Magnonic Devices: Numerical Modelling and Micromagnetic Simulation Approach 303122664X, 9783031226649

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SpringerBriefs in Materials C. S. Nikhil Kumar

Magnonic Devices Numerical Modelling and Micromagnetic Simulation Approach

SpringerBriefs in Materials Series Editors Sujata K. Bhatia, University of Delaware, Newark, DE, USA Alain Diebold, Schenectady, NY, USA Juejun Hu, Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA Kannan M. Krishnan, University of Washington, Seattle, WA, USA Dario Narducci, Department of Materials Science, University of Milano Bicocca, Milano, Italy Suprakas Sinha Ray , Centre for Nanostructures Materials, Council for Scientific and Industrial Research, Brummeria, Pretoria, South Africa Gerhard Wilde, Altenberge, Nordrhein-Westfalen, Germany

The SpringerBriefs Series in Materials presents highly relevant, concise monographs on a wide range of topics covering fundamental advances and new applications in the field. Areas of interest include topical information on innovative, structural and functional materials and composites as well as fundamental principles, physical properties, materials theory and design. Indexed in Scopus (2022). SpringerBriefs present succinct summaries of cutting-edge research and practical applications across a wide spectrum of fields. Featuring compact volumes of 50 to 125 pages, the series covers a range of content from professional to academic. Typical topics might include • A timely report of state-of-the art analytical techniques • A bridge between new research results, as published in journal articles, and a contextual literature review • A snapshot of a hot or emerging topic • An in-depth case study or clinical example • A presentation of core concepts that students must understand in order to make independent contributions Briefs are characterized by fast, global electronic dissemination, standard publishing contracts, standardized manuscript preparation and formatting guidelines, and expedited production schedules.

C. S. Nikhil Kumar

Magnonic Devices Numerical Modelling and Micromagnetic Simulation Approach

C. S. Nikhil Kumar Adam Mickiewicz University Pozna´n, Poland

ISSN 2192-1091 ISSN 2192-1105 (electronic) SpringerBriefs in Materials ISBN 978-3-031-22664-9 ISBN 978-3-031-22665-6 (eBook) https://doi.org/10.1007/978-3-031-22665-6 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to my parents, Naina, Supervisors Prof. Anil Prabhakar and Prof. Ashwin A. Tulapurkar

Preface

Magnonics—the study of spin waves—has seen a remarkable scope for the creation of magnetic field-controlled devices with properties tailored on the nanoscale. It aims to control and manipulate spin waves in ferromagnetic material. Spin wave can effectively carry and process information in magnetic nanostructures. The most controllability of their functioning is by an external magnetic field. The magnonic devices which are periodically modulated in space are seen as very promising devices because of the possibility of tunning their band structure. Periodically modulated magnonic devices can also be called as “magnonic crystal” magnetic counter part of photonic crystal, with spin wave acting as information carrier. Magnonic crystals are better candidates for miniaturization, since the wavelength of spin wave is several orders of magnitude shorter than that of electromagnetic waves of the same frequency. An example of one-dimensional magnonic crystal is a multilayered magnetic structure consisting of alternated ferromagnetic layers. The basic advantage of a such device is that frequency position and width of the band gap are tunable by an applied field and through material properties. The dispersion properties can be tuned by changing the dimensions of magnonic waveguide. Spectrum of magnonic crystal shows band gaps in which spin wave cannot propagate. Such band gaps have been experimentally studied using Brillouin light scattering spectroscopy [1, 2] and time-resolved scanning Kerr microscopy [3]. The study of magnonic crystal is first proposed in [4]. Puszkarski and Krawczyk have theoretically calculated the magnonic band structure of 2D MCs consisting of infinitely long cylindrical Fe rods periodically embedded in a yttrium iron garnet (YIG) background [5]. The materials Fe and YIG were selected for their large difference in magnetic properties, as it has been predicted that the larger the difference, the wider would be the band gap width. They investigated the position and width of band gaps in the spin wave spectrum versus period of the structure and magnetic properties. A popular method to investigate the band structure of periodic structure is plane wave expansion method. Plane wave method has been successfully implemented in both photonic [6], phononic [7] and magnonic crystals [5, 4, 8]. We follow the main idea behind the plane wave method to investigate the band structure of periodically modulated magnonic waveguide. The equation of motion for a periodically vii

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magnetic structure is transformed into reciprocal space using Fourier transformation and Bloch’s theorem. The equation of motion is then reduced to an eigenvalue problem. The numerical solutions were found out for such an eigenvalue problem by standard numerical routines. The solution yield eigenfrequencies that help to find the band structure and eigenvector yields spin wave profile inside the periodic material. One of the main objectives of this thesis is to follow the plane wave method in order to investigate the spin wave spectrum of the striped magnonic crystal in backward volume configuration. The result is published in [9]. We initially investigated the magnetostatic field inside the geometry using the idea presented in [10]. The obtained magnetostatic field is used in the governing Landau-Lifshitz equation. This helps us to reduce the equation in to an eigenvalue problem. This eigenvalue problem is solved numerically in order to investigate the spin wave spectrum. The eigenvector concept is used to investigate the spin wave profile inside the structure. This book describes the dispersion relation of dipolar spin wave in a magnonic curved waveguide. Walker’s equation in cylindrical coordinates is solved with appropriate boundary conditions. The dispersion of exchange spin waves is then calculated using perturbation theory. We validated our results by investigating the dispersion relation for a higher bending radius and compared it to that of a straight waveguide for higher bending radius. By introducing symmetry about the azimuthal direction, we also obtain analytical solutions for Walker’s equation and the mode profile characteristics of dipolar spin waves in a magnonic ring. Perturbation theory is used to investigate the mode profile characteristics of magnonic ring. Finally, analytical dispersion relation of curved waveguide is compared with micromagnetic simulation. We could see a reasonable agreement of fundamental mode between simulation and analytical results and attribute the differences to the large exchange in the simulation and the weak exchange in perturbation theory. This book also describes nanocontact-driven spin wave excitations in magnonic cavity. A spin-polarized electric current injected into Permalloy (Py) through a nanocontact exerts a torque on the magnetization, leading to spin wave (SW) excitation. We considered an array of nanocontacts on a Py film for an enhanced SW excitation. We designed an antidot magnonic crystal (MC) around the nanocontact to form a cavity. The MC was designed so that the frequency of the SW mode generated by the nanocontact lies in the band gap of the MC. The nanocontacts were placed in a line defect created in the MC by removing a row of antidots. The SW time series and power spectrum were observed at the output of the cavity. We observe that the SWs decay in the absence of the MC cavity, and when the nanocontacts are within the antidot MC cavity, the SW amplitude is amplified and stable. This is also reflected in the SW spectrum obtained at the output port. Finally, Q factor of the device is calculated using decay method and observed a high Q factor Q = 3.8 × 105 for a current of 7.8 mA. The proposed device behaves as a SWASER (spin wave amplification by the stimulated emission of radiation). Pozna´n, Poland

C. S. Nikhil Kumar

Preface

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References 1.

Tacchi S, Madami M, Gubbiotti G, Carlotti G, Tanigawa H, Ono T, Kostylev MP (2010) Anisotropic dynamical coupling for propagating collective modes in a two-dimensional magnonic crystal consisting of interacting squared nanodots. Phys Rev B 82:024401. 2. Tacchi S, Madami M, Gubbiotti G, Carlotti G, Goolaup S, Adeyeye AO, Singh N, Kostylev M.P (2010) Analysis of collective spin-wave modes at different points within the hysteresis loop of a one-dimensional magnonic crystal comprising alternative-width nanostripes. Phys Rev B 82:184408. 3. Kruglyak VV, Keatley PS, Neudert A, Hicken RJ, Childress JR, Katine JA (2010) Imaging collective magnonic modes in 2d arrays of magnetic nanoelements. Phys Rev Lett 104:027201. 4. Vasseur JO, Dobrzynski L, Djafari-Rouhani B, Puszkarski H (1996) Magnon band structure of periodic composites. Phys Rev B 54:1043. 5. Puszkarski H, Krawczyk M. (2003) Magnonic crystals-the magnetic counter part of magnonic crystal. Solid State Phenomena 94:125. 6. Joannopoulos JD, Johnson SG, Winn JN, Meade RD Photonic Crystals: Molding the Flow of Light (Second Edition). Princeton University Press, 2 ed., Feb. 7. Benchabane S, Khelif A, Rauch J-Y, Robert L, Laude V (2006) Evidence for complete surface wave band gap in a piezoelectric phononic crystal. Phys Rev E 73:065601. 8. Krawczyk M, Puszkarski H (2008) Plane-wave theory of three-dimensional magnonic crystals. Phys Rev B 77:054437. 9. Kumar N, Prabhakar A. (2013) Spin wave dispersion in striped magnonic waveguide. IEEE Transactions Magnetics 49(3):1024–1028. 10. Kaczer J, Murtinova L. On the demagnetizing energy of periodic magnetic distributions. Phys Status Solidi (a) 23(1).

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Backward Volume Spin Waves in a Rectangular Geometry . . . . . . . . . . 7 2.1 Solution to Walker’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Dispersion Characteristics for Dipolar Spin Waves . . . . . . . . . 10 2.1.2 Dispersion Characteristics of Exchange Spin Waves . . . . . . . . 10 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Magnetostatic Waves in Magnonic Crystals: A PWM Approach . . . . . 3.1 Magnetostatic Waves in 1D Magnonic Crystal . . . . . . . . . . . . . . . . . . . 3.2 Walker’s Equation in a One-Dimensional Magnonic Crystal . . . . . . . 3.3 Dispersion Characteristics of Backward Volume Spin Waves . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 17 17 20 22

4 Field Localization in Striped Magnonic Crystal Waveguide . . . . . . . . . . 4.1 Geometry and Method of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Static Demagnetizing Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Spin Wave Magnonic Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Spin Wave Localizations in Striped Magnonic Waveguide . . . . . . . . . 4.5 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 24 26 28 32 33 34 34

5 Walker’s Solution for Curved Magnonic Waveguide and Resonant Modes in Magnonic Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Spin Wave Dispersion for Curved Magnonic Waveguide . . . . . . . . . . 5.1.1 Geometry and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Dispersion Relation of Curved Magnonic Waveguide . . . . . . . 5.1.3 Validations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Modes in a Magnonic Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 38 42 44 46

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5.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6 Nanocontact-Driven Spin Wave Excitations in Magnonic Cavity . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Micromagnetic Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Method of Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Band Structure of Antidot MC . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Spin Wave Injection on Py Film Using an Array of Nanocontacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Fabry-Perot Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 SW Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Quality Factor Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 52 53 54

7 Magnetic Field Feedback Oscillator: A Micromagnetic Study . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Feedback Oscillator with Magnetic Field Feedback: A Closed Loop System Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Micromagnetic Simulation Without Magnetic Field Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Free Layer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Free Layer Hysteresis Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Ferromagnetic Resonance Frequency Versus Applied Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.5 Current Dependence on Resonance Frequency . . . . . . . . . . . . 7.2.6 Spintronic Oscillators with Magnetic Field Feedback . . . . . . . 7.2.7 Spin Wave Dynamics with Magnetic Field Feedback . . . . . . . 7.2.8 Spin Wave Dynamics at 300 K . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.9 Spin Wave Spectra with Different Delays . . . . . . . . . . . . . . . . . 7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63

54 56 57 58 61 61

63 64 64 65 66 71 71 72 73 75 75 76

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Chapter 1

Introduction

The magnetic field phenomena on different materials are known for many years. Based on the response of the material to the applied field, we can classify the major type of magnetism: (1) diamagnetism, (2) paramagnetism and (3) ferromagnetism. Diamagnetism is the property of the material that causes it to create a magnetic field in opposition to an externally applied magnetic field. Its magnetic permeability is less than the permeability of free space. Diamagnetism substances are composed of atoms that have no net magnetic moments. Paramagnetic is the material that contains permanent magnetic moments but long-range order. In the absence of a magnetic field, the moments are randomly oriented so that no net magnetic moment is exhibited. The application of an external field causes a partial alignment of the moments. This will cause net magnetic moments. The third type of magnetic material is ferromagnets. In these materials, there are domains in which the magnetic fields of the individual atom align. In equilibrium, these domains orient themselves so as to minimize the net magnetic moment. When an external field is applied, the magnetic fields of the individual domains tend to align in the direction of this external field. Thus, all the magnetic moments will align in the direction of the field in order to minimize the energy of the ferromagnetic material. The deflection of the magnetic moment of a single atom will tend to change the directions of the magnetic moment of the neighboring atom. This is mainly caused due to exchange interaction between neighboring atoms. This change of direction of magnetic moment in neighboring atoms causes a wave-like motion. The changes in magnetic moment thus excite the changes in the magnetic moment of neighboring atoms. This collective excitation of magnetic moment is called spin waves. This was described by Bloch [1]. Considering a discrete model of ferromagnets, Holstein and Primakoff [2] and Dyson [3] introduced quanta of spin waves called magnons [4]. They predicted that magnons should behave as weakly interacting quasi particles obeying Bose-Einstein statics.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. S. Nikhil Kumar, Magnonic Devices, SpringerBriefs in Materials, https://doi.org/10.1007/978-3-031-22665-6_1

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1 Introduction

Spin wave can be regarded as a magnetic analogue of a light wave. Spin wave exhibits wave-like behavior such as reflection and refraction [5–8] and interference [9]. These properties are well exploited in the field of magnonics. Magnonics—the study of the spin wave has seen a remarkable scope for the creation of magnetic fieldcontrolled devices with properties tailored on the nanoscale [10–12]. It aims to control and manipulate spin waves in ferromagnetic material. Spin waves can effectively carry and process information in magnetic nanostructures. The most controllability of their functioning is by an external magnetic field. Magnonic devices are easily controlled by the applied magnetic field. The magnonic devices which are periodically modulated in space are seen as very promising devices because of the possibility of tunning their band structure. Periodically modulated magnonic devices can also be called as “magnonic crystal” magnetic counterpart of photonic crystal, with spin wave acting as information carrier. Magnonic crystals are better candidates for miniaturization since the wavelength of the spin wave is several orders of magnitude shorter than that of electromagnetic waves of the same frequency. Based on the periodicity in space, the magnonic crystal can be divided into one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D). An example of one-dimensional magnonic crystal is a multilayered magnetic structure consisting of alternated ferromagnetic layers. The basic advantage of a such device is that the frequency position and width of the band gap are tunable by an applied field and through material properties. The dispersion properties can be tuned by changing the dimensions of the magnonic waveguide. A theoretical investigation of the magnonic band structure of 1D MC is studied in [13]. Magnonic crystal formed by periodically varying the width of the magnetic stripe is studied in [14]. The concept of 2D magnonic devices were implemented in [15]. The position and width of the band structure were investigated in these references as a function of filling fractions of the material. The band structure analysis of 3D magnonic crystals is investigated in [16]. Thin ferromagnetic rings are of interest for both fundamental studies of magnetization reversal and also for their potential in various applications. Thin film magnetic rings have generated a great deal of interest in recent years due to the existence of multiple energy states and their possible application in memory, logic and sensing devices [17–20]. Useful states such as “vortex” and “onion” states have been identified in ferromagnetic rings [21–23]. The ferromagnetic ring also helps to understand the effects of interference of spin wave [24]. The dispersion characteristics of the curved magnonic waveguide are not exploited well. The dispersion characteristics really help to understand the supporting frequency for each wave vector. This can be obtained by solving Walker’s equation in cylindrical coordinates with proper boundary conditions. The same analysis can be extended to investigate the mode profile characteristics of the magnonic ring structure by applying the symmetry conditions along the azimuthal direction. The plane wave method proved to be a convenient tool to investigate the spin wave spectrum in magnonic crystals. But the main drawback of this method was the assumption of the infinite thickness of the devices. So it is very difficult to model more practical devices. We investigated the magnonic band structure of a striped

1 Introduction

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magnonic waveguide having finite width and thickness using the plane wave expansion method in backward volume configuration. The backward volume configuration exhibits localized mode behavior in magnonic crystals, which promises to build novel nanoscale magnonic devices. In numerical modeling of magnonic devices, we investigate the dispersion characteristics and spin wave dynamics of magnonic straight as well as curved waveguide. The plane wave method proved to be a convenient tool to investigate the spin wave spectrum in magnonic crystals. But the main drawback of this method was the assumption of the infinite thickness of the devices. So it is very difficult to model more practical devices. We investigated the magnonic band structure of a striped magnonic waveguide having finite width and thickness using the plane wave expansion method in backward volume configuration. We follow the main idea behind the plane wave method to investigate the band structure of periodically modulated magnonic waveguides. The equation of motion for a periodical magnetic structure is transformed into reciprocal space using Fourier transformation and Bloch’s theorem. The equation of motion is then reduced to an eigenvalue problem. The numerical solutions were found out for such an eigenvalue problem by standard numerical routines. The solution yields eigenfrequencies that help to find the band structure and eigenvector yields a spin wave profile inside the periodic material. We also investigated the dispersion relation of dipolar spin waves in a magnonic curved waveguide by solving Walker’s equation in cylindrical coordinates with appropriate boundary conditions. The appropriate Maxwell’s equations are expressed in terms of magnetostatic potential (ψ) which satisfies Walker’s equation inside the material and the Laplace equation outside the material. The dispersion of exchange spin waves is then calculated using perturbation theory. We validated our results by investigating the dispersion relation for a higher bending radius and compared it to that of a straight waveguide for a higher bending radius. The solutions to Walker’s equation also yield mode profile characteristics in a magnonic ring by applying symmetry conditions along the azimuthal direction, for both dipolar and exchange spin waves. Finally, analytical dispersion relation of the curved waveguide is compared with micromagnetic simulation. We could see the reasonable agreement of fundamental mode between simulation and analytical results. The book is organized as follows. The dispersion characteristics of the rectangular thin film are derived in Chap. 3. The derived dispersion characteristics are compared with that of the infinite thin film. The exchange as well as dipolar coupled dispersion characteristics are explained. Dispersion characteristics of the infinite sheet of ferro-non ferromagnetic material forming superlattice is discussed in Chap. 4. The plane wave method (PWM) is discussed to derive the dispersion characteristics. In Chap. 5, using PWM we have numerically calculated the magnon band structure in a Co-Py striped thin film waveguide, biased in a backward volume configuration. The linearized Landau-Lifshitz equation was reduced to a matrix form, where the eigenfrequencies distribution yielded the dispersion relation ω(k). The eigenmodes of the matrix yield the magnetization amplitude in the waveguide. The dispersion diagrams show evidence of band gaps, where the waveguide does not support spin wave propagation. Instead, small signal excitations within a band gap result in standing spin

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wave patterns, with zero group velocity, and the waveguide acts as a microwave filter. The frequency and size of a gap can be controlled by varying the relative size of the Co and Py stripes. We have also observed that the dispersion diagrams are influenced by the angle between the wavevector and the external applied field. Thus, magnonic crystals offer a rich platform to control and explore spin wave excitations. In Chap. 6, we derived an analytical dispersion relation for a curved magnonic waveguide. The dispersion relation for dipolar spin waves is obtained by solving Walker’s equation in cylindrical coordinates. The appropriate Maxwell’s equations are expressed in terms of a magnetostatic potential (ψ) which satisfies Walker’s equation inside the material and Laplace’s equation outside the material. The equations are solved analytically with proper boundary conditions for the dipolar case. This is then extended to exchange spin waves, using perturbation theory, for different bending radii. As expected, for a large bending radius, the dispersion relation approaches that of a straight waveguide. By introducing symmetry about the azimuthal direction, we also obtain analytical solutions for Walker’s equation and the mode profile characteristics of dipolar spin waves in a magnonic ring. And then extended using perturbation theory to arrive at the modes in exchange spin waves. Finally, we compared the dispersion characteristics of curved waveguide (fundamental mode) with micromagnetic simulation. In Chap. 7, we have designed an antidot MC cavity to obtain sustained SW propagation in permalloy. The SWs were generated by an array of nanocontacts by injecting a spin-polarized current in them. The nanocontacts are placed in a line defect in the MC which was created by removing a row of antidots. The SWs obtained at the output of the MC cavity do not show signs of decay over a large time scale. This is attributed to the multiple SW reflections and amplification which occur in the MC cavity. The Q factor of the device is calculated for different currents and observed a high Q factor for a particular current. Our observations support the feasibility of a “SWASER” (spin wave amplification by stimulated emission of radiation) using nanocontacts confined within a magnonic crystal cavity. In Chap. A, we simulated the spin torque nano-oscillator with magnetic field feedback. The static and dynamic characterization were analyzed in detail. With magnetic field feedback, we observed an improvement in the linewidth of the spectrum. The effect of thermal noise can be nullified using magnetic field feedback.

References 1. F. Bloch, Theory of ferromagnetism. Z. Phys. 61, 206–219 (1930) 2. T. Holstein, H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet. Phys. Rev. 58, 1098 (1940) 3. F.J. Dyson, General theory of spin-wave interactions. Phys. Rev. 102, 1217–1230 (1956) 4. C. Kittel, Introduction to Solid State Physics, 6th edn. (Wiley, New York, 1986) 5. Yu.I. Gorobets, S.A. Reshetnyak, Reflection and refraction of spin waves in uniaxial magnets in the geometrical-optics approximation. Tech. Phys. 43 (1998) 6. S.A. Reshetnyak, Refraction of surface spin waves in spatially inhomogeneous ferrodielectrics with biaxial magnetic anisotropy. Phys. Solid State 46 (2004)

References

5

7. A.V. Vashkovsky, E.H. Lock, Properties of backward electromagnetic waves and negative reflection in ferrite films. Phys.-Uspekhi 49(4), 389 8. S.-K. Kim, S. Choi, K.-S. Lee, D.-S. Han, D.-E. Jung, Y.-S. Choi, Negative refraction of dipoleexchange spin waves through a magnetic twin interface in restricted geometry. Appl. Phys. Lett. 92(21), 212501 (2008) 9. S. Choi, K.-S. Lee, S.-K. Kim, Spin-wave interference. Appl. Phys. Lett. 89(6), 062501 (2006) 10. V.V. Kruglyak, S.O. Demokritov, D. Grundler, Magnonics. J. Phys. D: Appl. Phys. 43(26), 264001 11. S. Neusser, D. Grundler, Magnonics: spin waves on the nanoscale. Adv. Mater. 21(28), (2008) 12. Magnonics: experiment to prove the concept. J. Magn. Magn. Matl. 306(2), 191 (2006) 13. M. Krawczyk, Magnetostatic waves in one-dimensional magnonic crystals with magnetic and nonmagnetic components. IEEE Trans. Magn. 44(11), 2854–2857 (2008) 14. S.-K. Kim, K.-S. Lee, D.-S. Han, A gigahertz-range spin-wave filter composed of widthmodulated nanostrip magnonic-crystal waveguides. Appl. Phys. Lett. 95(8), 082507 (2009) 15. M. Krawczyk, H. Puszkarski, Magnonic crystal theory of the spin-wave frequency gap in low-doped manganites. J. Appl. Phys. 100(7), 073905 (2006) 16. M. Krawczyk, H. Puszkarski, Plane-wave theory of three-dimensional magnonic crystals. Phys. Rev. B 77, 054437 (2008) 17. F. Montoncello, L. Giovannini, F. Nizzoli, Spin mode calculations in nanometric magnetic rings: localization effects in the vortex and saturated states. J. Appl. Phys. 103(8), 083910 (2008) 18. F. Montoncello, L. Giovannini, F. Nizzoli, H. Tanigawa, T. Ono, G. Gubbiotti, M. Madami, S. Tacchi, G. Carlotti, Magnetization reversal and soft modes in Nanorings: transitions between onion and vortex states studied by Brillouin light scattering. Phys. Rev. B 78, 104421 (2008) 19. F. Giesen, J. Podbielski, D. Grundler, Mode localization transition in ferromagnetic microscopic rings. Phys. Rev. B 76, 014431 (2007) 20. X. Zhu, M. Malac, Z. Liu, H. Qian, V. Metlushko, M.R. Freeman, Broadband spin dynamics of permalloy rings in the circulation state. Appl. Phys. Lett. 86(26), 262502 (2005) 21. M. Steiner, J. Nitta, Control of magnetization states in microstructured permalloy rings. Appl. Phys. Lett. 84(6), 939–941 (2004) 22. P. Vavassori, O. Donzelli, M. Grimsditch, V. Metlushko, B. Ilic, Chirality and stability of vortex state in permalloy triangular ring micromagnets. J. Appl. Phys. 101(2), 023902 (2007) 23. M. Klaui, J. Rothman, L. Lopez-Diaz, C.A.F. Vaz, J.A.C. Bland, Z. Cui, Vortex circulation control in mesoscopic ring magnets. Appl. Phys. Lett. 78(21), 3268–3270 (2001) 24. J. Podbielski, F. Giesen, D. Grundler, Spin-wave interference in microscopic rings. Phys. Rev. Lett. 96, 167207 (2006)

Chapter 2

Backward Volume Spin Waves in a Rectangular Geometry

The dispersion relation, ω(k), gives us an insight into the characteristics of propagating spin waves and helps us build functional magnonic devices. The dispersion relation for an infinite thin film was studied in [1]. The discussed structure was infinite along the length and width of the waveguide with finite thickness and approximated as a parallel plate waveguide. We modify the geometry to that of a rectangular thin film waveguide, finite in thickness and width, but infinite in length. We use Maxwell’s equation and the LL equation of motion for the magnetization, consistent with the proper boundary condition to find the dispersion relation in a rectangular geometry. Magnetostatic waves, whose wave number lies between those of the electromagnetic waves and exchange spin waves, propagate in a magnetized ferromagnetic medium due to dipole interactions. The magnetostatic wave in a rectangular thin film can be classified into two groups according to the direction of the applied field. We call the magnetostatic waves as backward volume waves when the direction of propagation is along the applied field. We consider the propagation of guided waves along the +z-direction in the geometry of Fig. 2.1, with k parallel to HDC .

2.1 Solution to Walker’s Equation Let ψi and ψe be the potential inside and outside the sample. The potential inside the sample must satisfy  (1 + χ)

∂ 2 ψi ∂ 2 ψi + 2 ∂x ∂ y2

 +

∂ 2 ψi = 0. ∂z 2

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. S. Nikhil Kumar, Magnonic Devices, SpringerBriefs in Materials, https://doi.org/10.1007/978-3-031-22665-6_2

(2.1)

7

8

2 Backward Volume Spin Waves in a Rectangular Geometry

Fig. 2.1 Semi-infinite thin film structure. The width (w) and thickness (d) of the material are 100 nm and 10 nm, respectively

Outside the sample, we must satisfy Laplace’s equation ∂ 2 ψe ∂ 2 ψe ∂ 2 ψe + + = 0. ∂x 2 ∂ y2 ∂z 2

(2.2)

The continuity of h requires ψi |x=±d/2 = ψe |x=±d/2 .

(2.3)

The boundary condition on the normal component B requires that Bxi |x=±d/2 = Bxe |x=±d/2 .

(2.4)

We know that h = −∇ψ. Hence, the normal component of B can be written as Bx = μ0 (1 + χ)

∂ψ ∂ψ − iμ0 κ , ∂x ∂y

(2.5)

and the boundary condition becomes  μ0 (1 + χ)

∂ψi ∂ψi − iμ0 κ ∂x ∂y

 |x=±d/2 =

∂ψe . ∂x

(2.6)

In rectangular coordinates, the potential is separable and may be written as ψ(X, Y Z ) = X (x) Y (y) Z (z).

(2.7)

Let Yi and Z i be the solution inside the sample and Ye and Z e be the solution outside the sample. The boundary condition (2.3) says that Yi = Ye = Y and Z i = Z e = Z . Thus, the potential is rewritten as ψi = X i Y Z

(2.8)

ψe = X e Y Z X e = c exp(−k xe x) x > d/2 X e = e exp(+k xe x) x > d/2

,

(2.9)

2.1 Solution to Walker’s Equation

9

where k xe is a positive, real number. Inside the sample X i = [a sin (k xi x) + b cos (k xi x)] .

(2.10)

k xi may be either real or imaginary. Y has the form Y = cos (k y y).

(2.11)

Z inside the sample represents a traveling wave. So the Z takes the form Z = exp (−ik z z).

(2.12)

Outside the sample, the Z dependence can be represented as an exponential decaying function. Substituting these expressions in (2.1) and (2.2), we get the relations 2 + k 2y ) + k z2 = 0, (1 + χ)(k xi

(2.13) 2 − k 2y + k z2 = 0, k xe

k xi , k xe , k y and k z are unknown variables. Applying the boundary condition (2.3), c exp(−k xe d/2) = a sin (k xi d/2) + b cos (k xi d/2), (2.14) e exp(−k xe d/2) = −a sin (k xi d/2) + b cos (k xi d/2). Adding and subtracting the two equations in (2.14), we get a=

(c−e) exp(−k xe d/2) 2 sin (k xi d/2)

b=

(c+e) exp(−k xe d/2) . 2 sin (k xi d/2)

(2.15)

Applying the boundary condition (2.6), we get the following relations:     μ0 (1 + χ) k xi a cos(k xi d/2) − b k xi sin (k xi d/2) + iμ0 κ k y sin (k y y) = c k xe exp (−k xe d/2),     μ0 (1 + χ) k xi a cos(k xi d/2) + b k xi sin (k xi d/2) + iμ0 κ k y sin (k y y) = e k xe exp (−k xe d/2).

(2.16) Substitute the values of a and b from (2.15) and subtract (2.16) from (2.14), to eliminate the constants, and get a relation:  k xi tan

k xi s 2

 = k xe .

(2.17)

10

2 Backward Volume Spin Waves in a Rectangular Geometry

Eliminating k y from (2.13) gives  kz = k xi

  −(1 + χ) k xi d . sec 2 (2 + χ)

(2.18)

If we set k xi = k xe , we get a relation k xi d 2

= (4n + 1) π4 n = 0, 1, 2, ......

(2.19)

For the rectangular thin film, the width is finite. This introduces wave number quantization along y. Then the wave vector k for semi-infinite thin film is replaced by

nπ 2 2 k z + w . The expression for an infinite thin film is [1]  kz d = −(1 + χ). tan √ 2 −(1 + χ) 

(2.20)

The final expression for a rectangular thin film is   kx d . k 2 + χ = k x −(1 + χ) sec 2

(2.21)

2.1.1 Dispersion Characteristics for Dipolar Spin Waves As expected for BVSWs, the dispersion characteristics have a negative slope as shown in Fig. 2.2. This says that the group and phase velocity point in opposite directions. Figure 2.2 also compares the dispersion characteristics of a rectangular thin film with that of an infinite thin film.

2.1.2 Dispersion Characteristics of Exchange Spin Waves The inclusion of exchange increases the complexity of the problem, and a closedform analytic solution eludes us. Hence, we resort to using perturbation theory. The expression for χ can be written as x=

ω0 ωm . ω02 − ω 2

(2.22)

In the presence of exchange, the term ω0 is replaced with [1] ω0 + λex ωm k 2 .

(2.23)

Reference

11

Fig. 2.2 Dispersion characteristics of rectangular and infinite thin film

Fig. 2.3 Dispersion characteristics of semi and infinite thin film structures for low and high vectors with exchange included

We investigated the dispersion behavior of exchange spin waves for both rectangular as well as infinite thin film structures. It should be noted that with the inclusion of exchange, the dispersion characteristics show backward volume behavior only for a small value of k. For λex ωm k 2 > ω0 , the exchange dominates the dipolar term. This is clearly seen in Fig. 2.3.

Reference 1. D.D. Stancil, A. Prabhakar, Spin Waves Theory and Applications, 1st edn. (Springer, New York, 2008)

Chapter 3

Magnetostatic Waves in Magnonic Crystals: A PWM Approach

The band structure of excitations in periodically modulated materials, including electronic, photonic, phononic and magnonic crystals, can be calculated by the plane wave method. In the classical approach, the dynamics of the magnetization vector M (r, t) with negligible damping is described by the Landau-Lifshitz equation   ∂M α ∂M = −γμ0 [M (r, t) × Heff (r, t)] + , M× ∂t Ms ∂t

(3.1)

where γ is the gyromagnetic ratio (γ > 0), μ0 is the permeability of vacuum and Heff (r, t) denotes the effective magnetic field acting on the magnetic moments. The effective magnetic field Heff (r, t) is in general the sum of several components. On the assumption that the magnetocrystalline anisotropy is negligible,   Heff (r, t) = H0 zˆ + h (r, t) + ∇.λ2ex ∇ M (r, t) ,

(3.2)

 A with λex = μ02M 2 . H0 is the applied dc field, h (r, t) be the dipolar field and the last s term represents the exchange field. The total magnetization vector is given by M (r, t) = M S zˆ + m (r, t) ,

(3.3)

where M S is the saturation magnetization along z-axis and m (r, t) being the dynamic component which lies in (x, y)-plane. In a linear approximation, we can assume Mz = Ms . Since only monochromatic spin waves are considered, the dynamic component of M (r, t) can be written as m (r, t) = m (r) eiωt , h (r, t) = h (r) eiωt ,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. S. Nikhil Kumar, Magnonic Devices, SpringerBriefs in Materials, https://doi.org/10.1007/978-3-031-22665-6_3

(3.4)

13

14

3 Magnetostatic Waves in Magnonic Crystals: A PWM Approach

where ω is the spin wave frequency. The dynamical dipolar field satisfies ∇ × h = 0,

∇. (h + m) = 0.

(3.5)

The magnetostatic potential is related by h = −∇ .

(3.6)

We must solve (3.4) and (3.6) with proper boundary conditions.1 By inserting (3.2), (3.3), (3.4) and (3.6) in (3.1), we get the following relation: im x (r) +

1 H0

  Ms ∇.λ2ex ∇ m y (r) − m y (r ) −

1 m H0 y

  (r ) × ∇.λ2ex ∇ Ms +

Ms ∂ψ H0 ∂ y

+iαm y (r) = 0, (3.7) im y (r) −

1 H0

  Ms ∇.λ2ex ∇ m x (r) − m x (r ) +

1 m H0 x

  (r ) × ∇.λ2ex ∇ Ms −

Ms ∂ψ H0 ∂x

−iαm x (r) = 0, (3.8) where  is the reduced frequency defined by =

ω . |γ|μ0 H0

(3.9)

Similarly substituting (3.6) in (3.5) and expanding, we get  ∂m x (r) ∂m y (r) + . ∇ ψ(r) − ∂x ∂y 

2

(3.10)

The material parameters Ms and λ2ex are periodic functions of position vector r with period equal to the lattice vector a. This can be written as Ms (r + a) = Ms (r) , λ2ex (r + a) = λ2ex (r) . Considering the periodicity of our structure in the (x, y)-plane, λ2ex (r)and Ms (r) can be expanded in Fourier series: Ms (r) =



Ms (G) eiG.r ,

(3.11)

G

λ2ex (r) =



λ2ex (G) eiG.r ,

(3.12)

G

1

The detailed analysis of the dipolar fields for a striped magnonic waveguide is described in Chap. 5.

3 Magnetostatic Waves in Magnonic Crystals: A PWM Approach

α (r) =



15

α (G) eiG.r ,

(3.13)

G

  where G = G x , G y is a reciprocal lattice vector and r is the position vector. We use the plane wave method to solve the system of Eqs. (3.7), (3.8) and (3.10). The method is based on Bloch’s theorem. According to Bloch’s theorem, the solution of a differential equation with periodic coefficients can be represented as a product of a plane wave envelope function and a periodic Bloch function. The small signal magnetization and magnetostatic potential can be represented in plane wave form as m (r) =



mk eik.r , ψ (r) =



k

ke

ik.r

k

.

(3.14)

By considering only one wave vector and applying Bloch’s theorem, we get mk (r) eik.r =

mk (G) ei(k+G).r ,

(3.15)

G

k

(r) eik.r =

ψk (G) ei(k+G).r ,

(3.16)

G

where mk (r + a) = mk (r) , ψk (r + a) = ψk (r) .

(3.17)

k = [K x , K y ] being a two-dimensional vector in the first Brillouin zone of the reciprocal lattice. Including (3.16) and (3.17) in (3.10), we get ψk (G) = −i

  (k x + G x ) mxk (G) + k y + G y myk (G) (k + G)2

,

(3.18)

where k x , k y and G x , G y denote the x- and y-components of the wave vector k and reciprocal lattice vector G, respectively. In the Cartesian system of coordinates, mxk (G) and myk (G) are the Fourier coefficients in the expansion (3.18) of the dynamic magnetization components. After some straightforward algebra, (3.7) and (3.8) reduce to     A G, G mk , (3.19) M˜ G, G m = i k

where

   A G, G =

G

   δGG,   α G − G , δGG, −α G − G

(3.20)

  where δGG, is the Kronecker delta and the component of M˜ G, G is given by

16

3 Magnetostatic Waves in Magnonic Crystals: A PWM Approach 



  (k y +G y )(k x +G x ) ˜ M(G, G )x x = Ms (G − G ) H (k+G  2 ) 0

 ˜ M(G, G )x y =δGG, +

1

H0  G



     (k + G ) · (k + G ) − (G − G ) · (G − G ) ×





(k y +G y )2 H0 (k+G )2



Ms (G − G )λ2ex (G − G) +  ˜ M(G, G ) yx = −δGG, −

1

H0  G 





Ms (G − G )

     (k + G ) · (k + G ) − (G − G ) · (G − G ) ×



(k y +G y )2 H0 (k+G )2



Ms (G − G )λ2ex (G − G) −



Ms (G − G ) 



  (k y +G y )(k x +G x ) ˜ M(G, G ) yy = −Ms (G − G ) 2 H0 (k+G )

(3.21)

with the eigenvector   mT k = mx,k (G1 ) , ......, mx,k (G N ) , m y,k (G1 ) , ......, m y,k (G N ) .

(3.22)

The solution of (3.19) yields both eigenfrequencies and eigenvectors mk , the latter being the Fourier coefficient of the dynamic magnetization components. The eigenvalue gives us spin wave frequency for each wave vector. The analysis of spin wave spectra is explained in Sect. 3.3. The Fourier coefficient Ms (G) for one-dimensional magnonic waveguide can be calculated by using the formula Ms (G) =

1 V0

Ms (r) exp (−iG.r) dr

(3.23)

V0

where V0 is the area of the unit cell and Ms (r) = M S A S (r) + M SB [1 − S (r)] .

S (r) =

Ms (G) =

⎧ ⎪ ⎨1 ⎪ ⎩

|r| ≤ra (3.25)

0

 1  M S A − M SB V0

|r| ≥ra

S (r) ex p (−iG.r) dr. V0

This gives us

(3.24)

(3.26)

3.2 Walker’s Equation in a One-Dimensional Magnonic Crystal

Ms (G) =

Similarly λ2ex (G) =

⎧ ⎪ ⎨ ⎪ ⎩ ⎧ ⎪ ⎨ ⎪ ⎩

S D 2 GD

 M S A + M SB 1 − 



for G = 0 (3.27)

   M S A − M SB sin Gs for G = 0. 2

S 2 λ D ex A 2 GD

S D

17

 + λ2ex B 1 −

S D



for G = 0

 2    λex A − λ2ex B sin Gs for G = 0 2

,

(3.28)

where S/D is the filling fraction. In the following chapter, we will use the plane wave method to analyze the frequency characteristics of different magnonic structures.

3.1 Magnetostatic Waves in 1D Magnonic Crystal A photonic crystal is a periodic composite of materials with different dielectric constants. Magnonic crystals (MCs) were first proposed as the magnetic counterpart of photonic crystals, with spin waves acting as the information carrier [1]. The spectrum of spin waves in these composite materials has been investigated in experimental studies [2, 3]. Like photonic crystals, MCs are expected to possess interesting properties arising from their frequency band gaps. The plane wave method can be used to find a solution to Walker’s equation for the magnetostatic potential in a MC [4]. We apply the PWM to the case of 1D MCs composed of alternating layers of a ferromagnet and nonmagnetic materials. The resulting band structure includes bands of backward magnetostatic waves. The wave spectrum contains forbidden zones determined by the parameters of both the structure and external magnetic field. We consider a superlattice consisting of alternating sublayers of a magnetic and nonmagnetic material and analyze the variation in the band gap as we vary the relative thickness of the adjacent thin films.

3.2 Walker’s Equation in a One-Dimensional Magnonic Crystal In inhomogeneous magnetic materials, the magnetostatic potential and dynamic field are related by [4] h = −∇ ψ,

(3.29)

18

3 Magnetostatic Waves in Magnonic Crystals: A PWM Approach

Fig. 3.1 Infinite sheets of ferromagnetic-nonmagnetic material forming a periodic superlattice. Applied field H0 and static magnetization vectors M1 is parallel to the z-axis. The value of M2 is being zero as it is air

ψ should satisfy the Maxwell equation:   ∇.b =0 =⇒ ∇ · μ¯¯ · ∇ψ = 0,

(3.30)

μ¯¯ being the permeability tensor: ⎛

1+

0  M 20 −ω 2

M −i ω 0 2 −ω 2

⎞

⎟ ⎟ 0⎟, ⎠ 1

(3.31)

 M = −γμ0 M0 , 0 = −γμ0 H0 .

(3.32)

⎜ ⎜ M μˆ = ⎜ i ω 2 2 0 −ω ⎝ 0

0

1+

0  M 20 −ω 2

0

and We have assumed an external magnetic field, H0 applied along the z-axis. M0 , γ, μ0 and ω are the static magnetization, gyromagnetic ratio, permeability and frequency of the spin wave, respectively. Let us consider a superlattice consisting of alternating sublayers of different magnetic materials having magnetization vectors M1 and M2 , in the respective sublayers of thickness d A and d B , respectively. The system under discussion is shown in Fig. 3.1. Since the material is infinite along the applied magnetic field, the effect of the static demagnetization or that of any stray field will be neglected. We solve Walker’s equation by applying Bloch’s theorem. According to Bloch’s theorem ψk (G) ei(k+G).r , (3.33) ψ (r) = ψk eik.r = G

where k is the wave vector and G denotes a reciprocal lattice vector. Since the magnetization is a function of position, we write

3.2 Walker’s Equation in a One-Dimensional Magnonic Crystal



M (r) =

19

M (G) eiG.r .

(3.34)

G

Let us concentrate on a 1D MC, as shown in Fig. 3.1. The above equations reduce to ψ (r) =

ψG ei(kx +G)x eik y y eikz z ,

(3.35)

G



M (x) =

M (G) ei Gx ,

(3.36)

G

where G is a 1D reciprocal lattice vector: G = 2π/d (d = d A + d B ). The gradient of the scalar potential can be written as ∇ψ =

∂ψ ∂ψ ∂ψ aˆx + aˆy + aˆz ∂x ∂y ∂z

(3.37)

and Walker’s equation becomes ⎛

0  M 20 −ω 2

1+

⎜ ⎜ M ∇. ⎜ i ω 2 2 0 −ω ⎝ 0

M −i ω 0 2 −ω 2 0

1+

0  M 20 −ω 2

0

⎞⎛

⎟⎜ ⎟ 0⎟⎜ ⎠⎝ 1

∂ψ aˆ ∂x x ∂ϕ aˆ ∂y y ∂ψ aˆ ∂z z

⎞ ⎟ ⎟ = 0. ⎠

(3.38)

In (3.38),  M will be position dependant. We expand (3.38) and get ∂2ψ ∂ ∂2ψ ∂2ψ 0 + + + 2 2 2 ∂x ∂y ∂z 2 20 − ω ∂x

where

∂2 ψ ∂x 2

      −iω iω ∂ψ ∂ ∂ψ ∂ ∂ψ M + 2 M + 2 M = 0, 2 2 ∂x ∂y ∂x 0 − ω ∂x 0 − ω ∂ y

(3.39)

= − (k x + G)2 ψG ei(kx +G)x eik y y eikz z , G ∂2 ψ ∂ y2

= −k 2y

∂2 ψ ∂z 2

= −k z2



ψG ei(kx +G)x eik y y eikz z ,

G



(3.40)

ψG ei(kx +G)x eik y y eikz z .

G

Similarly, the last three terms in (3.39) can be expanded as 0

∂ 2 2 0 − ω ∂x

 M

∂ψ ∂x



  (k x + G )(k x + G) M (G − G )ψG ei(k x +G)x eik y y eikz z . =− G



G

(3.41)

20

3 Magnetostatic Waves in Magnonic Crystals: A PWM Approach

−iω ∂ 20 − ω 2 ∂x

  ∂ψ  M =− k y (k x + G) M (G − G )ψG  ei(kx +G)x eik y y eikz z . ∂y  G G

(3.42)     iω ∂ ∂ψ i(k +G)x ik y x y k y (k x + G ) M (G − G )ψG  e e eikz z . M =− ∂x 20 − ω 2 ∂ y  G G (3.43)

3.3 Dispersion Characteristics of Backward Volume Spin Waves Consider magnetostatic waves propagating in (x, z)-plane, i.e., k = (k x , 0, k z ); this configuration corresponds to backward volume magnetostatic waves propagating in a magnetic film. Then (3.39) becomes       (20 − ω 2 ) (k x + G)2 + k z2 ψG + 0  M (G − G ) (k x + G )(k x + G) ψG  = 0. G



(3.44) Equation (3.44) can be written in a more general form as 

ω 0



2 ψG = ψG +

G

     M (G − G ) (k x + G )(k x + G) ψG    0 (k x + G)2 + k z2

.

(3.45)

and can be treated as an eigenvalue problem. If we substitute (3.32), (3.34) reduces to 

   (G − G ) (k x + G )(k x + G)  2 ω  G   ψG = ψG + M0 (G − G )ψG  . (3.46) 2 2 0 H0 (k x + G) + k z M0 (G) is then calculated by taking the inverse Fourier transform of M (r). The Fourier coefficient is given by   Gd A dA . (M1 − M2 )sinc M0 (G) = d A + dB d A + dB

(3.47)

Consider a superlattice consisting of alternating sublayers of a magnetic (M1 = 0) and a nonmagnetic material (M0 = 0). The solution of (3.45) for different values of

3.3 Dispersion Characteristics of Backward Volume Spin Waves

21

Fig. 3.2 Magnonic band structure of a infinite sheets of Ferromagnetic-Nonmagnetic material forming a periodic superlattice with d A /d = 0.5 (top left), d A /d = 0.75 (top right), d A /d = 0.9 (bottom left), d A /d = 0.95 (bottom right)

wave vector components k x and k z yields the desired dispersion relation of magnetostatic waves. Equation (3.46) can be formulated as an eigenvalue problem and represented as AX = λX, (3.48) where X is the eigenvector

and λ

⎤ ψ1 ⎢ ψ2 ⎥ ⎥ X =⎢ ⎣ ψ3 ⎦ ψ4 ⎡

(3.49)

 ω ω0

is the corresponding eigenvalue. We can formulate the matrix A from 

(3.48) by varying G and G . Each row and column corresponds to different G,  G terms, respectively. For each value of k z , k x yield a different matrix, and the eigenvalues of each matrix represent ω/ω0 . Finally, we plot the magnonic band structure. Figure 3.2 shows the normalized magnetostatic wave frequencies plotted versus k z for different values of k x . As expected frequency of the bands decreases by increasing the wave vector (k z ) in a backward volume magnonic band.

22

3 Magnetostatic Waves in Magnonic Crystals: A PWM Approach

Fig. 3.3 Magnonic band structure of a one-dimensional magnonic crystal with d A /d = 1

From Fig. 3.2, it is clear that the band gap decreases as we reduce the air gap thickness.2 When d A = d, the structure becomes a single homogeneous material and the band gap reduces to zero. All the ranges√of frequency values are  available to the magnetostatic waves, i.e., for ω ∈ γ0 H0 , γ0 H0 (γ0 H0 + γ0 M0 ) (Fig. 3.3).

References 1. J.O. Vasseur, L. Dobrzynski, B. Djafari-Rouhani, H. Puszkarski, Magnon band structure of periodic composites. Phys. Rev. B 54, 1043 (1996) 2. Y.V. Gulyaev, S. Nikitov, L. Zhivotovskii, A. Klimov, P. Tailhades, L. Presmanes, C. Bonningue, C. Tsai, S. Vysotskii, Y.A. Filimonov, Ferromagnetic films with magnon bandgap periodic structures: magnon crystals. J. Exp. Theor. Phys. Lett. 77(10), 567–570 (2003) 3. Z.K. Wang, V.L. Zhang, H.S. Lim, S.C. Ng, M.H. Kuok, S. Jain, A.O. Adeyeye, Observation of frequency band gaps in a one-dimensional nanostructured magnonic crystal. Appl. Phys. Lett. 94(8), 083112 (2009) 4. M. Krawczyk, Magnetostatic waves in one-dimensional magnonic crystals with magnetic and nonmagnetic components. IEEE Trans. Magn. 44(11), 2854–2857 (2008)

2

The color is used to separate the eigenvalues and has no other significance.

Chapter 4

Field Localization in Striped Magnonic Crystal Waveguide

The wavelengths of magnon in MCs are orders of magnitude shorter than those of photons in photonic crystals, at the same frequency. This allows the fabrication of nanoscale magnonic devices. A magnonic crystal waveguide, the most attractive magnonic device, is used to guide information in the form of spin waves. The spectrum of magnonic crystal waveguide exhibits band gaps in which spin waves do not propagate and have potential applications such as in microwave resonators, filters and switches [1]. The band gaps of such devices can be observed experimentally by Brillouin light scattering [2]. One advantage of MC waveguide is that the frequency and width of the band gap become tunable by an external applied field, or by tailoring of material properties. Sokolovsky and Krawczyk have theoretically studied a MC consisting of a periodic array of alternating cobalt (Co) and permalloy (Py) nanostripes using the plane wave expansion method and validated their predictions against experimental results [3]. Their investigations focused on a surface spin wave configuration. We investigate a similar periodically modulated waveguide, with alternate strips of Co and Py, in a backward volume configuration [4]. The structure is finite in width and thickness, infinite in length, with spin wave propagation along the direction of the applied magnetic field, in the plane of the waveguide. The spin wave dispersion relation for a MC waveguide in the backward volume configuration exhibits frequency bands that support spin wave propagation. The dispersion properties can be tuned by changing the dimensions of magnonic waveguide. In addition, there is the possibility of altering the band structure by changing the angle between the wave vector and the external bias field. We discuss the field localization at the end of this chapter.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. S. Nikhil Kumar, Magnonic Devices, SpringerBriefs in Materials, https://doi.org/10.1007/978-3-031-22665-6_4

23

24

4 Field Localization in Striped Magnonic Crystal Waveguide

4.1 Geometry and Method of Analysis The magnonic waveguide under the investigation is shown in Fig. 4.1. A bias magnetic field, H0 , is applied along the length of the waveguide, along yˆ . The parameter values assumed in calculation of spin wave spectra are given in Table 4.1. In [5], the plane wave method was employed to model 3D magnonic crystals. We modified and used this approach to analyze the geometry shown in Fig. 4.1. We initially calculate the static and dynamic components of the demagnetizing field inside the geometry. Let ψi and ψo be the potential inside and outside the waveguide, (y) respectively. Assuming saturation along y, the potential ψi due to M y should satisfy [6] (y) (y) (y) ∂ψi ∂ψi ∂ψi + + = −M y . (4.1) ∂x 2 ∂ y2 ∂z 2 The material is periodic along the y-direction, and we write the static magnetization as  M y (r) = Ms (G) ei Gy , (4.2) G

where G = 2πn/d. (d is the period). The Fourier coefficient Ms (G) is given by

Ms (G) =

⎧ ⎪ ⎨ ⎪ ⎩

s D 2 GD

 M1 + M2 1 − (M1 − M2 ) sin

s D



 Gs  2

for G = 0 (4.3) for G = 0.

Similarly, the potential inside the material has the same periodicity, i.e.,

Fig. 4.1 Magnonic waveguide under investigation. The thickness of the Co and Py nanostripes is 2a = 30 nm. The length of the magnonic waveguide is assumed to be infinite. The width (w), the lattice constant (s) and period (d) are assumed to be 50, 250 and 500 nm, respectively Table 4.1 Parameter values used in calculations Py A (10−11 J/m) M s (106 A/m) A λex (Ao )= μ 2M 2 0 s

1.11 0.658 63.8

Co 2.88 1.15 58.87

4.1 Geometry and Method of Analysis

25

(y)

ψi (r ) =

 ψi G (x) ei Gy eikz z ,

(4.4)

G

assuming pinning at the lateral edges where we introduce a dependence on k z = nπ w of the waveguide [7]. Substituting (4.2) and (4.3) in (4.1) and looking only at one mode, we get

 4π Ms (G) −ikz z , ψi G (x) = Aei G I I x + Bei G I I x + e G 2I I

(4.5)

where G I I = G 2 + k z2 . Pick a trial potential function outside the material which will satisfy Laplace’s equation. Its solution yields k x = ±G I I , such that ψo(y) (r ) =

 C e−G I I x ei Gy .

(4.6)

G

Finally, the boundary conditions (y)

ψi

= ψo(y) |x=±a

(y)

∂ψi ∂x

(4.7)

(y)

=

∂ψo ∂x

|x=±a

yield the constants, to give us the magnetostatic potential

Fig. 4.2 Variation in the magnetostatic potential across the waveguide thickness

(4.8)

26

4 Field Localization in Striped Magnonic Crystal Waveguide

ψi (r ) =

 Ms (G)  1 − cosh (|G I I |x) e−|G I I |a ei Gy . 2 |G I I | G

(4.9)

The potential profile for the lowest even mode in the Co-Py magnonic stripe waveguide, with a period of 50 nm, is shown in Fig. 4.2.

4.2 Static Demagnetizing Field We now investigate the static demagnetizing field based on the idea presented in [6]. The demagnetizing field is the negative gradient of the magnetic scalar potential H (r ) = −∇ϕ (r ) ,

(4.10)

where the potential ϕ (r ) due to elementary dipoles Mdv  is given by  ϕ (r ) = −

  M r  .∇



 1 dv  . | r − r |

(4.11)

 1 dv  , | r − r |

(4.12)

Combining (4.10) and (4.11), we obtain  H (r ) = ∇

  M r  .∇



which represents the demagnetizing field of a known dipole distribution inside and outside a body. Let ψ α (r ) be the  potential of a body due to the distribution of magnetization component Mα r  , α = x, y, z. For ψ α (r ), we can write   Mα r  dv  . | r − r |



α

ψ (r ) =

(4.13)

Equation (4.11) and (4.13) are related by the set of equalities 

  ∂ Mx r  ∂x

(y)



  1  M y r  ∂∂y |r −r  | dv ,

(z)



  ∂ 1 Mz r  ∂z dv  . |r −r  |

(x)

ϕ(x) = − ∂ψ∂x = − ϕ(y) = − ∂ψ∂ y = − ϕ(z) = − ∂ψ∂z = −

1 dv  , |r −r  |

(4.14)

      We can find the potential ϕ(x) , ϕ(y) , ϕ(z) due to Mx r  , M y r  , Mz r  , respectively. Thus, each of the three components of H (r ), i.e., Hx , Hy , Hz can be resolved into three parts, each of them being caused by one of the respective component Mx ,

4.2 Static Demagnetizing Field

27

Fig. 4.3 Demagnetizing field variation across the waveguide thickness

Fig. 4.4 Spatial variation of demagnetizing field H y

M y , Mz . In general, the α component of the demagnetizing field produced by the magnetization component Mβ is Hα(β) . Hx(x) =

∂ 2 ψ (x) , ∂x 2

Hy(x) =

∂ 2 ψ (x) , ∂ y∂x

Hz(x) =

∂ 2 ψ (x) . ∂z∂x

(4.15)

The static demagnetizing fields inside the sample are obtained from ψi as Hy =





Ms (G) 1 − cosh (|G I I |x) e−|G I I |a ei Gy ,

(4.16)

G

Hx =





Ms (G) sinh (|G I I |x) e−|G I I |a ei Gy .

G

These fields show the bulk and edge mode characteristics, in Fig. 4.3.

(4.17)

28

4 Field Localization in Striped Magnonic Crystal Waveguide

The demagnetizing field also varies along the length of the material, as shown in Fig. 4.4. The peak amplitude of this field is different in the Co and Py regions.

4.3 Spin Wave Magnonic Band We are interested in harmonic solutions that correspond to monochromatic spin wave excitations, and we assume the forms m (r, t) = m(r) exp (i!t) for the small signal variations. With an external magnetic field along yˆ , we look at the small signal variations in the (x, z)-plane. (mx , mz ) are solutions to the differential equations ∂ψi(x) ∂x 2

+

∂ψi(x) ∂ y2

+

∂ψi(z) ∂ y2

+

∂ψi(x) ∂z 2

+

∂ψi(z) ∂z 2

= −mx , (4.18)

∂ψi(z) ∂x 2

= −mz ,

and also have the Bloch form mx =



m x (G) ei (k y +G ) y ei(kx x+kz z) ,

G

mz =



(4.19)

m z (G) ei (k y +G ) y ei(kx x+kz z) .

G

The trial potential function of choice is ψi (r ) =



ψi G (x) ei (k y +G ) y eikz z .

(4.20)

G

Substituting (4.19) and (4.20) in (4.18) and applying the appropriate boundary conditions, we get  m x (G)  1− k2

cosh(|k|x) e|k|a

  1− = m zk(G) 2

cosh(|k|x) e|k|a

ψi(x) = ψi(z)

G

G



ei (k y +G ) y eikz z , (4.21)

 e

i (k y +G ) y ik z z

e

,

where |k| = (k y + G)2 + k z2 . The dynamic magnetization at the lateral edges of a thin, axially magnetized magnetic element with finite in-plane size can be described by effective “pinning” boundary conditions [7]. Thus, in practical calculations, it is usually assumed that the boundary conditions at the lateral edges of a stripe or waveguide are either of the “free spins” type or of the “magnetic wall” type. These two assumptions lead to a quantization condition for the wave vector along the width (k z ),

4.3 Spin Wave Magnonic Band

29

kz = r

π , w

(4.22)

with r = n or n + 1 for free spin and magnetic wall, respectively. The small signal demagnetizing field is then obtained from (4.21) as

   m z (G) 2  m x (G) −|k|a −|k|a −i h z (x, y, z) = − k 1 − cosh (|k|x) e sinh (|k|x) e × |k|2 z |k|2 G

G

ei (k y +G y ) y eikz z eiωt ,

 h x (x, y, z) = − m x (G) cosh (|k|x) e−|k|a G

(4.23)   z (G) −|k|a × − i m|k| k sinh e (|k|x) z 2

ei (k y +G y ) y eikz z eiωt .

G

(4.24) We can now use these Fourier forms for the fields in the plane wave method. Substituting (4.23) and (4.24) for the dipolar fields in the linear form of (3.1), we obtain the eigenvalue problem [5]. ˆ k = iωmk , Mm

(4.25)

with the eigenvector

 mT k = mx,k (G1 ) , ......, mx,k (G N ) , mz,k (G1 ) , , .........mz,k (G N ) .

(4.26)

The coefficient matrix comprises of submatrices such that ⎛ Mˆ = ⎝

Mˆx x Mˆx z Mˆ

zx

⎞ ⎠

Mˆzz

(4.27)

with the submatrices defined as Mˆixjx = Mˆizzj = 0,

(4.28)

        Mˆixjz = −γμ0 H0 + l (Gi − Gl ) Gi − G j − k + G j . (k + Gl ) Ms (Gi − Gl ) λ2ex Gl − G j

    k z2 −γμ0 1 − exp −|k + G j |a 2 Ms Gi − G j , |k|

        Mˆizxj = γμ0 H0 − l (Gi − Gl ) Gi − G j − k + G j . (k + Gl ) Ms (Gi − Gl ) λ2ex Gl − G j

    k z2 +γμ0 exp −|k + G j |a 2 Ms Gi − G j . |k|

(4.29) We use standard numerical procedures to find the eigenvalues of (4.25), but again using a finite number of reciprocal vectors. All the frequencies in a band appear to converge satisfactorily when we use 30 reciprocal lattice vectors. The spin wave

30

4 Field Localization in Striped Magnonic Crystal Waveguide

Fig. 4.5 Spin wave spectra of magnonic thin film waveguide

Fig. 4.6 Spin wave profile for (ω, k) marked in Fig. 4.7 as A (left), B (right)

spectra are shown in Fig. 4.5. (The color of a line has no physical significance. It merely helps differentiate between adjacent bands.) When there is a non-uniform demagnetizing field along the length of the waveguide, low-frequency spin waves will concentrate their amplitude in the region of the low internal magnetic field. These spin waves can be interpreted as standing waves, with zero group velocity, observed as flat bands in Fig. 4.5 below 20 GHz. At higher frequencies, spin waves have finite group velocities and we see frequency gaps in the spectrum. This behavior is better visualized in Fig. 4.6, where we plot the spin wave profiles (|mx |) at the frequencies marked A and B in Fig. 4.5 We proceeded to analyze how gap widths depend on the filling fraction (s/d). Figure 4.7 shows this dependence for the lowest energy band gap in the magnonic spectrum. All the gap widths are found to vary with the filling fraction (s/d). We also investigated the dependence of the spin wave spectra on the angle between the applied field and wave vector (k). The flat bands can be eliminated by changing the angle between the wave vector and the applied field. We analyzed the dispersion characteristics for a specific angle of 20◦ . As we observe in Fig. 4.8, the dispersion curves around 20 GHz now suggest propagating waves with non-zero group veloc-

4.3 Spin Wave Magnonic Band

31

Fig. 4.7 Spin wave gap width versus filling fraction

Fig. 4.8 Spin wave dispersion in the Co-Py stripe waveguide, calculated for an angle of 20◦ between H0 and k

ities. Thus, angular variations in the wave vector k, relative to the applied field H0 , can be used to control the propagation of low-frequency spin wave excitations. We finally introduce an empty space (of air) instead of cobalt to understand the dipolar interaction between neighboring Py stripes. The calculation is done by setting (Ms,Co  0). The spin wave spectra of the corresponding geometry are shown in Fig. 4.9. As expected, the band structure shows a zero group velocity for spin waves at all frequencies. We calculated and plotted the amplitude of the spin wave profile with an arbitrary chosen frequency (marked at points A and B). The spin wave amplitudes corresponding to frequencies marked at points A and B show localized behavior as shown in Fig. 4.10.

32

4 Field Localization in Striped Magnonic Crystal Waveguide

Fig. 4.9 Spin wave dispersion in a magnonic thin film waveguide with strips of Py separated by an air gap

Fig. 4.10 Spin wave profile for (ω, k) marked in Fig. 4.10 as A (left), B (right)

4.4 Spin Wave Localizations in Striped Magnonic Waveguide The Ms,Co is 1.15×106 A/m, and Ms,py is 0.658×106 A/m. Now we apply an external magnetic field along the length of the 1D striped magnonic waveguide, which is periodic. Then the demagnetizing field will be opposite to the applied field. This demagnetizing field will be -(Ms,Co -Ms,py ). The demagnetizing field is not uniform. It will be more negative in Co than in the Py region. Thus, the internal demagnetizing field can be modeled as a potential well problem. The internal field will be more negative in Co. So the depth of the well is higher in the Co region than in Py. If we take a non-periodic stripe, there will be a uniform demagnetizing field opposite the propagation direction. But in our case, we will have inhomogeneity because of a different material. Thus, the internal field causes a non-uniform demagnetizing field. So spin wave amplitude will be more where the opposing field is low.

4.5 Convergence

33

So the precession will be more in Py than Co region. So we will get spin waves with different amplitudes in Co and Py regions. It is basically because of the non-uniform demagnetizing field. In the low-frequency region, we have a fully localized mode. The spin wave is fully concentrated in the Py region. These modes are localized. Considering the potential well problem, the low-energy (low-frequency) spin wave is trapped in the well and cannot escape from this well. So low-frequency(low-energy) spin wave is completely localized or trapped in the well. But for a higher frequency, we have propagating behavior. This is because spin waves have enough energy to cross the barrier.

4.5 Convergence To obtain a potential profile, we had to truncate the Fourier series in (4.9). Figure 4.11 is a test of convergence, and we observe satisfactory convergence for 20 reciprocal lattice vectors. We test the convergence of (4.16) by plotting its maximum, Hy (x = a), versus number of reciprocal vectors, shown in Fig. 4.12. Hy (x = a) is an damped oscillating function of the number of lattice vectors. An exponential curve fit yields a damping coefficient of 0.019, suggesting reasonable convergence when we use 150 reciprocal lattice vectors.

Fig. 4.11 Convergence of (2.15) at a large number of lattice vectors

34

4 Field Localization in Striped Magnonic Crystal Waveguide

Fig. 4.12 The demagnetizing field, Hy (x = a), has a damped oscillating dependence on the number of reciprocal vectors used in evaluating (4.16)

4.6 Conclusion The plane wave method is a suitable tool to explore the behavior of small signal excitations in magnonic crystals. Using this method, we have numerically calculated the magnon band structure in a Co-Py striped thin film waveguide, biased in a backward volume configuration. The linearized Landau-Lifshitz equation was reduced to a matrix form, where the eigenfrequencies distribution yielded the dispersion relation ω(k). The eignemodes of the matrix yield the magnetization amplitude in the waveguide. The dispersion diagrams show evidence of band gaps, where the waveguide does not support spin wave propagation. Instead, small signal excitations within a band gap result in standing spin wave patterns, with zero group velocity, and the waveguide acts as a microwave filter. The frequency and size of a gap can be controlled by varying the relative size of the Co and Py stripes. We have also observed that the dispersion diagrams are influenced by the angle between the wavevector and the external applied field. Thus, magnonic crystals offer a rich platform to control and explore spin wave excitations.

References 1. P. Bruno, V.K. Dugaev, Equilibrium spin currents and the magnetoelectric effect in magnetic nanostructures. Phys. Rev. B 72, 241302 (2005) 2. Z.K. Wang, V.L. Zhang, H.S. Lim, S.C. Ng, M.H. Kuok, S. Jain, A.O. Adeyeye, Nanostructured magnonic crystals with size-tunable bandgaps. ACS Nano 4(2), 643–648 (2010) 3. M. Sokolovskyy, M. Krawczyk, The magnetostatic modes in planar one-dimensional magnonic crystals with nanoscale sizes. J. Nanoparticle Res. 13, 6085–6091 (2011) 4. N. Kumar, A. Prabhakar, Spin wave dispersion in striped magnonic waveguide. IEEE Trans. Magnet. 49(3), 1024–1028 (2013)

References

35

5. M. Krawczyk, H. Puszkarski, Plane-wave theory of three-dimensional magnonic crystals. Phys. Rev. B 77, 054437 (2008) 6. J. Kaczer, L. Murtinova, On the demagnetizing energy of periodic magnetic distributions,. Phys. Status Solidi (a) 23(1), (1974) 7. Guslienko, K.Y. et al., Effective dipolar boundary conditions for dynamic magnetization in thin magnetic stripes. Phys. Rev. B 132402 (2002)

Chapter 5

Walker’s Solution for Curved Magnonic Waveguide and Resonant Modes in Magnonic Ring

5.1 Spin Wave Dispersion for Curved Magnonic Waveguide In addition to magnonic rings [1, 2], curved magnonic waveguides could also present interesting opportunities for the design of magnonic devices. Thin film magnetic rings also have generated a great deal of interest in recent years due to the existence of multiple energy states and their possible application in memory, logic and sensing devices [2–5]. Useful states such as “vortex” and “onion” states have been identified in ferromagnetic rings [6–9]. The ferromagnetic ring also helps to understand the effects of interference of spin wave [10]. Recently generation of chaotic spin wave in ferromagnetic ring was experimentally reported in [11, 12]. Bulk and surface spin waves in nanotubes and their dispersion characteristics were previously obtained by solving Maxwell’s equations with appropriate boundary conditions [13]. We focus on finding an analytical dispersion relation for a curved magnonic waveguide. The dispersion relation for dipolar spin waves is obtained by solving Walker’s equation in cylindrical coordinates. The appropriate Maxwell’s equations are expressed in terms of a magnetostatic potential (ψ) which satisfies Walker’s equation inside the material and Laplace’s equation outside the material. The equations are solved analytically with proper boundary conditions for the dipolar case. This is then extended to exchange spin waves, using perturbation theory, for different bending radii. As expected, for large bending radius, the dispersion relation approaches that of a straight waveguide. By introducing symmetry about the azimuthal direction, we also obtain analytical solutions for Walker’s equation and the mode profile characteristics of dipolar spin waves in a magnonic ring. As then extended using perturbation theory to arrive at the modes in exchange spin waves.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. S. Nikhil Kumar, Magnonic Devices, SpringerBriefs in Materials, https://doi.org/10.1007/978-3-031-22665-6_5

37

38

5 Walker’s Solution for Curved Magnonic Waveguide …

5.1.1 Geometry and Analysis The analytical forms for the field and the propagation constant in a bent slot optical waveguide were investigated in [14], where Maxwell’s equations were solved in cylindrical coordinates with appropriate boundary conditions, and the solution is represented as the linear combination of Bessel functions. In a similar manner, we consider a bent magnetic waveguide with outer and inner radii R1 and R2 , respectively. The thickness of the material is assumed to be out of the plane, along the directions of the magnetic field, H0 . We assume that spin waves propagate inside the ring structure, in the (r, φ)-plane in forward volume configuration (Fig. 5.1). If H0 is the applied dc field, the total magnetic field is H = H0 + h (t),

(5.1)

where we assume that the small signal amplitude | h | R1 ,

(5.12)

(5.13)

40

5 Walker’s Solution for Curved Magnonic Waveguide …

where √ the frequency dependence comes in through the dimensionless quantity ξ = 1/ 1 + χ. We find the constants (A, B, C and D) by applying the appropriate boundary conditions. • Tangential h must be continuous ψin |r =R1 = ψout |r =R1 , (5.14) ψin |r =R2 = ψout |r =R2 , or equivalently,

 AIkφ (ξk z R1 ) + B K kφ (ξk z R1 ) = C K kφ (k z R1 ) ,



 AIkφ (ξk z R2 ) + B K kφ (ξk z R2 ) = D Ikφ (k z R2 ) .

(5.15)

¯¯ • With b = μ.h, the normal component of b becomes br = (1 + χ)

jκ ∂ψ ∂ψ − . ∂r r ∂φ

(5.16)

Continuity of br at the surfaces r = R1 and r = R2 yields

   (1 + χ) Aξ Ikφ (ξk z R1 ) + Bξ K kφ (ξk z R1 ) +

κkφ R1



  (1 + χ) Aξ Ikφ (ξk z R2 ) + Bξ K kφ (ξk z R2 ) +

κkφ R2



  AIkφ (ξk z R1 ) + B K kφ (ξk z R1 ) = C K kφ (k z R1 ) ,



  AIkφ (ξk z R2 ) + B K kφ (ξk z R2 ) = D Ikφ (k z R2 ) ,

(5.17) where the primes represent the first derivative from (5.16). Rewriting in matrix form, we have ⎡

Ikφ (ξk z R1 ) K kφ (ξk z R1 ) −K kφ (k z R1 )

0

⎤⎡

A

⎤

⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎢ Ikφ (ξk z R2 ) K kφ (ξk z R2 ) 0 −Ikφ (k z R2 ) ⎥ ⎢ B ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = 0, ⎢ ⎥⎢ ⎥ ⎢  ⎥⎢C ⎥ ⎢ x x −K R 0 (k ) 1 2 z 1 kφ ⎥⎢ ⎥ ⎢ ⎦⎣ ⎦ ⎣  D x3 x4 0 Ikφ (k z R2 ) where x1 , x2 , x3 , x4 are

(5.18)

5.1 Spin Wave Dispersion for Curved Magnonic Waveguide Table 5.1 Parameter values Parameter

41

Value 8.6 ×105 A/m 1.3 ×10−11 J/m 0 175.86 ×109 Hz/T 0 10.1 kOe

Saturation magnetization (Ms ) Exchange constant (A) Anisotropy constant (K ) Gilbert gyromagnetic ratio (γ) Damping coefficient (α) DC bias field (H0 )

  x1 = (1 + χ) ξ Ikφ (ξk z R1 ) +

  x2 = (1 + χ) ξ K kφ (ξk z R1 ) +

  x3 = (1 + χ) ξ Ikφ (ξk z R2 ) +

  x4 = (1 + χ) ξ K kφ (ξk z R2 ) +

κkφ I R1 k φ κkφ R1

(ξk z R1 ) ,

K kφ (ξk z R1 ) ,

κkφ I R2 k φ R κkφ R2

(ξk z R2 ) ,

(5.19)

K kφ (ξk z R2 ) .

We set the determinant of the coefficient matrix in (5.18) to zero to obtain the dispersion relation. Table 5.1 lists the parameters used for our calculation. The dynamic magnetization at the lateral edges of a thin, axially magnetized magnetic element with finite in-plane size can be described by effective “pinning” boundary conditions [15]. Thus, in calculations, it is usually assumed that the boundary conditions at the lateral edges of a stripe or waveguide are either of the “free spins” type, or of the “magnetic wall” type. We apply a quantization rule [15] along the thickness of the ring. The quantized wave vector k z is given by   2 (m + 1) π 1− , kz = w d ( p) p = L/w and d ( p) =

2π . p [1 + 2ln (1/ p)]

(5.20)

(5.21)

In our geometry, the values of L (thickness) and w (width) are 5 nm and 100 nm, respectively, such that the quantized wave vector k z = 0.027 × 109 (rad/m). We shall assume this value for the rest of our analysis. We varied kφ from 0 to 62 and find the frequency for which the determinant of the coefficient matrix in (5.18) to zero. We also did the same analysis for exchange spin waves using perturbation theory (explained in 6.2). Figure 5.2 represents kφ versus f for dipolar and exchange spin waves.

42

5 Walker’s Solution for Curved Magnonic Waveguide …

Fig. 5.2 kφ versus f

5.1.2 Dispersion Relation of Curved Magnonic Waveguide We initially investigated the dispersion relation of dipolar dominated spin waves neglecting exchange interaction. The values of ω0 and ω M are given in Table 5.2. The frequency corresponding to each kφ is extracted by setting the determinant of the coefficient matrix in (5.18) to zero. Figure 5.3 shows the dispersion characteristics of spin waves with and without exchange. We included exchange interaction using perturbation theory as described below. Assume that the magnetostatic potential and spin wave function are of the form ψ = ψ0 + ψ1 + ψ2 + ...., (5.22) m = m0 + m1 + m2 + ...., where ψ0 , ψ1 , ψ2 are the zeroth-, first-, second-order correction to the magnetostatic potential state and m0 , m1 , m2 are the zeroth-, first-, second-order correction to the spinwave function in a perturbation series. We initially solve Walker’s equa tion ∇. μ0 .∇ψ = 0 to obtain the ground state potential wave function ψ0 . The corresponding spin wave amplitude can be obtained using m0 = χ(ω0 ).∇ψ0 . In the Table 5.2 Larmor precession frequency

Parameter

Value (GHz)

ω0 = γμ0 H0 ω M = γμ0 Ms

28.43 28.75

5.1 Spin Wave Dispersion for Curved Magnonic Waveguide

43

Fig. 5.3 Dispersion relation for forward volume spin waves, with and without exchange interaction. Circles represent modes in a ring, to first order in exchange, as described in Sect. 5.7

presence of exchange, the term ω0 in the permeability tensor (5.5) is replaced with [16] (5.23) ω0 → ωex = ω0 − ωM λ2ex ∇ 2 m0 . The change to ω0 can be calculated by finding ωM λ2ex ∇ 2 m0 . The addition of ¯ ωM λ2ex ∇ 2m0 willchange  the permeability tensor to μ¯ 1 . We again solve Walker’s equation ∇. μ¯¯ 1 .∇ψ = 0 for this modified permeability tensor to get ψ = ψ0 + ψ1 and ¯¯ ex ).∇ψ1 . To calculate second-order correction to ψ, we let then calculate m1 = χ(ω ω0 → ωex = ω0 − ωM λ2ex ∇ 2 (m0 +m1 ).

(5.24)

This our   permeability tensor and we again solve Walker’s equation  will change (∇. μ¯¯ 2 .∇ψ = 0 to obtain ψ = ψ0 + ψ1 + ψ2 . Following this iterative procedure, we obtain the dispersion relations with first- and second-order exchange corrections.1 To explain the dispersion of dipolar and exchange spin waves, we used a normalized wave vector κ = kφ R0−1 (1/nm). Figure 5.3 shows a comparison between the dispersion relation of dipolar and exchange coupled spin wave at low wave numbers. As expected, the dispersion relation shows forward volume characteristics [16], which are altered by the introduction of exchange interaction. The inclusion of higher order terms does not significantly alter the dispersion relation for small values of k. Dispersion relation of exchange spin wave for different bending radii.

μ0 is the zeroth-order tensor defined in (5.5). μ1 and μ2 are similarly defined, but include correction term due to (5.23) and (5.24).

1

44

5 Walker’s Solution for Curved Magnonic Waveguide …

5.1.3 Validations One possible test for our solution is the behavior of a curved waveguide for increasing radii. we used a normalized wave vector κ = kφ R0−1 for curved waveguide, in order to compare with straight waveguide. The general expression for potential profile is given by   (5.25) ψi (r, kφ , k z ) = AIkφ (ξk z r ) + B K kφ (ξk z r ) ei (kφ φ) where kφ is a dimensionless quantity. We want to analyze the dispersion of curved waveguide for different bending radii and compare with that of straight waveguide. Now multiply and divide the term kφ φ in (6.25) by R0 , where R0 is the bending radius. Then (6.25) becomes   ψi (r, κ, k z ) = AIκR0 (ξk z r ) + B K κR0 (ξk z r ) ei(κR0 φ) ,

(5.26)

where κ = kφ /R0 has a unit of 1/nm, and κφ has the unit of rad/nm. The potential profile for straight waveguide is described by   ψ k x , k y , k z = ψ0 eikz z ,

(5.27)

where the unit of k z is rad/nm. In order to compare the dispersion relation of straight and curved waveguide, we must satisfy the following relation: eikz z = eiκR0 φ .

(5.28)

To match κ and k z , we must satisfy z = R0 φ.

(5.29)

Assume that the length (L) of straight waveguide is 2000 nm (2 µm). Table 5.3 represents the combination of R0 and φ for L = 2000 nm. We expect the dispersion relation for large R to approach that of a straight waveguide [16]. These results are shown in Fig. 5.4. A second validation is possible by comparing the analytic results to those obtained from micromagnetic simulations, using the finite element package Nmag [17]. Table 5.3 List of arc radius and corresponding arc angle for waveguide length L = 2000 nm

Arc radius R0 (nm)

Arc angle φ (radian)

550 1000 1500 2000

3.63 2 1.33 1

5.1 Spin Wave Dispersion for Curved Magnonic Waveguide

45

Fig. 5.4 Dispersion characteristics of magnonic waveguide with different bending radii

Fig. 5.5 The dispersion relation obtained using simulations. The black dots correspond to the dispersion relation, for the fundamental mode, obtained by solving Walker’s equation

Figure 5.5 shows the dispersion relation obtained using the simulation.2 The different parabolic curves correspond to the dispersion relation, ω (k), for the different width modes in the ring corresponding to different values of m in (5.20). The analytic results for the m = 0 mode, and the lowest ω (k) curve in the simulation, are in reasonable agreement with some caveats. • At low values kφ , exchange effects are not captured accurately by the simulations [18]. This is due to the finite length of the stripe in the simulation which limits resolution as kφ → 0. • For large kφ , where exchange dominates over dipolar interactions, the group velocfrom the simulation is greater than that estimated by perturbation theory. ity ∂ω ∂k This is to be expected as the inclusion of higher order terms in the series expansion 2

Similar as done by G. Venkat.

46

5 Walker’s Solution for Curved Magnonic Waveguide …

showed an increase in the slope, Fig. 5.5. Despite these shortcomings, the perturbation method yields a reasonable estimate for the dispersion of spin waves in a bent magnonic waveguide.

5.1.4 Modes in a Magnonic Ring Magnonic ring exhibits resonance when the ring circumference equals an integer number of spin wavelengths. Magnonic ring exhibiting multiple resonances has been reported in [19]. The resonance frequency can be found from dispersion characteristics by finding the integer number of wavelengths. Using Walker’s equation in cylindrical coordinates from Chap. 6, we impose periodic boundary conditions along the azimuthal directions in order to find the resonant frequency for each mode. Consider a magnonic ring structure with outer and inner radii R1 and R2 , respectively, as shown in Fig. 5.6. The thickness of the material is assumed to be out of the plane, along the directions of the magnetic field, H0 . We assume that spin waves propagate inside the ring structure, in the (r, φ)-plane. The φ dependence of the potential in Walker’s equation is now written as 1 d 2φ = −n 2 . φ dφ2

(5.30)

ψ (φ) ∼ einφ = ein(φ+2π) ,

(5.31)

This leads to with the wave number n, along φ being an integer. The frequency corresponding to each value of n can be extracted by setting the determinant of the coefficient matrix in (5.18) to zero. Each n gives a different field pattern or mode inside the structure. So n can also be interpreted as a mode number.

Fig. 5.6 Magnonic ring structure with inner and outer radii are 500 and 600 nm, respectively. The thickness (w = 5 nm) of the ring is out of the plane (along z-axis) and R is assumed to be 550 nm

5.1 Spin Wave Dispersion for Curved Magnonic Waveguide

47

Fig. 5.7 Spatial profile m φ (φ) for dipolar and exchange spin wave for n = 2 mode(left), n = 5 mode (right)

Fig. 5.8 Mode frequency characteristics of dipolar and exchange dominated spin wave

Figure 5.7 shows the spatial profile m φ (φ) of exchange as well as dipolar spin waves. The number of lobes increases as we increase the mode number. The frequencies for each mode number sit on the dispersion characteristics of the curved magnonic waveguide as shown in Fig. 5.8. This will be true for dipolar dominated spin waves also. For a given mode, we can also study the potential ψ(r) and magnetization amplitude |mr (r)|. These are shown in Fig. 5.9 and where the dots made the first-order corrections due to λex . We use (5.3) to estimate |mr (r)| and observe that the angular mode number (n) influences the radial variation. This is evident in (5.12). Consequently, the maxims in ψ(r) shift toward the inner radius for large mode numbers. To understand the effect of exchange on the location of the maxima of |ψ|, we increased the exchange constant A from 1.3 ×10−11 J/m to 2.6 ×10−11 J/m with a corresponding increase in λex .

48

5 Walker’s Solution for Curved Magnonic Waveguide …

Fig. 5.9 Magnetostatic potential (left) and spin wave profile (right) for dipolar and exchange spin wave

Fig. 5.10 Location of maxima versus mode number for different exchange lengths. (λex = 0.00, 3.23 × 10−17 , 6.46 × 10−17 m2 )

The results in Fig. 5.10 are not fully understood. For both low and high mode numbers, the location of the maxima in |ψ| moves toward higher radii, suggesting competing effects. To better visualize this, we plot the quadratures of m. The time dependence of the spin wave profile is influenced by eiωt , where ω can be extracted from mode frequency characteristics. The total small signal magnetization is given by 

ˆ iωt . (5.32) m(r, t) = Re (m r rˆ + jm φ φ)e If we take the real part of (5.32), we get ˆ m(r, t) = m r cos (ωt) rˆ + m φ sin (ωt + φ) φ.

(5.33)

Consider m (r, t) for the n = 2 mode. The corresponding frequency is obtained from the mode frequency characteristics diagram from Fig. 5.8. The phase change φ between m r and m φ is taken to be 450 . Figure 5.11 shows the small signal variation of m as a function of space and time for a dipolar spin wave, which resembles a flower pot. The tip of the magnetization vector describes an ellipse for a fixed value of radius. The functional behavior of m(r, t) for lower order exchange spin waves is similar to that of dipolar spin waves.

References

49

Fig. 5.11 m (r, t) for dipolar spin wave for n = 2 mode (left) and n = 5 mode (right)

5.2 Conclusion The dispersion relation of the bent magnonic waveguide is derived for dipolar and exchange spin wave. We showed that the dispersion characteristics approach with that of the straight waveguide for a higher bending radius. We used the analytical solution of bent waveguide to investigate the mode characteristics of magnonic ring structure by applying symmetry conditions along the azimuthal direction. The mode frequency characteristics of the magnonic ring structure are then analyzed. The perturbation method is employed for obtaining the mode frequency characteristics of exchange dominated spin waves. The mode frequency characteristics clearly indicate the possible supporting modes and the corresponding frequencies inside the structure. The mode profile for exchange and dipolar dominated spin waves is plotted and compared. This gives us a clear picture of the spatial distribution of spin waves inside the structure for different modes.

References 1. Magnon modes in permalloy nanorings. J. Magn. Magn. Mater. 286(0), 366 – 369 (2005) 2. F. Giesen, J. Podbielski, D. Grundler, Mode localization transition in ferromagnetic microscopic rings. Phys. Rev. B 76, 014431 (2007) 3. F. Montoncello, L. Giovannini, F. Nizzoli, Spin mode calculations in nanometric magnetic rings: localization effects in the vortex and saturated states. J. Appl. Phys. 103(8), 083910 (2008) 4. F. Montoncello, L. Giovannini, F. Nizzoli, H. Tanigawa, T. Ono, G. Gubbiotti, M. Madami, S. Tacchi, G. Carlotti, Magnetization reversal and soft modes in nanorings: transitions between onion and vortex states studied by Brillouin light scattering. Phys. Rev. B 78, 104421 (2008) 5. X. Zhu, M. Malac, Z. Liu, H. Qian, V. Metlushko, M.R. Freeman, Broadband spin dynamics of permalloy rings in the circulation state. Appl. Phys. Lett. 86(26), 262502 (2005) 6. M. Steiner, J. Nitta, Control of magnetization states in microstructured permalloy rings. Appl. Phys. Lett. 84(6), 939–941 (2004) 7. P. Vavassori, O. Donzelli, M. Grimsditch, V. Metlushko, B. Ilic, Chirality and stability of vortex state in permalloy triangular ring micromagnets. J. Appl. Phys. 101(2), 023902 (2007)

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8. P. Vavassori, R. Bovolenta, V. Metlushko, B. Ilic, Vortex rotation control in permalloy disks with small circular voids. J. Appl. Phys. 99(5), 053902 (2006) 9. M. Klaui, J. Rothman, L. Lopez-Diaz, C.A.F. Vaz, J.A.C. Bland, Z. Cui, Vortex circulation control in mesoscopic ring magnets. Appl. Phys. Lett. 78(21), 3268–3270 (2001) 10. J. Podbielski, F. Giesen, D. Grundler, Spin-wave interference in microscopic rings. Phys. Rev. Lett. 96, 167207 (2006) 11. S. Grishin, Y. Sharaevskii, S. Nikitov, D. Romanenko, Generation of chaotic microwave pulses in ferromagnetic film ring oscillators under external influence. IEEE Trans. Mag. 49(3), 1047– 1054 (2013) 12. S. Grishin, Y. Sharaevskii, S. Nikitov, E. Beginin, S. Sheshukova, Self-generation of chaotic dissipative soliton trains in active ring resonator with 1-d magnonic crystal. IEEE Trans. Mag. 47(10), 3716–3719 (2011) 13. Magnetostatic modes and magnetic polaritons in ferromagnetic and antiferromagnetic nanorings. J. Magn. Magn. Mater. 310(2), Part 3, 2183 (2007) 14. K.R. Hiremath, Analytical modal analysis of bent slot waveguides. J. Opt. Soc. Am. A 26, 2321–2326 (2009) 15. K.Y. Guslienko et al., Effective dipolar boundary conditions for dynamic magnetization in thin magnetic stripes. Phys. Rev. B 132402 (2002) 16. D.D. Stancil, A. Prabhakar, Spin Waves Theory and Applications, 1st edn. (Springer, New York, 2008) 17. T. Fischbacher et al., A systematic approach to multiphysics extensions of finite-element-based micromagnetic simulations: Nmag. IEEE Trans. Magn. 43, 2896–2898 (2007) 18. G. Venkat, D. Kumar, M. Franchin, O. Dmytriiev, M. Mruczkiewicz, H. Fangohr, A. Barman, M. Krawczyk, A. Prabhakar, Proposal for a standard micromagnetic problem: Spin wave dispersion in a magnonic waveguide. IEEE Trans. Mag. 49(1), 524–529 (2013) 19. T.D. Poston, D.D. Stancil, A new microwave ring resonator using guided magnetostatic surface waves. J. Appl. Phys. 55(6), 2521 (1984)

Chapter 6

Nanocontact-Driven Spin Wave Excitations in Magnonic Cavity

6.1 Introduction Spin waves (SWs) are a collective excitation of magnetic moments in magnetic materials [20]. Magnonics, the study of spin waves, aims to provide control of spin waves and to carry and manipulate information. The wavelength of SWs is several orders of magnitude shorter than that of electromagnetic waves and magnonics offers better prospects for the miniaturization of devices. Various SW-based devices have now been proposed and investigated. These include SW couplers, multiplexers, filters and transistors [2–6]. Magnonic crystals (MCs) have attracted a lot of interest because of their abilities to tune the SW band gap [7–9]. It was recently reported that MCs can also be used for getting collimated spin waves [10]. Magnonic crystals (MCs) with defects have also been studied [11, 12] and the SWs propagating in these defects have shown an increase in group velocity. The name spin wave amplification by stimulated emission of radiation (SWASER) [13, 14] has already been suggested for different structures. However, a MC cavitybased SWASER has never been exploited so far. In this paper, we simulate an array of nanocontacts within an MC cavity. An electrical current is injected into a Py thin film through nanocontacts to excite spin waves [15–18]. An antidot MC with a line defect is used to create a cavity around the nanocontacts. We obtain sustained oscillations with the MC cavity with an increase in SW amplitude. The simulations have been performed using version 3 of MuMax [19], an open-source computational tool for micromagnetic simulations implemented on graphical processing units (GPU).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. S. Nikhil Kumar, Magnonic Devices, SpringerBriefs in Materials, https://doi.org/10.1007/978-3-031-22665-6_6

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6 Nanocontact-Driven Spin Wave Excitations in Magnonic Cavity

6.2 Micromagnetic Simulations The geometry considered for the study is shown in Fig. 6.1. The material parameters used for the simulation are Ms (Py) = 800 × 103 A/m for the saturation magnetization, and A (Py) = 13 × 10−12 J/m for the exchange stiffness constant. We assumed the typical Py damping of α = 0.01. Nanocontacts were added on the Py film and a spin polarized current was injected into it. The radius of the nanocontact is assumed as 20 nm and the center to center separation between them is 100 nm [20]. The thickness of the Py film is assumed as 5 nm. We considered an antidot MC cavity around the nanocontacts. The period of the MC is assumed as 100 nm and the radius of antidot is 20 nm. The cell size was taken as 4 ×4 × 4 nm3 with the in-plane cell dimensions lesser than the exchange length for Py which is lex = 5.7 nm. The micromagnetic simulation is performed at (0 K). We begin by applying a bias magnetic field B0 = 0.5 T along xˆ and allow m to relax to its ground state. We consider a two-stage simulation for the relaxation. In the first stage, we set a high damping of α = 0.4 and obtain an intermediate ground state. We use this ground state as the initial magnetization configuration for the next stage of the simulation in which we specify α = 0.01 which is the normal damping of Py. This two-stage simulation leads to faster convergence [21]. In the dynamics simulation, we start with the last state from the relaxation simulation and inject a spin-polarized current I = 7.7 mA at t = 0 s to excite SWs in the Py film. This current was chosen such that the frequency of SWs generated from an array of nanocontacts lies in the band gap of the magnonic crystal. To reduce the SW reflections from the edges, we apply an absorbing boundary layer (ABL) with

Fig. 6.1 The geometry of the device. MC is formed by antidots of radius 20 nm included in the Py film. The thickness of the Py film is 5 nm. The entire structure is surrounded by absorbing boundary layer (ABL). The diameter of nanocontact is 40 nm and the separation between them is 100 nm

6.3 Method of Calculation

53

Fig. 6.2 Snap shots of magnetization at different time instants. The white region in (c) is the region where magnetization is probed

a polynomial damping profile [22] of length 200 nm near all the boundaries of the structure. The damping was of the form α (x) = ax n −    so that α (xstart ) = 0.0 and α xstop = 1.0 where xstart and xstop mark the beginning and end of the ABL, respectively. This damping profile shows a 12 dB reduction in energy in the first 100 nm of the ABL without reflections from the interface at the beginning of the ABL. This is in contrast to using an ABL with a constant damping profile [23] which leads to significant reflections. We observed the spin wave dynamics for a time duration of 30 ns. The snap shots of m y (x, y, t) at different time instants are shown in Fig. 6.2. Each nanocontact emits spin waves which get reflected by the magnonic crystal and get amplified in the cavity. The SWs get coupled to the line defect and output is collected at the output port. This is analogous to a laser cavity with the nanocontacts acting as the gain medium and the MC acting as the reflective mirrors.

6.3 Method of Calculation To realize a MC cavity-based SWASER, we perform the following steps: • Investigate the band structure of an antidot MC using the plane wave method (PWM) and observed the band gap. The radius and lattice constant are assumed as 20 nm and 100 nm, respectively. • Choose an electric current such that the frequency of SWs excited using an array of nanocontacts lies in the band gap of MC. • Calculate the length of the MC cavity using Fabry-Perot model.

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6 Nanocontact-Driven Spin Wave Excitations in Magnonic Cavity

• An array of nanocontacts is placed inside the MC cavity which acts as the SW source. • Investigate the SWs spectrum and quality factor of the array of nanocontacts within the MC cavity.

6.3.1 Band Structure of Antidot MC We initially calculated the SWs spectrum in antidot MC (without defects). To investigate the SWs spectrum in antidot MC, we solve the linearized Landau-Lifshitz equation in the frequency domain using plane wave expansion method [24, 25]. The obtained SWs spectra for the antidot MC with lattice constant a = 100 nm and radius r = 20 nm are shown in Fig. 6.3. The antidot MC cavity supports the range of frequencies that lie in the band gap of antidot MC. We excite SWs whose frequency lies in the second band gap of antidot MCs (27 to 30 GHz), shown by the dotted line in Fig. 6.3.

6.3.2 Spin Wave Injection on Py Film Using an Array of Nanocontacts An array of nanocontacts is first considered on the Py film without considering the magnonic crystal cavity. Spin waves were excited in the Py film by injecting current

Fig. 6.3 Band diagram of antidot MC (without line defect) using PWM

6.3 Method of Calculation

55

Fig. 6.4 Snap shot of magnetization at t = 500 ps

Fig. 6.5 Spin wave power spectrum of nanocontacts on a Py film

through the nanocontacts. The current is assumed to be 7.7 mA such that the frequency of excited SWs lie in the band gap of MCs. Simulations are performed for a duration of 30 ns and data is saved for every 1ps. The snap shot of magnetization |m| at particular time instant is shown in Fig. 6.4. We probe the magnetization at the region marked by the vertical line in Fig. 6.4 and average the magnetization vertically, to get m y (x0 , < y >, t) and take its Fourier transform to get the spin wave spectrum. The obtained spin wave power spectrum from a time series of 30 ns with a discretization of 1ps is shown in Fig. 6.5. The SW propagation frequency of 28.15 GHz, obtained from Fig. 6.5, lies in the band gap of the MC (see Fig. 6.3). A line defect in the MC allows the SW of this frequency to propagate [26]. To extract the wavelength, we probe the magnetization m y (x, y, t), average it over the width (along y), and time, to get m y (x, < y >, < t >). We then

56

6 Nanocontact-Driven Spin Wave Excitations in Magnonic Cavity

Fig. 6.6 m y (x, yo , < t >) (left), |m y (k) |2 (right)

take its Fourier transform to extract m y (k). The obtained magnetization profile m y (x, < y >, < t >) and its Fourier transform are shown in Fig. 6.6. The main peak occurs at k0 = 0.0321 rad/nm, with a corresponding wavelength, λ0 ≈ 200 nm.

6.3.3 Fabry-Perot Model We interpret the spin wave confinement in a resonator with a Fabry-Perot model which consists of approximating the cavity mode as a stationary pattern formed by two counter propagating defect modes bouncing between two mirrors. The antidot lattice on two sides of the cavity acts as reflecting mirrors with reflection coefficient r = |r | exp (iφ). When the defect guided mode impinges onto the mirror, it is back reflected into the counter propagating guided mode, which is again back reflected  onto the second mirror. The plane P and P are used as phase references for the model reflectively used in the Fabry-Perot model (see Fig. 6.7). Within the FabryPerot cavity, a resonance at a wavelength λ0 (2π/k0 ) results from a phase matching condition for the defect mode. The total phase delay T (λ0 ) experienced by the guided mode along one-half cavity cycle has to be equal to a multiple of π T (λ0 ) = k0 L + φ (λ0 ) = nπ.

(6.1)

The antidot lattice on the left side of the source acts as a mirror with a reflection coefficient r = |r | exp (iφ). We investigate the reflection coefficient of the antidot lattice based on ideas presented in [27, 28]. We approximated the antidot MC as 1D binary grating and calculated the reflection coefficient. The phase change φ (λ0 ) caused due to reflection from two mirror is 0.3π radian. By substituting for λ0 , φ (λ0 ) in Eq. 6.1, we get   nπ − 0.3π L= . (6.2) k0

6.3 Method of Calculation

57

Fig. 6.7 Schematic top view of the investigated cavity formed by filling holes in two directions of a 2D MC composed of a rectangular lattice of air holes (lattice constant a = 100 nm) etched into a Permalloy slab. The picture holds for a few missing holes. The slab thickness is 5 nm and the hole radius is 20 nm

We chose a reasonably higher length (L = 800 nm) as compared to the wavelength λ0 by fixing n = 6 in Eq. 6.2. After fixing the cavity length, eight anti dots are removed along the length of the MC. The width of the cavity is taken as 200 nm.

6.3.4 SW Spectra We now run the simulation and monitor the magnetization at the output port (averaged vertically along the straight line in Fig. 6.1). We first study SWs without antidot MC cavity. The resulting time series is shown in Fig. 6.8a. The SWs get damped over a time scale of ≈ 30 ns. We then add the antidot MC (with line defect) as shown in Fig. 6.1 and obtain the SW time series shown in Fig. 6.8b. We observed that SWs are amplified and stable when the nanocontacts are within the MC cavity. The power spectral density (PSD) was estimated using Welch’s overlapped segment averaging estimator. The SW spectrum with and without the MC cavity with a time step of 1 ps is shown in Fig. 6.9. A Hanning window has been applied so that artifacts of a finite duration simulation are reduced. We see an increase of 20 dB in the SW amplitude on introducing the MC cavity. This is due to the repeated SW reflections which occur in the cavity. Moreover, SW spectrum (with MC cavity) shows different peaks at n f 0 , where f 0 = 28.15 GHz and n is an integer. This shows that our system has a similarity to a multi-mode laser system.

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6 Nanocontact-Driven Spin Wave Excitations in Magnonic Cavity

Fig. 6.8 m y (t)averaged across the width for a duration of 30 ns, without MC cavity (left), with cavity (right)

Fig. 6.9 SW spectra with and without antidot lattice

6.3.5 Quality Factor Calculation Quality factor of this device is calculated using the decay method [29], an idea developed in finding the Q factor of photonic crystal cavities. The Quality factor is defined as ωr (6.3) Q= FWHM where ωr is the resonant frequency and FWHM is the full width half max of the resonance intensity spectrum. The time domain signal of the resonance is described by (6.4) m (t) = e−t(α−iωr ) u (t) .

6.3 Method of Calculation

59

Fig. 6.10 slope m of log10 m y (x0 , < y0 >, t)

The Fourier transform of m (t) is given by |m (ω) |2 =

1 α2 + (ω − ωr )2

.

(6.5)

The maximum value of |m (ω) |2 is clearly 1/α2 at ω = ωr . FWHM = 2α. Therefore, ωr . In order to relate α to Q, we must determine how the slope of the time Q= 2α signal decay is related to Q. We must take the log of the time signal to make the envelope a linear function. log10 (|m (t) |) = −

ωr t log10 (e) = mt 2Q

(6.6)

where m is the slope of the log of the time signal envelope. Solving for Q, we get Q=−

ωr log10 (e) . 2m

(6.7)

Initially, the Q factor is calculated by not considering any MC cavity around the array of nanocontacts. We probe the magnetization at the vertical line marked in Fig. 6.1 nearer the output port. The slope m of the log of the envelope of time signal and the Q factor for different currents is shown in Figs. 6.10 and 6.11. Quality factor analysis is extended by considering the MC cavity around the nanocontacts. We monitored the magnetization at the vertical line marked in Fig. 6.1,

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6 Nanocontact-Driven Spin Wave Excitations in Magnonic Cavity

Fig. 6.11 Quality factor using (6.7)

Fig. 6.12 Slope of envelope for different currents

Fig. 6.13 Quality factor calculated for different currents using (6.7)

nearer to the output port. The slope and its quality factor for different current are shown in Figs. 6.12 and 6.13. The Q factor can be tuned by changing the current. The slope m is decreased with an increase in current which causes an increase in quality factor. We are getting a stabilized  at a certain current (I = 7.8 mA) which gives a high Quality  magnetization factor Q ≈ 3.8 × 105 .

References

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6.4 Conclusion A spin-polarized electric current injected into Permalloy (Py) through a nanocontact exerts a torque on the magnetization, leading to spin wave (SW) excitation. We considered an array of nanocontacts on a Py film for an enhanced SW excitation. We designed an antidot magnonic crystal (MC) around the nanocontact to form a cavity. The MC was designed so that the frequency of the SW mode generated by the nanocontact lies in the band gap of the MC. The nanocontacts were placed in a line defect created in the MC by removing a row of antidots. The SW time series and power spectrum were observed at the output of the cavity. We observe that the SWs decay in the absence of the MC cavity, and when the nanocontacts are within the antidot MC cavity, the SW amplitude is amplified and stable. This is also reflected in the SW spectrum obtained at the output port. Finally, Q factor of the device  is calculated using decay method and observed a high Q factor Q = 3.8 × 105 for a current of 7.8 mA. Our proposed device behaves as a SWASER (spin wave amplification by the stimulated emission of radiation).

References 1. D. Stancil, A. Prabhakar, Spin Waves: Theory and Applications (Springer, 2009) 2. Z.K. Wang, V.L. Zhang, H.S. Lim, S.C. Ng, M.H. Kuok, S. Jain, A.O. Adeyeye, Observation of frequency band gap in a one dimensional nanostructured magnonic crystals. Appl. Phys. Lett. 94, 083112 (2009) 3. J. Ding, M. Kostylev, A.O. Adeyeye, Broadband ferromagnetic resonance spectroscopy of permalloy triangular nanorings. Appl. Phys. Lett. 100, 062401 (2012) 4. H. Yu, G. Duerr, R. Huber, M. Bahr, T. Schwarze, F. Brandl, D. Grundler, Omnidirectional spin-wave nanograting coupler. Nat. Commun. 4, 2702 (2013) 5. K. Vogt, F. Fradin, J. Pearson, T. Sebastian, S. Bader, B. Hillebrands, A. Hoffmann, H. Schultheiss, Realization of a spin-wave multiplexer. Nat. Commun. 5, 3727 (2014) 6. A.V. Chumak, A.A. Serga, B. Hillebrands, Magnon transistor for all-magnon data processing. Nat. Commun. 5, 4700 (2014) 7. M. Krawczyk, J.-C. Lévy, D. Mercier, H. Puszkarski, Forbidden frequency gaps in magnonic spectra of ferromagnetic layered composites. Phys. Lett. A 282(3), 186–194 (2001) 8. M. Krawczyk et al., Magnonic band structures in two-dimensional bi-component magnonic crystals with in-plane magnetization. J. Phys. D, Appl. Phys. 46(49), 495003 (2013) 9. M. Krawczyk, D. Grundler, Review and prospects of magnonic crystals and devices with reprogrammable band structure. J. Phys. Condens. Matter 26(12), 123202 (2014) 10. D. Kumar, A.O. Adeyeye, Broadband and total autocollimation of spin waves using planar magnonic crystals. J. Appl. Phys. 117, 143901 (2015) 11. V.V. Kruglyak, M.L. Sokolovskii, V.S. Tkachenko, A.N. Kuchko, Spin-wave spectrum of a magnonic crystal with an isolated defect. J. Appl. Phys. 99(8), 08C906 (2006) 12. T. Schwarze, D. Grundler, Magnonic crystal wave guide with large spin-wave propagation velocity in CoFeB. Appl. Phys. Lett. 102(22), 222412 (2013) 13. L. Berger, Multilayers as a spin-wave emitting diode. J. Appl. Phys. 81, 4880 (1997) 14. M. Tsoi et al., Generation and detection of phase-coherent current-driven magnons in magnetic multilayers. Nature 406, 46–48 (2000) 15. J.C. Slonczewski, Current-driven excitation of magnetic multilayers. J. Magn. Magn. Mater. 159, L1–L7 (1996)

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16. L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current. Phys. Rev. B 54, 9353–9358 (1996) 17. V.E. Demidov, S. Urazhdin, S.O. Demokritov, ’ Direct observation and mapping of spin waves emitted by spin-torque nano-oscillators. Nature Mater. 9, 984 (2010) 18. B. Madami, etc all., Nature 6, Nanotechnol. 635 (2011) 19. A. Vansteenkiste, J. Leliaert et al., The design and verification of MuMax3. AIP Adv. 4 107133 (2014) 20. F.C. MaciÃ, F.C. Hoppensteadt, A.D. Kent, Spin wave excitation patterns generated by spin torque oscillators. Nanotechnology 25, 045303 (2014) 21. G. Venkat et al., Proposal for a standard micromagnetic problem: Spin wave dispersion in a magnonic waveguide. IEEE Trans. Magn. 49(1), 524–529 (2013) 22. G. Venkat, A. Prabhakar, Absorbing layer for spin waves, presented at the, Joint MMM-Intermag conference (San Diego, USA, 2016) 23. M. Dvornik, A.N. Kuchko, V.V. Kruglyak, Micromagnetic method of s-parameter characterization of magnonic devices. J. Appl. Phys. 109, 07D350 (2011) 24. M. Krawczyk, H. Puszkarski, Plane-wave theory of dimensional magnonic crystals. Phys. Rev. B 77, 054437 (2008) 25. N. Kumar, A. Prabhakar, Spin wave dispersion in striped magnonic waveguide. IEEE Trans. Magnet. 49(3) (2013) 26. G. Venkat, N. Kumar, A. Prabhakar, Micromagnetic and plane wave analysis of an antidot magnonic crystal with a ring defect. IEEE Trans. Magnet. 50(11) (2014) 27. Y. Gorobets, S. Reshetnyak, Reflections of spin waves from a ferromagnetic multilayer with interfacial coupling. Cent. Eur. J. Phys. 6(1) (2008) 28. V.K. Ignatovich, Usp. Fiz. Nauk, Étude on the one-dimensional periodic potential. Sov. Phys. Usp. 29 879 (1986) 29. A. Fushimi et al., Fast calculation of the quality factor for two-dimensional photonic crystal slab nanocavities. Opt. Exp. 22.19 (2014)

Chapter 7

Magnetic Field Feedback Oscillator: A Micromagnetic Study

7.1 Introduction Spintronic oscillators based on either the fully metallic GMR spin valves or magnetic tunnel junctions (MTJ) are very attractive for potential applications in nanoscale devices. These oscillators are working based on the spin transfer torque (STT) effect [1, 2]. It has been predicted in [1, 2] that direct electric current passing through a magnetized magnetic layered structure becomes spin-polarized, and, if the current density is sufficiently high, this spin-polarized current can transfer sufficient spin angular momentum between the magnetic layers to alter the static equilibrium orientation of magnetization in the thinner (free layer). These phenomena can lead either to the magnetization switching or to the excitation of magnetization precession. The frequency of precession depends on injected current and lies in the microwave range.

7.2 Feedback Oscillator with Magnetic Field Feedback: A Closed Loop System Study The major problem that arises during the development and application of spintronic oscillators is the small output power and large linewidth. At zero temperature, and in absence of any noise, the linewidth is expected to be zero. At non-zero temperature or in the presence of any other noise, the linewidth is non-zero. In most spintronic oscillator configurations, these linewidths are too large and great efforts are currently undertaken in order to understand the origin of linewidth broadening and to define a configuration to reduce it. Some work has been already done to improve the output power and oscillation linewidth [3–5]. Spintronic oscillator with magnetic field feedback provides a better linewidth [7]. The schematic diagram of the oscillator is shown in Fig. 7.1. The RF output voltage is fed back to the coplanar waveguide which lies just above the MTJ-free layer. As a result of the feedback, a RF magnetic field © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. S. Nikhil Kumar, Magnonic Devices, SpringerBriefs in Materials, https://doi.org/10.1007/978-3-031-22665-6_7

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7 Magnetic Field Feedback Oscillator: A Micromagnetic Study

Fig. 7.1 Spintronic oscillators with magnetic field feedback

is generated around the waveguide which influences the free layer magnetization fluctuation. We realize the spintronic magnetic field feedback oscillators through micromagnetic simulation. The simulations have been undertaken by using version 3 of MuMax [8], an open-source computational tool for micromagnetic simulations implemented by graphics processing units (GPU). We perform the micromagnetic simulation of oscillators with and without magnetic field feedback and analyzed the spectrum in both cases.

7.2.1 Micromagnetic Simulation Without Magnetic Field Feedback The schematic diagram of the oscillator is shown in Fig. 7.2. The easy direction of the free layer is assumed to be along the x-axis, in-plane hard direction along the y-axis and out-of-plane hard direction along the z-axis. The pinned layer magnetization is assumed to be along the x-axis. In equilibrium, both the free and fixed layers are along the x-axis. The external magnetic field is applied along the y-axis. This is done to make a non-zero angle between free and fixed layers. If we apply the magnetic field more than the coercive field of the free layer, the magnetization of the free layer will tend to orient along the y-axis. The oscillator sits on the top of the waveguide which is electrically insulated from the devices. The top and bottom electrodes of the oscillator are connected through a dc source as shown in Fig. 7.2. A dc current, passing through a MTJ, gets polarized by the pinned layer, exerts a STT on the free layer and causes fluctuations in magnetization in the free layer. We investigate the spin wave dynamics in the free layer by micromagnetic simulation.

7.2.2 Free Layer Model The sample is assumed as an ellipse with dimensions 500 × 300 × 3 (all are in nm) as shown in Fig. 7.3. The disc was discretized into cells of 5 × 5 × 1 nm3 . The material parameters used for the simulation are the following: Ms (Co40 Fe40 B20 ) = 1100 ×

7.2 Feedback Oscillator with Magnetic Field Feedback: A Closed Loop System Study

65

Fig. 7.2 Schematic of STNO. The top layer of the device shows the free layer, middle layer shows the spacer and bottom layer shows the pinned layer. The MTJ rests on the top of a waveguide which is electrically insulated from the MTJ. The waveguide is terminated into a resistance RT as shown Fig. 7.3 Ellipse with dimensions 500 × 300 × 3 (all are in nm)

103 A/m for the saturation magnetization and A (Co40 Fe40 B20 ) = 13 × 10−12 J/m for the exchange stiffness constant. The damping factor is initially assumed as zero to extract the ferromagnetic resonance characteristics. Temperature is assumed as 300 K.

7.2.3 Free Layer Hysteresis Loops We simulate the in-plane and out-of-plane hysteresis loop of the model. Initially, we sweep the magnetic field along the major axis (along x). The obtained hysteresis loop is shown in Fig. 7.4. We observed the state of magnetization at the reach region as marked as A,B, C, D as shown in Fig. 7.4. The static magnetization states at each region are shown in Figs. 7.5 and 7.6. At regions A and C, magnetization is oriented along x- and −x-axis, respectively. At regions B and D, magnetization starts rotating and tend to align along −x and +x, respectively. The hysteresis loop obtained by sweeping the external field along the y-direction is shown in Fig. 7.7. The initial magnetization is along the major axis. This is mainly due to the shape anisotropy field. We observed that (from a few states at region A) magnetizations get tilted and get aligned along y-direction as we increase the field (along y). At Regions B and D, magnetization gets aligned along the minor axis of the elliptical sample. This is shown in Fig. 7.8. Magnetization starts rotating from positive saturation to negative saturation at region C. The reverse scenario happened at region E (Fig. 7.9).

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7 Magnetic Field Feedback Oscillator: A Micromagnetic Study

Fig. 7.4 In-plane hysteresis loop

Fig. 7.5 Static magnetization at regions A and C

The out-of-plane hysteresis measured is shown in Fig. 7.10. The magnetization gets saturated at 1.4 × 104 Oe. Thus, the demagnetizing field |Hd | =1.4 × 104 Oe. The magnetization at regions A and B is shown in Fig. 7.11. Initially, magnetization lies along the easy axis. We need a large out-of-plane external field to pop out the magnetization. At point B, magnetization is completely oriented along z-axis.

7.2.4 Ferromagnetic Resonance Frequency Versus Applied Field We observed the ferromagnetic resonance frequency for different applied fields. The damping factor is assumed as zero for extracting the resonance frequency. We initially applied a field along the hard axis (along y) and relax it for some time and run for a finite duration (50 ns). We analyzed the spectrum of the full 50 ns time series and

Fig. 7.6 Static magnetization at regions B and D

7.2 Feedback Oscillator with Magnetic Field Feedback: A Closed Loop System Study

67

68

7 Magnetic Field Feedback Oscillator: A Micromagnetic Study

Fig. 7.7 Hysteresis loop by sweeping the magnetic field along y-direction (left). Few magnetization states at region A(right)

Fig. 7.8 Magnetization states at B and D

noted the main peak in the spectrum. This is repeated for different applied fields. The resonance frequency obtained for different fields is plotted and compared with experimentally measured values.

7.2.4.1

FMR Versus Applied Field for Different In-plane and Out-of-Plane Anisotropy

We initially fixed a small in-plane anisotropy field, H//,y = 10 Oe to fix the coercive field along x as 50 Oe exactly. FMR versus applied field is observed for different out-of-plane anisotropy fields. The obtained result is shown in Fig. 7.12 (left). For non-zero H⊥ , the minimum frequency is shifted to 110 Oe.

7.2.4.2

FMR Versus Applied Field for Different Out-of-Plane Anisotropy

We observed the FMR for various out-of-plane anisotropy fields. The obtained FMR curve for different out-of-plane anisotropy fields is shown in Fig. 7.12 (right). We obtained a reasonable agreement with experimentally measured values.

Fig. 7.9 Few magnetization states at region region C and E

7.2 Feedback Oscillator with Magnetic Field Feedback: A Closed Loop System Study

69

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7 Magnetic Field Feedback Oscillator: A Micromagnetic Study

Fig. 7.10 Out-of-plane hysteresis loop

Fig. 7.11 Magnetization at regions A and B

Fig. 7.12 FMR versus applied field for different anisotropy

7.2 Feedback Oscillator with Magnetic Field Feedback: A Closed Loop System Study

71

Fig. 7.13 Current dependence of f 0 with Hy = 60 Oe

7.2.5 Current Dependence on Resonance Frequency The effect of electric current on FMR frequency is studied. In our geometry, the angle between fixed and free layer magnetization is 90◦ and the current is injected along z-direction. I observed the FMR frequency for different currents. The thermal effect was neglected for simplicity. Figure 7.13 shows the current dependence of resonance frequency f 0 for Hy = 60 Oe. Current dependent shifts in the resonant frequency are due to non-linear effects. For Hy = 60 Oe, f 0 shifted to low-frequency side (frequency red shift effects).

7.2.6 Spintronic Oscillators with Magnetic Field Feedback We introduce a feedback mechanism in MTJ. As shown in the Fig. 7.14, a delay element is present in the circuit to adjust the phase of the current. The ac current flowing through the feedback wire at a position below the MTJ is given by Eq. 7.1 Iac (t) = −0.5

R Idc < m x (t − t) . > RT + RMTJ

(7.1)

The ac current flowing below the MTJ depends on the value of m x at time (t − t) due to delay element and cables connected in the circuit. The value of RT and RMTJ is 100 and 50 , respectively. R is taken to be 40 . A fluctuating voltage across the MTJ creates an ac current and this current is going through the feedback wire. We assume that the feedback wire is oriented along the y-axis. The current through the wire creates an Oersted field around the sample as

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7 Magnetic Field Feedback Oscillator: A Micromagnetic Study

Fig. 7.14 Spintronic oscillators with magnetic field feedback Fig. 7.15 Oersted field around the sample

shown in Fig. 7.15. We need to add an additional ac magnetic field along x- and z-directions. The resulting magnetic field acting on the free layer is given by h ac ≈ Iac /2w,1 where w is the width of the feedback wire. Thus, magnetic field acting on the free layer is given by h ac (t) = − f Idc < m x (t − t) >, (7.2) where the factor f is defined as f = R/ [4w (RT + RMTJ )].

7.2.7 Spin Wave Dynamics with Magnetic Field Feedback We observed the spin wave dynamics with magnetic field feedback. We initially put the temperature as zero to understand the effect of damping alone. The schematic diagram is shown in Fig. 7.16. We set a non-zero damping factor, α = 0.005 in this analysis. We initially observed the spin wave dynamics without magnetic field feedback. Magnetization decays quickly due to finite damping. We turn on the feedback signal after 5.0 ns and observed the spin wave dynamics. The obtained spatially averaged magnetization < m x (t) > with and without magnetic field feedback is shown in 1

ac current creates an Oersted magnetic field which varies with distance. In real, the Oersted field varies as 1/r . This approximation corresponds to assuming zero distance between the feedback wire and the free layer. A more accurate expression involves the distance between the free layer and feedback wire.

7.2 Feedback Oscillator with Magnetic Field Feedback: A Closed Loop System Study

73

Fig. 7.16 MTJ with magnetic field feedback

Fig. 7.17 Spatially averaged magnetization with and without magnetic field feedback

Fig. 7.17. We observed that with feedback, the amplitude of the spin wave increases gradually and reaches sustained oscillations.

7.2.8 Spin Wave Dynamics at 300 K We included thermal effects in this analysis by adding a thermal field Btherm according to Brown [6]:  2μ0 αk B T → , (7.3) Btherm = − η (step) Bsat γLL V t where α is the damping factor, k B the Boltzmann constant, T the temperature, Bsat the saturation magnetization expressed in Tesla, γLL the gyromagnetic ratio (1/Ts), V → the cell volume, t is the time steps and − η (step) a random vector from a standard normal distribution whose value is changed after every time step. The thermal effects can be included as defined in the script Sect. A.3.

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7 Magnetic Field Feedback Oscillator: A Micromagnetic Study

Fig. 7.18 MTJ with feedback at room temperature

Fig. 7.19 Spin wave spectra with and without magnetic field feedback

The schematic diagram of the setup with finite temperature is shown in Fig. 7.18. Temperature is gradually increased in order to reach the system in a noisy steady state quickly. Initially, simulation is performed with T = 0 K for 20 ns. The magnetization state is saved every 10 ps. The magnetization state at 20th ns is given as the ground state for 100 K simulation and again performed the simulation for 100 ns. The data saved at 100th ns is given as a ground state for T = 200 K simulation and again performed the simulation for 100 ns. This is repeated for 300 K and performed the simulation for a longer duration (500 ns) to ensure that a noisy steady state is reached. The obtained spin wave spectra(with and without feedback) for full 500 ns time series is shown in Fig. 7.19 The value of magnetization is saved for every 0.01ns, for a total simulation of 500 ns and the data is broken up into 5.26 ns pieces. The obtained spectrogram (with and without magnetic field feedback) is shown in Fig. 7.20.

7.2.8.1

Observations

From Fig. 7.20, it is observed that oscillations get improved by magnetic field feedback. The effect of thermal noise can be nullified by the feedback mechanism. The main peak occurs at a resonance frequency. If we see the spin wave spectra (without magnetic field feedback), it is hard to extract the linewidth. I think we can say that there is a significant improvement in the linewidth of the spectrum with magnetic

7.3 Conclusion

75

Fig. 7.20 Spectrogram of magnetization with and without feedback signal

field feedback. The linewidth of spin wave spectra will be larger (if we take the envelope of the spectrum) as compared to the linewidth of spin wave spectra with magnetic field feedback.

7.2.9 Spin Wave Spectra with Different Delays We observed the spin wave spectra with different delays (see Fig. 7.21). The resonance frequency does not change with delay time. But for a certain delay, we get sharp peaks with less noise. In all cases, we obtain multiple peaks with the main peak at a resonance frequency. We need to properly fix the delay time for better response. If the delay is an integer multiple of time period (nT)of signal, the feedback signal will add to the existing signal constructively. In that case, we would obtain a better response with less noise. As the time period of the signal is in ns, the delay also should be in nanosecond.

7.3 Conclusion We simulated the spin torque nano-oscillator with magnetic field feedback. The static and dynamic characterization were analyzed in detail. With magnetic field feedback, we observed an improvement in the linewidth of the spectrum. The effect of thermal noise can be nullied using magnetic field feedback.

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7 Magnetic Field Feedback Oscillator: A Micromagnetic Study

Fig. 7.21 Spin wave spectra with different delays

References 1. F.C. MaciÃ, F.C. Hoppensteadt, A.D. Kent, Spin wave excitation patterns generated by spin torque oscillators. Nanotechnology 25, 045303 (2014) 2. Z.K. Wang, V.L. Zhang, H.S. Lim, S.C. Ng, M.H. Kuok, S. Jain, A.O. Adeyeye, Observation of frequency band gap in a one dimensional nanostructured magnonic crystals. Appl. Phys. Lett. 94, 083112 (2009) 3. J. Ding, M. Kostylev, A.O. Adeyeye, Broadband ferromagnetic resonance spectroscopy of permalloy triangular nanorings. Appl. Phys. Lett. 100, 062401 (2012) 4. H. Yu, G. Duerr, R. Huber, M. Bahr, T. Schwarze, F. Brandl, D. Grundler, Omnidirectional spin-wave nanograting coupler. Nat. Commun. 4, 2702 (2013) 5. K. Vogt, F. Fradin, J. Pearson, T. Sebastian, S. Bader, B. Hillebrands, A. Hoffmann, H. Schultheiss, Realization of a spin-wave multiplexer. Nat. Commun. 5, 3727 (2014) 6. J.C. Slonczewski, Current-driven excitation of magnetic multilayers. J. Magn. Magn. Mater. 159, L1–L7 (1996) 7. D. Dixit, K. Konishi, C.V. Tomy, Y. Suzuki, A.A. Tulapurkar, Spintroics oscillator based on magnetic field feedback. Appl. Phys. Lett. 101, 122410-1-4 (2012) 8. A. Vansteenkiste, J. Leliaert et al., The design and verification of MuMax3. AIP Adv. 4, 107133 (2014)

Appendix

A.1 Derivation of Permeability Tensor In the forward volume configuration, the applied static field is assumed to be along z and small signal variations are in the (r, φ)-plane. Thus, the total magnetic field H is written as H = Hz a z +h r ar + h φ aφ .

(A.1)

This field produces a total magnetization M = Mz a z +m r ar + m φ aφ

(A.2)

whose dynamics are governed by dM = −γ μ0 [M × H] . dt

(A.3)

Substituting (109) and (110) in (111) gives us equations of motions   dm r = −γ μ0 Hz m φ − Mz h φ , dt

(A.4)

  dm φ = −γ μ0 Mz h r − Hz m r , dt

(A.5)

dm r = −ω0 m φ + ωm h φ , dt

(A.6)

dm φ = −ωm h r + ω0 m r , dt

(A.7)

which are simplified as

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. S. Nikhil Kumar, Magnonic Devices, SpringerBriefs in Materials, https://doi.org/10.1007/978-3-031-22665-6

77

78

Appendix

where ω0 = γ μ0 Hz and ωm = γ μ0 Mz . Differentiating once more yields dh φ d 2mr = −ω0 [−ωm h r + ω0 m r ] + ωm , 2 dt dt

(A.8)

  dh r d 2mφ . = +ω0 −ω0 m φ + ωm h φ − ωm dt 2 dt

(A.9)

If the small signal excitation h has a eiωt time-harmonic dependence, the AC steady state form of (116 and 117) reduces to   2 ω0 − ω2 m r = ω0 ωm h r + jωωm h φ , 

 ω02 − ω2 m φ = ω0 ωm h φ − jωωm h r .

(A.11)

¯ = χ¯¯ ·h, m

(A.12)

This is rewritten as

or

(A.10)

⎛

mr

⎞

⎛

jωωm ω0 ωm ω02 −ω2 ω02 −ω2

⎟ ⎜ − jωωm ⎜ ⎝ mφ ⎠ = ⎜ ⎝ ω02 −ω2 mz 0

ω0 ωm ω02 −ω2

0

0

⎞⎛

hr

⎞

⎟⎜ ⎟ 0⎟ ⎠ ⎝ hφ ⎠ . 1

(A.13)

hz

A.2 General Solution of Walker’s Equation Inside the material, we must satisfy Walker’s equation, 

∇ · μ¯¯ · ∇ψin = 0,

(A.14)

written in cylindrical coordinates as (1 + χ )



1 ∂ ∂ 2 ψin ∂ψin 1 ∂ 2 ψin + r + 2 = 0. r ∂r ∂r r ∂φ 2 ∂z 2

(A.15)

Outside the material, we must satisfy Laplace’s equation ∇ · (∇ψout ) = 0,

(A.16)



∂ψout 1 ∂ 2 ψout ∂ 2 ψout 1 ∂ r + 2 + = 0. r ∂r ∂r r ∂φ 2 ∂ 2 z2

(A.17)

or in cylindrical coordinates

Appendix

79

Using variable separation, ψout = R (r ) (φ)Z (z).

(A.18)

Substituting (A.18) in (A.17), we get Z d r dr

  dR R Z d 2 d2 Z r + 2 + R = 0, dr r dφ 2 dz 2

(A.19)

dividing (A.19) by RZ , we get 1 d Rr dr

  dR 1 d 2 1 d2 Z r + + = 0. dr r 2 dφ 2 Z dz 2

(A.20)

The φ dependence and z dependence are given by 1 d2  dφ 2

= −kφ2 ,

1 d2 Z Z dz 2

−k z2 .

(A.21) =

The solution of (A.21) will be in the form Z (z) = eikz z ,  (φ) = eikφ φ ,

(A.22)

where k z has the unit of rad/nm and kφ is a dimensionless quantity. Substituting (A.21) in (A.20) and multiplying by r 2 , we get r2

 d2 R d R  2 2 − k R = 0. + r + r (k ) φ z dr 2 dr

(A.23)

This equation is in the form of a modified Bessel function [1] r2

 d2 R dR  2 2 − m R = 0. + r + r dr 2 dr

(A.24)

The general solution for (A.24) will be either Im (r ) or K m (r ) (modified Bessel function). Im (r ) will be an exponentially increasing and K m (r ) will be an exponentially decaying function. Now solution for (2.16) is given by ψout

 C Ikφ (k z r ) r < R2 , = D K kφ (k z r ) r > R1 .

(A.25)

80

Appendix

A.3 Micromagnetic Simulation Script package main import (

"github.com/mumax/3/cuda" . "github.com/mumax/3/engine"

) import "fmt" import "math" func main() { cuda.Init(0) InitIO("With_feedback.mx3", "With_feedback.out", true) GoServe(":35367") defer Close() var x [50002]float64

// Array Declaration for storing .

SetGridSize(100, 60, 1)

//Set grid size

SetCellSize(500e-9/100, 300e-9/60, 3e-9/1)

//Set cell size

SetGeom(Ellipse(500e-9, 300e-9) )

//Set Geometry

DefRegion(1, Ellipse(500e-9, 300e-9)) //Material parameters Msat.Set(1100e3) Aex.Set(13e-12) Alpha.Set(0.005) //Spin torque parameter Lambda.Set(1) Pol.Set(0.5669) EpsilonPrime.Set(0) FixedLayer.SetRegionFn(1, fixed) J.SetRegionFn(1, current) M.Set(Uniform(0.0, 0.9, 0.1)) Relax()

//Assume some initial magnetization //Set anisotropy field and direction

Ku1.Set(5.0e3) AnisU.SetRegionFn(1,anisVect); AutoSave(&M, 1e-11) TableAutoSave(1e-11) B_ext.Set(Vector(0.00, 0.008, 0))

//Save Space dependant magnetization ovf format //Save spatially averaged magnetization //External field of 80 Oe is set along y direction

i:=0 for t:=0.0; t