Probing the limits of the rigid-intensity-shift model in differential-phase-contrast scanning transmission electron microscopy

The rigid-intensity-shift model of differential-phase-contrast imaging assumes that the phase gradient imposed on the transmitted probe by the sample causes the diffraction pattern intensity to shift rigidly by an amount proportional to that phase gradient. This behavior is seldom realized exactly in practice. Through a combination of experimental results, analytical modeling and numerical calculations, using as case studies electron microscope imaging of the built-in electric field in a p-n junction and nanoscale domains in a magnetic alloy, we explore the breakdown of rigid-intensity-shift behavior and how this depends on the magnitude of the phase gradient and the relative scale of features in the phase profile and the probe size. We present guidelines as to when the rigid-intensity-shift model can be applied for quantitative phase reconstruction using segmented detectors, and propose probe-shaping strategies to further improve the accuracy.

Figure 1

DPC-STEM imaging. A probe is focused onto a thin phase object and scattered onto the detector below. (a) When $\nabla \varphi =\vec{0}$, the diffraction pattern of the probe is unshifted. (b) When $\nabla \varphi =\vec{c}$, the diffraction pattern is rigidly shifted across the detector, with deflection angle $\beta \propto \left|\nabla \varphi \right|$. (c) When $\nabla \varphi$ varies across the width of the probe, the diffraction pattern intensity is not simply rigidly shifted, but rather redistributed across the detector in a more complex manner.

Figure 2

(a) Comparison of Airy (teal, solid) and Gaussian (lilac, dashed) intensity profiles. (b) Comparison of enclosed energy for the Airy and Gaussian probes. These plots assume $2\pi k_0\alpha =1{\mathrm{nm}}^{-1}$; for suitable scaling via Eq. (3) these distributions can apply to any $k_0$ and $\alpha$ combination.

Figure 3

Plots of the p-n junction phase profile, underlaid by a color map of the probe amplitude profile, extended vertically to aid visual comparison between the variation in curvature of the junction phase profile and spatial extent of the probe amplitude profile, for the probe-forming aperture semiangles (a) $\alpha =133\mu \mathrm{rad}$, (b) $\alpha =426\mu \mathrm{rad}$, and (c) $\alpha =852\mu \mathrm{rad}$. (d) Experimental p-n junction DPC-STEM profiles as imaged with the three different probes. These profiles were obtained by taking the difference between the STEM images from two diametrically opposed detector segments (depicted in the figure inset) under the edges of the bright field disk and integrating the result along the length of the junction.

Figure 4

Line profiles and 2D diffraction patterns comparing experiment and simulation for three different convergence semiangles, illuminating the p-n junction specimen. The line profiles compare results on-junction (teal, solid) with those off-junction (lilac, dashed).

Figure 5

For (a) the piecewise p-n junction model of Eq. (5), (b) the $D\to \infty$ limit, and (c) the $D\to 0$ limit we plot the (i) real-space transmission function phase, (ii) diffraction pattern, and (iii) reciprocal-space transmission function. In (a)-i the piecewise p-n junction model phase (teal, solid), is compared to the continuous equivalent function (lilac, dashed). In parts (ii), the diffraction pattern on-junction (teal, solid) is compared to that off-junction (lilac, dashed). In (a)-iii the reciprocal-space transmission function (teal, solid) is compared to the second term of Eq. (8) (lilac, dashed) and the third term of Eq. (8) (mustard, solid), with vertical black lines positioned at $\pm 1/D$ as guides to the eye.

Figure 6

(a) Real-space transmission function phase, (b) diffraction pattern, and (c) the reciprocal-space transmission function for the piecewise object of Eq. (5) for parameters taken for a transmission function of a magnetic domain taken from Ref. [31]. The line styles are as per Fig. 5. Note that (b) and (c) are plotted with different horizontal scales, because the features in (b) are an order of magnitude smaller than those in Fig. 5.

Figure 7

(a) Phase profile imposed by the magnetized specimen ($1\mu \mathrm{m}\times 0.5\mu \mathrm{m}$), with color bar in radians. The crosses mark the probe positions for which the diffraction patterns in (b) are calculated, assuming a $133\mu \mathrm{rad}$ probe [the intensity profile of which is plotted above the scale bar in (a)]. The intensity scale for each diffraction pattern has been set independently to better visualize the fine structure therein.

Figure 8

(a) Illustration of location of the edges of the off-sample bright-field disk (white, dashed line) compared to the segmented detector. (b) $\partial \varphi /\partial y$ of the phase profile in Fig. 7. Color bar is in terms of the probe deflection angle $\beta$ in $\mu \mathrm{rad}$. Difference between the calibrated segmented detector DPC-STEM estimate for $\partial \varphi /\partial y$ and the true value for convergence semiangles of (c) $90\mu \mathrm{rad}$ and (d) $133\mu \mathrm{rad}$, with the corresponding probe intensity profiles plotted to scale in the lower left corners. Color bars for (c) and (d) given in terms of percentage error from maximum $\beta$ in (b).

Figure 9

Line profiles of diffraction intensity from 426 $\mu \mathrm{rad}$ convergence semiangle Airy probes apodized at the (a) seventh, (b) fifth, (c) third, and (d) first radial minimum, for the probe off (lilac, dashed) and on (teal, solid) the p-n junction described by Eq. (4).

Figure 10

Difference maps in deflection angle between the true phase gradient and that calculated with the look-up table, for different $\alpha$ (rows) and apodization (columns). The differences decrease with increasing $\alpha$, as expected, but decrease more strongly with increasingly narrow apodization, the first Airy apodization cases (right-hand column) all giving similar difference measurements. The color maps are restricted to $\pm 6\mu \mathrm{rad}$ to visualize the details more clearly. Difference values outside this range have been set to white.