1 Introduction

While the piezoelectric and spontaneous polarization, and consequently the field, in an InGaN quantum well (QW) depends primarily on strain and composition, the potential drop and wave function overlap also depends on QW thickness. The reduced wave function overlap is the cause for the low radiative recombination rate and low internal quantum efficiency (IQE) usually observed for wide InGaN QWs. This would suggest inefficient devices; however, it was shown that laser diodes (LDs) QW thickness over 10 nm can be more efficient than similar LDs with thin QWs. Wide QWs have a large modal gain because of their high confinement factor and high wave function overlap between the excited states.^{[1, 2]} In addition, it was shown that the internal fields can be screened at high carrier density. Furthermore, lasing and a large differential gain were demonstrated.^{[1, 3, 4]} Other ways to compensate the high internal field were demonstrated, too. Doping of the barriers resulted in a reduction of the piezoelectric field^[5] and increased radiative recombination coefficient.^[6] An increased wave function overlap and recombination through excited states was also demonstrated for 9 nm wide InGaN QWs with doped barriers.^[7] Screening of the internal field by barrier doping was also applied to deep UV LEDs.^[8] Another route to decrease the internal field is the insertion of AlInN interlayers between InGaN QWs and GaN barriers.^[9]

We want to investigate the impact of the electrical field in the QW on carrier recombination rate and wavelength shift. In laser operation, the carrier density and consequently the screened field are clamped. However, below threshold, a highly nonlinear dependence of carrier density on a constant driving current is expected because of the strong influence of electric field on carrier recombination rate. To approach this, we want to investigate carrier recombination after the trailing end of a short pulse. The pulse with forward bias injects carriers into the QW. Then the device is switched to a reverse bias at a time scale fast compared to the recombination time. This largely decouples carrier density at the beginning of the decay, which is given by injection during forward bias, and the field, which is given as superposition of the field due to the built-in potential of the p–n junction and the piezoelectric and spontaneous field of the InGaN QW.

This effect was already demonstrated in green LEDs with narrow QWs.^{[10, 11]} Here, we repeat this investigation for wide InGaN QWs, which are expected to exhibit a much stronger wavelength shift and decay time variation. A fast pulse generator and a streak camera for temporal and spectral resolution are used to investigate these diodes.

2 Experimental Section

In this study, blue laser diodes grown by plasma-assisted molecular beam epitaxy (PAMBE) on high-quality c-plane Ammono-GaN substrates are analyzed. The structure of the LDs is presented in Figure 1 and starts with a 700 nm Al${}_{0.065}$Ga${}_{0.935}$N:Si followed by a 100 nm GaN:Si, both doped to a level of 2$\times {10}^{18}{\text{\hspace{0.17em}cm}}^{-3}$. The waveguide consists of two symmetrical undoped 110 nm In${}_{0.04}$Ga${}_{0.96}$N layers placed on both sides of a QW. The QW consists of In${}_{0.17}$Ga${}_{0.83}$N with a thickness of 10.4 or 25 nm. At the end of the waveguide, a 20 nm Al${}_{0.13}$Ga${}_{0.87}$N:Mg (Mg: 2$\times {10}^{19}{\text{\hspace{0.17em}cm}}^{-3}$) is placed and acts as an electron blocking layer. This is followed by a 100 nm GaN:Mg (Mg: 5$\times {10}^{18}{\text{\hspace{0.17em}cm}}^{-3}$) and a 600 nm Al${}_{0.02}$Ga${}_{0.98}$N:Mg (Mg: 1$\times {10}^{19}{\text{\hspace{0.17em}cm}}^{-3}$). The structure is capped with an InGaN:Mg heavily doped contact layer. It is important to note that, thanks to the use of PAMBE, all the layers, especially the QW, are fully strained to GaN substrate. No signs of InGaN relaxation were found in the studied structures. X-ray diffraction reciprocal space maps show that the QWs and all the other layers are fully strained to GaN substrate. No V-pit formation was observed. Additionally, no extended defects originating from the QW were revealed by defect-selective etching. The laser diodes were processed into ridge waveguide devices with oxide isolation. The resonators were cleaved and the mirrors were left uncoated. The resonator length and the ridge width were 800 $\mu \mathrm{m}$ and 2.2 $\mu \mathrm{m}$ for the 10.4 nm and 1000 $\mu \mathrm{m}$ and 3 $\mu \mathrm{m}$, respectively. Details on epitaxial growth and processing of the devices can be found elsewhere.^[12]

The laser diodes are driven with voltage pulses generated by a Standford Research Systems DG645 delay generator. The pulses are delivered to the laser diode via 50 Ω cables and stripe lines. A 22 Ω resistor is in series with the laser diode for current sensing. The distance from the resistor to the laser diode is 1 cm. The current pulses are measured with a 1 GHz bandwidth sampling oscilloscope. At the beginning of the pulse, the influx of charge carriers is mainly limited by the low current (4 mA, corresponding to 130 A cm${}^{-2}$). We account for the slow charging of the device by using a long pulse of 800 ns and start measuring only after over 700 ns. By that time the carrier density has reached a steady state, as observed by a constant electroluminescence intensity. The driving scheme is described in ref. 13 and was used to drive (Al,Ga,In)N laser diodes with voltage pulses of below 10 ns,^[14] where we observed delay times for the onset lasing down to below 1 ns for laser diodes with a factor of 2 smaller active areas. We conclude that the switching time of the field in the p–n transition is small compared to the decay observed below threshold in the measurements discussed here. The width of the depletion layer is large compared to the QW width. The carriers from the depletion region are extracted by a current, while carriers from the QW are confined and recombine radiatively and nonradiatively. Only for larger reverse bias a carrier transport by tunneling out of the QW becomes significant, as demonstrated with carrier lifetimes in the order of 100 ps for a −15 V reverse bias by Weig et al.^[15] In the current situation of not too large negative bias voltages and fast switching time, the carrier density in the QW should remain nearly constant during the change of the field.

3 Theory

In this study, wide QW laser diodes are operated below threshold. Therefore, a detailed consideration of the fields inside the InGaN QW is necessary. InGaN consists of group-III-nitrides, which have a biaxial compressive strain due to the crystal structure causing a high piezoelectric field.^[16-20] This field pulls the electron and hole wave functions to the opposite interfaces of the QW, resulting in a reduced overlap and transition matrix element.^{[10, 19]} The crystal structure also leads to a spontaneous electrical polarization directed along the c-axis of the hexagonal structure.^[21]

The piezoelectric field increases with indium content. For a given piezoelectric field, the potential drop across the QW increases with the width of the QW. Laser diodes in this work have an indium content of 17%; therefore, a large piezoelectric field of ≈2.5 MV cm⁻¹ is generated. Consequently, wide QWs have a very weak emission at low carrier densities and forward bias due to extremely low electron and hole wave function overlap. To achieve a strong emission in such QWs, the piezoelectric field needs to be screened, which can be done by high charge carrier densities in the QW. To analyze the spontaneous emission below the laser threshold in tilted QWs, the interplay of the piezoelectric field and built-in potential on carrier recombination in dependence on an external bias voltage must be studied.^{[10, 22]} Due to the quantum confined Stark effect (QCSE), this interplay can be investigated. The QCSE describes spatial separation of electron and hole wave functions as well as reduced energy difference between states of electrons and holes in the case of band tilting.^[22-25] The built-in potential of the p–n junction is in the order of 100 kV cm⁻¹ in the studied LDs, which is an order of magnitude lower than the piezoelectric field inside the QW. It depends on an external bias voltage and therefore the transition wavelength and matrix element also depend on the external bias voltage.^[26] The built-in potential points toward the substrate in the opposite direction than the piezoelectric field, resulting in a partial compensation of the piezoelectric field.^{[26, 27]} This potential increases with increasing reverse bias, but in the case of forward bias it becomes flat.

Overall, the QW tilt depends on the applied bias and carrier density, which is shown in Figure 2. This figure shows the different QW potential tilts over the course of a driving pulse. At the beginning, no bias is applied and the QW is tilted due to the crystal structure and the piezoelectric field. If a low forward bias is applied, the tilt slightly increases due to an external field. Therefore, the overlap and energy difference is smaller. With increasing forward bias, the tilt decreases because it gets partially screened by injected carriers. After the pulse has ended, the bias voltage goes back to zero or negative bias, resulting in a decreasing tilt and therefore a high overlap and maximal energy difference. This difference is approximately the bandgap energy due to a flat QW potential. In this situation, the piezoelectric field is compensated by both the built-in potential and screening by charge carriers, which results in a minimal tilt and a strong emission. If a stronger reverse bias is used, the tilt is reduced even more and both overlap and the blueshift are further increased.^[10] The switching of the QW tilt with bias voltage can be relatively fast compared to the carrier decay time. Therefore, the carrier density is approximately constant during the switching from forward to zero or negative bias. After the fast switching, the slower decay of carrier density is observed by intensity decrease and redshift due to vanishing screening. Afterward the QW is back to its normal tilted potential.

Another reason for the spatial separation in InGaN-based LDs is charge carrier localization in the InGaN alloy, which can be calculated using atomistic tight-binding theory^[28] or localization landscape theory^[29] and have a length scale on the order of nanometers. For the thick QW, a localization in perpendicular direction could therefore be possible. Reduced wave function overlap due to lateral separate localization will also reduce the radiative recombination rate. However, in our experiment we see and interpret the effect of the bias field on photoluminescence (PL) intensity, decay time, and wavelength. We do not observe a characteristic effect due to localization. Therefore, we are not including localization in our discussion, but mention that a quantitative interpretation of the measured decay times should also include these effects.

4 Results

4.1 Streak Camera Images

Measurements include two different QW thicknesses of 10.4 and 25 nm. For the measurements, a constant voltage is set for each diode during the pulse. Consequently, the pulse height remains on one value for each diode and only the external bias voltage before and after the pulse is changed. Therefore, the change from forward voltage during the pulse to bias voltage after the pulse is different for each measurement. For illustrating the driving scheme, Figure 3 shows the voltage curves for the laser diode with a 25 nm wide QW recorded with an oscilloscope. The voltage during the pulse is at a value of ≈3.3 V. Using the I–V characteristics, a current of 4 mA and a current density of 130 A cm${}^2$⁻¹ are calculated from the pulse voltage, corresponding to 3% of the laser threshold. Due to the constant pulse current for each measurement series, the carrier density at the end of the pulse is also constant. The pulse length is set to 800 ns because in this case a steady state is reached at the end, where the measurements are done.

A streak camera is used in the measurements to achieve temporal and spectral resolution. Figure 4 shows exemplary streak camera images of the measurements. The left image shows the emission below threshold for a 10.4 nm wide QW and the right two images for the 25 nm wide QW for different reverse bias. The results show roughly the last 100 ns of the 800 ns long pulse.

For the 10.4 nm wide QW laser diode, a constant emission during the pulse and an intensity increase at the end can be observed. Therefore, a fast increase in the recombination rate is present at the end of the pulse. The 25 nm wide QW laser diode shows no detectable intensity during the pulse. Light emission and intensity is only visible at the end of the driving pulse, where the recombination rate is increased. While comparing both laser diodes, it appears that the 10.4 nm shows intensity during the pulse, meaning that there is enough screening of the piezoelectric field to achieve light emission. In case of the 25 nm, the thickness of the QW is enlarged to more than twice the size of the 10.4 nm. Consequently, the spatial separation of electron and hole wave functions is increased as well. Therefore, stronger screening is necessary to achieve an emission.

Overall, the most important effects are visible at the trailing edge of the driving pulse, which are explained in the further context of this work.

4.2 Intensity Rise

At first, the sharp increase in intensity is explained in more detail. It is possible to observe a sharp intensity rise at the trailing edge of the driving pulse. To better visualize this effect, the intensity distribution over time for all measurements with different bias voltages is shown in Figure 5.

For the 10.4 nm wide QW, an increasing intensity rise can be observed for decreasing bias voltage. However, the highest reverse bias does not have the largest intensity rise. In this case, an abnormality is observed, which is visible by the double peak structure of the intensity trend. The 25 nm wide QW shows the same trend, an increasing intensity peak for decreasing bias voltage. In this case, a delay of the peak at the highest reverse bias is observed. This behavior could in both cases be a consequence of different bound states of the QW being involved in the light emission.

The intensity rise starts at the trailing edge of the driving pulse where the pulse and therefore the forward bias ends. At this moment, the built-in potential compensates the piezoelectric field, which leads to an increase of the electron and hole wave function overlap and therefore the recombination rate of electrons and holes rises. The intensity peak increases with decreasing bias voltage (i.e., increasing reverse bias voltage) because the built-in potential depends on a reverse bias and a larger built-in potential results in a better compensation of the piezoelectric field. This leads to a smaller tilt of the QW and lower spatial separation of the electron and hole wave functions.

In the logarithmic plot, the stretched exponential character of the decay is clearly visible. The initial PL decay times are ${\tau }_{1/e}=8.6\text{\hspace{0.17em}ns}$ and ${\tau }_{1/e}=5.8\text{\hspace{0.17em}ns}$ for the $10.4$ and $25\text{\hspace{0.17em}nm}$ wide QW, respectively. Assuming that photoluminescense intensity is depending on the carrier density squared ($I_{\text{PL}}=B{n}^2$), the carrier decay time is twice the PL decay time. With decreasing carrier density, the QW is tilted and decay rate decreases. The PL decay time at about $80\text{\hspace{0.17em}ns}$ after the driving pulse is ${\tau }_{1/e}=83\text{\hspace{0.17em}ns}$ and ${\tau }_{1/e}=273\text{\hspace{0.17em}ns}$, respectively. These fits are for a bias of $0.76\text{\hspace{0.17em}V}$ and $0.0\text{\hspace{0.17em}V}$, as shown by the dashed lines in the logarithmic plot in Figure 5. The initial decay for both QWs is similar and only weakly depending on a bias voltage. The partially compensated field can be screened by the injected carriers, and recombination is from a mostly flat QW. In general, the PL decay times 80 ns after the driving pulse have a larger value than normally observed. The extended decay times are caused by the wide QW structure in these diodes. It has already been shown that such broad QWs can have decay times in the range of several $\text{μs}$ and that it is possible to achieve an intensity increase within the decay time if a short reverse bias pulse is applied.^[30]

4.3 Wavelength Shift

The other effect is a wavelength shift, which is presented for the 10.4 nm wide QW in Figure 6. In case of the 25 nm wide QW, no shifts could be calculated due to multiple peaks in the spectrum. Wavelength shifts were also seen in pulsed-bias actuation of InGaN LEDs and used to design monolithic broadband LEDs. This shift, also called chirp, in the emission spectra is caused by the turn-on dynamics, which is determined by the transient QCSE.^[31]

For all measured bias voltages, a redshift is visible. This shift is caused by the charge carrier decay, which leads to an increase of the QW tilt. The piezoelectric field gets screened by the charge carriers during the current pulse. After the driving pulse ends, the charge carrier density decreases. Therefore, the screening decreases and the tilt of the QW increases again, which results in a redshift. At high reverse bias, it is possible to observe another shift just before the redshift and therefore at the trailing edge of the driving pulse. This is a shift to shorter wavelengths (blueshift), resulting from a reduction of the QW tilt due to the disappearance of the external field. This external field is caused by the forward voltage, which vanishes at the end of the driving pulse. The blueshift is only visible in the case of sufficiently high reverse bias, which generates stronger external fields. Consequently, the impact of the QW tilt decrease is higher and the blueshift gets visible. The disappearance of the external field is a rather quick process compared to the charge carrier decay; therefore, the blueshift is visible first, followed by the redshift.

Thin QW laser diodes also showed a wavelength shift; however, a significantly larger blueshift up to 20 nm could be observed in earlier works.^[10] Laser diodes with thinner QWs are screened significantly less, so the QW has a stronger tilt at the end of the driving pulse, which means that the switch from forward to reverse bias has a much greater effect. This results in a larger change of the QW potential, leading to a strong blueshift. In the case of the wide QW, only a small blueshift of about 1 nm was observed, which is due to the QW being screened to a point where it is barely tilted. Therefore, switching the bias has little or no effect on the QW. For thin QWs, no redshift was observed because the intensity was already too low to be detectable.^[10] However, for the wide QWs, the redshift due to the slow carrier decay is dominant.

5 Discussion and Conclusion

In this work, the interplay between the piezoelectric field and built-in potential of the p–n junction on carrier recombination is demonstrated on blue laser diodes with QW thicknesses over 10 nm. The laser diodes are analyzed in below threshold operation with an external bias. The investigations are possible because the switching time between forward and zero/reverse bias at the end of the driving pulse is much shorter than the recombination time. Due to this fact it is also possible to investigate the influence of QCSE on the transition energy and also the decay time for a given carrier density.

At the trailing edge of the driving pulse, it is possible to observe two different effects: a sharp intensity rise and wavelength shifts. At this moment, it switches between forward and zero/negative voltage, causing the built-in potential to rise rapidly. It compensates the piezoelectric field as it is pointing in the opposite direction. The intensity increase at the trailing edge of the driving pulse is visible because the QW tilt decreases and therefore a high overlap between the electron and hole wave functions is achieved. Furthermore, a dominant redshift is observed due to the carrier decay and the resulting increase in the QW tilt. For high reverse bias and therefore a high voltage step, a small blueshift can be seen. It results from the disappearing of the external field and a decreasing QW tilt. The effects are more noticeable at higher reverse bias voltages because the built-in potential rises with the reverse bias voltage, resulting in higher compensation of the piezoelectric field.

In comparison to previous measurements on thin QWs, a clear difference in the wavelength shift was noted. For thin QWs, the blueshift was dominant^[10] and for wide QWs, the redshift. This is explained by more effective screening during operation in forward bias in case of the wide QW.

Conflict of Interest

The authors declare no conflict of interest.