1. Introduction

Micro-Light Emitting Diode ($\mathrm{\mu}$-LED) arrays are a promising light engine for a wide range of display applications. In particular, $\mathrm{\mu}$-LEDs as small as one micrometer would be needed for future high-resolution augmented reality (AR) displays [1,2]. The requirements such as pixel size, pixel density, brightness, contrast ratio, lifetime, refresh rate and power efficiency for AR applications are highly demanding and cannot be fulfilled by currently available technologies like organic LEDs (OLEDs) or liquid-crystal displays (LCDs). Besides the superiority of $\mathrm{\mu}$-LED arrays for near-eye displays in many of the previously named criteria, also the possibility to build compact system devices is a main advantage [3–5].

Single $\mathrm{\mu}$-LEDs in the order of 10 $\mathrm{\mu}$m and larger can be individually transferred to the display substrate using a pick-and-place method. However, this transfer technology currently is difficult for $\mathrm{\mu}$-LEDs smaller than 10 $\mathrm{\mu}$m down to 1 $\mathrm{\mu}$m required for AR applications [6,7]. Therefore, a more advanced solution is wafer-to-wafer bonding of the $\mathrm{\mu}$-LED wafer and a silicon (Si) wafer with integrated complementary metal-oxide-semiconductor (CMOS) logic. This combination builds a promising platform for micro-displays as the Si wafer serves as the display substrate and at the same time contains CMOS-based integrated circuits which function as an electrical driver for single $\mathrm{\mu}$-LEDs and allows compact integration and mass production scalability [7–11].

Furthermore, the removal of the epitaxial growth substrate after bonding to the Si wafer offers high potential for compact integration with system optics such as waveguides and allows to improve light outcoupling via further processing on the exposed III-V semiconductor surface. For example, it allows to pattern microlens structures [12], metasurface structures [13] or deposit quantum dots for color conversion [14]. This so-called thin film chip (or also known as thin-film flip-chip) architecture is a mature technology for large LEDs for many years. After processing of the epitaxial LED stack on the p-side, the wafer is bonded to a Si substrate where the bonding metalization layer contains a metal mirror serving as light reflector and as electrical p-contact. After separation of the the epitaxial layer stack from the initial growth substrate the n-side with a thickness of only a few micrometer can be processed and used for light outcoupling. For larger LEDs the introduction of thin film technology resulted in a big boost of the light extraction efficiency (LEE) [15].

Despite the potential impact of thin film-based $\mathrm{\mu}$-LEDs on technological progress, their fundamental optical properties have not been studied in great detail so far. Recently published experimental studies dealing with optical properties including light extraction efficiency and far field behavior of $\mathrm{\mu}$-LEDs comprise predominantly so-called sapphire chips (without growth wafer removal and with p-side light emission) or flip chips (with wafer bonding step and n-side emission through the non-removed sapphire growth substrate). Furthermore, the focus is mainly on larger pixel sizes [16–21].

From a theoretical point of view, the initial studies on devices in the vicinity of thin film structures are promising [12,22–25]. However, there is still a considerable knowledge gap and many unanswered questions that need to be addressed. Not only for thin film-based $\mathrm{\mu}$-LEDs, but also for $\mathrm{\mu}$-LEDs in general. For example, ray tracing methods have been used for modeling $\mathrm{\mu}$-LED structures with feature sizes in the single micrometer size regime and smaller [16,26,27]. This could be problematic as ray tracing relies on the assumption that the wavelength of the light is small compared to the features of the environment and furthermore, the wave nature of light including phenomena such as interference and diffraction cannot be accounted for [28]. Other studies did use finite-difference time-domain (FDTD) simulations, but only with a single dipole source at the pixel center [29–31]. It is questionable, whether the optical properties can realistically be described by that approach, as the light is generated in the entire active area, which extends laterally over the full pixel [25]. Another aim of this study is therefore to compare wave-optical simulation data with experimentally measured properties of $\mathrm{\mu}$-LEDs.

We investigated thin film-based $\mathrm{\mu}$-LED arrays with pixel sizes as small as 1$\times$1 $\mathrm{\mu}$m² up to 8$\times$8 $\mathrm{\mu}$m² and focused primarily on optical properties such as light extraction efficiency and far field characteristics. Therefore, we fabricated nitride-based $\mathrm{\mu}$-LED arrays on chips without integrated CMOS logic, but with blank Si-substrates as common p-contact where all pixels are operated in parallel for better statistics. We performed measurements on external quantum efficiencies (EQEs) in an integrating sphere and far field characterization of mounted chips in a goniometer setup. Finally, we compare two different 1 $\mathrm{\mu}$m blue emitting InGaN $\mathrm{\mu}$-LEDs with 40 nm or 80 nm thick aluminium oxide both in experiment and in wave optical simulations.

2. Materials and methods

2.1 Sample design and fabrication

Thin film-based $\mathrm{\mu}$-LED arrays with six different pixel sizes from 1 µm to 8 µm were fabricated for optical characterization. Figures 1(a)–(c) depict the main processing stages with (a) the p-side pixelation and passivation, (b) the bonding step and growth substrate removal and (c) the final chip after n-side processing.

 

Fig. 1. Sample design and fabrication. Subfigures (a) to (c) show schematic cross sections of the device in three different processing states: (a) Structured InGaN $\mathrm{\mu}$-LEDs on sapphire and processed dielectric stack of Al$_2$O$_3$ and SiO$_2$ with a p-contact opening. (b) Deposition of a silver mirror, diffusion barrier and carrier bonding material on the p-side, before the wafer is bonded to a Si substrate and the sapphire growth substrate is removed. (c) Final thin film chip with processed n-contact on Si wafer. White circuit lines indicate the current spreading and parallel driving of all pixels. Light is emitted via the n-side. (d) Schematic cross section of two pixels shown in (a) with relevant dimensions defining the $\mathrm{\mu}$-LED array and a legend showing the corresponding materials. (e) $J$-$V$ characteristics of a $\mathrm{\mu}$-LED array with 1 $\mathrm{\mu}$m pixel size. (f) Microscope image of a final chip showing the n-contact bonding pad. (g) Electroluminescence image at 100 A/cm² of a bonded chip under an optical microscope (h) Cross-sectional FIB cut of the final thin film chip with 1 $\mathrm{\mu}$m pixels according to state (c). (i) Image of a mounted 1 mm² chip during operation at 100 A/cm².

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The pixel geometry is fully defined after the pixelation and passivation. Table 1 shows the pixel sizes manufactured together with the pixel pitch and the width of the p-contact opening encoded in the photo mask. These three parameters, schematically drawn in Fig. 1(d), define the pixel geometry together with other quantities such as the mesa etch depth and Al$_2$O$_3$ thicknesses.

Table 1. Dimensions of different $\mathrm{\mu}$-LED variants fabricated within this study.^a

View Table

The pixels pitch was chosen rather high to avoid crosstalk effects between pixels and amounts four times the pixel size. Therefore, $\mathrm{\mu}$-LED arrays of given outer dimensions but with differently sized pixels in total have approximately the same active area. The $\mathrm{\mu}$-LED array is embedded within a 1 mm² chip. For example, in case of the 1.0 $\mathrm{\mu}$m pixels one chip includes more than 30,000 $\mathrm{\mu}$-LEDs with a common n- and p-contact. All chips were fabricated on a single wafer and chips with $\mathrm{\mu}$-LED arrays of different sizes were processed next to each other to avoid process variations or variations during epitaxial growth over the wafer.

A standard blue InGaN LED structure was grown on a 150 mm sapphire wafer by metalorganic chemical vapor deposition. The semiconductor stack comprises a buffer layer for strain relief and for reduction of crystal defects, another Si-doped n-GaN layer for current spreading, the active region with a InGaN/GaN multiple quantum well (MQW) emitting at a wavelength of 440 nm and p-type layers consisting of Mg-doped AlGaN/GaN with a thickness of approximately 100 nm.

After growth and activation of the p-type GaN, indium tin oxide (ITO) was deposited and annealed to form an ohmic contact. Square pixels with different sizes were defined by conventional photo lithography and then dry etched with chlorine-based plasma with an etching depth of approximately ${600}\;\textrm{nm}$. The exposed sidewalls of the mesa were then treated with concentrated KOH solution to remove plasma damage and to passivate the surface. Figure S1 (Supplement 1) shows a test field including the six differently sized pixels next to each other after this processing stage. A double dielectric stack consisting of Al$_2$O$_3$ and SiO$_2$ with a p-contact opening was processed, which represents a standard surface passivation of the mesa [32]. Furthermore, it serves as the electrical isolation of the n-type GaN and p-type bonding material. Samples with 40 nm and 80 nm thick Al$_2$O$_3$ were fabricated to investigate its impact on the optical properties. The silver mirror was processed to cover the structured mesa and to make contact with the ITO in the p-contact opening, before a metallic diffusion barrier was deposited. Afterwards the sapphire wafer with structured $\mathrm{\mu}$-LED arrays was eutectically bonded to a Si substrate using a AuSn-based metallic solder. The sapphire growth substrate was then removed by laser lift-off. After removal of a few hundreds of nanometers via an initial polishing step, n-contacts were processed and finally, chips with an area of 1 mm² were singulated and mounted for measurements.

As illustrated in Fig. 1(c) the metallic silver mirror, the bonding metalization layer stack as well as the Si substrate are highly conductive and allow lateral current spreading and simultaneous driving of all pixels. The global n-contact is placed next to the pixel array close to the chip edge as shown in Fig. 1(c) (side view) and Fig. 1(f) (top view). The rather thick n-GaN layer in the order of 5 µm allows sufficient lateral current spreading over the chip. This is confirmed by the $J$-$V$ characteristics of the chip with 1 $\mathrm{\mu}$m pixels presented in Fig. 1(e) and the uniform electroluminescence image at 100 A/cm² shown in Fig. 1(g). Figure 1(h) shows a cross-sectional FIB cut of a final chip with 1 $\mathrm{\mu}$m pixels and Fig. 1(i) shows an image of a final singulated and mounted chip while being operated at 100 A/cm². In addition, microscope images of final chips with the six different sizes can be found in Fig. S2 (Supplement 1).

2.2 Characterization

Final devices were first measured in an integrating sphere under pulsed operation to avoid any heating of the sample. In order to compare different devices, the current densities $J=I/A$ were calculated using the pixel area $A$ obtained from size measurements during processing and the number of pixels per chip. The obtained EQE curves from different pixel sizes were rescaled by a common factor in a way that the EQE of the 1 $\mathrm{\mu}$m pixel at a current density of 500 A/cm² amounts 1.0. This allows to relatively compare the EQEs of different pixel sizes.

Afterwards the far field characteristics were measured in a goniometer system. Here the sample is mounted in a rotatable holder and light is collected in a distance of approximately ${1}\;\textrm{m}$ via an optical fiber. It is guaranteed that during rotation and tilting of the sample, the center is always on the same spot aligned with the fiber. The far field was measured within the full hemisphere in $3^\circ$ steps for both $\theta$ the polar angle and $\phi$ the azimuthal angle. It was recorded at a current density of 100 A/cm².

For further analysis and discussion two-dimensional FDTD simulations were performed in the same manner as described in previous theoretical studies [22,25]. The simulation models reflecting the samples were implemented using geometries and dimensions obtained during processing or afterwards from thickness measurements, SEM, AFM and cross-sectional FIB data. An exemplary cross section of an implemented model is shown in the appendix A which has a realistic p-contacting scheme and double dielectric stack in contrast to a simplified model used in a previous theoretical study [25]. For one device, several individual simulations were performed with different dipole source orientations and lateral placements over the entire active region. The weighting of the different dipole orientations was done according to an empirical ratio best suited for such nitride-based LEDs where the TM dipole polarization is double weighted compared to the TE dipole polarization. A geometric weighting was done assuming a rotational symmetry of the pixel. More details on the FDTD simulations can be found in Sec. A. The main simplifications are the usage of a single QW and the assumption of a circular pixel which allows to use rotational symmetry, whereas the faricated pixels have squared features in the litho masks.

3. Results and discussion

3.1 EQE measurements

The upper graph of Fig. 2 shows the obtained EQE curves of pixels with different sizes from 1.0 µm to 8.0 µm and 40 nm thick Al$_2$O$_3$. The forward voltage at 10 A/cm² was 2.7 V for all samples and they showed very similar $J$-$V$ characteristics.

 

Fig. 2. Size effect: The upper graph shows measured EQEs for different thin film-based $\mathrm{\mu}$-LEDs with pixel sizes from 1 µm to 8 µm and 40 nm thick Al$_2$O$_3$ with focus on the droop regime, where the IQE is rather independent of the pixel size. The lower graph shows EQEs at 500 A/cm² and simulated LEEs versus pixel size. The dashed line is a guide for the eye and defined by a power law. The higher EQEs of smaller pixel sizes is explained by a higher LEE of those.

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At high current densities $J> {100}\;\textrm{A}/\textrm{cm}^{2}$ the EQE of all pixel sizes decreases with higher $J$ due to Auger recombination. In this so-called droop regime the internal quantum efficiency (IQE) mainly depends on the $B$ and $C$ parameters given in the $ABC$-model [33]:

The IQE drop caused by non-radiative surface recombination on pixel sidewalls is negligible in the droop regime. This is reflected in the fact that the effective Shockley–Read–Hall term $An$, which contains a size-dependency for $\mathrm{\mu}$-LEDs, drops out in Eq. (1). For the parameters $B$ and $C$, on the other hand, it can be assumed that they are independent of the pixel size. [33,34] Therefore, the higher EQE of smaller pixel sizes in the droop regime can be explained by a higher LEE. This has been previously reported for other chip designs or theoretically [16,17,22,25]. In this work it is experimentally shown for thin film-based $\mathrm{\mu}$-LEDs. The lower graph of Fig. 2 shows the EQEs measured at 500 A/cm² plotted versus the pixel size. Furthermore, simulated LEEs of those devices are shown as grey triangles and also rescaled in a way that the LEE of the 1 $\mathrm{\mu}$m pixels is unity. The measurement data reveals that 1 µm pixels have a 34% higher EQE compared to the 8 µm pixels. The EQE improvement is 21% and 5% for a 2 µm and 5 µm pixel respectively when compared to the 8 µm pixel. The black dashed line represents a power law and underlines the increasing importance of the sidewalls acting on the light extraction efficiency. For smaller pixel sizes there is a good agreement between the simulated LEE trend and the measured size trend of the EQE. For pixels larger than 2 µm the simulations seem to overestimate the LEE and deviate. One possible reason for this difference is the usage of 2D simulations in combination with the usage of rotational symmetry. It seems that this leads to a higher deviation of the LEE, especially for larger pixel sizes, and that the 2D-based simulations provide higher values compared to the measured efficiencies. Other reasons for these deviations could be processing imperfections such as the roughness and quality of the silver mirror surface, which could have a greater impact at larger pixel sizes.

The samples with 80 nm thick Al$_2$O$_3$ showed overall a similar size trend with a higher EQE of approximately $30{\%}$ at all pixel sizes. This can be accounted most likely to a significant change in the optical properties and enhanced LEE which we will discuss in the following chapters.

3.2 Far field characteristics

Figure 3 shows the radiant intensity $I_e$ measured at 100 A/cm² in the full hemisphere for the 1 $\mathrm{\mu}$m pixels with 40 nm thick Al$_2$O$_3$ exemplary. Smallest pixels showed a slight current dependency of the far fields referring for different current regimes (low-current regime, high current regimes). However, for $J$ greater than 10 A/cm² no measurable change in far field was observed for all pixel sizes. Note that the plot shows a $\cos \theta$-scaled axis from center to edge to represent the projection of the hemisphere.

 

Fig. 3. Far field of a $\mathrm{\mu}$-LED array comprising 1 $\mathrm{\mu}$m pixels and 40 nm thick Al$_2$O$_3$. The radiant Intensity $I_e$ was measured in the full hemisphere with an optical fiber at a distance of 1 m at 100 A/cm². $\theta$ and $\phi$ denote the polar and the azimuthal angle respectively.

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The radiation behaviour shown in Fig. 3 is slightly asymmetric. This can be explained by a small asymmetry in the pixel due to misalignment of photolayers during processing. We observe that the optical properties of $\mathrm{\mu}$-LEDs are very prone to small photolithographic misalignments, such as an offset of only 25 nm. Here it seems that the alignment in the $x$-direction was quite good, as the far field is symmetrical to the $y$-axis in the first order. Conversely, the alignment along the $y$-direction appears to be comparatively less accurate. The maximum of the radiant intensity $I_e$ is not located at $\theta = {0}^{\circ}$, but at approximately $\theta = {30}^{\circ}$ and emphasizes the presence of wave optical effects inside the pixel, which cause such a far field to deviate from a simple Lambertian emitter profile, where the maxium is located at $\theta = {0}^{\circ}$. Another indicator of wave optical effects and the importance of pixel sidewalls on the optical properties of $\mathrm{\mu}$-LEDs is given by the 8-fold symmetry pattern around the azimuthal angle. This could originate by an octagonal cross section of the 1 $\mathrm{\mu}$m pixel as a result of a rounding effect at the corners during processing. For some projection applications only a certain fraction of the light around $\theta = {0}^{\circ}$ can be used and consequently it is key to understand the wave optical effects inside the pixel which lead to the observed far field.

Therefore, far field patterns obtained from pixels with different size and for two different Al$_2$O$_3$ thicknesses of 40 nm and 80 nm have been investigated in more detail and are presented in Fig. 4. For reasons of clarity and simple comparison we show the normalized radiant intensity $I_e$ versus $\theta$ for a single azimuthal angle that is $\phi ={0}^{\circ}$. Hence, the most left upper graph of Fig. 4 shows a linear plot of the data presented in Fig. 3.

 

Fig. 4. Normalized far field patterns for different pixel sizes at $\phi =0^\circ$ and 100 A/cm². The pixel size increases from the left (1.0 $\mathrm{\mu}$m) to the right (8.0 $\mathrm{\mu}$m), while the top and bottom rows show pixels with 40 nm and 80 nm Al$_2$O$_3$ around the pixel mesa, respectively. Gray dashed lines represent a Lambertian emitter profile given by $I(\theta )=\cos \theta$. In addition to the expected change with pixel size, the oxide thickness also plays a key role on the optical characteristics of $\mathrm{\mu}$-LEDs, even for ‘larger’ pixel sizes.

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As predicted in simulations, the far field of thin film-based $\mathrm{\mu}$-LEDs changes significantly when the pixel size becomes smaller [25]. None of the $\mathrm{\mu}$-LED arrays with larger pixel sizes shows a clean Lambertian emitter profile. Only the 8 $\mathrm{\mu}$m pixel with 80 nm thick Al$_2$O$_3$ approaches the cosine distribution $I(\theta )=\cos \theta$. As the pixel size decreases, various lobes of the far field become more pronounced while others disappear and thus completely change the overall characteristics. A common feature of all $\mathrm{\mu}$-LEDs is the kink at ±66° and similar emission profile for $|\theta |> {66}^{\circ}$. Elsewhere, the specific shape and size effect depends on wave optical effects inside the pixel. This is further evidenced by the fact that pixels with different oxide thicknesses significantly smaller than the wavelength of light exhibit substantially different far field characteristics. Regarding the larger pixels (5 $\mathrm{\mu}$m and 8 $\mathrm{\mu}$m) the difference between 40 nm and 80 nm thick Al$_2$O$_3$ clearly demonstrates the impact of pixel sidewalls on the optical properties even for ’larger’ $\mathrm{\mu}$-LEDs. Here, most of the p-side area is covered with ITO and silver mirror, without Al$_2$O$_3$, which is only present on the mesa corners and sidewalls (compare with Tab. 1). Comparing the fundamental different far field characteristics of the 1 $\mathrm{\mu}$m pixels with 40 nm and 80 nm thick Al$_2$O$_3$, it becomes clear that there is still a lot of room for optimizing a certain radiation behavior. This may involve different oxide thicknesses, but also different material stacks with different optical properties. At the same time, it reveals the challenges of manufacturing such small $\mathrm{\mu}$-LED arrays, as the smallest changes in dimensions, thicknesses or slight misalignment drastically affect the optical properties from one $\mathrm{\mu}$-LED to another.

The measured far field characteristics of thin film-based $\mathrm{\mu}$-LEDs differ significantly from ones without removed growth substrate reported in Ref. [16]. There, all pixel sizes showed a Lambertian emitter profile. However, the earlier study also used a ray tracing method to calculate the far field patterns for pixel sizes down to 2 $\mathrm{\mu}$m, which is not expected to properly describe the wave optical effects such as interference and diffraction inside the pixel [28]. Furthermore, the authors claimed that a 20 nm thick Al$_2$O$_3$ passivation layer can be omitted in the simulation model, which is in contradiction to our findings. To prove the importance of wave optical effects for $\mathrm{\mu}$-LEDs and to show that the measured far field changes between samples with 40 nm and 80 nm thick Al$_2$O$_3$ originate indeed from this thickness change, we calculated the far field patterns for the 1 $\mathrm{\mu}$m pixels using FDTD simulations.

3.3 Role of wave optical effects and modeling

Figure 5 shows the measured far fields originating from the 1 $\mathrm{\mu}$m pixels (see Fig. 4) together with the simulated far fields. The simulation data was scaled such that the maximum of $I_e^\text {80 nm, sim}$ is equal to one, while $I_e^\text {40 nm, sim}$ has the same scaling factor. The measured $I_e^{p,\text {exp}}$ were scaled in a way that the integrated intensities are equal to those of the simulated ones $\int I_e^{p,\text {exp}}(\theta )\;\text {d}\theta = \int I_e^{p,\text {sim}}(\theta )\;\text {d}\theta$ for $p = \{ \text {40 nm}, \text {80 nm}\}$. This allows a comparison between the data obtained from experiment and simulation or a relative comparison of different Al$_2$O$_3$ thicknesses.

 

Fig. 5. Far field characteristics for 1 $\mathrm{\mu}$m pixels with 40 nm and 80 nm thick Al$_2$O$_3$ around the pixel mesa obtained from measurements and simulations. Fundamentally different far field patterns resulting form variation of the oxide thickness could be reasonably reproduced by FDTD simulations and explained by wave optical effects inside the pixel. The scaling was done in such a way that the red and blue $I_e$ curves can be compared relative to each other. The experimental data was scaled to the calculated $I_e$ so that the integrated intensities match.

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It appears that the experimental data is smeared out or smoothed compared to the simulation data and not all features of the simulation are reflected in the experimental data. Reasons for this could be an averaging effect due to the high number of single pixels in the measured devices and/or a simplification of the simulation model which will be discussed in more detail later. However, the data from FDTD simulations generally correlates with the experimental values and several features can be found in both data sets. One example is the dominant peak at $\theta = {\pm 30}^{\circ}$ for the pixel with 40 nm thick Al$_2$O$_3$ (blue curves).

Both pixels with 40 nm and 80 nm thick Al$_2$O$_3$ exhibit almost equal $I_e$ for angles $|\theta |>{66}^{\circ}$ (experiment and simulation). This could indicate that the IQE of the two samples is comparable and only the angular dependent LEE is different. However, this hypothesis requires that the $I_e$ measured at high angles is less affected by wave optical effects inside the pixel depending on the Al$_2$O$_3$ thickness. The far field data reveals further that pixels with 80 nm thick Al$_2$O$_3$ have a higher total LEE or absolute light intensity along with a more directional light outcoupling. In numbers, the integrated intensity of the 1 $\mathrm{\mu}$m pixels with 80 nm thick Al$_2$O$_3$ is 28% higher than for the pixels with 40 nm thick Al$_2$O$_3$ which is in line with the factor obtained from the integrating sphere measurement. In the following, we would like to examine the portion of light emission within a cone spanning from $\theta =-{15}^{\circ}$ to $\theta = {15}^{\circ}$ in relation to light emission in the full hemisphere. In general the amount of light $\beta (\theta _1)$ within a cone defined by $\theta _1$ can be calculated by

Comparing the 1 $\mathrm{\mu}$m pixels with 80 nm and 40 nm Al$_2$O$_3$, $\beta (15^\circ )$ is equal to 6.2% and 4.7% respectively, which corresponds to a directionality improvement of 32%. This motivates future FDTD simulations that can help to find an optimized pixel geometry to maximize $\beta (\theta _1)$ with $\theta _1$ depending on the particular application.

In a direct comparison of the two far fields, it seems that a higher portion of the light from the pixel with 40 nm thick Al$_2$O$_3$ cannot escape the $\mathrm{\mu}$-LED structure. One possible explanation for this is destructive interference of quasi-standing waves that occur between the pixel sidewalls, as can be speculated in Fig. 6. Analogous to waveguide modes, there could be comparable phenomena present in the smallest $\mathrm{\mu}$-LEDs. In addition, the optical role of the dielectric passivation stack in combination with the Ag mirror needs to be further investigated, where the optical path length of the dielectric stack $\Lambda = n_{\text {Al}_2\text {O}_3}\cdot t_{\text {Al}_2\text {O}_3} + n_{\text {SiO}_2}\cdot t_{\text {SiO}_2}$ is certainly of importance, with $n_i$ and $t_i$ being the refractive indices and thicknesses of the respective media. For the pixels with 40 nm thick Al$_2$O$_3$, the optical path length is $\Lambda = 0.50\lambda _\text {vac}$. This could lead to an unfavorable confinement of light between the GaN semiconductor and the metallic reflector. This light is then finally absorbed by the GaN material or at the metallic mirror surface. Moreover, this value of $\Lambda$ could affect known phenomena such as the Purcell effect, which is still a comparatively unexplored area for $\mathrm{\mu}$-LEDs. We suggest that this effect generally plays an important role in $\mathrm{\mu}$-LEDs with vertical sidewalls and see in simulations that these effects are also present for pixels with other mesa etch depths from 600 nm up to 2000 nm (Sec. 2 Supplemental). A deeper understanding and modeling of the underlying physics could help to optimize the LEE of $\mathrm{\mu}$-LEDs.

 

Fig. 6. Two-dimensional electric field distribution for the simulated $\mathrm{\mu}$-LED with 40 nm thick aluminum oxide. Here the field was calculated with a linear superposition of several individual FDTD simulations for different dipoles distributed across the active area and different dipole orientations. Black lines indicate interfaces between different materials.

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One possible reason why not all lobe-like features can be resolved in the experimental data is the fact that an array is measured, rather than a single pixel. This leads to an averaging of the far field over 30,000 individual emitters, in contrast to the simulated single emitter. Here, even the smallest processing fluctuations can lead to measurable differences in the far field, which ultimately lead to the observed smearing effect.

On the other hand, simplifications of the simulation model also lead to deviations. We found that the far field patterns obtained from the simulations are very sensitive to the thickness of the p-GaN and thus to the vertical position of the active region with respect to the Ag mirror. Finally, the best matching results were obtained when the modeled single QW was vertically placed in the center of the MQW of the real sample. We anticipate that implementing real MQWs would yield more precise results, elucidating the underlying smearing effect. However, the number of simulations would multiply by the number of QWs and the weighting of the various QWs is unknown a priori. Further simplifications are perfect edges and corners in the simulation model, while we observe rounding in the manufactured $\mathrm{\mu}$-LEDs, besides the misalignment as already discussed in section 3.2. Of course, 3D simulations could improve the accuracy of the simulation result, but would at the same time increase the computational effort significantly. Parameter studies for optimization purposes would then be extremely time consuming and costly, which was out of scope for this first principle investigation.

Taking all these simplifications into account, the data reproduced from simulations describe the measurement data remarkably well. At the same time, the main advantage of these 2D-based FDTD simulations is that the overall simulation time is manageable despite a fine mesh grid and a high number of individual simulations per model. Matching the experimental data with simulation data paves the way for parameter studies and design improvements of nitride-based $\mathrm{\mu}$-LED arrays for practical display applications.

4. Conclusion

We successfully fabricated thin film-based $\mathrm{\mu}$-LED arrays with different sizes down to 1 µm and reported on the fundamental optical properties of those arrays with special focus on the far field pattern. The importance of wave optical effects within $\mathrm{\mu}$-LEDs was shown and supported by FDTD simulations.

We reported on size-dependent EQE trends for $\mathrm{\mu}$-LEDs with removed growth substrate and found an increasing EQE with decreasing pixel size from 8 µm to 1 µm at high current densities and explain this by a higher LEE as previously predicted by wave optical simulations of such $\mathrm{\mu}$-LEDs. Furthermore, we addressed the far field characteristics, which change significantly as pixels become smaller. The comparison of pixels with 40 nm and 80 nm thick Al$_2$O$_3$ around the pixel mesa reveals that the oxide thickness plays a key role on the optical properties, even for the ‘larger’ 8 $\mathrm{\mu}$m pixels. This reflects the strong impact of the pixel sidewalls on the radiation behavior of $\mathrm{\mu}$-LEDs with vertical sidewalls. For 1 $\mathrm{\mu}$m pixels, the 80 nm thick Al$_2$O$_3$ leads to an approximately $30{\%}$ higher light extraction efficiency compared to 40 nm thick Al$_2$O$_3$. This could also be reasonably reproduced in 2D-based simulations and underlines the importance of FDTD simulation for further understanding and optimization.

In contrast to prior work, a particular focus of this study was on the optical characterization of InGaN $\mathrm{\mu}$-LED arrays in a thin film chip architecture, where a metal mirror on the p-side reflects the light, which is coupled out via the n-side with the growth substrate removed. Although several previous studies have shown results on InGaN $\mathrm{\mu}$-LEDs with other pixel designs and, in particular, larger pixels, our results provide fundamental optical properties and trends of thin film-based $\mathrm{\mu}$-LEDs down to 1 $\mathrm{\mu}$m that are highly relevant for practical applications such as AR displays.

After revealing the size effect and far field change purely caused by pixel size and oxide thickness, future investigations could also include other factors. Depending on the application, it is important to explore ways to optimize the light extraction. This may involve advanced pixel designs, dielectric layer stacks and light extraction structures.