defect states in type-ii strained-layer superlattices

Defect states in type-II strained-layer superlattices

Michael E. Flatte and Craig E. Pryor

Department of Physics and Astronomy and Optical Science and Technology Center, Universityof Iowa, Iowa City, Iowa 52242, USA

ABSTRACTThe electronic structure of isoelectronic defects, donors and acceptors is calculated within a full superlatticepicture for InAs/GaSb and InAs/GaInSb superlattices. The wavefunctions associated with these states extendbeyond a typical layer width for the superlattices. Thus band alignments between the layers as well as interfaceproperties are predicted to dramatically change these defects’ binding energy as well as their influence on su-perlattice electronic, optical and transport properties. Defect properties are also substantially modified by theirlocation within a superlattice layer.

Keywords: infrared detectors, superlattices, deep levels, interfaces, carrier lifetimes

1. INTRODUCTIONThe large band offsets of constituent materials (InAs, GaSb, AlSb, and alloys) in type-II strained-layer superlat-tices (T2SL) offer novel opportunities for design of superlattice electronic structure and material properties.1,2The key final states for nonradiative carrier recombination processes lie one band gap above the conductionminimum (for electron Auger) and one band gap below the valence maximum (for hole Auger) near the Brillouinzone center. As those offsets exceed three times the band gap in the mid-wavelength infrared (MWIR), the finalstates for nonradiative processes in both the conduction and valence manifold can be modified to increase carrierlifetimes. In the long-wavelength infrared (LWIR), the smaller band gap provides sufficient flexibility to entirelyeliminate the final states at those energies for a broad range of crystal momenta, leading to calculated1 carrierlifetimes many orders of magnitude longer than in bulk alloys. This elimination is achieved by splitting the first(V1) and second (V2) valence subbands by an energy much greater than the T2SL bandgap. The V1-V2 splittingcan be traced back to the large band offsets of the T2SL constituent materials. Experimental measurementshave supported the reliability of Auger calculations which predict a reduction in the Auger rate from electronicstructure engineering of T2SLs.3–5

The long carrier lifetimes possible in T2SLs imply that intrinsically-limited T2SL detectors should outperformMCT detectors.6 The performance of T2SL and MCT detectors can be compared via the normalized dark current,or the dark current per unit of absorption7

eGth = enmin

ατmin(1)

where e is the charge of the electron, Gth is the normalized thermal generation rate, nmin is the minority carrierdensity, α is the absorption coefficient, and τmin is the minority carrier lifetime. Results are shown in Fig. 1 foran 11 micron cutoff wavelength superlattice, indicating that the T2SL should operate at 30K higher temperaturewith the same normalized dark current if the SRH lifetime is much longer than the intrinsic Auger lifetime.Improvements may still be possible by changing the doping level from the 1015 cm−3 level.

The calculations shown in Fig. 1 for these superlattices use a fourteen-band K · p envelope-function theory8

that has been used for accurate calculations of carrier lifetimes in a variety of narrow-gap materials.5 To simplifythe calculations α = 1000 cm−1 is assumed for both the T2SLs and for MCT. The minority carrier density forthe T2SLs is calculated from the superlattice electronic structure for a p-type region with doping 1015 cm−3.The minority carrier density for MCT is computed9 for a doping of 1015 cm−3. The minority carrier lifetime forthe T2SLs is computed from the superlattice electronic structure and includes the radiative lifetime10 and thehole-hole (Auger-7) lifetime for the superlattice.5 The minority carrier lifetime for MCT is computed by adding

Further author information: E-mail: michael [email protected]

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the radiative time11 to the Auger-7 time for p-type material with γ = 60 or the Auger-1 time for n-type material,both with |F1F2| = 0.2, as described by Blakemore.12

Published results on superlattice devices, however, show shorter minority carrier lifetimes than theoreticallimits.13–15 It has been suggested,6 on the basis of recent atomic-scale theory and measurements of InAs defectlevels,16 that shallow acceptors and deep levels would behave differently in superlattice structures from bulk InAs.The calculations in Fig. 1 show that the normalized dark current in T2SLs, at relevant temperatures and carrierdensities of interest, is dominated by the SRH lifetime. Electronic structure calculations that have been performedto optimize the superlattice detector performance by band structure engineering do not consider the defects andhow they may modify the behavior of the superlattices. There may be certain superlattice compositions or layerthicknesses that reduce the overall effect of defects on the SRH lifetime, or on the tunneling current throughdefects. Optimization of superlattices based on these considerations requires a detailed understanding of thenature of the defects in T2SLs, both through experimental measurements and theoretical calculations.

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LWIR T2SL will operate at 30K higher temperature with same dark current

11 micron cutoff wavelength

Figure 1. Shown is the normalized dark current for T2SLs with current 35 ns SRH lifetimes and ! µs SRH lifetimes,along with current MCT for a 11 micron cutoff wavelength superlattice. T2SLs are shown with and without photonrecycling (solid versus dashed curves). Photon recycling only makes a significant difference for the T2SL with the ! µsSRH lifetime.

Recent experiments have shown that shallow acceptors in InAs can be directly imaged in cross-sectional STM,and that the acceptor wavefunctions are highly anisotropic and distort in the presence of strain. Fig. 2 showscross-sectional scanning tunneling microscopy images16 at 4K of InAs intentionally doped with Mn acceptors.Mn is a substitutional acceptor in InAs with a binding energy of 28 meV, which makes it nearly shallow (theshallow acceptor binding energy is 17 meV). Also shown on the right of Fig. 2 is, on the same scale, a cross-sectional STM image of an InAs/GaSb superlattice with layer widths corresponding to an ∼ 11 micron cutoffwavelength.17 The images have been rotated so they share the same orientation and sized so they are on thesame scale. The light gray lines extending across the figure show the width of the InAs layer and permit directcomparison6 with the spatial extent of the acceptor wave function in InAs. The acceptor wave function is clearlybig enough to span the InAs layer width. It is not known at this time what role acceptors (or other defects thatplace a state within the gap) play in the Shockley-Read-Hall (SRH) minority-carrier lifetimes in T2SLs, althoughdefect-assisted recombination usually is the dominant contribution to SRH lifetimes.

Calculations of the defect properties in InAs have, to date, been done16 using a tight-binding theory18 basedon the deep level model of Vogl and Baranowski.19 In this model the interaction of the acceptor occurs viaT2-symmetric states that hybridize with the dangling bonds of the nearest-neighbor As. In order to calculate thedefect wave functions the Koster-Slater technique20 is used to calculate the Green’s function whose imaginary

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28 meV binding energy deep levels in InAs(intentionally doped, viewed with STM)

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deep levels also contribute to trap-assisted tunneling by spanning the superlattice layers

Size of deep level wave functions comparable to thickness of InAs layer(where electrons are conned)

Figure 2. (Left) Acceptor states in InAs as measured by cross-sectional scanning tunneling microscopy. Reproduced withpermission from.16 (Right) Cross-sectional STM of InAs/GaSb superlattice. Reproduced with permission from.17

part gives the local density of states. This method has been proven to give the correct chemical trend of impuritylevels in semiconductors.21 Starting with the tight-binding Hamiltonian H0(k) of homogeneous GaAs, one firstcalculates the retarded Green’s function, G0(k, ω) = [ω − H0(k) + iδ]−1. To obtain a good description of thevalence band structure, the sp3 model including spin-orbit interaction22 is used for the host semiconductor.Then, by Fourier transforming G0(k, ω), the homogeneous Green’s function is calculated in coordinate space,G0(Ri,Rj , ω), where Ri and Rj label the zincblende lattice sites. The final Green’s function is obtained bysolving the Dyson’s equation,

G(ω) = G0(ω) +[1 − G0(ω)V

]−1G0(ω) , (2)

where G(ω) is the full matrix representation using all atomic orbitals at all lattice sites. The LDOS at each siteRi is given by

A(Ri, ω) = − 1π

Im[trα G(Ri,Ri, ω)

], (3)

where the trα is taken with respect to the orbitals of the α atom, depending on which type of atom is actuallylocated at the site Ri.

Here we describe atomistic real-space envelope-function calculations of shallow acceptors and donors, in athree-dimensional eight-band formalism that accounts for inhomogeneous strain. We shall begin by describingthe method of calculating the electronic structure of the T2SL in the presence of dopants, followed by resultsobtained. We find that the binding energy of dopants depends sensitively on the superlattice layer where theyreside; donors in GaSb layers and acceptors in InAs layers in a T2SL have greatly reduced binding energiescompared to the corresponding dopant in a bulk material corresponding to the T2SL layer, and also relative tothe dopants in the other T2SL layers. The shifts in local superlattice material layers are sufficient to changefrom band-like transport at T ∼ 40K for holes to hopping transport.

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2. CALCULATION METHOD FOR DOPANTS IN T2SLThe technique used to calculate the electronic structure of the dopants is a three-dimensional eight-band real-space envelope-function approach originally developed23 for strained semiconductor quantum dots. The calcula-tion is performed on a cubic grid with periodic boundary conditions. First, the strain is calculated using linearcontinuum elasticity theory. The strain energy is computed using a finite differencing approximation, and thenminimized using a conjugate gradient approach.

The electronic structure is obtained with a real-space envelope function approach using an eight-band Hamil-tonian including strain.24,25 The parameters of the Hamiltonian23 are obtained from Ref.26 The energies andwave functions are calculated by replacing derivatives with finite differences on the same cubic grid as used forthe strain calculation. The material parameters and strain thus vary from site to site. The resulting real-spaceHamiltonian is a sparse matrix, which can be efficiently diagonalized using the Lanczos algorithm.

Shown below in Fig. 3 are the wave functions of acceptors in bulk GaSb (left) and bulk InAs (right), as wouldbe visible at the cleavage surface of the material, corresponding to (110). As the valence band masses of the twomaterials are comparable, the Bohr radii of the acceptors are very similar. A slight difference is apparent in theanisotropy of the wave function at distances of ∼ 5 nm from the acceptor; the greater spin-orbit interaction inthe valence band of GaSb produces a larger anisotropy in the wave function, closer to the “bow-tie” apparent inFig. 2.

Figure 3. (Left) Shallow acceptor wave function in GaSb. (Right) Shallow acceptor wave function in InAs. Lengthsmeasured in units of nm.

The wave functions of shallow donors, shown in Fig. 4, in bulk GaSb (left) and bulk InAs(right), are muchmore symmetric. The slight asymmetry in the wave functions near the donor itself are slight artifacts of the grid(the grid spacing is 0.5 nm in all directions).

Shown in Fig. 5 are the band-edge wave functions for a 4 nm InAs/2 nm GaSb T2SL. The band gap ofthe superlattice at T = 0K is calculated to be 12 µm. The conduction band-edge wave function is extendedthroughout the material, whereas the valence band-edge wave function is concentrated almost entirely in theGaSb layers.

Now that the ability of the calculations to calculate for both T2SLs and bulk shallow dopants has beendemonstrated, the properties of dopants in T2SLs will be explored.

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Figure 4. (Left) Shallow donor wave function in GaSb. (Right) Shallow donor wave function in InAs. Lengths measuredin units of nm.

Figure 5. (Left) Conduction band-edge wave function for a 4 nm InAs/2 nm GaSb superlattice (12 µm band gap). (Right)Valence band-ege wave function for the same superlattice.

3. ELECTRONIC PROPERTIES OF DOPANTS IN T2SLRelative to the vacuum energy, the energy of an acceptor in GaSb and in InAs differ greatly, due to the morethan half a volt offset between their valence bands. In a superlattice, even if the acceptor is located in the InAsregion, the wave function of the shallow state will predominately reside in the GaSb regions. This effect is shownbelow for the same shallow acceptor, placed in two different locations in the InAs/GaSb T2SL shown in Fig. 5.The acceptor is either placed in the center of the GaSb layer (Fig. 6, left) or in the center of the InAs layer(Fig. 6, right). In both cases the wave function resides predominately in the GaSb regions. When the acceptoris placed in InAs the acceptor wave function is evenly split between the two GaSb layers on either side of theInAs layer containing the acceptor. The region of real space considered in the calculation contains three repeatsof the superlattice unit cell in the growth direction, and the wave function of the acceptor shows no amplitude inthe third GaSb layer considered (which lies an additional InAs layer away from the acceptor’s atomic position).Thus the acceptor state is well-localized in the neighboring GaSb layers of the InAs layer containing the dopant.

A difference is also apparent in the binding energy for the acceptor. When placed in the center of the GaSblayer the binding energy is 17 meV, whereas in the center of the InAs layer the binding energy is only 7 meV.This is less than half the binding energy of an acceptor placed in bulk InAs (also 17 meV). Thus the acceptors

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are considerably more shallow when doped into the InAs layer of a T2SL, which should assist in reducing carrierfreezeout at the relevant temperatures for detector operation.

Figure 6. (Left) Valence-edge wave function for a shallow acceptor placed in the center of the GaSb layer of a 4 nmInAs/2 nm GaSb T2SL. (Right) Valence-edge wave function for a shallow acceptor placed in the center of the InAs layerof the T2SL.

Figure 7. (Left) Conduction-edge wave function for a shallow acceptor placed in the center of the GaSb layer of a 4 nmInAs/2 nm GaSb T2SL. (Right) Conduction-edge wave function for a shallow acceptor placed in the center of the InAslayer of the T2SL.

A similar difference is seen in the effect of the acceptor on the conduction band-edge wave functions forthe two situations. Shown in Fig. 7 are the conduction band-edge wave functions for the same two situationsshown in Fig. 6. The amplitude of the conduction-band wave function is strongly reduced in the vicinity ofthe (positively-charged) acceptor. When the acceptor is placed in the GaSb layer the conduction-band wavefunction is strongly reduced in the two InAs layers on either side, whereas when the acceptor is placed in theInAs layer the conduction-band wave function is only strongly reduced in the InAs layer containing the acceptor.This suggests that different effects are likely to be seen for the acceptor on the tunneling current in a biasedsuperlattice depending on whether the acceptor is in the center of the GaSb or the InAs layer. Nevertheless,there is non-zero band-edge probability density to the wave function in each of the three T2SL layers consideredin the calculation, indicating that the presence of the acceptor does not completely eliminate vertical transport.

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The situation is very different for donors acting on hole transport. Shown in Fig. 8 are the valence band-edge wave functions for the case of a shallow donor placed in the center of the GaSb layer or the InAs layer.The energy splittings between the superlattice hole states are substantial, as indicated by the complete lackof probability density in two of the GaSb layers for the ground-state hole wave function with a shallow donor.When the donor is placed in the center of the InAs layer the ground-state hole corresponds to one in the mostdistant GaSb well, and the region that is depleted of hole wave function in the other two layers is at least 30 nmlaterally. The energy splitting over this lateral distance exceeds 5 meV. Thus at relevant temperatures for LWIRdetector operation, corresponding to T ∼ 40K, a single donor per 30 nm laterally can completely destroy verticalhole mobility. The areal density corresponding to this is 1011 cm−3, or a volume density of ∼ 1016 cm−3 in thesuperlattice. We expect that these differences in hole energies will cause the low-temperature hole transport tobe dominated by hopping for these doping densities. Further studies to examine the doping densities where thiseffect can destroy the vertical hole mobility are underway.

Figure 8. (Left) Valence-edge wave function for a shallow donor placed in the center of the GaSb layer of a 4 nm InAs/2 nmGaSb T2SL. (Right) Valence-edge wave function for a shallow donor placed in the center of the InAs layer of the T2SL.

Further unusual features are apparent in the conduction band-edge wave functions for shallow donors, shownin Fig. 9. Although the ground state wave functions for electrons do not become disconnected, as the hole wavefunctions become, the properties are very different. When the donor is placed in the center of the InAs layer thebinding energy of the donor is similar to InAs bulk and the wave function is not too unusual. However, whenthe donor is placed in the center of the GaSb layer the bound state resides almost entirely in the InAs layerson either side. This is a similar phenomenon to the behavior of the bound state of a hole around an acceptorwhen the acceptor is placed in InAs, as shown above. The dramatic reduction in the bound state energy for theelectron bound to the donor may also assist in reducing carrier freezeout of n-type carriers.

4. CONCLUSIONSThe ideal performance of T2SL detectors surpasses that of MCT, although the observed performance is lessdue to the SRH lifetime. Lengthening the SRH lifetime requires further knowledge of the properties of defectstates in T2SLs. This initial exploration of the properties of shallow donors and acceptors in T2SL show thatthe ground-state binding energies of electrons and holes bound to donors and acceptors differ by over a factor oftwo depending on where the donors and acceptors are located in the superlattice unit cell. The energies of holestates also change dramatically depending on their proximity to donors, leading to the probable emergence ofhopping transport for holes at low temperature.

ACKNOWLEDGMENTSWe would like to acknowledge conversations with T. F. Boggess, C. H. Grein, J. Pellegrino, and M. Weimer.

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Figure 9. (Left) Conduction-edge wave function for a shallow donor placed in the center of the GaSb layer of a 4 nmInAs/2 nm GaSb T2SL. (Right) Conduction-edge wave function for a shallow donor placed in the center of the InAs layerof the T2SL

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defect states in type-ii strained-layer superlattices

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