The role of photon recycling in perovskite light-emitting diodes

Perovskite light-emitting diodes have recently broken the 20% barrier for external quantum efficiency. These values cannot be explained with classical models for optical outcoupling. Here, we analyse the role of photon recycling (PR) in assisting light extraction from perovskite light-emitting diodes. Spatially-resolved photoluminescence and electroluminescence measurements combined with optical modelling show that repetitive re-absorption and re-emission of photons trapped in substrate and waveguide modes significantly enhance light extraction when the radiation efficiency is sufficiently high. In this manner, PR can contribute more than 70% to the overall emission, in agreement with recently-reported high efficiencies. While an outcoupling efficiency of 100% is theoretically possible with PR, parasitic absorption losses due to absorption from the electrodes are shown to limit practical efficiencies in current device architectures. To overcome the present limits, we propose a future configuration with a reduced injection electrode area to drive the efficiency toward 100%.

nm was integrated assuming that the dipole has an emission spectrum the same as the measured photoluminescence. The figure shows the calculated mode ratios as a function of wavelength in a PeLED having 50 nm-thick perovskite. Because of the re-absorption loss, the direct outcoupling ratio is shown to be higher in the red region, while photons in blue region have further chances to be recycled through the relatively large re-absorption (A act ) of perovskite. Accordingly, the spectrum of outcoupled light is shown to be slightly red-sifted relative to the internal emission.
PR for PeLEDs with various thickness used in the manuscript, a-b, changing refractive index of perovskite to a, 1.6 + ik 0 (λ) or 2.8 + ik 0 (λ) and b, n 0 (λ) + i(0.5k 0 (λ)) or n 0 (λ) + i(4k 0 (λ)), and c, changing ITO thickness to 0 ~ 300 nm. d, Those for the best device having refractive index of 1.6 + i(4k 0 (λ)) without ITO. See Supplementary Note 6 for further details.  Figure 3c) and without PR (grey line). Since A act in the substrate mode is small for thin perovskite in Figure 3b, reduction of the maximum EQE is relatively smaller for 20 nm thickness when the substrate mode is excluded.

Reliability issues of the spatially resolved photoluminescence (PL) measurement
Since the spatially resolved PL and EL detect weak signals at long distances from the excitation, the analysis must be exceptionally careful to avoid possible misinterpretation. While we attributed the origin of broad spatial PL and EL shown in Figure 2 to recycling and scattering of trapped photons, we performed additional control measurements to exclude other possible pathways that might cause such a broad signal.
First, as shown in Supplementary Figure 2a and its inset, we excited a perovskite region near the edge of a perovskite LED pixel covered with a 100 nm-thick Al electrode, and collected the signal as a function of distance. In the uncovered region, while the detected laser profile rapidly decays with a full-width at half-maximum (FWHM) of 3 μm, that of PL measured for 515 (±5) nm is shown to be broader, possibly due to the nonlinear relationship between PL and excitation intensity shown in Fig. 1c. In the Al-covered region, the PL signal starts to decay rapidly (ten-fold per 4.6 μm), which can be regarded as the lateral resolution of the detection system as no PL underneath the focal point can outcouple through the Al electrode. This spatial resolution supports our attribution of PL signal to PR in Fig. 2b, where we observe that PL at 505 nm decays by only a factor of 3 moving from an excitation-collection separation of 210 μm to 310 μm. The sharp detection resolution is also shown in the rapid decay of 505 nm signal near separations of 230 μm, at the edge of the scratched region, in absorb the laser, a small portion of the laser may remain and circulate in the system by Fresnel reflection or scattering. Since PL from the diffused laser has the same spectrum as that from PR, there is no way to spectrally distinguish the signals from diffused laser and PR. For example, with our excitation lens having an (NA) value of 0.6, the 405 nm photons have a propagation angle up to 24º in the substrate (n = 1.5) and can reach the lateral distance of ~0.87 T if they bounce on the top and bottom surfaces of the substrate once by Fresnel reflection (T = substrate thickness). This is why we chose a thin PET substrate (T = 23 μm) and long distance (>200 μm) for the analysis in Fig. 2b-c to minimise the influence of such laser diffusion. In order to gain further confidence that the PL signal originates from PR rather than diffused laser light, we measured PL at 125 μm over time and observed degradation of in the emissivity of the film after laser exposure for 50 minutes. Here, we assume that density of absorbed photons at 125 μm is low enough to prevent degradation and the PLQE in this region is not changed (i.e. only the material near the direct illumination undergoes degradation) during the measurement as our perovskite film is stable in dark conditions for several days. In the result, Supplementary Figure 2b shows that the signal at 125 μm is clearly reduced over the whole spectrum with increasing exposure time. This indicates that the signal in this region is subject to the PL intensity at the excited region, which undergoes degradation due to strong light exposure, rather than the diffused laser intensity, which is constant throughout the measurement. Moreover, unlike the signal from the diffused laser, of which spectrum should be the same as "O" in Fig. 2c, the signal from trapped photons (i.e. PR + scattering) should have a broader and red-shifted spectrum similar as "C" in Fig. 2c. Hence, if the diffused laser had a meaningful influence on the signal, the signal would be expected to blue-shift with exposure time, as the signal from trapped photons is more sensitive to the degradation of the originally excited region. Therefore, the maintained spectral characteristic without such blue-shift shown in Supplementary Figure 2b verifies that the side effects such as laser diffusion are negligible compared to scattering and recycling of the trapped photons we studied.

Spectra of photons trapped in the waveguide mode
In Figure 2c, we discussed the highly red-shifted emission from the edge side (signal B at 290 μm) of the perovskite film, which came from the photons trapped in the waveguide mode. For the deeper investigation, we measured PL at edges with shorter distances from the excitation as shown in Supplementary Figure 4a. Supplementary Figure 4b clearly shows that, while the PL shape at the blue region is little changed, additional peaks appear at the red region and tend to be red-shifted along the distance variation of 13, 35, and 65 μm, Unlike Figure 2, this measurement was performed on a glass substrate much thicker (100 μm) than the investigated distances. Then, PR from substrate mode can be mostly ignored and the result can be explained with photons trapped in the waveguide mode (i.e. propagating inside the film), which can be scattered out at the edges. The photons in the waveguide mode undergo intensive re-absorption during propagation, especially for short wavelengths with higher absorption coefficient. This measurement experimentally demonstrates such a red-shift in the waveguide mode, more clearly than the signal B in Figure 2c, which is mixed with other signals related to substrate mode. It should be noted that the quantitative ratio between original PL (near 515 nm) and scattered signal (> 520 nm) in Supplementary Figure 4b does not have a physical meaning as the edge-scattering efficiency is not consistent for each measurement.

Supplementary Note 3 Quantitative estimation of the diffused emission in practical LEDs
Unlike the classical LED model, in PeLED, our spatial EL measurement has demonstrated that photons can be outcoupled even at long distances from the excited pixel, as shown in Figure 2d.
While PR of the waveguide mode has only a short propagation distance on a scale of 1/α ~ 1 μm (α: re-absorption coefficient of perovskite), the measured long-range diffused emission can be attributed to recycling of photons trapped in the substrate, either by total internal reflection (TIR) or Fresnel reflection. Knowing the dimension of such diffused emission would be helpful in the design of device architecture and precise measurement of efficiencies.
Using the measured spatial EL, we roughly estimated the profile of diffused emission. For estimation, we assume that (i) the emission intensity at a given spot is determined by the shortest distance (D min ) to the excited pixel and (ii) the contour integration along the same D min is proportional to exp(-α eff × D min ), with a fitted effective decay constant α eff . Then, the spatial emission intensity can be calculated by: I(x,y) = I 0 exp(-α eff × D min (x,y)) (pixel boundary length) / (contour length of the constant D min ) = I 0 exp(-α eff × D min (x,y)) (2W pixel + 2L pixel ) / (2W pixel + 2L pixel + 2π D min (x,y)) (1) for a rectangular excited pixel with a width of W pixel and length of L pixel . According to Figure Figure 5, the region out of the pixel is shown to add total emission as D margin increases, and the enhancement saturates at 9.04% for infinite D margin . In practical cases, additional emissions of 6.38%, 8.28%, and 9.02% (relative to the original emission) are expected from the region out of the pixel for D margin s of 1T substrate , 2T substrate , and 5T substrate , respectively. In our device having a 1.2 cm × 1.2 cm size substrate, D margin is approximately within 2T substrate ~ 6T substrate depending on the direction, and only small portion of trapped photons would escape the device before being re-absorbed.

Supplementary Note 4 Refractive index of perovskite used in the optical simulation
The re-absorption coefficient plays a key role in accurate calculation of LED structures with small Stokes shifts. While typical optical measurements such as ellipsometry and UV-visible spectroscopy may provide some information about absorption coefficients, the PL excitation (PLE) measurement shown in Figure 1a provides optical characterisation with high sensitivity even in the region of reabsorption with a low absorption coefficient. Therefore, we obtained the extinction coefficient (k) of the refractive index from the PLE spectrum (PLE(λ)) by assuming that the measured PLE is proportional to the film absorption, at least in the narrow spectral range of our interest, as shown below: The real part (n) of refractive index can be obtained from k, by applying a closed-form Kramers-Kronig relation fixing a single value at an arbitrary wavelength (n(λ 0 ) = n 0 ). 1 Then, the whole spectrum of n(λ) + ik(λ) can be represented by only two fitting parameters of P 0 and n 0 . We performed the transfer-matrix formalism (TMF) for a single 50 nm-thick perovskite film coated on a glass substrate optimising P 0 and n 0 to match the calculated and measured transmissions. The resulting n, k values and corresponding transmission are shown in Supplementary Figure 6. We obtained the precise k value in the re-absorption region with a sensitivity below 10 -3 as shown in Supplementary Figure 6b.

Supplementary Note 5
Virtual photon recycling: non-radiative self-coupling of dipole As described in the manuscript, we performed transfer-matrix formalism (TMF) calculations assuming uniformly distributed dipoles over finite meshes as shown in Supplementary Figure 7a. For a Hertzian dipole immersed in an arbitrary medium, it is known that the total power radiated outward measured from a sphere of radius R has a simplified form of: 2 where α and β are imaginary and real parts of the propagation constant of βjα. Therefore, while Unlike plasmon loss in metal, self-coupling with nearby emissive material generates another excited electron and there exists no net energy loss of the emissive layer if the newly excited electron decays radiatively. Therefore, by assuming a perfect recycling of non-radiatively self-coupled energy (white region in Supplementary Figure 7c), we can exclude this mode from the relative mode analysis for the dipole and the simulation becomes stabilised even for small mesh sizes as shown in Supplementary  Figure 7d. In systems such as quantum dots and organic molecules, each emissive/absorptive component is spatially separated. Then, the non-radiative self-coupling rate (~R -6 ) can be quantified with a finite R and it is also known as Förster resonance energy transfer (FRET). On the other hand, it is more complicated to define the finite self-coupling rate in a continuous medium such as perovskite, and the point dipole approximation may not be appropriate for this purpose. Due to such complexity, we did not consider the diffusive motion of dipoles caused by subsequent non-radiative re-excitations in the nearby region in the calculation and assumed a uniform dipole distribution as discussed in the next section. The 100% recycling of the non-radiatively self-coupled dipole is rationalised by the infinite Purcell factor corresponding to the divergent mode. The possible losses occurring during this process can be included in η rad as an electrical factor, rather than optical efficiency.

Supplementary Note 6 PR effect and LED efficiency for the future design of devices.
To provide more details to formulate design rules, Supplementary Figure  shown in Figure 3c, those for the smaller (n = 1.6) and larger (n = 2.8) real part refractive indices without PR converge to higher and lower values, respectively, due to the changed ray-optical limit of 1/2n 2 . However, even for n = 2.8, the local maxima of EQE were shown to be >20%, when it is assisted by strong microcavity effect in thin perovskite or PR in thick perovskites. For n = 1.6, in addition to its high EQE without PR, the EQE with PR is shown to further increase for thicker perovskite. In this device, because the refractive index is similar to that of the substrate (n = 1.5), most of the trapped photons propagate in the substrate mode rather than the waveguide mode. The thick perovskite layer screens Al and suppresses parasitic absorption; hence, the PR effect is maximised. On the other hand, Supplementary Figure 10b shows that a large imaginary part (k) of the refractive index is always beneficial in increasing the PR effect, despite the reduced EQE without PR. Compared to Figure 3c, the maximum EQE at 160 nm was enhanced from 20.2% to 25.1% with four times increased k, due to the enhanced PR contribution.
The PR effect is limited by the ratio between absorptions of perovskite and parasitic layers. While the non-perfect reflectivity of the metal layer is difficult to avoid, the parasitic absorption of transparent conductive electrode (TCE), which is ITO in our device, can be controlled by changing device geometry. Supplementary Figure 10c shows that the EQE with PR sharply increases as the ITO becomes thinner. Especially, when ITO is removed and no parasitic layer exists at the front side, although this may not be realistic, the maximum EQE was enhanced to 32.2 % at a perovskite thickness of 160 nm, and further increased to 34.3 % at 450 nm. In practical devices, there exists a trade-off between the transmission and conductivity of TCEs. Therefore, the optimal thickness of ITO needs to be found to minimise both parasitic absorption loss and electrical resistivity, considering not only the direct transmission, but also the recursive PR efficiency. Adoption of novel nanomaterials such as conductive metal oxides, metal grids, metal nanowires, and graphene can be also an alternative approach to reduce the parasitic absorption loss and maximise the PR effect.
As an upper-limit of the present PeLED architecture, the best properties of Fig. S10a-c were combined and its EQE was calculated as shown in Supplementary Figure 10d. The maximum EQE of a PeLED having reduced n of 1.6 and four-times enhanced k without ITO reached 49.3% at 500 nm thickness, integrating the positive effects of the increased ray-optical limit, the metal screening effect, increased re-absorption coefficients, and reduced parasitic absorption.