Magnetism I: from the atom to
the solid state
Preface
The Lecture on “Magnetism I: from the Atom to the solid state” is an introduction to the fundamental concepts in magnetism. It consists of two parts:
the first one (by D. Pescia) deals with magnetic effects in atoms (diamagnetism, paramagnetism, formation of magnetic moments in atoms) and with
the occurrence of magnetic order in the ground state of a solid in virtue of
the exchange interaction. The second part (by A. Vindigni) treats the occurrence of magnetism at finite temperatures, the role of small interactions
such as the dipolar interaction, and presents the essential facts about the
statistical physics of magnetism. A very extended introduction in modern
magnetism can be found in the book by J. Stöhr and H.C. Siegmann ” Magnetism: from fundamentals to the Nanoscale dynamics”, Springer-Verlag,
Berlin Heidelberg 2006.
Zürich, September 2012
D. Pescia
ii
Contents
Preface
ii
1 Magnetism in Atoms
1.1 Magnetism in classical physics . . . . . . . . . . . . . . . . .
1.2 Magnetism in quantum mechanics . . . . . . . . . . . . . . .
1.2.1 Free electrons in a magnetic field . . . . . . . . . . .
1.2.2 Electron in a magnetic field and a central potential .
1.3 The formation of the magnetic moment in atoms . . . . . . .
1.3.1 Paramagnetism in Atoms . . . . . . . . . . . . . . . .
1.4 Exchange interaction and the Heisenberg-Dirac-Van Vleck
Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Magnetism in solids
2.1 Stoner-Wohlfahrt model . . . . . . . . . . . . . . .
2.2 Friedel-Oscillations . . . . . . . . . . . . . . . . . .
2.2.1 The interatomic exchange interaction . . . .
2.2.2 RKKY oscillations . . . . . . . . . . . . . .
2.2.3 Anhang: mathematical details of the model
3 Magnetic order at finite temperature
3.1 Coupled effective spins: an N -body problem
3.2 Mean-field approximation (MFA) . . . . . .
3.3 Mean-field universality class . . . . . . . . .
3.4 The Landau approach . . . . . . . . . . . .
3.5 Classical spin models . . . . . . . . . . . . .
3.6 Correlation functions . . . . . . . . . . . . .
3.7 Landau theory of correlations . . . . . . . .
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4 Magnetic domains and domain walls
71
4.1 Magnetic anisotropy . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Domain walls in the classical Heisenberg model . . . . . . . . 74
iii
iv
CONTENTS
4.3
4.4
4.5
4.6
4.7
4.8
Continuum formalism . . . . . . . . . . .
Beyond the Mean-Field Approximation .
Finite size and superparamagnetic limit .
Dipolar interaction . . . . . . . . . . . .
Dipolar interaction in extended systems .
Origin of magnetic domains . . . . . . .
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77
80
88
90
94
100
1
CONTENTS
Part I
D. Pescia
Chapter 1
Magnetism in Atoms
1.1
Magnetism in classical physics
The experimental facts about magnetism in solids are as old as the history
of mankind: some materials have the property of producing a sizable magnetic field that attracts or repel other materials. One of the first references
to the magnetic properties of what we know now to be magnetite F e3 O4
(lodestone) is by 6th century BCE Greek philosopher Thales of Miletus. The
name ”magnet”may come from the lodestones found in Magnesia. In China,
the earliest literary reference to magnetism lies in a 4th century BC book
called Book of the Devil Valley Master: ”The lodestone makes iron come or
it attracts it”. The lodestone based compass was used for navigation in medieval China by the 12th century. The main observation about the origin of
the magnetic field originates with the experiments of Ampere and Oersted in
the early decades of the 19th century, demonstrating that i a current is able
to influence a magnetic needle (Oersted) and ii a mechanical force exists between two wires injected with current (Ampere). Later, Faraday completed
our knowledge of magnetic field by discovering that time dependent magnetic fields can produce a magnetic current. J.C. Maxwell gave a complete
description of electromagnetic fields that is still very precise (Maxwell equations). The origin of the magnetism in matter remained debated: Ampere
postulated that magnetism in atoms originates from the existence of a closed
atomic-sized current. Poisson and later Maxwell, instead, favored magnetic
charges that appear always coupled into dipoles as the source of the magnetic field. Distinguishing between the two hypothesis is a subtle problem,
as a paper in 1977 by J.D. Jackson show. Following Ampere hypothesis, the
magnetic field produced by a current circulating within a small loop C is
2 ~
given in term of the magnetic moment vector m
~ = 2e ~x × ~ẋ = 2m
L, which
2
CHAPTER 1. MAGNETISM IN ATOMS
3
Figure 1.1: Drawing of the magnetic field of the Earth by René Descartes,
from his ”Principia Philosophiae”, 1644. This was one of the first drawings
of a magnetic field.
. R
for a closed loop amounts to m
~ = I S ~nds, where S is the surface within the
loop C and ~n is the vector normal to S.
~
~ r ) = µ0 ∇
~ × (∇
~ × m
B(~
)
4π
| ~r |
µ0
1
~
µ0 ~ ~ m
)−
m△
~
∇(∇ ·
=
4π
| ~r |
4π
| ~r |
µ0 ~
~ 1 ) + µ0 · m
=
~ · δ(~r)
∇(m
~ ·∇
4π
| ~r |
i
h µ 3~r(~r · m)
~ − m~
~ r 2 µ0
0
− m
~ · δ(~r) + µ0 m
~ · δ(~r)
=
4π
| ~r |5
3
The first component of the magnetic field produced by the small current
~ field produced by an electric dipole moment.
loop is formally identical to E
This lead Poisson and later Maxwell to state that ” ...we may regard the
magnet,..., as made up of small particles, each of which has two equal and
opposite poles” (magnetic charges). This idea, which was alternative to the
one of Ampere, describes almost anywhere the magnetic field correctly. At
the position of the loop, however, – the second term, given by the Dirac delta
function – the picture of magnetic charges would produce a field amounting
to − µ30 m
~ · δ(~r). Experiments aimed at measuring the ”contact” term –
such as the hyperfine line of atomic hydrogen (relevant in astrophysics) –
CHAPTER 1. MAGNETISM IN ATOMS
4
Figure 1.2: Pierre Pelerin de Maricourt (French), Petrus Peregrinus de Maricourt (Latin) or Peter Peregrinus of Maricourt was a 13th century French
scholar who conducted experiments on magnetism and wrote the first extant treatise describing the properties of magnets. His work is particularly
noted for containing the earliest detailed discussion of freely pivoting compass needles, a fundamental component of the dry compass soon to appear
in medieval navigation.
can ultimately discriminate between the two hypotheses: the hypothesis by
Ampere that the magnetic moment is due to atomic current rather than
atomic charges is now well accepted.
While Maxwell equations and the postulate of Ampere are very exact
in describing atomic magnetic moments and their magnetic fields, the very
existence of magnetic moments in classical physics is challenged by a famous
theorem, the Bohr-van Leuwen theorem.
Theorem: given that the classical Hamilton function for an electron in an
~ writes H = 1 (~p − eA)
~ 2 + eφ it follows that the
applied magnetic field B
2m
canonical average of the atomic magnetic moment ≺ m
~ ≻ vanishes exactly
and any finite temperature.
Proof:
1. When one writes down explicitly the equations of motion in a uniform
5
CHAPTER 1. MAGNETISM IN ATOMS
field one can construct an integral of the motion – the total energy – which
reads
1 ˙2 ~
m~x − ∇φ(~r)
(1.1)
2
~ – indicating that the
Remarkably, the total energy is independent of B
Lorentz force is not doing any work, being perpendicular to the velocity.
Accordingly, when the statistical average based on the canonical Gibbs distribution is computed, the energy of the various classical states over which
one integrate does not contain the magnetic field and any partial derivative
with respect to the magnetic field – such as the average magnetic moment –
must vanish.
2. Considering that
∂H
≺m
~ ≻= − ≺
≻
(1.2)
~
∂B
one obtains
− ≺ mz ≻ =
R
1 (~
~ 2 −eφ(~
p−eA)
x)
− 2m
kB T
d~pd~x −(px − 2e Bz y)y + (py + 2e Bz x)x e
(1.3)
1 (~
~ 2 −eφ(~
p−eA)
x)
− 2m
R
kB T
d~pd~xe
Using the variable transformation

p′x x p′y
1

0
x′ = 
 0
0
0
1
eB
2
0
0
0
1
0


px


0 
 x 
0   py 
y
1
−eB
2
(1.4)
one can easily convince oneself that the integral – and with it the average
magnetic moment – is vanishing – a result known as Bohr van Leuwen
theorem of classical statistical physics.
1.2
1.2.1
Magnetism in quantum mechanics
Free electrons in a magnetic field
It was first Landau (1930) who explicitly produced an average magnetic
moment and thus determined the ”birth” of magnetism in matter. Landau
solved the problem of a free electron moving in a uniform magnetic field
using quantum mechanics to describe the motion, i.e. breaking away from
6
CHAPTER 1. MAGNETISM IN ATOMS
Lorentz force. For a simple case of one free electron, the energy levels write
(see textbooks in QM)
En = ~ωc (n + 1/2) +
~2 kz2
2m
(1.5)
), their degeneracy being (not including the spin of the
(n = 0, 1, 2..., ωc = eB
m
L2 eB
electron) 2π~ (L: linear dimension of the system). The partition function
reads
L2 eB X X
−En
Q(T, B, L) =
exp
2π~ k n
kB T
z
3
X −n~ω
L eB p
c
− ~ω
2
e kB T
2πk
T
e
=
B
(2π~)2
n
L3 eB p
1
=
2πkB T
~ωc
2(2π~)2
sinh( 2k
)
bT
(1.6)
The magnetization (per particle) amounts to
kB T
kB T
~ωc
~ωc
∂lnQ
=
−
coth(
)
∂B
B
2B
2kb T
(1.7)
For small magnetic field we have the simple result
mz = −
e 2 ~2 B
12m2 kB T
(1.8)
mz being the average magnetic moment of the free electron. This result
sanctions the appearance of a non-vanishing magnetic moment. Typically,
one deals with N free electrons, each energy level being occupied by two
electrons at the most, up to the Fermi energy. Considering that only electrons
within an energy range kb T below the Fermi energy can be excited, the
magnetization (magnetic moment per unit volume) induced by the magnetic
field amount to (a= lattice constant)
e2 ~2 B · ρ(EF )kB T
e 2 ~2
. mz
Mz = 3 = −
=
−
ρ(EF )
a
12m2
kB T
12m2
(1.9)
Here is ρ(EF ) the density of states at the Fermi level, which for a typical free
electron metal amounts to 0.805 · 1022 eV −1 cm−3 . The final result is that the
induced magnetization in a free electron gas (without considering the spin
of the electron) is independent of the temperature. Defining the magnetic
susceptibility as
∂Mz
.
χz = µ0
(1.10)
∂Bz
CHAPTER 1. MAGNETISM IN ATOMS
7
as a way of comparing the various magnetic responses, we find a value for
χz in the case of free electrons of about 10−6 , which means that an external
field of the order of 1 Tesla produces a magnetic field µ0 Mz of the order of
1 microtesla. Notice that letting ~ go to zero produces the vanishing of M
required by classical physics. This result shows that quantum mechanics is
the key for the appearance of a magnetic moment in matter, albeit weak.
1.2.2
Electron in a magnetic field and a central potential
In an atom, the Hamilton operator in the presence of a uniform magnetic
~ = − 1 (~r × B)
~ reads
field A
2
2
2
~ r) + V (~r) = [− ~ △ + V ]
p~ − eA(~
2m
~
− B·µ
~ → (a)
2
e ~2
+
A → (b)
2m
~
.
=
~ = −µB · L~ is the operator representing the magnetic moment (µB |e|~
µ
2m
−23 J
0.927410 T is the Bohr magneton).
The strength of the various terms can be computed e.g. on hydrogen orbitals.
We obtain
~2
<s|−
△ + V | s >= −E0 = −13.6eV
2m
~ · L)
~ | l = 1, ml = −1 >= −µB · B
(a) →< l = 1, ml = −1 | −µB (B
e2 ~ 2
A →| s >
< s | 2m
B
4
= 10−12 ( )2
(b) →
E0
3
T
In the last line B in supposed to be given in T .
Term a. This term is called the Zeeman term, the corresponding energy
is the Zeeman energy. Its contribution is non-vanishing when the atomic
orbital has a finite z-component of the angular momentum. It shows that
an atomic state with appropriate z-component of the angular momentum (in
the present case −~ lowers the total energy of the atomic level by an amount
µB · B, which, for a magnetic field of 1T corresponds to about −10−4 eV.
Notice that, at finite temperatures, levels with Lz = 0, +1 are also partially
occupied, decreasing the total average magnetic moment, which becomes
strongly temperature dependent, as we will discuss at the end of this chapter.
CHAPTER 1. MAGNETISM IN ATOMS
8
Term b. This term is much smaller than the Zeeman energy, almost temperature independent, and only observable if the total magnetic moment of the
atoms is exactly vanishing: it builds the so called diamagnetic contribution
to the energy. From the expression for the diamagnetic energy one can deD
fine an effective diamagnetic moment per atom through µD = − ∂E
and
∂B
2
∂ ED
−11
a diamagnetic susceptibility χD = −µ0 · a3 ∂B 2 amounting to ≈ −10 eV /T
0
and −10−6 , respectively, in a magnetic field of 1T . Thus, the diamagnetic
moment per electron points antiparallel to the applied magnetic field.
Figure 1.3: The top of the figure describes some alternative definitions of the magnetic
susceptibility. The Table on the left shows values of the diamagnetic susceptibility for some
substances (to find χD multiply the values by 10−5 ). The figure on the right illustrates
e2 ~ 2
A →| s > to the diamagnetic
a straightforward generalization of the expression < s | 2m
µ0 e2 P
2
susceptibility of many in an atom, which writes χD = − 6m
i < ri > the sum extending
over all electrons building the magnetic response (including the core electrons). Therefore,
the diamagnetic susceptibility should scale with Z· < ra2 >. This is confirmed by the plot
bottom right, where the horizontal axis shows the total number of electrons Z multiplied
by the square of the ionic radius < ra > in units of A2 , for noble gases and ions of atoms
with filled atomic shells, for which no net magnetic moment results.
CHAPTER 1. MAGNETISM IN ATOMS
1.3
9
The formation of the magnetic moment
in atoms
In the previous section, we have shown that the magnetic field couples with
the operator of the (negative) orbital angular momentum by means of the
coupling constant µB . We have now to take into account that
• each electron have an intrinsic spin that adds up in some way to the
orbital angular momentum to produce a total angular momentum quantum number J, which results from the addition of the orbital quantum
number of the electron and its spin.
• atoms have generally many interacting electrons: there spins and orbital angular momenta must be added in some way to find the total
ground state angular momentum which results from the addition of the
total angular momentum and the total spin of all electrons!
We are looking for a set of rules that allow angular momenta to be added
in quantum mechanics and to a set of rules that allow to select, among the
many possible angular momenta, those proper to the atomic ground state
of an atom. In other words: the key problem we have to solve is: given N
electrons in the configuration (ni , li , lz,i , s = 1/2), (i = 1...N ), 1st what are
the possible values of the total spin S, total orbital angular momentum L and
the total angular Momentum J (Russel Sounders symbol for the electronic
configuration: (2S+1) LJ ?) 2nd: which configuration has the lowest energy
(ground state)?
The first rule underlying all steps we are going to introduce to finding the
electronic configuration of an atom, is a central theorem of quantum mechanics (known as Clebsch-Gordan series) about adding angular momentum
operators:
Theorem: given an angular momentum operator P~ with z components
~ with z-components [q, q − 1, ..., −q], the total angular
[p, p − 1, ... − p] and Q
~ can assume the values
momentum operator T~ = P~ + Q
P + Q, P + Q − 1, P + Q − 2, ...., | P − Q |
The total spin: The total spin of an assembly of spin 12 particles is given by
the Clebsch-Gordan sum of all spins. For instance, for N = 2 we have the
possible values for S of 0, 1, carrying singlet (χs ) respectively triplet (χt ) spin
CHAPTER 1. MAGNETISM IN ATOMS
10
eigenfunctions:
r
1
1
√ (u1/2 u−1/2 − u−1/2 u1/2 ) =
(|↑>|↓> − |↓>|↑>)
2
2
u1/2 u1/2 = |↑↑>
r
1
1
√ (u1/2 u−1/2 + u−1/2 u1/2 ) =
(|↑>|↓> + |↓>|↑>)
2
2
u−1/2 u−1/2 = |↓>|↓>
The total orbital angular momentum Again, it is given by the ClebschGordan sum of all orbital angular momenta.
The total angular momentum. It is given by the Clebsch-Gordan sum of total spin and total angular momentum.
Pauli principle (exchange interaction part I) There is a restriction on the
possible configurations: the total wave function of Fermions must be antisymmetric (Pauli principle). A theorem by Weyl helps implementing the
Pauli principle in the electronic structure of a many electron system.
Theorem (H. Weyl): All eigenfunctions of Sz have the same symmetry with
respect to particle permutation: the symmetry property of the eigenfunction
of Sz under permutation are called ”Spinrasse”. This is e.g. clearly visible in the two-electron system, where the spin singlet is antisymmetric with
respect to particle permutations and the spin triplet is symmetric. This theorem divides the electronic structure of many-electrons atoms into separated
thermal schemes, according to their total spin (see e.g. ”para” and ”ortho”
He): in fact, because of the requirement that the total wave function be
antisymmetric, each spin rasse determines uniquely the symmetry properties
of the orbital wave functions under permutation. For instance, the energy
levels of the singlet thermal scheme can only have symmetric orbital wave
functions:
1
√ (un1 ,l1 (~r1 )un2 ,l2 (~r2 ) + un1 ,l1 (~r2 )un2 ,l2 (~r1 ))
2
while the triplet spin state carries only antisymmetric wave functions:
1
√ (un1 ,l1 (~r1 )un2 ,l2 (~r2 ) − un1 ,l1 (~r2 )un2 ,l2 (~r1 ))
2
The Pauli principles implies that, among the possible configurations arising
from the Clebsch-Gordan sum, some cannot be realized because orbital
wave functions of the proper symmetry do not exist. For instance, starting
from two inequivalent s electrons [n, s],[n′ , s], one can obtain the two configurations 3 S1 ,1 S0 . Starting e.g. from two equivalent s electrons [n, s],[n, s],
CHAPTER 1. MAGNETISM IN ATOMS
11
the triplet state cannot be constructed as the requirement of antisymmetric
orbital wave function makes it vanishing. In other words: The Pauli
principle is responsible that two identical spin 1/2 particles can only be
in the same orbital state (n1 , l1 , lz,1 = n2 , l2 , lz,2 ) if they form a singlet. In
other words: more than two electrons cannot be in the same orbital state:
if they are, they (sloppy) ”must have opposite spin”. Therefore, although
the Hamilton operator does not contain any spin dependent term (and the
two electrons might not be even interacting), the non distinguishibility
postulate of QM removes the degeneracy of some states and even forbids a
well defined spin state to be a possible configuration. The Pauli principle
acts as some sort of effective interaction that removes some degeneracy and
distinguishes between spin states. This interaction is know in the literature
with the broad terminology of ”exchange interaction”. It is a purely QM
effect and completely disappears together with the spin upon transition to
classical mechanics. In the present case the exchange interaction dictates
e.g. that the ground state of two identical spin 1/2-particles is a singlet. The
Pauli principle and the exchange interaction are key results explaining the
stability of matter and, for our purposes, are responsible for the formation
of magnetic moments in atoms.
Hund’s rules: exchange interaction part II
The Pauli principle acts to remove some configurations among the set arising
from the Clebsch-Gordan sum but, for instance, does not lift the degeneracy
between triplet and singlet states of electrons with different orbital states.
F. Hund (1925) and Russel and Saunders (Astrophysics Journal, 61, 38,
1925) formulated on the base of spectral data, a set of empirical rules that
allows a further analysis of the various configurations.
1st Hund rule. Provided there is sufficient degeneracy that non-equivalent
orbital wave functions can be constructed, the configuration realizing the
lowest energy state corresponds to a state of maximum spin number. In
other words: if the orbital states involved have different quantum numbers,
the filling of the electronic states with parallel spins produces the lowest
energy electronic configuration. Thus, provided orbitally degenerate
states exist, the triplet state is energetically favored. If degeneracy is
absent, then the singlet state is energetically favored. Accordingly, the
formation of a finite total spin requires orbital degeneracy.
There is an intuitive explanation for this result. The Coulomb energy is large when the
two electrons are closer to each other. In a triplet spin state the antisymmetric orbital
wave functions the two electrons takes care that the two electrons are as far as possible
from each other, an this reduces the Coulomb repulsion with respect to the symmetric
orbital wave function, where the two electrons are, on the average,allow to be closer to
each other.
CHAPTER 1. MAGNETISM IN ATOMS
12
2nd Hund rule. The second Hund rule states that, after having established
the maximum S, the lowest energy state correspond to the configuration
that maximizes L (again, the Coulomb energy is reduced for high values of
L).
3rd Hund rule. The third rule states that the total angular momentum
quantum number J minimizing the energy is | L + S | if the shell is more
than half full, | L − S | if the shell is less than half full. This last rule
minimizes the energy arising from spin-orbit interaction.
Of course, other electronic configurations exist, which have empirically
higher energies. These rules, together with the Pauli principle, determine
completely the total values of J, L, S for the ground state and its electronic
symbol 2S+1 LJ .
We are now ready to discuss the magnetic moment operator arising from
the electronic configuration 2S+1 LJ . As electrons do have both orbital and
spin angular momentum, the magnetic coupling with a magnetic field must
be extended to include the magnetic moment arising from the spin. We
~ with gS = 2
know from the Dirac theory of the electron that ~µS = −gS µB S,
(in contrast to gL = 1). Quantum electrodynamic corrects this value to
gS = 2.0023, which is extremely close to the experimental value. Accordingly,
the operator describing the interaction of a magnetic moment with a magnetic
field becomes
~ + g S)
~ ·B
~ = −µB (J~ + (g − 1)S)
~ ·B
~
HZ = −µB (L
~ S
~
(let us call gS g, for simplicity). HZ is called the Zeeman operator. J~ = L+
~ S
~ and J~ are dimensionless
is the operator of the total angular momentum. L,
angular momenta. Let us now assume that we have determined (by solving
the many body electron problem and using the Hund’s rules) the total orbital
angular momentum quantum number L of the ground state and its total spin
angular momentum quantum number S. Notice that the possible eigenvalues
of J~2 also labels the eigenspaces of an Hamiltonian that contains the spin
~ ·L
~ interaction, as both J~2 and the spin orbit interaction are scalar
orbit S
under rotation and therefore have, according to the Wigner-Eckart theorem,
the same eigenspaces. Notice that, again because of the W.-E. theorem, Jz ,
Lz and Sz have also common eigenspaces, as both are the z-components of
a vector operator under rotations. We consider Hz to be a perturbation of
the fine structure energy levels of an atom, i.e. those energy level resulting
from Hund’s rules and carrying a well defined quantum number J. To find
the first order eigenvalues of HZ we solve the eigenvalue problem of HZ
CHAPTER 1. MAGNETISM IN ATOMS
13
within the 2J + 1 dimensional space containing the symmetry adapted wave
functions to this J-value: un,l,j,mj . We seek the first order correction, i.e. the
magnetic field is small enough so that the Zeeman splitting is smaller than
the level splitting between the various multiplet components J arising from
~ = (0, 0, B):
the spin-orbit coupling. We choose B
HZ = µB · B · (Jz + (g − 1) · Sz )
The eigenvalue problem of Jz is simply solved, because
(un,l,j,mj , Jz un,l,j,mj ) = mj
(1.11)
mj = J, J − 1, ..., −J. The eigenvalue problem of Sz is simplified by the
Wigner-Eckart theorem, which states that
(un,l,j,mj , Sz un,l,j,mj ) = τLSJ mj
(1.12)
τLSJ is common to all mj and QM shows that
τLSJ =
J(J + 1) + S(S + 1) − L(L + 1)
2J(J + 1)
(1.13)
Accordingly, we obtain
(un,l,j,mj , Hz un,l,j,mj ) = µB · B · mj · gLSJ
with the Lande factor
gLSJ = 1 + (g − 1) ·
J(J + 1) + S(S + 1) − L(L + 1)
2J(J + 1)
(1.14)
Accordingly, a magnetic field lifts the 2J + 1-degeneracy of the fine structure
level completely. The Zeeman splitting is symmetric around the unperturbed
level Enj . The distance between two consecutive Zeeman levels is
△EZ = µB · B · gLSJ
(1.15)
i.e. it is proportional to B. These results are known as the anomalous
Zeeman effect, to be compared with the normal Zeeman effect, with g = 1
and gLSJ = 1.
CHAPTER 1. MAGNETISM IN ATOMS
14
CHAPTER 1. MAGNETISM IN ATOMS
15
Figure 1.4: Top: Effective magnetic moment of ions of rare earths atoms as
a function of the number of f -electrons and their electronic configurations.
CHAPTER 1. MAGNETISM IN ATOMS
1.3.1
16
Paramagnetism in Atoms
The statistical mechanics of an ensemble of non interacting N identical atoms
each carrying the same quantum number J allows to compute the partition
function as
hX
−gLSJ · µB · mj · B iN
ZN =
exp(
(1.16)
kB · T
m
j
the total free energy f (T, B) per atom as
hX
−gLSJ · µB · mj · B ]
−kB · T ln
exp(
kB · T
m
j
and the mean magnetic moment per atom as
−∂f (T, B)
= gLSJ · µB · J · BJ (α)
∂B
2J + 1
2J + 1
1
α
BJ (α) =
· coth[
α] −
coth
2J
2J
2J
2J
. gLSJ · µB · J · B
α=
kB · T
< µz >=
(1.17)
For the particular case J = 1/2 (a spin 1/2, s-state Atom) we obtain the
simple equation
µB · B
< µz >= µB tanh[
]
(1.18)
kB · T
For small arguments α, BJ (α) ≈
)
< µ(P
>≈
z
J+1 α
J 3
and we obtain the Curie law
(gLSJ )2 · J(J + 1) · µ2B
·B
3kB · T
(1.19)
which contains the purely quantum mechanical quantity (gLSJ )2 · J(J + 1).
The paramagnetic susceptibility in this limit amounts to
C
(1.20)
χP ≈ µ0
T
with the Curie constant
N (gLSJ )2 · J(J + 1) · µ2B
(1.21)
C=
V
3kB
At room temperature is χP ≈ 10−3 . Notice that the determination of the
Curie constant is a key experiment to access the quantities J and gLSM determining the ground state electronic configuration of atoms and thus provide
a reliable test of our quantum mechanical approach to the ground state configuration in atoms.
17
CHAPTER 1. MAGNETISM IN ATOMS
Figure 1.5: Temperature dependence of
1
χ
for some Cupric salts
CHAPTER 1. MAGNETISM IN ATOMS
18
19
CHAPTER 1. MAGNETISM IN ATOMS
1.4
Exchange
interaction
and
the
Heisenberg-Dirac-Van Vleck Hamiltonian
An estimate of the triplet-singlet splitting in atoms is often obtained by
explicitly computing the lowest energy states of the simplest many electrons atomic system: the He-atom. The full Hamiltonian in the BornOppenheimer approximation amounts to H0 + V (1, 2) with (e2 standing for
e2
)
4πǫ0
−~2 ~ 2 ~ 2
Ze2 Ze2
( ∇1 + ∇2 ) −
−
2m
r1
r2
2
e
V (1, 2) =
r12
H0 (1, 2) =
Let us first neglect V12 . The energy levels of H0 (i) are Eni =
(1.22)
−Z 2 e2
,
2·a·n2i
a being
Rni ,li
r
· Yli ,mI (ϑ, ϕ).
the Bohr radius. The eigenfunctions of Eni are ϕni li mi =
The ground state of H0 corresponds to the state in which both electrons are
2 2
in a 1s-orbital and has energy E0 = 2 · E1 = −Za e . Its wave function is
ψ0 = ϕ1s (1)ϕ1s (2) ⊗ χ(1, 2)
ϕ1s (1)ϕ1s (2) =
1 Z 3 − Z (r1 +r2 )
( )e a
π a
and
1
χ(1, 2) = √ (u1/2 u−1/2 − u−1/2 u1/2 )
2
The wave function ϕ1s (1)ϕ1s (2) is symmetric with respect to change of the
coordinate vectors and the ground state of He is a singlet, as required by
Pauli principle. The net magnetic moment is vanishing: S = 1 is prohibited
in the ground state.
Let us now introduce V12 as a perturbation. The energy of the ground state
is modified to EG = E0 + Q with
Z
e2
Q = dV (1)dV (2)ϕ21s (1) ϕ21s (2)
r12
In order to explicitly compute this last integral, one develops 1/r12 in spherical harmonics:
1
4π X 1
r2
∗
=
( )l · Yl.m
(ϑ1 , ϕ1 )Yl,m (ϑ2 , ϕ2 ) ⇐⇒ r1 > r2
| ~r1 − ~r2 |
r1
2l + 1 r1
l,m
1
| ~r1 − ~r2 |
=
4π X 1
r1
∗
( )l · Yl,m
(ϑ1 , ϕ1 )Yl,m (ϑ2 , ϕ2 ) ⇐⇒ r1 < r2
r2
2l + 1 r2
l,m
20
CHAPTER 1. MAGNETISM IN ATOMS
Inserting this development in Q and using the orthogonality of spherical harmonics we
obtain
Z r1
Z
Z ∞
2Zr2
2Zr
2Zr
4e2 Z 6 ∞
2
− a1 1
2 − a2
dr2 r2 e− a ]
dr1 r1 · e
dr2 r2 e
( )
[
+
Q=
π a
r
1 0
0
r1
Partial integration leads to Q =
same order of magnitude as E0 .
5Ze2
8a
and positive. Notice that the correction Q is of the
Summarizing : the two He-electrons have a ground state configuration (1s)2
2
with energy EG = − Zea (Z − 58 ).
The first excited state of the H0 -operator corresponds to the electronic
configuration (1s)1 (2s)1 , with antisymmetric wave function
1
ψS=0 = √ [ϕ1s (1)ϕ2s (2) + ϕ1s (2)ϕ2s (1)] ⊗ χs
2
1
ψS=1 = √ [ϕ1s (1)ϕ2s (2) − ϕ1s (2)ϕ2s (1)] ⊗ χt
2
The S = 0 wave function belongs to the parahelium thermal scheme, the
S = 1-wave functions belong to the orthohelium thermal scheme. Parahelium
shows diamagnetism, orthohelium is a paramagnet. Without consideration of
V12 para and ortho (1s)(2s) states are degenerate. In the absence of spin-orbit
coupling, optical transitions between the two thermal schemes are absolutely
forbidden. Should one be able to pump a He atom gas (by electrons excitation
or other mechanism) into the triplet excited state, then it will practically
stay forever (many months) in that state. Notice that because of spin orbit
splitting the triplet states degeneracy is lifted and a fine structure appear in
the excitation spectrum of ortho helium.
We show now that the result of introducing the Coulomb interaction is the
lifting of the degeneracy between triplet and singlet states, leading, in some
circumstances, to the formation of a magnetic moment, at least in one excited
state 1s2s with different orbital quantum numbers. We consider the four
states eigenspace of H0 to (1s)(2s) and solve the eigenvalue problem of H0 +
V12 within this space. The Hamiltonian matrix reads

E1s + E2s + Q + J

0


0
0
0
E1s + E2s + Q − J
0
0
0
0
E1s + E2s + Q − J
0
0
0
0
E1s + E2s + Q − J
mit
Z
e2
Q =
r12
Z
e2
∗
∗
J =
dV1 dV2 ϕ1s (1)ϕ2s (2) ϕ1s (2)ϕ2s (1)
r12
dV1 dV2 ϕ21s (1)ϕ22s (2)




CHAPTER 1. MAGNETISM IN ATOMS
21
from which the sougth for eigenvalues can be read out immediately. The integral Q is the Coulomb energy. The integral J is the result of the exchange
interaction and is called exchange integral. It provides the exchange energy contribution that arises from the correlation of the two electron as a
consequence of symmetrizing the wave functions according to the Pauli principle. The Coulomb interaction produces, via Pauli principle, a splitting of
the initial degeneracy of the (1s)(2s) configuration: the singlet state has the
energy E1s + E2s + Q + J, the triplet state has the energy E1s + E2s + Q − J.
In summary: there are two distinct thermal schemes for He, consisting of
para (singlet) - and ortho (triplet) states. Levels with the same quantum
numbers of the orbital wave functions (e.g. (1s)(1s)) are prohibited in ortho
helium. Furthermore, the splitting between ortho and para states depends on
the relative sign and strength of Q(n, l) and J(n, l). In the present example,
both integrals Qn,s and Jn,s are positive, so that the t-states level is lower
than the s-state. The first Hund’s rule states that the positivity of J when
degenerate non-equivalent states are involved is a general feature of atoms.
The strength of the splitting – the strength of the parameter J which we will
call the Hund exchange parameter JHu – is typically of the order of few eV ,
i.e. the same order of magnitude as the Coulomb interaction, from which the
exchange interatcion actually originates.
One can formally obtain the t − s splitting by caricaturing the exchange interaction (which is actually acting in the orbital space) with an effective spin
Hamiltonian: the Heisenberg-Dirac-Van Vleck operator. Dirac defined the
operator acting in spin space
HSpin = (E1s + E2s + Q) · E − J · P12
with the exchange operator P12
P12 | + >| + >=| + >| + >
P12 | + >| − >=| − >| + >
P12 | − >| + >=| + >| − >
P12 | − >| − >=| − >| − >
The eigenvalues of the operator Hspin when restricted to the 1s2s eigenspace,
are identical with the eigenvalues of the physical operator. A useful way of
writing P12 is
E + ~σ1 · ~σ2
1
~1 · S
~2 ]
P12 =
= [E + 4 · S
2
2
where ~σ are the two-by-two Pauli matrices and the product σi σj must be
taken as a Kronecker product of matrices. The spin Hamiltonian simulating
CHAPTER 1. MAGNETISM IN ATOMS
22
the original Hamiltonian acting in the orbital space reads
1
~r · S
~s ]
HSpin = (E1s + E2s + Q) · E − J · [ E + 2S
2
It can be formally generalized to many electrons:
HSpin =
X
r
Er +
X
1X
1 ~ ~
Q(r, s) −
J(r, s) · [ + S
r · Ss ]
2 r,s
4
r6=s
where r, s is a set of quantum numbers describing orbital wave functions and a
factor 21 has been placed in front of the sums to avoid double counting. Dirac
has shown with quite general arguments that the eigenvalues of this effective
spin Hamilton operator are correct within first order perturbation theory.
The difficulty is one of computing the various coupling constants forthcoming
in the operator, which ultimately requires knowledge of the orbital wave
functions.
Chapter 2
Magnetism in solids
In the first chapter we have shown how atomic magnetic moments are produced. On these grounds, diamagnetism and paramagnetism, which are the
main manifestation of atomic magnetism at finite temperature, are well understood. However, we still need to answer the broad question about ” why
Fe is ferromagnetic”. This means that we need to understand 1. what happens to an atomic magnetic moment when it is embedded into a ”see” of free
electrons, 2. how do magnetic moments (if they ”survive” the contact with
the free electrons) couple to align along the same direction (what makes the
triplet state between spins on different lattice sites energetically preferred
with respect to the single state) and what is the strength of the exchange energy distinguishing between triplet and single state on different lattice sites.
interaction
2.1
Stoner-Wohlfahrt model
Let us now introduce the Stoner-Wohlfahrt (SW) model of magnetism in
solids. In a solid, electrons have, in reality, wave functions which, for some
kind of inner shell electrons, like the d-electrons in transition metals, have
a strong component localized in the vicinity of the ion core (remember the
tight-binding model). So to speak, they spend a lot of time close to the ion
core: during this time they are almost atomic like and can feel the Hund rule
as we know it from atomic physics. We will see that this Hund rule provides
ultimately the net persistence of the net magnetic moment at each atom in
the solid state, although the contact with free electrons acts to destroy the
magnetic moment, in general. Although the atomic magnetic moment might
persist when the atom in embedded into a solid, there are two important
experimental features that distinguish atomic magnetism from magnetism
23
CHAPTER 2. MAGNETISM IN SOLIDS
24
in the solid state. First, in atoms the formation of the magnetic moments
is the result of the Hund rules that weight correctly, at least empirically,
the orbital angular momentum and the spin of the electrons into forming
the total angular momentum of the many electrons in one atom. The total
angular momentum produces the net atomic magnetic moment, which is
ultimately responsible for the occurrence of magnetism in matter. When
the magnetic moment per atom is measured in the solid state, it appears
that only the spin part of the total angular momentum is contributing to
it: one speaks of ”quenching” of orbital angular momentum, in particular
in transition metals, by the crystal field. Second, in angular momentum
quenched metals, magnetic moments per atom should be an integer multiple
of 2µB , i.e. their magneton number is an integer. This would lead, for
example, to atomic magnetic moments of 2,3, respective 4 µB for Ni, Co and
Fe. The experimentally measured value in bulk Fe is 0.616, 1.715 and 2.216
µB . The Stoner-Wohlfahrt-Slater model of magnetism provides the correct
framework to explain the occurrence of 1. finite magnetic moments in solids
and 2. the existence of non integer magneton numbers.
While the formation of a magnetic moment, because of the first Hund
rule, is almost the rule in atoms, it is a very rare event in solid, where
the electrons can be considered as delocalized and are therefore better
described by a band structure. The SW model in its simplest version
considers free electrons where energy levels are filled up to the Fermi radius
kF = (3π 2 N/V )1/3 , N/V being the electron density. In virtue of this filling
~2 k2
the electron gas has a total kinetic energy amounting to Ekin = N 35 2mF .
The non magnetic ground state foresees that all states up to EF are
filled with two electrons carrying opposite spins. Introducing an exchange
interaction shifts the energy levels of minority electrons to higher energies,
while the energy levels of majority spin electrons are shifted downwards.
This produces an energy gain that actually favors the relative shift of
energy bands and, ultimately, the formation of a magnetic moment. On
the other side, the radius of the Fermi sphere must be increased to host all
the electrons, as the double occupancy of each level is no longer possible.
This produces an increase of the total kinetic energy that goes against the
formation of a magnetic moment. Therefore, the formation of the magnetic
moment is the result of a delicate energy balance and is subject to strong
restrictions.
Let us now work out the Stoner criterion for ferromagnetism, i.e. for the
existence, in a Fermi gas subject to exchange interaction, of an imbalance
between spin up and spin down electrons. We start from a density of
CHAPTER 2. MAGNETISM IN SOLIDS
25
states n0 (E) common to both spin channels and change the energy levels
±
according to E~k,ν
= E~k,ν ∓ 21 I · M , where + refers to majority spins. I
is the intra atomic exchange interaction responsible for the Hund’s rule in
atoms, and P is the sought for spin imbalance, M = N↑ − N↓ . Accordingly,
the density of states separates out for the two spin channels according to
(
n± E) = n0 (E ∓ 21 I · M ). Integrating up to the (to be determined) Fermi
Figure 2.1: DOS for majority spins and minority spins are shifted by an
amount IM .
energy EF gives the number of electrons per unit cell and the total moment
per unit cell:
Z EF (M )
IM
IM
N=
[n0 (E +
) + n0 (E −
)]
2
2
0
Z EF (M )
IM
IM
) − n0 (E −
)]
(2.1)
M=
[n0 (E +
2
2
0
These are two equations for the sought for parameters EF (M ) and M . Solving the first one can obtain in principle EF (M ). Inserting this in the second
one we obtain an implicit equation for M :
M = F (M );
Z EF (M )
IM
IM
F (M ) =
[n0 (E +
) − n0 (E −
)]
2
2
(2.2)
CHAPTER 2. MAGNETISM IN SOLIDS
26
The function F (M ) has following important properties:
1. F (0) = 0
2. F (−M ) = −F (M ) , i.e. EF (−M ) = EF (M )
3. F (∞) = M∞ and −M∞ < F (M ) < +M∞
4.
dF
dM
|M =0 ≥ 0.
Proof: From
′
F (M )M =0
IM
IM
I 0
0
n (E +
) + n (E −
)
=
2
2
2
M =0
IM
IM
dEF
+ n0 (E +
) − n0 (E −
)
(2.3)
2
2
dM M =0
we obtain F ′ (0) = I · n0 (EF ) ≥ 0
M∞ is the largest magnetic moment by complete spin polarization of the
electron gas and corresponds to the first Hund’ rule magnetic moment. Under
these conditions, the graphical solution of the implicit equation for M has
two possible scenarios: either M = F (M ) has only the solution M = 0 for
F ′ (0) < 1, or it has two solutions: M = 0 and M finite but not necessarily
an integer for F ′ (0) > 1. In this case one can show that the M = 0 solution
maximizes the total energy, while the two solutions with opposite sign are
the sought for minima that establish a finite spin imbalance in the ground
state. Accordingly, the Stoner criterion for ferro magnetism reads
I · n0 (EF ) > 1
(2.4)
I is essentially an atomic quantity of the order of 0.7 eV for 3d atoms. The
tendency to ferro-magnetism therefore requires a high density of states of
the non-spin polarized band structure at the Fermi level. This can only be
achieved when the states close to the Fermi level are sufficiently localized,
i.e. their bandwidth is small enough, as is the case for d metals with partially
filled d shells. In the following figures we will illustrate the Stoner criterium
on a set of examples, starting with single magnetic impurities in metals.
CHAPTER 2. MAGNETISM IN SOLIDS
27
Figure 2.2: Graphical solution of the implicit equation for M . In A the only
solution is M = 0, in B a finite magnetization minimizes the energy.
Figure 2.3: Left: Local DOS of Mn in Ag according to LSDA computations
by R. Podloucky et al. Phys. Rev. B33, 5777 (1980). The spin splitting is
about 3 eV (experiment: 4 eV). Right: Computed and measured values of
3d- impurity atoms in Ag, Cu and Al. The highest moment appears always
in the middle of the 3d-row.
CHAPTER 2. MAGNETISM IN SOLIDS
Figure 2.4:
28
CHAPTER 2. MAGNETISM IN SOLIDS
29
Figure 2.5: FeRh: dispersion curves for the two spin directions; (a) spin+,
(b) spin(-). The broken line is the Fermi level.
CHAPTER 2. MAGNETISM IN SOLIDS
30
Figure 2.6: top: Band structure of F e along the ∆ direction. The minority
spin bands (broken lines) are shifted toward higher energies with respect to
majority spin bands (continuous lines). The symmetry label of the bands is
also shown. In contrast to the Stoner model, the shift is not exactly rigid but
depends slightly on ~k. On the top of the figure, the ~k-points selected by the
used photon energy are indicated. Bottom. On the left is an energy resolved
photo emission spectrum taken at normal emission from a (100)-surface of
Fe. In the middle, the same electrons are analyzed for their spin polarization.
on the right, the photo-emission intensity for the two spin channels is plotted
separately, by suitably compounding the total intensity and the polarization
data. The two peaks are identified as due to electrons originating from the
spin split bands close to the Γ-point, see top.
CHAPTER 2. MAGNETISM IN SOLIDS
2.2
2.2.1
31
Friedel-Oscillations
The interatomic exchange interaction
The long sought explanation for the origin of ferromagnet was provided after years of research and illustrated e.g. in a review article by M.B. Stearn,
Physics Today, April 1978, p.34. The first condition for ferromagnetism is
that we have some localized magnetic moments, and this conditions is met
in Fe by the localized d-electrons, which keep a part of their atomic magnetic
moment produced by the Hund-rule intra atomic exchange. The second
condition for ferromagnetism it that all these moments line up parallel to
each other, i.e. triplet state coupling between neighboring spins. However
we know that the chemical bonding between equivalent orbitals favors, in
line with the Pauli principle, the singlet state, so that we need a mechanism that acts ”against” the Pauli principle in order to get triplet coupling.
This alternative mechanism is provided, according to Stearns, by the indirect exchange between localized d-electrons through RKKY coupling with
the delocalized part of the d-wave functions (or with the s-like electrons).
This mechanism is therefore based on the existence of ”degenerate” states
in the band structure of solids. The RKKY coupling mechanism is also a
central one in modern research on magnetism and we want to illustrate its
peculiarities with a simple, computable exact model which is also relevant in
thin films coupling phenomena.
Notice that the singlet coupling underlying the chemical bond and the absence of magnetism associated with it in the ground state is a ”robust” result,
in the sense that there exists a very strong theorem by Lieb and Mattis that
states that in a linear arrangement of atoms the non-magnetic state, i.e. the
state with lowest total spin, is the ground state. One needs to go higher than
one dimension to escape this theorem, because in higher dimensions electrons
states with different symmetry – atomic orbitals with different quantum numbers – can hybridize: it is this degeneracy between orbital wave function with
different symmetry that provide a route to escape the strong Pauli principle that favor antiparallel alignment between orbital states with the same
quantum number. The situation is exactly the same as in atoms: only if
the electronic states participating to the formation of the magnetic moment
have different quantum number, the exchange interaction can act to lower
the energy of the triplet state. The situation can be therefore summarized
as follows. The chemical bond between same orbitals centered at different
atoms favors the antiparallel ground state, in virtue of Pauli principle. The
crystal potential, however, can act to mix different symmetries and different
orbitals into the wave functions forming the valence bands in solids. As in
CHAPTER 2. MAGNETISM IN SOLIDS
32
atoms, different symmetries might favor energetically the parallel coupling,
thus producing ferromagnetic alignment between neighboring atoms. However, it depends on the crystal potential and on the orbitals involved, whether
a total spin in the ground state is formed or not.
We point also out that the strength of the effective exchange energy that
favors the triplet coupling between two different sites (the interatomic exchange interaction) is one to two orders of magnitude smaller that the onesite (intra-atomic) exchange interaction (which amounts to about 3 − 5 eV).
This means rotating one spin in the presence of the other ones needs much
less energy that suppressing the magnetic moment. It is the interatomic exchange interaction which is relevant for determining the temperature scale
at which collective ferromagnetic order vanishes (in the next chapter, the so
called Curie temperature).
2.2.2
RKKY oscillations
The presence of a more or less localized magnetic moment, made of d-wave
functions creates a potential sink with the strength of the s − d exchange
interaction at the location of the magnetic moment for majority (spin up)
s-electrons, in virtue of the atomic Hund-rules. The minority spin down
electrons can be considered as non-affected by the impurity. A local perturbation in one spin channel produces an oscillating density in the affected
spin channel (see the Anhang at the end of this chapter), while the other
spin remains uniformly distributed. This produces a local spin polarization
of the electron gas surrounding the impurity
P=
2· < Sz >
ρ+ − ρ−
κ cos 2kF x
= +
≈ O(
)
−
~
ρ +ρ
kF x
(2.5)
that propagates far away from the perturbing magnetic moment. At some
location x within the spin polarized s-electron gas a spin imbalance appears.
This spin imbalance acts as an effective exchange field for d-waves functions
and tends to align a d-derived magnetic moment at that location parallel
to itself: A magnetic moment at the location x would lower his energy by
aligning along the direction of P. In this way, the exchange interaction
can propagate, oscillating between positive and negative depending on the
position x and can couple spins which are quite distant from each other.
CHAPTER 2. MAGNETISM IN SOLIDS
33
Figure 2.7: a): the left-hand side shows a typical hysteresis curve (M versus magnetic
field H) recorded for exchange-coupled Co films. At the shift field H = Hj the magnetizations of the individual films are aligned to the direction specified by the external magnetic
field. The critical field Hj is measured as a function of the Cu spacer thickness τ by
scanning a focused laser beam over a wedge-like multi layered structure, shown schematically on the right-hand side. b): Hj versus τ for a room temperature grown wedge-like
multi layered structure. A finite shift field means AFM coupling in the ground state. A
vanishing shift field means FM coupling. The thickness of the Co films are 13.2 ML and
15.8 mono layers, respectively. Inset, the Fourier transform, the two peaks corresponding
to the two periodicities 2.4 Ml and 5.4 ML. The long period dominates. c): as b) but with
the Cu wedge and the final Co film deposited and measured at 160 K. The short period
now dominates, see Fourier transform).
34
CHAPTER 2. MAGNETISM IN SOLIDS
2.2.3
Anhang: mathematical details of the model
We introduce in a 1d free electron gas a perturbing potential localized at the
origin and look for the total charge density produced by it at a location x
– essentially a continuation of the Anderson impurity model to. In order to
simplify the mathematics, we consider a Dirac-Delta like perturbation potential of strength λ located at the origin of a one dimensional solid filled with
a free electron gas. As we are interested here at the behavior of the charge
(or spin) density far away from the location of the perturbing potential, we
do not use the more realistic but also more cumbersome traditional potential
well with finite width Let us consider a segment extending from −L/2 to
+L/2 along the x-axis and establish in it a potential V (x) = λ · aδ(x). a
represents the width of an hypothetical well and λ its strength. We refer to
the segment with x < 0 as the left-hand side l and to the segment with x > 0
as the right-hand side r. We have to solve the Schr”odinger equation
[
−~2
+ V (x)]ψ(x) = Eψ(x)
2m
(2.6)
under the boundary conditions for a δ like potential
ψl′ (0) − ψr′ )0) −
ψl (0) = ψr (0);
2mλa
ψ(0) = 0
~2
(2.7)
In the two regions l and r the respective solutions have to fulfill the SE
−~2
ψ(x) = Eψ(x)
2m
(2.8)
We distinguish two cases: E < 0 and E > 0.
For E < 0 the SE away from the singularity has two solutions, one
growing exponentially toward ±∞, the other decaying exponentially toward
±∞. This last solution is the only physical one as it has a finite norm and
produces a bound state with energy amounting to
Eb = −
.
(κ =
λam
)
~2
~2 κ 2
mλ2 a2
=−
2m
2~2
(2.9)
and wave function
ψlb (x) =
√
κeκx
ψrb (x) =
√
κe−κx
(2.10)
CHAPTER 2. MAGNETISM IN SOLIDS
35
In the range E > 0 we have free electrons moving left and right, and
the solutions in each range l, r, under periodic boundary conditions at
±L/2, are
~2 k 2
2π
; k=
·n
2m
L
r
r
r
r
2
2
2
2
sin kx + Bl
cos kx);
ψr (x) = Ar
sin kx + Br
cos kx)
ψl (x) = Al
L
L
L
L
both basis functions being normalized to 1/2 in the range x ∈ [± − L/2, 0].
The boundary conditions at x = 0 read
2mλa
Bl = Br ; k · (Al − Ar ) −
Bl = 0
(2.11)
~2
E=
There are two classes of wave functions fulfilling these conditions. One class
has
A = Ar ⇒
r
rl
2
2
u
u
sin kx;
ψr (x) =
sin kx
ψl (x) =
L
L
where the total wave function is normalized to 1 over the segment with
length L.
Bl = 0;
.
The second class has Bl = Br = B 6= 0 and must be even under
change of sign, so that Ar = −Al = A and B = Ak
. This type of wave
κ
functions read
r
r
2
k 2
g
sin kx +
cos kx)
ψl (x) = A[
L
κ L
r
r
2
k 2
g
ψr (x) = A[−
sin kx +
cos kx)
L
κ L
A must be chosen so that the entire wave function is normalized to 1 in the
range x ∈ [−L/2, L/2]. This means
κ
(2.12)
A= √
κ2 + k 2
and
r
r
1
2
2
g
ψl (x) = √
[
κ · sin kx +
k · cos kx)
2
2
L
L
κ +k
r
r
1
2
2
g
ψr (x) = √
[−
κ · sin kx +
k · cos kx)
L
L
κ2 + k 2
36
CHAPTER 2. MAGNETISM IN SOLIDS
Using these wave functions, we compute the total charge density at a
point, e.g. x ≥ 0:
Z kF h
i
(κ sin kx − k cos kx)2
2 L
2
−2κx
·
dk
+ sin kx
ρ(x) = κe
+ ·
L 2π 0
κ2 + k 2
Z kF h
1
κ2 cos 2kx + κk sin 2kx i
−2κx
= κe
+ ·
dk 1 −
π 0
κ2 + k 2
Z ∞ h 2
κ cos 2kx + κk sin 2kx i
1
kF
dk
−
= κe−2κx +
π
π 0
κ2 + k 2
Z ∞ h 2
κ cos 2kx + κk sin 2kx i
1
dk
+
π kF
κ2 + k 2
One can prove by complex integration that
i
kF
κ·i −2κx h
ρ(x) =
+
e
· E1 (−2κx + 2ikF x) − E1 (−2κx − 2ikF x)
π
2π
E1 (z) being the exponential integral. Notice that the contribution of
the bound state to the total charge density cancels out with part of
the contribution of the free electron states. The following figure plots
the charge density as a function of the variable x for some characteristic
values of κ and k.
Some particular limits are worked out now.
(E1 (z) ≈ −γ − lnz)
For small x we have
kF
κ
kF
kF
κ
+ (π − arctan ) − O(x) ≈
+ − O(x)
π
π
κ
π
2
For large z we have
e−z
E1 (z) ≈
z
and accordingly
ρ(x) ≈
e2κx+2ikF x
e2κx−2ikF x
−
=
−2κx + 2ikF x −2κx − 2ikF x
h
i
4ie2κx
κx
sin
2k
x
−
k
x
cos
2k
x
F
F
F
4κ2 x2 + kF2 x2
and in the limit of large x (and small κ) we obtain
ρ(x) ≈
κ cos 2kF x
kF
+ O(
)
π
kF x
(2.13)
CHAPTER 2. MAGNETISM IN SOLIDS
37
Figure 2.8:
This result underlines the formation of a Friedel-like oscillation of the charge
density away from a localized perturbation. The wave-length of the oscillation is kπF . The oscillation decays as the inverse of the distance from the the
perturbation (this is in 1d: in 2d we have a decay as the square, in 3d as the
third power of the inverse distance.
38
CHAPTER 2. MAGNETISM IN SOLIDS
Part II
A. Vindigni
Chapter 3
Magnetic order at finite
temperature
In the first two Chapters
• we have shown how magnetic moments are created at the atomic level
according to the Hund’s rules (intra-atomic exchange interaction);
• we commented on how atomic magnetic moments, deduced assuming spherically symmetric surrounding (Hund’s rules), generally reduce
when the atom is “put” in a crystal, which lowers the symmetry of its
environment (Stoner-Wohlfahrt model);
• we have shown how an interatomic exchange interaction can arise in
a metal by means of the RKKY interaction;
• we defined the conditions under which a metal may show ferromagnetic
coupling between different magnetic moments.
Already at that level, it was clear that ferromagnetism is not the rule but
rather an exception, in the sense that many factors that are encountered
in ordinary materials usually prevent the formation of magnetic moments or
that of a ferromagnetic interatomic coupling. All the above-mentioned properties1 have been deduced neglecting the temperature or, in other words, they
are ground-state properties. In this Chapter we discuss the consequences of
introducing the temperature. The general trend is that thermal fluctuations
destroy the ground-state ferromagnetism (when present). In the same line
as before, we will define some conditions under which ferromagnetism can
“survive” at finite temperatures as well.
1
apart from the Brillouin function.
39
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
3.1
40
Coupled effective spins: an N -body problem
In the previous Chapters we have considered the conditions under which an
isolated atom possesses a finite magnetic moment. Hund’s rules allow computing the ground-state multiplet, characterized by the total angular momentum (orbital plus spin contribution) which results from all the unpaired
electrons. When dealing with coupled magnetic moments, we will indicate
the atomic total angular momentum with S i) in order to avoid confusion with
the exchange interaction (J) and ii) because – in this context – people often
speak about “spin” or “effective spin” to indicate the total single-atom angular momentum. As far as a single isolated atom is concerned, its magnetic
moment at finite temperature is well described by the Brillouin function:
gµB S B
z
m = −gµB hŜ i = gµB S BS
,
(3.1)
kB T
with
α
1
2S + 1
α −
coth
2S
2S
2S
gµB S B
.
kB T
(3.2)
In the derivation of this function, we have implicitly used the knowledge of
i) the eigenstates of the atom in an external applied field and ii) the way
of performing thermal averages for a quantum system (see Appendix). Note
that the intra-atomic and the Zeeman interaction have been treated on a
different ground: we have considered only the ground-state multiplet (which
minimizes the intra-atomic exchange interaction) but we have applied Boltzmann statistics to the levels of this multiplet in case they have been split
by an external field (Zeeman interaction). The reason for such a different
treatment reside in the characteristic energy scales of the two interactions in
relationship with the thermal energy kB T . In fact, the intra-atomic exchange
energy is of the order of 4 − 10 eV∼ 105 Kelvin, while the Zeeman splitting
is roughly 0.1 meV ∼ 1 Kelvin for one-Tesla applied field.
Further on, we have seen how a ferromagnetic interatomic exchange interaction is necessary for the occurrence of ferromagnetism in a solid. Under
specific and relatively strict conditions, this goal is attained by means of the
RKKY interaction2 . The order of magnitude of the RKKY interaction is
10 − 50 meV ∼ 100 − 500 Kelvin. Thus, depending on the material and the
2S + 1
coth
BS (α) =
2S
2
and α =
Other mechanisms are responsible for exchange interaction, e.g. super-exchange or
direct exchange, in insulators.
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
41
Typical exchange energies and magnetic moments
Fe
Co
Ni
Gd
Dy
EuO
EuS
Tc [K]
1043
1395
629
289
87
69.4
16.5
C[K]
2.22
2.24
0.588
4.68
3.06
µ[µB ]
2.22
1.71
0.605
7.1
J[eV]
0.012
0.015
0.013
0.00025
J[k]
139
174
151
2.9
Ms [Gauss]
1746
1446
0.510
2.060
2920
1930
1240
Table 3.1: Some typical values for the energy scales, Curie temperature, Curie
constant and saturation magnetization Ms = gµB S/a3 (a lattice constant).
Often it turns out useful to express the exchange interaction in Kelvin units:
1 eV ≃ 1.16 × 104 K.
temperature range of interest, a statistical-mechanics treatment is required
for the RKKY interaction as well. The competition between this interatomic
exchange interaction and thermal fluctuations is indeed responsible for the
loss of ferromagnetism above a certain temperature, called Curie temperature TC . Table 3.1 reports the values of the Curie temperature, the exchange
interaction and other relevant parameters for few typical magnetic materials.
Let us come back to the formal treatment of magnetism at finite temperatures, restricting ourselves to ferromagnetic exchange interactions. A system
of coupled magnetic moments arranged in a lattice can then be described by
the Hamiltonian
X
X
1
H=− J
Ŝ(n) · Ŝ(n′ ) + gµB B
Ŝ z (n) .
(3.3)
2
′
n
|n−n |=1
The dimension of the Hilbert space associated with this quantum many-body
problem scales as (2S + 1)N , N being the number of magnetic moments
(spins) in the lattice. Due to such an exponential dependence on N , the
exact treatment of a system of many coupled spins becomes intractable –
even numerically – as far as the number of spins approaches that of realistic
extended systems3 . In practice, one can try to circumvent this problem in
several ways:
3
Some effective zero-dimensional structures (magnetic clusters or nanoparticles) are also
studied in the context of nanomagnetism. For some of these systems, exact diagonalization
of the associated quantum problem is still feasible numerically and makes it possible to
describe their magnetic behavior at any temperature.
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
42
1. Reduce the many-body problem to a single-particle problem. This
corresponds to the mean-field approximation (MFA).
2. Simplify the problem replacing the quantum-spin operators by classical
vectors.
3. Take advantage of specific symmetries in the problem under investigation and use a Hamiltonian which can easily be diagonalized.
4. Consider only a selected family of excitations of the ground state, which
can have either local (domain walls) or non-local (spin waves) character.
3.2
Mean-field approximation (MFA)
The goal of the Mean-Field Approximation (MFA) is to reduce the manybody problem (3.3) to the a single-particle problem. This means to get
rid – somehow – of terms which directly involve two-spin operators such as
Ŝ(n) · Ŝ(n′ ). In this context, we understand a paramagnet as the reference
single-particle problem. We will first make use of the Brillouin function (3.1)
Many-body
problem
Single-particle
problem
MFA
effective
field
Figure 3.1: Sketch of the idea behind the mean-field approximation.
to write down the MF equation of state heuristically and discuss its relevant
implications. Further on, within the more rigorous Landau approach, we
will prove that the MFA is actually the best approximation of the Hamiltonian (3.3) in terms of a single-particle Hamiltonian.
Equation of state
Referring to the sketch in Fig. 3.1, we may think that the actual field experienced by each spin in a ferromagnetic sample contains a contribution
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
43
arising from the interaction with its neighbors, besides the typical Zeeman
term (due to the interaction with the external, applied field). More explicitly,
we assume that the physics of each spin can be described by a single-particle
Hamiltonian of the form
Hsp (n) = gµB (B + BW ) Ŝ z (n)
(3.4)
where gµB BW = −z̄JhŜ z (n)iHsp . The origin of the Weiss field BW is the
interatomic exchange interaction, whose effect is taken into account only
as an average and not rigorously. Such an average is performed using the
Hamiltonian (3.4) itself and z̄ indicates the number of nearest neighbors of
each spin. As anticipated, the Hamiltonian Hsp is equivalent to the one of
a paramagnetic atom in a magnetic field B t = B + BW so that the thermal
average of the Ŝ z (n) projection is given by the Brillouin function:
with
hŜ z (n)iHsp = −SBS (α)
(3.5)
gµB S B − z̄JShŜ z (n)iHsp
gµB S B t
=
.
α=
kB T
kB T
(3.6)
Since the average hŜ z (n)iHsp is also contained in α, i.e. the argument of the
Brillouin function, Eq. (3.5) is actually a self-consistent equation. To write
Eq. (3.5) in a more transparent way, we exploit the relation between the
average of the spin component along the field and the associated magnetic
moment m = −gµB hŜ z (n)iHsp . The MF equation of state finally reads:
z̄J S m
gµB S B
.
(3.7)
+
m = gµB S BS
kB T
gµB kB T
In order to visualize the solution of Eq. (3.7) graphically, it is convenient to
set
m
= BS (α)
gµB S
gµB B
kB T
α−
.
σ=
2
z̄JS
z̄JS
( σ=
(3.8)
Let us list the most remarkable facts arising from the graphical analysis of
solutions depending on the external parameters T and B.
1. When B = 0, there exists a non-trivial solution with σ 6= 0 only if
the slope of the Brillouin function exceeds that of the straight line (the
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
44
second one of Eqs. (3.8)). In particular, expanding the former around
α ≃ 0 yields
S+1
α + ...
(3.9)
BS (α) ≃
3S
so that a spontaneous magnetization (σ 6= 0) only arises for T < TC
with
S + 1 z̄JS 2
TC =
.
(3.10)
3 S kB
One can show that if solutions with σ 6= 0 exist, they have a lower free
energy than the solution corresponding to σ = 0.
2. For T < TC , the system of Eqs. (3.8) admit two graphical solutions of
opposite sign in the region B ∈ [−Bc , Bc ]. Outside of this interval the
solution is unique and with σ > 0 (σ < 0) for B > 0 (B < 0).
3. For T > TC and small α the Brillouin function can again be linearized
and the system of Eqs. (3.8) takes the simplified form
(σ = S + 1 α
3S
kB T
gµB B
σ=
α−
;
2
z̄JS
z̄JS
(3.11)
by using the definition of TC given in Eq. (3.10), the solution of the
previous set of equations can be written as
T
gµB
1−
σ=−
B
(3.12)
TC
z̄JS
or equivalently (using again Eq. (3.10))
1
(gµB )2 TC
(gµB )2 S(S + 1)
B=
B.
z̄J T − TC
3kB
T − TC
(3.13)
The pre-factor of B on the right-hand side is the susceptibility (computed in B = 0)
C
χ=
(3.14)
T − TC
which is the well-known Curie-Weiss law with
m = gµB Sσ =
C=
(gµB )2 S(S + 1)
3kB
(3.15)
being the Curie constant (already encountered when discussing paramagnetism).
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
45
Equation (4.49), together with other MF predictions, is not expected to hold
true in the vicinity of TC . In fact, in this critical region the neglected terms
(fluctuations) play a major role. This statement should sound clearer at the
end of this chapter. Keeping in mind the limitations of the MFA, it is still
interesting to investigate how different observables should behave according
to the MFA as a reference framework for introducing critical phenomena.
3.3
Mean-field universality class
Now we will discuss some consequences of this equation of state (3.7) in the
vicinity of TC . Since the results presented in this section (3.3) do not depend
on the value of the effective spin S, we will deduce some scaling relations for
the simplest case: S = 1/2. For this particular case, BS (α) = tanh(α) and
TC = z̄J/4kB . Additionally, we assume that S = 1/2 refers to the spin of an
electron4 so that g = 2. Under these hypotheses, Eqs. (3.8) can be written
in the compact form
µB B T c
σ = tanh
+ σ .
(3.16)
kB T
T
By using the fact that tanh(α) ≃ α − 31 α3 for α ≃ 0, Eq. (3.16) can be
expanded for small σ and B as follows:
µB B T c
1
σ=
+ σ − σ 3 + O(Bσ 2 )
(3.17)
kB T
T
3
which, for T ≃ TC and neglecting higher infinitesimal than σ 3 , becomes a
polynomial of the reduced temperature τ = (T − TC )/TC :
1
1
µB B
T − TC
σ + σ3 = τ σ + σ3 .
=
(3.18)
kB T
TC
3
3
Equation (3.18) is suitable for deriving some critical exponents.
Mean-field critical exponents
1. Setting B = 0, one has
4
T − Tc
Tc
1
σ + σ3 = 0
3
In fact, any system whose ground state is two-fold degenerate with a degeneration that
can be removed by the application of an external field can be thought of as possessing an
effective spin 1/2, with g 6= 2 in general.
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
whose solutions are


σ=0


β MF

√
1
T
σ ≃ 3 1−
∝ (−τ ) 2

T


| {z c }

46
for T ≥ Tc
for T < Tc .
:=−τ
This result provides the value of the critical exponent β within the
mean-field approximation: β MF = 1/2.
2. Now we want to evaluate the behavior of the susceptibility around TC .
First, let us recall the proportionality relation
∂M ∂σ
χ(T, B = 0) =
.
∼
∂B B=0 ∂B
Then the derivative ∂σ/∂B can easily be put in relationship with the
reduced temperature and σ by differentiating both sides of Eq. (3.18):
∂σ
∂σ
T
µB
·
+ σ2
∼− 1−
kB TC
TC
∂B
∂B
Since the relevant infinitesimal quantity is the reduced temperature τ ,
we have identified T = TC on the left-hand side of the equation above.
For T > TC , we can further neglect the term containing σ 2 so that
∂σ
1
∼ µB
∂B
T − TC
⇒
χ(T ) ≃
C
T − TC
for T > TC . (3.19)
This is nothing but the Curie-Weiss law deduced in an alternative way
in Eq. (4.49). For T > TC , instead, we have to take into account that
σ 2 ≃ −3τ . In this case one has
∂σ
µB
1
∼
∂B
2 TC − T
⇒
χ(T ) ≃
C
1
2 TC − T
for T < TC .(3.20)
The Eqs. (3.19) and (3.20) give the mean-field prediction for another
critical exponent: γ = 1.
Critical exponents in general
The fact that these observables behave like powers of the reduced temperature τ close to the transition point is not an artifact of the MFA. On the
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
47
contrary, this feature defines the condition of criticality. Other critical exponents can be deduced similarly to β and γ. Below we recall the definition of
some of them. Letting τ = (T − TC )/TC be the reduced temperature, α, β,
γ and δ critical exponents are defined as follows:
C(τ, B = 0) ∼ |τ |−α
M (τ, B = 0) ∼ (−τ )β ,
χ(B = 0, τ ) ∼ |τ |−γ
τ <0
|M (τ = 0, B)| ∼ |B|1/δ .
Another important feature captured by the MFA is that critical exponents
do not depend on the details of the model, e.g., on J and the details of
the lattice (dimensionality or z̄). However, this universality of mean-field
critical exponents is somewhat exaggerated: for instance the correct critical
exponents do depend on the dimensionality of the lattice.
In the following table the critical exponents obtained within the MFA
(classical values of the critical exponents) are compared with the numerical
values obtained for the 3d Heisenberg model (see the following sections):
α
β
γ
δ
MFA
3d-Heisenberg
0 (Jump) −0.11 ± 0.006
0.5
0.365 ± 0.002
1.0
1.386 ± 0.004
3.0
4.46
Experimentally, critical exponents are independent of the values of Ms , S,
g, TC , J, etc. which are specific of a given magnetic material. They rather
depend on more general symmetries of the experimental system under investigation. The concept of universality class is associated with this property.
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
3.4
48
The Landau approach
There is no univocal way of introducing it nor the literature is consistent in
what it is meant by Landau approach to critical phenomena. The procedure
proposed here aims at highlighting how the MFA is a drastic simplification
of the more general theoretical-field approach based on functional integrals.
We will try to avoid confusion between the Landau free-energy functional
and the MF (Gibbs) free energy. Finally, we should be able to discuss some
qualitative arguments whose validity goes beyond the mean-field approach.
To avoid useless complications, we refer to S = 1/2 operators and assume
that only their z component enters the spin Hamiltonian:
1
H=− J
2
X
|n−n′ |=1
Ŝ z (n) Ŝ z (n′ ) + gµB B
X
Ŝ z (n) .
(3.21)
n
Later on, we will discuss the Ising Hamiltonian (3.21) in some more details.
For now we use only the fact that the corresponding energy levels can be
written in terms of two-valued classical variables σi = ±1:
H [{σ}] = −
X
1X
1
σi ,
σi Ji,j σj + gµB B
2 i,j
2
i
(3.22)
where σ = (σ1 , σ2 , . . . σN ) and Ji,j is a symmetric matrix describing the (exchange) coupling among different spins in the lattice5 . The partition function
associated with the Ising Hamiltonian reads
(
"
#)
X
1 X
(3.23)
σi Ji,j σj − h
σi
Z (B, T ) = T r exp β
{σ}
2 i,j
i
with β = 1/kB T and h = βgµB B/2. The summation over all the configurations {σ} can be performed analytically only for the one-dimensional lattice
(Ising chain) and in 2d for B = 0. Before proceeding, it is useful to make a
mathematical digression and recall the well-known Gaussian identity
r
Z
2π s2 /2κ
2 −sη
η
−κ
dη =
e 2
e
(3.24)
κ
ℜ
which can easily be obtained from the integral of a Gaussian by completing
the square at the exponent. With some more efforts, the above result can be
5
Referring to Hamiltonian (3.21) the non-zero terms of Ji,j equal J/4. However, the
matrix Ji,j can describe a more general coupling among spins.
49
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
generalized to N variables η = (η1 , η2 , . . . ηN ) ∈ ℜN :
s
(2π)N
1 T
1 T −1
dη1 , dη2 . . . exp − η Mη − s · η dηN =
exp
s M s .
2
det(M )
2
(3.25)
Apart from a constant pre-factor, the right-hand side of Eq. (3.25) becomes
equal to the term involving spin pairs in Eq. (3.23) when M−1
i,j = βJi,j and s =
σ are chosen. The partition function of the Ising model (in any dimension)
can, thus, be written as
Z
−η T J−1 η/2β −σ·η −σ·h
e
dηN
(3.26)
e
dη1 , dη2 . . . e
Z (B, T ) = N T r
Z
{σ}
where6 h = (h, h, . . . h) and N is a constant irrelevant for magnetic observables. The trace over the variables {σ} appearing in Eq. (3.26) can now be
performed analytically
)
(
X
− P (h+ηi )σi Tr e i
e−(h+ηi )σi = 2N Π cosh(h + ηi ) .
(3.27)
=Π
i
{σ}
σi =±1
i
In order to write the partition function in (3.26) in a more transparent way,
we make the linear change of variables η = βJφ:
!
Z
X
1
Z (B, T ) = N ′ dφ1 . . . exp − βφT Jφ 2N Π cosh h + β
Ji,j φj dφN .
i
2
j
(3.28)
The latter is usually expressed in a more compact form
Z
Z (B, T ) = D[φ] e−βL[{φ}]
where the symbol
R
D[φ] ∝
R
(3.29)
Π dφi stands for the functional integral and
i
"
!#
X
X
1X
ln cosh h + β
Ji,j φj
φi Ji,j φj −β −1
−β −1 N ln(2)
L {φ} =
2 i,j
i
j
(3.30)
for the Landau free-energy functional. In this representation Z has been
rewritten as a Gaussian average over the auxiliary fields φ of the partition
6
The vector h has this simple form because a uniform B has been assumed but – in
principle – different sites could experience different external fields.
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
50
function of a paramagnet, experiencing the external field plus the auxiliary
fields themselves. By doing this, we have somehow “traded” the original
spin-spin interaction with the coupling (in principle site-dependent) of each
spin with a set of auxiliary fields φj . Note that no assumption has been made
on such fields which can, thus, span all over ℜN .
From Eqs. (3.29) and (3.30) an implicit form for the averaged spin projections
hσi i can be deduced. This is obtained straightforwardly if we let the field h
be site dependent:
!
X
∂ ln(Z)
1 ∂Z
hσi i = −
(3.31)
=−
= −htanh h + β
Ji,j φj i{φ} ,
∂hi
Z ∂hi
j
where h. . . i{φ} stands for average over the auxiliary fields. We will come
back to this result in the following. Generally, performing the functional
integral in Eq. (3.29), i.e. tracing over the auxiliary fields {φ}, is far from
being trivial. The simplest approximation which can be made to evaluate Z
is replacing the functional integral by the maximum value of the integrand,
namely
Z
n
o
−βL[{φ}]
−βL[{φ}] MFA
= exp −βMin L {φ}
= Max e
Z (B, T ) = D[φ] e
{φ}
{φ}
(3.32)
which is known as saddle-point approximation. This is equivalent to the
mean-field approximation. In fact, by requiring ∂L/∂φi = 0 for φi = φ̄i , the
following equation is obtained
!
X
φ̄i = tanh h + β
Ji,j φ̄j .
(3.33)
j
As we are considering nearest-neighbor ferromagnetic exchange coupling, the
solution to the previous equation turns out to be independent of the site index
i, meaning that the field which minimizes the Landau free-energy functional
is spatially homogeneous. Consequently, Eq. (3.33) is equivalent to the MF
equation of state (3.16). It is worth remarking that, within this framework,
only
when
it is evaluated in its minimum the Landau free-energy functional
L {φ} acquires the meaning of Gibbs free energy:
MFA
F (B, T ) = −β −1 ln (Z) = Min L {φ} = L {φ̄} .
{φ}
(3.34)
The equivalence between the MFA and the saddle-point approximation
of the functional integral (3.32) allows establishing that the average of the
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
51
auxiliary fields φ is proportional to local magnetic moments. In other words,
the average φ̄ plays the role of the Weiss field BW (apart from constant
factors). As far as the critical behavior (T ≃ TC ) is concerned, it makes
sense to expand the Landau free-energy functional for small values of the
fields φ. To this aim we make use of the Taylor expansion
1
1
ln [cosh(x)] = x2 − x4 + O(x6 ) for
2
12
x ≃ 0.
(3.35)
After some algebra and taking the continuum limit φi → φ(x) one obtains
Z 1
b 2 λ 4 d
2
LGL [φ] =
(3.36)
J (∇φ) + φ + φ d x − β −1 N ln(2)
2
2
4
with
(
b = z̄J 1 − 41 β z̄J = z̄J 1 − TTC ≃ z̄Jτ
4
TC 3
kB TC ≃ 16
k T
λ = 13 β 3 12 z̄J = 16
3
T
3 B C
(3.37)
where we have used the fact that the MF Curie temperature is TC = z̄J/4kB
for a system of spins one-half (see Eq. (3.10)). (The subscript in LGL stands
for Ginzburg-Landau). Note that in both Eqs. (3.30) and (3.36) the paramagnetic limit L = −kB T N ln(2) is recovered at high temperature, when
βJ → 0 and φ → 0 (kB ln(2) being the entropy of an isolated spin one-half).
Limiting – for the time being – ourselves to homogeneous fields φ(x),
we can set the gradient term to zero. First, we remark that λ appearing
in Eq. (3.36) is always positive. On the contrary, τ can change its sign
originating two different free-energy landscapes. For τ > 0, the Landau
functional LGL has a minimum for φ = 0 only, which clearly corresponds
to the magnetically disordered phase. For τ < 0, the Landau functional
displays the typical Mexican-hat shape with two minima occurring at some
finite φ = ±φ̄ (see Fig. 3.2). These minima are degenerate in the absence
of an external field and correspond to the non-trivial
solution of the MF
equation of state for T < TC . When φ = φ̄, LGL {φ̄} acquires the meaning
of Gibbs free energy. Then, from the knowledge of the Landau free-energy
functional the MF critical exponent α related
to the specific heat can be
deduced. For T > TC (τ > 0), we have LGL {φ̄} = −kB T N ln(2) so that
2
2
the specific
heat C = −T ∂ LGL /∂T2 = 0. For T < TC (τ < 0), instead,
LGL {φ̄} = −kB T N ln(2) + z̄JO(τ ). Therefore the specific heat is finite
when TC is approached from lower temperatures. This discontinuity implies
that α = 0 within the MF theory.
Note that only even powers of φ appear in the functional LGL in Eq. (3.36).
This fact is not accidental and reflects the symmetry σn ↔ −σn intrinsic to
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
52
Figure 3.2: Sketch of the Landau free-energy functional: the landscape
changes from a Mexican-hat shape to a single-minimum function when passing from the ferromagnetic to the paramagnetic phase.
the problem. Landau developed his theory of phase transitions starting from
the idea that the effective free energy should be an analytic function of the
order parameter (which needs to be identified with φ in our approach), consistently with the requirements of symmetry of the considered problem. In
fact, all the critical properties (critical exponents, etc.) derived in the previous section could have been obtained just postulating the form of Eq. (3.36)
for the Landau free-energy functional. For the Ising model discussed here,
postulating a form for LGL was not necessary since we could carry out the
calculation from first principles (i.e., starting from Hamiltonian (3.22)). For
problems characterized by less trivial symmetries and, e.g., vectorial order
parameters, being able to write the Landau free-energy functional on the
basis of symmetry arguments alone is often very useful. Then, performing a
saddle-point approximation analogous to Eq. (3.32) one can normally deduce
MF critical exponents with little mathematical efforts. Such exponents are
named classical, or mean-field, critical exponents and depend only on the
symmetry of the problem reflected in the functional LGL . However, a functional built with the same symmetry criteria can be used as a starting point
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
53
for more sophisticated mathematical treatments, like the renormalizationgroup approach. A successful example is the theory of critical phenomena
for which Kenneth G. Wilson was awarded the Nobel Prize in 1982.
3.5
Classical spin models
If the conditions to have magnetic moments coupled ferromagnetically among
them are fulfilled in the ground state, MF theory predicts that magnetic order is retained up to some finite temperature TC . Above this temperature
ferromagnetism is lost. This scenario describes a phase transition which is
indeed observed in real ferromagnets. However, the Curie temperature given
by formula (3.10) is almost always an overestimate compared to the values
computed with more sophisticated methods or observed in experiment. This
is because fluctuations, neglected in MF theory, tend to have a disordering effect, and therefore suppress the true TC value. In sufficiently low dimensions,
this suppression can lead to total loss of magnetic order at any temperature.
We will discuss this phenomenon for the simplest collective spin model: the
Ising model.
From what discussed about the MF critical exponents in the previous section,
it should be clear that they only depend on the powers of the order parameter appearing in the Landau free-energy functional of Eq. (3.36). More
generally, MF critical exponents depend on the symmetry of the considered
problem but not, e.g., on the dimensionality of the lattice. The fact that
such exponents are independent of the dimensionality is another artifact of
the MF approximation. On the contrary, two facts remain true beyond the
MF approximation: i) critical exponents do not depend on some details of
the system such as the strength of interactions while ii) they do depend on
the symmetry of the considered problem.
In the following some of these issues will be clarified in the context of classical
spin models.
Spins with continuous symmetry
The substitution of the quantum-spin operators in Hamiltonian (3.3) by classical spins is somewhat justified in the limit S → ∞, that is when the relative spacing between levels inside each multiplet S(n) becomes smaller and
smaller. Moreover, when correlations among spins develop, cooperative effects create a sort of collective large spin which behaves classically. Then,
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
one has:
~
Ŝ(n) → S(n)
≡ S0 (sin θ cos ϕ, sin θ sin ϕ, cos θ)
54
(3.38)
where S02 = S (S + 1) (more often S0 = 1 is assumed and the term S02 =
S (S + 1) is re-absorbed into the definition of the other constants, J and g,
in Eq. (3.3)). The Hamiltonian given in Eq. (3.3) is modified into
X
X
1
~ ′ ) + gµB B
~
· S(n
S z (n) .
(3.39)
H=− J
S(n)
2
′
n
|n−n |=1
and, accordingly, the partition function becomes
Z
Z
Z
~
Z = dΩ1 dΩ2 . . . dΩN e−βH({S(n)}) ,
(3.40)
with dΩn = sin θn dθn dϕn being the solid-angle element of the spin located at
the site n.
In some cases, due to the symmetry of the problem, it is more realistic to
describe each spin with a two-component vector (in-plane), which can thus
be parameterized with just one angle
~
S(n)
≡ S0 (cos ϕ, sin ϕ) .
(3.41)
Vocabulary of classical models with continuous symmetry:
~
• three-component S(n)
≡ S0 (sin θ cos ϕ, sin θ sin ϕ, cos θ):
classical Heisenberg model
~
• two-component S(n)
≡ S0 (cos ϕ, sin ϕ):
classical planar or XY model.
The Landau free-energy functionals associated with these two types of classical spins are generally different, between them and from the one given in
Eq. (3.36) for the Ising model (see below). This means that the number of
components of the order parameter determines the Landau free-energy functional and eventually the critical behavior of a system. This is one of the
features characterizing a specific universality class.
Spins with discrete symmetry: the Ising model
When consistent with the symmetry of the problem, two-value classical spins,
S z , can be assumed:
X
X
1
S z (n) .
(3.42)
S z (n) S z (n′ ) + gµB B
H [{S z (n)}] = − J
2
′
n
|n−n |=1
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
55
The configurations of the classical Hamiltonian (3.42) correspond to the spectrum of eigenvalues of the quantum Hamiltonian
1
H=− J
2
X
Ŝ z (n) Ŝ z (n′ ) + gµB B
X
Ŝ z (n) ,
(3.43)
n
|n−n′ |=1
which is, indeed, diagonal on the basis
|ϕi i ≡ |σ1 σ2 . . . σN i
1
with Ŝ z |σn i = σn |σn i and σn = ±1 .
2
(3.44)
More often, the Ising model is introduced directly assuming the Hamiltonian
X
X
1
σn
H [{σn }] = − J ′
σn σn′ − h′
2
′
n
(3.45)
|n−n |=1
with classical variables σn = ±1. Note that the Hamiltonian (3.45) is equivalent to the one given in Eq. (3.22) in which the nearest-neighbor coupling is
expressed in a matrix form. The model (3.45) corresponds to the exact spectrum of eigenvalues relative to the Hamiltonian of coupled quantum spins
one-half given in Eq. (3.43) provided that:
(
J ′ = 14 J
(3.46)
h′ = − 21 gµB B .
However, the Ising model is applied in many different contexts rather than
magnetism, ranging from biophysics to social sciences.
Magnetic order and lattice dimensionality
Probably one of the most striking failure of MF theory is the prediction of
a magnetic phase transition for d=1. In fact, for one-dimensional systems
rigorous proofs exist which forbid the occurrence of a magnetically ordered
phase at finite temperature in the sole presence of short-range coupling between spins. But let us clarify first what is meant by lattice dimension in
this specific context. The dimension d corresponds to the number of directions along which the exchange coupling propagates indefinitely. In practice,
this dimension may also be different from the actual dimensionality of the
considered solid. If the latter is D, in general one has d≤D.
The lattice dimensionality d is another fundamental feature characterizing a
specific universality class.
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
56
Limiting ourselves – for now – to the Ising model, we start considering the
case d=1.
d=1
An Ising chain composed of N spins can be represented schematically as in
Fig. 3.3. Following an argument due to Landau, we evaluate the variation
E 2, S 2
E 1, S 1
-
J
2
J
DF = F2 - F1 = - kBT ln N
2
DE = E2 - E1 =
DS = S2 - S1 = kB ln N
Figure 3.3: Sketch representing the free energy difference between a uniform
state, with all the spins parallel to each other, and a configuration consisting
of two domains with opposite spin alignment (one domain wall).
of the free energy associated with the creation of a domain wall in a configuration with all the spins parallel to each other. Creating a domain wall
increases the exchange energy by a factor J/2. However, such a domain wall
may occupy N different positions in the spin chain, so that this set of configurations has an entropy ∼ kB ln(N ). The free-energy difference between
the two configurations sketched in Fig. 3.3 is given by
∆F =
J
− kB T ln(N ) .
2
Thus, splitting the ground state into domains is
(
convenient
if ln(N ) > 2kJB T ⇒ N > eJ/2kB T
inconvenient if ln(N ) < 2kJB T ⇒ N < eJ/2kB T .
(3.47)
(3.48)
The inequalities written above suggest an estimate of how many consecutive
aligned spins can be found at finite temperature in an Ising chain. In particular, when the thermodynamic limit N → ∞ is taken, one immediately
realizes that it is always convenient to split the system into groups of parallel spins (magnetic domains), i.e., ferromagnetism is destroyed at any finite
temperature.
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
57
d=2
A similar, but more rough argument can be given for d=2 as well. In this
case we should refer to the possibility of reversing a cluster of spins enclosed
in a perimeter of l lattice sites and embedded in a region of spins all pointing
in the same direction. We consider for simplicity a square lattice. The total
cost in terms of exchange energy is of the order ∼ lJ/2. To estimate the
entropy we can think of a self-avoiding random walk: at each step the walker
has at most three choices of which way to go, since it has to avoid itself.
Thus, we expect the number of closed loops corresponding to the perimeter
l to be of the order pl , with p < 3. As a result, the free-energy variation
associated with the flip of a cluster delimited by a perimeter l is roughly
∆F = lJ/2 − kB T l ln p. Therefore, for T < J/(2KB ln p) the ordered phase
should be stable against the formation of large domains of reversed spins.
This argument for the existence of a phase transition in the 2d Ising model
was first given, in more precise terms, by Peierls.
Rigorous results
The Ising model represents a particularly lucky case in which the heuristic
arguments given above can be checked by solving the problem analytically.
Even if we will not derive these results, it is useful to recall which crucial
steps should be followed to prove rigorously whether a model is consistent
with a phase with spontaneous magnetization (finite magnetization in zero
external field) for T 6= 0 or not. To this end, one has to compute:
1. the partition function
z
Z = T r e−βH[{S (n)}]
(3.49)
where the trace is obtained by letting each discrete variable take the
two possible values S z (n) = ±1/2 (Z is a sum with 2N terms!)
2. the average magnetic moment
m(T, B) = −
1 ∂F
1 1 ∂ ln Z
=
N ∂B
N β ∂B
(3.50)
3. the limit
m(T, 0) = lim+ m(T, B)
B→0
(3.51)
and evaluate if there exists a temperature TC below which the
limit (3.51) takes a non-zero value.
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
58
This procedure can be carried out analytically for d=1 or d=2 only, producing
different results:
• for d=1, no spontaneous magnetization is possible at finite temperatures
• for d=2, a spontaneous magnetization appears for T < TC ≃ 2.27 4kJB .
Indeed these exact results show that the MF approximation overlooks some
important features as it predicts the occurrence of a phase with spontaneous
magnetization independently of the dimension d.
d=1: spin chains with uniaxial anisotropy
To fix the ideas, we take S = 1/2. As anticipated, in this case
#
"
1 X z
m(T, 0) = lim+ m(T, B) = −gµB lim+ lim
hS (n)i = 0 . (3.52)
B→0
B→0 N →∞ N
n
For the 1d case, two-spin correlations can also be computed:
r
1
βJ
1
z z
hSi Si+r i = tanh
= e−r/ξ
4
4
4
with
ξ=−
1
ln tanh
βJ
4
.
(3.53)
(3.54)
ξ is called correlation length and it is a fundamental quantity in the study
of critical phenomena. The correlation length of the 1d Ising model is characterized by an exponential divergence at low temperatures:
ξ ∼ eJ/2kB T .
(3.55)
By comparing the inequalities in Eq. (3.48) with the formula for the correlation length it is clear that ξ gives the order of magnitude of the average
size of groups of correlated spins. The existence of such a correlation, marks
a major difference between a 1d system of coupled spins and a paramagnet.
This is evidenced by the differential susceptibility at B = 0:
χ(T, B = 0) =
In practical cases
∂m
ξ
∼
∂B
T
(3.56)
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
59
Figure 3.4: Example of a molecular spin chain with uniaxial anisotropy which
– in a proper range of temperature – behaves as a 1d Ising chian. C. Coulon et
al., Physical Review B 69 p.132408 (2004). At low temperature, χ T saturates
because of the presence of non-magnetic impurities and 3d interactions with
the other spin chains in the crystal.
• the plot of 1/χ versus T highlights deviations from the paramagnetic
behavior (Curie-Weiss law)
• the plot of ln [χ T ] versus 1/T highlights an 1d Ising-like behavior (when
experimental points at low temperature lie on a line).
The behavior of two-spin correlations for the 1d Ising model is plotted in
Fig. 3.5.
d=2: ultrathin magnetic films with uniaxial anisotropy
The 2d Ising model was solved for the first time by Lars Onsager in 1944.
Such a solution is a “veritable mathematical tour de force” (M. Le Bellac). To
our purposes, it is enough to recall the formula which gives the spontaneous
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
60
magnetization7 for T < TC :
#
"
1 X z
hS (n)i
m(T, 0) = lim+ m(T, B) = −gµB lim+ lim
B→0
B→0 N →∞ N
n
81
βJ
−4
= gµB 1 − sinh
2
and the definition of TC itself
J
=1 ⇒
sinh
2kB TC
TC =
J
J
2
√
≃ 2.27
.
4kB
1 + 2 4kB
(3.57)
(3.58)
As anticipated at the beginning of this Chapter, MF theory typically overestimates the transition temperature. The specific value reported in Eq. (3.58)
has to be compared with the MF Curie temperature given by Eq. (3.10) for
a spin one-half and for z̄ = 4 (square lattice): TCMF = J/kB .
Expanding the spontaneous magnetization m(0) close to TC yields
1
m(T, 0) ∼ (TC − T ) 8 .
(3.59)
Thus, for the 2d Ising model β = 1/8 at odds with the MF value β MF = 1/2.
1
4
MFA
Ising
Figure 3.5: Two-spin correlation function for the ferromagnetic Ising chain
(adapted from Quantum and Statistical Field Theory, M. Le Bellac).
7
We identify the magnetization with the average magnetic moment per magnetic atom
(or molecule), while in the SI (Système international d’unités) it is the average mangetic
moment per unit volume.
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
3.6
61
Correlation functions
Figure 3.6: Group of people in the “paramagnetic” (a) and in the “ferromagnetic” (b) phase. (Taken from L. J. de Jongh and A. R. Miedema Adv.
Phys. 50 p. 947-1170 (2001)).
In the following sections we will try to render more quantitative the effect of neglecting fluctuations in the mean-field approximation. A pictorial
idea of what happens when a system passes from the paramagnetic phase
to the ferromagnetic phase is sketched in Fig. 3.6. In the picture on the
left-hand side, people walk in the street without conditioning each other, like
magnetic moments do in the paramagnetic phase. In the picture on the righthand side, instead, a strong feed-back mechanism is present so that if one of
the individuals is attracted, e.g., by a window all the others are conditioned
and end up staring at the same thing. This situation can be assimilated to
spontaneous symmetry breaking occurring in a magnet below TC . However,
everyday experience offers also intermediate degrees of correlation in which
such a feed-back mechanism involves a limited number of people. Think,
for instance, of a road artist playing music in a subway station: the majority of people will be more concerned of not missing the train rather than
listening at his/her music. Nevertheless, there will still be a sort of shortrange correlation among the people whose train is not departing soon and
whose attention is captured by the musician. This last situation resembles
short-range correlations in magnetic systems.
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
62
Figure 3.7: Magnetic specific heats of the S=1/2 Ising model for d=1,2,3 and
within the MF approximation. d=2 is given by the Onsager solution for the
square lattice while d=3 is obtained by high-temperature series expansion
for a simple cubic lattice. All temperatures are expressed in units of the
corresponding MF transition temperature, θ, with z̄ = 2, 4, 6 for the d=1,2,3
respectively. (Taken from L. J. de Jongh and A. R. Miedema Adv. Phys. 50
p. 947-1170 (2001)).
Specific heat tail
As already noticed, the mean-field theory predicts a finite discontinuity in the
specific heat at the critical temperature. On the contrary, in experimental
systems showing a magnetic phase transition the specific heat diverges at TC .
This behavior is reproduced by more sophisticated models. In Fig. 3.7, the
behavior of the specific heat is plotted for the Ising model. There the MF
prediction is compared with exact results for d=1,2 and high-temperature
series expansion for d=3 (no exact results solution is available in this case
yet).
First, we notice that the true TC is lower than the MF value in each case.
In particular, TC shifts at lower temperature as the lattice dimensionality is
reduced, down to TC = 0 for the Ising spin chain (d=1). This is an indication
that the effect of thermal fluctuations becomes progressively more dramatic
as the lattice dimensionality is reduced.
Second, apart from the MF calculation, all the models show a hightemperature tail in the specific heat. Also this feature is more enhanced
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
63
the lower the lattice dimensionality is. Such tail is due to the presence of
short-range correlations above TC that are not taken into account in the MF
theory. Both calculations for d=2 and d=3 show the expected singularity at
TC , while the specific heat does not diverge at any temperature for d=1, not
even at T = 0.
The most natural way to characterize short-range correlations is by studying the behavior of the correlation function.
The fluctuation-response theorem
Here we show how two-spin correlations are related to the susceptibility.
From the definition of the magnetization itself, it follows that
1 ∂F
1 1 ∂ ln Z
=
N ∂B
N β ∂B
("
)
#
X
1 X z
1 1 1
Tr
−βgµB
=
S z (n) e−βH = −gµB
hS (n)i
NβZ
N n
n
m(T, B) = −
(3.60)
where the trace is taken over all the possible values of the N variables S z (n).
With the definition of Eq. (3.60), the magnetization equals the average magnetic moment. This value can be converted to any other unit to compare
with experiments (Bohr magneton per atom, emu/mol, A/m, etc.).
The susceptibility is the derivative of the magnetization with respect to the
applied field





X
∂m(T, B)
1 1
S z (n) · S z (n′ ) e−βH
χ(T, B) =
=
T r β(gµB )2


∂B
NZ
nn′
#
))2
(
("
X
1
1
S z (n) e−βH
− β
Tr
gµB
N
Z
n


!2
2
X
X
β(gµB ) 
S z (n) i − h
S z (n)i2  ,
=
h
N
n
n
(3.61)
2 P
P
z
S
(n)
where we have used the fact that
= nn′ S z (n) S z (n′ ).
n
Defining the correlation function as
Gnn′ = hS z (n)S z (n′ )i − hS z (n)i hS z (n′ )i ,
(3.62)
64
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
a relation between Gnn′ and the magnetic susceptibility can be deduced
χ(T, B) =
β(gµB )2 X
Gnn′ ,
N
nn′
(3.63)
also known as fluctuation-response theorem.
Remarkably, according to the definition of the correlation function given in
Figure 3.8: Qualitative behavior of the correlation function for T > TC (left)
and T < TC (right) assuming an exponential decay Gij ∼ exp (−rij /ξ) (taken
from Quantum and Statistical Field Theory, M. Le Bellac).
Eq. (3.62), when hS z (n′ )i 6= 0 (i.e. for T < TC or for B 6= 0) the assertion
that “two spins are uncorrelated” means hS z (n) S z (n′ )i = hS z (n)i hS z (n′ )i ∝
m2 . In other words, the correlation function Gnn′ only measures the degree
of short-range correlation. The term relating to long-range order as been
eliminated by subtracting hS z (n)i hS z (n′ )i (see Eq. (3.62)).
Susceptibility for the different magnetic phases at B = 0
2d-3d systems T ≃ TC
χ(T, 0)
χ∼
Γ±
|T −TC |γ
1d-system
χ∼
ξ
T
Paramagnet
χ=
C
T
From the previous table and from the theorem (3.63), it is clear that the
differential susceptibility is strictly related to the degree of correlation of
fluctuations at the considered temperature:
χ(T, B) =
β(gµB )2 X
Gnn′ .
N
nn′
(3.64)
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
65
Note that even for the ordered phase (occurring for d≥2) the susceptibility
diverges only at T = TC and tends to zero as T → 0 (in a perfectly ordered
system fluctuations are not allowed).
Fourier transform of correlations
The Fourier transform (G̃) of the correlation function can be accessed experimentally, for instance through neutron scattering. At T = TC , G̃(q) is found
to behave as
1
G̃(q) ∼ 2−η .
(3.65)
q
A simple argument based on dimensional analysis suggests that
Z
1
at T = TC .
G(r) ≃ G̃(q)dd q ⇒ G(r) ∼ d−2+η
r
(3.66)
From Eq. (3.66) we learn that at the critical point spatial correlations decay
with power law. In other words, the system is not characterized by any typical length scale and possesses the property of being self-similar at different
spatial scales8 . Such a scale invariance is the key ingredient of the theory of
critical phenomena and it is, indeed, strictly related to the divergence of the
correlation length for T → TC . Notice that the fluctuation-response theorem
may also be written as
χ(T, B) ∼ G̃(0) .
(3.67)
Since in practice η < 2 always, Eq. (3.65) implies that G̃(q = 0) diverges for
T → TC and so does the susceptibility.
In the next section we will see that within the Landau theory of critical
phenomena the Fourier transfrom of the correlation function is given by
G̃(q) ∼
q2
1
.
+ ξ −2
(3.68)
The inverse Fourier transform of Eq. (3.68) gives the asymptotic behavior
G(r) ∼
e−r/ξ
r(d−1)/2
for T far away from TC .
(3.69)
In this case the decay of the correlation function is characterized by the
typical length scale ξ, the correlation length. The different behavior of the
8
See the paper C. H. Back, et al., Nature 378, p. 597. The authors report on the
experimental check of the scaling hypothesis on a Fe film which behaves as the 2d Ising
model.
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
66
correlation function above and below TC and both in real and reciprocal
space is sketched in Figs. 3.8 and 3.9.
It is worth remarking that we are considering short-range correlations as the
unique source of broadening of the quasi-elastic peaks. Of course, experimentally this is not true and the correlation length can be deduced only after
removing the other sources of broadening such as experimental resolution,
etc.
Figure 3.9: Qualitative behavior of the correlation function in the real space
(b) and in the Fourier space (a). (Taken from Quantum and Statistical Field
Theory, M. Le Bellac).
3.7
Landau theory of correlations
Before deducing correlations within the Laundau theory, let us recall some
basic concepts that have been discussed some sections ago. First, we showed
that the calculation of the partition function of the Ising model (for any
dimension d) can be recasted into the following problem:
Z
Z (B, T ) = D[φ] e−βL[{φ}]
(3.70)
from which, in principle (but not always in practice!), the whole thermodynamics can be deduced. Two possible independent approximations can be
made to tackle the problem stated by Eq. (3.70):
1. the saddle-point approximation, which consists in evaluating the partition function only in the minimum of the functional L [φ] (see
Eq. (3.32));
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
67
2. a Taylor expansion of L [φ] itself for small auxiliary fields φ, which holds
in the critical region (i.e., for T ≃ TC ) and gives the Ginzburg-Landau
functional LGL [φ].
The condition 1. requires to set to zero the functional derivative of L [φ]
with respect to φ; this leads to a self-consistent equation for the average
magnetic moment that turns out to be equivalent to the MF equation of state,
for every T . As a consequence, by making both approximations 1. and 2.
the MF critical behavior can be studied. More concretely, one can start
from the Ginzburg-Landau functional in Eq. (3.36) (which we recall here for
convenience)
Z 1
b 2 λ 4 d
2
LGL [φ] =
(3.71)
J (∇φ) + φ + φ d x − β −1 N ln(2) ,
2
2
4
set the gradient term to zero (because we seek for spatially homogeneous
solutions) and minimize the integrand with respect to φ. This leads to the
equation
φ̄ bφ̄ + λφ̄3 = 0
(3.72)
whose solutions are given by
(
φ̄ = 0
φ̄2 = − λb
for τ > 0
for τ < 0 ,
(3.73)
with b ≃ z̄Jτ and λ ≃ 16kB TC /3.
To evaluate correlation functions we need to go slightly beyond the crude
saddle-point approximation (equivalent to the MFA). In practice, we allow
the field φ to deviate slightly from the MF solution obtained for τ < 0 (the
treatment for τ > 0 is analogous):
φ = φ̄ + δφ ,
(3.74)
δφ being a small, random field. The integrand (fGL ) of the Ginzburg-Landau
functional in Eq. (3.71) takes the form
b
λ
1
fGL [φ̄ + δφ] = J (∇δφ)2 + (φ̄ + δφ)2 + (φ̄ + δφ)4
2
2
4
1
b 6λ 2
2
= J (∇δφ) +
+ φ̄ δφ2 + fGL [φ̄] + (. . . )
2
2
4
1
= J (∇δφ)2 − b δφ2 + fGL [φ̄] + (. . . )
2
(3.75)
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
68
where (. . . ) stands for constants, O(δφ4 ) and odds terms in δφ which vanish
after spatial or thermal averaging. The first two terms in the last line of
Eq. (3.75) are associated with fluctuations around the MF solution φ̄. Thus,
at this level of approximation, the partition function reads:
Z
−βLGL [φ̄]
Z (B, T ) = e
+ D[δφ] e−βLfl [δφ]
(3.76)
where the first term on the right-hand side corresponds to the saddle-point
approximation while
Z 1
2
2
(3.77)
Lfl [δφ] =
J (∇δφ) − b (δφ) dd x .
2
is the functional associated with the fluctuation field δφ(x). The underlying
strategy of the mathematical passages described above aims at considering
corrections to the saddle-point approximation which are described by a sort
of quadratic Hamiltonian with respect to the fluctuation field. In fact, the
functional (3.77) is formally equivalent to the potential energy of set of coupled harmonic oscillators, described in the continuum formalism (remember
that b < 0 for τ < 0): Thermal averages of the fluctuation field δφ(x) can be
computed similarly to average displacements in a system of harmonic oscillators. The gradient term in Lfl [δφ] effectively couples the fluctuation fields
δφ(x) defined at different points in space, at different locations x. However,
the functional (3.77) can be decoupled (diagonalized) passing to the Fourier
space:
Z
1
1
2
2 d
˜
Jq
−
2b
|δφ(q)|
Lfl [δφ] =
d q.
(3.78)
d
(2π)
2
If we forget the parametric9 dependence on temperature of b, Eq. (3.78) has
the form of a quadratic Hamiltonian with respect to the independent degrees
˜
Now, equipartition theorem can be applied to get
of freedom δφ(q).
1
1
2
˜
Jq 2 − 2b h|δφ(q)|
ifl = kB T
2
2
⇒
2
˜
h|δφ(q)|
ifl =
kB T
,
Jq 2 − 2b
(3.79)
where the subscript reminds that h. . . ifl represents an average over the fluctuation field δφ.
Before proceeding, it is useful to establish a contact between the correlation
9
This implicit temperature dependence, which may look strange at first sight, comes
from having expanded LGL [φ] around its minimum: it is reasonable that coefficients of the
expansion contain information about the saddle point φ̄, corresponding to the minimum.
69
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
function defined in Eq. (3.62) and the fluctuation field δφ. In the formalism
of the present section we shall write the correlation function as
G(x, x′ ) = hσ(x)σ(x′ )i − hσ(x)i hσ(x′ )i
= h φ̄ + δφ(x) φ̄ + δφ(x′ ) ifl − φ̄2
= hδφ(x)δφ(x′ )ifl .
(3.80)
By comparing Eq. (3.79) with Eq. (3.80) we obtain the following result
G̃(q) =
kB T
.
Jq 2 − 2b
(3.81)
Note that this result corresponds to the case in which τ < 0 and hence b < 0.
For τ > 0, the same calculation would yield
G̃(q) =
kB T
.
Jq 2 + b
(3.82)
Summarizing, within the Landau theory, the correlation function takes the
Ornstein-Zernicke form:
G̃(q) =
1
kB T
,
2
J q + ξ −2
(3.83)
with ξ ∼ |τ |−1/2 . As discussed at the end of Section 3.6, ξ has the meaning
of correlation length. The corresponding classical critical exponent ξ ∼ |τ |−ν
is ν cl = 1/2.
CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE
70
Literature
• A. Aharoni, Introduction to the Theory of Ferromagnetism
Oxford University Press
(Chapter IV: Magnetization vs. Temperature)
• J. Cardy, Scaling and Renormalization in Statistical Physics
Cambridge University Press
(Chapter VI: Low dimensional systems)
• M. Le Bellac, Quantum and Statistical Field Theory
Oxford University Press
(Chapter I: Introduction to critical phenomena. Chapter II: Landau
theory)
• G. Morandi, F. Napoli, E. Ercolessi, Statistical Mechanics
World Scientific Singapore
(Chapter III: Spin Hamiltonians I: Classical)
• L. D. Landau and E. M. Lifshitz, Statistical Physics
Oxford Pergamon Press
• D. C. Mattis, The Theory of Magnetism II
Springer Series in Solid-State Science
(Advanced)
Chapter 4
Magnetic domains and domain
walls
4.1
Magnetic anisotropy
Let us go back to consider a single magnetic center. For the atom embedded
in a spherically symmetric environment Hund’s rules generally succeed
in predicting the observed magnetic moment. When this scenario holds, the
spin “points” with the same probability along any spatial direction in the
absence of an external magnetic field. Due to the reduced symmetry of the
surrounding, the situation is generally different for an atom in a solid. As
already seen in Part I, a first consequence is that magnetic moments are
generally smaller in solids with respect to those predicted by Hund’s rules
DS2
J
0
p
2
p
Figure 4.1: Schematic representation of the energy landscape associated with
a uniaxial-anisotropy term as a function of the polar coordinate θ of a classical
spin.
71
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
72
(remember the quenching of the angular momentum). Another implication
is that magnetic moments (effective spins) prefer to lie along some crystallographic directions. This tendency is taken into account by introducing a
magnetic anisotropy energy which is function of the effective-spin projections
along the crystallographic axes. The simplest anisotropy term that can be
considered in single-spin Hamiltonian is
HA = −D(Ŝ z )2 .
(4.1)
Notice that the symmetry Ŝ z → −Ŝ z is not broken by such an anisotropy
term. Additional terms which combine higher powers of the single-spin operators may arise according to the symmetry of the lattice in which the
magnetic atom is embedded (or according to the symmetry of the substrate
for adatoms). For example, in the case of a crystal lattice with cubic symmetry the first non-zero anisotropy term is a fourth-order combination of the
spin operators; thus in this case the term in Eq. (4.1) vanishes.
The physical mechanism which couples the spin degrees of freedom with the
spatial degrees of freedom is the spin-orbit interaction.
Magnetic anisotropy away from the bulk
Figure 4.2: Ab-initio calculation of the magnetic anisotropy energy, DS 2 ,
and the magnetic moment per Co atom on Pt(111). Values in brackets have
been computed with a different computational method. Remember that 1
meV ≃ 11.6 K (C. R. Physique 6 p. 75 (2005)).
As stated above, a crucial ingredient for magnetic anisotropy to arise is the
reduced symmetry of the surrounding, “seen” by a magnetic atom in a solid,
with respect to the spherical symmetry (Hund’s rules). It is not surprising
that a further increase of the anisotropy is observed when the symmetry
of the environment is further reduced. This happens, e.g., when magnetic
atoms are arranged in clusters (0d) or in 1d and 2d nanostructures. In Fig 4.2
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
73
Figure 4.3: Experimental results: magnetic anisotropy energy DS 2 (right,
(b)), and magnetic moment per Co atom on Pt(111) (left, (a)) C. R. Physique
6 p. 75 (2005).
theoretical predictions (from ab initio calculations) for different aggregates
of Co atoms on a Pt(111) surface are reported. Notice that when passing
from a single atom to five atoms the value of the magnetic anisotropy per
atom already decreases of one order of magnitude. The magnetic moment
per atom also decreases with increasing the number of atoms. This fact
is instead associated with the degree of hybridization of magnetic orbitals,
which becomes more and more significant when approaching the bulk limit.
The theoretical predictions of Fig 4.2 are in qualitative agreement with the
experimental results reported in Fig 4.3. Indeed, the fact that the magnetic
anisotropy increases up to a factor 103 when approaching the atomic scale is
a good trend in view of magneto-storage applications.
Classical approximation
If the operator in Eq. (4.1) is substituted by a classical spin the anisotropy
energy reads
HA = −DS 2 cos2 θ .
(4.2)
Depending on the sign of D, the energy (4.2) has either one minimum for
θ = π/2 (D < 0) or two minima for θ = 0, π (D > 0), which describes the two
physical situations
D<0
D>0
easy plane
easy axis / uniaxial .
(4.3)
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
74
For the easy-axis case, D > 0 (see Fig. 4.1), only few configurations around
θ = 0 or θ = π will be statistically relevant for kB T ≪ DS 2 . In other words,
the spin will spend about half of the time visiting configurations for which
θ ≃ 0 and half of the time around θ ≃ π. For kB T ≪ DS 2 , the escape rate
2
from each one of the two wells is ν = ν0 e−DS /kB T , so that the relaxation
time diverges exponentially as:
DS 2
τA ∼ e k B T .
(4.4)
This time represents the average time it takes the system to jump from one
minimum of Fig. 4.1 to the other.
4.2
Domain walls in the classical Heisenberg
model
In chapter 2 we gave a justification for the use of the Heisenberg exchange
interaction which is isotropic. If we add to the classical Heisenberg Hamiltonian (Eq. (3.3)) an anisotropy term like the one in Eq. (4.1) we get
1
H=− J
2
X
|n−n′ |=1
Ŝ(n) · Ŝ(n′ ) + gµB
X
n
~ · Ŝ(n) − D
B
X
(Ŝ z (n))2 .
(4.5)
n
When D becomes large with respect to |J|, the model described by Hamiltonian (4.5) can be replaced with the two following models
D
→ +∞
|J|
D
→ −∞
|J|
Ising model
(4.6)
XY / planar model .
Domain walls: discrete lattice
In the following, we consider the Hamiltonian (4.5)
taking:
• D > 0, uniaxial anisotropy
• J > 0, ferromagnetic exchange interaction (parallel alignment of
nearest-neighboring spins is favored).
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
75
The study of the model described by the Hamiltonian (4.5) can be significantly simplified by substituting the quantum spin operators Ŝ(n) with clas~
sical vectors S(n).
This simplification is justified by thinking that a sort of
“collective” spin can be associated with a group of spins coupled ferromagnetically (J > 0). Such groups can emerge in a magnetic system due to either
long-range order or short-range correlations. In the latter case, the correlation length needs to be large enough. In both situations the collective spin
can be so large that its quantum-mechanical character becomes negligible1 .
In the appropriate temperature regimes, the results are then the same if the
classical approximation is assumed at the level of single effective spins.
For many theoretical and applicative aspects of magnetism, domain walls,
i.e. the boundaries between regions with opposite magnetization, play a
crucial role. In particular, their structure and the energy associated with
1
Remarkably, in this sense the classical-spin approximation is more justified for low
temperatures than for high ones. In the paramagnetic limit (kB T /J >> 1) one has to
recover a behavior described by the Brillouin function, in which the quantum nature of
each spin is relevant (S 2 → S(S + 1)).
2.0
1.5
Broad DW
1.0
Sharp DW
0.5
0.01
0.1
1
10
Figure 4.4: One-wall energy in J units vs D/J: minimum energy solution of
the non-linear equation (4.10) computed numerically (solid line); continuum
limit solution (dashed line). Inset: spin profile vs lattice distance: sharp wall
(low-right) and broad wall for D/J = 10−2 (up-left).
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
76
the creation of a domain wall in a uniformly magnetized configuration are
relevant. These features can be evaluated letting the spin direction vary
only along one spatial direction. This effectively reduces the problem to a
mono-dimensional one:
Nx h
i
X
~i · S
~i+1 + D (S z )2 ,
HH = −
(4.7)
JS
i
i=1
~i are classical spins and the constants J and D have to be thought of
where S
per unit length or per unit surface if the dimensionality of the original lattice
was d=2 or d=3 respectively.
With the Hamiltonian (4.7), the domain wall can be larger than one lattice
spacing. In fact, spreading the wall over more than one lattice spacing reduces
the global exchange-energy cost. On the other hand, the anisotropy term
would favor configurations with as less spins misaligned to the easy axis, z,
as possible. The domain-wall profile results from the competition between
these two energies (two opposite limits are reported in the insets of Fig. 4.4).
The lowest-energy deviations from the uniform state can be parameterized
through the angle that each spin forms with the z axis, θ, as
EH =
Nx
X
i=1
J − J cos (θi+1 − θi ) + D sin2 θi .
(4.8)
The energy cost for creating a domain-wall in a uniformly magnetized configuration is given by the spin profile which fulfills the boundary conditions
(
θ1
=π
(4.9)
θN x = 0
and minimizes the energy (4.8) with respect to θi :
∂EH
D
(4.10)
= sin (θi − θi−1 ) − sin (θi+1 − θi ) + sin (2θi ) = 0 .
∂θi
J
Eq. (4.10) can be solved numerically and the solution provides the spin profile
with respect to which the energy (4.8) is stationary. The true lowest-energy
profile can be obtained comparing different solutions, among which the sharpwall profile (see lower-right inset of Fig. 4.4):
(
θi = π
for 1 ≤ i < N2x
(4.11)
θi = 0
for N2x ≤ i ≤ Nx
which is also a solution of (4.10). In Fig. 4.4 the resulting energy (solid line)
is compared with that obtained from a continuum limit calculation (dashed
line) – that we are going to present in the next paragraph – as a function of
the ratio D/J.
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
4.3
77
Continuum formalism
Referring to the classical version of Hamiltonian (3.3), we rewrite in a different way the exchange-interaction term:
XX
X
1
~ ′ ) = −J
~ + eµ )
~
~
(4.12)
Hexch = − J
· S(n
· S(n
S(n)
S(n)
2
′
n
µ
|n−n |=1
where µ = x, y, z (spatial directions) and eµ is the unit vector along µ. Notice
that
2 2
~
~ 2 ~
~
~
~ + eµ ) . (4.13)
· S(n
S(n) − S(n + eµ ) = S(n) + S(n + eµ ) − 2S(n)
~
With the hypothesis that the direction along which each classical spin S(n)
is pointing varies smoothly from one lattice site to the other (index n), one
~
can describe S(n)
as a vector field which is a smooth function of a continuum
spatial variable r = an, a being the lattice spacing. This approximation is
justified
• In the classical isotropic Heisenberg chain (D = 0) at low temperatures. In fact, the lowest lying excitations – which actually destroy
ferromagnetism for d≤2 – are spin-waves with very long wavelength
(in the following we will show that for small wave vectors q → 0 the
spectrum of fluctuations is gapless).
• When the walls separating domains with opposite spin directions are
broad enough. Further on, we will render this statement quantitative.
In the presence of ferromagnetic (J > 0) exchange interaction and uniaxial anisotropy (D > 0), such a requirement is fulfilled for J ≫ D.
Thus one has,
~
~
~ + eµ ) − S(n)
~ + aeµ ) − S(r)
~
S(n
≃ S(r
≃ a∂µ S(r)
(4.14)
where in the first passage we have taken the continuum limit and in the
second one we have performed a Taylor expansion. Combining Eq. (4.13)
with Eq. (4.14), the exchange interaction between the spin located in r and
half of its nearest neighbors is obtained
X
~ + eµ )
~
· S(n
−J
S(n)
µ
2
X 1 2 X ~ 2
~
≃ Ja
∂µ S(r) − J
S(r)
2
µ
µ
X
z̄ X ν
1
(∂µ S ν (r) · ∂µ S ν (r)) − J
(S (r) · S ν (r))
= Ja2
2
2
µ,ν
ν
(4.15)
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
78
with z̄ number of nearest neighbors and ν = x, y, z label of the spin components. Normally, the first term is written as
~ 2 X
∇
S(r)
(∂µ S ν (r) · ∂µ S ν (r)) =
=
µ,ν
2
= (∂x S (r)) + (∂y S x (r))2 + (∂z S x (r))2
x
(4.16)
+ (∂x S y (r))2 + (∂y S y (r))2 + (∂z S y (r))2
+ (∂x S z (r))2 + (∂y S z (r))2 + (∂z S z (r))2 .
Taking the usual continuum limit for the sum
Z
X
1
· · · ≃ d . . . dd x ,
a
(4.17)
the classical and continuum version of Hamiltonian Eq. (4.5) is finally
obtained
Z Z
z̄ 1
1 2−d ~ 2 d
~ 2 d
d
x
−
J
S(r)
∇
S(r)
H = Ja
d x
2
2 ad
Z
Z
(4.18)
1
1
2 d
d
z
~
~
− D d |S (r)| d x + µB g d B · S(r)d x .
a
a
~
Within the continuum model the field S(r)
can be simplified as a twocomponent vector field or as a scalar field (see the two limits (4.6)). However,
some additional constraints or effective energy terms are normally
introduced
~ 2
in place of the stringent constraint on the spin modulus S(r) = S 2 . Of
course, the latter condition is automatically fulfilled if each spin is parameterized with polar coordinates

x

 S (r) = S sin(θ(r)) cos(ϕ(r))
(4.19)
S y (r) = S sin(θ(r)) sin(ϕ(r))


S z (r) = S cos(θ(r)) .
Broad domain walls: continuum limit
To the aim of computing the domain-wall energy in the continuum limit, we
let the polar angles (4.19) be a function of one spatial variable only, say x.
For B = 0, the Hamiltonian (4.18) can then be written as
2 #
Z " 2
dθ
dϕ
1
+ sin2 (θ(x))
dx
H = JNy Nz aS 2
2
dx
dx
(4.20)
Z
2
21
cos (θ(x))dx + const
− DNy Nz S
a
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
79
where we have implicitly assumed the integration domain to be a parallelepiped Nx Ny Nz a3 . The functional Eq. (4.20) can be minimized with respect to the functions θ(x) and ϕ(x). The corresponding Euler-Lagrange
equation is
(
dϕ 2
dθ
Ja sin2 (θ(x)) ddxϕ2 + 2Ja sin(θ(x)) cos(θ(x)) dx
=0
dx
dϕ 2
d2 θ
D
Ja dx2 − Ja sin(θ(x)) cos(θ(x)) dx − 2 a sin(θ(x)) cos(θ(x)) = 0
(4.21)
The solution to Eq. (4.21) with boundary conditions
(
limx→−∞ θ(x) = π
(4.22)
limx→+∞ θ(x) = 0
and corresponding to the minimum energy is
(
cos(θ(x)) = tanh xδ
ϕ(x)
= const.
(4.23)
q
J
with δ = a 2D
. Such a solution was proposed by Landau and Lifshitz in
1935.
√
√
The energy density associated with the spin profile (4.23) is Ew = 2 2S 2 DJ
(per unit length for d=2 and per unit surface for d=3). In Fig. 4.4 the
domain-wall energy obtained numerically for the discrete-lattice calculation
(solid line) is compared with that obtained in the continuum limit (dashed
line) as a function of the ratio D/J. The agreement is already good for ratios
D/J < 0.3. In the opposite limit, the discrete lattice calculation recovers the
domain-wall energy of the Ising model Ew = 2JS 2 (sharp domain wall defined
by Eqs. (4.11)).
In those relevant limits one has
(
J ≪D⇒δ=a
and Ew = 2S 2 J
q
√
√
(4.24)
J
J ≫ D ⇒ δ = a 2D
> 1 and Ew = 2 2S 2 DJ .
For J ≪ D, the wall-energy cost equals the Ising case and follows from having
δ = a. Concerning J ≫ D, Ew is one-soliton energy 2 . As one can appreciate
in Fig. 4.4, the two regimes are very well recovered and the transition region,
where none of the two limits (4.24) is expected to hold, is surprisingly narrow.
2
Sometimes the domain-wall
q
q width is defined with some numerical factors of difference
J
with respect to δ: π 2D
or 2J
D for instance.
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
80
The crossover between the sharp-wall (δ = a) and the broad-wall (δ > a)
regime3 occurs at D/J = 2/3.
In summary, the conditions (4.24) provide a criterion for the simplification
of the classical Heisenberg model in terms of
(
Ising model
D/J ≥ 2/3
(4.25)
continuum limit
D/J ≤ 0.3 .
Typically, metallic nanowires of technological relevance fall in the broad-wall
regime. In fact, materials like Co, Ni, Fe or Permalloy are characterized
by D ≃ 1 − 10 K (∼ 0.1 − 1 meV) and J ≃ 100 − 500 K (∼ 10 − 50 meV)
corresponding to a domain-wall width of the order 10 − 100 nm.
4.4
Beyond the Mean-Field Approximation
The argument used in chapter 3 to state that the Ising model does not show
a magnetically ordered phase at finite temperature for d=1 holds also for
the Heisenberg chain with uniaxial anisotropy, provided that the appropriate domain-wall energy is considered (remember Fig. 3.3). Similarly, one
concludes that the same model can sustain ferromagnetism at finite temperatures in d=2. Different arguments are, instead, needed to provide a
conclusive statement about the existence or not of magnetic order at finite
temperature in systems with continuous symmetry. For the last ones, it will
turn out that linear excitations are able to destroy ferromagnetism both in
d=1 and d=2. For the Heisenberg model, these linear excitations can be
identified with spin waves. Spin waves are usually introduced as linear solutions to the Landau-Lifshitz equation of motion. However, the capability
of these type of excitations to destroy magnetic ordering for d≤2 in systems
with continuous symmetry can be evidenced without the need of introducing
dynamics. We prefer to follow this way because it is straightforward to apply
a unique argument to both the XY and the Heisenberg model.
3
The crossover ratio D/J = 2/3 can be obtained analytically by analyzing the stability
of the sharp-wall profile, Eqs. (4.11), against small deviations between successive angles
θi (B. Barbara, Journal de Physique 34, p. 139 (1973)).
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
81
Linear excitations in models with continuous symmetry
We rewrite for convenience the classical spin Hamiltonian introduced in chapter 3, Eq. (3.39):
1
H=− J
2
X
|n−n′ |=1
~ ′ ) + gµB B
~
· S(n
S(n)
X
S z (n) .
(4.26)
n
The minimal energy is obtained by letting all the magnetic moments be
aligned along the direction of the applied field (spins along negative z direction). We consider how the energy increases due to small deviations from
this configuration. Our goal is to simplify the original problem by means of
an effective Hamiltonian that is formally equivalent to the one describing a
system of coupled harmonic oscillators. To this end, we may write
s
X
1 X α
z
S (n) = − 1 −
(S α (n))2 ≃ −1 +
(S (n))2
2
(4.27)
α=x,y
α=x,y
with the hypothesis
(S α (n))2 ≪ (S z (n))2 .
Note that for the planar (or XY) model α takes just one value and two values
for the Heisenberg model. From now on, we will not specify the number of
extra components but z represented by the index α; while doing so, we are
going to derive results that apply to both models. The approximation in
Eq. (4.27) reflects in the Hamiltonian as follows:
"
# "
#
X
X ′
1X α
1
1
(S α (n′ ))2
1−
H≃− J
(S (n))2 × 1 −
2
2
2
α
α′
|n−n′ |=1
#
"
X
X
X
X
1
1
1−
S α (n)S α (n′ ) − gµB B
(S α (n))2
− J
2
2 α
n
|n−n′ |=1 α


X
X
X
X
1 1
1
′
(S α (n))2 +
(S α (n′ ))2 
= − z̄N J − gµB BN + z̄J 
2
2 2 n α
n ′ α′
1
− J
2
X X
|n−n′ |=1 α
≃Eg.s. + Hh.o.
XX
1
S α (n)S α (n′ ) + gµB B
(S α (n))2 + O (S α )4
2
n
α
(4.28)
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
82
P P
P P
where, by nothing that the double summations
α′ are
n
α and
n′
actually the same, we have defined
X X
1 XX α
1
Hh.o. = z̄J
S α (n)S α (n′ )
(S (n))2 − J
2
2
α
n
|n−n′ |=1 α
(4.29)
XX
1
+ gµB B
(S α (n))2
2
n
α
and the constant ground-state energy
1
Eg.s. = − z̄N J − gµB BN .
(4.30)
2
The Hamiltonian Hh.o. , written in Eq. (4.29), is equivalent to the Hamiltonian of N coupled harmonic oscillators which can be decoupled by the usual
Fourier transform in the discrete space:
(
P
S α (n) = √1N q S̃ α (q) e−iq·n
(4.31)
P
S̃ α (q) = √1N n S α (n) eiq·n
with orthogonality relation
X
′
ei(q−q )·n = N δq,q′ .
(4.32)
n
For simplicity we assume unitary lattice constant. It is convenient to evaluate
the two relevant summations appearing in the Hamiltonian of Eq. (4.29)
separately. The first summation reads
X
X
1 XX α
′
S̃ (q)S̃ α (q ′ ) e−i(q+q )·n =
(S α (n))2 =
|S̃ α (q)|2 .
(4.33)
N n q,q′
n
q
This is nothing but the Parseval’s formula for the discrete-lattice Fourier
transform. For what concerns the second summation on the right-hand side
of Eq. (4.29), we first rewrite it as
X
XX
S α (n)S α (n′ ) =
S α (n)S α (n + δ)
(4.34)
n
|n−n′ |=1
δ
where δ is a vector connecting the site n with its nearest neighbors. For
simplicity, we will consider just a linear, square and simple-cubic lattice for
d=1, 2 and 3, respectively. Passing to the Fourier space one finds
XX
XX 1 X
′
′
S̃ α (q)S̃ α (q ′ ) e−i(q+q )·n e−iq ·δ
S α (n)S α (n + δ) =
N q,q′
n
n
δ
δ
(4.35)
X
XX
X
α
2 −iq·δ
α
2
=
|S̃ (q)| e
=
|S̃ (q)|
2 cos(q · δ) ;
δ
q
q
{δ>0}
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
83
the notation {δ > 0} means that the summation extends over half of the
nearest neighbors of the spin located at site n: it consists of z̄/2 terms.
Eqs. (4.33) and (4.35) enable us to decouple the elastic Hamiltonian given
in Eq. (4.29), which then reads


X
X
X
1
z̄ −
Hh.o. = J
2 cos(q · δ) |S̃ α (q)|2
2 q α
1
+ gµB B
2
=
{δ>0}
XX
q
α
|S̃ α (q)|2
(4.36)
1 XX
Γ(q)|S̃ α (q)|2 ,
2 α q
with
Γ(q) = J[z̄ −
X
{δ>0}
2 cos(q · δ)] + gµB B .
(4.37)
Figure 4.5: Sketch of a spin-wave excitation in a Heisenberg ferromagnetic
spin chain.
Indeed, for the Heisenberg model, the linear excitations associated with
the quadratic Hamiltonian in Eq. (4.29) are spin waves with dispersion relation ~ω(q) = Γ(q). Spin waves are collective excitations analogous to
phonons. Similarly to phonons, spin waves are also quantized and the specific dependence of Γ(q) on the wave vector (especially for q ≃ 0) determines
the behavior of the magnetization at low temperature (in the absence of
anisotropy). The dispersion curve Γ(q) can be measured, e.g., by inelastic
neutron scattering.
Coming back to our goal, we proceed by evaluating the average of fluctuations, namely those terms in Eq. (4.27) that we have assumed to be
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
84
small for linearizing the Hamiltonian (4.26). The approximated Hamiltonian, Eq. (4.36), consists of N independent quadratic degrees of freedom so
that equipartition theorem applies:
1
1
Γ(q) h|S̃ α (q)|2 ith = kB T
2
2
⇒
h|S̃ α (q)|2 ith =
kB T
;
Γ(q)
(4.38)
h. . . ith denotes thermal average performed using the Hamiltonian Hh.o. in
Eq. (4.36). Thermal averages of the squared transverse components in real
space read
1 X α
1 X α
′
h(S α (n))2 ith =
hS̃ (q)S̃ α (q ′ )ith e−i(q+q )·n =
h|S̃ (q)|2 ith ,
N q,q′
N q
(4.39)
where we have used the fact that transverse components fluctuate randomly
so that hS̃ α (q)S̃ α (q ′ )ith = δq,q′ h|S̃ α (q)|2 ith . Note that the right-hand side of
Eq. (4.39) is independent of the lattice site, thus the label n will be dropped
henceforth from h(S α (n))2 ith . In order to evaluate whether the considered
linear excitations are able or not to destroy ferromagnetism, we shall let the
field B → 0+ . First, we approximate the summation on the right-hand side
of Eq. (4.39) with an integral
Z d
kB T
d q
α 2
.
(4.40)
h(S ) i ≃
d
(2π)
Γ(q)
Since what matters is the behavior for small values of q (i.e., the effect of
fluctuations at large spatial scales), the denominator of the integral can be
linearized as
X
1
z̄ 1
(1− qµ2 )+gµB B = J z̄−2J( − q 2 )+gµB B = Jq 2 +gµB B
Γ(q) ≃ J z̄−2J
2
2 2
µ
2
with µ=1. . . d and q =
P
(4.41)
2
µ qµ ,
which yields
Z
dd q
kB T
α 2
.
h(S ) i ≃
(2π)d
Jq 2 + gµB B
(4.42)
When taking the limit B → 0+ , the integral in Eq. (4.42) has an infrared
divergence4 for d≤2. The consequences of such a divergence can be appreciated more effectively by setting a lower bond to the integral: qmin = π/Nα ,
4
A possible ultraviolet divergence does not matter i) because the lattice unit sets a
physical upper limit to large values of q ii) because we are interested in fluctuations acting
on large spatial scales corresponding to q ∼ 0.
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
85
with Nα being of the order of the linear size of the system in lattice units.
Depending on the dimensionality of the lattice we have


h(S α )2 i ∼ kBJ T Nα
Z
 d=1
d−1
k
T
q
dq
B
h(S α )2 i ∼
⇒
d=2
h(S α )2 i ∼ kBJ T ln(Nα ) (4.43)
2

J
q
qmin

d=3
h(S α )2 i < ∞
In order to understand what a divergence with increasing Nα means, it is
convenient to rephrase the mathematical steps that we followed according to
their physical sense:
• We assumed the system to be in a ferromagnetic state at T = 0, namely,
with all the spins aligned along the same direction.
• We let each spin deviate by a small amount from its direction of alignment, z.
• We built an effective linear Hamiltonian, describing these family of
small excitations, which can easily be decoupled passing to the Fourier
space.
• We calculated thermal averages of such small excitations (transverse
spin components) in the Fourier space.
• We transformed those averages back to the real space.
• We evaluated if the initial hypothesis stated in Eq. (4.27) remains valid
at finite temperature.
The set of Eqs. (4.43) allows stating that in the thermodynamic limit, Nα →
∞, the hypothesis of small deviations fails for d=1, 2 at any finite T . This
fact suggests that spontaneous magnetization is not stable against thermal
fluctuations for d≤2. On the contrary, according to Eqs. (4.43), it seems
possible to have ferromagnetism up to some finite temperature for d=3. This
scenario is indeed confirmed by more rigorous proofs such as the MerminWagner theorem.
At this point we are in the position to state that for both the isotropic
(D = 0) Heisenberg and XY classical model with short-range interactions
the lower critical dimension is d=2 (the highest dimensionality for which
magnetic order cannot occur at any finite temperature). This result marks a
major difference between the universality class of classical spin models with
continuous or discrete symmetry (Ising). As already noticed, in systems with
continuous symmetry the effects of thermal fluctuations are more severe and
manage to destroy ferromagnetism more easily.
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
86
The effect of uniaxial anisotropy
The presence of uniaxial anisotropy stabilizes a system against the linear
excitations considered above. Due to this interaction an additional term like
X
Hm.a. = −D
(S z (n))2
(4.44)
n
appears in the Hamiltonian in Eq. (4.26). The anisotropy energy indeed
favors configurations in which spins lie along a specific axis regardless of the
sign of their projections. In this case we choose theP
same axis as the one along
which the field is applied. Because of the equality α (S α (n))2 +(S z (n))2 = 1,
the Hamiltonian (4.44) may also be written as
XX
(S α (n))2 .
Hm.a. = −D + D
(4.45)
n
α
The summation appearing above transforms into the Fourier space according
to the Parseval’s formula in Eq. (4.33). Finally, the uniaxial anisotropy
provides a term into the energy spectrum Γ(q) formally equivalent to the
magnetic field:
X
Γ(q) = J[z̄ −
2 cos(q · δ)] + gµB B + 2D .
(4.46)
{δ>0}
For small values of q and B = 0, we get
Γ(q) ≃ Jq 2 + 2D .
The average of transverse fluctuations is modified as follows
Z
Z
kB T
q d−1 dq
dd q
α 2
h(S ) i ≃
∼
k
T
.
B
(2π)d
Jq 2 + 2D
Jq 2 + 2D
(4.47)
(4.48)
Henceforth, let us refer only to the thermodynamic limit Nα → ∞, consistent
with qmin = 0. Clearly, the introduction of uniaxial anisotropy removes the
infrared divergence4 from the average of fluctuations independently of the
lattice dimensionality. The consequences of this result have to be understood
as follows: “The considered linear excitations alone are not able to destroy
ferromagnetism at any finite temperature”. This statement does not exclude:
1. that ferromagnetism may be destroyed by some other type of excitations
2. that these linear excitations play any role in the “suppression” of ferromagnetism at finite temperature.
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
87
An obvious counterexample, supporting the comment 1., is represented by
the 1d Ising model. The latter can be considered as a limit case of the Heisenberg model with uniaxial anisotropy (the one we are discussing about here)
for D ≫ J. For d=1, as for any d, the integral in Eq. (4.48) is convergent
so that linear excitations are not able to destroy ferromagnetism for every
T 6= 0. However, we have seen that domain walls manage to destroy ferromagnetism at any temperature in the 1d Ising model. In conclusion, even if
the absence of an infrared divergence in the integral on the right-hand side of
Eq. (4.48) would allow for ferromagnetism at finite T , we know that a phase
with spontaneous magnetization does not occur.
Limitations of the mean-field approximation
To conclude this part about the critical aspects of magnetism at finite temperature we summarize the artifacts produced by the MFA around the critical
region, T ≃ TC .
1. The MFA predicts a the occurrence of a magnetic phase transition at
finite temperature independently of the lattice dimensionality, d. However, a phase with spontaneous magnetization is not encountered in
the 1d Ising model. The Mermin-Wagner theorem forbids the occurrence of spontaneous magnetization (spontaneous symmetry breaking)
in classical models with short-range interactions and with continuous
symmetry for d≤ 2. This fundamental theorem applies to both the
Heisenberg and the XY model.
2. The transition temperature is generally overestimated within the MFA.
3. The classical values of the critical exponents, i.e., those given by the
MFA, are generally not correct even when a phase with spontaneous
magnetization exists at finite temperature. Depending on the model,
classical critical exponents are wrong for d larger than the lower critical
dimension, dl , and smaller that the upper critical dimension (du =4 for
systems with short-range interactions). For the Ising model dl =1, while
it is dl =2 for the Heisenberg and XY models.
4. The classical critical exponents turn our to be exact for d≥4 in systems
with short-range interactions (Ginzburg criterion). Strictly speaking, only the critical exponents are exact for d≥4. However, the MFA
is expected to give a more appropriate description of finite-temperature
properties for a given model when the number of spins with which each
spin interacts increases. This number increases with increasing d or
88
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
with increasing the range of interactions. The best realization of the
MFA is an ideal case in which every spin interacts with any other one
with the same intensity. This is – of course – unrealistic when magnetic
systems are considered. For the model
H=−
J X z
Ŝ (n) · Ŝ z (n′ ) ,
N n6=n′
(4.49)
with the summation extended over all the different couples, the Mean
Field Approximation is exact. The model described by the Hamiltonian
in Eq. (4.49) is called Curie-Weiss model but sometimes also “meanfield” model. However, one should not confuse this model with the
Mean Field Approximation which does not assume from the very beginning an all-to-all interaction, like in the Hamiltonian of Eq. (4.49)5 .
4.5
Finite size and superparamagnetic limit
Figure 3.3 and the following discussion about the absence of ferromagnetism
at finite temperature in the 1d Ising model represent the analogous of the arguments given in the present chapter for systems with continuous symmetry,
Eqs. (4.43). For the 1d Ising model, through the inequalities in Eq. (3.48),
we commented that for small enough system sizes ferromagnetism – possibly
present at T = 0 – is stable against thermal fluctuations. Indeed, Eqs. (4.43)
allow drawing similar conclusions for system with continuous symmetry: both
for d=1 and d=2 ferromagnetism is not destroyed at finite temperature if the
system is small enough. Under this condition, the averages of transverse spin
components do not necessarily diverge and the inequality h(S α )2 i ≪ h(S z )2 i
may be fulfilled.
Bistability is a crucial property for most of the applications of nanosized
magnets (nanomagnetism). Thus an important question to be addressed is:
“what do we understand for bistability when dealing with a real nanomagnet?” Rephrasing what we have just stated about small enough systems, we
can answer that when a magnetic lattice does not extend indefinitely correlations – either of short- or long-range nature – may always develop; the system
as a whole then behaves like a giant classical spin. In the presence of uniaxial anisotropy, similar arguments as for a single classical spin (macrospin),
described by Eq. (4.2), then apply. In particular, the most relevant quantity
is the average escape rate from the minima of the total anisotropy energy, located at θ = 0, π (see Fig. 4.1). One possible way to pass from one of the two
5
The exchange interaction is divided by N to guaranty extensivity of the energy.
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
89
configurations to the other one is a rigid (coherent) rotation of all the spins6 .
The relaxation time for such a mechanism is nothing but a generalization of
Eq. (4.4):
vDS 2
τ ∼ e kB T ,
(4.50)
with v = Nx Ny Nz . Often the magnetic anisotropy is defined per unit volume
Kv = D/a3 so that the usual volume V = a3Nx Ny Nz can be used in Eq. (4.50).
Due to the exponential dependence on the system size, the characteristic time
given in Eq. (4.50) can become very large even for nanomagnets. Referring
again to Fig. 4.1, assume to magnetize the system by means of an external
field, thus lowering the energy of one of the two minima corresponding to
opposite magnetization. This essentially allows preparing the system in a
chosen state. Then remove the external field. Now, due to the exponential
divergence of Eq. (4.50), the system may behave as if it had undergone a
magnetic phase transition, i.e. it may show a remanent magnetization.
But such a situation corresponds to
• a metastable state
• which is not an equilibrium state (one cannot associate to it a free
energy F in the same meaning as, e.g., in chapter 3).
• If one could wait long enough, t ≫ τ , an average zero magnetization
would be obtained (in the absence of an external field).
• A similar scenario is recovered irrespectively of the dimensionality of
the lattice d as far as ξ ≫ Nν for all ν = x, y, z.
Superparamagnetic limit
For magnetic memory manufacturing, the quest to increase the density of
data storage calls for reducing the linear dimensions of nanomagents. Even if
reducing the linear dimensions of a magnetic unit prevents the occurrence of a
magnetic phase transition, one can just require that bistability holds for “long
enough”. The required time over which one can reasonably assume that a
nanomagnet remains in the desired metastable state depends on the practical
application it is supposed to be used for. But, according to Eq. (4.50)
and more general approaches, the relaxation time decreases when reducing
the linear size of a nanomagnet. As a consequence, when the total volume
becomes too small bistability is lost. This intrinsic constraint to the linear
dimensions of a bistable magnetic unit is called superparamagnetic limit.
6
For mesoscopic systems, processes which involve non-uniform magnetization reversal
may be more convenient.
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
4.6
90
Dipolar interaction
Assume that we are considering a material which fulfills all the groundstate requirements to give rise to ferromagnetism, as defined in the first
two chapters and summarized at the beginning of chapter 3. Assume that
ferromagnetism has also “survived” at finite temperatures, meaning that
there is a phase with spontaneous magnetization, which requires:
• the dimensionality of the magnetic lattice be d≥2 for systems with
uniaxial anisotropy
• the dimensionality of the magnetic lattice be d=3 for systems with
continuous symmetry (Heisenberg or XY).
In the former case, ferromagnetism is destroyed by thermally excited domain
walls for T > TC (with TC = 0 for d=1).
In the latter case, ferromagnetism is destroyed by thermally excited spin
waves (or linear excitations in general) for T > TC (with TC = 0 for d=1,
2).
Is there another mechanism which can destroy or frustrate the “surviving”
ferromagnetism? The answer is “yes”. In fact, the dipolar interaction of
magnetostatic origin, neglected so far but always present, may play such a
role.
Magnetostatic dipole-dipole interaction
So far we have not considered the contribution of due to the magnetostatic
dipole-dipole interaction, which arises directly from Maxwell equations:
µ0 ~µ1 · ~µ2
(~µ1 · r12 ) (~µ2 · r12 )
Hdd =
−3
.
(4.51)
3
5
4π
r12
r12
For our purposes, the pointlike dipoles are the magnetic moments of each
~i (i = 1, 2)7 and
magnetic atom in the classical approximation ~µi = −gµB S
r12 = r1 − r2 . Since thermal effects may be considered at different levels of
approximation, we prefer to distinguish the ground-state magnetic moment
~i from its thermal average m
~µi = −gµB S
~ i = h~µi ith introduced in the previous
chapter. The typical strength of the dipole-dipole energy is generally small
compared to the exchange energy. However, its characteristics impose to
handle the dipolar interaction with extreme caution.
7
For other applications µ
~ i could be, e.g., the electron or the nucleus magnetic moments.
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
91
1. Consider just two pointlike magnetic moments which interact via the
dipole-dipole interaction (4.51): The sign and the intensity of the dipolar interaction strongly depend on the relative orientation of the two
interacting magnetic moments and on their relative spatial position r12 .
2. In contrast to the exchange interaction, the dipolar interaction couples spins located indefinitely far from each other and the decay of its
strength with the distance is relatively slow: 1/r3 (long-ranged).
3. In a 3d solid, the dipolar interaction introduces a dependence of the
total energy on the shape of the sample itself (shape anisotropy).
Equally spaced dipoles
E =+ 1
a)
E =- 1
b)
r12
E =- 2
c)
E =+ 2
d)
r12
r12
r12
Figure 4.6: Different sample configurations of two dipoles ~µi (i = 1, 2), placed
at a fixed distance r12 . The different values of the interaction energy, E, are
µ0 µ 2
.
given in units Edd = 4π
r3
12
To fix the ideas about point 1), let us consider two dipoles at a fixed
µ 0 µ2
distance so that the relevant energy scale is given by Edd = 4π
3 . Referring
r12
to the configurations in Fig. 4.6, it is clear that when for some reasons (other
energies or geometrical constraints)
• two interacting magnetic moments are forced to lie perpendicularly to
the direction of ~r12 , then the antiparallel alignment is favored by the
dipolar interaction (cases a) and b) in Fig. 4.6);
• two interacting magnetic moments are forced to lie along the direction of ~r12 , then the dipolar interaction favors their parallel alignment
(cases c) and d) in Fig. 4.6).
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
92
In an absolute sense, the configuration c) has the lowest energy of those
reported in Fig. 4.6.
Parallel dipoles
In order to evaluate the dependence of the energy (4.51) on the relative
orientation in space of the two point-like dipoles, ~µ1 and ~µ2 , it can be useful
to set the origin of the spatial coordinates in ~µ1 , with the z axis parallel to
the direction of ~µ1 itself. Then choose ~µ1 k ~µ2 . The resulting energy only
1.0
0.5
E
0.0
-0.5
ferro
antiferro
ferro
-1.0
-1.5
-2.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
q
Figure 4.7: Plot of the interaction energy (4.52) as a function of the polar
µ0 µ1 µ2
angle θ (rad). The energy values, E, are given in units Edd = 4π
. Regions
3
r12
with E < 0 correspond to ferromagnetic coupling, while regions with E > 0
correspond to antiferromagnetic coupling.
depends on the polar angle θ defined as the angle that the vector r12 forms
with the z axis of our reference frame or, equivalently, with any of the two
parallel magnetic moments, ~µ1 and ~µ2 . Within this geometry, the interaction
energy (4.51) reduces to
µ0 µ1 µ2
2
1
−
3
cos
θ
.
(4.52)
Hdd =
3
4π r12
Eq. (4.52) is very interesting because it shows that the dipole-dipole interaction is
• ferromagnetic for θ ∈ [0, θM ] and θ ∈ [π − θM , π]
93
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
• antiferromagnetic for θ ∈ [θM , π − θM ]
where θM is the magic angle such that cos2 (θM ) = 1/3 (see Fig. 4.7). Exactly
at the magic angle the dipolar interaction vanishes, meaning that in this
geometrical configuration the two magnetic moments don’t “feel” each other,
for what concerns the dipolar interaction.
?
a)
r13
r23
r12
F
b)
S
S
F
S
c)
S
S
F
S
S
Figure 4.8: Dipolar-frustrated configurations. a) The vectors ~r12 , ~r23 and ~r13
lie all on the same plane and all the magnetic moments are constrained to
be aligned along the indicated direction: up (green) or down (violet). b) and
c) All the four dipoles are assumed to lie onto the same plane; the red “F”
indicates the frustrated bonds and black “S” satisfied bonds.
Fig. 4.8 evidences how the dipolar interaction easily introduces frustration
as far as more than two magnetic moments are considered. The configuration
a) in Fig. 4.8 represents three magnetic moments magnetized out of plane
lying at the vertices of an equilateral triangle. The spins at the bottom
of the sketch (up-green and down-violet) minimize their interaction energy
by aligning antiparallelly to each other, as in the case of Fig. 4.6 b. Then,
according to Fig. 4.7, the two bonds r13 or r23 correspond to ferromagnetic
coupling because for these specific cases θ = π/6 < θM . Thus both states
up or down of the third spin – the one located at the upper vertex – will
produce the frustration of one of the two bonds r13 or r23 .
Fig. 4.8 b) and c) refer to a situation in which the four magnetic moments lie
onto the same plane. Due to the fact that the configuration c) of Fig. 4.8 is
to the global minimum of the two-dipole interaction, the vertical bonds are
first fulfilled (ferromagnetic). The horizontal bonds (antiferromagnetic) and
the diagonal bonds (ferromagnetic) cannot be satisfied at the same time, so
that some frustration is introduced anyway: frustrated bonds are highlighted
with red. A detailed calculation shows that, eventually, the configuration in
Fig. 4.8 c has a lower energy for a square lattice.
The triangle (a) and the square (b) of Fig. 4.8, can be thought of as unit cells
of a 2d triangular and square lattice respectively: one can easily imagine that,
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
94
when passing to the thermodynamic limit, many different configurations will
have the same, or nearly the same, energy. This fact typically gives rise to
very complex behaviors, such as glassiness, metastability, order induced by
disorder, spin-ice behavior, etc.
4.7
Dipolar interaction in extended systems
In extended systems, the dipolar interaction (4.51) is always present and it
involves all the magnetic moments. The resulting contribution to the total
energy is
"
#
~µn · rnn′ ~µn′ · rnn′
~ n · ~µn′
1 µ0 X µ
Hdip =
(4.53)
−3
3
5
2 4π n6=n′
rnn
rnn
′
′
where the sum is extended over all the different couples (note the factor 12 !)
labeled by n and n′ , and rnn′ = a(n − n′ ) (of course the modulus is rnn′ =
a |n − n′ |). Evaluating the term (4.53) is usually complicated analytically
and computationally expensive in numerical calculations (due to the longrange character of the dipolar interaction). Thus, in practice, one tries to
neglect the dipolar contribution or simplify it taking advantage from the
fact that the dipole-dipole interaction normally has a much smaller strength
than the exchange interaction. With the help of Table 3.1 one can easily get
convinced of this. The strength of the nearest-neighbor dipolar interaction
µ0 (gµB S)2
is 4π
= M02 a3 (if the saturation magnetization M0 is expressed in the
a3
Gauss system, M02 has the units erg/cm3 ). Putting the proper numbers one
finds that this energy is of the order of few Kelvins or smaller. However, there
are cases in which the dipolar interaction may affect crucially the macroscopic
behavior of a magnetic system. In the following we will give some examples.
2d systems with uniaxial anisotropy
Here we start again from the 2d Ising model, in which spontaneous magnetization is encountered at finite T , and consider the qualitative effect of the
dipolar interaction. Typically, real systems with uniaxial anisotropy can possibly be experimental counterparts of the Ising model. We should also keep
in mind that the exchange interaction has usually a much larger strength
than the dipolar interaction (a factor 102 − 103 ). From what stated in the
previous sections, it is clear that in such systems the dipolar interaction will
play a different role depending on the direction of the easy axis:
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
95
• magnetization in plane: the dipolar interaction is globally satisfied
(see Fig. 4.8 b)
• magnetization out of plane: the dipolar interaction is frustrated and
tends to “destroy” ferromagnetism (see Fig. 4.6 b).
Figure 4.9: Experimental check of the 2d-Ising scaling behavior, observed in
Fe/W(110) films magnetized in plane. C. H. Back et al., Nature 378 p.
597.
The first case is realized, for instance, in Fe/W(110) ultrathin films. These
films are model realizations of the 2d Ising model as they obey the predicted
scaling behavior for T ∼ TC over eighteen orders of magnitude (see Fig 4.9).
As pointed out in Nature 378 p. 597, neither the dipolar interaction nor
other effects which are neglected in the ideal model seem to affect the 2dIsing critical behavior observed in Fe/W(110).
An example of ultrathin film magnetized out of plane is represented
by Fe/Cu(001). Here the competition between the ferromagnetic exchange
interaction – originating ferromagnetism for T < TC – and the dipolar interaction – frustrating ferromagnetism on a larger scale – produces a sort
of phase separation between regions of positive and negative magnetization
perpendicular to the film plane (see the scheme in Fig. 4.10). In other words,
ferromagnetism is limited to some spatial regions in which all the magnetic
moments point along the same direction. Such regions are called magnetic
domains. The whole scenario holds below the Curie temperature, thus magnetic domains need not be confused with the spatial regions defined by the
correlation length ξ. In Fig. 4.11 some images of different magnetic-domain
patterns observed in Fe/Cu(001) films are shown.
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
96
Figure 4.10: The competition between exchange interaction and the dipolar
interaction originates a modulated phase in magnetic films magnetized out
of plane.
Other dimensions
Since a spontaneous magnetization does not exist at any finite temperature,
the effect of dipolar interaction is usually less dramatic in 1d. However, the
dipolar interaction may still affect the elementary excitations which destroy
ferromagnetism
• spin-waves in isotropic spin chains (delocalized excitations)
• domain wall in spin chains with uniaxial anisotropy (localized excitations)
and finally modify the behavior of ξ(T ) (e.g. how it diverges at low T ).
The calculation of the dipolar interaction energy of one magnetic moment
with all the others involves an integral (see next section) like
Z
dd r
(4.54)
3
Nd r
which is convergent for d<3. For d=3 the total magnetostatic energy is
conditionally convergent and, as anticipated, depends on the shape of the
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
97
Figure 4.11: SEMPA images of magnetic domains in Fe/Cu(001) films: dark
and light gray regions correspond to domains with opposite out-of-plane magnetization.
sample (shape anisotropy). In 3d, both frustration effects and magneticdomain formation occur in a similar way to what discussed for d=2.
Continuum limit
The continuum version of the dipolar energy given in Eq. (4.53) can be
~n followed by the usual substitution
obtained by setting ~µn = −gµB S
Z
X
1
· · · ≃ d . . . dd r ,
(4.55)
a
which yields
Z ~
~ ′)
S(r) · S(r
Hdip
dd r ′ −
d r
|r − r′ |3
ih
i
h
′
Z
Z  S(r)
~ ′ ) · (r − r′ ) 
~
2
)
S(r
·
(r
−
r
µ0 (gµB S) 1
dd r ′ +
−3
dd r
5
3
2d−3
′


8π
a
a
|r − r |
Z
Z
µ0 (gµB S)2 1
d
~ ′ )δ (r − r′ ) dd r′
~
+
· S(r
d r S(r)
6
a3
a2d−3
(4.56)
µ0 (gµB S)2 1
=
8π
a3
a2d−3
Z
d
The unit lengths a have been grouped in such a way that the characteristic
µ0 (gµB S)2
is separated from the
energy scale of the dipolar interaction Ω = 4π
a3
geometrical terms. The term δ (r − r′ ) arises from a detailed magnetostatic
calculation. It essentially accounts for the fact that the pointlike-dipole approximation has to include this term to compensate the divergence at short
distances, r ≃ r′ . Thus the total magnetostatic energy is not ill-defined.
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
98
It is also useful to recall that the magnetization is introduced in elemen~ (r)
tary courses as a coarse-grained quantity. Ideally, the magnetization M
already represents an average over an elementary volume whose magnetic
~ (r)d3 r. The magnetostatic energy then reads
moment is8 ~µ = M
Z
Z
µ0
µ0
3
Edip =
ρm (r)φm (r)d r +
σm (r)φm (r)d2 r ,
(4.57)
2 V
2 ΣV
where
~ ·M
~
ρm = − ∇
and
~ · n̂
σm = M
(4.58)
are the volume and surface charge density respectively (n̂ normal to the
surface ΣV ), while
Z
Z
1
1
ρm (r′ ) 3 ′
σm (r′ ) ′
φm (r) =
d
r
+
dΣ
(4.59)
4π V |r − r′ |
4π ΣV |r − r′ | V
is the scalar magnetic potential (see, e.g., “Classical Electrodynamics”, J. D.
Jackson).
Note that, within this formalism, the total magnetostatic energy is the
Coulomb energy of an effective Coulomb charge distribution ρm (r). Such
an analogy provides a very helpful “rule of thumb” for analyzing realistic
situations:
the magnetostatic energy is minimized by configurations for which there are
as less magnetic charges as possible.
Even if derived in a coarse-grained context, the validity of this “rule of
thumb” is quite general, i.e., it provides the correct hints also in the discretelattice formalism.
Bloch and Néel domain walls
In section 4.3 of the present chapter we have investigated how domain walls
with a finite width δ emerge from the competition between the anisotropy
and the exchange energy. The “compromise” which minimizes the domainwall energy is represented by the solution (4.23) which we recall here for
convenience:
(
cos(θ(x)) = tanh xδ
(4.60)
ϕ(x)
= const.
All the solutions with constant ϕ(x) give the same energy if inserted in the
Hamiltonian (4.20). Now we ask ourselves whether the introduction of the
~ (r)d3 r, but for the present purposes it is convenient
More precisely, it is m
~ = h~
µith = M
to keep considering the dipolar interaction at T = 0, for which m
~ =µ
~.
8
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
99
dipolar energy term (4.56) may favor one specific value of ϕ(x). The two
extreme cases are named
(
ϕ(x) = π2 ∀x
Bloch domain wall
(4.61)
ϕ(x) = 0 ∀x
Néel domain wall .
More specifically the magnetization for these two cases will be
(
~ = M0 (0, sin(θ(x)), cos(θ(x)))
M
Bloch domain wall
~
M = M0 (sin(θ(x)), 0, cos(θ(x)))
Néel domain wall .
(4.62)
with θ(x) given by Eq. (4.60). In the bulk case (3d), it is evident that the
Figure 4.12: From Introduction to the Theory of Ferromagnetism by A. Aharoni. Energy per unit wall area, γ, (solid curves) as a function of the thickness for a permalloy film magnetized in plane. Dashed curves display the
domain-wall width (q ∝ δ in our notation).
Bloch wall has always a lower energy since
(
~ ·M
~ = ∂ x M x + ∂ y My + ∂ z Mz = 0
∇
6= 0
Bloch domain wall
Néel domain wall ,
(4.63)
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
100
so that only the Néel domain wall produces some magnetic charges ρm =
~ ·M
~ . This rule-of-thumb prediction is confirmed by detailed calculations
−∇
and experiment.
The situation is analogue for thin films (2d) with the easy axis pointing out
of plane.
For thin films magnetized in the xz plane (with easy axis parallel to z), the
surface charges σm produced by a Bloch wall (My needs to point out of plane)
can be so large that the Néel wall becomes energetically more convenient. In
this case, surface charges (Bloch wall) are replaced by volume charges ρm .
The solid curves in Fig. 4.12 display the domain-wall energy corresponding
to a Bloch and a Néel wall, computed with the phenomenological parameters
of permalloy (alloy of ≃ 20% Fe and ≃ 80% Ni), as a function of the film
thickness. Indeed, for this specific material with this specific geometry, up
to thicknesses of the order of 60 nm the Néel wall has a lower energy.
4.8
Origin of magnetic domains
The scenario described schematically in Fig. 4.10 is just a particular case in
which the competition between the exchange and dipolar energy gives rise
to a configuration with zero global magnetization. Such a configuration is
the compromise which produces as less magnetic charges as possible with
the minimum frustration of the exchange interaction. What results from the
competition between these two energies is generally different depending
• on the easy-axis direction
• on the geometry of the sample
so that each case needs to be evaluated on its own.
As a simple example, let us consider – again – a 2d system magnetized
out of plane and evaluate the energy variation associated with the creation
of a domain wall from a uniformly magnetized state (see Fig 4.13). Both
the exchange energy and uniaxial anisotropy contribute to the domain-wall
energy Ew , thoroughly discussed in the previous sections and whose values
are summarized schematically in Eq. (4.24). When deriving those results
we assumed that the spin profile was a function of one spatial variable only.
Now we consider a film of finite thickness t and a domain wall developing
indefinitely along the y direction, so that the total increase of the exchange
and anisotropy energy is Ny Nz Ew , with Nz = t/a. For a film magnetized out
of plane and with thickness t of few monolayers the dipolar energy (4.56) can
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
101
be approximated as
Hdip
1 t2
= Ω 3
2 a
Z
2
dr
Z
S z (r)S z (r′ ) 2 ′
dr .
|r − r′ |3
(4.64)
Splitting the xy plane into two half-planes and reversing the magnetization
(along z) of one of the two half-planes produces a decrease of the dipolar
Figure 4.13: Sketch of the two configurations corresponding to the energy difference evaluated in Eq. (4.65). Arrows represent magnetic moments pointing
out of plane.
energy by a factor
∆Edip
t2
= 2Ny Ω 2
a
Z
Nx a/2
dx
δ
Z
0
dx
−Nx a/2
′
Z
+∞
−∞
1
[(x −
x′ ) 2
+ y 2 ]3/2
dy ; (4.65)
the integral over dx is performed starting from a length scale equal to the
domain-wall width δ in order to avoid an unphysical divergence. In other
words, spins located inside the domain wall have been ideally “removed”
from the calculation of the dipolar energy. Note that ∆Edip in Eq. (4.65)
represents the variation of the interaction energy between the two half-planes;
the magnetostatic self-energy of the two half-planes remains the same. The
integral in Eq. (4.65) can be performed analytically and gives
t2
Nx a + 2δ
∆Edip = 4Ny Ω 2 ln
.
(4.66)
a
4δ
The condition ∆Edip = Ny Nz Ew gives the minimum linear dimension Nx that
a slab should have in order that splitting the uniform state into domains
becomes favorable. In the realistic limit of Nx a ≫ δ, one finds that for Nx
larger than the threshold value
4δ
Ew a
N̄x ≃
exp
(4.67)
a
4Ωt
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
102
it is convenient for the system to split into domains of opposite out-of-plane
magnetization. From this rough calculation one expects the typical size of
domains to be Leq ≃ N̄x a/2. The exponential dependence on the ratio Ew /Ω
is typical of d=2. For the 3d case, a further integration would be involved
in the evaluation of ∆Edip which, eventually, would result in a much weaker
dependence of the domain size Leq on the ratio Ew /Ω.
Striped pattern
Figure 4.14: Schematic view of the striped ground state of a ferromagnetic
film magnetized out of plane in the presence of the dipolar interaction and
B = 0. Stripes of different colors represent regions of opposite out-of-plane
magnetization. The typical stripe width is Leq .
In spite of the crude approximations that have been performed to obtain
Eq. (4.67), the predicted scaling with Ew /Ω and with the film thickness t
match with the optimal stripe width for an ideal stripe pattern at T = 0 (see
Fig. 4.14). This pattern corresponds to the ground state of a film magnetized
out of plane in the presence of dipolar interaction and zero external field.
Detailed calculations9 yield for Ω ≪ D ≪ J
10
Ew a
(4.68)
δ exp
Leq =
3π
4Ωt
and for Ω ≪ J ≪ D (Ising domain wall)
Leq = L0 exp
9
Ja
2Ωt
(4.69)
Case Ω ≪ D ≪ J adapted from S. A. Pighı́n et al., Journal of Magnetism and
Magnetic Materials 322 p. 3889 (2010). Case Ω ≪ J ≪ D (Ising) adapted from A. B.
MacIsaac et al., Physical Review B 51 p. 16033 (1995).
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
103
with L0 = 0.871a. Remembering that in the second case Ew = 2J and δ = a,
both Eqs. (4.68) and (4.69) confirm the scaling predicted by Eq. (4.67) on
the basis of heuristic arguments.
Characteristic length scales emerge from competing interactions
The typical domain size Leq is the second characteristic length scale that we
have encountered arising from the competition between different energies
(
J versus D ⇒ δ domain-wall width
(4.70)
J versus Ω ⇒ Leq domain width .
Notice that calculations to obtain both δ and Leq have been performed at
T = 0. Since the exchange interaction, the anisotropy energy and the dipolar
interaction involve different number of spins and different components of
each spin (e.g., uniaxial anisotropy contains only a single-site term (S z )2 ),
in general, they will be affected differently by thermal fluctuations. This
fact, may finally favor one of two competing interactions, thus introducing
an effective dependence on temperature in the characteristic length scales
δ and Leq .
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
Literature
• A. Aharoni, Introduction to the Theory of Ferromagnetism
Oxford University Press
• L. D. Landau and E. M. Lifshitz, Statistical Physics
Oxford Pergamon Press
• J. D. Jackson, Classical Electrodynamics
John Wiley and Sons
• C. Kittel, Introduction to Solid State Physics
John Wiley and Sons
Dr. Alessandro Vindigni
Laboratorium für Festkörperphysik
Wolfgang-Pauli-Str.16
ETH Hönggerberg, HPT C 2.2
Tel. +41 44 633 2077
Fax. +41 44 633 1096
e-mail: vindigni@phys.ethz.ch
104
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
105
Appendix
Averages at finite temperatures
Classical models
In the canonical ensemble, the partition function is given by
Z 3N 3N
d qd p −βH(p,q)
Z=
e
,
(2π~)3N
H being the Hamiltonian of the system and β =
free energy, F , via the general relation
1
.
kB T
Z is related to the
1
F = − ln Z .
β
The average of any observable O (p, q) can be computed as
Z 3N 3N
1
d qd p
hOi =
O (p, q) e−βH(p,q) .
Z
(2π~)3N
Classically, the trace operator is defined as
Z
d3N qd3N p
,
T r = ...
(2π~)3N
which allows defining
Z = T r e−βH(p,q)
and hOi =
(4.71)
1
T r O (p, q) e−βH(p,q) .
Z
(4.72)
(4.73)
(4.74)
(4.75)
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
106
Quantum models
Assume that |ψα i be a complete basis of the Hilbert space on which the
Hamiltonian of the model is defined. Quantum-mechanically, the trace is
then given by
X
Tr=
hψα | . . . |ψα i .
(4.76)
α
By analogy with (4.75), the partition function and thermal averages are
accordingly defined
X
hψα |e−βH |ψα i
Z = T r e−βH =
α
1
1 X
hOi = T r Oe−βH =
hψα |Oe−βH |ψα i .
Z
Z α
(4.77)
In few advanced computations one stops at this level. Generally, the trace is
evaluated on a complete basis of eigenstates of H:
H|ϕi i = E i |ϕi i .
(4.78)
The computation of (4.77) is, consequently, simplified:
X
X
i
Z=
hϕi |e−βH |ϕi i =
e−βE
i
i
1
hOi = T r Oe
Z
−βH
1 X i
i
=
hϕ |O|ϕi ie−βE .
Z i
(4.79)
Spin models
Limiting ourself to a Hamiltonian of the type
1
H=− J
2
X
|n−n′ |=1
Ŝ(n) · Ŝ(n′ ) + gµB B
X
Ŝ z (n)
(4.80)
n
one possible choice for the basis of the Hilbert space is the following one:
|ψα i=|M1 , M2 , . . . MN i=|M1 i ⊗ |M2 i · · · ⊗ |MN i with Ŝ z (n)|Mn i=Mn |Mn i
and n label for the lattice site. Note that the Hamiltonian in Eq. (4.80) is
not diagonal on this basis. Thermal averages are computed according to Eqs.
(4.79).
For many problems in magnetism, substituting the quantum-mechanical operators Ŝ(n) by classical vectors is legitimate:
~
≡ S0 (sin θ cos ϕ, sin θ sin ϕ, cos θ)
Ŝ(n) → S(n)
(4.81)
CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS
107
where S02 = S (S + 1) (more often S0 = 1). The partition function then reads
Z
Z
Z
~
Z = dΩ1 dΩ2 . . . dΩN e−βH({S(n)}) ,
(4.82)
with dΩn = sin θn dθn dϕn being the solid-angle element of the spin located at
the site n.
Both in the quantum and the classical case Z depends on T and B. Therefore,
the free energy associated with the canonical averages is the Gibbs free energy
of macroscopic thermodynamics:
1
G(B, T ) = − ln [Z(B, T )] .
β
(4.83)
Often in the literature the letter F is used also when dealing with spin models
to stress the fact that this type of averages are performed in the canonical
ensemble (constant number of particles). We will follow this convention.