1. IPCMS-GEMME, BP 43, 23 rue du Loess, 67034 Strasbourg Cedex 2
Characterization of thin films and bulk materials using x-ray and electron scattering
V. Pierron-Bohnes
IPCMS-GEMME, BP 43, 23 rue du Loess, Strasbourg Cedex 2
Plan : lattices
x-ray and electron - matter interaction
real lattice and reciprocal lattice in 3D and 2D samples
experimental set-ups
studies on single crystals
multilayers
strains measurements using x-ray scattering and TEM
powder scattering measurement
texture analysis
reflectometry
chemical analysis
short and long range order measurements
This chapter will contain:
orientation of crystals
maps in the reciprocal space
coherence lengths
Laue diffraction

2. Orientation of single crystals:
surface | to the j axis
with a goniometer inside the j cradle j
sample holder
specimen
To make sure that the specimen surface is perpendicular to the phi rotation axis, a usual mean is to let a laser beam reflect on the surface, when phi is rotated, the laser impact on a screen has to be unchanged. If it moves, the sample surface has to be modified until the beam impact is i the center of the ellipse described by the beam. This is the necessary condition to be able to find easily a peak from another peak position.
laser
screen

3. orientation of single crystals: at last 3 non aligned peaks Plan (100)j
orientation
of single
crystals:
at last 3
non aligned
peaks
To determine exactly the position of the single crystal in the space, we need at least 3 non-aligned Bragg peaks. For each,
we calculate alpha and theta from the indices of the peak and the lattice parameters of the structure. The counter is positioned at 2theta, either omega is positioned at theta +/- alpha (possible if |theta|>|alpha|), or chi is positioned at alpha. Phi is rotated by steps while monitoring the intensity. If 360° is scanned, there should be as many peaks as expected from the crystal symmetry.
Note that the two geometries omega = theta + alpha and omega = theta - alpha are not equivalent: the intensity registered is higher if the incidence angle is high and the exit angle is small.
Exercise: in CoPt with the orientation described in the graph, how many 024 peak are expected when rotatinf Phi by 360°? Idem for 013 for an x-variant, for 004.
at a and q, rotation in j until the peak
is found + optimization in (c, j), w, q/2q

4. orientation of single crystals: at last 3 non aligned peaks
Plan (1-10)
orientation
of single
crystals:
at last 3
non aligned
peaks
222
002
112
111
221
001
220
000
110
To determine exactly the position of the single crystal in the space, we need at least 3 non-aligned Bragg peaks. For each,
we calculate alpha and theta from the indices of the peak and the lattice parameters of the structure. The counter is positioned at 2theta, either omega is positioned at theta +/- alpha (possible if |theta|>|alpha|), or chi is positioned at alpha. Phi is rotated by steps while monitoring the intensity. If 360° is scanned, there should be as many peaks as expected from the crystal symmetry.
Note that the two geometries omega = theta + alpha and omega = theta - alpha are not equivalent: the intensity registered is higher if the incidence angle is high and the exit angle is small.
Exercise: in CoPt with the orientation described in the graph, how many 024 peak are expected when rotatinf Phi by 360°? Idem for 013 for an x-variant, for 004.

5. Coherence lengths: grains with half size L// in plane
If the crystal is not infinite, the Bragg peaks have a non-zero width. Typically, the peak width in the reciprocal space is equal to 2pi divided by the average width (coherence length) of the grains in the real space (typically sin(2piNxqxax) / sin(2piqxax) for the direction x in the case of an orthorhombic lattice). In the scans theta/2theta the coherence length perpendicularly to the specimen surface is measured. In rocking curve on a specular peak (on the surface normal), the coherence length in a parallel to the specimen surface is measured. For another peak it is a combination of both coherence lengths. In thin films deposited using different methods, these coherence length are interesting and a good indication of the film quality. For a very thin film, the perpendicular width of the peaks is large.
grains with half size L// in plane
and L | perpendicular to the plane

6. Coherence lengths: with mosaïcity (fluctuations in grain orientations)
varies with the peak order
slope
slope
mosaïcity and
coherence length
can be separated
by measuring several
order peaks
Most of samples present a mosaïcity: a distribution of the crystallographic directions around the most probable direction. The intensity of the specimen is the sum of the intensities of each grain. This enlarges the peaks by rotations of the reciprocal space: the peaks are enlarged in rocking curve and the farer the peak the bigger the enlargement. When measuring several peaks width in rocking curve at different orders, it is possible to separate both contributions.

7. Coherence lengths: with parameter fluctuations (distribution of grain
concentrations,
interface
interdiffusion,
inhomogeneous
strains)
varies with the peak order
slope :
If the specimen present a distribution of lattice parameter (concentration, strains…) or if the beam presents a wavelength spread, the peaks are enlarged by dilatation of the reciprocal space. When measuring several peaks width in theta/2theta at different orders, it is possible to separate both contributions.

8. Maps in the reciprocal space: why?
Verify that there is only one peak
Verify epitaxy
Measure the lattice parameter in-plane
Evidence stacking faults
Evidence fcc and hcp stacking
It is often useful to measure maps in the reciprocal space to verify that there is only one peak, to verify the epitaxy (a map containing both the film and the substrate peaks), to measure the lattice parameter in-plane on out of normal peaks, to evidence stacking faults (see Co/Mn), to evidence fcc and hcp stacking presence… The maps can be measured using theta/2theta scans or rocking curves. The choice depends on the direction in which the most numerous points are necessary (most often it is in theta/2theta thus several theta/2theta curves are measured).
maps
in rocking
curves:
same a at
start and end
maps
in q/2q:
same q at
start and end
measured
in 3 zones

9. Bragg peak corrected intensity
peak forms
experimental intensity = integral of the q/2q curve x width of the rocking curve
assumes that the crystal is not too perfect (no extinction effects)
coherence
length along x
Many information are deduced from the comparison of different Bragg peak intensities. It is thus necessary to correct the measured intensity to have access to the true intensity. There are several factors changing the measured intensity and that have to be corrected.
The incident beam can be easily be kept constant (but take care when comparing measurements made after demounting and mounting the sample: the geometry may have changed a little, and the tube emission decreases when they become older; it is more cautious to compare peaks measured in a single measurement run).
The second factor is the form factor characteristic of the basis. It is generally the information searched for.
The third term is due to the lattice periodicity. It can most often be approximated by a Gaussian peak with an integral equal to the number of atoms in coherence.
The 4 last terms are detailed in the next slides.
approaching
gaussian
half-height width
for most of the forms

10. Lorentz, polarisation, absorption corrections
Bragg peak corrected intensity
Lorentz, polarisation, absorption corrections
Lorentz factor
for a single crystal
geometrical factor due to the integration in the (w,q)
space instead of the Q space
polarisation correction :
depends on the nature of monochromator
- no monochromator: 1
-a monochromator with angle a0:
cos22a0 for each crystal
The Lorentz factor is applied due to the fact that the integration is made in Theta and in omega and not in Qx and Qz. The polarization correction is due to the difference of refelctivity of a monochromator depending on the polarization of the beam.
The absorption correction compensates for the reduced diffracting volume due to the absorption of the beam within some micrometers.
Exercise: Show that the formula for thin films is changed (1/ sin omega) instead of (1/sin theta) if omega is not equal to epsilon.
absorption correction :
I0 decreases inside the crystal as exp(-2mz/sinw)
thick samples
thin samples
number of atoms in the beam

11. Debye-Waller correction
Bragg peak corrected intensity
Debye-Waller correction
Form of the Debye-Waller correction:
- dynamical part: due to the vibration of atoms
around their equilibrium position
different for fundamental peaks (<x2>)
and superstructure peaks (<x12-x22>)
harmonic approximation:
static part: due to the strains: the equilibrium positions of atoms are not at the lattice sites
The Debye-Waller factor has two contributions: the vibration of atoms around their equilibrium position and the static departure of the atoms due to strains or to the difference in size between the elements in an alloy. The Debye-Waller factor is related to the average quadratic displacements of atoms. Plotting the Log(I) as a function of q2 it is possible to estimate this attenuation factor through the slope. In the ordered alloys, to the fundamental peaks corresponds to the displacements of all atoms in average, whereas the superstructure peaks correspond to the relative displacements of the two different atoms. The slope is thus different for both types of peaks. To have a good estimation of this parameter, a measurement with a very short wavelength and high flux is necessary. The results shown here were obtained on the French synchrotron LURE.
CoPt thin film

12. Orientation of a crystal (bulk)
Laue diffraction
The beam is not monochromatized but collimated
(Mo cathode, high background)
All l are present
→ all grains are in diffraction position for a given l
There is some intensity at sin(2x) if the plane normal makes
an angle x with the incident beam
collimator
When nothing is known on a crystal (single crystal as grown), it is necessary to have an idea of its orientation before making x-ray diffraction (the probability to get intensity in a random direction is almost zero for a single crystal). An usual way to orient a crystal is the Laue diffraction. Using a beam containing a continuous distribution of wavelength, it is possible to make all crystallographic planes diffract. The planes give some diffracted intensity in a direction with an angle double with the incident beam. There is intensity at sin(2xi) if the plane normal makes an angle xi with the incident beam. There is a spot on the film for those near enough to the incident beam.

13. Orientation of a crystal (bulk)
Laue diffraction
Example of Laue diffraction
made on a FePd single crystal
on a (001) face using an imaging plate.
This is an example of Laue diffraction made on a FePd single crystal on a (001) face using an imaging plate. The white circle is the position of the incident beam (the aperture holder is metallic). The two white lines are due to the imaging plate holder that contains metallic wires.

14. stereographic projection
specimen surface normal
scan in Φ at fixed Ψ
projection plane
To represent
stereographic projection
on normal
in the plane

15. Example of Laue diffraction: centering the pattern with the Wulff diagram – points are placed and shifted on the stereographic projection
incident
beam
position
The Wulff diagram (on the right) is a stereographic projection of all the directions in an hemisphere (used in astronomy to describe the stars in the sky). The perpendicular (vertical in astronomy ) is in the middle. The top point corresponds to the in plane in one direction. In astronomy top, down, left and right correspond to the 4 cardinal points. The angles are measured in two directions: towards the north and the east for example. Each graduation corresponds to 2°.
In a first step, the experimental points are transformed in a stereographic representation using the correspondence: D sin(2xi) -> xi in the same radial direction.
The highest symmetry point is chosen and all the figure translated in order to get the highest symmetry point in the middle of the figure. The angles are then readable on the figure directly. Here it is easy to recognize a [001] axis Laue.

16. Laue diffraction The centered stereographic projections are
compared with
well known patterns
(fcc, bcc, hcp…).
Only the point
symmetry group
is important.
impossible for thin
films → only TEM
The two crosses at 45° have now to be attributed either to <100> or to <110> types directions. The positions and intensities of the peaks are then considered: the most intense peaks correspond to the smallest indices planes. The intermediate reflections are also considered.
Some figures with the Laue stereographic projections can be found in the literature, they can also be calculated using a software (as CarIne).

17. Laue diffraction simulated with CarIne for FePd: construct the lattice
calculate the diffraction
calculate the stereographic projection
This are examples of what can be done with CarIne.
Exercise: I have CarIne on my computer. You can simulate a lattice, calculate diffraction, and stereographic projection of a structure of your choice.