1 Introduction

Gallium nitride-based light-emitting diodes (LEDs) show that gradual quantum efficiency decreases when the operation current moves toward the high current density range, a phenomenon dubbed as efficiency droop (see ref. [1] for a short review). Indeed, quantum efficiency can be affected by several unwanted processes such as self-heating [2], carrier overflow [3,4,5], electron leakage [6], strain effect [7], and Auger processes [1]. Recent electron emission spectroscopy investigation gave a direct evidence that Auger is the dominant non-thermal loss mechanism in GaN/InGaN LEDs [8]. The simplest way to deal with this matter is given by the ABC model that balances the density of the current injected in the quantum well (QW) and the carrier recombination rates, according to the following equation:

$${J_{{\text{QW}}}}/qd=A\,n+B\,{n^2}+C\,{n^3},$$
(1)

where \(q\) is the carrier charge, \(d\) the QWs thickness, \(n\) the non-equilibrium carrier density, A, B, and C the probability factors of Shockley–Read–Hall (SRH), and radiative and Auger recombinations, respectively [1]. Equation (1) assumes that non-equilibrium densities of electrons and holes are equal. The internal quantum efficiency (IQE), under the assumption of unitary injection efficiency, is defined as follows:

$${\eta _i}=\frac{{B\,{n^2}}}{{An\,+B\,{n^2}+\,C\,{n^3}}}.$$
(2)

The major criticism about the ABC model concerns the assumption that the QW is charge neutral [9,10,11,12,13], which is exceedingly restrictive, especially in the strong non-equilibrium regime where the confined carrier densities could be highly imbalanced. In particular, hole density is expected to be lower than the electron density due to poor hole injection, typical for GaN-based LED [9]. Moreover, the built-in polarization field, which contributes to differentiate carrier confinement inside the QW, and the occurring mitigating screening effect [10, 11] could affect the recombination rates. Indeed, it was shown that these effects also make unreliable the parameter figures retrieved by fitting operations [9], at least in the high current regime. On the other hand, the recent theoretical investigation showed that also more advanced models are far from to be conclusive on this matter [9], since specialized software packages are very CPU demanding, and require large information on the actual system, data which often are not accessible.

Nevertheless, it was found in many cases that Eq. (2) remains reasonably close to experiments, although it underestimates the efficiency droop at the high current densities [1]. It was proposed that terms of higher order than third one should be added to the balance equation to better match the efficiency curve [1, 14, 15]. This solution, of course, implies the use of a larger set of free parameters. Alternatively, it was suggested that the phase space filling (PSF) effect can produce the requested match. Indeed, the PSF gives, to band-to-band recombination probability factors, a semi-empirical dependence on carrier density, like \({B_0}/{(1+n/n*)^\gamma }\) and \({C_0}/{(1+n/n*)^\gamma }\), where \({B_0}\) and \({C_0}\) are the corresponding probability factors in the low carrier density limit, \(n*\) and \(\gamma\) are proper fitting parameters [16, 17]. However, the role of the PSF effect is debated because of its prediction on IQE temperature dependence due to the hole degeneration, especially in QW. The experimental results on the IQE temperature dependence are contrasting and there is not a clear indication, thus casting some doubts on the PSF effect [1, 18]. Moreover, also by considering PSF effect, the number of free parameters in the model is increased.

Since the carrier imbalance is highly probable in GaN-based LEDs, at least in the high carrier density regime, the above remarks suggest the possibility of revisiting the ABC model in a different perspective, that is in a way that could relax the QW charge neutrality assumption, thus leading back the accent on the physical phenomenon (the carrier imbalance) but keeping the original equation structure.

In this paper, we show that, by introducing a proper effective carrier density, the balance equation can be reduced in a way that incorporates Auger recombination, thus leading to a reduction of the number of free parameters. This could be advantageous in retrieving probability coefficients, since it reduces uncertainties rising from the Auger term. In Sect. 2, we will show that charge imbalance can be dealt with by the effective carrier density by changing the band-to-band recombination rates in a similar way as the PSF effect, the corresponding density parameter being defined as \({n_R}=B/\tilde {C}\), where \(\tilde {C}\) stands for a modified Auger probability factor. In Sect. 3, data available in the literature are used to compare the new model with the ABC and more sophisticated package-based models, showing that the new model guarantees the same descriptive efficiency of ABC and even better at larger current densities. Finally, the new model allows retrieving the dependence of carrier lifetime on current density that, when SRH recombination is negligible, can be expressed as \(\tau \approx \sqrt {qd/(4B{J_{QW}})}\).

2 The modified ABC model

To modify the ABC model accounting for the carrier density unbalance, we replace Eq. (1) with a simplified form of the balance equation where electron (n) and hole (p) densities are explicated [9,10,11]:

$${J_{{\text{QW}}}}/qd=A\,n+B\,np+C\,{n^2}p.$$
(3)

The SRH term is kept unchanged with respect to Eq. (1) as our main concern is the high current density range and possible mechanisms of carrier imbalance relevant in the low-density range are neglected [13]. The used Auger term corresponds to the electron–electron–hole processes that are found to be dominating, at least, in blue LED structures [1, 8, 13].

At the high current density range, the SRH recombination can be neglected, leading to the following:

$${J_{{\text{QW}}}}/qd \simeq B\,np+C\,{n^2}p.$$
(4)

Since we are interested in the comparison with the ideal Auger-free system, the injection current density can also be written as follows:

$${J_{{\text{QW}}}}/qd \simeq B\,{n_i}{p_i},$$
(5)

where \({n_i}\) and \({p_i}\) are the carrier densities that one could expect in the ideal case. Let be \({n_{{\text{eff}}}}=\,\sqrt {{n_i}{p_i}}\) an effective carrier density, so that \({n_i}=\alpha \,{n_{{\text{eff}}}}\) and \({p_i}={n_{{\text{eff}}}}/\alpha\), where \(\alpha\) is a proper numerical factor; due to different injection efficiency \({n_i}>{p_i}\), so that \(\alpha>1\). On the contrary, in a real system, Auger processes are active and we expect a decrease of both carrier densities with respect to the ideal case, in particular \(n={n_i}/\beta\) with \(\beta>1\). Both numerical factors are dependent on the injection current, since many unpredictable effects can increase or decrease the carrier imbalance [10, 11]. Thus, in the approximation dealt with, it holds the following:

$$B\,{n_{{\text{eff}}}}^{2}\,=\,B\,\frac{\alpha }{\beta }{n_{{\text{eff}}}}p+C\,{\left( {\frac{\alpha }{\beta }} \right)^2}{n_{{\text{eff}}}}^{2}\,p$$
(6)

that allows for

$$p=\frac{{\left( {\beta /\alpha } \right)\,{n_{{\text{eff}}}}}}{{1+\left( {\alpha /\beta } \right)C\,{n_{{\text{eff}}}}/B}}.$$
(7)

Thus, the radiative and Auger components of the injected current can be expressed as follows:

$$\frac{{{J_{{\text{rad}}}}}}{{qd}}=\frac{{B\,{n_{{\text{eff}}}}^{2}}}{{1+{n_{{\text{eff}}}}/{n_R}}}\,\,\,\,\,\,\,\,\,\,\frac{{{J_{{\text{Auger}}}}}}{{qd}}=\frac{{\tilde {C}\,{n_{{\text{eff}}}}^{3}}}{{1+{n_{{\text{eff}}}}/{n_R}}},$$
(8)

where \(\tilde {C}\) stands for a modified Auger probability factor and \(\tilde {C}=\left( {\alpha /\beta } \right)\,C\) with \({n_R}=B/\tilde {C}\). Now, the model can be completed by merging the above high-density range modification with the low-density range ABC model, where the Auger can be discarded, that is

$${J_{{\text{QW}}}}/qd=A\,{n_{{\text{eff}}}}+B\,{n_{{\text{eff}}}}^{2}.$$
(9)

We maintain the usual ABC assumptions that the equilibrium electron–hole density product is much smaller than np in QW under operating conditions and that the defect parameters in the SRH rate equation are neglected, as well [9, 19]. Thus, by extending Eq. (7) to the low-density range, so that approximation \(p \propto {n_{eff}}\) can be used, the proportional dependence of SRH on \({n_{eff}}\) is explained. Accordingly, the SRH coefficient in Eq. (9) acquires a dependence on the \(\alpha /\beta\) ratio that in turn could lead to a dependence on the carrier densities. We will not be concerned on the details of SRH recombination in GaN/InGaN QWs which have been deeply investigated in a recent paper [19], especially in connection with the dependence of recombination coefficients on the overlap of the electron and hole wavefunctions. As for the high-density range, the one we are concerned with, we will also consider how polarization and screening effects could affect the fit parameters in the next section.

Equations (8), (9) do not exclude the possibility of considering the band filling effect (PSF effect). In this case, the use of \({n_{eff}}\) implicitly changes n* in the same way as \(\tilde {C}\). However, as the role of the PSF effect in experimental IQE curves is not definitely established, in the next section, we will eventually consider both the cases of finite and infinite n*. Moreover, according to Eq. (9), the carrier density, which can be determined from the knowledge of lifetime \(\tau\); that is [16, 20]:

$$n=\int_{0}^{G} {\tau \,{\text{d}}G} ,$$
(10)

where \(G={J_{{\text{QW}}}}/qd\), which will be interpreted as the effective carrier density; that is:

$$\frac{1}{\tau }=\frac{{dG}}{{d{n_{eff}}}}.$$
(11)

3 Discussion

Equations (8) and (9) still do not contain additional assumptions with respect to the ABC model; on the contrary, the criticized charge neutrality assumption has been removed. However, this leaves us with a parameter \(\tilde {C}\) which, in general, is dependent on the injected current. In the following, we will analyse experiments by assuming \(\alpha /\beta\) to be constant. This is reasonable in the case of weak dependence on the injected current. On the contrary, since \({n_{{\text{eff}}}}\) is only dependent on \({n_i}\) and \({p_i}\), if \(\alpha /\beta\) increases significantly as the carrier densities increase, the above assumption leads to an underestimation of the IQE droop, and vice versa.

As a first test, we use data extracted from ref [16], where differential carrier lifetime experiments where performed on a GaN/InGaN double hetero-structure LED under varying electroluminescence injection conditions. Figure 1 shows the differential carrier lifetime (squares) and the corresponding radiative component (circles) vs carrier density, the latter being determined by means of Eq. (10): we prefer to deal with carrier density rather than current density to easy handle the PSF effect (at the end of this section, we will consider other lifetime data vs current density). As a matter of fact, these data were early fitted by an ABC model with \(A=2 \times {10^7}{{\text{s}}^{ - 1}}\), \({B_0}=7 \times {10^{ - 11}}\,{\text{c}}{{\text{m}}^{\text{3}}}\,{{\text{s}}^{ - 1}}\), \({C_0}={10^{ - 29}}\,{\text{c}}{{\text{m}}^{\text{6}}}{{\text{s}}^{ - 1}}\), and \(n^{*}=0.5 \times {10^{19}}\,{\text{c}}{{\text{m}}^{ - 3}}\) (\(\gamma =1\)) (dashed lines). To emphasize how PSF impacts on data interpretation, Fig. 1 also shows the curves corresponding to the case \(n^{*}=\infty\) (dash–dot lines). It can be noted that the early found \(n^{*}\) is much lower than the expected order of magnitude [1], but it is close to ratio \(B/C\), that is to \({n_R}\). Thus, for comparison, we consider convenient to maintain the above figures as A, \({B_0}\), and \({\tilde {C}_0}\) in Eqs. (8) and (9). Accordingly, based on Eq. (11) and by taking into account the PSF modification, the inverse differential carrier lifetime becomes the following:

Fig. 1
figure 1

Experimental curves of the differential carrier lifetime (squares) and the differential radiative lifetime (circles) versus carrier density: adapted from David and Grundmann (2010) (ref. [16]), with the permission of AIP publishing

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$$\frac{1}{\tau }=A+\frac{{{B_0}\,{n_{{\text{eff}}}}}}{{1+{n_{{\text{eff}}}}/n^{*}}}\left( {1+\frac{1}{{1+{n_{{\text{eff}}}}/n^{*}}}} \right).$$
(12)

Using Eq. (12) to fit experimental data, we consider two cases, that is \(n^{*}=10\,{n_R}\) (solid line; the parameter was obtained as the best fit to the B curve vs carrier density [16] here not shown) and \(n^{*}=\infty\) (dotted line). It is clear that, even using \(n^{*}=\infty\), the reduced model allows for a curve fit comparable with the early ABC model. Moreover, there is not a significant improvement in using a finite \(n^{*}\).

A more significant deviation is observed for the radiative component of the lifetime, where an inversion of the curve slope was reported in the high-density regime and the early ABC model largely fails to fit experimental data. It is not clear if this feature is real or an experimental artefact. However, by keeping this in mind, the fitting curves (solid and dotted lines as above) are obtained by calculating the inverse radiative lifetime as follows:

$$\frac{1}{{{\tau _{{\text{rad}}}}}}=\frac{{\text{d}}}{{{\text{d}}{n_{{\text{eff}}}}}}\frac{{{J_{{\text{rad}}}}}}{{qd}}=A+\frac{{{B_0}{n_{{\text{eff}}}}}}{{\left( {1+{n_{{\text{eff}}}}/n^{*}} \right)\left( {1+{n_{{\text{eff}}}}/{n_R}} \right)}}\left( {\frac{1}{{1+{n_{{\text{eff}}}}/{n_R}}}+\frac{1}{{1+{n_{{\text{eff}}}}/n^{*}}}} \right),$$
(13)

where we must deal with both the imbalance and PSF effects. Equation (6) shows that a change in the kinetic order is expected when carrier density becomes larger than \({n_R}\); that is, \({J_{{\text{rad}}}}/qd\sim {B_0}{n_{{\text{eff}}}}{n_R}\). Note that the only carrier imbalance would lead to a constant \({\tau _{{\text{rad}}}}\) in the high-density range (dotted line). This was also early predicted by considering the PSF effect (dashed line) [11], and essentially, this is the reason why PSF was considered as a correction to the early ABC model under these conditions. Instead, from Eq. (6), it is expected that \({J_{{\text{rad}}}}/qd\sim Bn^{*}{n_R}\) and, in turn, an increase of the differential lifetime, supporting the observed curve-slope inversion. It is evident that even if PSF is not occurring in this case, the reduced model with \(n^{*}=\infty\) is sufficient to explain the experimental data. Thus, to sum up the analysis of the first case, the modified ABC model is more efficient in describing differential lifetime and differential radiative lifetime than the early ABC model even without PSF effect. If PSF effect is also considered, the modified ABC model produces the best fit of the experimental data.

The modified ABC model can also be tested against IQE curve vs injected current. Indeed, using Eq. (9), the IQE function (ηi) can be easily handled in the density current domain; that is (as usual, it is assumed that \({J_{{\text{QW}}}}=J\)):

$${\eta _i}=\frac{{B{n_{{\text{eff}}}}^{2}/(1+{n_{{\text{eff}}}}/{n_R})}}{{A{n_{{\text{ef}}f}}+B{n_{{\text{eff}}}}^{2}}}=\frac{{{J_L}{{\left( {\sqrt {1+J/{J_L}} - 1} \right)}^2}}}{{J\left( {1+\sqrt {{J_L}/{J_R}} \left( {\sqrt {1+J/{J_L}} - 1} \right)} \right)}},$$
(14)

where \({J_L}/qd={A^2}/4B\) and \({J_R}/qd={B^3}/{\tilde {C}^2}\). Equation (13) can be eventually converted in current domain, by easy swap \(J \to I\). The features of the IQE curve are now uniquely determined by the two density current parameters, being the IQE maximum calculated as follows:

$${\eta _M}=\frac{Q}{{Q+2+1/Q}},$$
(15)

where \(Q=B/\sqrt {A\tilde {C}} ={({J_R}/4{J_L})^{1/4}}\) stands for the quality factor [1]. Note that Eq. (15) differs from the one predicted by the ABC model by the presence of the term \(1/Q\). The current density at maximum is calculated as follows:

$$\frac{{{J_M}}}{{qd}}=A\left( {\sqrt {\frac{A}{{\tilde {C}}}} +\frac{B}{{\tilde {C}}}} \right)=2\,\frac{{\sqrt {{J_L}{J_R}} }}{{qd}}\left[ {1+{{\left( {\frac{{4{J_L}}}{{{J_R}}}} \right)}^{1/4}}} \right].$$
(16)

To test the IQE prediction of our model (Eq. 14), we use experimental data extracted from ref [10], reported in Fig. 2 with the expected results of the modified ABC model (solid line). The best fit parameters are \({I_L}=0.005\,{\text{mA}}\)and \({I_R}=10.480\,{\text{mA}}\) that allow for \({I_M}=0.577\,{\text{mA}}\) and for \({\eta _M}=0.68\). The fitting curve presented in Fig. 2 is scaled by a factor 1.09, since the determination of the IQE maximum is model-dependent [1, 14]. For comparison, it is shown the ABC curve determined as in ref [1] (dashed line) with a proper scaling factor of 1.15. The plot displays a better agreement with the experimental data for the modified ABC model as compared to the early ABC model, in particular concerning the high current regime. In addition, the retrieved ABC quality factor is Q = 3.50, whilst a larger Q = 4.78 is obtained with the modified model. Indeed, the need of correction to the early ABC model was already proposed, but including a non-radiative fourth-order term; in that case, a better agreement was obtained (scaling factor 1), with a curve perfectly overlapping the one of our model reported in Fig. 2 [15]. We point out, however, that, in the present case, the best fit was achieved with a reduced model where only two parameters are required, whilst, in the latter case, the addition of a fourth-order term in the balance equation was required to achieve the same result.

Fig. 2
figure 2

Experimental IQE curve (squares): adapted from Dai e al. (2011) (ref. [15]), with the permission of AIP publishing. The solid curve is obtained by means of Eq. (14); the dashed curve is obtained by means of the ABC model

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In the first test of the model, we have proven that Eq. (9) is well performing in fitting lifetime data as Eq. (1) even if using only A and B parameters. We will dwell upon this point using data from ref [8]. The spectrum of electrons emitted from a GaN/InGaN LED was measured vs the injected current: a linear correlation was found between the electron current and the droop current [8]. Electron current and optical output power data were used for the determination of \(\left( {{J_{{\text{Auger}}}}/{J_{{\text{Rad}}}}} \right)\) vs the injected current. Since \(\left( {{J_{{\text{Auger}}}}/{J_{{\text{Rad}}}}} \right)=n/{n_R}\) (with \(n\)to be replaced with \({n_{{\text{eff}}}}\) in the case of the reduced model), we can re-write Eqs. (1) and (9) as follows:

$${J_{{\text{QW}}}}=a\left[ {{x^{1/2}}+{Q^2}\left( {x+b\,{x^{3/2}}} \right)} \right],$$
(17)

where \(x={\left( {{J_{{\text{Auger}}}}/{J_{{\text{Rad}}}}} \right)^2}\), \(a\) is a fitting parameter, \(b=1\) for the ABC model, and \(b=0\) for the reduced model. Experimental data are shown in Fig. 3 (squares), with Auger and radiative currents normalized with respect to the droop current [8]. We discarded lowest current data because of poor quality. We note that the two models give back two almost overlapping fits with retrieved quality factors \(Q=0.99\) and \(Q=1.69\) for the ABC and reduced models, respectively.

Fig. 3
figure 3

Injected current vs the square of the Auger to radiative current densities ratio versus (symbols). Data have been extracted from ref. [8] (see text). Fitting curves, obtained from Eq. (17), represent the ABC (dashed line) and the modified ABC (solid line) models

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By considering the only perspective of improving fitting on experiments, the above example shows that the use of the Auger term in the balance equation does not guarantee significant improvements. On the contrary, as the polynomial fit functions are very flexible, retrieved parameters are affected by uncertainties that increase as the number of free parameters increases. Thus, the third test underlines that the reduced model, where Auger recombination is hidden, can achieve fitting results for injected current vs \({\left( {{J_{{\text{Auger}}}}/{J_{{\text{Rad}}}}} \right)^2}\)as good as the early ABC model. To get a little deeper on this point, it could be useful a comparison with more detailed models. On this purpose, we use data from ref [9], which also proposes numerical fit with APSYS and Quatra/cels package-based models. These models account for the QW deformation, the built-in polarization field, and the carrier separation in 3D and 2D recombination model, respectively. To test the reduced ABC model against these ones, we considered the IQE and differential lifetime data obtained from experiments on a single quantum well (3 nm) GaN-based LED [9]; data are shown in Figs. 4 and 5 (downward triangles), respectively. From IQE data (Fig. 4), we obtained \({J_L}=0.03\,{\text{A/c}}{{\text{m}}^2}\), \({J_R}=152.24\,{\text{A/c}}{{\text{m}}^2}\), and consequently, \({J_M}=4.99\,{\text{A/c}}{{\text{m}}^2}\) and \({\eta _M}=0.73\). Once again, we highlight that the reduced model achieved very good fit results, better than early ABC model in the high current regime and equivalent to the ones of more sophisticated models. Combined IQE and differential lifetime analysis allows for the ABC figures reported in Table 1, where also analysis results from ref [9] relative to data shown in Fig. 4 are presented. According to Eq. (9), the inverse differential lifetime has been calculated as follows:

Fig. 4
figure 4

Internal quantum efficiency vs current density: adapted from Piprek et al. (2015) (ref. [9]), with the permission of AIP publishing. The solid curve is obtained by means of Eq. (14)

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Fig. 5
figure 5

Carrier lifetime vs current density: adapted from Piprek et al. (2015) (ref. [9]) with the permission of AIP publishing. The solid curve is obtained by means of Eq. (14)

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Table 1 Recombination parameter sets
Full size table
$$\tau =\frac{1}{{A\sqrt {1+J/{J_L}} }}.$$
(18)

We stress that, in the actual current range, that is, for \(J>>{J_L}\), we can use the following:

$$\tau =\sqrt {\frac{{qd}}{{4BJ}}} ,$$
(19)

As, indeed, we made in fitting lifetime data. One can see that Eq. (19) is more constraining than the ABC model in fitting data. Indeed, the latter deals with three free parameters that make easy fit accommodation (eventually reduced to two in the approximation dealt with). On the contrary, the reduced model applies only one free parameter to fit the data. Nevertheless, the small deviation of curve (19) from experiment (see Fig. 5) can be easily accounted for without calling for the Auger term. The comparison with the Quatra/cels-based model, with respect to which we note a closer correspondence, could suggest some insights. The fit outcome of ref [9] and reported in Table 1 are relative to calculations that do not consider the escape from the QW of hot electrons. It was found that the inclusion of the electron loss reduces Auger probability factor, but leaves unchanged radiative probability factor. On the contrary, by removing the polarization effect, we have a significant increase of both the parameters [9]. A reduction of the QW polarization is expected from charge screening at high injection level [10, 11]; consequently, an increase of both B and \({n_R}\) is to be expected, as well. On the other hand, electron loss and QW polarization are sensitive to the actual device fabrication [21], so that their weight cannot be established a priori but only recognized by comparison with a well-assessed model. However, an exploration within a larger range of current densities is required for a further investigation.

We stress again that, in spite of the used oversimplifying assumptions, the canonical ABC model is often considered prototypical in interpreting experiments and, even if it is generally acknowledged that many unpredictable effects are leaved out, it is still used to investigate some specific features of GaN-based LED [22,23,24]. On the other hand, we have shown that a reduction of that model, even if more constraining, does fit equally well experiments. In this connection, Figs. 2, 4 put in evidence how the fitting models can, in some cases, work better than in other cases [22].

Finally, to conclude this section, we show how the charge imbalance affects the canonical ABC model. To this purpose, let us define a new density \(\xi ={n_{{\text{eff}}}}/\sqrt {1+{n_{{\text{eff}}}}/{n_R}}\), so that Eq. (9) becomes for \(\xi <2\,{n_R}\):

$${J_{{\text{QW}}}}/qd=A\,\xi +B{\xi ^2}+C{\xi ^3}\left\{ {1+\frac{\xi }{{2{n_R}}}+O\left[ {{{\left( {\frac{\xi }{{2\,{n_R}}}} \right)}^2}} \right]} \right\},$$
(20)

where expansion of the SHR term has been discarded. The choice of \(\xi\) is necessary to maintain the canonical form of the radiative recombination rate, as opposed to the expression suggested to account for PSF in refs [1, 15]. From this expression, it is still possible to extract the optical power variable (proportional to \(B{\xi ^2}\)) to plot the IQE curves [1, 22]. We underline that Eq. (20) accounts for the extended form of the ABC model, thus including higher than third-order terms [1, 15, 25] .

4 Conclusion

We have proposed a simple and intuitive reduction scheme for the ABC balance equation, accounting for the carrier imbalance in GaN-based LED quantum wells. The reduction is imposed by considering an ideal Auger-free system, with respect to which we consider the internal quantum efficiency droop. Thus, by introducing a proper effective carrier density, the balance equation is converted to a second-order polynomial. The comparison with ABC and more detailed package-based models shows that the above reduction does not cause a loss of efficiency in fitting experiments and in retrieving recombination coefficients. On the contrary, the reduced model is more efficient in matching the LED efficiency droop represented in the IQE curves with respect to the standard ABC model. Moreover, it is capable to account for the carrier lifetime dependence on current density by means of a simple law that, in the medium–high current density range, exploits the only radiative recombination coefficient as a rate parameter.