spin density wave in chromium - ifw dresden · addition we allow for an external magnetic eld...

Leibniz-Institut für Festkörper- und Werksto�forschung DresdenInstitut für Theoretische Festkörperphysik

Spin Density Wave in Chromium

Diplomarbeitzur Erlangung des akademischen Grades

Diplom-Physiker

vorgelegt von

Mathias Bayergeboren am 28.02.1981 in Böblingen

Institut für Theoretische PhysikFachrichtung Physik

Fakultät für Mathematik und Naturwissenschaftender Technischen Universität Dresden

2008

1. Gutachter: Prof. Dr. Helmut Eschrig2. Gutachter: PD Dr. Peter Zahn

Eingereicht am 20. März 2008

Kurzfassung

Der magnetische Grundzustand von Chrom zeigt eine inkommensurableSpindichtewelle (SDW) mit einer Periode von etwa 21 Doppellagen. Kürz-liche Berechnungen der elektronischen Struktur fanden einen antiferromagne-tischen Grundzustand bei der experimentellen Gitterkonstante. In dieser Ar-beit wiederholen wir die Berechnungen mithilfe des full-potential local-orbitalCodes FPLO in der LSDA Näherung, welcher uns eine gute Genauigkeit inder Gesamtenergie ermöglicht. Wir konnten Probleme, die auf ungenügendeDichte des k-Netzes zurückzuführen sind, ausschlieÿen. Weiterhin haben wirdie E�ekte einer an die SDW gekoppelten Verzerrungswelle (SW) studiert,und konnten keinen Ein�uss auf die Gesamtenergie entdecken. Die Ergeb-nisse werden mit Experimenten und theoretischen Daten aus der Literaturverglichen.

Abstract

The magnetic ground state of chromium displays an incommensurate spindensity wave with a period of about 21 atomic double layers. Recent elec-tronic structure calculations for Cr found an antiferromagnetic ground stateat the experimental lattice constant. We have used the full-potential local-orbital code FPLO, which has allowed us a high accuracy of the total energyin LSDA. We have been able to rule out the in�uence of insu�cient k pointsampling. Further, a lattice distortion has been applied to simulate the ef-fects of the concomitant strain wave (SW). We have found no in�uence ofthe SW onto the total energy. The results are compared with experimentaland theoretical data in the literature.

Contents

1 Introduction and Motivation 3

2 Theory 52.1 Density Functional Theory . . . . . . . . . . . . . . . . . . . . 52.2 Spin Density Waves . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Literature 133.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 SDW with higher harmonics . . . . . . . . . . . . . . . 163.1.2 CDW . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.3 SW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Theoretical descriptions . . . . . . . . . . . . . . . . . . . . . 173.3 Band structure calculations . . . . . . . . . . . . . . . . . . . 18

4 Calculations 214.1 Technical details . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.1.1 Dependence on k-point number . . . . . . . . . . . . . 224.1.2 Scalar vs full relativistic version . . . . . . . . . . . . . 24

4.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 264.2.1 NM and AFM Cr . . . . . . . . . . . . . . . . . . . . . 264.2.2 SDW . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Summary and Outlook 37

Chapter 1

Introduction and Motivation

Density functional theory (DFT) has been used on a variety of metals andcompounds to successfully describe their ground state properties.

This work is focused on the transition metal chromium. Cr is the primeexample of an itinerant antiferromagnet, its incommensurate spin densitywave (SDW) is a truly many-body e�ect. The origin of the SDW is commonlyattributed to the nesting properties of the Fermi surface, which determinesthe wave vector q of the SDW.

Bcc Cr and fcc Fe seem to be the only 3d transition metals for which highprecision DFT calculations predict the wrong bulk magnetic ground state.This is in contrast to earlier band structure studies that have managed todescribe the Fermi surface of Cr including the nesting feature.

This work pursues three issues where there may have been shortcomingsin the past works and which might help solving the puzzle. One item is to geta really accurate resolution of the Fermi surface. By the help of the processingpower of our computer cluster we were able to reach a high resolution in theBrillouin zone integrations which has not been reached in this �eld yet. Wehave analysed if Fermi surface nesting performs.

The second point is the consideration of a lattice distortion that occurs inthe real metal and is potentially responsible for the stability of the SDW. Suchstrain waves (SW) have been examined within a large range of amplitudesaround the experimental value.

The last point examines the validity of neglecting the in�uence of spin-orbit coupling.

After a presentation of the basics of density functional theory, this thesisgives a short overview about the literature status of the SDW in Cr, boththeoretical and experimental, followed by our main results. Finally, the workis summarised and an outlook is given.

Chapter 2

Theory

Density functional theory is nowadays a crucial method for the study ofelectronic structure. We give a short introduction into the fundamentalsserving us as a working basis. Details about density functional theory can befound superiorly re�ected in the literature [1], [2]. Subsequently, the originof the spin density wave is discussed, its explanation is embedded into thepicture proposed by Peierls. We close the chapter by a short remark aboutthe generalised Stoner condition, being a test for instability against evolutionof antiferromagnetic (AFM) order.

2.1 Density Functional Theory

An ideal metal is a crystal with atoms around their equilibrium positionsand their electrons surrounding them itinerantly. They in�uence each othervia Coulomb interactions. We consider the non-relativistic Coulomb Hamil-tonian for a �xed number of atoms Na with masses Mν and charges Zν ,together with their Ne electrons, in natural units1:

H = Ha + He

Ha = −Na∑µ

∇2µ

2Mµ

+1

2

Na∑µ 6=ν

ZµZν

|Rµ −Rν |

He = −Ne∑i

∇2i

2−

Ne∑i

Na∑µ

Zµ

|ri −Rµ|+

1

2

Ne∑i6=j

1

|ri − rj|. (2.1)

The Hamiltonian He is expressed through the operators of the kinetic1h = m = |e| = 1, energy unit is 1 a.u. = 1 Hartree

6 CHAPTER 2. THEORY

energy T , the potential of the external �eld V , and the two-particle interac-tion W :

He = T + V + W . (2.2)

The external �eld consists of the part created by the nuclei from (2.1), inaddition we allow for an external magnetic �eld acting on spin only, with V =∑Ne

i=1 v(xi). We have combined electronic spin and real space coordinates intoxi = (ri, si), with ∫

dxi ≡+1/2∑

si=−1/2

∫d3ri . (2.3)

The �rst important approximation is the neglect of the in�uence of ionicmotion onto electron energy levels. We let the total ground state energy, andthe electronic wave function, parametrically depend on the nuclear positionsRµ. Electrons are light compared with the nuclei, so they move more rapidlyand can thus follow the motions of the nuclei quite unhesitatingly. We candecouple the full Schrödinger equation from the nuclear motion and assumethat electrons are always in the ground state with respect to the motion ofthe positive ions. This is called the adiabatic approximation [1].

This brings us to

HeΨ(x;R) = Ψ(x;R)E(R) , (2.4)

where we have employed the short notation x = {xi}, i = 1 . . . Ne andR = {Rµ}, µ = 1 . . . Na. The Hamiltonian (2.1) is not spin-dependent, butwe have to ful�l the Pauli principle when constructing the wave function Ψ.

Unfortunately, the Schrödinger equation for our problem (2.4) has a di-mension of the order of the Avogadro constant. The following steps arededicated to greatly simplify the problem.

Due to the large number of particles, for macroscopic systems, our Hamil-tonian produces energy levels having tiny distances no experiment could everresolve. One accessible quantity of interest is the ground state energy as areference, together with Ψ0 as the ground state wave function and the groundstate density

n0(x) =

∫Ψ∗

0(x, x2 . . . xNe)Ψ0(x, x2 . . . xNe)dx2 . . . dxNe . (2.5)

The essential insight is to make a change from (x1 . . . xNe)-dependence ofthe ground state properties and introduce the density n = n(x) as a newbasic variable. Hohenberg and Kohn [3] showed that this entails no lossof information. In general, we can write the energy and wave function asa functional of n, where n is a single particle density, as in (2.5). This

2.1 Density Functional Theory 7

functional E[n, v] possesses a minimum at the ground state density n0, andis equal to the ground state energy E0[v] at this density. In order to �nd theground state energy, we have to apply a variational principle. We follow theway given by Levy [4] and Lieb [5], the constrained search method. We �xa trial density n, and vary over the class of wave functions Ψn, which yieldthat trial density. In a second step, we vary over the densities n,

E0[v] = minΨ

⟨Ψ

∣∣∣He

∣∣∣ Ψ⟩

(2.6)

= minn

{FHK[n] +

∫dx nv

∣∣∣∣ ∫dx n = Ne

}. (2.7)

Ψn has to ful�l the criterion of being a normalised fermionic wave func-tion of density n. The unknown ingredient is the Hohenberg-Kohn functionalFHK[n], which is independent of the external potential V , and is thus a univer-sal functional, valid for any many-electron system. This constrained searchfunctional introduced by Levy has shortcomings and was replaced by Lieb bya more general functional based on Legendre transforms. Since both lead tothe same form of the Kohn-Sham equation and have to be modelled anyhow,we do not dive into these subtleties.

FHK[n]def= min

Ψn

{⟨Ψn

∣∣∣T + W∣∣∣ Ψn

⟩|Ψn 7→ n

}(2.8)

As we are allowed to consider an arbitrary interaction W , we can in parti-cular set W = 0. The ground state kinetic energy functional of an interaction-free electron gas T0[n] can be written as a single slater determinant, assuringthe correct fermionic antisymmetry with respect to permutation of particles:

T0[n] = min{φi}7→n

{Ne∑i=1

⟨φi

∣∣∣∣−∇2

2

∣∣∣∣ φi

⟩}, 〈φi | φj〉 = δij, {φi} 7→ n , (2.9)

where the sum runs over the Ne lowest orthonormal single particle eigen-states φi .

Kohn and Sham [6] perceived that it is of advantage to decompose theHohenberg-Kohn functional, to extract the noninteracting ground state ki-netic energy and the interaction energy between the electron and the densityas they can solely be expressed through the density,

FHK[n] = T0[n] +1

2

∫d3rd3r′

n(r)n(r′)

|r− r′|+ EXC[n] . (2.10)

8 CHAPTER 2. THEORY

The term in the middle on the ride hand side of equation (2.10) is theHartree energy EH, which describes a time-averaged repulsion through anelectrostatic �eld created by all electrons. The unknown parts are reducedto EXC, the exchange and correlation energy.

The variational principle is now carried out, ful�lling the boundary con-ditions contained in (2.9) by the method of Lagrangian multipliers,

(−∇

2

2+ v(x) +

∫d3r′

n(r′)

|r− r′|+

δEXC[n]

δn(x)

)φi(x) = φi(x)εi . (2.11)

The result is a Schrödinger equation in an e�ective one particle picturewhere particles are considered in an e�ective potential. This e�ective poten-tial consists of the external potential, the Hartree potential vH =

∫d3r′ n(r′)

|r−r′|and the general but unknown exchange potential vXC, which is the last termin (2.11). The density is obtained by summing up over the Ne lowest states,

n(x) =Ne∑i=1

|φi(x)|2 . (2.12)

The Kohn-Sham equations (2.11) are nonlinear and must be solved iter-atively. Each step puts out a new density, which again forms a new Hamilto-nian. The occurring eigenenergies can in a weakly correlated case be regardedas the approximants of the band energies.

To yield an expression for the total ground state energy, we multiply(2.11) by φ∗i (x), integrate, sum over i and consider (2.9)

E[v] =Ne∑i=1

εi≤EF

εi − EH[n]−∫

dx nvXC + EXC[n] . (2.13)

We need to add the nucleus-nucleus interaction term to (2.13), to �x thediverging electrostatic energy of the electrons.

One step further towards applicability is to draft a model for the unknownpotential vXC. An approved one is called the local spin-density approxi-mation (LSDA) [7]. It starts from a homogeneous electron liquid, includinginteraction. An external magnetic �eld evokes a spin split with the degree ofspin polarisation ζ = n(+)−n(−)

n. We take the exchange and correlation energy

density of a homogeneous Fermi liquid as a basis:

EXC[n, ζ] ≈ ELSDAXC [n, ζ]

def=

∫d3r n(r) ε(n(r), ζ(r)) . (2.14)

2.2 Spin Density Waves 9

This is a good approach if the density varies slowly in space. It is notrealistic in an inhomogeneous solid, where we have nuclei and space in bet-ween, but experience shows that it yields good results, particularly for 3dtransition metals, which are exclusively in the focus of this work. One reasonfor the good behaviour of the LSDA describing metals is the exchange andcorrelation energy being small compared to EH and Ekin.

It is known that LSDA tends to the so-called �over-binding�, which meansequilibrium lattice constants are predicted too small, compared with experi-ment.

The exchange and correlation potential is taken from analytical �ts byPerdew and Wang [7] to numerical simulations by Ceperley and Alder [8].

2.2 Spin Density Waves

We intend to explain the origin of the spin density wave (SDW) by using thepicture of the one-dimensional Peierls distortion. For a detailed introduction,we refer to [9]. We consider a linear chain of equidistant atoms with distancea. The basic cell in the reciprocal space is the interval −π

a≤ k ≤ π

a. If a

distortion is introduced, for example by displacing every rth atom by a smallamount in the same direction, translational symmetry is reduced. We thenhave to regard a new unit cell containing r atoms, while the reciprocal spaceshrinks to the interval − π

ra≤ k ≤ π

ra. More general, we apply a periodic

potential U to our system which opens energy gaps that can be expressed insecond order perturbation theory. The energy level of the unperturbed freeelectron state e0

k−K1is connected with the energy level of the perturbed state

e with a correction of order U2,

e = e0k−K1

+∑K′

|UK′−K1|2

e0k−K1

− e0k−K′

. + O(U3) (2.15)

This equation holds for states which exhibit no degeneracy. K′ = 2πra

neK′ ,where n integer, is a reciprocal lattice vector, Uq is the qth Fourier componentof the potential. K1 is a particular reciprocal lattice vector, for which thefree electron energy e0

k−K1is far from the values e0

k−K′ compared with U. By�xing the potential to U0 = 0, we incorporate the condition K′ 6= K1 in thesum.

Now let us consider the case where two electron levels e0k−K1

, e0k−K2

arewithin order U of each other, but far compared with U from all others. Weneed to apply degenerate perturbation theory to express the energy gap.With k−K1 = K and K2 −K1 = Q,

10 CHAPTER 2. THEORY

e =1

2(e0

K + e0K−Q)±

[(e0K − e0

K−Q

2

)2

+ |UQ|2]1/2

. (2.16)

This has the e�ect of splittings occurring at (kn = nπra

, n = 1, 2, . . . , r−1).The distortion separates two states which have been formerly close togetherin energy. When we neglect the interaction with more distant states, themean value of the two energies does not change. In other words, the energydoes not change in lowest order. The splitting becomes only important whenoccurring at the Fermi level.

For states K and K−Q, having nearly the same energy, the magnitudeof the gap is ∼ 2 |UQ|. The gain in energy is greatest when r is a smallnumber. If kF = π

2a, which holds for a half-�lled band, every second atom

will be displaced.The energy gain, however, may be compensated by deformation energy.

In three dimensions, the additional degrees of freedom raise the cost in elasticenergy, so that the energy gain becomes less favourable [10].

In the case of the SDW, the periodic potential is caused by a sinusoidalvariation of the spin polarisation P(R0

j),

P(R0µ) = P1 cos(q ·R0

µ) = P1 cos(qzµ) . (2.17)

This describes a linear polarisation with wave vector q parallel to thez-axis. The direction of P1 is arbitrary, but it turns out, that it is alsothe z-axis in our case, as chromium has a longitudinal polarisation at lowtemperatures (Fig. 3.3). For further details we refer to [11]. It turns out thatq is incommensurate with the lattice in chromium.

A linearly polarised SDW is the superposition of two helically polarisedSDWs with ±q. Each ±q, as in Fig. 2.1, connects two states at K andK ± q, opening a gap at ±q

2in the parabolic bands. The linear SDW, as

a superposition, opens both gaps, and is thus energetically preferred. Theenergy gain is considerable high, if the splitting occurs at EF , so that thestates raised in energy are unoccupied. The electron and hole surfaces areshown in greater detail in Fig. 2.2.

The gap is broadened, if many states are involved in the mixing process.This is called Fermi surface nesting. The mechanism works with large parallelparts of the Fermi surface, which can be translated into each other by q [13].The e�ect is again illustrated in Fig. 2.1, in the middle section. It can beunderstood by taking a look at the q-dependent static susceptibility, whichcan be derived from second order perturbation theory [2],

2.2 Spin Density Waves 11

-q +q

hole hole

energy

k

electron

Figure 2.1: This sketch shows a cut through the Fermi surface of Cr. Theshaded area in the �rst part is an electron surface, the unshaded are holesurfaces. The picture in the middle depicts the Fermi surface nesting: largeparts of the electron and hole surfaces are parallel, and can be connected bythe nesting vector q, this causes the SDW instability. As the electron areais slightly larger than the hole area, q is a little bit smaller than half of thereciprocal lattice vector a∗: q = (1−δ) · a∗

2. Below is shown how the free hole

bands at e0K±q are folded back into the �rst Brillouin zone and open energy

gaps at ±q2. The picture is inspired by [12].

12 CHAPTER 2. THEORY

χ0(q) =∑Kµν

[f(εKν)− f(εK−qµ)]

εK−qµ − εKν + iδ· |〈Kν|eiqr|K− qµ〉|2 . (2.18)

A large area, containing many states which are separated by q in the BZ,enhances the instability.

With the susceptibility, one can de�ne a generalised Stoner condition,

Iχ0(q) ≥ 1 . (2.19)

When the product of the q-dependent static susceptibility and the Stonerexchange constant I, exceeds unity, then the paramagnetic state is unstabletowards antiferromagnetic (AFM) spin polarisation with wave vector q. Op-posed to the Stoner criterion for ferromagnetism where simply the density ofstates (DOS) at the Fermi level (EF ) �gures, it is demanded that the sus-ceptibility is large for the SDW instability to occur. This is the case for Cr,exhibiting a half-�lled d-band.

Figure 2.2: Cut through the Fermi surface, containing the points Γ and H.There is an electron surface centred at Γ, and a hole surface centred at H.The picture reveals more details of Fig. 2.1, it is taken from a nonmagneticcalculation.

Chapter 3

Literature

We give a short summary over experimental facts forming the fundament ofour work. As a source we refer to the excellent review article by Fawcett [14].

3.1 Experiments

The �rst neutron scattering experiments on Chromium by Néel (1936) [15]found that the magnetic ground state is described through antiferromagnetic(AFM) order. Upon a deeper view though, Shull and Wilkinson (1953) [16]detected that the ground state is rather a spin density wave (SDW), meaningthat the moment is varied sinusoidally. This is a generalisation of the AFMstate, which can be interpreted as a special case of a commensurate SDWstate with wave vector of q = a∗

z

2, a∗z = 4π

aez being the unit vector in z

direction in the reciprocal space. Today we know that below T = 78 K,Cr shows a longitudinal incommensurate SDW state with a wave vector ofq = (1 − δ)a∗

z

2= (0, 0, 0.952)2π

a(Fig. 3.2), which corresponds to about 21

double layers (dl) of length. The amplitude of the moment is 0.62 µB at theexperimental lattice constant of aexp = 2.884 Å .

There is a spin-�ip transition at TSF = 123 K (Fig. 3.3), where thepolarisation changes from longitudinal with P ‖ q below TSF to transversewith P ⊥ q above. For T < TSF, we have (0, 0, 1−δ), above we have (0, 1, δ).At the Néel temperature TSF = 311 K, there is a phase transition from theincommensurate SDW to a paramagnetic state.

14 CHAPTER 3. LITERATURE

(a) SDW

1 2 3 4 5 5 16 4 3 2

(b) mirror planes

Figure 3.1: (a) is a sketch of the variation of the magnetic moments alongthe direction of the wave vector q. The sketched unit cell contains n atoms,or n

2double layers in a bcc lattice. As n

2is not an integer the node of the

SDW will not rest on an atom. Below T = 78 K there is a longitudinalstate with the magnetisation vector parallel to the wave vector P ‖ q, withq = 0.952 · a∗

z

2, according to Fig. 3.2. For sake of perspicuity, the sketch

illustrates a transversal state. (b) shows the symmetry of the SDW supercell.We can assume one mirror plane at the centre, two at the edges.

q

Figure 3.2: temperature dependence of the wave vector q, q is given in unitsof a∗

2; slightly modi�ed from [14]

3.1 Experiments 15

(0,1, )d

(0,0,1- )d

Figure 3.3: phase transition from longitudinal to transverse SDW phase.Plotted are the magnetic satellite re�ections in AFM Cr. I is the integrateddensity under the Bragg peaks at the satellite positions; slightly modi�edfrom [14].

|P /P |3 1

Figure 3.4: the ratio (absolute value) of the amplitude of the third-harmonicSDW over that of the fundamental SDW, |P3/P1|, plotted over δ. The circlesindicate measurements taken from alloys, the solid line comes from a modelcalculation, according to [14] (slightly modi�ed).


3.1.1 SDW with higher harmonics

Parts of the Fourier spectrum of the spin density have been determined ex-perimentally. Higher harmonics in�uence the shape of the SDW. The twoanti-nodes of the SDW are symmetric to each other, resulting in a mirrorsymmetry at the centre point. This property cancels the even terms in theFourier expansion. Let us consider the �rst higher harmonic, P3. A negativeP3 will result in a more rectangular waveform, whereas a positive coe�cientwill cause a triangular waveform. The ratio of the third harmonic SDW overthe fundamental SDW P3/P1 has been measured to be negative with a valueof −2.1 · 10−2 by Tsunoda 1985 at 144 K. More experimental details arecollected in Fig. 3.4. The Fourier expansion, knowing that the polarisationpoints in the z direction, can be written as

P (zµ) = P1 cos(qzµ) + P3 cos(3qzµ) (3.1)= P1 cos(πµδ) + P3 cos(3πµδ) . (3.2)

We have rewritten the last equation using cos[qzµ] = cos[(1 − δ)πµ] =(−1)µ cos(δπµ) and have considered only the envelope. zµ = µa

2are the �xed

lattice positions in the unit cell of the bcc lattice, µ is an integer countingthrough the atoms in the unit cell, as in Fig. 3.1.

3.1.2 CDW

There is a periodic perturbation of the electron charge density in addition,a charge-density-wave (CDW), with half the period of the envelope of theSDW. We may express the CDW as a variation of occupation numbers

N(zµ) = N0(zµ)±N2(z0µ) cos(2δzµ) , (3.3)

with N2(zµ) as the amplitude of the CDW, and N0(zµ) as the charge aroundthe site zµ without CDW. The CDW has been determined experimentally tobe N2 ≈ 0.3 ∼ 2.0 · 10−2.

3.1.3 SW

A magneto-volume e�ect in Cr causes the atomic positions to be shiftedsinusoidally, called strain wave (SW). The result is the lattice spacing beingmaximum when the SDW amplitude is maximum. The period of the latticedistortion is again half the period of the SDW, as in Fig. 3.5. We treat onlyshifts in the z direction and write the SW as

3.2 Theoretical descriptions 17

z′µ = zµ ± A2 sin(2qzµ) . (3.4)

Experimentally observed values for the SW amplitude are, relative tothe lattice spacing, A2/a ≈ 1.7 · 10−3 and A2/a ≈ 3.5 · 10−3 for 293K and130K, respectively. We prefer the lower temperature value, resulting in A2 ∼4.5 · 10−3 Å.

(a) without SW

(b) with SW

Figure 3.5: (b) depicts the periodic distortion as is caused by the magneto-volume e�ect. It is compared to the �xed lattice (a)

3.2 Theoretical descriptions

Overhauser (1962) [11] called a static SDW in account for the AFM state inCr, instead of a modulation of localised spins. A SDW wave vector q shouldconnect two pieces of the Fermi surface. A linearly polarised SDW shouldbe lower in energy than a helical SDW, as described in 2.2. This result isin contrast to the isotropic Heisenberg model, used for localised spins, wherethe ground state is helical [17].

Lomer (1962) [13] found that the stability of the SDW state is enhancedif the Fermi surface has the so-called nesting property, i.e. electron and holesurfaces can be superposed by translation through the nesting vector q. Theshape of the Fermi surface, where the electron area is slightly larger than thehole area, is responsible for the incommensurate SDW.


3.3 Band structure calculations

Early band structure calculations were performed by Asdente and Friedel(1961) [18], Loucks (1965) [19], Asano and Yamashita (1967) [20], Rath andCallaway (1973) [21] and Laurent et al. (1981) [22], on non-spin-polarisedCr.

Early spin-polarised calculations, assuming a commensurate SDW state(AFM), were carried out by Switendick (1966) [23], Kübler (1980) [24],Skriver (1981) [25], Kulikov and Kulatov (1982) [26], Ukai and Mori (1982)[27] and Nakao et al. (1986) [28].

Considerable importance can be assigned to the calculation by Marcuset al. (1998) [29]. They used the augmented spherical-wave (ASW) method(LSDA) with the exchange-correlation potential by von Barth-Hedin. Theyfound that LSDA predicts a nonmagnetic (NM) ground state for Cr, butwhen expanding the lattice by a small portion one obtains an AFM groundstate. This detail is delicate as we know that LSDA tends to over-binding,predicting a too small lattice constant. They suggested that a strain-wavedistortion (SW), caused by a magneto-volume e�ect, could stabilise the SDWground state.

SDW with larger unit cells have �rst been considered by Hirai (1998)[30]. He used the KKR method (LSDA), non-relativistic, with 9 × 10 ×2 k points and the exchange-correlation potential by Janak, Williams andMoruzzi [31]. He performed calculations for supercells with sizes 12, 16, 18,20 and 22 (dl), and found a SDW ground state for 20 dl with ESDW −EAF =−5 µHartree/atom.

The picture seemed to �t, until new calculations by Bihlmayer and Cot-tenier emerged. Bihlmayer et al. (2000) [32] applied the FLAPW method(GGA), with the parametrisation by Perdew 91 [33], employing 6 × 6 × 1k points. They con�ned their calculations to 12 and 14 dl, due to compu-tational limitations. They obtained ESDW − EAF = +360 µHa/atom and−180 µHa, respectively, at the GGA lattice constant of 2.85 Å. They con-cluded that, by �tting the energy with a function quadratic in q, they wouldobtain a SDW ground state with q ≈ 18

19, and thus having reproduced the

Fermi surface well enough.Cottenier et al. (2002) [34] used the WIEN2k program with GGA, with

the parametrisation by Perdew et al. 96 [35], and a k-mesh of 6×6×2 points.They found an AFM ground state, with ESDW−EAF = +250 µHartree/atomfor 20 dl, thus contradicting the results by Hirai. Vanhoof (2006) [36] againused the WIEN2k code with LSDA+U, with U=0.15 Hartree, and the para-metrisation by Perdew and Wang 92 [7]. She improved the k point samplingto 21× 21× 1 and obtained ESDW −EAF = +300 µHartree/atom, again for

3.3 Band structure calculations 19

20 dl, thus con�rming the value by Cottenier et al.Those last three authors found an AFM ground state, although they ob-

tained Fermi surfaces with the correct nesting feature. This led Demangeat(2006) [37] to the conjecture, that, assuming that the DFT calculations werecorrect, nesting could not be the �ultimate driving mechanism for the occur-rence of the SDW in Cr�. Part of this work will be the question, if the valuesobtained by these authors are coarse due to limited k sampling, and if thisproblem can be discarded.

Chapter 4

Calculations

After the presentation of experimental details, this chapter focuses on ourresults. It starts with the explanation of calculational details and proceedswith the justi�cation of the speci�c parameters used. Finally the results arepresented and discussed.

4.1 Technical details

FPLO 6 is used, with the parametrisation of the exchange and correlationpotential by Perdew and Wang 92 [7]. The calculations have been performedscalar relativistically. To model the magnetic ground state of Cr, we haveemployed 3 di�erent pictures:

First, nonmagnetic (NM) Cr in a body centred cubic (bcc) unit cell con-taining one atom (space group 229, I3M3). Second, the antiferromagnetic(AFM) con�guration uses a CsCl unit cell with 2 atoms (space group 221,PM3M) having opposite collinear spins with the same absolute value. Forobtaining zero total magnetisation, a �xed spin moment procedure (FSM)has been employed. We have set up a high initial spin split (±1 µB per atom),to ensure convergence to a spin split solution with high local moments.

Third, the SDW con�guration uses supercells varying from 15 to 23 dl(space group 123, P4MMM) in order to model the SDW wave length. Closestto experiment is the wavelength of 21 dl. The moments are still collinear, butvary sinusoidally. Spin-orbit coupling has been neglected, so the direction ofthe spin moment relative to the crystal lattice has no e�ect. This is discussedin detail in section 4.1.2. The magnetisation was again stabilised by the FSMmethod. We set up a sinusoidal variation of the moments with an initialamplitude of 1 µB.

Further speci�c details concern the convergence criterion of FPLO (6.00-

22 CHAPTER 4. CALCULATIONS

24): in all three cases the density+energy convergence criterion has beenapplied. While in the NM and AFM case the energy has converged with anaccuracy of 10−8 Hartree and the density to 10−6, we were confronted withproblems in the SDW case. Energy convergence was put to 10−6 Hartree,density to 10−3 only.

4.1.1 Dependence on k-point number

In the process of calculating the total energy, one imposes an equidistantmesh onto each direction of the Brillouin zone (BZ), whereupon the bandenergies are computed. To ensure that the selection of the k-mesh is �neenough, one has to consider the behaviour of the energy with respect tovariation of the k-mesh. For the NM and the AFM calculations, the numberof k points nk has been chosen isotropically in each direction, so the totalnumber of k-points nNM

k,total = nAFMk,total is n3

k. We have considered the NM casefor checking the convergence, and applied the same nk in the AFM case. Atnk of 48 the energy convergence is su�cient at about 10−6 Hartree (Fig. 4.2).

In the SDW case where we have the nesting feature, it is important thatthe k space is sampled accurately. The BZ of the SDW states is a slice ofthe simple cubic BZ of the AFM state, we have an extremely �at z-direction.Additionally, due to the tetragonal symmetry of the SDW states, we have amirror plane in that direction. As the bands have to have zero gradient atBZ boundaries in our case and in between they have to be continuous, theyare in a good approximation cosine like. The points are Γ and H. We havechecked the convergence for nk = 32, and there is only 0.2 µHartree energydi�erence between sampling the z direction with 2 and 4 points. We haveconcluded that it is enough to sample with 2 points, nSDW

k,total = 2n2k. The k

convergence has only been considered in the 21 dl case. At nk = 24, we �ndthe energy to be accurate enough with an error of ±10 µHartree, which isenough for our considerations and limits the computational e�ort reasonably.

4.1 Technical details 23

(a) bcc unit cell (b) BZ

Figure 4.1: (a) shows the extended bcc unit cell as it is used in the supercellsetup. (b) sketches the Brillouin zone of the supercell. Notice the �at zdirection

-630

-625

-620

-615

-610E

- E

0NM

[µH

artr

ee]

NM

0 25 50 75 100n

k

-350

-325

-300

-275

-250

E -

E0SD

W [

µHar

tree

]

SDW

Figure 4.2: Energy Di�erence over k point number. The black line shows theconvergence behaviour for the NM calculation at a = 2.791 Å, the red line isthe SDW calculation with 21 dl at a = 2.884 Å. ESDW

0 = −1048.880 Hartree,ENM

0 = −1048.883 Hartree. Note that both graphs are separated from eachother and scaled di�erently.


4.1.2 Scalar vs full relativistic version

We compare the band structures for the full relativistic (fr) and the scalarrelativistic (sr) calculation to justify the use of the CPU time saving sr ver-sion. The di�erence between the band structures in Fig. 4.3 and Fig. 4.4 isonly slight. The degeneracy of one of the bands going from Γ to H is liftedand causes a small split of that band into two which are separated by anenergy in the order of a tenth of an eV. This value is explained by the atomicspin-orbit split of the Cr 3d states, E3d3/2

−E3d5/2= 78 meV. The band split

causes one band to move up in energy, whereas the other band declines, so inlowest order of energy, the split is compensated. The total energy di�erencebetween fr and sr, on the other hand, amounts to Efr − Esr ≈ −1 mHartreeonly, and a large part of this value contains a constant shift with respectto di�erent magnetic states. But the value of the split is comparable to thesplit occurring at the Fermi level in the density of states (DOS) (Fig. 4.13).Changes in higher orders of energy might result in a shift of the Fermi level.Additionally, the isotropy in the fr version is broken. It could be possiblethat the SDW shows a higher anisotropy than the AFM state. We can putthis argument forward by considering an AFM state to be a special case ofa SDW with wavelength λ → ∞, but also with λ = a

2, where λ = a

δ. In

between, if not constant, the anisotropy can be either lower or higher. Asthere is a reorientation transition in Fig. 3.3, the anisotropy should be rathersmall though. No band gaps are opened at the Fermi level in Fig. 4.4, andthe value of the band split is small. This is the reason why we have decidedto employ the sr version.

4.1 Technical details 25

Γ H N Γ P N P−2.0

−1.0

0.0

1.0

2.0

Ene

rgy

εn(

k) [

eV]

Figure 4.3: bands for NM Cr at a = 2.884 Å, scalar relativistic

Γ H N Γ P N P−2.0

−1.0

0.0

1.0

2.0

Ene

rgy

εn(

k) [

eV]

Figure 4.4: bands for NM Cr at a = 2.884 Å, full relativistic


4.2 Results and Discussion

This section starts with the two cases NM and AFM Cr, which are moderatelydemanding with respect to computational e�ort. It continues with the SDWand concomitant e�ects.

4.2.1 NM and AFM Cr

In the following the selection of the lattice parameter and the reference energyof the NM and AFM ground state is determined for further consideration.

Energy Minimum

We are interested in the LSDA ground state for Cr. As the Kohn-Shamequations depend parametrically on the nuclear positions, we have to �nd itsequilibrium volume.

We �nd a NM ground state at the equilibrium lattice parameter of a =2.791 Å, the LSDA underestimates the lattice constant by approximately3 % (Fig. 4.5). From about a = 2.86 Å on, there exists an AFM state, andfrom a = 2.87 Å on, it is lowest in energy. The energy di�erence betweenthe NM and the AFM state at the experimental lattice constant is about0.1 mHartree.

The check with literature values gives quite good agreement, Vanhoof(2006) [36], obtained a = 2.79 Å (LSDA), and Cottenier et al. (2002) [34]gave a = 2.794 Å (LSDA).

4.2 Results and Discussion 27

-4

-3

-2

-1

0

E-E

0 [mH

artr

ee]

a LSDA

= 2.791 Å a exp

= 2.884 Å

AFMNM

2.75 2.8 2.85lattice parameter [Å]

-0.2

-0.1

0

EA

FM

-EN

M [m

Har

tree

]

∆

Figure 4.5: Energy over lattice constant at nk = 48 on the left hand scale.Energies are normalised with E0 = 1048.880 Hartree per atom and �tted witha 4th order polynomial. The right hand scale shows the energy di�erencebetween the AFM and NM state.

2.8 2.85 2.9lattice parameter [Å]

0

0.5

1

mag

netic

mom

ent [

µ B]

a exp

m exp

m LDA

Figure 4.6: Magnetic Moment of AFM Cr at nk = 48


Volume dependence of the magnetic moment

Below a = 2.85 Å, a small magnetic moment appears in the AFM calculationdue to numerical �uctuation. The evaluated moment at the equilibriumlattice constant is about 0.001 µB, which corresponds to an exchange energyof 10−8 Hartree, due to ∆EXC ≈ I

2M2

µ2B

= 0.5M2

µ2B

eV [38]. This can be assumedzero as it is just at the edge of accuracy. In principle, the states we obtainbelow a = 2.85 Å, are the same in both AFM and NM calculations. Startingfrom a = 2.86 Å the solution splits into two branches when larger momentsevolve.

The magnetic moments in Cr show considerable dependence on the vol-ume. From Fig. 4.6 it is apparent that Cr at its LSDA equilibrium volumelies marginal to magnetism. Having a nonmagnetic ground state at the equi-librium LSDA volume, it becomes magnetic when increasing the volume byabout 2 %.

We �nd a moment of 0.924 µB at the experimental lattice constant. Wegive literature values, which cannot quite be compared to our results, sincethe absolute value of an antiferromagnetic moment depends on the volume ofintegration. Di�erent cuto� radii may result in di�erent evaluated moments.

Vanhoof (2006) [36] obtained 0.45 µB (LSDA), Cottenier et al. (2002) [34]gave 0.60 µB (LSDA).

Band structure and DOS

The reason for the AFM con�guration being favourable becomes clear whenwe look at the band structure. In comparison to Fig. 4.7, Fig. 4.8 shows twosplittings at the Fermi level in directions Γ → R and Γ → M . These splitslower the energy of the occupied states and raise it for the unoccupied states.There is further insight into this splittings when we take a look at the DOS(Fig. 4.10). At the Fermi level, a considerable amount of states is shifted byabout 1

20th eV towards lower energies. For higher resolution of the DOS, we

have used nk = 96 points.Fig. 4.9 shows a comparison of the DOS of NM Cr in bcc NM and CsCl

AFM con�gurations. Both densities should be the same, but the small de-viation near the Fermi level points to high sensitivity of the DOS regardingk-point sampling. Obviously even a value of 96 was not enough here.

From the DOS we can also probe for instability against ferromagneticpolarisation by checking the Stoner criterion: IN = 0.25 < 1 at 2.791 Å and0.28 < 1 (NM) at 2.884 Å, where we have taken the value by Kübler, I =14 mHartree [2]. We see that in no case it is ful�lled.


Γ X M Γ R M X R−2.0

−1.0

0.0

1.0

2.0

Ene

rgy

εn(

k) [

eV]

Figure 4.7: Bands of NM Cr in CsCl geometry at a = 2.884 Å

Γ X M Γ R M X R−2.0

−1.0

0.0

1.0

2.0

Ene

rgy

εn(

k) [

eV]

Figure 4.8: Bands of AFM Cr at a = 2.884 Å


-4 -2 0 2Energy [eV]

0

1

2

3

DO

S [s

tate

s/eV

]NMAFM

Figure 4.9: DOS at a = 2.791 Å

-4 -2 0 2Energy [eV]

0

1

2

3

DO

S [s

tate

s/eV

]

AFMNM

Figure 4.10: DOS at a = 2.884 Å


4.2.2 SDW

The calculation of truly incommensurate SDW states is beyond band struc-ture calculation, as there is no translational symmetry along the directionof the SDW propagation. So we have set up commensurate states with aperiod close to the experimental wave vector, which is approximately 21 dl.For comparison with the energies of the NM and AFM states, we have re-peated the NM and AFM calculations in the supercell geometry. It couldbe possible that the values deviate in a supercell geometry from their valuesin the primitive cells. We have found that the values remained satisfactorilyconstant, so we give only the values for the primitive cells. It is assumed thatthe SDW shows similar energy over volume behaviour to the AFM as theyboth lie energetically close. We did not perform another energy over volumetest and we have chosen the experimental lattice constant in order to forcea magnetic ground state.

SDW of di�erent periods

We have been able to perform calculations for di�erent supercell sizes upto 23 dl. The energy decreases with increasing wavelength, but it neverreaches the energy of the AFM state. The AFM state is still the groundstate solution, with an energy di�erence of ESDW − EAFM = +40 µHartreefor 21 dl (Fig. 4.11). We �nd a local energy minimum for the 21 dl state, soFermi surface nesting seems to be intact.

Puzzling is the glance at the band structure plot in Fig. 4.12. We seenumerous gaps opening at the Fermi level in directions Γ → X, Γ → Mand Z → R. This picture is con�rmed by the DOS in Fig. 4.13, where theshift towards lower energies is even greater in the SDW case, over 1

10th of an

eV. Nevertheless, this does not give us the necessary energy gain, so we haveto assume that this gain in band energy is overcompensated by Coulomb orexchange contributions.

Analysis of SDW moment distribution

The following section is dedicated to compare the shape of our SDW stateswith experimental and theoretical values from the literature. This is a goodtest for the accuracy of our calculations. We try to answer the question whichrole higher harmonics play. For the analysis, the distributions of momentsof SDWs with di�erent wave vectors have been �tted according to (3.2).Fig. 4.14(a) shows the results. Our value of P3/P1 = −7.1 · 10−2 (21 dl)is in the region of Tsunodas measurement of −2.1 · 10−2 (see 3.1.1). Hirai(1996) [30] obtained −3.7 · 10−2, Cottenier et al. (2002) [34] gave −20 · 10−2,


0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

q / (a*/2)

-380

-360

-340

-320

-300E

- E

0SDW

[µ

Har

tree

]15 17 19 21 23

double layers

AFMSDW with n/2 ∈ N

Figure 4.11: energy per atom for varying supercell sizes, q in units of a∗

2

Γ X M ΓZ R A−2.0

−1.0

0.0

1.0

2.0

Ene

rgy

εn(

k) [

eV]

Figure 4.12: Bands of 21 dl SDW


-1 -0.5 0 0.5 1Energy [eV]

0

1

2

3

DO

S [s

tate

s/eV

]

AFMNMSDW

Figure 4.13: DOS of 21 dl SDW determined at ntotalk = 48× 48× 2

both for the 20 dl SDW. It is hard to tell if our calculations reproduce Fig. 3.4.From 15 to 21 dl, it follows quite well, but the 23 dl deviates. Our calculationsdeliver a quite sinusoidal shape, without much in�uence of higher harmonics.We can state that the amplitude of the SDW is well enough determined bythe coe�cient P1 up to an accuracy of a few %. The coe�cients P1 have thetendency to raise in magnitude towards larger wavelengths, except for onedrop at 23 dl (Fig. 4.14(b)). In the AFM case, we observe a high moment of∼ 0.972 µB. This value is somewhat di�erent from the value given in section4.2.1, which was 0.924 µB, obtained from the primitive cell. Nevertheless,the energies in both cases only di�er by 1 µHartree.

CDW

Deviations from a �normal� distributed density, which allocates the samecharge around each atom, are present in the SDW case, a sinusoidal modu-lation of the charge density (CDW) emerges. Motivated by the CDW, wehave analysed the behaviour of the occupation numbers, and found smallsinusoidal variations. Our value for the SDW without SW is N2 ≈ 2 · 10−3,this compares quite well with Cotteniers et al. (2002) [34] of N2 ≈ 2.6 · 10−3,and furthermore with the experimental value, given in section 3.1.2. Hirai


0.04 0.045 0.05 0.055 0.06 0.065 0.07 δ

0.03

0.04

0.05

0.06

0.07|P

3 / P 1|

1517192123double layers

(a) |P3/P1| over δ

0.94 0.95 0.96 0.97 0.98 0.99 1

q / (a*/2)

0.7

0.8

0.9

1

P1

[µB]

15 17 19 21 23double layers

(b) P1 over q

Figure 4.14: (a) shows the ratio (absolute value) of the 3rd harmonic overthe fundamental SDW, |P3/P1| over δ. The error bars are from the �ttingprocess, and have further been derived under consideration of propagationof uncertainty: P3 and P1 are uncorrelated as their referring functions areorthogonal, so the relative error is ∆(P3/P1)

P3/P1= ∆P3

P3− ∆P1

P1. (b) displays the

amplitudes of the fundamental SDW P1 over q. The moment appears to beconstant from 15 to 21 dl, but then drops at 23 dl. Again, the error barsresult from �tting.


(1997) [39] obtained N2 ≈ 0.6 ·10−4. The variations seem to be compensatedin the SW case, as in Fig. 4.15.

0 0.1 1 10 100A

2 [10

-3 Å]

-0.005

-0.004

-0.003

-0.002

-0.001

0

N 2

Figure 4.15: Occupation number wave in 21 dl supercell geometry. Shownis the coe�cient N2 over di�erent strain wave amplitudes. In the AFMsupercell, we have also observed a small N2. We perceived this as numerical�uctuation and included it into the error. Further the error from the �ttingprocess enters the total error. The red data point has been calculated at zerodistortion, this value has been denoted as 0 on the logarithmic scale.

SDW with SW

We were hopeful that the application of a lattice distortion would stabilisethe SDW and provide additional energy gain.

Unfortunately there is no such energy gain in a large area around theexperimental value (Fig. 4.16(a)). Even the reversal of the sign of the ampli-tude, meaning small lattice spacing at a SDW anti-node has no big in�uence.It was expected that this should raise the energy. We rather see a symmetricshape. The DOS picture in Fig. 4.16(b) does not help out either, as the shiftsare marginal.


1 10 100 1000SW amplitude [10

-4Å]

-400

-350

-300

-250

-200

-150

-100E

nerg

y [µ

Har

tree

]

A2 exp

SW + 21 dlSW - 21 dl

(a) Energies of SW

-1 -0.5 0 0.5 1Energy [eV]

0

1

2

3

DO

S [s

tate

s/eV

]

SW 10.5SW 0.5SDW

(b) SW DOS

Figure 4.16: (a) shows the energies for di�erent SW amplitudes, for bothpositive (black) and negative (red) A2. The experimental SW value is denotedas Aexp. SW have only been considered in the 21 dl case. The error barsare due to k sampling. (b) compares the DOS for 2 SW cases, one with anamplitude of 10.5 ·10−3 Å, the other one at 0.5 ·10−3 Å, with the undisturbedDOS

Chapter 5

Summary and Outlook

We are left with the conclusion that the LSDA, and methods beyond, asthe literature points out [34], fails to describe the magnetic ground state ofchromium correctly. There are three main points, where this error couldbe addressed to. The �rst point is that the resolution of the BZ was notsu�cient. But the careful consideration of energy convergence behaviourwith respect to k-point number was able to rule that possibility out.

The second point, as earlier mentioned by [29], is the supporting mech-anism of the SW, which would be able to deliver an additional energy gain.Nevertheless, the calculations performed over a wide range of SW amplitudesdo not give any hint on such a mechanism.

The last point is the missing spin-orbit coupling in our calculations. Thejusti�cation was a little bit un�rm. Additionally, the magnitude of the split iscomparable to the energy shift at the Fermi level. These calculations requirea high amount of computing time, and the results have not yet been obtained.They are left open for future discussions.

It cannot be ruled out at the moment, that even full relativistic LSDAmay fail to yield the correct magnetic ground state of Cr. However, the localenergy minimum for 21 double layers shown in Fig. 4.11 indicates that Fermisurface nesting seems to be operative.

Bibliography

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[2] J. Kübler, Theory of itinerant electron magnetism, Oxford SciencePublication, New York, 2000.

[3] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).

[4] M. Levy, Phys. Rev. A 26, 1200 (1982).

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[8] D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).

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[10] R. E. Peierls, Quantum theory of solids, Oxford University Press, 1955.

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[12] C. Y. Young and J. B. Sokolo�, J. Phys. F-Metal Phys. 4, 1304 (1974).

[13] W. M. Lomer, Proc. Phys. Soci. 80, 489 (1962).

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[15] L. Néel, C. R. Acad. Sci. 203, 304 (1936).

[16] C. G. Shull and M. K. Wilkinson, Rev. Mod. Phys. 25, 100 (1953).

[17] A. Yoshimori, J. Phys. Soc. Jpn. 14, 807 (1959).

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[18] M. Asdente and J. Friedel, Phys. Rev. 124, 384 (1961).

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[20] S. Asano and J. Yamashita, J. Phys. Soc. Jpn. 23, 714 (1967).

[21] J. Rath and J. Callaway, Phys. Rev. B 8, 5398 (1973).

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[28] S. O. Nakao, H., J. Magn. Magn. Mater. 54-57, 951 (1986).

[29] P. Marcus, S. Qiu, and V. Moruzzi, J. Phys.-Cond. Mat. 10, 6541 (1998).

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[31] J. F. Janak, A. R. Williams, and V. L. Moruzzi, Phys. Rev. B 11, 1522(1975).

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BIBLIOGRAPHY 41

[38] M. Richter, Handbook of Magnetic Materials Vol. 13, Elsevier ScienceB.V., 2001.

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Acknowledgement

I am grateful to Prof. Dr. H. Eschrig, director at the �Leibniz Institute forSolid State and Materials Research Dresden� for o�ering me the possibilityto write my diploma thesis under his guidance, for the scienti�c managementand support.

I am deeply indebted to my supervisor Dr. M. Richter, head of the group�Numerical Solid State Physics and Simulation�, for always being open todiscussion, proposing ideas, and encouraging me.

I would like to thank all members of our working group, especially Dr.K. Koepernik for opening my mind to technical details, and U. Nitzsche whogave me Linux skills where there haven't been any before.

Finally I should mention my parents and friends who supported methroughout the years.

Eidesstattliche Erklärung

Ich versichere, dass ich die vorliegende Arbeit ohne unzulässige Hilfe Drit-ter und ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigthabe. Aus fremden Quellen übernommene Gedanken und Passagen sind alssolche kenntlich gemacht. Diese Arbeit wurde in gleicher oder ähnlicher Formnoch keiner anderen Prüfungsbehörde vorgelegt.

Mathias Bayer

spin density wave in chromium - ifw dresden · addition we allow for an external magnetic eld...

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