# Coherent X-ray diffraction imaging of magnetostriction in ferromagnetic nanocrystals

Alvaro Palomino

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Alvaro Palomino. Coherent X-ray diffraction imaging of magnetostriction in ferromagnetic nanocrystals. 2018. [hal-05013768](https://hal.science/hal-05013768v1).

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Álvaro PALOMINO LOPEZ

FAME MASTER
-Functional Advanced Materials and Engineering-

2018

SIMAP Laboratory of the University Grenoble Alpes
1300 Rue de la Piscine, 38000 Saint-Martin d'Hères

COHERENT X-RAY DIFFRACTION IMAGING OF
MAGNETOSTRICTION IN FERROMAGNETIC NANOCRYSTALS

from 12/02/18 to 31/07/18

Ecole nationale
supérieure de physique,
électronique, matériaux

**Phelma**
Bât. Grenoble INP - Minatec
3 Parvis Louis Néel - CS 50257
F-38016 Grenoble Cedex 01

Tél +33 (0)4 56 52 91 00
Fax +33 (0)4 56 52 91 03

http://phelma.grenoble-inp.fr
Under the supervision of :

Company supervisor : Guillaume BEUTIER, CNRS research officer.

guillaume.beutier@grenoble-inp.fr

Marc VERDIER, CNRS research director.

marc.verdier@grenoble-inp.fr

Phelma Tutor : Luc SALVO,

luc.Salvo@grenoble-inp.fr

Confidentiality : no
Acknowledgments.

I would like first to thank my master thesis advisors Guillaume BEUTIER and Marc VERDIER for the opportunity to work with them in this project and their constant good mood. Their closeness allowed me to feel very comfortable and learn a lot in this period. They allowed this paper to be my own work, but steered me in the right direction whenever they thought I needed it.

A special mention needs to be also given to Philippe DAVID, for the growth of Co samples, in NEEL Institute. Thanks for all the time given to this project, that was not little.
Also from NEEL Institute, I would like to express my gratitude to Vitoria BARTHEM and Dominique GIVORD, for the VSM measurements.

Thanks to Marc DE BOISSEAU, that spent a lot of time working with us on the experiment at the ESRF, and helped me to learn the technique. In the ESRF, the service given by Steven LEAKE was essential. Thanks for your advices on the usage of PyNX.

Thanks to Florence ROBAUT, for the EBSD measurements, Fabien VOLPI that made possible the e-beam deposition of Ni and Manoel JACQUEMIN for the further dewetting.

In general, I am very happy to have worked at SIMAP laboratory as all the people has demonstrated to be excellent both professionally and personally.
# SIMaP Laboratory.

This master thesis has been carried out in **SIMaP Laboratory**. SIMaP was created in 2007 by association of three former laboratories of the CNRS (the national center for scientific research), Grenoble INP (the group of graduate engineering schools) and Université of Grenoble Alpes.

SIMaP is localized on the « east campus » of Grenoble, in the city of Saint Martin d'Hères.

Researchers are in the fields of chemistry, physics, mechanical properties of materials and fluids and thermic and work on the conception of processes and synthesis of materials. SiMaP is one the leading laboratory for physical metallurgy and thermodynamics as well as architectural materials.

SIMaP is leader for in situ characterization (in temperature, under magnetic, electric field or in microgravity) and for the coupling of properties analysis and measurements (mechanical-electrical-chemical). In addition, SIMaP is part of the consortium managing and running the French "Collaborative Research Group" beamline D2AM of the European Synchrotron Radiation Facility (ESRF).

The research groups are the following:

* GPM2: Physical and mechanical properties

* PM: Physical metallurgy

* TOP: Thermodynamics, Modeling, process development

* SIR: Reactivity of Surface and Interfaces

* EPM: Synthesis with magnetic processing

My work was done on the group of Physics of Metals (PM). Activities focus on the physical metallurgy of alloys.

This project was performed in collaboration with Philippe DAVID from **Institut NEEL**, where most samples were grown. Magnetic characterisation by VSM was also performed there. The Institute is a laboratory for fundamental research in condensed matter physics, enriched by interdisciplinary activities at the interfaces with chemistry, engineering and biology. Different scientific topics are studied: superconductivity, quantum fluids, new materials, crystallography, surface science, quantum nanoelectronics, nanomechanics, nonlinear and quantum optics, spintronics, magnetism...
Gantt Diagram of the Internship period.

In the following diagram, the planning of the internship is given.

<figure><img src="image_2.png" /></figure>

Scheduled time for the internship
TABLE OF FIGURES

<table><tr><td>Figure 1. General procedure for the calculation of the magnetostriction.</td><td>13</td></tr><tr><td>Figure 2. Magnetic domains are strained as the applied field generates a preferred direction.</td><td>15</td></tr><tr><td>Figure 3. Scheme of the physical origin of magnetic anisotropy.</td><td>16</td></tr><tr><td>Figure 4. Magnetization curves for single crystals of nickel (a) and cobalt hcp (b).</td><td>17</td></tr><tr><td>Figure 5. Domain Structure of a uniaxial crystal.</td><td>17</td></tr><tr><td>Figure 6. Magnetostriction curves for single crystals of nickel (a) and cobalt hcp (b).</td><td>18</td></tr><tr><td>Figure 7. Comparison between (a) lens-based imaging system and (b) lensless imaging system.</td><td>21</td></tr><tr><td>Figure 8. Bragg condition schematized.</td><td>22</td></tr><tr><td>Figure 9. Strained region of a crystal by a vector u(r).</td><td>22</td></tr><tr><td>Figure 10. General overview of BCDI. The lost phase during diffraction is recovered by phase retrieval algorithms.</td><td>24</td></tr><tr><td>Figure 11. Different diffraction patterns produced by an incoherent X-ray beam (a) and a coherent beam (b), from Dupraz.</td><td>25</td></tr><tr><td>Figure 12. General process of a phase retrieval algorithm.</td><td>26</td></tr><tr><td>Figure 13. Process flow of the ER algorithm.</td><td>27</td></tr><tr><td>Figure 14. Evolution of the RMS error with the number of iterations using ER.</td><td>28</td></tr><tr><td>Figure 15. Process flow of the IO algorithm.</td><td>28</td></tr><tr><td>Figure 16. RMS error after a fixed number of iterations vs the algorithm parameter.</td><td>30</td></tr><tr><td>Figure 17. SEM picture of a region of the grid of 12 pitch where the Ni crystals are grown (a). AFM figure of an isolated Ni nanocrystal (b).</td><td>31</td></tr><tr><td>Figure 18. Schematic diagram of a PLD process. From Cheung [35].</td><td>32</td></tr><tr><td>Figure 19. Optical microscopy image of Co-3 (a) and Ni-4 (b). It can be observed in (b) the grid used for the isolated crystals growth.</td><td>35</td></tr><tr><td>Figure 20. SEM images of Co-3 (a) and Ni-4 (b). Scale bar shows 1 µm and 200 nm respectively.</td><td>36</td></tr><tr><td>Figure 21. Stacking fault on a single Co crystal of Co-10 imaged with AFM.</td><td>36</td></tr><tr><td>Figure 22. General scheme of an AFM in tapping mode set-up.</td><td>37</td></tr><tr><td>Figure 23. AFM figure of Ni crystals from Ni-4 sample. (a) corresponds to E1 (b) corresponds to A2.</td><td>37</td></tr><tr><td>Figure 24. 2θ-Ω Scan of Co-3 sample. Vertical axes gives the intensity (counts) while the horizontal axis is given in degrees (°).</td><td>38</td></tr><tr><td>Figure 25. 2θ-Ω Scan of Co-3 sample close to the hcp (004) reflection.</td><td>39</td></tr><tr><td>Figure 26. Phi Scan of Co-3 sample showing the six fold symmetry of the Co crystals.</td><td>39</td></tr><tr><td>Figure 27. fcc Co phase presents two possible in plane orientations, that leads to a six-fold in plane symmetry (a). Building units of the fcc unit cell that follows an ABCABC stacking sequence (b).</td><td>40</td></tr><tr><td>Figure 28. Schematic set-up for EBSD measurements. Notice the tilt of the stage at typical angle about 70°.</td><td>40</td></tr></table>
Figure 29. EBSD results (inverse pole figure) of two crystals of Co-10, that can be observed on Figure X. (a) shows a hcp (0001) orientation while (b) is fcc with a (111) orientation.

Figure 30. SEM image of a region of Co-10 sample.

Figure 31. SEM figure of a particular region of Co-3.

Figure 32. Magnetic Force Microscopy measurements of Co-3.

Figure 33. MFM contrast of a fcc dot with height 33 nm (a) simulated MFM contrast on and fcc Co (111) dot of 50 nm(b).

Figure 34. VSM curves showing the magnetization behaviour of Ni-4 at 300 K.

Figure 35. Experimental set-up on the beamline ID01.

Figure 36. Kirkpatrick Baez mirror pair.

Figure 37. (a) X-ray beam reconstruction. (b) X-ray beam intensity profile.

Figure 38. Experimental set-up on the sample support for the application of the magnetic field.

Figure 39. (a) Fast map of a certain region of the grid. Horizontal and vertical axes are given in pixel number. (b) SEM picture of the crystal C3 missing on the fast map, confirming that its [111] axis is not parallel to the substrate.

Figure 40. Rocking curve for E1 crystal. Vertical axis gives the intensity while the horizontal axis is the eta angle (°).

Figure 41. 2Theta values (°) of the A1 Nickel crystal at three different states: state 1 (without magnet), state 2 (with magnet) and state 3 (magnet removed).

Figure 42. Rocking curves for a Ni-4 crystal (E1 in the grid) for the (111) reflection, before the magnet is mounted (OFF) (a), with the magnet (ON) (b), and once it is removed (c).

Figure 43. Diffraction patterns for the three different states. measured around g= 111.

Figure 44. Rocking curve for E1 crystal for the (311) reflection

Figure 45. Diffraction patterns for the three different states. measured around g= 113.

Figure 46. Difference between the diffraction patterns of the magnetized state and unmagnetized state of the crystal E1.

Figure 47. SEM (a) and AFM (b) images of a crystal (E1) of Ni-4 sample.

Figure 48. E1 Nickel crystal reconstruction displaying (a) a combination of phase and amplitude (b) phase (c) amplitude information. A longitudinal slide along the crystal is shown here.

Figure 49. Slices shown the phase retrieved by a homemade MATLAB code.

Figure 50. Longitudinal coherence, ξ, between two waves.

Figure 51. Transverse coherence, ξ, t. From Dupraz

Figure 52. (a) Frank-van der Merwe (FM) growth compared with (b) Volmer-Weber (VW) growth. (c) Perfectly 2D Ag film with Ag (111) lattice constant grown by depositing 25 ML Ag onto Pt (111) at 600 K and subsequent annealing to 800 K. (d) 3D Pb islands grown on highly oriented pyrolytic graphite (HOPG) imaged with a SEM. [33]

Figure 52. EBSD measurements of Co-10 nanocrystals.
TABLE OF CONTENTS

<table><tr><td>1. Introduction.</td><td>11</td></tr><tr><td>2. Magnetostriction.</td><td>15</td></tr><tr><td>2.1 Definition of magnetostriction.</td><td>15</td></tr><tr><td>2.2 Magnetic anisotropy and its physical origin.</td><td>16</td></tr><tr><td>2.3 Saturation magnetostriction for Co and Ni crystals.</td><td>17</td></tr><tr><td>3. Bragg Coherent Diffraction Imaging</td><td>21</td></tr><tr><td>3.1 Basics of Coherent Diffraction Imaging (CDI).</td><td>21</td></tr><tr><td>3.2 Coherence from synchrotron sources.</td><td>24</td></tr><tr><td>3.3 Phase retrieval algorithms.</td><td>25</td></tr><tr><td>3.3. a) Error-reduction algorithm</td><td>27</td></tr><tr><td>3.3. b) Input-Output, Output-Output and Hybrid Input-Output algorithms</td><td>28</td></tr><tr><td>4. Samples growth</td><td>31</td></tr><tr><td>4.1 a) Nickel crystals grown by Solid State Dewetting</td><td>31</td></tr><tr><td>4.1 b) Cobalt crystals grown by Pulsed Laser deposition</td><td>32</td></tr><tr><td>5. Samples Characterization</td><td>35</td></tr><tr><td>5.1 Optical microscopy</td><td>35</td></tr><tr><td>5.2 Scanning Electron Microscopy (SEM)</td><td>35</td></tr><tr><td>5.3 Atomic Force Microscopy (AFM)</td><td>36</td></tr><tr><td>5.4. X-ray diffraction (XRD).</td><td>38</td></tr><tr><td>5.5. Electron backscattering diffraction (EBSD)</td><td>40</td></tr><tr><td>5.6 Magnetic Force Microscopy (MFM).</td><td>43</td></tr><tr><td>5.7 Vibrating Sample Magnetometry (VSM).</td><td>44</td></tr><tr><td>6. BCDI Measurements</td><td>45</td></tr><tr><td>6.1. Set up</td><td>45</td></tr><tr><td>6.2. Measurements</td><td>47</td></tr><tr><td>6.3. Results</td><td>49</td></tr><tr><td>- Rocking curve analysis.</td><td>49</td></tr><tr><td>-Comparison between diffraction patterns</td><td>50</td></tr><tr><td>-Reconstructions</td><td>53</td></tr><tr><td>-Difference Fourier Method.</td><td>55</td></tr><tr><td>7. Conclusions</td><td>57</td></tr><tr><td>APPENDIX A. Longitudinal and transverse coherence.</td><td>59</td></tr><tr><td>APPENDIX B. SEM of Ni-4 Crystals.</td><td>61</td></tr><tr><td>APPENDIX C. Models of Thin Film Growth Modes</td><td>63</td></tr><tr><td>APPENDIX D. EBSD additional measurements of Co-10.</td><td>65</td></tr><tr><td>Bibliography</td><td>66</td></tr></table>
# 1. Introduction.

As technology advances, there is a clear trend to a dimensional shrink of devices. A new field of study known as nanotechnology has emerged as a consequence, as visualized by Richard Feynman almost 60 years ago, on his talk 'There's Plenty of Room at the Bottom' at Caltech. [1]

At the nanoscale, the materials exhibit a different behaviour from their bulk counterpart, as reduced melting points, enhanced reactivities or significantly modified band structures. [2]

The study of magnetism at small scale is a hot topic on research as a consequence of the need for high density data storage or magnetic logic among others. Spintronics is a very interesting approach to solve the problem, and new discoveries, as the magnetic skyrmions at ambient room temperature and the ability to manipulate them by electrical current, shows the importance of its research. The study of magnetism at small scale becomes therefore crucial for its development, and large scale facilities are expected to be fundamental tools. [3,4]

In the nanoscale, from the mechanical point of view, pure elastic effects are expected to follow the classical theory down to very small sizes, but this is not the case for plasticity. For crystals smaller than a micrometer, a 'smaller is stronger trend' appears, leading to an increase of the yield stress on smaller crystals.

The study on the strain induced in nano-objects is extensive, studying the influence of not only the size but also the shape as shown by D. Alloyeau or L. Yang. [5,6]

The strain origin in a crystal does not necessarily has to be mechanical, but can arise from chemical, electrical or magnetic forces from the crystal environment.

In this study, the effect of an external magnetic field on nickel nanocrystals is investigated. When the crystals are exposed to a magnetic field, strain will appear due to a phenomenon known as 'magnetostriction'.

Strain is a physical parameter used to describe the structural change of a crystal. It is defined as the spatial derivative of the displacement of the material from an ideal lattice, therefore it has a tensorial nature. [2]

To observe the resulting deformation on the Ni crystals, a lensless imaging technique known as Bragg Coherent diffraction imaging (BCDI), that takes advantage of the special properties of the x-rays produced on synchrotron facilities has been used. It has been already demonstrated that this technique is useful to study the inhomogeneity of crystalline order such as strain and defects in 3D. For instance, a dislocation loop inside a gold nanocrystal, nucleated by an AFM tip, was revealed by Bragg Coherent Diffraction Imaging (BCDI) [7].

An amplitude resolution of the order of a few 10's of nm can be achieved [8]. A picometer sensitivity to strain has been already demonstrated. [9,10]

Diffraction is a common way to characterize the average strain of a crystal by measuring the position of its diffraction peaks. However, strain inhomogeneity is encoded in the intensity distribution around the diffraction peaks. A detailed image can be obtained from the phase of the diffracted wave, when using a coherent beam. Unfortunately, the phase can not be measured by the detector, only the diffracted intensity.
To recover the phase lost during the measurement, iterative phase retrieval algorithms are needed, and they will be introduced later. The strategy to follow is mainly to perform Fourier transforms connecting real and reciprocal space arrays of data on an iterative way, reducing the error until convergence is achieved. From the diffraction pattern produced by the sample, intensity values will be used as inputs, in addition to a 3D 'support' volume which constrains the extent of the sample.

Although BCDI is appropriate to measure the strain distribution on sub-micron crystals, other methods as transmission electron microscopy (TEM) can achieve even higher resolution by direct imaging. However TEM techniques are limited by the weak penetration depth of electrons in matter, being therefore BCDI more suitable for a 3D analysis. [2, 7, 11]

An introduction to the 'magnetostrictive effect' is given in the next section, followed by an explanation of the BCDI technique that was mainly used for the strain imaging. Phase retrieval algorithms are then introduced, as they are the tools used to obtain the strain information from the diffraction pattern of the crystals.

A description of the samples preparation follows, where pulsed laser deposition (PLD) was used for Co crystals and solid state dewetting after Electron Beam physical vapour deposition (EBPVD) for the growth of Ni crystals.

Different techniques as optical microscopy, scanning electron microscopy (SEM), atomic force microscopy (AFM), conventional x-ray diffraction (XRD) and electron back scattering diffraction (EBSD) were used complementary to BCDI. The information gathered was useful to have a better understanding of the samples to be characterized by BCDI. In addition, when performing the phase retrieval, a previous knowledge of the size of the crystal is very useful to define the support used, which can be deduced by SEM and AFM. That allows to achieve a more reliable reconstruction.

XRD and EBSD techniques were used in order to determine the orientation of the crystals obtained during the growing process, as well as the lattice structure of the Co crystals as they can be present on its hcp or fcc form. A previous knowledge of the crystals orientation is preferential to ease the BCDI measurements. In addition, as seen later, the crystal orientation is useful to understand the magnitude of the effect to be measured, as magnetostriction is an anisotropic phenomena.

The main strategy followed for the magnetostriction determination by BCDI is sketched in Figure 1. BCDI allow us to get the diffraction pattern of the crystal and by phase retrieval algorithms it is possible to recover the phase of the diffracted wave. The phase of the complex electron density is related to the displacement field by $\phi(\vec{r}) = \vec{g} \cdot \vec{u}(\vec{r})$, where $\vec{g}$ is the diffraction vector $\vec{g} = k_f - k_i$ and therefore related to the strain in the crystal, $\varepsilon(\vec{r})$. The magnetostriction can then be computed as the difference in strain between unmagnetized and magnetized states.
<figure><img src="image_3.png" /><figcaption>Figure 1. General procedure for the calculation of the magnetostriction.</figcaption></figure>
## 2. Magnetostriction.

### 2.1 Definition of magnetostriction.

Magnetostriction can be defined as a dimensional change (strain) experienced by a ferromagnetic material when its magnetic state is modified (it is magnetized).

There are two types of magnetostriction, spontaneous and field-induced. The spontaneous magnetostriction occurs from the ordering of magnetic moments into domains when passing through the Curie temperature. In our case, the second effect is studied, domains that have spontaneous magnetostriction are reoriented under the action of the externally applied magnetic field, H.

The dimensional change, λ, is associated to a shift in the magnetic domains distribution inside the material, as a result of a reduction of the total energy (magnetic and elastic energy).

$$ \lambda = dl/l \qquad (1) $$

λ is simply a dimensional fractional change.

The opposite effect can also appear, an applied stress can modify the magnetic state of the crystal, known as the Villari effect (1865). [13]

Figure 2 schematizes the strain induced λ (might be also a compression), when a magnetic field H, is applied.

<figure><img src="image_4.png" /><figcaption>Figure 2. Magnetic domains are strained as the applied field generates a preferred direction.</figcaption></figure>

The magnetic domains are modified so they get oriented parallel to the magnetic field, with a resulting strain on the crystal. Note that the strain symbol is here λ, instead of the most commonly used ε. That notation is given to highlight the fact that the strain is produced by a magnetic field and not a mechanical force.

Once the concept of magnetostriction is clear, its anisotropic behaviour must be introduced. Magnetostriction depends on the magnetization, which in turn depends on the applied magnetic field. Both the magnetic susceptibility (response of magnetization to a magnetic field) and the
magnetostriction (strain response to magnetization) are anisotropic and depend on the crystal directions.

## 2.2 Magnetic anisotropy and its physical origin.

The magnetization properties of a material strongly depends on the direction of the applied magnetic field. This phenomenon is known as ‘magnetic anisotropy’, and is a consequence of spin-orbit coupling.

When applying an external magnetic field, for the reorientation of the spin of the electron, the orbit of that electron also tends to be reoriented. However, the orbit is strongly coupled to the lattice, being the main obstacle to shift the magnetization from the easy direction.

The direction of easy magnetization of a crystal is defined as the direction of spontaneous domain magnetization in the demagnetized state. [14]

An energy, known as the ‘anisotropy energy’, is required to overcome the spin-orbit coupling and rotate the spin away from the easy direction.

To summarize, magnetization is anisotropic because the spin reorientation is coupled with the orbit that depends on the crystal lattice of the material, as sketched in Figure 3.

<figure><img src="image_5.png" /><figcaption>Figure 3. Scheme of the physical origin of magnetic anisotropy.</figcaption></figure>

Although magnetization changes for the different crystallographic directions, there is not an easy relation between the arrangement of the atoms and which direction is easier to magnetize. For instance, in the case of nickel (face-centered cubic), <110> is the closest packed direction between neighbouring atoms but it is an axis of medium hard magnetization while for iron (bcc) the closest packed direction is <111>, being the hard axis.

Figures 4 (a) and (b) show the magnetization as a function of the applied field for different crystallographic directions on the case of Ni and Co hcp.

As shown in Figure 4 (a), for Ni, the saturation magnetization is achieved at smaller fields on the easy direction, <111>.

However, the maximum strain might not occur on that particular direction, so a smarter strategy is to apply a field high enough on the direction of maximum strain, so the observation of the strain will be more attainable (even if it is not the easy direction). It is not our case, since we used the [111] direction, perpendicular to the substrate due to the experimental set-up geometry.

However, it does not changes dramatically in the case of nickel as observed in Fig. 6 (a), only about a factor of 2.
<figure><img src="image_6.png" /></figure>

Figure 4. Magnetization curves for single crystals of nickel (a) and cobalt hcp (b). Vertical axes shows Magnetization M (emu/cm³), while horizontal axes displays the applied field H (Oe). [14]

An example of this directional dependence is given on Fig 6 (b). For an hcp Co crystal there is no strain on the easy direction 0001 (c-axis), when a magnetic field is applied parallel to that direction. However, the maximum effect appears at 60°. The lack of strain when the field is applied parallel to the c-axis is explained because only 180° wall motion is involved, as Co hcp is a uniaxial crystal.

<figure><img src="image_7.png" /></figure>

Figure 5. Domain Structure of a uniaxial crystal. [14]

The magnetostrictive effect is usually small; even in strongly magnetic substances, in which the so called saturation magnetostriction, $\lambda_s$ is typically of the order of $10^{-5}$.

That magnitude is therefore comparable to a temperature change of about 1K for pure Ni and Co which have thermal expansion coefficients of $13 \cdot 10^{-6}$ and $12-14 \cdot 10^{-6}$ per K, respectively.

[15]

## 2.3 Saturation magnetostriction for Co and Ni crystals.

Figures 6 (a) and 6 (b) show the evolution of the magnetostrictive effect with the applied field for different crystallographic directions for the case of nickel and cobalt (in its hcp form) respectively.
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Figure 6. Magnetostriction curves for single crystals of nickel (a) and cobalt hcp (b). [14]</figcaption></figure>

It should be noticed the minuteness of the effect to measure, a relative strain of the order of 10⁻⁵.
It can be seen on Table 1 that different measuring methods have been already used.
<table><caption>Table 1. Previously measured values of the magnetostriction of Nickel on the [111] direction at room temperature. From Arakawa [16].</caption><thead><tr><td>10<sup>6</sup> x λ<sub>III</sub></td><td>Methods and references</td></tr></thead><tbody><tr><td>-27.1</td><td>Combination of mechanical and optical levers (λ<sub>||</sub>); Masiyama. [17]</td></tr><tr><td>-14.4</td><td>Combination of mechanical and optical levers (converted from λ<sub>⊥</sub>); Masiyama. [17]</td></tr><tr><td>-22.6 ± 1.4</td><td>Strain gauge extensometer (converted from h<sub>2</sub> and h<sub>5</sub>); Bozorth and Hamming. [18]</td></tr><tr><td>-23.0</td><td>Capacitance bridge (λ<sub>||</sub>); Corner and Hutchinson. [19]</td></tr><tr><td>-13 ± 4</td><td>X-ray diffraction (converted from λ<sub>⊥</sub>); [16]</td></tr></tbody></table>

The measurement of the longitudinal magnetostriction has been commonly made by electrical-resistance commercial strain gages cemented to the sample, capacitance, inductance or optical measuring systems. In our case, synchrotron radiation will be used to study the effect. In particular BCDI is used, introduced on the following section.
# 3. Bragg Coherent Diffraction Imaging

## 3.1 Basics of Coherent Diffraction Imaging (CDI).

Coherent Diffraction Imaging (CDI) is a generic microscopy technique to obtain an image of the sample based on its diffraction pattern. The lens are replaced by the use of a coherent beam and “oversampled” measurements of the diffraction patterns [20] and the image is formed by applying phase retrieval algorithms to the diffraction patterns.

Coherent X-ray Diffraction (CXD) is the name used for the technique that provides the scattering pattern of a particular crystal, while Coherent Diffraction Imaging (CDI) is used when also phase retrieval algorithms are used to create a real space reconstruction from the scattered pattern provided by CXD. CDI therefore implies the use of CXD, and it is a lensless imaging technique as sketched in Figure 7. Instead of using lenses to create the image, the diffraction pattern is used as an input for the image reconstruction.

<figure><img src="image_10.png" /><figcaption>Figure 7. Comparison between (a) lens-based imaging system and (b) lensless imaging system. [21]</figcaption></figure>

Actually, the first use of CXD was X-Ray Photon Correlation Spectroscopy (XPCS), where fluctuations in the coherent scattering provide information about the material dynamics. [22]

In our case, the technique is used in Bragg geometry, which provides sensitivity to crystal strain. To achieve the Bragg condition, the following relation needs to be fulfilled (known as the Bragg's law):

$$n = 2d \sin\theta \qquad (2)$$

So twice the interplanar spacing, d, times the sine of the incident angle, θ, must be an integer number of the wavelength of the beam, λ. The condition can be visualized on Figure 8, and it is possible to observe that the waves will be in phase (creating constructive interference) only when the condition marked by Equation (2) is fulfilled.
<figure><img src="image_11.png" /><figcaption>Figure 8. Bragg condition schematized.</figcaption></figure>

It is straightforward that if the interplanar distance, d, changes (strain is produced), the bragg condition is compromised and the diffraction pattern will be in consequence distorted.

As can be observed in Figure 9, when a certain strain $\vec{u}$ is created, the lattice is distorted. The phase of the X-ray beams scattered is shifted relative to that of the reference crystal by

$\varphi = \vec{k}_f \cdot \vec{u} - \vec{k}_i \cdot \vec{u} = \vec{q} \cdot \vec{u}$.

<figure><img src="image_12.png" /><figcaption>Figure 9. Strained region of a crystal by a vector $u(r)$. [2]</figcaption></figure>

Where the scattering vector $\vec{q}$ is the vectorial difference between the incoming wavevector $\vec{k}_i$ and the outgoing wavevector $\vec{k}_f$. The magnitudes of $\vec{k}_i$ and $\vec{k}_f$ are equal to $2\pi/\lambda$. [11]

The centro-symmetry of the diffraction pattern is broken by the presence of strain fields. The Fourier transform of a real function is centro-symmetric, otherwise not. The electronic density is real, but the modified electronic density as defined for BCDI is not.

The strain can arise for instance, from the epitaxial contact forces inevitable appearing when nanocrystals are grown on a substrate. The presence of strain will reveal therefore additional information on the diffraction pattern, breaking its centro-symmetry. [8]

The electron density of the crystal is given by the amplitude of the scattering vector and its phase $\varphi(\vec{r})$ represents the projection of the displacement from the ideal crystal lattice, $\vec{u}(\vec{r})$, onto the reciprocal lattice vector, $\vec{q}$, of the Bragg peak used for the diffraction measurement.
[23]

The intensity distribution measured by the detector is given in the kinematical approximation of diffraction by Equation 3.
$$I(\vec{q}) = |A(\vec{q})|^2 = |FT\{f(\vec{r})\}|^2 \qquad (3)$$

where $f(\vec{r})$ is the scattering factor (the electronic density in most cases) and FT stands for the Fourier transform

$$|FT\{f(\vec{r})\}|^2 = \left|\int f(\vec{r}) \cdot e^{2i\pi \vec{q} \cdot \vec{r}} d\vec{r}\right|^2 \qquad (4)$$

and the integration is performed over the volume illuminated by the beam, shown later in our case on the setup section.

As it can be observed from Equation 4, when a displacement of the atoms from the perfect crystal lattice occurs, $\vec{r} = \vec{r}_0 + \vec{u}(\vec{r})$, the phase factor is given by

$$\vec{q} \cdot \vec{r} = \vec{q} \cdot \vec{r}_0 + \vec{g} \cdot \vec{u}(\vec{r}) + (\vec{q}-\vec{g}) \cdot \vec{u}(\vec{r}) \qquad (5)$$

where $\vec{g}$ is the diffraction vector for a perfect lattice.

If we now assume that $|(\vec{q}-\vec{g}) \cdot \vec{u}(\vec{r})| \ll 1$, which occurs in Bragg geometry (Takagi's approximation[24]), equivalent to assuming small distortions of the lattice and a restricted extent of the reciprocal space. The resulting modified scattering factor is given by Equation (6),

$$\hat{f}(\vec{r}) = f(\vec{r}) \cdot e^{2i\pi \vec{g} \cdot u(\vec{r})} \qquad (6)$$

where the atomic scattering factor is essentially the electronic density $\rho(r)$, which is a real scalar, and the modified scattering factor is referred as the complex electronic density $\hat{\rho}(\vec{r})$: its modulus is the real electronic density and its phase encodes the projection of the displacement field $u(\vec{r})$ onto the diffraction vector $\vec{g}$.

Figure 10 sketches the procedure of the technique, where the x-ray beam is diffracted by the crystal producing an scattering pattern in reciprocal space. The pattern is then used to reconstruct the object in real space by iterative algorithms as will be explained later.
<figure><img src="image_13.png" /><figcaption>Figure 10. General overview of BCDI. The lost phase during diffraction is recovered by phase retrieval algorithms. [7]</figcaption></figure>

The use of phase retrieval algorithms is required, because the x-ray detectors record only the intensity of the scattered wave, that is the square root of the amplitude. However, the phase information is lost. Therefore, a simple Fourier transform of the measured intensity is not enough for retrieving the image. This is known as the “phase problem” and requires the use of phase retrieval algorithms that set some constraints for the object reconstruction.

Revealing the structural defects or strain by analyzing the diffraction pattern is not an easy task. Dupraz et al. [7] have studied numerically the effect of dislocations and stacking faults on the diffraction patterns of gold nanocrystals. They showed how, for instance, in the case of a screw dislocation, a typical ring shaped diffraction peak with a lower intensity in the center is produced, although the study is complex and differs if for instance the dislocation is dissociated. [25]

The best way to approach the problem is then using computational methods to invert the pattern to real space.

## 3.2 Coherence from synchrotron sources.

The technique requires X-ray beams of high coherence, which have been developed at 3rd generation synchrotron facilities. This highly coherent source can be found on some synchrotron facilities as the one we used, the beamline ID01 at the European Synchrotron Radiation Facility in Grenoble, France.

Coherence is a property of a beam that describe its ability to produce interference. Technically speaking, it can be described as the existence of a phase relationship between different positions in the beam.

The use of coherent rays is necessary to ensure the well defined phase relation of the wavefront, being that preserved during scattering and therefore ensuring a high sensitivity to the position of the single scatterers. [26]

Synchrotron x-rays are not directly emitted with a high degree of coherence. Spatial filters, as slits or pinholes are applied to obtain the spatial coherence, while monochromators are used for obtaining the temporal coherence. The coherence of a beam is anisotropic in space, the propagation and transverse directions have different coherence properties. Both spatial and temporal coherence are defined in Appendix A.
A compromise between flux and coherence has to be reached for the particular experiment, with the knowledge that one limits the other. That is the reason why third generation synchrotron sources are necessary, although the intensity reduction, consequence of the appliance of those filters, the brilliance is sufficiently high to afford the lost in intensity.

Moreover, the Extremely Brilliant Source (ESRF-EBS) project will provide an improvement of 100 times the brilliance and coherence. This sets high expectations for future measurements at this unprecedented level of accuracy. [27]

In addition, as explained by Dupraz [7] and illustrated in Figure 11, using incoherent x-ray beams, it is only possible to obtain average information of multiple domains, as the lattice spacing. The incoherent beams average the intensity of a large number of coherence volumes, as in conventional X-ray scattering. However, by the use of coherent x-rays, information on single domains can be exploited, leading to a speckle pattern as shown in Figure 11 (b). Therefore, the information comes from the single crystal and not from a larger volume.

<figure><img src="image_14.png" /><figcaption>Figure 11. Different diffraction patterns produced by an incoherent X-ray beam (a) and a coherent beam (b), from Dupraz [7].</figcaption></figure>

In the semiconductor field for instance, lattice strain on nanostructures could be previously inferred only globally from broadening of Bragg peaks, while now it is possible to map out the strain field of a single semiconductor nanowire. [28]

While incoherent beams provide only statistically averaged properties of a material, coherent beams provide a detailed characterisation of their inhomogeneity

### 3.3 Phase retrieval algorithms.

It is possible to reconstruct a real space image from the intensity pattern obtained from the scattered x-rays by using phase retrieval algorithms. In Figure 12 a general view of the iterative process followed is shown, where FFT stands for Fast Fourier Transform and $g_i$ is an estimate of the object $f(x)$. Note the use of uppercase letters for the Fourier Transforms of the functions represented by lowercase letters ($G_i$ is FFT of $g_i$).
<figure><img src="image_15.png" /><figcaption>Figure 12. General process of a phase retrieval algorithm. [29]</figcaption></figure>

Phase retrieval basically consists on reconstructing the object $f(x)$ or equivalently retrieve the phase $\phi$ of its fourier transform $F$, whose modulus is measured.

At each step, constraints are applied in the reciprocal space (the modulus of the Fourier transform) and in the direct space (a finite support).

An idea of the shape and size of the object is beneficial to achieve convergence, as it helps to define the support constraint. In the support region the electron density of the retrieved image is allowed to be non-zero while outside the support the density is set as zero. [21]

Error Reduction (ER), Hybrid Input-Output (HIO) and Shrink Wrap (SW) algorithms were used together to complete the phase retrieval as have been demonstrated to be the most successful methods to achieve convergence according to Fienup, Pfeifer, Vartanyants and Marchesini. [11,29,30,43,44]

The SW method is sometimes conveniently used to determine the support, that should fit the shape of the object. Once the support size is evaluated, SW is not used anymore. A procedure combining ER and HIO is then used.

The two mainly used iterative methods (ER and HIO) are explained in the following section.
### 3.3. a) Error-reduction algorithm

The method is divided in 4 steps and schematized on Figure 13:

(1) Fourier transform an estimate of the object g (an estimate of random phase), resulting in G.

(2) Replace the modulus of the resulting computed Fourier transform with the measured Fourier modulus to form an estimate of the Fourier transform, G'.

(3) Inverse Fourier transform the estimate of the Fourier transform, giving g'.

(4) A complex density distribution in real space is obtained, generally with non-zero values in the region outside the support. The outside part of the support is then set to zero, keeping the inside part the same as calculated. Then one Fourier transforms to reciprocal space again as in (1).

<figure><img src="image_16.png" /><figcaption>Figure 13. Process flow of the ER algorithm. [29]</figcaption></figure>

Using the method, one can only reduce the error (or stay the same) at each iteration. However, the convergence is slow after a first few iterations.

The constraints used can be stated as follows:

$$g_{k+1}(x) = g'_{k}(x), \quad x \in S \qquad (7)$$

$$g_{k+1}(x) = 0 \quad \text{otherwise}$$

Figure 14 gives an example of the error reduction after k iterations. It should be noted the logarithmic scale and the presence of two different plateaus, after reaching 0.16 and 0.02 before it reaches 0.003 where the reduction is then negligible.
<figure><img src="image_17.png" /><figcaption>Figure 14. Evolution of the RMS error with the number of iterations using ER.</figcaption></figure>

### 3.3. b) Input-Output, Output-Output and Hybrid Input-Output algorithms

The input-output algorithm differs from the error-reduction only in the object-domain operation. The operation of the method can be observed in Figure 15. and consist, as the three first steps of the ER method, on (1) the Fourier transform of the object g, (2) satisfying the Fourier-domain constraints, (3) and inverse Fourier transforming the result.

In this method, the input $g(x)$ does not necessarily satisfy the object domain constraints. It does not have to be the current best estimate of the object but is just a driving function for the next output $g'(x)$. The flexibility of choice of the input allows to converge more rapidly.

<figure><img src="image_18.png" /><figcaption>Figure 15. Process flow of the IO algorithm.</figcaption></figure>
The basic input-output algorithm is performed as follows,

$$ 
\begin{aligned}
g_{k+1}(x) &= g_k(x) + \beta \Delta g_k(x) \\
&= g_k(x), & x \notin \gamma \\
&= g_k(x) - \beta g'_k(x) & x \in \gamma
\end{aligned} 
\qquad (8) $$

where $\gamma$ is the set of points at which $g'_k(x)$ violates the object-domain constraints. It can be observed that there is no change of the output when the constraints are satisfied, that is $\chi \in \gamma$. On the other hand, a change is required if the output does not satisfy the constraints, and it is the negative output at those points ($\Delta g_k(x) = -g'_k(x)$).

If an output $g'$ is used as an input, its output will be itself. When $g'$ is Fourier transformed, it satisfies the Fourier-domain constraints, so it is not changed and then when being inverse Fourier transformed, the result is itself. The system is therefore as follows,

$$ 
\begin{aligned}
g_{k+1}(x) &= g'_k(x) + \beta \Delta g_k(x) \\
&= g'_k(x), & x \notin \gamma \\
&= g'_k(x) - \beta g'_k(x) & x \in \gamma
\end{aligned} 
\qquad (9) $$

And this systems is referred as the output-output algorithm.

By mixing both systems mentioned above, one can get the hybrid input-output algorithm.

$$ 
\begin{aligned}
g_{k+1}(x) &= g'_k(x), & x \notin \gamma \\
&= g_k(x) - \beta g'_k(x) & x \in \gamma
\end{aligned} 
\qquad (10) $$

The hybrid input-output algorithm is a better solution than the output-output algorithm since it helps to avoid a stagnation problem that appears on the last, where the output does not change under successive iterations. [29]

HIO has the advantage over ER that is much faster and it can climb out of local minima at which ER stagnates.

It is based on making changes on the input that will lead to changes on the output on that same direction. The expected change of the output is a constant $\alpha$ times the change on the input. Although the particular change on the output due to an input change can not be exactly predicted, due to the presence of additional nonlinear terms on the output, it can be pushed in the general direction desired. For instance, to change the output by $\Delta g(x)$, the strategy to follow would be to change the input by $\beta \Delta g(x)$, being $\beta$ equal to $\alpha^{-1}$.

In our case, we used a $\beta=0.9$, in accordance with the better results obtained on the previous work of Fienup, for which a smaller RMS error $E_0$ was obtained by using a $\beta$ value close to 1.
In Figure 16 a comparison of the RMS error for different methods is given. On the case of HIO, line E, it can be observed how an algorithm parameter close to 1 is appropriate.

<figure><img src="image_19.png" /></figure>

Figure 16. RMS error after a fixed number of iterations vs the algorithm parameter. Curve A: steepest-descent method (O); B: conjugate-gradient method (●); C: basic input-output algorithm (▲); D: output-output algorithm (▼); E: hybrid input-output algorithm (■). The result using the error-reduction algorithm is indicated by a large circle. [29]
# 4. Samples growth

Two types of samples were grown for the purpose of this project: Ni nanocrystals obtained by solid state dewetting and Co nanocrystals grown by Pulsed laser Deposition.

## 4.1 a) Nickel crystals grown by Solid State Dewetting

Three different samples of nickel crystals were grown by depositing a nickel layer of 50 nm by e-beam deposition on a sapphire (0001) substrate in an argon atmosphere. A mask consisting of a grid of holes of 2 µm diameter and 12 µm pitch was used, in order to obtain isolated objects. The deposition was followed by solid state dewetting at different temperatures, as shown in Table 2, with an Ar flow of 800 sccm (Standard Cubic Centimeters per Minute) and 200 sccm of H₂.

<table><caption>Table 2. Main experimental parameters for the growing of Ni nanocrystals.</caption><thead><tr><td>Sample name</td><td>Atmosphere for e-beam deposition</td><td>Annealing Temperature (°C)</td><td>Time at temperature (min)</td></tr></thead><tbody><tr><td>Ni-1</td><td>Ar+H₂</td><td>1300</td><td>60</td></tr><tr><td>Ni-2</td><td>Ar+H₂</td><td>1100</td><td>10</td></tr><tr><td>Ni-4</td><td>Ar+H₂</td><td>1200</td><td>60</td></tr></tbody></table>

The P(O₂) levels were well below the threshold for Ni oxidation or Ni-spinel formation (i.e. P(O₂) < 3x10⁻⁹ atm.)[31]. A SEM image of part of the grid can be observed on Figure 17 (a). The metallic layer deposited at room temperature dewets at those temperatures forming crystals of about 300 nm height as seen in Figure 17 (b).

<figure><img src="image_20.png" /><figcaption>Figure 17. SEM picture of a region of the grid of 12 pitch where the Ni crystals are grown (a). AFM figure of an isolated Ni nanocrystal (b).</figcaption></figure>

The dewetting occurs as a consequence of the difference on the surface energy between the film and the sapphire substrate, forming the sub-micrometer sized crystals. This is possible after the annealing process, as the temperature enhances the mobility of the surface atoms towards an
equilibrium state. By keeping the temperature below the melting point, dewetting occurs in the solid state and the resulting crystals retain their initial orientation with respect to the substrate.

### 4.1 b) Cobalt crystals grown by Pulsed Laser deposition

Cobalt crystals were grown by pulsed laser deposition (PLD). Previous work has demonstrated the feasibility of the method. [32]

The technique is based on the use of pulsed laser radiation to vaporize a target material to be deposited, as a thin film in a vacuum chamber. The process is sketched in Figure 18.

<figure><img src="image_21.png" /><figcaption>Figure 18. Schematic diagram of a PLD process. From Cheung [35].</figcaption></figure>

The surface of the target is heated by the laser, that forms a plasma that expands forming a 'plume' directed to the substrate.

The main parameter for evaporation is the fluence F, i.e. the energy per shot and per unit area of the target. The fluence can be adjusted with both the laser power and a focusing lens located outside the chamber on the beam path. [36]

Some advantages of the technique are that the technique favours the epitaxial growth, or the versatility to deposit different materials (refractory, rare earth, ...). However, droplets can be formed at high energy and the deposition rate is slow (about 1 Å/min).

The substrate material was sapphire (11-20). Two bcc buffer layers of Mo and W of about 7.5 Å and 9 nm were grown respectively as buffer layers. The tungsten layer was then annealed followed by the Co deposition. A 2 nm gold layer is finally deposited for preventing oxidation. A plain mask was sometimes used to make two different depositions on the same substrate (letters "a" and "b" in the Table 3).
<table><caption>Table 3. List of Co samples grown with the substrate temperature used.</caption><thead><tr><td>Sample name</td><td>Substrate Temperature (°C)</td></tr></thead><tbody><tr><td>Co-3</td><td>500</td></tr><tr><td>Co-5 MBE</td><td>500</td></tr><tr><td>Co-6</td><td>590</td></tr><tr><td>Co-7 (a)</td><td>550</td></tr><tr><td>Co-7 (b)</td><td>520</td></tr><tr><td>Co-9 (a)</td><td>520</td></tr><tr><td>Co-9 (b)</td><td>500</td></tr><tr><td>Co-10</td><td>500</td></tr></tbody></table>

On Co-9, after the deposition of Co, a 1h annealing at 800°C was performed. During the growing process, different parameters need to be controlled as the deposited thickness, deposition rate, the substrate temperature or the annealing temperature of the W layer, as they will affect the kinetics of film growth. In Appendix A, the three thin film growth models are discussed.

In the case of Co grown by PLD, the islands follow a Stranski-Krastanov (SK) growth mode. For Ni, the island formation was not achieved during the deposition process but by further annealing as previously explained.
# 5. Samples Characterization

In this section an overview of the characterization techniques used complementary to BCDI is given. Microscopic, diffraction and magnetic measurements were performed.

## 5.1 Optical microscopy

Optical microscopy was the first step for the sample characterization, in order to identify if the islands were formed or in order to find some possible regions of interest. In Figure 19 (a) for instance, a region where the crystals were more separated in Co-3 was observed by optical microscopy. In Figure 19 (b) a part of the grid in Ni-4 sample is shown.

<figure><img src="image_22.png" /><figcaption>Figure 19. Optical microscopy image of Co-3 (a) and Ni-4 (b). It can be observed in (b) the grid used for the isolated crystals growth. Scale bar shows 50 µm and 200 µm respectively.</figcaption></figure>

## 5.2 Scanning Electron Microscopy (SEM)

SEM allowed a more detailed analysis of the samples one they were first observed in the optical microscope. SEM was used in all samples to find well-formed crystals that could be used for BCDI.

SEM images on the crystals of the Ni-4 sample analyzed by BCDI can be observed on Appendix A. An example of such a Ni crystal can be observed in Fig 20 (b). It can be seen in Figure 20 (a) how in the cobalt samples the crystals are grown very close to each other.
<figure><img src="image_23.png" /><figcaption>Figure 20. SEM images of Co-3 (a) and Ni-4 (b). Scale bar shows 1 µm and 200 nm respectively.</figcaption></figure>

## 5.3 Atomic Force Microscopy (AFM)

Atomic force microscopy was used due to its ability to resolve features on a surface with nanometer precision. [40]

The technique is based on the movement of a tip attached to a small cantilever that scans the region of interest, the position of the tip is recorded, obtaining the height information of the sample. In Figure 21 it is possible to observe a possible stacking fault or twin boundary on a Co crystal.

<figure><img src="image_24.png" /><figcaption>Figure 21. Stacking fault on a single Co crystal of Co-10 imaged with AFM.</figcaption></figure>

The images were taken in Tapping mode of operation, where the tip oscillates at or near its resonance frequency, with an usual amplitude range of 20 to 100 nm. The tip contacts slightly the surface at the bottom of its “swing”. The feedback loop keeps the oscillation amplitude constant, by using a signal received on a photodiode detector. The height is therefore recorded by the variation of the position of the tip during the scan, in order to keep the oscillation amplitude constant. Figure 22 gives a general overview of the set-up.
<figure><img src="image_25.png" /><figcaption>Figure 22. General scheme of an AFM in tapping mode set-up. [41]</figcaption></figure>

As can be observed from Figures 23 (a) and (b), the dimensions of the crystal can be obtained by using AFM, what is quite convenient for a more efficient use of the phase retrieval algorithms (defining the support), as stated before.

<figure><img src="image_26.png" /><figcaption>Figure 23. AFM figure of Ni crystals from Ni-4 sample. (a) corresponds to E1 (b) corresponds to A2.</figcaption></figure>
It is interesting to notice how in this case both crystal have almost identical dimensions, about 700 nm lateral size and about 300 nm height.

## 5.4. X-ray diffraction (XRD).

XRD measurements were performed in the Co-3 sample to analyze the presence of hcp and fcc phases.

The growth of Co dots by PLD has been previously reported to happen in both fcc and hcp structures. Although the most stable crystalline structure of Co below 425°C is hcp, fcc Co is also found in thin films and nanostructures [32]. The difference between FCC and HCP is just the sequence of stacking of the dense atomic planes (ABCABC for FCC, ABABAB for HCP) [37].

In our case we could also notice the coexistence of hcp and fcc Co, by using both X-ray Diffraction and Electron Back Scattering Diffraction (EBSD). The effect appears as consequence of the high temperature and stress related to low dimensions. [32]

In Figure 24 a specular 2θ-Ω scan of the sample can be observed. Several peaks can be observed. Sapphire at 2θ=37.77° and W (110) at 2θ=40.06° were easily identified. The Mo peak might be hidden by the other peaks or even is too thin (7.5 Å) to yield the diffraction peak.

In the case of cobalt, due to the proximity of the hcp (002) and (004) reflections to the fcc (111) and (222) it is difficult to tell whether we have hcp phase or not from the 2θ-Ω scan. The values are described on the table of figure 24.

In any case, it is clear that cobalt is oriented with its fcc (111)/hcp (002) plane parallel to the substrate.

<figure><img src="image_27.png" /><figcaption>Figure 24. 2θ-Ω Scan of Co-3 sample. Vertical axes gives the intensity (counts) while the horizontal axis is given in degrees (°). Specific crystal planes are described by the Miller indices (hkl).</figcaption></figure>

<table><thead><tr><th>Phase</th><th>Reflection (hkl)</th><th>Theoretical 2th Angle(°)</th><th>Experimentally observed 2th Angle (°)</th></tr></thead><tbody><tr><td>Co fcc</td><td>111</td><td>44.151</td><td>44.29</td></tr><tr><td>Co fcc</td><td>222</td><td>97.468</td><td>97.66</td></tr><tr><td>Co hcp</td><td>002</td><td>44.501</td><td>?</td></tr><tr><td>Co hcp</td><td>004</td><td>98.455</td><td>?</td></tr></tbody></table>

A smaller scan close to the region of hcp 004 reflection shows some contribution of the hcp phase as shown in Figure 25. A similar scan measured at a different azimuth shows a different intensity ratio between the fcc(222) and the hcp(004) reflections (~50%-50%). The discrepancy remains unclear, preventing from a precise determination of the proportions of both phases. We can however say that the fcc phase is dominant, but the proportion of phase is not negligible.
<figure><img src="image_28.png" /><figcaption>Figure 25. 2θ-Ω Scan of Co-3 sample close to the hcp (004) reflection. Vertical axes gives the intensity (counts) while the horizontal axis is given in degrees (°).</figcaption></figure>

Figure 26 shows a phi scan (azimuthal scan) of the sample that presents a six-fold symmetry for fcc reflections (220) and (311) as well as the hcp (101) reflection of the Co phase. This scan thus confirms the presence of the hcp phase. In addition, it shows that the crystals have a precise orientation not only out of plane, but also in plane.

<figure><img src="image_29.png" /><figcaption>Figure 26. Phi Scan of Co-3 sample showing the six fold symmetry of the Co crystals.</figcaption></figure>

The six fold symmetry is coherent as we have two possible variants on the fcc structure, each variant with three-fold symmetry, leading to a six-fold symmetry (see Figure 27). In hcp structure the six peaks come from the same single structure due to its geometry.
<figure><img src="image_30.png" /></figure>

Figure 27. fcc Co phase presents two possible in plane orientations, that leads to a six-fold in plane symmetry (a). Building units of the fcc unit cell that follows an ABCABC stacking sequence (b). [37]

This analysis informs us that they have the same in plane orientation.

To sum up, with XRD we identified the species we had and served us to demonstrate that we have both fcc and hcp phases. EBSD results further confirmed this hypothesis.

**5.5. Electron backscattering diffraction (EBSD)**

EBSD measurements served us to determine the crystallographic orientation of the Co crystals. On the case of Nickel, only EBSD measurements were made on Ni-1, as the sapphire substrate provoques charging effects that makes the measurement extremely complicated and a carbon coating (graphite) was therefore required. However, the measurement in Ni-1 confirmed a majority of fcc crystals with a (111) normal orientation.

Figure 28 shows a typical arrangement of an EBSD experiment. An electron backscattering pattern is produced by the finely focused electron beam, directed to a tilted single-crystal target. The formed pattern of lines and bands are related to the crystal structure. Different models discussing the origin of the EBSD were discussed by Wells. [38]

<figure><img src="image_31.png" /></figure>

Figure 28. Schematic set-up for EBSD measurements. Notice the tilt of the stage at typical angle about 70°. [39]
The advantage of EBSD over conventional XRD measurements is that we can target single crystals instead of obtaining an average information as with XRD.

In Figures 29 (a) and (b), the inverse pole figure of two different crystals from the Co-10 sample are shown.

<figure><img src="image_32.png" /><figcaption>Figure 29. EBSD results (inverse pole figure) of two crystals of Co-10, that can be observed on Figure X. (a) shows a hcp (0001) orientation while (b) is fcc with a (111) orientation.</figcaption></figure>

The two crystals shown before corresponds to (a) and (b) in Figure 30.
<figure><img src="image_33.png" /><figcaption>Figure 30. SEM image of a region of Co-10 sample.</figcaption></figure>

As can be deduced from the color code of the inverse pole figures, crystal (a) corresponds to hcp structure with a (0001) normal orientation while (b) is an fcc crystal with (111) normal orientation. The analysis was carried out for 8 different crystals with the result that only 2 were hcp with the same orientation, (0001), while the other 6 crystals where fcc (111) oriented (only one of them presented a twin boundary).

In Appendix D, the patterns for the other 6 crystals can be observed.

An interesting growth difference can be observed coming back to Figure 30, it can be easily noticed in the center of the picture a big particle that might come from contamination. However, the presence of that particle might have created a thermal gradient that induced a different growing of the surrounding crystals. Bigger and more isolated crystals appear, as compared to other regions of the sample where the growth is fairly homogeneous.

In fact, that phenomena was also observed on other samples as shown in Figure 31, from Co-3, and it is actually a benefit in our case, as we aim to have crystals as isolated as possible, in order to be characterized by the coherent x-ray beam of the synchrotron source.
<figure><img src="image_34.png" /><figcaption>Figure 31. SEM figure of a particular region of Co-3.</figcaption></figure>

As due to the coherence of the beam, single crystals can be characterized, their growth as isolated as possible is beneficial to avoid possible interference from neighbouring crystals.

**5.6 Magnetic Force Microscopy (MFM).**

MFM measurements were performed on Co-3, the result MFM contrast can be observed in Figure 32.

<figure><img src="image_35.png" /><figcaption>Figure 32. Magnetic Force Microscopy measurements of Co-3. The analysis of a single crystal might indicate a flux-closure magnetization (b).</figcaption></figure>

As a matter of fact, Fruchart has previously reported the presence of Co crystals that display a flux closure-magnetization pattern as observed in Figure 33 (a).
<figure><img src="image_36.png" /><figcaption>Figure 33. MFM contrast of a fcc dot with height 33 nm (a) simulated MFM contrast on and fcc Co (111) dot of 50 nm(b). [32]</figcaption></figure>

**5.7 Vibrating Sample Magnetometry (VSM).**

VSM measurements were made on Ni-4. The results (Figure 34) show that magnetization is preferentially in plane (blue line), with no remanence. The in plane magnetization is in accordance with the flux-closure domains observed by Fruchart on Co crystals. The lack of remanence indicates that the magnetic states before mounting the magnet and after removing it are probably identical.

<figure><img src="image_37.png" /><figcaption>Figure 34. VSM curves showing the magnetization behaviour of Ni-4 at 300 K. Perpendicular (red) and parallel (blue) components are shown.</figcaption></figure>

However, we can not tell if it favours a vortex structure or not. The point on the perpendicular component in Figure 34 shows the applied magnetic field on the BCDI experiment. It can be observed that with the magnet on, we do not saturate (≈80% from saturation).
# 6. BCDI Measurements

## 6.1. Set up

The x-rays are generated on the European Synchrotron Radiation Facility (ESRF) located in Grenoble. The experiment was performed on the line ID01 of the ESRF, line dedicated to microdiffraction imaging. Accelerated electrons produce electromagnetic waves tangential to the direction of movement. By using an ondulator, as the source of X-rays, the emission spectra can achieve peak brilliances in the order of $10^{20}$ ph.s$^{-1}$ .mm$^{-2}$.mrad$^{-2}$/0.1%BW. Figure 35 gives an scheme of the set-up used.

<figure><img src="image_38.png" /><figcaption>Figure 35. Experimental set-up on the beamline ID01.</figcaption></figure>

Once the brilliant x-rays are emitted, a channel cut double crystal Si (111) monochromator provides a beam with a monochromaticity of $\Delta\lambda/\lambda \sim 10^{-4}$ at 8 keV. After that, the coherence slits of 60x300 µm$^2$ (H x V) provide a nearly fully coherent beam. KB mirrors (Kirkpatrick-Baez) are then used to focus the X-ray beam to a spot size of the order of the micron or less.

<figure><img src="image_39.png" /><figcaption>Figure 36. Kirkpatrick Baez mirror pair.</figcaption></figure>

A K-B mirror system consists of two curved mirrors placed at right angles to each other as observed in Figure 36. A parabolic mirror shape will focus a collimated beam to a point, while an elliptical mirror shape will focus a point to another point. [42]

The resulting beam size depends on the opening of the coherence slits. In our case we did not want a too small beam, in order to fully illuminate the crystals, and we thus opened the slits to
50 µm (H) x 30 µm (V), which resulted in a beam size at focus of 900 nm x 750 nm., and a final flux of 10⁹ ph.s⁻¹ is obtained.

Figure 37 shows a reconstruction of the X-ray beam on the sample. It can be observed in (a) that it has a constant phase, so that not important corrections need to be done when performing the phase retrieval. In (b) the beam central lob is smooth and roughly Gaussian, while being big enough to illuminate the crystal area (the size of the beam at the sample position is roughly 911 nm x 746 nm FWHM).

<figure><img src="image_40.png" /><figcaption>Figure 37. (a) X-ray beam reconstruction. (b) X-ray beam intensity profile.</figcaption></figure>

A precision Huber 3+2 circle diffractometer is used in this line, with 3 sample axes and 2 detector axes, allowing various diffraction geometries. We performed diffraction in the vertical plane only, but the 3 sample axes allowed to reach non specular reflections such as the {113}. A 3-axes piezo stage mounted on the most inner axis of the diffractometer supports the sample. Above the stage, an optical microscope helps the positioning of the crystals.

For the detection, a 2D Maxipix pixel detector of 516 x 516 pixels, of 55 µm per pixel is used. The sample-detector distance is 1 m, so that the pixel size provides enough resolution to oversample the smallest diffraction features.

The sample stage (including rotation axes) is installed on a granite table for stability requirements and is decoupled from the detector arm.

Figure 38 shows the sample located above a bridge designed to locate afterwards the magnet underneath, when the magnetic field is applied. The magnetic field was applied with a NdFeB permanent magnet that was mounted at a fixed position under the sample.
<figure><img src="image_41.png" /><figcaption>Figure 38. Experimental set-up on the sample support for the application of the magnetic field.</figcaption></figure>

The set-up is shown before the magnet was mounted for the magnetic field application. As the magnet was mounted under the sample, the resulting field was perpendicular to the substrate surface, with a resulting magnetic field of 47mT on the Ni sample studied. The diffraction measurements were performed in three different states: before magnetization, with the magnet mounted and once the magnet was removed.

## 6.2. Measurements

The beam and the crystal must be carefully aligned at the center of rotation of the diffractometer. Once the alignment is done the diffractometer axes are moved such that the geometry satisfies the Bragg condition for each reflection.

We studied the (111), (113) and (311) reflections in some cases. Once the bragg condition is set, a fast map of a portion of the sample is performed as can be observed in Figure 39 (a). This allows to identify the crystals of interest, which were selected from the lab characterisation, and to select them for the individual measurement.

It can be clearly observed how there are some missing crystals, as we are in Bragg condition for those crystals (111) oriented normal to the surface. Missing crystals in the fast map are therefore differently oriented as can be seen for the case of C3 in Figure 39 (b) that is not (111) oriented normal to the surface.
<figure><img src="image_42.png" /><figcaption>Figure 39. (a) Fast map of a certain region of the grid. Horizontal and vertical axes are given in pixel number. (b) SEM picture of the crystal C3 missing on the fast map, confirming that its (111) axis is not parallel to the substrate. The SEM scale measures 200 nm.</figcaption></figure>

Once the position of the crystal of interest is selected, a rocking curve is measured over a width of 1° with 0.005° step size, what means that the sample is tilted stepwise (rock it) and for every angle step the diffracted intensity is recorded. You will obtain a distribution of intensities over tilting angles as shown in Figure 40. For each angle, a 2D diffraction pattern is recorded in the area detector, resulting in a 3D measurement of the reciprocal space.

<figure><img src="image_43.png" /><figcaption>Figure 40. Rocking curve for E1 crystal. The vertical axis gives the intensity (counts) while the horizontal axis is the eta angle (°).</figcaption></figure>

It can be observed on the figure different colours, as the rocking curves were usually performed various times to check that the alignment was correct during the measurement.

The Bragg reflections were taken not only for the (111) planes but also for (113) and (311) in some cases.

The (111) diffraction component is useful to get the longitudinal strain produced on the crystals.
$$ \epsilon_{zz}(\vec{r}) = \frac{\partial u_{111}}{\partial z} \qquad (11) $$

However, the (113) and (311) reflections will allow us to compute the in plane strain as the phase $q(r)$ represents the projection of the displacement from the ideal crystal lattice, $u(r)$, onto the reciprocal lattice vector, $G$, of the Bragg peak used for the diffraction measurement. As (113) has an horizontal component, the in plane strain can be deduced.

## 6.3. Results

Although both Ni and Co crystals were synthesized, due to time constraints during the beamtime, only the analysis for the Ni-4 sample, that was selected as the best candidate for the BCDI treatment was performed. The selection of Ni over Co was made for two main reasons: the magnetostrictive effect occurs at smaller fields in Ni than in Co (as seen before in Figure 6), and secondly because the Ni samples were grown within a grid with 12 µm pitch, resulting in well isolated crystals, as shown before in Figure 17 (a), being easier for the measurements and preventing interference effects from surrounding crystals.

- *Rocking curve analysis.*

A first approach to treat the phenomena has been to evaluate if an average change of lattice parameter could be observed upon application of magnetic field, by measuring the 2theta value of the diffraction peak. Because of the strain inhomogeneity in the crystal, the diffraction patterns are fairly broad and a precise measurement of the 2theta value is difficult to obtain, although the experimental geometry (pixel size and detector distance) provides a theoretical resolution of 0.003°. An homogeneous magnetostrictive effect of 10⁻⁵ would cause a shift of 0.0005° in 2theta. We used the center of mass of the diffraction pattern to evaluate the 2theta value, which provides a resolution better than the pixel value by a factor around 10. We are thus just at the limit of detection. The analysis was performed for the (111) direction on several crystals. In Figure 41, the 2theta values for one of the crystals can be observed. State 1 stands for the 2theta value obtained before the magnet was mounted (OFF state), the state 2 stands for the 2theta value obtained when the magnet was mounted (ON state), while state 3 is the state where the magnet has been removed.

<figure><img src="image_44.png" /><figcaption>Figure 41. 2Theta values (°) of the A1 Nickel crystal at three different states: state 1 (without magnet), state 2 (with magnet) and state 3 (magnet removed).</figcaption></figure>
From state 1 to state 2 we would expect a certain shift of the 2theta value, coming back to the original one (in state 3) as we saw there was no coercivity in the magnetic analysis. However, we see that in some cases, there is more shift between the same state (2 in this case) than the shift among different states. Therefore, this analysis is probably too simplistic, and the resolution we can achieve is not sufficient to analyze magnetostriction.

-Comparison between diffraction patterns

If the magnetostrictive effect is not homogeneous (for instance by changing the magnetic state from an inhomogeneous state such as a vortex to a saturated state), differences could appear in the detailed diffraction pattern.

First we try to see the effect of the applied magnetic field on the diffraction pattern measured by the detector. As stated before, the phase information is lost. Figure 42 shows the rocking curves for a Ni-4 crystal (E1 in the grid), before the magnet is mounted (OFF), with the magnet (ON), and once it is removed.

<figure><img src="image_45.png" /><figcaption>Figure 42. Rocking curves for a Ni-4 crystal (E1 in the grid) for the (111) reflection, before the magnet is mounted (OFF) (a), with the magnet (ON) (b), and once it is removed (c).</figcaption></figure>

Dupraz et al. observed from simulations that the visibility of the hexagonal shape of the particle in the CXD pattern at the (qx,qy) slice is an indication of the absence of strain. In our case however the diffraction pattern is highly distorted, suggesting a high presence of residual strain (see Figure 43).
<figure><img src="image_46.png" /></figure>

Figure 43. Diffraction patterns for the three different states. measured around g= 1 1 1. Slices of the reciprocal space, from left to right: (qx,qy), (qx,qz) and (qy,qz) for each state: before the magnet is mounted (OFF), with the magnet (ON), and once it is removed.

There are not great changes, but is visible close to the Bragg peak position between the OFF and
Removed states (qx,qy).

On the (qy, qz) slices the fringes close to the Bragg peak seem to be more intense.

On the (qx,qz) slices the contrast of the fringes after removing the magnet is enhanced as compared to the unmagnetized sample. The contrast is also better in (qx,qy).

For the (113) reflection, the following information is obtained before the magnet is placed (OFF state) and with the magnet on (ON state).
<figure><img src="image_47.png" /><figcaption>Figure 44. Rocking curve for E1 crystal for the (311) reflection (a) before the magnet was mounted and (b) with the magnet mounted.</figcaption></figure>

O
F

<figure><img src="image_48.png" /></figure>

O
N

<figure><img src="image_49.png" /><figcaption>Figure 45. Diffraction patterns for the three different states. measured around g=113. Slices of the reciprocal space, from left to right: (qx,qy), (qx,qz) and (qy,qz) for two states: before the magnet is mounted (OFF), and with the magnet (ON).</figcaption></figure>

Again, although there are some intensity changes around the bragg peak, they are not very noticeable.
We can therefore conclude here that the simple observation of the diffraction patterns is not giving us enough details to identify magnetostriction.

Another strategy we are working on is to subtract the values for the diffraction patterns. An example is given in Figure 46, that corresponds to the difference between diffraction patterns in detector space between the unmagnetized and magnetized states for the E1 crystal, for the (111) reflection. It can be noticed that there is a clear difference between both states.

<figure><img src="image_50.png" /><figcaption>Figure 46. Difference between the diffraction patterns of the magnetized state and unmagnetized state of the crystal E1.</figcaption></figure>

### -Reconstructions

A further step appart to the simple comparison of the diffraction pattern is the retrieval of the phase as previously explained.

As stated before, in order to perform the reconstructions a previous understanding of the particle dimensions is important to define the support that will set the constraints for the phase retrieval.

In Figures 47 (a) and (b), both SEM and AFM pictures of a Ni crystal can be observed, in which that structural information appears, being AFM even more suitable as it allows us to get the particle height. That characterization has been therefore quite useful for the phase retrieval.

<figure><img src="image_51.png" /><figcaption>Figure 47. SEM (a) and AFM (b) images of a crystal (E1) of Ni-4 sample. The scale-bar for SEM shows 200 nm.</figcaption></figure>
By using BCDI, and the PyNX library for the phase retrieval the reconstruction in Figure 48 was achieved for the E1 crystal on Ni-4, the same observed in the previous Figures 47 (a) and (b). PyNX stands for Python tools for Nano-structures Crystallography [45]. (see http://www.esrf.eu/computing/scientific/PyNX/README.html, for more details).

For the reconstruction with PyNX, not only ER and HIO are used but also SW method.
In Figure 48 (a) and (b) the phase is given by the color code bar at the right part of the figure in
radians. Figure 48 (c) displays only the amplitude of the diffracted wave (the electron density).

<figure><img src="image_52.png" /><figcaption>Figure 48. E1 Nickel crystal reconstruction displaying (a) a combination of phase and amplitude (b) phase (c) amplitude information. A longitudinal slide along the crystal is shown here.</figcaption></figure>

The reconstruction of the crystal on a homemade MATLAB code, using a combination of different iterations between ER and HIO methods, is shown in Figure 49.
<figure><img src="image_53.png" /><figcaption>Figure 49. Slices shown in the phase retrieved by a homemade MATLAB code. Three different slices are displayed here: xz (a) yz (b) and xy (c).</figcaption></figure>

It can be observed there are divergences between the reconstruction performed in PyNX and that one performed in MATLAB. However, this can be due to the fact that the MATLAB version does not consider yet the partial coherence of the beam.

-Difference Fourier Method.

An interesting approach for the problem of comparing the possible changes between the different magnetization states is to use the 'difference Fourier' method. The method was used before by Watari et al. [23] to study the stress induced by thiol adsorption on gold nanocrystals. The method assumes that the reciprocal-space phases of the diffraction pattern are the same before and after a small change is made in the structure. As small differences in the patterns before are observed, the Fourier difference method should be valid.

The main idea is to calculate the Fourier transform of the difference between the diffraction patterns.

This allows to calculate the difference Fourier density, the difference between the two complex density functions, $\rho_2(\vec{r}) - \rho_1(\vec{r})$.
$$ \Delta\rho(\vec{r}) = \rho_2(\vec{r}) - \rho_1(\vec{r}) = \int [\sqrt{I_2(\vec{q})} - \sqrt{I_1(\vec{q})}] e^{i\alpha(q)} e^{iq \cdot r} d^3\vec{q} \quad (12) $$

By using the phased diffraction of one of them as the common phase function, $\alpha(q)$. This step therefore allows to avoid the phase retrieval of both states, as only one of them is used for both.

In addition, $q = Q - G$, is the deviation of the momentum transfer, $Q$, from the reciprocal lattice vector, $G$, of the Bragg peak used. Therefore, the calculated difference $\Delta\rho(\vec{r})$ should give us an image of the changes between the two states, where the phase will mainly change due to a small strain, while the modulus, representing the electron density, hence the shape will remain nearly constant.

In our case therefore, the idea is then to get a Difference Fourier Map (DFM), subtracting the data from the unmagnetized state, using the phase $\alpha(q)$, so we can see the changes in crystal strain. This is possible as we are mainly interested on the changes that appear when the crystal is magnetized, not the previously present strain and so the method is ideal for that purpose.

The analysis has not been done yet, as the BCDI data was taken only three weeks before the submission of this report, but it will be started at the end of the internship period.
# 7. Conclusions

An analysis of the magnetostrictive effect on Nickel nanocrystals has been performed by BCDI. It has been seen here how the information obtained from the diffraction patterns and the phase retrieval algorithms is difficult to analyze and relate with magnetostriction. The diffraction patterns show some changes between the unmagnetized and magnetized states, but those are considerably small.

Another approach that will be used for the study of the effect is the Difference Fourier Method. With this approach, we expect to observe some changes on the phase that allow us to identify magnetostriction, but unfortunately, the analysis has not been possibly done yet.

Due to the previous succeed of the technique to image strain, we expect some positive results after the data analysis is carefully performed, although the small magnitude of magnetostriction.
# APPENDIX A. Longitudinal and transverse coherence.

## Longitudinal coherence

The longitudinal coherence (also called temporal coherence) of a light source is related to the beam monochromaticity, and it defines the coherence along its propagation direction as seen in Fig 50.

If two sources emit waves of slightly different wavelengths ($\lambda$ and $\lambda + \Delta\lambda$), from the same point, the distance over which they will complete dephase is known as longitudinal coherence, $\xi_l$.

Therefore, after the longitudinal coherence distance, the two waves will be out of phase and they will be again in phase after twice that distance.

It is given by equation 13.

$$ \xi_l \approx \frac{1}{2} \frac{\bar{\lambda}}{\Delta\lambda} \bar{\lambda} \qquad (13) $$

From equation 13, it can be observed how the temporal coherence length is inversely proportional to the bandwidth $\Delta\lambda$ of the source.

[11]

<figure><img src="image_54.png" /><figcaption>Figure 50. Longitudinal coherence, $\xi_l$, between two waves.</figcaption></figure>
## Transverse coherence

If now we assume two waves (of same wavelength) with slightly different direction of propagation, separated by $\Delta\theta$, the transverse coherence length, $\xi_t$, (also known as spatial coherence) is the distance along a wavefront over which there is a complete dephasing between the two waves.

<figure><img src="image_55.png" /><figcaption>Figure 51. Transverse coherence $\xi_t$. From Dupraz</figcaption></figure>

This arises because all sources have a finite size, with a non zero divergence $\Delta\theta$. [46]
It is given by Equation 14.

$$ \xi_t \approx \frac{\lambda L}{2S} \qquad (14) $$

Where S is the linear source size, and L the distance along which the dephasing is completed. [11]

As observed on Equation 14 the spatial coherence depends not only on the wavelength of the incident radiation but also on the distance between the source and the sample, L and the source size, S.

Therefore, it can be increased by reducing the source size, and increasing the distance.

The lateral extension of the source therefore can lead to incoherence in the transversal direction.
APPENDIX B. SEM of Ni-4 Crystals.

<table><caption>Table 4. SEM images of the crystals measured at the ESRF on Ni-4.</caption><thead><tr><th></th><th>H</th><th>G</th><th>F</th><th>E</th><th>D</th><th>C</th><th>B</th></tr></thead><tbody><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>
<figure><img src="image_56.png" /></figure>

A

8

7

6

5

4

3

2

1

<figure><img src="image_57.png" /></figure>
# APPENDIX C. Models of Thin Film Growth Modes

The growing of the islands on an epitaxial process can be described as follows: when the arriving atoms reach the crystalline homogeneous surface, they have a certain amount of energy to move across the surface, depending on the substrate temperature. When two atoms are next to each other, the simplest model assumes that the complex does not diffuse anymore, nucleating an island. Those islands will grow by aggregation of incoming atoms. The resulting distance between islands and island size will depend therefore on the deposition flux of atoms and the substrate temperature. However, there is usually a large island size distribution. [32]

Epitaxial growth has been divided into three growth modes:

* The Frank-van der Merwe (FM) growth mode (layer by layer).

* The Volmer-Weber (VW) growth mode (island growth).

* The Stranski-Krastanov (SK) growth mode (after a critical thickness, layer plus island growth).

A very intuitive explanation is given by Brune [33], it introduces the macroscopic surface tensions, i.e. the free energy per unit area between the three possible interfaces: overlayer/vacuum ($\gamma_o$), overlayer/substrate ($\gamma_i$) and substrate/vacuum ($\gamma_s$).

Depending on the values of the free energies for the specific system, a particular growth mode will be preferential. The two opposite cases of FM and VW growth can be observed in Figure 52, where the FM is preferred in case that the surface tensions $\gamma_o$ and $\gamma_i$ are lower compared to $\gamma_s$, i.e. $\gamma_o(n) + \gamma_i(n) \le \gamma_s$.
<figure><img src="image_58.png" /></figure>

Figure 52. (a) Frank-van der Merwe (FM) growth compared with (b) Volmer-Weber (VW) growth. (c) Perfectly 2D Ag film with Ag (111) lattice constant grown by depositing 25 ML Ag onto Pt (111) at 600 K and subsequent annealing to 800 K. (d) 3D Pb islands grown on highly oriented pyrolytic graphite (HOPG) imaged with a SEM. [33]

A possible explanation stated by Brune, of the change from a 2D layer growth to a 3D structure is a relief on the strain induced by the lattice misfit between the substrate and overlayer. [33]
APPENDIX D. EBSD additional measurements of
Co-10.

<figure><img src="image_59.png" /><figcaption>Figure 52. EBSD measurements of Co-10 nanocrystals.</figcaption></figure>
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Abstract:

This report contains a first approach to the study of the magnetostrictive effect in ferromagnetic nanocrystals by using Bragg Coherent Diffraction Imaging (BCDI). This technique, known to be sensitive to the strain produced on crystals would allow to observe the effect. Two different types of crystals were synthesized: Nickel by solid state dewetting and Co by pulsed laser deposition. The beam-time obtained for the experiment in the ESRF, as the technique uses the high degree of coherence of the X-rays produced in those type of 3d generation large scale facilities, allowed us to study the nanocrystals of Nickel. Both samples were characterized by different techniques for preparing the experiment at the ESRF. Microscopy measurements (Optical microscopy, Scanning Electron Microscopy, Atomic Force microscopy), diffraction measurements (X-ray Diffraction, Electron Backscatter Diffraction) and magnetic measurements (Magnetic force Microscopy, Vibrating Sample Magnetometry) provided us an useful prior knowledge of the samples.

**Keywords:** Bragg Coherent Diffraction Imaging, Nanomaterials, Magnetostriction