Interface tomography of GaInAs/AlInAs quantum cascade laser active regions

We present a high-resolution electron tomography study of buried ultra-thin layers and their interfaces from the active region of a (Ga,In)As/(Al,In)As quantum cascade laser (QCL) test structure. Using a high-angle annular dark-field scanning transmission electron microscopy image series, a three-dimensional (3D) reconstruction of the complex layer structure is obtained. From this 3D information, we determine quantitative values for the chemical width and, simultaneously and independently, the root mean square roughness (rms) and lateral correlation length of the individual interfaces of a cascade using topographic height maps. The interfacial widths are comparably small for all interfaces within a cascade and the layer thicknesses show only a small standard deviation of less than one monolayer. The rms roughness is systematically lower at the direct (Ga,In)As-on-(Al,In)As interface compared to its inverse. In addition, using the one-dimensional height–height correlation function, different lateral correlation lengths along the two perpendicular ⟨011⟩ in-plane directions are detected indicating a distinct anisotropy of the interface morphology. Our accurate and comprehensive results on the QCL test structure will serve as feedback to evaluate the growth process and help to assess the performance of corresponding future laser devices in detail.


Introduction
In the epitaxial growth of artificial semiconductor heterostructures such as quantum cascade lasers (QCLs), the control of growth down to the atomic level plays an essential role in device quality. For example, the active regions (ARs) within a cascade, which in turn is repeated 30-40 times in a laser, * Author to whom any correspondence should be addressed.
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consist of a complex sequence of quantum wells and barriers of various chemical compositions and thicknesses of the layers, typically in the range of 1-5 nm. Since the emitted wavelength depends on the thickness and composition to be precisely adjusted in the particular material system, the smallest variations in these parameters can already cause changes in the emission characteristics [1]. In addition, the structural perfection of each individual interface has to be taken into account, since interfacial roughness scattering is a major cause of broadening in the emission spectrum as well as performance reductions [2][3][4][5][6][7].
The main quantities used to describe an interface are the interface width W describing the chemical intermixing as well as the root mean square (rms) roughness σ rms and lateral correlation length ξ || , both of which characterize the pure morphology [8,9]. These values are an essential prerequisite for modelling device performance, considering e.g. electron scattering, tunnelling processes, etc. However, it is a great challenge to determine the rms roughness and correlation length of buried interfaces experimentally in a direct way. For example, when fitting gain spectra of the lasers, only the product σ rms • ξ || needs to be considered. Thus, no definite conclusions can be drawn about the individual factors. On the other hand, the surface roughness of the device can be measured by using atomic force microscopy, but this only allows an indirect and very unprecise evaluation of the interfaces underneath [10,11].
Tomographic analysis methods should in principle be able to resolve the three-dimensional (3D) structure and provide chemical and morphological data of interfaces with sufficient spatial resolution. However, there are very few studies of this kind in the literature so far. For example, atom probe tomography (APT) has recently been adapted to study epitaxial interfaces in the Si-Ge material system and was used to determine all relevant structural parameters [12][13][14]. Besides, APT studies on the roughness of interfaces and layer thickness in strained (Ga,In)As/(Al,In)As superlattices and QCLs, respectively, are also available [15,16]. In all these cases, the tomography investigations have been correlated with scanning transmission electron microscopy (STEM) or high-resolution x-ray diffraction results.
Electron tomography is another method that allows to study buried layers and heterointerfaces in 3D with high spatial resolution [17][18][19]. But only recently it could be demonstrated that it is also capable of quantitatively determining the interfacial parameters introduced above and over a relatively large sample volume [19]. In tomography, a 3D image or tomogram of the sample is reconstructed from a series of twodimensional projections from different viewing angles. Here we use high-angle annular dark-field (HAADF) images from STEM, which, due to their predominantly incoherent image formation, yield intensities that change monotonically with density, thus fulfilling the so-called projection requirement [20]. To maintain the latter for all viewing angles, a needle is prepared from the sample, since with conventional crosssection specimens an intensity drop and thus a monotonic break would occur at large projection lengths. An additional advantage of the HAADF contrast, which is transferred to the tomogram, is its chemical sensitivity, which in turn allows to determine the chemical interface width and to extract the interfaces for roughness analysis.
In this article, we present the application of this electron tomography method to the highly complex structure of a (Ga,In)As/(Al,In)As QCL grown by molecular beam epitaxy (MBE). We are dealing here with a test structure that contains only the AR of a QCL laser and is used to explore new growth processes or systems. Consequently, the results of our quantitative analysis of the interfaces serve as feedback on the growth parameters and thus as a basis for further optimising the growth process and should ultimately help to increase the QCL device efficiency. In this respect, we demonstrate how a high-spatial-resolution tomogram can be obtained using suitable preparation of the sample and reconstruction of the raw STEM data. Quantitative values can then be determined for the chemical widths as well as for the rms roughnesses and lateral correlation lengths of the individual interfaces of the AR.

Sample structure and specimen preparation
The AR of the QCL device emitting in the mid-infrared region has been grown on InP (100) substrate by MBE. The utilized MBE is a RIBER MBE49 multiwafer-MBE, capable of growing up to 4 × 4 ′′ wafers in one run. Growth temperatures have been controlled by pyrometry and band edge measurements and set to be 565 • C during the growth. Additionally, the wafers have been rotated to obtain good homogeneity across all wafers. The investigated sample is a strain-compensated Ga x In 1−x As/Al y In 1−y As QCL test structure consisting of 35 cascades with a nominal period length of 51.76 nm, and a 200 nm thick buffer and capping layer, respectively, each of Ga 0.47 In 0.53 As lattice-matched to InP. One cascade contains an injection region and an AR, which in turn are made up of Ga 0. 28 Figure 1(a) shows a cross-sectional HAADF micrograph of the entire structure. The periodic sequence of cascades is clearly visible. Due to the Z-contrast conditions, the internal structure of a cascade concerning the well-barrier sequences is clearly recognizable and can be exploited for reconstruction purposes (see the enlarged area in figure 1(b)). The (Ga,In)As wells appear brighter compared to the (Al,In)As barriers due to their higher average atomic number.
For the electron tomographic investigation, a needleshaped specimen with a diameter of about 180 nm was prepared using focused (Ga + ) ion beam milling (FIB). The needle axis coincides with the growth direction of the structure. The procedure is described elsewhere in more detail [19,21]. Subsequently, the needle was further thinned to about 140 nm diameter with Ar + -ions of 200 eV. This Ar + -ion polishing also reduced the damaged layer attributed to FIB milling with Ga + -ions from about 20 nm to a few nm. Figure 2 shows the final needle structure, the red rectangle highlights the area of interest where the tomography was performed.

Acquisition of the tilt series and reconstruction
The tilt series for tomography was recorded on a Cs-corrected JEOL ARM 200F microscope operated at an accelerating voltage of 200 kV in STEM mode. The signals of the annular dark-field (ADF) detector were recorded using the STEM   recorder software from System In Frontier Inc. This software was also used for semi-automatic specimen tracking. The acceptance angles of the ADF detector were set to 68-280 mrad for generating HAADF Z-contrast. Additionally, a condenser aperture of 30 µm diameter was inserted to achieve sufficient focus depth.
A total of 86 STEM micrographs were taken between −85 • and +85 • in 2 • steps, each with a size of 2048 × 2048 pixels and a pixel dwell time of 20 µs. The alignment of the HAADF images to each other was done with the IMOD software package [22]. The number of pixels in the images was reduced by 4 × 4 binning before reconstruction resulting in a pixel size of 0.326 nm × 0.326 nm. The 3D reconstruction was then calculated using the simultaneous iterative reconstruction technique algorithm with 30 iterations using the ASTRA Toolbox [23,24]. The Amira-Avizo™ software package was utilized for the visualisation of the 3D data, see figure 3. Figure 3(a) shows the rendered volume of the reconstructed needle fragment, highlighting its surface morphology. In addition, the plot allows easy extraction of thin sections in any crystallographic orientation and at any position of the reconstructed volume (cf orthogonal slices in figure 3(b)). In figure 3(c), isosurfaces, i.e. surfaces with the same voxel intensity, based on intensity gradients between the (Ga,In)As and (Al,In)As materials are used to represent layer stacking in cascades. All individual cascade layers can be identified in this way, and all wells and barriers are clearly recognizable and applicable for further analysis. It should be noted that nonlinear thickness dependence of the HAADF contrast is considered in the reconstruction resulting in the so-called cupping artefact that is extracted according to the procedure described by Wolf et al [25]. In short, assuming that the HAADF intensity follows Lambert-Beer's law and estimating the needle thickness from an initial reconstructed tomogram, the incidence intensity on the sample is fitted, which is then used to correct the nonlinear HAADF contrast in the measured projections. In addition, the reconstruction was accelerated by the 4 × 4 binning mentioned above and the noise in the tomogram itself was reduced with a non-local mean filter, which preserves fine structures and edges compared to other known filters [26].

Interface widths and layer thicknesses of the AR
The 3D reconstruction is used to determine the interfacial width and thickness of each layer with high precision. For this purpose, a thin section is axially cut out of the reconstructed volume parallel to the growth direction and perpendicular to the interfaces. Figure 4(a) shows such a slice with a thickness of one voxel, i.e. 0.326 nm. Two complete cascades are visible, the corresponding ARs are marked by green brackets (AR 1 and AR 2). The intensity of the voxels is directly linked to the contrast of the HAADF STEM images and thus sensitive to the chemical composition. In this respect, the (Ga,In)As wells are bright, and the (Al,In)As barriers are dark. Due to this contrast variation, an intensity profile along the interfaces is extracted to analyse the layer thicknesses as well as the chemical widths of the interfaces (cf red arrow in figure 4(a)). Note that the definite relation between (voxel) intensity and composition is not known. Under the reasonable assumptions of stoichiometry conservation, coherent interfaces, and a linear relationship between composition and HAADF intensity, the interface profile can be evaluated based on contrast. Figure 4(b) reveals the experimental data of the intensity profile along AR 1 laterally averaged over 60 unit cells to improve the signal-tonoise ratio. The voxel intensities of the first three (Ga,In)As wells I to III reach almost identical plateau values, whereas the intensity of the thinnest well IV is slightly reduced. Similarly, the voxel intensities of (Al,In)As barriers II to IV are almost constant, while the intensity of the thicker barriers I and V, respectively, is lower. These variations in the measured intensities of the layers with nominally identical chemical composition can be explained by the fact that in the case of the thinnest layers the plateau values are not achievable due to the inherent interfacial broadening. In other words, the intensity profile of these thinnest layers indicates that the final ternary composition of the barrier or well is not accessible and that these layers are in fact quaternary alloys, with the composition changing continuously along the growth direction. This finding might provide additional useful information for the QCL designers, since a QCL structure is conventionally modelled by assuming abrupt ternary interfaces. The actual interface profiles, with shapes as shown in figure 4(b), in particular the formation of quaternary interfaces for interfaces 7 and 8, result in shallower quantum wells compared to ideally abrupt interfaces, and consequently in smaller energy separations in the quantum wells and thus longer emission wavelength of the QCL.
To assess the extent of chemical intermixing at the interfaces and to extract a quantitative measure of the compositional interface width between the well-barrier pairs, a sum of sigmoidal functions is used to fit the measured voxel intensity profile [27,28]: Here, i counts each well in the growth direction. The parameter w l (w u ) indicates the lower (upper) inflection point and defines the position of the interface. The distance between these adjacent points indicates the measured thickness of the well or barrier. The parameter L in equation (1) analogous determines the value of interface width. According to the 10%-90% criterion, i.e. the change of composition between 10% and 90% of the total variation, the interface width equals W ≈ 4.4 L. The composition c 0 corresponds to that belonging to the nominally widest barrier (following interface 8) and is used here as a reference value for the parameters p i , which in turn describe the difference to c 0 .
Thus, several profiles as in figure 4(b) are evaluated from mutually parallel tomogram slices for AR 1 (not presented). Hence, the results for the respective interface widths and layer thicknesses in figure 5 come from an averaging over these profiles. The error bars given consequently correspond to the standard deviation. Disregarding statistical errors, the widths of all interfaces are between 1.1 and 1.3 nm only, with exception of interface 8, whose value is about 0.4 nm wider, which is roughly equivalent to one monolayer ( figure 5(a)). These results demonstrate the formation of gradual interfaces over a range of 1.2 nm on average, which is consistent with predictions and measurements from the literature for very smoothly grown or abrupt III-V heterointerfaces, such as (Al,Ga)As/GaAs, (In,Ga)As/GaAs or (In,Ga)N/GaN [29][30][31]. This is of even greater importance for the present case, where the change in concentration of three group III atomic species-In, Ga and Al-at the same time must be considered across the interface. In this context, it is remarkable that the width almost does not vary from interface to interface within an AR, suggesting particular growth control and demonstrating the high structural quality of the cascades.
With the exactly defined position of the interface as the inflection point of the profile, the thickness of various layers within the AR is accurately determined. The results of the measurement evaluation for the layer thicknesses are summarized in figure 5(b) together with the nominal values for comparison. The very good agreement of better than one monolayer again indicates a precise control of the MBE growth process and high reproducibility of the cascade formation. This statement is also supported by the fact that the results of AR 1 do not differ from those of the neighbouring AR 2 (not presented). The thickness of the thinnest barriers and well are below 1.3 nm and thus in the same range as the interface width, indicating the presence of quaternary alloys in these layers as discussed above. Also, the small thickness fluctuations determined by standard deviation (error bars in figure 5(b)) in the range of 0.02-0.23 nm are in the same range as measurements from the literature for (Ga,In)As/(Al,In)As QCLs grown with MBE [1].   figure 4(b)). The thicknesses are plotted together with the nominal values. The data correspond to mean values of the respective widths and thicknesses from several evaluated profiles from the tomogram. The error bars correspond to the associated standard deviation.

Interface roughness and lateral correlation length
The roughness and lateral correlation length of the well/barrier interfaces are quantitatively investigated using isosurfaces of the 3D reconstruction. Since a voxel of the tomogram corresponds to a concentration value in 3D, the isosurfaces can be interpreted as iso-concentration surfaces. As stated in the previous section, due to chemical intermixing and resulting interface broadening, the interface position is defined at the point where the change in chemical composition has reached half of the differential value between well and barrier. At this point, the iso-concentration surface is calculated, and the rasterization of this surface into a 2D topographic map thus represents the height variations of the interface at constant composition, i.e. the interface roughness [19]. Figure 6 represents the topographic maps of the eight interfaces located in AR 1 (cf marker in figure 4(a)). The height maps are sorted by direct (a) and inverse (b) interfaces, where the transition from (Al,In)As to (Ga,In)As layer is defined as direct. It is remarkable that all topographic maps in figure 6 exhibit very low roughness over an area averaging about roughly 50 nm × 100 nm. The overall height variations are in the range of ±0.15 nm for all interfaces. Shallow, terrace-like formations are seen on some of the maps, extending along the crystallographic ⟨011⟩ direction (marked by dashed lines in figure 6), but without the obvious emergence of distinct terrace steps.
For a quantitative analysis of the interface morphology, the rms roughness and lateral correlation length are determined using the one-dimensional height-height correlation function (HHCF) defined by [32,33] where z is the height of the interface at position x and τ is the distance between two interfacial points. The HHCF is often phenomenologically described by [34] H (τ ) = 2σ 2 where σ rms is the rms roughness, ξ || the lateral correlation length and h the Hurst parameter with 0 < h ⩽ 1. In figure 7 the data for interface 1 is plotted as an example in a double logarithmic graph to estimate the parameters σ rms and ξ || . For this purpose, two distinct regimes are visible in the graph: a linear increase where the HHCF strongly depends on the value Figure 6. The topographic maps of four direct (a) or inverse (b) interfaces of the active region 1 (cf figure 4(a)). The numbering follows the direction of growth (cf figure 4(b)). Direct interfaces are defined as the transition from (Al,In)As to (Ga,In)As. The very irregular shape of the topography maps is because of edge effects caused by damage during sample preparation and/or surface relaxation, which are removed during reconstruction. Terrace-like features are visible at some of the interfaces, which are longer in the ⟨011⟩ than in the ⟨011⟩ direction. The dashed lines serve as a guide to the eye. of τ , implying a correlation between two interface points, and a regime where the correlation function flattens out and follows a constant course. The point of crossover between the two regimes defines the lateral correlation length ξ || . For τ > ξ || there is no correlation between the height differences, and the values settle at 2(σ rms ) 2 (horizontal dashed line), indicating the global height variance with σ rms . The 1D HHCF data in figure 7 correspond to the ⟨011⟩ (red) and 011 (blue) directions, respectively. First, the determined rms roughness values of 0.117 nm along the ⟨011⟩ and 0.113 nm along the perpendicular direction are almost identical. Due to this directional independence, which was also found for the other studied interfaces, the rms roughness values averaged over both directions are plotted in figure 8 for interfaces 1-8 of AR 1. With σ rms values between 0.095 and 0.158 nm, the impression from the 2D topography maps regarding very smooth interfaces is confirmed. Moreover, apart from interface 1, all direct (Ga,In)As/(Al,In)As interfaces are smoother than the subsequent inverse interfaces. One reason could be the strain-compensation effect of the oppositely strained layers concerning InP substrate: Starting from a compressively grown Ga 0.28 In 0.72 As layer with a roughened interface to Al 0.70 In 0.30 As, a (partially) strain-compensating grown Al 0.70 In 0.30 As layer under tensile stress could flatten the growing surface, resulting in a smoother interface to the following Ga 0.28 In 0.72 As layer. Overall, there is a slight increase in roughness in the AR along the growth direction. It is also noteworthy that interface 8 with the highest roughness also has the largest interface width. Figures 9(a) and (b) depict the correlation lengths for the ⟨011⟩ and ⟨011⟩ directions, respectively, for both AR 1 and 2. The values confirm the anisotropy observed in the 2D maps: the correlation lengths in the ⟨011⟩ direction show a constant average length of 9 nm. On the other hand, ξ || is larger in Figure 9. The lateral correlation lengths ξ || for both ARs in the 011 direction (a) and ⟨011⟩ direction (b). A constant correlation length of 9 nm is present for 011. In the ⟨011⟩ direction, the values fluctuate more around the mean value of 27 nm. The variations occur because ξ || is of the same order of magnitude as the extent of the topography maps in this direction. the perpendicular ⟨011⟩ direction for each interface. Most values of both ARs are in the range between 20 and 33 nm, but the individual values are subject to significant variation. The reason for this is most likely the combination of varying topographic map sizes in this direction (cf figure 6) and a correlation length within the same order of magnitude. Thus, depending on the given section of the interface, ξ || might be underestimated. Nevertheless, an average correlation length of 27 nm can be obtained. This difference in correlation length in the two perpendicular directions is confirmed by the formation of atomically flat terraces elongated along ⟨011⟩ leaving, however, the overall roughness unaffected. Furthermore, it is observed that each HHCF has two different linear regimes. This would imply that the regime τ < ξ || with correlated heights splits itself into two [35], the one that is just described and a very short isotropic correlation length of only 1.1 nm. This short length should be viewed as holes or islands of 1-2 unit-cell-size superimposed on the flat, extended terraces.

Conclusion
The AR of a (Ga,In)As/(Al,In)As QCL test structure grown on InP (100) substrate was comprehensively analysed by an electron tomography study based on a chemical sensitive HAADF STEM image series. Using orthogonal slices of one voxel thickness from the centre of the reconstructed volume, line scans can be taken to record and model the interface profiles. This allows to accurately determine the chemical interface widths between the various (Ga,In)As wells and (Al,In)As barriers as well as the layer thickness of all individual layers within an AR. In addition, it is possible to generate isoconcentration surfaces of the interfaces, which are used to create topographic height maps of the buried interfaces. Thus, it is possible to qualitatively determine the rms roughness as well as the lateral correlation length over an area of about 50 nm × 100 nm separately from each other employing a onedimensional HHCF. The results demonstrate the directional dependence of these interface characteristics. This roughness analysis could be performed for all interfaces of an AR.
The separate and direction-dependent determination of the roughness parameters sheds new light on the use of the product of rms roughness and correlation length σ rms • ξ || , as it has been applied in the literature so far. For (Ga,In)As/(Al,In)As QCLs, on the one hand, products of about 1 nm 2 are reported in the literature [5,6,[36][37][38], values comparable to those obtained in the present study. On the other hand, our results raise the question whether an adaptation or extension of the previously isotropic interface description in device simulations might be necessary, respectively, or whether this can lead to a more comprehensive device characterization.
The presented data demonstrated that interface tomography and corresponding topographic mapping are powerful methods to determine structural properties of individual, buried interfaces without unintentional averaging due to projected imaging. The demonstrated successful application of the method to complex QCL structures offers a further step towards understanding the formation of interfaces and their influence on the functionality of devices. In a next step, a complete QCL device structure will be investigated.

Data availability statement
The data cannot be made publicly available upon publication because they are owned by a third party and the terms of use prevent public distribution. The data that support the findings of this study are available upon reasonable request from the authors.