Analytical calculation of plasmonic resonances in metal nanoparticles: A simple guide

Advances in nanotechnology and photonics go hand in hand, from super-resolution microscopy helping us visualize life at the nanoscale,¹ to extreme-UV lithography enabling the creation of nanoscale chips needed to analyze it.² At the intersection of photonics and nanotechnology lies the topic of plasmonics.³ Plasmonics involves the creation, study, and manipulation of signals embedded in optical-frequency oscillations of surface electrons along metal-dielectric interfaces. Plasmonics confines optical-frequency signals to subwavelength-size volumes, thereby providing the interface between optical electromagnetic fields and nanoscopic devices and circuitry.

When an electromagnetic field interacts with a metal nanoparticle, it will lead to charge oscillations in the metal. These collective oscillations, known as plasmons, are excited when the frequency of the electromagnetic field matches the resonant frequency of the metal nanoparticle. The electrons are free to move within the boundaries of the particle but are ultimately confined to its surface. For these reasons, the effect is called a localized surface plasmon (LSP). Given that the oscillation is at the same frequency as the incoming field, the effect is interchangeably called plasmonic resonance.

Designing metal nanoparticles to answer questions in nanophysics, nanochemistry, and nanobiology requires understanding the plasmonic resonance. Due to quantum mechanical effects, the properties of nanoscale objects often cannot be explained intuitively. Fortunately, the most prominent plasmonic effects can be explained within a classical framework. We will begin with a historical perspective. Then, to model the plasmonic resonance in metal nanoparticles, we will need a framework to describe three distinct features: the metallic characteristic, the size (the “nano”), and the shape of the particle. We will introduce the Drude model, a reasonable description of electrons in metals. The quasi-static approximation will then let us take into account the nanoscale size of the particles. Additionally, the generalization of the Clausius–Mossotti relation will let us consider a particle embedded in a dielectric medium and the effect of its shape. By combining these aspects, we are able to build a classical model for the plasmonic resonance in metal nanoparticles. We will then deal with some of the limitations of the quasi-static approximation by introducing final corrections to the model, to extend its validity as far as possible. As our assumptions are all based on classical electromagnetism, the final model will be classical too.

While nanotechnology emerged as a field only a few decades ago, peculiar optical phenomena due to nanoparticles have intrigued humankind since ancient times. Witnessing and harnessing such “technologies” far preceded any possible scientific explanation: photonic crystals shape light, thus creating spectacular iridescent colors in butterflies,⁴ lead-based quantum dots have been involved in black hair dyes manufactured by ancient Greeks and Romans,⁵ and copper nanoparticles were employed in red opaque glass production in Egypt and Mesopotamia.⁶ 

One of the most impressive pieces of glasswork incorporating metal nanoparticles is the Lycurgus cup (Figs. 1(a) and 1(b)). Dated around the 4th century, this Roman cage cup is made up of dichroic glass, so that an observer sees it red if light passes through it, but green if light is reflected to them. Recent analysis showed that the dichroism is due to the presence of colloidal gold and silver nanoparticles dispersed throughout the glass. The embedded particles have diameters around 70 nm, meaning that they are invisible to optical microscopy and require transmission electron microscopy (TEM) to be seen.⁷ 

Michael Faraday is credited with performing the first scientific experiments on the optical properties of nanoparticles, focusing on gold colloids in the 1850s. He was puzzled by the ruby red color of the solutions he synthesized (Fig. 1(c)), far removed from the aureate color of bulk gold. A satisfying classical explanation of this phenomenon came only in 1904, when Maxwell Garnett combined the new Drude theory of metals with Lord Rayleigh's description of electromagnetic properties of small spheres.⁸ Shortly after, in 1912, Richard Gans successfully extended the description of optical phenomena to oblate and prolate spheroids.⁹ However, these theories were all purely based on the optical properties of bulk metals, and it was not until 1970 that they were modified by Uwe Kreibig and Peter Zacharias to take their nanoscale size into account.¹⁰ For the first time, they explained the electronic and optical response of silver and gold nanoparticles in terms of localized surface plasmon excitations. It became clear then that size, shape, arrangement, medium, and temperature all have a crucial role in controlling the intensity and frequency of plasmonic resonances.

From the early 2000s on, the interest in plasmonics boomed, thanks to novel nanofabrication techniques, commercialization of simulation software and the plethora of biological and biomedical applications.^11,12 To summarize 50 years of research and more than a century of theoretical modeling, we start by examining the object under study, the metal nanoparticle, and its defining features.

In Table I, we report the Drude parameters for common plasmonic metals, fitted from experimental data in Ref. 14. Note that the experimental and fitted data do not match completely, as the Drude model is an approximation of the actual complex dielectric function, which can include interband and intraband transitions. Throughout the paper, we will take gold as a typical metal for plasmonics.

With these concepts in hand, we can provide a simple yet rigorous derivation of plasmonic resonance in metal nanoparticles.

The careful reader might point out that the wavelength inside the nanoparticle may not be the same as the one in the dielectric medium. The concept of wavelength itself may even be ill-defined if the field decays exponentially inside the metal. A relevant comparison would then be between the size of the nanoparticle and the skin depth δ of the metal. For common metals like gold, silver, copper, and aluminum, the skin depth at optical frequencies is on the order of a few tens of nanometers. If $d ≲ δ$⁠, then the field can penetrate the nanoparticle completely and the uniformity of the polarization can still be assumed. Refer to the supplementary material⁴³ for more details regarding skin depth.

It is important to note that particles larger than the wavelength may produce a plethora of effects that cannot be explained as the simple radiation of a dipole, due to the occurrence of multipolar effects. Eventually, extremely large particles will exhibit the optical properties of bulk metal.

Let us now consider an isolated ellipsoid having a complex dielectric function ϵ_m and semiaxes, respectively, a_x, a_y, and a_z, immersed in a dielectric material (from this point on referred to as the “host”) having a real and positive dielectric constant ϵ_h (Fig. 3(a)).

To calculate the polarizability, we, therefore, need to find the relation between the dipole moment and the electric field.

The depolarization factors arise from the solution of Laplace's equation in ellipsoidal coordinates. A full derivation can be found in Ref. 15.

It is important to note that $α ↔$ is a diagonal matrix. In other words, if the incoming field is aligned with one of the axes of the ellipsoid, the polarization will be parallel to it. This considerably simplifies the calculations.

Thanks to this normalization property, some easy geometries (see Table II) can be treated without calculating the integrals explicitly. Anisotropy is then reflected by the matrix nature of $α ↔$⁠, eventually leading $p → n p$ to not be parallel to $E → 0$⁠.

We now have all the tools to calculate the plasmonic resonance. However, what does it mean for a nanoparticle to have a plasmonic resonance? Why do we talk about a plasmonic peak? The simplest way to picture such an effect is by directly plugging the result of the Drude model (Eq. (3)) in the calculation of the polarizability (Eq. (13)). For a sphere with gold-like Drude parameters, put either in vacuum or water, we get the typical wavelength dependence of the polarizability shown in Fig. 4.

The functional form for $| α |$ is a sharp peak. At the angular frequency where the polarizability is maximal, the electrons oscillate with a higher amplitude. Given such a sharp feature in the frequency, we refer to the peak position as the resonance frequency.

Unfortunately, only a few materials satisfy Eq. (18) in the optical range. For the resonance to exist, the imaginary part of the dielectric function must be sufficiently low. For this reason, the most significant plasmonic materials are also the most conductive: silver, gold, copper, and aluminum. Other less commonly used metals are palladium, platinum and nickel.²¹ An invaluable contribution to the investigation of plasmonic candidates was given by Eadon and Creighton, in their review of the ultraviolet/visible spectrum of 52 different metal nanospheres, in vacuum and in water.²² Nowadays, the search for novel plasmonic candidates focuses on metallic alloys, (doped) semiconductors, and metamaterials.²³ 

One interesting effect can be noted in Fig. 5: along the direction orthogonal to the oscillation, the radiated field causes destructive interference with the incoming field, leading to areas where the field is quenched (⁠ $| E | 2 / | E 0 | 2 < 1$⁠) instead of enhanced. This is possible because, in Eq. (20), for some $r →$ and at a certain frequency, $r → · p → n p = 0$ and $| 1 − ( α / 4 π r 3 ) | 2 < 1$⁠.

The finite-difference time-domain simulations in Fig. 5 solve Maxwell's equations in discretized space and time, characterizing each portion of space with the complex dielectric function ϵ and complex magnetic permeability μ. These simulations introduce minimal approximations and accurately compute the electric and magnetic fields, presenting a case very close to reality. Using this simulation, the plasmonic resonance occurs at 540 nm, instead of 520 nm, as shown previously in Fig. 4. This shift is mainly explained as an effect of radiation damping,²⁷ which shows that the quasi-static approximation does not always provide accurate answers. This is why we need to look beyond it.

The quasi-static approximation describes nanoscale processes fairly well, such as the absorption of certain colors or the local field enhancement in the proximity of metal nanoparticles. However, it fails to predict other effects related to particle size: in the quasi-static approximation, particle size is irrelevant, as long as it is smaller than the incoming wavelength. Can we extend this theory to include the particle's size and shape?

The first correction we will introduce takes into account the collisions of the electrons with the nanoparticle boundaries, which have implicitly been neglected until now. Then, the second correction will take us closer to the multipolar regime by involving retardation effects. Considering the classical nature of this extended model, it is impressive how well it fits experiments for such nanoscopic objects, as we will see.

Up to now, we have implicitly considered the damping frequency γ to be dominated by the collisions of electrons with other electrons, lattice nuclei or phonons. At a very small scale, however, electrons will also impact the particle boundary. This is notably the case when the dimensions of the particle are smaller than the mean free path, that is, the average distance traveled by the electron between two consecutive collisions. In other words, if the particle is small enough, the electrons will impact the boundary much more often than they collide with other objects.¹⁵ The empirical model developed hereafter aims at taking this effect into account, and well matches experimental data. In the end, it is very similar to the exact one obtained via semiclassical calculations.^28,29

For a sphere of radius r, Kreibig¹⁰ used the linear relation $λ boundary = 4 3 r$⁠, but coefficients between 1 and 4 have been used by other authors.^30–34 A simple and intuitive motivation for a coefficient of $4 3$ is provided hereafter, based solely on geometrical considerations.

The dipolar approximation is valid as long as the dimensions of the metal nanoparticle are such that $d ≪ λ$⁠. Otherwise, variations in the incoming field will not be negligible, and multipolar modes will eventually be excited. Between this dipolar treatment and the brute-force computational solution of Maxwell's equations lies the so-called modified long-wavelength approximation (MLWA), a correction to the polarizability obtained in the quasi-static approximation (QSA) that includes retardation effects.¹⁷ 

The MLWA treats each atom in the nanoparticle as a dipole emitter and takes into account that its electric field propagates at the speed of light c (and not instantly), causing a retarded dipolar field. While in the QSA we imposed $k → · r → = 0$ (Eq. (4)), in the following derivation of the MLWA equations, the dipole radiation is expanded in a Taylor series up to the third order $( k r ) 3$⁠. The infinitesimal electric field is integrated over the volume of the particle, and finally a corrected polarizability α_MLWA is defined.

For sufficiently small nanoparticles, we retrieve the QSA (⁠ $α MLWA , j ≈ α j$⁠), as expected. For relatively small yet finite volumes, the imaginary term proportional to k³ can be neglected (⁠ $α MLWA , j ≈ 4 π α j / 4 π − α j k 2 / a j$⁠). For larger volumes, the imaginary term dominates the denominator.

The term $α j ( k 2 / a j )$ in Eq. (38) is called dynamic depolarization, because it is obtained in a dynamic calculation (k > 0), and its coefficient is real, corresponding to a change in the effective particle depolarization factor. Let us recall that α_j is proportional to particle volume (⁠ $α j ∝ a x a y a z$⁠). As $( a x a y a z / a j k 2 )$ increases, a more negative value of ϵ₁ is necessary to meet the resonance condition. At small but finite particle volumes, this effect enhances the plasmonic resonance; at larger volumes, it is responsible for the shift of the resonance peak.

The term $− i α j 2 3 k 3$ in Eq. (38) is called radiative damping, because it arises from the spontaneous emission of radiation by the induced dipole. The term grows rapidly with particle volume. Being imaginary, it contributes heavily to ϵ₂ and, therefore, to the resonance damping. For relatively large particle volumes, it accounts for the damping by radiative losses and results in a broadening and a strong decrease in the plasmonic resonance (also counterbalancing the enhancement due to dynamic depolarization just mentioned).

According to this approximation, an increase in particle size (a_j) will result in a red-shifted, broadened and less intense plasmonic resonance. This is indeed what we notice in Fig. 6(b) in the calculated peaks for a nanoparticle with gold-like parameters immersed in water. The MLWA usually works well for nanoparticles of dimensions up to $≈ 200$ nm,³⁶ above which multipolar resonances cannot be neglected, and computational methods are required.

The complete analytical model combines the corrected Drude model, the generalized Clausius–Mossotti formula for polarizability, the surface damping effect and the MLWA. In Fig. 7, an example for a gold elongated ellipsoid having $a x / a y = a x / a z = 2$ is presented against a gold sphere of r = 25 nm, of the same volume. The cross sections are calculated in the dipolar limit using Eq. (19). Agreement between the finite-difference time-domain (FDTD) simulation and the model is remarkable, both in the peak positions and the ratio of cross section components. Experimental evidence also corroborates this result.³⁷ 

Starting from a general model for metals and from the polarization of a particle immersed in a dielectric host, we explored the effects of several features on the final resonance peak induced by an external oscillating electric field.

The analytical approach took us far into the understanding of the plasmonic resonance. However, as already hinted at the end of Sec. IV, the corrections to the quasi-static approximation have their own limitations. In fact, neglecting key features in the shape of the particle like spikes, neat edges, flat sides, or amorphous protrusions will result in incorrect predictions. Yet where the analytical approach fails, numerical simulations can be used. The most common numerical techniques include generalized Mie theory, finite-difference time-domain (FDTD), discrete dipole approximation (DDA), finite-element method (FEM), and boundary element method (BEM).^38,39 Each simulation technique has its own advantages and drawbacks.

Should this material be used as didactic reference, the authors strongly suggest the incorporation of at least some examples of applications from state-of-the-art research. These examples should be tailored depending on the interest of the course and can be directed towards optics, biophysics, or condensed matter physics. To learn more about light-matter interaction or expand knowledge about state-of-the-art plasmonics and emerging applications, the authors recommend Fox's⁴⁰ and Maier's³ books, which provide valuable insight into the field and are very accessible for lecturers and students alike.

The properties of plasmonic resonances depend on material shape and intrinsic properties. The analytical approach supplied here is limited to ellipsoidal geometries but already takes into account virtually all relevant material properties (in a classical approximation). However, while only simulations will provide reliable descriptions of the plasmonic effects in realistic nanoparticles, the outlined fundamental approach, no matter its flaws, provides an understanding of why and how plasmonic resonances emerge in nanoparticles, and thus gives an intuitive basis and plausibility check for numerical design.

D.B. acknowledges support by an ERC Starting Grant (No. 850818-MULTIVIsion), an NWO Start-up Grant (No. 740.018.018), and an NWO XS (No. CENW.XS2.033). M.L. thanks professors Maurizio Canepa and Francesco Bisio for inspiration.

The authors have no conflicts to disclose.