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We investigate the dependence of various efficiencies in GaN-based vertical blue light-emitting diode (LED) structures on the thickness and doping concentration of the n-GaN layer by using numerical simulations. The electrical efficiency (EE) and the internal quantum efficiency (IQE) are found to increase as the thickness or doping concentration increases due to the improvement of current spreading. On the contrary, the light extraction efficiency (LEE) decreases with increasing doping concentration or n-GaN thickness by the free-carrier absorption. By combining the results of EE, IQE, and LEE, wall-plug efficiency (WPE) of the vertical LED is calculated, and the optimum thickness and doping concentration of the n-GaN layer is found for obtaining the maximum WPE.
©2013 Optical Society of America
1. Introduction
Recently, white light sources based on light-emitting diodes (LEDs) have attracted great attention as high energy-efficiency and environment-friendly lighting technologies [1–6]. Over the last decade, there has been considerable progress in the development of high-power and high-efficiency LEDs. However, the application of LEDs toward general lighting requires even higher brightness and efficiency. In the development of LEDs, there has been increasing demand on the high internal quantum efficiency (IQE) and light extraction efficiency (LEE), low operation voltage, and thermally stable operation. Recently, thin-film vertical LED structures have been introduced for high-power and high-efficiency lighting applications [6–10]. In the vertical LED structure, epitaxial layers are placed on a receptor substrate with high thermal conductivity. This LED structure is expected to be a prominent solution to overcome the limitations of current LED structures due to low thermal resistance, uniform current spreading, and high LEE.
The vertical LED structure is typically manufactured by bonding a conventional LED chip on a receptor substrate and subsequently removing the sapphire substrate of the LED chip by the laser lift-off process [7]. It has several advantages over the conventional planar LED structure with lateral current injection geometry. High thermal conductivity of the metal receptor substrate improves thermally-stable operation. Current is injected vertically between top and bottom contacts, which enables more uniform current distribution and reduced operation voltage. It has been reported that the uniform current spreading is also advantageous for reducing efficiency droop at high current density [11, 12]. Thus, relatively high IQE can be achieved at large current density by improving current spreading. In addition, the vertical LED is expected to show high LEE by combining the high-reflectance p-type electrode mirror and light extracting structures in the n-GaN surface such as surface roughening [13] and photonic crystals [14, 15].
Therefore, the vertical LED is expected to show high power-conversion efficiency or wall-plug efficiency (WPE) through the improvement in the electrical efficiency, IQE, and LEE. WPE () of LEDs is defined as
where Pout is optical output power emitted out of the LED. I and V are applied current and voltage, respectively.The electrical efficiency (), IQE (), and LEE () are respectively defined as
where is the average frequency of emitted light and e is the elementary charge. Pint is the optical power emitted from active region. By combining Eqs. (1) and (2), WPE is written asIn the vertical LED, the electrical efficiency (EE) and IQE can be increased by improving current spreading. Current can flow more uniformly as the thickness or doping concentration of the n-GaN layer increases. However, increasing the thickness or doping concentration may result in increased optical loss by the increasing contribution of the free carrier absorption, which leads to the decrease in LEE. Thus, in vertical LED structures, it is important to optimize the thickness and doping concentration of the n-GaN layer for obtaining the highest WPE. Although the thickness and doping concentration of the n-GaN layer may have significant influence on the various efficiencies of vertical LED structures, there have been few reports to study the effect of these parameters on LED efficiencies.
In this work, we systematically investigate the dependence of the efficiencies of InGaN/GaN vertical blue LED structures on the thickness and doping concentration of the n-GaN layer by using numerical simulations. For the simulation of EE and IQE, an advanced device simulation software, APSYS is employed [16]. The APSYS program self-consistently solves QW band structures, radiative and nonradiative carrier recombination, and the drift and diffusion equation of carriers. It has been widely used for the simulation of device characteristics of GaN-based LEDs. For the simulation of LEE, the finite-difference time-domain (FDTD) method is employed. The FDTD method is advantageous for simulating photonic structures with micro- or nano-scale patterns. Thus, it has been applied to the LEE simulation of LEDs with photonic crystal patterns or textured surface [17–20]. Finally, WPE in Eq. (3) is obtained by combining the results of EE, IQE, and LEE. Then, the dependence of WPE on the thickness and doping concentration of the n-GaN layer will be discussed.
2. Methods of simulation
2.1 Simulation of I-V and IQE curves
A cross section of the vertical LED structure for the APSYS simulation is schematically drawn in Fig. 1 . LED layer structures are basically composed of an n-GaN layer, multiple-quantum-well (MQW) active layers, a p-AlGaN electron-blocking layer (EBL), and a p-GaN layer. MQW layers consist of five 25-Å-thick In0.16Ga0.84N QWs separated by 80-Å-thick GaN barriers. All QW and barrier layers are undoped. The thickness and In composition at the QWs corresponds to the peak emission wavelength of ~0.45 μm. The thickness of EBL and the p-GaN layer is 20 and 150 nm, respectively. The Al composition of EBL is 15%. The hole concentration at both EBL and the p-GaN layer is set at 5 × 1017 cm−3. The dimension of the simulated vertical LED chip is 1 mm × 1mm, which is the typical dimension of high-power vertical LEDs operating with 1-W input power. The entire surface of the p-GaN is covered with a p-type electrode, which also acts as a high-reflectance mirror. N-type electrodes are formed on the surface of the n-GaN layer. Each n-type electrode has a stripe-shape with the dimension of 50 μm × 1mm, and the electrodes are equally spaced on the n-GaN surface.
In the carrier recombination model of APSYS, the Schockley-Read-Hall (SRH) recombination lifetime (τSRH) is set at 50 ns. The radiative recombination rate is calculated by integrating the spontaneous emission spectrum with a Lorentzian line-shape function. The Auger recombination in InGaN QWs has been one of the central issues in the debate of efficiency droop in InGaN LEDs [21–24]. The efficiency droop can be conveniently modeled by using the Auger recombination in the ABC carrier recombination model [25–28]. Theoretically calculated Auger recombination coefficients (C coefficients) by indirect Auger processes have been ~10−31 cm−3 [29, 30]. The C coefficient can be obtained by the fit of a measured efficiency curve, and the C coefficient of the order of 10−31 ~10−29 cm6/s has been reported experimentally [21, 23, 31]. However, no consensus has been reached yet regarding the Auger recombination coefficient. Here, we choose a recently reported value of 3.5 × 10−31 cm6/s as the C coefficient of the simulation [31].
IQE is defined as the ratio of radiative recombination current to total injection current. We also include built-in electric fields induced by spontaneous and piezo-electric polarizations at the hetero-interfaces of InGaN/GaN and AlGaN/GaN [32]. The strength of built-in electric fields at the interfaces of the InGaN QW and the GaN barrier is set at approximately 1 MeV/cm. Using these material parameters, the voltage versus current (I-V curve) and the IQE versus current (IQE curve) relations will be calculated when doping concentration of the n-GaN layer is varied from 5 × 1017 to 1 × 1019 cm−3 for three n-GaN thicknesses of 2, 3, and 4 μm. The electron concentration at the n-GaN layer is assumed to be the same as the doping concentration. In the simulation, electron current leakage from active layers to the p-GaN layer was not observed in all cases. Therefore, a possibility of IQE droop by the carrier overflow is excluded in this work.
2.2 Simulation of LEE
In the simulation of LEE, the three-dimensional (3-D) FDTD method based on the Yee’s algorithm with a perfectly-matched layer (PML) boundary condition is employed [33]. The cross sectional view of the FDTD computational domain is schematically drawn in Fig. 2 . LED layer structures are basically the same as those shown in Fig. 1. In vertical LEDs, textured patterns on the n-GaN surface are formed by wet chemical etching and typically show a cone-like shape. Thus, we introduce cone patterns on the surface of the n-GaN layer in the FDTD simulation. It is assumed that the bottom of each cone pattern is in contact with adjacent cones and the height of cones is the same as the bottom diameter of cones, and the cone patterns are formed as a periodic array of square lattice in the simulation. Here, the height and the bottom diameter of cones are fixed at 1.2 μm. In this dimension of cone patterns, LEE of vertical LEDs was found to be nearly the highest by separate simulations.
Fig. 2 Computational domain of the vertical LED structure for finite-difference time-domain (FDTD) simulations.
In the FDTD simulation, a single dipole source is positioned at the center of x-y plane of the computational domain and also in the middle of the MQW active region in the vertical direction. The dipole source is polarized in the in-plane direction for the excitation of transverse-electric (TE) polarized light that is primarily emitted radiation at InGaN QWs. In the spectrum of emitted light, the center wavelength and the full-width at half maximum of the spectrum are set to be 450 nm and 20 nm, respectively. LEE is defined as the fraction of emitted power out of the LED surface to the total emitted power, which is determined by the ratio of Poynting vectors integrated over the extraction surface to total integrated Poynting vectors. Owing to the limitation of computational capacity, the FDTD simulation domain is much smaller than the size of actual LEDs. Lateral dimension of the computational domain is only ~8 μm. In order to truncate the virtually infinite lateral dimension of actual vertical LED structures, perfect mirrors with 100%-reflectance have been placed along the four lateral boundaries of the simulated LED structure as shown in Fig. 2. This perfect-mirror configuration has been frequently employed for the LEE simulation of LED structures [18–20, 34, 35].
In vertical LEDs, a high-reflectance p-type electrode reflector exists below the p-GaN layer. In the simulation, the electrode reflector is assumed to be composed of a thin dielectric absorbing layer and a perfect mirror with 100% reflectance because using metal materials in the FDTD simulation increases computational burden considerably. By varying the absorption coefficient of the absorbing layer above the perfect mirror, reflectance of the electrode reflector can be adjusted. Here, the absorption coefficient of the absorbing layer is chosen so that the reflectance of the p-type electrode can be 95% for normal incidence. The refractive index of GaN, InGaN, and AlGaN layers is set at 2.52, 2.58, and 2.48, respectively. The LED chip is assumed to be encapsulated by an epoxy material with the refractive index of 1.5.
LEE in the vertical LED is strongly dependent on the absorption coefficient of materials. Many groups have investigated the absorption coefficient in the GaN-based materials. However, there have been large variations of reported values and it is difficult to know the actual values of the absorption coefficients. Liu et al. well summarized the absorption coefficients in GaN, InGaN, and AlGaN materials [36]. Here, the absorption coefficient of InGaN QW layers is chosen to be 1000 cm−1 according to Ref [36]. Total thickness of InGaN QW layers is 10 nm. Light absorption for sub-bandgap energy is mainly affected by defects, dopants, and the Urbach tail [36–38]. In early studies, the absorption coefficient of GaN was reported to be higher than 50 cm−1 [39–41], which is attributed to poor material quality. More recent studies reported lower absorption coefficient values of 1~20 cm−1 [42–44]. In highly-doped GaN, free-carrier absorption can be significant because it increases with increasing doping concentration. The absorption coefficient by the free-carrier absorption was reported to be 10~20 cm−1 depending on the doping concentration [38, 45].
In this work, we model the absorption coefficient of GaN and AlGaN layers as below.
where αb is the background absorption coefficient independent of doping concentration and αf is the concentration-dependent absorption coefficient by the free-carrier absorption. In the LEE simulation, we consider two cases of αb: high material quality (αb = 5 cm−1) and relatively low material quality (αb = 20 cm−1). In order to find αf in Eq. (4) as a function of the doping concentration, an empirical formula for the free-carrier absorption in GaN materials is used [38]. In Ref [38], a relation between the electron concentration, ne and the absorption coefficient at 0.6 eV, αf (0.6 eV) is expressed asBy solving Eq. (5), the absorption coefficient at 0.6 eV is obtained as a function of electron concentration. The absorption coefficient at 2.75 eV, which corresponds to the photon energy of blue light, can be known from αf (0.6 eV) by using the fact that the absorption coefficient by the free-carrier absorption is proportional to the square of light wavelength [37, 38]. In Fig. 3 , the absorption coefficient at 2.75 eV is plotted as a function of electron concentration. Figure 3 shows that αf increases from 3.4 to 17.7 cm−1 as electron concentration increases from 5 × 1017 to 1019 cm−3. This value of the absorption coefficient in the GaN layer is much lower than that in InGaN QW layers. However, since the n-GaN layer is much thicker than other epitaxial layers, the thickness and doping concentration of the n-GaN layer play an important role in optical absorption and hence LEE of the vertical LED. Using the model in Eqs. (4) and (5), LEE of the vertical LED will be calculated as the thickness and doping concentration of the n-GaN layer varies.Fig. 3 Absorption coefficient of blue light by the free-carrier absorption as a function of electron concentration.
3. Simulation results
3.1 EE and IQE
First, we numerically investigate the effects of the thickness and doping concentration of the n-GaN layer on carrier distribution in QW plane. As the thickness or doping concentration of the n-GaN layer increases, improvement in the current spreading is expected, which should result in more homogeneous distribution of carriers in the plane of QWs. Figure 4 shows the distribution of electron carrier concentration in the horizontal direction of the central QW in the MQW active region. In Fig. 4(a), electron concentration in the plane of the QW from 0 to 250 μm is shown for several doping concentration from 1018 to 1019 cm−3 when the thickness of the n-GaN layer is 3 μm. Electron concentration in the region below the n-electrode is flat and it decreases as the region becomes more distant from the electrode, showing nonuniform carrier distribution in the plane of QWs. The electron concentration in the QW plane becomes more uniform as the doping concentration increases from 1018 to 1019 cm−3. In Fig. 4(b), the electron concentration in the plane of the QW is shown for the n-GaN thickness of 2, 3, and 4 μm when doping concentration at n-GaN is 5 × 1018 cm−3. The electron concentration in the QW plane becomes uniform as the n-GaN thickness increases.
Fig. 4 (a) Electron concentration in the horizontal direction of the QW plane for several doping concentration from 1018 to 1019 cm−3 when the thickness of the n-GaN is 3 μm. (b) Electron concentration in the horizontal direction of the QW plane for the n-GaN thickness of 2, 3, and 4 μm when doping concentration at the n-GaN is 5 × 1018 cm−3.
The current spreading and the electron distribution in the plane of QWs may be dependent on the electron mobility of the n-GaN. In the simulation, the mobility model of Ref [46, 47]. was used, where the electron mobility of the n-GaN is ~500 cm2/Vs. We tested the effect of the electron mobility on the electron distribution in the horizontal direction in separate simulations, and found that the electron distribution curves in Fig. 4 were not affected appreciably unless the electron mobility was lower than 100 cm2/Vs. Therefore, it is expected that the uncertainty in the electron mobility does not have significant influence on the electron distribution, and hence the I-V and the IQE characteristics that will be shown below.
As the current spreading improves, series resistance in the n-GaN layer will decrease, which results in the decrease of the forward voltage. In Fig. 5(a) , I-V curves are shown for several doping concentration from 1018 to 1019 cm−3. Here, the n-GaN thickness is fixed at 3.0 μm. At sufficiently large injection current, forward voltage becomes to decrease as the doping concentration of the n-GaN increases due to the improvement in the current spreading. In Fig. 5(b), using the simulated data of I-V curves, EE at 350 mA is plotted as the doping concentration at the n-GaN increases from 5 × 1017 to 1019 cm−3 for three n-GaN thicknesses of 2, 3, and 4 μm. EE is observed to increase as the thickness or doping concentration of the n-GaN layer increases. EE at 350 mA is increased by ~1% as the doping concentration increases from 1018 to 1019 cm−3 for the three n-GaN thicknesses.
Fig. 5 (a) Voltage versus current relations (I-V curves) for several doping concentration from 1018 to 1019 cm−3 when the n-GaN thickness is 3 μm. (b) Electrical efficiency (EE) at 350 mA is plotted as a function of doping concentration at the n-GaN layer for three n-GaN thicknesses of 2, 3, and 4 μm.
Uniform carrier distribution in the QW plane can also be advantageous for reducing the efficiency droop in InGaN LEDs [11, 12]. It has been well known that inhomogeneous carrier distribution in MQW active region could lead to significant efficiency droop because of increased carrier loss at the region of high carrier density [48–50]. The concept of a uniform carrier distribution to reduce the efficiency droop effects can also be applied to the carrier distribution in the horizontal direction of QWs. In this sense, carriers should be uniformly distributed in the plane of QWs in order to reduce efficiency droop and hence to increase IQE at high current density [12].
Figure 6(a) shows IQE curves for several doping concentration from 1018 to 1019 cm−3 when the n-GaN thickness is 3.0 μm. The IQE curves exhibit typical efficiency droop behaviors: IQE begins to decrease substantially when injection current is larger than 150 mA. IQE droop becomes less significant as doping concentration of the n-GaN layer increases. Although the peak IQE value is almost the same for different level of doping concentration, the difference of IQE between the high and the low doping concentration becomes large as current increases. The reduction in the IQE droop for high doping concentration is attributed to the improvement in the uniformity of carrier distribution in the plane of QWs. In Fig. 6(b), IQE at 350 mA is plotted as doping concentration of n-GaN increases from 5 × 1017 to 1019 cm−3 for three n-GaN thicknesses of 2, 3, and 4 μm. IQE at 350 mA increases as the thickness or doping concentration of the n-GaN layer increases due to the reduction in the efficiency droop. IQE at 350 mA is increased by 1~2% as doping concentration increases from 1018 to 1019 cm−3 for the three n-GaN thicknesses. By further improving the current spreading and uniformity in carrier distribution in QWs through the optimization of LED structures, efficiency droop will be reduced and IQE at high injection current will be increased even more.
Fig. 6 (a) Internal quantum efficiency (IQE) versus current relations for several doping concentration from 1018 to 1019 cm−3 when the n-GaN thickness is 3 μm. (b) IQE at 350 mA is plotted as a function of doping concentration at the n-GaN layer for three n-GaN thicknesses of 2, 3, and 4 μm.
3.2 LEE and WPE
Then, 3-D FDTD simulation is performed to calculate LEE of vertical LEDs for different doping concentration at the n-GaN layer. Here, Eq. (3) and the data of free carrier absorption in Fig. 3 are used for obtaining the absorption coefficient of the n-GaN layer for the background absorption coefficient αb of 5 and 20 cm−1. In Fig. 7 , LEE is plotted as a function of doping concentration from 5 × 1017 to 1019 cm−3 for three n-GaN thicknesses of 2, 3, and 4 μm. Figures 7(a) and 7(b) show the results when αb is 5 and 20 cm−1, respectively. As the thickness or doping concentration of the n-GaN layer increases, LEE decreases significantly owing to the increased contribution of free-carrier absorption. The simulated LEE lies in 73~80% for αb of 5 cm−1 and in 68~75% for αb of 20 cm−1, respectively. This range of LEE well corresponds to the LEE of current vertical LEDs. When the material quality of GaN is low (αb = 20 cm−1), LEE is lower by ~5% compared with the case of high material quality (αb = 5 cm−1). When αb is 5 cm−1 and doping concentration is low, the difference in LEE between the n-GaN thickness of 2 and 4 μm is only ~2%. However, it is increased to 3~4% when αb is 20 cm−1 and doping concentration is high. This implies that LEDs with a thick n-GaN layer may undergo significant reduction in LEE when the background absorption and the doping concentration at the n-GaN layer is high.
Fig. 7 Light extraction efficiency (LEE) at 350 mA is plotted as a function of doping concentration at the n-GaN layer for three n-GaN thicknesses of 2, 3, and 4 μm when αb is (a) 5 and (b) 20 cm−1.
Finally, WPE in Eq. (3) is calculated by combining the results of EE, IQE, and LEE. As one can see from Figs. 5, 6, and 7, the dependence of EE, IQE, and LEE on the thickness and doping concentration was monotonic: both EE and IQE increases, and LEE decreases with increasing thickness or doping concentration. However, the dependence of WPE on structural parameters is not simple when these results are combined. In Fig. 8 , WPE at 350 mA is plotted as a function of doping concentration from 5 × 1017 to 1019 cm−3 for three n-GaN thicknesses of 2, 3, and 4 μm. Figures 8(a) and 8(b) show the results when αb is 5 and 20 cm−1, respectively. The range of WPE in the simulated range is 43 ~48.5%, which is similar to the WPE of high-performance vertical LEDs [3, 5].
Fig. 8 Wall-plug efficiency (WPE) at 350 mA is plotted as a function of doping concentration at the n-GaN layer for three thicknesses of 2, 3, and 4 μm when αb is (a) 5 and (b) 20 cm−1.
In Fig. 8, there exists the maximum value of WPE at specific doping concentration. The optimum doping concentration for obtaining the maximum WPE exists around 1~3 × 1018 cm−3. At low doping concentration of <1 × 1018 cm−3, WPE increases with doping concentration due to the rapid increase of EE and IQE with doping concentration as shown in Figs. 5 and 6. At high doping concentration of >3 × 1018 cm−3, WPE decreases with doping concentration because the influence of the decreasing LEE becomes more significant than that of the increasing EE and IQE as doping concentration increases. For αb of 5 cm−1, the maximum WPE for three n-GaN thicknesses is almost the same when the doping concentration is 1~2 cm−3. For αb of 20 cm−1, however, the WPE decreases as the n-GaN thickness increases, indicating the significant influence of the LEE on the WPE when the optical absorption at the n-GaN is large.
The results in Fig. 8 show that the maximum WPE exists when the n-GaN thickness is 2~3 μm and the doping concentration is 2~3 × 1018 cm−3. In addition, it does not seem to be desirable to increase the n-GaN thickness too thick or doping concentration too high at the n-GaN layer. In this way, the optimum values of the thickness and doping concentration of the n-GaN layer are found by combining the simulation results of EE, IQE, and LEE. However, these optimum values will change when different material and structural parameters are used in the simulations.
4. Conclusion
We theoretically investigated the dependence of various efficiencies in InGaN/GaN vertical blue LED structures on the thickness and doping concentration of the n-GaN layer. The electrical efficiency (EE) and the internal quantum efficiency (IQE) were calculated by the APSYS program and the light extraction efficiency (LEE) was calculated by 3-D FDTD simulations. As the thickness or doping concentration of the n-GaN layer increased, EE and IQE were found to increase due to the improvement of current spreading. However, LEE was decreased with increasing thickness or doping concentration as a result of increasing contribution of the optical absorption in the n-GaN layer. The wall-plug efficiency (WPE) of the vertical LED was calculated from the simulated results of EE, IQE, and LEE. In the simulated LED structure, the optimum thickness and doping concentration of the n-GaN layer for achieving the maximum WPE was found to be 2~3 μm and 1~3 × 1018 cm−3, respectively.
Acknowledgments
This work was supported by National Research Foundation of Korea Grant funded by the Korean Government (2011-0003376) and by the Ministry of Knowledge Economy (MKE), Korea, under the Information Technology Research Center (ITRC) support program supervised by the National IT Industry Promotion Agency (NIPA) (NIPA-2012-H0301-12-1010)
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