Strain in a silicon-on-insulator nanostructure revealed by 3D x-ray Bragg ptychography

Abstract

Progresses in the design of well-defined electronic band structure and dedicated functionalities rely on the high control of complex architectural device nano-scaled structures. This includes the challenging accurate description of strain fields in crystalline structures, which requires non invasive and three-dimensional (3D) imaging methods. Here, we demonstrate in details how x-ray Bragg ptychography can be used to quantify in 3D a displacement field in a lithographically patterned silicon-on-insulator structure. The image of the crystalline properties, which results from the phase retrieval of a coherent intensity data set, is obtained from a well-controlled optimized process, for which all steps are detailed. These results confirm the promising perspectives of 3D Bragg ptychography for the investigation of complex nano-structured crystals in material science.

Figure 1. The crystalline silicon-on-insulator (SOI) structure as seen by scanning electron microscope.

(a) Front view. (b) Same as (a), top view; The direction of the ptychography translation (black arrow) and the scanning step are indicated. The projected direction of propagation of the incident beam (white arrow) together with the beam footprint (FWHM of intensity, dotted ellipse) are shown. The gray ellipse corresponds to the beam footprint at the next beam-to-sample position. The ( orthogonal laboratory frame is given.

Figure 2. Experimental considerations for 3D Bragg ptychography: beam profile and Bragg diffraction geometry.

(a) Intensity pattern of the over-focused direct beam (arbitrary units) measured with a high-resolution camera. (b) Color rendition of the complex-valued beam profile, retrieved from the inversion of (a) and shown in the Fresnel zone plate focal plane. The brightness and color correspond to the linear scale amplitude and to the phase , respectively. (c) Description of the 3D Bragg diffraction geometry, including the Bragg angle (), the incident and exit wave vectors (, respectively) and the Bragg vector (). The 3D non-orthogonal () detection frame is defined in agreement with the detection acquisition modality, corresponding to the 2D detector plane and to the rocking curve direction. The () laboratory frame is also shown.

Figure 3. 3D Bragg ptychography data set.

(a) Intensity patterns extracted from the 4D coherent Bragg diffraction measurements. For each frame, measured in the plane, only the central part of the pattern is shown. The steps along correspond to , relatively to the most central position while along the ptychography translation , the steps are (m⁻¹ and nm). The intensity values in have been increased by a factor of 4 for sake of clarity. The white arrows emphasize the stronger intensity lobes, which are arising from the structure edges. (b) Intensity integrated along the direction, for a fixed , identical to (a). The dotted ellipses emphasize the missing intensity along the vertical streaks. In (a) and (b), the vertical and horizontal zero intensity lines correspond to blind pixels in the detector. The common logarithmic photon scale is shown in (b).

Figure 4. Presence of a crystalline displacement field: numerical studies.

Estimation of the expected diffraction patterns calculated for different 3D strained crystals with shape similar to the SOI structure. (a) Common 3D iso-surface rendering of the synthetic object together with the incoming beam shape (FWHM of intensity) for . The laboratory frame is given; the length of the black lines is 100 nm. (b) Three synthetic models, corresponding to three different strain states and their corresponding diffraction patterns. The 2D sample description is shown in the plane indicated in (a) while the diffraction patterns are taken at the same and values as the ones of (Figure 2, left column). (c) Intensity integrated along the direction, for the same value. The specific features of the calculated diffraction patterns are emphasized by the white arrows and the dotted ellipse. The three strain states are as followed: (Left) The 3D strain-free crystal case. A 2D cut through the 3D amplitude is shown in (a). Note the assymetry in the spatial scale, which is underlined by the white lines, representing a 100 nm length. (Middle) Same calculation, obtained for a strained crystal: a displacement field with a radial symmetry is introduced at the edge of the structure. A 2D cut through the corresponding sample phase is shown at the top. (Right) Same as before with the simultaneous introduction of the displacement field at the edges and at the interface. This last model produces diffraction patterns in good agreement with the experimental ones.

Figure 5. Optimizing the inversion scheme.

(Left) The 3D synthetic model object used to test the inversion procedure. It corresponds to the model shown on the right column of Figure 4. (Second column) Retrieved image using a conjugate gradient optimization of the Bouman and Sauer maximum likelihood, initialized with the shape of the object. (Third column) Same as before introducing an additional regularization term to constrain the sample support. (Right) Same as before, initialized with the true synthetic object. The top, middle and bottom rows are different cuts of the 3D object, as defined on the 3D isosurface plot rendition on the right. The assymetric spatial scale is given on left (y,z) and (x,y) cuts. Each line corresponds to a 100 nm length. The sample density and the displacement field color scales are indicated on the right. The excellent agreement observed between the two last retrieved solutions shows that the found inversion process is optimum.

Figure 6. 3D x-ray Bragg ptychography image of the SOI structure.

(a) 3D Isosurface plot rendition of the retrieved crystalline SOI structure density, shown in the laboratory frame (threshold at 30%). The length of the frame black lines corresponds to 0.1 μm. (b) Same as (a), other view. (c) Orthogonal 2D cuts of the density. (d) Orthogonal 2D cuts of the displacement field component . The color scale used to plot the images is given at the bottom. (e) 2D cut in the (y,z) plane extracted from (d). The specific behavior of is emphasized in the 1D cuts taken along the colored lines in (f) and (g).

Figure 7. Transmission electron microscopy at the Si/SiO₂ interface (a) Transmission electron microscopy image of the Si <110>/SiO₂ interface measured on the un-patterned SOI wafer.

The crystallographic directions of the Si layer are indicated. (b) The strain component extracted from (a), in absolute units. The inset shows the mean value of as a function of the distance to the interface, calculated in the region delimited by the black rectangle. An increase of is observed near the interface.