# Effect of Die Shape and Size on Performance of III-Nitride Micro-LEDs: A Modeling Study

^{*}

## Abstract

**:**

## 1. Introduction

^{2}was recently demonstrated for 10 × 10 μm

^{2}devices and light extraction to silicone with the refractive index of 1.41 [3]. Those μ-LEDs utilized profiled sapphire as the substrate for LED structures, and minimized the area of metallic electrodes on the emitting surface of the devices, in order to improve their LEE (see Figure 1a for schematic design of the μ-LED die).

## 2. Simulation Approach and μ-LED Design

_{i}(IQE) on the current density j, and (iii) a current density dependence of sheet carrier concentration n

_{2D}injected into the active region assumed to be nearly the same for electrons and holes. In order to obtain the LED structure characteristics j(U), η

_{i}(j), and n

_{2D}(j), direct simulations can be applied [10], provided that temperature-dependent recombination coefficients are known with sufficient accuracy. Below, we will show how the above dependences can be extracted from characterization data of large-size LEDs, where the impact of surface recombination on LED characteristics is considered as negligible.

#### 2.1. Approximation of LED Structure Characteristics

_{f}is the forward voltage, j is the current density, i.e., the ratio of the LED operating current to the area of the active region, ρ

_{S}is the specific series resistance, m is the ideality factor, k is the Boltzmann constant, q is the elementary charge, and ${j}_{S}(T)={j}_{0}\mathrm{exp}\text{\hspace{0.17em}}(-{E}_{a}/mkT)$ is the saturation current density with temperature-independent specific current density j

_{0}and activation energy E

_{a}.

^{2}MQW LED reported in [18]. Figure 2a demonstrates excellent fitting of the experimental characteristic corresponding to 300 K by Equation (1) and shows the parameters j

_{S}, ρ

_{S}, and m extracted from the fitting. Processing of the data reported in [18] for temperatures varied from 200 K to 440 K, provides the ideality factor m = 1.8 ± 0.1, which is practically constant in the above temperature range. Also, the saturation current density j

_{S}varies exponentially with the prefactor j

_{0}= 5 kA/cm

^{2}and activation energy E

_{a}= 3.06 eV. The activation energy is higher than the energy gap of InGaN QW in the LED active region, but it is lower than the bandgap of GaN barriers cladding the QWs.

_{i}on the current density j is derived by using the ABC recombination model considering three main recombination channels: SRH non-radiative recombination with the coefficient A, radiative recombination with the coefficient B, and Auger recombination with the coefficient C. The model neglects the electron leakage from the LED active region to p-layers of the LED structure, and assumes the concentrations of electrons and holes in the active region to be nearly equal to each other. Within the ABC-model, current density and IQE of an LED can be calculated by the expressions [19,20]:

_{out}at a certain current to the power P

_{m}at the EQE peak, j

_{m}is the current density corresponding to the EQE peak, and Q is the dimensionless combination of the recombination constants A, B, and C. Using conventional characterization data, i.e., dependence of EQE on the LED operating current, one can directly obtain P

_{m}and j

_{m}, and extract Q-factor by a special procedure suggested in [18]. The data of [18] provide the following approximations for the blue MQW LED:$Q(T)={Q}_{0}\mathrm{exp}(-T/{T}_{Q})$, where Q

_{0}= 110 and T

_{Q}= 134 K, and ${P}_{m}(T)={P}_{0}\mathrm{exp}(T/{T}_{P})$, where P

_{0}= 0.17 mW and T

_{P}= 67 K. Using them, it is possible to estimate the current density j

_{m}by the following expression: ${j}_{m}(T)={Q}^{-1}(Q+2){P}_{m}/{\eta}_{ext}{E}_{ph}S$, where S = 0.01 cm

^{2}is the active region area, E

_{ph}= 2.74 eV is the mean energy of emitted photons, and η

_{ext}is LEE, linearly decreasing from 71% to 68% as the temperature varied from 200 K to 440 K [18]. Finally, the IQE dependence on j corresponding to a particular temperature can be obtained from Equation 2 by varying the normalized power p in an appropriate range. It has been shown in [18] that Equation (2) fits, very accurately, the experimental EQE dependence on operating current/output power of the LED in a wide range of its variation.

_{2D}, injected in the LED active region, can also be obtained within the ABC-model. In terms of the variables discussed above,

_{m}and Q. The temperature dependence of A-coefficient has been reported recently for blue and green LEDs [15]; both data can be well approximated by the expression $A(T)={A}_{0}\mathrm{exp}\text{\hspace{0.17em}}(T/{T}_{A})$

**,**where A

_{0}= 8000 s

^{−1}and T

_{A}= 65 K. Combination of Equations (2) and (3) provides a parametric dependence of n

_{2D}on the current density j.

#### 2.2. Evaluation of Surface Recombination Velocity

_{S}is known for InGaN materials with insufficient accuracy. In order to evaluate its value more accurately, we used the recent data [22] on the size-dependent effective SRH recombination coefficient A’, which is related to the perimeter $\mathcal{P}$ and area Σ of the LED active region, via an approximate equation valid for sufficiently small devices [10,23]:

_{S}= 7.5 × 10

^{3}cm/s. This value is 7.5 times higher than that assumed in our previous simulations [10]. Due to the lack of data on the temperature dependence of surface recombination velocity, we used the above value in the whole range of temperature variation.

#### 2.3. Design of μ-LED Dice and Material Parameters

_{b}, a lateral size of InGaN-based active region d

_{AR}, a lateral size of the p-electrode d

_{p}, a mesa depth M, and a width of the n-contact w

_{n}. Variation of the mesa facet inclination angle θ, used in our simulations, did not change the dimensions of active region (d

_{AR}) and μ-LED die (d

_{b}, d

_{p}, and w

_{n}). The thicknesses of the n-GaN contact layer h

_{n}= 5 μm, and p-GaN contact layer h

_{p}= 0.2 μm, were also fixed in the simulations.

_{p}= 1 μm, d

_{AR}= 5 μm, d

_{b}= 15 μm, and w

_{n}= 3 μm, and large μ-LEDs with those dimensions enlarged by a factor of four, i.e., d

_{p}= 4 μm, d

_{AR}= 20 μm, d

_{b}= 60 μm, and w

_{n}= 12 μm. Besides, in the case of small μ-LEDs, we considered two kinds of mesas: a shallow mesa with M = 1.0 μm and a deep mesa with M = 2.4 μm. The size of p-electrode was intentionally chosen much smaller than d

_{AR}, in order to prevent carriers injected into the LED active region from surface recombination at its edges.

_{D}= 1×10

^{19}cm

^{−3}(ionization energy of donors is 13 meV), and to have a temperature-independent mobility of 100 cm

^{2}/V·s. The p-GaN contact layer was assumed to be doped with Mg up to the acceptor concentration N

_{A}= 3×10

^{19}cm

^{−3}(ionization energy of acceptors is 170 meV), and to have the hole mobility decreasing with temperature from 10 cm

^{2}/V·s at 300 K to 6 cm

^{2}/V·s at 600 K. Temperature dependence of both electron and hole concentrations in the contact layers was accounted for in our simulations. The specific resistances of n- and p-contacts were chosen to be 10

^{−5}Ω·cm

^{2}and 10

^{−4}Ω·cm

^{2}, respectively.

_{2}insulating film, respectively, corresponding to the emission wavelength of 453 nm. The main substrate was assumed to be removed after growth of LED structure. No intentional surface texturing was assumed on free semiconductor surfaces, including back surface of the n-GaN contact layer, so that the surfaces provided Fresnel reflection of light dependent on its polarization.

^{5}W/K·m

^{2}. Temperature-independent heat conductivity of 120 W/K·m was chosen for all GaN-based contact layers.

## 3. Results

#### 3.1. Optimization of Insulating Layer Thickness

_{2}. This film influences, remarkably, the reflectivity of light emitted from the active region. Figure 5 shows the reflection coefficients of photons with the wavelength of 453 nm versus their incident angle calculated for SiO

_{2}films of various thicknesses placed on bulk Ag, which is the basic material of n-electrode. The calculations were carried out for two polarizations of the incident light: transverse-electric (TE) and transverse-magnetic (TM), accounting for interference of photons reflected from the SiO

_{2}/GaN and Ag/SiO

_{2}interfaces. At small SiO

_{2}thickness (d

_{SiO2}= 50 nm), the reflectivity of TE-polarized light is higher than 90% in the whole range of the incident angles. However, there is a remarkable dip in the reflectivity of TM-polarized light at the angles close to the Brewster angle of silver. As photons traveling inside the μ-LED die change their incidence angle and polarization from one reflection to another, the above dip results in higher optical losses caused by incomplete reflection and, eventually, in a lower LEE.

_{2}film is increased up to the value, comparable with the light wavelength in SiO

_{2}, i.e., about 305 nm (see Figure 5), the dip magnitude is reduced considerably at the expense of slight reduction in the reflectivity of TE-polarized light. This occurs due to a constructive interference of TM-polarized photons reflected from SiO

_{2}/GaN and Ag/SiO

_{2}interfaces. As a result, LEE of the μ-LED dice with thicker SiO

_{2}film tends to rise. Increasing the thickness of the film above 250 nm no longer leads to further LEE improvement. Therefore, d

_{SiO2}= 250 nm has been regarded as the optimal value permanently used in subsequent simulations.

#### 3.2. Optimization of μ-LED Die Shape

^{19}cm

^{−3}provides, here, the room-temperature free-electron absorption coefficient of 10.7 cm

^{−1}. The losses by this mechanism consume from 10% (at optimal shape of the die) to 25% of emitted photons, depending on particular μ-LED design, with a rather weak dependence on the inclination angle.

#### 3.3. Current Density-Dependent Light Extraction Efficiency

_{AR}= 5 μm, M = 2.4 μm, and θ = 13° for small, and d

_{AR}= 20 μm, M = 2.4 μm, and θ = 8° for large devices.

_{ext}), with the mean current density, can be well approximated by the function

_{0}is the low-current LEE, Δη is the magnitude of the LEE reduction at high currents, j

_{c}is the current density at which current crowding reduces LEE by the half-value of Δη, and γ is the specific exponent. We have found η

_{0}= 54.4%, Δη = 4.3%, j

_{c}= 230 A/cm

^{2}, and γ = 1.5 for small μ-LEDs, and η

_{0}= 42.6%, Δη = 8.3%, j

_{c}= 26 A/cm

^{2}, and γ = 0.9 for large μ-LEDs. Comparison of the parameters shows that current crowding starts to affect LEE at lower current densities, and produces a higher LEE reduction in large devices. This conclusion is in line with those made in our previous study on scaling μ-LED dimensions [10].

#### 3.4. Output Characteristics of μ-LEDs

^{2}. Analysis of the electric potential distributions in the μ-LED dice has shown that the major contribution to the specific series resistance of the devices comes from the p-contact. Indeed, because of the small area of p-electrode (see Section 2.3 for description of μ-LED design), the current density at the p-electrode is 25 times higher than the mean current density in the μ-LED active region. Therefore, at the specific contact resistance of 10

^{−4}Ω·cm

^{2}assumed in our simulations, this provides the μ-LED-specific series resistance of 2.5 mΩ·cm

^{2}, which is slightly less than the value estimated directly from the current density–voltage characteristics shown in Figure 8a.

^{2}. Below 1 kA/cm

^{2}, the thermal effects, including the thermal efficiency droop, may be regarded as negligible.

## 4. Discussion

^{2}) and 29.8% (achieved at the current density of 50.2 A/cm

^{2}), respectively. In the case of light extraction to silicone, peak EQE values become equal to 43.1% for large, and 40.4% for small μ-LEDs. These values are close to each other, due to the fact that the remarkable difference in IQEs of large and small μ-LEDs (see Section 3.4) is compensated by the opposite difference in their LEEs. It is important that the peak EQE of small μ-LED is achieved at the current density an order of magnitude higher than in the large device, which is advantageous for some μ-LED applications. Also, the EQE predicted for small μ-LEDs encapsulated with silicone is comparable with that reported in [3] for 10 × 10 μm

^{2}devices. This means that controlling the shape of μ-LED dice, as an approach to EQE improvement, is at least as effective as that based on the use of patterned sapphire substrates.

_{p}= 1.0 μm) p-electrode aimed at suppression of surface recombination by localization of current far from the active region edges. However, this results in a high specific series resistance of the μ-LED leading, eventually, to a lower WPE of the device. In order to understand whether it is possible to optimize the p-electrode dimensions, we have simulated characteristics of μ-LEDs with d

_{p}= 2.0 μm and 3.2 μm. The most important results of the simulations are shown in Figure 9.

^{2}to 0.33 mΩ·cm

^{2}. Comparison of WPEs of the μ-LEDs, given in Figure 9b, shows the following trends. First, the current density corresponding to WPE peak is shifted to higher values, and the peak WPE lowers at larger d

_{p}, which is evidence for higher carrier losses for surface recombination. On the other hand, high-current WPE grows when d

_{p}is enlarged. The latter effect can be explained by reduction of the μ-LED series resistance at larger d

_{p}, thus lowering the operating voltage of the device. Hence, the optimal p-electrode dimensions depend on desirable μ-LED operation conditions. If the device is assigned for operation at peak WPE, then the use of small p-electrode is preferable. By contrast, high-current operation of the μ-LED requires utilizing larger p-electrode sizes. In particular, μ-LEDs with d

_{p}= 3.2 μm are predicted to have WPE higher than in the case of d

_{p}= 1.0 μm, by a factor of two, at 1 kA/cm

^{2}(see Figure 9b).

^{2}and 20 × 20 μm

^{2}active regions, show that this is not the case. This can be clearly seen, in particular, from Figure 7, demonstrating essentially different LEE dependences on the current density for small and large μ-LEDs. Since the dependence originates from localization of the injected carriers under poorly reflective p-electrode enhanced with current, this effect can be unambiguously attributed to different current crowding in small and large devices. Another signature of the valuable current crowding is different self-heating of small and large μ-LEDs at high current densities, producing a corresponding thermal droop of the emission efficiency (see Figure 8b). Hence, current crowding still remains an important factor, which should be accounted for in further developments of efficient μ-LEDs.

_{S}= 7.5 × 10

^{3}cm/s, obtained in [22] for vertical side walls of the mesa likely formed by non-polar facets of wurtzite crystal. On the other hand, the use of small angles, θ, of the facet inclination, which is beneficial for high LEE, may alter the surface recombination velocity. Indirect evidence for this is a large scatter in the data on V

_{S}in InGaN and GaN, reported for various crystal orientations (see [21] for a more detailed literature review on this issue). Therefore, experimental investigations into the crystal orientation dependence of surface recombination velocity are quite desirable for future developments of μ-LEDs. If the dependence of V

_{S}on the orientation of the active region facets is found to be strong enough, the proper choice of the mesa-facet orientation/inclination may become an additional degree of freedom for optimization of μ-LEDs with embedded micro-reflectors.

## 5. Summary

^{2}active region at the parameters borrowed from characterization of real large-size LEDs. At the peak efficiency, the carrier losses in μ-LEDs caused by surface recombination do not exceed 15–20%. For light extraction to silicone, the projected total EQE approaches 43% (37% for bottom emission), which is higher than experimental EQEs reported to date. These results show that shaping of the μ-LED dice is an effective way for improvement of their emission efficiency.

^{2}predicted for optimized design of the μ-LED with 5 × 5 μm

^{2}active region, seems to be large enough for practical applications.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic μ-LED design from Ref. [3] where triangles indicate the features of the profiled sapphire substrate and insulating film serves as an omnidirectional reflector; (

**b**) Schematic design of a flip-chip μ-LED with internal micro-reflector and removed substrate. Blue arrows show selected pathways of emitted photons.

**Figure 2.**(

**a**) Current–voltage characteristic of a blue LED at 300 K reported in [18] (symbols) and the solid line presents its fitting by Equation (1); (

**b**) Dependence of saturation current density j

_{S}on inverse temperature (symbols) and its approximation by an Arrhenius curve (line).

**Figure 3.**Dependence of the effective Shockley–Read–Hall (SRH) recombination coefficient A’ on the size of square-shaped blue μ-LEDs. Symbols are experimental points borrowed from Ref. [22], line is their approximation by Equation (4).

**Figure 4.**Schematic design of μ-LED chips considered in this study, with most important geometrical parameters: lateral size of the active region d

_{AR}, lateral dimension of the base d

_{b}, size of the p-electrode d

_{p}, mesa depth M, width of the n-contact w

_{n}, thickness of the n-GaN contact layer h

_{n}, and mesa facet inclination angle θ.

**Figure 5.**Reflectivity of the SiO

_{2}/Ag stack as a function of photon incident angle calculated for various thicknesses of SiO

_{2}film and different light polarizations: TE (dash-dotted lines) and TM (solid lines). The wavelength of light is 453 nm.

**Figure 6.**Light extraction efficiency (LEE) to the bottom hemisphere into air computed for small and large μ-LEDs with different mesa depths. Insets show, schematically, the shapes of the μ-LED dice and selected photon pathways corresponding to the first and second maxima of LEE.

**Figure 7.**LEE to the bottom hemisphere into air as a function of mean current density computed for small and large μ-LEDs, with M = 2.4 μm and optimized facet inclinations; symbols are simulation results, lines are approximations by Equation (5). Insets show 2D distributions of sheet carrier concentration in the active region of large μ-LED corresponding to low and high current densities indicated by arrows; inner squares correspond to the p-electrode positions. The color scale spreads from zero to a maximum concentration at every particular current density.

**Figure 8.**(

**a**) Current density–voltage characteristic; (

**b**) average temperature of the active region; (

**c**) EQE; (

**d**) wall-plug efficiency (WPE) as a function of mean current density simulated for small (solid lines) and large (dash-dotted lines) μ-LEDs of optimized shapes.

**Figure 9.**(

**a**) Current density–voltage characteristics; (

**b**) WPE dependence on the current density simulated for small μ-LEDs with p-electrodes of various lateral sizes.

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## Share and Cite

**MDPI and ACS Style**

Bulashevich, K.A.; Konoplev, S.S.; Karpov, S.Y.
Effect of Die Shape and Size on Performance of III-Nitride Micro-LEDs: A Modeling Study. *Photonics* **2018**, *5*, 41.
https://doi.org/10.3390/photonics5040041

**AMA Style**

Bulashevich KA, Konoplev SS, Karpov SY.
Effect of Die Shape and Size on Performance of III-Nitride Micro-LEDs: A Modeling Study. *Photonics*. 2018; 5(4):41.
https://doi.org/10.3390/photonics5040041

**Chicago/Turabian Style**

Bulashevich, Kirill A., Sergey S. Konoplev, and Sergey Yu. Karpov.
2018. "Effect of Die Shape and Size on Performance of III-Nitride Micro-LEDs: A Modeling Study" *Photonics* 5, no. 4: 41.
https://doi.org/10.3390/photonics5040041