Spin torque and interactions in ferromagnetic semiconductor domain walls

Transcription

1 University of Iowa Iowa Research Online Theses and Dissertations Summer 211 Spin torque and interactions in ferromagnetic semiconductor domain walls Elizabeth Ann Golovatski University of Iowa Copyright 211 Elizabeth Ann Golovatski This dissertation is available at Iowa Research Online: Recommended Citation Golovatski, Elizabeth Ann. "Spin torque and interactions in ferromagnetic semiconductor domain walls." PhD (Doctor of Philosophy) thesis, University of Iowa, Follow this and additional works at: Part of the Physics Commons

2 SPIN TORQUE AND INTERACTIONS IN FERROMAGNETIC SEMICONDUCTOR DOMAIN WALLS by Elizabeth Ann Golovatski An Abstract Of a thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Physics in the Graduate College of The University of Iowa July 211 Thesis Supervisor: Professor Michael Flatté

3 1 ABSTRACT The motion of domain walls due to the spin torque generated by coherent carrier transport is of considerable interest for the development of spintronic devices. We model the charge and spin transport through domain walls in ferromagnetic semiconductors for various systems. With an appropriate model Hamiltonian for the spin-dependent potential, we calculate wavefunctions inside the domain walls which are then used to calculate transmission and reflection coefficients, which are then in turn used to calculate current and spin torque. Starting with a simple approximation for the change in magnetization inside the domain wall, and ending with a sophisticated transfer matrix method, we model the common π wall, the less-studied 2π wall, and a system of two π walls separated by a variable distance. We uncover an interesting width dependence on the transport properties of the domain wall. 2π walls in particular, have definitive maximums in resistance and spin torque for certain domain wall widths that can be seen as a function of the spin mistracking in the system when the spins are either passing straight through the domain wall (narrow walls) or adiabatically following the magnetization (wide walls), the resistance is low as transmission is high. In the intermediate region, there is room for the spins to rotate their magnetization, but not necessarily all the way through a 36 degree rotation, leading to reflection and resistance. We also calculate that there are widths for which the total velocity of a 2π wall is greater than that of a same-sized π wall. In the double-wall system, we model how the system reacts to changes in the separation of the domain walls. When the domain walls are far apart, they act as a spin-selective resonant double barrier, with sharp resonance peaks in the transmission profile. Brought closer and closer together, the number and sharpness of the peaks decrease, the spectrum smooths out, and the domain walls brought together have a

4 2 transmission spectrum with many of the similar features from the 2π wall. Looking at the individual walls, we find an interesting interaction that has three distinct regimes: 1) repulsion, where the left wall moves to the left and the right wall to the right; 2) motion together, where the two walls both move to the right, even at the same velocity for one special value of separation; and 3) attraction, where the left wall moves to the right and the right wall moves to the left. This speaks to a kind of natural equilibrium distance between the domain walls. This is of major interest for device purposes as it means that stacks of domain walls could be self-correcting in their motions along a track. Much experimental work needs to be done to make this a reality, however. Abstract Approved: Thesis Supervisor Title and Department Date

5 SPIN TORQUE AND INTERACTIONS IN FERROMAGNETIC SEMICONDUCTOR DOMAIN WALLS by Elizabeth Ann Golovatski A thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Physics in the Graduate College of The University of Iowa July 211 Thesis Supervisor: Professor Michael Flatté

6 Copyright by ELIZABETH ANN GOLOVATSKI 211 All Rights Reserved

7 Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVAL PH.D. THESIS This is to certify that the Ph.D. thesis of Elizabeth Ann Golovatski has been approved by the Examining Committee for the thesis requirement for the Doctor of Philosophy degree in Physics at the July 211 graduation. Thesis Committee: Michael Flatté, Thesis Supervisor Craig Pryor Thomas Boggess Yannick Meurice Hassan Raza

8 To my husband, who helped me pass the qual...barely ii

9 ACKNOWLEDGMENTS I want to start by thanking my advisor, Michael Flatté, without whom none of this would be possible. I have said, in utmost honesty, to anyone who would listen, that I ve had the best advisor in the world during my stay at the University of Iowa, and I stand by that statement. Besides being more brilliant than I could ever aspire to be, he s been an incredible source of help, encouragement, and humor (especially humor) during this long strange journey. That I ve had someone willing to guide me, willing to get angry on my behalf when the world didn t make sense, and willing to give me the free reign to have Nope, I m doing fine right now be the entirety of our status meetings has made all the difference in my being able to get where I am today. I also need to thank my fellow grad student/co-worker/friend Victoria Kortan. Not only because we were the only women in the group, but because she was always there to listen, to engage in office gossip, and to engage in endless sessions of What did Michael mean when he said that?. Having someone around, if only part-time, who understood my work, my world, and my sense of humor, was the best thing that could have happened to me in grad school. Plus, who else would come all the way to Iowa just to be my moral support? Thanks also to my parents, for their continuing support as I wrap up my umpteenth consecutive year of schooling. Mom said that she knew when I was five that I would end up with a PhD, and now here I am. And finally, cryptically, thanks to J. Michael Straczynski. Because Faith Manages. iii

10 TABLE OF CONTENTS LIST OF FIGURES vi CHAPTER 1 INTRODUCTION Motivations Ferromagnetism Domain Structure Domains Domain Walls Transport Resistance Effects Domain Wall Resistance GMR and TMR Spin Torque Spin Torque in Metallic Layers Spin Torque in Domain Walls Theoretical Approaches DOMAIN WALL MODEL Motivation Model and Formalism Exchange Field Schrödinger Equation Rotation of Local Basis Outside Regions Coefficients Probabilities Currents and Torques Definitions: Charge Current Density, Spin Current Density, and Spin Torque Population Integration Material Parameters and Notes LINEAR MODEL OF DOMAIN MAGNETIZATION Motivation Methods and Calculations Setup of the Schrödinger Equation Rotation of Local Basis Boundary Conditions Results Transmission and Reflection Probabilities Charge Current Density Energy Dependent Spin Torque iv

11 3.3.4 Total Spin Torque Width Dependence of the Total Spin Torque Domain Wall Velocity CALCULATING SPIN TORQUE WITH TRANSFER MATRICES Motivation Introduction to calculating wavefunctions with transfer matrices Introduction Example: tunnel barrier as a piecewise potential Transfer Matrices for Domain Wall Calculations Magnetization angle Wavefunctions and transfer matrix setup Results and Comparison with Linear Model Transmission and Reflection Spin Torque Domain Wall Velocity Conclusions MODULATION OF SPIN TORQUE FROM SPIN TRANSPORT THROUGH TWO NEARBY DOMAIN WALLS Motivation Model and Schematics Center of Mass Properties Transmission and Reflection Probabilities nm walls Spin Torque - Energy Dependence Total Spin Torque Comparison for Other Widths of Domain Walls Individual Wall Properties Spin torque on individual walls nm walls Analysis of 1.8 nm walls in ˆx direction Comparison for other domain wall widths CONCLUDING REMARKS AND OUTLOOK Summary π walls vs. 2π walls, Round One: the linear approximation π walls vs. 2π walls, Round Two: the transfer matrix model Interactions in a double domain wall system Outlook and future ideas Internal Forces Dynamics and the LLG Equation BIBLIOGRAPHY v

12 LIST OF FIGURES Figure 1.1 The Pauli exclusion principle and ferromagnetism Unpaired electrons in Ni, Co, and Fe Magnetization in a Bloch, Néel, and Vortex domain wall Example of two perpendicularly-oriented domains meeting in a π/2 wall. The red line indicates the position of the domain wall Schematic representation of a)π/2, b)π and c)2π θ Néel walls Illustrations of CIP GMR (a,b), CPP GMR(c,d for a nonmagnetic metal spacer), and TMR (c,d for an insulating tunnel barrier spacer) Schematic representation of a layered spin torque structure Schematic representations of Néel type domain walls with total change in magnetization equal to φ = (a) π and (b) 2π Schematic representations of Néel (a) π and (b) 2π-domain walls. Charge transport is assumed to be by holes Probabilities for transmission with spin flip (T sf ), transmission without spin flip (T nf ), reflection with spin flip (R sf ) and reflection without spin flip (R nf ) for π(a-d) and 2π(e-h) Néel walls with widths of.1 nm, 1 nm, 5 nm, and 1 nm Probability of transmission without spin flip for a 2π wall over the full range of domain wall widths for a set hole energy of / Charge current as a function of bias voltage for π(a) and 2π(b) walls. Curves correspond to different domain wall widths Spin torque components as a function of hole energy for π(a-d) and 2π(e-h) walls Spin torque components as a function of bias voltage for π(a,b) and 2π(c,d) walls Spin torque components and magnitudes as a function of domain wall width for a π and 2π wall at V = 5 mv Calculated peak domain wall velocities for a π, 2π, 3π and 4π wall.. 54 vi

13 4.1 An arbitrary potential for a < x < b A tunnel barrier with height V for a < x < b Angle of magnetization θ(x) for a π domain wall. The red line is the analytic value of the magnetization angle. From Ref. [48]. The blue line is our piecewise linear approximation of the magnetization angle Probabilities for transmission with spin flip (T sf ), transmission without spin flip (T nf ), reflection with spin flip (R sf ) and reflection without spin flip (R nf ) for π(a-d) and 2π(e-h) Néel walls with widths of.1 nm, 1 nm, 5 nm, and 1 nm Transmission and reflection probabilities from Fig. 4.4, compared with the linear calculations. Dashed lines are linear calculations. Solid lines are transfer matrix calculations Spin torque components as a function of hole energy for π(a-d) and 2π(e-h) walls. Dashed walls are for a linear rotation calculation. Solid lines are for the transfer matrix calculation Total spin torque as a function of domain wall width for π and 2π walls for linear (dashed lines) and transfer matrix (solid lines) Calculated peak domain wall velocities for a π, 2π, 3π and 4π wall. Dashed lines are for linear calculations. Solid lines are for transfer matrix calculations Ratio of the spin torque in a 2π wall to the spin torque in a π wall. The dashed line shows the torque above which the 2π wall will have a higher velocity Schematic for a system with two Néel-type π walls, separated by a domain in which the magnetization is antiparallel to that in the leads Magnetization angle as a function of position for two 1.8 nm π walls, separated by a 5 nm gap Magnetization angle as a function. The solid line is two 1.8 nm π walls with a separation of nm. The dashed line is a 3.6 nm 2π wall nm domain walls: Reflection and transmission probabilities, with and without spin flip, for four values of separation between the domain walls. Inset: The same transmission spectrum for a single 2π wall of double the width Spin torque as a function of incoming carrier energy, for four values of separation between the domain walls. Inset: The same transmission spectrum for a single 2π wall of double the width vii

14 5.6 Spin torque through the entire two wall system (solid lines). Dashed lines are the same calculated value for a single 2π wall of 3.6 nm. Dotted lines are twice the calculated value for a single π wall of 1.8 nm nm domain walls: Reflection and transmission probabilities, with and without spin flip, for four values of separation between the domain walls. Inset: The same transmission spectrum for a single 2π wall of double the width nm domain walls: Reflection and transmission probabilities, with and without spin flip, for four values of separation between the domain walls. Inset: The same transmission spectrum for a single 2π wall of double the width nm domain walls: Reflection and transmission probabilities, with and without spin flip, for four values of separation between the domain walls. Inset: The same transmission spectrum for a single 2π wall of double the width Spin torque through the entire two wall system (solid lines) for four sizes of the π walls. Dashed lines are the same calculated value for a single 2π wall of 3.6 nm. Dotted lines are twice the calculated value for a single π wall of 1.8 nm Spin torque for each individual domain wall (Each π wall is 1.8 nm) Repulsion between the two domain walls for a small domain wall separation Both domain walls moving in the +ˆx direction. At the point shown, the walls have the same velocity Attraction between the two domain walls for a large domain wall separation Spin torque for each individual domain wall (Each π wall is.12 nm) Spin torque for each individual domain wall (Each π wall is.5 nm) Spin torque for each individual domain wall (Each π wall is 4 nm).. 94 viii

15 1 CHAPTER 1 INTRODUCTION 1.1 Motivations The spin of an electron plays a vital role in solid state physics. The discovery of Giant Magnetoresistance (GMR) in 1988 by Fert [1] and Grünberg [2] showed that the relative magnetization orientation of magnetic layers can have a dramatic effect on the electrical resistance of a device, jump-starting the field of spin-dependent electronics, more commonly known as spintronics. The fundamental goal of spintronics is to understand how spins interact with their environment. Transport, relaxation, injection, and detection of spin are all problems the spintronics community seeks to understand. Magnetoelectronic devices are already a large part of the technology industry today. IBM replaced anisotropic magnetoresistance-based read heads in its commercial hard drives with read heads based on GMR in 1997, and today, read heads based on the similar Tunneling Magnetoresistance (TMR) are the industry standard. Magnetic random access memory (MRAM) that uses the resistance from a magnetic tunnel junction is currently available commercially from Everspin [3]. Research and development of such effects and devices is of great interest due to the potential advantages of such devices, including he non-volatility of MRAM memory devices, increased speed, increased scalability, and integration with existing semiconductor technologies, particularly when devices made with semi-magnetic or dilute magnetic semiconductors [4] like GaMnAs are considered. Spin torque generated by spin transport through inhomogeneous magnetic systems - a direct manifestation of the conservation of the angular momentum associated with spin - underlies both unresolved fundamental questions and potential applications, including fast, localized switching of magnetic moments or domains [5 9], depinning and transport of domain walls [1 16], electrical driving of ferromagnetic

16 2 resonance [17 2], and controlled generation of coherent magnons [16, 21]. Structures showing spin torque are commonly domain walls between two regions whose magnetization orientation differs by an angle θ, called θ-domain walls. Although spin torque on a π wall has garnered much experimental and theoretical attention [1,11,22 31], little has been done to explore spin torque in 2π walls, which are known to be stable in many metallic systems [32], and have been seen experimentally [33]. The difference between 2π wall behavior and π behavior might be most marked when ballistic transport across the domain wall is possible, such as for magnetic semiconductor domain walls (whose π walls are predicted to have highly non-linear dependencies of spin current and charge current on voltage [34, 35]). Understanding 2π domain walls may also lead to novel spin torque devices, such have been predicted for π walls [13, 36, 37]. In addition, such devices will inevitably require the manipulation of more than one domain wall with a single current. Understanding how two π walls interact with each other at various distances - from far away to the almost-2π-wall limit - could lead to the ability to move around stacks and patterns of domain walls, giving rise to more specialized and complicated devices. 1.2 Ferromagnetism Ferromagnetism, the spontaneous alignment of spins in certain materials, is something that comes about solely from quantum mechanics - it has no real classical explanation. A simple way of looking at it (from Ref [38], and from the theory of Heisenberg [39]) follows. Suppose that two electrons associated with some atom approach each other. If the two electrons have the same spin orientation, the Pauli exclusion principle forbids them from sharing the same orbital state. If the two electrons have opposite spin orientations, they can share the same orbital state, but their close proximity to each other considerably increases the coulomb repulsion between them. As the lower

17 3 energy state, having two electrons with the same spin orientation would tend to be the preferred state. This is called the exchange interaction. Large Coulomb Repulsion Small Coulomb Repulsion Figure 1.1: The Pauli exclusion principle and ferromagnetism We can expand on this idea further by looking at the wavefunctions of a pair of electrons in an atom [38]. An atomic wavefunction has both a spatial coordinate r(x, y, z) and spin variable σ, which we can represent together by the general coordinate q. Consider a wavefunction for an atom with two electrons. We can approximate the two-electron wavefunction ψ(q 1, q 2 ) by ψ(q 1, q 2 ) = ψ 1 (q 1 )ψ 2 (q 2 ) (1.1) Since the electrons are indistinguishable, this means that ψ 2 (q 1, q 2 ) must be invariant for exchange of the electrons. This means that one of the following must be true: ψ(q 2, q 1 ) = ψ(q 1, q 2 ) (1.2) OR ψ(q 2, q 1 ) = ψ(q 1, q 2 ). (1.3)

18 4 The Pauli exclusion principle excludes the first possibility, as it allows electrons to be in the same state: ψ 1 (q 2 )ψ 2 (q 1 ) = ψ 1 (q 1 )ψ 2 (q 2 ) is perfectly reasonable under the requirement in Eq The second possibility is the correct one under the Pauli exclusion principle, as the only way for ψ 1 (q 2 )ψ 2 (q 1 ) = ψ 1 (q 1 )ψ 2 (q 2 ) to be true is if ψ 1 (q 2 )ψ 2 (q 1 ) =. Eq. 1.3 being true means that the wavefunction for a two-electron system must be antisymmetric under the exchange of electrons. If we decompose the two-electron wavefunction into a spatial and spin component: ψ(q 1, q 2 ) = ϕ(r 1 r 2 )χ(σ 1, σ 2 ) (1.4) then ψ(q 2, q 1 ) = ϕ(r 2 r 1 )χ(σ 2, σ 1 ) = ψ(q 1, q 2 ) = ϕ(r 1 r 2 )χ(σ 1, σ 2 ) (1.5) which means that either the spatial component of the wavefunction or the spin component of the wavefunction has to be anti-symmetric, but not both. This means that the symmetry of the spin variable directly affects the spatial part of the wavefunction, resulting in a change in the electrostatic interaction. For two electrons with antiparallel spins, the wavefunction must be symmetric for the exchange of spatial coordinates: ϕ s (r 1, r 2 ) = 1 2 [ϕ 1 (r 1 )ϕ 2 (r 2 ) + ϕ 2 (r 1 )ϕ 1 (r 2 )] (1.6) while for electrons with parallel spins the wavefunctions must be antisymmetric for the exchange of spatial coordinates: ϕ a (r 1, r 2 ) = 1 2 [ϕ 1 (r 1 )ϕ 2 (r 2 ) ϕ 2 (r 1 )ϕ 1 (r 2 )]. (1.7) The energy of the system is calculated by U = ϕ HϕdV (1.8)

19 5 where H = H 1 +H 2 +H 12 with H 1 and H 2 the hamiltonians for the individual electrons kinetic energy and interaction with the nucleus, and H 12 = e 2 /r 12 the interaction between the two electrons. The total energy in Eq. 1.8 is given by: U = I 1 + I 2 + K 12 ± J 12 (1.9) where I 1 and I 2 are the energies of the two electrons, and K 12 is the Coulomb interaction between them. J 12, on the other hand, with the form: J 12 = ϕ 1(r 1 )ϕ 2(r 2 )H 12 ϕ 2 (r 1 )ϕ 1 (r 2 )dv 1 dv 2, (1.1) has no classical analogue. Called the exchange integral, it represents the energy produced by the exchange of the electrons between the two orbitals. In Eq. 1.9, the plus sign on the J 12 term is for antiparallel spins and the minus sign is for parallel spins. Thus, as long as J 12 > then the antisymmetric spatial state with the parallel spins will have a lower energy. (Materials in which J 12 < are anti-ferromagnetic. As we move away from single-molecule materials and into crystals, this preference for spin alignment diminishes, as the kinetic energy scales much more rapidly with the density of electrons than the exchange energy. But for a series of transition metals, the loosely bound, partially-filled d-orbitals give rise to ferromagnetic materials. Fig. 1.2 shows a d-orbital configuration for the three most common elemental ferromagnets - Nickel ([Ar]4s 2 3d 8 ), Cobalt([Ar]4s 2 3d 7 ) and Iron([Ar]4s 2 3d 6 ). The figure shows how the outer shell electrons will fill each orbital state first with electrons of one spin orientation before adding any of the opposite configuration, thus reducing the coulomb repulsion for the system. In addition to standard ferromagnetic materials and their alloys, we note just one of many other magnetic systems: Dilute Magnetic Semiconductors (DMS) are so named

20 6 Fe Co Ni Figure 1.2: Unpaired electrons in Ni, Co, and Fe because they are made by doping a regular semiconductor with magnetic dopants (see an overview of this topic by Ohno in Ref. [4]). These dopants bring additional, unpaired carriers into the system. The extra carriers will align ferromagnetically. An advantage to a DMS is the ability to interface with semiconductors that are already in use in devices. Perhaps the most common DMS, GaMnAs [4], is readily compatible with GaAs, which is widely used in electronic devices. An additional advantage is the fact that the ferromagnetism is induced by the addition of carriers to the system, opening up the possibility for spin-polarized transport.

21 7 1.3 Domain Structure Domains The exchange interaction discussed in the previous section is strong in the short range - the interaction requires overlapping wavefunctions, and the antisymmetric wavefunctions previously discussed only have that overlap at short range.. Thus, close clusters of electrons will tend to align together and point the same way. In the long range, however, the classical tendency of dipoles to anti-align with each other will come to dominate the exchange interaction. The result is the formation of small regions, called domains, in which the spins are aligned. The long-range dipole interaction causes adjacent domains to anti-align. Just an idea without a name when it was introduced by Pierre Weiss in 197 [41] and first hinted at experimentally by Barkhausen in 1919 [42], it wasn t until 1926 that the spin-aligned regions in ferromagnetic materials were given the name domains [43]. True experimental confirmation of the existence of these domains came in the early 193s [44, 45]. These experiments measured the stray fields above domain patterns by way of decoration with a fine magnetic powder - a method even now of some use for its sensitivity to fine changes in domain structure and the simplicity of its preparation. The theory followed not far behind, with first Bloch [46], and then Landau and Lifshitz [47] looking at the relationship between the minimization of the energy in a ferromagnetic system and the formation of domains Domain Walls An abrupt change in magnetization at the boundary of two anti-aligned domains is not a favorable energy condition. Domain walls form between such domains as a means of minimizing the energy of the two anti-aligned domains. Domain walls are transition layers in which the magnetization changes gradually from one magnetization to another. This gradual change prevents the large increase in exchange energy that

22 8 would accompany an abrupt change in the magnetization angle. The size and geometric orientation of the domain walls is governed by the relevant energies in the ferromagnetic system. We can estimate the energies by [38]: E = E exch + E anis + E field + E mag (1.11) Exchange Energy E exch is the exchange energy. It is proportional to the gradient of the magnetization squared, and take the form: E exch = A( M) 2, where A is the exchange constant for the material. This energy prefers to have adjacent spins aligned. A minimum exchange energy would have an infinitely long domain wall to transition from one domain orientation to the other. Anisotropy Energy E anis is the anisotropy energy. This can be defined as the energy cost to align the magnetization from one crystallographic axis to another. This energy is lowest when all the spins are aligned with the crystal axes. If the domains themselves are aligned in an easy-axis direction, then the changing moments inside the domain wall are going to often be oriented along a hard direction. A minimum anisotropy energy would have an infinitely short domain wall to avoid having many moments along a hard-axis direction. In a system with cubic anisotropy, this energy takes the form: E a,c = K c (m 2 1m m 2 1m m 2 2m 2 3) where m 1, m 2 and m 3 are the magnetization components along the cubic axes. For a system with uniaxial anisotropy, the anisotropy energy takes the form (to second order terms) of E a,u = K u sin(θ) where θ is the angle between the anisotropy axis and the magnetization direction. Field Energy E field is the energy that aligns spins along an external magnetic field. If such a field is applied, minimizing this energy would have all the spins aligned along the direction of the external field. This energy takes the form: E field = µb M where B

23 9 is the applied field, and M is the magnetization. Magnetostatic Energy E mag is the magnetostatic energy caused by the surfaces of the domain structure. A single domain produces surface free poles which raise this energy. Minimizing this energy requires the alteration of the spin distribution. The alteration of spin distribution, however, tends to increase E exch, E anis, or E field. The stable solutions are found by minimizing the whole energy, which requires some sort of balance between the individual energies Planes of domain wall rotation The rotation of the magnetization inside the domain wall can take several forms. These depend largely on the geometry of the system and the anisotropy in the ferromagnetic material. Common domain wall geometries include: Bloch walls These are the most common domain walls in bulk systems. The magnetization in a Bloch wall rotates in the plane that is parallel to the domain wall/domain interface. Néel walls While less common in bulk materials, Néel walls have a lower energy in thin films than Bloch walls, and are commonly found there. The Néel wall s magnetization rotates in the plane perpendicular to the domain wall/domain interface. This is due to the shape anisotropy in the thin films. It requires a huge energy cost to rotate the spins out of plane as happens in a Bloch wall. The anisotropy thus forces the magnetization to stay in the plane of the film. Vortex walls Vortex walls are another thin-film domain wall. The magnetization in a vortex wall rotates in the same plane as a Néel wall, but the local magnetization is wrapped around a single vortex point.

24 1 Illustrations of all three walls are show in Fig Bloch wall Néel wall Vortex wall Figure 1.3: Magnetization in a Bloch, Néel, and Vortex domain wall

25 θ-domain walls Two domains whose magnetizations differ by an angle θ are separated by what is called a θ-domain wall. Three types of θ-domain walls are discussed here and shown in Fig. 1.5 π walls A π or 18 wall is found between domains that have antiparallel magnetizations. The magnetization undergoes a 18 rotation from one side of the domain wall to the other. This is a commonly seen type of domain wall. Minimizing the total energy shown above leads to an equation giving the dependence of the angle of magnetization inside the domain wall on the position inside the wall [48]. The lowest energy configuration for a π wall is θ(x) = sin 1 [tanh(x/λ)] (1.12) where λ is a parameter equal to the square root of the ratio of the exchange stiffness constant and the uniaxial anisotropy constant, (A/K). This parameter is what determines the width of the domain wall. π/2 walls A π/2 or 9 wall is another kind of naturally occurring domain wall that exists between domains that have perpendicular magnetizations. The magnetization goes through a 9 rotation inside the domain wall. This is much rarer than the π wall, however, as the anisotropy energy would be raised in an unfavorable way to have one domain pointed along an easy axis, and its adjacent domain along a hard axis. These generally occur at the corner of a sample, as shown in Fig π walls: A 2π wall, existing between two parallel domains, are less common, usually formed from one or more π walls during a magnetization reversal process. The

26 12 Figure 1.4: Example of two perpendicularly-oriented domains meeting in a π/2 wall. The red line indicates the position of the domain wall. magnetization in a 2π wall undergoes a full 36 rotation between one side of the wall and the other. The formation of 2π walls is best understood by looking at π walls as pairs of topological charges with different winding numbers. Two π walls approaching each other will have their topological charges either annihilate each other, eliminating both walls, or repel each other and form a stable 2π wall. A good micromagnetic study of this effect is in Ref. [49]. 1.4 Transport The spin-dependent, spatially varying potential inherent to a domain wall has interesting consequences for the transport of carriers. The energy of the incoming carrier is not the only factor that determines whether or not it passes to the other side of the domain wall the spin also must be taken into account. Since each spin orientation experiences a different potential, the two reflection and coefficients found in standard tunneling calculations are replaced with two of each - one for each spin orientation.

27 13 a) π/2 domain wall z x -d b) π domain wall z x -d c) 2π domain wall z x -d Figure 1.5: Schematic representation of a)π/2, b)π and c)2π θ Néel walls. This spin-dependent transport forms the basis for the resistance and torque effects that follow. Several researchers have studied this effects in ferromagnetic metals [5 53]. Later efforts involved the same kind of transport in semiconductors [34, 35] and were

28 14 later seen experimentally [24, 54]. 1.5 Resistance Effects Domain Wall Resistance The study of the resistance in ferromagnetic structures happened in close concert with the discovery of ferromagnetic domains. Perhaps the first to look at how resistance might or might not change with the addition of domain structures was Gerlach in 1932 [55], but Gerlach did not detect any change in resistance due to the the domain structure, nor did Steinberg in 1933 [56]. A change in resistance was finally detected in a system with domain walls by Heaps in 1934 [57], but this was attributed to anisotropic magnetoresistance(amr) rather than to the presence of the domain walls. The 196s and early 197s proved a period of great interest in the magnetoresistance of ferromagnetic metals and their alloys. Studies of nickel alloys and polycrystals [58, 59], as well as cobalt, gadolinium and iron [6] were done around this time, with an important series of experiments on iron whiskers showing non-ohmic resistance was possible in these materials [61 63]. Experiments later in the 197s in thin films showed the magnetoresistance to be closely tied to the domain structure [64, 65]. The theory of this magnetoresistance came somewhat after the experiments that showed it. The first to calculate resistances in domain walls were Cabrera and Falicov, who published a two-paper series on the resistance in Bloch walls [66, 67]. They calculate reflection coefficients for electronic wavefunctions through both a narrow(abrupt) and a wide(adiabatic) Bloch wall modeled by a series of potential steps, and examining the change in resistance. In 1978, Berger treated the concept of the adiabatic changing of spins by likening them to microwaves propagating along a twisted waveguide in a paper [68] that not only treated the low-field magnetoresistance of Bloch type domain walls but also the concept of domains being dragged along by the forces involved, and the idea that

29 15 the rotation of spins inside the domain wall would apply a torque to it. The 199s brought additional theories, including a semiclassical method using a pseudo-larmor precession of spin around the exchange field by Viret et. al [69], and the first fully quantum mechanical model by Levy and Zhang [7]. Others produced models with fuller bandstructures than the simple two-band system many had used before [5, 51]. An alternative approach using the Drude method was developed by van Gorkom, Brataas and Bauer [71], and yet another method, carefully tracking spin and charge accumulation during transport was developed by Dugaev et al. [71] At the same time, others were developing theories that did not favor increased resistance in the presence of domain walls [72, 73]. Tatara and Fukuyama published multiple papers [74 76] using a linear response theory arguing that the spatial inhomogeneities of the magnetization would contribute to decoherence, thus reducing, rather than increasing the resistance in the presence of a domain wall GMR and TMR The discovery of giant magnetoresistance (GMR) in 1988 is widely considered to be the birth of the field of spintronics. Discovered independently by the groups of Peter Grünberg [2] and Albert Fert [1], the discovery of the differing resistance between aligned and anti-aligned ferromagnetic layers led to the two physicists sharing the 27 Nobel Prize in Physics. The configuration used for the initial discovery of GMR is shown in Fig. 1.6, (a) and (b). Two ferromagnetic layers (iron in this case) are placed with a thin non-magnetic metal spacer (cobalt) between them. If the ferromagnetic regions are oriented with their magnetizations parallel(a), one spin orientation is blocked and the other is allowed to transport freely, producing a relatively low resistance state. If the ferromagnetic regions are oriented with their magnetizations antiparallel(b), then each ferromagnetic region effectively blocks one of the spin orientations, producing a

30 16 CIP GMR (a) Low resistance (b) High resistance FM Spacer FM FM Spacer FM X X X Current Current CPP GMR or TMR (c) Low resistance (d) High resistance FM Spacer FM FM Spacer FM Current X Current X X Figure 1.6: Illustrations of CIP GMR (a,b), CPP GMR(c,d for a nonmagnetic metal spacer), and TMR (c,d for an insulating tunnel barrier spacer) high-resistance state. The configuration first studied in 1988 and shown in (a) and (b) is called current-in-plane or CIP GMR. The discovery of GMR in the experimentally simpler current-perpendicular-to-plane configuration followed shortly thereafter [77, 78], and was shown to have a much larger difference in resistance between the parallel and antiparallel states than the CIP geometry. The CPP configuration is shown in Fig. 1.6, (c) and (d).

31 17 A closely related effect called tunneling magnetoresistance(tmr), achieved by replacing the nonmagnetic metal spacer in a CPP GMR configuration with an insulating tunnel barrier, was first hinted at in 1975 [79], but the result was very small, a less than 1% difference in resistance between the parallel and antiparallel states, and the effect wasn t revisited until 1995 [8]. After the theoretical postulations that MgO might be a better barrier [81,82], initial 2% [83,84] results gave way quickly to results as high as 6% at room temperature and over 1% at low temperature [85]. 1.6 Spin Torque Spin torque is a direct manifestation of the conservation of angular momentum associated with spin quanta. When a carrier encounters some kind of spatial variation in magnetization this could be caused by magnetic layering, or by the rotating magnetization in a domain wall the carrier will experience a torque to bring its spin magnetic moment into alignment with the local magnetization. An additional torque is required by the conservation of angular momentum this torque, from the carriers to the region of magnetization, is called the spin torque. If a large number of highly polarized carriers pass through the same magnetic region, the spin torque can be substanital. This effect can be seen most directly by looking at the spin current density Q. This tensor quantity (it has directions both in real space and in spin space) represents the flow of spin in the way the charge current density represents the flow of charge. Assuming a one-dimesional transport model, with a carrier moving along the x axis with a wavefunction ψ, the we can write the charge current density as: J = e h 2im [ψ ( x ψ) ( x ψ ) ψ]ˆx. (1.13) Because the transport is one dimensional, we can write the flow of spin along the ˆx direction as a vector with spin space components:

32 18 Q = h 2im [ψ S ( x ψ) ( x ψ ) S ψ]. (1.14) The spin torque per unit area can then be defined as the amount of spin current absorbed by the non-uniform region [86]. N = Q in + Q r Q t. (1.15) Spin Torque in Metallic Layers FM1 FM2 θ r t z x Figure 1.7: Schematic representation of a layered spin torque structure In a layered metallic device like the simple model shown in Fig. 1.7, there is a fixed magnetic layer (FM1) and a free magnetic layer (FM2), separated by some kind of non-magnetic spacer (this could be metallic, or insulating, depending on the device). The fixed layer acts as a polarizer, resulting in a spin-polarized current passing into the spacer layer. If the spacer layer is sufficiently thin, the polarization persists until

33 19 reaching the spacer/free layer interface. If the magnetization of the fixed layer and the free layer are not in the same direction, the spin current in the spacer layer can be broken up into components that are parallel to and perpendicular to the magnetization in the free layer: Q in = Q +Q. The component of the spin current that is parallel to the free layer s magnetization will pass through the layer: Q t = Q. To a good approximation, it can be said that no transverse spin current flows away from the spacer/free layer interface [86], so Q r in this case is zero. Thus the spin torque in a system such as this can be approximated as: N layer = Q in + Q r Q t = Q + Q Q = Q. (1.16) The resulting torque on the magnetization of the free layer can be used to change - and eventually reverse - the magnetization of the layer. The study of spin torque in metallic layers began in earnest in 1996, with Slonczewski [5] and Berger [6] independently developing theories on spin torque in multilayer devices. Traces of this effect were seen in 1998 by Tsoi et al, looking at excitations and changes in resistance in Co/Cu multilayers [7, 87]. Confirmation of the ability of this spin torque to switch the magnetization in a ferromagnetic layer in the late 199s from the Buhrman group at Cornell, using both multilayer point contacts [8] and nanopillars [88]. This feat was repeated in 21 by the Orsay group in Co/Cu/Co trilayers [89]. In 25, Slonczewski published a theory treatment of the spin-torque induced switching in a magnetic tunnel junction [9], something that was seen experimentally around the same time [91 93].

34 Spin Torque in Domain Walls Domain walls were the motivation for the beginning of spin torque research going back to the late 197s and 198s, when Berger predicted the motion of domain walls due to spin torque [68,94]. This was experimentally verified soon after by his group [95 97]. But because of the overly large currents required using this experimental setup, this didn t generate much excitement, and for years, the focus in spin transfer physics stayed on layered structures. Interest was renewed near the turn of the century, with Gan et al. [98] publishing results showing domain wall motion in 2µm permalloy wires. With large enough current densities, current pulses could induce a domain wall to move distances on the order of a micron. Grollier et al. went on to study domain walls in a spin valve configuration, using the motion of the domain wall to switch the spin valve from one state to the other [12] These experiments were notable in that the reversibility of the effects under the reverse of of the current direction prove that they are caused by spin transfer rather than by any other applied fields or by Joule heating. Also notable is the comparably low critical current density needed for moving the the domain walls far below the currents originally needed in Berger s experiment. An experiment by Tsoi et al. [99] and related experiments by Vernier et al. [1] and Allwood et. al. [11] show that it is possible to reversibly move the domain wall with just a spin polarized current without any external magnetic field. Measurements of the velocity of this domain wall motion came from Yamaguchi et al in 24 [23, 12]. Of particular interest today are domain walls in Dilute Magnetic Semiconductors like GaMnAs [25, 31]. These systems have a unique carrier-based magnetism. The first to show current-induced domain wall motion in a DMS was Yamanouchi et. al. in GaMnAs. [1, 22]. These experiments were remarkable for the much lower critical current density needed to move a domain wall in GaMnAs as opposed to other ferromagnetic materials a full order of magnitude smaller. It is speculated that this

35 21 may have something to do with the much smaller saturation magnetization in dilute magnetic semiconductors, where the magnetic moments come from scattered dopants as opposed to the entirety of the material. This means that there are fewer spins to flip in the space of the domain wall before it can move. Spin Torque induced domain wall motion opens up a host of possibilities for applications. One of the most prevalent ideas, currently under development at IBM, is the idea of putting a long line of domain walls on a racetrack and moving them back and forth as part of a memory device [13]. This would require the motion of multiple domains with a single current pulse, and while this is theoretically possible, much work need to be done to make this an experimental reality. Other memory devices have been proposed using domain walls, including one by Everspin [3], which already produces a magnetic memory device. Another possibility is the use of a domain wall in a microwave frequency oscillator. In this device, a partially pinned domain wall would be rotated underneath a fixed layer, and GMR used to extract the oscillating electrical signal. [37] 1.7 Theoretical Approaches Berger s pioneering work on the interaction between current and domain walls focused on a phenomenon called hydromagnetic drag [13, 14]. He focuses on the displacement of current as it passes through a domain wall. With an s-d exchange interaction potential V (x) = gµ B (s H sd (x) + H sd 2 ) (1.17) that has a non-zero gradient, a force on the domain wall can be found. Most of the work on spin torque in domain walls is based on ideas formed by Slonczewski [5] concerning spin torque in layered geometries. Slonczewski was the first

36 22 to incorporate additional terms representing the spin torque into the Landau-Lifshitz- Gilbert equation for the dynamics of magnetization: M t = γm H eff γα M M (M H eff). (1.18) This novel idea of incorporating spin torque into the LLG equation has itself been tincorporated into many other models. In layered geometries, Stiles and collaborators examined the angular dependence of this torque [15, 16], Wegrowe examined it in a thermokinetic approach [17], and Sun used it to examine the injection of a spin polarized current into a nanomagnet [18]. Bazaliy, Jones, and Zhang [19] took the concept further, modifying the LLG equation to incorporate effects of a spin-polarized current in a magnetic system, and the resulting spin transfer. Starting with the case of a spin-polarized current passing from a normal metal into a semi-infinite magnet, they go on to develop a solution for a moving Bloch wall, in the absence of pinning. Zhang and Li, in their 24 paper [11], develop a form for the spin torque based on the spatial variation the magnetization, as especially appropriate approach for domain walls. They use a commonly-seen equation for the spin torque: τ = b j M 2 s M ( M (Ĵ e ) M), (1.19) where b j = P j e µ B /em s, P is the spin polarization, j e is the charge current density, µ B is the Bohr magneton, and M s is the saturation magnetization. This torque is formally identical to Bazaliy et. al s conclusion, and also matches up with the work of Slonczewski. Zhang and Li added this term to a micromagnetic simulation code similar in function to NIST s OOMMF code [111], and used it to simulate Néel walls between head-to-head domains. They found an out-of-plane component to the spin torque that appeared to cause damping. This was further investigated in another 24 paper [112], working out the difference between the adiabatic and non-adiabatic torque

37 23 contributions. This leads to an even longer magnetization dynamics equation: M t = γm H eff + α M M M s t b ( j M M M ) c j M M M s x x, (1.2) with slightly changed coefficient b j = P j e µ B /em s (1 + ξ 2 ) and new coefficient c j = ξb j. The b j term represents the adiabatic Slonczewski torque from their previous paper, and the c j term represents the non-adiabatic spin torque, which they attribute to spin-mistracking in the domain wall. Thiaville, Nakatani, and Miltat [113] use a modified LLG equation - using just the adiabatic Slonczewski torque term - and study micromagnetic simulations of a domain wall in a thin permalloy strip. They also derive equations of motion for the domain wall position, angle, and width, pointing toward a potential deformation of the domain wall in the process of running a current across it. They find that while the model correctly reproduces much of the relevant physics, the currents required to produce domain wall motion in this model are ten times the values found by experiment. They conclude that more elaborate models must be necessary. They continue in 25 with a paper [26] that introduces the non-adiabatic torque to the equation, using an LLG equation equivalent to Zhang and Li s. They find that the introduction of the non-adiabatic term, at least in perfect wires, eliminates the threshold current requirement for moving a domain wall. They reintroduce this threshold current by studying rough wires, leading to the conclusion that the threshold current for domain wall motion is an extrinsic property. Waintal and Viret [11] start with a simple quantum mechanical model for the transport to find the transmission matrix, which is then used in a modified Landauer formula to find the conductance of the system {[ g = e2 h Tr 1 + P ] } I P N σ z tt. (1.21) 1 P N This conductance figures into their equation for a spin torque per unit voltage: