Full length article Influence of nuclear quantum effects on frozen phonon simulations of electron vortex beam HAADF-STEM images André Löfgren, Paul Zeiger, Vancho Kocevski 1 , Ján Rusz n Department of Physics and Astronomy, Uppsala University, P.O. Box 516, 75120 Uppsala, Sweden article info Article history: Received 22 October 2015 Received in revised form 22 January 2016 Accepted 28 January 2016 Available online 4 February 2016 Keywords: HAADF STEM Electron vortex beams Multislice Frozen phonons Nuclear quantum effects abstract We have evaluated atomic resolution high-angle annular dark field images with ordinary beams and electron vortex beams for thin crystals of bcc iron, explicitly considering the atomic vibrations using molecular dynamics. The shape of the image representing an atomic column depends on the orbital angular momentum, sample thickness and temperature. For electron vortex beams we observe char- acteristic doughnut-shaped images of atomic columns. It is shown how the thermal diffuse scattering reduces the depth of their central minima, which get further smeared by finite source size effects. In addition, it is shown that in calculations of HAADF-STEM images at low temperatures one has to ex- plicitly consider the nuclear quantum effects (zero point vibrations), otherwise the effect of atomic vi- brations is strongly underestimated. & 2016 Elsevier B.V. All rights reserved. 1. Introduction Electron vortex beams [1–3] (EVBs) are electron probes that carry a nonzero orbital angular momentum (OAM) thanks to a specific phase distribution in their wavefront. It was suggested [3] that thanks to OAM they can be useful for measuring electron magnetic circular dichroism [4] (EMCD) at high spatial resolution in a scanning mode. Later it was argued theoretically [5–7] that this can only work in an atomic resolution mode. This is a strong motivation for generating clean EVBs of atomic size. Several ap- proaches have been proposed, for example forming the beam with fork [2,3,8] or spiral apertures [9,10], use of magnetized needle [11,12] or phase masks [13]. That it is a non-trivial task was shown in a recent work with spiral apertures [14], where a focused vortex component of 3Å full-width at half-maximum (FWHM) was generated, however no statistically significant EMCD could be detected. There it was suggested that the defocused beam com- ponents with opposite OAM are still present and interacting with the sample, effectively suppressing the magnetic signal. That in- dicates that EMCD may require a clean probe with well-defined OAM, which remains a challenge. Due to the phase singularity in the middle of the EVB, the amplitude is necessarily zero at the center of the beam, giving it a characteristic doughnut profile. For that reason it is expected that a high-angle annular dark field (HAADF) image, measured with a scanning transmission electron microscope (STEM), will be com- posed of doughnut shaped rings at the position of atomic columns. Qualitatively this can be understood in the following way: if the EVB is perfectly centered on an atomic column, Fig. 1a, the ring with intensity maximum passes around the column. Scattering to high angles will not happen very often, Fig. 1b, because the probe wavefunction passes around the divergence of the electrostatic potential. However, when the beam is slightly displaced from the atomic column, so that the ring with maximal intensity passes through the nuclei, Fig. 1c, the scattering to larger angles sub- stantially increases, Fig. 1d. This should result in a characteristic doughnut-shaped HAADF image of an atomic column. This qualitative picture is however rather incomplete, because it has not taken into consideration the atomic vibrations. Average atomic displacements can easily reach substantial fraction of the interatomic distances, depending on the measurement tempera- ture and melting temperature of the material. For example in bcc iron, as simulated in Section 2.2, atoms at 300 K have a mean squared displacement of 70.13 Å, which is about 5% of the nearest neighbor distance. In general this can be a non-negligible value when compared to the FWHM of atomic size EVBs, which is ty- pically between 1 and 3 Å. Atomic vibrations will most likely soften the central minimum in the expected doughnut shape of the image of atomic columns. In this paper we have calculated the atomic resolution HAADF- STEM images with ordinary beams and EVBs. We have considered the effects of convergence angle, sample thickness, temperature Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ultramic Ultramicroscopy http://dx.doi.org/10.1016/j.ultramic.2016.01.007 0304-3991/& 2016 Elsevier B.V. All rights reserved. n Corresponding author. E-mail address: jan.rusz@fysik.uu.se (J. Rusz). 1 Present address: Department of Materials Science and Engineering, North- western University, Evanston, IL 60208, USA. Ultramicroscopy 164 (2016) 62–69