Figure B.1: Estimation of the height of the Siemens Star 30 nm big structures using phase contrast:
a) phase reconstruction of Siemens star and background correction image; b) calculated thickness of the Siemens Star from the phase image.
Figure B.2: Edge artifacts: a) phase image ofLiFePO4nanoplate obtained by ptychography show-ing halo artifact; b) line profile correspondshow-ing to the blue line and c) line profile correspondshow-ing to the red line in the image a).
Another possible artifact leading to incorrect phase image projection is a missing or incorrect unwrapping of the phase. The phase difference bigger thanπresults in incorrect phase unwrapping. The phase wrapping is shown in figure B.3 a) that is projected as a black area with big negative values in the region supposedly having positive phase shift.
The figure B.3 b) is a corrected phase image obtained by changing the values which have jumps greater thanπto their2πcomplement. This process is called phase unwrapping that is free from 2πjumps and returns the wrapped phase to a continuous phase signal.
The image was corrected using Python programming language.
Another source of error is differential phase clipping which occurs when the phase changes within one pixel. Mainly it happens at the edges of highly scattering objects, when the change of the phase from background to the particle is comparatively high.
Figure B.3: Phase images ofLiFePO4 nanoplate: a) shows the wrapping of the phase to negative values that cause incorrect color projection, b) corrected phase image.
Appendix C
Methods for resolution evaluation
Spatial resolution is a fundamental parameter of any imaging technique. As it was dis-cussed in the previous chapters resolution of a reconstructed ptychographic image depends on many experimental factors: scattering power of sample, coherence, numerical aperture of detector, as well as external noises and sample deformations. Therefore resolving power of ptychography must be analyzed using the reconstructed image itself. Beside the con-ventional methods of resolution evaluation, e.g. imaging of test targets or measurement of step profile of the regions with different contrast, there are methods which examine image resolution in the Fourier space, described in literature [201, 202, 203, 204]. In the current work power spectral density (PSD) and Fourier ring correlation (FRC) are used.
Power spectral density (PSD)shows the strength of the signal in reciprocal space as a function of frequency. In this case resolution of the image is estimated as a signal power depending on the spatial frequency in Fourier space. It derived as following:
P SD=F T[ρ(r)], (C.1)
whereρ(r) = x(r)·x(−r)is an auto-correlation function of the signalx(r). The 2D PSD function is presented as a 1D radial profile spectrum over allϕin reciprocal space.
The spatial frequency analysis indicates the achievable reconstruction resolution that is determined as a double level of statistical noise. The flat profile at the highest frequencies end of PSD curve can be regarded as the noise level.
Fourier ring correlation (FRC)is a method that initially was used for the evalu-ation of resolution in electron microscopy [202] . However it became commonly used for diffraction imaging techniques. FRC estimates the similarity of two independent pty-chographic reconstructions of the same sample in the reciprocal space. It determines the resolution threshold at which both reconstructions are consistent. To calculate the corre-lation a diffraction data set of one image has to be split into two sets and reconstructed
into two independent images, which then are Fourier transformed and multiplied. The normalized average correlation is computed forNrconcentric rings of increasing radius, which correspond to increasing spatial frequencies in the Fourier space, and expressed as following:
F RC(R) = Σi∈RI1(ri)I2(ri)∗
p(Σi∈R|I1(ri)2|)·(Σi∈R|I2(ri)2|), (C.2) whereR- ring number,ri- the individual pixel at radiusR,I1andI2Fourier transforms of two ptychography images.The number of rings determines the binning of the FRC curve.
Determination of resolution in FRC is still controversial issue [205]. Currently three basic threshold methods are used:
• The1/2−bitthreshold is a method when the intersection of the threshold line with the FRC curve defines the point where information per pixel is equal to 1/2 bit.
• The fixed threshold of 0.143 [206] that is an estimated value representing the av-erage signal-to-noise ratio in a reciprocal space resolution ring and the error in the structure factors.
• The threshold ofσ-factor aims to determine the high-frequency limit, where the signal stays significantly above the random-noise level. So the intersection of the FRC curve with this threshold line corresponds to the value where the FRC begins primarily represent high frequency noise.
The described methods are used for resolution evaluation in this thesis.
Appendix D
Magnetic properties of skyrmion multilayer with different
thickness of ferromagnetic layer
Multilayer films Ta(5)/[CoFeB(1.1 nm wedge)/MgO(2)/Ta(3)]15 (numbers are thickness of layer in nm) with total CoFeB layer thicknesses between 11.55 and 21.75 nm have been magnetically investigated by SQUID magnetometry at different temperatures from 5 to 350 K. The samples were numbered from 1 to 7 in accordance with the FeCoB single layer thickness as following: No.1=0.8 nm, No.2=0.9 nm, No.3=1.0 nm, No.4=1.1 nm, No.5=1.2 nm, No.6=1.3 nm, No.7=1.4 nm.
The intrinsic magnetic material parameters were determined: saturation polarization Js(or saturation magnetizationMs = Js/µ0), exchange constantAand magnetocrys-talline anisotropy constantK1as it is shown in figures D.1 and D.2.
The saturation polarizationJsis directly obtained with the approach to saturation of the magnetic polarization in the OOP hysteresis loops shown in figures D.3, D.4, D.5. The samples with higher CoFeB layer thicknesses have largerJsvalues for a given temperature as it is seen from figure D.1 a). The exchange constantArelates to the strength of inter-action between neighboring magnetic moments due to exchange interinter-action and cannot be determined directly. It is evaluated as a function of temperature from the temperature dependence of the saturation polarization, that is given as following:
A(T) = Js(T)Dsp
2gµ0µB =Js(T)kBTC
2gµ0µB
0.117µ0µB
Js(0) 2/3
, (D.1)
where the parametergis theLand´eg-factor,Dspis the spin wave stiffness constant,µB=
9.274×10−24 A/m is the Bohr magneton, Js(0) describes the spontaneous magnetic polarization atT = 0K,TCis the Curie temperature andkB is the Boltzmann constant.
The parametersJs(0)andTCcan be determined by plottingJs(T)versus(T /T0)3/2and extrapolating it to the axes. The exchange constant continuously increases with increasing of CoFeB thickness as it is seen from figure D.1 b).
The anisotropy describes the energy required to rotate the magnetization vector from easy to hard direction in ferromagnetic. The temperature dependent magnetocrystalline anisotropy constantK1(T)for skyrmion samples have been determined by using follow-ing expression:
K1(T) =K1(0)
Js(T) Js(0)
3
(D.2) Initially the K1(0) values have been obtained by inserting the experimentally derived K1(RT)andJs(RT)values into equation D.2. Opposite to the behavior observed for the saturation polarization Js and exchange constantA the magneto-crystalline anisotropy constant decreases with the increase of CoFeB thickness as it is shown in figure D.2.
Curie temperatureTccan strongly depend on the thickness of ferromagnetic layer. It is found to drop significantly for the Ta systems as it is shown in figure D.6.
Figure D.1: Temperature dependent magnetic properties of Ta/CoFeB/MgO/Ta skyrmion samples with different thickness of ferromagnetic layer: a) polarization saturationJsand b) exchange stiff-nessA.
Figure D.2: Temperature dependent anisotropy constantK1of Ta/CoFeB/MgO/Ta skyrmion sam-ples with different thickness of ferromagnetic layer.
Figure D.3: Temperature dependent hysteresis loops for skyrmion samples No. 1 and 3.
Figure D.4: Temperature dependent hysteresis loops for skyrmion samples No. 4 and 5.
Figure D.5: Temperature dependent hysteresis loops for skyrmion samples No. 6 and 7.
Figure D.6: Dependance of Curie temperature on the thickness of CoFeB layer in skyrmion multi-layer.
Appendix E
Magnetisation simulation of nanoparticle Ni shell
E.1 Cell size of the simulations
The cell size requirement for the simulation is defined by the exchange length of Ni, which is 9.9 nm. Since the thickness of Ni core is 15 nm the cell size of 5 nm was chosen that assures the complete coverage of the surface of simulated volume with square subunits.
The size of the cell determines the processing time of simulations, since the total amount of calculated points is scaled with a cube of the cell size. Amount of simulated cells is also limited by computational power of the GPUs: the maximum total particle size is 255×255×255for 1 GPU.
The simulated shell volume was calculated as a difference of the total volume of the particle and the core volume. In order to estimate the simulated shell volume let’s consider icosahedron shape. The volume of icosahedron shell is determined as:
V =5(3 +√ 5)
12 (a3part−a3core), (E.1) whereapartandacoreare the edge lengths of the whole particle and the core, respectively.
If particle size is 700 nm andapart=368 nm the simulated volume is1.27·107nm3. Using 5 nm cell size it results in1.02·105simulated cells in total. All the values related to the core of the particle are set to zero.