1. Introduction

Surface-enhanced Raman spectroscopy (SERS) has received much of attention recently due to a very broad range of its applications ranging from optics and plasmonics to biochemistry and medicine [1–3]. High scattering intensities within narrow spectral bands reduce the probability for spectral overlapping to allow for better recognition of multiple markers, making SERS one of the most efficient optical sensing techniques. With the development of advanced nanomaterials, various SERS substrates are demonstrated [4–10]. However, there is still a need for inexpensive substrates of improved sensitivity and signal reproducibility, which require clear understanding of the underlying scattering mechanisms to be developed.

In general, the SERS effect originates from the resonance increase of the induced transition dipole moment of an atomic or molecular scatterer when positioned in the near-surface zone of a metallic structure. This can be for two reasons. They are: (i) due to quasi-static electric fields associated with resonance plasmon excitation in metallic structures, and/or (ii) due to the electron polarizability increase associated with the charge transfer between the substrate and the scatterer. It is generally agreed to distinguish between the electromagnetic (EM) and chemical SERS effects, accordingly [1–3].

Most of the applications of carbon nanotubes (CNs) to increase the Raman scattering signal have been to decorate them with metallic nanoparticles [8, 9], in order to obtain local EM field enhancement generated by the spatially confined plasmon modes of the nanoparticles with CNs only used as a network to support the particles. Only very recently, the chemical SERS effect with CNs alone, without any combination with metallic nanoparticles, was first reported by Andrada et al. in [10], where molecules covalently bound to single wall CNs demonstrated the resonantly increased Raman signal that was even stronger than that of the nanotube itself.

In this article, a quantum electrodynamics (QED) theory of the resonance Raman scattering is presented for an atom near a carbon nanotube, to demonstrate that individual CNs are capable of providing the electromagnetic SERS effect as well. Nanotubes offer extraordinary stability, flexibility and precise tunability of their EM properties on-demand by simply varying their diameters and/or chiralities. CNs of different diameters and chiralities feature similar electronic band structure peculiarities, yet shifted in frequency relative to one another [11–13]. This yields similar EM properties over a broad range of excitation frequencies both in the far- and in the near-field zone, originating from exciton and plasmon excitations, respectively. Excitons and plasmons are different in their physical nature, but originate from the same circumferentially quantized electronic transitions. Due to the circumferential quantization of the longitudinal electron motion, real axial (along the CN axis) optical conductivities of single wall CNs consist of series of peaks E ₁₁ ,E ₂₂ ,…, representing the 1st, 2nd, etc. excitons, respectively [see Fig. 1(a)]. Imaginary conductivities are linked with the real ones by the Kramers-Kronig relation, and so real inverse conductivities show the resonances P ₁₁ ,P ₂₂ ,… [Fig. 1(a)] next to their excitonic counterparts. These are inter-band plasmons that were theoretically demonstrated quite recently to play the key role in a variety of new interesting surface EM phenomena with CNs [16–26], including exciton-plasmon coupling [16, 17] and plasmon generation by excitons [18, 19], exciton Bose-Einstein condensation in individual single wall CNs [20], Casimir attraction in double wall CNs [17, 21, 22], resonance optical absorption [23] and atomic entanglement in hybrid systems of extrinsic atoms/ions doped into CNs [24–26], to mention a few, — all of direct relevance to conceptually new tunable optoelectronic device applications with carbon nanotubes [27, 28]. Experimental evidence for these low-energy (~1–2 eV) weakly-dispersive plasmon modes in CNs was first reported by Pichler et al. in [29]. Inter-band plasmons are standing charge density waves due to the periodic opposite-phase displacements of the electron shells with respect to the ion cores in the neighboring elementary cells of the CN [18, 19]. When excited, their (plasmon-induced) quasi-static electric fields can be strong enough to result in the enhanced Raman scattering effect by atomic type species (extrinsic atoms, ions, molecules, or semiconductor quantum dots) in the CN vicinity. This work derives and analyzes the differential cross-section for such scattering.

 

Fig. 1 (a) Fragment of the energy dependence of the dimensionless (normalized by $e^2/2\pi \overline{h}$) axial surface conductivities σ_zz for the semiconducting (6,4), (10,0) and (11,0) nanotubes of increasing diameter. Peaks of Reσ_zz represent excitons (E ₁₁, E ₂₂, …); peaks of Re(1/σ_zz) are inter-band plasmons (P ₁₁, …). (b) Photonic DOS functions for the CNs in (a) with the TLS placed at the distance r_A=R_cn +2b (see inset). Dimensionless energy is [Energy]/2γ ₀. Conductivities are obtained using the (k · p)-scheme developed by Ando [12]. DOS functions are calculated as described by Bondarev and Lambin in [14, 15]. See text for notations.

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2. The Hamiltonian

In absence of external EM radiation, an atom, modeled here by a two-level system (TLS) positioned at the point r _A near an infinitely long single wall CN, interacts with the quantum EM field of the CN via an electric transition dipole moment $d_z=〈u\left|{\widehat{d}}_z\right|l〉$ between the TLS lower and upper states, |l〉 and |u〉, respectively, with the z-quantization axis being the CN symmetry axis [Fig. 1(b), inset]. Transverse dipole orientations can be neglected due to the strong transverse depolarization effect in individual CNs [30–34]. The full QED second quantized Hamiltonian for such a coupled CN–TLS quantum system was earlier formulated by Bondarev and Lambin in [14, 15] to have the following form

The matrix elements of the CN field interaction with the TLS are of the form

Functions g^(±)(r _A, R ,ω) have the property as follows

Hamiltonian (1) involves only two standard approximations, the electric dipole and two-level approximation [14, 15], while conveniently representing the coupled TLS–CN system in terms of the relative distance dependent DOS functions ξ ^⊥(||)(r _A,ω). For short TLS–CN separation distances EM retardation effects play no role [15], and so one has ξ ^⊥ = ξ^||= ξ(r _A,ω) for the DOS functions in Eq. (2). Figure 1(b) shows examples of ξ (r_A,x) as functions of the dimensionless energy $x=\overline{h}\omega /2{\gamma }_0$, where γ ₀ =2.7 eV is the C-C overlap integral, calculated for the three semiconducting CNs of increasing diameter with the TLS positioned at r_A = R_cn+ 2b, b=1.42 Å being the C-C interatomic distance. Details of these calculations and similar graphs for other CN–TLS separation distances can be found in [14, 15, 34]. We see the sharp single-peak resonances originating from the inter-band plasmons of the respective CNs [cf. Figs. 1(a) and 1(b)]. These are responsible for the CN–TLS coupling in the near-field. The coupling is due to the virtual (vacuum-type) EM energy exchange between the TLS and the CN to create and annihilate plasmon excitations on the CN surface with the TLS de-excited and excited, respectively, as described by the second line in the Hamiltonian (1).

3. Eigen states spectrum

To proceed with the Raman scattering cross-section calculations, it is necessary to determine the spectrum of the eigen states of the Hamiltonian (1). In the linear coupling regime, quite generally, the coupled CN–TLS system can be represented as a four-level system with the eigenvectors of the Hamiltonian (1) of the form

Here, |{0}〉 and |{1(R ,ω)}〉 are, respectively, the vacuum and single-quantum excited states of the CN field subsystem, and $C_{u,l}^{(1,2)}$ are unknown mixing coefficients for the non-radiative spontaneous decay transition |u〉|{0}〉 →|l〉|{1(R ,ω)〉 to excite one plasmon of frequency ω at point R of the CN surface with simultaneous de-excitation of the TLS [15,34]. These mixing coefficients can be found by solving the eigenvalue problem for the Hamiltonian (1) in the basis (3). Similar mixing of the |l〉|{0}〉 and |u〉|{1(R ,ω)}〉 states, known to be responsible for the long-range dispersive van der Waals interaction [14, 15], is neglected here for simplicity.

Solving the eigenvalue problem for the Hamiltonian (1) in the basis (3), one obtains the energy eigenvalues

Here, the second equation gives

Here, ε_i = E_i/2γ ₀ with i=0,1,2,3 and $({\tilde{x}}_A,x_p,\mathrm{\Delta }{x}_p)=\overline{h}({\tilde{\omega }}_A,{\omega }_p,\mathrm{\Delta }{\omega }_p)/2{\gamma }_0$ are dimensionless energies, $\delta ={\tilde{x}}_A-x_p$, $X=(\overline{h}/2{\gamma }_0){\left[2\mathrm{\Delta }{\omega }_0{\mathrm{\Gamma }}_0({\omega }_p)\left(1+{\omega }_A^2/{\omega }_p^2\right)\xi \left({\mathbf{r}}_A,{\omega }_p\right)\right]}^{1/2}$, and Δx_p is added to phenomenologically account for the finite half-width of the plasmon resonance (finite plasmon lifetime), as seen in Fig. 1(a), which is assumed to be much broader than the excited atomic level natural half-width dropped here on this account for simplicity.

The mixing coefficients in the eigenvectors |1〉 and |2〉 in Eq. (3) can be found straightforwardly by using the normalization condition

Equations (3), Eq. (8) and (9) represent the complete solution to the eigenvalue problem for the Hamiltonian (1) of the coupled CN–TLS system in absence of external EM radiation. They are valid both in resonance, where δ ~0 and so X ² /δ ²≫1, and out of resonance where X ² /δ ²≪1, and give different easily derivable asymptotical expressions in these two regimes of relevance to strong and weak CN–TLS coupling, respectively. The eigen energy level structure given by Eq. (8) is sketched in Fig. 2. We see the anti-crossing behavior of the eigen energy levels ε ₁ and ε ₂, a characteristic of the strong coupling regime, that is controlled by the parameters X and Δx_p. When in resonance, the actual coupling regime, strong or weak, depends on the relation between X and Δx_p. The CN–TLS system will only be coupled strongly if the parameter X, which plays the role of the vacuum-field Rabi splitting here [15, 23, 34], is much greater than the plasmon resonance broadening Δx_p as shown in Fig. 2. When X ≫ Δx_p, the Rabi splitting of the levels ε ₁ and ε ₂ is not hidden by the plasmon resonance broadening, and so the strong CN–TLS coupling regime is realized. Otherwise, if X ≪ Δx_p, the levels ε ₁ and ε ₂ are smeared, showing no clear anti-crossing behavior, and so no strong coupling can be realized.

 

Fig. 2 Schematic of the energy level structure as given by Eq. (8) for the coupled four-level CN–TLS system. Thick red lines show the eigen energy levels as functions of x_p. Thin red dashed lines indicate their broadening due to finite Δx_p. Horizontal dotted lines are to show Rabi-splitting and the in-resonance strong-coupling solutions given by Eq. (8) with δ = 0.

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4. Raman scattering cross-section

Under the assumption that the coupled CN–TLS system with the eigen states (3), (8) and (9) is initially in the ground state, the inelastic scattering of external EM radiation by this system only involves transitions between levels |0〉, |1〉 and |2〉, as shown in Fig. 3, top, due to the dipole moment selection rule restrictions. The entire scattering process includes three sequential steps. They are:

 

Fig. 3 Schematic of the Raman scattering process (top) in terms of the inter-level transitions (levels sketched in Fig. 2) of the coupled CN–TLS system given by Eqs. (3), (8) and (9), and the Feynman diagrams (bottom) for the scattering cross-section calculations.

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There are four Feynman diagrams, shown in Fig. 3, bottom, to contribute to this three-step process. They are two for plasmon emission (bottom left) and two for plasmon absorption (bottom right), to represent two indistinguishable ways for emission and absorption to occur. Two types of the emission (absorption) diagrams should be summed up and squared, followed by adding the emission and absorption contributions together [37], to result in the Fermi Golden Rule transition rate

Matrix elements in here can be consistently evaluated within the Lorentzian approximation for the DOS function ξ (r _A,ω). Substituting $C_l^{(1)}$ out of Eq. (6), with E replaced by E ₁ per Eq. (8), into Eq. (11) and performing exactly the same integral evaluation as was done in Eq. (7), one has

To obtain the differential scattering cross-section, one has to multiply Eq. (12) by the density of final states ${\left(\overline{h}{\omega }_s\right)}^{2}d\left(\overline{h}{\omega }_s\right)d{\Omega }_s/{\left(2\pi \overline{h}\right)}^3$ for photons scattered within the solid angle dΩ_s, and then integrate it over $\overline{h}{\omega }_s$

, the scattered radiation energy [36]. With Eqs. (8), (13) and (14), this eventually results in the differential Raman scattering cross-section as follows

${\delta }_{\pm }=\delta \pm \sqrt{{\delta }^2+X^2}$, and only the resonant terms are left to represent the contributions from plasmon emission and absorption, respectively, while (insignificant, quasi-constant back ground) non-resonant terms of the transition rate (12) are dropped for brevity.

5. Discussion

Each term in Eq. (16) has the product of two resonance energy denominators that include x_i and x_s, incident (incoming) and scattered (outgoing) photon energies. This is what makes the Raman scattering cross-section (15) resonant. In addition to that, there is an important pre-factor there, A(δ, X, Δx_p), given by Eq. (17). This comes from Eq. (13), which can be viewed as $[d_z{E}_z({\omega }_i)][d_z{E}_z({\omega }_s)]{[d_z{E}_z^{(loc)}({\mathbf{r}}_A)]}^2$ with $d_z{E}_z^{(loc)}({\mathbf{r}}_A)~X\propto {[{\mathrm{\Gamma }}_0({\omega }_p)\xi ({\mathbf{r}}_A,{\omega }_p)]}^{1/2}$, thus bringing the local-field enhancement factor ${[d_z{E}_z^{(loc)}({\mathbf{r}}_A)]}^4\propto {\xi }^2({\mathbf{r}}_A,{\omega }_p)$ into the Raman cross-section due to plasmon generated quasi-static electric fields (see [18,19] for more details) at the TLS location r _A when in the CN near-surface zone r_A~R_CN [Fig. 1(b)]. This factor is significant when δ ~0 and X ≫ Δx_p simultaneously, a regime whereby the CN–TLS system couples strongly by means of the virtual EM energy exchange between the TLS and the nanotube, corresponding to non-exponential spontaneous decay dynamics with Rabi oscillations of the TLS excited state [15, 34], and Rabi splitting of the TLS optical absorption line profile [23].

Maximum Raman intensity is controlled by the ratio $A\left(\delta ,X,\mathrm{\Delta }{x}_p\right)/\mathrm{\Delta }{x}_p^4$ as can be seen from Eq. (16). This is a slightly asymmetric function of δ peaked at $\delta \approx X/\sqrt{3}$ when X ≫ Δx_p, not at δ =0 as one would expect, which can be easily shown by testing it for maximum analytically. Bondarev and Vlahovic have shown earlier in [23] by analyzing the absorption line shape profile that X, the Rabi splitting, may typically be as large as ~0.01–0.1, as, whereas Δx_p ~0.005–0.01 can be seen from Fig. 1(a). Figure 4, left and right, shows the enhancement factor $A/\mathrm{\Delta }{x}_p^4$ as a function of δ and X at Δx_p = 0.005, and as a function of δ and Δx_p at X = 0.05, respectively. We see that as long as X ≫ Δx_p, whereby the CN–TLS system is in the strong coupling regime, the maximum intensity decreases by a factor ~ 1.5 within the detuning window ~0.08, thus providing the spectral band as large as 0.08×2γ ₀ .~ 0.43 eV for a significant plasmon enhanced Raman scattering effect to occur. This spectral band can be shifted both to the red and to the blue by shifting the inter-band plasmon resonance energy, as can be seen in Fig. 1, which can merely be achieved by using CNs with different diameters and/or chiralities, thereby offering the flexibility and precise tunability in designing CN based SERS substrates with parameters required on-demand.

 

Fig. 4 Enhancement factor $A(\delta ,X,\mathrm{\Delta }{x}_p)/\mathrm{\Delta }{x}_p^4$ as given by Eq. (17) for Δx_p = 0.005 (left) and for X =0.05 (right) to show the influence of the detuning $\delta (={\tilde{x}}_A-x_p)$ on the maximum intensity of the plasmon enhanced Raman scattering effect.

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Returning back to Eq. (16) we see that for each x_i = x_p +δ_±/2 only one term contributes, resulting in either Stokes scattering with x_s=x_p+δ₋/2<x_i=x_p+δ ₊/2 and a plasmon created in the CN, or in anti-Stokes scattering with x_s=x_p+δ ₊/2 > x_i=x_p+δ₋/2 and a plasmon absorbed from the CN. The absolute value of the Raman shift is $\left({\delta }_{+}-{\delta }_{-}\right)/2=\sqrt{{\delta }^2+X^2}$, yielding a quantity ∼X in resonance, where δ ∼ 0 whereby X ² /δ ²≫1, and that ∼δ out of resonance with X ² /δ ²≪1. In the latter case, A(δ≫X, Δx_p) ∝X ⁴ /δ ⁴, according to Eq. (17), totally ruling out the probability P(x_i,x_s) of the scattering process. When in resonance, on the other hand, the P(x_i,x_s) maximum value goes as $A\left(0,X,\mathrm{\Delta }{x}_p\right)/\mathrm{\Delta }{x}_p^4\propto \left(X^4/\mathrm{\Delta }{x}_p^4\right)/\left(X^2+\mathrm{\Delta }{x}_p^2\right)$, being strongly suppressed under the weak CN–TLS coupling where $X^2/\mathrm{\Delta }{x}_p^2\ll 1$, and being dramatically enhanced in the case where $X^2/\mathrm{\Delta }{x}_p^2\gg 1$ so that the CN–TLS coupling is strong. The scattering enhancement factor is about square of that for resonance absorption by atomically doped CNs [23], the way it should be for scattering as a two-step process of absorption followed by emission (viewed as ”reversed absorption”).

Figure 5 shows an example of the numerical calculations for the scattering probability P(x_i,x_s) as given by Eq. (16) for $x_p=0.35[P_{11}^{(6,4)}$ plasmon in Fig. 1(a), corresponding to the energy 0.35×2γ ₀ =1.89 eV (red spectral line)] with X and Δx_p being varied independently [rows (a) and (b), respectively], to see the role of the DOS resonance variation due to the local field enhancement/dehancement effect as the TLS–CN separation distance changes and the plasmon decoherence effect, respectively. As discussed above, Raman scattering is seen to be very sensitive to the strong CN–TLS coupling, blowing up by a factor of over 10³ for X/Δx_p ~10 and totally vanishing when X/Δx_p ~1. Raising X increase both the Raman shift and the intensity, while greater Δx_p quench the intensity with no Raman shift change.

 

Fig. 5 Raman scattering probability as a function of the incident x_i and scattered x_s photon energies as given by Eq. (16) with $\delta =X/\sqrt{3}$ (maximum of $A/\mathrm{\Delta }{x}_p^4$ in Fig. 4, see text) for $x_p=0.35[P_{11}^{(6,4)}$ plasmon in Fig. 1(a)], with X and Δx_p being varied independently [rows (a) and (b)]. TLS–plasmon coupling strength is represented by the ratio X/Δx_p being greater or less than unity for strong and weak coupling, respectively. Raman scattering is seen to be manifestly indicative of the strong TLS–plasmon coupling, dramatically increasing as X/Δx_p goes much greater than unity and disappearing when it is comparable with unity.

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The QED theory of the plasmon enhanced Raman scattering developed here applies to chemically inactive atoms, ions, or even organic molecules and semiconductor quantum dots that are physisorbed on the nanotube walls, whereby there is no local electronic orbitals hybridization between the nanotube and the atomic transition levels involved. The theory can be tested experimentally by using rear-earth ion complexes, Eu³⁺ ions, in particular [38–40]. These are known to be excellent probes for near-field effects in spatially confined systems, owing to the dominant narrow, easily detectible 5D ₀ →7F ₂ electric dipole transition of the wavelength ~614 nm between two deep-lying electronic levels (f-shell) of europium that essentially create an ideal TLS. Corresponding transition energy is 2.02 eV, falls within the 0.43 eV spectral band of the first inter-band plasmon resonance $P_{11}^{(6,4)}$ of the (6, 4) nanotube (see Fig. 1, x_p=0.35 corresponding to E_p =1.89 eV) whose calculated Raman spectrum is shown in Fig. 5 and was discussed above. Pre-alignment of europium doped CNs is desirable to facilitate the excitation efficiency, but is not crucially important.

6. Conclusion

In this article, the QED theory of the resonance Raman scattering is developed for a two-level dipole emitter — TLS coupled to a weakly-dispersive low-energy (~1–2 eV) inter-band plasmon resonance of a carbon nanotube. The theory applies to atomic type species such as atoms, ions, molecules, or semiconductor quantum dots that are physisorbed on the nanotube walls. The analytical expression derived for the Raman cross-section covers both weak and strong TLS–plasmon coupling, and shows dramatic enhancement in the strong coupling regime. Such resonance scattering is a manifestation of the general electromagnetic SERS effect, in which the enhancement is due to the plasmon-induced near-fields that affect the TLS in a close proximity to the CN surface, given that the TLS transition energy is within the spectral band of ~0.43 eV of the corresponding inter-band plasmon resonance energy of the nanotube.

The QED scattering model used here belongs to a broad class of driven four-level quantum systems analyzed by Delgado and Gomez Llorente in [41]. There is an important distinction though, that instead of being driven by the strength of the coupling to an external periodic field, the scattering properties of the four-level CN–TLS system considered here are controlled by the strength of the (vacuum-type, intrinsic) coupling of the TLS to the plasmon-induced quasi-static electric fields in the near-surface zone of a carbon nanotube. Similar models are used to describe the peculiarities of EM interaction processes in a variety of quantum systems. These include both cavity-type systems such as atoms in optical traps [42], matter excitations in one-dimensional [43–45] or zero-dimensional [46, 47] nanostructures embedded in solid-state nanocavities, and general molecular [48] and semiconductor systems [37]. In the latter case, the phenomenon of the resonance Raman scattering originates from lambda-type transitions, such as vibrational transitions of a molecule or electron band transitions in a semiconductor, within a subset of states of the same system. In the case considered here, the resonance scattering effect becomes possible due to the formation of the hybridized (”dressed”) states when two different systems, the TLS and the CN, enter the strong coupling regime.

This theoretical work provides a unified description of the near-field plasmon enhancement effects. The work will help establish new design concepts for future generation CN based nanophotonics platforms for single molecule/atom/ion detection, precision spontaneous emission control, and optical manipulation, which will benefit from the extraordinary stability, flexibility and precise tunability of the electromagnetic properties of carbon nanotubes by means of their diameter and chirality variation.