# Magnetic Characterization of Direct-Write Free-Form Building Blocks for Artificial Magnetic 3D Lattices

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Geometry of Nano-Cubes and Nano-Trees

#### 2.2. Temperature-Dependent Magnetization Switching

#### 2.3. Angular-Dependent Magnetization Switching—Experiment and Micromagnetic Simulation

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Focused Electron Beam Induced Deposition

#### 4.2. Micro-Hall Magnetometry

#### 4.3. Micromagnetic Simulations

**Nano-tree**- Geometrical dimensions: stem diameter ${D}_{s}=100$ nm (cylindrical) and length ${L}_{s}=185$ nm, edge diameters ${D}_{e,1}=60$ nm and ${D}_{e,2}=48$ nm (elliptical) at a length of ${L}_{e}=340$ nm. The longer semi-axis with diameter ${D}_{e,1}$ is roughly in the beam direction (see [19] for details). Material parameters: saturation magnetization ${M}_{S}=1.5\times {10}^{6}\phantom{\rule{0.166667em}{0ex}}$A/m and exchange constant $A=1.5\times {10}^{-11}\phantom{\rule{0.166667em}{0ex}}$J/m by averaging the respective value for Fe and Co [36,38].
**Nano-cube**- Geometrical dimensions: stem diameter ${D}_{s}=100$ nm (cylindrical) and length ${L}_{s}=185$ nm, edge diameter ${D}_{e}=60$ nm (cylindrical) at a length of ${L}_{b}=340$ nm. Material parameters: see material parameters for nano-tree.

- The positions of the four nano-trees and nano-cubes on the Hall sensor area were determined from SEM images.
- For each voxel 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 $n\times n$ positions of the sensor array area (roughly $5\times 5\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$m${}^{2}$) in the $xy$-plane at the z-position of the 2DEG sensor $115$ nm below the substrate surface.
- The resulting four averaged stray fields were added to obtain the full averaged stray field of the four nano-trees/cubes.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

3D (2D) | Three (Two)-Dimensional |

FEBID | Focused Electron Beam Induced Deposition |

SEM | Scanning Electron Microscopy |

TEM | Transmission Electron Microscopy |

EELS | Electron Energy Loss Spectroscopy |

## References

- Kruglyak, V.V.; Demokritov, S.O.; Grundler, D. Magnonics. J. Phys. D Appl. Phys.
**2010**, 43, 264001. [Google Scholar] [CrossRef] - Pulizzi, F. Spintronics. Nat. Mater.
**2012**, 11, 367. [Google Scholar] [CrossRef] [PubMed] - Demokritov, S.O.; Slavin, A.N. (Eds.) Magnonics: From Fundamentals to Applications; Springer: Berlin, Germany, 2013. [Google Scholar]
- Fernández-Pacheco, A.; Streubel, R.; Fruchart, O.; Hertel, R.; Fischer, P.; Cowburn, R.P. Three-dimensional nanomagnetism. Nat. Commun.
**2017**, 8. [Google Scholar] [CrossRef] [PubMed] - Piraux, L.; Renard, K.; Guillemet, R.; Mátéfi-Tempfli, S.; Mátéfi-Tempfli, M.; Antohe, V.A.; Fusil, S.; Bouzehouane, K.; Cros, V. Template-Grown NiFe/Cu/NiFe Nanowires for Spin Transfer Devices. Nano Lett.
**2007**, 7, 2563–2567. [Google Scholar] [CrossRef] [PubMed] - Donnelly, C.; Guizar-Sicairos, M.; Scagnoli, V.; Holler, M.; Huthwelker, T.; Menzel, A.; Vartiainen, I.; Müller, E.; Kirk, E.; Gliga, S.; et al. Element-Specific X-Ray Phase Tomography of 3D Structures at the Nanoscale. Phys. Rev. Lett.
**2015**, 114. [Google Scholar] [CrossRef] [PubMed] - Bedanta, S.; Barman, A.; Kleemann, W.; Petracic, O.; Seki, T. Magnetic Nanoparticles: A Subject for Both Fundamental Research and Applications. J. Nanomater.
**2013**, 2013, 22. [Google Scholar] [CrossRef] - Williams, G.; Hunt, M.; Boehm, B.; May, A.; Taverne, M.; Ho, D.; Giblin, S.; Read, D.; Rarity, J.; Allenspach, R.; et al. Two-photon lithography for 3D magnetic nanostructure fabrication. Nano Res.
**2018**, 11, 845–854. [Google Scholar] [CrossRef] - Chern, G.W.; Reichhardt, C.; Nisoli, C. Realizing three-dimensional artificial spin ice by stacking planar nano-arrays. Appl. Phys. Lett.
**2014**, 104. [Google Scholar] [CrossRef] - Perrin, Y.; Canals, B.; Rougemaille, N. Extensive degeneracy, Coulomb phase and magnetic monopoles in artificial square ice. Nature
**2016**, 540, 410–413. [Google Scholar] [CrossRef] [PubMed] - Fowlkes, J.D.; Winkler, R.; Lewis, B.B.; Stanford, M.G.; Plank, H.; Rack, P.D. Simulation-Guided 3D Nanomanufacturing via Focused Electron Beam Induced Deposition. ACS Nano
**2016**, 10, 6163–6172. [Google Scholar] [CrossRef] [PubMed] - Teresa, J.M.D.; Fernández-Pacheco, A.; Córdoba, R.; Serrano-Ramón, L.; Sangiao, S.; Ibarra, M.R. Review of magnetic nanostructures grown by focused electron beam induced deposition (FEBID). J. Phys. D Appl. Phys.
**2016**, 49, 1–24. [Google Scholar] [CrossRef] - Córdoba, R.; Sharma, N.; Kölling, S.; Koenraad, P.M.; Koopmans, B. High-purity 3D nano-objects grown by focused-electron-beam induced deposition. Nanotechnology
**2016**, 27. [Google Scholar] [CrossRef] [PubMed] - Fernández-Pacheco, A.; Serrano-Ramón, L.; Michalik, J.M.; Ibarra, M.R.; De Teresa, J.M.; O’Brien, L.; Petit, D.; Lee, J.; Cowburn, R.P. Three dimensional magnetic nanowires grown by focused electron-beam induced deposition. Sci. Rep.
**2013**, 3. [Google Scholar] [CrossRef] [PubMed] - Pablo-Navarro, J.; Sanz-Hernández, D.; Magén, C.; Fernández-Pacheco, A.; de Teresa, J.M. Tuning shape, composition and magnetization of 3D cobalt nanowires grown by focused electron beam induced deposition (FEBID). J. Phys. D Appl. Phys.
**2017**, 50. [Google Scholar] [CrossRef] - Sanz-Hernández, D.; Hamans, R.F.; Liao, J.W.; Welbourne, A.; Lavrijsen, R.; Fernández-Pacheco, A. Fabrication, Detection, and Operation of a Three-Dimensional Nanomagnetic Conduit. ACS Nano
**2017**, 11, 11066–11073. [Google Scholar] [PubMed] - Huth, M.; Porrati, F.; Dobrovolskiy, O. Focused electron beam induced deposition meets materials science. Microelectron. Eng.
**2018**, 185–186, 9–28. [Google Scholar] [CrossRef] - García-Cervera, C.J.; Gimbutas, Z.; Weinan, E. Accurate numerical methods for micromagnetics simulations with general geometries. J. Comput. Phys.
**2003**, 184, 37–52. [Google Scholar] [CrossRef] - Keller, L.; Mamoori, M.K.I.A.; Pieper, J.; Gspan, C.; Stockem, I.; Schröder, C.; Barth, S.; Winkler, R.; Plank, H.; Pohlit, M.; et al. Direct-write of free-form building blocks for artificial magnetic 3D lattices. Sci. Rep.
**2017**, arXiv:1709.05847. [Google Scholar] - Melko, R.G.; den Hertog, B.C.; Gingras, M.J.P. Long-Range Order at Low Temperatures in Dipolar Spin Ice. Phys. Rev. Lett.
**2001**, 87, 067203. [Google Scholar] [CrossRef] [PubMed] - Wang, R.F.; Nisoli, C.; Freitas, R.S.; Li, J.; McConville, W.; Cooley, B.J.; Lund, M.S.; Samarth, N.; Leighton, C.; Crespi, V.H.; et al. Artificial spin ice in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature
**2006**, 439, 303–306. [Google Scholar] [CrossRef] [PubMed] - Ladak, S.; Read, D.E.; Perkins, G.K.; Cohen, L.F.; Branford, W.R. Direct observation of magnetic monopole defects in an artificial spin-ice system. Nat. Phys.
**2010**, 6, 359–363. [Google Scholar] [CrossRef] - Mengotti, E.; Heyderman, L.J.; Rodriguez, A.F.; Nolting, F.; Hugli, R.V.; Braun, H.B. Real-space observation of emergent magnetic monopoles and associated Dirac strings in artificial kagome spin ice. Nat. Phys.
**2011**, 7, 68–74. [Google Scholar] [CrossRef] - Zhang, S.; Gilbert, I.; Nisoli, C.; Chern, G.W.; Erickson, M.J.; O’Brien, L.; Leighton, C.; Lammert, P.E.; Crespi, V.H.; Schiffer, P. Crystallites of magnetic charges in artificial spin ice. Nature
**2013**, 500, 553–557. [Google Scholar] [CrossRef] [PubMed] - Nisoli, C.; Moessner, R.; Schiffer, P. Artificial spin ice: Designing and imaging magnetic frustration. Rev. Mod. Phys.
**2013**, 85, 1473–1490. [Google Scholar] [CrossRef] - Gilbert, I.; Lao, Y.; Carrasquillo, I.; O’Brien, L.; Watts, J.D.; Manno, M.; Leighton, C.; Scholl, A.; Nisoli, C.; Schiffer, P. Emergent reduced dimensionality by vertex frustration in artificial spin ice. Nat. Phys.
**2016**, 12, 162–165. [Google Scholar] [CrossRef] - Drisko, J.; Marsh, T.; Cumings, J. Topological frustration of artificial spin ice. Nat. Commun.
**2017**, 8. [Google Scholar] [CrossRef] [PubMed] - Winkler, R.; Schmidt, F.P.; Haselmann, U.; Fowlkes, J.D.; Lewis, B.B.; Kothleitner, G.; Rack, P.D.; Plank, H. Direct-Write 3D Nanoprinting of Plasmonic Structures. ACS Appl. Mater. Interfaces
**2017**, 9, 8233–8240. [Google Scholar] [CrossRef] [PubMed] - Winkler, R.; Lewis, B.B.; Fowlkes, J.D.; Rack, P.D.; Plank, H. High-Fidelity 3D-Nanoprinting using a Focused Electron Beam: Growth characteristics. ACS Appl. Nano Mater.
**2018**. under review. [Google Scholar] - Wirth, S.; Anane, A.; von Molnár, S. Thermally activated magnetization reversal in nanometer-size iron particles. Phys. Rev. B
**2000**, 63. [Google Scholar] [CrossRef] - Pohlit, M.; Porrati, F.; Huth, M.; Ohno, Y.; Ohno, H.; Müller, J. Magnetic stray-field studies of a single Cobalt nanoelement as a component of the building blocks of artificial square spin ice. J. Magn. Magn. Mater.
**2016**, 400, 206–212. [Google Scholar] [CrossRef] - Pohlit, M.; Porrati, F.; Huth, M.; Ohno, Y.; Ohno, H.; Müller, J. Nanocluster building blocks of artificial square spin ice: Stray-field studies of thermal dynamics. J. Appl. Phys.
**2015**, 117. [Google Scholar] [CrossRef] - Wernsdorfer, W.; Orozco, E.B.; Hasselbach, K.; Benoit, A.; Barbara, B.; Demoncy, N.; Loiseau, A.; Pascard, H.; Mailly, D. Experimental Evidence of the Néel-Brown Model of Magnetization Reversal. Phys. Rev. Lett.
**1997**, 78, 1791–1794. [Google Scholar] [CrossRef] - Wernsdorfer, W.; Doudin, B.; Mailly, D.; Hasselbach, K.; Benoit, A.; Meier, J.; Ansermet, J.P.; Barbara, B. Nucleation of Magnetization Reversal in Individual Nanosized Nickel Wires. Phys. Rev. Lett.
**1996**, 77, 1873–1876. [Google Scholar] [CrossRef] [PubMed] - Li, Y.; Xiong, P.; von Molnár, S.; Ohno, Y.; Ohno, H. Magnetization reversal in elongated Fe nanoparticles. Phys. Rev. B
**2005**, 71. [Google Scholar] [CrossRef] - Porrati, F.; Huth, M. Diagram of the states in arrays of iron nanocylinders. Appl. Phys. Lett.
**2004**, 85, 3157–3159. [Google Scholar] [CrossRef] - Porrati, F.; Pohlit, M.; Müller, J.; Barth, S.; Biegger, F.; Gspan, C.; Plank, H.; Huth, M. Direct writing of CoFe alloy nanostructures by focused electron beam induced deposition from a heteronuclear precursor. Nanotechnology
**2015**, 26. [Google Scholar] [CrossRef] [PubMed] - Pohlit, M.; Stockem, I.; Porrati, F.; Huth, M.; Schröder, C.; Müller, J. Experimental and theoretical investigation of the magnetization dynamics of an artificial square spin ice cluster. J. Appl. Phys.
**2016**, 120. [Google Scholar] [CrossRef] - Müller, J.; Körbitzer, B.; Amyan, A.; Pohlit, M.; Ohno, Y.; Ohno, H. Noise spectroscopy studies of GaAs/AlGaAs Hall devices for optimizing micro- and nano-scale magnetic measurements. In Proceedings of the 2015 International Conference on Noise and Fluctuations (ICNF), Xi’an, China, 2–6 June 2015; pp. 1–4. [Google Scholar]
- Müller, J.; Li, Y.; von Molnár, S.; Ohno, Y.; Ohno, H. Single-electron switching in Al
_{x}Ga_{1−x}As/GaAs Hall devices. Phys. Rev. B**2006**, 74. [Google Scholar] [CrossRef] - Cornelissens, Y.G.; Peeters, F.M. Response function of a Hall magnetosensor in the diffusive regime. J. Appl. Phys.
**2002**, 92. [Google Scholar] [CrossRef] - Cerchez, M.; Heinzel, T. Correction factor in nondiffusive Hall magnetometry. Appl. Phys. Lett.
**2011**, 98. [Google Scholar] [CrossRef] - Vansteenkiste, A.; Leliaert, J.; Dvornik, M.; Helsen, M.; Garcia-Sanchez, F.; Waeyenberge, B.V. The design and verification of MuMax3. AIP Adv.
**2014**, 4. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**(

**a**) SEM image of $2\times 2$ array of non-coupled nano-trees. Note the varying cross section of the branches emanating from the vertical stem; (

**b**) SEM image of $2\times 2$ array of non-coupled nano-cubes.

**Figure 2.**(

**a**) Temperature dependence of the CoFe nanocubes’ magnetic hysteresis loop. Inset shows an SEM micrograph of the corresponding Hall cross and the Hall measurement configuration; (

**b**) ’Area’ of the hysteresis loop (up- minus down-sweep) for the nano-cubes and -trees at selected temperatures; (

**c**) Temperature dependence of the switching field (

**left**) and zero-field stray field magnitude (

**right**) corresponding to coercivity and remanence, respectively, of both nano-cubes and -trees.

**Figure 3.**(

**a**) Temperature and magnetic field protocol for thermal activation of a single switching event within the nano-cubes’ hysteresis loop taken at $\theta ={0}^{\circ}$. A: Saturation at negative fields at $T=25$ K. B: Preparation of the state at ${\mathsf{\mu}}_{0}{H}_{\mathrm{ext}}=14$ mT. C: Two major thermally activated switching events are observed (red arrows) while warming up to $T=40$ K in a fixed field of 14 mT. D: Cool-down and read out of the accessed state by completing the hysteresis loop at $T=25$ K; (

**b**) Time dependence of a repeatedly prepared state being thermally unstable at a temperature of 30.866 K (selected curves for switching event 1); (

**c**) Occupation number $P\left(t\right)$ of the so-prepared state. The dots mark the time after which each single switching event of a total of 50 consecutively identically prepared states occurs. Only events that were observed within 13 min are recorded.

**Figure 4.**Angular dependence of the magnetic hysteresis loops at $T=30$ K shown as ${R}_{H}\equiv {V}_{H}/I$ vs. ${\mathsf{\mu}}_{0}{H}_{\mathrm{ext}}$ for CoFe nano-cubes, obtained for the indicated values of the angle $\theta $ of the external applied magnetic field relative to the surface normal of the Hall sensor. Note the varying scales of stray field and external field axes. Black and red arrows indicate down- and up-sweep curves, respectively, following initial saturation at positive field values. Blue and green arrows indicate characteristic switching events and magnetization configurations described in the text.

**Figure 5.**Angular dependence of the magnetic hysteresis loops shown as the magnetic stray field $\langle {B}_{z}\rangle $ vs. ${\mathsf{\mu}}_{0}{H}_{\mathrm{ext}}$ calculated from micromagnetic simulations for the same angles as in Figure 4. Blue and green arrows indicate characteristic switching events and magnetization configurations described in the text.

**Figure 6.**(

**a**) Simulated hysteresis loop for $\theta =-{85}^{\circ}$; (

**b**–

**e**) Magnetization distribution represented by the y-component of the magnetization (red color: +1, blue: $-1$) at different positions in the hysteresis loop indicated by colored discs.

**Figure 7.**(

**a**,

**c**) polar plots of the angular dependences of the remanent magnetization (stray field ${B}_{r}$) comparing the experimental and simulated results (scaled to the values at $\theta ={0}^{\circ}$) for the nano-cubes and nano-trees, respectively; (

**b**,

**d**) linear plots of the angular dependences of the measured and simulated ’coercive fields’ ${\mathsf{\mu}}_{0}{H}_{c}$ determined by evaluating the ’area’ curves (down-sweep minus up-sweep) and taking half the value of the full-width-at-half-maximum for both nano-cubes and -trees. Lines are guides to the eyes. (At $\theta =-{30}^{\circ}$, this method yields a too small value for the simulated hysteresis, likely due to the configuration being ’trapped’ in a metastable state. Reading just the value in the demagnetized state for the down-sweep curves, circle in (

**c**), results in a better correspondence with the experiment.)

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Al Mamoori, M.K.I.; Keller, L.; Pieper, J.; Barth, S.; Winkler, R.; Plank, H.; Müller, J.; Huth, M. Magnetic Characterization of Direct-Write Free-Form Building Blocks for Artificial Magnetic 3D Lattices. *Materials* **2018**, *11*, 289.
https://doi.org/10.3390/ma11020289

**AMA Style**

Al Mamoori MKI, Keller L, Pieper J, Barth S, Winkler R, Plank H, Müller J, Huth M. Magnetic Characterization of Direct-Write Free-Form Building Blocks for Artificial Magnetic 3D Lattices. *Materials*. 2018; 11(2):289.
https://doi.org/10.3390/ma11020289

**Chicago/Turabian Style**

Al Mamoori, Mohanad K. I., Lukas Keller, Jonathan Pieper, Sven Barth, Robert Winkler, Harald Plank, Jens Müller, and Michael Huth. 2018. "Magnetic Characterization of Direct-Write Free-Form Building Blocks for Artificial Magnetic 3D Lattices" *Materials* 11, no. 2: 289.
https://doi.org/10.3390/ma11020289