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Chapter 8

Nanoscale Magnetism

8.1 Characteristic length scales

8.2 Thin films

8.3 Thin film heterostructures

8.4 Wires and needles

8.5 Superparamagnetism

8.6 Bulk nanostructures

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One nanoscale dimension: Thin films

Two nanoscale dimensions: Nanowires and acicular particles

Three nanoscale dimensions: Nanoparticles

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Fig 8.1 Magnetostriction in iron thin films

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8.1 Characteristic length scales

Exchange length Hardness parameter

Spin diffusion length !sd >> ! mean free path

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8.2 Thin films

surface

interface

substrate

Intrinsic magnetic properties Ms, TC, K1, !s can be

significantly different in thin films and in the bulk.

Epitaxial filmsOriented films

Lattice parameters are influenced by the substrate, when the difference is < 4%

Seed and cap layers.

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2.1 Magnetization and Curie Point

Some metals become ferromagnetic in thin film form (V, Rh) although they are not magnetically ordered in the bulk;Others become ferromagnetic when deposited on a ferromagnetic substrate (Pd on Ni)

Magnetism of iron is especially sensitive to structure and lattice parameters.

Fe has moment of 4 µB as an isolated atom;

3.3 µB in a chain,

3.0 µB as a plane

2.2 µB in the bulk

bcc iron has a surface layer wth a moment 20% greater than the bulk..

Moment enhancement is due to band narrowing related to reduction in the number of nearest-neighbours

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Curie temperature of thin films of 3d transition metals on various substrates

Number of planes

TC is weakened in ultra-thin films by the reduction in the number of exchange bonds,

Also by surface spin waves, structural relaxation.

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A uniformly-magnetized thin film produces no stray field.

B" = µ0 (H"+ M")= 0. B" is continuous Hence H" = 0 outside

-N M"; N =1

H|| = -NM|| .; N= 0. H|| is continuous. Hence H|| = 0 outside

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2.2 Anisotropy and domain structure

Extra ‘3s’ contributions to the anisotropy of a thin film: – shape; surface; strain.’

The demagnetizing factor for a uniformly magnetized film is N = (0, 0, 1)

The anisotropic contribution to the self energy in the demagnetizing field is -(1/2)µ0MHd

#

– Shape

E = -Ku sin2# where Kshape = -(1/2) µ0Ms

2

Fe -1.85 MJ m-3

Co -1.27

Ni -0.15

This has to be overcome by some other form of anisotropy if we want to make a true permanent magnet.

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– Surface

8.4 Surface anisotropy per unit area cobalt thickness for CoPd multilayers

Intercept gives Esurface ! 1 mJ m-2

Surface anisotropy often leads to perpendicularanisotropy in films about one nm thick.

Monolayer thickness is about 0.25 nm; This surface anisotropy corresponds to 4 MJ m-3, as in L10 compounds

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– Strain

Fig 8.5 Strain anisotropy induced by epitaxy.The strain in Ni layers on Cu is relaxed beyond4.5 nm

1050

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Fig 8.6 Twist of magnetization due to surface anisotropyKs (mJ m-2+)

Euler equation

Out of plane with a twist when Kv > (1/2)µ0Ms2

Magnetic structure of thin films

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Fig 8.7 Magnetization and domain structure in a film with perpendicular anisotropy

Maze domains

Bubble domains

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Magnetization of thin films. Q = -Ku/Kd where Kd = (1/2)µ0Ms2

Perpendicular anisotropy for Q > 1; Maze domains

In plane when Q < 1 and t < 2$w

Fig 8.8 Magnetic structure of a thin film as a function of Q and thickness t

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Fig 8.9. Magnetization curves and surface domain structure for a 200 nm film of Ni.Magnetization curves show the magnetization is largely in-plane. The MFM image of strayfield at the surface picks up the small perpendicular component.

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8.3 Thin film heterostructures

3.1 Direct exchange coupling; exchange bias

A magnetic multilayer is a stack of alternating magnetic and nonmagnetic layers.

A bilayer is a pair of layers of different magnetic materials

A superlattice is an epitaxial multilayer

FM2

FM1

FM2

AF

FM

F1

YCo2

GdCo2

Field-controllable domain wall

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Exchange Bias.

Discovered by Mieklejohn and Bean; 1956 ‘A new type of magnetic anisotropy has been discovered, which is bestdescribed as exchange anisotropy. This anisotropy is a result of an interaction between an antiferromagnetic materialand a ferromagnetic material’

Fig 8.11 Rotational hysteresis of the same particles

Fig 8.10 Shifted hysteresis loop of Co particles measured on field cooling in 1 T to 77 K

CoO

Co

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Exchange bias of thin films.

Néel 1964

AFM

FM

Fig 8.13

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It is as if an effective field Heff = H + Hex is acting on the film; Hex ! 4 kA/m; µ0Hex ! 50 mT

H

%M

Hex = Kex/µ0M2

x

y

z

The energies are better written per unit area of film as exchange bias scales with the area.

Kex = &/tp The energy per unit area is:

The corresponding field is EA/µ0Mptp

Minimize EA

Switching occurs when %='/2; H = Hex = -&/Mptp Perpendicular anisotropy field Ha = (&+2Kutp )/µ0Mptp

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Dependence on layer thickness

Hex! 1/tp

Fig 8.14

There is a threshold taf necessary for exchange bias to become effective; tcritKas ! &; tcrot = 10 nm, Kaf = 20 kJ m-3 & ! 0.2 mJ m-2

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Table 8.2. Antiferromagnetic Materials for Exchange Bias

Exchange bias only becomes effective below a blocking temperature Tb which is considerably lower than TN

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Models for exchange bias

*Atomically flat antiferomagnetic surface.

A) could be spin compensated; & = 0; B) could present one ferromagnetic plane; & = A/d ! 200 mJ m-2

*Only about 1/1000 of the spins seem to participate in the exchange coupling.

Surface is inevitably rough. Regions of dimension L contain (L/a)2 atoms. Uncompensated moment is that of ((L/a)2

atoms. Hence L ≈ 1000a ! 200 nm. OK But these regions will themselves add randomly.

* Exchange bias may arise from defects of grain boundaries where there are frustrated spins

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Models for exchange bias

Fig 8.16

* Interfacial coupling leads to perpendicular fm and afm axes. Coupling energy will be similar to that in a 90degree domain wall; (1/2)((KAaf) ! 0.4 mJ m-2

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Fig. 8. 17

Fig. 8. 18

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3.2 Indirect exchange coupling

Fig. 8. 17

Fig 8.21 Oscillations of the exchange coupling betweenferromagnetic layers as a function of the rutheniumspacer thickness

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Fig 8.20 The aliasing effect

FM

FM

Artificial Antiferromagnet

FM

FM

Artificial Ferrimagnet

Best for af coupling is 0.8 nm Ru

Ru

Ru

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3.3 Dipolar coupling

A perfectly smooth film createsno stray field. Correlatedroughness leads to orange-peelcoupling

With tn,the spacer thickness - 5nm, roughness $ = 1 nm, period l = 20 nm, the coupling is 0.03 mJ m-2

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3.4 Giant magnetoresistance

The GMR effect was discovered by Fert et al in 1988

Magnetoresistance in an Fe/Cr multilayer was as high as 80 %at low temperature and in high fields

Much greater than AMR - hence the name.

First understanding in terms of the Mott two-current model.The ) and * channels conduct in parallel, with no spin-flipscattering. +)and +* are the resistivities of the twochannels. , = +)/+*

Fig 8.22 Derivation of GMR in the two-current model

Fig 8.23 GMR of an Fe/Cr multilayer

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Fig 8.24

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3.5 Spin valves

FM

FM

(pseudo) Spin valve

Cu

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3.6 Magnetic tunnel junctions

FM

FM

Magnetic tunnel junction

AlOx

I

V

I = GV + -V3 E

!

w

•Nonlinear I:V

• Current decreasesexponentially with thickness w

• Little temperature dependenceV

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I

Julière formula:MR = 2P1P2/(1 - P1P2)

If P1 = P2

MR = 2P2/(1 - P2)

Taking P = 45%, MR = 51%

magnetic tunnel junction

Parallel ))

Antiparallel )*

Juliere formula for TMR

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Fig 8.25

500 % TMR

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•Majority channel tunneling is dominated by thetransmission through a !1 (sp) state

•!1 state decays rapidly in anti-parallel configuration

Calculation of tunelling through a an Fe/MgO/Fe crystalline tunnel barrier

100 200 .

R/R

%

µ0H (mT)

Exchange-biased MgOmagnetic tunnel junction

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A thin layer of ferromagnetic insulator can act as a spin filter, producing a spin-polarized tunnel current. An N/F/N structure. F = EuO, NiFe2O4, CoFe2O4…

The spin-split barrierfavours)electron tunneling.

Evac

EF

*

)w1

w2

.ex

t

%

I

))

t

%%

***

Spin filter

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Metal/Insulator/Superconductor junctions

Fig 8.27

Tederov-Meservey experiment

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8.4 Wires and needles

Acicular particles 30x30x300 nm are used inmagnetic recording.

N < 0.1

Shape anisotropy

Kshapa = [(1-3N)/4]µ0Ms2

For a long wire Kshapa = (1/4]µ0Ms2

Maximum anisotropy field 2Kshape/µ0Ms = Ms/2

The coercivity cannot exceed Ms/2 – not enoughfor a permanent magnet.

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Sophisticated nanostructures with spinodal nanostructureof oriented acicular Fe-Co in a nonmagnetic Al-Ni matrix,developed mainly in the 1930s.

Shape anisotropy: Ea = (1/4)µ0(1- 3N)Ms2sin2# = "1 sin

2#

Anisotropy field: Ha = 2K1/µ0Ms = (1/2)(1- 3N)MS

HC < -HA

Coercivity due to shape anisotropy < Ms/2

Insufficient for a permanent magnet!

FeCo

NiAl

Alnico

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8.5 Superparamagnetism

Energy landscape of a superparamagnetic particle

1//0 ! 1 GHz

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T

0 blocked Tb superparamagnetic TC paramagnetic

Blocking is not a phase transition, but an exponential variation of fluctuation tims

Superparamagnetic behaviour of cobalt nanoparticles

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When an ensemble of superparamagnetic particles is cooled through Tb in a magnetic field, it acquires athermoremanent magnetization.

Igneous rocks (basalts) contain superparamagneticmagnetite particles. They acquire a TRM as they cool inthe Earth’s magnetic field.

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5.1 Magnetic viscosity

coercivity

spontaneous magnetization

remanence

major loop

virgin curveinitial susceptibility

The entire hysteresis loop reflects metastable states. The magnetization at any point evolves with time

M(t) = M(0) - S ln t

M

Ln t

Viscosity coefficient

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8.6 Bulk nanostructures

Fig 8 .29

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6.1 Single-phase nanostructures

In single-phase nanostructures the bulk anisotropy can be greatly reduced by exchange coupling ofnanocrystallites with different anisotropy axes. Exchange-averaging occurs when 1. Crystallites aresingle-domain with D << $w and 2. There is exchange coupling across grain boundaries.

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Coercivity vs. grain size for a range of soft

magnetic materials.

Fig 8.30

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Remanence enhancement:

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6.2 Two-phase nanostructures

Two-phase nanostructures can be produced by partial recrystallization of an amorphousmaterial

If vc is the volume fraction of the crystalline phase, which has anisotropy K1, and theamorphous phase has no anisotropy.

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Recrystallization of amorphous Fe-Cu-Nb-Si-B to obtain a two-phase crystalline/amorphous soft nanocomposite

Finemet is a near-ideal soft magnetic material

with high polarization ! 1.6 T

zero magnetostriction

Minimal anisotropy

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Hard/soft nanocomposite;

Nd2Fe14B/Fe

SmCo5/Co35Fe65

$w is too short to average away anisotropy,When the size of the soft region is <! 2$w the

soft and hard phases are exchange coupled. andbehave in an averaged way,

In this way it is possible to obtain a hardmaterial with a magnetization greater than anysingle-phase hard magnet.

Fig 8.33

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Fig. 8.33

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Fig 8.35