Structure retrieval with fast electrons using segmented detectors

We introduce an algorithm for the reconstruction of the complex transmission function of a specimen using segmented detectors in scanning transmission electron microscopy geometry. The phase of the transmission function can be related to magnetic and electric fields within the specimen and is sensitive to lighter elements. The technique is demonstrated for simulated data and also using experimental datasets taken from a ${\mathrm{MoS}}_2$ monolayer and a ${\mathrm{SrTiO}}_3$ crystal. We present an extension to the algorithm to account for uncertainties in the illuminating probe. The algorithm can be implemented using fast Fourier transforms, and this provides the possibility of reconstructing specimen transmission functions in real time.

Figure 1

Reconstruction of a model transmission function using a focused coherent probe for the imaging conditions described in the text. (a) Model transmission function intensity (top) and phase (bottom). (b) Segments in the bright field are used to construct the images in (c). (d) The band-pass-limited transmission function intensity (top) and phase (bottom) can be compared with the retrieved transmission function in (e) after 10 and 75 CGLS iterations and also with the latter as corrected for neglect of the nonlinear term in Eq. (5).

Figure 2

Fidelity as a function of CGLS iteration number for the results shown in Fig. 1. The “$\times$” indicates the fidelity after correction for neglect of the nonlinear term.

Figure 3

Results for the ${\mathrm{MoS}}_2$ sample. (a) Detector segments in the bright field used to synthesize the images in (b). (c) Retrieved phase of the transmission function. The method to deconvolve the blur associated with spatial incoherence (estimated to be 0.45 Å) described in Sec. 2 was employed to achieve the phase map shown in (c). The adjacent plot compares a line scan taken from the phase map (Exp 1), as indicted by the (red) arrow, with a line scan from a result (not shown) where spatial incoherence is not taken into account (Exp 2) and a transmission function simulated using $\mu \mathrm{STEM}$ with one sulfur atom per unit cell (Sim). Both Exp 1 and Sim were convolved with the Gaussian associated with the spatial incoherence for comparison with Exp 2. (d) Plot of the fidelity, where the reference transmission function is Sim, and the L-curve as a function of CGLS iteration. (e) Plot of the phase of the transmission function calculated in $\mu \mathrm{STEM}$ assuming both 1 (Sim 1S) and 2 (Sim 2S) sulfur atoms per primitive cell to that retrieved in (c) convolved with a Gaussian associated with the spatial incoherence for the sake of comparison.

Figure 4

(a) Before and (b) during ADF images for the ${\mathrm{MoS}}_2$ sample. Arrows in images (a) and (b) indicate sites of sulfur atoms. For the uppermost arrow, the sulfur intensity is not appreciably reduced between images (a) and (b), while for the bottommost arrow, the change in intensity indicates sputtering of one or both sulfur atoms.

Figure 5

Results for 35.1- and 78.1-Å-thick ${\mathrm{SrTiO}}_3$ samples. (a) Detector segments in the bright field used to synthesize the images in (b) for the 35.1-Å-thick ${\mathrm{SrTiO}}_3$ sample. (c) Retrieved phase with a line scan as indicated by the (red) arrow. Sim is the simulated projected transmission function, Exp is the retrieved transmission function, and Cor is the retrieved transmission function corrected for neglect of the nonlinear term. The method to deconvolve the blur associated with spatial incoherence (estimated to be 0.5 Å) described in Sec. 2 was employed to achieve the phase map shown in (c). The results in (c) are convolved with this blur for the purposes of comparing to the simulated transmission function in the line scan. (d) A bright-field image synthesized from the same dataset using a detector with the same radius as the bright field disk. The results displayed in (e)–(g) are the same as those displayed in (b)–(d) except that they are for the 78.1-Å-thick sample.

Figure 6

Application of the $M$-plane inversion to the ${\mathrm{SrTiO}}_3$ dataset. For the 35.1-Å-thick sample, the retrieved amplitude (a) without using the $M$-plane inversion and (b) using the $M$-plane inversion. A line scan comparing the retrieved intensity to the intensity of the simulated transmission function for a layer of ${\mathrm{SrTiO}}_3$ with a thickness of one unit cell (3.905 Å) is shown in (c). The results without $M$-plane inversion are labeled “Exp” and the results with $M$-plane inversion are labeled “Cor.” Parts (d)–(f) are the same as (a)–(c) except that they are for the 78.1 Å case. The intensity of the simulated transmission function for a layer of ${\mathrm{SrTiO}}_3$ with a thickness of one unit cell is shown in (g) for comparison. The effect of the $M$-plane inversion on the retrieved phase is displayed in (h) for the 35.1-Å-thick sample of ${\mathrm{SrTiO}}_3$ and in (i) for the 78.1-Å-thick sample of ${\mathrm{SrTiO}}_3$.