Enhanced Internal Quantum Efﬁciency of Bandgap-Engineered Green W-Shaped Quantum Well Light-Emitting Diode

Featured Application: Since ‘green-gap’ is a challenging issue which is of great interest to the community, we present and analyze bandgap-engineered W-shaped quantum well, as a solution, to improve the optoelectronic performance of InGaN-based green light-emitting diodes. Abstract: To improve the internal quantum efﬁciency of green light-emitting diodes, we present the numerical design and analysis of bandgap-engineered W-shaped quantum well. The numerical results suggest signiﬁcant improvement in the internal quantum efﬁciency of the proposed W-LED. The improvement is associated with signiﬁcantly improved hole conﬁnement due to the localization of indium in the active region, leading to improved radiative recombination rate. In addition, the proposed device shows reduced defect-assisted Shockley-Read-Hall (SRH) recombination rate as well as Auger recombination rate. Moreover, the efﬁciency rolloff in the proposed device is associated with increased built-in electromechanical ﬁeld.


Introduction
Enhancing the internal quantum efficiency (IQE) of GaInN-based light-emitting diodes (LEDs) has been discussed in the literature for potential solid-state lighting applications [1][2][3]. There are multiple reports to improve the IQE of LEDs mainly by using nonpolar substrates [4,5] and bandgap engineering [6][7][8][9]. Salient bandgap engineering approaches include triangular-shaped quantum wells [10,11], staggered quantum wells [12], graded quantum wells [8,13], polarization-matched quaternary barriers [14,15], quaternary electron blocking layer (EBL) [16], modification of EBL [17,18], InGaN barriers [19,20], as well as last quantum barrier [21], coupled quantum wells [22], and recently proposed all-quaternary device with and without electron-blocking layer [23,24]. In comparison to the conventional rectangular quantum wells, it has been shown using triangular, staggered, dip-shaped, and trapezoidal wells that the electron-hole wavefunction overlap is significantly improved in the quantum wells (QWs), which leads to improved optoelectronic output of the device [11,[25][26][27]. In addition, indium fluctuations in the GaN-based LEDs are known to influence the optoelectronic properties of the device [28][29][30][31][32]. Most of the reported work on bandgap engineering is based on influence the optoelectronic properties of the device [28][29][30][31][32]. Most of the reported work on bandgap engineering is based on LEDs in the blue emission range [11,22,27]. However, because of the 'green gap' challenge, it is of great interest to improve the performance of green LEDs [32,33]. Therefore, in this work, we present bandgap-engineered green W-shaped quantum well and discuss its influence on the improved optoelectronic performance of the green light-emitting diode by numerical simulation.

Device Structure
Two device structures are numerically employed labeled as R-LED and W-LED. Both the structures have similar configuration, except the quantum well shape. Two quantum wells are used in the active region. 2 nm thick undoped GaN layer is followed by 4.5 nm thick n-GaN layer (doping = 5 × 10 18 cm −3 ). 20 nm thick Al0.1GaN (doping = 5 × 10 17 cm −3 ) EBL is followed by 15 nm thick p-GaN layer (doping = 1 × 10 18 cm −3 ). The quantum well composition in the two structures is In0.31GaN and In0.34GaN/In0.28GaN/In0.34GaN for R-LED and W-LED, respectively. The indium composition in the quantum wells of the devices is such that the peak output wavelength of R-LED and W-LED is ~518 and ~513 nm, respectively. The SRH and Auger recombination coefficients are 20 ns and 1.5 × 10 30 , which are similar to the reported literature [34]. Both of the recombination coefficients are reported to be smaller in the graded quantum wells than the regular quantum wells by roughly a factor of half [13,35]. Other simulation parameters, such as mobility, temperature, and background losses are the same as reported in the literature [23]. Figure 1 shows the schematic of the two devices and the respective indium compositions.

Methodology
To conduct our study, we have used a commercial simulator APSYS to simulate our devices [36]. The energy band is computed as [37]: where Eg is the bandgap, A and B are the two binary elements, the bowing parameter C account for deviations from the linear behavior and is given in Ref. [38]. The current densities ⃗ and ⃗ of both the electrons and holes are originated by the electrostatic field ⃗ , as well as by the gradient of concentration of electrons ∇n and holes ∇p

Methodology
To conduct our study, we have used a commercial simulator APSYS to simulate our devices [36]. The energy band is computed as [37]: where E g is the bandgap, A and B are the two binary elements, the bowing parameter C account for deviations from the linear behavior and is given in Ref. [38]. The current densities where q is elementary charge, n and p represent carrier densities, and µ n,p are their corresponding mobilities, respectively. The heterointerface charge density is estimated using reported nonlinear model [39,40].
The spontaneous, Shockley-Read-Hall, and Auger recombination rates, represented by R sp , R SRH and R Aug respectively, are given by [41]: where nonradiative SRH lifetimes for electrons (n), and holes (p) is given by τ SRH n,p , n i is intrinsic density is C n and C p are Auger coefficients, depending on the type of material. For both the electrons as well as holes, τ SRH n,p is assumed to be same. Fermi statistics as well as thermionic emission of carriers at heterointerfaces is estimated, as given in [42]. A typical value of offset ratio ∆E c /∆E v is assumed to be 0.7 in our work [43,44]. Other details are given in [45]. The internal quantum efficiency (IQE), η i , is estimated as [46]:

Results and Discussion
Energy band diagram is shown in Figure 2. The R-LED and W-LED energy band diagrams are shown. The shaded areas are highlighting the quantum well region. The energyband in both the cases is strongly tilted towards the p-side because of the built-in electromechanical field direction. The W-shape well can utilize the carrier localization within the well to improve the carrier confinement and consequent radiative recombination. Strong band-bending in W-LED is observed because of the increased electrostatic field, owing to the indium fluctuation i.e., 34%-28%-34%, which also hinders carrier transport in the W-LED apart from the indium localization [28,29,31,32,45,47,48]. The influence of the electrostatic field and carrier transport in our proposed device is presented in detail in the following discussion. The inclined triangular wells in the reference structure are known to reduce theelectron-hole wavefunction overlap, therefore reducing the overall device performance [9,32,47]. This overlap is further reduced in green LEDs due to increased built-in electrostatic field arising from the increased indium composition and lattice mismatch [32]. The wells are also inclined in the W-LED but the W-shaped indium composition in the well improves the electron-hole wavefunction overlap, due to its peculiar composition fluctuations, leading to the improved device performance. Figure 3 shows the estimation of carrier concentrations in the quantum wells of the two devices. The electron concentrations in Figure 3a are quite identical in the two devices, which means that the W-LED has no significant influence on the electron current transport in the quantum wells. The electron concentration peaks towards the p-side of the well. In contrast, the hole concentration in W-LED in Figure 3b shows significant enhancement in comparison to W-LED, specifically in the quantum well towards p-side, which means that significant radiative recombination is expected because of the improved confinement of the holes in the active region. The hole concentration peaks towards the n-side of the quantum well, which also suggests that radiative recombination is expected to occur towards the p-side of the quantum well. The hole concentration tends to decrease in each quantum well from p-side to n-side in the reference structure, which is consistent with the reported results [47]. Since strong localization of holes is known to occur due to the fluctuations in indium composition [28,29,31,32], this results in most of the holes being confined in the last QW of the proposed W-LED in comparison to the first QW, where negligible holes exist, in addition to the effect of increased band-bending coming from the comparatively higher electrostatic field in W-LED.  The inclined triangular wells in the reference structure are known to reduce theelectron-hole wavefunction overlap, therefore reducing the overall device performance [9,32,47]. This overlap is further reduced in green LEDs due to increased built-in electrostatic field arising from the increased indium composition and lattice mismatch [32]. The wells are also inclined in the W-LED but the W-shaped indium composition in the well improves the electron-hole wavefunction overlap, due to its peculiar composition fluctuations, leading to the improved device performance. Figure 3 shows the estimation of carrier concentrations in the quantum wells of the two devices. The electron concentrations in Figure 3a are quite identical in the two devices, which means that the W-LED has no significant influence on the electron current transport in the quantum wells. The electron concentration peaks towards the p-side of the well. In contrast, the hole concentration in W-LED in Figure 3b shows significant enhancement in comparison to W-LED, specifically in the quantum well towards p-side, which means that significant radiative recombination is expected because of the improved confinement of the holes in the active region. The hole concentration peaks towards the n-side of the quantum well, which also suggests that radiative recombination is expected to occur towards the p-side of the quantum well. The hole concentration tends to decrease in each quantum well from p-side to n-side in the reference structure, which is consistent with the reported results [47]. Since strong localization of holes is known to occur due to the fluctuations in indium composition [28,29,31,32], this results in most of the holes being confined in the last QW of the proposed W-LED in comparison to the first QW, where negligible holes exist, in addition to the effect of increased band-bending coming from the comparatively higher electrostatic field in W-LED. The inclined triangular wells in the reference structure are known to reduce theelectron-hole wavefunction overlap, therefore reducing the overall device performance [9,32,47]. This overlap is further reduced in green LEDs due to increased built-in electrostatic field arising from the increased indium composition and lattice mismatch [32]. The wells are also inclined in the W-LED but the W-shaped indium composition in the well improves the electron-hole wavefunction overlap, due to its peculiar composition fluctuations, leading to the improved device performance. Figure 3 shows the estimation of carrier concentrations in the quantum wells of the two devices. The electron concentrations in Figure 3a are quite identical in the two devices, which means that the W-LED has no significant influence on the electron current transport in the quantum wells. The electron concentration peaks towards the p-side of the well. In contrast, the hole concentration in W-LED in Figure 3b shows significant enhancement in comparison to W-LED, specifically in the quantum well towards p-side, which means that significant radiative recombination is expected because of the improved confinement of the holes in the active region. The hole concentration peaks towards the n-side of the quantum well, which also suggests that radiative recombination is expected to occur towards the p-side of the quantum well. The hole concentration tends to decrease in each quantum well from p-side to n-side in the reference structure, which is consistent with the reported results [47]. Since strong localization of holes is known to occur due to the fluctuations in indium composition [28,29,31,32], this results in most of the holes being confined in the last QW of the proposed W-LED in comparison to the first QW, where negligible holes exist, in addition to the effect of increased band-bending coming from the comparatively higher electrostatic field in W-LED.   Figure 4a shows the estimation of the radiative recombination rates in the two devices. It can be observed that the radiative recombination rate in W-LED is more than twice than that of R-LED. This is due to the significant accumulation of holes in the well towards the p-side, as discussed in Figure 3. Since the holes are concentrated more towards the n-side in each QW, the radiative recombination in each quantum well is also concentrated towards the n-side because of the better electron-hole wavefunction overlap towards the n-side of the well. Because of the reduction of hole concentration from p-side towards the n-side of the device, owing to the reduced hole transport, the radiative recombination is asymmetrically reduced in the quantum wells of both devices [49]. Figure 4b,c show that W-LED has reduced Shockley-Read-Hall and Auger recombination rates in comparison to R-LED. The defect-assisted Shockley-Read-Hall (SRH) recombination rate is reduced by more than half and the Auger recombination rate is estimated to be negligible in comparison to the R-LED. The comparatively reduced SRH recombination and Auger recombination rates of W-LED is supported by earlier experimental and theoretical reports [13,35]. The rate of Auger recombination has been reported to be higher in rectangular wells than softened or graded wells, hence reducing the device efficiency [35].
Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 10 Figure 4a shows the estimation of the radiative recombination rates in the two devices. It can be observed that the radiative recombination rate in W-LED is more than twice than that of R-LED. This is due to the significant accumulation of holes in the well towards the p-side, as discussed in Figure  3. Since the holes are concentrated more towards the n-side in each QW, the radiative recombination in each quantum well is also concentrated towards the n-side because of the better electron-hole wavefunction overlap towards the n-side of the well. Because of the reduction of hole concentration from p-side towards the n-side of the device, owing to the reduced hole transport, the radiative recombination is asymmetrically reduced in the quantum wells of both devices [49]. Figure 4b,c show that W-LED has reduced Shockley-Read-Hall and Auger recombination rates in comparison to R-LED. The defect-assisted Shockley-Read-Hall (SRH) recombination rate is reduced by more than half and the Auger recombination rate is estimated to be negligible in comparison to the R-LED. The comparatively reduced SRH recombination and Auger recombination rates of W-LED is supported by earlier experimental and theoretical reports [13,35]. The rate of Auger recombination has been reported to be higher in rectangular wells than softened or graded wells, hence reducing the device efficiency [35].  The internal quantum efficiency comparison is shown in Figure 5. By using W-LED, we are able to increase the internal quantum efficiency of the device significantly. Improved IQE peak and droop ratio in W-LED is observed in comparison to R-LED i.e., droop ratios of R-LED and W-LED are ~65% and ~57%, respectively. The IQE peak of R-LED and W-LED occurs at ~0.18 µA·cm −2 and ~0.17 A·cm −2 respectively. The IQE peak in W-LED is shifted to higher current density, which is typically a case in graded LEDs [8,13]. This shift of IQE peak or droop onset is desirable in improving the device performance [2]. Similarly, the light output power of W-LED is improved by ~116% than that of the R-LED at 100 A·cm −2 . The internal quantum efficiency comparison is shown in Figure 5. By using W-LED, we are able to increase the internal quantum efficiency of the device significantly. Improved IQE peak and droop ratio in W-LED is observed in comparison to R-LED i.e., droop ratios of R-LED and W-LED are~65% and 57%, respectively. The IQE peak of R-LED and W-LED occurs at~0.18 µA·cm −2 and~0.17 A·cm −2 respectively. The IQE peak in W-LED is shifted to higher current density, which is typically a case in graded LEDs [8,13]. This shift of IQE peak or droop onset is desirable in improving the device Appl. Sci. 2019, 9, 77 6 of 10 performance [2]. Similarly, the light output power of W-LED is improved by~116% than that of the R-LED at 100 A·cm −2 .
Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 10 The reason of the droop in W-LED can be understood by the built-in field comparison of the two structures in Figure 6. It can be observed that the average built-in field in W-LED is ~24% higher than the R-LED. Despite the reduced Auger recombination rate, the internal quantum efficiency roll-off in Figure 5 can be associated with the comparatively higher built-in field in the proposed W-LED. The emission spectra of the two devices is shown in Figure 7. The peak wavelength of R-LED and W-LED is ~518 nm and ~513 nm, respectively. The broad emission spectrum is the characteristic of the indium fluctuations in the quantum well region [50]. The full width half maximum (FWHM) of R-LED and W-LED is 25 nm and 28 nm, respectively at the peak wavelength. With increasing injection current and higher indium fluctuations, FWHM, is reported to be broader in green light-emitting diodes [51]. In addition, indium-rich regions, regardless of high dislocation density, have been attributed to improve radiative recombination because of localization of carriers [51][52][53]. At peak wavelength, the emission of W-LED is ~93% improved in comparison to the R-LED. The reason of the droop in W-LED can be understood by the built-in field comparison of the two structures in Figure 6. It can be observed that the average built-in field in W-LED is~24% higher than the R-LED. Despite the reduced Auger recombination rate, the internal quantum efficiency roll-off in Figure 5 can be associated with the comparatively higher built-in field in the proposed W-LED. The reason of the droop in W-LED can be understood by the built-in field comparison of the two structures in Figure 6. It can be observed that the average built-in field in W-LED is ~24% higher than the R-LED. Despite the reduced Auger recombination rate, the internal quantum efficiency roll-off in Figure 5 can be associated with the comparatively higher built-in field in the proposed W-LED. The emission spectra of the two devices is shown in Figure 7. The peak wavelength of R-LED and W-LED is ~518 nm and ~513 nm, respectively. The broad emission spectrum is the characteristic of the indium fluctuations in the quantum well region [50]. The full width half maximum (FWHM) of R-LED and W-LED is 25 nm and 28 nm, respectively at the peak wavelength. With increasing injection current and higher indium fluctuations, FWHM, is reported to be broader in green light-emitting diodes [51]. In addition, indium-rich regions, regardless of high dislocation density, have been attributed to improve radiative recombination because of localization of carriers [51][52][53]. At peak wavelength, the emission of W-LED is ~93% improved in comparison to the R-LED. The emission spectra of the two devices is shown in Figure 7. The peak wavelength of R-LED and W-LED is~518 nm and~513 nm, respectively. The broad emission spectrum is the characteristic of the indium fluctuations in the quantum well region [50]. The full width half maximum (FWHM) of R-LED and W-LED is 25 nm and 28 nm, respectively at the peak wavelength. With increasing injection current and higher indium fluctuations, FWHM, is reported to be broader in green light-emitting diodes [51]. In addition, indium-rich regions, regardless of high dislocation density, have been attributed to improve radiative recombination because of localization of carriers [51][52][53]. At peak wavelength, the emission of W-LED is~93% improved in comparison to the R-LED. Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 10 Figure 7. Emission spectra of R-LED and W-LED. The inset shows the peak wavelength comparison of the normalized emission spectra of the two devices.

Conclusions
The proposed W-LED structure has significant improvement in the device output in comparison to the reference rectangular LED. The enhancement in the light output power and the internal quantum efficiency are associated with the improved radiative recombination rate and reduced defect-assisted SRH, and the Auger recombination rates contribute in the proposed device. The efficiency rolloff in the proposed device is associated with the higher built-in field in comparison to the reference structure. It is also to be noticed that by increasing the number of quantum wells in W-LED, the overall device performance could further be improved.

Conclusions
The proposed W-LED structure has significant improvement in the device output in comparison to the reference rectangular LED. The enhancement in the light output power and the internal quantum efficiency are associated with the improved radiative recombination rate and reduced defect-assisted SRH, and the Auger recombination rates contribute in the proposed device. The efficiency rolloff in the proposed device is associated with the higher built-in field in comparison to the reference structure. It is also to be noticed that by increasing the number of quantum wells in W-LED, the overall device performance could further be improved.