Plasmonic nanotechnology for photothermal applications – an evaluation

1 Introduction

With an ever-increasing demand for energy and the inevitable reduction in the dependency on fossil fuels, global energy demand looks to solar power to be a significant provider for its needs, with various solar power conversion technologies in place and rapidly progressing [1]. Electromagnetic radiation, when interacting with a material can transfer energy to its atoms, eventually converted to heat through a series of energy-loss processes. This conversion of electromagnetic energy into heat is called the photothermal (PT) effect. Early stages of the PT effect were initially observed in semiconductors [2], after which researchers started to explore various material phenomena other than bandgap absorption for heat generation in nanoparticles (NPs), leading to a rapid proliferation of materials for the same. For example, organic materials undergo rapid internal relaxation by the PT effect and are often desired in cancer treatment research as they cause little damage to adjacent healthy tissues due to extremely localized heating [3]. Generally, the reduction of material dimensions to the nanoscale, such as in graphene, carbon nanotubes (CNT) and polymers, leads to an enhancement of the PT effect due to factors such as improved thermal conductivity, tunability of materials for realizing broadband energy absorption, appearance of new mechanisms of photon absorption, and improved prospects of preserving material properties [4-6]. Nanoparticle heating can result also due to the conversion of optical absorption by plasmons into heat. This phenomenon of surface plasmon resonance results from the interaction between electromagnetic radiation and typically high-valence materials, leading to oscillations of the free electrons in it. The decay of these collective oscillations into heat is the plasmonic photothermal (PPT) effect. The absorption characteristics such as the wavelength in plasmon resonance can be tuned and controlled by the properties of the nanoparticle such as size, shape, proximity to other particles, as well as the surrounding medium [6]. Indeed, advancements in such manipulation at the nanoscale has aided the use of plasmonic materials [7,8], such as Au nanoparticles (AuNPs), in photodynamic therapy [9-11]. Metal nanoparticles in general have been extensively explored in PPT applications due to their high free electron density and the possibility of intricate tuning of light absorption [12]. Noble metal nanoparticles with resonances in the UV–vis–IR part of the electromagnetic spectrum are especially researched on for PT applications [13], with excellent reviews on materials for mid-IR applications [14], cancer treatment [15], antibacterial research [16], solar-driven vapour evaporation [16], solar collectors [11,17,18], catalysis [19], clean water production [15], and wearable heaters [20,21], to name a few. This review is on PPT nanoparticle research spanning the conventional options (metals and alloys) as well as materials with induced plasmonic properties, with a special emphasis on their stability in terms of temperature and reactivity. With broad applications in therapy [22,23], laser combined imaging, solar vapour generation [24], and biosensors [25], the global market for PT devices is expected to be a multimillion dollar enterprise by 2025 [26]. This review will focus on concepts such as the theoretical aspects of PPT energy conversion (which influence material selection and design), studies on different classes and morphologies of nanomaterials that have been investigated for different applications of PT conversion, and the thermal and chemical stability of PT nanomaterials, which need to be considered prior to making the final choice. We conclude with a broad perspective on current research, challenges that remain to be solved, as well as prospects in terms of material design and deployment for better exploitation of such nanostructures for PT energy conversion.

2 Plasmonics in PT conversion

Of the incident radiation from the sun, 8% UV, 42.4% visible light, and 49.6% infrared radiation reach the earth's surface. Applications such as steam generation from solar power can evidently benefit from the use of materials that can absorb as much as possible of the entire spectrum of solar radiation. In this regard, plasmonic nanomaterials with tunable energy absorption can help, and the tunability of such materials only manifests at the nanoscale as changes in the absorption of incident radiation. This tunability is of utmost benefit for PT applications as the region of the electromagnetic spectrum that is not absorbed by generic PT materials can be utilized for absorption and eventual conversion into heat by incorporation of plasmonic nanoparticles of appropriate sizes and shapes. Plasmons, that is, collective electron excitations, are either excited in the bulk of the material (volume plasmons) or on the surface through excitations of the conduction electrons (surface plasmons) as shown in Figure 1. Such excitations, when occurring in nanoparticles, are termed localized surface plasmon resonances (LSPRs) as they are confined within the boundaries of the nanoparticle (in the case of continuous films, they are propagating oscillations termed surface plasmon polaritons (SPPs)). A plasmon in LSPR can be visualized as a quasiparticle confined to the volume of the nanoparticle [27]. The resulting confinement of the absorbed incident electromagnetic radiation within the nanoparticle thus means an effective localization of the incident photon energy, and the decay of this oscillation (through phenomena such as electron–electron, electron–phonon, and electron–surface scattering) releases the absorbed energy into the lattice as heat (or as photons), often making them efficient tunable PT energy materials [24,25]. Interaction of electromagnetic radiation with a material can lead to absorption, transmission, or scattering. Regarding scattering, elastic and inelastic scattering are the major classifications. Elastic scattering means conservation of the photon energy, in inelastic scattering, there are processes other than complete absorption through which photon energy can be transferred to a material. Elastic scattering is not relevant for PT applications as there is no transfer of energy into the material for heating. Absorption/inelastic scattering of electromagnetic radiation can lead to electronic, translation, vibration, and rotational transitions. The interaction time period of electromagnetic radiation with electrons is around 10−14 to 10−15 s. SPR falls within the regime of electronic transitions and, generally, electronic transitions can be interband as well as intraband transitions. When the energy of the photons is greater than the bandgap, interband transitions are observed. As an example of the energies at which interband transitions [28] occur, Cu, Au, Ag exhibit them at 2.25, 2.4, and 4 eV, respectively, and threshold energy levels of interband transitions are 1.6–1.8 eV for Cu, Au, and Al, as well as 3.5 eV for Ag [29]. Concerning PT applications, radiative transitions such as luminescence and scattering imply inefficiency, as this scattered energy is not converted to heat. Hot electron generation and subsequent thermalization are consequences of SPR absorption that can lead to heat generation, depending on whether the decay of the SPR is through radiative or non-radiative processes. Many metals show plasmonic properties, but for PT applications there is a specific set of requirements including, but not limited to, broadband absorption of electromagnetic radiation, specifically in the UV–vis range (as infrared is already applied for heating), efficient of conversion of the absorbed energy into heat (in contrast to scattering), chemical and physical stability of the nanoparticles (e.g., against agglomeration), ease of synthesis, and low cost. Coinage metals, such as Au, Ag, and Cu, with high densities of free electrons exhibit plasmon resonances in the visible region suitable for PT applications [30]. The subsequent part of this review highlights the parameters that influence various properties of plasmonic materials relevant to PT energy conversion. We first derive the equation quantifying the absorption frequency of plasmons, followed by a discussion on the changes to this frequency that can be induced by changing the governing parameters of this equation, and conclude with a few examples that model the optical scattering properties of generic morphologies such as spheres and nanorods.

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Figure 1: Schematic representation of surface plasmon resonance (SPR) excitation. (a) SPR wave or surface plasmon polariton (SPP). (b) Localized SPR (LSPR) in a spherical nanoparticle and the associated absorbance spectrum. Figure 1a,b was used with permission of The Royal Society of Chemistry, from [31] (“Portable and field-deployed surface plasmon resonance and plasmonic sensors”, J.-F. Masson, Analyst, vol. 145, issue 11, © Copyright 2020); permission conveyed through Copyright Clearance Center, Inc. This content is not subject to CC BY 4.0.

2.1 The plasmonic oscillation frequency

The optical response of plasmonic nanoparticles, such as AuNPs, to incident electromagnetic radiation depends on their size, shape, morphology, proximity to one another, as well as the surrounding medium [32]. The vast changes in absorbance due to changes morphology stem predominantly from changes to the directionality of the LSPR (due to changing curvature and dimensionality of the nanoparticle) and changes to the dynamics between the restoring and exciting forces of the plasmons, such as the mean free path, the relative contributions to the plasmon damping of different scattering phenomena, the different scattering processes of the oscillating plasmons, and the screening between the plasmons and the restoring nuclear forces [33,34]. The proximity among LSPR-active nanoparticles is also a major factor. Indeed, combined effects of proximity as well as morphology influence considerably the LSPR properties, for example, in Au nanorods and nanospheres. In contrast, nanospheres and nanorods exhibited considerable tunability of the LSPR due to changes to the localized electromagnetic field of the plasmons due to changing curvature [35]. Finally, changes to the material composition, such as through doping or vacancy processing, can affect the LSPR because of changes in the free electron density, the electron effective mass, and the electronic band structure in general [36,37]. An understanding of the changes in absorbance with respect to changing parameters of the material under consideration can be developed with a few examples of the theoretically arrived optical cross sections of a few generic morphologies of nanoparticles and will be discussed next for the case of nanospheres, nanorods, and nanomatryushkas.

The arrival at the expression for the LSPR frequency of a free electron cloud (as is typically assumed to be present in metals) starts with the relations between the dielectric displacement (D) of the electron gas in relation to the incident electric field (E) which it is [38] subjected to, given by

[2190-4286-14-33-i1](1)

wherein P is the polarization density. P can be arrived at by solving the equation of motion for a single electron as

[2190-4286-14-33-i2](2)

Hence the expression relating the dielectric displacement (D) and the external electric field can be obtained as

[2190-4286-14-33-i3](3)

where [Graphic 1] is the natural frequency of oscillation of the electron cloud. Comparing Equation 3 to the general constitutive relation for a linear isotropic material given by Equation 4, we get the relation in Equation 5. εr is the relative permittivity of the material and ε0 the permittivity of free space.

[2190-4286-14-33-i4](4) [2190-4286-14-33-i5](5)

For frequencies close to ωp, the temporal duration of damping (quantified by the product of ωτ, where τ is the relaxation time of the free electron gas) is much higher than unity, thus leading to an approximation that there is no damping. Hence ignoring the damping term in Equation 5, we get

[2190-4286-14-33-i6](6)

It follows also that under plasmon resonance conditions ε1 < −εm, εm is the dielectric constant of the surrounding medium and hence the LSPR frequency can be arrived at as [39]:

[2190-4286-14-33-i7](7)

Thus, for any morphology of a plasmonic nanoparticle, the LSPR frequency is intimately tied to the free electron density and the dielectric constant of the surrounding matrix. These factors thus decide the shape, position, and width of the plasmonic absorption and will be further elaborated in the subsequent sections.

Plasmon absorption is also determined by the nanoparticle shape, which, although it does not appear in the plasmonic frequency equation, manifests as a shape/size factor in the calculations of the extinction spectra of nanostructures. Illustrative examples of the absorbance spectra for different morphologies of Ag nanoparticles are shown in Figure 2, elucidating the influence of the same. The extinction spectra (the summation of absorption and scattering spectra) from which the shape effects of different morphologies on plasmon excitation can be understood are hence crucial for assessing the PT properties of nanomaterials.

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Figure 2: Optical spectra (absorption – red, scattering – blue, and extinction – black) of different morphologies of Ag nanoparticles representing shape effects of (a) nanosphere, (b) nanocube, (c) tetrahedron, (d) octahedron, and (e, f) core–shell structures with different shell thicknesses. Figure 2a–f was reprinted with permission from [40], Copyright 2006 American Chemical Society. This content is not subject to CC BY 4.0.

2.1.1 Tunability of the plasmon frequency – changes to the dielectric constant. In 1834, William Whewell coined the term dielectric [41]. In a dielectric material, positive charges are arranged in the direction of electric field and negative charges opposite to the field, causing a polarisation. The wavelength-dependent electric field and dipole moment determine the dielectric property of the material and also the plasmon absorption (by affecting the polarizability, as was shown in prior sections) [42].

The real (εr) and imaginary part (εi) of the dielectric constant relate to the refractive index (n) and extinction coefficient (k), respectively. Energy from a time-varying incident electric field is dissipated in part as heat, termed as dielectric loss. It can be envisioned that this dielectric loss is hence an important attribute to be considered for PT applications. The dielectric loss (δ) can be expressed as:

[2190-4286-14-33-i8](8)

where εi is the imaginary part of the dielectric function, and εr is the real part of the dielectric function [43]. Polarization occurs for charge carriers in a number of orbitals or band, which then very often overlap, leading to multiple transitions even during plasmonic resonance. For example, metals such as Au, Ag, and Cu have d-band electrons close to the Fermi surface, and the polarization (P) in the presence of an electric field (i.e., the plasmon oscillation) has contributions from interband transitions as well. The dielectric function must account for this, and thereby the conventional definition given as [44]:

[2190-4286-14-33-i9](9)

changes (for metals at optical frequencies, for example) to

[2190-4286-14-33-i10](10)

χ∞ is the susceptibility arising from the core electron polarizability (causing interband transitions), and χD is the corresponding susceptibility of the conduction electrons (modelled through the Drude assumption of a free electron “sea”). Hence, PT applications targeting plasmonic materials must account for the contributions to the dielectric function of the terms mentioned in Equation 10. A broad absorption of wavelengths is a reinforcing attribute of plasmonic materials for PT applications. However, increased broadening is associated with a reduced absorbance due to an increased scattering of the plasmon oscillations. An important phenomenon that broadens the plasmon line width is the scattering of plasmons at the surface for nanoparticles of sizes approaching the mean free path of electrons. Apart from the minor contribution to the linewidth arising from the disparity in particle sizes (when considering the absorption of a cluster of nanoparticles), the linewidth is controlled solely by surface scattering in such nanoparticles [45]. For free electron metals, the frequency of electron scattering (γ) is equal to the width of the plasmon resonance (Γ) [46]:

[2190-4286-14-33-i11](11)

In order to account for surface scattering, the linewidth needs to be modified as,

[2190-4286-14-33-i12](12)

where νF is the Fermi velocity of electrons, Γ0 is the plasmon linewidth of the particles or damping constant, a is the radius of the spherical metal particles, and A is a parameter that depends on the scattering process. The final expression for the dielectric function taking into account the inter- and intraband transitions as well as the changed linewidth due to surface scattering (as appropriate), is

[2190-4286-14-33-i13](13)

Multiple inferences can be made from this formulation. In addition to linewidth broadening contributions to the dielectric function for smaller nanoparticles, electron orbit contractions (due to the majority of electrons being in proximity to the surface) result in an increased Coulombic force of restoration and hence a shift in the dielectric function [47]. Similarly, with a decrease in grain size, due to the fact that there is an increase in the volume fraction of grain boundaries compared to the grains, and since the dielectric strength of a grain is lower than a grain boundary, the dielectric permittivity decreases with decreasing grain size [48]. Moreover, the interaction between plasmonic nanoparticles and substrates on which they are deposited cannot be ignored. The polarization of charges in the nanoparticles induces dipoles in the substrate atoms in proximity of this polarization field, which in turn affects the nanoparticle resonance. This has been observed to induce higher-order resonances when the mismatch between the permittivity of the substrate and the surrounding of the nanoparticle increases, as well as to (depending on the orientation of the applied field with respect to the induced fields in the nanoparticle and the substrate) increase or decrease in the absorption intensity [49-51].

It follows from the discussion that considerable effects to the plasmon resonance are affected by the dielectric surrounding the plasmonic nanoparticle, apart from the permittivity of the nanoparticle itself. The shift in resonance on incorporation of the nanoparticle into a dielectric will decide its absorbance and hence the efficiency of conversion of the resonance into heat. This has been verified experimentally in multiple studies, wherein plasmon absorption peak shifts of up to 150 nm [52] have been observed with just a unit change in refractive index of the surrounding medium. As important as the dielectric function, the free electron density of the nanoparticle is also crucial for PT applications. This factor decides multiple attributes of a material, such as the quantum of this value that makes a given material plasmonic, the shifts in absorption and linewidth, and the ratio of radiative to non-radiative to radiative damping. A discussion on the manifestation of the free electron density on the plasmonic performance of different nanomaterials follows and is illustrated in Figure 3.

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Figure 3: Manifestation of the governing factors of SPR. Shown are the changes to the peak position while considering only the shape (red circles), the shape as well as the electron density (blue squares), and the combination of shape, electron density, and deformation potential (green diamonds). Figure 3 was reprinted with permission from [53], Copyright 2017 American Chemical Society. This content is not subject to CC BY 4.0.

2.1.2 Tunability of the plasmon oscillation frequency – changes to the free carrier density: Free carrier densities of electrons are in the range of 1022 to 1023 cm−3 for plasmonic metals such as Au, Ag and Cu [54]. The free electron density is tied to the effective mass and determines the resonant plasma frequency (ωp), given by

[2190-4286-14-33-i14](14)

where N is the free electron density, m is the effective mass, and e is the elementary charge. The LSPR effect is not present in most of the semiconductors because of their lack of the required free carrier concentration. Similar to how the free carrier density of metals can be tuned by size, morphology, and refractive index of the nanomaterial, the free carrier density of semiconductors can be easily tuned by doping, temperature variations, or by phase transitions. LSPR in semiconductors in the NIR–mid-IR region is possible when the free carrier concentrations lies between 1016 and 1019 cm−3[54].

As an example of how the free electron density influences the plasmon resonance when materials with different work functions are combined, metals in contact with semiconductor metal oxide nanoparticles exhibit the spill-over effect, which alters the plasmonic absorption and spectral width of the plasmonic nanoparticles integrated in dielectric matrices. This spill-over effect, however, decreases with a decrease in electron density. For nanoparticles with low electron density (typically for radii less than 10 nm), the omnidirectional diffuse scattering dominates the resonance and the spill-over effect can be safely neglected. The spill-over effect and diffusive scattering can be related by [55]:

[2190-4286-14-33-i15](15)

m* is the effective mass of an electron, rs is the radius of on-electron, dT is the complex length, [Graphic 2] is the classical Mie LSPR frequency of the sphere = [Graphic 3], and the Bohr radius is [Graphic 4]

The free electron density can be tuned to make non-plasmonic materials plasmonic, which is useful for PT applications that require materials with properties that are not present in conventional plasmonic materials, such as a higher melting point, alloying capabilities, as well as possibly lower reactivities. Morphologies with optimal PT properties in terms of optical absorbance can benefit well from prior knowledge of the same, which can be obtained quite well through modelling efforts. Modelling of the absorbance of a plasmonic nanoparticle is done by calculation of its optical cross sections and specifically the extinction cross section, which includes the absorption as well as the scattering cross section. A few examples of the same will be discussed to compare the differences that need to be accounted for regarding different morphologies, essential for obtaining the accurate results in Figure 4.

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Figure 4: Dissipation ratio of electron–hole pair loss vs phonon loss of the surface plasmon excitation for different metal oxides. Figure 4 was reprinted with permission from [56], Copyright 2015 by the American Physical Society. This content is not subject to CC BY 4.0.

2.2 Extinction properties of nanomaterials

2.2.1 Nanospheres: The interaction of light with a particle is in one way simplistically modelled using the so-called quasi-static approximation where the incident electric field is assumed to be spatially uniform. This assumption is valid only for wavelengths much larger than the particle size. The extinction cross section, which is the result of this modelling and which is an expression summing up absorption and scattering of the incident radiation, is derived starting from Laplace’s equation with an electric potential (ϕ),

[2190-4286-14-33-i16](16)

Calculation of the resulting scalar potentials inside and outside the particle leads to the expression for the polarizability (α) of the particle,

[2190-4286-14-33-i17](17)

εm is the dielectric constant of the surrounding medium. The extinction coefficient (σext) is introduced as:

[2190-4286-14-33-i18](18) [2190-4286-14-33-i19](19)

where Cext is the extinction cross section. On substituting for the polarizability, the final expression for the extinction cross section [57] for a spherical particle interacting with light is arrived at as

[2190-4286-14-33-i20](20)

where a and εm are the particle size and dielectric constant of the surrounding medium, respectively, and ε1 and ε2 are the real and imaginary parts of the dielectric function of the material.

For larger particles (above approx. 40 nm), field-retardation effects affect the resonance position due to increasingly higher radiative damping, with significantly different extinction spectra arising also from the excitation of multipole resonances, which is not captured by the quasi-static approximation as shown in Figure 5.

[2190-4286-14-33-5]

Figure 5: Extinction efficiencies of gold nanospheres calculated through the quasi-static approximation vs using Mie theory. The expected trends in the plasmonic peak shifts with particle size post the quasi-static limit can be seen to be predicted better by Mie theory. Figure 5 was reprinted by permission from Springer Nature from [58] ("Optical Properties of Metal Nanoparticles" by N. Harris et al., in Encyclopaedia of Nanotechnology, 2nd edition, Springer Dordrecht 2016, pp. 3027–3048), Copyright 2016 Springer Nature. This content is not subject to CC BY 4.0.

The Mie theory solution of the extinction cross section is used to account for this through the inclusion of a size parameter (x = 2πa/λ) as well as through a coefficient for covering partial electric and magnetic waves (for the multipole orders). The extinction cross section is then given by

[2190-4286-14-33-i21](21)

where

[Graphic 5]

[Graphic 6]

[Graphic 7]

[Graphic 8]

[Graphic 9]

Here, ψ and ξj are the Riccatti–Bessel functions, and J and Y are Bessel functions of the first and second order, respectively, m is the ratio of nm and n, the real refractive index of the surrounding medium and the complex refractive index of the spherical nanoparticles, and j is an integer representing the order of scattering (dipole, quadrupole and so on). These equations for the extinction cross section more accurately capture the absorbance for particles larger than 40 nm. Even by accounting only for the dipole mode in such particles, depending on the order to which ψj and ξj are expanded, the shift as well as the changes to the peak broadening can be captured. For example, expanding the function to the order of x2 yields [58]:

[2190-4286-14-33-i22](22)

This equation captures well the redshift in the plasmon resonance for increasing particle diameters, not observable with the quasi-static approximation alone, as shown in Figure 6 for the case of Au nanospheres. Redshift and broadening of the plasmon resonance for the simple spherical morphology, allowing tunability of the absorption characteristics, are also evident from this discussion.

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Figure 6: Absorption spectra of Au nanospheres of different diameters showing the shift of excitation wavelengths. The broadening of the SPR curve due to damping effects is evident. Figure 6 was reprinted from [59] (© 2018 H. S. Kim and Y. Lee, distributed under the terms of the Creative Commons Attribution 4.0 International License, https://creativecommons.org/licenses/by/4.0).

A common analogue to the spherical morphology are concentric nanoparticles, wherein a core nanoparticle is surrounded by a shell layer of specified thickness and composition. The change in refractive index of the shell in comparison to the core leads to a multitude of interesting effects such as altered/enhanced absorption and stability [60]. Hence core–shell nanoparticles can be tuned very effectively to the desired wavelength range by manipulating the thickness and/or composition of the shell in addition to the tunability prospects of the core itself, such as size and composition.

2.2.2 Nanorods: Nanorods in the shapes of cylinders or rectangles are a very common morphology explored in applications of sensing, catalysis, and plasmonics. Their attributes, including aspect ratio (ratio of the length to width), curvature, homogeneity, dimer formation, and placement, make them very interesting for multiple applications such as communication, hot-carrier enhanced catalysis, high-temperature sensing [61], and electronics applications such as transistors. Their extinction coefficient is often calculated using Gans theory for randomly oriented nanorods (with the rod geometry assumed as prolate spheroids with three principal axes) in the dipole approximation:

[2190-4286-14-33-i23](23)

where εm is the dielectric constant of the surrounding medium, Pj is the depolarization factor along the A, B, and C axes, and R is the aspect ratio of the nanorods (B/A or width/length). For example:

[Graphic 10]

[Graphic 11]

[Graphic 12]

Because of the distinct and significant changes to the polarization of the electron cloud with respect to the elongation (aspect ratio) of the rods and any such rod-like nanostructures, considerable changes to the absorption properties are observed. This is because in addition to the polarization changes, there are also changes in the scattering processes (radiative vs non-radiative) of the plasmon oscillations. It follows that elongated nanostructures of plasmonic nanoparticles are more conducive for PT applications than spherical particles due to the precise and disparate manipulations possible in the absorbance of the former.

2.2.3 Matryushka: Multilayered nanoparticles have interestingly different optical characteristics than their single or non-layered pristine counterparts. Taking multiple forms such as core–shell, sandwich structures, and films, such multilayered nanostructures exhibit tunable plasmon resonance bands, resulting in additional shifts due to plasmon hybridization [62]. As an example, the interaction and hybridization of the plasmons of two separate metal shells (nanomatryushka) and two core dielectrics constitute the nanomatryushka (simplistically viewed as a ring within a ring). The coupling strength and energy between the plasmons on the inner and outer shells will determine the plasmon properties, which can be tuned by altering the dielectric spacer layer as well as the thickness of the specific nanoshells. Because of this intensified interaction between inner-core and outer-shell plasmons, PT energy transduction is significantly more effective.

The absorption coefficient (Cabs) of the nanomatryushka (NM) can be calculated by varying the volumetric factor of the different layers of the nanostructures and the refractive index of the surrounding medium [63]:

[2190-4286-14-33-i24](24)

PLSPR is the power absorbed by the nanoparticle, ε0 and μ0 are, respectively, the permittivity and permeability of vacuum, nm is the refractive index of the surrounding medium, and [Graphic 13] is the amplitude of the transverse incident wave of the laser source. Equation 24 can be simplified in the quasi-static limit as:

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