Direct-write of free-form 3D nanostructures with controlled magnetic frustration
Building nanotechnological analogues of naturally occurring magnetic structures has proven to be a powerful approach to studying topics like geometry-induced magnetic frustration and to provide model systems for statistical physics. Moreover, it practically allows to engineer novel physical properties by realizing artificial lattice geometries that are not accessible via natural crystallization or chemical synthesis. This has been accomplished with great success in the field of two-dimensional artificial spin ice systems with important branches reaching into the field of magnetic logic devices. Although first proposals have been made to advance into three dimensions (3D), established nanofabrication pathways based on electron beam lithography have not been adapted to obtain free-form 3D nanostructures. Here we demonstrate the direct-write fabrication of freestanding ferromagnetic 3D nano-architectures with full control over the degree of magnetic frustration. By employing micro-Hall sensing, we have determined the magnetic stray field generated by our free-form structures in an externally applied magnetic field and we have performed micromagnetic and macro-spin simulations to deduce the spatial magnetization profiles in the structures and analyze their switching behavior. Furthermore we show that the magnetic 3D elements can be combined with other 3D elements of different chemical composition and intrinsic material properties.
Nanomagnetic structures are ubiquitous, as they form the basic functional elements in various applications, such as in magnetic storage and information processing, magnonics and spintronics, see e. g. ; ; . Nanomagnetic structures are traditionally planar, but recent work is expanding nanomagnetism into three dimensions and it has been generally recognized that in three-dimensional nanomagnets complex magnetic configurations with unprecedented properties become possible, see  for a recent review. In the narrower sense of magnetic information storage and processing, the advantages of extending the typically 2D structures into the third dimension for higher integration density have already been realized and have lead to developments such as the racetrack memory . Further on, in so-called artificial spin ice systems ; ; ; ; ; ; ; , that are currently subject of intensive research efforts, the actual limitation to lithographically defined 2D arrays of interacting ferromagnetic nano-islands prevents investigations of novel phases that can emerge from the more complex ground states of frustrated lattices in 3D. This is why first steps into 3D artificial spin ice systems are now being taken by combining multilayer techniques with sophisticated electron beam lithography (EBL) . However, standard lithography techniques are intrinsically designed for 2D pattern formation and, consequently, they are barely suitable for the fabrication of free-form 3D nanostructures. Focused electron beam induced deposition (FEBID) follows a different approach to overcome this EBL-related limitation. It represents a highly flexible direct-write fabrication method which allows for excellent control in creating 3D structures very much like 3D printing on the nanometer scale. FEBID uses precursor gases which, being adsorbed on a surface, are dissociated in the focussed electron beam to form the deposit. In most cases the resulting structures are not simple phase-pure metals or oxides. Instead, the resulting nanostructures typically contain significant amounts of carbon, which is predominantly part of the precursor species. Also, FEBID is a highly complex process in which many parameters, such as electron beam energy and current, precursor flow and adsorption characteristics, precursor diffusion and beam steering strategy all influence the final deposit shape and the deposit’s composition ; ; . However, intensive research over the last decade has pushed the capabilities of FEBID in two important areas. It is now possible to obtain fully metallic nanostructures of Fe, Co and FeCo-alloys  and also of Au and Pt ; ; ; ; . In addition, very recently the simulation-guided nano-manufacturing of 3D structures has matured to such a degree that even complex 3D objects can now be fabricated under controlled conditions . For pillar-like Fe and Co structures this has recently been demonstrated in a detailed investigation of the sample composition . The next important scientific development is now the synergistic combination of these two developments towards the realization of free-form magnetic 3D nanostructures ; . Here, we demonstrate this next step by showing different examples of free-form 3D magnetic nano-architectures with a focus on magnetic frustration effects. The structures have been directly written on a high-resolution micro-Hall sensor. We followed the magnetization switching behavior by measuring the associated magnetic stray field during external field sweeps using a sensor with dimensions adapted to the size of the magnetic structures. With the help of micromagnetic and macro-spin model simulations we are able to explain the observed complex switching behavior.
ii.1 Geometry and microstructure
For the deposition of magnetic 3D nanostructures by FEBID (see Fig. 1(a) for FEBID principle) we chose the recently introduced precursor HCoFe(CO), as this was shown in our previous work to yield deposits of high CoFe metal content under beam conditions which are suitable for writing high-resolution structures, i. e. high beam voltage and low beam current (see methods section for details) . In view of very recent findings by Cordoba et al.  it should be possible to obtain similar structures than the ones shown here with high metal content from the precursors Fe(CO) and Co(CO) under suitable beam conditions.
In Fig. 1 we show two scanning electron microscopy (SEM) images taken directly after the writing of arrays of Fe-Co nano-trees (b) and nano-cubes (c) onto the top Au gate of a GaAs/AlGaAs micro-Hall sensor (see methods section for details). These represent magnetic nanostructures with vertices that are connected to three and four neighboring vertices, respectively.
In order to determine the microstructure and element distribution in the deposits we have performed transmission electron microscopy (TEM) experiments on nano-cubes grown onto metallic TEM grids. As the growth in 3D very sensitively depends on the precursor flux distribution , special care was taken to reproduce the nano-cube geometry as obtained on the micro-Hall sensor. Microstructure and element distribution contain the essential information for developing an appropriate micromagnetic simulation model, as described later. In Fig. 1(d) we present a TEM bright field image of one of the nano-cube edges. The 3D edges reveal a homogeneously distributed nano-granular structure consisting of nano-crystallites of about nm diameter on average. Additional chemical information is extracted from electron energy loss spectroscopy (EELS) and energy dispersive X-ray spectroscopy (EDXS). The latter reveal an overall composition of CoFeCO, corresponding to individual contents of at metal, at oxygen, and at carbon. Such a small carbon content can in fact be caused by unavoidable deposition of amorphous carbon during the EDXS analysis. This is a consequence of electron beam induced deposition of hydrocarbons adsorbed or chemisorbed on the sample surface and from the residual gas. During deposition the main carbon source is the precursor which contains a substantial amount of carbonyl groups. For planar deposits we found that the carbon to oxygen ratio is close to one . This indicates that carbonyl groups remain largely intact after dissociation and become part of the deposit, if their desorption is not sufficiently fast. Considering the small carbon content in the 3D structures in conjunction with the enhanced oxygen content, we argue that the oxygen is the result of a post-growth oxidation effect that occurred during sample transport and storage under ambient conditions before the TEM investigations were performed. Two further observations support this argument. First, scanning TEM EELS analysis of the oxygen content along the cross section of one of the nano-cube 3D edges reveals an enhanced oxygen content towards the surface (see 2(b)).
However, as EELS signal intensities do not directly provide quantitative concentration data we have to deconvolve the signal. We therefore start from the following relation for quantitative elemental analysis (see, e. g. )
for a given sample thickness, where denotes the EELS intensity of the element of interest with areal density in the energy range beyond the element-specific threshold. is the integral of the low-loss spectrum up to , including the entire zero-loss peak, is a partial cross section, and is the collection semiangle. The oxygen content may vary depending on the distance from the sample surface and the sample thickness will depend on the position of the EELS line scan. Tilt series via both, TEM and high-resolution SEM, reveal elongated instead of circular edge cross sections for 3D elements in agreement with previous studies ; . To approach the real situation with a simplified analytical expression, we assume an elliptical cross-section of radii and for the two main axes of any of the nano-cube edges. From Eq. 1 we derive the following expression for the EELS intensity at any given position along the line scan shown in Fig. 2(a) by integrating along the beam direction in the limits and defined by the respective sample thickness at scan position
with , being the volume density or concentration of oxygen, and nm the inelastic mean free path. The form of the oxygen concentration function is taken to be exponentially decaying from the surface towards the bulk, as Fig. 2(b) indicates, so that the oxygen concentration decreases from the surface into the bulk. Correspondingly, is the vertical distance of the point inside the cube edge from the surface. The result of this fit, assuming different oxygen contents, is shown as solid lines in Fig. 2(b). The measured line scan can be reproduced quite well by the fit and clearly indicates a very low oxygen content (below at) in the center of the nano-cube edge that increases towards the surface to about at with a characteristic length of nm. Given this, we assume a similar oxidation profile for the corresponding nano-cube and nano-tree structures used for the magnetic measurements.
A second, independent observation supports our assumption of a nearly at metal content in the bulk of our 3D nanostructures. Atom probe tomography on Fe- and Co-nano-pillars performed by Cordoba and collaborators showed that mass transport limited 3D growth provides favorable conditions for the complete desorption of carbonyl groups after dissociation .
In order to elucidate the degree of crystallinity of the deposits we performed selective area electron diffraction (SAED). The smallest field of view accessible in our setup is indicated in 2(c) by the shaded area (green circle). Consequently, we were not able to discriminate between the near-surface and bulk regions. This has to remain for future investigations. In Fig. 3 we show a diffraction image as measured and reference the diffraction rings with the corresponding scattering vectors. Apparently, the deposits are crystalline and we can attribute all diffraction rings to the -phase of the Co-Fe binary system (bcc) and a Co-rich spinel phase of ferrimagnetic CoFeO, which is expected to contain either amorphous or cubic cobalt-oxide phase contributions for metal ratio retention with regard to the precursor composition of . This result corresponds well to our previous observations for planar Fe-Co deposits .
Concluding this part on the microstructural characterization, we are led to consider the following microstructure concerning the magnetic properties of the deposits: a metallic -CoFe core is surrounded by a metal-oxide sheath which is a ferrimagnetic spinel phase. Next we turn to the results of the magnetic measurements on the arrays of nano-trees and nano-cubes.
ii.2 Magnetic stray field measurements and macro-spin simulations
Magnetically frustrated interactions at the three- and four-edge vertices associated with nano-cube and nano-tree structures are expected to lead to non-trivial spatial magnetization profiles and, correspondingly, rather complex distributions of the magnetic stray field vectors. Micro-Hall magnetometry, as sketched in Fig. 4(a), is particularly well suited for measuring such stray fields of individual or small arrays of magnetic micro- and nanostructures ; ; ;  in a wide range of temperatures and external magnetic fields applied under various angles with respect to the magnetic structures. In Figs. 4 and 5 we show stray field measurements of the nano-cube and nano-tree arrays for selected inclination angles of the external magnetic field (see Fig. 4(a) for the definition of the angle). The -component of the stray field, , emanating from the magnetic nanostructures is calculated from the measured Hall voltage difference , where is the applied current and the product of charge carrier density of the GaAs/AlGaAs Hall sensor and the electron charge. denotes the average over the active area of the Hall cross and the in situ subtraction of the Hall signal of an empty reference cross in a gradiometry setup, see SI and methods section for details.
We first discuss the magnetic nano-cubes. If applied along the -axis (), the external field causes a magnetization reversal which appears to proceed rather continuously, see Fig. 4(b). A closer look, however, reveals step-like features in the stray field response. It is therefore instructive to compare the experimental stray field curves with a simple macro-spin approach, where stem and edges of the nano-cubes are represented by a single macro-spin based on the assumption that all microscopic magnetic moments point in the same direction and rotate collectively. Since such an approach omits the existence of form anisotropy this has to be modeled by an additional uniaxial anisotropy for each macro-spin, see methods section for more details and the parameters used, see the SI for a discussion of the deviations between the measured and simulated absolute stray field signals. The model is lattice-based and mesh-free, which makes it very efficient for computing the mutual dipolar interactions of stems and edges within the nano-cube and -tree arrays. The almost vertical decrease or increase of of the simulated up- and down-sweep curves, respectively, at shown in the right panel of Fig. 4(b) are associated with the flipping of the four stems whereas the smaller steps are caused by the almost simultaneous flipping and canting of the edge macro-spins. Although the qualitative agreement is satisfying, the more rounded loop with smaller area and coercive field observed in the experiment points to a non-uniform magnetization switching of the stems dominated by multi-domain switching events.
A pronounced step-like switching behavior and better agreement of the measurements with the model is observed for a tilt angle of as shown in Fig. 4(c). Again, the large steps are associated with the flipping of the stems. Smaller steps and the finite slopes in between are connected with the flipping of edge spins and the rotation of their magnetization direction towards the external field. The closer the direction of the external field is to the anisotropy axis of an edge spin the larger is the coercive field resulting in the observed stair-case shape of the hysteresis.
A remarkable qualitative correspondence of the measured and simulated curves is seen for the complex and strongly pinched hysteresis loop at shown in Fig. 4(d), where is almost perpendicular to the anisotropy axis of the stems. Upon lowering the absolute value of the field, e.g. from negative saturation, all macro-spins relax towards their anisotropy axes. However, when approaching , the edge spins are not in the lowest energy state but their total moment has a large component parallel to the external field axis whereas the stem spins point upwards parallel to the anisotropy axis. This remains the case even for small positive fields and causes the stem spins to suddenly rotate by then pointing downwards. This causes the sharp peaks in the hysteresis which are observed in the measurements as well, however smeared out due to finite temperature and because the magnetization reversal mechanism of stem and edges are more complicated than the coherent rotation assumed in a single-domain macro-spin model. Nevertheless, although idealized, macro-spin simulations allow for identifying the relevant switching scenarios that occur for different inclination angles.
Next, we focus on the comparison of the measured nano-trees’ magnetization reversal with macro-spin simulations followed by a more sophisticated theoretical investigation based on micromagnetic simulations (see next section). As for the nano-cubes discussed above, the angular dependence of the measured stray fields, exemplarily shown for three angles , and shown in Fig. 5(b), (c) and (d), respectively, demonstrates a rich variety of reversal processes determined by the field angle with respect to the stem and edges of the 3D nano-trees. If the external field is applied parallel to the stem (), the continuous progression of the stray field upon decreasing from saturation indicates a gradual rotational change of the magnetization direction followed by a switching process comprising a substantial part of the sample volume. The experimental hysteresis is qualitatively reproduced by the macro-spin model, which identifies the gradual change in magnetization with an umbrella-like magnetic canting and rotation of the edges towards the direction of the applied field, while a fast switching sequence of the stems accounts for the sudden change of the magnetization, see the sequence of macro-spin configurations in the lower panel of Fig. 5(a) corresponding to two field values marked in the hysteresis loop of Fig. 5(b).
At an incident field angle of the field is nearly parallel to one of the edges. Here, a distinctly different magnetic hysteresis which narrows at zero applied field and exhibits a broad minimum and maximum at negative and positive fields for the down- and up-sweep curves, respectively, is observed. This behavior is only partly reproduced by the macro-spin model. The hysteresis loop shows maxima in the field cycle, but no narrowing of the loop occurs around zero field. Such a behavior requires a more finely grained effective spin model and is well reproduced by micromagnetic simulations considering a non-uniform reversal mechanism in the presence of the thin metal-oxide sheath, as shown in Fig. 6 below. In contrast, certain features of the stray field hysteresis for being less pinched and exhibiting sharper peaks are found in the macro-spin model. The pronounced peaks correspond to the rotation of the stems’ magnetization relative to the edges. At large fields, the magnetization of the edges and the stem are aligned along the field direction. Compensating stray fields from oblique magnetization angles of edges and stems lead to the small kinks near the peaks at smaller fields. A stronger compensation, which would reproduce the experimentally observed narrowing over a wide range of external fields at and is not found within the macro-spin model. Indeed, micromagnetic calculations, which are described in the next section, show a much more complex switching behavior beyond the limits of a fixed magnetic moment and a single axis anisotropy assumed in a macro-spin model.
ii.3 Micromagnetic simulations
Quite generally, micromagnetic simulations are invaluable for obtaining a deeper understanding of hysteresis effects by visualization of the magnetization reversal process on a microscopic scale . At the same time, they are computationally much more demanding than macro-spin simulations, which still limits their application depending on the size of the simulation volume. However, using the parallel computing power of high-end graphics cards has led to about an order of magnitude faster code execution (see, e. g. ) and the development of multi-scale and multi-physics of micromagnetic solvers is already foreseeable ; . Here we present results of micromagnetic simulations for the reversal process of the nano-trees in order to provide insight into the reasons for the discrepancy between the micro-Hall data and the macro-spin simulations. We also shed light onto the importance of taking the different magnetic behavior of the near-surface oxide into account. We focus on the nano-trees, because for these, the discrepancies compared to macro-spin simulations are most pronounced and they can still be treated by micromagnetic simulations if a core-shell structure consisting of metallic CoFe (core) and a ferrimagnetic spinel phase of CoFeO (shell) is assumed. Details on the simulation parameters are given in the methods section. For results of micromagnetic simulations of the nano-cubes we refer to the SI.
In Fig. 6(b), (c) and (d) we present the results of the micromagnetic simulations at the same angles which have been shown before. We discuss two different material composition scenarios. The first scenario assumes an all-metal CoFe nano-tree, wheres the second scenario assumes an oxide shell of the ferrimagnetic spinel phase CoFeO covering an all-metal CoFe core, as is schematically indicated in Fig. 6(a). For the spinel phase, the saturation magnetization is assumed to be a factor of smaller than that of the CoFe core (see methods section for details where it is also explained how the average stray field was calculated.).
For the all-metal micromagnetic model we find the same qualitative behavior as in the macro-spin simulations. The overall correspondence with the micro-Hall data for is good and a reasonable agreement can be stated for . This indicates that the macro-spin model catches the main features of the magnetization reversal processes in these cases, however see also the SI for a more detailed presentation of the spatial magnetization distribution as obtained from the micromagnetic simulations. In contrast to this, for the micro-Hall data are not reproduced by the all-metal micromagnetic or macro-spin simulations. However, if the core-shell structure with a spinel shell of reduced saturation magnetization is taken into account, we find for all angles a very good correspondence with the data. In particular, the pinched hysteresis form for is very well reproduced, and the shapes of the stray-field hysteresis for and are also very similar to the ones obtained by micro-Hall magnetometry. In addition, we observe that the coercive fields correspond quite well to the measured values. With regard to the remaining differences one has to take into account that our micromagnetic simulations assume K, whereas the micro-Hall data shown here were taken at K.
We have shown that the main features of the magnetization switching of the 3D nano-architectures, as monitored by micro-Hall magnetometry, are already quite well reproduced by a fast and scalable macro-spin approach. If complemented by carefully designed micromagnetic simulations, the correspondence becomes very satisfying also in those cases for which the macro-spin model is less successful. In addition we note that from our micromagnetic simulations it becomes also quite apparent that for the presented case of frustrated interactions through a vertex, which is magnetic itself, the magnetization distribution inside the magnetic 3D structures can be rather complex (see SI for details). In view of a prospective application of the presented building blocks towards 3D artificial spin-ice systems it may be desirable to reduce this level of complexity in the magnetization distributions. For such arrays, micromagnetic simulations will not be feasible, and it will be exceedingly difficult to acquire a satisfying understanding of all details of the array’s switching behavior. Thus, we consider replacing the vertices in the nano-elements by non-magnetic material and demonstrating that the nano-elements can be arranged in 3D array structures by our FEBID approach. Fig. 7 illustrates first results in these directions for the nano-tree geometry. Figure parts (a) and (b) show top and tilted views of a 3D nano-tree array employing again the precursor HCoFe(CO). In this case the FEBID writing strategy has to be carefully adapted to compensate for precursor gas flux shadowing effects  and anisotropic growth or proximity bending of growing nano-elements. If done properly, the nano-trees’ shape reproducibility and the placement accuracy are very good, as shown here. In figure parts (c) and (d) we demonstrate that it is furthermore possible to replace the vertex segment in the nano-tree by non-magnetic material, in our case nano-granular Pt, using MeCpMePt(IV) as precursor.
We have demonstrated that FEBID provides a powerful and flexible way to realize free-form magnetic 3D nano-elements and arrays of such structures. Our TEM-based characterization clearly indicates that HCoFe(CO) is a particularly well-suited precursor, as it leads to 3D deposits with nominally pure metallic character under beam conditions which are perfectly suitable for high-resolution structuring. The microstructure and composition analysis allowed us to pinpoint the essential features for suitable micromagnetic modeling (core-shell structure) and we were able to reproduce the most important stray field effects, as observed by micro-Hall magnetometry. In macro-spin model calculations, which are favorable because of their scaling behavior towards larger arrays, several observations in the switching behavior of the 3D nano-magnets can already be well reproduced. Replacing the vertex segment of the nano-elements by a non-magnetic material and the arrangement of nano-elements in 3D arrays was successfully demonstrated employing the FEBID approach. This will be of advantage for future work on 3D artificial spin-ice which is but one example of various other possible application fields of 3D magnetic FEBID structures on the single-element and array basis. Mesoscopic 3D arrays using this approach may allow for experimentally realizing and studying classical Ising or Heisenberg model systems . A chiral geometry of the nano-elements in 3D arrays is of high interest with regard to magneto-optical properties (see, e. g. ). Finally, single 3D nano-magnetic elements with increasing geometrical complexity are conceivable and will pave a new way for bringing topology-design elements into micromagnetic research.
Samples were fabricated using a dual beam SEM/FIB (FEI, Nova NanoLab 600), equipped with a Schottky electron emitter operating at a base pressure of about mbar. The precursors HCoFe(CO) and MeCpMePt(IV) (Me: methyl, Cp: cyclopentadienyl) were injected in the SEM by means of a capillary with an inner diameter of 0.5 mm. The distance capillary-surface was about m and the tilting angle of the injectors was . The crucible temperature of the gas injection system (GIS) was set to C and C for the Fe-Co and Pt precursor, respectively. The electron beam parameters used during deposition were 20 keV for the acceleration voltage and 13 pA for the beam current. The dwell time was set after optimization of the 3D growth to ms. The pitches depend on the inclination angle of the 3D structures and have to be adapted to the precursor and gas flow conditions . Concerning the synthesis of the Co-Fe precursor we refer to .
TEM investigations were carried out on a Tecnai F20 from FEI with a Schottky Field Emitter operating at kV. Images were taken with a post-column energy filter (Gatan Imaging Filter, GIF) using an energy slit of eV. The images were recorded zero-loss filtered (i. e., elastically scattered electrons only) on a 2K charge coupled device. For the image recording and processing (Fourier transformation) the software DigitalMicrograph from Gatan was used.
v.3 Micro-Hall magnetometry
The basic principle of operation of micro-Hall magnetometry is schematically sketched in Fig. 4(a). When a magnetic sample is placed on top of the sensor, in first approximation, the measured Hall voltage, , is proportional to the -component of the sample’s magnetic stray field averaged over the active area of the Hall cross, , via . Here, denotes the charge carrier concentration, is the applied current, and is the electron charge. A more detailed account, including a discussion of background subtraction and magnitude of the measured stray fields, is given in the SI.
The homebuilt Hall sensor is fabricated from an MBE-grown AlGaAs/GaAs heterostructure hosting the two-dimensional electron gas (2DEG) as the sensitive layer. In a first step, standard UV-lithography followed by wet chemical etching is employed to form six adjacent Hall-crosses of nominal size. Then the sensor structure is electrically contacted by annealed AuGe/Ni contact pads and gold wire bonding. Subsequently the sensor is covered with a Cr/Au top-gate which also serves as substrate for the 3D nano-cubes and -trees directly written by FEBID, see the SEM micrograph shown in Fig. (b) in the SI. The gate is grounded during the measurements. The sensor can be operated in a wide magnetic field and temperature range, but is optimized for K.
After writing of the 3D magnetic nanostructures the Hall sensor has been transferred in less than one hour to a cryogenic system equipped with a superconducting solenoid essentially free of magnetic flux jumps. The sample can be rotated with respect to the applied magnetic field, where for field angles perpendicular to the sensor plane.
v.4 Macro-spin simulations
Each element of the nano-trees and nano-cubes is modeled by a single macro-spin. The form anisotropy of each macro-spin is accounted for by a single uniaxial anisotropy. For the determination of the strength of this uniaxial anisotropy we assume that each edge and stem can be associated with a prolate ellipsoid. This allows us to calculate the anisotropy constant by means of the Stoner-Wohlfarth model . Assuming an average magnetization of kA/m the corresponding uniaxial anisotropy values lead to coercive fields much higher than the experimental ones. Therefore, we have kept the calculated ratio between the stem and edge anisotropy of 1:1.3 and fitted the anisotropy constant to the experimental coercive field at an inclination angle of . With a stem anisotropy constant of eV and a edge anisotropy constant of eV significant features of the experimental stray field hysteresis curves can be reproduced. This allows us to identify the underlying switching behavior of the different elements.
In order to study the dynamics of our macro-spin model we numerically solve the stochastic Landau-Lifshitz–Gilbert equation
Eq. 3 describes the motion for a macro-spin of unit length at site caused by an effective field which is generated by the interactions given in the Hamiltonian Eq. 6 below. In Eq. 3 denotes the gyromagnetic ratio and a phenomenological damping factor in front of the so-called Landau-Lifshitz damping term that drives the macro-spin towards a full alignment with the effective field , i. e. a local or global minimum configuration. Eq. 3 also allows to study finite-temperature effects by setting
where is the amplitude of the fluctuations and defined by
The evolution of the trajectories obtained by solving Eq. 3 and 4 leads to the stationary Gibbs distribution for a given temperature . The macro-spin calculations are used to study the dynamic switching behavior of the nano elements. As the temperature of such non-equilibrium processes within a macro-spin model does not correspond to a physical temperature, we have performed the macro-spin simulations at K.
The Hamiltonian describing the interactions among all macro-spins is given by
Here, the first term describes a uniaxial anisotropy, where is the anisotropy constant and is the unit vector pointing into the anisotropy direction. For this term describes an easy-axis anisotropy. The second term is the dipole-dipole interaction, where describes the effective magnetic moment per macro-spin for the stem or the edge, respectively. The direction between two interacting macro-spins is given by the unit vector and the distance is given by . The last term in Eq. 6 is the Zeeman term which describes the interaction of the macro-spins with the external magnetic field .
We obtained our hysteresis curves by linearly ramping up and down the external field in the range of using time steps of length ps. Using a damping constant of such calculations are very fast and just need minutes on a single core of a common computer processor. The calculation of the stray fields was done in the following way. (1) The positions of the four nano-trees and nano-cubes on the Hall sensor area where determined from SEM images. (2) The cumulated stray field contributions of all macro-spins of the nano-tree/cube have been averaged over positions in the -plane of the sensor array area (roughly m) at the -position of the 2DEG nm below the substrate surface.
v.5 Micromagnetic simulations
Zero temperature micromagnetic simulations were performed by numerically solving Eq. 3 for a single nano-tree consisting of CoFe (scenario 1), CoFe / CoFeO core/shell-structure (scenario 2) or a single nano-cube of CoFe (see SI for nano-cube). We used the GPU-accelerated micromagnetic simulation program MuMax3  running on a Linux notebook with Intel Core i7-7700HQ processor, 32 GB random access memory and NVidia GeForce GTX 1060 graphics card. Using cubic voxels of edge length nm for the finite difference discretization in MuMax3 the simulations for a typical external field cycle at a step size of 0.0033 T took about 60 hrs for one nano-tree. Simulations for the nano-cube with core/shell structure were not attempted, as the simulation volume and voxel number was expected to lead to simulation time of more than 260 hrs per external field cycle. Nano-cube simulations assuming full metallic CoFe as material were performed within 6 hrs for a typical field cycle. The simulation parameters were chosen as follows:
Geometrical dimensions: stem diameter nm (cylindrical) and length nm, edge diameters nm and nm (elliptical) at a length of nm, thickness of oxide spinel shell (scenario 2). Material parameters: the saturation magnetization of the shell was set to A/m using experimental data from . For the CoFe-core we used A/m and the exchange constant J/m by averaging the respective value for Fe and Co ; . As we could not find a reference for the exchange constant of the spinel we used the same exchange constant as for the core material.
Geometrical dimensions: stem diameter nm (cylindrical) and length nm, edge diameters nm (assumed cylindrical) at a length of nm. Material parameters: the saturation magnetization was set to A/m and the exchange constant to J/m.
The nano-grain microstructure of the deposits leads to an averaging of the magnetic anisotropy, which is why we have omitted any anisotropy energy contributions in our simulations. In order to guarantee sufficiently fast convergence we set the damping parameter to , used the full relaxation of MuMax3  at the initial field value and then the conjugated gradient method for quicker convergence at all other field values of each cycle with a stop criterion of .
The simulation data on the orientation of the magnetic moment vectors within the volume elements of the nano-tree or nano-cube for each external field was used to calculate the stray field at the sensor layer in the following way. (1) The positions of the four nano-trees and nano-cubes on the Hall sensor area where determined from SEM images. (2) For each volume element of the nano-tree/cube the associated simulated magnetic moment was used to calculate the corresponding dipolar stray field. The stray field contributions of all moments of the nano-tree/cube set to one of the four positions were averaged over positions of the sensor array area (roughly m) in the -plane at the -position of the 2DEG nm below the substrate surface. (3) The resulting four averaged stray fields were added to obtain the full averaged stray field of the four nano-trees/cubes.
Appendix A Acknowledgements
M. H. and L. K. acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Centre SFB/TR 49. I. S. acknowledges financial support from the Swedish Government Strategic Research Area in Materials Science on Functional Materials at Linköping University (Faculty Grant SFOMatLiU No 2009 00971). M. A. M. acknowledges financial support from the Faculty of Materials Engineering, University of Babylon, Babylon, Iraq, and from the Deutscher Akademischer Austauschdienst (DAAD) within the doctoral program for research studies in Germany. The MBE-grown high-mobility wafer material that was used to build the Hall magnetometer was kindly provided by Dr. Jürgen Weis, Max-Planck-Institute for Solid State Research, Stuttgart, Germany. H. P. thanks Prof. Ferdinand Hofer, Prof. Werner Grogger and Prof. Gerald Kotleitner for support. This work was conducted within the framework of the COST Action CM1301 (CELINA).
Appendix B Author contributions
M. H. and J. M. devised the project. L. K. performed the FEBID process with support by R. W. and H. P.. M. A. M. and J. P. performed the micro-Hall measurements. M. P. prepared the micro-Hall sensor and supported the stray-field data analysis. I. S. and C. S. performed the macro-spin simulations. Micromagnetic simulations were done by M. H.. S. B. synthesized the precursor. C. G. did the TEM experiments and analysis, supported by H. P.. M. H. and J. M. wrote the manuscript with input from all authors.
Appendix C Additional information
Appendix D Competing financial interests
The authors declare no competing financial interests.
Appendix E Micro-Hall magnetometry
e.1 Principle of measurement
The magnetic stray field of a sample – which is directly linked to its magnetization [] – is measured by detecting the Hall voltage generated in the sensor plane formed by the 2DEG at the interface of the AlGaAs/GaAs heterostructure, see the schematics in SI-Fig. 8(a). In first approximation, the detected -component of the stray field averaged over the active area of the Hall-cross is directly proportional to the measured Hall voltage . Since the integrated stray field of the arrays of nano-cubes or -trees grown on top of the Hall sensor, see SI-Fig. 8(b), is more than two orders of magnitude smaller than the applied external field , a so-called gradiometry measurement is performed, where the large signal, which is linear in is cancelled in situ by applying opposite currents across two Hall crosses, one decorated with magnetic particles and one empty, see Fig. 8(a). Then, the stray field contribution of the magnetic nanostructures is given by
where denotes the carrier density of the sensor and the applied currents, which have been . Data of vs. shown in this work have been corrected by subtracting a small linear background caused by slight differences between the two crosses in the gradiometry setup.
e.2 Stray field magnitude in measurement and simulation
Figures 4, 5 and 6 of the main paper compare the magnetic stray fields measured by micro-Hall magnetometry with macro-spin and micromagnetic simulations. In all cases, the magnitudes of the simulated -components of the stray fields are larger than the experimental values for . For example, the measured remanent field for the nano-cubes at shown in Fig. 4 is a factor of 4.5 smaller than the values simulated in the macro-spin model. Here, we briefly discuss possible reasons.
In [] it is pointed out that introducing an effective Hall coefficient and a Hall response function in the ballistic and diffusive transport regime, respectively, allows for taking account of the specific geometry of the Hall bar, such as circular corners and asymmetry in the probes. At K, where the measurements shown in this work have been performed, the transport properties of the 2DEG are considered in the diffusive regime, and therefore:
Here, is the stray field detected in the plane of the 2DEG buried about 115 nm below the Cr/Au top-gate onto which the magnetic structures are grown. is the active area of the Hall cross and is the stray field of the -th nano-cube/-tree in the respective -array detected in the 2DEG plane.
In this work, we have assumed a constant resulting in . Although calculating requires extensive numerical simulations for each magnetization configuration, we can roughly estimate that the expected decrease of the Hall response may be up to 30 % for the present geometry []. Another large effect is the increase of the effective Hall cross area due to the rounding of the corners. For both the macro-spin and micromagnetic simulations an idealized quadratic shape have been used with channel widths taken from the SEM microgrpahs, i. e. neglecting round corners and edge depletion effects. Assuming a realistic increase of by a factor of about 1.3 reduces the calculated stray fields by another 50 % for . Other factors that may contribute to the discrepancy between the measured and calculated stray fields are a small uncertainty in determining the distance to the 2DEG (an effect, however, less than 5 %), and possible uncertainties in determining the magnetic volume of the sample and the exact value of the saturation magnetization of the material. Finally, the calculation of the stray field from the measured differential Hall voltages do not account for possible inhomogeneous current distributions in the active area of the Hall cross.
Appendix F Comparison of micromagnetic and macro-spin simulations
The MS and MM simulations without the core/shell structure, shown in SI-Fig. 9, are very similar. One is therefore led to assume that the macro-spin model contains the essential features of the magnetization reversal process. However, the MM simulations show quite clearly that this assumption is premature. For the prominent stray field states (see solid- and dashed-line circles in the figure) selected, vortex-like magnetization profiles are visible at the terminal faces of the cylinder-shaped edges and also at the bottom of the stem. A closer inspection of different cross section through the nano-tree (not shown) reveals that these vortex structures are not threading throughout the full sample volume on any given cylindrical element. This is caused by the magnetic vertex segment joining the three edges and the stem of the nano-tree.
The situation is even more complex, if the nano-tree’s core-shell structure is taken into account. In this case, the distribution of magnetization orientations seen in the MM simulations exhibits even stronger spatial inhomogeneities in the magnetization direction. At the end caps of the cylindrical edges hedgehog-like structures occur, whereas the vortex-like magnetization profiles observed for the all-metal nano-tree micromagnetic model are virtually absent. We note that the thickness of the spinel outer layer only contains two voxel cells. A fully satisfying account on the details of the magnetization distribution in the shell would require a voxel edge length significantly below the used 5 nm. However, even at a moderate reduction to 3 nm the simulations could not be performed anymore on the hardware available to us (see method section of main text).
Appendix G Comparison of magnetic stray field measurements and micromagnetic simulations for 3D nano-cubes
In SI-Fig. 11 we show the results of micromagnetic simulations at the same angles which are shown in Fig. 4 of the main paper. The all-metal micromagnetic model already shows a very good qualitative agreement with the measured stray fields. For example, the experimentally observed crossing of the up- and down-sweep curves for are well reproduced by the simulations. As we have shown for the nano-trees, taking into account a core/shell structure with an outer metal-oxide layer we would expect an even better agreement with the measurements. However, for the nano-cube micromagnetic model the simulation volume is significantly larger than for the nano-tree. Even at the rather large 5 nm edge length for the cubic voxels used in our simulations, a core/shell structure simulation could not be performed anymore on the hardware available to us (see method section of main text).
- V V Kruglyak, S O Demokritov, and D Grundler. Magnonics. Journal of Physics D: Applied Physics, 43(26):264001, 2010.
- Fabio Pulizzi. Spintronics. Nat Mater, 11(5):367–367, 05 2012.
- Sergej O. Demokritov and Andrei N. Slavin, editors. Magnonics: From Fundamentals to Applications. Springer Berlin Heidelberg, Berlin, Heidelberg, 2013.
- Amalio Fernández-Pacheco, Robert Streubel, Olivier Fruchart, Riccardo Hertel, Peter Fischer, and Russell P. Cowburn. Three-dimensional nanomagnetism. Nature Communications, 8:15756 EP –, 06 2017.
- Stuart S. P. Parkin, Masamitsu Hayashi, and Luc Thomas. Magnetic domain-wall racetrack memory. Science, 320(5873):190–194, 2008.
- Roger G. Melko, Byron C. den Hertog, and Michel J. P. Gingras. Long-range order at low temperatures in dipolar spin ice. Phys. Rev. Lett., 87:067203, Jul 2001.
- R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville, B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H. Crespi, and P. Schiffer. Artificial spin ice in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature, 439(7074):303–306, 01 2006.
- S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford. Direct observation of magnetic monopole defects in an artificial spin-ice system. Nat Phys, 6(5):359–363, 05 2010.
- Elena Mengotti, Laura J. Heyderman, Arantxa Fraile Rodriguez, Frithjof Nolting, Remo V. Hugli, and Hans-Benjamin Braun. Real-space observation of emergent magnetic monopoles and associated dirac strings in artificial kagome spin ice. Nat Phys, 7(1):68–74, 01 2011.
- Sheng Zhang, Ian Gilbert, Cristiano Nisoli, Gia-Wei Chern, Michael J. Erickson, Liam O/’Brien, Chris Leighton, Paul E. Lammert, Vincent H. Crespi, and Peter Schiffer. Crystallites of magnetic charges in artificial spin ice. Nature, 500(7464):553–557, 08 2013.
- Cristiano Nisoli, Roderich Moessner, and Peter Schiffer. Artificial spin ice: Designing and imaging magnetic frustration. Rev. Mod. Phys., 85:1473–1490, Oct 2013.
- Ian Gilbert, Yuyang Lao, Isaac Carrasquillo, Liam O/’Brien, Justin D. Watts, Michael Manno, Chris Leighton, Andreas Scholl, Cristiano Nisoli, and Peter Schiffer. Emergent reduced dimensionality by vertex frustration in artificial spin ice. Nat Phys, 12(2):162–165, 02 2016.
- Jasper Drisko, Thomas Marsh, and John Cumings. Topological frustration of artificial spin ice. Nature Communications, 8:14009 EP –, 01 2017.
- Gia-Wei Chern, Charles Reichhardt, and Cristiano Nisoli. Realizing three-dimensional artificial spin ice by stacking planar nano-arrays. Applied Physics Letters, 104(1):013101, 2014.
- W. F. van Dorp and C. W. Hagen. A critical literature review of focused electron beam induced deposition. Journal of Applied Physics, 104(8):081301, 2008.
- Michael Huth, Fabrizio Porrati, Christian Schwalb, Marcel Winhold, Roland Sachser, Maja Dukic, Jonathan Adams, and Georg Fantner. Focused electron beam induced deposition: A perspective. Beilstein Journal of Nanotechnology, 3:597–619, 2012.
- Jason D. Fowlkes, Robert Winkler, Brett B. Lewis, Michael G. Stanford, Harald Plank, and Philip D. Rack. Simulation-guided 3d nanomanufacturing via focused electron beam induced deposition. ACS Nano, 10(6):6163–6172, 2016. PMID: 27284689.
- J M De Teresa, A Fernández-Pacheco, R Córdoba, L Serrano-Ramón, S Sangiao, and M R Ibarra. Review of magnetic nanostructures grown by focused electron beam induced deposition (febid). Journal of Physics D: Applied Physics, 49(24):243003, 2016.
- Barbara Geier, Christian Gspan, Robert Winkler, Roland Schmied, Jason D. Fowlkes, Harald Fitzek, Sebastian Rauch, Johannes Rattenberger, Philip D. Rack, and Harald Plank. Rapid and highly compact purification for focused electron beam induced deposits: A low temperature approach using electron stimulated h2o reactions. The Journal of Physical Chemistry C, 118(25):14009–14016, 2014.
- Roland Sachser, Heiko Reith, Daniel Huzel, Marcel Winhold, and Michael Huth. Catalytic purification of directly written nanostructured pt microelectrodes. ACS Applied Materials & Interfaces, 6(18):15868–15874, 2014. PMID: 25111450.
- E Villamor, F Casanova, P H F Trompenaars, and J J L Mulders. Embedded purification for electron beam induced pt deposition using mecpptme 3. Nanotechnology, 26(9):095303, 2015.
- Mostafa M. Shawrav, Philipp Taus, Heinz D. Wanzenboeck, M. Schinnerl, M. Stöger-Pollach, S. Schwarz, A. Steiger-Thirsfeld, and Emmerich Bertagnolli. Highly conductive and pure gold nanostructures grown by electron beam induced deposition. Scientific Reports, 6:34003, 09 2016.
- Robert Winkler, Franz-Philipp Schmidt, Ulrich Haselmann, Jason D. Fowlkes, Brett B. Lewis, Gerald Kothleitner, Philip D. Rack, and Harald Plank. Direct-write 3d nanoprinting of plasmonic structures. ACS Applied Materials & Interfaces, 9(9):8233–8240, 2017. PMID: 28269990.
- Rosa Córdoba, Nidhi Sharma, Sebastian Kölling, Paul M. Koenraad, and Bert Koopmans. High-purity 3d nano-objects grown by focused-electron-beam induced deposition. Nanotechnology, 27(35):355301, 2016.
- Amalio Fernández-Pacheco, Luis Serrano-Ramón, Jan M. Michalik, M. Ricardo Ibarra, José M. De Teresa, Liam O’Brien, Dorothée Petit, Jihyun Lee, and Russell P. Cowburn. Three dimensional magnetic nanowires grown by focused electron-beam induced deposition. Scientific Reports, 3:1492 EP –, 03 2013.
- Javier Pablo-Navarro, Dédalo Sanz-Hernández, César Magén, Amalio Fernández-Pacheco, and José María de Teresa. Tuning shape, composition and magnetization of 3d cobalt nanowires grown by focused electron beam induced deposition (febid). Journal of Physics D: Applied Physics, 50(18):18LT01, 2017.
- F Porrati, M Pohlit, J Müller, S Barth, F Biegger, C Gspan, H Plank, and M Huth. Direct writing of cofe alloy nanostructures by focused electron beam induced deposition from a heteronuclear precursor. Nanotechnology, 26(47):475701, 2015.
- Robert Winkler, Jason Fowlkes, Aleksandra Szkudlarek, Ivo Utke, Philip D. Rack, and Harald Plank. The nanoscale implications of a molecular gas beam during electron beam induced deposition. ACS Applied Materials & Interfaces, 6(4):2987–2995, 2014. PMID: 24502299.
- E. F. Egerton. Electron energy-loss spectroscopy in the tem. Report on Progress in Physics, 72:016502, 2009.
- P. Das, F. Porrati, S. Wirth, A. Bajpai, M. Huth, Y. Ohno, H. Ohno, and J. Müller. Magnetization dynamics of a cro2 grain studied by micro-hall magnetometry. Applied Physics Letters, 97(4):042507, 2010.
- Merlin Pohlit, Fabrizio Porrati, Michael Huth, Yuzo Ohno, Hideo Ohno, and Jens Müller. Nanocluster building blocks of artificial square spin ice: Stray-field studies of thermal dynamics. Journal of Applied Physics, 117(17):17C746, 2015.
- Merlin Pohlit, Fabrizio Porrati, Michael Huth, Yuzo Ohno, Hideo Ohno, and Jens Müller. Magnetic stray-field studies of a single cobalt nanoelement as a component of the building blocks of artificial square spin ice. Journal of Magnetism and Magnetic Materials, 400:206 – 212, 2016. Proceedings of the 20th International Conference on Magnetism (Barcelona) 5-10 July 2015.
- Merlin Pohlit, Paul Eibisch, Maryam Akbari, Fabrizio Porrati, Michael Huth, and Jens Müller. First order reversal curves (forc) analysis of individual magnetic nanostructures using micro-hall magnetometry. Review of Scientific Instruments, 87(11):113907, 2016.
- Josef Fidler and Thomas Schrefl. Micromagnetic modelling - the current state of the art. Journal of Physics D: Applied Physics, 33(15):R135, 2000.
- Arne Vansteenkiste, Jonathan Leliaert, Mykola Dvornik, Mathias Helsen, Felipe Garcia-Sanchez, and Bartel Van Waeyenberge. The design and verification of mumax3. AIP Advances, 4(10):107133, 2014.
- M. Dvornik, Y. Au, and V. V. Kruglyak. Micromagnetic Simulations in Magnonics, pages 101–115. Springer Berlin Heidelberg, Berlin, Heidelberg, 2013.
- Sahand Eslami, John G. Gibbs, Yvonne Rechkemmer, Joris van Slageren, Mariana Alarcón-Correa, Tung-Chun Lee, Andrew G. Mark, Geert L. J. A. Rikken, and Peer Fischer. Chiral nanomagnets. ACS Photonics, 1(11):1231–1236, 2014.
- Merlin Pohlit, Irina Stockem, Fabrizio Porrati, Michael Huth, Christian Schröder, and Jens Müller. Experimental and theoretical investigation of the magnetization dynamics of an artificial square spin ice cluster. Journal of Applied Physics, 120(14):142103, 2016.
- I. P. Muthuselvam and R. N. Bhowmik. Structural phase stability and magnetism in cofeo spinel oxide. Solid State Sciences, 11:719, 2009.
- F. Porrati and M. Huth. Diagram of the states in arrays of iron nanocylinders. Applied Physics Letters, 85(15):3157–3159, 2004.
- Y. G. Cornelissens and F. M. Peeters. Response function of a hall magnetosensor in the diffusive regime. Journal of Applied Physics, 92:2006, 2002.