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Isotherm Theoretical Study of the Al_{x}Ga_{1−x}As_{y}Sb_{1−y} Quaternary Alloy Using the Regular Solution Approximation for Its Possible Growth via Liquid-Phase Epitaxy at Low Temperatures

^{1}

^{2}

^{3}

^{4}

^{5}

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## Abstract

**:**

_{x}III

_{1−x}V

_{y}V

_{1−y}. In particular, the isotherm diagrams for the Al

_{x}Ga

_{1−x}As

_{y}Sb

_{1−y}quaternary alloy at low temperatures were calculated (500 °C, 450 °C, 400 °C, and 350 °C). The Al

_{x}Ga

_{1−x}As

_{y}Sb

_{1−y}quaternary alloy was formed from four binary compounds such as GaAs, AlAs, AlSb, and GaSb, all with direct bandgaps. The regular solution approximation was used to find the quaternary isotherm diagrams, represented in four linearly independent equations, which were solved using Parametric Technology Corporation Mathcad 14.0 software for different arsenic and antimony atomic fractions. The results support the possible growth of layers via liquid-phase epitaxy in a range of temperatures from 500 °C to 350 °C, where the crystalline quality could be improved at low temperatures. These semiconductor layers could have applications for optoelectronic devices in photonic communications, thermophotovoltaic systems, and microwave devices with good crystalline quality.

## 1. Introduction

_{x}Ga

_{1−x}Sb alloy were obtained via LPE, which were tuned with a wavelength of 1.55 µm (0.80 eV), where the silica fibers have their lowest losses [7]. Figure 1 shows the components and chamber of a liquid-phase epitaxy system [2]. The lower part of Figure 1a shows a graphite boat, in which it is possible to observe the “niches” where the metallic solids that make up the mixture are placed, while the upper part shows the cavities for substrates. Figure 1b shows the sliding graphite boat and its manipulator on the quartz support, where the mixture and the substrate do not have contact at the beginning of the growth. However, when the system reaches the liquidus temperature using the quartz manipulator, the upper rule is slipped, initiating contact between the mixture and the substrate. At this moment, the temperature is slightly lowered to break the equilibrium of the solution and deposit the excess solution on the substrate (depending on the growth technique). On the other hand, Figure 1c shows the chamber of the liquid-phase epitaxy system, where the heater covered by a layer of gold can be observed to concentrate the emitted radiation and generate a flat area large enough to not have temperature gradients during growth. It is important to mention that the processes in LPE growth are carried out in an ultrahigh-purity hydrogen environment.

_{x}Ga

_{1−x}As

_{y}Sb

_{1−y}quaternary alloy lattice-matched to GaSb substrates is an attractive material in optoelectronic devices such as photodetectors and thermophotovoltaic systems [5]. The importance of this material is due to its wide bandgap as a function of the x–y composition (1.0–1.8 µm in wavelength), which was previously grown via LPE at 605 °C [8]. In another study [9], a type p Al

_{0.6}Ga

_{0.4}As

_{0.05}Sb

_{0.9}layer lattice-matched to GaSb substrates was also grown via LPE at 635 °C to obtain a p–n junction. Toginho et al. reported a study of the photoluminescence properties in undoped and Te-doped AlGaAsSb alloys synthesized by MBE [10]. On the other hand, Hassan et al. employed density functional theory (DFT) to study the structural, electronic, optical, and thermodynamic properties of Al

_{x}Ga

_{1−x}As

_{y}Sb

_{1−y}quaternary alloys [11], while Pessetto et al. calculated the phase diagrams using the delta lattice parameter (DLP) model to grow the Al

_{x}Ga

_{1−x}As

_{y}Sb

_{1−y}quaternary alloy via LPE at temperatures above 700 °C [12]. However, Ito showed that the essence of the DLP model uses the bonding energy proportional to a

^{−2.5}(a = lattice parameter) in semiconductors to calculate the mixing enthalpies with good results, principally in A

_{x}B

_{1−x}C ternary alloys with x = 0.5 [13]. Then, different studies on Al

_{x}Ga

_{1−x}As

_{y}Sb

_{1−y}quaternary alloy showed growth temperatures above 500 °C, which could produce structural defects during layer deposition at high temperatures. This work presents the isotherm diagrams obtained for Al

_{x}Ga

_{1−x}As

_{y}Sb

_{1−y}quaternary alloy, indicating the possible growth of layers via liquid-phase epitaxy lattice-matched to GaSb substrates at temperatures below 500 °C. The theoretical analysis of isotherm diagrams was conducted using the regular solution approximation, which has been extensively studied and proven in quaternary alloys such as AlGaPAs [14,15]. The possible obtention of AlGaAsSb layers at temperatures below 500 °C could enhance the crystalline quality during the growth, thereby decreasing structural defects. Another application of AlGaAsSb layers with a good crystalline quality might be in their use as substrates or buffer layers for the investigation of nitride (III) materials via nitridation [16].

## 2. Theoretical Model

_{i}is the atom fraction of component i in the solid [14]. The chemical potentials (${\mu}_{i}^{l}$) can be expressed as a function of temperature and total pressure, which is given by the following equation:

^{−1}), R the universal gas constant (8.314472 J·K

^{−1}·mol

^{−1}), T is the temperature (K), and ${a}_{i}^{l}$ is the activity of each element in the liquid.

_{ij}can be obtained; therefore, Equation (4) can be represented in the following form:

_{x}III

_{1−x}V

_{y}V

_{1−y}were obtained, which were used in particular for the Al

_{x}Ga

_{1−x}As

_{y}Sb

_{1−y}quaternary alloy to evaluate its possible growth via LPE at low temperatures, according to the following equations:

_{x}Ga

_{1−x}As

_{y}Sb

_{1−y}quaternary alloy, the interaction parameters for the binaries, fusion entropies, fusion temperatures, and interaction parameters of the ternaries were searched in the literature, assigning the conditions Al = 1, Ga = 2, As = 3, and Sb = 4 [14,15,17,18]. Therefore, Equations (9a)–(12a), which are linearly independent, were solved numerically using the PTC Mathcad 14.0 software. The numerical values of the parameters mentioned above are shown in Table 1. It is important to mention that, when the temperatures were entered into the software, a conversion from kelvin (K) to degrees Celsius (°C) was performed before plotting the graphs.

## 3. Results and Discussion

^{−8}for 350 °C to 10

^{−5}for 500 °C. On the other hand, as the aluminum, arsenic, and antimony atomic fractions were small, the quaternary alloy was in the gallium-rich region as the solvent, while the other elements were solutes in the metallic solution. The aluminum fraction values in Figure 2 could indicate that, upon continuing to increase the saturation temperature of the Al

_{x}Ga

_{1−x}As

_{y}Sb

_{1−y}quaternary alloy, the aluminum atomic fraction would continue to decrease; accordingly, the antimony and arsenic atomic fractions would continue to increase until reaching the GaAsSb ternary alloy. Furthermore, when the temperature decreased, the arsenic fraction also decreased until reaching the AlSbGa ternary alloy, whereas the GaAsSb ternary alloy could be obtained upon increasing the temperature.

_{x}Ga

_{1−x}P

_{y}As

_{1−y}quaternary alloy (see graph 1 in their study) [15]. This behavior agrees with the results obtained in our study, suggesting their accuracy.

^{−4}mol, along with an aluminum atomic fraction in the liquid of 0.002 mol, and an antimony atomic fraction in the liquid of approximately 0.03 mol (in the gallium-rich region). Figure 3b shows that the atomic fractions in solid x and y began to separate, shifting toward smaller values of the arsenic atomic fraction in the liquid (approximately 10

^{−5}), with values for the atomic fractions of aluminum and antimony in the liquid of 0.0025 mol and 0.01 mol (450 °C), respectively. Figure 3c shows a greater separation between the atomic fractions in solid x and y, for an approximate value of 10

^{−6}mol for the arsenic atomic fraction in the liquid, with values for the atomic fractions of aluminum and antimony of 0.0035 mol and 0.004 mol, respectively, in the liquid at 400 °C. Lastly, Figure 3d shows the maximum separation between the atomic fractions in solid x and y, for a value of approximately 10

^{−7}mol for the arsenic atomic fraction in the liquid, with values for the atomic fractions of aluminum and antimony of 0.0041 mol and approximately 0.001 mol (in the gallium-rich region), respectively, in the liquid at 350 °C.

## 4. Conclusions

_{x}Ga

_{1−x}As

_{y}Sb

_{1−y}quaternary alloy were calculated using a regular solution approximation for possible growth of layers at temperatures below 500 °C, via liquid-phase epitaxy. A similar behavior in Al

_{x}Ga

_{1−x}P

_{y}As

_{1−y}quaternary alloy has been reported, agreeing with the results obtained in our study and suggesting their accuracy. Isotherm diagrams showed that, when the temperature increased from 350 °C to 500 °C, the aluminum atomic fraction decreased from 0.0041 mol to 0.002 mol, whereas the arsenic and antimony atomic fractions increased. On the other hand, graphs of the atomic fractions in solids x and y, as a function of the arsenic atomic fraction in the liquid, showed that, at low temperatures, the x fraction increased, while the y fraction decreased. Furthermore, the arsenic atomic fraction maintained a very small value until reaching the AlGaSb ternary alloy.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

^{†}and Javier Martínez Juárez for their valuable guidance during his first years of training in scientific research.

## Conflicts of Interest

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**Figure 1.**Liquid-phase epitaxy system: (

**a**) sliding graphite boat; (

**b**) quartz manipulator; (

**c**) LPE system chamber.

**Figure 2.**Isotherm diagrams of the Al

_{x}Ga

_{1−x}As

_{y}Sb

_{1−y}quaternary alloy at low temperatures indicating its possible growth via liquid-phase epitaxy.

**Figure 3.**Atomic fractions in solids x and y as a function of the arsenic atomic fraction in the liquid ${x}_{As}^{l}$ for different temperatures: (

**a**) 500 °C, (

**b**) 450 °C, (

**c**) 400 °C, and (

**d**) 350 °C.

**Table 1.**Interaction parameters, fusion entropies, and fusion temperatures of binary compounds, along with interaction parameters for the ternary compounds [8].

Interaction Parameters in Liquid Phase (cal·mol ^{−1}) | Fusion Entropies (cal·mol ^{−1}·K^{−1}) | Fusion Temperatures of Binary Compounds (K) | Interaction Parameters for the Ternary in Solid Phase (cal·mol ^{−1}) | |||||||
---|---|---|---|---|---|---|---|---|---|---|

${\alpha}_{12}\left(T\right)$ | ${\alpha}_{AlGa}\left(T\right)$ | 104 | $\Delta {\mathrm{S}}_{\mathrm{AlAs}}^{\mathrm{F}}$ | ${\mathsf{\delta}}_{13}$ | 15.6 | ${\mathrm{T}}_{\mathrm{AlAs}}^{\mathrm{F}}$ | ${\mathrm{T}}_{13}$ | 2043 | ${\mathrm{AlAs}}_{\mathrm{x}}{\mathrm{Sb}}_{1-\mathrm{x}}$ | 4200 |

${\alpha}_{13}\left(T\right)$ | ${\alpha}_{AlAs}\left(T\right)$ | −12T+600 | ||||||||

${\alpha}_{14}\left(T\right)$ | ${\alpha}_{AlSb}\left(T\right)$ | 4539–9.16T | $\Delta {\mathrm{S}}_{\mathrm{AlSb}}^{\mathrm{F}}$ | ${\mathsf{\delta}}_{14}$ | 14.74 | ${\mathrm{T}}_{\mathrm{AlSb}}^{\mathrm{F}}$ | ${\mathrm{T}}_{14}$ | 1338 | ${\mathrm{GaAs}}_{\mathrm{x}}{\mathrm{Sb}}_{1-\mathrm{x}}$ | 4000 |

${\alpha}_{23}\left(T\right)$ | ${\alpha}_{GaAs}\left(T\right)$ | 5160–9.16T | $\Delta {\mathrm{S}}_{\mathrm{GaAs}}^{\mathrm{F}}$ | ${\mathsf{\delta}}_{23}$ | 16.64 | ${\mathrm{T}}_{\mathrm{GaAs}}^{\mathrm{F}}$ | ${\mathrm{T}}_{23}$ | 1511 | ${\mathrm{Al}}_{\mathrm{x}}{\mathrm{Ga}}_{1-\mathrm{x}}\mathrm{As}$ | 0 |

${\alpha}_{24}\left(T\right)$ | ${\alpha}_{GaSb}\left(T\right)$ | 4700–6T | $\Delta {\mathrm{S}}_{\mathrm{GaSb}}^{\mathrm{F}}$ | ${\mathsf{\delta}}_{24}$ | 15.8 | ${\mathrm{T}}_{\mathrm{GaSb}}^{\mathrm{F}}$ | ${\mathrm{T}}_{24}$ | 985 | ${\mathrm{Al}}_{\mathrm{x}}{\mathrm{Ga}}_{1-\mathrm{x}}\mathrm{Sb}$ | 0 |

${\alpha}_{34}\left(T\right)$ | ${\alpha}_{AsSB}\left(T\right)$ | 750 |

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**MDPI and ACS Style**

Gastellóu, E.; García, R.; Herrera, A.M.; Ramos, A.; García, G.; Robles, M.; Rodríguez, J.A.; Ramírez, Y.D.; Carrillo, R.C.
Isotherm Theoretical Study of the Al_{x}Ga_{1−x}As_{y}Sb_{1−y} Quaternary Alloy Using the Regular Solution Approximation for Its Possible Growth via Liquid-Phase Epitaxy at Low Temperatures. *Entropy* **2022**, *24*, 1711.
https://doi.org/10.3390/e24121711

**AMA Style**

Gastellóu E, García R, Herrera AM, Ramos A, García G, Robles M, Rodríguez JA, Ramírez YD, Carrillo RC.
Isotherm Theoretical Study of the Al_{x}Ga_{1−x}As_{y}Sb_{1−y} Quaternary Alloy Using the Regular Solution Approximation for Its Possible Growth via Liquid-Phase Epitaxy at Low Temperatures. *Entropy*. 2022; 24(12):1711.
https://doi.org/10.3390/e24121711

**Chicago/Turabian Style**

Gastellóu, Erick, Rafael García, Ana M. Herrera, Antonio Ramos, Godofredo García, Mario Robles, Jorge A. Rodríguez, Yani D. Ramírez, and Roberto C. Carrillo.
2022. "Isotherm Theoretical Study of the Al_{x}Ga_{1−x}As_{y}Sb_{1−y} Quaternary Alloy Using the Regular Solution Approximation for Its Possible Growth via Liquid-Phase Epitaxy at Low Temperatures" *Entropy* 24, no. 12: 1711.
https://doi.org/10.3390/e24121711