Introduction

Antimony (Sb)-containing ternary systems, such as InAs1−xSbx, have garnered immense excitement and have been recognized as the material of choice for high-frequency optoelectronics devices. In particular, the fact that the magnitude of fundamental band gap energy can be tuned to absorb infrared (IR) radiation in the entire medium wavelength (MWIR: 3.4-5.2 μm), long wavelength (LWIR: 7.5-14 μm), and very long wavelength (VLWIR: > 14 μm) atmospheric transmission windows, makes this system almost perfect and ideal material in designing infrared photodetectors (IR-PDs).1,2,3 Furthermore, due to its appealing features, including the weak ionic character4 of its covalent bonding contribution, high optical quality, and good uniformity of epitaxial growth,5 InAs1−xSbx provides a prevalent alternative candidate to their II–VI competitors, mainly Hg1−xCdxTe (MCT) alloys, in terms of implementation in uncooled advanced IR-PDs.6,7,8,9 Although MCT is a crucial material for MWIR and LWIR bands, there are still some challenges in terms of a lack of stability and non-uniformity over a large composition area due to the Hg vapor pressure. In addition, MCT-based IR-PDs work only under cryogenic conditions and are high-cost materials.

In practical design and simulation of InAs1−xSbx -based optoelectronics devices, a serious challenge lies in the control of optical cut-off wavelength, and hence its spectral response for a desired application. However, it is well known that the optical cut-off wavelength depends mainly on material composition and crystalline quality. Moreover, a high-quality semiconductor crystal is required to construct IR-PDs with excellent performances, and this often requires the PDs epitaxial layers to be grown with minimal defects and smooth interfaces, and be compositionally homogeneous. Despite the fact that several III–V substrates can be used for growing InAs1−xSbx, the huge lattice mismatch leads to a poor crystal quality, drastically limiting the range of material composition. As a result, many properties of InAs1−xSbx alloys are still unknown due to the increased difficulties in growing high-quality materials and the decreased emission efficiency with increasing Sb composition. Hence, for device applications, improving the material quality and understanding the nonlinear optical properties of InAs1−xSbx in a wider composition range are greatly required.

In the recent past, there has been widespread interest toward the study of the temperature and compositional dependence of the electro-optical properties of InAs1−xSbx semiconductor alloys.10,11,12,13,14 However, the observed significant spread among the reported data implies variations in the measured band gap energy of InAs1−xSbx alloys with the same composition, and reflects inaccuracies related to the difficulty in determining the actual material composition. Moreover, the difficulties in the synthesis of InAs1−xSbx semiconductor alloys and their structural and optical quality are strongly correlated with the mismatch between its parent elements directly reflected by the so-called bowing parameter. Hence, this parameter carries important information about the character of a given alloy and is most often compositionally independent. However, when the two binaries indium arsenide (InAs) and indium antimonide (InSb) are combined to form the InAs1−xSbx ternary alloy, the optical band gap energy versus x shows a distinct bowing behavior, rather than a linear relationship. Typically, it has been asserted that the deviation from a linear dependence, predicted by the virtual crystal approximation,15 can be accounted for by including the effects of disorder on the conduction and valence band edges. Furthermore, in a more advanced treatment of this issue, it was assumed that the band gap bowing parameter can be routinely decomposed into three parts, originating from volume deformation, charge transfer effects, and structural relaxation.16 Concurrently, it was also argued that a single bowing parameter can describe the band gap energy behavior in the whole composition range. In consequence, a large spread in the reported values of InAs1−xSbx band gap energy relies on accurate knowledge of the material band gap bowing parameter.

Moreover, after a comprehensive literature review, we have noticed a large spread in the reported values of InAs1−xSbx band gap bowing parameter.17,18,19,20,21 The commonly accepted value is 670 meV, recommended by Vurgaftman et al.22 However, no agreement has been reached on the magnitude of the band gap bowing parameter of InAs1−xSbx alloys, or even on the issue of whether a single bowing parameter can describe the band gap width over the entire composition range. It is also worth noting that the large reported values of the band gap bowing parameter are rather unusual for the band gap energies in regular semiconductor alloys such as InAs1−xSbx. However, it has been clearly shown that the bowing parameter varies versus temperature in the range of bg(low T) = 938 meV17 to bg(300 K) = 590 meV.22 Meanwhile, the dependence of the bowing parameter on the composition in the aforementioned works is weak.

The purpose of this paper is to extend the theoretical description of the optical band gap energy and related parameters versus temperature in the whole Sb composition range (0 ≤ x ≤ 1). The results of this study provide several values of band gap-related parameters for InAs1−xSbx, which should prove very useful for enable suitable applications, as well as for the correct determination of epitaxial growth conditions. Moreover, it is well known that the study of electronic transport and optical properties of semiconductor materials requires a sound knowledge of the non-negligible coupling between electrons and lattice phonons. Therefore, it is clearly of considerable value to provide information on the basic vibration properties in the alloys under investigation. The average phonon energy and the corresponding electron–phonon interaction strength were also investigated in a broader composition range.

Calculation Method

The first investigation of the nonlinearity behavior in the optical properties of the InAs1−xSbx alloys is indicated in the fundamental band gap energy, εg, compositional dependence, which can be modeled using a second-order polynomial relationship between the band gaps of two terminal binary compounds with a quadratic term proportional to the band gap bowing parameter:

$$ \varepsilon_{g}^{InAsSb} (x, \, T) = x\varepsilon_{g}^{InSb} (1, \, T) + (1 - x)\varepsilon_{g}^{InAs} (0, \, T) - x(1 - x)b_{g} (T) $$
(1)

This relationship is valid not only for the fundamental band gap energy but also for other energy positions of the band structure with distinct bowing parameters. In this paper, our proposed band gap bowing parameter versus the lattice temperature is determined from:

$$ b_{g} (T) = - 0.47T + 803 \, (meV) $$
(2)

This equation has been benchmarked against experimental measurements, where we find that it can well reproduce the measured band gap energy versus Sb composition and temperature by introducing the temperature dependence of the InAs and InSb band gap energies. For the absorption layer, the relationship between the band gap energy of the material and its optical cut-off wavelength, λcutoff, is conventionally expressed as:

$$ \lambda_{cutoff} \left( {x, \, T} \right) \times \varepsilon_{g} \left( {x, \, T} \right) = 1240 \, \left( {\mu m\; \times \;meV} \right) $$
(3)

The temperature effect on the fundamental band gap energy of InAs1−xSbx is important for device design, as most IR-PDs are cooled below room temperature to minimize the effects of irradiative recombination. Furthermore, it is well known that the total changes of the band gap energy in semiconductor materials under temperature follows from a shift in the relative positions of the conduction and valence bands due to the temperature dependence of (1) dilatation of the lattice, and (2) electron–phonon interaction, which forms the major contribution to the variation. In the present work, the effect of temperature on the fundamental band gap energy of InAs1−xSbx is described by using the Varshni23 and Bose–Einstein24 three-parameter thermodynamic models:

$$ \varepsilon_{g} (T) = \varepsilon_{gv} (0) - \alpha_{v} \left( {\frac{{T^{2} }}{{\beta_{v} + T}}} \right) $$
(4)
$$ \varepsilon_{g} (T) = \varepsilon_{gBE} (0) - \frac{{2a_{BE} }}{{\left( {e^{{\frac{{\theta_{BE} }}{T}}} - 1} \right)}} $$
(5)

where εgi(0) (meV) is the 0 K band gap energy, αv(meV/K) and βv(K) are material-dependent parameters, and T(K) is the temperature in Kelvin. The value of αv(meV/K) is related to the thermal expansion of the lattice, and the value of βv(K) is thought to be similar to the Debye temperature, θD(K). Usually, above θD(K), the band gap energy changes almost linearly with temperature due to linear changes of the lattice parameters and electron–phonon interaction. The parameter θBE(K) describes the mean frequency of the phonons involved and aBE(meV) describes the strength of the electron–phonon interaction. It is worth noting that the Bose–Einstein model retains physical meaning and enables an accurate extraction of average phonon energy, < hν > (meV), for the material under investigation. However, these two models are related as follows:

$$ \langle h\nu \rangle \approx k_{B} \theta_{BE} \approx 2k_{B} \beta_{v} $$
(6)

The band structure parameters of the two terminal binary compounds, InSb and InAs, have been investigated and refined over the years. A survey of the adopted values used in this study is summarized in Table I.

Table I Band structure parameters of InAs and InSb parents adopted in the present work, Ʌ is the valence band offset between the valence band maxima near the centre Г of the first Brillouin zone of InAs and InSb
Full size table

Results and Discussion

The InAs1−xSbx cut-off wavelength versus Sb composition is depicted in Fig. 1 for both low (Fig. 1a) and room (Fig. 1b) temperatures. The continuous and short dashed lines are the calculated results obtained using our band gap bowing parameter given by Eq. (2) and that adopted by Murawski el al.,26 respectively. Also shown for comparison are the available experimental data reported in the literature. As presented in Fig. 1, the cut-off wavelengths of InAs1−xSbx under investigation vary rapidly at the dilute Sb composition side, but changes slowly at the relatively large Sb composition side. A close inspection of Fig. 1 demonstrates that the incorporation of dilute Sb composition into the InAs host material has been shown to enhance the cut-off wavelength by a linearized average rate of about 1.90 μm, and by 1.41 μm per 10% Sb content at room and low temperatures, respectively. This rapid increase of cut-off wavelengths, with the bowing parameter spanning the range of 662–801 meV, implies that the Sb incorporation significantly perturbs the band structure of the InAs pristine compound. Also, Fig. 1 shows the existence of InAs1−xSbx alloys capable of reaching the LWIR regime with relatively lower bowing values than previously reported. Interestingly, reaching the beginning of the LWIR regime at approximately 7.5 μm needs a bowing parameter of 662 meV for only 25% Sb content and 801 meV for 35% Sb content at room and liquid helium temperatures, respectively.

Fig. 1
figure 1

InAs1−xSbx optical cut-off wavelengths versus the Sb composition at (a) low temperature and (b) room temperature. The available experimental data were taken from selected works as indicated by the legend. MBE molecular beam epitaxy technique.

Full size image

The proper correlation between our theoretical results and the available experimental ones was reached for both As-rich (x < 0.5) and Sb-rich (x > 0.5) sides. In a moderate composition range, especially at room temperature, no such credible data are available. Furthermore, in the low-temperature range, our results deviate from the photoluminescence predictions substantially in the range of 0.40–0.65 Sb content. Additionally, the data determined from the optical transmission measurements, for samples grown by the metal organic chemical vapor deposition technique on lattice mismatched GaAs substrates, fits well with our bowing parameter at the Sb-rich side. It is believed that the relatively higher cut-off wavelengths (lower band gap energies) observed in the experimental reported values were masked partly by electron filling of the conduction band due to background doping, as these samples were grown with various degrees of residual strain and relaxation. Consequently, the slight deviation of our data from the experimental ones can be attributed to the copper–platinum (CuPt)-type atomic ordering of the material, hypothesized to be caused mainly by residual strain.

The size and core electronic structure of Sb (Z = 51, Ratom = 140 pm) are significantly different from those of As (Z = 33, Ratom = 120 pm). As expected, the substitution of As by Sb implies an average increase in the unit cell volume. Thus, a compressive strain develops in the crystal resulting in a significant modification in the electronic structure and related electro-optical properties of the pristine alloys. Furthermore, the different behaviors of the cut-off wavelengths observed on both the As- and Sb-rich sides suggest different origins of the band gap bowing which could be explained as follows (Fig. 2). First, as depicted in Fig. 2a for the As-rich side, incorporating a small amount of Sb into the InAs can introduce a localized Sb-level lying below the valence band maximum (ГVBM) of InAs near the first Brillouin zone Г center. This induced localized Sb-level has a strong effect on the ГVBM of the InAs host material. Consequently, the coupling interaction between them can lead to the upward movement of the ГVBM of InAs1−xSbx, thus enhancing the cut-off wavelengths when the Sb content is relatively high. Second, Fig. 2b clearly shows that, at the Sb-rich side, an increasing trend of the cut-off wavelengths with increasing As content can be observed. The reason is because a localized As level can be introduced after incorporating a small amount of As into the InSb host material. This localized As level is located above the Г conduction band minimum (ГCBm) of the InSb. When the As content increases, the coupling interaction strength between the As level and the ГCBm of the InSb is enhanced. Furthermore, this enhanced coupling interaction strength is able to push down the ГCBm of the InAs1−xSbx, leading to the cut-off wavelengths increase in the Sb-rich range.

Fig. 2
figure 2

Schematic showing the Sb composition dependence of energy positions of the conduction band minimum (ГCBm) and valence band maximum (ГVBM) band edges for InAs1−xSbx. The ГVBM was chosen as Ʌ = − 590 meV for InAs and 0 meV for InSb. (a) The InAs band structure and the effect of the localized Sb level in the InAs0. 65Sb0.35 As-rich alloy. (b) The conduction band edge InSb host material shift below As level in the InAs0.35Sb0.65 Sb-rich alloy.

Full size image

To further validate the obtained values, Fig. 3 depicts that the satisfactory agreement between our calculation and the experiment, over the entire composition range, shows confidence in the measuring procedures used in the selected works, as well as the success of our data analysis method, and offers assurance for investigating other effects due to Sb incorporation. Although, among other effects, some tensile strain is believed to exist in the InAs1−xSbx epitaxial reference samples, their band gap energies and the corresponding cut-off wavelength values are either in agreement or only slightly below the results reported in this study. Furthermore, Fig. 3 clearly shows that some measured values are found to be larger than the corresponding theoretically predicted ones. This may be attributed to errors in experimental measurements and/or the dominance of the disordered excited states in the samples. Meanwhile, it is worth noting that, for a given concentration, the band gap energy of the ground states is generally larger than the excited states energy.

Fig. 3
figure 3

Calculated versus the available measured InAs1−xSbx cut-off wavelengths in the entire Sb composition range.

Full size image

It is plausible to investigate the complete optical cut-off wavelength temperature dependence in the entire Sb composition range. However, due to the relatively large band gap energies, a wide range of temperature is very much needed for an accurate description of the temperature effect. As seen from Fig. 4, for a fixed composition value, the cut-off wavelength followed an increasing trend in quite similar ways with increasing the sample temperature. Moreover, in the As-rich side, the enhanced cut-off wavelength with Sb content is roughly 3.51 μm in the MWIR band for pure InAs and 8.86 μm in the LWIR band for InAs0.65Sb0.35 at room temperature. Simultaneously, at liquid helium temperature, the cut-off wavelength decreases to 2.97 μm for pure InAs and to 7.26 μm for InAs0.65Sb0.35, enabling it to cover both MWIR and LWIR atmospheric windows. The total cut-off wavelength red shift follows from the lowering in the band gap energy variation from 63.16 meV in InAs to only 30.86 meV in InAs0.65Sb0.35 as the temperature increases.

Fig. 4
figure 4

Simulated contour maps of the dependence of the optical cut-off wavelength on both Sb composition and lattice temperature in the InAs1−xSbx alloy system. The calculations were carried out using our proposed band gap bowing parameter temperature dependence.

Full size image

The present findings confirm that the two conventional InAs0.40Sb0.60 and InAs0.35Sb0.65 samples exhibit sufficiently high possible cut-off wavelengths implying longer absorption in the LWIR and VLWIR atmospheric windows. Furthermore, Fig. 5 clearly indicates that the longest cut-off wavelengths that can be quenched by the 662-meV bowing parameter is about 14.40 μm in the VLWIR atmospheric windows at room temperature and approximately 10.73 μm in the LWIR atmospheric band with 801 meV at liquid helium temperature. In consequence, the longest cut-off wavelength values also decline formidably when the lattice temperature decreases. The corresponding total red shift is about 1.20 μm in InAs0.40Sb0.60 for T < Tc and 2.43 μm for InAs0.35Sb0.65 beyond Tc = 186.1 K.

Fig. 5
figure 5

Longest cut-off wavelengths and the corresponding band gap bowing parameter as a function of lattice temperature for InAs0.40Sb0.60 (solid curve for T < Tc = 186.1 K) and InAs0.35Sb0.65 (dashed curve for temperatures greater than Tc = 186.1 K).

Full size image

The compositional dependence of the Varshni parameters, αv(meV/K) and βv(K), for InAs1−xSbx , is difficult to determine because of the observed scatter in the available data. Furthermore, the alloy composition was limited to a narrow region. The αv(meV/K) and βv(K) parameters determined previously for the InAs1−xSbx samples are listed in Table III. The observed discrepancy in the corresponding values is probably from sample-dependent features such as strain and defects. Also, this is presumably caused by the difference in the incorporation scheme of Sb atoms into InAs1−xSbx and relevant localized states at a microscopic scale.

The temperature-induced band gap energy red shift for InAs0.65Sb0.35 and InAs0.35Sb0.65 together with that of InAs and InSb are depicted in Fig. 6. Based on the Varshni model, the total red shift of the band gap energies between 4 K and 300 K are 63.16 meV, 30.84 meV, 30.27 meV, and 61.25 meV for x = 0, 0.35, 0.65, and 1, respectively. Accordingly, with increasing Sb content, the optical band gap energy is less sensitive to the temperature changes. Such a functional dependence of the optical band gap energy on Sb composition can be commonly given in terms of the Varshni and Bose–Einstein parameters. Fitting εg(T) in the entire composition range, using the Levenberg–Marquardt method, expresses the nonlinearity composition dependence of the Varshni and Bose–Einstein coefficients with the following general equation:

$$ P_{{{\text{InAsSb}}}} (x, \, T) = xP_{{{\text{InSb}}}} + (1 - x)P_{{{\text{InAs}}}} - x(1 - x)b_{i} $$
(7)
Fig. 6
figure 6

Fundamental band gap energies as a function of lattice temperature for InAs0.65Sb0.35 and InAs0.35Sb0.65 ternary alloys together with that of InAs and InSb. The solid and dashed curves correspond to the Varshni and Bose–Einstein models, respectively. A list of the extracted fitting parameters using the Levenberg–Marquardt method are shown for each case.

Full size image

The bowing parameter values bi offering the best fits are listed in Table II.

Table II Calculated bowing parameters values for the three Varshni and Bose–Einstein parameters
Full size table

Figure 7 shows that the 0-K band gap energy of InAs is 182 meV higher than that of InSb. One can see that the incorporation of Sb composition from 0 to 0.6 in the InAs host material causes a 0-K band gap energy red shift with a linearized average rate of 5 meV per 1% Sb content. Furthermore, the 0-K band gap energy minimum occurs for InAs0.40Sb0.60 implying longer absorption cut-off wavelengths. Interestingly, the temperature coefficient, αv(meV/K), of the InAs1−xSbx alloys is appreciably smaller compared with that of the InSb. This is reflected by a shift by approximately 57% in InAs0.40Sb0.60 compared with that of InSb. This result is hoped to offer an improvement on InAs1−xSbx optoelectronics-based device stability. However, no clear explanation of the observed effects has yet been offered. Additionally, for x < 0.6, the band gap temperature coefficient of InAs1−xSbx lies in a narrow range of 0.260–0.158 meV/K. It appears that, in this composition region, the band gap energy is less sensitive to the temperature and that incorporating 60% Sb has a relatively little effect on the temperature dependence of the band gap energy. These results illustrate a promising feature of the Sb-containing III–V alloys for the temperature-insensitive semiconductor materials for optoelectronic devices.

Fig. 7
figure 7

Compositional dependence of the (a) fundamental 0-K band gap energies for the two models adopted here, (b) band gap temperature coefficient, (c) average phonon energy, and (d) the corresponding strength of electron–phonon interaction in InAs1−xSbx; the hashed gray area represents the corresponding upper–lower confidence limits, and the solid green lines are fitted curves drawn using the Levenberg–Marquardt method (Color figure online).

Full size image

The average phonon energy < hν > (meV) and the corresponding electron–phonon interaction strength, aBE (meV), as a function of Sb composition are depicted in Fig. 7c, d, as well as the corresponding upper and lower confidence limits curves. One can see that < hν > and aBE versus x are found to exhibit a strong correlation with a negative bowing parameter, as listed in Table II. Additionally, the average phonon energy in the pseudo-binary InAs0.10Sb0.90 alloy is comparable to the thermal energy at room temperature, and the electron–phonon interaction is significantly strong. Moreover, the Varshni parameters determined for InAs1−xSbx alloys lie in between values of those parameters for InAs and InSb. Concurrently, the average phonon energy and the corresponding strength of electron–phonon interaction in InAs1−xSbx are higher than that of InAs and InSb.

Table III summarizes the presently obtained Varshni and Bose–Einstein parameters for InAs1−xSbx together with the previously available values for similar compositions. To further validate our obtained values for InAs1−xSbx ternary alloys, the direct comparison can be made for InAs and InSb binary compounds because the effects due to compositional fluctuation are absent. Obviously, satisfactory agreement between the present values and the corresponding experimental data is observed with a relatively slight average deviation. These results make us conclude that the irregular changes in the reported values may be sample-dependent, in the sense that they may be related to imperfections in the samples, such as, impurities, built-in strain, stress, or defects. Moreover, for the Varshni parameters, the rather different values of αv(meV/K) and βv(K) listed in Table III lead to almost identical band gap energy results over a significant temperature range. In addition, a reliable determination of band gap energy versus temperature can be obtained with a suitable combination of αv(meV/K) and βv(K).

Table III Comparison of the three Varshni and Bose–Einstein parameters with the available results for similar compositions in InAs1−xSbx ternary alloys
Full size table

Our results regarding εg(0) and (α/β) for InAs1−xSbx are shown in Fig. 8. Also shown for comparison are the available data obtained exploiting the εg(T, x) relationship reported in Ref. 32. Apparently, our predicted results overlap the results of Ref. 32 over a significant composition range. The deviation of our value of (α/β) from the experimental ones is less than 39% for x < 0.1.

Fig. 8
figure 8

Comparison of the composition dependence of (a) α/β and (b) 0-K fundamental band gap energy of InAs1−xSbx.

Full size image

Conclusions

Despite the progress in current research on the electro-optical properties of the InAs1−xSbx alloy system, uncertainties remain regarding the fundamental basis of their band structure-related properties, ultimately rooted in its temperature and compositional dependence. Moreover, for an effective design and simulation of InAs1−xSbx -based infrared photodetectors, as well as the correct determination of epitaxial growth conditions, it is clearly of considerable value to investigate the band gap energy dependence on temperature and composition over a significant range. In this work, we have reported on the dependence of the fundamental band gap energy of InAs1−xSbx, and the corresponding cut-off wavelength, on temperature with Sb compositions spanning the entire range. The results reveal entirely different origins of the high cut-off wavelength bowing for the Sb-rich and As-rich sides. The relatively weaker band gap temperature coefficients reported in this work clearly illustrate promising features of Sb-containing alloy systems for the temperature-insensitive semiconductor materials optimized for infrared technology. Also, a correlation between the average phonon energy and the corresponding electron–phonon coupling interaction strength were observed over the entire Sb composition range. The reported results are likely to serve as a useful reference for improving the photodetection parameters of InAs1−xSbx -based devices, not only by improving the material quality, through a precise determination of epitaxial growth conditions, but also by very accurate designing of their structures. The satisfactory agreement with the available results yields confidence in our estimating procedure as well as our data analysis.