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Abstract
21st-century studies in the field of epitaxy brought observable progress in a field of heterogeneous integration of III-V materials onto silicon photonic systems. BGaAs/GaP quantum wells (QWs) are a new material system, which can be grown on GaP/Si templates and thereby can be integrated with a Si platform. This work presents calculations of the material optical gain spectra for the QWs modified by an incorporation of group III and V elements, in both the QWs region and its barriers. The gain spectra are calculated based on the 8-band k · p model and Fermi’s Golden Rule. Incorporations of indium and phosphorus are analyzed for the thin film of the QWs, and boron, aluminum, indium, and arsenic for its barriers. Since an energy of gain peak in the BGaAs/GaP QWs can be tuned within a range of 690-730 nm and the positive optical gain is presented for the QWs with these modifications, this system is promising for further development of red-light lasers integrated with Si.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Silicon technology is well applied in the field of optoelectronics [1–3] with an exclusion of highly efficient light-generating structures. Its main assets are: scalability, natural easiness of surface passivation and well developed methods of making waveguides. However, the indirect gap of silicon averts having both easily accessible and easy to integrate light sources, and therefore, obstructs manufacture of cheap lab-on-a-chip devices. Many methods, such as utilising Si bulk crystals, Si nanoparticles, Er-doped Si, GeSn crystals grown on Si, III-V structures bonded to Si based structures, etc. [4–10], have already been tested or are still under development. Yet, none of available methods are satisfactory enough to be recognized as a mature technology of Si integrated lasers. Other new and promising structures that may be proposed for such devices are BGaAs/GaP quantum wells (QW) which can be grown on advanced templates such as GaP/Si [11–13]. These QWs are also proposed in our previous work [14]. Even though we do not focus on the technological aspect of these QWs in either publication, some of recent advances in the field of their growth are worth mentioning.
An addition of boron to a GaAs host by both the molecular beam epitaxy [15,16] and the metalorganic chemical vapor epitaxy [17–27] has been reported in several papers so far. In all the works, BGaAs layers or QWs were grown on GaAs substrate with a low incorporation of boron. The highest fractions of boron (8%) in BGaAs films that are grown on GaAs are achieved by Hamila et al. [18]. A larger incorporation of boron leads to a high tensile strain, which is why, crystal growers are not interested in BGaAs layers grown on GaAs with high mole fractions of boron. But a replacement of GaAs substrate with GaP allows growing BGaAs layers coherently strained to it, when at higher mole fraction of boron. The discussed layers were reported recently at a boron mole fraction of up to 17.5% [28]. So, as expected, BGaAs with a larger mole fractions of boron may be grown on GaP substrate. Since GaInAs and GaPAs can be achieved in the full content range [29], an incorporation of In or P into such QW system may be easily anticipated. And so, a growth of BGaAs/GaP QWs is feasible. A possible challenge may lay in obtaining an optical quality of the B-containing layers. That is because boron is a small atom in comparison to other group III or V elements and may be incorporated in interstitial positions, thereby being a source of a non-radiative recombination. However, we believe that the optimization of the growth conditions (growth temperature, etc.) as well as the post-growth treatments (annealing temperature and time) would allow obtaining the BGaAs/GaP QWs gain medium with a high quantum efficiency. It is vital to investigate what kind of effect an incorporation of other elements from groups III and V, i.e. isovalent dopants, may have on the material optical gain. Therefore, the aim of this paper is to explore the BGaAs/GaP QWs, focusing on optical gain sensitivity to a few selected modifications of constituent layers.
As P is present in the barriers and In is widely used for growth of III-V structures in band gap engineering or alternation a thin film quality, analogously to emitters made of III-nitrides, the first part of presented analysis examines cases of P and In substitution of Ga and As respectively. Other important and typical for lasers parts of optimization of QWs are modifications of barriers in order to obtain proper waveguides. Therefore, the second part of the paper focuses on effects of substitutions in the barriers on the gain, namely Ga with B, Al, or In, and P with As. The scheme of the analysis is presented in Fig. 1.
Fig. 1. A scheme presenting the admixtures of the thin-film and barriers of B$_x$Ga$_{1-x}$As/GaP quantum wells that are considered in this work.
All band structures are computed based on the 8-band $\boldsymbol{k} \cdot \boldsymbol{p}$ model with an envelope function approximation. The applied procedure includes self-consistent approach to the Poisson and Schrödinger equations with parabolic approximation of indirect valleys. Such approach is necessary for a reasonable description of the free carrier distributions in this system. The optical gain spectra are computed based on Fermi’s Golden Rule. For the details on the $\boldsymbol{k}\cdot \boldsymbol{p}$ Hamiltonian and gain calculation see our previous papers [30–32]. The critical thickness of each layer of the analyzed structures is estimated based on the Matthews-Blakeslee model [33].
2. Results and discussion
BGaAs/GaP QWs are expanded by adding either In or P to the BGaAs thin film material and B, Al, In or As to the GaP barrier material. As a result, BGaInAs and BGaPAs alloys for QWs thin film and BGaP, AlGaP, GaInP, and GaPAs alloys for barriers are used to describe the QWs in this work. All material parameters used for description of these alloys are presented in the Appendix A. In the course of this study, the modification of QWs thin films is considered first and followed by the modification of GaP barriers analysis.
2.1 Critical thickness
Since the critical thickness of the modified BGaAs thin layer and GaP barriers is crucial in the context of QWs’ growth, a short discussion on the topic is presented in this work. Critical thicknesses of the BGaAs, BGaPAs with 10% of P, and BGaInAs with 10% of In are shown in the Fig. 2(a). In the thin film of the QWs, all considered reasonable B fractions vary from around 15 up to around 30%. In such fraction range, the critical thickness of the layer is higher than around 15 nm. When creating a compound of BGaAs with 10 $\%$ of either P or In in the thin layer, the critical thickness can be decreased down to around 10 nm in some extreme cases. For B fractions from 20 up to 26%, which are close to conditions of lattice matching a GaP substrate, the critical thickness of admixed material is greater than around 20 nm. The admixed barriers are considered to be strained to the GaP substrate as it is more reliable technologically to have a pure GaP crystal as a substrate compared to any other alloy. A number of various situations occur among considered admixtures of barriers showed in the Fig. 2(b), where alloys: GaInP, GaPAs, BGaP, and AlGaP are considered. Because the lattice parameter of BP differs from the lattice parameter of GaP by about 16.7$\%$, alloying 10$\%$ of this material results in its critical thickness of only around 10 nm. And it is not enough for a reasonable barrier. The lattice parameters of InP and GaAs differ from the lattice parameter of GaP by about 7.7 and 3.7$\%$ respectively, alloying 10$\%$ of these materials results in their critical thicknesses of around 30 and 90 nm. These thicknesses are sufficient to form barriers of QWs. In a typical semiconductor QW, barriers of around 20 nm are enough for the wave function of confined electron to vanish. A more favorable situation occurs in the case of an addition of 10% of Al into GaP, which does not change the lattice parameter substantially. Lattice parameters of AlP differ from the lattice parameter of GaP by about 0.3 $\%$, which further results in the critical thicknesses around 1250 for incorporation of 10$\%$ of Al into GaP. Therefore, simulations of the QWs with AlGaP barriers correspond directly to the hypothetical structures that should be possible to grow. Simulations of the QWs with GaInP or GaPAs barriers correspond to the structures that also should be possible to grow, but with notably limited thicknesses of the barriers. The case of the QWs with BGaP barriers is presented with such fractions only for a systematic presentation of the admixing effects.
Fig. 2. (a) Critical thicknesses of B$_x$Ga$_{1-x}$As, B$_x$Ga$_{1-x}$P$_{0.1}$As$_{0.9}$, and B$_x$Ga$_{0.9-x}$In$_{0.1}$As grown on a GaP substrate. (b) Critical thicknesses of Ga$_x$In$_{1-x}$P, GaP$_x$As$_{1-x}$, B$_x$Ga$_{1-x}$P, and Al$_x$Ga$_{1-x}$P grown on a GaP substrate.
2.2 P and In incorporation into BGaAs
The conduction and the valence band extremes of BGaPAs and BGaInAs when strained to GaP substrate are shown in Fig. 3(a) and (b). The majority of changes caused by such incorporations are visible in the energies of conduction bands at $\Gamma$ point ($CB^\Gamma$) and at $\Delta$ line ($CB^\Delta$). The energies of a heavy-hole (HH) and a light-hole (LH) bands shift to a lower extent. All changes of band energies at L point ($CB^L$) are negligible. It is observed that the main result of admixing P and BGaAs is the shift of $CB^\Gamma$ toward higher energies. To be precise, at P mole fraction of approximately 3% the energy of $CB^\Gamma$ passes a $CB^\Delta$ energy of GaP substrate and QWs barriers, and at 4% direct-indirect transition of the admixture occurs. It is slightly different when incorporating In instead of P. In such case, while energy of $CB^\Gamma$ increases with an addition of In, the energy of $CB^\Delta$ decreases. Whereas significant changes in energies of conduction bands are also observed, these are smaller than in the previously described circumstances. In this case, In at 7% results in an indirect-gap character of the thin film. The direct-indirect transition occurs below the $CB^\Gamma$ energy of GaP. Slight changes visible in the valence bands are mostly negligible, yet still, taking part in the changes of the band gap. Additional effects of these changes may appear at B fractions at around 23%, where strain-free or slightly strained materials are formed. In such cases, any change of valence band energies may cause additional modifications of polarization due to an occurring interchange of relative energies of HH and LH bands. Therefore, the most important effects on the system that are induced by an incorporation of In or P into BGaAs thin film are present in the conduction bands.
Fig. 3. (a) Conduction and valence band extremes of B$_x$Ga$_{1-x}$P$_y$As$_{1-y}$ with 0%, 1%, 2%, and 3% admixtures of P. (b) Conduction and valence band extremes of B$_x$Ga$_y$In$_{1-x-y}$As with 0%, 1%, 2%, and 3% admixtures of In.
The results of the energy changes of the bands in the gain spectra are presented in Fig. 4(a)-(d). As shown, the gain values decrease substantially in both cases of the admixing. If compared, an addition of P causes bigger reduction of the gain. Hence, such alloying significantly influences the polarization of the emitted light. Whether transverse electric (TE) or transverse magnetic (TM) polarization remains present depends on the B fraction in the QWs thin film. The fractions for which the gain spectra disappear are lower than the fractions of the direct-indirect transitions, due to values of the ground-state energies of electrons at $CB^\Gamma$. Moreover, for the admixtures of up to 3$\%$ of additional elements the gain spectra take negative values in all presented cases. In line with our previous work [14], selected fractions and thicknesses of the QWs are optimal or almost optimal for a given radiation polarization. Therefore, it is expected that the optical gain will predominantly vanish at lower % of P or In in the calculations performed for other thicknesses and fractions of B in the thin layers.
Fig. 4. Material gain of 10 nm wide QWs in a temperature of 300 K and with two-dimensional carrier concentration equal to $12\cdot 10^{12}$ cm$^{-2}$. (a),(b) The gain of B$_x$Ga$_{y}$In$_{1-x-y}$As/GaP QWs with fraction of In substituting Ga. (c),(d) The gain of B$_x$Ga$_{1-x}$P$_{y}$As$_{1-y}$/GaP QWs with fraction of P substituting As. (a),(c) The gain of TE polarization. (b),(d) The gain of TM polarization.
2.3 B, Al, In, and As incorporation into GaP
The elements can be also incorporated into other parts of the structure - the barriers of the QW. There are four cases to be considered: admixtures of B, Al, In, and As. The extremes of the conduction bands in GaP$_x$As$_{1-x}$, B$_x$Ga$_{1-x}$P, Ga$_x$In$_{1-x}$P, and Al$_x$Ga$_{1-x}$P that are strained to GaP are shown in Fig. 5(a)-(d) respectively. The results of such alloying have various effects on $CB^\Gamma$, all of which are negligible within a wide fraction range for 1.5 eV deep electron quantum confinement at $\Gamma$ point. However, a decrease of the energy connected to the $CB^\Delta$ minimums is observed in every case.
Fig. 5. (a)-(d) Conduction band extremes of B$_x$Ga$_{1-x}$P, Al$_x$Ga$_{1-x}$P, Ga$_x$In$_{1-x}$P, and GaP$_x$As$_{1-x}$. (e)-(h) Material gain of 10 nm wide QWs in a temperature of 300 K and with two-dimensional carrier concentration equal to $12\cdot 10^{12}$ cm$^{-2}$: (black lines) the gain for B$_{0.23}$Ga$_{0.77}$As/GaP QWs, (color lines) the gain for QWs with 10% admixtures of B, Al, In, and As respectively.
An addition of B to GaP crystal reduces the lattice parameter of the alloy, what results in notable tensile strain effects visible in the energies of these valleys. While the average energy of $CB^\Delta$ is increasing due to hydrostatic deformations, the lift of a degeneracy causes an energy decrease in one third of the $\Delta$ valleys. When Ga is substituted with Al, the lattice parameter of GaP is changed negligibly so there is almost no strain effects present. The decrease is, therefore, a pure effect of the interpolation of respective values related to GaP and AlP crystals. Hence, all $\Delta$ valleys experience a decrease in energy. A replacement of Ga with In or P with As increases the lattice constant of the alloy. A strain related splitting of the $\Delta$ valleys causes a decrease in the energy in two thirds of them. The decrease in the energy of the $\Delta$ valley dominantly determines changes in the gain values, despite having various numbers of the valleys involved in each case. Since a density of states of $\Delta$ valleys is around 2 orders of magnitude greater than the one of $CB^\Gamma$, the slowly decreasing valleys always bring a great number of states that are free to be filled and cause comparable electron leakages in the structure. Any changes in electron and hole confinements at $\Gamma$ point, varied among the described cases, are negligible and do not cause any additional leakages. Therefore, an addition of B to GaP crystal affects the gain values to the lowest degree and addition of As to the highest. This is because changes in the energy of $\Delta$ valleys are the smallest in the first case and the biggest in the last case.
3. Conclusions
In this paper, we discuss the effects of BGaAs/GaP QWs modifications on optical gain values obtained by admixing constituent layers with B, Al, In, P, and As. It is interesting as some isovalent dopants may be profitable during the epitaxy, and so, understanding what is the effect on the optical gain seems valuable. All effects observed in the gain values were mainly caused by the energies of $\Delta$ valleys. An addition of In or P to BGaAs thin film may be favorable only at a low mole fraction regime, since even a small increase of their fractions can significantly reduce the optical gain. However, alloying GaP barriers with other materials should affect the gain in lower degree, which then may allow a greater flexibility of the structure design. Finally, it is shown that the presented tuning is not a direct way to improve light emitting properties of the structures by increasing the material gain. At the same time, alternative advantages (growth quality, wave guiding, etc.) of such tuning may occur and result in the increase of modal gain in real structures, especially when the admixtures do not affect the material gain substantially. All these results show that BGaAs/GaP QWs grown on the GaP/Si virtual substrates are rising structures for gain medium of red-light lasers and can be additionally, and with caution, tuned by alloying with other III-V semiconductors.
Appendix A
Parameters of materials
Parameters of the ternary alloys (BGaP, AlGaP, GaInP and GaPAs) are interpolated on the basis of Vegard’s law with bowing parameters, as it is well established method. In the system discussed there are two different types of quaternary alloys, and parameters for them are interpolated based on respective weighted sums, including a correction for one of them as reported by M. Guden, et al. [34]. All material parameters of 7 binary compounds (BP, AlP, GaP, InP, BAs, GaAs, and InAs) needed for the interpolation are listed in Table 1. Parameters that are not available (N.A.) are approximated to the parameters of GaAs, when applied to the description of the thin film, or GaP, when evaluating the parameters of the barriers. All known bowing parameters for the ternary alloys (BGaP, AlGaP, GaInP, GaAs, BInAs, GaInAs, BPAs, and GaPAs) are listed in Table 2. In this work, all bowing parameters that are not included in the table are posited as zero.
Table 1. Material parameters of binary compounds.
Table 2. Non-zero bowing parameters of ternary alloys.
Parameters of the binary compounds
As AlP, GaP, InP, GaAs and InAs are well known materials, parameters used to describe them are mostly taken from review work of Vurgaftman et al. [29] or semiconductor handbooks [35,36]. Excluded from the group are: hydrostatic deformation potentials of GaAs, taken from a more reliable experimental work [37]; and Varshni parameters of GaP and InP, not included in the table since other than empirical Varshni form was used in the calculations [29]. As BAs and BP are not as well known binary compounds, a complete sets of theirs parameters are not yet available in the literature. For BaAs we used the set of the parameters as gathered in our previous paper [14], see authors commentary on some parameters gathered below. For BP we gathered a necessary set of parameters in the course of this study, to be presented further.
Several parameters describing BAs are obtained from literature indirectly. The valence band offsets (VBO) for BAs/GaAs interface are taken from work of Hart & Zunger [38]. Since the VBO of BAs is taken from other source than VBO of other binary compounds, its value is consistently shifted. Hydrostatic deformation potentials are not available separately for conduction and valence bands at $\Gamma$ point. Therefore, in this work, the hydrostatic deformation potential for the top valence band is 0 eV and for the conduction band equals to the hydrostatic deformation potential of the direct gap of BAs, which is taken from a theoretical work of Chimot et al. [39]. The effective mass of electrons in the $\Gamma _{1c}$ band is estimated by fitting a parabola to the band structure calculated by Hart & Zunger [38]. The remaining parameters are taken directly as reported.
BP is a binary compound for which complete set of material parameters is not yet available. For the purpose of this work, some of the parameters are estimated from related data. Hence both directly and indirectly obtained parameters are gathered together and discussed in this paper. The VBO of $-3.24$ eV is taken from a theoretical work [40], where it’s calculation is based on local density approximation for various phosphides, including GaP with $-4.46$ eV. In the Table 1 VBO of GaP has a different value ($-1.27$ eV), therefore the value of VBO of BP is shifted to $-0.23$ to keep it consistent. The spin-orbit splitting is posited as zero because it is not yet reported. Additionally, it is an usual for III-V semiconductors made of atoms from the periods 2 and 3 to have small spin-orbit splittings. The energy gap at $\Gamma$ point of 4.25 eV is measured with photoreflectance and electroreflectance methods [41]. The energy gap at $\Delta$ point is obtained by various electro-optical measurements [42,43]. The energy gap at L point, not yet measured, is calculated based on linear combination of atomic orbitals (LCAO) by Huang & Ching [44] and takes value of 1.98 eV. The Varshni parameters are not available, and therefore approximated as described before. Electron effective mass is taken directly from LCAO calculations [44]. DOS effective mass of electrons in $\Delta$ valley is calculated using longitudal 1.125 and transversal 0.204 masses in this valley [44] with a standard formula [35]. DOS effective mass of electrons in L valley is not found, and therefore approximated as described before, depending on the major compound in the given layer of an analyzed structure. In the same theoretical work [44], heavy-hole and light-hole effective masses are given for specific directions: $m_{lh,[100]}=0.150,\,m_{hh,[100]}=0.375,\,m_{lh,[111]}=0.108$, and $m_{hh,[111]}=0.926$. Based on the masses presented and their relations with Luttinger’s coefficients [45], we found and selected: $\gamma _{1}=5.170,\, \gamma _{2}=1.251$, and $\gamma _{3}=2.045$. The three picked coefficients reproduce the masses: 0.130, 0.375, 0.107, and 0.925 respectively, such that only $m_{lh,[100]}$ is slightly underestimated. The deformation potentials: $a_c^\Gamma , a_v^\Gamma$, and $b^\Gamma$ for band extremes at $\Gamma$ point are cited directly from theoretical estimations [35]. Reliable deformation potentials $\Delta$ and L valleys are not yet available, and therefore estimated in the same manner as DOS effective mass in the L valley. Lattice constant of 4.5383 Å is measured with X-Ray diffraction method [46]. The elements of stiffness tensor $C_{11}$ and $C_{12}$ are taken from Brillouin scattering measurements [47]. The Kane parameter $E_p=24$ eV is estimated based on a formula arising from the first-order 8-band Kane model [45] which connects it with $m_{e}^{\Gamma }$, $E_{g}^{\Gamma , 0}$, and $\Delta _{so}$. The relative permittivity is taken from capacitance measurements [48].
Bowing parameters of the ternary alloys
The majority of bowing parameters gathered in Table 2 are taken from a review publication of Vurgaftman & Meyer [29], with GaInP and GaInAs described the best. The advantage in detail of description of these two compounds derives from the fact that they are widely used in red-light and infrared emitting devices. The remaining relatively well described materials are AlGaP and GaPAs, for which the most crucial gap bowing parameters are available in the same reference. AlGaP is a well known ternary compound of AlGaInP alloy commonly used in red, orange and yellow light emitting diodes, and GaPAs is used in low cost, low brightness red light sources. A group of materials containing B atoms is not described in comparable detail since these are not yet as well developed for light emitting applications. This is why there are only three bowing parameters of BGaAs and BInAs listed in the Table 2. These are: 3.7 eV for the direct energy gap of BInAs (a result taken from ab-initio calculation to be published by Polak), 3.5 eV for the direct energy gap of BGaAs (see Ref. [38]), and 0.06 for the spin-orbit splitting energy of BGaAs (based on DFT calculations presented in work of Kudrawiec et al. [28]). BPAs and BGaP ternary alloys are not included in the table as there are no reliable bowing parameters for these materials available. For the calculation purposes, all of the missing bowing parameters are posited as zero.
Funding
Narodowe Centrum Nauki (SONATA BIS 3 no. 2013/10/E/ST3/00520); Narodowe Centrum Badań i Rozwoju (WPC/130/NIR-Si/2018).
Acknowledgments
RK acknowledges for financial support from NCBiR within the polish-china grant.
Disclosures
The authors declare no conflicts of interest.
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