Conference-ISSS-7-Nonequilibrium Green ’ s Function Theory of Scanning Tunneling Microscope-Induced Light Emission from Molecule Covered Metal Surfaces : Effects of Coupling between Exciton and Plasmon Modes

Light emission from interface plasmons and molecular excitons induced by the tunneling current of a scanning tunneling microscope is theoretically investigated using the nonequilibrium Green’s function method. Luminescence property of the system is found to be well interpreted in terms of superposition of two coupled modes that form due to coupling between a molecular excitonic mode and an interface plasmon mode. The coupled mode whose energy is closer to the energy of the molecular excitonic mode (the interface plasmon mode) leads to a sharp (broad) peak structure in luminescence spectra of interface plasmon. Interference between these coupled modes gives rise to suppression or enhancement of luminescence intensity depending on the energy region. Especially, in the energy region near the molecular excitonic mode, the interference leads to strong suppression of luminescence from interface plasmons. Thus novel aspects of interpretation of the luminescence spectra of the system are obtained by using the analytical approach presented here. [DOI: 10.1380/ejssnt.2015.385]


I. INTRODUCTION
Optical properties of the system including both metal nanostructures (MNS) and adsorbed molecules have attracted much attention because the system provides fascinating new physics and has potential applications in highresolution microscopy, sensitive sensors, high-efficient organic optoelectronic devices, and so forth [1][2][3][4][5][6][7].The most striking characteristic of the system is the generation of intense electromagnetic fields associated with the excitation of interface plasmons of MNS.Many of conventional studies focus on the use of electromagnetic fields generated by interface plasmons for enhancing luminescence intensities and/or optical responses of target molecules.In addition to this aspect, the interaction between the electromagnetic fields of interface plasmons and the transition moment of intramolecular electronic excitations results in coupling between a molecular excitonic mode and an interface plasmon mode.This exciton-plasmon coupling gives rise to peculiar phenomena; e.g., occurrence of characteristic features in optical absorption spectra of the system due to quantum interference between the coupled modes [8][9][10][11][12].Understanding details of optical properties of the MNS-molecule system within the framework, in which molecular excitons and interface plasmons are considered as quantum mechanically coupled systems, will open up novel perspectives for interpretation of complex optical phenomena observed in the MNS-molecule system and provide a new possibility for developing future nanophotonics.
Luminescence properties of nanomaterials can be investigated using scanning tunneling microscope (STM)induced light emission (STM-LE) with atomic spacial resolution, where the luminescence is induced by the electron tunneling between the tip of an STM and a substrate [13][14][15][16][17][18][19][20].STM-LE from a clean metal surface can be attributed to radiative decay of interface plasmons localized near the tip/vacuum/substrate interface [21][22][23][24].When a molecule is placed near the tip-substrate gap region, optical processes of molecules and interface plasmons can strongly influence each other.Enhancement of molecular luminescence intensity using intense electromagnetic fields generated by interface plasmons have been investigated in previous theoretical [25][26][27][28][29] and experimental studies [30][31][32][33][34][35].Moreover, recent experimental results have suggested that the dynamics of molecule (e.g.energy absorption and emission of molecules) affect a luminescence spectral profile of interface plasmons [36].Since the dynamics of molecules and interface plasmons affect each other, the interplay between their dynamics can lead to peculiar phenomena that arise from quantum many-body effects.
We have recently constructed a theory of luminescence from interface plasmons and molecular excitons induced by the tunneling current of STM [37][38][39][40].Nonequilibrium Green's function method [41] has been applied to investigate the luminescence properties of the system in two cases: STM-LE from a clean metal surface using a molecule-covered tip and from molecular layers on a metal substrate using a clean metallic tip.Prominent peak and dip structures observed in the recent experiment [36] have been interpreted by the theory developed by us [39,40].Energy transfer from molecular excitons to interface plasmons and energy absorption by the creation of molecular excitons lead to the peak and dip structures in luminescence spectra, respectively.An additional peak structure near the energy of the molecular excitonic mode is attributed to interference between the processes of energy absorption by the creation of molecular excitons and the re-emission of energy in interface plasmons [39,40].Focusing on the coupled modes that form due to the coupling between a molecular excitonic mode and an interface plasmon mode, luminescence spectra of the system can be analyzed in terms of superposition of these coupled modes.Therefore the model analysis of exciton-plasmon interactions, where for simplicity we ignore vibrational dynamics that are considered in previous models [37][38][39][40], is expected to give clearer understanding about origins of some of the peak and dip structures.To obtain the profound knowledge as well as to interpret the calculation results and experimental findings without concerning complex optical processes of the system, it is crucial to examine the mechanism for occurrence of the peak and dip structures from the viewpoint where the contribution of the coupled modes are considered.

II. MODEL AND METHODS
In this paper, we calculate optical properties of the system with the aid of the nonequilibrium Green's function method and analyze the results in terms of superposition of the coupled modes.A schematic illustration of the model is presented in Fig. 1.A molecular exciton is coupled to interface plasmons that are localized near the tip-substrate gap region.Creation of molecular excitons is induced by the absorption of energy of interface plasmons that are excited by the tunneling current of STM.The Hamiltonian of the system which includes couplings of molecular exciton and interface plasmons to photons reads: Here d † (d) denotes the creation (annihilation) operator for the molecular excitonic mode with the energy ϵ ex .d and d † are assumed to obey boson commutation rules.a † (a) is the creation (annihilation) operator for the interface plasmon mode with the energy ) are for the eigen modes β and β ′ in a thermal bath, respectively, and α † l (α l ) is for a photon mode l with the energy ℏν l .The parameters ℏg, U β , U β ′ , U dl , and U al describe exciton-plasmon coupling, coupling between molecular exciton and thermal bath, coupling between interface plasmon and thermal bath, coupling between molecular exciton and photons, and coupling between interface plasmons and photons, respectively.The model Hamiltonian H can be utilized to analyze STM-LE from molecular layers on a metal substrate using a clean metallic tip [33] and from a clean metal surface using a molecule-covered tip [36] because the values of parameters included in H are expected to be on the same order in both cases.
Luminescence spectra of interface plasmon J P and molecular exciton J L are expressed as [40]: where P < and L < are the lesser projections of the Green's function of interface plasmon P and molecular exciton L on the Keldysh contour, respectively.These Green's function are defined as where τ is the Keldysh contour time variable and the operator T C denotes the time ordering along the contour [41].
Although the two kinds of radiative processes, i.e., luminescence from molecular excitons and interface plasmons, contribute to light emission detected experimentally, dominant contribution would vary depending on the configuration of the system.In the case of STM-LE from a clean metal surface using a molecule-covered tip [36], it is expected that the luminescence spectra detected experimentally correspond to J P , since U al can be considerably larger than U dl [40].For STM-LE from molecular layers on a metal substrate using a clean metallic tip [33], the transition dipole moment of intramolecular electronic transition can be strongly enhanced [25] and therefore J L should be observed.
Effects of the tunneling current of STM, which drive the dynamics of the system by exciting interface plasmons, are introduced in the following manner.By assuming the condition of stationary current, the distribution function N pl of interface plasmons excited with the inelastic electron tunneling (IET) between the tip and substrate is given by where T pl is a coefficient related to the IET amplitude [42], e is the elementary charge, and V bias is the bias voltage.The lesser projection of P (0) , which is the plasmon Green's function for exciton-plasmon coupling ℏg = 0, is obtained from the relation, where P (0),r is the retarded projection of P (0) .Then the Green's functions are calculated by considering the effects of exciton-plasmon coupling ℏg.The integral equations for the Green's function of interface plasmon P and molecular exciton L read Here L (0) is the Green's function of molecular exciton for ℏg= 0.
To interpret the optical property of the system, we introduce the operators Here coefficients c 1 and c 2 satisfy the relations The Hamiltonian H ex−pl given by Eq. ( 2) can be diagonalized as , where ϵ A = (ϵ ex + ℏω p + R) /2 and ϵ B = (ϵ ex + ℏω p − R) /2 are energy of coupled modes |A⟩ and |B⟩, respectively.Luminescence spectra of interface plasmon J P and molecular exciton J L can be expressed by superposition of these coupled modes.The lesser projections (P < and L < ) of Green's function of interface plasmon and molecular exciton, that give J P and J L , Eqs. ( 5) and ( 6), are divided into four terms as where is the lesser projection of the Green's function defined as G OO ′ (τ, τ ′ ) = (1/iℏ)⟨O(τ )O ′ (τ )⟩ H . Also, we define the relative phase θ AB between |A⟩ and |B⟩ as The parameters used in the calculations correspond to the STM-LE experiment with tetraphenylporphyrin (TPP) molecules, a silver tip and a silver surface for which ϵ ex = 1.89 eV [36].The energy of the interface plasmon mode ℏω p is set to 2.05 eV and U β ′ is set to give a plasmon lifetime of 4.7 fs for ℏg = 0 which are consistent with experiments [36,43] and earlier simulation results [44,45].The relaxation constant of the molecular exciton for ℏg = 0 is given by ℏΓ ex = ∑ β |U β | 2 πδ (ℏω − ℏω β ) and set to 0.005 eV [40].

III. RESULTS AND DISCUSSION
Figure 2(a) shows luminescence spectra J P of interface plasmon.J P has a sharp peak structure near 1.84 eV and a dip structure near 1.89 eV.The result is interpreted in terms of the dynamics of molecular excitons and interface plasmons.As shown in Refs.[39,40], the sharp peak structure is attributed to the interference between the energy absorption processes of molecular exciton and interface plasmon.The suppression of luminescence intensity of interface plasmon at the energy of the molecular excitonic mode ϵ ex = 1.89 eV is attributed to energy absorption by the creation of molecular excitons.In the energy range near 1.89 eV, only the luminescence of molecular excitons is excited and luminescence spectra J L of molecular exciton [Fig. 2  In the energy region ϵ B < ℏω < ϵ A , where θ AB ≈ π, |A⟩ and |B⟩ interfere destructively.When θ AB ≈ 0, these modes interfere constructively.The interference effects on J P and J L are investigated using Eqs.( 15) and (16).
To analyze J P , the contributions of each term on the right hand side (RHS) of Eq. ( 15) are shown in Fig. 3.The first and fourth terms (green dashed and magenta dashed-dotted lines, respectively) on the RHS of Eq. ( 15) represent the contribution of the coupled modes |A⟩ and  15).The red solid line is the product of −1/π and the imaginary part of P < , which gives luminescence spectra JP of interface plasmon, Eq. ( 5).The green dashed, blue dotted, magenta dasheddotted lines are contribution from 1st, 2nd+3rd, and 4th terms on the RHS of Eq. ( 15), respectively.Gray dashed lines indicate the energy of the coupled modes, ϵA and ϵB.
Exciton-plasmon coupling is ℏg = 0.10 eV and the energy of the plasmon mode is ℏωp = 2.05 eV.The relaxation constants of interface plasmon ℏΓ pl and molecular exciton ℏΓex are ℏΓex = 0.005 eV and ℏΓ pl = 0.14 eV, respectively.The bias voltage is V bias = 3.0 V.
|B⟩, respectively.The second and third terms (blue dotted line) describe effects of interference between |A⟩ and |B⟩.In the energy range ϵ B < ℏω < ϵ A , where the relative phase θ AB is almost inverted, the interference gives a negative contribution to plasmon luminescence intensity.That is, the interference leads to suppression of luminescence intensity in J P .On the other hand, when the energy is higher or lower than both ϵ A and ϵ B , in which θ AB ≈ 0, the interference enhances the luminescence intensity in J P .Thus the results show that the luminescence spectral shape of J P can be well interpreted in terms of superposition of the coupled modes.Spectral profiles of the coupled modes |A⟩ and |B⟩ can be approximated to Lorentzian functions with the widths of ( , respectively.The Green's function G AB as well as G BA are proportional to the product of the Green's functions of the coupled mode |A⟩, G AA , and |B⟩, G BB .According to Eqs. ( 5) and (15), luminescence spectra J P of interface plasmon can be expressed by the linear combination of these Green's functions.Such a simple expression of J P will help in interpretation of luminescence properties of interface plasmon.
The contributions of each term on RHS of Eq. ( 16) are shown in Fig. 4 to analyze the spectral shape of J L .The coupled mode |B⟩ gives a dominant contribution (magenta dashed-dotted line).Interference between |A⟩ and |B⟩ gives positive or negative contribution depending on the energy range.For ϵ B < ℏω < ϵ A , where the relative phase θ AB ≈ π, the interference gives a positive contribution to luminescence intensity of molecular exciton.On the other hand, when θ AB ≈ 0, the interference suppresses the luminescence intensity of molecular excitons.
When the strength of exciton-plasmon coupling ℏg inhttp://www.16).The red solid line is the product of −1/π and the imaginary part of L < , which gives luminescence spectra JL of interface plasmon, Eq. ( 6).The green dashed, blue dotted, magenta dasheddotted lines are contribution from 1st, 2nd+3rd, and 4th terms on the RHS of Eq. ( 16), respectively.Gray dashed lines indicate the energy of the coupled modes, ϵA and ϵB.
Exciton-plasmon coupling is ℏg = 0.10 eV and the energy of the plasmon mode is ℏωp = 2.05 eV.The relaxation constants of interface plasmon ℏΓ pl and molecular exciton ℏΓex are ℏΓex = 0.005 eV and ℏΓ pl = 0.14 eV, respectively.The bias voltage is V bias = 3.0 V.
creases to ℏg ≥ ℏΓ pl , J L exhibits two peak structures.These features are consistent with those obtained in earlier study investigating the strong exciton-photon coupling system [46].Figures 2-4 demonstrate that luminescence spectral profile of interface plasmon and molecular exciton are greatly modified by exciton-plasmon coupling ℏg.Especially the results show that the detailed investigation of variation in spectral profiles of interface plasmon due to energy exchange with molecular excitons requires a quantum mechanical treatment of interface plasmons.The theory developed by us is suitable for such studies, compared to previous theories of STM-LE [25,28,29,42,44,45,47].Moreover, analysis of the luminescence property of the system from the viewpoint of superposition of the coupled modes provides simple description of luminescence spectral profile of the system, which aids in interpretation of the calculation results as well as experimental results.Although the statistics of molecular exciton are different for the model presented here, the classical model of two coupled harmonic oscillators [48,49], and the single fermion model [50,51], essential parts of the results presented here are equivalent to those obtained for the others.The developed theory can be extended to more complicated systems including more multiple quanta, e.g.quantum of molecular vibrations [37][38][39][40] and electron spins [9], as well as electrons tunneling between the tip and substrate [52].The nonequilibrium Green's function method is appropriate to investigate dynamical properties of the system where multiple quanta couple to each other and exchange energy.

IV. CONCLUSION
In conclusion, we have analyzed luminescence properties of the system, where luminescence of molecular excitons and interface plasmons are excited by the tunneling current of STM.The coupling between a molecular excitonic mode and an interface plasmon mode can strongly affect the luminescence spectral profile of the system.A sharp (broad) peak structure in luminescence spectra of interface plasmon is attributed to the coupled mode, the energy of which is closer to the molecular excitonic mode (the interface plasmon mode).In the energy region near the molecular excitonic mode, luminescence spectra of interface plasmons show great suppression of intensity, and luminescence from molecular exciton is selectively excited.The mechanism of this phenomenon can be interpreted in terms of quantum interference between the coupled modes.The developed theory and the obtained results should facilitate the understanding of the optical properties of the system including both molecules and metal nanostructures as well as their dynamics in photoelectric conversion processes such as current-induced light emission processes.

FIG. 1 .
FIG. 1.(a) Schematic picture of the exciton-plasmon coupled system.Coupling between a molecular excitonic mode and an interface plasmon mode gives rise to coupled modes |A⟩ and |B⟩.(b) Schematic picture of the process in which an interface plasmon is excited by the tunneling current of an scanning tunneling microscope.
Figure2(a) shows luminescence spectra J P of interface plasmon.J P has a sharp peak structure near 1.84 eV and a dip structure near 1.89 eV.The result is interpreted in terms of the dynamics of molecular excitons and interface plasmons.As shown in Refs.[39,40], the sharp peak structure is attributed to the interference between the energy absorption processes of molecular exciton and interface plasmon.The suppression of luminescence intensity of interface plasmon at the energy of the molecular excitonic mode ϵ ex = 1.89 eV is attributed to energy absorption by the creation of molecular excitons.In the energy range near 1.89 eV, only the luminescence of molecular excitons is excited and luminescence spectra J L of molecular exciton [Fig.2(b)]show a sharp peak structure.Next, we interpret the optical properties of interface plasmon and molecular exciton in terns of superposition