3.1 Shape of nanostructures

To describe the morphology of the nanostructures formed after GLAD, their shape has to be analyzed. For this purpose, SEM images were taken from a broken edge of the silicon substrates with the deposited film. In this way, the structure (spirals, screws or columns) can be identified and the layer thickness can be measured.

Figure 2 shows a series of thin layers of tantalum deposited at a deposition angle of α=80^∘ but differing in growth pitch P_ω. Starting from a low rotational speed and a high deposition rate, which corresponds to a high growth pitch, the deposition rate and rotational speed have been adjusted according to Table 1 in order to achieve a decreasing growth pitch.

Figure 2(a)–(e) Layer structure and (f)–(j) their magnified view of a series of tantalum structures deposited at a deposition angle α of 80^∘ with decreasing growth pitch P_ω. The growth pitch can be used directly to adjust the microstructure.

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With decreasing growth pitch a significant transformation of the microstructure takes place, whereby different ranges can be identified (Table 2).

Table 2Ranges of the growth pitch P_ω and the obtained shape of the deposited structures.

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According to Table 1, the deposition rate in Fig. 2a and b is high, whereas the rotational speed is low. This results in a high growth pitch, which produces spirals. With decreasing growth pitch, the spiral cross section is reduced (b) until growth pitch approaches the column width w, thus producing screws (c). If the speed of rotation is further increased, screw-like columns (d) and eventually columns form (e).

3.2 Dominant intercolumn spacing

With the Fourier analysis periodic signals can be examined for their frequency spectrum. Therefore, it is useful to analyze the recurring spatial pattern of GLAD thin films.

A top-view SEM image of a GLAD layer serves as input for the Fourier analysis. The gray values of the individual pixels can be interpreted as a location-discrete, equidistant signal, which is transformed into a discrete frequency spectrum by the discrete Fourier transform. For an efficient calculation, the fast Fourier transform (FFT) is used as an algorithm, which is given for a SEM image with n_x pixels in the x direction and n_y pixels in the y direction by

where I(x, y) is the grayscale intensity of the image at the (x, y) position, and x_f and y_f are the spatial frequency coordinates in the x and y directions. To quantify the dominant spatial frequencies, the power spectral density (PSD) (Zhao et al., 2001) is then obtained from the transformed image with the relation

An efficient calculation of the dominant spatial frequencies is achieved when the dimensions of the SEM images in pixels are a power of 2. Therefore, image sections with the pixel dimensions 2048×2048 have been used. The evaluation of the SEM images is done with the free program Fiji (version 1.52e) (Schindelin et al., 2012) and the plugin Parallel FFTJ (version 1.4), which implements the calculations from Eqs. (3) and (4).

Figure 3 illustrates an example for the application of the Fourier analysis on the quantitative analysis of the periodic structure of a GLAD thin film. Fig. 3a shows the top view of an SEM image of a tantalum layer deposited at α=80^∘ with a growth pitch P_ω of 6.2 nm per revolution. The resulting two-dimensional PSD, as shown in Fig. 3b, has a diffuse ring-shaped structure, indicating the presence of a dominant spatial frequency and thus a dominant distance between adjacent columns. This spatial frequency lies at the ring radius where the PSD is maximum.

Figure 3Procedure to evaluate the dominant spatial frequency from an SEM image. (a) The starting point is an SEM image of the top view of tantalum columns produced with GLAD; (b) power spectral density of the top view according to Eq. (4), where high gray values are colored red and the ring-shaped pattern is revealed; (c) plot of the angular-averaged power spectral density according to Eq. (5) with marking of the dominant spatial frequency $f_{\mathrm{r}}^{*}$. The range of variation $f_{\mathrm{r},\max }^{*}-f_{\mathrm{r},\min }^{*}$ of the dominant spatial frequency is defined by the half-width (dashed line) (Buzea et al., 2005).

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The rotational symmetry of the annular structure, derived from an image with the same number of pixels in the x and y direction, shows that vertical columns deposited by glancing angle deposition and fast substrate rotation have an isotropic distribution. In particular for such distributions, it is recommended to calculate the angular-averaged power spectral density R_PSD¹ (Zhao et al., 2001) by

with the number of pixels n_r on the circumference of the circle with the radius f_r given by

All points on the circumference of this circle, whose center lies in the center of the PSD image and thus also in the center of the ring-shaped structure, represent a constant spatial frequency f_r. The angle between a point on the circumference and the center of the circle represents a direction in the original image. Equation (5) thus averages the power spectral density across all directions in the original image so that the dominant spatial frequency $f_{\mathrm{r}}^{*}$ can be read from the maximum of the direction-averaged power density as a function of the spatial frequency (Fig. 3c). Thus, the dominant column spacing r^* is

In Fig. 3c the spacial frequency $f_{\mathrm{r}}^{*}$ amounts to 13.8 µm⁻¹, and thus the dominant column spacing r^* between the structures is approximately 72 nm.

Figure 4(a)–(d) SEM top view and (e)–(h) side view images of tantalum GLAD films deposited at P_ω=6.2 nm per revolution and the labeled deposition angles to produce column structures. The top-view images serve as input for the Fourier analysis.

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Figure 5Dominant column spacing r^* versus growth pitch for different deposition angles.

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Figure 5 shows the analysis of the dominant column spacing as a function of growth pitch for the different deposition angles. For the evaluation of the dominant intercolumn spacing at the growth pitch P_ω of 6.2 nm per revolution the top-view SEM images in Fig. 4 have been used. As can be seen the Fourier analysis does not produce reasonable results for the top-view SEM images of the spiral structures (P_ω>170 nm per revolution). A possible explanation for this might be the insufficient resolution of the top-view SEM images of the spiral structures.

The results in Fig. 5 indicate that the dominant column spacing for α=80 and 85^∘ shows only small variations depending on growth pitch. For the other deposition angles a similar tendency can be observed, but here the number of samples is too small to give a general statement. For α=85^∘ the variation of the dominant column spacing as a function of growth pitch is particularly low for the screw and column structures investigated.

Figure 6Dominant intercolumn spacing r^* versus deposition angle α. The bars show the range of variation as related to growth pitch.

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With an increasing deposition angle the dominant column spacing increases strongly. At α=80^∘ the mean value amounts to $r^{*}\approx \mathrm{80}$ nm and reaches $r^{*}\approx \mathrm{289}$ nm at α=87^∘ (Fig. 6).

Figure 7Sample preparation to determine the pixel density dependent on the height above the substrate. (a) SEM image of the sample, (b) SEM image of the FIB sliced sample (inverted), (c) segmented FIB-SEM image, and (d) calculated pixel density from an FIB-SEM image.

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Figure 8Pixel density profiles across the layer thickness for column structures (P_ω=6.2 nm per revolution) at different deposition angles as a function of (a) height h_z above the substrate and (b) deposition angle α. The bars indicate the variation along the height coordinate of the layer, with 95 % of the η values lying within the bars.

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3.3 Porosity distribution

To reveal the cross-sectional area of the samples (Fig. 7a), a small volume is cut out by using focused ion beam (FIB) technology. The result is a smooth-cut surface that can easily be imaged with SEM. To prevent sputtering from the sample's upper part, it is covered with carbon from above. Since the SEM images have a high depth of field, the front of the sample is coated with carbon as well. The intersection surface is then polished again by an ion beam so that the carbon deposited on the interface is removed again. After such a preparation SEM images (Fig. 7b) are recorded.

Firstly, the FIB-SEM image is post-processed to separate the intersection from the image background. Due to the frontal deposition of carbon, the prepared intersection surface is distinguishable from underlying structures, which are still covered with carbon and appear darker. By applying a threshold, the intersection area can now be separated from the image (Fig. 7d). The threshold is automatically determined in Fiji using the Otsu algorithm (Otsu, 1979) to eliminate the user's subjectivity and improve reproducibility.

The pixel density η is a quality measure for the layer porosity and can be used to compare the different layers with each other. It is calculated line-by-line by counting the white pixels n_W per line and putting this number in relation to the total number n_tot of pixels in the latitudinal direction (Fig. 7d):

Figure 9Pixel density across the layer thickness for different layer structures with different growth pitch values P_ω for a deposition angle α of 80^∘ as a function of (a) height h_z above the substrate and (b) pixel density η. C – columns, S – screws, H – spirals. The bars indicate the variation along the height coordinate of the layer, with 95 % of the η values lying within the bars.

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Figure 10Cross section of tantalum spiral structures deposited at α=80^∘ and P_ω=162 nm per revolution. (a) SEM image of the FIB-sliced sample; (b) segmented FIB-SEM image.

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Plotting the pixel density of column structures (P_ω=6.2 nm per revolution) as a function of height above the substrate for the different deposition angles (Fig. 8a), the ratio of material to air across the entire layer thickness is nearly constant regardless of the deposition angle. This fact confirms the assumption that the incident tantalum particles settle where they are deposited because their surface mobility is strongly limited by substrate cooling.

From Fig. 9 it becomes clear that the growth pitch, and thus the layer structure, has a significant influence on the pixel density profile. As soon as the screw structures start to transform into spirals, there is a significant reduction in η with increasing growth pitch, and the pixel density profile suddenly shows a periodic pattern. This reduction in η can be explained by the fact that the spiral structures, due to their extraordinary geometry, grow into adjacent spiral structures, which in turn produces a denser layer structure (Fig. 10).

The period of the pixel density profile seems to depend on the growth pitch. From Fig. 9 a period $\stackrel{\mathrm{̃}}{h}$ of approximately 100 nm at P_ω=254 nm per revolution and of 66 nm at P_ω=162 nm per revolution can be estimated. It is therefore assumed that the ratio of growth pitch and the period $\stackrel{\mathrm{̃}}{h}$ is constant:

Applying Eq. (9) to the spiral structures, it follows $P_{\mathit{\omega }}/\stackrel{\mathrm{̃}}{h}\approx \mathrm{2.54}$ per revolution for P_ω=254 nm per revolution and $P_{\mathit{\omega }}/\stackrel{\mathrm{̃}}{h}\approx \mathrm{2.45}$ per revolution for P_ω=162 nm per revolution. Under the assumption that the ratio is 2.5 per revolution, a period length $\stackrel{\mathrm{̃}}{h}$ of approximately 36 nm follows for a growth pitch P_ω=90 nm per revolution, which means approximately five to six periods for each 200 nm interval in Fig. 9a. Counting the peaks in the 200 nm intervals in Fig. 9a, one can actually obtain this value between 5 and 6. For the column-like structures with P_ω≈33 per revolution a period length $\stackrel{\mathrm{̃}}{h}$ of ca. 13 nm follows, which means approximately 15 periods per 200 nm interval. This large number of periods can no longer be resolved from the pixel density curve.