1. Introduction

A measure for the state of knowledge on a certain semiconductor material system is the degree to which microscopic, parameter–free simulations are able to predict the behavior of real–world devices. For GaInAs, InGaPAs, and GaAsSb the degree of predictive power of the highest–level microscopic simulations is striking [1]. One reason for this success is the high material quality and epitaxial growth control. Inhomogeneous broadening in these devices is typically low and nonradiative recombination negligible. The group III–nitrides are far from this level of understanding, because of the complexity of the system. Quantum well (QW) fluctuations are intrinsic to this material class due to indium segregation. Strong internal piezoelectric and spontaneous fields cause the reduction of the electron and hole wave function and a red shift (quantum confined Stark effect, QCSE) in all structures grown along the crystallographic c–axis. The role of threading dislocations and cause of other nonradiative centers in the wavelength range from UV to green is the topic of intense discussions. The claim that Auger processes can be neglected in this wide–bandgap material is more a hope than a proven statement. Certainly (Al,In)GaN light emitting diodes (LED) and laser diodes (LD) occupy a huge market segment in optoelectronic devices, which will further grow when LEDs for general and automotive lighting and LDs for full color laser projection become available. On the theory side, gain spectra have been successfully compared to microscopic simulations [2, 3]. Yet nobody is able at the moment to predict or optimize a full (Al,In)GaN LD or LED structure without close feedback with experiments and at least some assumptions in form of adjustable parameters, most notably inhomogeneous broadening.

Here we present gain spectra for a full set of (Al,In)GaN laser diodes ranging from UV (375nm) to aquamarine (470nm), thus spanning the whole range currently reachable with current injection, continuous wave (cw) (Al,In)GaN laser diodes. Beyond this wavelength range the threshold current density rises rapidly, both at the UV [4] and blue–green side [5]. The provided data is of immediate importance to adjust and test microscopic models. We demonstrate by comparison with basic simulations that different models for the inclusion of in–plane transport lead to the prediction of very different values of inhomogeneous broadening of the gain spectra, average band–gap energy, and different carrier densities. We also show that the gain model based on the assumption of global constant quasi–Fermi levels predicts laser threshold carrier densities which are under realistic threshold conditions independent of the inhomogeneous broadening parameter. This is in agreement with the experimental observation of a nearly equal threshold current density in all laser diodes from UV to aquamarine.

 

Fig. 1. Optical gain spectra. The gain spectra of laser diodes with lasing wavelengths of 375 nm, 405 nm, 440 nm, and 470 nm from near UV to aquamarine spectral range show an increasing inhomogeneous broadening for longer wavelengths.

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2. Gain spectra and laser diodes structural data

The (Al,In)GaN diodes were grown at Nichia Corporation by metalorganic chemical vapor deposition (MOCVD) on bulk GaN substrates with a low threading dislocation density of 10⁵ to 10⁶cm^-2. The near UV laser diode, which emits at around 375nm, has one 10nm thick quantum well with an estimated indium composition of 2 % to 3 %. The violet (405 nm), blue (440 nm) and aquamarine (470 nm) laser diodes are designed with double quantum wells of In_0.08Ga_0.92N/In_0.02Ga_0.98N, In_0.15Ga_0.85N/In_0.02Ga_0.98N and In_0.20Ga_0.80N/In_0.02Ga_0.98N composition, and 7nm, 3 nm, and 3 nm width respectively. The confinement factors are 0.025, 0.03, 0.016, and 0.016 from UV to aquamarine. The geometry of the waveguide ridge was optimized to decrease the operating current density at higher output power. Waveguide facets were formed by cleaving along the (11̅00) face. Cavity length was 650 μm for the near UV and blue laser diodes and 675 μm for the violet and aquamarine ones. A detailed description of the laser diodes, including the waveguide and cladding layer dimensions, can be found elsewhere [4, 5, 6].

For the optical gain spectroscopy we employ the method first described by B. W. Hakki and T. L. Paoli, where the modal gain is derived from the modulation depth of individual longitudinal modes [7]. To resolve the longitudinal modes with an extremely narrow spacing of the order of about 0.04 nm, we use a 80 cm double monochromator with a 1800 lines/mm grating in second order, equipped with a cooled CCD detector [8]. Sets of gain spectra for four different LDs with lasing wavelengths of 375 nm, 405 nm, 440 nm, and 470 nm are shown in Fig. 1. All gain spectra presented in Fig. 1 are included as ASCII data files in the supplementary material. The method is very precise when compared to the more common stripe length method, yet it is limited to carrier densities up to threshold. The laser diodes are operated under realistic conditions, i.e. cw electrically driven. The current is provided by a stabilized laser diode driving circuit, and the heat sink temperature is precisely stabilized to T = 298 K. We compared gain spectra for the 405 nm and 470 nm LDs measured by the Hakki–Paoli method at two completely different experimental setups and found quantitative agreement [9]. Also for each peak wavelength four different laser diodes were measured, giving similar results for the gain spectra. The driving currents for the experimental spectra (thin black curves) are 5 to 35 mA in 5 mA steps for the 375 nm and 405 nm laser diodes, 5 to 40 mA in 5 mA steps for the 440nm laser diode, and [2,4,6,8,10,15,20,25,30,35,40] mA for the 470nm laser diode. The current densities given for the highest currents in Fig. 1 lie slightly (few %) below threshold, as a consequence of the Hakki–Paoli method. The internal losses α_i of the 405 nm to 470 nm are in the range of 20 cm^-1 to 26 cm^-1. For the 375 nm laser diode the internal losses are slightly higher and increase with increasing wavelength. Therefore no stable plateaux is formed at the long wavelength side of the gain spectrum. We estimate the internal losses α_i = 35 cm^-1 from the gain value in the energy range of 3.28 to 3.3 eV in the limit of low driving currents.

 

Fig. 2. Gain at the maximum of the gain spectra. (a) peak modal gain for four (Al,In)GaN laser diodes with lasing wavelength of 375nm, 405nm, 440nm, and 470nm. (b) corresponding differential gain.

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Modal and differential gain is significantly different for the different LDs (see Fig. 2). For the 375 nm LD differential gain increases with current density, for the 405 nm it is constant, and for the longer wavelength LDs it decreases. A striking feature is the crossing of all modal gain curves in a current density range between 2kA/cm² and 2.5kA/cm², caused by the sublinear (440 nm and 470 nm), linear (405 nm), and superlinear (375 nm) dependency of the peak gain on current density.

3. Optical gain simulations

The theoretical gain spectra were calculated in the free carrier, single band effective mass approximation. Homogeneous broadening was included by a secans hyperbolicus (sech) function with Γ_hom = 25 meV [2, 10], which is standard to omit spurious absorption at long wavelengths [11, 12]. We consider the impact of QW fluctuations on the inhomogeneous broadening of the gain spectra in two different ways, starting from a free carrier gain model in the effective mass approximation. First, in the standard way by a convolution of the gain spectra with a Gaussian distribution [13]. Because the homogeneously broadened spectra are in this model independent of the QW band gap energy E_G this model corresponds to a constant carrier density. The quasi–Fermi levels in the valence and conduction band then fluctuate parallel to the energies of the respective band gap edges. Second, we assume global constant quasi-Fermi levels for valence and conduction band [14]. Then the local carrier density and thus the local gain spectra vary according to the difference between the constant quasi–Fermi levels and the fluctuating band gap energy. The full details and parameters for the two gain models are described in [15].

The physical origin of the difference between both models is the redistribution of carriers in the complex energy landscape of the QW. In the constant carrier model one assumes that the recombination time is much shorter than the time for in–plane redistribution, thus a quasi–Fermi level is established locally. In the second model in-plane redistribution is fast compared to recombination and thus globally constant quasi-Fermi levels are established.

 

Fig. 3. Calculated peak material gain. Material gain as function of carrier density for different values of inhomogeneous broadening corresponding to the gain spectra of the individual LDs for the (a) constant carrier density and (b) constant Fermi level model.

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From recent measurements of the temperature dependency of optical gain spectra of a 405nm laser diode and a detailed comparison with many–body simulations we know that the contributions of inhomogeneous broadening (Γ_inh = 31meV) and homogeneous broadening (Γ_hom = 25meV) are similar at room temperature [2, 10]. The gain spectra in Fig. 1 show a large variation in inhomogeneous broadening: Γ_inh is nearly negligible in the UV laser diode, while it is the dominant contribution to the width of the gain spectra for the 440nm and 470nm laser diodes. For the 3nm narrow QWs of the two long wavelength LDs homogeneous broadening is smaller than for the two LDs with 10nm (375nm) and 7nm (405nm) thick QWs.

The experimental gain spectra can be fitted by both models using appropriated parameters. Therefore it is impossible to distinguish between the two models just from the shape of the gain spectra. Yet the two models predict different differential gain, inhomogeneous broadening, and gap energy for identical gain spectra. In Fig. 3 we plot the peak gain as function of carrier density for both models. For the constant carrier density model the gain decreases with increasing inhomogeneous broadening for all carrier densities, as can be expected from the convolution of the homogeneously broadened gain spectra with a Gaussian distribution. For the constant Fermi level model the gain increases with increasing E_G fluctuations in the regime of low carrier densities. The origin of this behavior is the lower density of states of regions of low E_G which provide the major contribution to optical gain. For higher carrier densities the effect of the lower density of states diminishes and the broadening of the gain spectra with increasing Γ_inh dominates, causing a decreasing gain. Thus the constant Fermi level model can naturally explain two experimentally observed features, namely the increasing gain with increasing QW fluctuations at lower carrier density, and the cross over of gain spectra. Neither observation can be explained by the constant carrier density model.

To reach for a quantitative agreement between Fig. 2 and Fig. 3 it is necessary to relate carrier and current density, considering nonradiative recombination, spontaneous emission, and Auger scattering. We also want to stress that we neither included the quantum confined Stark effect (QCSE), nor many–body effects in our simulations. The former would cause an increasing matrix element with increasing carrier density and contributes to the very high differential gain at low carrier densities and thus may cause the sublinear behavior of the 470nm LD. The latter cause band–gap renormalization, Coulomb enhancement, excitonic effects, and allow for a parameter–free inclusion of homogeneous broadening [1, 2, 3]. Because the crossover in the optical gain curves is caused by the basic effect of lowering the effective density of states in the constant Fermi model, we expect this effect to survive a transition from our basic simulations to more realistic ones.

 

Fig. 4. Dependency of threshold carrier density on carrier transport model. Contour plot of equal material gain for (a) the constant carrier density and (b) the constant Fermi level model. The horizontal dashed lines mark the inhomogeneous broadening for the four LDs. The threshold material gain of all four LDs, extrapolated to a standard LD geometry, is marked by red circles. Note that the inhomogeneous broadening scale in both panels is different: a value Γ_inh = 100meV in the constant carrier density model predicts gain spectra of the same width as a corresponding value Γ_inh = 140meV in the constant Fermi level model.

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To carry the comparison further and access the threshold behavior, we refer to the simulation of optical gain for both models as a function of carrier density and inhomogeneous broadening in Fig. 4. The lower density of states in the constant Fermi level model causes an increased gain in the regime of low carrier densities. The four LDs have a different cavity length, mirror reflectivity, and therefore mirror losses. To compare the four LDs we assumed a standard LD geometry of 650μm ridge width and mirror reflectivities R ₁ = 0.98 and R ₂ = 0.50, corresponding to mirror losses α_m = 5.5cm^-1. With internal losses α_i =[35; 26; 24; 26]cm^-1 the threshold model gain is g_th = α_i + α_m =[40.5; 31.5; 29.5; 31.5]cm^-1, and the threshold material gain is G_th = g_th/Γ =[1.62; 1.05; 1.84; 1.97]×10³ cm^-1 for the [375; 405; 440; 470]nm LD, respectively. From the curves of the modal gain in Fig. 2(a) we interpolate resp. extrapolate the threshold current densities [3.1; 2.8; 3.4; 3.0]kA/cm^-2 corresponding to the threshold modal gain for the standard geometry.

The threshold material gain for the standard geometry is marked as red circles in Fig. 4. In the constant carrier density model the carrier density needed to reach this threshold material gain is significantly increasing for the longer wavelength LDs (see Fig. 4(a)). In contrast to this behavior, threshold carrier density is nearly insensitive to the very different inhomogeneous broadening of all four LDs in the constant Fermi level model (see Fig. 4(b)). The carrier lifetime for all four LD structures at high carrier densities is 0.2ns, as measured by time–resolved photoluminescence spectroscopy. Thus the insensitivity of material gain on carrier density in the constant Fermi level model explains the similar threshold current density observed for all four laser diodes over the whole spectral range accessible now for electrically pumped (Al,In)GaN laser diodes.

To push laser diodes towards longer wavelength it is mandatory to decrease internal losses to a level, where the LD gains from the reduced density of localized states. This becomes difficult with increasing indium content level due to deteriorating crystal quality. The constant Fermi level model also predicts, why it is difficult to make aquamarine lasers with lower mirror reflectivities to generate high optical output power: For higher carrier densities optical gain does not benefit from the lower density of states, but inhomogeneous broadening decreases the optical gain strongly.

4. Conclusion

In conclusion, we provide optical gain spectra for (Al,In)GaN laser diodes over the whole wavelength range accessible for current injection, continuous wave operation. Motivated by our basic simulations we state that a correct microscopic description of in–plane transport of carriers in the complex energy landscape of the QW will be mandatory for predictive simulations of (Al,In)GaN laser diodes. A lower density of states caused by indium fluctuations causes an increased gain under the assumption of global constant quasi–Fermi levels. This is in agreement with the insensitivity of measured threshold current density and simulated threshold carrier density on inhomogeneous broadening over the whole spectral range from near UV to aquamarine.