1. Introduction

Breakthrough in development of group-III nitride epitaxial technology has led to mass production of high-efficiency blue, green, and white light-emitting diodes (LEDs) suitable for general lighting, display, automotive, and other applications [1–5]. A lot of efforts has been made in recent decade in order to improve substantially the efficiency of blue and green nitride LEDs and to extend their emission wavelengths towards the yellow-red spectral range. The main directions and achievements of the extensive activity can be found in the reviews [6–9]. Despite the lack of native substrates, resulting in high threading dislocation densities in the epitaxial structures [10], a peak external quantum efficiency (EQE) of more than 80% has been attained with blue LEDs (see data from various LED manufacturers collected in Ref. [11] and references therein). This value exceeds the record EQE of about 70% of red LEDs based on AlGaInP alloys lattice-matched with GaAs substrate [12]. Analysis of datasheets from such leading manufacturers of red LEDs, as Osram OS, Lumileds, Nichia, and Epistar, has shown that EQEs of commercial devices at their operating currents do not typically exceed 40% with two exceptions of 46% and 49%. The wall-plug efficiencies (WPEs) of the commercial AlGaInP-based LEDs are even lower by 6–7 absolute percent on average. This contrasts with III-nitride LEDs that allow potentially an opposite relationship between EQE and WPE [13].

A reason for the lower efficiency of red LEDs compared to blue ones is a lower light extraction efficiency (LEE) [11], which cannot be simply explained by the difference in the refractive indexes of nitride and phosphide semiconductors. Therefore, the origin of the lower LEE of red LEDs should be looked for among the features of their designs. Another reason especially important at high currents is the electron leakage into p-side of an LED structure [14], leading to EQE droop similar to that occurring in nitride LEDs but having different nature. Detailed characterization of a commercial red LED manufactured by Osram OS has demonstrated a strong impact of LED self-heating on the electron leakage [15,16]. In particular, pumping of the LED with sub-microsecond current pulses and comparing results with continuous-wave (CW) operation enabled reliable identification of the onset of thermal EQE droop dependent on the current pulse duration. A reduced empirical ABC-model was applied to interpret the evolution of LED characteristics with current [15,16].

Unfortunately, the interpretation did not take into account specific design features of the studied LED where enhanced light extraction was provided by an embedded micro-reflector formed by facets of the inclined mesa etched in the epitaxial structure (see Refs. [12,17] for more detail). In this case, surface recombination might come into play, affecting the LED efficiency, which was not considered in the simplified analysis carried out in Refs. [15,16]. As the carrier losses caused by surface recombination depended substantially on the current crowding inside the LED die [18], their effect on EQE might be rather non-trivial.

The goal of this study is a detailed analysis of the red LED from Refs. [15,16] aimed at identifying and understanding the most critical aspects of its design and operation. For this purpose, we have applied a fully coupled electrical-thermal-optical numerical model [19]. In the absence of detailed specification of the LED structure and chip, their plausible designs were generated on the basis of open sources, retaining the most important features of the actual designs. Within this framework, we were able to identify the principal factors limiting LEE, controlling the EQE evolution with current, and affecting the spectral characteristics of the LED.

The paper is organized as follows. In Sec. 2, the plausible LED structure and chip geometry and design accepted for our simulations are described in detail. The most important parameters and material properties used in the simulations are discussed in Sec. 3. Analysis of simulation results and suggestions aimed at further improvement of the LED performance are presented in Sec. 4. Main conclusions coming from our simulations are summarized in Sec. 5.

2. Plausible LED structure and chip designs

Schematic structure of the red AlGaInP/GaAs LED is given in Refs. [15,16]. Unfortunately, many parameters of the structure were not disclosed in the papers. Therefore, in the lack of detailed information we have generated a plausible heterostructure, which would possess typical properties of those normally utilized in red LEDs. To do this, we tried to follow general ideas underlying development of such devices.

First of all, materials lattice-matched with the GaAs substrate were chosen for the plausible structure. According to Refs. [15,16], the LED active region included 20 quantum wells (QWs). As the aluminum-free materials were preferable for their using as light-emitting layers, the Ga_0.51In_0.49P alloy was chosen as the QW material. The width of every undoped QW of 4.1 nm was determined from the fitting of the predicted peak emission wavelength to its experimental value of 622 nm. The width of (Al_0.5Ga_0.5)_0.52In_0.48P barriers between the QWs equal to 9.9 nm were estimated from the known total active region thickness. The 100 nm n- and p-type Al_0.52In_0.48P confinement layers doped with the donor and acceptor concentrations of 5×10¹⁷ cm⁻³ and 1×10¹⁸ cm⁻³, respectively, were chosen to be sufficiently thick for effective suppression of the carrier leakage from the LED active region at room temperature. A 4 µm n-(Al_0.5Ga_0.5)_0.52In_0.48P with the donor concentration of 5×10¹⁷ cm⁻³ and a 1.52 µm p-Al_0.8Ga_0.2As with the acceptor concentration of 2×10¹⁸ cm⁻³ served as cladding/emitter layers. The compositions of barriers separating QWs and n-type cladding layer were chosen in such a way, as to provide the maximum conduction band offsets with the QW material, which was aimed at suppression of the electron leakage into p-side of the LED structure. The thickness of the top p⁺-GaAs layer doped up to the acceptor concentration of 1×10¹⁹ cm⁻³ was chosen as that reproducing the experimental LEE of the LED (see Sec.4.2 for more detail).

According to Refs. [12,17], the 1×1 mm² LED chip consisted of ten stripes fabricated as embedded micro-reflectors with inclined mesa sidewalls and common n-electrode formed to the back surface of the n-cladding layer after removing the GaAs substrate. In the micrograph of the operating chip in Fig. 1(a) one can see many pairs of dark spots in every stripe, being tentatively attributed to circular p-electrodes with estimated diameter of 10 µm. Figure 1(b) shows a plausible cross-section of a single stripe where double micro-reflector was formed by the inclined mesa sidewalls etched in the LED structure. The mesa inclination was assumed to correspond to [111] facets typically emerging during selective wet etching. The mesa depth of 2 µm provided the micro-reflector dimensions indicated in Fig. 1(b). The surface of the micro-reflector beyond the p-electrodes was assumed to be covered with SiO₂ insulating film and highly reflective metal, presumably silver, aimed at reducing the optical losses caused by incomplete reflection of emitted photons from the micro-reflector surfaces. Highly reflective electrodes (also Ag-based) were assumed to be formed to both n-(Al_0.5Ga_0.5)_0.52In_0.48P cladding and p⁺-GaAs contact layers.

 

Fig. 1. Micrograph of operating red LED chip with dark dots tentatively corresponding to circular p-electrodes (stripe selected for simulations is marked by white rectangle) (a); plausible cross-section of a single stripe with double micro-reflector formed by inclined mesa sidewalls and active region shown schematically by thick red line (b) and top and bottom views of the single stripe (c).

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In view of essentially multi-scale overall chip geometry and a large number of small units in the chip design, we decided to simulate the operation of a single stripe, i.e. central one, assuming the operation conditions not to change remarkably from stripe to stripe. Referring to the micrograph given in Fig. 1(a), we estimated the width of 100 µm and average length of 900 µm of the single stripe, as shown in Fig. 1(c). Random texturing of the bottom surface of n-(Al_0.5Ga_0.5)_0.52In_0.48P cladding layer free from n-electrode was applied in order to maximize LEE from the LED die. In this study, we emulate the random texturing with a regular one utilizing rectangular pyramids with a height of 400 nm and the bases of neighboring ones of 500 nm, closely adjacent to each other.

3. Simulation model and key parameters

Simulation of LED operation was carried out by a hybrid approach [19] implemented in the SimuLED software package [20]. The approach combined 1D modeling of carrier injection and recombination in the LED active region with 3D current spreading in the quasi-neutral contact layers. Here, 3D distribution of the electric potential and current density vector was found from the differential Ohm law, whereas 1D simulations served as the local non-linear boundary conditions matching the electric potential difference at the outer interfaces of the active region with the normal component of the current density, continuous in this region. The heat transfer in the LED structure was described by conventional equations. Photon propagation, absorption, and extraction in/from the LED chip was simulated by 3D ray tracing. The lateral distribution of the emission intensity in the active region was obtained here from the current spreading results. Other details of simulations are discussed below.

The use of the above approach is the key factor enabled performing fast and operative coupled electrical-thermal-optical simulations of the devices with realistic complex geometry, which is especially important to the red LED considered in our study (see the complex LED design discussed in Sec.2). Previously, the approach has been successfully validated and applied to design and optimization of III-nitride LEDs (see, e.g., Ref. [21]).

3.1 Carrier injection, transport, and recombination in the active region

Simulation of carrier transport and recombination and spontaneous photon emission was done within 1D approximation using SiLENSe 6.3 simulator, which is a part of the SimuLED package. The simulator implements an isothermal drift-diffusion model [22] with quantum-mechanical corrections accounted for by a quantum (effective) potential [23]. The use of the corrections was found to be necessary for a more realistic prediction of the electron leakage into the p-side of the LED structure at high currents. Occupation of all three conduction-band valleys (Γ, X, and L) was considered for an accurate prediction of the electron density in direct/indirect AlGaInP alloys; possible ordering of the alloys was neglected. Mobilities of electrons and holes in every particular layer of the LED structure were borrowed from Refs. [24,25].

Three main recombination channels are accounted for in the model: non-radiative Shockley-Read-Hall (SRH) recombination, spontaneous radiative recombination, and non-radiative Auger recombination. CHCC-process was assumed to dominate over other microscopic Auger processes, providing the recombination coefficient C_n = 10⁻³⁰ cm⁶/s [25] (C_p ≈ 0). As the square of matrix element between the ground-state electron and hole wave functions in each GaInP QWs was found to exceed 0.95, the radiative recombination constant B was approximated by its bulk value found by extrapolating the data on B-constants from Ref. [26] to the bandgap of Ga_0.51In_0.49P. The extrapolation provided the value B = 5.2×10⁻¹⁰ cm³/s at room temperature.

SRH lifetimes of electrons and holes, τ_n and τ_p, were supposed to be equal to each other and estimated by fitting to the experimental EQE values at the onset of its saturation occurring in the current density range of 0.5–10 A/cm² (see Fig. 3(a)). For this purpose, the theoretical EQE was determined as the product of experimental LEE reported in Refs. [15,16] and internal quantum efficiency (IQE) calculated within 1D approximation (dash-dotted line in Fig. 3(a)). The fitting provided the carrier lifetimes of 80 ns. One can see from Fig. 3(a) that EQE calculated in such a way overestimates remarkably EQE at the current densities j < 0.5 A/cm², which is caused by neglecting surface recombination at the intersection of the mesa sidewalls and the active region edges (see Sec. 3.2 for more detailed discussion).

 

Fig. 2. Room-temperature (25°C) band diagrams with carrier concentrations (a,b), radiative and non-radiative recombination rates (c,d), and partial current densities of electrons and holes (e,f) computed for two current densities: 7 A/cm² (a,c,e) and 160 A/cm² (b,d,f). Grey shadow marks the energy gap.

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Fig. 3. Comparison of the measured [15,16] EQE as a function of current density with those predicted by 1D simulation of IQE (thin dash-dotted line) and by 3D simulations without (thin blue line) and with (thick pink line) account of the LED self-heating (a); fractions of carriers lost by surface recombination (solid red line) and by electron leakage (dash-dotted blue line) (b).

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To calculate emission spectra of the GaInP QWs, Schrödinger equations for electrons and holes were solved by neglecting the valence sub-band mixing. Using the obtained electron and hole wave functions, the spectra were calculated in a conventional way, accounting for different electron and hole concentrations in every particular QW and assuming the uniform spectral broadening to be of 2 meV. Such a broadening has provided good agreement of the predicted spectrum widths with the experimental ones (see Sec. 4.4).

Numerous band-structure parameters, including deformation coefficients, band offsets between various materials, and mechanical properties necessary for simulation of LED structure operation were included in the internal database of material properties of the SimuLED package. The database was largely formed on the basis of the data collected in Refs. [24] and [27], using physics-based interpolation of the data for semiconductor alloys.

3.2 Current spreading, photon propagation and extraction, and heat transfer

Coupled current spreading in quasi-neutral contact layers and heat transfer were simulated in 3D approximation using the Ohm law and conventional heat-transfer equation, respectively [19]. At that, results of 1D simulations discussed in Sec. 3.1 served as non-linear boundary conditions for the current-spreading problem, on the one hand [19], and as the source of non-equilibrium carriers for 2D analysis of lateral carrier diffusion and surface recombination in the LED active region, on the other hand [28].

Thermal conductivities of dielectrics and metals were borrowed from Ref. [29] whereas those of semiconductors are taken from Ref. [24]. The heat was assumed to be released through the whole upper surface of the LED die (see Fig. 1(b)) and modelled by a unified heat-transfer coefficient irrespective of a particular type of the surface piece. Its value of 3 W/cm²·K was chosen as that reproducing best of all the experimental EQE droop observed at high currents at CW LED operation (see circles in Fig. 3(a)). The lateral carrier transport in the active region was modelled using the electron and hole diffusion coefficients estimated via Einstein’s relationship from the bulk carrier mobilities in GaInP borrowed from Ref. [24]. The surface recombination velocity (SRV) of 1500 cm/s and the corrected SRH carrier lifetimes τ_n = τ_p = 140 ns in the LED active region were obtained as those providing the best fit of the experimental EQE values in the current density range of 0.1-10 A/cm².

A common in-plane unstructured grid was employed for both electrical and thermal simulations. In order to resolve properly the current density distribution near/under the circular p-electrodes, a 3D grid with 241 000 cells was generated.

3D ray tracing was applied to estimate the optical losses and LEE in the LED die with account of light interference in the SiO₂/Ag multilayer reflector on the top of the die. Here, non-uniform distribution of photon emission intensity over the active region obtained from the 3D simulation of current spreading and heat transfer was taken into account. The principal mechanism of optical losses in the bulk of n- and p-contact layers was free-carrier absorption with the absorption coefficients α_n and α_p estimated from the respective carrier mobilities µ_n and µ_p [30]:

4. Results

4.1 Operation of LED structure

In order to provide necessary input data for 3D coupled electrical-thermal-optical modeling, 1D isothermal simulations of LED structure operation were carried out in the temperature range of 300–700 K with the step of 10 K. The results corresponding to 298 K (25°C) are presented in Fig. 2.

Figures 2(a) and 2(b) compare room-temperature band diagrams and distributions of non-equilibrium carrier concentrations in the LED structure at low, 7 A/cm², and high, 160 A/cm², current densities. The conduction or valence band edges correspond here to the valley/valence sub-band with the minimum/maximum energy, respectively. One can see the conduction and valence band profiles in the MQW active region to be flat, except for the space-charge regions formed next to the confinement-layer interfaces. The QW profiles in both conduction and valence bands are very close to rectangular ones with the square of the overlap integral between the ground-state wave functions of electrons and holes exceeding 0.95. At low current density, a non-uniformity in the concentrations of injected electrons and holes is predicted near the active region borders, which becomes less pronounced at high current density. The carrier concentration varies between 2×10¹⁷ and 8×10¹⁷ cm⁻³ in most of QWs at the current density ranged between 7 and 160 A/cm². At such concentrations, Auger recombination produces a negligible contribution to the non-radiative recombination, unlikely to InGaN-based LEDs. Therefore, the IQE of the quantum wells is entirely controlled by competition of the radiative and non-radiative SRH recombination channels.

Distributions of radiative and non-radiative recombination rates at low and high current densities are shown in Figs. 2(c) and 2(d), respectively. Despite the aforementioned non-uniformities in the carrier concentrations, the recombination rates are rather uniformly distributed among the QWs, so that each of them produces nearly the same contribution to the overall photon emission rate.

At room temperature, computed distributions of partial electron and hole current densities (Figs. 2(e) and 2(f)) demonstrate negligible electrons leakage into the p-side of the LED structure irrespectively of the operating current density. The dependence of EQE on the current density estimated from the isothermal 1D simulation (dash-dotted line in Fig. 3), best of all reproducing the onset of the efficiency saturation with τ_n = τ_p = 80 ns, predicts no EQE droop up to the current density of about 1 kA/cm². This means that the p-AlInP confinement layer provides a barrier for electrons, which is sufficiently high to suppress their leakage into p-layers. The barrier is formed by a negative charge of ionized acceptors induced in the space-charge region of the p-AlInP layers next to the active region border. The electron leakage becomes significant, however, at elevated temperatures, which will be discussed below in detail. Therefore, the electron leakage is essentially a thermo-stimulated process closely related to the LED self-heating.

4.2 External quantum efficiency and carrier losses

Dependence of EQE on the mean current density is presented in Fig. 3(a). Three variants of simulations are shown in the figure in order to elucidate the roles of LED self-heating, surface recombination, and electron leakage: (i) 1D isothermal simulation of the heterostructure at 298 K, (ii) 3D isothermal (298 K) simulation of the heterostructure and current spreading in the LED chip, including the surface recombination, and (iii) 3D coupled thermal-electrical-optical simulation of both the heterostructure and LED chip. Experimental data from Refs. [15,16] measured at CW and sub-microsecond-pulse operation are presented for comparison where the current density is estimated as the ratio of the LED operating current to the total chip area. In order to estimate EQE from the 1D computations of IQE, we have applied the value of LEE recommended in Ref. [15].

One can see from Fig. 3(a) that 1D simulations do not fit properly the experimental EQE at low current densities where thermal effects are out of play. The reason for this is neglecting surface recombination at the active region edges. This means, in particular, that the surface recombination cannot be accounted for by a simple shortening of the electron and hole lifetimes, i.e. regarding it as an additional channel of SRH recombination. Account of surface recombination in 3D simulations has required to specify the surface recombination velocity and to revise the electron and hole lifetimes initially determined in Sec.3.2. This enabled a much more close agreement between the coupled simulations and experiment (see Fig. 3(a)).

Dramatic difference in the EQE droop at high currents observed at CW and sub-microsecond-pulse operation revealed a crucial role of self-heating in limiting the overall LED efficiency [15,16]. Coupled simulation of the current spreading and heat transfer has confirmed the importance of this factor. Comparison of the data obtained at pulsed operation with simulations carried out with and without account of the LED self-heating suggests that even reduction of the current pulse duration below 1 µs does not avoid completely the thermal effect (see Fig. 3(a)). This conclusion agrees with that made in Refs. [15,16] on the basis of optical characterization.

The carrier losses caused by surface recombination and carrier leakage into p-side of the LED structure obtained by coupled 3D simulations are shown in Fig. 3(b) as a function of the current density. One can see that surface recombination leads to almost 100% losses at the current densities lower than 0.02 A/cm², as the carrier diffusion length is longer than the active region width in a single stripe. Increase in the current density, i. e. in the carrier concentrations in the active region, shortens the carrier lifetime thus decreasing their diffusion lengths. As a result, carrier losses by surface recombination decrease rapidly and become less than 1% at the current density of about 200 A/cm². In contrast, the losses caused by thermo-stimulated electron leakage are as low as 3–4% up to the current density of 10 A/cm² and then grow rapidly up to 40% at the current density of about 200 A/cm². The onset of the leakage rise corresponds to the onset of the dramatic EQE droop (see Fig. 3(a)), emphasizing the thermal enhancement of the leakage.

4.3 J-V characteristics, current spreading, and self-heating

Simulations of current-voltage characteristic with and without device self-heating are practically indistinguishable from each other, at least, up to the current densities of 100 A/cm², which is in line with the experimental data (Fig. 4(a)). They fit quite accurately the experimental points at the current densities j > 10⁻³ A/cm². At lower j, the simulations underestimate the current density at a given voltage. There are three possible mechanisms capable of generating such an extra current: (i) surface leakage, (ii) shunting of p-n junction by extended defects like threading dislocations, and (iii) trap-assisted tunnelling of carriers. It is interesting that j = 10⁻³ A/cm² is just the current density at which the built-in electric field in the active region vanishes and the band alignment becomes nearly flat, providing domination of the injection current in the LED structure.

 

Fig. 4. Experimental and simulated current density-voltage characteristics of the LED (a); 2D distributions of the total sheet carrier concentrations in the active region at the mean current densities of 1 A/cm² (b), 10 A/cm² (c), and 100 A/cm² (d). In every case, one third of the whole stripe is shown, adjacent to the left edge of the LED die.

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At the current density greater than 10 A/cm², the contact resistances at the metal/semiconductor interfaces become a key factor controlling the total series resistance of the LED chip. The n-contact resistance of ρ_n = 1×10⁻⁵ Ω·cm² was chosen in accordance with the data reported for GaInP [31]. Typical resistances of contacts formed to p⁺-GaAs (ρ_p) are well below 1×10⁻⁵ Ω·cm² [32]. Attempts to simulate the current density-voltage (J-V) characteristics of the red LED with such ρ_p resulted in underestimating the total series resistance. Only when ρ_p was increased up to 2×10⁻³ Ω·cm², our simulations provided a good agreement with the experimental J-V curve (see Fig. 4(a)). A possible explanation attributes the high ρ_p to the small thickness of the p⁺-GaAs contact layer (see Sec. 4.5), which is beneficial for minimizing the optical losses in the layer but may be non-optimal for p-contact formation, normally exploiting thermal annealing. Alternative explanations, invoking (i) a much lower carrier density in the space-charge regions formed near the abrupt interfaces of the contact layers and (ii) considerable reduction of electron concentration in the n-AlGaInP contact layer caused by DX-centre formation, were excluded by special simulations.

Two-dimensional distributions of the effective sheet (2D) carrier concentration in the LED active region, which is the geometric mean of the electron and hole sheet concentrations [28], is shown in Fig. 4 for different current densities. At low current density of 1 A/cm² (Fig. 4(b)), the sheet concentration is quite uniformly distributed over the whole active region except for the narrow border adjacent to the stripe edges whose width nearly corresponds to the carrier diffusion length. At the intermediate current density of 10 A/cm² (Fig. 4(c)), the current crowding leads to localization of carriers under circular p-electrodes. Further increase of the current density up to 100 A/cm² (Fig. 4(d)) results not only in the localization enhancement but also in the development of a longitudinal non-uniformity in the carrier concentration caused by a noticeable variation of the local p-n junction bias and temperature along the stripe. Such a non-uniformity originates from different conditions of the heat release from the LED die in its center and on the periphery.

Predicted maximum and mean temperatures in the LED active region are plotted in Fig. 5(a) by lines. One can see that their difference is rather small compared to the mean value, which is the evidence for a high uniformity of the temperature distribution in every stripe of the LED chip. The experimental data shown in Fig. 5(a) were derived from the temperature-dependent shifts of the LED emission spectra [15]. It is seen that simulations slightly overestimate the device self-heating, especially at high currents. This can be explained by a simplified model of the heat removal from the LED chip where 3D heat transfer in the heat sink is not regarded and the heat release is modeled by a unified heat-transfer coefficient.

 

Fig. 5. Mean and maximum temperatures of the active region (a), peak wavelength (b), and FWHM of the emission spectra (c) as a function of the mean current density. Lines are simulations with and without self-heating and, symbols are experimental points.

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4.4 Spectral parameters of LED

The heat transfer coefficient is fitted in our simulations in such a way, as to reproduce the experimental onset of the EQE droop at CW operation (see Sec. 3.2), which also provides a good agreement between the predicted and experimental peak wavelengths of the emission spectra as a function of the current density (Fig. 5(b)). The agreement is not so good at j = 2–20 A/cm² where simulations do not reproduce a small red shift of the wavelength observed experimentally. The reason for the discrepancy is neglecting in the simulations the many-body bandgap shrinking at a high concentration of carriers injected into the active region, which was discussed earlier in Ref. [16]. Isothermal simulations made for 25°C predict at high currents a small blue shift of the emission spectra caused by the Burstein-Moss effect. It is interesting that the peak wavelength measured at sub-microsecond pulse LED operation takes an intermediate position between the simulation curves obtained with and without account of the device self-heating. This confirms once again the conclusion that the sub-microsecond current pulse duration is insufficient for complete avoiding of the self-heating.

Simulated and experimental dependencies of the spectral full width at half maximum (FWHM) on the current density are compared in Fig. 5(c). A reasonable agreement between them is observed either at low, less than 0.2 A/cm², or at high, greater than 80 A/cm², current densities. In the intermediate current-density range, the experimental FWHM is remarkably larger than the theoretical one. The discrepancy may be explained by attributing the spectral broadening to the thermal effects only in our model. On the other hand, additional mechanisms may provide a comparable contribution to the broadening of emission spectra. One of such mechanisms, which seems to be valuable, is filling the density-of-state tails originated from the width/composition fluctuations in the GaInP QWs.

4.5 Light extraction and emission pattern

Figure 1 shows the most part of the LED die to be covered by an insulating SiO₂ film with highly reflective bulk metal (Ag) deposited on top of the film. In order to diminish optical losses caused by incomplete reflection of emitted photons from the SiO₂/Ag reflector, it is necessary to optimize the thickness of the SiO₂ film.

Calculated reflection coefficients of transverse-electric (TE-) and transverse-magnetic (TM-) polarized light as a function of the incident angle are plotted in Figs. 6(a) and 6(b) for various SiO₂ thicknesses. One can see that the reflectivity of TE-polarized light (R_TE) is always greater than 90%, whereas that of TM-polarized light (R_TM) exhibits a deep downfall mainly at the angles corresponding to the onset of total internal reflection from the p-AlGaAs/SiO₂ interface. In order to characterize the reflectivity by a unified parameter, we have introduced an averaged reflection coefficient (R_av) obtained by averaging the incident-angle dependent R_TE and R_TM over all the angles and polarization type (the weight function, $\sin \theta $, corresponds to the assumption on a uniform distribution of the ray directions over the solid angle):

 

Fig. 6. Reflection coefficients of TE- (a) and TM- (b) polarized light from SiO₂/Ag reflector as a function of photon incident angle calculated at various SiO₂ thicknesses and averaged reflection coefficient vs. thickness of SiO₂ film (c).

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Figure 6(c) shows that R_av exhibits a local (about 98%) and an absolute (about 99%) maxima at the thicknesses of the SiO₂ film of 100 and 1000 nm, respectively. The deposition of a 1000 nm dielectric film seems to be unsuitable from the technological point of view. Therefore, the thickness of 100 nm, providing rather high R_av can be regarded as an optimal solution, minimizing optical losses caused by incomplete reflection of emitted photons. So, further simulations were carried out with this SiO₂ thickness.

A typical layout of optical losses in the LED die obtained by ray tracing is given in Fig. 7(a). One can see that strong light absorption in the p⁺-GaAs contact layer (more than 50% of emitted photons) is the dominant mechanism of losses. Since the actual thickness of the layer was unknown, it was varied in such a way, as to reproduce the experimental maximum EQE value. As a result, LEE of 31.3% has been obtained at the thickness of 11.5 nm, being in close agreement with the value of 32.3% estimated in Refs. [15,16]. In order to estimate a possible room for LEE improvement, we have made a similar ray-tracing analysis, assuming (i) the absence of highly absorbing p⁺-contact layer and (ii) light extraction into a silicone resin with the refraction index of 1.7 instead of extraction into air.

 

Fig. 7. (a) Layout of optical losses in the LED die. (b) Computed far-field emission pattern of LED in various azimuthal directions. Grey line shows a Lambertian emission pattern for comparison.

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As a result, a LEE of 84.9% has been predicted. For comparison, the experimental LEE of an AlGaInP red LED with the record EQE of 72% at 650 nm [12] is estimated to be of about 80%. These results demonstrate the availability of a large potential for LEE improvement by excluding the highly absorbing p⁺-GaAs contact layer from the LED structure. In practice, a transparent p-GaP/p⁺-GaP window layer is frequently used for this purpose instead of p-AlGaAs/p⁺-GaAs one [33]. The Ohmic contact is formed to p⁺-GaP either directly or via an intermediate ITO [33], AZO [34], or graphene [35] spreading layer. Despite some increase in the p-contact resistance, the overall improvement of the LED efficiency due to removal of the absorbing p⁺-GaAs contact layer is feasible in those cases.

The predicted far-field emission pattern exhibits a weak azimuthal anisotropy at the observation angles smaller than 30° (Fig. 7(b)), which is attributed to the assumed texturing of the back n-AlGaInP contact-layer surface with a regular-pyramid array. In practice, random texturing is usually exploited. Therefore, no anisotropy should actually be observed in the actual emission pattern. Despite the azimuthal anisotropy, all the emission patterns can be well approximated by a Lambertian one (see Fig. 7(b)). This fact is in agreement with the datasheets of red LEDs packaged in the Golden Dragon cases [36].

5. Conclusion

In this paper, coupled electrical-thermal-optical simulations of an AlGaInP-based red (622 nm) LED packaged in the Golden Dragon case are presented. In the lack of detailed information on the LED structure and chip design, the use of plausible ones combined with adjustment of a number of recombination and thermal parameters enabled quite reasonable fitting of the numerous characterization data, published in Refs. [15,16] and some others. The good agreement between the theory and experiment attained simultaneously for current-voltage characteristics, efficiency as a function of current density, and thermal and spectral data indicated the principal specific features of the LED to be caught properly by its plausible designs. As a result, the simulation model allowed identification of the most critical factors limiting the LED performance.

The thickness of the p⁺-GaAs contact layer absorbing a considerable portion of the emitted photons is found to be the principal factor limiting LEE and, thus, EQE of the LED. With the chosen p⁺-GaAs thickness of 11.5 nm, the predicted efficiency of light extraction from the LED die into air is of 31.3%. In our opinion, there is a big room for the LEE improvement. In particular, excluding the p⁺-GaAs layer from the LED structure enables the LEE increase up to 84.9%, if the light is extracted into a silicone resin with the refractive index of 1.7. Such a big potential improvement has become possible due to (i) application of embedded micro-reflectors in the chip design and (ii) optimization of the dielectric film thickness in the Ag/SiO₂ micro-reflector. In practice, substitution of p-AlGaAs/p⁺-GaAs window/contact layers with p-GaP one has been already tested, though a certain optimization of the p-contact resistance is still required.

The next factor limiting EQE at high currents is the thermally stimulated electron leakage. Our simulations have demonstrated that the leakage is negligible at room temperature but it increases substantially with temperature. So, the problem of heat-sink optimization is quite critical for further EQE improvement.

Because of relatively small lateral dimensions of individual mesa stripes in the LED chip and large ambipolar diffusion length of carriers, the negative impact of surface recombination at the active region edges on the LED efficiency is found to be rather important at low currents. Based on simulations, our estimation of SRV on the SiO₂/GaInP QW interface has provided the value of 1500 cm/s, which is considerably lower than 2–5×10⁴ cm/s reported for GaInP surfaces non-passivated and passivated by a (NH₄)S_x treatment [37,38]. This means that passivation of the active region free surface with SiO₂ dielectric is well optimized in the actual red LEDs. The conclusion on possibility of SRV reduction on the dielectric/active region interface is very important for development of AlGaInP micro-LEDs. However, we should note that such a conclusion is based on indirect estimates and its confirmation by additional experiments would be quite desirable.

One more result also important for AlGaInP micro- and mini-LEDs is that the surface recombination is just the factor substantially reducing the low-current emission efficiency. As our simulations have shown, the effect of surface recombination cannot be accounted for by a simple reduction of the SRH electron and hole recombination lifetimes. The reason for that is a non-linear dependence of the total carrier lifetimes on their concentration in the LED active region. As the total lifetimes control the carrier movement to the active region edges via diffusion lengths, they affect substantially the integral surface-recombination rate. Therefore, only a combined analysis of the current spreading in the LED die, injection of electrons and holes into the active region followed by their drift/diffusion to its edges is capable of correct accounting for the surface recombination effects. On the other hand, surface recombination is considerably suppressed at high-current density operation due to shortening of the carrier diffusion length. So, surface recombination is expected to contribute little to the carrier losses in micro-LEDs operating at high currents, like in communication systems.

Our simulations have revealed a strong current crowding near circular p-electrodes occurring at high currents. This leads to the current density localization under and around the electrodes, which enhances, in turn, the electron leakage into p-layers of LED structure. In addition, longitudinal non-uniformity of the current density is produced in every stripe, caused by the LED self-heating dominated in the centre of the stripe. The thermally stimulated electron leakage is the reason for the negative impact of current crowding on EQE of LED and its droop at high currents.

Modelling of spectral LED characteristics, namely peak wavelength and FWHM of the emission spectra, has shown that the major factor affecting the characteristics is the device self-heating. A close correlation between the peak emission wavelength and mean temperature of the active region is revealed by simulations, justifying estimations of the LED temperature made in Refs. [15,16] using optical techniques. Accounting for self-heating enabled quite accurate prediction of the red shift of peak wavelength with the LED operating current. Less accurate are the theoretical results obtained for the FWHM of the emission spectra. A possible reason for this is neglecting in the simulation model many-body effects and extra spectral broadening originated from composition/width fluctuations in the GaInP QWs served as the active region material. Including the mechanisms into simulation model is a task for future research.