Ø A brief survey of the early history of magnetism Ø Hard and Soft magnets --- what are they for? Ø Materials abundance issues in the best hard magnets … the

“rare earth crisis” Ø Origins of Ferromagnetism, key role of demagnetising fields Ø  Figures of merit: magnetic energy, moment, exchange, and

anisotropy Ø  Relation of key magnetic properties to electronic structure Ø Domain walls, relation to anisotropy and demagnetising

fields, and hysteresis Ø What causes anisotropy? What makes it large? Ø  Reducing rare earth content in permanent magnets

Permanent Magnets and the “Rare Earth crisis” Mark van Schilfgaarde, King’s College London

Early History of Magnetism Magnets have been around a long time: Thales of Miletus (6th century BC) is credited

with discovering lodestone's attraction to iron (lodestone is mostly magnetite).

Even earlier: a grooved magnetic bar was found from artifacts of the Olmecs in Central America ~1000 years before Thales

Lodestone

Sir William Gilbert did experiments on magnetism and wrote De Magnete (1600). He concluded that the Earth was magnetic. Some credit him as the father of electricity and magnetism. Gilbert also studied magnetic properties of carbon steel.

Uses of Magnetism Until recently magnets were mostly

confined to specialised applications as electromagnetic energy converters (motors and generators).

Until the 1950’s magnets tended to assume “interesting” shapes, because they were not strong enough to overcome the demagnetising field (later)

Neodymium magnet

Today magnets are used everywhere are (a typical household has ~100 magnets, e.g. as microwave generators) though we are often not aware of many of them.

Hard and Soft Magnets

A magnet is typically classified by “hardness---” the amount of hysteresis it exhibits.

Cycle the magnet around a loop ⇒ energy dissipated in proportion to loop area.

The ideal hard magnet forms a square hysteresis loop. Both hard and soft magnets have widespread uses.

Coercive field Hc

Remanence Mr

Saturation Ms

Applications of Hard and Soft Magnets

Soft magnetic materials •  Used when the magnetization is cycled rapidly

(transformer coils, generators and some motors) •  Low hysteresis minimizes energy loss. •  Popular magnet: Permalloy (Ni/Fe alloy), Hc ~ 2 . 10−7 Tesla Hard magnets (or permanent magnets) •  Used in applications where you don’t want material to

demagnetise e.g. loudspeakers, motors, magnetic recording •  The hardest magnets contain rare earths, e.g. Nd2Fe14B,

with a coercive field Hc ~ 1.2 Tesla. •  Rare earths make the best magnets, but issues of

materials abundance become important (later).

Inexpensive Hard Magnets

Ferrite family: e.g. BaFe12O19, SrFe12O19, (the seal on your refrigerator, or the magnetron in your microwave oven). Comprises most hard magnets, but only ~1/3 by cost. First developed in the Netherlands in [J. Went, G.W. Rathenau, E.W. Gorter, G.W. van Oosterhout, Phys. Rev. 86, 424 (1951)] and marked the onset of the modern era of magnetic materials.

A big drawback of the Ferrites is that the net magnetic moment M is moderate because some spins align antiferromagnetically.

MR44CH10-McCallum ARI 28 March 2014 15:29

SrFe12O19

12k4f14f22a2b

12k112k26h

4f4e

2dSr:

Fe:

O:

Figure 9The crystal structure of ferrite magnets. The high ordering temperature of these magnets stems from thestrong indirect exchange between Fe atoms mediated by O. This interaction is antiferromagnetic, such thatthe net moment of the material is a result of unequal moments on the Fe sites.

In addition to structural studies, atom probe measurements have allowed for detailed chemicalcharacterization of the phase evolution, including of metastable transitional phases, accompanyingspinodal decomposition of the alnico solid solution. Recently, researchers accurately measured onan atomic scale the composition of the transitional phases during spinodal decomposition of alnico(L. Zhou, M.K. Miller, P. Lu, L. Ke, R. Skomski, et al., submitted) and the role of the appliedmagnetic field on the formation of the spinodal (54). These advances have improved characteri-zation of the constituent phase properties in alnico and have fostered new avenues for increasingcoercivity.

Ferrite MagnetsPrior to the 1950s, permanent magnets were based on magnetic steels. Even alnico resulted fromstudies to improve the properties of those steels. In 1952, a new class of permanent magnet basedon a class of oxides with a hexagonal crystal structure was discovered (Figure 9) (55). Thesecompounds adopt a hexagonal crystal structure and are described by the formula MO-6Fe2O3

or MO-2Fe2O4-6Fe2O3, where M is Ba, Sr, or Pb and O is oxygen. For toxicological reasons,Pb-based compounds are not used as commercial magnets. The Ba-containing composition is ahighly successful permanent magnetic material known as barium ferrite or barium hexaferrite; itis often simply referred to as a ceramic magnet.

The ferrite family of oxides exhibits high coercivity but rather low magnetization. Unlikemetallic magnets, the hexaferrites are ferrimagnetic, with both ferromagnetic coupling and anti-ferromagnetic coupling between Fe magnetic moments. This magnetic coupling scheme is deter-mined by the specific positions within the oxide sublattice of the metallic cations, and by associated

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Neodymium Magnets

Rare earth family: (esp Nd2Fe14B) developed simultaneously in Japan and the U.S. [M. Sagawa, S. Fujimura, N. Togawa, H. Yamamoto, Y. Matsuura, J. Appl. Phys, 55, 2083 (1984); J.J. Croat, J.F. Herbst, R.W. Lee, F.E. Pinkerton, J. Appl. Phys. 55, 2079, (1984)].

The combination of Fe (a transition metal with large moment and large exchange coupling) and Nd (a 4f element with very large anisotropy) make this a nearly perfect magnet!

MR44CH10-McCallum ARI 28 March 2014 15:29

Nd2Fe14B

4g

16k1

16k2

8j1

8j2

4c

4e

4f

4fNd:

Fe:

B:

Figure 3The crystal structure of Nd2Fe14B showing high-moment Fe layers separated by high-anisotropy,Nd-containing layers.

As these inner-shell electrons do not interact with the lattice-bonding electrons, the total REangular momentum remains unquenched (LRE ̸= 0). Consequently, the very strong spin-orbitcoupling ensures that the lanthanide 4f electron charge clouds, described by Stevens coefficients,are rigidly coupled to the d electron spins of the TM (24). In this manner, the corresponding REmagnet anisotropy energy is equal to the electrostatic interaction energy difference as a functionof orientation between the RE ions and the anisotropic crystal field.

By contrast, the magnetocrystalline anisotropy of permanent magnets that do not contain REelements, referred to here as TM magnets, reflects the interplay between the crystal-field andspin-orbit interactions of the d electrons. In a TM magnet, the physical origin of magnetocrys-talline anisotropy is derived from the electrostatic energy of the 3d electron cloud. This crystal-field or electrostatic energy depends on the electron cloud’s orientation relative to that of thebonds that define the crystal axes. The spin-orbit coupling in TM-based magnets is derived from3d electrons that execute a circular motion with an axis of rotation parallel to the electron spindirection in space. The corresponding 3d electron cloud may be oblate or prolate, depending onthe involved orbitals, but it is the spin-orbit coupling phenomenon that links this cloud’s axis ofrevolution to the spin direction. Figure 4 illustrates the basic physical mechanism underlying themagnetocrystalline anisotropy in chemically ordered magnets such as L10-type (τ-phase) MnAl,in which the magnetocrystalline anisotropy originates from the Mn electrons.

In Figure 4a, the Mn charge clouds are predominantly located in the Mn planes, whereas inFigure 4b, these charge clouds reach deep into the Al layers. The crystal field in the L10-orderedmagnets results from the different electronic structures of the chemically distinct atomic layersthat constitute this crystal. Tetragonally strained cubic magnets yield conceptually similar butmuch weaker crystal fields (20), which lead to magnetoelastic anisotropy. In turn, pseudocubicL10 magnets with axial ratios a ∼ b ∼ c generally exhibit large anisotropies associated with thechemical constituents in the structure.

www.annualreviews.org • Modern and Future Permanent Magnets 10.7

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Pure Nd2Fe14B has limited applicability because hysteresis falls of rapidly with To. Nd2Fe14B can be stabilised up to ~200C by replacing 3–10wt.% of Nd by Dy (or Tb)

Breakdown of the Hard Magnet Market

Ferrites are more commonly used because of low cost (<$5/kg). But energy density is low (<38 kJ/m3)

Nd2Fe14B is a vastly superior magnet: some lab magnets have energy density ~470 kJ/m3; commercial magnets a bit lower.

O. Gutfleisch et al. Adv. Mat. 23, 821 (2011)

But even though Nd2Fe14B can be stabilised by subsituting some Nd for Dy, Dy is much more rare and costly than Nd!

Cost is a big issue, because of the Nd (next slide)

The SmCo market is aimed mainly at high-To applications where the Nd magnets do not perform well.

The more Dy, the better the magnet Why Dy is so useful. From McCallum et al Annu. Rev. Mater. Res. (2014)

MR44CH10-McCallum ARI 28 March 2014 15:29

Magnetgrade suffix

HcJ and Br as a function of approximate Dy contentHcJ and Br as a function of approximate Dy content

10

11

12

13

14

15

Br (kG

)

Br

HcJ

HDDs, CDs,DVDs, sensors

Holding,sensors

General-purposemotors, some wind power generators

High-performancemotors and generators,

wind power

Superhigh-performancemotors and generators,

auto traction drives

10,000

80°C

15,000

20,000

25,000

30,000

35,000

HcJ

(Oe)

0 2 4 6 8 10 12

Dy (%)

M

H

SH

UH

EH

AH

100°C

120°C

150°C

180°C200°C

220°C

Figure 1The intrinsic coercivity (HcJ) and remanence, or induction (Br), as a function of alloy Dy content in weight percent for commercialNdFeB-based magnets. The ovals contain the grade designations M, H, SH, UH, EH, and AH based on the maximum operatingtemperature noted for that grade. Typical uses for each grade are shown along the top columns. Data supplied by Arnold MagneticTechnologies, http://www.arnoldmagnetics.com.

Automotive traction drive motors have even more demanding magnet size and weight con-siderations than those of wind turbines, especially in hybrid vehicles in which there is lim-ited space for the electric motor between the internal combustion engine and the wheel wells.Although induction motors remain an option for all-electric vehicles, considerations of superiorpower conversion efficiency, desired high torque over a wide rpm range, and a favorable mass-to-power ratio clearly favor the use of a high-performance, permanent magnet–based motor formany applications (19). However, the advantages of permanent magnet–based machines disap-pear if lower-energy-product magnets, such as current non-RE permanent magnets, are used.Even though most non-RE permanent magnetic materials developed prior to the invention ofRE permanent magnets are still commercially produced, the development of these older ma-terials has been neglected since the end of the 1970s. We discuss a few of the more promisingnon-RE-containing systems with the potential to replace RE-based permanent magnet alloys inelectric machines. Additionally, we articulate challenges to improving the magnetic and structuralproperties of these select systems to meet the demanding and escalating performance criteria forpermanent magnet machines.

MAGNETIC METRICS AND PHENOMENAThe properties and performance of a permanent magnet depend both on the characteristics of thebase compound (i.e., intrinsic properties) and on the attributes of the microstructure (i.e., extrinsic

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Volatility of the RE Market

The price of Nd and Dy underwent a huge spike in 2011. China adopted a new policy that it would no longer export raw material. Dy peaked at $1400/kg in 2011; it is ~$500/kg today, apparently

O. Gutfleisch et al. Adv. Mat. 23, 821 (2011)

L Lewis and F. Jimenez-Villacorta, Metall and Mat. Trans. A 44, 2 (2013)

largely as a result of illegal production in China (China mines 99% of worldwide production, ~100 tonnes/year). This solved the immediate crisis, but …

RE’s are Emerging Critical Materials Regardless of Chinese policies (China has only ~30% of all RE reserves worldwide), a critical shortage of Dy is shaping up, owing to anticipated large increases in demand for motors in electric cars and wind turbines.

~100 grams of Dy go into in every electric car motor. If Toyota were to manufacture 2 million cars per year, the available Dy supply would be exhausted. According to the US DOE, Dy is the single most critical element for emerging clean energy technologies, since no good alternative has yet emerged.

L Lewis and F. Jimenez-Villacorta, Metall and Mat. Trans. A 44, 2 (2013)

Basic Science of a Ferromagnet: origin of FM

But ferromagnetism does not originate from circulating e− …

In classical magnetism we associate a magnetic moment µ with a current loop. An electron at point

r generates a magnetic moment

!µ = − e

2r × v = (−e / 2m)

γ"#$ %$r × p

L&≡ γ L

Classical gyromagnetic

ratio

Instead, an e− has an intrinsic angular momentum (spin) which emerges from the intersection of relativity & quantum theory. Magnetism is a consequence if the fact that e− must obey the Pauli exclusion principle. Formally: the many-particle wave function must be antisymmetric when two particles are exchanged.

Origin of FM II

The total wave function ψ consists of a product of orbital part r and spin part σ. Consider a two-particle ψ(r1σ1, r2σ2) under particle exchange r1σ1 ⇄ r2σ2. It is not difficult to show that: • if σ1 and σ2 are parallel (↑,↑). the orbital part ψ of must be antisymmetric •  if σ1 and σ2 are antiparallel (↓,↑). the orbital part ψ of must be symmetric Let φ be the orbital part of ψ. Consider the symmetric case when r1 → r2. We must have simultaneously:

1 2 1 2 1 2 1 22 1 21 2 1 2 1lim ( , ) lim ( , ) and lim ( , ) lim ( , )φ φ φ φ

→ → → →= − = +

r r r r r r r rr r rrrr r r

These can both be true only when φ(r1,r2) →0.

Origin of FM III

This implies that when the spins are aligned parallel (↑,↑) , the probability of two e− meeting each other vanishes. The exclusion principle imposes no equivalent constraint when spins are antiparallel (↓,↑). Thus, e− with (↑,↑) spins tend to stay away from each other while e− with (↓,↑) spins have no such built-in correlation. Remember that e− are charged and repel each other. Thus, correlations keeping e− apart reduce their repulsion: e− can lower their energy by aligning their spins. This interaction is called the “exchange interaction” and can be cast in terms of an effective magnetic field called the “exchange field” or “exchange-correlation field” Bxc .

Last slide: these can both be true only when φ(r1,r2) →0.

Demagnetising Fields True magnetic dipolar fields fields do play an important role in real magnets. They are tiny compared to Bxc. But in a

The total and external fields are related: B = µ0(H+M).

From E&M we know ∇⋅B = 0, so ∇⋅H = −∇⋅M. M originates from the e− through Bxc.

macroscopic magnet Avogadro's number of them combine to generate a field whose energy scales in proportion to the volume. The effects of this field become important at the boundary of the magnet.

M changes abruptly at the surface and therefore ∇⋅M and ∇⋅H become very large. The resultant field, called the demagnetizing field Hd, is present both inside and outside the magnet, opposing the internal field.

Stray Fields Inside the magnet Hd opposes B, destabilising the ferromagnetism. Now Hd depends strongly on the shape in complicated ways (see later). Until the discovery of the Ferrite magnets in the 1950’s they had to be formed in funny shapes.

Outside the magnet Hd are “stray fields.” For a magnetic memory device, they interfere with neighbouring bits and the trick is to avoid them as much as possible. For motors and generators, they are essential as they supply the torques which couple mechanical and electrical power.

Influence of Magnet Shape on Energy Density

Hd is a very complicated function of the shape, so it is often parameterised by a factor Ν with Hd ≈ −Ν M. (Note: N should be a function if r; usually ignored)

30

2d VE d rµ= ⋅∫ dB HThe energy stored by the demagnetising

field is given by

Consider an ideal hysteresis loop. Use B = µ0(Hd+Ms) and Hd ≈ −Ν Ms :

2 20 )2d sVE µ= −M (N N

Thus there is no energy contained in the magnet if no demagnetising field (N = 0) or if it approaches the internal field (N = 0)

Energy Figure of Merit For the optimum energy density, maximise wrt N:

200 1 2 ) 1/ 22

ds

dE Vd

µ= = − ⇒ =M ( N NNIt turns out that a cylinder with radius/height ≈ 2 yields N = Hd/M ≈ 1/2.

From Sunshine Magnets

Modern commercial magnets are often in this shape.

The standard figure of merit is |BH|max ~ 2 Ed / V As the hysteresis loop deviates from ideal, |BH|max drops.

Many Figures of Merit

(1) Large maximum energy product |BH|max (2) high saturation Ms (3) nearly rectangular hysteresis loop with Mr ≈ Ms and Hc > 1/2 Ms (4) very high uniaxial magnetocrystalline anisotropy energy K (see soon) (5) Low cost

(5) Significant structural and/ or composition fluctuations ⇒ high Hc from domain wall nucleation or pinning (7) High Curie temperature Tc (8) Good temperature stability (9) Corrosion resistance (10) Mechanical strength (11) Machinability

In practical magnets, many factors to consider. Some key figures of merit are:

No magnet has good figures of merit in all of these categories!

Local-Moments picture of Magnetism

Suggests that magnetism can be well described in term of a local-moment picture, an ensemble of spins on a lattice that interact with each other.

The spin density (n↑ - n↓) originates from d or f orbitals, and tends to be rather localized near the nucleus.

A spin of size Si resides at site i canted by some angle.

Heisenberg model : each Si interacts with each Sj through an interaction J:

Jij can be determined from electronic structure theory

θij Ri Rj

Jij Si . Sj = Jij Si Sjcos(θij)

Temperature Dependence

If J=0, an ensemble of local moments will be randomly oriented.

Application of external field H provides a driving force to align spins, which competes against entropy (favoring disorder)

M =〈 µ 〉= 0 M

H

Interactions J effectively act as an internal field, analagous to H, causing the material to spontaneously magnetise.

M drops with To, and vanishes at the Curie temperature.

Electronic Structure and Critical Temperature

Mean-field theory gives a simple (reasonably reliable) estimate of Tc. For one kind of atom Tc is:

To make Tc large, S and J should be large. S and J are determined by the electronic structure.

20

02 , ,

3ˆ ˆ ˆ

B c ij

ii jj i

j

j

J Sk T J J

H J

= =

= − ⋅

∑∑ SS

For J to be large, atomic wave functions must strongly overlap with neighbors.

Rare earths, with their 4f electrons, can have very large S, (Gd has S=7), but the wave functions are atomic-like: they weakly overlap with (and couple to) neighbors. Tc =292K for Gd

The middle of the 3d series (half filled d bands have pretty large S and large J. Tc can be quite high (Tc = 1400K for Co)

High Tc is a necessary but not sufficient condition If Tc and Ms were the only materials properties that mattered, transition metals would make excellent magnets

Material Tc Fe 1043 Co 1400 Ni 627 Gd 292 EuO 69 Dy 88 MnBi 630 MnSb 587 CrO2 386 Fe2O3 948 Nd magnets 583-673 SmCo magnets 993-1073

A third critical property is the uniaxial magnetocrystalline anisotropy K. Without anisotropy, there is no hysteresis.

Origin of Magnetocrystalline Anisotropy

Originates from coupling between spin and orbital angular momentum (a consequence of relativity). Spin-orbit coupling links the electron spins to the crystalline lattice: the energy of the spin system depends on its crystallographic orientation. As the spin system rotates between two axes of low energy (easy axes) they must pass through a harder axis. There is energy barrier to rotating the collective system, called the uniaxial Magnetocrystalline Anisotropy (MAE).

Easy

Hard

Easy

Crystals possess a magnetic “easy axis” and a “hard axis.”

Anisotropy can arise from other sources (esp shape anisotropy) but MAE is the most important one.

Making MAE large

Roughly, ΔE(θ) ~ Kcos(θ) .

MAE is a complicated business! No one single parameter characterizes it. It originates from a relativistic effect that couples spin and orbital angular momentum (λL.S). λ~Z2 (Landau) because e− move fastest near nuclei. Larger Z, faster e− accelerate near nucleus. But … the MAE is not E(λL.S), but the change ΔE(λL.S) as the crystallographic orientation rotates through angle θ. Usually ΔE(λL.S) ≪ E(λL.S) --- it is several orders of magnitude smaller in high-symmetry systems.

Rules of thumb: Heavier elements have larger E(λL.S): helps to increase ΔE(λL.S) Lower symmetry tends to increase ΔE(λL.S)

Demagnetizing Fields Create Magnetic Domains The “demagnetizing energy” (or the “dipolar” energy) Ed is positive and increases with volume. Ed can be lowered by splitting up the magnetic element into smaller regions because the dipolar fields from region of opposing M partially cancel.

A region where the moments all point in a single direction is called a domain or a Weiss domain. The small region separating domains is called a domain wall.

The domain wall has an energy cost 2J in proportion to the wall area A. 2JA competes with the reduction in Ed : it is a complicated function of the shape of the domains. The globally optimum domain shape and thus global minimum energy structure depends on the shape of the specimen.

Size of a Domain Wall

Energy cost to rotate neighbouring spins

The three energies, exchange coupling J, MAE K, and dipolar fields Hd give rise to rich magnetic behavior. Dipolar fields depend on sample shape and can give rise to remarkable domain patterns. Example: Interplay between J and K governs energy, mobility, shape of domain wall.

Net rotation : angle π over N sites ⇒ θ =π/N.

−2JS1 . S2 = −2J S2 cosθ ≈ JS2θ2 if θ is small.

Energy / area: 2 2

2 2

JSN a

πσ =Competes with MAE. Energy of

layer i depends on angle φi as Ei ≈ K sin2 φi

z φi Sum MAE for N sites

2sin2ii

NKK ϕ ≈∑Minimize

2NKσ + ⇒ DW width 2JNa S

Kaδ π= =

Hysteresis and Domain Walls

Application of external field H causes the crystal to favor regions where spins are aligned with H.

Walls will move enabling domains parallel to the field to grow. But their motion is restricted because there are barriers to motion, because of the anisotropy. Barriers are the reason for hysteresis.

Barriers depend in a complicated way on the relative strengths of exchange coupling J, MAE K, and dipolar fields Hd, and also by inhomogeneities (e.g. defects), which can “pin” them. Thus domain-wall motion is intricately linked to microstructure.

MAE in MnBi: a detailed theoretical study MnBi is one possible replacement for Neodymium magnets. A study of its magnetic properties highlights how challenging it is. Up to 628K MnBi is ferromagnetic and forms in the NiAs structure (hexagonal). a and c/a vary significantly with To .

Calculations from Antropov et al, PRB 90, 054404 (2014)

MAE in MnBi The MAE K, depends strongly on T, even changing sign. Around room To K is respectable (~0.9mJ/m3) but its To – dependence makes it unsuitable

K originates from pockets of energy bands just above and below the (very complicated) Fermi surface.

Pockets contribute to K with varying signs. As To ↑, c and a change, which modifies pocket shapes and their individual contributions to K.

PRB 90, 054404

What makes K large? What controls K is a rather involved question, even for a simple two-level model system. For a discussion of K in a model context, see the following paper:

PHYSICAL REVIEW B 92, 014423 (2015)

Band-filling effect on magnetic anisotropy using a Green’s function method

Liqin Ke1,* and Mark van Schilfgaarde2

1Ames Laboratory U.S. Department of Energy, Ames, Iowa 50011, USA2Department of Physics, King’s College London, Strand, London WC2R 2LS, United Kingdom

(Received 18 April 2015; revised manuscript received 5 July 2015; published 28 July 2015)

We use an analytical model to describe the magnetocrystalline anisotropy energy (MAE) in solids as a functionof band filling. The MAE is evaluated in second-order perturbation theory, which makes it possible to decomposethe MAE into a sum of transitions between occupied and unoccupied pairs. The model enables us to characterizethe MAE as a sum of contributions from different, often competing terms. The nitridometalates Li2[(Li1−xTx)N],with T = Mn, Fe, Co, Ni, provide a system where the model is very effective because atomiclike orbital charactersare preserved and the decomposition is fairly clean. Model results are also compared against MAE evaluateddirectly from first-principles calculations for this system. Good qualitative agreement is found.

DOI: 10.1103/PhysRevB.92.014423 PACS number(s): 71.70.Ej, 75.30.Gw, 71.20.−b

I. INTRODUCTION

Magnetocrystalline anisotropy is a particularly importantintrinsic magnetic property [1]. Materials with perpendicularmagnetic anisotropy are used in an enormous variety ofapplications, including permanent magnets, magnetic randomaccess memory, magnetic storage devices, and other spintron-ics applications [2–5].

Modern band theory methods have been widely used toinvestigate the magnetocrystalline anisotropy energy (MAE)in many systems [6,7]. The MAE in a uniaxial system canbe obtained by calculating the total-energy difference betweendifferent spin orientations (out of plane and in plane). However,MAE is usually a small quantity and a reliable ab initio calcu-lation requires very precise, extensive calculations. Moreover,MAE is, in general, harder to interpret from the electronicstructure than other properties, such as the magnetization.MAE often depends on very delicate details of the electronicstructure [8]. Using perturbation theory, the MAE can bedecomposed into virtual transitions between different orbitalpairs. In practice, the d bandwidth is large enough that itis nontrivial to meaningfully resolve the MAE into orbitalcomponents and predict its dependence on band filling.

The magnetocrystalline anisotropy originates from spin-orbit coupling (SOC) [9] or, more precisely, the changein SOC as the spin-quantization axis rotates. Including therelativistic corrections to the Hamiltonian lowers the systemenergy and breaks the rotational invariance with respect to thespin-quantization axis. Here we refer to the additional energydue to the relativistic correction as SOC energy or relativisticenergy Er . MAE is a result of the interplay between SOC andthe crystal field [10]. The MAE and change in orbital momenton rotation of the spin-quantization axis are closely related. Wedescribe this below and denote them as K and KL, respectively.Without the SOC, the orbital moment is totally quenched bythe crystal field in solids. Except for very heavy elements suchas the actinides, SOC usually alleviates only a small part ofthe quenching and induces a small orbital moment relative tothe spin moment. For 3d transition metals, SOC is often muchsmaller than the bandwidth and crystal-field splitting, and thuscan be neglected in a first approximation. While the Er is

*Corresponding author: liqinke@ameslab.gov

generally small, its anisotropy with respect to spin rotation isoften even orders of magnitude smaller.

Recently, it had been found that a very high magneticanisotropy can be obtained in 3d systems such as lithiumnitridoferrate Li2[(Li1−xFex)N] [11–14], which can be viewedas an α-Li3N crystal with Fe impurities. As found bothin experiments [15] and calculations [12,13] using densityfunctional theory (DFT), the Li2(Li1−xFex)N system possessesan extraordinary uniaxial anisotropy that originates from Feimpurities. The linear geometry of Fe-impurity sites resultsin an atomiclike orbital and then a large MAE. As found inboth x-ray absorption spectroscopy [11] and DFT calculations[11–13], 3d ions T have an unusually low oxidation state (+1)in Li2(Li1−xTx)N for T = Mn, Fe, Co, and Ni. Recently, Jescheet al. [16] developed a single-crystal growth technique forthese systems and directly observed that the MAE oscillateswhen progressing from T = Mn → Fe → Co → Ni [16].Electronic structure calculations also show that the atomiclikeorbital features are preserved for different T elements. Con-sidering the rather large MAE and well-separated density ofstates (DOS) peaks in this system, it provides us with a uniqueplatform to investigate the MAE as a function of band filling.

Li and N are very light elements with s and p elec-trons, respectively. They barely contribute to the MAE inLi2[(Li1−xTx)N]; rather, MAE is dominated by single-ionanisotropy from impurity T atoms, especially for lower Tconcentration where T -T atoms become well separated. In thiswork, we investigate the magnetic anisotropy with differentT elements based on second-order perturbation theory byusing a Green’s function method. Lorentzians are used torepresent local impurity densities of states and calculate theMAE as a continuous function of band filling. First-principlescalculations of MAE are also performed to compare with ouranalytical modeling.

The present paper is organized in the following way. InSec. II, we overview the general formalism of the single-ion anisotropy [17,18] with Green’s functions and second-order perturbation approach [19–24]. Analytical modeling andcalculational details are discussed. In Sec. III, we discuss thescalar-relativistic electronic structure of these systems. Theband-filling effect on MAE in Li2[(Li1−xTx)N], with T = Mn,Fe, Co, and Ni, is examined within our analytical model andresults are compared with first-principles DFT calculations.The results are summarized in Sec. IV.

1098-0121/2015/92(1)/014423(9) 014423-1 ©2015 American Physical Society

PHYSICAL REVIEW B 92, 014423 (2015)

Band-filling effect on magnetic anisotropy using a Green’s function method

Liqin Ke1,* and Mark van Schilfgaarde2

1Ames Laboratory U.S. Department of Energy, Ames, Iowa 50011, USA2Department of Physics, King’s College London, Strand, London WC2R 2LS, United Kingdom

(Received 18 April 2015; revised manuscript received 5 July 2015; published 28 July 2015)

We use an analytical model to describe the magnetocrystalline anisotropy energy (MAE) in solids as a functionof band filling. The MAE is evaluated in second-order perturbation theory, which makes it possible to decomposethe MAE into a sum of transitions between occupied and unoccupied pairs. The model enables us to characterizethe MAE as a sum of contributions from different, often competing terms. The nitridometalates Li2[(Li1−xTx)N],with T = Mn, Fe, Co, Ni, provide a system where the model is very effective because atomiclike orbital charactersare preserved and the decomposition is fairly clean. Model results are also compared against MAE evaluateddirectly from first-principles calculations for this system. Good qualitative agreement is found.

DOI: 10.1103/PhysRevB.92.014423 PACS number(s): 71.70.Ej, 75.30.Gw, 71.20.−b

I. INTRODUCTION

Magnetocrystalline anisotropy is a particularly importantintrinsic magnetic property [1]. Materials with perpendicularmagnetic anisotropy are used in an enormous variety ofapplications, including permanent magnets, magnetic randomaccess memory, magnetic storage devices, and other spintron-ics applications [2–5].

Modern band theory methods have been widely used toinvestigate the magnetocrystalline anisotropy energy (MAE)in many systems [6,7]. The MAE in a uniaxial system canbe obtained by calculating the total-energy difference betweendifferent spin orientations (out of plane and in plane). However,MAE is usually a small quantity and a reliable ab initio calcu-lation requires very precise, extensive calculations. Moreover,MAE is, in general, harder to interpret from the electronicstructure than other properties, such as the magnetization.MAE often depends on very delicate details of the electronicstructure [8]. Using perturbation theory, the MAE can bedecomposed into virtual transitions between different orbitalpairs. In practice, the d bandwidth is large enough that itis nontrivial to meaningfully resolve the MAE into orbitalcomponents and predict its dependence on band filling.

The magnetocrystalline anisotropy originates from spin-orbit coupling (SOC) [9] or, more precisely, the changein SOC as the spin-quantization axis rotates. Including therelativistic corrections to the Hamiltonian lowers the systemenergy and breaks the rotational invariance with respect to thespin-quantization axis. Here we refer to the additional energydue to the relativistic correction as SOC energy or relativisticenergy Er . MAE is a result of the interplay between SOC andthe crystal field [10]. The MAE and change in orbital momenton rotation of the spin-quantization axis are closely related. Wedescribe this below and denote them as K and KL, respectively.Without the SOC, the orbital moment is totally quenched bythe crystal field in solids. Except for very heavy elements suchas the actinides, SOC usually alleviates only a small part ofthe quenching and induces a small orbital moment relative tothe spin moment. For 3d transition metals, SOC is often muchsmaller than the bandwidth and crystal-field splitting, and thuscan be neglected in a first approximation. While the Er is

*Corresponding author: liqinke@ameslab.gov

generally small, its anisotropy with respect to spin rotation isoften even orders of magnitude smaller.

Recently, it had been found that a very high magneticanisotropy can be obtained in 3d systems such as lithiumnitridoferrate Li2[(Li1−xFex)N] [11–14], which can be viewedas an α-Li3N crystal with Fe impurities. As found bothin experiments [15] and calculations [12,13] using densityfunctional theory (DFT), the Li2(Li1−xFex)N system possessesan extraordinary uniaxial anisotropy that originates from Feimpurities. The linear geometry of Fe-impurity sites resultsin an atomiclike orbital and then a large MAE. As found inboth x-ray absorption spectroscopy [11] and DFT calculations[11–13], 3d ions T have an unusually low oxidation state (+1)in Li2(Li1−xTx)N for T = Mn, Fe, Co, and Ni. Recently, Jescheet al. [16] developed a single-crystal growth technique forthese systems and directly observed that the MAE oscillateswhen progressing from T = Mn → Fe → Co → Ni [16].Electronic structure calculations also show that the atomiclikeorbital features are preserved for different T elements. Con-sidering the rather large MAE and well-separated density ofstates (DOS) peaks in this system, it provides us with a uniqueplatform to investigate the MAE as a function of band filling.

Li and N are very light elements with s and p elec-trons, respectively. They barely contribute to the MAE inLi2[(Li1−xTx)N]; rather, MAE is dominated by single-ionanisotropy from impurity T atoms, especially for lower Tconcentration where T -T atoms become well separated. In thiswork, we investigate the magnetic anisotropy with differentT elements based on second-order perturbation theory byusing a Green’s function method. Lorentzians are used torepresent local impurity densities of states and calculate theMAE as a continuous function of band filling. First-principlescalculations of MAE are also performed to compare with ouranalytical modeling.

The present paper is organized in the following way. InSec. II, we overview the general formalism of the single-ion anisotropy [17,18] with Green’s functions and second-order perturbation approach [19–24]. Analytical modeling andcalculational details are discussed. In Sec. III, we discuss thescalar-relativistic electronic structure of these systems. Theband-filling effect on MAE in Li2[(Li1−xTx)N], with T = Mn,Fe, Co, and Ni, is examined within our analytical model andresults are compared with first-principles DFT calculations.The results are summarized in Sec. IV.

1098-0121/2015/92(1)/014423(9) 014423-1 ©2015 American Physical Society

Generically speaking you need : 1. Large λL.S ⇒ favors heavy elements (remember λ~Z2) 2. Large orbital moment. λL.S provides the driving force to induce orbital moments (it breaks symmetry between m and -m) but it tends to be strongly quenched by the crystal field. 3. Large crystal field is needed to generate anisotropy in λL.S , i.e. change in λL.S as spins rotate --- comes from anisotropy of the crystal field.

Easy Hard Easy

Why do f shell elements have large K? f elements can have large anisotropy (generically) because: (1)  they are heavy (large λL.S) (2) the crystal field is large enough to interact with λL.S but (3) it is not large enough to quench the orbital moments. Hard to beat!

MR44CH10-McCallum ARI 28 March 2014 15:29

Fe

NiNi

Figure 10The crystal structure of tetrataenite (L10-ordered FeNi). The layered structure results in high anisotropy.

(58). Tetrataenite has relatively high magnetocrystalline anisotropy and a magnetization equal tothat of Nd2Fe14B and is predicted to be an excellent candidate for permanent magnet applications.Unfortunately, in the FeNi binary system, the chemically ordered L10 structure is only marginallymore stable than the disordered (A1-type face-centered cubic) structure, which is entropically sta-bilized above the equilibrium chemical order-disorder temperature (Torder-disorder) of 320◦C. TheFe-Ni interdiffusion rates for T < Torder-disorder are so slow that, even for the very limited diffusionalmotion required to achieve local chemical ordering, the L10 structure does not form under normallaboratory conditions but requires astronomically slow cooling rates (on the order of millions ofyears), the same timescale as that for meteorites.

The main challenge that must be overcome to realize a tetrataenite-based permanent magnet isdevelopment of a kinetic pathway to form the hard magnetic FeNi L10 structure in an industriallyviable process on terrestrial timescales. The mere existence of this phase suggests that all possiblepermutations of Fe-based systems have not been fully exploited. Thermodynamic calculations,in-depth first-principles computational studies, and more comprehensive screening tools (suchas combinatorial analysis to assess phase formation and stability) will be required to more fullyinvestigate this phase space. Advanced, far-from-equilibrium processing will be needed to explorenew phase space for compounds to realize high magnetocrystalline anisotropy with reasonablemagnetization via chemical substitutions and unconventional processing methods, as well as toinduce chemical ordering in this RE-free system (59).

α′-Fe16N2 (iron nitride). The technologically important magnetic properties of ferromagnetic,TM-containing compounds may also be enhanced by inserting N into interstitial lattice sites toform TM nitrides (60–62). Interstitial N alters the TM interatomic spacing, thereby altering themagnetization, magnetocrystalline anisotropy, and Tc of the parent compound. The only commer-cialized nitride permanent magnet is based on the ferromagnetic RE compound Sm2Fe17, whichillustrates both the advantages and limitations of this type of material. In these materials (abbrevi-ated here as SmFeN), interstitial N in the lattice changes the Fe-Fe interatomic spacing, increasesTc, and provides uniaxial anisotropy. The combination of these effects produces a high-energy-product magnet powder that is used to form bonded magnets. However, the near-100% densityrequired for magnets used in traction motor or wind turbine applications cannot be obtained in thiscompound because the interstitial N content is unstable during sintering; Sm2Fe17N3 decomposesinto SmN and Fe phases at temperatures lower than those required for sintering (60, 63).

Shifting our discussion to RE-free parent materials with the potential for permanent mag-net behavior, some TM-N compounds have large magnetic moments and may be amenable for

10.18 McCallum et al.

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Anisotropic 3d compounds can yield respectable K, e.g. FeNi can be made to form in an ordered L10 structure (AKA tetrataenite) with K=1.3 mJ/m3. So do CoPt, FePt, and MnAl with roughly similar K.

Compare to K=4.7 mJ/m3 (Nd2Fe14B) and K=6.5 mJ/m3 (YCo5)

Reducing Rare Earths in Hard Magnets Strategy 1: Reduce the Nd and Dy/Tb content in Neodymium magnets while retaining its good properties (e.g. preserving Hc up to 200C)

It may be possible to reduce the Nd and Dy content by ~50% largely by processing that better aligns grain boundaries (G.H. Yan et al, J. Phys. Conference Series 266, 012052 (2011)).

Domain wall motion and pinning are closely connected to inhomogeneities (microstructure) --- extremely complicated as small long-range forces interact with strong short-ranged ones. Optimise through Edisonian trial-and-error approach

MR44CH10-McCallum ARI 28 March 2014 15:29

Nd2Fe14B

4g

16k1

16k2

8j1

8j2

4c

4e

4f

4fNd:

Fe:

B:

Figure 3The crystal structure of Nd2Fe14B showing high-moment Fe layers separated by high-anisotropy,Nd-containing layers.

As these inner-shell electrons do not interact with the lattice-bonding electrons, the total REangular momentum remains unquenched (LRE ̸= 0). Consequently, the very strong spin-orbitcoupling ensures that the lanthanide 4f electron charge clouds, described by Stevens coefficients,are rigidly coupled to the d electron spins of the TM (24). In this manner, the corresponding REmagnet anisotropy energy is equal to the electrostatic interaction energy difference as a functionof orientation between the RE ions and the anisotropic crystal field.

By contrast, the magnetocrystalline anisotropy of permanent magnets that do not contain REelements, referred to here as TM magnets, reflects the interplay between the crystal-field andspin-orbit interactions of the d electrons. In a TM magnet, the physical origin of magnetocrys-talline anisotropy is derived from the electrostatic energy of the 3d electron cloud. This crystal-field or electrostatic energy depends on the electron cloud’s orientation relative to that of thebonds that define the crystal axes. The spin-orbit coupling in TM-based magnets is derived from3d electrons that execute a circular motion with an axis of rotation parallel to the electron spindirection in space. The corresponding 3d electron cloud may be oblate or prolate, depending onthe involved orbitals, but it is the spin-orbit coupling phenomenon that links this cloud’s axis ofrevolution to the spin direction. Figure 4 illustrates the basic physical mechanism underlying themagnetocrystalline anisotropy in chemically ordered magnets such as L10-type (τ-phase) MnAl,in which the magnetocrystalline anisotropy originates from the Mn electrons.

In Figure 4a, the Mn charge clouds are predominantly located in the Mn planes, whereas inFigure 4b, these charge clouds reach deep into the Al layers. The crystal field in the L10-orderedmagnets results from the different electronic structures of the chemically distinct atomic layersthat constitute this crystal. Tetragonally strained cubic magnets yield conceptually similar butmuch weaker crystal fields (20), which lead to magnetoelastic anisotropy. In turn, pseudocubicL10 magnets with axial ratios a ∼ b ∼ c generally exhibit large anisotropies associated with thechemical constituents in the structure.

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Finding Replacements for Neodymium Magnets

Strategy 2: Find a “good enough” replacement that doesn’t contain rare earths or other rare elements such as Pt.

Nd2Fe14B is already nearly ideal, so probably not realistic to equal it.

Transition Metal based Hard Magnets

R. Skomski, J.M.D. Coey, Scripta Materialia (2015)

A zoo of alternatives have been studied. Each is unique. Some (YCo5, Mn2Ga, FePt) have materials abundance issues Others (Fe16N2, MnBi) have stability issues.

R. Skomski, J.M.D. Coey, Scripta Materialia (2015)

Conclusions A great deal is understood generically about what ingredients are needed to make a hard magnet. The rare earths combined with Fe or Co are special because they have an optimum mix of high magnetisation Ms, essential for high |BH|max large λL.S combined with enough, but not too strong, coupling to crystal fields to make possible large MAE K. TM provides strong magnetic exchange interactions ⇒ high Tc. RE free magnets are not likely to be fully replaced soon. The search is on for intermediate quality magnets with good combinations of (Ms , |BH|max, K, Tc). Many promising candidates, each with its own unique set of challenges. No magic bullet yet!