Theory of Crystal Fields and Magnetism of f-electron Systems

Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 976

Theory of Crystal Fields andMagnetism of f-electron Systems

BY

MASSIMILIANO COLARIETI TOSTI

ACTA UNIVERSITATIS UPSALIENSISUPPSALA 2004

ananni,mamma,kiae minlillablomma

List of papers

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

I First-principles theory of intermediate valence f -electron systemsM. Colarieti-Tosti, M.I. Katsnelson, M. Mattesini, S. I. Simak, R.Ahuja, B. Johansson, C. Dallera and O. ErikssonSubmitted to Phys. Rev. Lett..

II Electronic Structure of UCx Films Prepared by Sputter Co-DepositionM. Eckle, R. Eloirdi, T. Gouder, M. Colarieti-Tosti, F. Wastin, J. Re-bizantJ. Nucl. Mater. (accepted).

III Crystal Field Excitations as Quasi-ParticlesM. S. S. Brooks, O. Eriksson, J. M. Wills, M. Colarieti-Tosti and B.JohanssonPsi-k Scientific Highlights Oct. 1999, URL: http://psi-k.dl.ac.uk/index.html?highlights (1999).

IV Approximate Molecular and Crystal Field Excitation Energies de-rived from Density Functional TheoryM. Colarieti-Tosti, M.S.S. Brooks and O. ErikssonIn manuscript.

V Crystal Field Levels in Lanthanide SystemsM. Colarieti-Tosti, O. Eriksson, L. Nordstrom, M.S.S. Brooks and J.M.WillsJ. Magn. Magn. Mater. 226-230, 1027 (2001).

i

VI Crystal Field Levels and Magnetic Susceptibility in PuO2

M. Colarieti-Tosti, O. Eriksson, L. Nordstrom, M.S.S. Brooks and J.M.WillsPhys. Rev. B 65, 195102 (2002).

VII On the Structural Polymorphism of CePt2Sn2; Experiment andTheoryHui-Ping Liu, M. Colarieti-Tosti, A. Broddefalk, Y. Andersson E. Lid-strom and O. ErikssonJournal of Alloys and Compounds 306, 30 (2000).

VIII Origin of Magnetic Anisotropy of Gd MetalM. Colarieti-Tosti, S.I. Simak, R. Ahuja, L. Nordstrom, O. Eriksson,D. Aberg, S. Edvardsson and M.S.S. BrooksPhys. Rev. Lett. 91, 157201 (2003).

IX Theory of the Temperature Dependence of the Easy Axis of Mag-netization in Gd MetalM. Colarieti-Tosti, O. Eriksson, L. Nordstrom and M.S.S. BrooksSubmitted to Phys. Rev. B.

Reprints were made with permission from the publishers.

The following papers are co-authored by me but are not included in the Thesis• Theory of the Magnetic Anisotropy of Gd Metal

M. Colarieti-Tosti, S.I. Simak, R. Ahuja, L. Nordstrom, O. Eriksson andM.S.S. BrooksJ. Magn. Magn. Mater. in press, available on line (2004).

• Magnetic Anisotropy from Electronic Structure CalculationsL. Nordstrom, T. Burkert, M. Colarieti-Tosti and O. ErikssonJ. Magn. Magn. Mater. (accepted).

• Magnetic x-ray scattering at relativistic energiesM. Colarieti-Tosti and F. SacchettiPhys. Rev. B 58, 5173 (1998).

Comments on my contributionIn 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 participation in the analysis.

ii

Contents

List of papers i

Introduction 3

1 Density functional theory 51.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Local density approximation and the Kohn-Sham equations . . 51.3 The success of DFT in local approximations . . . . . . . . . . 7

2 Solving the DFT equations: the MTO method 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 LMTO in the atomic sphere approximation . . . . . . . . . . 102.3 Full-potential LMTO . . . . . . . . . . . . . . . . . . . . . . 14

3 Crystalline electric field 193.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Crystalline electric field, the standard theory . . . . . . . . . . 19

3.2.1 CEF parameters evaluation from first principles . . . . 243.3 Total energy calculations of CEF splitting . . . . . . . . . . . 26

3.3.1 Obtaining the CEF charge density . . . . . . . . . . . 273.3.2 Total energy of a CEF level: symmetry constrained

LDA calculations . . . . . . . . . . . . . . . . . . . . 313.3.3 Generalisation to the magnetic case . . . . . . . . . . 333.3.4 Applications . . . . . . . . . . . . . . . . . . . . . . 35

4 Valence stability of f -electron systems 394.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Comparing volumes . . . . . . . . . . . . . . . . . . . . . . . 404.3 The Born-Haber cycle . . . . . . . . . . . . . . . . . . . . . . 414.4 Adding correlation effects . . . . . . . . . . . . . . . . . . . 42

iii

4.4.1 Application to Yb metal under pressure . . . . . . . . 43

5 Magnetocrystalline anisotropy in Gd metal 475.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 The force theorem . . . . . . . . . . . . . . . . . . . . . . . . 485.3 The anomaly of Gd . . . . . . . . . . . . . . . . . . . . . . . 49

Summary and Outlook 53

iv

Introduktion

Under de senaste aren har den kondenserade materiens teori blommat upp,speciellt inom omradet berakningsfysik. Fysiker kan nufortiden, med hjalp avdatorsimulationer, forutse egenskaper av extremt komplexa system i minutiosdetalj. Denna kapacitet, att simulera komplexa system, har tyvarr inte alltidfoljts av en djupare forstaelse av dessa system. Datorsimulationer kan ses somvirtuella experiment som darfor behover en forklaring, eller en modell. Vibegriper fysikaliska fenomen genom att bygga modeller utifran vara experi-mentella undersokningar. I vissa fall ar vi tvungna att valja alltfor forenklademodeller for att kunna losa dom men oftare anvander vi oss av numeriskt ap-proximerade losningar av analytiskt olosbara modeller.

I denna avhandling sambandet mellan modeller och datorsimulationer aren aterkommande ingrediens. Modeller anvandes for att generera en inputfor datorsimulationer i vara studier av elektriska kristallfalt som presenterasi kapitel 3. I detta fall har en modell hjalpt oss att utvidga ramen av elek-tronstrukturberakningar. I var studie av blandade valens system har vi daremotanvant elektronstrukturberakningar for att utvinna inputparametrar till en mod-ell som kan hjalpa oss att battre forsta fenomenet. Aven i studiet av den magne-tokristallina anisotropin av Gd har vi forsokt att forsta det underliggande skaletbakom det markliga beteendet av den latta magnetiseringsaxeln med hjalp aven enkel modell.

Vi vill avsluta denna introduktion med de ord som A. M. N. Niklasson skrevi sin Doktors avhandling:

”Det verkligt mystiska ar sambandet mellan verklighet och modell. Det enavet vi ingenting om och den andra har vi hittat pa sjalv.”

1

2

Introduction

In recent years the field of computational condensed matter physics has experi-enced a blossom. Physicists can nowadays, with computer simulations, predictminute properties of extremely complex systems. This improved capability ofsimulating complex systems is unfortunately not always correlated to a betterunderstanding of those systems. Computer simulations can be seen as virtualexperiments and therefore need an explanation, or a model. We understandphysical phenomena by constructing models out of our experimental investi-gations. Sometimes we are forced to choose oversimplified models in orderto be able to solve them but more and more often we resort to numerical ap-proximate methods in order to solve the analytically unsolvable equations of amodel.

In this Thesis the interplay between models and computer simulations is anunderlying feature. Models are used to generate an input to computer simula-tions in our crystalline electric field studies presented in chapter 3. In this casea model helped us in widening the range of applicability of electronic structurecalculations. In our study of intermediate valence systems, instead, we usedelectronic band structure calculations in order to obtain input parameters for amodel that can give us a better insight in the phenomenon. Also, in the studyof the magnetocrystalline anisotropy of Gd, we tried to understand, with thehelp of a simple model, the reason for the peculiar behaviour of the easy axisof magnetisation.

We would like to conclude this introduction with the words that A. M. N.Niklasson wrote in his PhD thesis

”What is really mysterious is the relation between reality and models. We knownothing about the former and we made up the latter.”

3

4

Chapter 1Density functional theory

1.1 IntroductionIt is remarkably simple to show that for an interacting electron gas in an exter-nal potential v(r)

”there exists a universal functional of the density, F [n(r)], independent of v(r),such that the expressionE ≡ ∫ v(r)n(r)dr+F [n(r)] has as its minimum valuethe correct ground state energy associated with v(r).”

The quote is taken from the abstract of the article in Ref. 1 by P. Hohenberg andW. Kohn that, despite the simplicity, was one of the main reasons behind theawarding of the Nobel prize in Chemistry to W. Kohn in 1998. The demon-stration (see Ref. 1) is done in two steps: First the fact that the total energyis a unique functional of the density is proved and then it is shown that thisfunctional has its minimum for the correct ground-state density. The work ofHohenberg and Kohn laid the basis for a transition from a quantum theory ofsolids based on wave-functions to one based on the density with an impressivedrop of the number of variablesa. In the following we will show how, based onthis theorem, a new way of treating many body systems could be devised.

1.2 Local density approximation and the Kohn-Shamequations

The functionalF introduced in the previous section contains the kinetic energy,

T ≡ 12

∫∇Ψ∗(r)∇Ψ(r)dr,

aThis at the cost of restricting ourselves to the ground state.

5

Chapter 1. Density functional theory

and the interaction between the electrons

U ≡ 12

∫Ψ∗(r)Ψ∗(r′)Ψ(r′)Ψ(r)

|r − r′| drdr′,

where Ψ(r) is the (unknown) wave function of the entire system and atomicunits are used. A more convenient expression for the ground-state energy ofan interacting inhomogeneous electron gas in a static potential v(r) is

E ≡∫v(r)n(r)dr +

12

∫n(r)n′(r)|r − r′| drd

′r +G[n], (1.1)

where the long range Coulomb potential is separated out from the functionalF [n(r)]. Since the functional G[n] is not yet specified there is no problem inhaving the self-interaction term not explicitly excluded in the electron-electronCoulomb interaction. Namely one could have a cancellation of the doublecounting term present in the Coulomb contribution by a corresponding term inG[n] as it happens in Hartree-Fock theory.2

Now the question to address is how to find the unknown universal func-tional G[n]. W. Kohn and L. J. Sham in Ref. 3 made the first proposal foran approximate functional leading to a set of self-consistent equations for thedetermination of E and n(r). They divided G in two parts

G[n] ≡ Ts[n] + Exc[n], (1.2)

where Ts[n] is the kinetic energy of the non interacting electron gas andExc[n]is the exchange-correlation energy, of which Eq. (1.2) is the definition. There-foreExc[n] will also contain the difference between the real kinetic energy andTs, that is, Txc ≡ T − Ts. Supposing now a slowly varying density, one canwrite3

Exc[n] ∫n(r)εxc[n(r)]dr. (1.3)

Then, once an expression for εxc[n(r)] is given, it is possible to find the energyand the density by solving the constrained variational problem

δE[n(r)]δn(r)

= 0∫n(r)dr = N

where N is the total number of electrons. One then finds that E and n(r) areobtained solving self consistently

(−∇2 + V [n(r)])ψi = εiψi

n(r) =∑

i

|ψi|2 (1.4)

6

1.3. The success of DFT in local approximations

with V being the sum of the external potential v(r), the electron Coulombpotential and the exchange-correlation potential, µxc[n]. The latter is definedas

µxc[n(r)] ≡ δExc[n(r)]δn(r)

= εxc[n(r)] + n(r)δεxc[n(r)]δn(r)

.

Then, the total energy functional can be written in the form,

E =∑

i

εi − 12

∫n(r)n′(r)|r − r′| drd

′r +∫

εxc[n(r)] − µxc[n(r)]dr, (1.5)

by observing that the kinetic energy for the non interacting electrons can beobtained for the KS equations (1.4) as

Ts[n(r)] =∑

i

εi −∫n(r)V [n(r)]dr.

1.3 The success of DFT in local approximationsIn Ref. 1 it is shown that in the limit of constant density one recovers theThomas-Fermi theory4, 5 and therefore, basically, εxc[n(r)] ∼ n1/3(r). In thepast 50 years there have been tremendous efforts to find the exact, or at leastthe best possible, functional for the exchange-correlation term. The approx-imation in Eqns. (1.2) and (1.3) is called local density approximation (LDA)and, together with the generalisation to magnetic systems, the local spin den-sity approximation6 (LSDA), it is the most commonly used approximation toDFT. Modern calculations commonly use a εxc[n(r)] that is parametrised withthe help of quantum Monte Carlo simulations on the homogeneous electrongas.7–10 The approximation of a slowly varying density can be improved bytaking into account gradient corrections. This is what has led to the generalisedgradient approximation11, 12 (GGA). Another correction to any local approxi-mation to the exchange-correlationb has originated from the observation that,with a local exchange, the term with r′ = r in the Coulomb contribution is notexactly cancelled by an analogous term of opposite sign in the exchange termas it happens in the Hartree-Fock theory. For this reason the self interactioncorrection13 (SIC) has been devised. However, a part of the self interaction(SI) present in the Hartree contribution is actually cancelled by a local approx-imation to the exact exchange. This poses a problem for the SIC of Ref. 13that removes correctly the SI of the Hartree part of the potential but fails incompletely removing the SI part of the LDA or GGA exchange.14

Surprisingly, improvements on the LDA approximation have not alwaysresulted in a better predictive capability and up to date there is still no known

bGGA is also to be considered a local approximation.

7

Chapter 1. Density functional theory

way to systematically improve on the exchange-correlation functional in orderto obtain theoretical results that have the requested accuracy. This is, probably,the biggest deficit of DFT. In very recent times J. Perdew and co-authors havemade an effort in trying to provide such a systematic scheme of approximations(see Ref. 15 and references therein).

Jones and Gunnarsson in Ref. 16 attributed the success of LDA to the factthat LDA reproduces the integral properties of the LDA exchange-correlationfunctional. In particular they demonstrated that the integral of the ‘exchange-correlation hole’ depends only on its spherical component, that is the one com-ponent that LDA describes well. The GGA was therefore designed to obey thesum rule of exchange-correlation hole. Nevertheless, the number of systems,properties and phenomena that LDA has been able to explain has always beena puzzle to physicists. Expecially puzzling is the relationship between KS or-bitals, i.e. the ψi of Eq. (1.4) and the eigenstates of real systems. In principlethere is no relation between them17 but experience shows that absorption andemission spectra often compare well with calculated density of states (DOS).Examples of such an agreement are given in paper I and in paper II.

8

Chapter 2Solving the DFT equations: theMTO method

2.1 IntroductionIn this chapter we will present the basic concepts underlying a common wayof solving the KS equations in a solid. We will focus on linear methods and, inparticular, the linearised muffin tin orbital method (LMTO) ideated by O. K.Andersen.18 Our description is going to be far from complete and is intendedonly as an introduction. In choosing which algebra to show and which to omitwe have tried to give preference to equations that have an underlying newconcept or that are particularly significant. We refer the reader, for example, toRefs. 19, 20 and references cited therein, for a comprehensive review.

The basic idea behind the development of the Muffin Tin Orbital (MTO)method for solving the DFT problem is that in solids the potential seen by elec-trons can be divided in two parts: An atomic-like, almost spherical, deep neg-ative potential in the region close to the atomic lattice sites and a free-electronlike, almost flat, potential in the region between lattice sites. If the interstitialpotential is approximated by a flat potential and the potential inside the atomicspheres approximated by a spherical potential, the total potential becomes amuffin tin (MT) potential. In this case solutions to the wave equation may beobtained to arbitrarily high accuracy by expanding the wave function insidethe atomic spheres in terms of partial waves, φRL(E,r) = φRl(E,r)Ylm(r)

Ψj(r) =∑RL

φRL(Ej , r)cjRL + Ψi(Ej , r) (2.1)

where Ψi is the wave function in the interstitial, R labels the lattice site, L isa short notation for l,m and the coefficients cj are chosen such that the partialwaves from different spheres join continously and differentiably. The result is

9

Chapter 2. Solving the DFT equations: the MTO method

the set of homogeneous, linear equations

M(E)cj = 0. (2.2)

The secular matrix M depends upon energy in a complicated, non-linear, man-ner. Therefore Eq. (2.2) is solved by finding the roots of the determinant of M.Eq. (2.1) is an example of a single-centreexpansion.

An alternative methodology is to use fixed basis functions (as in the LCAOmethod) as trial functions for the solid and to use the variational principle,leading to the Rayleigh-Ritz equation

(H − EjO)cj = 0, (2.3)

where H and O are the Hamiltonian and overlap matrices and the wave func-tion has been expanded in the multi-centreexpansion

Ψj(r) =∑RnL

χRnL(rR)cjRnL. (2.4)

The solutions to the algebraic eigenvalue problem, Eq. (2.3), are easier to ob-tain than the root searching involved in solving Eq. (2.2). Another advantage ofthe fixed basis set method is that it may be used for potentials of general shapewhereas its disadvantage is that the basis set required to obtain high accuracymay be very large.

Linear methods combine some of the better properties of partial waves andfixed basis set expansions by using envelopefunctions and augmentation. A setof envelope functions, which is reasonably complete in the interstitial region,is chosen and then augmented inside the atomic spheres by functions related topartial waves inside the atomic spheres. The result is an algebraic eigenvalueproblem, Eq. (2.3), with a far smaller basis set than atomic orbitals since thebasis set is better adapted to the solid state as it is derived from partial waves.The type of method depends upon the chosen envelope functions. For example,envelope functions that are plane waves give rise to the linear augmented planewave method (LAPW) whereas envelope functions that are Hankel functionsgive rise to the linear muffin tin orbital method (LMTO).21

Andersen and his collaborators, developed the method we will briefly de-scribe in this chapter that goes by the name linearised muffin tin orbital (LMTO)18

or LMTO in the atomic sphere approximation (LMTO-ASA).

2.2 LMTO in the atomic sphere approximationThe simplest approximation to the real potential is to take, overlapping, spacefilling spheres inside which the potential is taken to be spherically symmetric.

10

2.2. LMTO in the atomic sphere approximation

Outside these atomic spheres instead, the potential is approximated with a con-stant value (sometimes set to zero). This is the atomic sphere approximation.The flatness of the potential in the interstitial region allows one to approximatethe basis functions with the solutions to the radial Helmholtz equation:(

d

drr2d

dr+ r2k2 − l(l + 1)

)R = 0. (2.5)

The solutions to this are

jl =(kr)l

(2l + 1)!!

nl =(2l − 1)!!(

1kr

)l+1

. (2.6)

Note that jl is regular at the origin while nl is irregular. It is convenient toscale those functions as follows

Jl ≡ jl(2l − 1)!!2(ks)l

=1

2(2l + 1)

(rs

)l

Kl ≡ nl(ks)l+1

(2l − 1)!!=(sr

)l+1(2.7)

Here s is a scaling length usually taken to be the average Wigner-Seitz radius.Inside the atomic spheres (AT), the basis function is the solution, φ, of the

stationary Schrodinger equation

(−∇2 + V − E)φ = 0. (2.8)

For reason of foresight let us write here also the energy derivative of (2.8)

(−∇2 + V − E)φ = φ. (2.9)

We will have to match, continously and differentiably the function φwith a lin-ear combination of the regular and irregular solutions to Eq. (2.5). In general,matching a function, φ, continously and differentiably to a linear combinationof two others, J and K, at some point, s, is done by solving for the constantsa and b, the coupled equations

φ(s) =aJ(s) + bK(s)φ′(s) =aJ ′(s) + bK ′(s). (2.10)

The prime here indicates a derivative with respect to r. The resulting matchedfunction is then

φ(s) =Wφ,KJ(s) −Wφ, JK(s)

WJ,K , (2.11)

11


where the Wronskian, W , is defined as

WJ,K ≡ s2(JK ′ −KJ ′), (2.12)

and J,K, J ′,K ′ are evaluated at s. The Wronskian between the functionsdefined in (2.7), WK(r), J(r)|r=s = s/2 is readily evaluated since the onebetween the spherical Bessel and Neumann functions, Wn(kr), j(kr) = 1(here the derivative is with respect to x = kr).

The solutions inside the AT are energy dependent. Andersen21 observedthat it is possible to approximate to first order φ(E) with a linear combinationof φ(E) and φ(E), where E is a linearisation energy chosen according to thewindow of energies E of interest. The other Wronskian that is needed is,therefore, the one between the solution to the wave-equation and its energyderivative, Wφ, φ. This can be calculated observing that from Eqns. (2.8)and (2.9) one can write

0︷︸︸︷∫φ(−∇2 + V − E)φdr−

1︷︸︸︷∫φ(−∇2 + V − E)φdr = −1, (2.13)

and using Green’s second identity for integrating the above expression, oneobtains

Wφ, φ = 1. (2.14)

Next, one introduces a way to write a function on a crystal that explicitlydistinguishes between contributions inside any given AT sphere and the onesin the interstitial. Also this notation is due to Andersen and co-authors anduses the vectorial bra-ket notation

|fR >∞= |fR >︸︷︷︸sphere at R

+|fR >i −∑R′

|fR′ >︸︷︷︸sphere at R′

TR′;R, (2.15)

where the superscript ∞ indicates that the function, f , extends over the entirecrystal, the superscript i means that the function is defined only in the inter-stitial region and no superscript means that the function is truncated outsidethe AT region corresponding to the site indicated as a subscript. The matrixTR′;R represents a generalised structure constant matrix that is unknown untilthe envelope function (see below) is specified. Eq. (2.15) highlights the factthat a function belongingto site R can be written as an expansion around a siteR′ inside the AT centred on R′.

One can write the envelope function of the MTO’s in this notation

|K0RL >

∞= |KRL >︸︷︷︸sphere at R

+|K0RL >

i −∑R′,L′

|J0R′L′ >︸︷︷︸

sphere at R′

S0R′,L′;R,L. (2.16)

12

2.2. LMTO in the atomic sphere approximation

In this case the coefficients S0R′,L′;R,L are well known and are referred to as

the energy independent structure constants (as they are determined only by thecrystal structure). Schematically this is simply written

|K >∞= |K > −|J > S. (2.17)

We have here dropped the part of the wave-function that explicitely corre-sponds to the interstitial region. This was done because LMTO-ASA usesoverlapping, space filling, atomic spheres.

The solutions inside each sphere are then matched to an envelope functionof the form of the one in Eq. (2.17). First, let us write the solution to thewave-equation in the same form as (2.17), remembering that, because of lin-earisation, this solution will be a linear combination of φ and φ

|χ >∞= |φ > −| ˙φ > h, (2.18)

where ˙φ = φ+oφ and o =< φ| ˙φ >. The matrix h is related to the Hamiltonianby

< χ|H − Eν |χ >= h(1 + oh) (2.19)

and the overlap matrix is

< χ|χ >= (1 + ho)(1 + oh) + hph (2.20)

with p =< φ|φ >. At the AT sphere boundary |K >∞ is matched to |χ >∞,

|K >∞→ −|φ > WK, ˙φ

− | ˙φ > [−W K,φ +W J, φS]. (2.21)

Since o = −Wφ, J

/W φ, J, then

WK, ˙φ

W J, φ = W

K, φ

W J, φ + oW K,φW J, φ

=S

2(2.22)

and

|K >∞ → −WK, ˙φ

|φ > +| ˙φ >

−W K,φ

WK, ˙φ

+W J, φWK, ˙φ

S

|K >∞ → −WK, ˙φ

×|φ > +| ˙φ >

−W K,φ

WK, ˙φ

+2SW J, φSW J, φ

.

13


Since(−W

K, ˙φ

)−1is the normalisation factor arising from the augmen-

tation performed to obtain linearisation, a comparison to (2.18) yields

h =W K,φWK, ˙φ

− 2SW J, φSW J, φ (2.23)

for the Hamiltonian.In Ref. 22 Andersen and co-authors proposed an interesting idea: Since the

envelope function can be chosen according to one’s needs, one can choose it ina way that resembles the grounding of spheres with an electrostatic potentialin order to get short range or screenedwave-functions.22 Let us consider, asan example, the case in which one wants to minimise the overlap o instead.Let us take a modified function Jα

l (r) = J0l (r)− αlKl(r) to which −α of the

irregular solution has been added to the regular solution. Observe, then, that

|J > = |Jα > +α|K >

|K >∞ = |K > −[|Jα > +α|K >]S|K >∞ = |K > [1 − αS] − |Jα > S

|K >∞ [1 − αS]−1 = |K > −|Jα > S[1 − αS]−1

|Kα >∞ = |K > −|Jα > Sα (2.24)

where

|Kα >∞ = |K >∞ [1 − αS]−1

Sα = S[1 − αS]−1. (2.25)

The functions φα = φ+oφ and Jα have the same logarithmic derivative at the

atomic sphere boundary therefore Wφα, Jα

= 0. One has then that

o = −Wφ, Jα

W φ, Jα = 0 (2.26)

for the orthogonal representation characterised by

α =

φ, J

φ, J . (2.27)

2.3 Full-potential LMTOThe LMTO method can be efficiently used also to solve the DFT equationswithout making any assumption on the shape of the potential and of the charge

14

2.3. Full-potential LMTO

density in both the interstitial and the MT region. In this case the calculationscheme is, quite unimaginative, called full potential (FP). The last generationLMTO-ASA implementations20 are already accurate but FP-LMTO removes,at the cost of computational time, also the small influence of the sphericallyassumed potential inside each ASA sphere. In the FP method the division intoMT and interstitial regions carries no approximation as in the LMTO-ASAcase. It is then important to understand the role of MT radii choice in FP meth-ods. MT radii do not necessarily reflect any physical property of the potential.The division in MT and interstitial region is only a matter of computationalconvenience. Of course, the physical properties of the system at hand can helpalso in making an efficient choice for the MT radii but it is important to keepin mind that this convenient geometrical division of the crystal does not influ-ence the shape of the calculated potential or charge density. A good choiceof the MT radii can result in a speed-up of the calculations. Nevertheless, ifthe series expansion representing the solution inside the MT region and in theinterstitial are taken to convergence the total energy is virtually insensitive tothe choice of the MT radii. In this case, the MT radii can be regarded as varia-tional parameters. In fact, their influence on the total energy is just through thedifference in the functional dependence of the basis function inside and outsidethe MT region. The minimisation of the total energy with respect to the MTchoice gives the optimal FP-LMTO solution to DFT equations.

Another important thing that should be mentioned is the fact that in somecases the Hamiltonian is not the same inside and outside the MT region. Thisis, for example, the case for relativistic calculations where spin-orbit couplingis commonly considered only inside MT spheres. Then the choice of the MTradii requires some care and physical insight.23

The FP-LMTO implementation that is used throughly in this Thesis is theone of Ref 24. We will therefore follow the notation of that reference, whichis reported in the following for convenience. The spherical harmonics relatedfunctions are

Ylm(r) =ilYlm(r) (2.28)

Clm(r) =

√4π

2l + 1Ylm(r) (2.29)

Clm(r) =ilClm(r) (2.30)

15


Also the following functions will be used

Kl(k, r) = − kl+1

nl(kr) − ijl(kr) k2 < 0nl(kr) k2 > 0

(2.31)

KL(k, r) =Kl(k, r)YL(r) (2.32)

Jl(k, r) =jl(kr)/kl (2.33)

JL(k, r) =Jl(k, r)YL(r) (2.34)

where L is a short notation for lm and nl and jl are the spherical Neumannand Bessel functions, respectively.

Inside a MT of radius, sτ , at site τ , it is convenient to define a local coordi-nate system referred to the centre of the MT, rτ ≡ Dτ (r−τ). The latter definesthe rotation Dτ that brings global crystal coordinates, r, to the site-local ones,τ . Schematically, inside a MT sphere, the basis functions, the charge density,n, and the potential, V, are

ψτ (k, r)∣∣∣r<sτ

≡∑lm

χlm(k, rτ )Ylm(rτ ) (2.35)

nτ (r)|r<sτ≡∑

h

nτ (h,rτ )Dh(rτ ) (2.36)

Vτ (r)|r<sτ≡∑

h

Vτ (h,rτ )Dh(rτ ) (2.37)

whereDh(rτ ) = Dh(D(r)) =

∑mh

α(h,mh)Clhmh. (2.38)

The last functions, called spherical harmonics invariants, make clear the reasonwhy a rotation to site-local coordinates is made. If the latter are chosen so thatthere exists a transformation T such that Dτ ′ = DτT

−1 if Tτ = τ ′, then thespherical harmonics invariant will depend only on the global symmetry andare site independent. In addition, by this choice, the number of terms in theexpansions (2.36) and (2.37) is minimised.

In the interstitial region the basis functions, the charge density and the po-tential are instead expressed as Fourier series. Schematically, that is written

ψ(k, r)∣∣∣r>sτ

≡∑

g

ei(k+g)·rψ(k + g) (2.39)

n(r)|r>sτ≡∑Z

nI(Z)∑gεZ

eig·r (2.40)

V (r)|r>sτ≡∑Z

V I(Z)∑gεZ

eig·r (2.41)

16

2.3. Full-potential LMTO

where Z indicates the symmetry stars of the reciprocal lattice. To be moreprecise, the basis functions in the interstitial will be Bloch sums of sphericalHankel or Neumann functions

ψi(k, r)∣∣∣r>sτ

=∑R

eik·RKli(ki, |r − τi − R|)Ylimi

(Dτi(r − τi − R)).

(2.42)The index i here identifies the lattice site. At the MT spheres the envelopefunction can be written as (for site R = 0)

ψi(k, r)∣∣∣r=sτ

=∑R

eik·R∑

L

YL(Dτ rτ ) (Kl(ki, sτ )δ(R, 0)δ(τ, τi)δ(L,Li)

+JL(ksτ )BL,Li(ki, τ − τ ′ − R))

=∑L

YL(Dτ rτ ) (Kl(ki, sτ )δ(τ, τi)δ(L,Li)

+JL(ksτ )BL,Li(ki, τ − τ ′ − k)), (2.43)

where rτ = r − τ and B are equivalent to the KKR structure constant.25, 26

Defining the two-component vectors

Kl(k, r) ≡ (Kl(k, r), Jl(k, r)) (2.44)

SL,L′(k, τ − τ ′,k) ≡(

δ(τ, τ ′)δ(L,L′)BL,L′(k, τ − τ ′,k)

)(2.45)

the value of a basis function at a MT boundary can be compactly written

ψi(k, r)∣∣∣r=sτ

=∑L

YL(Dτ rτ )Kl(ki, sτ )SL,L′(ki, τ − τ ′,k) (2.46)

now the index i includes τ, l,m, k.The radial part of a basis function inside the MT is a linear combination of

atomic-like functions φ augmented with their energy derivative φ. This willbe continously and differentiably matched at the MT boundary to the radialfunction K of Eq. (2.46). Let

U(E, r) ≡(φ(E, r), φ(E, r)

), (2.47)

then, the matching conditions can be schematically written

U(E, s)Ω(E, k) = K(k, s) (2.48)

U ′(E, s)Ω(E, k) = K ′(k, s) (2.49)

17


with Ω being the 2× 2 coefficient matrix. A basis function inside a MT sphereis then

ψi(k, r)∣∣∣r<sτ

=l≤lmax∑

L

UtL(Ei,Dτrτ )Ωtl(Ei, ki)SL,Li(ki, τ − τ ′,k) (2.50)

withUtL(Ei, r) ≡ YL(r)UtL(Ei, r). (2.51)

Something has to be said here about the energy parameters Ei. In general abasis set must be able to describe accurately energy levels corresponding to thesame quantum number l but differing by the principal quantum number n. Thisis particularly important in actinide (An) systems where, for example, 6p and7p states give raise to the infamous problem of the so called ghost bands. It iscustomary, in fact, whenever there is the need of considering semi-corestates(like the 6p’s in An) to have two different energy panels for valence and semi-core states. Whenever those energy panels overlap non physical ghost bandsappear in the calculated electronic structure. The FP-LMTO method24 that wehave used in this Thesis, is different on that regard. The radial dependence of abasis inside a MT is calculated with a single, fully hybridising, basis set, whereone uses atomic-like functions, φ and φ calculated with energies Enl withdifferent principal quantum number, n. Thus, Ei in Eq. (2.50) indicates theenergy parameterEnl corresponding to the principal quantum number assignedto the basis i.

Another important parameter in the expression (2.50) is the angular mo-mentum cut-off, lmax. This has to be determined from time to time dependingon the system at hand. Typical values for which most systems are convergedare lmax =6-8. Now, using Eqns. (2.43) –or (2.46)–, (2.42) and (2.50) onecan evaluate the potential, the overlap matrix and the charge density matrixelements in the entire crystal

18

Chapter 3Crystalline electric field

3.1 IntroductionIn a free atom the quantum number, J , corresponding to the total angular mo-mentum, is what is called a good quantum number. This reflects the fact thata free atom has a complete rotational symmetry and its ground state is degen-erate with degeneracy equal to 2J + 1. If one imagines to put a free atom in acrystal this situation will no longer be true. The ions surrounding (called lig-ands) our ”formerly free atom” (called central atom) produce an electric fieldto which the charge of the central atom adjusts and the 2J + 1 degeneracywill be removed. We refer to this field as the crystalline electric field (CEF) orsimply the crystal field. The new eigenstates are eigenstates of the CEF (linearcombinations of the original 2J + 1 eigenstates) and their corresponding den-sities are non spherical (see Fig. 3.1). The evaluation of the effects of the CEFin f -electon systems is the object of the present chapter.

3.2 Crystalline electric field, the standard theoryThe potential due to a charge distribution is

V (r) =∫

ρ(r′)|r − r′|

dr′. (3.1)

After expanding the denominator in the integrand of the above equation inspherical harmonics function, Ylm, one can write27

V (r) =∑lm

4π2l + 1

Ylm(r)∫

rl<

rl+1>

ρ(r′)Y ∗lm(r′)dr′, (3.2)

19

Chapter 3. Crystalline electric field

Figure 3.1: Charge densities for an f shell with 2 electrons in cubic symmetry. Thetop figure corresponds to a Γ1 symmetry CEF state while the bottom one is a Γ4 state.

20

3.2. Crystalline electric field, the standard theory

or, separating the r< and the r> integrals

V (r) =∑lm

rnYlm(r)4π

2l + 1

∫ ∞

rρ(r′)

(1r′

)l+1

Ylm(r′)dr′

+∑lm

(1r)l+1Ylm(r)

4π2l + 1

∫ r

0(r′)l

ρ(r′)Y ∗lm(r′)dr′.

(3.3)

In a MT geometry, one has a natural separation between the contribution orig-inating from the ions surrounding the central atom (r ≥ SMT ) and the on-sitecontribution (r ≤ SMT ). In conventional crystal field theory the latter is usu-ally neglected as it is assumed that the crystal field at an ion site arises fromthe ions around that site. In this case one can simply consider r = SMT andthe crystal field potential can be written

V (r) =∑lm

AlmrlYlm(r) =

∑lm

Blm(r)C lm(r) (3.4)

where, the crystal field parameters Alm are

Alm =4π

2l + 1

∫ ∞

Sdr′ρ(r′)(

1r′

)l+1Ylm(r′), (3.5)

and∫∞S indicates integration outside the given site. Eq. (3.4) also defines the

parameters

Blm(r) =(

2l + 14π

)1/2

rlAlm (3.6)

and

C lm(r) ≡ Ylm(r) ≡

(4π

2l + 1

)1/2

Ylm(r) (3.7)

is a tensor operator.In general, however, the on-site contribution need not be negligible. If the

charge distribution in the solid is known both integrals in Eq. (3.3) may becalculated. Then one should replace Eq. (3.4) with

V (r) =∑lm

[AlmrlYlm(r) +A′

lmr−(l+1)Ylm(r)] =

∑lm

Blm(r)C lm(r) (3.8)

where

A′lm =

4π2l + 1

∫ r

0dr′ρ(r′)(r′)lYlm(r′) (3.9)

and

Blm(r) =(

2l + 14π

)1/2 [rlAlm + r−l−1A′

lm

]. (3.10)

21


The crystal field potential is now expressed as a linear combination of prod-ucts of spherical harmonics and crystal field parameters. The ground statedensity of the system at hand can therefore be used to calculate the Alm andthe radial integrals < f |rl|f > and < f |(1/r)l+1|f >.

Localised states are most naturally expressed as |JM > manifolds and arecharacterised by the total orbital momentum, L, the total spin momentum, S,and the total angular momentum, J . In order to evaluate the CEF matrix el-ements for such localised states the latter have to be re-expressed in terms ofthe operator equivalents of spherical harmonics. This is done via the Wigner-Eckart theorem that factors the angular dependence of the matrix elements ofa spherical tensor operator, Xk

q , as follows

< αjm|Xkq |αj′m′ >= (−1)j−m < αj||X(k)||αj′ >

(j k j′

−m q m′

)(3.11)

where < αj||X(k)||αj′ > is the reduced matrix element. What we need isa recipeto decompose the N-electron state |JM > in a single electron basisilRlYlmχ

σ = |lmσ, r >, where Rl is a radial function and χσ a spinor, or,equivalently, a definition for the electron creation operator a†i . Once that isknown, one can write

|JM >= a†1......a†N |0 > . (3.12)

Then, the matrix elements of the CEF potential can be written as

< m′σ|Bkq (r)Ck

q (r)|mσ > =∫Y ∗

lm′(r)Ckq (r)Ylm(r)d(r)

∫r2R2

l (r)Bkq (r)dr (3.13)

≡< Bkq >< m′|Ck

q |m >,

where the approximation that the radial wave function, Rl, is the same for alllocalised electrons of given l, has been made. It remains to calculate

Akq =

∑mm′σ

a†m′σ < m′σ|Ckq |mσ > amσ (3.14)

in the basis

|JM > = (2J + 1)1/2∑

MLMS

|LMLSMS >

×(−1)L−S+M

(L S J

ML MS −M

)

22


The first step is made by observing that

< JM ′|Akq |JM >=

(2J + 1)∑

M ′LM ′

S

∑MLMS

(−1)M+M ′(

L S J

ML MS −M

)(3.15)

×(

L S J

M ′L M ′

S −M ′

)< LM ′

LSM′S |Ak

q |LMLSMS > .

The problem is now reduced to the evaluation of the matrix elements of Akq in

the LS basis. Since Akq is a tensor operator and is spin independent

< LM ′LSM

′S |Ak

q |LMLSMS >= (3.16)

(−1)L−MLδMS ,MS′ < L||A(k)||L >(

L k L

−ML q ML′

).

The reduced element, < L||A(k)||L >, may be evaluated by observing firstlythat the state |LLSS > is a single Slater determinant

|LLSS >= a†l+1−N↑.....a†l↑|0 > (3.17)

(this is the recipewe needed). Therefore

< LLSS|Akq |LLSS >=∑

mm′σ

< lm′σ|Ckq |lmσ >< LLSS|a†m′σamσ|LLSS >

=l+1−N∑

m=l

< lmσ|Ckq |lmσ > (3.18)

=l+1−N∑

m=l

δq,0(−1)l−m < l||C(k)||l >(

l k l

−m 0 m

).

where the reduced matrix element < l||C(k)||l > is

< l||C(k)||l >= (−1)l(2l + 1)

(l k l

0 0 0

). (3.19)

Since < LLSS|Akq |LLSS > may also be written in terms of the reduced

matrix element, < L||A(k)||L >

< LLSS|Akq |LLSS >=< L||A(k)||L >

(L k L

−L 0 L

)δq, 0 (3.20)

23


one can write

< L||A(k)||L >= (2l + 1)

(l k l

0 0 0

)(

L k L

−L 0 L

) (3.21)

×l+1−N∑

m=l

(−1)m

(l k l

−m 0 m

).

Now the matrix elements of the crystal field are easily evaluated

< JM ′|Akq |JM >= (−1)L+S−M ′+k(2J + 1)

× < L||A(k)||L >J J K

L L S

(J k J

−M ′ q M

). (3.22)

It is customary to replace the matrix elements of the CEF with operatorequivalents. The matrix elements of the Racah operator equivalent, Ok

q (J), forthe manifold |JM > are, from the Wigner-Eckart theorem

< JM ′|Okq (J)|JM >= (−1)J−M ′

(J k J

−M ′ q M

)< J ||Ok

q (J)||J >(3.23)

where the reduced matrix element of Okq (J) is given by

< J ||Okq (J)||J >=

12k

[(2J + k + 1)!

(2J − k)!

]1/2

, (3.24)

consequently, the ratios,

fkN =

< LSJM ′|Akq |LSJM >

< JM ′|Okq (J)|JM >

= (3.25)

(−1)L+S+J+k(2J + 1)

J J K

L L S

< L||Ak||L >< J ||Ok||J >,

are the Stevens factors used to replace the crystal field matrix elements byoperator equivalents.

3.2.1 CEF parameters evaluation from first principlesStarting from the work of Schmitt28, 29 and continuing with the work of Refs. 30–37 and references cited therein, there have been a number of calculations of

24


CEF parameters in rare-earth elements and compounds, using first-principlestheory. The basic procedure in those works is as follows: The ground statecharge density of the system at hand is calculated by first principles and thenthe integrals in Eqns. (3.5) and (3.9) are evaluated. In the more recent calcula-tions both the Coulomb, VC = VN + VH and the exchange correlation poten-tial, µex, are contributing to the CEF potential, while in earlier works only theCoulomb potential was considered. The expression for the CEF parameters de-rived in the previous section is easily generalised, the potential V = VC + µex

is expanded in spherical harmonics instead of VC alone. Then the procedure isabsolutely analogous.

For the calculation of the total charge density the f charge density is con-strained to be spherical. This is a natural choice when the on-site contributionto the CEF is neglected. When this is not the case this choice is not fully justifi-able since any given symmetry of the f charge density will result in a differentvalence density. However, the screening effect due to the polarisation of theconduction electrons by the non spherical part of the central atom charge den-sity (see Ref. 38 and paper III) is outside the framework of the standard CEFmodel that we have just described. In fact, calculating CEF parameters startingfrom different constrained symmetries of the f charge density may, in princi-ple, give different values of those parameters. The model is then valid anduseful only if these changes are negligible.

Fahnle and Co-authors39, 40 tried a different, elegant approach to the prob-lem of evaluating CEF parameters, considering the energy change due to therotation, via an applied magnetic field, of the charge density of the f shell inthe frozen potential calculated self consistently with the unrotated f shell. Inpresence of a molecular field the f shell will be a mixture of various densitiescorresponding to different CEF states. It is therefore possible, by calculat-ing the energy change in chosen rotations, to obtain the CEF parameters. InRef. 39 it is observed that, to first order in perturbation theory, there is a can-cellation of the kinetic and potential contributions if a given mixture of CEFcharge densities is rigidly rotated in a frozen (apart from the rotating charge)potential. This is because the total energy of the unperturbed system is at avariational minimum. The rotation can be seen as a change δn(θ, φ) (with∫Ω δn(θ, φ)dΩ = 0 ) in the density n(r), which corresponds to a change ofO(δn2) in the energy. If one divides the energy of the system in the sum of theenergy originating from the unperturbed density n(r), E(n(r)), and the oneoriginated from δn(θ, φ), E(δn), to first order one can write

δE(n(r)) = 0. (3.26)

Then, if the change in the kinetic energy of the f electrons is disregarded,

25


when the charge density is rotated, one obtains that

δE[n(r)] = δE[δn(θ, φ)] =∫V (n(r))δn(θ, φ)dr. (3.27)

Since the new rotated charge density will correspond to a linear combinationof CEF charge densities (the CEF eigenstates span the entire subspace of thegiven symmetry), one can use Eq. (3.27) to estimate the CEF parameters.However, in Ref. 40 it is claimed that calculating CEF parameters with thisprocedure, starting from self consistent solutions corresponding to differentmolecular field mixtures of CEF charge densities, gives substantially differentvalues for the calculated CEF parameters. This fact could throw some doubtson the validity of the CEF model, leading one to believe that the screeningcontribution that is left out by the model is not negligible. In Ref. 40 it isalso observed that CEF parameters obtained with the above described rotationmethod in a frozen potential that has been calculated with a spherically con-strained f charge density are in agreement with CEF parameters calculatedindependently. The apparent dependence of the CEF parameters on the statechosen for evaluating the self-consistent potential is explained by noting thatEq. (3.27) is only valid to first order in δn. Contributions of order δn2 can besubstantially different for different CEF states. By choosing the frozen poten-tial corresponding to the spherically constrained f -charge density one obtainsa less biasedtreatment of the contributions of second order in δn. The effectof choosing a transition state on second order corrections is analysed in moredetails in paper IV.

The problem of taking into account the effect on the CEF splitting by thescreening from the conduction band remains an open question. In the nextsection we present an approach41 that can solve this problem.

3.3 Total energy calculations of CEF splittingThe ideal experiment of taking a free atom and plugging it into a lattice witha hole is in strong analogy to what is done in most of the implementationsof LDA-DFT and certainly in the LMTO method that is used throughout thisThesis. At each single site, focusing only on electrons (nuclei provide a back-ground, external potential since the Born-Oppenheir approximation is usuallyadopted), one divides the electrons of the atom occupying the site in core andvalence or band electrons. Core electrons are not allowed to hybridise withother states in the system and are considered to have a spherical symmetry.Those electrons are regarded to occupy atomic-like states. Band electrons areinstead let free to interact with the rest of the crystal and their hybridisationis self consistently evaluated. Any adjustment of the charge density of the

26

3.3. Total energy calculations of CEF splitting

core electrons to the crystal potential is disregarded. The total angular mo-mentum J is still considered to be a good quantum number for them and theirfull rotational symmetry is maintained. This is a very good approximation inmost cases and is especially good for filled shells because they will have atotal angular moment that is equal to zero and will not be influenced by thecrystalline electric field. For band electrons the interaction with the crystal po-tential is taken into account in a natural way since those electrons are describedby Bloch states and no assumption is made on the shape of their density.

The situation is somewhat more complex for f electrons. Experience hasshown that in some case they are well described by treating them as band elec-trons while in many cases they are better described if treated as core electrons.In the latter eventuality one has a problem with the CEF: Apart for the case inwhich the f shell is filled or half filled, the above described standard approachwill not properly account for the f -electron charge density interaction with theelectric field generated by the surrounding crystal lattice. The evaluation ofthis, usually neglected contribution, is the object of this section.

Under the hypothesis that the f states have a negligible hybridisation withany of the other electronic states of the system at study and that the CEF con-stitutes a small perturbation to the Russel-Saunders coupling scheme, i.e.

exchange coupling spin-orbit coupling (SOC) CEF,

a parameter-free scheme for calculating CEF splitting of the lowest J multiplethas been devised.38 Whenever the above is true one can start by consideringJ as a good quantum number and the CEF as a perturbation. The situation isthe one schematically depicted in Fig. 3.2. It is crucial to the method that wewill describe in the following that the CEF does not mix levels belonging todifferent J-multiplets.

3.3.1 Obtaining the CEF charge densityLet us, from now on, focus on the lowest-energy or ground-state (GS) J-multiplet. In the following, the 2J+1 CEF levels in which the GS J-multipletis split by the CEF will be referred to as the CEF levels. Let G ≤ 2J + 1 bethe number of non degenerate CEF levels. Since we restrict ourselves to theGS J-multiplet and since the CEF does not mix states belonging to differentJ’s, the eigenvectors |JGSMJGS

> with MJGS= −JGS ,−JGS + 1, ...., JGS

can be used as a basis set in which the CEF levels of interest can be expanded(since they span the entire subspace of the lowest J-multiplet)

ΨCEFi =

∑MJ

CMJ|JMJ > . (3.28)

Note that the suffix GS has been dropped for the sake of a lighter notation.

27


J'

J''

J

Figure 3.2: Schematic representation of the limit in which the method of section 3.3for CEF calculations is valid. The left hand side levels are the Russel-Saunders Jmultiplet when the CEF splitting is disregarded. The levels on the right hand side arethe CEF levels in which the former are split by the CEF.

28


We will in the following show how to write the charge density correspond-ing to the CEF level of Eq. (3.28) in the form

nσf (r) = Rf (r)

∑h

ασhDh(r), (3.29)

where Rf (r) is a radial function common to all the 2J + 1 levels belongingto the GS J-multiplet, Dh(r) are the spherical harmonic invariants defined inEq. (2.38) and ασ

h are expansion coefficients. The procedure involves tedioustensorial algebra in which the powerful Wigner-Eckart theorem plays the mainrole and is analogous to what we have shown in section 3.2. We start by writingthe density matrix

nσ(r, r′) = Rl(r)Rl(r′)∑m,m′

Ylm′(r′)χσY ∗lm(r)χσ†

< a†mσam′σ′ > . (3.30)

Our task is to evaluate the expectation value < a†mσam′σ′ > in the |JMJ >basis in which the CEF states are easily expressed. In order to do that, let usdefine the operator

T (mσ,m′σ′) = a†mσ b†m′σ′ (3.31)

where bmσ = (−1)l+m+ 12+σa−m−σ is the hole creation operator. It is easy to

show that

< a†mσam′σ′ >= (−1)l+m+ 12−σ′〈T (mσ,−m′ − σ′)〉. (3.32)

Since we want to obtain an expression for the density in terms of the sphericalharmonic invariants, instead of spherical harmonics, we want to rewrite thetensor operator, T , in theDh(r) basis, denoted in the following with a subscripth. This is done with the standard Clebsch-Gordan coefficient technique42

T (lhmh, shσh) =∑

mm′σσ′T (mσ,m′σ′)(lmlm′|lllhmh)(

12σ

12σ′|1

212shσh).

(3.33)Inverting Eq. (3.33) one obtains

< a†mσam′σ′ >=

= (−1)l−m+ 12−σ∑lh

√2lh + 1

(l l lh

m −m′ m−m′

)(3.34)

×∑sh

√2sh + 1

(12

12 sh

σ −σ′ σ − σ′

)〈T (lhm−m′, shσ − σ′)〉.

29


We now invoke the Wigner-Eckart theorem in order to step down to an LSbasis

〈LSJM |a†mσam′σ′ |LSJM ′〉 =

= (−1)M+M ′(2J + 1)

∑MS ,M ′

S ,ML,M ′L

(L S J

ML MS −M

)(3.35)

×〈LMLSMS |a†mσam′σ′ |LM ′LSM

′S〉(L S J

M ′L M ′

S −M ′

).

Now the problem is transformed via Eq. (3.34) to the evaluation of

〈LMLSMS |T (lhmh, shσh)|LM ′LSM

′S〉 =

= (−1)L−ML

(L lh L

ML mh −M ′L

)(LSMS ||T (lh, shσh)||LSM ′

S)

(3.36)

= (−1)L−ML

(L lh L

ML mh −M ′L

)(S sh S

MS σh −M ′S

)

×(LS||T (Lh, sh)||LS).

The reduced matrix element (LS||T (lh, sh)||LS) can be calculated observingthat, for less than half filling, Eq. (3.36) can be written for the particular case

〈LLSS|T (lh0, sh0)|LLSS〉 =∑mm′σσ′

〈LLSS|LLSS〉a†mσ bm′σ′(lmlm′|lllh0)( 12

σ 12

σ′| 12

12

sh 0 )

=l+1−N∑

m=l

(−1)l−m(lml −m|lllh0)( 12

12

12− 1

2| 12

12

sh 0 ) (3.37)

=√

(2lh + 1)(2sh + 1)

(12

12 sh

12 −1

2 0

)l+1−N∑

m=l

l(−1)l−m

(l l lh

m −m 0

)

and therefore

(LS||T (lh, sh)||LS) =

√(2lh + 1)(2sh + 1)

(S sh S

−S 0 S

)−1(L lh L

−L 0 L

)−1

(3.38)

×(

12

12 sh

12 −1

2 0

)l+1−N∑

m=l

(−1)l−m

(l l lh

m −m 0

).

30


For more than half filling all needed matrix elements are obtained just inter-changing the electron and the hole creation operator.

Combining Eqns. (3.32) to (3.38), we are finally able to associate to anygiven CEF level a one-electron like charge density in the desired form givenin Eq. (3.29) and therefore we are able to calculate its total energy, ECEF

i , bymeans of DFT. This is so since any CEF level in which the lowest J multipletis split will be the GS of a given symmetry.43, 44 Then, by constraining thef -charge density to be the one of the ith CEF levels as calculated above, wecan evaluate its total energy, ECEF

i . We can do this for all the CEF levelsbelonging to the GS multiplet and calculate in this way the CEF splitting ofthe lowest J-multiplet.

In passing let us note that, when taking total energy differences, we will beable to exploit a convenient cancellation of errors that will make our resultsmore reliable. In fact, the error generated by the use of the approximated LDAexchange-correlation potential will be very closely the same in all the CEFlevels, hence their differences will be closer to the exact values.

3.3.2 Total energy of a CEF level: symmetry constrained LDAcalculations

Constraining the symmetry of the f shell to a CEF charge density calculatedas in the previous section requires some care. The LDA calculated total energyof a free atom with its f -charge density constrained to the one correspondingto the CEF level Γi will differ from that of the level Γj . This is not a truephysical phenomenon, of course. In absence of a CEF the two configurationsmust have the same energy since the lowest J-multiplet has fully rotationalsymmetry and is 2J + 1 degenerate. The energy difference comes from thefact that LDA is an approximation to the true exchange-correlation functional.A way to recover the full degeneracy of the GS-CEF levels in the absence of aCEF is to correct the LDA total energy functional by removing the interactionof the non-spherical part of the f -charge density with itself. To do this, let usdivide the f -charge density, nf (r) as follows:

nf (r) = nf (r) + nnsf (r),

where nnsf (r) is the non-spherical part. The total density can be divided con-

sequently, n(r) = n(r) + nnsf (r). Then the Hartree energy should be

EH [n] =12

∫n(r)n(r′)drdr′

|r − r′|+∫n(r)nns

f (r′)drdr′

|r − r′|, (3.39)

31


where a third term containing nnsf (r)nns

f (r′) has been excluded. The exchange-correlation part of the total energy functional,

Exc[n] =∫n(r)εxc(n)dr =

∫ n(r) + nns

f (r)εxc[n+ nns

f ]dr, (3.40)

also contains O(nnsf

2(r)) interactions which have to be dropped. Since thenon spherical part of the f -charge density is relatively small one can expandthe Exc[n] as follows

Exc[n] =∫

n(r) + nnsf (r)

εxc[n] + nns

f (r)δεxc[n]δn

|n=n

dr. (3.41)

Eliminating contributions of second order in the non spherical f density, oneobtains

Exc[n] =∫n(r)εxc[n]dr +

∫nns

f (r)εxc[n] + n(r)

δεxc[n]δn

|n=n

dr

(3.42)or, alternatively

Exc[n] =∫n(r)εxc[n]dr +

∫nns

f (r)µxc[n]dr. (3.43)

Also, the wave equation

(−12∇2 + VN + VH + µxc)ψi = εiψi (3.44)

should be modified, with the first two terms remaining unchanged, whereas

VH =∫

n(r) + nnsf (r)

dr

|r − r′|(3.45)

for non f states, and

V fH =

∫n(r)dr

|r − r′|(3.46)

for f states. Finally the exchange correlation potential should be modified

µxc[n] = µxc[n] + nnsf (r)

δµxc[n]δn

(3.47)

for non f states, andµf

xc[n] = µxc[n] (3.48)

32


for f states. Then the kinetic energy, Ts, becomes

Ts[n] =∑

i

niεi −∫VN (r)n(r)dr

−∫VH(r)n(r)dr −

∫V f

H(r)nnsf (r)dr (3.49)

−∫

µxc[n] + nnsf (r)

δµxc[n]δn

n(r)dr −

∫µxc[n]nns

f (r)dr

The sum of Eqns. (3.39), (3.43) and (3.49) yields the expression we use forcalculating the total energy of CEF levels

E =∑

i

niεi − 12


|r − r′| −∫n(r)nns

f (r′)drdr′

|r − r′|

+∫εxc [n] − µxc [n]n(r)dr −

∫nns

f (r)δµxc [n(r)]δn(r)

n(r)dr. (3.50)

3.3.3 Generalisation to the magnetic caseIf the system one wants to study is magnetic the functional of the density de-rived in the previous section needs to be changed. To the CEF Hamiltonianone has to add a term corresponding to a magnetic potential generated by aneventual external field, Hext, and by the strong, internal exchange field, B.The latter is non local and depends, in general, on the magnetisation density ofall CEF levels. We chose to approximate this complex behaviour with a simplemean field approximation. The internal field will be the Weiss field generatedby the magnetic moment of the GS CEF level.

One can decompose this mean field magnetic contribution in orbital (L) andspin (S) parts as follows:

Emagn = L ·Hext + 2S ·Hext + 2S ·B. (3.51)

Let us now write the magnetic contribution, Emagn, in (3.51) on the samefooting as the crystal field part using the properties of angular momenta andthe Russel-Saunders scheme:

Emagn =∑

µ

Hext−µ

∑

ML,M ′L

⟨LMLSMS |12Lµ|LM ′

LSM′S

⟩

+∑

MS ,M ′S

⟨LMLSMS |Sµ|LM ′

LSM′S

⟩ (3.52)

+kBTc

µB

∑µ

e−µ

∑MS ,M ′

S

MS + 1MS

⟨LMLSMS |Sµ|LM ′

LSM′S

⟩

33


where the Weiss field has been expressed in the form:

B =kBTc(S + 1)

µBSS

with Tc indicating the Curie temperature, µB the Bohr magneton and kB theBoltzmann constant.The tensor eµ stands for

eµ =

x− iy, µ = −1√2z, µ = 0

x+ iy, µ = 1

(3.53)

Lµ, Sµ and Hextµ are defined analogously.The matrix elements in Eq. (3.52) can be evaluated using once again the

Wigner-Eckhart theorem⟨LMLSMS |Lµ|LM ′

LSM′S

⟩=

δMSM ′S

(L||Lµ||L

)(−1)L−ML

(L 1 L

−ML −µ M ′L

). (3.54)

The reduced matrix elements is evaluated observing that

〈LLSS|L0|LLSS〉 =(L||Lµ||L

)( L 1 L

−L 0 L

)

therefore (L||Lµ||L

)=

L(L 1 L

−L 0 L

) .The procedure is completely analogous for Sµ and the expressions for the spinangular momentum can be obtained from the ones for the orbital angular mo-mentum just interchanging L↔ S, ML ↔MS and M ′

L ↔M ′S .

So, substituting the last two expressions in the expression for Emagn, oneobtains:

Emagn =∑

µ

Hext−µ

1

2

∑ML,M ′

L

(−1)(L−ML)L

(L 1 L

−ML −µ M ′L

)(

L 1 L−L 0 L

) δMS ,MS′

+∑

MS ,M ′S

(−1)(S−MS) S(S 1 S−S 0 S

) ( S 1 S−MS −µ M ′

S

)δML,ML′

(3.55)

+kbTc

µb

∑µ,MS ,MS′

e−µMS + 1MS

(−1)(S−MS)S

(S 1 S

−MS −µ M ′S

)(

S 1 S−S 0 S

) δML,ML′ .

34


If |ΨCEFi > is one of the eigenstates that diagonalise the CEF Hamiltonian

with the above magnetic contribution, the charge and the magnetisation densitycorresponding to that particular CEF level will be

nCEFi = < ΨCEF

i ||ΨCEFi >

mCEFi = < ΨCEF

i |σ|ΨCEFi >,

where σ represents the vector of the Pauli matrices. The change in the expres-sion for the total energy is slightly more complicated than in the non magneticcase. The derivation follows the same line as the non magnetic case and wereport here only the final result.

E =∑σi

nσi ε

σi − 1

2


|r − r′| −∫n(r)nns

f (r′)drdr′

|r − r′|

+∑

σ

∫ [εxc

[n↑(r), n↓(r)

]− µσ

xc

[n↑(r), n↓(r)

]]nσ(r)dr (3.56)

−∫nnsσ

f (r)δµxc

[n↑(r), n↓(r)

]δnσ(r)

nσ(r)dr

.

Also the exchange correlation potential has to be modified for non f elec-trons,

µσxc[n

↑(r), n↓(r)] = µnsσ

xc [n↑(r), n↓(r)] (3.57)

+∑σ′nnsσ′ δ2

δnσδnσ′

(n(r)εxc[n↑(r), n↓(r)]

)∣∣∣n↑=n↑,n↓=n↓

where

µnsσ

xc =δ

δnσ(r)nσ(r)εxc

[n↑(r), n↓(r)

](3.58)

is the potential seen by f electrons.

3.3.4 ApplicationsWe have applied the total energy method here described to a number of systemswith some success. In particular in paper V we have investigated the stabil-ity of the method with respect to parameters from which the standard CEFmodel is extremely dependent as the MT radii and the boundary conditionsimposed for the radial part of the charge density of the f electrons. We foundthat our calculated CEF splitting in PrSb does not change significantly whenchanging any of these two parameters. The reason is, in our opinion, twofold.We are evaluating total energy differences between CEF levels. Each level is

35


characterised by the change of the angular dependence (or symmetry) of the felectron density. The boundary conditions for the radial part are the same inall CEF levels, therefore eventual changes cancel out when taking total energydifferences. Moreover, our method does not recur to a division in on-site andlattice contribution that will naturally lead to a coupling between the CEF sizeand the chosen geometry. In fact, the integrals for evaluating CEF parame-ters are evaluated only inside the MT spheres in the standard first-principlesCEF method. It is therefore very likely, in that case, that CEF parameters willstrongly depend on the MT radii.

Another factor we investigated in paper V is the nature of the screening fromvalence electrons. In particular we wanted to see whether the re-adjustment ofthe valence charge density to the particular CEF state in which the f chargedensity was constrained, were a local effect or not. If the CEF state on site Ri

can influence the valence charge density surrounding site Rj , CEF splittingswill not only depend on the symmetry of the lattice but also on the distributionof CEF charge densities on that lattice. In order to check this we performedsupercell calculations in which we compared the splitting obtained in PrSbwhen all rare earth ions in the supercell were constrained to be in the sameCEF state to the splitting obtained when the f charge density at one site ofthe super cell was constrained to the one corresponding to the CEF level ofwhich the energy was to be evaluated, while the f electron charge density atthe remaining sites was put in another CEF state. No significant difference wasobtained.

We also applied our total energy method to selected An systems in whichthe 5f ’s are believed to be localised. The fact that the 5f radial density ismore extended than the 4f one and the overall less atomic-like character ofthe 5f electrons compared to the 4f ’s puts An systems on the border of valid-ity of the CEF model. Therefore we regarded our calculations on An systemsas exploratory, the legitimacy of which had to be proven a posteriori. Theusefulness of being able to calculate CEF splittings from first principles isdemonstrated in paper VI. There, we were able to solve an apparent incon-sistency between experimental values for the CEF splitting of PuO2 obtainedwith different techniques. The only transition that is allowed to inelastic neu-tron scattering is the Γ1 → Γ4 and for that Kern and co-authors45 measuredan excitation energy of 123meV. The flatness of the magnetic susceptibility,χ, as a function of temperature46, 47 (see Fig. 3.3) tends to indicate that thenon-magnetic Γ1 state is the GS and a simple two-level analysis of χ finds theexcited state, Γ4, about 300meV higher in energy. An f2 configuration in acubic symmetry, if Russel-Saunders coupling is assumed valid, has the lowerJ = 4 multiplet split in four CEF levels, Γ1,Γ3,Γ4 and Γ5. We were able tocalculate the energy differences between all of these levels and, consideringthe presence of an antiferromagnetic exchange in PuO2, in analogy to what is

36


200 400 600 800

4,0x10-4

8,0x10-4

1,2x10-3

1,6x10-3

Γ14

Γ14

C E F (99)

C E F (123)

Figure 3.3: The magnetic susceptibility of PuO2. The measurements are the tempera-ture independent straight dotted line and the calculated bare susceptibility with a soleΓ1 → Γ4 excitation energy of 284 meV which fits the data at T = 0 is the dashedline labelled Γ14(284). The corresponding calculated bare susceptibility with a soleΓ1 → Γ4 excitation energy of 123 meV which fits the neutron scattering data is thedotted line labelled Γ14(123). Adding calculated additional crystal field transitions tothe 123 meV transition produces the improvement shown by the solid line labelledCEF(123) whereas replacing the measured Γ1 → Γ4 excitation energy by the calcu-lated 99 meV transition produces the solid line labelled CEF(99). The effect of usingthe antiferromagnetic molecular field deduced from that of UO2 to enhance the lattertwo bare susceptibilities results in the full curves labelled CEF+I.

37


observed in UO2, we were able to show that a CEF splitting of about 100meV(that is our calculated value) for the Γ1 → Γ4 transition could be consistentwith a flat magnetic susceptibility over a range of temperature of about 300K.

38

Chapter 4Valence stability of f -electronsystems

4.1 IntroductionThe lanthanide (RE) and actinide (An) series differ from the other series con-stituted by a row in the Periodic Table: Going along the row one adds oneelectron not always to the chemically active valence band (as is the case, forexample, in the transition metal 3d series) but to a shell of which the degreeof participation in the bonding is not known. The determination of the valenceof RE and An in compounds and even in elemental solids is then an inter-esting question to address. Even if the lanthanide contractionas well as theCurie-Weiss behaviour of the magnetic susceptibility constitutes an evidencefor the picture of a chemically inert 4f shell in RE-metals48 – Ce excluded –,intermediate valence (IV) phases can be induced by pressure (see paper I andreferences therein). The situation is more complex in An systems where, withincreasing atomic number, the light An elements show a volume dependencesuggesting that the 5f electrons are in the valence band –similar to the 3d, 4dand 5d transition metals– while the heavier An elements have volumes that fol-low a pattern similar to the lanthanide contraction, suggesting a localisation ofthe f ’s.48 Pu, being on the border between these two situations has a very richphase diagram (see for example Ref. 49 and references therein). The Wigner-Seitz radius, RWS , as a function of atomic number in lanthanide and actinideelements, is shown in Fig. 4.1 where also the RWS of the 5d transition metalsare reported for comparison a. Fig. 4.1 shows clear similarities between theheavy actinides and the lanthanides, and between the light actinides and thetransition metals.

aThe Wigner-Seitz radius, RWS , is defined as the radius of the sphere that has the samevolume as the volume per atom in a solid.

39

Chapter 4. Valence stability of f -electron systems

R(Å

)W

S

Lanthanides (4f )

Actinides (5f )

Transition metals

Figure 4.1: Wigner-Seitz radii for lanthanides (squares), actinides (circles) and 5dtransition metals (triangles). Lines are guides for the eye.

In this chapter we will present different theoretical methods that we haveused to evaluate the valence of RE and An systems. In section 4.2 we willdescribe the simplest possible approach to try to predict whether or not the felectrons are participating in the bonding, that is comparing LSDA calculatedvolumes to experimental volumes. A second, more refined method for evalu-ating the valence of an f -element, first proposed by Johansson in Ref. 50, 51is described in section 4.3. Finally, in section 4.4, we will present a new ap-proach, designed for intermediate valence (IV) systems, that we have appliedto Yb metal under pressure.

4.2 Comparing volumesIf the experimental volume of the compound of interest is known experimen-tally, then the simplest approach to the valence determination consists in com-paring the theoretical volumes obtained in a straightforward LSDA calcula-tion, when the f electrons are considered as valence or core, to the experi-mental one. Some care is needed in this procedure and an example is given inpaper VII where the character of the 4f electron of Ce in CePt2Sn2 was inves-tigated. A direct comparison to the experimental volume would suggest a de-localised scenario for the f -electron of Ce. Comparing, though, the differencebetween theoretical and experimental volumes in other Ce and RE systems weconcluded that the f electron of Ce in CePt2Sn2 is essentially localised. Thelimitations of such an approach are evident. One needs experimental volumes

40

4.3. The Born-Haber cycle

f n

fn+1

subl

imat

ion

Solid f n Solid f n+1

solidification

atomic promotion en.

freea

tom

solid

Figure 4.2: Born-Haber cycle for the evaluation of the energetically favourable con-figuration of f -electron ions in a solid compound.

and, most of all, standard deviationsbetween theory and experiment have tobe considered in a less well defined way.

Another limitation, that is of general character for band structure calcula-tions, is the legitimacy of opposing a fully itinerant to a fully localised scenariofor f shells. In nature electrons will most probably oscillate between these twobehaviours and one can just say that on the average a certain number, z, of thef electrons will occupy atomic-like states while the remaining part, v, willhave a delocalised character. In standard DFT-LDA there is no possibility oftreating part of a shell as localised and part as delocalised. This problem hasbeen addressed with success52–60 by the self interaction corrected local spinapproximation (SIC-LSDA).61

4.3 The Born-Haber cycleJohansson50, 51, 62 observed that the difference in energy in a solid with An orRE as constituents, when the number of the localised f electrons changes fromn to n + 1 can be evaluated using the Born-Haber cycle depicted in Fig. 4.2.The cycle involves the sublimation energy from the bulk to free atoms in thefn configuration and a solidification energy by atoms in the fn+1 configura-tion to a solid with ions in the same state. The other quantity involved is the

41


atomic promotion energy that is the link between the two different configu-ration one wants to compare. The sublimation and solidification energies inFig. 4.2 are calculated in two steps: LSDA is capable of giving accurate val-ues for cohesive energies when the f electrons are in the grand barycentreofthe f multiplet, that is the average over the levels constituting the multiplet.This generalised cohesive energy (approximated by the difference in energybetween an atom and the bulk both in a paramagnetic state) must be correctedin order to evaluate the energy gain when atoms in the lowest level of the fn

multiplet solidify, remaining in the very same f configuration. The fact thatthe configuration of the f multiplet does not change is one key factor for usingthe cycle of Fig. 4.2: Since we are going to take energy differences we do notneed to estimate f -intra-shell contributions. Those are simply eliminated in thedifference between the energy of free atoms and bulk. The coupling betweenthe f shell and the conduction band (mainly the d-band) is instead differentin an atom and in the crystal for the very reason that band states are differ-ent from atomic states. This difference can be precisely evaluated combiningatomic calculations and spectroscopic experimental data. This has been donefor the RE elements in Refs. 50, 51. There are no such precise estimations forthe An series, the reasons being that elemental An have an itinerant characterfor half of the series and the lack of experimental data. However, we note thatit is possible to approximately reproduce the values obtained for RE elementsin Refs. 50, 51 using a simple Stoner-Exchange model for the f -d coupling.63

Exchange integrals, Il,l′ , for the RE and for the An series have been calculatedby Brooks and co-authors.63, 64 It is then possible to calculate the f -d couplingfor actinides as well. In the same way it is possible to evaluate spin and orbitalpolarisation energies.

Within the framework of LDA(GGA) we are not able, unfortunately, to cal-culate the atomic promotion energy with the required accuracy. This parame-ter is therefore taken from experiments.65, 66 It is important here to stress thatthe very same parameter, for example the promotion energy from f2s2pd tofs2pd2 in a certain RE can be used to analyse the valence of all compounds inwhich that particular RE assumes those two configurations.

4.4 Adding correlation effectsThere are two features of the method described in the previous section that wehave tried to improve on in paper I. The first one is the possibility to treat IV

states in a computationally efficient though accurate way. The second one isto include the effect of the f -d hybridisation and of the Coulomb attractionbetween the hole left behind in the f shell and the electron promoted to the va-lence band.67, 68 This we have done by considering the Kimball-Falicov modeltheory of IV phenomena developed in Refs. 69–71 in combination with the

42

4.4. Adding correlation effects

approach described in the previous section. The difference in energy betweentwo valence configurations as calculated with the cycle of Fig. 4.2 is the zerothorder solution, ∆0, to the coupled equations

x =

EF +|∆(x)|∫EF

dEN (E)

∆ (x) =∆0 +Gx

(4.1)

where x is the valence change, ∆(x) is the renormalised energy differenceand N(E) is the density of states per energy, E. The parameter G measuresthe Coulomb attraction between the f hole and the promoted (fraction of an)electron. G can be simply evaluated as the derivative of the position of thecentre of mass of the valence band as a function of the f -shell filling. Thebasic idea underlying this model is simple. The first observation is to note thatthe energy ∆0 obtained from a Born-Haber cycle is not always sufficient fora complete promotion of an f electron to the conduction band. The effectiveamount of charge transferred from the f -shell to the conduction band is there-

fore calculated as the integralEF +|∆0|∫

EF

dEN (E). When this amount of charge

is promoted, a corresponding hole is left in the f -shell. Now there will be aCoulomb attraction between this hole and the promoted charge. The simplestway to model this attraction is to renormalise the energy difference ∆0 via theparameter G as in (4.1) (for further details see paper I).

A further refinement of the valence can be obtained by considering the ef-fects of the electron promotion on the f − band hybridisation parameter

veff

1 −G

∫dEN (E)√

(E − Y )2 + 4v2eff

= v, (4.2)

where Y = EF + |∆ (x)| is the renormalised Fermi energy and v is the barehybridisation parameter calculated from first principles as suggested in Ref. 72.while veff is the renormalised one. Then one has to solve Eqs. (4.1) and (4.2)self-consistently.

4.4.1 Application to Yb metal under pressureRecently the x-ray absorption spectrum of Yb under pressure has been mea-sured in Partial Fluorescence Yield (PFY-XAS).73 The experiments probedthe L-edge by detecting the Partial Fluorescence from the 3d-2p decay. Thismeans that the unoccupied 5d-density of states (DOS) was mapped with a re-duced lifetime broadening of the 3d core-hole with respect to traditional L-edge XAS.74 The experimental spectrum, taken at a pressure of 20GPa, is

43


0 10 20

Photon energy (eV)

0

0.2

0.4

0.6

0.8

1

Inte

nsity

(ar

b. u

nits

)

measureddivalentcalculated

Yb XAS at 20GPa

Figure 4.3: Measured (full line) and calculated x-ray absorption spectrum of elemen-tal bcc Yb at a pressure of 20 GPa. The dotted line shows the calculated spectrumof the IV state, whereas the broken line shows the spectrum obtained for a divalentconfiguration. The first peak is aligned and the zero of the photon energy scale isarbitrarily chosen.

44

4.4. Adding correlation effects

shown in Fig. 4.3. If a comparison to a 5d-projected DOS of the divalentstate (lower curve of Fig. 4.3) is made, a rather large disagreement with ex-perimental data is observed. Although the two peaks at lower energies arereproduced with some accuracy by this calculation, there is a substantial partof the experimental spectrum, at 10eV and above, that is not present at all in thecalculated DOS of the purely divalent configuration. A comparison of the DOSof a purely trivalent configuration with the PFY-XAS data in Fig. 4.3 gives anequally poor agreement (data not shown in the figure). The other calculatedspectrum shown in Fig. 4.3 (middle curve) is a superposition of contributionsto the spectrum both from divalent Yb atoms and from trivalent Yb atoms ascalculated from an intermediate valence state, obtained in the coherent poten-tial approximation (CPA). The concentration of divalent and trivalent atoms inthe CPA alloy is chosen accordingly to the calculated valence at 20GPa, whichis 2.57. This value for the valence was obtained with the model described insection 4.4. In the theoretical IV spectrum in Fig. 4.3 there is an applied shiftof 5.7 eV in order to account for the chemical shift of the core states betweenthe two atom types. The IV spectrum reproduces much better the features ofthe experimental spectrum both in terms of the three peak positions as wellas regarding their relative intensities. We regard this as an indication that ourcalculated valence is accurate. In paper I, we have also evaluated the equationof state of Yb obtaining a good agreement with experimental observations alsofor this properties, which is another indication that the devised model is sound.

45


46

Chapter 5Magnetocrystalline anisotropy inGd metal

5.1 IntroductionIn a ferromagnet, below the Curie temperature, the energy is found to be de-pendent on the orientation of the magnetisation. Three physical mechanismsare responsible for such a phenomenon that goes by the name of magnetocrys-talline anisotropy. One is of classical origin and relies in the multipole (mainlydipole) interaction between the moments localised at lattice points.27, 75, 76 Thesecond one is related to the interactions between the charge-density multipolesof a given shell and the CEF. This term is of electrostatic origin and is thereforeoften separated out and treated as a CEF effect for localised shells (for exam-ple, the 4f electrons in RE metals) where it can be the dominant contribution.It is, instead, smaller for band-like electrons and therefore often disregarded inband structure calculations. The last one is instead related to the orientationof the spin axis and is of pure relativistic character appearing only when thespin-orbit interaction is taken into account.77 The SO coupling provides themechanism that couples the spin moment to the crystal generating thereby adependence of the energy from the spin axis.

The magnetocrystalline anisotropy energy (MAE) is defined as the differ-ence in energy:

MAE ≡ E(hard axis) − E(easy axis).

Sometimes, since the hard and easy axes are not known in advance, with MAEone refers to the difference between two crystallographic axes. This pragmaticdefinition of the MAE is the one we used in papers VIII and IX, where, in ahcp crystal, we considered

MAE ≡ E(a axis) − E(c axis).

47

Chapter 5. Magnetocrystalline anisotropy in Gd metal

Further, one can define an anisotropy energy, or the energy dependence fromthe magnetisation orientation, EA(θ) as

EA(θ) ≡ E(θ) − E(ref. axis),

where ref. axisindicates the axis chosen as reference (typically the easy axisor a symmetry axis of the crystal) and θ is the angle measured from it. Theanisotropy energy can also be expanded as

EA(θ) = K1sin2θ +K2sin

4θ + (K3 +K4cosφ)sin6θ + ... (5.1)

where Ki are the anisotropy constants. Even though this is the expansion mostcommonly used, a more physically transparent one is

EA(θ) = A(cos2θ − 1) +B(cos4θ − 1) + ..... (5.2)

where the constants A,B,C... are increasingly small.

5.2 The force theoremThe fact that the SO interaction can be introduced as a perturbation to scalarrelativistic systems can be exploited in order to speed up the evaluation ofthe MAE. The way to do so is given by the force theorem for band structurecalculations.78 Consider an unperturbed system a with its total energy givenby

E = ENN +TS [n(r)]+∫VN (r)n(r)dr+

12


|r − r′|+Exc[n(r)].

(5.3)By switching on a perturbation, one introduces a change in the total energy, tothe first order in the charge density change, δn, equal to

δE = δTS [n(r)] +∫VN (r)δn(r)dr

+∫n(r)δn(r′)drdr′

|r − r′|+∫µxc[n(r)]δn(r)dr +O(δn2) (5.4)

≡ δTS [n(r)] +∫V (r)δn(r)dr +O(δn2)

aThe perturbation will be, for our purposes the SO interaction, but, the theorem is of moregeneral nature.

48

5.3. The anomaly of Gd

where the change in the nuclei has been disregarded and the identities

Exc ≡∫εxc[n(r)]n(r)dr

n(r)δεxc[n(r)]δn(r)

= µxc[n(r)] − εxc[n(r)]

V ≡ VN + VH + µxc

have been used. The kinetic energy can be written in the form

TS [n(r)] =∑

i

niεi −∫V (r)n(r)dr, (5.5)

therefore, its change is (also to the first order in the charge density change)

δTS [n(r)] = δ∑

i

niεi−∫δV (r)n(r)dr−

∫V (r)δn(r)dr+O(δn2). (5.6)

Thus, if the potential is kept frozen, a substitution of Eq. (5.6) in (5.4), yields

δE = δ∑

i

niεi, (5.7)

which is the force theorem we wanted to derive and is valid to order O(δn).The reason why we wanted to show here the derivation of Eq. (5.7) is that itis interesting to see that some changes in the single contributions to the totalenergy are not zero but they partially cancel each other to first order.

Since the change in total energy in a frozen potential is equal to just thechange in the eigenvalue sum, one can calculate this latter, less computation-ally demanding quantity, in order to obtain the former. A large number ofevaluations of MAE via the force theorem in various elements and compoundshave been carried out in the past 20 years (see, for example, Refs. 79–82 andreferences therein), showing that contributions of orderO(δn2) are most of thetime negligible and that the change in the eigenvalue sum is very close to thetotal energy change.

5.3 The anomaly of GdGd metal is in the middle of the RE series and its f -shell is half filled. Thismeans that, in a Russel-Saunders (RS) scheme, no orbital moment is to be ex-pected from the f -electron shell. As a consequence of that, one expects no CEFcontribution to the anisotropy and indeed the MAE of Gd (∼ 30µeV/atom) istwo orders of magnitude smaller than the MAE of the other RE metals (∼meV/atom). Given the sphericity of the 4f shell one may ask where does

49


the observed MAE of Gd originate from? In paper VIII we addressed thisquestion and we found that an interplay between the dipole interaction of thelarge, localised, 4f spin moments and the, SO driven, MAE of the conductionband (mainly 5d) can well explain the observed anisotropy energy, EA(θ).In particular an easy axis of magnetisation at 20 off the c axis is predictedin excellent agreement with experiments.83 Let us note that the fact that the5d band anisotropy plus the dipole anisotropy from the localised 4f momentreproduces the observed MAE is also an indirect demonstration of the locali-sation of the 4f ’s in Gd. When the 4f electrons are considered as part of theband we obtain a value for the MAE that is about twenty times larger than theobserved one since the SO is in this case affecting also the f ’s.

In Fig. 5.3 we report the calculated band anisotropy of Gd for different k-points in the kx-ky plane (kz=0). It is clear that the MAE is strongly influencedby the states closer to the Fermi surface. This is consistent with the observationmade by Franse and Gersdorf83 that 21 parameters are required in the expan-sion of the torque curve in sinusoidal functions in order to obtain a satisfactoryfit but that this number could be reduced to only four if one took into accountFermi surface effects with a simple model as proposed in the same paper. Itis also consistent with the very large number of k-points needed to convergeour calculations and with the fact that using the Gaussian smearing Brillouinzone integration scheme of Ref. 84 gives different results compared to usingthe quadratic interpolation of the band structure of Ref. 85.

We continued our investigations on the MAE of Gd in paper IX where westudied also the peculiar temperature dependence of the easy axis of magneti-sation. This we attributed mainly to the reduction of the effective magnetisa-tion with increasing temperature. We analysed this effect by means of bandstructure calculations in which we constrained the 4f moment to chosen re-duced values. A qualitative agreement between our calculated temperature de-pendence of the easy axis and the observed one was obtained. In paper IX alsothe dependence of the anisotropy from the c/a ratio was investigated, show-ing that the size of the MAE and the orientation of the easy axis can be tunedvia this parameter. As shown in Fig. 5.2, the ratio between the band electronMAE and the dipolar contribution to the magnetic anisotropy can be changedby changing c/a. While the dipole anisotropy favours the c axis for a c/a ratioless than the ideal (

√8/3) and the in-plane orientation for larger c/a ratios,

the band electron MAE has the more complex behaviour reported in Fig. 5.2.The temperature driven c/a ratio change is of about 0.3% in the temperaturerange going from zero to the Curie temperature of Gd,86 therefore we believethat its role in the temperature dependence of the easy axis is negligible.

Let us note in passing that just the sphericity of the 4f shell of Gd makesthis metal a perfect playgroundfor testing the accuracy of band structure cal-culations of the MAE. In fact, the CEF contribution that is not easily accounted

50

5.3. The anomaly of Gd

Figure 5.1: Calculated EbandA of Gd metal for different k-points in the kx-ky plane

(kz=0) (top left). The central figure reports contributions from all states with energyε: EF − 1.714eV ≤ ε ≤ EF while the top-right figure refers to contributions fromstates with energy −∞ ≤ ε ≤ EF − 1.714eV . EF indicates the Fermi energy. Thecolour scale (blue, green, brown, yellow) represents an increasing MAE going from≤ −5000µeV/atom (blue) to ≥ +5000µeV/atom (yellow). The transition fromnegative to positive contributions occurs between green and brown colouration.

for in standard electronic structure calculations for localised open shells, is ab-sent in Gd and only weakly correlated conduction electrons have to be properlytreated. The other prospective candidate for a similar investigation is thereforeCm, the actinide analogous of Gd b

Finally, let us comment on the anisotropy constant expansion for Gd. It is inmost cases sufficient to consider just the first two terms in the expansion (5.2)and this we have done in papers VIII and IX in order to obtain, from a limitednumber of spin axis orientations, the entire EA(θ) curve. However, we mustadd that it is possible to show87 that the magnetic anisotropy constants of ordern are proportional to the ratio between the SO splitting parameter, ξ, over theband width, W , roughly in the form ξn

W n−1 . Therefore, the expansion (5.2)converges rapidly for transition metals where W is considerably larger thanthe ξ but has a slower convergence for Gd where these two quantities are ofcomparable size. This observation is also consistent with the, observed andcalculated, non-monotonic behaviour of EA(θ).

bCm crystallises in the more complex dhcp structure and it is likely that the more extended5f shell is not in a pure S-state. Hence the analogy has to be taken with some care.

51


1.55 1.6 1.65c/a

-20

-10

0

10

20

E(9

0o ) -E

(0o )

(µeV

)

dipoleband

Figure 5.2: Calculated dipolar anisotropy (open squares) and EbandA (90) −

EbandA (0) (open circles) of Gd metal for different values of the c/a ratio. Lines

are guides for the eye.

52

Summary and Outlook

In chapter 3 we presented a DFT-LDA total energy method for the evaluationof the CEF splitting of the lowest Russel-Saunders J-multiplet. A direct eval-uation of the splitting has the advantage of taking into account in a naturalway the screening from the valence electrons that adjust to the f charge den-sity. With the standing increase in computational power a systematic study ofCEF splittings with a total energy approach has become possible. A compar-ison with the results obtained with a standard CEF model approach would bemost valuable. A question that could be answered by such a study is whetheror not the screening from the valence electrons can be viewed as a correctionto the standard CEF model, i.e. whether it is possible to model CEF effectsor one is doomed to electronic structure calculations. A model of the CEF,apart from giving us a better physical insight, could be used to derive, in asimple and physically transparent way, the CEF splittings or parameters of acompound by the knowledge of a similar one (and will define the meaning ofsimilar in the context). Our present impression is that this can be possible buta new parameter must be added to the CEF model in order to represent thedegree of mobility of the conduction electrons. Devising such a screened CEFmodelcould be an interesting project for the future. On a more modest levelthere are a number of systems where the labelling of CEF levels , or even thesize of the CEF splitting, is not easily determined experimentally and wheresuch informations could be valuable. As an example, the four phase transi-tions that UPd3 exhibits below 8K have been recently explained88 assuming acertain CEF splitting. A first-principles calculation of the CEF splitting at thequasi-cubic U site would be able to confirm the assumptions of Ref. 88.

The MAE of Gd has been investigated in details. The number of k-pointsneeded to converge our calculations has proven to be huge, nevertheless webelieve that in the near future it will be possible to calculate the anisotropy en-ergy of Gd including SO effects in a self consistent manner and without havingto resort to the force theorem. The interplay between dipolar interactions andconduction band anisotropy giving raise to the peculiar MAE of Gd could also

53


be present in Cm. A study of the MAE of this actinide metal would be aninteresting theoretical challenge. It would also be interesting to see whetheror not our predictions on the c/a ratio dependence of the MAE of Gd have anexperimental confirmation. Gd can be grown with c/a ratios different fromits natural one (which is 1.6 at low temperature) in a multilayer structure byusing ad hocchosen substrates. In order to describe this geometry, our dipolarcalculations should be slightly changed.

The proposed approach to IV system of chapter 4.4 has been applied to Ybbut is of completely general character. A number of systems could be attackedin the same way and we are looking forward to doing so. In addition, it wouldbe valuable to make the method totally first-principles based by being able tocalculate the zeroth order energy difference ∆0 of the model. This could bedone, possibly, by means of accurate configuration interaction calculations forthe atomic promotion energy and for the atomic polarisation energies.

54

Acknowledgements

This is going to be long. I am indebted for an enormous number of things toOlle ’det blir bra’ Eriksson. For an equally large number of things (totally dif-ferent ones, though) I have to thank Mike ’det blir nog aldrig bra’ Brooks. LarsNordstrom has spent a lot of time in teaching me many things and in pullingmy legs. Borje Johansson is, in every single thesis produced in this group,acknowledged for providing a pleasant working atmosphere. It is true and thisthesis will be no exception to the rule. I only hope that the last performances ofLeksand will not change his good mood. I have learned a lot from Misha Kat-snelson in the past two years. And from Igor Abrikosov during my first year asa student in Uppsala. I want to thank Gerry Lander for letting me spend a lotof time in Uppsala. Something that I owe also to Jean-Pierre Michel, RolandSchenkel, Maurice Leroy and Jean Rebizant. The administrative amount ofwork for a student in Karlsruhe has been less overwhelming thanks to the helpof Sabine Yael and Frank Wastin who never said no when I needed help. Herein Uppsala there is no administrative work for students because there is Elis-abeth Bill. To John Wills I want to say thank you for the three visits to LosAlamos National Laboratory, for spending time with my crystal field problemsand for letting me (mis)use your code. Manuel Richter has always found anencouraging or a stimulating word about my work and is also acknowledgedfor inviting me for two weeks in Dresden. I am grateful to Ingrid Brooks formany meals, a lot of nice chats and for coping with me being around her homeduring some weekends. Till Burkert deserves all thanking for the work he isdoing in keeping John’s FP-LMTO code working at its best on our machinesand for always providing a slightly better script for just about everything. Iwant to give my thanks to Kay Dewhurst for letting me use his EOS code andJorge Osorio-Guillen for advices about EOS fitting. Rajeev Ahuja and SergeiSimak are acknowledged for being able of co-authoring more than one paperwith me without being forced. Weine Olovsson, Lars Bergvist, Per Anders-son, Urban Lundin, Anders Ekstrom, Bjorn Skubic, Thomas Dziekan, CarlosOrtiz and Erik Holstrom are acknowledged for keeping our computers going.

55


I also want to thank Lars Fast, Anders Niklasson Peter James and Anna Delinfor introducing me to the group and to Swedish. Elisabeth Sjostedt and JonasFransson have been insisting in the tremendous task of talking with me on adaily basis and they well deserve my gratitude for that. I want to thank Ve-limir Meded for never discussing anything serious with me. Adrian Taga forhelping me with a Linux installation on my lap-top and Leonid Pourovskii forsome late-night chess games. Erik Holmstrom for helping me in getting anapartment and in getting rid of it and for helping me with the LaTeX packagesfor this thesis. Alexei Grechnev for always being able to turn a boring con-versation into something funny. I am grateful to Pavel Korzhavi for letting meuse his office in Stockholm from time to time. Maurizio Mattesini and OlivierBengone are acknowledged for keeping my Italian from disappearing from mymind while I want to thank Biplab Sanyal for his inspiring storytelling aboutcricket and how Pakistan is the best team in the world. I am grateful to CristinaSchippa for many vital things and for the fact that she helped me in translat-ing the introduction to this thesis in Swedish in order to fulfill a brilliant newrule of this department. All the other people in Fysik4 are acknowledged forsomething, for making me laugh, for teaching me something, for an interestingdiscussion or for a silly one. I have surely forgotten someone in this late nightbefore the deadline falls. Please forgive me for not putting all your names inthis final note.

Uppsala, April 2004

Massimiliano Colarieti Tosti

56

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61

Acta Universitatis UpsaliensisComprehensive Summaries of Uppsala Dissertations

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ISSN 1104-232XISBN 91-554-5963-3

A doctoral dissertation from the Faculty of Science and Technology, UppsalaUniversity, is usually a summary of a number of papers. A few copies of thecomplete dissertation are kept at major Swedish research libraries, while thesummary alone is distributed internationally through the series ComprehensiveSummaries of Uppsala Dissertations from the Faculty of Science and Technology.(Prior to October, 1993, the series was published under the title “ComprehensiveSummaries of Uppsala Dissertations from the Faculty of Science”.)

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