J. Phys. Chem. C 2008, 112, 15721–15728
15721
Field Enhancement around Metal Nanoparticles and Nanoshells: A Systematic Investigation Katsuaki Tanabe* Institute of Industrial Science, UniVersity of Tokyo, Tokyo 153-8505, Japan ReceiVed: June 03, 2008; ReVised Manuscript ReceiVed: July 15, 2008
Metal nanoparticles and nanoshells consisting of metal shells and dielectric cores are known to significantly enhance incident electromagnetic fields around themselves due to surface plasmons. The field enhancement factors were calculated for spherical metal nanoparticles and nanoshells in the quasistatic limit using empirical wavelength-dependent dielectric constants. Dependence of the field enhancement factor on various parameters, such as wavelength, distance from the nanoparticle/nanoshell, metal element, dielectric core material, surrounding medium, and diameter ratio between the core and the shell, was investigated. Interestingly the peak field enhancement factor becomes largest with core-to-shell diameter ratio at ∼0.9 for any combination of noble metal shell and dielectric core. As a consequence of the parameter optimization, it was found that an Ag nanoshell with a Teflon core with a core/shell diameter ratio of 0.88 exhibits a peak field enhancement factor of 1400 in its vicinity in water. I. Introduction Metal nanoparticles are known to exhibit distinctive optical characteristics, such as surface-enhanced Raman scattering (SERS) and second harmonic generation, relative to the bulk form of metals.1-7 Representatives of the use of metal nanoparticles are biomolecular manipulation, labeling, and detection with SERS.8-11 Other optoelectronic fields inspired by metal nanoparticles are also emerging, such as multiphoton absorption and fluorescence excitation for microscopy, microfabrication and optical data storage,12,13 all-optical nanoscale network,14,15 and surface plasmon enhanced light absorption for photovoltaic materials.16,17 One of the most important factors for all of those applications is the enhancement of the electromagnetic field intensities around subwavelength-size metal particles due to the coupling between the incident photons and collective oscillation of free electrons at the metal surface. It also has been recently known that nanoshells, concentric nanoparticles consisting of a dielectric core and a metallic shell, exhibit attractive features due to the hybridization of the plasmons supported by the nanoscale sphere and a cavity in the surrounding medium.18,19 By adjusting the relative core and shell dimensions, we can widely tune the resonance frequency, or the peak wavelength for field enhancement, of a nanoshell, while the optical resonance is essentially a fixed frequency resonance, almost independent of their particle sizes for metal nanospheres. Also, nanoshells can exhibit significantly larger field enhancement than that for nanoparticles according to conditions. In this paper, the field enhancement factors, defined as the intensity ratio between the fields around the object and those under no existence of the object or the original incident fields, for spherical metal nanoparticles and nanoshells in the air and in water were calculated based on the classical electromagnetic field theory in the quasistatic limit using empirical wavelengthdependent dielectric constants. The electrostatic calculations conducted in this paper are valid for particle sizes in the range of 10-100 nm, where the field enhancement will be largest and metal nanoparticles and nanoshells therefore become most * E-mail:
[email protected].
Figure 1. Schematic configuration of a metal nanoparticle or nanoshell in an electric field.
applicable, as discussed in section II. Dependence of the field enhancement factor on various parameters, such as wavelength, distance from the nanoparticle/nanoshell, metal element, dielectric core material, surrounding medium, and diameter ratio between the core and the shell, was investigated. The formalism in this paper is relatively simple, but the systematic collection of data and the analysis given will be an effective guide for the design of emerging nanoparticle/nanoshell-enhanced optoelectronic devices. It is known that metal nanoellipsoids can provide more field concentration around their tips than nanospheres do because the field concentration is dependent on the curvature of the metal/ dielectric interface.20,21 Spherical nanoparticles/nanoshells are however more practical for fabrication while it is not trivial to control the geometrical aspect ratio and the alignment of a group of ellipsoids. Also with the aim of this paper to give a conceptual understanding for the nature of the concentration of electromagnetic fields around nanoparticles/nanoshells, the case for a single stand-alone spherical nanoparticle/nanoshell geometry with high symmetry should be more suitable to deal with due to its relatively straightforward computation and analysis. The shape and interparticle/shell effects will be discussed elsewhere. II. Theory and Computational Method The intensities of electromagnetic fields around subwavelength-size metal nanoparticles and nanoshells can be described by the formalism below in the quasistatic limit.22 Consider a homogeneous, isotropic sphere placed in a medium in which there exists a uniform static electric field b E0 ) E0eˆz, as schematically depicted in Figure 1. If the permittivities or
10.1021/jp8060009 CCC: $40.75 2008 American Chemical Society Published on Web 09/13/2008
15722 J. Phys. Chem. C, Vol. 112, No. 40, 2008 dielectric constants of the sphere and the medium are different, a charge will be induced on the surface of the sphere. The initially uniform field will be therefore distorted by the introduction of the sphere. The electric fields inside and outside E2, respectively, are derivable from scalar the sphere, b E1 and b potentials Φ1(r,θ) and Φ2(r,θ):
b E1 ) -∇ Φ1 b E2 ) -∇ Φ2
| || | a3
R
2
η ) 1 + 2 3β ) 1 + r 2πr3 using a factor
β≡
(1)
where
2
Tanabe
ε1 - εm ε1 + 2εm
(10)
(11)
defined for convenience, and the polarizability of the sphere
∇ Φ1 ) 0 (r < a) ∇ Φ2 ) 0 (r > a) 2
2
(2)
where a is the radius of the sphere. Because of the symmetry of the problem, the potentials are independent of the azimuthal angle φ. At the boundary between the sphere and the medium, the potentials must satisfy
Φ1 ) Φ2
ε1
∂Φ1 ∂Φ2 ) εm ∂r ∂r
(r ) a)
limΦ2 ) -E0r cos θ ) -E0z
(4)
rf∞
that is, the electric field is the unperturbed applied field at large distances from the sphere. It can be derived that the potentials
3εm E r cos θ ε1 + 2εm 0
Φ2 ) -E0r cos θ + a3E0
(5)
ε1 - εm cos θ ε1 + 2εm r2
(6)
satisfy eqs 2-4. These solutions for the potentials could be also derived rigorously using Legendre polynomials. From eqs 1 and 6, the electric field outside the sphere can be expressed as
{(
) (
a3 ε1 - εm ˆ b E2 ) E0 1 + 2 3 er cos θ + r ε1 + 2εm -1 +
)
}
a3 ε1 - εm ˆ eθ sin θ (7) r3 ε1 + 2εm
where eˆr and eˆθ are the unit vectors to r and θ directions, respectively, and thus
| |
(|
2 3 ε -ε m b2|2 ) |E b0|2 1 + 2 a 1 |E cos2 θ + r3 ε1 + 2εm
-1 +
|
)
a3 ε1 - εm 2 2 sin θ (8) r3 ε1 + 2εm
The electric field intensity will be therefore maximized at the direction θ ) 0, π for most cases and the field enhancement factor is defined as
η≡
b2|2 |E
|
a3 ε1 - εm ) 1 + 2 b0|2 |E r3 ε1 + 2εm
|
2
(9)
Note that this field enhancement factor is defined as the ratio of field intensities and not field magnitudes. Equation 9 can be alsowritten as
ε1 - εm ) 4πa3β ε1 + 2εm
(12)
For the case of a concentric spherical core-shell structure consisting of an inner spherical core of radius a1, dielectric constant ε1 and an outer spherical shell of radius a2, dielectric constant ε2, the polarizability can be written as
(3)
where ε1 and εm are the permittivities or dielectric constants of the sphere and the surrounding medium, respectively. It is additionally required that
Φ1 ) -
R ) 4πa3
R ) 4πa32
(ε2 - εm)(ε1 + 2ε2) + f 3(ε1 - ε2)(εm + 2ε2) (ε2 + 2εm)(ε1 + 2ε2) + f 3(2ε2 - 2εm)(ε1 - ε2)
) 4πa32β
(13)
where f ≡ a1/a2 and β)
(ε2 - εm)(ε1 + 2ε2) + f 3(ε1 - ε2)(εm + 2ε2) (ε2 + 2εm)(ε1 + 2ε2) + f 3(2ε2 - 2εm)(ε1 - ε2)
(14)
Note that the dielectric constants ε1, ε2, and εm are generally complex functions of wavelength and are expressed as εj ) εj′ + iεj″ (j ) 1, 2, m), where both of εj′ and εj′′are real. As seen in eq 10, the field enhancement factor for subwavelength-size spherical nanoparticles or nanoshells depends simply on the relative distance from the center of the particle or the shell, r/a, and on the diameter ratio of the inner to the outer material, f, for a nanoshell, but not on the absolute size of the particle or the shell. The calculation of the field enhancement factors η were carried out for spherical metal nanoparticles of 11 kinds of metals, Ag, Al, Au, Co, Cr, Cu, Ni, Pd, Pt, Sn, and Ti, with eqs 11 and 12 and for spherical metal nanoshells of the same 11 metals with dielectric cores of SiO2, ZnS, Al2O3, Si3N4, CaF2, and Teflon (polytetrafluoroethylene) with eqs 13 and 14. For the core materials, in addition to the common inorganic materials, Teflon was adopted due to its small dielectric constants, ε′ and ε′′. The wavelength-dependent optical constant data of Palik23 was used for the computation in this paper, except ε′ ) 1.7 and ε′′ ) 0 throughout the entire optical wavelengths for Teflon referring ref 24. While the Drude model has been widely used for dielectric functions of metals for electromagnetic calculations on nanoparticles/nanoshells,25,26 polynomial fitting to the measured data in ref 23 was adopted in this paper, as shown in the Appendix. This is because it is difficult to fit the formalism of Drude model to the dispersion of dielectric functions of realistic metallic materials for entire frequencies though the ultraviolet (UV) to the infrared (IR), specifically for the higher energy region due to the bound-charge contribution.22 Figure 2a and b shows the polynomial fit used in this study for the real and imaginary parts, respectively, of the dielectric functions for Ag, Al, Au, and Cu. It should be noted that the calculations shown in this paper are valid for the particles and shells smaller than the wavelengths where the phase retardation is negligible throughout the objects. This electrostatic approximation is empirically known to be valid
Metal Nanoparticles and Nanoshells
J. Phys. Chem. C, Vol. 112, No. 40, 2008 15723
Figure 3. Dependence of the field enhancement factors, η, around a spherical Ag nanoparticle in the air on the relative distance from the center of the particle.
Figure 2. (a) Real and (b) imaginary parts of the dielectric functions for Ag, Al, Au, and Cu. The dots are the empirical values from ref 23, and the lines are the polynomial fit used for the calculations in this study. The fitting equations are found in the Appendix.
for spherical particles with diameters 2a up to 0.1λ.27 Also, the dielectric functions of materials used for the computation in this paper were empirical values for bulk materials, whose validity will be debatable when the materials sizes become smaller than 10 nm due to the electron mean free path limitation or scattering of conduction electrons off particle surfaces.21,22,28,29 Therefore, the computation results for the optical wavelengths in this paper are valid for metal particles and shells with diameters of 10-100 nm. Both smaller and larger metal particles than these limits exhibit broader plasmon resonances and smaller field enhancements due to surface scattering losses and radiative losses or electrodynamic damping, respectively.21,28,29 The choice of particle sizes, 10-100 nm, in this paper is therefore most suitable for plasmon-enhanced photonic applications due to the largest field enhancements. III. Metal Nanoparticles
Figure 4. Calculated field enhancement factors, η, of spherical metal nanoparticles of 11 metals (a) in the air and (b) in water.
Figure 3 shows the calculated field enhancement factor η around a Ag nanoparticle in the air depending on the relative distance of the observing point from the center of the nanoparticle to the radius of the nanoparticle, r/a. We can see that the field enhancement factor η drastically changes according to the distance from the metal particle, as also seen in eq 10. This strong dependence stems from the fact that the field enhancement factor η is roughly proportional to (a/r)6 provided η is larger than 1, as seen in eq 10. The largest field enhancement is thus obtained immediately adjacent to the metal particle surface, r ) a, and therefore, the calculation in this paper focuses on this point of observation. The field enhancement factors shown hereafter in this paper are all computed for this point, r ) a, for both metal nanoparticles and nanoshells. It should be also noted that η converges to the value (1 + 2(a/r)3)2 at longer
wavelengths as the factor β converges to unity since the modulus of the real and imaginary parts of dielectric constant of Ag become much larger than those of the air, as seen from eqs 10 and 11. Figure 4a,b shows the computed field enhancement factors η of nanoparticles of 11 metals surrounded by air (εm ) 1) and water, respectively. Their horizontal axes represent the wavelengths in vacuo. The peaks seen in these figures are associated with the resonance or surface mode, characterized by internal electric fields with no radial nodes. Among these 11 metal elements, particularly the noble metals, Ag, Al, Au, and Cu, show distinctively higher peaks of the field enhancement factor than the other metals due to their high conductivities. The peak positions were independent of the particle size (see eq 4), which
15724 J. Phys. Chem. C, Vol. 112, No. 40, 2008
Tanabe
is consistent with the experimental results in ref 30. The peak positions for Ag, Au, and Cu in Figure 4a has a good agreement with the experimental data in ref 31 with slight blue-shifts by ∼10% supposedly due to the particles’ shape difference, which strongly affects resonant frequencies in contrast to the particle size.22,30 The peak position of ∼550 nm for Au nanoparticles is also consistent with the absorption spectra measured in ref 32. It should be noted that for both cases of nanoparticles and nanoshells the field enhancement factors for all kinds of metals converge to 9 at longer wavelengths, as seen in Figure 4a and b, because the modulus of the real and imaginary parts of dielectric functions of metals becomes much larger than those of the core dielectric materials such as SiO2 and the surrounding media such as air and H2O, and thus, the factor β converges to unity. (See eqs 11 and 14.) Generally, the peak position of the field enhancement factor is around the wavelength where
ε1 ′ ) -2εm ′
Figure 5. Calculated field enhancement factors, η, of a spherical Al nanoparticle surrounded by an Al2O3 shell in the air with various factors f ) a1/a2, where a1 and a2 are the radius of the Al sphere and the Al2O3 shell, respectively.
(15)
is satisfied by minimization of the denominator of the factor β (see eq 11), which corresponds to maximization of the polarizability R of the metal particle in the surrounding medium.22 As seen in Figure 4b relative to Figure 4a, the peak positions generally move toward red when the metal nanoparticles are in water from the case in the air since εm′ of water is larger than that of the air and therefore eq 15 will be satisfied at longer wavelengths with decreasing ε1′ of metals. The maximum field enhancement factor or the peak height also becomes larger or smaller roughly depending on (1) whether ε1′′ got smaller or larger by the change of the peak wavelength. (2) The factor that εm of water (∼1.8) is significantly larger than εm of the air ()1) also gives a rise to η by enlarging the numerator of β due to negative ε1 of metals at most wavelengths as seen in Figure 2a. For example, the values of ε1′′ at the η peak wavelengths in the air and in water are quite similar to each other (∼0.7) for Ag in terms of the factor 1, but factor 2 makes a higher peak in water. For Al, influence from factor 1 with ε1′′ ∼0.2 and ∼0.5 at each η peak wavelength in the air and in water, respectively, exceeds the effect of (2) leading to a lower peak in water. Both factors 1 and 2 give a rise to the peaks in water for Au and Cu. As seen in Figure 4a and b, Al has a distinctively high peak field enhancement factor in the UV region, with the factor’s maximum up to ∼1100. This is due to the fact that the ε1′′ at the peak wavelength is as small as 0.18 (0.46) for Al nanoparticles in the air (in water), relative to those of 0.65 (0.72) for Ag, 2.1 (1.8) for Au, and 2.5 (2.3) for Cu. However, engineers might not be able to simply refer these data for device designing since Al surface can be easily oxidized to form Al2O3 in the atmosphere. Figure 5 shows the dependence of the field enhancement factor η on the f factor, or the radius ratio of the Al core to the Al2O3 shell, for a spherical Al-core/Al2O3-shell nanostructure to examine the influence of the existence of aluminum oxide Al2O3 layer surrounding the Al particles. We can see from Figure 5, for instance, the field enhancement factor η would reduce from ∼1100 into ∼650 with its peak slightly red-shifted provided a Al2O3 layer of 1 nm thick is covering a Al particle with a diameter of 100 nm. This argument also is a good example that the formalism for nanoshells described in this paper can be also used for analyses of the oxidization effect of metal particle surface on field concentration taking surfaceoxidized metal particles as a form of metal/dielectric core-shell structures.
Figure 6. Dependence of the field enhancement factor, η, on the f factor () a1/a2) for Ag nanoshells with SiO2 core in the air.
IV. Metal Nanoshells Figure 6 shows the calculated dependence of η on the f factor () a1/a2) for Ag nanoshells with the SiO2 core in the air. The resonant or the peak wavelength is found to be quite sensitive to the f factor, and we can widely tune the peak wavelength through the UV region to the IR by adjusting the f factor. The peak η becomes larger as the factor f becomes larger until the Ag shell gets too thin to support strong plasmons. This result can be understood by considering the strong interaction between the sphere and the cavity plasmons for thin shells.18 The maximum peak η is ∼730 with f ∼ 0.9, and peak η larger than 500 can be obtained in a wide range of wavelengths through 600-1100 nm. Also, the full width of the half-maximum (fwhm) is as large as 800 nm (1.5 eV) ranging from 500 (2.5 eV) to 1300 nm (0.95 eV). However, uniform nanoshells with a single f can exhibit a fwhm less than 100 nm (
