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Abstract
This study investigated the temperature dependence of the Auger recombination coefficient (C) in an InGaN/GaN blue multiple-quantum-well (MQW) light-emitting diode structure at temperatures between 20 and 100°C. The temperature dependence of C was determined by fitting the measured external quantum efficiency (EQE) data using an analytical model or numerical simulation. In the analytical model, the carrier density in InGaN MQWs was assumed to be constant and independent of temperature. In contrast, the inhomogeneous carrier distribution in MQWs and its temperature-dependent redistribution were included in the numerical simulation. When the analytical model was employed to fit the EQE curve, C decreased with increasing temperature. On the other hand, when the numerical simulation was employed, C increased steadily by ∼31% as the temperature was increased from 20 to 100°C. We found that the temperature-dependent carrier distribution is important to consider when determining the temperature dependence of the Auger recombination coefficient in InGaN MQW structures.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Over the last two decades, there has been remarkable progress in the development of InGaN/GaN-based light-emitting diodes (LEDs) for use in solid-state lighting and display applications owing to their high efficiency and eco-friendliness [1–3]. Although high efficiencies have been achieved at relatively low current densities, GaN-based blue LEDs suffer from a significant decrease in efficiency as the current density is increased [4–7]. In addition to this current droop, the temperature-dependent decrease in efficiency, known as thermal droop, has attracted increasing attention [8–13]. It is important to understand the mechanisms of thermal droop and current droop of InGaN LEDs for their use in high-power and temperature-stable lighting applications. For this purpose, many studies have investigated the temperature dependence of the recombination coefficients in InGaN quantum wells (QWs) [14–23].
In general, the carrier recombination rate in QWs can be denoted as An + Bn2 + Cn3, where A, B, and C represent the Shockley–Read–Hall (SRH), radiative, and Auger recombination coefficients, respectively, and n is the carrier density in the QW active region. In semiconductors, A and C generally increase and B decreases with increasing temperature, leading to a decrease in efficiency with increasing temperature. InGaN QWs have also shown an increase in A and a decrease in B with increasing temperature. On the other hand, inconsistent temperature-dependent trends in C have been reported in InGaN QWs. Theoretical studies have shown that C increases slowly with increasing temperature, corresponding to indirect Auger recombination processes [14–17]. Experimentally, Galler et al. also reported an increase in C with increasing temperature in an InGaN single QW structure: a ∼35% increase in C from 25 to 100°C [18]. However, the majority of groups have shown experimentally that C showed decreasing behaviors with increasing temperature at least for temperatures higher than 300 K [19–23]. Moreover, the mechanism for this counter-intuitive temperature dependence of C has not been clearly understood.
Experimentally, the recombination coefficients in the QW active region have usually been obtained by fitting the measured efficiency as a function of current using the ABC recombination model [18–23]. The conventional ABC model assumes that the carrier density is constant through InGaN/GaN multiple-quantum-well (MQW) structures. On the other hand, it is well known that the distribution of the carrier density and recombination rates could be significantly inhomogeneous in InGaN MQWs [24–30]. Therefore, the experimentally extracted recombination coefficients based on the ABC model with a constant carrier density could deviate significantly from the actual values. Moreover, the carrier distribution in MQWs could be strongly dependent on the temperature as a result of thermally enhanced carrier transport through the MQW layers [31–33]. Consequently, if the conventional ABC model with constant carrier density was employed, the temperature-dependent recombination coefficients would be determined incorrectly. LEDs with a single QW layer do not undergo such carrier distribution problems encountered in MQW structures. We notice that Refs. [19–23] used InGaN/GaN MQW structures, whereas Ref. [18] employed an InGaN single QW structure, which might have led to the opposite temperature dependencies of C between Ref. [18] and Refs. [19–23]. This suggests that the descending behaviors of C with temperature in Ref. [19–23] could be due to the assumption of a constant carrier density through MQWs independent of temperature.
In this study, we investigate the temperature dependence of the Auger recombination coefficient in an InGaN/GaN blue MQW LED structure considering the temperature-dependent carrier distribution in MQWs. The external quantum efficiency (EQE) of the LED sample was measured as temperatures between 20 and 100°C. The EQE versus current relation for each temperature was fitted with the internal quantum efficiency (IQE) curve given by the analytical model or numerical simulation. We developed the analytical IQE model based on the ABC carrier recombination equation to fit the EQE data with minimum assumption of parameters. In the analytical model, the carrier density was assumed to be constant through the MQWs independent of temperature. In the numerical simulations, however, the inhomogeneous carrier distribution in MQWs was included when obtaining the temperature dependence of C. For the simulation, we employed a commercial software, APSYS, which self-consistently solves the QW band structures, radiative and nonradiative carrier recombination, and the drift and diffusion equation of the carriers [34]. By comparing the results obtained using these two methods, it is expected that the origin of the peculiar temperature dependence in C reported in previous works can be clarified, and the temperature-dependent C can be determined reliably.
2. Experiments
The epitaxial layers used for this study were grown on a c-plane sapphire substrate by metal-organic chemical vapor deposition. The layer structure consisted of a Si-doped n-GaN layer, MQW active region, a 15-nm-thick p-type Mg-doped AlGaN electron-blocking layer, and 150-nm-thick Mg-doped p-GaN layer. The MQW active layers were composed of five 3-nm-thick InGaN QWs separated by 8-nm-thick GaN barriers. The peak emission wavelength was ∼450 nm at 20°C. The LED chip was fabricated as a vertical-injection structure by using wafer bonding and laser lift-off processes. The exposed n-GaN surface was roughened by a KOH solution for efficient light extraction. The chip dimension was 1 × 1 mm2. The fabricated LED chip was encapsulated with epoxy resin and mounted in a ceramic package as a type of surface-mount device. Then, the LED package was soldered on a copper block with a temperature controlled by a thermo-electric cooler (TEC).
The optical characteristics were measured using an LED characterization system with a calibrated integrating sphere when the TEC temperature was varied from 20 to 100°C. The LED sample was operated under pulsed current injection with a pulse width of 0.5 ms and the duty cycle of 1% to minimize self-heating effects. The light output power (LOP) was measured as the injection current increased up to 350 mA for each temperature. The EQE is determined using the following formula [32]:
where λc is the centroid wavelength of an emission spectrum, Pout is LOP, and I is current injected into the LED sample. q, h, and c are the elementary charge, Planck constant, and the speed of light in vacuum, respectively.Figure 1 presents the EQE determined using Eq. (1) as a function of the injection current up to 350 mA at temperatures of 20, 40, 60, 80, and 100°C. The peak EQE decreased from 0.544 to 0.464 and the current corresponding to the peak EQE increased from 55 to 70 mA as the temperature increased from 20 to 100°C. The measured EQE curve at each temperature will be used to fit the IQE using either the analytical model or numerical simulation.
Fig. 1. External quantum efficiency (EQE) of a measured LED sample as a function of the current at temperatures of 20, 40, 60, 80, and 100°C.
3. Results and discussion
3.1 Analytical model: constant carrier density in MQWs
In this subsection, analytic expressions on the relationships between the carrier recombination coefficients were developed to determine the temperature dependence of C under a constant carrier density in MQWs. This procedure is basically similar to those reported elsewhere on the temperature-dependent recombination coefficients in InGaN QWs [18–23].
In the ABC recombination model, the current, I, injected into active region is expressed as
where V is the active volume of MQW layers. Here, V is the product of the area of the QW planes and the total thickness of the MQW layers, and is assumed to be constant independent of temperature. The carrier density n is also assumed to be constant through all QW layers. The IQE of an LED is defined as In the IQE versus current relation, which is often referred to as the IQE curve, the peak IQE, ηp and the corresponding injection current, Ip are related to the coefficients A, B, and C as follows [35]: Using Eqs. (2)–(4), the IQE curve can be obtained by solving the following quadratic equation on I [36]:Figure 2 shows the IQE curves along with measured data at 20, 40, 60, 80, and 100°C. For the IQE fitting using above equations, the EQE data around the peak EQE were used. Therefore, the IQE fitting was performed up to 150 mA. Good agreements between the measured data and the IQE fit curves were observed up to 150 mA for all temperatures. As the temperature was increased from 20 to 100°C, ηp decreased from 0.656 to 0.557. From the EQE and IQE data in Figs. 1 and 2, the light extraction efficiency (LEE), which is defined as the ratio of EQE to IQE, was obtained to be ∼0.83. This LEE value is similar to that reported in other vertical LED structures [37–40]. From the results in Fig. 2, the data set of ηp(T) and Ip(T) can be obtained for each temperature T.
Fig. 2. Theoretical fit (lines) of the internal quantum efficiency (IQE) using Eq. (4) to the measured data (solid dots) at temperatures of 20, 40, 60, 80, and 100°C.
From Eq. (4), coefficients A and C are related to the coefficient B through ηp and Ip:
The coefficient B can be obtained from the spontaneous emission rate in QWs. The spontaneous emission rate at the angular frequency ω for the transition from the i state in the conduction band to the j state in the valence band is written as [9–11]
Fig. 3. Theoretical radiative recombination coefficient (B) obtained by using Eq. (8) is shown as a function of temperature. The red dotted line represents the fit to the calculated B using the inverse power law with an exponent of 1.24.
The temperature dependence of B has often been described as an inverse power law of the form 1/(Tδ ) with the single exponent δ which can be simply expected to be 1.5 and 1.0 for the bulk and QW semiconductors, respectively [22,41]. On the other hand, δ is dependent on the detailed band structures and the carrier density, resulting in a more complicated temperature dependence [41]. In many cases, the reported δ values of the InGaN QW LEDs have been between 1.0 and 1.5 [19–21,42]. The calculated B(T) was fitted with the inverse power law, which is shown as the dotted line in Fig. 3. The figure shows good agreement between the calculated data and the fitted curve. Here, δ of 1.24 was obtained. This value almost corresponds to the median value of the reported range of δ in B.
With the obtained information of ηp(T), Ip(T), and B(T), the temperature dependence of A and C were determined using Eqs. (7) and (8), respectively. Figure 4 presents the relative recombination coefficients A(T)/A(T0) and C(T)/C(T0) as a function of the temperature T. The reference temperature T0 was set to 20°C. Here, three cases of temperature dependent B(T) were considered: the theoretical B(T) in Fig. 2 and the B(T) following the inverse power law dependence with δ equal to 1 and 1.5. A(T) was found to increase with increasing temperature for all three cases, as expected. Applying the Arrhenius model (${\propto} \exp [ - {E_a}/{k_B}T]$) with an activation energy Ea to A(T) in Fig. 4(a), Ea of 41 to 48 meV was obtained. This is reasonably close to the previously reported activation energy values of 50 meV in Ref. [18] and 54 meV in Ref. [22].
Fig. 4. (a) Relative SRH recombination coefficients (A(T)/A(T0)) as a function of temperature. (b) Relative Auger recombination coefficients (C(T)/C(T0)) as a function of temperature. The reference temperature T0 was set to 20°C. Three cases of B(T) were considered: the theoretical B(T) in Fig. 3 and the inverse power law dependence with δ = 1.0 and 1.5. On the right axes, the absolute values of A(T) and C(T), which were obtained using B(T) in Fig. 3, are also shown.
In contrast, C(T) has a mostly decreasing trend with increasing temperature, as shown in Fig. 4(b). As the temperature was increased from 20 to 100°C, C(T) decreased by ∼20% for δ = 1.5 and less than 5% for δ = 1.0. For δ = 1.0, C(T) was almost constant when T was higher than 40°C. When the theoretical data of B(T) in Fig. 3 were used, C(T) decreased steadily by ∼12% in the temperature range of 20 to 100°C. As δ in B increased, C(T) decreased with increasing temperature. As δ smaller than 1 has not been reported, it can be regarded that the C(T) for δ = 1.0 is an upper-limit result, and hence, it can be regarded that C(T) decreased with increasing temperature overall. This descending behavior in the coefficient C with increasing T has also been demonstrated in several previous works [19–23]. That is, the present analytical model also showed a similar temperature dependence of C to those reported previously.
In Fig. 4, the absolute values of coefficients A(T) and C(T) are also shown on the right axis of each figure. These absolute values were obtained from Eq. (6) using the data of B(T) in Fig. 3. As the temperature was increased from 20 to 100°C, A(T) increased from 8.7 × 106 to 12.7 × 106 s−1 and C(T) decreased from 3.51 × 10−29 to 3.38 × 10−29 cm6/s. These temperature dependent A and C values will be compared with the results of numerical simulation in the next subsection.
3.2 Numerical simulation: inhomogeneous carrier distribution in MQWs
In this subsection, the measured EQE curves in Fig. 1 are fitted with the IQE curves obtained by numerical simulations using the APSYS program. Here, the effects of an inhomogeneous carrier distribution in MQWs on the IQE curve were included, and temperature-dependent redistribution of carriers in MQWs was also considered in simulations. In the simulation, the conduction band offset of the InGaN/GaN and AlGaN/GaN layers was set to 0.7 [43]. The Mg doping concentration of the p-GaN and p-AlGaN layers was 1 × 1019 cm−3, and the Si doping concentration of the n-GaN layer was 5 × 1018 cm−3. The incomplete ionization of Mg acceptors and the field ionization model were included, and the AlGaN acceptor energy was scaled linearly from 170 meV in p-GaN to 470 meV in p-AlN [44,45]. As the temperature was increased from 20 to 100°C, the actual hole concentration in p-GaN was found to increase from 1.3 × 1017 to 3.3 × 1017 cm−3. The internal electric field induced by spontaneous and piezo-electric polarizations at the hetero-interfaces, InGaN/ GaN and AlGaN/GaN were also included using the model in Ref. [46]. Assuming 50% compensation of theoretical polarization fields [43,47], the strength of polarization fields at the interfaces of the InGaN QW and the GaN barrier was calculated to be approximately 1 MeV/cm. As to the carrier mobility, the mobility model and parameters of Refs. [48–50] was adopted to model the temperature and doping concentration dependent mobility. The electron mobility in GaN was found to decrease from 380 to 260 cm2/Vs as the temperature was increased from 20 to 100°C. The hole mobility in the AlGaN, InGaN, and GaN layers was assumed to be 5 cm2/Vs [50].
In the simulation, the radiative recombination rate was calculated by integrating the spontaneous emission spectrum with a Lorentzian line-shape function. That is, B(T) in Fig. 3 was used. The IQE curve for a given temperature was simulated varying the coefficients A and C. The simulated IQE curves for various sets of A and C were compared with the measured EQE data in Fig. 1 and the best fitting results were obtained. Figure 5 shows the simulated IQE curves that exhibited the best fit to the measured EQE data at temperatures of 20, 40, 60, 80, and 100°C. Good agreements between the measured data and the fit curves are observed for all temperatures. In the simulation results, electron leakage current from MQW layers to the p-GaN layer was found to be negligible for all temperatures from 20 to 100°C. The absence of electron leakage in the simulation of this study is consistent with the previous report using the similar simulation parameters [51]. ηp decreased from 0.655 to 0.555 as the temperature was increased from 20 to 100°C, which is quite similar to the results obtained using the analytical model in Fig. 2.
Fig. 5. Internal quantum efficiency (IQE) versus current relation obtained by numerical simulations fitted to the measured data (solid dots) at 20, 40, 60, 80, and 100°C.
Figure 6 plots the coefficients A and C used for the IQE fitting in Fig. 5 as a function of temperature. The coefficient A increased with increasing temperature as in the case of the analytical model in Fig. 4(a). It increased from 7.6 × 106 to 9.8 × 106 s−1, as the temperature was increased from 20 to 100°C, which is similar to that obtained by the analytical model in Fig. 4(a). However, the increase rate of A was ∼28% in the temperature range from 20 to 100°C, which was somewhat lower than those obtained in Fig. 4(a). In addition, the temperature dependence of A could not be fitted well to the Arrhenius model in this case, which was attributed to the temperature-dependent carrier distribution in InGaN MQWs.
Fig. 6. Coefficients (a) A and (b) C used for the IQE fitting in Fig. 5 are plotted as a function of temperature from 20 to 100°C. On the right axes, the absolute values of A and C are shown. On the left axes, A and C relative to the value at 20°C is shown.
As shown in Fig. 6(b), the coefficient C also increased steadily from 2.07 × 10−30 to 2.72 × 10−30 cm6/s with increasing temperature from 20 to 100°C. This absolute value of C obtained by numerical simulation was found to be more than 10 times lower than that obtained by the analytical model in Fig. 4(b). This large difference in C values resulted from the fact that the analytical ABC model assumed the carrier density is constant inside a QW and uniform throughout MQW layers. In numerical simulation, on the contrary, the separation of electron-hole wavefunctions and the inhomogeneous carrier distribution were considered, which resulted in significant reduction in effective active volume and hence the larger value of C [52–55]. In addition, the coefficient C increased by 31.4% over this temperature range. This temperature dependence of C is in strong contrast to the result of the analytical model in Fig. 4(b). This is also opposite to the results of other experimental studies on InGaN MQWs, where C decreased with temperature [19–23].
The reason for these conflicting results on the temperature dependence of C is believed to be related to temperature-dependent distributions of the carrier density and recombination rate in MQWs. To address this aspect, we conducted simulations on carrier distribution and recombination rates in InGaN MQWs as the temperature was varied. Figure 7 shows the electron and hole concentration distribution in five InGaN QWs at temperatures 20, 60, and 100°C when the injection current was 100 mA. The electron concentration did not change significantly with temperature. On the other hand, the hole concentration showed stronger temperature dependence. At 20°C, the distribution of hole concentration is quite inhomogeneous, decreasing rapidly as hole carriers move from the p-side to the n-side QW. This inhomogeneity in hole distribution resulted from inefficient hole transport through QWs caused by low hole mobility. As the temperature was increased from 20°C, the hole concentration distribution became increasingly homogeneous, resulting from thermally enhanced hole transport from the p-side to the n-side QW [31–33].
Fig. 7. Distribution of (a) electron and (b) hole concentration at five QWs of the simulated InGaN/GaN MQW structure at 20, 60, and 100°C.
Figure 8 shows the distribution of the radiative recombination rate (Rrad) and the Auger recombination rate (RAuger) in InGaN MQWs at 20, 60, and 100°C when the current was 100 mA. At 20°C, the distributions of both Rrad and RAuger were significantly inhomogeneous, decreasing rapidly from the p-side to the n-side QW. In particular, RAuger was dominant at the QW nearest n-GaN because RAuger is proportional to the cube of carrier density. With increasing temperature, Rrad and RAuger became homogeneous owing to thermally-enhanced hole transport. In Fig. 9, average recombination rates in each QW for radiative and Auger recombination are plotted as a function of the QW number for temperatures from 20 to 100°C. The QW numbers of 1 and 5 correspond to the QW nearest the n-GaN and the p-GaN layer, respectively. As the temperature was increased, both recombination rates become increasingly homogeneous through the MQW layers. As a measure of the uniformity in the distribution, we calculated the coefficient of variation (CV), which is defined as the ratio of the standard deviation to the mean of the recombination rate distribution at five QWs. As the temperature was increased from 20 to 100°C, the CV of Rrad and RAuger decreased from 1.22 to 0.33 and from 1.63 to 0.38, respectively. The improvement in the homogeneity with temperature is more pronounced for the Auger recombination. At 20°C, the ratio of the highest RAuger at the 5th QW (nearest p-GaN) to the lowest RAuger at the 1st QW (nearest n-GaN) was as large as 40. At 100°C, this ratio was reduced considerably to only ∼2.
Fig. 8. Distribution of (a) the radiative and (b) the Auger recombination rate at five QWs of the simulated InGaN/GaN MQW structure at 20, 60, and 100°C.
Fig. 9. Average recombination rate at each QW for (a) the radiative and (b) the Auger recombination is plotted as a function of QW number for temperatures from 20 to 100°C. The QW numbers of 1 and 5 correspond to the QW nearest the n-GaN and the p-GaN layer, respectively.
The simulation results in Figs. 8 and 9 show that the carrier recombination occurs mainly at the 5th QW nearest the p-side layers at 20°C. In this case, the effective active volume should be reduced significantly compared to the physical volume of MQWs [52–55]. As the temperature is elevated, more QW layers begin to act as an effective active region. At 100°C, where carrier recombination rates among QWs are reasonably uniform, the effective active volume of MQWs is expected to be similar to the physical volume of MQWs. That is, the effective active volume increases with increasing temperature. On the other hand, in the analytical model in Section 3.1, the volume V of the QW active region in Eqs. (2) and (6) was constant and independent of temperature. Because the effective volume of MQWs at relatively low temperatures is much smaller than the physical volume V, the coefficient C could be overestimated if the physical volume of all MQW layers was used as V in Eq. (6). That is, the coefficient C determined by the analytical model could be significantly larger than the actual C value at low temperatures. As the temperature increases, this difference in C will be reduced because the effective active volume approaches the physical volume. Therefore, C could decrease with increasing temperature as a result of an overestimation of C at low temperatures when the analytical ABC model with a constant carrier density was employed. On the other hand, when a numerical simulation was used to fit the measured EQE data, the temperature-dependent change in the carrier and recombination distribution was considered. Hence, C increased steadily with increasing temperature as shown in Fig. 6(b).
A comparison of the results of the analytical model in Fig. 4(b) and the numerical simulation in Fig. 6(b) showed that the decrease in C with temperature, which was reported in many experimental studies on InGaN MQWs [19–23], could result from the assumption of a constant carrier density through MQWs independent of temperature. Therefore, the temperature dependence of C can be determined correctly when the temperature-dependent redistribution of carriers in MQWs is considered. In Fig. 6(b), the coefficient C increased by ∼31% as the temperature was increased from 20 to 100°C. This increasing rate is slightly lower than the experimental result in the InGaN single QW structure [18] and approximately double the theoretically calculated one based on indirect Auger processes assisted by phonon coupling and alloy scattering [14,17]. The data in Fig. 6(b) could not be fitted with the Arrhenius model well, implying that the direct band-to-band Auger recombination mechanism may be excluded. Further theoretical and experimental studies on the temperature dependence of C are expected to reveal the mechanism of Auger recombination processes in the InGaN QW.
4. Summary
Up to now, the temperature dependence of C in InGaN QWs has been reported to be inconsistent. Theoretical studies showed weakly increasing behaviors in C with temperature, whereas experimental studies mostly reported that C in InGaN MQWs decreased with increasing temperature. In this study, to clarify this issue, the temperature dependence of C was investigated by fitting the EQE data of an InGaN blue MQW LED using an analytical model or numerical simulation. In the analytical model, where the carrier density in InGaN MQWs was assumed to be constant independent of temperature, C decreased with increasing temperature, as in the case of previous experimental works on InGaN MQWs. On the other hand, the numerical simulation included the effects of an inhomogeneous carrier distribution in MQWs and its temperature-dependent redistribution. When the numerical simulation was employed to fit the EQE curve, C was found to increase by ∼31% as the temperature was increased from 20 to 100°C. This opposite temperature dependence in C between two methods was attributed to the temperature-dependent carrier distribution in InGaN MQWs. As the temperature increases, the carrier distribution in MQWs becomes increasingly homogeneous. At relatively low temperatures, carriers accumulated mainly in the p-side QW layers, leading to an overestimation of C in the analytical model of a constant carrier density, and hence the descending behaviors in C with temperature. In this study, we found that the temperature dependence of C can be determined correctly when the temperature-dependent carrier distribution in MQWs is considered. The properly determined temperature-dependent C is also expected to provide insight into the Auger recombination mechanisms in InGaN materials.
Funding
National Research Foundation of Korea (NRF-2016R1D1A1B03932092, NRF-2019R1A2C1010160); Korea Institute of Science and Technology (2E30100-20-038).
Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science and ICT (NRF-2019R1A2C1010160) and the Ministry of Education (NRF-2016R1D1A1B03932092), and by the KIST Institutional Program (2E30100-20-038).
Disclosures
The authors declare no conflicts of interest.
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