no slide title - fisica...certain crystallographic directions it is easier to magnetize the crystal,...

P. Vavassori [email protected] I www.nanogune.eu 1

Paolo Vavassori

Ikerbasque, Basque Fundation for Science and CIC

nanoGUNE Consolider, San Sebastian, Spain.

MagneticMagnetic nanostructuresnanostructures


• Introduction.

• Fabrication of artificial magnetic nano-structures.

• Magnetism and reduced dimensionality.

• Micromagnetics of nano-shaped magnetic elements.

• Dynamic properties

• Experimental techniques for studying the reversal of artificial

magnetic nano-structures.

•Probing magnetic dynamic.

OutlineOutline

Part IPart I

Part IIPart II

Part IIIPart III

P. Vavassori [email protected] I www.nanogune.eu

Every material which is put in a magnetic field H, acquires a magnetic moment.

In most materials M = m H (M magnetic dipole per unit volume, magnetic susceptibility).

Basics: diamagnetism and Basics: diamagnetism and paramagnetismparamagnetism

M

H

paramagnetism

M

H

diamagnetism

Each atom has a non-zero magnetic moment The moments are randomly oriented (T);

H arranges these moments in its own direction.

Each atom acquires a moment caused by

the applied field H and opposed to it

(Larmor frequency).

= 0 e.g., noble gas.

Eappl = - 0 M . H temperature kbT

= - B(L + gS) orbital and spin angular momenta

In soilds ≈ - gBS (crystal field)

Low T

High H

3


M = m H

B = 0(H + M) -> B = H

= 0 (1+ m)

Cgs System

B = (H + 4p M) 0 = 1

= (1+ 4p m)

cgs SI

H units Oe A/m

B units Oe T

M units emu /cm3 A/m

Conversions:

For H 1Oe = 103/ 4p A/m = 79,58 A/m

For B 1T = 104Oe

For M 1emu/cm3 = 103 A/m

Magnetic moment 1 Am2 = 103 emu

Constitutive equations and unitsConstitutive equations and units

4


FerromagnetismFerromagnetism

H

MH

Limiting hysteresis curve: all the points

enclosed in the loop are possible

equilibrium states of the system.

With an appropriate history of the

applied field one can therefore end at

any point inside the limiting hysteresis

loop.

There are materials in which M is NOT proportional to H.

M may be, for example, non-zero at H = 0.

M in these materials is not even a one-valued function of H, and its value depends

on the history of the applied field (hysteresis).

remanence

coercive field

saturation magnetization MS

Fe, Co, Ni, alloys also with TM , C, and RE

5


PhasePhase transitiontransition ferromagnetferromagnet →→paramagnetparamagnet

Ms (T)

MS

T TC

Above a critical temperature called

Curie temperature (TC) all ferromagnets

become regular paramagnets → MS = 0 at

H = 0

Since

This temperature for anti-ferromagnets is called Néel temperature

(TN)

MS (TC-T) T < TC

= ½ mean field theory (identical average exchange field felt by all spins)

6


In ferromagnetic materials the magnetic moments of the individual atoms interact

strongly with each other creating an order against the thermal fluctuations.

Origin of hysteresisOrigin of hysteresis

The interaction between the magnetic moments is not dipolar (too weak);

Weiss internal or “molecular “ field (1907) -> OK for Ms(T) but for hysterisis?

Weiss assumed the existence of domains

From Tc: the molecular field is of the order of 106 Oe (102T) but hysteresis loops

tell us that with 1-103 Oe we can rearrange and even eliminate the

domains……?

7



The interaction between the magnetic moments is not dipolar (too weak);

it is electrostatic (Coulomb) determined by correlaction effects (Quantum

mechanics):

symmetry of the electrons wavefunction and Pauli principle → Hund’s rule

Eex = -(1/V) ij Jij Si . Sj (Heisember hamiltonian H = - ij Jij Si . Sj)

(short range interaction ij is over the nearest neighbors and V is the unit cell

volume)

Jij is the exchange integral Jij > 0 ferromagnetic order

Jij < 0 anti-ferromagnetic order

8


The magnetization of a sample may be split in many domains.

Each of these domains is magnetized to the saturation value Ms but the direction

of the magnetization vector may vary from one domain to the other at H = 0.


9


WhyWhy magneticmagnetic domainsdomains??

The field created outside the magnet in cases a) and b) costs B2/20 Joules/m3, thus case c)

is the one energetically favoured. This is due to the finite size of the magnet, so it is an effect

of lateral confinement.

a) b) c)

Energy densities

In vacuum

u = B2/20

Inside a

material

u =1/2 0 Ms2

Total energy

U = ∫∫∫udt All

space

10


The magnetic field H in presence of a magnetization M can be split into two components, the applied filed

Hext and the magnetostatic or demagnetizing field Hd coming only from the magnetization M (viz. j = 0).

Since j = 0, from Maxwell equations we can write for Hd: d .

The most general solution of this equation is: d U(r)

Since the condition ( ) holds, by substituting d U(r) in this equation we obtain

that U(r) is solution of a Poisson’s equation 2U =

with the boundary conditions (H|| and B ┴ have to be continuous on the boundary of two materials) at the

sample surface Uin = Uout and

where n is the surface normal taken to be positive in the outward direction.

Under these conditions and imposing that

, U(r) is given by: U(r) =

surfacesample

volumesample

dSd 'r'r

)M(r'nτ'

r'r

)M(r''

4

1

p

MagnetostaticMagnetostatic fieldfield..

Mnnn

outin UU

0)(lim

rUr


MagnetostaticMagnetostatic energyenergy..

Magnetostatic energy is potential energy of magnetic moments in the field Hd

they have created themseves. Once Hd is known from previous eq.

d U(r)

the magnetostatic energy em can be evaluated as:

em= 0

( ( ( rdrHrMrdrH d

samplespaceall

d

3

0

32

02

1

2

1


MagnetostaticMagnetostatic energyenergy..

= 0 ( is uniform everywhere)

and n = 0 ( parallel to the borders)

d 0 em is minimum +

+ + -

- - OK, surface charges

at domain’s boundaries

compensate

+ + + +

+ + Not OK, surface charges

at domain’s boundaries

do not compensate

Principle of poles avoidance

d can be calculated like a field in electrostatics. The only difference is that the magnetic

charges (bulk - and surface n ) never appear isolated but are always balanced by

opposite charges

surfacesample

volumesample

dSdrU 'r'r

)M(r'nτ'

r'r

)M(r''

4

1)(

pd U(r)


MagnetostaticMagnetostatic energyenergy: : examplesexamples

• M = 0 (M is uniform everywhere)

and n • M = 0 (M parallel to the borders) Hd = 0

Em ≈ 0

+ +

+ - -

-

d

n • M > 0 n • M < 0

Infinite ferromagnet

uniformly magnetized • M = 0 (M is uniform everywhere)

and n • M = 0 (no borders) Hd = 0 Em = 0

Hd= -M ≠ 0

Em > 0

• M = 0 (M is uniform everywhere)

but n • M ≠ 0 (borders)


MagnetostaticMagnetostatic energyenergy and domain and domain structuresstructures..

?

Magnetostatic energy is not the only ingredient to

determine the actual domain structure.

Anisotropy energy plays a role:

structure a) is expected with cubic anisotropy;

structure b) with uniaxial anisotropy with EA along x;

structure c) with uniaxial anisotropy with EA along y.

But still, what decides the number of subdivisions,

for instance?

Somwhere ther is also Exchange energy stored.

Where?

a)

b)

c)

x

y


WhereWhere M inside a domain M inside a domain isis pointingpointing toto??

The direction of the magnetization inside each domain is NOT arbitrary.

For instance, the crystal structure is not isotropic so it is expected that along

certain crystallographic directions it is easier to magnetize the crystal, along others

it is harder (confirmed by experiments).

The exchange energy term introduced so far (Heisemberg) is isotropic.

We have to introduce a phenomenological expression for this additional term Eanis.

There are several types of anisotropy, the most common of which is the

magnetocrystalline anisotropy caused by the spin-orbit interaction (the electron

orbits are linked to the crystallographic structure and by their interaction with the

spins they make the latter prefer to align along well-defined crystallographic axes.

In this case Eanis will be a power series expansions that take into account the

crystal symmetry.

16


Anisotropy sources (a)Anisotropy sources (a)

• Magnetocrystalline anisotropy: dependence of internal energy on the

direction of sposntaneous magnetization respect to crystal axis. It is due to

anisotropy of spin-orbit coupling energy and dipolar energy. Examples:

- Cubic Eanis = K1 (ax2ay

2 + ay2az

2 + az2ax

2) + K2 ax2ay

2 az2 +….

- Uniaxial Eanis = K1 sin2q + K2 sin4q +… ≈ -K1(n . M)2

• Surface and interface anisotropy: due to broken translation symmetry

at surfaces and intefaces. The surface energy density can be written:

- Esurf = Kp af2 - Ksan

2; where an and af are the director cosines respect to the

film normal and the in plane hard-axis.

[K] = J/m3

17


Anisotropy sources (b)Anisotropy sources (b)

• Strain anysotropy: strain distorts the shape of crystal (or surface) and,

thus can give rise to an uniaxial term in the magnetic anisotropy.

Es = 3/2 ls sin2q; where l is the magnetostriction coefficient (positive or

negative) along the direction of the applied stress s and q is the angle

between the magnetization and the stress direction.

• Growth induced anisotropy: preferential magnetization directions can

be induced by oblique deposition or by application of an external magnetic

field during deposotion.

18


MagnetoMagneto--crystallinecrystalline anisotropyanisotropy: :

spinspin--orbitorbit couplingcoupling..

-

-

-

-

Orbital motion is highly hindered

-

-

-

-

Orbital motion is less hindered

H

H

S

S

<Lz> = -1, -2

<Lz> = 1, 2

d-orbital momentum in an atom

H

H

Spin-orbit coupling tends

to induce an orbital motion

as sketched…but there is

the crystal field potential.


SpinSpin--orbit coupling energy.orbit coupling energy.

The magnetude of the magnetocrystalline anisotropy depends on

the ratio between the crystal filed energy (bandwidth, typically a

few eV) and the spin-orbit energy (in the order of 10 meV)

Spin-orbit interaction is very small ~ 100 eV/atom

compared to:

exchange interaction ~ 100 meV/atom

cohesive energy ~ a few eV/atom

3d bandwidth a few eV.


Shape anisotropyShape anisotropy

Shape anisotropy is due primarily to dipole-dipole interaction and it is

related to the lateral confinement (reduced dimensionality).

We have seen that: Ed = -1/2 0 ∫V Hd . M dV ; with Hd the so-called dipolar

field, previously defined, is the magnetic field produced by the magnetic body itself.

If for simplicity we assume that M is uniform inside the body, . M = 0, the

integral above becomes a surface integral where Hd can be thought as produced by

surface magnetic charges ss = M . n and the energy Ed depends solely on the

shape of the body.

The uniformity condition can be realized only for isotropic ellipsoids and for

such special cases Hd = -N M, where N is a tensor called demagnetizing tensor.

Referring to the ellipsoid semi-axes the tensor becomes diagonal and Nx, Ny, Nz are

called demagnetizing factors and Nx + Ny + Nz = 1


with Ks = 1/2 0 Ms2 (Nx - Ny) [1/2 4p Ms

2 (Nz - Ny) in CGS units],

Magnetostatic self interaction for an ellipsoid (referring to the ellipsoid semi-

axes ) Ed = 1/2 0(Nx Mx2 + Ny My

2 + Nz Mz2).

For a flat ellipsoid (to which an elliptical disk can be approximated), where M

should lie in the yz plane and q is the angle of M respect to the x axis,

we can write :

Mz = 0: Ed = 1/2 0(Nx Mx2 + Ny My

2) and Mx2 = Ms

2 cos2q ; My2 = Ms

2 (1-cos2q )

Ed = Ks cos2q

DescriptionDescription ofof shapeshape anisotropyanisotropy in in termsterms ofof

anisotropyanisotropy constantsconstants

x

y

M q Ks<0


EffectsEffects ofof shapeshape anisotropyanisotropy..

Hd = 0

Nz = 1 -> Hd = - M

favoured

unfavoured

favoured

unfavoured

Hd = - Nz M

Hd = - Nx M

Nz < Nx


Domain wall energy, per unit of surface:

with A=nJS2/a, the exchange stiffness constant, where n is the number of sites in the unit

cell, J is the average exchange integral value, S is the spin number and a is the unit cell

edge.

Domain Domain boundariesboundaries

Bloch domain wall

Neél domain wall

(thin films)

So to set up a domain structure and reduce the magnetostatic energy there is a price to pay:

an excess of anisotropy and exchange energy has to be stored in the boundaried between

domains, the domain walls (there is also some magnetostatic energy).

1AKw s

1KAw Domain wall width:

w

[A] = J/m

24


( DaMaa

a w

wwB

222

21 p

ss

( DaaDMa

w

w

awN

222

1 p

ss

In a thin film, Nèel demonstrated (approximating a domain wall to an ellipsoid)

that the total energy of a Bloch and Nèel domain wall can be expressed to first

order as:

Energies and widths of domain boundariesEnergies and widths of domain boundaries (d): (d):

filmsfilms

D

a

26


Energies and widths of domain boundariesEnergies and widths of domain boundaries

Bloch

Nèel

Threshold

between Bloch

and Nèel walls

in a Fe film

nm

27


Energies and widths of domain boundariesEnergies and widths of domain boundaries (e)(e)

Bloch

Nèel

Threshold between

Bloch and Nèel walls

in a typical soft film

(sligthly anysotropic)

nm

28


Summary of energy contributionsSummary of energy contributions

Etot = Eappl + Eex + Eanis + Em

Eappl is the Zeeman energy related to the spin alignment in the external magnetic film H.

Eappl = - 0 M . H

Eex is the interatomic exchange interaction favoring parallel atomic moments alignment

(short range).

Eex = -(1/V) ij Jij Si . Sj

(ij is over the nearest neighbors and V is the unit cell volume)

Eanis is the magnetic anisotropy energy associated to preferential magnetization directions.

For a preferential axes n : Eanis = -K1(n . M)2

Em is the magnetostatic self-interaction due to the long-range magnetic dipolar coupling.

Responsible for domain formation in bulk- and film-like specimens

Em = -1/2 0 Hd . M

29


To determine hysteresis loops it is necessary to trace the local magnetization M(r) as a function of

the applied field. The starting point is the energy functional (micromagnetic free energy on a

continuum level)

30


Infinite 3D, 2D and 1D Infinite 3D, 2D and 1D systemssystems free free energyenergy

Etot = Eappl + Eanis

E

q K1V

E

q K1V H

p

Bistable one-dimensional potential: uniaxial anisotropy

Etot = K1 sin2f - 0MsHcosq


0.5

0.25

0

0.25

0.5

Field kOe

2

0

2

Angle rad.

0.5

0.25

0

0.25

0.5

Energy

0.5

0.25

0

0.25

0.5

Field kOe

-3 -2 -1 1 2 3

0.02

0.04

0.06

0.08

0.1

-3 -2 -1 1 2 3

-0.1

-0.05

0.05

0.1

-3 -2 -1 1 2 3

-0.2

-0.1

0.1

0.2

-3 -2 -1 1 2 3

-0.3

-0.2

-0.1

0.1

0.2

0.3

H = 0

H = -0.1

H = -0.2

H = -0.3

ReversalReversal: : coherentcoherent rotation rotation modelmodel


sMo

KoH

12

E = K1 sin2f - 0MsHcosq

StonerStoner and and WohlfarthWohlfarth modelmodel

Stoner and Wohlfarth model: coherent

rotation of an uniaxial particle uniformly

magnetized. Ms

H

Easy axis

q

f

Free energy for unit volume


-1000 -500 0 500 1000

-1.0

-0.5

0.0

0.5

1.0

-400 -200 0 200 400

-1.0

-0.5

0.0

0.5

1.0

1.5

-1000 -500 0 500 1000

-1.0

-0.5

0.0

0.5

1.0

-400 -200 0 200 400

-1.0

-0.5

0.0

0.5

1.0

Film Patterns

Field (Oe)

Pattern 1

Pattern 2

Pattern 3

Pattern 1

Pattern 2

Pattern 3

H || Fe(110)

Hard axis

M/M

sa

t

Field (Oe)

H || Fe(100)

Easy axis

M/M

sat

H || Fe(100)

H || Fe(110)

A “A “realreal” 2D system: Fe(100) ” 2D system: Fe(100) thinthin filmfilm

Measured EA loop Expected EA loop (anisotropy field 282 Oe:

K = 48000 J/m3 Ms 1.7 106 A/m ->28.2 mT)

-0.75 -0.5 -0.25 0 0.25 0.5 0.75

-1

-0.5

0

0.5

1

Why this difference? Different reversal process: reversed domains nucleation


The “The “realreal” 2D system ” 2D system isis confinedconfined

Etot = Eappl + Eanis + Ed

favoured

unfavoured

more favoured


MagnetizationMagnetization reversalreversal and and domainsdomains

Ideally, reversal through domain walls motion does not cost energy because the wall energy

necessary at the new position is released at the previous position (reversible).

The annihilation of DWs costs energy, of course.

In the case of domain wall pinning at local defects (non-magnetic impurities, voids, grain

boundaries…) some activation energy is necessary to release the domain wall from the pinning

centre (abrupt displacement, Barkhausen jumps, viscosity due to Lenz law, energy dissipation ->

irreversible process -> hysteresis).

36


MagnetizationMagnetization reversalreversal and and domainsdomains

Energetically, coherent rotation means moving the magnetization inside the

entire volume away from the easy axis defined by crystalline anisotropy. This

cost an energy over the 3D space.

Reversal through domain walls motion does not cost energy because the wall

energy necessary at the new position is released at the previous position

(reversible).

Only in the case of domain wall pinning at local defects (non-magnetic impurities,

voids, grain boundaries…) some activation energy is necessary to release the

domain wall from the pinning centre (abrupt displacement, viscosity due to Lenz

law, energy dissipation -> irreversible process -> hysteresis).

The introduction of a new domain wall costs only the domain wall energy, i.e.

anisotropy and exchange energy over a 2D space. This is much less costly than

coherent rotation even though the activation energy for this process, in a defect-

free sample, is high because the magnetization of the complete volume of the

new domain has to be inverted.


rotation

(high fields) domain walls

displacement

(low fields)

Barkhausen jumps

Domain wall motion is the preferred

way of changing the magnetization at

low fields.

With increasing field strength, first

domain walls will move and increase

the size of domains with a

magnetization component parallel to the

field (with the magnetization in every

domain being parallel to an easy axis).

Therefore some misalignment with the

applied field remains if the field is not

aligned with one of the easy axes.

At high fields the domain walls are

removed and the magnetization is

rotated coherently towards the field

direction.

LetLet’s ’s seesee a “a “realreal” ” exampleexample

38


Exchange and dipolar interactions….Exchange and dipolar interactions….

short and long range.short and long range.

Exchange interaction is a very short-range force: it affects nearest or at most next-nearest neighbours only; it is a

very strong interaction responsible of magnetic ordering at room temperature (and above). This is clear.

Dipolar magnetostatic interaction is usually defined as a long-range force but what this means is less clear.

Let’s consider a uniformi magnetized ellipsoid. The field measured at a point inside the ellipsoid is given by Hd = -

N M with N determined by the ratio of its axes. The absolute size does not enter. Suppose the ellipsoid inflates in

such a way that its size increases, but its shape is held the same, i.e. N is kept constant. Then the demagnetizing

field Hd is the same as it was for the small ellipsoid. If the inflation continues to infinity, Hd in the ellipsoid

remains unvaried even though the surface charges are now at an infinite distance away. Thus the “long-range” in

the present context means that this range actually extends all the way to infinity. In ferromagnetism there is always

a surface even for an infinite crystal, and it is the surface that is responsible for the subdivision into domains.

Therefore the calculations ignoring the surface introduce always an error. This error would not be important if the

magnetostatic energy was small. But it is not. It is often pointed out, to justify the neglection of the surface, that the

exchange energy density is order of magnitude larger than the magnetostatic energy density. However, the physical

system is governed by the total energy and not by its density. The exchange forces are effective inside the unit cell

of the crystal so that the total exchange energy is of the order of its density integrated over the volume of a unit

cell. The magnetostatic density energy is small, but having a long range, it is integrated over the whole volume of

the crystal. For a sufficiently large crystal it is never negligible.

Exchange energy controls the microscopic properties, as in the inside of a domain wall (Tc etc.), but it is the

magnetostatic energy term that mostly (also anisotropy) detemines the magnetization distribution over the crystal.


SummarySummary ofof energyenergy contributionscontributions

Etot = Eappl + Eex + Eanis + Em

Eappl is the Zeeman energy related to the spin alignment in the external magnetic film H.

Eappl = - 0 M . H

Eex is the interatomic exchange interaction favoring parallel atomic moments alignment

(short range).

Eex = -(1/V) ij Jij Si . Sj

(ij is over the nearest neighbors and V is the unit cell volume)

Eanis is the magnetic anisotropy energy associated to preferential magnetization directions.

For a preferential axes n : Eanis = -K1(n . M)2

Em is the magnetostatic self-interaction due to the long-range magnetic dipolar coupling.

Responsible for domain formation in bulk- and film-like specimens

Em = -1/2 0 Hd . M

40


The expressions of the single energy terms, in the semiclassical continuum approximation

and in reduced units (i.e., normalized to the saturation magnetization Ms), are (for unit

volume):

eappl = - 0 m . h

eex = -A [(mx)2 +(my)2 + (mz)2]

eanis = -K1(n . m)2

ed = -1/2 0 hd . m

where, m = M/Ms, h = H/Ms, A denotes a macroscopic exchange constant specific for the

material (exchange stiffness constant : A = c J S2 / a [J/m3], with c = 1, 2, 4, 22, for sc,

bcc, fcc, and hcp structures, respectively), K1 is the first-order anisotropy constant, hd =

Hd/Ms with Hd the so-called dipolar field is the magnetic field produced by the

magnetic body itself (see slide 15).

MagneticMagnetic configurationsconfigurations and and reversalreversal: :

MicromagneticMicromagnetic energyenergy functionalfunctional

( mi)2= 2

2

2

2

2

2

z

m

y

m

x

m iii


To determine hysteresis loops it is necessary to trace the local magnetization M(r) as a function of

the applied field. The starting point is the energy functional (micromagnetic free energy on a

continuum level)

MagneticMagnetic configurationsconfigurations and and reversalreversal: :

MicromagneticMicromagnetic energyenergy functionalfunctional

42


reversal

HN

IncoherentIncoherent vs. vs. coherentcoherent reversalreversal

2K1

oMs

2K1

oMs

For single domain particles the reversal process can be still incoherent, in a way

different from doman wall displacement: curling mode (Brown).

43


Equation of motion of a magnetic dipoleEquation of motion of a magnetic dipole

If L is the angular momentum associated to a magnetic moment M,

according to quantum theory: M = -g L where g = g B/ħ is the gyromagnetic

ratio, g is the Landé factor, close to 2 (moment due mostly to electrons spin),

B = eħ/2me is the Bohr magneton).

Second equation of dynamics

The dynamic equation of an angular moment L subjected to a torque t is:

dL/dt = t with the equilibrium condition t = 0.

H M

E= -M•H

t = M х H


Equation of motion of a magnetic dipoleEquation of motion of a magnetic dipole

Since t = M х H, with H the effective field acting on M, the dynamic equation for M is

thus: dM/dt = -g M х H

If H is only the applied field Ha: dM/dt = -g M х Ha= -M х g Ha= w х M

And therefore the equation above describes a rotation of the vector M with angular

velocity w = g Ha: this is the familiar Larmor precession of a magnetic top about the

field direction (counter-clockwise as seen from above and g ≈ 18.5 106 rad/(Oe s)

and n = w/2p ≈ 2.95 GHz/kOe). H M

dM/dt = -g t

t = M х H

H dM/dt

No damping (dissipation) → no

alignement but only precession

counter-clockwise


A damping of the precessional motion is necessary to establish the

experimentally observed magnetization reversal in an applied field.

The simples way to introduce a damping is via an ohmic type dissipation term,

viz., dM/ dt → a dM/ dt .

The action of damping must be a torque forcing the magnetization to reduce the

radius of its precessional motion.

It must be perpendicular to the plane of M and dM/ dt

and thus the term to add is: (M/Ms) х (a dM/ dt)

so that the dynamic equation becomes:

dM/dt = -g M х H + (a/Ms) M х dM/dt

Gilbert dynamic equation: damped motionGilbert dynamic equation: damped motion

46


dM/dt = -g M х H + (a/Ms) M х dM/dt

This is the Gilbert dynamic equation.

As a consequence of the damping term the

Magnetization spirals down to a stable

configuration with M || H.

The stable state is achieved the faster, the higher

the damping parameter a.

Gilbert dynamic equation: damped motionGilbert dynamic equation: damped motion

Low damping

High damping

47


Coherent rotation accomplishes magnetization reversal much faster than inhomogeneous and

domain walls displacement mechanisms.

Magnetization rotation : 100 ps to 1 ns

Domain walls displacement : 100 ns up to 100 ms

Magnetization reversal is a dynamic processMagnetization reversal is a dynamic process

Magnetization reversal through coherent rotation takes place with magnetization

precession (N.B., always counter-clockwise looking from +z). If the precession is

effectively dumped the reversal can be very fast.

48


Damping a 0.2

1700 emu x (10nm)3

H 5kOe : reversal time 646 ps

H 500 Oe: reversal time 5950 ps

Damping and field Damping and field dependancedependance of the reversal of a of the reversal of a

magnetic momentmagnetic moment

H

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

-1.0-0.5

0.00.5

1.0

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

-1.0-0.5

0.00.5

1.0

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

-1.0-0.5

0.00.5

1.0

My

Mx

Mz

My

Mx

Mz

My

Mx

Mz

Damping a 0.05

1700 emu x (10nm)3

H 5 kOe: reversal time 2121 ps


Damping a 1.0

1700 emu x (10nm)3

H 5 kOe: reversal time 185 ps


t = 0

t t

49


MagnetizationMagnetization dynamicsdynamics: : timetime scalescale


LandauLandau--LifshitzLifshitz--Gilbert dynamic equationGilbert dynamic equation

The dynamic Equation introduced in the previous slides, namely:

dM/dt = -g M х H + (a/Ms) M х dM/dt is due to Gilbert.

There is an equivalent older form of Landau and Lifshitz that can be derived as follows:

Both sides of the equation are multiplied by M ·. The right-hand side vanishes and thus:

M · dM/dt =0.

This means that dM2/dt = 0 and thus that M2 remains constant, equal to Ms2, during the motion.

Both sides of the equation are multiplied by M , using the rule of the cross product of a cross

product (abc=[a·b]c+a[b·c]) and exploiting the previous result one obtains:

M dM/dt = -g M х (M х H) -aMs dM/dt.

Substituting in the Gilbert Equation this expression for M dM/dt one gets:


LandauLandau--LifshitzLifshitz--Gilbert dynamic equationGilbert dynamic equation

This is called Landau-Lifshitz-Gilbert Equation.

When the damping is small (a 0.1) 1/(1+a2) ≈ 1 and the Equation becomes simply:

that is the form of LLG Equation usually employed.

( (

HMMHM

M

sMdt

d agg

a 21

1

( HMMHMM

sMdt

d agg


The element is discretized in a grid of, usually cubic, small cells. The size of each cell

should be below the exchange length lex. The magnetization distribution is then computed

with the Landau-Lifshitz-Gilbert (LLG) equation:

where

Mi is the magnetization of the ith cell (A/m),

Heff, i is the local effective field in the ith cell (A/m),

gis the gyromagnetic ratio (gB/, g is the ‘Landé factor’)

a is the damping coefficient (dimensionless) < 1.

The effective local field is defined as:

( ieffii

s

ieffii

Mdt

d,, HMMHM

M

agg

i

ii,eff

E

MH

1

0

LLG LLG MicromagneticsMicromagnetics simulationssimulations of of nanoelementsnanoelements

Convergence:

Mi x Heff,i <

Ei is the energy of ith cell determined

by the configuration of all the other cells

external torque damping torque


Scale length parametersScale length parameters

The relative strength of the anisotropy and magnetostatic (1/20 Ms2) energies (per unit

volume) can be expressed by the dimensionless parameter: k = 2K1/0Ms2.

k provides a quantitative definition of the conventional distinction between soft (k <<1,

i.e., dipolar effects dominate over anisotropy ones) and hard (k >1) materials.

The competition between exchange and dipolar energy is expressed in terms of the

exchange length: lex = 202 sMA

The comparison between exchange and anisotropy may be expressed through the length:

lw = kexlKA 1

A

(J/m)

Ms

(A/m)

K

(J/m3)

lex

(nm)

lw

(nm) k

(adim.)

Fe 2110-12

1.7106 4810

3 3.4 20.9 2.610

-2

Co 3010-12

1.42106 52010

3 4.9 7.6 4.210

-1

Ni 910-12

0.49106 -5.710

3 7.5 39.7 3.710

-2

Py 1310-12

0.86106 0 5.3 0


Critical size for single domain Critical size for single domain

It is useful to introduce the length scale is ld related to the Bloch domain wall energy,

per unit of surface:

k exsd lMAKl 2012

Spherical element: RSD≈ 4lex in the case of a soft magnetic material (k <<1)

and RSD ≈ 18 ld for a hard one (k >1).

Cubic element: RSD ≈ 7lex and RSD ≈ 24 ld in the case of a soft and hard

magnetic material, respectively.

For spherical particles one finds that RSD for Co is 70 nm, whereas for Fe is 15 nm,

and for Ni 55 nm [“hard” magnetic particles (Co) are more stable than “soft”

magnetic ones (Fe, Ni, Py)].

RSD increases for elongated particles.


- 150 - 100 - 50 0 50 100 150

- 1.0

- 0.5

0

0.5

1.0 (e)

(c)

(d)

(b) (a)

Mag

net

izati

on

, M

/Ms

Field, mT

Micromagnetic simulations

H

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

-2000 -1000 0 1000 2000

-1.0

-0.5

0.0

0.5

1.0

M/M

sat

Field (Oe)

Vortex

displacement

Vortex

annihilation

Vortex

nucleation


FerromagneticFerromagnetic resonanceresonance

On the previous pages, we saw how a single magnetization vector, or a collection of spins, underwent precessional

motion when disturbed from equilibrium. The next natural question is: how does one disturb the magnetic

moment(s)? In spin waves the disturbance is provided by thermal energy.

A good way to move the magnetization vector is to apply another, smaller magnetic field, perpendicular to the static

H field. The precession can then be driven by varying the strength of this small field over time ( "pump field“). For

typical materials and operating conditions, the pump field is readily supplied by the magnetic component of

microwave radiation.

The situation is analogous to a damped, driven, simple harmonic oscillator (a mass-on-a-spring), driven by a

harmonically-varying force. If the frequency of the driving force is varied, one will find that the amplitude of the

responding motion changes. At resonance, the amplitude is maximum.

Ferromagnetic resonance (FMR) is similar. If a magnetization vector is

subject to a static field and a perpendicularly applied pump field, resonance

will occur at a frequency more or less proportional to the strength of the static

field. The microwave power absorbed by the magnetic sample as a function

of frequency will typically be a lorentzian centered at resonance. The power is

proportional to the square of the amplitude of the precession.

In practice, however, it is difficult to vary the frequency of the

microwave field during the experiment. It is much easier to keep

the frequency fixed and instead vary the strength of the static

field very slowly.

Videos: http://www.cc.gatech.edu/scivis/education/demo1c.html

http://www.cc.gatech.edu/scivis/education/demo1c.html


DampedDamped harmonicharmonic oscillatoroscillator

58


The amplitude of the solution is therefore,

( 2'22

220

0

b

mFA

www

m

k0w

m

bb ' Lorentzian lineshape

DampedDamped harmonicharmonic oscillatoroscillator

wb’ Lorentzian half width

Video: http://www.acs.psu.edu/drussell/Demos/SHO/mass-force.html

59


SpinSpin waveswaves: :

eccitationseccitations ofof a lattice a lattice ofof coupledcoupled spinsspins

( kaJS cos14 w1

1

2

j

N

j

jJU SS

U0 U1 =U0+8JS2

U’1 <U1

i

jjU Hμ2

1jBj g Sμ ( ( 112 jjBj gJ SSH

jj

j

dt

dHμ

S

3 non-linear equationsi for Si; Sxi, S

yi << S, thus Sz

i=S and dSzi/dt = 0 -> linearization

( tjkaixj ueS w ( tjkaiy

j veS w ( kaJS cos14 w e v = -iu Quantiztion

Felix Bloch law ( 23

0T

M

M

60

no slide title - fisica...certain crystallographic directions it is easier to magnetize the crystal,...

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