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An improved rate equation model for GaN-based LEDs considering the effective volume of the active region is proposed. Through numerical simulations, it is confirmed that the IQE, especially efficiency droop is related with small effective volume. Also, we confirmed that the effective volume is controlled by polarization charge, the barriers between the quantum wells, and current density. We also developed a fast and reliable method for extracting the recombination coefficients and the IQE of the GaN-based LEDs by measuring transient characteristics and considering the effective volume.
©2014 Optical Society of America
1. Introduction
During the last several years, GaN-based light emitting diodes (LEDs) have gradually substituted for conventional fluorescent lights in the fields of backlights of display and general lighting applications [1]. In order to reduce electric power consumption and increase light output power of the GaN-based LEDs, it is important to extract and analyze recombination coefficients and internal quantum efficiency (IQE) which is the relative probability of radiative recombination over the total recombination of injected carriers. There have been many researches to analyze the IQE of the GaN-based LEDs and explain the origin of ‘efficiency droop’ which is the decrease of the IQE when current density increases [2–10]. However, there is no consensus on this problem yet [11, 12].
In this paper, we propose the concept of the effective volume of the active region and apply the effective volume to the rate equation of the LEDs. In addition, the recombination coefficients and the IQE of the GaN based LEDs are extracted by analyzing transient characteristics with the rate equation considering the effective volume of the active region.
2. Effective volume of the active region
In order to increase radiative recombination of the electron-hole pairs, GaN-based LEDs have employed multiple quantum well (MQW) structure [13]. However, the carrier concentration and recombination process of the GaN-based LEDs are not uniformly distributed in the MQW. Because of the polarization of nitride-based materials, the wave-function of electron and hole is not overlapped exactly in the quantum well [14]. Furthermore, as the effective mass of hole is much larger than that of electron [15] and activation energy of Mg acceptor is high [16], most of the injected holes are concentrated in the single quantum well near the p-GaN layer. Because of these characteristics of nitride-based materials, most of the recombination processes are concentrated in the small volume of the active region. As carrier concentration severely affects the IQE of the LEDs, we have to consider carrier distribution in the active region when we analyze and calculate the IQE of the LEDs [17].
In addition, carrier and recombination distribution in the active region change when current density increases. Figure 1(a) shows normalized recombination rates in the five quantum wells of the GaN-based LEDs calculated by numerical simulation tools at current density of 2 A/cm2 and 200 A/cm2 [18]. In this simulation, the physical parameters of nitride-based materials are based on the research of I. Vurgaftman, et al [19]. Surface charges induced by spontaneous and piezoelectric polarizations are also considered [20]. Shockley-Read-Hall (SRH) recombination coefficient (A) and radiative recombination coefficient (B) are kept constant at 5.0 × 106 s−1 and 1.0 × 10−11 cm3s−1, which are in the range of generally accepted values [11]. In the case of Auger recombination coefficient (C), we used theoretically calculated value, 2.0 × 10−31 cm6s−1 [21]. The active region of the structure consists of five 3-nm-thick undoped In0.2Ga0.8N QWs and four 6-nm-thick n-GaN (n-doping = 2 × 1017 cm−3) barriers. The structure also includes 17.5-nm-thick p-Al0.2Ga0.8N (p-doping = 5 × 1019 cm−3) electron blocking layer (EBL) to avoid electron leakage from the MQW.
Fig. 1 (a) Calculated recombination rates (normalized) in five MQW of GaN-based LEDs at current density of 2 A/cm2 and 200 A/cm2. (b) Calculated electron and hole concentration in the quantum well near the p-GaN layer.
As can be seen in Fig. 1(a), the recombination process is more crowded in a very small volume when current density increases. In order to analyze this modification of the recombination profile, we confirmed electron and hole concentration in the quantum well near the p-GaN layer and plotted them in Fig. 1(b). When current density is low (2 A/cm2), both of electrons and holes are concentrated at the each side of the quantum well because of the polarization field, and this leads to uniform recombination rate in this quantum well. As current density becomes higher (200 A/cm2), electrons spread in the quantum well uniformly, but holes are still concentrated at the right side of the quantum well because of heavy effective mass. Therefore, carriers are concentrated at the right side of the quantum well and recombination rate, especially Auger recombination (Cnn2p + Cpnp2) caused by high hole concentration, increases abruptly in this small volume.
We defined and calculated this ‘effective volume’ which as the Eqs. (1) and (2).
The volume correction factor and the IQE of polar c-plane GaN-based LED having 5 MQW are calculated by a numerical simulation. As shown in Fig. 2, effective volume is reduced as current is increased. Because of the reduced effective volume of active region, carrier concentration becomes crowded and IQE is greatly reduced as Auger recombination increases. For comparison, the volume correction factor and the IQE of non-polar (m-plane) MQW and SQW structure are also calculated and plotted in Fig. 3. Both structures have the exactly identical physical volume of the active region with the polar (c-plane) MQW structure. In the case of non-polar MQW structure, since there is no polarization charge in the active region, the recombination process is distributed more uniformly in the single quantum well near the p-type GaN layer and the volume correction factor and the IQE are improved as shown in Figs. 3(a) and 3(b). Figures 3(c) and 3(d) show the volume correction factor, the IQE and recombination rate of non-polar SQW structure. As the barriers between quantum wells blocking the transport of holes are removed, holes are injected more efficiently into the active region. As a consequence, the volume correction factor and the IQE of non-polar SQW structure are greatly improved even though the volume correction factor could not reach 100% because of poor hole diffusion length. This result confirms that the IQE is influenced by the effective volume of the active region. In addition, the effective volume of the GaN-based LEDs can be changed by polarization charge, the barriers between quantum wells, and current density.
Fig. 2 Calculated volume correction factor and the IQE of polar c-plane GaN-based LED using 5 MQW structure.
Fig. 3 Effects of the polarization field and barriers of the MQW on the volume correction factor and the IQE. In order to confirm the effect of the polarization field, the calculated effective volume and the IQE of non-polar m-plane GaN-based LED using five MQW structure are analyzed (a), and the distribution of the recombination rate in the MQW is shown (b). To analyze the effect of the barriers, the same simulations are conducted for non-polar m-plane GaN-based LED using single quantum well structure (c), (d).
3. Optical characteristics of the GaN-based LEDs considering the effective volume of the active region
3.1 Extraction of recombination coefficients
In order to find the relationship between the effective volume of the active region and the recombination process of the carriers, the recombination coefficients are extracted by measuring transient characteristics of the optical power. Samples for the experiment were grown on c-plane sapphire substrates. Five InGaN/GaN QWs were grown, emitting light with a wavelength of 450 nm. The area of the active region was approximately 600 × 1000 μm2. The structure of the prepared sample is exactly identical with the simulated structure shown in Figs. 1 and 2.
The current continuity equation of carrier concentration injected into the active region can be written as Eq. (3a) while Eq. (3b) shows the solution of Eq. (3a). Since the initial carrier concentration (n0) can be calculated from Eq. (4) and the optical power is proportional to VactiveBn2, time-carrier concentration characteristics can be obtained by measuring the transient characteristics of the optical power. In addition, as Eq. (3b) shows the decaying characteristics of the carrier concentration (n), Shockley-Read-Hall (SRH) recombination coefficient (A), radiative recombination coefficient (B), and Auger recombination coefficient (C) of GaN-based LED can be extracted by fitting this equation to time-carrier concentration characteristics. In order to calculate the carrier concentration at each current from Eq. (4), the effective volume of the active region should be known. Through a numerical simulation, the effective volume of the sample is calculated as shown in Fig. 4. As already discussed, the effective volume is much smaller than the physical volume of the active region and reduced as carrier concentration increases ().
Fig. 4 Calculated volume correction factor by using a numerical simulation when carrier concentration in the active region of the GaN-based LED increases.
(n0 and t0 represent initial carrier concentration and time, respectively)
Figure 5 illustrates the experimental setup for the transient characteristics in optical power measurement. At first, current pulse (20 A/cm2 (3.0 V) amplitude, 4μs duration and 6μs period) generated from 81110A pulse generator is applied to the prepared GaN-based LED sample. In order to reduce the change of the band structure and carrier sweep-out from the active region, the ‘off’ state bias is set to 2.4 V instead of 0 V. When the LED sample emits the optical signal caused by the current pulse, a fast response photo receiver collects this optical signal and converts the optical signal to a voltage signal. At last, this voltage signal is recorded by an oscilloscope. Since the transient characteristics can be obtained by applying a very short current pulse to a sample, recombination coefficients can be extracted rapidly by this method. And, as it is measured by applying operating current at room temperature, recombination coefficients are extracted under real operating circumstances.
Transient characteristics of the light output power obtained by this experiment are illustrated in Fig. 6(a). As already mentioned, carrier concentration vs. time characteristics can be obtained by measuring the light output power as a function of time. Figure 6(b) shows the carrier concentration vs. time characteristics of an LED sample for two cases: (1) the physical volume or (2) the effective volume is used as the volume of the active region. By fitting the transient characteristics of carrier concentration to Eq. (3b), recombination coefficients are extracted as can be seen in Table 1. When the physical volume is used for calculating carrier concentration, the extracted Auger recombination coefficient is 5.1( ± 0.2) × 10−30 cm6 s−1, much higher than the theoretically calculated Auger recombination coefficient [21]. In contrast, when the effective volume is applied, the calculated carrier concentration becomes much larger, and the extracted Auger recombination coefficient is 1.4( ± 0.2) × 10−31 cm6 s−1 which is similar to the theoretically calculated value. In addition, the extracted SRH and radiative recombination coefficients are also in the range of generally accepted values [11]. This result shows that the proposed model considering the effective volume of the active region is reasonable and more accurate than the conventional model using the physical volume.
Fig. 6 Measured transient characteristics of the light output power (a) and the calculated carrier concentration (b) after the current pulse is applied to the LED sample. When the effective volume is used, the calculated carrier concentration becomes much larger.
Table 1. Calculated recombination coefficients by measuring transient characteristics
3.2 Internal quantum efficiency and droop
Figure 7 shows the calculated IQE using the physical volume or the effective volume of the active region and the IQE measured by light output power measurement with correctly calibrated light extraction efficiency. Recombination coefficients extracted by transient measurements (Table 1) are used to calculate the IQE. The calculated IQE based on the physical volume shows significant difference from the result of the light output power measurement. In contrast, the calculated IQE based on the effective volume shows good agreement with the result of the light output power measurement.
Fig. 7 Calculated IQE applying the physical volume or the effective volume of the active region and the IQE obtained by light output power measurement.
The proposed model considering the effective volume of the active region can also explain the symmetry issue of the IQE and the origin of the efficiency droop. Figure 8 shows the IQE as a function of the square root of light output power, which is plotted in logarithmic scale. When the conventional rate equation is used for calculating the IQE of the LEDs, the square root of light output power is proportional to the carrier concentration (n) and the IQE should show symmetry about the peak IQE line. However, the experimental IQE shows significant asymmetry, unlike the result that the conventional equation predicts [22]. This symmetry problem of the IQE can be explained by considering the effective volume when the carrier concentration increases. As we already confirmed in Fig. 4, the effective volume is not constant and reduced as the carrier concentration increases. Therefore, as the light output power increases, injected carriers are more crowded in a small volume and the carrier concentration becomes higher than that predicted by the conventional rate equation. This higher carrier concentration in the effective volume is the origin of the asymmetry of the IQE and the severe efficiency droop.
Fig. 8 Calculated IQE applying the effective volume of the active region as a function of the square root of light output power.
4. Conclusion
We have proposed a modified rate equation model of GaN-based LEDs considering the effective volume of the active region. It is confirmed that polarization charge, quantum well barriers, and high current density reduce the effective volume and cause the efficiency droop. We have also introduced a simple method to extract recombination coefficients and IQE by using transient analysis of GaN-based LED considering the effective volume of the active region. By analyzing the transient characteristics of the LED sample, we have confirmed that the proposed model can extract the recombination coefficients and IQE of GaN-based LEDs correctly.
Acknowledgments
This work was supported by Samsung Electronics and the Center for Integrated Smart Sensors funded by the Ministry of Education, Science and Technology as Global Frontier Project (CISS-2012M3A6A6054186).
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