Study of the spectral and power characteristics of In0.2Ga0.8N/GaN superluminescent light-emitting diodes by taking into account the piezoelectric polarization fields

In this study, the effects of the piezoelectric polarization field have been investigated on the spectral and power characteristics of In0.2Ga0.8N/GaN superluminescent light-emitting diodes. The Schrödinger and Poisson equations, the rate equations in the multiple quantum well active region and separate confinement heterostructure layers, and the optical propagating equations have been solved in the presence of the piezoelectric field. The results have been compared with results of the case of without piezoelectric field. According to the results, in the presence of piezoelectric field, the red-shift occurs in the spectra, and the width of spectrum increases. Also, the piezoelectric field decreases the peak intensity of spectrum and modal gain of the device.


Introduction
The suitable design of the electronic devices based on InGaN/GaN heterojunctions requires an accurate investigation of the effects of polarization on the quantum well (QW) structures. The performance of devices based on InGaN/GaN can be improved by increasing the information on internal electric fields. The piezoelectric (PZ) polarization is proportional to the strain induced by the lattice mismatch [1]. Furthermore, the spontaneous polarization (SP) is an intrinsic property resulting from the internal cell parameters c/a of the wurtzite materials [2]. In the QW structures, PZ and SP polarizations are the sources of an internal electric field in the range of MVcm -1 which causes the so-called quantum-confined Stark effect (QCSE) [3]. This field can affect the spectral and the power characteristics of the device [4].
A superluminescent light-emitting diode (SLD) is an edge emitter semiconductor optical source which contains regions of QWs, barriers, separate confinement heterostructure (SCH), and cladding layers. As with a laser diode (LD), a SLD is based on a p-n junction that operates optically when it is exposed to the forward bias, causing amplified spontaneous emission (ASE) in a wide range of wavelengths. On the other hands, the SLD has a low optical feedback, and there is no laser performance on it. A SLD combines the low temporal coherence of light-emitting diodes (LEDs) with high spatial coherence of LDs. The key parameters for describing SLDs are output power, optical gain, spectrum bandwidth, coherence length, spectrum modulation, and coherence function. The SLD has been developed as a key element in fiber-optic gyroscopes, optical coherence tomography for medical imaging, and fiber-optic communication devices. Other applications include optical fiber sensors, display screens for projectors, optical test instruments, and spectroscopy [5][6][7][8][9][10].
The effects of the PZ field on the output of LEDs have been reported in articles. The PZ field in InGaN/GaN MQW LEDs using the electric field-dependent electroreflectance spectroscopic method has been investigated by W. Z. Tawfik et al. [11]. Most produced SLDs are based on GaAs and InP. In recent years, studies have been conducted on GaN-based SLDs. A high optical power blue-emitting SLD grown on semi-polar bulk GaN substrate has been described by C. Shen et al. [12]. Also, A. Kafar [16]. The effects of internal electric field on the spectral characteristics of blue GaN-based SLDs have been studied by H. Absalan et al. [17]. In the present study, we examine the effects of the polarization on the spectral and power characteristics of In 0.2 Ga 0.8 N/GaN multiple quantum well (MQW) blue SLD via a self-consistent solution of the Schrödinger and Poisson equations, the rate equations in MQW active region and SCH layers, and the optical propagating equations.

Theory
The polarization in the SLD has an excellent effect on the recombination of carriers. Therefore, it is important to have a precise description of the polarization. The PZ coefficients of In x Ga 1-x N vary linearly with indium mole fraction [18]. In the In x Ga 1-x N/GaN structure, which is under the influence of biaxial compressive strain, the PZ polarization is in the (0001) direction; therefore, the SP and PZ polarizations oppose each other [19]. In this structure, the PZ field is much bigger than the SP polarization, so the total internal field value is close to that of the PZ field. If a material is grown as pseudomorphically on a substrate with different lattice constants, the grown layer is related to PZ polarization. The PZ polarization in the grown layer is expressed as follows [1] where e xx is the basal strain that defined as, where a GaN is the lattice constant of substrate, and a InGaN is the lattice constant of grown layer. The PZ field is expressed as follows where j and e 0 are the dielectric constant of the semiconductor and vacuum permittivity, respectively. The material parameters of In0.2Ga0.8N are listed in Table 1.
In order to design an appropriate SLD, it is necessary to have information about the band structure of the active region as well as the effects of the internal electric field on it. This is done by solving the Schrödinger and Poisson equations separately for conduction and valence bands based on effective mass approximation under an electric field. The polarization can alter the optical and electrical characteristics of the structure. Hence, it should also be taken into account when solving the Schrödinger and Poisson equations. We must calculate the subband energies of the QWs in active region. In our study, this is done by the finite difference method (FDM). A schematic of the SLD structure is shown in Fig. 1. In this device, the QWs produce the light, while the SCH and cladding layers play the role of confinement and waveguides of the light, respectively [27]. Figure 2 (black lines) shows the schematic profile of the studied MQW in without PZ field case. This structure consists of four In 0.2 Ga 0.8 N QWs (3 nm thickness) separated by three GaN barrier layers (10 nm thickness). The two AlGaN SCH layers (100 nm thickness) are located on the up and down of the active region. The temperature of active region is supposed constant (300K). The conduction and valence bands of MQW are strongly influenced by the PZ field. The PZ field can change the lineup of conduction and valence bands. Therefore, the depth of the well in the conduction and valence bands increases and decreases, respectively [28]. Also, the PZ field can tilt the structure of Piezoelectric constant e 31 Cm À2 À0:34 [25] Piezoelectric constant e 33 Cm À2 0:54 [21,25] Spontaneous polarization P SP Cm À2 À0:041 [20] Hydrostatic deformation potential Fig. 1 Schematic of the SLD structure the bands. In this case, the structure of the well becomes to a triangular well. This can be seen in Fig. 2 (red lines). In our theoretical model, the rate equations are written, while taking into account the effects of carriers escape from MQW to the SCH layer and the transport of carriers from SCH layer to MQW. The optical propagating equations are also written according to spectral power densities, i.e., P AE ðz; kÞ. We assume that the injection current is applied from the cladding layer to the SCH layer. Thus, regardless of the leakage effect from SCH layer, the equations are In above equations, J is the current density, N is the carrier density per unit volume of QWs, and N SCH is the carrier density per unit volume of the SCH layers. s n is the carrier recombination lifetime in the QW, s nðSCHÞ is the carrier recombination lifetime in SCH layers, s s is the carrier transport time in the SCH layers, and s e is the carrier escape time from the well to SCH layer. r sp ðkÞ is the spontaneous emission rate of photons, R st is the stimulated recombination of carriers caused by ASE [16,29], and S is the cross-section of active region of SLD that defined as multiplication of the thickness and width of the active region. Other parameters are listed in Table 2. The parameters without reference have been calculated for this research, and general parameters are available in Ref. [16].  Crystal-field split energy One-third of spin-orbit split-off energy The carrier reduction rates, due to the radiative and nonradiative recombination, are s À1 respectively, where A,B,C,A SCH ,B SCH , and C SCH are the Shockley-Read-Hall, spontaneous radiative recombination, and Auger recombination coefficients in QW and SCH layers, respectively [30,31]. By considering the transition model without k-selection rule, the spectral gain as a function of wavelength is expressed by [16] gðkÞ ¼ GLn and n r is the refractive index of QW, F C and F V are quasi-Fermi levels in the conduction and valence bands, respectively. E h1 and E e1 are the first energy subbands for heavy holes (HHs) and electrons in QW, respectively. In the presence of the PZ field, these values are changed, and therefore, the value of gain is changed. m Ã hw and m Ã ew are the effective mass of HHs in the QW valence band and effective mass of electrons in QW conduction band, respectively [29]. The momentum matrix elements M j j 2 represent the stimulated electron transitions that depend on the angle between the electron wave vector and the optical field vector. The momentum matrix elements are anisotropic, and the gain depends on the optical polarization. Usually, there are two distinct polarization modes, in which either the electric field (TE mode) or the magnetic field (TM mode) lies within the QW transversal plane (x-y plane) [34][35][36].
The electric field caused by PZ polarization affects the overlap integral and reduces it. In TE mode, the momentum matrix elements are written as, [16] where O ij is the overlap integral, and m Ãt ew is the electron effective mass in transversal direction relative to the hexagonal c-axis. The D 1 ,D 2 and D 3 parameters represent the diagonal and non-diagonal elements of 6 Â 6 Hamiltonian that explain the three valance bands [33]. The strain changes the band gap energy. In this case, the band gap energy is [26]: where D is the interband hydrostatic deformation potential, and c 11 and c 12 are the elastic stiffness constants. Our selfconsistent calculations have been carried out in both strained and unstrained cases for comparison.
In steady-state case, by analyzing the equation (6) and assuming exp½CgðkÞ À aÞL111, the total output power of the SLD in a given current on the right facet of the SLD cavity (z ¼ L) in direct propagation is defined as where n SP is the spontaneous emission factor [16]. By assuming that the z-axis is along the [0001] direction, the Schrödinger equation is defined as, where h is the Planck's constant divided by 2p, m Ã ðzÞ is the electron effective mass at the band edge, wðzÞ is the wave function of electron, E is the subband energy of electron, and VðzÞ is the potential energy: where DE c is conduction band gap discontinuity. The Poisson's equation for the electrostatic potential uðzÞ is expressed as follows where nðzÞ and pðzÞ are electrons and HHs distribution in the wells and barriers, respectively [37]. The process of the simulation method used in this study is presented as flowchart in Fig. 3. As shown, by choosing the minimum current density, and taking into account the initial carrier density and quasi-Fermi Levels equations, the Schrödinger, Poisson, and gain equations are solved in the presence of PZ field, and the electronic structure is obtained. Then, the rate equations are solved self-consistently in stationary state. In the case of convergence, the next current density is considered and the program is repeated. Otherwise, the new carrier density is selected and the program is updated. After final current density, the diagrams of the light output power as a function of current, the spectral radiation power as a function of wavelength, and the modal gain as a function of wavelength are plotted.

Results and discussion
In order to investigate the PZ polarization field effects on the optical properties of the studied device, the light output power as a function of injection current is plotted in Fig. 4 in the cases of with and without PZ polarization. Unlike LDs, the light output power intensity of SLD does not show a sharp threshold, but it gradually rises with increasing of injection current. The soft curvature in the output power curve as a function of the injection current represents the transition from a situation influenced by spontaneous emission (SE) to a situation influenced by ASE. It can be seen from Fig. 4 that as the injection current increases, the output power in both cases (without PZ and with PZ) grows. Comparison of two diagrams shows that at low current injections, there are no significant differences in output power of two cases but after that, the output power decreases in the presence of PZ field. The reason for the decline in the output power at high injection currents is that the presence of the PZ field leads to a gradual shrinkage in the spatial overlap between the electron and hole wave functions, where the probability of the radiative recombination diminishes. At the applied currents of 110, 270, and 540 mA, the differences between the output power in without and with PZ cases are 0.046, 0.136, and 1.081 mW, respectively. At the injection current of 600 mA, the maximum powers in without and with PZ field cases are 13.31 and 11.79 mW, respectively (8.86% dropped power). Figure 5 shows the spectral radiation power as a function of wavelength under different injection currents (400, 500, and 600 mA) in without and with PZ field cases. According to the diagrams, the obtained spectra are in the form of a Gaussian function. The obtained diagrams are in good agreement with the diagram of the experimental work of Tawfik et al. on the LED [38]. The reason for this comparison is that there is no experimental work in this field on SLDs, and therefore, comparison is not possible. The peak intensity, the width, and the peak wavelength of spectrum in SLD depend on the composition of the active region and the value of injection current. As seen, upon the rise of the injection current, the electroluminescence (EL) spectra show significant luminescence enhancements because of the photo-electric conversion. As can be seen, in   The spectral radiation power as a function of wavelength under different injection currents in without PZ field and with PZ field cases both cases (without PZ and with PZ), with elevation of the current, the peaks of the spectrum are transmitted to shorter wavelengths. For example, in with PZ field case, as the injection current increased from 400 to 600 mA, the peak wavelength of the spectra shifted from 425.3 to 424.4 nm. This is mainly due to the screening of internal fields [24]. In ternary nitride alloys grown on GaN, similar to the studied structure, the internal fields significantly affect the properties of QW. Upon augmentation of the injection current into the QW, charge screening is expected to reduce the polarization field effects. This effect has a great influence on the band structure and carrier dynamics in QW. The screening of the electric field inside the QW reduces the polarization effects, with this reduction in QW causing improved efficiency of SLD.
Polarization has an important influence on recombination mechanisms in QWs. The surface polarization charges are somewhat screened by defects of charges, and only about half of them contribute to the internal polarization field [39]. By applying the polarization, the internal field in the p-n junction of the QW is negative, and the holes accumulate at the n-side of junction. In without polarization case, the situation reverses and the holes accumulate on the p-side of the QW. In both of the above cases, the electron distribution peak is near the center of the QW. At low injection currents, the spectra are affected by the SE process. By increasing the injection current, the ASE process dominates the spectra, thus elevating their bandwidth. This can be clearly seen in Fig. 5. On the other hands, it can be observed that in the presence of the PZ field, the spectra shift toward the higher wavelengths (red-shift). This is attributed to the recombination of carriers at the deep energy levels of QW and to the renormalization of the band gap [38,40].
The useful parameters that are related to the distribution of spectral power density at different wavelengths are the optical bandwidth, the peak wavelength, and the peak intensity of the spectrum. The first parameter is the full width at half maximum (FWHM) of spectral power curve versus wavelength, while the second parameter depends on the wavelength with the highest intensity. Table 3 provides the values of the peak wavelength, FWHM, and the peak intensity of the spectra across different injection currents. It can be observed that in both cases (with and without PZ field), as the injection current rises, the peak wavelength of the spectra is declined. The PZ field reduces the transition energy in QW, and therefore, the red-shift occurs in the emission wavelength [41,42].
It can be seen that the PZ field increases the width of the spectrum. The broadening of the spectrum can be attributed to the QCSE and band filling effect [39,43,44]. The band filling effect is due to of the occupation of low energy states of bands by the carriers. Consequently, transitions between the occupied states in the valence and conduction bands are prohibited. Note that a strong PZ field, by tilting the band structure in QW, can reduce the effective width of the well and hence the carrier capture rate in QW. On the other hands, the PZ field, by reducing the overlap of the electron and hole wave functions, prolongs the lifetime of the carriers in the QW. Hence, the probability of carrier escape rate diminishes, and as a result, the peak intensity of the spectra decreases. At injection current of 400 mA, the peak intensity in without PZ field and with PZ field cases is 0:926 Â 10 18 and 0:921 Â 10 18 s À1 , respectively. At 600 mA, these values are 2:229 Â 10 18 and 1:828 Â 10 18 s À1 , respectively. It can also be seen that the red-shift has occurred in the spectra. The results show that in the presence of PZ field, the change in the peak wavelength of the spectra is about 2 nm, which has a good agreement with the results of the Refs. [45] and [46]. At 400 mA, the spectrum peak in cases of with PZ field and without PZ field has occurred at wavelengths of 425.3 and 423.3 nm, respectively. At 600 mA, these values are 424.4 and 422.6 nm, respectively. Figure 6 shows the FWHM of spectra as a function of injection current in without PZ field and with PZ field cases. In Fig. 5, due to the slight difference in the spectral radiation powers of the two cases at injection currents of 100 to 300 mA, only the currents of 400 to 600 mA are considered. In Fig. 6, in order to better compare the FWHM changes in the two cases, the injection currents of 100-300 mA are also considered. As can be seen from Fig. 6, in the presence of the PZ field, the bandwidth of the spectra increases. Elevation of the FWHM of spectra is one of the important features of SLDs. Since the FWHM is dependent on the wavelength and the gain, with increase in Table 3 The peak wavelength, FWHM, and the peak intensity of the spectra the injection current, the gain of device grows, causing enhanced FWHM [12]. In InGaN/GaN MQW structures, the QCSE created by the PZ field separates the electrons and holes spatially. This spreads out the occupancy of the carriers in the joint density of states, whereby the FWHM increases. As previously mentioned, upon elevation of the injection current, the influence of ASE is intensified, and the spectra are expanded. The maximum value of FWHM in without PZ field obtained in the injection current of 600 mA is 5.1 nm. In the presence of PZ field, as the current injection increases from 100 to 600 mA, the FWHM values rise from 5.6 to 6.0 nm.
In SLDs, the optical gain is one of the main recombination mechanisms and can be effective on the optical and spectral properties of the structure. Therefore, a careful examination of it is necessary. Figure 7 shows the modal gain as a function of wavelength in the injection currents of 400, 500, and 600 mA in without PZ field and with PZ field cases. It is observed that in both cases, with increasing the injection current, the modal gain increases. As the carrier density increases in the active region of the device, the distance between the quasi-Fermi surfaces increases, and thus, more surfaces can contribute to the optical gain. As a result, the gain increases. On the other hands, it is observed that the PZ field reduces the gain. This is why in SLDs the internal field caused by the PZ polarization causes the spatial separation between the electron-hole wave functions. This reduces the overlap of electron and hole wave functions, and as a result, the interband optical matrix elements are reduced. Table 4 presents the peak modal gain values and their corresponding wavelengths at 400, 500 and 600 mA injection currents in without PZ field and with PZ field cases. As can be seen from the table, by increasing the injection current, the peak modal gain increases in both cases. Also, the position of the gain peaks shifts to shorter wavelengths (blue shift). As the carrier density increases, in addition to the band filling effect, the screening electric field also increases. It can be seen that with increasing injection current from 400 to 600 mA, the blue shift of wavelength in without PZ field and with PZ field cases is 1.0 and 1.1 nm, respectively. On the other hands, it is observed that the PZ field reduces the peak modal gain and increases the wavelength at which the peak modal gain appears (red-shift). These red-shift values are 2.1, 2, and 2 nm in the 400, 500, and 600 mA injection currents, respectively.

Conclusions
The effects of the strain-induced PZ polarization field investigated on the spectral and power characteristics of In 0.2 Ga 0.8 N/GaN SLD. The calculations were based on the self-consistent solution of the Schrödinger and Poisson Fig. 6 The FWHM of spectra as a function of injection current in without PZ field and with PZ field cases Fig. 7 Modal gain as a function of wavelength at different injection currents in without PZ field and with PZ field cases. equations, the rate equations, and the optical propagating equations. The results showed that the presence of the PZ field in the structure led to lose of the modal gain of the device and the peak intensity of spectrum, red-shift in spectra, and increase the FWHM and the peak wavelength of spectra. In the presence of PZ field, under 600 mA injection current, the output power has dropped 8.86%. The FWHM was also increased from 5.6 to 6.0 nm.