We extend the model of Chance, Prock and Silbey and analytically determine the Poynting vector in the direction perpendicular to the plane of a multilayer organic device. The result is used to predict the spatial profile of Förster energy transfer, the radiative output of an organic light emitting device, and to calculate the efficiency of surface plasmon polariton-mediated energy transfer across a thin silver film.
© 2007 Optical Society of America
Organic semiconductors are typically characterized by strong optical absorption but relatively poor charge transport. Taken together, these properties oblige the use of thin organic device structures, typically on the order of the wavelength of visible light. Thus, the design of organic photovoltaics (OPVs) and light emitting devices (OLEDs) must consider near field phenomena and the photonic mode density. The traditional approach relies on the dipole model of Chance, Prock and Silbey (CPS).  In CPS theory an exciton within the organic device is modeled as a point dipole whose electric field is described by dyadic Green’s functions. This technique calculates the dipole lifetime and the wavevector distribution of its decay. But it does not directly describe interlayer energy transfer, and in many cases, a quantitative understanding of interlayer energy transfer is impossible. Thus, it is common to couple the exact CPS model with a quantum mechanical model that compares the probability of exciton decay as a function of wavevector to the various non-radiative and radiative modes of the multilayer stack.  Such hybrid approaches have been used, for example, to calculate the fraction of photons emitted by an OLED in the viewing direction, [3,4] but they suffer from complexity and are innately approximate. In this work we eliminate the need for the quantum mechanical appendage to CPS by analytically determining the Poynting vector. The solution is compact and easily implemented in existing CPS models. We demonstrate its accuracy by: (i) calculating the spatial profile of Förster energy transfer within a thin film consisting of a mixture of donor and acceptor molecules,  (ii) calculating the angular emission profile and the radiative output of a typical OLED, and (iii) calculating the efficiency of surface plasmon polariton (SPP)-mediated energy transfer across a thin silver film.
Spontaneous emission from a dipole depends on the local density of states at the position of the dipole. Following the formulation of Tai and the model of CPS, we calculate the interaction between the dipole and electromagnetic field using dyadic Green’s functions. Figure 1 shows a general multilayer structure, where εj, represents the complex dielectric function and dj the thickness of each layer. The first and last layers are semi-infinite. The Green function coefficients cj, and fj will be explained below. The randomly-oriented dipole resides in the sth layer, which is arbitrarily placed in the multilayer stack. Each layer is assumed to be isotropic and higher order multipole radiation is neglected.
We begin the calculation with the expression for the electric field due to an oscillating current in the MKS unit system
where ω is the oscillation frequency, μ 0 is the magnetic permeability, J(R′) is the current and G(R|R′) is the dyadic Green’s function, which incorporates the boundary conditions. For a two-dimensionally-symmetric multilayer stack the Green’s function can be described using two independent sets of eigenfunctions in cylindrical coordinates:
where j is the layer index, κ and h are the amplitudes of the parallel and perpendicular components of the propagation vector k, and Jn refers to a Bessel function of the first type of order n. Even and odd eigenfunctions are represented by e and o. Using the eigenfunctions M and N we write the Green’s functions for the source and scattering: 
where primed eigenfunctions are functions of R′, s denotes the source layer, j denotes the jth layer, and the dipole position is taken as z = 0. In the convention we use, with the primed eigenfunctions in transpose form, G takes the form of a 3×3 matrix to be multiplied by the 3 × 1 J (current) vector, giving the integrand in Eq. (1).
The coefficients c, f and c′, f′ correspond to the left and right traveling eigenfunctions, respectively. Solving Maxwell’s equations at the interfaces, the relations between these coefficients can be determined: 
In the absence of external radiation sources, we begin the calculation of the coefficients of scattering Green’s functions in each layer by setting c 1′ = f 1′ = 0 and cN = fN = 0. Next, using the interface equations we numerically calculate the ratios of the coefficients starting at the outer layers. Arriving at the dipole layer, we determine the individual coefficients from the calculated ratios, noting the addition of the non-scattering Green’s function (G 0). Using these calculated coefficients and once again applying the interface equations, we calculate the coefficients for each layer from dipole layer to outermost layer. Once all the coefficients are determined, the value of the Green’s function can be calculated at every point in the stack.
To calculate the dipole energy transfer efficiency, the real part of the time-averaged divergence of the complex Poynting vector must be normalized by the dipole decay rate. We begin with calculation of the dipole decay rate, b. Following CPS, it is found by incorporating the effect of the reflected field on the dipole by the following equation: 
where the expression outside the brackets is the natural decay rate in vacuum, b 0; E 0 is the magnitude of the electric field at the dipole position, e is the electron charge, m is the reduced mass of the exciton, ε is the permittivity and q is the quantum yield of the emitting state. Due to the anisotropy of the electric field in Eq. (10), b is calculated for surface-parallel and perpendicular dipoles separately. Since there are two axes parallel to the layer plane and one axis in the perpendicular direction, the isotropic decay rate is biso =b ⊥/3 + 2b ∥/3. Expanding the field in terms of the Green’s functions we get the perpendicular and parallel components of b:
Next, we calculate the divergence of the Poynting vector. Because the structure is assumed to be infinite in the radial dimension, the presence of loss in any layer allows the divergence to be simplified as
We start with the expression of Sz in terms of general E field components in cylindrical coordinates
where * indicates the complex conjugate. Using Eq. (1), we restate the electric field components in terms of the spatial components of the eigenfunctions and the current. Assigning the dipole position to the origin we can define the current as J(R′) = -iω p 0 δ(R′) where p 0 is the dipole vector.
As in the calculation of dipole decay rates, we consider surface-parallel and perpendicular dipole orientations separately. Using the fact that Mz, J ⊥ r,ϕ, and J ∥ z are zero, we have two cases for the primed eigenfunctions at the origin: (1) when the Bessel function index n = 1, M′r = -M′ϕ = κ/2, N′r = -N′ϕ= κhs/2ks; (2) when the Bessel function index n = 0, N′z= κ 2/ks. For both cases, other components of the primed eigenfunctions are zero. Thus, E ⊥ jϕ is zero and we get the following expressions for the non-zero electric field components at each layer j,
where p 0 is the dipole moment in the absence of reflected field on the dipole.
Next, we calculate the final form of the Poynting vector perpendicular to the stack, Sz*, to determine the power flow. We insert the expressions, Eq. (15) through Eq. (19), for the electric field into the equation for Sz* and integrate over the surface area. Products of Bessel functions of different indices are orthogonal. In the integration, the remaining Bessel functions add up to κ for J 0 terms, and κ/2 for J 1 terms. To simplify the wavevector components, we normalize the wavevector, u = κ/ks and use two identities, κ 2 + h 2 = k 2 and kj 2 = εjks 2. Finally, we normalize with respect to the total energy of the free dipole (b 0), given by mp 0 2 ω 2/2e 2 (in Joules), and we obtain
Equations (20) and (21) are the central result of this work. The dipole energy transfer efficiency to an individual layer as a unitless percentage of total power emitted is found by taking the difference of the magnitude of this flux found at both boundaries of the layer and then dividing it by b ⊥/b 0 or b ∥/b 0; see Eq. (11)–(13).
Using the analytical results of the previous section and the optical constants of each material, we first calculate the spatial profile of Förster energy transfer in a thin film consisting of a mixture of donor and acceptor molecules. Then we calculate the angle-dependent emission of an OLED and the fraction of power ‘outcoupled’ to air. These results are compared to the previous experimental work of Segal et al.  Lastly, we predict the efficiency of SPP-mediated energy transfer and the emission spectra of the organic-metal-organic structure of Andrew and Barnes.
(i) Förster energy transfer
Förster energy transfer occurs when the evanescent near field of a donor dipole couples with the evanescent near field of an acceptor molecule. Since the near fields of both dipoles fall off as 1/R 3, the overall rate of energy transfer falls off like 1/R 6. In addition, if the donor is to transfer energy E, it is necessary that the acceptor possess an allowed transition to a state of energy E above the ground state. Although no real photon is emitted in Förster transfer, it is common to express this latter requirement in terms of the overlap between the absorption spectrum of the acceptor and the emission spectrum of the donor. 
In Fig. 2(a), we show the energy transfer spectrum from an excited Alq3 molecule at the origin. The excited molecule is embedded within an infinite film of 1% copper phthalocyanine (CuPC) in Alq3. Since the absorption of CuPC overlaps the Alq3 fluorescent spectrum, we expect Förster energy transfer from Alq3 to CuPC. The rate of energy transfer is plotted as a function of u, which is the wavevector component parallel to the surface (k ∥) normalized by the wavevector magnitude in the dipole layer (k 0). As expected for evanescent coupling, the spectrum is dominated by short range energy transfer through modes with very large k ∥. The z dependence of the normalized energy transfer rate is shown in Fig. 2(b). In cylindrical coordinates, the typical 1/R 6 dependence of the energy transfer rate, bET, becomes
where R 0 is the Förster radius, a measure of the strength of the coupling, and ρ is the density of acceptor molecules. Thus, we expect the rate of Förster transfer to decay as 1/z 4, consistent with the result in Fig. 2(b). The Förster radius is calculated to be R 0 = 38Å.
This calculation demonstrates that Eqs (20) and (21) can be used to determine whether Förster transfer is enhanced in complex planar structures.[11,12] Clearly, however, the cavity must influence evanescent modes if near field energy transfer is to be enhanced.
(ii) OLED outcoupling
To test the model, we simulate the OLED of Segal et al. This structure was chosen because its outcoupling fraction, ηc, has been accurately measured using a reverse bias technique. In brief, the PL efficiency of the emissive layer is measured within the OLED by applying reverse bias. The applied field dissociates some excitons, and the decrease in PL is compared to the induced photocurrent. This yields the product of PL efficiency and outcoupling efficiency, since the emissive layer is measured within the OLED structure. Then by normalizing to the free-space PL efficiency, the outcoupling fraction is found. Using this technique, Segal et al. obtained ηc = (24±4)%. As with most OLEDs of this structure, the emission profile is approximately Lambertian.
The OLED has the following structure: the substrate is glass precoated with a 1600Å-thick layer of indium tin oxide (ITO) substrate. Next, a 300Å-thick layer of poly(3,4-ethylenedioxythiophene):poly(4-styrenesulphonate) (PEDOT:PSS) is spun on. The organic layers are 500Å of N,N‵-diphenyl-N,N‵-bis(3-methylphenyl)-[1,1‵-biphenyl]4,4‵-diamine (TPD), 200Å of tris(8-hydroxyquinoline) aluminum (Alq3), and 500Å of 2,9-dimethyl-4,7-diphenyl-1,10-phenanthroline (BCP). The cathode is a 1000Å-thick Mg:Ag layer with 60:1 Mg:Ag ratio with a 200Å-thick Ag cap on top of everything. The device structure and the measurement setup are shown in Fig. 3. 
To simulate the device, we obtain the optical constants of each layer using a spectrophotometer (Aquila Instruments® nkd8000). This measures the reflection and transmission (RT) from a thin film of the desired material on a glass substrate at a certain angle (30°) under both s and p polarizations. The dielectric function is then determined iteratively by matching the RT calculation to the measurement. Then using Eqs (20) and (21) we calculate the angular dependence of power flow into a semi-infinite glass substrate. Subsequent energy transfer from glass to air is determined using classical ray optics.
Figure 4 shows our calculation for the angular emission profile for this OLED. Each red or blue curve corresponds to the angular emission profile into air and into the glass layer, respectively, for 10 different dipole positions spaced 20Å apart in the Alq3 layer. The curves with maxima at larger angles correspond to the dipoles nearer to the metal cathode. Figure 4(a) shows the angular profile of perpendicularly-oriented dipole emission. The strength at acute angles preferentially couples perpendicular dipoles to photonic and plasmonic waveguide modes. The parallel dipoles [Fig. 4(b)] dominate the radiated emission due to their strength around the normal. Hence the parallel and isotropic [Fig. 4(c)] angular distributions turn out to be very similar. The overall angular distribution of the emission of this OLED resembles a Lambertian emission profile as expected, which means the intensity is equal in all directions.
The dipole energy outcoupling and absorption as a function of the distance from the dipole position and normalized surface-parallel wavevector (u) is shown in Fig. 5(a) and Fig. 5(b) for parallel and perpendicular dipoles, respectively. In the calculation, the emission wavelength is λ = 535 nm and the dipole is placed at the center of the Alq3 layer. Once again we normalize the surface-parallel wavevector (k ∥) using the magnitude of the wavevector in the dipole layer (k 0). The parallel dipole shows an absorption peak at u ~ 1.07 exponentially decaying in the metal layer, corresponding to a SPP mode, and two absorption peaks around ~ 0.90 and u ~ 0.96 mainly in the ITO layer, corresponding to the waveguide modes in the ITO and organics. The perpendicular dipole also couples to an ITO/organic waveguide mode, but it displays almost ten times stronger SPP absorption. The green curves on top of Fig. 5(a) and Fig. 5(b) show the corresponding energy flux through the ITO-glass interface. These curves share the same u axis with the main figure and they can be divided into three regions, shown by dashed gray lines on the figure. The first region extends from u = 0 to u = 0.58. This is the set of wavevectors that outcouple into air. The second region extends from u = 0.58 to u = 0.87. This region contains the wavevectors that are guided in the glass. Organic/ITO waveguide modes are found for 0.87 < u < 1.
Integrating the function in Fig. 5, we obtain the energy transfer efficiencies, which are shown as a function of the distance from Alq3-BCP interface in Fig. 6(a). A simplified model of the energy flow is shown in Fig. 6(b). The total losses in the organic and ITO layers are largely independent of the dipole position; however, the glass waveguide coupling increases while energy transfer to the metal decreases with the increasing distance from the metal cathode. Averaged over the entire Alq3 emissive layer we obtain ηc = 22%, in agreement with the experimental result .
(iii) Surface plasmon polariton-mediated energy transfer
Next, as an example of layer-to-layer dipole energy transfer, we have calculated the emission spectra of the structure experimentally studied by Andrew and Barnes.  This structure is formed a glass substrate by first spin coating a 60-nm-thick donor film of polymethylmethacrylate (PMMA) doped with 4% Alq3 by weight, then thermally evaporating a 60-nm-thick silver film, and finally spin coating a 60-nm-thick acceptor PMMA film doped with 1.6% rhodamine-6G (R6G) by weight. The samples are pumped by a laser on the donor side at a wavelength of λ = 408 nm. The excitation approximately corresponds to the Alq3 absorption maximum and R6G absorption minimum. During photoexcitation the photoluminescent spectrum is recorded on the acceptor side of the sample. In the calculation, we integrated the contribution of dipoles throughout the donor and acceptor films. The result was found to be similar to the case where the dipoles are located at two thin strips at the middle of each PMMA film. The quantum yields (q) of the dipoles are taken to be[13,14] 25% and (95±1.5)% for Alq3 and R6G molecules, respectively.
Figure 7 shows the energy absorption in the silver and R6G-doped PMMA layers as a function of the normalized surface-parallel wavevector, u. Once again, the wavelength used for this calculation is λ = 535 nm. The SPP peak at u ~ 1.1 dominates the absorption and is strongly evident in both the silver and the acceptor film. Thus, we conclude that the energy transfer to the R6G molecules occurs mainly via the SPP mode, although there is significant loss in the silver film. The coupling to SPP modes is best for perpendicular dipoles. As in the OLED simulation, parallel dipoles outcouple better to the air. The radiated modes have normalized surface-parallel wavevectors smaller than u = 0.67. Parallel wavevectors between u = 0.67 and u = 1 are guided in the glass and PMMA. (Note that the refractive index of PMMA is only slightly lower than that of glass). The amount of radiated power directly from Alq3, however, is small, due to the thick silver layer. Thus, the measured light emission from this structure is dominated by the R6G emission, which in turn, gains its energy predominantly from the SPP-assisted energy transfer from the Alq3 dipoles. For completeness, we note that the Alq3 dipoles also radiate into the glass substrate; see the blue curves in Fig. 7. The radiated power in the glass substrate is about 2000 times larger than the power radiated into the air on the acceptor side.
The spectral distribution of the outcoupled energy fraction with respect to the total dipole energy is shown in Fig. 8(a). It compares well with the experimental data of Andrew and Barnes; see Fig. 2(b) in Ref. . Calculation of the emission spectra is done by multiplying the outcoupling fractions at each wavelength by the normalized intrinsic emission spectra extracted from Fig. 1(d) of Ref. . We also calculate the total energy transfer efficiency by normalizing to the energy of an Alq3 dipole. Figure 8(b) shows an exponential decrease in the transfer efficiency as the silver thickness is increased. The maximum transfer efficiency is approximately 6%. Energy transfer can be enhanced by increasing the concentration of R6G molecules in the PMMA layer. Relative to Förster transfer between point dipoles, mediation by the SPP enables energy transfer over much longer distances. The limitation for SPP-mediated energy transfer is typically the decay length of the evanescent SPP field in the donor and acceptor dielectrics. This may be on the order of 100nm, as compared to a typical Förster radius for point dipoles of < 5nm.
We present an analytic model of dipole energy transport in a multilayer stack based on the dyadic Green’s function formalism. We use it to simulate Förster transfer, calculate the angular emission profile and outcoupling efficiency of an OLED, and determine the efficiency of surface plasmon polariton (SPP)-mediated energy transfer through a silver film. These simulations match the corresponding experimental data. The absence of significant approximations makes the technique especially useful for the optimization of OLED outcoupling, which is the remaining fundamental limitation to OLED quantum efficiencies. Quantifying interlayer energy transfer allows the prediction of the impact of surface plasmons and metal quenching in organic photovoltaics and OLEDs.
References and links
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