A designing principle for low dark-current strained layer superlattices

For over three decades, there has been a constant search for alternative materials and designs to replace HgCdTe in photodiodes.^1–4 The type II superlattice (T2SL) in which the valence band of one of the constituents lies above or near the conduction band of the other constituent is an exciting possibility. The associated flat bands, photon recycling, and high-quality molecular-beam-epitaxy growth may, respectively, suppress the Auger, radiative, and Shockley-Read-Hall (SRH) recombinations and promise a performance far superior to HgCdTe-based photodiodes.⁵ However, it is now generally accepted that lifetimes in type II strained layer superlattice (SLS) systems—such as the InAs-GaSb system—are limited by SRH mechanisms.⁶ While the lifetimes in Ga-free SLS mid-wave infrared (MWIR) and long-wave infrared (LWIR) wavelength regions are a few μs⁷ to a few hundred ns,⁸ the lifetimes in InAs-GaSb SLS are still only a few tens of ns.^9,10 Although it is believed that the native point defect (NPD) of GaSb origin is responsible for the short lifetimes, very little is known regarding the type of NPD and its location in the band gap. A better understanding could enable the defect mitigation and thus a better performance.

There has been a considerable effort to understand the defect levels and their origins in the SLS system from empirical Hamiltonian approaches^11,12 or by systematically analyzing the measured lifetime values.^13–16 Although there are extensive first principles studies of defect levels in bulk solids, very few studies¹⁷ are being carried out for SLS systems, owing to computational complexities arising from much larger supercells and the inaccurate band gap. Unless the Green's function (GF) approach is used, the calculations would be limited to a high concentration of NPDs due to the size of the supercell. The existing GF-based calculations typically use empirical Hamiltonians but with an inaccurate defect potentials.^11,18,19

In this letter, we employ a first principles Hamiltonian for reliable defect potentials, a long-range tight-binding Hamiltonian for accurate band gaps, and a GF approach for studying the isolated NPD in bulk materials and ten InAs-GaSb SLS systems in both the MWIR and LWIR spectral regions. In the SLS systems, we consider the defects at the interface region and in the bulk region (away from the interface). We observe that the energy levels of NPDs in the bulk region of the SLS share the same origin with the corresponding NPDs in the respective bulk constituents. In addition to a good agreement between our calculations and measurements¹⁵ in a SLS, we show the possibility of designing the SLS system with fewer defect states in the gap by shifting the SLS mini-bands to an energy region in which the bulk defect states are not present. This design principle narrows the choice of SLS structures to be grown without an extensive computation.

We use the GF formalism to study the states from isolated defects. The GF for the lattice with defects, G^m is given by²⁰ 

where G^0k is the GF of the unperturbed lattice with Hamiltonian H⁰ and overlap matrix S⁰. ΔV is the defect potential, and Z is the complex energy at which the GF is evaluated. The eigenstate of the Hamiltonian with the defect is given by Z for which the (G^m)⁻¹ is zero. We see from Eq. (1) that the defect-induced resonance levels, whether they are located in the gap or in the bands, are uniquely given by the roots of M.

As described in detail in a recent publication,²⁰ we calculate H⁰ using long-range, but sp³ local orbitals-based, hybrid pseudopotential tight binding (HPTB) Hamiltonian²¹ and ΔV from first principles SIESTA²² also with sp³ local basis. This approach has been used²⁰ to study the defects in bulk materials such as InAs, InSb, GaAs, and GaSb. The averaging of interatomic interactions across the interface is chosen so that HPTB Hamiltonian consistently produces band gap at 77 K for a number of SLS systems in fairly good agreement with measurements^23–25 and other calculations.^26,27 We now further expand this approach to study the defects in SLS systems, and the results are discussed here. In our calculations, the effect of temperature appears only through the band gap of the underlying Hamiltonian.

We consider Type II SLS systems lattice-matched to GaSb with each unit cell of the superlattice containing n monolayer (ML) of InAs and m ML of GaSb. Each ML contains one anion and one cation layer, and the SLS growth direction is along the (100). The in-plane lattice constant of SLS is matched to that of GaSb. For the calculation of band structures, the lattice constant in the growth direction is modified by the ratio of the elastic constants −2C₁₂/C₁₁. All term values in the TB Hamiltonian of GaSb and InAs are rigidly shifted so that the valance band edge (E_v) of bulk GaSb and InAs are, respectively, −0.03 eV and −0.60 eV.²⁸ Since the same term values are used in the construction of SLS Hamiltonian, the conduction band edge (E_c) and E_v can be obtained in the common energy scale for all designs by diagonalizing the Hamiltonian at Brillouin zone center. In the defect potential calculations, the atoms are allowed to relax around the defect in the growth direction. The potential change due to defect is found to die off nearly completely at the 2nd neighbor shell. Hence, all local matrices used in our calculations—G, ΔV, S, ΔS—are 136 × 136 in size. (17 atoms each with four orbitals per spin.) We studied 10 SLS systems with band gaps ranging from MWIR to LWIR spectral regions. For each SLS, we calculated the defect energy levels in the gap for eight NPDs—cation vacancy (V_Ga, V_In), anion vacancy (V_Sb, V_As), cation antisite (Ga_Sb, In_As), and anion antisite (Sb_Ga, As_In)—in the bulk region (near the middle of InAs and GaSb region). For a few selected SLS structures, the energy levels of NPDs in the 1st ML at the interface are also calculated. The SLS structures, the band gap (E_g), E_v, E_c, and defect energy (E_t) levels for various SLS systems are summarized in Table I. All energies are in meV and the E_t values are measured from the valence band edge of the SLS miniband. There is paucity of experimental data on InAs-GaSb SLS for comparison. The only detailed analysis based on photoluminescence (PL) measurements for the InAs-GaSb SLSs reported¹⁵ in literature are on the 10/10 structure, where a defect level at 140 meV above the valence band edge of the SLS miniband is identified. Our calculation (Table I) for this structure predicts a defect level, arising from V_As, at 133 meV above the SLS valence band, in a very good agreement with this measurement.

A few observations can be made from Table I. First, except for 8/8 SLS, most of the defect states have origins in GaSb. This is consistent with the measurements that considerably increased lifetimes can be achieved in Ga-free SLS systems.⁸ For the 8/8 SLS, there are many defect states associated with InAs as well. Second, contrary to conventional conclusions, we find that the number of defect states decreases with the band gap. Third, more defect states are created in the gap when the defects move from the bulk region to the interface. Fourth, as the defects are moved from the bulk region to interface, the defect states move closer to the mid gap. The mid-gap states are the most damaging SRH centers and could lead to high dark currents. Finally, of the designs considered, the 17/8 design is predicted to have no lifetime-killing mid gap states, for defects locating both in the bulk and at the interface.

To understand these observations, we compared the change in defect potential, ΔV, calculated at the defect and the neighbor sites in various SLS systems with that in the constituent bulk material. We notice that ΔV changes very little from that in the bulk constituent material, and fluctuates weakly among the different SLS structures. Noting that the roots of M (of Eq. (2)) depend on both ΔV and G^0 k, we calculate the defects in bulk constituent materials to explore their correlation to the corresponding defect states in the SLS. Since the GF approach can be used to calculate the resonant defect states even within the bands, we obtain all defect states in both InAs and GaSb in the energy region between −0.6 eV (valence band edge of InAs) and 0.8 eV (the conduction band edge of GaSb). The InAs has a gap of 0.42 eV, and the valence band of GaSb is assumed to be 0.57 eV above that of InAs.²⁸ The results are shown in Fig. 1.

The conduction band (dashed line), valence band (dotted line), and the calculated defects states (short dash) in both InAs and GaSb are shown in Fig. 1. Superimposed on this band diagram are the calculated lowest conduction and highest valence band of 6/4 SLS (thin line) and 17/8 (thick line) with corresponding band gaps of 350 and 97 meV. We first consider the 6/4 case to verify the correlation of defect levels between the bulk compounds and the SLS structure. We see that defect levels of V_As, In_As for bulk InAs and all four NPDs for bulk GaSb are present within the SLS band gap region. Similarly, for the case of 17/8 design, owing to a small gap, we see from Fig. 1 that there are no defect states of InAs origin and only a few defect states of GaSb origin near the band edges. We then compare these results with those from full calculations, displayed in Fig. 2 for 6/4 (thin dash) and 17/8 (thick dash) SLS. The detailed calculations indicate the presence of the same defects in the SLS gap. In the case of 17/8, no states of InAs origin and only two states of GaSb origin are predicted to be in the gap. Similar agreement is observed for two different SLS designs (24/4 and 16/8) with an identical band gap of 120 meV (see supplementary material). Owing to different wavefunctions and energy band structures of bulk constituents and of the SLS (used in the calculation of G^0k), the defect energy levels in SLS do not always agree with the corresponding bulk defect levels. However, for all the SLS designs studied here, we find a fairly strong correlation between them. The correlation is particularly strong when a SLS has a highly uneven composition between InAs and GaSb. In such a case, the wavefunctions in the bulk region of the dominant compound in the SLS closely resemble those in the corresponding bulk constituent. The correlation is poor when the compound layer thickness is either small or the thicknesses of the two compound layers are equal. In both cases, the wavefunction in the SLS is considerably different from that of bulk constituent materials. The observed correlation between the SLS and bulk constituents suggests that the location of bulk defect states with the band gap region is indicative of defect levels in the SLS. Hence it is possible to identify a design with no mid gap states at longer wavelengths by shifting the SLS miniband locations. This approach can be used to quickly screen of SLS designs for growth without requiring an extensive computation.

To validate our predictions regarding the above design principle, we identify two studies^29,30 in the literature that considered the 17/7 structure, which is close to the predicted best design of 17/8. Although those experimental studies differed considerably on the evaluated band gap, both sensors yielded very high dark-current density—∼4 A/cm² and ∼1 A/cm²—in an apparent disagreement with our prediction. For an accurate comparison, we carry out the band-gap and defect-level calculations for the 17/7 design. Our calculated band gap of 113 meV is in close agreement with one of the studies.³² The calculated energy levels arising from defects in the bulk regions are few and comparable to those of 17/8 design as shown in Table I. Then, for both designs, we calculate the energy levels from defects at the interface. The values are given in Table I and displayed in Fig. 3. We see the defect levels for 17/8 (thick dash) have moved slightly into the gap with one state from V_Ga occurring near 0.33 E_g and that from V_Sb at ∼ 0.3 E_g. However, in the case of 17/7 (thin dash), there are two states near the mid gap (0.54 E_g for V_Ga and 0.42 E_g for V_Sb). If these defects were preferred from thermodynamic conditions,³¹ it could explain the measured short lifetimes in this system. Based on the defects considered here in the bulk and at the interface, we expect 17/8 design to perform better than the 17/7 design.

We include a few cautionary notes herein. First, our calculations predict the defect levels if the defects are present. Since we did not calculate the probability of any of the defects being present, the SLS systems with mid-gap states may still be attractive if the corresponding defect formation energies are so high that these defects are unlikely to be present. Only detailed calculations—not attempted or described here—would confirm that. However, the approach developed in this study can be used to identify the SLS candidates without mid-gap states, thus alleviating the critical requirement of maintaining low defect densities. Second, we consider only the effect of NPDs at the abrupt interface. In certain growth patterns, As and Sb are known to diffuse across the interface. The effect of interface diffusion needs to be evaluated. Third, we have assumed that the lattice relaxes around the defect. If the growth rate is faster than the typical relaxation rate (that is, growing in a meta-stable state), the calculations and conclusions need to be revisited. Fourth, we have yet to consider the effect of interstitials and isoelectronic impurities arising from inter-layer diffusion. The most common defect in this class appears to be As_Sb. Finally, the approach developed here is ideally suited for more accurate calculations in which our HPTB Hamiltonian is replaced with a SIESTA Hamiltonian and hybrid functionals^32–34 to obtain a correct gap. That is, both ΔV and G^0k are calculated using the gap-corrected SIESTA Hamiltonian. However, those calculations are time consuming as they require a basis set considerably larger than sp³,³⁴ and an extension of hybrid functionals to SLS has not been accomplished. In spite of these qualifiers, this work demonstrates a design principle—keep the SLS mini-gap away from defect states of constituent bulk defect states, inside or outside the bulk bands—to potentially eliminate the effect of defects in the bulk region of SLS and thus enhance the lifetimes considerably. The method developed here can be easily applied to Type I SLSs such as GaAs-AlAs (Type I) and, in principle, to other T2SL such as InAs-InAsSb. However, the calculation of ΔV for defects in alloys is not straightforward because the value will be dependent on the environment, requiring configurational averaging.

In summary, we have used a hybrid method of first-principles Hamiltonian, long-range empirical tight-binding Hamiltonian, and Green's function approach to study NPDs in a number of SLS structures. The energy levels arising from the defects in the bulk region of the SLS have strong correlations to the defect levels—both in the gap and within the bands—of constituent compounds. Using the approach described, it is possible to design the SLS candidate systems that have no defect states in the band gap. The knowledge of bulk defect states and mini-band location can guide the choice of SLS structures for growth without performing detailed calculations of defect levels in these SLSs.

See supplementary material for further discussion on design principles as applied to two SLS designs (24/4 and 16/8) with an identical band gap of 120 meV that is provided.

The authors are grateful for funding from the United States Air Force (USAF Contract No. FA8650-11-D-5800/TO 0008) through a Universal Technology Corporation Subcontract (No. 14-S7408-02-C1).