## Abstract

The electrical-to-optical power conversion efficiencies of the light-emitting devices based on gallium nitride (GaN) are seriously limited by electron leakage currents due to the relatively low mobility and activation ratio of holes. However, there have been few theoretical models on the behavior of the leakage current with an increasing total current. We develop an Ohmic-law-like method to describe the transport behaviors of the systems with electron and hole currents simultaneously. Based on reasonable assumptions, the ratio of the leakage current to the total current is related to the differential resistances of the devices. Through the method, we develop analytical models of the leakage currents in GaN-based laser diodes (LDs) and light-emitting diodes (LEDs). The ratios of the leakage currents with total currents in LDs and LEDs are shown to increase, which explains the sublinear behaviors of the luminescence-current (LI) curves of the devices. The theory agrees well with the numerical simulation and experimental results in larger current ranges in comparison to the traditional ABC model. The above analytical model can be used to fast evaluate the leakage currents in GaN-based LDs and LEDs.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

Over the past three decades, tremendous progress has been made on light-emitting diodes and laser diodes based on gallium nitride. The electrical-to-optical power conversion efficiencies of GaN-based LEDs and LDs have achieved up to 80% [1] and 40% [2] respectively, but are limited at only a small current density. A dramatic reduction of the internal quantum efficiency (IQE) with an increasing current density has been generally observed in LEDs [3], well known as the efficiency droop phenomenon [4]. Although several interpretations of the droop phenomenon have been made, the real dominant factor is still under debate. Current leakage [5–10], nonradiative recombination [3,11,12], and self-heating effect [13] are the most probable candidates for the origin of the droop effect. Among these, the nonradiative recombination theory has been adapted to fit the efficiency-current curve of LEDs in many works [14–16], known as the ABC model [17] by introducing three parameters respectively describing the Shockley-Read-Hall (SRH) recombination (with coefficient *A*), radiative recombination (with coefficient *B*) and Auger recombination (with coefficient *C*) in a simple expression:

*R*is the total recombination rate per unit volume,

*n*is the carrier density in the active region. However, the radiative recombination coefficient

*B*and the Auger coefficient

*C*obtained by fitting the LED efficiency curve deviate severely from those calculated by the first principle [18,19]. More specifically, coefficient

*B*is underestimated [19,20] while coefficient

*C*is overestimated [3,18–20]. The abnormally large value of

*C*could only be explained by considering alloy-disorder- or phonon-assisted Auger scattering [21,22]. On the other hand, Lin et al. showed that no efficiency droop would be found if an LED is pumped optically instead of electrically, indicating a critical role of the transport properties in the droop phenomenon [8]. Dai et al. provided a phenomenological method to describe the injection-dependent leakage current by expanding its contribution into the Taylor series with respect to

*n*[23]. Nevertheless, few reports have considered the change of the ratio of leakage current to the total current with the injection level, but treat it as a constant.

Another problem of the ABC model is that it fails to explain the efficiency droop phenomenon in LDs. The reductions of efficiencies were also observed in lasing LDs [2,24], where the LI curve shows a sublinear behavior. It is not likely that the reduction of the efficiency originates from the nonradiative recombination because the carrier concentrations are clamped above the threshold by a steady-state condition:

*g*is the material gain, $\Gamma $ is the modal confinement factor, ${\alpha _i}$ is the optical loss of the mode by scattering and absorption [25], ${\alpha _m}$ is the mirror loss of the mode,

*L*is the cavity length, and ${R_{m1,2}}$ are the reflectivities of the two facets of the cavity. For a specific lasing mode, $\Gamma $, ${\alpha _i}$,

*L*, and ${R_{m1,2}}$ can be treated as constants above the threshold, making

*g*a constant as well. Since the material gain $g$ is the function of the carrier concentrations in the active region [26], the carrier concentrations are fixed above the threshold. Thus, according to the ABC model, the total nonradiative recombination rate ($An + C{n^3}$) should be a constant above the threshold, making it impossible to contribute to the droop effect. It seems that the leakage current effect must be taken into consideration seriously.

Casey developed a theory of leakage current in semiconductor heterojunction lasers by calculating the diffusion current around the turn-on voltage, where the drift electric field can be neglected [27]. Without considering the drift current, it is not capable to predict the leakage current at a higher injection level. Moreover, the lengths of the p-type area are typically several hundred nanometers in GaN-based LEDs or LDs, in the same order of the magnitude of the electron diffusion length [28,29]. Therefore, the drift current is the dominant component under such a condition, especially when the current density is large. This was confirmed by experiments by Meyaard et al., where the onset of the efficiency droop was found to be strongly correlated with the onset of the drift electric field in the p region [10].

In this paper, we present analytical models of leakage currents in GaN-based LDs and LEDs, in which the long-neglected drift current is well considered. The LI curves of LDs and LEDs from numerical simulation and experimental results are fitted by the expressions from the models. The fitting results of the models are discussed and compared with the traditional ABC model.

## 2. Theoretical model

Both LDs and LEDs have p-type III-nitride layers with a total length of several hundred nanometers following the active region. We focus on the p-type areas, which start at the edge of the active regions and end at the anodes. The energy band diagram of the area is schematically shown in Fig. 1(a). Some common characteristics are suitable for both LDs and LEDs in such regions. Basically, the total current in the area consists of two types: electron and hole currents. Thus, the p area is treated as a parallel circuit. As an example, the equivalent circuit of the p area of an LD is shown in Fig. 1(b). The case for LEDs is similar. The electron current in the p area is defined as the leakage current. The hole leakage current is neglected because of the large effective mass and poor mobility of a hole [7,18]. From theories of semiconductor physics [26], the electron and hole current densities can be written as functions of carrier concentrations and the derivatives of the quasi-Fermi levels:

where ${j_n}$ and ${j_p}$ are electron and hole current densities,*n*and

*p*are carrier concentrations, ${\mu _n}$ and ${\mu _p}$ mobilities, and ${E_{fn}}$ and ${E_{fp}}$ the quasi-Fermi levels of electrons and holes.

*x*is the distance from the active region. Note that Eq. (3) and (4) have the same form of Ohm’s law if ${E_{fn}}/e$ and ${E_{fp}}/e$ are regarded as quasi-electric potentials, where

*e*is the elementary charge.

The vital parameter that identifies an equivalent circuit is its quasi-resistance for each branch. Similar to the normal Ohm’s law, an identical form of the real differential resistivity associated with the carrier concentration is obtained when the recombination in the p area is negligible [26] (discussed in Supplement 1):

where Σ denotes the sum of all the layers in the p area and*l*is the thickness of a layer.

Since the quasi-resistance may change with the total current due to the nonlocal behaviors of carriers in p-n-diodes, it is necessary to analyze the systems with a differential form of Ohm’s law. The increments (denoted with $d$) of the drops (denoted with $\mathrm{\Delta }$) of the quasi-electric potentials on the p area can be regarded as the same between electrons and holes when a tiny increment of the total current is applied. This assumption relies on the fact that the changes of (${E_c} - {E_{fn}}$) or (${E_{fp}} - {E_v}$) at boundaries are negligible compared to the changes of the conduction band edge ${E_c}$ and valence band edge ${E_v}$, which can be several electron volts in the range of the current being studied. As a result, the increment of the drops of the quasi-electric potentials on the whole p area of either electrons or holes is approximately the same as the increment of the real voltage drop (more details are discussed in Supplement 1):

*U*is the real voltage drop. Once the differential resistances of electrons and holes are determined, the ratio of the leakage current and the total current can be determined:

^{2}) of electrons and holes of the whole p area. By calculating ${r_n}$ and ${r_p}$ with respect to carrier concentrations for LDs and LEDs separately, the expressions of the leakage current against the total current will be derived through Eq. (8).

#### 2.1 LD case

A typical GaN-based LD structure is composed of, from bottom to top, an n-doped aluminum gallium nitride (AlGaN) lower cladding layer, an n-doped GaN lower waveguide layer, (an) indium gallium nitride (InGaN) quantum well(s) as the active region, a p-doped GaN upper waveguide layer, and a p-doped AlGaN upper cladding layer. AlGaN has a larger bandgap and a lower refractive index than GaN. Therefore, the two cladding layers confine carriers and photons simultaneously. In earlier studies, an AlGaN electron blocking layer (EBL) with a high composition of aluminum (Al) was applied closely behind the last quantum barrier of the active region [30] to suppress the electron overflow to the p-waveguide region. The technique was modified when Kawaguchi et al. displayed a new structure [2] which is widely used now. It is known that the major optical loss in GaN-based LDs comes from the photon absorption associated with the inactive magnesium (Mg) acceptor impurities in the p-waveguide area [25]. Kawaguchi et al. suggested an unintentionally doped upper-waveguide to minimize the loss [2]. To compensate for the large resistance arising from the undoped GaN waveguide layer, they switched the positions of the EBL and the upper waveguide to accumulate electrons bounced back by the EBL into the upper waveguide layer. Our research is based on a structure similar to Kawaguchi’s but being applied a low level of Mg doping in the upper waveguide as a tradeoff between the optical loss and electric performance without losing generality. The structure being studied is schematically shown in Fig. 2.

Since holes are native carriers in the p area, the total hole resistance of the p area is treated as a constant ${r_p}$ like in normal bulk materials. Since the EBL (as well as the contact layer) is typically only several nanometers, its resistance is neglected. However, the large band offset between the upper waveguide and the EBL has a significant impact on the electron resistance of the upper waveguide, which will be discussed later. The electron resistance ${r_n}$ of the p area is thus the sum of the electron resistances of the upper waveguide layer ${r_{nw}}$ and the upper cladding layer ${r_{nc}}$. Since the electrons in the p area are nonlocal and flow from the active region, the electron concentration at the beginning of the p area should be analyzed. The case for LDs is simple because above the threshold, the carrier concentrations are clamped in the active region by the steady-state condition [Eq. (2)]:

As a consequence, the electron resistance ${r_{nw}}$ of the upper waveguide should be constant according to Eq. (5). The validity of Eq. (5) in such situations will be discussed later.

The last undetermined variable is the electron concentration in the upper cladding layer for calculating ${r_{nc}}$. The electrons drifting in the cladding layer (as well as the EBL) are supplied by the waveguide layer at the heterojunction between them, through a thermal emission mechanism [26,31,32]. Only those electrons with higher energy than the heterojunction barrier can diffuse into the cladding layer with a thermal velocity ${v_{th}}$. According to the thermal emission theory, the electron current and the electron concentration at the cladding-side of the heterojunction is related [31,32]:

where ${v_{th}} = \sqrt {{k_B}T/2\pi {m_n}} $ is a constant only determined by the material and the temperature, in which ${k_B}$ is the Boltzmann constant,*T*is the temperature, and ${m_n}$ is the effective mass of an electron. ${n_c}$ is the electron concentration at the barrier side of the heterojunction, and a constant concentration ${n_{eq}}$ provides a constant reverse thermal current ${j_{eq}}$ to guarantee a zero net current at equilibrium (or at turn-on voltage when electron current is negligible compared to the current being interested in). From Eq. (10), the electron concentration in the upper cladding layer ${n_c}$ as well as the electron resistance ${r_{nc}}$ can be represented by the leakage current ${j_n}$.

Eventually, the total electron differential resistance per unit area in the p area can be written as:

Equation (12) should be integrated from the threshold where the clamping assumption is satisfied. Nevertheless, it is reasonable to extend Eq. (12) to zero current where both ${j_n}$ and *j* are zero and the integral constant in Eq. (13) is determined to be zero. The physical basis behind such an approach is that the Fermi levels are nearly pinned well below the threshold like above the threshold. Interestingly, the pinning, as well as the domination of the stimulated radiation far below the threshold, were observed experimentally in GaN-based LDs in our previous work [33].

As revealed by Eq. (13), the asymmetry of current injections between electrons and holes originates from the poor mobility of holes as well as the large activation energy of the Mg impurities [34,35], which limits the hole concentration in the p area. These lead to a relatively large ${r_p}$ and cause a non-negligible leakage current ${j_n}$ when the total current *j* is given.

#### 2.2 LED case

For most LEDs, there are no AlGaN cladding layers in the p region because there’s no need to support an internal optical mode. Thus, the whole p area can be treated as a simple bulk material [36]. Similar to the LD case, the resistance for native carrier ${r_p}$ in an LED is still treated as a constant.

For ${r_n}$, however, the carrier concentration in the active region will increase with an increasing injection current. As a result, more electrons will flow into the p area and cause a reduction in the electron resistance of the p area. This is in contrast to LDs, where the carrier concentration is clamped by the stimulated radiation. Therefore, the fundamental problem is to relate the total electron resistance to the carrier concentration in the active region because it is observable through the recombination current (the emitting power). According to the thermal emission model [26], the electron concentration in the p area follows the relation:

where ${E_c}$ is the conductive band edge at the beginning of the p area, ${E_{fnw}}$ is the electron quasi-Fermi level at the active region, which is lifted by an increasing total current. Since the electron resistance of the p area is reversely proportional to the electron concentration, it can be written as:Equation (15) is related to experiments by relating ${E_{fnw}}$ and *n* to the observable recombination current. We neglect the hole leakage current as before and assume that the total recombination current is equal to the hole current. The recombination current is decided by carrier concentrations in the active region:

*B*coefficient in the ABC model if the effective length of the active region

*u*is known [14]:

Since ${n_w}$ is decided by ${E_{fnw}}$, the latter can be related to the observable recombination current. It should be aware that the carriers in the quantum well obey the Fermi-Dirac distribution rather than the Maxwell-Boltzmann distribution because of the degeneration [26]:

Therefore,

## 3. Results and discussions

#### 3.1 LD case

To testify our theory, we compare our analytical results for LDs with the numerical results from a finite element method. We used the commercial software *Lastip*, which solves the coupled Poisson-Schrodinger equation, photon rate equation, optical wave equation, and current continuity equation self-consistently. The software is generally used in the studies on the leakage current as well as other effects in GaN-based LDs [18,37], as it contains the polarization model, valence band mixing model, etc*.* The self-heating model is turned off to avoid a disturbance. The conductive band offset between GaN and AlGaN is set to be 0.58 [37,38]. The screening factor of the polarization charges due to defects is set to be 0.2. The LD structure we use is shown in Fig. 2. The cavity length is 1000µm and the reflectivities of the cavity facets are 0.1 and 0.9 respectively. The internal optical loss is set to be 20cm^{-1}. We set Neumann boundary conditions for solving the optical wave equations at the lateral boundaries of the LDs as shown in Fig. 2, thus the problem is one-dimensional to avoid a lateral current spreading [39,40]. Other parameters in the simulation can refer to our previous work [30]. Equation (13) can be modified to fit the LI curve by replacing ${j_n}$ with the laser power ${L_{laser}}$:

Here, the leakage current (several A/cm^{2}) at the threshold is neglected, ${j_{th}}$ is the threshold current density, and ${\eta _d}$ is the external quantum efficiency above the threshold:

*h*is the Planck constant, $\nu $ is the frequency of the lasing photons,

*S*is the injection area.

We fit the LI curve with three adjustable parameters ${r_{nw}}$, ${r_p}$ and ${j_{eq}}$. The other parameters in Eq. (13) and Eq. (24) are fixed to that from the software.

The fitting result for a typical LD with a 10% Al-component EBL is shown in Fig. 3(a) and (b). The fitting parameters are shown in Table 1. As shown in Fig. 3(a) and (b), Eq. (13) perfectly explains the sublinear LI curve of the LD from a numerical result in a large current density range up to 20 kA/cm^{2}. The average (root mean square) relative error by fitting is 0.49% as listed in Table 1. The fitting ${r_{nw}}$ indicates an equivalent electron concentration of about 6×10^{15}/cm^{3}, corresponding to that at the beginning of the upper waveguide from the simulation result. The fitting ${r_p}$ indicates an equivalent hole concentration of about 4×10^{16}/cm^{3} and is also consistent with the doping concentration in the simulation. The fitting ${j_{eq}}$ is on the same order of magnitude as the leakage current around the turn-on voltage from simulation.

We investigate a series of LDs with different Al components in the EBLs. The results are shown in Fig. 3(c) and Table 1. The ratios of the leakage current are also shown in Fig. 3(d). The hole resistance ${r_p}$ increases with the EBL component, probably due to the blocking of holes from the highly-doped upper cladding layer by the EBL. As expected, ${j_{eq}}$ decreases with the EBL components because fewer electrons diffuse into the upper cladding layer with a higher energy barrier of the EBL. Surprisingly, we find that the electron resistance of the upper waveguide ${r_{nw}}$ is strongly modulated by the EBL component. It is the larger ${r_{nw}}$ that mainly determines a less leakage current for those LDs with higher Al-component EBLs.

To find out the reason, we calculate the energy band of the LD structure. The results are shown in Fig. 4(a) and (b). The electron Fermi level is lifted along the direction of the electron flow relative to the band edge in the waveguide layer. This is due to a bounce-back effect of the EBL proposed by Kawaguchi et al. [2]. Only a few high-energy electrons diffuse into the cladding layer by thermal emission, while the other electrons are reflected by the EBL and gain a reverse speed against the external electric field applied. Finally, the reversely drifting electrons slow down due to the electric field and accumulate around the heterojunction between the waveguide and the EBL. A high electron concentration at the heterojunction is necessary to form a large electron current in the cladding layer according to Eq. (10). A nearly identical amount of holes are induced at the waveguide to satisfy the condition of neutrality as shown in Fig. 4(c). These additional holes will contribute to a positive diffusion current and improve the electric performance of the device. The large difference in concentrations between electrons and holes at the interface is due to the polarized surface charge, where the Debye length is about 2 nm [41]. The anomalous direction of the concentration gradient of holes cannot be explained by the polarized charge but by the neutralization of the bounced-back electrons. More details of a drift-diffusion interpretation are discussed in Supplement 1. Smowton and Blood measured the spontaneous emission intensities at the wavelength corresponding to the quantum wells and the waveguide respectively above the threshold in GaInP-based LDs. Interestingly they found that the spontaneous emission intensity stayed constant in the quantum wells due to the clamping effect while increasing slowly with the injection current in the waveguide region. The results can be explained by our theory which predicts an increase in the carrier concentration (thus the spontaneous emission) in the waveguide region [42].

We suggest that the effective electron quasi resistance of the waveguide is different from that of a trivial bulk layer without a heterojunction as the boundary. Specifically, if the probability for an electron to cross the EBL is denoted as ${P_T}$, the concentration of electrons that actually contribute to the resistance should be multiplied by a factor of ${P_T}$ compared to the bulk one. Thus, the effective resistance should be divided by a factor of ${P_T}$. Due to the exponential nature of the Boltzmann distribution of electrons in the p area, the transmittance ${P_T}$ of the EBL should be proportional to the exponent of the conductive band barrier between the waveguide and the EBL without considering a tunneling effect [43] and the difference in the density of states (discussed in Supplement 1):

Thus,

#### 3.2 LED case

For the LED case, we fabricate two LED samples with the same epitaxial structure shown in Fig. 2. The samples are grown on n-GaN free-standing substrates by metal-organic chemical vapor deposition (MOCVD). Each sample has one single emitting InGaN quantum well as the active region with its In component 17% and thickness 4 nm. The Al component of the 7-nm thick EBL is 10% and 20% for sample 1 and sample 2 respectively. The samples are etched into the shape of a cylinder with a radius of 80 micrometers. The LI curve is measured for both samples under a pulsed current with a 10µs pulse width and a 30 Hz frequency up to a current of 500 mA (about 2.5 kA/cm^{2}) to exclude the self-heating effect (discussed in Supplement 1). Light is detected by a half integrating sphere on the p-type sides of the samples.

We modify Eq. (22) to fit the experimental results by introducing an external quantum efficiency ${\eta _{ext}}$:

where*L*is the measured luminescence power of the LEDs. Since the total current is approximately equal to the recombination current at a small current limit where the leakage current is negligible, ${\eta _{ext}}$ is determined with Eq. (28) by the slope of the experimental LI curve at a small current density of about 60A/cm

^{2}. This current density is however much larger than 10A/cm

^{2}, below which the SRH recombination current may have a significant contribution [44]. The extraction efficiencies are determined to be 8.74% and 5.48% respectively for samples 1 and 2. The final extraction efficiencies are low and different between the samples because the extraction of light is not deliberately optimized. Finally, only two independent parameters are adjusted to fit the LI curve, ${j_T}$ and $\gamma = 2{r_p}/{r_{n0}}$, where ${j_T}$ is related to the

*B*coefficient and $\gamma $ is related to the

*C*coefficient in the ABC model, which will be discussed later.

The experimental and fitting results of the two LED samples by Eq. (22) are shown in Fig. 5. ${j_T}$ are fitted to be 75.65A/cm^{2} and 74.69A/cm^{2} respectively for samples 1 and 2. The $\gamma $ coefficient is fitted to be 0.27 and 0.07. The average (root mean square) relative error by fitting is 3.92% and 1.73% respectively. The ratios of the leakage currents to the total currents are shown as well.

We have proven that there’s an intrinsic parameter ${j_T}$ independent from the EBL component but playing an important role in the sublinear shape of the LI curve. From Eq. (23) we know that it is only determined by the active region itself. The $\gamma $ parameter is significantly different between the two samples, which is only determined by the transport properties of the p area (determined by the EBL components in our case). Thus, the effects of the active region and the p-area transport are decoupled into ${j_T}$ and $\gamma = 2{r_p}/{r_{n0}}$ in Eq. (22). Since the two samples are all the same in the active region, they have very close ${j_T}$ values fitted from experiments, but quite different $\gamma $ values. To minimize the leakage current, ${j_T}$ should be as large as possible, which demands a larger density of states *g* in the active region, a larger active length *u* (by increasing the number of the wells*,* e.g.), and a larger recombination coefficient *B*. On the other hand, $\gamma $ should be as small as possible by decreasing the hole resistance of the p area and increasing ${r_{n0}}$ through EBLs.

The radiative recombination coefficient *B* can be estimated by ${j_T}$ through Eq. (23). We take $T = 300$ K, $u = $ 4 nm as the thickness of the quantum well, and ${m_n} = 0.21{m_e}$, where ${m_e}$ is the electron mass, for calculating the density of states [45,46]. As a result, *B* is estimated to be $4.07 \times {10^{ - 11}}$ cm^{3}s^{-1} and $4.01 \times {10^{ - 11}}$ cm^{3}s^{-1} respectively for the two samples. Notice that these results are one-magnitude-order larger than that estimated by the ABC model [20], but very close to the value originally predicted by the first principle [18,19] where $B$ is about $2 \times {10^{ - 11}}$ cm^{3}s^{-1}. The difference of a factor of two is possibly due to a quantum confinement effect which reduces the effective width *u* of the quantum well from its physical thickness.

It should be noticed that the injection-dependent leakage current will contribute to the ABC coefficients as well as higher-order terms. We expand the expression of the leakage current with Eq. (16), (17), and (22) to obtain its power series, where the SRH term is neglected:

Usually, the third-order term with respect to carrier concentration in the ABC model is written as [14]:

Therefore, the contribution of the leakage current to the third-order term coefficient is:

${C_{leakage}}$ is estimated to be $6.66 \times {10^{ - 31}}$ cm^{6}s

^{-1}and $1.84 \times {10^{ - 31}}$ cm

^{6}s

^{-1}for samples 1 and 2, very close to those

*C*coefficients reported in the literature [3,20]. In the literature, the

*C*coefficients are attributed to Auger recombination, but we propose that they are more likely attributed to the expansion term of the leakage current. This also explains the large difference in the reported

*C*values between works because the leakage currents may be very different. The real Auger coefficient may be negligible as originally expected in GaN-based devices [18]. Also, higher-order terms must be taken into consideration when the current density is large enough. Figure 6 shows the fitting results of the LI curve of sample 2 with the ABC model as well as the theory in this work. The $A$,

*B*, and

*C*coefficients are best fitted as $1.01 \times {10^8}$ s

^{-1}, $1.68 \times {10^{ - 12}}$ cm

^{3}s

^{-1}, and $8.66 \times {10^{ - 31}}$ cm

^{6}s

^{-1}respectively, close to the results in previous works [20], but rather different from the results interpreted by the leakage current. It can be seen that although the ABC model fits well at the current below 300 mA (about 1.5 kA/cm

^{2}, which is seldom achieved in previous works based on the ABC model [14,20,23]), it fails to fit at a higher injection level, unlike that by Eq. (22).

There’s another problem with the Auger recombination. Feng et al. [47] estimated the ratio of the Auger recombination current to the total current in LDs with the ABC coefficients from experiment [20] by Scheibenzuber et al*.* It turned out that the Auger current can contribute to over 50% of the total current around the threshold and its ratio decreases rapidly above the threshold because of the clamping effect and an increasing stimulated radiation recombination current. If it is true, there should be an increase in the slope efficiency of LDs above the threshold, while that is contradictory to the experimental results [24]. If the third-order term in the ABC model is considered to be from the leakage current, the slope efficiency does not necessarily decrease, because the leakage current will not be clamped but even increase above the threshold.

## 4. Conclusion

In summary, we have derived expressions of the leakage currents against the total currents in LEDs and LDs through a method of the quasi-differential Ohm’s law. The heterojunction between the upper waveguide layer and the upper cladding layer is found to be responsible for the efficiency droop in LDs above the threshold. We introduce a special transmittance to describe the increase of the effective quasi electron resistance of the upper waveguide layer due to the bounce-back effect by the EBL and thus parametrically clarify the working mechanism of the EBL. We reveal that the difference between the thermal distribution of carriers in the active region and the p area plays an important role in the leakage current in LEDs. An intrinsic thermal recombination current ${j_T}$ is found by fitting the experimental results with our theory, from which the radiative recombination coefficient *B* is estimated to be close to that by the first principle. Finally, we suggest that the leakage current may be responsible for the large *C* coefficient in the ABC model rather than the Auger recombination. More experiments are expected to examine the theory and to determine the band offset directly in devices.

## Funding

National Natural Science Foundation of China (61874004); National Key Research and Development Program of China (2017YFB0405000, 2017YFB0405001); Beijing Municipal Science and Technology Commission (Z201100004520004); Beijing Nova Program (Z201100006820081, Z201100006820137).

## Acknowledgments

The authors thank Prof. Weikun Ge and Prof. Yahong Xie for fruitful discussions.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

## Supplemental document

See Supplement 1 for supporting content.

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