of the host dielectric is as small as 2. The latter range comprises many common organic and inorganic dielectric materials, such as polymethyl methacrylate, plexit, quartz, silica, MgF, ZnS, TiO2, and silicon. It is expected that CPs of other shapes, such as nanoprisms, nanocubes, or nanoboxes, would exhibit even narrower line widths and higher sensing potentials. We hope that our work will stimulate further research into the exciting subject of CPs. Acknowledgment. I thank A. van Blaaderen for a careful reading of the manuscript and useful suggestions. References and Notes (1) Neeves, A. E.; Birnboim, M. H. J. Opt. Soc. Am. B 1989, 6, 787– 796. (2) Oldenburg, S. J.; Averitt, R. D.; Westcott, S. L.; Halas, N. J. Chem. Phys. Lett. 1998, 288, 243–247. (3) Klar, T.; Perner, M.; Grosse, S.; von Plessen, G.; Spirkl, W.; Feldmann, J. Phys. ReV. Lett. 1998, 80, 4249–4252. (4) Link, S.; El-Sayed, M. A. J. Phys. Chem. B 1999, 103, 3073–3077. (5) Link, S.; El-Sayed, M. A. J. Phys. Chem. B 1999, 103, 8410–8426. (6) Schultz, S.; Smith, D. R.; Mock, J. J.; Schultz, D. A Proc. Natl. Acad. Sci. U.S.A. 2000, 97, 996–1001. (7) Dubertret, B.; Calame, M.; Libchaber, A. J. Nat. Biotechnol. 2001, 19, 365–370. (8) Lakowicz, J. R. Anal. Biochem. 2001, 298, 1–24. (9) Graf, C.; van Blaaderen, A. Langmuir 2002, 18, 524–534. (10) So¨nnichsen, C.; Franzl, T.; Wilk, T.; von Plessen, G.; Feldmann, J.; Wilson, O.; Mulvaney, P. Phys. ReV. Lett. 2002, 88, 077402. (11) Sun, Y.; Xia, Y. Science 2002, 298, 2176–2179. (12) Jackson, J. B.; Westcott, S. L.; Hirsch, L. R.; West, J. L.; Halas, N. J. Appl. Phys. Lett. 2003, 82, 257–259. (13) Raschke, G.; Kowarik, S.; Franzl, T.; So¨nnichsen, C.; Klar, T. A.; Feldmann, J.; Nichtl, A.; Ku¨rzinger, K. Nano Lett. 2003, 3, 935–938. (14) McFarland, A. D.; Van Duyne, R. P. Nano Lett. 2003, 3, 1057– 1062. (15) Raschke, G.; Brogl, S.; Susha, A. S.; Rogach, A. L.; Klar, T. A.; Feldmann, J.; Fieres, B.; Petkov, N.; Bein, T.; Nichtl, A.; Ku¨rzinger, K. Nano Lett. 2004, 4, 1853–1857. (16) Moroz, A. Chem. Phys. 2005, 117, 1–15. (17) Chen, K.; Liu, Y.; Ameer, G.; Backman, V. J. Biomed. Opt. 2005, 10, 024005-024110. (18) Sherry, L. J.; Chang, S.-H.; Schatz, G. C.; Van Duyne, R. P.; Wiley, B. J.; Xia, Y. Nano Lett. 2005, 5, 2034–2038. (19) Sherry, L. J.; Jin, R.; Mirkin, C. A.; Schatz, G. C.; Van Duyne, R. P. Nano Lett. 2006, 6, 2060–2065. (20) Pillai, S.; Catchpole, K. R.; Trupke, T.; Green, M. A. J. Appl. Phys. 2007, 101, 093105. (21) Pecharroma´n, C.; Pe´rez-Juste, J.; Mata-Osoro, G.; Liz-Marza´n, L. M.; Mulvaney, P. Phys. ReV. B 2008, 77, 035418. (22) Hao, F.; Sonnefraud, Y.; Van Dorpe, P.; Maier, S. A.; Halas, N. J.; Nordlander, P. Nano Lett. 2008, 8, 3983–3988. (23) Verellen, N.; Sonnefraud, Y.; Sobhani, H.; Hao, F.; Moshchalkov, V. V.; van Dorpe, P.; Nordlander, P.; Maier, S. A. Nano Lett. 2009, 9, 1663–1667. (24) Yang, W.-H.; Schatz, G. C.; Van Duyne, R. P. J. Chem. Phys. 1995, 103, 869–875. (25) Garnett, J. C. M. Phil. Trans. R. Soc. London 1904, 203, 385–420. (26) Dolgaleva, K.; Boyd, R. W. J. Opt. Soc. Am. B 2007, 24, A19– A25.
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