Non-Planar Geometrical Effects on the Magnetoelectrical Signal in a Three-Dimensional Nanomagnetic Circuit

Expanding nanomagnetism and spintronics into three dimensions (3D) offers great opportunities for both fundamental and technological studies. However, probing the influence of complex 3D geometries on magnetoelectrical phenomena poses important experimental and theoretical challenges. In this work, we investigate the magnetoelectrical signals of a ferromagnetic 3D nanodevice integrated into a microelectronic circuit using direct-write nanofabrication. Due to the 3D vectorial nature of both electrical current and magnetization, a complex superposition of several magnetoelectrical effects takes place. By performing electrical measurements under the application of 3D magnetic fields, in combination with macrospin simulations and finite element modeling, we disentangle the superimposed effects, finding how a 3D geometry leads to unusual angular dependences of well-known magnetotransport effects such as the anomalous Hall effect. Crucially, our analysis also reveals a strong role of the noncollinear demagnetizing fields intrinsic to 3D nanostructures, which results in an angular dependent magnon magnetoresistance contributing strongly to the total magnetoelectrical signal. These findings are key to the understanding of 3D spintronic systems and underpin further fundamental and device-based studies.


S1 Geometrical details of the nanobridge inferred from scanning electron microscopy (SEM) images
The projection of the probe region on the substrate plane is approximately 350 × 2 = 700 nm.
By tilting the sample stage by 45º (Figure S1b), we can calculate the angle, α, between the bridge and substrate. We have measured the projected angle α′ to be 62º ± 0.5º and 60º ± 0.5º from the SEM image, which are the 45° projection to the substrate plane. Hence, the real angle can be calculated as α = tan !" -#$%('() *+,(-.°) . to be 69.4 ± 0.56º and 67.8 ± 0.56º. From this analysis we determine that there is a 1.6 ± 0.8º difference in the angle formed by the legs and the x-axis.
From the top view perspective in Figure S1a, we further estimate the symmetry of the structure by considering the boundary lines of the probed region (green dash lines) which are parallel to each other, indicating that any deviation of the structure from the x-axis is below the SEM resolution. As a result, we conclude that the degree of asymmetry is very small in our structure.
With the known a, the length of one side of the probed region is estimated to be 1023 nm.
Finally, the thickness of the measured bridge is estimated from another broken bridge ( Figure   S1c) that is grown under the same condition, and where the thickness is about 120 nm for the main bridge and 370 nm for the apex region. Figure S2 êThe macrospin model. a, The applied field Ha, the demagnetising field Hd and the effective field H are considered for a nanowire in the local coordinate system. b-c, Three nanowires forming the nanobridge, as considered in the model.

S2.1 Determination of M in a single domain nanowire at high fields
To calculate the magnetoresistance of the 3D nanobridge, we first need to determine its magnetisation. A macrospin model is used here to determine the magnetisation in the nanobridge at high fields (± 4 T) in a simple, computationally efficient way. Before studying the 3D nanobridge, we first consider how to determine the magnetisation in a single domain nanowire with a known applied field by minimising the sum of Zeeman and demagnetising energies.
As shown in Figure S2a, the magnetisation M can be written within the local coordinate system x'y'z' as, N y' N z' ], which are associated with the three-principal axes of the geometry, and they obey the general constraint 4( + 5( + 6( = 1. So, the demagnetising field can be written as The applied field can be written as ] is the known unit vector for the applied field.
The total energy E can be written as: By solving the two partial differential equations below, we can determine , and hence the direction of the magnetisation vector M.

S2.2 Determination of the demagnetising factor for each section of the bridge
To apply the macrospin model of a single nanowire to the nanobridge, we assume the bridge is made up of three single-domain sections as shown in Figure S2 b, c. To do this, we need to determine the demagnetising factor for each section. To get a reasonable estimation of demagnetising factor for each section, we have simulated the demagnetising field ( ) for fields applied from = 0° to = 90° using a finite element method based on the magnum.fe package. 2 The saturation magnetisation, M s , used in this simulation is 1.67 T, 3 and the results are shown in Figure S3. The grey arrows indicate the direction of and the colour scale represents the x-component of . We also plot the average magnitude of in each section of the bridge in Figure S4.  For Section 1 ( Figure S4a), there is a minimum in | | at = 70°. As shown in Figure   S3h, when the field is applied at = 70°, is parallel to the long axis (easy) of the section 1, and hence gives the smallest . On the contrary, for Section 2, | | peaks at = 20° ( Figure S4b), where, as shown in Figure S3c, is parallel to the hard axis of the Section 2, leading to a maximum in . Due to the unusual geometry of section 3, it is not immediately clear which is the easy or hard axis. However, from Figure S4c, we see | | decreases with increasing , which indicates that its easy axis is aligned with the = 90° direction. Figure S5 çComparison of the averaged x, y, z components of the demagnetising field, , obtained from macrospin and micromagnetic simulations a, Section 1. b, Section 2. c, Section 3. For each section, the demagnetising field of micromagnetic simulations is given in red and used to identify the optimal demagnetising factor for the structure.
To determine the demagnetising factor for each section, we plot the x, y, z components (in the global coordinate system) of the simulated obtained from micromagnetic simulations, against the applied field angle , as a thick red line in Figure S5. We also plot the simulated from the macrospin model described in For Section 1 and Section 2 ( Figure S5 a,

S3.1 Boundary conditions
To calculate the MR signal, we assume a constant current supplied to the bridge and we solve the electric potential u in ∇ • [ (∇ )] = 0, using FEM simulations. The resistivity tensor ( !" ) in the simulation is described in the main paper, and the boundary conditions are described here. First, we assume a constant current of 1 A flowing into the bridge through Face 1 and out of the bridge through Face 2, as shown in Figure S6a. Thus, the current flowing in and out of the nanobridge is set using Neumann boundary conditions as B⃗ • ( ∇ ) = − 9: ;<= , respectively. Here B⃗ is the outward unit normal and 9: = ;<= = 1A/(500 nm × 100nm) is the current density at Face 1 and 2. Secondly, we set the electric potential to be 0 at Face 2, as a Dirichlet boundary condition = 0. Finally, for all other Faces in this model, we set B⃗ • ( ∇ ) = 0, as no current flows in or out of the model through other Faces.

S3.2 Materials properties used and unintended deposition around the 3D nanobridge
The 3D nanobridge is deposited by FEBID with 30 kV acceleration voltage and 0.34 nA beam current using Co 2 (CO) 8 as precursor. Similar growth conditions as the ones used here for 2D deposits lead to a metallic nanocrystalline material, formed by cobalt crystals with typical sizes around 5-10 nm, and atomic percentages of ≈ 90-95% Co. 3,4 In our case, the larger beam currents in combination with a 3D geometry both enhance the local heating, 5 which is likely to promote higher Co purity via autocatalytic effects, 6-11 enhancing the electrical conduction properties of the material. Local heating during 3D growth is also likely to enhance the Co content, crystallinity, and magnetic properties. 12 Taking into account the lowest resistivity value reported so far for FEBID, equal to 26 µΩcm in reference, 6 and the typical resistivity of polycrystalline cobalt thin films deposited by conventional physical vapour deposition methods, [13][14][15][16] we thus expect the resistivity of the probed region of the nanobridge to be in the range of 11-26 µΩcm. [13][14][15][16] We first substitute the resistivity Co =11 µΩcm into the bridge model shown in Figure S6c, which gives as a result a simulated voltage across the side contacts of about 10 V, which is 10 times larger than the resistance experimentally measured. Since the resistivity of FEBID Co is not likely to be smaller than the resistivity of polycrystalline cobalt thin films, this small measured resistance is attributed to an unintended deposition ('halo') around the desired 3D nanostructure. This 'halo' effect is caused by precursor dissociation by secondary and backscattered electrons reaching distances far beyond the primary electron beam, a common effect in FEBID. 17 This indicates the FIB milled trenches explained in the main manuscript did not completely prevent the influence of this 'halo'. Since this parasitic deposit is reported to have low cobalt concentration, [18][19][20] here it is modelled as a non-magnetic, round thin film as shown in Figure S6a. With this 'halo' included in the FEM model, the simulated resistance is reduced by approximately a factor of 10, and we reach a good quantitative agreement between experiments and simulations ( Figure S6b). To determine the reproducibility of our results, in addition to the sample used in the paper ( Figure S7a), we also measured a smaller subset of data for a second structure with similar geometry as shown in Figure S7b. Specifically, we compare the = 0° and = 90° hysteresis loops for both samples for a field range of -3 T to 3 T. The sample presented in the paper ( Figure S7c) and the second sample ( Figure S7d) exhibit similar resistance and a very similar trend of the data for the = 0° and = 90° cases, demonstrating that the realisation of the 3D nanomagnetic circuits are reproducible. Figure S7 ç a-b, The SEM images taken for the sample described in the paper and the supporting sample, respectively. c-d, The θ = 0° and θ = 90° measurements for the sample described in the paper and The supporting sample, respectively. In the main paper, the sum of the ordinary and anomalous Hall effect is shown in Figure 3b.

S3.3 Ordinary Hall effect and anomalous Hall effect
Here, we compare the contribution of each effect separately. The simulated results of AHE, OHE and their sum are plotted in Figure S8, where it is clear that AHE is the dominant effect.
Values of @AB = 5.6 × 10 !C Ωm and DAB = −1.2 × 10 !"0 Ωm/T for the anomalous Hall resistivity and the ordinary Hall coefficient, respectively, are used to produce the best fit to the data. Temperature gradients are known to lead to electrical and spin transport effects. 22 Here, we simulate the temperature increase of the 3D nanobridge due to joule heating, as 3D

S4 Influence of heat
nanostructures cannot dissipate heat to the substrate as well as 2D structures do. Here, we use the same 3D model as the one detailed above, finding the steady-state solution for the general heat equation, where r is the density, E is the specific heat capacity, k = 3 W/mK 23 is the heat conductivity, Q is the internal heat source. The first term can be ignored in our case, as we only evaluate the steady-state solution. The source of heat generation here is the joule heating and can be written as = |∇ | 8 .
In the experiment, a constant current of 0.6 µA is supplied, which is equivalent to a current density of 2.7´10 7 A/m 2 . We simulate the voltage drop across the side contacts with this current density and then obtain the heat generation. The halo is set to have a constant temperature of 180 K, and convection though cool helium gas used in experiments, and radiation, are ignored.
As shown in Figure S9, the temperature increase obtained for our 3D geometry is only 0.5 mK, which is negligible for our magnetotransport studies.