Quantifying Epitaxial Growth using a Purely Topographical Signal

FiAbstract—Thin films are ubiquitous, with uses ranging from optoelectronics to antibacterial coatings. Unfortunately, precisely quantifying how the choice of substrate influences epitaxial growth remains an unsolved problem. Here, a novel thin film of holmium oxide with record-high paramagnetic saturation was grown on a variety of substrates. Conventional attempts to extract epitaxial information to characterize the growth mechanism were ineffective, due to the unique size regime of the product. Instead, a signal-processing inspired Fourier method was used to elucidate information on epitaxial ordering from purely topographical data, avoiding the pitfalls of atomic-level diffraction. Further, we define and utilize an inner product-based metric termed a q-score that can quantify the relative degree of ordering of epitaxial crystallites. The q-score provides a direct measure of epitaxy, enabling more quantitative future studies of thin film growth.

Introduction
Magnetic resonance force microscopy (MRFM) is an emerging technology that offers the promise of single nucleon detection in individual biological samples or nanodevices [1]. Such a device functions like a nanoscale magnetic resonance imaging (MRI) machine, applying a radiofrequency (RF) current and measuring response with a magnetic cantilever, all against the background of a strong uniform magnetic field [2,3]. Effective MRFM demands a large signal to noise ratio (SNR). From [3], this is:

Since the SNR is quadratically dependent on the magnetic field gradient ∂B/∂x, developing strong, new magnetic materials is important for optimizing MRFM. Recently, a new crystal phase of holmium oxide thin film with record-high paramagnetic saturation (above 2 Tesla) was synthesized using a thermal physical vapor deposition technique [4]. Holmium films grown on different substrates (A-plane sapphire, C-plane sapphire, and amorphous quartz) under otherwise identical deposition conditions have different morphological properties, suggesting that the substrate controls film growth. This direction is not a chemical process, as the two sapphire substrates are chemically identical, and growth occurs in a regime well below the vaporization of all the substrates [5,6]. Rather, the different crystal structures and orientations of the substrates visually appear to direct holmium growth in different ways. For example, in Figure 1, we see the holmium oxide film grows into crystalline triangles regardless of surface, but on the Sapphire C substrate (Fig. 1b), they seem to be aligned with one another.

When thin films grow, individual atoms land on the surface and then move around until they lose their kinetic energy and reach a thermodynamic minimum [7]. This can occur in different ways depending on how the landing atoms align with the substrate crystal atoms. When films grow on a substrate, the substrate can direct their growth to align with its own crystal structure, creating a periodic result [8]. This is termed epitaxial growth.
To determine whether different substrates are directing growth of different crystal phases of holmium oxide, we attempted a variety of standard diffraction techniques, including coarse x-ray diffraction (XRD) and more fine-grained elastic recoil detection (ERDA) and transmission electron microscopy (TEM). Most of these techniques determine bulk crystal structure and elemental composition, ultimately determining that we have a new phase of holmium oxide. However, they operate at the wrong size scale to determine whether the crystallites themselves are oriented. XRD and ERDA are too coarse to detect individual crystal grains reliably [9,10]. Available TEM only operates at a small domain size (~50 nm), while the crystallites in question are on the order of 100 nm. Although TEM has been successful at studying epitaxial growth in the past, it has traditionally been done on much thinner and smaller nanostructures [11] or been focused on defects [12]. In other words, no techniques were successful at revealing the orientation of individual crystal grains relative to each other, which is key to understanding the influence of using different substrates on holmium oxide growth and designing future experiments. 

Fortunately, techniques from signal processing can be used to uncover periodicity that is otherwise not apparent in images. In particular, the 2-dimensional Fourier transform is a promising tool and is defined by the relation in Equation (2). While the most common transforms in signal processing work by converting between time and frequency domains, the 2D transform here operates between two spatial regimes: real, physical space, parametrized by x,y, and inverse k-space, parametrized by u,v. By converting the image from real space into k-space using an orthogonal basis of complex exponentials, patterns emerge that can be used to quantify how ordered the substrate is [13]. In particular, the discrete Fourier transform (DFT, defined by Equation (3)) of an image is readily implemented using an FFT algorithm in MATLAB, as described in [14].

Note that in (3), X is the total number of pixels in the horizontal direction, and likewise for Y in the vertical direction. The discrete transform sums over all available pixels and sets f[x,y]=0 outside the observed X by Y region.
Quantifying epitaxy has applications beyond holmium oxide thin films. Thins films are ubiquitous [15], with applications ranging from optoelectronics as in [16] to antibacterial coatings as in [17]. In general, quantifying the influence of substrate on thin film growth, termed epitaxial growth, is a challenging problem [8,18]. Most literature relies on expensive atom-level measurements and can only detect a few layers of atoms (~1 nm) worth of epitaxial growth [11,12,19], while thin films with practical applications are often over 100 nanometers thick [20].

Here, we develop and test a facile classification scheme that quantifies how ordered thin film growth is using only topological data. This approach relies on transforming topographical atomic force microscopy (AFM) data into k-space and comparing the result to the predicted Fourier transform of perfect thin film epitaxy, as well as measuring maintenance of periodicity over large regions of film.
The following notation will be used throughout the manuscript:  f[x,y] denotes the z-height/brightness at a given spatial coordinate of an AFM image F[u,v] denotes the transform of f(x,y); this is the FFT in all cases except in section V.

<A,B> denotes a Frobenius inner product
A^* denotes the complex conjugate of A

Topographical Crystallinity

It is well known that using different substrates can result in the growth of different types of crystal grains. There are three main cases of thin film growth, reviewed extensively in [8] and [15].

Epitaxial crystalline growth. In the case of epitaxial growth, the crystal grains align perfectly with the substrate’s crystal structure, and the edges of crystal grains also align with the substrate and thus align with each other. Topographical data of this type of growth presents as a perfect tessellation of crystal edges. The shape of the grains depends on the shape of the substrate’s exposed crystal plane.

Non-epitaxial crystalline growth. In the case of non-substrate directed crystalline growth, nucleation and growth occurs in random directions. Although crystals with hard edges can still form, they will not be aligned with each other. This can be modeled as an assortment of randomly oriented crystals, where the shape of the crystal depends mostly on the thin film material’s intrinsic crystalline properties.

Random growth. In the case of non-epitaxial, non-crystalline growth, the thin film grows haphazardly on an unordered substrate, forming unordered structures. This case is largely non-instructive, and has no consistent morphological ordering.

In this experiment, holmium oxide was grown on two different crystal planes of sapphire, A-plane and C-plane, depicted in Fig. 2. The A-plane atomic cross section is a rectangular lattice, while the C-plane cross-section contains a hexagonal lattice.

Meanwhile, holmium naturally grows into a hexagonal lattice [21] that is a close match in terms of lattice strain to C-cut sapphire, so it is expected that the C-plane sapphire will clearly direct its growth, whereas holmium will grow into randomly oriented triangular or hexagonal crystallites on the A-plane sapphire. The amorphous quartz substrate is not crystalline, and thus cannot epitaxially direct growth.

Fig. 2. Different sapphire crystal planes. C-plane sapphire is a hexagonally symmetric lattice, while A-plane sapphire is more rectangular.

Since there should be one main set of periodic components in the first case, and many rotated sets of the same kind of triangle in the second case, the 2-dimensional Fourier transform (FT) offers a way to extract the magnitude of different periodic components and quantify underlying periodicity. The 2D FT is based on the same principles as a one-dimensional Fourier transform, using complex exponential functions of x and y as the basis set.

Since we have discrete images of 512×512 datapoints, we have to use the DFT described in Equation (3) rather than the FT described in Equation (2), with X=Y=512. Note that this transform works theoretically because the AFM signal is viewed as a multiplication of the real thin film surface with the bed-of-nails function in Equation (4), which samples out 262,144 discrete datapoints from the true, underlying, continuous thin film surface.

Implementing the FFT in MATLAB on a representative set of potential crystal patterns, it is apparent that the FFT of disordered triangles contains the superposition of many rotated transforms of individual triangles, ultimately creating a messy (and, ad infinitum, uniform and circularly symmetric) Fourier transform (Fig. 3a-b). 

Meanwhile, any set of ordered or tessellated triangles results in six main lines originating from the zero-order center, rotated depending on the initial orientation of the triangles (Fig. 3c-h). The hexagonal symmetry emerging from a central node is most apparent when multicolored triangles are used (Fig. 3e-h). These multicolored images are more realistic representations of crystal grain growth, individual grains grow to varying sizes and thus are different heights in AFM micrographs. 

The hexagonal ordering and Fourier pattern appear to be independent of variations in amplitude (color) or size variations in the initial triangle tessellation, suggesting the Fourier transform may be an effective way to classify ordering in crystal growth. Now that it is clear that the different theoretical growth regimes of crystals should be distinguishable using the Fourier transform, the given holmium oxide AFM data can be transformed into k-space and qualitatively compared to the theoretical transforms in Fig. 3. 

Qualitatively Investigating Holmium Oxide Thin Film Growth

Taking the 2-dimensional FFT of the holmium oxide thin film data yields k-space images that immediately reveal orientation differences between the films grown on A-plane sapphire, C-plane sapphire, and amorphous quartz.

In particular, both films grown on sapphire substrates have considerable spread from the zero-order lines in the Fourier plane (Fig. 4a-d), while the film grown on quartz is clearly disordered optically and has no visible triangles (Fig 4e). The Fourier transform of the film grown on amorphous quartz is focused around the zero-order lines, suggesting more random growth (Fig. 4f). 

Note that in the initial AFM image, while it is clear to the eye that the crystallites are triangular on both A sapphire (Fig. 4a) and C sapphire (Fig. 4c), whether these triangles are aligned with each other is an open question. However, the slight qualitative differences in the Fourier domain suggest an answer. While the transform of the holmium oxide film grown on Sapphire A is largely circularly symmetric (Fig. 4b), suggesting some amount of sharp triangular edges but not ordered triangles (similar to the pattern in Fig. 3b), the transform of the holmium oxide film grown on C-plane sapphire has higher brightness along a hexagonal set of contours (Fig. 4d), akin to the theoretical transforms in Fig. 3f and 3h. 

While this qualitative comparison of Fourier transforms to the theoretical transforms of tessellations given in Fig. 3 suggests a slight difference in epitaxial ordering between the Sapphire A- and Sapphire C-grown films, it is unclear if there is some fine degree of overall threefold ordering on Sapphire A (which would be possible if, for example, each triangular crystallite constrained the orientation of the neighboring crystallite; in other words, the holmium could self-direct its own growth.) Thus, quantifying the perceived difference between Fig. 4b and Fig. 4d is important for understanding how ordered the thin films are.