High resolution X-ray diffraction (HRXRD) is employed in this work to characterize the basic structural properties of the ferromagnetic films. HRXRD is a powerful tool for non-destructive ex situ investigations of epitaxial layers. From HRXRD, information about the composition and uniformity of the epitaxial layers, layer thickness, strain and strain relaxation, as well as the presence of defects such as dislocations can be obtained. There are several text books and reviews [1,2,3,4] on the analysis of epitaxial layers by HRXRD. Here only a brief description of some relevant topics will be presented.

Geometries of asymmetric reflections   The two major kinds of scans in HRXRD, namely w-2q scan and w scan, are sensitive to different properties of the layer. The w-2q scan is sensitive to information such as strain, lattice constant and composition, whereas the w scan is sensitive to defects such as dislocations and mosaic spread. The w-2q type scans can be used for both symmetric and asymmetric reflections resulting in quite different scattering geometries. The geometry of the w-2q scan for an asymmetric reflection (hkl) making an angle j with the sample surface is schematically defined in Fig. 1. There are three measurement geometries: grazing-incidence (GI), grazing-exit (GE) and skew. In the skew geometry (a quasisymmetric configuration), the sample is tilted with respect to its surface normal by the lattice plane inclination j. Because of this tilting requirement, a four-circle diffractometer is required for the measurements in skew geometry.

Figure 0.1: Definitions of diffraction geometry for an asymmetric Bragg diffraction.(a) grazing-incidence geometry (b) grazing-exit geometry and (c) skew-geometry for the plane (hkl) making an angle j to the sample surface. q_B is the Bragg angle.

Determination of the strain and composition of the epitaxial layers   The lattice constant of a thin film that grows coherently on a single crystalline substrate is modified parallel to the growth direction. From X-ray diffraction, the information about lattice constant of the layer is obtained, which in principle is determined by Bragg's law:

where d_hkl is the spacing of the lattice planes with Miller indices (hkl) and q_B is the corresponding Bragg angle. From symmetric X-ray diffraction, information on the lattice constant of the layer perpendicular to the film plane a_^^L can be obtained. However, this is the strained lattice constant of the layer. For a tetragonal distortion of the layer, the unstrained lattice constant of the layer a₀^L is related to the strained lattice constant by the following equation [4]:

where the constants C₁₁ and C₁₂ are the elastic stiffness of the layer and a₀^s is the relaxed lattice constant of the substrate. This unstrained lattice constant of the layer is used to determine the composition of the layer. For binary and ternary alloys, the composition is often determined from Vegard's law, which simply states that the lattice constant of a solid solution alloy varies linearly with composition, following a line drawn between the values for the pure constituents.

Determination of thickness of the layers   A very accurate way to determine the layer thickness can be achieved from the so called thickness fringes, which are small periodic oscillations around the layer peak in an w-2q scan. These fringes originate from the interference of the wave fields. The measurement of these interference peak separation, Dq_B, provides the thickness t [1]:

where l is the wavelength, and j is the angle between the reflecting plane and the surface. Positive sign applies to the GI geometry and negative sign to the GE geometry. This is a very useful method, since the above equation does not contain anything about the material or diffraction conditions other than the Bragg angle and geometry. In practice, a more accurate computer simulation is usually employed to extract the layer thickness and other parameters. In this study, a computer program called MadMax was employed which uses a dynamical X-ray diffraction formalism using the Takagi-Taupin formalism [5].

Reciprocal space map (RSM)   As the name suggest, the RSM refers to the intensity contour maps, keeping one of the Miller indices, e.g., l in the reciprocal lattice, fixed, while the other two indices h and k are varied by �Dh and �Dk, respectively. This is achieved with a triple axis diffractometer, where an analyzer is placed behind the specimen and in front of the detector. The analyzer is mounted on an axis concentric with the specimen and is scanned independently of the sample. The experimenter can then map the intensity distribution with respect to the direction of the radiation scattered by the instrument. Usually, one measures several w-2q scans for different w offsets with the analyzer crystal. The w-2q scan in fact corresponds to a scan along the vector S_x[hkl] reflection in reciprocal space whereas the w scan is perpendicular to S_x[hkl] as shown in Fig. 2(a). The conversion of a peak position (w, 2q) to reciprocal space co-ordinates S_x and S_y can be obtained from Fig. 2(b) [2]:

Usually, the vector components S_x and S_y refer to directions perpendicular and parallel to the growth plane. The region that is accessible in reciprocal space depends on the geometry, the wavelength and the lattice constants of the epitaxial layers. In a RSM, the strain influences only the S_y direction whereas the mosaic spread or tilt is observed along the S_x direction in reciprocal space. Thus, using RSM, the strain and strain gradients can be separated from structural imperfections such as tilts and mosaicity.

Figure 0.2: (a) Scans in reciprocal space for two different reciporcal lattice points (hkl) (symmetric) and (h₁k₁l₁) (asymmetric). (b) Reciprocal space construction for the asymmetric reflection (h₁k₁l₁). k_s and k₀ are the wave vectors for the diffracted and the incident X-rays, respectively. S=k₀+k_s is the scattering vector.

Bibliography

[1]

D. K. Bowen and B. K. Tanner, High Resoultion X-ray Diffractometry and Topography (Taylor and Francis, London, 1998), 1st ed.

[2]

P. F. Fewster, X-ray Scattering from Semiconductors (Imperial College Press, London, 2005), 2nd ed.

[3]

P. F. Fewster, Rep. Prog. Phys. 59, 1339 (1996).

[4]

A. Krost, G. Bauer, and J. Woitok, in Optical Characterization of Epitaxial Semiconductor Layers, edited by G. Bauer and W. Richter (Springer-Verlag, New York, 1996).

[5]

O. Brandt, P. Waltereit, and K. H. Ploog, J. Phys. D 35, 577 (2002).

Last Updated Wednesday October 26, 2005