Unique features of plasmonic absorption in ultrafine metal nanoparticles: unity and rivalry of volumetric compression and spill-out effect

1 Introduction

Plasmonics has emerged as a pivotal field in nanoparticle technologies, showcasing a diverse range of applications in biomedical sensing, cellular imaging, cancer therapy, sensing probes, detection of molecules within living cells, nanoparticle-assisted bioimaging, plasmon-enhanced fluorescence, and Raman spectroscopy (SERS, TERS) for the precise identification of individual molecules, among other notable uses [1], [2], [3], [4], [5], [6], [7]. Plasmonic nanoparticles (NPs) enhance the electromagnetic field near the metal surface at nanoscale distances, making them indispensable in imaging technologies for probing local environments. Ultrafine plasmonic NPs possess an extraordinary capacity to achieve unparalleled light localization. This exceptional property allows us to delve into the realm of quantum plasmon effects and to reveal the full extent of their potential [3]. In the context of these tasks, it is important to highlight the biological applications of ultrafine plasmonic NPs, which demonstrate, for instance, a remarkable capability to efficiently traverse ion channels in cell membranes. This enables precise laser hyperthermia targeting of malignant cells, leading to their death [8], [9]. Investigating the relationship between the resonance properties of NPs and their size, shape, and surrounding environment has, therefore, emerged as a critical research objective in this dynamic field. Given their importance in applications, it is essential to have precise models that can accurately predict and interpret the optical properties of gold and silver plasmonic NPs across a broad size range with high predictive capabilities that would facilitate the possibility of creating ultrafine plasmonic structures with predefined properties to expand their applications.

In addressing the mentioned issues, a crucial aspect lies in exploring the interdependence between the (long-wavelength) redshift of plasmon resonance as the NP size grows, along with the properties of the medium in which the NPs are immersed. Below we will follow the rule of “resonance shifts with increasing particle size” adhering to the classical consideration when registering the pattern of the size-dependent plasmon resonance shift [10].

In the size range of plasmonic NPs between 10 and 30 nm, a subtle, nearly linear size-dependent trend with a negligible redshift is detected in experimental observations, transforming into a monotonically increasing long-wavelength shift with further growth of particle size outside this range (larger than 30 nm) due to retardation effects [10]. However, in the size range below 10 nm, this dependence changes drastically. A remarkable display of the anomalous behavior of the plasmonic maximum (λ _max) in the ultrafine size range is showcased through the experimental relationship between λ _max and NP size, illustrated, e.g., in [11] or in [12], focusing specifically on gold NPs. The review of works related to this topic includes numerous publications [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29] among them, size dependence of redshift in ultrafine plasmonic NPs, e.g., [30], [31], [32], [33].

Theoretical interpretations of experimental data on the size-dependent plasmon resonance position of spherical NPs are particularly challenging in the ultrafine size range of 2.5–10 nm. In this range, classical Mie theory predicts that the plasmonic maximum position should be independent of the particle size, leading to significant deviations from experimental observations, as shown in studies such as [17], [34], [35] (silver NPs) and [11], [12] (gold NPs). Experimental data for both gold and silver NPs indicate an anomalous behavior in the ultrafine size range, which is dominated by evidence of a considerable redshift of the plasmon resonance with growing particle size up to 10 nm, as observed, for example, in silver NPs [17], [34], [35], [36]. The concept of redshift of the surface plasmon resonance in ultrafine metallic NPs with increasing size is discussed in many papers, e.g., [11], [12], [19], [30], [35], [37], [38], [39], including early publications from the 1970s, e.g., [13], [14], [40].

However, the ultrafine size range continues to present a significant challenge in terms of prediction and exploration, despite extensive research efforts over the past few decades and the use of a wide range of analytical tools. These tools encompass classical Mie theory including its “core–shell” modification [41], the quantum hydrodynamic theory, and TD-DFT method [29]; furthermore, models based on Thomas-Fermi theory [42], [43], the Feibelman method [44], and a combination of the Mie core–shell model with the Feibelman method [22], discrete interaction models like DDA [45], [46] and Ex-DIM [31] among others. In Ref. [31] interpretation of anomalous red shifts is discussed in terms of chaotization and local anisotropy of environment in Ex-DIM.

Our research endeavors to investigate the underlying patterns influencing the spectral positioning of plasmon resonance in ultrafine NPs by considering critical factors such as size, morphology, variations in the dielectric properties of the particle material, and the radial distribution of these properties, including the surface layer. Our goal is to construct a comprehensive model with robust predictive capabilities across a broad size range that effectively explains changes in the NP resonance characteristics as NP size approaches 10 nm and below. Specifically, we aim to address the notably rapid redshift of the plasmon resonance observed as particle diameter increases within this size range by taking into consideration the spill-out effect, volume compression, and their combined manifestation under different experimental condition.

2 Model

2.3 Volumetric compression

A distinctive characteristic of ultrafine metal NPs is their capacity to shrink and occupy a reduced volume in comparison to a fragment of a bulk sample containing an equivalent number of atoms. This phenomenon arises from crystal lattice contraction, which depends on the particle size: the smaller the particle, the more pronounced the lattice contraction, e.g., [17], [30], [55].

In the general case Δ V / V = R 0 3 − R 3 / R 0 3 , ΔV = V ₀ − V, where R is the radius of a real particle with contracted lattice, V is its real (reduced) volume, and R ₀ and V ₀ are the radius and volume of a particle with the uncompressed bulk crystal lattice and with the same number of atoms (ΔR = R ₀ − R).

The particle compression results in the growth of electron density and, consequently, in an increase in plasma frequency. The high sensitivity of the plasmon resonance position to the particle size in the ultrafine range can be attributed to this effect.

The size-dependent particle compression is just one component of the overall process, which should be considered along with the electron spill-out effect discussed above. This phenomenon inevitably accompanies particle compression and transforms ultrafine particles into core–shell structures. These NPs exhibit a surface layer with a smoothly varying electron density. This layer is produced due to the spill-out effect, where conduction electrons dynamically leave and return to the particle surface, potentially leading to a time-averaged reduction in electron density within this layer [25], [36], [56]. Such a layer, in general, is present in any particles regardless of size, and also near the flat interface of bulk samples. The surface layer exhibits different dielectric constants compared to a bulk sample.

The thickness of the surface layer is typically of the order of the lattice parameter. For particles in the tens of nanometers size range, the relative volume of the surface layer approaches zero, making its impact negligible. However, when the volume of the surface layer is comparable to that of the particle itself [31], its impact becomes significant and warrants consideration. The Mie model featuring a core and multilayer shell serves as a powerful tool for illustrating the substantial impact of the surface layer.

In our research, we investigate the impact of two interplaying processes: the size-dependent volumetric compression of ultrafine Ag NPs and the spill-out effect. Figure 1 provides a schematic representation of the key processes involved and their impact on the spectral position of the plasmon resonance in particles of different sizes. It demonstrates that a consideration of these processes (particle compression and spill-out effect) leads to substantial modifications in the electron density distribution across the transverse direction of the particles, as opposed to neglecting them (dashed contours). The top row illustrates an increasing contribution to the electron density (Δn _e ) as the particle size decreases, while the bottom row depicts the blurring of the electron density function at the particle boundaries, which is independent of their sizes.

Figure 1:

Schematic illustration of two essential processes in ultrafine metal NPs: the size-dependent volumetric compression and the spill-out effect, as well as the influence of these processes on the electron density distribution within the particles and the spectral position of the localized surface plasmon resonance (LSPR) maximum.

3 Results and discussion

Figure 2 illustrates the rapid compression of Ag NPs as their size decreases. In particular, the relative volume compression (ΔV/V) of an Ag particle with a radius R = 1.5 nm reaches 0.075 (using experimental data from [30], [55]) with respect to an Ag particle with the same number of atoms but with a bulk crystal lattice (lattice parameter is 0.4086 nm).

Figure 2:

Size dependencies of the relative volume of ultrafine Ag NPs under crystal lattice contraction strengthening as the NP diameter (2R) decreases (using experimental data from [30], [55], where NPs are embedded in argon and glass matrices); ΔV/V = 0 corresponds to a bulk. In the inset, the dashed circle schematically represents the boundary of the region within the crystal lattice of bulk containing an equivalent number of atoms as the NP depicted at the center of the circle.

Interestingly, under identical conditions in a glass matrix, an Ag particle with a radius of 1.5 nm experiences significantly less shrinkage, reducing by a factor of only 1.048 compared to free particles in an argon environment, with which the interaction is negligible. The vertical dashed line corresponds to the minimum Ag particle radius for which experimental data on lattice contraction in different environments have been obtained [30], [55]. The relation 1 + ΔV(R)/V =  n e ( R ) / n 0 ( b u l k ) represents the multiplier to the square of plasma frequency ω p 2 in the Drude model in Eqs. (2,3) used for computations in particles with different radii.

The considerations outlined above address the experimental conditions for registration of changes in lattice parameters resulting from the volumetric compression of NPs using high-resolution transmission electron microscopy. These conditions encompass a range of scenarios, including free NPs in gas or liquid environments, as well as NPs embedded in solid dielectric matrices like glass. In the latter case, observations indicate an increased resistance of NPs to compression, potentially leading to complete cessation of contraction or its noticeable suppression [17], [30], alongside a lack of size-dependent variations in the plasmon resonance shifts [30], [34]. This phenomenon is attributed to the interaction between atoms of metal and matrix atoms, where stronger interactions result in reduced NP shrinkage, accompanied by a decrease in the local anisotropy of the environment of surface atoms.

Consideration should also be given to the potential occurrence of the reverse process associated with partial stretching of particles and their deformation within the matrix, particularly if they have previously experienced shrinkage before being immersed in this medium. This is determined by the specific matrix type, by the synthesis method of the composite medium, and by the process of incorporating NPs into this medium. A discussion regarding these issues is touched upon in Refs. [17], [57].

The contraction of the crystal lattice within a metal NP can be increased, particularly near the boundary layer of atoms. Even a minor reduction in lattice parameters additionally alters the dielectric properties of the surface layer by redistributing the conduction electrons to this region. However, as a general trend, smaller plasmonic particles experience stronger compression [30], [55], [58].

Figure 3 shows the variation of optical constants n and κ in NPs of different sizes from ultrafine range, considering the increase in electron density in them. In addition, the Drude model (Eqs. (2,3)) for ϵ′ and ϵ″ used for bulk silver (regarding the relations n =  ϵ ′ 2 + ϵ ′ ′ 2 + ϵ ′ 1 / 2 / 2 , κ =  ϵ ′ 2 + ϵ ′ ′ 2 − ϵ ′ ) 1 / 2 / 2 was verified by the experimental data [49] and showed satisfactory agreement.

Figure 3:

Spectral dependencies of the Ag optical constants (n, κ) are the real and imaginary parts, respectively) in the wavelength range of the plasmon band for Ag NPs with radii from 1.45 to 8.9 nm as a result of the compression effect n _e (R) (calculation with the Drude model using data from [30], [55]). Verification of the Drude model for a bulk sample by comparison with experimental data by McPeak (McP) [49]. Color bar shows the values of the particle radius.

Figure 4 shows characteristic radial dependencies of electron density in ultrafine Ag NPs, considering the spill-out effect at the interface and the compression effect with different values of electron density in the particle center (r = 0). These results reveal that the blurred part of the electron density function remains consistent regardless of the NP radius.

Figure 4:

Blurring of the electron density distribution in the radial direction n _e(r) near the boundaries of Ag NPs (vertical dashed lines) with radii from 1.5 to 4.0 nm due to the spill-out effect accompanying compression of the particle.

Figure 5 shows a series of plasmonic extinction spectra of ultrafine Ag NPs ranging to 8 nm, considering particle size-dependent compression and spill-out effect accompanying compression. In addition to the pronounced increase in the plasmonic maximum, Figure 5 demonstrates a significant redshift with increasing particle size. The computations used the Mie “core-multishell” model [41].

Figure 5:

Evolution of plasmonic absorption spectra of Ag NPs in vacuum ranging in radius (R) from 1.5 to 4 nm, considering the particle size-dependent compression and spill-out effect accompanying compression (the maxima are marked with circles).

Figure 6 summarizes the primary findings of our study, illustrating the relationship between the plasmonic maximum position of free Ag NPs and their size within the ultrafine range. This figure shows the individual quantitative contributions of various influencing factors to this correlation, as well as their cumulative impact (black solid circles). This resulting curve shows good agreement with the experimental data (Figure 7), while the classical Mie computations exhibit significant divergence from the measured values. The compression factor in Figure 6 is marked with red triangles, the spill-out effect corresponds to circles, and green rhombi show a calculation with the Mie model.

Figure 6:

Dependence of the spectral position of the plasmon maximum in ultrafine Ag NPs on their size in vacuum, considering the joint impact of the compression and the spill-out effect, and demonstrating the individual contribution of each of these factors: the particle compression without spill-out effect and the spill-out effect ignoring the compression.

Figure 7:

Experimental (dots) and computed (solid red lines) dependencies of the plasmon maximum in ultrafine Ag NPs on their size in argon and glass matrices [34] (refractive indices are 1.27 [59] and 1.52 [34], correspondingly). The redshift values are in the same size range are Δλ _(gl) = 5.2 nm in glass matrix, and Δλ _(Ar) = 10.86 nm in Ar matrix. Experimental data on lattice contraction are taken from Ref. [30], [55]. The vertical dashed lines denote the minimum (2R = 2.9 nm) and maximum (2R = 10.0 nm) values of the particle size range for which experimental evidence supports a reduction in lattice parameter dependent on the particle size in both media.

The redshift, shown in Figure 6, reveals a significant correlation between the particle size and the observed spectral behavior. The redshift value of the plasmon resonance determined in the investigated size range Δλ _max = 12 nm aligns with experimental data for ultrafine gold NPs within the same size range obtained in hydrosols using optical spectroscopy in Ref. [12] and in Ref. [11] with the value of redshift Δλ _max = 10 nm. An additional factor that could potentially contribute to an increase in the calculated redshift value within the ultrafine size range is the strengthening lattice contraction in the radial direction toward the NP boundary.

These results should be considered when comparing with the experimental data and their interpretation. In particular, [36] gives contradictory examples of red- and blueshifts of the plasmon resonance in ultrafine Ag NPs due to the spill-out effect, with a statement that this effect can produce a redshift as well. However, Figure 6 shows that the exclusive contribution of this effect leads only to a blueshift. The same conclusion is drawn in [29], where a blueshift of the plasmon resonance was demonstrated as the ultrafine particle size increased, accounting for the spill-out effect. The computations in this paper were carried out using two methods – quantum hydrodynamic theory and TD-DFT – yielding nearly identical results for particle sizes below 10 nm.

Figure 7 shows measured dependencies of the plasmonic maximum in ultrafine Ag NPs on their size in argon and glass matrix [34] and a comparison with the results of our computations. The primary outcome highlighted in the computations reveals a 50 % reduction in the redshift value within the glass matrix when compared to the argon matrix, aligning closely with the observed trend in the experimental data. Unfortunately, the experimental data showcased in Figure 7 and utilized in the computations originate from diverse sources, resulting in a conspicuous dispersion of the compared values. The spectral dependencies are from [34], while the data on the particle lattice contraction in matrices are from [30], [55]. Note, that there is a lack of available literature that provides data on both dependencies under the same experimental conditions.

The paper by Cai et al. [30] provides evidence supporting the capacity of an argon matrix to facilitate unobstructed compression of Ag particles, equating them in such a medium to free particles, and considering the local porosity of solid argon, which is interpreted as an ideal “vacuum shell” [60]. Based on the observed dependency in an argon medium, the authors of the paper [30] conclude that the particles may be considered free. It is also important to take into account that the shear modulus of solid argon is 29 times smaller than that of glass [61]. Additionally, it is notable that the optical constants of silver experience minimal alterations when subjected to cryogenic temperatures significantly below 300 K [62] with a close to zero temperature-dependent frequency shift of plasmon resonance [63].

The notable decrease in the correlation in Figure 7 between the spectral peak position and particle size within a glass matrix, as opposed to the conditions in argon, implies a reduced impact of particle compression and a more substantial role of the spill-out effect. In certain scenarios, these processes can effectively offset each other.

Considering the significance of the issue at hand, we employed an additional alternative approach in our research based on a combination of the Mie core–shell model with the Feibelman method to compute the plasmon spectra of ultrafine NPs based on data from [22].

Figure 8 illustrates the relationship between the plasmonic maximum and the size of Ag NPs, as computed by the Feibelman method embedded in the Mie model (the curve marked with solid circles). Note that this analysis does not incorporate the size-dependent compression of NPs as it was done in Figure 6. The depicted results reveal a discrepancy with the anticipated size-dependent redshift observed in the experimental data. The computed dependence shows a weak correlation with the particle size in the ultrafine range and a slight nonmonotonicity.

Figure 8:

Relationship between the plasmon peak of ultrafine Ag NPs and their size in solid argon medium: evaluation using the Feibelman method incorporated into the Mie core–shell model. The Feibelman parameters were taken from Ref. [22] were used. We compare between computations without considering nanoparticle compression (solid circles) and those accounting for compression (hollow circles), n = 1.27.

While the Feibelman method [44] and its combined version [22] offers an alternative approach to account for the processes near the surface of ultrafine plasmonic NPs compared to the Mie core–shell model, its application falls short in replicating experimental outcomes. This inconsistency stems from overlooking a crucial accompanying process in the analysis, specifically particle size-dependent compression. In particular, considering the influence of size-dependent compression results in a modification of the relationship marked with the hollow circles in Figure 8 makes it resemble the curve represented by triangles in Figure 6.

The discrepancy between the experimental findings demonstrating a size-dependent redshift and the results obtained through the quantum hydrodynamic theory and TD-DFT analysis in Ref. [29], which showcases a blueshifted plasmon resonance under the same conditions, is probably also due to the neglect of particle compression.

It is important to emphasize that available experimental data for silver NPs measured by EELS [35], [36], [39] consistently demonstrate the same trend in redshift in the plasmon resonance as in Figures 6 and 7 when the size of ultrafine Ag NPs increases. However, all experimental discrete values obtained exhibit an equal shift toward longer wavelengths [35], [36], [39]. This shift can be attributed to the influence of the dielectric substrate made of silicon nitride [19] on which the NPs are located in the experiments, which has a rather high refractive index n ≈ 1.7–2 [64] (see Figure 1 in the Supplementary Material). This results in a redshift in the plasmon band [65], although the substrate effect [66] is less pronounced compared to the impact of a solid matrix with a similar refractive index.

4 Summary and concluding comments

To summarize our study, we here outline its primary outcomes:

1. Abnormally rapid redshifts of the plasmon resonance in ultrafine metal NPs that occur as their size increases can be attributed to changes in the electron density in the volume of the entire particle, and locally near the surface, which ultimately affects the dielectric constants of the particle material;

2. Ultrafine metallic NPs shrink considerably compared to the volume occupied by an equivalent number of atoms in a bulk crystal lattice with the same number of atoms. The contraction of their crystal lattice results in an increase in the electron density and the plasma frequency;

3. The phenomenon causing a reduction in electron density within the surface layer is the spill-out effect. This effect results in the transformation of the particle into a core–shell structure, whose resonance properties depend on the dielectric constants of each component of the particle. An important feature of the spill-out effect is its independence of the particle size when the surface layer thickness remains constant due to the Debye screening;

4. The impact of the surface layer on the plasmonic properties of NPs intensifies as their size decreases;

5. Both processes – the particle compression and the spill-out effect play crucial role in ultrafine plasmonics. The interplay between these phenomena dictates the spectral position of the plasmon resonance and highlights the intricate relationship between size, electron density, and dielectric constants determining the resonance properties of NPs;

6. In scenarios where the electron density in ultrafine NPs is weakly correlated with their size, the spill-out effect can compete with volumetric compression and even dominate, reversing the pattern of size dependence of plasmon resonance on the particle size. However, an essential inquiry arises regarding the relative impact of the compression and spill-out effects and their combined influence on the NP resonance properties. Our study has quantified the distinct contributions of these competing processes and has determined how the rebalancing between them may impact the plasmon resonance;

7. The increased contraction of the crystal lattice of smaller metal NPs always results in a redshift of the plasmon resonance. However, if the lattice is resistant to compression, the spill-out effect can lead to decrease in electron density in the surface layer, potentially causing a blueshift or no-shift as the particle size grows. An instance illustrating the occurrence of compression resistance in ultrafine metallic particles is when they are embedded within solid dielectric matrices, as opposed to liquid or gaseous media. At the same time, it is quite possible that the spill-out electrons from the surface layer of a metallic particle can penetrate into the dielectric matrix, as in vacuum, at the average interatomic distance, forming near the particle boundary a shell with reduced electron density. These conditions can introduce discrepancies in the experimental data on the size dependence of the spectral shift of the resonance when experimentally studying the same object. It is notable that the decrease in the lattice parameter of ultrafine NPs primarily arises from the existence of uncompensated interatomic bonds near the nanoparticle surface. Surface tension forces have a negligible impact on the compression process [67], [68], as only surface atoms are involved, influencing the particle shape rather than its volume. Compression occurs when all the atoms, including those beneath the surface, interact to produce forces directed toward the particle center;

8. When analyzing the optical characteristics of plasmonic NPs below 10 nm in size, it is crucial to incorporate the additional size correction factor to the dielectric constant. This correction should address not only the contribution of the surface scattering effects in the electron damping constant (ΔΓ(R) ∼ v _F /R) but also account for the size-dependent variation of the electron density within the nanoparticle (Δn _e (R));

9. The Ex-DIM method described in our paper [31] revealed the presence of a surface layer in ultrafine gold plasmonic NPs. In addition, it revealed a redshift of Δλ _max(R) = 13 nm of the plasmon resonance with increasing particle size from 3 to 10 nm, which was discussed in terms of dipole chaotization and local anisotropy of the environment of light-induced atomic dipoles near the surface. However, additional research was necessary to define the physical mechanisms driving this transition, the task that we have pursued in the study presented herein;

10. In our research, we concentrated on the dominant factors influencing the plasmonic properties of the NPs, forming the foundational basis of the phenomenon at hand. Through a meticulous examination of these factors, we successfully established a satisfactory correlation with the experimental data.