Direct Evidence of Topological Defects in Electron Waves through

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Direct evidence of topological defects in electron waves due to nanoscale localized magnetic charge Charudatta Phatak, and Amanda Petford-Long Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b02915 • Publication Date (Web): 22 Oct 2018 Downloaded from http://pubs.acs.org on October 22, 2018

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Direct evidence of topological defects in electron waves due to nanoscale localized magnetic charge Charudatta Phatak∗,† and Amanda Petford-Long‡,¶ †Materials Science Division, Argonne National Laboratory, Lemont, IL 60439, USA ‡Materials Science Division, Argonne National Laboratory, Lemont, IL 60439, USA. ¶Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA. E-mail: [email protected] Phone: +1-630-252-5379

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Abstract Topological concepts play an important role in, and provide unique insights into, many physical phenomena. In particular topological defects have become an active area of research due to their relevance to diverse systems including condensed matter and the early universe. These defects arise in systems during phase transitions or symmetry-breaking operations that lead to a specific configuration of the order parameter that is stable against external perturbations. In this work, we experimentally show that excitations or defects carrying magnetic charge in artificial spin ices introduce a topological defect in incident coherent electron waves. This results in the formation of a localized electron vortex beam carrying orbital angular momentum that is directly correlated with the magnetic charge. This work provides unique insight into the interaction of electrons with magnetically-charged excitations, and the effect on their topology, thereby opening new possibilities to explore exotic scattering and quantum effects in nanoscale condensed-matter systems. Keywords: Nanoscale artificial spin ices, Vortex beams, Lorentz transmission electron microscopy

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Topological defects are critical to understanding a wide range of physical phenomena because they can carry energy and momentum. 1,2 Moreover, they are not only stable against small external perturbations but they also cannot easily decay or be un-entangled since they cannot be continuously transformed into a “trivial” form. For example, the study of topology and defects for understanding the physics of materials has given rise to a new class of materials known as topological insulators that possess conducting surface states that are protected due to topology. 3 Monopoles can also be considered to be zero dimensional topological defects that arise due to breaking of spherical symmetry. The quest for magnetic monopoles has been a hot topic of research spanning many areas of physics including particle physics, 4,5 cosmology, 6 and condensed matter physics. 7–9 In condensed matter systems, the emergence of magnetic monopole excitations arising from fractionalization of magnetic dipoles connected by Dirac strings can be shown to emulate Maxwell electromagnetism, 10 and recent reports have also shown the emergence of quantum magnetic monopoles as topological defects in Bose-Einstein condensates, 11 and as gapless heavy quasiparticles in three-dimensional spin liquids. 12 Understanding their behavior, and the novel physics arising from these topological defects critically requires utilizing their inherent topological quantum mechanical properties to observe how they interact with their local environment. Phase singularities are a ubiquitous feature of wave topology and have been studied extensively in order to understand both their fundamental behavior and their potential applications. The study of phase singularities in optical, X-ray and electron wavefronts, both quantized and classical, has been an extensive area of research over the past decade. 13–17 These phase singularities, or topological defects, can also be characterized using the topological charge, or winding number, most commonly denoted by l and defined as the circulation H of the phase gradient about the singularity as C ∇ϕ · ds = 2πl. Here, ϕ is the phase and s is the unit vector tangential to the closed path C around which the loop integral is taken. Due to the nature of the resulting wavefront, these waves carry orbital angular momentum and are characterized as vortex beams. For a circularly-symmetric electron wave, l also

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determines the orbital angular momentum (OAM) of the beam in units of l¯ h per electron. This effect is similar to the behavior of topological defects in crystal lattice such as screw dislocations. At present, the most common methods for generating electron vortex beams is using phase masks, or diffraction gratings. However, these methods reduce the effective intensity of the beam carrying OAM due to beam splitting. In this work, we bring together two condensed matter systems to understand the behavior of topological defects and their interaction with electron waves. We have explored the quantum mechanical properties of magnetic monopole excitations in artificial spin ices, in which the excitations are localized to specific sites in the lattice. We show that the presence of monopole excitations with magnetic charge introduces a topological defect or phase singularity to the propagating coherent electrons that constitute the electron beam in a transmission electron microscope, thereby resulting in local electron vortex states. We show that the chirality and topological charge of the local electron vortex states is directly correlated with the sign and magnitude of the local magnetic charge, and by utilizing the magnetic charge of the monopole excitations, we can produce complex sculpted electron wavefronts without significant loss of intensity. 18,19 Our experiments demonstrate that various theories about angular momentum, scattering, and quantum mechanical phenomena can easily be studied in such mesoscale systems. When relativistic electrons in a transmission electron microscope (TEM) interact with electromagnetic fields or potentials, they acquire a phase shift given by the Aharonov-Bohm effect. 20 Recent work has shown that electron vortex beams can be generated using single magnetic islands by exploiting this effect. 18,21 However, there has not been a direct correlation established between the magnetic charge and the topological charge of the vortex beam. In addition, to date only single magnetic islands have been explored, for which a monopole-like field near the end of the island enables the generation of a vortex beam with fixed OAM. Here we have fabricated artificial spin ice lattices, which consist arrays of magnetic islands lithographically patterned on a lattice. Artificial spin ices mimic the behavior of their bulk

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counterparts with the advantage that the monopole excitations and corresponding Dirac strings are observed at room temperature and can be controlled using external magnetic fields. 22–24 In Dirac’s original paper, he had proposed the emergence of a phase singularity in the quantum wavefunction along the so-called Dirac string that ends in a magnetic monopolelike object. 25 We show that due to the variation in the net magnetic charge at the nodes at which the islands meet across the entire lattice, we can generate topological defects in the electron wave with different values of OAM from the same sample. We have lithographically patterned a square lattice of magnetic islands of Co50 Fe50 of length 2 µm, width 160 nm, and thickness of 25 nm. The lattice was patterned on the electron-transparent SiN membrane window of a TEM grid. The sample was coated with carbon to avoid any charging of the SiN membrane. The magnetic flux of the sample and its effect on the electron wave were observed using an aberration-corrected Lorentz TEM (LTEM) operating at 200 kV, which allows imaging under field-free conditions. Figure 1(a) shows a LTEM image of part of the artificial spin ice lattice, consisting of magnetic islands that are closely spaced at the nodes. Figure 1(b) shows the experimental magnetic induction color map obtained from the same region using the transport-of-intensity equation (TIE) formalism. 26 The color indicates the direction of magnetization in the islands as given by the inset color wheel. In a square lattice, at a node where four islands come together, the magnetic charge can be considered to be quantized with values ranging between 0, ±2, ±4 depending upon the orientation of magnetization in each island (+1 is the magnetization points towards the node, and −1 if it points away). Specific nodes in the lattice at which magnetic monopole excitations with net magnetic charge of ±2 and ±1 (where three islands meet at the edge of the lattice) are present, as shown in Figure 1(b). A selected area aperture was introduced in the electron beam path in order to explore the interaction between these monopole excitations and the incident plane electron wave (as indicated by the red dash line in Figure 1(a)). A through-focus series (∆f = ±10 µm) of far-field electron diffraction patterns were recorded as shown in Figure 1(c). Figure 1(d) shows the simulated far-field

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Figure 1: (a) In-focus TEM image showing a region of the artificial spin ice and (b) the corresponding magnetic induction color map with inset colorwheel. The locations of magnetic monopole defects with the corresponding magnetic charges are identified in (b). (c) shows defocused electron diffraction patterns from the region enclosed by the dashed red circle in (a) as a function of objective lens defocus. (d) shows the simulated diffraction patterns for similar defocus values. The electron vortex states, which are produced only at the precise locations of the magnetic charges, can be clearly observed. (e) shows the magnified view of the region highlighted in (c) indicating the chirality of the observed electron vortex states. (f) line plots of the intensity across the dark region of the electron vortex state at the location of magnetic monopole defects with charge +1 (red line) and +2 (blue line) indicated in (c). diffraction patterns computed using the Fresnel propagator 27 and the magnetization of the islands that is shown in Figure 1(b). Additional details about the simulated diffraction patterns are given in the supplementary material. It should be noted that the diffraction pattern images for negative defocus are inverted with respect to the real-space images. There is excellent agreement between the experimentally-observed diffraction patterns and the

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simulated ones. We can identify the presence of topological defects present as dark regions in the electron wave at the location of the nodes carrying net magnetic charge in the diffraction pattern. We focus on the presence of dark regions of destructive interference that are clearly seen in the far field diffraction pattern, but only at nodes that have magnetic charge (i.e. at which there are magnetic monopole excitations). Such dark regions are a indicative of a phase singularity as expected for vortex states. 16,17 A closer analysis of the dark region shows that there is an offset in the position of the region of destructive interference in the diffraction pattern with respect to the node position, but only for nodes that carry magnetic charge. This results in the appearance of a sense of rotation that is clockwise for nodes with positive magnetic charge and counter-clockwise for nodes with negative magnetic charge as shown in Figure 1(e) for negative defocus. It should be noted that this sense of rotation reverses with the sign of defocus. This observation is a direct indication of the chirality of the vortex state. Furthermore, the area, or size, of the dark region for a given defocus value is dependent on the magnetic charge: for monopole excitations with magnetic charge ±2, the size of the dark region is larger than that observed for the excitations with magnetic charge ±1. This is shown in the line plot of intensity across the dark region from experimental (symbols) and simulated (dashed line) images in Figure 1(f). There is good agreement between the widths of the dark region in the experimental and simulated images for a given defocus value and the same trend was observed for other values of defocus. Therefore, we can think of the vortex states being locally formed at the positions of the magnetic monopole defects, with Dirac strings that connect pairs of negative- and positive-charged monopoles and that carry the magnetic flux. It should be noted that the presence of local vortex states is seen in the direct transmitted beam and not for the scattered beams as is often the case for diffraction gratings. This results in higher intensity being maintained for the OAM states. We measured the total integrated intensity of the diffraction patterns as a function of defocus as shown in Figure 1(c) to be about 70% at the in-focus condition.

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Figure 2: (a) and (b) Decomposition of the experimental phase shift into OAM eigenmodes for the corresponding nodes labeled in Figure 1(b). The dashed line indicates a Gaussian fit to the distribution. (c)-(d) show the interference holograms in the far-field diffraction pattern for Nodes D,B,E, and A identified in Figure 1(b). The dashed red circle shows the region of dark region of the electron vortex state. In order to provide a more quantitative relation between the magnetic charge of the monopole excitations and the topological charge of the localized electron vortex state, we performed electron interference experiment to directly show the presence of OAM. A biprism excited with negative voltage was used to create an interference hologram in the far-field diffraction pattern around various nodes in the artificial spin ice lattice. The resulting effect on the hologram fringes for four different node configurations is shown in Figure 2(a) (additional details and low magnification images are shown in Supplementary Material). The interference holograms were also used to retrieve the local phase shift of the electrons, which is displayed as a 3D surface plot in Figure 2(b). The presence of a dislocation in the fringes for nodes D, B, and E, and the absence of a dislocation for node A confirms the local presence of OAM in the electron beam. The dislocation fringe which is related to a phase jump is similarly seen in the phase shift for nodes D, B and E only. Moreover, the direction of the dislocation fringes for node D and node B shows that they impart opposite sign to the OAM, which correlates with the magnetic charge at the nodes. This is manifested 8


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as a variation of the phase shift from low to high value around the phase jump, and can be seen to be clockwise for node D and counter-clockwise for nodes B and E. In addition, nodes D and B show two additional fringes as compared to a single additional fringe at node E, which also correlates with the value of the magnetic charge at each of the nodes. This can be quantitatively seen in the phase jump value of 6π for nodes D and B which is twice as big when compared to the value of 3π for node E. This quantitatively shows that the interaction of a defect-free electron plane wave with a topological defect carrying magnetic charge, imprints a topological defect into the electron wave. This topological defect carries approximately the same topological charge as the magnetic charge with which it interacted.

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Figure 3: Simulations of (a) color magnetic induction maps, overlaid with cosine of the phase shift to indicate the lines of magnetic flux, for the different node configurations in a square spin ice lattice (magnetization direction indicated by the colorwheel). (b) Phase of the electron wave in the diffraction plane, for a defocus value of 5 µm. The spiral nature of the phase can be clearly seen for Nodes 3 and 4, which carry magnetic charge. (c) Simulations of the corresponding far-field diffraction patterns. The electron vortex states produced for Nodes 3 and 4 can be seen at the node position (center). In order to gain a deeper understanding of the correlation between the topological charge

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of the electron vortex state and the net magnetic charge of the monopole excitation, we performed simulations to calculate the electron wave phase shift and to simulate a corresponding far-field diffraction pattern for the various node configurations. Each island was taken to be a rectangular prism with uniform magnetization. The saturation magnetic induction for CoFe was taken to be 1.75 T. Using a shape amplitude formalism, the electron wave phase shift was calculated for each node configuration. 28 Figure 3(a) shows magnetic induction color maps for the four node configurations possible in a square artificial spin ice lattice overlaid with the cosine of the phase shift, which indicates the lines of magnetic flux. The exit wave phase shift was then calculated for an objective lens defocus of ∆f = 5 µm in the far-field diffraction plane. Figure 3(b) shows color plots of the phase of the electron wave in the far-field, scaled between ±π, and Figure 3(c) shows the corresponding simulated far-field diffraction images. There is a distortion in the phase due to the presence of the magnetic islands, which carry the magnetic flux for all four node configurations. However, what is striking is that for Node 3 and Node 4, at which there are magnetic monopole excitations with net magnetic charge of −2 and +4 respectively, we can see a clear spiraling of the phase. The number of phase spirals emanating from the central point is directly proportional to the winding number, or the topological charge, of the wave, which is l = 2 and l = 4 for Node 3 and Node 4 respectively. The far-field diffraction images again show the presence of a dark region at the node center for the nodes that carry net magnetic charge. The relative difference in the size of the dark region (smaller for node 3 compared to node 4) as a function of magnetic charge is also clearly observed, thus supporting the experimental observations. We have shown direct evidence for the influence of local magnetic charge on the topology of coherent incident electrons, with a one-to-one correspondence between magnetic charge and the topological charge (or winding number) of the vortex state formed in the electron wave. These effects were observable even though the monopole excitations were surrounded by nodes with no net magnetic charge, suggesting that these experiments could be extended to bulk spin ices containing atomic scale magnetic monopoles such as in pyrochlore mate-

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Figure 4: (a),(b) and (c) show experimentally recovered magnetic induction color maps for nodes with 3, 5 and 6 interacting bars respectively, and (d), (e), and (f) show the corresponding far-field diffraction patterns showing the presence of localized vortex states at the center of the nodes as well as ends of the magnetic bars. rials. 7 Our analysis can easily be extended to other lattice geometries such as rewriteable artificial spin ices. 29 Figure 4 shows the experimental realization of localized vortex states for various nodes with 3,5 and 6 interacting magnetic bars. The non-uniform color for magnetic induction map in some of the islands (Figure 4(a) and (c)) is due to artefacts resulting from TIE reconstruction. So long as there is a net magnetic charge associated with the monopole excitation, an electron vortex state can be generated and observed. This opens new areas for the exploration of spatially-modulated programmable arrays of magnetic islands for electron beam sculpting. One limitation of the proposed method is that the OAM can only be tuned to the few specific values of magnetic charge that can be produced in an artificial spin ice system for a given lattice geometry. However, these experiments could be extended to micron-scale solenoids fabricated in a hexagonal or octagonal arrangement instead of magnetic islands, similar to the hexapole or octapole lenses currently used in elec11


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tron optics. Then by dynamically changing the magnetic flux through each of the solenoids, the topological charge of the electron wave could be directly controlled, enabling the possibility to create either a scanning TEM probe carrying OAM (device in condenser plane of TEM) or a reconfigurable phase-plate to observe novel chiral effects in the sample (device in the back-focal plane of the objective lens). A similar approach for programmable phase plate using electric field was recently demonstrated. 30 A fully OAM-tunable electron-optic column would prove to be an ideal platform for performing the next generation of quantum scattering experiments. Finally, this work also paves the way to understanding behavior of multiple, localized topological defects in propagating electron waves which can have significant implications towards novel imaging and sensing applications using structured quantum electron waves. Methods.Sample preparation. The square artificial spin ice lattice was fabricated using electron beam lithography directly onto silicon TEM grids with SiN (50 nm) electron transparent windows. The islands were fabricated to be 2.1 µm × 160 nm in size. A thin film of Ta(2 nm)/Co50 Fe50 (25 nm)/Ta(2 nm) was sputter deposited at the rate of 1.0 ˚ A/s, followed by lift-off and resist removal. The sample was then coated with amorphous carbon to minimize charging of the SiN membrane under the electron beam. The sample was then directly inserted into the TEM without exposure to any external magnetic fields. Lorentz transmission electron microscopy. The far-field diffraction images were acquired using a Tecnai F20 TEM operating at 200 kV in Lorentz (field-free) mode. The highest camera length available (31 m) was used to record the diffraction patterns on a CCD (chargecoupled device) camera. The defocus of the imaging lens was varied in steps of ∆f = ±2 µm as indicated on the microscope software. It should be noted that this value of defocus was not calibrated. Phase retrieval. A through-focus series of bright-field images was acquired using a JEOL 2100F operating at 200 kV, and equipped with a dedicated low-field objective lens and an imaging Cs corrector. The sample was then inverted about its axis by 180◦ and another

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through-focus series of images was recorded of the same area. The defocus used for recording the images was ∆f = ±273 µm. Image post-processing was performed to remove high intensity pixels resulting from x-ray photons landing on the CCD array, and to remove the effect of the CCD modulation transfer function. All the through-focus series images were then aligned with respect to each other, in order to correct for rotation and magnification changes that occur due to the defocus change and inversion of the sample. The registered images were then used to recover phase maps by solving the Transport-of-Intensity equation given as: 2π ∇ · [I(r⊥ , 0)∇φ] = − λ

∂I(r⊥ , 0) ∂z

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where I(r⊥ , z) represents the image intensity observed at a given defocus z, φ is the phase and λ is the electron wavelength (2.508 pm at 200 kV). The image intensity derivative with respect to z on the right-hand side of the equation was computed using the through-focus series of images by a central difference method. A Tikhonov regularization filter was employed to eliminate slowly-varying phase features, caused by low frequency noise, 31 from appearing in the reconstructed phase. Image symmetrization was used to avoid errors due to periodic boundary conditions that may arise because of the use of discrete Fourier transforms to solve the Transport of Intensity equation. The phase recovery was performed using procedures implemented in the Interactive Data Language programming environment. 32 The magnetic part of the phase shift was then recovered by subtracting the phase shift of the inverted sample from the phase shift of the upright sample. The gradient of the phase shift was then computed to obtain the thickness integrated magnetic induction maps.

Acknowledgement This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division. Use of Center for Nanoscale Materials was supported by the U.S. Department of Energy, Office of Science, Office of Basic 13


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Energy Sciences, under contract no. DE-AC02-06CH11357.

Supporting Information Available Comparison of experimental and simulated phase shift of magnetic islands; details about eletron-optical simulations; high magnification images of spin ice lattice; defocused diffraction pattern showing interference hologram; and comparison of phase of transfer function for ideal vortex beams and those generated by ASI nodes.

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(22) Ladak, S.; Read, D. E.; Perkins, G. K.; Cohen, L. F.; Branford, W. R. Nature Physics 2010, 6, 359–363. (23) Mengotti, E.; Heyderman, L. J.; Rodr´ıguez, A. F.; Nolting, F.; H¨ ugli, R. V.; Braun, H.B. Nature Physics 2010, 7, 68–74. (24) Phatak, C.; Petford-Long, A.; Heinonen, O.; Tanase, M.; De Graef, M. Physical Review B 2011, 83, 174431. (25) Dirac, P. A. M. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 1931, 133, 60–72. (26) Volkov, V.; Zhu, Y.; De Graef, M. Micron 2002, 33, 411–416. (27) De Graef, M. Introduction to conventional transmission electron microscopy; Cambridge University Press: Cambridge, U.K., 2003. (28) Beleggia, M.; Zhu, Y. Philosophical Magazine 2003, 83, 1045–1057. (29) Wang, Y.-L.; Xiao, Z.-L.; Snezhko, A.; Xu, J.; Ocola, L. E.; Divan, R.; Pearson, J. E.; Crabtree, G. W.; Kwok, W.-K. Science 2016, 352, 962–966. (30) Verbeeck, J.; Béché, A.; M¨ uller-caspary, K.; Guzzinati, G.; Anh, M.; Hertog, M. D. Ultramicroscopy 2018, 190, 58–65. (31) Ishizuka, K.; Allman, B. J Electron Microsc (Tokyo) 2005, 54, 191–7. (32) The Interactive Data Language (Version 7). ITT Visual Information Solutions, Boulder, CO, 2007.

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Direct Evidence of Topological Defects in Electron Waves through

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