magnetism i: from the atom to the solid state - microstructure research

Magnetism I: from the atom tothe solid state

Preface

The Lecture on “Magnetism I: from the Atom to the solid state” is an intro-duction to the fundamental concepts in magnetism. It consists of two parts:the first one (by D. Pescia) deals with magnetic effects in atoms (diamag-netism, paramagnetism, formation of magnetic moments in atoms) and withthe occurrence of magnetic order in the ground state of a solid in virtue ofthe exchange interaction. The second part (by A. Vindigni) treats the oc-currence of magnetism at finite temperatures, the role of small interactionssuch as the dipolar interaction, and presents the essential facts about thestatistical physics of magnetism. A very extended introduction in modernmagnetism can be found in the book by J. Stohr and H.C. Siegmann ” Mag-netism: from fundamentals to the Nanoscale dynamics”, Springer-Verlag,Berlin Heidelberg 2006.

Zurich, September 2012

D. Pescia

ii

Contents

Preface ii

1 Magnetism in Atoms 21.1 Magnetism in classical physics . . . . . . . . . . . . . . . . . . 21.2 Magnetism in quantum mechanics . . . . . . . . . . . . . . . . 5

1.2.1 Free electrons in a magnetic field . . . . . . . . . . . . 51.2.2 Electron in a magnetic field and a central potential . . 7

1.3 The formation of the magnetic moment in atoms . . . . . . . . 91.3.1 Paramagnetism in Atoms . . . . . . . . . . . . . . . . . 16

1.4 Exchange interaction and the Heisenberg-Dirac-Van VleckHamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Magnetism in solids 232.1 Stoner-Wohlfahrt model . . . . . . . . . . . . . . . . . . . . . 232.2 Friedel-Oscillations . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.1 The interatomic exchange interaction . . . . . . . . . . 312.2.2 RKKY oscillations . . . . . . . . . . . . . . . . . . . . 322.2.3 Anhang: mathematical details of the model . . . . . . 34

3 Magnetic order at finite temperature 393.1 Coupled effective spins: an N -body problem . . . . . . . . . . 403.2 Mean-field approximation (MFA) . . . . . . . . . . . . . . . . 423.3 Mean-field universality class . . . . . . . . . . . . . . . . . . . 453.4 The Landau approach . . . . . . . . . . . . . . . . . . . . . . 483.5 Classical spin models . . . . . . . . . . . . . . . . . . . . . . . 533.6 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . 613.7 Landau theory of correlations . . . . . . . . . . . . . . . . . . 66

4 Magnetic domains and domain walls 714.1 Magnetic anisotropy . . . . . . . . . . . . . . . . . . . . . . . 714.2 Domain walls in the classical Heisenberg model . . . . . . . . 74

iii

CONTENTS iv

4.3 Continuum formalism . . . . . . . . . . . . . . . . . . . . . . . 774.4 Beyond the Mean-Field Approximation . . . . . . . . . . . . . 804.5 Finite size and superparamagnetic limit . . . . . . . . . . . . . 884.6 Dipolar interaction . . . . . . . . . . . . . . . . . . . . . . . . 904.7 Dipolar interaction in extended systems . . . . . . . . . . . . . 944.8 Origin of magnetic domains . . . . . . . . . . . . . . . . . . . 100

CONTENTS 1

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 historyof mankind: some materials have the property of producing a sizable mag-netic field that attracts or repel other materials. One of the first referencesto the magnetic properties of what we know now to be magnetite Fe3O4

(lodestone) is by 6th century BCE Greek philosopher Thales of Miletus. Thename ”magnet”may come from the lodestones found in Magnesia. In China,the earliest literary reference to magnetism lies in a 4th century BC bookcalled Book of the Devil Valley Master: ”The lodestone makes iron come orit attracts it”. The lodestone based compass was used for navigation in me-dieval China by the 12th century. The main observation about the origin ofthe magnetic field originates with the experiments of Ampere and Oersted inthe early decades of the 19th century, demonstrating that i a current is ableto influence a magnetic needle (Oersted) and ii a mechanical force exists be-tween two wires injected with current (Ampere). Later, Faraday completedour knowledge of magnetic field by discovering that time dependent mag-netic fields can produce a magnetic current. J.C. Maxwell gave a completedescription of electromagnetic fields that is still very precise (Maxwell equa-tions). The origin of the magnetism in matter remained debated: Amperepostulated that magnetism in atoms originates from the existence of a closedatomic-sized current. Poisson and later Maxwell, instead, favored magneticcharges that appear always coupled into dipoles as the source of the mag-netic field. Distinguishing between the two hypothesis is a subtle problem,as a paper in 1977 by J.D. Jackson show. Following Ampere hypothesis, themagnetic field produced by a current circulating within a small loop C isgiven in term of the magnetic moment vector ~m = e

2~x× ~x = 2

2m~L, which

2

CHAPTER 1. MAGNETISM IN ATOMS 3

Figure 1.1: Drawing of the magnetic field of the Earth by Rene Descartes,from his ”Principia Philosophiae”, 1644. This was one of the first drawingsof a magnetic field.

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.

~B(~r) =µ0

4π~∇× (~∇× ~m

| ~r |)

=µ0

4π~∇(~∇ · ~m

| ~r |)−µ0

4π~m 1

| ~r |=

µ0

4π~∇(~m · ~∇ 1

| ~r |) + µ0 · ~m · δ(~r)

=[µ0

4π

3~r(~r · ~m)− ~m~r2

| ~r |5 − µ0

3~m · δ(~r)

]

+ µ0 ~m · δ(~r)

The first component of the magnetic field produced by the small currentloop is formally identical to ~E field produced by an electric dipole moment.This lead Poisson and later Maxwell to state that ” ...we may regard themagnet,..., as made up of small particles, each of which has two equal andopposite poles” (magnetic charges). This idea, which was alternative to theone of Ampere, describes almost anywhere the magnetic field correctly. Atthe position of the loop, however, – the second term, given by the Dirac deltafunction – the picture of magnetic charges would produce a field amountingto −µ0

3~m · δ(~r). Experiments aimed at measuring the ”contact” term –

such as the hyperfine line of atomic hydrogen (relevant in astrophysics) –


Figure 1.2: Pierre Pelerin de Maricourt (French), Petrus Peregrinus de Mari-court (Latin) or Peter Peregrinus of Maricourt was a 13th century Frenchscholar who conducted experiments on magnetism and wrote the first ex-tant treatise describing the properties of magnets. His work is particularlynoted for containing the earliest detailed discussion of freely pivoting com-pass needles, a fundamental component of the dry compass soon to appearin medieval navigation.

can ultimately discriminate between the two hypotheses: the hypothesis byAmpere that the magnetic moment is due to atomic current rather thanatomic charges is now well accepted.

While Maxwell equations and the postulate of Ampere are very exactin describing atomic magnetic moments and their magnetic fields, the veryexistence of magnetic moments in classical physics is challenged by a famoustheorem, the Bohr-van Leuwen theorem.Theorem: given that the classical Hamilton function for an electron in anapplied magnetic field ~B writes H = 1

2m(~p − e ~A)2 + eφ it follows that the

canonical average of the atomic magnetic moment ≺ ~m ≻ vanishes exactlyand any finite temperature.Proof:1. When one writes down explicitly the equations of motion in a uniform


field one can construct an integral of the motion – the total energy – whichreads

1

2m~x2 − ~∇φ(~r) (1.1)

Remarkably, the total energy is independent of ~B – indicating that theLorentz force is not doing any work, being perpendicular to the velocity.Accordingly, when the statistical average based on the canonical Gibbs dis-tribution is computed, the energy of the various classical states over whichone integrate does not contain the magnetic field and any partial derivativewith respect to the magnetic field – such as the average magnetic moment –must vanish.2. Considering that

≺ ~m ≻= − ≺ ∂H

∂ ~B≻ (1.2)

one obtains

− ≺ mz ≻ =

∫d~pd~x

[−(px − e

2Bzy)y + (py +

e2Bzx)x

]e

−1

2m (~p−e ~A)2−eφ(~x)

kBT

∫d~pd~xe

−1

2m (~p−e ~A)2−eφ(~x)

kBT

(1.3)

Using the variable transformation

(p′x x p′y x′

)=

1 0 0 −eB2

0 1 0 00 eB

21 0

0 0 0 1

pxxpyy

(1.4)

one can easily convince oneself that the integral – and with it the averagemagnetic moment – is vanishing – a result known as Bohr van Leuwentheorem of classical statistical physics.

1.2 Magnetism in quantum mechanics

1.2.1 Free electrons in a magnetic field

It was first Landau (1930) who explicitly produced an average magneticmoment and thus determined the ”birth” of magnetism in matter. Landausolved the problem of a free electron moving in a uniform magnetic fieldusing quantum mechanics to describe the motion, i.e. breaking away from


Lorentz force. For a simple case of one free electron, the energy levels write(see textbooks in QM)

En = ~ωc(n+ 1/2) +~2k2z2m

(1.5)

(n = 0, 1, 2..., ωc =eBm), their degeneracy being (not including the spin of the

electron) L2eB2π~

(L: linear dimension of the system). The partition functionreads

Q(T,B, L) =L2eB

2π~

∑

kz

∑

n

exp

(−En

kBT

)

=L3eB

(2π~)2

√

2πkBTe− ~ωc

2

∑

n

e−n~ωkBT

=L3eB

2(2π~)2

√

2πkBT1

sinh( ~ωc

2kbT)

(1.6)

The magnetization (per particle) amounts to

kBT∂lnQ

∂B=

kBT

B− ~ωc

2Bcoth(

~ωc

2kbT) (1.7)

For small magnetic field we have the simple result

mz = − e2~2

12m2

B

kBT(1.8)

mz being the average magnetic moment of the free electron. This resultsanctions the appearance of a non-vanishing magnetic moment. Typically,one deals with N free electrons, each energy level being occupied by twoelectrons at the most, up to the Fermi energy. Considering that only electronswithin an energy range kbT below the Fermi energy can be excited, themagnetization (magnetic moment per unit volume) induced by the magneticfield amount to (a= lattice constant)

Mz.=mz

a3= − e2~2

12m2

B · ρ(EF )kBT

kBT= − e2~2

12m2ρ(EF ) (1.9)

Here is ρ(EF ) the density of states at the Fermi level, which for a typical freeelectron metal amounts to 0.805 · 1022eV −1cm−3. The final result is that theinduced magnetization in a free electron gas (without considering the spinof the electron) is independent of the temperature. Defining the magneticsusceptibility as

χz.= µ0

∂Mz

∂Bz

(1.10)


as a way of comparing the various magnetic responses, we find a value forχz in the case of free electrons of about 10−6, which means that an externalfield of the order of 1 Tesla produces a magnetic field µ0Mz of the order of1 microtesla. Notice that letting ~ go to zero produces the vanishing of Mrequired by classical physics. This result shows that quantum mechanics isthe key for the appearance of a magnetic moment in matter, albeit weak.

1.2.2 Electron in a magnetic field and a central poten-tial

In an atom, the Hamilton operator in the presence of a uniform magneticfield ~A = −1

2(~r × ~B) reads

(

~p− e ~A(~r))2

+ V (~r) = [− ~2

2m+ V ]

− ~B · ~µ→ (a)

+e2

2m~A2 → (b)

~µ.= −µB · ~L

~is the operator representing the magnetic moment (µB

|e|~2m

=0.927410−23 J

Tis the Bohr magneton).

The strength of the various terms can be computed e.g. on hydrogen orbitals.We obtain

< s | − ~2

2m+ V | s >= −E0 = −13.6eV

(a) →< l = 1,ml = −1 | −µB( ~B · ~L) | l = 1,ml = −1 >= −µB ·B

(b) → < s | e2

2m~A2 →| s >E0

=4

310−12(

B

T)2

In the last line B in supposed to be given in T .Term a. This term is called the Zeeman term, the corresponding energyis the Zeeman energy. Its contribution is non-vanishing when the atomicorbital has a finite z-component of the angular momentum. It shows thatan atomic state with appropriate z-component of the angular momentum (inthe present case −~ lowers the total energy of the atomic level by an amountµB · 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 partiallyoccupied, decreasing the total average magnetic moment, which becomesstrongly temperature dependent, as we will discuss at the end of this chap-ter.


Term b. This term is much smaller than the Zeeman energy, almost temper-ature independent, and only observable if the total magnetic moment of theatoms is exactly vanishing: it builds the so called diamagnetic contributionto the energy. From the expression for the diamagnetic energy one can de-fine an effective diamagnetic moment per atom through µD = −∂ED

∂Band

a diamagnetic susceptibility χD = −µ0 · ∂2ED

a30∂B2 amounting to ≈ −10−11eV/T

and −10−6, respectively, in a magnetic field of 1T . Thus, the diamagneticmoment per electron points antiparallel to the applied magnetic field.

Figure 1.3: The top of the figure describes some alternative definitions of the magneticsusceptibility. The Table on the left shows values of the diamagnetic susceptibility for somesubstances (to find χD multiply the values by 10−5). The figure on the right illustrates

a straightforward generalization of the expression < s | e2

2m~A2 →| s > to the diamagnetic

susceptibility of many in an atom, which writes χD = −µ0e2

6m

∑

i < r2i > the sum extendingover all electrons building the magnetic response (including the core electrons). Therefore,the diamagnetic susceptibility should scale with Z· < r2a >. This is confirmed by the plotbottom right, where the horizontal axis shows the total number of electrons Z multipliedby the square of the ionic radius < ra > in units of A2, for noble gases and ions of atomswith filled atomic shells, for which no net magnetic moment results.


1.3 The formation of the magnetic moment

in atoms

In the previous section, we have shown that the magnetic field couples withthe operator of the (negative) orbital angular momentum by means of thecoupling constant µB. We have now to take into account that

• each electron have an intrinsic spin that adds up in some way to theorbital angular momentum to produce a total angular momentum quan-tum number J , which results from the addition of the orbital quantumnumber of the electron and its spin.

• atoms have generally many interacting electrons: there spins and or-bital angular momenta must be added in some way to find the totalground state angular momentum which results from the addition of thetotal angular momentum and the total spin of all electrons!

We are looking for a set of rules that allow angular momenta to be addedin quantum mechanics and to a set of rules that allow to select, among themany possible angular momenta, those proper to the atomic ground stateof an atom. In other words: the key problem we have to solve is: given Nelectrons in the configuration (ni, li, lz,i, s = 1/2), (i = 1...N), 1st what arethe possible values of the total spin S, total orbital angular momentum L andthe total angular Momentum J (Russel Sounders symbol for the electronicconfiguration: (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 theelectronic configuration of an atom, is a central theorem of quantum me-chanics (known as Clebsch-Gordan series) about adding angular momentumoperators:Theorem: given an angular momentum operator ~P with z components[p, p− 1, ...− p] and ~Q with z-components [q, q− 1, ...,−q], the total angularmomentum operator ~T = ~P + ~Q can assume the values

P +Q,P +Q− 1, P +Q− 2, ...., | P −Q |

The total spin: The total spin of an assembly of spin 12particles is given by

the Clebsch-Gordan sum of all spins. For instance, for N = 2 we have thepossible values for S of 0, 1, carrying singlet (χs) respectively triplet (χt) spin


eigenfunctions:

1√2(u1/2u−1/2 − u−1/2u1/2) =

√

1

2(|↑>|↓> − |↓>|↑>)

u1/2u1/2 = |↑↑>1√2(u1/2u−1/2 + u−1/2u1/2) =

√

1

2(|↑>|↓> + |↓>|↑>)

u−1/2u−1/2 = |↓>|↓>

The total orbital angular momentum Again, it is given by the Clebsch-Gordan sum of all orbital angular momenta.The total angular momentum. It is given by the Clebsch-Gordan sum of to-tal spin and total angular momentum.Pauli principle (exchange interaction part I) There is a restriction on thepossible configurations: the total wave function of Fermions must be an-tisymmetric (Pauli principle). A theorem by Weyl helps implementing thePauli principle in the electronic structure of a many electron system.Theorem (H. Weyl): All eigenfunctions of Sz have the same symmetry withrespect to particle permutation: the symmetry property of the eigenfunctionof Sz under permutation are called ”Spinrasse”. This is e.g. clearly visi-ble in the two-electron system, where the spin singlet is antisymmetric withrespect to particle permutations and the spin triplet is symmetric. This the-orem divides the electronic structure of many-electrons atoms into separatedthermal schemes, according to their total spin (see e.g. ”para” and ”ortho”He): in fact, because of the requirement that the total wave function beantisymmetric, each spin rasse determines uniquely the symmetry propertiesof the orbital wave functions under permutation. For instance, the energylevels of the singlet thermal scheme can only have symmetric orbital wavefunctions:

1√2(un1,l1(~r1)un2,l2(~r2) + un1,l1(~r2)un2,l2(~r1))

while the triplet spin state carries only antisymmetric wave functions:

1√2(un1,l1(~r1)un2,l2(~r2)− un1,l1(~r2)un2,l2(~r1))

The Pauli principles implies that, among the possible configurations arisingfrom the Clebsch-Gordan sum, some cannot be realized because orbitalwave functions of the proper symmetry do not exist. For instance, startingfrom two inequivalent s electrons [n, s],[n′, s], one can obtain the two config-urations 3S1,

1S0. Starting e.g. from two equivalent s electrons [n, s],[n, s],


the triplet state cannot be constructed as the requirement of antisymmetricorbital wave function makes it vanishing. In other words: The Pauliprinciple is responsible that two identical spin 1/2 particles can only bein the same orbital state (n1, l1, lz,1 = n2, l2, lz,2) if they form a singlet. Inother words: more than two electrons cannot be in the same orbital state:if they are, they (sloppy) ”must have opposite spin”. Therefore, althoughthe Hamilton operator does not contain any spin dependent term (and thetwo electrons might not be even interacting), the non distinguishibilitypostulate of QM removes the degeneracy of some states and even forbids awell defined spin state to be a possible configuration. The Pauli principleacts as some sort of effective interaction that removes some degeneracy anddistinguishes between spin states. This interaction is know in the literaturewith the broad terminology of ”exchange interaction”. It is a purely QMeffect and completely disappears together with the spin upon transition toclassical mechanics. In the present case the exchange interaction dictatese.g. that the ground state of two identical spin 1/2-particles is a singlet. ThePauli principle and the exchange interaction are key results explaining thestability of matter and, for our purposes, are responsible for the formationof magnetic moments in atoms.Hund’s rules: exchange interaction part IIThe Pauli principle acts to remove some configurations among the set arisingfrom the Clebsch-Gordan sum but, for instance, does not lift the degeneracybetween 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 thatallows a further analysis of the various configurations.1st Hund rule. Provided there is sufficient degeneracy that non-equivalentorbital wave functions can be constructed, the configuration realizing thelowest energy state corresponds to a state of maximum spin number. Inother words: if the orbital states involved have different quantum numbers,the filling of the electronic states with parallel spins produces the lowestenergy electronic configuration. Thus, provided orbitally degeneratestates exist, the triplet state is energetically favored. If degeneracy isabsent, then the singlet state is energetically favored. Accordingly, theformation 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.


2nd Hund rule. The second Hund rule states that, after having establishedthe maximum S, the lowest energy state correspond to the configurationthat maximizes L (again, the Coulomb energy is reduced for high values ofL).3rd Hund rule. The third rule states that the total angular momentumquantum number J minimizing the energy is | L + S | if the shell is morethan half full, | L − S | if the shell is less than half full. This last ruleminimizes the energy arising from spin-orbit interaction.Of course, other electronic configurations exist, which have empiricallyhigher energies. These rules, together with the Pauli principle, determinecompletely the total values of J, L, S for the ground state and its electronicsymbol 2S+1LJ .

We are now ready to discuss the magnetic moment operator arising fromthe electronic configuration 2S+1LJ . As electrons do have both orbital andspin angular momentum, the magnetic coupling with a magnetic field mustbe extended to include the magnetic moment arising from the spin. Weknow from the Dirac theory of the electron that ~µS = −gSµB

~S, with gS = 2(in contrast to gL = 1). Quantum electrodynamic corrects this value togS = 2.0023, which is extremely close to the experimental value. Accordingly,the operator describing the interaction of a magnetic moment with a magneticfield becomes

HZ = −µB(~L+ g~S) · ~B = −µB( ~J + (g − 1)~S) · ~B

(let us call gS g, for simplicity). HZ is called the Zeeman operator. ~J = ~L+ ~S

is the operator of the total angular momentum. ~L, ~S and ~J are dimensionlessangular momenta. Let us now assume that we have determined (by solvingthe many body electron problem and using the Hund’s rules) the total orbitalangular momentum quantum number L of the ground state and its total spinangular momentum quantum number S. Notice that the possible eigenvaluesof ~J2 also labels the eigenspaces of an Hamiltonian that contains the spinorbit ~S · ~L interaction, as both ~J2 and the spin orbit interaction are scalarunder 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 ofa vector operator under rotations. We consider Hz to be a perturbation ofthe fine structure energy levels of an atom, i.e. those energy level resultingfrom Hund’s rules and carrying a well defined quantum number J . To findthe first order eigenvalues of HZ we solve the eigenvalue problem of HZ


within the 2J + 1 dimensional space containing the symmetry adapted wavefunctions to this J-value: un,l,j,mj

. We seek the first order correction, i.e. themagnetic field is small enough so that the Zeeman splitting is smaller thanthe level splitting between the various multiplet components J arising fromthe spin-orbit coupling. We choose ~B = (0, 0, B):

HZ = µB ·B · (Jz + (g − 1) · Sz)

The eigenvalue problem of Jz is simply solved, because

(un,l,j,mj, Jzun,l,j,mj

) = mj (1.11)

mj = J, J − 1, ...,−J . The eigenvalue problem of Sz is simplified by theWigner-Eckart theorem, which states that

(un,l,j,mj, Szun,l,j,mj

) = τLSJmj (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, Hzun,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 structurelevel completely. The Zeeman splitting is symmetric around the unperturbedlevel 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 anomalousZeeman effect, to be compared with the normal Zeeman effect, with g = 1and gLSJ = 1.


Figure 1.4: Top: Effective magnetic moment of ions of rare earths atoms asa function of the number of f -electrons and their electronic configurations.


1.3.1 Paramagnetism in Atoms

The statistical mechanics of an ensemble of non interacting N identical atomseach carrying the same quantum number J allows to compute the partitionfunction as

ZN =[∑

mj

exp(−gLSJ · µB ·mj ·B

kB · T]N

(1.16)

the total free energy f(T,B) per atom as

−kB · T ln[∑

mj

exp(−gLSJ · µB ·mj ·B

kB · T)

]

and the mean magnetic moment per atom as

< µz >=−∂f(T,B)

∂B= gLSJ · µB · J ·BJ(α)

BJ(α) =2J + 1

2J· coth[2J + 1

2Jα]− 1

2Jcoth

α

2J

α.=gLSJ · µB · J ·B

kB · T (1.17)

For the particular case J = 1/2 (a spin 1/2, s-state Atom) we obtain thesimple equation

< µz >= µB tanh[µB ·BkB · T ] (1.18)

For small arguments α, BJ(α) ≈ J+1J

α3and we obtain the Curie law

< µ(P )z >≈ (gLSJ)

2 · J(J + 1) · µ2B

3kB · T · B (1.19)

which contains the purely quantum mechanical quantity (gLSJ)2 · J(J + 1).

The paramagnetic susceptibility in this limit amounts to

χP ≈ µ0C

T(1.20)

with the Curie constant

C =N

V

(gLSJ)2 · J(J + 1) · µ2

B

3kB(1.21)

At room temperature is χP ≈ 10−3. Notice that the determination of theCurie constant is a key experiment to access the quantities J and gLSM deter-mining the ground state electronic configuration of atoms and thus providea reliable test of our quantum mechanical approach to the ground state con-figuration in atoms.


Figure 1.5: Temperature dependence of 1χfor some Cupric salts


1.4 Exchange interaction and the

Heisenberg-Dirac-Van Vleck Hamilto-

nian

An estimate of the triplet-singlet splitting in atoms is often obtained byexplicitly computing the lowest energy states of the simplest many elec-trons atomic system: the He-atom. The full Hamiltonian in the Born-Oppenheimer approximation amounts to H0 + V (1, 2) with (e2 standing fore2

4πǫ0)

H0(1, 2) =−~

2

2m(~∇2

1 +~∇2

2)−Ze2

r1− Ze2

r2

V (1, 2) =e2

r12(1.22)

Let us first neglect V12. The energy levels of H0(i) are Eni= −Z2e2

2·a·n2i, a being

the Bohr radius. The eigenfunctions of Eniare ϕnilimi

=Rni,li

r· Yli,mI(ϑ, ϕ).

The ground state of H0 corresponds to the state in which both electrons arein a 1s-orbital and has energy E0 = 2 · E1 =

−Z2e2

a. Its wave function is

ψ0 = ϕ1s(1)ϕ1s(2)⊗ χ(1, 2)

ϕ1s(1)ϕ1s(2) =1

π(Z

a)3e−

Za(r1+r2)

and

χ(1, 2) =1√2(u1/2u−1/2 − u−1/2u1/2)

The wave function ϕ1s(1)ϕ1s(2) is symmetric with respect to change of thecoordinate vectors and the ground state of He is a singlet, as required byPauli principle. The net magnetic moment is vanishing: S = 1 is prohibitedin the ground state.Let us now introduce V12 as a perturbation. The energy of the ground stateis modified to EG = E0 +Q with

Q =

∫

dV (1)dV (2)ϕ21s(1)

e2

r12ϕ21s(2)

In order to explicitly compute this last integral, one develops 1/r12 in spherical harmonics:

1

| ~r1 − ~r2 | =4π

r1

∑

l,m

1

2l + 1(r2r1

)l · Y ∗

l.m(ϑ1, ϕ1)Yl,m(ϑ2, ϕ2) ⇐⇒ r1 > r2

1

| ~r1 − ~r2 | =4π

r2

∑

l,m

1

2l + 1(r1r2

)l · Y ∗

l,m(ϑ1, ϕ1)Yl,m(ϑ2, ϕ2) ⇐⇒ r1 < r2


Inserting this development in Q and using the orthogonality of spherical harmonics weobtain

Q =4e2

π(Z

a)6∫

∞

0

dr1r2

1 · e−2Zr1

a [1

r1

∫ r1

0

dr2r2

2e−

2Zr2

a +

∫∞

r1

dr2r2e−

2Zr2

a ]

Partial integration leads to Q = 5Ze2

8aand positive. Notice that the correction Q is of the

same order of magnitude as E0.

Summarizing : the two He-electrons have a ground state configuration (1s)2

with energy EG = −Ze2

a(Z − 5

8).

The first excited state of the H0-operator corresponds to the electronicconfiguration (1s)1(2s)1, with antisymmetric wave function

ψS=0 =1√2[ϕ1s(1)ϕ2s(2) + ϕ1s(2)ϕ2s(1)]⊗ χs

ψS=1 =1√2[ϕ1s(1)ϕ2s(2)− ϕ1s(2)ϕ2s(1)]⊗ χt

The S = 0 wave function belongs to the parahelium thermal scheme, theS = 1-wave functions belong to the orthohelium thermal scheme. Paraheliumshows diamagnetism, orthohelium is a paramagnet. Without consideration ofV12 para and ortho (1s)(2s) states are degenerate. In the absence of spin-orbitcoupling, optical transitions between the two thermal schemes are absolutelyforbidden. Should one be able to pump a He atom gas (by electrons excitationor other mechanism) into the triplet excited state, then it will practicallystay forever (many months) in that state. Notice that because of spin orbitsplitting the triplet states degeneracy is lifted and a fine structure appear inthe excitation spectrum of ortho helium.We show now that the result of introducing the Coulomb interaction is thelifting of the degeneracy between triplet and singlet states, leading, in somecircumstances, to the formation of a magnetic moment, at least in one excitedstate 1s2s with different orbital quantum numbers. We consider the fourstates 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 00 E1s + E2s +Q− J 0 00 0 E1s + E2s +Q− J 00 0 0 E1s + E2s +Q− J

mit

Q =

∫

dV1dV2ϕ21s(1)ϕ

22s(2)

e2

r12

J =

∫

dV1dV2ϕ∗1s(1)ϕ

∗2s(2)

e2

r12ϕ1s(2)ϕ2s(1)


from which the sougth for eigenvalues can be read out immediately. The in-tegral Q is the Coulomb energy. The integral J is the result of the exchangeinteraction and is called exchange integral. It provides the exchange en-ergy contribution that arises from the correlation of the two electron as aconsequence of symmetrizing the wave functions according to the Pauli prin-ciple. The Coulomb interaction produces, via Pauli principle, a splitting ofthe initial degeneracy of the (1s)(2s) configuration: the singlet state has theenergy 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 ofpara (singlet) - and ortho (triplet) states. Levels with the same quantumnumbers of the orbital wave functions (e.g. (1s)(1s)) are prohibited in orthohelium. Furthermore, the splitting between ortho and para states depends onthe 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 lowerthan the s-state. The first Hund’s rule states that the positivity of J whendegenerate non-equivalent states are involved is a general feature of atoms.The strength of the splitting – the strength of the parameter J which we willcall 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 theexchange interatcion actually originates.One can formally obtain the t− s splitting by caricaturing the exchange in-teraction (which is actually acting in the orbital space) with an effective spinHamiltonian: the Heisenberg-Dirac-Van Vleck operator. Dirac defined theoperator 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 ofwriting P12 is

P12 =E + ~σ1 · ~σ2

2=

1

2[E + 4 · ~S1 · ~S2]

where ~σ are the two-by-two Pauli matrices and the product σiσj must betaken as a Kronecker product of matrices. The spin Hamiltonian simulating


the original Hamiltonian acting in the orbital space reads

HSpin = (E1s + E2s +Q) · E − J · [12E + 2~Sr · ~Ss]

It can be formally generalized to many electrons:

HSpin =∑

r

Er +1

2

∑

r,s

Q(r, s)−∑

r 6=s

J(r, s) · [14+ ~Sr · ~Ss]

where r, s is a set of quantum numbers describing orbital wave functions and afactor 1

2has been placed in front of the sums to avoid double counting. Dirac

has shown with quite general arguments that the eigenvalues of this effectivespin Hamilton operator are correct within first order perturbation theory.The difficulty is one of computing the various coupling constants forthcomingin the operator, which ultimately requires knowledge of the orbital wavefunctions.

Chapter 2

Magnetism in solids

In the first chapter we have shown how atomic magnetic moments are pro-duced. On these grounds, diamagnetism and paramagnetism, which are themain manifestation of atomic magnetism at finite temperature, are well un-derstood. However, we still need to answer the broad question about ” whyFe is ferromagnetic”. This means that we need to understand 1. what hap-pens to an atomic magnetic moment when it is embedded into a ”see” of freeelectrons, 2. how do magnetic moments (if they ”survive” the contact withthe free electrons) couple to align along the same direction (what makes thetriplet state between spins on different lattice sites energetically preferredwith respect to the single state) and what is the strength of the exchange en-ergy 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 insolids. In a solid, electrons have, in reality, wave functions which, for somekind of inner shell electrons, like the d-electrons in transition metals, havea strong component localized in the vicinity of the ion core (remember thetight-binding model). So to speak, they spend a lot of time close to the ioncore: during this time they are almost atomic like and can feel the Hund ruleas we know it from atomic physics. We will see that this Hund rule providesultimately the net persistence of the net magnetic moment at each atom inthe solid state, although the contact with free electrons acts to destroy themagnetic moment, in general. Although the atomic magnetic moment mightpersist when the atom in embedded into a solid, there are two importantexperimental 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 momentsis the result of the Hund rules that weight correctly, at least empirically,the orbital angular momentum and the spin of the electrons into formingthe total angular momentum of the many electrons in one atom. The totalangular momentum produces the net atomic magnetic moment, which isultimately responsible for the occurrence of magnetism in matter. Whenthe magnetic moment per atom is measured in the solid state, it appearsthat only the spin part of the total angular momentum is contributing toit: one speaks of ”quenching” of orbital angular momentum, in particularin transition metals, by the crystal field. Second, in angular momentumquenched metals, magnetic moments per atom should be an integer multipleof 2µB, i.e. their magneton number is an integer. This would lead, forexample, to atomic magnetic moments of 2,3, respective 4 µB for Ni, Co andFe. The experimentally measured value in bulk Fe is 0.616, 1.715 and 2.216µB. The Stoner-Wohlfahrt-Slater model of magnetism provides the correctframework to explain the occurrence of 1. finite magnetic moments in solidsand 2. the existence of non integer magneton numbers.

While the formation of a magnetic moment, because of the first Hundrule, is almost the rule in atoms, it is a very rare event in solid, wherethe electrons can be considered as delocalized and are therefore betterdescribed by a band structure. The SW model in its simplest versionconsiders free electrons where energy levels are filled up to the Fermi radiuskF = (3π2N/V )1/3, N/V being the electron density. In virtue of this filling

the electron gas has a total kinetic energy amounting to Ekin = N 35

~2k2F2m

.The non magnetic ground state foresees that all states up to EF arefilled with two electrons carrying opposite spins. Introducing an exchangeinteraction 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 ofenergy bands and, ultimately, the formation of a magnetic moment. Onthe other side, the radius of the Fermi sphere must be increased to host allthe electrons, as the double occupancy of each level is no longer possible.This produces an increase of the total kinetic energy that goes against theformation of a magnetic moment. Therefore, the formation of the magneticmoment is the result of a delicate energy balance and is subject to strongrestrictions.Let us now work out the Stoner criterion for ferromagnetism, i.e. for theexistence, in a Fermi gas subject to exchange interaction, of an imbalancebetween spin up and spin down electrons. We start from a density of


states n0(E) common to both spin channels and change the energy levelsaccording to E±

~k,ν= E~k,ν ∓ 1

2I · M , where + refers to majority spins. I

is the intra atomic exchange interaction responsible for the Hund’s rule inatoms, and P is the sought for spin imbalance, M = N↑ −N↓. Accordingly,the density of states separates out for the two spin channels according ton(±E) = n0(E ∓ 1

2I ·M). Integrating up to the (to be determined) Fermi

Figure 2.1: DOS for majority spins and minority spins are shifted by anamount IM .

energy EF gives the number of electrons per unit cell and the total momentper unit cell:

N =

∫ EF (M)

0

[n0(E +IM

2) + n0(E − IM

2)]

M =

∫ EF (M)

0

[n0(E +IM

2)− n0(E − IM

2)] (2.1)

These are two equations for the sought for parameters EF (M) and M . Solv-ing the first one can obtain in principle EF (M). Inserting this in the secondone we obtain an implicit equation for M :

M = F (M);

F (M) =

∫ EF (M)

[n0(E +IM

2)− n0(E − IM

2)] (2.2)


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. dFdM

|M=0≥ 0.Proof: From

F ′(M)M=0 =I

2

[

n0(E +IM

2) + n0(E − IM

2)

]

M=0

+

[

n0(E +IM

2)− n0(E − IM

2)

]dEF

dM M=0(2.3)

we obtain F ′(0) = I · n0(EF ) ≥ 0

M∞ is the largest magnetic moment by complete spin polarization of theelectron gas and corresponds to the first Hund’ rule magnetic moment. Underthese conditions, the graphical solution of the implicit equation for M hastwo possible scenarios: either M = F (M) has only the solution M = 0 forF ′(0) < 1, or it has two solutions: M = 0 and M finite but not necessarilyan integer for F ′(0) > 1. In this case one can show that the M = 0 solutionmaximizes the total energy, while the two solutions with opposite sign arethe sought for minima that establish a finite spin imbalance in the groundstate. 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. Thetendency to ferro-magnetism therefore requires a high density of states ofthe non-spin polarized band structure at the Fermi level. This can only beachieved 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 partiallyfilled d shells. In the following figures we will illustrate the Stoner criteriumon a set of examples, starting with single magnetic impurities in metals.


Figure 2.2: Graphical solution of the implicit equation for M . In A the onlysolution is M = 0, in B a finite magnetization minimizes the energy.

Figure 2.3: Left: Local DOS of Mn in Ag according to LSDA computationsby R. Podloucky et al. Phys. Rev. B33, 5777 (1980). The spin splitting isabout 3 eV (experiment: 4 eV). Right: Computed and measured values of3d- impurity atoms in Ag, Cu and Al. The highest moment appears alwaysin the middle of the 3d-row.


Figure 2.4:


Figure 2.5: FeRh: dispersion curves for the two spin directions; (a) spin+,(b) spin(-). The broken line is the Fermi level.


Figure 2.6: top: Band structure of Fe along the ∆ direction. The minorityspin bands (broken lines) are shifted toward higher energies with respect tomajority spin bands (continuous lines). The symmetry label of the bands isalso shown. In contrast to the Stoner model, the shift is not exactly rigid butdepends slightly on ~k. On the top of the figure, the ~k-points selected by theused photon energy are indicated. Bottom. On the left is an energy resolvedphoto emission spectrum taken at normal emission from a (100)-surface ofFe. 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 plottedseparately, by suitably compounding the total intensity and the polarizationdata. The two peaks are identified as due to electrons originating from thespin split bands close to the Γ-point, see top.


2.2 Friedel-Oscillations

2.2.1 The interatomic exchange interaction

The long sought explanation for the origin of ferromagnet was provided af-ter 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 isthat we have some localized magnetic moments, and this conditions is metin Fe by the localized d-electrons, which keep a part of their atomic magneticmoment produced by the Hund-rule intra atomic exchange. The secondcondition for ferromagnetism it that all these moments line up parallel toeach other, i.e. triplet state coupling between neighboring spins. Howeverwe know that the chemical bonding between equivalent orbitals favors, inline with the Pauli principle, the singlet state, so that we need a mecha-nism that acts ”against” the Pauli principle in order to get triplet coupling.This alternative mechanism is provided, according to Stearns, by the indi-rect exchange between localized d-electrons through RKKY coupling withthe delocalized part of the d-wave functions (or with the s-like electrons).This mechanism is therefore based on the existence of ”degenerate” statesin the band structure of solids. The RKKY coupling mechanism is also acentral one in modern research on magnetism and we want to illustrate itspeculiarities with a simple, computable exact model which is also relevant inthin films coupling phenomena.Notice that the singlet coupling underlying the chemical bond and the ab-sence 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 thatstates that in a linear arrangement of atoms the non-magnetic state, i.e. thestate with lowest total spin, is the ground state. One needs to go higher thanone dimension to escape this theorem, because in higher dimensions electronsstates with different symmetry – atomic orbitals with different quantum num-bers – can hybridize: it is this degeneracy between orbital wave function withdifferent symmetry that provide a route to escape the strong Pauli princi-ple that favor antiparallel alignment between orbital states with the samequantum number. The situation is exactly the same as in atoms: only ifthe electronic states participating to the formation of the magnetic momenthave different quantum number, the exchange interaction can act to lowerthe energy of the triplet state. The situation can be therefore summarizedas follows. The chemical bond between same orbitals centered at differentatoms favors the antiparallel ground state, in virtue of Pauli principle. Thecrystal potential, however, can act to mix different symmetries and differentorbitals into the wave functions forming the valence bands in solids. As in


atoms, different symmetries might favor energetically the parallel coupling,thus producing ferromagnetic alignment between neighboring atoms. How-ever, it depends on the crystal potential and on the orbitals involved, whethera total spin in the ground state is formed or not.We point also out that the strength of the effective exchange energy thatfavors the triplet coupling between two different sites (the interatomic ex-change interaction) is one to two orders of magnitude smaller that the one-site (intra-atomic) exchange interaction (which amounts to about 3− 5 eV).This means rotating one spin in the presence of the other ones needs muchless energy that suppressing the magnetic moment. It is the interatomic ex-change interaction which is relevant for determining the temperature scaleat which collective ferromagnetic order vanishes (in the next chapter, the socalled Curie temperature).

2.2.2 RKKY oscillations

The presence of a more or less localized magnetic moment, made of d-wavefunctions creates a potential sink with the strength of the s − d exchangeinteraction at the location of the magnetic moment for majority (spin up)s-electrons, in virtue of the atomic Hund-rules. The minority spin downelectrons can be considered as non-affected by the impurity. A local per-turbation in one spin channel produces an oscillating density in the affectedspin channel (see the Anhang at the end of this chapter), while the otherspin remains uniformly distributed. This produces a local spin polarizationof the electron gas surrounding the impurity

P =2· < Sz >

~=ρ+ − ρ−

ρ+ + ρ−≈ O(

κ cos 2kFx

kFx) (2.5)

that propagates far away from the perturbing magnetic moment. At somelocation x within the spin polarized s-electron gas a spin imbalance appears.This spin imbalance acts as an effective exchange field for d-waves functionsand tends to align a d-derived magnetic moment at that location parallelto itself: A magnetic moment at the location x would lower his energy byaligning along the direction of P . In this way, the exchange interactioncan propagate, oscillating between positive and negative depending on theposition x and can couple spins which are quite distant from each other.


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 magnetiza-

tions 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 schemat-

ically 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).


2.2.3 Anhang: mathematical details of the model

We introduce in a 1d free electron gas a perturbing potential localized at theorigin 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 tosimplify the mathematics, we consider a Dirac-Delta like perturbation poten-tial of strength λ located at the origin of a one dimensional solid filled witha 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, wedo not use the more realistic but also more cumbersome traditional potentialwell 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). arepresents the width of an hypothetical well and λ its strength. We refer tothe segment with x < 0 as the left-hand side l and to the segment with x > 0as the right-hand side r. We have to solve the Schr”odinger equation

[−~

2

2m+ V (x)]ψ(x) = Eψ(x) (2.6)

under the boundary conditions for a δ like potential

ψl(0) = ψr(0); ψ′l(0)− ψ′

r)0)−2mλa

~2ψ(0) = 0 (2.7)

In the two regions l and r the respective solutions have to fulfill the SE

−~2

2mψ(x) = Eψ(x) (2.8)

We distinguish two cases: E < 0 and E > 0.

For E < 0 the SE away from the singularity has two solutions, onegrowing exponentially toward ±∞, the other decaying exponentially toward±∞. This last solution is the only physical one as it has a finite norm andproduces a bound state with energy amounting to

Eb = −~2κ2

2m= −mλ

2a2

2~2(2.9)

(κ.= λam

~2) and wave function

ψbl (x) =

√κeκx ψb

r(x) =√κe−κx (2.10)


In the range E > 0 we have free electrons moving left and right, andthe solutions in each range l, r, under periodic boundary conditions at±L/2, are

E =~2k2

2m; k =

2π

L· n

ψl(x) = Al

√

2

Lsin kx+ Bl

√

2

Lcos kx); ψr(x) = Ar

√

2

Lsin kx+ Br

√

2

Lcos kx)

both basis functions being normalized to 1/2 in the range x ∈ [± − L/2, 0].The boundary conditions at x = 0 read

Bl = Br; k · (Al − Ar)−2mλa

~2Bl = 0 (2.11)

There are two classes of wave functions fulfilling these conditions. One classhas

Bl = 0; Al = Ar ⇒

ψul (x) =

√

2

Lsin kx; ψu

r (x) =

√

2

Lsin kx

where the total wave function is normalized to 1 over the segment withlength L.

The second class has Bl = Br.= B 6= 0 and must be even under

change of sign, so that Ar = −Al = A and B = Akκ. This type of wave

functions read

ψgl (x) = A[

√

2

Lsin kx+

k

κ

√

2

Lcos kx)

ψgr (x) = A[−

√

2

Lsin kx+

k

κ

√

2

Lcos kx)

A must be chosen so that the entire wave function is normalized to 1 in therange x ∈ [−L/2, L/2]. This means

A =κ√

κ2 + k2(2.12)

and

ψgl (x) =

1√κ2 + k2

[

√

2

Lκ · sin kx+

√

2

Lk · cos kx)

ψgr (x) =

1√κ2 + k2

[−√

2

Lκ · sin kx+

√

2

Lk · cos kx)


Using these wave functions, we compute the total charge density at apoint, e.g. x ≥ 0:

ρ(x) = κe−2κx +2

L· L2π

·∫ kF

0

dk[(κ sin kx− k cos kx)2

κ2 + k2+ sin2 kx

]

= κe−2κx +1

π·∫ kF

0

dk[

1− κ2 cos 2kx+ κk sin 2kx

κ2 + k2

]

= κe−2κx +kFπ

− 1

π

∫ ∞

0

dk[κ2 cos 2kx+ κk sin 2kx

κ2 + k2

]

+1

π

∫ ∞

kF

dk[κ2 cos 2kx+ κk sin 2kx

κ2 + k2

]

One can prove by complex integration that

ρ(x) =kFπ

+κ·i2π

e−2κx ·[

E1(−2κx+ 2ikFx)− E1(−2κx− 2ikFx)]

E1(z) being the exponential integral. Notice that the contribution ofthe bound state to the total charge density cancels out with part ofthe contribution of the free electron states. The following figure plotsthe charge density as a function of the variable x for some characteristicvalues of κ and k.

Some particular limits are worked out now. For small x we have(E1(z) ≈ −γ − lnz)

ρ(x) ≈ kFπ

+κ

π(π − arctan

kFκ)−O(x) ≈ kF

π+κ

2−O(x)

For large z we have

E1(z) ≈e−z

zand accordingly

e2κx−2ikF x

−2κx+ 2ikFx− e2κx+2ikF x

−2κx− 2ikFx=

4ie2κx

4κ2x2 + k2Fx2

[

κx sin 2kFx− kFx cos 2kFx]

and in the limit of large x (and small κ) we obtain

ρ(x) ≈ kFπ

+O(κ cos 2kFx

kFx) (2.13)


Figure 2.8:

This result underlines the formation of a Friedel-like oscillation of the chargedensity away from a localized perturbation. The wave-length of the oscilla-tion is π

kF. 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 thethird power of the inverse distance.


Part II

A. Vindigni

Chapter 3

Magnetic order at finitetemperature

In the first two Chapters

• we have shown how magnetic moments are created at the atomic levelaccording to the Hund’s rules (intra-atomic exchange interaction);

• we commented on how atomic magnetic moments, deduced assum-ing spherically symmetric surrounding (Hund’s rules), generally reducewhen the atom is “put” in a crystal, which lowers the symmetry of itsenvironment (Stoner-Wohlfahrt model);

• we have shown how an interatomic exchange interaction can arise ina metal by means of the RKKY interaction;

• we defined the conditions under which a metal may show ferromagneticcoupling between different magnetic moments.

Already at that level, it was clear that ferromagnetism is not the rule butrather an exception, in the sense that many factors that are encounteredin ordinary materials usually prevent the formation of magnetic moments orthat of a ferromagnetic interatomic coupling. All the above-mentioned prop-erties1 have been deduced neglecting the temperature or, in other words, theyare ground-state properties. In this Chapter we discuss the consequences ofintroducing the temperature. The general trend is that thermal fluctuationsdestroy the ground-state ferromagnetism (when present). In the same lineas before, we will define some conditions under which ferromagnetism can“survive” at finite temperatures as well.

1apart from the Brillouin function.

39

CHAPTER 3. MAGNETIC ORDER AT FINITE TEMPERATURE 40

3.1 Coupled effective spins: an N-body prob-

lem

In the previous Chapters we have considered the conditions under which anisolated atom possesses a finite magnetic moment. Hund’s rules allow com-puting the ground-state multiplet, characterized by the total angular mo-mentum (orbital plus spin contribution) which results from all the unpairedelectrons. When dealing with coupled magnetic moments, we will indicatethe atomic total angular momentum with S i) in order to avoid confusion withthe exchange interaction (J) and ii) because – in this context – people oftenspeak about “spin” or “effective spin” to indicate the total single-atom an-gular momentum. As far as a single isolated atom is concerned, its magneticmoment at finite temperature is well described by the Brillouin function:

m = −gµB〈Sz〉 = gµB S BS

[gµB S B

kBT

]

, (3.1)

with

BS (α) =2S + 1

2Scoth

(2S + 1

2Sα

)

− 1

2Scoth

( α

2S

)

and α =gµB S B

kBT.

(3.2)In the derivation of this function, we have implicitly used the knowledge ofi) the eigenstates of the atom in an external applied field and ii) the wayof performing thermal averages for a quantum system (see Appendix). Notethat the intra-atomic and the Zeeman interaction have been treated on adifferent ground: we have considered only the ground-state multiplet (whichminimizes the intra-atomic exchange interaction) but we have applied Boltz-mann statistics to the levels of this multiplet in case they have been splitby an external field (Zeeman interaction). The reason for such a differenttreatment reside in the characteristic energy scales of the two interactions inrelationship with the thermal energy kBT . In fact, the intra-atomic exchangeenergy is of the order of 4− 10 eV∼ 105 Kelvin, while the Zeeman splittingis roughly 0.1 meV ∼ 1 Kelvin for one-Tesla applied field.Further on, we have seen how a ferromagnetic interatomic exchange inter-action is necessary for the occurrence of ferromagnetism in a solid. Underspecific and relatively strict conditions, this goal is attained by means of theRKKY interaction2. The order of magnitude of the RKKY interaction is10− 50 meV ∼ 100− 500 Kelvin. Thus, depending on the material and the

2Other mechanisms are responsible for exchange interaction, e.g. super-exchange ordirect exchange, in insulators.


Typical exchange energies and magnetic moments

Tc[K] C[K] µ[µB] J [eV] J [k] Ms[Gauss]Fe 1043 2.22 2.22 0.012 139 1746Co 1395 2.24 1.71 0.015 174 1446Ni 629 0.588 0.605 0.013 151 0.510Gd 289 7.1 0.00025 2.9 2.060Dy 87 2920EuO 69.4 4.68 1930EuS 16.5 3.06 1240

Table 3.1: Some typical values for the energy scales, Curie temperature, Curieconstant and saturation magnetization Ms = gµBS/a

3 (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 requiredfor the RKKY interaction as well. The competition between this interatomicexchange interaction and thermal fluctuations is indeed responsible for theloss of ferromagnetism above a certain temperature, called Curie tempera-ture TC . Table 3.1 reports the values of the Curie temperature, the exchangeinteraction and other relevant parameters for few typical magnetic materials.Let us come back to the formal treatment of magnetism at finite tempera-tures, restricting ourselves to ferromagnetic exchange interactions. A systemof coupled magnetic moments arranged in a lattice can then be described bythe Hamiltonian

H = −1

2J∑

|n−n′|=1

S(n) · S(n′) + gµBB∑

n

Sz(n) . (3.3)

The dimension of the Hilbert space associated with this quantum many-bodyproblem scales as (2S + 1)N , N being the number of magnetic moments(spins) in the lattice. Due to such an exponential dependence on N , theexact treatment of a system of many coupled spins becomes intractable –even numerically – as far as the number of spins approaches that of realisticextended systems3. In practice, one can try to circumvent this problem inseveral ways:

3Some effective zero-dimensional structures (magnetic clusters or nanoparticles) are alsostudied in the context of nanomagnetism. For some of these systems, exact diagonalizationof the associated quantum problem is still feasible numerically and makes it possible todescribe their magnetic behavior at any temperature.


1. Reduce the many-body problem to a single-particle problem. Thiscorresponds to the mean-field approximation (MFA).

2. Simplify the problem replacing the quantum-spin operators by classicalvectors.

3. Take advantage of specific symmetries in the problem under investiga-tion and use a Hamiltonian which can easily be diagonalized.

4. Consider only a selected family of excitations of the ground state, whichcan 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 many-body problem (3.3) to the a single-particle problem. This means to getrid – somehow – of terms which directly involve two-spin operators such asS(n) · S(n′). In this context, we understand a paramagnet as the referencesingle-particle problem. We will first make use of the Brillouin function (3.1)

MFA

Single-particle

problem

Many-body

problem

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 relevantimplications. Further on, within the more rigorous Landau approach, wewill prove that the MFA is actually the best approximation of the Hamilto-nian (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 ex-perienced by each spin in a ferromagnetic sample contains a contribution


arising from the interaction with its neighbors, besides the typical Zeemanterm (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-particleHamiltonian of the form

Hsp(n) = gµB (B + BW) Sz(n) (3.4)

where gµBBW = −zJ〈Sz(n)〉Hsp . The origin of the Weiss field BW is theinteratomic exchange interaction, whose effect is taken into account onlyas an average and not rigorously. Such an average is performed using theHamiltonian (3.4) itself and z indicates the number of nearest neighbors ofeach spin. As anticipated, the Hamiltonian Hsp is equivalent to the one ofa paramagnetic atom in a magnetic field Bt = B + BW so that the thermalaverage of the Sz(n) projection is given by the Brillouin function:

〈Sz(n)〉Hsp = −SBS(α) (3.5)

with

α =gµB S B

t

kBT=gµB S B − zJS〈Sz(n)〉Hsp

kBT. (3.6)

Since the average 〈Sz(n)〉Hsp is also contained in α, i.e. the argument of theBrillouin function, Eq. (3.5) is actually a self-consistent equation. To writeEq. (3.5) in a more transparent way, we exploit the relation between theaverage of the spin component along the field and the associated magneticmoment m = −gµB〈Sz(n)〉Hsp . The MF equation of state finally reads:

m = gµB S BS

[gµB S B

kBT+

zJ S m

gµB kBT

]

. (3.7)

In order to visualize the solution of Eq. (3.7) graphically, it is convenient toset

σ =m

gµB S= BS(α)

σ =kBT

zJS2α− gµBB

zJS.

(3.8)

Let us list the most remarkable facts arising from the graphical analysis ofsolutions depending on the external parameters T and B.

1. When B = 0, there exists a non-trivial solution with σ 6= 0 only ifthe slope of the Brillouin function exceeds that of the straight line (the


second one of Eqs. (3.8)). In particular, expanding the former aroundα ≃ 0 yields

BS(α) ≃S + 1

3Sα + . . . (3.9)

so that a spontaneous magnetization (σ 6= 0) only arises for T < TCwith

TC =S + 1

3S

zJS2

kB. (3.10)

One can show that if solutions with σ 6= 0 exist, they have a lower freeenergy than the solution corresponding to σ = 0.

2. For T < TC , the system of Eqs. (3.8) admit two graphical solutions ofopposite sign in the region B ∈ [−Bc, Bc]. Outside of this interval thesolution 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 linearizedand the system of Eqs. (3.8) takes the simplified form

σ =S + 1

3Sα

σ =kBT

zJS2α− gµBB

zJS;

(3.11)

by using the definition of TC given in Eq. (3.10), the solution of theprevious set of equations can be written as

(

1− T

TC

)

σ = − gµB

zJSB (3.12)

or equivalently (using again Eq. (3.10))

m = gµBSσ =(gµB)

2

zJ

TCT − TC

B =(gµB)

2S(S + 1)

3kB

1

T − TCB .

(3.13)The pre-factor of B on the right-hand side is the susceptibility (com-puted in B = 0)

χ =C

T − TC(3.14)

which is the well-known Curie-Weiss law with

C =(gµB)

2S(S + 1)

3kB(3.15)

being the Curie constant (already encountered when discussing para-magnetism).


Equation (4.49), together with other MF predictions, is not expected to holdtrue 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 theend of this chapter. Keeping in mind the limitations of the MFA, it is stillinteresting to investigate how different observables should behave accordingto 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 thevicinity of TC . Since the results presented in this section (3.3) do not dependon the value of the effective spin S, we will deduce some scaling relations forthe simplest case: S = 1/2. For this particular case, BS(α) = tanh(α) andTC = zJ/4kB. Additionally, we assume that S = 1/2 refers to the spin of anelectron4 so that g = 2. Under these hypotheses, Eqs. (3.8) can be writtenin the compact form

σ = tanh

(µB B

kBT+TcTσ

)

. (3.16)

By using the fact that tanh(α) ≃ α − 13α3 for α ≃ 0, Eq. (3.16) can be

expanded for small σ and B as follows:

σ =

(µB B

kBT+TcTσ

)

− 1

3σ3 +O(Bσ2) (3.17)

which, for T ≃ TC and neglecting higher infinitesimal than σ3, becomes apolynomial of the reduced temperature τ = (T − TC)/TC :

µB B

kBT=

(T − TCTC

)

σ +1

3σ3 = τσ +

1

3σ3 . (3.18)

Equation (3.18) is suitable for deriving some critical exponents.

Mean-field critical exponents

1. Setting B = 0, one has(T − TcTc

)

σ +1

3σ3 = 0

4In fact, any system whose ground state is two-fold degenerate with a degeneration thatcan be removed by the application of an external field can be thought of as possessing aneffective spin 1/2, with g 6= 2 in general.


whose solutions are

σ = 0 for T ≥ Tc

σ ≃√3

(

1− T

Tc

)

︸︷︷︸

:=−τ

βMF

∝ (−τ) 12 for T < Tc .

This result provides the value of the critical exponent β within themean-field approximation: βMF = 1/2.

2. Now we want to evaluate the behavior of the susceptibility around TC .First, let us recall the proportionality relation

χ(T,B = 0) =

(∂M

∂B

)∣∣∣∣B=0

∼ ∂σ

∂B.

Then the derivative ∂σ/∂B can easily be put in relationship with thereduced temperature and σ by differentiating both sides of Eq. (3.18):

µB

kBTC∼ −

(

1− T

TC

)

· ∂σ∂B

+ σ2 ∂σ

∂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 σ2 so that

∂σ

∂B∼ µB

1

T − TC⇒ χ(T ) ≃ C

T − TCfor T > TC . (3.19)

This is nothing but the Curie-Weiss law deduced in an alternative wayin Eq. (4.49). For T > TC , instead, we have to take into account thatσ2 ≃ −3τ . In this case one has

∂σ

∂B∼ µB

2

1

TC − T⇒ χ(T ) ≃ C

2

1

TC − Tfor T < TC .(3.20)

The Eqs. (3.19) and (3.20) give the mean-field prediction for anothercritical exponent: γ = 1.

Critical exponents in general

The fact that these observables behave like powers of the reduced temper-ature τ close to the transition point is not an artifact of the MFA. On the


contrary, this feature defines the condition of criticality. Other critical expo-nents can be deduced similarly to β and γ. Below we recall the definition ofsome 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, τ < 0

χ(B = 0, τ) ∼ |τ |−γ

|M(τ = 0, B)| ∼ |B|1/δ .

Another important feature captured by the MFA is that critical exponentsdo not depend on the details of the model, e.g., on J and the details ofthe lattice (dimensionality or z). However, this universality of mean-fieldcritical exponents is somewhat exaggerated: for instance the correct criticalexponents 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 numericalvalues obtained for the 3d Heisenberg model (see the following sections):

MFA 3d-Heisenbergα 0 (Jump) −0.11± 0.006β 0.5 0.365± 0.002γ 1.0 1.386± 0.004δ 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 ratherdepend on more general symmetries of the experimental system under inves-tigation. The concept of universality class is associated with this property.


3.4 The Landau approach

There is no univocal way of introducing it nor the literature is consistent inwhat it is meant by Landau approach to critical phenomena. The procedureproposed here aims at highlighting how the MFA is a drastic simplificationof the more general theoretical-field approach based on functional integrals.We will try to avoid confusion between the Landau free-energy functionaland the MF (Gibbs) free energy. Finally, we should be able to discuss somequalitative arguments whose validity goes beyond the mean-field approach.

To avoid useless complications, we refer to S = 1/2 operators and assumethat only their z component enters the spin Hamiltonian:

H = −1

2J∑

|n−n′|=1

Sz(n) Sz(n′) + gµBB∑

n

Sz(n) . (3.21)

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 bewritten in terms of two-valued classical variables σi = ±1:

H [σ] = −1

2

∑

i,j

σi Ji,j σj +1

2gµBB

∑

i

σi , (3.22)

where σ = (σ1, σ2, . . . σN) and Ji,j is a symmetric matrix describing the (ex-change) coupling among different spins in the lattice5. The partition functionassociated with the Ising Hamiltonian reads

Z (B, T ) = T rσ

exp

[

1

2β∑

i,j

σi Ji,j σj − h∑

i

σi

]

(3.23)

with β = 1/kBT and h = βgµBB/2. The summation over all the configura-tions 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 amathematical digression and recall the well-known Gaussian identity

∫

ℜe−

κ2η2−sη dη =

√

2π

κes

2/2κ (3.24)

which can easily be obtained from the integral of a Gaussian by completingthe square at the exponent. With some more efforts, the above result can be

5Referring to Hamiltonian (3.21) the non-zero terms of Ji,j equal J/4. However, thematrix Ji,j can describe a more general coupling among spins.


generalized to N variables η = (η1, η2, . . . ηN) ∈ ℜN :

∫

dη1, dη2 . . . exp

(

−1

2ηTMη − s · η

)

dηN =

√

(2π)N

det(M)exp

(1

2sTM−1s

)

.

(3.25)Apart from a constant pre-factor, the right-hand side of Eq. (3.25) becomesequal 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 (B, T ) = NT rσ

∫

dη1, dη2 . . . e−ηTJ−1η/2β e−σ·η e−σ·h dηN

(3.26)

where6 h = (h, h, . . . h) and N is a constant irrelevant for magnetic observ-ables. The trace over the variables σ appearing in Eq. (3.26) can now beperformed analytically

T rσ

e−

∑

i(h+ηi)σi= Π

i

∑

σi=±1

e−(h+ηi)σi

= 2NΠicosh(h+ ηi) . (3.27)

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 (B, T ) = N ′∫

dφ1 . . . exp

(

−1

2βφTJφ

)

2NΠicosh

(

h+ β∑

j

Ji,jφj

)

dφN .

(3.28)The latter is usually expressed in a more compact form

Z (B, T ) =

∫

D[φ] e−βL[φ] (3.29)

where the symbol∫D[φ] ∝

∫Πidφi stands for the functional integral and

L[φ]=

1

2

∑

i,j

φi Ji,j φj−β−1∑

i

ln

[

cosh

(

h+ β∑

j

Ji,jφj

)]

−β−1N ln(2)

(3.30)for the Landau free-energy functional. In this representation Z has beenrewritten as a Gaussian average over the auxiliary fields φ of the partition

6The vector h has this simple form because a uniform B has been assumed but – inprinciple – different sites could experience different external fields.


function of a paramagnet, experiencing the external field plus the auxiliaryfields themselves. By doing this, we have somehow “traded” the originalspin-spin interaction with the coupling (in principle site-dependent) of eachspin with a set of auxiliary fields φj. Note that no assumption has been madeon such fields which can, thus, span all over ℜN .From Eqs. (3.29) and (3.30) an implicit form for the averaged spin projections〈σi〉 can be deduced. This is obtained straightforwardly if we let the field hbe site dependent:

〈σi〉 = −∂ ln(Z)

∂hi= − 1

Z∂Z∂hi

= −〈tanh(

h+ β∑

j

Ji,jφj

)

〉φ , (3.31)

where 〈. . . 〉φ stands for average over the auxiliary fields. We will comeback to this result in the following. Generally, performing the functionalintegral in Eq. (3.29), i.e. tracing over the auxiliary fields φ, is far frombeing trivial. The simplest approximation which can be made to evaluate Zis replacing the functional integral by the maximum value of the integrand,namely

Z (B, T ) =

∫

D[φ] e−βL[φ] MFA= Max

φ

e−βL[φ]

= exp

−βMinφ

(L[φ])

(3.32)which is known as saddle-point approximation. This is equivalent to themean-field approximation. In fact, by requiring ∂L/∂φi = 0 for φi = φi, thefollowing equation is obtained

φi = tanh

(

h+ β∑

j

Ji,jφj

)

. (3.33)

As we are considering nearest-neighbor ferromagnetic exchange coupling, thesolution to the previous equation turns out to be independent of the site indexi, meaning that the field which minimizes the Landau free-energy functionalis spatially homogeneous. Consequently, Eq. (3.33) is equivalent to the MFequation of state (3.16). It is worth remarking that, within this framework,only when it is evaluated in its minimum the Landau free-energy functionalL[φ]acquires the meaning of Gibbs free energy:

F (B, T ) = −β−1 ln (Z)MFA= Min

φ

(L[φ])

= L[φ]. (3.34)

The equivalence between the MFA and the saddle-point approximationof the functional integral (3.32) allows establishing that the average of the


auxiliary fields φ is proportional to local magnetic moments. In other words,

the average φ plays the role of the Weiss field BW (apart from constantfactors). As far as the critical behavior (T ≃ TC) is concerned, it makessense to expand the Landau free-energy functional for small values of thefields φ. To this aim we make use of the Taylor expansion

ln [cosh(x)] =1

2x2 − 1

12x4 +O(x6) for x ≃ 0 . (3.35)

After some algebra and taking the continuum limit φi → φ(x) one obtains

LGL [φ] =

∫ [1

2J (∇φ)2 + b

2φ2 +

λ

4φ4

]

ddx− β−1N ln(2) (3.36)

with

b = zJ(1− 1

4βzJ

)= zJ

(1− TC

T

)≃ zJτ

λ = 13β3(12zJ)4

= 163

(TC

T

)3kBTC ≃ 16

3kBTC

(3.37)

where we have used the fact that the MF Curie temperature is TC = zJ/4kBfor a system of spins one-half (see Eq. (3.10)). (The subscript in LGL standsfor Ginzburg-Landau). Note that in both Eqs. (3.30) and (3.36) the para-magnetic limit L = −kBTN ln(2) is recovered at high temperature, whenβJ → 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 appearingin Eq. (3.36) is always positive. On the contrary, τ can change its signoriginating two different free-energy landscapes. For τ > 0, the Landaufunctional LGL has a minimum for φ = 0 only, which clearly correspondsto the magnetically disordered phase. For τ < 0, the Landau functionaldisplays the typical Mexican-hat shape with two minima occurring at somefinite φ = ±φ (see Fig. 3.2). These minima are degenerate in the absenceof an external field and correspond to the non-trivial solution of the MFequation of state for T < TC . When φ = φ, LGL

[φ]acquires the meaning

of Gibbs free energy. Then, from the knowledge of the Landau free-energyfunctional the MF critical exponent α related to the specific heat can bededuced. For T > TC (τ > 0), we have LGL

[φ]= −kBTN ln(2) so that

the specific heat C = −T∂2LGL/∂T2 = 0. For T < TC (τ < 0), instead,

LGL

[φ]= −kBTN ln(2) + zJO(τ 2). Therefore the specific heat is finite

when TC is approached from lower temperatures. This discontinuity impliesthat 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


Figure 3.2: Sketch of the Landau free-energy functional: the landscapechanges from a Mexican-hat shape to a single-minimum function when pass-ing from the ferromagnetic to the paramagnetic phase.

the problem. Landau developed his theory of phase transitions starting fromthe idea that the effective free energy should be an analytic function of theorder parameter (which needs to be identified with φ in our approach), con-sistently with the requirements of symmetry of the considered problem. Infact, all the critical properties (critical exponents, etc.) derived in the previ-ous 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 thecalculation from first principles (i.e., starting from Hamiltonian (3.22)). Forproblems characterized by less trivial symmetries and, e.g., vectorial orderparameters, being able to write the Landau free-energy functional on thebasis of symmetry arguments alone is often very useful. Then, performing asaddle-point approximation analogous to Eq. (3.32) one can normally deduceMF critical exponents with little mathematical efforts. Such exponents arenamed classical, or mean-field, critical exponents and depend only on thesymmetry of the problem reflected in the functional LGL. However, a func-tional built with the same symmetry criteria can be used as a starting point


for more sophisticated mathematical treatments, like the renormalization-group approach. A successful example is the theory of critical phenomenafor 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 amongthem are fulfilled in the ground state, MF theory predicts that magnetic or-der is retained up to some finite temperature TC . Above this temperatureferromagnetism is lost. This scenario describes a phase transition which isindeed observed in real ferromagnets. However, the Curie temperature givenby formula (3.10) is almost always an overestimate compared to the valuescomputed with more sophisticated methods or observed in experiment. Thisis because fluctuations, neglected in MF theory, tend to have a disordering ef-fect, 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: theIsing 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 param-eter appearing in the Landau free-energy functional of Eq. (3.36). Moregenerally, MF critical exponents depend on the symmetry of the consideredproblem but not, e.g., on the dimensionality of the lattice. The fact thatsuch exponents are independent of the dimensionality is another artifact ofthe MF approximation. On the contrary, two facts remain true beyond theMF approximation: i) critical exponents do not depend on some details ofthe system such as the strength of interactions while ii) they do depend onthe symmetry of the considered problem.In the following some of these issues will be clarified in the context of classicalspin models.

Spins with continuous symmetry

The substitution of the quantum-spin operators in Hamiltonian (3.3) by clas-sical spins is somewhat justified in the limit S → ∞, that is when the rela-tive spacing between levels inside each multiplet S(n) becomes smaller andsmaller. Moreover, when correlations among spins develop, cooperative ef-fects create a sort of collective large spin which behaves classically. Then,


one has:S(n) → ~S(n) ≡ S0 (sin θ cosϕ, sin θ sinϕ, cos θ) (3.38)

where S20 = S (S + 1) (more often S0 = 1 is assumed and the term S2

0 =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

H = −1

2J∑

|n−n′|=1

~S(n) · ~S(n′) + gµBB∑

n

Sz(n) . (3.39)

and, accordingly, the partition function becomes

Z =

∫

dΩ1

∫

dΩ2 . . .

∫

dΩNe−βH(~S(n)) , (3.40)

with dΩn = sin θndθndϕn being the solid-angle element of the spin located atthe site n.In some cases, due to the symmetry of the problem, it is more realistic todescribe each spin with a two-component vector (in-plane), which can thusbe parameterized with just one angle

~S(n) ≡ S0 (cosϕ, sinϕ) . (3.41)

Vocabulary of classical models with continuous symmetry :

• three-component ~S(n) ≡ S0 (sin θ cosϕ, sin θ sinϕ, cos θ):classical Heisenberg model

• two-component ~S(n) ≡ S0 (cosϕ, sinϕ):classical planar or XY model.

The Landau free-energy functionals associated with these two types of clas-sical spins are generally different, between them and from the one given inEq. (3.36) for the Ising model (see below). This means that the number ofcomponents of the order parameter determines the Landau free-energy func-tional and eventually the critical behavior of a system. This is one of thefeatures characterizing a specific universality class.

Spins with discrete symmetry: the Ising model

When consistent with the symmetry of the problem, two-value classical spins,Sz, can be assumed:

H [Sz(n)] = −1

2J∑

|n−n′|=1

Sz(n)Sz(n′) + gµBB∑

n

Sz(n) . (3.42)


The configurations of the classical Hamiltonian (3.42) correspond to the spec-trum of eigenvalues of the quantum Hamiltonian

H = −1

2J∑

|n−n′|=1

Sz(n) Sz(n′) + gµBB∑

n

Sz(n) , (3.43)

which is, indeed, diagonal on the basis

|ϕi〉 ≡ |σ1σ2 . . . σN〉

with Sz|σn〉 =1

2σn|σn〉 and σn = ±1 .

(3.44)

More often, the Ising model is introduced directly assuming the Hamiltonian

H [σn] = −1

2J ′

∑

|n−n′|=1

σn σ′n − h′

∑

n

σn (3.45)

with classical variables σn = ±1. Note that the Hamiltonian (3.45) is equiv-alent to the one given in Eq. (3.22) in which the nearest-neighbor coupling isexpressed in a matrix form. The model (3.45) corresponds to the exact spec-trum of eigenvalues relative to the Hamiltonian of coupled quantum spinsone-half given in Eq. (3.43) provided that:

J ′ = 14J

h′ = −12gµBB .

(3.46)

However, the Ising model is applied in many different contexts rather thanmagnetism, ranging from biophysics to social sciences.

Magnetic order and lattice dimensionality

Probably one of the most striking failure of MF theory is the prediction ofa magnetic phase transition for d=1. In fact, for one-dimensional systemsrigorous proofs exist which forbid the occurrence of a magnetically orderedphase at finite temperature in the sole presence of short-range coupling be-tween spins. But let us clarify first what is meant by lattice dimension inthis specific context. The dimension d corresponds to the number of direc-tions along which the exchange coupling propagates indefinitely. In practice,this dimension may also be different from the actual dimensionality of theconsidered solid. If the latter is D, in general one has d≤D.The lattice dimensionality d is another fundamental feature characterizing aspecific universality class.


Limiting ourselves – for now – to the Ising model, we start considering thecase d=1.

d=1An Ising chain composed of N spins can be represented schematically as inFig. 3.3. Following an argument due to Landau, we evaluate the variation

-2 1

2

JE E ED = - =

2 1 lnB

S S S k ND = - =

2 1 ln2

B

JF F F k T ND = - = -

,E2 S 2 ,E1 S 1

Figure 3.3: Sketch representing the free energy difference between a uniformstate, with all the spins parallel to each other, and a configuration consistingof two domains with opposite spin alignment (one domain wall).

of the free energy associated with the creation of a domain wall in a con-figuration with all the spins parallel to each other. Creating a domain wallincreases the exchange energy by a factor J/2. However, such a domain wallmay occupy N different positions in the spin chain, so that this set of con-figurations has an entropy ∼ kB ln(N). The free-energy difference betweenthe two configurations sketched in Fig. 3.3 is given by

∆F =J

2− kBT ln(N) . (3.47)

Thus, splitting the ground state into domains is

convenient if ln(N) > J2kBT

⇒ N > eJ/2kBT

inconvenient if ln(N) < J2kBT

⇒ N < eJ/2kBT .(3.48)

The inequalities written above suggest an estimate of how many consecutivealigned spins can be found at finite temperature in an Ising chain. In par-ticular, when the thermodynamic limit N → ∞ is taken, one immediatelyrealizes that it is always convenient to split the system into groups of paral-lel spins (magnetic domains), i.e., ferromagnetism is destroyed at any finitetemperature.


d=2A similar, but more rough argument can be given for d=2 as well. In thiscase we should refer to the possibility of reversing a cluster of spins enclosedin a perimeter of l lattice sites and embedded in a region of spins all pointingin the same direction. We consider for simplicity a square lattice. The totalcost in terms of exchange energy is of the order ∼ lJ/2. To estimate theentropy we can think of a self-avoiding random walk: at each step the walkerhas 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 perimeterl to be of the order pl, with p < 3. As a result, the free-energy variationassociated with the flip of a cluster delimited by a perimeter l is roughly∆F = lJ/2 − kBT l ln p. Therefore, for T < J/(2KB ln p) the ordered phaseshould be stable against the formation of large domains of reversed spins.This argument for the existence of a phase transition in the 2d Ising modelwas first given, in more precise terms, by Peierls.

Rigorous results

The Ising model represents a particularly lucky case in which the heuristicarguments given above can be checked by solving the problem analytically.Even if we will not derive these results, it is useful to recall which crucialsteps should be followed to prove rigorously whether a model is consistentwith a phase with spontaneous magnetization (finite magnetization in zeroexternal field) for T 6= 0 or not. To this end, one has to compute:

1. the partition function

Z = T re−βH[Sz(n)] (3.49)

where the trace is obtained by letting each discrete variable take thetwo possible values Sz(n) = ±1/2 (Z is a sum with 2N terms!)

2. the average magnetic moment

m(T,B) = − 1

N

∂F

∂B=

1

N

1

β

∂ lnZ∂B

(3.50)

3. the limitm(T, 0) = lim

B→0+m(T,B) (3.51)

and evaluate if there exists a temperature TC below which thelimit (3.51) takes a non-zero value.


This procedure can be carried out analytically for d=1 or d=2 only, producingdifferent results:

• for d=1, no spontaneous magnetization is possible at finite tempera-tures

• for d=2, a spontaneous magnetization appears for T < TC ≃ 2.27 J4kB

.

Indeed these exact results show that the MF approximation overlooks someimportant features as it predicts the occurrence of a phase with spontaneousmagnetization independently of the dimension d.

d=1: spin chains with uniaxial anisotropyTo fix the ideas, we take S = 1/2. As anticipated, in this case

m(T, 0) = limB→0+

m(T,B) = −gµB limB→0+

limN→∞

[

1

N

∑

n

〈Sz(n)〉]

= 0 . (3.52)

For the 1d case, two-spin correlations can also be computed:

〈Szi S

zi+r〉 =

1

4tanh

[(βJ

4

)]r

=1

4e−r/ξ (3.53)

with

ξ = − 1

ln[tanh

(βJ4

)] . (3.54)

ξ is called correlation length and it is a fundamental quantity in the studyof critical phenomena. The correlation length of the 1d Ising model is char-acterized by an exponential divergence at low temperatures:

ξ ∼ eJ/2kBT . (3.55)

By comparing the inequalities in Eq. (3.48) with the formula for the corre-lation length it is clear that ξ gives the order of magnitude of the averagesize of groups of correlated spins. The existence of such a correlation, marksa 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) =∂m

∂B∼ ξ

T(3.56)

In practical cases


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 etal., Physical Review B 69 p.132408 (2004). At low temperature, χT saturatesbecause of the presence of non-magnetic impurities and 3d interactions withthe other spin chains in the crystal.

• the plot of 1/χ versus T highlights deviations from the paramagneticbehavior (Curie-Weiss law)

• the plot of ln [χT ] versus 1/T highlights an 1d Ising-like behavior (whenexperimental points at low temperature lie on a line).

The behavior of two-spin correlations for the 1d Ising model is plotted inFig. 3.5.

d=2: ultrathin magnetic films with uniaxial anisotropyThe 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). Toour purposes, it is enough to recall the formula which gives the spontaneous


magnetization7 for T < TC :

m(T, 0) = limB→0+

m(T,B) = −gµB limB→0+

limN→∞

1

N

[∑

n

〈Sz(n)〉]

= gµB

[

1− sinh−4

(βJ

2

)] 18

(3.57)

and the definition of TC itself

sinh

(J

2kBTC

)

= 1 ⇒ TC =2

1 +√2

J

4kB≃ 2.27

J

4kB. (3.58)

As anticipated at the beginning of this Chapter, MF theory typically overes-timates 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) fora spin one-half and for z = 4 (square lattice): TMF

C = J/kB.Expanding the spontaneous magnetization m(0) close to TC yields

m(T, 0) ∼ (TC − T )18 . (3.59)

Thus, for the 2d Ising model β = 1/8 at odds with the MF value βMF = 1/2.

1

4

MFA

Ising

1

4

Ising

Figure 3.5: Two-spin correlation function for the ferromagnetic Ising chain(adapted from Quantum and Statistical Field Theory, M. Le Bellac).

7We identify the magnetization with the average magnetic moment per magnetic atom(or molecule), while in the SI (Systeme international d’unites) it is the average mangeticmoment per unit volume.


3.6 Correlation functions

Figure 3.6: Group of people in the “paramagnetic” (a) and in the “ferro-magnetic” (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 ef-fect of neglecting fluctuations in the mean-field approximation. A pictorialidea of what happens when a system passes from the paramagnetic phaseto the ferromagnetic phase is sketched in Fig. 3.6. In the picture on theleft-hand side, people walk in the street without conditioning each other, likemagnetic moments do in the paramagnetic phase. In the picture on the right-hand side, instead, a strong feed-back mechanism is present so that if one ofthe individuals is attracted, e.g., by a window all the others are conditionedand end up staring at the same thing. This situation can be assimilated tospontaneous symmetry breaking occurring in a magnet below TC . However,everyday experience offers also intermediate degrees of correlation in whichsuch a feed-back mechanism involves a limited number of people. Think,for instance, of a road artist playing music in a subway station: the major-ity of people will be more concerned of not missing the train rather thanlistening at his/her music. Nevertheless, there will still be a sort of short-range correlation among the people whose train is not departing soon andwhose attention is captured by the musician. This last situation resemblesshort-range correlations in magnetic systems.


Figure 3.7: Magnetic specific heats of the S=1/2 Ising model for d=1,2,3 andwithin the MF approximation. d=2 is given by the Onsager solution for thesquare lattice while d=3 is obtained by high-temperature series expansionfor a simple cubic lattice. All temperatures are expressed in units of thecorresponding MF transition temperature, θ, with z = 2, 4, 6 for the d=1,2,3respectively. (Taken from L. J. de Jongh and A. R. Miedema Adv. Phys. 50p. 947-1170 (2001)).

Specific heat tail

As already noticed, the mean-field theory predicts a finite discontinuity in thespecific heat at the critical temperature. On the contrary, in experimentalsystems showing a magnetic phase transition the specific heat diverges at TC .This behavior is reproduced by more sophisticated models. In Fig. 3.7, thebehavior of the specific heat is plotted for the Ising model. There the MFprediction is compared with exact results for d=1,2 and high-temperatureseries expansion for d=3 (no exact results solution is available in this caseyet).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 isreduced, down to TC = 0 for the Ising spin chain (d=1). This is an indicationthat the effect of thermal fluctuations becomes progressively more dramaticas the lattice dimensionality is reduced.Second, apart from the MF calculation, all the models show a high-temperature tail in the specific heat. Also this feature is more enhanced


the lower the lattice dimensionality is. Such tail is due to the presence ofshort-range correlations above TC that are not taken into account in the MFtheory. Both calculations for d=2 and d=3 show the expected singularity atTC , while the specific heat does not diverge at any temperature for d=1, noteven at T = 0.

The most natural way to characterize short-range correlations is by study-ing 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

m(T,B) = − 1

N

∂F

∂B=

1

N

1

β

∂ lnZ∂B

=1

N

1

β

1

ZT r[

−βgµB

∑

n

Sz(n)

]

e−βH

= −gµB1

N

∑

n

〈Sz(n)〉

(3.60)

where the trace is taken over all the possible values of the N variables Sz(n).With the definition of Eq. (3.60), the magnetization equals the average mag-netic moment. This value can be converted to any other unit to comparewith experiments (Bohr magneton per atom, emu/mol, A/m, etc.).The susceptibility is the derivative of the magnetization with respect to theapplied field

χ(T,B) =∂m(T,B)

∂B=

1

N

1

ZT r

β(gµB)2∑

nn′

Sz(n) · Sz(n′)

e−βH

− 1

Nβ

1

ZT r[

gµB

∑

n

Sz(n)

]

e−βH

2

=β(gµB)

2

N

〈(∑

n

Sz(n)

)2

〉 − 〈∑

n

Sz(n)〉2

,

(3.61)

where we have used the fact that(∑

n Sz(n)

)2

=∑

nn′ Sz(n)Sz(n′).

Defining the correlation function as

Gnn′ = 〈Sz(n)Sz(n′)〉 − 〈Sz(n)〉〈Sz(n′)〉 , (3.62)


a relation between Gnn′ and the magnetic susceptibility can be deduced

χ(T,B) =β(gµB)

2

N

∑

nn′

Gnn′ , (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) (takenfrom Quantum and Statistical Field Theory, M. Le Bellac).

Eq. (3.62), when 〈Sz(n′)〉 6= 0 (i.e. for T < TC or for B 6= 0) the assertionthat “two spins are uncorrelated” means 〈Sz(n)Sz(n′)〉 = 〈Sz(n)〉〈Sz(n′)〉 ∝m2. In other words, the correlation function Gnn′ only measures the degreeof short-range correlation. The term relating to long-range order as beeneliminated by subtracting 〈Sz(n)〉〈Sz(n′)〉 (see Eq. (3.62)).

Susceptibility for the different magnetic phases at B = 0

2d-3d systems T ≃ TC 1d-system Paramagnet

χ(T, 0) χ ∼ Γ±

|T−TC |γ χ ∼ ξT

χ = CT

From the previous table and from the theorem (3.63), it is clear that thedifferential susceptibility is strictly related to the degree of correlation offluctuations at the considered temperature:

χ(T,B) =β(gµB)

2

N

∑

nn′

Gnn′ . (3.64)


Note that even for the ordered phase (occurring for d≥2) the susceptibilitydiverges only at T = TC and tends to zero as T → 0 (in a perfectly orderedsystem fluctuations are not allowed).

Fourier transform of correlations

The Fourier transform (G) of the correlation function can be accessed exper-imentally, for instance through neutron scattering. At T = TC , G(q) is foundto behave as

G(q) ∼ 1

q2−η. (3.65)

A simple argument based on dimensional analysis suggests that

G(r) ≃∫

G(q)ddq ⇒ G(r) ∼ 1

rd−2+ηat T = TC . (3.66)

From Eq. (3.66) we learn that at the critical point spatial correlations decaywith power law. In other words, the system is not characterized by any typ-ical length scale and possesses the property of being self-similar at differentspatial scales8. Such a scale invariance is the key ingredient of the theory ofcritical phenomena and it is, indeed, strictly related to the divergence of thecorrelation length for T → TC . Notice that the fluctuation-response theoremmay also be written as

χ(T,B) ∼ G(0) . (3.67)

Since in practice η < 2 always, Eq. (3.65) implies that G(q = 0) diverges forT → TC and so does the susceptibility.In the next section we will see that within the Landau theory of criticalphenomena the Fourier transfrom of the correlation function is given by

G(q) ∼ 1

q2 + ξ−2. (3.68)

The inverse Fourier transform of Eq. (3.68) gives the asymptotic behavior

G(r) ∼ e−r/ξ

r(d−1)/2for T far away from TC . (3.69)

In this case the decay of the correlation function is characterized by thetypical length scale ξ, the correlation length. The different behavior of the

8See the paper C. H. Back, et al., Nature 378, p. 597. The authors report on theexperimental check of the scaling hypothesis on a Fe film which behaves as the 2d Isingmodel.


correlation function above and below TC and both in real and reciprocalspace is sketched in Figs. 3.8 and 3.9.It is worth remarking that we are considering short-range correlations as theunique source of broadening of the quasi-elastic peaks. Of course, experimen-tally this is not true and the correlation length can be deduced only afterremoving 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 FieldTheory, M. Le Bellac).

3.7 Landau theory of correlations

Before deducing correlations within the Laundau theory, let us recall somebasic concepts that have been discussed some sections ago. First, we showedthat the calculation of the partition function of the Ising model (for anydimension d) can be recasted into the following problem:

Z (B, T ) =

∫

D[φ] e−βL[φ] (3.70)

from which, in principle (but not always in practice!), the whole thermody-namics can be deduced. Two possible independent approximations can bemade to tackle the problem stated by Eq. (3.70):

1. the saddle-point approximation, which consists in evaluating the par-tition function only in the minimum of the functional L [φ] (seeEq. (3.32));


2. a Taylor expansion of L [φ] itself for small auxiliary fields φ, which holdsin the critical region (i.e., for T ≃ TC) and gives the Ginzburg-Landaufunctional 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 averagemagnetic 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 startfrom the Ginzburg-Landau functional in Eq. (3.36) (which we recall here forconvenience)

LGL [φ] =

∫ [1

2J (∇φ)2 + b

2φ2 +

λ

4φ4

]

ddx− β−1N ln(2) , (3.71)

set the gradient term to zero (because we seek for spatially homogeneoussolutions) and minimize the integrand with respect to φ. This leads to theequation

φ(bφ+ λφ3

)= 0 (3.72)

whose solutions are given by

φ = 0 for τ > 0

φ2 = − bλ

for τ < 0 ,(3.73)

with b ≃ zJτ and λ ≃ 16kBTC/3.To evaluate correlation functions we need to go slightly beyond the crudesaddle-point approximation (equivalent to the MFA). In practice, we allowthe field φ to deviate slightly from the MF solution obtained for τ < 0 (thetreatment for τ > 0 is analogous):

φ = φ+ δφ , (3.74)

δφ being a small, random field. The integrand (fGL) of the Ginzburg-Landaufunctional in Eq. (3.71) takes the form

fGL[φ+ δφ] =1

2J (∇δφ)2 + b

2(φ+ δφ)2 +

λ

4(φ+ δφ)4

=1

2J (∇δφ)2 +

(b

2+

6λ

4φ2

)

δφ2 + fGL[φ] + (. . . )

=1

2J (∇δφ)2 − b δφ2 + fGL[φ] + (. . . )

(3.75)


where (. . . ) stands for constants, O(δφ4) and odds terms in δφ which vanishafter spatial or thermal averaging. The first two terms in the last line ofEq. (3.75) are associated with fluctuations around the MF solution φ. Thus,at this level of approximation, the partition function reads:

Z (B, T ) = e−βLGL[φ] +

∫

D[δφ] e−βLfl[δφ] (3.76)

where the first term on the right-hand side corresponds to the saddle-pointapproximation while

Lfl[δφ] =

∫ [1

2J (∇δφ)2 − b (δφ)2

]

ddx . (3.77)

is the functional associated with the fluctuation field δφ(x). The underlyingstrategy of the mathematical passages described above aims at consideringcorrections to the saddle-point approximation which are described by a sortof quadratic Hamiltonian with respect to the fluctuation field. In fact, thefunctional (3.77) is formally equivalent to the potential energy of set of cou-pled harmonic oscillators, described in the continuum formalism (rememberthat b < 0 for τ < 0): Thermal averages of the fluctuation field δφ(x) can becomputed similarly to average displacements in a system of harmonic oscil-lators. The gradient term in Lfl[δφ] effectively couples the fluctuation fieldsδφ(x) defined at different points in space, at different locations x. However,the functional (3.77) can be decoupled (diagonalized) passing to the Fourierspace:

Lfl[δφ] =1

(2π)d

∫1

2

(Jq2 − 2b

)|δφ(q)|2ddq . (3.78)

If we forget the parametric9 dependence on temperature of b, Eq. (3.78) hasthe form of a quadratic Hamiltonian with respect to the independent degreesof freedom δφ(q). Now, equipartition theorem can be applied to get

1

2

(Jq2 − 2b

)〈|δφ(q)|2〉fl =

1

2kBT ⇒ 〈|δφ(q)|2〉fl =

kBT

Jq2 − 2b, (3.79)

where the subscript reminds that 〈. . . 〉fl represents an average over the fluc-tuation field δφ.Before proceeding, it is useful to establish a contact between the correlation

9This implicit temperature dependence, which may look strange at first sight, comesfrom having expanded LGL [φ] around its minimum: it is reasonable that coefficients of theexpansion contain information about the saddle point φ, corresponding to the minimum.


function defined in Eq. (3.62) and the fluctuation field δφ. In the formalismof the present section we shall write the correlation function as

G(x, x′) = 〈σ(x)σ(x′)〉 − 〈σ(x)〉〈σ(x′)〉= 〈(φ+ δφ(x)

) (φ+ δφ(x′)

)〉fl − φ2

= 〈δφ(x)δφ(x′)〉fl .(3.80)

By comparing Eq. (3.79) with Eq. (3.80) we obtain the following result

G(q) =kBT

Jq2 − 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

G(q) =kBT

Jq2 + b. (3.82)

Summarizing, within the Landau theory, the correlation function takes theOrnstein-Zernicke form:

G(q) =kBT

J

1

q2 + ξ−2, (3.83)

with ξ ∼ |τ |−1/2. As discussed at the end of Section 3.6, ξ has the meaningof correlation length. The corresponding classical critical exponent ξ ∼ |τ |−ν

is νcl = 1/2.


Literature

• A. Aharoni, Introduction to the Theory of FerromagnetismOxford University Press(Chapter IV: Magnetization vs. Temperature)

• J. Cardy, Scaling and Renormalization in Statistical PhysicsCambridge University Press(Chapter VI: Low dimensional systems)

• M. Le Bellac, Quantum and Statistical Field TheoryOxford University Press(Chapter I: Introduction to critical phenomena. Chapter II: Landautheory)

• G. Morandi, F. Napoli, E. Ercolessi, Statistical MechanicsWorld Scientific Singapore(Chapter III: Spin Hamiltonians I: Classical)

• L. D. Landau and E. M. Lifshitz, Statistical PhysicsOxford Pergamon Press

• D. C. Mattis, The Theory of Magnetism IISpringer Series in Solid-State Science(Advanced)

Chapter 4

Magnetic domains and domainwalls

4.1 Magnetic anisotropy

Let us go back to consider a single magnetic center. For the atom embeddedin a spherically symmetric environment Hund’s rules generally succeedin predicting the observed magnetic moment. When this scenario holds, thespin “points” with the same probability along any spatial direction in theabsence of an external magnetic field. Due to the reduced symmetry of thesurrounding, the situation is generally different for an atom in a solid. Asalready seen in Part I, a first consequence is that magnetic moments aregenerally smaller in solids with respect to those predicted by Hund’s rules

DS2

J

p0 2p

Figure 4.1: Schematic representation of the energy landscape associated witha uniaxial-anisotropy term as a function of the polar coordinate θ of a classicalspin.

71

CHAPTER 4. MAGNETIC DOMAINS AND DOMAIN WALLS 72

(remember the quenching of the angular momentum). Another implicationis that magnetic moments (effective spins) prefer to lie along some crystal-lographic directions. This tendency is taken into account by introducing amagnetic anisotropy energy which is function of the effective-spin projectionsalong the crystallographic axes. The simplest anisotropy term that can beconsidered in single-spin Hamiltonian is

HA = −D(Sz)2 . (4.1)

Notice that the symmetry Sz → −Sz is not broken by such an anisotropyterm. Additional terms which combine higher powers of the single-spin op-erators may arise according to the symmetry of the lattice in which themagnetic atom is embedded (or according to the symmetry of the substratefor adatoms). For example, in the case of a crystal lattice with cubic sym-metry the first non-zero anisotropy term is a fourth-order combination of thespin operators; thus in this case the term in Eq. (4.1) vanishes.The physical mechanism which couples the spin degrees of freedom with thespatial 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, DS2,and the magnetic moment per Co atom on Pt(111). Values in brackets havebeen computed with a different computational method. Remember that 1meV ≃ 11.6 K (C. R. Physique 6 p. 75 (2005)).

As stated above, a crucial ingredient for magnetic anisotropy to arise is thereduced symmetry of the surrounding, “seen” by a magnetic atom in a solid,with respect to the spherical symmetry (Hund’s rules). It is not surprisingthat a further increase of the anisotropy is observed when the symmetryof the environment is further reduced. This happens, e.g., when magneticatoms are arranged in clusters (0d) or in 1d and 2d nanostructures. In Fig 4.2


Figure 4.3: Experimental results: magnetic anisotropy energy DS2 (right,(b)), and magnetic moment per Co atom on Pt(111) (left, (a)) C. R. Physique6 p. 75 (2005).

theoretical predictions (from ab initio calculations) for different aggregatesof Co atoms on a Pt(111) surface are reported. Notice that when passingfrom a single atom to five atoms the value of the magnetic anisotropy peratom already decreases of one order of magnitude. The magnetic momentper atom also decreases with increasing the number of atoms. This factis 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 theexperimental results reported in Fig 4.3. Indeed, the fact that the magneticanisotropy increases up to a factor 103 when approaching the atomic scale isa good trend in view of magneto-storage applications.

Classical approximation

If the operator in Eq. (4.1) is substituted by a classical spin the anisotropyenergy reads

HA = −DS2 cos2 θ . (4.2)

Depending on the sign of D, the energy (4.2) has either one minimum forθ=π/2 (D<0) or two minima for θ=0, π (D>0), which describes the twophysical situations

D<0 easy plane

D>0 easy axis / uniaxial .(4.3)


For the easy-axis case, D > 0 (see Fig. 4.1), only few configurations aroundθ = 0 or θ = π will be statistically relevant for kBT ≪ DS2. In other words,the spin will spend about half of the time visiting configurations for whichθ ≃ 0 and half of the time around θ ≃ π. For kBT ≪ DS2, the escape ratefrom each one of the two wells is ν = ν0e

−DS2/kBT , so that the relaxationtime diverges exponentially as:

τA ∼ eDS2

kBT . (4.4)

This time represents the average time it takes the system to jump from oneminimum 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 exchangeinteraction which is isotropic. If we add to the classical Heisenberg Hamilto-nian (Eq. (3.3)) an anisotropy term like the one in Eq. (4.1) we get

H = −1

2J∑

|n−n′|=1

S(n) · S(n′) + gµB

∑

n

~B · S(n)−D∑

n

(Sz(n))2 . (4.5)

When D becomes large with respect to |J |, the model described by Hamil-tonian (4.5) can be replaced with the two following models

D

|J | → +∞ Ising model

D

|J | → −∞ XY / planar model .

(4.6)

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 ofnearest-neighboring spins is favored).


The study of the model described by the Hamiltonian (4.5) can be signifi-cantly simplified by substituting the quantum spin operators S(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 ferromag-netically (J > 0). Such groups can emerge in a magnetic system due to eitherlong-range order or short-range correlations. In the latter case, the correla-tion length needs to be large enough. In both situations the collective spincan be so large that its quantum-mechanical character becomes negligible1.In the appropriate temperature regimes, the results are then the same if theclassical 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 acrucial role. In particular, their structure and the energy associated with

1Remarkably, in this sense the classical-spin approximation is more justified for lowtemperatures than for high ones. In the paramagnetic limit (kBT/J >> 1) one has torecover a behavior described by the Brillouin function, in which the quantum nature ofeach spin is relevant (S2 → S(S + 1)).

0.01 0.1 1 10

0.5

1.0

1.5

2.0

Broad DW

Sharp DW

Figure 4.4: One-wall energy in J units vs D/J : minimum energy solution ofthe non-linear equation (4.10) computed numerically (solid line); continuumlimit solution (dashed line). Inset: spin profile vs lattice distance: sharp wall(low-right) and broad wall for D/J = 10−2 (up-left).


the creation of a domain wall in a uniformly magnetized configuration arerelevant. These features can be evaluated letting the spin direction varyonly along one spatial direction. This effectively reduces the problem to amono-dimensional one:

HH = −Nx∑

i=1

[

J ~Si · ~Si+1 +D (Szi )

2]

, (4.7)

where ~Si are classical spins and the constants J and D have to be thought ofper unit length or per unit surface if the dimensionality of the original latticewas d=2 or d=3 respectively.With the Hamiltonian (4.7), the domain wall can be larger than one latticespacing. In fact, spreading the wall over more than one lattice spacing reducesthe global exchange-energy cost. On the other hand, the anisotropy termwould favor configurations with as less spins misaligned to the easy axis, z,as possible. The domain-wall profile results from the competition betweenthese two energies (two opposite limits are reported in the insets of Fig. 4.4).The lowest-energy deviations from the uniform state can be parameterizedthrough the angle that each spin forms with the z axis, θ, as

EH =Nx∑

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 con-figuration is given by the spin profile which fulfills the boundary conditions

θ1 = π

θNx = 0(4.9)

and minimizes the energy (4.8) with respect to θi:

∂EH∂θi

= sin (θi − θi−1)− sin (θi+1 − θi) +D

Jsin (2θi) = 0 . (4.10)

Eq. (4.10) can be solved numerically and the solution provides the spin profilewith respect to which the energy (4.8) is stationary. The true lowest-energyprofile can be obtained comparing different solutions, among which the sharp-wall profile (see lower-right inset of Fig. 4.4):

θi = π for 1 ≤ i < Nx

2

θi = 0 for Nx

2≤ i ≤ Nx

(4.11)

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 (dashedline) – that we are going to present in the next paragraph – as a function ofthe ratio D/J .


4.3 Continuum formalism

Referring to the classical version of Hamiltonian (3.3), we rewrite in a differ-ent way the exchange-interaction term:

Hexch = −1

2J∑

|n−n′|=1

~S(n) · ~S(n′) = −J∑

n

∑

µ

~S(n) · ~S(n+ eµ) (4.12)

where µ = x, y, z (spatial directions) and eµ is the unit vector along µ. Noticethat∣∣∣~S(n)− ~S(n+ eµ)

∣∣∣

2

=∣∣∣~S(n)

∣∣∣

2

+∣∣∣~S(n+ eµ)

∣∣∣

2

− 2~S(n) · ~S(n+ eµ) . (4.13)

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 continuumspatial variable r = an, a being the lattice spacing. This approximation isjustified

• In the classical isotropic Heisenberg chain (D = 0) at low tempera-tures. In fact, the lowest lying excitations – which actually destroyferromagnetism 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 arebroad enough. Further on, we will render this statement quantitative.In the presence of ferromagnetic (J >0) exchange interaction and uni-axial anisotropy (D>0), such a requirement is fulfilled for J≫D.

Thus one has,

~S(n+ eµ)− ~S(n) ≃ ~S(r + aeµ)− ~S(r) ≃ a∂µ~S(r) (4.14)

where in the first passage we have taken the continuum limit and in thesecond one we have performed a Taylor expansion. Combining Eq. (4.13)with Eq. (4.14), the exchange interaction between the spin located in r andhalf of its nearest neighbors is obtained

− J∑

µ

~S(n) · ~S(n+ eµ)

≃ 1

2Ja2

∑

µ

∣∣∣∂µ~S(r)

∣∣∣

2

− J∑

µ

∣∣∣~S(r)

∣∣∣

2

=1

2Ja2

∑

µ,ν

(∂µSν(r) · ∂µSν(r))− J

z

2

∑

ν

(Sν(r) · Sν(r))

(4.15)


with z number of nearest neighbors and ν = x, y, z label of the spin compo-nents. Normally, the first term is written as

∣∣∣∇~S(r)

∣∣∣

2

=∑

µ,ν

(∂µSν(r) · ∂µSν(r)) =

= (∂xSx(r))2 + (∂yS

x(r))2 + (∂zSx(r))2

+ (∂xSy(r))2 + (∂yS

y(r))2 + (∂zSy(r))2

+ (∂xSz(r))2 + (∂yS

z(r))2 + (∂zSz(r))2 .

(4.16)

Taking the usual continuum limit for the sum

∑

· · · ≃ 1

ad

∫

. . . ddx , (4.17)

the classical and continuum version of Hamiltonian Eq. (4.5) is finallyobtained

H =1

2Ja2−d

∫ ∣∣∣∇~S(r)

∣∣∣

2

ddx− Jz

2

1

ad

∫ ∣∣∣~S(r)

∣∣∣

2

ddx

−D1

ad

∫

|Sz(r)|2 ddx+ µBg1

ad

∫

~B · ~S(r)ddx .(4.18)

Within the continuum model the field ~S(r) can be simplified as a two-component vector field or as a scalar field (see the two limits (4.6)). However,some additional constraints or effective energy terms are normally introduced

in place of the stringent constraint on the spin modulus∣∣∣~S(r)

∣∣∣

2

= S2. Of

course, the latter condition is automatically fulfilled if each spin is parame-terized with polar coordinates

Sx(r) = S sin(θ(r)) cos(ϕ(r))

Sy(r) = S sin(θ(r)) sin(ϕ(r))

Sz(r) = S cos(θ(r)) .

(4.19)

Broad domain walls: continuum limit

To the aim of computing the domain-wall energy in the continuum limit, welet 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

H =1

2JNyNzaS

2

∫[(

dθ

dx

)2

+ sin2(θ(x))

(dϕ

dx

)2]

dx

−DNyNzS2 1

a

∫

cos2(θ(x))dx+ const

(4.20)


where we have implicitly assumed the integration domain to be a paral-lelepiped NxNyNza

3. The functional Eq. (4.20) can be minimized with re-spect to the functions θ(x) and ϕ(x). The corresponding Euler-Lagrangeequation is

Ja sin2(θ(x))d2ϕdx2 + 2Ja sin(θ(x)) cos(θ(x))

(dθdx

) (dϕdx

)= 0

Ja d2θdx2 − Ja sin(θ(x)) cos(θ(x))

(dϕdx

)2 − 2Dasin(θ(x)) cos(θ(x)) = 0

(4.21)The solution to Eq. (4.21) with boundary conditions

limx→−∞ θ(x) = π

limx→+∞ θ(x) = 0(4.22)

and corresponding to the minimum energy is

cos(θ(x)) = tanh(xδ

)

ϕ(x) = const.(4.23)

with δ = a√

J2D

. Such a solution was proposed by Landau and Lifshitz in

1935.The energy density associated with the spin profile (4.23) is Ew = 2

√2S2

√DJ

(per unit length for d=2 and per unit surface for d=3). In Fig. 4.4 thedomain-wall energy obtained numerically for the discrete-lattice calculation(solid line) is compared with that obtained in the continuum limit (dashedline) as a function of the ratio D/J . The agreement is already good for ratiosD/J < 0.3. In the opposite limit, the discrete lattice calculation recovers thedomain-wall energy of the Ising model Ew = 2JS2 (sharp domain wall definedby Eqs. (4.11)).In those relevant limits one has

J ≪ D ⇒ δ = a and Ew = 2S2J

J ≫ D ⇒ δ = a√

J2D

> 1 and Ew = 2√2S2

√DJ .

(4.24)

For J ≪ D, the wall-energy cost equals the Ising case and follows from havingδ = a. Concerning J ≫ D, Ew is one-soliton energy 2. As one can appreciatein 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.

2Sometimes the domain-wall width is defined with some numerical factors of difference

with respect to δ: π√

J2D

or√

2JD

for instance.


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 simplificationof the classical Heisenberg model in terms of

Ising model D/J ≥ 2/3

continuum limit D/J ≤ 0.3 .(4.25)

Typically, metallic nanowires of technological relevance fall in the broad-wallregime. In fact, materials like Co, Ni, Fe or Permalloy are characterizedby 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 showa magnetically ordered phase at finite temperature for d=1 holds also forthe Heisenberg chain with uniaxial anisotropy, provided that the appropri-ate domain-wall energy is considered (remember Fig. 3.3). Similarly, oneconcludes that the same model can sustain ferromagnetism at finite tem-peratures in d=2. Different arguments are, instead, needed to provide aconclusive statement about the existence or not of magnetic order at finitetemperature in systems with continuous symmetry. For the last ones, it willturn out that linear excitations are able to destroy ferromagnetism both ind=1 and d=2. For the Heisenberg model, these linear excitations can beidentified with spin waves. Spin waves are usually introduced as linear so-lutions to the Landau-Lifshitz equation of motion. However, the capabilityof these type of excitations to destroy magnetic ordering for d≤2 in systemswith continuous symmetry can be evidenced without the need of introducingdynamics. We prefer to follow this way because it is straightforward to applya unique argument to both the XY and the Heisenberg model.

3The crossover ratio D/J = 2/3 can be obtained analytically by analyzing the stabilityof the sharp-wall profile, Eqs. (4.11), against small deviations between successive anglesθi (B. Barbara, Journal de Physique 34, p. 139 (1973)).


Linear excitations in models with continuous symmetry

We rewrite for convenience the classical spin Hamiltonian introduced in chap-ter 3, Eq. (3.39):

H = −1

2J∑

|n−n′|=1

~S(n) · ~S(n′) + gµBB∑

n

Sz(n) . (4.26)

The minimal energy is obtained by letting all the magnetic moments bealigned along the direction of the applied field (spins along negative z direc-tion). We consider how the energy increases due to small deviations fromthis configuration. Our goal is to simplify the original problem by means ofan effective Hamiltonian that is formally equivalent to the one describing asystem of coupled harmonic oscillators. To this end, we may write

Sz(n) = −√

1−∑

α=x,y

(Sα(n))2 ≃ −1 +1

2

∑

α=x,y

(Sα(n))2

with the hypothesis (Sα(n))2 ≪ (Sz(n))2 .

(4.27)

Note that for the planar (or XY) model α takes just one value and two valuesfor the Heisenberg model. From now on, we will not specify the number ofextra components but z represented by the index α; while doing so, we aregoing to derive results that apply to both models. The approximation inEq. (4.27) reflects in the Hamiltonian as follows:

H ≃− 1

2J∑

|n−n′|=1

[

1− 1

2

∑

α

(Sα(n))2

]

×[

1− 1

2

∑

α′

(Sα′

(n′))2

]

− 1

2J∑

|n−n′|=1

∑

α

Sα(n)Sα(n′)− gµBB∑

n

[

1− 1

2

∑

α

(Sα(n))2]

= −1

2zNJ − gµBBN +

1

2zJ

1

2

∑

n

∑

α

(Sα(n))2 +∑

n′

∑

α′

(Sα′

(n′))2

− 1

2J∑

|n−n′|=1

∑

α

Sα(n)Sα(n′) +1

2gµBB

∑

n

∑

α

(Sα(n))2 +O((Sα)4

)

≃Eg.s. +Hh.o.

(4.28)


where, by nothing that the double summations∑

n

∑

α and∑

n′

∑

α′ areactually the same, we have defined

Hh.o. =1

2zJ∑

n

∑

α

(Sα(n))2 − 1

2J∑

|n−n′|=1

∑

α

Sα(n)Sα(n′)

+1

2gµBB

∑

n

∑

α

(Sα(n))2(4.29)

and the constant ground-state energy

Eg.s. = −1

2zNJ − gµBBN . (4.30)

The Hamiltonian Hh.o., written in Eq. (4.29), is equivalent to the Hamilto-nian of N coupled harmonic oscillators which can be decoupled by the usualFourier transform in the discrete space:

Sα(n) = 1√N

∑

q Sα(q) e−iq·n

Sα(q) = 1√N

∑

n Sα(n) eiq·n

(4.31)

with orthogonality relation∑

n

ei(q−q′)·n = Nδq,q′ . (4.32)

For simplicity we assume unitary lattice constant. It is convenient to evaluatethe two relevant summations appearing in the Hamiltonian of Eq. (4.29)separately. The first summation reads

∑

n

(Sα(n))2 =1

N

∑

n

∑

q,q′

Sα(q)Sα(q′) e−i(q+q′)·n =∑

q

|Sα(q)|2 . (4.33)

This is nothing but the Parseval’s formula for the discrete-lattice Fouriertransform. For what concerns the second summation on the right-hand sideof Eq. (4.29), we first rewrite it as

∑

|n−n′|=1

Sα(n)Sα(n′) =∑

n

∑

δ

Sα(n)Sα(n+ δ) (4.34)

where δ is a vector connecting the site n with its nearest neighbors. Forsimplicity, we will consider just a linear, square and simple-cubic lattice ford=1, 2 and 3, respectively. Passing to the Fourier space one finds∑

n

∑

δ

Sα(n)Sα(n+ δ) =∑

n

∑

δ

1

N

∑

q,q′

Sα(q)Sα(q′) e−i(q+q′)·n e−iq′·δ

=∑

δ

∑

q

|Sα(q)|2 e−iq·δ =∑

q

|Sα(q)|2∑

δ>02 cos(q · δ) ;

(4.35)


the notation δ > 0 means that the summation extends over half of thenearest neighbors of the spin located at site n: it consists of z/2 terms.Eqs. (4.33) and (4.35) enable us to decouple the elastic Hamiltonian givenin Eq. (4.29), which then reads

Hh.o. =1

2J∑

q

∑

α

z −∑

δ>02 cos(q · δ)

|Sα(q)|2

+1

2gµBB

∑

q

∑

α

|Sα(q)|2

=1

2

∑

α

∑

q

Γ(q)|Sα(q)|2 ,

(4.36)

withΓ(q) = J [z −

∑

δ>02 cos(q · δ)] + gµBB . (4.37)

Figure 4.5: Sketch of a spin-wave excitation in a Heisenberg ferromagneticspin chain.

Indeed, for the Heisenberg model, the linear excitations associated withthe quadratic Hamiltonian in Eq. (4.29) are spin waves with dispersion re-lation ~ω(q) = Γ(q). Spin waves are collective excitations analogous tophonons. Similarly to phonons, spin waves are also quantized and the spe-cific dependence of Γ(q) on the wave vector (especially for q ≃ 0) determinesthe behavior of the magnetization at low temperature (in the absence ofanisotropy). The dispersion curve Γ(q) can be measured, e.g., by inelasticneutron scattering.Coming back to our goal, we proceed by evaluating the average of fluc-tuations, namely those terms in Eq. (4.27) that we have assumed to be


small for linearizing the Hamiltonian (4.26). The approximated Hamilto-nian, Eq. (4.36), consists of N independent quadratic degrees of freedom sothat equipartition theorem applies:

1

2Γ(q) 〈|Sα(q)|2〉th =

1

2kBT ⇒ 〈|Sα(q)|2〉th =

kBT

Γ(q); (4.38)

〈. . . 〉th denotes thermal average performed using the Hamiltonian Hh.o. inEq. (4.36). Thermal averages of the squared transverse components in realspace read

〈(Sα(n))2〉th =1

N

∑

q,q′

〈Sα(q)Sα(q′)〉th e−i(q+q′)·n =1

N

∑

q

〈|Sα(q)|2〉th ,

(4.39)where we have used the fact that transverse components fluctuate randomlyso that 〈Sα(q)Sα(q′)〉th = δq,q′〈|Sα(q)|2〉th. Note that the right-hand side ofEq. (4.39) is independent of the lattice site, thus the label n will be droppedhenceforth from 〈(Sα(n))2〉th. In order to evaluate whether the consideredlinear excitations are able or not to destroy ferromagnetism, we shall let thefield B → 0+. First, we approximate the summation on the right-hand sideof Eq. (4.39) with an integral

〈(Sα)2〉 ≃ kBT

(2π)d

∫ddq

Γ(q). (4.40)

Since what matters is the behavior for small values of q (i.e., the effect offluctuations at large spatial scales), the denominator of the integral can belinearized as

Γ(q) ≃ Jz−2J∑

µ

(1−1

2q2µ)+gµBB = Jz−2J(

z

2−1

2q2)+gµBB = Jq2+gµBB

(4.41)with µ=1. . . d and q2 =

∑

µ q2µ, which yields

〈(Sα)2〉 ≃ kBT

(2π)d

∫ddq

Jq2 + gµBB. (4.42)

When taking the limit B → 0+, the integral in Eq. (4.42) has an infrareddivergence4 for d≤2. The consequences of such a divergence can be appreci-ated more effectively by setting a lower bond to the integral: qmin = π/Nα,

4A possible ultraviolet divergence does not matter i) because the lattice unit sets aphysical upper limit to large values of q ii) because we are interested in fluctuations actingon large spatial scales corresponding to q ∼ 0.


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

〈(Sα)2〉 ∼ kBT

J

∫

qmin

qd−1 dq

q2⇒

d=1 〈(Sα)2〉 ∼ kBTJNα

d=2 〈(Sα)2〉 ∼ kBTJ

ln(Nα)

d=3 〈(Sα)2〉 <∞(4.43)

In order to understand what a divergence with increasing Nα means, it isconvenient to rephrase the mathematical steps that we followed according totheir 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 align-ment, z.

• We built an effective linear Hamiltonian, describing these family ofsmall excitations, which can easily be decoupled passing to the Fourierspace.

• We calculated thermal averages of such small excitations (transversespin 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 validat 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 . Thisfact suggests that spontaneous magnetization is not stable against thermalfluctuations for d≤2. On the contrary, according to Eqs. (4.43), it seemspossible to have ferromagnetism up to some finite temperature for d=3. Thisscenario is indeed confirmed by more rigorous proofs such as the Mermin-Wagner 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 interactionsthe lower critical dimension is d=2 (the highest dimensionality for whichmagnetic order cannot occur at any finite temperature). This result marks amajor difference between the universality class of classical spin models withcontinuous or discrete symmetry (Ising). As already noticed, in systems withcontinuous symmetry the effects of thermal fluctuations are more severe andmanage to destroy ferromagnetism more easily.


The effect of uniaxial anisotropy

The presence of uniaxial anisotropy stabilizes a system against the linearexcitations considered above. Due to this interaction an additional term like

Hm.a. = −D∑

n

(Sz(n))2 (4.44)

appears in the Hamiltonian in Eq. (4.26). The anisotropy energy indeedfavors configurations in which spins lie along a specific axis regardless of thesign of their projections. In this case we choose the same axis as the one alongwhich the field is applied. Because of the equality

∑

α(Sα(n))2+(Sz(n))2= 1,

the Hamiltonian (4.44) may also be written as

Hm.a. = −D +D∑

n

∑

α

(Sα(n))2 . (4.45)

The summation appearing above transforms into the Fourier space accordingto the Parseval’s formula in Eq. (4.33). Finally, the uniaxial anisotropyprovides a term into the energy spectrum Γ(q) formally equivalent to themagnetic field:

Γ(q) = J [z −∑

δ>02 cos(q · δ)] + gµBB + 2D . (4.46)

For small values of q and B = 0, we get

Γ(q) ≃ Jq2 + 2D . (4.47)

The average of transverse fluctuations is modified as follows

〈(Sα)2〉 ≃ kBT

(2π)d

∫ddq

Jq2 + 2D∼ kBT

∫qd−1 dq

Jq2 + 2D. (4.48)

Henceforth, let us refer only to the thermodynamic limit Nα → ∞, consistentwith qmin = 0. Clearly, the introduction of uniaxial anisotropy removes theinfrared divergence4 from the average of fluctuations independently of thelattice dimensionality. The consequences of this result have to be understoodas follows: “The considered linear excitations alone are not able to destroyferromagnetism at any finite temperature”. This statement does not exclude:

1. that ferromagnetism may be destroyed by some other type of excita-tions

2. that these linear excitations play any role in the “suppression” of fer-romagnetism at finite temperature.


An obvious counterexample, supporting the comment 1., is represented bythe 1d Ising model. The latter can be considered as a limit case of the Heisen-berg 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 convergentso that linear excitations are not able to destroy ferromagnetism for everyT 6= 0. However, we have seen that domain walls manage to destroy ferro-magnetism at any temperature in the 1d Ising model. In conclusion, even ifthe absence of an infrared divergence in the integral on the right-hand side ofEq. (4.48) would allow for ferromagnetism at finite T , we know that a phasewith spontaneous magnetization does not occur.

Limitations of the mean-field approximation

To conclude this part about the critical aspects of magnetism at finite tem-perature we summarize the artifacts produced by the MFA around the criticalregion, T ≃TC .

1. The MFA predicts a the occurrence of a magnetic phase transition atfinite temperature independently of the lattice dimensionality, d. How-ever, a phase with spontaneous magnetization is not encountered inthe 1d Ising model. The Mermin-Wagner theorem forbids the occur-rence of spontaneous magnetization (spontaneous symmetry breaking)in classical models with short-range interactions and with continuoussymmetry for d≤ 2. This fundamental theorem applies to both theHeisenberg 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 theMFA, are generally not correct even when a phase with spontaneousmagnetization exists at finite temperature. Depending on the model,classical critical exponents are wrong for d larger than the lower criticaldimension, dl, and smaller that the upper critical dimension (du=4 forsystems with short-range interactions). For the Ising model dl=1, whileit is dl=2 for the Heisenberg and XY models.

4. The classical critical exponents turn our to be exact for d≥4 in systemswith short-range interactions (Ginzburg criterion). Strictly speak-ing, only the critical exponents are exact for d≥4. However, the MFAis expected to give a more appropriate description of finite-temperatureproperties for a given model when the number of spins with which eachspin interacts increases. This number increases with increasing d or


with increasing the range of interactions. The best realization of theMFA is an ideal case in which every spin interacts with any other onewith the same intensity. This is – of course – unrealistic when magneticsystems are considered. For the model

H = − J

N

∑

n 6=n′

Sz(n) · Sz(n′) , (4.49)

with the summation extended over all the different couples, the MeanField Approximation is exact. The model described by the Hamiltonianin Eq. (4.49) is called Curie-Weiss model but sometimes also “mean-field” model. However, one should not confuse this model with theMean Field Approximation which does not assume from the very be-ginning 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 ferromagnetismat finite temperature in the 1d Ising model represent the analogous of the ar-guments 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 – possiblypresent at T = 0 – is stable against thermal fluctuations. Indeed, Eqs. (4.43)allow drawing similar conclusions for system with continuous symmetry: bothfor d=1 and d=2 ferromagnetism is not destroyed at finite temperature if thesystem is small enough. Under this condition, the averages of transverse spincomponents do not necessarily diverge and the inequality 〈(Sα)2〉 ≪ 〈(Sz)2〉may be fulfilled.Bistability is a crucial property for most of the applications of nanosizedmagnets (nanomagnetism). Thus an important question to be addressed is:“what do we understand for bistability when dealing with a real nanomag-net?” Rephrasing what we have just stated about small enough systems, wecan answer that when a magnetic lattice does not extend indefinitely correla-tions – either of short- or long-range nature – may always develop; the systemas a whole then behaves like a giant classical spin. In the presence of uniax-ial anisotropy, similar arguments as for a single classical spin (macrospin),described by Eq. (4.2), then apply. In particular, the most relevant quantityis the average escape rate from the minima of the total anisotropy energy, lo-cated at θ=0, π (see Fig. 4.1). One possible way to pass from one of the two

5The exchange interaction is divided by N to guaranty extensivity of the energy.


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 ofEq. (4.4):

τ ∼ evDS2

kBT , (4.50)

with v=NxNyNz. Often the magnetic anisotropy is defined per unit volumeKv=D/a

3 so that the usual volume V = a3NxNyNz can be used in Eq. (4.50).Due to the exponential dependence on the system size, the characteristic timegiven in Eq. (4.50) can become very large even for nanomagnets. Referringagain to Fig. 4.1, assume to magnetize the system by means of an externalfield, thus lowering the energy of one of the two minima corresponding toopposite magnetization. This essentially allows preparing the system in achosen state. Then remove the external field. Now, due to the exponentialdivergence of Eq. (4.50), the system may behave as if it had undergone amagnetic 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 freeenergy F in the same meaning as, e.g., in chapter 3).

• If one could wait long enough, t ≫ τ , an average zero magnetizationwould be obtained (in the absence of an external field).

• A similar scenario is recovered irrespectively of the dimensionality ofthe 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 ofdata storage calls for reducing the linear dimensions of nanomagents. Even ifreducing the linear dimensions of a magnetic unit prevents the occurrence of amagnetic phase transition, one can just require that bistability holds for “longenough”. The required time over which one can reasonably assume that ananomagnet remains in the desired metastable state depends on the practicalapplication it is supposed to be used for. But, according to Eq. (4.50)and more general approaches, the relaxation time decreases when reducingthe linear size of a nanomagnet. As a consequence, when the total volumebecomes too small bistability is lost. This intrinsic constraint to the lineardimensions of a bistable magnetic unit is called superparamagnetic limit.

6For mesoscopic systems, processes which involve non-uniform magnetization reversalmay be more convenient.


4.6 Dipolar interaction

Assume that we are considering a material which fulfills all the ground-state requirements to give rise to ferromagnetism, as defined in the firsttwo chapters and summarized at the beginning of chapter 3. Assume thatferromagnetism has also “survived” at finite temperatures, meaning thatthere is a phase with spontaneous magnetization, which requires:

• the dimensionality of the magnetic lattice be d≥2 for systems withuniaxial anisotropy

• the dimensionality of the magnetic lattice be d=3 for systems withcontinuous symmetry (Heisenberg or XY).

In the former case, ferromagnetism is destroyed by thermally excited domainwalls for T > TC (with TC = 0 for d=1).In the latter case, ferromagnetism is destroyed by thermally excited spinwaves (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 ofmagnetostatic origin, neglected so far but always present, may play such arole.

Magnetostatic dipole-dipole interaction

So far we have not considered the contribution of due to the magnetostaticdipole-dipole interaction, which arises directly from Maxwell equations:

Hdd =µ0

4π

[~µ1 · ~µ2

r312− 3

(~µ1 · r12) (~µ2 · r12)r512

]

. (4.51)

For our purposes, the pointlike dipoles are the magnetic moments of eachmagnetic atom in the classical approximation ~µi = −gµB

~Si (i = 1, 2)7 andr12 = r1 − r2. Since thermal effects may be considered at different levels ofapproximation, we prefer to distinguish the ground-state magnetic moment~µi = −gµB

~Si from its thermal average ~mi = 〈~µi〉th introduced in the previouschapter. The typical strength of the dipole-dipole energy is generally smallcompared to the exchange energy. However, its characteristics impose tohandle the dipolar interaction with extreme caution.

7For other applications ~µi could be, e.g., the electron or the nucleus magnetic moments.


1. Consider just two pointlike magnetic moments which interact via thedipole-dipole interaction (4.51): The sign and the intensity of the dipo-lar interaction strongly depend on the relative orientation of the twointeracting magnetic moments and on their relative spatial position r12.

2. In contrast to the exchange interaction, the dipolar interaction cou-ples spins located indefinitely far from each other and the decay of itsstrength with the distance is relatively slow: 1/r3 (long-ranged).

3. In a 3d solid, the dipolar interaction introduces a dependence of thetotal energy on the shape of the sample itself (shape anisotropy).

Equally spaced dipoles

12r

1E =+

12r

1E =+

12r

1E =-

12r

1E =-

a) b)

12r

2E =-

12r

2E =-

c)12r

2E =+

12r

2E =+

d)

Figure 4.6: Different sample configurations of two dipoles ~µi (i = 1, 2), placedat a fixed distance r12. The different values of the interaction energy, E, aregiven in units Edd = µ0

4πµ2

r312.

To fix the ideas about point 1), let us consider two dipoles at a fixed

distance so that the relevant energy scale is given by Edd = µ0

4πµ2

r312. Referring

to the configurations in Fig. 4.6, it is clear that when for some reasons (otherenergies or geometrical constraints)

• two interacting magnetic moments are forced to lie perpendicularly tothe direction of ~r12, then the antiparallel alignment is favored by thedipolar interaction (cases a) and b) in Fig. 4.6);

• two interacting magnetic moments are forced to lie along the direc-tion of ~r12, then the dipolar interaction favors their parallel alignment(cases c) and d) in Fig. 4.6).


In an absolute sense, the configuration c) has the lowest energy of thosereported in Fig. 4.6.

Parallel dipoles

In order to evaluate the dependence of the energy (4.51) on the relativeorientation in space of the two point-like dipoles, ~µ1 and ~µ2, it can be usefulto set the origin of the spatial coordinates in ~µ1, with the z axis parallel tothe direction of ~µ1 itself. Then choose ~µ1 ‖ ~µ2. The resulting energy only

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

E

q

antiferro ferroferro

Figure 4.7: Plot of the interaction energy (4.52) as a function of the polarangle θ (rad). The energy values, E, are given in units Edd = µ0

4πµ1 µ2

r312. Regions

with E < 0 correspond to ferromagnetic coupling, while regions with E > 0correspond to antiferromagnetic coupling.

depends on the polar angle θ defined as the angle that the vector r12 formswith the z axis of our reference frame or, equivalently, with any of the twoparallel magnetic moments, ~µ1 and ~µ2. Within this geometry, the interactionenergy (4.51) reduces to

Hdd =µ0

4π

µ1 µ2

r312

(1− 3 cos2 θ

). (4.52)

Eq. (4.52) is very interesting because it shows that the dipole-dipole interac-tion is

• ferromagnetic for θ ∈ [0, θM] and θ ∈ [π − θM, π]


• antiferromagnetic for θ ∈ [θM, π − θM]

where θM is the magic angle such that cos2(θM) = 1/3 (see Fig. 4.7). Exactlyat the magic angle the dipolar interaction vanishes, meaning that in thisgeometrical configuration the two magnetic moments don’t “feel” each other,for what concerns the dipolar interaction.

12r

13r 23r

?a)

S S

b)

F

F

S

F

F

SS S

c)

F

S

S

Figure 4.8: Dipolar-frustrated configurations. a) The vectors ~r12, ~r23 and ~r13lie all on the same plane and all the magnetic moments are constrained tobe aligned along the indicated direction: up (green) or down (violet). b) andc) 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 frustrationas far as more than two magnetic moments are considered. The configurationa) in Fig. 4.8 represents three magnetic moments magnetized out of planelying at the vertices of an equilateral triangle. The spins at the bottomof the sketch (up-green and down-violet) minimize their interaction energyby 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 ferromagneticcoupling because for these specific cases θ = π/6 < θM. Thus both statesup or down of the third spin – the one located at the upper vertex – willproduce 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 lieonto the same plane. Due to the fact that the configuration c) of Fig. 4.8 isto the global minimum of the two-dipole interaction, the vertical bonds arefirst fulfilled (ferromagnetic). The horizontal bonds (antiferromagnetic) andthe diagonal bonds (ferromagnetic) cannot be satisfied at the same time, sothat some frustration is introduced anyway: frustrated bonds are highlightedwith red. A detailed calculation shows that, eventually, the configuration inFig. 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 cellsof a 2d triangular and square lattice respectively: one can easily imagine that,


when passing to the thermodynamic limit, many different configurations willhave the same, or nearly the same, energy. This fact typically gives rise tovery complex behaviors, such as glassiness, metastability, order induced bydisorder, spin-ice behavior, etc.

4.7 Dipolar interaction in extended systems

In extended systems, the dipolar interaction (4.51) is always present and itinvolves all the magnetic moments. The resulting contribution to the totalenergy is

Hdip =1

2

µ0

4π

∑

n 6=n′

[

~µn · ~µn′

r3nn′

− 3

(~µn · rnn′

) (~µn′ · rnn′

)

r5nn′

]

(4.53)

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 analyticallyand computationally expensive in numerical calculations (due to the long-range character of the dipolar interaction). Thus, in practice, one tries toneglect the dipolar contribution or simplify it taking advantage from thefact that the dipole-dipole interaction normally has a much smaller strengththan the exchange interaction. With the help of Table 3.1 one can easily getconvinced of this. The strength of the nearest-neighbor dipolar interaction

is µ0

4π(gµBS)2

a3= M2

0a3 (if the saturation magnetization M0 is expressed in the

Gauss system, M20 has the units erg/cm3). Putting the proper numbers one

finds that this energy is of the order of few Kelvins or smaller. However, thereare cases in which the dipolar interaction may affect crucially the macroscopicbehavior 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 magne-tization is encountered at finite T , and consider the qualitative effect of thedipolar interaction. Typically, real systems with uniaxial anisotropy can pos-sibly be experimental counterparts of the Ising model. We should also keepin mind that the exchange interaction has usually a much larger strengththan the dipolar interaction (a factor 102 − 103). From what stated in theprevious sections, it is clear that in such systems the dipolar interaction willplay a different role depending on the direction of the easy axis:


• magnetization in plane: the dipolar interaction is globally satisfied(see Fig. 4.8 b)

• magnetization out of plane: the dipolar interaction is frustrated andtends to “destroy” ferromagnetism (see Fig. 4.6 b).

Figure 4.9: Experimental check of the 2d-Ising scaling behavior, observed inFe/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. Thesefilms are model realizations of the 2d Ising model as they obey the predictedscaling 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 norother effects which are neglected in the ideal model seem to affect the 2d-Ising critical behavior observed in Fe/W(110).

An example of ultrathin film magnetized out of plane is representedby Fe/Cu(001). Here the competition between the ferromagnetic exchangeinteraction – originating ferromagnetism for T < TC – and the dipolar in-teraction – frustrating ferromagnetism on a larger scale – produces a sortof phase separation between regions of positive and negative magnetizationperpendicular 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 magneticmoments point along the same direction. Such regions are called magneticdomains. The whole scenario holds below the Curie temperature, thus mag-netic domains need not be confused with the spatial regions defined by thecorrelation length ξ. In Fig. 4.11 some images of different magnetic-domainpatterns observed in Fe/Cu(001) films are shown.


Figure 4.10: The competition between exchange interaction and the dipolarinteraction originates a modulated phase in magnetic films magnetized outof 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, thedipolar interaction may still affect the elementary excitations which destroyferromagnetism

• spin-waves in isotropic spin chains (delocalized excitations)

• domain wall in spin chains with uniaxial anisotropy (localized exci-tations)

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

∫

Nd

ddr

r3(4.54)

which is convergent for d<3. For d=3 the total magnetostatic energy isconditionally convergent and, as anticipated, depends on the shape of the


Figure 4.11: SEMPA images of magnetic domains in Fe/Cu(001) films: darkand light gray regions correspond to domains with opposite out-of-plane mag-netization.

sample (shape anisotropy). In 3d, both frustration effects and magnetic-domain 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

obtained by setting ~µn = −gµB~Sn followed by the usual substitution

∑

· · · ≃ 1

ad

∫

. . . ddr , (4.55)

which yields

Hdip =µ0

8π

(gµBS)2

a31

a2d−3

∫

ddr

∫ ~S(r) · ~S(r′)|r − r′|3

ddr′−

−3µ0

8π

(gµBS)2

a31

a2d−3

∫

ddr

∫

[

~S(r) · (r − r′)] [

~S(r′) · (r − r′)]

|r − r′|5

ddr′+

+µ0

6

(gµBS)2

a31

a2d−3

∫

ddr

∫

~S(r) · ~S(r′)δ (r − r′) ddr′

(4.56)

The unit lengths a have been grouped in such a way that the characteristic

energy scale of the dipolar interaction Ω = µ0

4π(gµBS)2

a3is separated from the

geometrical terms. The term δ (r − r′) arises from a detailed magnetostaticcalculation. It essentially accounts for the fact that the pointlike-dipole ap-proximation has to include this term to compensate the divergence at shortdistances, r ≃ r′. Thus the total magnetostatic energy is not ill-defined.


It is also useful to recall that the magnetization is introduced in elemen-tary courses as a coarse-grained quantity. Ideally, the magnetization ~M(r)already represents an average over an elementary volume whose magneticmoment is8 ~µ = ~M(r)d3r. The magnetostatic energy then reads

Edip =µ0

2

∫

V

ρm(r)φm(r)d3r +

µ0

2

∫

ΣV

σm(r)φm(r)d2r , (4.57)

whereρm = −~∇ · ~M and σm = ~M · n (4.58)

are the volume and surface charge density respectively (n normal to thesurface ΣV ), while

φm(r) =1

4π

∫

V

ρm(r′)

|r − r′|d3r′ +

1

4π

∫

ΣV

σm(r′)

|r − r′|dΣ′V (4.59)

is the scalar magnetic potential (see, e.g., “Classical Electrodynamics”, J. D.Jackson).Note that, within this formalism, the total magnetostatic energy is theCoulomb energy of an effective Coulomb charge distribution ρm(r). Suchan analogy provides a very helpful “rule of thumb” for analyzing realisticsituations:the magnetostatic energy is minimized by configurations for which there areas less magnetic charges as possible.Even if derived in a coarse-grained context, the validity of this “rule ofthumb” is quite general, i.e., it provides the correct hints also in the discrete-lattice formalism.

Bloch and Neel domain walls

In section 4.3 of the present chapter we have investigated how domain wallswith a finite width δ emerge from the competition between the anisotropyand the exchange energy. The “compromise” which minimizes the domain-wall energy is represented by the solution (4.23) which we recall here forconvenience:

cos(θ(x)) = tanh(xδ

)

ϕ(x) = const.(4.60)

All the solutions with constant ϕ(x) give the same energy if inserted in theHamiltonian (4.20). Now we ask ourselves whether the introduction of the

8More precisely, it is ~m = 〈~µ〉th = ~M(r)d3r, but for the present purposes it is convenientto keep considering the dipolar interaction at T = 0, for which ~m = ~µ.


dipolar energy term (4.56) may favor one specific value of ϕ(x). The twoextreme cases are named

ϕ(x) = π2

∀x Bloch domain wall

ϕ(x) = 0 ∀x Neel domain wall .(4.61)

More specifically the magnetization for these two cases will be~M =M0 (0, sin(θ(x)), cos(θ(x))) Bloch domain wall~M =M0 (sin(θ(x)), 0, cos(θ(x))) Neel 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. Aha-roni. Energy per unit wall area, γ, (solid curves) as a function of the thick-ness for a permalloy film magnetized in plane. Dashed curves display thedomain-wall width (q ∝ δ in our notation).

Bloch wall has always a lower energy since

~∇ · ~M = ∂xMx + ∂yMy + ∂zMz

= 0 Bloch domain wall

6= 0 Neel domain wall ,(4.63)


so that only the Neel domain wall produces some magnetic charges ρm =−~∇ · ~M . This rule-of-thumb prediction is confirmed by detailed calculationsand experiment.The situation is analogue for thin films (2d) with the easy axis pointing outof plane.For thin films magnetized in the xz plane (with easy axis parallel to z), thesurface charges σm produced by a Bloch wall (My needs to point out of plane)can be so large that the Neel wall becomes energetically more convenient. Inthis case, surface charges (Bloch wall) are replaced by volume charges ρm.The solid curves in Fig. 4.12 display the domain-wall energy correspondingto a Bloch and a Neel wall, computed with the phenomenological parametersof permalloy (alloy of ≃ 20% Fe and ≃ 80% Ni), as a function of the filmthickness. Indeed, for this specific material with this specific geometry, upto thicknesses of the order of 60 nm the Neel wall has a lower energy.

4.8 Origin of magnetic domains

The scenario described schematically in Fig. 4.10 is just a particular case inwhich the competition between the exchange and dipolar energy gives riseto a configuration with zero global magnetization. Such a configuration isthe compromise which produces as less magnetic charges as possible withthe minimum frustration of the exchange interaction. What results from thecompetition 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 magnetizedout of plane and evaluate the energy variation associated with the creationof a domain wall from a uniformly magnetized state (see Fig 4.13). Boththe exchange energy and uniaxial anisotropy contribute to the domain-wallenergy Ew, thoroughly discussed in the previous sections and whose valuesare summarized schematically in Eq. (4.24). When deriving those resultswe 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 developingindefinitely along the y direction, so that the total increase of the exchangeand anisotropy energy is NyNzEw, with Nz = t/a. For a film magnetized outof plane and with thickness t of few monolayers the dipolar energy (4.56) can


be approximated as

Hdip =1

2Ωt2

a3

∫

d2r

∫Sz(r)Sz(r′)

|r − r′|3d2r′ . (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 dif-ference evaluated in Eq. (4.65). Arrows represent magnetic moments pointingout of plane.

energy by a factor

∆Edip = 2NyΩt2

a2

∫ Nxa/2

δ

dx

∫ 0

−Nxa/2

dx′∫ +∞

−∞

1

[(x− x′)2 + y2]3/2dy ; (4.65)

the integral over dx is performed starting from a length scale equal to thedomain-wall width δ in order to avoid an unphysical divergence. In otherwords, 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. Theintegral in Eq. (4.65) can be performed analytically and gives

∆Edip = 4NyΩt2

a2ln

(Nxa+ 2δ

4δ

)

. (4.66)

The condition ∆Edip = NyNzEw gives the minimum linear dimension Nx thata slab should have in order that splitting the uniform state into domainsbecomes favorable. In the realistic limit of Nxa ≫ δ, one finds that for Nx

larger than the threshold value

Nx ≃ 4δ

aexp

(Ewa4Ωt

)

(4.67)


it is convenient for the system to split into domains of opposite out-of-planemagnetization. From this rough calculation one expects the typical size ofdomains to be Leq ≃ Nxa/2. The exponential dependence on the ratio Ew/Ωis typical of d=2. For the 3d case, a further integration would be involvedin the evaluation of ∆Edip which, eventually, would result in a much weakerdependence of the domain size Leq on the ratio Ew/Ω.

Striped pattern

Figure 4.14: Schematic view of the striped ground state of a ferromagneticfilm magnetized out of plane in the presence of the dipolar interaction andB = 0. Stripes of different colors represent regions of opposite out-of-planemagnetization. The typical stripe width is Leq.

In spite of the crude approximations that have been performed to obtainEq. (4.67), the predicted scaling with Ew/Ω and with the film thickness tmatch with the optimal stripe width for an ideal stripe pattern at T = 0 (seeFig. 4.14). This pattern corresponds to the ground state of a film magnetizedout of plane in the presence of dipolar interaction and zero external field.Detailed calculations9 yield for Ω ≪ D ≪ J

Leq =10

3πδ exp

(Ewa4Ωt

)

(4.68)

and for Ω ≪ J ≪ D (Ising domain wall)

Leq = L0 exp

(Ja

2Ωt

)

(4.69)

9Case Ω ≪ D ≪ J adapted from S. A. Pighın 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).


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) onthe basis of heuristic arguments.

Characteristic length scales emerge from competing in-teractions

The typical domain size Leq is the second characteristic length scale that wehave encountered arising from the competition between different energies

J versus D ⇒ δ domain-wall width

J versus Ω ⇒ Leq domain width .(4.70)

Notice that calculations to obtain both δ and Leq have been performed atT = 0. Since the exchange interaction, the anisotropy energy and the dipolarinteraction involve different number of spins and different components ofeach spin (e.g., uniaxial anisotropy contains only a single-site term (Sz)2),in general, they will be affected differently by thermal fluctuations. Thisfact, may finally favor one of two competing interactions, thus introducingan effective dependence on temperature in the characteristic length scalesδ and Leq.


Literature

• A. Aharoni, Introduction to the Theory of FerromagnetismOxford University Press

• L. D. Landau and E. M. Lifshitz, Statistical PhysicsOxford Pergamon Press

• J. D. Jackson, Classical ElectrodynamicsJohn Wiley and Sons

• C. Kittel, Introduction to Solid State PhysicsJohn Wiley and Sons

Dr. Alessandro VindigniLaboratorium fur FestkorperphysikWolfgang-Pauli-Str.16ETH Honggerberg, HPT C 2.2Tel. +41 44 633 2077Fax. +41 44 633 1096e-mail: [email protected]


Appendix

Averages at finite tempera-tures

Classical modelsIn the canonical ensemble, the partition function is given by

Z =

∫d3Nqd3Np

(2π~)3Ne−βH(p,q) , (4.71)

H being the Hamiltonian of the system and β = 1kBT

. Z is related to thefree energy, F , via the general relation

F = − 1

βlnZ . (4.72)

The average of any observable O (p, q) can be computed as

〈O〉 = 1

Z

∫d3Nqd3Np

(2π~)3NO (p, q) e−βH(p,q) . (4.73)

Classically, the trace operator is defined as

T r =∫

. . .d3Nqd3Np

(2π~)3N, (4.74)

which allows defining

Z = T re−βH(p,q)

and 〈O〉 = 1

ZT rO (p, q) e−βH(p,q)

. (4.75)


Quantum modelsAssume that |ψα〉 be a complete basis of the Hilbert space on which theHamiltonian of the model is defined. Quantum-mechanically, the trace isthen given by

T r =∑

α

〈ψα| . . . |ψα〉 . (4.76)

By analogy with (4.75), the partition function and thermal averages areaccordingly defined

Z = T re−βH =

∑

α

〈ψα|e−βH|ψα〉

〈O〉 = 1

ZT rOe−βH =

1

Z∑

α

〈ψα|Oe−βH|ψα〉 .(4.77)

In few advanced computations one stops at this level. Generally, the trace isevaluated on a complete basis of eigenstates of H:

H|ϕi〉 = Ei|ϕi〉 . (4.78)

The computation of (4.77) is, consequently, simplified:

Z =∑

i

〈ϕi|e−βH|ϕi〉 =∑

i

e−βEi

〈O〉 = 1

ZT rOe−βH =

1

Z∑

i

〈ϕi|O|ϕi〉e−βEi

.(4.79)

Spin modelsLimiting ourself to a Hamiltonian of the type

H = −1

2J∑

|n−n′|=1

S(n) · S(n′) + gµBB∑

n

Sz(n) (4.80)

one possible choice for the basis of the Hilbert space is the following one:|ψα〉=|M1,M2, . . .MN〉=|M1〉 ⊗ |M2〉 · · · ⊗ |MN〉 with Sz(n)|Mn〉=Mn|Mn〉and n label for the lattice site. Note that the Hamiltonian in Eq. (4.80) isnot diagonal on this basis. Thermal averages are computed according to Eqs.(4.79).For many problems in magnetism, substituting the quantum-mechanical op-erators S(n) by classical vectors is legitimate:

S(n) → ~S(n) ≡ S0 (sin θ cosϕ, sin θ sinϕ, cos θ) (4.81)


where S20 = S (S+1) (more often S0 = 1). The partition function then reads

Z =

∫

dΩ1

∫

dΩ2 . . .

∫

dΩNe−βH(~S(n)) , (4.82)

with dΩn = sin θndθndϕn being the solid-angle element of the spin located atthe site n.Both in the quantum and the classical case Z depends on T andB. Therefore,the free energy associated with the canonical averages is the Gibbs free energyof macroscopic thermodynamics:

G(B, T ) = − 1

βln [Z(B, T )] . (4.83)

Often in the literature the letter F is used also when dealing with spin modelsto stress the fact that this type of averages are performed in the canonicalensemble (constant number of particles). We will follow this convention.

magnetism i: from the atom to the solid state - microstructure research

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