1 Introduction

Shuji Nakamura predicted in his Nobel lecture that GaN-based laser diodes may replace light-emitting diodes (LEDs) in solid state lighting (SSL) applications (Nakamura 2014). InGaN/GaN lasers are already utilized in the headlights of some high-end cars. However, the main driving force behind most SSL applications is the promise of high energy efficiency. Some GaN-based blue LEDs achieve more than 80 % electrical-to-optical power conversion efficiency ηPCE (Hurni et al. 2015) but the highest ηPCE reported for GaN-based laser diodes is still below 40 % (Kawaguchi et al. 2016; Raring 2016). We here investigate the reasons for the efficiency limitation of GaN-lasers by numerical analysis of measured high-power laser characteristics.

The power conversion efficiency ηPCE is defined as ratio of light output power P to electrical input power IU (I—current, U—bias). It can be separated into the electrical efficiency ηU = hν/qU (hν—photon energy, q—electron charge) and the external quantum efficiency (ηPCE = ηU·ηEQE). The latter is the ratio of emitted photon number to injected number of electron–hole pairs. It is typically split up (Crump et al. 2013) into a current-averaged differential slope efficiency ηS = P(q/hν)/(I−Ith) and the threshold current efficiency ηth = (I−Ith)/I (Ith—threshold current, ηEQE = ηS·ηth). Note that ηS is unitless and differs from the common slope efficiency P/(I−Ith) given in W/A as well as from the differential efficiency dP/dI, especially with sub-linear P(I). It is also important to point out that the threshold current is considered constant in this model. Our analysis will show that such approach delivers misleading results if the laser experiences relevant self-heating. Rising temperatures reduce the optical gain and cause the threshold carrier density as well as the threshold current to grow with higher current. We propose an alternative approach which is also applicable without any simulation.

2 Models and parameters

The numerical laser model is based on the LASTIP simulation software (Crosslight 2015). It self-consistently computes carrier transport, the wurtzite electron band structure of strained InGaN quantum wells (QWs), stimulated photon emission, wave guiding, and heat flow. Schrödinger and Poisson equations are solved iteratively in order to account for the QW deformation with changing device bias (quantum-confined Stark effect). The transport model includes drift and diffusion of electrons and holes, Fermi statistics, built-in polarization and thermionic emission at hetero-interfaces, as well as all relevant radiative and non-radiative recombination mechanisms. More details on the employed laser models can be found elsewhere (Piprek 2003).

Our analysis focuses on recently published experimental results for 405 nm GaN-based Fabry–Perot laser diodes exhibiting a light output power of P = 7.2 W at I = 4 A input current and U = 6.3 V bias in continuous-wave (CW) operation at room temperature (Kawaguchi et al. 2016). The authors claim that this is the highest power among single-stripe InGaN/GaN laser diodes reported thus far. It is based on an optimized ridge-waveguide structure with low modal loss (αi = 2/cm) as well as on a low thermal resistance of about 6 K/W (Nozaki et al. 2016). The ridge is 12 μm wide and the cavity is 1200 μm long. Facet reflectivities are 0.056 and 0.95, respectively.

Vertical waveguide structure and lasing mode are illustrated in Fig. 1. The low modal loss was accomplished by using thick undoped GaN waveguide layers which reduce the modal overlap with the highly absorptive p-doped layers. The p-Al0.36GaN electron blocking layer (EBL) was moved to the p-side edge of the waveguide in order to minimize the bias (Kawaguchi et al. 2016). The In0.066GaN/In0.008GaN multi-quantum well (MQW) active region features 7.5 nm thick quantum wells with an optical confinement factor of 3.8 %. MQW energy band diagram and quantum levels are shown in Fig. 2 and demonstrate the deformation by the built-in polarization which separates electrons and holes and which is still not fully screened despite the high injection current density of 28 kA/cm2.

Fig. 1
figure 1

Vertical profiles of the refractive index and the simulated lasing mode (normalized, MQW—multi-quantum well, EBL—electron blocking layer)

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Fig. 2
figure 2

Energy band diagram of the two InGaN/GaN quantum wells (QW) with quantum level wave functions (blue) as calculated for the reference laser structure at I = 4A. (Color figure online)

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Key material parameters are obtained by simulating the measured laser performance (Fig. 3). The threshold current is reproduced by using QW Auger coefficients of Cn = 65 × 10−31 cm6/s and Cp = 6 × 10−31 cm6/s at room temperature. This large Cn/Cp ratio is in agreement with recent measurements which find ratios of up to 12 (Nirschl et al. 2016). Theoretical QW Auger coefficients for the dominating indirect transitions have not been published yet, but calculations for bulk InGaN also suggest a large ratio Cn/Cp (Bertazzi et al. 2012). Implementation of the reported modal loss αi = 2/cm gives good agreement with the measured P(I) characteristic. Our simulation also reproduces the injection efficiency of ηi = 92 % measured in pulsed operation without self-heating (Kawaguchi et al. 2016). The calculated MQW temperature rise in CW operation is ΔT = 110 K at I = 4A corresponding to a thermal resistance of ΔT/(IU−P) = 6 K/W as measured (Nozaki et al. 2016). For comparison, Fig. 3 also shows simulation results without self-heating (dashed lines). The P(I) sub-linearity in CW operation is obviously caused by the temperature rise which reduces the QW gain and therefore requires an increasing QW threshold carrier density (Piprek and Nakamura, 2002). This leads to enhanced QW Auger recombination which is the dominating carrier loss mechanism in our case (see below).

Fig. 3
figure 3

Comparison between measurements (dots) and simulations (lines). The dashed lines are simulated without self-heating

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The bias-current simulations shown in Fig. 3 with and without self-heating reveal a thermal reduction of the series resistance. This phenomenon was also observed experimentally (Raring et al. 2010; Becerra et al. 2015) and it is mainly caused by the thermal activation of holes in the p-Al0.026GaN cladding layer (Piprek 2015). The Mg acceptor ionization energy is 178 meV in that layer and the acceptor density was calibrated to match the reported density of 1018 cm−3 free holes (Kawaguchi et al. 2016). Hole mobility (μp = 2 cm2/Vs) and contact resistivity (Rc = 0.05 mΩ cm2) were then adjusted to achieve agreement with the bias measurement in Fig. 3. Both numbers are not reported for this laser but quite reasonable (Yonkee et al. 2016).

3 Analysis and discussion

Figure 4 plots the current fractions consumed by different physical mechanisms inside the laser. Stimulated emission starts at the initial threshold current of Ith = 227 mA which is mainly controlled by QW Auger recombination and by spontaneous photon emission. Leakage current is negligible at that point and so is the Shockley–Read–Hall (SRH) recombination, despite the relatively short SRH carrier lifetime of 20 ns assumed for the QWs. Even at high current, the latter remains irrelevant. Electron leakage into the p-side waveguide layer rises to about 4 % of the total current while QW Auger recombination consumes 22 % of all carriers at I = 4 A. Adding up all these contributions leads to a threshold current Ith(I) = 1.2 A at I = 4 A. The increase in threshold current is responsible for the P(I) sub-linearity in Fig. 3 since the optical loss is considered temperature independent. Carrier leakage from the waveguide layers remains negligible.

Fig. 4
figure 4

Simulated fractions of the total injection current as consumed by different recombination mechanisms. Recombination outside the quantum wells (QWs) is considered leakage

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Our efficiency analysis applies the separation ηPCE = ηU·ηth·ηS as outlined in the introduction. However, in contrast to the typical assumption of a fixed threshold current Ith and a fixed photon energy hν, we here include the simulated changes Ith(I) and hν(I) with rising MQW temperature. The results are plotted as solid lines in Fig. 5. The slope efficiency ηS is the least limiting mechanism in the reference laser, based on the record-low optical loss. The threshold efficiency ηth is dominated by QW Auger recombination and it limits the PCE more strongly, especially at low power. But at higher power, the declining electrical efficiency ηU becomes the most restrictive factor. This is mainly attributed to the poor hole conductivity in the p-cladding layer which is related to the large ionization energy of Mg acceptors. The required high Mg density not only reduces the hole mobility by scattering but also contributes to photon losses by band-tail absorption (Piprek et al. 2007). The photon-energy dependent relationship between Mg density and absorption has not been clearly identified (Kioupakis et al. 2010) so that it is difficult to find the ideal balance between hole conductivity and optical loss theoretically. Several alternative p-side cladding concepts are currently explored, such as indium-tin-oxide cladding layers (Pourhashemi et al. 2015) and tunnel-junctions (Yonkee et al. 2016).

Fig. 5
figure 5

Efficiency versus current: ηPCE—power conversion efficiency (dots from measurements), ηU—electrical efficiency, ηth—threshold current efficiency, ηS—slope efficiency (solid/dashed: with/without temperature sensitivity of threshold current and emission wavelength)

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For comparison, Fig. 5 also shows the results with fixed Ith = 227 mA and hν = 3.066 eV as dashed lines. The calculated red-shift hν(I) turns out to have a negligible influence on the electrical efficiency ηU = = hν/qU since QW bandgap shrinkage and band filling effects almost compensate each other. The slope efficiency ηS ∝ P/(I−Ith) is always larger with rising threshold current than with fixed threshold current. But most importantly, the significant difference between solid and dashed lines for ηth(I) demonstrates that the thermally induced change in threshold current Ith(I) should not be neglected. An approximate function Ith(I) can be directly extracted from the measurement by keeping the slope efficiency constant at the threshold value, i.e., by shifting the dashed blue line in Fig. 3 parallel to the current axis so that a value Ith(P) can be assigned to every power P(I) in CW operation. However, the change in threshold current is smaller if the self-heating leads to increased absorption (Piprek et al. 2000). Thus, the solid and dashed lines in Fig. 5 represent the two extreme cases of constant slope efficiency and constant threshold current, respectively.

Figure 6 demonstrates the effect of excluding efficiency limiting mechanisms from the simulation. The separation of electrons and holes shown in Fig. 2 is reduced by eliminating the built-in polarization and it raises the peak efficiency to 46 % (purple curve). As expected from Fig. 5, the optical loss of αi = 2/cm is already very low and its complete elimination only increases the peak efficiency to 48 % (green curve) since the QW threshold carrier density is slightly lower and therefore also the Auger recombination. Without QW Auger recombination, the threshold current drops significantly and hardly changes with increasing current, so that ηPCE peaks at 67 % (red curve). But the bias remains almost unchanged and drives the efficiency below 40 % at high power. Some bias reduction can be achieved by eliminating the small contact resistance Rc but this only raises the peak power conversion efficiency to 43 % (orange curve). Finally, we simulate the effect of an alternative p-cladding layer with 50-times higher hole mobility of μp = 100 cm2/Vs (dark yellow curve, Rc = 0). The efficiency is significantly enhanced up to 57 % and remains near this peak value even with rising current, while all other simulated exclusions are unable to surpass the apparently magic number of ηPCE = 40 % at high power.

Fig. 6
figure 6

Power conversion efficiency versus current for various parameter variations as discussed in the text (dots from measurements)

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4 Summary

In summary, we identified the poor hole conductivity of the Mg doped cladding layer as main reason for the low energy conversion efficiency of high-power InGaN/GaN lasers. However, restrictions due to increasing Auger recombination cannot be neglected. We proposed an improved efficiency analysis of laser diodes by considering the thermally induced rise of the threshold current.