Electronic, optical, and mechanical properties of Al_xIn_1−xP alloys under temperature and pressure

Abstract

In this work, the optoelectronic and mechanical properties of Al_xIn_1−xP combinations in the zinc-blende structure were studied for different Al concentrations. The energy band gaps $\left( {{\text{E}}_{{\text{g}}}^{{\text{L}}} ,\,{\text{E}}_{{\text{g}}}^{\Gamma } ,\,{\text{E}}_{{\text{g}}}^{{\text{X}}} } \right)$, refractive index (n), high frequency and static dielectric constants (ɛ_∞, ɛ_o), elastic parameters (C₁₁, C₁₂, C₄₄) were investigated. Other mechanical properties such as bulk (B_u), shear (C_s), Young’s (Y₀) moduli, Poisson ratio (σ), linear compressibility $(C_{o} )$, Cauchy $(C_{a} )$ ratio, isotropy factor (A), bond stretching parameter $(\alpha \,)$, bond-bending force parameter $(\beta \,)$, internal-strain parameter $(\xi )$, and the transverse effective charge $(e_{T}^{*} )$ were calculated. Also, the temperature and pressure dependences of these properties were studied. Our estimations were made with the empirical pseudo-potential method combined with the virtual crystal approximation incorporated the compositional disorder impact. There was a reasonable agreement between our determined outcomes and the accessible experimental values for the binary materials AlP and InP which give help for the consequences of the ternary combinations.

1 Introduction

The alloys of III–V zinc-blende semiconductor compounds are of vital importance because these materials are potentially good for the application of optoelectronic and high-speed electronics (Ohnuma et al. 2000). The Al_xIn_1−x P is a wide bandgap III–V semiconductor alloy which is motivating due to its least index of refraction between other As or P semiconductor composites (Ishitani et al. 1997; Munns et al. 1993). It very well may be used for photovoltaic devices, light-emitting diodes in the visible spectral range, and high-performance laser diodes (Bour et al. 1987; Christian et al. 2013; Dai et al. 2015; Lin et al. 2015; Tukiainen et al. 2006; Zhang et al. 2010). Also, Al_xIn_1−xP is regularly utilized as a window layer of semiconductor solar cell combinations (Abdollahi et al. 2016; Corkish and Green 1993). The alloying technique prompts critical variations in the lattice parameters, distribution of electronic charge.

The study of semiconductor behavior under high temperatures has become a critical subject showing important development (Degheidy and Elkenany 2015). Various authors have reported the temperature dependence of the energy gaps for certain semiconductors, both theoretically (Degheidy et al. 2012; Fan 1951; Keffer et al. 1970; Tsang and Cohen 1971) and experimentally (Pandey and Phillips 1974; Skelton et al. 1972; Walter and Cohen 1970, 1969). The examined alloy Al_xIn_1−xP has consisted of two binary materials AlP and InP. Probably the simplest approaches to change the electronic and structural properties of the ternary combination are changeable of the compositional content, going from 0 to 1. A few techniques have been created to figure the band structure of semiconductors, between them is the EPM (Chelikowsky and Cohen 1976; Degheidy et al. 2012, 2021a, b; Elkenany 2021a, b, c; Elkenany and Othman 2021; Al Maaitah and Elkenany 2022; Pandey and Phillips 1974; Phillips and Pandey 1973). There is also a wonderful method for calculating the electronic, optical, and mechanical properties of semiconductor materials which is the density functional theory (DFT) (Al-Douri et al. 2015; Bouhemadou et al. 2019; Moakafi et al. 2008; Ouahrani et al. 2010; Reshak et al. 2011). The examination of pressure and temperature reliance of electronic, optical, and mechanical properties in semiconductors has been the object of numerous investigations (Chen and Ravindra 2012; Degheidy et al. 2018a, 2017, 2012; Degheidy and Elkenany 2012, 2013a, b, c, 2017; Elkenany 2016; Jappor et al. 2010; Saib et al. 2008; Wang et al. 2012).

In the current study, the electronic properties such as the energy bandgaps $\left( {{\text{E}}_{{\text{g}}}^{{\text{L}}} {,}\,{\text{E}}_{{\text{g}}}^{{\Gamma }} \;{\text{E}}_{{\text{g}}}^{{\text{X}}} } \right)$, optical properties such as refractive index, dielectric constants (n, ɛ_∞, ɛ_o) and mechanical properties such as elastic constants (C₁₁, C₁₂, C₄₄), bulk, shear and Young's moduli (B_u, C_s, Y₀) for Al_xIn_1−xP alloys have been calculated. The Poisson ratio (σ), linear compressibility $(C_{o} )$, Cauchy ratio $(C_{a} )$, anisotropy factor (A), bond stretching parameter $(\alpha \,)$, bond-bending force parameter $(\beta \,)$, internal-strain parameter $(\xi )$, and the transverse effective charge $(e_{T}^{*} )$ for Al_xIn_1−xP alloys were determined. The temperature, pressure, and composition dependence of the considered alloy for the studied properties has been studied. Our calculations were matched with the available published data especially for binary AlP and InP compounds and showed very good agreement which gives support for the results of the Al_xIn_1−xP ternary alloys.

2 Theory and calculations

The electronic structure of the alloys Al_xIn_1−xP was determined using the EPM. The alloy potential was calculated using the $V_{VCA} ({\mathbf{r}})$ beside $V_{dis} ({\mathbf{r}})$ due to the compositional disorder (Bouarissa and Aourag 1995; Lee et al. 1990)

$$V({\mathbf{r}})_{alloy} = V_{VCA} ({\mathbf{r}}) + V_{dis} ({\mathbf{r}})$$

(1)

where the potentials in Eq. (1) have the formats (Bouarissa and Aourag 1995; Lee et al. 1990)

$$V_{VCA} ({\mathbf{r}}) = xV_{AlP} ({\mathbf{r}}) + (1 - x)V_{InP} ({\mathbf{r}})$$

(2)

$$V_{dis} ({\mathbf{r}}) = - \Omega \sqrt {x(1 - x)} [V_{AlP} ({\mathbf{r}}) - V_{InP} ({\mathbf{r}})]$$

(3)

where x is the Aluminum concentration (Al) and Ω is treated as an adjustable parameter. The x-dependent form factors of the considered ternary alloys at a constant T and p could be given in the form

$$W^{S,A} = x W_{AlP}^{S,A} + \left( {1 - x} \right)W_{InP}^{S,A} - \sqrt {x\left( {1 - x} \right)} \left[ {W_{AlP}^{S,A} - W_{InP}^{S,A} } \right]$$

(4)

where $W_{AlP}^{S,A}$, $W_{InP}^{S,A}$ are the symmetric and anti-symmetric form factors of the AlP and InP, respectively. The lattice constant of Al_xIn_1−xP alloy was acquired by the Vegard’s relation (Vegard 1921)

$$a_{alloy } = \left( {1 - x} \right)a_{InP} + xa_{AlP}$$

(5)

where $a_{AlP} and a_{InP}$ are the lattice constants of the associated compounds AlP and InP, respectively. Knowing the lattice constants and the form factors of Al_xIn_1−xP alloy at a constant composition parameter, the energy eigenvalues $E_{{n{\mathbf{k}}}} (x)$ were calculated by solving numerically the secular determinant (Fan 1951; Keffer et al. 1970)

$$\left\| {\frac{1}{2}\left| {\vec{k} + \left. {\vec{G}^{\prime}} \right|} \right.^{2} - } \right.\left. {E_{nk} + \sum\limits_{{\vec{G} \ne \vec{G}^{\prime}}} {V(\left| {\Delta \vec{G})} \right.} } \right\| = 0$$

(6)

where

$$V(\left| {\Delta \vec{G}} \right.) = W^{s} (\Delta \vec{G})\cos (\Delta \vec{G} \bullet \vec{\tau }) + iW^{a} \,(\Delta \vec{G},x)\sin (\Delta \vec{G} \bullet \tau )$$

is the pseudo-potential and $\tau$ is the atomic position and equals to $\frac{a}{8} \left( {1, 1, 1} \right).$ The calculated values of the fundamental energy band gaps of Al_xIn_1−xP alloy were utilized to determine n (Moss 1950) and the ε_∞ by applying the Samara relation (Samara 1983). The gained $W^{S}$ and $W^{A}$ at G(1,1,1) were utilized to determine the ${\upalpha }_{{\text{p}}}$ using Vogl’s relation (Vogl 1978) as

$$\alpha_{p} = - \frac{{ w_{3}^{A} }}{{w_{3}^{S} }}$$

(8)

The elastic constants at constant T and p were given by Refs. (Bouarissa 2003; Munns et al. 1993; Shen 1994). The knowledge of the C₁₁, C₁₂, and C₄₄ permitted us to calculate $B_{u}$, $C_{s}$ and $Y_{o}$ of Al_xIn_1−xP alloys (Cahn and Cohen 1970; Pandey and Phillips 1974; Walter and Cohen 1969). Other significant parameters as the ratio σ, $C_{o}$, $C_{a}$ and A were also successfully calculated (Bouarissa 2003; Shen 1994). Finally, the $\alpha \,$, $\beta \,$, $\xi$ and $e_{T}^{*}$ for the studied alloys have been determined (Bouarissa 2003; Walter and Cohen 1969).

3 Results and discussions

3.1 Effect of Al concentration

The variation of the calculated ${\text{E}}_{{\text{g}}}^{{\text{L}}} {,}\,{\text{E}}_{{\text{g}}}^{{\Gamma }}{,} \;{\text{E}}_{{\text{g}}}^{{\text{X}}}$ with Al content in Al_xIn_1−xP is recorded in Table 1 and shown in Fig. 1. It was seen that ${\text{E}}_{{\text{g}}}^{{\Gamma }}$ and ${\text{E}}_{{\text{g}}}^{{\text{L}}}$ are raised with different rates with increasing Al content, however, ${\text{E}}_{{\text{g}}}^{{\text{X}}}$ is increased with a weaker rate. We noted that Al_xIn_1−xP is a direct semiconductor and changed to an indirect one at about x = 0.36. This result is in good accord with that mentioned by Vurgaftman et al. (Vurgaftman et al. 2001) who proposed that this material has a direct energy bandgap for x < 0.44. The utilization of the improved VCA in our computations of Al_xIn_1−xP introduced the optical bowing parameters as − 1.6 eV, − 0.49 eV, and 0.27 eV at L, Γ, and X, respectively. Our calculated values of the optical bowing parameters are in excellent agreement with the published values of − 1.7 eV, − 0.48 eV, and 0.3 eV, respectively as in Refs. (Mezrag and Bouarissa 2018; Vurgaftman et al. 2001), justifying the reliability of our calculated method and results. The studied ${\text{E}}_{{\text{g}}}^{{\text{L}}} {,}\,{\text{E}}_{{\text{g}}}^{{\Gamma }}{,} \;{\text{E}}_{{\text{g}}}^{{\text{X}}}$ were compared with the published ones and indicated fantastic agreement (Adachi 2005, 1987; Tiwari and Frank 1992; Vurgaftman et al. 2001). The variation of these energy gaps with concentration could be shaped by the following polynomials:

$${\text{E}}_{{\text{g}}}^{{\text{L}}} = \, 2.0552 + \, 3.2277 \, x - 1.6015 \, x^{2}$$

(9)

$${\text{E}}_{{\text{g}}}^{{\Gamma }} = \, 1.358 + \, 2.7848 \, x - 0.4931 \, x^{2}$$

(10)

$${\text{E}}_{{\text{g}}}^{{\text{X}}} = \, 2.2421 + \, 0.0438 \, x + 0.2742 \, x^{2}$$

(11)

Table 1 The E ^L_g , ${\text{E}}_{{\text{g}}}^{{\Gamma }}$ and E ^X_g  of Al_xIn_1−xP at T = 300 K and p = 0 Kbar for different x

Fig. 1

Energy band gaps of Al_xIn_1-xP at T = 300 K& p = 0 Kbar as function of concentration

Figure 2 displays the variation of refractive index (n), optical and static dielectric constants (ɛ_∞, ɛ_o) of Al_xIn_1−xP with composition at T = 300 K and p = 0 Kbar. It was seen that the refractive index is slightly linear decreased with enhancing composition. It was also found that the optical and static dielectric constants (ɛ_∞, ɛ_o) are decreased nonlinearly with increasing the Al content. This is due to the inverse relationship between the energy band gap and the refractive index.

Fig. 2

The refractive index (n), optical and static dielectric constants (ɛ_∞, ɛ_o) of Al_xIn_1−xP as function of Al content at T = 300 K & p = 0 Kbar

The polarities ${\upalpha }_{{\text{p}}}$, C₁₁, C₁₂, C₄₄, B_u, C_s, and Y₀ of Al_xIn_1−xP at various concentrations are recorded in Table 2 and offered in Fig. 3. A good agreement was obtained between our results and the published data as in Refs. (Bour et al. 1987; Christian et al. 2013; Keffer et al. 1970; Munns et al. 1993; Ohnuma et al. 2000). We noted that the elastic parameters (C₁₁, C₁₂, and C₄₄) are enhanced nonlinearly with increasing the Al concentration. From Fig. 3, we observed that the change rate with the composition of C₁₂, C₄₄ is small compared to that with C₁₁. It was observed that the change of the elastic moduli B_u, C_s, and Y₀ with composition has similar behavior to the elastic constants. This means that the bulk, shear, and Young's moduli are increased with enhancing the Al content.

Table 2 The polarity and mechanical parameters (C₁₁, C₁₂, C₄₄, B_u, Y₀ and C_s) in (10¹¹ dyn/cm²) of AlxIn1−xP at T = 300 K& p = 0 Kbar for different x

Fig. 3

The mechanical parameters (C₁₁, C₁₂, C₄₄, B_u, Y₀ and C_s) of Al_xIn_1−xP as function of Al content at T = 300 K& p = 0 Kbar

The study of mechanical properties enables us to classify and identify the materials. Also, it defines a material's extent of usefulness and determines the service life that can be predicted. The Poisson ratio (σ), linear compressibility $(C_{o} )$, Cauchy $(C_{a} )$ ratio, Born ratio (B₀), anisotropy factor (A), bond stretching parameter $(\alpha \,)$, bond-bending force parameter $(\beta \,)$, internal-strain parameter $(\xi )$, the transverse effective charge (e_T*), and effective charge $\left( {Z^{*} } \right)$ of Al_xIn_1−xP at different values of Al content are listed in Table 3. The comparison between our calculations and the available published data is found to be very good agreement (Bour et al. 1987; Christian et al. 2013; Ishitani et al. 1997; Keffer et al. 1970; Ohnuma et al. 2000; Tan et al. 2010; Tsang and Cohen 1971). It was observed that the α and β are enhanced with enhancing Al concentration, C₀ is reduced with raising Al content, while the other parameters in the table are very marginally influenced with Al concentration in Al_xIn_1−xP alloy.

Table 3 The (σ), (C₀), (C_a), (B₀), (A), (α), (β), $(\xi )$, (e_T*) and (Z*) of Al_xIn_1−xP at T = 300 K&p = 0 Kbar for various compositions

3.2 Influence of pressure and temperature

In this section, the influence of temperature (0–600 K) and pressure (0–120 Kbar) on the considered properties of Al_xIn_1−xP with a constant concentration (x = 0.5) were determined. The ${\text{E}}_{{\text{g}}}^{{\text{L}}} {,}\,{\text{E}}_{{\text{g}}}^{{\Gamma }} {\text{ and E}}_{{\text{g}}}^{{\text{X}}}$ of Al_0.5In_0.5P versus temperature and pressure are recorded in Table 4 and exhibited in Fig. 4. We found that all the band gaps of Al_0.5In_0.5P are decreased with enhancing temperature. We also found that the ${\text{E}}_{{\text{g}}}^{{\text{X}}}$ is linearly decreased, while ${\text{E}}_{{\text{g}}}^{{\Gamma }}$ and ${\text{E}}_{{\text{g}}}^{{\text{L}}}$ are increased with increasing pressure. It was observed that the Al_0.5In_0.5P is an indirect semiconductor alloy within the entire range of temperature and pressure.

Table 4 The $\left( {{\text{E}}_{{\text{g}}}^{{\text{L}}} {\text{, E}}_{{\text{g}}}^{{\Gamma }} {\text{, and E}}_{{\text{g}}}^{{\text{X}}} } \right)$ in eV of Al_0.5In_0.5P for different values of temperature and pressure

Fig. 4

The energy band gaps of Al_0.5In_0.5P as function of temperature and pressure

The determined polarities ${\upalpha }_{{\text{p}}}$, C₁₁, C₁₂, C₄₄, B_u, C_s, and Y₀ of Al_0.5In_0.5P at different estimations of temperature and pressure are listed in Table 5 and presented in Fig. 5. It was found that the C₁₁, C₁₂, C₄₄, B_u, C_s, and Y₀ are slightly influenced by temperature. Also, it was found that these parameters are increased with enhancing pressure. Due to the lack of published data, our calculated data of C₁₁, C₁₂, C₄₄, B_u, C_s, and Y₀ of Al_0.5In_0.5P alloy at high temperatures and pressures may serve as a prediction for future experimental works. The analysis of the elastic constants could help to obtain data about the stability of the crystal. The mechanical stability requirements of a cubical crystal were given by Ref. (Bouarissa 2003) as (C₁₁ + 2C₁₂) > 0, C₄₄ > 0, (C₁₁− C₁₂) > 0. Our study confirms that the conditions are satisfied at the numerous values of temperatures (0–600 K) and pressures from (0–120 Kbar) which indicates the stability of Al_xIn_1−xP in its structure.

Table 5 Polarity and elastic moduli (C₁₁, C₁₂, C₄₄, B_u, Y_0, and C_s) in 10¹¹ dyn/cm² of Al_0.5In_0.5P at different values of T and p

Fig. 5

The elastic moduli (C₁₁, C₁₂, C₄₄, B_u, Y₀ and C_s) of Al_0.5In_0.5P as function of temperature and pressure

The calculated Poisson ratio (σ), linear compressibility (C₀), Cauchy ratio (C_a), Born ratio (B₀), anisotropy factor (A), bond stretching parameter $(\alpha \,)$, bond-bending force parameter $(\beta \,)$, internal-strain parameter $(\xi )$, effective charge (Z*) and the transverse effective charge $(e_{T}^{*} )$ of Al_0.5In_0.5P at various values of temperature and pressure are listed in Table 6 and displayed in Fig. 6. Our calculated values of σ, C₀,${\text{C}}_{{\text{a}}}$, ${\text{B}}_{{\text{o}}} {,}$ A, α, β, ξ, ${\text{e}}_{{\text{T}}}^{*}$ and ${\text{Z}}^{*}$ of Al_0.5In_0.5P alloy at the higher values of temperature and pressure may serve as a prediction for future work. This is due to the lack of published data. It was found that every one of these quantities is marginally influenced by temperature. Moreover, with pressure increasing, both α and β increase monotonously, C₀ decreases, while the σ,${\text{C}}_{{\text{a}}}$, ${\text{B}}_{{\text{o}}} {,}$ A, ξ, ${\text{e}}_{{\text{T}}}^{*}$ and ${\text{Z}}^{*}$ have weak influence with pressure.

Table 6 The (σ), (C₀), (C_a), (B₀), (A), (α), (β), $(\xi )$, (Z*) and (e_T*) of Al_0.5In_0.5P alloy at various temperatures and pressures

Fig. 6

Mechanical parameters (σ, C₀, ${\text{C}}_{{\text{a}}}$, ${\text{B}}_{{\text{o}}} ,$ A, α, β, ξ, ${\text{e}}_{{\text{T}}}^{*}$ and ${\text{ Z}}^{*} { }$) of Al_0.5In_0.5P as function of temperature and pressure

4 Conclusions

Based on (EPM) within VCA, the mechanical properties of zinc-blende Al_xIn_1−xP ternary semiconductor alloys were studied under the influence of temperature and pressure. The optoelectronic and mechanical properties of Al_xIn_1−xP were successfully determined. All quantities were studied for various estimations of temperature from 0 to 600 K and pressure from 0 to 120 Kbar. Our calculations at p = 0 Kbar and T = 300 K for the AlP and InP were found in excellent accord with the accessible theoretical and experimental data which may be a help for the outcome of the studied ternary alloys. The determined properties in this study may give helpful for optoelectronic applications.