1 Introduction

During the last couple of years, much research in the near-infrared (NIR), mid-infrared (MIR), and far-infrared (FIR) regions of the electromagnetic spectrum has been focused on revealing the potential of novel promising semiconductor materials and semiconductor devices [1,2,3,4,5,6,7]. Undoubtedly, the emergence of the so-called highly mismatched alloys (HMAs) has accelerated the studies conducted for semiconductor-based applications in the infrared (IR) region [1]. Of the members of this class, Bismuth (Bi)-containing alloys of HMAs has become the center of interest thanks to the high tunability of the bandgap modifying valence band (VB) structure without compromising the electronic charge transport in the conduction band (CB) [8,9,10,11,12,13]. The strong Bi-dependence of all VBs (heavy and light hole and spin-orbit bands) has revealed that the potential of Bi-containing alloys not only suppress Auger recombination in semiconductor lasers but also to be used in NIR and MIR applications [14]. Being the newest member of Bi-containing alloys such as quaternary InxGa1−xAs1 − yBiy /InP: Fe has become attractive with its superior properties and its potential to be utilized in devices for the NIR region [15, 16]. So far, it has been presented electronic band structure, optical bandgap, and growth conditions of the InxGa1−xAs1 − yBiy [17,18,19]. It has been witnessed that the effect of the Bi atoms on the electronic band structure of InxGa1−xAs alloys can be explained by the VBAC model, which results in the splitting of each VB, such as heavy and light holes and spin split-off band. In contrast, the CB is unaffected, leaving the electronic transport unaffected, which is already known from previously discovered Bi-containing alloys [15, 20]. The narrowing of the bandgap without affecting electron transport-related parameters (electron effective mass, CB discontinuity, etc.) allows for the extension of the operation wavelength to the IR region without losing the performance of the devices [13, 21, 22]. In addition to Bi-induced band structure modification, free carrier (electron/hole) density and electron/hole mobility in the InxGa1−xAs1 − yBiy are determined using optical (photoluminescence and absorption measurements) and classical Hall effect measurements at room temperature [18, 19, 23, 24]. Furthermore, the electron effective mass is determined from optical transition energies at room temperature for highly doped InGaAsBi alloys without considering electron-optic phonon coupling (Fröhlich coupling), which may affect the effective mass values [17]. In order to eliminate the Fröhlich coupling effect to obtain the actual value of the electron effective mass, the experiments have to be conducted at low temperatures to suppress the optical phonon effect on effective mass. By introducing real effects of the Bi on electronic transport parameters in InxGa1−xAs1 − yBiy, one can show the true potential of InxGa1−xAs1 − yBiy in technological applications.

The electronic charge transport under magnetic fields strongly influences the electrical and optical properties of semiconductors. Analyzing the changes under an applied magnetic field gives unique and accurate information on optical and electrical properties. The electrical transport has a classical HE characteristic under \({\mu }_{t}B\ll 1\) condition (Ohmic behavior), where \({\mu }_{t}\) and \(B\) are transport or Hall mobility and magnetic field, respectively [25]. However, the HE and MR characteristics switch from classical to quantum mechanical behavior under \({\mu }_{t}B\gg 1\) condition. According to the quantum theory for the two-dimensional (2D) system under a sufficiently high magnetic field (\({\mu }_{t}B\gg 1\)), free carriers (electrons or holes) are forced to follow an orbit with a discrete set of the Landau levels (LL) and increasing magnetic field-induced LLs cross-over the Fermi level. Crossing LLs over the Fermi level results in oscillations in the longitudinal resistivity that oscillations are called SdH. The analysis of the SdH oscillations gives accurate results for crucial transport parameters such as electron effective mass, Fermi level, and quantum mobility [26, 27]. A few LLs are occupied at the high magnetic field, and the 2D free carriers enter the quantum Hall regime, which appears as the Hall plateau in Hall resistivity and is called the quantum Hall Effect (QHE). These characteristic behaviors (SdH oscillations in longitudinal resistivity and quantum Hall oscillation/plateau in transverse resistivity) of the QHE have been widely observed in low dimensional structures [28]. On the other hand, the observation of the QHE in 3D materials has also been reported and explained under certain conditions as the presence of the Fermi surface instability coming from a periodic modulation of electron density, formation of charge-density-wave, and/or spin-density wave, etc., which creates the quasi-mobility-gap as in integral (integer) QHE [29, 30]. Recently, the formation of the quasi-mobility-gap or QHE has been reported for bulk InSb, HgCdTe, and InAs semiconductors [31, 32].

In this study, it is the first time we report an analysis of the MR and HE measurement results of InxGa1−xAs1 − yBiy alloys with various do** densities at low-temperature regimes to study transport parameters. The electronic band structure of the sample is calculated using FEM in Comsol Multiphysics environment. The transport mechanism is determined by a combination of experimental and FEM results. The transport parameters are determined by analyzing the temperature-dependent SdH oscillations. The VBAC model is exploited to explain the effect of Bi incorporation in the host materials (InGaAs here) on electron transport. The high do** effects on the transport are expressed using a non-parabolic band model.

2 Experimental methods

n-type (Si-doped) InxGa1−xAs1 − yBiy alloys were grown by molecular beam epitaxy (MBE) on double-side polished Fe-doped InP substrate to have a semi-insulating (SI) substrate. Prior to the growth process, the sample surface was cleaned by heating the substrate to 565 °C with arsenic overpressure and it is confirmed by changes in the reflection high energy electron diffraction image. The growth was begun by a 90 nm undoped (UD) In0.48Ga0.52As layer, and then a 500 nm thick InxGa1−xAs1 − yBiy layer was grown at 300 ℃. The In composition is slightly adjusted, so the Ga composition is changed to keep the lattice-matched InxGa1-xAs1-yBiy layer to the In0.48Ga0.52As buffer layer [33]. The composition of the InxGa1−xAs1 − yBiy alloys is given in Table 1. Dopant density was adjusted by varying the temperature of the Si cell. Nominal Si do** densities are between 1 × 1017 cm− 3 and 7 × 1018 cm− 3. The structural properties of the sample were verified by high resolution X-ray diffraction (HR-XRD). The HR-XRD results agree with the Rutherford backscattering spectroscopy (RBS) results. In addition, RBS channeling measurements were also conducted to confirm that Bi was substituted in the crystal lattice. Further growth details can be found in Ref [17, 19, 34].

Table 1 (a) Information about sample structure with nominal growth parameters and (b) the layered structure of the sample
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The changes in alloy composition may affect the electronic band structure of the InxGa1−xAs1 − yBiy alloys. The bandgap energy and lattice constant are calculated using the equations given below for ternary (host) and quaternary alloys [35, 36],

$$\begin{aligned} &{E}_{G}\left(I{n}_{x}G{a}_{1-x}As\right)=x{E}_{G}\left(InAs\right)+\left(1-x\right)\\&{E}_{G}\left(GaAs\right)-0.477x\left(1-x\right)\end{aligned}$$
(1a)
$$\begin{aligned} &a\left(I{n}_{x}G{a}_{1-x}A{s}_{y}B{i}_{1-y}\right)=xya\left(InAs\right)+x\left(1-y\right)\\&a\left(InBi\right)+y\left(1-x\right)a\left(GaAs\right)+ \left(1-x\right)\left(1-y\right)a\left(GaBi\right)\end{aligned}$$
(1b)

where \({E}_{G}\), \(a\), and \(x/y\) are bandgap energy, lattice constant, and alloy compositions, respectively. According to the VBAC model, incorporation of Bi atom into host InGaAs alloys can only change the VB states but does not CB. In order to predict the effect of Bi on bandgap energy, we first calculate the bandgap of the host InGaAs alloys. The bandgap of the host (InxGa1−xAs) alloy was calculated using GaAs and InAs semiconductor parameters taken from Ref [36] in Eq. 1a and are given in Table 2. The changes in net bandgap energy are lower than 10 meV considering In0.48Ga0.52As buffer layer bandgap energy [15]. The lattice constants of the InxGa1−xAs1 − yBiy alloys are calculated using semi-metallic GaBi and InBi, GaAs, and InAs parameters taken from the literature [37, 38] in Eq. 1b. As seen in Table 2, the strain value between InGaAs and InGaAsBi layer is small, hence it is omitted [39].

Table 2 Lattice constant of binary semiconductors (\({a}_{Binary}\)) and calculated bandgap of the host InGaAs alloys (\({E}_{G}\)), the lattice constant of quaternary InxGa1−xAs1 − yBiy alloy (\({a}_{I{n}_{x}G{a}_{1-x}A{s}_{y}B{i}_{1-y}}\)), and strain value between host and InxGa1−xAs1 − yBiy alloys
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In order to carry out electrical measurements, the samples were fabricated in Hall Bar geometry with a wet etching process in a clean room environment [40, 41]. Ohmic contact was obtained through thermal evaporation of Au (200 nm)/Cr (20 nm) metals. The formation of the Ohmic contact was verified by conducted current-voltage measurements (not shown here) [42]. Orthodox HE and MR measurements were conducted in order to identify the nature of the electron transport parameters and characteristics. A DC magnetic field was applied perpendicularly to the sample surface (or along the growth direction) and the constant DC current. A Bruker 2 Tesla (T) electromagnet for 77–300 K and Oxford Instruments superconducting 18 T magnet system equipped with a Heliox insert for ~ 4.3 K and 70 K were utilized in HE and MR measurements. In HE measurements, the magnetic field was kept at 0.5 T. In addition to these experiments, photoconductivity measurements were conducted to determine the optical bandgap (bandgap) as a complementary experiment to calculate bandgap nonparabolicity.

3 Results and discussions

Figure 1 shows the temperature dependence of Hall mobility and free electron density in the samples. The highest electron mobility is obtained for the Bi-13 sample at the lowest temperature. The temperature dependence of the electron mobility in Bi-01(the lowest doped) sample exhibits the conventional characteristic observed in tree-dimensional (3D) structures, in which electron mobility increases then decreases with increasing temperature, due to the ionized impurity scattering (IIS) at low temperature regime and phonon scattering (PS) at intermediate and high-temperature regime, respectively [43]. As seen in Fig. 1a, at a low-temperature regime between ~ 4.3 and 70 K, the temperature dependence of the electron mobility in the sample with the lowest do** density (Bi-01) exhibits IIS-limited characteristic, which has a \({T}^{3/2}\) dependence and detail can be found elsewhere [43,44,45]. However, the electron mobility in the other samples behaves almost independently of the temperature because of the degeneracy of the samples, which is a well-known characteristic of the degeneracy of the samples. Temperature dependence of electron mobility in degenerate III-V semiconductors can be analyzed with modified IIS and PS, electron-electron interaction, considering carrier density-dependent effective mass and bandgap nonparabolicity, and details can be found elsewhere [43,44,45]. Therefore, analytical modeling of IIS, PS, and electron interaction is not repeated here. However, (carrier density dependent) effective mass and bandgap nonparabolicity is one of the arguments of this paper, which will be discussed later.

The sheet electron density in all samples is almost independent of the temperature and agrees with the nominal growth values, indicating that the Si atoms’ ionization energy is not affected by Bi incorporation in the InGaAs host semiconductors [46] because all donor atoms are ionized at the lowest temperature.

Fig. 1
figure 1

Temperature-dependent (a) Hall mobility and (b) free electron density in InxGa1−xAs1 − yBiy samples. Symbols depict the experimental results. Solid lines show the fitted 𝑇3/2 dependence of electron mobility at the low-temperature region

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Figure 2a shows temperature-dependent raw experimental MR data for all samples. The amplitude of the SdH oscillations decreases slightly as the temperature is increased. We observed clear SdH oscillations above ~ 5 T for As-15 (Bi-free), Bi-13, and Bi-70 samples with almost monotonic MR background. Even the onset of the appearance of the SdH oscillations depends on the free electron density; the SdH oscillations are suppressed in the Bi-01 sample, which is the lowest-doped sample. In order to analyze SdH oscillations, we eliminate the background in raw MR data. There are two analytic methods to eliminate background: the negative of the second derivative of the raw MR data (or derivative method) and the polynomial fitting procedures. It is shown that neither method affects the characteristics of the SdH oscillations [10, 47, 48]. The polynomial fitting methods were applied for As-15 (Bi-free), Bi-13, and Bi-70 as there are almost negligible backgrounds, as seen in Fig. 2a, while the second derivative method is used for Bi-01 samples because of the existence of non-linear background in MR.

Fig. 2
figure 2

Temperature-dependent (a) raw MR and (b) background removed MR results for all samples. The different colors represent the different temperatures

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In background removed MR results (Fig. 2b), there are no SdH oscillations in the Bi-01 sample at the high magnetic field region. This interesting behavior can be explained by the criteria to observe SdH oscillations, (i) \(\hslash {\omega }_{0}\gg {k}_{B}T\), (ii) \(\mu B \left(\text{o}\text{r} {\omega }_{0}{\tau }_{q}\right)\gg 1\), and (iii) \({E}_{F}\gg \hslash {\omega }_{0}\) (or \(\varDelta E={E}_{F}-\hslash {\omega }_{0}\gg 0\)) criteria should be satisfied [26]. Here, \({\omega }_{0} (=eB/{m}_{e}^{*})\), \({E}_{F}\), and \({\tau }_{q}\) are oscillation frequency, Fermi level, and quantum relaxation time, respectively. The first condition is satisfied since experiments were conducted at a low temperature. The magnetic field strength at the experiments is enough to satisfy the second condition,\(\mu B \left(\text{o}\text{r} {\omega }_{0}{\tau }_{q}\right)\gg 1\), considering electron mobility, presented in Fig. 3a. It is clear that the third condition, \(\varDelta E\), is satisfied in all samples except for Bi-01. The energy difference between the \({E}_{F}\) and the nearest magnetic field-dependent LL becomes larger for Bi-01 above ~ 8 T because the lower electron effective mass (calculated later) induced a significant difference in each quantized LL, and the next LLs are, hence, far from the Fermi level. Hence, SdH oscillation was not observed for the Bi-01 sample at the higher magnetic field region.

Fig. 3
figure 3

Calculated energy difference between the \({E}_{F}\) and the nearest LL for all samples

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The number of oscillation frequencies (periods) was determined by Fast Fourier Transform fitting (FFT) to background removed MR results [40, 48], and the FFT results are given in Fig. 4. As seen in Fig. 4, there is a single peak, which validates the single-band (also single channel) occupancy of the electron, contributing to the formation of the SdH oscillations. It is worth noting that in the case of occupation of more than one band or transport channel, MR is composed of the SdH oscillations with different frequencies and FFT result has different peaks [28]. The peak position of the FFT changes for all samples because the SdH extrema appears in different magnetic field values due to the different electron density and effective mass values.

Fig. 4
figure 4

Normalized FFT of the background eliminated experimental MR data of all sample at the lowest temperature

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In order to clarify the nature of electron transport, we need to resolve the electron density in MR, which provides information for the transport mechanism. The electron density can be found from the experimental period of the SdH oscillations. The relation between the period of SdH and carrier density is calculated by the analytical equation given as [26, 49],

$${\varDelta }_{i}\left(\frac{1}{{B}_{i}}\right)=\frac{e}{\pi \hslash {n}_{SdH}}$$
(2)

where \({n}_{SdH}\) is 2D electron density. Figure 5 shows the plot of the \(1/{B}_{n}\)versus peak number. It is worth noting that both local maximum and minimum points of the SdH oscillations are contemplated for Bi-01 because of an insufficient number of the SdH oscillations to be able to analyze the results. However, it does not deviate from the fundamental characteristic of \(1/{B}_{n}\) versus peak number, as seen in Fig. 5.\({\varDelta }_{i}\left(\frac{1}{{B}_{i}}\right)\) is determined from the slope of the results given in Fig. 5. The calculated electron densities from MR (\({n}_{SdH}\)) and HE (\({n}_{Hall}\)) results at low temperatures are given in Table 3. The \({n}_{SdH}\) electron density for all samples follows similar trends as in \({n}_{Hall}\) results.

Fig. 5
figure 5

Plot of numbered local maxima of the SdH oscillations versus peak number results. The symbols are ascribed for experimental results. The straight-line results are fitting results to experimental data

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However, the \({n}_{SdH}\) is much lower than \({n}_{Hall}\). The reason for this drastic difference between \({n}_{SdH}\) and \({n}_{Hall}\) can originate from either distribution of the electrons in the different layers as observed in layered heterostructure or a mechanism such as magnetic freeze-out, anomalous HE, etc., which suppress the electrons not to contribute transport [29, 50,51,52,53].

Table 3 Experimental transport parameters of the investigated sample at low temperatures
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In order to determine electron distribution through the layer of the samples from surface to substrate, we calculate electronic band structure using FEM in Comsol Multiphysics environment under thermal equilibrium conditions [5]. In the calculation, we use the Fermi level pinning at the surface due to the degeneracy of the samples [54] and utilize the well-known parameters for In0.47Ga0.53As [36]. In0.48Ga0.52As alloy is intrinsic with ~ 1015 cm− 3 free electron density, and the substrate is a semi-insulating InP semiconductor. The calculated FEM results for all samples’ CB are given in Fig. 6. As seen in Fig. 6, the electrons can reside in the near-surface region along ~ 200 nm because of the band bending, but the accumulation of electrons at the interface between the InGaAs buffer and InP layers is not allowed due to the n-type nature of the undoped InGaAs buffer layer because the Fermi level lies below CB of the buffer layer and of the absent of the strong attractive force for electron. However, Tsubaki et al. reported that for the undoped InGaAs layer with 400 nm thickness grown on SI-InP substrate, the electron accumulation occurs at the interface between the UD InGaAs and SI-InP [55]. In addition, it has been shown that the growth of the InGaAs layer on the n-type or p-type InP layer, whether InP is used as a buffer or substrate, increases the density of the electrons at the interface [55, 56]. Also, Fe do** of the InP substrate has no role in the free carrier accumulation at the interface [56]. The accumulation of electrons (hole) forms the two-dimensional electron gas at the interface, and the transport characteristic substantially differs from the ones in bulk structure [50, 51]. In order to validate our FEM calculation, we calculate the electronic band structure of the sample as in Ref [55] and given in Fig. 6b. We obtain the same electronic band structure and formation of the 2DEG at the interface as presented in Ref [55]. In light of the calculated CB profile and electron distribution and our findings, electron transport occurs in a single channel (Fig. 4); we exclude the possibility of electron distribution in the other layer.

Fig. 6
figure 6

Calculated conduction band profiles (left) and electron distribution (right) (a) in the samples studied here and (B) in the sample as in Ref [55]

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To reveal another mechanism, such as magnetic freeze out, anomalous HE, etc., causing the large difference between \({n}_{SdH}\) and \({n}_{Hall}\), QHE is reported as a useful tool. It has been shown that the QHE can also be observed in 3D semimetals, bulk semiconductors, and Dirac semimetals when certain experimental and structural conditions are satisfied. The reason for the appearance of the QHE in 3D semiconductors is explained by magnetic freeze-out (MF), spin-density wave (SDW), and charge-density wave (CDW) mechanism [29]. Recently, Zhang et al. showed that the QHE is observed in Dirac semimetal because of the unusual electronic band structure of Cd3As2 [57]. Linhart and Kudrewiec showed that the change in the electronic band structure of In0.53Ga0.47AsyBi1−y alloys is explained with the VBAC model as in the other highly mismatched alloys, and there is no unusual electronic band structure as in Cd3As2 [58]. It has been depicted experimentally for narrow bandgap InSb that the experimental evidence of the magnetic freeze-out is the magnetic field dependence of the free electron density [52, 59], in which free carrier density becomes lower with magnetic field. As seen in Fig. 7a, the free electron density for all samples is independent of the magnetic field and is the same as presented in Fig. 1b; therefore, this is not the case for our samples. Keyes and Sladek showed that the magnetic freeze-out occurs in low do** density for narrow bandgap InSb under a high magnetic field, and electron density decreases with increment of the magnetic field [60]. However, as seen in Fig. 7a, the magnetic freeze-out is not observed in Bi-01 (with lowest electron density). In addition, the anomalous Hall effect (or Lifshitz transition) is not observed, as seen in Fig. 7b [52]. Hence, the reasons for lower electron density in MR are not related to magnetic freeze-out, anomalous Hall effect, and unusual electronic band structure. Wawrzynczak et al. disseminated the theoretical and experimental results for bulk InAs (with low electron density) that the weak oscillations or plateau-like feature in Hall resistance, QHE, is observable for III-V semiconductors with low carrier density [31]. In addition, they observed anomalous behavior in temperature dependent longitudinal resistance. Li et al. [61] mentioned that one of the experimental evidence of CDW is appearance of the SdH oscillations in Hall resistivity (or resistance) then formation of the QHE plateau under magnetic field. Here, we observed suppressed oscillations in Hall resistance, as seen in Fig. 7c. When the monotonic background is eliminated from Hall resistivity, suppressed oscillation becomes pronounced, as shown in Fig. 7d. The oscillations were detected in all samples. However, we do not observe plateau formation because of the samples’ lower mobility and more ionized impurity scattering centre in our sample. The carrier density fluctuates in the CDW mechanism in samples, which makes it position-dependent but does not in the SDW mechanism [62]. In addition, it is shown that CDW and SDW mechanisms occur in low carrier density conditions at high magnetic field regimes [53, 63]. Because of these facts, we did not address the unique reason for two-dimensional characteristics and observed the lower carrier density in MR results because there is no appropriate theoretical model for degenerated semiconductors whose transport properties are more complicated to model analytically.

Fig. 7
figure 7

(a) Magnetic field-dependent electron density, (B) temperature-dependent resistivity, (C) Hall resistivity (𝜌𝑋𝑌), and (D) linear background removed Hall resistivity of the samples at ~ 4.3 K. The results are multiplied and shifted for clarity. The multiplication factors are given in legends

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The determination of the electron mass from the SdH oscillations in longitudinal resistivity is independence of the nature of the electron transport mechanism as bulk or 2DEG [26, 28, 64]. Hence, we can determine the electron mass from conventional analysis of SdH oscillations. Analytic evaluation of temperature dependence of the SdH oscillations under a constant quantum lifetime of the electron can be used to determine the electron effective mass. The temperature dependence of the SdH oscillation amplitude is given as [10, 47, 64]

$$\frac{A(T,{B}_{n})}{A\left({T}_{0},{B}_{n}\right)}=\frac{Tsinh\left(\frac{2{\pi }^{2}{k}_{B}{T}_{0}{m}_{e}^{*}}{\hslash e{B}_{n}}\right)}{{T}_{0} sinh\left(\frac{2{\pi }^{2}{k}_{B}T{m}_{e}^{*}}{\hslash e{B}_{n}}\right)}$$
(3)

where \({T}_{0}\), \(T\), \({m}_{e}^{*}\), and \({B}_{n}\) are the lower lattice temperature, the higher lattice temperature, electron effective mass, and magnetic field value at maxima, respectively. \(A({T}_{0},{B}_{n})\) and \(A(T,{B}_{n})\) are the SdH oscillation amplitude in the background removed results (Fig. 2b). Using experimental values in Fig. 2b, we have calculated the electron effective mass for all samples and tabulated them in Table 3. As seen in Table 3, the value of the electron effective mass increases with electron concentration. The lowest effective mass is determined for Bi-01 samples and the highest for Bi-70. The electron effective mass in Bi-free (As-15) is higher than that of the well-known value [50, 56, 65]. The origin of these results may be related to high carrier density. In order to clarify the effect of the free electron density on the electron effective mass, the position of the Fermi level must be found. The Fermi level can be found from the experimental period of the SdH oscillations. The period can be calculated using [40, 41, 47, 48],

$${\varDelta }_{i}\left(\frac{1}{{B}_{i}}\right)=\frac{e\hslash }{{m}_{e}^{*} ({E}_{F}-{E}_{C\left(i\right)})}$$
(4)

where \({E}_{F}-{E}_{C\left(i\right)}\) is the energy difference between the Fermi level and CB edge. The calculated Fermi levels are given in Table 3. As seen in Table 3, Fermi level energy increases with increasing the electron density. The location of the Fermi levels in the CB indicates the degeneracy of the samples. The effect of the high electron density on the effective mass can be described using electron energy in the CB and given as [66, 67]

$$\frac{1}{{m}_{e }^{*}\left(k\right)} =\frac{1}{{m}_{e}^{*}(k\approx 0)} \left(1-\alpha \frac{{\varDelta E}_{C}\left(k\right)}{{E}_{G}(k\approx 0)}\right)$$
(5)

where \(\alpha\) is the nonparabolicity parameter and is calculated by using InGaAs parameters. \({m}_{e}^{*}(k\approx 0)\) and \({E}_{G}(k\approx 0)\) are the band edge electron effective mass and bandgap energy, respectively. \(\varDelta {E}_{C}\left(k\right)\) is the electron wave vector-dependent bandgap energy, which is described as [66, 67]

$${\varDelta E}_{C}\left(k\right)=\frac{{\hslash }^{2}{k}^{2}}{2{m}_{0}}+\frac{1}{2}\left(\sqrt{{E}_{G}{\left(k\approx 0\right)}^{2}+4{E}_{P}\frac{{\hslash }^{2}{k}^{2}}{2{m}_{0}}}-{E}_{G}(k\approx 0)\right)$$
(6)

where, \({E}_{P}\) is the momentum matrix element of the alloy. \(k\) is the electron wave vector for longitudinal transport and can be found using \(\sqrt{2\pi {n}_{SdH}}\) relation. For the sake of simplicity, the thickness of the electron accumulated layers is taken as reciprocal thickness (\(d=2\pi /k\) ). The first and second terms in Eq. (6) correspond to band-filling and nonparabolicity effects. The optical bandgap,\({E}_{G}(k\approx 0)\), is determined from the lowest doped Bi-01 samples by in-plane photoconductivity measurements as 0.68±0.04 eV at 30 K (not shown here). Figure 8 shows the electron density versus effective mass results. As seen in Fig. 8, only the band-filling effect is not enough to explain the electron density dependency of the electron effective mass. It is clear that a better match can be obtained using bandgap nonparabolicity to determine electron density-dependent electron effective mass.

Fig. 8
figure 8

Electron density-dependent electron effective mass results. The open symbols and lines show the experimental and calculated results, respectively

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The extracted effective mass coefficient is 0.040 ± 0.003 for In0.48Ga0.52AsBi0.016 alloys, and the determined values match with well-known value of In0.48Ga0.52As alloys (here, host material) [36]. According to the VBAC model, the CB is not affected by incorporating Bi atoms into the host materials. The extracted effective mass value validates the prediction of the VBAC model. However, the sample with the lowest do** density, Bi-01, does not obey the calculated trend. The origin of this may be the observation of fewer SdH oscillations, which may give rise to errors in the analysis.

Electronic quantum transport parameters of electrons, quantum lifetime (\({\tau }_{q}\)), and quantum mobility (\({\mu }_{q}\)) can be found from the magnetic field dependence of the SdH oscillations at a given lattice temperature using,

$$\text{ln}\left(\frac{A\left(T,{B}_{n}\right){B}_{n}^{-\frac{1}{2}}{sinh}\left(i\chi \right)}{i\chi }\right)=C-\frac{\pi }{{\mu }_{q}}\frac{1}{{B}_{n}}$$
(7)

where C is the constant. \({\mu }_{q}\) is \(e{\tau }_{q}/{m}_{e}^{*}\). To calculate \({\tau }_{q}\) and \({\mu }_{q}\), the effective mass value is taken from Table 3. Figure 9 shows the Dingle plots for quantum Hall mobility (QHM). The QHM values are given in Table 3. As seen in Table 3, the QHM becomes lower with increasing do** density due to the increment of the density of the scattering center and is slightly lower than the HE mobility. The ratio \({\mu }_{Hall}\)/\({\mu }_{q}\) also gives information about the carrier scattering mechanism [68].

Fig. 9
figure 9

Dingle plots. The solid lines are ascribed for fitted results using Eq. 5. The symbols show the experimental results

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The electrons and dopant atoms are in the same space; the transport occurs in the channel where the donor atoms also locate. Hence, we expect the interaction mechanism between electrons and ionized dopant atoms to be similar in both HE and MR, even though the electron density in both HE and MR is not equal. The \({{\mu }_{Hall}/\mu }_{q}\) is slightly larger than the unit, which indicates that the electron-ionized donor atom interaction mechanism may be in a long-range scale [68, 69].

4 Conclusions

The electronic transport properties of n-type (Si-doped) InxGa1−xAs1 − yBiy alloys are investigated by conducting Hall effect and magnetoresistance measurements. We showed that all free electrons do not contribute to transport in MR even though the two-dimensional electrons are not formed at the interface, which is a substantial result for 3D semiconductors. We observed SdH oscillations in both the longitudinal and transverse resistivity of the samples. A quasi-two-dimensional electron gas characteristic in 3D InGaAsBi alloys is achieved under a high magnetic field regime, which may give new inspiration for metrology. The electron effective mass is not affected by the incorporation of the Bi atoms into the InGaAs host, which shows the validity of the VBAC model for InGaAsBi alloys. The high electron density effects on the electron effective mass are explained by nonparabolicity in the CB. High do** effects also dominate the characteristic of quantum transport under high magnetic fields.