Method of electron affinity evaluation for the type-2 InAs/InAs1−xSbx superlattice

The type-2 InAs/InAs1−xSbx superlattices on GaAs substrate with GaSb buffer layer were investigated by comparison of theoretical simulations and experimental data. The algorithm for selection of input parameters (binary and ternary materials) for simulations is presented. We proposed the method of the bandgap energy extraction of the absorption curve. The correct choice of the bulk materials and bowing parameters for the ternary alloys allows to reach good agreement of the experimental data and theoretical approach. One of the key achievements of this work was an electron affinity assessment for the device’s theoretical simulation. The detectivity of the long-/very long-wave InAs/InAs1−xSbx superlattice photoconductors at the level of ~ 8 × 109 cm Hz1/2/W (cutoff wavelength 12 µm) and ~ 9 × 108 cm Hz1/2/W (cutoff wavelength 18 µm) at a temperature 230 K confirmed the good quality of these materials.


Introduction
The InAs/GaSb and InAs/InAs 1-x Sb x superlattices (SLs) have been considered as an alternative to HgCdTe. These materials were first presented by Smith et al. in 1987 initiating the process of the HgCdTe use for infrared (IR) applications [1][2][3][4]. Recently, researchers have focused on the fact that InAs/InAs 1-x Sb x SLs are more flexible in the optimization of device performance. The type-2 superlattices (T2SLs) InAs/InAs 1-x Sb x are used for the fabrication of barrier detectors to include nBn and pBn design [5,6], low-noise interband cascade infrared photodetectors (ICIP) [7,8], and avalanche photodiodes (APD) [9], for wide range of applications in the field of science, medicine, technology, safety, industry, medical diagnostic imaging, night vision devices, and spectroscopy. Figure 1 shows the bandgap (E g ) dependence on the lattice constant (a) for bulk A 3 B 5 materials (InAs, InSb, GaSb, and GaAs) being used for SLs fabrication. In the case of the ternary materials, Vegard's law was used to determine lattice constant [10][11][12].
The GaSb has a similar lattice constant to InAs and InSb, while the GaAs exhibits a large lattice mismatch in comparison with the mentioned materials leading to the high strains. When growing a SL, it was important to reduce the tension between the SL constituting layers and the substrate on which the SL was deposited. That is why it is important to choose the buffer layer and minimal thickness (with reference to the thickness of the SL) in which the tension between the SL and the buffer layer was averaged. As a result, the proper buffer layer selection reduced the dislocations density having a great influence on the characteristics of the processed detectors. We decided to apply the interfacial misfit array (IMF) technique in order to get a defect-free buffer layer.
In this work, a numerical modeling by APSYS platform of medium wavelength infrared (MWIR) and long wavelength infrared (LWIR) T2SLs InAs/ InAs 1-x Sb x IR structure is presented. In the simulation, the strains created by GaSb buffer in InAs and InAs 1-x Sb x layers were considered. It was shown that the use of T2SLs InAs/InAs 1-x Sb x with the ''strainbalanced'' condition makes it possible to grow highquality SLs for IR applications.

T2SL InAs/InAs 12x Sb x structure
The InAs/InAsSb T2SLs were grown on semi-insulating GaAs (001) substrates with 2°offcut toward \110[ in RIBER Compact 21-DZ solid-source molecular beam epitaxy (MBE) system. The substrates were thermally deoxidized, and a 250-nmthick GaAs layer was deposited at 665°C in order to smooth the surface after deoxidization. Then, a 1.2lm-thick GaSb layer was grown using IMF technique to reduce the large lattice mismatch between GaAs substrate and the SLs. Finally, a 300-450 periods, non-intentionally doped InAs/InAs 1-x Sb x T2SL absorber layer, were deposited at 425°C. The growth processes have been monitored in situ by RHEED system. The T2SL InAs/InAs 1-x Sb x structure is presented in Fig. 2. The n describes the number of SL periods, and L is the thickness of SL components. Detailed description of the T2SLs growth procedure is given in [13].
High-resolution X-ray diffraction (PANalytical X'Pert) was utilized to assess the structural and crystallographic properties of the grown samples. The thickness of the several layers in SLs could be determined by the distance between the satellite peaks appearing on the radial scan. In order to know the antimony composition in InAsSb layer, the secondary ion mass spectrometry (SIMS) was used. The PL emission was analyzed using a Bruker Vertex v70 Fourier transform infrared (FTIR) spectrometer to determine the bandgap energy of the InAs/InAsSb T2SLs absorber. Table 1 presents the basic parameters of the grown samples.
All samples were checked for compliance with the ''strain-balanced'' condition [14,15]. This means that the strain occurring between the components of the SL (InAs and InAs 1-x Sb x ) and the buffer layers (GaSb) was small, not causing a large number of dislocations. Hence, the conclusion was made that these strains will not have a significant impact on the characteristics of the device.
The several expressions for designation of the compressive (L InAsSb ) and tensile (L InAs ) layer thickness have been used. An average lattice method arose from the notion that a ''strain-balanced'' structure could be reached by averaging of the compressive and tensile lattice parameters. This could be derived assuming identical elastic material properties (material elastic constants), and the balance equation is a form: where a i is the lattice constant of ''i'' materials.  The thickness of a ''strain-balanced'' structure for the samples presented in Table 1 was determined by the equation [14,16]: where L is the thickness of one SL period. The next approach suggested that the ''strain-balanced'' structure could be reached by assuming equal thickness for the tensile and compressive layers. It could be assumed that the structure is balanced if the following condition is met [15][16][17][18][19]: is the stresses related to the buffer layer. In order to express the strain, the following equation was used: Comparison of these two equations suggested that the discrepancy in the thickness of the InAsSb layer was less than 5%, not causing significant changes in the devices performance. The sample satisfaction of the condition ''strain-balanced'' is marked by in the Table 1.

Theoretical simulation of the SLs optical characteristics
The commercial APSYS platform allowed to study the SLs optical characteristics. The 4 bands method (kp 898) with periodic boundary was used, which is well described in the papers [20][21][22]. In addition, to obtain symmetric wave functions W e(h) , the Kane parameter, F = 0 was assumed [22]. The standard value of k = 0.06 was used in simulations with the exception for the determination of the effective masses, for which the smaller interval of the vector k = 0.03 was applied. Up to date, many different values of the parameters required for the band structure theoretical modeling have been used. The parameters implemented in our calculations are presented in ''Appendix'' in Table 3 [20,[23][24][25][26][27][28][29][30][31][32].
The strain could also influence the band structure of the material. In our calculations, the solid-theory model was used [23,24]. Figure 3 shows the SLs InAs/InAs 1-x Sb x band structure for unstrained and strained conditions. Electron affinity for unstrained materials InAs and InSb is presented in Table 3. Under the influence of strain, an electron affinity changed and the valence band splitted on the heavy and light holes subbands.
The set of dispersion curves E = f (k, q), allowing to estimate electron and hole effective masses (heavy and light holes) for the SL, was simulated. The effective masses were consistent with those presented in [21]. Figure 4 shows the absorption coefficient (a) for three samples: Sample-A; Sample-B, and Sample-D with selected absorber thicknesses and x Sb .  The presented ''strain-balanced'' structures, Sample-A; Sample-B; and Sample-D, were optimized for 12 lm, 18 lm, and 7 lm cutoff wavelengths, respectively. In comparison with the MWIR devices where a & 6000 cm -1 , the LWIR detector absorption coefficient was less than 1500 cm -1 . The sharp change in the absorption characteristics (e.g., Sample-D) was caused by two transitions: c 1 -hh 1 and c 1 -lh 1 . This was presented in more detail in [21].
The correct determination of the SL energy gap could be reached by a derivative of the absorption coefficient which was shown for Sample-C at T = 300 K in Fig. 5. In this case, a derivative of the absorption coefficient on energy (this graph is shown upper left) had to be calculated in order to determine the maximum value. Figure 6 represents a theoretical simulation of the absorption coefficients for Sample-C at selected temperatures 120 K, 180 K, 230 K, and 300 K. It can be seen that when temperature increased, the width of the bandgap decreased and the cutoff wavelength increased, correspondingly. Also, the maximum value of the absorption coefficient decreased with temperature.
In order to determine the T2SLs electron affinity v (position of the conductivity band E c ), the band structure of one of the SL samples with period L = 18.3 nm and selected thicknesses of the InAs and InAs 1-x Sb x layers were simulated. Since APSYS platform gives only the relevant values of E c , the simulated value was adjusted to that extracted from the linear relationship for the thickness L InAsSb of the layer InAs 1-x Sb x corresponding to the ''strain-balanced'' structure. Figure 7 shows the linear dependence of the SL electron affinity v (red line) versus the thickness of the InAsSb layer and the theoretical simulation (black curve) intersecting at the point L InAsSb = 3.95 nm corresponding to the balanced SL (see eq. 2). The different slope of these curves could be explained by additional strains between the layers when their thickness deviates from the thickness in the ''strainbalanced'' structure. For thicker InAsSb layers, the electron affinity was reduced, while for thinner ones it was higher than the linear averaging. Even though these changes were low, a 20% deviation from the ''strain-balanced'' thickness resulted in a reduction in electron affinity by 0.002 eV.

Results
In order to compare the simulated absorption curves (Sample-C at temperature T = 220 K) with the experimental data, two characteristics, photoluminescence (PL) and absorption coefficient, were compared  Figure 6 The simulated absorption coefficient versus temperature for Sample-C.
( Fig. 8). This figure illustrates the difference between the energy corresponding to 100% a cutoff and the energy corresponding to 50% a cutoff, being mostly determined. It is shown that energy corresponding to the 50% PL maximum correlated with the width of the bandgap (the definition was described above), but the maximum of the PL curve agreed with the half of the absorption coefficient value allowing to draw the conclusion that the bandgap energy extracted from PL should be determined very carefully.
We determined bandgap based on the calculated absorption curve and PL measurements using the method specified above. Figure 9 shows the temperature dependence of the energy gap for Sample-C. The green and blue lines correspond to the theoretical calculations, while the purple squares correspond to the experimental results (reading PL data from the maximum and from the 50% cutoff). It is worth mentioning that a proper coincidence between theory and experiment was reached by accepting the temperature dependence of almost all the necessary parameters for theoretical simulations. Table 2 presents a comparison of simulated and measured bandgap energy for the samples described in Table 1.
The detector's performance was fully confirmed by our calculations. Particularly interesting were the results for photoconductors made of Sample-B optimized for VLWIR range. After the growth, standard photolithography technique with wet-etching was used to define the active (75 9 85 lm) and contact areas of the IR photoconductors. The vacuum evaporation of Au/Ti electroplating was applied to fabricate ohmic contacts. At the end of the processing, the monolithic GaAs immersion lenses were formed to increase optical area of the detectors [13]. Figure 10 shows the relative responsivity characteristics (R i ) normalized to unity depending on the wavelength at the temperatures 210 K, 230 K, and 300 K. The R i temperature dependence in arbitrary   Comparing these data with the previous results (estimated convergence of the energy gap by photoluminescence measurement and theoretical simulations), we saw a fairly high agreement of the theory and experimental data. At 300 K, c-1h 1 transitions were more decisive than c-hh 1 , whereas at low temperatures the situation was reverse. Comparison of the simulated absorption coefficient (see Fig. 4, Sample-B at T = 230 K) and the experimentally measured R i (Fig. 10) gave a good bandgap match. Figure 11 presents the detectivity (D * ) of the two samples (Sample-A and Sample-B). It should be noted that the D * , as well as the absorption coefficient, depend on the spectral range of the detectors. At T = 230 K, D * reached * 8 9 10 9 cm Hz 1/2 W -1 (Sample-A) and * 9 9 10 8 cm Hz 1/2 W -1 (Sample-B) for V = -0.5 V. Those experimental data were similar to the one presented in [6].

Conclusions
This work presents the characterization of the T2SL absorber made of the InAs and InAs 1-x Sb x materials on GaAs substrate with GaSb buffer layer. In our simulations, two options were considered: when the SL layer thickness corresponded to the ''strain-balanced'' condition and when this requirement was not met. By the ''strain-balance,'' we meant to achieve zero average in-plane stress in the tensile/compressively strained layer of the SL and of the buffer layer, such that no shear force was generated at the interfaces between layers of the SL. By producing the SL structures that were ''strain-balanced'' to the buffer layer, we eliminated the possibility of the strain and dislocations occurrence suppressing the device performance. The experimental data's confirmed our theoretical simulations. The proposed method of determining the energy gap from the composition of the PL and absorption characteristic gave results consistent with the data from the responsivity detectors characteristic.

Acknowledgments
This work has been completed with the financial support of the Grant No. TECHMATSTRATEG1/ 347751/5/NCBR/2017.

Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of interest.
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Appendix: Choice of the SL-based materials parameters
In this appendix, material parameters of binary InAs, InSb, GaSb, and ternary InAs 1-x Sb x materials used in the theoretical simulation the type-2 superlattice are presented. We have tested these parameters more than once in theoretical modeling of superlattices. The obtained simulation results gave good agreement with the experimental data.
The approach for selecting the necessary parameters for InAs 1-x Sb x bulk material is presented in Table 3. In order to determine the InAs 1-x Sb x ''Y'' band parameter, the following equation was used: where bow Y was a bowing for Y parameter. In special cases, another non-linear interpolation was used as described in the table below. The description of the theoretical calculation of the conductivity band offset (CBO) was presented previously [20,[23][24][25][26][27][28][29][30][31][32][33][34]. The CBO value was defined as the difference between the energies of the subbands of the conduction band for the materials constituting the SL.
where E 0 c are presented in Table 3. a c is the conduction band deformation potential and DX=X ¼ xx þ yy þ zz . In our case, the stresses in the xx, yy, and zz directions were referenced to the GaSb buffer layer. For example, at the temperature T = 300 K the stresses for InAs/GaSb system were assumed by: where D 001 ¼ À2 Á C 12 =C 11 .
The bandgap for ternary InAs 1-x Sb x (1 [ x [ 0) was determined based on binary compounds InAs and InSb according with Eq. (5). The temperature dependencies for binary compounds InAs: E g ¼ E g0 À 2:76 Á 10 À4 Á T 2 ðTþ93Þ and InSb: E g ¼ E g0 À 3:20 Á 10 À4 Á T 2 ðTþ170Þ were used. It should be noted that, the bowing parameter, bow E g , was temperature dependent, too. For example, at the T = 0 K, the bowing parameter was 0.9486 eV, and at the T = 300 K, this parameter equaled 0.67 eV [35][36][37][38]. This dependence could be approximately represented by the linear equation. We wanted to point out the fact that the bowing parameter bow Eg was the sum of the bowing parameters for the electron affinity energy and the bowing parameters for the valence band energy.
The partition ratios of the band-edge discontinuities ''fractional band offsets'' Q c and Q v were important parameters for assessing the conductivity and valence band. Moreover, the fulfillment of the Q c ? Q v = 1 condition was important.
The temperature dependence of the electron mass and heavy/light hole mass in bulk materials was estimated by change in the energy gap with temperature [39]. The Table 3 shows masses at temperature T = 0 K.
In the case of SL optical properties simulation, the Luttinger coefficients (c 1 , c 2 , and c 3 ) were believed to be important parameters [12,21,22,30]. These coefficients were related to the effective masses of holes (light and heavy), bandgap, and spin-orbit energies. Figure 12 shows the theoretical calculations of the Luttinger coefficients versus the Sb molar composition (x Sb ).
Since we were interested in a specific x Sb = 0.25-0.45, the whole range of dependence (from 0 to 1) was not presented. The results shown in Fig. 12 directly indicated that in this range, Luttinger parameters were nearly linear. Such assumption of the linear dependence of the Luttinger coefficients was not allowed when calculating ternary InAs 1-x Sb x in the whole range 0 \ x Sb \ 1. In order improve the readability of Fig. 12, the c 1 coefficient was presented in relation to the left, while c 2 , c 3 to the right axis. The difference between the coefficients c 2 and c 3 did not exceed one.  Figure 12 The Luttinger parameters versus the Sb molar composition.