Non-collinear Magnetism in d- and f-electron Systems168187/... · 2009-02-14 · Non-collinear Magnetism in d- and f-electron Systems ... When applying a magnetic ﬁeld B to a system

ACTAUNIVERSITATISUPSALIENSISUPPSALA2006

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 169

Non-collinear Magnetism ind- and f-electron Systems

RAQUEL LIZÁRRAGA JURADO

ISSN 1651-6214ISBN 91-554-6540-4urn:nbn:se:uu:diva-6812

To Bandi

The illustration in the front page is afree interpretation of a non-collineararrangement of spins, out there, inthe free space. It was designed anddrawn by Constanza Bertolone Hojasand Ariel Fernández Luna.

iii

List of Publications

This thesis is based on the collection of papers given below. Each article willbe referred to by its Roman numeral.

I Noncollinear magnetization density on VAu4

R. Lizárraga, L. Nordström, E. Sjöstedt and O. ErikssonInter. J. Quant. Chem. 90, 1610, (2002).

II A crystal and magnetic structure investigation ofTbNi5−xCux (x = 0,0.5,1.0,1.5,2.0);Experiment and TheoryR. Lizárraga, A. Bergman, T. Björkman, H-P. Liu, Y. Andersson,T. Gustafsson, A. G. Kuchin, A. S. Ermolenko, L. Nordströmand O. Eriksson(submitted to Phys. Rev. B)

III Studies of the incommensurate magnetic structure of a heavyfermion system: CeRhIn5

R. Lizárraga, M. Colarieti-Tosti, A. Bergman, T. Björkman, O.Eriksson, L. Nordström and J. M. Wills(in manuscript)

IV First principles calculations of multiple-k magneticstructures, crystal field levels and the oxygen distortion inUO2

R. Lizárraga, M. Colarieti-Tosti, T. Björkman, O. Eriksson, L.Nordström and J. M. Wills(in manuscript)

V Crystal and magnetic structure of Mn3IrSiT. Eriksson, R. Lizárraga, S. Felton, L. Bergqvist, Y. Andersson,P. Nordblad, and O. ErikssonPhys. Rev. B 69, 054422 (2004).

iii

iv

VI Conditions for noncollinear instabilities of ferromagneticmaterialsR. Lizárraga, L. Nordström, L. Bergqvist, A. Bergman, E.Sjöstedt, P. Mohn and O. ErikssonPhys. Rev. Lett. 93 107205 (2004).

VII Noncollinear magnetism in γ-Fe within the local spin densityapproximationR. Lizárraga, E. Sjöstedt and L. Nordström.(in manuscript)

VIII Theoretical and experimental study of the magnetic structureof TlCo2Se2

R. Lizárraga, S. Ronneteg, R. Berger, A. Bergman, P. Mohn,O. Eriksson and L. NordströmPhys. Rev. B 70, 024407 (2004).

IX On the bonding situation in TlCo2Se2

M. V. Yablonskikh, R. Berger, U. Gelius, R. Lizárraga, T. B.Charikova, E. Z. Kurmaev and A. Moewes.J. Phys.:Condens. Matter 18, 1757 (2006).

X On the magnetic structure of TlCo2Se2

R. Lizárraga, S. Ronneteg, R. Berger, P. Mohn, L. Nordströmand O. Eriksson,J. Magn. Magn. Mater. 272-276, 557, (2004) (Conferenceproceedings. ICM2003.)

XI Non-collinear states in TlCo2Se2−xSx alloys; theoryR. Lizárraga, L. Nordström and O. Eriksson(in manuscript)

XII Non-collinear magnetism in the high-pressure phase of ironR. Lizárraga, L. Nordström, O. Eriksson and J. M. Wills(in manuscript)

Reprints were made with permission from the publishers.

iv

v

The following papers are co-authored by me but are not included in thisthesis;

• Electronic structure calculations of electronic and structuralproperties of plutonium 115 compoundsJ. M. Willis, R. Lizárraga, J. J. Joyce, T. Durakiewiicz, J. L. Sarrao, L.Morales and O. Eriksson.(preprint)

• Crystal structure and magnetic properties of the new phase Mn3IrSiT. Eriksson, S. Felton, R. Lizárraga, O. Eriksson, P. Nordblad and Y.AnderssonJ. Magn. Magn. Mater. 272-276, 823, (2004) (Conference proceedings.ICM2003.)

• Local and global magnetism in random FeV alloysE. Holmström, R. Lizárraga, S. Shallcross and I. A. Abrikosov(in manuscript)

Comments on my participation

In the papers where I am the first author I am responsible for the main part ofthe work, from ideas to the finished papers. Concerning the other papers I havecontributed in different ways, such as ideas, various parts of the calculationsand the analysis.

v

CONTENTS vii

Contents

List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Magnetism in condensed matter . . . . . . . . . . . . . . . . . . . . . . . . 22 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 The many-body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Exchange-correlation energy functionals . . . . . . . . . . . . . . . . . 82.4 Spin density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Non-uniformly magnetized systems . . . . . . . . . . . . . . . . . . . . . 102.6 The self-interaction correction . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1 The secular equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 The linear augmented plane wave method . . . . . . . . . . . . . . . . 173.3 APW with local orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Non-collinear Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.1 Spin spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Origin of magnetic ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2.1 Itinerant electron theory (Stoner criterion) . . . . . . . . . . . . 244.2.2 The static nonuniform magnetic susceptibility . . . . . . . . . 254.2.3 Fermi surface nesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.4 Conditions for non-collinear states . . . . . . . . . . . . . . . . . . 284.2.5 Iron at high pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Localized States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.1 Hund’s rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2 Crystal field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.3 Magnetic structure and distortion in UO2 . . . . . . . . . . . . . . . . . 395.4 Spin spirals in rare earths systems . . . . . . . . . . . . . . . . . . . . . . 425.5 TbNi5 and CeRhIn5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

vii

1

1. Introduction

The discovery of magnets lies back in ancient times. The legend of a shepherdnamed Magnes, who found that his iron tipped crook and the nails of his bootswere attracted to the ground on the slopes of the mount Ida in Crete, is prob-ably the earliest account we have concerning magnets. The magical powersof magnetite, as the stone Magnes had stepped on was called later, are men-tioned in the writings of the Roman encyclopedist Pliny the Elder (23-79 AD).For many years after its discovery, magnetite was surrounded by superstition.The powers of healing the sick, frightening away evil spirits, and attractingand dissolving ships made of iron were some of the prodigies associated withmagnetite. As in many other subjects of human knowledge, supernatural in-fluences and divine intervention were left behind when the work of scientistslike Oersted (1777-1851), André Marie Ampère (1775-1836), Michael Fara-day (1791-1867) and James Clerk Maxwell (1831-1879) cast light on the phe-nomenon of magnetism and electromagnetism. Since the Chinese compass(3er BC), which is the first known application of magnetism, technological in-terest in magnetic materials has grown immensely. Storage media like the harddisks in our computers, floppy disks, tapes and permanent magnets in electricmotors are some of the practical uses of magnetism in our everyday lives. Thisis why magnetism has been of so much interest in the field of condensed mat-ter in recent decades. The manifestation of magnetism in solid state physics isa matter of importance in this thesis.

Magnetic materials are often found to be ferromagnets or antiferromagnets,i.e. magnetic moments pointing parallel or anti-parallel to a certain globalquantization axis. However, on some occasions, the moments are orientatedin such a manner that there is no global quantization axis. The latter is callednon-collinear magnetism. Why the spins choose to order either in a collinearway like in ferromagnets or in a non-collinear way like in spin spirals is aquestion that has not found a thorough answer yet. In this thesis a seriousattempt to find an answer to these questions was made. Before embarking onthe discussion of this issue some basis facts of magnetism will be revisited,going all the way from a single electron until it is placed with other electronsin an atom or in a solid.

1

2 CHAPTER 1. INTRODUCTION

1.1 Magnetism in condensed matterThe fundamental object in magnetism is the magnetic moment, which in clas-sical electromagnetism is defined as

dµ = Ida (1.1)

where I is a current around an elementary oriented loop of area |da|. The di-rection of the vector da is normal to the loop and determined by the directionof the current around the elementary loop (the screw rule). A current is pro-duced by the motion of one or more electrical charges which are associatedwith particles that have mass. Therefore, there is an orbital motion of mass aswell as charge in the current loop. The magnetic moment is connected withthe angular moment.

When applying a magnetic field B to a system of interacting electrons,an induced magnetization appears. We could try to calculate the net magneticmoment of this system in a classical manner and then complete the descriptionwith the appropriate quantum mechanical corrections. However, as Bohr andVan Leeuwen showed, magnetism can not be understood in the framework ofa classical theory based on the magnetism of moving charges. Consequently, afully quantum mechanical description is necessary in order to give any accountof magnetism. This includes the intrinsic angular momentum or spin of theelectrons, which is characterized by the spin quantum number s. The spinangular momentum is associated with an intrinsic magnetic moment µs =−gµBs, g is a constant known as the g-factor and µB is the Bohr magneton.The total electron magnetic moment contains contributions from the angularand spin magnetic moments.

The Hamiltonian [1, 2] that describes an atom with Z electrons moving ina potential V due to the nucleus is

H =Z∑

i=1

12m

p2i + V (ri) +

12

∑i=j

e2

|ri − rj |. (1.2)

An external magnetic field B will change the momentum of each electronpi to (pi + eA/c), where A is the magnetic potential1. The energy of themagnetic moment in B is −µ · B, so the Hamiltonian of the system in thepresence of B can be expressed as

H =Z∑

i=1

(1

2m

(pi +

e

cA)2

+ V (ri) + gµBB · si

)+

12

∑i=j

e2

|ri − rj |.

(1.3)From quantum mechanics we know that everything we could know about the

1In a purely classical theory this would be the only effect of the field.

2

1.1. MAGNETISM IN CONDENSED MATTER 3

system, described by the Hamiltonian in Eqn (1.2), can be found by solvingthe Schrödinger equation

HΨ = EΨ, (1.4)

where Ψ is the total wave function of the whole system. This represents amany-body problem. The problem of three bodies is already unsolvable andtherefore this approach is intractable for atoms with the exception of the hy-drogen atom unless we incorporate some approximations. Hartree introducedan important concept which appeals to the variational principle of quantummechanics. The principle establishes that the total energy,

E = 〈Φ|H|Φ〉 =∫

Φ∗H Φ dr, (1.5)

is stationary with respect to variation of Φ, and that E is always an upperbound to the ground state energy. In Eqn (1.5) Φ is an approximate but nor-malized wave function that has the appropriate form of the electron systemunder investigation. Clearly if Φ was the exact ground state wave functionthen E would be the ground state energy. By writing the wave function Φ as adeterminant of single-particle wave functions φi, which takes into account theantisymmetry of the wave function, the variation ofE (Eqn (1.5)) with respectto the single-particle wave functions leads to the so-called Hartree-Fock (HF)equations2,(

− 2

2m∇2 + V (r)

)φn(r) +

N∑j=1

∫φ∗j (r

′)φj(r

′)

e2

|r − r′ |dr′φn(r)

−N∑

j=1

∫φ∗j (r

′)φn(r

′)

e2

|r − r′ | dr′φj(r) δSjSn = εn φn(r).

(1.6)

In deriving Eqn (1.6) the Hamiltonian in Eqn (1.2) was used. The last term onthe left hand side of Eqn (1.6) is called the exchange potential and even thoughit is Coulombic, its origin is quantum mechanical. If we neglect the last termin Eqn (1.6) which singles out those electrons with spins of state j parallelto the state n, we obtain the Hartree equations. They represent an electronmoving in an effective or averaged potential due to the all other electrons.The HF approximation has turned out to give very accurate agreement withexperiments for atoms.

However, in solids, the HF approximation becomes less helpful. The band-width obtained by this approach is considerably larger than the experimentalvalues for the simple metals Li, Na, Be, Hg and Al. Moreover the velocity at

2The derivation of the Hartree-Fock equations can be found in any solid state book such us Refs.[1, 3, 4].

3

4 CHAPTER 1. INTRODUCTION

the Fermi surface diverges, in clear contradiction with experimental observa-tions. The reasons for these failures can be attributed to the unscreened longrange Coulomb interactions.

First principles calculations based on density functional theory (DFT) pro-vide an accurate and reliable way to obtain ground state properties of solids.The essential point is to replace the complication of calculating the total wavefunction of the many-body problem by the problem of finding the ground statedensity3. DFT adds effects of exchange and correlation to the Hartree-typeCoulomb terms to describe electron-electron interaction. This theory has beenextended to the spin-polarized case which permits us to investigate magneticsystems. This is the avenue we will follow in this thesis in order to investigatemagnetism in various systems.

3A more detailed description of density functional theory will be given in the next chapter.

4

5

2. Density Functional Theory

DFT is a theory of correlated many-body systems. It provides a way of dealingwith the many-body problem by replacing it with an auxiliary independent-particle system in which all the interaction and correlation effects are includedin an exchange-correlation functional. As such, DFT has become the primarytool for calculation of electronic structure, magnetism and other properties incondensed matter. The remarkable successes of the local density approxima-tion (LDA) and the generalized gradient approximation (GGA) functionalswithin the Kohn-Sham scheme have led to widespread interest in DFT as themost promising approach for accurate, practical methods in the study of realmaterials. In the following sections the basic ideas behind DFT will be out-lined.

2.1 The many-body problemThe fundamental equation that governs a non-relativistic, time-independentquantum system is the Schrödinger equation

HΨ = EΨ. (2.1)

The total energy of such a system is E, Ψ is the total wave function whichcontains information of the whole system and H is the Hamiltonian, that foran arrangement of electrons and nuclei can be expressed as

H =∑

µ

[−

2

2Mµ∇2

µ +∑ν>µ

VI(Xµ − Xν)]

+∑

i

[−

2

2mi∇2

i

+∑j>i

e2

|ri − rj |+∑

µ

Ue−I(ri − Xµ)], (2.2)

where Xµ and Mµ are the coordinates and masses of the nuclei and ri andmi are the corresponding quantities for the electrons. The first and the thirdterms in the Hamiltonian given in Eqn (2.2) are the kinetic energies of thenuclei and the electrons, respectively. The quantity VI(Xµ −Xν) is the inter-action potential of the nuclei with each other, and Ue−I(ri − Xµ) represents

5

6 CHAPTER 2. DENSITY FUNCTIONAL THEORY

the interaction between the electrons and the nuclei. The simplicity of Eqn(2.1) is deceptive, because it is not possible to solve for solids, in particularmetals, whose density of conduction electrons is very high (∼ 1023/cm3).The Born-Oppenheimer approximation provides us with a way of simplifyingthe Hamiltonian of Eqn (2.1), because the electrons are much lighter than thenuclei they move much more rapidly and can follow the slower motions ofthe nuclei quite accurately. This fact allows us to discuss the motion of theelectrons separately from the motion of the nuclei. The Born-Oppenheimerapproximation leaves us with an electronic Hamiltonian, in which the nu-clear coordinates enter only as parameters. Thus the Hamiltonian becomesless complicated; nevertheless the complexity of the electron-electron termstill remains and makes the Schrödinger equation unsolvable. The intricacyof the many-body problem then forces us to find another route towards theunderstanding of solids.

2.2 Kohn-Sham equationsA significant reduction of the complicated many-body problem was suppliedby DFT, which was developed by Hohenberg and Kohn [5] and Kohn andSham [6]. The essential point of DFT is the realization that the ground statedensity is sufficient to calculate all physical quantities of interest. Therefore,instead of calculating the many-body wave function Ψ, the knowledge of theground-state density becomes crucial. DFT is based in the following two the-orems established by Hohenberg and Kohn:

Theorem 1 The total ground state energy of a many-electron system is a func-tional of the density

n(r) = N

∫· · ·∫

|Ψ(x1,x2, ...,xN )|2ds1 dx2...dxN . (2.3)

where the coordinates xi = ri, si.

Theorem 2 The functional E[n] = 〈Ψ|H|Ψ〉 of a many-electron system has aminimum equal to the the ground state energy at the ground state density, E0.

The proof of these theorems as well as v-representability issues will not be dis-cussed here but the interested reader can find detailed information in Ref. [7, 8,9]. Unfortunately these theorems provide no information about the form of thefunctional E[n] and therefore the applicability of DFT relies upon our abilityto find accurate approximations. Kohn and Sham (1965) used the variationalprinciple implied by the second theorem to derive single-particle Schrödinger

6

2.2. KOHN-SHAM EQUATIONS 7

equations. Following their approach, we proceed by writing the total energyfunctional E [n] as1

E [n] = T [n] +∫n(r)vext(r) dr +

∫∫n(r)n(r′)|r − r′| drdr′ + Exc [n] , (2.4)

which consists of the kinetic energy, the external potential which in the Born-Oppenheimer approximation is the potential due to the ions, the Hartree com-ponent of the electron-electron energy, and the exchange-correlation energy.The last term in Eqn (2.4) contains the non-classical contributions to theelectron-electron interaction, namely the exchange and Coulomb correlationeffects. Since we know the expression for the kinetic energy of non-interactingparticles T0 [n], it is convenient to split up the kinetic energy term in Eqn (2.4)into two terms T = T0 + Txc, where Txc stands for the exchange-correlationpart of the kinetic energy and is simply included in Exc. Although explicitforms of Txc and Exc are not known in general, we can use the variationalprinciple on the total energy functional to write

δE [n]δn(r)

+ µδ(N −

∫n(r) dr)

δn(r)= 0, (2.5)

where µ is a Lagrange multiplier which takes care of particle conservation.Finally by using the density 2

n(r) =N∑

i=1

|ψi(r)|2 , (2.6)

where the sum extends over the lowestN occupied states, we are able to deter-mine the functional derivatives in Eqn (2.5). This procedure leads to effectivesingle-particle equations called the Kohn-Sham (KS) equations 3[

−∇2 + veff (r) − εi

]ψi(r) = 0, (2.7)

which are Schrödinger equations4 where the external potential has been re-placed by an effective potential defined by

veff(r) = vext(r) + 2∫

n(r′)|r − r′| dr′ + vxc(r), (2.8)

1Hereafter atomic Rydberg units (a.u.) will be used.2Here we will simply assume that we can determine single-particle wave functions ψi(r)which permit us to express the density as in Eqn (2.6) leaving further discussions to the special-ized literature.3A full derivation of the KS equations will not be given here but it can be found in Ref. [7, 8, 9]4The Lagrange multiplier µ becomes εi in Eqn (2.7).

7


with the exchange-correlation potential

vxc(r) =δExc [n]δn(r)

. (2.9)

In Eqn (2.9), Exc now contains the exchange-correlation part of the kinetic en-ergy Txc. The eigenvalues εi obtained above are not in general simply relatedto measured quantities and their physical meaning is still controversial [4]. Itshould be noted here that if Exc and vxc were known the KS approach wouldresult in the exact ground state energy.

2.3 Exchange-correlation energy functionalsDFT as outlined above supplies a scheme to map the many-body probleminto a Schrödinger-like effective single-particle equation provided that we in-troduce an approximation to the exchange-correlation functional. The localdensity approximation (LDA) achieves this task by writing the exchange-correlation energy functional as

Exc [n(r)] =∫n(r)εxc(n(r)) dr, (2.10)

where εxc(n(r)) is the exchange-correlation energy per particle of a homoge-neous electron gas of density n(r). The performance of LDA can be summa-rized as follows;• The equilibrium lattice constants are generally accurate within 0.1 Å, usu-

ally predicting too small values.• The binding energies are often better than 1 eV, although there are cases in

which the overbinding is greater.• There is a 10-20% error in vibrational frequences.• Charge densities can be obtained within a 2% error.• Geometries are frequently correct.• Physical trends are generally correct [10].

There are attempts to refine LDA, for instance the generalized gradient ap-proximation (GGA) and the weighted density approximation (WDA). An ex-pression similar to that shown in Eqn (2.10) is used in GGA but in this case εxc

is a function of the gradient of the density |∇n(r)| as well as the density n(r)[11, 12, 13]. WDA [14, 15, 16] is a more sophisticated approach that incor-porates truly non-local information through Coulomb integrals of the densitywith model exchange correlation holes. Although WDA improves greatly thepredicted energies of atoms it is more computationally demanding than LDAor GGA and therefore there are very few reports in the literature of WDAapplied to solids. In contrast, GGA has been widely used in first principles

8

2.4. SPIN DENSITY FUNCTIONAL THEORY 9

calculations but despite its success in predicting the bcc ground state of iron,5

it has not been found to improve significantly LDA calculations in metallicmagnets, at least not the ground state magnetic properties.

2.4 Spin density functional theorySo far we have discussed DFT for non-spin-polarized systems and since thework presented in this thesis pertains to magnetic materials we now turn intothe description of the spin density functional theory (SDFT). We will empha-size those features that are typical for SDFT, in particular its applications tothe study of non-collinear magnets,6 and omit details since they are similar towhat was discussed in earlier sections.

In 1972 von Barth and Hedin [9] extended the DFT to the spin polarizedcase. They used a 2 × 2 matrix formalism to represent the density and theexternal potential instead of single variables,

n(r) =⇒ ρ(r) =

(ρ11 ρ12

ρ21 ρ22

)(2.11)

vext(r) =⇒ vext(r) =

(v11 v12

v21 v22

). (2.12)

We begin our discussion by noticing that the wave functions will take the formof spinors,

ψi(r) =

(φiα(r)φiβ(r)

), (2.13)

where φiα and φiβ are the two spin projections. In the non-spin polarized casewe defined the density (see Eqn (2.6)) as the sum of |ψi|2 extended over thelowest N-occupied states. We now write the density matrix ρ in Eqn (2.11) as

ρ(r) =N∑

i=1εiα,εiβ≤EF

(|φiα(r)|2 φiα(r)φ∗iβ(r)

φ∗iα(r)φiβ(r) |φiβ(r)|2

), (2.14)

which generally can be expanded in terms of the density n(r) and the magne-

5LDA favors instead a nonmagnetic fcc ground state for Fe in contradiction with experiments.6Although SDFT was formulated completely general, its earlier implementations were mostlydone for the special case of diagonal matrices, i.e. collinear magnetism.

9


tization density m(r), that is naturally a vector density

ρ(r) =12

[n(r)+ m(r) · σ] , (2.15)

where is the 2 × 2 unit matrix and σ = (σx, σy, σz) are the Pauli matrices.The next step is to write down the total energy that now is a functional of ρ

E [ρ] = T0 [ρ] + Vext[ρ] +∫∫

n(r)n(r′)|r − r′| drdr′ + Exc [ρ] . (2.16)

The external potential Vext [ρ] in Eqn (2.16) is the potential due the ions spec-ified by ∑

αβ

∫ραβ(r) vext

βα (r) dr . (2.17)

By applying the variational principle on the total energy functional inEqn (2.16), and proceeding in the same way as in the non-polarized case wederive the single-particle equations which constitute the KS equations for aspin system, ∑

β

(−δαβ∇2 + veff

αβ (r) − εiδαβ

)φiβ(r) = 0, (2.18)

where no assumption of collinearity, i.e. all spin being parallel or anti-parallelto a global quantization axis, has been made. The effective potential matrixelements in Eqn (2.18) can be written down as

veffαβ(r) = vext

αβ (r) + 2δαβ

∫n(r′)|r − r′| dr′ + vxc

αβ(r), (2.19)

with the exchange-correlation potential matrix elements

vxcαβ(r) =

δExc[ρ]δρβα(r)

. (2.20)

2.5 Non-uniformly magnetized systemsThe application of SDFT as well as its non-spin-polarized partner requiresthe introduction of an approximation for the exchange-correlation functional.LDA can be readily generalized to the spin-polarized case (LSDA) [9] bydefining spin-up and spin-down densities. However the aim of this section isto include in our discussion also those cases where there is no global spinquantization axis, i.e. non-collinear magnetization. Hence the discussion pre-sented here will continue in a general way.

10

2.5. NON-UNIFORMLY MAGNETIZED SYSTEMS 11

In the previous section we learned that SDFT uses a 2×2 matrix formalism.The density matrix (see Eqn (2.14)) elements were defined as

ραβ(r) =N∑

i=1εiα,εiβ≤EF

φiα(r)φ∗iβ(r) . (2.21)

From the definition in Eqn (2.15) it is clear that the electron and the magneti-zation density can be expressed as

n(r) = Tr(ρ(r)) =N∑

i=1

|ψi|2 and m(r) =N∑

i=1

ψi†σψi, (2.22)

where the sums in Eqn (2.22) extend over the lowest occupied states. In thesimplest case where the spins are arranged in a collinear way, the densitymatrix is diagonal and therefore the magnetization becomes,

mz(r) =N∑

i=1

[|φiα|2 − |φiβ|2

]=

N∑i=1

[n↑(r) − n↓(r)] . (2.23)

In Eqn (2.23) the global magnetization axis was assumed to be in the z-direction and the elements of the diagonal density matrix to be n↑ (spin-updensity) and n↓ (spin-down density). The exchange-correlation energy in Eqn(2.10) then depends on both spin densities, εxc(n↑, n↓), and corresponds to theexchange-correlation energy density for a spin-polarized homogeneous elec-tron gas. However, in a more general case where non-diagonal matrices areconsidered the exchange-correlation functional may be given by

Exc [n(r),m(r)] =∫n(r)εxc(n(r),m(r)) dr, (2.24)

which allows us to determine the exchange-correlation potential (see Eqn(2.20))

vxcαβ(r) =

δExc[ρ]δρβα(r)

=δExc[ρ]δn(r)

δn(r)δρβα(r)

+δExc[ρ]δm(r)

δm(r)δρβα(r)

. (2.25)

The first term constitutes a non-magnetic contribution to the exchange-correlation potential whereas the second term is a magnetic potential whichadopts the form of a magnetic field,

b(r) =δExc[ρ]δm(r)

=δExc[ρ]δm(r)

δm(r)δm(r)

=δExc[ρ]δm(r)

m . (2.26)

As follows from Eqn (2.26), in LSDA the magnetic potential is always par-

11


allel to the magnetization density everywhere. The effective potential matrixelements (Eqn (2.19)) can now be written as

veffαβ(r) = vext

αβ (r) + 2δαβ

∫n(r′)|r − r′| dr′ +

δn(r)δραβ(r)

δExc[ρ]δn(r)

+b(r)δm(r)δραβ(r)

. (2.27)

By using the fact that δn(r)/δραβ(r) = and δm(r)/δραβ(r) = σ in Eqn(2.27) we can write the effective potential matrix as the sum of two contribu-tions, a non-magnetic term7 and the magnetic potential,

veff(r) = vnm(r)+ b(r) · σ. (2.28)

Finally, the KS Hamiltonian matrix in LSDA can be written as

H = (−∇2 + vnm)+ b(r) · σ. (2.29)

The non-magnetic part of the effective potential is diagonal. In the specialcase of a collinear system with a global magnetization axis chosen along thez-direction, the magnetic part of the potential becomes(

bzσz 00 −bzσz

). (2.30)

Therefore a collinear system can be treated as two separate electron systems,each moving in an effective potential veff± = vnm±bzσz . An example of a non-collinear magnetization density for VAu4 is shown in paper I (Fig. 3). VAu4 isa ferromagnet, therefore the magnetic potential is diagonal as in Eqn (2.30).However the presence of spin-orbit coupling8 produces a mixing between thedifferent spinor components of the wavefunction in Eqn (2.13) so that an intra-atomic non-collinearity appears, i.e. the direction of the magnetization densityvaries on the length scale of an atom. The general implementation of SDFTthat we have presented here allows us to study magnetism in systems such asVAu4.

In concluding this section, we point out that the KS equations, in their non-spin and spin-polarized versions, lead to a self-consistent cycle, i.e. a densitymust be found that produces an effective potential that once inserted in theKS equations yields single-particle wave functions that reproduce the density.This will be discussed to some extent in the next chapter.

7This term is equivalent to the effective potential in Eqn (2.8) where magnetism was not con-sidered.8In our discussion we did not introduce the effect of spin-orbit coupling in the Hamiltonian.Details of the implementation of the spin-orbit coupling can be found in Ref. [17].

12

2.6. THE SELF-INTERACTION CORRECTION 13

2.6 The self-interaction correctionSo far we have discussed exchange-correlation functionals that are based onthe homogeneous electron gas. We could anticipate that such approximationswould not do well in materials in which the electrons tend to be localized andstrongly interacting, like in transition metal oxides and rare earth elementsand compounds. In the Hartree-Fock scheme (Eqn (1.6)), the Hartree energyrepresents the response of a particular electron to the electron density whichis due to all electrons, including the particular electron (the term j = i is in-cluded). This spurious self-interaction energy is exactly canceled out by theself-exchange energy. Unfortunately, LSDA9 achieves only a partial cancella-tion. For a metal, this self-interaction is not a terrible disaster since a givenelectron is only a small part of the vast conduction electron sea. However fora localized state, it can be very harmful. Perdew and Zunger [18] developedthe self-interaction correction (SIC). They pointed out that the Hartree energy(see Eqn (2.4))

U [n] =∫∫

n(r)n(r′)|r − r′| drdr′ (2.31)

must cancel the exchange-correlation energy of a single, fully occupied orbitalψα, i.e.,

U [nασ] + Exc [nα,σ, 0] = 0. (2.32)

We note here that a single orbital is fully spin-polarized e.g., nα↑ = n andnα↓ = 0. In order to satisfy this condition the total energy functional E [n](Eqn (2.4)) is corrected for the Hartree and the exchange-correlation energyof each of the occupied electron states,

ESIC [ψα] =∑α

〈ψα| − ∇2|ψα〉 +∫n(r)vext(r) dr + U [n]

+ ELDAxc

[n↑, n↓

]−∑α

U [nασ] + ELDAxc [nασ, 0].

(2.33)

Here, ESIC is a functional of a set of N occupied orthonormal single elec-tron wavefunctions ψα. The last term in Eqn (2.33) corresponds to the self-interaction correction, where for each occupied orbital ψα, the term U [nα]+ELDA

xc [nασ, 0] of the corresponding single-electron spin density nασ is sub-tracted. If the orbitals in the kinetic energy term are taken to be KS orbitalsand the SIC term is omitted we recover the LSDA total energy functional. Ina paper by Lundin and Eriksson [19], it was shown that SIC in Eqn (2.33) re-moves only partially the self interaction due to the non-linear behavior of the

9Here we discuss the collinear case, so that the exchange-correlation functional depends onspin-up and -down densities, Exc[n↑, n↓].

13


electron density for the exchange correlation term. SIC-LSDA has been usedwidely [20] and in this thesis it was used to describe the localized 4f states inTbNi5 in paper II, CeRhIn5 in paper III and the 5f states in UO2 in paper IV.

14

15

3. Computational Methods

This chapter is devoted to the application of DFT to real solids. In the previouschapter we found that DFT reduces the complexity of the many-body problemto an effective single-particle theory. In this framework, a set of KS equationsare formulated and their solution entails a self-consistent cycle. This meansthat a density is used to determine an effective potential, which in turn, isinserted into the KS equations, whose solution produces single-particle wavefunctions ψi called KS orbitals. The new set of ψi yields a new startingdensity. This process is repeated until the difference between the densities atthe beginning and the end of a cycle is substantially small. Then it is said thatself-consistency is achieved. The procedure as outlined above normally leadsto large oscillations and bad convergence of the self-consistent cycle. Hencethe resulting density is always mixed in some way with the initial density toproduce a new density to start the process again. The self-consistent cycle isillustrated in Fig. 3.1.

This cycle has been implemented in many codes. Since all calculationsin this thesis have been performed using a relatively new linearized form ofthe augmented planewave (APW) method, the so-called APW with local or-bitals (APW+lo) method, we shall describe here both the traditional linearaugmented planewave method (LAPW) and APW+lo.

3.1 The secular equationAlthough it is not necessary to define a basis to construct the KS orbitals ψiwhen solving the KS equations1, it has been customary in DFT-based methodsto expand ψi in a certain basis set χj with coefficients cij ,

ψi(r) =∑

j

cij χj(r). (3.1)

In Eqn (3.1) we have assumed that the KS orbitals can be accurately describedby the basis set χj. Unless the chosen basis set is infinitely large, this cannever be achieved. Consequently the optimal cij must be obtained through avariational procedure. Thus, the KS orbitals as defined in Eqn (3.1) are in-

1 For instance, it is possible to solve the differential equations numerically on grids.

15

16 CHAPTER 3. COMPUTATIONAL METHODS

KS EquationsSolve Single Particle

Determine E F

Calcule ρout (r)Mix ρ inρout

,

k point loop

k point loop

ρ in

Compute Veff (r)

Figure 3.1: Schematic flow-chart for self-consistent density calculations.

serted in the KS equations (Eqn 2.7) which in turn are multiplied from the leftby ψ∗ and integrated. Finally, the resulting expression is varied,

δ∑jk

c∗ij cik(∫

χ∗j (r)H χk(r) dr − εi

∫χ∗

j (r)χk(r) dr)

= 0, (3.2)

which produces∑k

(∫χ∗

j (r)H χk(r) dr − εi

∫χ∗

j (r)χk(r) dr)cij = 0. (3.3)

Eqn (3.3) is called the secular equation and in a matrix representation it iswritten as

(H− εiO)ci = 0, (3.4)

where H and O are the Hamiltonian and the overlap matrices respectively andci are vectors containing as many coefficients as the number of basis functionsthat have been included in Eqn (3.1). This equation has to be solved for eachk point in the irreducible wedge of the Brillouin zone.

16

3.2. THE LINEAR AUGMENTED PLANE WAVE METHOD 17

3.2 The linear augmented plane wave methodThe LAPW method [21] is a slight modification of the APW method of Slater[22]. Consequently, we shall first establish the essence and motivation of theSlater method as follows: The potential and wave functions in the vicinity ofa nuclei vary strongly and are nearly spherical. In contrast, they are smootherbetween the atoms. These observations lead to the division of space into tworegions where different kinds of basis functions are used to represent the den-sities and potentials. Inside the non-overlapping, atom-centered spheres (S)radial solutions of the Schrödinger equation are used to describe the wavefunctions, whereas planewaves constitute a suitable basis in the remaining in-terstitial region,

ψ(r) =

1√Ω

∑K

aK ei(K+k)·r r ∈ Interstitial,

∑lm

blm ul(r)Ylm(r) r ∈ S.(3.5)

In Eqn (3.5), Ω is the cell volume, aK and blm are expansion coefficients, Ylm

are spherical harmonics and ul is the regular solution of(− d2

dr2+l(l + 1)r2

+ V (r) − El

)r ul(r) = 0. (3.6)

In Eqn (3.6) El is assumed to be a variable, not an eigenvalue and V (r) is thespherical component of the potential in the sphere.

In order for the kinetic energy to be well defined the double representationof the basis in Eqn (3.5) must be continuous at the sphere boundary. This is ac-complished in the APW method by defining the coefficient blm in terms of aKand using the spherical harmonic expansion of the plane waves. Subsequently,by matching each coefficient blm at the sphere boundary, the coefficients blmbecome

blm =4πil√Ωul(R)

∑K

aK jl(|K + k|R)Y ∗lm(K + k), (3.7)

where the quantities jl(kr) are spherical Bessel functions of order l and Ris the sphere radius. The coefficients blm are completely determined by theplanewave coefficients and the energy variables El.

The planewaves in the interstitial region that are matched to the radial func-tions in the spheres constitute what we call the augmented planewaves orAPWs. They are the solution of the Schrödinger equation inside the spheresfor a given El. They do not posses the freedom to allow the wave function toadapt itself as the band energy deviates from the reference El. Therefore, the

17

18 CHAPTER 3. COMPUTATIONAL METHODS

El must be set equal to the band energy ε, which makes the APWs energy de-pendent functions. Thus, the searching for the roots of the non-linear energydependent2 secular determinant

det [H(El) − El O(El)] = 0, (3.8)

must be achieved. This is a much more computationally demanding procedurethan the single diagonalization of the secular matrix in Eqn (3.4) as it wouldbe if the El were fixed parameters.

In the LAPW method3, the exact solutions ul for the spherical potentialinside the spheres are replaced by linear combinations of ul(r) and its energyderivative ul(r). The wave functions written in terms of this basis are

ψ(r) =

1√Ω

∑K

aK ei(K+k)·r r ∈ Interstitial,

∑lm

[dlm ul(r) + glm ul(r)] Ylm(r) r ∈ S.(3.9)

Here, the ul are defined exactly as in APW, i.e. they are the solutions of theradial Schrödinger equation (see Eqn (3.6)), with a fixed El, and the energyderivatives ul satisfy(

− d2

dr2+l(l + 1)r2

+ V (r) − El

)r ul(r) = r ul(r). (3.10)

As in APW, the wave functions must be continuous at the boundary and there-fore the functions ul and u are matched to the respective values and derivativesof the planewaves at the boundary.

The inclusion of ul in the expansion of the wave function inside the spheresupplies more flexibility to the basis set, which enables it to represent alleigenstates in a region around El. For instance, if El differs slightly from theband energy ε, the radial function ul at the energy band will be reproduced bya linear combination,

ul(ε, r) = ul(El, r) + (ε− El)ul(r) + O((ε− El)

2). (3.11)

In this way, all valence bands may be treated with a single set of El, leadingto an energy independent basis set. Therefore, the secular equation becomeslinear in energy and only a single diagonalization is needed to obtain accurateenergy bands. The last fact constitutes a great advantage of LAPW comparedto APW. However the price that LAPW pays due to its flexible basis is the

2The APW basis are energy dependent, so are the H and the O matrices represented in suchbasis.3A detailed account of the LAPW method can be found in Ref. [17].

18

3.3. APW WITH LOCAL ORBITALS 19

loss of the optimal physical form of ul inside the sphere, which implies that alarger number of basis functions is required to achieve convergence.

3.3 APW with local orbitalsAn alternative linearization of APW was developed by Sjöstedt et al. [23], theso-called APW+lo method. One of the problems of the APW basis set is thatit can only describe eigenstates with energies in the immediate vicinity of El.The situation improves in LAPW when modifying the augmented planewaveinside the sphere, which adds more flexibility to the basis set. However thewave function inside the sphere is not as well represented as it was in theAPW.

APW+lo keeps the original APW basis set (Eqn (3.5)) with the exact radialsolutions ul of the Schrödinger equation inside the spheres evaluated at El butadds a complementary basis set, namely, a set of local orbitals [17, 24],

χlo(r) =

0 r ∈ Interstitial,(olm ul(r) + plm ul(r)) Ylm(r) r ∈ S.

(3.12)

This extra basis set introduces the desired variational freedom. Local orbitalswere initially brought into the LAPW method to deal with semi-core states.Their local character comes from the fact that they are entirely confined to thespheres. Moreover, they have a specific lm character and are independent of kand K. As in the former methods, the basis functions must be continuous andconsequently local orbitals are matched at the boundary to zero, so we obtain,

plm

olm= −ul(R)

ul(R), (3.13)

where R is the sphere radius.The APW+lo method uses both bases in such a manner that ul is included

as in APW to describe eigenstates whose eigenenergies are close to El anda linear combination of ul and ul for eigenstates of eigenenergies far awayfrom El. Therefore the number of basis functions is considerably smaller thanin the LAPW method yielding a faster convergence. Of course, for perfectlyconverged calculations both methods produce the same result.

19

21

4. Non-Collinear Magnetism

Solids may contain magnetic moments that can act cooperatively, leading to abehavior which is quite different from what would be observed if all magneticmoments were isolated from each other. This collective behavior and the di-versity of types of magnetic interactions produce a surprisingly rich variety ofmagnetic properties in real systems. Magnetism is a large field in condensedmatter. In this chapter we will concentrate on non-collinear magnetic order-ings and its origin within an itinerant picture. We will start by describing spinspirals (see Fig. 4.1) and a way of treating them efficiently within DFT. Theorigin of magnetism will be considered as a predecessor for a discussion onconditions for non-collinear states to exist. The concepts of a q-dependentsusceptibility and Fermi surface nesting will be introduced as indicators ofmagnetic instabilities.

4.1 Spin spiralsDifferent types of magnetic ground state order can be found as a direct con-sequence of the different types of exchange interactions that operate betweenthe magnetic moments in a solid. Some of them are illustrated in Fig. 4.1. Thefirst two spin arrangements are commonly called collinear structures, for thereis a global spin quantization axis along which all the spins in the structure arealigned, either parallel or anti-parallel. The last three structures in Fig. 4.1, he-lical, spin spiral and spin glass, are characterized by the lack of such a globalaxis and they are called non-collinear structures.

A spin spiral structure as depicted in Fig. 4.1d) can be defined by expressingthe Cartesian coordinates of the magnetization density as

m(r) = m(r) [cos(q · t + ϕ) sin θ, sin(q · t + ϕ) sin θ, cos θ] , (4.1)

where m is the magnitude of the magnetic moment, ϕ and θ are polar angles,t is a lattice vector and q is the wavevector that characterizes the spin spiral. Adistinctive property of the spiral structure is the lack of translational symmetryalong the direction of the wavevector q (see Fig. 4.1d)). However, two atomsof the spin spiral, separated by a lattice vector t, become equivalent if a spinrotation about the spiral axis with the proper angle is applied on the magnetic

21

22 CHAPTER 4. NON-COLLINEAR MAGNETISM

b)

a) c) d) e)n

Figure 4.1: Various spin arrangements in ordered systems: (a) ferromagnet, (b) anti-ferromagnet, (c) helical structure, (d) spin spiral and (e) spin glass.

moment.As was first pointed out by Herring [25] and later by Sandratskii [26, 27],

transformations Tφ combining a translation Tt by a lattice vector t and a spinrotation R(φ) about the spiral axis1 n (see Fig. 4.1d)) by an angle φ = q · t

R(φ) = e−iφσz/2 (4.2)

leave the spiral structure invariant,

Tφ m(r) = m(r). (4.3)

The symmetry operators that describe these transformations belong to a spin-space group (SSG)[28]. Three important properties of these generalized trans-lations can be stated as follows: i) spinors transform according to

Tφ ψ(r) =

(e−iq·t/2 0

0 eiq·t/2

)ψ(r − t), (4.4)

ii) the generalized translations commute with the Kohn-Sham Hamiltonian(see Eqn (2.29)) of a spin spiral structure, and iii) they form an Abelian groupisomorphic to the group of ordinary space translations Tt. Therefore, theyhave the same irreducible representation, which constitutes the Bloch theoremin the case of ordinary space translations. The generalized Bloch theorem is

1In writing Eqn (4.2) the spiral axis n was taken along the z-direction. In the absence of thespin-orbit term, this can be done without any loss of generality.

22

4.1. SPIN SPIRALS 23

then established asTφψk(r) = e−ik·t ψk(r). (4.5)

These properties enable us to express the generalized Bloch spinors [25] thatdiagonalize the spiral Hamiltonian H as

ψk(r) = eik·r(e−iq·r/2αk(r)eiq·r/2 βk(r)

), (4.6)

where αk and βk are the periodic functions for the spin-up and spin-downcomponents respectively. These generalized spinors will produce the samecharge density as the ordinary spinors and the magnetization density as de-fined in Eqn (4.1). The generalized symmetry group permits us to solve theKohn-Sham equations (Eqn (2.18)) in the presence of spin spirals withoutusing supercells. In this way, we have recovered the symmetry of the chemi-cal unit cell. However, the fast Fourier transforms that full-potential methodsbased on planewaves use require translationally invariant potentials and den-sities. Nordström [29] developed a slightly different scheme, which naturallyincorporates the spin spiral symmetries. Two new complex quantities are in-troduced, u and h, with the spin axis taken in the z-direction,

u(r) = e−iq·r (mx(r) + imy(r)) (4.7)

h(r) = e−iq·r (bx(r) + i by(r)) , (4.8)

where mx and my are the perpendicular components of the magnetizationdensity and bx and by are the corresponding magnetic field components. Thenew quantity h is used now to re-write the Kohn-Sham Hamiltonian matrix(see Eqn (2.29)) as

H = (−∇2 + vnm)+12

(eiq·r h(r)σ− + e−iq·r h†(r)σ†−

)+ bzσz (4.9)

withσ− = σx − iσy. (4.10)

Finally, we can easily obtain the density u from the generalized Bloch spinorsas defined above,

u(r) =∑j,k

ψ†j,k

(e−iq·r σ†−

)ψj,k. (4.11)

These quantities are invariant under ordinary space translations by construc-tion, and hence they can be implemented in full-potential methods with nor-mal Fourier transforms.

23


4.2 Origin of magnetic ordering

The nature of the magnetic ordering in solids is not understood on a micro-scopic level despite decades of research. In particular it is not known why inNature one most often observes collinear magnets, non-collinear magnetic or-dering appearing less frequently. Before this question is addressed we shallbriefly analyze the conditions for magnetism to exist at all. The important roleof the magnetic susceptibility as an indicator of magnetic instabilities will beintroduced. The concept of Fermi surface nesting will be discussed as a crucialfactor in the stabilization of the spin spiral structure.

4.2.1 Itinerant electron theory (Stoner criterion)

In this section, we will derive the Stoner condition in the context of the free-electron model, however this criterion as well as the enhanced magnetic sus-ceptibility can be obtained in a more sophisticated way from DFT. In somecases the magnetic moments in metals are associated with the conduction elec-trons, i.e. itinerant magnetism. In the Stoner theory of itinerant magnetism[30, 31] a molecular field which contains all exchange interactions is intro-duced. In analogy with the Weiss model2 the molecular magnetic field is writ-ten as

Hm = Iζ,

where I denotes the molecular field constant and ζ the magnetization givenby the difference of the number of up and down spins that Hm generates.The total energy E(ζ) of electrons moving in the molecular field containscontributions from the kinetic and the field energies [4, 32]

E(ζ) =∫ ε+

0εN (ε)d ε+

∫ ε−

0εN (ε)d ε− I

∫ ζ

0ζ′d ζ

′

=9

20N0

[(1 + ζ)5/3 + (1 − ζ)5/3

]− I

2ζ2, (4.12)

where N0 is the density of states at the Fermi energy. A detailed inspectionof the total energy (Eqn (4.12)) reveals that as the value of N0I grows largerthan one, a minimum in the total energy for a finite value of ζ develops, i.e. aspontaneous magnetization appears in the system when

N0I > 1. (4.13)

2A complete description of Weiss model can be found in [32].

24

4.2. ORIGIN OF MAGNETIC ORDERING 25

This relation is called the Stoner condition and is fulfilled when the densityof states at the Fermi energy is large3. Therefore the phase transition from thenon-magnetic state to the ferromagnetic state is usually caused by a peak inthe density of states at the Fermi energy.

The Stoner condition manifests itself also in the enhanced susceptibility,which can be obtained from the total energy in the non-magnetic limit ζ → 0as

χ = limζ→0

(d2E

dζ2

)−1

=N0

1 − IN0. (4.14)

The enhanced susceptibility in Eqn (4.14) can be rewritten as the product ofthe Pauli susceptibility χp and an enhancement factor which represents thefact that the susceptibility of an interacting electron system can not be deter-mined merely by the Pauli susceptibility alone. The enhanced susceptibilitybecomes very large for metals near a magnetic instability. Hence a peak in thesusceptibility indicates that the investigated system may be close to a mag-netic phase transition. In the framework of DFT, an expression is derived forthe Stoner factor I , which is now seen as an exchange-correlation integral, andit is simply called the Stoner exchange constant. 4

4.2.2 The static nonuniform magnetic susceptibility

In the following sections we shall explore the conditions of magnetic instabil-ities that could lead to spin spiral structures. The magnetic susceptibility is anexample of a more general concept, namely, linear response functions. Herewe are interested in particular in the magnetic response of an inhomogeneoussystem to an external static magnetic field that is non-uniform in space. If themagnitude of the field is small, we can assume the response χ to be linear. Ingeneral, we can write the induced magnetization as

δm(r) =∫χ(r, r′) δB(r′) dr′. (4.15)

Note that δm responds not just to the field at r, but to a weighted average overnearby values. The problem can be addressed from DFT [33] by specifying amagnetic field whose spatial variation is characterized by the vector q,

∆B(r) = ∆B∑

n

(cos(q · R), sin(q · R), 0) Θ(|r − R|), (4.16)

3It has been found that the molecular field constant is an atomic property and of the same orderof magnitude for most metals [32].4A detailed derivation of the Stoner exchange constant can be found in Ref. [4, 32].

25


Figure 4.2: Illustration of a paramagnetic Fermi surface cross-section for a materialthat has a spin spiral magnetic structure with a very small q-vector. The arrow showsthe q-vector that separates the Fermi surface into two parallel portions.

where Θ(r) is the unit step function which is 1 for r smaller than the atomicsphere radius and zero otherwise. The magnetic field specified above favorsa spiral with θ = 90 (see Eqn (4.1)). After some calculations, the enhancedsusceptibility5 for static magnetic fields parallel to the magnetization densityat each point (see Eqn (2.26)) is found to be

χ =χ0(q)

1 − I(q)χ0(q), (4.17)

where χ0(q) is the susceptibility of a non-interacting uniform electron gas.In Eqn (4.17) we can see that the enhanced q-dependent susceptibility willdiverge if I(q)χ0(q) > 1, which means that the non-magnetic state becomesunstable against a magnetic structure characterized by the vector q. This is thegeneralized Stoner condition.

4.2.3 Fermi surface nestingWe shall now examine a Fermi surface feature of fundamental interest, nest-ing. Fermi surface nesting can largely influence the magnetic susceptibilityand therefore it constitutes a decisive factor in the study of magnetic instabil-ities.

Let us start from Eqn (4.17). In order to obtain χ0(q), the so-called Lind-hard expression in its non-magnetic form is used, which in this case describes

5Details of this calculation are given in Ref. [4].

26


the response function for non-interacting electrons6 as

χ0(r, r′) =∑kµ

∑k′ν

Θ(εF − εkµ) − Θ(εF − εk′ν)εkµ − εk′ν

×ψkµ(r)ψ∗kµ(r′)ψk′ν(r′)ψ∗

k′ν(r). (4.18)

Here Θ(x) is the unit step function, ν and µ are band indices, and k and k′ arewave vectors that label Bloch states. By applying the Fourier transform twiceon this equation we obtain (see Ref. [3]),

χ0(q,q′) =∫∫

e−iq·re−iq′·r′χ0(r, r′) drdr′

= δq,q′χ0(q) (4.19)

where

χ0(q) =∑kµν

[f(εkν) − f(εk−qµ)]εk−qµ − εkν + iδ

· |〈kν|eiq·r|k − qµ〉|2. (4.20)

The expression in Eqn (4.20) was generalized by using the Fermi distributionf(ε) instead of the Θ(x) function and an infinitesimal iδ. The denominator inEqn (4.20) becomes very small if there exist parallel portions of the electronand hole Fermi surfaces that, by a rigid shift defined by a vector q, can bemade to coincide (see Fig. 4.2). This feature of the Fermi surfaces is callednesting. It should be noted that in the case of strong nesting and low tempera-ture, χ0(q) becomes very large since the denominator in Eqn (4.20) is almostzero for many values of k in the sum and the numerator is close to one. There-fore, the susceptibility χ0(q) can become very large for the value of q thatdefines the Fermi nesting shift. The generalized Stoner condition

Iχ0(q) ≥ 1, (4.21)

is then fulfilled. Fig. 4.2 depicts an illustration of the Fermi surface cross-section, where the q-vector that separates the portions of electron and holeFermi surfaces has been identified. It is worth noticing that the magnitude ofq in Fig. 4.2 is less than a reciprocal lattice vector magnitude, which meansthat the adopted spin spiral structure will not be commensurate with the lattice.Fermi surface nesting has been proved to have a large effect in the suscepti-bility, which in turn, provides us with an indicator of a magnetic instability.

6 A derivation of the Lindhard expression can be found in Ref. [3, 4].

27


Figure 4.3: Schematic energy band structures of a hypothetical element. The blue andred straight lines stand for two orthogonal spin down and up bands respectively in theferromagnetic state. The presence of a non-collinear coupling allows the hybridizationof the two spin channels and the hybridized bands become the color-graded paraboliccurves. The horizontal line represents the Fermi energy. Two different possibilities areshown in a) and b).

4.2.4 Conditions for non-collinear statesThe most common theoretical explanations for non-collinear magnetic order-ing involve either magnetic frustration in materials (see paper V) with crys-tal structures that have anti-ferromagnetic exchange interactions [4] or near-est neighbor ferromagnetic interactions that are of similar size to next near-est neighbor anti-ferromagnetic interactions [34]. For the late rare earths theRKKY interaction has also been argued to cause non-collinear states [35]. Inpaper VI of this thesis, two criteria are identified which determine whether amagnetic metal may be stabilized in a collinear or non-collinear arrangement.In the following discussion, only spin spirals will be considered as examplesof non-collinear states. Nevertheless the results presented here hold for moregeneral cases as well. To illustrate these ideas, schematic energy bands of ahypothetical element are displayed in Fig. 4.3. In a ferromagnetic state, spinup and spin down bands are orthogonal and therefore they can not hybridize.This is represented by red and blue straight lines in Fig. 4.3. However, aswas discussed in chapter 2, in the more general non-collinear formulation theone-electron states do not possess pure spin up or down character and thewave functions are described by two component spinors (see Eqn (2.13)). Inthe case of spirals, the generalized Bloch spinors that diagonalize the spiralHamiltonian in Eqn (2.29) were given in Eqn (4.6). Consequently, since theterm b · σ in the spiral Hamiltonian is not diagonal, the two spin compo-

28


nents hybridize. Such hybridization is depicted in Fig. 4.3 as the color-gradedparabolic curves. These two bands experience repulsion which opens up anenergy gap. If the hybridization gap occurs at the Fermi energy, the band thathas been pushed down will lower the total energy whereas the band that hasbeen pushed up will not contribute to the total energy, since it is now empty.Fig. 4.3b presents a similar situation but the mechanism that lowers the totalenergy is not as efficient as in Fig. 4.3a. Two criteria for the formation of spinspirals can be deduced from the mechanism sketched above. First, the Fermienergy should cut through both spin up and down states. This condition im-mediately excludes strong ferromagnets such as bcc Fe, hcp Co and fcc Ni,where the spin up band is filled. In contrast, spin spirals are expected to beobserved in fcc Fe, bcc Mn and fcc Mn at ambient conditions. Secondly, themechanism discussed in Fig. 4.3 is extra efficient if there is nesting betweenspin up and spin down states (see Fig. 4.2 and the discussion around the fig-ure). This guarantees that many k-points are involved in the energy loweringprocess discussed above. It is clear that a larger gap causes a larger effect inlowering the total energy.

To investigate the above suggested scenario we carried out first principlestheoretical calculations for several transition metals. In order to meet the de-mands of our first condition, the exchange splitting was tuned either by mod-ifying the volume or by using the so called fixed spin moment method [36].All elements that have been investigated, bcc V; bcc and fcc Mn; bcc Fe; bccand fcc Co; and bcc and fcc Ni, are either non-magnetic or adopt a collinearmagnetic structure at normal conditions. We did not include here the most ob-vious case of spin spiral, fcc Fe, since it has been extensively studied (see Ref.[37] and references therein) and it has also been discussed in paper VII of thisthesis. We show our results in Fig. 4.4, where the energy versus the wave vec-tor q = (0, 0, q) that describes the spin spiral structure is plotted for severalelements. The ferromagnetic state is represented by q=0 and q=1 describesan antiferromagnetic ordering; all other q values correspond to different spinspirals. In Fig. 4.4, the equilibrium volumes of bcc V, bcc Mn, bcc Fe, fcc Co,bcc Ni and fcc Ni were modified; whereas in fcc Mn and bcc Co the mag-netic moment was fixed to the values shown in the figure. The data in Fig. 4.4shows that fulfilling the conditions for non-collinear order discussed above isequally well accomplished by changing the lattice constant as by fixing themagnetic moment. In the case of bcc Co, we fixed the moment to be 0.7 µB

and 0.15 µB . In the former case a ferromagnetic state has the lowest total en-ergy, while an almost flat curve with several local minima is observed for thelatter case. The variations observed in this curve are smaller than the accu-racy of the calculations, and it is difficult to conclude if one q-vector is morestable than the other. However, one can conclude that there is a competitionbetween non-collinear and collinear interactions that are of almost the same

29


Figure 4.4: Calculated energies as a function of q-value for several elements, Ni, Fe,Co, Mn and V, in the bcc and fcc crystal structures. The wave vector q was consideredfor simplicity along the (001) except in the case of bcc Fe, where the (110) directionwas investigated.

30


Figure 4.5: Band structures of TlCo2Se2. The upper panel shows the band structuresof the ferromagnetic state (solid lines) and a spin spiral with q = (0, 0, 0.6)2π/c(dashed line). In the lower panel, the solid line corresponds to the band structure ofthe ferromagnetic state and the dashed line indicates the band structure of the antifer-romagnetic state.

size. In all the cases presented in Fig. 4.4, except for bcc Co, we have foundthat non-collinear states are favored against collinear magnetism. Hence ourfirst principles results are consistent with the analysis of Fig. 4.3.

In paper VIII, the stability of the non-collinear and the antiferromagneticstates over the ferromagnetic ordering of the TlCo2Se2 compound was an-alyzed from the electronic structure (the bonding situation of TlCo2Se2 isdiscussed in paper IX). A powder neutron diffraction study showed that thisquasi two-dimensional system has an incommensurate helix running along thec-axis with a turn angle of ∼ 121. The magnetic structure can be seen in Fig.1 of paper VIII. The magnetic moments of the cobalt atoms are ferromagnet-ically ordered within each layer and perpendicular to the c-axis. The helicalwave vector was found to be (0, 0, q) with q ∼ 0.6 and the moment on Coatoms was 0.46µB . However, LSDA first principles calculations showed thatthe antiferromagnetic configuration was more stable than the helical struc-ture. Here we would like to point out that our linear muffin-tin orbitals withthe atomic sphere approximation LMTO-ASA [38] calculations in paper Xshowed that a spin spiral state was more stable than the antiferromagnetic

31


configuration in accordance with experiments. However, in paper VII, the dis-crepancy between the two methods (APW+lo and LMTO-ASA) was ascribedto the less accuracy of the atomic spherical approximation when dealing withopen crystal lattices [39], especially considering the very small energies thatare involved for resolving the correct magnetic structure. In order to studythe magnetic structure the band structure of TlCo2Se2 was investigated. Theupper panel of Fig. 4.5 displays the ferromagnetic (dark solid line) and thespin spiral bands with q = (0, 0, 0.6)2π/c (dotted-dashed line), while in thelower panel of the figure the ferromagnetic bands (dark solid line) are com-pared with the bands of the antiferromagnetic configuration (dotted-dashedline). The Fermi level is indicated in both panels by a thin solid line. It isworth noticing that the ferromagnetic bands in the upper and the lower part ofthe figure do not look the same. This is because of the way we have chosento represent the spin spiral wavefunction (see Eqn (4.6)) in which the spin upand spin down components are shifted ±q/2, respectively. Therefore in orderto represent a ferromagnetic state with this wavefunction, either the wavevec-tor q = 0 is used or the polar angles θ and φ that describe the direction ofthe magnetization density (see Eqn (4.1)) are set to zero with an arbitrary q.Thus, in the upper panel, the ferromagnetic state was represented by q = 0.6and the polar angles θ = φ = 0. This was done in order to compare with theband structure of the spin spiral of wavevector q = 0.6. In a similar way theferromagnetic state in the lower panel of the Fig. 4.5 was described by q = 1and θ = φ = 0 to compare with the band structure of the antiferromagnetic(q = 1) configuration. Hence the band structures of the ferromagnetic statein both panels have a different q-shift which makes them appear different. Inthe upper panel of Fig. 4.5, there are two bands of the ferromagnetic systemclose to the Fermi energy that cross each other twice. In contrast, these twobands in the spin spiral configuration have hybridized and split. In the lowerpanel of Fig. 4.5, the antiferromagnetic bands resemble the spin spiral bandsof the upper panel. According to Fig. 4.5, the lowering energy mechanismdescribed above (see discussion around Fig. 4.3) would be more effective forthe antiferromagnetic bands than for the spin spiral configuration, since theband that has been pushed up is completely empty in the former case. Thiswould explain the antiferromagnetic ground state found by our first principlescalculations (see paper VIII).

The study of the substitutional alloys TlCo2Se2−xSx gives us the opportu-nity to investigate the transition between the ferromagnetic state in TlCo2S2

to the antiferromagnetic state in TlCo2Se2 via a serie of non-collinear states.Neutron diffraction experiments on the substitutional alloys TlCo2Se2−xSx

for x = [0 − 2.0] were performed by Ronneteg et al. [40]. The distance be-tween the Co layers (inter-layer distance) and the distance between the Coatoms within a layer (intra-layer distance) were found to decrease almost lin-

32


0 0.1 0.2 0.3 0.4 0.5(0 , 0 , q) 2π /c

-0.8

-0.6

-0.4

-0.2

0

0.2

E(q

)-E

(0)

(mR

y)

x=0.0x=2.0c/2=6.406c/2=6.567c/2=6.489

TlCo2Se

2-xS

x

(x=1.0)

(x=2.0)

(x=1.5)

Figure 4.6: The calculated total energy of TlCo2Se2 (open squares), TlCo2S2 (opencircles) as a function of (0, 0, q)2π/c. The total energies of TlCo2Se2 for differentvalues of the inter-layer distance c/2 = 6.406 Å (filled diamonds), c/2 = 6.567 Å(filled circles) and c = 6.489 Å (filled up triangles) are also displayed. The energyis given with respect to the energy of the ferromagnetic state (q = 0) of each curve.The values of the corresponding concentration on the TlCo2−xS2 alloys are given inparenthesis.

33


early with the concentration. Spin spirals are formed in TlCo2Se2−xSx withdifferent turn angles in the inter-layer distance range of 6.49 ≤ c/2 ≤ 6.71 Å,which corresponds to the concentration 0 ≤ x ≤ 1.5. Finally, at c/2 = 6.44Å (x = 1.75), a ferromagnetic phase appears, which persists in TlCo2S2

(see Fig. 1 in paper XI). As S takes the place of Se in the crystal structurethe distance between the Co layers is reduced and the turn angle becomessmaller until it totally vanishes at x = 1.75. In paper XI we performed to-tal energy calculations using GGA for TlCo2S2 and TlCo2Se2. Our resultsare plotted in Fig. 4.6. TlCo2Se2 was found to be have a energy minimumat q = (0, 0, 1/2)2π/c which corresponds to an antiferromagnetic configu-ration. This result agrees with our previous LSDA findings in paper VIII. Weconclude that the small discrepancy between theoretical and experimental val-ues of the magnetic structure of TlCo2Se2 is not resolved by using GGA. Theenergy curve for TlCo2S2 (open circles) shows a minimum at q = 0 whichindicates that TlCo2S2 is a ferromagnet in accordance with neutron diffrac-tion experiments [40]. In order to investigate whether or not the transition to aferromagnetic phase when S replaces Se is an effect caused by the differencein chemistry of these two atomic species, or if it is merely a geometrical effect(the change in the interlayer distance), we have calculated the total energyof TlCo2Se2 for different values of the inter-layer distance. Hence we alsopresent in Fig. 4.6 the calculated total energy of TlCo2Se2 for various val-ues of the inter-layer distance that correspond to different concentrations ofthe substitutional TlCo2Se2−xSx alloys. It may be seen that the energy vs. qcurve for TlCo2Se2 with a inter-layer distance that has the same length as thatof TlCo2S2 (6.406 Å, (filled diamonds)), is very similar to the correspond-ing curve for TlCo2S2 (open circles). This shows that geometrical effects, i.e.distance between Co planes, is the most important factor for determining themagnetic structure of these material. We note in Fig. 4.6 that for values ofthe inter-layer distance intermediate between that of TlCo2Se2 and TlCo2S2

we obtain a gradual change in the magnetic structure, from ferromagnetic vianon-collinear to anti-ferromagnetic, as a function of increasing c lattice con-stant.

The band structures of TlCo2Se2 and TlCo2S2 (see Fig. 4 of paper XI) in aferromagnetic configuration were analyzed in connection with the mechanismthat favors non-collinear states. Two almost parallel bands around the Fermienergy were identified in the TlCo2Se2 band structure. In a spin spiral statethese two bands can hybridize and repel each other, as discussed earlier inthis section. Since the bands lie in the vicinity of the Fermi energy, an energygap arises, lowering the total energy. If strong nesting exists between spin-upand -down Fermi surfaces there may be an effective energy gain large enoughto favor the spin spiral state over the ferromagnetic configuration. These twobands were also observed in the TlCo2S2 band structure, but they were located

34


Figure 4.7: The calculated total energy of hcp Fe in the AFM (II) (see text) con-figuration (open circles), in the paramagnetic state (filled squares) and in the non-symmetrical spin spiral ( q = (0.56, 0.22, 0)2π/a) (open triangles) are plotted as afunction of the volume.

well above the Fermi level. In a spin spiral configuration, the hybridization ofthese two bands will not lower the total energy for TlCo2S2. As a matter offact the spin-mixing of the spin spiral state may even increase the energy sincethe magnetic moments decrease, which increases the intra-atomic exchangeenergy. Hence, the mechanism discussed above will not favor a spin spiralstate in TlCo2S2, which agrees with the experimental findings [40].

4.2.5 Iron at high pressuresIron is an interesting element in many aspects, being maybe the most remark-able, the richness of its magnetism. At ambient conditions Fe is a ferromagnetin the bcc structure. Non-collinear states are observed in fcc Fe (see paperVII) and fcc FexNi1−x Invar alloys [41, 42]. When applying pressure (∼ 13GPa) Fe undergoes a phase transition by adopting the hexagonal close-packed(hcp) structure which is believed to be a non-magnetic phase [43]. In thishigh-pressure phase, Shimizu et al. [44] reported the occurrence of supercon-ductivity in a very narrow range of pressure, between 15 and 30 GPa (132a.u.3 < V < 145 a.u.3). Recently, evidence has been found that suggests thatFe, would develop an antiferromagnetic ground state [45] or a non-collinearmagnetic order [46] in the hcp phase under pressure. Spin fluctuations havebeen discussed to play a role on the appearance of superconductivity in Fe un-der pressure [47]. Calculations of the static paramagnetic spin susceptibility

35


under pressure [48] at finite temperatures in the high-pressure phase of ironshowed that the dominant magnetic fluctuations are incommensurate AFM,characterized by the wave vector q = (0.56, 0.22, 0)2π/a.

In paper XII, several non-collinear structures such as spin spirals, two typesof AFM structures denoted by AFM (I) and (II) (see Fig. 1 in paper XII) andthe ferromagnetic configuration were studied. The calculated total energy ofhcp Fe in the AFM (II) configuration (open circles) and in the paramagneticstate (filled squares) are plotted as a function of the volume in Fig. 4.7. Theparamagnetic curve has a minimum at 70 a.u.3 and it lies lower in energy(∼ 10 mRy) than the AFM (II) configuration whose equilibrium volume isfound at 70.7 a.u.3. This result does not agree with previous first principlescalculations (LAPW) [45]. It is worth pointing out here that the magnetic mo-ments calculated in the present investigation (see Fig. 2 in paper XII) are inexcellent agreement with those in Ref. [45]. In Fig. 4.7 the energies for thespin spiral with the non-symmetric wave vector (0.56, 0.22, 0)2π/a are alsodisplayed. From this figure it is clear that this spin spiral is lowest in energyof all considered magnetic structures. This finding is in accordance with thesusceptibility calculations by Thakor et al. [48].

36

37

5. Localized Magnetic Moments

In the previous chapter, we discussed the so-called band or itinerant mag-netism in metals, where the electrons that are responsible for magnetism formbands and move more or less freely in the solid. Here, we shall consider mag-netism in systems that have localized magnetic moments. Such is the case ofthe rare earth (RE) metals and some actinide compounds where the magneticelectrons are confined to the vicinity of the nucleus. When a large number ofRE atoms are brought together to form a solid, the 4f electrons generally re-main localized and are shielded by the 5s and 5p states. Instead, the 5d and 6selectrons become delocalized into Bloch states, and compose the conductionband. The case in the actinides is a little more complex because of the greaterspatial extension of the atomic wavefunction of the 5f electrons. The phenom-ena observed in the actinide compounds are related to the peculiarities of the5f electrons which sometimes behave as localized electrons and sometimesappear to be itinerant. As a starting point we will review Hund’s rules; eventhough they were established for atoms and ions, they have proven to give agood description of the ground state of a RE ion in a lattice provided that thecrystal field effects can be neglected.

5.1 Hund’s rulesIn a typical atom, the electrons arrange themselves in filled shells with no netangular moment. However, it could be that some shells are incompletely filledso the electrons in these shells can combine to give a non-zero spin and orbitalangular momentum. An atom can thus have a net spin and orbital angularmomentum, S and L, respectively. The number of ways that the electrons canbe combined is given by

(2L+ 1) (2S + 1) .

The configuration that minimizes the total energy can be estimated by usingHund’s rules [1]. They are only applicable to the ground state configurationand assume that there is only one unfilled shell. These three rules state thefollowing:1. The total spin S should be maximized in order to minimize the Coulomb

37

38 CHAPTER 5. LOCALIZED STATES

energy in accordance with the Pauli exclusion principle, i.e. the exchangeenergy.

2. The total orbital angular momentum L is maximized according to the valuedetermined for S in the first rule. This can be understood here by imaginingthat electrons rotating in orbits in the same direction can avoid each othermore efficiently and thus reduce the Coulomb repulsion between them.

3. The third rule states that if the shell is less than half full then J = |L− S|and J = |L+ S| if it is more than half full. The third rule is caused by thespin-orbit energy.In condensed matter, Hund’s rules may still be obeyed in systems where the

electrons in unfilled shells are confined within the vicinity of the nucleus so,for instance, they do not participate in the bonding. With the exception of Ceand Yb compounds, RE metals can be regarded as such systems.

5.2 Crystal fieldThe Hamiltonian of a free atom is invariant under all rotations and reflectionsin space which leave the position of the nucleus unchanged. The degeneracyof its ground state is 2J + 1. Now, if we imagine placing this free atom at alattice site the point group of the lattice will determine the symmetry-induceddegeneracies of the atomic levels. The electric field that our atom (the centralatom) will experience due to the charge distribution of the surrounding atomsis called the crystal field (CF)1. The size and nature of CF effects dependcrucially on the symmetry of the local environment. The effect of the CF onthe 4f electrons in the RE atom is very small since these electrons are confinedin the vicinity of the nucleus and are shielded from the rest of the crystal by the5s and 5p states. The 3d electrons in the transition metals, in the other hand,are more extended and hence the effect of the CF is stronger on them. Thesituation for 5f electrons is somewhere between that of the 4f and 3d states.The contribution of the CF to the potential energy is

vcf(r) =∫

n(R)|r − R|dR, (5.1)

where n(R) is the charge density of the surrounding electrons and nuclei. Ifthis density does not penetrate the charge cloud of the central atom, vcf is asolution of Laplace’s equation, and may be expanded in spherical harmonics2

1A good account on CF theory can be found in Ref. [34].2Note that the full multipole expansion has not been used since the on-site contribution has beenneglected due to the fact that only the charge surrounding the central atom is considered.

38

5.3. MAGNETIC STRUCTURE AND DISTORTION IN UO2 39

asvcf =

∑lm

Aml r

lYlm(r), (5.2)

with

Aml = (−1)m 4π

2l + 1

∫n(R)Rl+1

Yl−m(R) dR. (5.3)

Stevens [49, 50] pointed out that provided that we remain within a manifoldof constant J , in this case the ground state multiplet, the matrix elements ofvcf(r) are proportional to those of operator equivalents written in terms of theJ operators [34]. We may then write the CF Hamiltonian by using the Racahoperators Olm(J) as

Hcf =∑

i

∑lm

Aml αl〈rl〉

(2l + 1

4π

)1/2

Olm(Ji), (5.4)

or equivalently, the CF Hamiltonian can be expressed in terms of the Stevensoperators Om

l (J),Hcf =

∑i

∑lm

Bml O

ml (Ji). (5.5)

The Hamiltonian in Eqn (5.5) lifts the degeneracy of the atomic |JMJ〉states and since it is expressed in terms of the J operators, it may be easilydiagonalized to produce the CF energies and eigenfunctions.

In paper IV the effect of the CF on the U ion, due to the surrounding Oions, is very important, even though the 5f electrons in this compound areconsidered to be well localized. The occurrence of a distortion of the oxygensublattice at TN lifts the degeneracy of the CF, lowering the total energy.

5.3 Magnetic structure and distortion in UO2

Complex magnetic structures are common among the actinide compounds.For instance multi-k arrangements are often observed in these materials [51].These structures may be found in highly symmetric lattices, such as the fcclattice, in which the star of the wave vector k of the magnetic structure con-tains several members. If three members of the star are needed to describe themagnetic structure, the magnetic ordering is called 3k and is non-collinear.Such is the case in UO2, which crystallizes in the fcc CaF2 type structure (seepaper IV).

Unfortunately, neutron diffraction does not distinguish single-k (collinear)3

3A simple antiferromagnet is described as a single-k or 1k structure.

39


a

Figure 5.1: The 3k magnetic structure of UO2 (longitudinal) is shown (for the sakeof clarity, only the U ions are depicted). The U magnetic moments are arranged suchthat they point in the 〈1 1 1〉 direction.

from multi-k (non-collinear) structures because they both give the samediffraction pattern. The only way to discriminate between the two structuresis by performing neutron diffraction on a single crystal under uniaxial stressor an applied magnetic field.

In paper IV we investigated the magnetic structure of UO2 (see Fig. 5.1), anionically bonded semiconductor. In this compound the U ions are tetravalentwith the 5f2 levels well localized which has been confirmed by optical spec-troscopy and photoemission results [52]. There is a first order phase transitionat TN = 30.8 K [53] from the paramagnetic phase towards an antiferromag-netic phase with a magnetic moment µ = 1.74µB [54]. Associated with themagnetic phase transition there is an oxygen distortion that preserves the cu-bic symmetry [55]. Small volume discontinuities and anomalies in the elasticconstants were also found to occur at TN [56], indicating a coupling betweenmagnetic and lattice degrees of freedom.

First principles calculations of the total energy of the 3k magnetic structuresin their transversal and longitudinal forms [57] with and without the associated3k oxygen distortion proposed in Ref. [57] were performed in paper IV. The1k oxygen distortion suggested in Ref. [58] was also considered together witha 1k antiferromagnetic structure. The 3k magnetic structures are essentiallydegenerate within the accuracy of our method, and the 1k is almost alwaysfavored against the 3k structure.

In order to analyze further the structural distortion that accompanies the ex-perimental 3k structure, an analysis similar to the one described in Ref. [58]was done. The total energy contains two components, an elastic energy whichcorresponds to the chemical binding between U and O ions, and a Jahn-Teller(JT) contribution from the CF levels, E= Eelast + EJT . Experimentally it is

40

5.3. MAGNETIC STRUCTURE AND DISTORTION IN UO2 41

Figure 5.2: Calculated elastic energy (solid line and open circles) and values of ∆ asa function of the magnitude of the oxygen distortion. The sum of the two contributionsto the total energy, Eelast+EJT , is also shown (dashed-dotted line). Since two differentvalues of EJT have been used (calculated and estimated from experiments (dashedlines)), two different total energies are shown in dashed-dotted lines.

known that the lowest level, the Γ5 triplet, splits up into one singlet and onedoublet, which lowers the total energy of the system with an amount ∆ (seeFig. 4 in paper IV). EJT was considered in this thesis to be equal to ∆. The CFenergies were evaluated from a method based on DFT described in paper IV.From this calculations the correct ordering of the levels comes out correctlyby the theory, with the Γ5 triplet being lowest in energy followed by the Γ3

doublet, Γ4 triplet and Γ1 singlet. The energy splittings between the differentCF levels are in good agreement with observations. The calculated value ofEJT (from the CF level calculations) was added to a calculated elastic energy,to obtain the total energy of UO2 as a function of the oxygen distortion of the3k structure. The elastic energy follows readily from our total energy calcula-tions, and it is shown in Fig. 5.2 as a function of magnitude of the distortion ofthe oxygen atoms of the 3k structure. Since two different values of the Jahn-Teller energy were used (one calculated from CF levels and one estimatedfrom experiments), two total energy curves are displayed in Fig. 5.2. Bothcalculations give a minimum in the total energy for finite values of oxygendistortions and the correct order of magnitude of the distortion is obtained; itis 0.02 Å based on the calculated Jahn-Teller energy and 0.003 Å based on theestimated Jahn-Teller energy, whereas the experimental value is 0.008 Å. Theabsolute value of the oxygen distortion is not quite reproduced in the calcula-tions, and of the two calculations the one based on an estimated Jahn-Tellerenergy seems to agree somewhat better with observations.

41


Figure 5.3: The helical magnetic structure of TbNi5. The magnetic moments are lo-calized on the Tb atoms. Tb atoms are represented by gray balls and the two Ni sitesare depicted as white and black balls.

5.4 Spin spirals in rare earths systemsTwo-ion couplings are mainly responsible for cooperative effects and mag-netic ordering in the RE. The indirect exchange or RKKY (Ruderman, Kittel,Kasuya and Yosida) interaction, being one of the most important, can be medi-ated by the conduction electrons. A localized magnetic moment spin-polarizesthe conduction electrons and this polarization in turn couples to a neighboringlocalized magnetic moment. This interaction takes the Heisenberg form [34]

H = −12

∑ij

Jij Si · Sj , (5.6)

where

Jij =1N

∑q

J (q)eiq·(ri−rj) and J (q) =V

Nµ2B

|I(q)|2χ(q). (5.7)

In Eqn (5.7), I(q) is the Fourier transform of the exchange integral and χ(q)is the conduction electron susceptibility. A maximum in J (q) will favor a qvector which corresponds to a spin spiral or helical structure.

5.5 TbNi5 and CeRhIn5

In paper II, the magnetic structure of TbNi5 was investigated by first princi-ples theory. This RE-transition metal compound crystallizes in the hexagonalCaCu5-type structure (space group P 6/mmm) [59] with one formula unit perunit cell. There are one Tb (1a) and two non-equivalent Ni sites, Ni1(2c) and

42

5.5. TBNI5 AND CERHIN5 43

0 0.1 0.2 0.3 0.4 0.5q c/2π

-0.5

0

0.5

1

1.5

2

2.5

3

E(q

)-E

(0)

(m

Ry)

0 0.04 0.08 0.12-0.1

0

0.1

0.2

0.3

0.4

Figure 5.4: The calculated total energy as function of the spin spiral wave vectorq = (0, 0, q) in units of c/2π (gray dots, the line is just a guide to the eye). In the insetwe show a section of the energy curve around the low-q region.

Ni2(3g). The neutron powder diffractogram at 10 K of TbNi5 showed the for-mation of an incommensurate helimagnetic ordering. The diffraction patternswere fitted with a helimagnetic structure which was resolved into a helix anda ferromagnetic component. The satellite reflections were indexed by a prop-agation vector q ∼ (0, 0, 0.02). The preliminary results indicate that the ferro-magnetic component of the magnetic moment was 7.16(1) µB/Tb atom in theab-plane and the helical part 4.5(1) µB/Tb atom. The total magnetic moment,8.4 µB/Tb atom, was obtained with the half-cone-angle (32) to the [1 0 0]helical axis, and the spins are rotated 7.0(1) between successive magneticsheets projected on the ab-plane. The Ni atoms do not carry any significantmoment.

In our study, we treated the 4f electrons in Tb as part of the core whichreflects their localized nature, while they are still able to produce an exchangefield that acts on the valence states by imposing Hund’s first rule (see subsec-tion 5.1). In oder to accomplish this, the 4f states have been spin-polarizedamong the core states [60]. The LSD-SIC approximation was used (see chap-ter 2). The spin occupation numbers are determined by applying the Russel-Saunders coupling scheme to the 4f shell. The spin spiral symmetry as dis-cussed in chapter 4, was adopted to handle the incommensurate magneticstructure of TbNi5. In Fig. 5.4 the calculated total energy is displayed and aminimum can be seen at q ∼ (0, 0, 0.06)2π/c. Unfortunately, the agreementwith the experimental value, q = 0.02, is not perfect. The calculated magnetic

43


moments are observed to be mainly placed on the Tb atoms, µ ∼ 9 µB andantiparallel to those of the Ni atoms, µ = 0.04 µB. To explain the origin ofthe observed magnetic spin spiral structure we calculated the Fermi surface ofTbNi5 (see paper II), in order to analyze possible nesting features of the va-lence band electronic structure. We found a very small shift between the spinup and down Fermi surface which is expected to result in very strong nestingfeatures in the z-axis of reciprocal space. We concluded from this analysis thatthe strong nesting along the z-axis as discussed in chapter 4 may be responsi-ble of the occurrence of a spin spiral in TbNi5.

CeRhIn5 belongs to the family of heavy fermion superconductors in whichsuperconductivity (SC) develops out of a normal state where electronic corre-lations produce a large enhancement of the effective mass of the conductionelectrons. CeRhIn5 is an antiferromagnet [61] at ambient pressure with a Néeltemperature of 3.8 K and it becomes a superconductor below Tc=2.1 K whenpressure is applied (1.65 GPa). This compound crystallizes in the tetragonalHoCoGa5 type of structure which is also common to the ambient-pressuresuperconductors CeIrIn5 [62], CeCoIn5 [63] and PuCoGa5 [64]. A neutrondiffraction study [65] performed on CeRhIn5 revealed an incommensuratemagnetic structure with wave vector q ∼ (1/2, 1/2, 0.3) and a staggered mo-ment of 0.75(2) µB per Ce ion [66] at 1.4 K. The nearest-neighbor momentson the tetragonal basal plane align antiferromagnetically.

A slightly different scheme was used in paper III to treat the 4f electron inCeRhIn5. CF effects seem to play an important role in this material. Hence,in order to evaluate the moment of the Ce ions, the ground state charge andmagnetization density were calculated by diagonalizing a CF Hamiltonian asdescribed in paper II and III.

Non-collinear density functional theory was applied on CeRhIn5 as de-scribed in chapter 4. The calculated magnetic moment of the Ce ion was foundto be 0.7 µB. The 4f spin density was found to induce a very small valence mo-ment 0.03 µB and the total moment in the cell was 0.73 µB. A Fermi surfacecross-section and the band structure were investigated in order to elucidatethe origin of the magnetic structure. Unfortunately the very small energy be-tween the different magnetic structures investigated made it impossible to beconclusive.

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45

6. Summary and outlook

Density functional theory is a very powerful tool in understanding con-densed matter. In this thesis it has been extensively used to investigate non-collinear magnetism, i.e. a spin arrangement in a material where there is noglobal spin quantization axis for the whole spin system. Two conditions wereidentified that favor a non-collinear state over the ferromagnetic arrangementof the spins. These conditions involve the presence of spin-up and -downbands at the Fermi energy and strong nesting between spin-up and -downFermi surfaces. Paramagnetic materials such as bcc V and well-known fer-romagnets such bcc Fe and fcc Ni were studied in order to illustrate the appli-cability of these criteria. All the investigated materials adopted a spin spiralstate after modifying either the volume or fixing the spin moment, which wasdone in order to meet the conditions described above. The conclusion wasthat, most likely, any metal could be turned into a non-collinear state providedthat the tuning of the exchange-splitting is done properly. Experimentally, thiscan be observed after applying pressure on materials. It has been widely spec-ulated that this would be the case of Fe in its high-pressure phase (hcp). Al-though magnetism has not been observed in this material, first principles cal-culations predict that when Fe is being compressed an antiferromagnetic or anon-collinear state develops. The questions of whether or not Fe is magneticat all under pressure and if this could be somehow related to the supercon-ducting phase that arises at almost the same volume as the hcp phase appears,are still open.

We investigated more complex materials such as the substitutionalTlCo2Se2−xSx alloys. From the theory we concluded that the nature ofmagnetism in the TlCo2Se2−xSx alloys changes from ferromagnetic forlow interlayer distances (distance between Co layers) to anti-ferromagneticfor large distances, via a set of non-collinear structures for intermediateinterlayer distances. We also investigated the band structure of TlCo2Se2 andTlCo2S2 and demonstrated that the condition that both spin channels shouldbe present at the Fermi energy was no longer fulfilled in the TlCo2S2 caseand therefore a ferromagnetic state becomes more stable. An experimentalstudy for TlCo2Se2−xTex is proposed to clarify the magnetic ordering inthese systems.

The relation between superconductivity and magnetism has also been dis-

45

46 CHAPTER 6. SUMMARY AND OUTLOOK

cussed in heavy Fermion systems where unconventional superconductivity ap-pears to coexist with non-collinear magnetism. Such is the case on CeRhIn5.A theoretical study of the evolution of the magnetic structure as a function ofpressure is still missing.

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47

Sammanfattning på Svenska

Den fasta materien runt omkring oss består som bekant av atomer som lig-ger tätt ihoppackade. Atomer är också väldigt små och består i sin tur av enpositivt laddad kärna som är omgiven av ett negativt laddat "moln" av elek-troner. Faktum är att trots att elektronerna är mycket mindre än neutronernaoch protonerna i kärnan så är det elektronerna och deras växelverkningar medomgivningen som ger ett material dess flesta egenskaper. Till exempel berorfärgen på saker och ting på hur elektronerna reagerar på ljusstrålning ellervärme och att metaller känns varmare eller kallare än en träbit vid sammatemperatur beror på att elektronerna i en metall leder värme mycket bättre änelektronerna i de kolföreningar som utgör träbiten. För att förstå hur mångaatomer det finns i ett fast material kan man jämföra med Saharas öken: Omman antar att Sahara är c:a 100 meter djup så finns det uppskattningsvis likamånga sandkorn där som det finns atomer i 300 gram järn, d.v.s. 1cm järnvägeller en sjättedels stekpanna.

Den teori som jag har använt i avhandlingen baserar sig på kvantmekanik.Det som är speciellt med kvantmekanik i det här sammanhanget är att det är ettgenerellt sätt att beskriva hur atomerna känner av sin omgivning som endastbaserar sig på fundamentala principer. Med det menas att man alltid användersamma matematiska ekvation för att beskriva ett material utan att justera denberoende på vad man vill räkna ut. Problemet är bara att ekvationen är såkomplicerad att den bara har lösts exakt för två! partiklar (t.ex. en elektronoch en proton som i väteatomen). Alltså har vi å ena sidan ett problem avstorleken 3 Sahara öknar och andra sidan en väldigt bra teori som bara går attanvända exakt för en väteatom...

Lösningen är dels att förenkla teorin så att den går att använda för störreproblem och dels att använda datorer för att snabbare göra beräkningarna. Dethar visat sig att om man gör en klyftig förenkling av de kvantmekaniska ek-vationerna med hjälp av symmetrier är det enklast att göra beräkningar föroändligt stora problem. Det svåra ligger då i att göra beräkningar för situa-tioner där man inte kan utnyttja symmetrier. Alla beräkningar i avhandlingenär gjorda med datorhjälp och olika versioner av grundläggande teori som äranpassad för att användas på datorer.

Mina undersökningar har varit inriktade på magnetiska egenskaper hosolika material och speciellt på hur magnetismen ser ut inuti materialen. Det

47

48 CHAPTER 6. SAMMANFATTNING

b)

a) c) d) e)n

Figure 6.1: Olika typer av magnetisk ordning hos ett material: (a) ferromagnetism, (b)antiferromagnetism, (c) helix struktur, (d) spin-spiral struktur och (e) spinglas.

enklaste exemplet är en järnkristall som är magnetisk i vanlig bemärkelseeftersom järn består av atomer som är magnetiska och alla pekar åt sammahåll, vilket resulterar i ett magnetiskt fält på utsidan av järnbiten. Dettafält användes t.ex. i generatorer, kreditkort och vanliga kylskåpsmagneter.Den magnetisering som järn har är dock inte den enda i naturen. Andramaterial kan innehålla atomer där varannan har en magnetisering som pekarupp medan de andra pekar ner. Det betyder att det för varje "upp" finnsen atom med magnetisering "ner" och i genomsnitt har materialet ingenmagnetisering. Det finns även situationer där atomernas magnetiseringar ärnästintill parallella med varandra så när som på en liten vinkel. Eftersomvarje magnetiseringsriktning då är förskjuten i förhållande till sin granneuppstår det en spiralform av den interna magnetiseringen i materialet. Dentotala magnetiseringen på utsidan beror då på summan av alla atomer ispiralen. Den summan kan vara både så att totala magnetiseringen är nolleller ha ett värde så att materialets magnetisering liknar järnets.

De studier som sammanfattas här i avhandlingen har varit inriktade på attförsöka förstå varför dessa spiralstrukturer uppkommer och vilka grundläg-gande fysikaliska principer som ligger bakom. I figur 6.1 illustreras olika typerav magnetisk ordning. Flera olika material har studerats och det viktigaste re-sultatet som framkommit är identifieringen av två kriterier som ett materialmåste uppfylla för att en spiralstruktur ska kunna uppstå. Dessa kriterier ärbaserade på den exakta fördelningen av elektronhastigheter och elektronernaseget spinn inuti en metall och är beskrivna i detalj i kapitel 4.

48

49

Acknowledgments

Despite their importance, acknowlegments are always left behind to be writ-ten in the last minute... or at least I did so. So I hope I do not forget anyone.Erik, I think you how important you are for me in all the aspects of my life,Thank you for everything! I would like to thank my supervisors Olle Eriks-son and Lars Nordström for their support, help, confidence and for never stopencouraging me. I have learned a lot from you. I really appreciate the effortBörje Johansson and Olle Eriksson make in providing us with the physicaltools, computers, books, etc., but most importantly, a nice work atmosphere.Do you know that friday’s fikas have been exported to T1 in Los Almos? Itis getting contagious. I would also like to thank everyone in Fysik 4 and The-orethical Magnetism, Ph. D students, my old roommates (Anders B, Carlosand Torbjörn), seniors, system administrators and people who already left thegroup. I am very grateful to Lars N and Bettan, who spent a lot of time withme teaching me all the tricks of the APW+lo code. Specially, I would like tothank Weine, Björn, Lars B, Carlos, Thomas, Alexei, Peter, Anders Ekströmand anyone who has ever had to spend some time in the computer’s room.Thanks to my experimentalist collegues, Sabina Ronneteg, Therese Eriksson,Yvonne Andersson and Rolf Berger. Jag vill också tacka hela Eriks familj förert stöd och kärlek.

I have spent the last year of my Ph. D. working in T1 at Los Alamos Na-tional Laboratory. I would like to thank John Wills for his help, encourageand for always having time to have scientific and sometimes not so scientificconversations! Eric Chisolm, thank you, thank you very much for revising mymanuscript. I really appreciate that. Nick thank you, for keeping the comput-ers up and my husband alive (just kidding! but if you would not keep an eyeon the number of cup of coffe he drinks...uf!). Thank you all for your friend-ship, Nick, Eric, Julia, JeeYon, Brad, Sven, Travis, Anders N., Angelica andeveryone in T1. Angelica, I know that everytime Erik and I travel you get aserious headache. Thank you for helping us with the paper work!!!

♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥

¡Uf! ¡Que largo va a hacer esto!. Ya, voy a empezar. Gracias a mi mami y ami papi por dejarme hacer y seguir todas mis ideas locas sin protestar mucho.

49

50 ACKNOWLEDGMENTS

Gracias a Roxana, a Eduardo y a mis hermanitos, Jorge y Cristian, por elapoyo, los emails y el entusiasmo compartido. Gracias a todos mis amigos ya toda mi familia en Chile. Gracias por todos estos años de amistad, a miseternas amigas Caro, Marilen y Mónica que sin sus consejos, risas y largasconversas a altas horas de la madrugada (en realidad a toda hora, pero lasconversaciones que se entablan tarde en la noche son las más divertidas y lasmás profundas también) yo ni siquiera estaría aqui. Especialmente me gustaríaagradecerle a la Cony y al Ariel por la paciencia con esta clienta regodiona queno les pago ni un veinte por los servicios de diseño gráfico. Muchas gracias alos dos; la portada quedo linda, quizas es lo más chori de la tesis. ¡Ya ahi estala fama prometida, Ariel! (Esto se está pareciendo a ’Gracias a la vida’ perobueno sigamos...) Muchas gracias a Juan por su apoyo técnico computacional,por esas divertidas horas de estudio (... o bueno, esa era nuestra intención) peroaún más importante: Gracias por abstenerse de comer ricos pastelitos, ñoquisy demases para que la Cony se pudiese concentrar en la figura de la portada.Gracias a la Pauli y a Mauro, mis pilares en Suecia, gracias a los dos por esasoncecitas que compartimos juntos y las partidas de carioca... que yo siempregané por supuesto. Gracias a Jorge y Gloria que ya no están en Suecia.

Gracias a todos ustedes a quienes amo por darme el empujoncito necesariocuando la consternación me invade.

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