Since the early 2000s, the experimental and theoretical studies of photothermal effects in plasmonics have been mainly oriented toward systems composed of nanoparticles, mostly motivated by applications in biomedecine, and have overlooked the case of plasmonic resonances of nanoholes in metal layers (also called nanopores or nano-apertures). Yet, more and more applications based on plasmonic nanoholes have been reported these last years (e.g., optical trapping, molecular sensing, and surface-enhanced Raman scattering), and photothermal effects can be unexpectedly high for this kind of systems, mainly because of the very large amount of metal under illumination, compared with nanoparticle systems. Nanoholes in metal layers involve a fully different photothermodynamical picture, and few of what is known about nanoparticles can be applied with nanoholes. A plasmonic nanohole mixes localized and surfaces plasmons, along with heat transport in a two-dimensional highly conductive layer, making the underlying photothermodynamical physics particularly complex. This Tutorial is aimed to provide a comprehensive description of the photothermal effects in plasmonics when metal layers are involved, based on experimental, theoretical, and numerical results. Photothermal effects in metal layers (embedded or suspended) are first described in detail, followed by the study of nanoholes, where we revisit the concept of absorption cross section and discuss the influences of parameters such as layer thickness, layer composition, nanohole size and geometry, adhesion layer, thermal radiation, and illumination wavelength.
I. INTRODUCTION
Metal layers and nanoparticles, embedded in dielectric media, are the two main families of objects widely used in plasmonics. Layers exhibit surface plasmon resonances (SPRs), while nanoparticles undergo localized plasmon resonances (LPRs).1,2 On the one hand, nanoparticles (typically <100 nm) and their associated LPR have attracted strong interest for applications in bio-medicine,3 SERS (surface-enhanced Raman scattering),4 molecular (bio)sensing,5,6 nanochemistry,7–9 or solar light harvesting,8,10–12 among others. On the other hand, plasmonic layers (typically 30–100 nm thick) and their associated SPRs have led to much fewer applications, albeit at the basis of the most accomplished application of plasmonics, that is, SPR biosensing.13,14 The interest of plasmonic layers can be further enriched when drilled with nanoholes (called also nanopores or nano-apertures), either single or as a periodic array,15,16 for applications in molecular optical sensing,17,18 single molecule detection,19 optical trapping,19–25 DNA manipulation,26,27 extraordinary transmission through nanohole arrays,28,29 photovoltaics,30 SERS,31,32 nanochemistry,33 and thermoplasmonics.34–36
In all the plasmonic systems described above, the unavoidable heat generation in the metal due to light absorption is a natural concern. Depending on the application, photothermal effects can be either detrimental (for optical sensing, SERS, etc.) or, on the contrary, the targeted mechanism (cancer therapy,37 nanochemistry,38 etc.). When dealing with photothermal effects, the literature on plasmonic nanoparticles is now abundant and constitutes the field of thermoplasmonics, born in the early 2000s.39–42 Many theoretical and experimental studies have been reported on the photothermal effects of nanoparticles of many compositions, under cw or pulsed illumination, over many ranges of wavelengths and powers, and for numerous applications, in particular for bio-medicine, bioimaging, microfluidics, and nanochemistry. In contrast, photothermal effects in metal layers have drawn much less attention. In the mid-2000s, Passian and co-workers used heating of plain plasmonic layers in the Kretschman configuration as a means to modulate a light beam intensity43,44 and to study the dynamics of liquid droplets45 on a plain metal layer. Several studies have also shown that nanoholes can enhance light absorption compared with plain metal layers.34,35 However, the common belief is that any temperature increase is supposed to be seriously damped by the presence of a metal layer, acting as an efficient heat sink due to its high thermal conductivity.46 This idea held until very recently when observable temperature increases have been demonstrated in optical trapping experiments with nanoholes illuminated with a focused laser beam.47 This observation led the authors to revisit the actual mechanisms into play in plasmonic trapping experiments. Today, there is no consensual and comprehensive description of the photothermodynamics of plasmonic layers with or without nanoholes.
This lesser interest in SPR vs LPR photothermal effects is mainly due, we believe, to four reasons. First, for most SPR applications, photothermal effects are not the main process of interest. They are even detrimental. Second, they have been assumed to be insignificant,46 but this has been questioned recently.47,48 Third, photothermal effects of metal layers and nanoholes are much more complicated to address, predict, and model numerically. While very simple closed-form expressions exist to rapidly estimate the temperature increase of a nanoparticle,49,50 no such expressions have been derived so far for a metal layer. Finally, all the temperature microscopy techniques developed for plasmonics have been used in the case of nanoparticles40 and were not demonstrated for the case of metal layers, except very recently.47 This recent interest in photothermal effects for SPR application pressed a deeper description of the underlying photothermodynamics of plasmonic layers.
In this Tutorial, we intend to provide a self-consistent and comprehensive description of the underlying physics of heat generation in metal layers without and with single nanoholes. The two first parts are dedicated to the case of a uniform metal layer under far-field illumination. We first describe the optical absorption physics, and then the heat conduction and temperature increase, discussing the effects of the (adhesion) layer thickness and composition. In the third part, we introduce the theory of the thermoplasmonics of nanoholes based on circular nanohole geometries. We define the concept of absorption cross section for nanoholes and explain the LPR/SPR origin of heat generation. All the results are mainly considering gold as the metal, since it is used in most applications, but the formalism introduced in this Tutorial can be extended to other materials. In particular, several Matlab and Comsol codes are provided in supplementary material for this purpose.
II. ABSORPTION OF LIGHT BY A THIN LAYER
In this introductory section, we recall the optical properties of a metal layer. Due to the simplicity of the system, the physics can be described by simple closed-form expressions based on Fresnel coefficients. We start from the simple case of a single interface and generalize the formalism in the case of a multiple interface system.
A. Fresnel coefficients for a single interface
B. Fresnel coefficients for a two-interface system
These expressions are both valid for TE (s) and TM (p) polarized beams by selecting the appropriate s or p coefficients in Eq. (1).
C. Fresnel coefficients for a m-layer system
D. Transmittance, reflectance, and absorbance
In the case where n1 sin(θ1)/nm > 1 (total internal reflection), there is no real solution for θm. However, all the above equations are still valid using the exponential definition of the cosine function and the logarithmic definition of the arcsine function, which are both defined for complex numbers.
E. Heat generation in a metallic layer
(a) Skin depth of gold. (b) SPP wavelength of a gold/dielectric interface as a function of the free-space wavelength λ0 = 2πc/ω. The cases of glass and water are represented. The dotted line represents the free-space wavelength λ0, as a comparison. (c) Propagation length of a SPP of a water/Au/glass system as a function of the wavelength for four different Au layer thicknesses. (d) and (e) Absorbance of a gold layer on glass for TE and TM polarization at λ = 900 nm, as a function of the layer thickness and incidence angle. Calculations conducted using Eqs. (1)–(9). Optical constants for the gold layer are taken from Ref. 53.
(a) Skin depth of gold. (b) SPP wavelength of a gold/dielectric interface as a function of the free-space wavelength λ0 = 2πc/ω. The cases of glass and water are represented. The dotted line represents the free-space wavelength λ0, as a comparison. (c) Propagation length of a SPP of a water/Au/glass system as a function of the wavelength for four different Au layer thicknesses. (d) and (e) Absorbance of a gold layer on glass for TE and TM polarization at λ = 900 nm, as a function of the layer thickness and incidence angle. Calculations conducted using Eqs. (1)–(9). Optical constants for the gold layer are taken from Ref. 53.
F. Surface plasmon resonance in a metal layer
A localized plasmon resonance (LPR) is a kind of resonance in illumination wavelength that occurs in plasmonics when the free electrons in the metal are spatially confined, i.e., typically with nanoparticles. The displacement of the free charges from the equilibrium position following the electric field of the incoming light creates a restoring force, attracting the electron back to equilibrium, just like with a mass–spring system, hence the resonance at a particular light frequency.
Experimentally, this theoretical expression of the propagation length matches well experimental measurements, provided the metal layer is crystalline. If the metal layer is polycrystalline/amorphous or rough, then this propagation length is reduced.56–58 For instance, for gold, Kuttge et al. used cathodoluminescence imaging spectroscopy to excite and measure ℓSPP for crystalline and polycrystalline gold layers.56 A reduction of the propagation length of around a factor of 3 was evidenced in the infrared range.
Close to the resonance angle θSPR, the system demonstrates reduced reflectivity of the light beam and an increase in light absorption. Figures 2(d) and 2(e) present numerical simulations related to a textbook case in plasmonics: a gold layer on glass and in contact with water, where the angle of illumination and the thickness of the layer are varied. Only with a TM, polarization and at a specific angle of incidence, a surface plasmon can be excited, as noticed by an increase in the absorbance [Fig. 2(e)]. The excited SPP propagates on the metal/water interface, while the light illuminated the glass/water interface. No SPP excitation can be achieved when illuminating the system from the lower refractive index medium (air in our case). The reduced reflectivity at a very specific angle due to absorption is at the basis of a famous application called SPR biosensing, dating from the 1990s,59 one of the rare applications of plasmonics that achieved widespread commercialization.
G. Effect of the adhesion layer
Gold poorly adheres on glass. To achieve stable plasmonic samples made of gold on glass substrates, a thin adhesion layer is often required to avoid detachment of the gold structures or layer during use. This adhesion layer usually consists of a few-nanometer-thick layer of chromium or titanium.60 Unfortunately, such metals have poor optical properties, damping the plasmon resonance, and exhibit significant optical absorption in the whole visible-IR range, stronger than gold, as demonstrated in Figs. 3(a)–3(c). The simulations are here conducted at normal incidence for the sake of simplicity and because it is sufficient to derive the main conclusions. In particular, Fig. 3(b) shows that Cr and Ti layers absorb around four times more than gold in the visible range at equal thickness. This ratio increases even more in the infrared, as shown in Fig. 3(c). In any case, the absorption of Ti, Cr, and Au layers saturates above 10 nm thickness. Figures 3(d) and 3(e) are striking, as they show that the absorption of a Cr–Au layer is quite constant, around 50% for 5 nm Cr, no matter the thickness of gold that is put on top, meaning that the absorption is mainly due to the Cr layer and that the Au layer has no effect on the absorption. This statement is less true in the IR range, as shown in Fig. 3(f). The absorption is no longer constant upon adding gold on top of a 5 nm Cr layer. However, surprisingly, the addition of gold on top of a 5 nm thick Cr layer reduces the absorption, down to a factor of 2. This phenomenon of adding the material but decreasing the absorbance is also observed, albeit less pronounced, with single metallic layers, as can be seen in Figs. 3(b) and 3(c), where the absorption of a gold, chromium, or titanium layer shows a local absorbance maximum for a layer thickness around 10 nm. This observation means that the absorption behavior of a metallic layer is not intuitive and requires modeling and computation even for simple systems.
(a) Schematic of a single-metal-layer system related to (b) and (c). (b) Absorption of a single-layer system at λ = 600 nm. (c) Absorption of a single-layer system at λ = 900 nm. (d) Schematic of a two-metal-layer system related to (e) and (f). (e) Absorbance of a two-layer system, at λ = 600 nm, as a function of the Cr layer thickness (dark gray curve, the Au layer thickness being maintained at 100 nm) and as a function of the Au layer thickness (yellow curve, the Cr layer thickness being maintained at 5 nm). (f) Same as (e) for λ = 900 nm. (g) Schematics of the systems related to images (h) and (i): (1) A 5-nm thick chromium layer, (2) a 100-nm thick gold layer, and (3) and 100-nm thick gold layer on top of a 5 nm thick chromium layer. (h) Absorbance as a function of the wavelength for the three systems, when the upmost layer is air, and for upward and downward illumination. (i) Same as (h) when the upmost layer is water.
(a) Schematic of a single-metal-layer system related to (b) and (c). (b) Absorption of a single-layer system at λ = 600 nm. (c) Absorption of a single-layer system at λ = 900 nm. (d) Schematic of a two-metal-layer system related to (e) and (f). (e) Absorbance of a two-layer system, at λ = 600 nm, as a function of the Cr layer thickness (dark gray curve, the Au layer thickness being maintained at 100 nm) and as a function of the Au layer thickness (yellow curve, the Cr layer thickness being maintained at 5 nm). (f) Same as (e) for λ = 900 nm. (g) Schematics of the systems related to images (h) and (i): (1) A 5-nm thick chromium layer, (2) a 100-nm thick gold layer, and (3) and 100-nm thick gold layer on top of a 5 nm thick chromium layer. (h) Absorbance as a function of the wavelength for the three systems, when the upmost layer is air, and for upward and downward illumination. (i) Same as (h) when the upmost layer is water.
Figures 3(g)–3(i) address the effect of the illumination side. Whether the illumination impinges on the adhesion layer side or on the opposite side changes much the absorption, as expected. When illuminating from above (the opposite side of the adhesion layer), the adhesion layer has no effect on the absorbance property of the system.
To give an idea of the temperature increase, a 5 nm Cr + 100 nm Au thin layer excited with a 1 mW focused laser beam experiences a 3.6 °C temperature increase when excited at 600 nm and 1.4 °C when excited at 900 nm, which can easily become an issue for SPR-based sensors61 [calculations obtained using the Temperature_layer.mph Comsol code provided in the supplementary material (see Sec. III B for more details)]. A temperature increase can be detrimental for some applications. There exist some possibilities to limit this temperature increase. The first one is to reverse the excitation side and shine the sample from the water/air side. In this case, the gold layer blocks the incident light and prevents it from heating the adhesion layer, as shown in Fig. 3.
In case excitation has to come from the glass side, to excite surface plasmon polaritons (SPPs), for example, it is possible to use other adhesion layers such as TiO2, Cr2O3,62 and Ge63 in order to reduce the absorbance of the adhesion layer64 or even totally suppress it using a mercaptosilane molecular layer.65,66
III. HEAT TRANSFER IN A METAL LAYER
Section II was focused on how much light is absorbed by a metal layer system. This section rather focuses on how this amount of thermal energy diffuses in this system. The presence of a metal layer, that is, a highly thermally conductive film, makes the thermal diffusion highly anisotropic and more complicated to describe compared with the usual case in thermoplasmonics of a nanoparticle.
In this section, we consider a thin, infinite metal layer (subscript label “2”) of thickness h sandwiched between two semi-infinite media, one labeled “1” located at z < 0 and the other labeled “3” located at z > h. We consider the problem invariant by rotation around the (Oz) axis. We shall use the polar coordinates: r = (r, θ, z) = (ρ, z) and define ρ = |ρ|.
A. Exact solution of the steady-state heat diffusion equation
(a) Schematics of three geometries heated by a point-like source of heat. (b) Numerical simulation of the radial profile of the temperature increase for the three cases depicted in (a). (c) Same data as (b) in log–log scale. (d) Numerical simulation of the temperature distribution in the (ρ, z) plane within a 50-nm-thick gold metal layer, glass below, water on top, illuminated by a focused 1 mW laser beam, at λ = 1064 nm, focused by a 0.7 NA objective. (e) Associated heat power density qlayer(ρ, z). Simulations conducted using the three-layer Green’s function formalism [Eq. (30)].
(a) Schematics of three geometries heated by a point-like source of heat. (b) Numerical simulation of the radial profile of the temperature increase for the three cases depicted in (a). (c) Same data as (b) in log–log scale. (d) Numerical simulation of the temperature distribution in the (ρ, z) plane within a 50-nm-thick gold metal layer, glass below, water on top, illuminated by a focused 1 mW laser beam, at λ = 1064 nm, focused by a 0.7 NA objective. (e) Associated heat power density qlayer(ρ, z). Simulations conducted using the three-layer Green’s function formalism [Eq. (30)].
Figure 4(d) displays the result of a temperature distribution calculated using Eq. (35). This profile corresponds to a 50-nm gold film between glass and water, illuminated at 1064 nm (a common laser wavelength used in nanohole plasmonics68) by a focused laser beam (NA 0.7). These calculations evidence that the temperature spreads much in the z direction within the metal layer, making it almost uniform in z, although the heat source is not.
B. Using Comsol for thermoplasmonics simulations
Using Comsol can also be an efficient approach to model such a system, provided some pitfalls are avoided, in particular, regarding the boundary conditions. The 1/r decay of the temperature in a dense medium, which is a slow decaying function, implies that the temperature usually does not exactly reach zero in the boundary of the modeled system, no matter how far the boundary is set.
The case of a nanoparticle in a uniform medium is instructive, although it is not the subject of interest of this Tutorial. If the nanoparticle is spherical, using Comsol is not necessary to compute any temperature profile because the large thermal conductivity of the nanoparticle yields to the assumption that the heat source density can be considered as uniform within the nanoparticle of volume V and equal to q = σabsI/V, and the problem has a closed-form expression in the surrounding medium that reads T(r) = σabsI/4πκr.39,49 The use of Comsol becomes relevant for nanoparticles of arbitrary geometries. To compute the temperature profile using Comsol, one can still assume q = σabsI/V uniform. The critical feature concerns the boundary conditions. The natural idea consists in placing the boundary far enough and set it to room temperature. However, the slowly decreasing temperature profile 1/r causes problem. Even when setting the boundaries at a distance that is ten times the radius of the particle, it yields a temperature underestimation of 10%. Another possible solution, a priori, consists in setting the outer flux instead of the temperature at the boundary of the domain because in the steady state, the integral of this flux over the boundary must be equal to the power delivered by the particle qV. However, in practice, it does not converge as even an infinitely small difference between qV and the integrated flux yields a divergence. We rather propose two methods to properly set the boundary conditions in thermal simulation of heated nanoparticles. Method 1 consists in defining a spherical system boundary with temperature boundary conditions set as uniform and equal to Tboundary = Tambient + σabsI/4πκRd, where Rd is the radius of the spherical domain. Indeed, far from the nanoparticle (couple of nanoparticle sizes), the temperature profile varies as 1/r and is no longer affected by the particular geometry of the nanoparticle. Method 2 requires the heat transfer module (not the built-in one), which offers the use of infinite domains. Such domains have to be placed at the boundary of the system.69
Let us now focus on COMSOL thermal computation involving infinite metal layers. The slow 1/r decay problem mentioned in the previous paragraph is even more significant here due to the fast heat diffusion through the metal layer, which makes the temperature decay even slower [see Fig. 4(b), case 2]. Let us explain to what extent the two above-mentioned methods can be applied to metal layers. Method 1: The plasmonic system necessarily reaches here the boundary of the system, making Method 1 a priori ineffective. However, Fig. 4(c) suggests an efficient trick to make it effective. Since the temperature eventually reaches the uniform bulk temperature at large ρ, it suffices to consider the boundary far enough, at least at ρ = 100h, make it spherical (with a radius Rd), and set its temperature to , where . Using this caution, Comsol simulations become accurate and perfectly correspond to Green’s function simulations. Figure 5 compares Green’s function simulations, COMSOL simulations for a 100 nm thick gold layer, with a Cr adhesion layer, heated at 790 nm from the top (water side) or from below (glass slide). In any case, the match between Comsol and Green’s function is perfect. The Comsol and Matlab programs are provided in the supplementary material (Temperature_layer.mph, Fig4.m). Method 2, requiring the use of the heat transfer module, was found also effective with metal layers. Meshing infinite domains has to be done with caution, but it enables one to use much smaller computation domains. The code Temperature_nanohole.mph provided in the supplementary material makes use of infinite domains.
Comparison between numerical simulation (COMSOL and Green’s functions) and experimental measurements of the temperature profile of a gold film under illumination by a uniform laser beam at λ = 790 nm. (a) Schematic of the system, consisting of a 100 nm gold layer, supported on a 5 nm thick Cr adhesion layer, illuminated either from the top or from below. (b) Temperature profiles for the case of top illumination, with a beam diameter of 25 µm. In that case, the Cr layer has no effect on the light absorption. The models did not consider it. Experiments were conducted with a 40× water immersion objective. (c) Temperature profiles for the case of bottom illumination, with a beam diameter of 9.5 µm. The Cr layer thickness was used as a free parameter, set to 6.05 nm, a value fully consistent with the estimation. Experiments were conducted with a 100× oil-immersion objective.
Comparison between numerical simulation (COMSOL and Green’s functions) and experimental measurements of the temperature profile of a gold film under illumination by a uniform laser beam at λ = 790 nm. (a) Schematic of the system, consisting of a 100 nm gold layer, supported on a 5 nm thick Cr adhesion layer, illuminated either from the top or from below. (b) Temperature profiles for the case of top illumination, with a beam diameter of 25 µm. In that case, the Cr layer has no effect on the light absorption. The models did not consider it. Experiments were conducted with a 40× water immersion objective. (c) Temperature profiles for the case of bottom illumination, with a beam diameter of 9.5 µm. The Cr layer thickness was used as a free parameter, set to 6.05 nm, a value fully consistent with the estimation. Experiments were conducted with a 100× oil-immersion objective.
In Fig. 5, we also take the opportunity to show experimental measurements performed on this exact system. The correspondence looks good, supporting the effectiveness of the numerical approaches we propose here. The only free parameter was for Fig. 5(c) the Cr layer thickness, which was not known precisely, and had to be set to 6.05 nm, a value within the experimental uncertainty of 5 ± 2 nm. This good match also confirms that using the bulk gold permittivity53 and gold thermal conductivity is still justified, despite the nanometric dimension of the system and the presence of interfaces. It has been shown in plasmonics that using the bulk permittivity of gold was fine for nanoparticles, provided their dimensions are not smaller than a few nanometers.70–72 Below a few nanometers, surface effects70,71 and nonlocal effects72 may become dominant, making the use of the bulk permittivity in simulation not accurate. The same rule certainly applies for gold layers and their thicknesses. In addition, the thermal conductivity of the adhesion layer may strongly differ from the bulk conductivity due to the very small thickness and roughness of the layer. However, the adhesion layer is so small compared with the gold layer that its contribution to thermal diffusion is negligible anyway.
C. What is the temperature increase of a metal film under illumination?
The question raised in the title of this subsection is what eventually matters, but the answer is not unique. Many parameters come into play, such as the beam size, beam power, multilayer materials, wavelength, layer thickness, and angle of incidence. In the frame of this subsection, we will focus on the most typical case of a gold layer on top of glass, covered with water, illuminated using a focused laser beam of P = 1 mW, at normal incidence. This configuration lets us with only two free parameters: the layer thickness h and the wavelength λ, and offers thus the possibility to answer the initial question with 2D maps, as shown in Fig. 6. Any other configuration than the one presented here can be modeled using the codes provided in the supplementary material (Fig6def.m and Fig6ghi.m). Figure 6 evidences the strong absorption of gold layers for wavelengths typically below 550 nm due to the onset of interband absorption. Above this wavelength threshold, photon absorption is strongly reduced because of the low energy photon that prevents authorized (interband) transitions and because of the perfect translation invariance of the system. Above a certain thickness, on the order of the skin depth (see Fig. 2), the fraction of absorbed light is quite constant. However, the temperature markedly drops upon thickening the layer, as the heat has to spread over a larger volume. The main conclusion from these data is that heating a gold layer in the red-infrared range, even upon focalizing the beam, remains unlikely when using reasonable laser powers. Importantly, this conclusion holds as long as an adhesion layer is not introduced: Figs. 3(h) and 3(i) display results related to the introduction of a Cr adhesion layer between the glass and the gold substrates. The effect of an adhesion layer is to strongly raise the absorption in the red-infrared range, where the absorption used to be almost nonexistent with a pure gold layer. Experimentally, when temperature increase is to be avoided, it is thus preferable to shine the other face of the metal layer (see Fig. 3).
(a)–(c) Three layer systems studied numerically, namely, a single Au layer (a), a Au/Cr layer (b), and a Cr layer (c), deposited on glass and covered with water and illuminated by a focused laser of power P = 1 mW. (d) Absorption of the layer as a function of the wavelength and layer thickness, corresponding to the geometry (a). (e) Same as (d) for the geometry (b). (f) Same as (d) for the geometry (c). (g) Maximum temperature increase in the layer as a function of the wavelength and layer thickness, corresponding to the geometry (a). (h) Same as (h) for the geometry (b). (i) Same as (g) for the geometry (c).
(a)–(c) Three layer systems studied numerically, namely, a single Au layer (a), a Au/Cr layer (b), and a Cr layer (c), deposited on glass and covered with water and illuminated by a focused laser of power P = 1 mW. (d) Absorption of the layer as a function of the wavelength and layer thickness, corresponding to the geometry (a). (e) Same as (d) for the geometry (b). (f) Same as (d) for the geometry (c). (g) Maximum temperature increase in the layer as a function of the wavelength and layer thickness, corresponding to the geometry (a). (h) Same as (h) for the geometry (b). (i) Same as (g) for the geometry (c).
D. Suspended layer in the steady state
E. Suspended layer and transient heat transfer
Illuminating a suspended layer does not necessarily yield catastrophic photothermal effects. When considering a single pulse of light, the temperature increase can be moderate if the pulse duration is short enough. Let us investigate in detail this question.
Let us derive some orders of magnitude, with the example of a suspended 100-nm thick gold layer. Equation (9) (i.e., code Fig2.m in the supplementary material) gives A = 0.0819 for a wavelength of λ = 800 nm. Assuming S = λ2 to model a focused laser, giving a thermal diffusion time τD = 5 ns, a laser power of Plaser = 1 mW, and a pulse duration of τlaser = 1 ns, one finds a temperature increase of T(0, 0) = 0.5 °C. When using a pulsed laser, Plaser is the peak power, not the average power that would be measured with a powermeter. When using a pulsed laser with a given repetition rate, then the layer will have to release a continuing input of light energy, and a higher temperature increase compared with the T(0, 0) can be achieved if the repetition rate is high enough. To determine this temperature increase, the time average laser power (the one measured with a powermeter) has to be considered and the discussion of Sec. III D has to be applied as if the laser were in a continuous mode.
IV. THERMOPLASMONICS OF NANOHOLES
The system now consists of a multilayer, as in the previous part, where the middle, metal layer is pierced by a cylindrical nanohole. For a plain metal layer, we have seen that a surface plasmon resonance (SPR) can be excited only with a large angle of incidence (Fig. 2) and only with an illumination from the glass side (not from the water side). These restrictions stems from mismatch of vector components of light and surface plasmons at the interfaces. However, when a defect is present, that is, any breakage of the translational symmetry, then propagating plasmons (SPP) can be excited by the wave vector components of the defect’s near-field, even at normal incidence. Such defects can be nanoholes or even particles deposited on the metal layer. Moreover, a localized plasmon resonance (LPR) can also occur at the defect location, in our case the nanohole. The plasmonic resonance of a nanohole is supposed to match the one of the inverse geometry following the Babinet’s principle.16,76–79 These features make a nanohole system much richer and valuable for the numerous applications cited in the Introduction, compared with nanoparticles. However, it also makes the system more complex. Figure 7 shows a scanning electron microscope (SEM) image of a nanohole in a Au/Cr/glass layer system, fabricated by focused ion-beam lithography. For this kind of system, a relevant question is how much light is absorbed compared with a uniform metal layer as treated in the previous part. There is less absorbing material, but the possible excitation of localized and propagating plasmons may increase the absorption. So less material does not necessarily mean here less absorption, a priori. In this section, we investigate this question in detail.
Scanning electron microscope images of an experimental 300 nm diameter nanohole etched using focus ion beam on a 100 nm thick gold layer. (a) Tilted view and (b) cross section. The chromium adhesion layer is clearly visible as a thin gray layer between the gold layer (white) and the glass substrate (black). Scale bars: 150 nm. Courtesy of Jean-Benoit Claude and Jerome Wenger.
Scanning electron microscope images of an experimental 300 nm diameter nanohole etched using focus ion beam on a 100 nm thick gold layer. (a) Tilted view and (b) cross section. The chromium adhesion layer is clearly visible as a thin gray layer between the gold layer (white) and the glass substrate (black). Scale bars: 150 nm. Courtesy of Jean-Benoit Claude and Jerome Wenger.
A. Definition of the absorption cross section of nanohole
For the sake of simplicity, we discuss only in this Tutorial the case of cylindrical nanoholes, but these newly defined quantities σabs and Cabs can be applied to any nanohole shape and dimension or even to any localized defect on a metal layer.
B. Numerical calculation of the absorption cross section
Unlike with nanoparticles, no closed-form expression exists for the expression of the absorption cross sections of nanoholes, which have to be calculated numerically.
σabs, as defined by Eq. (48), consists of a difference between two terms, the first one related to the pierced layer and the second one related to the plain layer. Importantly, these two terms should not be considered as absorption cross sections, which could be calculated separately because each of these terms is infinite. This is due to the fact that an infinite metal layer illuminated by a plane wave absorbs an infinite power, hence the consideration of a limit and a subtraction of the plain layer system to avoid a divergence and get a finite quantity.
Evolution of the absorption cross section of a 200 nm diameter nanohole on a 50 nm thick gold layer with the radius of integration for an excitation at (a) 450, (b) 650, and (c) 950 nm. At 450 nm, gold cannot support surface plasmons; therefore, the loss of material in the nanohole induces a decrease in the absorption. At 650 and 950 nm, plasmon-induced absorption enhancement induces a positive absorption cross section. The exponential increase in the absorption cross section with the distance is due to the progressive absorption of the SPP launched at the nanohole. High frequency oscillations can be attributed to interactions between SPP at the water–glass interface and either the transmitted light (λSPP) or the SPP at the glass–gold interface (λbeat).
Evolution of the absorption cross section of a 200 nm diameter nanohole on a 50 nm thick gold layer with the radius of integration for an excitation at (a) 450, (b) 650, and (c) 950 nm. At 450 nm, gold cannot support surface plasmons; therefore, the loss of material in the nanohole induces a decrease in the absorption. At 650 and 950 nm, plasmon-induced absorption enhancement induces a positive absorption cross section. The exponential increase in the absorption cross section with the distance is due to the progressive absorption of the SPP launched at the nanohole. High frequency oscillations can be attributed to interactions between SPP at the water–glass interface and either the transmitted light (λSPP) or the SPP at the glass–gold interface (λbeat).
C. Origin and characteristics of heat dissipation with nanoholes
The results of the previous part, displayed in Fig. 8, dedicated to discuss the proper choice of the integration radius R for the estimation of σabs, are also the occasion to depict the physics of energy dissipation when illuminating a nanohole in a metal layer. With a nanohole pierced in a metal layer, two kinds of plasmons can be excited, a LPR, which locally exists at the vicinity of the hole, and SPPs, the excitation of which from the far field are made effective, thanks to the presence of the nanohole that breaks the invariance by translation throughout the metal layer. These two transduction pathways lead to two energy dissipation pathways. The first one is localized at the vicinity at the nanohole, while the other one, only possible in the IR range, can extend over several tens of micrometers. The relative effects of LPR and SPP can be observed in Fig. 8(c). At short R, one can notice a first offset, of the absorption, which results from the LPR dissipation, and a progressive increase in the estimated absorption as a function of R arising from the SPP spatial extension.
Aside from these global variations, two high frequency oscillations can be observed. The wavelength of the higher frequency one is equal to the SPP wavelength at the glass/gold interface. It can be attributed to an interference phenomenon between the incident light field and the propagative surface plasmons on the water side. The lower frequency beating ωbeat is assigned to the frequency difference between the SPPs at the water–gold (ωSPP–water) and glass–gold (ωSPP–glass) interfaces, following the relation ωbeat = ωSPP–glass − ωSPP–water (see Sec. 4 of the supplementary material for a detailed calculation).
The other interesting feature illustrated by Fig. 8(a) is the possible occurrence of negative values of the absorption cross section for nanoholes, a feature that has no counterpart in the case of nanoparticles. The possibility to have negative values of σabs for nanoholes arises from its definition as a difference between two terms, the absorption of the pierced layer and the absorption of the plain layer. Negative values of σabs mean that the presence of the nanohole reduces light absorption. As discussed further on, this effect only occurs at short wavelengths (below 500 nm) or, of course, above a certain nanohole size, when too much material is removed.
D. Effect of nanohole diameter and the layer thickness
In order to discuss the geometrical parameters of a nanohole affecting the light absorption, we computed the absorption cross section for four hole diameters and two layer thicknesses (Fig. 9).
(a) Absorption cross sections of nanoholes with diameters ranging from 50 to 300 nm in a 50-nm thick gold layer. (b) Same as (a) for a 100-nm thick gold layer. (c) and (d) Same as (a) and (b) where the cross sections have been normalized by the area of the nanohole.
(a) Absorption cross sections of nanoholes with diameters ranging from 50 to 300 nm in a 50-nm thick gold layer. (b) Same as (a) for a 100-nm thick gold layer. (c) and (d) Same as (a) and (b) where the cross sections have been normalized by the area of the nanohole.
Even if slight differences can be observed between the case of the 50 nm [Figs. 9(a) and 9(c)] and the 100 nm [Figs. 9(b) and 9(d)] thick layers, the overall shapes of the curves are similar, meaning that the layer thickness only has a marginal effect on the absorption of the nanohole.
The effect of the nanohole diameter is, however, much more pronounced, with an absorption maximum matching the plasmonic resonance frequency of the nanohole. The normalized absorption cross section [Figs. 9(c) and 9(d)] reaches values that can be higher than 1, meaning that the power absorbed by a pierced layer and converted into heat is higher than the power of the light geometrically reaching the nanohole. This behavior is similar to what can be observed with plasmonic nanoparticles80,81 and predicted by Mie theory.82
Below 550 nm, however, the absorption cross section is low and can even become negative. As already discussed, this is because in this wavelength range, gold behaves as a simple absorber, characterized by a strong increase in the imaginary part of the permittivity due to interband transitions. This feature cancels any plasmonic effect, and the dominant rule becomes simple: less material, less absorption. This rule of thumb originates from the balance between two scaling laws: First, an absorption decrease related to the removal of material scaling as −πa2 (the surface of the nanohole); second, an absorption increase arising from the edge of the nanohole scaling as 2πa. Thus, for sufficiently large nanohole radius a, the first contribution dominates, making σabs negative.
Figures 10(a)–10(c) plots results of Comsol simulations of the temperature and heat power density associated with a 100-nm nanohole in a 100-nm thick gold film, illuminated with a focused laser at λ0 = 650 nm and a power of 1 mW. This wavelength corresponds to the LPR resonance. The temperature map is similar to the one of a plain layer (see Fig. 4). However, the heat power density looks localized at the boundary of the nanohole. These simulations were obtained using the Temperature_nanohole.mph Comsol program provided in the supplementary material.
Comsol simulations of the heat power density and the temperature distributions in a glass/metal/water system pierced with a nanohole and illuminated by a focused laser beam at λ0 = 650 nm. (a) Temperature distribution for a gold layer, 100-nm thick and 100-nm nanohole. (b) Associated heat power density. (c) Same as (b) in log colorscale. (d)–(f) Same as (a)–(c) for a 95-nm thick gold layer adhering on a 5-nm thick Cr layer. In (e), a zoom window highlights the heat generation confined in the thin Cr adhesion layer.
Comsol simulations of the heat power density and the temperature distributions in a glass/metal/water system pierced with a nanohole and illuminated by a focused laser beam at λ0 = 650 nm. (a) Temperature distribution for a gold layer, 100-nm thick and 100-nm nanohole. (b) Associated heat power density. (c) Same as (b) in log colorscale. (d)–(f) Same as (a)–(c) for a 95-nm thick gold layer adhering on a 5-nm thick Cr layer. In (e), a zoom window highlights the heat generation confined in the thin Cr adhesion layer.
E. Effect of the wavelength
We now consider a gold layer deposited on glass with a chromium adhesion layer. As discussed in Sec. II G, light absorption can be markedly enhanced by the presence of an adhesion layer, leading the community to usually illuminate the sample from the water side for trapping applications, for instance.64 In the following, both illumination directions will be considered from the water and the glass sides [Fig. 11(a)]. In this section, we consider only a hole diameter of 200 nm. The COMSOL program provided in the supplementary material can be used to compute any other configuration.
(a) Geometries considered by the COMSOL calculations of absorption cross sections and temperature increases of nanohole systems. The nanoholes are 200 nm in diameter, the Ay layer is 50 nm thick, and the Cr layer is 5 nm thick. (b) Absorption cross sections of nanoholes calculated at λ0 = 450 nm. (c) Same as (b) for λ0 = 700 nm. The dotted line represents the area of the hole. (d) Calculation of the temperature increase for the eight geometries depicted in (a) and for λ0 = 450 nm, for an incident power of 1 mW, and a Gaussian spot with a full-width half-maximum of λ0/2n. (e) Same as (d) for λ0 = 530 nm. (f) Same as (d) for λ0 = 700 nm.
(a) Geometries considered by the COMSOL calculations of absorption cross sections and temperature increases of nanohole systems. The nanoholes are 200 nm in diameter, the Ay layer is 50 nm thick, and the Cr layer is 5 nm thick. (b) Absorption cross sections of nanoholes calculated at λ0 = 450 nm. (c) Same as (b) for λ0 = 700 nm. The dotted line represents the area of the hole. (d) Calculation of the temperature increase for the eight geometries depicted in (a) and for λ0 = 450 nm, for an incident power of 1 mW, and a Gaussian spot with a full-width half-maximum of λ0/2n. (e) Same as (d) for λ0 = 530 nm. (f) Same as (d) for λ0 = 700 nm.
Figures 11(b) and 11(c) compare the effect of the wavelength. At 450 nm, gold behaves as a simple absorber, like chromium [Figs. 3(h) and 3(i)]. No surface plasmon can be excited. In any case, the absorption cross section is slightly negative when using an excitation wavelength below 500 nm, in agreement with Fig. 9. The amount of light absorbed, with or without nanohole, is mainly dictated by the size of the metal surface under illumination. Without nanohole, this surface just corresponds to the beam profile. The presence of a nanohole removes a surface of the layer, decreasing the absorption, but adds another area 2πRlaserh, that is, the inner wall surface of the hole, which increases absorption accordingly. The balance of these two contributions can be considered to explain the behavior observed in Figs. 11(b) and 11(c). The reason why σabs remains very small (compared with the 700 nm case discussed later on) is that the two surface and 2πRlaserh are almost equal: the removed surface of the layer is replaced by the nanohole inner walls. The addition of the chromium layer increases the absorption (makes it less negative), no matter the illumination side, mainly because it increases the layer thickness and thus the total amount of the metal. Shining from the water side also increases the absorption of the nanohole. This effect comes from the absorbance A of a continuous gold layer that is about 15% larger when excited from the glass side (see Fig. 3), increasing qlayer of Eq. (48).
Let us now discuss the more complicated case of the excitation in the red–infrared region, where absorption is also affected by plasmonic effects. At 700 nm [Fig. 11(c)], all the absorption cross sections are positive, meaning that the presence of the hole increases the amount of absorbed energy compared with a plain layer. Let us start with the comparison of a gold layer without the adhesion layer (cases 1 vs 3). The absorbances of a plain Au layer excited either from the glass or from the water side are similar [see Eq. (9) for the calculation]: 3.62% from the water side and 4.05% from the glass side, i.e., 10% variation. Therefore, the strong difference in absorption cross section observed in Fig. 11(c) (case 1 vs case 3) is not due to variations of plain layer absorbance. The difference comes from a more efficient excitation of surface plasmons from the glass side due to a better match of the wave vectors: kSPP > kglass > kwater.
F. Effect of the adhesion layer
Let us now focus on the effect of the adhesion layer and start with the glass side illumination, still at 700 nm [Fig. 11(c), cases 1 vs 2]. Counter-intuitively, the presence of a Cr layer strongly reduces the absorption cross section. This counterintuitive effect comes from the definition of the absorption cross section, given by Eq. (48): the subtracted term qlayer is larger in the presence of chromium, leading to a reduced calculated absorption cross section. One has to keep in mind that the absorption cross section as defined in this Tutorial is a comparison with a plain layer, considered as a reference. Thus, two absorption cross sections cannot be really compared with each other if they correspond to different layer geometry. In particular, they cannot be compared to predict relative temperature increases, as illustrated by Fig. 11(f). At the same wavelength of 700 nm, while a hole in a gold layer (case 1) has a larger absorption cross section than with a Cr adhesion layer, it, however, yields a weaker temperature increase. An example of temperature and heat power density distribution is given in Figs. 10(d)–10(f).
To end this section, we wish to notice that all the descriptions and the theory described so far could be applied to nanostructures on top of metal film, not only for nanoholes. Nanostructures would also have LPR that could couple to SPP, for which absorption cross section could also be defined by Eq. (48). Addressing such a system is out of the scope of this Tutorial. However, it could straightforwardly be modeled by adapting the Comsol program Temperature_nanohole.mph provided in the supplementary material.
V. CONCLUSION
This Tutorial intends to provide a comprehensive description of the photothermal effects that can occur when illuminating a metal layer, without and with the presence of nanohole. The aim is to tackle all the aspects of the problem, from the physics, to the numerical approaches. We investigate and discuss the influence of parameters such as the layer composition, thickness, the presence of an adhesion layer, the illumination wavelength, incidence angle, polarization, and the nanohole size. We also define the absorption cross section of a nanohole and discuss its meaning in the case of a 50-nm thick gold layer, with or without chromium adhesion layer.
We also provide in the supplementary material all the numerical codes used in this Tutorial, namely, (i) the Matlab code for Fresnel coefficients calculations, (ii) the Mathematica code for the calculation of Green’s function for a three-layer system, (iii) the Matlab code for Green’s function calculation of the temperature profile in a metal layer, (iv) the Comsol program for the calculation of the temperature profile in a metal layer under focused illumination, and (v) the Comsol program for the calculation of the temperature profile in a metal layer pierced with a nanohole.
Thermoplasmonics led to numerous applications based on desired photothermal effects of plasmonic nanoparticles. However, photothermal effects may also be unexpected and unwanted. In the latter case, careful attention to the temperature increase has to be paid to avoid sample damage or misleading interpretation of the results, like recently evidenced in plasmonic-assisted nanochemistry83 or optical trapping.47,64 Considering the increasing amount of studies involving metal films in plasmonics and nanoapertures (and no longer nanoparticles), photothermal effects will also have to be carefully considered in layer geometries, albeit much more complex.
SUPPLEMENTARY MATERIAL
See the supplementary material for access to all the Matlab, 1093 Comsol, and Mathematica numerical codes that have been used to com pute the data presented in this Tutorial.
ACKNOWLEDGMENTS
This work was supported by the Agence Nationale de la Recherche (SeqSynchro, Grant No. ANR-18-CE42-0013).
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
DATA AVAILABILITY
The data that support the findings of this study are available within the article and its supplementary material.