Long wavelength Infrared Detection, Bands Structure and effective mass in InAs/GaSb Nanostructure Superlattice

. We have investigated in the bands structure and the effective mass, respectively, along the growth axis and in the plane of InAs (d 1 =48.5Å)/GaSb(d 2 =21.5Å) type II superlattice (SL), performed in the envelop function formalism. We studied the semiconductor to semimetal transition and the evolutions of the optical band gap, E g (Γ), as a function of d 1 , the valence band offset Λ and the temperature. In the range of 4.2–300 K, the corresponding cutoff wavelength ranging from 7.9 to 12.6 µm, which demonstrates that this sample can be used as a long wavelength infrared detector. The position of the Fermi level, E F = 512 meV, and the computed density of state indicates that this sample is a quasi-two-dimensional system and exhibits n type conductivity. Further, we calculated the transport scattering time and the velocity of electrons on the Fermi surface. These results were compared and discussed with the available data in the literature.


Introduction
The atmosphere absorbs weakly in the near and the long wave infrared (LWIR) wavelength range, which offer a high potential for long-range observation sensors. The detection of infrared radiations is very important for a wide variety of activities. LWIR photodetectors have a several specialized applications [1]. For instance, in astronomy to detect celestial objects, for night vision, in the military missile tracking, environmental sensing, in medical diagnosis, in meteorology and climatology, and, can be used also in remote sensing systems [2].
Despite the mercury cadmium telluride (MCT) consider as the most widely used infrared detector, it is not a perfect material system, due to the intrinsic shortcomings such as the poor uniformity of large area [3], the difficulty of alloy component adjustment, and contains toxic elements. Recently, the InAs/GaSb superlattice with a type-II band gap alignment, is a promising alternative to MCT technology in the LWIR detection in view of its various advantages [4], such as the adjustable band gap energy [5][6], the low Auger recombination process (due to the separation of heavy hole hhi and light hole lhi bands) [7], the directbandgap absorption, high absorption coefficient, high carrier lifetime and the greater stability due to strong bonds in the III-V materials [8], the large electron effective mass, in addition to the convenience of material growth and device preparation process [9]. The dark current density is an important indicator for measuring the performance of infrared detectors. This indicator can be reduced by proper the band engineering. Indeed, the fundamental band gap of InAs/GaSb superlattice can be designed with varying the thickness of the constituent's layers for strong broad band absorption.
The study of such semiconductor heterostructues properties requires the investigation of bands structure that describes all the subband behavior [10]. The commonly used methods is the k•p method, that has been successfully used to explain many physical phenomena. It is more efficient to provide the electronic structure [11].
In this paper, we have focused on the investigation of the electronic bands structure properties, energy subbands and carrier's effective mass in the growth direction and in-plan of InAs(d1=48.5)/GaSb(d2=21.5Å), based on the envelope function formalism of k.p theory and the effective mass approximation [12]. We have discussed the effect of the well width on the band gap energy of SL, the carrier effective masses and the effect of temperature on the optoelectronic parameters. In addition to the density of state and Fermi level energy calculations.

Computational details
The general expression of dispersion relation of type-II SLs for light particles (electrons and light holes) subbands is given by [13][14]: With kz and kp(kx,ky) the wave vector in the growth direction and in-plane of the superlattice respectively. We chose the origin of the energy E at the top of InAs valence band as shown in the Fig.1.
For a given energy, in each host material, the Kane model [15] gives the wave vector (ki 2 +kp 2 ). Then, the energy E of the light particles (electron and light hole) is given by: The expression of  and r in the Eq. (1) are: From the same equation (1), the type II superlattice heavy holes' mini-bands can be calculated with the following relations: The valence band offset that we have adopted was 510 meV, that we calculated using the model given by [16]. P is the Kane matrix element whose values are 1.36 10 6 J/kg for InAs and 1.41 10 6 J/kg for GaSb [17]. Using these values of Λ, P and the input basic parameters of bulk materials the gap energy i reported by [18] and the effective mass of heavy holes m * HH (m0) for each material InAs(0.41m0) and GaSb(0.40m0) given by [19], we have solved the general dispersion relation (Eq.1) in order to compute the bands structure for a fixed ratio R= d1/d2=2.26 of this SL at 300 K.

Results and discussion
We calculated the effect of InAs well's layer thickness, d1, on subbands energies of light particles and heavy holes at the center (kz= 0) and the limit (kz= π/d) of the first Brillouin zone (FBZ) as shown in Fig. 2. As we see, when d1 increases, Ei decreases whereas E(HHi) and E(lhi) increases. Thus, the optical band gap energy Eg()= E1()-HH1() decreases until the crossover between E1 and HH1 that take place at the transition point Tc(dc= 81 Å; Ec = 462 meV) where Eg=0meV.
The conductivity of the system is converted from semiconductor (SC with Eg0) to Semimetal (SM with Eg small and negative). The electrons are moving from the top of GaSb into the bottom of the InAs [20]. Such charge transfer yields a rather high mobility in the  semimetallic samples since the electrons do not depend too much at low temperature of impurity scattering. This transfer process is similar to the well-known modulation doping in GaAs-GaA1As SL [21]. The semimetallic character and the semiconductor-semimetal transition of InAs/GaSb SL was studied experimentally well by transport and far infrared magneto-optical measurements by guldner et al. [22]. We used the measured electron carrier density n as a function of the inverse of temperature by haugan et al. [23], we determine the experimental band gap in the intrinsic domain by plotting nT -3/2 as a function of 1000/T. We found Eg= 95 meV, which is in agreement with 98 meV calculated at 300 K.
We illustrate in Fig. 3, the band gap Eg as a function of band offset. The calculated Eg decreases as the valence band offset Ʌ increases. Eg decreases to zero at the transition semiconductor-semimetal and becomes negative accusing a semimetallic conduction beyond Λc = 635.35 meV.
In Fig. 4, we plotted the evolution of coordinates of transition point for various temperature. The transition goes to a higher d1c when the temperature decreases.
The effect of temperature on the band gap is presented in Fig. 5. At very low temperatures, the band gap is constant. Then, it decreases as the temperature increase from 188.6 meV at 4.2 K to 98 meV at 300 K. Such behavior is due to the increased thermal energy and expansion of lattice constant which increase the amplitude of atomic vibrations and the interatomic spacing increase, then the potential seen by the electrons decreases, leading to a smaller band gap [24].
In order to determine the type of infrared detector of this sample, we calculated the cutoff wavelength using this formula: As shown in Fig. 5, in the investigated temperature range of 1.2-300 K the corresponding cutoff wavelength is 7.9 ≤ λc(m ) ≤ 12.6. Thus, this system can be employed for long wavelength infrared detection. Even, InAs/InxGa1-xSb SLs have been examined for infrared detector applications in the l -19 (m) wavelength range for very long-wave infrared detection [7].
In Fig 6, we show the calculated SL bands structures along the growth direction kz and in plane kp. We observe the dispersion curves of E1 and E2 bands along kz, with a width |Ei(kz=π /d)-Ei(kz=0)| of E1 and E2 of 138 meV and 198 meV respectively. The presence of these dispersion indicates a three dimensional electrons gas.
We calculated the carrier's effective masses using the dispersion of energy curves (Fig.  6), turns out to be a tensor with nine components, and its matrix elements along i and j directions given by Kittel   Fig. 7. Calculated carrier's effective mass along the growth direction and in plan of the SL.
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 [27]: The generalization of Eq. (6) required a sum over all mini-bands: In the absence of magnetic field, we calculated DOS of the two lowest conduction bands Ei and the first valence subband HH1 and lh1 at 300 K. As seen in Figure 8, the DOS is quantized in term of m*/πћ 2 d. The observed dispersion in Ei indicates the strong interaction between InAs wells.   (8) where kF 2D and kF 3D are the two dimensional and three dimensional Fermi wave vector respectively and n is the measured concentration of electron charge in the n(T) type sample [23]. The effective mass of electrons at the Fermi wave vector kF is m*E1(kF)=0.0284 m0.
Using the formula 8, we calculate the Fermi level EF, at the center  of the first Brillouin zone, where the bands are parabolic as seen in Figure 6. In Figure 8 EF is on the large E1 band which demonstrates n type and quasi-bidimensional (Q2D) conduction behavior. Although, the full first valence subband HH1 has a very narrow band width indicating 2D-like behavior.
In Figure 9, we plotted the energy bands of light (lh1), heavy (HH1) holes, and E1 electrons as a function of temperature which permits us to precise the dimensionality of system. The E(HH1) and E(lh1) still almost constant. Whereas The Fermi level EF(3D) decreases with E1 when T increases. This indicates a three-dimensional electron gas behavior. While the energy of Fermi level EF(2D) is constant.
For a T<Tc=42 K, the electron gas behavior is Q2D while for T  Tc it's converted to 3D. We assist to an electronic conductivity transition Q2D-3D. Using the Hall mobility measured by [23] and our calculated effective mass of electrons m*E1, we calculated the transport scattering time p=H m*E1/e, we found 0.27 ps at 300 K. This value of p can be compared to the quantum relaxation time q= 0.5 ps and p=2.34 ps measured in AlGaN/GaN twodimensional electron gas [28]. We calculated also the velocity of electrons on the Fermi surface vF = ћkF/ m * EF, it is about 3.97 10 4 m/s. This is a clear demonstration of faster collisions between electrons and phonon and high transport performances of this SL at high temperature.