Home Extracting information from X-ray diffraction patterns containing Laue oscillations
Article Publicly Available

Extracting information from X-ray diffraction patterns containing Laue oscillations

  • Aaron M. Miller , Mellie Lemon , Marisa A. Choffel , Sarah R. Rich , Fischer Harvel and David C. Johnson EMAIL logo
Published/Copyright: March 28, 2022
Download Article (PDF)
Become an author with De Gruyter Brill
Author Information Explore this Subject
Zeitschrift für Naturforschung B
From the journal Zeitschrift für Naturforschung B Volume 77 Issue 4-5

Abstract

The presence of Laue oscillations in a film grown on a solid surface is broadly taken as indicating a high quality, crystallographically aligned film of the targeted compound. In this paper we briefly review the origins of both Laue oscillations and Kiessig fringes and show how they can be used together to determine if extra thickness exists above or below the coherently diffracting domains. The differences between experimental and “ideal” films are discussed and the effect of structural features (roughness, different thickness coherently diffracting domains and thickness in addition to the coherently diffracting domains) are illustrated with experimental and simulated data for metal and mixed-metal chalcogenide films of titanium, bismuth, vanadium/iron, and bismuth/molybdenum. Examples are given showing how quantitative information can be extracted from experimental diffraction patterns.

Keywords: modeling; structural analysis; thin films; X-ray diffraction; X-ray reflectivity

1 Introduction

Laue oscillations result from the incomplete destructive interference of a finite number of unit cells and occur when a sample consists of domains with the same number of unit cells across most of the area being probed. First predicted by Max von Laue, the Laue interference function relates the number of unit cells in the diffracting crystal to the distribution of diffracted intensity [1]. Generally, the presence of Laue oscillations are taken as confirmation that grown films are of high quality, homogenous, contain only the targeted compound, and have smooth and planar top and bottom interfaces [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. The presence of Laue oscillations is frequently used as evidence of “the high crystallinity of samples” [12], “the uniformity of the film and smoothness of the interfaces” [13], or that “the out-of-plane order is high and coherent over the entire film thickness” [14].

While it is true that Laue oscillations are a qualitative indicator of sample quality, the presence of Laue oscillations also provides an opportunity to gain significant structural information about films. The most common quantitative analysis of Laue oscillations utilizes an equation derived from the Laue interference function to extract the total thickness of the crystalline phase [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. The thickness obtained in this manner is often taken to be the total film thickness, which assumes that there is no additional thickness from amorphous or non-crystallographically aligned layers present above and/or below the diffracting crystal. There are only a few reports in the literature where both the oscillations in the X-ray reflectivity (XRR) at low diffraction angles and the Laue oscillations observed in the vicinity of a Bragg reflection are used to detect potential excess material. In these reports, differences between the total film thickness calculated from Kiessig fringes in the XRR data and the thickness of the crystalline layers obtained via the Laue oscillations were found [26, 27]. Furthermore, the intensity of experimental Laue oscillations often differs from those predicted from the Laue function. The Laue function results in symmetric intensities of satellite reflections on either side of the Bragg maxima, but an asymmetric distribution of intensities on each side is also frequently observed [9, 26–28]. In addition, the number of Laue oscillations observed on either side of the Bragg reflection varies significantly from sample to sample [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]. The extraction of structural information from Laue oscillations has been challenging due to the lack of a discussion of all relevant physical phenomena in a single reference that relates structural parameters to Laue intensities and provides examples illustrating the development of structural models from experimental data.

This paper addresses this challenge by presenting a summary of the relevant physical phenomena, showing how structural features in films impact the intensity and number of both Kiessig fringes and Laue oscillations that are observed, and provides examples of developing a structural model from experimental data. The first example illustrates an approach to simultaneously model reflectivity and diffraction patterns when these two phenomena are relatively uncoupled. The second example involves a more complex example where these two phenomena are both important in the same angular regions. Further efforts are required to create simulation software that enables the development of atom-level structural descriptions of films using the intensities of Kiessig fringes and Laue oscillations.

2 Results and discussion

We begin this section with a short review of the physical origin of Kiessig fringes and show how increasing the structural complexity of films affects the intensity of the fringes with increasing angle using simulations. Examples show how to extract structural information from experimental data. Next, we discuss the origin of Laue oscillations and use examples to demonstrate how Laue oscillation intensities calculated using the Laue function differ from experimental patterns. We illustrate how the interaction between reflectivity and diffraction effects cause the asymmetry in Laue intensities around the central Bragg reflection and use simulations to show the impact of structural imperfections on the intensities of Laue oscillations. We conclude by developing structural models from two experimental data sets where the total film thickness (calculated from the period of the Kiessig oscillations) is different from the thickness derived from the period of the Laue oscillations (defined as the product of the number of coherently diffracting unit cells and the c-axis lattice parameter).

2.1 X-ray reflectivity (XRR) and Kiessig fringes

Figure 1 shows XRR patterns for an experimental TiSe2 film, along with simulated reflectivity patterns for 5.00 nm (blue) and 25.00 nm (purple) TiSe2 films assuming a uniform electron density slab using the BedeREFS software [29]. The oscillations observed at low angles in these patterns are known as Kiessig fringes, first reported in 1931 [30], which result from interference between X-rays reflected off the top air/sample interface and those reflected off the sample/substrate interface. The position and spacing of the maxima (or minima) can be used to quantitatively determine the film thickness using a modified version of Bragg’s law that includes a correction for refraction, as shown in Eq. (1) [31].

(1) sin 2 θ i = θ c 2 + ( n i + Δ n ) 2 λ 2 4 t 2
Figure 1: Experimental X-ray reflectivity (XRR) pattern of a 50.10(5) nm (black/gray) TiSe2 film. Simulated reflectivity patterns from 50.1 nm (green), 25 nm (purple), and 5 nm (blue) TiSe2 films are also shown. The period of the Kiessig fringes is inversely related to the film’s thickness.
Figure 1:

Experimental X-ray reflectivity (XRR) pattern of a 50.10(5) nm (black/gray) TiSe2 film. Simulated reflectivity patterns from 50.1 nm (green), 25 nm (purple), and 5 nm (blue) TiSe2 films are also shown. The period of the Kiessig fringes is inversely related to the film’s thickness.

In this equation, θi is the angle of the observed Kiessig fringe maxima, θc is the critical angle, ni is the index of the observed Kiessig fringe maxima, λ is the wavelength of the radiation utilized, and t is the total film thickness. As shown in Figure 1, the period of the observed Kiessig fringes is inversely related to the thickness of the films. The thickness of the experimental TiSe2 film calculated using this equation is 50.10(5) nm (black trace).

An important point is that Kiessig fringes result solely from reflectivity phenomena – their presence and period does not depend on the crystallinity of the sample. Kiessig fringes will be observed in the reflectivity pattern for any thin enough samples with sufficiently smooth planar air/sample and sample/substrate interfaces, provided there is a difference in the index of refraction between the sample and the substrate. The intensity of the Kiessig fringes scales with the difference in electron density between the substrate and the film, with a larger difference producing more intense oscillations. Additionally, because the critical angle for total internal reflection (θc) scales with electron density, the angle of each Kiessig fringe shifts depending on the average electron density of the film (Eq. (1)).

The rate of decay in the intensity of Kiessig fringes with increasing angle depends on interface roughness. Simulated reflectivity patterns from models containing atomically abrupt interfaces have Kiessig fringes that continue throughout the angular range, decreasing in intensity as the angle increases. The Kiessig fringes in the experimental pattern become unresolvable at an angle of ∼7° 2θ. Parratt showed that the angle where Kiessig fringes are no longer visible depends on the sample’s average top and bottom surface roughness and derived the relationship:

(2) Δ t = λ 4 ( θ m 2 − θ c 2 ) 1 / 2

where θm is the angle of the last observed Kiessig maxima and θc is the critical angle [32]. The roughness of both the top and bottom interfaces controls the angle to which the Kiessig interference pattern will be visible. Figure 2 demonstrates how different amounts of roughness in the bottom sample/substrate (σsubstrate) and the top air/sample (σsample) interfaces reduce the intensity of the Kiessig fringes. The slope of the initial decay of the intensity of the Kiessig fringes is different for the bottom and top interfaces. The shape of the intensity envelope can therefore be used to distinguish between roughness at the top or bottom interfaces. Kiessig fringes observed out to 7° 2θ correspond to an interfacial roughness of about 5 Å according to the Parratt relationship, Eq. (2).

Figure 2: Simulated reflectivity patterns from a 30.19 nm TiSe2 film on a Si substrate with different amounts of interfacial roughness. The amount of interfacial roughness determines the maximum angle at which the Kiessig interference fringes are visible. The shape of the intensity decay differs depending on whether the roughness is at the top or at the bottom interface.
Figure 2:

Simulated reflectivity patterns from a 30.19 nm TiSe2 film on a Si substrate with different amounts of interfacial roughness. The amount of interfacial roughness determines the maximum angle at which the Kiessig interference fringes are visible. The shape of the intensity decay differs depending on whether the roughness is at the top or at the bottom interface.

Further information about a sample’s structure can be extracted from deviations in the shape of its XRR pattern from that expected for a single layer [29]. The reflectivity from a film with a single constituent and perfect, planar interfaces is described by the Fresnel equations and the intensity decay is smooth, even, and continues to the angle at which the average scattered intensity is less than the background intensity [33]. If there are two layers in the film with different electron densities, Kiessig fringes from the two layers will both be apparent in the XRR scan. Figure 3 illustrates this effect, showing separate simulated reflectivity patterns for 2.42 nm TiO2 and 50.05 nm TiSe2 films, along with the calculated reflectivity patterns for a film containing a 2.42 nm TiO2 layer on top of a 50.05 nm TiSe2 film on a silicon (Si) substrate. The presence of oscillations with two different frequencies indicates that a second layer of material with a unique electron density and thickness is present in a film. Although somewhat subtler, the experimental pattern in Figure 1 also shows this effect, with a weak, low frequency oscillation apparent under the higher intensity oscillations from the total film thickness. The large period of the underlying oscillation suggests that the additional layer is much thinner than the TiSe2 layer.

Figure 3: Simulated X-ray reflectivity (XRR) patterns of 2.42 nm TiO2 (blue) and 50.05 nm TiSe2 (green) films, along with the simulated pattern for a film containing 2.42 nm TiO2 on top of 50.05 nm TiSe2 (cerise).
Figure 3:

Simulated X-ray reflectivity (XRR) patterns of 2.42 nm TiO2 (blue) and 50.05 nm TiSe2 (green) films, along with the simulated pattern for a film containing 2.42 nm TiO2 on top of 50.05 nm TiSe2 (cerise).

2.2 Laue oscillations

If the sample consists of a crystallographically aligned film or contains a repeating sequence of deposited amorphous layers with different electron densities, the interference caused by the periodic changes in electron density results in Bragg reflections at angles given by Bragg’s Law:

(3) n λ = 2 d     sin ( θ )

where n is an integer, and nλ is the difference in distance traveled by a wave scattered by repeating planes of equal electron density that are a distance (d) apart. The resulting evenly spaced set of reflections can be indexed as a one-dimensional crystal. Using Bragg’s Law, the thickness of the layers (or the size of the unit cell of crystals orientated perpendicular to the substrate) can be extracted from the diffraction pattern. Figure 4 shows the XRD pattern for a 79-layers thick TiSe2 film, displaying four evenly spaced Bragg maxima, which can be indexed as 00l reflections consistent with those expected for a unit cell with a c-axis lattice parameter of 6.036(1) Å. The inset of Figure 4 expands the intensity and angular scale about the 001 reflection. The weak subsidiary maxima seen on the sides of this Bragg reflections are Laue oscillations. The positions and intensities of these satellite reflections are predicted by the Laue interference function:

(4) I ( Q ) ∝ sin ( N 2 Q c ) 2 sin ( 1 2 Q c ) 2

where c is the relevant lattice parameter, Q is the scattering vector, and N is the integral number of coherently diffracting unit cells [23, 33]. Because Laue oscillations originate from the incomplete destructive interference of a finite number of diffracting unit cells between Bragg reflections, their presence suggests a low defect density.

Figure 4: Experimental X-ray reflectivity (XRR) (gray) and X-ray diffraction (XRD) (black) patterns of a 79-layers crystalline TiSe2 film. The four Bragg reflections can be indexed as 00l reflections yielding a c-axis lattice parameter of 6.036(1) Å. The inset highlights the Laue oscillations observed on the 001 reflection.
Figure 4:

Experimental X-ray reflectivity (XRR) (gray) and X-ray diffraction (XRD) (black) patterns of a 79-layers crystalline TiSe2 film. The four Bragg reflections can be indexed as 00l reflections yielding a c-axis lattice parameter of 6.036(1) Å. The inset highlights the Laue oscillations observed on the 001 reflection.

Kiessig fringes and Laue oscillations provide complementary structural information about the sample. The period of the Kiessig fringes determines the total film thickness (tsample), inclusive of any impurity layers or amorphous material, while the period of the Laue oscillations determines the number of coherently diffracting unit cells in the film (N), which can be multiplied by the c-axis lattice parameter (c) to obtain the thickness of the coherently diffracting crystal.

Structural defects typically prevent experimental diffraction data from exactly matching that expected for a perfect film. For example, the total amount of material in a film is difficult to control so the total film thickness is typically not equal to the thickness of the coherently diffracting crystal. Thus, the angular positions of the Kiessig fringes will differ from those calculated from the thickness of the coherently diffraction domain obtained from the Laue oscillations. The experimental amplitude of Laue oscillations is also typically much smaller than calculated, often asymmetric with respect to the Bragg reflection, and the rate of decay of the oscillations as one moves away from the main Bragg reflection varies significantly from sample to sample. These differences in amplitude are not often discussed when Laue oscillations are observed.

The Laue interference function predicts a symmetric distribution of satellite reflections centered on each Bragg maximum, as shown in Figure 5a (blue trace), but experimentally the intensity of the Laue oscillations is often different on either side of the Bragg reflection. Asymmetry of intensities was present in slightly more than half of the 27 representative reports we examined in a non-exhaustive literature search. This asymmetry occurs whether the Bragg reflection is dominated by the substrate, as in epitaxially grown films, or if the Bragg reflection is caused only by the film itself. The TiSe2 film in Figure 5a (black trace) illustrates a typical intensity asymmetry around a Bragg reflection. The cause of the asymmetry in the intensity in this sample is the changing phase relationship between Kiessig fringes and Laue oscillations as the diffraction angle moves through that of a Bragg reflection. Figure 5b illustrates the effect of this changing phase relationship in a simulated diffraction pattern of a structurally perfect 301.8 nm thick film containing 50 6.036 Å thick TiSe2 layers. Here we use the approach of Zwiebler et al., approximating the unit cell structure with slabs of the appropriate element in the simulation [34]. For the TiSe2 layers, equal thickness slabs of Se, Ti, and Se were used totaling the thickness of the c-axis of the unit cell. Before the 001 reflection, the Kiessig and Laue effects are constructively interfering. Between the 001 and 002 reflections, the two are destructively interfering, resulting in the much lower average intensity between these reflections. The average intensity between the 2nd and 3rd reflections increases because the Kiessig and Laue effects are again constructively interfering. The bottom XRR pattern (black trace) in Figure 5 shows an expanded view of the oscillations around the 001 reflection, which are asymmetric. Asymmetry caused by interference of the Kiessig and Laue effects is most likely to be observed for Bragg reflections at smaller 2θ values due to the decay of Kiessig fringe intensities as 2θ increases.

Figure 5: Plots showing the significance of asymmetry in Laue oscillations. (a) Comparison of the experimental Laue oscillations observed on either side of the 001 reflection of a TiSe2 film with that calculated from the Laue interference function and a simulation that includes reflectivity using the BedeREFS simulation software. (b) A simulated X-ray reflectivity (XRR)/X-ray diffraction (XRD) pattern of a 50-layer Se|Ti|Se film with a c-axis lattice parameter of 6.036 Å. The changing phase relationship between Kiessig fringes and Laue oscillations as the scattering angle moves through that of Bragg reflections is apparent in the lower average intensity between the 1st and 2nd Bragg reflections.
Figure 5:

Plots showing the significance of asymmetry in Laue oscillations. (a) Comparison of the experimental Laue oscillations observed on either side of the 001 reflection of a TiSe2 film with that calculated from the Laue interference function and a simulation that includes reflectivity using the BedeREFS simulation software. (b) A simulated X-ray reflectivity (XRR)/X-ray diffraction (XRD) pattern of a 50-layer Se|Ti|Se film with a c-axis lattice parameter of 6.036 Å. The changing phase relationship between Kiessig fringes and Laue oscillations as the scattering angle moves through that of Bragg reflections is apparent in the lower average intensity between the 1st and 2nd Bragg reflections.

Experimentally observing the shift in phase between the Kiessig and Laue interference effects through Bragg reflections requires a film with extremely smooth interfaces, which is challenging to prepare experimentally. Figure 6 shows an experimental diffraction pattern where the changing sign relation between the two interference effects is clearly visible. This pattern also shows how the changing relative intensity of the Kiessig and Laue effects can cause a very weak Bragg reflection, resulting from the location of atoms in the unit cell, to appear split as the relative phase changes moving through the center of the 001 reflection.

Figure 6: Experimental diffraction data of a (BiSe)0.97(Bi2Se3)1.26(BiSe)0.97(MoSe2) heterostructure.
Figure 6:

Experimental diffraction data of a (BiSe)0.97(Bi2Se3)1.26(BiSe)0.97(MoSe2) heterostructure.

The intensities of both Kiessig and Laue oscillations also depend on the abruptness and smoothness of the interfaces in the film. The relative magnitude of intensities of Kiessig oscillations depends on the smoothness of interfaces and the magnitude of the electron density differences between the constituents. The intensity of Laue oscillations depends on the percentage of the film that contains the dominant thickness of coherently diffracting crystalline domains, the distribution of thickness of the crystalline domains (a form of roughness, as the entire area measured might contain several thicknesses), and the inherent intensity of the Bragg reflections, which depend on the location of atoms within the unit cell. Figure 7 shows the effects of increasing the substrate and sample interfacial roughness on the appearance of the interference pattern around the 001 reflection in simulated diffraction patterns of TiSe2. The simulations approximate roughness by replacing an abrupt change in electron density at interfaces with a smooth gradient of width σ. Kiessig interference fringes are damped out as the magnitude of the roughness at the interfaces increases. Increasing roughness can damp out the Kiessig fringes enough that a symmetrical distribution of satellite reflections occurs around the Bragg maxima at higher angles. This infers that samples with an asymmetric distribution of the intensity of Laue oscillations have smooth interfaces.

Figure 7: Simulated X-ray reflectivity (XRR) patterns of a 50-layers TiSe2 film on a Si substrate illustrating how the roughness affects the symmetry of the satellite reflections around the Bragg reflections. Increasing roughness of the substrate and/or surface damps the intensity of the Kiessig fringes, making the Laue oscillations more symmetric.
Figure 7:

Simulated X-ray reflectivity (XRR) patterns of a 50-layers TiSe2 film on a Si substrate illustrating how the roughness affects the symmetry of the satellite reflections around the Bragg reflections. Increasing roughness of the substrate and/or surface damps the intensity of the Kiessig fringes, making the Laue oscillations more symmetric.

The experimentally observed decrease in the intensities of Laue oscillations as one moves further away from the central Bragg maxima is typically much faster than predicted by the Laue oscillation function. While some of this intensity decrease is caused by substrate and/or surface roughness of “extra” material, a distribution in the thickness of the coherently diffracting domain also contributes to this accelerated decrease in intensity. Figure 8 contains several simulations, where the percentage of coherent domains of different thicknesses was varied. If there are only two different thicknesses present, the interference pattern between the two different Laue oscillation functions is evident and the fringes closest to the Bragg maxima yield the average value of the coherent diffracting domain thicknesses. The fringes close to the Bragg maxima also yield the average value for the thickness for broader distributions, but the intensity of the oscillations decreases as one moves away from the Bragg maxima. These simulations suggest that the further out from the Bragg maxima that Laue oscillations are observed, the narrower the size distribution of coherently diffracting domains.

Figure 8: Simulations of diffraction patterns from different distributions of coherently diffracting domains. The top simulation is from a sample with 44 TiSe2 layers. The simulations below this are from different percentages of film area with the indicated number of TiSe2 layers in the coherently diffracting domain.
Figure 8:

Simulations of diffraction patterns from different distributions of coherently diffracting domains. The top simulation is from a sample with 44 TiSe2 layers. The simulations below this are from different percentages of film area with the indicated number of TiSe2 layers in the coherently diffracting domain.

2.3 Developing structural models from Kiessig Fringes and Laue oscillations

We conclude with two examples demonstrating how to systematically construct structural models using extracted structural information from XRR and XRD data on samples with both Kiessig fringes and Laue oscillations. The first example is an Fe-doped VSe2 film whose XRR and XRD scans are shown in Figure 9. The total thickness of the film can be extracted from the Kiessig oscillations using Eq. (1), yielding a film thickness of 271.0(2) Å. The Laue oscillations are used to determine the number of unit cells in the coherently diffracting domain from their positions. The Laue oscillations are consistent with 44 unit cells in the diffracting domain. The positions of the 00l Bragg reflections are used to determine the c-axis lattice parameter of 6.088(3) Å. The product of the number of unit cells (44) and the c-axis lattice parameter (6.088(3) Å) yields the thickness of the coherently diffracting domain −267.9(1) Å. The Kiessig derived thickness is 3.1 Å larger, indicating that there is a thin layer of excess material. Independent corroborating evidence for a small amount of excess material was obtained from the absolute number of atoms/Å2 of each element determined from X-ray fluorescence (XRF) data, which indicated that the excess material is vanadium oxide [35].

Figure 9: Comparison of experimental XRR (gray) and XRD (black) patterns of a 271.0(2) Å thick crystalline Fe x V1−xSe2 film with simulated XRR patterns utilizing various structural models. A model of a 267.9 Å Fe x V1−xSe2 block with 3.1 Å of oxide on top allows estimation of interfacial roughness (teal). Splitting the Fe x V1−xSe2 into 44 unit cells results in a good fit to the experimental Laue oscillations around the 001 reflection, but the intensities are too high (cerise). Adding the interacial roughness damps these intensities to provide a better fit (green), although further damping would continue to improve the fit, such as that resulting from a broader distribution of coherently diffracting domains.
Figure 9:

Comparison of experimental XRR (gray) and XRD (black) patterns of a 271.0(2) Å thick crystalline Fe x V1−xSe2 film with simulated XRR patterns utilizing various structural models. A model of a 267.9 Å Fe x V1−xSe2 block with 3.1 Å of oxide on top allows estimation of interfacial roughness (teal). Splitting the Fe x V1−xSe2 into 44 unit cells results in a good fit to the experimental Laue oscillations around the 001 reflection, but the intensities are too high (cerise). Adding the interacial roughness damps these intensities to provide a better fit (green), although further damping would continue to improve the fit, such as that resulting from a broader distribution of coherently diffracting domains.

A structural model of the film to simulate the diffraction data below 20° 2θ was created from the data derived from the Kiessig and Laue oscillations. A model with a 267.9(1) Å thick Fe x V1−xSe2 layer and top 3.1 Å thick surface layer of vanadium oxide was used to determine the top and bottom roughness of the film. Figure 9 shows the simulated pattern with interfacial roughness of σVSe2 = 5.75 Å, σoxide = 5 Å, and σsubstrate = 2.5 Å, which reasonably matches the low angle experimental reflectivity pattern. Dividing the 267.9(1) Å thick Fe x V1−xSe2 layer into 44 explicit unit cells of Fe x V1−xSe2 by using elemental slabs as discussed earlier provides a good fit to the experimentally observed positions of the Laue oscillations around the 001 reflection (see Figure 9). The intensities of the Laue oscillations, however, are too large, as the actual sample probably does not contain exactly 44 unit cells across the entire area probed by the X-ray beam, and we need to add the effect of substrate roughness. Including the substrate roughness determined from the Fe x V1−xSe2 slab model does a reasonable job of matching the experimental pattern except that the Laue intensities are still too intense. The intensities of the Laue oscillations can be reduced by assuming that the film consists of regions that contain thinner coherently scattering domains, as discussed above. However, this will not yield a unique structural model for the film.

Diffraction patterns and their analysis become increasingly complicated as the Kiessig and Laue intensities interact across a large angular range. The XRR/XRD patterns collected of a (BiSe)0.97(Bi2Se3)1.26(BiSe)0.97(MoSe2) heterostructure, displayed in Figure 10, illustrates these challenges [36]. The extracted total film thickness from the Kiessig fringes is 309.6(5) Å. From the period of the Laue oscillations at higher angles, it was determined that there are 10 unit cells in the coherently diffracting domains. The product of 10 unit cells times the c-axis lattice parameter (27.97 (10) Å) gives a crystal thickness of 279.7 Å. The difference between these two values is 29.9 Å. The question is how does one divide this thickness between the top and bottom of the crystalline domains? Figure 10 contains several simulated XRR patterns from models that distribute the 30 Å of excess material between the top and/or the bottom of the (BiSe)0.97(Bi2Se3)1.26(BiSe)0.97(MoSe2) diffracting domain (Se was used as the excess material in these models). The simulated patterns are very sensitive to the exact distribution of the excess material between the front and back of the film. While we assumed in these simulations that the composition of excess material at the bottom and the top were the same, this is not necessarily true, which adds another unknown parameter to potential models. This experimental pattern does not contain an explicit feature that allows us to estimate or separate the roughness of the substrate, the film, or the excess material. The large number of potential variables makes it currently impossible to extract additional information through simulations. Additional information, for example from HAADF-STEM images of film cross sections, is needed to limit the parameter space.

Figure 10: Experimental X-ray reflectivity (XRR) (grey) and X-ray diffraction (XRD) (black) patterns of a (BiSe)0.97(Bi2Se3)1.26(BiSe)0.97(MoSe2) heterostructure annealed at 350 °C, along with simulated patterns for models that consist of 10 unit cells of the targeted heterostructure, plus 30 Å of additional material distributed between either the top and/or the bottom of the heterostructure. No interfacial roughness was added to these models.
Figure 10:

Experimental X-ray reflectivity (XRR) (grey) and X-ray diffraction (XRD) (black) patterns of a (BiSe)0.97(Bi2Se3)1.26(BiSe)0.97(MoSe2) heterostructure annealed at 350 °C, along with simulated patterns for models that consist of 10 unit cells of the targeted heterostructure, plus 30 Å of additional material distributed between either the top and/or the bottom of the heterostructure. No interfacial roughness was added to these models.

3 Conclusions

This manuscript shows how to extract quantitative structural information from Laue oscillations and Kiessig fringes. The thickness of the coherently diffracting domain can be calculated from the product of the c-axis lattice parameter and the number of unit cells present determined from the Laue oscillations. If there is extra material in the film, there will be a difference between total film thickness from the period of the Kiessig fringes and the thickness of the coherently diffracting crystal domain. When the Kiessig fringes damp out before Laue oscillations are observed, it is possible to extract the roughness of the substrate and of the deposited layers. Samples with large differences between the total film thickness and crystal thickness are challenging to analyze, because the simulated patterns vary considerably as the extra thickness is partitioned above or below the coherently diffracting domain. Also challenging are samples with large angular regions where both the Laue and Kiessig interference effects contribute significantly. In films where Laue oscillations occur around reflections at high angles, however, the approach presented provides valuable additional information. Additional simulation tools need to be developed to get access to the additional structural information present in Laue oscillations obtained from experimental data.

4 Experimental

Films for this study were prepared using a custom high vacuum physical vapor deposition (PVD) chamber. Artificially layered precursors for binary thin films were prepared by repeatedly depositing M|Se bilayers, where ideally the number of atoms/Å2 deposited in each bilayer is identical to the number required to form one unit cell of the targeted compound. Similarly, precursors for heterostructures were prepared by repeatedly depositing M|Se|M′|Se layers in the same manner. Metal layers were deposited using electron-beam guns, while Se was deposited with a Knudsen effusion cell. All precursors were deposited onto ⟨100⟩ Si substrates with a native SiO2 layer. A pressure of less than 1 × 10−7 Torr was maintained during the deposition. In-house deposition software was used to control and monitor the amount of material deposited in each layer using pneumatic-controlled shutters and quartz crystal microbalances. After deposition, the precursors were removed from the vacuum chamber, briefly exposed to atmosphere, and brought into a dry-box (N2 with <0.2 ppm O2) where they were heated for 30 min at 500 °C and 350 °C for the Fe-doped VSe2 and the (BiSe)0.97(Bi2Se3)1.26(BiSe)0.97(MoSe2) films, respectively.

Structural characterization was carried out via XRR and XRD, while composition was determined using XRF. XRR and specular XRD patterns were collected on a Bruker D-8 Discover diffractometer. All diffraction measurements utilized a Cu Kα radiation source. Special care must be taken when aligning each sample to the diffractometer, as the positions and intensities of reflectivity/diffraction features are extremely sensitive to sample alignment. To confirm the alignment of the sample in the goniometer, rocking curve scans were collected at two different small 2θ values. The maxima in both rocking curve scans occurring when the incident and exit angles are equal are evidence that the sample is correctly aligned in the center of the goniometer.

The absolute amount of each element deposited was determined using XRF data collected on a Rigaku ZSX Primus II with a rhodium source. Previously published calibration curves were used to relate the background-corrected integrated raw intensity to the atoms/Å2 for each element [35]. Table 1 contains the calculated amount of each element required for the crystalline domains that contribute to the observed Laue oscillations along with the total number of atoms per unit area (areal density) of each element, as determined from the XRF measurements of each film. Estimates of the target atoms/Å2 for each heterostructure were obtained from the quotient of the number of atoms of each element per unit cell and the basal plane area of the unit cell.

Table 1:

Experimental and target atomic areal density for the Fe-doped VSe2 and (BiSe)0.97(Bi2Se3)1.26(BiSe)0.97(MoSe2) films as determined from XRF measurements.

Exp. V atoms/Å2 Exp. Fe atoms/Å2 Exp. Se atoms/Å2 Target V atoms/Å2 Target Fe atoms/Å2 Target Se atoms/Å2 Target # of unit cells
3.43(7) 1.4(1) 10.2(3) 3.43 1.71 10.29 44
Exp. Bi atoms/Å 2 Exp. Mo atoms/Å 2 Exp. Se atoms/Å 2 Target Bi atoms/Å 2 Target Mo atoms/Å 2 Target Se atoms/Å 2 Target # of unit cells
3.88(8) 1.15(2) 6.5(3) 3.37 1.18 6.8 10

Simulated XRR/XRD patterns were created using the BedeREFS software, which incorporates both reflectivity and diffraction physics that are required to accurately simulate thin film X-ray patterns [29]. BedeREFS uses “slabs” of electron density that are appropriately scaled to match the thickness and electron density of each layer in the film’s structure. Unless otherwise specified, the “slab” models created to simulate the reflectivity patterns, shown in Figures 1–3, 5, and 7–10, had no added interfacial roughness, and were generated using a Si substrate with 10–20 Å of SiO2.

Dedicated to Professor Christian Näther on the occasion of his 60th birthday.

Corresponding author: David C. Johnson, Department of Chemistry & Biochemistry, University of Oregon, Eugene, OR 97403, USA, E-mail:

Funding source: U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES)

Award Identifier / Grant number: Award #DE-SC0020095

Funding source: Alexander von Humboldt Foundation

  1. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: DCJ acknowledges a research Humboldt Research Award from the Alexander von Humboldt Foundation. This work was partially supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) under Award #DE-SC0020095.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

1. James, R. W. The Optical Principles of the Diffraction of X-Rays; Ox Bow Press: Woodbridge, CT, 1982.Search in Google Scholar

2. Miyadera, T., Auchi, Y., Yamamoto, K., Ohashi, N., Koganezawa, T., Yaguchi, H., Yoshida, Y., Chikamatsu, M. Insights into microscopic crystal growth dynamics of CH3NH3PbI3 under a laser deposition process revealed by in situ X-ray diffraction. ACS Appl. Mater. Interfaces 2021, 13, 22559–22566; https://doi.org/10.1021/acsami.1c04488.Search in Google Scholar PubMed

3. Forst, M., Caviglia, A. D., Scherwitzl, R., Mankowsky, R., Zubko, P., Khanna, V., Bromberger, H., Wilkins, S. B., Chuang, Y. D., Lee, W. S., Schlotter, W. F., Turner, J. J., Dakovski, G. L., Minitti, M. P., Robinson, J., Clark, S. R., Jaksch, D., Triscone, J. M., Hill, J. P., Dhesi, S. S., Cavalleri, A. Spatially resolved ultrafast magnetic dynamics initiated at a complex oxide heterointerface. Nat. Mater. 2015, 14, 883–888; https://doi.org/10.1038/nmat4341.Search in Google Scholar PubMed

4. Du, C., Wang, H., Yang, F., Hammel, P. C. Systematic variation of spin-orbit coupling with d-orbital filling: large inverse spin Hall effect in 3d transition metals. Phys. Rev. B 2014, 90, 1–5; https://doi.org/10.1103/PhysRevB.90.140407.Search in Google Scholar

5. Tsuchiya, Y., Norota, K., Watabe, Y., Kuroda, T., Iwata, N., Hashimoto, T., Yamamoto, H. Growth difference of LaFeO3 thin films by pulsed laser deposition method using the targets prepared by Pechini and conventional solid solution methods. Trans. Mater. Res. Soc. Japan 2012, 37, 369–372; https://doi.org/10.14723/tmrsj.37.369.Search in Google Scholar

6. Iwata, N., Watabe, Y., Tsuchiya, Y., Norota, K., Hashimoto, T., Huijben, M., Rijnders, G., Blank, D. Growth and evaluation of [AFeOx/REFeO3] (A = Ca, Sr, RE = La, Bi) superlattices by pulsed laser deposition method using high density targets prepared by Pechini method. Mater. Res. Soc. Symp. Proc. 2012, 1477, 61–66; https://doi.org/10.1557/opl.2012.Search in Google Scholar

7. Grelet, E., Dardel, S., Bock, H., Goldmann, M., Lacaze, E., Nallet, F. Morphology of open films of discotic hexagonal columnar liquid crystals as probed by grazing incidence X-ray diffraction. Eur. Phys. J. E 2010, 31, 343–349; https://doi.org/10.1140/epje/i2010-10586-2.Search in Google Scholar PubMed

8. Krauss, T. N., Barrena, E., Zhang, X. N., De Oteyza, D. G., Major, J., Dehm, V., Würthner, F., Cavalcanti, L. P., Dosch, H. Three-dimensional molecular packing of thin organic films of PTCDI-C 8 determined by surface X-ray diffraction. Langmuir 2008, 24, 12742–12744; https://doi.org/10.1021/la8030182.Search in Google Scholar PubMed

9. Abe, Y., Kawamura, M., Sasaki, K. Highly textured (100) RuO2/(001) Ru multilayers prepared by sputtering. Jpn. J. Appl. Phys., Part 1 Regul. Pap. Short Notes Rev. Pap. 2002, 41, 6857–6861; https://doi.org/10.1143/JJAP.41.6857.Search in Google Scholar

10. Naito, M., Karimoto, S., Yamamoto, H., Nakada, H., Suzuki, K. Production of double-sided large-area high-Tc wafers by molecular beam epitaxy. IEEE Trans. Appl. Supercond. 2001, 11, 3848–3851; https://doi.org/10.1109/77.919904.Search in Google Scholar

11. Brangham, J. T., Meng, K. Y., Yang, A. S., Gallagher, J. C., Esser, B. D., White, S. P., Yu, S., McComb, D. W., Hammel, P. C., Yang, F. Thickness dependence of spin Hall angle of Au grown on Y3Fe5O12 epitaxial films. Phys. Rev. B 2016, 94, 6–11; https://doi.org/10.1103/PhysRevB.94.054418.Search in Google Scholar

12. Shu, X., Zhou, J., Liu, L., Lin, W., Zhou, C., Chen, S., Xie, Q., Ren, L., Xiaojiang, Y., Yang, H., Chen, J. Role of interfacial orbital hybridization in spin-orbit-torque generation in Pt-based heterostructures. Phys. Rev. Appl. 2020, 14, 1; https://doi.org/10.1103/PhysRevApplied.14.054056.Search in Google Scholar

13. Xu, J., Katoch, J., Ahmed, A. S., Pinchuk, I. V., Young, J. R., Johnston-Halperin, E., Pelz, J., Kawakami, R. K. Growth of uniform CaGe2 films by alternating layer molecular beam epitaxy. J. Cryst. Growth 2017, 460, 134–138; https://doi.org/10.1016/j.jcrysgro.2016.12.102.Search in Google Scholar

14. Song, Y., Li, Z., Li, H., Tang, S., Mu, G., Xu, L., Peng, W., Shen, D., Chen, Y., Xie, X., Jiang, M. Epitaxial growth and characterization of high quality Bi2O2Se thin films on SrTiO3 substrates by pulsed laser deposition. Nanotechnology 2020, 31; https://doi.org/10.1088/1361-6528/ab6686.Search in Google Scholar PubMed

15. Switzer, J. A., Hill, J. C., Mahenderkar, N. K., Liu, Y. C. Nanometer-thick gold on silicon as a proxy for single-crystal gold for the electrodeposition of epitaxial cuprous oxide thin films. ACS Appl. Mater. Interfaces 2016, 8, 15828–15837; https://doi.org/10.1021/acsami.6b04552.Search in Google Scholar PubMed

16. Watabe, Y., Iwata, N., Oikawa, T., Hashimoto, T., Huijben, M., Rijnders, G., Yamamoto, H. Fabrication and crystal structure of [ABO3/REMO3] (A = Ca, La, B = Fe, Mn, RE = Bi, La, M = Fe, Fe0.8Mn0.2) superlattices grown by pulsed laser deposition method. Jpn. J. Appl. Phys. 2014, 53, 05FB12-1–05FB12-8; https://doi.org/10.7567/JJAP.53.05FB12.Search in Google Scholar

17. Onbasli, M. C., Kehlberger, A., Kim, D. H., Jakob, G., Kläui, M., Chumak, A. V., Hillebrands, B., Ross, C. A. Pulsed laser deposition of epitaxial yttrium iron garnet films with low gilbert damping and bulk-like magnetization. APL. Mater. 2014, 2, 106102; https://doi.org/10.1063/1.4896936.Search in Google Scholar

18. Peters, B., Alfonsov, A., Blum, C. G. F., Hageman, S. J., Woodward, P. M., Wurmehl, S., Büchner, B., Yang, F. Y. Epitaxial films of Heusler compound Co2FeAl0.5Si0.5 with high crystalline quality grown by off-axis sputtering. Appl. Phys. Lett. 2013, 103, 162404; https://doi.org/10.1063/1.4825338.Search in Google Scholar

19. Hauser, A. J., Soliz, J. R., Dixit, M., Williams, R. E. A., Susner, M. A., Peters, B., Mier, L. M., Gustafson, T. L., Sumption, M. D., Fraser, H. L., Woodward, P. M., Yang, F. Y. Fully ordered Sr2CrReO6 epitaxial films: a high-temperature ferrimagnetic semiconductor. Phys. Rev. B 2012, 85, 4–7; https://doi.org/10.1103/PhysRevB.85.161201.Search in Google Scholar

20. Switzer, J. A., Gudavarthy, R. V., Kulp, E. A., Mu, G., He, Z., Wessel, A. J. Resistance switching in electrodeposited magnetite superlattices. J. Am. Chem. Soc. 2010, 132, 1258–1260; https://doi.org/10.1021/ja909295y.Search in Google Scholar PubMed

21. Asanuma, S., Fukunaga, M., Uesu, Y., Haumont, R., Dkhil, B., Malibert, C., Kiat, J. M. Fabrications of Pb(Sc1/2Nb1/2)O3/xPbTiO3 thin films and their structural characterizations. Jpn. J. Appl. Phys., Part 1 Regul. Pap. Short Notes Rev. Pap. 2004, 43, 6581–6584; https://doi.org/10.1143/JJAP.43.6581.Search in Google Scholar

22. Dürr, A. C., Schreiber, F., Münch, M., Karl, N., Krause, B., Kruppa, V., Dosch, H. High structural order in thin films of the organic semiconductor diindenoperylene. Appl. Phys. Lett. 2002, 81, 2276–2278; https://doi.org/10.1063/1.1508436.Search in Google Scholar

23. Ern, C., Donner, W., Dosch, H., Adams, B., Nowikow, D. Temperature-dependent interfacial stiffness of the disorder layer in a thin Cu3Au alloy film. Phys. Rev. Lett. 2000, 85, 1926–1929; https://doi.org/10.1103/PhysRevLett.85.1926.Search in Google Scholar

24. Mahenderkar, N. K., Chen, Q., Liu, Y., Duchild, A. R., Hofheins, S., Chason, E., Switzer, J. A. Epitaxial lift-off of electrodeposited single-crystal gold foils for flexible electronics. Science 2017, 355, 1203–1206.10.1126/science.aam5830Search in Google Scholar

25. Lu, W., Yang, P., Song, W. D., Chow, G. M., Chen, J. S. Control of oxygen octahedral rotations and physical properties in SrRuO3 films. Phys. Rev. B 2013, 88, 1–11; https://doi.org/10.1103/PhysRevB.88.214115.Search in Google Scholar

26. Xu, S. J., Shi, Z., Zhou, S. M. Clear evidence of interfacial anomalous Hall effect in epitaxial L10 FePt and FePd films. Phys. Rev. B 2018, 98, 024413; https://doi.org/10.1103/PhysRevB.98.024413.Search in Google Scholar

27. Sojková, M., Dobročka, E., Hutár, P., Tašková, V., Pribusová Slušná, L., Stoklas, R., Píš, I., Bondino, F., Munnik, F., Hulman, M. High carrier mobility epitaxially aligned PtSe2 films grown by one-zone selenization. Appl. Surf. Sci. 2021, 538, 147936; https://doi.org/10.1016/j.apsusc.2020.147936.Search in Google Scholar

28. Lee, A. J., Brangham, J. T., Cheng, Y., White, S. P., Ruane, W. T., Esser, B. D., McComb, D. W., Hammel, P. C., Yang, F. Metallic ferromagnetic films with magnetic damping under 1.4 × 10-3. Nat. Commun. 2017, 8, 1–6; https://doi.org/10.1038/s41467-017-00332-x.Search in Google Scholar

29. Wormington, M., Panaccione, C., Matney, K. M., Bowen, D. K. Characterization of structures from X-ray scattering data using genetic algorithms. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 1999, 357, 2827–2848.10.1098/rsta.1999.0469Search in Google Scholar

30. von Kiessig, H. Interferenz von Röntgenstrahlen an Dünnen Schichten. Ann. Phys. 1931, 402, 769–788; https://doi.org/10.1002/andp.19314020702.Search in Google Scholar

31. Huang, T. C., Gilles, R., Will, G. Thin-film thickness and density determination from x-ray reflectivity data using a conventional power diffractometer. Thin Solid Films 1993, 230, 99–101; https://doi.org/10.1016/0040-6090(93)90499-F.Search in Google Scholar

32. Wainfan, N., Parratt, L. G. X-ray reflection studies of the anneal and oxidation of some thin solid films. J. Appl. Phys. 1960, 31, 1331–1337; https://doi.org/10.1063/1.1735837.Search in Google Scholar

33. Schierle, E. Antiferromagnetism in Thin Films Studied by Resonant Magnetic Soft X-Ray Scattering. Dissertation, Freie Universität Berlin, Berlin, 2006.Search in Google Scholar

34. Zwiebler, M., Hamann-Borrero, J. E., Vafaee, M., Komissinskiy, P., Macke, S., Sutarto, R., He, F., Büchner, B., Sawatzky, G. A., Alff, L., Geck, J. Electronic depth profiles with atomic layer resolution from resonant soft X-ray reflectivity. New J. Phys. 2015, 17, 083046; https://doi.org/10.1088/1367-2630/17/8/083046.Search in Google Scholar

35. Hamann, D. M., Bardgett, D., Cordova, D. L. M., Maynard, L. A., Hadland, E. C., Lygo, A. C., Wood, S. R., Esters, M., Johnson, D. C. Sub-monolayer accuracy in determining the number of atoms per unit area in ultrathin films using X-ray fluorescence. Chem. Mater. 2018, 30, 6209–6216; https://doi.org/10.1021/acs.chemmater.8b02591.Search in Google Scholar

36. Choffel, M. A., Gannon, R. N., Göhler, F., Miller, A. M., Medlin, D. L., Seyller, T., Johnson, D. C. Synthesis and electrical properties of a new compound (BiSe)0.97(Bi2Se3)1.26(BiSe)0.97(MoSe2) containing metallic 1T-MoSe2. Chem. Mater. 2021, 33, 6403–6411; https://doi.org/10.1021/acs.chemmater.1c01623.Search in Google Scholar
Received: 2022-03-04
Accepted: 2022-03-10
Published Online: 2022-03-28
Published in Print: 2022-05-25

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. In this issue
  3. Laudatio/Preface
  4. Christian Näther zum 60. Geburtstag gewidmet
  5. Research Articles
  6. Bismuth-rich bimetallic clusters (CuBi8)3+ and [MBi10]4+ (M = Pd, Pt) from ionothermal synthesis
  7. Crystal structure of phenanthrenide salts stabilized by 15-crown-5 and 18-crown-6
  8. Structure and properties of two new heteroleptic bismuth(III) dithiocabamates of the general composition Bi(S2CNH2)2X (X = Cl, SCN)
  9. Synthesis and structural characterization of three new mixed ligand alkaline-earth metal picrates
  10. Dimorphism of MnHAsO4(H2O): natural monoclinic krautite and its synthetic triclinic modification
  11. Synthesis, crystal structure, and topology of a polycatenated bismuth coordination polymer
  12. The unexpected crystal structure of thallium(I) tricyanomethanide Tl[C(CN)3]
  13. Synthesis, structure characterization and properties of a new oxidovanadium(IV) coordination polymer incorporating bridging (MoO4)2– and (Mo8O26)4– ligands
  14. Crystal structure of Dy11Ge4.33In5.67 and Tm11Ge4In6 from X-ray single-crystal and powder data
  15. Crystallisation of phosphates revisited: a multi-step formation process for SrHPO4
  16. Oxygen evolving reactions catalyzed by different manganese oxides: the role of oxidation state and specific surface area
  17. Synthesis and structural characterization of a new heterometallicmolybdate coordination polymer based on a µ3-bridging amino alcohol
  18. Chemically and Light-Driven Coordination-Induced Spin State Switching (CISSS) of a nonheme-iron complex
  19. Extracting information from X-ray diffraction patterns containing Laue oscillations
  20. Gadolinium trisilicide − a paramagnetic representative of the YbSi3 type series
https://doi.org/10.1515/znb-2022-0020
Keywords for this article
modeling; structural analysis; thin films; X-ray diffraction; X-ray reflectivity

Articles in the same Issue

  1. Frontmatter
  2. In this issue
  3. Laudatio/Preface
  4. Christian Näther zum 60. Geburtstag gewidmet
  5. Research Articles
  6. Bismuth-rich bimetallic clusters (CuBi8)3+ and [MBi10]4+ (M = Pd, Pt) from ionothermal synthesis
  7. Crystal structure of phenanthrenide salts stabilized by 15-crown-5 and 18-crown-6
  8. Structure and properties of two new heteroleptic bismuth(III) dithiocabamates of the general composition Bi(S2CNH2)2X (X = Cl, SCN)
  9. Synthesis and structural characterization of three new mixed ligand alkaline-earth metal picrates
  10. Dimorphism of MnHAsO4(H2O): natural monoclinic krautite and its synthetic triclinic modification
  11. Synthesis, crystal structure, and topology of a polycatenated bismuth coordination polymer
  12. The unexpected crystal structure of thallium(I) tricyanomethanide Tl[C(CN)3]
  13. Synthesis, structure characterization and properties of a new oxidovanadium(IV) coordination polymer incorporating bridging (MoO4)2– and (Mo8O26)4– ligands
  14. Crystal structure of Dy11Ge4.33In5.67 and Tm11Ge4In6 from X-ray single-crystal and powder data
  15. Crystallisation of phosphates revisited: a multi-step formation process for SrHPO4
  16. Oxygen evolving reactions catalyzed by different manganese oxides: the role of oxidation state and specific surface area
  17. Synthesis and structural characterization of a new heterometallicmolybdate coordination polymer based on a µ3-bridging amino alcohol
  18. Chemically and Light-Driven Coordination-Induced Spin State Switching (CISSS) of a nonheme-iron complex
  19. Extracting information from X-ray diffraction patterns containing Laue oscillations
  20. Gadolinium trisilicide − a paramagnetic representative of the YbSi3 type series
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 14.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/znb-2022-0020/html
Scroll to top button