Thermal response of electronic, optical, mechanical properties, phonon frequencies, and sound velocity of InP_xAs_ySb_1−x−y/InAs quaternary semiconductor system

Abstract

The temperature dependence of acoustic velocities, thermal properties, phonon frequencies, mechanical, electronic, and optical properties for the InP_xAs_ySb_1−x−y/InAs system has been studied. The physical properties of the binary components InSb, InP, and InAs that constitute the quaternary alloy were used in this research. The study has been done using the empirical pseudo-potential method (EPM) under the virtual crystal approximation (VCA). The thermal properties, phonon frequencies, and acoustic velocities for the InP_xAs_ySb_1−x−y/InAs system under the effect of temperature have not been fully studied. Therefore, we have focused on these properties under the influence of temperature. Due to the lack of published theoretical and experimental values on these properties, our findings will provide a significant reference for future experimental work.

1 Introduction

Among all semiconductor materials, the III–V semiconductors family is of great importance due to the variety of its applications. The III–V family is used in making very efficient optoelectronic devices of high-speed electron mobility, small binding energy, and direct energy band gaps (Adachi 2005, 2009). Quaternary alloys built using the III–V semiconductors are highly important, this is because they are methods to tune the value of band gaps to enhance the workings of the high-speed optoelectronic devices (Adachi 2005, 2009). This alloy is used to manufacture high-quality optical fibers in the range of 2–4 μm and TPV cells that are working in the range of 0.35 to 0.5 eV (Adachi 2005, 2009). Manufacturing the InP_xAs_ySb_1−x−y can be done using lattice matching with either InAs or GaSb as substrate [2]. The use of this class of alloys has drawn attention since the invention of light-emitting diodes. The invention of transistors and the discovery of the minority carrier led to the development of light-emitting diodes and laser diodes. Along with the importance of InP_xAs_ySb_1−x−y alloys in optoelectronics, this opened the way for today's more advanced III–V photonic integrated circuits. The InP_xAs_ySb_1−x−y alloys are utilized in optical communications optoelectronic devices.

Numerous methods have been utilized to study the band-structure of semiconductors; among them is the so-called EPM (Chelikowsky and Cohen 1976; Degheidy et al. 2017, 2021; Degheidy and Elkenany 2017; Elkenany 2021a, 2021b, 2021c; Pandey and Phillips 1974). The temperature and pressure dependence of the physical properties for the alloys have been recorded theoretically and experimentally by several works (Chen and Ravindra 2012; Degheidy et al. 2013, 2017; Degheidy and Elkenany 2015a, 2015b, 2016; Elkenany 2015; Al Maaitah and Elkenany 2022; Othman 2020b, 2020a; Saib et al. 2008; Sherzad Othman et al. 2021; Wang et al. 2012; Yang et al. 2021). This area of research is still active and growing. Although we do understand the physical concepts behind these quaternary semiconductors, there is a great need to have more accurate values for the parameters used in practical applications. The III–V quaternary alloy under study in this research is based on the binary compounds InP, InAs, and InSb.

In this study, we adopted InAs as a substrate to make a theoretical and computational study using the EPM technique within the validity of the VCA to give a prediction of the temperature dependence of the physical parameters of this quaternary alloy. The input parameters of the calculations can be derived from the corresponding values of the binary compounds that form the quaternary alloy using the appropriate interpolation scheme. The basic parameters of the InP_xAs_ySb_1−x−y quaternary alloy, such as the energy band gaps, mechanical parameters, thermal properties, phonon frequencies, speed of sound and optical properties, and their temperature dependence have been studied. To the best of our knowledge, this work has not been well-reported in the literature, so this research is an attempt to add value to the literature in that arena.

2 Computational method

The electronic properties of the quaternary alloy system of interest were calculated by using the (EPM) within (VCA) (Chelikowsky and Cohen 1976; Degheidy et al. 2017; Degheidy and Elkenany 2017; Pandey and Phillips 1974). The primary advantage of using pseudopotentials is that only valence electrons must be considered. The valence electrons are considered to travel in a weak one-electron potential (Harrison 2005). The core's electrons are treated as though they were frozen in an atomic state. Even though there were other methods for predicting electronic band structures at the time, the pseudopotential method gives remarkably precise results for the amount of computer time and effort necessary. To make the problem even simpler, model pseudopotentials are used instead of actual pseudopotentials. Empirical parameters, such as the symmetric and anti-symmetric form factors, are changed to meet the experimental energy band gaps of the examined alloy's parent binary components, InP, InAs, and InSb. According to the (VCA) and Vegard’s law (Vegard 1921), the lattice constants and pseudo-potential form factors for the quaternary alloys InP_xAs_ySb_1−x−y could be created in the form of the linear combinations,

$$ {\text{a}}_{{{\text{InP}}_{{\text{x}}} {\text{As}}_{{\text{y}}} {\text{Sb}}_{{{1 - x - y}}} }} {\text{ = xa}}_{{{\text{InP}}}} {\text{ + ya}}_{{{\text{InAs}}}} {{ + (1 - x - y) a}}_{{{\text{InSb}}}} $$

(1)

$$ {\text{W}}_{{{\text{InP}}_{{\text{x}}} {\text{As}}_{{\text{y}}} {\text{Sb}}_{{{1 - x - y}}} }} {\text{ = xW}}_{{{\text{InP}}}} {\text{ + yW}}_{{{\text{InAs}}}} {{ + (1 - x - y) W}}_{{{\text{InSb}}}} $$

(2)

The lattice match condition for InP_xAs_ySb_1−x−y on InAs substrate was given as follows (Adachi 2005):

$$x=0.68933-0.68933{\text{y}},\quad 0\le x \le 1$$

(3)

The elastic constants C₁₁, C₁₂, C₄₄ of the studied alloys and their related parameters such as bulk B_u, shear S_h, and Young's Y₀ moduli could be determined by knowing the polarities (Baranowski 1984; Bouarissa 2003; Harrison 1980; Shen 1994). The density of crystal (g) and the elastic constants (C_ij) were utilized to determine the velocity of sound as (Adachi 2005):

$$ {\text{v = }}\sqrt {\frac{{c_{ij} }}{g}} $$

(4)

The refractive index, high frequency, and static dielectric constants could be determined as in Adachi (2005, Davydov and Tikhonov (1998), Moss (1950), and Samara (1983).

The temperature-dependent refractive index can be calculated as follow (Moss 1950):

$$\text{n}(\text{T})=\sqrt[4]{\frac{108}{{\text{E}}_{\text{g}}(T)}}$$

(5)

The temperature-dependent high-frequency dielectric constant can be determined as follow (Samara 1983):

$${\upvarepsilon }_{\infty }(\text{T})={\text{n}}^{2}(T)$$

(6)

The temperature-dependent static dielectric constant can be calculated as follow (Davydov and Tikhonov 1998):

$$\frac{{\upvarepsilon }_{\text{o}}(\text{T})-1}{{\upvarepsilon }_{\infty }(\text{T})-1}=1+\frac{{{\propto }_{\text{p}}}^{2}\left(\text{T}\right)[1+2{{\propto }_{\text{c}}}^{2}\left(\text{T}\right)]}{{2{\propto }_{\text{c}}}^{4}(\text{T})}$$

(7)

where ${\propto }_{\text{p}}$ is the polarity, ${\propto }_{\text{c}}$ is the covalency.

The optical phonon frequencies of the alloy of interest ${\upomega }_{\text{LO}}$ and ${\upomega }_{\text{TO}}$ could be obtained as in Refs. (Davydov and Tikhonov 1998; Kittel and McEuen 1976)

$$\frac{{\omega }_{TO}^{2}}{{\omega }_{LO}^{2}}=\frac{{\varepsilon }_{\infty }}{{\varepsilon }_{s}}$$

(8)

$${\omega }_{LO}^{2}-{\omega }_{TO}^{2}=\frac{4\pi {{e}_{T}^{*}}^{2}{e}^{2}}{M{\Omega }_{o}{\varepsilon }_{\infty }}$$

(9)

where $\text{M}$ is twice of reduced mass and ${\Omega }_{\text{o}}$ is the volume occupied by one atom. By knowing the longitudinal phonon frequency, Debye temperature ${\theta }_{D}$, the heat capacity could be determined as in Refs. (Adachi 1985, 2005).

3 Results and discussions

The electronic energy band gaps of InP_xAs_ySb_1−x−y alloys lattice-matched to InAs were calculated throughout the high-symmetry points L, Γ, and X in the Brillouin zone. The dependence of the energy gaps of the InP_xAs_ySb_1−x−y/InAs system on temperature from 0 to 500 K at specified values of the y-composition parameter (0.2, 0.6, 1) is listed in Table 1 and displayed in Fig. 1. Our findings are in excellent agreement with the published values at room temperature at y = 1 (InAs) (Boucenna et al. 2014; Degheidy et al. 2018c). At a higher temperature, our results may serve as references for future experimental work. It can be seen that the energy gaps at the Γ, L, and X points are decreased by enhancing temperature at various compositions (0.2, 0.6, 1). Also, we have noticed that the energy band gaps $E_{g}^{L} , \, E_{g}^{\Gamma } , \, E_{g}^{X}$ are shifted downward with the composition from 0.2 to 1. This means that they $E_{g}^{L} , \, E_{g}^{\Gamma } , \, E_{g}^{X}$ are decreased by enhancing composition in the temperature range (0–500 K). This is because rising temperature raises the lattice constant while lowering the potential energy. The curves in Fig. 1 can be fitted by the following polynomials:

Table 1 The energy gaps of InP_xAs_ySb_1−x−y/InAs system at various temperatures at different values of y-composition parameter (0.2, 0.6, 1)

Fig. 1

The energy gaps of InP_xAs_ySb_1−x−y/InAs system as function of temperature at different values of y-composition parameter (0.2, 0.6, 1)

For y = 0.2,

$$ \left\{ \begin{gathered} E_{g}^{L} (T) = E_{g}^{L} (0) - ({0}{{.0008 - 4*10}}^{{ - 8}} \, T) \, T \hfill \\ E_{g}^{\Gamma } (T) = E_{g}^{\Gamma } (0) - ({0}{{.0004 - 8*10}}^{{ - 8}} \, T) \, T \hfill \\ E_{g}^{X} (T) = E_{g}^{X} (0) - ({0}{{.0011 - 6*10}}^{{ - 8}} \, T) \, T \hfill \\ \end{gathered} \right\} \, $$

(10)

For y = 0.6,

$$ \left\{ \begin{gathered} E_{g}^{L} (T) = E_{g}^{L} (0) - ({0}{{.0008 - 5*10}}^{{ - 8}} \, T) \, T \hfill \\ E_{g}^{\Gamma } (T) = E_{g}^{\Gamma } (0) - ({0}{{.0003 - 2*10}}^{{ - 9}} \, T) \, T \hfill \\ E_{g}^{X} (T) = E_{g}^{X} (0) - ({0}{{.0011 - 1*10}}^{{ - 7}} \, T) \, T \hfill \\ \end{gathered} \right\} \, $$

(11)

For y = 1,

$$ \left\{ \begin{gathered} E_{g}^{L} (T) = E_{g}^{L} (0) - ({0}{{.0008 - 1*10}}^{{ - 7}} \, T) \, T \hfill \\ E_{g}^{\Gamma } (T) = E_{g}^{\Gamma } (0) - ({0}{{.0003 - 4*10}}^{{ - 8}} \, T) \, T \hfill \\ E_{g}^{X} (T) = E_{g}^{X} (0) - ({0}{{.001 - 5*10}}^{{ - 8}} \, T) \, T \hfill \\ \end{gathered} \right\} \, $$

(12)

In designing and manufacturing optoelectronic devices such as solar cells and waveguides detectors, the refractive index and optical dielectric constants are very essential. The variation of refractive index, static and high-frequency dielectric constants of the InP_xAs_ySb_1−x−y/InAs system with the temperature for compositions (0.2, 0.6, 1) is displayed in Fig. 2. It is observed that all the studied optical properties are increased by enhancing temperature for the compositions (0.2, 0.6, 1). Moreover, the refractive index, static and high-frequency dielectric constants of the InP_xAs_ySb_1−x−y/InAs system are increased with enhancing composition over the whole region of temperature. Since the energy band gaps are decreased by increasing the temperature and the refractive index has an inverse relation with the electronic band gaps, also, the direct relationship between the high-frequency dielectric constant and the refractive index. So, the high-frequency dielectric constant is enhanced with enhancing temperature.

Fig. 2

The refractive index, high frequency and static dielectric constants of InP_xAs_ySb_1−x−y/InAs system as function of temperature at compositions (0.2, 0.6 and 1)

The mechanical moduli play a vital role in giving useful data on the structural stability and anisotropic binding characteristics. Table 2 and Fig. 3 provide the dependence of elastic constants (C₁₁, C₁₂, C₄₄, B_u, S_h, and Y₀) on temperature for the InP_xAs_ySb_1−x−y/InAs system at different values of y (0.2, 0.6, and 1). For y = 1 (InAs) at T = 300 K, the calculated C₁₁, C₁₂, C₄₄, B_u, S_h, and Y₀ are in excellent accord with the published values (Algarni et al. 2018). The calculated C₁₁, C₁₂, C₄₄, B_u, S_h, and Y₀ at the other values of temperature for y = 0.2 and 0.6 can be taken as references for the experimental data. It is noticed that C₁₁, C₁₂, C₄₄, B_u, S_h, and Y₀ are decreased by increasing temperature. Besides, these moduli are increased with increasing compositions over the temperature range from 0 to 500 K. This is because the polarity is decreased by increasing temperature and the nearest-neighbor distance is increased with increasing temperature. The conditions of stability were given by knowing the elastic constants as in Ref. (Zhang et al. 2012). Our results display that C₁₁(T)–C₁₂(T) > 0, C₁₁(T) > 0, C₄₄(T) > 0, C₁₁(T) + 2C₁₂(T) > 0, which achieve the stability of the InP_xAs_ySb_1−x−y/InAs system in its structure over the studied range of temperature. Pugh (Pugh 1954) introduced an important formula that distinguishes between the ductile and brittle properties of materials, i.e. the material acts ductility if B_u/S_h > 1.75; otherwise the material acts in a brittle way. From our results, the InP_xAs_ySb_1−x−y/InAs system has a ductile behavior over the temperature range (0–500 K). This is because the calculated values of B_u/S_h over the temperature range (0- 500 K) for compositions (0.2, 0.6, and 1) are greater than 1.75.

Table 2 Mechanical parameters (C₁₁, C₁₂, C₄₄, B_u, S_h and Y₀) in (10¹² dyn/cm²) of InP_xAs_ySb_1−x−y/InAs system at various temperatures for compositions (0.2, 0.6 and 1)

Fig. 3

Mechanical moduli (C₁₁, C₁₂, C₄₄, B_u, S_h and Y₀) of InP_xAs_ySb_1−x−y/InAs system as function of temperature at constant y-composition parameter (0.2, 0.6 and 1)

The variation of the sound velocities in the different directions of propagation and planes of polarization of the InP_xAs_ySb_1−x−y/InAs system with the temperature at different values of compositions (0.2, 0.6, and 1) are recorded in Table 3 and plotted Fig. 4. At the room temperature and y = 1 (InAs), there is a good accord between the calculated values and the published data by Adachi (Adachi 2005). All the components of the sound velocity are slightly decreased by increasing temperature as shown in Fig. 4. This is because the elastic constants decrease by increasing temperature and the crystal density rises with raising the temperature. Also, it can be seen that the sound velocities in the major directions [100], [110], and [111 are decreased with enhancing composition. Due to the lack of experimental data for the sound velocities of the InP_xAs_ySb_1−x−y/InAs system at different temperatures with compositions (0.2, 0.6), our results are predictions for future experimental work.

Table 3 The sound velocities in (10⁵ cm/s) in the different directions of propagation and planes of polarization of InP_xAs_ySb_1−x−y/InAs system at different values of compositions (0.2, 0.6, and 1)

Fig. 4

The sound velocity in the different directions of propagation and planes of InP_xAs_ySb_1−x−y/InAs system as a function of temperature at different values of compositions (0.2, 0.6, and 1)

The optical phonon frequencies are very useful in the study of the transport and optical properties of semiconductors. The calculated longitudinal and transversal optical phonon frequencies ω_LO and ω_TO of InP_xAs_ySb_1−x−y/InAs system at various temperatures for concentrations (0.2, 0.6, and 1) are documented in Table 4 and displayed in Fig. 5. Also, the calculated values of the Debye temperature θ_D and the molar specific heat C_p of InP_xAs_ySb_1−x−y/InAs system at various temperatures with concentrations (0.2, 0.6, and 1) are listed in Table 4. At room temperature and y = 1 (InAs), the optical phonon frequencies, Debye temperature, and the molar specific heat are found in reasonable accord with the available data in the literature (Degheidy et al. 2018c; Kittel and McEuen 1976). Figure 6 shows the variation of Debye temperature and the molar specific heat for the InP_xAs_ySb_1−x−y/InAs system with the temperature at certain values of compositions (0.2, 0.6, and 1), respectively. It will be noted that the Debye temperature is decreased by increasing temperature and composition. From Fig. 6, it can be perceived that the C_p is enhanced by increasing temperature. Also, it is seen that the C_p is increased by increasing the composition. The other calculated results can be served as a reference for future experimental work. It is observed that both ω_LO and ω_TO are decreased by increasing temperature and composition. The curves in Fig. 5 can be fitted by the following polynomials:

Table 4 The longitudinal and transversal optical phonon frequencies (ω_LO and ω_TO) in 10¹³ Hz, Debye temperature, and the molar specific heat (J/Kg.Mol) of InP_xAs_ySb_1−x−y/InAs system for various temperatures at different compositions (0.2, 0.6, and 1)

Fig. 5

The longitudinal optical phonon frequency ω_LO and the transversal optical phonon frequency ω_TO of InP_xAs_ySb_1−x−y/InAs system as a function of temperature at certain values of compositions (0.2, 0.6, and 1)

Fig. 6

The Debye temperature and the molar specific heat for InP_xAs_ySb_1−x−y/InAs system with temperature at certain values of compositions (0.2, 0.6, and 1)

At y = 0.2,

$$ \left\{ \begin{gathered} \omega_{{{\text{LO}}}} (T) = \omega_{{{\text{LO}}}} (0) - ({0}{{.0008 - 6*10}}^{{ - 7}} \, T) \, T \hfill \\ \omega_{{{\text{TO}}}} (T) = \omega_{{{\text{TO}}}} (0) - ({0}{{.0003 + 2*10}}^{{ - 7}} \, T) \, T \hfill \\ \end{gathered} \right\} \, $$

(13)

At y = 0.6,

$$ \left\{ \begin{gathered} \omega_{{{\text{LO}}}} (T) = \omega_{{{\text{LO}}}} (0) - ({0}{{.0002 + 3*10}}^{{ - 7}} \, T) \, T \hfill \\ \omega_{{{\text{TO}}}} (T) = \omega_{{{\text{TO}}}} (0) - ({0}{{.0007 - 5*10}}^{{ - 7}} \, T) \, T \hfill \\ \end{gathered} \right\} \, $$

(14)

At y = 1,

$$ \left\{ \begin{gathered} \omega_{{{\text{LO}}}} (T) = \omega_{{{\text{LO}}}} (0) - ({0}{{.0002 + 8*10}}^{{ - 8}} \, T) \, T \hfill \\ \omega_{{{\text{TO}}}} (T) = \omega_{{{\text{TO}}}} (0) - ({0}{{.0005 - 3*10}}^{{ - 7}} \, T) \, T \hfill \\ \end{gathered} \right\} \, $$

(15)

4 Conclusion

A study of some optoelectronic properties, thermal properties, phonon frequencies, mechanical properties, and acoustic velocities for the InP_xAs_ySb_1−x−y/InAs system has been studied. The variation of the studied quantities with temperature from (0–500 K) for composition (0.2, 0.6, and 1) has been determined. Our calculations for the considered alloy lattice-matched to InAs substrate are in good agreement with the available experimental values. The calculated results in the present study seem likely to be convenient as a reference for future experimental work, especially at high temperatures.