For the sample without the coverage of Ag nanostructures, the excitons generated in SiN _x films by optical or electrical pumping are terminated by radiative or nonradiative recombination with the internal quantum efficiency (η _int) determined by the ratio of the radiative recombination rates (κ _rad) to the nonradiative recombination rates (κ _nrad) as η _int = κ _rad / (κ _rad + κ _nrad) = τ _rad ⁻¹/(τ _rad ⁻¹ + τ _nrad ⁻¹). After the addition of Ag nanostructures, the interaction between LSPs and excitons can be treated as a two-step process (shown in Figure 2). When the energy (ћω _ex) of excitons is close to the electron vibration energy (ћω _SP) of LSPs, the excitons transfer their energies to the LSP modes with the radiative recombination rates (κ _rad) enhanced by the Purcell factor F _p 4 11 . This process will compete with the nonradiative decay and enhance the PL decay rate (κ _PL) significantly due to the large electromagnetic fields introduced by the high mode density in LSPs 21 . For the second step, the energy from the LSP mode will be out-coupled to radiated photons with the rate γ _rad, as shown in Figure 2. This coupling will compete with the nonradiative loss (γ _nrad) due to the absorption of Ag nanostructures, where the coupling efficiency can be defined as η _c = γ _rad / (γ _rad + γ _nrad) 22 . This efficiency (η _c) can be optimized via the modulation of the parameters of metal nanostructures, such as the size, shape, density, and the distance between metal nanostructures 22 .

As has been mentioned above, the Purcell factor F _p can be used to characterize the efficiency of coupling into the SP mode 11 12 22 estimated as the ratio of the effective density of the SP modes ρ _SP to that of the radiation components ρ _rad 23 . For the dipole placed at the distance d from the particle bottom surface and oriented in the z direction normal to the surface, the value of ρ _SP is given by 22 23

ρ SP = L ω V eff r 0 r 0 + d 6

where the normalized line shape of the dipole oscillation L(ω) is

L ω = Im ε M ω + 2 ε D − 1 ∫ Im ε M ω + 2 ε D − 1 dω

with ε _M and ε _D being the dielectric constant of the metal and surrounding media, respectively, which can be acquired from the ellipsometric data. A simple model is used for the estimation of the effective volume V _eff of SP modes, where the shape of the Ag nanostructures is a sphere in approximate with the radii (r ₀) equal to 45 and 110 nm for Ag40 and Ag80, respectively. The values of V _eff can be obtained via the following equation 22 :

V eff = ∫ ∫ ∫ r < r 0 1 2 ε 0 ∂ ω ε M ∂ ω E in 2 d 3 r + ∫ ∫ ∫ r > r 0 1 2 ε 0 ε D E out 2 d 3 r 1 2 ε 0 ε D E max 2 = 4 3 π r 0 3 1 + 1 2 ε 0

where the dipole field inside (E _in) and outside (E _out) of the metal sphere with radius r ₀ can be obtained by solving the Laplace equation with proper boundary conditions at r = r ₀ 24 . The maximum dipole field (E _max) occurs at the surface of the metal nanostructures for this model. The value of ρ _rad is described as follows 22 :

ρ rad = 1 3 π 2 2 π λ D 1 ω

with λ _D being the emission wavelength in the SiN _x matrix. Combining Equations 1 to 4, the Purcell factor can be obtained by

F p ω = ρ SP ρ rad = L ω V eff r 0 r 0 + d 6 1 3 π 2 2 π λ D 3 1 ω − 1

Besides the parameters of the Ag nanostructures which will alter the values of ρ _SP, both the distance (d) between the metal nanostructures and the excitons in SiN _x and the emission wavelength (λ _D) of the luminescence matrix will influence the values of F _p. Therefore, the influences of λ _D and d accompanied with the size (r ₀) of the Ag nanostructures on the values of F _p are simulated, as shown in Figure 3. By fixing the value of d, the influence of λ _D on F _p for Ag40 is compared with that for Ag80, as shown in Figure 3a,b. Obviously, the values of F _p are increased with the emission wavelength following the relationship of F _p ∝ λ _D ³ in both the cases. It means that the Purcell enhancement may be more conspicuous for the luminescence matrix with longer emission wavelength. A larger value of F _p for Ag40 is obtained compared to that for Ag80, which may result from the predomination of the volume term r ₀ ³ in Equation 5 over the factor r 0 r 0 + d 6 here. It is clear that more effective LSP-exciton coupling can be achieved for Ag40 compared to Ag80. The influences of d on F _p are also checked in both Ag40 and Ag80 by setting the emission wavelength at 600 nm, as shown in Figure 3c,d. The values of F _p are increased with the decrease of d, and this increase is more obvious in Ag40 than that in Ag80. The influence of d may be much more significant than that of λ _D on F _p for the SiN _x matrix investigated here.

From the discussions above, we can estimate the average position of excitons in the luminescence matrix with unconfined structures (the confined structure referred to the one with the active matrix inserted between two barrier layers) and optimize their luminescence properties conveniently based on this model. These matrixes may be SiO _x , SiN _x , Si nanocrystals within SiO _x or SiN _x , ZnO, (In)GaN, and so on. For the estimation of the average position of excitons in our SiN _x matrix, PL spectra are measured for the determination of the value of λ _D, as shown in Figure 4a. Broadband emission range from approximately 400 nm to approximately 850 nm with the central wavelength of approximately 600 nm could be observed. By dividing the PL intensity of Ag40 or Ag80 to that of Ag0, the PL enhancement factor (F _PL) can be obtained. As can be seen in Figure 4b,c (left axis), two obvious peaks of the F _PL can be resolved, where the peak with shorter wavelength is induced by the improvement of light extraction from the active layer due to the minimum absorption of the Ag nanostructures. The peak with longer wavelength results from the coupling between LSPs and excitons in SiN _x confirmed by the extinction spectra, shown in Figure 4b,c (right axis), due to the consistence between this peak and the one of the dipole resonance, which is enhanced by the Purcell factor F _p 11 . Obviously, the dipolar resonance peak of the Ag nanostructures redshifts gradually with the increase of their sizes. Correspondingly, the F _PL is located at approximately 750 nm with the value of approximately 2.5 for Ag40 and at approximately 780 nm with the value of approximately 2.0 for Ag80.

To find out the relationship between F _p and F _PL, the originations of these two factors are considered. Both of them mainly result from the enhancement of the spontaneous emission rate, where the value of F _p can be rewritten as F _p(ω) = κ _PL ^*(ω)/κ _PL(ω), with κ _PL(ω) and κ _PL ^*(ω) standing for the original and enhanced PL decay rates, respectively 25 . Consequently, the approximation relation between F _p and F _PL (F _P ≈ F _PL) can be employed for the estimation of the average position of excitons (d) in SiN _x by plotting the curves of F _p vs. d at the wavelength where the enhancement of dipole resonance occurred, as shown in Figure 4d. The average position of excitons is located at 43 to 45 nm beneath the top surface of the SiN _x films. Consequently, the optimized luminescence properties can be achieved via the optimization of parameters in the Equation 5 by the modulation of the sizes and shape of the Ag nanostructures. Both the optimal PL and electroluminescence efficiency of a SiN _x -based light-emitting device are achieved by the addition of Ag nanostructures with the radii of approximately 50 nm from our experimental results (not shown here). Further attentions can be paid to the decrease of the distance between LSPs and excitons, which may be achieved via the optimization of the thickness of the luminescence matrix as well as the appropriate design of the luminescence structure. Multilayers with the active matrix inserted between barrier layers may be also an available choice.