Revising of the Purcell effect in periodic metal-dielectric structures: the role of absorption

Periodic metal-dielectric structures attract substantial interest since it was previously proposed that the spontaneous emission amplification rates (the Purcell factor) in such structures can reach enormous values up to 105. However, the role of absorption in real metals has not been thoroughly considered. We provide a theoretical analysis showing that absorption leads to diminishing values of Purcell factor. We also suggest that using emitting organic compounds such as CBP (4,4-Bis(N-carbazolyl)-1,1-biphenyl) can lead to a moderate increase of about an order of magnitude in the Purcell factor. Defining the experimentally measured Purcell factor as a ratio between the excited state lifetimes in bare CBP and in periodic structure, this increase in the fabricated periodic structure is demonstrated through a 4–8 times decrease in excited state radiative lifetime compared to a bare organic material in a wide emission spectrum.


Results and Discussion
First, we have calculated the probability density of spontaneous emission for infinite idealized structures, for which we have assumed absence of absorption in silver. Such approximation is necessary for investigating of electromagnetic modes structure of infinite periodic structure. The period of the structure consisted of two layers: 15 nm of silver and 15 nm of vacuum. We limit our consideration to the case of TM polarization because SP and its associated features in the dispersion relations responsible for the increase of DOS and spontaneous emission rate can be realized only for TM polarization of light. The dispersion relations hω(K) and the spatial profile of the electromagnetic mode can be calculated using Bloch theorem (see chapter calculation technique of the section "Methods"). The resulting band structure is shown in Fig. 1. Plasmonic bands, positioned outside the light cone near the frequencies of SP of single silver-vacuum interface are clearly seen. SP and associated plasmonic band are waveguided modes, therefore, formally, there are solution for any wavevector k ⊥ . At the same time, since concentration of electrons in the plasma is finite, we should confine the interval of wave vectors, which are considered by the value corresponding to the mean inverse distance between electrons in the plasma, which is about one third of inverse nanometer for silver. Based on the dispersion relation and the spatial profile of electromagnetic modes, one can calculate the probability density of spontaneous emission (Equation 10) for specific wavevector, and then, integrating it over k ⊥ one can get the Purcell factor. Figure 1b,c,e,f,h,i show the dependence of the probability density on the wavevector k ⊥ by color pattern. It can bee seen, that major contributions to the Purcell factor is provided by the states with high values of k ⊥ . Integrating the probability density of spontaneous emission, and normalizing it to the probability of the emission in the free space, one can obtain the Purcell factor, demonstrating pronounced peaks near the frequencies of SP on single interface. Integral Purcell factor (a,d,g) spontaneous emission probability density shown by color pattern (b,c,e,f,h,i) calculated for an infinite periodic structure of 15 nm Drude-modelled silver with no absorption/15 nm vacuum. The position of the emitting dipole is as follows: middle of the vacuum layer (top row); middle of the silver layer (middle row); layers interface (bottom row). Dipole orientation is as follows: parallel to the layers (red curve and middle column); perpendicular to the layers (blue curve and right column).
www.nature.com/scientificreports www.nature.com/scientificreports/ To analyze the effect of absorption, we compare here results obtained for silver, which were used in our experimental studies and in previous theoretical work 22 .
The dielectric constant in metals according to the Drude theory can be written as follow: Where for silver ε b = 4.96, ω = .
8 98 p  eV and γ = . 0 018  eV 27 . However, the Drude theory fails within the frequency range which is particularly important for plasmon phenomena. Figure 2 illustrates the comparison of the dielectric constant approximated by the Drude model with the dielectric constant measured experimentally for real silver 28 . It is clear that the Drude model is inappropriate for the higher frequency range. The main difference is not related to the refractive index but to the definition of the coherence length: www.nature.com/scientificreports www.nature.com/scientificreports/ Here A is the absorption coefficient. The coherence length for real silver is between 10 and 100 nm in the energy range of 3-4 eV. These values are much smaller than predicted by the Drude model, which leads to a significant reduction of any collective effects on scales beyond that length. Moreover, the energy density in metals is proportional to ε + ωdε/dω, whose value (plotted in Fig. 2d) differs dramatically for real silver.
The difference between the Drude model and experimental values becomes even more drastic for the plasmon dispersion curve. Indeed, from the dispersion equation we get the plasmon wavevector: Here ε 1 and ε 2 are dielectric constants of materials forming the interface. Comparison between a dispersion curve calculated using the Drude model and for real silver is shown in Fig. 3. We point out here that ignoring absorption in the Drude model leads to a situation when states with any wavevector k ⊥ are available; at the same time, there is a forbidden zone for energy: The situation changes completely when absorption is taken into account. Now, the spectrum is continuous in the whole frequency range, but there is a limit on the values of the wavevector k ⊥ . Numerically, its value doesn't exceed 0.1 nm −1 and 0.05 nm −1 for the Drude-modelled and experimentally measured silver, respectively.
For experimental structure, it is important to choose a suitable material that will serve as an emitter instead of vacuum layers. Currently, organic emitting materials are considered as very promising in the field of photonics 29 . CBP (4,4-Bis(N-carbazolyl)-1,1-biphenyl) is one of the most popular host materials for different types of organic light emitting diode (OLED) systems including thermally activated delayed fluorescence devices due to its relatively large band-gap and ambipolarity 30,31 . We have chosen CBP as the dielectric emitting organic material due to the fact that the interface between CBP and silver allows SP within the CBP emission region (see Fig. 4a for the CBP refractive index, PL spectrum and molecular structure scheme; the calculated plasmonic band structure is shown in Fig. 4b). Co-occurrence of the spectral position of the SP on the silver/CBP interface and the emission band of CBP could be used for development of efficient organic light-emitting diodes for the ultraviolet region.
It should be noted that attempts to use organic compounds as light emitters in periodic structures have been reported previously, however, without success in achieving high values of the Purcell factor 32 . The possible reason for this result can be related to the fact that the emitting material was only deposited on the surface of the structure. Contrary, we use an emitting organic material as the dielectric layer in each period of the structure.
Moreover, theoretical calculations of the Purcell factor for a single metal-dielectric interface have been performed for three different configurations: vacuum/silver (according to the Drude model), vacuum/silver (after experimental work) 28 and, finally, for CBP/silver (after experimental work) 28 . The results are presented in Fig. 5. It can be seen that calculations with real silver accounts for a dramatic decrease in the Purcell factor (by more than two orders of magnitude), while using CBP has several effects such as a shift of the peak amplification away from a small coherence length region, an increase of the Purcell factor value to about 10 and broadening the spectrum.
In order to approve our theoretical analysis, we have fabricated a series of test structures with the different CBP layer thicknesses (10 nm, 15 nm, 20 nm and 30 nm) and with a silver layer with a thickness of 15 nm deposited on Al 2 O 3 (sapphire) substrate (see Fig. 6a illustrating an example of the test structure). Cross-section scanning electronic microscopy (SEM) image of the 5 period 30 nm CBP/15 nm Ag is shown in Fig. 6b. To get SEM images, the sample was cracked, which affected interfaces; however, it is clear that the interfaces between CBP and silver achieved by this fabrication method are rather smooth. www.nature.com/scientificreports www.nature.com/scientificreports/ Room temperature photoluminescence spectra measured from the top of the aforementioned structure are presented in Fig. 7. The bare CBP layer shows a wide emission band in the frequency interval from 2.8 eV to 3.3 eV, and the emission spectrum almost does not depend on the emission angle. It can be seen that a periodic structure exhibits a trend of widening of the emission range compared to the bare emitting CBP while also growing distinct peaks which roughly correspond to plasmonic peaks in Fig. 4b. The positions of the basis peaks for different widths of CBP are nearly the same as in the bare CBP case (3.02 eV, 3.17 eV, 3.26 eV) but with a redistributed intensity according to the coupling with periodic structure.
Further, we measured the lifetimes of excited states at certain energies. Fluorescence decay of CBP material is determined by two lifetimes and biexponential behavior as shown in Fig. 8 -an example of intensity decay curve for 20 nm CBP/15 nm Ag structure. The CBP molecule is characterized by "fast" (τ fast ≈ 0.33 ns) and "slow"  www.nature.com/scientificreports www.nature.com/scientificreports/ (τ slow ≈ 2 ns) lifetimes and those parameters are constant within the error of measurement and approximation within the whole CBP emission range.
We define the experimentally measured Purcell factor as a ratio between the excited state lifetimes in bare CBP and in the periodic structure. In our case measurements showed that for every structure the ratios between "fast" lifetimes of a bare CBP and periodic structure are the same as the ratios between "slow" lifetimes within the range of the experimental error. Figure 7b demonstrates the distribution of lifetime ratios for various periodic structures at the energies corresponding to CBP emission range. Lifetime ratios' magnitude is from 4 to 8; relatively high experimental error, especially at the edge of the CBP is due to low intensity of CBP emission at energies  www.nature.com/scientificreports www.nature.com/scientificreports/ higher than 3.4 eV. This means moderate increase of spontaneous emission rate for the whole set of test structures. The interface between metal and CBP accounts for the majority of the decay, making it non-radiative. Taking this fact into account we can conclude that decrease in the excited state lifetime due to radiative decay in structures with various CBP thickness is still no less than 4.

Conclusions
In summary, we have modelled, fabricated and studied periodic metal-dielectric structures. We have shown that contrary to previous propositions the giant values of the calculated Purcell factor are only achievable when absorption in metals is not considered. Using experimentally obtained refractive index spectra for silver we have calculated that in a silver/vacuum structure the values can barely exceed unity. However, by tuning away from high DOS region and using organic materials that emit light in the plasmonic band one can achieve a moderate increase in the Purcell factor that is about 10 times higher than in simple metal/vacuum model. We have backed this claim with experimental study of a periodic structure that had CBP as active region. We have found that the excited state lifetime is 4-8 times shorter in metallic structures compared to bare CBP. Methods sample preparation. The set of periodic structure samples with different thickness of the CBP (10 nm, 15 nm, 20 nm and 30 nm) was fabricated using a Kurt J. Lesker ® Spectros II ™ system, an organic and metal thin film deposition system capable of reaching vacuum levels as low as 1 × 10 −6 mbar. 5 periods of CBP and silver layers were deposited on the Al 2 O 3 (sapphire) substrate. A set of calibration depositions for both materials was performed with greater accuracy. Also, a CBP layer of thickness 50 nm was grown on Al 2 O 3 substrate for comparison of bare CBP properties with periodic structures properties. optical measurements. Emission spectra of bare CBP layer and periodical structures were measured using a Jobin-Yvon Horiba ® Fluorolog ™ FL3-22 spectrometer (bare CBP and periodical structures were excited at 4.73 eV) at room temperature and atmospheric pressure.
Time-resolved measurements were performed using time-correlated single photon counting (TCSPC) measurement scheme at room temperature and atmospheric pressure. Structures was excited through the top silver layer at 262 nm wavelength, the third harmonic of the output from a Coherent ® Mira ™ 900F Ti:Sapphire oscillator tuned to 786 nm central wavelength, with a 76 MHz repetition rate. A 365 ± 10 nm bandpass filter with OD (optical density) >4 outside this region was used. Samples were excited at 45° angle to the substrate normal. Emission from samples was collected also at 45° angle to the substrate normal at various energies (from 3.02 eV to 3.4 eV which is inside the range of CBP molecule emission).

Calculation techniques.
To obtain the Purcell factor we start with deriving a formula for the emission probability. In an infinite structure with layers parallel to xy plane the coordinates in the k-vector space are (k x , k y , K), where K is the Bloch vector. The well-known equation for the emission probability is: where dW is the probability of a radiative transition between dipole states with energies E i and E f . The transition is characterized by the matrix element V fi , the photon energy ω  and the number of photon states dν that is proportional to a volume element in k-space and quantization volume V (which in an infinite structure spans one period): x y 3 www.nature.com/scientificreports www.nature.com/scientificreports/ Since we are deriving the frequency Purcell factor we have to calculate ∫ dW for a constant ω and then divide the obtained emission probability by that of a free space (where d is the absolute value of the dipole moment): This task is approached by replacing variables (k x , k y , K) with (ω, k ⊥ , ϕ) where k ⊥ is the value of wavevector in xy plane and ϕ is the angle in this plane. This gives 3 At this point it is important to consider the matrix element V fi :   In a finite structure such as a single interface, the procedure is very similar to the described above; however, we now deal with two dimensional k space, and after proper substitutions we obtain Here S is quantization space (as opposed to quantization volume). Then, the normalization condition can be expressed as follow: n z E z dz 4 ( ) ( ) 2 (13) 2 2 after that the ambiguity in choosing S is also resolved. Finally, the value of Purcell factor is 2 2 0 2

Data Availability
The experimental and modelling data are present in the paper. Additional data related to this paper may be requested from the authors.