Revising quantum optical phenomena in adatoms coupled to graphene nanoantennas

2.1 Hamiltonian without external illumination

In graphene, three of the four valence electrons at each carbon atom hybridize to create sp ² orbitals, forming σ-type bonds responsible for the hexagonal lattice structure. The tight-binding model accounts for electrons introduced by the remaining p _z orbital, oriented perpendicularly to the graphene sheet, and responsible for most of the physical properties of graphene near the Fermi energy [46]. The adatom is represented with a small number of eigenstates that correspond to atomic orbitals, modeled as point sites localized near the flake. The adatom orbitals can directly exchange electrons with selected carbon sites at the flake. It has been shown that particular adatom types on graphene interact with each other through p _z graphene orbitals, which justifies the use of the π-electron approximation [50, 51]. The Hamiltonian of a flake of N carbon sites and N _a adatom orbitals constructed within the tight-binding approximation reads

An electron can be exchanged between neighboring carbon atoms at the rate t. The symbol ⟨l, l′⟩ indicates that the summation goes only over nearest neighbor atomic sites l and l′. The adatom orbitals, labeled by α, have energies ϵ _α evaluated with respect to the onsite energy in graphene. The results presented throughout this paper are obtained for adatoms with two levels (a ground state |g⟩ and an excited state |e⟩) and assume that the adatom introduces only one electron to the system. Electrons can be exchanged between the adatom orbitals and selected flake sites with corresponding tunneling rates t _α,l . We will focus on armchair-edged graphene flakes, since they have a gap around the Fermi energy, which is interesting for optical effects and also makes them more stable. It is worth noting that armchair-edged nanoflakes do not support edge states. On the contrary, zig–zag terminated flakes support zero-energy edge states of topological nature. These states are located in the middle of the energy gap and would be interleaved on the energy scale with the considered adatom states. This constitutes a more complex case to be investigated in future studies. Adatoms in chains with such topological states have already been investigated in our previous work [45].

For symmetry reasons, three possible positions of adatoms are favorable for the adsorption above the graphene antenna: top, bridge, and hollow. It means that in the nearest-neighbor approximation, the last summation in Eq. (1) can span over one, two or six sites [52]. Independent on the adatom’s type, attaching it to graphene breaks the symmetries of the perfect lattice and, consequently, changes the optical and electronic properties of this material [53]. Throughout this paper, we consider the top position of the adatom, meaning that it will be located over and coupled to one particular graphene site. This corresponds to configurations that can be found in covalently functionalized graphene with hydrogen and fluorine adatoms and the hydroxyl group OH and covalently bond adsorbates [47]. Note that in this way, the adatom breaks the sublattice symmetry.

We use the hopping parameter t = 2.66 eV between neighboring sites in graphene [54]. The hopping rates between the adatom orbitals and carbon sites t _α,l generally depend on the adatom properties and its distance from the flake. We treat them as parameters and leave them unspecified to keep the approach general and characterize the scope of possible physical effects achievable within the model, rather than investigate specific cases. We vary the hopping between 0 and 2t which covers most of the reasonable realization scenarios, including atoms, molecules, quantum dots, and defects. Values for t _e, t _g that are larger than the graphene hopping t are feasible for fluorine atoms, OH groups, and some C radicals [47, 50]. On the other hand, values for t _e, t _g below t are feasible for metal adatoms [55]. These values have been verified using more accurate methods, like density functional theory (DFT) [47]. A comparison of results obtained with time dependent DFT and the tight-binding methods has been performed for graphene flakes in particular in Ref. [3].

In Appendix A, we provide the relation of the hopping rates with the distance between the nanoflake and the adatom.

Diagonalization of H _TB provides a set of energies E _j and eigenstates |j⟩. Note that, from now on, we can operate in two different bases to describe the hybrid system: (i) the real-space basis of sites |l⟩ and |α⟩, which we use to construct the tight-binding Hamiltonian, and (ii) the energy basis of eigenstates |j⟩ obtained from the diagonalization of the Hamiltonian. The two bases are related via a linear unitary transformation with coefficients a _jL :

where we have introduced a joint index L, which goes over both flake and adatom sites.

2.2 Induced Coulomb repulsion

Given the basic set of eigenstates, we construct the single-particle density matrix as a statistical mixture of density matrices corresponding to eigenstates with energies below or equal to the Fermi energy, according to the Aufbau principle (see Appendix B for details). This density matrix is assumed to represent an average state of one among N _e electrons. In isolated, undoped graphene flakes, this mixture describes a uniform charge distribution in real space. The presence of the adatom induces a charge nonuniformity on the flake, which due to induced Coulomb repulsion pushes the electrons back towards a uniform distribution ρ ⁰. However, the uniform distribution is not the lowest-energy state when the adatom is present. The equilibrium state of the system is a result of a trade-off between two effects, as the energy minimization in the noninteracting system and the Coulomb repulsion balance each other. This equilibrium state ρ ^sc can be found iteratively in a self-consistency loop, similarly as introduced in Ref. [56]. The self-consistency procedure is described in detail in Appendix B. It additionally provides a self-consistent Hamiltonian H ^sc, a basis set of eigenstates |j ^sc⟩, dressed by the charge-inhomogeneity-induced Coulomb interaction, and the corresponding energies E j sc . The procedure can be applied for an arbitrary number of N _e electrons in the system, which allows for taking into account doping with either electrons or holes. In this work, we do not include the exchange energy. In literature, it has been taken into account, e.g., by modification of the Coulomb interaction terms [57]. Similarly, adjusting Coulomb interactions may be used to mimic magnetic effects in certain nanostructures [58], which is also beyond the scope of this work.

Figure 1:

Visualization of the considered hybrid system consisting of a graphene nanoflake and an adatom attached over a particular carbon site.

The energy spectra of triangular armchair-edged graphene nanoflakes after the self-consistency procedure are shown in Figure 2. The color of the lines denotes the occupation probability |a _jg|² + |a _je|² of the adatom sites in the hybridized eigenstates j when the flake and adatom become increasingly coupled. This shows explicitly that the flake eigenstates that mix with the adatom eigenstates most strongly are also the ones that change their energies the most when the hopping parameters t _e and t _g are increased. Shifting up or down one of the adatom energy levels breaks the electron–hole symmetry in the energy spectrum of the system, as demonstrated in Appendix C. This has its consequence for the absorption spectra described in Section 2.5, in which the originally degenerate peaks would split, leading to a more complex optical response. The real-space representation of self-consistent density matrices ρ ^sc corresponding to the charge distributions N _e eρ ^sc for triangular graphene flakes of 18 and 126 atoms with adatoms are shown in Figure 3. Here, N e ρ l l sc is the expectation value of the number of electrons on site l. The impact of the adatom on the charge distribution is limited to the small region below 1 nm in size, in practice, including all the carbon atoms of the smaller flake. On the contrary, on a larger flake, the adatom affects the few nearest hexagons [Figure 3(b and d)]. The adatom modifies the symmetry of the structure which is decisive for optical properties. This intuitive result persists in the doped case, as shown exemplarily in Figure 4. With doping, the ground state ρ ^sc is no longer uniform, and the effect of the adatom on that structure may extend throughout the entire flake. For small doping, the influence from the adatom dominates the charge redistribution effect. But even for higher doping values the effect of the adatom is clearly visible in charge distribution as shown in the case of 15 doping electrons. As a consequence, the optical properties of the flake will be modified, relaxing the selection rules for optically driven transitions.

Figure 2:

Energy levels of hybrid systems consisting of an armchair triangular graphene flake with 18 atoms (0.71 nm side length, left) and 126 atoms (2.41 nm side length, right) and an adatom with energy levels E _e = 0.5 eV and E _g = −0.5 eV (inside the gap around the Fermi energy) located over the left top-most site of the flake. For the larger flake, a selected range of energies around the Fermi energy is presented. The line color indicates the occupation of the adatom sites in a given eigenstate |j⟩: |a _je|² + |a _jg|², where e and g denote the excited and ground adatom sites, and a _jL is defined via Eq. (2).

Figure 3:

(a and b) Charge distribution on a hybrid system consisting of an armchair triangular graphene flake with 18 atoms and an adatom. The adatom is located near one particular site at a position marked with X and coupled with t _e = t _g = 2.0 eV. The adatom level energies lie inside the gap around the Fermi level with E _e = 0.5 eV and E _g = −0.5 eV. The plots show the real-space charge distribution due to the self-consistent density matrix ρ ^sc (a), as well as the difference with respect to the density matrix corresponding to a uniform charge distribution ρ ^sc − ρ ⁰. (b) The occupation of the adatom levels is presented on the left side of the figure (upper dot denotes the excited state, lower dot – the ground state). (c and d) As in (a and b), but for the flake with 126 atoms. On panels (b and d) the area of the spots is proportional to the density matrix modification on a given site ( ρ sc − ρ 0 ) l l .

Figure 4:

Upper row: Charge distribution due to the self-consistent density matrix ρ ^sc of an armchair triangular graphene flake of 126 atoms doped with 5 electrons (left) or 15 electrons (right). Lower row: As in the upper one, but with an adatom.

2.3 Including electric field

An electric field E(r, t) can be coupled to the nanoflake through the interaction Hamiltonian

where ρ(t) is the time-dependent density matrix of the system evaluated based on the master Eq. (7) described in details in Section 2.4. The external potential φ ^ext(t) is given by

where r _L is the position of the given site and r ₀ – of the adatom, d _eg = e r _eg is the transition element of the dipole moment operator evaluated between the adatom states. In our model, the flake is described by one orbital per site. Therefore, φ ^ext is diagonal in the site basis, except for the elements which couple the adatom states. A transformation to the energy basis will then reveal nonzero off-diagonal elements responsible for electromagnetic coupling of the hybrid structure’s eigenstates. On the contrary, the internal geometrical structure of the adatom is beyond the level of approximation applied here, so it is explicitly included via the transition dipole moment d _eg.

As the electric field moves the electrons away from their equilibrium positions, Coulomb repulsion within the electron cloud arises, accounted for by the field-induced potential

where the Coulomb elements v _ll ′ have been evaluated in Ref. [59] and are listed in Appendix D.

The oscillating induced charge density generates an induced electric field [60] which reads as:

where Q l ( t ) = N e e ρ l l ( t ) − ρ l l sc is the charge induced on the lth site at time t and ϵ ₀ denotes the vacuum permittivity. In all our calculations, the electric field E(r, t) in Eq. (4) is the sum of an external electric field E _ext(r, t) and the induced electric field E _ind(r, t).

The induced field distribution near an 18-atom flake subject to a plane-wave illumination resonant with the adatom transition frequency of 1 eV is illustrated in Figure 5. In the blue regions, the induced field amplitude is lower than that of the illuminating field, while the red color indicates regions where the induced field dominates over the external one. Apparently, higher-order interactions could still be neglected for adatom distances of the order of carbon–carbon distance in graphene. Dipolar and quadrupolar coupling mechanisms may become comparable at short distances corresponding to dark red areas in Figure 5. Higher-order interactions would also be increased in even stronger fields or for illumination frequencies tuned to flake resonances.

Figure 5:

Distribution of the x-components (left) and y-components (right) of the electric field that is induced around an 18-atom graphene nanoflake due to an external illumination polarized either in x-direction (upper panels) or y-direction (lower panels).

2.4 Dynamics

The dynamics of the system is described with the single-electron master equation

The Hermitian Hamiltonian described in previous sections accounts for the reciprocal processes in the system, with the nonlinearity related to the inclusion of the Coulomb interactions. The phenomenological damping term reads as

where τ is the coherence lifetime known from experiments in bulk graphene. It characterizes the dissipation of the system adiabatically moving towards a predefined stationary state ρ ^stat. In practice, we choose ρ ^stat = ρ ^sc arising from the self-consistency procedure, so that the dissipation forces the system back into its equilibrium state. In general, different lifetimes could be assigned to specific elements of the density matrix, in particular, when considering the adatom and the flake. Such a modification would not make a significant impact in the cases investigated below.

2.5 Absorption spectra

Square roots of the optical absorption spectra for the 18-atom flake with the adatom are shown in Figure 6 (see the calculation method in Appendix E). The noninteracting absorption spectrum in the left panel corresponds directly to the tight-binding energy spectra from Figure 2. The spectra of a bare flake correspond to t _e, t _g = 0, while the impact of the adatom is revealed for nonzero hopping rates. The adatom introduces new low-energy resonances that arise from transitions involving its own states. Additionally, it influences the frequency and strengths of other transitions. Some transitions missing in the spectrum of a bare flake appear at presence of the adatom, which is caused by the flake symmetry breaking and the corresponding modification of selection rules. As evident from the energy spectra in Figure 2, only selected flake eigenstates couple to the adatom. This results in the splitting of resonances already visible in the bare flake spectrum. These observations are in line with our previous results [45] for linear atom chains with adatoms. The middle panel shows the dependence of the absorption spectra on the Coulomb interaction strength for fixed adatom–flake hopping rates t _e = t _g = −2 eV. The Coulomb scaling parameter on the horizontal axis multiplies the induced Coulomb interactions. The figure explains how the Coulomb interaction shifts optical resonance positions of the system and reveals the connection between the noninteracting and interacting absorption spectra. The right panel shows the interacting absorption spectrum including the Coulomb interaction with the scaling parameter equal to 1. Despite the fact that some resonances are significantly shifted, the conclusions drawn about the adatom impact in the noninteracting case remain valid.

Figure 6:

Optical absorption spectra for an 18-atom flake with an adatom having energy levels at E _e = 0.5 eV, E _g = −0.5 eV. Left panel: Noninteracting spectra. Middle panel: Dependence of the spectrum on the Coulomb interaction strength for t _e = t _g = −2 eV. Right: Interacting absorption spectra.

3.1 Modification of Rabi oscillations

As the generic model of quantum optics, the two-level atomic system subject to external illumination undergoes sinusoidal Rabi oscillations of population between its ground and excited states. The population oscillation amplitude is equal to 1 in the case of resonance, and the Rabi oscillation frequency reads as

where E is the field amplitude the system is subject to. For moderately detuned fields, in the range of applicability of the rotating wave approximation, this picture holds with the oscillation amplitude reduced and the Rabi frequency increased to [61]

where Δ is the detuning between the continuous wave illumination frequency and the transition frequency.

Below, we study how coupling the two-level emitter to a graphene nanoflake affects this phenomenon. We study the evolution of the system by solving the master Eq. (7) and find oscillatory behavior in the population dynamics of the HOMO and LUMO states of the coupled system. In this case, the system is subject to a field that includes the external illumination in the form of a plane wave modeled in the quasistatic limit E ext r , t = E ext t and the induced field E ind r , t evaluated according to Eq. (6). The induced field is strongly position-dependent and involves both x and y polarizations (Figure 5). When the resulting dynamics pertains oscillatory character, this leads to a corresponding position and polarization dependence of Rabi frequency as suggested by Eq. (9). For particular adatom positions, the Rabi oscillation frequency can be strongly enhanced (reddish regions in Figure 5). From the same figure it follows that it would be possible to drive transitions of a system whose dipole moment is oriented perpendicularly to the polarization of the incoming beam. This possibility arises due to the existence of all polarizations in the induced near field and is not present in free space.

We now focus on hybrid systems that consist of an undoped armchair-edged triangular nanoflake with 18 atoms (side length 0.71 nm) or 126 atoms (side length 2.41 nm) and a two-level adatom with energies 0.5 eV and −0.5 eV, characterized with a dipole moment d _eg = 7.5 D. The rather large value of the dipole moment assures relatively fast Rabi oscillations and, therefore, the demonstration of the effect requires relatively short simulation times. Moreover, for the chosen value of the external field, this dipole moment results in Rabi oscillations on a similar time-scale as the dissipation processes in bulk graphene. Here, we only consider adatom levels which are located in the energy gap around the Fermi energy. Moreover, we assume for simplicity that both adatom levels are coupled equally strongly to a carbon atom at the corner of the graphene flake, i.e., t _e = t _g for all simulations. We investigate hopping parameters t _e and t _g up to 3.5 eV, which corresponds to distances in the order of Angstroms and larger (see the model in Appendix A). The external field has an amplitude of 0.05 V Å and is polarized along the y-axis, as defined in Figure 7. Damping effects are neglected by setting D ( ρ ) = 0 in Eq. (7).

Figure 7:

Real-space distribution of the HOMO state in a triangular 18-atom flake with an adatom attached near the tip of the flake at coordinates (3, 7.5) Å with a coupling strength of (a) t _e = t _g = 0.5 eV, (b) t _e = t _g = 2.0 eV, (c) t _e = t _g = 3.5 eV. The black mark X denotes the site to which the adatom is coupled. The area of the spots is proportional to the HOMO contribution to the population of a given site in real space.

We first analyze the dynamics neglecting the electron–electron interactions in the system, i.e., setting φ _ind = 0 in Eq. (3). This allows us to understand the basic process that occurs as the adatom gets coupled to the flake with increasing strength t _e,g – the change of frequency of the Rabi oscillations between a pair of states.

In the absence of coupling, the adatom states |g⟩ and |e⟩ fit in the energy gap of the graphene flake (Figure 2). As the coupling strengths between the subsystems increase, the orbitals hybridize and shift in energies. In particular, the HOMO and LUMO states become localized on both subsystems with the dominant contributions originating from the adatom and a small part coming from the flake. There is no energy crossing involving the pair of orbitals, which originate from the adatom, that retain their HOMO/LUMO character. We illuminate the system with a monochromatic field resonant with the LUMO – HOMO transition energy, which now depends on t _e,g. For the set of parameters listed above, we find that the stronger the adatom is coupled to the graphene flake, the lower the frequency of Rabi oscillations it exhibits [Figure 8a–c].^[1] This can be explained by the fact that the transition dipole moment of an isolated adatom d _eg has been set to a relatively large value. On the other hand, the dipole moment operator elements d j j ′ = N e e | ⟨ j | r ̂ | j ′ ⟩ | of isolated flakes considered here are lower and up to 5 D. In the coupled case t _e,g ≠ 0, the states originally corresponding to the adatom acquire components localized on the flake [Figure 7] and vice versa. The corresponding transition dipole moment elements reflect this mixing effect: The dipole moment d _{jj ^′} between the j = LUMO and the j′ = HOMO states is decreased with respect to the original dipole moment in the adatom d _eg due to the admixture of the flake component. This leads to a reduction of the Rabi frequency as suggested by formula (9).

Figure 8:

Occupation dynamics with evident Rabi oscillations between the HOMO and LUMO states in the 18-atom flake with an adatom. Here, the Coulomb interaction is not included. In the “resonant illumination” case (a–c), the illumination frequency is always resonant to the HOMO–LUMO transition frequency. In the “off-resonant illumination” case (d–f), the illumination frequency is fixed and equal to 1 eV in each case.

Next, we fix the illumination frequency to be constant and equal to the frequency difference (ϵ _e − ϵ _g)/ℏ of an uncoupled two-level system. As the adatom is increasingly coupled to the nanoflake, the energy structure of the system is modified so that, in this case, detuned behaviour is expected – an increased Rabi frequency and a lowered amplitude of oscillations [61]. This is shown in Figure 8d–f.

Finally, we include the electron–electron interaction potential φ ind ρ ( t ) given by Eq. (5) and look again at the time evolution of the hybrid system (Figure 9a–c). The system is illuminated with the HOMO–LUMO transition frequencies evaluated self-consistently according to the procedure in Appendix B. On top of the Rabi frequency modification due to hybridization, we now note additionally a relatively fast modulation and a detuning effect that arise from the time-dependent Coulomb potential e φ ind ρ ( t ) , which adds to the system Hamiltonian and modulates the energy eigenvalues locally in time. We also repeat the same calculation including dissipation with a relaxation time τ determined from ℏτ ⁻¹ = 10 meV, which is a realistic value for graphene. The decay rate is rather slow compared to the timescale of Rabi oscillations, such that multiple cycles of the process can be clearly observed, as shown in Figure 9d–f.

Figure 9:

Occupation dynamics for the hybrid system illuminated at a frequency resonant to the HOMO–LUMO transition. This time, the Coulomb interaction is included.

(a–c) 18-atom flake without dissipation, (d–f) 18-atom flake with dissipation (ℏτ ⁻¹ = 10 meV), (g–i) 126-atom flake without dissipation. In panels g–i, the orange line corresponds to the occupation of the HOMO−2 state and the purple line to the LUMO+2 state, whose energies ∓0.82 eV happen to be separated from the HOMO/LUMO states by a difference almost resonant with the illumination frequency.

The results presented above concern the adatom positioned at a distance similar to the carbon–carbon distance in graphene. For the applied illumination frequency, the induced field has a moderate effect on the dynamics. In doped flakes, we anticipate a stronger effect exploiting plasmonic enhancement of light–matter interaction strength resulting in Rabi oscillations at considerably higher frequencies.

These effects appear in a similar form in larger nanoflakes (Figure 9g–i). However, due to the increased density of states in larger structures, it is generally more likely that additional states originating from the flake actively contribute to the energy exchange as transitions involving these states get close to resonance with the frequency of the illuminating field. This is illustrated in Figure 9h for t _e,g = 2.0 eV, where the orange and violet lines indicate transitions involving the states with energies ±0.82 eV, being the LUMO+2 and HOMO−2 states at approximately twice the value of the HOMO/LUMO energies. Here, the coupling with the adatom has lifted the degeneracy of these states with the LUMO+1 and HOMO−1 states, which remain uncoupled to the adatom. As a result, the symmetry of states that hybridize with the adatom is broken, as shown in Figure 10 for the HOMO, HOMO−1, and HOMO−2 states, which play the most important roles in interactions with light. The symmetry breaking modifies the selection rules allowing for optical transitions to states of arbitrary symmetry in general.

Figure 10:

Real-space distribution of the HOMO, HOMO−1 and HOMO−2 states in a 126-atom flake with the adatom attached with a coupling strength of t _e = t _g = 2.0 eV near the tip of the flake at coordinates (11, 22) Å. The black mark X denotes the location of the adatom in the XY plane.

3.2 Spontaneous emission

In an isolated atom, energy dissipation occurs radiatively through spontaneous emission [61]. A lot of efforts have been devoted to exploit traditional or nanophotonic cavities to control the emission rate via the so-called Purcell effect [28, 62, 63]. The control is possible by adjusting the geometrical parameters of the cavity to modify the density of electromagnetic states that can be accounted for with the electromagnetic Green’s tensor associated with the cavity [64]. The formalism has been applied to plasmonic nanoantennas playing the role of a cavity as well [65–67]. Here, we discuss an example of the same phenomenon near a graphene nanoflake, where, besides the optical coupling channel between the adatom and the flake, electron tunneling effects occur and may modify the result.

3.2.2 Influence of electron tunneling

For the homogeneous component that neglects the presence of scatterers we find in the limit of r′ → r that I m G H ( r , r , ω ) = ω 6 π c [65, 66] and the expression for the homogeneous part of the emission rate in Eq. (11) simplifies to the Weisskopf–Wigner formula. Even though the homogeneous part does not account for the optical interaction between the adatom and the flake, it can be influenced by electron tunneling, i.e., hopping between the adatom and the flake. This is due to the dependence of the transition dipole moment d _{jj ^′} and frequency ω _{jj ^′} on the values of the hopping rates t _e,g, corresponding to the adatom distance to the nearest carbon site on the flake.

The dependence of the resulting spontaneous emission rate on the distance is presented in Figure 11, where the result is normalized to the free-space rate of a decoupled atom Γ 0 = ω e g 3 | d e g | 2 3 π ϵ 0 ℏ c 3 . In fact, it is the hopping rate t _e,g that enters the calculations rather than the distance directly, while their relation is given in Appendix A with β = 1. However, for easier comparison with the following results we plot the emission rates as a function of the distance. For the y-orientation of the dipole we note the dip centered at the distance of about 9 Å. It arises due to a destructive interference of two dipole moment contributions: the transition dipole of the adatom d _eg and the dipole moment component arising on the flake. The two contributions cancel each other at the mentioned distance. This effect depends on the orientation of the dipole.

Figure 11:

Influence of electron tunneling effects on the spontaneous emission rate between LUMO and HOMO states of a hybrid system as a function of the distance between the adatom and the closest carbon atom in the nanoflake. The considered nanoflake is a triangular armchair-edged graphene flake with 36 atoms and the adatom has energies ±1 eV. The adatom is attached in the upper-left corner of the nanoflake and is marked in the subfigure as a purple dot, and moved away from the nanoflake in the y-direction. The dipole moment in the uncoupled adatom is set to d _eg = 0.01 eÅ ≈ 0.05 D to ensure that we stay in the linear response regime and is oriented in the x-direction or y-direction. The distance has been evaluated based on the coupling strengths t _e and t _g according to the formula in Appendix A (see Figure 13) with β = 1.

3.2.3 Influence of optical coupling

To evaluate the scattered Green’s tensor based on Eq. (12), we illuminate the nanostructure with a dipolar electric field from a source of amplitude and polarization in accordance with d _LH located at the position of the adatom r ₀. We propagate the entire system in time and calculate the scattered part of the field E _ind(r ₀, t) at the adatom’s position using Eq. (6). The electric field in Eq. (12) is then the Fourier component of E _ind(r ₀, t) corresponding to the transition frequency, and only its imaginary part is relevant for the spontaneous emission rate in Eq. (11).

The resulting dependence of the spontaneous emission rate Γ_S on the distance between the nanoflake and adatom is presented in Figure 12 for two dipole orientations in the x- or y-direction. The adatom dipole moment has been set to d _eg = 0.01 eÅ ≈ 0.05 D to assure the linear response regime. For the hopping rate t _e,g = 0, the adatom is electronically decoupled from the flake, and the spontaneous emission enhancement originates only from the optical coupling mechanism. This enhancement is shown with red solid lines as a function of adatom distance from the nearest carbon site. For the x-orientation of the dipole, the optical coupling with the flake is much more efficient and the calculated transition rates are 1–2 orders of magnitude larger. Please note that since only the scattered component of the Green’s tensor is taken into account in this calculation, the limiting value of the transition rate at large distances is zero for both orientations. At distances in the order of a few Å, the scattered part of the emission rate Γ_S is by far the dominant contribution, overcoming the homogeneous one evaluated in the previous section by more than 3 orders of magnitude. The value of Γ_S drops below the free-space value Γ₀ for distances on the order of 15–40 nm, depending on the dipole orientation.

Figure 12:

Dependence of the spontaneous emission rate on the distance between the nanoflake and the adatom for the electronically decoupled case with t _e,g = 0 (red solid line) and including the electronic coupling mechanism with t _e,g (blue solid line) being functions of distance, for the adatom transition dipole moment oriented in the x (a) or y direction (b). The considered nanoflake is a triangular armchair-edged graphene flake with 36 atoms and the adatom has energies ±1 eV. The dipole moment in the uncoupled adatom is set to d _eg = 0.01 eÅ ≈ 0.05 D. The dashed line presents an analytical prediction of how the resonant frequency and the HOMO–LUMO transition dipole moment in the coupled system changes depending on the distance between the nanoflake and the adatom. The expression in the y-label contains squared values of frequency and dipole moment such that they coincide with Eq. (11).