1. Introduction

In 1977, Van Duyne et al. and Creighton et al. reported strong enhancement to the Raman signal of pyridine molecules adsorbed on rough electrode when compared with those measured in solution [1,2]. This interesting phenomenon was later found to be caused by the strong plasmonic enhancement near the metal surface and named as surface enhanced Raman scattering (SERS) [3,4]. Compared with ordinary Raman spectra, SERS inherits all of the advantages of the Raman process, and has much higher sensitivity and better resolution which enables the detection of ultra-low concentration substances. Nowadays, SERS has found a wide range of applications in areas including single molecule imaging [5–9], chemical reaction tracking [10–12], surface characterization [13], food safety [14,15], social security [16–19] and biomedical research [20–25].

The underlying mechanism of the SERS effect has been extensively studied over the past few decades. Among the different enhancement mechanisms, the physical enhancement caused by the enhanced electromagnetic field and the chemical enhancement related to the charge transfer effect are widely recognized [26–29]. In the two mechanisms, the physical enhancement is especially interesting and can explain well the vast enhancement of the Raman signal as observed in the SERS experiment [27,30–32]. Such enhancement was a result of the enhanced electromagnetic near field due to the generation of surface plasmons which describes the collective oscillation of the free electrons near the surface. In particular, for the metal surfaces with fluctuations and curvatures, the plasmon resonance will be confined to a certain geometric range and lead to the localized surface plasmon resonance (LSPR) which can greatly enhance the electromagnetic near field. As a result, both the excitation field and the scattering field can be enhanced which lead to a Raman enhancement at the fourth power of the LSPR induced field enhancement [30,31,33].

In the present work, we aim to investigate the physical enhancement effect in the SERS process of a generic molecule placed inside a sub-nanometer nanogap between two metallic nanoparticles (NPs). A particularly interesting aspect of such sub-nanometer nanogaps, which has indeed been utilized in recent single-molecule Raman measurements [8], is that it could support significant quantum tunneling effect [34]. It has been shown by previous calculations that the quantum tunneling effect could affect the plasmonic enhancement in the cavity of metallic nanogaps [35,36]. Thus it is expected that such an effect could have significant influence on the SERS spectra of molecules confined in sub-nanometer nanogaps. To this end, we systematically studied the SERS properties of a model molecule inside the sub-nanometer metallic cavities by taking into account the quantum tunneling effect. The quantum corrected model (QCM) as proposed by Esteban et al. [36,37] was applied to describe the electron tunneling effect between the NPs. The SERS spectra of the molecule were computed by combining finite element method simulations and the density matrix method developed by Xu and Johansson et al. [30,31]. Such a combined method allows the efficient treatment of both the quantum corrected filed enhancement and the actual molecular dynamics and enables the detail analysis of the influences of the different factors that could affect the SERS spectra of the molecule. Two types of commonly used SERS substrates, gold and silver NP dimers, were considered in the simulations. The influence of the quantum tunneling effect as well as the enhanced molecular decay rates on the SERS processes were systematically analyzed based on the simulation results.

2. Model and theoretical method

2.1 Model for the SERS simulation

We used a model system shown in Fig. 1 to describe the SERS process. A generic molecule was assumed to be confined in the center of the nanocavity formed by a pair of metallic NPs with radius $R$ and separation $d$ [Fig. 1(a)]. The nanocavity was illuminated by the incident light propagating in the $z$ direction with electric field along the $x$ direction. For the sake of simplification, we assumed that the photon scattering process takes place between the molecular ground state and an excited state that was on resonance with the incident light, as demonstrated in Fig. 1(b). The vibrational degree of freedom was included by coupling each electronic state to a harmonic oscillator. The two harmonic potentials were displaced with each other to describe the electron-vibration coupling and the vibrational structures in the resulting spectrum. The possible scattering processes, including Rayleigh scattering (green line), Stokes Raman scattering (red line) and anti-Stokes scattering (blue line) are also demonstrated in Fig. 1(b). It is worth to mention that, under the resonant condition, Raman transitions other than the usual fundamental 0-1 transition are also allowed which could lead to additional peaks in the obtained SERS spectra.

 

Fig. 1. (a) A schematic diagram of the model used in the simulation. The shaded area in the center of the gap represents the effective medium of the QCM. (b) The energetic diagram of the possible light scattering processes.

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2.2 Theoretical method

The density matrix method as proposed by Xu and Johansson et al. [30,31] was applied to calculate the SERS spectra of the model molecule. It has been shown that the scattering cross sections of a molecule described by the model shown in Fig. 1 can be computed by evaluating the dipole-dipole correlation function $\left \langle {{P^ - }} \right.(r,0)\left. {{P^{\rm {\ +\ }}}(r,\tau )} \right \rangle$ with $P^ - (r,0)$ and $P^ + (r,\tau )$ corresponding to the absorption and emission processes, respectively. The correlation function can be obtained from the time evolution of the system density matrix based on the quantum regression theorem [31]. After the stationary state of the system was obtained, the molecular spectra that carries both the SERS and the fluorescence signals can be calculated as [31]:

In most of the previous studies, the electromagnetic enhancement factor M$_{e}$ was treated based on classical electromagnetic simulations and did not take into account the quantum tunneling effect. In the present work, we would like to exam the influence of such an effect on the SERS properties of molecular systems. In principle, a full quantum mechanical description will be needed to study the tunneling effect. However, for realistic systems with excessive number of electrons, such a quantum treatment could be too expensive for practical calculations. To this end, we have adopted the QCM as proposed by Esteban et al. [36,37] which incorporates the quantum tunneling into the classical electrodynamics by constructing an effective medium (shown as the red area in Fig. 1(a)) in the nanocavity. The QCM greatly reduces the computational difficulties and enables the efficient calculation of the electromagnetic enhancement with the consideration of the tunneling effect. The efficiency and accuracy of the QCM method have been systematically discussed in previous works [38].

Within QCM, the effective medium is assumed to be a special metal with dielectric constant ($\varepsilon _{\rm {g}}$) given by the Drude model [36]. Based on the wave-packet propagation method [39], the gap distance ($\ell$) dependent damping of the effective medium $\gamma _{\rm {g}}$ that takes into account the quantum tunneling effect can be obtained as $\gamma _{\rm {g}} =\gamma _p{{\rm {e}}^{\ell /{\ell _c}}}$ with $\gamma _p$ and ${\ell _c}$ being the damping parameter and the decay length of the damping in the gap, respectively. It can be found that the $\gamma _{\rm {g}}$ varies exponentially with the distance which is a result of the exponential dependence of the tunneling processes on the separation of the metallic NPs. On the basis of the Drude model, the dielectric constant of the effective medium can be obtained as [37]

It can be found from Eq. (2) that, as $\ell$ decreases, the two particles will come into contact, and the barrier at the dimer gap gradually disappears. As a result, the dielectric constant of the effective medium tends to the dielectric constant of the metal $\varepsilon _{\rm {m}}$. In contrast, when the dimer separation is wide enough, the dielectric constant of the effective medium will approach the dielectric constant of the surrounding environment.

2.3 Simulation details

In this work, we used a generic model with an oscillating point dipole with dipole length of $p_{0}$=1.2 Å and electronic excitation energy of 2.35 eV [30,31] to model the molecule confined in the nanocavity formed by the metallic NPs. A single vibrational mode with frequency of 0.16 eV was included in the calculations. In order to describe the electron-vibration coupling effect, the Huang-Rhys factor which is a measure of such coupling strength has been set to 0.25 following the works of Xu et al. [30,31]. Such a model resembles quite nicely the vibrational structures of a Rhodamine 6G (R6G) molecule and also gives a general description of a Franck-Condon active mode.

The incident light was assumed to be linearly polarized along the axis of the dimer with energy of 2.45 eV. Two types of noble metals that were commonly used as SERS substrate, gold and silver, were considered in the simulations. The distance $d$ between the two metallic NPs was set to vary in the range of 0.1$\sim$1 nm. Such small distances were considered because the tunneling effect is expected to play an important role only at such short distances. The dielectric function of the NPs was described by the Brendel-Bormann model proposed by Rakićet al. [40]. The decay lengths of the damping in the gap ($\ell _c$) and the $d$-electron contributions ($\ell _d$) have been set to 0.042 nm and 0.079 nm for Ag and 0.040 nm and 0.079 nm for Au [37], respectively. Finite-element method (FEM) simulations were performed to obtain the electric field enhancement factors for both the classical electromagnetic (CEM) and the quantum-corrected (described by the QCM) cases. The enhancement of the radiative and non-radiative transition rates were simulated by the generalized Mie theory with the GMM-Dip code [41]. The molecular SERS spectra were obtained from Eq. (1) after solving the density matrix equations.

3. Results and discussion

3.1 Electric field enhancement

Before the simulation of the molecular SERS spectra, we first investigated the electromagnetic enhancement effect that was experienced by the molecule in the metallic nanocavity. Both CEM and QCM calculations were performed to demonstrate the influence of the quantum tunneling effect on the localized near field.

Figure 2(a) and (b) show the relationship between the field enhancement at the center of the dimer and the radius of the Ag and Au NPs with a separation of $d$=0.1 nm. The incident wavelength has been set to 600 nm in this set of simulations. It can be found that the field enhancements are quite sensitive to the changes of the radius of the NPs for both the CEM and QCM cases. A dramatic decrease of the field enhancement was also evident for the QCM results which is as expected due to the involvement of the tunneling effect that could neutralize the charge densities in the nanogap [36]. Interestingly, there are also some differences between the profiles of the field enhancements obtained with the two type of models. Especially, the QCM enhancement experienced much faster decrease than the CEM counterparts with the increase of the radius (see also Fig. S1 in the Supplement 1 for the same plot with linear scale). Such a feature is caused by the different energy shift of the plamonic peaks obtained with QCM when compared to that predicted by CEM [34,36]. For both metals, a relatively large field enhancement appears near the radius $R$=40 nm except for the QCM case in Au. For the sake of simplicity, the radius for the NPs was then set to 40 nm in the following calculations.

 

Fig. 2. The electric field enhancement as a function of the radius of the NPs at the center of the Ag (a) and Au (b) dimers. The red and blue lines represent the CEM and QCM results, respectively. $d$=0.1 nm, $\lambda$ = 600 nm.

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The quantum tunneling effect is strongly dependent on the distance between the two NPs in the dimer. To this end, we also systematically investigated the gap size dependence of the electric field enhancement factors of the two types of metallic dimers as functions of the gap size and the excitation wavelength. The simulation results are demonstrated in Fig. 3. From the CEM results of Ag and Au NPs dimers, as respectively shown in Fig. 3(a) and (b), it can be found that there are two main plasmonic peaks in the considered energy range. Together with the decrease of the gap size, the plasmonic peaks underwent continued red-shift, which could be attributed to the Coulomb interaction between charge densities in the two NPs in the dimer [31,36,37]. Another feature of the CEM results is that the plasmonic peaks experienced a monochromatic increase with the decreased gap sizes. For the QCM results, as shown in Fig. 3(c) and (d) for the two types of metals, it can be clearly seen that the quantum effect can lead to strong suppressions to the local electric field and greatly reduce the field enhancements at shorter gap distances ($d<$ 0.34 nm). At larger separations, the contribution of the quantum tunneling effect decreases gradually, and the electric field distribution calculated by the QCM method is similar to, and eventually converges to, the CEM calculation results. These features agree well with previous calculations [36,37] and indicate that the quantum tunneling effect could have significant influence on the molecular spectra in the short gap-distance regime.

 

Fig. 3. Colour plots of the local electric field enhancement at the center of the Ag (a and c) and Au (b and d) dimers. (a) and (b) were obtained with CEM calculations. (c) and (d) were obtained with QCM calculations.

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Before we move on to the investigation of the molecular SERS properties, it is also interesting to exam the distribution of the electric field in the gap region. Figure 4 shows the electric field distribution in the range of the AgNPs dimer with a 0.1 nm gap, which is similar to the results of AuNPs dimer (Fig. S2 in the Supplement 1). The results obtained with CEM as shown in Fig. 4(a) and (c) show that the electric field reaches the maximum at the center of the dimer which should be attributed to the most intense plasmonic resonance at the center of the dimer where metal-metal distance is the shortest. For the QCM results, on the other hand, the electric field at the center of the nanocavity is strongly suppressed which leads to a ring-like structure for the field distribution as can be found in Fig. 4(b) and (d). This interesting phenomenon is the result of the tunneling effect which is strongly dependent on the gap distance and peaks near the center of the nanocavity. For the 0.1 nm gap, the tunneling effect is strong enough to allow the direct electron transfer between the two metallic NPs and reduce the field enhancement which causes the opposite plasmon response for QCM and CEM at the center of the gap (a 2d line plot along the $x$-axis for the two types of NPs can be found in Fig. S3 in the Supplement 1). For molecules located at the center of the nanocavity, such strong difference between the QCM and CEM results indicates that the quantum tunneling effect is expected to have significant influences on the Raman responses. The slight asymmetries of the field distributions as can be found in Fig. 4(c) and (d) are caused by the propagation direction of the field also the $z$ direction.

 

Fig. 4. The electric field distribution in the nanocavity of the Ag dimer as simulated with CEM (a and c) and QCM (b and d). (a) and (b) are side views and (c) and (d) are vertical section at the center of the dimer. $d$=0.1 nm, $\lambda$ = 600 nm.

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3.2 Molecular SERS spectra

On the basis of the above calculations on the electric field enhancement factors, the SERS cross sections of the model molecule with and without the consideration of the quantum tunneling effect were computed by the density-matrix method. The calculated SERS spectra as a function of the gap size are shown in Fig. 5. The spectra as computed based on the CEM and QCM field enhancements were presented as red and blue lines, respectively. For each spectrum, a series of scattering peaks can be found together with a rather broad fluorescence background. The peak located at 2.45 eV corresponds to the elastic Rayleigh scattering. The low-energy peaks located at 2.29 eV, 2.13 eV and 1.97 eV are the Raman peaks for the fundamental 0-1 transition, the 0-2 overtone and the 0-3 transition, respectively. It is worthwhile to note that the two Raman peaks associated with the 0-2 and 0-3 transitions are active because the resonant condition was reached at the incident energy of 2.45 eV.

 

Fig. 5. The Raman scattering cross section as simulated for the Ag (a) and Au (b) dimers. The red and blue lines are the results calculated based on the CEM and QCM enhancements, respectively. $\hbar \omega _{in}$=2.45 eV, $\hbar \omega _{mol}$=2.35 eV, $\hbar \omega _{vib}$=0.16 eV.

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A number of interesting features can be found from the results presented in Fig. 5. Firstly, it can be found that the CEM (red lines) and QCM (blue lines) results are essentially the same at larger gap sizes (0.55 nm and 0.65 nm for Ag and Au). Such a result is due to the fact that the quantum tunneling process is exponentially dependent on the size of the nanogap. At larger distances the tunneling effect would have negligible contribution and lead to the similar behaviors for QCM and CEM. In contrast, the scattering cross sections as obtained with QCM enhancements were significantly suppressed at smaller gap sizes where the quantum tunneling effect starts to strongly reduce the field enhancements. At the shortest gap distance of 0.1 nm that was considered, the quantum tunneling induced suppression to the fundamental Raman peak is 10$^{10}$ and 10$^{6}$ for Ag and Au, respectively. These results indicate that the quantum tunneling effect could have strong influences on the SERS spectra of molecules placed in sub-nanometer gaps and have to be taken into account when such effect is expected to take place.

Secondly, a change of the relative intensities of the Raman peaks with the change of the gap size can also be observed. Together with the decrease of the gap size, the 0-2 overtone peak gradually becomes the more intense Raman peak in both Ag and Au dimers. Such a feature should be attributed to the red-shift of the plasmonic peaks which resulted in the larger field enhancement to the peaks with lower energies.

Another interesting aspect is that, for both Ag and Au, a decay of the overall scattering cross section together with the decrease of the gap size can be observed in Fig. 5. Especially, for the case of Au dimer, a significant drop of the scattering intensity can be found in the considered gap regime except for a small increase at around 0.21 nm. Such changes are clearly demonstrated in Fig. 6(a) and (b) where we have plotted out the intensity of the fundamental Raman peak at 2.29 eV as a function of the gap size. To better understand such changes, the Raman enhancement as predicted solely from the field enhancements via the simplified relation of $|M_{exc}|^2|M_{sca}|^2$ ($M_{exc}$ and $M_{sca}$ represent the field enhancement factors at the excitation and scattering energies, respectively, as shown by the dashed lines in Fig. 3) as well as the decay rates of the molecular excited state in the nanocavity were also shown in Fig. 6(c) and (d).

 

Fig. 6. The fundamental Raman peak, the field enhancement factors $|M_{exc}|^2|M_{sca}|^2$ and the decay rates as a function of the separation between the NPs in the Ag (a and c) and Au (b and d) dimers. The red and blue lines are the results calculated based on the CEM and QCM enhancements, respectively. The green lines in (c) and (d) show the relationship between the decay rates and the gap sizes.

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From the results shown in Fig. 6, it can be found that the suppression of the scattering cross section is associated with the strongly enhanced decay rates in the sub-nanometer dimer cavity. For the case of Ag, the electric field enhancement is able to balance the increase of the decay rates together with the decrease of the gap size before the quantum tunneling effect starts to come into play. As a result, the scattering cross section did not experience significant decrease at relative larger gap distances. Further reducing the gap size starts to cause clear suppressions for both QCM and CEM spectra due to the faster enhancement of the molecular decay rate. For the QCM result, the involvement of the quantum tunneling effect is responsible for the sharp decrease to the Raman cross section at gap sizes smaller than about 0.34 nm. For Au dimers, the field enhancement does not show much increase because the incident and scattering photon are not on resonance with the intense plasmonic peaks. In fact, the red-shift of the plasmonic peak even causes a decrease to the field enhancement in the CEM case. As a result, a constant decrease to the Raman cross section was observed. The small recovery of the scattering cross section at gap size around 0.21 nm in the CEM case can also be attributed to the energy shift of the plasmonic peaks which starts to coincide with the energy of the incident photon in such distances. These interesting results highlight the importance of the quantum tunneling effect as well as the enhanced decay rates on the molecular SERS process and both should be considered when investigating SERS in highly confined nanocavities.

It is worth to note that the quantum tunneling effect as demonstrated in the present work could be difficult for larger molecules such as R6G because the tunneling current decays exponentially with the increase of the gap size. We expect such an effect to be significant for molecules that could be confined into sub-nanometer gaps such as molecules with planar configurations. One group of molecules that could fulfill such a purpose is porphyrin derivatives that was commonly used in the current state-of-the-art single-molecule Raman measurements [5,8,9]. For example, the Raman spectra and corresponding optical images of a signal Co(ii)–tetraphenyl porphyrin molecule at tunneling gaps with size smaller than 0.2 nm have been reported by Lee et al. recently [8] where the tunneling effect could play an important role. Another possibility is to anchor the molecule strongly to the metallic dimers to form a metal-molecule-metal tunneling junction as commonly used in the field of molecular electronics. In such cases, the strong coupling between the molecule and the metallic dimer could open up new electron tunneling channels. SERS spectra of molecular systems in such configurations have been reported previously [42,43], which could supply another possible route to experimentally exam the quantum tunneling effect as studied in the present work.

4. Conclusion

In this work, we systematically studied the effect of the quantum tunneling effect on the SERS properties of molecules placed inside metallic nanoparticle dimers. Calculation results show that with the decrease of the dimer separation, the plasmonic enhancement to the localized field in the dimer cavity firstly shows rapid increase until when the quantum tunneling effect starts to take place at short separation range ($d <$ 0.34 nm). In such regime, the field enhancement starts to be strongly suppressed because the tunneling effect reduces the collective oscillation of the charge carriers. A clear energy-shift to the plasmonic peaks was also observed which is consistent with previous calculations [36]. As a result, the quantum tunneling effect also greatly reduced the SERS intensity of the molecule. We also demonstrated that in the sub-nanometer nanocavities considered in the present work, both the quantum tunneling effect and the strongly enhanced molecular decay rates have to be considered to understand the possible changes to the molecular SERS spectra. These results could be helpful for the further investigation of the molecular SERS properties with highly confined plasmonic nanocavities.