Properties and dynamics of spin waves in one and two dimensional magnonic crystals

Transcription

1 University of Iowa Iowa Research Online Theses and Dissertations Summer 16 Properties and dynamics of spin waves in one and two dimensional magnonic crystals Glade Robert Sietsema University of Iowa Copyright 16 Glade Robert Sietsema This dissertation is available at Iowa Research Online: Recommended Citation Sietsema, Glade Robert. "Properties and dynamics of spin waves in one and two dimensional magnonic crystals." PhD (Doctor of Philosophy) thesis, University of Iowa, Follow this and additional works at: Part of the Physics Commons

2 PROPERTIES AND DYNAMICS OF SPIN WAVES IN ONE AND TWO DIMENSIONAL MAGNONIC CRYSTALS by Glade Robert Sietsema A thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Physics in the Graduate College of The University of Iowa August 16 Thesis Supervisor: Professor Michael E. Flatté

3 Copyright by GLADE ROBERT SIETSEMA 16 All Rights Reserved

4 Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVAL PH.D. THESIS This is to certify that the Ph.D. thesis of Glade Robert Sietsema has been approved by the Examining Committee for the thesis requirement for the Doctor of Philosophy degree in Physics at the August 16 graduation. Thesis Committee: Michael E. Flatté, Thesis Supervisor David Andersen Maxim Khodas Craig Pryor Markus Wohlgenannt

5 To my family ii

6 ACKNOWLEDGMENTS I would like to thank my advisor, Michael E. Flatté for his assistance and guidance throughout my graduate studies. Without him, none of this would have been possible. Thanks also to my family for their continued support through this long process. I would also like to thank my first physics teacher, Jeff Saueressig, for getting me interested in physics, as well as all the professors at Gustavus for expanding both my interest and knowledge in the field. Thanks also to everyone at Zion for all the wonderful years of music we had while I was in Iowa City, I will never forget it. iii

7 ABSTRACT Spintronics is a newly emerging field in physics, aimed at using the spin of electrons to carry information. One of the primary ways in which this could be done is through the use of spin waves. In order to do this, it will be necessary to have a complete understanding of spin waves and how they behave in various materials and structures. In this dissertation, we aim to create a thorough model of spin waves in both one-dimensional and two-dimensional magnonic crystals in an effort to understand and control their dispersion properties and propagation patterns. Using the Landau-Lifshitz-Gilbert equation, we have derived a model for spin waves in magnonic crystals that allowed us to calculate their dispersion and propagation properties. In the first part of this work we considered two-dimensional magnonic crystals consisting of magnetic cylinders arrange in a lattice and embedded in a second magnetic material. The dispersion relations were found to be heavily dependent on the magnetic properties of the two materials, with band gaps appearing more readily when the magnetization was larger in the cylinders than in the host. It was also found that the dipolar field reduced the symmetry of the results, with reflection symmetry not appearing in the dispersion relations even when it was present in the physical lattice. For the propagation of spin waves in two-dimensional magnonic crystals, we found that their directionality was highly dependent on changes in frequency. Propagation patterns varied from roughly isotropic for spin waves in the middle of a band level, to highly directional propagation along the x and y axes for a frequency near the edge of a band. The absence of propagation was also found for frequencies in a band gap. For spin waves in one-dimensional magnonic crystals, we investigated the effects of applying an electric field to the system. When a uniform electric field was applied to a magnonic crystal consisting of a periodic variation in magnetic materials, the band levels were found to shift downward in frequency, with the magnitude of the shift being iv

8 dependent on the strength of the electric field. While this method could move existing band gaps, it was not capable of creating a band gap in the dispersion relations. Creation of band gaps was found to occur when a periodically varying electric field was applied to a uniform magnetic material. This effect could be used to create a magnonic device where the dispersion properties can be dynamically controlled with an electric field. v

9 PUBLIC ABSTRACT A newly emerging field in physics is spintronics. While electronic devices use the charge of electrons, spintronic devices are aimed at using the spin of electrons to carry information. One way of doing this is with a spin wave, which is a propagation of a magnetic disturbance in a material. Unlike in electronic devices, utilizing spin waves does not require a transfer of charge, which could result in the power consumption of spintronic devices being much lower than their electronic counterparts. Structures with periodic variations in their magnetic properties, known as magnonic crystals, can be used to control the properties of spin waves. With twodimensional magnonic crystals, we demonstrate that the propagation direction of spin waves can be controlled by varying their frequency, and propagation can even be blocked if that frequency is in a band gap. We also demonstrate that an electric field can be used to further control spin waves in a one-dimensional magnonic crystal by opening up band gaps, frequency regions in which spin waves are forbidden from traveling. This effect could be used in the development of a magnon transistor, in which the propagation of a spin wave through a device could be controlled by turning an electric field on or off. vi

10 Contents LIST OF FIGURES ix CHAPTER 1 INTRODUCTION Magnonics Magnetic Materials Ferromagnetic Resonance and the Suhl Instability Spin Wave Modes in a Thin Film Non-reciprocity of Spin Waves Landau-Lifshitz Equation Origins of Spin Wave Damping Magnonic Crystals TWO-DIMENSIONAL MAGNONIC CRYSTALS.1 Magnonic Structure Theoretical Model Landau-Lifshitz-Gilbert Equation Dipolar Field Exchange Field Plane-Wave Expansion Method Summary SPIN WAVE DISPERIONS RELATIONS IN TWO-DIMENSIONAL MAGNONIC CRYSTALS Solutions of the LLG Equation Dispersion Relations and Band Gaps Contours Alternate Exchange Field Symmetry Breaking Dispersions without the Dipolar Field Comparison of Dipolar and Exchange Fields Convergence of Dispersion Relations Summary SPIN WAVE PROPAGATION PATTERNS IN TWO-DIMENSIONAL MAGNONIC CRYSTALS Response to Spin Wave Sources Green s Function Dynamic Spin Wave Response Spin Wave Propagation Propagation Patterns in the Frequency Domain Propagation Patterns in the Time Domain Convergence of Propagation Calculations ONE-DIMENSIONAL MAGNONIC CRYSTALS WITH APPLIED ELECTRIC FIELDS 6 vii

11 5.1 Theoretical Model LLG Equation Demagnetization Field Boundary Conditions Electric Field and the Dzyaloshinsky-Moriya Interaction Plane-Wave Expansion Results Periodic Material Periodic Electric Field Transverse Modes Summary CONCLUSION Summary The Landau-Lifshitz-Gilbert Equation Dispersion Relations Propagation Patterns Modifying Dispersion Relations with Electric Field Possible Extensions of this Work to Additional Situations Enhance Symmetry Breaking Large Differences in Magnetic Properties Two-Dimensional Magnonic Crystal Slab Tunability of Band Gap Dependence of Propagation on Cylinder Cross Section Spin Wave Propagation with Electric Fields BIBLIOGRAPHY viii

12 LIST OF FIGURES Figure 1.1 A propagating spin wave. The magnetic moments (red arrows) oscillate at small angles around the ground state magnetization The four categories of magnetic materials Ferromagnetic material in which all magnetic moments are aligned (left) and when domains are formed (right) Magnetic component of heat capacity of (a) Fe and (b) Co as a function of temperature. The dots indicate experimental results obtained from calorimetric data, and the lines are theoretical predictions Configurations for propagation of FVMSW (left), BVMSW (middle), and MSSW (right) Propagation of a FVMSW Propagation of a BVMSW Propagation of a MSSW Dispersions of the three different spin wave modes in a thin film using µ H =.7T, M = A/m, and d = 1 nm Demonstration of the Faraday effect Diagram of a Faraday circulator Example of an etched one dimensional magnonic crystal Example of a one dimensional width modulated magnonic crystal Example of a one-dimensional crystal created with layers of alternating cobalt and permalloy Example of a two-dimensional magnonic crystal Structure of the two-dimensional magnonic crystals studied Comparison of Fourier series calculation of saturation magnetization with (top) and without (bottom) using the Lanczos sigma factor with N 1 = ix

13 3.1 Empty square lattice band structure for Fe (a), Co(b), Ni(c), and YIG (d) Empty hexagonal lattice band structure for Fe (a), Co(b), Ni(c), and YIG (d) Square lattice band structures for Fe cylinders in Co (a), Ni (b), and YIG (c), and for Co (d), Ni (e), and YIG (f) cylinders in Fe Hexagonal lattice band structures for Fe cylinders in Co (a), Ni (b), and YIG (c), and for Co (d), Ni (e), and YIG (f) cylinders in Fe The lowest nine spin wave frequencies (in THz) for a square lattice magnonic crystal composed of Fe cylinders in Ni. The square figures show the entire Brillouin zone of the lattice The spin wave relaxation rates (in GHz) corresponding to the spin wave modes in Fig The square figures show the entire Brillouin zone of the lattice The lowest nine spin wave frequencies (in THz) for a hexagonal lattice magnonic crystal composed of Fe cylinders in Ni. The hexagonal figures show the entire Brillouin zone of the lattice The spin wave relaxation rates (in GHz) corresponding to the spin wave modes in Fig The hexagonal figures show the entire Brillouin zone of the lattice Empty square lattice band structure for Fe (a), Co(b), Ni(c), and YIG (d) obtained when using the exchange field in Eq Square lattice band structures for Fe cylinders in Co (a), Ni (b), and YIG (c), and for Co (d), Ni (e), and YIG (f) cylinders in Fe obtained when using the exchange field in Eq The lowest nine spin wave frequencies (in THz) for a square lattice magnonic crystal composed of Fe cylinders in Ni obtained when using the exchange field in Eq The square figures show the entire Brillouin zone of the lattice The spin wave relaxation rates (in GHz) corresponding to the spin wave modes in Fig The square figures show the entire Brillouin zone of the lattice The lowest nine spin wave frequencies (in THz) for a square lattice magnonic crystal composed of Fe cylinders in Ni obtained the dipolar field is set to zero. The square figures show the entire Brillouin zone of the lattice x

14 3.14 The spin wave relaxation rates (in GHz) corresponding to the spin wave modes in Fig The square figures show the entire Brillouin zone of the lattice Symmetry of dispersion relations calculated with the dipolar field Symmetry of dispersion relations calculated without the dipolar field Dipolar field (top row) and exchange field (bottom row) for a square lattice magnonic crystal composed of Fe cylinders in Ni obtained when using the derived exchange field in Eq Dipolar field (top row) and exchange field (bottom row) for a square lattice magnonic crystal composed of Fe cylinders in Ni obtained when using the alternate exchange field in Eq Convergence of Fourier series for saturation magnetization for N 1 = 6 (top left), N 1 = 1 (top right), and N 1 = (bottom) Convergence of spin wave modes for a square lattice of Fe cylinders in Ni. The top shows mode 1 and the bottom shows mode Dispersion (left) and linewidth (right) plots for a magnonic crystal consisting of Fe cylinders in a YIG host with lattice constant a = 5nm (top) a = 5nm (bottom). The horizontal lines indicate the three frequencies at which propagation patterns were calculated Dispersion and linewidth contours for the first three modes of Fig. (4.1) with a = 5nm Green s Functions, multiplied by distance from source, showing the steady-state spin wave response to a point source of various frequencies located at the center of one of the Fe cylinders in the magnonic crystal. The units are A s/m Spin wave propagation patterns resulting from a spin wave pulse in one of the Fe cylinders at ω 1 = 5.5GHz Spin wave propagation patterns resulting from a spin wave pulse in one of the Fe cylinders at ω = 8.1GHz Spin wave propagation patterns resulting from a spin wave pulse in one of the Fe cylinders at ω 3 = 31.7GHz One-dimensional magnonic crystal with a periodic variation in the magnetic properties xi

15 5. Schematic diagram of the YIG slab with interdigitated electrodes to generate a periodically-varying electric field. The magnetization (M) is in-plane, and the size of the slab along the ŷ and ẑ directions is small enough that the transverse modes do not influence the spin-wave propagation Direct exchange coupling between neighboring spins Superexchange coupling between spins through an intermediate nonmagnetic ion Dispersion relations and linewidths for a one-dimensional magnonic crystal with a periodic variation in magnetic properties and a uniform applied electric field of E = (left) and E = V/m (right) Spin wave dispersions of YIG with a = nm, E A =, and E B = (top) and E B = V/m (bottom) Density of States for zero electric field (bottom), uniform electric field( middle), and periodic electric field (top) Variation of band gap width and center between the first two modes at a = nm (top) and E = V/m (bottom) Modification of spin wave dispersion relations when k z = m Modification of spin wave dispersion relations when k y = m xii

16 1 CHAPTER 1 INTRODUCTION 1.1 Magnonics Magnonics is a relatively young field in solid state physics that focuses on the study of spin waves. Consider a homogeneous magnetic material in which the spins are all aligned. If one of the spins precesses about this alignment axis, the precession will propagate through the material (Fig. 1.1). This is one example of a spin wave, which was first conceived of by Bloch in 193 [1]. When considering the scale at which classical mechanics breaks down, there must be a quantized model of spin waves that satisfies quantum mechanics. The first theoretical model of these spin wave quanta, later to be called magnons, was laid out by Holstein and Primakoff []. From a model consisting of a ferromagnetic array of spins in an external magnetic field, they obtained the following Hamiltonian: H HP = C + A λ a λa λ + ( 1 B λa λ a λ + 1 ) B λa λa λ. (1.1) λ λ Where a λ and a λ are the creation and annihilation operators for spin waves. Their results consisted of a quadratic part, which alone would describe a theory of noninteracting spin waves, and a higher order part, describing interacting spin waves. This would indicate that the quadratic part could be looked at as a first approximation of a spin wave interaction model, with the non-quadratic part being treated with perturbation theory. It was later pointed out by Dyson that the Holstein-Primakoff Hamiltonian is only accurate for large external magnetic fields [3]. For low frequency spin waves in a weak external field, the non-quadratic part of the Hamiltonian will dominate over the quadratic part. The Hamiltonian derived by Dyson is H D = E + λ (L + ɛ λ ) a ja j 1 4 JN 1 λ,ρ,δ a σ+λa ρ λa ρ a σ Γ λ ρσ. (1.)

17 This model also showed that the interaction between spin waves is weak, and was grossly overestimated by the Holstein-Primakoff model. Figure 1.1: A propagating spin wave. The magnetic moments (red arrows) oscillate at small angles around the ground state magnetization. Griffiths performed the first experimental detection of spin waves for the case of uniform precession via ferromagnetic resonance (FMR) [4]. Resonant peaks with linewidths on the order of 1 Oe were found in Fe, Co, and Ni, and were indications of the uniform spin wave mode, that is, spin waves with a wave vector of zero. The width of these peaks comes from magnetic damping, and the mechanisms that bring this about include two-magnon scattering from surface imperfections, magnetic impurities, and coupling to phonons [5, 6]. These mechanisms, among others, will be discussed in further detail in section There have been several recent advances made in the concept and construction of magnonic devices. One area that has received relatively significant attention is the topic of spin wave logic devices. Research done here has focused on using the wave properties of magnonics to construct logic devices. One method of doing this involves utilizing the amplitude of the wave in a Mach-Zehnder interferometer configuration [7 9]. Information for logic devices could also be carried in domain walls between magnetic regions [1], or in the phase of the spin wave [11 14]. The latter method of carrying information also has the potential for use in spin wave interconnect devices [11, 1], allowing for an interface between electronic circuits and spintronic circuits. A more detailed analysis on the recent state of spin wave logic devices can

18 3 be found in references [15] and [16]. Tunable filters are another area of interest in the field of magnonics [17 19]. Such devices can use the properties of multi-layered microstrips or magnetic lattices to be able to filter out certain frequencies. The frequencies that are filtered can than be tuned by adjusting some parameter of the system, such as the strength of an external magnetic field. There are several advantages to using magnonic devices described above as opposed to their electronic counterparts. For example, in spintronics, information is carried in the spin wave itself, which requires no transfer of charge, unlike electronics. Without the need to transfer charge, the power consumption of magnonic devices may be much lower than in electronic devices. However, they do have some downsides, with higher signal attenuation caused by magnon-electron, magnon-phonon, and magnonmagnon scattering, and lower signal speed, which is limited by the strength of the exchange interaction [1]. There have been some methods proposed for counteracting the attenuation by amplifying the signal using either the magnetoelectric effect [] or the spin-torque effect [1] Magnetic Materials There are several different classifications of magnetism that can occur in a material: ferromagnetic, paramagnetic, antiferromagnetic, and ferrimagnetic (1.). The different categories relate to the structure of the magnetic moments in the material. For paramagnets, the spins are disordered unless they are in the presence of an applied magnetic field, in which case the spins will align with the field. In ferromagnets, the spins are aligned even in the absence of a magnetic field. Both antiferromagnets and ferrimagnets have neighboring spins with opposite alignment. The difference between them is in the relative magnitudes of the magnetic moments. In antiferromagnets, both moment directions have the same magnitude, resulting in no net magnetization

19 4 for the material, while in ferrimagnets, the moment in one direction is larger than the other. Figure 1.: The four categories of magnetic materials. Of these four magnetic categories, paramagnetism is the only one that requires the presence of an applied magnetic field to be in an ordered state. In this case, the degree of magnetization is determined by the strength of the applied field and by the material s paramagnetic susceptibility: M = χ p H. (1.3) Where the susceptibility, χ p, is dependent on temperature []: χ p = C T. (1.4) This relation is known as Curie s law, and C is the material dependent Curie constant. This expression was determined experimentally by Curie and only holds for high

20 5 temperatures or weak magnetic fields. While Curie s law is sufficient for describing the temperature dependence of susceptibility for paramagnets, it fails to do so for ferromagnets. Since ferromagnets exhibit magnetization even in the absence of an applied magnetic field, Eq. 1.3 needs to be modified to account for this. Weiss proposed that for ferromagnets, there was a large internal magnetic field acting on the magnetic moments. This is known as the Weiss molecular field, or exchange field: Adding this term in Eq. 1.3 gives where the new susceptibility is given by Curie-Weiss law: H ex = λm. (1.5) M = χh, (1.6) χ = C T T C. (1.7) Where T C = Cλ is the Curie temperature. A physical interpretation of the Curie temperature is that it describes temperature at which a material transitions from being ferromagnetic to being paramagnetic. In magnetic materials, there is a competition between the Weiss molecular field, which prefers alignment of the magnetic moments, and the disorder that is caused by high temperatures. Below the Curie temperature, the effect of the molecular field is stronger, and so the spins are all aligned and the material behaves like a ferromagnet. Once the temperature rises above T C, the temperature effects win out and the magnetic moments are oriented in random directions, resulting in a paramagnetic material. The exchange field, which favors alignment of the magnetic moments when below the Curie temperature, is not the only magnetic interaction that needs to be considered. There is also the interaction of magnetic dipoles, known as the dipolar interaction. While the exchange interaction is much stronger at short range, it drops off rapidly with distance. The dipolar interaction, on the other hand, has a stronger

21 6 Figure 1.3: Ferromagnetic material in which all magnetic moments are aligned (left) and when domains are formed (right). effect at long ranges. While having all the moments in a material aligned in the same direction minimizes the exchange energy, this configuration is not favorable for the dipolar energy. As a result, magnetic domains are formed in ferromagnets that greatly lower the dipolar energy while only moderately increasing the exchange energy (Fig. 1.3). The transition from ferromagnetic to paramagnetic is represented in Bloch s law [3], ( ( ) ) 3/ T M (T ) = M () 1, (1.8) which models the magnetization of a ferromagnet as a function of temperature and can be obtained directly from the Heisenberg model of a ferromagnet. The T 3/ law was shown by Dyson to be the first term in the expansion of spontaneous magnetization at low temperatures [4]. As the temperature in a ferromagnet rises from absolute zero, the excitation of spin waves lowers the overall magnetization. The temperature dependence of the magnetization has consequences for the thermodynamic properties of the material as well, like the heat capacity. For low temperatures, the heat capacity in a ferromagnet can be expressed by [5] c p (T ) = a 1 T + a T 3/ + a 3 T 3, (1.9) T C

22 7 Figure 1.4: Magnetic component of heat capacity of (a) Fe and (b) Co as a function of temperature [5]. The dots indicate experimental results obtained from calorimetric data, and the lines are theoretical predictions. where the three terms represent the electronic, magnetic, and lattice contributions, respectively. The magnetic contribution to the heat capacity rises as the temperature increases and reaches a maximum at the Curie temperature (Fig. 1.4) Ferromagnetic Resonance and the Suhl Instability When the magnetization in a material is excited at its fundamental frequency, it undergoes a process known as ferromagnetic resonance (FMR). It involves a uniform precession of the total magnetic moment around the axis of an external magnetic field, and the first experimental observation was performed by Griffiths [4]. One of its primary uses is in examining the FMR spectrum in order to determine various spin

23 8 wave properties of a material. However, one of the drawbacks is that there is a spin wave instability that occurs for large amplitudes of the ac field. The theory behind this instability was first developed by Suhl [6 8], where he showed that when that the ac field reached the instability threshold, magnons at the FMR frequency would be destroyed, with two magnons being created in its place. Magnons at the FMR frequency have zero momentum, and their energy is determined by the frequency. Both of these quantities must be conserved when the two new magnons are created. As a result, the magnons will be traveling in opposite directions with half the FMR frequency. It should be noted that this process can only occur if a magnon excitation exists at half the FMR frequency. As such, being able to control the band levels in that frequency range would allow for the ability to turn the spin wave instability on or off at will. We will propose one method of doing this later on in chapter Spin Wave Modes in a Thin Film When it comes to investigating spin waves in a thin film, there are three primary types of modes that can propagate depending on the direction of the magnetic bias field and its relative orientation to the spin wave propagation vector (Fig. 1.5). These three modes are known as forward volume magnetostatic spin waves (FVMSW), backward volume magnetostatic spin waves (BVMSW). and magnetostatic surface spin waves (MSSW). Forward and backward volume waves are so named because of properties of their dispersion relations. As we will see below, FVMSW have a group and phase velocity that are in the same direction, while BVMSW have a group and phase velocity that are in opposite directions. The MSSW differs from the volume waves in that it travels primarily on the surface of the material, while the volume waves are distributed throughout. Early work in this area was performed by Damon and Eshbach [9, 3], while a comprehensive summary of the results can be found in references [31 33].

24 9 Figure 1.5: Configurations for propagation of FVMSW (left), BVMSW (middle), and MSSW (right). Figure 1.6: Propagation of a FVMSW. FVMSW occur when the bias field is perpendicular to the plane of the slab and to the spin wave vector (Fig. 1.6). An approximation of the dispersion relations for this mode can be obtained from using perturbation theory on the magnetic equation of motion, and is given by [33, 34] ( f F V MSW = f H (f H + f M 1 1 )) e kd. (1.1) kd Where f H = 4πγµ H, and f M = 4πγµ M. For small values of k, this function starts out at the frequency f H (Fig. 1.9). The frequency increases as k increases, and approaches the limit of f F MR = f H (f H + f M ), which is the ferromagnetic resonance (FMR) frequency. This mode is also unique in that the wave vector can be in any direction. As long as it is in plane, it will be perpendicular to the magnetization, and will therefore result in FVMSW. When the bias field is in plane and parallel to the wave vector, we get BVMSW (Fig. 1.7) which have dispersion relations that are the complete opposite of those for

25 1 Figure 1.7: Propagation of a BVMSW. Figure 1.8: Propagation of a MSSW. f (GHz) f F MR f H FVMSW BVMSW MSSW k (m -1 ) Figure 1.9: Dispersions of the three different spin wave modes in a thin film using µ H =.7T, M = A/m, and d = 1 nm. the FVMSW [33, 34]: f BV MSW = ( ( )) 1 e kd f H f H + f M. (1.11) kd This starts at the FMR frequency for small k and approaches f H at large values of

26 11 the wave vector (Fig. 1.9). This behavior results in a negative slope of the dispersion relations, and is the reasoning for referring to these as backward spin waves. The phase velocity of a wave is given by its frequency divided by the wave vector: v p = f/k, while the group velocity is given by the derivative of the dispersion: v g = f/ k. The negative slope of the BVMSW implies that the phase and group velocity of the wave are in opposite directions. Lastly, we have the MSSW, which occur when both the propagation direction magnetization are in plane, but are perpendicular to each other (Fig. 1.8). The dispersion relation for this mode is given by [33, 34]: ( f MSSW = fh + f ) ( ) M fm e kd. (1.1) Both FVMSW and BVMSW are referred to as volume spin waves because the amplitude of the wave has a cosinusoidal dependence throughout the thickness of the material. The MSSW, on the other hand, is a surface wave, meaning that the spin wave propagates primarily on the surface of one side of the slab, and the amplitude decays exponentially the farther it is from the surface. When looking at spin waves in a thin film, it also important to consider the properties of the spin wave at the boundaries. One of the major effects on the boundary is known as surface spin pinning, and it received much interest in the period after the observation of spin waves [35 37]. One explanation of pinning was to assume that there was a thin layer on the film surface in which the magnetization differs from that in the body of the film [37]. This pinning at the surface leads to certain modes being suppressed. Only those modes that satisfied the boundary conditions would be allowed to exist. The degree to which the spins waves are pinned at the interface is expressed with the pinning parameter. The boundary conditions are highly dependent on this parameter, and its effects will be discussed in more detail in section

27 Non-reciprocity of Spin Waves As mentioned previously, the MSSW mode of thin film propagates primarily on the surface of one side of the film. Which side the spin wave propagates on is dependent on the direction it is traveling in. This gives the waves non-reciprocal properties and has also been shown to also appear in the volume waves [38, 39]. At a more fundamental level, non-reciprocal properties arise in materials due to magneto-optic effects. Magneto-optic effects describe situations in which an electromagnetic wave travels through a material that is influenced by a magnetic field. Such materials are called gyrotropic, and have the primary effect of modifying the permittivity, ɛ, or permeability, µ, tensors. These tensors describe the capability of a material to form an electric or magnetic field within it: D = ɛe (1.13) B = µh. (1.14) For isotropic materials these are scalar quantities, whereas for gyrotropic materials they are non-symmetric tensors. Here we will focus on one specific non-reciprocal effect that emerges in magnetooptics, the Faraday effect, and its application in the creation of optical isolators [4,41] and microwave circulators [4]. An optical isolator essentially functions as an optical diode, only allowing light to travel through it in one direction. It is commonly used in lasers to protect them from reflected light. The primary component in an isolator is the Faraday rotator [43], which itself utilizes the Faraday effect [44]. The Faraday effect is a magneto-optical effect that occurs when an electromagnetic wave is in the presence of a magnetic field in a material (Fig. 1.1). For circularly polarized light, the rotation of the electric field influences the motion of the electrons in the medium. Their circular motion will then result in the appearance of a magnetic field, whose direction depends on whether the light is left or right circularly polarized. This is what

28 13 B = B ẑ y E x k y β E x d y β E x k y E x B = B ẑ Figure 1.1: Demonstration of the Faraday effect. results in the permeability being a non-symmetric tensor, which in turn causes the two polarizations to travel at different speeds in the material. Therefore, when linearly polarized light, which is just a combination of left and right circularly polarized light, passes through the material, its polarization direction will be rotated (Fig. 1.1). The degree to which the light wave is rotated is given by β = vbd, (1.15) where v is the material dependent Verdet constant. Sending the same wave back through the material does not reverse the rotation, which makes this a non-reciprocal effect and is what allows this to be used in creating optical isolators. The Faraday effect can also be used in the construction of microwave circulators, another non-reciprocal device. A circulator is a multiple port device in which a signal travels cyclically from one port to the next. For example, a four-port circulator is shown in Fig The four ports consist of rectangular waveguides, with each

29 14 Figure 1.11: Diagram of a Faraday circulator []. subsequent waveguide being rotated by and angle of π/4. The magnetic field and the length of the ferrite are tuned so that the linear waves coming from the ports is rotated by that same angle. In this manner, an input signal from port 1 will be rotated and pass into port. Since the Faraday effect is non-reciprocal, the rotation of the wave polarization will be in the same direction for waves traveling in the opposite direction. This results in an input signal on port being sent to port 3. In a similar manner, inputs on port 3 get sent to port 4, and inputs on port 4 are sent back to port 1. By blocking ports 3 and 4, this device can operate as a microwave isolator, with signals only being allowed to travel in one direction. It also has applications as a duplexer, with a transmitter connected to port 1, a antenna to port, and a receiver to port 3. In this way the transmitter and receiver are isolated from each other, and can both utilize the same antenna.

30 15 1. Landau-Lifshitz Equation The Landau-Lifshitz (LL) equation gives a description of the dynamic magnetization of a system. It was first derived by Landau and Lifshitz [45] and can be written without a damping term as t M (r, t) = γµ M (r, t) H eff (r, t). (1.16) The effective magnetic field can be viewed as a torque that causes a precession of the magnetization. The damping term is typically added in as a phenomenological effect. During the time shortly after the derivation of the LL equation, there was much discussion on how to add in a relaxation term [45 57]. The first method was proposed by Landau and Lifshitz [45]: t M (r, t) = γµ M (r, t) H eff (r, t) + λ M s M (r, t) (M (r, t) H eff (r, t)). (1.17) This form can be obtained from the requirement that M remains constant, as it implies that M/ t must be perpendicular to M, and can be written as the sum of two vectors in the plane orthogonal to M [58]. While the Landau-Lifshitz formulation works well when the damping effects are small, it becomes inaccurate when the damping is large, such as with damping caused by eddy currents [46, 5]. To account for this, Gilbert proposed a different equation to represent the phenomenological damping [5]: t M (r, t) = γµ M (r, t) H eff (r, t) γηm (r, t) M (r, t). (1.18) t This is known as the Landau-Lifshitz-Gilbert (LLG) equation and can be obtained from a Lagrangian formulation with a dissipative force described by a Rayleigh dissipation function [53]. To compare Eqs and 1.18 we can write them using the Gilbert damping parameter, α: t M (r, t) = γµ M (r, t) H eff (r, t) + α (r) M s (r) M (r, t) M (r, t). (1.19) t However, the LL formulation will have a modified gyromagnetic ratio, γ = γ (1 + α ).

31 16 This implies that the precession rate will increase for large damping when using the LL equation. It should be noted that a third form of damping was proposed by Bloembergen [51]. This method simply employed the Bloch equations, which are typically used to describe paramagnetic systems. However, due to the better accuracy of the LL and LLG formulations, both of which have been studied in detail [45, 5, 53, 55, 58 63], it was never widely used Origins of Spin Wave Damping While it has been demonstrated experimentally that the LLG equation correctly models relaxation effects in materials [64, 65], the damping term was introduced as a purely phenomenological term since its physical origin at the time was not known. Since then, there has been much difficulty in determining its exact cause. This is in part due to the likelihood that the damping originates from multiple different mechanisms. In metallic films, the damping effects can arise from eddy currents, and its contribution can be obtained from Maxwell s equations [66]: α eddy = 1 ( ) 4π σd. (1.) γm s 6 c Where σ is the electrical conductivity, and d is the film thickness. This has a strong dependence on the thickness of the film, and becomes negligible for small values of d. For Fe films, damping from eddy currents is insignificant for d < 1 nm [66]. In order to determine the role of eddy currents in thicker films, the LL equation and Maxwell s equations must be solved self consistently [66, 67]. Another source of damping comes from scattering of magnons and phonons, also referred to as phonon drag. The phonon drag contribution is given by [68] α ph M s γ ( ) B (1 + ν) = η, (1.1) E

32 17 where η is the phonon viscosity, B is the magnetoelastic shear constant, E is Young s modulus, and ν is the Poisson ratio. The relaxation caused by phonon drag was found to negligible, its contribution being 3 times smaller than the intrinsic damping in Ni [66], and while it was predicted to have effects on the FMR linewidth in certain situations [69], experiments failed to find results that corroborated this. More significant contributions to spin wave relaxation have been shown to come from spin orbit calculations. It was first shown by Kamberský that damping could be treated using the spin-orbit interaction Hamiltonian [7]. The Hamiltonian consists of a three particle interaction involving the annihilation of an electron and a magnon, and the creation of an electron with the combined momenta of the two annihilated particles. This model was based on the fact that changing the direction of the magnetization in a material also changes the Fermi surface. Since a spin wave consists of time varying magnetization, the Fermi surface is constantly changing. As a result, the electrons continually try to repopulate the Fermi surface, but are delayed by a relaxation time. This delay then results in frictional damping. Calculations using this model were later performed by Kuneš and were shown to be in good agreement with experimental data [71]. More recently, it has been shown that the Gilbert term can be derived from first principles from a nonrelativistic expansion of the Dirac equation [7]. Here they have shown that the damping arises from relativistic corrections to the Hamiltonian that couple the spin and electric field. The effects of these magnetic damping mechanisms can also be seen in the linewidth of FMR peaks. The origins of the linewidths were highly investigated in iron garnets, particularly in YIG, where it has been shown to arise primarily from surface imperfections and magnetic impurities [5, 6]. YIG was of particular interest in part because of its small linewidth, which could be as low as. Oe, compared to other magnetic materials, which are on the order of 1-1 Oe.

33 Magnonic Crystals A magnonic crystal is a structure that has a periodic variation in its magnetic properties. This periodicity is designed to modify the properties of spin waves, and they can be looked at as the magnetic counterpart of photonic [73, 74] and phononic crystals [75, 76]. Construction of these crystals can be done with a single material by etching [77 8], dipolar coupling of strips or wires [81, 8], or through a periodic variation of the material s width [83 85]. Magnonic crystals can also consist of multiple magnetic materials, and can be constructed with periodicity in one dimension [86 94], two dimensions [95 11], and three dimensions [1 16]. The appearance and control of magnonic band gaps in these magnonic crystals has received much attention [77, 8 88, 9, 91, 94 97, 1 13], as they could be used to create lossless waveguides, which has already been demonstrated with photonic crystals [17]. Figure 1.1: Example of an etched one dimensional magnonic crystal [78]. Most of the studies done on magnonic crystals have been on the one-dimensional variant, as it has the simplest geometry and is the easiest to fabricate. As a result of this, several different types of periodic structures have been developed. One method involves etching out a series of strips in the ferromagnetic sample to create a periodic variation in the width (Fig. 1.1). Experimental studies performed by Chumak

34 19 consist of a one dimensional etched magnonic crystal with a periodicity of 3µm, and a groove depth that varied between 1nm and 1µm. With spin waves in the single GHz frequency range, they found band gap widths ranging from 5-3MHz, and demonstrated that deeper grooves resulted in a larger band gap. 3 nm z y x 5 nm P xn P W 1 =3nm W =4nm P 1 P 1 nm Figure 1.13: Example of a one dimensional width modulated magnonic crystal [85]. Another type of one dimensional magnonic crystals are created with width modulation (1.13). These are similar to the etched materials, except these a periodic variation in width instead of depth. Width modulated devices investigated by Kim [85] and Lee [8] had a considerably smaller periodicity, a = 18nm, and a slightly higher frequency range, GHz-1GHz, than the etched crystals studied by Chumak. Band gap widths in the range of 1GHz-3GHz were found, the widths being dependent on the length ration of the two modulated regions. x z y H Co Py s s a k Figure 1.14: Example of a one-dimensional crystal created with layers of alternating cobalt and permalloy [94].

35 Lastly, for the one-dimensional case, we consider multi-layered structures (Fig. 1.14) studied by Wang [88]. With a lattice constant of 5nm and spin waves in the single GHz range, they found that band gaps were influenced by an applied magnetic field, and varied from 1GHz-GHz in width. y Co α Fe a x (a) Figure 1.15: Example of a two-dimensional magnonic crystal [11]. Slightly more geometrically complex, are the two-dimensional magnonic crystals. The most common type of two-dimensional magnonic crystal studied consists of a lattice arrangement of magnetic cylinders embedded in a magnetic host (1.15). Research on these structures have looked at their dispersion relations and the dependence of band gaps on parameters such as the filling fraction [95], as well as the shape and orientation of the cylinder cross section [11]. Large differences in the magnetic properties of the two materials were also shown to be favorable to producing band gaps [96]. Spin wave frequencies for these cases were in the range of 5GHz-15GHz with band gaps on the order of 1GHz. There has also been a moderate amount of work done on three-dimensional magnonic crystals. These are typically structured in a similar manner as the twodimensional crystals, with periodic arrangement of two different magnetic materials. For spheroids arranged in a cubic lattice with a spacing of 1nm, Krawczyk [1]

36 1 found band gaps for spin wave frequencies in the single THz range with widths of about THz. As with the two-dimensional lattices, the presence of gaps was shown to be highly dependent on the difference in magnetic parameters between the two materials. They also investigated the band structure dependence on the shape of the scattering center, with spheres giving the maximum band gap width. In the following chapters we will look at both two-dimensional and one-dimensional magnonic crystals. In the two-dimensional materials, we will study spin wave propagation and how its directionality is heavily influenced by the lattice structure. For the one-dimensional materials, we will look at how an electric field can be used to control the appearance of band gaps in the spin wave dispersion relations.

37 CHAPTER TWO-DIMENSIONAL MAGNONIC CRYSTALS In this chapter we will lay the groundwork for examining the properties of spin wave dispersion relations in two-dimensional magnonic crystals. Magnonic crystals are the magnetic counterpart to photonic [73, 74] and phononic crystals [75, 76]. They consist of a structure that has a periodic variation in its magnetic properties, which results in a modification of the properties of spin waves in the crystal. Studies of spin waves in two-dimensional magnonic crystals is an area that has already been studied [95 11], however there are some details that have not been properly investigated or have seen little attention, such as the inclusion of damping effects [98, 18]. Obtaining the dispersion relations is also an important first step in calculating the propagation patterns for spin waves, which is an area of magnonics that has not been thoroughly investigated, and will be examined later on in chapter 4..1 Magnonic Structure The structures studied here are shown in Fig..1; they consist of a magnetic material, A, arranged in either a square or hexagonal lattice and embedded in a second magnetic material, B. The radius of the cylinders is R cyl and the lattice constant is a. The magnetization of both materials is saturated along the axis of the cylinders by a static external magnetic field. Figure.1: Structure of the two-dimensional magnonic crystals studied.

38 3. Theoretical Model..1 Landau-Lifshitz-Gilbert Equation The system of Fig..1 is described by the Landau-Lifshitz-Gilbert (LLG) equation, which is the magnetic equation of motion [53]: t M (R, t) = γµ M (R, t) H eff (R, t) + α (R) M s (R) M (R, t) M (R, t), (.) t where γ is the gyromagnetic ratio, M s (R) is the saturation magnetization, α (R) is the Gilbert damping parameter, and R is a two dimensional position vector in the x-y plane. The first term of the LLG equation describes the torque experienced by the magnetization due to the effective magnetic field, H eff (R, t) = H + h (R, t) + H ex (R, t). (.3) where H = H ẑ is an externally applied static magnetic field which is assumed to be uniform throughout the crystal and aligned along the axes of the cylinders. The other terms in Eq..3 are the dynamic dipolar field, h (R, t), and the exchange field, H ex (R, t), which will be discussed further in the following sections. The second term in Eq.. is the phenomenological damping term derived by Gilbert [5, 53]... Dipolar Field The dipolar field is included in the effective field of Eq..3 because it is needed to satisfy Maxwell s equations: h (R) =, (.4) (h (R) + m (R)) =. (.5) Note that we have left out the time dependence for simplicity. It will be reintroduced in section..4. Using Eq..4 we represent the dipolar field with a magnetostatic

39 4 potential, Ψ (R), such that h (R) = Ψ (R). (.6) We also note that Eq..4 will prevent some symmetries from appearing. Since h (R) is in the x-y plane and is independent of z, we have h y (R) x h x (R) y =. (.7) This breaks any symmetry under reflection about the x or y axes except for the case where both partial derivatives are zero...3 Exchange Field The exchange field is an effective magnetic field that is caused by the direct exchange interaction energy between neighboring spins in a ferromagnetic material. This interaction can be modeled by the Heisenberg Hamiltonian: Ĥ direct = J ij S i S j. (.8) Where J ij is the exchange coupling constant that describes the strength of the exchange interaction between neighboring spins. For a homogeneous material, the exchange energy can be obtained from Eq..8 and is given by [19] ˆ [( Mx (r)) + ( M y (r)) + ( M z (r)) ] dr, (.9) U ex [M (r)] = A M s where A is the exchange stiffness constant, and is directly related to the coupling constant. Following Gilbert [53], the exchange field is then obtained through a functional derivative of the exchange energy: H ex (r) = 1 µ δu ex [M (r)] δm (r) = A M (r). (.3) µ Ms Due to the inhomogeneity of the system considered here, both the exchange constant and saturation magnetization will be spatially dependent quantities: A (R) = A B + Θ (R) (A A A B ), (.31) M s (R) = M sb + Θ (R) (M sa M sb ), (.3)

40 5 where Θ (R) = 1 in material A and Θ (R) = in material B. Following the derivation in Kittel [19], while assuming spatially dependent exchange constant and saturation magnetization, we obtain the exchange energy for an inhomogeneous material: ˆ { [ ( )] [ ( )] [ ( )] } Mx (R) My (R) Mz (R) U ex [M (R)] = A (R) + + dr. M s (R) M s (R) M s (R) (.33) The total magnetization consists of both a static and dynamic component: M (R, t) = M s (R) ẑ + m (R, t). Using the linear magnon approximation, we assume that the dynamic term is small compared to the saturation magnetization and therefore we will only keep terms up to first order in m (R, t). With these assumptions, we can write the exchange field for an inhomogeneous material as H ex (R, t) = ) A (R) ( µ Ms (R) M (R, t) M (R, t) + µ M s (R) ( A (R) ) 1 M s (R) [ ] m (R, t) µ Ms (R) m (R, t) ( A (R) ) ẑ. (.34) M s (R) The exchange field enters the LLG equation only as a cross product with the magnetization M (R, t). The second term is parallel to M (R, t) and thus will not contribute to Eq. (.). The third term of Eq. (.34), which is proportional to m (R, t) and parallel to M s (R), will only produce terms of second order in m (R, t) in Eq. (.) and can safely be dropped. Therefore, we can approximate H ex (R, t) = 1 µ ( λ ex ) M (R, t), (.35) where λ ex (R) = A (R) / (M s (R)) is the exchange length. This produces a LLG equation from Eq. (.) that is correct to first order in m (R, t)...4 Plane-Wave Expansion Method The LLG equation is solved using the plane-wave expansion method. This is an often used method with magnonic crystals [95 98, 1, 11]. We begin by considering

41 6 magnons of a single frequency, ω, and so we write Using Eqs..6 and.35 the LLG equation becomes iωm x (R) + M s (R) [λ ex (R) m y (R) ] m y (R) H m (R, t) = m (R) e iωt, (.36) h (R, t) = h (R) e iωt. (.37) m y (R) M s (R) H H [λ ex (R) M s (R) ] Ψ (R) y iωm y (R) M s (R) [λ ex (R) m x (R) ] + m x (R) H +m x (R) + M s (R) H + iωα (R) m y (R) =, (.38) H [λ ex (R) M s (R) ] Ψ (R) x iωα (R) m x (R) =, (.39) where Ω = ω/( γ µ H ). By taking advantage of the crystal s periodicity, we can use Bloch s theorem to write the magnetization and magnetostatic potential as an expansion of plane waves: N m (R) = e ik R m= N N Ψ (R) = e ik R m= N N n= N m k (G mn ) e igmn R, (.4) N n= N Ψ k (G mn ) e igmn R. (.41) Here G mn = (πm/a) ˆx + (πn/a) ŷ represents a two dimensional reciprocal lattice vector of the crystal and k is a wave vector in the first Brillouin zone. Substituting Eqs..4 and.41 into Eq..5 from Maxwell s equations allows us to write the magnetostatic potential in terms of the magnetization: Ψ k (G) = i m x,k (G) (G x + k x ) + m y,k (G) (G y + k y ) (G + k). (.4)

42 7 The dipolar field can then be expressed as N h x (R) = e ik R m= N N h y (R) = e ik R N n= N (.43) m x,k (G mn ) (G x,mn + k x ) + m y,k (G i ) (G x,mn + k x ) (G y,mn + k y ) (G mn + k) e igmn R, m= N N n= N (.44) m x,k (G mn ) (G x,mn + k x ) (G y,mn + k y ) + m y,k (G mn ) (G y,mn + k y ) (G mn + k) e igmn R. Next, we need to be able to write the material properties M s (R), λ ex (R), and α (R) in reciprocal space. Since these have the same periodicity as the crystal lattice, this can be done with a Fourier series expansion: Q (R) = N 1 m= N 1 N 1 n= N 1 σ m σ n Q (G mn ) e igmn R, (.45) σ m = sin (πm/ (N 1 + 1)). (.46) πm/ (N 1 + 1) Here Q (R) is one of the three material properties mentioned above and σ m is the Lanczos sigma factor. The Lanczos sigma factor is used to reduce the Gibbs phenomenon at the boundaries between the two materials. As shown in Fig.., inclusion of this factor gives a much better approximation of the desired magnonic structure. The Fourier coefficients are obtained by an inverse Fourier transform: Q (G) = 1 ˆ Q (R) e ig R d R. (.47) S S where S is the area of the two-dimensional unit cell. Performing the integration for G = gives the average Q (G = ) = Q A f + Q B (1 f), (.48) where f is the fractional space occupied by a cylinder in the unit cell. For G, we have Q (G ) = (Q A Q B ) f J 1 ( G R cyl ) G R cyl. (.49) Here J 1 is a Bessel function of the first kind, and R cyl is the radius of the cylinders. The

43 8 a - a - a a Ms (R) (A/m) Ms (y = ) (A/m) a x a a - a - a a Ms (R) (A/m) Ms (y = ) (A/m) a x a Figure.: Comparison of Fourier series calculation of saturation magnetization with (top) and without (bottom) using the Lanczos sigma factor with N 1 = 1. following infinite system of equations in reciprocal space is obtained by substituting Eqs. (.4)-(.45) in Eqs. (.38) and (.39): iω j (m x,k (G i ) δ ij + α (G i G j ) m y,k (G j )) = { M s (G i G j ) (G x,j + k x ) (G y,j + k y ) H (G j + k) m x,k (G j ) + [δ ij + j M s (G i G j ) (G y,j + k y ) H (G j + k) + 1 ( Ms (G i G l ) λ ex (G l G j ) H l ((k + G j ) (k + G l ) (G i G j ) (G i G l )))] m y,k (G j )} (.5)

44 9 iω j (m y,k (G i ) δ ij α (G i G j ) m x,k (G j )) = { M s (G i G j ) (G x,j + k x ) (G y,j + k y ) H j (G j + k) m y,k (G j ) + [δ ij + M s (G i G j ) (G x,j + k x ) H (G j + k) + 1 ( Ms (G i G l ) λ ex (G l G j ) H l ((k + G j ) (k + G l ) (G i G j ) (G i G l )))] m x,k (G j )} (.51) Here we have re-indexed the reciprocal lattice vectors, G i, such that i = (m + N ) + (n + N ) (N + 1). m x,k (G ). δ iω ij α (G i G j ) m x,k (G N ) = α (G i G j ) δ ij m y,k (G ). m y,k (G N ) Bxx ij B yx ij B xy ij B yy ij m x,k (G ). m x,k (G N ) m y,k (G ). m y,k (G N ) (.5) B xx ij = B yy ij = M s (G i G j ) (G x,j + k x ) (G y,j + k y ) H (G j + k) (.53) B xy ij = δ ij + M s (G i G j ) (G y,j + k y ) H (G j + k) + 1 M s (G i G l ) λ ex (G l G j ) H l [(k + G j ) (k + G l ) (G i G j ) (G i G l )] (.54) B yx ij = δ ij M s (G i G j ) (G x,j + k x ) H (G j + k) 1 M s (G i G l ) λ ex (G l G j ) H l [(k + G j ) (k + G l ) (G i G j ) (G i G l )]. (.55) Where N = (N + 1) 1. The LLG equation is now reduced to finding the

45 3 eigenvalues and eigenvectors for the above equation. Note that we have chosen to use a different number of lattice vectors for the terms written as a Fourier expansion (Eq. (.45)) and those expressed with Bloch s theorem (Eqs. (.4)-(.41)). This is because the derivatives in the exchange term (Eq. (.34)) make it very sensitive to changes occurring in the boundary region, and it will fluctuate rapidly as the number of Fourier terms in the magnetization increases. Therefore, we choose a value of N 1 that gives a decent approximation of the magnonic structure, and then increase N until the dispersion relations obtained from Eq. (.5) are well converged. For all of the following calculations involving the two-dimensional lattice, we have used N 1 = 1 and N = Summary In this chapter I laid out the methods used for examining spin waves in magnonic crystals. I used the plane-wave expansion method to express the LLG equation in the reciprocal lattice space as a matrix equation. In the process, I derived the exchange field for an inhomogeneous material, and showed that the dipolar field will prevent some symmetries from appearing in solutions to the LLG equation. I have also shown that the solutions to the LLG matrix equation can be used to obtain the Green s function for the system, from which the dynamic response function for a spin wave source can be obtained.

46 31 CHAPTER 3 SPIN WAVE DISPERIONS RELATIONS IN TWO-DIMENSIONAL MAGNONIC CRYSTALS Having derived the matrix equation form of the LLG equation, we can now use it obtain the dispersion relations of spin waves in the two-dimensional magnonic crystals. We will use these results to examine the effects of the crystal s magnetic properties on the band structure. In particular, looking for situations that are favorable for producing band gaps. We will also further investigate the symmetry breaking caused by the dipolar field and compare the effects of using the alternate form of the exchange field that has been used in some studies. Lastly, we will perform an investigation into the convergence of the dispersion relations we have calculated to ensure that they are valid. 3.1 Solutions of the LLG Equation Dispersion Relations and Band Gaps From Eq. (.5) we calculate the complex eigenvalues ω corresponding to the frequencies of magnons in the two-dimensional magnetic superlattices of Fe, Co, Ni, and YIG (Y 3 Fe 5 O 1 ). YIG, which stands for yttrium iron garnet, is a ferrimagnetic material that is characterized by a small magnetization compared to ferromagnetic materials and a much smaller damping parameter. The real part of ω n (k) is the magnon frequency of branch n for the wave vector k and the imaginary part is the inverse spin wave lifetime. To focus on the dependence of these properties on magnetic material combinations we consider superlattices with a lattice constant a = 1nm, an external field µ H =.1T, and a filling fraction f =.5. The material properties, M s, A, and α, are listed in Table 3.1. First, we consider the empty lattice case, which is when both the cylinders and host are the same material. Figs. 3.1 and 3. show the dispersion relations for both the empty square and empty hexagonal lattices. The band structures are affected by both

47 3 M s (A/m) A(pJ/m) α Fe Co Ni YIG Table 3.1: Magnetic properties of Fe, Co, Ni, and YIG. the saturation magnetization and the exchange constant of the two materials, with the magnetization having a slightly stronger effect. Generally, as the magnetization of the material decreases, the frequency scaling of the dispersion relations increases, while retaining the same band structure. This is most evident when comparing the Fe and YIG calculations, since their magnetizations differ by roughly a factor of ten. Consider where the second and third modes meet at the Γ point for these two materials. In going from Fe to YIG, the scaling of this frequency increases by about a factor of five for both the square and hexagonal lattices. Changing the exchange constant has the opposite effect. This can be seen in the comparison of the dispersion relations for Co and Ni, where there is a slight decrease in the frequency despite Ni having a lower magnetization. It is difficult to discern exactly why the spin wave frequencies are affected in this manner, however we suspect that it is at least partially due to how these variables appear in the exchange field (Eq..35). Here the exchange constant is in the numerator, while the magnetization is in the denominator, which could account for how they affect the frequencies of the empty lattice in and inverse manner. In Figs. 3.3 and 3.4, we show the band structures for various combinations of Fe with Co, Ni, or YIG. The change in the spin wave dispersion relations from the empty lattice case is primarily dependent on the differences of the saturation magnetization between the two materials. Fe and Co, which have very similar magnetic properties,

48 33 Re(ω) (THz) 1 a) Re(ω) (THz) 3 1 c) Re(ω) (THz) 3 1 b) Re(ω) (THz) 1 5 d) Γ M X Γ Figure 3.1: Empty square lattice band structure for Fe (a), Co(b), Ni(c), and YIG (d). Re(ω) (THz) 1 4 a) Re(ω) (THz) 3 1 c) Re(ω) (THz) 3 1 b) Re(ω) (THz) 1 5 d) Γ M K Γ Figure 3.: Empty hexagonal lattice band structure for Fe (a), Co(b), Ni(c), and YIG (d).

49 34 differ little from from homogeneous case, with only a few small splittings of the band levels. However, the saturation magnetization of Fe and YIG differ by roughly a factor of ten, and so this combination yields results that differ greatly from the empty lattice case in Figs. 3.1 and 3.. Furthermore, multiple band gaps appear when the magnetization is larger in the cylinders than in the host. A square lattice of Fe cylinders in YIG results in four band gaps within the lowest nine spin wave modes, and a hexagonal lattice with the same materials produces five gaps in the lowest nine modes. However, when we look at YIG cylinders in Fe, neither the square nor hexagonal lattices produce any band gaps. Re(ω) (THz) 1 a) Re(ω) (THz) 1 d) Re(ω) (THz) 1 b) Re(ω) (THz) 1 e) Re(ω) (THz) 3 1 c) Γ M X Γ Re(ω) (THz) 3 1 f) Γ M X Γ Figure 3.3: Square lattice band structures for Fe cylinders in Co (a), Ni (b), and YIG (c), and for Co (d), Ni (e), and YIG (f) cylinders in Fe.

50 35 Re(ω) (THz) a) Re(ω) (THz) d) Re(ω) (THz) Re(ω) (THz) b) c) Γ M K Γ Re(ω) (THz) Re(ω) (THz) e) f) Γ M K Γ Figure 3.4: Hexagonal lattice band structures for Fe cylinders in Co (a), Ni (b), and YIG (c), and for Co (d), Ni (e), and YIG (f) cylinders in Fe Contours Here we show the detailed dispersion curves (Figs. 3.5 and 3.7) and spin wave relaxation rates (Figs. 3.6 and 3.8) for both square and hexagonal lattices of Fe cylinders in Ni. Plotted are the lowest nine spin wave modes (Re (ω)) in the entire first Brillouin zone as well as the corresponding inverse spin wave lifetimes (Im (ω)). When considering a system with a particular lattice structure, it might be expected that any results would have that same symmetry. However, in section.. it was shown that the dipolar field cannot have symmetry under reflections about the x or y axes. The consequences of this are most evident in Figs. 3.6 and 3.8, as there is a symmetry breaking that occurs in the linewidths. The reason for the small magnitude of the asymmetry, and how it can be enhanced will be discussed in the following sections.

51 Re(ω) Re(ω) Re(ω).7.6 Re(ω) Re(ω) Re(ω).8.7 Re(ω) Re(ω) Re(ω) Figure 3.5: The lowest nine spin wave frequencies (in THz) for a square lattice magnonic crystal composed of Fe cylinders in Ni. The square figures show the entire Brillouin zone of the lattice Im(ω) Im(ω) Im(ω) Im(ω) Im(ω) Im(ω) Im(ω) Im(ω) Im(ω) Figure 3.6: The spin wave relaxation rates (in GHz) corresponding to the spin wave modes in Fig The square figures show the entire Brillouin zone of the lattice.

52 Re(ω) Re(ω) Re(ω) Re(ω) Re(ω) Re(ω) Re(ω).75 Re(ω). Re(ω).3.5 Figure 3.7: The lowest nine spin wave frequencies (in THz) for a hexagonal lattice magnonic crystal composed of Fe cylinders in Ni. The hexagonal figures show the entire Brillouin zone of the lattice Im(ω) 6 Im(ω) Im(ω) Im(ω) 35 Im(ω) 8 Im(ω) Im(ω) 1 Im(ω) 3.5 Im(ω) 4 6 Figure 3.8: The spin wave relaxation rates (in GHz) corresponding to the spin wave modes in Fig The hexagonal figures show the entire Brillouin zone of the lattice.

53 Alternate Exchange Field Here we note that there have been some papers studying magnonic crystals that have used an expression for the exchange field that differs the one derived in section..3 [95, 98]. The difference in this form is where the spatially dependent material properties are placed in regards to the derivatives: ( H ex (R, t) = µ M s (R) A (R) M s (R) ) M (R, t). (3.56) Figs. 3.9 and 3.1 show the dispersion relations obtained when using this alternate form of the exchange field instead of the derived form of Eq..35. The band structures for the empty lattices (Fig. 3.9) are found to be the same as those in Fig This is expected because in this case both the saturation magnetization and exchange constant are independent of position, and so Eqs..35 and 3.56 are equivalent. However, when considering two different materials in the magnonic crystal, the dispersion relations obtained from the alternate exchange field (Fig. 3.1) differ greatly from those of the derived exchange field (Fig. 3.3). When using the alternate form, the effects of the differing magnetization between the two materials is greatly reduced. The combinations of Fe with Co and Fe with Ni both show very little deviation from the empty lattice structure. Only the large difference in magnetization between Fe and YIG is enough to see an appreciable difference, and even with this, there is only one small band gap observed between the sixth and seventh modes. We now show the lowest nine spin wave modes (Fig. 3.11) and corresponding linewidths (Fig. 3.1) for Fe cylinders in Ni obtained when using the alternate exchange field, for comparison with Figs. 3.5 and 3.6. Of particular interest are the linewidths, which show a much larger symmetry breaking in the lower modes than what was seen in Fig This provides a clue as to how we might be able to control the degree of symmetry breaking of the spin waves, and will be explored in the next section.

54 39 Re(ω) (THz) 1 a) Re(ω) (THz) 3 1 c) Re(ω) (THz) 3 1 b) Re(ω) (THz) 1 5 d) Γ M X Γ Figure 3.9: Empty square lattice band structure for Fe (a), Co(b), Ni(c), and YIG (d) obtained when using the exchange field in Eq Re(ω) (THz) 1 a) Re(ω) (THz) 1 d) Re(ω) (THz) Re(ω) (THz) b) c) Γ M X Γ Re(ω) (THz) Re(ω) (THz) e) f) Γ M X Γ Figure 3.1: Square lattice band structures for Fe cylinders in Co (a), Ni (b), and YIG (c), and for Co (d), Ni (e), and YIG (f) cylinders in Fe obtained when using the exchange field in Eq

55 Re(ω) Re(ω) Re(ω).8.6 Re(ω) Re(ω) 1.8 Re(ω) Re(ω) Re(ω) Re(ω) Figure 3.11: The lowest nine spin wave frequencies (in THz) for a square lattice magnonic crystal composed of Fe cylinders in Ni obtained when using the exchange field in Eq The square figures show the entire Brillouin zone of the lattice Im(ω) Im(ω) Im(ω) Im(ω) Im(ω) Im(ω) Im(ω) Im(ω) Im(ω) Figure 3.1: The spin wave relaxation rates (in GHz) corresponding to the spin wave modes in Fig The square figures show the entire Brillouin zone of the lattice.

56 41 3. Symmetry Breaking 3..1 Dispersions without the Dipolar Field In the previous sections, it was shown that the spin wave relaxation rates have a broken symmetry, and that the magnitude of the asymmetry was much larger when an alternate form of the exchange field was used. While the dipolar field is the cause of the symmetry breaking, it is apparent that the exchange field plays a role in this. First, we show that the dipolar field is the source of the asymmetry. This is done by solving the LLG equation with the dipolar field set to zero. The resulting lowest nine spin wave modes and relaxation rates are shown in Figs and 3.14 respectively. These results appear to have the same symmetry of the lattice, indicating that the dipolar field is the cause of the symmetry breaking. To be certain of this, we will take a closer look at the symmetry of the first spin wave mode both with and without the dipolar field. In Fig we show the dispersions and linewidths of the first mode, as well as an examination of their symmetries, in which we take the frequencies in one octant of the unit cell, and compare it to the other seven octants. This demonstrates that the reflection symmetry is indeed broken as was predicted in section 3.. When the dipolar field is turned off, this same examination shows that the symmetry is restored (Fig. 3.16), indicating that the symmetry breaking originates solely from the dipolar field.

57 Re(ω) Re(ω) Re(ω) Re(ω) Re(ω) Re(ω) Re(ω) 1.. Re(ω) 1.8 Re(ω) Figure 3.13: The lowest nine spin wave frequencies (in THz) for a square lattice magnonic crystal composed of Fe cylinders in Ni obtained the dipolar field is set to zero. The square figures show the entire Brillouin zone of the lattice Im(ω) Im(ω) Im(ω) Im(ω) Im(ω) Im(ω) Im(ω) Im(ω) Im(ω) Figure 3.14: The spin wave relaxation rates (in GHz) corresponding to the spin wave modes in Fig The square figures show the entire Brillouin zone of the lattice.

58 43 π a - π a - π π a a.4.3. Re(ω)(THz) π a - π a - π π a a % Difference % Difference Im(ω)(GHz) Figure 3.15: Symmetry of dispersion relations calculated with the dipolar field. π a Re(ω)(THz) π a Im(ω)(GHz) - π a - π π a a - π a - π π a a % Difference % Difference Figure 3.16: Symmetry of dispersion relations calculated without the dipolar field.

59 Comparison of Dipolar and Exchange Fields Looking back at the effective field (Eq..3) used in the LLG equation, we see that the only spatially dependent terms are the dipolar and exchange fields. In Fig a comparison of the dipolar and derived exchange fields is shown. The exchange field is shown to be roughly two orders of magnitude larger than the dipolar field. This accounts for the small symmetry breaking seen in Figs. 3.6 and 3.8, as the exchange field covers up most of the symmetry breaking effects caused by the dipolar field. Fig shows a comparison of the dipolar and alternate exchange fields. Here, the situation is reversed; the exchange field is now one order of magnitude smaller than the dipolar field. As a result, the effects of the dipolar field are much more apparent, as seen in Fig We also see a much larger asymmetry in the exchange field since its calculation is dependent on the magnetization vectors obtained from the LLG equation. This shows that if the relative strengths of the dipolar and exchange fields can be controlled, then the symmetry breaking effect could be enhanced.

60 45 a.6 a.6 - a - a a a hx (R) (A/m) Hexx (R) (A/m) - a - a a a hy (R) (A/m) Hexy (R) (A/m) - a - a a - a - a a Figure 3.17: Dipolar field (top row) and exchange field (bottom row) for a square lattice magnonic crystal composed of Fe cylinders in Ni obtained when using the derived exchange field in Eq..35. a - a - a a a hx (R) (A/m) Hexx (R) (A/m) a - a - a a a hy (R) (A/m) Hexy (R) (A/m) - a - a a - a - a a Figure 3.18: Dipolar field (top row) and exchange field (bottom row) for a square lattice magnonic crystal composed of Fe cylinders in Ni obtained when using the alternate exchange field in Eq

61 Convergence of Dispersion Relations Here we will show that the dispersion relations calculated in this section are well converged. There are two numerical parameters that influence the accuracy of the calculation. These are N 1, from Eq..45, which controls the number of terms in the Fourier series expansion of the material properties, and N, from Eq..4, which controls the number of terms in the Bloch function of the magnetization vectors. Increasing theses values gives a better approximation of the structure and its dispersion relations, but will also increase the time needed to complete the calculation. As such, we must find a balance between accuracy and computation time. These two variables are controlled separately because they influence the dispersion relations in different ways. The Fourier series impacts the band structure indirectly by modifying the appearance of the magnonic crystal. A larger number gives a closer approximation to the ideal case of discrete boundaries. The effect of this on the saturation magnetization for Fe cylinders in Ni is shown in Fig The saturation magnetization of Fe and Ni are A/m and A/m respectively. Outside of the boundary region, the Fourier series is converged on these values even at N 1 = 6. Increasing N 1 to 1 improves the approximation by decreasing the width of the boundary region by roughly half, giving a decent representation of the desired crystal structure (Fig..1). Further increasing it to continues to reduce the width of the boundary, but it is not worth the increased time it would take for the calculation to complete. While the Fourier series affects the appearance of the crystal structure, the Bloch functions directly impact the convergence of the dispersion relations. Increasing the number of terms in this function will first increase the size of the matrix in Eq..5. Since the number of eigenvalues depends on the size of the matrix, this will result in higher order modes being obtained from the LLG equation. The total number of modes will be equal to N (recall that N = (N + 1) 1). A larger N will not

62 47 a a Ms (R) (A/m) Ms (R) (A/m) a - a a - a - a a a Ms (R) (A/m) a - a a Figure 3.19: Convergence of Fourier series for saturation magnetization for N 1 = 6 (top left), N 1 = 1 (top right), and N 1 = (bottom). only give rise to additional modes, but will also improve the convergence of already existing modes. The convergence of two modes for increasing N is shown in Fig. 3. for Fe cylinders in Ni. The first mode changes by less than.1% as N increases from 6 to, and so would be well converged for any of these values. However, as we look at higher order modes the convergence rate slows down. For mode number 11, which is the highest order mode when N = 6, the frequency changes by roughly 4% when going from N = 6 to N =. This difference in convergence rates is most likely to due the fact that the higher order modes don t appear until N reaches a certain value. While mode 11 was first created at N = 6, mode 1 first appeared at N = 1 and had multiple iterations to converge by that point. The slower convergence of the higher order modes is not of particular importance for this chapter, however it will be necessary to take it into account in the following

63 48 chapter when we calculate the Green s functions and propagation patterns for the spin waves. Re (ω) (GHz) Re (ω) (GHz) N N Figure 3.: Convergence of spin wave modes for a square lattice of Fe cylinders in Ni. The top shows mode 1 and the bottom shows mode Summary In this chapter I have shown the results of my research on spin waves dispersion relations in two-dimensional magnonic crystals. I have shown that the spin wave band structure is largely dependent on the relative saturation magnetizations. While multiple band gaps appear when there is a large difference in magnetization between the two materials, this only occurs when the saturation magnetization is larger in

64 49 the cylinders than in the hosts. I have shown that the dipolar field prevents some symmetries from appearing in the dispersion relations and that the magnitude of the symmetry breaking is dependent on the relative strengths of the dipolar and exchange fields. I have also confirmed that the results of the dispersion calculations are well converged for the parameters used.

65 5 CHAPTER 4 SPIN WAVE PROPAGATION PATTERNS IN TWO-DIMENSIONAL MAGNONIC CRYSTALS We will now shift from the spin wave dispersion relations to focus on the properties of spin wave propagation. In order to fully understand spin waves in two-dimensional magnonic crystals, we need to know how a spin wave source placed in the structure will propagate. In this chapter we will begin by looking at the Green s function obtained from the solutions to the LLG matrix equation. These show the steady-state response of the system to a point source, and will be used to construct the response function, which will show the time-dependent propagation of a spin wave pulse. We will investigate the properties of these propagation patterns and compare them to the dispersion relations to see how they depend on frequency. 4.1 Response to Spin Wave Sources Green s Function Once the eigenvalues and eigenvectors are obtained from.5, they can be used to describe spin wave sources in the magnonic crystal. First, we construct the Green s function for this system: G (R, R, ω) = G xx (R, R, ω) G xy (R, R, ω) G yx (R, R, ω) G yy (R, R, ω) where each element is defined using an eigenfunction expansion: G rl (R, R, ω) = n,k, (4.57) m r,k (R) m l,k (R ) ω ω n,k, (4.58) where m r,k (R) = m x,k (R) + im y,k (R) and m l,k (R) = m x,k (R) im y,k (R). This element gives the counterclockwise response at position R of a point source at position R rotating clockwise at frequency ω. With N = 16 there would be 178 modes to sum over in the above equation. However, the higher order modes will not be well converged and as a result, some frequencies may have an imaginary component with

66 51 the wrong sign, which would lead to an exponential growth of the spin wave rather than an exponential decay. Therefore, we will limit the sum to include only the first 4 modes Dynamic Spin Wave Response The Green s function gives the steady-state response of a point source at a specific frequency. Here we obtain an expression that describes the timed response of the system to a spin wave pulse. To obtain the time dependent Green s function, we simply take the Fourier transform of Eq. 4.58: ˆ G rl (R, R ; t, t ) = G rl (R, R, ω) e iω(t t ) dω = m r,k (R) m l,k (R ) e iωn(t t ). n,k (4.59) The response function is then obtained by integrating over the spatial and temporal distribution of the spin wave source. We will choose a source with frequency ω that is Gaussian in both space and time: D (R, t ) = e R (t tc) σ σ e t e iω t. (4.6) Where σ and σ t are the widths of the Gaussian in space and time, respectively, and t c indicates the center of the spin wave pulse in time. Integrating the product of Eqs and 4.6 over the primed variables gives the response function: R rl (R, t) = i ˆ mr,k (R) m σ t e iωnt l,k [f (t) f ()] (R ) ω ω n n,k f (t) = F e R σ dr (4.61), ( ) σ (t tc) t (ω ω n ) i (t t c ) σ e t e i(ω ω n)t. (4.6) σt Where F (x) = ( π/) exp ( x ) erfi (x) is the Dawson function.

67 5 4. Spin Wave Propagation 4..1 Propagation Patterns in the Frequency Domain Here we will show the results of the Green s function obtained from Eqs This function show the steady-state propagation patterns of a spin wave point source at a given frequency, and is an intermediary step in obtaining the time-dependent spin wave propagation patterns. We will begin by considering magnonic crystals of Fe cylinders embedded in YIG arranged in a square lattice with a lattice constant a = 5nm, a filling fraction f =.4, and a bias field µ H =.1T. There are two main reasons we have chosen to use Fe and YIG in the magnonic crystal. The first is that they have a relatively large difference in saturation magnetization and exchange length, which previous calculations have shown to be favorable for producing band gaps [96, 97]. The second is because the spin wave damping in YIG is much smaller than most other materials. This results in a larger propagation distance and a smaller linewidth for the spin waves. In order to determine what lattice spacing to use for the propagation calculations, we show the dispersions and linewidths of the spin waves at a = 5nm and a = 5nm in Fig. (4.1). This shows the effect of switching from the exchange dominated regime to the dipolar dominated regime. For small lattice constants, the exchange field, a short range effect, dominates the dipolar field, a long range effect, in Eq. (.3). As the lattice constant increases, the magnitude of the exchange field decreases and the effects of the dipolar field become visible in the dispersion relations. This can be seen in the switching of the sign of the group velocity of the lowest mode near the Γ point from positive to negative. The other result of increasing the lattice constant is a smaller spin wave frequency, that is closer to the typical range of a spin-torque oscillator. We will therefore use a lattice constant of 5nm for the spin wave propagation calculations and show the dispersions and linewidths in the full Brillouin zone in Fig. 4..

68 53 Re(ω) (GHz) 15 Im(ω) (GHz).4.3. Γ M X Γ Γ M X Γ 35.6 Re(ω) (GHz) 3 ω 3 ω Im(ω) (GHz).4. 5 ω 1. Γ M X Γ Γ M X Γ Figure 4.1: Dispersion (left) and linewidth (right) plots for a magnonic crystal consisting of Fe cylinders in a YIG host with lattice constant a = 5nm (top) a = 5nm (bottom). The horizontal lines indicate the three frequencies at which propagation patterns were calculated Re(ω) (GHz) Im(ω) (MHz) Re(ω) (GHz) Im(ω) (MHz) Re(ω) (GHz) Im(ω) (MHz) Figure 4.: Dispersion and linewidth contours for the first three modes of Fig. (4.1) with a = 5nm.

69 54 G rr (r, r = ; ω = 5.5GHz) a a ω a a a a 1a a a a G rr (r, r = ; ω = 8.1GHz) a a ω G rr (r, r = ; ω = 3.6GHz) a 1a a 1 6 a a a G rr (r, r = ; ω = 6.4GHz) a a 1a a a a 1a 1a a a a G rr (r, r = ; ω = 8.8GHz) a G rr (r, r = ; ω = 31.GHz) a a a a a a 1 6 G rr (r, r = ; ω = 7.3GHz) a 1a 1a a a a 1a 1a a a a G rr (r, r = ; ω = 9.7GHz) a 1 5 1a a a a G rr (r, r = ; ω = 31.7GHz) a a ω 3 6 Figure 4.3: Green s Functions, multiplied by distance from source, showing the steadystate spin wave response to a point source of various frequencies located at the center of one of the Fe cylinders in the magnonic crystal. The units are A s/m

70 55 To get an idea for what the propagation patterns might look like, we first calculate the Green s functions for a range of frequencies (Fig. (4.3)). This is done because the time it takes to perform the Green s function calculations is not heavily dependent on the number of frequencies chosen. This allows us to examine a large number of frequencies, then select those with the most interesting properties for propagation calculations. Skipping straight to the propagation patterns would be very time intensive since those calculations scale almost linearly with the number of frequencies. These plots show that the directionality of the spin wave can vary greatly as the frequency changes. For example, when the source frequency is in the first band at ω 1 = 5.5 GHz (see Fig. 4.1), the spin wave propagation is roughly isotropic. Near the X-point of the second band (ω = 8.1 GHz), the spin wave travels almost exclusively along the x and y axes. A lack of propagation can also be seen when the frequency is in the band gap (ω 3 = 31.7 GHz). These three frequencies demonstrate how the directionality of the spin wave can vary greatly depending on the frequency of the source, and are therefore selected for further study by calculating their time-dependent propagation patterns. 4.. Propagation Patterns in the Time Domain Having found three frequencies demonstrating the varying directional properties of spin waves in this magnonic crystal, we now show the results of the time-dependent response function (Eq. 4.61). Here, we will choose a spin wave source that is centered on the origin at t c = 8 ns and has a Gaussian width of σ = R cyl /4 in space and σ t = ns in time. With these values, we can assume that the source distribution is effectively zero outside of the cylinder, and therefore limit the integration in Eq to this region. The first frequency we will look at is ω 1 = 5.5 GHz (Fig. 4.4). Here we see propagation of the spin wave in all directions. This agrees with the dispersion contours

71 56 of Fig. 4., which show wave vectors of multiple directions appearing at that frequency. While the spin wave is propagation in all directions, it is not perfectly isotropic since the velocity changes with the direction. This is easily accounted for by considering the group velocity of waves, which is v g = ω/ k. Looking at Fig. 4.1 we see that the slope of the dispersion curves at ω 1 is slightly smaller along the diagonal (Γ-M) than it is along M-X, which is closer to the horizontal/vertical direction. a a 1a ω a a t = 5ns 1 5 1a t = 1ns 1 4 a a 1a 1a a a a 1a 1a a a 1a 1a t = 15ns a a 1a 1a a a 1a 1a t = ns a a 1a 1a a Figure 4.4: Spin wave propagation patterns resulting from a spin wave pulse in one of the Fe cylinders at ω 1 = 5.5GHz. For the second frequency, we consider ω = 8.1 GHz (Fig. 4.5). This frequency exhibits highly directional propagation along the x and y axes, with hardly any propagation in the other directions. This is to be expected since this frequency is at the bottom edge of the second mode, where the only wave vector is at the X-point, which is along the x and y axes of the crystal lattice (Fig. 4.). Lastly, we look at the third frequency, ω 3 = 31.7 GHz (Fig. 4.6), which is in the band gap between the second and third modes. Compared to the other two frequencies,

72 57 a 1a ω a 1a a t = 5ns a t = 1ns a a 1a 1a a a a 1a 1a a a a a a a t = 15ns 1 5 1a t = ns 1 5 a a 1a 1a a a a 1a 1a a Figure 4.5: Spin wave propagation patterns resulting from a spin wave pulse in one of the Fe cylinders at ω = 8.1GHz. this one shows hardly any propagation of the spin wave, which is to be expected since there are no wave vectors at this frequency. From what little propagation the spin wave makes, it is clear that it is not symmetric under reflections. This is just the symmetry breaking caused by the dipolar field discussed in section 3.. It is most evident here, but is still present even in the previous two frequencies. At this frequency, we also note that the small damping constant of YIG is especially important here, as it results in a much smaller linewidth. In comparison to the band gap width, the linewidth is roughly ten times smaller, which allows us to see the gap when looking at the propagation patterns. If the linewidth were on the same order of magnitude as the gap, then even if the frequency was in the exact center, we would still see the effects of the band edges above and below.

73 58 a a 1a ω a a 1 5 1a t = 5ns t = 1ns a a 1a 1a a a a 1a 1a a a a 1 4 1a 1a t = 15ns a a 1a 1a a a 1a t = ns a a 1a 1a a Figure 4.6: Spin wave propagation patterns resulting from a spin wave pulse in one of the Fe cylinders at ω 3 = 31.7GHz Convergence of Propagation Calculations In the previous chapter we discussed the convergence of the dispersion relations obtained from the LLG equation. Here we continue the discussion by showing the convergence of the propagation calculations and how they are influenced by the dispersion relations. We begin with the equation for the Green s function in the frequency domain (Eq. 4.58). This equation involves the eigenvalues (frequencies) and eigenvectors (magnetization vectors) obtained from the LLG equation (Eq..5) in a summation over the modes and wave vectors. In performing the summation over the modes, we only include those that appear when N = 6. As shown in section 3.3, frequencies up to this mode are well converged for the parameters used. We do not include any of the higher modes because they converge more slowly. Also, the higher frequency terms in the Green s function are inversely proportional to the frequency. In Fig. 3.