Download M - Unife

Introduction to magnetism of confined systems
P. Vavassori
CIC nanoGUNE Consolider, San Sebastian, Spain;
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Basics: diamagnetism and paramagnetism
Every material which is put in a magnetic field H, acquires a magnetic moment.
In most materials M =  H (M magnetic dipole per unit volume,  magnetic susceptibility.
paramagnetism
diamagnetism
M
M
H
H
Each atom has a non-zero magnetic moment m
The moments are randomly oriented (T);
H arranges these moments in its own direction.
Eappl = - m0 M . H
temperature kbT
m= - mB(L + gS) orbital and spin angular
In soilds m≈
Each atom acquires a moment
caused by the applied field H and
opposed to it
(Larmor frequency).
m= 0
e.g., noble gas.
momenta
- gmBS (crystal field)
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Ferromagnetism
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).
saturation magnetization MS
remanence
Limiting hysteresis curve: all the points
enclosed in the loop are possible
equilibrium states of the system.
MH
coercive field
With an appropriate history of the
applied field one can therefore end at
any point inside the limiting hysteresis
loop.
H
Fe, Co, Ni, alloys also with TM and RE
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Origin of hysteresis
In ferromagnetic materials the magnetic moments of the individual atoms interact
strongly with each other creating an order against the thermal fluctuations.
The magnetization of a sample may be split in many domains.
Each of these domains is magnetized to the saturation value M s but the direction of
the magnetization vector may vary from one domain to the other at H = 0.
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
(short range interaction
(Heisember hamiltonian
H = - ij Jij Si . Sj)
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
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Phase transition ferromagnet →paramagnet
Exchange interaction provides the magnetic
order against the thermal fluctuations
Ms (T)
MS
Above a critical temperature called
Curie temperature (TC) all ferromagnets
become regular paramagnets → MS = 0
at H = 0
TC
T

MS  (TC-T)
Since T < TC
 = ½ mean field theory (identical average exchange field felt by all spins)
This temperature for anti-ferromagnets is called Néel temperature (TN)
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Why magnetic domains?
Energy densities
In vacuum
u = B2/2m0
Inside a material
u =1/2 m0 Ms2
Total energy
U = ∫∫∫udt
All
space
a)
b)
c)
The field created outside the magnet in cases a) and b) costs B2/2m0 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.
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Magnetostatic energy
It can be shown (Maxwell equations) that the dipolar magnetostatic energy Em can be
espressed as:
1
Em   m0
M r  H d r d 3r  0
2
sample

Where Hd is called, variously, demagnetizing field, magnetostatic field or dipolar field.
Energy due to the interaction of each dipole with the field Hd created by all the other dipoles.
The complication arises from the fact that Hd [M(r)].
Hd can be calculated like a field in electrostatics as due to the magnetic
charges (bulk -• M and surface n • M ). The only difference is that they
never appear isolated but are always balanced by opposite charges.
n•M>0
+ + + +
M
-
Hd
- - -
n•M<0
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Magnetostatic energy: examples
Infinite ferromagnet
uniformly magnetized
-
Hd = 0
Em = 0
n•M>0
n•M<0
-
• M = 0 (M is uniform everywhere)
and n • M = 0 (no borders)
-
d
+
+
+
• M = 0 (M is uniform everywhere)
but n • M ≠ 0 (borders)
• M = 0 (M is uniform everywhere)
and n • M = 0 (M parallel to the borders)
Hd= -M ≠ 0
Em > 0
Hd = 0
Em ≈ 0
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Where M inside a domain is pointing to? Anisotropy
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.
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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 (ax2ay2 + ay2az2 + az2ax2) + K2 ax2ay2 az2 +….
- Uniaxial Eanis = K1 sin2q + K2 sin4q +… ≈ -K1(n . M)2
3
[K] = J/m
• Surface and interface anisotropy: due to broken translation
symmetry at surfaces and intefaces. The surface energy density can be
written:
- Esurf = Kp af2 - Ksan2; where an and af are the director cosines respect
to the film normal and the in plane hard-axis.
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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.
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Summary 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 = - m0 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.
Eanis = -K1(n . M)2
For a preferential axes n :
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 m0
Hd . M
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Anisotropy and domain structures.
Magnetostatic energy is not the only ingredient to
determine the actual domain structure.
a)
Anisotropy energy plays a role:
y
?
b)
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.
x
But still, what decides the number of subdivisions, for
c)
instance?
Somwhere ther is also Exchange energy stored.
Where?
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Domain boundaries
Bloch domain wall
Neél domain wall
(thin films)
w
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).
s w  AK1
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.
[A] = J/m
Domain wall width:
 w  A K1
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Energies and widths of domain boundaries
Nèel
Threshold
between Bloch
and Nèel walls
in a Fe film
Bloch
nm
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Energies and widths of domain boundaries (e)
Threshold between
Bloch and Nèel walls
in a typical soft film
(sligthly anysotropic)
Bloch
Nèel
nm
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Magnetization reversal and domains
Hext
Hext = 0
Hext
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).
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Let’s see a “real” example
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.
rotation
(high fields)
domain walls
displacement
(low fields)
Barkhausen jumps
At high fields the domain walls are
removed and the magnetization is
rotated coherently towards the field
direction.
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Hysteresis due to magnetization rotation
Stoner and Wohlfarth model: coherent rotation of an
uniaxial particle uniformly magnetized.

E = K1 sin2 - mMsHcosq
Ms
q
H
Easy axis
Free energy
for unit volume
Ho 
2 K1
mo M s
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Ferromagnetic nano-structures
Ferromagnetic nano-structures offer a unique opportunity to investigate
properties at length scales previously unattainable.
Because intrinsic magnetic length-scales (e.g., exchange length or the
domain wall thickness) are comparable to the sample size, novel
physical properties can be expected, respect to bulk- and film-like
materials.
Surface effects become relevant (dominant).
Novel physical properties can be expected with respect to bulk- and
film-like materials (e.g., completely different magnetization reversal
mechanism).
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Applications
This field has attracted much attention because of its close ties
to potential technological applications.
• Nonvolatile Magnetic Random Access Memories (MRAM).
• Periodic bi-dimensional arrays of magnetic dots for future high
density magnetic storage media (1 Tbit/in2).
• Magnetoelectronic devices with new functionalities (sensors).
A key issue for such applications is to understand and control
the magnetic switching of small magnetic elements.
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Scale length parameters
The relative strength of the anisotropy and magnetostatic (1/2m Ms2) energies (per
unit volume) can be expressed by the dimensionless parameter:
k = 2K1/mMs2.
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 =
2 A m0 M s2
The comparison between exchange and anisotropy may be expressed through the length:
lw = A K1  lex k
DW width
Fe
Co
Ni
Py
A
(J/m)
2110-12
3010-12
910-12
1310-12
Ms
(A/m)
1.7106
1.42106
0.49106
0.86106
K
(J/m3)
48103
520103
-5.7103
0
lex
(nm)
3.4
4.9
7.5
5.3
lw
(nm)
20.9
7.6
39.7

k
(adim.)
2.610-2
4.210-1
3.710-2
0
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From 3D to 2D (infinite)
Etot = Eappl + Eexc + Eanis + Em
n •M>0
unfavoured
+ + + +
Hd = - M
M
-
favoured
- -
-
n•M<0
M
Hd = 0
“Real” 2D systems have a lateral finite size
more favoured
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Magnetostatic effects due to shape.
For uniformly magnetized bodies  • M = 0 (M is uniform everywhere) so that magnetostatic
energy depends only on surface magnetic charges n • M ≠ 0 → shape of the body)
favoured
Hd = 0
Hd = - M
unfavoured
Nz < Nx
Hd,x = - Nx M
Aspect ratio
Hd,z < Hd,x
unfavoured
Elongated particles
favoured
Hd,z = - Nz M
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Demagnetizing tensor : shape anisotropy.
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.
Magnetostatic self interaction for an ellipsoid (referring to the ellipsoid semi-axes )
Ed = 1/2 m(Nx Mx2 + Ny My2 + Nz Mz2).
Uniaxial anisotropy
y
M
Eanis = K1 sin2q
q
z
K1 = 1/2 m Ms2 (Ny - Nz)
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Lateral confinement and single domain state
Upon sufficient lateral confinement the size of a ferromagnet can
become so small that not even a single domain wall can be
accomodated inside the system. The magnetization configuration
inside the ferromagnet can become a single domain (this concept will
be defined more precisely later on).
Size
Multi-domains
structure
Single domain
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Single domain particles: the spherical case
Domains are favourable from the point of view of magnetostatics (minimize
dipolar energy) but they cost domain wall energy
s w  AK1
For a sphere of radius R of material
with A and K1, by equating the energy
of the single domain state with that of
two domains, one gets that the critical
radius below which the single domain
state is energetically stable is:
RSD =
36 AK1
m 0 M s2
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)].
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Too small !!
If the size becomes too small (as for RSD for Co, Fe, and Ni) the
magnetic moment of the single domain ferromagnet can fluctuate due
to thermal energy: superparamagnetic limit.
Size
Single domain Superparamagnet
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Thermal stability of the remanent state:
superparamagnetism
Competition between thermal and magnetic (anisotrpy) energy: thermal activated
reversal. This leads to a relaxation time at zero field (for uniaxial particle):
t  1010 e
K1V
k BT
K1 is the anisotropy constant V is the particle volume, kB the boltzmann’s
constant and T the absolute temperature.
t depends on the particle volume V: if the particle is too small it
becomes unstable at room temperature -> superparamagnetic limit.
E
kBT
K1V
0
p
q
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Retarding the onset of superparamagnetism
- Increase K1 as much as possible.
- Larger aspect ratio retards the onset of superparamagnetism by adding
“shape anisotropy” to the material intrinsic anisotropy.
K1 = 1/2 m Ms2 (Ny - Nz)
Shape anisotropy
hard
soft
“Material” effect (K1)
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Lateral confinement and remanent state: summary
Size effects
Single
Closure
domains
domain Superparamagnetic
Shape effects
Closure
domains
Single
domain
Aspect ratio effects
Closure
domains
Single
domain
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A caveat about “single domain” state in
confined systems
The uniformity condition can be realized ONLY for isotropic ellipsoids.
Real particles are never ellipsoids…..deviations from uniformity close to
the edges.
Flower state
Leaf state
In order to avoid these deviations from uniformity the size of the particle
(cubic) should be: d ~ lex which is of a few nm for Co,Fe and Ni.
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An interesting case: very soft material Fe20Ni80
Eg., a particle 10
nm thick and with
lateral size below
700 nm could not
accomodate even
a domain wall.
Nèel
Bloch
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Soft nano-scale disk and rings: vortex state
Magnetization configuration determined by magnetostatic and exchange energies
(no anisotropy).
Vortex core
The energy is almost all due to exchange.
Residual magnetostatic energy is confined in
the core.
The core is necessary to avoid the singularity
in the in-plane magnetization curling.
Systems developing vortes state
configurations are interesting because of
their reduced sensitivity to edge effects.
Stabilization of the vortex state by removing
the high energy core in a ring structure.
Magnetic force microscopy image
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Incoherent vs. coherent reversal
For single domain particles the reversal process can be still incoherent, in a
way different from doman wall displacement: curling mode.
reversal
2K1
moMs

HN
2K1
moMs
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Magnetization revrsal is a dynamic process
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
E = - mB
dJ
G
dt
G = m× B
m= -mBg J = - J
dm
Larmour precession (c = B)
  m  B
dt M motion is damped (a)
damping
my
Low damping
mx
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.
my
High damping
torque
mx
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Damping and field dependance of the reversal of a
magnetic moment
t  t
m
H
Damping a 0.2
Damping a 0.05
t=0
m 1700 emu x (10nm)3
m 1700 emu x (10nm)3
H 5kOe : reversal time 646 ps
H 5 kOe: reversal time 2121 ps
H 500 Oe: reversal time 5950 ps
H 500 Oe: reversal time 20481 ps
1.0
M
x
M
x
My
My
1.0
0.5
0.5
0.0
0.0
1.0 -0.5
1.0
1.0 -0.5
0.5
0.5
0.5 -1.0
-1.0
0.0
0.0
0.0
-1.0
-1.0
-0.5
-0.5
-0.5
-0.5
-0.5 0.0
0.0 0.5
0.0 0.5
0.5 1.0 -1.0
-1.0
-1.0
1.0
1.0
M
M
z
Mz
z
M
x
My
1.0
0.5
0.0
-0.5
-1.0
-1.0 -0.5
Damping a 1.0
m 1700 emu x (10nm)3
H 5 kOe: reversal time 185 ps
H 500 Oe: reversal time 1565 ps
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References and further reading
• S. Blundell, Magnetism in condensed matter, Oxford University Press, 2006.
• A. Aharoni, Introduction to the theory of Ferromagnetism, University Press,
Oxford 2000.
• W. F. Brown, Micromagnetics, Wiley, New York 1963.
• S. Chikazumi, Physics of Magnetism, Wiley, New York 1964.
• M. Prutton, Thin ferromagnetic films, Butterworth &Co. (Publishers) Ltd., London
1964.
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