Top

Hint

Swipe to navigate through the chapters of this book

Published in:

Read chapter Read first chapter

2022 | OriginalPaper | Chapter

3. Experimental and Computational Methods

Author : Dr. Jannis Lehmann

Published in: Toroidal Order in Magnetic Metamaterials

Publisher: Springer International Publishing

Abstract

This chapter provides an overview of techniques that I have used for this work. After introducing the fabrication of artificial crystals, the major part discusses magnetic force microscopy as one of the two main techniques that have been applied for studying and manipulating the magnetic state of nanomagnetic arrays. The chapter continues by providing background for the second main part—optical methods, mainly the magneto-optical Kerr effect, that are able to detect changes in the magnetisation, but eventually of the toroidisation as well, exploiting more sophisticated experimental settings. Since the as-grown magnetic state of the fabricated crystals cannot simply be changed by annealing procedures, a concept for a non-thermal relaxation of the samples is presented to disprove a possible pinning of the magnetic configuration. A section of this chapter discusses a statistical analysis scheme to relate the local energy landscape to the formation of particular micromagnetic states. Moreover, a self-written two-dimensional micromagnetic calculation script is presented that gives access to magnetic fields that cause correlation between the building blocks in magneto-toroidal arrays. Last, the MFM-revealed microscopic magnetisation pattern do not show the corresponding toroidal order directly. Therefore, the chapter closes with ideas for uncovering contrast between different toroidal domain states using suitable image post-processing.

To get access to this content you need the following product:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt 90 Tage mit der neuen Mini-Lizenz testen!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt 90 Tage mit der neuen Mini-Lizenz testen!

inform now

previous chapter Scientific Background

next chapter Tailoring of the Sample System

This might directly originate from ferro- or ferrimagnetism, or, for instance, from non-compensated magnetic moments at domain walls in antiferromagnets or from induced spin separations and accumulations.

Note that, due to the large refractive index of conductors, light refracts into a metal with almost normal incidence. The consequence is a generally smaller signal for LMOKE than for PMOKE.

The domain states do not necessarily have to form a strictly periodic alternating pattern, but should have a characteristic length scale of the order of the light wavelength.

It can be shown that two-dimensional magnetic structures exhibit a much weaker correlation as compared with three-dimensional ones. The integration of the dipole-interaction potential (\(\propto r^{-3}\)) over a three-dimensional volume of interacting magnetic moments yields \(\;\lim _{r' \rightarrow \infty }\int _0^{r'} r^{-3} \cdot r^2 dr = \ln (r') + \text {const.}\;\) and, thus, diverges for \(r' \rightarrow \infty \). In the case of a two-dimensional system, the area-integral yields \(\;\lim _{r' \rightarrow \infty }\int _0^{r'} r^{-3} \cdot r dr = -1/r' + \text {const.,}\) which gives finite value for \(r' \rightarrow \infty \).

Note that, for a quench from temperatures that correspond to a thermal energy of a magnitude that is comparable to the magnetic coupling energy, entropy provides a non-negligible contribution to the formation of short-range order. This in turn is the reason for the thermal population of vertex states of different energy.

Lehmann J et al (2019) Microdisplays as a versatile tool for the optical simulation of crystal diffraction in the classroom. Journal of Applied Crystallography 52(2):457–462 CrossRef

J. I. Martin et al. Ordered magnetic nanostructures: fabrication and properties. Journal of Magnetism and Magnetic Materials (2003), p. 53

Fitzsimmons MR, Silva TJ, Crawford TM (2006) Surface oxidation of Permalloy thin films. Physical Review B 73(1):014420. https://doi.org/10.1103/PhysRevB.73.014420 ADSCrossRef

H. Hopster et al., eds. Magnetic Microscopy of Nanostructures. Springer Berlin Heidelberg, 2005. http://dx.doi.org/10.1007/b137837

U. Hartmann. Magnetic Force Microscopy (1999), p. 36

E. Meyer, H. J. Hug, and R. Bennewitz. Scanning Probe Microscopy. Springer Berlin Heidelberg, 2004. https://doi.org/10.1007/978-3-662-09801-1

Zhu X et al (2005) Diffracted magneto-optical Kerr effects of permalloy ring arrays. Journal of Applied Physics 97(10):10J712 CrossRef

B. Bhushan and H. Fuchs. Applied scanning probe methods. Springer, 2009

Eaton P, West P (2010). Atomic Force Microscopy Oxford University Press. https://doi.org/10.1093/acprof:oso/9780199570454.001.0001 CrossRef

10.

O. Kazakova et al. Frontiers of magnetic force microscopy. Journal of Applied Physics 125 6 (2019), p. 060901. http://aip.scitation.org/doi/10.1063/1.5050712

11.

Y. Martin and H. K. Wickramasinghe. Magnetic imaging by “force microscopy” with 1000 Å resolution. Applied Physics Letters 50 20 (1987), pp. 1455-1457. http://aip.scitation.org/doi/10.1063/1.97800

12.

Saenz JJ et al (1987) Observation of magnetic forces by the atomic force microscope. Journal of Applied Physics 62(10):4293–4295 ADSCrossRef

13.

R. A. Buckingham. The classical equation of state of gaseous helium, neon and argon. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 168 933 (1938), pp. 264-283. http://dx.doi.org/10.1098/rspa.1938.0173

14.

Garcia R (2002) Dynamic atomic force microscopy methods. Surface Science Reports 47(6):197–301. https://doi.org/10.1016/S0167-5729(02)00077-8 ADSCrossRef

15.

Whangbo M-H, Bar G, Brandsch R (1998) Description of phase imaging in tapping mode atomic force microscopy by harmonic approximation. Surface Science 411(1):794–801. https://doi.org/10.1016/S0039-6028(98)00349-5 CrossRef

16.

Hartmann U (1989) The point dipole approximation in magnetic force microscopy. Physics Letters A 137(9):475–478. https://doi.org/10.1016/0375-9601(89)90229-6 ADSCrossRef

17.

U. Hartmann. Fundamentals and Special Applications of Non-Contact Scanning Force Microscopy. In: Advances in Electronics and Electron Physics. Vol. 87. Elsevier, 1993, pp. 49-200

18.

https://www.ntmdt-si.com/products/modular-afm/ntegra-ii. [Online content, accessed on 27.12.2019]

19.

Kerr J (1877) On rotation of the plane of polarization by reflection from the pole of a magnet. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 3(19):321–343. https://doi.org/10.1080/14786447708639245 CrossRef

20.

P. Weinberger. John Kerr and his effects found in 1877 and 1878. Philosophical Magazine Letters 88 12 (2008), pp. 897-907. https://www.tandfonline.com/doi/abs/10.1080/09500830802526604

21.

J. Zak et al. Fundamental magneto-optics. Journal of Applied Physics 68 8 (1990), pp. 4203-4207. http://aip.scitation.org/doi/10.1063/1.346209

22.

A. K. Zvezdin and V. A. Kotov. Modern magnetooptics and magnetooptical materials. Institute of Physics Pub, 1997

23.

S. Sugano and N. Kojima. Magneto-optics. Springer, 2000

24.

H. Kronmueller and S. Parkin, eds. Handbook of Magnetism and Advanced Magnetic Materials. 1st ed. John Wiley & Sons, Ltd, 2007. http://dx.doi.org/10.1002/9780470022184

25.

A. Hubert and R. Schaefer. Magnetic domains: the analysis of magnetic microstructures. Springer, 2009

26.

McCord J (2015) Progress in magnetic domain observation by advanced magneto-optical microscopy. Journal of Physics D: Applied Physics 48(33):333001. https://doi.org/10.1088/0022-3727/48/33/333001 CrossRef

27.

J. A. Arregi, P. Riego, and A. Berger. What is the longitudinal magneto-optical Kerr effect? Journal of Physics D: Applied Physics 50 3 (2017), 03LT01. http://dx.doi.org/10.1088/1361-6463/aa4ea6

28.

Hulme HR (1932) The Faraday Effect in Ferromagnetics. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 135(826):237–257. https://doi.org/10.1098/rspa.1932.0032 ADSCrossRefMATH

29.

Argyres PN (1955) Theory of the Faraday and Kerr Effects in Ferromagnetics. Physical Review 97(2):334–345. https://doi.org/10.1103/PhysRev.97.334 ADSCrossRef

30.

Freiser M (1968) A survey of magnetooptic effects. IEEE Transactions on Magnetics 4(2):152–161. https://doi.org/10.1109/TMAG.1968.1066210 ADSCrossRef

31.

Ebert H (1996) Magneto-optical effects in transition metal systems. Reports on Progress in Physics 59(12):1665–1735 ADSCrossRef

32.

Traeger G, Wenzel L, Hubert A (1992) Computer experiments on the information depth and the figure of merit in magnetooptics. Physica Status Solidi (a) 131(1):201–227. https://doi.org/10.1002/pssa.2211310131 ADSCrossRef

33.

Bader SD (1991) Smoke. Journal of Magnetism and Magnetic Materials 100(1):440–454 ADSCrossRef

34.

Z. Q. Qiu and S. D. Bader. Surface magneto-optic Kerr effect (2000), p. 14

35.

Krinchik GS, Artemjev VA (1968) Magneto-optic Properties of Nickel Iron and Cobalt. Journal of Applied Physics 39(2):1276–1278 ADSCrossRef

36.

Buschow K, van Engen P, Jongebreur R (1983) Magneto-optical properties of metallic ferromagnetic materials. Journal of Magnetism and Magnetic Materials 38(1):1–22. https://doi.org/10.1016/0304-8853(83)90097-5 ADSCrossRef

37.

G. Di and S. Uchiyama. Temperature dependence of Kerr rotation for Ni film. Journal of Applied Physics 75 8 (1994), pp. 4270-4272. http://aip.scitation.org/doi/10.1063/1.355968

38.

P. Oppeneer. Chapter 3 Magneto-optical kerr spectra. Handbook of Magnetic Materials. Vol. 13. Elsevier, 2001, pp. 229-422

39.

H. Ebert and G. Schuetz, eds. Spin-orbit-influenced spectroscopies of magnetic solids. 466. Springer, 1996

40.

J. F. Dillon and J. P. Remeika. Diffraction of Light by Domain Structure in Ferromagnetic CrBr \(_3\). Journal of Applied Physics 34 3 (1963), pp. 637-640. http://aip.scitation.org/doi/10.1063/1.1729321

41.

Boersch H, Lambeck M (1964) Zur Beugung des Lichtes an Magnetisierungsstrukturen. Zeitschrift fuer Physik 177(2):157–163. https://doi.org/10.1007/BF01375333 ADSCrossRef

42.

Telesnin RV et al (1972) Diffraction of light in magnetic stripe-structure. Physica Status Solidi (a) 12(1):303–306 ADSCrossRef

43.

B. Kuhlow and M. Lambeck. Light Diffraction By Magnetic Domains. Physica B+C 80 1 (1975), pp. 374-380. http://dx.doi.org/10.1016/0378-4363(75)90079-0

44.

B. Kuhlow. Light Diffraction By Magnetic Domain Gratings And Its Applications. Journal of Magnetism and Magnetic Materials (1980), pp. 391-394

45.

Geoffroy O et al (1993) TMOKE hysteresis loops in Bragg diffraction from 2D patterns. Journal of Magnetism and Magnetic Materials 121(1):516–519. https://doi.org/10.1016/0304-8853(93)91258-9 ADSCrossRef

46.

Souche Y, Schlenker M, Dos Santos A (1995) Non-specular magneto-optical Kerr effect. Journal of Magnetism and Magnetic Materials 140–144:2179–2180. https://doi.org/10.1016/0304-8853(94)00847-7 ADSCrossRef

47.

van Labeke D et al (1996) Diffraction of light by a corrugated magnetic grating: experimental results and calculation using a perturbation approximation to the Rayleigh method. Optics Communications 124(5):519–528. https://doi.org/10.1016/0030-4018(95)00598-6 ADSCrossRef

48.

Suzuki Y et al (1997) Simple model for the magneto-optical Kerr diffraction of a reg ular array of magnetic dots. Journal of Magnetism and Magnetic Materials 165(1):516–519. https://doi.org/10.1016/S0304-8853(96)00605-1 ADSCrossRef

49.

Vial A, Van Labeke D (1998) Diffraction hysteresis loop modelisation in transverse magneto-optical Kerr effect. Optics Communications 153(1):125–133. https://doi.org/10.1016/S0030-4018(98)00188-6 ADSCrossRef

50.

Vavassori P et al (1999) Magnetic information in the light diffracted by a negative dot array of Fe. Physical Review B 59(9):6337–6343. https://doi.org/10.1103/PhysRevB.59.6337 ADSCrossRef

51.

Grimsditch M et al (2001) Brillouin scattering and diffracted magneto-optical Kerr effect from arrays of dots and antidots (invited). Journal of Applied Physics 89(11):7096–7100 ADSCrossRef

52.

T. Schmitte et al. Magneto-optical Kerr effects of ferromagnetic Ni-gratings. Journal of Applied Physics 87 9 (2000), pp. 5630-5632. http://aip.scitation.org/doi/10.1063/1.1359789

53.

Schmitte T, Westerholt K, Zabel H (2002) Magneto-optical Kerr effect in the diffracted light of Fe gratings. Journal of Applied Physics 92(8):4524–4530. https://doi.org/10.1016/j.spmi.2004.01.004 ADSCrossRef

54.

Schmitte T et al (2003) The Bragg-MOKE: magnetic domains in Fourier space. Superlattices and Microstructures 34(1):127–136. https://doi.org/10.1016/j.spmi.2004.01.004 ADSCrossRef

55.

Vavassori P et al (2004) Magnetization reversal via single and double vortex states in submicron Permalloy ellipses. Physical Review B 69(21):214404. https://doi.org/10.1103/PhysRevB.69.214404 ADSCrossRef

56.

Westphalen A et al (2006) Magnetization reversal of micropattern Fe bar array: Combination of vector and Bragg magneto-optical Kerr effect measurements. Journal of Magnetism and Magnetic Materials 302(1):181–189. https://doi.org/10.1016/j.jmmm.2005.09.005 ADSCrossRef

57.

Vavassori P et al (2007) Chirality and stability of vortex state in Permalloy triangular ring micromagnets. Journal of Applied Physics 101(2):023902. https://doi.org/10.1103/PhysRevB.69.214404 ADSCrossRef

58.

T. Verduci et al. Fourier magnetic imaging. Applied Physics Letters 99 9 (2011), p. 092501. https://doi.org/10.1063/1.3630049

59.

Grimsditch M, Vavassori P (2004) The diffracted magneto-optic Kerr effect: what does it tell you? Journal of Physics: Condensed Matter 16(9):275–294. https://doi.org/10.1088/0953-8984/16/9/R01 ADSCrossRef

60.

Westphalen A et al (2007) Invited article: Vector and Bragg Magneto-optical Kerr effect for the analysis of nanostructured magnetic arrays. Review of Scientific Instruments 78(12):121301 ADSCrossRef

61.

Lee M-S et al (2008) Extended longitudinal vector and Bragg magneto-optic Kerr effect for the determination of the chirality distribution in magnetic vortices. Journal of Applied Physics 103(9):093913. https://doi.org/10.1063/1.2919160 ADSCrossRef

62.

B. Aktas, L. Tagirov, and F. Mikailov, eds. Magnetic nanostructures. 94. Springer, 2007

63.

Caloz C et al (2018) Electromagnetic Nonreciprocity. Physical Review Applied 10(4):047001. https://doi.org/10.1103/PhysRevApplied.10.047001 ADSCrossRef

64.

Baranova NB, Bogdanov YV, Zel’dovich BY (1977) New electro-optical and magneto-optical effects in liquids. Soviet Physics Uspekhi 20(10):870–877. https://doi.org/10.1070/PU1977v020n10ABEH005470 ADSCrossRef

65.

Ross HJ, Sherborne BS, Stedman GE (1989) Selection rules for optical activity and linear birefringence bilinear in electric and magnetic fields. Journal of Physics B: Atomic, Molecular and Optical Physics 22(3):459–473. https://doi.org/10.1088/0953-4075/22/3/011 ADSCrossRef

66.

Rikken GLJA, Strohm C, Wyder P (2002) Observation of Magnetoelectric Directional Anisotropy. Physical Review Letters 89(13):133005. https://doi.org/10.1103/PhysRevLett.89.133005

67.

T. Arima. Magneto-electric optics in non-centrosymmetric ferromagnets. Journal of Physics: Condensed Matter 20 43 (2008), p. 434211. https://doi.org/10.1088/0953-8984/20/43/434211

68.

Szaller D, Bordacs S, Kezsmarki I (2013) Symmetry conditions for nonreciprocal light propagation in magnetic crystals. Physical Review B 87(1):014421. https://doi.org/10.1103/PhysRevB.87.014421 ADSCrossRef

69.

Tomita S et al (2018) Metamaterials with magnetism and chirality. Journal of Physics D: Applied Physics 51(8):083001. https://doi.org/10.1088/1361-6463/aa9ecb ADSCrossRef

70.

Cheong S-W et al (2018) Broken symmetries, non-reciprocity, and multiferroicity. Npj Quantum Materials 3(1):19. https://doi.org/10.1038/nmat1804 ADSCrossRef

71.

Nisoli C (2012) On thermalization of magnetic nano-arrays at fabrication. New Journal of Physics 14(3):035017. https://doi.org/10.1103/RevModPhys.85.1473 ADSMathSciNetCrossRef

72.

X. Zhang et al. Understanding thermal annealing of artificial spin ice. APL Materials 7 11 (2019), p. 111112. https://doi.org/10.1063/1.5126713

73.

Ke X et al (2008) Energy Minimization and ac Demagnetization in a Nanomagnet Array. Physical Review Letters 101(3):037205. https://doi.org/10.1103/PhysRevLett.101.037205

74.

R. F. Wang et al. Demagnetization protocols for frustrated interacting nanomagnet arrays. Journal of Applied Physics 101 9 (2007), 09J104. https://doi.org/10.1063/1.2712528

75.

Rougemaille N, Canals B (2019) Cooperative magnetic phenomena in artificial spin systems: spin liquids, Coulomb phase and fragmentation of magnetism - a colloquium. The European Physical Journal B 92(3):62. https://doi.org/10.1140/epjb/e2018-90346-7 ADSCrossRef

76.

Morgan JP et al (2011) Thermal ground-state ordering and elementary excitations in artificial magnetic square ice. Nature Physics 7(1):75–79 ADSCrossRef

77.

Tannous C, Gieraltowski J (2008) The Stoner-Wohlfarth model of ferromagnetism. European Journal of Physics 29(3):475–487. https://doi.org/10.1088/0143-0807/29/3/008 ADSCrossRef

Title: Experimental and Computational Methods
Author: Dr. Jannis Lehmann
Publisher: Springer International Publishing
Book: Toroidal Order in Magnetic Metamaterials
Print ISBN: 978-3-030-85494-2

Electronic ISBN: 978-3-030-85495-9

Copyright Year: 2022
DOI: https://doi.org/10.1007/978-3-030-85495-9_3