The contribution of nonlocal electro-opto-thermal interaction to single molecule nonlinear Raman enhancement

Chao-Yi Tai; Wen-Hsiang Yu

doi:10.1364/OE.21.025026

1. Introduction

Surface Enhanced Raman Spectroscopy (SERS) is a practical technique for the identification and characterization of molecule fingerprints [1]. In contrast to conventional spontaneous Raman spectroscopy, vibrational signatures enhanced by over six orders-of-magnitude can be routinely obtained, which provides a sensing capability under low concentrations. In general, the origin of the huge enhancement factor G_EF can be attributed to two major mechanisms: (1) the long-range electromagnetic (EM) enhancement resulting from the excitation of surface plasmons (SPs) [2]; and (2) the short-range chemical enhancement due to charge transferring (CT) [3]. So far, the EM enhancement is widely recognized as the key contribution to the amplification of the measured Raman signals. Therefore, substantial efforts were devoted in developing versatile substrates where the manipulation of surface plasmon modes via the fabrication of self-assembled monolayers [4], nanoparticles [5], nanoislands [6], random metal nanostructures [7], and isolated metal nanoparticles [8], were carried out. In parallel to this, the investigation of the interaction between various molecules and metallic surfaces promoted the progress of chemical enhancement [9]. A dramatic breakthrough of obtaining a G_EF up to 10¹⁴ was soon realized by attaching Rhodamine 6G on isolated silver nanoparticles [10], raising the sensitivity of SERS further up to single molecular level. Although the ultrahigh G_EF was subsequently verified by experiments [11, 12], a number of phenomena accompany SERS remain to be controversial issues. One of the well-known phenomena is the spectral and intensity intermittency, or the so-called blinking effect. This phenomenon, exhibiting power-law statistics for the on and off state, has been observed as an universal feature in a wide range of isolated quantum emitters, such as quantum dot [13], fluorescent molecules [14], and molecular Raman scattering. Due to the fluctuation in the measured signal, blinking has become a major obstacle particularly in molecular image reconstruction. Intriguingly, most of the observed blinking behavior of SERS have the following common features: first, they were observed in molecules adsorbed on noble nanoparticles such as gold or silver. Secondly, the blinking period is inversely proportional to the global temperature [15]. Thirdly, the spatial average of the G_EF is proportional to the global temperature and the illumination intensity [16]. A comprehensive experiment conducted by S. R. Emory suggested that as the substrate temperature rises from 25°C to 80°C, the intensity of the Raman scattering is enhanced by over 30 times, and the blinking period is shortened from several seconds to milliseconds [15]. To explain these phenomena, a variety of models based on either theoretical or experimental results were proposed. To name a few, the charging and discharging mechanism [17] attributed blinking to the ionization of molecules under intense field irradiation; the diffusion model based on the motion of molecule drifting in and out of the hot spot [18]; reorientation model describes the blinking by the anisotropic motion of molecules, and others such as chemical reaction [19] suggests a completely distinct origin. Although the suggested models are still under debate, thermal activation is widely accepted to be the most probable and common trigger that responsible for the abovementioned mechanisms [20].

Recently, plasmonic heating has been investigated quantitatively in both theoretical [21] and experimental [22] works. Due to good thermal insulation between the isolated metal nanoparticles and the surrounding medium, significant temperature rise is expected under even very weak light illumination. Current theoretical works either completely ignore the contribution of the coupling between the electromagnetic field and the temperature field or simply use the non-iterative model to calculate the temperature rise. The interplay between the transport of electrons, thermal and EM fields are rarely considered simultaneously. To our knowledge, no iterative process was ever implemented in existing simulations or algorithms. Although previous works still provide reasonably good explanations for localized surface plasmons (LSPs) assisted photothermal effects, nonlocal hot-electron effect plays dominant role under certain conditions, particularly in the field of single molecule SERS (SM-SERS).

Essentially, EM field and temperature field are coupled. Electromagnetic absorption results from the excitation of LSPs can induce tremendous nonlocal hot-electrons. Upon oscillation, not only the EM response of metals but the spatial distribution of the EM field and the temperature field are altered. The significance of this study is that the nonlinear Raman enhancement and mitigation of Raman blinking can be simultaneously obtained. This has been accounted never possible in the past and proper models fall short till now. Through plasmonic inter-modal coupling, we show that this problem can be solved. To illustrate the concept, two prototype cases are considered: the isolated nanosphere and nanodimer. The formal case is used to illustrate the nonlinear enhancement of the G_EF and the latter is for the suppression of the temperature rise. The nonlinear relations between the SERS signal, illumination intensity and environmental temperatures were explored in both cases. To verify our modeling, results are comprehensively compared with those reported in literatures [15, 16, 23, 24], and close agreement was found. Our study provides a more accurate way to estimate the temperature in steady state [25], pointing out a potential inaccuracy in temperature determination by the intensity ratio of the Stokes to anti-Stokes scattering [26].

2. Model and numerical procedures

Our model is established by linking the Maxwell’s equations and the thermal diffusion equation which can be expressed as follows [27]:

\nabla \cdot [κ (\overset{⇀}{r}) \nabla T (\overset{⇀}{r})] = - ρ (\overset{⇀}{r})

T (\overset{⇀}{r})

represents the spatial distribution of the temperature field,

κ (\overset{⇀}{r})

represents the position dependent thermal conductivity, and

ρ (\overset{⇀}{r})

stands for the heat source density (HSD) as a consequence of EM absorption. For monochromatic wave with angular frequency ω₀,

ρ (\overset{⇀}{r})

can be expressed by Eq. (2),

ρ (\overset{⇀}{r}) = ω_{0} ε'' (\overset{⇀}{r}) < E {(\overset{⇀}{r}, t)}^{2} >

where

ε'' (\overset{⇀}{r})

stands for the imaginary part of the permittivity, and the bracket gives the time-average light intensity. Combining Eq. (1) and Eq. (2) can only obtain the temperature rise caused by the EM absorption and is widely used in most of the research up to date. However, according to the study of Wiedemann and Franz [28], the electrical conductivity

σ (\overset{⇀}{r})

is highly correlated to the temperature of the metal itself. The EM response thereby changes correspondingly as the temperature rises. This recurrence relation should be taken into account repeatedly until reaching steady-state in order to reveal material properties with high fidelity.

κ (\overset{⇀}{r}) / σ (\overset{⇀}{r}) T (\overset{⇀}{r}) = L

Eq. (3) is the well-known law proposed by Wiedemann and Franz, which is used to describe the relations between the temperature

T (\overset{⇀}{r})

, thermal conductivity

κ (\overset{⇀}{r})

, and the electrical conductivity

σ (\overset{⇀}{r})

of the metal in consideration. In Eq. (3), L is the so-called Lorenz number which is intrinsically material dependent. For noble metals such as gold and silver, L and

κ (\overset{⇀}{r})

can be treated as temperature-independent constants. Hence, the electrical conductivity is inversely proportional to the temperature. Once the incident light impinges on the structure, the electro-opto-thermal interaction is initiated. The risen temperature results in a reduced conductivity which in turn causes the redistribution of the EM fields according to the Ampère’s law, as shown in Eq. (4). Based on Eq. (2), this further affects the HSD, and the temperature field will be deviated correspondingly from the initial condition (according to Eq. (1)). This process repeats until the system reaches steady state.

\nabla \times \vec{H} (\overset{⇀}{r}, t) = σ \vec{E} (\overset{⇀}{r}, t) + j ω ε' \vec{E} (\overset{⇀}{r}, t)

The numerical procedures are detailed as follows: The transport behavior of hot-electrons is modeled by thermal diffusion equation under thermal equilibrium; the nonlocal hot-electrons contributed to the electrical conductivity is governed by the law of Wiedemann-Franz; and the heat source density (HSD) resulted from the electromagnetic absorption is governed by the interacted Maxwell’s equation. The full vectorial finite difference time domain (FDTD) method is used to simulate the enhanced localization of electromagnetic field around the isolated gold nanosphere and nanodimers, where the dispersion relation of gold is modeled by Lorentz-Drude model and the incident plane wave is introduced by total and scattering field boundaries. Twelve perfectly matched layers are allocated on each boundary to absorb the outgoing electromagnetic waves. Time and space grid set in FDTD are 8.34 × 10⁻¹⁹ s and 1.25nm × 1.25nm × 1.25nm respectively, which ensures the simulation yields convergent and accurate results.

The iterative finite difference method of the explicit form and the modified Gauss-Seidel algorithm were applied to model the distribution of the temperature field that accelerated the numerical procedure effectively. For the purpose of obtaining a realistic temperature distribution in a wider region, non-uniform grids are applied to extend the simulation domain by over 550 μm³. Numerically, the undisturbed steady-state electromagnetic fields are calculated first. Subsequently, the thermal diffusion equation was solved to obtain the undisturbed steady-state temperature field. The temperature field then feeds back to the electrical conductivity and the discrete Maxwell’s equations are solved again to acquire the redistributed electromagnetic fields. The HSD is renewed correspondingly as a result of the redistributed electromagnetic field, which is then substituted back into the thermal diffusion equation to obtain a renewed temperature field again. The iterative calculation between the discrete Maxwell’s equations and the thermal diffusion equation should be computed repeatedly until the system reaches self-consistence. Typically, fifteen to forty loops are included in this process. The flow chart of the numerical procedure is schematically shown in Fig. 1.

Fig. 1 The flow chart of the computational procedure.

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3. Results and discussion

In order to illustrate how the mutual interaction enhances the absorption cross section, an isolated gold nanosphere (50 nm in radius) immobilized on the glass substrate subject to the illumination of monochromatic plane wave was studied. When the localized surface plasmons are excited, intense EM fields are induced around the rim of the gold nanosphere.

In the conventional analysis, the absorption cross section is assumed to be independent on the illumination intensity. Here, we demonstrate the nonlinear characteristic, considering the electro-opto-thermal interaction. Based on the wavelength dependent absorption cross section as shown by the black dots in Fig. 2(a), we select the wavelength of λ₀ = 532nm near the LSPR to calculate the absorption cross section as a function of the illumination intensity. As shown by the red dots in Fig. 2(a), under extremely weak illumination, the absorption cross section degenerates to the non-interaction result, as predicted by the conventional analysis. With the increasing of the illumination, a larger absorption cross section was induced due to the increase of the resonant strength. Normally in far-field experiments, increased absorption cross section implies stronger modal coupling between the LSPs and the excitation, playing a role as a measure of the degree of mutual interaction. With the illumination intensity varies from 0.1mW/μm² to 1.0mW/μm², the maximum $G_{E F} (\vec{r})$ ( $\propto {| E / E_{i n} |}^{4}$ ) increases from 10⁴ to 10⁵, and the temperature rises correspondingly from 25 °C to 250 °C, as shown in Fig. 2(b). Unlike the EM field, the temperature field distributed uniformly within the gold nanosphere. This is attributed to the large contrast of the thermal conductivity between gold and the surrounding medium (Air) which provides a quasi-adiabatic outer layer to preserve heat. Good thermal insulation in combination of LSP induced EM absorption results in the significant temperature rise which in turn changes the electromagnetic response, leading to the magnification of $G_{E F} (\vec{r})$ by one order of magnitude under moderate illumination intensity of 1 mW/um². In our calculation, the thermal conductivity of gold, fused silica, and air are set to 310W/m-K, 1.38 W/m-K, and 0.025 W/m-K at 25 °C, respectively [29].

Fig. 2 The near and far-field characteristics of isolated gold nanosphere with/without electro-opto-thermal interactions. (a) The absorption cross section as functions of the incident wavelength and illumination intensity I. (b) The enhanced EM field and the temperature field as a function of the illumination intensity at the near-field region.

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In order to quantify the intensity dependent $G_{E F} (\vec{r})$ , the interaction enhancement factor $G_{I E F} (\vec{r})$ was introduced as follows:

G_{I E F} (\vec{r}) = \frac{{| E_{int} (\vec{r}) |}^{4}}{{| E_{n o n - int} (\vec{r}) |}^{4}}

E_{int} (\vec{r})

is the electric field considering the effect of electro-opto-thermal interaction while

E_{n o n - int} (\vec{r})

is that without interaction considered. Since

G_{I E F} (\vec{r})

is a function of coordinates,

{〈 G_{I E F} (\vec{r}) 〉}_{S u r}

, defining by the spatially-averaged

G_{I E F} (\vec{r})

on the surface of the gold nanosphere, is introduced to quantify the nonlinear amplification of the

G_{E F} (\vec{r})

.

Compared to the result calculated by the non-interaction model (the blue-dashed line in Fig. 3), ${〈 G_{I E F} (\vec{r}) 〉}_{S u r}$ is proportional to the illumination intensity, exhibiting a nonlinear enhancement due to plasmon heating. (the blue solid-line in Fig. 3). In contrast to ${〈 G_{I E F} (\vec{r}) 〉}_{S u r}$ , the surface temperature grows almost linearly with the increase of the illumination intensity. With electro-opto-thermal interaction considered, the surface temperature is slightly higher than that calculated without interaction, as shown by the red line in Fig. 3.

Fig. 3 The surface temperature and spatially averaged interaction enhancement factor as a function of the illumination intensity. The bottom level of each bar gives the degradation temperature for the noted molecules. The values on top axis give the degradation intensities.

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Interestingly, one may find that interaction enhancement accompany blinking was rarely reported in literatures, especially for bio-molecular systems. An intuitive interpretation is that light induced temperature rise may accelerate the degradation of bio-molecules, and the threshold degradation intensity is far below that required for observable enhancement or blinking. To illustrate how accurate our modeling defines the boundary which separates regimes between interaction enhancement and thermal degradation, the degradation temperature of some commonly used molecules were converted to the illumination intensity. As shown in Fig. 3, the degradation temperature for erythrocyte and nerve fiber are 45 °C and 60 °C [30, 31], and the corresponding degradation intensities are calculated to be 0.1 mW/μm² and 0.2 mW/μm², respectively. It should be noted that the degradation temperature for the abovementioned molecules lies below the curve of ${〈 G_{I E F} (\vec{r}) 〉}_{S u r}$ even under very weak illumination, indicating that the interaction enhancement is very difficult to be observed. In contrast to bio-molecules, the interaction enhancement is relatively easy to be observed for non-biomolecules as long as the illumination intensity is sufficiently moderate. Examples can be indeed found in literatures. For instance, the degradation temperature for polyvinylpyrrolidone (PVP), crystal violet (CV), polystyrene (PS), rhodamine 6G (R6G), and benzenethiol (BT) are around 150 °C, 200 °C, 240°C, 325°C, and 650°C, respectively [15, 32–34]. The corresponding degradation intensities were calculated to be 0.7 mW/μm², 0.9 mW/μm², 1.1 mW/μm², 1.4 mW/μm², and 1.5 mW/μm² respectively.

Since the tunability of resonance in metal nanosphere is limited, nanoparticles of various shapes, such as ellipsoid, prism, disk, dimer, trimer, etc. have been investigated to tailor the resonant wavelength so as to increase the resonant strength. Through structural engineering, the temperature rise due to plasmon heating may also be suppressed. Here an isolated gold nanodimer immobilized on glass substrate was considered to demonstrate the suppression of plasmonic heating. The same particle size (50 nm in radius) with an inter-particle distance of 10nm is assumed in the calculation. In comparison with the far field characteristics of gold nanosphere without electro-opto-thermal interaction, the resonant wavelength of the nanodimer is red shifted owing to strong inter-modal coupling, as shown by the black stars in Fig. 4(a). To calculate the intensity dependent absorption cross section, a commonly used wavelength (λ = 633nm) close to the LSPR was chosen. In contrast to the case of nanosphere, the absorption cross section drops with the increase of the illumination intensity, as shown by the red stars in Fig. 4(a). In the near field, the coupled plasmonic mode produces an intense electromagnetic field within the gap, yielding an EF as high as 10⁷ accompanied by a much suppressed temperature rise, as shown in Fig. 4(b). With the illumination intensity varied from 0.1 mW/μm² to 1.0 mW/μm², the $G_{E F} (\vec{r})$ changes insignificantly around 10⁷. Meanwhile, the surface temperature rises from 39 °C to 145 °C, which is lower than the degradation temperature for most of the analytes. In order to quantify the electromagnetic field and temperature field in the gap, values of the temperature and $G_{I E F} (\vec{r})$ at the coordinate center of the gap are extracted from the simulation. Figure 4(c) gives the results (solid lines) compared to those (broken lines) obtained without the electro-opto-thermal interaction. Since the electromagnetic energy is more concentrated in the dielectric gap region instead of on the metallic surface, the contribution from the coupling between the electromagnetic field and the temperature field to the enhancement factor is very limited. In addition, the nanoparticle dimer forms a structure similar to a tapered slit, mimicking the propagation characteristics of the extraordinary transmission (EOT) like behavior [35, 36]. In EOT, the electromagnetic energy is concentrated within the gap and the plasmonic gap mode is insensitive to the change of the material parameters of the consisting metals. As a result, the magnification of $G_{E F} (\vec{r})$ is only very moderate which is insufficient to counteract the decrease of the absorption coefficient (proportional to the σ) as in the case of the nanosphere. Hence $G_{I E F} (\vec{r})$ tends to saturate at high illumination intensities and the decrease of the absorption cross section is mainly attributed to the decrease of the absorption coefficient as a consequence of temperature increase. This result renders isolated nanodimer as an appropriate platform for molecular sensing without suffering severe thermal issues. The degradation intensities for bio- and nonbio-molecules are higher as compared to the case of nanosphere. For erythrocyte and nerve fiber, the degradation intensities are 0.15 mW/μm² and 0.35 mW/μm², respectively. For PVP, CV, PS, R6G, and BT, the degradation intensities are all higher than 1.2 mW/μm².

Fig. 4 Near and far-field characteristics of isolated gold nanodimer. (a) The absorption cross section as functions of the incident wavelength and illumination intensity. (b) The enhanced electromagnetic field and the temperature field as a function of the illumination intensity in the near-field region. (c) The nonlinear relationship of the gap temperature and the G_IEF as a function of the illumination intensity.

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For practical use, we provide fitting functions of G_IEF(I), G_IEF(T), and T(I) in Table 1. For the case of isolated nanosphere (nanodimer), the sign of the second-order terms are all positive (negative), indicating nonlinearly enhanced (suppressed) growth with the illumination intensity or the global temperature. Finally, to verify our modeling, results extracted from several other studies are compared. The magnification of $G_{E F} (\vec{r})$ as a function of light intensity is observed in [16]. Up to 10 times magnification were obtained which coincides with our calculation. In addition, the nonlinear magnification of $G_{E F} (\vec{r})$ as a function of the global temperature is consistent with [15], and the temperature rise as a function of the illumination intensity is also compared with [22], [27], and [37]. Note that in our result, the predicted temperature rise is a bit higher. This is because the surrounding medium is air, having relatively lower thermal conductivity as opposed to water in the given references. The agreements between our predictions and those reported in literatures are well within the same order of magnitude, showing that the present model considering the electro-opto-thermal interaction is very reliable and superior to many existing estimations of the enhancement factor for SERS.

Table 1. Fitting Functions of G_IEF(I), G_IEF(T), and T(I) near LSPR for Gold Nanosphere and Gold Nanodimer.

View Table

4. Conclusion

We developed a numerical technique to analyze the contribution of nonlocal electro-opto-thermal interactions to the enhancement of the electromagnetic field as well as the plasmonic heating, and is applicable to any nanostructures. For the gold nanosphere under weak light illumination (I<1.5 mW/μm²), the spatially averaged interaction enhancement factor ${〈 G_{I E F} (\vec{r}) 〉}_{S u r}$ was amplified by ~10 times and the surface temperature was raised to over 300°C, showing very close agreement with experimental observations in literatures. For the nanodimer, a large field enhancement factor of 10⁷ accompanied by much suppressed plasmonic heating was demonstrated, showing potential contribution to mitigate molecule degradation and blinking. We conclude that the interaction enhancement factor is more appropriate than the conventional definition for SERS, where the electro-opto-thermal interactions are all included to reveal the electromagnetic and temperature field with high fidelity.

Acknowledgments

This work was sponsored by the National Science Council, Taiwan (R.O.C.). The authors would like to thank for the grant support under contact number NSC100-2112-M-008-009-MY3.

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	Gold Nanosphere
G_IEF(I)	1 + 1.62I + 2.33I² + 0.70I³
G_IEF(T)	1 + 0.01ΔT + 3.61 × 10⁻⁵ΔT² + 9.51 × 10⁻⁸ΔT³
T(I)	25 + 202.2I + 27.5I²-11.8I³

	Gold Nanodimer
G_IEF(I)	1 + 0.50I-0.16I² + 0.03I³
G_IEF(T)	1 + 3.56 × 10⁻³ΔT-4.25 × 10⁻⁶ΔT² + 2.45 × 10⁻⁹ΔT³
T(I)	25 + 141.97I-25.66I² + 4.21I³

The contribution of nonlocal electro-opto-thermal interaction to single molecule nonlinear Raman enhancement

Abstract

1. Introduction

2. Model and numerical procedures

3. Results and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Tables (1)

Equations (5)

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