Waveguides in Magnetism and Spintronics

Abstract

In this chapter, we propose to review and discuss the use of waveguides in spintronic devices and other magnetic systems as well as some emergent phenomena linked to ultrafast magnetic dynamics. Spintronic devices are expected to replace the recent nanoelectronic memories and sensors due to their efficiency in energy consumption and functionality with scalability. In the field of spintronic devices, it is required the development of magnetic thin films with a wide range of magnetic properties. To achieve the characterization of magnetic thin films, several techniques are very useful, particularly in studying magnetization dynamics described by the phenomenological Landau-Lifshitz-Gilbert equation. These techniques are developed based on key phenomena such as spin pumping, ferromagnetic resonance, the recently observed terahertz oscillations and ultrafast switching. The coupling of waveguides and microwave generators to produce radio-frequency magnetic fields and integrate them with magnetic thin films is crucial to characterize the magnetization dynamics.

1. Introduction

This chapter shows the state of the art of the use of waveguides in different experiments in magnetism, spintronics and magnonics and gives a general view of the relevance of waveguides in the development of the areas of spintronics and magnonics. We also discuss perspectives on new technologies related to these elements.

The chapter is divided as follows: we give a brief introduction to magnetization dynamics, the phenomena of resonance and the basic characteristics of spinwaves. Afterward, we describe the type of waveguides that are commonly used in in order to excite spinwaves and generate resonance conditions in magnetic materials. Then, we describe some phenomena in spintronics where oscillating fields give rise to novel and interesting phenomena. Finally, we briefly give a non-exhaustive view of potential applications and concluding remarks.

2. Magnetic dynamics

In this section, we give a brief introduction to the principles of magnetization dynamics. We start by explaining the Landau-Lifshitz-Gilbert (LLG) equation since this expression describes the dynamics of magnetic moments generated by a magnetic field and in general, by a driving force.

We suppose that we have a ferromagnetic layer with magnetic moment per unit volume $\mathbf{M}$ and it is under an effective magnetic field ${\mathbf{H}}_{\mathit{eff}}$. The effective field has several components: the Zeeman energy $H_Z$, exchange interaction $H_{\mathit{\text{exch}}}$, magnetocrystalline anisotropy energy $H_{\mathit{\text{anis}}}$, and the demagnetization energy $H_{\mathit{\text{demag}}}$, among others, depending on the system [1]. All these terms sum the energy interactions of magnetic materials per unit volume. We find the lowest energy state or the equilibrium state when the magnetic moment points in the same direction as the effective field, ${\mathbf{H}}_{\mathit{eff}}$. On the other hand, if the magnetization vector $\mathbf{M}$ deviates from the equilibrium state, then a torque is exerted on the magnetic moment [2]:

$$\mathit{\tau }=\mathbf{m}\times {\mu }_0{\mathbf{H}}_{\mathbf{eff}}$$E1

where $\mathbf{m}=\left|\mathbf{M}\right|/\mathbf{M}$ is a unitary vector in direction of $\mathbf{M}$. Using the semiclassical approximation, we have that the relation between the magnetic moment $\mathbf{m}$ and the quantum angular momentum $\mathbf{L}$ is $\mathbf{m}=\left|\gamma \right|\mathbf{L}$ where $\gamma =g{\mu }_B/\mathrm{\hslash }$ is the gyromagnetic ratio, with $g$ is a dimensionless quantity that expresses the proportionality between the quantum angular momentum and the magnetic moment and ${\mu }_B$ is the Bohr magneton. Due to the conservation of angular momentum, the ratio between $\mathbf{L}$ and time $t$ is equal to the torque: $\frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t}=\mathit{\tau }$. Therefore, from Eq. (1) and the definition of torque, we have [2]:

$$\frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}=-\left|\gamma \right|{\mu }_0\left(\mathbf{m}\times {\mathbf{H}}_{\mathbf{eff}}\right)$$E2

which is the unitary version of the Landau Lifschitz (LL) equation with no damping [2]. The complete version of the dynamic description includes a damping term therefore the complete description is called the Landau-Lifschitz-Gilbert (LLG) equation:

$$\frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}=-\left|\gamma \right|{\mu }_0\left(\mathbf{m}\times {\mathbf{H}}_{\mathbf{eff}}\right)+\alpha \left(\mathbf{m}\times \frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}\right)$$E3

The second term describes the damping in the magnetization, where the precession decays continuously. This decaying depends on the $\alpha$ term, this is the dissipation term that encompasses all the energy losses [3]. Figure 1 shows the magnetization trajectory for different values of $\alpha$. Without this extra damping term, $\mathbf{m}$ would precess indefinitely. The dissipation causes $\mathbf{m}$ to align parallel to ${\mathbf{H}}_{\mathit{eff}}$ its state of minimum energy.

Figure 1.

Sketch of the trajectory followed by $\mathit{m}$ over time under ${\mathit{H}}_{\mathit{eff}}$ with (a) $\alpha =0$ and (b) $\alpha \ne 0$.

2.1 Ferromagnetic resonance

In this section, we describe the phenomenology of ferromagnetic resonance, which is a widely used technique to characterize ferromagnetic materials. The magnetic moments of a ferromagnetic material present an intrinsic magnetic moment that precesses at a given frequency (Larmor frequency) and couples with an electromagnetic wave, absorbing its power if the frequency of the wave is the same as the Larmor frequency. This phenomenon is known as ferromagnetic resonance (FMR) [4]. This effect has various applications such as in spectroscopic techniques and in the conception of microwave devices.

Let us consider a ferromagnetic sample under a uniform external magnetic field ${\mathbf{H}}_{\mathbf{0}}$ with an arbitrary shape. We can express the internal field ${\mathbf{H}}_{\mathbf{int}}$ as [3]:

$${\mathbf{H}}_{\mathbf{int}}={\mathbf{H}}_{\mathbf{0}}-\mathbf{N}\cdot \mathbf{M}$$E4

with $\mathbf{N}$ the tensor of the demagnetizing factors. If we choose the Cartesian coordinate system with the $x$, $y$, and $z$ directions along the symmetry axes of the sample as shown in Figure 2a, we have that the tensor is diagonal, and that its components satisfy [3]:

Figure 2.

General coordinate system $\left(x,y,z\right)$ and local coordinate system $\left(x^{\prime },y^{\prime },z^{\prime }\right)$ where the precessing magnetization $\mathit{m}$ is located. (a) Precession around the applied field ${\mathbf{H}}_{\mathit{ext}}$ perpendicular to the sample. (b) The transformation is performed by first rotating $\phi$ around $z=z^{\prime }$ and then rotating by ($\frac{\pi }{2}-\theta$) around the new $y^{\prime }$ axis [4]. Representation for the case when the external applied field in the sample plane, where we have the component of the magnetization $m_x$ [5].

$$N_x+N_y+N_z=1$$E5

If we consider an external uniform field ${\mathbf{H}}_{\mathit{ext}}$ applied along the $z$ axis and the magnetization precessing around it as shown in Figure 2a. From Eq. (4), we consider $m_x$ and $m_y$, the components of the magnetization equation of motion. Solving these equations, we obtain the general Kittel resonance frequency [3]:

$${\omega }_0={\mathit{\gamma \mu }}_0{\left[H_0+\left(N_x-N_{z}M\right)\right]}^{1/2}{\left[H_0+\left(N_y-N_z\right)M\right]}^{1/2}$$E6

The above equation shows that the FMR frequency depends on the shape of the sample and the direction of the applied field. For thin films we take the cases that are represented in Figure 2. For the case of the configuration of Figure 2a, the field ${\mathbf{H}}_{\mathit{ext}}$ is perpendicular to the sample plane, and the magnetization ${\mathbf{M}}_{\mathit{ext}}$ only acts on the $z$ axis, so the component $N_z=1$. Since it does not act on the $x,y$ plane, we have that $N_x=N_y=0$. For the configuration in Figure 2b, the magnetization acts on the plane of the thin film; for simplicity, we consider it acts on the direction of the $x$ axis, that is, $N_x=1$, and since it only acts on this axis, we have that for the axes $x$ and $y$ $N_y=N_z=0$. The solutions of these two configurations are represented by the following equations, respectively [3]:

$$f_{\mathit{res}}=\left(\gamma /2\pi \right)\left[H_0-{\mu }_0{M}_{\mathit{eff}}\right]$$E7

$$f_{\mathit{res}}=\left(\gamma /2\pi \right){\left[H_0\left(H_0+{\mu }_0{M}_{\mathit{eff}}\right)\right]}^{1/2}$$E8

With these resonance conditions, we can describe the magnetization motion by taking as reference the field with local coordinates already transformed, as in Figure 2b. We assume that $m_x$ is the saturated component along the external field and therefore $m_{x^{\prime }}=\frac{M_{x^{\prime }}}{M_s}=1$. If this is the case when we change the parametric description of $\mathbf{m}$ in the local coordinate system $\left(x^{\prime },y^{\prime }{z}^{\prime }\right)$ we perform two rotations [6]:

$$\left(\begin{array}{c}{m}_x\\ m_y\\ m_z\end{array}\right)=\left(\begin{array}{ccc}\mathit{\text{senθcosϕ}}& -\mathit{\text{senϕ}}& -\mathit{\text{cosθcosϕ}}\\ \mathit{\text{senθsenϕ}}& \mathit{\text{cosϕ}}& -\mathit{\text{cosθsenϕ}}\\ \mathit{\text{cosθ}}& 0& \mathit{\text{senθ}}\end{array}\right)\left(\begin{array}{c}{m}_{x^{\prime }}\\ m_{y^{\prime }}\\ m_{z^{\prime }}\end{array}\right)$$E9

We have that in Figure 2b, the dynamics are limited to the $y^{\prime }{z}^{\prime }$ plane, in this case $z^{\prime }=z$. If we rewrite the external field as the sum of the static field and the alternating field $H_{\mathit{ext}}=H+h\left(t\right)$, and replace this in the LLG equation, we can describe the dynamics of the magnetization. Moreover, we can write the magnetization components in the local coordinate system as a function of $h\left(t\right)$ through the susceptibility tensor that expresses the shape of the curve close to the resonance [5]:

$$\left(\begin{array}{c}{m}_{y^{\prime }}\\ m_z\end{array}\right)=\left(\begin{array}{cc}{\chi }_{y^{\prime }{y}^{\prime }}& {\chi }_{y^{\prime }z}\\ {\chi }_{z{y}^{\prime }}& {\chi }_{\mathit{zz}}\end{array}\right)\left(\begin{array}{c}{h}_{y^{\prime }}\\ h_z\end{array}\right)$$E10

The shape of absorbed power during the resonance is in general a combination of symmetric and antisymmetric Lorentzian functions. The components of the susceptibility tensor in Eq. (10) depend on the parameters of these Lorentzian functions [5].

The absorbed power in the resonance condition can be expressed as:

$$p_{\mathit{abs}}\left(H,H_0,\mathrm{\Delta }H\right)=A\left(F_S\left(H\right)\mathit{cos}\left(\epsilon \right)+F_A\left(H\right)\mathit{sin}\left(\epsilon \right)\right)$$E11

Where $A$ is an amplitude factor, $\epsilon$ is an angle that mixes the symmetric and antisymmetric Lorentzian functions $F_S$ and $F_A$_, respectively [6]:

$$F_S\left(H\right)=\frac{{\left(\mathrm{\Delta }H\right)}^2}{{\left(H-H_0\right)}^2+{\left(\mathrm{\Delta }H\right)}^2}$$E12

$$F_A\left(H\right)=\frac{\mathrm{\Delta }H\left(H-H_0\right)}{{\left(H-H_0\right)}^2+{\left(\mathrm{\Delta }H\right)}^2}$$E13

The line width $\mathrm{\Delta }H$ is half the width at half maximum in the symmetric Lorentzian curve, and the terms of the type $\chi$ denote the susceptibilities at resonance. The term $\mathrm{\Delta }H$ is the result only of the Gilbert damping parameter ${\alpha }_{\mathit{eff}}$ which is composed of the interaction of $H_{\mathit{eff}}$ with the resonance frequency given by the equation of Kittel as in the text [5, 6]:

$$\mathrm{\Delta }H=\frac{{\alpha }_{\mathit{eff}}\omega }{{\mu }_0\gamma }$$E14

Thus we have that our resonance condition FMR expresses the magnetization movement where the parameter of ${\alpha }_{\mathit{eff}}$ corresponds to the speed at which the spin angular momentum dissipates [6].

In order to detect the FMR signal, a microwave resonant cavity with an electromagnet is required. The resonant cavity is fixed at a frequency in the super GHz range, and a detector is placed at the end of the cavity to detect the microwaves. To achieve a full characterization of thin film materials, one needs to carry out FMR measurements at a wide range of microwave frequencies, and for this end, a cavity at a fixed frequency is not convenient. It has been shown that using a broadband microwave signal generator, a coplanar waveguide (CPW), and a broadband microwave detector, it is possible to obtain a good signal-to-noise ratio when a GHz current is injected into the CPW from one end [5, 7].

2.2 Spin waves

So far, we have considered only the case where the magnetization dynamics are homogeneous throughout all the samples. However, it is possible to find varying magnetization through the sample; the elementary excitations are known as spin waves (SW). The quanta of these excitations are known as magnons.

To describe this phenomenon, we consider a linear chain of $N$ classical spins, each with magnitude $S$ coupled with nearest-neighbor exchange interaction, uniformly spaced, and separated by a distance $a$. We consider only the Heisenberg interaction and the ground state to be where all the spins are parallel, therefore, ${\mathbf{S}}_i\cdot {\mathbf{S}}_{i+1}={\mathbf{S}}^2$ and the exchange energy becomes $U_0-2{\mathit{JNS}}^2$ [3].

In 1930 Felix Bloch showed that the low-lying excitations of the spin system consisted of non-localized, collective spin deviations that he called SW. This was a further conceptualization from the Pierre Weiss model that considered the first excited state to be one spin reversed in the spin chain described before [8]. The existence of low-energy excitations is related to the spin system having an infinite number of degenerate ground states with infinitesimally different spin orientations in the absence of an external field.

In the linear spin chain, the effective field arising from the exchange interaction is:

$${\mathbf{H}}_{\mathit{exc}}^{\mathit{eff}}=-\frac{2J}{g{\mu }_B{\mu }_0}\left({\mathbf{S}}_{i-1}+{\mathbf{S}}_{i+1}\right)$$E15

Using the first term of the LLG equation and considering that the spins are subject to an applied static magnetic field $\mathbf{H}$, we write the total field acting over the spins as ${\mathbf{B}}_T={\mu }_0\left(\mathbf{H}+{\mathbf{H}}_{\mathit{exc}}^{\mathit{eff}}\right)$. Therefore, the spin equation of motion is written as follows [3]:

$$\frac{d{\mathbf{S}}_i}{\mathit{dt}}=-{\mathit{\gamma \mu }}_0{\mathbf{S}}_i\times \left(\mathbf{H}+{\mathbf{H}}_{\mathit{exc}}^{\mathit{eff}}\right)$$E16

where $\gamma =g{\mu }_B/\mathrm{\hslash }$ is the gyromagnetic ratio. Developing these equations for each component and taking into account that the magnetization points in the opposite direction of the spin and therefore taking $\mathbf{H}=-\mathbf{z}H$, we find that the possible solutions are in the form of traveling waves:

$$S_i^x=A_x{e}^{{\mathit{kx}}_i-\mathit{\omega t}},S_i^{x}y=A_y{e}^{{\mathit{kx}}_i-\mathit{\omega t}}$$E17

with $\omega$ the angular frequency and $k$ the wave number. Substituting these solutions in the equations of spin motion for each component, we obtain the dispersion relation, in other words, the dependence of the spin wave frequency on the wave number:

$${\omega }_k={\mathit{\gamma \mu }}_{0}H+\frac{4\mathit{JS}}{\mathrm{\hslash }}\left(1-cos\mathit{ka}\right)$$E18

From here, we get the relation between the amplitudes of the spin components, and we can write the real parts of the transverse components as:

$$S_i^x=A_{0}cos\left({\mathit{kx}}_i-{\omega }_{k}t\right),S_i^y=A_{0}sin\left({\mathit{kx}}_i-{\omega }_{k}t\right),$$E19

From these equations, we can represent the classical picture of a SW in one dimension where the spins precess circularly around the equilibrium direction, as shown in Figure 3. If the wave travels along the $+x$ shortest distance between two spins that precess with the same phase is the wavelength corresponding the wave number $\lambda =2\pi /k$.

Figure 3.

Representation of a spin wave in a 1D chain propagating in the $x$ direction. In the upper part of the figure, we show the top view of the spins. The wavelength corresponds to the distance between two spins pointing in the same direction.

When $k=0$, the magnon energy is determined only by the magnetic field intensity $\mathrm{\hslash }{\omega }_0=\mathrm{\hslash }{\mathit{\gamma \mu }}_{0}H$, however, the general expression for the energy of one magnon in a 1D chain is:

$$\mathrm{\hslash }{\omega }_k=\mathrm{\hslash }{\mathit{\gamma \mu }}_{0}H+4\mathit{JS}\left(1-cos\mathit{ka}\right)$$E20

As the wave number increases, the phase difference of precession for neighboring spins. Typically, the Zeeman contributions ($\mathrm{\hslash }{\omega }_0=\mathrm{\hslash }{\mathit{\gamma \mu }}_{0}H$) of the applied magnetic fields are small in comparison with the exchange energy. For $\mathit{ka}=\pi$_, the exchange energy is maximum and its value is $8\mathit{JS}$_, and it is the energy of a linear chain when two consecutive spins are reversed.

At low temperatures, the modes with small wave numbers are dominant, and they are important for many magnonic phenomena. For $\mathit{ka}\ll 1$, we can rewrite Eq. (20) as:

$$\mathrm{\hslash }{\omega }_k\approx \mathrm{\hslash }{\mathit{\gamma \mu }}_{0}H+2{\mathit{JSa}}^2{k}^2$$E21

which is a quadratic dispersion relation with an energy gap. The frequencies of SWs with $\mathit{ka}\ll 1$ are in the microwave range. When $k\sim 1/a$ the spin wave frequencies are in terahertz range.

There are three types of SWs based on the propagation direction ($k$) with respect to the external magnetic field ($H_e\mathit{xt}$) and the magnetization ($M$): Damon-Eshbach (DE) or surface waves when $\mathbf{k}$ is perpendicular to the field $\mathbf{H}$ , but both are on the sample plane, backward volume waves (BV) when $\mathbf{k}$ is parallel to the field $\mathbf{H}$ and both are on the sample plane and forward volume waves (FV), when $\mathbf{k}$ is perpendicular to the field $\mathbf{H}$ but the field is perpendicular to the sample plane [9].

3. Coplanar waveguides in magnetic resonance experiments

Waveguides are important in the study of any high-frequency signal and since we have described the magnetization dynamics in previous sections, we continue to describe examples of applications in magnetism and spintronics. In ferromagnetic materials, the Larmor frequency usually falls in the range of a few tens of GHz. Thus, transmission lines, such as coplanar waveguides, provide a large frequency band in which the impedance is relevant in FMR-related experiments where ferromagnetic films are studied [10].

Coplanar waveguides are composed of a thin metal film layer that is the conductor with a width $W_{\mathit{sl}}$ with two ground planes parallel to the conductor with a width $W_{\mathit{gap}}$ separated by a distance $W_{\mathit{gl}}$ which in this case are composed of copper and have a thickness $t_{\mathit{TL}}$ where they are placed on a dielectric substrate with a thickness $t_{\mathit{\text{substrate}}}$ as shown in Figure 4 [11].

Figure 4.

Coplanar waveguide. The current $j$ flows inside and outside along the direction $y$, generating the transverse magnetic field lines $h\left(t\right)$. The most important parameters of a coplanar waveguide are indicated: the metal thickness $t_{\mathit{TL}}$, the substrate thickness $t_{\mathit{\text{substrate}}}$, $W_{\mathit{sl}}$, the width of the signal line, the width of the gap between the signal line and the ground lines $W_{\mathit{gap}}$, and the width of the ground line $W_{\mathit{gl}}$ [10].

One of the main features of coplanar waveguides is the different possible combinations of $W_{\mathit{gap}}$ and $W_{\mathit{sl}}$ to design circuits with a specific impedance and modulation of the electromagnetic field generated by the signal line and its two ground lines. This is of great interest to us because we use this property to control the magnetic field generated by the CPW at frequencies from $0\mathrm{GHz}$ to $20\mathrm{GHz}$, which corresponds to the study of the properties of ferromagnetic resonance (FMR) [12].

3.1 Theoretical model coplanar waveguide in magnetism

We give a brief theoretical description of the physics involved in the operation of CPWs. We start by assuming that electromagnetic waves propagate along the $y$ direction of the CPW, as shown in Figure 4. In this CPW system, transverse electromagnetic modes (TEM) are admitted, and the local electric ${\mathbf{E}}^{-}$ and magnetic field $\mathbf{H}$ are perpendicular to each other. The direction of propagation and the direction of the magnetic field $\mathbf{H}$ are indicated in Figure 4. Applying Maxwell’s equations to a general homogeneous lossy medium [10]:

$$\nabla \times \mathbf{E}=-\mathit{i\omega \mu }\mathbf{H}$$E22

$$\nabla \times \mathbf{H}=\mathit{i\omega \epsilon }\mathbf{E}+\sigma \mathbf{E}$$E23

where $\omega /2\pi$ is the frequency of the electromagnetic wave, $\mu ={\mu }^{\prime }+i{\mu }^{\prime }$ is the complex magnetic permeability, $\epsilon ={\epsilon }^{\prime }-i{{\epsilon }^{\prime }}^{\prime }$ is the complex electrical permittivity and $\sigma$ is the conductivity of the guide wave [10]. For the electric field, we have to [10, 13]:

$${\nabla }^2\mathbf{E}+{\omega }^2\mathit{\mu \epsilon }\left(1-i\frac{\sigma }{\mathit{\omega \sigma }}\right)\mathbf{E}=0$$E24

We define the complex propagation constant for the medium as follows [10]:

$$\gamma =\mathit{i\omega }\sqrt{\mathit{\mu \epsilon }}\sqrt{1-i\frac{\sigma }{\mathit{\omega \epsilon }}}$$E25

If we assume that the electric field acts only in the $x$ direction and, at the same time, remains uniform in the $\mathit{xy}$ plane, Eq. (24) reduces to [10]:

$$\frac{{\partial }^2{E}_x}{\partial z^2}-{\gamma }^2{E}_x=0$$E26

The general solution of this differential equation is given by [10]:

$$E_x\left(z\right)=E^{+}exp\left(-\mathit{\gamma z}\right)+E^{-}exp\left(\mathit{\gamma z}\right)$$E27

where the term on the right side describes a wave that travels toward the positive side of the $z$ axis and the left side of a wave that travels in the opposite direction in the $-z$ direction. The time-dependent propagation factor of the positively moving wave (in this case $-\mathit{\gamma z}$) is written as [10]:

$$\Re \left(exp\left(-\mathit{\gamma z}\right)exp\left(\mathit{i\omega t}\right)\right)=\Re \left(exp\left(-\mathit{\alpha z}\right)exp\left(-\mathit{i\beta z}\right)exp\left(\mathit{i\omega t}\right)\right)=exp\left(-\mathit{\alpha z}\right)cos\left(\mathit{\omega t}-\mathit{\beta z}\right)$$E28

For good conductors $\omega <<\sigma$, this means that the dielectric losses are small compared to the conductivity, and when the conductivity loss predominates, we have that Eq. (25) reduces to [10]:

$$\gamma =\mathit{i\omega }\sqrt{\mathit{\mu \epsilon }}\sqrt{\frac{\sigma }{\mathit{j\omega \epsilon }}}=\left(1+i\right)\sqrt{\frac{\mathit{\omega \mu \sigma }}{2}}$$E29

Where the real part of the above equation corresponds to $\alpha$ and this defines the skin depth ${\delta }_s$ of the electric field in the material with the relationship [10]:

$${\delta }_s=\frac{1}{\alpha }=\sqrt{\frac{2}{\mathit{\omega \mu \sigma }}}$$E30

where $\alpha =\mathit{i\omega }\sqrt{\mathit{\mu \epsilon }}$. As we can see from the above equation, the amplitude of the current density decreases exponentially on the scale of the characteristic length ${\delta }_s$ within conductive transmission lines [10].

This result is related to the geometric parameters of CPW. If $t_{\mathit{Tl}}<<{\delta }_s$ we approximate the current density to be homogeneously distributed along the $z$ axis in the CPW. In contrast, along the $x$ direction and due to $W_{\mathit{sl}}$, $W_{\mathit{gl}}>>{\delta }_s$ we cannot assume that the current density is homogeneous. Thus, to determine the local in-plane and out-of-plane amplitudes of the excitation fields, one has to consider the distribution of the current density layers as a function of frequency along the $x$ direction [10, 13].

Thus, we have that the change in the increase in current densities that goes toward the limits of the transmission lines is proportional to the frequency, where they are related through the depth of the skin $\delta$. Therefore, the frequency-dependent magnetic field within the CPW can now be calculated by summing all contributions from the current density paths using the Biot-Savart law. Thus, the field can be calculated at each point of the CPW [10].

3.2 Example of CPW compatible with FMR experiment

As explained in the previous section, a coplanar waveguide (CPW), as shown in Figure 4, consists of ground lines and conducting lines. The ground lines must be at ground potential, and the signal line must have relative voltage amplitude [10, 13]. We notice in Figure 4 that the magnetic field lines are parallel to axis $x$ and if we put a thin ferromagnetic film on top, these field lines would lie on the sample plane [10].

Now, we will discuss an example of a CPW design, starting with the choice of materials for an FMR experiment. The conductive material that was used is copper. This metal is commonly chosen due to its low cost, good electrical conductivity and low dielectric loss. It is also compatible with the transmission of the wave frequencies studied in an adequate way. The substrate we use is Rogers RO4350B, which is made of glass-reinforced woven hydrocarbon/ceramic with a dielectric constant of ${\epsilon }_r=3.48\pm 0.05$ [14]. According to the data we have, this material has a high frequency for applications of up to 18–22 GHz with a loss of $\pm 0.05$. Moreover, its dielectric constant does not change abruptly with temperature therefore is a suitable material for temperature variation experiments [14].

To design the waveguide, we use the Advancing the Wireless Revolution Design Environment (AWR) program, which is provided under license by Cadence Design System, Inc. [15].

The process to use this software is to delimit the parameters. For this purpose, we use the AWR transmission line calculator (TX-LINE) to get a first approximation of realistic parameters (considering an impedance of $50\mathrm{\Omega }$ and a loss close to $0$) [15]. Then, in circuit schematics, using the predefined designs and the dimensions of the CPW, we create an electromagnetic (EM) structure. At this stage, we design the CPW in three-dimensions (3D) and analyze how the electromagnetic fields act on the waveguide and how they affect the loss and power in it. What this procedure is broadly about is comparing the design in Figure 5b with the ideal dimensions that we consider in Figure 5a, finding the best configuration, following this procedure.

Figure 5.

The coplanar waveguide layout of our example designed in AWR is shown [15]. (a) Circuit diagram for the designed coplanar waveguide. (b) Return loss and power vs. frequency. The pink squares represent the loss of the CPW modeled in the first approximation with the TXline parameters, the brown diamonds represent the loss of the AWR-designed CPW, the blue triangles represent the power of the approximate CPW, and the red double triangles represent the power of the AWR-designed CPW.

Furthermore, the CPW is designed in 3D, taking into account the conductor and substrate mentioned above in the Axiem EM section of AWR [15]. This software simulates the electromagnetic fields in the CPW in order to calculate the loss and power of our reference CPW, which is shown in Figure 5b and compared to the loss and power in the CPW with the parameters obtained by TX-LINE. Thus, by the aforementioned procedure, we refine our design until we reach the one we have graphed in Figure 5a, which minimizes the loss and maximizes the power (Figure 5a).

We see how the loss of both the approximate CPW and that of the CPW designed in 3D is very close to zero, and the power of the CPW designed in 3D (EM Structure) is delimited below the CPW that we take as a base (Coplanar), this tells us that we have a suitable design for our purposes. Thus, after this process, we have the dimensions found for this example of CPW using the characteristics of Figure 4 are: $W_{\mathit{gl}}=3\mathit{mm}$, $W_{\mathit{sl}}=1\mathit{mm}$, $W_{\mathit{gap}}=0.2\mathit{mm}$, $t_{\mathit{TL}}=0.2\mathit{mm}$.

4. Spintronics and magnonics phenomena with waveguides

In this section, we describe experiments and physical phenomena that are intensely studied in the subareas of spintronics and magnetization dynamics. We will focus on potential applications based on the production of spin currents, phenomena related to spin-orbit torques and spin transport via spinwaves.

4.1 FMR, spin pumping and spin currents

The magnetization dynamics in thin magnetic multilayers, specifically in ferromagnetic films in contact with paramagnetic conductors, can produce a moving magnetization vector that causes “pumping” of spins into adjacent nonmagnetic layers [16]. This spin transfer affects the magnetization dynamics, similar to the Landau-Lifshitz-Gilbert phenomenology. This phenomenon is known as spin pumping, and it is the Onsager reciprocal to spin-transfer torque (STT) [17]. In both cases, there is transfer of spin angular momentum between a ferromagnet (F) and a metallic reservoir.

In the first one, a spin (polarized) current injected from the metal into the ferromagnet transfers angular momentum onto the magnetization $M$ of the latter, thereby inducing magnetization dynamics [18]. In contrast, for spin pumping, the nonequilibrium magnetization dynamics (magnons) in the ferromagnet act as the source for an angular momentum flow from the ferromagnet into the metal, inducing a spin current [19].

There are two different detection schemes for spin pumping in ferromagnetic/metallic (F/M) bilayers: the first one detects spin pumping spectroscopically by the increase of the (Gilbert) damping constant of coherent magnetization precession in the ferromagnet, in this case, the metal acts like an angular momentum sink. The second is the electrical detection of the spin current electrically via the inverse spin Hall effect in the metal, which acts as a spin current detector. The excedent of angular momentum can also flow out or be “pumped” across the interface from the ferromagnet into the normal metal, where both longitudinal and transversal angular momentum are pumped. The latter is the main contributor to damping for small precession angles $\theta$. This is schematically shown in Figure 6d. Since spin pumping increases the damping $\alpha$, one way to detect spin pumping is through the increase in the line width in FMR measurements, as compared in Figure 6a and b.

Figure 6.

(a) The absorption in the resonance condition corresponding to (b). The linewidth $\mathrm{\Delta }H$ is determined by all the relaxation mechanisms, (c) the magnetization vector of a ferromagnet out of its equilibrium position precessing around the magnetic field with an angle $\theta$ and excited via resonance produced by a microwave magnetic field $h_{\mathit{RF}}$, and (d) ferromagnet/normal metal bilayer. In this structure, there is an additional component to the damping mechanism due to the excess of angular momentum that broads the FMR line width. This can be understood as the generation of pure spin current $j_s$ with longitudinal spin-polarization; that is, the spin current is pumped from the ferromagnet in its resonance condition into the normal metal.

The spin current density (the spin current per interface area) pumped across the F/N interface by the spin-pumping mechanism is [20, 21, 22]:

$${\mathbf{j}}_s^{\mathit{\text{pump}}}=\frac{\mathrm{\hslash }}{4\pi }\left\{\mathit{Re}\left(g^{\uparrow \downarrow }\right)\left[\mathbf{m}\times \frac{d\mathbf{m}}{\mathit{dt}}\right]-\mathit{Im}\left(g^{\uparrow \downarrow }\right)\frac{d\mathbf{m}}{\mathit{dt}}\right\}.$$E31

where $g^{\uparrow \downarrow }$ is the spin mixing conductance that can be understood as the available spin transport channels at the interface, $\mathbf{m}=\mathbf{M}/\mid \mathbf{M}\mid$ is the magnetization unit vector and $\mathrm{\hslash }$ is the reduced Planck constant.

We can see from Eq. (31) that the current generated as a result of spin pumping across the F/N interface is equivalent to an additional Gilbert-like damping channel, increasing the FMR line width. The additional Gilbert-like damping arising from spin pumping can thus be expressed as an increase of $\alpha$ by ${\alpha }_{\mathit{SP}}$ [23, 24]:

$${\alpha }_{\mathit{SP}}=g{\mu }_B\frac{\mathit{Re}\left(g^{\uparrow \downarrow }\right)}{4\pi M_S}\frac{1}{t_F}\eta$$E32

where $g$ Landé $g$-value, the Bohr magneton ${\mu }_B$, the saturation magnetization of the ferromagnet $M_S$, the thickness of the ferromagnet $t_F$, and the backflow parameter $0\le \eta \le 1$, account for a possible spin current backflow into the ferromagnet.

Another way to detect spin currents arising from spin pumping is via electrical measurements through the inverse spin Hall effect (ISHE) [25, 26]. The charge current density can be written as:

$${\mathbf{j}}_c=-\frac{2e}{\mathrm{\hslash }}{\mathrm{\Theta }}_{\mathit{SH}}{\mathbf{j}}_s\times \mathbf{s}$$E33

where ${\mathbf{j}}_s$ is the spin current in conductors with nonzero spin-orbit coupling and $e$ is the electronic charge, $\mathbf{s}$ is the spin current polarization and represents the orientation of the spin flow in the direction of ${\mathbf{j}}_S$ and ${\mathrm{\Theta }}_{\mathit{SH}}$ is the spin Hall angle, which represents the spin-to-charge current conversion efficiency; for heavy metals, we expect large spin-to-charge conversion.

From Eq. (31), we can see that the spin current ${\mathbf{j}}_S$ has AC and DC components. The DC spin current component arises from the longitudinal nonequilibrium magnetization components and the AC spin current components from the transverse components.

From Eq. (31), we can see that the spin current ${\mathbf{j}}_S$ has AC and DC components. The DC spin current component arises from the longitudinal nonequilibrium magnetization components and the AC spin current components from the transverse components.

For the electrically detected spin pumping, we will consider only the DC part of ${\mathbf{j}}_s$. The DC ${\mathbf{j}}_s$ is pumped across the surface and, depending on the direction of $\mathbf{M}$ and according to Eq. (33), ${\mathbf{j}}_c$ lies along the $N$ plane, as shown in Figure 7. The pumped DC spin current ${\mathbf{j}}_s$ propagates across the F/N interface and is normal to it; this is the physical origin of the ISHE. From Eq. (33), we see that ${\mathbf{j}}_c$ lies in the plane of the N layer and perpendicular to $\mathbf{s}$. The pumped spin current into N decays at the same scale as the spin diffusion length ${\lambda }_{\mathit{sp}}$, usually between 1 and 100 nm.

Figure 7.

Schematic of the DC electrically detected spin-pumping experiments, the DC spin current ${\mathit{j}}_s$ pumped into the normal metal is converted into a charge current ${\mathit{j}}_c$ via the inverse spin Hall effect and then detected via the corresponding open-circuit voltage $V_{\mathit{DC}}$.

It is important to notice that ${\mathbf{j}}_s$ will be pumped into N within the resonance condition. Under these conditions, it is possible to detect $j_c=\mid {\mathbf{j}}_c\mid$ or $E_c=V_{\mathit{DC}}/L=j_c/{\sigma }_{\mathit{tot}}$ where $L$ is the distance between the electrical contacts and ${\sigma }_{\mathit{tot}}$ is the total conductivity of the F/N bilayer. The spin-pumping spin Hall voltage $V_{\mathit{DC}}$ is given by [27, 28]:

$$\frac{V_{\mathit{DC}}}{L}=\frac{2e}{\mathrm{\hslash }}{\mathrm{\Theta }}_{\mathit{SH}}\frac{j_s{\mathit{\eta \lambda }}_{\mathit{sd}}\mathit{tanh}\left(\frac{t_N}{2{\lambda }_{\mathit{sd}}}\right)}{{\sigma }_F{t}_F+{\sigma }_N{t}_N}$$E34

The DC electric field can also be generated by microwave rectification phenomena apart from ISHE. In this context, the resonant magnetization precession in FMR generates a modulation of the sample’s resistivity tensor at the microwave frequency. The resistivity modulations rectify microwave-frequency charge currents induced in the sample by the microwave signal. Since different components of the resistivity tensor (anisotropic magnetoresistance, anomalous Hall effect, etc.) can contribute to the rectification voltages, based on symmetry or line shape can be ambiguous the presence of an additional, magnetically ordered layer may provide an additional magnetization damping channel [29]. Furthermore, there are other effects that should be taken into account in microwave rectification or magneto-caloric effects such as the anomalous Nernst effect, which may then play a role in the proximity of polarized metal.

Note that the ferromagnetic layer in this type of experiment does not need to be metallic in order to produce a spin current in the adjacent layer. There has been experimental evidence both for and against static magnetic polarization in the Pt in F/Pt hybrids made from an electrically insulating ferromagnet such as yttrium iron garnet (YIG) [30, 31, 32].

Spin pumping, as well as other interesting phenomena well known in the spintronics areas such as spin Hall effect (SHE) [25, 33], Rashba effect [34, 35], and Dyaloshinskii-Moriya Interaction (DMI) are all known to have origin in spin-orbit coupling (SOC) [36].

Spin-orbit torques (SOT) originating from pure spin current are expected to be crucial to control the magnetization as well as SW much more efficiently than for instance conventional spin-transfer torque, since SOT spin-polarization arises from the carrier velocity difference, and there is no need for a magnetized layer that plays as a spin filter. SOT is capable of compensating Gilbert damping over extended regions, which has opened up a new avenue for on-chip SW communication devices. These applications will be discussed further in Section 5.

We have discussed spin-pumping experiments in which the magnetization was driven out of equilibrium via conventional FMR. In other words, the spin current across the F/N interface was excited by microwave photons. However, magnetization dynamics can be caused by other mechanisms: thermal gradients, acoustic strain fields, charge currents, and high-frequency elastic deformations.

4.2 Microwave techniques to study spin waves

In magnonic device technology, the main mechanisms under study are the ones implicated in three major blocks: generation, detection, and manipulation of the spin waves, which correspond to input, gate, and output. The magnetic waveguide in a magnonic device guides the spin wave signal of a given frequency , wave vector, phase, and amplitude, which are the key ingredients for wave-based computing [37].

An important breakthrough was the understanding of the charge current to spin current conversion and vice versa. The charge-to-spin current has been achieved via SHE, where a charge current is injected into a heavy metal with a large SOC, leading to an orthogonal spin current, as explained in the previous section. The inverse mechanism ISHE is useful to detect spin waves (Figure 7) [33, 38].

In order to propagate a magnon current, we need to be able to regulate the input, output, and gate. To do so, it is necessary to manipulate and control regimes of frequency, wavevector, and phase of the spin wave. We also need to be able to control the decay length ($\lambda$), which, as explained before, is the length over which the spin wave intensity drops by a factor of $e$. For this, we need materials with low damping as YIG, as we already have pointed out; other examples include Permalloy (${\mathit{Ni}}_{80}{\mathit{Fe}}_{20}$) [39], CoFeB [40], and Heusler Alloys [41].

A very common technique to study long spin waves in ferromagnets is the use of microwave radiation. For a monochromatic driving field with frequency $\omega$, the absorbed microwave power by a spin wave when ${\omega }_k=\omega$ is given by [3]:

$$P=\frac{1}{2}{\mu }_0\mathit{\omega \Re }\left[\int \mathbf{h}\cdot {\mathbf{m}}^{\ast }d\mathbf{r}\right]$$E35

where $\mathbf{h}$ and $\mathbf{m}$ are the driving field and the small signal magnetization with frequency $\omega$, they are both complex quantities. If we consider the driving field along the $x$ direction, we can write the power absorbed by a spin wave with magnetization $m_x\left(\mathbf{k}\right)exp\left(i\mathbf{k}\cdot \mathbf{r}\right)$ due to a driving field with a spatial variation $h_{x}exp\left(i\mathbf{q}\cdot \mathbf{r}\right)$ as:

$$P=\frac{1}{2}{\mu }_0\mathit{\omega V\Re }\left[h_x\left(\mathbf{q}\right)m_x^{\ast }\left(\mathbf{k}\right)\right]{\delta }_{\mathbf{q},\mathbf{k}}=\frac{1}{2}{\mu }_0\mathit{\omega V}{{\chi }^{\prime }}_{\mathit{xx}}^{\prime }{\left|h_x\left(\mathbf{k}\right)\right|}^2$$E36

where ${\delta }_{\mathbf{q},\mathbf{k}}$ is the Kronecker delta, equal to 1 for $\mathbf{q}=\mathbf{k}$ and zero for other values, ${{\chi }^{\prime }}_{\mathit{xx}}^{\prime }$ is the susceptibility tensor. This expression shows that a spin wave with a certain wave vector $\mathbf{k}$ can only be excited by a driving magnetic field with a Fourier component at the same wave vector.

In bulk materials with uniform internal magnetizing fields, it is only possible to excite the $k\sim 0$ magnon, that is, the case of ferromagnetic resonance.

In order to excite higher-order magnons, we need microwave radiation, fine wire antennas, and thin magnetic films. For instance, a metallic multielement antenna is deposited by lithography on the surface of the magnetic film [42].

Another possibility is to use the boundary conditions at the film surface in order to generate a nonuniform internal driving field. In this scheme, the driven spin waves are comparable to the film thickness [43]. In films, the first observations of spin wave resonances were measured in a Permalloy film, with the static field perpendicular to the film surface. In this experiment, plane spin waves with wave vectors perpendicular to the film plane generated standing waves between the two film surfaces [43].

The wavelength is given by $d=n\left(\lambda /2\right)$ where $d$ is the film thickness and $n$ is an integer. As the field increases, the wave number varies, and the spin wave resonances produce absorption peaks corresponding to a different wavelength, with index number $n$.

The first studies of spin waves were done in YIG bulk single crystals in the shape of disks, rods, or slabs. When the sample has these shapes, the applied uniform field produces a nonuniform internal field due to demagnetizing effects on the surfaces. For static fields, the wave frequency remains unchanged; however, the wave number varies since there is no momentum conservation. The experiment described in ref. [44], uses a YIG displaced in a cylindrical microwave cavity with a 9.4 GHz microwave input. In the experiment performed by Eshbach et al., the oscilloscope signal showed magneto-elastic wave echoes at various magnetic field values. This indicated that due to the nonuniform dc field, the microwave pulse drives the spin wave in the region where it has a wave number $k\sim 0$.

The wave packet with circular wavefront propagates toward the center of the disk with constant frequency. As the field decreases, the wave number increases and approaches a crossover region. In a uniform field, the two branches of magnetoelastic dispersion would correspond to normal modes. Since the field varies in space, the excitation can hop into another branch [45]. The hopping probability depends on the combination of magnetic field values at a given position and if, in that position, the magnon and phonon wave numbers are equal. On the other hand, if the field gradient is much smaller, the probability for branch hopping decreases, and in this case, the spin wave packet is converted very efficiently into an elastic wave, just like in the experiment performed in Ref. [44].

Magnon–phonon conversion in a time-varying field can be experimentally demonstrated in a YIG rod using a setup consisting of the microwave source, that produces rf bursts with frequency in the range of 1–2 GHz and duration $\sim 0.2\mathit{\mu s}$, and the receiving system, which consists of a broadband microwave amplifier, a diode detector or a narrowband superheterodyne system, composed of a local oscillator, mixer, amplifier and diode detector.

In this type of experiment, one can tune the amplifier and measure the pulse frequency by scanning the oscillator frequency in the narrowband. Then, an oscilloscope displays the pulses and the waveform of the current pulses produced by the time-varying field [46].

We have discussed the first experiments in order to measure spin wave frequency shift and magnon–phonon conversion under a time-varying magnetic field; however, there are other measuring techniques that have recently become very important in the study of spin waves, such as inelastic light scattering techniques that nowadays are crucial to study elementary excitations in solids, among which one of the most used ones is Brillouin light scattering (Figure 8) [3].

Figure 8.

Diagram of the setup used to observe spin wave frequency shift and magnon-phonon conversion in a magnetic field varying with time.

5. Potential applications

Today’s computing systems rely on information read and written by electric charge or voltage, and therefore, computation is performed by charge movements. The fundamental circuit element is the transistor, which plays the role of both a switch and an amplifier. Nowadays, large-scale integrated circuits are based on complementary metal–oxide–semiconductor (CMOS) field-effect transistors due to their high density, low power consumption, and low fabrication cost [47].

Logic gates can be built with CMOS transistors, such that they represent a full set of Boolean algebraic operations. The device density and the performance of the CMOS have been steadily improved in the last decades according to Moore’s law [48]. CMOS technology has driven and adapted to an exploding information technology market and therefore CMOS has consolidated as a preferred mechanism in the digital domain.

Spintronics, which uses the spin degree of freedom for information coding, is particularly promising as an alternative CMOS paradigm in computing technology due to the low intrinsic energies of magnetic excitations as well as their collective nature [49, 50, 51].

Several implementations of spintronic Boolean logic devices have been investigated such as magnetic semiconductors [52], spin currents [53], nanomagnets [54], domain walls [55, 56], and skyrmions [57]. Beyond logic gates, interconnects are the key elements of any circuit. Usually more energy is typically dissipated in interconnects by moving data around. The fraction of the power dissipated in interconnects may be even larger for spin-based logic if communication happens via spin currents, domain wall propagation, or by converting back-and-forth spin signals to electrical signals. In spin waves, there is no charge transported; therefore, using them as information carriers is a potential solution for energy dissipation [58].

For this reason, of the most novel spintronic logic device concepts have been based on spin waves as information carriers, and several spin-wave-based logic devices have been proposed [47] and experimentally demonstrated, for instance, via interferometer-based logic gates [59] and spin-wave majority gates [60]. However, in order to develop practical applications of such logic devices, a combination of circuits and computing systems containing both logic and memory is necessary.

The current challenge in magnonic circuits is to keep losses under control. If this is achieved, communication may not require extra energy costs. Magnonic circuits can scale to nanoscale dimensions, unlike optical waveguides. For now, some realistic proposals point toward a hybrid systems concept with local spin-wave islands embedded in a CMOS periphery.

Another advantage of spin waves is their relatively large coherence length $l_{\mathit{pl}}$, even at room temperature (up to 1 cm in YIG of micrometer thicknesses) as well as their broad frequency range. These characteristics make spin wave devices promising candidates for applications related to data transfer and processing for rf applications. Additionally, the recent miniaturization of spin-wave wavelength $\lambda$ below 100 nm that, for now, is limited to the lattice constant of the material gives further perspective to enable access to magnonics in the THz range [61].

Current technology covers the frequency range of up to 100 GHz, which is of special interest for the upcoming 5G communication systems. As a result, typical components for wireless communication (antennas, amplifiers, filters, mixers, modulators, and frequency synthesizers) will have to adapt to new frequency bands and communication protocols.

The development of ultrahigh-speed beyond–CMOS spin-wave-based technology for signal processing requires, as said before, long-range enhancement of spin waves, that is, enhanced spin wave coherence and propagation length. These can be addressed by spin-orbit torques (SOT) since, through these torques, it is possible to electrically control damping, resulting in significantly enhanced spin-wave propagation.

The theoretical concepts of SOC and spinwaves were both formulated separately. However, the interplay between these two topics has drawn attention recently due to its various advantages [61, 62]. There are several phenomena that originate from SOT, for instance, perpendicular magnetic anisotropy (PMA), SHE, Rashba effect, spin pumping and Dyaloshinskii-Moriya Interaction (DMI) [63, 64]. At the same time, it has recently and experimentally realized that SOT originates from pure spin current, as it is explained in Section 4.1.

Observation of chiral spin textures is caused, among other factors, by the presence of DMI in thin film heterostructures where a ferromagnetic layer is placed adjacent to a heavy metal layer or 2D material. The influence of DMI or antisymmetrical interaction in the spin wave dispersion has become a topic of interest. It has been theoretically predicted that spin wave asymmetry can be induced by DMI, and this can be measured by the frequency difference of counter-propagating spin waves in the Brillouin light scattering technique [62, 65].

SOTs provide a versatile tool to electrically manipulate the magnetization of all classes of magnetic materials, that is, metals, semiconductors, or insulators, as well as different types of magnetic order, including ferrimagnetic and antiferromagnetic structures. This versatility has led to a variety of experimental and conceptual results, which have greatly expanded the scope of spintronics in the last decade [50].

The demonstration of the current-induced switching of a single-layer ferromagnet, by injecting a current density $\sim 108A/{\mathit{cm}}^2$ into a Pt layer a few nm thick, adjacent to a Co dot with perpendicular magnetization [36]. Further experiments showed switching of three terminal magnetic tunnel junctions (MTJs) based on Ta/CoFeB/MgO or W/CoFeB/MgO stacks with either perpendicular or in-plane magnetization [66, 67], using $\mathit{ns}$ current pulses with extremely low error rates and without external fields. In MTJs, SOTs allow for the switching of the free layer without passing a current through the tunnel barrier. The separation of the write and read current paths further avoids write errors during readout and voltage breakdowns. For these reasons, SOTs are attracting increasing attention as a replacement for or addition to STT in magnetic random access memories (MRAMs).

The goal of the scientific challenges is to combine a number of spin-orbit effects with the magnonic crystals and therefore to the magnonic devices. For example, the simultaneous existence of PMA and DMI as well as patterning of 2D magnonic crystals may stabilize topological magnetic textures like skyrmions and might improve the control of their dynamics.

A new path for novel physical phenomena may be paved by the combined effect of current flow and magnon on skyrmions due to the competition between thermally driven magnon current, transverse magnon current, and electron flow. One of the main challenges of magnonics for applications is the large damping of spinwaves in metallic ferromagnets, which limits its propagation distance and efficiency. The properties of SOT may indeed dramatically reduce damping.

6. Conclusions

Waveguides in spintronic phenomena are crucial due to the nature of magnetization dynamics, which makes magnetic moments sensitive to time-varying fields, producing a wide variety of spin phenomena from spin waves to resonance phenomena which, with proper adjacent layers, produce pure spin currents due to SHE or Rashba effect. The optimization of waveguides so that the magnetic moments reach the resonant condition is a key ingredient in generating spin currents and observing phenomena like spin pumping, SHE, and ISHE, which arise from SOC and produce SOTs.

SOTs provide a variety of tools to electrically manipulate the magnetization of all classes of magnetic materials: metals, semiconductors, or insulators, as well as different types of magnetic order (ferrimagnetic and antiferromagnetic structures). This has produced an enormous amount of experimental results, which have greatly expanded the scope of spintronics in the last decade.

Spin waves are considered to be promising as information carriers in low-power-consuming computing and logic. Since, in the case of spin waves, there is no charge transport but purely spin transport, and it presents interesting advantages such as the absence of Joule heating and clock frequencies up to the THz frequency regime.

On the other hand, the wide variety of physical phenomena and properties of materials linked to SOTs open the door for innovative technology that can leverage the topological properties of certain materials in spintronic devices, spin logic, efficient reading and writing mechanisms, etc.

One of the current challenges in the field of spintronics-magnonics lies in implementing spin-orbit effects to modify the magnon properties in tailored magnetic materials. This achievement is very promising for energy-efficient high-frequency nanoelectronic devices.

Conflict of interest

The authors declare no conflict of interest.

Abbreviations

LLG	Landau-Lifshitz-Gilbert
FMR	ferromagnetic resonance
CPW	coplanar waveguide
DE	Damon-Eshbach
TEM	transverse electromagnetic
EM	electromagnetic
STT	spin-transfer torque
ISHE	inverse spin Hall effect
YIG	ytrium iron garnet
SHE	spin Hall effect
DMI	Dyaloshinskii-Moriya interaction
SOC	spin-orbit coupling
SOT	spin-orbit torque
SW	spin wave
CMOS	complementary metal-oxide-semiconductor
PMA	perpendicular magnetic anisotropy
MTJ	magnetic tunnel junction
MRAM	magnetic random access memory