1. Introduction
Since ancient times, people have mastered the art of producing colored decorative glass, which later revealed the presence of ultrafine particles of noble metals. In the early 20th century, Gustav Mie developed a theory that explained these intriguing color effects. It was discovered that fragmenting noble metals into nanoparticles imparts unique optical properties that are not found in their bulk counterparts. However, comprehensive studies on the optical characteristics of metallic nanoparticles only began in the late 20th century, coinciding with the advent of high-performance computing. By the 21st century, these particles had already found a wide range of applications. During this period, metallic nanoparticles were referred to as plasmonic ones due to their ability to excite collective oscillations of conduction electrons, akin to electron density waves in plasma media.
One of the most significant properties of nanoparticles is their behavior under light exposure. These nanoparticles, composed of hundreds or thousands of atoms, can be likened to giant atoms. When illuminated, the conduction electrons within these particles respond to the light by synchronously shifting in relation to the crystal lattice, with the frequency of the incident light wave. As we vary the frequency of the light, we observe that at a specific frequency, the amplitude of the electron oscillations within the nanoparticles sharply increases. This phenomenon occurs when we reach a condition known as optical resonance, where the light excites collective oscillations of electrons—referred to as surface plasmon resonances. At this resonant frequency, the nanoparticle acts as a source of an enhanced optical field that is significantly greater than the incident light. This enhanced field is concentrated near the surface of the nanoparticle. The ability to enhance the local electromagnetic field is a fundamental phenomenon that underpins plasmonics, one of the most crucial areas of modern nanooptics.
The frequency of plasmon resonance is influenced not only by the type of metal but also by several other factors, with size being one of the most significant. For nanoparticles larger than 10 nanometers, the plasmon resonance frequency remains relatively stable despite increases in size. However, once the nanoparticle size reaches approximately 30 nanometers, a notable trend emerges: further increases in size lead to a pronounced decrease in the resonance frequency, resulting in what is known as a “red shift.” This shift indicates a shift towards longer wavelengths, reflecting the complex interplay between particle size and plasmonic behavior.
2. Research Backgrounds
Investigating the relationship between the resonance properties of nanoparticles and their size, shape, and environment has emerged as a crucial research focus in this rapidly evolving field. Given their significance in various applications, it is essential to develop robust models that can accurately predict and interpret the optical properties of plasmonic gold and silver nanoparticles across a broad range of sizes. Such models will enable the design of ultrafine plasmonic structures with predetermined characteristics, thereby expanding their application potential.
Current theoretical frameworks, particularly the Mie theory—recognized as one of the most precise—successfully align with experimental data regarding the dependence of plasmon resonance frequency on nanoparticle size for particles larger than 10 nanometers. However, in the sub-10 nanometer size range, this relationship undergoes a dramatic shift, as experimental findings reveal a rapid, size-dependent resonance shift that existing theoretical models, whether classical or quantum mechanical, fail to account for. This specific size range of plasmonic nanoparticles has remained enigmatic for decades, presenting a significant challenge for prediction and analysis despite extensive research efforts and the application of various analytical techniques. These techniques encompass classical Mie theory, including its core-shell modifications, quantum hydrodynamic theory, time-dependent density functional theory (TD-DFT), and models based on Thomas-Fermi theory. In addition, researchers have applied the Feibelman method, hybrid approaches combining the Mie core-shell model with the Feibelman method, and discrete interaction models such as the discrete dipole approximation (DDA) and extended discrete dipole approximation (Ex-DIM), among others.
3. Innovative research
What lies behind the enigma of ultrafine metallic nanoparticles, particularly those smaller than 10 nanometers, that renders them inadequately described by existing models — both classical and quantum? The key issue stems from the fact that as nanoparticles decrease in size, their crystal lattice undergoes significant contraction. This contraction leads to a structure that markedly differs from that of a bulk material containing the same number of atoms. For instance, a 3-nanometer silver nanoparticle exhibits a compression of approximately 7-8 percent compared to its bulk counterpart (see Figure 1). Consequently, bulk gold and gold in nanoparticles are to some extent different materials, exhibiting distinct properties. Only when particles exceed sizes of 10 to 12 nanometers they revert to a crystal lattice structure comparable to that of conventional bulk material.
Figure 1: Size dependencies of the relative volume of ultrafine silver nanoparticles under crystal lattice contraction strengthening as the nanoparticle diameter (2R) decreases (using experimental data where nanoparticles 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 nanoparticle depicted at the center of the circle (ΔR is difference in radii).
Additionally, there is a phenomenon known as the spill-out effect. This occurs when conduction electrons near the surface of a nanoparticle — whether flat or spherical — due to the high value of its momentum and velocity can partially jump out of the geometric boundary of the nanoparticle, only to return afterward. As a result, a dynamic electron cloud exists above the particle’s surface, where electrons continuously leave and re-enter. In essence, the metal does not terminate at the outer layer of atoms; instead, its boundary becomes diffuse due to the spill-out effect, which also accompanies by a reduction in electron density in the surface layer of nanoparticles (see Figure 2). This alteration significantly affects the material characteristics of the surface layer compared to the core region of the particle, which itself is affected by size-dependent compression.Together, these factors exert a profound impact on the resonant plasmon frequency.
Figure 2: Blurring of the electron density distribution in the radial direction (ne) near the boundaries of silver nanoparticles (vertical dashed lines) with radii from 1.5 to 4.0 nm due to the spill-out effect accompanying compression of the particle.
It is important to note that the thickness of the electron-depleted surface layer in nanoparticles does not exceed the crystal lattice parameter and remains independent of nanoparticle size. While such a layer is also present in larger particles, its volume is negligibly small compared to that of the nanoparticle itself, rendering its influence on optical properties minimal. However, when the radius of an ultrafine nanoparticle approaches the thickness of this surface spill-out layer, it begins to significantly affect the resonance properties of the nanoparticles.
Consequently, we observe two opposing effects that strongly influence the plasmon resonance frequency of ultrafine plasmonic particles. As the particle size decreases, the compression effect results in an increase in resonance frequency, while the spill-out effect causes a decrease under similar conditions (see Figure 3).
Figure 3: Dependence of the spectral position of the plasmon resonance maximum (its wavelength) in ultrafine silver nanoparticles 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 4 illustrates the key processes involved and their impact on the spectral position of plasmon resonance across different particle sizes.
Figure 4: Schematic illustration of two essential processes in ultrafine metal nanoparticles: 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.
In our study, we investigated the interplay between these two competing processes and quantified their combined effect on the resonant frequency of ultrafine plasmonic nanoparticles as a function of size. Our findings indicate that the volumetric compression factor of the nanoparticles predominates over the opposing influence of the spill-out effect, which aligns with experimental observations.
The primary conclusion of our study is that neglecting these two processes in optical computations leads to results that are inconsistent with experimental findings. This underscores the importance of our computations, which align closely with the experimental data, as illustrated in Figure 5. Additionally, we would like to highlight that once the nanoparticle size exceeds 10 nanometers, the effects we have described can be considered negligible.
Figure 5: Experimental (dots) and computed (solid red lines) dependencies of the plasmon maximum in ultrafine silver nanoparticles on their size in argon and glass matrices (refractive indices are 1.27 and 1.52, correspondingly). The redshift values are in the same size range are Δl(gl)=5.2nm in glass matrix, and Δl(gl)=10.86 nm in Ar matrix. 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.
4. Application and prospects
As a result, we have proposed a model that adequately describes the rapid variation in the spectral position of the plasmonic maximum of ultrafine metallic nanoparticles characterized by its long-wave shift with increasing particle size in the range from 3 to 10 nanometers. The practical significance of this model lies in its ability to facilitate the meaningful application of ultrafine plasmonic nanoparticles in various applied problems, enabling accurate predictions of their performance under dominant effect conditions. The ultrafine size range is particularly noteworthy due to its non-trivial applications in fields such as biomedical sensing, cellular imaging, cancer therapy, and the detection of molecules within living cells. Additionally, these nanoparticles play a crucial role in nanoparticle-assisted bioimaging, plasmon-enhanced fluorescence, and optical spectroscopy. By enhancing the electromagnetic field near the metal surface at nanoscale distances, plasmonic nanoparticles become indispensable tools for localized environmental sensing technologies. Their small size endows ultrafine plasmonic nanoparticles with an exceptional capacity for achieving unprecedented 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. Moreover, it is essential to highlight the biological applications of these nanoparticles. They exhibit a remarkable ability to penetrate cells through ion channels in cell membranes, which enables precise targeting of malignant cells during laser hyperthermia, ultimately leading to their destruction.
These research results are published online with the title “Unique Features of Plasmonic Absorption in Ultrafine Metal Nanoparticles: Unity and Rivalry of Volumetric Compression and Spill-out Effect” for publication in Nanophotonics, 2024.
The authors of this article are: Daniil Khrennikov, International Research Center of Spectroscopy and Quantum Chemistry – IRC SQC, Siberian Federal University, Krasnoyarsk, 660041, Russia; Victor Labuntsov, Konstantin Ladutenko, Ivan Terekhov, School of Physics and Engineering, ITMO University, St. Petersburg 197101, Russia; Andrey Bogdanov, School of Physics and Engineering, ITMO University, St. Petersburg 197101, Russia; Qingdao Innovation and Development Center, Harbin Engineering University, Qingdao 266000, Shandong, China; Hans Ågren, Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden; Sergey Karpov, L. V. Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Krasnoyarsk, 660036, Russia; International Research Center of Spectroscopy and Quantum Chemistry – IRC SQC, Siberian Federal University. All authors contributed equally to this work.