Electron diffraction and Fast-Fourier transforms (FFTs) to determine

For a given accelerating voltage, an emitted electron beam has a corresponding

wavelength, λ. As electrons pass through a crystalline sample, they are scattered at some angle off of the optic axis with a corresponding phase shift. When this phase shift, or the path length difference, is equal to a multiple of λ (also known as meeting the Bragg condition), there is a constructive interference of diffracted beams. By altering the imaging conditions of the TEM with objective apertures and lens strength, a single diffracted beam can be projected as a dark field image, or each diffracted beam can be focused to create a spot pattern containing

information on crystal structure and lattice spacing.

Constructive and deconstructive interference will ultimately determine which diffracted beams appear in a diffraction pattern and their intensity. In general, a miller plane family that

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passes through all of the atoms of a unit cell will have a strong intensity in the diffraction pattern. There are 219 distinct space groups for crystals, and since a diffraction pattern is a two-

dimensional projection of a 3D scattering event, both the symmetry and the orientation (zone axis) of a crystal is necessary to extract information on crystal structure and lattice constants.

2DMs present an interesting case for electron diffraction. Large changes in crystal symmetry can occur for monolayer, bilayer, and trilayers of 2DMs. We take advantage of these changes in symmetry to identify monolayer 2DMs using a careful analysis of diffraction patterns and FFT and a comparison to electron simulation. Here I describe the two critical techniques to characterize the thickness of 2DM using diffraction:

1. Obtaining experimental diffraction patterns (DPs) and FFT:

TEM grids with dropcast suspensions of 2DMs were prepared as described in section 2.2.1. To obtain diffraction patterns, a properly aligned JEOL 100CX II TEM was used. The TEM was operated at 100 kV accelerating voltage and first samples were scanned for optimal flakes for interpretable diffraction patterns. An ‘optimal’ flake meets the following criteria: (i) isolated, to prevent double diffraction or polycrystalline patterns. (ii) low contrast, as described in section 2.3.1, contrast can be used to screen the sample for thin flakes quickly and efficiently. (iii) lateral size and position, flakes should be over vacuum and larger than the smallest selected area aperture (0.4 microns) to ensure coherent diffraction with detectable intensities. (iv) tilt, flakes should be lying almost perfectly flat so that the optical axis is perpendicular to the atomic layers. (v) defocus, samples should be imaged at Scherzer defocus so that all diffracted electrons are realigned to their point of origin.

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Once a suitable sample was found the smallest selected area aperture was inserted and positioned over the flake. The TEM was switched to SA diffraction mode and a diffraction pattern would then illuminate the phosphor screen. To capture the pattern, the camera was inserted and a 0.1 s exposure time and 83 cm camera length were used. Immediately after the image is taken, the camera was removed to prevent damage to the CCD. The image can then be analyzed using Gatan Digital Micrograph to confirm that flakes are crystalline and oriented perpendicular to the electron beam.

Liquid exfoliation can result in flakes that have lateral dimensions too small to reliably observe a diffraction pattern. To characterize the thickness of these samples, a high-resolution JEOL 2010F-FasTEM at 200kV accelerating voltage was used to obtain FFTs. In addition to the previously outlined criteria for sample selection, flakes exhibiting strong phase-contrast are highly crystalline and ideal for FFT analysis. FFTs are easily obtained by capturing the back focal plane during standard HRTEM imaging. The resulting DPs (FFTs) from low resolution (high resolution) TEM can then be compared to electron microscopy simulations.

2. Modeling Thickness Dependent FFT and DP using Java Electon Microscopy Simulation: I used Java Electron Microscopy Simulation (JEMS) software to predict variations in the DP and FFT of 2DMs as a function of thickness. Crystallographic information files (cif) were generated using Accelerys/BIOVIA Materials Studio for monolayer, bilayer, trilayer, etc, flakes. These files were then imported to JEMS and a zone axis perpendicular to the layered structure was chosen (typically [001]) for HRTEM imaging, DP, and FFT simulation.

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Diffraction patterns were generated in JEMS with a straightforward application of structure factor for diffracting crystals, however, simulations for FFT are sensitive to HRTEM components and imaging conditions. The most important aspect of modeling FFTs is an accurate depiction of the contrast transfer function (Eqn. 2.2, 2.3).

𝑇(𝑘⃑ ) = exp⁡[2𝜋𝑖𝜒(𝑘⃑ )] (2.2)

𝜒(𝑘⃑ ) = 0.25𝐶𝑠𝜆3𝑘4+ 0.5𝛥𝑧𝜆𝑘2 (2.3)

The CTF determines the contrast observed for elements of an image with a given spatial frequency (k), and relies on the accelerating voltage (wavelength) of the electron beam, the spherical aberration Cs, and the defocus value z. Typically, Scherzer defocus (Eqn. 2.4) is chosen as a value of z to minimize the number of nodes in the CTF, thus simplifying the interpretation of the FFT and maximizing the range of frequencies observed in the image.

∆𝑧 = (4₃𝐶_𝑠𝜆)1/2 _(2.4)

Specifically, I modeled the experimental imaging conditions of our JEOL 2010F- FasTEM. The accelerating voltage of the electron beam was 200 kV, which corresponds to a wavelength of 0.0027 nm. This HRTEM is equipped with a high-resolution polepiece with a Cs of 1.00 mm, and samples were imaged near Scherzer defocus (61.0 nm). Using these values, I modeled the FFT of different thicknesses of 2DM. I quantified the intensities of the diffraction plane families in the HRTEM FFT simulations and DP using signal integration image analysis to establish a trend in spot intensities. Finally, the observed experimental DP and FFTs can be compared to simulations to characterize the thickness of exfoliated flakes. An illustrative case for black phosphorus is presented in Chapter Three, where the ratio of the intensity for [101]:[200] plane families is zero for

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all even layered flakes, severely diminished (<1.0) for odd layered flakes greater with 3 or more layers, and strong (>>1.0) for a monolayer.