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Abstract
Hole injection is one of the fundamental limitations that affect the lasing power for GaN-based vertical-cavity surface-emitting lasers (VCSELs). In this report, a GaN-based VCSEL with a composition gradient quantum barrier (CGQB) structure is proposed and investigated. The designed InxGa1-xN quantum barrier has a linear gradient level of InN composition along the [0001] orientation, which is effective in reducing the energy band barrier height for holes. Furthermore, the polarization-induced bulk charges that are generated in the proposed quantum barriers can reduce the electric field magnitude in quantum wells, which is known as the polarization self-screening effect. Therefore, the hole injection and the electron-hole stimulated recombination rate can be both enhanced. We also find that although the hole injection can be enhanced and the polarization induced electric field in the quantum wells can be reduced, an increased gradient level of the InN composition for the polarization self-screened quantum barriers is not always favored. The reduced quantum barrier height will redistribute the energy subbands and make peak gain not coupled with the cavity resonance wavelength, which will decrease the lasing power. Hence, to avoid the substantial variation of the subbands, we suggest that the polarization self-screened active region shall possess properly thick quantum wells for maximizing the lasing power. Moreover, the optimized active region design can increase the 3dB frequency.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
III-V group semiconductor vertical-cavity surface-emitting lasers (VCSELs) have received significant research attention in recent years, owing to the low threshold current, excellent beam quality, small volume, high integration capacity when compared to edge-emitting lasers [1,2]. Currently, GaAs-based VCSELs have been successfully commercialized, while GaN-based VCSELs are expected to be the candidate for ultraviolet-to-green light source for the applications in communication, displays and lighting [3,4]. However, GaN-based VCSELs are still not very maturely developed at the current stage. Thus, there are essential obstacles worthy optimizing for GaN-based VCSELs. Epitaxial AlN/GaN distributed Bragg reflector (DBR) with high-quality reflectivity is still challenging for GaN-based VCSELs [5]. Therefore, dielectric DBRs have been alternatively proposed to achieve high optical reflection [6,7]. Nevertheless, neither epitaxial nor dielectric DBRs can support the electrical injection because of the non-conductive features for the two types of DBRs [8,9]. As a result, GaN-based VCSELs possess the lateral current injection scheme by using current aperture. In order to make the current flow into the aperture, different device structures have ever been proposed and demonstrated, such as utilizing PNP-GaN current spreading layer [10], reverse-biased PN-GaN junction [11], buried insulator with properly increased dielectric constant [12] and buried GaN-based tunnel junction [13,14]. After the holes are injected into the aperture, the holes have to enter the multiple-quantum-well (MQW) region by crossing over the p-type electron blocking layer (p-EBL) [15,16]. However, the valence band discontinuity between the p-EBL and the p-GaN layer can hinder the hole injection. Insufficient hole concentration in the active region can induce optical loss, thus sacrificing the threshold current and the lasing power [17]. Therefore, to promote the hole injection, different p-EBLs have been designed, e.g., composition-step-graded p-EBL and superlattice p-EBL [15,18]. By using the alternative p-EBL, the hole blocking effect by the bulk AlGaN-based p-EBL can be decreased. Then, the holes will transport across the MQWs, so that the interband gain can be obtained in each quantum well. For achieving that purpose, a cascaded tunnel junction is proposed to be inserted between each MQW stack, and by doing so, the carriers can be recycled [17]. In addition, the electron-hole stimulated recombination rate is also influenced by the electron-hole wave function overlap in the MQWs, which is strongly affected by the polarization induced electric field in the [0001] oriented GaN-based VCSELs [19]. The polarization induced electric field in the quantum well can be screened by using Si-doped quantum barriers (QBs) [20]. However, doping in active area can easily cause the optical loss for laser diodes [21]. Moreover, Si-doped quantum barriers will sacrifice the hole injection because of the increased valence band barrier height [20]. Hence, in this work, we propose the GaN-based VCSEL structure with composition gradient InxGa1-xN quantum barriers (CGQB), such that the InN composition for InxGa1-xN quantum barrier is graded along the growth direction as shown in Fig. 1. It has been proven that such quantum barrier design with gradient compositions can generate polarization induced bulk charges, which can partially screen the polarization sheet charges at the quantum barrier/quantum well interface, known as the polarization self-screening effect [22]. Therefore, the polarization induced electric field in the quantum wells can be reduced, which more favors the stimulated radiative recombination and enhances the lasing power. At the same time, the reduction for the valence band barrier height also helps to improve hole injection across the MQWs. In addition, we also find that such quantum barrier design also favors the increased 3dB bandwidth for GaN-based VCSELs.
Fig. 1. (a) Schematic diagram for investigated VCSEL. (b) and (c) are schematic valence bands for Devices R and A in the equilibrium state.
2. Device structures and physical parameters
The structural diagram for GaN-based VCSEL is shown in Fig. 1. The VCSELs possess 15 pairs of SiO2/Ta2O5 top DBRs and 20 pairs of AlN/GaN bottom DBRs, which sandwich a 7λ resonant cavity with the target wavelength of 438 nm. The laser cavity consists of 850 nm n-doping GaN layer with the doping concentration of 5×1018 cm-3, 5-pair In0.21Ga0.79N/GaN MQW as active region, 20-nm-thick Al0.18Ga0.82N p-EBL, 248-nm-thick p-GaN hole injection layer with doping concentration of 8×1017 cm-3 and 20-nm-thick ITO as current spreading layer. A 20-nm-thick SiO2 is buried in the p-GaN layer to reduce internal loss in which 8-µm-width aperture is formed [23]. Note, the SiO2 confinement layer is carefully designed to be at the node of standing wave and the MQW is located at the antinode for the optical gain. The schematic valence bands of the InGaN/GaN MQWs for the reference VCSEL is shown in Fig. 1(b). The schematic valence band of the proposed InGaN/InxGa1-xN with gradient InN composition is presented in Fig. 1(c).
In this study, Crosslight PICS3D is used for conducting numerical calculations and revealing the underlying device physics. The drift-diffusion equations, Schrödinger and Poisson’s equations are taken into consideration to compute electrical characteristics by setting properly boundary conditions [24]. The Auger coefficient is set to 5×10−31 cm6 s-1 and the Shockley-Read-Hall (SRH) recombination lifetime is set to 4.3×10−8 s [25,26]. The band offset ratio between the conduction band and the valence band defined as ΔEc/ΔEv is assumed to be 0.70:0.30 for InGaN/GaN MQWs [27]. The polarization level of 40% is considered so that the polarization induced charges can be included to model the spontaneous and piezoelectric polarizations at the lattice-mismatched interfaces [28]. The average optical background loss is 2000 m-1 [29,30]
3. Results and discussion
3.1 Proof of the advantage of the CGQB in improving carrier transport and recombination for GaN-based VCSELs
To investigate the effect of CGQB structure, Device R with 4-nm-thick GaN QBs and the proposed Device A consisting of CGQBs are firstly investigated. The InN composition in the 4-nm-thick InxGa1-xN quantum barrier is linearly graded change from 0 to 0.09 along the [0001] direction for Device A. Besides the polarization induced interface charges, the polarization induced bulk charge density in such CGQBs can be calculated by using the formula of $\rho _B^{Pol}(l )= \frac{{\partial P}}{{\partial x}}\ast \frac{{\partial x}}{{\partial l}}$, in which P represents polarization density, x stands for the composition of InN, and l denotes the length along the gradient direction [22]. Thus, the negative polarization induced bulk charge density is 8.28×1024 m-3. No polarization induced bulk charges are generated in the GaN QBs for Device R.
The calculated energy bands for the two VCSELs are shown in Fig. 2(a). When compared to Device R, the valence band offset between the quantum well and the quantum barrier is reduced to 146.8 meV for ΔEBA from 266.2 meV for ΔEBR when the GaN QB is replaced by the InxGa1-xN CGQB. Therefore, when the holes are injected from the p-GaN side, the holes can more easily penetrate through the quantum well and the hole injection uniformity can be improved. Furthermore, when we look into the energy band profile for the quantum well, we can see that the energy band for the quantum well is less titled for Device A, such that ΔEWR of 161.1 meV and ΔEWA of 136.6 meV are for Devices R and A, respectively. We attribute the less tilted energy band profile in the quantum well for Device A to the polarization self-screening effect, such that the polarization induced electric field in the quantum well can be suppressed by the polarization induced bulk charges in the CGQBs. To further prove that point, we show the electric field profile in Fig. 2(b). Note, to avoid the impact of the external bias on the electric field, we show the electric field profiles in the equilibrium state. Figure 2(b) clearly shows that the electric field intensity in the quantum wells for Device A is lower than that for Device R. The reduced polarization induced electric field intensity in the quantum wells can increase the overlap level for the electron-hole wave functions, e.g., carrier wave function overlap level is 88.18% in the second quantum well counted from n-GaN side for Device A which number is 80.74% for Device R. Such reduced quantum confined Stark effect (QCSE) helps to more favor the electron-hole stimulated recombination.
Fig. 2. (a) Valence band profiles for Devices R and A along the [0001] orientation and ΔEWR and ΔEWA denote the valence band tilted level in the quantum wells for Devices R and A, respectively; ΔEBR and ΔEBA represent the valence band offset between the quantum well and the quantum barrier for Devices R and A, respectively. (b) Electric field profile along the [0001] orientation for Devices R and A in the equilibrium state. The positive direction for the electric field is along the [0001] orientation.
Then, the hole concentration distributions (i.e., Hole Conc.) for Devices R and A are shown in Fig. 3(a). The non-uniform carrier injection for Device R is obviously observed such that the hole concentration gets decreased for those quantum wells close to the n-GaN side. However, the hole concentration distribution becomes more homogenized in the active region for Device A, which is ascribed to the reduced valence band offset between the quantum well and the quantum barrier. We also present the stimulated recombination (i.e., Sti. Recombination) for Devices R and A in Fig. 3(b). It shows that the stimulated recombination rate for Device A is more significantly enhanced than that for Device R. It is worth mentioning that in spite of the lower hole concentration in the last two quantum wells for Device A, the stimulated recombination there in for Device A is still larger than that for Device R. This is ascribed to the reduced polarization induced electric field that also promotes the electron-hole stimulated recombination. Hence, the promoted hole injection capability and the reduced QCSE in the quantum wells enable the improved stimulated recombination, which then helps to enhance the lasing power and reduce the threshold current as shown in Fig. 3(c). The increased lasing power indicates the reduced differential carrier lifetime. Thus, the 3 dB frequencies at the current of 15 mA for Devices R and A are 4.35 and 7.27 GHz, respectively.
Fig. 3. (a) Hole concentration and (b) stimulated recombination distribution in MQWs along the [0001] orientation for Devices R and A. (c) Output power in terms of the injection current for Devices R and A.
3.2 Impact of the InN gradient level in the CGQB on carrier transport and recombination for GaN-based VCSELs
Our previous discussion shows that the CGQB structure can enhance the hole injection and suppress the polarization induced electric field in the quantum wells. Therefore, the lasing power and the 3dB frequency can be improved accordingly. In this section, we will conduct detailed analysis regarding the impact of the composition level in the quantum well on valence band barrier height and the polarization effect for the active region, so that the optimization strategy can be summarized. Different VCSELs (i.e., Devices B1, B2, B3, B4 and B5) are designed. Device R is also used here as the reference structure. The information for the alloy composition and the polarization induced bulk charge density is shown in Table 1.
Table 1. Structural Information for Devices R and B1 to B5
The lasing power in terms of the injection current for Devices R, B1, B2, B3, B4 and B5 is presented in Fig. 4(a). The inset for Fig. 4(a) demonstrates the lasing power as a function of the InN gradient level in the quantum wells for the studied devices at the current of 15 mA. When compared to Device R, the lasing power increases first (i.e., Devices B1 to B3) and then decreases (i.e., Devices B4 and B5). The lasing power for Device B5 is even lower than that for Device R. As we have mentioned previously, the lasing power can be partially affected by the polarization effect in the quantum wells. Hence, the electric field profiles for Devices R, B1, B3 and B5 are selectively presented in Fig. 4(b) and Device R shows the largest electric field intensity in the quantum wells. The electric field intensity gets decreased for Devices B1, B3 and B5. For the purpose of demonstration, the overlap levels for the electron-hole wave functions in the second quantum well counted from n-GaN side are summarized in Table 2, which agrees well with polarization field profiles in Fig. 4(b). Moreover, the increased gradient level of the InN composition in the quantum barriers can produce more polarization induced bulk charges in the quantum barrier, which will further screen the polarization effect in the quantum well, i.e., the smallest electric field intensity and the largest overlap level for the carrier wave functions can be obtained in the quantum wells for Device B5. The increased lasing power for Devices B1, B2 and B3 can be partially caused by the decreased polarization effect in the quantum wells.
Fig. 4. (a) Lasing power in terms of the injection current for Devices R and B1 to B5. The inset shows the lasing power at the current of 15 mA for the studied devices. (b) Electric field profiles in the MQWs along the [0001] orientation for Devices R, B1, B3 and B5 in the equilibrium state.
Table 2. Wave Function Overlap Level φ for Devices R and B1 to B5
However, Table 2 and Fig. 4(b) cannot explain the decreased lasing power for Devices B4 and B5 in Fig. 4(a). Therefore, for further investigation, we show the hole concentration profiles selectively for Devices R, B1, B3 and B5 at the current of 15 mA in Fig. 5(a). It shows that as the InN gradient level increases in the quantum barriers, more holes can be injected into the quantum wells close to the n-GaN side, e.g., when we compare Devices R and B5. Figure 5(b) illustrates the stimulated recombination rate profiles selectively for Devices R, B1, B3 and B5 at the current of 15 mA. The promoted hole injection across the MQWs also makes the quantum wells close to the n-GaN side produce more stimulated emission, e.g., when we compare Devices B1 and B3. For even better comparing the impact of different quantum barriers on the hole transport and the stimulated recombination, we integrate the hole concentration and the stimulated recombination in the active region, which are demonstrated in Fig. 5(c). We can see that the integrated hole concentration gets increased as the InN gradient level increased. Nevertheless, the integrated stimulated recombination decreases for Devices B4 and B5, which agrees with Fig. 4(a). Hence, the reduced polarization effect and the enhanced hole injection efficiency do not increase the lasing power for Devices B4 and B5 for GaN-based VCSELs. Even more investigations shall be conducted.
Fig. 5. (a) Hole concentration distribution, (b) stimulated recombination rate along the [0001] orientation for Devices R, B1, B3 and B5 at 15 mA. (c) Integrated hole concentration and stimulated recombination rate for devices with InN gradient level. (d) and (e) are valence bands and energy subband levels in the second QW counted from n-GaN side for Devices B1 and B5.
The additional factor that affects the lasing power is the gain wavelength, which shall possess the resonance feature with the designed cavity so that the lasing power can be enhanced. However, the gain wavelength is strongly affected by the energy bands for the quantum wells. Then, Figs. 5(d) and (e) show the valence band profiles selectively for Devices B1 and B5, respectively. The energy subband (ψh) profiles show that the reduced average InN composition in the quantum barrier for Device B5 can shift the energy level for the subbands. When compared with Device B1, the transition energy bandgap for Device B5 is reduced by ∼31.16 meV. Therefore, if the targeted lasing wavelength of 438 nm is set as the reference, the wavelengths that correspond to the peak gain are 441.81 nm and 442.35 nm for Device B4 and B5, respectively, which will cause the less optical resonance in the cavity and lead to the reduced lasing power.
3.3 Impact of the quantum well thickness for the CGQB structural active region on carrier transport and recombination for GaN-based VCSELs
Our previous discussions indicate that besides the hole injection and the polarization effect in the MQWs, a stable gain wavelength is also essentially important in generating high-power lasing diodes. However, the impact of the energy band offset between the quantum barrier and the quantum well on the gain wavelength can be suppressed if the quantum well thickness can be properly increased [31–34]. Hence, GaN-based VCSELs with 3 nm and 4 nm thick quantum wells will be compared and investigated. The quantum barrier thickness is fixed at 4 nm. Detail structural information for the active region can be found in Table 3. Because of the increased quantum well thickness, the thicknesses for other layers are redesigned so that the cavity length is still 7λ with the target wavelength of 450.7 nm.
Table 3. Structural Information for Devices Rc and C1 to C5
The lasing power in terms of the injection current is shown in Fig. 6(a). The inset shows the lasing power at the current of 15 mA as the function of the InN gradient level in the quantum barriers. Unlike the results in Fig. 4(a), all the proposed VCSELs show the enhanced lasing power and the reduced threshold current when compared with Device Rc. Therefore, we can tentatively conclude that the polarization self-screened active region shall adopt properly thick quantum wells for maximizing the lasing power. Figure 6(b) selectively shows the electric field profiles in the MQWs for Devices Rc, C1, C3 and C5. The same conclusion can be obtained that the polarization self-screening effect can reduce the electric field intensity in the quantum wells, which partially favors the enhanced stimulated recombination between electrons and holes. For the purpose of demonstration, Table 4 also presents the overlap level of the carrier wave functions in the second quantum well counted from n-GaN side for the studied devices. The overlap level also gets enhanced as the InN gradient level increases. However, the comparison between Tables 2 and 4 also indicates that the increased thickness for the quantum wells will sacrifice the overlap levels for the carrier wave functions.
Fig. 6. (a) Lasing power in terms of the injection current for Devices Rc and C1 to C5. The inset shows the lasing power at the current of 15 mA for the studied devices. (b) Electric field profiles in the MQWs along the [0001] orientation for Devices Rc, C1, C3 and C5 in the equilibrium state.
Table 4. Wave Function Overlap Level φ for Devices Rc and C1 to C5
We show the profiles for the hole concentration and the stimulated recombination rate in the MQWs at the current level of 15 mA in Figs. 7(a) and (b), respectively. Figure 7(a) illustrates that the hole injection capability can be enhanced by using the polarization self-screened active region, e.g., if we compare Devices Rc and C5. The stimulated recombination rate in the quantum wells is partially influenced by the hole distribution as shown in Fig. 7(b). The conclusions made by Figs. 7(a) and (b) agree with that made by Figs. 5(a) and (b). To even better show the impact of different quantum barriers on the hole concentration and the stimulated recombination rate, we present the integrated the hole concentration and the stimulated recombination rate in terms of the InN gradient levels in Fig. 7(c). We find that the integrated stimulated recombination for Devices C1 to C5 is less affected by the InN gradient level than that for Devices B1 to B5 except that Device C5 shows the slightly reduced stimulated recombination rate. We next selectively shows the valence band profiles for Devices C1 and C5 in Figs. 7(d) and (e), respectively. When compared with Device C1, the energy level for the subband is reduced and effective transition energy bandgap reduces by ∼7.45 meV for Device C5. However, the ∼31.16 meV is obtained when we analyze Figs. 5(d) and (e). Hence, by increasing the quantum well thickness, the wavelength at the peak gain position is less affected by the InN gradient level for the CGQB structures, e.g., it is 452.8 nm for Device Rc and 454.5 nm for Device C5.
Fig. 7. (a) Hole concentration distribution and (b) stimulated recombination rate along the [0001] orientation for Devices Rc, C1, C3 and C5 at 15 mA. (c) Integrated hole concentration and stimulated recombination rate for devices with InN gradient level. (d) and (e) are valence bands and energy subband levels in the second QW counted from n-GaN side for Devices C1 and C5.
3.4 Comprehensively investigating the impact of different CGQB structural active region designs on lasing power and 3 dB frequency for GaN-based VCSELs
Our previous discussions indicate that the lasing power is coaffected by the polarization field, the hole injection and the energy levels of the subband in the MQWs. More importantly, we find that if the InN gradient level increases, the polariztion self-screened active region with thin quantum wells will shift the peak gain wavelength from the resonant value for the cavity. Hence, in this section, we will conduct a comprehensive investigation to optimize the design for the polarization self-screened MQWs. In the meanwhile, the 3dB frequency is also studied.
Figure 8(a) summarizes the lasing power for VCSELs with different InN gradient levels for quantum well thicknesses of 2 nm, 3 nm and 4 nm. It is worth mentioning that the cavity length is still fixed at 7λ. The InN gradient level of 0 means the adopting of GaN-based quantum barriers. The threshold current levels for the investigated devices are shown in the inset of Fig. 8(a). We can see that the lasing power increases as the quantum well becomes thick. Moreover, for the devices with 2 nm thick quantum wells, the lasing power decreases with InN gradient level beyond 9%. Nevertheless, the lasing power shows a slight reduction when the InN gradient level reaches 11% for the devices with 3 nm thick quantum wells. The lasing power increases and reaches a stabilized value with the InN gradient level for the devices with 4 nm thick quantum wells. The increase of the lasing power as the InN gradient level e.g., when the gradient level is smaller than 9% for the devices with 2 nm thick quantum wells, can be well attributed to the decreased QCSE and the enhanced hole injection capability in the MQWs. However, further increase the InN gradient level for the quantum barriers is speculated to cause the deviation of the peak gain wavelength from the resonant value especially for VCSELs with thin quantum wells [35]. Thus, Fig. 8(b) presents the effective transition energy bandgap for the investigate devices. We can see that, for VCSELs with e.g., 2 nm thick quantum well, the transition energy shows a very strong dependence on the InN gradient level. Then, Fig. 8(c) demonstrates the peak gain wavelength for all the investigated devices, which illustrates that the deviation from the peak gain wavelength becomes big as the InN gradient level increased for the 2 nm thick quantum wells. The conclusion in Fig. 8(c) agrees with our speculations.
Fig. 8. (a) Lasing power, (b) electron-hole transition energy and (c) peak gain wavelength in terms of the InN gradient level for the quantum barriers at the current of 15 mA. The quantum well thicknesses are set to 2 nm, 3 nm and 4 nm, respectively. The inset in (a) summarizes the threshold current value for each device, and the differences between peak gain wavelengths for GaN and maximum composition gradient InxGa1-xN quantum barriers in devices with 2 nm, 3 nm and 4 nm thick quantum well thickness are signed in (c).
The stimulated recombination rate is strongly associated with differential carrier lifetime, which also determines the frequency modulation for laser diodes. We then calculate and demonstrate the 3dB frequency for the investigated devices in Fig. 9. The 3dB frequency is affected by the lasing power and the threshold current [36], such that the increased lasing power or the reduced threshold current can produce the wide 3dB frequency, e.g., if we look into the devices with 2 nm thick quantum wells. Meanwhile, the 3dB frequency for devices with 3 nm and 4 nm thick quantum wells gets increased when compared with that for the devices with the 2 nm thick quantum wells. However, if we compare the devices with 3 nm and 4 nm thick quantum barriers within the probed range, the 3dB frequency does not show significant difference although the lasing power for the devices with 4 nm thick quantum wells is higher than that for the devices with 3 nm thick quantum well according to Fig. 9, and this is caused by increased threshold current according to the inset in Fig. 8(b). Note, the increased threshold current will delay the resonance condition [36].
Fig. 9. 3dB frequency bandwidth in terms of the InN gradient levels in the CGQBs for different VCSELs at the current of 15 mA.
4. Conclusions
In summary, GaN-based VCSELs with polarization self-screened active region are explored. The polarization self-screening effect is enabled by the polarization induced bulk charges in the quantum barrier that have the InN gradient composition. It shows that such polarization self-screened active region can reduce the polarization effect in the quantum wells and enhance the hole injection efficiency into the quantum wells close to the n-GaN side. Therefore, the lasing power can be enhanced. However, even insightful investigations indicate that further increase of the InN gradient level in the quantum barriers will decrease the lasing power especially for the VCSELs with thin quantum wells. We find that the reduced barrier height for the quantum barriers can shift the energy levels for the subbands, which will shift the wavelength of the peak gain position. Therefore, the optical feedback cannot be effectively obtained and causes the decreased lasing power. We therefore propose utilizing properly thick quantum wells when the polarization sell-screened active region is used. Finally, this work reports that the increased threshold current for VCSELs with thick quantum wells may sacrifice the 3dB frequency. Hence, a compromised design that can maintain both the enhanced lasing power and the wide 3dB frequency shall be made by using polarization self-screened active regions with properly thick quantum wells. We strongly believe that the findings in this work provide more device physical for GaN-based VCSELs, and will be useful for the community of optoelectronic devices.
Funding
National Natural Science Foundation of China (62074050); Research fund by State Key Laboratory of Reliability and Intelligence of Electrical Equipment, Hebei University of Technology (EERI_PI2020008); Natural Science Foundation of Hebei Province (F2020202030).
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.
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