Accurate Prediction of Optical Transitions in Epitaxial InGaAs/InAlAs Asymmetric Coupled Quantum-Well Structures

Abstract

Atomically resolved Z-contrast and strain mappings are used to extract a model of the composition of an InGaAs/InAlAs asymmetric coupled quantum-well structure grown on InP by using metal–organic vapor phase epitaxy. The model accounts for grading across multiple alloy interfaces and is used to compute intersubband absorption in the structure. The simulation accurately predicts the experimental absorption spectrum of the structure within only a few meV, an almost 10-fold improvement over simulations using a square-band profile with nominal alloy compositions. This work thus provides a significant step forward in accurate and predictive simulations of the optical properties of epitaxial heterostructures for emission, modulation, and detection in mid-infrared.

Introduction

Optoelectronic devices operating in the mid-infrared domain are required for several applications, including night vision, free-space communications, light detection and ranging (LIDAR), spectroscopy, and observational astronomy. Unipolar quantum devices, i.e., quantum cascade lasers (QCLs), detectors (QCDs), and quantum-well infrared detectors (QWIPs), as well as interband cascade lasers (ICLs) offer unique possibilities for room-temperature and high-frequency operation. These devices are now all reaching a level of maturity sufficient for the development of mid-infrared optoelectronic systems, , with recent examples including systems for free-space communication, combining a QCLmodulated either directly or using an external modulatorand a high-frequency quantum-well detector. ,

Further development of such systems hinges on a better understanding and modeling of the complex heterostructures that underpin the individual devices. Indeed, devices based on semiconductor multi quantum wells rely on designs involving several hundred wells and barriers, with some designs calling for layers only a fraction of an atomic monolayer and even minute deviations in the epitaxial structure induce large shifts of the operating wavelength. In this context, fine-tuning a given device design to improve overall system performance is a time-consuming and laborious process involving numerous cycles of epitaxy, processing, and characterization.

One of the key issues is understanding, modeling, and controlling interfaces. In most semiconductor devices, individual layers in structures are sufficiently thick to consider interfaces ideally abrupt and to model the potential in these crystals using square wells and barriers. Epitaxial interfaces, however, have a finite width, comparable to or greater than the width of certain layers in the design of unipolar devices, and, as a result, greatly impact their performance. Developments in characterization techniques recently made it possible to measure the characteristic interface length of epitaxial interfaces. −

These works highlight the importance of investigating and more accurately modeling the microstructure of epitaxial crystals used in unipolar devices. Nevertheless, the models proposed in these works exhibit a few shortcomings. First, with few exceptions (see for instance Reference), the studies were limited to heterostructures where the thinnest layer is a few nanometers thick, thicker than the reported interface width by as much as an order of magnitude. Furthermore, the studies often consider interfaces where only one element is exchanged, e.g., Ge/SiGe or GaAs/AlGaAs. ,, However, several key structures in unipolar devices rely on alloys where more than one element is exchanged across interfaces. Finally, previous studies have been limited to alternations between two materials, e.g., a Ge well in a SiGe of a single composition, or a GaAs well in an AlGaAs barrier of a single composition. , Heterostructures used in the active region of unipolar devices may contain three or more distinct materials. ,,−

Beyond the model, all previous studies have relied on characterization techniques that are limited in resolution, chemical precision, or both. Atom-probe tomography , (APT) is, for instance, used in References, , although chemically precise enough to detect dopants in semiconductor crystals, it is limited in resolution to 1 nm. Dark-field transmission electron microscopy (DF-TEM) achieves atomic-monolayer resolution, but it is limited in chemical precision to very few systems, such as AlGaAs/GaAs, that exhibit a strong variation of the form factor. In (In,Al,Ga)­As alloys, the 002 DF-TEM contrast is more difficult to interpret as it depends on the lamella thickness as well as on the concentrations of the different elements.

The present contribution aims to improve upon previous work on both the model and the characterization technique. A generalized expression for compositional grading for an arbitrary number of interfaces, spacings, and number of chemical elements in alloys is proposed. The use of the model is showcased in the design of asymmetric coupled quantum wells (ACQW) using (In,Al,Ga)As alloys lattice-matched on InP. The model is calibrated on experimental data obtained from atomically resolved quantified scanning transmission electron microscopy. The model is then used in numerical simulations and shown to accurately predict the absorption spectra of the ACQW structures.

Models and Methods

Compositional grading at interfaces can be modeled with a variety of mathematical sigmoid functions. Two such functions have been used in the literature, the logistic function − , and the error function. Both functions were shown to yield good agreement with experiment. The logistic function is chosen here for reasons discussed in detail in the following section. Following Reference, in a crystal for which the composition of different layers varies across an interface at position z _n , the variation due to grading at an interface is modeled using

$${\theta }_n(z)=1-\frac{1}{1+{\mathrm{e}}^{(z-z_n)/L_n}}$$ 1

where L _n is the characteristic length of transition between the two layers of different compositions on either side of the interface at z _n . Now, consider a crystal consisting of N layers, ordered along direction z, and made of various alloys. Let M be the number of total elements in the crystal and c _m,n the target composition far from any interface of alloy element m in layer n. The sum of the compositions over all alloy elements obeys

$$\sum _{m=0}^{M-1}{c}_{m,n}=1$$ 2

at every point in the crystal. Taking grading into account, the composition of element m, ${\mathcal{C}}_m(z)$ , at each position z is given by

$${\mathcal{C}}_m(z)=c_{m,0}+\sum _{n=0}^{N-1}(c_{m,n+1}-c_{m,n}){\theta }_n(z)$$ 3

i.e., the sum of contributions from all layers in the crystal, weighted by θ _n that accounts for grading across interfaces. This expression meets the criteria set out above: it accounts for multiple layers, arbitrarily spaced, and taking into account all elements in the structure. One can readily show that the sum over m of all ${\mathcal{C}}_m$ also equals unity at all z in the crystal. The parameters in this expression, the positions of the interfaces z _n , the characteristic lengths L _n , and the compositions c _m,n are determined by fitting the model to experimental data from scanning transmission electron microscopy.

To calibrate the model, we grew an ACQW structure in a Veeco D180 Turbodisc MOVPE reactor (sample A). The structure consists of two InGaAs wells that are 5.6 and 2.5 nm thick, separated by a 1.4 nm thick InAlAs barrier. This core structure is surrounded by 25 nm thick InAlAs barriers. Trimethylgallium, trimethylaluminum, and trimethylindium were used as organometallic precursors to elementary gallium, aluminum, and indium, respectively, and arsine was used as a hydride precursor to elementary arsenic. All alloys are lattice-matched on InP. The nominal compositions for lattice-matched InGaAs are 53.2 at% indium and 46.8 at% gallium; those of lattice-matched InAlAs are 52.2 at% indium and 47.8 at% aluminum. The thin InAlAs barrier was grown following the aluminum flow-modulation scheme introduced in Reference to compensate for the partial loss of aluminum in very thin barriers.

A lamella was prepared from sample A using ion milling and thinning, carried out in a FEI SCIOS dual-beam FIB-SEM. It was then observed in an aberration-corrected FEI TITAN 200 TEM-STEM operating at 200 keV. The convergence half angle of the probe was 17.6 mrad and the detection inner and outer half angles for HAADF-STEM were 69 and 200 mrad, respectively. The lamella was imaged along the ⟨110⟩ zone axis. Atomically resolved mappings of composition and strain were obtained from the HAADF-STEM micrograph following the algorithm detailed in Reference.

A separate sample was grown containing 30 repetitions of the ACQW structure, to carry out multipass absorption experiments (sample B). In this structure, the 5.6 nm InGaAs well was n-doped with Si, with a target concentration of 1.5 × 18¹⁸ cm^–3. Furthermore, a semi-insulating InP substrate was used in sample B to avoid free-carrier absorption. The absorption measurement is carried out with a Fourier-Transform Infrared (FTIR) spectrometer and a multipass absorption cavity formed by two gold mirrors placed on either side of the sample. The length of the cavity can be controlled to change the number of passes. The sample is inserted inside the cavity and positioned at the Brewster angle, and the experiment is carried out in a dry nitrogen atmosphere. The light generated by a globar is filtered with a linear TM polarizer before being injected. The detector in this setup is a broadband DLaTGS. The baseline is normalized by dividing the sample transmission by the measured transmission on the semi-insulating InP substrate alone.

Band-structure simulations were carried out using an in house code that calculates the band structure in the envelope function approximation and takes into account Poisson redistribution of charges and nonparabolicity, as described in Reference. The compositions from eq calibrated on quantified STEM data are used as inputs to describe the crystal structure. The potential is calculated self-consistently from this description, without the need for an additional approximation to specifically describe graded interfaces, such as the one proposed in Reference, at no additional computational cost. The absorption is calculated through Fermi’s golden rule, taking into account the difference in effective mass of each level as in Reference. The maximum absorption in this case is red-shifted with respect to the transition energies between levels computed in the structure. For the sake of brevity, the absorption maximum corresponding to the transition between levels E _i and E _j is noted E _ij . Finally, a depolarization shift is used to account for doping in the wells. All parameters relevant to band-structure simulations are taken from Reference.

Results and Discussion

An atomically resolved HAADF-STEM micrograph of the ACQW structure in sample A is shown in Figure (a). Following the algorithm in Reference, mappings of the Z-contrast and the in-plane (ε₁₁₀) and out-of-plane (ε₀₀₁) strain were computed from this micrograph and are shown in panels (b), (c), and (d), respectively. The mappings are obtained from a partition of the micrograph in panel (a) into cells. Each cell is the fraction of the Wigner–Seitz cell of the crystal that intersects the observation plane, i.e., the smallest observable fraction of the crystal. The cells are centered at the center-of-mass of the (In,Al,Ga)-As dumbbell. The contrast is the average over the cell, and the corresponding variation in composition is less than 1 at%. The position is determined precisely enough to limit the error in the determination of strain to within 0.005. Individual cells belonging to the same 110 atomic plane were averaged together to produce line profiles along the ⟨001⟩growth direction. The line profiles for the Z-contrast and ε₀₀₁ are shown in red in the middle and bottom plot of Figure . The Z-contrast $\mathcal{R}$ was modeled using

$$\mathcal{R}(z)=\frac{1}{Z_{\mathrm{ref}}}(\sum _{m=0}^2{Z}_m^{1.8}{\mathcal{C}}_m(z)+Z_3^{1.8})$$ 4

where Z _m is the atomic number of alloy element m and ${\mathcal{C}}_m$ its composition from eq , with elements indium, aluminum, gallium, and arsenic numbered m = 0, 1, 2, 3. The reference is taken to be InGaAs lattice-matched to InP, i.e., Z _ref = 0.53Z₀ + 0.47Z ₂ + Z ₃ . For pseudomorphic accommodation on the InP substrate, ε₀₀₁ follows

$${\epsilon }_{001}(z)=(1+\frac{C_{12}(z)}{C_{11}(z)})\frac{a_0(z)-a_{\mathrm{InP}}}{a_{\mathrm{InP}}}$$ 5

where C _ij (z) are the position-dependent, alloy elastic constants, a ₀(z) is the strain-free alloy lattice parameter, and a _InP is the lattice parameter of the InP substrate. The elastic constants and strain-free lattice parameters are linearly interpolated from the corresponding parameters of the end-point binaries and their respective compositions in the alloy, as shown in eq .

1.

(a) HAADF-STEM micrograph of the asymmetric coupled quantum-well structure. (b) Corresponding atomically resolved mapping of the aluminum concentration. (c) Displacement map along the in-plane ⟨110⟩ direction: no strain is observed, showing that the structure is pseudomorphically accommodated on the InP substrate. (d) Displacement map along the ⟨001⟩ growth direction. The displacement deviates from the lattice parameter of InP for the InAlGaAs barrier indicating a departure from lattice-matched compositions of the InAlGaAs barrier.

2.

Composition profiles computed using the sigmoidal model to analyze the Z-contrast and displacement observed in the mappings shown in Figure . The measured Z-contrast and displacement profiles are obtained by averaging the corresponding mappings for (110) planes along the growth direction. (top) Composition profiles of elementary indium, aluminum, and gallium from the sigmoidal model. (middle) Measured and modeled Z-contrast from the compositions in the (top) panel. (bottom) Measured and modeled displacement along the growth direction using the compositions in (top) and assuming pseudomorphic accommodation on InP.

The modeled curves that yield the best fit to experiment are shown in blue in the middle and bottom plots of Figure . The best fit was obtained for a single value for all characteristic lengths L _n , equal to 1.5(1) atomic monolayers or, equivalently, 0.44(3) nm, and of effective compositions {c _m,n } _M,N given in Table were used. As stated in Section 2, Models and Methods, index m represents elements: 0 for indium, 1 for gallium, and 2 for aluminum. Alloying occurs in (In,Al,Ga)As alloys on the column-III elements; the composition of element 4, arsenic, C ₄ is equal to 1 in all of the crystals and will be omitted in the discussion henceforth. Index n represents layers in the ACQW structure0 to 4 from left to right in the HAADF-STEM micrograph in Figure (a). The ordinates z _n of the four interfaces visible in the micrograph were obtained from multipeak fitting to the first derivative of $\mathcal{R}$ . They are positioned at the following ordinates z _n : 8.55(3), 14.67(3), 16.08(3), and 18.12(3) nm. Details on the fit are given in the accompanying supporting information document. The resulting models for the distribution of indium, aluminum, and gallium along the ⟨001⟩ direction for each element are shown in Figure (top).

1. Effective Compositions c _m,n used in eq to Model the Compositions of Each Layer in the HAADF-STEM Micrograph of the ACQW Structure Presented in Figure (a) .

		layer index along 001
	c m,n	0	1	2	3	4
[In]	0	0.52	0.53	0.45	0.50	0.52
[Al]	1	0.48	0.00	0.53	0.08	0.48
[Ga]	2	0.00	0.47	0.02	0.42	0.00

^aRow indexes indicate element0 for indium, 1 for aluminum, and 2 for gallium, and columns represent the distinct layers in the structure, numbered 0 to 4 from left to right in Figure (a). All values are given within ± 0.01. The best fit for the remaining parameter of the modelthe characteristic interface transition lengthwas found to be 1.5(1) monolayers, or 0.44(3) nm.

Several conclusions can be drawn from these composition profiles. First, the determined characteristic interface length of 0.44(3) nm is comparable to those reported in Reference for a variety of MBE-grown III–V semiconductors. Such short characteristic lengths indicate that grading is not due to diffusion in the solid state but is rather due to the kinetics of adsorption and exchange between the vapor phase and the solid crystal, typical in kinetically limited growth processes such as MOVPE and MBE. This value, therefore, a posteriori validates the choice of a logistic function over the error function, the latter being a solution to Fick’s equation and, consequently, more relevant when modeling diffusion. Nevertheless, Grange et al. showed that, mathematically, both solutions yield similar grading profiles. Antimonides, where diffusion of antimony has been observed and reported, may be a better system to differentiate between the error and logistic functions.

Beyond these considerations on the characteristic transition length and the underlying physical processes responsible for grading, the composition profiles show that the 1.4 nm thick InAlAs barrier deviates significantly from a nominally lattice-matched InAlAs alloy, containing at most only 25(1) at. % aluminum, 35 at. % gallium, and 40(1) at% indium. The subsequent 2.4(10) nm thick InGaAs well also contains up to 10(1) at% aluminum. The effect of these deviations from an ideally abrupt composition profile was evaluated by numerically computing the absorption for both the measured composition profile and an ideal square-potential equivalent, calculated for nominal compositions of the wells and the barrier, and comparing them to the experimental absorption spectrum. The computed band structure of the measured composition profiles is shown in Figure (a) and its square-potential equivalent in Figure (b). In both cases, the two quantum wells are tunnel coupled, with the wave function of the state E ₁ mostly localized in the 5.6 nm thick well and that of the state E ₂ mostly in the 2.5 nm thick well. The wave function associated with state E ₃ significantly differs in the two structures: in the actual structure E ₃ is above the barrier and its wave function distributed across the entire ACQW structure, whereas it is mostly localized in the 5.6 nm thick well in the case of the square-band potential. The differences in the band structures are slight. They do, however, yield a significant difference in the absorption spectra shown in Figure . The peak absorption for the E ₁₂ transition appears at 170(1) meV in the experimental spectrum (green solid). This peak is exactly reproduced by the sigmoidal profile (black, dot-dashed), whereas the square-band equivalent E ₁₂ is offset and appears at 144(1) meV. The sigmoidal profile also more accurately reproduces the E ₁₃ transition, predicting it at 278(1) meV for an experimental peak at 281(1) meV, versus 272(1) meV for the square-band equivalent.

3.

Band-structure simulations of the ACQW structure, carried out using (a) the sigmoidal profile described previously and (b) ideally abrupt, perfectly nominal wells and barrier. (c) Experimental absorption spectrum (green, solid line) compared against the simulated absorption using the sigmoid model (black, dot-dashed) and the ideally abrupt profile (red, dashed).

While the deviations of the square-band model from the experiment appear small, they are significant when designing unipolar devices emitting, modulating, or detecting in the mid-infrared. The difference between 170(1) and 144(1) meV in the E ₁₂ transition of the ACQW corresponds to a 1.3 μm shift in the operating wavelength. Furthermore, it is important to recall that the sample observed using HAADF-STEM is distinct from the one on which absorption measurements were carried out. This was in part done to ensure that damage left on Sample A after FIB preparation does not interfere with multipass absorption measurements. Nevertheless, the two samples were grown several months and several hundred epi runs apart as a test of the long-term validity of the model. The good agreement between the simulated and the measured absorption spectrum confirms the validity of the model. Therefore, the proposed empirical model and its demonstrated predictive strength can help one better design the target device and overcome limitations of epitaxial crystals. The authors have already used this information to produce a Stark-effect modulator used in Reference, that yielded a free-space communication mid-infrared channel at 9 μm with a record speed of 10 gbit s^–1.

Beyond the demonstrated utility in terms of device design, the proposed composition model provides a useful metric for the analysis of epitaxial crystals and their interfaces. In the present contribution, it was applied to (In,Al,Ga)As alloys and explains how the flow-modulation scheme introduced in Reference helps partially counteract grading. Indeed, if one considers the derivative of the composition at a given interface and for a given alloy, then one sees that it is proportional to (c _n+1–c _n )/4L _n . Hence, increasing the flow of aluminum at the barrier increases the slope by acting on the Δc = c _n+1–c _n . In the authors’ experience, this effect is much larger than that of temperature or growth stops on the characteristic interface length L. It would be interesting to apply eq to a variety of other alloys, particularly those that tend to exhibit dissymmetric interfaces. One particularly relevant example is InAs/InAsSb superlattices on GaSb, where memory effects for antimony occur and where diffusion also occurs and that are an ideal ground to test which sigmoid functionlogistic or error functionbetter describes experimental data in the case of diffusion. Another interesting system is InAs/GaSb superlattices, where exchanges happen across the interface for both the III and the V element. In principle, eq can be used in these cases as well to describe grading. Experimental data from quantified HAADF-STEM would, however, require additional developments of the quantification algorithm , to further separate the dumbbell into cation and anion sites within each cell. Such improvements are possible, given recent evolutions of TEM/STEM microscopes, particularly in terms of resolution and drift correction, but have yet to be implemented. Finally, additional data could be mined from the mappings in Figure , in particular, the distribution of elements within a single atomic monolayer. The standard deviation of this distribution is linked to random alloy disorder, a parameter responsible for spectral broadening. Measuring and modeling it would allow one to also predict the spectral line width of optical transitions in unipolar devices, following for instance the proposal in Reference.

Conclusions

A generalized empirical model for epitaxial semiconductor superlattices has been described. The model describes superlattices with an arbitrary number of interfaces and alloy elements. It accurately reflects finite grading at interfaces between wells and barriers, reproducing the microstructure of the crystal lattice, as observed in atomically resolved HAADF-STEM images. The model is used to obtain the band structure of semiconductor heterostructures and predict the absorption spectrum of an asymmetric coupled quantum-well structure, accurately reproducing the transitions observed in the experimental absorption spectrum.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsnanoscienceau.5c00156.

• Additional information on the methodology employed to determine the interface coordinates from experimental profiles and on the precision (PDF)

CRediT: Konstantinos Pantzas conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration, resources, supervision, validation, visualization, writing - original draft, writing - review & editing; Virginie Trinité conceptualization, formal analysis, methodology, software, validation, writing - original draft, writing - review & editing; Angela Vasanelli conceptualization, data curation, formal analysis, investigation, methodology, validation, writing - original draft, writing - review & editing; Carlo Sirtori funding acquisition, project administration, supervision, writing - review & editing; Gregoire Beaudoin investigation, resources, writing - review & editing; ISABELLE SAGNES project administration, resources, supervision, validation, writing - review & editing; Jean Luc Reverchon funding acquisition, project administration, supervision, validation; Gilles Patriarche conceptualization, data curation, investigation, supervision, validation, writing - original draft, writing - review & editing.

The authors declare no competing financial interest.