# Ferromagnetic Resonance Studies in Magnetic Nanosystems

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Vector Network Analyzer Ferromagnetic Resonance

## 3. Fundamental Theory of Spin Wave Dispersion in Ferromagnetic Structures

## 4. Results and Discussion

#### 4.1. Co/Ag Bilayers and Nanodots

#### 4.2. NdCo/Al/Py Layered Structures

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AGM | Alternating gradient magnetometry |

CPW | Coplanar waveguide |

dc | Direct current |

DE | Damon–Eschbach |

FMR | Ferromagnetic resonance |

IMA | In-plane magnetic anisotropy |

LLG | Landau–Lifshitz–Gilbert |

MFM | Magnetic force microscopy |

MOKE | Magneto-optical Kerr effect |

PMA | Perpendicular magnetic anisotropy |

PSSW | Perpendicular standing spin wave |

Py | Permalloy |

rf | Radio frequency |

SWR | Spin-wave resonance |

VNA | Vector network analyzer |

## References

- Bland, J.; Heinrich, B. (Eds.) Ultrathin Magnetic Structures III: Fundamentals of Nanomagnetism; Springer: Berlin/Heidelberg, Germany, 2005. [Google Scholar]
- Heinrich, B.; Bland, J. (Eds.) Ultrathin Magnetic Structures IV: Applications of Nanomagnetism; Springer: Berlin/Heidelberg, Germany, 2005. [Google Scholar]
- Hirohata, A.; Yamada, K.; Nakatani, Y.; Prejbeanu, I.L.; Diény, B.; Pirro, P.; Hillebrands, B. Review on spintronics: Principles and device applications. J. Magn. Magn. Mater.
**2020**, 509, 166711. [Google Scholar] [CrossRef] - Heinrich, B.; Bland, J. (Eds.) Ultrathin Magnetic Structures II: Measurement Techniques and Novel Magnetic Properties; Springer: Berlin/Heidelberg, Germany, 1994. [Google Scholar]
- Schmool, D.S.; Kachkachi, H. Chapter Four–Single-Particle Phenomena in Magnetic Nanostructures. In Solid State Physics; Academic Press: Waltham, MA, USA, 2015; Volume 66, pp. 301–423. [Google Scholar] [CrossRef]
- Schmool, D.; Kachkachi, H. Chapter One–Collective Effects in Assemblies of Magnetic Nanoparticles. In Solid State Physics; Academic Press: Cambridge, MA, USA, 2016; Volume 67, pp. 1–101. [Google Scholar] [CrossRef]
- Bland, J.; Heinrich, B. (Eds.) Ultrathin Magnetic Structures I: An Introduction to the Electronic, Magnetic and Structural Properties; Springer: Berlin/Heidelberg, Germany, 1994. [Google Scholar]
- Hillebrands, B.; Ounadjela, K. (Eds.) Spin Dynamics in Confined Magnetic Structures I; Springer: Berlin/Heidelberg, Germany, 2002. [Google Scholar] [CrossRef]
- Hillebrands, B.; Ounadjela, K. (Eds.) Spin Dynamics in Confined Magnetic Structures II; Springer: Berlin/Heidelberg, Germany, 2003. [Google Scholar] [CrossRef]
- Hillebrands, B.; Thiaville, A. (Eds.) Spin Dynamics in Confined Magnetic Structures III; Springer: Berlin/Heidelberg, Germany, 2006. [Google Scholar] [CrossRef] [Green Version]
- Kalarickal, S.S.; Krivosik, P.; Wu, M.; Patton, C.E.; Schneider, M.L.; Kabos, P.; Silva, T.J.; Nibarger, J.P. Ferromagnetic resonance linewidth in metallic thin films: Comparison of measurement methods. J. Appl. Phys.
**2006**, 99, 093909. [Google Scholar] [CrossRef] [Green Version] - Maksymov, I.S.; Kostylev, M. Broadband stripline ferromagnetic resonance spectroscopy of ferromagnetic films, multilayers and nanostructures. Phys. E Low-Dimens. Syst. Nanostruct.
**2015**, 69, 253–293. [Google Scholar] [CrossRef] [Green Version] - Cansever, H.; Lindner, J. Microresonators and Microantennas—Tools to Explore Magnetization Dynamics in Single Nanostructures. Magnetochemistry
**2021**, 7, 28. [Google Scholar] [CrossRef] - Ding, Y.; Klemmer, T.J.; Crawford, T.M. A coplanar waveguide permeameter for studying high-frequency properties of soft magnetic materials. J. Appl. Phys.
**2004**, 96, 2969–2972. [Google Scholar] [CrossRef] - Markó, D.; Schmool, D.S. Université Paris-Saclay, UVSQ, CNRS, GEMaC, Versailles, France. Unpublished work. 2019. [Google Scholar]
- Landau, L.D.; Lifshitz, E. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjet.
**1935**, 8, 153. [Google Scholar] - Gilbert, T. A Lagrangian formulation of the gyromagnetic equation of the magnetic field. Phys. Rev.
**1955**, 100, 1243. [Google Scholar] - Smit, J.; Beljers, H.G. Ferromagnetic Resonance Absorption in BaFe
_{12}O_{19}, a Highly Anisotropic Crystal. Philips Res. Rep.**1955**, 10, 113–130. [Google Scholar] - Vonsovskii, S.V. (Ed.) Ferromagnetic Resonance: The Phenomenon of Resonant Absorption of a High-Frequency Magnetic Field in Ferromagnetic Substances. Pergamon. 1966. Available online: https://www.sciencedirect.com/book/9780080110271/ferromagnetic-resonance (accessed on 19 August 2021).
- Rado, G.; Weertman, J. Spin-wave resonance in a ferromagnetic metal. J. Phys. Chem. Solids
**1959**, 11, 315–333. [Google Scholar] [CrossRef] - Maksymowicz, A. Spin-wave spectra of insulating films: Comparison of exact calculations and a single-wave-vector model. Phys. Rev. B
**1986**, 33, 6045–6053. [Google Scholar] [CrossRef] [PubMed] - Schmool, D.S.; Barandiarán, J.M. Ferromagnetic resonance and spin wave resonance in multiphase materials: Theoretical considerations. J. Phys. Condens. Matter
**1998**, 10, 10679–10700. [Google Scholar] [CrossRef] - Puszkarski, H. Theory of surface states in spin wave resonance. Prog. Surf. Sci.
**1979**, 9, 191–247. [Google Scholar] [CrossRef] - Herring, C.; Kittel, C. On the Theory of Spin Waves in Ferromagnetic Media. Phys. Rev.
**1951**, 81, 869–880. [Google Scholar] [CrossRef] - Kalinikos, B.A.; Slavin, A.N. Theory of dipole-exchange spin wave spectrum for ferromagnetic films with mixed exchange boundary conditions. J. Phys. C Solid State Phys.
**1986**, 19, 7013–7033. [Google Scholar] [CrossRef] - Farle, M. Ferromagnetic resonance of ultrathin metallic layers. Rep. Prog. Phys.
**1998**, 61, 755–826. [Google Scholar] [CrossRef] - Li, X.; Alkadour, B.; Chuang, W.C.; Marko, D.; Schmool, D.; Wu, J.C.; Manna, P.K.; Lin, K.W.; van Lierop, J. Temperature evolution of the magnetic properties of Ag/Fe nanodot arrays. Appl. Surf. Sci.
**2020**, 513, 145578. [Google Scholar] [CrossRef] - Eyrich, C. Exchange Stiffness in Thin-Film Cobalt Alloys. Master’s Thesis, Simon Fraser University, Burnaby, BC, Canada, 2012. [Google Scholar]
- Mergel, D.; Heitmann, H.; Hansen, P. Pseudocrystalline model of the magnetic anisotropy in amorphous rare-earth–transition-metal thin films. Phys. Rev. B
**1993**, 47, 882–891. [Google Scholar] [CrossRef] [PubMed] - Cid, R.; Alameda, J.M.; Valvidares, S.M.; Cezar, J.C.; Bencok, P.; Brookes, N.B.; Díaz, J. Perpendicular magnetic anisotropy in amorphous Nd
_{x}Co_{1}-x thin films studied by x-ray magnetic circular dichroism. Phys. Rev. B**2017**, 95, 224402. [Google Scholar] [CrossRef] [Green Version] - Markó, D.; Valdés-Bango, F.; Quirós, C.; Hierro-Rodríguez, A.; Vélez, M.; Martín, J.I.; Alameda, J.M.; Schmool, D.S.; Álvarez-Prado, L.M. Tunable ferromagnetic resonance in coupled trilayers with crossed in-plane and perpendicular magnetic anisotropies. Appl. Phys. Lett.
**2019**, 115, 082401. [Google Scholar] [CrossRef] - Ebels, U.; Buda, L.; Ounadjela, K.; Wigen, P.E. Ferromagnetic resonance excitation of two-dimensional wall structures in magnetic stripe domains. Phys. Rev. B
**2001**, 63, 174437. [Google Scholar] [CrossRef] - Cao, D.; Song, C.; Feng, H.; Song, Y.; Zhong, L.; Pan, L.; Zhao, C.; Li, Q.; Xu, J.; Li, S.; et al. Microwave excitations and magnetization dynamics of stripe domain films. arXiv
**2019**, arXiv:1903.00656. [Google Scholar] - Cao, D.; Pan, L.; Song, Y.; Cheng, X.; Feng, H.; Zhao, C.; Li, Q.; Xu, J.; Li, S.; Liu, Q.; et al. Influence of the Phase Structure on the Acoustic and Optical Mode Ferromagnetic Resonance of FeNi Stripe Domain Films. In Proceedings of the 2018 IEEE International Magnetics Conference (INTERMAG), Singapore, 23–27 April 2018. [Google Scholar] [CrossRef]
- Shull, R.; Kabanov, Y.; Gornakov, V.; Chen, P.; Nikitenko, V. Shape critical properties of patterned Permalloy thin films. J. Magn. Magn. Mater.
**2016**, 400, 191–199. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Components of a typical VNA-FMR spectrometer: A two-port VNA is connected to a CPW using coaxial cables and end launch connectors. The sample lies in the x–y plane face-down on top of the CPW with the x-axis along the CPW. An electromagnet (not shown) generates a static bias field ${\mathbf{H}}_{\mathrm{dc}}$, whereas the microwave current flowing through the CPW generates a weak oscillating field ${\mathbf{h}}_{\mathrm{rf}}$ along the y-axis. The four S-parameters that can be detected with such a two-port setup are schematically shown as well.

**Figure 2.**(

**a**) In-plane VNA-FMR data for a 50 nm thick Py film [15] showing the frequency–field characteristics. The color variation shows the absorption intensity. (

**b**) Extracted experimental data (points) from (

**a**) and corresponding fit (line).

**Figure 3.**Planar view SEM images of a Co/Ag nanodot array at (

**a**) 150×, (

**b**) 50,000×, and (

**c**) 200,000× magnification.

**Figure 4.**VNA-FMR spectra for the Co(50 nm)/Ag(30 nm) samples. Raw data for (

**a**) the bilayer sample and (

**b**) the nanostructured dots, extracted data and fits for (

**c**) the bilayer sample and (

**d**) the nanostructured dots. The solid lines are the fits obtained from Equation (23), using the wave vector profiles given in Equation (21).

**Figure 5.**In-plane frequency–field characteristics for VNA-FMR measurements of samples with composition $x=7.5$. The f–H characteristics are shown for the hard (

**a**) and easy (

**b**) axis of the sample. The corresponding insets illustrate the field configurations with respect to the in-plane hard and easy axis along with the virgin-state stripe domain pattern, after initial saturation at ${\mathbf{H}}_{\mathrm{dc}}=+0.9$ T along the in-plane hard or easy axis. (

**c**) Raw VNA-FMR data for the easy axis of sample X7.5T5 for the up-swept field, showing the absorption line discontinuity passing through zero-field.

**Figure 6.**Schematic illustration of the stripe domain pattern for the coupled Py layer illustrating the precessional configurations for the acoustic and optical modes. Black arrows indicate the direction of the x-component of the in-plane magnetization ${m}_{\mathrm{x}}$ in the coupled Py layer inside the stripe domains shown in orange and white color.

**Figure 7.**Schematic illustration of the stripe domain pattern of the coupled Py layer for increasing magnetic field, starting at negative saturation, $-{H}_{\mathrm{sat}}$, and increasing in the positive direction up to $+{H}_{\mathrm{c}}<H<+{H}_{\mathrm{sat}}$. Black arrows indicate the direction of the x-component of the in-plane magnetization ${m}_{\mathrm{x}}$ in the coupled Py layer, whereas their length is proportional to the magnitude of ${m}_{\mathrm{x}}$ inside the stripe domains. The change of the width of the stripe domains in the direction perpendicular to ${\mathbf{H}}_{\mathrm{dc}}$ is, however, largely exaggerated and in reality only very small.

**Figure 8.**(

**a**) Hard axis data of the X7.5 series. Red arrows indicate the field sweep directions with respect to the FMR line hysteresis around zero-field. The suggested modes corresponding to the different field regions of the f–H characteristic are also shown. (

**b**) Easy axis frequency–field characteristic of the X5T5 sample for the up-sweep field direction. Critical and coercive fields are also indicated.

**Figure 9.**Schematic illustrations of the in-plane magnetization and in-plane FMR hysteresis loops. In the lower panel, for the M–H loop, we indicate the stripe domain patterns, which schematically show the relative sizes of the oppositely aligned magnetic stripe domain structure. The reversible portions (black), as well as the up-sweep (red) and down-sweep (blue) branches of the magnetization reversal loop, are also shown. The corresponding color scheme is also used in the upper panel for the FMR hysteresis. Indicatively shown are the expected acoustic and optical modes.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Schmool, D.S.; Markó, D.; Lin, K.-W.; Hierro-Rodríguez, A.; Quirós, C.; Díaz, J.; Álvarez-Prado, L.M.; Wu, J.-C.
Ferromagnetic Resonance Studies in Magnetic Nanosystems. *Magnetochemistry* **2021**, *7*, 126.
https://doi.org/10.3390/magnetochemistry7090126

**AMA Style**

Schmool DS, Markó D, Lin K-W, Hierro-Rodríguez A, Quirós C, Díaz J, Álvarez-Prado LM, Wu J-C.
Ferromagnetic Resonance Studies in Magnetic Nanosystems. *Magnetochemistry*. 2021; 7(9):126.
https://doi.org/10.3390/magnetochemistry7090126

**Chicago/Turabian Style**

Schmool, David S., Daniel Markó, Ko-Wei Lin, Aurelio Hierro-Rodríguez, Carlos Quirós, Javier Díaz, Luis Manuel Álvarez-Prado, and Jong-Ching Wu.
2021. "Ferromagnetic Resonance Studies in Magnetic Nanosystems" *Magnetochemistry* 7, no. 9: 126.
https://doi.org/10.3390/magnetochemistry7090126