Efficiency and rate of spontaneous emission in organic ...

PHYSICAL REVIEW B 85, 115205 (2012)

Efficiency and rate of spontaneous emission in organic electroluminescent devices Mauro Furno, Rico Meerheim, Simone Hofmann, Björn Lüssem, and Karl Leo* Institut für Angewandte Photophysik, Technische Universität Dresden, George-Bähr-Straße 1, 01069 Dresden, Germany (Received 16 October 2011; revised manuscript received 11 February 2012; published 21 March 2012) We examine spontaneous emission in organic electroluminescent devices and investigate the influence of the local photonic mode density on the emissive properties of molecular emitters. Spontaneous emission in organic electroluminescent devices is modeled by means of an approximate closed-form solution for the exciton rate equation, which yields the efficiency of conversion of electrical charges into molecular excited states. The exciton decay rate and the efficiency of conversion of molecular excitation into far-field radiated photons are described using a state-of-the-art classical electromagnetic formalism suitable to model multilayered organic light-emitting diodes (OLEDs). We present an in-depth analysis of the influence of optical microcavities and the corresponding resonant modes on the luminescent properties of organic molecules. Near-field coupling and coupling to metallic reflectors are demonstrated as the main effects responsible for environment-induced modifications of the rate and efficiency of spontaneous emission. The extent to which the excitonic decay rate is modified by the optical microcavity (Purcell effect) is shown to be strictly dependent on the intrinsic luminescence quantum yield of the molecular emitter. The modeling formalism is successfully validated against experimental results obtained on three series of small-molecule p-i-n OLED samples, featuring phosphorescent or fluorescent molecular emitters, with a widely varying thickness of the optical microcavity. We demonstrate that, within its limits of validity, the theoretical treatment in this work provides a rigorous quantitative description of spontaneous emission in organic luminescent devices and allows for the identification of the factors determining the OLED internal and external quantum efficiencies. DOI: 10.1103/PhysRevB.85.115205

PACS number(s): 78.60.Fi, 78.66.Qn, 85.60.Jb, 71.35.−y

I. INTRODUCTION

In the last decade, spontaneous emission in organic semiconducting materials has been gaining vivid interest, driven by the fast development of novel technological approaches based on organic electroluminescent elements for backlightfree display applications and solid-state lighting. Indeed, the organic light-emitting diode (OLED) technology has been achieving outstanding experimental results, and laboratory prototypes have attained efficiency figures of merit,1–7 which are comparable to or superior than those of conventional approaches. In spite of the substantial maturity being reached by the OLED technology, an intimate understanding of the physical processes at the basis of solid-state organic luminescence has fallen apart as a consequence of the inherent complexity of the matter compared to ease of fabrication offered by the technologies based on organic semiconductors. Electroluminescence in organic solids is a multiparticle process, which involves the interplay of particles belonging to different nature.8,9 Under electrical operation, electrical charges are first injected in the organic luminescent device from its electrodes. These free charges tunnel into polaron levels in proximity of the electrical contacts,10–12 and the excitation is then driven to the active region of the device by the applied electric field.13 There, recombination events occur,14 leading to formation of excitons, molecular excitations consisting in the bound state of an electron and a hole.15 The spontaneous relaxation of the molecular excitation to its ground level may eventually lead to the conversion of the surplus energy into luminous radiation. The latter process, spontaneous emission, is inherently quantum mechanical and the probability of photon emission obeys Fermi’s golden rule16 ∝ |Mij |2 ρ(νij ). 1098-0121/2012/85(11)/115205(21)

(1)

According to this picture, the transition rate between the excited state i and the lower-energy state j is proportional on the matrix element Mij connecting the energy states i and j , and on the photonic mode density ρ(νij ) at the location of the luminescent center. The latter quantity describes the density of the optical field at the transition frequency νij and is determined by the electromagnetic properties of the optical environment in which photon emission occurs. As originally observed by Purcell in 1946 for radio frequencies17 and, successively, by Drexhage with fluorescence experiments,18 the excitonic decay rate is strongly altered by variations of the local mode density in electromagnetic cavities. Experimental evidence of the modification of the spontaneous emission rate in organic electroluminescent microcavities has also been recently reported by various research groups.19–22 In organic electroluminescent devices, the consequences of Fermi’s golden rule are not only limited to the variation in emissive decay rate of organic molecular emitters. The overall efficiency of radiative emission in OLEDs is controlled by three main factors: the exciton generation efficiency, the efficiency of radiative exciton decay, and the efficiency at which the internally generated photons escape the device generating far-field radiation.23,24 The photonic mode density has a direct influence on the two latter factors,25 whereas the exciton generation efficiency may be considered, in first approximation, to be left unchanged. By means of a classical equivalent of the quantum-mechanical model dating back to Chance et al.26,27 and Lukosz,28 Barnes and co-workers29–31 theoretically and experimentally demonstrated the effect of a microcavity environment on both the radiative lifetime and the efficiency of spontaneous emission in solids. Various research groups have then attempted to present a unified model to describe organic electroluminescence by exploiting similar

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©2012 American Physical Society

¨ FURNO, MEERHEIM, HOFMANN, LUSSEM, AND LEO


modeling techniques.32–41 The aim of this work is to show the application of state-of-the-art models based on classical electromagnetism to the analysis of spontaneous emission in multilayered organic electroluminescent devices. Particular emphasis is given to the identification of the validity limits of common literature modeling approaches and to the derivation of guidelines necessary for the design of experiments consistent with the necessary assumptions. We unequivocally show that extremely accurate and general agreement between theory and carefully designed experiments is achievable for stateof-the-art organic electroluminescent devices. This has two major implications. First, state-of-the-art modeling techniques are proven to be effective in quantitatively reproducing and, to a certain extent, predicting experimental results. Second, the extraction of intrinsic materials and device parameters, such as radiation quantum yield and electrical efficiency, is demonstrated to be possible. Section II presents an extensive overview of the modeling formalism used throughout this paper, along with its necessary simplifying assumptions. We describe in this section a selfconsistent modeling framework for solid-state electroluminescence derived from the existing literature on the topic. The starting point of the analysis is the exciton rate equation, which is solved in an approximate form to derive the exciton generation efficiency of organic electroluminescent structures. The derivation of an integral expression for the external quantum efficiency based on classical electromagnetism is then reviewed. In Sec. III, we describe the experimental details of this study. After presenting the guidelines employed to design three sets of OLED samples with appropriate electrical and optical characteristics, we provide the details of sample fabrication and characterization along with the main experimental results. Selected modeling results are then analyzed in Sec. IV. We investigate the optical properties of weak OLED microcavities and discuss the environmentinduced modifications in exciton lifetime and device optical efficiency. The resonant modes of organic microcavities are first analyzed as a function of the cavity length, and the corresponding variations in radiated power and outcoupling of generated photons are deeply discussed. Environment-induced radiative rate and efficiency modifications are then addressed. The extraction of model parameters from experimental data fitting is then presented in the last part of Sec. IV, together with the full validation of the modeling framework. We finally summarize our results and draw our conclusions in Sec. V.

II. THEORETICAL BACKGROUND A. Charge-to-photon conversion in organic electroluminescent devices

We consider the general case of a system characterized by a single excitonic species (either triplet or singlet excitons) characterized by a time- and position-dependent concentration per unit volume ne (z,t). The time evolution of the exciton concentration is governed by the rate equation15,42,43 ∂ne (z,t) ne (z,t) ∂ne (z,t) = ξe G(z,t) − ∗ − . ∂t τe (z) ∂t losses

(2)

In the above equation, G denotes the exciton generation rate due to electrical excitation, and the factor ξe accounts for the generation fraction of the exciton species due to spin statistics. The second term in Eq. (2) describes the decay of the excitonic species, characterized by an effective lifetime τe∗ or by its inverse e∗ , the effective exciton decay rate. In general, the decay rate e∗ describes the probability of monomolecular excitonic decay per unit time. As such, it includes the two competing channels molecular excitations can transfer their energy to, ∗ namely, radiative exciton decay, characterized by a rate rad,e , and nonradiative phonon-mediated monomolecular thermal relaxation processes, with a rate nrad,e (Refs. 41,44,45): 1 τe∗ (z)

∗ = e∗ (z) = rad,e (z) + nrad,e .

(3)

In writing the above equation, we have introduced an explicit ∗ , which is motivated by position dependence for the rate rad,e the modifications of the probability for spontaneous emission induced by the optical environment. The derivation of an explicit functional form for this term is detailed in Sec. II B. The last term in the exciton rate equation (2) accounts for exciton annihilation processes, such as annihilation due to the interaction with other excitons or quenching with polarons.46,47 Without loss of generality, we have neglected the exciton diffusion term48,49 in Eq. (2). As it will be further discussed, this choice is motivated by the characteristics of the devices that we shall analyze in this work, where molecular excitations are efficiently confined in a region narrower than the overall device thickness, thus making exciton diffusion unimportant in the present treatment. With the aim of getting a closed-form solution for Eq. (2), we shall consider in the remainder a luminescent system operated at low excitations levels. In these operating conditions, we assume exciton annihilation processes to be negligible with respect to the other terms in Eq. (2), namely, exciton generation and monomolecular decay. Consequently, the rate equation reads as ∂ne (z,t) ne (z,t) ξe G(z,t) − ∗ ∂t τe (z)

(4)

and we obtain the following simple expression for the exciton concentration in the steady state: ne (z) = ξe G(z)τe∗ (z).

(5)

According to the above equation, the position-dependent exciton concentration per unit volume ne can be formally evaluated in any complex luminescent system, provided that the position-dependent exciton generation profile G and effective lifetime τe∗ are known. The exact quantification of the generation profile G(z) in a multilayered organic luminescent system is a complex and challenging computational task. Indeed, G(z) is dependent on a number of physical and electrical parameters, related to the charge transport properties of different organic materials and to the physical properties of the interfaces between those.50,51 Since an exact quantification for G(z) is well beyond the scope of this work, we avoid this shortcoming by considering a spatially integrated total concentration of decaying excitons n˜ e per unit area and unit

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EFFICIENCY AND RATE OF SPONTANEOUS EMISSION . . .

time52 [with unit (s−1 m−2 )] as ne (z) n˜ e = dz. ∗ z τe (z)


(6)

en˜ e , (7) J where e denotes the elementary charge. For the sake of simplicity, we further assume the exciton generation region to be infinitely thin and described by the δ distribution γ =

(8)

where z = z0 denotes a generic location within the emitting layer of the device and g˜ = z G(z) dz. The substitution of Eqs. (5) and (8) into (6) yields the following trivial expression for n˜ e : n˜ e = ξe G(z) dz = ξe g˜ (9) z

and the electrical efficiency γ finally reads as e (10) γ = ξe g˜ . J We note in the two latter equations that the electrical behavior of the device is solely dependent on the number of recombination events occurring in the active layer of the device ˜ a priori unknown. On the other hand, n˜ e and γ do not show g, an explicit dependence on the effective exciton lifetime τe∗ in this approximate formulation. We apply now these considerations in the derivation of a closed-form equation for the external quantum efficiency of electroluminescent devices, i.e., the efficiency of conversion of electrical charges into externally radiated photons. As usual, we decompose the external quantum efficiency ηq into the product of three partial terms23,34,53,54 : ∗ ηout . ηq = γ ηrad,e

(11)

The electrical efficiency γ appears in Eq. (11) as a first factor, whereas the second one represents the emitter effective efficiency of radiation, or radiative quantum yield. This term describes the effective probability of radiative exciton decay in a microcavity environment and is obtained from the exciton decay rates via the usual formula36–38,40 ∗ ηrad,e =

∗ rad,e . ∗ nrad,e + rad,e

(13)

λ

By considering now electroluminescent operation at a bias current density J , we can define the electrical efficiency γ for an electroluminescent device as the unitless ratio between the total density of decaying excitons to the flux density of electrical charges injected by the external circuit, i.e.,

˜ G(z) = gδ(z − z0 ),

(Refs. 5,32,41) and ηq assumes the final form ∗ (λ)ηout (λ) dλ. ηq = γ sel (λ)ηrad,e

(12)

Finally, the last factor in Eq. (11) denotes the outcoupling efficiency ηout of the optical structure, i.e., the efficiency of conversion of internally generated radiation into far-field measurable optical power. The theory described up to this point requires a final generalization due to the broadening of the excitonic level in organic luminescent materials. To this aim, Eq. (11) is convoluted with the normalized luminescence spectrum sel (λ) of the emitting material [ λ sel (λ) dλ = 1]

We have introduced here an explicit wavelength dependence for the radiative and outcoupling efficiencies since these quantities are directly influenced, among other factors, by the optical properties of the various materials constituting the device stack and their refractive index dispersion (see further Sec. II B). According to Eq. (13), the external quantum efficiency ηq can be formally treated as the product of two independent terms. The first factor is the electrical efficiency γ for the conversion of electrical charges into decaying excitons. In the treatment presented above, γ is solely determined by the electrical properties of the luminescent structure via the number of useful recombination events g˜ as given by Eq. (10). The second factor contributing to external quantum efficiency ηq is represented by the integral in Eq. (13). This describes the efficiency of conversion of decaying excitons into farfield radiated photons and includes the emitter luminescence ∗ and the outcoupling efficiency of the quantum yield ηrad,e internally generated photons ηout , both dependent on the local density of photonic modes. It is important to keep in mind that this formulation for ηq strictly holds only if two conditions are satisfied: (i) devices are operated at low excitation levels and (ii) the exciton generation profile can be considered as a δ distribution. The first assumption is necessary to ensure that electrical processes can be treated independently. If considering device operation at current densities J at which exciton annihilation processes are important, Eqs. (5) and (10) lose their validity and the efficiency of conversion of electrical charges into decaying excitons should then include an explicit dependence on the exciton effective lifetime and on the rates of exciton annihilation processes. The second assumption requires that charge recombination, exciton decay, and light generation can be safely considered to occur within a region with extension much smaller than the total length of the optical microcavity. If this condition is not met and charge recombination extends in a larger portion of the device stack, a full treatment of the z-dependent problem is required, thereby ∗ including the explicit position dependence of G(z), ηrad,e (z), and ηout (z). In such a case, a more general formulation for the external quantum efficiency should instead be considered: e ∗ ηq = ξ e sel (λ)G(z)ηrad,e (λ,z)ηout (λ,z) dλ dz. (14) J λ z In the latter case, exciton diffusion should also be explicitly taken into account in the rate equation given at the beginning of this section. B. Electromagnetic model

The radiative decay of excitonic species in organic luminescent devices, along with the outcoupling of the internally generated photons, is conventionally treated classically in the scientific literature associating molecular excited states to an ensemble of classical electrical dipole antennas radiating electromagnetic power.25,27,28,55–57 The dipole model is conventionally used when describing the interaction of luminescent

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centers with a planar, multilayered structure. In this formalism, the local electromagnetic field at the location of the lightemitting molecules is obtained as a superposition of (i) a source term with a certain dipole transition moment, accounting for the actual molecular emission, and (ii) the component of the electromagnetic field parallel to the dipole moment eventually reflected by the optical structure, accounting for the local density of electromagnetic modes.58 The knowledge of the local field yields then the total electromagnetic power radiated at the dipole location, along with the momentum distribution of the radiated modes. A comprehensive review of the complete mathematical description of the dipole model applied in this work to organic multilayers is given in the Appendix. We quote in this section only some general considerations necessary for the quantification of rate and efficiency of spontaneous emission. The two relevant quantities to be computed in the framework of the dipole model are the total radiated power at the emitter location F (λ), also known as Purcell factor, and the fraction of power U (λ) outcoupled from the optical cavity. In the formalism applied here, the spectral power F (λ) quantifies the effect of the optical environment on the emissive properties of molecular species and therefore is the classical equivalent of the quantum-mechanical photonic mode density.25,26,29,58,59 The total radiated power F (λ) is obtained for the multilayered organic electroluminescent devices according to ∞ F (λ) = K(λ,u) du2 (15) 0 ∞ uK(λ,u) du. (16) =2 0

Here, u denotes the normalized transverse wave vector and is defined as u = kρ /k, with k total wave vector at the emitter location and kρ projection of the wave vector in the plane of the source. The spectral power density K(λ,u) per unit wavelength and unit normalized in-plane wave vector is obtained for a given optical structure from the optical constants and thickness values of the materials stack in which dipole emission occurs [see Eqs. (A1) to (A7)]. It shall be noted that the spectral power F (λ) is a dimensionless quantity, normalized with respected to the corresponding spectral power F0 (λ) emitted by the same emitter in an infinite cavity.25 In other words, F (λ) = 1 for emitters in free space and it is generally different from unity for radiation occurring in a multilayered optical environment. With reference to the formulas given in the Appendix for the calculation of the spectral power density F (λ), we would like to stress that we consider here only the case of molecular emitter characterized by isotropic transition dipole moment. Nevertheless, all model equations can be easily generalized to the case of an emitter with preferentially oriented dipole moment.56 For a molecular emitter characterized by a free-space radiative decay rate rad,e , the effective radiative decay rate ∗ in the optical cavity rad,e (λ) is commonly obtained for the wavelength λ as ∗ (λ) = F (λ)rad,e rad,e

and the effective total decay rate

e∗ (λ)

∗ (λ) rad,e ∗ e (λ) F (λ)ηrad,e = , 1 − ηrad,e + F (λ)ηrad,e

∗ ηrad,e (λ) =

(19) (20)

where ηrad,e = rad,e /(rad,e + nrad,e ) is the intrinsic emitter quantum yield. It becomes apparent here that the efficiency of the molecular emitter increases (decreases) in the case F (λ) > 1 [F (λ) < 1] as consequence of the cavity-induced variation of the radiative decay rate. It is as well worth mentioning that the effective radiative quantum yield for emitters inside a microcavity is generally a wavelength-dependent quantity as a clear consequence of the wavelength dependence of F (λ). The explicit expression for the external quantum efficiency ηq is obtained by considering the fraction of the spectral power F (λ) outcoupled from the optical structure and radiated to the far field. With reference to the formulas given in the Appendix, the power component U (λ) outcoupled at λ is calculated by integration of the spectrum of externally radiated power Kout (λ,u) per unit normalized in-plane wave vector u: ucrit (λ) Kout (λ,u) du2 (21) U (λ) = 0

ucrit (λ)

=2

uKout (λ,u) du.

(22)

0

The upper integration limit ucrit is the maximal value of the in-plane wave vector allowing for far-field propagation of the emitted photons. By considering light emission in a medium with refractive index ne (λ) and far-field propagation in a medium characterized by no (λ), ucrit is obtained by application of Snell’s law as ucrit (λ) =

no (λ) . ne (λ)

(23)

(17)

By combining the definition for U (λ) along with Eqs. (13) and (20), we finally obtain the following explicit form for the spectrally integrated external quantum efficiency: U (λ) ηrad,e F (λ) dλ. (24) ηq = γ sel (λ) 1 − ηrad,e + ηrad,e F (λ) F (λ) λ

(18)

In the latter formula, the influence of the optical environment on the emissive properties of the system is fully described by the electromagnetic power densities F (λ) and U (λ). It

reads as

e∗ (λ) = nrad,e + F (λ)rad,e .

Accordingly, the radiative decay rate of a molecular excited state in a cavity environment is directly proportional to the spectral radiated power F (λ), whereas the nonradiative decay rate stays unmodified.60 This implies that if the properties of the optical environment are such that the radiated power is large (e.g., large mode availability due to cavity resonances, proximity to a reflecting surface), the lifetime of the excited state shortens and the probability for photon emission is enhanced in comparison to the free-space case [for which F (λ) = 1]. Owing to a variation in radiative decay rate, the radiative efficiency of a luminescent center in the microcavity consistently varies. We can now write the general explicit expression for the effective radiative quantum yield for molecular emitters as the ratio between the radiative decay rate and the total decay rate in the optical cavity:

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is interesting to note as well that for a set of electroluminescence devices with the same electrical behavior and therefore identical electrical efficiency γ , Eq. (24) yields a fully quantitative evaluation for variations in ηq determined by modified properties of the optical cavity, i.e., strictly related to the spectral power densities U (λ) and F (λ). This peculiar property will be exploited in the following section to design appropriate experiments allowing for the validation of the OLED model and the extraction of the two unknown system parameters in Eq. (24), namely, the device electrical efficiency γ and the emitter radiation quantum yield ηrad,e . C. Overview of model assumptions

It is now worth reviewing the main assumptions of the modeling framework described above in order to define the general limits of validity of this approach and therefore derive suitable experimental conditions for its application. We can summarize the assumptions used in this section and in the Appendix as follows: (i) The emitting medium, the substrate material, and the far-field medium in the OLED multilayer are assumed to be nonabsorbing. (ii) The transition dipole moment of fluorescent and phosphorescent molecular emitters is assumed to be isotropic. (iii) OLED devices are operated at low excitation levels or, equivalently, at low exciton concentrations in the emitting layer. (iv) The extension of the exciton generation zone is small compared to the cavity length. In particular, we consider the limit of a δ-distributed exciton generation profile. The first assumption is an intrinsic limitation of the electromagnetic model we use in this work. Actually, the assumption of a real refractive index for the substrate and the far-field medium is not considered as a major source of error since light absorption in these media is usually rather weak. On the other hand, dissipation in the emitting medium was demonstrated by rigorous electromagnetic modeling to modify, viz., partially suppress, spontaneous emission from radiating dipoles.61 A relatively simple phenomenological treatment for emitting layers with complex-valued refractive index was recently proposed for polymeric OLEDs,62 consisting in the subdivision of the emitting layer into sublayers with real refractive if containing emitting dipoles and complex refractive index otherwise. For the systems we will analyze in the following sections, we verified that the effect of self-absorption in the emitting layer could be safely neglected due to the generally weak overlap between absorption peak and luminescence spectrum of the materials used in this study. The second of the assumptions in the list above, i.e., isotropic transition dipole moments, has been chosen due to the present lack of specific information about an eventual orientation of the emitting dipoles in molecular emitting systems. In recent studies, experimental evidence for the preferential in-plane orientation of the emissive dipole moment in the phosphorescent system NPB:Ir(MDQ)2 (acac) was reported.63,64 In particular, the ratio for parallel versus perpendicular emitting sites was reported to be equal to 2:0.67 and therefore larger than the commonly considered isotropic proportion 2:1. Our preliminary calculations indicated that

a slight variation in the distance between emitting centers and the metallic reflectors might determine a change in the macroscopic OLED optical characteristics similar to a variation in the parallel-to-perpendicular emitter ratio. Thus, we expect the information about the emitter orientation to lie within the accuracy limits of our device fabrication process and characterization uncertainties. Although beyond the scope of this paper, we firmly believe that an accurate determination of the orientation of molecular emitters is necessary for achieving a complete theoretical picture of the dynamics of molecular emission in organic electroluminescent diodes. The usage of the two latter model assumptions (low excitation level and δ-distributed generation profile) has been similarly made necessary to apply the theory to real devices and the corresponding measured data. These assumptions make it possible to overcome the missing quantitative knowledge of various internal microscopic quantities and distributions, for which physical models are unavailable or still under development, while retaining an accurate description for the physics in the organic electroluminescent devices under analysis. As we will further discuss in the next section, particular care has to be used in the design of experiment and in the choice of the characterization conditions to ensure that these two assumptions are satisfied, and thus experiment and theory can be meaningfully compared. We would like to emphasize that these assumptions on light generation profile and low excitation operation are not intrinsic limitations of our modeling framework, but merely necessary workarounds. Indeed, various extensions of the model can be envisaged provided that the necessary missing information is made available with a deeper knowledge and understanding of the intimate processes at the basis of OLED operation. It is apparent that the complementation of the theoretical framework presented here with other experimental or modeling techniques, e.g., exciton generation profile extraction,51,62 improved charge transport modeling,50,65–67 improved understanding of exciton diffusion,49 and annihilation processes,15,46 would offer a complete theory generally suitable to quantitatively describe the operation of organic electroluminescent devices and the relative efficiency. As an application example, we would like to mention that the full solution of the exciton rate equation (2) with appropriate parameters coupled with the predictions of the present spontaneous emission model would provide a powerful numerical tool for the analysis of roll-off effects in small-molecule OLEDs. Besides, minor modifications to the model and the inclusion of an effective exciton generation profile would also allow us to extend its validity to polymerbased OLEDs or, more in general, to luminescent devices with extended (i.e., non δ-distributed) light generation regions.

III. EXPERIMENT A. Design of experiment

We have demonstrated in Sec. II A that, provided that certain requirements are met, the electrical efficiency γ of an electroluminescent structure can be treated separately from the other factors influenced by the local photonic mode density in the external quantum efficiency formula [cf. Eqs. (13) or (24)]. The main guideline of the experiment we intend

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FIG. 1. (Color online) Structures of the fabricated bottom emission OLED samples. From left to right: red, green, and blue OLED stacks. The dashed lines denote the position of the emitting molecules assumed in theoretical calculations. Abbreviations are defined in the main text.

to perform is therefore the fabrication of series of samples in which the local photonic mode density at the location of molecular emitters can be finely tuned while letting unchanged the electronic characteristics of the device. To this aim, we fabricate different sets of small-molecule organic light-emitting diodes with layer stacks as shown in Fig. 1 featuring red, green, and blue molecular emitters. The devices are designed with a multilayered p-i-n architecture including transport and blocking layers.68–70 The usage of doped electron and hole transport layers (ETL and HTL) in the samples is very convenient as, by varying the thickness of the p- and n-doped transport layers, the intensity of the local electromagnetic field at the location of the emitting molecules is largely varied, while keeping the electrical properties of the device virtually unchanged.71 Moreover, electron and hole blocking layers (EBL and HBL, respectively) ensure charge and exciton confinement in the emitting layer (EML). This aspect is rather important here since, on the one hand, it further ensures that the internal distribution of microscopic variables is not affected by a variation in thickness of the transport layers and, on the other hand, that charge recombination and, successively, exciton formation and decay occurs within the EML only. Finally, as the thickness of the EMLs is considerably lower compared to the total length of the optical microcavities (see Fig. 1), the simplifying assumption of a delta-distributed exciton generation profile can be considered to be generally satisfied. All OLEDs are designed with weak microcavity structure in bottom emission configuration on structured glass substrates precoated with a 90-nm-thick layer of indium tin oxide (ITO) serving as hole injecting contact. An opaque metallic layer, with thickness 100 nm, is used in the OLEDs as electron injecting electrode and reflecting mirror. To induce a strong and controlled variation of the optical field at emitter location, the devices are designed by carefully adjusting the length of the optical microcavity and the location of the emitting molecules with respect to the metallic contact and the glass substrate. By considering a simplified system composed of an ideal metallic conductor (i.e., a material with pure imaginary refractive index), an organic layer with refractive index n(λ), and a glass substrate, we estimate the emitter-to-metal and the emitter-

to-substrate separation distances (d1 and d2 , respectively) yielding constructive interference at an angle θ = 0◦ and a wavelength λ from the following round-trip conditions72–76 : 4π n(λ)d1 − π = 2π m1 , (25) λ 4π n(λ)d2 = 2π m2 , (26) λ where m1 and m2 = 0,1, . . . denote the resonance order. Here, we assume a phase rotation angle equal to π at the organic/metal interface75 and a phase angle equal to 0 at the organic/glass interface. Furthermore, the total cavity length d = d1 + d2 is obtained from the combination of the latter equations as λ 1 (27) + m1 + m2 . n(λ)d = 2 2 In what follows, we fix the emitter-to-substrate separation to the constant reduced thickness d2 = λ/2 (m2 = 1) to accommodate a charge injection and transport layer and the transparent ITO contact. Cavity resonances are then obtained for d1 = λ/4 + m1 λ/2,77 corresponding to the two first resonant reduced cavity lengths d = 3λ/4 (m1 = 0) (Ref. 78) and 5λ/4 (m1 = 1). From similar considerations, it can also be demonstrated that destructive interference at the emitter location occurs for a reduced distance equal to λ/2,79 which corresponds to a total cavity length equal to λ (still keeping d2 = λ/2 as a constant parameter). Numerical calculations of the electromagnetic field profile in the multilayered stacks, obtained with the approach described in Refs. 80 and 81, confirm these considerations. This is apparent in Fig. 2, where the numerically calculated optical field is shown for the 3λ/4, λ, and 5λ/4 optical cavities (m2 = 1 for all device configurations). In the former and latter devices, the emitting molecules are located, respectively, at the first and second antinodes of the electromagnetic field from the metallic reflector. We can anticipate that, due to the large local availability of radiation modes,82,83 the efficiency of radiation of the emitter is expected to be high in these configurations and the same for the efficiency of the corresponding devices. At antiresonance (d = λ), the emitter is placed at a node of the electromagnetic field, where practically no modes are available

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FIG. 2. Calculated electromagnetic field intensity at λ = 610 nm and θ = 0◦ for metal/organic/ITO structures. From left to right: 3/4λ cavity with emitter located 70 nm from the metallic contact, λ cavity with emitter at 155 nm, 5/4λ cavity with emitter at 250 nm.

for the excitonic radiative decay, ultimately yielding extremely poor device emission characteristics. B. Device fabrication and characterization

The fabricated samples are deposited by thermal evaporation onto ITO-coated, structured glass substrates in an UHV chamber (Kurt J. Lesker Company) with a base pressure around 10−8 mbar. The devices are encapsulated immediately after preparation under a nitrogen atmosphere using epoxy glue and glass lids. As p-type hole injection and transport layer (HTL), we use NHT-5 as host material doped with 4 wt% NDP-2 (Novaled AG). This system has virtually equivalent transport properties to MeO-TPD doped with 2 wt% of 2,3,5,6-tetrafluoro-7,7,8,8-tetracyanoquinodimethane (F4TCNQ) with a conductivity of around 105 S cm−1 . The NDP-2 dopant is preferred due to the volatility of F4-TCNQ during device processing. Cs-doped 4,7-diphenyl-1,10-phenanthroline (BPhen) is used as n-type electron injection and transport layer (ETL), with a conductivity comparable to the that of the p-doped layers. For red OLEDs, the EML consists of the phosphorescent dye iridium(III)bis(2-methyldibenzo[f,h]quinoxaline)(acetylacetonate) [Ir(MDQ)2 (acac)] doped with a 10 wt% concentration into a matrix of N,N -di(naphthalen-2-yl)-N,N -diphenyl-benzidine (NPB) with 20-nm thickness. Finally, we use 2, 2 ,7,7 -tetrakis(N,N-diphenylamino)-9, 9 -spirobifluorene (STAD) as EBL and bis-(2-methyl-8-quinolinolato)-4-(phenylphenolato)aluminium-(III) (BAlq2 ) as HBL in the red system. We recently published the experimental results obtained for these devices in Ref. 6. For green OLEDs, we choose a double emitting layer structure, which has been proven to enhance charge carrier balance in the EML.54,84 The phosphorescent dye tris(2-phenylpyridine)iridium [Ir(ppy)3 ] is doped with concentration 8 wt% into an electron conducting matrix, 2,2 ,2 (1,3,5-benzenetriyl)tris1-phenyl-1H-benzimidazole (TPBi), and a hole conducting one, 4,4 ,4 tris(N -carbazolyl)-triphenylamine (TCTA), with thickness equal to 12 and 6 nm, respectively. As in red devices, BAlq2 and STAD are used as HBL and EBL, respectively. For blue OLEDs, the EML consists of a 10-nm 2-methyl-9,10-bis(naphthalen-2-yl)anthracene (MADN) doped with 1.5 wt% of the sky-blue fluorophor 2,5,8,11-tetra-tert-butylperylene (TBPe).85 Such an emitting system is preferred to a molecular blue phosphor for

stability reasons. Finally, BAlq2 and NPB are chosen as HBL and EBL, respectively, and Al is preferred to Ag for its better reflectivity at the short wavelengths of the visible spectrum. The OLEDs are electrically characterized by a Keithley source meter unit (SMU2400), and an absolute calibrated spectrometer (Instrument Systems GmbH CAS140) is used to record the spectral radiance in the direction perpendicular to the substrate. A spectrogoniometer, including an optical fiber and a calibrated Ocean Optics USB4000 miniature fiber optic spectrometer, is used to record the angular-dependent OLED emission patterns, which may generally deviate from those of ideal Lambertian radiators.86,87 OLED fingerprints such external quantum efficiency ηq , luminous efficacy ηe , or luminance L(θ ) for a given viewing angle θ are then calculated by integration of spectrally and angularly resolved measurements.7,24,54,86 The intrinsic luminescence spectra of the emitting molecules used in this work are extracted from photoluminescence characterization on thin film. The optical constants of the various organic materials to be used in numerical calculations are extracted from reflection and transmission measurements on pristine films with different known layer thicknesses by an iterative fitting algorithm.88 Optical constants for the glass substrate and the ITO contact are obtained by ellipsometry measurements performed at Fraunhofer IPMS, Dresden, Germany. Finally, the optical constants for Al and Ag are taken from commercially available databases and verified by previous reflection experiments on opaque thin films. C. Experimental results

We show in Fig. 3 the external quantum efficiency-current density characteristics of selected red, green, and blue OLED samples. For red OLEDs, ηq reaches a remarkably high value exceeding 21%. On the same figure, it can be noticed that the measured external quantum efficiency of the fabricated green OLEDs is slightly lower, peaking at 18%. As expected from the preliminary considerations in this section, a strong modulation in device performance with varying thickness of the ETL is observed for both red and green OLEDs. In particular, the external quantum efficiency is as low as 4% for the devices designed with dETL = 145 and 120 nm, for red and green, respectively. It can be finally noted that the most efficient devices have either thin or thick electron transport layer, corresponding to the 3λ/4 and 5λ/4 designs.

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FIG. 4. (Color online) Measured external quantum efficiency as a function of the electron transport layer thickness dETL for red, green, and blue OLEDs. Data are shown at forward current densities of 1.5, 0.1, and 3.0 mA cm−2 for red, green, and blue devices, respectively.

FIG. 3. (Color online) External quantum efficiency rolloff as a function of current density for selected OLED samples. From top to bottom: red, green, and blue OLEDs. The gray vertical lines at J = 1.51 , 0.1, and 3.0 mA cm−2 show the current density values considered in calculations.

From the data in Fig. 3, we further note a strong rolloff in the efficiency of both red and green OLED structures. The degradation of the performance of OLEDs based on phosphorescent Ir complexes when driven at large currents is a phenomenon well known in the scientific literature46,47,89 related to the bimolecular annihilation rates in Eq. (2). To ensure the validity of the low excitation assumption in the previous section, we shall identify a suitable value for the current density. The corresponding OLED fingerprints

(external quantum efficiency, luminous efficacy, luminance, and/or radiance) are then obtained by linear interpolation of the available measured data points. For the red devices, a forward current of 1.5 mA cm−2 is chosen as reference value since the ηq versus J characteristics of most of the red OLEDs peak at this current level. For the green OLEDs, the efficiency rolloff is considerably stronger, its onset being observed in the range 10−2 to 10−1 mA cm−2 . The reason for this enhanced rolloff, not consistent with previously published data,24,84 has not been clarified yet. We postulate it to be related to extremely efficient charge and exciton confinement performed by the double EMLs used in these OLEDs, but a more detailed understanding is beyond the scope of this work. For the blue device stacks with fluorescent molecular emitter, the efficiency versus current curves peak at J = 3 mA cm−2 . The interpolated ηq values at the chosen current density values are collected in Fig. 4 as a function of the thickness of the electron transport layer for the three sets of devices. Several considerations may be issued from the data. First of all, the quality of the fabricated OLED samples is remarkable both considering the absolute measured ηq values and the clear trends as a function of the thickness of the electron transport layer. Furthermore, the expected modulation in device efficiency due to the varied distance between the emitting molecules and the reflecting metallic contact is nicely visible on the figure. Due to the practically identical current-voltage characteristics of the three sample sets (not shown here), we can confidently attribute the modulation in device efficiencies in Fig. 4 solely to a varied photonic mode density. This conclusion is corroborated by the observation that the ETL thickness at which each set of ηq values shows a local maximum or minimum nicely scales with the emission wavelength in good qualitative agreement with Eq. (27). For green and red OLEDs featuring Ir-based phosphors and opaque Ag electrode, the 5λ/4 configuration of the optical cavity yields the best external quantum efficiency. On the other hand, the landscape is slightly different for the fluorescent blue OLEDs with Al cathode, for which the best absolute performance is recorded for a cavity with approximate reduced length of 3λ/4.

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IV. SELECTED MODELING RESULTS

By making use of the theoretical considerations outlined in Sec. II, we present in the following a detailed analysis of the efficiency of electroluminescence processes in the fabricated OLED samples with the final aim to provide a quantitative description for the device operation in terms of basic physical mechanisms. We begin our discussion by considering purely optical processes occurring in a set of luminescent devices (red OLEDs) by means of an electromagnetic analysis. Afterward, the analysis of the photon outcoupling efficiency of different device configurations is discussed, and optical environmentinduced modifications in efficiency and rate of spontaneous emission are dealt with. Finally, we present the validation of our theoretical framework against the experimental results measured on the fabricated set of OLED samples in the last part of this section. A. Optical modes in weak organic microcavities

Figure 5 shows the power dissipation spectra uK(λ,u) calculated for red OLED structures with selected ETL thickness values at a free-space wavelength λ = 610 nm. The analysis of these power spectra offers a useful tool for the identification of the optical environment-dependent decay routes to which the luminescent molecules can transfer energy to. The peaks in the power spectra in Fig. 5 represent the resonant modes

of the optical cavity surrounding the molecular emitters. The positions of the peaks yield information about the nature of the various modes, and the corresponding intensity allows for a quantitative determination of the coupling strength to each of the observed modes.25,55,86 Four distinct regions can be identified on each diagram in Fig. 5. Power components characterized by values of the in-plane component of the normalized wave vector larger than unity (u > 1) are evanescent in nature25 since, for these modes, the √ out-of-plane component of the normalized wave vector v = 1 − u2 is purely imaginary. In this region, the radiated power is transferred to intrinsically nonradiative modes, such as surface plasmon polaritons (SPPs) at metal/dielectric interfaces29 or lossy surface waves in absorbing media.55 Due to the evanescent nature of these modes, molecular emitters can couple to these modes only through the near field.90 For 0 u 1, power components possess a real out-of-plane wave vector (v ∈ R), and the corresponding modes are plane waves in the emitting medium. Molecular radiation may then emerge as useful far-field power depending on the associated wave vector u. The condition for far-field propagation in an outer medium with real refractive index no can be derived from Snell’s law: for molecular emitters embedded in a medium with real refractive index ne (ne > no ), far-field propagation in the outer medium is allowed for modes with u < no /ne . These power components can eventually contribute to the far-field emission of the

FIG. 5. The integrand of Eq. (16) calculated for different red OLED structures at λ = 610 nm. (a) dETL = 70 nm, 3/4λ cavity. (b) dETL = 250 nm, 5/4λ cavity. (c) dETL = 20 nm. (d) dETL = 160 nm, λ cavity. 115205-9



optical structure with transmission probabilities given by Eqs. (A9)–(A14). In case the structure includes a substrate with refractive index ns and no < ns < ne , modes can escape the emitting medium for u < us = ns /ne and will be confined in the substrate for u no /ns . Figure 5(a) shows the calculated power spectrum for the red OLED in 3λ/4 configuration [dETL = 70 nm (cf. Fig. 2)]. The peak at u 1.1 represents coupling of radiation to the SPP mode bound at the interface ETL/cathode. From the peak intensity, we can observe that coupling to this mode is rather strong and, consequently, optical losses due to plasmonic coupling are expected to be large for this device. As it shall be further demonstrated, coupling to this SPP mode is as well the main factor responsible for modifications in the exciton lifetime. Two guided modes are then observed at u 0.85 and 0.9. From the analysis of the emission components of the different dipole orientations86 and consistently with literature results,38 the former, here close to cutoff, is a mode with transversemagnetic (TM) polarization, whereas the latter is a transverseelectric (TE) mode. A broad resonance is finally observed for u < us , representing power contributions which can escape the emitting medium and eventually radiate either in the glass substrate or in air. If the thickness of the ETL is increased up to a total cavity length of 5λ/4, calculations yield the power spectrum in Fig. 5(b). The coupling strength to the plasmonic mode is observed to decrease by roughly one order of magnitude due to the increased separation between the molecular emitters and the metallic reflector.5,83,90 Due to the larger cavity length, power coupling to waveguide modes increases considerably. Furthermore, guided modes become increasingly localized in the emitting medium and the ITO electrode. Two broad (leaky) modes are now present as well for u < us . In particular, the second mode, which was not apparent in the 3λ/4 case, is the precursor for the third guided mode of the microcavity, but it still retains a leaky nature in this configuration. Overall, we observe a lower amount of power being radiated by the molecular emitters for u < uo (and u < us ) compared to the 3λ/4 cavity. If the ETL thickness is now reduced to 20 nm, i.e., the total cavity length is lower than 3λ/4, we obtain the power spectrum depicted in Fig. 5(c). For this device, the spacing between the emitting molecules and the metallic contact is such that emitters can not interfere constructively with their images at the metallic reflector.25,91 Consequently, power radiated in plane waves is considerably reduced. On the other hand, due to proximity of the molecular emitter and the metallic mirror, coupling to the SPP mode is strongly enhanced in comparison, e.g., to the situation in Fig. 5(a). It is finally interesting to analyze the power spectrum in Fig. 5(d) obtained for a λ cavity configuration. The most striking feature in the diagram is that virtually no power is radiated by the molecular emitters for u < uo . The reason for this is clarified by considering the electromagnetic field profiles in Fig. 2: for the λ cavity, destructive interference occurs at the emitter location and the resultant total EM field is practically zero. Consequently, the optical cavity strongly suppresses molecular radiation and, furthermore, few decay channels are available for the radiative decay of the excited species. Such a low availability of radiation modes negatively affects the charge-to-photon conversion efficiency, yielding the poor external quantum efficiency observed for this device in

Fig. 4. The deeper analysis from Flämmich et al.91 reveals that this conclusion is generally true only if considering molecules with in-plane emissive momentum: constructive interference does occur for emitters oriented perpendicularly to the device surface, eventually resulting in radiation enhancement. For an electroluminescent device featuring in-plane oriented emitters, the overall power radiated in the far field would be practically zero, and therefore the overall radiative efficiency. This is not the case for isotropic molecular emitters, as emitters with vertically oriented momentum yield a small, but nonzero, contribution to the measured external quantum efficiency. Calculations indeed show for the red OLED in λ configuration that the ratio between the far-field power radiated by the vertically oriented dipoles to the in-plane ones is as large as 3.31. For the 3/4λ and 5/4λ designs, this ratio is lower than 10−2 . B. Total radiated power and outcoupling efficiency

To study the global emissive properties of a luminescent device, it is instructive to consider the wavelength-resolved spectra F (λ) and U (λ) obtained according to Eqs. (16) and (21). These two spectral quantities yield, respectively, the total power radiated by the molecular emitters at a wavelength λ (or equivalently on the total number of photons internally generated on average at λ) and the fraction of the total power, which is actually far-field radiated. The results we obtain for the 3λ/4, λ, and 5λ/4 cavities featuring the red emitters are shown in Fig. 6. In the figure, the total power is normalized to the power radiated by the same molecular emitters in free space. Therefore, molecular radiation at λ is enhanced by the microcavity in case F (λ) > 1, whereas radiation is inhibited if the total power is lower than unity. We observe for the 3λ/4 configuration that the microcavity enhances the light emission rate in a broad spectral range extending from about 430 nm to the near-infrared region. The maximum of the curve is located close to λ = 600 nm since the cavity structure has been designed to maximize constructive interference at this wavelength. We also notice that the peak width of the calculated curve is comparably larger than the full width at half maximum of the intrinsic luminescence spectrum of the red emitter Ir(MDQ)2 (acac), which is about 80 nm.7 Due to large overlap between the cavity resonance peak and the emitter spectrum, radiative excitonic decay is favored at all wavelengths at which photon emission actually occurs. Consequently, we can expect a strong enhancement in the overall wavelength-integrated decay rate and emitter radiation quantum yield. The situation appears reversed for the λ cavity. The total radiated power is lower than unity in a broad region of the visible spectrum as the local electric field acts as radiation inhibitor. For the optimized 5λ/4 cavity, we finally observe a behavior that is somewhat in-between. Radiation is slightly enhanced for 525 < λ < 675 nm, with a peak equal to 1.1 at λ 600 nm, and slightly inhibited elsewhere in the visible spectrum. Overall, we can conclude that the effect of the local EM field on molecular radiation is observed to be stronger for the 3λ/4 cavity. The curves in Fig. 6(a) demonstrate that enhancement or inhibition of spontaneous emission in these weak microcavities (composed by a transparent contact and a reflecting one) is substantially an effect occurring in the

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FIG. 6. (Color online) Wavelength dependence of the power spectra calculated for some selected red OLEDs. (a) Total radiated power at the emitter location F (λ). (b) Outcoupled power U (λ).

near field of the emitting centers, and the distance between the molecular emitters and the metallic mirror determines the observed modifications. On the other hand, the narrowing of the spectrum obtained for the 5λ/4 configuration (as compared to the 3λ/4 one) is related to the increase in the number of modes sustained by the optical cavity, in turn related to the increase in cavity length. With increasing number, modes becomes more localized, both in space and in frequency, and the wavelength selectivity of optical structure increases accordingly. As far as the outcoupled power fraction U (λ) is concerned, calculation results are depicted in Fig. 6(b), exhibiting several similarities to the behavior of the F (λ) spectra. The curves calculated for the 3λ/4 and the 5λ/4 devices peak at λ 600 nm, as expected from the device design. The largest values for U (λ) are obtained for the device in 3λ/4 configuration due to the larger local field at the location of the emitting centers and, as observed previously, the width of the curves decreases with increasing cavity order. For the detuned λ cavity, the calculated outcoupled power tends to zero due to the aforementioned destructive interference at the emitter location. As it is apparent from the dashed line in Fig. 6(b), the λ cavity is tuned for emission at wavelengths longer or shorter than 600 nm, where the intensity of the intrinsic luminescence spectrum of the Ir-based phosphor is very low, owing to inefficient coupling of light to the leaky modes of the cavity. To allow for a more quantitative understanding about the effect of the local field on molecular spontaneous emission, we finally analyze the variation of the wavelength-integrated power densities as a function of the separation between the emitters and the metallic reflector. We now consider the overall total radiated power F , the outcoupled power fraction U , and , which we define as follows: the outcoupling efficiency ηout F =

This yields the curves shown in Fig. 7, which completely characterize the global electromagnetic properties of the luminescent devices under study. It is apparent that the total power F is strongly enhanced up to an ETL thickness of about 100 nm, due to near-field coupling between molecular emitter and the free charges at the surface of the metallic contact. With increasing

sel (λ)F (λ) dλ,

(28)

λ

U = sel (λ)U (λ) dλ, λ ηout = sel (λ)U (λ)/F (λ) dλ. λ

(29) (30)

FIG. 7. (Color online) (a) Total radiated power F and outcoupled power fraction U calculated for the red OLEDs. (b) Calculated for the same devices. outcoupling efficiency ηout

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separation, interference effects become considerably weaker. Owing to destructive or constructive interference for dETL > 100 nm, molecular radiation is still modulated, but to a much lower extent. On the other hand, the outcoupled power U and the outcoupling efficiency ηout exhibit a qualitatively similar behavior. A first maximum in both curves is observed for an ETL thickness corresponding to the 3λ/4 design, for which molecular radiation is enhanced by the optical structure and leaky modes are available for far-field coupling. A minimum is then observed for a cavity length λ, determined by two concurring effects: overall inhibition of molecular radiation (F < 1) and strong coupling of radiation to loss modes (mainly substrate and waveguided modes and plasmonic modes to a lower extent). A further increase in cavity length induces again an increase in outcoupling efficiency. It is interesting to note that the overall outcoupling efficiency is larger for the 5λ/4 cavity. This may appear counterintuitive since we have previously shown that molecular radiation is maximally enhanced in the 3λ/4 cavity [see Fig. 6(a)]. Photon radiation is very efficient in the 3λ/4 design due the larger availability of EM modes at the emitter location and, consequently, the outcoupled power is large. Nevertheless, the efficiency of the device is strongly limited by optical losses and, in particular, by the strong coupling of internally generated photons to the nonradiative SPP mode. This latter aspect is apparent from the power distribution spectrum in Fig. 5(a) and it has been previously discussed by the authors in a separate publication.6 For the 5λ/4 device, F and U are both lower (or, equivalently, the absolute probabilities for internal photon generation and successive outcoupling of radiated power photons are both lower), essentially due to a weaker environment-induced modification of spontaneous emission. However, the ratio between the quantities is larger and therefore the 5λ/4 cavity configuration exhibits the largest efficiency in light generation due to a more favorable modal distribution.5,6,38,83

C. Radiative rate modifications in weak organic microcavities

In the preceding parts of this section, we have presented a discussion concerning the purely electromagnetic properties of classical dipole sources radiating optical power in microcavities. Conversely to classical dipole emitters, which radiate continuously in time, a quantum-mechanical excited state only exists for a limited and finite period of time and decays with a rate determined both by the intrinsic properties of the molecular emitter and by the modification induced by the optical environment (the well-known Purcell effect).56 For the quantification of the radiation efficiency of a real luminescent device, the inclusion of the intrinsic decay properties leads to interesting changes compared to the classical EM picture.6,34,38 To illustrate these effects, we shall include in our discussion the nonideal radiative properties of the molecular emitters, quantified by the two intrinsic decay rates rad and nrad . We consider for molecular emitters in a microcavity the ratio between the total effective decay rate ∗ and the intrinsic rate in free-space , given by ∗ F rad + nrad = = F ηrad + 1 − ηrad . rad + nrad

(31)

FIG. 8. (Color online) Effective molecular decay rate calculated for the red OLEDs according to Eq. (31) as a function of the thickness of the ETL. Curves are shown for various values of the intrinsic molecular radiative efficiency (ηrad from 0.2 to 1, step 0.2). All values are normalized to the intrinsic molecular decay rate .

Figure 8 illustrates the variation of the normalized decay rate ∗ / for different values of the intrinsic quantum yield of the molecular emitter. It is interesting to note that the variation in effective decay rate with varying cavity length (or emitterto-metal separation) appears strongly modulated by the actual value of ηrad . For an efficient molecular emitter (rad nrad or ηrad → 1), the thickness dependence of ∗ / is essentially the same as in Fig. 7(a). Indeed, for such an efficient emitter, Eq. (31) can be approximated as follows: F rad ∗ = F. (32) rad nrad rad On the other hand, if the intrinsic quantum yield of the molecular emitter is very low (rad nrad or ηrad → 0), Eq. (31) modifies into nrad ∗ = 1. (33) rad nrad nrad

FIG. 9. (Color online) Calculated external quantum efficiency for the red OLEDs. Curves are shown for different values of the intrinsic molecular radiative efficiency ηrad and unitary device electrical efficiency γ .

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PHYSICAL REVIEW B 85, 115205 (2012) TABLE I. Parameter values extracted for the fabricated OLED sample series. The reference current values at which parameter extraction is performed are given in Sec. III C. Emitter Ir(MDQ)2 (acac) Ir(ppy)3 TBPe

γ (−)

ηrad (−)

e1 (−)

e2 (nm)

0.92 0.94 0.90

0.84 0.76 0.73

1.01 1.00 1.04

−5.08 −5.53 −8.47

In the limiting case of molecular emitter with negligible nonradiative decay rate (ηrad → 1), the substitution of Eq. (32) into the above equation yields the following expression for ηq : ηq rad nrad ηrad ηout , (35) i.e., the external quantum efficiency is solely determined by the intrinsic luminescence quantum yield and the outcoupling efficiency ηout of the optical structure. This finding has an intuitive explanation: for an emitter with zero nonradiative losses, each excited state emits a photon with unitary probability. Hence, no further enhancement in radiative quantum yield is possible, and the largest ηq value will be obtained for the cavity configuration with the largest outcoupling efficiency, i.e., for the 5λ/4 configuration [see Fig. 7(b)]. On the other hand, owing to the proportionality law (32), variations in total radiated power correspond to strong variations in lifetime of the excited states. In the opposite limiting case of an emitter with very low quantum yield, the external quantum efficiency is approximated by the following equation: ηq rad nrad ηrad ηout F. (36)

FIG. 10. (Color online) Red OLED efficiency fingerprints. Comparison of the measured (symbols) and calculated (lines) external quantum efficiency, luminous efficacy, and forward luminance at a forward current J = 1.51 mA cm−2 as a function of the electron transport layer thickness dETL .

It is apparent from the two equations above and from the curves in Fig. 8 that the effect of the local photonic mode density on the excitonic decay rate is considerably different in the two limiting cases. In the former, the effect of cavityinduced modifications is maximal, being the overall decay rate directly proportional to the Purcell factor F . In the opposite limiting case, nonradiative molecular transitions dominate and the total effective decay rate remains equal to its intrinsic value independently from cavity-induced modification. For values of the intrinsic quantum yield ηrad comprised between the two limiting cases, intermediate behaviors are observed. The normalized ratio ∗ / strongly influences as well the dependence of the external quantum efficiency ηq on the emitter radiative quantum yield. It is convenient to consider here the case of a monochromatic emitter emitting at λ with unitary electrical efficiency γ . We rewrite Eq. (24) in the form92 ηq = ηrad ηout

F . ηrad F + 1 − ηrad

(34)

It can be noticed that the influence of the optical environment on ηq is maximal in this second limiting case, and the overall charge to photon conversion efficiency becomes proportional to the Purcell factor F . Further rearrangement of the above equation indicates an apparent direct proportionality between the outcoupling efficiency and the outcoupled power fraction U (ηq ηrad U ). Consequently, the largest external quantum efficiency shall be expected for the device exhibiting the largest outcoupled power fraction and, according to the results in Fig. 7(a), the 3λ/4 design would be favorable for a low-efficiency emitter. These considerations are confirmed by the full calculation of Eq. (24). The external quantum efficiencies we obtain for different intrinsic luminescence quantum yield values are shown in Fig. 9 for the red OLED stacks under analysis in this work. It can be noticed that 5λ/4 cavities exhibit a larger ηq for ηrad 0.6. For lower-efficiency values, 3λ/4 devices exhibit the largest external quantum efficiency. As we shall show in the next section, this variation in ηq depending on the value of ηrad can be exploited for the extraction of the latter parameter from the experimental data sets. Further modeling results, which shall not be detailed here, indicate that similar considerations hold for the green OLEDs, with a tradeoff at ηrad = 0.8. For the fabricated blue OLEDs, featuring an Al cathode, a consistently different picture is obtained. For these devices, the ηout value obtained for the 3λ/4 cavity is only slightly lower compared to the corresponding value for the 5λ/4 configuration and the tradeoff in ηq maximum occurs

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FIG. 11. (Color online) Measured J (V ) characteristics of red OLEDs with different ETL thickness. The gray line at J = 1.51 mA cm−2 shows the current density value considered in simulations.

for ηrad 0.95. The reasons are related to the different loss mechanisms observed for light emission in the blue spectral region, where coupling to surface plasmons is considerably weaker and waveguide losses larger.

framework detailed in Sec. II and, furthermore, allows us to achieve quantitative predictions for the influence of the local electromagnetic field (or equivalently the photonic mode density at the emitter location) on spontaneous emission in the devices under study. The basic idea underlying the proposed parameter extraction method is to consider a set of organic light-emitting devices characterized by the same layer stack and different values for the electron transport layer thickness dETL . Provided that a change in dETL does not influence the electrical behavior of the device, we calculate for the ith device with ETL thickness dETL,i the external quantum efficiency ηcalc,i according to Eq. (24). We then compare the calculated value to the corresponding measured efficiency ηmeas,i at a fixed current density. In order to account for eventual processing errors and uncertainties on the thickness of the electron transport layer, we further define an effective ETL ∗ as thickness dETL,i ∗ dETL,i = e1 (dETL,i − e2 ).

(37)

The four unknown parameters γ , ηrad , e1 , and e2 are then extracted by means of a least-square fitting algorithm implemented to minimize the quantity N {ηmeas,i − ηcalc,i [γ ,ηrad ,e1 (dETL,i − e2 )]}2 ,

D. Model validation and parameter extraction

(38)

i=1

In this section, we provide an extraction methodology for the unknown model parameters in Eq. (24), namely, the emitter intrinsic quantum yield ηrad and the electrical efficiency γ of the electroluminescent device. Parameter extraction is mandatory for the validation of the proposed modeling

where N denotes the number of samples in each series. The comparison of the calculated and measured external quantum efficiency for the red OLEDs featuring the phosphorescent emitter Ir(MDQ)2 (acac) is shown in Fig. 10.6 The same figure also shows the comparison of the OLED luminous

FIG. 12. (Color online) Comparison of measurements (symbols) and calculations (lines) for red OLEDs. Top panel: Spectral radiant intensity per unit area Iλ (θ,λ) at viewing angles θ from 0◦ to 60◦ , step 20◦ (dark to light tones). The number of measured data points at each viewing angle has been reduced by a factor 20. Bottom panel: Radiance L(θ). The error bars denote a ±10% error on the experimental value. In both panels, data are shown for the devices with dETL = 55,130, and 250 nm (from left to right, respectively). 115205-14


FIG. 13. (Color online) Green OLED efficiency fingerprints. Comparison of the measured (symbols) and calculated (lines) external quantum efficiency, luminous efficacy, and forward luminance at a forward current J = 0.0065 mA cm−2 as a function of the electron transport layer thickness dETL .

efficacy ηv and luminance in the forward viewing direction Lv (θ = 0 deg), derived according to Eqs. (A16) and (A21), respectively. The values of the four extracted parameters are finally collected in Table I. We notice that our theoretical predictions agree remarkably well with the measured data for the whole set of samples. In particular, it is interesting to observe that not only the positions of the efficiency maxima and minima are correctly described, but also their intensities are nicely reproduced within a confidence interval ± 10 %. We further observe that the model correctly accounts for variations over more than two orders of magnitude as far as the forward luminance is concerned. It can be noticed that the luminous efficacy ηv is overestimated by the model for large values of dETL , whereas the fit on the other quantities is fully satisfactory. The observed discrepancy is undoubtedly related to an increase in Ohmic voltage drop for the devices with larger ETL thickness. As it is apparent from the curves in Fig. 11, the J (V ) characteristics of the red OLEDs show an increase in the bias voltage Va required to achieve the required current density level (J = 1.51 mA cm−2 for this


device) as the ETL thickness increases. This is attributable to an insufficient Cs-doping concentration in the BPhen matrix constituting the electron transport layer. Due to the low doping level of the ETL, the Ohmic voltage drop on the ETL increases slightly with the layer thickness, thus lowering the OLED luminous efficacy. No increase in bias voltage with dETL is observed for the green and blue OLED sample series for which the same nominal dopant concentration has been used. Considering the parameter values collected in Table I, the red emitting system NPB:Ir(MDQ)2 (acac) exhibits a rather large intrinsic luminescence quantum yield (ηrad = 0.84, the largest value among the materials considered in this study). The extracted electrical efficiency value γ = 0.92 is rather large as well, indicating that the p-i-n architecture used in the device design and the materials used as EBL and HBL are very effective in providing a good balance of electrons and holes in the active layer and efficient confinement of triplet excitons. Consequently, the devices are very efficient in the charge-to-exciton conversion process. Similar considerations have been previously published by Baldo et al.,23 who reported charge-balance factors close to 0.9 for three-layer Ir-based phosphorescent OLEDs, and by Krummacher et al.,40 who reported slightly lower values. Owing to efficient charge transport and exciton radiative decay, the resulting external quantum efficiency measured for our best OLED sample is excellent [ηq = 21% (Ref. 6)] and among the best values reported for similar devices.24 As far as the parameters e1 and e2 are concerned, we estimate from calculations a constant processing error of 5 nm in the thickness of the ETL and a relative 1% thickness variation. The latter values are a clear indication that the overall quality of our OLED fabrication process is extremely high. As a further validation of the theoretical framework we are presenting in this work, we show in Fig. 12 the comparison between the measured and calculated spectral radiant intensity per unit area Iλ (θ,λ) and radiance L(θ ) for three selected red OLEDs. The calculated data are obtained by making use of Eqs. (A19) and (A20) for the spectral radiant intensity and radiance, respectively, and the parameters collected in Table I. The quantitative agreement between theoretical predictions and the experiment is remarkably good given that no additional fitting parameters have been introduced. It is worth noticing that the extraction performed on the external quantum efficiency curve yields values for the four unknown model parameters, which allows us to model all observed spectral features (intensity, resonance wavelength, spectral width, angular dependence) for a series of devices with a variation in cavity length of more than 200 nm. For the radiance values, we observe that the discrepancies between theory and experiments lie within a 10% range. We note a somewhat lower accuracy for the spectral features of the device with dETL = 115 nm, which we attribute to the strong detuning of the optical cavity. We postulate that the assumption of a delta-distributed exciton generation zone does not strictly hold for detuned microcavities since slight variations of the emitter location in the optical cavity may induce a considerable change in the macroscopic characteristics of the device. Nonetheless, the quantitative agreement remains fully satisfactory even in the case of strongly detuned cavities.

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FIG. 14. (Color online) Same as in Fig. 12 but for selected green OLEDs. In both panels, data are shown for the devices with dETL = 60,135, and 210 nm (from left to right, respectively).

Figure 13 shows the comparison of calculations and experiments for the green OLEDs. We extract for these devices an intrinsic radiative efficiency ηrad = 0.76, i.e., a slightly lower value compared to the red emitting system. The extracted ηrad nicely agrees with the value obtained for the system TCTA:Ir(ppy)3 by Mladenovski et al.21 by transient electroluminescence analysis. The extracted electrical efficiency γ = 0.94 is even larger than the corresponding values we obtain for the red OLEDs. Such a large value for the charge-balance factor in the green OLEDs is ascribed to the double emitting layer architecture, which ensures efficient charge carrier transport in the electron- and hole-conducting matrices in which the Ir-based phosphor is dispersed. The overall ηq is lower compared to the red devices we have analyzed above, although we could achieve a remarkable ηv = 80 lm W−1 for the 5λ/4 cavity. The lower external quantum efficiency of the green OLEDs compared to the red emitting devices is essentially related to a lower efficiency in light outcoupling, in turn due to the nonoptimal thickness of the ITO anode and to the lower reflectivity of the Ag cathode in the green spectral region. The values of the parameters e1 and e2 are finally very similar to those we extracted for the red systems, indicating a similar quality in device fabrication and an overall extremely good reproducibility of the experimental conditions. The comparison of the measured and calculated spectral radiant intensity per unit area and radiance is shown in Fig. 14. We notice that the fit is very accurate and its quality is comparable to the red OLED case. We finally show the comparison of theoretical predictions and experiments for the blue OLEDs in Fig. 15. For the blue fluorophor TBPe, we consider a singlet-to-triplet ratio ξe = 0.25. The electrical and radiative efficiency values extracted for the blue OLEDs in Table I are lower compared to both the

red and green systems. The extracted γ = 0.90 indicates that charge transport and recombination is slightly less efficient in blue devices. Nonetheless, it is worth mentioning that this value may considerably vary in case the spin factor ξe is different from the theoretical 1:3 ratio, as often reported for fluorescent emitters.15,93 The electroluminescence quantum yield of the emitting system MADN:TBPe is finally estimated to be only slightly lower compared to the green emitting system we used in our experiment. Overall, we can conclude that the processes of singlet exciton generation and radiative decay are relatively efficient. The main efficiency limiting factor is therefore related to spin statistics, as the radiative decay of triplet excitons is forbidden in molecular flurophors. As one can notice from the data in Fig. 15, the largest efficiencies are now obtained for the 3λ/4 design. The reason for this behavior, opposite to the case of red and green OLEDs discussed above, is related to the different optical properties of the metallic reflector used in blue OLEDs (Al instead of Ag) and to the fact that (i) plasmonic losses are lower and (ii) waveguide losses are larger if one considers optical processes occurring at the short wavelengths of the visible spectrum. We finally observe that the extracted values for e1 and e2 are considerably larger compared to the red and green sample series. This may be related either to larger processing errors in this series or to the larger sensitivity of the characteristics of optical systems to thickness variations at short wavelengths, i.e., the same thickness variation has a larger impact in the blue spectral region compared to the green and red ones [cf. Eq. (27)]. Finally, we compare the measured and calculated spectral radiant intensity per unit area and radiance for some selected blue OLEDs in Fig. 16. As for the red and green emitting devices, the quantitative agreement between theory and experiment is remarkable.

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FIG. 15. (Color online) Blue OLED efficiency fingerprints. Comparison of the measured (symbols) and calculated (lines) external quantum efficiency, luminous efficacy, and forward luminance at a forward current J = 0.2 mA cm−2 as a function of the electron transport layer thickness dETL . V. SUMMARY AND CONCLUSION

We have presented an extensive theoretical and experimental study on spontaneous emission in organic electroluminescent devices. The theoretical treatment includes an approximate quantification of the concentration of generated excitons within the active layer of organic electroluminescent devices. This is complemented with a numerical electromagnetic model accounting for the influence of the local density of photonic modes on the properties of light-emitting molecules. The model has been applied to organic electroluminescent devices in weak microcavity configuration. After having identified the resonant modes of the optical structures, we have disclosed the main effects responsible for the modification in radiative rate and efficiency of molecular emitters. Namely, near-field coupling to the metallic reflector induces strong variations in the photonic mode density at the location of the emitting centers, yielding enhancement or inhibition of molecular radiative transitions. Interference effects have then been identified as a second factor determining the overall efficiency of spontaneous emission. The inclusion of the intrinsic


electroluminescence quantum yield into the electromagnetic modeling formalism finally completes the quantitative picture. We have demonstrated that the intrinsic luminescent properties of molecular emitters determines to what extent the photonic mode density effectively influences the macroscopic characteristics of organic luminescent devices. By considering three sets of small-molecule p-i-n OLED samples with identical electrical behavior and varied density of photonic and plasmonic modes at the location of the molecular emitter, we have demonstrated that classical electromagnetic modeling is applicable to state-of-the-art devices provided some requirements are met in the device design and characterization conditions. We have further shown that the modeling framework used here allows for the extraction of unknown material and device parameters, namely, the device electrical efficiency, the emitter radiative quantum yield. For all sets of samples, we have obtained electrical efficiencies close to or slightly exceeding 90% and radiative quantum yields in the range 73% to 84%. The predictions of the model for the three sets of samples analyzed here are in extremely accurate agreement with the experiment, with a relative degree of accuracy on average within ±10%. The accuracy we have demonstrated in this work allows us to be confident that all relevant physical processes have been correctly accounted for in our treatment and, consequently, the modeling framework is successfully applicable to gain quantitative access to internal, nonmeasurable quantities determining the electro-optical operation of organic devices. As an example, we have shown an intuitive application of our modeling approach in a previous publication6 dealing with the identification of the efficiency loss channels (hereby including electrical losses, nonradiative exciton decay, and optical losses) in red phosphorescent OLEDs in various configurations, such as bottom emission, top emission, and bottom emission on high refractive-index substrate. For the latter set of devices, we were able to report a remarkable ηq = 54%, corresponding to a record luminous efficacy ηv = 108 lm W−1 . This demonstrates the potentialities of the present modeling framework: it can not only be used as an effective design tool for the optimization of OLED structures, but it might also provide important guidelines for the successful development of the technology by identification of the main sources of efficiency loss. In further previous publications,4,85,94 we have demonstrated the application of an extension of the present model to the design and analysis of multiemitter white organic electroluminescent devices, along with the extraction of useful internal quantities from the experiment, such as the relative electrical efficiency for exciton formation on the different emitters and the efficiency for light outcoupling in the various spectral regions. The assumptions of the model have been thoroughly discussed to stress the limits of validity of the present treatment for spontaneous emission in organic luminescent devices and to provide guidelines for future extensions. In spite of the remarkable accuracy achieved here, further work is still required to extend the applicability of the modeling framework to the general case of complex organic luminescent devices featuring multiple emitters and operated at arbitrary bias

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FIG. 16. (Color online) Same as in Fig. 12 but for selected blue OLEDs. In both panels, data are shown for the devices with dETL = 40,170, and 230 nm (from left to right, respectively).

conditions. Nonetheless, we believe the results in this work will be of use in the assessment and development of solutions improving the efficiency of spontaneous emission in organic luminescent devices.

ACKNOWLEDGMENTS

This work was partially funded by the BMBF under Contract No. 13N11060, project acronym “R2FLEX.” Novaled AG, Dresden, is acknowledged for providing some of the materials used in this work. One of the authors, M.F., is indebted to R. Nitsche at sim4tec GmbH, K. Neyts at Electronics and Information Systems Department, Ghent University, and R. Scholz at Institut für Angewandte Photophysik, Technische Universität Dresden, for fruitful and inspiring discussions.

APPENDIX: FORMULATION OF THE DIPOLE MODEL FOR MULTILAYERED OLEDS

In this appendix, we describe the computational details of the state-of-the-art electromagnetic model used in this work. The implementation of the dipole model for a system composed of a multilayered stack and a thick, optically incoherent substrate is described first. Successively, we consider the necessary transformations of the calculated power densities for the comparison of theoretical results to measurable quantities, such as luminous power, spectral radiant intensity per unit solid angle, or luminance. We present in this section a complete overview on the actual implementation of the dipole model for the multilayered, planar optical cavities we consider in this work. This formulation is derived from various literature

sources and the model equations are given here in a unified and formally consistent notation. We consider the case of a vertical dipole source v, with orientation perpendicular to the planar multilayered system, and a horizontal dipole source h, oriented in the plane. The dipole sources are assumed to be embedded in a nonabsorbing emitting layer with real refractive index ne . The power densities radiated by the two dipole sources per unit wavelength λ and unit normalized transverse wave vector u are calculated as58,86 + − u2 (1 + aTM )(1 + aTM ) 3 , (A1) KTMv = Re √ 2 4 1 − aTM 1−u

+ − (1 − aTM )(1 − aTM ) 3 KTMh = Re , (A2) 1 − u2 8 1 − aTM + − 1 (1 + aTE )(1 + aTE ) 3 . (A3) KTEh = Re √ 2 8 1 − a 1−u TE In the above equations, Re[...] denotes the real part of the complex quantity enclosed by brackets. Furthermore, + + aTM,TE (A4) = rTM,TE exp 2j kz,e z+ , − − − aTM,TE = rTM,TE exp 2j kz,e z , (A5) + − + − and aTM,TE = aTM,TE aTM,TE . Here, rTM,TE (rTM,TE ) denotes the reflection coefficient for waves traveling from the EML in the upward (downward) direction for TM and TE polarized waves; z+ (z− ) is the distance of the emitting dipoles from the top (bottom) interface of the active layer. For an OLED stack consisting of an arbitrary number of layers of different materials, the wavelength- and direction-dependent reflection + − coefficients rTM,TE and rTM,TE are calculated by making use the

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transfer matrix approach.95 For an isotropic dipole distribution, the total spectral power K per unit normalized in-plane wave vector is finally obtained as K=

1 3

(KTMv + 2KTMh + 2KTEh ) .

(A6)

Calculations are performed for the wavelength λ of interest as a function of the normalized wave-vector component u. The total spectral power F (λ) radiated by the dipole emitter at λ is then obtained by integration over all directions in the wave-vector space, according to ∞ K(λ,u) du2 . (A7) F (λ) = 0

The calculation of the power radiated by the dipoles into the far field, namely, the power determining the useful OLED light emission, is performed by dividing the total radiated power K into different contributions. We consider the general case of an OLED emitting light through an optically thick, incoherent substrate with real refractive index ns and ns < ne . The outcoupled power fractions KTMv , KTMh , KTEh , and K radiated into the substrate for u < ns /ne are calculated as56,86 1 + a − 2 u2 3 TM + KTMv =

(A8) T , 8 1 − u2 1 − aTM 2 TM 1 − a − 2 3

TM + 2 KTMh = 1−u (A9) T , 16 1 − aTM 2 TM 1 + a − 2 1 3 TE +

(A10) KTEh = T , 16 1 − u2 1 − aTE 2 TE K =

1 + 2KTMh + 2KTEh ), (K 3 TMv

(A11)

+ + and TTE denote the energy transmittance of the where TTM top half stack for TM- and TE-polarized waves, respectively, according to 2 kz,s + + 2 ne TTM = |tTM | , (A12) ns kz,e + + 2 kz,e TTE = |tTE | . (A13) kz,s

In the above equations, kz,e and kz,s are, respectively, the z component of the photon wave vector in the emitting medium and in the substrate. The fraction of power Kout effectively radiated in a farfield medium with refractive index no (no < ne ) is finally derived by considering incoherent multiple reflections of field intensities in the optically thick substrate.96,97 Accordingly, the outcoupled power fraction is calculated as86 Kout = Kout

Ts,o , 1 − Rs,o Rc


TE- and TM-polarized waves with the appropriate reflection and transmission coefficients. The spectral power density U (λ) outcoupled at the wavelength λ is finally obtained by integrating Eq. (A14) over all directions in the light escape cone: ucrit (λ) Kout (λ,u) du2 , (A15) U (λ) = 0

where the upper integration limit is obtained by application of Snell’s law as ucrit (λ) = no (λ)/ne (λ). Other quantities than the external quantum efficiency ηq [see Eq. (13)] are of interest for validating the model against the experiment. In view of lighting applications of organic electroluminescent devices, the most used fingerprint is the luminous efficacy ηv , which is defined as the ratio of the total luminous power radiated by the device to the input electrical power. The conversion of the calculated photon flux [Eq. (13)] into luminous power by means of the eye-sensitivity (or standard luminosity) function V (λ) yields the following expression for ηv : γ sel (λ) ∗ η (λ)ηout (λ) dλ, (A16) Km hc V (λ) ηv = eVa λ rad,e λ where h denotes the Planck’s constant, c the speed of light in vacuum, Km = 683.002 lm W−1 is a constant, and Va the applied bias voltage. For the comparison of model results to measured angleresolved electroluminescence patterns, the above spectral power quantities per unit normalized in-plane wave vector can be converted in power densities per unit solid angle.56 This is accomplished by converting the outcoupled power fraction Kout (λ,u) into the outcoupled power Pout (λ,θ ) per unit wavelength and per unit solid angle. For viewing angles in the far-field medium θ lower than the critical angle θcrit (λ) = arcsin [ucrit (λ)], the power densities Kout (λ,u) and Pout (θ,λ) are related by the following integral relation: 2π Pout (λ,θ ) sin(θ ) dθ = Kout (λ,u) du2 .

By noting that u = no (λ)/ne (λ) sin(θ ) and du/dθ = no (λ)/ne (λ) cos(θ ), the outcoupled power spectrum per unit solid angle Pout (θ,λ) is obtained as no (λ) 2 cos(θ ) Kout [λ, sin(θ )]. Pout (λ,θ ) = (A18) ne (λ) π From the above definition, we can easily derive the spectral radiant intensity per unit area Iλ (θ,λ) of the electroluminescent organic light-emitting device by analogy with Eq. (A16): Iλ (θ,λ) = γ

(A14)

where Rs,o and Ts,o denote, respectively, the reflectance and the transmittance of the substrate outer medium interface, and Rc is the total reflectance of the OLED optically thin layers.98 The generalized phase term96,97 accounting for light absorption in the substrate is neglected here, as ns is assumed to be real. It is clear that the calculation of the far-field power fractions according to Eq. (A14) has to be performed separately for

(A17)

hc Pout (λ,θ ) ∗ (λ) sel (λ)ηrad,e . λ F (λ)

(A19)

The radiance L(θ ) and luminance Lv (θ ) can be finally derived, respectively, as Iλ (λ,θ ) dλ, (A20) L(θ ) = λ cos(θ ) Iλ (λ,θ ) Lv (θ ) = Km V (λ) dλ. (A21) cos(θ ) λ

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¨ FURNO, MEERHEIM, HOFMANN, LUSSEM, AND LEO *


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EFFICIENCY AND RATE OF SPONTANEOUS EMISSION . . . 63

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