4.magneticproperties:fromtraditionaltospintronic...

Magnetic Pro85

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4. Magnetic Properties: From Traditional to Spintronic

Charbel Tannous, Jacek Gieraltowski

This chapter reviews basic concepts used in thetraditional macroscopic magnetism in order tounderstand current and future developments ofsubmicronic spin-based electronics where in-terplay of electronic and magnetic properties iscrucial. Traditional magnetism is based on macro-scopic observation and physical quantities arededuced from classical electromagnetism. Physicalinterpretations are usually made with reference toatomic magnetism where localized magnetic mo-ments and atomic physics prevail, despite the factthat standard ferromagnetic materials such as Fe,Co, and Ni are not localized-type magnets (theyhave extended s and localized d electronic states).While this picture might be enough to understandsome aspects of traditional storage and electrome-chanics, it is no longer sufficient for the descriptionof condensed matter systems with smaller lengthscales progressing toward the nanometer range.The precise nature of magnetism (localized, free,or itinerant like Fe, Co, and Ni transition metals)with simultaneous presence of charge and spin ofcarriers should be considered. In addition, whenwe deal with thin films ormultilayers as in conven-tional electronics or with reduced dimensionalityobjects such as wires, pillars, dots, or grains, mag-netic properties are expected to be different fromthree-dimensional conventional bulk systems.

4.1 Traditional Magnetism.......................... 874.1.1 Fundamental Magnetic Quantities .......... 884.1.2 The Hysteresis Loop ............................... 894.1.3 Intrinsic Magnetic Properties

of a Material ........................................ 934.1.4 Traditional Types of Magnetism

and Classes of Magnetic Materials .......... 96

4.2 Nonconventional Magnetismand Progress Toward Spintronicsand Quantum Devices ........................... 99

4.2.1 Types of Exchange and Couplingin Magnetic Materials ............................ 99

4.2.2 Thin Magnetic Films ............................ 1004.2.3 Electronic Properties:

Localized, Free, Itinerant Magnetismand Spin-Polarized Band Structure ...... 101

4.3 Spintronicsand Quantum Information Devices ...... 104

4.3.1 Magnetic Signal Processing Devices....... 1064.3.2 Magnetic Quantum Dot Arrays .............. 1074.3.3 Magnetic Vortex Properties

and Applications ................................ 107

References ................................................... 109

Digital information technology is based on three majorconstituents:

� Processing of information (transistors, logic gates,CPU, RAM, digital signal processor (DSP), etc.)� Communication of information (networks,switches, cables, fibers, antennas, etc.)� Storage of information (hard disks, CD, DVD, Blu-ray, Flash, etc.).

Traditionally, magnetism has been confined to safestorage at the time of the development of bubble andferrite core technologies when RAM memory belonged

to the realm of magnetism. RAM memory is a specialtype of information storage called primary (nonperma-nent) storage to distinguish it from secondary (perma-nent) or mass storage such as hard disks, floppies, CDs,DVDs, Blu-rays, Flash, etc.

Presently, magnetism is undergoing a very deeptransformation because of several recent developments,among which is the progress toward the nanometerscale in integrated circuits. On that length scale, quan-tum effects become extremely important, and the spinof carriers becomes of interest since it might be con-served over that length scale and, consequently, can beused to carry and manipulate information.

© Springer International Publishing AG 2017S. Kasap, P. Capper (Eds.), Springer Handbook of Electronic and Photonic Materials, DOI 10.1007/978-3-319-48933-9_4

PartA|4

86 Part A Fundamental Properties

Moreover, magnetism is endowed with at least twoparticularly interesting features:

1. The absence of dipolar length scale stemming fromthe absence of monopoles in contrast with the elec-trical dipole, which leads to easy adaptation todevice scaling.

2. The atomic nature of dynamic response, which isequivalent to being universal and of potentially highspeed.

Those properties pave the way toward the fabrica-tion of new integrable and fast devices based on chargeand spin (spintronic devices) instead of charge only, likein traditional microelectronics. New types of junctions,transistors, logic gates, sensors, and devices can be builtin a way such that magnetism will be able to tackle pro-cessing of information along with its storage.

Quantum effects have already delivered many prod-ucts since they are the basis of the GMR effect (Ta-ble 4.1) that underlies the recent surge of extremelyhigher densities in hard disks; nevertheless they areslated for other deeper developments geared toward thebuilding of the basic parts of the quantum computer andquantum communication devices. The quantum com-

Table 4.1 Magnetic topics and corresponding selected references. Kittel [4.27] and Hurd [4.26] are recommended asgeneral introductory references

Topic Applications/Comments ReferenceAmorphous magnets Shielding, sensing, transformers, transducers, etc. Boll and Warlimont [4.1]Coherent rotation model Hysteresis loop Stoner and Wohlfarth [4.2]Coupling and exchange inmultilayers

Biquadratic exchange, exchange bias, etc. Platt et al. [4.3], Slonczewski [4.4],Koon [4.5]

Giant magnetoimpedance Sensing, detection, etc. Tannous and Gieraltowski [4.6]Giant magnetoresistance(GMR)

Sensors, spin filters, magnetic recording heads, etc. White [4.7]

Giant magnetostriction Smart wings, actuators, transducers, resonators, etc. Schatz et al. [4.8], Dapino et al. [4.9]Itinerant magnetism Magnetism in transition metals Himpsel et al. [4.10]Localized magnetism Atoms/molecules/ions/insulators Jansen [4.11]Losses in magnetic materials Eddy currents, hysteresis loss, etc. Goodenough [4.12]Magnetic recording Information storage technology Chap. 49 in this Handbook, Richter [4.13]Magnetic thin films Growth and characterization Himpsel et al. [4.10]Magnetoelastic effects Cantilevers, microelectromechanical systems, etc. Farber et al. [4.14], Dapino et al. [4.9]Microwave devices Communications, bubble memory, etc. Coeure [4.15], Wigen [4.16]Permanent magnets Relays, motors, transformers, etc. Gutfleisch [4.17]Quantum computing/communications

Quantum devices, magnetic RAM, etc. Burkard and Loss [4.18]

Sensors Field detector, probes, etc. Hauser et al. [4.19]Soft magnets Shielding, sensing, transformers, transducers, etc. Boll and Warlimont [4.1]Spintronics Spin diode, spin LED, spin transistor, magnetic RAM,

etc.Prinz [4.20], Zutic et al. [4.21],Wuet al. [4.22] and Dietl et al. [4.23]

Technology overview Permanent and soft magnets Kronmueller [4.24], Simonds [4.25]Types of magnetic order Ferromagnetism, antiferromagnetism, diamagnetism,

paramagnetism, etc.Hurd [4.26]

puter is based on the qubit or the quantum bit, whichis the basic unit of information replacing the classi-cal bit. If we consider a sphere, the classical bit canbe viewed as an object with two possible states at thenorth and the south pole of the sphere, whereas a qubitis an object that can sit anywhere on the surface ofthe sphere (called the Bloch sphere, see Nielsen andChuang [4.28]). A quantum computer can do massiveparallel computations since quantum mechanics is lin-ear. This means one is allowed at any time to access anysuperposition of states, in contrast to a classical Von-Neumann-type computer where one has access to onlya single state at any time (Table 4.1).

Quantum communications are extremely securesince a caller may build a coherent state with the calleeand any intrusion can perturb the coherence, providinga very efficient detection.

From the fundamental point of view, one may startwith a single magnetic moment representing a magneticmaterial and study its behavior before indulging intomany interacting moments that are the building blocksof magnetic materials. From an applied point of view,the orientation of the moment (left–right or up–down)defines the value of the bit. Once the orientation ofthe moment has been linked to a bit value, it becomes

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Magnetic Properties: From Traditional to Spintronic 4.1 Traditional Magnetism 87Part

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important to understand the physical processes, the en-ergetics, and the dynamics of the moment orientationchange (moment reversal from left to right or momentflip from up to down) in order to be able to control, alter,and predict the bit value.

This work is organized as follows. We present thetraditional elements of magnetism such as the hystere-sis loop, conventional types of magnetism and magneticmaterials, proceeding toward a brief description of nonconventional magnetism useful for understanding new

high-tech materials that will make the backbone of fu-ture spintronic and quantum information-based devices.In Sect. 4.3, we describe spintronics in more detail,drawing from analogies with standard electronics anddescribing magnetic quantum dot arrays with a singleapplication based on magnetic vortex devices in thecylindrical dot case. Table 4.1 contains topical worksfor readers who are interested in additional informa-tion that is briefly mentioned or not covered in thiswork.

4.1 Traditional Magnetism

Classical magnetism relates to magnetic moments andtheir behavior under external field or mechanical stressor some other perturbing effect. Basically, one writes anenergy for the magnetic moment and tries to understandits behavior with time. In order to describe the differ-ent magnetic energy terms that control the behavior ofa moment, we can start from a single isolated momentin vacuum, at zero temperature T D 0K, zero appliedmagnetic field (H D 0), and zero frequency .f D 0/. Itis obvious that the energy is zero from the classicalpoint of view (quantum mechanically, one might saythat even at T D 0K, there exist quantum fluctuationsthat might disorient, flip, or reverse the moment).

The next step is to apply a magnetic field (T D 0K,f D 0); then we obtain the Zeeman energy EZ D �M �H, with M representing the moment. If we place themoment in an unbounded (infinite sized) crystal, thereappears a magnetocrystalline anisotropic energy [4.27]characterized (in the uniaxial case such as cobalt) bya constant K (4.5) since the moment will point ina direction (called an easy axis direction) imposed bythe crystal’s internal symmetry, in contrast to vacuumwhere M is free to point in any direction. When thecrystal has a set of easy axes (or easy planes), the mo-ment will point in one of several directions (or anydirection in one or several planes). This occurs in fer-romagnetic materials such as iron or nickel possessingcubic anisotropy.

If the body containing the moment has a finite size,a new energy will appear: the demagnetization energy.Magnetic surface charges (poles) induced on the surfacebounding the body create a demagnetizing field inside it(outside, it is called a stray field). The demagnetizationenergy is also called shape anisotropy energy or magne-tostatic energy because it is expressed like anisotropyenergy as 2�N˛ˇM˛Mˇ , where N˛ˇ is a set of fac-tors (demagnetization coefficients) that depend on bodyshape (Einstein summation convention is used for re-

peated indices). In the ellipsoidal shaped body case, thelatter expression is exact.

Now suppose we include several local moments ina material. If sites i and j carrying momentsMi andMj,respectively, are close enough (say less than 1 nm), wehave an interaction energy called exchange energy (ofelectrostatic origin) �AijMi:Mj that will align momentsMi and Mj if Aij is positive (ferromagnetic interaction).If Aij is negative, the moments will align antiparallel toeach other (antiferromagnetic interaction). In a ferro-magnet, we have a nonzero net moment, whereas in anantiferromagnet the net moment is zero.

The above description considers localized mag-netism in distinct atoms (such in a gas), ions, molecules,or in special materials (like insulators or rare-earthsolid state compounds possessing external shell f elec-trons with a highly atomic-like character). If we havea conducting material with free electrons interactingwith localized atoms/ions/molecules, a different type ofmagnetism, called itinerant magnetism, appears (e.g.,transition metals with s and d electrons or magneticsemiconductors, etc.). Nevertheless, it is possible toextend the notion of magnetic moment to that case ac-counting for the combined effects of free and localizedcharges modeled as an effective number of Bohr mag-netons (Sect. 4.1.1 and Table 4.1).

The different physical mechanisms and types ofmagnetism briefly described above operate at differentlength scales. In order to put things into perspective andponder about what lies ahead in terms of possible devel-opments and hurdles, the summary in Fig. 4.1 displaysdifferent mechanisms, characterization methods, andmanufacturing processes along with the correspondinglength scale. Note that, on the nanometer scale, devicesize becomes comparable to most ranges of interactionsencountered in magnetic materials and this will triggerdevelopment of novel effects and fabrication of new de-vices (Sect. 4.3).

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Length (nm)10–1 100 101 102 103 104

I

Exchange

RKKY

Magnetism

Transport

Domain walls

Single domains

Exchange length

Domain sizes

Spin diffusion

Mean free path

Photolithography

Particle beam lithography

Holography

Self-assembly

Scanning probes

X-ray diffraction

Neutron diffraction

Electron microscopy

Magneto-optics

X-ray diffraction

Neutron diffraction

Electron microscopy

Scanning electron microscopy

Scanning probe microscopy

LSDA

Classical spin dynamics

Micromagnetics

II

III

IV

V

Fig. 4.1 Typical lengths (in nm) encountered in magneticmaterials and spintronics with corresponding basic mag-netic phenomena. Structures, interactions, and transportare gathered under group I. Group II is for growth andfabrication methods, whereas diagnostic techniques appearunder groups III and IV. Group III is for depth-profilingtechniques whereas group IV is for surface-probing tech-niques. Finally, group V is for theoretical analysis andcharacterization techniques. LSDA (local spin density ap-proximation) is the spin counterpart of the density functiontheory (DFT) used for traditional band structure calcula-tion. Since most interaction lengths are in the 1�100 nmrange, the nanoelectronics device size is comparable to theinteraction range. (After [4.29])

4.1.1 Fundamental Magnetic Quantities

Magnetization is the fundamental property exhibitedby a magnetic substance originating from its electrons,similar to the electric dipole moment. It can be in-trinsic (existing without any external field) or inducedby an external magnetic field. In atoms/molecules/ionsand rare-earth solids, magnetization is made from indi-vidual atomic-like localized magnetic moments. In thesolid state, the moment is generally defined from theband structure, and an effective moment can be definedin terms of a Bohr magneton �B D e„=2m, the mag-

Table 4.2 Selected ferromagnetic solids with their satura-tion magnetization Ms, effective Bohr magneton numbernB, and Curie temperature Tc. (After [4.27])

Substance Ms

(emu=cm3)at 300K

Ms

(emu=cm3)at 0 K

nB at0K

CurieTc

(K)Fe 1707 1740 2:22 1043Co 1400 1446 1:72 1388Ni 485 510 0:62 627Gd – 2060 7:63 292Dy – 2920 10:2 88MnAs 670 870 3:4 318MnBi 620 680 3:52 630CrO2 515 – 2:03 386EuO – 1920 6:8 69

netic moment carried by a single electron (e and m areelectron absolute charge and mass). Magnetization isa thermodynamic quantity that changes with tempera-ture, mechanical stress, and chemical processes such asalloying [4.30].

When a field is applied to a magnetic substance,the largest acquired magnetization measured along thedirection of the applied field is the saturation mag-netization Ms, implying that all moments are alignedparallel to the field. Since temperature T misalignsmoments, one defines a T D 0K saturation momentNnB�B, with N the number of ions atoms or molecules(for a mole N D NA Avogadro number) and nB the ef-fective number of Bohr magnetons �B. The saturationmagnetization Ms value is NAnB�B per unit volume(Table 4.2).

nB is different for atoms/ions/molecules and solidsand determined by the electronic state. For example,it is 5 for Fe3C ions (according to Hund rules, seeJansen [4.11]) since we have a 3d5 atomic configura-tion (without orbital contribution), whereas for Fe atomit is 2 (3d8 configuration). Trivalent rare earth ionswith highly localized 4f orbitals possess magnetic mo-ments determined by gJ

pJ.J C 1/, with gJ being the

total Landé factor and J the total angular momentum.Transition metal ions possess magnetic moments de-termined by 2

pS.SC 1/ with S being the total spin of

the ion (no orbital contribution because of orbital mo-ment quenching, see Kittel [4.27]). In the solid state,such as ferromagnetic metals, nB is determined from theband structure (Table 4.1). In lower dimensions (clus-ters, dots, thin films, etc.), nB tends to increase becauseof lower symmetry and coordination, since high sym-metry and coordination tend to decrease it. Among theelements, rare earth ions (Dy3C, Ho3C, and Er3C) pos-sess large values of nB on the order of 10, whereas insolids, Gd-based garnets [4.16] have nB on the order of15 (at T D 0K).


A|4.1

The moment due to an angular momentum isgiven in the localized case (atoms/ions/molecules) by�gL�BL (for an orbital angular momentum L/ withgL D 1 (the orbital Landé factor), or by �gS�BS (fora spin angular momentum S) with gS D 2 (the spinLandé factor). In the case of an atom/ion with a to-tal angular momentum J D LCS, the moment is givenby �gJ�BJ with

gJ D 3

2C ŒS.SC 1/� L.LC 1/�

Œ2J.J C 1/�(4.1)

the total Landé factor. In the solid state, the Landé fac-tor is determined from the band structure (spin splittingof electron or hole bands in the presence of a magneticfield). It may become negative, anisotropic (tensor),and depends on wavevector k. In semiconductors suchas InSb: g D �44, GaAs: g D 0:32, InAs: g D �12 atwavevector k D 0 [4.11, 21].

4.1.2 The Hysteresis Loop

When a magnetic field is applied to a ferromagnetic ma-terial [4.27], a magnetization change takes place. Thebest way to understand the underlying phenomena is toplot the value of the magnetization M projected alongthe direction of the applied field H. The locus of themagnetization M measured along the direction of theapplied magnetic field M.H/ versus the field and de-picted in the M–H plane is the hysteresis loop. Theterm hysteresis means that when the material is field cy-cled (i. e., the field H is increased then decreased), twodifferent nonoverlapping curves M.H/ are obtained.Another point of view involves rather the induction Bversus H since it is B that is, in fact, measured. Severalpoints in the M–H plane (Fig. 4.2) are worth men-tioning. Point d is where the magnetization reaches itslargest valueMs (saturation magnetization). Points i andf denote the magnetic fields ˙Hc for which the magne-tization is zero. Magnetic field Hc is called the coercivefield. It is large for hard materials and small for softmagnets. Points e and h denote remanent magnetizationMr, that is, the leftover magnetization after the appliedfield is switched off, thus it is the hallmark of permanentmagnets.

The ratio Mr=Ms, called squareness, is close to 1when the applied magnetic field is close to the easy axisand therefore the hysteresis loop is squarer. Once theeasy axis (or set of easy axes) is determined, the anglebetween the easy axis and the magnetic field (say �)is varied and the hysteresis loop is graphed for differ-ent angles. When the angle � is increased, the openingof the hysteresis loop is reduced; it is largest whenthe magnetic field is most parallel to the easy axis and

g

–Hsat

–Hc f

e

d

i

HsatHcO

Ms

Mr

–Ms

–Mrh

μini

H

Fig. 4.2 Hysteresis loop with the fundamental points inthe M–H plane such as saturation magnetization Ms, re-manent magnetizationMr, coercive fieldHc (at whichM D0), and the saturation field Hsat (for which M D Ms). Fora virgin (unmagnetized) material, the initial magnetizationcurve (dashed line) with permeability �ini are displayed.(After [4.31])

μmax

M (Magnetization)

H (Field)

Ms(saturation)

Hk (anisotropy)

Hc (coercive)

Mr (remanant)

Hr

Fig. 4.3 Hysteresis loop obtained for an arbitrary angle�, between the magnetic field and the anisotropy axis(easy axis). Associated quantities such as coercive fieldHc, remanent magnetization Mr, and field Hr (given bythe intersection of the tangent to the loop at �Hc and theMr horizontal line) are shown. The tangent to the loopat �Hc is called maximum differential permeability �max.The thick line is the hysteresis loop when the field is alongthe hard axis and (the anisotropy field Hk) is the field valueat the slope break where the magnetization reaches its sat-uration value Ms. Quantities such as Hc, Hr, Mr , and �max

depend on the angle �. When the field is along the easyaxis, the coercive field reaches its largest value (the hys-teresis loop is broadest) at which the magnetization jumps(at ˙Hc)

smallest when the magnetic field is most parallel to theso-called hard axis. As shown in Fig. 4.3, characteristicsof the hysteresis loop are depicted and for a given tem-perature and frequency of the applied field H, quantities

PartA|4.1


such as the remanent magnetization Mr, the remanentfield Hr, the coercive field Hc, and the maximum differ-ential permeability �max all vary with angle �.

In addition, when the temperature varies or the fre-quency of the field is varied, the hysteresis loop changesand might even disappear altogether above a given tem-perature (Curie temperature) or be seriously altered bythe frequency of the field (see below).

Magnetization Process Point of ViewWhen a magnetic field is applied to a material, a mag-netization process is initiated. That means a magneti-zation change can occur taking any of the followingroutes. At low applied magnetic fields (and low fre-quencies), domain boundaries bulge (this is a reversibleregime implying that if we decrease the field again,the magnetization will go back to the initial statefollowing the same path). This is called domain nu-cleation (high-slope path) or pinning process (smallslope path). At higher fields, the walls are depinnedand free to move. This is an irreversible regime (incontrast to the low field case) meaning that if we de-crease the field, the magnetization versus field variationwill not follow the same path. This regime corre-sponds to free domain wall movement. At higher fields(or frequencies), magnetization changes through mo-ment rotation and thus we have two possibilities: ifthe material is homogeneous, all happens as if wehad a single moment in the material. This is the co-herent rotation regime (well described by the Stoner–Wohlfarth model, see Table 4.1). On the other hand,if magnetization changes in a nonuniform way (dif-ferent points in the sample undergo different magne-tization changes), we have a curling process. Finally,if the magnetic field is applied along the anisotropyaxis (also called the easy axis) and suddenly reversed,the magnetization change is referred to as a switch-ing process. This means the moment jumps fromone value to its negative without any rotation pro-cess since magnetization is already along the easy axis(Fig. 4.4).

Energetic Point of ViewHysteretic behavior can be viewed as stemming fromthe motion of the energy minimum versus field. InFig. 4.5, the field is cycled in a clockwise fashion. Theenergy minimum is displayed for several values of thefield. Hysteresis is shown to arise from the asymmet-ric behavior of the energy versus magnetization as thefield is cycled. Hysteresis loss is given by the area cir-cumscribed by the hysteresis loop. Since soft materialshave thinner hysteresis loops (equivalent to smaller co-ercivity) with respect to hard materials, their losses aresmaller.

M

H

Ms

Mr

d

ce

fa

b

Depinning

Pinning centers–Hc

e

H

O

Fig. 4.4 Reversible and irreversible motion of the mag-netization M with the field H. For small fields, H < Hc

and initial magnetization, Rayleigh’s law is valid along a–b but for large H, saturation inM is induced by irreversibledomain boundary motion along b–c, followed by rotationof magnetization along c–d. When the field is reversedalong d–e–f toward �Hc, magnetization motions are notexactly those we had previously, and this is why a differ-ent M.H/ curve is obtained. Reversible movement at a lowfield means that domain walls are pinned by impurities andsimply bulge under the action of H (0–a), whereas jerky ir-reversible movement is associated with depinning of thedomain walls (a–b), creating Barkhausen noise. Free ir-reversible movement beyond pinning centers is depicted(right). (After [4.31])

Signal Processing Point of ViewConsider a time-dependent applied field H.t/ as aninput signal to the magnetic material and the magnetiza-tion M.t/ as the output. Then relation M.t/ D F.H.t//pertains to a nonlinear filter. A magnetic material actsas a nonlinear filter since M is not proportional to Hexcept at very low fields. A simple illustration of non-linearity is to observe M.t/ as a square signal witha sinusoidal input excitation H.t/ (Fig. 4.6). In addi-tion, the material imposes a propagation delay to thesignal proportional to the width of the hysteresis loop(twice the coercive field). Hysteretic behavior meansthat different values of the output are obtained when in-put excitation is varied (increasing or decreasing). Thisis generally exploited in control and power systems, forinstance.

Information Storage Point of ViewSince hysteresis means that different output behavior isobserved as the input is increased or decreased, this im-plies that one obtains a nonzero response (M ¤ 0/ atzero excitation (H D 0) called remanence (with valueMr when H D 0). The major advantage of remanence


A|4.1

(in addition to its usefulness in permanent magnets) isthat information (value ofMr) is stored without any ex-citation source (H D 0). Getting Mr when the signal isdecreased (say Mr1), we ought to have a different Mr

when the signal is increased (say Mr2) because of hys-teresis. Thus we have a bit value (0 for Mr1 and 1 forMr2). This contrasts sharply with (non-Flash) electronicstorage where a voltage has to be maintained in orderto keep the charge (representing information) in place(Chap. 49).

Electromagnetic Compatibilityand Frequency Synthesis Point of View

Hysteretic behavior can be viewed as a flux � (cor-responding to magnetization M) induced in a circuitwith an exciting current i.t/ at time t (corresponding tofield H). The resulting relationship �Œi.t/� is a nonlinearcharacteristic that can be expanded as

�Œi.t/� D a0 C a1i.tC �/C a2Œi.tC �/�2

C a3Œi.tC �/�3 C a4Œi.tC �/�4 C � � � ;(4.2)

where � accounts for the response delay of the flux (thelarger the delay, the broader the hysteresis curve). Or-dinary linear inductance of the circuit corresponds tofirst derivative [d�=di] that equals a1 for a short delay� . Higher order terms define a nonlinear inductor be-havior such that (for a short delay)

L.i/ D a1 C 2a2iC 3a3i2 C 4a4i

3 C � � � : (4.3)

A nonlinear inductor response is interesting in itselffrom the frequency synthesis point of view since it cangenerate harmonics and subharmonics off the excitingcurrent i.t/ (see, for instance, Chua [4.32]).

Time-Dependent Point of ViewIn spite of different points of view describing hysteresis,some experimentally observed phenomena cannot beexplained with the above, such as a particularly impor-tant phenomenon known as the magnetic after-effect.Experimentally, one observes a magnetization changewith time M.t/, despite the fact that the applied fieldH is kept constant in time, implying a time-dependenthysteresis loop stemming solely from M.t/ (Fig 4.7).Therefore, the material must contain units (single do-main grains) that respond differently in time, as if eachunit had its intrinsic delay; thus magnetization mightbe written in the form M.t/ D M0S.t/ ln.t/, where S.t/is called the magnetic viscosity. If the response timeof each grain is considered of the thermal activation

1

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–0.5

1–1 –0.5 0.5 10

5

0

–5

–1 10

M

H MM

H MM

H MM

H MM

H M

a) b)

Fig. 4.5 Hysteresis loop (a) and energy as a function of magneti-zation (b). When field H is varied in the hysteresis loop (indicatedby the round spot) in a clockwise fashion, the energy changes itsform and its absolute minimum indicated by the thick arrow movesaccordingly. Hysteresis is due to asymmetric energy behavior as thefield is cycled

type, that is, following the Néel–Arrhenius formula(Chap. 49) � D �0 exp. E=kBT), where �0 is an at-tempt time to cross some energy barrier E (Fig. 4.5)at a temperature T , then after-effect implies that wehave a distribution of �0 and E values. Calling thecorresponding probability density function g.�/ andM0

some scale initial magnetization, M.t/ is consequently

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1

0.5

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–0.5

–10 500 1000 1500 2000

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0.5

0

–0.5

–1–1 –0.5 0 0.5 1

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–1–1 –0.5 0 0.5 1

M

t H

H

M

t

H(t)

M(t)

M(t)

H(t)

a) b)

c) d)

Fig. 4.6a–d Hysteresis loop as an input–output relationship.(a,c) The magnetic field H.t/ and magnetization M.t/ are displayedexplicitly as functions of time t. (b,d) Resulting input–output rela-tionship (M–H characteristic) parametric curve M.H/. In (a,b) wehave an output .M.t//, undelayed and proportional to the input H.t/resulting in a paramagnetic (no hysteresis) M–H characteristic.(c,d) is an output M.t/, delayed and nonlinearly distorted ver-sion of the input H.t/ resulting in a hysteretic characteristic M–H.(After [4.33])

defined as

M.t/ D M0

1Z

0

g.t/exp� t

�

�d� : (4.4)

In the simple case where we have a single response time��, that is, g.�/ D 2ı.� � ��/,M.t/ behaves asM.t/ DM0 exp.�t=��).

After analyzing different aspects of the hysteresisloop, we move on to tackle physical processes affectingmagnetization behavior. Let us start with small mag-netic fields H < Hc with respect to the coercive field.The initial magnetization curve follows Rayleigh’s law:M.H/ D �0HC˛RH2, where �0 is the low-field suscep-tibility and ˛R the Rayleigh coefficient. Since this law isvalid for small fields, it describes the reversible motionof the magnetization.

We move next to description of a domain, which issome physical region inside a material where the mag-netization is oriented along some direction.

Typically, when the extent of the material is smallerthan the exchange length, one expects a single-domainstructure (that contains about 1012�1018 atoms, see alsoFig. 4.1).

In magnetic materials described by a uniaxialanisotropy K, the expression of the exchange lengthis `ex D p

A=K with A being the exchange stiffness

constant (typically, A � 10�6 erg=cm). In soft materi-als with anisotropy constant K � 0, the magnetostaticexchange length is used. It is defined as

`ex Ds

A

2�M2s

given the magnetostatic energy density is 2�M2s .

For instance, a recording medium is considered asbeing made of small grains made of single domains. Asthe recording density is increased, the grain size makingthe recording media decreases and if it is small enough,its magnetization becomes sensitive to thermal energy,that is, it can flip or reverse simply by the effect oftemperature. This is called the superparamagnetic ef-fect that traditionally limits longitudinal recording to100Gbit=in2 and (current) perpendicular recording to1 Tbit=in2. It was partially solved recently through theintroduction of antiferromagnetic coupling between thestorage layer and a stabilization layer. Practically, a thinfilm of ruthenium (7�9Å thick) is deposited betweenthe storage layer and the stabilization layer (Chap. 49).If the extent of the magnetic material is large, oneexpects to have a multidomain structure separated bydomain walls. A multidomain structure emerges in or-der to minimize the magnetostatic long-range dipolarinteraction energy between the different moments ex-isting in the different domains [4.34].

Since a domain is a piece of a material with the mag-netization along a given direction, generally a magneticmaterial contains many regions where the magnetiza-tion is along some direction in order to minimize themagnetostatic energy. Regions with different magneti-zation orientations can be close to one another, albeitwith a boundary called a domain wall (typically con-taining about 102�103 atoms, see Fig. 4.1). Saturationoccurs when all these regions align along some direc-tion imposed by the external applied field. The domainwall width (in CGS units) is equal to �

pA=K, where

A is the typical (nearest neighbor) exchange stiffnessand K the typical uniaxial anisotropy (Sect. 4.1.3). Itresults from competing exchange and anisotropy en-ergies. It is thinner for higher anisotropy or smallerexchange (in Fe it is about 30 nm, whereas in a hardmaterial such as Nd2Fe14B, it is only 5 nm). Its energyis equal to 4

pAK. For bulk materials, it is of the Bloch

type, whereas for thin films (i. e., with thickness on theorder of the exchange length) it is of the Néel type(Fig. 4.8), again with a width proportional to exchangelength

pA=2�M2

s . A single parameter Q D 2K=M2s al-

lows us to discriminate between simple .Q< 1/ andcomplex wall profiles .Q> 1/ [4.35].

During a magnetization process, the irreversiblejerky movements of domain walls due to local instabil-

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A|4.1

a) b)Domain wall (180)

N

M

S

N

S

S

N

c) d)

90 domain wallClosure domains Closure domains

NS

N

S

S

N

SN

Fig. 4.7a–d Possible domain structures and wall structure in a magnetic material. (a) Initial magnetic configuration witha large stray field (outside the material). (b) In order to reduce magnetostatic energy (and stray field), domains form, asin a highly anisotropic material. If n domains are formed, the magnetostatic energy is reduced by n. (After [4.27]) (c) Ina material with small anisotropy, 45ı closure domains are formed to minimize the magnetostatic energy. (d) As moredomains are formed, more walls are also formed until a compromise is reached since it is necessary to minimize the sumof the magnetostatic energy and wall energies. (After [4.31])

ities created by impurities, defects, inclusions, or inter-actions between domain walls lead to Barkhausen noisein a hysteresis loop cycle. In devices, it is preferred toinduce magnetization change by rotation processes (im-plying that the external magnetic field must be appliedalong hard axis) that are less noisy.

4.1.3 Intrinsic Magnetic Propertiesof a Material

The induction B in a linear isotropic material is re-lated to H through the relation B D �0.HCM/, where�0 is the free space permeability and M is the mag-netization of the material. Using the relation M D �Hrelating the magnetization to the applied field, onegets B D �0.1C�/H. This leads to the definition ofthe total permeability � D �0.1C�/, which yields theconstitutive relation B.H/ D �H. The quantity � iscalled the susceptibility. In general, nonlinear materialpossesses nonlinear constitutive equations M.H/ andB.H/. A constitutive relation is like the I.V/ charac-teristic of an electronic device. For a linear material,the permeability � is generally frequency dependent. Iflosses occur during propagation of an electromagneticwave in a material resulting from the absorption of mag-netic energy, it is possible to extend the definition ofpermeability to the complex plane (like in dielectrics).Losses are attributed to peaks in the imaginary part ofthe permeability at the absorption frequencies [4.12].At low frequencies, losses are attributed to domains,whereas at high frequencies, losses are attributed to

a)

b)

Fig. 4.8a,b Possible domain wall shapes in a magneticmaterial. (a) Bloch type for bulk materials with the mag-netization rotating in a vertical plane with the associatedpoles. (b)Néel type for thin films. For these walls, the mag-netization rotates in a horizontal plane when the width ofthe film is smaller than the exchange length. (After [4.36])

rotation processes. Since permeability relates the twovector quantities B and H, in a linear anisotropic mate-rial (such as a linear crystal), the relation B.H/ D �Hbecomes B˛ D �˛ˇHˇ (Einstein summation conven-tion) and �˛ˇ is a rank-2 (r D 2) permeability tensorthat has different components according to the relativedirections of B andH (nine components in total or 3r inthree dimensions). For a nonlinear anisotropic material,we have �˛ˇ D @B˛=@Hˇ .

When a material is run through a field cycle (hys-teresis loop) for the first time, the magnetizationM fol-lows a path different from the above M.H/ lines. Thatpath represents the initial magnetization and the associ-

PartA|4.1


ated permeability is the initial permeability (Fig. 4.2).If a magnetic material is subjected to a static magneticfield H and a small time-dependent field H.t/ (alternat-ing at frequency f ) parallel toH, the magnetic responseis called the longitudinal permeability (at frequency f ).This kind of excitation is encountered in ferromagneticresonance (FMR) problems [4.16].

Alternatively, one may describe magnetic behaviorin terms of susceptibility. It relates the magnetizationto the applied field in a linear material through M D�H. For a nonlinear isotropic material, we have � D@M=@H and for a nonlinear anisotropic material, wehave �˛ˇ D @M˛=@Hˇ with the M component alongdirection ˛ and theH component along ˇ. Any jump in� signals an important magnetic change in the material.When the temperature is decreased, the Curie tempera-ture signals the onset of ferromagnetic order from para-magnetic disorder, and the Néel temperature signalsthe onset of antiferromagnetic order from paramagneticdisorder.Magnetic susceptibility spans several orders ofmagnitude (from about 10�5 to 106 cm3=mol).

The magnetization settles along a direction in orderto minimize the sum of the magnetic anisotropy energyand the Zeeman energy provided by the applied field.Since the hysteresis loop changes its shape as we varythe applied field angle, it might reach its largest widthfor some angle. The field then indicates the easy axisdirection, also called the anisotropy axis. The extremevalue of the magnetic field is the coercive field ˙Hc atwhich the magnetization (switches) jumps to its satura-tion value ˙Ms (Fig. 4.9).

When the field is along the hard axis direction, ittakes a longer time to align the magnetization slowlyby rotation, in sharp contrast to fast switching that oc-curs when the field is along the easy axis direction. Itis easily determined with the hysteresis loop when it is

M

H–Hc Hc

Ms

–Ms

Fig. 4.9 Easy axis hysteresis loop. When the magneticfield is along the anisotropy axis (easy axis) of the mate-rial, the hysteresis loop is broadest, and the magnetizationjumps (switches) to ˙Ms at the coercive field ˙Hc

narrowest as the angle of the applied field is varied. Atslope break we have the value of the anisotropy field˙HK (Fig. 4.10).

Crystals are generally anisotropic because their mi-croscopic structure is not the same along different di-rections. Hence, different properties will show up alongdifferent orientations of the crystal. Magnetic propertiesare no special case, and there are special preferred di-rections along which the magnetization will point. Thismeans there exists some energy contribution that will beminimized when the magnetization settles along thesedirections. This is the anisotropy energy. On top of be-ing inherent to crystal structure, anisotropy can also beinduced by an external field, a change of symmetry(e.g., surface with respect to bulk), or a mechani-cal deformation. Microscopically, anisotropy originatesfrom spin–orbit coupling. Coercivity increases withanisotropy, that is, hardness increases with anisotropy(Fig. 4.11). Anisotropy energy can be evaluated fromthe hysteresis loop by determining the loop for variousangles between the applied field and the easy axis. Ingeneral, it is an expansion of the form

X

˛;ˇ

K˛ˇm˛mˇ CX

˛;ˇ;�;ı

K˛ˇ�ım˛mˇm�mı � � � ;

(4.5)

where K˛ˇ;K˛ˇ�ı are the anisotropy coefficients (sec-ond and fourth-order, respectively) and the coefficientsm˛ , mˇ , m� ; : : : are the components of M=Ms, themagnetizationM normalized by the saturation magneti-zationMs. ˛; ˇ; � are the indices along the crystal x; y; zaxes. This means the norm of m,

jmj DqŒm2

x Cm2y Cm2

z � D 1 :

M

H–HK HK

Ms

–Ms

Fig. 4.10 Hard axis hysteresis loop. When the magneticfield is along the hard axis of the material, the hysteresisloop is the thinnest and the magnetization rotates smoothlyfrom a value to another between ˙HK the anisotropyfield


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Uniaxial and Biaxial AnisotropiesAnisotropy energy in the case of crystals with a singledominant axis such as hexagonal (like cobalt), tetrago-nal, and rhombohedral crystals is called uniaxial. To thelowest approximation, it is given by K1 sin2 , whereK1 is the anisotropy strength, angle being the an-gle between the magnetization and the easy axis. Thenext higher approximation allows fourth-order terms tobe added to the K1 sin2 term. In hexagonal crystals,the fourth-order term is of the form K2 sin4 . Rhom-bohedral symmetry allows two fourth-order terms ofthe form K2 sin4 CK4 cos sin3 cos 3�. In tetrag-onal crystals, the fourth-order terms are of the formK2 sin4 CK4 cos sin4 cos 4�. The easy axis is takenalong the z-axis, the angle � is the magnetization polarangle in the x–y plane, and is the azimuthal angle.

Anisotropy energy in the case of crystals with twodominant axes such as monoclinic, triclinic, and or-thorhombic crystals is called biaxial. It also meansthat two anisotropy directions exist and are competing.Some other cases are given below (see also Table 4.1for other definitions).

Cubic Anisotropy. Anisotropy energy in the case ofthe most symmetric crystals, cubic crystals, is called assuch because it relates to the cube’s axes. The energy tofourth order is

K1.m21m

22 Cm2

1m23 Cm2

2m23/CK2m

21m

22m

23 ;

where the coefficientsm1, m2,m3 are the components ofthe normalized magnetizationM=Ms with respect to thecrystal axes. That is m1 D sin cos�, m2 D sin sin�,m3 D cos . The easy axis, as above, is taken along thez-axis and the angle � is the polar angle in the x–y plane.

Shape Anisotropy. A finite-size magnetic body (el-lipsoidal shape is assumed for simplicity) possessinga single uniform magnetization M (represented by itscomponents M˛/ contains a magnetic energy (alsocalled magnetostatic energy) given by 2�N˛;ˇM˛Mˇ

(Einstein summation). The N˛ˇ coefficients are the de-magnetization (akin to depolarization) coefficients ofthe body determined by its shape. The origin of theterminology is due to the resemblance to the familiaranisotropy energy of the form K˛ˇM˛Mˇ (Einsteinsummation). If a magnetic material contains N mo-ments (atoms, ions, or molecules each carrying a mo-ment �i, the energy originating from dipolar couplingenergy between the different moments is written as

Wdip D 1

2

NX

iD1

NX

jD1

�i � �j

r3ij�3

.�i � rij/.�j � rij/r5ij

: (4.6)

107

105

103

101

10–1

60

50

40

30

20

10

0

(BH)max (MGOe)480

400

320

240

160

80

0

(BH)max (kJm3)

1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Hc (A/m)

101 103 105 107 109

K1 (J /m3)

Steels FerritesAlnico

Sm-Co

Nd-Fe-B

Sm-Fe-N

Steels

Ferrite

Alnico

Sm-Co

Nd-Fe-B

Soft ferritespermalloynanocrystalline alloysamorphous alloys

Metals +transition metal alloy

Hard ferritesrecording media

Intermetallics rare-earth-trans. metals

a)

b)

Fig. 4.11 (a) Development of hard materials for permanent mag-nets indicated by the value of the energy density .BH/max. Belowprogress is indicated by equivalent volume change with fixed en-ergy density. (b) Correlation between coercivity Hc and anisotropycoefficient K1 for various materials, their magnetic hardness, andcorresponding applications. (After [4.17, 24])

where �i and �j are two moments (i ¤ j) in the materialseparated by a distance rij. This energy can be repre-sented by constant coefficients (N˛ˇ/ only in the casewhen the body has an ellipsoidal shape. Hence one canwrite

Wdip � 2�N˛ˇM˛Mˇ .Einstein convention/

where the single magnetization M is the sum of all in-dividual moments �i.

Surface Anisotropy. A finite-size magnetic body witha not too large bulk anisotropy will have a reorientationof the magnetization close to its surface in order to min-

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imize the magnetostatic energy. This can be describedby a surface anisotropy that is different from the bulkone. It originates from an abrupt change of symmetry atthe interface between the bulk and free surface.

Helical Anisotropy. In a magnetic wire, the magneti-zation may prefer to lie radially in a plane perpendicularto the wire axis. This is the case of a radial easyaxis (radial anisotropy). If it lies in the circumferen-tial direction along the tangent to circles in the planeperpendicular to the wire axis, then we have circularanisotropy. However, if it lies in a plane tangent to thelateral surface of the wire (parallel to the wire axis) andmaking some angle different from 0ı or 90ı with anyplane perpendicular to the wire axis, then we have heli-cal anisotropy.

Anisotropy from DemagnetizationDemagnetization energy is expressed with coefficientsthat describe the demagnetization field Hd insidea finite-size material and created by the bulk magne-tization against an applied external magnetic field. Thecomponents of the demagnetization field (in the ellip-soidal case) are given by (with Einstein summation)ŒHd�˛ D �2�N˛ˇMˇ (it is much like the depolariza-tion field in the electrical case). The constant coeffi-cients’ (N˛ˇ/ representation is valid only in the casethe body has an ellipsoidal shape. The coefficients de-pend on the geometry of the material. Usually one hasthree positive coefficients along three orthogonal di-rections .Nxx;Nyy;Nzz/ (assuming that the off-diagonalterms are all 0) for simple geometries such as wires,disks, thin films, and spheres. All three coefficients arepositive and smaller than 1, and their sum is equal to1. For a sphere, all three coefficients are equal to 1/3.For a thin film (or a disk), they are given by (0,0,1) ifthe z-axis is perpendicular to the film (or the disk) lyingin the x–y plane. For an infinite length cylindrical wirewith its axis lying along the z direction, the values are(1/2,1/2,0).

4.1.4 Traditional Types of Magnetismand Classes of Magnetic Materials

The main traditional types of magnetism are:

� Ferromagnetism� Antiferromagnetism� Ferrimagnetism� Paramagnetism� Diamagnetism.

However, a larger number of types are given in thereview by Hurd [4.26] and with promising advances inmaterials research, we expect new emerging classes as

described in Sect. 4.2 of this work dedicated to non con-ventional magnetic types (Table 4.1).

A ferromagnet is an assembly of magnetic momentsinteracting with a positive exchange integral that willminimize their energy by adopting a common paral-lel configuration with a net large value of the totalmagnetization. Such a definition is valid for localizedmagnetism and not for itinerant ferromagnets (suchas Fe, Ni, and Co transition metals) since one doesnot have distinct localized moments to define an ex-change integral. A ferromagnetic material (itinerant orlocalized) displays a characteristic hysteresis curve andremanence (M ¤ 0 for H D 0) when one varies the ap-plied magnetic field. When heated, the material loses,in general, this ordered alignment and becomes para-magnetic at the Curie temperature. Ferromagnets areusually metallic; however there are ferromagnetic in-sulators such as CrBr3, EuO, EuS, and garnets [4.27,37].

An antiferromagnet is made of an assembly of mag-netic moments interacting with a negative exchangeintegral that will minimize their energy by adopting anantiparallel configuration. Again, such a definition isnot valid for itinerant antiferromagnets (such as Cr andMn transition metals) since one does not have distinctlocalized moments; thus an exchange integral cannot bedefined. The net total magnetization is zero and, there-fore, we do not have hysteresis. In the localized mag-netism case, it is possible to consider the entire crystalas being made of two interpenetrating sublattices con-taining moments all parallel inside each sublattice butwith the total magnetization of each sublattice oppo-site to one another. When heated, the material loses, ingeneral, this alternately ordered alignment and becomesparamagnetic at the Néel temperature. Oxides are gen-erally antiferromagnetic insulators except EuO [4.31,37].

The intermediate order between ferro and antifer-romagnets is a ferrimagnet used in microwave de-vices [4.16]. In the localized case, one considers thecrystal as being made of two sublattices, as in the caseof an antiferromagnet, with total magnetizations oppos-ing one another. The magnitude of each magnetizationis different, resulting in a net total magnetization (calleduncompensated case) in contrast to an antiferromagnet.

Another intermediate case is a canted antiferro-magnet, also called a weak ferromagnet. This is anassembly of magnetic moments alternating in directionand making a small inclination (canting). The smallangle (Fig. 4.12d) between two neighboring momentsMi;MiC1 results in a small ferromagnetic momentyielding weak ferromagnetism.

Canting originates from the Dzyaloshinski–Moriyaexchange interaction, which is a vectorial exchange


A|4.1

M

A B

a) b) c) d)

Fig. 4.12a–d Magnetic moment arrangement in an ferri-magnet. Total net magnetization is different from zero.(a–d) Comparison of the magnetic arrangements betweena ferromagnet (a) an antiferromagnet (b), a ferrimagnet (c),and a weak ferromagnet (d). (After [4.31])

interaction between two neighboring localized mo-ments Mi and Mj of the form DijMi �Mj that contrastswith the scalar ordinary exchange interaction of theform AijMi �Mj. It is responsible for canting (producinga small inclination) two neighboring antiferromagneticmoments that are usually antiparallel in an ordinaryantiferromagnet. This is due to asymmetric spin–orbiteffects [4.26].

An assembly of magnetic moments oriented at ran-dom with a zero total magnetic moment is calleda paramagnet. A material of that sort does not displayhysteresis or remanence when one varies the appliedmagnetic field. The magnetic field simply imposessome alignment to the randomly oriented moments. Fora paramagnet, the susceptibility � is positive, manyorders of magnitude smaller than ferromagnets, andbehaves as 1=.T �Tc/, where Tc is the Curie tempera-ture. Some materials never order magnetically, not evenat low temperature (that implies Tc D 0K) where or-der prevails. On the other hand, some materials arecalled superparamagnetic with no hysteresis and no re-manence but with Tc D 1. This does not mean thatthey are always ordered but their individual momentis so large (a classical moment corresponds to a quan-tum angular momentum J D 1, resulting in Tc D 1/that it takes an infinite temperature to destroy it com-pletely [4.27, 37].

A material that combats the influence of a mag-netic field by trying to maintain zero induction (B D 0)in its midst is diamagnetic (a metal like copper is dia-magnetic and superconductors are perfect diamagnets).

Linguistically, the term diamagnetic is a misnomersince dia means across, implying the field gets acrossthe material, whereas the opposite is true. When a dia-magnetic substance such as a silicon crystal is placedin a magnetic field, the magnetization vector M in thematerial is in the opposite direction to the applied field.A negative susceptibility can be interpreted as the dia-magnetic substance trying to expel the applied fieldfrom itself. A substance exhibits diamagnetism when-ever the constituent atoms in the material have closedshells and subshells. This means that each constituentatom has no permanent magnetic moment in the ab-sence of an applied field. Covalent crystals and manyionic crystals are typically diamagnetic materials be-cause the constituent atoms have no unfilled subshells.Since the diamagnet tries to minimize the effect of B,it expels field lines that can be exploited in magneticlevitation. A superconductor is a perfect diamagnetand a metal at high frequencies is partially diamag-netic since the applied field can penetrate it within theskin-depth layer only. For a diamagnet, the suscep-tibility � is constant (no variation with temperature)and is slightly negative. Superconductors have � D �1(since B D 0 inside a superconductor below criticaltemperature due to the Meissner effect), whereas �values of semiconductors are typically (Si: � 0:26�10�6, Ge: �0:58�10�6, GaAs: �1:22�10�6 as takenfrom Harrison [4.38]) in cm3=mol at room tempera-ture.

Materials with a relatively small coercive field (typ-ically smaller than 1000A=m) employed in transformercores and magnetic read heads are called soft mag-netic materials. Typically, but not all, simple metals andtransition metals and their compounds are soft. Permal-loy, amorphous, and nanocrystalline alloys and someferrites are soft. Amorphous materials are soft sincetheir disordered structure does not favor any direction(no anisotropy energy) whereas nanocrystals possessanisotropy over a short length scale (albeit it can be en-hanced with respect to the bulk counterpart). Softness isalso measured by the maximum attainable permeability(Table 4.3, soft elements).

In the opposite case, a material with a relativelylarge coercive field (typically larger than 10 000A=m)employed in permanent magnets, motors, and magneticrecording media (disks and tapes) is called a hard mag-netic material. This means that stored data is not easilylost since it will require a large field to alter the magne-tization value. Typically, but not all rare-earth metals,their compounds, and intermetallics are hard. There arealso hard ferrites. Permanent magnets are used in powersystems such as in relays and in motors and audio–video equipments such as headphones, videotapes, etc.(Table 4.1 and Fig. 4.11).

PartA|4.1


Table 4.3 Examples of soft magnetic materials and theirhierarchy according to saturation magnetostriction coeffi-cient �s. The main composition is successively based onFe, NiFe, and finally Co. �max is the maximum differentialpermeability and Hc is the coercive field. Fe80 B20 is alsocalled Metglass 2605. Fe40 Ni40 P14 B6 is also called Met-glass 2826. The highest �max is attained by (Fe0:8 Ni0:2/78Si8 B14 reaching 2�106 after annealing. (After [4.37])

Hc �max �s

Alloy (mOe) (50Hz)Fe80 B20 40 320 000 � 30�10�6

Fe81 Si3:5 B13:5 C2 43:7 260 000Fe40 Ni40 P14 B6 7:5 400 000 � 10�10�6

Fe40 Ni38 Mo4 B18 12:5�50 200 000Fe39 Ni39 Mo4 Si6 B12 12:5�50 200 000Co58 Ni10 Fe5 .Si;B/27 10�12:5 200 000 � 0:1�10�6

Co66 Fe4 .Mo;Si;B/30 2:5�5 300 000

Ferrimagnetic ceramic-like alloys with composi-tion FeOFe2O3 are called ferrites. The ferrite definitionhas been extended to MOFe2O3 with M being a di-valent metal (note the trivalent state of Fe in Fe2O3).Ferrites are used in microwave engineering, record-ings, etc., because of their very low eddy currentloss [4.12] and they operate over a large frequencyinterval (kHz to GHz). Their resistivities may reacha ratio as high as 1014 with respect to typical metal-lic resistivities. They are made by sintering a mixtureof metallic oxides MOFe2O3 with M D Mn, Mg, Fe,Zn, Ni, Cd, etc. Ferrite read heads are nonetheless lim-ited to frequencies below 10MHz as far as switchingis concerned, and that is why several new types ofread heads (thin films, anisotropic magneto-resistance(AMR), GMR, spin valves, magnetic tunnel junctions)have been developed in order to cope with faster switch-ing (Chap. 49). Other conventional magnetic materialsclose to ferrites are discussed next.

SpinelsAlloys with a composition generalizing the ferrite one.MO/x.MO/1�x Fe2O3 with the structure of MgAl2O4

(origin of the word spinel) [4.16].

GarnetsOxides with composition initially issued from the spinelfamily of the form (3M2O3, 5Fe2O3) crystallize in thegarnet cubic structure (Ca3Fe2.SiO4/3). They are ferro-magnetic insulators of the general formula M3Fe5O12,where M is a metallic trivalent ion (e.g., M D Fe3C).

Garnets were been used in memory bubble technology,lasers, and microwave devices (because their ferromag-netic resonance linewidth versus field is small, being onthe order of a fraction of an oersted while the resonancefrequency is several tens of GHz, see Wigen [4.16]),on the basis of another generalization of the formulato M3Ga5O12. For instance, Gd3Ga5O12 called GGG(gadolinium gallium garnet) when produced as a thinfilm, a few microns thick, has perpendicular anisotropywith domains (bubbles) having their magnetization upor down (perpendicularly with respect to the thin filmsample plane).

Thus a bit can be stored with a bubble and can becontrolled with a small magnetic field. GGG is con-sidered as one of the most perfect artificially madecrystals since it can be produced with extremely fewdefects (less than 1 defect per cm2). Another nomencla-ture called the [cad] notation is used with rare-earth irongarnets having the general formula X3Y2Z3O12. This no-tation means dodecahedral (c site is surrounded by 12neighbors and represented by element X), octahedral(a site is surrounded by 8 neighbors and represented byelement Y), and tetrahedral (d site is surrounded by fourneighbors and represented by element Z). The most im-portant characteristic of these garnets is the possibilityto adjust their composition and, therefore, their mag-netic properties according to selected substitutions onthe c, a, or d sites. The element X is a rare earth, whereasY and Z are Fe3C. The magnetization is changed byplacing nonmagnetic ions on the tetrahedral d site; in-creasing the amount of Ga3C, Al3C, Ge4C, or Si4C

will decrease the magnetization. On the other hand, in-creasing the amount of Sc3C or In3C at the octahedrala site will increase the magnetization. Ion substitutioncan also be used to tailor other magnetic properties(anisotropy, coercivity, magnetostriction, etc.).

Garnets are typically grown with liquid-phase epi-taxy with a growth speed that easily reaches 1�mthickness in 1min with a very high yield. They werenot only in bubble materials but also in magneto-opticaldisplays, printers, optical storage, microwave filters,and integrated optics components, etc. In spite of allthese attractive properties, the ease of tunability, andthe very high yield, problems soon arose with the accesstime limitation in bubblememories, since switching fre-quency was found to be bounded to less than about10MHz (see Wigen [4.16]; Chap. 49). More recently,garnets are being used in spin–charge interconversionas discussed in the last section.

http://dx.doi.org/10.1007/978-3-319-48933-9_49

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Magnetic Properties: From Traditional to Spintronic 4.2 Nonconventional Magnetism 99Part

A|4.2

4.2 Nonconventional Magnetism and Progress Toward Spintronicsand Quantum Devices

4.2.1 Types of Exchangeand Coupling in Magnetic Materials

In conventional magnetic materials, the strength of themagnetic interactions between two neighboring local-ized moments i and j (as in atoms/ions/molecules andrare-earths) is described by a direct exchange interac-tion. The latter is essentially a Coulomb (electrostatic)interaction between the electrons at the i and j sites. Theword exchange is used since the corresponding overlapintegral describing this interaction involves wavefunc-tions with permuted (exchanged) electron coordinates(in order to respect the Pauli exclusion principle).The exchange energy between sites i and j is givenby AijMi Mj, where Mi and Mj are, respectively, themagnetization at the i and j sites. When Aij > 0 en-ergy is minimized when the moments are parallel andwhen Aij < 0 antiparallel configuration of the momentsis favored. The energy arising from exchange overa distance r in the continuum limit is approximatelyAM2=r2.

Surrounding the localized moments i and j by anelectron gas modifies the nature of their interactioninto an RKKY (Ruderman–Kittel–Kasuya–Yoshida,see Kittel [4.27]) coupling. The oscillatory interactionoccurring between the moments is mediated by the sur-rounding electron gas. It varies in three-dimensional(3-D) systems as cos.2kFr/=r3, where r is the distancebetween the moments and kF the Fermi wavevector ofthe electron gas. The finite value of kF is responsiblefor the oscillatory behavior since it produces a cut-off in the Fourier transform [4.27]. It was discoveredrecently that a two-dimensional (2-D) counterpart ofthe RKKY interaction exists between two magneticthin films separated by a metallic spacer [4.39]. TheRKKY-like interaction between the two magnetic filmsacross a metallic spacer is oscillatory with respect tothe spacer thickness z. Thus it becomes possible to pos-itively (ferro) or negatively couple magnetic layers bychanging the thickness z of the sandwiched metallicspacer (Fig. 4.1 for typical lengths). This is extremelyuseful for thin-film devices (Chap. 49 and Table 4.1).

In planar magnetic technology, the interest is inmagnetic thin films and their interactions. Informationstorage, sensing, spintronics, quantum computing, andother applications of magnetic thin-film devices are themain drive for understanding the nature and extent ofmagnetic exchange interactions and also coupling ef-fects arising between magnetic 2-D layers in order totailor appropriate devices. Novelty is expected since

the device size is comparable to the interaction range(Fig. 4.1).

The coupling strength of the interaction betweentwo neighboring magnetic films i and j can be mod-eled by a factor Jij. It is similar to the exchange integralAij between two neighboring localized moments; how-ever, it involves entire layers generally made fromitinerant magnets and not single moments like withAij. The exchange interaction is of the form JijMi:Mj,where Mi and Mj are the magnetization per unit sur-face of films i and j. The main interest of Jij liesin the fact that its range is longer in reduced dimen-sions (one-dimensional (1-D) and 2-D) with respect to3-D (for instance, an RKKY-like interaction betweentwo magnetic films across a metallic spacer is oscilla-tory with a longer range than 3-D since it varies like1=r2 instead of 1=r3) and its physics based on com-plex Fermi wavevectors [4.39] is entirely different fromthe standard RKKY interaction between localized mo-ments [4.39]. The sign of the interaction depends onthe thickness of the metallic spacer. Other types ofexchange exist between films such as biquadratic orhigher order ones with a generalized Heisenberg formŒJijMi �Mj�

n with n 2 (see The entry related to ex-change and coupling in multilayers in Table 4.1).

Interactions between two ferromagnetic (or othertype of magnetic) films are not limited to metallic spac-ers; they may occur across an insulating or semicon-ducting spacer (Fig. 4.13). Magnetic pinholes generally

Fig. 4.13 Top: ferromagnetic coupling arising from pin-holes existing across the spacer.Middle: exchange coupledlayers that might be ferro (left) or antiferromagnetic (right)depending on the spacer thickness. Bottom: magnetostaticdipolar coupling between the top and bottom layers. (Af-ter [4.10])

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Table 4.4 Surface energies � in J=m2 for magnetic and nonmagnetic materials, listed with respect to their atomic numberfor the low-energy cleavage surface. These are approximate values, which are difficult to measure in general and aredependent on surface orientation and reconstruction. (After [4.10])

Magnetic metal� (J=m2)

Cr2:1

Mn1:4

Fe2:9

Co2:7

Ni2:5

Gd0:9

Transition metal� (J=m2)

Ti2:6

V2:9

Nb3:0

Mo2:9

Ru3:4

Rh2:8

Pd2:0

Ta3:0

W3:5

Pt2:7

Simple or noble metal� (J=m2)

Al1:1

Cu1:9

Ag1:3

Au1:6

Semiconductor� (J=m2)

Diamond1:7

Si1:2

Ge1:1

GaP1:9

GaAs0:9

Insulator� (J/m2)

LiF0:34

NaCl0:3

CaF20:45

MgO1:2

Al2O3

1:4

mediate direct ferromagnetic coupling between neigh-boring multilayers. They are viewed as shorts acrossthe insulating or semiconducting spacer. Néel pro-posed that conformal roughness (orange-peel model,see Schulthess and Butler [4.40]) at interfaces can re-sult in ferromagnetic coupling for a moderate thicknessof spacer material. Magnetostatic dipolar coupling oc-curs between the roughness features or between domainwalls in the two magnetic layers. In this case, strayflux fields from walls in one film can influence themagnetization reversal process in the other. The Slon-czewski loose-spin (see the exchange and coupling inmultilayers in Table 4.1) model is based on angularmomentum change as experienced by spin-polarizedelectrons tunneling across an insulating spacer, result-ing in a magnetic exchange coupling. The magnitudeand sign of the coupling oscillate, depending on theinterfacial barrier properties (Himpsel et al. [4.10] andTable 4.1).

In the absence of a spacer, the exchange energyper unit surface of ferromagnetic and antiferromagneticfilms gives birth to a new phenomenon called exchangebias (a shift of the hysteresis curve of the ferromagneticfilm, see the entry in Table 4.1 related to exchange andcoupling in multilayers). An important consequence ofbias is pinning of the direction of the ferromagnetic mo-ment used in low noise read heads because it hindersnoisy (jerky) domain wall motion (Chap. 49). Exchangebias is also called anisotropic exchange and is still rela-tively poorly understood from the fundamental point ofview despite the fact that it was discovered more than50 years ago.

4.2.2 Thin Magnetic Films

A magnetic film is considered thin if its thickness issmaller than the (bulk defined) exchange length; inother words its thickness corresponds to the spatial ex-tent of a single domain [4.10]. Magnetic thin filmspossess very attractive and distinct physical properties

with respect to their bulk counterparts. For instance, re-duced dimensionality, coordination and symmetry leadto magnetocrystalline anisotropy energies that are twoor three orders of magnitude larger with respect tobulk (quenching of the orbital angular momentum asin transition metal ions is absent, see Kittel [4.27]). Sat-uration magnetizationMs is also enhanced with respectto bulk since the effective Bohr magneton number nBper atom/molecule is larger in lower dimensions (forinstance, in Nickel clusters nB might reach 1.8, whereasin the bulk it is only 0.6). Couplings in thin films arealso enhanced as in the case of the RKKY-like inter-action across a metallic spacer. Coupling between twomagnetic layers across a semiconducting spacer or aninsulator leads to entirely new interactions, such as theSlonczewski loose-spin model or the Bruno quantuminterference model. Other types of couplings or materi-als may be encountered besides exchange bias (betweena ferro and an antiferromagnetic layer) such as springmagnets fabricated from alternating hard and soft thinfilms (Table 4.1).

Considering the growth of magnetic materials asthin films, we have to recall that these possess distinctphysical properties with respect to the materials usedin conventional microelectronics such as surface ener-gies. A dominant role is played by surface free energies� and the interface. They determine the growth modesin thermal equilibrium; the morphology of material Bgrown on material A depends on the balance betweenthe free surface energies of substrate, overlayer, andinterface [4.10]. Transition-metal-based magnetic ma-terials exhibit a relatively high surface energy, owing totheir partially filled d shell. Noble metal substrates havesmaller surface energies and insulating substrates haveeven smaller ones (Table 4.4).

Additionally, when one performs epitaxial growthof materials, another concern is lattice matching for thedifferent underlayers, as displayed in the tables belowalong with typical quantities of interest in representativemagnetic materials (Table 4.5 and 4.6).

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Table 4.5 Lattice-matched combinations of magnetic materials, substrates, and spacer layers. There are two main groupsof lattice-matched systems with lattice constants close to 4:0 or 3:6Å, respectively, after taking 45ı rotations of the latticeor doubling the lattice constant. (After [4.10])

First group: magnetic metalap2 (Å) [a (Å)]

Cr (bcc)4:07 [2:88]

Fe (bcc)4:05 [2:87]

Co (bcc)3:99 [2:82]

Simple or noble metala (Å)

Al4:05

Ag4:09

Au4:07

Semiconductora=

p2 (Å) [a (Å)]

Ge3:99 [5:65]

GaAs4:00 [5:65]

ZnSe4:01 [5:67]

Insulatora (Å) [a=

p2 (Å)]

LiF4:02 [2:84]

NaCl5:65 [3:99]

MgO4:20 [2:97]

Second group: Materiala (Å)

Fe (fcc)3:59

Co (fcc)3:55

Ni (fcc)3:52

Cu3:61

Diamond3:57

Table 4.6 Magnetic properties of Fe, Co, and Ni transi-tion metals. Note that the domain wall (DW) thickness andenergy are approximate. This is a complement to Table 4.2

Magneticsubstance/property

Fe Co Ni

Exchange stiffness A(meV)

0:015 0:03 0:020

Anisotropy K1 (J=m3) at300K

4:8�104 45�104 �0:5�104

Lattice spacing (nm): aLattice spacing (nm): c

0:29 0:250:41

0:35

DW thickness (nm)DW thickness (latticeparameters)

40138

1536

100285

DW energy (J=m2) 3�10�3 8�10�3 1�10�3

4.2.3 Electronic Properties:Localized, Free, Itinerant Magnetismand Spin-Polarized Band Structure

Building a working device requires understanding notonly magnetic properties but also electronic propertiesand their interplay. The new devices are expected to bemade of a variety of magnetic (conventional and uncon-ventional) materials as well as others already knownin microelectronics. Insulating oxides (except EuO)and rare-earth compounds with well-localized externalshell f electrons are the solid-state representative ofatomic-like magnetism. Magnetism of atoms, ions, andmolecules deals with well-defined, localized orbitalsand individual moments arising from orbital, spin, or to-tal angular momentum. When these moments get closeto one another, as in the solid state, they interact Heisen-berg wise: AijMi:Mj [4.31]. The latter is going to bealtered by presence of the surrounding free electrongas. Therefore we have to understand magnetism ina free-electron gas, its counterpart arising from local-ized moments, and finally its nature when we have thehybrid case (itinerant magnetism) like in a transitionmetal (free s and localized d electrons). This problem

being complicated due to its many-body nature, we relyupon a one-electron approximation and spin-polarizedband structure [4.11].

In a free-electron gas, one assumes independentnoninteracting electrons; therefore many-electron andnonlocal effects (arising from exchange) are not to betaken into account. Magnetism in this case is due to in-dividual electron spin, and it is straightforward to estab-lish so-called Pauli paramagnetism [4.27]. In addition,orbital effects give rise to Landau diamagnetism. Thespin-polarized density of states is shifted by the fieldcontribution ˙�BB depending on the spin orientationwith respect to induction field B. The magnetization inthe free-electron Pauli case (simple metals) can be eval-uated (at T D 0K) from

M D �B.N"�N

#/ D �B

2

EFZ

�BB

dEN.EC�BB/

� �B

2

EFZ

BB

dEN.E��BB/ (4.7)

that yieldsM � �2BN.EF/B, where �B is the Bohr mag-

neton, N.E/ is the free electron, and N";#.E/ D N.E˙

�BB/ the spin-polarized density of states with respec-tive occupations N

";#. The Landau diamagnetism (or-bital) contribution is a lot more complicated to evaluate,nevertheless it is exactly minus one-third the paramag-netic expression [4.27].

In transition metals (itinerant magnets), states de-rived from the 3d and 4s atomic levels are responsi-ble for their physical properties. The 4s electrons aremore spatially extended (of higher principal quantumnumber) and determine, for instance, compressibility,whereas the 3d states determine magnetic properties.The 3d electrons propagate throughout the material,hence the term itinerant magnetism. The spin-polarizedband structure transition of metals is different fromthat of conventional electronic materials (see Figs. 4.14

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3

2

1

0–8 –6 –4 –2 0 2 4

2

1

0–8 –6 –4 –2

Density of states (eV–1)

Energy (eV)

EF

Density of states (eV–1)

Energy (eV)

EF

0 2

b)

a)

Fig. 4.14a,b Spin-polarized density of states for bcc Feand fcc Ni. The bands for the spin up (dark) electronsare separated from the spin down (light) band by the ex-change splitting energy (see text). The method used issuperior to the LSDA method that yields a noisy densityof states. The authors call it LDA C DMFT meaning lo-cal density approximation C dynamical mean field theory.(After [4.41])

and 4.15 for iron, nickel, and cobalt spin-polarized bandstructures).

Many metals have an odd number of electrons (inthe atomic state), thus one would expect a magnetic ma-terial from unpaired spins; however, only five of themare actually magnetic (Cr, Mn, Fe, Ni, and Co are alltransition metals with s–d hybridization). Note that cop-per is diamagnetic with � D � 0:77�10�6 cm3=mol atroom temperature, Fe, Ni, and Co are ferromagnets,whereas Cr and Mn are antiferromagnets.

Magnetism and related properties in transitionmetalalloys are not straightforward to model owing to theelectrons’ itinerant nature. In solids, magnetism arisesmainly from electrostatic (Coulomb) electron–electroninteractions, namely the exchange interactions, andin magnetic insulators these can be described rathersimply by associating electrons appropriately with par-ticular atomic sites so that the Heisenberg exchangeAijMi �Mj can be used to describe the behavior of suchsystems. In metallic systems, it is not possible to dis-tribute the itinerant electrons in this way, and suchsimple pairwise interactions between sites cannot beeasily defined. Metallic magnetism is a complicated

2

1

0

–1

–2

2

1

0

–1

–2

Density of states (1/eV)

Density of states (1/eV)

Co(hcp) d

d

s

s

Ni(fcc) d

d

s

s

a)

b)

Fig. 4.15a,b Symmetric double-sided representation ofthe spin-polarized density of states for (a) Co and (b) Ni(obtained with the LMTO method, i. e., linearized muffin-tin orbitals) with the Fermi level indicated. (After [4.42])

many-body effect and has attracted significant effortover a long period to understand and describe it [4.10].A widespread approach is to map this problem ontoone involving independent electrons moving in fieldsset up by all other electrons. It is this aspect whichgives rise to the spin-polarized band structure that isoften used to explain the properties of metallic mag-nets such as noninteger values in multiples of the Bohrmagneton �B. In addition, the compositional struc-ture plays a major role in itinerant magnetism, suchas, for instance, in the Ni–Pt alloy. It is an antifer-romagnet in the ordered case and ferromagnetic inthe disordered case. Itinerant magnetism is comprisedbetween two extreme limits: the localized (rare-earthand insulators) and the completely free electron case(more appropriate to alkalis) as described below. Asfar as transport is concerned, one may use the Stonertwo-band model, which is a two-fluid model (likeelectrons and holes in semiconductors) with a popu-lation of electrons with spin-up and another of spin-down with the corresponding spin-polarized densitiesof states [4.10].

In ordinary crystalline nonmagnetic materials, theelectronic band structure is built from the lattice (pe-


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Table 4.7 Spin polarization in several ferromagnetic ma-terials according to several authors. The discrepanciesbetween the different results stem from the various ap-proaches to estimate the density of states at the Fermi leveland point to the difficulty of getting a unanimous figure.Note that for a half-metal like CrO2 or NiMnSb, the polar-ization is 100%

Meservey–Tedrow [4.43]

Moodera–Mathon [4.44]

Monsma–Parkin [4.45]

Fe 40% 44% 45%Co 35% 45% 42%Ni 23% 33% 31%Ni80Fe20 32% 48% 45%Co50Fe50 47% 51% 50%

Table 4.8 Magnetic splitting ıkex, full width at half maxi-mum ık

"and ık

#(˙0:01Å�1/, and spin-dependent mean

free paths �"

and �#

for NiFe and NiCr alloys. (Af-ter [4.46])

ıkex(Å�1)

ık"

(Å�1)ık

#

(Å�1)�

"

(Å)�

#

(Å)Ni 0:14 0:046 0:046 > 22 > 22Ni0:9Fe0:1 0:14 0:04 0:10 > 25 10Ni0:8Fe0:2 0:14 0:03 0:22 > 33 5Ni0:93Cr0:07 0:09 0:096 0:086 11 10Ni0:88Cr0:12 � 0:05 0:12 0:11 8 9

riodic) potential seen by a single electron (havingany state of spin) in the structure. Mathematically, itamounts to solving the Schrödinger equation corre-sponding to that potential and calculating its eigenfunc-tions (spin-independent wavefunctions). The eigenval-ues versus wavevector are the ordinary bands. In a mag-netic material, one has to include electronic exchangeeffects (originating from the Pauli exclusion princi-ple applied to electrostatic interactions between pairsof electrons leading to nonlocal interactions) with ex-plicit accounting for spin in the wavefunctions. Severalmethods exist such as the LSDA (local spin densityapproximation, see Fig. 4.1), which is a spin exten-sion of the LDA (local density approximation) whereone builds a local approximation to the exchange in-teraction from the local density of electrons [4.41].The LSDA (Fig. 4.1) formalism provides a reliabledescription of magnetic properties of transition metalsystems at low temperatures. It also provides a mecha-nism for generation of the noninteger effective number

6

5

4

3

2

1

0

3 d magnetic splitting (eV)

Magnetic moment (μB/atom)

Mn in Ag

Fe atomCo atom

bcc Fe

Ni atom

hcp CoCr Cr + Mnfcc Ni

0 1 2 3 4 5 6

Fig. 4.16 Exchange splitting versus magnetic moment inthe Bohr magneton �B per atom in 3d transition metals.The diagonal line is 1eV=�B and the magnetic momentvalues (in Bohr magnetons) are given for the atom and thecorresponding crystal. (After [4.10])

nB of Bohr magnetons, together with a plausible ac-count of the many-body nature of magnetic momentformation at T D 0K. The obtained bands depend onspin as depicted in Figs. 4.14 and 4.15. From thespin-dependent band structure, one may obtain the spin-polarized density of states for each spin polarization(up " or down #, also called majority and minoritylike in ordinary semiconductors). This nomenclaturewill eventually be confusing when one starts deal-ing with metals and semiconductors simultaneously.For the time being, this nomenclature is accepted (Ta-ble 4.7), as long as we are dealing solely with magneticmetals and one defines a new type of gap (originatingfrom the exchange interaction) called exchange split-ting (Fig. 4.16 and Table 4.8) or the (spin) gap betweentwo spin-dependent bands (Fig. 4.14). This explainsthe existence of novel materials such as half-metalsthat ought to be contrasted with semimetals (graphite)where we have a negative electronic gap because of va-lence and conduction band overlap.

Half-metals (such as CrO2 and NiMnSb) possessone spin-polarized band (up, for instance) full and theother (down) empty. These materials are very impor-tant for spintronics and specially for injecting spin-polarized carriers.

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4.3 Spintronics and Quantum Information Devices

Presently, we are witnessing electronics extension to-ward the treatment of spin and charge instead of chargeonly, the realm of traditional electronics.

The reason spin becomes interesting and use-ful stems from the following. As device integrationincreases and feature length decreases toward thenanometer, the spin of individual carriers (electrons orholes) becomes a good quantum number. This meansthat the spin value is conserved over nanometric dis-tances (spin diffusion length is typically 5�50 nm).This was not the case in microelectronics; consequently,spin can be used in the nanometer regime to carry use-ful information. Henceforth carriers transport energy,momentum, charge, and additionally spin. The quan-tum nature of spin leads to the natural consequence thatquantum computers might be made successfully withspintronic components [4.28].

Spintronics, like microelectronics, is based on mate-rials and their physical properties (mechanical, thermal,electrical, magnetic, etc.) as well as the know-how ofgrowing/fabricating thin films and reduced dimension-ality objects (quantum dots, quantum wires, quantumpillars, clusters, etc.). The different classical processingsteps (epitaxial growth, deposition, oxidation, diffusion,doping, implantation, etching, passivation, thermal in-sulation, annealing, texturing, sputtering, patterning,etc.) that are well established in microelectronics haveto be extended and adapted to tackle magnetism-relatedelectronics. Magnetic interactions have an anisotropicvectorial character in contrast to scalar electrical inter-

Table 4.9 Spintronic effects viewed as spin extension of ordinary unpolarized bulk and 2DEG electronics

Spintronic effect or device Comments ReferenceSpin diode Diode with spin polarized carriers Merchant et al. [4.47]Spin transistor Bipolar magnetic junction transistor based on InMnAs (DMS)

with magnetic field control of amplificationRangaraju et al. [4.48]

Spin-Hall effect (SHE)and its inverse (ISHE)

Can be used to interconvert charge and spin currents Inoue et al. [4.49]

Spin LED and lasers LED and lasers with spin polarized carriers Holub et al. [4.50]Spin photovoltaic effect High efficiency (50%) can be reached in DMS (AlP:Cr) Olsson et al. [4.51]Spin photogalvanic effect(SPGE)

Optical excitation of 2DEG system in quantum well structures Ganichev et al. [4.52] andYang et al. [4.53]

Spin Raman and nonlinearoptical effects

Electrically induced Raman emission from 2DEG spin oscilla-tor with nonlinear optical frequencies tuned by magnetic fieldgradient and electron density

Nogaret [4.54](see also Dietl et al. [4.23])

Spin-Hall magnetoresistance(SMR)

Hall magnetoresistance with spin polarized carriers Sandweg et al. [4.55]

Spin-Seebeck effect (SSE) Thermal gradient effect with spin polarized carriers Bauer et al. [4.56]spin-Peltier effect (SPE) Spin currents can be used for cooling or heating Flipse et al. [4.57]Spin-Gunn effect (SGE) Instability at high electric fields against dynamic formation of

spin-polarized current pulsesQi et al. [4.58]

Spin breakdown Spin counterpart of Avalanche/Zener breakdown Johnston–Halperin et al. [4.59]

actions (based solely on charge) present in conventionalmicroelectronic devices. Magnetism introduces the no-tions of anisotropy (magnitude, nature, and direction),coercivity, saturation magnetization, etc. that ought tobe controlled during growth (magnetic field assistedgrowth or epitaxial growth has to be developed in or-der to favor magnetic anisotropy along one or severalselected directions).

For instance, in some devices, there is a need togrow amorphous metallic magnetic layers (in order tohave a very small anisotropy resulting in a magneti-cally soft layer), and these might be harder to growthan amorphous semiconductors or insulators (recallthat a cooling speed of one million degrees per secondis typically needed in order to produce an amorphousmetal).

Overall, temperature is a very important parameterin magnetism since above the Curie (or Néel) tempera-ture, ferromagnetism (antiferromagnetism) is lost.

Spin-polarized materials with novel proper-ties emerge, such as half-metallic materials (CrO2,NiMnSb, etc.), embodying carriers that are completelypolarized spin wise (all up or all down).

Since we are dealing mostly with s–d hybridizedtransition metals, we expect at least three types of spincurrents [4.22]: SPC (spin-polarized current) made offree (s-type) spin-polarized carriers, SWC (spin-wavecurrent) carried by undulating localized spins (d-type)(with no dissipative Joule effect produced), and STT(spin-transfer torque) current stemming from the s–d

Magnetic Properties: From Traditional to Spintronic 4.3 Spintronics and Quantum Information Devices 105Part

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(double) exchange between free and localized carri-ers [4.60].

Other types of spin current can be generated byshining circularly polarized light [4.52, 53] over quan-tum well structures [4.52, 53] over quantum well struc-tures exploiting a special effect originating from spin–orbit interaction in a 2DEG (two-dimensional electrongas) (Table 4.9) and called Rashba coupling [4.22,23]. A spin current can also be produced with lin-early polarized light [4.61] exciting antiferromagneticmaterials such as NiO, Fe2O3 (hematite), or bis-muth ferrite materials. The physical mechanism isbased on a nonlinear optical effect called shift current(or photovoltaic drift current) produced by second-order interaction with the light electric field in non-centrosymmetric materials such as perovskites [4.61](BaTiO3, LiNbO3, KNbO3, etc.) of the form Ji D�ijkEjEk (second-order extension of Ohm’s law Ji D�ijEj).

This is to be added to the wealth of magnetic inter-actions that can be based on para, dia, ferro, antiferro, orferrimagnetic materials, which can be simultaneouslyelectrically wise metallic (simple, transition, noble, orrare-earth), insulating, or semiconducting. As an ex-ample, multiferroics (see below) that are single-phasematerials possessing interconnected electrical, mag-netic, and elastic properties can be used as transducersin such multifunctional structures.

In analogy with ordinary electronics, spintronics isbased on seven operations:

� Spin injection: How can a nonequilibrium densityof spin-polarized carriers, that is, electrons withspin up n

"(or down n

#/ or holes with spin up p

"

(or down p#/ be created? Optical injection might

be viewed as the spin extension of the Haynes–Shockley experiment. Spin injection can also bedone with carbon nanotubes since they do not alterthe spin state over large distances (Chap. 49).� Spin filtering: Separating spin-polarized carriers isrequired in order to avoid spin-flips altering spinstates. Thin ferromagnetic layers as in spin valves(Chap. 49) and chiral materials such as mono-layers of double-stranded DNA molecules can beused [4.62].� Spin manipulation: How can the spin and chargestates of a nonequilibrium density of spin-polarizedcarriers be altered and controlled?� Spin coherence: How can the spin and charge statesof a nonequilibrium density of spin-polarized car-riers be maintained over an important propagationlength at room (and higher) temperature in the pres-ence of perturbing magnetic stray fields (terrestrialand other)?

� Spin accumulation: How can spin-polarized carriersbe coherently concentrated without destroying theirspin coherence and without inducing spin flips thatwill destroy the resulting spin value?� Spin detection: How can spin and charge ofa nonequilibrium density of spin-polarized carriersbe detected nondestructively?� Spin pumping: This is the spin counterpart of chargepumping that occurs by application of a nonlinear acsignal to a semiconductor in order to achieve cur-rent rectification [4.63]. In the spin-polarized case,it is translated into the question of spin angularmomentum transfer mode from magnetization pre-cession to conduction electrons. It is experimentallyfeasible [4.64] with a ferromagnet (F) containinga precessing magnetization and a neighboring nor-mal metal (N) into which a spin current is injectedthrough the F–N interface. In summary, the questionis related to ways of efficiently generating a contin-uous spin current from a time-evolving magnetiza-tion (viewed as a spin dynamo [4.65]).

In Table 4.9, we display the spin extension of sev-eral physical effects that are familiar in conventionalelectronics.

Many of the effects are simply the extension froma standard semiconductor or a metal to a ferromagnet.For instance, in the Seebeck effect, a thermal gradi-ent rT applied to a metal induces an electric voltagealong the direction of rT . In order to extend this phe-nomenon to the spin-Seebeck case, a thermal gradientrT is applied to a ferromagnet and a spin voltage isgenerated from thermal excitation of the magnetic mo-ments in the ferromagnet. Those excitations are used topump a spin current into a neighboring paramagneticmetal. Spin currents can be used for cooling or heating,and the spin-Peltier effect is based on the ability of thespin-up and spin-down currents to transport heat inde-pendently [4.57].

SMR (spin-Hall magnetoresistance) is even moreinteresting since it is related to conversion of chargeinto spin current (called interconversion) via SHE(spin-Hall effect). SHE implies that a current can betransferred between a noble metal (such as Pt or Au)and a magnetic insulator (such as yttrium iron garnet(YIG)) [4.22] making it the spin counterpart of thedisplacement current [4.22]. SHE leads to spin accu-mulations with opposite magnetizations at the edgesof a magnetic material perpendicularly to the chargecurrent and derives from the AHE (anomalous Halleffect) [4.22]. The latter is derived as follows. In anordinary metal, a galvanomagnetic effect is writtenin terms of resistivity tensor �ij that depends on theapplied magnetic field H and contains two contribu-

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tions: symmetric and antisymmetric like any rank-2tensor.

Symmetric terms contain even powers of the ap-plied magnetic field H, whereas antisymmetric termscontain odd powers of H. The lowest order antisym-metric terms are linear inH components and are termedHall effect terms, whereas the lowest order symmetricterms are quadratic in H components and are termedmagnetoresistive.

Let us suppose that �ij in terms of field H in an or-dinary metal has an expansion to lowest order (in Hcomponents) given by

�ij D aij C bijkHk C � � � : (4.8)

In contrast, when we have a ferromagnet, an additionalterm due to magnetization M should be added with re-spect to field H contribution. Thus one may extend theabove expansion to

�ij D aij C bijkHk C cijkMk C � � � : (4.9)

The ordinary Hall (H dependent) effect is thus extendedto the AHE (M dependent) present in ferromagnets.

Hall terms are given by cross products H� J (or-dinary Hall effect) or M � J (AHE), whereas magne-toresistance is related to scalar products betweenM andcurrent density J.

4.3.1 Magnetic Signal Processing Devices

Magnetic signal processing devices extend electronicsignal processing to include magnetic field action,meaning that arithmetic, logic, memory, and other op-erations that traditionally belong to the realm of elec-tronics will be performed by devices containing com-ponents that are sensitive to magnetic excitation.

An example given in Table 4.10 is an oscillatorwhose frequency is controlled by a magnetic field (thismight be called magnetic agility) instead of an electri-

Table 4.10 Magnetic logic and signal-processing devices along with microwave ferrite devices covering applicationsranging from L-band (0:5�1GHz) to W-band (75�100GHz)

Magnetic signal/device operation Comments ReferenceRandom access memory Electrical read–write operations in a (Ga,Mn)As (DMS) memory cell Mark et al. [4.66]Spin-wave guiding Signal processing with spin-waves in magnetic nanowires Choi et al. [4.67]Spin-based logic circuits Design of reconfigurable logic circuits based on spin such as to be

compatible with traditional electronic devicesKhitun et al. [4.68]

Spin oscillator Microwave oscillation driven by STT current Kiselev et al. [4.69]Nonlinear parametricgeneration of subharmonics

Spectral manipulation of an external microwave signal Kobljanskyjet al. [4.70]

Time-reversed wave-packetgeneration

Magnetic components allow tweaking of time-reversal symmetry Chumak et al. [4.71]

Microwave ferrite device Antennas, circulators, isolators, and inductors Wu et al. [4.22]

cal voltage (as in ordinary electronic agility) despite thefact electrical control is generally preferred.

Fabrication of practical magnetic signal processingdevices requires replacing magnetic field control by anelectrical one (whenever possible) since stray magneticfields might interfere with device operation. In addi-tion, it is technically difficult to produce large magneticfields over a nanometer length at present.

The simplest way of replacing magnetic field con-trol is to trigger coupling between electrical and mag-netic degrees of freedom by growing composite ma-terials containing both ferromagnetic and ferroelectricelements. Coupling between magnetic and electrical de-grees of freedom effects or magnetoelectric couplingis generated, for instance, through elastic interactionbetween a piezoelectric and a magnetostrictive sub-stance [4.72].

Structures allowing possible magnetoelectric cou-pling are magnetic superlattices (also called magnonicor metamaterials), the magnetic analog of semicon-ducting superlattices or photonic structures (with vary-ing dielectric constants) with modulated magnetiza-tion, anisotropy, or type (para, dia, ferro, antiferro,ferri).

Multiferroics are single-phase materials that re-spond to electrical, magnetic, and elastic excitations.They are more versatile than magnetoelectric compos-ites; however their stoichiometry and growth criticallyaffect the value substantially [4.72] of their magneto-electric coefficient.

Dilute magnetic semiconductors (DMS) containmagnetic elements that are sensitive to magnetic fieldsand therefore make good candidates for magnetoelec-tric coupling (Table 4.10).

Substituting magnetic field control by an electricalone is possible in semiconductors through spin–orbit in-teraction since it allows generation and manipulation ofcarrier spins by an electric field [4.73].

Spin–orbit interaction has also been shown to al-low electric control of spin-waves in single-crystal YIG

Magnetic Properties: From Traditional to Spintronic 4.3 Spintronics and Quantum Information Devices 107Part

A|4.3

waveguides, which paves the way to developing electri-cally tunable magnonic devices [4.74, 75].

Instead of coupling magnetic and electrical degreesof freedom or substituting magnetic field control by anelectrical one, interconverting charge and spin degreesof freedom [4.22, 23] through spin currents, such asSHE described above, will ultimately lead to electric-field control (Tables 4.9 and 4.10).

Recently, voltage-controlled magnetism of Co thinfilms adjacent to Gd2O3 gate oxides was demon-strated [4.76]. Using gate voltages of a few volts, theanisotropy field as well as the saturation magnetizationwere altered, producing a large magnetic anisotropy en-ergy change of up to 0:73 erg=cm2. Voltage-induced re-versible oxidation of the Co layer resulted from a largeinterfacial electric field and a high oxygen ion (O2�)mobility in Gd2O3.

4.3.2 Magnetic Quantum Dot Arrays

Submicron quantum dots based on traditional semi-conductor, insulator, or metallic materials are well de-veloped [4.77] with many applications. They can betailored with lithographical techniques and behave ina fashion similar to synthetic atoms. When arranged inarrays, they can be x–y addressed like random accessmemories or used in applications such as displays, cam-eras, sensors, etc.

The magnetic counterparts of quantum dot arraysare being developed with magnetic materials that pos-sess a significantly larger number of possibilities stem-ming from the wide panel of magnetic materials (seeHurd [4.26] for a classification of magnetic materialsprior to 1982 and Kronmueller and Parkin [4.78] formore recent developments).

Magnetic elements are specially adapted to scalingand scale integration since submicron-ordered mag-netic dot arrays possess properties such as moment,susceptibility, coercivity, and remanence that are signif-icantly enhanced with respect to their bulk counterparts,given the fact that they are closer to quantum atomicstructures with augmented magnetic properties whenfabricated on the nanometer scale.

In addition, present quantum dot array fabricationmethods are fast and inexpensive with electrochemicaltemplate depositionmethods, without the need for time-consuming processes based on sophisticated lift-off anddeposition techniques borrowed from microelectronicsoperations, adapted and scaled down to tackle magneticnanostructures.

In a disk-shaped magnetic quantum dot, several or-dered structures can occur such as single domain (within-plane or out-of-plane magnetization), multiple do-mains, and vortex excitations that are akin to solitons

0

5

10

15

h/ℓex

R/ℓex0 2 4 6 8

Out-of-plane(perpendicular)

In-plane magnetization

Vortex/multidomain

Fig. 4.17 Phase diagram h=èx versus R=èx displayingvalues for which vortex excitations appear in a disk-shapedquantum dot of thickness h and radius R. (After [4.79]) Oc-currence of single domains with in-disk plane orientationor perpendicular to it as well as multiple domain regionsare indicated

in higher dimension D 2 (Fig. 4.17). A cylindricalmagnetic quantum dot structure depends on two ratios:h=èx and R=èx, where h is the disk thickness and R theradius; èx is the exchange length (Sect. 4.1).

The phase diagram displaying the regions of occur-rence of single domains (with in-plane or perpendicularto plane orientation), multiple domains, and vortex ex-citations is depicted in Fig. 4.17.

Vortex structures are stable in a disk possessing a ra-dius of several hundred nanometers and a thickness ofa few tens of nanometers.

4.3.3 Magnetic Vortex Propertiesand Applications

A magnetic vortex possesses a central core of about10 nm diameter with magnetization perpendicular to thex–y plane of the disk. The core magnetization is de-scribed by a polarization p D ˙1 that can point in the˙z direction. Thus it can sense a vertical magnetic field.The core is surrounded by an x–y planar spin flow pat-tern making a flux-closure magnetization distribution inthe plane of the disk. It is characterized by a chiralityC D ˙1 that can sense an in-plane magnetic field. Thusa single vortex, being able to sense all spatial compo-nents of a magnetic field, is the perfect ingredient ofsensitive 3-D magnetometers.

The attractivity of vortex-based magnetic sensorsoperating at room temperature contrasts sharply withvortex-based superconductingmagnetometers that mustbe cooled at low temperature (helium-4 for traditional

PartA|4.3


Table 4.11 Magnetic vortex applications in storage, memory, logic, and oscillator arrays

Application Comments ReferenceProposal for magnetic logic Charge and spin operate simultaneously Cowburn [4.80]Nonvolatile RAM calledV-MRAM and AV-MRAM

Not to be confused with vertical MRAM based on stackedring-shaped GMR cells [4.20, 81]

Bohlens et al. [4.82], VanWaeyen-berge [4.83], Cowburn [4.84],Pigeau et al. [4.85]

Data storage and memory A vortex can store 2 bit simultaneously with its polarizationand chirality

Bohlens et al. [4.82, 86]

Oscillator arrays Coupled vortex oscillators in arrays of Permalloy disks Hanze et al. [4.87]Electrical switching of a vortexcore in a magnetic disk

Read–write magnetic logic operations Yamada et al. [4.88]

Vortex transistor A new type of magnetic transistor Kumar et al. [4.89]

superconductors and nitrogen for high-Tc superconduc-tors). Moreover, Yamada et al. [4.88] were recently ableto perform electrical switching of a vortex core (Ta-ble 4.11).

In many areas, magnetometry is needed, for in-stance, in biology, biomedicine, and geophysics wheretensor gradiometry is required in order to determineall 3-D field components and their variation alongdifferent spatial directions with respect to Earth coor-dinates.

Detection of vortices is a very versatile field of re-search since electron holography [4.90] has been used,as well as magnetoresistance measurement in layereddots with a current traversing the structure along the ax-ial direction (perpendicular to the vortex).

Some vortex detection techniques that performidentification of distinctive magnetization states, suchas vortex nucleation and excitation (or switch-ing [4.91]), domain wall motion, ferromagnetic reso-nance, and spin-wave excitations both in the quasistaticand dynamic regimes, are:

� Differential resistance dV=dI measurement [4.92,93]� Spin rectification effect [4.65, 94]� Inductive methods [4.95, 96]� RF/microwave transmission measurements [4.97–99]

Wang et al. [4.100] have shown that a resonantmagnetic field with 30mT amplitude stimulates strongaxially symmetric magnetization oscillation and forcesthe vortex core to stay at the center of the disk. Thecompression of the vortex core by spin-wave excitation

leads to fast core reversal [4.100] at 602�10�12 s rate.More recently, Dong et al. [4.101] managed to confinevortex cores in spin-wave potential wells and obtainedcore switching times well below 200�10�12 s.

The detection range of vortex-sensing magnetome-ters, being in the kOe range, is too strong for fieldsusually encountered in biology, biomedicine, and geo-physics. Thus we need a conditioning stage for thesensor in order to match its own detection range to an-other one usually encountered in the above-mentionedareas. The conditioning is made with IMC (integratedmagnetic concentrators) that at present possess a gainof 10�100 order of magnitude [4.102].

This gain, defined as the ratio between detectablefield in the sensor (kOe) and an external one originat-ing from biological or geophysical sources, is currentlybeing increased steadily with the use of softer mate-rials and better geometrical design to attain at least1000 [4.103].

In micro-Hall magnetometry where magnetic nan-odisks are placed on top of Greek cross-shaped [4.104]Hall plate sensors, vortex motion is detected from theirstray-field sensed by Hall voltage measurements.

Another route to vortex detection is through themagnetoelectric effect in layered structures consist-ing of multiferroic elements or composite elementsconsisting of alternating piezoelectric layers and giantmagnetostrictive elements.

A host of applications exist for vortices (and an-tivortices their magnetic mirror counterpart) rangingfrom data storage to magnetic RAM, logic, and infor-mation processing devices. More recently a new type oftransistor [4.89] based on vortices has been introduced(Table 4.11).

Magnetic Properties: From Traditional to Spintronic References 109Part

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