## Simplified calculation of dipole energy transport in a multilayer stack using dyadic Green’s functions

Optics Express, Vol. 15, Issue 4, pp. 1762-1772 (2007)

http://dx.doi.org/10.1364/OE.15.001762

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### Abstract

We extend the model of Chance, Prock and Silbey[

© 2007 Optical Society of America

## 1. Introduction

2. V. Bulovic, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov, and S. R. Forrest, “Weak Microcavity Effects in Organic Light Emitting Devices,” Phys. Rev. B **58**,3730–3740 (1998). [CrossRef]

3. M. H. Lu and J. C. Sturm, “External coupling efficiency in planar organic light-emitting devices,” Appl. Phys. Lett. **78**,1927–1929 (2001). [CrossRef]

4. M. H. Lu and J. C. Sturm, “Optimization of external coupling and light emission in organic light-emitting devices: modeling and experiment,” J. Appl. Phys. **91**,595–604 (2002). [CrossRef]

## 2. Theory

*ε*, represents the complex dielectric function and

_{j}*d*the thickness of each layer. The first and last layers are semi-infinite. The Green function coefficients

_{j}*c*, and

_{j}*f*will be explained below. The randomly-oriented dipole resides in the

_{j}*s*layer, which is arbitrarily placed in the multilayer stack. Each layer is assumed to be isotropic and higher order multipole radiation is neglected.

^{th}7. R. L. Hartman, “Green dyadic calculations for inhomogeneous optical media,” J. Opt. Soc. Am. A **17**,1067–1076 (2000). [CrossRef]

*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:[1]

*j*is the layer index,

*κ*and

*h*are the amplitudes of the parallel and perpendicular components of the propagation vector

**, and**

*k**J*refers to a Bessel function of the first type of order

_{n}*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: [1]

**R′**,

*s*denotes the source layer,

*j*denotes the

*j*layer, and the dipole position is taken as

^{th}*z*= 0.[8

8. R. L. Hartman, S. M. Cohen, and P. T. Leung, “A note on the green dyadic calculation of the decay rates for admolecules at multiple planar interfaces,” J. Chem. Phys. **110**,2189–2194 (1999). [CrossRef]

**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).

*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: [1]

*c*

_{1}′ =

*f*

_{1}′ = 0 and

*c*=

_{N}*f*= 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 (

_{N}**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.

*b*. Following CPS, it is found by incorporating the effect of the reflected field on the dipole by the following equation: [1]

*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

*b*=

_{iso}*b*

^{⊥}/3 + 2

*b*

^{∥}/3. Expanding the field in terms of the Green’s functions we get the perpendicular and parallel components of

*b*:[1]

*S*in terms of general E field components in cylindrical coordinates

_{z}**J**(

**R′**) = -

*iω*

**p**

_{0}

*δ*(

**R′**) where

**p**

_{0}is the dipole vector.

*M*,

_{z}*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}=

*κh*/2

_{s}*k*; (2) when the Bessel function index

_{s}*n*= 0,

*N*′

_{z}=

*κ*

^{2}/

*k*. For both cases, other components of the primed eigenfunctions are zero. Thus,

_{s}*E*

^{⊥}

_{jϕ}is zero and we get the following expressions for the non-zero electric field components at each layer

*j*,

*p*

_{0}is the dipole moment in the absence of reflected field on the dipole.

*S**, to determine the power flow. We insert the expressions, Eq. (15) through Eq. (19), for the electric field into the equation for

_{z}*S** 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

_{z}*κ*for

*J*

_{0}terms, and

*κ*/2 for

*J*

_{1}terms. To simplify the wavevector components, we normalize the wavevector,

*u*=

*κ*/

*k*and use two identities,

_{s}*κ*

^{2}+

*h*

^{2}=

*k*

^{2}and

*k*

_{j}^{2}=

*ε*

_{j}k_{s}^{2}. Finally, we normalize with respect to the total energy of the free dipole (

*b*

_{0}), given by

*mp*

_{0}

^{2}

*ω*

^{2}/2

*e*

^{2}(in Joules), and we obtain

*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).

## 3. Simulations

*et al*. [9

9. M. Segal, M. A. Baldo, R. J. Holmes, S. R. Forrest, and Z. G. Soos, “Excitonic singlet-triplet ratios in molecular and polymeric organic materials,” Phys. Rev. B **68**,075211 (2003). [CrossRef]

10. P. Andrew and W. L. Barnes, “Energy transfer across a metal film mediated by surface plasmon polaritons,” Science **306**,1002–1005 (2004). [CrossRef] [PubMed]

### (i) Förster energy transfer

*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. [5]

_{3}molecule at the origin. The excited molecule is embedded within an infinite film of 1% copper phthalocyanine (CuPC) in Alq

_{3}. Since the absorption of CuPC overlaps the Alq

_{3}fluorescent spectrum, we expect Förster energy transfer from Alq

_{3}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,

*b*, becomes

_{ET}*R*

_{0}is the Förster radius, a measure of the strength of the coupling,[5] 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.

11. P. Andrew and W. L. Barnes, “Forster energy transfer in an optical microcavity,” Science **290**,785–788 (2000). [CrossRef] [PubMed]

12. D. M. Basko, G. C. La Rocca, F. Bassani, and V. M. Agranovich, “Electronic energy transfer in a planar microcavity,” Physica Status Solidi A **190**,379–382 (2002). [CrossRef]

*evanescent*modes if near field energy transfer is to be enhanced.

### (ii) OLED outcoupling

*et al*.[9

9. M. Segal, M. A. Baldo, R. J. Holmes, S. R. Forrest, and Z. G. Soos, “Excitonic singlet-triplet ratios in molecular and polymeric organic materials,” Phys. Rev. B **68**,075211 (2003). [CrossRef]

*η*, 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

_{c}*et al*. obtained

*η*= (24±4)%.[9

_{c}9. M. Segal, M. A. Baldo, R. J. Holmes, S. R. Forrest, and Z. G. Soos, “Excitonic singlet-triplet ratios in molecular and polymeric organic materials,” Phys. Rev. B **68**,075211 (2003). [CrossRef]

*N*,

*N*‵-diphenyl-

*N*,

*N*‵-bis(3-methylphenyl)-[1,1‵-biphenyl]4,4‵-diamine (TPD), 200Å of tris(8-hydroxyquinoline) aluminum (Alq

_{3}), 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. [9

**68**,075211 (2003). [CrossRef]

*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.

_{3}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.

_{3}-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 Alq

_{3}emissive layer we obtain

*η*= 22%, in agreement with the experimental result [9

_{c}**68**,075211 (2003). [CrossRef]

### (iii) Surface plasmon polariton-mediated energy transfer

10. P. Andrew and W. L. Barnes, “Energy transfer across a metal film mediated by surface plasmon polaritons,” Science **306**,1002–1005 (2004). [CrossRef] [PubMed]

_{3}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 Alq

_{3}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

13. D. Z. Garbuzov, V. Bulovic, P. E. Burrows, and S. R. Forrest, “Photoluminescence efficiency and absorption of aluminum-tris-quinolate (Alq_{3}) thin films,” Chem. Phys. Lett. **249**,433–437 (1996). [CrossRef]

14. H. Mattoussi, H. Murata, C. D. Merritt, Y. Iizumi, J. Kido, and Z. H. Kafafi, “Photoluminescence quantum yield of pure and molecularly doped organic solid films,” J. Appl. Phys. **86**,2642–2650 (1999). [CrossRef]

15. D. Magde, R. Wong, and P. G. Seybold, “Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: Improved absolute standards for quantum yields,” Photochem. Photobiol. **75**,327–334 (2002). [CrossRef] [PubMed]

_{3}and R6G molecules, respectively.

*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 Alq

_{3}, 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 Alq

_{3}dipoles. For completeness, we note that the Alq

_{3}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.

10. P. Andrew and W. L. Barnes, “Energy transfer across a metal film mediated by surface plasmon polaritons,” Science **306**,1002–1005 (2004). [CrossRef] [PubMed]

**306**,1002–1005 (2004). [CrossRef] [PubMed]

_{3}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.[10

**306**,1002–1005 (2004). [CrossRef] [PubMed]

## 3. Conclusion

## References and links

1. | R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near metal interfaces,” in |

2. | V. Bulovic, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov, and S. R. Forrest, “Weak Microcavity Effects in Organic Light Emitting Devices,” Phys. Rev. B |

3. | M. H. Lu and J. C. Sturm, “External coupling efficiency in planar organic light-emitting devices,” Appl. Phys. Lett. |

4. | M. H. Lu and J. C. Sturm, “Optimization of external coupling and light emission in organic light-emitting devices: modeling and experiment,” J. Appl. Phys. |

5. | T. Förster, “Transfer mechanisms of electronic excitation,” Disc. Faraday Soc. |

6. | C.-T. Tai, |

7. | R. L. Hartman, “Green dyadic calculations for inhomogeneous optical media,” J. Opt. Soc. Am. A |

8. | R. L. Hartman, S. M. Cohen, and P. T. Leung, “A note on the green dyadic calculation of the decay rates for admolecules at multiple planar interfaces,” J. Chem. Phys. |

9. | M. Segal, M. A. Baldo, R. J. Holmes, S. R. Forrest, and Z. G. Soos, “Excitonic singlet-triplet ratios in molecular and polymeric organic materials,” Phys. Rev. B |

10. | P. Andrew and W. L. Barnes, “Energy transfer across a metal film mediated by surface plasmon polaritons,” Science |

11. | P. Andrew and W. L. Barnes, “Forster energy transfer in an optical microcavity,” Science |

12. | D. M. Basko, G. C. La Rocca, F. Bassani, and V. M. Agranovich, “Electronic energy transfer in a planar microcavity,” Physica Status Solidi A |

13. | D. Z. Garbuzov, V. Bulovic, P. E. Burrows, and S. R. Forrest, “Photoluminescence efficiency and absorption of aluminum-tris-quinolate (Alq |

14. | H. Mattoussi, H. Murata, C. D. Merritt, Y. Iizumi, J. Kido, and Z. H. Kafafi, “Photoluminescence quantum yield of pure and molecularly doped organic solid films,” J. Appl. Phys. |

15. | D. Magde, R. Wong, and P. G. Seybold, “Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: Improved absolute standards for quantum yields,” Photochem. Photobiol. |

**OCIS Codes**

(230.4170) Optical devices : Multilayers

(250.3680) Optoelectronics : Light-emitting polymers

(250.5230) Optoelectronics : Photoluminescence

(310.6860) Thin films : Thin films, optical properties

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: January 24, 2007

Revised Manuscript: February 9, 2007

Manuscript Accepted: February 11, 2007

Published: February 19, 2007

**Citation**

K. Celebi, T. D. Heidel, and M. A. Baldo, "Simplified calculation of dipole energy transport in a multilayer stack using dyadic Green’s functions," Opt. Express **15**, 1762-1772 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-4-1762

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### References

- R. R. Chance, A. Prock and R. Silbey, "Molecular fluorescence and energy transfer near metal interfaces," in Advances in Chemical Physics, I. Prigogine and S. A. Rice, eds. (Wiley, 1978), Vol. 37,pp. 1-65.
- V. Bulovic, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov and S. R. Forrest, "Weak Microcavity Effects in Organic Light Emitting Devices," Phys. Rev. B 58, 3730-3740 (1998). [CrossRef]
- M. H. Lu and J. C. Sturm, "External coupling efficiency in planar organic light-emitting devices," Appl. Phys. Lett. 78, 1927-1929 (2001). [CrossRef]
- M. H. Lu and J. C. Sturm, "Optimization of external coupling and light emission in organic light-emitting devices: modeling and experiment," J. Appl. Phys. 91, 595-604 (2002). [CrossRef]
- T. Förster, "Transfer mechanisms of electronic excitation," Disc. Faraday Soc. 27, 7-17 (1959).
- C.-T. Tai, Dyadic Green's functions in electromagnetic theory (IEEE Press, 1994).
- R. L. Hartman, "Green dyadic calculations for inhomogeneous optical media," J. Opt. Soc. Am. A 17, 1067-1076 (2000). [CrossRef]
- R. L. Hartman, S. M. Cohen and P. T. Leung, "A note on the green dyadic calculation of the decay rates for admolecules at multiple planar interfaces," J. Chem. Phys. 110, 2189-2194 (1999). [CrossRef]
- M. Segal, M. A. Baldo, R. J. Holmes, S. R. Forrest and Z. G. Soos, "Excitonic singlet-triplet ratios in molecular and polymeric organic materials," Phys. Rev. B 68, 075211 (2003). [CrossRef]
- P. Andrew and W. L. Barnes, "Energy transfer across a metal film mediated by surface plasmon polaritons," Science 306, 1002-1005 (2004). [CrossRef] [PubMed]
- P. Andrew and W. L. Barnes, "Forster energy transfer in an optical microcavity," Science 290, 785-788 (2000). [CrossRef] [PubMed]
- D. M. Basko, G. C. La Rocca, F. Bassani and V. M. Agranovich, "Electronic energy transfer in a planar microcavity," Physica Status Solidi A 190, 379-382 (2002). [CrossRef]
- D. Z. Garbuzov, V. Bulovic, P. E. Burrows and S. R. Forrest, "Photoluminescence efficiency and absorption of aluminum-tris-quinolate (Alq3) thin films," Chem. Phys. Lett. 249, 433-437 (1996). [CrossRef]
- H. Mattoussi, H. Murata, C. D. Merritt, Y. Iizumi, J. Kido and Z. H. Kafafi, "Photoluminescence quantum yield of pure and molecularly doped organic solid films," J. Appl. Phys. 86, 2642-2650 (1999). [CrossRef]
- D. Magde, R. Wong and P. G. Seybold, "Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: Improved absolute standards for quantum yields," Photochem. Photobiol. 75, 327-334 (2002). [CrossRef] [PubMed]

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