3.1.

Experimental Setup and Measurements

Reflectometry measurements were performed at the soft X-ray beamline^13,14 in the laboratory of the Physikalisch-Technische Bundesanstalt at the synchrotron radiation facility BESSY II in Berlin. It provides s-polarized ($>98\%$) monochromatic radiation ($E/\mathrm{\Delta }E\approx 400$) with low divergence ($<1\mathrm{mrad}$). The goniometer allows for precise six-axis alignment of the samples and features full lubricant-free mechanics to minimize contamination of the samples through hydrocarbons from the bearings.¹⁵ Radiation reflected off the sample is measured by a GaAsP photodiode, scanning the angle-of-incidence $\theta$ in the range of $1\dots 89°$ and the photon energy in the range of $80\dots 250\mathrm{eV}$. The raw measurement data are presented in Fig. 1(b), with an average relative measurement uncertainty of 0.8%. At grazing incidence $\theta \approx 90°$, the reflectivity of the sample approaches 1 while it drops to ${10}^{-3}\dots {10}^{-5}$ at near normal $\theta \approx 0°$, depending on the photon energy. Several interference fringes are visible throughout the data set, shifting with the photon energy. Around 100 eV, a sudden feature can be seen that stems from the silicon L-edges. A more detailed account of the measurement and the data fitting procedure is given elsewhere.^16–18

3.2.

Model Fit

We use a transfer matrix approach^16,19–21 to calculate the reflectivity of a specific sample as a function of its geometrical parameters and of the optical constants of the materials. This method is based on the Fresnel equations and Beer’s law to describe the reflectivity and transmission of the individual interfaces and layers. Diffuse scattering from the interfaces is taken into account by a Névot-Crocet/Debye-Waller factor that reduces the reflectivity according to Refs. 20 and 22

The model is used for a fit to the experimentally obtained data. There are two kinds of parameters in the model: global parameters, which are valid for all energies. These are the layer thicknesses $l_i$, interface parameters ${\sigma }_{i/i-1}$, the density of silicon, and a small offset in the angle of incidence $\theta$. Then, there are energy-dependent parameters, which are the optical constants $(\delta ,\beta )$ for the SiGe layers and the contamination. The model assumes that tabulated data for the optical constants of silicon, slightly scaled by the material density, can be used.²⁴ The full model consists of 18 individual layers and 19 interfaces. We assume that the 2 nm thin layers of silicon have a density ${\rho }_1$ that is slightly different from the density of the thicker layers and the substrate ${\rho }_2$; therefore, these two densities are fit parameters, too. Together with an offset of the angle of incidence, these are 40 global parameters. The model further assumes that all of the SiGe1 layers in the stack share identical optical constants and that the same is true for all the SiGe2 layers. At 85 measured energies, the number of energy-dependent parameters (${\delta }_i(E)$, ${\beta }_i(E)$) adds up to $2\times 2\times 85=340$.

While the energy-dependent optical constants were calculated using least square optimization,²⁵ we used a global optimization algorithm²⁶ to determine a set of global parameters that describe the data set well.¹⁶ These parameters were used as initial guess for a Markov-Chain Monte-Carlo sampling over the global parameters.^27,28 The statistics show that all parameters are sufficiently independent. The resulting fit to the data is presented in Fig. 2. The agreement of fit and data is overall excellent, deviations occur only at low angles of incidence for the higher photon energies, where the total reflectance is on the order of ${10}^{-5}$. Through this fit, a set of layer thicknesses $l_i$ and interface parameters ${\sigma }_{i/i-1}$ was determined for the blanket layer stack, as well as the optical constants of the two SiGe variants. The geometrical parameters are discussed in Sec. 5.1, and the optical constants are discussed in Sec. 5.2. The fit results for the silicon densities are: ${\rho }_1=2.345\mathrm{g}/{\mathrm{cm}}^3$, ${\rho }_2=2.362\mathrm{g}/{\mathrm{cm}}^3$, which is very close to the tabulated value of $\rho =2.329\mathrm{g}/{\mathrm{cm}}^3$ for crystalline silicon. The differences thereof probably reflect the accuracy of the fit rather than actual differences in the layers. The fitted offset of the angle of incidence amounts to 0.006 deg, which is plausible given the accuracy of the used goniometer axis.

Fig. 2

Fit results of the soft X-ray reflectometry data from Fig. 1(b) for selected photon energies throughout the measured spectrum. Since the blanket layer stack is complex, the measured and calculated curves show many features. The agreement of fit and data is overall excellent, and deviations occur only at low angles of incidence (aoi) for the higher photon energies, where the total reflectance is on the order of ${10}^{-5}$.

4.1.

Experimental Setup and Measurements

Lamellae of 35 to 45 nm thickness were prepared by means of a manually operated Helios5 UX FIB/SEM dual beam system. A protective capping layer of tungsten was deposited. STEM and STEM-EDX micrographs were acquired at ThermoFisher Scientific by means of a spectra ultra transmission electron microscope. The system was equipped with a monochromated X-FEG (not excited), a piezo stage, a PantherSTEM™ detector, and an UltraX™ EDX detector. Figure 1(c) presents STEM-EDX data of a single lamella cut out of the sample where the different layers are clearly visible. We observe that the transition between the individual layers are not atomically sharp but that there is a considerable transition region. Laterally, the interfaces show no sign of roughness on the length scales observed here. Although it is common practice to determine layers thicknesses by analyzing HAADF-STEM or TEM micrographs, we decided to focus on the chemical nature of the interfaces considered; therefore, we utilized the STEM-EDX signals for determining the thickness and the extent of the intermixing zones between the layers. Therefore, the STEM-EDX signal was processed through ThermoFisher Scientific’s Velox™ software. In this environment, the STEM-EDX map of the lamella is quantified over an $X$ by $Y$ window using an empirical model consisting in a three-parameter Bethe-Heitler function, which is used to fit the entire measured spectrum. Applying a background model, such spectrum based quantification was applied to a line scan over the entire length of the stack. The data acquired in this way are shown for the germanium atomic fraction as blue dots in Fig. 3. Two more lamellas of the same sample were used to generally verify the results but have not undergone the entire data evaluation procedure. The measured atomic fraction of germanium in SiGe1 is 18.7% and in SiGe2 40.5%.

Fig. 3

Measured germanium content through the sample cross section determined by STEM-EDX (blue dots). The silicon substrate begins at $x>115\mathrm{nm}$. The red line shows a model fit according to Eq. (2). The vertical gray lines point to the positions of the layer interfaces and the red shaded areas show the extent of the associated intermixing layers and correspond to $\pm 2\sigma$.

4.2.

Model Fit

To determine the layer thicknesses and to extract the interface sharpness, we use the following model equation to describe the EDX data:

5.1.

Comparison of Soft X-ray and STEM-EDX Results: Layer Thicknesses, Interface Sharpness, and Applicability

In Fig. 4, we present the geometrical parameters of the sample as determined by soft X-ray reflectometry and STEM-EDX. Figures 4(a) and 4(c) show the film thicknesses and their correlation where we find an excellent agreement between the two methods, which has also been verified on a second sample (data not shown). This demonstrates the general applicability of both, broadband soft X-ray reflectometry and STEM-EDX to the problem. Note that the measured thicknesses deviate from the design values, given in Fig. 1(a) by $\pm 1\mathrm{nm}$.

Fig. 4

Comparison of soft X-ray-based and STEM-EDX-based determination of film thicknesses and interface intermixing $\sigma$. In panels (a) and (c), it is shown that the film thicknesses correlate very well for all layers. In panels (b) and (d), it is visible that the interface intermixing parameters, determined through STEM-EDX follow the same trend as those determined through reflectometry but feature slightly higher values.

Figures 4(b) and 4(d) compare the retrieved interface sharpness parameters $\sigma$. We find that both methods show the same trend over the layer stack and that they compare well, although STEM-EDX retrieves generally higher values of $\sigma$ than reflectometry, which means that reflectometry detects slightly sharper layer transitions than STEM-EDX. The difference between the determined layer thicknesses and sharpness of both methods is in the range of a few angstroms. It is possible that these small differences originate from the fact that different sample positions of the 300 mm wafer were probed, where deviations on an angstrom-level can occur over lateral distances of centimeters. Furthermore, the STEM-EDX data represent only a small fraction on the sample in the range of tens of nanometers, whereas the reflectometry results represent a spatial average in the range of a few square millimeters, due to the nature of the measurements. Therefore, it cannot be excluded that the small observed differences stem from spatial inhomogeneities of the sample itself. The labels in Fig. 4(b) refer to the interfaces from top to bottom, such that “Si/SiGe1” refers to an interface where a silicon layer had been deposited on top of a SiGe1 layer. Detected by both methods are two remarkable trends in $\sigma$ that give insight into the details of the layer structure: first we see that SiGe on top of Si gives sharper interfaces (lower $\sigma$) than Si on top of SiGe. Second, the interfaces containing SiGe1 are generally sharper than those, containing SiGe2.

As explained earlier, the $\sigma$-value in reflectometry is a measure of interface sharpness and roughness.²⁰ This number is based on the theory that either the lateral displacement of the interface’s position varies stochastically or that there is a region of layer intermixing instead of perfectly well defined interfaces.²² Our TEM study shows that the interfaces between the layers can be well described by a smooth transition of the germanium/silicon content, following an error function. It further shows that lateral roughness can be neglected for the present case because its amplitude is far smaller than the extent of the intermixing region, visible in the dark field TEM image in Fig. 1. Therefore, we can directly compare the determined values of ${\sigma }_i^{\mathrm{refl}.}$ from Eq. (1) with those determined through STEM-DEX ${\sigma }_i^{\mathrm{EDX}}$ in Eq. (2), because they describe the same quantity.

In the present case, the overall quality of the blanket layer stacks was very high in the sense that the layers were crystalline, spatially homogeneous, not porous, and very smooth, due to the epitactic deposition process. This is advantageous both for TEM lamella preparation and the data evaluation of the reflectometry measurements and makes the samples ideally suited for this kind of comparison. Both measurement methods come with advantages and disadvantages. Soft X-ray reflectometry is a non-destructive method that gives insight into averaged sample properties. It is sensitive to the surface and buried layers and interfaces down to approximately 100 nm depth, depending on the materials and the wavelengths used. It requires a relatively large sample area due to the increased beam footprint at grazing incidence and can only provide layer thickness and interface properties through model-based reconstruction. This modeling either has to include a fit of the optical constants, as done in the present work, or needs precise knowledge of the properties of the materials in question. STEM-EDX, as an imaging technique, directly provides the sample geometry and the material distribution. It gives local, but high-resolution information about the sample. The method is destructive since lamellas must be cut out of the sample. Both methods require advanced equipment, but STEM-EDX is typically more easily accessible than synchrotron-based reflectometry.

5.2.

Optical Constants of SiGe

An additional result of the reflectance data fit were the optical constants of the two SiGe variants, which we present in Fig. 5 alongside a comparison to existing data of pure silicon and pure germanium.²⁴ Appendix A gives the full list of the retrieved optical constants for reference. Germanium’s extinction coefficient $\beta$ is monotonically decreasing with increasing energy, and its dispersion coefficient $\delta$ features a very broad maximum around 180.1 eV from the M3 edge.²⁴ Silicon, on the other hand, has a prominent absorption edge from its L-edges at 99.2 eV (L2) and 99.8 eV (L3),²⁴ visible around 100 eV in the data. This feature is also observed in the optical constants of both SiGe variants. Without further data evaluation, it is visible that SiGe1 (blue line) falls between the optical constants of pure silicon and pure germanium. When the optical constants of SiGe1 are fitted to a mixture of the displayed tabulated data of silicon and germanium, an atomic fraction of 17.7% germanium is determined at a RMSE of $5·{10}^{-4}$, showing that the optical constants of this material can be predicted reasonably well from the materials of its constituents. For SiGe2, this works only in a qualified sense. Here, the obtained atomic fraction is 40.3% at a RMSE of $1.4·{10}^{-3}$, which means that the prediction of the optical constants from tabulated data will not be as accurate. This is especially true for the spectral range around the silicon L-edge and underlines the need to determine optical constants for compound materials.¹⁶ In this regime, the independent atom approximation for the optical constants begins to fail and the electric states of the inner shells are influenced by their neighborhood.²⁹

Fig. 5

Optical constants of two variants of SiGe, compared to tabulated data for pure silicon and pure germanium.²⁴

5.3.

Atomic Fraction of Germanium in the SiGe Layers

Both methods determine the atomic mass fraction of the layers, i.e., they can determine the amount of germanium in the SiGe layers. For STEM-EDX, the quantification is straight-forward, whereas soft X-ray reflectometry depends on the comparison to tabulated data of the optical constants. Table 1 summarizes the atomic fraction of germanium as determined by the two complementary methods versus their design values. We find a good agreement and note that for SiGe1, the atomic fraction of germanium is lower than its design value.

Table 1

Comparison of the atomic fraction of germanium in the two SiGe variants.

5.4.

Comparison with Other Works and Methods

Many other works exist that study the quality of interfaces, especially for semiconductor materials. Often, the focus lies on comparative studies of interfaces with varying quality, such as in Manz et al.,³⁰ who used a combination of many methods, among them XRR and STEM-EDX, to arrive at the conclusion that they can all be utilized to study the interface sharpness. A similar comparison was done much earlier using Rutherford backscattering and TEM-EDX.³¹ Luneville at el.³² presented a comparison between XRR and EDX for different Cr/Si interfaces, based on the good material contrast for harder X-rays. However, providing a quantitative comparison of different measurement methods is still the exception, as it was usually sufficient to compare trends only. For the fabrication of quantum wells, the precise determination of the extent of intermixing regions is central due to its impact on valley splitting. Just recently, a number of studies were published that use atom probe tomography^4,5 or HR-STEM^2,3 to quantify the interface sharpness of Si/SiGe interfaces, reporting similar values to our work.