Theoretical Investigations of Selected Heavy Elements and Metal-hydrogen Systems by Means of Electronic Structure Calculations

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Comprehensive Summaries of Uppsala Dissertations

from the Faculty of Science and Technology 653

_____________________________

Theoretical Investigations of Selected

Heavy Elements and Metal-Hydrogen

Systems by Means of Electronic

Structure Calculations

BY

PER ANDERSSON

ACTA UNIVERSITATIS UPSALIENSIS

UPPSALA 2001

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University in 2000

Abstract

Andersson, P. 2001. Theoretical investigations of selected heavy elements and metal-hydrogen systems by means of electronic structure calculations. Acta Universitatis Upsaliensis. Comprehensive Summaries of Uppsala Dissertations from the Faculty of

Science and Technology 653. 80 pp. Uppsala. ISBN 91-554-5103-9.

Using ab initio electronic structure calculations based on density functional theory the crystal, electronic and magnetic structures of selected materials have been inves-tigated. The materials which are the subjects of these investigations can be divided into two groups. Parts of the investigations have concerned actinides and rare earths, heavy elements with an f -shell electronic configuration. Here the effects of delocal-ization on EuCo2P2 have been studied as well as the effect of including relativistic interactions when calculating the properties of thorium. For EuCo₂P₂ it was found that at a low pressure the valence state of Eu changes from divalent to trivalent with associated effects on the crystal structure and magnetic state.

The other group of materials investigated are the hydrogen and metal-lithium systems. Both of these have an important technological application in the form of batteries. Here the emphasis of the investigations has been the fundamental understanding of the mechanism of hydrogenation, and a novel theory explaining the driving force behind hydrogenation is suggested. Vanadium hydride, VH_x, has been examined in detail and the reason for the anomalous non-isotropic expansion is ex-plained. A scheme to make vanadium magnetic is also proposed.

Finally a method based on electron-hole coupled Green’s functions has been used to facilitate the comparison between calculated electronic structures and X-ray ab-sorption spectra. In connection to this a novel theory of charge transfer in the X-ray absorption process applied to transition metal oxides and lithium intercalated tran-sition metal oxides is presented.

Per Andersson, Department of Physics, Uppsala University, Box 530, SE-751 21 Uppsala, Sweden

c

Per Andersson 2001

ISSN 1104-232X ISBN 91-554-5103-9

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List of Publications

I. Hydrogen-induced changes of the electronic states in ultrathin single-crystal vanadium layers

L.-C. Duda, P. Isberg, P. H. Andersson, P. Skytt, B. Hj¨orvarsson, J.-H. Guo, C. S˚athe and J. Nordgren,

Phys. Rev. B 55, 12914 (1997).

II. Theoretical study of structural and electronic properties of VHx

P.H. Andersson, L. Fast, L. Nordstr¨om, B. Johansson and O. Eriksson, Phys. Rev. B 58, 5230 (1998).

III. Eﬀects of varying compressive biaxial strain on the hydrogen uptake of thin vanadium (001) layers

G. Andersson P.H. Andersson and B. Hj¨orvarsson, J. Phys. Cond. Matter 11, 6669 (1999).

IV. Eﬀect of hydrogenation on the magnetic state in cubic Pd₃Mn

P.H. Andersson, L. Nordstr¨om and O. Eriksson , Phys. Rev. B 60, 6765 (1999).

V. Spin-orbit coupling in the actinide elements: A critical eval-uation of theoretical equilibrium volumes

L. Nordstr¨om, J.M. Wills, P.H. Andersson, P. S¨oderlind and O. Eriks-son ,

Phys. Rev. B 63, 35103 (2001).

VI. The eﬀect of hydrogenation on the crystal structure and mag-netic state in Pd₃Mn

P. H. Andersson, O.Eriksson and L. Nordstr¨om, J. Magn. Magn. Mater. (accepted).

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VII. Geometric and electronic structure of PdMn bimetallic sys-tems on Pd(100)

A. Sandell, P. H. Andersson, E. Holmstr¨om, A.J. Jaworowski and L. Nordstr¨om,

Submitted to Phys. Rev. B

VIII. Theoretical investigation of a pressure induced phase transi-tion in EuCo₂P₂

P.H. Andersson, L. Nordstr¨om, P. Mohn and O. Eriksson

In manuscript.

IX. A theoretical model for the H-H interaction in metals

A. Grechnev, P.H. Andersson, R. Ahuja, O. Eriksson, M. Vennstr¨om and Y. Andersson

Submitted to Phys. Rev. Lett.

X. Electronic structure, structural phase stability and hydrogen-hydrogen interaction in Ti₃SnHx: theory

A. Grechnev, P.H. Andersson, R. Ahuja, O. Eriksson

In manuscript.

XI. Charge transfer eﬀect in x-ray absorption spectroscopy of transition metal oxides

P.H. Andersson, M. Klintenberg, A. Henningsson, H. Siegbahn and O. Eriksson

Submitted to Phys. Rev. Lett.

XII. Electronic structure and x-ray absorption spectroscopy of TiO₂ and LiTi₂O₄

P.H. Andersson, M. Klintenberg, A. Henningsson, H. Siegbahn and O. Eriksson

In manuscript.

XIII. Possible magnetic states of vanadium ﬁlms: theory

P.H. Andersson and L. Nordstr¨om

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CHAPTER

1

Introduction

Since the earliest days mankind has always tried to improve its understanding of the world which surrounds it, first perhaps only to survive but later, when the technological improvements like fire and stone working made the daily chores less of a burden, out of curiosity and a wish to make a better tool or a more beautiful ornament. There are two ways of improving a tool. Either you change the tool in a more or less random way and perform an experiment. If the new tool is better than the old one you keep it, otherwise you throw it way. You will eventually get a better tool but you will end up with a big pile of bad tools. Instead you could sit down, have a close look at your old tool and try to figure out how it actually works and from experience or inspiration you can figure out how you can improve the tool or maybe even invent a new use for your old tool. Maybe you have been using a hammer for a drill all the time. After a while you realize that you could try out your new tiger hunting spear on a deer the first time instead of on a dangerous tiger. You make an approximation.

Scientists have to make approximations all the time. Often they are made out of convenience but sometimes out of necessity. Nature in its full glory is very complicated and a lot of processes are intermixed. Some are very important and others not. To be able to comprehend a process or a result of an experiment we disregard some parts. The open question is which items we can disregard and which must be retained. You can make Swedish pea soup without thyme and even without meat but without the peas it simply is not pea soup any more. In this thesis we will among other things discuss some of the approximations made in the ﬁeld of condensed matter theory.

One of the earliest quests of the evolving science and technology was to make things move in a general sense. Magnetic materials have been used ﬁrst in the form of compasses to help us know where to move ourselfs, but later to move things in the form of electrical engines. Over time we have gathered a lot of knowledge of the eﬀects of magnetism and we can to some extent predict the behavior of magnetic materials, but we still have no good answer to the question ”what is magnetism at a fundamental level?”. The magnetic properties of a material can be connected to the electronic properties, so by

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examination of the electronic structure of a material, either by experiment or calculation, we can gain knowledge of the magnetic structure as well.

But if the electronic structure defines the magnetic structure, maybe we can change the magnetic properties by changing the electronic structure? This can be done in several different ways. Either the shape can be changed at both the microscopic and macroscopic level, or the thermodynamic environment can be changed. A third way is to change the chemical composition of the material. By the addition of hydrogen to a material both the structure and the chemical composition can be changed. Part of the work presented in this thesis concerns the effects of hydrogenation and intercalation of lithium on the magnetic and crystal structures of selected materials.

If hydrogen and its interaction with other elements is relatively easy to understand, the behavior of the heaviest elements is more complex. In Papers V and VIII we discuss the behavior of some selected materials where heavy elements are included.

During ﬁve years of graduate studies there are of course a lot of things that have to be done which are very important but can not be published in the same way as the actual research, but still need to be documented. This can include development of numerical methods or algorithms that will speed up calculations or make them possible at all. In the appendix we will present some unpublished material that hopefully will be useful for others. The material is of a more technical nature and maybe of less interest to the general audience. Feel free to skip it.

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CHAPTER

2

The many-body problem and how to avoid solving it

One of the fundamental ideas of modern physics is that the world around us is built from small building blocks, quanta. Luckily for us, as the eﬀects of this discretisation can be very counter intuitive, the scale of the granularity is very small. At the microscopic scale things are, at least conceptually, very simple. The only things that exist are particles and the interactions between particles. These interactions can be very complex and non-linear as in quantum chromo dynamics, the theory describing quarks and gluons, or deceivingly simple as the coulomb interaction between charged particles like electrons.

The solution of the equations of motion for a system consisting of two par-ticles is trivial, but as soon as we have three or more parpar-ticles the equations describing the system can no longer be solved analytically. This is analogous to the problem of solving the equations of motion for a gravitationally bound system with more than two bodies. As we will show in this chapter the com-putational requirements for an attack by brute force are beyond our reach for all but the smallest systems, so some ingenuity is needed.

The materials investigated in the papers presented in this thesis share a few common properties. They are all built from two building blocks that are of interest to us, negative electrons and positive nuclei. The nuclei are positioned in a repeated pattern which is gowerned by a set of symmetry operations. The discrete translational symmetry of the lattice inﬂuences the behavior of the electrons in a dramatic way as will be discussed below.

2.1 The Schr¨

odinger equation

The basic equation used to describe quantum systems is the time dependent Schr¨odinger equation [1] proposed by Schr¨odinger in 1926,

HΨ = i¯h∂

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where the Hamiltonian operator, for a system consisting of negative electrons and positive nuclei, can be expressed as

where Ze,M and R are charge, mass and position of a nucleus and e, m and

r are charge, mass and position of an electron. From here on in this thesis

Rydberg units will be used, i.e. ¯h = 2me = e2/2 = 1. The indices i and j

number the electrons and k and l the nuclei. The Hamiltonian can be divided in two parts, one describing the kinetics of the electrons and nuclei and the second part describing the Coulomb interactions between particles: electron-electron, electron-nucleus and nucleus-nucleus.

In the same way as Newton’s equations of motion can not be proved to be true, neither can the Schrödinger equation. Instead one can argue that the solutions to the Schrödinger equation should have certain properties, e.g. they must be linear and homogeneous to describe quantum particles and therefore the equation must have these properties. Equation 2.1 is often referred to as a wave equation as the solutions to it in the case of free electrons are plane waves, but it is rather a kind of diffusion equation as it is only first order in the time derivative. A result of this is that the Schrödinger equation does not behave correctly under covariant transformations, i.e. it is not relativistic. For that we either need a equation that is second order both in the space and time derivative, the Klein-Gordon equation [2, 3], or an equation that is first order in both, the Dirac equation [4]. The consequence of this broken symmetry between space and time in the Schrödinger equation is that it can not be used to describe relativistic effects except as corrections in the form of additional terms in the Hamiltonian, which can be derived from relativistic equations.

There exists a lot of diﬀerent methods [5] for solving Eq. 2.1, giving us information on the time evolution of e.g. molecules and clusters, like grid based methods. If only the ground state properties of a system are of interest the (in some sense) more simple time independent Schr¨odinger equation can be used,

HΨ = EΨ (2.3)

The Hamiltonian is the same as in Eq. 2.2 and E is the energy of the system. Further simplifications can be done if the motion of the ions can be disregarded, the Born-Oppenheimer approximation. This can be justified by looking at the time scales involved in the electronic and the ionic interactions and the veloci-ties of the electrons and ion [6]. The average time between two scattering events involving electrons is of the order of 10−14 seconds, four orders of magnitude smaller than for ionic interactions. In the same way, the average velocity of electrons is several orders of magnitude higher than for ions, 106m/s compared to 103m/s. This difference in time and speed scales is caused by the difference in mass between the nucleons and the electrons, the proton being almost 2000

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times heavier than the electron. This approximation is not valid at higher tem-peratures where the ionic motion is so strong that the crystal melts, except for in the adiabatic limit.

Under this assumption Eq. 2.2 simpliﬁes to a Hamiltonian describing only the behavior of the electrons,

H_e= i ∇2 r_i+ i_=j 1 | ri− rj | + Vext (2.4)

where Vext includes all electron-ion interaction and any external potential

e.g. the potential arising from an external magnetic ﬁeld.

As the solution to the Schr¨odinger equation must describe the whole quan-tum system with all interactions included we can not use the single particle wave function but the many-body wave function has to be used,

Ψ = Ψ(r₁s₁, r₂s₂, . . . , rNsN) (2.5)

where si is the spin coordinate for electron i and N is the total number of

electrons in the system. Solving Eq. 2.3 using the Hamiltonian from Eq. 2.4 and the many-body wave function turns out to be a Herculian task for anything but the simplest system, so some modiﬁcations are needed.

2.2 The Hartree and Hartree-Fock equations

From variational principles it can be shown that the solution to Eq. 2.3 is a wave function that, if we are looking for ground state properties, minimizes

HΨ =(Ψ, HΨ)_{(Ψ, Ψ)} (2.6)

where H is deﬁned in Eq. 2.4 and (Ψ, Ψ) is a short hand notation for inte-gration over all space coordinates and summation over all spins. Using the many body wave function (Eq. 2.5) it is still unsolvable. The perhaps most natural simpliﬁcation is to use a less complex wave function, e.g. a product of

N orthonormal one-particle wave functions

Ψ(r₁s₁, r₂s₂, . . . , rNsN) = Ψ1(r1s1)Ψ2(r2s2) . . . ΨN(rNsN). (2.7)

Minimizing Eq. 2.6 using Eq. 2.7 yields as a result the so called Hartree equations [7]  −∇2_{+ V}_ext_{(r) +} j_=i dr | Ψj(r)|2 1 | r − r _|   Ψi(r) = εiΨi(r). (2.8)

The Hartree equations are a set of N coupled non linear equations. The sum in the electron-electron term couples the equations for all electrons and makes computations somewhat diﬃcult. A possible simpliﬁcation would be to let the 13

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sum run over all j, including j = i. In the independent electron picture the charge density at site r can then be identiﬁed as

ρ(r) =−e

j

| Ψj(r)|2 (2.9)

and can be calculated once and for all and be used for calculating all Ψi. This

simpliﬁcation gives a equation somewhat similar, but in a fundamental way diﬀerent, to Eq. 2.8. The new equation is often also referred to as a Hartree equation, −∇2_{+ V} ext(r) + drρ(r) 1 | r − r_| Ψi(r) = εiΨi(r). (2.10)

As the charge density is summed over all electrons it also includes the contribu-tion from electron i when we are solving Eq. 2.10. This introduces a ﬁctitious self-interaction which can be substantial, depending on the system of interest. The prime example is the hydrogen atom where the electron-electron interac-tion would be zero without the self-interacinterac-tion. We will later see that this error will be cancelled by further corrections introduced. There have been several attempts to remove the self-interaction in a more systematic way. These meth-ods are referred to by the common name self-interaction corrections (SIC), see e.g. Ref. [8, 9]. The introduction of SIC into Hartree theory is rather straight forward but it is in theories presented later in this chapter more complicated due to the problem of double counting. We must only remove the self interac-tion once.

By inspection we can see that Eq. 2.10 describes the behavior of a non-interacting electron moving in the average potential created by the ions and the electrons. This description is oversimplified in several aspects. First, the exact configuration of the N− 1 other electrons is not taken into account, only the average configuration via the charge density. Second, the response by the other electrons to the motion of the i:th electron is not handled in a correct way. Neither the Pauli principle nor the effect called screening are accounted for but there are remedies for both of these shortcomings. The perhaps most elegant but often more cumbersome approach is the so-called many-body technique or Green’s functions based methods [10, 11]. The full description of these methods is beyond the scope of this thesis but some parts will be discussed later.

If the wave function in Eq. 2.7 is replaced by a wave function that in itself is antisymmetric, i.e. changes sign when electron i and electron j are exchanged, the Pauli principle will be obeyed. One simple choice of such a wave function is the Slater determinant produced by the linear combination of one-electron wave functions. If this new wave function is put into Eq. 2.6 the result after

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the variational step is (−∇2+ Vext(r))Ψi(r) + j dr| Ψj(r)|2 1 | r − r_|Ψi(r)− (2.11) j dr 1 | r − r _|Ψ∗j(r)Ψi(r)δsisjΨi(r) = HHartreeΨi(r)− j dr 1 | r − r_|Ψ∗j(r)Ψi(r)δs_is_jΨi(r) = εiΨi(r).

Eq. 2.11 is known as the Hartree-Fock equation [12] which is quite similar to the Hartree equation, Eq. 2.8, apart from an additional term, the exchange term. This term is zero when electron i and j are of diﬀerent spin so it takes into account the Pauli principle. What we also can see is that if the summation over

j includes i there will be a complete cancellation of the ﬁctitious self interaction

discussed above. Any further corrections to the Hamiltonian are collectively called correlation. This will be further discussed below.

The ground state formed by the electrons is gowerned by two principles, one giving the ground state as the conﬁguration yielding the lowest energy out of the allowed conﬁgurations, and the second, the Pauli principle, stating that only one fermion, the class of particles to which the electron belongs, can occupy any given point in phase-space. In a momentum representation this means that levels with higher and higher momentum will be occupied, and the momentum of the highest occupied level is called the Fermi momentum. The surface spanned by all Fermi momentum vectors kf is called the Fermi surface.

This surface can have a complex form as we will discuss in Chaps. 3 and 5. The energy of the highest occupied level is consequently called the Fermi energy or Fermi level.

2.3 Screening and Thomas-Fermi theory, a

some-what diﬀerent approach

As stated above, Hartree-Fock theory does not take into account the response of the electrons to the behavior of a probe electron. As the Coulombic interaction is of inﬁnite range, VCoulomb ∝ r−1, the derivative with respect to the wave

vector k of the solution to Eq. 2.11 for the free electron case will diverge for the highest occupied states, at k = kf, where kf is the Fermi momentum. This

is obviously unphysical.

If a positive charge is introduced into a sea of free electrons it will attract a cloud of electrons through Coulombic attraction, creating a volume around the positive charge with elevated electron density. It can be shown that the excess charge density sums up to exactly the same charge (but with opposite sign) as we introduced into the system. In the same way would an volume of lowered charge density be formed around a negative charge, creating a hole. This reorganization of charge will compensate for the Coulombic interaction arising 15

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from the introduced charge and screen it, and thereby remove the singular behavior of the free electrons at the Fermi level.

As there is no diﬀerence between the probe charge and any electron in the system when it comes to the electrostatic interaction, the electrons in the electron sea will behave in the same manner. If we allow the electrons to interact with each other there will be a hole surrounding each electron where the charge density is lower than the average charge density, the so called exchange-correlation hole.

The charge density in the presence of a probe charge is a function of the total potential, which is the sum of the external potential, i.e. the potential which has its origin in the probe charge, and the induced potential. If the external potential, and therefore the total potential, is only slowly varying as a function of the position r it can be shown [13] that the total charge density is a function of the energy of the highest occupied state, the Fermi energy, and the total potential,

ρ(r) = 1

3(Ef− V (r))

3

2. (2.12) Equation 2.12 is exact for a constant potential. The approximation of a slowly varying potential is called the Thomas-Fermi approximation and Eq. 2.12 is the basis for the Thomas-Fermi theory. For the general case V is non-local, i.e. the charge density ρ at position r is determined by the value of V for all r. Postponing this discussion it can be shown that the charge density given by

ρ(r) =

E_f

i

Ψ∗_i(r)Ψi(r) (2.13)

where Ψi(r) is the one-electron wave function, is a functional of V . As the

kinetic energy per particle for a free-electron gas is 3₅Ef and V = 0, using

Eq. 2.12 we get, T = 3 5(3π 2₎2 3 drρ53(r). (2.14) In the same way we can write the potential for the electron-electron interaction as

Vee=

_ρ(r₎

| r − r_|dr (2.15)

and the interaction energy

Eint=

drρ(r)[Vext(r) + Vee(r)] (2.16)

where Vext is, in the case of a crystal, the potential from the ions and any

external ﬁeld. It is therefore possible to write the total energy as a function of the charge density, which in turn can be calculated from the one-electron wave functions.

The Thomas-Fermi theory is based on a very crude approximation valid only in the limit of a slowly varying potential, although it should be noted that the theory is exact in the limit of inﬁnite charge density. There have been

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some attempts to improve the potentials used [14, 15] and the description of the kinetic energy [16].

As was seen earlier in Hartree-Fock theory the concept of exchange is very important but it is lacking in Thomas-Fermi theory. To remove this weakness Dirac [14] among others, introduced the Hartree-Fock exchange energy

Eexch=− 1 4 ρ(r, r)2 | r − r_|drdr, (2.17)

where ρ(r, r) is the Dirac density matrix. If the charge density is substituted with the free-electron charge density locally, the ﬁnal expressions for the ex-change energy is Eexch=− 3 4( 3 π) 1 3 [ρ(r)]43dr. (2.18) The one-electron potential is then

Vexch=−(

3

π) 1

3[ρ(r)]13. (2.19) This expression was later generalized by Hohenberg and Kohn who also pro-vided a solid foundation for the theory. This will be discussed below.

2.4 Density Functional Theory

As we could see from the Thomas-Fermi theory the charge density rather than the exact conﬁguration of electrons can be used in electronic structure calcu-lations, which is an enormous simpliﬁcation. The use of a functional based on the charge density was put in a more stringent form by Slater in 1951 [17]. Later, in 1964, Hohenberg and Kohn [18] proved that the total ground state energy is a unique functional of the charge density.

E[ρ] = Tn_−i[ρ] + drVext(r)ρ(r) + drdrρ(r)ρ(r) | r − r _| + Exc[ρ] (2.20)

The ﬁrst term, Tn−i, is the kinetic energy of non-interacting electrons but it

should be noted that it is a functional of the charge density for interacting elec-trons. The second term is the energy contribution from the external potential and the third is the Coulomb energy. Compared to e.g. Eq. 2.11 there are a few things missing. These, the exchange and correlation, are all collected in the last term, Exc where there are some contributions to the kinetic energy as

well. From the variational principle we know that the extremum of the total energy is given by the correct wave functions. Hohenberg and Kohn [18] and Kohn and Sham [19] proved that this is also true for the correct charge density. As we are only interested in physical, ground state properties we can disregard the solution given by the maximum of the energy functional. By variational techniques the so called Kohn-Sham equation [19] is obtained.

(−∇2+ Vef f(r))Ψn(r) = εnΨn(r) (2.21)

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where

Vef f(r) = Vext(r) +

dr ρ(r)

| r − r_|+ vxc(r). (2.22)

The last term is the exchange correlation potential deﬁned as

vxc(r) =

δExc[ρ]

δρ(r) . (2.23)

The index n in Eq. 2.21 can be traced to the fact that we have gone from an

N -electron equation to N independent one-electron equations and n is often

referred to as the orbital index. Eq. 2.21 can be solved by various techniques, which will be brieﬂy discussed in a later chapter, and the charge density can be calculated using Eq. 2.13. We can then extract the total energy from the obtained solutions as E = N n=1 εn− drdrρ(r)ρ(r) | r − r _| − drvxc(r)ρ(r) + Exc[ρ] (2.24)

where the sum is over all occupied states.

This is an exact expression for the total energy but a few difficulties still remain. First, we might be interested in extracting other properties of a ma-terial than the total energy, like transport properties or the interaction with electromagnetic fields. This is far from trivial as these are collective effects and the Kohn-Sham equation is a one-electron equation describing indepen-dent electrons moving in an average potential. There is also the problem of finding the exact expression for Excwhich is not known. We will discuss these

problems and some other topics concerning the failures and successes of density functional theory below.

2.5 Choosing the exchange-correlation functional

As was discussed above we saw that by using density functional theory we have gone from calculating a numerically diﬃcult non-local term in the Hartree-Fock equations, describing the exchange contribution to the Hamiltonian, to the problem of ﬁnding vxc(r), the exchange-correlation potential. As the potential

describes the contribution from the many-body interaction between all electrons to the total potential it can be argued that the exchange correlation should be non-local, i.e vxc(r, r), which would take us back to the original diﬃculty.

Although the exact potential or energy functional are not known there are a few things we can say about them. The potential has its physical origin in the fact that the electrons will correlate their motion, both to adhere to the Pauli principle and to screen the electric ﬁeld surrounding each electron. The exchange-correlation energy can be seen as the energy due to this interaction.

Exc[ρ] =

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V_xc (r, r) is an undeﬁned potential and ρxc(r, r) is the exchange-correlation

hole. Each electron is surrounded by a volume which is depleted of charge. From this we know that

ρxc(r, r) < 0 (2.26)

because absence of negative charge can be viewed as a surplus of positive charge. It can also be shown [20] that

drρxc(r, r) =−1 (2.27)

from the conservation of the number of particles. Finally it is also true that the exchange-correlation energy only depends on the spherical average of the exchange-correlation hole [21].

Armed with this knowledge we can make an educated guess how a simple but eﬀective exchange-correlation functional might look. From a numerical point of view it would be preferable if the potential was local. One possible functional is the one used in the local density approximation

E_xcLDA=

drρ(r)xc(ρ) (2.28)

where xc is the exchange-correlation energy per particle of a free electron

gas. There are a number of diﬀerent ways to calculate xc, see e.g. [8]. In

these parametrisations results from Monte-Carlo simulations are often used. Depending on its physical origin, LDA should only give good results for slowly varying potentials and charge densities. In a crystal however, the potential and therefore the charge density are varying rapidly close to the nucleus but for several reasons rather good results can be obtained using LDA. First, LDA obeys the sum rule, Eq. 2.27, forced on any exchange-correlation functional by the particle conservation. Second, a lot of physical properties like binding and transport are only inﬂuenced by the valence states, which are more or less free-electron like, especially in the case of the simple metals. There are cases where LDA should and does fail. The use of LDA results in a bad description of strongly correlated systems like CoO or high-Tc superconductors, or when

calculating properties that depend on more strongly bound electrons.

It should also be noted that so far we have not taken into account the possibility of a spin polarized system. This is a natural extension of LDA, called the local spin density approximation, LSDA [22], where the exchange-correlation functional of LDA, Eq. 2.28, is replaced by

E_xcLSDA=

drρ(r)xc(ρ_↓, ρ_↑). (2.29)

Although we have used the spin-up and spin-down densities in Eq. 2.29, some-times the total charge density and the magnetization density, the diﬀerence between spin-up and spin-down density, are used.

There exists a lot of improvements and extensions to LDA but we will only discuss the generalized gradient approximation, GGA, here. Among other 19

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methods we can mention the weighted density method [23], exact exchange [24], LDA+U [25], GW [26] and LDA+U+Σ [27], to name a few.

A free-electron gas is a rather crude approximation to a crystal so it should not come as a surprise that LDA also has some problems handling materials that are not strongly correlated as discussed above. LSDA improves the results for magnetic systems, but it is generally accepted that the use of LDA results in a slight overestimation of the bonding strength. This leads to an under-estimation of the equilibrium volume, usually to the order of 3-10 %, and an overestimation of the bulk modulus [28]. Another well known case where LDA fails is the ground state of Fe. The calculated ground state of Fe using LSDA is a low-volume paramagnetic fcc structure [22], whereas the GGA-result is a high-volume bcc ferromagnet [29], in good agreement with experiment. Early implementations of gradient corrections to LDA failed to improve on the ap-proximation because of the lack of adherence to the sum rule, Eq. 2.27. Later implementations [30] corrected this error. The basic principle behind GGA is to include non-local contributions to the exchange-correlation functional through terms which are functions of the gradient of the charge density as well as the charge density itself

E_xcGGA=

drρ(r)xc(ρ,∇ρ). (2.30)

LDA, and to some extent GGA, correspond to the random phase approx-imation, RPA [13], which is, simply put, an averaging scheme. Among more recent developments aiming to improve the GGA functional the main object has been to calculate corrections to RPA [31]. One advantage, or drawback depending on personal opinion, is that in this method there remains a free parameter which allows us to inﬂuence the ﬁnal results. The parameter can be chosen to improve the accuracy for either heavy or light atoms. This new functional is also reported to give better results for atoms and molecules.

2.6 Some ﬁnal thoughts on DFT

To replace the full many-body description of a material with a one-electron description and the exact exchange-correlation functional with a free-electron functional, LDA, ought to introduce some errors as both approximations are, at least at ﬁrst glance, rather crude. But still DFT-LDA gives extremely good results when used in a suitable way on suitable materials. How can that be so? Let us ﬁrst of all look at what we are actually calculating. The solutions to Eq. 2.21, εn, are really just numbers. They are often referred to as

one-electron energies, but can we really talk about such a thing? The eigenvalue of the highest occupied state necessarily coincides with the chemical potential, but the Kohn-Sham Fermi surface does not have the same shape as the true physical Fermi surface, although they do have the same volume [32]. The Coulomb interaction is strong and of long range, which should invalidate the independent-electron picture. As can be seen there are a lot of potential problems that each one could be fatal to DFT, but there is a simple but at the same time very subtle way out of this dilemma. The answer is the so-called Fermi liquid theory, below

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presented as formulated by Landau [33, 34, 35]. The theory was developed to explain the behavior of superﬂuid He₃, thereof the name.

So far we have used a one-electron description to describe what is actually a many-body system. The energy-momentum relation is changed in a drastic way but we still use one-electron levels and nomenclature in both the Hartree-Fock theory and DFT. Even in the simple case of the hydrogen molecule H₂ this description breaks down. But what if the electrons of interest, the ones close to the Fermi level, behave like independent electrons? The eﬀect of paramount importance that we are interested in here, in an interacting system, is the electron-electron scattering, which scatters electrons in and out of one-electron levels. It can be shown that the rate of the scattering is proportional to (ε−

εF)2, giving an inﬁnite lifetime for an electron at the Fermi surface at zero

temperature. Therefore we can use a relaxation time model and go on and pretend that the electrons do not interact. There is unfortunately one very big ”if” left. The argument above is only valid in a theory for weakly coupled electrons, but in a strongly coupled many-body system it is not valid.

What we have is however not bare electrons, but electrons and their ac-companying exchange-correlation holes, which screen the coulomb interaction. This combination is often referred to as a quasiparticle. These quasiparticles will only interact weakly, rendering the picture of independent quasiparticles valid, and we can use the above argument again. The solutions to Eq. 2.21, εn,

which are sometimes interpreted as one-electron energies, should more correctly be interpreted as Kohn-Sham quasiparticle energies. Although the underlying physical picture and the interpretations bulit from this picture is vague and ambigouos it is hard to argue against the success of DFT.

The term ”normal Fermi system” is often used for systems of interacting Fermi-Dirac particles of which DFT quasiparticles is one. Systems like super-conductors and strongly correlated systems fall outside of this description and are therefore in some sense not normal Fermi systems, but must be treated with a method based on a true many-body theory.

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CHAPTER

3

Magnetism and Fermi surface nesting

Mankind has been intrigued by magnetism since historical times and the use of magnetic materials goes back more than 1000 years. The oldest description of a compass is Chinese [36] but the use in Europe followed very soon [37]. In the 17th_{and 18}th _{centuries there was some progress in producing stronger}

magnets and natural philosophers, the physicists of the day, realized that only some very specific materials made good magnets. But the question of what magnetism really is remained unanswered. One modern interpretation is that a magnetic field is the relativistic transform of an electric field in motion, see e.g. Ref. [38],but that only takes us to the next question, that of what charge is. We can work with and use magnetism if we know what it does rather than what it is. In this chapter we will discuss some features of magnetism and especially how we can describe and understand the magnetic properties of materials.

Of the elemental metals there are only a few that prove to have a macro-scopic magnetic moment at room temperature, the majority are non-magnetic, or rather they have no global magnetization. At room temperature only Fe, Co, Ni, and Gd have global magnetization. At lower temperatures Tb, Dy, Er and Ho join the exclusive group. Chromium has a local moment but ordered in a complex magnetic structure yielding no global moment [39]. Although all electrons carry a spin moment, in most materials there is a cancellation of moments either locally at the atom or globally when all spins point in random directions. We will brieﬂy discuss why the ordered magnetic state is stabilized in some systems but not in others.

Magnetism in materials can be divided into two main categories often re-ferred to as localized magnetism and itinerant or band magnetism, respectively. Although the cause of both kinds is almost the same the properties of the lo-calized and itinerant magnetism are a bit diﬀerent, due to the exact nature of the electrons involved in the magnetic state.

In the case of ordered magnetic moments the most common cases are when the spins order in parallel or in antiparallel, called ferromagnetism and anti-ferromagnetism respectively, but fairly recently systems with a more complex spin structure were discovered. Chromium was the ﬁrst material to draw the

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attention to this behavior. The moments can e.g. form a helical structure, a spin spiral, or more complicated structures like planes ordered ferromagneti-cally pairwise and rotated between pairs. The driving mechanism behind the complex structure is electronic interactions at the Fermi surface when the Fermi surface has some special properties. This will also be discussed below.

3.1 Localized magnetism

Magnetism is a complicated phenomen so let us make it a bit simpler for a while and look at the atom or an ionic compound instead of a metal. All electrons carry an intrinsic spin of magnitude 1/2. If the direction of the spin was unimportant all the diﬀerent spin conﬁgurations would be degenerate and the average magnetic moment would be zero. Obviously that is not the case, so there must be a mechanism that can align the spin moments.

The Hamiltonian for the interaction of the spin with a uniform magnetic ﬁeld B is

Hspin = g0µBB· S, (3.1)

where g₀ is the electronic g-factor or the spin gyromagnetic ratio, a coupling constant between the intrinsic spin and the magnetic moment of the electron, and which to a good approximation is equal to 2. µB is the Bohr magneton.

From the equation of motion for an electron in a magnetic ﬁeld we get, in the coulomb gauge i.e. ∇·(r×B) = 0, a diﬀerent expression for the kinetic energy compared to the usual T₀= _ip2_i (see e.g. Ref. [40])

T = i (pi− 1 2ri× B) 2 _(3.2)

which, in the case of a magnetic ﬁeld in the z direction, B = B ˆz, can be written

as T = T₀+ µBL· B + 1 2B 2 i (x2_i + y_i2) (3.3) where L is the electronic orbital momentum deﬁned as

L =

i

ri× pi. (3.4)

Combining Eq. 3.1 and Eq. 3.3 gives

Hspin= µB(L + g₀S)· B + 1 2B 2 i (x2_i + y_i2). (3.5) This equation is only exact for atoms and ions, but it is a good approximation for solids where the ions are only slightly deformed. The change in energy for atomic level n can be calculated using second order perturbation theory.

∆En = µBn | L + g₀S| n · B + µB n_=n | n | (L + g0S)· B | n |2 En− En + 1 2B 2_{n |} i (x2_i + y2_i)| n (3.6)

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The index n runs over atomic (ionic) levels and the index i runs over the electrons.

The ﬁrst term in Eq. 3.6 is several orders of magnitude larger than the other two except for the case when | L =| S = 0 and when | En− En | is very

small. The first case can occur in atoms and ions with all shells filled. In atoms the electronic levels are well separated but in solids the states might be almost degenerate, so a different theory must be employed in that case. This will be discussed below.

If we look at an atom with all shells but one filled, we notice that there are a lot of different ways to fill that last shell. If the electron-electron interaction is weak the different configurations would be degenerate. This degeneracy is lifted by the Coulomb interaction, but the way it is done depends on the atom. We will not give any proofs of the schemes presented below. The interested reader can find it in any text book on quantum mechanics, e.g. the book by Atkins [41].

In the case of light atoms where the spin-orbit interaction (the interaction between the spin and the internal electric ﬁeld) is weak, all the intrinsic spins of the electrons in the unﬁlled shell form a total S and the orbital moments form a total L. The total moment formed by all moments in the atom is then given by J = L + S. This is called Russel-Saunders coupling or LS-coupling. If on the other hand the spin-orbit interaction is not negligible, which is the case for very heavy atoms, each si and li form a ji which then couples to common

J. This is referred to as JJ-coupling.

According to the Pauli principle there can be only one electron with a unique set of quantum numbers, i.e. in each orbital there can only be one electron with spin up and one with spin down. From Eq. 2.11 we see that the interaction between electrons with parallel spins will lower the total energy. Therefore the

n electrons will populate the 2(2+1) available levels so that S is maximized.

The orbitals will be populated in such a way that also L is maximized, still with the Pauli principle and the maximized S in mind. The spin-orbit interaction lifts the degeneracy of the possible combinations of S and L through a term in the Hamiltonian proportional to L· S, with a prefactor, proportional to the strenght of the spin-orbit interaction and which is positive for shells which are less than half full and negative for shells that are more than half full, giving the rule for choosing J as J =| L − S | for n ≤ 2 + 1 and J = L + S for n ≥ 2 + 1. These guidelines on how to choose J ,L and S are called Hund’s rules.

A useful measure of the magnetic state in a material is the magnetic sus-ceptibility χ, which tells us how a magnetization density in a material responds to an external magnetic ﬁeld and is deﬁned as

χ = ∂M ∂B =− 1 V ∂2F ∂B2 (3.7)

as the magnetization density is the derivative of the magnetic Helmholtz free energy F which is deﬁned as

F = nEnexp (−En/kBT ) nexp (−En/kBT ) . (3.8) 25

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In the case of J = 0 and low temperature so that the system is close to the magnetic ground state i.e. only the 2J + 1 lowest states are occupied, we can calculate χ. The free energy is

F = J J_z_=−Jg(J LS)µBBJzexp (−g(JLS)µBBJz/kBT ) J J_z=−Jexp (−g(JLS)µBBJz/kBT ) , (3.9) where g(J LS) is the Land´e g-factor describing the energy separation of the 2J + 1 non-degenerate levels. Using Eq. 3.7 and assuming g(J LS)µBB  kBT

we ﬁnally get χ = N V (g(J LS)µB)2 3 J (J + 1) kBT . (3.10) From this we see that the external magnetic ﬁeld tends to align the moments and that high temperature favors disorder. The inverse dependence of the susceptibility on temperature is known as Curie’s law.

The treatment of the magnetic properties discussed above is only valid for atoms and ions, but what will happen in an ionic solid or in an insulator? In a material where the magnetic atoms (ions) have an unfilled f shell, the rare-earths and actinides, that shell is inside of a partially filled d shell which will screen the interaction with the electrons from the surrounding atoms, leaving the f shell almost atomic-like with a very small f -f overlap. The case is different for a solid with an unfilled d shell as the outermost shell and no f electrons like transition metals. Here the electrons are interacting with the field generated by the rest of the electrons in the crystal, the so-called crystal field. For 3d systems the crystal field is stronger than the spin-orbit interaction, so the highly anisotropic crystal field will break the symmetry and lift the degeneracy of the L multiplet. The time reversal symmetry will then give L = 0 [42] leaving J = S. This is called crystal field quenching, and in such a system we can still use the theory discussed above but with the different J . For the heavier transition metals, 4d and 5d, where the spin-orbit interaction is stronger the situation is more complicated requiring a group-theoretical treatment beyond the scope of this thesis.

3.2 Itinerant magnetism

What would happen in a material where the orbitals overlap and form a metallic state, and the occupation number can be fractional? Obviously we can not use the theory from above in its present form. The short and very much simplified picture of why a metal can be magnetic, i.e. have different occupation numbers for the spin-up and spin-down states, can be described as follows. The shape of bands and therefore the density of states (DOS) is to a large extent decided by the symmetry of the system, and the energy of the highest occupied state, the Fermi energy, is decided by the the number of electrons in the system. By coincidence the Fermi level can be situated at a peak in the density of states, which is energetically very unfavorable. One solution for the material can be to lower the symmetry and change the crystal structure, a effect known as a

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Jahn-Teller distortion. Another way is to shift e.g. the spin-up states to a higher energy and the spin-down states to a lower energy conserving the total number of occupied states but changing the magnetization density n_↑−n_↓. An example of this eﬀect is seen in Fig. 3.1, showing the both the paramagnetic and spin-polarized DOS for EuCo₂P₂ in its high pressure phase, a material with a calculated moment of 1.2µB per formula unit originating from the Co

3d orbitals. Of course this kind of hand waving argument is not enough to

ï0.7 ï0.5 ï0.3 ï0.1 0.1 0.3 Energy (Ry), EF=0 ï100 ï50 0 50 100 DOS (states/Ry) paramagnetic spinïpol. up spinïpol. down

Figure 3.1: The total density of states for paramagnetic and spin-polarized EuCo₂P₂at high pressure.

interpret the behavior of a magnetic material, but it gives a hint about the nature of band magnetism.

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For a more accurate treatment let us start by applying an external magnetic ﬁeld to a non-interacting electron gas. This external ﬁeld will cause a shift of the spin-up and spin-down Fermi energies,

ε±= εF± µBBext. (3.11)

As both spin-up states and spin-down states have the same chemical poten-tial they will also have the same Fermi energy. Therefore electrons will be transferred from one spin state to the other leading to a shift of the spin-up and spin-down bands, compensating for the change in energy introduced in Eq. 3.11. The band energy will then be given by

E_bandmag = εF −∞ εg(ε)dε− εF ε− εg(ε)dε + εF −∞ εg(ε)dε + ε+ ε_F εg(ε)dε (3.12) where g(ε) is the density of states for one spin channel, i.e. half of the total density of states for the paramagnetic case. If g(ε) is assumed to be almost constant on an energy scale of µBBext around εF, Eq. 3.12 can be integrated

giving

Emag_band= Enon_band−mag+ g(εF)µ2BB2ext. (3.13)

Above we have used the theory for bands at T =0 K where all states below

εF are occupied and all states above εF are unoccupied. At T =0 K the band

energy is equivalent to the free energy and can therefore be used to calculate the susceptibility, which for the paramagnetic non-interacting electron gas is (from Eq. 3.13)

χnon= 2g(εF)µ2B. (3.14)

This theory was developed by Stoner [43], but he also used the assumption of a parabolic band.

We can compare Eq. 3.14 with the susceptibility of interacting electrons. The exchange interaction between electrons regulates the magnetic properties but is unnecessarily complicated for this discussion, so we will defer that theory to the next section. The effect of the exchange field in a solid can be viewed as an internal magnetic field often referred to as the Weiss molecular field [44]. The band energy of the non-interacting electron gas, Eq. 3.12, will then be corrected by an additional molecular field term giving the total energy for interacting electrons

Eint= Enon−

ISM2

2 . (3.15)

M is the total magnetic moment of the system induced by a ﬁeld B, M =−∂F

∂B, (3.16)

at constant temperature and volume. IS, the Stoner parameter or the Stoner

exchange integral, is a measure of the electron-electron exchange interaction and is deﬁned by the overlap of wave functions of diﬀerent angular momentum

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l and the angular momentum projected density of states at the Fermi level

normalized by the total density of states

IS = ill gil(εF)gil(εF) g(εF)2 drK(r)Ψl(εF, r)Ψl(εF, r), (3.17)

where the index i runs over atoms and K(r) is a function of the charge density [45]. To a good approximation it can be regarded as an atomic property with a very small dependence on bulk properties. Integration of Eq. 3.15, again under the assumption of constant density of states close to the Fermi level, gives

E = Enon_−mag+ g(εF)µ2BB2−

ISM2

2 (3.18) where Epis the paramagnetic contribution to the energy, resulting in a

suscep-tibility

χint=

χnon

1− 2µ2_BISg(εF)

. (3.19) The susceptibility has been enhanced by a factor (1− 2µ2_BISg(εF))−1, the

Stoner enhancement factor, which can result in an instability, a negative sus-ceptibility, leading to a spontaneous magnetic ordering when

2µ2_BISg(εF) > 1. (3.20)

This so-called Stoner criterion tells us that we can expect to see some kind of magnetic ordering in a material when either the Stoner factor is large (strong exchange interaction) or the density of states is high at the Fermi level, as was discussed above. For most materials the density of states is not high enough, with the exception of materials with narrow 3d bands. It is also in this group that we ﬁnd the elements with a spontaneous magnetic ordering: Fe, Co and Ni.

3.3 Magnetic ordering

In the above discussion we have assumed that all spins align to be parallel to one another, but this is not always the case. Depending on the properties of the exchange interaction, band structure and temperature eﬀects, other spin conﬁgurations can be more favorable. We will start by looking at the Heisen-berg model for localized moments, as it gives a good picture of the nature of the magnetic excitations.

3.3.1 The Heisenberg model

The starting point of the Heisenberg model [46] is a model Hamiltonian de-scribing the interaction between spins at diﬀerent lattice sites as well as the interaction between the spins and an external ﬁeld

HHeis=−J i,j SiSi+j− gµBBext i Sz,i (3.21) 29

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where J is the exchange integral, g is the Landé factor and the index i runs over all lattice sites and j over all nearest neighbors and sometimes next nearest neighbors to site i. Here we can directly see that (in the absence of a strong external field) a positive value of J favors a parallel alignment, ferromagnetism, and a negative value an anti-parallel alignment, antiferromagnetism, as the ground state configuration. The excitations of this Hamiltonian are spin waves, magnons, i.e. collective movements over the whole system.

The calculation of the excitation spectrum is simpliﬁed by the introduction of the step operators S+ = Sx+ iSy and S− = Sx− iSy. The eﬀect of these

operators is to ﬂip a spin, MS → MS+ 1 when applying S+ and MS → MS− 1

when applying S−. The Heisenberg Hamiltonian for small excitations can be written as HHeis=−J i,j Sz,iSz,i_+j+ 1 2(S + i Si−_+j+ Si−Si+_+j) − gjµBBext i Sz,i (3.22) using the step operators. These step operators, or ladder operators as they are often called, can be expressed as bosonic creation and annihilation operators in real space, a+_i and a−_i , or often more suitably as reciprocal space magnon operators b+_k and b−_k where b+_k (b−_k) will create (annihilate) a magnon with wave vector k.

After rather a lot of algebraic manipulations and fancy footwork the terms bilinear in the magnon operators can be collected from Eq. 3.22,

Hbi= k (2J Sz(1− γk) + gµBBext)b+_kb−_k = k ωkb+_kb−_k. (3.23) Here γk = 1_z

jexp(ikj). The conclusion of Eq. 3.23 is that the thermal

exci-tations of the spin degrees of freedom in a system are correlated into collective excitations, magnons.

3.3.2 Generalized spin structures

In the previous section we discussed the case of excitations in localized mo-ments resulting in collective motions of the momo-ments, magnons, which can be interpreted as spin waves with a defined wave vector. Is it possible that this kind of complex ordering could exist as a static ground state configuration? Let us start by looking at the uniform susceptibility χ, defined as the measure of the response of a system to a uniform magnetic field. A uniform field would align all spins to be parallel to each other resulting in ferromagnetic ordering. Consider instead a field that changes its direction with a periodicity equal to half the periodicity of the lattice. That would align the spins to be anti-parallel giving an antiferromagnetic ordering.

Depending on the shape of the bands close to the Fermi surface the total energy of the system may be lowered if spin-up states and spin-down states can hybridize. In this way the states below the Fermi level, the bonding states, can be lowered even more at the expense of the states above the Fermi level, the anti-bonding states, but as these states are unoccupied they do not contribute

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to the total energy. At the same time this means that as the anti-bonding states are being emptied, the balance between the two diﬀerent spin projections will change leading to an increased magnetic moment and a large susceptibility. As will be shown in the next chapter, one way of shifting the bands so that they can hybridize is to introduce a helical spin ordering that will break the symmetry of the system. A helical spin ordering will cause a rigid shift of the band structure, k→ k ± 1₂q, where the sign of the shift depends on the spin

direction i.e. spin-up states will be shifted in one direction and spin-down states in the other. This can be seen in Fig. 3.2, the band structure of paramagnetic

Figure 3.2: The band structure of paramagnetic Pd₃Mn

Pd₃Mn, and Fig. 3.3, the band structure of the same material subjected to a non-collinear spin ordering in the shape of a spin spiral with a wave vector of

q=0.13×(111), which is the calculated total energy minimum of this system, see

Paper IV. Notice how the bands which in the paramagnetic case are crossing each other close to the Fermi level at about three quarters of the way from the Γ-point to the M-point have hybridized and split for the spin spiral structure. The strength of this effect is of course influenced by the number of states that can be shuffled away from the Fermi level and the largest effect is achieved when large portions of the Fermi surface can be connected by common q’s called nesting vectors. A closer inspection of the Fermi surface of Pd₃Mn, Fig. 3.4, will reveal almost parallel sheets of different spin projections with the correct nesting vectors. A quantitative measure of this so-called Fermi surface nesting is given by the poles in the generalized susceptibility

χ(q) = 2µ2BF (q)

1− 2ISF (q)

(3.24) 31

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Figure 3.3: The band structure of Pd₃Mn subjected to a spin spiral with a wave vector of q=0.13×(1,1,1)

where IS is the Stoner factor and

F (q) = k n↑ k+1₂q− n ↓ k−1₂q ε↓_k₋1 2q− ε ↑ k+1₂q− ∆E (3.25) is the q-dependent equivalent to the unenhanced susceptibility χnon. ∆E is

the energy gap between the spin-up and spin-down bands.

When looking at the dynamics of the magnetic moments in itinerant sys-tems it can be noted that at elevated temperatures the number of excitations

Figure 3.4: A slice of the Fermi surface of ferromagnetic Pd₃Mn in the simple cubic symmetry. A cut through the Γ-point is made in a (110) plane. The solid and dotted lines represent the spin-up and spin-down states, respectively. The arrows indicates nesting vectors along the [111] direction with q=0.13×(111).

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will increase, leading to an increased disorder in the system. At a sufficiently high temperature,the so called Curie temperature for systems with ferro- or ferrimagnetic ordering and Néel temperature for antiferromagnetic ordering, the average magnetic moment will go to zero, causing a phase transition to a non-magnetic state. Early theories where the possibility of non-collinear spin ordering was not included, e.g. the Stoner model discussed above, gave Curie temperatures that were too high. The reason for this is that in this simpi-fied theory, the only possible excitation influencing the magnetic state is an electron-hole transition at a fixed k, changing the magnitude of the total mag-netic moment but not the direction of local magnetization density. When the magnetization density is free to vary both in magnitude and direction the en-ergy of the lowest excitations is decreased, increasing the number of excitations at a given temperature and therefore the disorder, resulting in a lower critical temperature.

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CHAPTER

4

Details on implementations

In Chapter 2 we discussed the properties of a Hamiltonian suitable for calculat-ing the ground state properties of a crystalline material, but that left us with the problem of solving the eigenvalue equation, the Kohn-Sham equation,

Ψ | HDF T − ε | Ψ = 0 (4.1)

where the DFT Hamiltonian can be of a number of diﬀerent kinds, e.g. LDA, GGA or LDA+SIC. To make the calculation eﬃcient some care must be used in the choice of the wave function. To get a good description of the physical system an expansion in a suitable basis set is used,

Ψ(r) =

i

Aiψi(r) (4.2)

where Aiare expansion coeﬃcients and ψi(r) are wave functions constituing a

basis set. This basis set is most of the time either optimized for accuracy or for simplicity. To get a good accuracy we can either choose a basis set that is ﬂexible and close to the geometry of the problem, or a simple one to allow us to use many basis functions. To be able to solve the largest systems, include complicated interactions or examine the dynamics of a system, we sometimes have to sacriﬁce some accuracy.

During the investigations presented in this thesis three different methods have been used: two very similar methods, Full Potential Linearized Muffin-Tin Orbitals (FP-LMTO) and Full Potential Linearized Augmented Plane Waves (FP-LAPW), for calculating crystallographic, electronic and magnetic prop-erties where the highest accuracy was needed, and one, Plane Wave Pseudo Potentials (PW-PP), used for calculating deviations from perfect lattice sym-metries. The charge density and potential in a normal system are varying rapidly close to the ions but are almost constant in-between, which leads to a natural division of space into two regions, one with a spherical symmetry surrounding each ion called muffin-tin spheres and the rest, called the intersti-tial. The geometry is used to define basis functions and to specify regions over which particular expansions are done. The potential and charge density are not

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restricted to this geometry. The muffin-tin geometry is used for the first two methods. The third method uses a completely different way of handling the dissimilar behavior in the separate regions by disregarding the behavior close to the ion completely.

Below we will try to describe the technical aspects of the diﬀerent methods without going into too much detail, and ﬁnally we will take a closer look at the calculation of magnetism in non-commensurate systems.

4.1 Full potentials and muﬃn tins

The basic idea behind the use of muﬃn tins is to divide space into two regions. Close to the ions we would like to use a basis set that has a spherical symmetry, as that would closely describe the (almost) localized electron states found there. In the interstitial region the electrons are almost free-electron like, which makes the spherically symmetric basis set very unsuitable. Due to the complex shape of this region we also need a basis set that converges for a small number of basis functions. The name ”full potential” is used to distinguish these methods from regular muﬃn-tin methods [47, 48], where the potential in the interstitial region is constant and a perfect spherical symmetry is assumed inside each sphere. In the full potential methods no shape approximations of charge density or potential have been made.

There are many features that the two methods share so we will start by describing FP-LMTO in some detail, and then we will focus on the diﬀerences in FP-LAPW.

4.1.1 Full Potential Linearized Muﬃn-Tin Orbitals

The implementation of FP-LMTO that has been used here is the result of many years of development by mainly one man, John Wills [49, 50] at the Los Alamos National Laboratory, although many more have contributed during the development of the code. The regular LMTO method was introduced by O.K. Andersen [47].

The introduction of muffin tins in principle adds a new variational param-eter, the muffin-tin radius, but if the basis set is complete enough the result should not depend on the choice of radius. The exception is when the Hamil-tonian differs between the muffin-tin and interstitial regions. This is the case when spin-orbit interaction (see Chap. 3) is included. A discussion on some of the problems connected to this can be found in Paper V.

In the interstitial region the basis functions of choice are Bloch sums of spherical Hankel and Neumann functions

ψi(k, r) =

R

exp (ik· R)Kl_i(κi,| r − τi− R |)Yl_im_i(Dτ_i(r− τi− R)). (4.3)

The sum over R, integer multiples of the basis vectors, is in principle inﬁnite but the interstitial basis functions are only used as Fourier transformed functions.

Theoretical Investigations of Selected Heavy Elements and Metal-hydrogen Systems by Means of Electronic Structure Calculations

Comprehensive Summaries of Uppsala Dissertations

from the Faculty of Science and Technology 653

_____________________________

_____________________________

Theoretical Investigations of Selected

Heavy Elements and Metal-Hydrogen

Systems by Means of Electronic

Structure Calculations

BY

PER ANDERSSON

ACTA UNIVERSITATIS UPSALIENSIS

UPPSALA 2001

Abstract

To G, O and X

G¨

or man inget s˚

a blir inget

gjort.

List of Publications

CONTENTS

CHAPTER

1

Introduction

CHAPTER

2

The many-body problem and how to avoid solving it

2.1

The Schr¨

odinger equation

2.2

The Hartree and Hartree-Fock equations

2.3

Screening and Thomas-Fermi theory, a

some-what diﬀerent approach

2.4

Density Functional Theory

2.5

Choosing the exchange-correlation functional

2.6

Some ﬁnal thoughts on DFT

CHAPTER

3

Magnetism and Fermi surface nesting

3.1

Localized magnetism

3.2

Itinerant magnetism

3.3

Magnetic ordering

3.3.1

The Heisenberg model

3.3.2

Generalized spin structures

CHAPTER

4

Details on implementations

4.1

Full potentials and muﬃn tins

4.1.1

Full Potential Linearized Muﬃn-Tin Orbitals