di huong tu_5

8/7/2019 di huong tu_5

1/36

Magnetic anisotropy


2/36

Magnetic anisotropy energy: definition

1 cos =

2 sin cos =3 sin sin =

cubic system:

( ) ( )aE E z E z = M MCan be defined as: magnetic anisotropy energy per atom (eV/atom)

magnetic anisotropy energy per unit volume (MJ/m3, erg/cm3)

uniaxial system:

MagneticN

anostructuresP.Gambardella


3/36

Two types of magnetic anisotropy

Spin-Orbit Coupling:

MAGNETOCRYSTALLINE ANISOTROPY

i i

L

l

s.o.H

= =

S

s

Dipolar (magnetostatic) Interaction:

SHAPE ANISOTROPY

1 2 1 2

3 5

1 ( )( )3

4dipH

r r

=

m m m r m r

r

m1

m2

MagneticN


classical electromagnetic (Maxwell) theory quantistic + relativistic treatment ofS and L

Formation of magnetic domains Preferred orientation of themagnetization in crystalline materials

Dynamic relaxation of the magnetization

Stability of the orientation of the magnetization in bulk as well as nanomagnets(remanence, coercivity, superparamagnetic relaxation)

Size and shape of magnetic domains


4/36

- +

Magnetic dipolar interaction is intrinsically anisotropic

0 1 2 1 2

3 5

( )( )3

4dipE

r r

=

m m m r m rr

m1

m2

1 2

0

1

4Coulomb

q qE

r=

isotropic

anisotropic

MagneticN



5/36

Dipolar interaction

Parallel alignment is favored for C= 54.74

0 1 2 1 2

3 5

( )( )

34dipE r r

=

m m m r m r

r

m1

m2

( ) 20 1 23 m m 1 3cos4dipE

r

=

If only up and down directions are considered:

Cone of alignment :

Conclusions:

in a planar structure, e.g,, a thin film,in-plane direction is favored

in an elongated nanoparticle:long axis is favored

MagneticN



6/36

Dipolar interaction and magnetostatic energy

0

B = H + M

L0

H antip. Hdemag stray

d =

H li

MagneticN


( )

0

0

outside: 0,

1inside: 0,

stray

demag

= =

=

BM H

M H B M

B

B

M

0 demag

1 M H2

demag demag V

E dV= 0B = H + M

( )0B = H + M0

M H2demag demag VE dV

=

SI !! vedi Mencuccini e Silvestrini cap. VI,Fisica II e incubi vari

Di l i t ti d t t ti


7/36

Dipolar interaction and magnetostatic energy

0 M H2ms demag

E dV

=

H H -M / 3

M

demag dip=

= D

Magnetostatic self-energy:mutual magnetostatic interaction of

the elementary dipoles of which amagnet is composed

D

D

D

=

D

Demagnetizing field:

field created by the magnet itself

For an ellipsoid of revolution with rotational symmetry and M = Mz:

z

20

2msE D M V

=

Sphere:

isotropic

1

3D D = =

Infinite thin film:

In-plane orientation favored

0, 1D D = =

N.B.: D depends on the direction ofM w/r to the symmetry axes of the ellipsoidMagneticN


M t t ti f ti f ti d i


8/36

Magnetostatic energy formation of magnetic domains

Pay domain wall energy(exchange + MCA)

MagneticN


E amples of magnetic domain config rations in microstr ct res


9/36

Examples of magnetic domain configurations in microstructures

MagneticN


Field direction

Type of domains


10/36

Type of domains

Total energy = Magnetostatic Energy + Wall Energy

C. Kittel, Phys. Rev. 70, 965 (1946)

MagneticN


Atomically thin magnetic domain walls in Fe nanostripes on W(110)


11/36


O. Pietzsch, Phys. Rev. Lett. (2000); M. Bode, Rep. Progr. Phys.(2003)

1.3 monolayersFe / steppedW(110)

STM topography Spin-polarized STM

Real-space observation of dipolar induced antiparallel domain orientation

Atomically narrow domain walls

MagneticNanostructuresP.Gambardella

Atomically thin magnetic domain walls in Fe nanostripes on W(110) *


12/36


Pratzer et al., Phys. Rev. Lett. 87, 127201 (2001)

Magnetizationp

rofile

Micromagnetics: 2 /w A K= 11

4 3

exchange stiffness

5 10 anisotropy constant

2.5

10 J/m,

J/m ,

eV/atom

bulk

mcbulkK

A

=

=

=

30 nmw =

2 12/ 2 4 10 J/m,

where =8.7 meV from of 1 Fe monolayer

nn

C

A JS a

J T

= =

exp 0.6 nmw =

Wrong! factor 50 larger than measured!

6 3

4.2

20 10

meV/atom

J/m ,mcmonolayer

K

=

= Working out Kmc from wexp:

In a Fe/W(110) monolayer:

But is magnetics valid close to the atomic scale?

w


*

Single-domain nanoparticles


13/36

Single domain nanoparticles

Domain Wall Energy:

20

2 30

2

1 4

2 3 3

msE D M V

r

=

=2 30 4 1 2, where =

6 3ms

rE M r n

n d

=

d

r

Single-domain n-domains

( )2 2W rE rd

=

( )0

ms WE E

d

+

=

where domain wall en./unit area

Minimization of total energy:

=

mJ m-2

rsdnm

Fe 2.6 6

Co 9.3 34

SmCo5 78 764

2

2

0

18 rd

=

2

0

922

sd sd r d r

= =Critical radius:


Multi-domain to single-domain transition


14/36

Multi domain to single domain transition

Holographic interference fringes of electronsscattered off Co platelets showing lines of flux.

T. Matsuda et al., J. Appl. Phys. 53, 5444 (1982)

55 nm thick

15 nm thick


with thickness with lateral dimensions

3 monolayers Fe islands on Cu(100)

Scanning electron microscopywith spin polarization analysis

C. Stamm, PhD thesis, ETHZ (2000).

Magnetocrystalline anisotropy in bulk metals


15/36

easy axis: (100) easy axis: (111)easy axis: (0001)

5 3

1 4.1 10 /45 /

K J meV atom

= =

3 3

1 5.5 10 /0.3 /

J meV atom

= =

4 3

1 4.8 10 /2.4 /

J meV atom

= =

Fe bcc Co hcp Ni fcc

Magnetic field (Oe)

Magnetization(emu/cm3)

Magnetic field (Oe)

Magne

tization(emu/cm3)

Magneti

zation(emu/cm3)

Magnetic field (Oe)

Magnetocrystalline anisotropy in bulk metals

S. Kaya, Sci. Reports Tohoku Univ. 17, 639 (1928)MagneticNanostructuresP.Gambardella


16/36

Temperature dependence of magnetic anisotropy energy constants


17/36

p p g py gy

EASY

EASYPLANE


Effective anisotropy constants in magnetic thin films


18/36

W. J. M. de Jonge et al., in Ultrathin Magnetic Structures I,J. A. C. Bland and B. Heinrich eds., Springer (1994)

Au/Co(t)/Au T= 10 K

Chappert and Bruno, J. Appl. Phys. 64, 5736 (1988)

( ) 22eff 2

Co Volume Surfac Coe0K K M K /t= +

KV tlayers

KS

KS

MagneticNanostructures

P.Gambardell

a

Competition between dipolar and magnetocrystalline anisotropy


19/36

Co wedge

Pt substrate

20 m

Magn

etization(a.u.)

thickness (ML)

out-of-planeMCA predominates

in-plane, shape anisotropy M2

Vpredominates

g y y

Orientation and shape

of Co magnetic domains

Rusponi, Gambardella et al., X-ray photoemission electron microscopy, SIM beamline @ Swiss Light Source

Reorientation of the magnetization due to dipolar anisotropy above threshold thickness


P.Gambardella

Dependence of the magnetic anisotropy on the substrate crystallographic orientation


20/36

Co/Pt superlattices grown by MBE with

[001], [110], [111] orientation:

Epitaxy along these differentorientations can clearly induce defectstructures and local lattice distortions

that may result in different values of themagnetocrystalline anisotropy.


P.Gambardella

Orbital moment and magnetocrystalline anisotropy in 3dmetals


21/36

crystal field electron orbits fixes L relative to the crystal lattice

Different L values along different crystal directions

Direction with thelargest component ofL

Lowest spin-orbit energyeasy direction

of magnetization

see, e.g., P. Bruno, PRB 39, 865 (1989);H. A. Drr et al., Science 277, 213 (1997).

L

L


P.Gambardella

Perturbation theory in quantum mechanics *


22/36

0 0 0

0

0

unperturbed Schroedinger equation: H

small perturbation: (H V)

gnd gnd gnd E

E

=

+ =(0) (1) 2 (2)

(0) (1) 2 (2)

+ ...,

+ ...

E E E E

= + +

= + +

(1) (0) (0)

(0) (0)

(1) (0)

(0) (0)exc gnd

= gnd gnd

exc gnd

exc

exc gnd

E V

V

E E

=

2(0) (0)

(2)

(0) (0)exc gnd

exc gnd

exc gnd

VE

E E

=

V = L S(0) (0)

gnd

(0) (0)

(0) (0)

(0) (0)exc gnd

= 0

= 0

gnd gnd

exc gnd

gnd exc

exc gnd E E

=

L L

L SL L L


P.Gambardella

Spin-orbit interaction, orbital moment anisotropy, and MCA *


23/36

MODEL:Bruno, PRB 1989 Only unoccupied states matter

B4 4a z x L LL L m m

= =


P.Gambardella

Onset of magnetic anisotropy in single atoms: Co1/Pt(111)


24/36

Free magnetic atom:

- isotropy of space

K= 0

Small magnetic particles:

- broken symmetry- increased complexity

K= ?

P. Gambardella et al., Science 300, 1130 (2003)

Factors that determinethe magnetic anisotropy:

Angular dependence

- atomic symmetry

Magnitude

- 3dbandwidth- orbital moment- spin-orbit coupling

Kdepends on theatomic coordination:

KCo1/Pt= 200 KCo bulk


P.Gambardella

Magnetic moments and magnetic anisotropy: from the atom to the bulk


25/36


P.Gambardella

Finite-sized particles: the rise and fall of magnetic anisotropy


26/36

Co particles on Pt(111) with average size n

magneticanisotropy energy

orbital magneticmoment

n = 3Tdep = 10 K

n = 1Tdep = 10 K

n = 8Tdep = 83 K

20 40 40

out-of-plane

in-plane

P. Gambardella et al., Science 300, 1130 (2003)MagneticNanostructures

P.Gambardella

Ferromagnetism Superparamagnetism - Paramagnetism


27/36

FerromagnetismT< TC

SuperparamagnetismT< TCT> TB

Paramagnetism

T> TC

T< TB


P.Gambardella

Superparamagnetism


28/36

Ferrite particles


P.Gambardella

( )1

x coth(x)x

L =

0

B

M M ,kL T

=

Mag

netization

B/T [Tesla/K]

J. Crangle, The Magnetic Properties of Solids,Edward Arnold (1977)

i

M(B,T) qualitatively similar to paramagnetismbut the fluctuating moment is the total moment of each particle

= where is the sum of the atomic moments in a particle

Superparamagnetism


29/36

depends on B, K, TM sum of atomic mag. MomentsVparticle volume

H

Mz

H

Mz

texp >

0Kexp VkT

=

texp <

8 11

0 10 10 s = where

E KV > kT

E


P.Gambardella

2

H K cosV = M B

Hamiltonian:

easyaxis

Magnetization relaxation timeNel-Brown model:

Superparamagnetism: blocking temperature TB


30/36

H

Mz

H

Mz

T > TB < texp

0

Kexp

V

kT =

T < TB

> texp

where


P.Gambardella

easyaxis

Temperature required to observe a stablemagnetization on the timescale of the experiment

easyaxis

1

0

KlnB

VT

k

=

expt >>

an ill-defined quantity

e.g., stability criterion for magnetic recording requires:texp ~ 10 years at T= 300 K

Shape-dependent magnetic reversal of Fe nanoparticles on a Mo surface


31/36

Bode et al., Phys. Rev. Lett.92, 067201 (2004).

STM topography Spin-polarized STM

elongated

compact

Magnetic domains nucleate more easily in elongated particles (see, e.g., H.B. Braun, JAP 1999)


P.Gambardella

Magnetisation reversal processes


32/36


P.Gambardella

Hc = HA = 2K/M

Perfect materials, very small particles:Coherent rotation

Hc


33/36


P.Gambardella

-1

0

1

-1.5 -1 -0.5 0 0.5 1 1.5

M

h

010

3045

70

900

0.2

0.4

0.6

0.8

1

0

30

60

90

120

210

240

270

300

330

hsw

B

M

E. Bonet-Orozco et al. PRL (2000)

Coherent rotation: Stoner-Wohlfart model

Length of vector =

amplitude of Hc

The Stoner-Wohlfart Model for single-domain particles

B h i f h i i f i l d i i l h i fi ld i li d h i l


34/36


P.Gambardel

la

a

b HIs

Easy axis

U D =1

20

Is2 (N a cos

2 + N b sin2 )V =

1

40

Is2 (N b N a )Vcos2 + const .

U H = HIsVcos

U

=1

20

(N b N a )Is2cos 2 + H Is sin = 0

1

2sin 2 ( ) + h sin = 0

h =0H

Nb

Na( )Is

= + , < 0

- Behaviour of the magnetization of a single domain particle when a magnetic field is applied: hysteresis loops.

- hypotheses: ellipsoidal single-domain particle with only shape anisotropy and negligible magnetocrystalline anisotropy. The

magnitude of the particle magnetization Is remains constant for all values of the applied field (coherent rotation).

The demagnetization energy is given by

The magnetization I will point along a direction that makes U=UD+UH a minimum:

It is convenient to rewrite this equation as where

We wish to solve this equation for as a function of h and

Suppose a magnetic field H is applied. At equilibrium the magnetization Is will lie in the plane defined by the direction of

the field and the polar axis (a) of the ellipsoid. The Zeeman energy term is

It is difficult to obtain direct solutions and only the general nature of the results will be given here.

The Stoner-Wohlfart Model for single-domain particles


35/36


P.Gambardella

To plot hysteresis curves it is convenient to use reduced

units: the value of the reduced magnetization along the field

IH/Is = cos is plotted against h.

Assume = /2, i.e., H applied parallel to the hard axis,then the component IH is proportional to the field until h=1

(see figure). The rotation of the magnetization is fully

reversible. (Fig. 1)

Assume =0, H parallel to the easy axis but opposite Is;the anisotropy will maintain the initial orientation of Is untilabove the critical field h=1 a small perturbation will induce

an irreversible 180 jump of Is. There is a discontinuity in IHand hysteresis occurs. Energy is dissipated as heat. (Fig. 2)

Assume 0


36/36


P.Gambardella

- Assembly of particles with easy axes oriented at random:

the coercive force of the individual particles ranges from h=0 to h=1 and it is reasonable to expect a mean coercive force for

the system close to 0.5. The remanence (I at H=0) is Ir==0.5.

.

N.B. Incoherent rotationsIn reality, the magnitude of the magnetization |Is| doesn't need to stay constant as a magnetic field is applied to the sample,

as assumed in the S.-W- model. It turns out that reversible magnetization changes occur coherently while irreversible ones

occur incoherently. The nucleation field for an incoherent reversal is lower than the corrsponding critical field of the Stoner-

Wolfarth theory. As a result a lower cohercive force is predicted.

di huong tu_5

Documents