Strain in materials is one of the important factors affecting physical properties of the materials such as carrier mobility, dielectric property, magnetism and so on. In semiconductor industry, strain engineering has been playing a primary role for the improvement of the device performance. Measurement of strain has also been very important as a key technique supporting the strain engineering. So far, the strain measurement by diffraction technique has been done mainly by measuring the positions of diffraction peaks and by fitting the experimental peak positions to simulated ones. This method implicitly assumes that strain in the volume contributing to diffraction intensities is uniform. Strain in real materials, however, is not always uniform and varies over the diffraction volume of the specimen. In the present study, we applied convergent-beam electron diffraction (CBED) to determine such non-uniform strain, whose lattice displacement vector varies along the incident direction of the electron beam.

Lattice scattering amplitude of reflection **g**, *φ*** _{g}**, is given as a sum of scattered waves from each lattice point, which can be expressed by the Fourier transform of the lattice. If the lattice has a displacement field

**R**(

**r**), a phase factor of 2π

**g**·

**R**(

**r**) has to be taken into account for

*φ*

**, where**

_{g}**r**is a positional vector for a point in the crystal. In the case of convergent-beam electron diffraction,

**R**(

**r**) is unchanged in the direction perpendicular to the incident direction of the beam because the probe diameter is sufficiently small, and thus,

**R**(

**r**) can be written as

**R**(z), where z indicate a coordinate along the incident direction. The Fourier transform of the phase factor of exp(2π

*i*

**g**·

**R**(z)) can be written as

*φ*

**(s) with an excitation error s, which is a conjugate variable of z. We applied the Fourier iterative phase retrieval technique to restore the phase part of**

_{g}*φ*

**, and to determine the phase factor of the lattice displacement 2π**

_{g}**g**·

**R**(

*z*). In the present study, the modulus of

*φ*

**(**

_{g}*s*) was measured from a rocking curve profile observed by a CBED pattern.

A Ge_{92.9}Sn_{7.1} layer of 200 nm was deposited on a Ge (001) substrate by the chemical vapor deposition method in a ultra-high vacuum. Cross section samples for electron microscopy were prepared by mechanical polishing and ion-beam thining. Rocking curves were obtained by the CBED technique at an incidence inclined by about 10 degree from the [110] direction. CBED experiment was conducted by using a transmission electron microscope operated at an acceleration voltage of 200 kV. CBED patterns were taken by using Gatan imaging fileter with an energy window of 5 eV to remove inelastic scattering mainly by the plasmon loss.

Figure 1(a) shows a cross-section TEM image of the specimen. CBED patterns were taken from the positions indicated as 1 to 6 in the Ge substrate. Figure 1(b) shows a whole CBED pattern used in the phase retrieval. Figures 2(a), 2(b) and 2(c) show enlarged disks of the -26-8, -553 and -317 reflections and their rocking curve profiles, respectively. Figures 2(d), 2(e) and 2(f) respectively show phase profiles of 2π**g**·**R**(*z*) of the -26-8, -553 and -317 reflections as a function of the *z*-coordinate determined by the present study. From these phase profiles, the lattice displacements in the [001], [110] and [110] directions are determined as shown in Figures 3(a), 3(b) and 3(c), respectively. It is clearly seen that the displacement field in the [001] direction is of mirror symmetry about the center of the specimen, which is consistent to the elasticity theory. The displacements field determined by the present method is quantitatively compared to the simulated values obtained by the finite element method.

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**EMC Abstracts** - https://emc-proceedings.com/abstract/determination-of-a-3d-displacement-field-at-a-vicinity-of-a-gesnge-interface-by-the-phase-retrieval-of-electron-rocking-curves/