Study of the Natural Auger Suppression Mechanism in

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Study of the Natural Auger Suppression Mechanism in Heterostructures through Heteroboundary Engineering Andre Slonopas* and David Tomkinson U.S. Army Night Vision and Electronic Sensors Directorate, Fort Belvoir, Virginia 22060, United States ABSTRACT: Planar superlattice devices revolutionized our approach to solid-state technology by reducing the Shockley−Read−Hall losses to negligible levels. Despite these achievements, significant efficiency losses are found in current devices presumably caused by the Auger recombinations. This work present the theoretical considerations of the Auger recombination suppression through heterostructure engineering. It is found that Auger recombinations are suppressed through the heterobarrier-carrier interactions. It is shown that a minima in Auger recombinations exists in type-II and III heterostructures, and can be reached through proper conduction and valence band alignments. Furthermore, the careful consideration of the heterostructure enables natural Auger suppression for high operating temperatures. Dark current based on the optimized heterostructure was computed and found to be over an order of magnitude below the currently reported measurements for the superlattice and QD devices. This research provides crucial information about the underlying physics behind the Auger recombination, enabling future superlattice and quantum dot device optimization. lattice.16 This success in the Auger suppression is attributed to the Umklapp processes, and not the carrier-heteroboundary interaction. Previous work, however, has shown significant differences in the Auger recombination mechanisms in the bulk of the material as compared to the heterostructure.17 The interaction of the carriers with the perpendicular heteroboundary causes the conservation law of momentum normal to the heteroboundary to remain unsatisfied. Therefore, no threshold for the Auger recombination mechanism exists and the recombination mechanism becomes a power function of temperature. The temperature dependence of the Auger recombination rates found in the bulk of the material, on the other hand, has been shown to be an exponential function.17 Considering the above discussion, an ideal heterostructure can exist where the Auger recombination rates are suppressed by the heteroboundary. The band structure of the narrow bandgap (Eg) semiconductors has been shown to be highly alloy and doping concentration dependent.18,19 Furthermore, extensive work in the metalorganic chemical vapor deposition (MOCVD) of narrow bandgap semiconductors has demonstrated a possibility of an extremely accurate doping control over a broad range of concentrations.20,21 This enables precise positioning of the p−n junction within the heterostructure for various alloy compositions.22,23 Given these achievements in the deposition techniques, it is of interest to investigate the effects of the

1. INTRODUCTION Strained layer superlattice devices based on narrow bandgap semiconductors have brought about a small revolution in solidstate technology. The high performance of these devices make them ideal candidates for various applications across a broad band of the electromagnetic spectrum.1−3 The manufacturing limitations in the narrow bandgap semiconductors, however, diminish the performance characteristics of the conventional detectors. The performance is mainly limited by the difficulty of controlling the doping levels and defect concentrations of the small forbidden gap, thus performance is diminished through the defects.4 Alternative barrier structures for various semiconductor types have been proposed and demonstrated.5,6 The introduction of a barrier, blocks majority carriers, thus rectifying the flow of the excitons, while simultaneously minimizing the Shockley−Read−Hall (SRH) generation-recombination (G-R) rates.7−10 A significant improvement in the dark current characteristics has been shown in these structures.11,12 Despite the numerous successes, a limit in performance optimization has been reached in this architecture. The dissipative Auger GR arising at the grown layer interfaces dominates the dark current characteristics, limiting the real world applications.13 Thus, there is great interest in an alternative designs for the suppression of Auger recombinations (AR). Auger suppression has been proposed through the anisotropy exploitation of the electronic structure.14 The Auger recombinations, however, have proven to be largely insensitive to dimensionality for a given carrier concentration.15 Thus, the modification of the band structure by the use of strain provides a limited success in Auger suppression. Successful Auger suppression has been demonstrated through a type-II superThis article not subject to U.S. Copyright. Published XXXX by the American Chemical Society

Received: August 3, 2017 Revised: September 18, 2017 Published: September 26, 2017 A

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heterostructure on the Auger suppression and the optimal heterostructure conditions for maximum Auger suppression. This work demonstrates the efficiency of Auger suppression in Type-I, II, and III heterostructures by restricting the free motion of carriers. More precisely, this work reveals that a minima of the Auger recombination exists in all heterojunction types. This indicates the possibility of an ideal structure and a natural method of suppressing Auger recombination through band structure engineering. An “ideal” band architecture is elucidated and is shown to produce acceptable Auger recombination rates. In practice the Auger recombination has been suppressed through cryogenic cooling of the narrow bandgap semiconductor. Such practice, however, is extremely cost-inefficient, thus limiting the device applications. This work reveals that it is possible to achieve comparable Auger recombination rates at 100 and 150 K. This is confirmed by the computed dark current of an optimized heterostructure that is nearly an order of magnitude below the current experimental results for an Hg1−xCdxTe superlattice and InAs quantum dots.

where |Mif | is the matrix element of carrier (electron) interactions, Ei and Ef are the energies of the initial and final stages, respectively, and f(Ek) is the Fermi−Dirac distribution function for the kth particle. Accurately estimating the |Mif | matrix poses a significant challenge. This is mainly due to the fact that photoexcitation energy is transferred as momentum, ℏk, among trions. The wavenumber and the magnitude of the momentum varies with the type of the heterostructure. Regardless of the case, the equivalent wave function becomes oscillating.26 Thus, the |Mif | matrix may be approximated from the corresponding wave functions for the initial and final states27 |Mif | =

∫∫

ψi*(ri)ψ f*(r f )

e2 ψ (ri)ψf (r f ) d3ri d3r f ε|ri − rf | i (2)

where ψm denotes the quantum well wave functions. Substituting the Fourier representation of Coulomb potential gives

2. COMPUTATIONAL DETAILS The band-edge energies of a typical Hg1−xCdxTe detector illustrate the complexity and diversity of the heterostructure types in the device, Figure 1a.21 The methodology of producing

|Mif | =

e2 επ 2

∫ q12 [∫ ψi*(ri)ψi(ri) eiqr d3r

∫ ψ f*(r f )ψf (r f ) e−iqr d3r] d3q

(3)

where q is the carrier longitudinal momenta. The form of the quantum well wave function must be defined in order to proceed with the solution of |Mif |. Assuming the general case in which the well depth will be much greater than the wave function’s decay length outside of the well, the evanescent parts of the wave function may be ignored. Thus, the ground state wave function is defined as ψ (r ) =

CN [u+(x) eik·x + u−(x) e−ik·x](e−iκ·χ ) A

(4)

for −L/2 < x < L/2 and 0 outside of these bounds. The wavevector k is obtained from the sub-band energy or assigned π/L for further simplification. CN is the normalization constant, and u±(x) denotes the periodic parts of the Bloch functions. κ and ρ are the two-dimensional wavevector and the corresponding position vector, respectively.28,29 The wave functions become the antisymmetrized product of the single carrier function by restricting their interactions to rigidly confined regimes.30 This enables the initial and the final state wave functions to be expressed in terms of the singlecarrier ground state, ψ0(xn). Assigning Λ and Β as the basis vectors of the states having a spin of ±1/2, the initial two carrier wave function can be written as

Figure 1. (a) Band-edge diagram of an MOCVD grown Hg1−xCdxTe detector with visible types II and III heterostructures21 and (b) position of the valence and conduction bands used for computation of the valence/conduction alignment ratio.

such complex devices has been refined and no longer poses a formidable challenge. Most importantly, however, this work illustrates that it is possible to fine-tune band alignment and place the band energy at desired locations within the MCT device. Thus, it is possible to control the offset ratio of the conduction-valence bands, that is, VC/VV, Figure 1b. The band alignment tunability enables performance optimization by providing a natural Auger suppression mechanism.21,24 Assuming an active area A, parallel to the x−y plane with the origin at the well center, constant effective masses, m*, and dielectric constant, ε, the Auger recombination rate per unit area can then be derived from first order perturbation theory in electron−electron interactions:25 2π |Mif |2 δ(E i − Ef ) dℜf × f (Ek ) AR = (1) ℏA

ψi(x1 , x 2) =

[Λ1Β2 − Λ 2Β1]ψh0(x1)ψh0(x 2) 2

(5)

Because of the involvement of third particle in the Auger recombination, a wave function of third particle is introduced to the final wave function. ψf (x1 , x 2) =

∫

[Λ1Β2 − Λ 2Β1][ψe0(x1)ψe0(x3) + ψe0(x 2)ψe0(x3)] 2 (6)

B

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heterostructure the matrix is computed for the excited state nearest to the energy conservation requirement. Noticeable dips occur when the excited state no longer meets the criteria, that is, a migration to the next excited state. Type-II heterostructure suppresses the |Mif |2 matrix. This effect is profoundly noticeable in the computed Auger recombination rates, discussed later in the manuscript. Several expressions for the alloy, temperature, and layer thickness dependence of the Hg1−xCdxTe band structure have been proposed. The expressions were elucidated by fitting the experimental data for 0 ≤ x ≤ 0.6 and 4.2 ≤ T ≤ 300 K.37,38 This dependence is shown in Figure 2b. At a fixed layer thickness the bandgap increases with x. This behavior is almost entirely due to the decreased strain and an increase of the lattice constants.19 Computed Auger recombination rates for the types I, II, and III heterojunctions and alloy compositions in Hg1−xCdxTe are presented in Figure 3. The Auger recombinations converge to a

The third particle has a chance of being excited to a higher subband. Thus, the four-band Kane model was used for the multiband wave function approximation.31 The simplified fourband model, however, provides only a qualitative idea on the Auger suppression in HgCdTe. A more accurate PidgeonBrown model must be used for accurate quantitative analysis of the material.32 The differences between the four- and eightband models have been shown to be over five percent near the edges of the Brillouin zone.33 These differences are not expected to have a significant effect on the overall findings of this work. Given the multiband nature of the model, the integration procedure requires continuity of wave function envelopes, and current conservation. These requirements provide for the following boundary conditions: ψi(x1 , x 2) = ψf (x1 , x 2)

1 ⎛ dfi ⎞ 1 ⎛⎜ df f ⎜ ⎟= ⎜ mi* ⎝ dx ⎠ m*f ⎝ dx

⎞ ⎟⎟ ⎠

(7)

(8)

where f is the envelope function, utilized in lieu of the wave function for simplicity.34,35 All of the solutions satisfied the definite boundary conditions on the interface.28,36 Integrating eq 3 over the carrier momenta in the x direction and simplifying using Cauchy’s integral provides a complex plane of the two contributing momentums. First the contribution from the Coulombic interactions of the excitons. Second, contribution from the transferred momentum. The final |Mif | matrix is a summation of both of these contributions. The two contributing factors to the matrix are caused by the carrier-boundary interactions and are absent in homogeneous material. From here it is concluded that an “ideal” heterostructure must exist in which the Auger recombinations are naturally suppressed.

Figure 3. Log of the Auger recombination rate versus alloy composition of Hg1−xCdxTe at 100 K for (a) Type-I, (b) Type-II, and (c) Type-III heterostructures. Results were acquired by the integration of the first-order perturbation theory for the electron− electron interaction in the three distinct heterostructure types.

3. RESULTS AND DISCUSSION Equation 3 was solved numerically by utilizing eqs 45. Computed |Mif |2 for heterostructure types and band alignment ratios is shown in Figure 2a. A carrier interaction matrix is shown to be type and placement dependent. In each

certain minimum in the Type-I heterostructure. This minimum is significantly larger than the observed minima in the latter cases. This behavior is caused mainly by the accumulation of holes and electrons at the respective lowest energy states from which the Auger recombinations occur, inset Figure 3a. A distinct minimum in the recombination rate is observed in Type-II heterojunction. This minimum is both conductionvalence offset and alloy composition dependent. Various HgCdTe material properties, including bandgap, the dielectric constant, and carrier mobility, are known to be effected by the alloy composition. Alloy composition, however, has been shown to have the greatest effect on the band structure of the narrow bandgap materials.39,40 Given that the computations are done utilizing the Kane approximations, it is thus assumed that the band structure of the material plays a crucial role in the location of the Auger recombination minimum with respect to the valence-conduction band alignment ratio. Recombination can take place either with a hole tunneling through a heterojunction to recombine with a confined electron, or an electron traveling through the heterojunction to recombine with a confined hole, inset Figure 3b. Both of these possibilities have an equal chance of occurring, and both contribute to the |Mif | with opposite signs. The opposite signs in the electron−electron interactions are mandated by the Fermion commutation to achieve ordering.41 These two distinct paths interfere destructively thus diminish-

Figure 2. (a) |Mif |2 matrix versus the band alignment ratio (b) conduction and valence bands of Hg1−xCdxTe as a function of layer thickness and alloy composition. C

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ing the Auger recombination rate. Two minimums are observed in the Type-III heterostructure, Figure 3c. As in the previous case, band structure of the material determines the location of the minimum as a function of valence-conduction band alignment ratios. The first minimum is governed by the probability of the carrier to tunnel through the heterobarrier. The probability of tunneling is affected directly by the magnitude of the conduction band alignment.42,43 This phenomenon is explained through the analysis of the changes in the band structure, Figure 4a−c. In type-III heterostructure

Figure 4. Calculated band structure of a Type-III 1 μm thick Hg1−xCdxTe/Hg1−xCdxTe superlattice with valence-conduction band alignments of (a) 1.5, (b) 2.0, and (c) 2.5. ESPLIT represents the peak energy of the split-band at Γ. Figure 5. Auger recombination as a function of valence and conduction band alignments in Hg1−xCdxTe with x = 0.3, at 100 and 150 K.

the band alignment ratio of 1.5 causes the zone center split energy to decrease below the band gap energy, Figure 4a. This allows for a greater volume of phase space in the light-hole band allowing for the energy-momentum conservation to be satisfied with ease, thus increasing Auger recombination rates. An opposite effect is observed at the band alignment ratio of 2, Figure 4b. As per the previous conversation the energymomentum criteria are not as easily met in this case. Thus, when a particular state is no longer closest in energy, an abrupt jump to another energy level takes place. This discontinuous jump is mandated by the conservation of energy, causing a narrow region of order-deep decays in Auger rates. The second, more profound, minimum is reached where the bands offset also forces longer travel paths for the carrier recombinations, inset Figure 3c. As previously discussed the Auger recombination rate is expected to be a power function of temperature. Given the minima in the Auger recombination rates found in the heterostructures, it is of interest to study the effects of temperature on the recombination rates. Two cases of 100 and 150 K are considered, Figure 5. Type-I heterostructure shows an almost uniform distribution of the Auger recombination regardless of the band architecture. On the contrary, a noticeable reduction in the Auger recombination rates is observed under certain valence and conduction band offsets in Type-II and Type-III heterostructures. Furthermore, at specific band alignments the Auger recombination rates are found to be comparable in both the 100 and 150 K regimes. These results validate that an effective Auger suppression is possible solely through heterostructure engineering. As was mentioned earlier the orthogonal carrier-heteroboundary interaction prevents the fulfillment of the energymomentum conservation requirements. Therefore, the Auger recombination is a power function of temperature, that is, AR = CTp. Where C and p are constants, and T is the temperature. The power factor, p, was elucidated to be 0.20, 0.19, 0.17 in types I, II, and III heterostructures, respectively. The power factor of ∼0.07 was found in the minimia locations of Type-II

and Type-III junctions. The latter types of the heterojunctions possess superior qualities for the Auger suppression. Furthermore, these results reveal a region where proper band alignment facilitates Auger suppression for high operating temperatures. Computed dark currents densities based on the optimized Auger recombination rates along with the experimental measurements for the nBnN superlattice and QD InAs devices are shown in Figure 6. There have been no systematic experimental studies of the Auger suppression through heterostructure engineering. Clear evidence of Auger suppression from heterobarrier−carrier interactions, however, has been reported widely in the literature.44−46 These experimental results unambiguously demonstrate a reduction in the dark current by realigning the heterostructures through alloy and doping compositions. Dark current is dominated by the thermal generation associated with the Auger mechanism in the narrow bandgap semiconductors.47 The computed dark currents for the type-I heterostructures are found to be ∼10−5 A cm−2. This is slightly above the current achievements in the superlattice and QD technology. The dark currents are found to be ∼10−7 and ∼10−6 A cm−2 for Type-II and Type-III heterostructures, respectively. Type-II heterostructures are thus shown to be most effective in Auger suppression. This architecture may, however, prove to be more difficult to manufacture, especially in the narrow bandgap semiconductors. In either case the dark current density is found to be ∼1−2 orders of magnitude below the current technological achievements. The methods discussed in this paper enable further Auger and dark current reductions to below the current achievements. This will render the superlattice and QD narrow bandgap devices useful for near room temperature cost-effective applications. D

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Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the U.S. Army Night Vision and Electronics Sensors Directorate for the financial support.

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Figure 6. Computed dark current for an Hg1−xCdxTe superlattice operating at 100 and 150 K as compared with the experimental dark current of an HgCdTe nBnN superlattice and InAs QD.12,48 The values were computed for the band alignment ratios where the AR values were found to be smallest.

4. CONCLUSION In summary, this work investigates the Auger suppression through heterostructure engineering. Auger recombination may be minimized through proper valence and conduction band alignment in the Type-II and Type-III heterostructures. The Auger recombinations are also found to be highly susceptible to the material band structure. Thus, proper consideration of the band structure and band alignments is required for an effective Auger suppression. This work shows that the proper heterobarrier engineering allows the Auger recombinations at 150 K to be at par with those found at 100 K. It is also found that the proper heterobarrier architecture ensure a rather slow increase in the G-R rates with temperature, ∼0.07 power of temperature. Such structures of narrow bandgap superlattice and QD devices allow for real world high operating temperature applications. Lastly, the computations show that dark-current densities of the optimized structures are minimized by ∼1−2 orders of magnitude over those currently reported. In principle this allows “ideal” superlattice and QD structures with improved performance.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Andre Slonopas: 0000-0002-9128-9553 E

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