Exciton diffusion and annihilation in nanophotonic Purcell landscapes

Excitons spread through diffusion and interact through exciton-exciton annihilation. Nanophotonics can counteract the resulting decrease in light emission. However, conventional enhancement treats emitters as immobile and noninteracting. Here, we go beyond the localized Purcell effect to exploit exciton dynamics. As interacting excitons diffuse through optical hotspots, the balance of excitonic and nanophotonic properties leads to either enhanced or suppressed photoluminescence. We identify the dominant enhancement mechanisms in the limits of high and low diffusion and annihilation to turn their detrimental impact into additional emission. Our guidelines are relevant for efficient and high-power light-emitting diodes and lasers based on monolayer semiconductors, perovskites, or organic crystals.

For excitonic emitters, however, the picture of emission arising from non-interacting dipoles at fixed positions is incomplete. In a variety of semiconductors, excitons are mobile and spread to large diffusion lengths compared to nanophotonic scales (10 -500 nm).
A photonic modification of the radiative decay rate could decrease the effective diffusion length, thus improving performance.
Another important aspect of exciton dynamics is exciton-exciton annihilation 25,30,31 . At high exciton densities, this nonlinear process contributes to and even dominates non-radiative losses, degrading the performance of light-emitting devices at high powers 32,33 and potentially preventing lasing. Annihilation thus curtails the advantages of nanophotonic intensity cient excitonic-nanophotonic systems. Our guidelines for tailoring nanophotonic structures to diffusing and annihilating excitons will aid the design of efficient light-emitting devices.

Exciton dynamics and nanophotonic enhancement
Excitons evolve in nanophotonic environments under the combined influence of incident intensity, radiative enhancement, non-radiative decay, diffusion, and annihilation ( Figure 1a).
We consider excitonic emitters in ultrathin films on top of nanodisk arrays with a negligible variation of electromagnetic fields across the film thickness. The two-dimensional exciton density evolves according to the exciton dynamics equation 20 : (1) ∂n(r, t) ∂t = I(r, t)σ − [Γ r (r) + Γ nr,0 ] n(r, t) + D∇ 2 n(r, t) − γn 2 (r, t) , where n(r, t) is the exciton density at point r at time t, I is the nanophotonically enhanced local excitation intensity at r, σ is the absorption coefficient, Γ r (r) is the spatially varying radiative decay rate, Γ nr,0 is the intrinsic non-radiative decay rate, D is the diffusion constant, and γ is the annihilation constant. In the absence of nanophotonic structures, the intrinsic decay rates are Γ r,0 = η 0 Γ 0 and Γ nr,0 = (1 − η 0 )Γ 0 , where Γ 0 is the total decay rate and η 0 is the intrinsic quantum yield. The exciton decay time is τ 0 = 1/Γ 0 and the diffusion length is L D = √ Dτ 0 . We assume that the nanostructures do not modify the non-radiative decay rate Γ nr,0 and the diffusion constant D, although our model can incorporate such changes as well. We also neglect saturation of absorption at high power.
To compare systems with different excitonic and nanophotonic properties and extract the universal behavior of nanophotonic systems in the presence of exciton dynamics, we non-dimensionalize Equation (1). We identify physically relevant scales of exciton density, incident power, length, and time in the system to scale the variables n, I, r, and t with these values: • n = n/n 0 , where n 0 = Γ 0 /γ is the exciton density at which Γ 0 n (the intrinsic total decay, which includes radiative and non-radiative rates but not annihilation) equals γn 2 (the density-dependent annihilation), • I = I/I 0 , where I 0 = Γ 0 n 0 /σ is the incident continuous-wave power at which Iσ (the exciton generation) equals Γ 0 n 0 (the intrinsic decay at n = n 0 ), • r = r/P , where P is the period of the nanophotonic structures, • t = t/τ 0 , where τ 0 is the exciton decay time.
The characteristic scales depend on both excitonic and nanophotonic properties. Note that we perform scaling with respect to intrinsic properties (in the absence of nanostructures), except the length scale, which is the period of the array. Expressing Equation (1) in terms of the primed variables, we obtain the non-dimensionalized exciton dynamics equation where F ex and F em are the local nanophotonic excitation and radiative rate enhancements, and F decay = η 0 F em + 1 − η 0 is the localized total decay rate enhancement. The nondimensionalized diffusion constant D = Dτ 0 /P 2 is related to the diffusion length and the period of the nanophotonic structures by D = (L D /P ) 2 . The annihilation rate γ does not appear explicitly and is part of the characteristic incident power I = Iγσ/Γ 2 0 , demonstrating the utility of non-dimensionalization in comparing different systems. Excitonic materials that differ only in the annihilation rate γ will behave identically in a nanophotonic system except for scaling of incident power. We perform electromagnetic simulations using the surface integral equation (SIE) method and solve the non-dimensionalized exciton dynamics equation (2) to demonstrate the impact of exciton diffusion and annihilation on nanophotonic photoluminescence enhancement. We illustrate the diverse behavior and the possible scenarios of exciton dynamics using a variety of nanostructures.
First, we analyze how diffusion affects it by spreading excitons out. We study excitonic emitters consisting of orientationally averaged dipoles above an array of silicon nanodisks. Exciton density of emitters with intrinsic quantum yield η 0 = 1 above an array of silicon nanodisks with radius R = 100, height H = 75, and period P = 365 nm. A low-power, continuous plane wave is incident from below the disks at λ = 553 nm. Scale bar 100 nm. (b) The nanophotonic photoluminescence enhancement of the nanodisk array decreases due to diffusion. The blue area indicates enhancement due to excitation only. (c) In the absence of nanophotonic structures, photoluminescence scales sublinearly with incident power due to annihilation.
Although exciton density and diffusion are two-dimensional because the exciton film is thin, the exciton dipole moment can have components out of the plane -therefore, we average the orientations in all three dimensions 34 . We assume perfect collection efficiency of the emission. We illuminate with a continuous-wave source at low power so that annihilation is initially negligible.
For low diffusion constants, the exciton density concentrates near the edge of the nanodisk, corresponding to the local excitation profile of an electric dipole in the nanodisk ( Figure 2a). As the non-dimensionalized diffusion constant D increases, the exciton density distribution expands, eventually becoming uniform over the unit cell. In this case, diffusion suppresses the total photoluminescence enhancement by taking excitons from regions of high radiative enhancement to positions of low enhancement (Figure 2b). The total photoluminescence is a combination of excitation and emission enhancements. Immobile emitters with low intrinsic quantum yield benefit from both excitation and emission enhancements, whereas only the increased excitation is relevant for emitters with high quantum yield 35,36 .
The Purcell effect can be strongly modified upon diffusion, while the excitation is unaffected by diffusion in the absence of saturation effects. The impact of diffusion is the strongest for emitters with low quantum yield because of their larger contribution from radiative enhancement. For emitters with high intrinsic quantum yield, the Purcell effect enhances only the decay rate and not the photoluminescence. Diffusion reduces this enhancement (Supporting Figure S1a).
In the absence of nanostructures, exciton-exciton annihilation suppresses photoluminescence by opening an additional non-radiative channel at high excitation powers and exciton densities ( Figure 2c). Compared to continuous-wave excitation, pulsed excitation creates higher instantaneous exciton densities, thereby reducing emission even further. Nanophotonic structures can ameliorate this deterioration of emission, as we shall discuss later. Additionally, the quick initial decay due to the high exciton density shortens the decay time considerably (Supporting Figure S1b).

Enhancing emission through diffusion
Generally, the deterioration of nanophotonic enhancement with diffusion is due to losing the advantage of spatial overlap between excitation and emission enhancements. To understand the contribution of each process and their overlap, we solve the non-dimensionalized exciton dynamics equation (2) analytically for limiting cases of quantum yield, diffusion, and annihilation (Supporting Section S1). We list the photoluminescence enhancement for these extremes in Table 1 and depict the contributions from excitation and emission and their overlap in Figure 1b, where the levels indicate the increasing role of annihilation. When the incident power is much lower than I 0 = Γ 0 n 0 /σ, annihilation is negligible compared to Table 1: Nanophotonic photoluminescence enhancement (F tot ) in the presence of exciton dynamics: limiting cases of diffusion, quantum yield, and incident power. x represents the spatial average of the quantity x in the unit cell. Spatial averages of products of excitation and emission enhancements such as F ex · F em indicate the benefit of their spatial overlap. Diffusion decouples the enhancements spatially, turning the expressions into products of spatial averages such as F ex F em .
Diffusion Quantum yield Incident power (with annihilation) intrinsic decay (bottom level in Figure 1b). In this regime of negligible annihilation (I → 0) and diffusion length much smaller than the period (D → 0), the total enhancement for emitters with poor efficiency (η 0 → 0) is F ex (r) · F em (r) , which is the spatial average of the product of local enhancements in the unit cell. Hence, in the absence of diffusion, we obtain high total enhancement if the excitation and emission significantly overlap.
In the regime of high diffusion, however, the total enhancement becomes F ex (r) F em (r) , which is the product of the average values of excitation and emission in the unit cell. The spatial overlap of the enhancement factors is then no longer of benefit. As a result, the photoluminescence enhancement typically worsens with diffusion for low-efficiency emitters ( Figure 3a). Although emitters with high quantum yield do not suffer a similar loss of enhancement because their emission efficiency does not change, their decay rate deteriorates with increasing diffusion (Figure 3a). Surprisingly, diffusion can modify the decay rate of emitters even in the limit of zero quantum yield (Supporting Figure S2).
Diffusion can also improve emission by removing the spatial overlap between enhancement contributions. By controlling the excitation conditions such as the angle of incidence, polarization, or wavelength, we can lift the requirement of spatial overlap for maximum photoluminescence. As a first example, we spatially decouple excitation and emission by  and Supporting Figure S3). The excitation profile shows a strong front-back asymmetry because the high refractive index of silicon causes retardation of electromagnetic fields along its height 5 . We illuminate the nanodisks from above to benefit from this asymmetry. The photoluminescence increases with diffusion at low η 0 , and so does the decay rate enhancement at high η 0 . Nanostructures designed under the assumption of immobile emitters can thus behave differently with diffusing excitons. Whether the impact of diffusion on enhancement is beneficial or detrimental depends on the nanophotonic system. Nanostructures aiming at

Overcoming annihilation through nanophotonic enhancement
So far, we have only considered the effects of diffusion on nanophotonic enhancement. Next, we add exciton-exciton annihilation, which typically suppresses photoluminescence. As the incident power of a continuous-wave source increases, photoluminescence enhancement usually decreases for all quantum yields in the absence of diffusion ( Figure 5a for the array in Figure 3a). At high power, the total nanophotonic enhancement falls even below the excitation enhancement (blue line). Exciton-exciton annihilation increases nonlinearly with power, suppressing the effect of excitation enhancement and reducing the steady-state exci-ton density enhancement (Figure 5c). At low power, the photoluminescence enhancement is due to F ex for high-η 0 emitters, whereas it arises from the product of F ex and F em for low-η 0 emitters. In contrast, at high power, the photoluminescence enhancement is the product of √ F ex and F em independent of the quantum yield because exciton-exciton annihilation becomes the dominant non-radiative decay channel ( Table 1).
The suppression of photoluminescence is even stronger for pulsed excitation (Figure 5b Next, we demonstrate that it is possible to improve performance even as annihilation becomes dominant. At low incident power, high-η 0 emitters benefit only from excitation enhancement whereas at high power, the effect of excitation diminishes and emission enhancement becomes dominant (Table 1). Therefore, nanophotonic structures with emission enhancement comparable to or higher than excitation enhancement offer improved photoluminescence enhancement with increasing power. This counter-intuitive behavior arises from the increasing benefit of emission enhancement as a strong non-radiative decay channel opens at high exciton densities. We exemplify such a case with an array of silver nanoparticles which has significantly higher emission enhancement compared to excitation enhancement (Supporting Figure S4). Indeed, as the incident power increases, emitters above the array of silver nanoparticles benefit from increasing photoluminescence enhancement. (Figure 6b). Although we have shown improved performance here for plasmonic nanoparticles, dielectric nanostructures with strong emission enhancements behave similarly (Supporting Figure S5).
High-η 0 emitters with exciton-exciton annihilation are important for light-emitting devices, which suffer from efficiency loss at high powers. The ability to reduce the impact of annihilation on emission through the combination of nanophotonic design and diffusion is thus of practical interest.

Nanophotonic enhancement in relevant excitonic materials
Finally, we show that the non-dimensionalized limits of diffusion and annihilation become significant for light emission from realistic excitonic emitters. We tabulate reported excitonic parameters for representative materials and calculate their characteristic exciton density n 0 , incident power I 0 and the non-dimensionalized diffusion constant D (Supporting Table S1).
For transition metal dichalcogenide monolayers and two-dimensional perovskites, D is of the order of unity. As a result, the excitons spread through the entire unit cell before they decay.
Monolayer semiconductors with low quantum yield thus benefit from additional enhancement in nanostructures designed for diffusive excitons (Supporting Figure S6). Emitters with high quantum yield also suffer from strong annihilation . The incident power at which annihilation becomes dominant, I 0 , is very low (of the order of nW/µm 2 -µW/µm 2 ), especially for transition metal dichalcogenides (Supporting Table S1). As a result, they benefit from additional enhancement in nanophotonic systems designed for materials with high annihilation (Supporting Figure S7).
Although we focused our analysis on excitons in thin films, the general principles also apply to other geometries. Nanowires support one-dimensional diffusion and can be placed along directions of high excitation and emission enhancements to obtain stronger photoluminescence (Supporting Figure S10). In the case of thick excitonic materials around nanophotonic structures, exciton diffusion will be three-dimensional, and the decay of the evanescent near field away from the nanostructure plane will also play a role. In addition to photonic enhancement, material interfaces can modify intrinsic decay, diffusion, and annihilation through doping, dielectric screening, or phonons 29,33,39-43 . Our model can accommodate such effects through the use of locally varying excitonic parameters modified by the environment. It is also possible to prevent such environmental modification of excitonic parameters with a thin dielectric spacer. At high exciton densities in transition metal dichalcogenide monolayers, exciton-phonon effects modify diffusion, resulting in halo formation [44][45][46]

Electromagnetic simulations
We perform the electromagnetic simulations using the surface integral equation (SIE) method for periodic nanostructures 55,56 . We use the permittivities of silicon and silver from Green 57 and Johnson and Christy 58 . We set a homogeneous relative permittivity r = 1.5 for the background medium as the geometric mean of air and glass to approximate the effect of a substrate. We apply a realistic rounding radius of 20 nm to the sharp edges of the nanodisks.
We treat the emitters as electric dipole sources lying on a plane 5 nm above the nanodisks.
To compute the excitation enhancement F ex , we illuminate the system with a plane wave under normal incidence from above or below depending on optimal excitation conditions and evaluate the electric field E on the plane above the nanostructure. The excitation enhancement for dipolar emitters orientationally averaged in three dimensions is where E 0 is the electric field in the absence of the nanostructures.
The emission enhancement F em is the integral of the power radiated in all directions (θ, φ), normalized to the same quantity in the absence of nanostructures. We compute the dipole radiation in a given direction (θ, φ) using electromagnetic reciprocity 59 by evaluating the field intensity at the location of the emitter under illumination by a plane wave incident from the same direction 60-62 . This method assumes no absorption losses in the nanodisks that might reduce antenna radiation efficiency. The emission enhancement of a dipole depends on its orientation. We compute the average emission enhancements for emitters along all possible orientations in three dimensions, integrating total photoluminescence in all directions.
For emitters such as transition metal dichalcogenides where the dipoles are oriented in the plane, excitation enhancement requires using the in-plane projection of the electric field.
Additionally, orientational averaging should then be performed in two dimensions. With these modifications, our treatment applies to two-dimensional excitonic materials as well and does not change the results qualitatively (Supporting Figure S8).

Numerical solution of exciton dynamics
To solve the non-dimensionalized exciton dynamics equation (2) numerically, we discretize the exciton density into a grid with non-dimensionalized coordinates: n (x i , y j ) where x i = i/2N, y j = j/2N for (i, j) ∈ {−N, . . . N }. We choose the value of N in each simulation to obtain 5 nm spatial resolution. As a result of periodicity, the exciton densities are equal at opposite edges of the unit cell (indices −N and N ). In the limiting case of low incident power, the quadratic annihilation term vanishes, and we obtain a linear differential equation in n .
Under continuous-wave illumination, in the steady state, we have [F decay − D ∇ 2 ] n = F ex I , where we have dropped the explicit spatial dependence. Taking the spatial Fourier transform, where the quantities with a tilde (such as n ) are Fourier transforms of the real-space quantities, F decay is the circular convolution matrix for F decay , and the matrix q 2 is the squared momentum in the Fourier transform of the discrete Laplace operator, with elements q 2 l,m = 4π 2 (l 2 + m 2 ). Inverting the matrix on the left-hand side of Equation (3) gives the steady-state exciton density. Its eigenvalues describe the time evolution under pulsed illumination, providing the total decay rate enhancement.
In the presence of annihilation, we can no longer use linear methods. Hence we let the system evolve explicitly according to Equation (2) using the forward Euler method until T = 10τ 0 . Under continuous-wave illumination, the system reaches a steady state by this time. We model pulsed excitation using an ultrashort impulse I δ(t ) so that the exciton density instantaneously becomes n (r , 0) = F ex (r )I . We ensure that the high initial decay rates do not result in numerical errors by using an adaptive time step that limits the maximum relative change in exciton density at a location within a time step to one percent.
The total photoluminescence from the unit cell is then We can also calculate the decay time as the mean lifetime of emission from the temporal decay of photoluminescence: The total decay rate enhancement is the ratio of the decay time without the nanostructure to the average decay time in the presence of the nanostructure.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author. The photoluminescence is then PL(r ; I → 0, D = 0) = η 0 I F ex (r )F em (r ) F decay (r , η 0 ) .
By spatially averaging the photoluminescence in the unit cell and normalizing by its value in the absence of enhancements, we obtain the photoluminescence enhancement Under the limiting cases of quantum yield (η 0 → 0) and η 0 = 1, this expression reduces to F ex (r )F em (r ) and F ex (r ) , respectively.
In the infinite diffusion limit (D → ∞), the F ex (r )I excitons immediately migrate through the unit cell making the exciton density uniform. Similarly, diffusion smoothens out any variations due to unequal decay rates. Hence, the exciton density becomes n (r ; I → 0, D → ∞) = I F ex (r ) F decay (r , η 0 ) , giving the total photoluminescence enhancement Under the limiting cases of quantum yield η 0 → 0 and η 0 = 1 , we obtain total enhancements F ex (r ) F em (r ) and F ex (r ) , respectively.
At high incident powers for which annihilation becomes significant, we need to retain the quadratic term. Under continuous-wave illumination and in the absence of diffusion, in the steady state we have n 2 (r ; D = 0; CW) + F decay (r , η 0 )n (r ) − F ex (r )I = 0 , from which we can solve for n (r , t ) n (r ; D = 0; CW) = F decay (r , η 0 ) 2 + 4I F ex (r ) − F decay (r , η 0 ) 2 .
In the limiting case of very high incident power (I → ∞), where exciton-exciton annihilation dominates emission and non-radiative decay, we have n (r ; I → ∞, D = 0; CW) = I F ex (r ) , giving a total photoluminescence enhancement independent of quantum yield The limit of infinite diffusion (D → ∞) follows from similar arguments as before, Under pulsed illumination, we have an initial exciton density n (r , 0) = F ex (r )I . Solving the non-dimensionalized exciton dynamics equation analytically for this initial condition under zero diffusion yields S1 n (r , t ; I → ∞, Multiplying the exciton density with the radiative decay rate and integrating it over time gives the total photoluminescence from the pulse, PL(r ; I → ∞, D = 0; pulsed) = F em (r )η 0 log 1 + F ex (r )I F decay (r , η 0 ) . The total photoluminescence enhancement in the limit I → ∞ is then The same enhancement occurs for infinite diffusion.

S2 Modification of decay rates by diffusion and annihilation
Diffusion affects the modification of decay rate when the film is placed on the nanodisk array ( Figure S1a). The bare film in the absence of the nanostructure shows a mono-exponential photoluminescence decay. In the limit of ultra-low quantum yield (η 0 → ∞), both when diffusion is very high or very low, placing the film on the nanostructure modifies only the intensity of photoluminescence but not the decay rate. This is because in this limit, the local decay rate enhancement F decay = η 0 F em + 1 − η 0 → 1. As a result, the slope of the photoluminescence decay remains the same on the array. However, for excitons with high quantum yield, the decay rate changes on the array. In the absence of diffusion, the emitters at different locations decay with different rates, resulting in photoluminescence decay which is no longer mono-exponential. In the limit of infinite diffusion, the instant redistribution of exciton density makes the decay mono-exponential again. The increased decay rate results, however, in a different slope compared to the bare exciton film.
Exciton-exciton annihilation modifies the decay rate at high exciton densities even in the absence of nanostructures ( Figure S1b). At low incident powers, the photoluminescence decay is mono-exponential because low exciton densities make annihilation negligible. As the incident power increases, annihilation results in significant decay at short time scales.
S3 Decay rate modification in the limit of ultra-low quantum yield due to diffusion In the presence of diffusion, even the decay rates of emitters with ultra-low quantum yield change near nanostructures ( Figure S2a). Both in the limit of very low and very high diffusion, there is no decay rate enhancement, as previously seen -it is only in the intermediate diffusion regime that we observe a modification of decay rate. The reason is that, S5 Improving photoluminescence enhancement at high power using emission-dominant enhancement Emitters above the silver nanodisk array in Figure 6b show improved photoluminescence enhancement with increasing incident power. The array has a sharp resonance at the emission wavelength λ em = 600 nm whereas the excitation wavelength λ ex = 450 nm falls on a different resonance ( Figure S4a). Emission enhancement of the array is higher than the excitation enhancement ( Figure S4b), resulting in the nanophotonic enhancement improving at high  Figure S4: Nanophotonic structures with emission enhancement comparable to or higher than excitation offer improved photoluminescence enhancement with increasing power. (a) Reflectance from, and (b) maps of excitation enhancement at λ ex = 450 nm and emission enhancement at λ em = 600 nm above, for emitters above the array of silver nanodisks in Figure 6b. Scale bar 100 nm. powers.

Dielectric nanostructures in which emission enhancement dominates excitation can also
show improved photoluminescence enhancement at high incident powers ( Figure S5).

PL enhancement
Non-dimensionalized diffusion constant, D'  Figure S5: Dielectric nanostructures can also offer improved photoluminescence enhancement with increasing power. Photoluminescence enhancement as a function of incident power under continuous-wave illumination from above for emitters with η 0 = 1 above an array of silicon nanodisks with R = 110, H = 85, P = 400 nm. Table S1: Representative reported parameters for excitonic materials. Intrinsic quantities are decay rate Γ 0 , quantum yield η 0 , diffusion constant D, annihilation constant γ, and absorption constant σ. Exciton density n 0 and incident power I 0 are the characteristic values used to non-dimensionalize the exciton dynamics equation (1). D is the non-dimensionalized diffusion constant for period P = 365 nm. The quantum yield of the MoS 2 monolayer on the quartz substrate is unity due to treatment with bis(trifluoromethane)sulfonimide (TFSI). S2 Transition metal dichalcogenide monolayers

S6 Nanophotonic enhancement in realistic materials
Diffusion and annihilation strongly modify the photoluminescence enhancement of realistic excitonic materials in nanostructured landscapes. We tabulate the intrinsic excitonic parameters of some transition metal dichalcogenide monolayers and two-dimensional perovskites from various references in Table S1. As the sources do not provide the absorption constant σ directly, we calculate it from the absorptance by assuming that every absorbed photon generates an exciton. S4 Along with the intrinsic parameters, we also show the characteristic exciton density n 0 and the characteristic power I 0 used for non-dimensionalization, as well as the non-dimensionalized diffusion constant D for P = 365 nm.
The non-dimensionalized diffusion constants for the all the TMD monolayers are of the order of unity (Table S1). As a result, the silicon array in Figure 4c optimized for highly diffusive excitons provides strong photoluminescence enhancement for TMD monolayers with low 5 6 7

PL enhancement
Quantum yield, η  Figure S6: Realistic excitonic materials are highly diffusive, which modifies their photoluminescence enhancement. Excitonic parameters for TMD monolayers and 2D perovskites, overlaid on the photoluminescence enhancement of the array of silicon nanodisks (Figure 4c).
quantum yield ( Figure S6). Treating the excitons as conventional immobile emitters would significantly underestimate their photoluminescence enhancement in this case. Emitters with high quantum yield such as 2D perovskites and TFSI-treated MoS 2 do not see a significant modification of photoluminescence enhancement on account of their high diffusion.
Typical excitonic emitters with high quantum yield also suffer from strong exciton-exciton annihilation (Table S1). Due to the low intrinsic decay rate and high annihilation in MoS 2 monolayers treated with TFSI, annihilation effects become significant at very low powers of the order of a few nW/µm 2 ( Figure S7). At typical powers of the order of µW/µm 2 , the silver nanodisk array in Figure 6b

S7 Dimensionality of emitter orientations
As excitons in typical materials do not have a preferential orientation, we compute the average excitation enhancement in the main text from the total electric field intensity (see Methods). Similarly, we computed the emission enhancement by averaging emitters along all possible orientations in three dimensions. However, the direct transitions in TMD monolayers occur through in-plane electric fields only. As a result, their excitation enhancement only depends on the in-plane electric field intensity S10 and their emission enhancement will be the average enhancement of dipoles along all two-dimensional orientations.
Although the dimensionality changes the numerical value of enhancements, the qualitative behavior remains the same ( Figure S8). Physical effects we discussed in the main text such as diffusion decoupling excitation and emission enhancements and annihilation suppressing the relevance of excitation are independent of the whether we consider two-dimensional  Figure S8: Nanophotonic enhancements of dipolar emitters orientationally averaged in two dimensions have a similar qualitative dependence with exciton dynamics as emitters orientationally averaged in three dimensions. Enhancements of photoluminescence and total decay rate under excitation at λ ex = 461 nm and emission at λ em = 553 nm for the silicon nanodisk array in Figure 3a for emitters orientationally averaged in (a) three dimensions (corresponding to Figure 4c), and (b) in two dimensions.
or three-dimensional emitters.

S8 Emission enhancement counteracts diffusion
In systems with high emission enhancement, the reduced lifetime of the exciton lowers the effective diffusion length as well. As a result, increasing the emission enhancement can turn an emitter in the D → ∞ limit to one in the D → 0 limit.
As an example, consider a system where excitation and emission enhancements are both For an excitonic material with D = 1 and η 0 = 0.2, increasing the Purcell factor takes PL enhancement from the infinite diffusion limit to the zero diffusion limit.
Gaussians at the centre of the unit cell with half-widths equal to 25% of the periodicity. We take the exciton parameters D = 1 and η 0 = 0.2. Keeping the height F ex,max of the excitation enhancement Gaussian fixed at 10, we vary the height F em,max of the emission enhancement Gaussian. At low values of emission enhancement, excitons can diffuse throughout the unit cell, making the PL enhancement similar to the case of infinite diffusion ( Figure S9).
However, when the emission enhancement increases, the excitons decay quicker, reducing the distance they can diffuse. As a result, the PL enhancement becomes similar to the limit of zero diffusion.

S9 Photoluminescence enhancement in nanowires
Although we have discussed the interplay of exciton dynamics and nanophotonic enhancement in the context of excitonic films, the underlying principles can be extended to other geometries as well. Nanowires are ubiquitous one-dimensional excitonic structures. We con- is replaced by a long nanowire along the diagonal of the unit cell, the dependence of PL enhancement (λ ex = 461 nm, λ em = 553 nm) on diffusion remains the same qualitatively.
sider a long nanowire along the diagonal of the silicon nanodisk array in Figure 3a, spanning many unit cells so that we can treat it as infinitely long. We neglect scattering effects at the two ends of the nanowire and compute the PL enhancements at the zero and infinite diffusion limits for λ ex = 461 nm, λ em = 553 nm (corresponding to Figure 4c). As the average excitation and emission enhancements are higher on the diagonal compared to other regions ( Figure 4b), PL enhancement is higher for the nanowire than the film ( Figure S10). Qualitatively, both the film and the wire show similar behavior of PL enhancement increasing with diffusion and decreasing with quantum yield.