Controlling magnetic domain wall oscillations in confined geometries

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domain wall oscillations

in confined geometries

Doctoral thesis by André Bisig

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Universität Konstanz

Fachbereich Physik

Controlling magnetic domain wall oscillations in confined geometries

Dissertation

zur Erlangung des akademischen Grades:

Doktor der Naturwissenschaften

vorgelegt von

André Bisig

2012

Konstanzer Online-Publikations-System (KOPS) URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-228348

URL:https://kops.ub.uni-konstanz.de/xmlui/handle/urn:nbn:de:bsz:352-228348

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Tag der mündlichen Prüfung: 18. Februar 2013 Referent: Prof. Dr. Mathias Kläui

Referentin: Prof. Dr. Gisela Schütz

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Introduction

Today (2012), I’m writing this introduction on a book sized tablet computer, yet its compu- tational power exceeds that of the Apollo spacecraft guidance computer used for navigation and control by orders of magnitude [1]. Our current miraculous “smart”-devices are the result of ongoing miniaturization of their electronic components and improvement of their performance, following the exponential Moore’s Law, which states that the “[c]omplexity of integrated circuits has approximately doubled every year since their introduction” [2], which has proven to be correct since it was stated in 1975. The field of spintronics has played a principal role of increasing the data storage density over the last decade, after the discovery of the Giant Magnetoresistance (GMR) effect by Fert and Grünberg, for which they were awarded the 2007 Nobel Prize in physics. This effect is based on the sensitive change of the resistance with respect to small magnetic fields in magnetic multilayer sys- tems with a thickness of only a few nanometers and is used to boost the sensitivity of the hard disk read-out heads. Spintronics corresponds to the recent development of electronics, whose functionality is based not only on the charge of the electrons but also on its spin.

Currently, spintronic nanoscale technologies are intensely investigated in the pursuit of novel, faster and smaller magnetic storage, sensing, and memory devices. Such new technologies include spin-transfer oscillators (STO) [3], which can be used to generate oscillations in the microwave frequency range and have applications in radio electronics. They are based on the transfer of spin angular momentum of spin-polarized electrons passing through a “free” magnetic layer, inducing non-resonant oscillation of the device’s resistivity. The oscillation frequency can be directly controlled by the current, allowing for fast switching of the transmission frequency, which is a big advantage in encryption technologies over today’s resonant RC-electronics. Another example is the Magnetic Random Access Memory (MRAM), which combines the speed of conventional RAM with the non-volatility of magnetic storage. Recently, the Racetrack Memory has been proposed as a new con-

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cept for magnetic storage and memory. It is based on the controlled motion of magnetic domains by magnetic fields [4] and spin currents [5]. Both the MRAM and the Racetrack memory do not require regular refreshing of the stored information, a process which is time and energy consuming, and they could allow computers to boot-up instantaneously.

A commercially available device that senses the rotation of a magnetic field is theMultiturn Counter [6], which is based on the motion of magnetic domain walls in combination with detection of the domain wall position using the GMR effect.

As the last examples illustrate, domain walls and domain wall phenomena are explored for spintronics applications. Magnetic domain walls can be interpreted as quasi like particles exhibiting a mass [7]. This leads to resonant domain wall oscillations [7–12] in an external potential and to domain wall inertia [12,13], due to domain wall energy that is stored in the deformation of the domain wall spin structure [12]. By exploiting the fast precessional behaviour of magnetic domain walls, the performance of spintronic devices can be highly improved. For instance, by accurately tuning the shape of a current pulse to move magnetic domain walls from one position to another, the domain walls can be resonantly excited, which strongly reduces the current density needed to depin and move the domain wall [9,10].

Key to operation of devices based on the manipulation of magnetic domain walls is a thorough understanding of the mechanisms that lead to precession and oscillation of the domain wall spin structure.

While much has been achieved in the field of spintronics, much remains to be understood before the full capability of these concepts can be explored. In this work, we add to our understanding of the role of internal domain wall dynamics to the kinetic behaviour of domain walls subject to external excitation. In particular, we investigate the oscillations of magnetic domain walls in confined geometries and the dynamic interaction of the domain walls with fast magnetic fields and electric currents. By using state-of-the-art magnetic X-ray microscopy techniques, we are able to unravel the physical mechanisms that are responsible for spin dynamics at the 10⁶ − 10⁹ Hz frequency range. In particular, we employ scanning transmission X-ray microscopy (STXM) which combines high spatial and temporal resolution necessary to image the fast dynamics of inhomogeneous magnetization configurations in nanostructures. The scope of this thesis covers specific types of domain wall oscillations:

ã We propose a novel type of spin-torque domain wall oscillator (DWO), which is based on the precessional oscillation of a magnetic domain wall that is trapped in a geometrical constriction in a nanowire with perpendicular magnetic anisotropy [14]. The full rotation of the spins within the domain wall, in combination with a GMR device that is fabricated on top of that constriction [15], can lead to strongly enhanced microwave emission, in contrast to conventional STO, where the spins only precess by a small angle [3].

ã As the continuous miniaturization leads to progressively smaller nanostructures, their size and shape become the dominant parameters that determine the magnetic spin configuration. In particular, magnetic domain walls can form closed magnetic induction loops to minimize stray fields, i.e., the magnetization forms a vortex where the magnetization

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3 curls in the plane around a central point, called thevortex core. We employ time-resolved magnetic X-ray microscopy to image the resonant oscillation of a magnetic vortex domain wall trapped in an artificial confinement potential [16]. By studying the vortex core gyration at resonance, we probe the shape of the pinning potential, which is determined by the specific shape of the geometrical constriction.

ã The operation speed for spintronic devices based on the displacement of magnetic domain walls, depends directly on the domain wall velocity. In particular, when spin- polarized currents are used, the speed of magnetic domain walls depends on the transfer mechanism of spin angular momentum from the conduction electrons to the local magnetization (adiabatic and non-adiabatic spin-transfer torque) [17–19]. The origin of the non-adiabaticity is still controversial and under debate. To test existing theories [19–23], we investigate the resonant oscillations of a magnetic vortex domain wall under ac-current excitation, employing time-resolved STXM.

ã Irrespective of the driving mechanism, domain walls traveling at high domain wall velocities, above the so-called Walker breakdown [24], exhibit precessional domain wall motion. In this work, we demonstrate that the same phenomenon occurs also below the Walker breakdown by direct visualization of the domain wall propagation in circular nanowires. Hence, we conclude that intrinsic oscillations of the domain wall velocity occur below and above the Walker breakdown and present therefore a universal feature in circular geometries. Our experiments demonstrate the precise control of the domain wall position at high domain wall velocities, enabling reliable and reproducible high speed domain wall manipulation for magnetic logic and sensing devices.

This thesis is organized as follows:

Chapter 1 introduces the theory of micromagnetism in ferromagnetic nanostructures, including the equations of motion of a one-dimensional domain wall and the steady-state motion of a vortex core.

Chapter 2 details a novel type of spin-torque magnetic domain wall oscillator (DWO) in materials with perpendicular magnetic anisotropy, which is based on the precessional oscillation of a magnetic domain wall trapped within a geometrical constriction in a nanowire.

Chapter 3 gives an overview of the experimental methods used in this work, including the description of the sample fabrication and explaining time-resolved STXM, which is used to directly image magnetization dynamics on the nanoscale.

Chapter 4 We directly image the resonant oscillation of a trapped magnetic vortex domain wall, employing time-resolved STXM. The vortex core trajectory at resonance re- veals the shape of the asymmetric pinning potential, which is determined by the shape of the geometrical constriction.

Chapter 5 To test existing theories on the transfer mechanism of spin angular momentum from spin-polarized conduction electrons to the local magnetization, we study the resonant

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oscillations of a vortex domain wall, induced by alternating currents injected through the nanowire.

Chapter 6 presents the first experimental visualization of oscillating domain wall propagation below and above the Walker breakdown in ring-shaped nanowires, using a rotating magnetic field to control the domain wall position and velocity. By direct imaging of the domain propagation, we probe directly the interplay between moving domain walls and the local pinning potential, revealing intrinsic velocity variations due to intrinsic oscillations of the domain wall spin structure and extrinsic velocity variations due to pinning of the domain wall at defects and impurities present in the nanowire.

Parts of the results presented in this thesis have been published in peer reviewed journals and a list of publications is given in page 162.

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CHAPTER 1 Micromagnetic theory of ferromagnetic nanostructures

Ferromagnetism is purely quantum mechanical origin, as shown by Bohr and van Leeuwen in the early 20th century [25,26]. They find that in a classical system, the total magnetization vanishes in thermal equilibrium and in the absence of external fields. It is the exchange interaction, due to the spatial overlap of the electron wavefunctions and Pauli’s exclusion principle, that leads to a short-ranged ferromagnetic coupling of neighboring atomic moments. The band model of itinerant ferromagnetism describes magnetism in terms of a competition between direct exchange interaction between the localized d-electrons and the kinetic energy of the itinerant electrons in the 3d-metals (Mn, Fe, Co and Ni). How- ever, even the smallest nanomagnets are made of a very large number atoms and therefore the quantum mechanical theory is impractical for realistic calculations of their properties.

Therefore we employ a semiclassical theoretical framework calledmicromagnetism, describ- ing the magnetic properties of nanoscale ferromagnets, where the magnetic configuration is largely determined by the size and the shape of the element, in contrast to macroscopic

“bulk” magnets.

At the nanoscale there are two competing interactions, the exchange interaction and the dipole-dipole interaction, that determine the magnetism in soft ferromagnetic materials, where the crystal anisotropy is negligible. In ferromagnets, the exchange interaction favours parallel alignment of neighboring atomic moments on a short length scale, while the dipole- dipole interaction favours closed magnetic loops with zero net magnetic moment to minimize stray fields. Although the dipole-dipole interaction is much weaker than the exchange interaction, due to its long-range nature (decaying as 1/r²) it can compete with the exchange at sizes much larger than the so-called exchange length, l_ex = ^qA/µ₀M_S². The

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a b c

𝒙

𝒚

Figure 1.1: Magnetization configurations in ferromagnetic nanostructures. Micromagnetic simulations of a) a nearly monodomain S-state in a100×100 nm²,20 nmthick nanostructure, b) a Landau configuration in a 250×250 nm²,20 nmthick square element and c) a vortex structure in a 20 nmthick disc with125 nm radius. The magnetization direction is indicated by the color code as illustrated on the bottom left.

interplay between the exchange- and dipolar interaction leads to many possible magnetization configurations in ferromagnetic nanostructures, depending on the size, shape and material. [27]

In very small structures < 100 nm, the short-ranged exchange interaction dominates, re- sulting in a uniformly magnetized monodomain state, see figure 1.1a [28]. The magnetostatic energy is minimal when the magnetostatic poles at the sample boundaries are reduced, meaning that the magnetization prefers to align parallel to the sample boundaries. This effect of the magnetostatic interaction is called shape anisotropy, since the shape of the sample that tends to define the magnetic easy axes. In particular, in magnetic thin-films, the shape anisotropy strongly forces the magnetization to point in the sample plane. In larger structures, the cumulative dipolar interaction is strong enough to compete with the exchange interaction, favouring flux closed magnetization configuration, which effectively eliminates all stray fields. Examples are thevortex orLandau configurations [29], where the magnetization curls in-plane around a central vortex core, see figure 1.1b and 1.1c. At this very center, however, to avoid an a large increase in the exchange energy, the magnetization turns out of the sample plane “up” or “down”, depending on the vortex core polarity. The existence of the vortex core was predicted by Feldkeller and Thomas [30], but due to its small size it is difficult to observe directly and could only be imaged recently by magnetic force microscopy (MFM) [31] and by Lorentz microscopy [32]

, while employing high-resolution spin-polarized scanning tunneling microscopy it became possible to resolve the internal spin structure of the vortex core [33].

In this chapter we will provide the basic theoretical background relevant for this work.

After a brief review of commonly used micromagnetic quantities (section 1.1) we provide an overview of the different contributions to the Landau free energy in the micromagnetic framework (section 1.2) and discuss the formation of the equilibrium magnetization con-

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1.1. Definitions of micromagnetic quantities 7 figuration by minimizing the Landau free energy. In particular, we calculate the ground state of the vortex structure and the Bloch domain wall in perspective to the description of the one-dimensional domain wall, see section 1.3. The dynamics of the magnetization are governed by the Landau-Lifshitz-Gilbert (LLG) equation in interaction with external fields and spin-polarized currents, as described in section 1.4. Finally, we derive within the Thiele formalism the equations of motion of a one-dimensional 180^◦ magnetic domain wall and the Thiele equation for the steady-state vortex core gyration, sections1.6and1.7.

More information about the quantum mechanical origin of the magnetism and the theory of micromagnetism can be found for instance in [25,26].

1.1 Definitions of micromagnetic quantities

The micromagnetic theory describes the magnetization configuration in a ferromagnet at each positionr as a continuous vector fieldM(r). This continuum approach is justified, as the magnetization only varies on length scales much larger than the interatomic distances, because of the short-ranged exchange interaction. This characteristic exchange length is typically in the order of a few nanometers [34], see subsection 1.2.2. Some micromagnetic quantities as used throughout this work are listed in table 1.1.

Magnetic moment µ Am²

Magnetization M A/m

Saturation magnetization M_S A/m M_S =|M|

Normalized magnetization m dimensionless m=M/MS

Magnetic field H A/m

Magnetic induction B T =V s/m² B=µ₀(H+M) Spin drift velocity u m/s

Magnetic anisotropy K J/m³

Gilbert damping parameter α dimensionless

Exchange stiffness A J/m

Vacuum permeability µ₀ 4π·10⁻⁷ Vs/Am

Gyromagnetic ratio γ 2.21·10⁵ m/As (in radians) Table 1.1: Commonly used micromagnetic quantities in SI units.

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1.2 Micromagnetic description and Landau free energy

The energy of a magnetic system is given by the thermodynamical Gibbs free energy potential:

G(T,M) = U(T,M)−T S−µ₀

Z

MHdV, (1.1)

where T is the temperature, M is the spatially varying magnetization configuration, S and U are the entropy and the internal energy of the magnetic system, respectively. We consider only the case of an open system at T = 0, such that we can neglect the entropy S. The Gibbs free energy can now be expressed by theLandau free energy [35]:

E_tot = E_ext

| {z }

−µ0MH

+E_ex+E_dem+E_ani

| {z }

:=U(T = 0,M)

, (1.2)

which is the sum of the Zeeman energy E_ext from the external field, the exchange energy E_ex, the magnetostatic energy E_dem and the anisotropy energy E_ani. The Landau free energy is minimal when the local magnetization in equilibrium M_eq aligns parallel to the effective field H_{ef f} and the total energy can be written as

E_tot =−µ₀

Z

M(r) ·H_{ef f}(r) dV, (1.3) and inversely the effective field can be calculated from the total energy density E_tot [36]

H_{ef f} =− 1 µ₀

δE_tot

δM. (1.4)

The relevant energy terms contributing to the total energy, the Zeeman energy, the ex- change energy, the magnetostatic energy and the anistotropy energy terms are discussed in the following subsections.

1.2.1 Zeeman energy

The Zeeman energy is the energy of the magnetic moments in an external field H_ext [28]:

E_ext=−µ₀

Z

M(r) ·H_ext(r)dV. (1.5) For a uniform magnetic field, the Zeeman energy only depends on the angle θ between the net magnetic moment and the external field and it is minimal when the magnetization aligns with the external field, E_ext=−µ₀^R M_S|H_ext|cosθ dV.

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1.2. Micromagnetic description and Landau free energy 9

1.2.2 Exchange energy

The exchange interaction is at the origin of ferromagnetism, aligning the magnetic moments of neighbouring atoms. In the Heisenberg model [37], the exchange energy in the micromagnetic framework is given by:

E_ex =A

Z

|∇m|²dV, (1.6)

where |∇m|² = (∇m_x)²+ (∇m_y)²+ (∇m_z)² is the squared gradient of the magnetization m=M/M_S and the parameterA is the exchange stiffness, indicating the strength of the exchange interaction. The equation holds for small anglesφbetween neighbouring magnetic moments, which is the case in soft magnetic materials. The effective field associated with the exchange interaction is given by the derivative of the exchange energy densityE_ex with respect to M, equation (1.4):

H_ex = 2A∆m, (1.7)

where ∆m is the Laplacian of the magnetization:

∆m= ∂²mx

∂x² + ∂²mx

∂y² +∂²mx

∂z² ,∂²my

∂x² + ∂²my

∂y² +∂²my

∂z² ,∂²mz

∂x² + ∂²mz

∂y² +∂²mz

∂z²

!

. (1.8)

1.2.3 Magnetostatic energy

The magnetostatic energy is the interaction energy of the magnetic moments in the dipolar field created by the magnetization itself. The magnetostatic energy is associated with the demagnetization field H_dem which tends to minimize the global magnetization of a system, also referred to as stray field originating from the net magnetic moment.

According to the Gauss’s law for magnetism ∇·B = 0 there are no “magnetic charges”

and B is given by

B=µ₀(H_dem+M), (1.9)

and hence

∇·H_dem =−∇·M. (1.10)

Assuming further that∇×H_dem = 0, according to Ampéres circuital law, we can transform this equation to a Poisson equation:

∆φ(r) = −ρ(r), (1.11)

whereH_demderives from the scalar magnetic potentialH_dem=−∇φandρ=−∇Mis the magnetic charge density. To obtain the demagnetization field, the Poisson equation must be solved, which involves an integration over the entire volumeV of the magnetic structure.

The magnetostatic energy is then calculated assumingH_dem is an external magnetic field:

E_dem=−µ₀ 2

Z

V

H_dem(r) ·M(r) dV, (1.12) where the factor 1/2 is introduced to avoid counting each volume element twice.

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a ^K¹^{> 0} b ^K¹^{< 0}

−1

−0.5 0 0.5 1

−1

−0.5 0

0.5 1

−1

−0.5 0 0.5 1

y−axis x−axis

z−axis

−1

−0.5 0 0.5 1

−1

−0.5 0

0.5 1

−1

−0.5 0 0.5 1

y−axis x−axis

z−axis Anisotropy energy density [a.u.]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 1.2: Uniaxial anisotropy energy density. The uniaxial energy density iso-surfaces are plotted for: a)K₁>0where the energy is minimal along the z-axis and b)K₁<0where the energy is minimal within the x-y-plane. The easy axis points along the z-axis.

1.2.4 Anisotropy energy

Because of the spin-orbit interaction, the direction of the local magnetic moments is coupled to the orientation of the electronic orbitals and hence the crystal structure of a material can lead to preferred directions of the magnetization, the so calledeasy axes. Phenomeno- logically, the anisotropy energy only depends on the direction of the magnetization with respect to the easy axes, which depends on the symmetry of the crystal [28]. If only one easy axis exists, the uniaxial anisotropy energy density is usually expanded into the first significant term:

E_ani=K₁sin²(θ), (1.13)

where K₁ is the phenomenological anisotropy constants and θ is the angle between the easy axis and the magnetization. Equivalently, the anisotropy energy can be calculated by integrating over the sample volume

E_ani=^Z

V

E_ani(m) dV. (1.14)

ForK₁ >0 the anisotropy energy is minimal when the magnetization aligns with the easy axis (θ = 0), while forK₁ <0 the energy is minimal when the magnetization points in the easy plane perpendicular to the easy axis (θ =π/2), as illustrated in figure1.2, where the easy axis is pointing along the z-axis. In soft magnetic materials, such as permalloy, K₁ is very small and can be neglected [38]. On the other hand, in CoPt multilayer materials, the uniaxial anisotropy perpendicular to the sample plane is stronger than the shape anisotropy of a thin film [39,40]:

K₁ > µ₀M_S²

2 , ⇒K_{ef f} =K₁− µ₀M_S²

2 >0, (1.15)

and is therefore a material with perpendicular magnetic anisotropy.

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1.3. Static magnetization configurations 11

Bloch domain wall Néel domain wall

𝒚 𝒙 𝒛

a b

Figure 1.3: The one-dimensional domain wall. Schematic illustration of a) a Bloch domain wall and b) a Néel domain wall. The magnetization direction is indicated by the arrows. (Inspired by [42])

1.3 Static magnetization configurations

In equilibrium, the magnetizationM_eqminimizes the Landau free energyEtotand in general M_eq is inhomogeneous. The goal of this section is to derive the equilibrium magnetization of the one-dimensional 180^◦-Bloch domain wall and of the vortex structure, by minimizing the Landau free energy by variational calculus.

1.3.1 The one-dimensional 180^◦-Bloch domain wall

In 1932 Bloch introduced the concept of domain walls and has shown that within the region between two ferromagnetic domains the magnetization changes continuously [41,42]. It is the imbalance between the exchange interaction and the magnetocrystaline or shape anisotropy that leads to domain walls with finite width.

In wider domain walls the exchange energy decreases, while on the other hand the anisotropy energy increases, therefore the minimum of the total energy is achieved for a finite domain wall width. Néel has shown that domain walls with vanishing magnetostatic energy, so calledBloch domain walls, are energetically most favourable [43], while for ∇M 6= 0 the domain wall is calledNéel domain wall.

To calculate the equilibrium magnetization configuration of a 180^◦ Bloch domain wall we express the magnetization in spherical coordinates. The magnetization only varies along the x-direction and rotates within the y-z-plane (ϕ=π/2), such that

M =M_Scos (θ(x)) ˆe_z+M_Ssin (θ(x)) ˆe_y. (1.16) The magnetostatic energy of this magnetization configuration is constant, because∇M= 0 and thus ∇H_dem = 0 according to equation (1.10). The boundary conditions are

θ(x=−∞) = 0, and θ(x= +∞) = π. (1.17)

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−4 −3 −2 −1 0 1 2 3 4

−1

−0.5 0 0.5 1

δDW

Mz(x) =−MStanh µ x

δDW

¶

Real−space coordinate x/δ

DW

Magnetization [MS]

Magnetization M_z Magnetization M_y

Figure 1.4: Domain wall profile of a 180^◦-Bloch domain wall. TheMz

andMy magnetization components are plotted as a function of the normalized real-space coordinatex/δDW. The magnetization rotates within the y-z- plane and in the center of the wall (x=0) the magnetization points along the y- axis. (Inspired by [42])

Now we can write the total energy per unit square section L² using equations (1.2), (1.6) and (1.14):

E_tot L² =^Z

"

A ∂θ(x)

∂x

!₂

| {z }

Exchange energy

+K₁sin²(θ(x))

| {z }

Anisotropy energy

#

dx. (1.18)

Minimizing this energy with respect toθ(x) yields the Euler equation and the solution gives the equilibrium magnetization configuration of the one-dimensional 180^◦-Bloch domain wall:

θ(x) = cos⁻¹−tanh x δ_DW

, (1.19)

where δ_DW =^q_K^A₁ is the domain wall width [42] and M_z(x) =−M_Stanh x

δDW

. (1.20)

The domain wall profile of the equilibrium magnetization of a 180^◦-Bloch domain wall is plotted in figure1.4. The energy per unit square cross section can be calculated by putting (1.19) into (1.18):

E_Bloch

L² = 4^qAK₁. (1.21)

1.3.2 The vortex structure

In larger nanostructures, the cumulative dipolar interaction is strong enough to compete with the exchange interaction, favouring flux closed magnetization configuration, which minimizes the magnetostatic energy. The simplest configuration is the vortex structure, where the magnetization curls in-plane around a central vortex core. The sense of cir- culation the magnetization defines the vortex chirality c, clockwise (CW) (c = −1) or counter-clockwise (CCW) (c = +1). At the very center, however, the magnetization

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1.3. Static magnetization configurations 13

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.2 0.4 0.6 0.8 1

Mz(r) =−MSexp µ r²

ρ²_{V C}

¶ θ(r) = cos⁻¹

· exp

µ

−r² 2

µ₀M_S² 2A

¶¸

ρ_VC

Real−space coordinate r/ρ VC Magnetization [MS]

0 30 60 90

Angle [deg]

Magnetization M_z Function θ(r)

Figure 1.5: Magnetization profile of the vortex core. The magnetization componentM_zand the function θ(r) are plotted as a function of the distance r to the center of the vortex, where the magnetization points out-of- plane. The out-of-plane magnetization profile of the vortex core follows in good approximation a Gaussian peak [28].

turns out-of-plane, defining the vortex core polarity p, pointing either “up” (p = +1) or “down”(p=−1). Thus the vortex structure has four possible configurations.

We calculate the equilibrium magnetization following Feldtkeller and Thomas [30], assuming cylindrical symmetric vortex structure:

M=M_Ssin (θ(r)) ˆe_ϕ+M_Scos (θ(r)) ˆe_z. (1.22) The goal is to find a function θ(r) that minimizes the Landau free energy. Neglecting the anisotropy energy we have

E_tot =

R_V

Z

0 2π

Z

0 t

Z

0

A





∂θ

∂r

!₂

+sin²θ r²





| {z }

Exchange energy

+ µ₀M_S 2 cos²θ

| {z }

Magnetostatic energy

rdrdϕdz, (1.23)

where t and RV are the thickness and the radius of the vortex structure, respectively. To solve the variational Euler differential equation the ansatzm_z = exp(−r²/2ρ²_{V C}) is chosen, or equivalently

θ(r) = cos⁻¹

"

exp − r² 2ρ²_{V C}

!#

, (1.24)

as shown in 1.5. The radius of the vortex core ρ_{V C} is then given by ρV C =

s 2A

µ₀M_S², (1.25)

in permalloy it is typically in the order of ρ_V ≈5 nm [30,33,34]. It has been shown that the profile of the vortex core follows in good approximation a Gaussian profile [28]. A more thorough calculation can be found in [28] and a study of the vortex core profile using micromagnetic simulations can be found in [44].

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1.4 Field and current induced magnetization dynamics

The goal in this section is to describe how the magnetization evolves in time as a result of external stimulation by magnetic fields and spin-polarized currents. The magnetization dynamics are governed by the generalized Landau-Lifshitz-Gilbert (LLG) equation, which includes the adiabatic and non-adiabatic spin-torque terms [45]:

dM

dt =−γ⁰(M×H_{ef f})

| {z }

Precession

− α

M_SM×(M×H_{ef f})

| {z }

Damping

−(u·∇)M

| {z }

Adiabatic spin-torque

+ β

M_SM×(u·∇)M

| {z }

Non-adiabatic spin-torque

, (1.26) where α is the damping parameter,γ⁰ is the renormalized gyromagnetic ratio, β describes the strength of the non-adiabaticity and u is the spin drift velocity, associated with the spin polarized current. Throughout this thesis we use γ₀ =γ⁰/µ₀, the renormalized gyromagnetic ratio that is normalized to the magnetic inductionµ₀H rather than the magnetic field H.

The first term describes the precession of the magnetization around the effective magnetic field. The second (damping) term introduces phenomenologically energy dissipation through relaxation of the magnetization towards the effective magnetic field. The last two terms describe the spin-transfer torque from spin-polarized currents to the local magnetization. It is well known that the electrical resistivity of a sample depends on the magnetization configuration, for instance due to the anisotropic magneto resistance (AMR) [46] or the giant magnetic resistance effect (GMR) [47,48]. Here, we are interested in the inverse effect, the influence of an injected current on the local magnetization. It has been found that spin-polarized currents can interact with the local magnetization and transfer their spin-angular momentum to the local magnetization [17]. This spin-torque term was the first to be integrated in the framework of a macrospin model into the LLG equation [49,50].

While only recently the non-adiabatic spin-transfer torque was introduced [19,45].

In the following subsections we will derive the LLG equation step by step.

1.4.1 Landau-Lifshitz equation

The dynamics of a magnetic moment, induced by an effective magnetic fieldH_{ef f} are given by the Landau-Lifshitz (LL) equation [29]:

dM

dt =−γ(M×H_{ef f})−α_LL

M_SM×(M×H_{ef f}), (1.27) The first term corresponds to the precession of the magnetization around the effective magnetic field, it can be derived from the spin commutator relations and the Ehrenfest theorem [26]. This term conservesM·H_{ef f} and thus the energy of the system is conserved.

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1.4. Field and current induced magnetization dynamics 15 The precession rate (Larmor frequency) is proportional to the field strength |H_{ef f}| with the gyromagnetic ratio γ as proportionality factor

ω=γ|H_{ef f}|. (1.28)

The second term of the LL equation models phenomenologically energy dissipation, due to Faucoult currents, spin-phonon scattering etc. and leads to relaxation of the magnetization Mtowards the direction of the effective fieldH_{ef f}, as illustrated in figure1.6. The damping parameter α_LL determines the strength of the damping.

𝐌 𝐇_𝒆𝒇𝒇

𝐌 × 𝐇_𝒆𝒇𝒇

𝐌 𝐇_𝒆𝒇𝒇

𝐌 ×𝑑𝐌 𝑑𝑡

Precession Damping

a b

Figure 1.6: Magnetization dynam- ics of the Landau-Lifshitz-Gilbert equation. Schematic illustration of the precession term (a) and the damping term (b) of the LL and LLG equation.

The precession term preserves the energy in the system µ₀M·H_{ef f} while the damping is always perpendicular to Mand ^dM_dt.

Gilbert introduced a slightly different viscosic damping term [51]:

dM

dt =−γ(M×H_{ef f}) + α_LLG MS

M× dM

dt . (1.29)

The implicit Gilbert equation (1.29) can be transformed to the explicit Landau-Lifshitz- Gilbert (LLG) equation:

dM

dt =− γ 1 +α²_LLG

| {z } :=γ⁰

(M×H_{ef f})− γα_LLG (1 +α²_LLG)

| {z }

:=α

1 MS

M×(M×H_{ef f}), (1.30)

with γ⁰ = γ/(1 +α²_LLG) and α = γα_LLG/(1 +α²_LLG). Note that the magnitude of the magnetization|M|is preserved, since the damping term is always perpendicular toMand

dM

dt . Mathematically, the LL equation (1.27) and the LLG equation (1.30) are equivalent, but physically their behaviour is different for large damping:

dM

dt → ∞, for (αLL → ∞), dM

dt →0, for (αLLG→ ∞), (1.31) hence the torque on the magnetization diverges for high damping in the case of the LL equation. This means that the LLG equation reflects the magnetization dynamics physically better than the Landau-Lifshitz equation [52,53].

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1.4.2 Adiabatic spin-transfer torque

Electrons traveling through ferromagnets with inhomogeneous magnetization, act on the local magnetization via the transfer of spin-angular momentum. We distinguish between two types of electrons; the spin-polarized conduction s-electrons near the Fermi level and the localized d-electrons defining the local magnetization [19]. The spinsS of the localized d-electrons are substituted by the classical magnetization S/S =−M/M. This semiclassical approximation holds, because generally the magnetization dynamics are much slower than the spin dynamics of the itinerant s-electrons.

Adiabatic spin-transfer torque means that the spin s of the conduction electrons adiabatically align with the local magnetizationM. When electrons travel through a region where the magnetization direction changes, the electron spin direction adapts to the direction of the magnetization and their spin changes by ds, while angular momentum is transferred to the local magnetization to conserve the angular momentum, as shown in figure 1.7.

Local magnetization 𝐌

𝒆

⁻

d𝒔 d𝑴

𝒙

𝒔(𝑥)

𝐌(𝑥) 𝐌 𝑥 + 𝑑𝑴

𝒔 𝑥 + d𝒔 Injected electrons

Figure 1.7: Illustration of the origin of the adiabatic spin-transfer torque.

The spin s(x) of the conduction electrons adiabatically aligns to the local magnetization. When the electron spin changes by ds, angular momentum is transferred to the local magnetization, inducing a change in the direction of M(x)bydM(after [44])

The total amount of angular momentum transferred by an electron current I to the local magnetization is

ds=−~P I

e dt·∇_xmdx, (1.32)

where the first term in the product is the spin per unit cross sectionAcarried by the electron current, taking the spin polarization P of the conduction electrons into account [54]. The second term is the change of the direction of the electrons spin, when it adiabatically adapts to the local magnetization. Note that only the perpendicular spin component can be transferred, because the magnetization |M|=const.has to be preserved, therefore the calculation is only valid for slow varying magnetization.

The change of the magnetization dM is then directly proportional to the change of the angular momentum per unit volume V =Adx:

dM= gµ_B 2~ · ds

Adx =−gµ_BP 2eMS

j_e

| {z } :=u

∇_xMdt, (1.33)

wherej_e =I/Ais the electron current density,uis thespin drift velocity andm=M/M_S.

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1.4. Field and current induced magnetization dynamics 17

In vector notation the adiabatic spin-transfer torque reads:

T_adj = dM

dt =−(u·∇)M, (1.34)

which is consistent with the spin-torque found by Thiaville et al. [45,55] and the results obtained by Li and Zhang [56].

1.4.3 Non-adiabatic spin-transfer torque

The non-adiabatic spin-transfer torque deals with electrons whose spin do not point in the direction of the local magnetization. This happens for instance when large magnetization gradients are present and the spin of the injected conduction electrons cannot adapt adiabatically to the local magnetization. This creates an additional torque on the magnetization, since the electron spin s has a component perpendicular to M [44]. The effective field of these electrons induces precession of the local magnetic field, similar to an external magnetic field. The torque is perpendicular toManddMand hence proportional to M×dM, see figure 1.8. For this reason the non-adiabatic spin-transfer torque term is also called field-like term. The strength of the non-adiabatic torque is proportional to the spin drift velocity u, such that

T_na = β

M_SM×(u·∇)M, (1.35)

where β is a proportionality constant, measuring the degree of non-adiabaticity.

Local magnetization 𝐌

𝒆

⁻

𝒙 𝐇_𝑒𝑓𝑓

𝑑𝐌

𝐌 d𝐌

d𝑡 = 𝐌 × d𝐌 Injected electrons

Figure 1.8: Illustration of the origin of the non-adiabatic spin-transfer torque. The effective field of the conduction electrons, whose spin does not point in the direction of the local magnetization, induces precessional motion of the magnetizationM. The torque is perpendicular todMandM.

Thiavilleet al.[45] introduced this term phenomenologically to explain differences between experimental observations of current induced domain wall motion and the theory. However, the origin of the non-adiabaticity is still controversial and under debate. Non-adiabatic current flow can be generated by different mechanisms, such as spin-relaxation due to spin flip scattering [19], or spin mistracking in narrow domain walls [20–22]. Recently, the role of the spin diffusion as an additional mechanism contributing to the non-adiabaticity has been presented [23]. A more thorough discussion of the non-adiabatic spin-transfer torque is given in chapter 5.

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1.5 Steady-state motion of magnetic domains

In order to simplify the equations of motion of the magnetization according to the LLG equation, Thiele [57] derived an integrated equation to approximate steady-state domain wall motion. It assumes that magnetic features move while conserving their internal shape.

Hence the system M(x, t) at position x and time t, can be described by the displacement X(t) and the initial magnetization configuration M(x,0):

M(x, t) = M(x−X(t),0), (1.36) dX

dt = v, (1.37)

dM

dt = −(v∇)M. (1.38)

Assuming further that the saturation magnetization is constant |M|= const., the Thiele equation of steady-state motion can be written as [57]:

F+G×v+αDv= 0. (1.39)

The Thiele equation describes the steady state motion of a magnetic system or a quasi particle by “forces” acting on this particle. In the case of steady-state motion, the force F can be separated into two terms [58]:

F=F_int+F_ext. (1.40)

The internal forceF_intcontains all the forces from the internal energy, including the anisotropy, exchange and the magnetostatic energies. The internal force vanishes when summing over the internal energy contribution, because the internal magnetization configuration is conserved. Therefore, we only consider the external energy contribution to the force,

F=F_ext=^Z

V

∇W dV =^Z

V

∂W

∂θ (∇θ) + ∂W

∂ϕ (∇ϕ)

!

dV, (1.41)

where W is the external energy density. In particular, in the case of field induced magnetization dynamics, the external energy is given by W = −µ₀H_ext·M. The vector G is the total gyrocoupling vector or simply gyrovector and D is the dissipation tensor:

G = −M_S γ₀

Z

V sinθ(∇θ× ∇ϕ) dV, (1.42)

D = −M_S γ₀

Z

V

∇θ∇θ+ sin²θ∇ϕ∇ϕ dV, (1.43)

where we expressed the magnetization in spherical coordinates,

M=M_Ssinθcosϕeˆ_x+M_Ssinθsinϕˆe_y+M_Scosθˆe_z. (1.44)

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1.6. One-dimensional domain wall model 19 In order to take the magnetization dynamics induced by spin-polarized currents into account, Thiaville et al. [45] expanded the Thiele equation to include the adiabatic and non-adiabatic spin-transfer torque, as in the LLG equation (1.26):

F+G×(v−u) +D(αv−βu) = 0. (1.45) In the following sections we will apply the Thiele formalism to derive the steady-state equations of motion of the one-dimensional domain wall and of the vortex core in a magnetic disc.

1.6 One-dimensional domain wall model

In section 1.3 we derived the static equilibrium magnetization configuration of a one- dimensional 180^◦-Bloch domain wall. Now we apply the Thiele formalism to derive the steady state motion of the one-dimensional domain wall, described by the dynamics of the collective coordinates, the domain wall position q and the polar angle ψ, as illustrated in figure 1.9.

𝜽 𝒛

𝒙

𝒚

Polar angle Domain wall position

𝝍 𝒒

Figure 1.9: Collective coordinates of the one-dimensional domain wall.

The domain wall center position q and the polar angleψare the collective coordinates of the one-dimensional domain wall. Forψ= 0it is a Néel domain wall and for ψ = π/2 it is a Bloch domain wall. The angleθ is the angle between the vertical axis and the magnetization.

The magnetization profile of the one-dimensional domain wall in spherical coordinates is given by equation (1.19)

θ(x, t) = cos⁻¹

"

−tanh x−q(t)

∆(t)

!#

, (1.46)

φ(x, t) = ψ(t), (1.47)

where ∆ is the domain wall width. By placing the domain wall profile into equations (1.41) - (1.43), we find that the gyrovector is zero G = 0 and the dissipation tensor has only one non-zero component D_xx =−2M_S/γ₀∆ [58]. The external force per cross-sectionS is F/S= 2µ₀HM_Seˆ_x, assuming an external fieldH=Hˆe_z and a spin drift velocity along the x-directionu =uˆe_x. With the Thiele equation (1.45) we immediately find the steady-state domain wall velocity:

˙

q= γ₀∆

α µ₀H+β

αu. (1.48)

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Following Thiaville et al. [59] by inserting the one-dimensional domain wall profile into the LLG equation (1.30) and applying the Lagrange formalism, we find the equations of motion of the collective coordinates of the one-dimensional domain wall [14,45,55,60]

∆ ˙ψ+αq˙=γ₀∆µ₀H+βu,

˙

q−α∆ ˙ψ =u+γ₀∆µ₀H_K

2 sin(2ψ),

˙∆ = 12γ₀ αµ₀MSπ²

A

∆−∆K_l+K_tcos²ψ

,

(1.49)

where H_K is the effective anisotropy field and K_l and K_t being the effective longitudinal and transverse anisotropies, respectively. The domain wall width ∆(t) relaxes relatively fast, with a relaxation time in the order of ≈<1 ps [59], towards the equilibrium domain wall width ∆0(ψ)

∆₀ =

s A

K_l+K_tcos²ψ, (1.50)

Therefore we assume the domain wall width is always in equilibrium ∆ = ∆₀(ψ). For a Bloch domain wallψ =π/2, the magnetostatic anisotropy is zero and hence the equilibrium domain wall with is^q_K^A₁, which is consistent with equation (1.19). Note that in materials with perpendicular magnetic anisotropy, the longitudinal anisotropy K_l here corresponds to the anisotropy constant K₁ for uniaxial anisotropy parallel to the z-axis, while in thin films K_t= 0.

Let us now discuss the field and current induced motion of the one-dimensional domain wall. Eliminating ˙q in equations (1.49) provides the evolution of the polar angle ψ:

ψ˙ = 1 1 +α²

"

βu

∆0 − αu

∆0 +γ₀µ₀H−αγ₀µ₀H_K

2 sin(2ψ)

#

. (1.51)

For field induced domain wall motion below the Walker breakdownH < H_W,H_W =αH_K/2 is the Walker field, the domain wall propagates with constant polar angleψ^∗and the domain wall velocity scales linearly with the external field:

ψ^∗ = 1

2sin⁻¹ 2H αH_K

, q˙=γ₀∆0(ψ^∗)µ₀H

α . (1.52)

For fields higher than the Walker field, H > H_W, the anisotropy cannot compensate for the torque on the polar angle and ψ starts to precess. The velocity-field relation becomes non-linear, because of the variation of the domain wall width [59]:

<ψ >˙ _t= γ₀

1 +α²µ₀H, q˙= v_max (1 +α²)





(1 +α²) H HW

−

v u u t

H

HW

2

−1





, (1.53) with the maximum velocity v_max = γ₀∆0µ₀H_K/2, which is independent of the damping α, and < ψ >˙ _t is the average angular velocity of the precession of the polar angle. The

Controlling magnetic domain wall oscillations in confined geometries

domain wall oscillations

in confined geometries

Doctoral thesis by André Bisig

Universität Konstanz

Fachbereich Physik

Controlling magnetic domain wall oscillations in confined geometries

Dissertation

zur Erlangung des akademischen Grades:

Doktor der Naturwissenschaften

André Bisig

Contents

Introduction

CHAPTER 1

Micromagnetic theory of ferromagnetic nanostructures

a b c

1.1 Definitions of micromagnetic quantities

1.2 Micromagnetic description and Landau free energy

a b

1.3 Static magnetization configurations

1.4 Field and current induced magnetization dynamics

a b

𝒆

𝒆

1.5 Steady-state motion of magnetic domains

1.6 One-dimensional domain wall model