Current Induced Domain-Wall Motion in Magnetic

Current Induced Domain-wall Motion in MagneticNanowires

Luc Thomas and Stuart ParkinIBM Almaden Research Center, San Jose, CA, USA

1 Introduction 1

2 Theory and Simulations 2

3 Experiments 14

4 Outlook 35

Acknowledgments 36

References 37

1 INTRODUCTION

Over the past several years, there has been renewed interest inunderstanding the interaction between spin-polarized currentand magnetic domain walls (DWs), a phenomenon that wasfirst studied more than 20 years ago in macroscopic magneticthin films. Although there are several possible ways inwhich current can interact with domain walls, perhaps themost interesting interaction is that in which spin angularmomentum from spin-polarized current can result in motionof the domain wall. Current passing through almost anymagnetic material readily becomes spin polarized throughspin-dependent electron scattering processes. Since motionof a domain wall requires reversal of magnetic moments,and, since spin angular momentum is conserved, the transferof spin angular momentum from the current to the magneticsystem can result in domain-wall excitation or movement,both precessional and translational.

Handbook of Magnetism and Advanced Magnetic Materials. Editedby Helmut Kronmuller and Stuart Parkin. Volume 2: Micromag-netism. 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-02217-7.

Advances in lithographic techniques now allow the studyof domain walls in confined magnetic nanostructures withlateral dimensions as small as a few tens of nanometers.Moreover, the magnetic configurations of such structurescan be engineered by properly shaping them. Thus, whereasearlier studies involved complex patterns of several domainwalls, it is now possible to study the current-inducedmotion of a single domain wall by injecting a wall into ananopatterned magnetic device.

In this chapter, we review recent theoretical developmentsand experiments related to the current-driven motion ofdomain walls in magnetic nanowires. This is a challengingtask, since this field of research is still in its infancy andyet the field is developing at a rapid pace. Note that severalaspects of the current-induced motion of domain walls fromspin momentum transfer are closely related to that of thecurrent-induced excitation and switching of magnetizationin spin-valve and magnetic tunnel junction structures. Theselatter phenomena are reviewed extensively elsewhere in thisencyclopedia.

In the first part of this chapter, we briefly review thestructure of DWs in magnetic nanowires. We then discusstheoretical models, which have been developed to describethe interaction of spin-polarized current with such DWs, anddiscuss some of the consequences and predictions of thesemodels.

The second part of this chapter is devoted to firstly, areview of various experimental procedures and techniques,including the fabrication of magnetic nanodevices and usefulDW detection techniques. Secondly, the manipulation ofDWs using magnetic fields is discussed in some detailsince this is very helpful in appreciating the significantlydifferent consequences of their manipulation by current.

2 Magnetic configurations in small elements, magnetization processes and hysteretic properties

Finally, we review experiments on the current-driven motionof DWs.

In the final part of this chapter, we briefly discuss possibleapplications of the phenomena described in this chapter topotential magnetic memory and logic devices.

2 THEORY AND SIMULATIONS

2.1 Domain walls in nanowires

Magnetic DWs have been studied in bulk samples andthin films for nearly a century. DWs have structures whichrange from the simplest Bloch wall, which can be describedanalytically by a one-dimensional model, to complex two-and three-dimensional structures. There are several excellentreviews of this topic including, for example, Hubert andSchafer’s comprehensive textbook (Hubert and Schafer,2000). In this section, we discuss the most common DWstructures found in soft magnetic nanowires, the so-calledhead-to-head (or tail-to-tail) DWs. We also describe Blochwalls, which can be found in nanowires formed frommaterials with large perpendicular magnetic anisotropy.

2.1.1 Head-to-head domain walls in soft magneticnanowires

Macroscopic magnetic structures will typically form flux-closed magnetic domain structures, which lower their energy.In sufficiently narrow nanowires made from soft mag-netic materials, for example, submicron-wide permalloy(Ni81Fe19) wires, flux-closed domain structures are no longerenergetically favored. Rather, owing to the nanowire’s mag-netic shape anisotropy, magnetic domains are aligned alongthe nanowire’s length, with magnetizations pointing toward(or away) from each another. These domains are separatedby head-to-head (or tail-to-tail) DWs.

The structure of head-to-head DWs was first studiedusing micromagnetic simulations by McMichael and Don-ahue (1997). They found two distinct DW structures: thetransverse (T) wall and the vortex (V) wall. Simulated mag-netization maps for these two wall structures are shown inFigure 1(g) and (h). Which of these two DWs has the low-est energy depends on the width w and thickness t of thewire: the (w, t) boundary between these two states was foundnumerically to be given by:

t · w = Cδ2

where the exchange length δ is given by δ2 = A

µ0M2s

(1)

For permalloy (Ms = 800 emu cm−3, A = 1.3 × 10−6 ergcm−1), the exchange length is very short at about δ = 4 nm

and the numerical constant C was determined to be 128. Thisphase diagram has been refined recently by Nakatani, Thiav-ille and Miltat (2005), who have identified an intermediatemagnetic state between the T and V walls, as an asymmetrictransverse wall.

Note that the relative stability of T and V walls was alsostudied by calculating the energy of the V wall as a functionof the position of the vortex core with respect to the centerof the nanowire (Youk et al., 2006). When the V wall is thelowest energy state, its energy is minimized when its coreis at the center of the wire, whereas when the T wall hasthe lower energy, the V wall’s energy is minimized when itscore is at one edge of the nanowire. Interestingly, the authorsalso identified metastable configurations in which the vortexcore is offset from the center of the wire.

The DW width is a critical parameter for both field andcurrent-driven DW motion. However, this quantity is not welldefined, since the magnetization of both T and V walls variessignificantly across the width of a nanowire, as shown inFigure 1. In Nakatani, Thiaville and Miltat (2005), the DWwidth parameter was estimated by fitting the DW magneti-zation profile with that derived for a 1D Bloch wall, namely:

θ(x) = 2 arctan[ex/]

mx = cos[θ(x)] = tanhx

(2)

my = sin[θ(x)] = 1

cosh x

where mx,y are the two in-plane components of the magneti-zation normalized to the saturation value, and θ is the anglebetween the local magnetization direction and the nanowire’slong axis x (i.e., the easy magnetization direction). Thisprofile describes the transverse wall form quite well, pro-viding that is allowed to vary across the nanowire’swidth (i.e., the direction y), as shown by the solid linesin Figure 1(a)–(c). By contrast, these expressions do notaccount for the V wall’s profile. Nakatani, Thiaville and Mil-tat (2005) extracted a width for the V wall by fitting theprofile of the longitudinal magnetization averaged across thewidth of the nanowire (Figure 1c and f). For both V andT walls, the DW width parameter was found to dependonly weakly on the wire thickness, but to scale with the wirewidth w, according to the approximate relations (Nakatani,Thiaville and Miltat, 2005):

TW = w

πVW = 3w

4(3)

The V wall is significantly wider than the T wall. Note thatthe actual length scale over which most of the magnetizationchange occurs is π, which is much larger than that givenby equation (3) (Malozemoff and Slonczewski, 1979).

Current induced domain-wall motion in magnetic nanowires 3

Transverse wallw = 100 nm t = 5 nm

1.5

1

−1

0.5

−0.5

−1.5

0

1.5

1

−1

0.5

−0.5

−1.5

0

1.5

1

−1

0.5

−0.5

−1.5

0

Side

Side

Averaged

0 0.2 0.4 0.6 0.8 1

M/M

sM

/Ms

M/M

s

Position (µm)

Vortex wallw = 100 nm t = 5 nm

Middle

Averaged

Side

0 0.2 0.4 0.6 0.8 1

Position (µm)

(a) (d)

(e)

(f)

(b)

(c)

(g)

(h)0.1 (µm)

Figure 1. Calculated structures of transverse (T) and vortex (V) head-to-head DWs derived from micromagnetic simulations. Profiles ofthe longitudinal (dark gray) and transverse (medium gray) magnetizations along the nanowire’s length are shown at different positionsacross the nanowire’s width: T wall: top and bottom edges; V wall: topside and center. The profiles of the magnetization averaged over thewidth of the nanowire for both the T and V walls are also shown. Fits to the analytical 1D Bloch wall, equation (2), are shown in all casesfor the T wall, but only for the averaged longitudinal magnetization for the V wall. For the T wall, the magnetization profile has roughlythe same form across the width of the wire, although the DW width varies significantly. By contrast, the magnetization profiles are quitedifferent for the V wall at the edges and at the center of the wire.

The DW width can be deduced in several other ways. Forexample, as discussed in Section 3.4.3, the DW velocity isproportional to in small magnetic fields. The dynamicalDW width derived in this way is in good agreement withthat obtained from the 1D DW expression of equation (2) for

the transverse wall, but this is not the case for the V wall.Indeed, since the V wall moves more slowly than the T wall,the dynamical DW width is actually smaller for the V thanfor the T wall. As shown by Nakatani, Thiaville and Miltat(2005), a better agreement with the dynamical DW width is


(a)

(b)

(c)

Figure 2. (a) Magnetization distribution of a two-vortex wallcalculated from micromagnetic simulations of a permalloy nanowire(200 nm wide, 40 nm thick). The middle map (b) shows thedivergence of the magnetization for comparison with the MFMimage (c) of such a two-vortex wall measured in a 40-nm-thick,300-nm-wide permalloy nanowire. The dotted lines show the edgesof the nanowires as determined by AFM.

found by using a definition of the DW width proposed byThiele (1973a), in which the inverse DW width is given by:

−1 = 1

2wtM2s

∫V

(∂ M∂x

)2

dV (4)

Another experimental method of estimating the DW width,which is relevant to many of the experimental results dis-cussed in this chapter, is from the anisotropic magnetoresis-tance (AMR) of the DW (see Section 3.3.1).

More complex head-to-head DW structures than either theT or the V DW can also be found in magnetic nanowires,depending on their size and the aspect ratio of their crosssection. Figure 2 shows an example of a DW found insufficiently thick nanowires. Flux-closed domain structures,which can be seen in the figure at the upper edge of thenanowire, form so as to reduce the magnetostatic energy ofthe DW. Two-vortex DWs of opposite chirality can be seenin the figure. We have observed this wall structure usingmagnetic force microscopy (MFM) imaging in permalloynanowires more than 20 nm thick. Florez, Krafft and Gomez(2005) have reported similar structures in permalloy wires,40 nm thick.

In narrow wires with circular or square cross sections, avortex wall can also appear, with the vortex core parallel tothe long axis of the wire. This structure has been describedby Thiaville and Nakatani as a Bloch-point wall (Thiavilleand Nakatani, 2006).

An interesting methodology for categorizing these differ-ent DW structures was proposed recently by Youk et al.

(2006), Tchernyshyov and Chern (2005), and Chern, Youkand Tchernyshyov (2006). They describe the DWs ascomposite objects built from a certain number of topolog-ical defects characterized by a winding number n. Vor-tices in the bulk of the wire have n = 1 and antivortices,n = −1, whereas edge defects present in the transverse wallhave winding numbers n = ±1/2. The vortex DW wall isbuilt from one bulk defect (the vortex itself) and two edgedefects n = −1/2, and the transverse wall from two edgedefects n = ±1/2. For any DW structure, the total topolog-ical charge, including both edge and bulk defects, must bezero. The authors propose that the DW dynamics can bedescribed by the creation, propagation, and annihilation ofthese topological defects.

As an illustration, magnetic domain structures in U-shapednanowires of various widths, formed from 10- and 20-nm-thick CoFe films are shown in Figure 3. These imagesare measured in zero magnetic field by photoemissionelectron microscopy (PEEM). The gray scale reflects theprojection of the nanowire’s in-plane magnetization along thevertical axis of the image (see Section 3.3 for more details).The nanowire’s magnetization is first saturated along thisdirection with a large magnetic field (1 kOe). After this fieldis reduced to zero, the magnetization is aligned along thearms of the U-shaped structure so that a DW is nucleated inthe curved portion of the U. In the narrowest wires (200 nmwide), the DWs have a T structure, clearly identified by thesingle black triangle-shaped region. For intermediate widths,the PEEM images clearly show an asymmetric transversewall structure (see, e.g., the 400-nm-wide/20-nm-thick orthe 600-nm-wide/10-nm-thick nanowires). When the widewidth is increased, V walls can clearly be identified by thealternating black/white regions (e.g., the 600-nm-wide/20-nm-thick wire). Note that for the same 600 nm width,the 10-nm-thick wire still exhibits an asymmetric T wallstructure, whereas the 20-nm-thick wire already shows the Vstructure, following the trend shown by the phase boundaryof equation (1). The 800-nm-wide wires show more complexdomain patterns, with ripples starting to appear along thearms of the wire.

Similar experiments have been reported by Klaui et al.(2004) who used PEEM microscopy to study the DWstructure in cobalt rings, whose thicknesses were variedbetween 2 and 38 nm, and whose widths were varied from100 to 730 nm. A clear phase boundary between T and Vwalls was observed as a function of the ring dimensions.However, this transition did not agree with that anticipatedby equation (1). The T wall was observed for dimensionswhere the V wall should be more stable. This discrepancy isa consequence of the metastability of the two wall structures.Even though the V has lower energy, an energy barrierprevents the transformation of the T wall into the V wall.


800 nm

800 nm

600 nm

600 nm

400 nm

400 nm

200 nm

200 nm

Hsat

Figure 3. PEEM images of the magnetic domain structure of U-shaped CoFe nanowires with different widths and a thickness of 10 nm(upper) or 20 nm (lower). The samples were first saturated along the direction indicated by the arrow, and the images were taken inremanence.

It turns out that the T wall is favored by the DW injectionprocess (using a large transverse field), so T walls areobserved for a wide range of ring sizes.

2.1.2 Other types of walls

Head-to-head DWs are only stable in nanowires withstrong shape anisotropy and/or uniaxial anisotropy along thenanowire’s direction. In the presence of significant crys-talline anisotropy in the in-plane transverse or out-of-planedirections, other domain patterns can be observed even forsubmicron wires. For example, Schrefl, Fidler, Kirk andChapman (1997) have reported flux-closed domain patternsin NiFe elements as narrow as 200 nm in which the domains’magnetization is aligned transverse to the nanowire’s direc-tion. This was unexpected in nominally soft NiFe, but wasexplained by a strong stress-induced anisotropy. For mate-rials such as Pt/Co/Pt trilayers with strong perpendicularanisotropy (∼107 erg cm−3), the domains’ magnetization isoriented along the out-of-plane anisotropy direction. In suchnanowires, the DWs are nearly ideal Bloch walls (Wunder-lich et al., 2001; Cayssol et al., 2004), with widths as smallas 5 nm. In the case of (Ga,Mn)As epilayers, in-plane cubicanisotropy has been reported (Tang, Kawakami, Awschalomand Roukes, 2003), giving rise to 90 DWs (Holleitner et al.,2004; Honolka et al., 2005; Tang et al., 2004; Tang andRoukes, 2004) albeit in large structures (100 µm wide).

2.2 Theoretical models of current-drivendomain-wall motion

2.2.1 Early work

The interplay between magnetization and charge carriers inmetallic ferromagnets has been studied for several decades.The first reported interaction of current on DWs in suchmaterials was due to eddy-current losses (Williams, Shockley

and Kittel, 1950). The DW mobility in macroscopic samplesat low fields was found to be almost two orders of magnitudesmaller than that expected from calculations by Landau andLifshitz, which took into account magnetic relaxation.

The influence of an electric current flowing within aferromagnet, or in the vicinity of it, was first studied inthe 1970s by Carr (1974a,b), Emtage (1974), and Charap(1974), and independently by Berger (1974), Partin, Karne-zos, deMenezes and Berger (1974). These authors found thatin materials for which the current flow is sensitive to themagnetization, for example, because of the Hall effect ormagnetoresistance, the presence of a DW would, in turn,affect the current distribution. For example, in the case ofa 180 DW, the reversal of the magnetization is associatedwith reversal of the Hall electric field. The nonuniform cur-rent distribution can be modeled by a uniform current onwhich an eddy-current loop concentric with the center ofthe DW is superimposed. This current loop thus creates amagnetic field, which exerts a net force on the DW in thedirection of the drift velocity of the carriers and, thereby, canlead to DW motion. This mechanism was called self-inducedDW drag or hydromagnetic domain drag. The force on theDW can be written, following Berger’s notation, as:

Fx = 2Msµ−1e (R1J − vw) µe = π3ρ

8.4tMs(5)

where J is the current density perpendicular to the wall, R1

is the anomalous Hall constant, vw is the wall velocity, µe

is the wall mobility, as limited by eddy currents, (which isdifferent from the intrinsic wall mobility, which is related todamping), and t is the film thickness. Since the net force onthe DW increases with the sample thickness it vanishes forvery thin wires.

Because magnetic fields extend over fairly long distances,the DW drag mechanism can also occur if the current doesnot flow directly through the DW, but rather through a


neighboring over or underlayer, such as a semiconducting(Carr, 1974b; Charap, 1974) or a permalloy (Carr, 1974a;Emtage, 1974) layer. This is particularly applied to themotion of DWs in magnetic bubble materials, which weremade of high-resistivity oxides, and which were the subjectof much of this early work.

Note that all these theories describe 180 DWs, for whichthe current is perpendicular to the magnetization directionboth in the domains and in the DW. In this case, the Hallelectric field reverses across the DW, irrespective of theDW structure. This is not the case for head-to-head DWs,for which current and magnetization are parallel everywhereexcept within the DW.

This first mechanism derives solely from electromagneticeffects. The influence of the carrier spins was recognizeda few years later by Berger (1978). He proposed that thes–d exchange interaction between the conduction electronsand the localized magnetic moments could influence theDW dynamics in two different ways. The first contribution,which Berger called s –d exchange drag (Berger, 1984), is aviscous force on the DW which is proportional to the current.This term arises from the difference between the spin-dependent reflection coefficients of the conduction electronsat the DW. The second contribution is an ‘exchange torque’related to the transfer of spin angular momentum fromthe s conduction electrons to the localized magnetization(Berger, 1978, 1986). This mechanism is analogous to thespin-transfer torque proposed by Slonczewski in magneticheterostructures in which magnetic layers are separated bythin metal or insulating layers (Slonczewski, 1996).

2.2.2 Recent results: two types of torques

In the past few years, new experimental results on the inter-action between electric current and magnetic DWs havetriggered a flurry of theoretical studies (Shibata, Tatara andKohno, 2005; Bazaliy, Jones and Zhang, 1998; Tatara andKohno, 2004; Tatara et al., 2006; Zhang and Li, 2004; Liand Zhang, 2004a,b; He, Li and Zhang, 2005; Barnes andMaekawa, 2005; Thiaville, Nakatani, Miltat and Vernier,2004; Thiaville, Nakatani, Miltat and Suzuki, 2005; Waintaland Viret, 2004; Xiao, Zangwill and Stiles, 2006; Dugaevet al., 2006; Tserkovnyak, Skadsem, Brataas and Bauer,2006; Tatara, Vernier and Ferre, 2005; Ohe and Kramer,2006). In most cases, theorists follow a two-step approach.First, they calculate the current-induced torque on the mag-netization from the spin-polarized s conduction electrons inthe limit of static magnetic moments, since the magnetizationdynamics are slow compared to those of the electrons. Sec-ond, the influence of the current-related torque on the DWdynamics is studied by solving the Landau–Lifshitz–Gilbert

(LLG) equation of motion, usually by approximating the DWstructure, so as to obtain analytical expressions.

The influence of the current on the magnetization dynam-ics is often treated by including two spin-torque terms pro-portional to the gradient of the magnetization in the LLGequation. In the case of homogeneous magnetic material, andassuming the current is flowing in the x direction, the LLGequation can be written as (Thiaville, Nakatani, Miltat andSuzuki, 2005):

∂ m∂t

= −γ m × H + α m × ∂ m∂t

− u∂ m∂x

+ βu m × ∂ m∂x

(6)

where m is the magnetization normalized to the saturationvalue, H is the micromagnetic effective field, γ is thegyromagnetic factor, and α is the Gilbert damping constant.For permalloy films, α is of the order of 0.01 (Nibarger,Lopusnik and Silva, 2003; Nibarger, Lopusnik, Celinski andSilva, 2003)

The first two terms on the right-hand side of equation (6)are the usual precessional and damping terms, respectively,and the last two terms describe the interaction with thecurrent.

The first current contribution is derived in the adiabaticlimit. In the adiabatic limit, which, a priori, is justifiedfor sufficiently wide DWs, the conduction electrons spinorientation follows the local magnetization direction. Themagnitude of the adiabatic spin torque, which can be deriveddirectly from the conservation of spin angular momentum, isgiven by:

u = gµBJP

2eMs(7)

where g is the Lande factor (∼2), J is the current den-sity, P is the spin polarization of the current, µB =0.927 × 10−20 emu, the Bohr magnetron, and e = 1.6 ×10−19 C, the electron charge. For permalloy, for which Ms =800 emu cm−3, and assuming P = 0.4, u = 1 m s−1 whenJ = 3.5 × 106 A cm−2. Note that the polarization is some-times replaced by the Slonczewski function (Slonczewski,1996) g(P ).

The second contribution of the current is often dubbedthe nonadiabatic spin-torque or β term. As shown byequation (6), it behaves as a spatially varying magnetic fieldwhich is proportional to the gradient of the magnetization.The magnitude of the nonadiabatic term is given by thedimensionless constant β, which is of the order of thedamping constant α. Both the origin and the magnitude ofthe ‘β term’ is under much debate.

Zhang and Li (2004) have proposed a model in which thereis a slight mistracking between the electron spin and the localmagnetization direction. This generates a nonequilibrium


spin accumulation across the DW, which relaxes by spin-flipscattering toward the magnetization direction. This modelleads to both adiabatic and nonadiabatic spin-torque terms(there are also two small terms proportional to the timederivative of the magnetization, which slightly modify thegyromagnetic ratio and the damping constant). The mag-nitude of the nonadiabatic term is written as β = τ ex/τ sf,where τ ex is the relaxation time associated with the s–dexchange energy Jex(τ ex = /SJex) and τ sf is the spin-fliprelaxation time. Numerical estimates are obtained by assum-ing that Jex ∼ 1 eV, S = 2, and τ sf ∼ 1 ps, giving β ∼ 0.01.Note that this model assumes the DW width to be muchlarger than the length scale of the transverse spin accumu-lation (only a few nanometers), such that the nonadiabaticcontribution does not vanish even for very wide DWs.

Xiao, Zangwill and Stiles (2006) cast doubt on theexistence of this nonadiabatic contribution. They find thatalthough there is a transverse spin accumulation across theDW, the spin current follows the magnetization adiabaticallyunless the DW width is extremely small. In the narrow walllimit, the nonadiabatic spin torque is nonlocal and oscillatoryin space.

Tatara and Kohno (2004) have also calculated the twocurrent interaction terms shown in equation (2), in both thenarrow and wide DW limits. In their model, the relevantlength scale is the Fermi wavelength (a few angstrom). Inthe wide DW limit, they obtain a spin-transfer-torque termdubbed spin transfer with the same form as that shown inequations (6) and (7). In the narrow DW limit, they derive aforcelike term called momentum transfer, which plays thesame role as the β term integrated over the DW. Thismomentum transfer is very similar to the s–d exchangedrag proposed by Berger (1984) and is a function of theDW resistance RDW. The momentum transfer per unit cross-sectional area of the DW is written as:

Fel = neRDWJ (tw) = RDW

R0J (tw) = ρDW

R0J (8)

where J is the current density, t and w are the wirethickness and width, respectively, n is the electron density,R0 is the ordinary Hall coefficient (1/R0 = ne), and ρDW =RDWtw/, is the DW resistivity, where is the DW width.

The linak between the nonadiabatic spin torque and DWresistance is also mentioned by Zhang and Li (2004).Although they do not explicitly discuss this relationship, it isinteresting to consider the relationship between the momen-tum transfer force (8) and the β term. It can be seen from theLLG equation (6) (and more obviously from the integratedform (13)) that the β term indeed plays the same role as themagnetic field force term. It follows that the β-term-related

force (per unit cross-sectional area) can be written as:

Fβ = 2Ms

γβu (9)

By equating this with equation (8), it follows that

β = γ2e

gµBP

ρDW

R0(10)

Interestingly, if the DW resistivity decreases as 1/2, aspredicted by the Levy–Zhang model on DW magnetoresis-tance (Levy and Zhang, 1997), β will be roughly constant,independent of the DW width. In order to obtain a numer-ical estimate, we consider the DW resonance experimentsof Saitoh, Miyajima, Yamaoka and Tatara (2004), whichwill be discussed later in this chapter. The sample is apermalloy nanowire, with a thickness t = 45 nm and a widthw = 70 nm. The DW has a transverse structure such that theDW width can be estimated as ∼ 22 nm (as discussed inSection 2.1). We assume a spin polarization P ∼ 0.5, anda Hall resistance R0 ∼ 1.3 × 10−10 C m−3 for permalloy (asmeasured, e.g., by Freitas and Berger (1985)). The DW resis-tance is reported to be RDW = 0.26 m, such that the DWresistivity is about 3.7 × 10−11 m. From equation (10), thisleads to β ∼ 0.4.

In a recent paper, Berger has developed the relationshipbetween nonadiabatic spin torque (exchange drag in histerminology) and DW resistance further so as to describe in asingle scaling plot experimental results for DW velocity, DWresistance, and critical current for DW depinning (Berger,2006).

In both Tatara and Kohno and Zhang and Li’s models,in the absence of the β term, there is an intrinsic thresholdcurrent for irreversible DW motion, even in an ideal wirewithout any DW pinning sites. Below this threshold value,the DW only moves short distances while the current isapplied, but moves back to its original position after thecurrent is turned off (note that this is not the case if thereis an irreversible deformation of the DW’s structure or ifDW pinning is taken into account). Barnes and Maekawa(2005) argue that this intrinsic pinning does not exist (Barnes,2006; Tatara Takayama and Kohno, 2006). In their model,the ground state in the presence of current correspondsto the DW moving at a constant velocity u, without anytilt or distortion. This corresponds to massless motion andin the ideal case (no roughness), there is no thresholdcurrent for DW motion: the DW starts moving, albeit veryslowly, as soon as the current is nonzero. The discrepancyarises from the means of introducing damping into the

LLG equation. Whereas, both the Gilbert(α m × ∂ m

∂t

)and

the Landau–Lifshitz(

αγ

1+α2 m × m × H)

formulations are


equivalent in the absence of current, this is no longer the casewhen the current interaction terms are included. When theadiabatic spin-torque term is added to the LLG equation inits Gilbert or Landau–Lifshitz forms, the resulting equationdiffers by a term αu m × ∂ m

∂x. In other words, in the second

case, the adiabatic spin torque includes a β term, with β = α.In a recent extension of their work, Tatara et al. (2006)

have made a distinction between the nonadiabatic spin torque(i.e., momentum transfer) defined by equation (8) and the β

term. The implication of both the adiabatic and nonadiabaticspin torques on the DW dynamics will be discussed in moredetail in Section 2.2.3, in the framework of the semianalyticalone-dimensional model.

Several recent papers cast doubts about the previousdescription and find that the current-induced torques cannotbe written as a simple function of the magnetization gradient.Xiao, Zangwill and Stiles (2006) find that in the very narrowDW limit, the nonadiabatic torque oscillates across the DW,and extends over a length scale much larger than the DWwidth. Other authors find that the nonadiabaticity of theconduction electrons lead to the precession of their spins(Waintal and Viret, 2004), in turn generating spin waveswhich influence the DW motion (Ohe and Kramer, 2006). Arecent study focused on GaMnAs also shows that the spin-orbit coupling increases the reflection of the carriers (holesin this case) at the DW, leading to spin accumulation andthe enhancement of the nonadiabatic spin torque (Nguyen,Skadsem and Brataas, 2007).

Although thermal effects are not included in most theories,a few authors have addressed thermally assisted processes.Tatara, Vernier and Ferre (2005) find that in a rigid wallapproximation, thermally activated wall motion occurs belowthe zero-temperature threshold value, and the DW velocityvaries exponentially with the spin current. They also findthat the velocity exhibits a ‘universal behavior’ in the sensethat it does not depend on the pinning potential or thematerial parameters. In their model, thermal fluctuationsare simply introduced as the transition rate over an energybarrier, which is derived from the transverse anisotropy.Recently, Duine, Nunez and Mac Donald (2007) havedeveloped a more complete description of thermally activatedprocesses by deriving the Langevin equations of the nonzerotemperature motion of a rigid DW. They find that at nonzerotemperatures, the DW velocity varies linearly with current,even without a β term.

2.2.3 Analytical descriptions: one-dimensional modeland vortex model

Although DWs are complex three-dimensional objects, it isvery useful to develop models that allow for an analyticalor semianalytical description of the DW dynamics. The most

widely used approach is the so-called one-dimensional (1D)model, which was developed in the 1970s and has beendescribed in detail by Malozemoff and Slonczewski in theircomprehensive book about magnetic bubbles (Malozemoffand Slonczewski, 1979). The model assumes that the DWhas the 1D profile given by equation (2), such that themagnetization varies only in the direction perpendicular tothe DW (here the x direction). It is also assumed that thestatic profile is essentially preserved during the DW motion,although the DW width is allowed to change. The dynamicalDW structure is described by (in spherical coordinates):

θ(x, t) = ±2 arctan(

ex−q(t)

)ψ(x, t) = (t) (11)

where q is the position of the DW center, is the tilt angleof the DW magnetization away from its equilibrium position(see Figure 4), and is the DW width parameter. The latteris written as:

0 =√

A

K0 = 0√

1 + KK0

sin2( )(12)

where A is the exchange constant (erg cm−1). K0 isthe magnitude of the uniaxial anisotropy that defines themagnetization direction in the magnetic domains, and K

is the magnitude of the uniaxial transverse anisotropy. Inthe head-to-head configuration, the transverse anisotropy isthat within the plane perpendicular to the wire’s long axis.The magnitude of the transverse anisotropy can also bewritten as an anisotropy field Hk = 2K/Ms, where Ms isthe saturation magnetization of the material. This profilewas originally derived for the 1D Bloch wall. It alsogives a good description of the Neel wall and the head-to-head transverse wall, provided that the axes are properlydefined (see Figure 4 for the T wall). Qualitative agreementwith experiments is also obtained for more complex wallstructures such as the head-to-head vortex wall (Thomaset al., 2006).

z

y

x

q

ψ

Figure 4. Transverse DW showing the definition of the variablesof the 1D model: q (position of the center of mass of the DW) and (tilt angle of the DW’s magnetization out of the plane of thenanowire).


In this model, the LLG equation (6) can be integratedover the static DW profile and the dynamics are describedby two time-dependent variables q (DW position) and

(domain distortion). The DW width depends on suchthat it is also time dependent unless K0 K . For simplicity,we assume that is not time dependent in the followingdiscussion.

The LLG equation including the current-related terms canbe rewritten as:

(1+α2) = − γ

2Ms

(∂σ

∂q

)− γα

2Hk sin(2 )+ (β−α)u

(1+α2)q = −αγ

2Ms

(∂σ

∂q

)+ γ

2Hk sin(2 )+(1+αβ)u

(13)The DW potential energy σ(q) includes the contributionsfrom an external field and from position-dependent energyterms arising, for example, from defects in the nanowire.The field term is simply written as σ(q) = −2MsHq. Theinfluence of roughness or pinning may be approximated bypotential wells or barriers, which depend only on q.

It was pointed out long ago (Malozemoff and Slonczewski,1979) that equation (13) resembles Hamilton’s equations ofmotion for two canonical conjugate variables q and 2Ms /γ ,that is the position and its conjugate momentum. Followingthis analogy, the DW mass (Doring mass) can be defined as:

mD = 2Ms

γ 2HkS (14)

S is the cross section of the nanowire. These equationscan be solved analytically in many cases, yielding usefulexpressions for both the critical current and the DW veloc-ity. As discussed earlier, the physics of the current-drivenDW motion depend strongly upon the presence or absenceof the β contribution. Without the β term and in zero mag-netic field (∂σ/∂q = 0), there is an intrinsic critical current,below which the DW only moves transiently. This criticalcurrent is readily calculated by finding stationary solutionsof equation (13). The stationary solution, such that = 0and q = v = 0, only exists if u is smaller than a criticalvalue, given by:

uc = γHk

2(15)

Note that uc depends only on the magnetic materialparameters, unless there is extremely large pinning (Tataraand Kohno, 2004). In permalloy nanowires (or other softmaterials where crystalline anisotropy can be neglected),the transverse anisotropy field Hk is proportional to theshape anisotropy in the plane perpendicular to the longaxis of the wire. Note that this anisotropy field is also the

key parameter in models of field-driven DW motion. Forexample, the Walker breakdown field is directly proportionalto Hk (Malozemoff and Slonczewski, 1979). If the current issmaller than this critical value, the velocity decreases rapidly(over a few nanoseconds) and the DW stops. However, whenthe current is reduced to zero (for example, at the end of acurrent pulse), the DW moves back to its initial position.On the contrary, for currents higher than this critical value,the wall can move irreversibly over long distances. In thisregime, the instantaneous velocity oscillates strongly in time,in a way reminiscent of the field-driven DW motion abovethe Walker limit. The average velocity can be written as:

v =√

u2 − u2c

1 + α2(16)

The situation is very different when the β term is takeninto account. Indeed, for an ideal wire without pinning,irreversible DW motion occurs as soon as the current isnonzero, and the terminal velocity is simply v = βu/α.Therefore, in this case, the critical current becomes extrinsic,and is dependent on the magnitude of any pinning (i.e., willbe related to roughness and defects).

The two key parameters of the model are the DW width and the transverse anisotropy field Hk. It is essential toestimate realistic values in order to attempt a quantitative(or even qualitative) description of experiments or simula-tions. The DW width is readily defined for a head-to-headtransverse wall, as described in Section 2.1.1. Its definitionbecomes slightly more ambiguous for a vortex wall, wherethe ‘physical’ DW width, largely dominated by the tails ofthe DW does not account for the field-driven DW dynam-ics. Much better agreement is obtained using the so-calleddynamical DW width defined by Thiele (see Section 2.1.1),in which the (small) vortex core has larger ‘weight’ than the(large) tails of the DW. The transverse anisotropy Hk, whichprevents the rotation of the DW’s magnetization out of theplane of the wire, is somewhat more delicate. In permalloynanowires with head-to-head transverse walls, the transverseanisotropy is directly related to the shape anisotropy of thecross section of the wire. For small values of (only smallout-of-plane rotation of the wall’s magnetization), calculatingHk from the shape anisotropy appears reasonable. However,for large values of , the DW’s magnetization does notrotate coherently out of plane. Lower energy paths are pos-sible, which involve deformations of the wall structure, forexample, by nucleation of an antivortex. In such a case, theeffective value of Hk is smaller than that calculated from theshape anisotropy. Thus, if the dynamics involves high val-ues, the value of Hk obtained from the shape anisotropy doesnot provide a good quantitative description. For example, theWalker breakdown field, which is directly proportional to Hk


in the 1D model, is significantly overestimated. Another strat-egy is to find other methods to estimate Hk, for example,from the value of the Walker breakdown field found frommicromagnetic simulations. However, a contrario, this valuemight not account for the low dynamics. The definition ofHk becomes even more problematic in the case of a vortexwall, for which the 1D approximation is certainly not cor-rect. However, we have shown (Thomas et al., 2006) that the1D model can still be useful in describing the DW dynam-ics, provided that Hk can be estimated from micromagneticsimulations.

Examples of DW trajectories obtained by numerical inte-gration of equation (13) are shown in Figure 5. We chose val-ues of the parameters to match the properties of the nanowiredescribed in the following paragraph using micromagneticsimulations ( = 48 nm, Hk = 1600 Oe). The damping con-stant is α = 0.01. The first two panels show the time evolu-tion of the variables q and for a constant current corre-sponding to u = 100 m s−1, calculated for different valuesof the β term. Since u is smaller than the critical valueuc = 675.8 m s−1 given by equation (15), the DW motionstops after a transient time for β = 0. Similarly, satu-rates to a constant value given by sin(2 ) = −2u/(γHk)

(stationary solution of equation (13)). If β = 0, the DWmoves at a constant velocity after a transient time, theterminal velocity is given by βu/α and saturates at a value

1000

1000

800

800

600

600

400

400

200

200

0

0

Vel

ocity

(m

s−1

)

u (m s−1)

b = 5a b = a

b = 0

b = a /5

Figure 6. Averaged DW velocity calculated within the 1D modelfor the same set of parameters as in Figure 4, for different valuesof the ratio β/α.

sin(2 ) = −2u(1 − β/α)/(γHk). Figure 5 shows the timeevolution of q and for β = 0 and different values of u. Forvalues smaller than the critical current, the behavior is thatdescribed in the preceding text. Above the critical value (seecurve for u = 800 m s−1), the DW keeps moving continu-ously and its motion becomes precessional, in a fashion sim-ilar to the field-driven motion mechanism above the Walkerbreakdown field. The angle increases as the DW magneti-zation rotates continuously form one in-plane direction to the

5

4

3

2

1

00 20 40 60 80 100

t (ns)

q (µ

m)

b = 5ab = a

b = 0

b = a /5

0 20 40 60 80 100t (ns)

20

15

10

5

0

−5

−10

ψ (

°)

t (ns)

q (µ

m)

100

150 200100500

80

60

40

20

0

−20

u = 800 m s−1

u = 600 m s−1

ψ (

°)

200

0

−200

−400

−600

−800

−1000

t (ns)

150 200100500

b = 0 a = 0.01 ∆ = 48 nm H k = 1600 Oe

u = 100 m s−1 a = 0.01 ∆ = 48 nm H k = 1600 Oe

Figure 5. Time dependence of the DW position q and tilt angle calculated using equation (13) for the parameters indicated in thefigure.


0 20 40 60

60

50

40

30

20

10

0

−1080 100 120

Current pulse length (ns)

0 20 40 60 80 100 120

Field pulse length (ns)

Fin

al D

W p

ositi

on (

µm) u = 800 m s−1 ~ 1.18 uc H = 9.5 Oe ~ 1.18 H WB

Figure 7. Final DW position after a current or field pulse, calculated using the same parameters as in Figure 4, for β = 0.

other, and the DW velocity oscillates accordingly. Note thatthe DW velocity is a maximum when the DW magnetizationrotates across the out-of-plane direction ( = ±π/2), con-trary to the field case, for which the velocity drops (and evenreverses). Time-averaged terminal velocity curves are shownin Figure 6 as a function of u, for different ratios of β/α.

An interesting feature of the current-driven motion forzero-β is that for u > uc, the DW displacement after currentpulses appears to be quantized. The DW moves only bymultiples of π/α. This peculiar property follows fromthe DW’s relaxation toward its equilibrium state, = 0,after the current is turned off. The same DW relaxation alsooccurs at the end of a field pulse, leading to a discontinuityof the DW position as a function of the pulse lengthrather than a quantized position. This can be understoodfrom equation (13). Let us assume that the DW moves inthe stationary regime with either field (q = v = γH/α,sin(2 ) = 2H/(αHk)) or current (q = v = 0, sin(2 ) =−2u/(γHk)). After the field and current are turned off, theequations of motion are the same. The DW velocity is simplyproportional to sin(2 ). Therefore, in the field case, since

is positive, the DW velocity remains positive and decreases tozero. On the contrary, in the current case, sin(2 ) is negative,such that the velocity become negative when the current isturned off, and the DW moves back to its original position.If u is larger than the critical current uc, the relaxation afterthe pulse will be either backward (if < π/2 (mod π )) orforward (if > π/2 (mod π)). The DW’s final position isshown in Figure 7 as a function of the length of a current orfield pulse, for the same parameters used previously.

Note that because of this relaxation mechanism, the criticalcurrent given by equation (15) is only correct for dc currents.For current pulses, irreversible DW motion only occurs if thepulse is long enough for to exceed π/2. If not, the DWgoes back to its original position even if u > uc. As a result,the critical current varies as the inverse of the pulse length.

The addition of a magnetic field and/or pinning poten-tial to equation (13) yields many interesting results. Forexample, the depinning from a potential well is very different

depending on the depth of the pinning potential, the valueof the β term and the external magnetic field. This leads todifferent expressions for the depinning current in differentregimes, as described by Tatara et al. (2006).

Thiaville and Nakatani (2006) have studied in detailthe comparison between the 1D model and micromagneticsimulations for the case of field-driven DW dynamics.To paraphrase their conclusions, ‘it is very helpful for aqualitative understanding, and it becomes quantitative atreally small sizes, a few exchange lengths.’

A different approach is clearly needed to describe complexwall structures such as the head-to-head vortex wall. Amore general description of the steady state DW motionhas been proposed by (Thiele, 1973a,b) and extended byThiaville, Nakatani, Miltat and Suzuki (2005) to include bothcontributions from the current. Thiele’s approach has beenused successfully to describe quantitatively the dynamicsof a vortex located in a small elliptical disk, driven eitherby magnetic field (see, e.g., the recent papers Novosadet al., 2005; Buchanan et al., 2005) or spin-polarized current(Shibata et al., 2006). However, a quantitative descriptionof a moving DW using this framework is still lacking,although first attempts have been published recently (He, Liand Zhang, 2006a).

2.2.4 Micromagnetic simulations

Micromagnetic simulations are extremely useful to exploreDW dynamics because realistic DW structures can be studiedwithout limitations, and in particular, transformations ofthe DW structure can be described without approximation.Moreover, nonuniform fields (such as the Oersted fieldfrom the current) or nonuniform current distributions can bereadily included. In addition, roughness and pinning can beeasily introduced.

Several authors have published results of micromagneticsimulations of current-driven DW motion for both vortexand transverse DW structures (Thiaville, Nakatani, Miltat andVernier, 2004; Thiaville, Nakatani, Miltat and Suzuki, 2005;


0.5

0

−0.5

−1

−1.5

−2

−2.5

−30 5 10 15 20

Time (ns)

0 5 10 15 20

Time (ns)

DW

pos

ition

(µm

) 3 mA b = 03 mA b =

0.004

0.003

0.002

0.001

−0.001

−0.002

0

Out

-of-

plan

e m

agne

tizat

ion

3 mA b = a12 mA b = 012 mA b = a

3 mA b = 03 mA b = a12 mA b = 0

(e)

(a) (b) (c) (d)

(f)

t (ns) 0 1 2 3 4 0 1 2 3 4 0 1 2 3 45 6 0 1 2 3 4 5 6 7 8 9 10

Figure 8. Micromagnetic simulations of the motion of a transverse DW in a permalloy nanowire, 150 nm wide, and 5 nm thick, for I = 3and 12 mA. Top panels, from left to right: magnetization maps for I = 3 mA, β = α (a, left), I = 12 mA, β = α (b), I = 3 mA, β = 0 (c)and I = 12 mA, β = 0 (d, right). Bottom panels: Time dependence of the DW position (e) and the out of plane magnetization (f) in allfour cases.

He, Li and Zhang, 2006a,b). There are also a few recentreports of the field-driven DW motion in similar structures(Thiaville and Nakatani, 2006; Nakatani, Hayashi, Ono andMiyajima, 2001; Nakatani, Thiaville and Miltat, 2003; Porterand Donahue, 2004).

Examples of micromagnetic simulations of the current-driven propagation of DWs for different nanowires (5 and20 nm thick, 150 nm wide, 4 µm long) are shown in Figures 8and 9. These calculations were performed using the LLGmicromagnetic simulator code developed and commercial-ized by Mike Scheinfein. Standard parameters for permalloyare used, and the Gilbert damping constant is set at α = 0.01.The cell size is 5 × 5 × 5 and 5 × 5 × 10 nm3 for 5- and 20-nm-thick wires, respectively. Fixed boundary conditions areapplied at both ends of the nanowire to pin the magnetizationalong the wire axis. Note that the Oersted field is includedin the simulations.

The first example (Figure 8) shows the time evolution ofa transverse wall (the most stable structure for a nanowire ofthese dimensions), for two different current values (3 and12 mA), and for the cases β = 0 (adiabatic torque only,b) and β = α (a). Note that the current is turned oninstantaneously at time zero. In the latter case (β = α), asdiscussed earlier in the framework of the 1D model, the DWpropagates without distortion, at a constant velocity, directlyproportional to the current.

On the contrary, when β = 0, there is an intrinsic thresholdcurrent (whose value is given by equation (15)). Below thisvalue, at I = 3 mA, the DW only moves during the firstnanosecond after the current is turned on. The DW velocityis a maximum at t = 0 and then drops progressively tozero as an out-of-plane magnetization component develops(proportional to in the one-dimensional model) as shownin the bottom panel of Figure 8. The DW displacement


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14t (

ns)

t (ns

)

0 3 6 9 12 15 18 21 24 27 30

(a) (b) (c)

Figure 9. Micromagnetic simulations of the motion of a vortex DW in permalloy nanowires (a and b) width 150 nm, thickness 5 nm,current 3 mA, β = α and β = 0, respectively (c) width 150 nm, thickness 20 nm, current 9 mA, β = 0.

is completely reversible: if the current is turned off, theDW moves back to its initial position (not shown). On thecontrary, for I = 12 mA (above the threshold current), theDW motion is irreversible. The first stage of the motionis qualitatively similar to the low-current case: the DWstarts moving at its maximum velocity and then slows down.However, the displacement is much larger than in the low-current case. More importantly, the out-of-plane componentis also larger, and the DW is not in dynamical equilibrium:an antivortex is nucleated at the right side of the wire, and itspropagation across the width of the nanowire is associatedwith a strong increase of the DW velocity (note that justabove the threshold current, the nucleation of the antivortexis extremely slow, on the order of tens of nanoseconds).Once the antivortex is expelled at the left side of the wire,the DW recovers the same structure as its initial state, exceptits magnetization has been rotated by 180. If the current ismaintained then the DW motion is periodic, and the DWstructures oscillates between T walls with left- and right-handed orientations, and the DW velocity oscillates betweenhigh and low values as antivortices penetrate and are expelledfrom the nanowire. If the current is turned off, the DWwill relax to an equilibrium state without current, namelyan undistorted left or right-handed T wall, such that it canmove either forward or backward after the current is turnedoff, depending upon its state at this time. Note that the currentdensity needed to reach this regime is very high, on the orderof 1.6 × 109 A cm−2.

The second example (Figure 9) shows the case of a vortexwall for I = 3 mA, and for β = α (a) and β = 0 (b). Resultsare very similar to the T wall case. In the first case (β = α),

the DW moves at roughly constant velocity. Contrary to thefield-driven motion, the DW velocity is not related to the DWstructure, and the V and T walls move at the same velocityv = βu/α. In the second case (β = 0), the DW structure isdistorted as the vortex core is progressively pushed to theside and finally expelled from the nanowire. Note that forsmaller currents, a dynamical equilibrium would be reached,in which the V wall would stop moving, as described belowfor the 20-nm-thick wire case. However, in the case shownhere, the current is larger than the threshold for the V wall,but smaller than that for the T wall; therefore, the V wall istransformed into a T wall and then follows the same behavioras that described before. Since the V wall is very unstable forthese dimensions (see equation (1)), this transformation canoccur at very low current (<1 mA, about 1.3 × 108 A cm−2).Similar results have been described by He et al. for a 128-nm-wide, 8-nm-thick nanowire (He, Li and Zhang, 2006a,b).They concluded that the V wall has a smaller critical currentthat the T wall.

For wire dimensions such that the V wall is the stablestructure, the threshold current of the V wall increases, andmay even exceed that of the T wall. As shown in Figure 9(c)for a 20-nm-thick wire (with β = 0), at 9 mA (current densityof 3 × 108 A cm−2) a dynamical equilibrium is reached andthe DW stops moving. Larger current (∼15 mA) are requiredto overcome this dynamical equilibrium and achieve irre-versible motion. Interestingly, if the initial state is a T wall,a current of 9 mA is sufficient to drive the T wall ∼2 µm,before it is transformed into a V wall and eventually stops(not shown). Note that the timescale is much slower for thislarger structure.


Note that the details of the propagation mechanism abovethe threshold current depend on both the size of the structureand the magnitude of the current, in the same way as theDW propagation mechanism for fields larger than the Walkerbreakdown field. When the current (or similarly the magneticthe field) increases, higher energy routes become possible. Inthe example shown in Figure 8, an antivortex is nucleated.For larger structures, more complex scenarios can occur(reversal of the vortex core magnetization, nucleation, andannihilation of one or more vortices, etc.).

3 EXPERIMENTS

3.1 Device fabrication

3.1.1 Lithography techniques

Submicron-sized magnetic wires can be fabricated by variouslithography techniques, such as focused ion beam (FIB),optical lithography, and electron-beam lithography. Othermethods, which are not discussed here include nanoimprintand interference lithography.

FIB lithography is essentially a one-step process: thenanostructure is drawn from a plain film by milling the mag-netic material locally with a high-energy ion beam (typically,Ga+ ions accelerated at ∼30 keV). The ion current can betuned to achieve high resolution (Xiong, Allwood, Cookeand Cowburn, 2001). Large currents (a few nanoamperes)are used to mill wide areas, whereas much smaller currents(a few picoamperes) are used to define finer features of thestructure. This multistep approach ensures that critical cutsare made in as short a time as possible, thus reducing blurringdue to stage drift. Resolutions down to about 20 nm can beachieved. Fairly complex structures can be fabricated, suchas tracks for DW logic (Allwood et al., 2002b). However, itis not simple to make electric contacts to the nanostructurewithout adding an additional electron-beam lithography step.One method is to use the magnetic material itself as a contact,by leaving small bridges between the nanostructures them-selves and a larger outer structure. In the example shown inFigure 10(a), a large area permalloy element (10 × 0.1 mm2)was first fabricated by magnetron sputter deposition througha metal shadow mask. The nanowire was then patterned usinga multistep FIB milling process. A 6.6 nA beam was used tocut two 5-µm-wide trenches perpendicular to the wire’s longaxis (see inset of Figure 10a). Much lower current beams (11and 4 pA) were then used to precisely shape the nanowire.This method has a major disadvantage for studies of DWpropagation. Since the nanowires are connected at both endsby magnetic material, it is difficult to control the nucleationand injection of a single DW. To solve this problem, the

(a)

(b)

(c)

Figure 10. Examples of nanowires fabricated by FIB (a and b) andelectron-beam lithography (c). (Courtesy of Charles Rettner.)

magnetic nanowire can be isolated completely form the outerstructure during milling, both electrically and magnetically.The ends of the nanowires can be shaped to favor or sup-press DW nucleation. Electrical connections can be restoredby using the FIB to deposit a metal locally by reduction of anorganometallic vapor (Thomas et al., 2005). In the exampleshown in Figure 10(b), small Pt contacts were deposited atboth ends of the wire.

FIB lithography allows the fabrication of only one deviceat a time, and thus it is not practical for systematic studiesof nanowires with different sizes and shapes. It is also worthnoticing that FIB lithography is not suitable for all materials.For example, the giant magnetoresistance (GMR) of spin-valve structures patterned by FIB is strongly reduced, whichis likely related to damage at the edges of the device (Katine,Ho, Ju and Rettner, 2003).

Conventional optical and electron-beam lithography tech-niques are more convenient to fabricate nanowires and theirelectric contacts in successive lithographic steps. Both liftoffand ion milling methods have been successfully used tofabricate magnetic nanowires. An example of a nanowirefabricated by electron-beam lithography and ion milling is


shown in Figure 10(c). The nanowire was first patterned froman extended film. Its ends were tapered to favor the nucle-ation and propagation of a DW from one end of the device.The nonmagnetic metallic contacts at either end were addedin a second step.

These film-based methods usually produce ‘flat’ nanowireswhose widths are much larger than their thicknesses. Thelayout in the sample plane can be changed easily, andcomplex structures have been fabricated. Nanowire shapesare most often designed to accommodate constraints suchas the measurement technique (Hall crosses for Hall-effectdetection), the control of the DW injection and pinning (L-or U-shaped structures, rings, etc.).

3.1.2 Nanoporous templates – electroplating

Electroplating is also a powerful (and inexpensive) techniqueto fabricate nanowires. In this approach, the first step isthe fabrication of a template with pores of appropriatediameter. These templates can be track-etched polycarbonatemembranes, anodized alumina films (Whitney, Jiang, Searsonand Chien, 1993) or photoresist patterned by electron-beamlithography techniques (Duvail et al., 1998). The pores inthe template are then filled by electrodeposition. This leadsto an array of nanowires that can be studied in the templateor after dissolution of the template. Several methods havebeen reported to make electric contacts to a single nanowire,either in the template or after dissolution. These methodsinclude monitoring the resistance during the deposition soas to stop after only one nanowire becomes electricallyconnected (Wegrowe et al., 1998) and depositing contacts ina later step using electron-beam or optical lithography (Vila,Piraux, George and Faini, 2002).

Nanowires prepared by electrodeposition are usually cylin-ders, with a roughly circular cross section and potentiallyvery high aspect ratios (length/diameter up to ∼1000). Bentnanowires have also been fabricated by centrifugation, afterdissolution of the membrane and suspension in a liquid(Tanase, Silevitch, Chien and Reich, 2003).

Although nanowires fabricated by electrodeposition havebeen used to study DW magnetoresistance (Ebels et al.,2000) or magnetization reversal (Wernsdorfer et al., 1996),there are no reports as yet of current-driven DW motionin such samples. However, multilayered nanopillars madeby electrodeposition have been used to study current-drivenmagnetization reversal (Wegrowe et al., 2004, 1999).

3.2 Materials

The list of materials used for current-driven DW motionstudies is surprisingly short. The majority of experiments

use permalloy (Fe81Ni19). There are also a few reports onother soft magnetic materials (CoFe and Ni), on hard mag-netic materials with perpendicular magnetization (epitax-ial Pt/Co/Pt multilayers), on diluted magnetic semiconduc-tors (epitaxial GaMnAs) as well as one or two studies onnanowires formed from spin-valve structures.

There are several requirements for materials suitable forthe study of current-driven DW motion in nanowires. First,their magnetic properties must be well behaved, so as toallow for well-defined DWs on the submicron scale. Softmagnetic materials are well suited, since long exchangelengths and large wall widths make them less sensitive tolocal structural defects. The crystalline quality becomes moreimportant for harder materials. Second, the resistivity mustbe small to limit losses and Joule heating. Third, the spinpolarization must be high so as to enhance spin-transferefficiency. Fourth, in order to detect the DW motion byelectrical means, materials with significant magnetoresistance(AMR, GMR, or anomalous Hall resistance) are most useful.

These constraints have so far precluded the exploration ofvery many materials. This means that the detailed influenceof material parameters, such as the saturation magnetizationand the magnetic anisotropy, have not yet been studied indetail.

3.3 Detection of domain walls in nanowires

In this section, we describe some of the experimental tech-niques that have been used to probe DWs in nanowires.General principles and technical details can be found else-where. Here we emphasize the main aspects of these differentmeasurement techniques in the specific context of probingDWs in nanowires, and address some of their advantagesand drawbacks. It is important to note that different tech-niques are useful for measuring different DW properties. Forexample, some measurement methods such as the magneto-optical Kerr effect (MOKE) and GMR (Parkin and Wessels,1995) are sensitive to the magnetization of the magneticdomains, from which the presence of DWs and their loca-tion can be inferred. By contrast, other methods, such as theAMR probe the presence of the DW itself but lack sensitiv-ity to its position. Therefore, depending on the characteristicsand the sensitivity of the measurement technique, modifica-tions of the DW structure or motion over small distances canbe detected or be completely overlooked. This can result insignificant ambiguities, for example, in the definition of thecritical current for DW motion.

Another key aspect of the measurement method is thecapability to acquire enough statistics to account for stochas-tic variations in the motion of the DW. For example, manymicroscopy techniques are slow. Thus, even though these


techniques have sufficient spatial resolution to access thestatic DW structure before and after its motion, the lim-ited number of experiments which can be performed in areasonable time may not allow for the measurement of allpossible configurations. On the other hand, techniques suchas time-resolved MOKE require a large number of repetitionsof nominally the same DW motion to achieve adequate signalto noise, and so can lead to misleading results if fluctuationsin this motion are important.

3.3.1 Electrical measurements

Magnetoresistance measurements are very convenient todetect DWs in nanowires. The AMR and GMR effects aremost commonly used (Parkin and Wessels, 1995). Thereare also several experimental studies based on the ordi-nary and anomalous Hall effects (HE). These techniques,which are based on electrical measurements, are fast, allow-ing for extensive and systematic studies of current-drivenDW motion.

Anisotropic magnetoresistance (AMR)The resistivity of ferromagnetic metals typically exhibitsAMR, whereby it depends on the angle θ between themagnetization and the electric current direction in the mate-rial, according to the relation:

ρ(θ) = ρ⊥ + δρ cos2(θ) with δρ = ρ‖ − ρ⊥ (17)

The normalized AMR ratio is defined as δρ/ρav =δρ/(ρ‖/3 + 2ρ⊥/3). This effect, which arises from spin-orbit coupling, can exceed 5% for some Ni–Fe and Co–Nialloys at room temperature (and can be much larger at lowertemperatures). The AMR values of many 3d transition-metalalloys are listed in the review paper of McGuire and Potter(1975).

The presence of a DW in a nanowire, which exhibits AMR,changes the nanowire’s resistance because the magnetizationwithin the DW deviates from the wire’s long axis and thusfrom the current direction. In order to estimate the magnitudeof the signal, consider the case of two head-to-head domainsseparated by a DW of width with a magnetization profilefollowing the 1D Bloch wall of equation (2). When no DW ispresent, and the magnetization lies parallel to the nanowire’slong axis, the resistance is simply:

Rsat = ρ‖Lwt

(18)

where L, w, and t are the length, width and thickness ofthe wire, respectively. When a DW is present within the

nanowire, its resistance becomes:

RDW =∫ L/2

−L/2ρ[θ(x)]

dx

wt(19)

Thus, the contribution of the DW to the nanowire’s resis-tance, in the limit L is:

RDW − Rsat = −δρ2

wt(20)

The signal from the DW is directly proportional to the AMRratio and the DW width. In permalloy, to a first approx-imation, the DW width scales with the nanowire’s width ∝ w (see Section 2.1.1). Therefore, the DW contributionto the resistance is of the order of δρ/t and is independent ofthe wire width. Note that since the AMR signal arises fromthe DW itself, its relative contribution to the total nanowireresistance decreases as 1/L, and can become quite small inlong wires. It can also be easily washed out by drifts in thenanowire resistance over time (e.g., owing to small changesin temperature).

An important characteristic of the AMR signal is that itdoes not depend on the position of the DW (in uniform,homogeneous nanowires). Thus, the AMR signal detects thepresence, or absence, of a DW between the electrical contactsused to probe the nanowire’s resistance.

An example of the magnetoresistance (MR) hysteresisloop of a straight NiFe nanowire, 300 nm wide and 10 nmthick, is shown in Figure 11(a). The distance between theelectrical contacts is 4 µm. One end of the nanowire has atapered tip, to prevent DW nucleation, and the other endis attached to a large nucleation pad to assist DW injection(see Section 3.4.1). Note that these pads have very differentshapes in Figure 11(a) and (b). A triangular notch is patternedon one side of the wire, which acts as a pinning center for theDW. A magnetic field H is applied parallel to the nanowire’slong axis. When the magnetization is saturated along thisaxis, so that it is parallel to the current flow, the resistanceis highest. When the field is decreased to H = −25 Oe aDW is injected into the nanowire and is trapped at thepinning site. This results in a sharp drop of resistance by∼0.2 . The resistance level decreases further by ∼30% asthe field becomes more negative. This may be because theDW is moved further into the notch or because the DWbecomes wider as it is stretched away from the notch. Thiswill depend on the detailed structure of the DW (whetherV or T and its chirality). When the field exceeds ∼−25 Oe,the DW is driven from the pinning site to the other end ofthe nanowire outside the region between the contacts andthe resistance reverts back to the saturation value. Note thatsince the AMR is only sensitive to the presence of the DWbetween the contacts, no signal is detected in these quasistatic


4 µm

474

473.8

473.6

473.4

473.2

473−300 −200 −100 0 100 200 300

Field (Oe)

Res

ista

nce

(Ω)

High-R

High-R

Low-R

−300 −150−250 −200 −100 −50 0

Field (Oe)

Res

ista

nce

(Ω)

500

498

496

494

492

490

488

486

High-R

Low-R

Intermediate(a) (b)

Figure 11. Examples of magnetoresistance curves obtained when the propagation of a DW along the nanowire is probed by AMR in aNiFe nanowire (a) or GMR in a spin-valve nanowire, which has the basic structure of NiFe/Cu/CoFe/Ru/CoFe/IrMn (b).

0

−0.05

−0.1

−0.15

−0.2

−0.25

−0.3

−0.35

−0.40 10 20 30 40 50

Thickness (nm)

0 10 20 30 40 50

Thickness (nm)

∆R (

Ω)

1

0.8

0.6

0.4

0.2

0

∆ w

Transverse wallVortex wallTwo-vortex wall

3/4

1/p

(a) (b)

Figure 12. (a) DW resistance, for three different DW structures, in a permalloy nanowire of width 20 nm, as a function of the nanowirethickness t , calculated from micromagnetic simulations. (b) Ratio of the DW widths, obtained from equation (20), to the nanowire width(symbols), and comparison with values obtained from fits of the magnetization profile to the 1D Bloch wall model profile.

measurements if the DW is not trapped between the contactsafter injection but rather propagates directly to the end ofthe wire.

Although the signal is rather small, the AMR is highly sen-sitive to details of the DW structure. For example, since theAMR signal depends on the DW width, V and T walls canbe distinguished. The AMR signals calculated from micro-magnetic simulations for three different DW structures in100-nm-wide NiFe wires are shown in Figure 12, as a func-tion of the wire thickness. Values of resistivity of 30 µcmand an AMR ratio of 1% are used. Interestingly, the V wallexhibits a signal about 30% larger than the T wall and thedouble-V wall shows an even larger signal. The calculatedvalues vary inversely with nanowire thickness, as expectedfrom equation (20). The DW widths can be estimated fromthis same equation (Figure 12b). Interestingly, the ratio ofthe DW width to the nanowire width (/w) is nearly inde-pendent of thickness for the V and T walls. For the T wall,/w is ∼ 1/π , as proposed by Nakatani, Thiaville and Mil-tat (2005). However, for the V wall, /w is ∼0.4, which is

much smaller than that obtained by fitting the magnetizationprofile to the 1D Bloch wall profile, that is, 3/4.

Experimentally, we find that the AMR signal does notdepend much on the wire width, in good agreement withequation (20). The comparison between nanowires with dif-ferent thicknesses is less conclusive since both the nanowireresistivity and the AMR ratio vary with thickness. How-ever, different wall structures can be well resolved (Hayashiet al., 2006b). The AMR signatures of four different DWstrapped at a notch in a 300-nm-wide, 10-nm-thick permalloynanowire are shown in Figure 13.

In summary, AMR is a powerful technique to probethe presence of DWs in magnetic nanowires. The maindrawbacks of AMR measurements are the small outputsignal, the lack of sensitivity to DW displacement, and thelimited choice of magnetic materials with large enough AMRratios.

Note that AMR is not the only contribution to theresistance of the DW. Spin-dependent scattering acrossthe DW can also contribute to its resistance (Levy and


900

600

300

0

Cou

nts

469.7 469.8 469.9 470

Resistance (Ω)

No DW

Vor

tex

cloc

kwis

e

Tra

nsve

rse

cloc

kwis

e

Vor

tex

antic

lock

wis

e

Tra

nsve

rse

antic

lock

wis

e

Figure 13. Histogram of measurements of the resistance of apermalloy nanowire (10 nm thick and 300 nm wide) after injectionof a DW in a series of repeated experiments. The four peaks inthe histogram correspond to four different DW structures. The DWstructure, corresponding to each resistance value, was determinedby MFM imaging and reproduced by micromagnetic simulations.For all four DW structures are shown the MFM image (top), thecorresponding magnetization map from micromagnetic simulation(bottom) and the map of the divergence of the magnetization(middle). (Reproduced from Masamitsu Hayashi, Luc Thomas,Charles Rettner, Rai Moriya, Xin Jiang, and Stuart S.P. Parkin,Phys. Rev. Lett. 97, 207205 (2006), copyright by the AmericanPhysical Society, with permission from APS.)

Zhang, 1997). This latter effect is often termed the intrinsicDW resistance. Although the resistance of DWs has beenextensively studied in the past decade (Marrows, 2005), theintrinsic contribution is usually smaller than the AMR in 3dtransition-metal nanowires (∼0.1 to 1% of the resistivity).Indeed, in permalloy nanowires, we find that the AMR canlargely account for the DW resistance in wires wider than∼100 nm. Large intrinsic DW effects have been reportedin nanoconstrictions in permalloy nanowires although it isdifficult to rule out contributions from magnetoelastic effects.Intrinsic contributions to the DW resistance in GaMnAswires have also been reported (Ruster et al., 2003).

Giant magnetoresistanceGMR is observed in multilayered structures in which two ormore ferromagnetic layers are separated by thin nonmagneticmetallic spacer layers (Parkin and Wessels, 1995; Parkin,1994). The current can flow either parallel or perpendicularto these layers. These geometries are often termed current-in-plane (CIP) and current perpendicular to plane (CPP),

respectively. Few GMR measurements of DW motion havebeen reported although both the CIP (Ono, Miyajima, Shigetoand Shinjo, 1998; Ono et al., 1999a; Grollier et al., 2002,2003, 2004; Lim et al., 2004.) and CPP (Zambano and Pratt,2004) geometries have been used, typically with a spin-valvestructure (Parkin and Wessels, 1995; Parkin et al., 2003). Onemagnetic layer, nominally in a single domain state, acts as areference layer, and the DW motion is observed in a secondmagnetic layer, the free layer, where the two layers areseparated by a thin Cu layer (Parkin, 1994). Although GMRvalues of more than 70% are found in Co/Cu multilayersat room temperature (Parkin, Li and Smith, 1991), typicalGMR effects in spin-valve structures useful for DW studiesare much smaller (∼1–5%). However, the GMR signals aremuch larger than the AMR signal from a DW in a permalloynanowire, since the GMR arises from the magnetic domainsand not from the DW alone. This also means that the GMRsignal is sensitive to the precise position and thus the motionof a DW along a nanowire.

A typical example is illustrated in Figure 11(b), in whichthe resistance hysteresis loops of a 200-nm-wide spin-valve nanowire formed from a NiFe/Cu/CoFe/Ru/CoFe/IrMnstructure are shown. In the field range shown, only themagnetization of the free layer (a 20-nm-thick NiFe layer)reverses. When the free layer’s moment is completelyswitched from being parallel to antiparallel to the referencelayer (an NiFe layer exchange biased by an antiferromagneticIrMn layer) the resistance of the device changes by ∼12 ,which is nearly 100 times larger than the AMR signal (0.2 )for a permalloy nanowire device with similar resistance.In the example shown in Figure 11(b) the nanowire wasfabricated with three notches of different depths along oneside of the nanowire (as shown in the scanning electronmicrograph (SEM) in Figure 11b) so that the DW is trappedsuccessively at these pinning sites. This accounts for theresistance plateaus in the figure.

While GMR appears well suited to the study of the motionof DWs in nanowires, the necessary use of additional mag-netic and nonmagnetic layers causes severe problems. Inparticular, the shunting of current through the nonmagneticspacer layer, which is typically much more conducting thanthe other layers in the stack, results in an inhomogeneouscurrent distribution through the stack. This may lead to Oer-sted fields from the current within the magnetic layer inwhich the DWs move. A second, perhaps even more impor-tant problem, is the interaction of the fringing fields fromthe DWs with the reference layer moment through magneticdipolar interactions (Gider, Runge, Marley and Parkin, 1998;Thomas, Samant and Parkin, 2000; Thomas et al., 2000).Moreover, any inhomogeneities in the reference layer (eithermagnetic or structural) will likely lead to pinning centers forthe DWs in the free layer, again through dipolar fields. A


third significant problem is that the current densities neededto move DWs in transition-metal ferromagnetic materials areoften so high that the device becomes significantly heated,so that the exchange bias is weakened (typical blocking tem-peratures are ∼500 K) and the reference layer moment is nolonger magnetically stabilized. Notwithstanding these prob-lems, GMR has a particular advantage in that, since theGMR effect derives largely from spin-dependent interfacescattering (Parkin, 1992, 1993), many different free layerferromagnetic materials can be studied by inserting ultrathinlayers at the interface with the Cu spacer layer which givehigh GMR (Parkin, 1993).

Hall effectThe Hall effect is also a powerful technique to study DWmotion. For materials with magnetization perpendicular tothe sample plane, for example, CoPt alloys or multilayersand GaMnAs, the anomalous hall effect (AHE) provides adirect measurement of the DW position (Wunderlich et al.,2001; Yamanouchi, Chiba, Matsukura and Ohno, 2004).Indeed, just as for GMR, the AHE is proportional to the netmagnetization (within the Hall cross), although the signal canbe even larger. For example, in epitaxial GaMnAs nanowires,a change in resistance of more than 400 has been reported(although at low temperatures) (Wunderlich et al., 2001).This large signal allows the detection of DWs moving overdistances as short as 10 nm (Ravelosona et al., 2005).

The Hall effect is also observed for samples magnetizedin plane when the magnetization is not exactly parallel tothe current, as a consequence of the AMR. Although theeffect is quite small in metallic samples (Berger, 1991a;Sato et al., 2000; Gopalaswamy and Berger, 1991), a giantplanar Hall effect has been reported in GaMnAs filmsmagnetized in plane (Tang, Kawakami, Awschalom andRoukes, 2003) and this effect has been used to detecta single DW (Honolka et al., 2005; Tang et al., 2004;Tang, Masmanidis and Kawakami, 2004). Hall-effect-basedmeasurements require the device to be patterned in a Hall-cross geometry, thus limiting the flexibility of the devicestructure. Moreover, DWs can be strongly pinned at the cross,as shown by Wunderlich et al. (2001).

3.3.2 Optical measurements

The magneto-optical Kerr effect (MOKE) has been usedto probe DW motion (Lee et al., 2000; Cowburn, All-wood, Xiong and Cooke, 2002; Allwood et al., 2002a;Atkinson et al., 2003; Vernier et al., 2004; Beach et al.,2005, 2006; Yamanouchi, Chiba, Matsukura and Ohno,2004; Yamanouchi et al., 2006). However, using conven-tional optics, the spatial resolution of MOKE is limited bythe wavelength of the light used, typically to about 0.5 µm.

This makes it difficult to detect DWs directly. However, itis possible to detect the magnetization of magnetic domains,and thereby detect the presence and position of a DW innanowires even as narrow as 120 nm (Vernier et al., 2004).Even though the size of the light source is diffraction lim-ited, the motion of DWs can be detected over much shorterdistances, that is, within the light spot, when the signal tonoise ratio of the Kerr detection scheme is high enough,for example, by using pump-probe MOKE experiments. Themain advantages of MOKE detection are its application to awide range of magnetic materials and, perhaps most impor-tantly, the fact that it does not perturb the magnetic structure.

3.3.3 Magnetic microscopy techniques

High-resolution magnetic microscopy is extremely useful tounambiguously measure a DW’s structure. As discussed inparagraph 3.3.1, resistance measurements can be very sensi-tive to the DW structure, but can only be fully interpretedin conjunction with magnetic microscopy images. The Kerreffect can be used to image magnetic structures, either byscanning a beam of light across the sample or by using anoptical microscope. The latter has been used to study injec-tion and pinning of DWs in permalloy nanowires, as narrowas 500 nm wide (Yokoyama et al., 2000).

MFM is an elegant and simple technique, with a resolutionof ∼25 nm, which is good enough to well resolve thestructure of many DWs. However, it has very seriousdrawbacks. First, by contrast with the MOKE effect, MFMprobes not the magnetization but rather the dipole fieldsgenerated by inhomogeneities in the magnetization. Forsamples magnetized in plane these fields are approximatelyproportional to the divergence of the magnetization. Thismeans that it can be difficult to resolve the structure ofcomplex DW structures from an MFM image without the aidof micromagnetic simulations. Second and more importantis the interaction between the MFM magnetic tip and thesample, which is often large enough to perturb the magneticstructure of the sample. The field from the tip can both distortand cause transformations in the structure of DWs (Lacouret al., 2004) and can also drag DWs along a nanowireeven if the DWs are trapped at pinning centers. Clearly,it is preferable to use MFM magnetic tips with the lowestpossible magnetic moments, as discussed in several studies(Saitoh, Miyajima, Yamaoka and Tatara, 2004; Yamaguchiet al., 2004). Examples of MFM images of T and V DWstrapped at a notch are shown in Figure 13. Also shown aremaps of the divergence of the magnetization computed forthese DW structures using micromagnetic simulations, whichare in good agreement with the experimental measurements.Another example is shown in Figure 2 in which the MFMimage of a double-V structure is compared with the simulated


w = 600 nm w = 500 nm w = 400 nm

Figure 14. MFM images of a vortex DW in 40-nm-thick FeCoNinanowires with widths of 400, 500, and 600 nm.

structure. Figure 14 shows MFM images of V walls locatedin curved nanowires with different widths. In these cases thenanowires are relatively thick (40 nm), so that the vortex corepolarity can be resolved even though the core dimensions(∼5 nm radius) are much smaller than the experimentalresolution. Note that the core appears as the white signalin the middle of the V wall.

In homogeneous nanowires with smooth edges and sur-faces DWs can be moved in fields of just a few oersteds sothat even low moment MFM tips can destabilize such DWs.Even though a DW may appear stable in consecutive MFMimages, the structure observed may have been modified bythe proximity of the tip even before the first image was taken.Moreover, the signal measured with low moment MFM tipsis quite small, particularly for thin magnetic layers (<10 nm).MFM provides the most useful results for strongly pinnedDWs or DWs in nanowires with large propagation fields.

Scanning tunneling microscopy (STM) has atomic lateralresolution and does not perturb the magnetic structure. STMhas been used successfully to image the vortex core in Feislands (Wachowiak et al., 2002) and has potential for thestudy of DW motion.

Other magnetic imaging techniques include PEEM andvarious X-ray microscopy techniques, which are typicallymost useful when carried out at a synchrotron, which can pro-vide energy resolved photon fluxes of high intensity. PEEMis sensitive to the projection of the magnetization in the X-rayincidence plane, and has a spatial resolution below 100 nm.Examples of PEEM images of magnetic domain structuresare shown in Figures 3 and 15. V and T wall structurescan readily be identified. A particular advantage of PEEMis its elemental selectivity, which can be used to detect themagnetization of individual layers in multilayered structures.Other advantages are that PEEM is nonperturbative, and itis suitable for time-resolved measurements of magnetizationdynamics in submicron structures (Choe et al., 2004). PEEMcan be used to study samples grown on any substrate, pro-vided the capping layer is thin enough. However, PEEMis sensitive only to the surface of the sample (∼10–20 A),which can be both good and bad. For example, PEEM canmeasure very thin samples but, on the other hand, sur-face contamination can significantly affect the measurement.Since PEEM is an electron microscopy technique, imaging

in the presence of magnetic fields is not possible. Moreover,large voltages (15 to 20 kV) are required which risks elec-trical discharges, which can damage the sample. This meansthat it is also difficult to apply electric currents to the samplesin situ.

Transmission X-ray microscopy and scanning transmissionX-ray microscopy (STXM) can also be used for imagingmagnetic nanostructures both in the time and spatial domains(Puzic et al., 2005; Van Waeyenberge et al., 2006). Thespatial resolution is higher than that of PEEM and it ismuch easier to apply magnetic fields and electric currentto the sample. However, these techniques require specialsamples grown on thin transparent membranes. This can bean issue for current-driven DW motion studies, for which thedissipation of heat through the substrate from Joule heatingof the wire may be important.

Electron microscopy techniques are also suitable for imag-ing DWs in nanowires with high spatial resolution. Transmis-sion electron microscopy in the Lorentz mode has been usedsuccessfully to study magnetization processes in permalloyand Co nanowires with widths as small as 100 nm (Schrefl,Fidler, Kirk and Chapman, 1997). Spin-resolved scanningelectron microscopy (spin-SEM, or SEMPA) is also non-perturbative and has very high lateral resolution (∼20 nm).This technique has been used successfully to image theDW structure in nanoconstrictions (Jubert, Allenspach, andBischof, 2004) and the DW displacement induced by currentpulses in nanowires (Klaui et al., 2005b, Jubert et al., 2006).By combining high-resolution images of the two in-planecomponents of the magnetization, Klaui et al. (Klaui et al.,2005b) have shown the progressive distortion of a vortexwall and its eventual transformation into a transverse wallas a result of the injection of several current pulses. Notethat spin-SEM has the same drawbacks as other electronmicroscopy based techniques such as PEEM, for example thesurface sensitivity (such that capping layers cannot be used orneed to be removed in-situ) and the difficulty of using mag-netic fields while imaging. Recently, high-resolution electronholography was used to study the structure of DWs trappedin constrictions (Klaui et al., 2005a).

3.3.4 Time-resolved experiments

Time-resolved measurements are extremely useful to studythe dynamics of DW propagation. Measurements using theKerr effect (Atkinson et al., 2003; Beach et al., 2005), GMR(Ono et al., 1999a,b), and AMR (Hayashi et al., 2006a)have been reported. In all cases reported so far, reasonablesignal to noise ratios could only be achieved by signalaveraging over very many repeated measurements. However,with improved detection sensitivity all of these techniqueshave the potential for single-shot detection. Other techniques


offer the potential for time-resolved measurements of current-driven DW motion, for example, PEEM (Choe et al., 2004),STXM (Puzic et al., 2005), or inductive measurements(Silva, Lee, Crawford and Rogers, 1999). However, all thesemethods rely upon a pump-probe approach, which is ratherinconvenient for the measurement of irreversible processessuch as DW motion. Indeed, the sample must be reset toan identical initial state before each measurement, and themotion must be reasonably reproducible in order to obtainuseful results. An important technical point is that the samplereset must be relatively fast to collect data in a reasonableamount of time. This can be difficult if even modest fieldsare required for this process.

3.4 Field-driven domain-wall motion in magneticnanowires

Understanding and controlling the field-driven motion ofDWs in nanowires is an important preliminary step tostudying their current-driven dynamics.

The simplest way to create a DW in a magnetic nanowireis to first saturate its magnetization along its long axis andthen in a second step to apply a smaller magnetic field inthe opposite direction. The magnetization reversal processhas been studied both experimentally (Schrefl, Fidler, Kirkand Chapman, 1997) and numerically (Nakatani, Hayashi,Ono and Miyajima, 2001) in nanowires with square ends. Inthese cases the magnetization reversal originates from closuredomains at the ends of the nanowire, which expand as themagnetization rotates rotate toward the applied field. Thefield associated with the nucleation of the reversed domaincan be large, since it must overcome the shape anisotropy ofthe wire. This field decreases as the inverse of the wire widthand can reach several hundred oersteds in submicron-widepermalloy or Co nanowires (Shigeto et al., 2000).

Since the field required to nucleate a DW in a nanowireis generally much larger than the field needed to propagatea DW along the wire, once a DW is created it will beswept along the length of the wire. Different schemes canbe used to circumvent this problem and generate a singleDW in a nanowire whose field- and current-driven motioncan subsequently be studied. All these schemes enable theindependent control of the three critical steps of field-drivenDW motion (i) DW nucleation, (ii) DW injection into thenanowire, and (iii) DW propagation along the nanowire.These will be discussed in the following paragraphs.

3.4.1 Nucleation and injection of domain walls

Three methods have been proposed to nucleate DWs in softmagnetic nanowires.

Nucleation padShigeto, Shinjo and Ono (1999) first introduced the techniqueof using a nucleation pad attached to one end of a nanowireto generate a DW in the wire. The pad’s lateral dimensionsare much larger than the wire width, typically of the orderof or larger than 1 µm. Various pad shapes have been usedincluding squares, diamonds, and ellipses. Owing to the pad’ssize and shape, the field required to nucleate a DW insidethe pad is smaller than that needed to nucleate a DW withinthe nanowire itself. If the shape of the pad favors a flux-closed structure, a DW may reside in the pad at remanence.Cowburn et al. (Cowburn, Allwood, Xiong and Cooke, 2002)studied a 1-µm-sized square pad attached to a 100-nm-wide,5-nm-thick permalloy wire. DW propagation was probedby MOKE at different positions along the nanowire. Thereversal field of the nanowire alone, when terminated withsquare ends, exhibits a reversal field of ∼180 Oe. When thenucleation pad is attached to the nanowire, the reversal fieldis reduced significantly to ∼40 Oe. The same principle meansthat by shaping the end of the nanowire, for example toa point, so as to increase the local shape anisotropy, thenucleation field for a DW can be significantly increased(Schrefl, Fidler, Kirk and Chapman, 1997).

The nucleation of a DW in the pad does not guarantee thatit can be injected into the nanowire. The junction betweenthe wire and the pad is a local energy minimum, and amagnetic field is required to move the DW into the wire. Thisinjection field depends on the dimensions and shape of thenucleation pad and the nanowire. The injection field typicallydecreases for wider wires. However, it can be quite largefor wire widths below 500 nm. For example, in experimentsreported by Yokoyama et al. (2000), the injection field witha diamond-shaped nucleation pad is ∼60 Oe for a 500-nm-wide, 20-nm-thick permalloy wire. Without the nucleationpad, the injection field is slightly higher (about 85 Oe). For2-µm-wide nanowires, these fields are reduced to ∼25 and∼35 Oe, respectively.

The relatively high value of the injection field makes it dif-ficult to inject a DW into a nanowire without the DW movingalong the length of the nanowire. The DW will move alongthe nanowire when the field exceeds a critical propagationfield, which is determined by any defects in the nanowire,such as edge or surface roughness. For smooth wires, thepropagation field may be as small as a few oersteds, whichis much smaller than the injection field in most cases. There-fore, nucleation pads do not solve the problem of the con-trolled injection of a DW into the nanowire, unless there issignificant pinning along the nanowire. One way to solve thisproblem is to fabricate an artificial pinning center within thenanowire. The distinction between nucleation, injection, andpropagation fields is illustrated in Figure 15, in which PEEMimaging was used to study field-driven DW injection in a


Remanence

H > nucleation field

H > injection field

H > propagation field

Figure 15. PEEM images of the domain structure of a 20-nm-thick,250-nm-wide permalloy nanowire, measured at remanence afterapplication of magnetic fields of various amplitudes (Thomas et al.,2005). The elliptical pad is ∼1.7 µm long, 1 µm wide. (Reproducedfrom L. Thomas, C. Rettner, M. Hayashi, M.G. Samant, S.S.P.Parkin, A. Doran, and A. Scholl: ‘Observation of injection andpinning of domain walls in magnetic nanowires using photoemissionelectron microscopy’, Applied Physics Letters 87, (2005) copyright 2006 American Institute of Physics, with permission from theAIOP.)

permalloy nanowire (Thomas et al., 2005). Note that DWscan also be injected from a nucleation pad into a nanowirewithout an artificial pinning site if sufficiently short fieldpulses are used, such that the DW does not have time to fullypropagate along the nanowire (Beach et al., 2005). This isonly possible if the applied field can be changed on a shorttimescale determined by the DW propagation speed.

Nonlinear nanowire shapeThe second scheme for DW nucleation uses a shapednanowire whose remanent state after saturation along a par-ticular orientation of an external magnetic field exhibitsone or several DWs. Numerous shapes are possible includ-ing U- or L-shaped wires (Thomas et al., 2006; Yam-aguchi et al., 2004, 2006a), zig-zag wires (Klaui et al.,2005b; Taniyama, Nakatani, Namikawa and Yamazaki, 1999;Taniyama, Nakatani, Yakabe and Yamazaki, 2000), and rings(Rothman et al., 2001; Klaui et al., 2005b; Laufenberg et al.,2006b). An example of DW nucleation is shown in Figure 16for an L-shaped wire. In this case a saturation field, highenough to overcome the nanowire’s shape anisotropy, isapplied at ∼45 to the two straight legs of the nanowire. Themagnetization in both legs of the nanowire rotates toward thefield to reach an angle defined by the ratio of the externalfield and the shape anisotropy field. When the field is reducedto zero, the magnetization rotates back toward the wire’s axis(the easy direction of the shape anisotropy), so nucleating aDW near the bend. This method is attractive because it gen-erates a DW highly reproducibly, even in smooth nanowireswithout any pinning centers. However, its main drawback isthat it requires a large external field, big enough to overcomethe wire’s shape anisotropy.

(a)

(b)

Figure 16. Schematic of the field-induced injection of a DW intothe bend of a L-shaped nanowire by applying an external field at∼45 to the two straight legs of the nanowire. MFM measurementsat remanence after the injection of a DW into 20-nm-thick permalloynanowires with widths of (a) 300 and (b) 100 nm, respectively.Magnified images of the DWs are shown in the inset.

Nucleation lineThe third technique to nucleate a DW in nanowire uses a sep-arate conducting wire situated above or below and approx-imately perpendicular to the magnetic nanowire (Hayashiet al., 2006a,b; Himeno, Kasai and Ono, 2005; Himeno et al.,2005). A current is passed through this wire, which generatesa localized Oersted field which is used to reverse the magne-tization of a section of the nanowire immediately adjacent tothe conducting wire (see Figure 17). Thus two DWs are cre-ated. The Oersted field will cause the DWs to move along thenanowire away from the conducting wire a short distance tillthe Oersted field is decreased below the propagation field ofthe nanowire. This method is very attractive because it allowsthe local application of significant local fields of several hun-dred oersteds on the nanosecond timescale at a very highrepetition rate compared to an external electromagnet. On theother hand the current needed to nucleate the DW may bequite high, particularly for thick nanowires with square aspectratios, so that heating or electromigration may be an issue.

3.4.2 Domain-wall pinning and depinning from localpinning sites

Once a DW is nucleated and injected into a nanowire, it willmove along the wire if an applied magnetic field exceedsthe propagation field. The propagation field arises fromlocal pinning centers associated, for example, with roughnessor defects in the nanowire. For permalloy, it is strongly


−1.5 −1 −0.5 0 0.5 1 1.5Position (um)

Local field (Hl)

Total field

50 Ω

50 Ω

50 Ω

HSAT

HBIAS

HBIAS

Pulsegenerator

Pulsegenerator

Pulsegenerator

Figure 17. A schematic diagram showing DW injection using a local Oersted field generated in a nearby current lead. (Courtesy ofMasamitsu Hayashi.)

dependent on the quality of the samples, and can takevalues from a few (Faulkner et al., 2004) to tens of oersteds(Yamaguchi et al., 2004). Note that the propagation fieldtypically depends on the DW position along the nanowire.This means that the observed propagation field for motion ofthe DW over a given length will depend both on the initialDW position and this length.

Artificial pinning sites can be provided by fabricatingappropriately shaped local modifications of the nanowire.In soft nanowires, quite a variety of designs can be usedto successfully pin DWs. These include notches, humps,constrictions, or crosses, which lead to potential wells(attractive) or potential barriers (repulsive) of various widthsand depths. Pinning centers can also be provided by creatinglocal modifications of the nanowire using, for example, ionbombardment (Holleitner et al., 2004) or possibly localizedoxidation, localized thermal annealing treatments or by usingatomic force microscopes to physically or chemically modifythe nanowire (Schumacher et al., 2001). It is also possibleto use magnetic materials deposited under, on, or near thenanowire to change the magnetic properties of the nanowirein localized regions. One example is the use of magneticallyhard CoSm pads on permalloy nanowires (Nagahama, Mibuand Shinjo, 2000). DW pinning has been carried out byvarying the thickness of GaMnAs nanowires (Yamanouchi,Chiba, Matsukura and Ohno, 2004) and by using a Hall-cross geometry (Pt/Co/Pt layers by Wunderlich et al. 2001)for perpendicularly magnetized nanowires.

The depinning field from a symmetric V-shaped notchis roughly proportional to the notch depth (Yokoyamaet al., 2000; Himeno et al., 2003; Faulkner et al., 2004;Klaui et al., 2005a). For permalloy wires, depinning fieldscan reach several hundred oersteds. Values as high as∼1 kOe have been reported at low temperature in permalloynanorings (Klaui et al., 2003a) and in CoFe nanowires atroom temperature (Tsoi, Fontana and Parkin, 2003).

A very important point is that the depth of a pinningpotential depends on details of the DW’s structure and may,for example, be quite different for T and V walls and evenfor the same wall if it is of clockwise or counterclockwisechirality. Moreover, since the structure of a DW may evolvewith time, for example, because of forces on the DW fromcurrent or field, the pinning potential itself may changewith time and even during the transit time of a DW acrossthe potential. This makes the notion of DW pinning verycomplex!

Another point is that the depinning field of a DW froma notch (or protuberance) may depend on the directionof motion of a DW. For example, the propagation fieldof DWs in nanowires with highly asymmetrically shapedprotuberances (Himeno, Kasai and Ono, 2005; Allwood,Xiong and Cowburn, 2004) was found to differ by a factorof two depending on the direction of DW motion. However,even if the pinning site is symmetric, for example a V-shapednotch, the DW depinning fields to either side of the notchmay be quite different. This can be because the DW may not


1000 × 100 × 5 nm3 Transverse wall

Exchange energy

Magnetostatic energy

−2 × 10–5 −1 × 10–5 1 × 10–5 2 × 10–50

Total DW energy

−2 × 10–5 −1 × 10–5 1 × 10–5 2 × 10–50−1

−0.8

−0.6

−0.4

−0.2

0

0.2

Ene

rgy

(erg

cm

–2)

Wall center (cm) Wall center (cm)

Figure 18. Micromagnetic simulations of the energy landscape of a DW pinned at a notch.

reside in the center of the notch but rather to one side, asdiscussed later in more detail.

Micromagnetic calculations are very helpful in understand-ing the pinning of DWs in pinning sites such as notches (see,e.g., a symmetric notch in Figure 18). The pinning potentialis determined as follows. First, a stable DW structure is pre-pared at a specific location, either inside or outside the notch.This state is relaxed in zero field until equilibrium is reached.This equilibrium state is in general a metastable state. Sec-ond, a magnetic field is applied so that the DW moves alongthe nanowire, either away from or across the notch. Themagnitude of the field is chosen to be large enough to over-come the pinning potential but is as small as possible so asto ensure that the DW follows the lowest energy trajectory.The latter is encouraged by using a large damping constant(e.g., α = 1). The potential energy landscape for the DW isthen obtained from the total energy of the system minus theZeeman energy. The DW position is then calculated fromthe net longitudinal magnetization of the nanowire. In gen-eral, boundary conditions are used to fix the direction of themagnetization of the nanowire at either end along the longaxis of the nanowire. Thus the DW energy will be modifiedwhen too close to the ends of the nanowire. For the exampleshown in Figure 18 the nanowire has a width and thicknessof 100 and 5 nm, respectively, so that the stable state of theDW in the nanowire is a T wall. There are two minima inthe pinning potential at either side of the notch. By consid-ering the contributions of the exchange and magnetostaticenergies to the total potential, it is clear that the DW pinningis associated with two main factors (i) the reduction of theDW volume within the notch (i.e., due to the reduced widthof the nanowire) and (ii) the reduction of the DW energywhen the nanowire’s edges are not parallel to the nanowire’slong axis. The exchange energy is minimized when the DWvolume is reduced. However, the magnetostatic energy plays

the more dominant role and, in particular, accounts for theenergy minima when the DW sits within either of the twowedged portions of the nanowire, which define the edges ofthe notch. It should be emphasized that the magnetostaticenergy is the dominant contribution to the energy for both Vand T head-to-head DWs, accounting for ∼80–90% of thewall’s energy (McMichael and Donahue, 1997). Thus, thepinning potential is dominated by magnetostatic rather thanvolumetric effects in the head-to-head DW case, in contrastto the theoretical prediction for a Bloch wall trapped at ananconstriction (Bruno, 1999).

The pinning potential becomes more complex for thickerwires, as illustrated in Figure 19. The nanowire width andnotch shape are the same as before, except that the nanowire

−2 × 10–5 −1 × 10–5 1 × 10–5 2 × 10–50−10

−8

−6

−4

−2

0

2

1000 × 100 × 20 nm3

Vortex or transverse wall

Ene

rgy

(erg

cm

–2)

Wall center (cm)

Total DW energy

Figure 19. Micromagnetic calculations of the energies of a V(medium gray) and a T (dark gray) DW pinned at a notch of thesame form as that shown in Figure 18. The black curve shows theenergy of the DW when it transforms from a V to a T wall as itenters the notch.


thickness is increased from 5 to 20 nm. In this case, the stablestate outside the notch is a V wall. In these simulations, theinitial state was either a V or a T wall centered in the notchand the DW was moved away from the notch to the leftor right with magnetic fields to calculate the potential. Thepinning potential exhibits the same features as for the thinnerwire, with two minima at either side of the notch, for both theV and T DWs. However, there is an additional shallow localminimum in the middle of the notch for the V wall. Thisexample is interesting because the structure of the DW withthe lower energy is different within (T wall) and outside thenotch (V wall). This leads to a change of the wall structurewhen the V wall moves across the notch, as illustrated bythe black curve in Figure 19. The energy barrier for thistransformation is indeed smaller than that needed to movethe vortex wall across the notch.

It is quite typical that there are significant variationsin the DW pinning and depinning processes in repeatedexperiments, leading to, for example, variations in depinningfields. As discussed earlier, one reason is that there maybe several metastable structures of a DW trapped at anotch, each with a different pinning potential (Hayashiet al., 2006b). A particularly interesting example is shownin Figure 20 for the case of a DW injected and trapped at anartificial pinning site in a permalloy nanowire, 10 nm thickand 300 nm wide. Four distinct DW states are found, whose

structures, as determined from MFM images, correspond totwo V walls and two T walls, one of each with a clockwiseand the other with a counterclockwise chirality. Note thatthe DWs are pinned at the left side of the notch in theimages because they were injected from the left side. Each ofthese DWs is a metastable state with a corresponding pinningpotential and associated depinning fields, which can be wellreproduced by micromagnetic calculations. Interestingly,even though the notch is symmetric, the pinning potentialis quite asymmetric for one chirality of the T and V DW.Similar results have been reported by Miyake et al. (2005)for a nanocontact much narrower (∼15 nm) than the exampleshown in Figure 20.

3.4.3 Time-resolved experiments

Field-driven DW dynamics were studied in detail in the1970s with regard to magnetic bubble memory applications,as extensively reviewed by Malozemoff and Slonczewski(1979). As mentioned in paragraph 2.2.3, a 1D theory ofDW motion was developed to account for Bloch-like DWs.Many of the results of this theory, however, also apply tomore complex DW structures.

In an ideal situation without pinning, the motion of aDW along a magnetic nanowire has distinct characteristicsdepending on the strength of the applied field. The motion

Ene

rgy

R (

Ω)

R (

Ω)

R (

Ω)

R (

Ω)

Field (Oe)Position

0 40 80 120−40−80−120−0.5 0

00.10.20.30.4

Ene

rgy

00.10.20.30.4

Ene

rgy

00.10.20.30.4

Ene

rgy

00.10.20.30.4

0.5

Micromagnetic calculation

Figure 20. Measured depinning fields from a notch for DWs with metastable T and V structures, in a 10-nm-thick, 300-nm-wide permalloynanowire. Also shown are the corresponding energy landscapes calculated from micromagnetic simulations. (Reproduced from MasamitsuHayashi, Luc Thomas, Charles Rettner, Rai Moriya, Xin Jiang, and Stuart S.P. Parkin, Phys. Rev. Lett. 97, 207205 (2006), copyright bythe American Physical Society, with permission from APS.)


can be described analytically within the 1D model. In a firstregime, at low fields, the DW velocity increases at short timesto attain a steady value. In this stationary regime, the velocityincreases linearly with field, according to the relation:

v(H) = γ

αH (21)

When the field exceeds a certain threshold value, the so-called Walker breakdown field, the average DW velocitydrops sharply. Above this field, the DW moves in a complexmanner, and its velocity oscillates in time. When the field isincreased further, the average velocity is again proportionalto the field, although the wall mobility is strongly reducedcompared to the low-field value. The velocity in this secondregime depends on field according to the relation:

v(H) = γ

α + α−1H (22)

When pinning is included, the DW only moves alongthe nanowire when the field exceeds a critical value Hc.Depending on the nature of the potential (whether it isconservative or nonconservative), the DW velocity can bewritten (Malozemoff and Slonczewski, 1979; Baldwin andCuller, 1969), for |H | > Hc, as:

v(H) ∝ (H 2 − H 2c )

H

or v(H) ∝ H − Hc respectively (23)

Only a few experimental studies have explored the time-resolved motion of DWs in nanowires. Ono et al. (1999b)have used GMR measurements to probe the propagation of aDW in a trilayer structure NiFe (40 nm)/Cu (20 nm)/NiFe(5 nm). The nanowire was 500 nm wide and extremelylong (2 mm). The DW motion in the 40-nm-thick layerwas probed on the microsecond timescale, for magneticfields between 100 and 150 Oe. The DW velocity wasfound to increase linearly with field above a threshold field(∼35–40 Oe). The wall velocity v was between 150 and250 m s−1, corresponding to wall mobilities v/H of about2.6 m s−1 Oe−1. Atkinson et al. (2003) performed a similarstudy on a single layer permalloy nanowire (5 nm thick,200 nm wide) using time-resolved MOKE experiments. Thewire was much shorter (about 50 µm) than that used in thestudy of Ono et al. and the DW motion was probed on muchshorter timescales (20–500 ns). These authors also foundthat the DW velocity increased linearly with field for fieldsabove ∼25 Oe. However, the DW mobility was more than10 times higher (∼38 m s−1 Oe−1). These two experimentswere performed in relatively high fields, compared to theexpected Walker breakdown fields. Moreover, in both cases

the propagation fields were large. Nevertheless, these authorsinterpreted their data using the theory in the low-field regime,(equation (21)) in which the velocity is proportional to field.They deduced extraordinarily high values of the Gilbertdamping constant, namely, α = 0.63 (Ono et al., 1999b) andα = 0.053 (Atkinson et al., 2003), in both cases much highervalues than those found in thin permalloy films (0.01–0.02)(Nibarger, Lopusnik and Silva, 2003; Nibarger, Lopusnik,Celinski and Silva, 2003).

Recent experiments by Beach et al. (2005) and our work(Hayashi et al., 2006a) suggest that these surprisingly highdamping constants (i.e., small DW mobilities) more likelyindicate DW motion in the high-field regime above theWalker threshold field, in which equation (21) is not valid.Indeed, in the high-field regime, the velocity also increaseslinearly with field, but is lowered by a factor α2 comparedto the low-field regime (see equations (21) and (22)).Clear evidence of the Walker breakdown was observed inthe work of both Beach et al. (2005) and Hayashi et al.(2006a), with a field dependence of the velocity following thetextbook behavior (as shown, e.g., in Figure 22). The Walkerbreakdown field was found to be only a few oersteds, andthe data are consistent with reasonable values of the Gilbertdamping constant. Note that we have recently reported theexperimental observation of periodic oscillations of the DWstructure, as predicted above the Walker breakdown (Hayashiet al., 2007b).

The experiments discussed above address the ballistic(or viscous) regime for DW motion, for which the DWdynamics can be described by the Landau–Lifshitz–Gilbertequation at zero temperature. Thermally activated processesallow for DW motion below the critical field Hc. Thishas been observed in nanowires made with thin Pt/Co/Pttrilayers and two regimes have been identified. The thermallyactivated regime is characterized by a wall velocity writtenas (Wunderlich et al., 2001; Ravelosona et al., 2005):

v(H, T ) = v0 exp

(−2MsV (Hc − H)

kBT

)(24)

where kB is the Boltzmann constant and T the temperature,H and Hc are the applied and propagation fields, respec-tively, and V is the activation volume. In the creep regime,the DW velocity is written as (Cayssol et al., 2004):

v(H, T ) = v0 exp

(−Uc(Hc/H)1/4

kBT

)(25)

with Uc and Hc are the scaling energy and the critical field,respectively. In this creep regime, the wall velocity wasfound to vary as the inverse of the wire width, and to decreasefor larger wire roughness (Cayssol et al., 2004). Both regimesare characterized by very small DW velocities (1 m s−1).


3.5 Current-induced domain-wall motion

3.5.1 Experimental considerations

The first studies of current-driven DW motion were per-formed by Berger et al. in extended permalloy films. In thesestudies (and also in the more recent work of Gan et al.(2000)), the DWs have 180 Bloch, Neel, or cross-tie DWstructures, the current flows in the direction perpendicularboth to the wall and the magnetization within the domains,and there is no pinning besides the local defects responsi-ble for the film’s coercivity. Several DWs are probed at thesame time, in contrast to recent experiments on magneticnanowires, which probe single DWs. For nanowire sampleswith perpendicular anisotropy, the current is also perpen-dicular to both the wall and to the domain magnetizationdirection, as in extended films. However, for head-to-headDWs in nanowires, the current is perpendicular to the wallbut parallel to the magnetization in the domains. These dif-ferent geometries must be taken into account in experiments,which probe the current-induced motion of DWs since, forexample, the role of the self-field (Oersted field) and thehydromagnetic domain drag are different.

Current-induced DW experimental studies can be sepa-rated into two broad categories, current-induced DW propa-gation and current-induced DW depinning. In the first cate-gory, the position of the DW in a nanowire is probed beforeand after the application of the current. Thus, the criticalcurrent for DW motion and the DW displacement directionand distance can be inferred. When current pulses are used,the DW velocity can also be determined. In the second cate-gory, the field to depin DW is measured as a function of theapplied current.

3.5.2 Current-induced domain-wall motion in zerofield

From an experimental viewpoint, current-induced DW motionoccurs above a critical current density Jc. As mentionedearlier, this is an important parameter both for comparingexperiment with theory and for potential technological appli-cations. It is unfortunate that the magnitude of Jc is oftenambiguous, making comparison between data and theory dif-ficult.

1. Uncertainties with regard to the definition of the criticalcurrent(a) What is the experimental definition of the critical

current?The experimental definition of the critical currentdepends on the type of measurement and the sen-sitivity of the experimental technique. Indeed, thecritical current for DW motion over micron-sized

distances can be quite different from that needed tochange the DW structure within a notch or movea DW over very short distances between neighbor-ing pinning sites. For example, in the early work bySalhi and Berger (1994), wall motion was detectedby Kerr microscopy, such that ‘the critical currentdensity is determined by finding the smallest valuefor which a sequence of 600 pulses produces adetectable wall displacement’. Given the resolutionof the setup, this corresponds to an overall displace-ment of about 12 µm (or 20 nm/pulse, assuming allpulses induce the same DW displacement, which isfar from certain). By contrast, in a recent report byRavelosona et al. (2005), DW motion over a fewtens of nanometers can be detected directly in aHall cross, owing to the high sensitivity of the AHEeffect. In summary, the experimental definition ofthe critical current is as vague as ‘the minimumcurrent for some measurable change to take place’.

(b) Joule heatingThe current densities used in most experiments arelarge (see Table 1) and Joule heating is significant.This temperature increase can play a significant rolein current-induced DW dynamics, as discussed later.Here we only address experimental consequencesof the heating of the nanowire. In particular, cur-rent pulse generators often deliver constant voltagepulses rather than constant current pulses so thatthe current delivered depends on the load resis-tance. While, for dc or long current pulses this iseasily corrected by directly measuring the actualcurrent delivered to the nanowire, this correctionbecomes more difficult for short current pulses. Thetemperature of a nanowire can increase by manyhundreds of degrees on the nanosecond timescaledepending on the amount of Joule heating. This isobviously more significant the larger the current andthe higher the resistivity of the nanowire (and theelectrical contacts to the nanowire). Thus, the resis-tivity of permalloy nanowires, for example, caneasily increase by a factor of 2 or 3 in 1 or 2 ns,and similarly decrease on very short timescales sothat these temperature changes cannot be detectedby dc measurements after the current pulses areapplied. Thus, it is very important to carry out time-resolved resistance measurements. The change inthe nanowire resistance due to heating can lead toincorrect values of critical currents for DW motion,if it is not properly taken into account. For example,this effect accounts for a reduction of the criticalcurrent density by a factor of 2 between the valuespublished by Yamaguchi et al. (2004, 2005).


Table 1. Compilation of values of the critical current density for DW motion in various nanowires. Note that some experiments areperformed at low temperature.

Thickness Width Wall type Current Type of Critical currentMaterial (nm) (nm) (dc or pulse length) experiment Jc(107 A cm−2) References

NiFe 28–45 3.5 mm 180 NW <2 µs P 1–1.4 (nc) Freitas and Berger (1985)NiFe 14 < t< 86 3 mm 180 1 µs P 2.3–360 (nc) Hung and Berger (1988)NiFe 120 <t < 740 3.5 mm 180 50–300 ns P 0.01–1 (nc) Salhi and Berger (1994)NiFe 102–161 20 µm 180 BW 10 ns (rt) P 2.5–4.3 (nc) Gan et al. (2000)NiFe 40 200 HH dc S 1 (with field) Kimura et al. (2003)NiFe 34 400 HH dc S 7 (with field) Klaui et al. (2003b)NiFe 5 120 HH dc S 7 Vernier et al. (2004)NiFe 30 <1 µm HH dc S 1.1 Lepadatu and Xu (2004)Ni 30 2.2NiFe 10 240 HH 0.3–1 µs P 11 (nc) Yamaguchi et al. (2004)

6.7 (c) Yamaguchi et al. (2006b)NiFe 40 300 HH 20 ns (rt) P 7.5 (nc) Florez, Krafft and Gomez

(2005)NiFe 5–35 200 HH 20 µs S 05–13 (nc) Klaui et al. (2005c)NiFe 10 500 HH 10 µs P 22 (nc) Klaui et al. (2005b)CoFe 10 100 HH dc S 1 (with field) Tsoi, Fontana and Parkin

(2003)NiFe 10–113 110–275 HH 5 µs P 3.9–7.7 (c) Yamaguchi et al. (2006a)NiFe 35 110 HH 10 µs S 27.5 (c) Laufenberg et al. (2006b)NiFe 10 300 HH 10–100 ns P 10 (c) Hayashi et al. (2007a)NiFe 10 300 HH 4 ns S 30 (c) Hayashi et al. (2006b)SV 1000 HH dc P–S 2.6 Grollier et al. (2002)SV 10 300 HH dc P–S <0.8 Grollier et al. (2003)SV 5 300 HH 0.4–2 ns P–S <0.8 (full sw) Lim et al. (2004)

<0.27 (2 µm)MnGaAs 17–25 20 000 180 BW 100 ms P 0.008 Yamanouchi, Chiba, Mat-

sukura and Ohno (2004)Pt/Co/Pt 0.5 200 180 BW dc P <1 Ravelosona et al. (2005)

P: DW propagation experiment; S: switching experiment; dc: dc current; rt: rise time of the current pulse for exponentially shaped pulses; c/nc: valuescorrected/ not corrected for resistance change induced by Joule heating.

There is another more subtle consequence of theresistance increase during the application of cur-rent pulses to a nanowire. Indeed, both the rateof change and the magnitude of the temperaturerise from Joule heating depend strongly on thethermal conductivity of the nanowire to the sub-strate. The timescale of the resistance increase canvary by 1 or 2 orders of magnitude dependingon the thermal conductivity of the seed layers onwhich the nanowire is prepared, the quality ofthe nanowire/seed layer interface and the substratematerial. For short pulses compared to this ther-mal timescale, the current becomes time dependent.Therefore, the actual value of the critical current forDW motion can depend on the relative timescalesof heating and that of the DW dynamics.

(c) Inhomogeneous current flowFor nanowires with varying width or for multilay-ered structures, such as spin valves, the definitionof the current density is ambiguous. In the case of

nanowires with notches, the current density reachesits maximum value in the narrowest part of the wire.However, in many cases, the DW is not trapped inthe middle of the notch, but rather to one side ofthe notch, so that it is not clear which nanowirewidth should be used to calculate the critical cur-rent density. For the case of spin-valve stacks, alarge fraction of the current flows in the nonmag-netic spacer. The structure studied by Grollier et al.is one example (Grollier et al., 2003). The spin-valve stack used was composed of CoO (3)/Co(7)/Cu (10)/NiFe (5)/Au (3), where the thicknessesare given in nanometers and DW motion was stud-ied in the NiFe layer. If the current density isassumed to be uniform throughout all the layersthen the critical current density reported by Grollieret al. in zero field was ∼8 × 106 A cm−2. On theother hand, if one assumes reasonable values of theconductivity of each layer, the critical current den-sity in the NiFe layer is only ∼1.8 × 106 A cm−2,


which is nearly five times smaller! It is interestingto note that, in either case these critical current den-sities are surprisingly low compared to other studiesof current-induced DW motion. Similarly, when amagnetic nanowire is prepared on a seed layer (e.g.,for improved film quality) or with a cap layer (e.g.,to prevent oxidation or to aid in the fabrication ofnanowire devices), some of the current may flowthrough these layers and affect the determinationof the critical current density in the magnetic layeritself.

(d) Parasitic magnetic fields and self-fieldThe field dependence of the critical current forDW motion will be discussed in more detail inSection 3.5.4. However, for weakly pinned DWsand nanowires with small propagation fields,parasitic magnetic fields such as the remanent fieldof an electromagnet, or the self-field from currentpassing through the electrical leads to the samplecan easily influence critical current measurements.The self-field from the current passing through themagnetic nanowire itself can also play a significantrole. For straight nanowires, its influence is oftenneglected because there is no net field along thenanowire direction, assuming uniform current flow.However, there will be strong highly nonuniformcomponents of the currents self-field in directionstransverse to the nanowire. Even though the averagevalue should be zero in the ideal case, it is diffi-cult to rule out ‘wall streaming’ effects, which weredescribed by Berger in several publications (Salhiand Berger, 1994; Berger, 1992): the local inhomo-geneous fields associated with a current pulse canshake the DW significantly and assist its motion byeffectively reducing the local pinning. In any case,for inhomogeneous current flows, for example, inspin-valve stacks, as mentioned above, there willbe a net nonzero contribution of the Oersted field inthe direction transverse to the nanowire within themagnetic layer in which the DW is moved.

2. Critical current density in zero fieldValues of the critical current density measured in zerofields for various magnetic nanowires are complied inTable 1. Most of these data were obtained for permalloynanowires. Very few experimental results have beenreported for other materials. It is interesting that muchsmaller current densities have been reported in nanowiresformed from GaMnAs, Pt/Co/Pt and spin-valve-basedstructures than for permalloy nanowires, although all ofthese experiments are subject to detailed interpretation.Results reported for permalloy nanowires by differ-ent groups vary significantly, perhaps not surprising

1 10 100 1000

Thickness (nm)

103

104

105

106

107

108

109

J c (

A c

m−2

)

Salhi 94Freitas 85Gan 00Yamaguchi 04Florez 05Klaui 05

Klaui 05Vernier 04Parkin (unpublished)Yamaguchi 06Laufenberg 06Hayashi 06

Figure 21. Critical current for DW motion in permalloy structuresreported in the literature as listed in Table 1, plotted as a functionof the sample thickness.

considering the wide variety of techniques and geome-tries used in these studies and the uncertainty in themeasurement of Jc. Some general conclusions can bedrawn from these data, which are plotted in Figure 21as a function of the sample thickness:(a) Shorter current pulses allow the use of higher cur-

rent densities before breakdown of the nanowire.The dependence of the critical current densityon pulse length has been investigated by severalgroups. Pulse lengths between 50 and 300 ns wereexplored in Salhi and Berger (1994) and between0.4 and 2 ns in Lim et al. (2004). In both cases, nosignificant variation in Jc was found.

(b) The wire width does not seem to play a significantrole. In particular, 180 DWs in large structuresand head-to-head walls in narrow wires exhibit verysimilar critical current densities. However, in allthese experiments, the aspect ratio of the nanowire’scross section is about the same, the width alwaysbeing much larger than the thickness.

(c) The wire thickness seems to play a more importantrole. For films thicker than about 50 nm, Salhi andBerger (1994) have observed that Jc decreases as1/t2 (in the 120–740 nm range). This dependenceis explained by current-induced distortions of theBloch wall. For wires thinner than about 40 nm, Jc

is between 5 and 30 × 107 A cm−2, if the currentis corrected for the resistance increase during thepulse. There is no clear dependence of Jc in thisthickness range when considering all the availableexperimental data. However, Klaui et al. (2005c)have reported a significant increase, from 5.0 ×107 A cm−2 for 5-nm-thick to 1.3 × 108 A cm−2 for35-nm-thick permalloy nanowires. Note that theseexperiments were performed at low temperature.


(d) There is no clear correlation between Jc and the wallstructure. In particular, head-to-head DWs in narrowwires and 180 DWs in large structures exhibit verysimilar values.

(e) Surprisingly, the magnitude of the DW pinning doesnot appear to play any significant role on the crit-ical current density. Data summarized in Figure 21show results for both strongly pinned DWs (withdepinning fields up to 200 Oe) and extended films,with propagation fields of a few oersteds. Differenttheories of DW motion arrive at different conclu-sions as to whether Jc is intrinsic or extrinsic. Todate experimental results seem rather inconclusive.Indeed, there have been two apparently contradic-tory attempts to find a scaling rule for differentmagnetic nanowire samples. In the first case (Yam-aguchi et al., 2006a), Ono’s group plot Jc versusthe transverse anisotropy times the DW width Hk

for permalloy nanowires. A roughly linear varia-tion is observed, as given by equation (15) whichdescribes an intrinsic critical current, in the pres-ence of adiabatic spin torque, without a β term.By contrast, Berger presents a different scaling law(Berger, 2006), in which the DW mobility is plottedversus the aspect ratio of the wire (itself a functionof the DW width) and includes different magneticmaterials. His conclusion is that exchange drag (i.e.,momentum transfer or nonadiabatic spin torque) canlargely account for the various results, which meansthat the critical current is in fact extrinsic (i.e., onlyrelated to coercivity or pinning).

3. DW propagation direction and distanceExperimental results for samples made with 3d metals(NiFe, CoFe, Ni, Pt/Co/Pt) show consistently that DWsmove along the electron flow (or opposite to the currentdirection) at zero or low fields, in agreement with thespin-torque model. The direction of DW motion is alsoopposite to the current direction for GaMnAs, eventhough the carriers in this system are holes (Yamanouchi,Chiba, Matsukura and Ohno, 2004). Two other importantexperimental findings agree with spin-torque models.First, the DW motion direction reverses with the currentpolarity. Second, head-to-head and tail-to-tail DWs movein the same direction with current, so that simple field-driven mechanisms can be ruled out.

Even though the overall propagation is along the elec-tron flow, the propagation distance can vary significantly,both for a single wall over repeated experiments and forseveral walls during the same current excitation. This wasfirst reported by Berger, who observed that only a sub-set of walls in the field of view of a Kerr microscopemoved during a current pulse (Freitas and Berger, 1985).

This behavior was explained by local variations of thecoercivity and/or the presence of a parasitic field. Ganet al. (2000) also observed significant variations of wallmotion in large structures (20 µm wide). The distributionof the DW propagation distance was significantly differ-ent, and in some cases, DWs moved backwards (oppositeto the electron flow). Moreover, portions of the same wallsometimes moved in opposite directions.

Experiments on current-induced DW motion in nano-wires exhibit similarly strong variations. For example,Yamaguchi et al. (2004) report that five identical currentpulses applied to a V wall result in motion of the DWof 2.2, 0.9, 0, 1.9, and 1.1 µm, respectively. In ourown experiments we have also observed wide variationsin DW motion direction and distance in many cases.Occasionally, DWs are found to move opposite to theelectron flow, particularly when DWs are depinned froma notch.

These fluctuations could be related to local variationsof the DW pinning strength, as was invoked for largerstructures. However, DWs can be moved repeatedlyacross exactly the same position in a nanowire and thus,it should be possible to correlate the DW displacementinduced by a current pulse with position in the nanowire.There is no evidence of such an effect in Yamaguchi’spaper. Another possibility is that the DW structure mayevolve during motion, as was observed in experimentsby (Klaui et al., 2005b; Jubert et al., 2006), who reporta gradual decrease of the displacement of a V DWwhen consecutive pulses are injected across it. TheDW eventually stops after a few pulses. High-resolutionimaging with SEMPA revealed that the DW structure wasmodified by the current pulses, which caused the DW toevolve from a V to a T wall structure. Such a systematicvariation of the DW displacement with successive pulseshas not been observed in other studies (Yamaguchi et al.,2004; Gan et al., 2000).

Many other results are not fully understood. Forexample, Lim et al. (2004) have observed a reversal ofthe DW propagation direction when the current increasesabove some threshold value.

3.5.3 Domain-wall velocity

The first measurements of current-driven DW velocity wereobtained by simply dividing the observed DW displacementafter the current pulsed had been applied by the current pulselength, for microsecond long current pulses. However asdiscussed in the preceding text, the DW displacement oftenexhibits significant fluctuations over repeated experiments. Itis unclear whether such estimates of the DW velocity aremeaningful.


In several reports, it was shown that the DW displace-ment scales approximately with the pulse length (Yamaguchiet al., 2004; Klaui et al., 2005b; Jubert et al., 2006) givinga more reliable measurement of the wall velocity. In bothstudies, the current density was slightly higher than Jc andlong pulses were used, between 0.3 and 1 µs in Yamaguchiet al. (2004), and 1 order of magnitude larger (10 µs) in Yam-aguchi et al. (2004) and Klaui et al. (2005b). The measuredvelocities were very small, of the order of 3–5 m s−1 and0.1–0.5 m s−1, respectively.

Interestingly, DW velocities estimated by dividing theDW displacement by the current pulse length are muchhigher when short pulses (nanoseconds long) are used. Forexample, Lim et al. (2004) reported switching a spin-valvenanowire in zero field with pulses as short as 0.4 ns andconcluded that the DW moves >20 µm in less than 1 ns. Thecorresponding DW velocity of about 20 km s−1 seems muchtoo high to be meaningful. It may suggest a more complexmechanism of DW motion, perhaps involving DW motionlong after the end of the current pulse. Such overshootingand a DW inertia effect have been observed in the contextof field-driven motion of magnetic bubbles (Malozemoff andSlonczewski, 1979). We have also performed experimentsusing pulse lengths between 0.5 and 100 ns. In some cases,we have observed DW displacements of ∼1 µm for currentpulse lengths of 1 ns, with an associated ‘velocity’ of up to1000 m s−1. Importantly, we do not observe any clear linearscaling of the DW displacement with the pulse length forthese long pulses (note that the data vary widely). This alsosuggests that the mechanism of DW motion is more complexfor short excitations.

Recently, the DW velocity in permalloy nanowires hasbeen measured more accurately by time-resolved measure-ments, in the presence of a combination of field and current(Hayashi et al., 2006a; Beach et al., 2006) as well as cur-rent alone (Hayashi et al., 2007a). Both measurements were’time-of-flight’ experiments, in which the velocity is obtainedby dividing a distance traveled by the DW by the time theDW takes to move between two points along the nanowire.In both cases, data were averaged over a large number of rep-etitions. The distance was between different positions of thelaser spot in the Kerr measurements by Beach et al. (2006)and between two electrical contacts in the AMR studies byHayashi et al. (2006a, 2007a).

In the presence of field and current, the DW velocity isaffected differently depending on the relative magnitude ofthe field and current and the current polarity (Hayashi et al.,2006a) (see Figure 22). When the electron flow is alongthe direction of the field-driven motion, the DW velocityis enhanced by up to ∼100 m s−1. The Walker breakdownfield is not significantly affected, at least not for the currentdensities investigated. By contrast, when the electron flow

00

100

200

300

400

20 40 60 80

−2.8 V

2.8 V

Field (Oe)

DW

vel

ocity

(m

s–1

)

Figure 22. DW velocity determined by time-resolved AMR mea-surements in a 10-nm-thick, 200-nm-wide permalloy nanowire, inthe presence of current flowing in both directions. The currentdensity is ∼1.4 × 108 A cm−2. (Reproduced from figure 2(c) inM. Hayashi, L. Thomas, C. Rettner, R. Moriya, and S.S.P. Parkin,‘Direct observation of the coherent precession of magnetic domainwalls propagating along permalloy nanowires’, Nature Physics 3,21 (2007).)

is opposite to the direction of field-driven motion, the DWvelocity is reduced. The DW motion can even be stopped (orslowed below the experimental detection limit) – when thefield is smaller than ∼25 Oe (2.5 times the Walker breakdownfield). Note that in the case of a moving DW, there does notseem to be a critical current: the velocity varies continuouslywith the current density, and even small currents influencethe DW velocity.

DW motion driven by current only was also reported. Inthis case, the DW motion was only detected for currentdensities higher than ∼7–10 × 107 A cm−2. Motion wasalong the electron flow. The DW velocity increases withthe current, from ∼80 m s−1 at the threshold current up toabout 150 m s−1. Note that the DW velocity saturates forthe highest current densities investigated, although the originof this effect is not clear. Interestingly, such high velocities(more than 2 orders of magnitude larger than those reportedby Yamaguchi and Klaui) approach and may even exceedthe rate of spin angular momentum transfer given by theparameter u (equation (15)) (Hayashi et al., 2007b).

Finally, we briefly mention experiments performed on thecurrent-induced motion of DWs in GaMnAs nanowire struc-tures by Ohno’s group (Yamanouchi, Chiba, Matsukura andOhno, 2004; Yamanouchi et al., 2006) (similar data werealso published recently by Tang, Kawakami, Awschalom andRoukes, 2006) but only in combination with a magneticfield). Ohno et al. report DW motion, driven by current only,with DW velocities spanning more than 5 orders of magni-tude up to 22 m s−1. Two different regimes are observed: theDW velocity increases roughly linearly with the current den-sity for currents above a threshold current (between 3.5 and5.5 × 105 A cm−2 depending upon the temperature). The data


are consistent with the spin-transfer efficiency expected fromequation (15) (albeit slightly larger), once the temperaturedependence of the magnetization and the spin polarizationare taken into account. A second subthreshold regime isobserved, in which the DW velocities are between 10−5 and1 m s−1, which is interpreted as DW creep.

3.5.4 Field dependence of the critical current

The field dependence of the critical current has been studiedby several groups. In most of these experiments, the DW isfirst pinned at either a defect (Thomas et al., 2006; Laufen-berg et al., 2006b; Grollier et al., 2004; Lim et al., 2004),or a notch (Florez, Krafft and Gomez, 2005; Hayashi et al.,2006b; Klaui et al., 2005c) or at a cross (Ravelosona et al.,2005). In one report (Vernier et al., 2004), a quite differentexperimental method was used: the propagation field alonga nanowire is measured while dc currents of both polaritiesare applied. All these studies show some consistent trends:

• The critical current for DW depinning decreases whenthe field is increased.

• DW motion occurs along the field direction, except forvery small fields.

• The polarity dependence of the depinning current isweak. The strongest effect of the current is a reduction ofthe depinning field independent of the current polarity.

• Both linear and nonlinear field dependences of thecritical current have been reported.

We have studied the field dependence of the criticalcurrent for depinning a DW from a notched pinning site,with widely varying pinning strengths, in NiFe wires withvarious different dimensions (see e.g., Hayashi et al., 2006b).Our results show very little dependence on the currentpolarity, in agreement with other studies. In most cases,the critical current exhibits a strongly nonlinear variationwith field. We observe two distinct regimes. In a low-fieldregime, the critical current depends only weakly on the notchpinning strength (as measured by the depinning field) and themagnetic field. By contrast, in a high-field regime, the fielddependence is much stronger. An illustration of these tworegimes is shown in Figure 23, for permalloy nanowires withdifferent notches. Note that for these experiments, two DWswere nucleated on either side of the wire. Current-drivendepinning most likely causes annihilation of these two DWs.

3.5.5 Subcritical domain-wall transformations andmetastability

The role of metastable DW states was first noted by Berger(1992). He pointed out that in some thickness range, Bloch

0 10 20 30 40 50 60

Magnetic field (Oe)

0

5

10

Crit

ical

cur

rent

l sw (

mA

)

Figure 23. Field dependence of the critical dc current for depinningDWs in 18-nm-thick, 315-nm-wide, permalloy nanowires.

∼3 × 108 A cm−2/7 ns ∼1 × 108 A cm−2/5–10 µs

(a) (b)

Figure 24. DW transformations measured by MFM in 20-nm-thick,300-nm-wide permalloy nanowires upon successive application ofcurrent pulses.

and Neel walls are nearly degenerate. Thus, only a smalltorque is needed to rotate the wall’s magnetization from oneDW state to the other, resulting in a reduced critical current.

Current-driven DW changes can take place below thecritical value or depinning. For example, in the work ofFlorez, Krafft and Gomez (2005), it was observed thatthe wall structure was modified even though the DW wasstill trapped within the notched pinning site. We havealso observed this behavior, both using MFM and AMRexperiments. Examples of the modification of the DWstructure upon application of current pulses are shown inFigure 24. The NiFe nanowire is 300 nm wide and 20 nmthick. In the first set of experiments (a), the DW retains itsV structure, but the V chirality is reversed. In the secondset of images (b), the original V DW is transformed intoa T wall, which remains stable for a few pulses beforetransforming back to into a V wall (with a reversed chirality).The detailed mechanisms responsible for this behavior arenot fully understood. However, these experiments suggestthat the electric current allows the magnetic system to accessmetastable states, even if the energy barriers separating thesestates are often very high (for example, ∼6 × 10−11 erg in the


example of Figure 19). Thermally activated transformationsare also possible if the energy barrier is smaller. For example,transformation from a T to a V DW was observed at ∼550 Kby Laufenberg et al. (2006a) in 7-nm-thick, 730-nm-widepermalloy nanorings. This corresponds to an energy barrierof ∼ 8.0 × 10−14 erg.

Current-driven depinning also exhibits significant stochas-tic variations, particularly in low fields. In repeated experi-ments, the probability for DW depinning varies significantly.In other words, the critical current at a given field can varywidely. This can partly be related to the multiple metastableDW states at a notch.

3.5.6 Dynamical effects

Most experiments published so far have used dc currentsor long pulses (microseconds). Little is known on possibledynamical effects driven by shorter pulses or alternatingcurrent. Until recently, there were only a few experimentalreports on the influence of the pulse length for nanosecond-long current pulses. Salhi and Berger (1994) studied theinfluence of a series of square pulses with lengths between50 and 300 ns on the critical current, and they did not observeany dependence. The same insensitivity to pulse length hasbeen reported by Lim et al. (2004) for a spin-valve nanowire,for much shorter pulses (between 0.4 and 2 ns).

However, Berger has proposed several interesting dynam-ical effects related to current pulses or ac currents. Forexample, in the case of pulses with fast rise-times and slowfall-times, he proposed a ‘wall streaming’ mechanism inwhich the DW can escape a pinning potential on the trailingedge of the pulse (Salhi and Berger, 1994; Berger, 1992).He also proposed several other notions such as the exis-tence of a ‘Ferro–Josephson’ effect (Berger, 1986) wherethe precession of the DW driven by the current generatesa dc voltage across the DW, following a mechanism quitesimilar to the Josephson effect in superconducting junctions.In a further development, he showed that the combinationof a hard-axis, high-frequency field and a dc current couldlead to the locking of the DW precession frequency with theac field. This would give rise to steps in the current depen-dence of this induced voltage, similar to Shapiro steps inJosephson junctions (Berger, 1991b). Finally, he also showedthat ac currents could generate oscillations of a DW, with aresonance frequency of ∼10–100 MHz (Berger, 1996).

In a recent experiment, the influence of a small ac excita-tion has been studied by Saitoh et al. in a U-shaped permalloynanowire (Saitoh, Miyajima, Yamaoka and Tatara, 2004).The authors observed a peak in the device resistance as afunction of the frequency of the ac current in the MHz range.The peak frequency could be tuned between ∼5 and 25 MHzby varying the applied magnetic field. Interestingly, the

peak was observed even though the excitation is extremelysmall (about two orders of magnitude smaller than the dccritical current). These data are understood in terms of acurrent-driven resonance of the DW in the parabolic potentialcreated by the applied field. This interpretation is supportedby a model by Tatara, Saitoh, Ichimura and Kohno (2005).The authors propose that this behavior results from momen-tum transfer (or nonadiabatic spin torque) rather than spintransfer. Even though the momentum transfer (proportionalto the DW resistance in Tatara’s model) is very small, it isamplified at resonance and becomes dominant. The authorsalso suggest that the amplitude of the DW displacement isvery large (∼10 µm), although there is no direct evidence tosupport this claim.

We have reported recently that the current-driven motionfrom a local pinning potential is indeed very sensitive to thepulse length (Thomas et al., 2006). We have shown that theprobability of depinning a DW from a local pinning site oscil-lates with the length of the current pulse, with a period ofa few nanoseconds (see Figure 25). Importantly, both head-to-head and tail-to-tail DWs exhibited the same behavior.These oscillations of depinning probability lead to strongoscillations in the critical current. We have shown that thisbehavior is related to the current-induced precession of theDW trapped in a pinning potential. When the pulse length isclose to a multiple of half the precession period, the ampli-tude of the DW oscillation is amplified after the end of thecurrent pulse, in turn leading to DW depinning. Importantly,this is a subthreshold behavior, which occurs for currentssmaller than the dc critical current. Moreover, in this depin-ning regime, we showed that the DW displacement is againstthe electron flow, opposite to the propagation direction abovethreshold. This behavior can be accounted for within theframework of the 1D model described in Section 2.2.3. Eventhough the DW in the experiment was a vortex wall, wewere able to achieve a good qualitative description of theexperimental results Yamanouchi et al., (2006).

3.5.7 Thermal effects: current-induced heating andnucleation of domain walls

As already emphasized, the high critical current densityrequired to achieve current-induced DW motion in mostsamples induces significant Joule heating. The effect of tem-perature on the current/ DW interaction is not fully under-stood. One obvious effect is the decrease of magnetiza-tion and anisotropy of the magnetic material. According toequation (15), this should help to reduce the intrinsic crit-ical current, since Jc is proportional to MsHk, and Hk isan anisotropy field which is itself proportional to Ms. How-ever, the polarization of the current should also be reduced,which should compensate at least partially the gain from


tp = 6 ns

−50 0 50

q (nm)

tp = 8 ns

−50 0 50

q (nm)

−50 0 50−10

0

10

20 tp = 5 ns

q (nm)

Ψ (°

)

−60

−3

−40 −20 0 0 20

10

50

40 60u (m s−1)u (m s−1)

t p (n

s)

10

50t p

(ns)

0 0 3

Vp (V) Vp (V)

0 0.5 1

Depinning probability

(a)

(b)

(c)

Figure 25. Oscillatory depinning of a DW as a function of the current pulse length. Experiments (a), calculations (b), and schematicexplanation of the mechanism (c). (From Thomas et al. (2006).)

the reduction of Ms. Moreover, spin-wave excitations can beamplified by the spin torque and lead to chaotic behavior,thus preventing DW motion.

In a recent report on permalloy nanorings, Laufenberget al. (2006b) observed a slight increase of the criticalcurrent with temperature between 4 and 300 K, from ∼2 to2.8 × 108 A cm−2. This variation is opposite to that of theDW depinning field, which decreases with temperature. Theauthors conclude that the spin-transfer efficiency decreaseswith temperature, although the origin of such a variationis not clear. In this temperature range, far below theCurie temperature for permalloy, the magnetic propertiesexhibit little variation. The authors propose to explain thisdecrease in efficiency by thermally excited spin waves,which couple with the current flow, effectively absorbing

some spin angular momentum. Results are opposite forGaMnAs structures for which the critical current decreasesby ∼50% when the temperature increases between 100and 107 K, as the spin-torque efficiency increases owing tothe temperature dependence of the magnetization and spinpolarization (Yamanouchi et al., 2006).

Temperature effects were also addressed by Yamaguchiet al. (2005) in a much higher temperature range. It wasshown that spontaneous DW nucleation occurs in permal-loy nanowires when the current density exceeds 7.5 ×107 A cm−2. The authors suggest that this phenomenonoccurs because the temperature exceeds the Curie temper-ature for NiFe during the pulse, thus leading to the break-down of the nanowire into a multidomain state. We haveobserved similar effects in many nanowires. Interestingly


Figure 26. MFM images measured from a 10-nm-thick, 300-nm-wide permalloy nanowire, upon successive application of 3-ns-longcurrent pulses, with a current density of ∼7 × 108 A cm−2 (valuenot corrected for resistance increase due to heating).

in the example shown in Figure 26, DWs are nucleated,whereas preexisting DWs are essentially unaffected by thecurrent pulse. This suggests that DW generation can occurbelow the Curie temperature, for example, because of thereversal of the magnetization in one or several portions ofthe nanowire, thereby generating pairs of DWs. In the manycases where we have observed this behavior, the ends of thenanowires are not reversed, and the DWs always appear inpairs, thus supporting this interpretation. Micromagnetic sim-ulations that include thermal effects also suggest that the spintorque amplifies magnetization fluctuations and can therebyeventually lead to the complete reversal of parts a nanowire.

Interestingly, Shibata et al. find by theoretical calculationsthat, above a current threshold, the ground state of a fer-romagnet becomes a multidomain state with several DWs(Shibata, Tatara and Kohno, 2005). The threshold for nucle-ation is given by (using the notation of Section 2.2.3):

unuc = γHk0

(1 +

√Hk

Hk0

)(26)

where Hk0 and Hk are the longitudinal and transverseanisotropy fields, respectively. If unuc < uc (uc the intrinsiccritical current given by equation (13)), that is if Hk > 8Hk0,the multidomain state is stable, and this could appear as thespontaneous nucleation of multiple DWs in a nanowire, asobserved experimentally. An estimate of the current densityrequired to reach this nucleation regime is found to be∼3 × 109 A cm−2 for Co. For permalloy, this value couldbe lower, particularly since Joule heating would contributeto a reduction of the magnetization, since permalloy has alower Curie temperature than Co.

4 OUTLOOK

The observation that current can induce motion in DWs has along history dating back more than 25 years. Indeed, it wasappreciated long ago that there are several ways in whichcurrent can interact with DWs, including the notion that thetransfer of spin angular momentum from current to DWs canperturb a DW. However, there were few experiments on thisimportant interaction perhaps because it is only dominant inmagnetic structures on the nanoscale, and is thus is obscuredby other effects in larger structures. As the capability to makeever smaller high quality structures has evolved over the pastyears it has become possible to fabricate structures in whichsingle DWs can be controllably injected and whose motioncan be probed following or even during the application ofnanosecond-long current and field pulses. These experimentalstudies have clearly demonstrated that spin-polarized currentdoes result in the motion of DWs along magnetic nanowires.Moreover, short current pulses, whose length is matched withthe precessional frequency of a DW, can resonantly excitethe precession of the DW. While there are a number ofopen questions, as discussed briefly at the end of the lastsection, the notion of moving DWs with current is not onlyof considerable scientific interest, but suggests a number ofinteresting technological applications.

More than a quarter of a century ago there was consid-erable interest in the possibility of building magnetic mem-ory devices by storing information in the form of magneticdomains or bubbles in single-crystalline garnet thin films.In this original planar magnetic memory the bubbles weredefined and moved by magnetic fields created on chip. Muchcreative and imaginative work was expended to develop inge-nious structures to reliably manipulate magnetic domains inthese materials, as discussed in several review articles andpapers (see, e.g., Malozemoff and Slonczewski, 1979; deLeeuw, Van Den Doel and Enz, 1980; Bar’yakhtar, Ivanovand Chetkin, 1985; Schneider, 1975; Pugh, Critchlow, Henleand Russell, 1981; Eschenfelder, 1980). In this chapter wefocused the discussion on the motion of head-to-head DWsin magnetic nanowires. One problem with manipulating suchtypes of DWs by magnetic fields is that head-to-head andtheir counterpart tail-to-tail walls move in opposite direc-tion in the presence of a magnetic field. This problem canbe solved by using spin momentum transfer from current tomove DWs since the current becomes repolarized over veryshort distances (e.g., the spin relaxation length can be as shortas ∼5 nm in permalloy (Bass and Pratt, 2007)). Thus currentmoves head-to-head and tail-to-tail DWs in permalloy andother transition metals in the same direction. This distinctivefeature of current-induced DW motion allows the possibilityof interesting new device technologies not previously pos-sible with field manipulation. One of the more interesting


Positivecurrent pulse

Negativecurrent pulse

String ofabout 100 bits

Bit sringdomain walls

move right

Bit sringdomain walls

move left

Read deviceWrite device

Figure 27. Schematic of the magnetic racetrack memory device.

of these devices is the magnetic racetrack memory (MRM)(Parkin, 2004).

The magnetic racetrack memory promises a nonvolatilesolid-state data-storage device that combines the very lowcost of conventional magnetic hard-disk drives with the highreliability and high performance of nonvolatile solid-statememory. The MRM concept is that of a magnetic shiftregister in which individual spintronic reading and writ-ing elements, formed using conventional CMOS technology,are married to magnetic ‘racetracks’ in which 10 to 100nanoscale bits are stored as magnetic domains (schematicallyshown in Figure 27). The racetracks are vertical columns ofmagnetic material, formed in the third dimension, perpendic-ular to a plane of CMOS logic elements and devices formedon a silicon substrate. Each of these columns is addressedby a single read and write element whereby the magneticdomains in the racetrack are moved, by nanosecond-longcurrent pulses using current-induced motion of DWs, to theread and write devices. Since the racetracks are formed in thethird dimension the footprint of each racetrack in the CMOSlogic plane is very small and therefore the cost should below. Indeed the goal of the MRM is to store ∼100 data bitsin the same area of a silicon wafer that one data bit in con-ventional solid-state memories would occupy. Since the costof a CMOS memory or logic device is largely determinedby the area of the silicon wafer the proposed magnetic shiftmemory has the potential to be ∼100 times cheaper per bitthan conventional solid-state memory and so comparable incost to a HDD. At the same time the proposed solid-statememory will have the reliability and integrity of conventionalsolid-state memories. Thus, the proposed spintronic magneticshift memory will have a previously unrealized combinationof extremely low cost, high reliability, and excellent perfor-mance as compared to flash drives and HDDs.

The magnetic racetrack is just one example of a possiblenew technology which takes advantage of the new physicsof current-driven DW motion, which has just begun to befully appreciated and understood in the past few years. Asdiscussed throughout this chapter, however, many aspectsof current-driven DW motion remain unanswered. The mostfundamental question is the detailed theoretical description ofthe current-magnetization interaction. Of particular interest isthe understanding of the role of nonadiabatic contributions tothe current–magnetization interaction. There is considerabledebate over the existence and nature of this contribution,and, consequently whether there is an intrinsic thresholdcurrent for DW motion in uniform, homogeneous magneticnanowires.

Another important question is the maximum possiblespeed of DWs driven by current alone. Many experimen-tal studies, mostly in permalloy but also in GaMnAs, haveindicated extremely low current-driven DW velocities. Thesevelocities are so small that they are difficult to rationalizewithin today’s theoretical models. Clearly, further exper-imental and theoretical studies are called for. Systematicstudies of the relationship between the critical current neededto move a DW as well as the DW velocity and the intrinsicmaterial parameters of the magnetic material (e.g., magneti-zation, anisotropy, and spin polarization of the current) willlikely lead to a much improved understanding of the basicmechanisms involved in the current–DW interaction.

A particularly interesting question is the magnitude ofthe current required to sustain the motion of a DW alonga nanowire and whether this is largely determined by thespin polarization of the current and the magnetization ofthe magnetic domains or whether other factors may play asignificant role. The answer to this important question willdetermine how useful current-induced DW motion will befor technological applications. From both a technologicaland a scientific perspective, the understanding and controlof the DW structure and the interaction between DWs isboth challenging and interesting.

The renewed interest in DW dynamics in magnetic nanos-tructures has already revealed new physical phenomena andthe considerable activity in this field promises exciting futurediscoveries.

ACKNOWLEDGMENTS

We would like to thank our colleagues at the AlmadenResearch Center for their contribution to this work:Masamitsu Hayashi, Rai Moriya, Charles Rettner, Xin Jiang,Yaroslaw Bazaliy, Bastiaan Bergman, Brian Hughes andKevin Roche. We are also grateful to Pierre-Olivier Jubertand Michael Scheinfein for their helpful comments on the


manuscript. This work was partly supported by the DefenseMicroelectronics Activity.

REFERENCES

Allwood, D.A., Vernier, N., Xiong, G., et al. (2002a). Shifted hys-teresis loops from magnetic nanowires. Applied Physics Letters,81, 4005.

Allwood, D.A., Xiong, G., Cooke, M.D., et al. (2002b). Submi-crometer ferromagnetic NOT gate and shift register. Science, 296,2003.

Allwood, D.A., Xiong, G. and Cowburn, R.P. (2004). Domainwall diodes in ferromagnetic planar nanowires. Applied PhysicsLetters, 85, 2848.

Atkinson, D., Allwood, D.A., Xiong, G., et al. (2003). Magneticdomain-wall dynamics in a submicrometre ferromagnetic struc-ture. Nature Materials, 2, 85.

Baldwin, J.A. and Culler, G.J. (1969). Wall-pinning model ofmagnetic hysteresis. Journal of Applied Physics, 40, 2828.

Barnes, S.E. (2006). Comment on theory of current-driven domainwall motion: spin transfer versus momentum transfer. PhysicalReview Letters, 96, 189701.

Barnes, S.E. and Maekawa, S. (2005). Current-spin coupling forferromagnetic domain walls in fine wires. Physical ReviewLetters, 95, 107204.

Bar’yakhtar, V.G., Ivanov, B.A. and Chetkin, M.V. (1985). Dynam-ics of domain walls in weak ferromagnets. Soviet PhysicsUspekhi, 28, 563.

Bass, Jack. and Pratt, William.P. Jr. (2007). Spin-diffusion lengthsin metals and alloys, and spin-flipping at metal/metal interfaces:an experimentalist’s critical review, Journal of Physics: Con-densed Matter, 19, 183201.

Bazaliy, Y.B., Jones, B.A. and Zhang, S-C. (1998). Modification ofthe Landau-Lifshitz equation in the presence of a spin-polarizedcurrent in colossal- and giant-magnetoresistive materials. Physi-cal Review B, 57, R3213.

Beach, G.S.D., Knutson, C., Nistor, C., et al. (2006). Nonlineardomain-wall velocity enhancement by spin-polarized electriccurrent. Physical Review Letters, 97, 057203.

Beach, G.S.D., Nistor, C., Knutson, C., et al. (2005). Dynam-ics of field-driven domain-wall propagation in ferromagneticnanowires. Nature Materials, 4, 741.

Berger, L. (1974). Prediction of a domain-drag effect in uniaxial,non-compensated, ferromagnetic metals. Journal of Physics andChemistry of Solids, 35, 947.

Berger, L. (1978). Low-field magnetoresistance and domain drag inferromagnets. Journal of Applied Physics, 49, 2156.

Berger, L. (1984). Exchange interaction between ferromagneticdomain-wall and electric-current in very thin metallic-films.Journal of Applied Physics, 55, 1954.

Berger, L. (1986). Possible existence of a Josephson effect inferromagnets. Physical Review B, 33, 1572.

Berger, L. (1991a). Galvanomagnetic voltages in the vicinity ofa domain-wall in ferromagnetic thin-films. Journal of AppliedPhysics, 69, 1550.

Berger, L. (1991b). Prediction of Shapiro steps in the dc voltageacross a magnetic domain-wall traversed by a dc current andexposed to high-frequency magnetic-fields. Journal of AppliedPhysics, 69, 4683.

Berger, L. (1992). Motion of a magnetic domain-wall traversed byfast-rising current pulses. Journal of Applied Physics, 71, 2721.

Berger, L. (1996). Current-induced oscillations of a Bloch wallin magnetic thin films. Journal of Magnetism and MagneticMaterials, 162, 155.

Berger, L. (2006). Analysis of measured transport properties ofdomain walls in magnetic nanowires and films. Physical ReviewB, 73, 014407.

Bruno, P. (1999). Geometrically constrained magnetic wall. Physi-cal Review letters, 83, 2425.

Buchanan, K.S., Roy, P.E., Grimsditch, M., et al. (2005). Soliton-pair dynamics in patterned ferromagnetic ellipses. Nature Physics,1, 172.

Carr, W.J. (1974a). Force on domain-wall due to perturbationof current in a magnetoresistive overlayer. Journal of AppliedPhysics, 45, 3115.

Carr, W.J. (1974b). Propagation of magnetic domain-walls by aself-induced current distribution. Journal of Applied Physics, 45,394.

Cayssol, F., Ravelosona, D., Chappert, C., et al. (2004). Domainwall creep in magnetic wires. Physical Review Letters, 92,107202.

Charap, S.H. (1974). Interaction of a magnetic domain-wall withan electric-current. Journal of Applied Physics, 45, 397.

Chern, G.W., Youk, H. and Tchernyshyov, O. (2006). Topologicaldefects in flat nanomagnets: the magnetostatic limit. Journal ofApplied Physics, 99, 08Q505.

Choe, S.B., Acremann, Y., Scholl, A., et al. (2004). Vortex core-driven magnetization dynamics. Science, 304, 420.

Cowburn, R.P., Allwood, D.A., Xiong, G. and Cooke, M.D. (2002).Domain wall injection and propagation in planar Permalloynanowires. Journal of Applied Physics, 91, 6949.

Dugaev, V.K., Vieira, V.R., Sacramento, P.D., et al. (2006).Current-induced motion of a domain wall in a magnetic nanowire.Physical Review B: Condensed Matter and Materials Physics, 74,054403.

Duine, R.A., Nunez, A.S. and Mac Donald, A.H. (2007). ThermallyAssisted Current-Driven Domain Wall Motion. Physics ReviewLetter, 98, 056605.

Duvail, J.L., Dubois, S., Piraux, L., et al. (1998). Electrodepositionof patterned magnetic nanostructures. Journal of Applied Physics,84, 6359.

Ebels, U., Radulescu, A., Henry, Y., et al. (2000). Spin accumu-lation and domain wall magnetoresistance in 35 nm Co wires.Physical Review Letters, 84, 983.

Emtage, P.R. (1974). Self-induced drive of magnetic domains.Journal of Applied Physics, 45, 3117.

Eschenfelder, A.H. (1980). Magnetic Bubble Technology, Springer-Verlag: Berlin.


Faulkner, C.C., Cooke Michael, D., Allwood, D.A., et al. (2004).Artificial domain wall nanotraps in Ni[sub 81]Fe[sub 19] wires.Journal of Applied Physics, 95, 6717.

Florez, S.H., Krafft, C. and Gomez, R.D. (2005). Spin-current-induced magnetization reversal in magnetic nanowires withconstrictions. Journal of Applied Physics, 97, 10C705.

Freitas, P.P. and Berger, L. (1985). Observation of s-d exchangeforce between domain walls and electric current in very thinPermalloy films. Journal of Applied Physics, 57, 1266.

Gan, L., Chung, S.H., Aschenbach, K.H., et al. (2000). Pulsed-current-induced domain wall propagation in Permalloy patternsobserved using magnetic force microscope. IEEE Transactionson Magnetics, 36, 3047.

Gider, S., Runge, B-U., Marley, A.C. and Parkin, S.S.P. (1998). Themagnetic stability of spin-dependent tunneling devices. Science,281, 797.

Gopalaswamy, S. and Berger, L. (1991). Observation of galvano-magnetic voltages at a magnetic domain-wall in Ni-Fe films.Journal of Applied Physics, 70, 5825.

Grollier, J., Lacour, D., Cros, V., et al. (2002). Switching themagnetic configuration of a spin valve by current-induced domainwall motion. Journal of Applied Physics, 92, 4825.

Grollier, J., Boulenc, P., Cros, V., et al. (2003). Switching a spinvalve back and forth by current-induced domain wall motion.Applied Physics Letters, 83, 509.

Grollier, J., Boulenc, P., Cros, V., et al. (2004). Spin-transfer-induced domain wall motion in a spin valve. Journal of AppliedPhysics, 95, 6777.

Hayashi, M., Thomas, L., Bazaliy, Y.B., et al. (2006a). Influenceof current on field-driven domain wall motion in Permalloynanowires from time resolved measurements of anisotropicmagnetoresistance. Physical Review Letters, 96, 197207.

Hayashi, M., Thomas, L., Rettner, C. et al. (2006b). Dependenceof current and field driven depinning of domain walls on theirstructure and chirality in Permalloy nanowires. Physical ReviewLetters, 97, 207205.

Hayashi, M., Thomas, L., Rettner, C., et al. (2007a). Current drivendomain wall velocities exceeding the spin angular momentumtransfer rate in Permalloy nanowires. Physical Review Letters,98, 037204.

Hayashi, M., Thomas, L., Rettner, C., et al. (2007b). Directobservation of the coherent precession of magnetic domain wallspropagating along Permalloy nanowires. Nature Physics, 3, 21.

He, J., Li, Z. and Zhang, S. (2005). Current-driven domain-walldepinning. Journal of Applied Physics, 98, 016108.

He, J., Li, Z. and Zhang, S. (2006a). Current-driven vortex domainwall dynamics by micromagnetic simulations. Physical ReviewB, 73, 184408.

He, J., Li, Z. and Zhang, S. (2006b). Effects of current on vortex andtransverse domain walls. Journal of Applied Physics, 99, 08G509.

Himeno, A., Kasai, S. and Ono, T. (2005). Current-driven domain-wall motion in magnetic wires with asymmetric notches. AppliedPhysics Letters, 87, 243108.

Himeno, A., Okuno, T., Kasai, S., et al. (2005). Propagation ofa magnetic domain wall in magnetic wires with asymmetricnotches. Journal of Applied Physics, 97, 066101.

Himeno, A., Ono, T., Nasu, S., et al. (2003). Dynamics of amagnetic domain wall in magnetic wires with an artificial neck.Journal of Applied Physics, 93, 8430.

Holleitner, A.W., Knotz, H., Myers, R.C., et al. (2004). Pinning adomain wall in (Ga,Mn)As with focused ion beam lithography.Applied Physics Letters, 85, 5622.

Honolka, J., Masmanidis, S., Tang, H.X., et al. (2005). Domain-wall dynamics at micropatterned constrictions in ferromagnetic(Ga,Mn)As epilayers. Journal of Applied Physics, 97, 063903.

Hubert, A. and Schafer, R. (2000). Magnetic Domains, Springer-Verlag: Berlin.

Hung, C.Y. and Berger, L. (1988). Exchange forces betweendomain-wall and electric-current in Permalloy-films of variablethickness. Journal of Applied Physics, 63, 4276.

Jubert, P.-O., Allenspach, R. and Bischof, A. (2004). Magneticdomain walls in constrained geometries. Physics Review B, 69,220410.

Jubert, P.-O., Klaui, M., Bischof, A., et al. (2006). Velocity ofvortex walls moved by current. Journal of Applied Physics, 99,08G523.

Katine, J.A., Ho, M.K., Ju, Y.S. and Rettner, C.T. (2003). Patterningdamage in narrow trackwidth spin-valve sensors. Applied PhysicsLetters, 83, 401.

Kimura, T., Otani, Y., Yagi, I., et al. (2003). Suppressed pinningfield of a trapped domain wall due to current injection. Journalof applied Physics, 94, 7266.

Klaui, M., Ehrke, H., Rudiger, U., et al. (2005a). Direct observa-tion of domain-wall pinning at nanoscale constrictions. AppliedPhysics Letters, 87, 102509.

Klaui, M., Jubert, P.O., Allenspach, R., et al. (2005b). Directobservation of domain-wall configurations transformed by spincurrents. Physical Review Letters, 95, 026601.

Klaui, M., Vaz, C.A.F., Bland, J.A.C., et al. (2005c). Controlled andreproducible domain wall displacement by current pulses injectedinto ferromagnetic ring structures. Physical Review Letters, 94,106601.

Klaui, M., Vaz, C.A.F., Bland, J.A.C., et al. (2003a). Domain wallpinning and controlled magnetic switching in narrow ferromag-netic ring structures with notches (invited). Journal of AppliedPhysics, 93, 7885.

Klaui, M., Vaz, C.A.F., Bland, J.A.C., et al. (2003b). Domain wallmotion induced by spin polarized currents in ferromagnetic ringstructures. Applied Physics Letters, 83, 105.

Klaui, M., Vaz, C.A.F., Bland, J.A.C., et al. (2004). Head-to-headdomain-wall phase diagram in mesoscopic ring magnets. AppliedPhysics Letters, 85, 5637.

Lacour, D., Katine, J.A., Folks, L., et al. (2004). Experimentalevidence of multiple stable locations for a domain wall trappedby a submicron notch. Applied Physics Letters, 84, 1910.

Laufenberg, M., Backes, D., Buhrer, W., et al. (2006a). Observationof thermally activated domain wall transformations. AppliedPhysics Letters, 88, 046602.

Laufenberg, M., Buhrer, W., Bedau, D., et al. (2006b). Temperaturedependence of the spin torque effect in current-induced domainwall motion. Physical Review Letters, 97, 046602.


Lee, W.Y., Yao, C.C., Hirohata, A., et al. (2000). Domain nucle-ation processes in mesoscopic Ni80Fe20 wire junctions. Journalof Applied Physics, 87, 3032.

de Leeuw, F.H., Van Den Doel, R. and Enz, U. (1980). Dynamicproperties of magnetic domain walls and magnetic bubbles.Reports on Progress in Physics, 43, 689.

Lepadatu, S. and Xu, Y.B. (2004). Direct observation of domainwall scattering in patterned Ni80Fe20 and Ni nanowires bycurrent-voltage measurements. Physical Review Letters, 92,127201.

Levy, P.M. and Zhang, S.F. (1997). Resistivity due to domain wallscattering. Physical Review Letters, 79, 5110.

Li, Z. and Zhang, S. (2004a). Domain-wall dynamics driven byadiabatic spin-transfer torques. Physical Review B, 70, 064708.

Li, Z. and Zhang, S. (2004b). Domain-wall dynamics and spin-waveexcitations with spin-transfer torques. Physical Review Letters,92, 207203.

Lim, C.K., Devolder, T., Chappert, C., et al. (2004). Domain walldisplacement induced by subnanosecond pulsed current. AppliedPhysics Letters, 84, 2820.

Malozemoff, A.P. and Slonczewski, J.C. (1979). Magnetic DomainWalls in Bubble Material, Academic Press: New York.

Marrows, C.H. (2005). Spin-polarised currents and magneticdomain walls. Advances in Physics, 54, 585.

McGuire, T.R. and Potter, R.I. (1975). Anisotropic magnetoresis-tance in ferromagnetic 3d alloys. IEEE Transactions on Magnet-ics, 11, 1018.

McMichael, R.D. and Donahue, M.J. (1997). Head to head domainwall structures in thin magnetic strips. IEEE Transactions onMagnetics, 33, 4167.

Miyake, K., Shigeto, K., Yokoyama, Y., et al. (2005). Exchangebiasing of a Neel wall in the nanocontact between NiFe wires.Journal of Applied Physics, 97, 014309.

Nagahama, T., Mibu, K. and Shinjo, T. (2000). Electric resistanceof magnetic domain wall in NiFe wires with CoSm pinning pads.Journal of Applied Physics, 87, 5648.

Nakatani, Y., Hayashi, N., Ono, T. and Miyajima, H. (2001).Computer simulation of domain wall motion in a magnetic stripline with submicron width. IEEE Transactions on Magnetics, 37,2129.

Nakatani, Y., Thiaville, A. and Miltat, J. (2003). Faster magneticwalls in rough wires. Nature Materials, 2, 521.

Nakatani, Y., Thiaville, A. and Miltat, J. (2005). Head-to-headdomain walls in soft nano-strips: a refined phase diagram. Journalof Magnetism and Magnetic Materials, 290, 750.

Nguyen, A.K., Skadsem, H.J. and Brataas, A. (2007). Giant current-driven domain wall mobility in (Ga,Mn)As. Physics ReviewLetter, 98, 146602.

Nibarger, J.P., Lopusnik, R., Celinski, Z. and Silva, T.J. (2003).Variation of magnetization and the Lande g factor with thicknessin Ni-Fe films. Applied Physics Letters, 83, 93.

Nibarger, J.P., Lopusnik, R. and Silva, T.J. (2003). Damping asa function of pulsed field amplitude and bias field in thin filmPermalloy. Applied Physics Letters, 82, 2112.

Novosad, V., Fradin, F.Y., Roy, P.E., et al. (2005). Magnetic vortexresonance in patterned ferromagnetic dots. Physical Review B, 72,094442.

Ohe, J. and Kramer, B. (2006). Dynamics of a domain wall andspin-wave excitations driven by a mesoscopic current. PhysicalReview Letters, 96, 027204.

Ono, T., Miyajima, H., Shigeto, K., et al. (1999a). Propagationof the magnetic domain wall in submicron magnetic wireinvestigated by using giant magnetoresistance effect. Journal ofApplied Physics, 85, 6181.

Ono, T., Miyajima, H., Shigeto, K., et al. (1999b). Propagationof a magnetic domain wall in a submicrometer magnetic wire.Science, 284, 468.

Ono, T., Miyajima, H., Shigeto, K. and Shinjo, T. (1998). Mag-netization reversal in submicron magnetic wire studied byusing giant magnetoresistance effect. Applied Physics Letters,72, 1116.

Parkin, S.S.P. (1992). Dramatic enhancement of interlayer exchangecoupling and giant magnetoresistance in Ni81Fe19/Cu multilayersby addition of thin Co interface layers. Applied Physics Letters,61, 1358.

Parkin, S.S.P. (1993). Origin of enhanced magnetoresistance ofmagnetic multilayers: spin-dependent scattering from magneticinterface states. Physical Review letters, 71, 1641.

Parkin, S.S.P. (1994). In Ultrathin Magnetic Structures, Heinrich,B. and Bland, J.A.C. (Eds.), Springer-Verlag: Berlin, p. 148,Vol. II.

Parkin, S.S.P. (1995). In Annual Review of Materials Science,Wessels, B.W. (Ed.), Annual Reviews: Palo Alto, p. 357,Vol. 25.

Parkin, S.S.P. (2004) USA Patent No. 6,834,005 (2003-06-10(filed).

Parkin, S.S.P., Jiang, X., Kaiser, C., et al. (2003). Magneticallyengineered spintronic sensors and memory. Proceedings of theIEEE, 91, 661.

Parkin, S.S.P., Li, Z.G. and Smith, D.J. (1991). Giant magnetore-sistance in antiferromagnetic Co/Cu multilayers. Applied PhysicsLetters, 58, 2710.

Partin, D.L., Karnezos, M., deMenezes, L.C. and Berger, L.(1974). Nonuniform current distribution in neighborhood of aferromagnetic domain-wall in cobalt at 42k. Journal of AppliedPhysics, 45, 1852.

Porter, D.G. and Donahue, M.J. (2004). Velocity of transversedomain wall motion along thin, narrow strips. Journal of AppliedPhysics, 95, 6729.

Pugh, E.W., Critchlow, D.L., Henle, R.A. and Russell, L.A.(1981). Solid state memory development in IBM. IBM Journalof Research and Development, 25, 585.

Puzic, A., Van Waeyenberge, B., Chou, K.W., et al. (2005). Spa-tially resolved ferromagnetic resonance: imaging of ferromag-netic eigenmodes. Journal of Applied Physics, 97, 10E704.

Ravelosona, D., Lacour, D., Katine, J.A., et al. (2005). Nanometerscale observation of high efficiency thermally assisted current-driven domain wall depinning. Physical Review Letters, 95,117203.


Rothman, J., Klaui, M., Lopez-Diaz, L., et al. (2001). Observationof a bi-domain state and nucleation free switching in mesoscopicring magnets. Physical Review Letters, 86, 1098.

Ruster, C., Borzenko, T., Gould, C., et al. (2003). Very largemagnetoresistance in lateral ferromagnetic (Ga,Mn)As wires withnanoconstrictions. Physical Review Letters, 91, 216602.

Saitoh, E., Miyajima, H., Yamaoka, T. and Tatara, G. (2004).Current-induced resonance and mass determination of a singlemagnetic domain wall. Nature, 432, 203.

Salhi, E. and Berger, L. (1994). Current-induced displacements ofBloch Walls in Ni-Fe films of thickness 120-740 Nm. Journal ofApplied Physics, 76, 4787.

Sato, H., Hanada, R., Sugawara, H., et al. (2000). Local magne-tization rotation in NiFe wire monitored by multiple transverseprobes. Physical Review B, 61, 3227.

Schneider, J. (1975). Amorphous GdCoCr films for bubble domainapplications. IBM Journal of Research and Development, 19, 587.

Schrefl, T., Fidler, J., Kirk, K.J. and Chapman, J.N. (1997). Domainstructures and switching mechanisms in patterned magneticelements. Journal of Magnetism and Magnetic Materials, 175,193.

Schumacher, H.W., Ravelosona, D., Cayssol, F., et al. (2001).Propagation of a magnetic domain wall in the presence of AFMfabricated defects. IEEE Transactions on Magnetics, 37, 2331.

Shibata, J., Nakatani, Y., Tatara, G., et al. (2006). Current-inducedmagnetic vortex motion by spin-transfer torque. Physical ReviewB, 73, 020403.

Shibata, J., Tatara, G. and Kohno, H. (2005). Effect of spin currenton uniform ferromagnetism: domain nucleation. Physical ReviewLetters, 94, 076601.

Shigeto, K., Okuno, T., Shinjo, T., et al. (2000). Magnetizationswitching of a magnetic wire with trilayer structure using giantmagnetoresistance effect. Journal of Applied Physics, 88, 6636.

Shigeto, K., Shinjo, T. and Ono, T. (1999). Injection of a magneticdomain wall into a submicron magnetic wire. Applied PhysicsLetters, 75, 2815.

Silva, T.J., Lee, C.S., Crawford, T.M. and Rogers, C.T. (1999).Inductive measurement of ultrafast magnetization dynamics inthin-film Permalloy. Journal of Applied Physics, 85, 7849.

Slonczewski, J.C. (1996). Current-driven excitation of magneticmultilayers. Journal of Magnetism and Magnetic Materials, 159,L1.

Tanase, M., Silevitch, D.M., Chien, C.L. and Reich, D.H. (2003).Magnetotransport properties of bent ferromagnetic nanowires.Journal of Applied Physics, 93, 7616.

Tang, H.X., Kawakami, R.K., Awschalom, D.D. and Roukes, M.L.(2003). Giant planar hall effect in epitaxial (Ga,Mn)As devices.Physical Review Letters, 90, 107201.

Tang, H.X., Kawakami, R.K., Awschalom, D.D. and Roukes, M.L.(2006). Propagation dynamics of individual domain walls in Ga1-xMnxAs microdevices. Physical Review B, 74, 041310.

Tang, H.X., Masmanidis, S., Kawakami, R.K., et al. (2004). Nega-tive intrinsic resistivity of an individual domain wall in epitaxial(Ga,Mn)As microdevices. Nature, 431, 52.

Tang, H.X. and Roukes, M.L. (2004). Electrical transport acrossan individual magnetic domain wall in (Ga,Mn)As microdevices.Physical Review B, 70, 205213.

Taniyama, T., Nakatani, I., Namikawa, T. and Yamazaki, Y. (1999).Resistivity due to domain walls in Co zigzag wires. PhysicalReview Letters, 82, 2780.

Taniyama, T., Nakatani, I., Yakabe, T. and Yamazaki, Y. (2000).Control of domain structures and magnetotransport properties inpatterned ferromagnetic wires. Applied Physics Letters, 76, 613.

Tatara, G. and Kohno, H. (2004). Theory of current-driven domainwall motion: spin transfer versus momentum transfer. PhysicalReview Letters, 92, 086601.

Tatara, G. and Kohno, H. (2006). Comment on Theory of current-driven domain wall motion: spin transfer versus momentumtransfer – reply. Physical Review Letters, 96, 189702.

Tatara, G., Saitoh, E., Ichimura, M. and Kohno, H. (2005). Domain-wall displacement triggered by an ac current below threshold.Applied Physics Letters, 86, 232504.

Tatara, G., Takayama, T., Kohno, H., et al. (2006). Thresholdcurrent of domain wall motion under extrinsic pinning, beta-termand non-adiabaticity. Journal of the Physical Society of Japan,75, 064708.

Tatara, G., Vernier, N. and Ferre, J. (2005). Universality of thermallyassisted magnetic domain-wall motion under spin torque. AppliedPhysics Letters, 86, 252509.

Tchernyshyov, O. and Chern, G.W. (2005). Fractional vortices andcomposite domain walls in flat nanomagnets. Physical ReviewLetters, 95, 197204.

Thiaville, A. and Nakatani, Y. (2006). In Spin Dynamics in Con-fined Magnetic Structures III, Springer-Verlag: Berlin, pp. 161,Vol. 101.

Thiaville, A., Nakatani, Y., Miltat, J. and Suzuki, Y. (2005).Micromagnetic understanding of current-driven domain wallmotion in patterned nanowires. Europhysics Letters, 69, 990.

Thiaville, A., Nakatani, Y., Miltat, J. and Vernier, N. (2004).Domain wall motion by spin-polarized current: a micromagneticstudy. Journal of Applied Physics, 95, 7049.

Thiele, A.A. (1973a). Steady-state motion of magnetic domains.Physical Review Letters, 30, 230.

Thiele, A.A. (1973b). Excitation spectrum of magnetic domain-walls. Physical Review B, 7, 391.

Thomas, L., Hayashi, M., Jiang, X., et al. (2006). Oscillatorydependence of current-driven magnetic domain wall motion oncurrent pulse length. Nature, 443, 197.

Thomas, L., Luning, J., Scholl, A., et al. (2000). Oscillatory decayof magnetization induced by domain-wall stray fields. PhysicalReview Letters, 84, 3462.

Thomas, L., Rettner, C., Hayashi, M., et al. (2005). Observationof injection and pinning of domain walls in magnetic nanowiresusing photoemission electron microscopy. Applied Physics Let-ters, 87, 262501.

Thomas, L., Samant, M.G. and Parkin, S.S.P. (2000). Domain-wallinduced coupling between ferromagnetic layers. Physical ReviewLetters, 84, 1816.


Tserkovnyak, Y., Skadsem, H.J., Brataas, A. and Bauer, G.E.W.(2006). Current-induced magnetization dynamics in disordereditinerant ferromagnets. Physical Review B, 74, 144405.

Tsoi, M., Fontana, R.E. and Parkin, S.S.P. (2003). Magnetic domainwall motion triggered by an electric current. Applied PhysicsLetters, 83, 2617.

Van Waeyenberge, B., Puzic, A., Stoll, H., et al. (2006). Magneticvortex core reversal by excitation with short bursts of analternating field. Nature, 444, 461.

Vernier, N., Allwood, D.A., Atkinson, D., et al. (2004). Domainwall propagation in magnetic nanowires by spin-polarized currentinjection. Europhysics Letters, 65, 526.

Vila, L., Piraux, L., George, J.M. and Faini, G. (2002). Mul-tiprobe magnetoresistance measurements on isolated magneticnanowires. Applied Physics Letters, 80, 3805.

Wachowiak, A., Wiebe, J., Bode, M., et al. (2002). Direct observa-tion of internal spin structure of magnetic vortex cores. Science,298, 577.

Waintal, X. and Viret, M. (2004). Current-induced distortion of amagnetic domain wall. Europhysics Letters, 65, 427.

Wegrowe, J.E., Dubey, M., Wade, T., et al. (2004). Spin transfer byspin injection between both interfaces of a Ni nanowire. Journalof Applied Physics, 96, 4490.

Wegrowe, J.E., Gilbert, S.E., Kelly, D., et al. (1998). Anisotropicmagnetoresistance as a probe of magnetization reversal in indi-vidual nano-sized nickel wires. IEEE Transactions on Magnetics,34, 903.

Wegrowe, J.E., Kelly, D., Jaccard, Y., et al. (1999). Current-inducedmagnetization reversal in magnetic nanowires. Europhysics Let-ters, 45, 626.

Wernsdorfer, W., Doudin, B., Mailly, D., et al. (1996). Nucleationof magnetization reversal in individual nanosized nickel wires.Physical Review Letters, 77, 1873.

Whitney, T.M., Jiang, J.S., Searson, P.C. and Chien, C.L. (1993).Fabrication and magnetic-properties of arrays of metallic nano-wires. Science, 261, 1316.

Williams, H.J., Shockley, W. and Kittel, C. (1950). Studies ofthe propagation velocity of a ferromagnetic domain boundary.Physical Review, 80, 1090.

Wunderlich, J., Ravelosona, D., Chappert, C., et al. (2001). Influ-ence of geometry on domain wall propagation in a mesoscopicwire. IEEE Transactions on Magnetics, 37, 2104.

Xiao, J.A., Zangwill, A. and Stiles, M.D. (2006). Spin-transfertorque for continuously variable magnetization. Physical ReviewB, 73, 054428.

Xiong, G., Allwood, D.A., Cooke, M.D. and Cowburn, R.P. (2001).Magnetic nanoelements for magnetoelectronics made by focused-ion-beam milling. Applied Physics Letters, 79, 3461.

Yamaguchi, A., Nasu, S., Tanigawa, H., et al. (2005). Effect of Jouleheating in current-driven domain wall motion. Applied PhysicsLetters, 86, 012511.

Yamaguchi, A., Ono, T., Nasu, S., et al. (2004). Real-space observa-tion of current-driven domain wall motion in submicron magneticwires. Physical Review Letters, 92, 077205.

Yamaguchi, A., Ono, T., Nasu, S., et al. (2006a). Erratum: real-space observation of current-driven domain wall motion insubmicron magnetic wires (vol 92, pg 077205, 2004). PhysicalReview Letters, 96, 179904.

Yamaguchi, A., Yano, K., Tanigawa, H., et al. (2006b). Reduc-tion of threshold current density for current-driven domain wallmotion using shape control. Japanese Journal of Applied PhysicsPart 1-Regular Papers Brief Communications and Review Papers,45, 3850.

Yamanouchi, M., Chiba, D., Matsukura, F. and Ohno, H. (2004).Current-induced domain-wall switching in a ferromagnetic semi-conductor structure. Nature, 428, 539.

Yamanouchi, M., Chiba, D., Matsukura, F., et al. (2006). Velocityof domain-wall motion induced by electrical current in the fer-romagnetic semiconductor (Ga,Mn)As. Physical Review Letters,96, 096601.

Yokoyama, Y., Suzuki, Y., Yuasa, S., et al. (2000). Kerr microscopyobservations of magnetization process in microfabricated ferro-magnetic wires. Journal of Applied Physics, 87, 5618.

Youk, H., Chern, G.W., Merit, K., et al. (2006). Composite domainwalls in flat nanomagnets: the magnetostatic limit. Journal ofApplied Physics, 99, 08B101.

Zambano, A.J. and Pratt, W.P. (2004). Detecting domain-walltrapping and motion at a constriction in narrow ferromag-netic wires using perpendicular-current giant magnetoresistance.Applied Physics Letters, 85, 1562.

Zhang, S. and Li, Z. (2004). Roles of nonequilibrium conductionelectrons on the magnetization dynamics of ferromagnets. Phys-ical Review Letters, 93, 127204.

Current Induced Domain-Wall Motion in Magnetic

Documents

Transcript of Current Induced Domain-Wall Motion in Magnetic