1. Introduction

Organic light-emitting diode (OLED) performance is significantly limited by poor light outcoupling efficiency, g_out (20% - 33.7%) [1–6] manifesting from: (1) non-radiative exciton decay within the organic semiconductor as heat to one of the electrodes (5.3% loss) [2,5]; (2) guided photonic modes within the emitter layer (9.3% - 35.2% loss) [1,3–5]; and, (3) significantly, to tightly bound surface plasmon polariton (SPP) modes at the metal electrodes (up to 46.8% loss) (see Fig. 1(a)) [1,2,4,6]. The major approaches taken to minimize optical energy loss to SPP modes in OLEDs include: (i) introduction of Bragg gratings or other wavelength-scale corrugated microstructures to fulfill the energy and momentum conservation of trapped modes and light propagating in air [1,3,5,6]; (ii) anisotropic orientation of the molecular dipole emitters of the active organic layer to an in-plane configuration that avoids coupling to bound SPP modes (limited to extended or anisotropic molecules) [1,3]; and (iii) using high index prisms in to extract SPPs and wave-guided modes [1]. Other approaches include developing electrodes fabricated from metals with SPP energies larger or smaller than the device operating energies so that they do not couple with photons emitted by the device [2,6], and adding spacer layers to control the degree of coupling to SPP modes [6,7]. The aforementioned strategies have potential to reduce SPP losses but may introduce new leakage or optical dissipation channels, require additional and potentially complex fabrication, and significant changes to the device architecture [6].

 

Fig. 1 Illustrations of (a) a generic OLED exhibiting a tightly-bound SPP mode at the metal electrode propagating in the z-direction, and (b) the F8BT-Ag-SiO₂ SMI waveguide studied in this work . (c-f) Schematic representations of |H_y|versus x cross-sections for: SB, (c) SL, (d) AB, (e) and AL (f) for an antisymmetric IMI waveguide (where$0<\left(\left(\left|\epsilon \text{cover-}\epsilon \text{substrate}\right|/\epsilon \text{substrate}\right)\times 100\right)<1\%$). Blue dotted line in (e, f) represents H_y versus x cross-sections of the AB and AL modes, respectively.

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Compelling evidence suggests that if the SPP mode character and leakage into the semiconductor can be controlled rather than suppressed entirely, SPPs may boost performance in a host of visible-light-based optoelectronic devices including organic photovoltaics, organic lasers, and OLEDs [7–14]. Here, we have taken a theoretical approach towards mitigating optical loss to tightly bound SPPs in light-emitting organic optoelectronics by solving a set of three dispersion relations for a semiconductor-metal-insulator (SMI) asymmetric planar waveguide, which serves as a direct analogue to a metal electrode/organic semiconductor interface. Through calculations of mode effective index, propagation length, penetration depth and magnetic field confinement, we demonstrate that by simply tuning metal film thickness and emission regime, a significant fraction of a SPP mode amplitude can leak into the organic semiconductor emitter layer, thereby decreasing the amount of optical energy lost to parasitic coupling to tightly bound SPP modes.

2. Methods

Extensive theoretical work has been conducted on understanding SPP mode propagation and mode symmetries that can be supported by insulator-metal-insulator (IMI) waveguides as a function of cover and substrate dielectric constants, metal type, and metal film thickness, towards the realization of chip-scale SPP propagation lengths [15–20]. We use this work as the basis for much of the theoretical calculations reported for the SMI structure presented here. While some of the seminal theoretical IMI studies consider the cover dielectric medium to have a complex dielectric constant ((i.e., ${\epsilon }_{\mathrm{cov}er}={\epsilon }_{\mathrm{cov}er}^r+ i{\epsilon }_{\mathrm{cov}er}^i$, where${\epsilon }_{\mathrm{cov}er}^r$ and $i{\epsilon }_{\mathrm{cov}er}^i$are the real and imaginary components), many of these studies assume both the cover and substrate materials dielectric constants to be purely real (i.e., the imaginary component is neglected; ${\epsilon }_{\mathrm{cov}er}={\epsilon }_{\mathrm{cov}er}^r$and ${\epsilon }_{substrate}={\epsilon }_{subsatrate}^r$), and only the metal film is taken to have a complex dielectric constant (i.e., ${\epsilon }_{metal}={\epsilon }_{metal}^r+ i{\epsilon }_{metal}^i$) [15,16,19]. This was done to minimize the computational intensity of solving the dispersion relation to yield the complex surface plasmon polariton wavevector, k. In some systems, because low loss dielectric media such as SiO₂ were utilized, neglecting the imaginary component of the complex dielectric constant (i.e., loss) and taking the dielectric constant to be a purely real was a valid simplifying assumption [15]. However, in making this assumption, SPP propagation lengths could be somewhat overestimated.

In this study, SPP propagation is in the z-direction with the electric and magnetic field components decaying in the x-z and x-y planes, respectively (see Fig. 1). The SMI waveguide is comprised of a silver film of thickness, t, which extends from 0-t along the x-axis and two infinite dielectric/semiconductor slabs: a SiO₂ substrate, which extends from t to + x along the x-axis and an F8BT (Poly[(9,9-di-n-octylfluorenyl-2,7-diyl)-alt-(benzo [2,1,3]thiadiazol-4,8-diyl)]) cover, which extends 0 to −x, along the x-axis (see Fig. 1(b)). Dispersion relations were solved at a fixed wavelength, λ, of 576 nm (which correlates to the peak gain wavelength of F8BT [21]) for three dielectric constant cases, where the imaginary component of the complex dielectric constants (ε), (where$\epsilon ={\epsilon }^r+i{\epsilon }^i$and ${\epsilon }^r$ and ${\epsilon }^i$ are the real and imaginary components) of the SiO₂ substrate and F8BT cover were varied. The complex dielectric constant of silver at 576 nm for all three cases was fixed at $-12.5681+i0.854591$ [23]. The complex dielectric constant for F8BT at 576 nm under steady state excitation was calculated using empirically-derived expressions for the standard critical point exciton model from Campoy-Quiles et al. (see Eq. (1) in [21] and Table 1 in [21] and [22]). The first dielectric constant case, defined as Purely Real Steady State (PRSS) was solved to establish the characteristics of a system without absorption losses in the substrate or cover (i.e., lacking an imaginary dielectric constant component:$\epsilon ={\epsilon }^r$) and where the F8BT undergoes steady state (i.e., spontaneous) emission. For the PRSS case, ${\epsilon }_{Si{O}_2}=2.13$and${\epsilon }_{F8BT}=3.6446$. The second case, defined as Complex Steady State case (CSS), was also solved for when the F8BT undergoes spontaneous emission but the cover and substrate media were considered to have complex dielectric constants ($\epsilon ={\epsilon }^r+i{\epsilon }^i$), which accounted for absorption losses in both media. For the CSS case, ${\epsilon }_{Si{O}_2}=2.13+i0.0015$ and${\epsilon }_{F8BT}=3.6446+i0.2358$. The third case, defined as Complex Gain (CG), took account for absorption loss in the SiO₂ substrate but was solved for when the F8BT undergoes stimulated emission (i.e., a population inversion in the F8BT is achieved) and can provide loss compensation (gain) to the propagating SPP modes. The intrinsic gain of F8BT measured in aqueous solution can be ≥ 10,000 cm⁻¹, but in solid-state waveguide geometries this value is often lower due to losses associated with the structure [24–26]. For the CG dispersion relation, a net gain of 47 cm⁻¹ was assumed - a relatively modest assumption for F8BT in planar thin film geometries, where net gains upwards of 79 cm⁻¹ have been measured [24–26]. Gain can be mathematically reflected in the complex dielectric constant of a material by a negative imaginary component (${\epsilon }^i$) [27]. To relate the value of ${\epsilon }^i$ for F8BT to a net gain of 47 cm⁻¹ at 576 nm, the following expression was used [27]: ${\gamma }_{net}=-k_0{\epsilon }^i/{({\epsilon }^r)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ where ${\gamma }_{net}$is the net power gain coefficient, $k_0$ is the free space wavenumber ($k_0=2\pi /\lambda$). Thus, for the CG case, ${\epsilon }_{Si{O}_2}=2.13+i0.0015$and${\epsilon }_{F8BT}=3.6446-i0.00083$.

Dispersion relations (Eqs. (1-4) for the three dielectric constant cases were solved iteratively by bounding the real ($k^r$) and imaginary ($k^i$) components of the complex plasmon polariton wavevector, $k$, using the Wolfram Mathematica® 8 “NMinimize” function which finds the global minimum of a problem with constraints (see Appendix for Wolfram Mathematica® dispersion relation code and boundary conditions used for the solutions). NMinimize can return values indicating a solution but which are not physically reasonable or accurate. Thus, only values for $k^r$ and $k^i$ were accepted when the left hand side of the dispersion relation (Eq. (1) had magnitudes $<{10}^{-15}$ [15]:

From the real ($k^r$) and imaginary ($k^i$) components of the complex surface plasmon polariton wavevector, the mode effective index, N_eff, and characteristic SPP mode propagation length, L, were calculated for all four modes of each of the three dielectric constant cases according to [15]:

3. Results and discussion

For asymmetric IMI waveguides studied in the literature, two mode symmetries (symmetric and antisymmetric) and two mode leakage types (bound and leaky) are typically observed, giving rise to four distinct mode types: symmetric bound (SB), antisymmetric bound (AB), symmetric leaky (SL), and antisymmetric leaky (AL) [15–20]. These four modes occur when the dielectric constants of the substrate and cover are not identical $({\epsilon }_{\mathrm{cov}er}\ne {\epsilon }_{substrate})$and arise because the decay constants ($\alpha \text{'}\text{s}$) are only defined as their squares (Eqs. (2-4), thus having both positive and negative roots [16]. In symmetric structures$({\epsilon }_{\mathrm{cov}er}={\epsilon }_{substrate})$the bound and leaky modes are degenerate and only two modes are supported: symmetric and antisymmetric modes [15–17]. By assigning which root is used in the dispersion relation equation for each of the three media SiO₂, Ag and F8BT, four solutions arise. The SB and AB modes were found when all three decay constants were taken to be positive (${\alpha }_0,{\alpha }_1,{\alpha }_2>0$). The SL mode was found when${\alpha }_2<0$, ${\alpha }_0,{\alpha }_1>0$and the AL mode was found when ${\alpha }_0<0$, ${\alpha }_1,{\alpha }_2>0$. No solutions were found when the negative root of the metal decay constant (${\alpha }_1$) was used in the dispersion relation solution, nor was a solution found when both ${\alpha }_0$ and ${\alpha }_2<0$ [15].

SPP mode symmetry is typically defined by the character of the cross-sectional profile of the magnetic field (H_y) or magnitude of the magnetic field (|H_y|) in the metal film and is schematically represented in Figs. 1(c)-1(f) for a structure with only slightly broken symmetry (i.e., 1% difference between ${\epsilon }_{\mathrm{cov}er}$ and ${\epsilon }_{substrate}$) [16,18]. Symmetric modes have H_y fields that do not change sign across the metal film (Figs. 1(c) and 1(d)), while antisymmetric modes exhibit a zero or nodal point within the metal film (Figs. 1(e) and 1(f)) [16]. Thus, H_y fields of antisymmetric modes result in opposite charge on either side of the metal film and, hence, a significant fraction of the mode power resides within the metal, which leads to less leakage into the surrounding media and shorter SPP propagation lengths (Figs. 1(e) and 1(f)) [17,19]. ‘Leaky’ modes are less confined to the metal and in turn generally penetrate further into the surrounding media than the ‘bound’ modes (Figs. 1(d) and 1(f)). In cases where the difference between the cover and substrate dielectric constants is larger (~71% for the SMI waveguide proposed here), the magnetic field profiles can maintain their characteristic behavior, but leakage is expected to occur more significantly into the higher refractive index insulator/semiconductor medium [15,17].

For the F8BT-Ag-SiO₂ SMI structure studied here, mode effective index (N_eff) versus metal film thickness (t) is shown in Figs. 2(a)-2(c) for the three dielectric constant cases (see Eq. (5). For all three cases, the two antisymmetric modes had greater N_eff values (ranging from 2.26 to 8.78) than the two symmetric modes (ranging from 1.60 to 1.64) over all calculated metal film thicknesses, and N_eff increased with decreasing t. A greater N_eff indicated that the majority of the mode energy was confined within the material with the highest dielectric constant, i.e., the metal [15]. Since the symmetric modes had the lowest N_eff values, it was expected that the modes penetrated much further into the insulator/semiconductor media than the antisymmetric modes. Bound modes generally had slightly higher N_eff values than leaky modes suggesting they are more tightly confined to the metal-insulator/semiconductor interface [15]. This was observed more clearly at $t<40$ nm, where the disparity in N_eff between bound and leaky modes for both the symmetric and antisymmetric cases increased with decreasing t.

 

Fig. 2 (a-c) N_eff and (d-f) Log₁₀(L) versus t calculated for (a, d) the PRSS case, (b, e) the CSS case and (c, f) the CG case for all four SPP modes supported by the F8BT-Ag-SiO₂ SMI waveguide

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For all three dielectric constant cases, symmetric modes exhibited mode propagation length (L) values significantly larger than antisymmetric modes over the range of t (see Figs. 2(d)-2(f) and Eq. (6). The shorter L values of the antisymmetric modes were attributed to greater mode confinement within the metal. With increasing t, the bound and leaky mode solutions of the symmetric and antisymmetric branches became degenerate in L [15]. The SB mode, which had the longest L$$values for 45 nm ≤ t ≤ 47 nm, was calculated to propagate 217 μm for the CSS case, 271 μm for the PRSS case, and 1357 μm for the CG case. It is notable that the PRSS case overestimated L by a factor of 1.25 over the CSS case and that the CG case showed a L enhancement of 6.25 over the CSS case. The L values of the SB mode decreased significantly from its maximum for t < 45 nm and t > 47 nm. For all three dielectric constant conditions, at larger metal film thicknesses, e.g., t = 80 nm (selected because for t > 60 nm, L of all four modes were almost unchanged) L values were reduced significantly and converged to values in the range 8.26 - 8.82 μm for the SB mode and 6.08 - 7.47 μm for the SL mode. The antisymmetric modes exhibited maximum L at t > 60 nm, and at t = 80 nm the CSS case resulted in AB and AL mode L values of 0.68 μm, i.e., more than a factor of 4 shorter than the PRSS (AB: 2.88 μm, AL: 2.93 μm) and CG (AB: 2.92 μm, AL: 2.96 μm) cases, indicating they are more lossy and, hence more tightly bound modes relative to the symmetric modes. These results show that stimulated emission in the semiconductor can enhance SB mode propagation length significantly and also underscore the importance of treating the cover and substrate as media with complex dielectric constants, so that absorption loss is taken into account and, in turn, L is not overestimated. For PRSS and CG cases, solutions for the SB mode for L below t = 47 nm and 45 nm, respectively, were not included in Fig. 2(d) and 2(f) because the imaginary component of the complex plasmon polariton wavevector $k^i$ became negative, indicating that the structure no longer supported the SB mode. It is known that the SB mode can exhibit a mode cutoff at a certain metal thickness (t_cutoff), below which $k^i$ becomes negative and the mode either ceases to propagate or changes character from a bound to a growing or leaky mode [18,19]. The existence of a SB mode t_cutoff is highly dependent on the values of the real and imaginary parts of the complex dielectric constants of the cover and substrate media [18,19]. t_cutoff is expected to decrease as the values of the real parts of the cover and substrate dielectric constants converge [18]. Positive imaginary components of the cover and substrate dielectric constants also results in a decrease in t_cutoff because the attenuation introduced by the positive imaginary component forces the mode to be more tightly confined to the metal-dielectric/semiconductor interface as opposed to leaking into the cover and/or substrate media. For the F8BT-Ag-SiO₂ SMI waveguide, the PRSS case had no imaginary dielectric constant components for either the cover or substrate$(i{\epsilon }_{Si{O}_2}^i=i{\epsilon }_{F8BT}^i=0)$; the CG case had a small negative imaginary component for F8BT to reflect gain $(i{\epsilon }_{F8BT}^i=-0.00083)$ and only a small positive imaginary component for SiO₂$(i{\epsilon }_{Si{O}_2}^i=0.0015)$. Based on these considerations we conclude that the SB mode exhibited a metal-film-thickness-dependent mode cutoff under the PRSS and CG conditions and the SMI structure did not support the SB mode below t_cutoff~45 and 47 nm, respectively. Conversely, under the CSS condition, which had both a cover and substrate with positive imaginary components $(i{\epsilon }_{Si{O}_2}^i=0.0015$and$i{\epsilon }_{F8BT}^i=0.2358)$, the SB mode is supported down to a metal film thickness of less than 30 nm, well below the cutoff thickness of the PRSS and CG cases.

Mode penetration depth, D, into both cover and substrate for the four modes and three dielectric constant cases were calculated over a metal film thickness range of 25-100 nm (see Figs. 3(a)-3(h) and Eq. (7). For all modes and dielectric constant cases, D was much greater in the medium with the larger dielectric constant (F8BT), particularly for symmetric modes. The symmetric modes in Figs. 3(a)-3(d), both exhibited significantly greater D values in both SiO₂ and F8BT because they were not as tightly bound to the metal as the antisymmetric modes. As a result, the dielectric constant changes for the three cases had significant impact on D. The SB mode exhibited relatively constant D values (D ≈150 nm) in SiO₂ as a function of t but penetration into F8BT increased significantly with decreasing film thickness to a value of 176 μm for the PRSS case, 0.8 μm for the CSS case and 74 μm for the CG case at a thickness of 47 nm, i.e., the approximate film thickness that yielded the longest propagation length; see Figs. 2(d)-2(f), 3(a), and 3(b). D of the SB mode in F8BT for the CSS case was over two orders of magnitude less than the PRSS and the CG cases. When comparing the PRSS case and the CG case, the PRSS case penetrated only slightly farther in to the F8BT.

 

Fig. 3 D as a function of t, in F8BT (a,c,e,g) and SiO₂ (b,d,f,h) for: (a,b) SB; (c,d) SL; (e,f) AB; (g,h) AL for three SMI waveguide dielectric constant cases.

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The SL mode exhibited a constant D of approximately 138 nm in SiO₂ for t > 30 nm for the dielectric constant three cases. However, for t < 30 nm the PRSS and CG cases showed an increase in D and the CSS case exhibited a decrease in D with decreasing t (see Figs. 3(c) and 3(d)). D of the SL mode in F8BT for the PRSS and CG cases exhibited a trend of decreasing D with decreasing thickness for t < 65 nm, with the PRSS case having greatest overall D value of 4.956 μm at t = 100 nm compared to the CG case with a maximum D value of 4.859 μm at t = 100 nm (see Figs. 3(c) and 3(d)). At t < 40 nm, D of the SL mode in F8BT for the CSS case deviated slightly from the PRSS and CG cases and exhibited a local maximum at t = 23.5 nm and then rapidly decreases with decreasing t (see Figs. 3(c) and 3(d)).

For the AB and AL modes (Fig. 3(e)-3(h)), D values were small (<55 nm in SiO₂ and <80 nm in F8BT) and were virtually identical for the three dielectric constant cases. Small penetration depths relative to the SB and SL modes further suggest that these were more tightly bound lossy modes that were largely confined to the metal-insulator/semiconductor interfaces. As a result, large changes in the imaginary part of the complex dielectric constants of the cover/substrate media did not appear to have a notable impact on the penetration depth of antisymmetric modes into the SiO₂ and F8BT. D was relatively constant for both the AB and AL modes for all three dielectric constant cases for t > 40 nm with average D values of 52 nm in SiO₂ and 74 nm in F8BT. Below t = 40 nm, D decreased with decreasing film thickness.

Two representative film thicknesses (t = 47 nm, where the maximum D and L values were observed; and t = 80 nm, selected because for t > 60 nm, D and L values for all four modes were almost unchanged) were chosen to illustrate how cover and substrate dielectric constant and F8BT emission regime (i.e., spontaneous or stimulated) affect H_y and|Hy| leakage of the four modes (see Figs. 4(a)-4(g) and Eqs. (8-10). At t = 80 nm, |Hy| at the SiO₂/silver interface for all four modes, especially the SB and SL modes, was greater than that at t = 47 nm due to decreased field leakage into the cover and substrate. |Hy|leaked significantly further into the higher dielectric constant medium (F8BT) and the SB mode exhibited virtually identical magnetic field profiles in SiO₂ and silver for the three dielectric constant cases at both t values. However, the SB mode, which had the greatest field amplitude of the four SPP modes, exhibited a strong dependence of Hy and|Hy| leakage into the F8BT on the dielectric constant case (see Figs. 4(a) and 4(b), respectively). At t = 47 nm, the PRSS and CG cases showed extreme leakage into the F8BT and did not exhibit the characteristic exponential decay into the cover and substrate media (see Figs. 4(a) and 4(b)). At 75% of the amplitude at the F8BT/silver interface, the magnetic field of the SB mode for the PRSS case extended more than 50 μm into the F8BT and the CG case extended over 21 μm, whereas the CSS case only extended 0.5 μm. This extreme leakage enhancement is depicted in the schematic inset in Fig. 4(b). The SB mode supported by all three cases was still a bound mode according to convention (see Fig. 1(c)), albeit weakly as the metal thickness was almost at the cutoff value at which the mode character changes to a “leaky” mode and/or ceases to propagate in the z-direction. Large SB mode field leakages at small t values have been reported in prior studies of IMI waveguides and while the historical naming convention of the mode does not accurately reflect this leaky behavior, it is congruent with the literature [15,18,19]. At t = 80 nm and at 75% of the amplitude at the F8BT/silver interface, the SB mode |Hy| profiles for all three cases exhibited shorter magnetic field decay lengths into the F8BT (~5.5 μm for the PRSS and CG cases and 0.96 μm for the CSS case) (see Fig. 4(c)). This substantially diminished leakage is depicted in the illustration inset in Fig. 4(c). This is attributed to increased mode confinement within the thicker metal slab and diminished influence by the surrounding media. The SL mode followed similar trends to that of the SB mode in that |Hy| was virtually identical for the three dielectric constant cases in both the silver and SiO₂, and in that leakage into the F8BT was greater and followed the same trend (|Hy|_PRSS >|Hy|_CG > (|Hy|_CSS) (see Fig. 4(d) and 4(e)). The antisymmetric modes exhibited much lower |Hy| values than the symmetric modes and their field profiles were largely unaffected by film thickness and dielectric constant case (see Figs. 4(f) and 4(g)). The PRSS and CG antisymmetric mode solutions exhibited virtually identical |Hy| versus x profiles at both t values. However, the CSS case exhibited greater confinement to the metal and more pronounced leakage into the SiO₂ (see Fig. 4(d) and 4(e)).

 

Fig. 4 (a) H_y and (b-g) |H_y| calculated at t = 47 nm (a,b,d,f) and t = 80 nm (c,e,g) for the SB (a-c), SL (d,e) and AB (f,g) modes for all three SMI waveguide dielectric constant cases. Schematic illustrations of the F8BT-Ag-SiO₂ SMI waveguide inset in (b,c) represent the |Hy| of the highly radiative ‘leaky’ SB mode at t = 47 nm and the tightly-bound SB mode t = 80 nm, respectively. AL mode not included.

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4. Conclusion

In conclusion, this theoretical demonstration of enhanced surface plasmon field leakage through the variation of metal film thickness and introduction of gain may lead to the ability to mitigate surface plasmon losses at metallic electrodes in organic optoelectronic devices such as OLEDs and organic lasers. Our theoretical results may vary somewhat from experiment due to metal film quality and the fact that our calculations do not take into account a finite thickness of the cover and substrate nor the F8BT dipole orientation (which is expected to play a critical role in the degree of coupling to SPP modes at optical frequencies) [8,28–31]. A dipole oriented parallel to the plane of the polymer-metal interface is expected to be damped by its own image on the metal surface, while a dipole oriented perpendicular to the plane is expected to couple to SPP modes [8,31]. Conversely, because the calculations here were carried out at a single wavelength coinciding with the peak gain wavelength of the F8BT and at a conservative net gain value for F8BT, the results do not fully account for optical gain that can occur over a broader spectral range (typically 552-600 nm) and for the higher F8BT gain values that could be achieved in practice [24,31]. Ongoing experimental work is expected to address these factors and determine the impact of the predicted surface plasmon leakage effects on the emissive properties of organic semiconductors waveguides as a function of metal film thickness, polymer dipole orientation and emission regime.

5. Appendix

Dispersion relations (Eqs. 1-4) for the three dielectric constant cases (PRSS, CSS, and CG) were solved iteratively in Mathematica using an N-minimize^® function and setting boundary conditions for the real part of the complex surface plasmon polariton wavevector, $k^r.$ The code in Lines 1-8 takes in the real and imaginary parts of the complex plasmon polariton wavevector, $k,$ plus all of the parameters to find the absolute value of the homogenous function (kmode). The function kplot[kr,ki] is used to calculate values of the homogenous equation for the standard inputs for the empirical dielectric constants for ${\epsilon }_0$ (${\epsilon }_{Si{O}_2},$substrate), ${\epsilon }_1$ (${\epsilon }_{Ag},$film) and ${\epsilon }_2$ (${\epsilon }_{F8BT},$cover). ${\epsilon }_{Ag}$ was fixed for the three dispersion relation conditions but ${\epsilon }_{Si{O}_2}$and ${\epsilon }_{F8BT}$ varied for the three dispersion relation conditions (see Appendix Table 1). Excitation wavelength was fixed at 576 nm and initial metal film thickness was fixed at 20 nm.

Table 1. ε values for the substrate (SiO₂) and cover (F8BT) media for the three dispersion relation conditions

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Dispersion Relation Solution Code

Ln[1] = apow2[k_,k0_,ε_]:=k^2-k0^2*ε

Ln[2] = k0[λ_]:=2*Pi/(λ*10^-9)

Ln[3] = apos[asqrd_]:=Sqrt[asqrd]

Ln[4] = aneg[αsqrd_]:=-Sqrt[αsqrd]

Ln[5] = kmode[kr_,ki_,ε0_,ε1_,ε2_, λ_,t_]:= Simplify[Abs[((apos[apow2[kr + I*

ki,k0[λ],ε1]]*ε2)/(apos[apow2[kr+I*ki,k0[λ],ε2]]*ε1))+((apos[apow2[kr+I* ki,k0[λ],ε1]]*ε0+apos[apow2[kr+ I* ki,k0[λ],ε0]]*ε1)+(apos[apow2[kr + I* ki,k0[λ],ε1]]*ε0-apos[apow2[kr+I*ki,k0[λ],ε0]]*ε1)* ε-2*apos[apow2[kr+I*ki,k0[λ],ε1]]*t*10^-9)/((apos[apow2[kr+I* ki,k0[λ],ε1]]*ε0+apos[apow2[kr + I* ki,k0[λ],ε0]]*ε1)-(apos[apow2[kr +I*ki,k0[λ],ε1]]*ε0-apos[apow2[kr+I*ki,k0[λ],ε0]]*ε1)* ε-2*apos[apow2[kr + I* ki,k0[λ],ε1]]*t*10^-9)]]

Ln[6] = e0 = 2.13; e1 = $-12.5681+i0.854591$ *I; e2 = 3.6446; λ1=576; t1= 20;

Ln[7] = kplot[kr_,ki_]:=kmode[kr,ki,e0,e1,e2,λ1,t1]

Ln[8] = kneff[neff_,lexp_]:=kmode[k0[λ1]*neff,10^(-lexp),e0,e1,e2,λ1,t1]

To solve for all four distinct mode solutions for this structure, either the positive or negative roots of $\alpha$ are selected for the desired condition from Lines 2 and 4 and inputted accordingly into Line 5 to solve for a particular mode. The solution step size and metal film thickness range over which the solution is to be solved for are inputted (Lines 9-11). For these solutions, a fixed step size of 1 nm was used and solutions were searched for over a range of 5-100 nm (Lines 10 and 11). Line 12 takes the real and imaginary parts of the complex surface plasmon polariton wavevector,$k$, and calculates the effective mode index and Log of $1/k^i$ (which is equal to the propagation length, L) over a range of metal film thickness (5-100 nm). Lines 13 and 14 generate tables of the real and imaginary parts of the complex surface plasmon polariton wavevector.

• Ln [9] = stepsize = 1;
• Ln [10] = tstart = 5;
• Ln [11] = tend = 100;
• Ln [12] = kthick = {}; Do[sol = Quiet@NMinimize [{kmode[kr,ki,e0,e1,e2,λ1,t],
• Kr>A*k0[λ1],kr<B*k0[λ1]}, {kr,ki}, WorkingPrecision $\to$100, MaxIterations$\to$5000];
• AppendTo[kthick,sol],{t,tstart,tend,stepsize}];
• Ln [13] = kthicklist = Table[{tstart + stepsize(t-1), kthick[[t,2]][[1,2]]/k0[λ1]}, {t, 1,
• Length[kthick]}]
• Ln [14] = kthickimaglist = Table[{tstart + stepsize(t-1), Log[10,1/kthick[[t,2]][[2,2]]]}, {t, 1,
• Length[kthick]}];

The most critical step in the finding physically reasonable mode solutions are the $k^r$ boundary conditions (denoted by “A” and “B” in Line 12 of code). For a given mode, multiple distinct boundary conditions may be required over the 5-100 nm metal film thickness range of interest. Appendix Table 2 summarizes the boundary conditions used to solve for the four distinct modes (SB, SL, AB, and AL) for the three dispersion relation conditions (PRSS, CSS, and CG).

Table 2. Boundary conditions (A and B) for the real part of the complex surface plasmon polariton wavevector,$k^r,$at specific metal film thickness ranges (t_start to t_end) for the four mode solutions (SB, SL, AB, and AL) for the three dispersion relation conditions (PRSS, CSS, and CG). Metal film thicknesses are entered into Lines 10 and 11 and boundary conditions (A and B) are entered into Line 12 of the code.

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Acknowledgments

The authors wish to acknowledge D. D. C. Bradley and M. Campoy-Quiles for providing the optical constants used to calculate the complex dielectric constant of F8BT. The authors acknowledge support from The Nanotechnology for Clean energy NSF IGERT Program (grant no. 0903661) and partial support from National Science Foundation (DMR-1309459).

References and links

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