Correlation Between Bands Structure and Quantum Magneto Transport Properties in InAs/GaxIn1−xSb Type II Superlattice for Infrared Detection

We have used the envelope function formalism to investigate the bands structure of LWIR type II SL InAs (d1 = 2. 18d2)/In0.25Ga0.75Sb(d2 = 21.5 Å). Thus, we extracted optical and transport parameters as the band gap, cut off wavelength, carriers effective mass, Fermi level and the density of state. Our results show that the higher optical cut-off wavelength can be achieved with smaller layer thicknesses. The semiconductor-semi metal transition was studied as a function of temperature. Our results permit us the interpretations of Hall and Shubnikov-de Haas effects. These results are in agreement with experimental results in literature and a guide for engineering infrared detectors.


INTRODUCTION
There are a several efforts in developing infrared materials for Very Long Wavelength Infrared (VLWIR) photodetection with the optical cut-off wavelengths λ c ≥ 7.5 µm in the atmospheric window. Currently, InAs/In 1−x Ga x Sb type-II superlattices (SLs) are still the most promising candidate. In this area, Smith and Mailhiot [1] have shown that this material system provides distinctive advantages suitable for VLWIR photodetection.
The key lies in the fact that by increasing indium composition in In 1−x Ga x Sb layer, the corresponding lattice constant increases. As a result, the tension in the InAs layer lowers the conduction subbands (E i ) whereas the compression in the In 1−x Ga x Sb layer raises the heavy hole subband (HH i ). Therefore, a very smaller band gap can be optimized with a reasonably thinner period which is not possible in InAs/GaSb SL [2]. The most important advantage of this SL lies in the fact that the presence of strain within this SL layers reduces the Auger recombination process and improves significantly the carrier lifetime and detectivity [3]. We note that in the previous years, a very few works were reported to verify the theoretical performance of InAs/InGaSb SLs for LWIR and VLWIR photodetection.
The applications of this technology are in military, automobile, aircraft safety, medical diagnosis, and in remote sensing systems [4].
The theory of band structure is necessary for describing the effects of external excitations on transport parameters and properties of optoelectronics devices [5]. Owing to this lack of studies, we have investigated the bands structure of presented LWIR type II SL InAs (d 1 = 47 Å)/Ga 0.75 In 0.25 Sb(d 2 = 21.5 Å) with d 1 /d 2 = 2.186 along the wave vector k z in the growth direction and k p (k x ,k y ) in-plane of the superlattice. We extracted useful electro-optical parameters like the optical cutoff wavelength, the effective mass of electrons and holes and their evolution with temperature. We found the point of semiconductor (SC) to Semimetal (SM) transition. We calculated the density of states without and with applied magnetic field and deduced the dimensionality and the limit of the quantification of the energy of electron gas.

THEORY AND BAND STRUCTURE
The general expression of dispersion relation of type-II SLs, within the envelope function formalism and effective mass approximation, is given by [6][7][8][9]: In each host material, we used the six bands Kane Hamiltonian (6 * 6 matrix) describing the k.p interaction of Ŵ 6 doublet [ S, m s J = ± 1/2 for electrons) and Ŵ 8 quadruplet ( P, m p J = ± 1/2 for light holes and P, m p J = ± 3/2 for heavy holes]. The spin-orbit coupling is too large, compared to the hosts gaps ε i , so the Ŵ 7 doublet is neglected. With the help of the Bloch condition, the continuity of the envelope functions and their derivatives at the interfaces we obtain a system with solution Equation (1). The inferior limit of the validity of this model is d 1 < 10 Å and d 2 < 10 Å as indicated by Bastard [6], Andlauer and Vog [7], Masur et al. [8]. Here k z and k p (k x ,k y ) are the wave vector in the growth direction and in-plane of the superlattice. The origin of the energy E has been chosen at the top of InAs valence band as shown in the Figure 1. In each host FIGURE 1 | Energy bands alignment in InAs/In 1−x Ga x Sb SL along the growth axis z. (E c ) i , (E v ) i , and ǫ i where i = 1 and 2 are conduction and valence band edges and band gap of bulks, respectively. E v = ∧ is the valence band offset. material and for a given energy, the two-band Kane model [10] gives the wave vector (k 2 i + k 2 p ). Therefore, the energy E of the light particles (electron and light hole) is given by: The expression of ξ and r in the Equation (1) are: From the same Equation (1), the superlattice heavy holes minibands can be calculated with the following relations: The direct band gaps parameters ε i (i = 1 and 2) used here, as a function of temperature, are given by the empirical expressions reported in Matossi and Stern [11] and Roth et al. [12]. P i is the Kane matrix element. The heavy holes effective masses of InAs and In 0.25 Ga 0.75 Sb are, respectively (m * hh ) 1 = 0.41 m 0 and (m * hh ) 2 = 0.42 m 0 , as given in Levinshtein et al. [13]. Here m 0 is the free electron mass.
In this work, we have used the valence band offset of 510 meV between heavy holes sub-bands edges of InAs and In 0.25 Ga 0.75 Sb [14]. The origin of energy is taken on the top of VB in InAs (Figure 1).
The computation of band structures consists of solving the general dispersion relation with the reported parameters. In the Equation (1), state energy exists if the right-hand side lies in the range −1 to 1. In other words, −1≤ cos (k z d) ≤1 which implies -π/d ≤ k z ≤ π/d in the first Brillouin zone, with d = d 1 +d 2 is the superlattice period. In the study of energy E as a function of wave vector in the direction of growth k z (k p = 0), the solving procedure consists of going with small steps of energy through the studied range. Then finding, for a given E, the value of k z which is root of the dispersion relations. The same procedure is used for studying E vs. the in-plane wave vector k p (k z = 0 and k z = π/d).

RESULTS AND DISCUSSIONS Bands Structures, Band Gap, Cut Off Wavelength, and Carriers Effective Mass
In the Figure 2, we have plotted the energy of conduction (E i ), light-hole (h i ), and heavy-hole (HH i ) subbands at the center Ŵ(k z = 0) and the limit (k z = π/d) of the first Brillouin zone (FBZ) as a function of d 1 = 2.18 d 2 at 1.2 K. These results show that when d 1 increases, the width of E i and HH i deceases. In addition, we can notice that the fundamental band gap energy E g (Ŵ) = E 1 (Ŵ)-HH 1 (Ŵ) decreases when d 1 increases. Therefore, a crossover between E 1 and HH 1 takes place at the semiconductor (SC) to Semimetal (SM) transition point T c (d 1c = 115.43 Å, E 1c = HH 1c = 485.25 meV).
In Figure 3, at a given temperature the E g decreases when the valence band offset ∧, between the top of the heavy hole bands in the two host materials, increases. The transition SC-SM occurs when E g goes to zero.
The evolution of the cut off wavelength |λ c | as a function of d 2 is shown in Figure 4. These results were calculated by using the expression: The dependencies of |λ c (Ŵ)| on d 2 and temperature (T) are shown in Figure 4. The figure divides itself into two regimes: (i)    1 and the SL shows an inverted bands structure with a SM conductivity. In this regime a reverse behavior was observed. The inset figure shows the sensitivity of E g on T. This result shows that when T increases E g shifts to lower energies with 0.21 meV/K. This is due to the fact that the thermal energy and electronphonon interaction increase with T which lead to a reduction of the potential seen by electrons, leading to a reduced E g . In the investigated temperature range, the corresponding 7.46 ≤ λ c (µm) ≤ 11.94 situates this sample in the LWIR region. The SL bands structure shown in Figure 5 was computed at 1.2 K along k z and k p the growth and in-plan directions, respectively. Along k z , the widths of E 1 and E 2 ,   were found to be 129 and 193 meV, respectively. The presence of such dispersion indicates three dimensional like behavior of electrons which is in agreement with the reported results of Mitchel et al. [15]. The energy as a function of k 2 z and k 2 p at 1.2 K shows that: along k z , the bands E i and h i of the light particles are non-parabolic and the heavy holes bands HH i are parabolic; but along k p , E 1 and HH i are parabolic but h 1 is non-parabolic. The theoretical effective mass in general turns out to be a tensor with nine components defined as [16]: From Figure 5 we calculated carrier's effective mass of light particles: electrons E 1 , light holes h 1 and heavy holes HH 1 , in the direction of growth (k z ) and in plane (k p ) of this sample.

Interpretations of Hall Effect, the Density of States, and the Dimensionality of Carrier's Charges
We used the carrier density n as a function of the inverse of temperature measured in Mitchel et al. [18] and Haugan et al. [19]. In order to determine the band gap with accuracy, we plotted nT −3/2 as a function of 1,000/T. This lead to an experimental band gap of E g = (95 ± 8) meV in the intrinsic domain, which is in agreement with our 104 meV calculated at 300 K. The observed difference of 8.87 meV is due to the reduction of the gap by strain [1] and the input parameters considered here as well as fluctuations during the growth of SL which could affect E g . Note that Szmulowicz [20] and Szmulowicz et al. [21,22] [19] vs. temperature in logarithmic scales. This shows that when T increases, µ decreases as T −1.5 indicating the interaction of electrons with phonons in the intrinsic regime. This result confirms a tridimensional electron gas.
For the superlattice i th mini-band, with energy width E(i) = E(i) max -E(i) min , the density of states (DOS) can be expressed as [23]: The generalization of Equation (6) required a sum over all mini-bands: From the dispersion curves E(k z ) in Figure 5, we have calculated the SL density of states (DOS) of the two lowest conduction minibands E i and the first valence subband HH 1 at 1.2 K. As seen in Figure 7, the DOS is quantized in term of m * /πh 2 d, with the presence of large dispersion in E i . This shows a significant interaction between InAs wells justified by the ratio d 1 = 2.18 d 2 in this sample. At given temperature the Fermi level energy was obtained by the formula: Frontiers in Physics | www.frontiersin.org where k 2D F and k 3D F are the two dimensional and three dimensional Fermi wave vector, respectively and n is the measured concentration of carrier charge in the n(T) type sample [18,19]. The effective masse of electrons at the Fermi wave vector k F is m * E1 (k F ) = 0.0887 m 0 from Figure 6. The formula (8) permit as the determination of E F , at the center Ŵ of the first Brillouin zone, where all the bands are parabolic as seen in Figure 5.
In Figure 7, the calculated Fermi level energy E F is on the large E 1 band indicating an n type (in agreement with [18,19]) and quasi-bidimensional (Q2D) conduction behavior. Whereas, the full first valence subband HH 1 has a very narrow band width indicating 2D-like behavior. The observation of Mitchel et al. [15] indicated a tridimensional conduction (3D).
In order to precise the dimensionality of carriers charges we plot, Figure 8 which shows the energy bands of light (h 1 ), heavy (HH 1 ) holes, and E 1 electrons as a function of temperature. As can be seen, the band gap E 1 -HH 1 decreases as the temperature increases. The Fermi level E F (3D) decreases with E 1 when T increases. This is the behavior of a three-dimensional electron gas. For a two-dimensional electron gas the energy of Fermi level E F (2D) is constant i.e., independent of temperature as seen on the top of Figure 8. One can argue that the electron gas is Q2D up to T c = 40 K and 3D for T ≥ T c . We are witnessing a Q2D to 3D transition. In the following, we will consider that the electron gas is three-dimensional at temperature domain under consideration.

Interpretation of the Shubnikov-de Haas Effect and the Density of States With Magnetic Field
We have calculated the Fermi level in this SL by using the magnetoresistance results reported in Mitchel et al. [15].
It is well-known that the oscillations of longitudinal magnetoresistance at very low temperatures and under high  magnetic fields, due to the Shubnikov-de Haas effect (SdH), are periodic with respect to 1/B [24].
The inverse of the minima 1/B m as a function of the entire N, Landau level index, is given by the formula: The straight line slope of (9) gives the period, (1/B) = 1.16 T −1 .
The relation between (1/B) and the electron concentration n 3D for a 3D system is given by: So at T = 1.2 K the SL electron density is n 3D = 4.53·10 15 cm −3 .
The later permits us to estimate the Fermi energy by: Here, m * E1 (k F ) = 0.09 m 0 is the electron effective mass at the Fermi wave vector, extracted from Figure 6. As seen in Figures 5,  7, E F = 576.87 meV and the SL has an n type conductivity at 1.2 K.
In their work of InAs/InGaSb SL magnetotransport properties properties, Levinshtein et al. [13] have shown that such SL shows Shubnikov-de Haas (SdH) effect. In the Figure 9 we show the calculated Landau levels (LL) energies of E 1 in the investigated SL. These results are obtained by transposing the quantification rule [25] of the wave vector in the plane of the SL in E(k p ) plotted in Figure 5: with N (i) the Landau index of i th mini-band with N SL = 73 is the SL number of periods. The Hall resistance has been calculated with the formulate: R xy ( ) = h/νN SL e 2 and ν is the filling factor [26]. In the Figure 9, when the magnetic field increases, the crossover between the calculated LL energy and the Fermi level gives the same minima (at B = 5.71 T, 4.13 T, 3.40 T, 2.73 T and 2.34 T), of the Shubnikov-de Haas effect (SDH) oscillations of the transverse magneto-resistance measured by Mitchel et al. [15].
The Figure 10 represents the computed 3D density of states (DOS) of the first conduction mini-band of the investigated SL by using the following expression [24]: As can be seen in the figure the DOS is inversely proportional to the square root of total energy and the spacing between LL increases with the applied magnetic field.
A condition for the observation of the SDH effect is that the separationhω c (ω c = eB/m * E1 is the cyclotron pulsation) between LL must be superior to activation thermal energy k B T. So here the magnetic field must be B > k B m * E1 T/he = 0.9 T for T = 1.2 K and m * E1 = 0.09 m 0 . When B < 0.9 T, the LLs disappear and E 1 is a continuum. This condition of quantum limit agrees well with our calculated DOS (E 1 , B) in Figure 10.
We also calculated the transport scattering time τ p = µ H m * E1 /e = 0.52 ps for Hall mobility µ H = 1.02·10 4 cm 2 /Vs measured at 1.2 K by Mitchel et al. [18] and Haugan et al. [19] and our m * E1 = 0.09 m 0 . This value of τ p can be compared to the quantum relaxation time τ q = 0.5 ps and τ p = 2.34 ps measured in AlGaN/GaN two-dimensional electron gas [27].
Fuchs et al. [28] reported for the SL1 InAs(d 1 = 34.8Å)/Ga 0.8 In 0.2 Sb(d 2 = 29Å) with d 1 /d 2 = 1.2, 8 µm ≤ λ c ≤ 12 µm corresponding to 103 meV ≤ E g ≤ 155 meV from electroluminescence for 10 K ≤ T ≤ 240 K. In the studied sample SL2 InAs(d 1 = 47Å)/Ga 0.75 In 0.25 Sb(d 2 = 21.5Å) with d 1 /d 2 = 2.19, the insert of Figure 4 shows λ c (T) with 7.46 ≤ λ c (µm) ≤ 10.5 and 118 meV ≤ E g ≤ 166 meV for the same investigated temperature range. In particular at low temperature, E g (77K, SL1) = 150 meV from measured dynamical impedance increases to Eg(77K, SL2) = 157 meV in agreement with the fact that E g increases when d 1 /d 2 increases at a given temperature as seen in Figure 11. In the SL1, an electron effective mass of 0.03 m 0 for transport perpendicular to the SL plane was used in the band-to-band tunneling model. This effective mass is in agreement with our 0.026 m o ≤ m * E1 (k z ) ≤ 0.028 m o for 1.2 K ≤ T ≤ 300 K near the center Ŵ of the first Brillion zone in Figure 6. All these transport parameters are in the same order of transport characteristics of the investigated SL.

CONCLUSION
Using the envelope function formalism, we have investigated the electronic and optical properties of LWIR InAs(d 1 = 2.18 d 2 )/In 0.25 Ga 0.75 Sb (d 2 = 21.5 Å) type II SL. The dependencies of the band gap and the corresponding cut-off wavelength on layers' thicknesses and temperature were studied and interpreted. Our results show that a very smaller band gap can be achieved with reasonably thinner layers. The SC-SM transition was found to go to lower d 1c and/or d 2c when the temperature increases. Transport properties of this sample indicate a tridimensional conduction behavior. Whereas, the first valence subband HH 1 shows 2D-like behavior. We interpreted the Shubnikov-de Haas effect and the density of states without and as a function of magnetic field. The condition of quantum limit agrees well with our calculations. In the intrinsic regime, our results agree well with experimental observations of W. C. Mitchel et al. These results are a guide for engineering infrared detectors.

DATA AVAILABILITY STATEMENT
All datasets generated for this study are included in the article/supplementary material.

AUTHOR CONTRIBUTIONS
NB: corresponding author doing the most of the article. AN: the supervisor is correcting the article. DB: help in the band gap calculation. AB, MB, SM, E-SE-S, and FC: help in calculation of masses effective and magnetotraport. All authors have contributed to this work and approved it for publication.