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Analytical analysis of internal quantum efficiency with polarization fields in GaN-based light-emitting diodes
Superlattices and Microstructures 135 (2019) 106271
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Analytical analysis of internal quantum efficiency with polarization fields in GaN-based light-emitting diodes Muhammad Usman *, Abdur-Rehman Anwar, Munaza Munsif, Shahzeb Malik, Noor Ul Islam Faculty of Engineering Sciences, Ghulam Ishaq Khan Institute of Engineering Sciences and Technology, Topi, 23460, Khyber Pakhtunkhwa, Pakistan
A B S T R A C T
This paper presents the analytical model for the analysis of internal quantum efficiency as well as light output power of GaN-based light-emitting diodes by introducing the polarization factor, which accounts for the polarization fields in the active region, in the standard ABC model. With the increase of the polarization, effective volume decreases which causes strong localization of carriers at lower potentials causing the carrier density to increase. This effect leads to loss mechanisms such as carrier leakage and Auger loss. Our theoretical analysis for internal quantum efficiency and light output power shows good agreement with the experimental results for both the blue and green InGaN light-emitting diodes.
1. Introduction Gallium Nitride (GaN) has dramatically revolutionized the area of solid-state lightning due to their broad wavelength emission spectrum [1,2]. In comparison to conventional light sources, solid-state lighting sources are highly energy-efficient and environmental friendly [3]. That is why, globally, solid-state lightning has been replacing conventional light sources [4,5]. But the critical issue which still needs attention is the efficiency improvement in GaN-based light emitting diodes (LEDs) at high current density [6–8]. IQE is the ratio of photons emitted from the active region to the electrons injected in the LED [9]. The actual root mechanism of IQE at high current density is still unidentified [10], albeit some of the reported reasons behind the droop include Auger non-radiative recom bination [11–13], leakage of electrons [14,15], lack of carrier localization [16–18], and internal field in the quantum well (QW) [14, 19,20]. The piezoelectric field (Pz), which is induced due to lattice misalignment, reduces the effective volume for the radiative recombination [21,22]. Due to Pz, the regular shape of QW is distorted as a result probability of recombination is reduced owing to the quantum-confined Stark effect (QCSE) [1,23–26]. Various analytical models have been proposed to analyze the efficiency of light-emitting diodes. Some of the reported methods include peak efficiency fitting [27], phase-space filling effect [28] and incor poration of leakage mechanisms [29–32]. In addition to the aforementioned reported methods, a detailed review discusses the strengths and weaknesses of the ABC model [33]. In this paper, we report a methodology by introducing a polarization factor to model the degradation of internal quantum efficiency and light output power at high current density. 2. Methodology The ABC model is used, as a standard model, for describing the carrier recombination processes i.e. radiative as well as nonradiative recombination mechanisms as a function of carrier concentration as [33],
* Corresponding author. E-mail address: [email protected] (M. Usman). https://doi.org/10.1016/j.spmi.2019.106271 Received 27 July 2019; Received in revised form 9 September 2019; Accepted 11 September 2019 Available online 14 September 2019 0749-6036/© 2019 Elsevier Ltd. All rights reserved.
Superlattices and Microstructures 135 (2019) 106271
M. Usman et al.
Fig. 1. Influence of polarization (P) on the effective volume in the active region (a) No polarization (P ¼ 0%) (b) Maximum Polarization (P → 100%).
Fig. 2. (a) Influence of polarization (P) on internal quantum efficiency as a function of current density (b) Effect of “C” on IQE at P ¼ 0.
IQE ¼
BN 2 AN þ BN 2 þ CN 3
(1)
2
Superlattices and Microstructures 135 (2019) 106271
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Fig. 3. Effect of incorporating polarization factor on (a) current density (b) carrier overflow ratio and (c) LOP as a function of injection current.
where A and C are both non-radiative recombination coefficients which are termed as Shockley-Read-Hall (SRH) and Auger recom bination coefficients respectively, while B is radiative recombination coefficient and N represents carrier concentration. At high current density, the active region volume is reduced which means that active region for radiative recombination decreases [34]. The polarization in the active region is reported to significantly affect the active region and thus device operations [35]. Therefore, to account for the effect of polarization in the ABC model, we modify Eq. (1) as, � �2 B pNffiγ IQE ¼ � � (2) � �2 � �3 A pNffiγ þ B pNffiγ þ C pNffiγ
3
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where, γ¼
Ve Va
(3)
and P¼1
(4)
γ
where γ is a ratio of effective volume (Ve) to active volume (Va) and P is polarization factor to account for the influence of polarization fields in the active region. From Eq. (4), it can be calculated that, theoretically, P will be 100% when γ ¼ 0 whereas P will be 0% when γ ¼ 1. The polarization factor depends on the effective volume such that by decreasing Ve, P increases. Fig. 1 shows the illustration of the aforementioned phenomenon. Fig. 2a shows the behavior of IQE with respect to current density at maximum and minimum polarization by using Eq. (2). In Eq. (2), the values of parameters A, B and C have been used from the reported literature [36]. Looking at Fig. 2a, it can be observed that at polarization P ¼ 0, the droop in the IQE is lowest in comparison to the polarization of 50% or 90%. However, despite P ¼ 0 the ef ficiency droop can be associated with the Auger recombination at higher current density (see Eq. (1)). Fig. 2b shows the influence of both the Auger recombination coefficient (C) as well as P (or γ) in our proposed model i.e. Eq. (2). At P ¼ 0, the case when whole active volume is available for carrier recombination, varying values of C have been used to see the effect of Auger on the efficiency droop. The values of C, which have been used in this work, are reported in Ref. [37]. By reducing the value of C, the degradation of IQE is observed to be significantly mitigated with respect to the increasing current density. It may be noted that our results are consistent with the reported literature [10,37]. In Eq. (2), Auger coefficient has cubic dependence on the carrier concen tration. This results in significant influence of non-radiative Auger recombination at higher current densities. We also know that ABC rate equation model in terms of injected current I in the active region is given as [33]. � I ¼ qVe AN þ BN 2 þ CN 3 (5) Rearranging, we get, I ¼ AN þ BN 2 þ CN 3 qVe By substituting above relation in Eq. (1) IQE ¼ Ve
BN 2 I=q
Rearranging the above equation, carrier density will be pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I � IQE N ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q � B � Ve
(6)
Substituting Eq. (3) in Eq. (6) then, N n’ ¼ pffiffi γ
(7)
By using Eq. (7), Fig. 3a shows the relationship between injection current and carrier density in the active region. It can be observed that the polarization greatly influences the carrier density in the active region. The carrier density can be observed to increase with the increasing polarization. High polarization causes reduced effective volume for radiative recombination leading to strong localization of carriers at deeper potential. This results in overall enhanced carrier concentration in the active region leading to increased Auger recombination rate [21,38] and degraded IQE [36]. The carrier overflow (leakage) ratio can be calculated by Ref. [39]. ILeakage 1 ηint ¼ IQW ηint
(8)
We can modify Eq. (8) by incorporating the polarization factor because it plays key role in carrier leakage [14]. In green LEDs, the probability of carrier leakage is high as compared to blue LEDs because of strong built-in polarization [14]. With the increase in the concentration of indium in the active region, the polarization increases and the effective volume for carrier recombination decreases. This results in the carrier leakage out of the active region. The modified Eq. (8) can be written as ILeakage 1 1 ηint ¼ � γ IQW ηint
(9)
By using Eq. (9), Fig. 3b shows the behavior of carrier leakage ratio with respect to injection current at different values of po larization. At low polarization (P ¼ 0), there is minimum ratio of carrier leakage because maximum effective volume for carrier recombination is available. Similarly, by increasing the value of polarization the leakage ratio also increases due to decrease in the 4
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Fig. 4. (a) Comparison of IQE between different models and our proposed model for blue LED (b) Comparison of IQE between different models and our proposed model for green LED. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
effective volume. The light output power (LOP) of LEDs is also affected due to the influence of polarization. The LOP is reported to increase almost linearly with the increase in injection current at low current density. At higher current density, the LOP is reported to saturate [40]. With increase of polarization (or decrease of effective volume of the active region) the LOP degrades. We can estimate LOP by using the relation [41]. LOP ¼ EQE � I
hc eλ
(10)
where EQE and λ are external quantum efficiency and average emitted wavelength respectively. We can modify Eq. (10) by incor porating effective volume to active volume ratio for better accuracy as LOP ¼ EQE � I � γ
hc eλ
(11)
By using Eq. (11), Fig. 3c shows that with increasing polarization, the LOP tends to decrease. At P ¼ 50%, LOP reduces by more than half in comparison to P ¼ 0. In case of maximum polarization (P ¼ 90%), the LOP is significantly reduced in comparison to P ¼ 0 and P ¼ 50%. 3. Result and discussion In this study, we choose experimental internal quantum efficiency data of blue and green LEDs reported elsewhere [20] to test our model. In comparison to ABC model and rate equation model [41], our proposed modified model, by incorporating the polarization factor i.e. Eq. (2), shows better approximation of IQE with experimental IQE of blue LED in Fig. 4a and green LED in Fig. 4b. The ABC 5
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Fig. 5. Comparison between experimental and modelled LOP for blue and green LEDs. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
model misses the experimental data both at low and high current density. On the other hand, rate equation model fits the peak of IQE but its approximation deviates at high current density. For blue LED, using published parameters A, B and C [42] and P ¼ 15% in Eq. (2), a good agreement with experimental result is observed both at low as well as high current density, as shown in Fig. 4a. Similarly for the green LED, by considering parameters A, B and C as reported in Ref. [20] and P ¼ 35%, in Eq. (2), we get the estimation of various analytical models, as shown in Fig. 4b. In comparison to the reported analytical models, our proposed model shows better agreement with the experiment. Our higher estimation of P (in case of green LED) in contrast to the lower estimation of P (in case of blue LED) not only shows best approximation with the experimental data but also our proposed estimation is consistent with the literature [43]. When composition of indium in InGaN LEDs is relatively low, the carriers are weakly localized at deeper potentials which means the carriers have higher chances for radiative recombination, as a result more effective volume can be utilized. Conversely, at high indium composition the carriers are strongly localized at deeper potentials which means the carriers have higher chances of non-radiative recombination, as a result lesser effective volume is utilized. This situation facilitates non-radiative Auger recombination [44]. As effective volume decreases, both the SRH and radiative recombination rates decrease but rate of Auger recombination is enhanced [10]. Due to high concentration of indium in green LEDs, strong Pz is created which reduces the over lapping probability of electron-hole wavefunction as compared to blue LEDs [39]. The overlap probability reduces due to strong tilting of QWs in green LEDs. Therefore, green LEDs have reduced radiative efficiency and high efficiency droop ratio in comparison to blue LEDs [45,46]. The non-radiative recombination rate increases abruptly at high current density, therefore, the degradation of IQE as well as LOP becomes distinct in GaN-based LEDs. InGaN blue LEDs have comparatively greater LOP than green LEDs because of strong Pz in green LEDs which reduces the effective volume for radiative recombination. Without the above-mentioned proposed modification of Eq. (10), the numerical estimation of the LOP of our devices can be very inaccurate. Fig. 5 shows the comparison between the experimental LOP and our proposed model. By incorporating the polarization factor in Eq. (10), good estimation of the experimental data can be observed. By using Eq. (11), with λ ¼ 470 nm and EQE ¼ 50% [47], blue LED shows best agreement at P ¼ 15%. The light output power of blue LED is ~380 mW at about 30 A cm 2. Similarly for green LED, with λ ¼ 530 nm and EQE ¼ 30% [47] in Eq. (11), a good agreement at P ¼ 35%. is observed. However, overall light output power in green LED is less than blue LED, i.e. ~260 mW power at about 30 A cm 2 for the reasons mentioned earlier. It may be noted that the same value of polarization factor has been used to estimate the IQE as well as LOP of both the blue and green LEDs. 4. Conclusion We have proposed modified ABC model by incorporating polarization in the model. Our results give better approximation of in ternal quantum efficiency in comparison to other reported models. Our proposed model shows good agreement with experimental results of both the blue and green LEDs in comparison to other reported models. Not only our model shows good agreement with the experimental internal quantum efficiency but also with the light output power of the devices. From the discussion above it can be concluded that limiting the influence of polarization, not only increases the effective volume but also mitigates the droop ratio. In addition, the nonradiative effects such as Auger recombination and carrier leakage dominate due to the reduction of effective volume. Acknowledgments The authors are obliged to Higher Education Commission of Pakistan and Semiconductor Photonics Laboratory, Hanyang Uni versity, South Korea for lending technical support for this work. The authors would like to thank Ms. Nabila Nawaz and Prof. Khasan 6
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Karimov for the useful discussion. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.spmi.2019.106271.
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[40] A. Laubsch, M. Sabathil, J. Baur, M. Peter, B. Hahn, High-power and high-efficiency InGaN-based light emitters, IEEE Trans. Electron Devices 57 (1) (2009) 79–87. [41] H.-Y. Ryu, G.-H. Ryu, S.-H. Lee, H.-J. Kim, Evaluation of the internal quantum efficiency in blue and green light-emitting diodes using the rate equation model, J. Korean Phys. Soc. 63 (2) (2013) 180–184. [42] M. Meneghini, N. Trivellin, G. Meneghesso, E. Zanoni, U. Zehnder, B. Hahn, A combined electro-optical method for the determination of the recombination parameters in InGaN-based light-emitting diodes, J. Appl. Phys. 106 (11) (2009), 114508. [43] J. Piprek, Chapter 9 - nitride light emitters, in: Semiconductor Optoelectronic DevicesBoston, Academic Press, 2003, pp. 187–211. [44] J. Edmond, et al., High efficiency GaN-based LEDs and lasers on SiC, J. Cryst. Growth 272 (1–4) (2004) 242–250. [45] A. David, M.J. Grundmann, Influence of polarization fields on carrier lifetime and recombination rates in InGaN-based light-emitting diodes, Appl. Phys. Lett. 97 (3) (2010), 033501. [46] Y.-D. Lin, et al., Characterization of blue-green m-plane InGaN light emitting diodes, Appl. Phys. Lett. 94 (26) (2009), 261108. [47] F. Akyol, D.N. Nath, E. Gür, P.S. Park, S. Rajan, N-polar III–nitride green (540 nm) light emitting diode, Jpn. J. Appl. Phys. 50 (5R) (2011), 052101.
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Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Analytical analysis of internal quantum efficiency with polarization fields in GaN-based light-emitting diodes Muhammad Usman *, Abdur-Rehman Anwar, Munaza Munsif, Shahzeb Malik, Noor Ul Islam Faculty of Engineering Sciences, Ghulam Ishaq Khan Institute of Engineering Sciences and Technology, Topi, 23460, Khyber Pakhtunkhwa, Pakistan
A B S T R A C T
This paper presents the analytical model for the analysis of internal quantum efficiency as well as light output power of GaN-based light-emitting diodes by introducing the polarization factor, which accounts for the polarization fields in the active region, in the standard ABC model. With the increase of the polarization, effective volume decreases which causes strong localization of carriers at lower potentials causing the carrier density to increase. This effect leads to loss mechanisms such as carrier leakage and Auger loss. Our theoretical analysis for internal quantum efficiency and light output power shows good agreement with the experimental results for both the blue and green InGaN light-emitting diodes.
1. Introduction Gallium Nitride (GaN) has dramatically revolutionized the area of solid-state lightning due to their broad wavelength emission spectrum [1,2]. In comparison to conventional light sources, solid-state lighting sources are highly energy-efficient and environmental friendly [3]. That is why, globally, solid-state lightning has been replacing conventional light sources [4,5]. But the critical issue which still needs attention is the efficiency improvement in GaN-based light emitting diodes (LEDs) at high current density [6–8]. IQE is the ratio of photons emitted from the active region to the electrons injected in the LED [9]. The actual root mechanism of IQE at high current density is still unidentified [10], albeit some of the reported reasons behind the droop include Auger non-radiative recom bination [11–13], leakage of electrons [14,15], lack of carrier localization [16–18], and internal field in the quantum well (QW) [14, 19,20]. The piezoelectric field (Pz), which is induced due to lattice misalignment, reduces the effective volume for the radiative recombination [21,22]. Due to Pz, the regular shape of QW is distorted as a result probability of recombination is reduced owing to the quantum-confined Stark effect (QCSE) [1,23–26]. Various analytical models have been proposed to analyze the efficiency of light-emitting diodes. Some of the reported methods include peak efficiency fitting [27], phase-space filling effect [28] and incor poration of leakage mechanisms [29–32]. In addition to the aforementioned reported methods, a detailed review discusses the strengths and weaknesses of the ABC model [33]. In this paper, we report a methodology by introducing a polarization factor to model the degradation of internal quantum efficiency and light output power at high current density. 2. Methodology The ABC model is used, as a standard model, for describing the carrier recombination processes i.e. radiative as well as nonradiative recombination mechanisms as a function of carrier concentration as [33],
* Corresponding author. E-mail address: [email protected] (M. Usman). https://doi.org/10.1016/j.spmi.2019.106271 Received 27 July 2019; Received in revised form 9 September 2019; Accepted 11 September 2019 Available online 14 September 2019 0749-6036/© 2019 Elsevier Ltd. All rights reserved.
Superlattices and Microstructures 135 (2019) 106271
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Fig. 1. Influence of polarization (P) on the effective volume in the active region (a) No polarization (P ¼ 0%) (b) Maximum Polarization (P → 100%).
Fig. 2. (a) Influence of polarization (P) on internal quantum efficiency as a function of current density (b) Effect of “C” on IQE at P ¼ 0.
IQE ¼
BN 2 AN þ BN 2 þ CN 3
(1)
2
Superlattices and Microstructures 135 (2019) 106271
M. Usman et al.
Fig. 3. Effect of incorporating polarization factor on (a) current density (b) carrier overflow ratio and (c) LOP as a function of injection current.
where A and C are both non-radiative recombination coefficients which are termed as Shockley-Read-Hall (SRH) and Auger recom bination coefficients respectively, while B is radiative recombination coefficient and N represents carrier concentration. At high current density, the active region volume is reduced which means that active region for radiative recombination decreases [34]. The polarization in the active region is reported to significantly affect the active region and thus device operations [35]. Therefore, to account for the effect of polarization in the ABC model, we modify Eq. (1) as, � �2 B pNffiγ IQE ¼ � � (2) � �2 � �3 A pNffiγ þ B pNffiγ þ C pNffiγ
3
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where, γ¼
Ve Va
(3)
and P¼1
(4)
γ
where γ is a ratio of effective volume (Ve) to active volume (Va) and P is polarization factor to account for the influence of polarization fields in the active region. From Eq. (4), it can be calculated that, theoretically, P will be 100% when γ ¼ 0 whereas P will be 0% when γ ¼ 1. The polarization factor depends on the effective volume such that by decreasing Ve, P increases. Fig. 1 shows the illustration of the aforementioned phenomenon. Fig. 2a shows the behavior of IQE with respect to current density at maximum and minimum polarization by using Eq. (2). In Eq. (2), the values of parameters A, B and C have been used from the reported literature [36]. Looking at Fig. 2a, it can be observed that at polarization P ¼ 0, the droop in the IQE is lowest in comparison to the polarization of 50% or 90%. However, despite P ¼ 0 the ef ficiency droop can be associated with the Auger recombination at higher current density (see Eq. (1)). Fig. 2b shows the influence of both the Auger recombination coefficient (C) as well as P (or γ) in our proposed model i.e. Eq. (2). At P ¼ 0, the case when whole active volume is available for carrier recombination, varying values of C have been used to see the effect of Auger on the efficiency droop. The values of C, which have been used in this work, are reported in Ref. [37]. By reducing the value of C, the degradation of IQE is observed to be significantly mitigated with respect to the increasing current density. It may be noted that our results are consistent with the reported literature [10,37]. In Eq. (2), Auger coefficient has cubic dependence on the carrier concen tration. This results in significant influence of non-radiative Auger recombination at higher current densities. We also know that ABC rate equation model in terms of injected current I in the active region is given as [33]. � I ¼ qVe AN þ BN 2 þ CN 3 (5) Rearranging, we get, I ¼ AN þ BN 2 þ CN 3 qVe By substituting above relation in Eq. (1) IQE ¼ Ve
BN 2 I=q
Rearranging the above equation, carrier density will be pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I � IQE N ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q � B � Ve
(6)
Substituting Eq. (3) in Eq. (6) then, N n’ ¼ pffiffi γ
(7)
By using Eq. (7), Fig. 3a shows the relationship between injection current and carrier density in the active region. It can be observed that the polarization greatly influences the carrier density in the active region. The carrier density can be observed to increase with the increasing polarization. High polarization causes reduced effective volume for radiative recombination leading to strong localization of carriers at deeper potential. This results in overall enhanced carrier concentration in the active region leading to increased Auger recombination rate [21,38] and degraded IQE [36]. The carrier overflow (leakage) ratio can be calculated by Ref. [39]. ILeakage 1 ηint ¼ IQW ηint
(8)
We can modify Eq. (8) by incorporating the polarization factor because it plays key role in carrier leakage [14]. In green LEDs, the probability of carrier leakage is high as compared to blue LEDs because of strong built-in polarization [14]. With the increase in the concentration of indium in the active region, the polarization increases and the effective volume for carrier recombination decreases. This results in the carrier leakage out of the active region. The modified Eq. (8) can be written as ILeakage 1 1 ηint ¼ � γ IQW ηint
(9)
By using Eq. (9), Fig. 3b shows the behavior of carrier leakage ratio with respect to injection current at different values of po larization. At low polarization (P ¼ 0), there is minimum ratio of carrier leakage because maximum effective volume for carrier recombination is available. Similarly, by increasing the value of polarization the leakage ratio also increases due to decrease in the 4
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Fig. 4. (a) Comparison of IQE between different models and our proposed model for blue LED (b) Comparison of IQE between different models and our proposed model for green LED. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
effective volume. The light output power (LOP) of LEDs is also affected due to the influence of polarization. The LOP is reported to increase almost linearly with the increase in injection current at low current density. At higher current density, the LOP is reported to saturate [40]. With increase of polarization (or decrease of effective volume of the active region) the LOP degrades. We can estimate LOP by using the relation [41]. LOP ¼ EQE � I
hc eλ
(10)
where EQE and λ are external quantum efficiency and average emitted wavelength respectively. We can modify Eq. (10) by incor porating effective volume to active volume ratio for better accuracy as LOP ¼ EQE � I � γ
hc eλ
(11)
By using Eq. (11), Fig. 3c shows that with increasing polarization, the LOP tends to decrease. At P ¼ 50%, LOP reduces by more than half in comparison to P ¼ 0. In case of maximum polarization (P ¼ 90%), the LOP is significantly reduced in comparison to P ¼ 0 and P ¼ 50%. 3. Result and discussion In this study, we choose experimental internal quantum efficiency data of blue and green LEDs reported elsewhere [20] to test our model. In comparison to ABC model and rate equation model [41], our proposed modified model, by incorporating the polarization factor i.e. Eq. (2), shows better approximation of IQE with experimental IQE of blue LED in Fig. 4a and green LED in Fig. 4b. The ABC 5
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Fig. 5. Comparison between experimental and modelled LOP for blue and green LEDs. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
model misses the experimental data both at low and high current density. On the other hand, rate equation model fits the peak of IQE but its approximation deviates at high current density. For blue LED, using published parameters A, B and C [42] and P ¼ 15% in Eq. (2), a good agreement with experimental result is observed both at low as well as high current density, as shown in Fig. 4a. Similarly for the green LED, by considering parameters A, B and C as reported in Ref. [20] and P ¼ 35%, in Eq. (2), we get the estimation of various analytical models, as shown in Fig. 4b. In comparison to the reported analytical models, our proposed model shows better agreement with the experiment. Our higher estimation of P (in case of green LED) in contrast to the lower estimation of P (in case of blue LED) not only shows best approximation with the experimental data but also our proposed estimation is consistent with the literature [43]. When composition of indium in InGaN LEDs is relatively low, the carriers are weakly localized at deeper potentials which means the carriers have higher chances for radiative recombination, as a result more effective volume can be utilized. Conversely, at high indium composition the carriers are strongly localized at deeper potentials which means the carriers have higher chances of non-radiative recombination, as a result lesser effective volume is utilized. This situation facilitates non-radiative Auger recombination [44]. As effective volume decreases, both the SRH and radiative recombination rates decrease but rate of Auger recombination is enhanced [10]. Due to high concentration of indium in green LEDs, strong Pz is created which reduces the over lapping probability of electron-hole wavefunction as compared to blue LEDs [39]. The overlap probability reduces due to strong tilting of QWs in green LEDs. Therefore, green LEDs have reduced radiative efficiency and high efficiency droop ratio in comparison to blue LEDs [45,46]. The non-radiative recombination rate increases abruptly at high current density, therefore, the degradation of IQE as well as LOP becomes distinct in GaN-based LEDs. InGaN blue LEDs have comparatively greater LOP than green LEDs because of strong Pz in green LEDs which reduces the effective volume for radiative recombination. Without the above-mentioned proposed modification of Eq. (10), the numerical estimation of the LOP of our devices can be very inaccurate. Fig. 5 shows the comparison between the experimental LOP and our proposed model. By incorporating the polarization factor in Eq. (10), good estimation of the experimental data can be observed. By using Eq. (11), with λ ¼ 470 nm and EQE ¼ 50% [47], blue LED shows best agreement at P ¼ 15%. The light output power of blue LED is ~380 mW at about 30 A cm 2. Similarly for green LED, with λ ¼ 530 nm and EQE ¼ 30% [47] in Eq. (11), a good agreement at P ¼ 35%. is observed. However, overall light output power in green LED is less than blue LED, i.e. ~260 mW power at about 30 A cm 2 for the reasons mentioned earlier. It may be noted that the same value of polarization factor has been used to estimate the IQE as well as LOP of both the blue and green LEDs. 4. Conclusion We have proposed modified ABC model by incorporating polarization in the model. Our results give better approximation of in ternal quantum efficiency in comparison to other reported models. Our proposed model shows good agreement with experimental results of both the blue and green LEDs in comparison to other reported models. Not only our model shows good agreement with the experimental internal quantum efficiency but also with the light output power of the devices. From the discussion above it can be concluded that limiting the influence of polarization, not only increases the effective volume but also mitigates the droop ratio. In addition, the nonradiative effects such as Auger recombination and carrier leakage dominate due to the reduction of effective volume. Acknowledgments The authors are obliged to Higher Education Commission of Pakistan and Semiconductor Photonics Laboratory, Hanyang Uni versity, South Korea for lending technical support for this work. The authors would like to thank Ms. Nabila Nawaz and Prof. Khasan 6
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Karimov for the useful discussion. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.spmi.2019.106271.
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