# Application of GISAXS in the Investigation of Three-Dimensional Lattices of Nanostructures

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{2}O

_{3}) matrix. These materials were deposited as thin films by the magnetron sputtering deposition system K.J. Lesker Company (Jefferson Hills, PA, USA), CMS-18, available in the Thin films laboratory of the Ruđer Bošković Institute (Zagreb, Croatia), on Si(100) substrates. The base pressure was in the range of 10

^{−6}–10

^{−5}Pa, while the Ar pressure was 0.66 Pa in a continuous flow (13 cm

^{3}∙min

^{−1}). The substrate was rotated (with 1 rpm) during the depositions to ensure a homogeneous thickness of the layers. The 3D ordering of the NOs was achieved by self-assembled growth type, using surface morphology effects during their nucleation (see References [8,10,11]). The sputtering power of the alumina target was 150 W for all depositions. All depositions were controlled by the software of the KJLC CMS-18 system, ensuring the same deposition conditions for each layer.

_{2}O

_{3}) × 20 multilayer deposition at 400 °C with an addition of nitrogen (6 at.%). Pure Ge and Al

_{2}O

_{3}(99.995%), produced by K.J. Lesker Company were used as targets in Direct Current (DC) (9 W) and Radio Frequency (RF) (150 W) magnetron discharge modes, respectively, with a duration of 45 s for Ge and 110 s for alumina in each layer. The second type, Ge NWs (denoted by NW1 and NW2), were prepared by continuous Ge + Al

_{2}O

_{3}co-deposition with duration of 30 min, at 400 °C for NW1 and 200 °C for NW2, while the sputtering powers for the Ge target were 2.5 W and 10.0 W for the films NW1 and NW2, respectively. The third material type considered (GeSi1), consisting of Ge/Si core/shell QDs, was prepared by deposition of (Ge/Si/Al

_{2}O

_{3}) × 20 multilayer at 300 °C. Pure Ge, Si, and Al

_{2}O

_{3}(99.995%) were used as targets in DC (9 W), DC (50 W), and RF (150 W) magnetron discharge modes, with a duration of 50, 60, and 120 s respectively. Details of the preparation can be found in References [10,11].

## 3. Results

#### 3.1. General Properties of GISAXS Intensity Maps

**R**and

**R’**are the position vectors of the NOs,

**Q**is the scattering vector,

**q**is the complex scattering vector corrected to refraction at the vacuum–substrate interface, and Ω

**(**

_{R}**q**) is the shape factor of a NO occurring atposition

**R**; the shape factor is derived from the function that is unity in the NO volume and zero outside it; t

_{i}and t

_{f}are the Fresnel transmittivities [20] of the substrate surface corresponding to the primary and scattered waves, respectively (${\left|{t}_{i}{t}_{f}\right|}^{2}$ gives the Yoneda wing in a GISAXS map). The brackets 〈 〉 indicate averaging over the positions and shapes of the NOs. In this paper, we consider the case in which the NO sizes are not correlated with their positions (decoupling approximation); for details about different approximations, please see Reference [1].

**Q**) space. The key parameter for the successful application of Equation (1) is to make good models for the NO shape (shape factor) and NO ordering (structure factor), as mentioned previously. The shape factor Ω(

**q**) is the Fourier transform of the NO shape; therefore, if we know the shape precisely, it is usually not difficult to calculate it. However, for the determination of the structure factor, the summation should be performed over all pairs of the NO positions

**R**and

**R**’ within the probed volume. This is the most important task in the determination of the structure factor, and, for that purpose, we must use approximations for the definitions of the NO positions, which make the calculation of the sum and its averaging in Equation (1) easier.

**R**

_{n}) can be expressed as a sum of basis vector

**a**of an ideal (undisturbed) lattice and deviation vector: ${\mathit{R}}_{n}=n\mathit{a}+{\mathit{D}}_{n}$, where

**D**

_{n}denotes the total deviation of a NO with index n from its ideal position (n

**a**). The description of

**D**

_{n}depends on the type of NO ordering, as described in detail in the next section.

#### 3.2. Ordering Types in Nanostructured Materials

**R**

_{n}), the positions of the 3D regularly ordered NOs deviate from the ideal positions that they would have in a perfect crystal lattice defined by basis vectors

**a**

_{1}–

**a**

_{3}. While the ideal positions define the position of peaks in the GISAXS map of a certain nanomaterial, the deviations are very important because they are closely related to the intensity distribution of each peak in the map. Therefore, the statistical properties of the NO deviations from the ideal positions (vectors

**D**

_{n}) are crucial for the successful analysis of a GISAXS map. The mathematical description of the statistical properties of the deviations is sometimes complicated. As mentioned before, we use a 3D paracrystal model to describe the NO positions and their deviations. This model treats each component of the basis vectors (

**a**

_{1}–

**a**

_{3}) and the related deviations (

**D**

_{n}

^{1}–

**D**

_{n}

^{3}) as independent variables.

**a**

_{1}–

**a**

_{3}and we assume that the x, y, z components (δ

_{x;y;z}) of the random deviation

**δ**

_{n}are normally distributed with zero mean and root-mean-square (r.m.s.) dispersion σ

_{x}, σ

_{y}, and σ

_{z}, respectively. Hence, to describe the system of NOs, we need to know the component of the basis vectors that characterize their ordering ${a}_{x,y,z}^{1-3}$ (we interchanged the upper and lower indices of the standard notation

**a**

_{1}–

**a**

_{3}to include the space x, y, z components and to have the same notation as in Reference [3]), number of periods in each space direction N

_{1}–N

_{3}, the disorder parameters in a particular space direction${\sigma}_{x,y,z}^{1-3}$, and, of course, the NO shape parameters.

**f**

_{1}and

**f**

_{2}are the components of the shape factor, calculated as follows:

_{1}–N

_{3}are the numbers of the QD position (

**R**

_{n}

_{1,n2,n3}) indices (number of periods) in the x, y and z directions, respectively.

_{3D}(

**q**) is a product of three 1D correlation functions ${G}_{ot}^{\left(i\right)}$(

**q**); i = 1, 2, 3 and they depend on the ordering type in the particular space direction.

**a**

_{1}and

**a**

_{2}are in the x, y plane, while the third basis vector may have all three components, depending on the ordering type. Therefore, only the equations related to the basis vector

**a**

_{3}(${G}_{ot}^{\left(3\right)}$(

**q**);) have the refraction effects included (the others contain only the x and y components; thus, there is no need).

_{3D}(Equation (14)) using 1D functions G(

**q**) (Equations (7)–(13)). We must be careful to choose the correct functions for the description, because a GISAXS map strongly depends on the ordering type. Actually, we must know the main ordering properties of the NOs before starting the GISAXS analysis. These properties usually can be determined from the production process of the nanomaterial. If the material is produced by some kind of self-assembly process, then usually the separations fluctuate; therefore, we have to use the SRO model. However, if the positions are predefined from some reason (pre-pattering of the substrate, predefined layer thicknesses (vertical positions of NOs), ion-beam-induced nucleation, ion tracks, etc.), the LRO model should be applied. In some cases, we have a combination of LRO and SRO related to one basis vector. Most usually, this appears in films composed on multilayers with predefined thicknesses, i.e., vertical positions, that contain NOs. If these NOs are ordered by some self-assembly process, then their lateral (x and y) components show SRO, while the z component has LRO. In that case, we use Equation (11), i.e., ${G}_{MIX}^{\left(3\right)}\left(\mathit{q}\right)$, for the description of ordering along basis vector

**a**

_{3}. We sometimes have to combine several models in different special directions, as illustrated in the examples given later.

**Q**-space, the SRO and LRO differ by the behavior of the correlation peaks: the width of the peaks increases with the order of the peak for the SRO, while the peak width is constant for the LRO model and its intensity decreases with the order of the peak (for details, please see the Reference [3]). Hence, the peaks in the GISAXS map show the same basic behavior as the distribution of the NO positions when measured from the origin (Equations (2) and (3)). In Section 4, we examine three examples to illustrate the effects of ordering of NOs and their shape.

## 4. Examples

#### 4.1. 3D Ge Quantum Dot Lattices

_{2}O

_{3}); thus, the z position of each layer should be predefined (assuming a constant deposition speed). However, the QDs in this material form by diffusion-mediated nucleation, and their positions within the layers (x, y plane) are defined by a self-assembly process (see Reference [4]). Therefore, for this particular case, we should use a combination of LRO in the z direction and SRO in the x and y directions. The correct correlation function is

_{y}direction and two peaks in the Q

_{z}direction. Here, we must be careful with the conclusion that there is no more than one peak in the Q

_{y}direction, and that the ordering is worse in that direction, because the intensity visible in GISAXS maps also depends on the shape factor. Where the shape factor is zero, no intensity will be visible in the GISAXS map; thus, the shape factor can “make” higher orders not visible.

**a**

_{1}–

**a**

_{3};

**a**

_{1}and

**a**

_{2}are placed in the x, y plane and have the same length |

**a**

_{1}| = |

**a**

_{2}| = a, while the vertical stacking of the QDs is defined by the vector

**a**

_{3}(see Figure 3a). As explained before, the multilayer has a predefined position of the single layers; thus, the z components of the QD positions (

**a**

_{3}

^{z}; |

**a**

_{3}

^{z}|= c) have LRO, as illustrated in Figure 3b. The QDs within the layers show SRO (see Figure 3c), as they grow by a self-assembly process. It is also important to note that the regular ordering appears in domains randomly rotated around the z axis; thus, azimuthal averaging must be taken into account. The parameters of the analyzed QD lattice obtained by fitting of the experimental map from Figure 2a are given in Table 1, and the simulation is shown in Figure 2b. The shape of the QDs is assumed to be spheroidal with the lateral and vertical radii R

_{x}

_{,y}and R

_{z}, respectively, with the distribution having the standard deviation σ

_{R}. We chose a gamma size distribution because it is very easy to get the analytical expressions for the averaged sizes using it.

#### 4.2. 3D Networks of Ge Nanowires

_{2}O

_{3}co-deposition. In Figure 4, we present GISAXS maps and their simulations for two films differing by the NW length and the parameters of the NW lattice. The parameters of the simulations are given in Table 2. Details of their preparation are given in Section 2. For this case, there are no predefined positions for the network nodes; thus, all components of the G

_{3D}function are SRO, as shown below.

**a**

_{3}as shown in the Figure 5c. R

_{x}

_{,y}denote the radii of the NWs, and R

_{z}is the half of their length. Here, we have to perform averaging over all four equivalent configurations of the basis vectors

**a**

_{3}(connection of the center of the BCT lattice to the four neighboring nodes in the

**a**

_{1}–

**a**

_{2}plane), as well as azimuthal averaging.

#### 4.3. Lattices of Core/Shell Quantum Dots

_{2}O

_{3}multilayer deposition (for details, please see Section 2). Due to such a deposition process, each Ge QD is covered by a thin (about 1 nm thick) Si shell, and this shell is not symmetric around the core. This process is analyzed in detail in References [10,11]. Hence, the shape and ordering type of the QDs is the same as in Section 4.1, but the internal structure of the QDs is slightly different. It seems reasonable that a different internal structure of the QDs leads to different GISAXS intensity distributions, and no additional story about it is necessary. However, here, we want to point out that a very tiny difference in the QD structure, like a shell with the thickness of a few angstroms that is often invisible by microscopy, can cause very strong effects and cause problems in the GISAXS analysis (please see Reference [10] for an example). In many cases, we are not even aware that such a shell exists around NOs that we wish to investigate, because a chemical reaction easily occurs at the interface between the NO and the matrix that surrounds it to form a shell. Therefore, this example illustrates how a thin shell affects a GISAXS intensity distribution.

_{z}) are different from in the experiment. Here, we see it because of the logarithmic scale for the intensity distribution; for the linear case, it could be easily overlooked. If we use larger radius values, then we cannot get any good agreement for the peak shape and intensities. Thus, a very careful analysis is needed. The effect depends strongly on the material of the core and shell, i.e., on their electron density contrast. Generally, the strongest difference should have a stronger effect on the GISAXS intensity distribution.

## 5. Discussion

_{y}= 0 axis. This region contains a contribution of the coherent scattering that is not taken into account in the used simulations. Additionally, it also contains the contribution from the interface roughness in the multilayer samples, (not taken into account in the simulations). If we take into account all these contributions, the number of fitting parameters will be huge, and the fit would be completely unreliable. Therefore, in the GISAXS analysis, we do not take this central part into consideration. In addition, the paracrystal-based models that we use for the simulations do not work well in this region because the shape effects of the entire QD lattice become significant there. Therefore, cross-like structures appear in this part in the simulations, instead of sheets (please see the GISAXS simulations in the given examples). Only the central position (at the Q

_{y}= 0 plane) of this cross-structure is correct and we should take it into account. An illustration of how to select properly the region of interest for GISAXS is shown in the Figure 7.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Overview of different models for the description of nano-object (NO) ordering in one dimension (1D). The NOs are indicated by the numbers 1–3. The red arrows and gray lines above them illustrate the probability of a NO position.

**Figure 2.**(

**a**) Experimental and (

**b**) simulated grazing-incidence small-angle X-ray scattering (GISAXS) map of Ge quantum dots (QDs) embedded in alumina matrix. (

**c**,

**d**) Intensity profiles taken along the lines 1 and 2, respectively, indicated in the inset of (

**d**).

**Figure 3.**(

**a**) Basis vectors

**a**

_{1}–

**a**

_{3}of the body-centered tetragonal (BCT) QD lattice used for the GISAXS modeling. (

**b**) Arrangement of QDs within the layers that have long-range ordering (LRO) in the z direction. The positions of the QDs within each layer fluctuate around predefined z positions. The position of the n-th QD within the layer

**R**

_{n1, n2, n3}are denoted in the figure. (

**c**) Model of the QD ordering within a single layer of the multilayer stack. The QDs have short-range ordering (SRO) within each layer due to the self-assembly mechanism of their growth, and no predefined positions.

**Figure 4.**GISAXS maps of nanowire (NW) networks NW1 and NW2 (

**a**,

**c**) and their simulations (

**b**,

**d**), respectively.

**Figure 5.**(

**a**) Model of ideal three-dimensional (3D) nanowire networks. They consist of cylinders connected in the nodes of a BCT lattice. (

**b**) Transmission electron microscopy image of the film cross-section made perpendicular to the substrate. (

**c**) Schematic presentation of the model used in the GISAXS simulation. The QDs used in the previous example are replaced by elongated QDs tilted along the basis vector

**a**.

_{3}**Figure 6.**(

**a**) Experimental GISAXS map of Ge/Si core/shell quantum dots in alumina [11]. (

**b**) Simulation of the experimental map using the shape factor for core/shell QDs with shifted origin [10]—Sim 1. (

**c**) Simulation of the experimental map using the shape factor for spheroidal QDs—Sim 2. (

**d**) Transmission electron microscopy image of the same film demonstrating the real size of the QDs. R

_{L}denotes the radius parallel to the multilayer, and R

_{V}denotes that perpendicular to it. Model of the structure properties of QDs obtained from (

**e**) Sim 1 and (

**f**) Sim 2.

**Figure 7.**Example how to choose the region of interest for fitting of a GISAXS experimental map. The light-blue shaded areas should be included in the fit, while the positions of the sheets in the yellow shaded areas should be adjusted to fit the position of the crosses in the simulated maps.

**Table 1.**Parameters of the simulation shown in Figure 2b. QD—quantum dot.

Parameter (nm) | a | c | σ_{1,2}^{x}^{,y} | σ_{1,2}^{z} | σ_{3}^{x}^{,y} | σ_{3}^{z} | R_{x}_{,y} | R_{z} | σ_{R} |
---|---|---|---|---|---|---|---|---|---|

QD1 | 5.1 | 4.2 | 1.6 | 0.2 | 0.7 | 0.2 | 1.6 | 1.0 | 0.4 |

**Table 2.**Parameters of the simulation shown in Figure 4b,d. NW—nanowire.

Parameter (nm) | a | c | σ_{1,2}^{x}^{,y} | σ_{1,2}^{z} | σ_{3}^{x}^{,y} | σ^{z} | R_{x}_{,y} | R_{z} | σ_{R} |
---|---|---|---|---|---|---|---|---|---|

NW1 | 5.5 | 6.3 | 1.2 | 1.8 | 0.8 | 1.6 | 1.0 | 2.3 | 0.4 |

NW2 | 2.9 | 1.9 | 0.9 | 0.9 | 1.2 | 3.7 | 0.4 | 0.8 | 0.1 |

**Table 3.**Parameters of the two simulations shown in Figure 6b,c for the film GeSi1; t is the shell thickness, and d is the shift of the core with respect to the center of the shell.

Parameter (nm) | a | c | σ_{1,2}^{x}^{,y} | σ_{1,2}^{z} | σ_{3}^{x}^{,y} | σ^{z} | R_{x}_{,y} | R_{z} | σ_{R} | t | d |
---|---|---|---|---|---|---|---|---|---|---|---|

Sim1 | 5.2 | 6.3 | 1.5 | 0.2 | 1.4 | 0.6 | 2.6 | 2.0 | 0.3 | 0.9 | 0.3 |

Sim2 | 5.2 | 6.5 | 1.9 | 0.3 | 1.1 | 0.5 | 1.8 | 1.1 | 0.3 | --- | --- |

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**MDPI and ACS Style**

Basioli, L.; Salamon, K.; Tkalčević, M.; Mekterović, I.; Bernstorff, S.; Mičetić, M.
Application of GISAXS in the Investigation of Three-Dimensional Lattices of Nanostructures. *Crystals* **2019**, *9*, 479.
https://doi.org/10.3390/cryst9090479

**AMA Style**

Basioli L, Salamon K, Tkalčević M, Mekterović I, Bernstorff S, Mičetić M.
Application of GISAXS in the Investigation of Three-Dimensional Lattices of Nanostructures. *Crystals*. 2019; 9(9):479.
https://doi.org/10.3390/cryst9090479

**Chicago/Turabian Style**

Basioli, Lovro, Krešimir Salamon, Marija Tkalčević, Igor Mekterović, Sigrid Bernstorff, and Maja Mičetić.
2019. "Application of GISAXS in the Investigation of Three-Dimensional Lattices of Nanostructures" *Crystals* 9, no. 9: 479.
https://doi.org/10.3390/cryst9090479