1. Introduction

The ever-increasing demand for high-efficiency light-emitting diodes (LEDs) fuels multidisciplinary efforts in material discoveries, device structural optimization, and mechanism elucidation. Substantial progress has been made in materials engineering, evidenced by booming organic emitters [1–9] and recently the most sought-after perovskite family [10–14]. They share similarities of high photoluminescence quantum yields (PLQYs), readily tunable bandgaps, low-cost solution processability, etc. The external quantum efficiency (EQE) of planar organic LEDs (OLEDs) achieved 30% with the close-to-unity PLQY [11], while that of planar PeLEDs soared to 20% [15,16] from less than 1% [17] in a few years through dimensional engineering [18–20], interface and defect engineering [21–24] and so forth. The theoretical EQE limit of PeLEDs is 1.6 times higher than that of OLEDs, being 75% [20] and 46% [11], respectively, due to the significant photon recycling effect of perovskite materials with small Stokes shifts [25,26]. Nevertheless, experimentally achieved EQEs of OLEDs and PeLEDs are far below the theoretical prediction, mainly restricted by the total internal reflection at the interface between active materials and adjacent layers, where a big refractive index difference exists. To address this problem, the strategy of incorporating nanostructures has been taken for light outcoupling to improve energy utilization, which can be categorized into either the internal scheme with nanostructured emissive layers or the external scheme with planar emissive materials and patterned substrates [23].

To gain a significant light outcoupling enhancement, nanostructured interfaces need to be created between materials with high refractive index contrasts. The internal scheme is advantageous by embedding nanostructures between high-refractive-index active layers and low-refractive-index claddings. The structure can be directly patterned on an initially planar emissive layers [27,28] or through conformal coating of emissive materials on patterned substrates [29–32]. The latter approach precludes concerns over both the protection of deposited active layers throughout patterning steps and the introduction of extra high-refractive-index materials, which is usually involved in the external scheme [33,34].

Nonetheless, numerical modeling of internal-scheme PeLEDs (IS-PeLEDs) is more challenging than that of external-scheme ones because the nanostructured emissive layer exhibits position-dependent carrier density and emitting dipole distribution. Well-developed optical simulation tools, such as the Lumerical finite-difference time-domain (FDTD) solver, are widely utilized to model the light-matter interaction and to optimize nanostructures for LEDs with planar emissive layers [33,34], but few reports show quantitative simulation-assisted optical analysis of IS-PeLEDs [28,35]. This is because the setting of dipole sources is easy to determine via experimental characterization of the recombination zone in planar emissive layers [10,14] but remains uncertain in nanostructured perovskite materials since the nanostructure-disturbed local electric field alters the dipole distribution [26]. Therefore, accurate quantitative optical simulation of IS-PeLEDs is impossible without knowing the spatial distribution of the emissive dipoles.

Here, we accomplished reliable device performance analysis of IS-PeLEDs by calibrating light source setting in FDTD with dipole distribution extracted from electrical simulations using Silvaco TCAD. The electrical modeling provides information about the local electric field distribution and the intrinsic radiative recombination rate (RRR) profile within nanostructured emissive layers. The RRR denotes the number of excitons generated per unit volume per unit time, and thereby its profile represents the distribution of excitons over the examined region. Notice that the Purcell effect is not considered for the RRR calculated from electrical simulations but included in the optical modeling. By integrating the RRR over the entire volume of the emissive material, the exciton generation rate is obtained. The excitons in the electrical simulation serve as electric dipole light sources in the optical simulation, so the emitting dipole distribution can be determined accordingly to achieve a concrete analysis of the IS-PeLED performance concerning radiated power enhancement (the ratio of radiated power from IS-PeLEDs to that from the planar device) and far-field patterns. Light extraction efficiency (LEE) and EQE can also be estimated, revealing the light outcoupling effect of nanostructures. This hybrid simulation method can serve as a universal analyzing platform for device performance analysis of LEDs with nanostructured emissive layers.

2. Method

2.1 Device configuration

IS-PeLEDs with nano-gratings on patterned substrates are investigated in this paper as a typical demonstration, and two-dimensional simulations are carried out on the cross-section perpendicular to the grating. Nano-gratings are patterned on fused silica substrates, with a sidewall slope of approximately 80° [36,37]. A 100-nm-thick ITO anode, a 40-nm-thick PEDOT:PSS hole transport layer (HTL), a 100-nm-thick CsPbBr₃ emissive layer, a 40-nm-thick 2,2’,2"-(1,3,5-Benzinetriyl)-tris(1-phenyl-1-H-benzimidazole) (TPBi) electron transport layer (ETL), and a 100-nm-thick LiF (1 nm)/Al (100 nm) cathode are sequentially deposited on the nanostructured fused silica. Figure 1(a) shows the architecture of the IS-PeLED, whose energy diagram is illustrated in Fig. 1(b). The functional layers possess nanostructured profiles conformal with the substrate corrugation as reported [30,31,33].

 

Fig. 1. Device structure and RRR profiles of Pe-LEDs. (a) Device structure of the IS-PeLED. (b) Energy diagram of the PeLED. RRR “hotspots” appear at the corners of the nano-grating in the (c) 900-nm-pitch IS-PeLED, and they merge into one in the (d) 200-nm-pitch device. The RRR profile is uniform along the perovskite/HTL interface in (e) planar PeLEDs. (c), (d), and (e) share the same color bar. (f) Integrated RRR of planar and IS-PeLEDs with the pitch ranging from 200 nm to 900 nm reveals increased exciton numbers resulted from introduction of nanostructures. In the legend, “PN” stands for “Pitch = N nm”, where N = 200, …, 900.

Download Full Size | PDF

2.2 Determination of exciton profiles from electrical simulations

The Silvaco simulation result of the IS-PeLED shows that the electric field in the perovskite layer is concentrated at the thinnest part (Fig. S1), consistent with the report that the local electric field intensity increases at the thinner part of the active layer in internal-scheme OLEDs (IS-OLEDs) as it is almost inversely proportional to the distance between electrodes [30]. As a result, the RRR in the perovskite layer is concentrated at the corner of the nano-ridge near the perovskite/HTL interface in the IS-PeLED, as shown in Fig. 1(c). These two RRR “hotspots” gradually merge into one when the pitch of nanostructures decreases from 900 nm to 200 nm, with the grating duty cycle (DC) fixed at 0.5 (Fig. 1(d)). The evolution of RRR profiles with nanostructure pitches from 800 nm to 300 nm is demonstrated in Fig. S2. The RRR profile indicates that the exciton distribution is not uniform along the interface, which leads to varied weights of dipole light sources at different positions in optical simulations. In contrast, the RRR in the planar configuration is uniform along the interface (Fig. 1(e)). It is noticeable that the nanostructure-dependent RRR phenomenon is generally applicable to thin-film LEDs, and another example of PeLEDs with a 30-nm-thick perovskite layer is demonstrated in Fig. S3. Aside from the difference in exciton distribution, the quantity of excitons also changes with different nanostructures. Figure 1(f) plots the integrated RRR over the volume of planar and IS-PeLEDs with an effective area of 1 × 1 cm². The integrated RRR of IS-PeLEDs is significantly larger than that of the planar counterpart, with the largest value from the 200-nm-pitch IS-PeLED tripling that of the planar device, indicating that more excitons are generated in nanostructured devices and the enhancement increases with the decrease in the grating pitch.

2.3 Dipole weight calibration for optical analysis

The dipole distribution determines far-field radiation patterns since dipoles at different locations in the IS-PeLED contribute to distinguished components in the far field. Here we compared three dipoles at distinct positions along the perovskite/HTL interface: the center of the nano-ridge (position 1), the middle of the nano-grating slope (position 2), and the center of the nano-trench (position 3), as annotated in Fig. 2(a). Far-field patterns from dipoles at these three positions are depicted in Fig. 2(b), calculated by averaging calculations of orthogonal dipole components incoherently (Fig. S4). The angular emission pattern shows strong position dependence. The outcoupling peak becomes sharper as the dipole moves from the nano-ridge to the nano-trench, and the angular peak shifts from around 8.6° to 11.5° from the normal. Figure 2(b) also reveals distinguished radiated power of dipoles at different positions, indicating the significance of dipole weight calibration for reliable prediction of device performance.

 

Fig. 2. Case study of dipoles at different positions and illustration of three dipole weight calibration methods. (a) RRR profile of a 300-nm-pitch IS-PeLED. The red dashed line indicates the position of electric dipoles. Points 1, 2, and 3 specify three representative locations of the dipoles. (b) Far-field radiation patterns of the three dipoles at position 1, 2, and 3 in (a). (c) The red curve illustrates the RRR along the red dashed line in (a), as a function of the dipole position; the discrete blue dots show the value of dipole weights calculated by different methods.

Download Full Size | PDF

To reveal the light emission properties of the IS-PeLED under working conditions, a string of dipoles is placed 5 nm above the perovskite/HTL interface, with dipole intervals of 1 nm. Dipole positions are indicated by the red dashed line in Fig. 2(a). Due to the symmetry of the structure, FDTD simulations are conducted with dipoles only in a half of the unit cell, and the result is unfolded during post-processing. With the center of the nano-ridge being the origin, the length of the interface in a half unit cell of the structure in Fig. 2(a) (pitch = 300 nm, DC = 0.5, structural depth = 120 nm) is approximately 250 nm. In conventional FDTD simulations, dipoles along the interface share equal weights [35], and the weights add up to 1, as shown in Fig. 2(c). For clarity of the figure, dipole intervals are set as 5 nm in Fig. 2(c). This uniform dipole weight allocation is referred to as the Conventional method in this paper. In this case, there are totally 50 dipoles along the interface in a half cell, with dipoles located at p_n= 2.5 + 5*(n-1) (nm) (n = 1,2,…,50), and each dipole is assigned a weight of 0.02. Alternatively, the RRR Calibration method is proposed, where the dipole weights are adjusted according to the RRR profile (exciton distribution) along the interface, with the sum of dipole weights unchanged. For instance, the RRR along the interface in Fig. 2(a) is illustrated by the red curve in Fig. 2(c), and the dipole weight is calculated proportionally to the RRR value, marked by blue triangles. Apart from the concern of exciton distribution, the calculation of exciton numbers is of vital importance as more excitons lead to larger radiated power. The sum of dipole weights is further modified according to the amplification of exciton numbers, which equals to RRR integration enhancement. For the device illustrated in Fig. 2(a), the RRR integration is 1.86 times larger than that of the planar counterpart under the voltage of 3 V, so the dipole weight is multiplied by a factor of 1.86. This modified version of the RRR Calibration method for dipole weight calibration is named as the Integrated Calibration method. Details about the electrical and optical simulation mechanisms of the hybrid modeling method are enclosed in Method S1 and Method S2, respectively.

3. Results and discussion

3.1 Effects of structural parameters

Maximizing the power conversion efficiency of nanostructured PeLEDs and OLEDs through optimizing the structure pitch is a prevalent approach to improve device performance [28,36–38]. The calculated relation between the current density and the applied voltage (Fig. 3(a)) implies that the introduction of nanostructures results in a lower turn-on voltage for IS-PeLEDs, similar to the case of IS-OLEDs [30]. In the following discussion, the applied voltage is fixed at 3 V as it provides a relatively high internal quantum efficiency (IQE) (Fig. S5) and a satisfactory luminance for both planar and IS-PeLEDs. The radiated power enhancements calculated using different methods are shown in Fig. 3(b). While the Conventional method claims little difference in the enhancement among various pitches, the Integrated Calibration method reveals that the enhancement increases when the pitch decreases. Despite the lack of experimental measurement of PeLEDs identical to the simulation models, there are reported results of IS-OLEDs that evidence the influence of nanostructures on current densities and luminance (radiated power) [37]. Specifically, device performance of planar and IS-OLEDs with pitches of 277 nm, 417 nm, and 833 nm is compared, showing that a smaller pitch results in a higher current density and larger luminance. It should be noted that although the absolute values of enhancement calculated using the Integrated Calibration method might be overestimated because material defects, interfacial trapping, and thermal degradation are neglected in the electrical modeling, the result is still instructive to nanostructure effect analysis. The result calculated by the RRR Calibration method also differs from that by the Conventional method with less deviation, indicating that the radiated power is more affected by the exciton quantity. Figure 3(c) compares the calculated LEE and EQE of planar and IS-PeLEDs, illustrating that the LEE of PeLEDs is largely improved by incorporating nanostructures into the device. The EQE of IS-PeLEDs is also improved owing to enhancement in both IQE and LEE. It is worth noting that the EQE obtained under the applied voltage of 3 V are not the maximal EQE values, and therefore the maximal EQE enhancement is out of reach from these data because devices with different nanostructures achieve the maximal EQE under distinct voltages [21,39]. The calculation method is described in Method S3. The estimated LEE of the planar device is only 5.9% due to the super-high refractive index of the vapor-deposited perovskite layer [40], which leads to a low EQE of 1.7%. This value is comparable to the measured maximal EQE (1.55%) of the planar PeLED composed of CsPbBr₃ perovskite deposited by thermal evaporation despite overestimated EQE [34], which is attributed to the enhancement in measured EQE from photon recycling in perovskites that cannot be counted in FDTD optical simulations [24]. Although the RRR (and thereby LEE and EQE) of planar PeLEDs is underestimated in simulations due to photon recycling, the hybrid modeling of IS-PeLEDs can be close to reality because the contribution to LEE (and thus EQE) from light scattering dominates over photon recycling as re-absorption and re-emission of photons are significantly suppressed with the nanostructure-boosted photon escape probability [24,41]. Note that Conventional method cannot reflect the change of LEE with increasing voltage and therefore gives an inaccurate calculation of the EQE droop, while Integrated Calibration method is able to (Fig. S6). Figure 3(d) and (e) illustrate angular radiation patterns of IS-PeLEDs with various pitches, calculated using Conventional method and Integrated Calibration method, respectively. The dipole weight calibration leads to distinguished far-field patterns, as much sharper angular peaks are observed with the Integrated Calibration method. Herein, dipole weight calibration is a necessity for reliable predictions on the device behavior.

 

Fig. 3. Electrical and optical simulations of planar and IS-PeLEDs with various pitches. (a) Current density – voltage curves of planar and IS-PeLEDs. (b) Calculated radiated power enhancement of IS-PeLEDs using different dipole weight calibration methods. (c) Calculated LEEs and EQEs of planar and IS-PeLEDs. Estimated far-field radiation patterns of IS-PeLEDs with (d) the Conventional method and (e) the Integrated Calibration method are distinct. (d) and (e) share the same legend in (e). In the legend, “PN” stands for “Pitch = N nm”, where N = 200, …, 900.

Download Full Size | PDF

The depth of the nanostructure also plays a crucial role in the performance of IS-PeLEDs. The deeper the structure, the larger the partial reduction of perovskite thickness on the nanostructure sidewall and thus the stronger the local electric field enhancement (Fig. S7). When the depth increases, the device features a smaller turn-on voltage (Fig. 4(a)), a larger integrated RRR (more excitons) (Fig. 4(b)), and more considerable radiated power enhancement (Fig. 4(c)). It is worth noting that since the electrical simulation does not include the thermal effect, the modeling result cannot reflect the increasing risk of device burnout with decreasing partial thickness arising from deepening the nanostructure. However, the simulation result elucidates the strategy to improve the device performance by increasing the structural depth, until problems with short circuiting and burnout occur. Figure 4(d) plots the calculated LEE and EQE of IS-PeLEDs with different structural depths, showing a downturn when the depth exceeds 90 nm. In terms of far-field patterns, the Integrated Calibration method gives predictions of enhanced radiated power and sharpened radiation peaks as the structural depth increases (Fig. 4(e)), while the Conventional method indicate subtle different between different devices (Fig. 4(d)). Simulation results from the hybrid modeling method are highly consistent with the experimental measurement of IS-OLEDs, where obvious evolution of radiation peaks presents with the change of structural depth [33,36].

 

Fig. 4. Electrical and optical simulations of planar and IS-PeLEDs with different structural depths. (a) Current density and (b) integrated RRR as functions of the applied voltage. (c) Radiated power enhancement and (d) LEEs and EQEs of IS-PeLEDs with various structural depths. Simulated far-field patterns of IS-PeLEDs with different depths show extinguished evolution using (e) the Conventional method and (f) the Integrated Calibration method. (e) and (f) share the same legend in (f). In the legend, “DN” stands for “Depth = N nm”, where N = 60, 90, and 120.

Download Full Size | PDF

We also compared three samples with DCs of 0.25, 0.5, and 0.75 to further elucidate the structural dependence of the device performance. Intriguingly, while a larger DC helps to lower the turn-on voltage (Fig. 5(a)), the device with the smallest DC generates the most excitons (Fig. 5(b)). However, generating more excitons does not guarantee a better device performance. It is the device with a DC of 0.75 that provides the largest radiated power enhancement (Fig. 5(c)). This is because the radiated power depends not only on the number of excitons but also the dipole distribution profile. Specifically, dipoles on the nano-ridge deliver less radiated power into the far field than those in the nano-trench (Fig. S8(a)). The IS-PeLED with a DC of 0.25 possesses the most excitons, but most of them lie on the nano-ridge (Fig. S8(b)) and therefore contribute less radiated power. By contrast, PeLEDs with larger DCs own fewer excitons, yet many of them are located at the nano-trench, where the emitted light can be better received in the far field, resulting in a larger radiated power enhancement. Distinct predictions on LEEs and EQEs are made using Conventional and RRR Calibration methods (Fig. 5(d)), emphasizing the significance of selection of proper modeling methods for reliable calculations. Far-field radiation patterns predicted by Conventional (Fig. 5(e)) and Integrated Calibration (Fig. 5(f)) methods are distinctive regarding both angular distribution and intensity.

 

Fig. 5. Electrical and optical simulations of planar IS-PeLEDs with various DCs. (a) Current density and (b) integrated RRR as functions of the applied voltage. (c) Radiated power enhancement and (d) calculated LEEs and EQEs in terms of DCs. Distinct far-field patterns of IS-PeLEDs are estimated using (e) the Conventional method and (f) the Integrated Calibration method. (e) and (f) share the same legend in (f).

Download Full Size | PDF

3.2 Effects of material deposition methods

Thermal evaporation and spin-coating are two major material deposition processes used in PeLED device fabrication. These two fabrication processes typically introduce different morphology profiles of the device layers. While the evaporated perovskite layer is conformal with the corrugated HTL, the spin-coated perovskite tends to fill the nano-trench and flatten the sags and crests. The Integrated Calibration method can offer more accurate predictions on device behaviors and reveal the influence of fabrication methods when the morphological change is considered. Note that a more reliable comparison needs to take the electrical and optical property difference of the perovskite into account because the material property is dependent on deposition methods. Here, we define the filling factor (FF) as the ratio of the perovskite thickness in the nano-trench (t_t) over that on the nano-ridge (t_r) (FF = t_t/t_r), as illustrated in Fig. 6(a). FFs of spin-coated samples are determined by multi-parameters such as precursor solution concentration and spin-coating speed, but always larger than unity because the material lying in the nano-trench is thicker than that on the nano-ridge, whilst the FF of evaporated perovskites equals to 1. The electrical properties of the planar control and IS-PeLEDs with FF = 1, 1.35, and 1.65 are compared. Surprisingly, the turn-on voltage of the spin-coated device with FF = 1.65 is even larger than that of the planar device (Fig. 6(b)) owing to increased electrical resistance resulted from the thickened perovskite layer. Improved RRR is still achieved in spin-coated PeLEDs (Fig. S9), because the devices always benefit from nanostructure-induced local electric field enhancement, though smaller than that of the evaporated sample (Fig. S10). As a result, IS-PeLEDs always show strengthened radiated power than their planar counterpart whatever fabrication processes, but the enhancement of spin-coated samples is lower than that of the evaporated device, as shown in Fig. 6(c). Calculations of LEEs and EQEs (Fig. 6(d)) also implicate a superior device performance of the IS-PeLED with FF = 1. The Integrated Calibration method clearly shows that a smaller FF contributes to a larger radiated power enhancement, while the Conventional and RRR Calibration methods fail to reveal the influence of the FF. Figure 6(e) and Fig. 6(f) plot the simulated angle-resolved radiated power from PeLEDs with different FFs using Conventional and Integrated Calibration methods. In both plots, the angular distribution profile is not sensitive to the FF. However, the Integrated Calibration method reveals that the intensity declines as the FF increases, which is not observable in the Conventional simulation.

 

Fig. 6. Electrical and optical simulations of IS-PeLEDs with different fabricated methods. (a) Device structure diagram of IS-PeLEDs. (b) Current density, (c) radiated power enhancement and (d) LEE and EQE calculations of IS-PeLEDs with different filling factors. Far-field radiation patterns are predicted with (e) the Conventional and (f) the Integrated Calibration methods. (e) and (f) share the same legend in (f).

Download Full Size | PDF

4. Conclusion

In summary, we have developed a hybrid modeling method that combines electrical simulations of carrier dynamics and optical simulations of far-field light emission to achieve performance analysis of PeLEDs with nanostructured emissive layers. Specifically, we developed the Integrated Calibration method to calibrate dipole weights using the exciton distribution and quantity acquired from electrical modeling for optical calculations of LEEs, EQEs, radiated power, and radiation patterns. The Integrated Calibration method proves to be capable of making predictions on nanostructure-altered radiated power and far-field patterns consistent with experimental results, providing a reliable means to help optimize the nanostructure for efficient devices.