Structural characterization

Scanning Electron Microscopy (SEM) is used to determine the thin films structure. SEM is a kind of electron microscope to image the sample surface by scanning with a high energy electron beam. During measurement, the electrons interact with electrons in the sample, which produce a secondary electron signals that can be detected and distinguished. Information on the sample surface topography and composition can be obtained. Film growth and film thickness analysis can also be performed using cross section SEM. The SEM work in this thesis was done on a Zeiss-1550 HRSEM, operated between 0.2 and 30 kV.

Page | 23 The crystal structure of PZT thin films was analyzed by X-ray diffraction (XRD) measurements. The measurements were performed on either Bruker D8 Discover or PANalytical X’Pert diffractometer, using Cu Kα1 radiation. The films and substrates were analyzed using θ-2θ scans, rocking curve scans and reciprocal space map, to determine the crystal structure, roughness, growth orientations domain structure and lattice tilting. Using a reciprocal space map (HL scan and HK scan), the in-plane and out-of-plane lattice parameters, and the domain tilt can be determined. More details about reciprocal space measurement results will be discussed in Chapter 3.

2.4.1.1

Basic principles of reciprocal space mapping

Reciprocal space refers to a space in which the Fourier transform of a wave function is represented35_{. Before talking about reciprocal space, we first go back}

to the basic and essential formula, Bragg’s law, = 2

sin

Where n is an integer, λ is the wavelength of the incident wave, d is the spacing between the equivalent lattice planes, and θ is the angle between the incident beam and the scattering planes. When using the Bragg’s law, we consider diffraction in terms of the crystallographic planes hkl. Moreover, the vector perpendicular to the planes hkl is introduced to define the orientation of the plane, as shown in figure 2.6. , 2 and 3 are the reciprocal vectors of the real

space lattice vector 1, 2 and 3. Since Hhkl is perpendicular to the hkl planes,

the d spacing can be written as

= 1

When all the hkl vectors are drawn for all values of the indices hkl, the terminal

points form a new lattice. This lattice is called the reciprocal lattice. And the reciprocal lattice of a reciprocal lattice is the original real space lattice.

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Bragg’s law can also be expressed by using the vector hkl. As shown in figure

2.6, if 0 and are the unit vectors in the directions of the primary and diffracted

beams, angle θ is the angle between the diffraction planes and vector 0. Then

the relation between vector and vector is, −

= When we take into account angle θ,

−

=2

In this case, the equation becomes,

= 1 =2

sin

which is equivalent to the Bragg’s law. Figure 2.6 is a simple graphical, which tells us that by satisfying Bragg’s law, one obtains the reciprocal lattice. And the reciprocal lattices turn out to be extremely useful in the analysis of crystal diffraction studies.

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2.4.1.2 Ewald sphere

As we shown in the previous section, the reciprocal lattice provides a simple graphical representation of the satisfying the Bragg’s law. Another powerful and useful way to satisfy Bragg’s law is the Ewald sphere construction.

The reciprocal lattice is represented schematically in two dimensions, see figure 2.7 (a). The direction of the primary beam is shown. A vector of length 1/λ terminates on the origin of the reciprocal lattice. A sphere of radius 1/λ centered on the original crystal. For a diffraction point in reciprocal space to be in diffraction condition, it must lie on the surface of the Ewald sphere. The calculations of Bragg’s law in the sphere reflection is in the same condition as the reciprocal lattice point hkl, which falls on the surface of the sphere. Even though this schematic drawing is in two dimensions, the Ewald sphere is valid in three dimensions.

Here, we should also notice that not all the hkl planes can be visible in XRD. It depends on how much the crystal can rotate in order to reach that diffraction plane. For thick samples, absorption of either the incident beam or the diffracted beam will restrict access to the reflection geometry in reciprocal space. The forbidden areas are shown in (0kl) plane, see figure 2.7 (b), where each red cross corresponds to a crystal plane. During the measurements, different values in the reciprocal lattice can be chosen, which define different reflection planes. For an ideal perfect epitaxial thin film, a single spot can be expected in reciprocal space mapping, which corresponds to a sharp peak in a one dimensional theta- 2theta scan. If the epitaxial films contains some defects, an elongated spot can be expected in the reciprocal space map. This is different with a textured film. In a textured film, the out of plane is aligned, while the in plane orientation is random. Thus a parts of a circle can be expected in two dimensional reciprocal map, while no peak will be shown in phi scan. Phi angle is the in-plane rotation angle, more details about phi scan, see figure 3.7. If the sample is a randomly

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orientated polycrystalline, a full range of circle ring will be detected in 2 dimensional reciprocal space.

Figure 2.7 Schematic drawing of 2 dimensional (a) Ewald sphere, (b) reciprocal lattice mapping with the forbidden areas.