1 Introduction

UV-emitting semiconductor optoelectronic devices are of tremendous interest for a wide range of applications, including solid-state lighting [11]. These leaked electrons may then recombine with the holes in the p-region, which will overcome the injection of holes in active region [12]. The recent proposed designs have overcome these challenges, which include the superlattice design of last quantum barrier [13], quaternary superlattice last barrier [14], double tapered EBL [15], step do** in waveguide and cladding layer [16], compositional Al-grading of silicon-doped layers [17], AlGaN-based polarization doped layers without impurity do** [18], step-graded quantum barriers with graded EBL [19] and inverse trapezoidal EBL [1. Both LDs have a cavity length of 1500 μm. Both end mirrors have a reflectance of 0.9. The design parameters are tabulated in Table 1. The following parameters were used in the designs, i.e., electron and hole mobilities are 100 cm2V−1 s−1 and 10 cm2V−1 s−1, respectively, while the value of the Auger coefficient is 1 × 10–30 cm6/s. We analyzed our structures numerically using SiLENSe™ 6.3. SiLENSe™ is a one-dimensional module that simulates the behavior of LD heterostructures made up of direct-bandgap wurtzite semiconductors like group-III nitrides and group-II oxides. It is based on a 1D drift–diffusion model that takes into account some of III-Nitride's unique features such as direct-bandgap wurtzite semiconductor materials like TDD, low efficiency of acceptor activation, possible spontaneous electric polarization and piezoelectric polarization. The SiLENSe™ simulates the LD energy band diagrams as a function of voltage (bias), electron and hole momentum inside the LD heterostructure, and radiative and non-radiative recombination that causes light emission.

Fig. 1
figure 1

Structures of LD-A and LD-B

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Table 1 Design parameters
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3 Theory and methodology

SiLENSe™ enables the user to design various layers of III-Nitride LDs by using bandgap engineering principles [21,22,23]. If Jn and Jp are the current densities of electrons and holes, respectively, then using the Golden Fermi’s rule, we can determine:

$$ J_{n} = q\mu_{n} nE + qD_{n} \nabla n $$
(1)
$$ J_{p} = q\mu_{p} pE + qD_{p} \nabla p $$
(2)

Electron and hole mobility are represented by μn and μp, while carrier diffusion coefficients are denoted by Dn and Dp. The relation of net current density J = Jn + Jp must follow the continuity equation:

$$ \nabla \left( {J_{n} { } + { }J_{p} } \right){ } + { }q{ }\frac{\partial }{{\partial t}}\left( {p{ } - { }n} \right) = { }0{ } $$
(3)

Also

$$ \nabla J_{n} - q\frac{\partial n}{{\partial t}} = + qR $$
(4)
$$ \nabla J_{p} + q \frac{\partial p}{{\partial t}} = - qR $$
(5)

The net recombination (R) is given as,

$$ R = R_{n} - G_{n} \;\;\;\;{\text{for electrons}} $$
(6)
$$ R = R_{p} - G_{p} \;\;\;\;{\text{for holes}} $$
(7)

here Gn, Gp and Rn, Rp denote the generation and recombination rate for n-type and p-type non-equilibrium carriers. If the total recombination rate across an active region having width w is constant, then

$$ J_{n} = qwR = qw(R_{n} - G_{n} ) $$
(8)
$$ J_{p} = qwR = qw(R_{p} - G_{p} ) $$
(9)

The chemical rate equation of generation rate and recombination rate is given as,

$$ \begin{array}{*{20}c} c \\ {n + p \rightleftharpoons \gamma } \\ e \\ \end{array} $$
(10)

here c and e represent the capture and emission rate of electron–holes pairs, while γ signifies a photon created. So, the recombination and generation rates are:

$$ R_{n} = R_{p} = cnpG_{n} = G_{p} = e $$
(11)

From the absorption of each photon, an electron–hole pair is created. At an equilibrium, the net rate will be zero such as:

$$ \begin{aligned} & R_{n} = cn_{i} p_{i} - e = 0 \\ & where\;e = cn_{i} p_{i} \\ \end{aligned} $$
(12)

hence the resultant recombination rate is then,

$$ R = c(np - n_{i} p_{i} ) $$
(13)

4 Results and discussion

At a current density of 10.7 kA/cm2 in Fig. 2a, LD-B has a higher internal quantum efficiency than LD-A. The maximum IQE of LD-A is 44%, whereas LD-B has a peak IQE of 73%. The efficiency droop is reduced from 50 to 5%. The emission spectra of both structures are represented in Fig. 2b. Figure 2b shows that at a current density of 10.7 kA/cm2, the emission peak of LD-B is higher than that of LD-A. Both LDs emit in the DUV range, i.e., 255–275 nm. The high intensity of LD-B is due to the high radiative recombination rate in the multiquantum wells which will be discussed further.

Fig. 2
figure 2

a IQE vs current density profiles and b emission intensity vs wavelength profile of LD-A and LD-B

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The IQE profile of varied thicknesses of the quaternary layer (QL) in LD-B is illustrated in Fig. 3a. The IQE reduces as the thickness of the QL increases from 3 to 8 nm. This is because a thick p-doped quaternary layer, i.e., 8 nm, results in limited hole transport toward the active region, lowering the radiative recombination rate and hence lowering the IQE. Thus, 3-nm thickness is the optimized thickness in the given device. Figure 3b depicts the IQE profile with different Al concentrations in QL. At 70% Al composition, the IQE is maximum with the lowest efficiency droop. IQE decreases as the Al content is increased further. LD-B with a 65% Al concentration in QL has a lower IQE than LD-B with a 70% Al content in QL. Similarly, LD-B with 80% Al content in QL has lower IQE as well as higher efficiency droop than LD-B with 70% Al content in QL. Thus, 70% aluminum is the optimized composition in the given device. The high Al concentration (80%) results in increased barrier height for holes due to which the hole injection in MQWs decreases. The number of electrons is high as compared to the number of holes, which results in the asymmetrical distribution of carriers in the active region. This reduces the device efficiency and increases the efficiency droop as shown in Fig. 3b. Similarly, low Al concentration (0.65) results in high electron leakage from the active zone that contributes to non-radiative recombination. The non-radiative recombination results in reduced IQE and power [24]. The optimum QL with 3 nm thickness and 70% Al concentration reduces the lattice mismatch between the EBL and the active layers due to which the built-in polarization is reduced. The asymmetry between electrons and holes is minimized, which leads to a high radiative recombination rate. An optimized thickness and Al concentration in the quaternary layer are needed to enhance the IQE of the proposed device [25]. Thus, we have employed the optimized QL in LD-B.

Fig. 3
figure 3

IQE versus current density profiles with a different QL thicknesses and b different Al contents

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The enhancement in IQE is due to optimum hole and electron concentration and their recombination in quantum wells (QWs). The electron and hole concentration profiles are shown in Fig. 4. It is found that the carrier concentration in the active region close to the p-side of the proposed LD-B is higher than the reference LD-A. The hole concentration is enhanced by 35% compared to LD-A as depicted in Fig. 4a. And the electron concentration is increased by 37%, as shown in Fig. 4b. Consequently, LD-B has a higher IQE than LD-A.

Fig. 4
figure 4

Carrier concentration in MQWs. a Hole concentration. b Electron concentration. c Radiative recombination rate profile

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The high recombination rate in LD-B MQWs is responsible for the increase in IQE. The radiative recombination rate profiles of both LDs as a function of position are shown in Fig. 4c. LD-B has a higher rate of radiative recombination than LD-A. This is because many electrons and holes recombine in MQWs in LD-B. The presence of a p-doped quaternary layer between the p-EBL and the p-waveguide lowers the barrier for holes due to high p-do** in the quaternary layer, allowing many holes to get into the active region. The radiative recombination rate is improved by 74% as compared to LD-A.

Carrier current densities of LD-A and LD-B are plotted in Fig. 5. The electrons are injected into the active zone from the n-region and recombine with holes in quantum wells, resulting in a lower electron current density in the quantum well profile [26]. Also, because more carriers contribute to the radiative recombination rate, the drop in electron current density grows from n-region to p-region as illustrated in Fig. 5a. Electron leakage in LD-B is smaller than LD-A at the p-side of the last quantum well, indicating that more carriers recombined in quantum wells of LD-B. Similarly, the hole current density in LD-B is higher than in LD-A on p-side as illustrated in Fig. 5b.

Fig. 5
figure 5

a Electron flux and b hole flux profiles of LD-A and LD-B

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Figure 6a illustrates the energy band profiles of LD-A and LD-B. The conduction band barrier height in LD-B is 335 meV, which is much larger than the barrier height of 297 meV in the reference LD-A as depicted in the figure. The increased barrier height prevents electron leakage on the p-side. The energy barrier height for holes in the valence band is reduced from 643 to 556 meV. So, the hole injection is larger in the active zone of the proposed LD (LD-B), which results in a high recombination rate.

Fig. 6
figure 6

Energy band diagram of a LD-A and b LD-B

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The power (P) vs current (I) plot is depicted in Fig. 7. The threshold current of LD-A is about 662 mA, whereas the threshold current density of LD-A is 21 kA/cm2. For LD-B, the threshold current is comparatively reduced to about 186 mA, whereas the threshold current density is 6.13 kA/cm2. So, LD-B has a lower threshold current as compared to LD-A.

Fig. 7
figure 7

P-I characteristics of LD-A and LD-B

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5 Conclusion

The influence of a quaternary layer on the optical performance of UV-C AlGaN-based laser diodes (LDs) is demonstrated in this study. The recombination of carriers increases in multiple quantum wells due to a 35% increase in hole concentration and a 37% increase in electron concentration, resulting in a 74% increase in the radiative recombination rate. The efficiency droop is reduced from 50% to 5%, and the IQE is improved by 73%. The emission intensity has greatly improved in our proposed LD-B. Therefore, our proposed LD-B could lead the researchers toward attaining high-efficiency AlGaN-based DUV LDs with little or no droop.