In conventional hexagonal InGaAlN multiple-quantum-well (MQW) (h-) light-emitting diodes (LEDs), carrier leakage from QWs is the main source of internal quantum efficiency (IQE) degradation without contributing to the LED efficiency droop. Our analysis based on the newly developed Open Boundary Quantum LED Simulator indicates that radiative recombination is hampered by the poor electron–hole wavefunction overlap induced by strong internal polarization for which QW carriers mostly recombine via Auger scattering rather than by radiative processes. By contrast, in non-polar h-LEDs, the IQE peak doubles its value compared to conventional h-LEDs while quenching the efficiency droop by 70% at current density of 100 A/cm2. Those effects are further enhanced in cubic InGaAlN MQW (c-) LEDs for which the IQE peak increases by an additional 30%, and the efficiency droop is further reduced by 80% compared to non-polar h-LEDs, thanks to the larger optical transition matrix element and the strong electron–hole wavefunction overlap in c-LEDs. Overall, a c-LED with a low efficiency droop of 3% at 100 A/cm2 is anticipated, paving a clear pathway toward ultimate solid-state lighting.

InGaAlN multiple-quantum-well (MQW) light-emitting diodes (LEDs) are driving the solid-state lighting (SSL) market with a compound annual growth rate of 14.4% (2018–2023) due to increasing utilization in consumer electronics and commercial, industrial, and residential usages. Yet, performances of LED-based SSL are far below their theoretical limit because traditional LEDs suffer from quantum efficiency rollovers under high current densities called efficiency droop. This shortcoming imposes a design trade-off between light output power, efficiency, and costs.1 

Numerous mechanisms, such as Auger recombination, carrier leakage, internal polarization, and last but not least, phase-space filling, have been proposed to explain the efficiency droop in conventional hexagonal InGaAlN MQW (h-) LEDs; yet, none of them explains the efficiency droop alone.2 In some of the latest experimental works, hot-carrier emission from the active region of h-LEDs strongly points to Auger recombination,3,4 while other experiments attribute the mixture of mid- and high-energy peaks in the electron emission spectra to the coexistence of secondary Auger-electron escaping the QWs and carrier leakage in h-LEDs.5,6 The same ambiguity persists in theoretical works where some simulations show Auger current outweighing carrier leakage current in h-LEDs,7–9 whereas other works not only claim the opposite, but propose to quench the efficiency droop experimentally by growing h-LEDs on non-/semi-polar substrates,10,11 increasing barrier doping,12 using the electron blocking layer (EBL) that has an internal polarization matching to GaN, or band-engineering the EBL.13,14 In some respects, the remedies against the efficiency droop have been shown to reduce carrier leakage, but their effects on Auger recombination have not been investigated simultaneously, thereby leading to potential misinterpretations of the inherent cause of the efficiency droop. Very recently, exceptionally strong Auger recombination in h-LEDs has been explained by the coexistence of strong internal polarization, large QW hole effective mass, and small active volume, thereby correlating internal polarization with Auger recombination.15,16 This interpretation disregards the role of carrier leakage in the efficiency droop and casts doubt on the actual droop reduction mechanism in non-polar h-LEDs.

In this article, the interplay between Auger recombination, carrier leakage, and internal polarization and their respective contributions to the efficiency droop in polar h-LEDs, non-polar h-LEDs, and cubic InGaAlN MQW (c-) LEDs are investigated by computational modeling. It is found that the carrier leakage in h-LEDs degrades the internal quantum efficiency (IQE) peak by 64% but does not account for the efficiency droop, since most QW carriers end up recombining via the Auger process. On the opposite, by considering non-polar h-LED to eliminate the internal polarization, the IQE peak doubles, and the efficiency droop is quenched by 70%, which is attributed to the lower carrier densities and the weakened Auger recombination compared to polar h-LEDs. Finally, c-LEDs with larger optical transition matrix elements compared to h-LEDs are found to increase the IQE peak by an additional 30%, while quenching the efficiency droop further 80% with respect to non-polar h-LEDs.

In our work, we simulate a polar h-LED, a non-polar h-LED, and a c-LED with the following structure configuration reported in the experiment:11 160-nm thick n-type GaN:Si (Si concentration of 3 × 1018 cm−3)/6 periods of 2.7-nm-thick i-In0.22Ga0.78N QW with 11-nm-thick i-GaN barrier/20-nm p-Al0.1Ga0.9N:Mg (Mg concentration of 1018 cm−3)/200-nm-thick p-GaN:Mg (Mg concentration of 1018 cm−3). Later in the analysis, the polar h-LED was found to have significant carrier leakage; hence, it is labeled as “polar leaking h-LED.” To eliminate QW carrier leakage, another polar h-LED with an increased p-Al0.1Ga0.9N doping density (1019 cm−3) is simulated. Compared to the polar leaking h-LED, this modified structure enhances electron blocking and hole injecting capabilities, thereby preventing carrier leakage, and is identified as “polar non-leaking h-LED.” Schematics of the four LED device structures are shown in Fig. S1 in the supplementary material.

For the implementation of our Open Boundary Quantum LED Simulator (OBQ-LEDsim),2,17 we use variational electron and hole wavefunctions in the ground state of individual QWs with finite barriers and electric fields,

ψ(x)=β22παsin(απβ)eα(x+γ)cosh[β(x+γ)],
(1)

where α, β, and γ are variational parameters accounting for the wavefunction symmetry, width, and position, respectively. While perfectly consistent with the k · p technique,2 this formalism has the advantage of removing artificial boundaries between QW and classical continuum, allowing inter-well carrier interaction and recombination at arbitrary positions outside QWs to be modeled with high accuracy. Details of the numerical methods and the material parameters were reported in earlier works.2,17–20 To include phase-space filling effects and increase the predictive capability of OBQ-LEDsim, spontaneous emission rates in QWs and classical continuums are calculated by Fermi's golden rule,21 

Rsp=nrq2π2c3ϵ0m020(Egeff+E)|Mchh|2Dr(E)f(Ec,0+qϕn+mrE/mekBT)f(Ev,0qϕp+mrE/mhkBT)dE,
(2)

where nr, q, , c, ϵ0, m0, kB, T, Egeff, Ec,0(Ev,0), ϕn(ϕp), mr, and me(mh) are refractive index, elementary charge, reduced Planck constant, speed of light, vacuum permittivity, free electron mass, Boltzmann constant, temperature, effective bandgap, ground-state energy in the conduction (valence) band, quasi-Fermi potential for electrons (holes), reduced effective mass, and in-plane electron (hole) effective mass, respectively. Dr and f are reduced density of states and Fermi–Dirac function, respectively. |Mchh|2 is the optical transition matrix element between the conduction and valence bands obtained by generalizing Kane's model.21,22 For the sake of comparison, the calculated radiative coefficients of bulk hexagonal and cubic In0.22Ga0.78N in non-degenerate conditions are 1.85 × 10−11 and 6.09 × 10−11 cm3 s−1, respectively. The former agrees with experiments,23 while the latter being three times larger than the former is consistent with the prior theoretical study24 as well as with experimental data obtained in bulk cubic GaN,25 cubic GaN quantum dot,26 and cubic InGaN QW.27 It should be noted that in comparing our theoretical results with experiment in the polar leaking h-LED and the non-polar h-LED, we had to turn off our OBQ-LEDsim routine accounting for incomplete dopant ionization because the experimental literature omitted the structure doping profile to only provide the QW carrier concentrations.11 The code, however, accounts for carrier degeneracy over the whole LED structures.28 

In our theoretical analysis, we use a Shockley–Read–Hall (SRH) nonradiative lifetime of 50 ns, an ambipolar Auger coefficient of 3.00 × 10−30 cm6 s−1, and an Auger electron–hole asymmetry of 0.4. The simulations agree very well with the experimental results conducted on the polar leaking h-LED and the non-polar h-LED as shown in Fig. 1. For consistency, these nonradiative parameters are assumed in the polar non-leaking h-LED and the c-LED. It should be acknowledged that the Auger coefficients in c-LEDs are still unavailable, but first-principles calculations indicate that they are smaller than their h-LED counterparts.29 Assuming the same Auger coefficient for both phase LEDs, thus, underestimates the c-LED performances. In Fig. 1, a recent experimental work on eight-QW polar h-LED is simulated and compared with the measurement using the same parameters, except that the SRH nonradiative lifetime of 12 ns is used.30 This again validates the OBQ-LEDsim capability in modeling today's LEDs.

FIG. 1.

Experimental (solid symbols) and simulated (open symbols) normalized internal quantum efficiency (IQE) data as a function of current density for polar and non-polar InGaAlN MQW h-LEDs. The polar leaking (gray hexagons) and non-polar (blue hexagons) h-LEDs are from Ref. 11 and the polar h-LED (green hexagons) is from Ref. 30. A schematic of the device structures is shown in the bottom inset.

FIG. 1.

Experimental (solid symbols) and simulated (open symbols) normalized internal quantum efficiency (IQE) data as a function of current density for polar and non-polar InGaAlN MQW h-LEDs. The polar leaking (gray hexagons) and non-polar (blue hexagons) h-LEDs are from Ref. 11 and the polar h-LED (green hexagons) is from Ref. 30. A schematic of the device structures is shown in the bottom inset.

Close modal

In Fig. 2, we display the histogram of the (a) electron sheet charge densities (n2D), (b) hole sheet charge densities (p2D), and (c) electron–hole wavefunction overlaps (ψe|ψeψhψh) in each individual QWs for all four LEDs operated under 100 A/cm2 current density. The corresponding band diagrams, including band edges, quasi-Fermi levels, and wavefunctions, and position-dependent carrier concentrations are shown in Fig. S2 in the supplementary material. One notices that in the polar leaking h-LED, the electrons and holes are captured and recombined primarily in the last two (5th and 6th) QWs because of strong internal polarization. In this case, many electrons that are not captured by the last QW contribute to overall carrier leakage efficiency degradation. In the polar non-leaking h-LED, the large EBL doping density of 1019 cm−3 increases (decreases) the effective AlGaN barrier height for electrons (holes) in the conduction (valence) band. This configuration enhances electron QW capture and hole injection across the LED active region. Therefore, both n2D and p2D, particularly in the last two QWs, increase by 22% and 73% relative to the polar leaking h-LED corresponding QWs. It should be emphasized that internal polarization remains unaffected in the polar non-leaking h-LED as evidenced by the same electron–hole wavefunction overlap of ∼0.3 in each QW. This observation implies that the polar non-leaking h-LED clearly singles out the effect of carrier leakage from the influence of internal polarization on the efficiency droop. Now, in the non-polar h-LED, as the internal polarization effect is non-existent, electrons (and holes) are captured mostly in the first (and last) QW, while spreading uniformly among the remaining wells. When compared to the polar non-leaking h-LED, the electron–hole wavefunction overlaps in the non-polar h-LED surge to 0.96 and enables homogenous recombination across all QWs (rather than in the last two QWs), which reduces the n2D (and p2D) peaks from 5.91 × 1012 to 2.58 × 1012 cm−2 (and from 3.78 × 1012 to 1.62 × 1012 cm−2). In the c-LED, the n2D and p2D concentrations in each QW are further decreased by ∼20% on average relative to the non-polar h-LED. These density reductions are attributed to the fact that the c-LEDs have a smaller hole effective mass than the h-LEDs, thereby resulting in a greater electron–hole wavefunction overlap of 0.98.15 Overall, this comparison among the four structures elucidates the influence of carrier leakage and internal polarization on carrier densities and wavefunction overlaps distributed among the six QWs; it emphasizes the pivotal roles played by the redistribution of carrier densities in recombination processes and efficiency droop.

FIG. 2.

(a) Electron sheet charge density (n2D), (b) hole sheet charge density (p2D), and (c) electron–hole wavefunction overlap (ψe|ψeψhψh) in each individual quantum wells at 100 A/cm2 for the polar InGaAlN MQW leaking h-LED, the polar non-leaking h-LED, the non-polar h-LED, and the InGaAlN MQW c-LED.

FIG. 2.

(a) Electron sheet charge density (n2D), (b) hole sheet charge density (p2D), and (c) electron–hole wavefunction overlap (ψe|ψeψhψh) in each individual quantum wells at 100 A/cm2 for the polar InGaAlN MQW leaking h-LED, the polar non-leaking h-LED, the non-polar h-LED, and the InGaAlN MQW c-LED.

Close modal

Figure 3 displays the individual contributions of the current densities resulting from (a) carrier leakage (Jleak), (b) SRH recombination (JSRH), (c) radiative recombination (Jrad), and (d) Auger recombination (JAuger) as a function of current density for all four LEDs. As shown in the polar leaking h-LED in Fig. 3(a), QW carrier leakage is the main contributor to the total current density due, on the one hand, to the small effective EBL barrier height for electrons and, on the other hand, to poor hole injection. This dominance of carrier leakage over the total current density is reduced either by increasing EBL doping density or by quenching internal polarization. The former simply enhances carrier densities in QWs leading to higher recombination current. Hence, the SRH, radiative, and Auger current densities in the polar non-leaking h-LED are higher than those in the polar leaking h-LED as shown in Figs. 3(b)3(d). As for the effect of internal polarization quenching, it not only prevents carrier leakage from QWs, but results in carrier redistribution and affects optical transition matrix elements, as will be discussed in the comments of Figs. 3(b)3(d). In Fig. 3(b), the SRH current density in the absence of internal polarization (i.e., in the non-polar h-LED and the c-LED) is higher than that in its presence (i.e., in the polar h-LEDs). This is because the SRH recombination rate is dominated by minority carrier density, which is higher in the absence of internal polarization. However, one notices that the radiative current density offsets the SRH current density at a low current density of ≤0.2 A/cm2 for all four LEDs, thereby weakening the effect of SRH recombination on the efficiency droop. For a total current density of 100 A/cm2, one can see that the polar leaking h-LED, usually taken as the structure reference, suffers from a large carrier leakage current density of 69 A/cm2 and a poor electron–hole wavefunction overlap of 0.3, resulting in the smallest radiative current density of 13 A/cm2 among the four LEDs, as shown in Fig. 3(c). However, in the polar non-leaking h-LED, the carrier leakage is suppressed; therefore, the radiative recombination contribution to the total current density of 100 A/cm2 surges to 29 A/cm2. Still, the optical transition matrix element in this latter structure is limited by the poor electron–hole wavefunction overlap, for which most carriers recombine via the Auger process as discussed later in Fig. 3(d). The non-polar h-LED featuring a large electron–hole wavefunction overlap of 0.96 boosts the radiative current density to 55% of the total current density of 100 A/cm2. This is further increased to 80 A/cm2 in the c-LED because the band-to-band optical transition matrix element of cubic In0.22Ga0.78N is higher than that of hexagonal In0.22Ga0.78N. Additionally, the hole effective mass of 0.84 m0 in cubic In0.22Ga0.78N is smaller than its hexagonal counterpart of 1.80 m0, thereby enhancing the electron–hole wavefunction overlap to 0.98.15 In Fig. 3(d), it can be seen that, in the polar leaking h-LED, the Auger current density contributes for 18 A/cm2 to the total current density of 100 A/cm2 because most electrons escape easily from the QWs and do not participate in recombination processes. In the polar non-leaking h-LED, the Auger current density surges from 18 to 70 A/cm2 as most carriers are captured by QWs enhancing the overall recombination rate. The polar non-leaking h-LED does not experience such a high carrier leakage current density of 69 A/cm2 as shown in Fig. 3(a) but ends up adding to the SRH, radiative, and Auger current densities by 0.5, 16.5, and 52 A/cm2 as shown in Figs. 3(b)3(d), respectively. The carriers captured by the QWs recombine mostly via the Auger process, instead of radiatively, because of the poor electron–hole wavefunction overlap. In the non-polar h-LED, the Auger current density decreases to 39 A/cm2 because the structure averts carrier leakage while quenching internal polarization. In the c-LED, this value is further reduced to 15 A/cm2. In addition to the absence of both carrier leakage and internal polarization, the quenching of the Auger process is a consequence of the small hole effective mass, coinciding with enhanced radiative recombination boosted by the large band-to-band optical transition matrix element in c-LEDs. From this analysis on the behavior of the four different types of InGaAlN LEDs, one concludes that the removal of internal polarization becomes a pre-condition for designing best-performing LEDs because the internal polarization either causes large carrier leakage current density or induces substantial Auger current density.

FIG. 3.

Individual contributions of the current densities resulting from (a) carrier leakage (Jleak), (b) SRH recombination (JSRH), (c) radiative recombination (Jrad), and (d) Auger recombination (JAuger) as a function of the current density. Gray solid hexagons and red open hexagons refer to the polar InGaAlN MQW leaking h-LED and the polar non-leaking h-LED; blue hollow hexagons refer to the non-polar h-LED. Green solid squares represent the InGaAlN MQW c-LED.

FIG. 3.

Individual contributions of the current densities resulting from (a) carrier leakage (Jleak), (b) SRH recombination (JSRH), (c) radiative recombination (Jrad), and (d) Auger recombination (JAuger) as a function of the current density. Gray solid hexagons and red open hexagons refer to the polar InGaAlN MQW leaking h-LED and the polar non-leaking h-LED; blue hollow hexagons refer to the non-polar h-LED. Green solid squares represent the InGaAlN MQW c-LED.

Close modal

Figures 4(a) and 4(b) display the IQE and efficiency droop as a function of current density for all four LEDs. In the polar leaking h-LED, the IQE at 100 A/cm2 and IQE peak are pretty low at 13% and 23%, respectively, as seen in Fig. 4(a), which accounts for the large efficiency droop of 46% in Fig. 4(b) due to the coexistence of carrier leakage and internal polarization in the structure. In the polar non-leaking h-LED, suppression of the carrier leakage does increase the IQE at 100 A/cm2 and IQE peak to 29% and 65%, respectively [Fig. 4(a)], but the efficiency droop nevertheless increases from 46% to 55% [Fig. 4(b)]. In polar non-leaking h-LEDs, prior studies reported a similar efficiency droop increase when the carrier leakage was suppressed by increasing the EBL bandgap (rather than the EBL doping density) that lifts the effective AlGaN barrier height in both conduction and valence bands.31 Hence, the enhanced efficiency droop was explained by poor hole injection.31 In our study, however, the polar non-leaking h-LED experiences better hole injecting capability compared to polar leaking h-LEDs since increasing the EBL doping density lowers the effective AlGaN barrier height for holes in the valence band. This suggests that the enhancement of the efficiency droop is not caused by the poor hole injection, but because most carriers captured by QWs recombine via the Auger process as explained in Fig. 3(d). In the non-polar h-LED, the IQE at 100 A/cm2 and IQE peak surge to 55% and 63%, respectively, as shown in Fig. 4(a), which results in a low efficiency droop of 13% as shown in Fig. 4(b). Switching from the non-polar h-LED to the c-LED further enhances the IQE to 80% at 100 A/cm2, whereas the IQE peaks to 82% [Fig. 4(a)], thereby quenching the efficiency droop down to 3% only [Fig. 4(b)]. It should be pointed out that such low efficiency droop in c-LEDs is achieved by the combination of the suppression of internal polarization, the lower hole effective mass, and the larger band-to-band optical transition matrix element than in h-LEDs, even though the Auger coefficients of the former LEDs are assumed to be as large as those in the latter LEDs. By complementing our earlier study,15 these findings clarify the ambiguous correlation between Auger recombination, carrier leakage, and internal polarization as well as their respective contributions to the efficiency droop. Ultimately, c-LEDs are anticipated to be a practical low-droop solution for the next-generation SSL.

FIG. 4.

(a) Internal quantum efficiency (IQE) and (b) normalized IQE (left y-axis) and efficiency droop (right y-axis) as a function of current density. Gray solid hexagons and red open hexagons refer to the polar InGaAlN MQW leaking h-LED and the polar non-leaking h-LED, whereas blue hollow hexagons refer to the non-polar h-LED. Green solid squares represent the InGaAlN MQW c-LED.

FIG. 4.

(a) Internal quantum efficiency (IQE) and (b) normalized IQE (left y-axis) and efficiency droop (right y-axis) as a function of current density. Gray solid hexagons and red open hexagons refer to the polar InGaAlN MQW leaking h-LED and the polar non-leaking h-LED, whereas blue hollow hexagons refer to the non-polar h-LED. Green solid squares represent the InGaAlN MQW c-LED.

Close modal

In conclusion, our analysis on the performances of four different InGaAlN MQW LEDs indicates there is no single cause to IQE degradation and the efficiency droop in conventional h-LEDs. The competition between Auger recombination and carrier leakage in IQE degradation and the efficiency droop is not only associated with the device design (e.g., EBL), but also depends on the crystal structure (i.e., internal polarization, hole effective mass, and band-to-band optical transition matrix element). Nevertheless, the primary factor for IQE degradation and droop is the internal polarization since it either enhances Auger recombination or increases carrier leakage. Both effects cannot be averted by modifying the device design alone. Switching from hexagonal to cubic phase LEDs enhances the IQE peak from 23% (in the polar leaking h-LED) to 82% (in the c-LED), while quenching the efficiency droop at 100 A/cm2 from 46% (in the polar leaking h-LED) to 3% (in the c-LED). Our c-LED analysis paves a clear pathway toward the ultimate low-droop SSL.

See the supplementary material for the schematics and band diagrams of the four LEDs.

This work was supported by the National Science Foundation Faculty Early Career Development (CAREER) Program under Award No. NSF-ECCS-16–52871. The authors acknowledge the computational resources allocated by the Extreme Science and Engineering Discovery Environment (XSEDE) with No. TG-DMR180075.

The authors have no conflicts to disclose.

Ethics approval is not required.

The data that support the findings of this study are available within the article and its supplementary material.

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