nuclear part one can determine the stoichiometric fluctuations, film roughness and substrate roughness.
In case of a non-saturated film, the magnetization can be canted away from the polarization direction of the neutron spin. The component of the magnetization perpendicular to the neutron polarization will cause SF scattering, while the parallel component leads to NSF scattering as shown in Fig.3.8. For such cases, one has to measure UD and DU channels as well along with UU and DD. Hence, when a neutron with spin↑will interact with a magnetic component perpendicular to it, the SF scattering will occur and this spin-flipped neutron ↓will be able to pass through analyzer, given analyzer spin selection is ↓ resulting in SF signal. In this way one can determine the magnetic depth profile using Polarized Neutron Reflectometry.
3.7 Diffraction
The crystallinity of a film is determined by the diffraction technique where the incident beam results in an interference pattern after encountering a structure. With the resulting interference pattern one can obtain information about the internal lattice of a crystalline structure, unit cell dimensions, site-ordering and depending on the probe (neutrons) magnetic ordering as well. X-rays and neutrons have the wavelength in an order of 10−10m and therefore, can be used to probe the crystal structures. In this thesis work, X-ray diffraction has been used to determine the crystallinity of the thin films. A crystal lattice in three dimensions can be described by three translational vectors a⃗1, ⃗a2 and a⃗3. Using a linear combination, one can address all the lattice points as
⃗a =u ⃗a1+v ⃗a2+w ⃗a3 (3.37) where u,v and w are arbitrary integers. The diffraction pattern depends on the crystal structure and the wavelength of the probe. W.L. Bragg gave a simple ex-planation of diffraction from crystal. When an incident wave strikes a crystal with parallel planes of atoms, it is reflected specularly from each plane with αi =αf =θ. These reflected beams interfere constructively yielding a diffraction pattern. As pre-sented in the Fig.3.9, with parallel atomic planes distance ,d, apart then the path difference is 2dsinθ between the reflected rays from adjacent atomic planes.The con-structive interference will occur when the path difference is an integral number n of the wavelength λ, yielding Bragg’s law
2dsinθ=nλ (3.38)
The periodicity of the lattice is the base for Bragg’s law. Now, with X-ray diffrac-tion, the lattice parameters of the epitaxially grown thin film can be deduced and for that one needs to model the scattered intensity for a single crystalline layer us-ing the electron density ρe of the thin film. In this thesis work, the out-of-plane lattice parameter of the film is parallel to one of the crystallographic axis. This
Figure 3.9: Schematic of Bragg’s law 2dsinθ =nλ, where d is the space between the parallel atomic planes and 2πn is the phase difference between the reflections from successive planes. Inspired from [59]
means that the layer consisting ofN,M and P unit cells along x,y and z direction, the z will be the out-of-plane direction. The overall structure of a thin film can be described mathematically as a convolution of periodic lattice and the unit cell struc-ture. The Fourier transform of ρe can be written as a product of Fourier transform of lattice functionρL times the Fourier transform of the unit cell functionρu.c. using convolution theorem:
ρL ∝
N−1
X
n=0 M−1
X
m=0 P−1
X
p=0
δ(⃗r−(n⃗a+m⃗b+p⃗c)) (3.39) where ⃗a,⃗b and ⃗c are lattice parameters. The Fourier transform of eq.3.39 yields Laue-function for point like scatterers
I(Q⃗)∝ sin212N ⃗Q⃗a
sin212Q⃗a⃗ · sin212M ⃗Q⃗b
sin212Q⃗b⃗ ·sin212P ⃗Q⃗c
sin212Q⃗c⃗ (3.40)
Figure 3.10: Plot of Laue function along the lattice direction⃗cwith 5 and 10 periods
3.7 Diffraction
The Laue function along lattice direction ⃗c is plotted in Fig.3.10. The maxima occur at the positions Q=n· 2πc and are also known as Bragg reflections obtained for scattering from a crystal lattice . The intensity of Bragg reflections scale to the square of the number of periods N2 and the width of 2πN. With the increasing number of periodsN, the intensity of Laue oscillations becomes negligible compared to the intensity of the Bragg reflections. From the positions of Bragg peaks, it is possible to determine the lattice constants and the unit cell angles.
4 Scattering with Electrons
This chapter discusses the scattering processes that take place when electrons inter-act with a solid matter and the techniques based on electron scattering used in this thesis. The literature is based on the textbook written by D.B. Williams and C.B.
Carter: Transmission Electron Microscopy [60].
4.1 Interaction of electrons with matter
It is known that electron exhibits wave-particle duality based on de-Broglie’s ideas, where the particle momentum, p, is related to its wavelength, λ, through Planck’s constant, h: λ = hp. Now in TEM, the momentum is imparted to an electron by accelerating it through a potential drop, V, giving it a kinetic energy of eV. This potential energy must be equal to the kinetic energy, which yields in non-relativistic approximation:
eV = m0ν2
2 (4.1)
where m0 is the mass of an electron and ν is the velocity. Thus, using eq. 4.1, the momentum can be equated as
p=m0ν = (2m0eV)1/2 (4.2)
Furthermore using eq. 4.2 in de-Broglie’s relation, one gets
λ= h
(2m0eV)1/2 (4.3)
From this equation, one can see that, by increasing the accelerating voltage, the wavelength of electrons decreases. Until now, the relativistic effects have been ig-nored, but for energies higher than 100 keV, the relativistic effects have to be taken into account. This is because the velocity of the electrons (as particles) becomes greater than half the speed of light. Therefore, eq. 4.3 can be modified as
λ= h
h2m0eV 1 + 2meV0c2
i1/2 (4.4)
When electrons passes through a thin sample, different scattering processes take place: elastic (no loss of energy) and inelastic (change in energy) scattering, coherent and incoherent scattering as shown in Fig. 4.1. The coherently scattered electrons stay in-phase while the incoherently scattered electrons lose the phase relationship after the scattering event. Elastic scattering usually occurs at low angles in the range from 1° - 10°. As the angle becomes larger than 10°, the elastic scattering becomes more incoherent, whereas for inelastic scattering, it always incoherent and occurs typically at angles lower than < 10°.
Figure 4.1: Different kind scattering events with electrons passing through a thin sample. Inspired from [60].
Electrons can scatter more than once with increasing scattering angle and the more the scattering events, the more difficult it gets to interpret the images and the spectra obtained. Therefore, to avoid the multiple scattering events, one needs to thin down the sample so that it becomes electron transparent, which is the basis of the TEM.
The specimen thickness for TEM is usually in the range from a monolayer (e.g.
graphene) to a few hundred nanometer depending from the material and acceleration voltage of the TEM (typically 60 to 300 kV). With TEM, unless one has a thick sample which will result in multiple scattering, the single-scattering assumption is plausible for thin enough specimens. Therefore it is important to have very thin samples for TEM experiments. The electrons that are scattered in forward direction, parallel to the incident beam, form the direct beam, which is used for most of the elastic scattering, diffraction, refraction and the inelastic scattering. The scattering events depend on various factors such as thickness, density, crystallinity, atomic number of the scattering atom and the angle of sample with respect to the incident beam.