Acknowledgement

First of all I would like to thank Dr. Bongjae Kim for his support and I hope that hecan pursue his further career surrounded by his family. I also want to thank MicheleReticcioli, Michael Poeltl and of course Prof. Franchini also for his support and thesuper topic.

In addition I would like to thank all my friends and the whole Dobrovit'schen com-munity. I especially hope that the newest members who joined the club recently willkeep their interests in nature and science and �nally I want to mention the personI am grateful the most, Roberto.

1

Abstract

Transition metal oxides are often subject of studies and there are several goodreasons for that. The physics is extremely rich including multiferroicity, colossalmagnetoresistance and even high-Tc superconductors are possible. But an ongo-ing challenging task are certainly metal-to-insulator transitions. Especially whenit comes to heavy transition metal oxides with enhanced spin orbit coupling novelquantum states appear that are totally di�erent from the already well analyzedones. This work focus on NaOsO3 which is a magnetic Lifshitz insulator and herethe possibility to induce an insulator-to-metal transition upon doping is studied. Thecalculations are fully ab initio and by the gradual substitution of sodium with mag-nesium atoms, an insulator-to-metal transition at zero temperature was induced.To analyze the e�ect of additional electrons the band structure for di�erent dop-ing concentrations was computed. To maintain the periodicity of compounds withimpurities the supercell approach is required. But as the supercell in reciprocalspace is smaller than the original one, the obtained band structure is often hardto interpret and di�cult to compare with angle resolved photo emission (ARPES)results. Hence, all band structures are �unfolded� to allow a clear interpretation. Foreach doping concentration several arrangements have been examined to guaranteemeaningful results. Finally, the insulator-to-metal transition is discussed in terms ofmagnetism, structural features and spin orbit coupling.

2

Zusammenfassung

Verschiedenste Übergangsmetalloxide sind oftmals Gegenstand wissenschaftlicherUntersuchungen, da ihre physikalischen Eigenschaften sehr vielfältig sind. Dazuzählen: Mulitferroika, Materialien, die den kolossalen Magnetowiderstand aufweisen,aber auch Hochtemperatursupraleiter sind möglich. Obwohl bereits groÿe Erfolgebei der Untersuchung genannter Eigenschaften erzielt wurden, bleiben die Metall-Isolator Übergänge eine immer währende Herausforderung. Insbesonders spannendsind Übergangsmetalle mit schweren Atomen, da hier die Spin-Bahn Kopplung er-höht ist und die Quantenphysik komplett neue Eigenschaften zum Vorschein bringt.

In dieser Arbeit wird untersucht, ob das Übergangsmetalloxid NaOsO3, das zurKlasse der magnetischen Liftshitz-Isolatoren gehört, sich durch die sukzessive Erset-zung von Natriumatomen mit Magnesiumatome von einem Isolator in einen Leiterüberführen lässt. Um festzustellen, wie zusätzliche Elektronen die Eigenschaftendes Festkörpers verändern, wurde im Grundzustand die Bandstruktur berechnet.Nachdem durch die Substitution mit Magnesiumatomen die Periodizität der ur-sprünglichen Zelle gestört wurde, musste eine Superzelle erstellt werden, die allerd-ings den Nachteil besitzt, im reziproken Raum kleiner zu sein als die ursprünglicheprimitive Zelle. Die Bandstruktur lässt sich daduch nur sehr schwer mit experi-mentellen Ergebnissen vergleichen, welche zum Beispiel mit ARPES erzielt wurden.Mit Hilfe der Unfolding-Technik ist es jedoch möglich, die Bandstruktur der Su-perzelle in die der primitiven Zelle zu entfalten, wodurch eine einfache Interpretationmöglich war.

Wie sich die zusätzlichen Elektronen im System auf den Isolator-Metall Übergangauswirken wird in Hinsicht auf Magnetismus, Struktur der Zelle sowie Spin-BahnKopplung diskutiert, weshalb pro Doping-Konzentration verschiedene Anordnungender Magnesiumatome analysiert wurden.

3

Contents

1 Introduction 6

2 Hubbard model 8

2.1 The strong coupling limit . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 The half-�lled system . . . . . . . . . . . . . . . . . . . . . 10

2.1.2 The not half-�lled system . . . . . . . . . . . . . . . . . . . 10

2.2 The weak coupling limit . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 The Hubbard subbands . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 The Mott-Hubbard transition . . . . . . . . . . . . . . . . . . . . . 13

3 Density Functional Theory 17

3.1 The Many Body Problem . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Hohenberg-Kohn theorem . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 The Kohn-Sham scheme . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.1 Self consistent Kohn-Sham scheme . . . . . . . . . . . . . . 26

3.4 The Exchange-Correlation potential . . . . . . . . . . . . . . . . . . 27

3.4.1 Local density approximation (LDA) . . . . . . . . . . . . . . 27

3.4.2 Perdew-Burke-Ernzerhof (PBE) functional . . . . . . . . . 28

3.5 GGA+U method (Dudarev) . . . . . . . . . . . . . . . . . . . . . . 28

3.6 DFT in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 The Vienna Ab initio Simulation Package 31

4.1 Pseudo potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Bloch theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Plane wave basis set . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3.1 Kohn-Sham equations with plane wave basis set . . . . . . . 39

4.4 Electronic ground-state . . . . . . . . . . . . . . . . . . . . . . . . 40

4.5 Smearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Supercell approach 43

5.1 Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . 43

5.2 Reciprocal space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4

6 E�ective band structure 46

6.1 Folding and unfolding of wave vectors . . . . . . . . . . . . . . . . 47

6.2 Folding and unfolding of states . . . . . . . . . . . . . . . . . . . . 48

7 NaOsO3 51

7.1 Structural features . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7.2 Electronic features . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8 Computational setup 57

9 Results 59

9.1 Undoped case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

9.2 Doped samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

9.3 Band structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

9.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

10 Conclusion 71

5

1 Introduction

Transition metal compounds form certainly an interesting class of materials. Eachdi�erent combination of atoms might exhibit di�erent physics. Some of them aregood metals, others insulators and then there are ones where a metal-to-insulatortransition occurs e.g. by varying the temperature. Also magnetism plays an im-portant role in transition metals which leads to e�ects like high temperature super-conductors, multiferroicity or colossal magnetoresistance. Until present the focusfor applications lied on these magnetic properties but recently also the electronicproperties slipped into the center of attention. The description of the electronicstructure needs a speci�c treatment di�erently from conventional metals, insula-tors or semiconductors in band theory. In order to model the realistic situation itis inevitable to include the electron correlation, called the on-site Coulomb poten-tial because the peculiar characteristics are often directly linked to these electronicfeatures from the atoms that build the solid.

Among transition metals quite often the perovskite structure appears, de�ned asABX3. A is a large cation, B is also a cation but smaller than A and X is an anionlike oxygen. The perovskite structure can also be distorted leading to a modi�cationof electronic and magnetic features. For analyzing the functionalities the band gapis essential and for proper results of course the exchange and correlation energyhas to be included. The exchange energy and Pauli's exclusion principle was �rstcorrectly implemented within the Hartree-Fock approximation but a drawback is thatthe correlation energy is not comprised. It was the Hohenberg-Kohn theorem thatseemed to be the loophole for calculations with a vast amount of electrons as it statesthat the knowledge of the ground-state energy is su�cient to determine all otherphysical quantities. The Kohn-Sham scheme �nally provides the practical schemefor realistic calculations and also includes the correlation energy in an approximatedmanner. Therefore density functional theory is a widely used and powerful toolthese days.

For a �rst hint about the relationship between the correlation strength, bandwidthand spin orbit coupling a look into the periodic table is advisable.

6

Figure 1: D-block in the periodic table.

Transition metal elements are arranged in the d-block in the periodic table, (see�gure 1 taken from ref. [1]) with 3d metals starting from the left side on the topof the block and �nishing with 5d metals on the right side of the bottom. Thevalue of the correlation strength is following the same path. 3d metals have astrong on-site Coulomb repulsion and a small bandwidth, whereas 5f metals showthe opposite behaviour, a small on-site Coulomb repulsion and a large bandwidth.For 5d elements to whom belongs NaOsO3 and which are located in the middle ofthe block an on-site Coulomb repulsion that is in the same range of the bandwidth istypical. First of all 5d-orbitals are in general more extended compared to 3d-orbitalsand secondly the core orbitals screen the nuclear charge, both e�ects leading to ahigher degree of freedom. Plus, going down the periodic table means an ascendingatomic number which results in a stronger spin orbit coupling. The balance betweenthese quantities: on-site Coulomb repulsion, bandwidth and spin orbit coupling arecrucial for the band structure and replacing one atom with another one in thecompound might change this balance and hence the band structure. Especiallyborderline cases like NaOsO3 are suitable candidates for doping.

The theory behind the on-site interaction is typically explained within the Hubbardmodel but to incorporate it within band theory a step beyond density functionaltheory is necessary where the value of the correlation strength is comprised byadding it separately for several electrons (DFT+U sheme after Dudarev). The thesisis in the following way organized. The Hubbard model (chapter 2) is followed by anintroduction to density functional theory and beyond (chapter 3). As all calculations

7

herein were performed with the Vienna Ab initio Simulation Package (VASP) it willbe represented in chapter 4. The methodologies are explained in chapter 5 and 6.In chapter 7 the features of NaOsO3 are discussed, including a small insight into theproposed insulator-to-metal transitions, the computational setup is brie�y describedin chapter 8. In chapter 9 the results i.e the band structures are analyzed includingthe insulator-to-metal transition and �nally in chapter 10 summary and conclusionsare provided.

2 Hubbard model

Albeit all calculations herein are based on density functional theory (DFT) or ratherbeyond DFT, it is worth to have a look into the model Hamiltonian for strongcorrelated materials where the interaction between fermions are factored in. Hencein the following sections will be outlined not only the model Hamiltonian but alsohow to interpret the density of states and how this knowledge can be used tointroduce a metal-to-insulator transition. Last but not least some practical examplesare given.

The Hubbard model [2] is the appropriate model Hamiltonian to get an insightinto the richness of phenomena materials can exhibit from this class of compounds.Here, the many body problem gets simpli�ed with only considering a few bandsclose to the Fermi energy and the Coulomb interaction is screened i.e. only very�short-range� interaction is comprised. This is called the on-site interaction, the�price� unlike electrons have to pay when sitting on the same site. The other partof the Hubbard model contains the single particle part and represents the kineticenergy electrons can get due to the delocalization when hopping on a lattice. Thisis called the hopping part. For an illustration let's imagine a grid like in �gure 2(taken from ref. [3]) which is quite a simpli�cation of the problem.

Figure 2: Electron hopping in the Hubbard model.

The spin down and spin up particles are the red and blue arrows respectively, thevertical arrows imply the direction the electron can jump to. During the hoppingprocess the spin does never change. The nearest neighbours from the spin downparticle in the middle of �g. 2 could make a jump to the left or to the right butthe two situations are di�erent because on the right side, in addition the on-site

8

��ne� is due. Whether a compound is a metal or an insulator depends on the ratiobetween the hopping and the on-site parameter as well as the lattice which de�nesbasically the magnetic properties. For a larger t than U the electrons are hoppingand the state is metallic (�gure 3 left panel). For a larger U than t the electronswill remain in their single position (see �g. 3 right panel) and the state is insulating(both �g. taken from ref. [3]).

Figure 3: The left panel shows a metallic state within the Hubbard model the rightone an insulating state.

From a mathematical point of view, the Hubbard model is similar to the tightbinding model [4]. Even though the model looks simple with the Hamiltonian

H = −t∑〈i,j〉σ=↑,↓

(c†σicσj + c†σjcσi

)+ U

∑i

n↑in↓i, (1)

analytical solutions are only available in the one dimensional case with the Betheansatz and in in�nite dimensions with the dynamical mean-�eld theory [5, 6]. Inthe �rst part of eq. (1) t denotes the hopping parameter between adjacent sites iand j, the probability that an electron on site i is hopping to the adjacent site j.c†iσ creates an electron on site i with spin σ, ciσ destroys on site i an electron withspin σ. Therefore, c†iσcjσ describes the electron hopping between the sites j and i,whereas c†jσciσ describes the hopping between the sites i and j. nσ is the numberoperator

nσ (r) = c†σicσi (2)

and 〈, 〉 denotes from now on the nearest neighbour sites. Due to Pauli's principlethe number operator is nσ = 0, 1 or n2σ = nσ and hence there exist four possiblestates at each site: |0〉, | ↑〉, | ↓〉 and | ↑↓〉, i.e. the two states, |0〉 and | ↑↓〉 arespin singlets with S = 0. Following relations are satis�ed:

{cσi, c

†σ´j

}= δσσ′δij (3)

{cσi, cσ′j} ={c†σi, c

†σ′j

}= 0. (4)

9

The two limits of the one dimensional Hubbard model are considered more closely:

1. The strong coupling limit, i.e. U � ∞

2. The weak coupling limit, i.e. U � t.

2.1 The strong coupling limit

An important special case represents the half-�lled system for U>0.

2.1.1 The half-�lled system

An in�nitely large U maximises also the spin S = |σi,j | cf. eq. (1) because theelectrons are forced into a single occupied state like in �g. 4 (taken from ref. [5])left panel. That is why in the special case of the half-�lled system real hoppingis forbidden and only virtual hopping is allowed (middle panel of �g 4). In theseintermediate states there are spin up electrons hopping to the site of a spin downelectron and the other way round leaving some sites empty. The double occupiedstates are a magnitude U higher in energy than the single states. Eventually thismeans that the state with altering spin ups and spin downs are lower in energy andcan therefore be expected as ground-state. The hopping part is interpreted as asecond order perturbation leading to the e�ective Hamiltonian:

Heff = const.+4t2

U

∑〈i,j〉

Si · Sj . (5)

Therefore the half-�lled system leads always to an antiferromagnetic insulating state.

Figure 4: Virtual electron hopping.

2.1.2 The not half-�lled system

When U�t it costs a lot of energy to have two electrons on the same site andhence these states are forbidden. On the contrary, holes which carry charge but no

10

spin can move through the lattice. In this case, the e�ective Hamiltonian is the t-jmodel

H = −t∑〈i,j〉σ=↑,↓

(c†σicσj + c†σjcσi

)+

2t2

|U |∑〈i,j〉

Si · Sj . (6)

The condition ∑σ=↑,↓

c†σicσi = ni = 0, 1 (7)

guarantees that double occupied states do not appear. As already seen in �gure 4if a hole is moving through a lattice the spin con�guration can change and a longrange order might be destroyed when enough holes exist.

2.2 The weak coupling limit

In the weak coupling limit U�t the focus lies on the hopping part which is re-sponsible for the band structure. The bandwidth W is not only proportional to thehopping parameter t but also to the coordination number z, the number of nearestneighbours.

Now let's imagine that there are M sites with n electrons with orbital energy ε onthe grid as in �gure 5 (taken from ref. [7]), the initial energy for the system is

Ei = Mnε. (8)

When one electron jumps from one site dn−1 to another one dn+1 cf. �gure 5, the�nal state is

Ef = (M − 2)nε+ 1 (n− 1) ε+ 1 (n+ 1) ε. (9)

The energy di�erence is4E = Ef − Ei = 0 (10)

because U is zero and therefore hopping does not cost any energy leading to ametallic state! This situation is called the Fermi sea.

11

Figure 5: Electron hopping between two sites.

2.3 The Hubbard subbands

When the hopping t as well as the intra Coulomb force U are both switched on,interesting cases can occur! For a simpli�cation the situation of a so-called Fermisea, where one electron was hopping leaving a hole on its inital site and payingthe price of U on the new site, is assumed. This leads to the density of states asdepicted in �gure 6 (taken from reference [7]).

Figure 6: Hubbard subbands.

The Hubbard subbands consists of a lower (LHB)- and an upper (UHB)- Hubbardband. The gap is denoted by Eg, the distance between the middle of both subbandsis U. The reason for this can be understood by looking into the initial and �nal states

12

on each site when there is a U included. For one electron on each site the initialand �nal states are:

Ei = (M − 1)

(nε+

n (n− 1)

2U

)+

((n− 1) ε+

(n− 1

2

)U

)(11)

Ef = (M − 1)

(nε+

n (n− 1)

2U

)+

((n+ 1) ε+

(n+ 1

2

)U

)(12)

and the energy di�erence isEf − Ei = U. (13)

This result does also not change even if there are n electrons on each site, theenergy di�erence is still U. The left panel of �g. 6 shows an energy gap Eg = U-W.If U∼W, Eg∼U and the band gap is vanishing. Uc denotes the critical value for Uwhen the bands are touching each other. Also, as the hopping is described by bandtheory the width W of the bands gets broaden ∼2zt, z is the amount of nearestneighbours. Without band theory the energy spectra would be discrete.

Even though this example was quite a simpli�cation of the Hubbard bands it can beused for all kinds of combination with spin up, spin downs and existing paramagneticmixtures of them.

Finally it is to mention that the gap in �gure 6 has nothing in common with thegap from a semiconductor within band theory. Both bands have only space forL electrons with L denoting the number of lattice sites. If and only if the lowersubband has more than L spins, the upper subband becomes occupied because twoelectrons are sharing a site. A band insulator can accommodate 2L electrons withunlike spins according to Pauli's principle and a half �lled band means a metallicstate. Hence the Hubbard subbands clearly describe a correlation e�ect and is notan independent electron theory.

2.4 The Mott-Hubbard transition

The Mott-Hubbard transition, where a metal-to-insulator transition is observed,happens at U/t∼1 or rather can be interpreted in terms of the bandwidth W. Atexactly the half-�lled case, W=2zt the bands are touching each other. To inducenow an arti�cial metal-to-insulator transition two things can be done:

1. decrease U

2. increase t.

The �rst option can be realized by carrier doping. Due to additional electrons U isexpected to decrease [8]. Secondly the hopping can be enhanced. Sometimes thiscan be done by applying pressure or strain to reduce the lattice constant. Hence,

13

the overlap is higher and the hopping as well. Even though the second methoddoes more change the kinetic part in eq. (1) there might also be a decrease of Usimultaneously but this e�ect is still a topic of current research.

When doping a transition metal oxide also a possible involvement of the oxygenatoms have to be checked. This can be done by looking into the density of states.Figure 7 (taken from ref. [7]) shows three density of states diagrams for insulators:�gure (B) the Mott-Hubbard insulator, �g. (C) the charge-transfer insulator and�g. (E) depicts a covalent insulator. The �gure shows the energies on the verticalline, the broader density of states belong to the oxygen band. Furthermore theoxygen and the lower Hubbard band are in all scenarios full. The charge-transferenergy is denoted as ∆ and de�nes the energetic di�erence between the oxygenand d-orbitals. If the oxygen atom is a contributive factor or not depends on thequantities ∆, U and W.

Figure 7: Density of states of various metals and insulators.

Mott-Hubbard Insulator (�g. 7 B): ∆ > U

Figure 8 (taken from ref. [3]) shows the virtual hopping between the Hubbard bandsand oxygen. As ∆ is larger than U the ground-state of the system is the exchange

14

between the oxygen and the lower Hubbard band. Therefore the lowest excitedstate is, if the electron gets transfered straight from one transition metal atom tothe other one. The gap is determined by the factor U and described in eq. (1). IfW>U the insulator turns into a d-metal like depicted in �gure 7 (A). LaMnO3 is anexample for a Mott-Hubbard insulator [9].

.

Figure 8: Virtual hopping Mott-Hubbard insulator.

Charge-transfer insulator (�g. 7 C): ∆ < U

Figure 9: Virtual hopping charge-transfer insulator.

The upper and lower Hubbard bands are far away from each other in such a way thatthe oxygen 2p states are in between. The virtual hopping for a half-�lled system in�gure 9 (taken from ref. [3]) shows that to reach a conducting state an electrontransfer from the oxygen to the transition metal must be achieved. The gap isdetermined by the factor ∆ and not by U. If W becomes larger than ∆ the oxygenstates are overlapping with the upper Hubbard band, �g. (D). Here, the transfervia the oxygen can of course not be neglected that is why the Hubbard-Hamiltonianneeds to be equipped with the p-d interaction. The ground-state would be thevirtual hopping between the lower Hubbard and the oxygen band which leads inthe half-�lled case also to an antiferromagnet. Even though the ground-state ofthe charge-transfer insulator is probably the same as it is for the Mott-Hubbardinsulators, the transport properties are di�erent. An example for a charge-transferinsulator is LaMO3 [9].

15

Covalent insulator (�g. 7 E): ∆ < U

It might happen that the compound shows insulating behaviour even though theparameters are in the range W>∆<U. Here, there are no free electrons for theconduction left because they all get shared by covalent bonds.

16

3 Density Functional Theory

Density functional theory established by Hohenberg and Kohn argues that all prop-erties of an interacting particle system are determined by a functional of the ground-state density. This matter of fact enables a solution to the complicated many bodyproblem with more than a few of electrons. How to approximate these functionalsfrom a computational point of view is written in the Kohn-Sham ansatz. The ansatzintroduces an auxiliary system with non-interacting particles in order to replace theoriginal interacting one. All interactions are con�ned in an exchange-correlationfunctional. In addition the self-consistent Kohn-Sham scheme is represented whichis ready to be implemented in routines. Strong correlated materials, a feature thatis often observed in transition metal oxides, require in density functional theorya special treatment to achieve results that concur with experiments. Finally, thischapter will be closed with a short practical guide what parameters are importantfor calculations.

3.1 The Many Body Problem

For solids with a maximum of one-two thousands of atoms single-electron theoriescan be used. These single-particle approaches have to be embedded into the manybody Hamiltonian H to solve the Schrödinger equation

EΨ = HΨ (14)

with the wave function Ψ and energy E. The many body Hamiltonian for a systemwith n electrons and m nuclei reads as follows:

H = Tn + Tp + Vee + Vext + Vpp. (15)

The operators in eq. (15) on the right hand side are: the kinetic energy for eachelectron Tn, the kinetic energy for each nuclei Tp, the electron-electron repulsionpotential Vee, the potential for the Coulomb attraction between electrons and nucleiVext and the Coulomb repulsion between the nuclei Vpp. This number of terms isindeed problematic and needs a simpli�cation because analytical solutions are justfor a few cases available [9]. The Born-Oppenheimer approximation is often anadequate choice in order to simplify calculations [10]. It is also called the adiabaticapproximation because electrons respond to oscillations of the nuclei quite instan-taneous and hence the movement between electrons and protons can be decoupled.The nuclei is in its rest frame and the motion of the electrons will be dependenton the position from the nuclei. All in all, three out of the �ve terms in eq. (15)are left by virtue of these assumptions. The second term, the kinetic energy of thenuclei is zero due to the slow dynamic of the nuclei and the �fths term dependsonly between the distance of the nuclei and contibutes only a �xed value. Theseassumptions reduce the Hamiltonian to

17

H = T + Vee + Vext, (16)

the n in the kinetic part for the electrons is from now on dropped. The speci�coperator for the kinetic energy is

T = − ~2

2m

n∑i=1

∇2i , (17)

i refers to the number of the electron and n is the total amount of electrons. Thelast term in eq. (16) Vext de�nes the electron interaction within a uniform �eld ofnuclei

Vext = − 1

4πε0

m∑l=1

n∑i=1

Zle2

|ri − rl|(18)

here again, the number i declares the number of each electron and l the numberof each nuclei, the distance between them is in the denominator. The interactionbetween the electrons and the nuclei is caused by an external quantity, the ion, thatis why it is denoted as Vext.

Now comes the tricky part of the Hamiltonian, the electron-electron interaction

Vee =1

2

n∑i,j=1;i 6=j

e2

|ri − rj |. (19)

The factor 1/2 in eq. (19) comes from the fact that two electrons interacting geteach punished by the half of repulsion energy. Unfortunately eq. (19) includesneither that fermions are indistinguishable nor Pauli's exclusion principle which willboth turn out to have a large e�ect on determining the properties of a solid. TheHartree-Fock approximation deals with these problems [4]. The electrons are stillinteracting via Coulomb repulsion with each other and also the uniform backround ofions is kept but it incorporates the possibility that during an interaction the electronscan change places or not. Upon exchange of the electron coordinates the wavefunction must get a minus sign to ensure that the wave function is antisymmetric.That is why the interactions have to be summed up. The essence of the Hartree-Fock approximation is the way the single particle wave function |Ψ〉 is written. It isthe product of an orbital part |Φ〉 and a spin part |σ〉

|Ψ〉 = |Φ〉|σ〉. (20)

To create a many-particle wave function the single particle wave functions are writ-ten as Slater determinant

18

Ψ =1√n!Det

∣∣∣∣∣∣∣∣∣∣Ψ1(1) Ψ2(1) · · · Ψn(1)

Ψ1(2). . .

......

. . ....

Ψ1(n) · · · · · · Ψn(n)

∣∣∣∣∣∣∣∣∣∣. (21)

The prefactor in (21) ensures the normalization of the wave function. With theSlater determinant the expectation value in �rst quantization without Vext can becalculated:

E = 〈Ψ|H|Ψ〉

=∑i,j

〈Ψi|−~2∇2

2m|Ψj〉+

1

2

∑i,j

〈Ψi|〈Ψj |e2

|ri − rj ||Ψj〉|Ψi〉︸ ︷︷ ︸

direct energy

−1

2

∑i,j

〈Ψj |〈Ψi|e2

|ri − rj ||Ψj〉|Ψi〉.︸ ︷︷ ︸

exchange energy

(22)

The last term in eq. (22) is the exchange energy where the fermions exchangedplaces and is a pure quantum mechanical e�ect. But there is more. The orbital part|φ〉 of the wave function can be even or odd. An even orbital requires an antisym-metic spin function |σA〉 and an odd orbital requires a symmetric spin function |σS〉.Figure 10 (taken from ref. [12]) shows the probability density of an even |Φeven|2(upper panel) and odd orbital part |Φodd|2 (lower panel) of the wave function. Thedots represent the nuclei.

Figure 10: Molecule with two protons.

19

In the upper panel in �g. 10 the electrons are more located between the nucleileading to an antisymmetric spin function whereas in the lower panel the electronsare located closer to the nuclei, here the spin function is even. The energy isreduced for an even orbital because the two electrons attract protons more towardseach other which compensates the electron-electron repulsion better. The orbitalsare also more overlapping. All in all this tends to result in an antiferromagneticordering because the spins are �locked� into opposite directions. Upon doping andtherefore changing the electrostatic forces with extra or less electrons the system canbe manipulated. For example electrostatics forces the orbital to be antisymmetricto lower its energy and therefore the spin part has no other choice then to besymmetric, making the material a ferromagnet. This spin magnetization is done bythe exchange interaction alone without any magnetic �eld!

Even though the exchange term is accurate in the Hartree-Fock approximation, thecorrelation energy is missing. The motion of each atom depends on the actualposition of surrounded electrons, they are correlated with each other but not by astatic average �eld of electrons. This leads to wrong results. But there is another bigdrawback. To solve the problem for n interacting electrons in an external potentialrequires quite a large number of plane wave states nk. The size of the Hilbert spaceis given by how many number of ways n electrons can be put into nk �boxes�. Thatresults in a factor

nk!

n! (nk − n)!(23)

growing factorially. Therefore this wave function method can become very cumber-some when calculating the expectation values in �rst quantization. Another strategyis welcome in order to include the electron-electron interaction. This is providedwith the density functional theory.

3.2 Hohenberg-Kohn theorem

In last section it was explained why a new strategy is necessary in order to solve themany-body problem in an accurate way. This section starts to introduce the conceptof density functional theory which is based on the Hohenberg-Kohn theorem [10].This theorem looks simple but it circumvents that the three dimensional Schrödingerequation dependent on 3n variables, for n electrons, has to be calculated. Instead,the electron density is used as a basic variable. This simpli�es the problem immenselybecause the density is a scalar with three variables. Also it turns out that theknowledge of a systems ground-state density is su�cient to determine all otherphysical quantities. Starting point for the Hohenberg-Kohn theorem is again theHamiltonian from eq. (16),

H = T + Vext + Vee

again with the operator for the kinetic energy T for all electrons i

20

T = − ~2

2m

n∑i

∇2i ,

the interaction between the electron and the nuclei Vext

Vext = −m∑l=1

n∑i=1

Zl|ri − rl|

and the electron-electron interaction

Vee =1

2

∑i6=j

e2

|ri − rj |.

Now let's imagine that the Schrödinger equation (eq. (14)) for this Hamiltonian issolved and the eigenstates Ψi are known. Then the expectation value 〈Ψi|A|Ψi〉for any operator A can also be determined. But it is a matter of fact, that theHamiltonian determines the expectation value of all subsequent calculations. Theeigenstates Ψi can be replaced by the density operator ni to calculate any expecta-tion value 〈ni|A|ni〉. As a result the Hamiltonian must also de�ne the ground-statedensity. This is basically what the Hohenberg-Kohn theorem states: the expectationvalue of any operator A is a unique functional A(n0(r)) of the ground-state densityn0. Since the kinetic energy operator T and the electron-electron interaction Veeare the same for all systems, the di�erence between ground-state densities is de�nedby the external potential. The point of the Hohenberg-Kohn theorem is that thisis also valid for the other way round: once the ground-state density for an externalpotential is found the according ground-state Hamiltonian H0 is determined. Eventhough it is a ground-state theory it can also provide information on the excitedstates but without a formal mathematical and physical basis. What is need to beproofed is to show that the solution of the Hamiltonian in the Schrödinger equation

H0|Ψ0〉 = Vext|Ψ0〉 = ε0|Ψ0〉 (24)

is unique, meaning that di�erent external potentials lead always to di�erent ground-state densities.

The formal proof of the Hohenberg-Kohn theorem goes by contradiction. The �rstassumption is that two potentials Vext and V

′

ext lead to di�erent ground-state wavefunctions Ψ0 and Ψ

′

0. The two Schrödinger equations are

(T + Vext + Vee) Ψ0 = ε0Ψ0 (25)

(T + V

′

ext + Vee

)Ψ

′

0 = ε′

0Ψ′

0. (26)

21

For the contradiction it is assumed that these two di�erent potentials lead to thesame ground-state wave function Ψ0. That is why eq. (26) is subtracted from eq.(25), yielding:

(Vext − V

′

ext

)Ψ0 = (ε0 − ε′0) Ψ0. (27)

Comparing the left and right hand side of eq. (27) reveals that the potentialsdi�er only by a real number and as the reference point of the potential can bechosen freely this goes against the contradiction. Yielding that if Vext 6= V

′

ext thenΨ0 6= Ψ′0. Now, it has to be shown that this also applies for the ground-statedensities n0 (r) 6= n′0 (r) with

no (r) =

ˆΨ∗0 (r1, . . . , rn)

n∑i

δ (r − ri) Ψ0 (r1...rn) dr1...drn. (28)

The ground-state energy ε0 is the expectation value and can be written using eq.(28)

〈Ψ0|Vext|Ψ0〉

=

ˆΨ∗0 (r1, ..., rn)

n∑i

Vext (ri) Ψ0 (r1, ..., rn) dr1, ..., drn

=

ˆΨ∗0 (r1, ..., rn)

n∑i

δ (rp − ri)Vext (rp) Ψ0 (r1, ..., rn) dr1, ..., drndp

=

ˆΨ∗0 (r1, ..., rn)

n∑i

δ (r − ri) Ψ0 (r1, ..., rn)

N∑i

δ (rp − ri)Vext (rp) dr1, ..., drndp

=

ˆn0 (r)Vext (r) dr.

(29)

The next step of the proof is, if Vext 6= V′

ext and Ψ0 6= Ψ′0 then it follows for theground-state density n0 (r) 6= n′0 (r). The proof goes again by contradiction. TheRayleigh-Ritz variational principle states [13]

ε0 = 〈Ψ0|H|Ψ0〉 < 〈Ψ′0|H ′|Ψ′0〉. (30)

It follows

22

ε0 < 〈Ψ′0|H|Ψ′0〉= 〈Ψ′0|H ′|Ψ′0〉+ 〈Ψ′0|H −H ′|Ψ′0〉

= ε′0 +

ˆn′0 (r) (Vext − V ′ext) dr. (31)

And of course for the other way round:

ε′0 < 〈Ψ0|H ′|Ψ0〉= 〈Ψ0|H|Ψ′0〉+ 〈Ψ0|H ′ −H|Ψ0〉

= ε0 −ˆn0 (r) (Vext − V ′ext) dr. (32)

Now we sum eq. (31) and eq. (32), yielding

ε0 + ε′0 < ε0 + ε′0 (33)

which is obviously not true! That was the proof of the Hohenberg-Kohn theorem.Two di�erent non-degenerate external potentials are leading to two di�erent ground-state densities. Hence the expectation value of any operator A is a unique functionalof the ground-state density n0(r). Now, a variational principle can be used. Theenergy functional can be rewritten as

ε[n] ≡ 〈Ψ0[n]|T + Vee + Vext|Ψ0[n]〉 (34)

but, it is the external potential Vext with ground-state energy n0(r) that gives theHamiltonian the �uniqueness�. Therefore if a density n(r) equals the ground-statedensity n0(r) then ε [n]=ε0 [n] and the Rayleigh-Ritz principle

ε0 < ε[n] for n 6= n0. (35)

can be used. Eventually, the ground-state energy can be found by varying thedensity as long as the form of the functional ε [n] is known. The ground-stateenergy functional can also be written as

ε [n] = FHK [n] +

ˆVext (r)n (r) dr (36)

with

FHK [n] = 〈Ψ[n]|T + Vee|Ψ[n]〉. (37)

23

Again, FHK is a unique functional with density n(r) and once determined it can beapplied to all systems. Here, the discussion is reduced to the �non degenerate� casebut it can be extended to �degenerate� ones as well as �spin polarized� cases. Forthe sake of completeness it has to be mentioned that the proof consists of two parts.The �rst one is the proof that there exists a unique ground-state electron densitythat imagines exactly the ground-state wave function for many particles. And thesecond step is that the ground-state energy indeed minimizes the total energy of thewhole system. Therefore some literature refers to two Hohenberg-Kohn theorems[10].

3.3 The Kohn-Sham scheme

Even though the Hohenberg-Kohn theorem states that the density is an option to�nd the ground-state energy for n interacting electrons the exact form of ε [n] ineq. (34) is still not known and therefore not solvable. The solution to this problemis adressed in the Kohn-Sham scheme [10]. The idea is the following: instead of aninteracting system, a non interacting �reference� system is the starting point. As anext step an e�ective potential or �reference potential� Vr needs to be found thathas the same ground-state density as the interacting real system. Once this ground-state density is known it can be plugged into eq. (34) or into an approximation ofit. It will turn out that the reference potential Vr is also density dependent and theequations have to be solved self-consistently. That is why in practical calculationsan initial density is assumed. This gives a potential Vr which allows to calculate thesingle-particle states. Then again, from these a new density as well as a new Vr isobtained. This process is repeated until the input and output densities are in oneiteration su�cient close to each other. The Hamiltonian Hr for the non interactingreference system with n electrons is:

Hr = Tr + Vr. (38)

The Hohenberg-Kohn theorem states that there is a unique energy functional

εr [n] = Tr [n] +

ˆVr (r)n (r) dr (39)

but it should be emphasized that the term Tr in eq. (39) is not the same functionalas FHK in formula (36) because these n electrons are not interacting with eachother. The ground-state density of the reference system are the n occupied singleparticle states or the so-called Kohn-Sham orbitals Φi

nr (r) =

n∑i

|Φi (r) |2 (40)

which satisfy the equation[−~2

2m∇2 + Vr (r)

]Φi (r) = εiΦi (r) ε1 ≤ ε2 ≤ .... (41)

24

Now, Vr needs to be determined to make the ground-state density of the noninteracting system the same as n interacting electrons in an external potential VextThe interacting energy functional ε [n] in eq. (36) is rewritten as

ε[n] = Tr [n] +

(T [n]− Tr [n] + V [n]− e2

2

ˆ ˆn (r)n (r′)

|r − r′|drdr′

)+

e2

2

ˆ ˆn (r)n (r′)

|r − r′|drdr′ +

ˆn (r)Vext (r) dr

≡ Tr +e2

2

ˆ ˆn (r)n (r′)

|r − r′|drdr′ +

ˆn (r)Vext (r) dr + εxc. (42)

The Hartree term (second term in the last line of eq. (42)) which arises from theelectrostatic potential of the electron density and the non interacting kinetic energyfunctional Tr gets added and subtracted. The sum of terms in the brackets arede�ned as the exchange-correlation energy functional εxc and it can be also writtenas

εxc ≡ FHK [n]− e2

2

ˆ ˆn (r)n (r′)

|r − r′|drdr′ − Tr [n] .

All unknown electron interactions are now summarized in the exchange-correlationenergy functional εxc. The Hohenberg-Kohn theorem states that the ground-statedensity is minimizing εxc and therefore the variation of eq. (42) with respect to theparticle density can be caluclated

δε [n]

δn (r)=δTr [n]

δn(r)+e2

2

ˆn (r′)

|r − r′|dr′ + Vext (r) + vxc [n (r)] = 0 (43)

with the exchange-correlation potential

vxc [n (r)] ≡ δεxc [n]

δn (r). (44)

From the non interacting system and its Schrödinger equation it turns out that

δTrδn (r)

+ Vr (r) = 0. (45)

Comparing eq. (45) with eq. (43) shows that the potential Vr (r) must satisfy

Vr (r) = Vext (r) +e2

2

ˆn (r′)

|r − r′|dr′ + vxc (r) . (46)

Summarized, the interacting picture got connected with the non interacting one byadding an additional exchange-correlation functional giving all ingredients to obtainresults with the self consistent Kohn-Sham scheme.

25

3.3.1 Self consistent Kohn-Sham scheme

Now, everything is ready to implement the self consistent Kohn-Sham scheme. Firstof all a trial density n (r) is chosen and substituted into eq. (46) yielding a trialreference potential Vr(r). The result is plugged into the Kohn-Sham equation (41)to solve for the single particle wave functions. Then, a new trial charge density iscalculated with eq. (40). These steps are as long as repeated until two subsequentcharge densities are the same, yielding the ground-state energy eventually. Figure11 (taken from ref. [11]) shows the work �ow diagram.

Figure 11: Self consistent Kohn-Sham scheme.

26

3.4 The Exchange-Correlation potential

Notwithstanding that the exchange function vxc is only 10 % of the energy, withouta good approximation the whole calculation would fail. As already mentioned insection 3.3, the exchange-correlation potential includes all quantum mechanicalinteraction terms, i.e. the spin-polarization, the bonding etc. [14]. Therefore, inthe exchange-correlation energy the exchange hole and correlation hole which formtogether the exchange-correlation hole have to be considered.

Exchange hole

Pauli's exclusion principle does not allow two electrons with same spin in the sameorbital. Electrons with like spin have to occupy orthogonal orbitals with a spatialseparation. The hole that arises is called the exchange hole. The reduced repulsionbetween electrons in di�erent orbitals leads to a lower energy of the system.

Correlation hole

Pauli's exclusion principle is not violated for two electrons with di�erent spins onthe same orbital but they still do repel each other and create a correlation hole inthe electron density. But at the same time there is a small attractive force due tothe di�erent spins.

Exchange-Correlation hole

The exchange-correlation hole in density functional theory depends on both, theexchange- and correlation hole. Depending on whether the electron density is highor low the exchange hole or correlation hole respectively is the dominant part. Fur-thermore both of them are only an approximation which reduces computational time.The widely used methods are the local density approximation, generalized gradientapproximation and hybrids. Since the used form in this work is the generalized gra-dient appoximation which is the descendant of the localized density approximationthe following discussion is reduced to these two.

3.4.1 Local density approximation (LDA)

The �rst approximation available was the LDA [10]. This approximation is basedon the fact that the exchange-correlation potential depends only on the local spindensity, n = n↑ + n↓ and furthermore varies very slowly. The exchange term takesthe form

ELDAxc [n↑, n↓] =

ˆεunifxc (n↑, n↓)n (r) dr,

27

εunifxc is the exchange-correlation energy per particle of a homogenous system ofdensity, usually the homogenous electron gas. In the LDA the density is neutraland in the thermodynamic limit the solution of the exchange-correlation energy isknown analytically. Albeit the ansatz is simple the LDA leads very often to goodresults, sometimes even in cases with varying densities. For the disadvantages ofLDA please refer to next section.

Generalized gradient approximation (GGA) It was already brie�y mentionedthat there are some shortfalls within LDA. The GGA scheme should correct someof them like to improve total energies, atomization energies, energy barriers andstructural energy di�erences [15]. The general form of the GGA exchange-correlationenergy is

EGGAxc =

ˆεGGAxc (n↑, n↓,∇n↑,∇n↓)n (r) dr

and �rst attempts to create GGA's were based on the idea to make an expansion inpowers of �rst- and higher order gradients of the density but the results were ratherdisappointing. There exists di�erent GGA's for di�erent problems, the one that wasused in this work is the PBE (Perdew, Burke and Ernzerhof).

3.4.2 Perdew-Burke-Ernzerhof (PBE) functional

The PBE functional keeps some features of the LDA but improves the non localitiesincluding the gradient of the electron density. The PBE potential came straightafter the Perdew-Wang 1991 (PW91) and has no more empirical elements [14, 15].Its computational e�ciency makes it the most used functional these days. Thereason for this is that the PW91 was created to satisfy as much real condititions aspossible but the PBE does only match the ones that are energetically signi�cant.

3.5 GGA+U method (Dudarev)

Even though the implementation of GGA instead of LDA is an improvement fortransition metals, the partially �lled d- (also p- or f-) shells, how they often occurin transition metal oxides like NaOsO3, are still causing troubles. In these stronglocalized orbitals an additional on-site repulsion to �x two electrons on the same siteis needed. Hence a step beyond DFT is necessary. Like often in antiferromagneticcases, there occurs indeed a gap with GGA but the value of it is smaller thanexperiments show. The reason for this lies in the broken symmetry of the crystal.The usual form of a transition metal oxide is a metal ion with partially �lled d-orbitals surrounded by oxygen ions leading to e.g. octahedral complexes. By virtueof the Jahn-Teller e�ect the degeneracy of the states might be lifted to gain energywhich is the so-called ligand �eld splitting and its strengths is dependent from thearrangement of the ligands around the metal ion. If the ligand �eld splitting is largerthan Coulomb's repulsion the system will end up in a low-spin state. But for the

28

other way round the system will be in a high-spin state with localized electrons.Therefore these electrons must be treated with a potential jump if the number ofelectrons do increase by an integer number because the electrons have to overcomeCoulomb's force to be in the same orbital. This leads to the GGA+U method wherea correction of the localized d- or f- orbitals is implemented. While delocalizedorbitals are still described by GGA, the localized ones are treated with a Hubbard-like term 1

2Ueff∑i 6=j ninj separating the electrons into two subsystems. In the

Dudarev approach is Ueff = U − J meaning that only the di�erence between theCoulomb repulsion U and the Hubbard parameter J is taken into account. ni,j areorbital occupation numbers. Let's assume that the d electrons are delocalized, theshell is a �uctuating system and therefore described by GGA. The Coulomb energyfor the d-d interaction is E =

Ueff

2 n (n− 1). As this term is already included in theDFT functional as a mean �eld value it is essential to subtract it if the Hubbard liketerm is added. Otherwise these electrons are double counted. The energy functionalturns out as

EGGA+U [n] = EGGA [n]− Ueff2

n (n− 1) +Ueff

2

∑i 6=j

ninj . (47)

For the orbital energies εi the derivative of eq. (47) is needed:

εi = ∂EGGA+U

∂ni= εi,GGA + Ueff

∑i 6=j

nj −Ueff

2(n− 1)− Ueff

2n

= εi,GGA + Ueff (n− ni)− Ueff +Ueff

2

= εi,GGA + Ueff

(1

2− ni

)(48)

The one electron energy εi in eq. (48) gets shifted by a factor of εi,GGA − Ueff

2

for an occupied state ni = 1 and by a factor of εi,GGA +Ueff

2 for an unoccupiedstate ni = 0, resulting in a Mott-Hubbard gap (cf. section 2.3), provided theright value of Ueff is known. As it is very di�cult to gain Ueff from experiments,nowadays it is common to compute Ueff fully ab initio with the constrained randomphase approximation [16]. The GGA+U by Dudarev implemented in VASP uses anrotationally invariant form [17]

EGGA+U [n (r) , n] = EGGA [n (r)] + EU (n)− Edc (n) . (49)

Again, the �rst term on the right hand side is the GGA functional, EU (n) denotesthe electron-electron interaction for the localized electrons and last term is the dou-ble counting correction term considered because this interaction is already includedin the �rst term. n is the occuption matrix with spin-diagonal and spin-non-diagonalterms.

29

3.6 DFT in Practice

This section is a short recap and summary what choices one has to make to generategood results within DFT when one is using VASP, the Vienna Ab initio SimulationPackage. There are the following important choices to make:

1. The exchange potential vxc, already introduced and discussed in section 3.3

2. The basis set for the expansion of the wave function or Kohn-Sham orbitalsas well as the pseudo potential see sections 4.1-4.3

3. Relaxation procedure, see section 4.4.

30

4 The Vienna Ab initio Simulation Package

This chapter examines VASP, the Vienna Ab initio Simulation Package [9]. The�rst part is a general introduction then, a closer look at pseudo potentials aregiven. These are representing the atomic potentials which have not only an impacton electrons travelling through the solid, it also reveals why plane waves are stilla convenient choice. The last three sections are emblazing VASP from a practicalpoint of view and also chosen tags from VASP are explained.

All calculations conducted in this work were performed with VASP. The packageincludes among other things electronic structure calculations and quantum mechan-ical molecular dynamics from �rst principles and can be used to model materialson an atomic scale. The many-body Schrödinger equation can be solved withinDFT or within the Hartree-Fock approximation using iterative matrix diagonaliza-tion techniques (RMM-DISS and blocked Davidson [17]) to �nd the ground-state.For approximations within DFT the Kohn-Sham equations are solved, within theHartree-Fock approximation the Roothaan equations [18]. All calculations withVASP are controlled by four �les: POTCAR, KPOINTS, POSCAR, INCAR and inthe next following paragraphs a brief description of each is given.

POTCAR

The POTCAR �le includes the potential of each sort of atom in the unit cell anddi�erent approximations are possible. The basic idea is to separate the nuclei andcore electrons by a cut-o� radius from the valence electrons. The VASP packageincludes two di�erent types of pseudo potentials:

1. Norm-conserving pseudo potential

In this approach the pseudo and all electron charge densities are the samewithin the cut-o� radius and often it is constructed to meet the criterion

ˆ rc

0

|ΨPP (r) |2dr =

ˆ rc

0

|ΨAE |2dr. (50)

ΨPP (r) denotes the pseudo potential wave function and ΨAE (r) the allelectron wave function [19].

2. Ultra-soft pseudo potential

Here the pseudo wave function is smoothened with lesser nodes and a largercut-o� radius rc [20].

Both of them can be combined in the PAW method �rst introduced by Blöchl [21].This will be all explained in detail in section 4.1.

31

KPOINTS

In the KPOINTS �le the number of k points and the mesh for the k point grid tosample the Brillouin zone can be chosen [17]. The k points can be entered explicitlyas well as a path for band structure calculations or as automatic k-mesh. In thelatter case only the input of the grid in all directions of the Brillouin zone is required.

POSCAR

The POSCAR �le includes the sort of atoms as well as their positions [17]. Basicallyit contains all structural information, the lattice vectors and the atomic basis i.e. theatomic coordinates. The input can be carried out in direct- or cartesian- coordinates.

INCAR

The INCAR �le holds the information about what exactly is going to be calculated[17]. This is for example: spin polarized, spin unpolarized, band structure, volume-and/or electronic- relaxation etc..

4.1 Pseudo potentials

For plane wave calculations the valence wave functions are expanded as Fouriercomponents. Even though the electrons close to the core are inert because they donot contribute to electric current, the orbitals are sustained to heavy oscillations.Fig. 12 (taken from ref. [22]) shows wave functions of the radial Schrödingerequation of platinum with the bound states and the d valence states dependentfrom the radius of the nucleus.

32

Figure 12: Radial wave functions of the platinum atom.

To incorporate for each calculation these wiggles is for plane waves not very handybecause a lot of plane waves are required and they all would be treated as valenceelectrons. The idea is to separate the core electrons and the nuclei from the valenceelectrons and �nd a scheme to calculate e�ciently the wave function within thesetwo areas and connect them. Three methods: the LAPW, pseudo potentials andPAW to tackle this task are brie�y discussed within the following paragraphs.

Linearized augmented plane wave (LAPW) method

The ab initio LAPW method [23] is based on the augmented plane wave method(APW) [10]. Both of them are �mu�n-tin� approximations. The crystal gets dividedinto augmented regions and the interstitial area. The mu�n-tin potential which isaround the atom, is considered as spherical symmetric and the interstitial regionunderlies a constant potential. That is why within the mu�n-tin sphere the wavefunctions are expanded as spherical harmonics and the interstitial region in planewaves. As these two areas are independent from each other the core states areorthogonal to the valence states and at the mu�n-tin sphere these two regionsare connected. This is already quite a simpli�cation because the two areas canbe evaluated separately. Unfortunately the matching conditions do turn out ascumbersome calculations because there occurs a non-linearity in the independentparticle Schrödinger equation which prevents to calculate many eigenstates withinone calculation �run�. This non-linear phase shift in the scattering process is curedwith the LAPW method. Fig. 13 (taken from ref. [24]) shows the oscillations insidethe mu�n-tin areas, orange and green balls (MT), matching with plane waves atthe spheres.

33

Figure 13: Interstitial and mu�n-tin radius in the APW and LAPW method.

This accurate method can be used for calculations including high pressures becausethe core states and semi core states are directly included in all calculations.

Pseudo potentials

The pseudo potential approach is di�erent from the LAPW method in that waythat the wiggles within a sphere around the core are smoothend [23]. The valenceelectrons of the outer shells feel a strong attractive force due to the nucleus eventhough it is screened by the core electrons. On the other hand there is a strongrepulsive force between the valence and core electrons. All in all these two oppositee�ects of attraction and repulsion do compensate each other leaving just a smallpotential left, shifting the orthogonal valence electrons from the core electrons up toa chosen energy range. This is the basic idea of pseudo potentials and can be seenin �g. 14 (taken from ref. [25]). The �gure shows the all electron wave function(blue dashed upper line), the Coulomb potential blue dashed lower line, the pseudowave function and pseudo potential in red. Near the cut-o� radius rc both functionsare in agreement.

34

Figure 14: Pseudo potential of a 1s wave function and potential inside and outsidethe cut-o� radius rc.

As the atom now appears as a small perturbation plane waves can be used again.VASP o�ers two types of pseudo potentials, both of them have in common thefrozen core method. The core electrons are calculated separately in an atomicenvironment and kept frozen, which means constant in all subsequent calculations.The two types VASP o�ers are norm-conserving or ultra soft pseudo potentials [17].

Norm conserving pseudo potentials

The norm-conserving condition

Qlm = 4π

ˆ Rc

0

r2φ∗l (r, ε)φm (r, ε) dr} = 4π

ˆ Rc

0

r2Ψ∗l (r, ε) Ψm (r, ε) dr (51)

with the all electron wave function Ψ and pseudo wave function φ guarantees thatthese two wave functions generate the right charge density outside the core [19]. Inthis approach the scattering properties are correct.

Ultrasoft pseudo potentials.

Norm-conserving pseudo potentials are accurate but only moderately smoother com-pared to the all electron wave function inside the core region [20]. Ultrasoft pseudopotentials are much smoother because the norm-conserving condition is relaxed witha non zero di�erence in the norms

Qlm =

ˆ Rc

0

(φ∗l (r, ε)φm (r, ε) dr −Ψ∗l (r, ε) Ψm (r, ε) dr) . (52)

35

The advantage is that each pseudo wave function can be found independently. Thepseudo wave function must only match the value of the all electron wave functionat the sphere

Φlm (Rc) = Ψlm (Rc) .

Also, they are more transferable than norm-conserving ones i.e. it is better adaptablein di�erent chemical environments because the cut-o� radius can be reduced.

Projector augmented wave (PAW) method

The PAWmethod combines the LAPWmethod, where the all electron wave functionis accessable with the pseudo potential approach, to replace the full potential witha weaker one, in a very elegant way [21]. Again the crystal gets separated intomu�n-tin areas and the interstitial region. Also, the frozen core approximation iskept. In the PAW method the pseudo wave functions in the augmented regions arenodeless and smooth whereas the interstitial region can be described with a coupleof plane waves or rather on a coarse grid. The idea is to connect the pseudo wavefunction Ψn by a linear transformation τ with the real wave function Ψn by

|ψn〉 = τ |Ψn〉, (53)

n denotes the states one is interested in e.g. all- or only the valence-states. Thetransformation with the all electron wave function on the left hand side turns outas

|Ψn〉 = |Ψn〉+

sites∑lmε

(|ϕlmε〉 − |ϕlmε〉

)〈plmε|Ψn〉 (54)

with the partial wave ϕ and partial pseudo wave ϕ. Inside the spheres Ψ and Ψ are

|Ψn〉 = (≈)

sites∑lmε

|ϕlmε〉clmε (55)

|Ψn〉 = (≈)

sites∑lmε

|ϕlmε〉clmε. (56)

(in practice often approximated in order to reduce computational time) where l andm in eq. (54) denote the quantum numbers l and m, ε is a reference energy andplm a projector function. The projector function

clmε = 〈plmε|Ψn〉 (57)

that obeys〈plmε|ϕl′m′ε′〉 = δll′δmm′δεε′ (58)

36

gives back the radial character inside the augmented region because ϕ and ϕ aresolutions of the radial Schrödinger equation. All in all, the information written ineq. (54) is that the real wave function is the sum of the smooth wave function andthe sum over all atoms, at each atom there is one atom all electron minus one atompseudo. This principle is also valid for the charge density, exchange-correlation,energy etc. Outside the spheres the real wave function is the same as the pseudowave function. In this work the PAW potentials were used as it is recommended bythe VASP team [17].

4.2 Bloch theorem

When an electron is travelling through a solid it does not bump into other electronsbecause of Pauli's exclusive principle but it feels the attractive potential of the nucleieven though the force is small because of the screening e�ect from the electronsclose to the core. This was discussed in the last section and that is why the movingelectrons are considered to be as nearly free but the residual periodic potential ofthe solid with lattice vector R has to be taken into account. It is also a fact thatthe e�ective number of electrons can easily go to in�nity due to periodic boundaryconditions [26]. With the Bloch theorem [27] it turns out that is su�cient to reducethe calculations to a few of them. Only propagating waves with almost the wavelength of the period are disturbed by the potential and underly a tuning condition.First of all the periodic potential V(r) obeys

V (r) = V (r + R) . (59)

The time dependent Schrödinger equation for the periodic potential is(− ~2

2m∇2 + V (r)

)Ψ (r) = EΨ (r)

and it implies that also the energy, the wave function and all other quantitiesdependent from r exhibit this periodicity. The wave function is characterized by thewave vector k and can always be written as

Ψk (r) = vk (r) fk (r) (60)

with vk obeying the same condition as eq. (59). Of course also the charge density|Ψk (r) |2 is periodic leading to

|fk (r + R) |2 = |fk (r) |2 (61)

which is satis�ed for all R if fk (r) = exp(ikr). Therefore the Bloch function turnsout as

Ψk (r) = uk (r) eikr (62)

37

and all wave functions that satisfy this condition are so called Bloch waves. Fromhere it is easy to derive the Bloch theorem

Ψk (r + R) = Ψk (r) eikR. (63)

The meaning of it is basically that the wave functions at the positions r and R arethe same apart from a phase factor exp(ikR). Now it is clear why all caclulations arereduced to the Wigner-Seitz cell. Each electron occupies a speci�c k state but thereis only a minor di�erence in the wave functions if the states are close to each otherand therefore it is su�cient to evaluate the wave functions on certain k points.

The real space and reciprocal space are linked by the Fourier transform, yielding forthe periodic function vk (r)

vk (r) =∑G

ck (G) eirG (64)

so at each G (the reciprocal lattice vector) a plane wave is travelling orthogonal toG through the solid. With eq. (64) the wave function in reciprocal space can bewritten as

Ψk (r) = vk (r) eikr

=∑G

ck (G) ei(k+G)r (65)

Looking at eq. (65) reveals that also in reciprocal space the calculations can bereduced to �rst Brillouin zone and a k points grid with results converging withrespect to the number of k points. Of course all other quantities like the chargedensity are transformed following the same way

n (r) =∑G

n (G) eiGr.

4.3 Plane wave basis set

The Kohn-Sham orbitals cf. eq. (40) do not necessarily need to be expanded as asuperposition of plane waves, there are also other options like a Gaussian basis set[9]. Nevertheless there are several good reasons why to use them, namely

φ = ceikr. (66)

Very convincing is its computational e�ciency. The real and reciprocal space arelinked by the Fast Fourier transform which is easy to implement in algorithms.Plane waves are especially suitable for periodic systems because the Bloch theorem

38

is satis�ed. In addition due to the non locality because the plane wave is scatteredover the whole crystal, there is no Pulay force [17]. In combination with pseudopotentials or PAW potentials the electrons created with plane waves are manageableand the energy cut-o� can be reduced. In general one is more interested in lowerenergies than in the higher unbound electrons. The energy cut-o� separates thesetwo kinds of electrons by a speci�c kinetic energy value,

Ekin ≤1

2|k + G|. (67)

Only electrons with a kinetic energy below this value are considered. The ones withhigher energies are neglected. This of course, further reduces the required amountof k points.

4.3.1 Kohn-Sham equations with plane wave basis set

To �nally solve the Kohn-Sham equation with a plane wave basis set, VASP isexpanding all quantities with plane waves leading to:(−1

2∇2 + Veff

)∑G

cn (k + G) ei(k+G)r = εnk∑G

cn (k + G) ei(k+G)r (68)

withEkin =

1

2|k + G| (69)

andVeff (r) =

∑G

Veff (G) eiGr (70)

[9]. Eq. (68) is integrated over the Brillouin zone and extended with the factorexp[−i (k + G′)] from the right hand side. The next two calculations are the rightand left hand side, respectively(

−1

2∇2 + Veff

)∑G

cn (k + G) ei(k+G)r =

ˆ (e−i(k+G)r

(−1

2∇2 + Veff

)∑G

cn (k + G) ei(k+G)r

)d3r =

ˆ ∑G

(1

2|k + G|2ei(G−G

′)r + Veffei(G−G′)r

)cn (k + G) d3r =

∑G′

(1

2|k + G|2δGG′ + Veff (G−G′)

)cn (k + G′) (71)

39

εnk∑G

cn (k + G) ei(k+G)r =

εnk

ˆ ∑G

(e−i(k+G′)rcn (k + G) ei(k+G)r

)=

εnk∑G

cn (k + G) δGG′ =

εnk∑G

cn (k + G) (72)

yielding eventually

∑G′

(1

2|k + G|2 + Veff (G−G′)

)cn (k + G′) = εnkcnk (k + G) (73)

with

Veff (G−G′) =

ˆVeffe

i(G−G′)rdr. (74)

Of course this can be written in matrix form

Hc = εc (75)

which is yielding �rst the eigenvectors cn (k,G) and �nally allows to calculate theKohn-Sham orbitals φ and the energies εnk. The index n denotes the band index.

4.4 Electronic ground-state

As di�erent pseudo potentials and PAW potentials can be selected, to get properresults with VASP the �rst step should be to relax the structure. In an inner loop theelectronic relaxation by diagonalizing the Hamiltonian with the blocked Davidsonalgorithm is performed to �nd the ground-state [17]. In an outer loop the structuralrelaxation including ionic, volume and shape can be chosen. In the ionic relaxationthe forces which are acting on each ion are calculated and driven into the directionof steepest descent [17]. The relaxation is complete when the force on each ionis close to zero, approximately 0.01 eV/Å, this was also the reference value in thiswork. The volume relaxation alters the volume such that the total energy is in itsminimum. The appropriate �ags to choose in the INCAR �le are brie�y introduced.

� ISIFcontrols if the stress tensor is calculated or not and what kind of relaxationshould be conducted [17]. To simulate reality as close as possible the ISIF=3tag was chosen, including the calculation of the stress tensor and forces ofeach ion. In addition, ions, cell shape and cell volume are relaxed.

40

� IBRIONde�nes the method for the ionic relaxation and for a smooth calculation theconjugate gradient algorithm, with IBRION=2 was chosen [17].

� POTIMdenotes the step size of iterations and for IBRION=2 the VASP manual sug-gests POTIM=0.5.

� NSWdenotes the amount of ionic steps and was set to 150 because the breakcondition is implemented anyway.

� EDIFFis the break condition for the inner self-consistent loop. It denotes the al-lowed error in the total energy and was kept at the default value of 10-4, alsorecommended by the VASP team .

� EDIFFGis the break condition for the outer loop, if the change in total energy isbelow the break condition the calculation stops. EDIFFG can also be set toa negative value which is according to the VASP team the more convenientchoice. In this case the relaxation stops if the forces are smaller than thetoken value. This was also set to -1*10-4.

The results of the relaxation can be checked in the OSZICAR �le.

4.5 Smearing

For a periodic system due to Blochs theorem the calculation is reduced to the�rst Brillouin zone, i.e. all k points belong to �rst Brillouin zone. In order to getresults like the density of states, charge density, matrix elements, energy band etc.the integral over the �rst Brillouin zone is necessary. To put this into executioncomputationally the sum over these k points with weights is performed and forconverging results a su�cient dense k-mesh appropriate to the system is crucialcf. section 4.3.1. But it is not necessarily a given that the integrated function iscontinuous. For example metals, the occupied and unoccupied areas by electronsare separated by the Fermi surface. The function drops abruptly at the Fermi levelfrom 1 to 0 at 0 K. For converging results this would imply that a very dense kmesh is necessary. VASP o�ers two methods to circumvent this problem, both ofthem create a �ctitious temperature. Firstly, in the tetrahedron method discrete kpoints are used to create tetrahedra that �ll the reciprocal space. Then the integralwithin each tetrahedron will be calculated on every point by interpolation. Secondly,another approach to deal with the discontinuity are smearing methods [17]. Thediscontinuity is smeard out by replacing the step function with a smoother one, suchas the Fermi-Dirac Distribution or a Gaussian-type function with the associatedsmearing energy σ [17]. Figure 15 (taken from ref. [28] ) shows in (a) the Diracstep function whereas in (b) the smoother Gauss function is depicted [17].

41

Figure 15: Comparision of the Dirac step function, (a) and the Gauss function f,(b).

In VASP both methods are controlled with the ISMEAR tag in the INCAR �le. TheVASP manual recommends for relatively large cells the Gaussian function that iswhy the ISMEAR=0 with σ=0.01 eV was chosen[17].

42

5 Supercell approach

In section 4.2 it was outlined why it is su�cient to reduce all calculations to Brillouin-zone and that only a few k points are necessary to sample it. By virtue of the dopingissue within this thesis the supercell approach for all band structure calculations wasapplied [14, 26]. After a short introduction why supercells are appropriate for dopingand an explanation about the boundary conditions, the focus lies on how routines,like VASP, reduces the amount of in�nite particles to a manageable size.

A supercell constists of several unit cells and after transformed into reciprocal spacereduced to a �rst Brillouin zone. This zone is especially important because on theboundaries, Bragg-re�ection occurs [26]. The wave functions are again de�ned bythe wave vector k, reciprocal lattice vector G and Blochs theorem is valid. Becauseof the inverse lenght unit in reciprocal space the supercell in k space is smallerthan the unit cell. This approach in combination with periodic boundaries allowalso to treat solids with impurities because if the supercell is large enough arti�cialperiodicity is preserved on a larger scale and the dopants are not disturbed by eachother. Also, it contains all information of the solid. For supercells which are notneutral in charge a correction for the dipol moment has to be added. To get a visualimpression please look at �gure 16 (taken from ref. [14]). The vacancies v of thetwo supercells have an interaction distance of four lattices.

Figure 16: Two supercells with vacancies v and periodic boundary conditions.

The disadvantage of supercells is that the band structure might be hard to comparewith experimental results because the bands are calculated in the Brillouin zone ofthe supercell and not in the conventional Brillouin zone associated with the primitveunit cell. Please see next section.

5.1 Periodic boundary conditions

To create a bulk material and to avoid surface e�ects periodic boundaries are im-posed. The initial box is repeated in all directions and if an electron moves outon the right hand side it immediately enters into it on the left hand side. These

43

are the Born-Karman boundary conditions [26]. There are no forces between anatom and its copy of it. If a position of an atom is needed in the adjacent box acoordinate transformation can be used. Periodic boundaries are easy to implementand contribute to reduce the calculation time.

5.2 Reciprocal space

In section 4.2. it was explained that in real space all calculations can be reduced tothe Wigner-Seitz cell according to Bloch's therom. In addition it is easy to switchto reciprocal space with the Fourier transform and as a result all output quantitiesare in reciprocal space. The scheme that explains the mapping from the real toreciprocal space can be seen in �g. 17 (taken from ref. [14]).

Figure 17: The mapping from real to reciprocal space.

Starting point is the left side on the top. A bulk material is built by creating a super-cell in real space consisting of several unit cells with periodic boundary conditions.As a next step the supercell is transformed into reciprocal space and limited to �rstBrillouin zone where all properties are contained (top right panel). Based on sym-metry operations the Brillouin zone can be reduced to a small wedge, the irreducibleBrillouin zone. All wave vectors in the �rst Brillouin zone can be constructed fromthe wave vectors in the irreducible Brillouin zone by symmetry operations. Each kpoint in the irreducible Brillouin zone becomes a weight depending on the numberof times it was folded from the Brillouin zone into the irreducible one. E.g. the zonecenter is counted just once. For converging results su�cient enough k points, oftenuniform grids, have to be chosen. This is shown in the last panel on the right handside. Band structure calculations are performed along so-called symmetry paths, anillustration for an orthorhombic cell is given in �gure 18 (taken from ref. [30]).

44

Figure 18: Brilliouin zone of an orthorhombic lattice with possible high symmetrylines.

The Monkhorst-Pack and Γ point

This famous method uses a grid with equal spacing for the irreducible Brillouinzone and is in most programs that model materials implemented [31]. The Γ pointdenotes the central k point (k = 0) where the real and reciprocal lattice coincide.

45

6 E�ective band structure

As all band structures in the second part of the thesis are conducted with thismethod the next two sections will give an insight into the theory behind the unfoldingprocedure and also why its use is advisable.

The reason for supercells was already discussed in chapter 5. What distincts theprimitive cell from the supercell is the following: Whereas the primitive cell is merelyconstituent of basic vectors and atomic positions, i.e. a lattice, the supercell is arepetition of many unit cells in order to accomodate a certain concentration ofsubstitutional atoms. That leads not seldom to band structures hard to interpretbecause of their complexity and also the comparision with for example angle-resolvedphoto emission spectroscopy (ARPES) is more or less impossible and requires thee�ective band structure scheme [32]. As the supercell is smaller in reciprocal spacethan the primitive cell the dispersion relation becomes hard to recover because ofthe zone folding. To �translate� the supercell band structure into the primitive basisrepresentation the spectral weight is introduced. Each band becomes a weight andwhen plotting the band structure unwanted translated points are removed.

Figure 19 (taken from ref. [33]) shows what happens to the band structure if thesupercell approach is applied. For a tiny size of the supercell �at bands appear whichhave nothing in common with the original band structure or results from ARPES.

Figure 19: Zone folding dependent from the size of the supercell.

The band structure of the primitive cell E (k) is reconstructed from the band struc-ture of the supercell E (K) using the unfolding technique. From now on smallletters are used for the primitive cell and big ones for the supercell.

46

The basis vectors of the supercell can be created with a transformation matrix Mof the primitive cell by A1

A2

A3

=

m11 m12 m13

m21 m22 m23

m31 m32 m33

a1a2a3

, (76)

mij are integers with mij ∈ Z. The transformation matrix M must indeed beinvertible but not necessarily diagonal. If M is diagonal, the primitive vectors ofthe two cells are collinear leading to the easiest case of unfolding. For a supercellthat is twice as large as the primitive cell, an example for M is

M =

1 0 −10 1 01 0 1

. (77)

For �nding the basis vectors for the primitive cell one simply has to �nd the inverse

M−1A = a. (78)

6.1 Folding and unfolding of wave vectors

The folding and unfolding depends only on the relation between the symmetry andgeometry of the Brillouin zones of the supercell and the underlying primitive cell[32]. Since the Brillouin zone of the supercell is in k-space smaller than the onefrom the primitve cell, the k-vector k is fold into one K-vector if a reciprocal latticevector G0 exists, by

K = k −G0, (79)

whereas one K vector is unfold in ki vectors if reciprocal lattice vectors Gi existwith i = 1, ..., Nk, by

ki = k + Gi. (80)

The connection between the volumes of the two cells is

Vsc = Vpcdet (M) . (81)

47

Figure 20: left: supercell and primitve cell in direct space, right: Brillouin zones inreciprocal space and folding/unfolding of k-vectors.

For a visualization of the direct- and the reciprocal space see �gure 20 (taken fromref. [34]) left- and right-panel respectively. The transformation matrix betweenthese two reciprocal basis vectors is de�ned as

B =(M−1

)Tb (82)

with the reciprocal lattice vectors:

gn =∑i

pibi, pi ∈ Z (83)

Gn =∑i

qiBi, qi ∈ Z. (84)

For an illustration of the folding and unfolding of the k-vectors see again right panel�g. 20.

6.2 Folding and unfolding of states

The calculated eigenvectors |kn〉 and |Km〉 (n, m are band indices) from theSchrödinger equation are as suitable as the wave vectors to perform the foldingor unfolding to obtain the e�ective band structure. With the linear combinationof eigenvectors |kin〉 (i = 1, ...Nk) the eigenvector of the supercell |Km〉 can bewritten as

|Km〉 =

Nk∑i=1

∑n

F (ki, n;K,m) |kin〉. (85)

48

With the folding and unfolding two things can be calculated. First of all, theeigenvectors |kin〉 as well as the contribution F (ki, n;K,m) to the eigenstates|Km〉. And secondly, the dispersion relation E (k) from E (K). For the unfoldingof a periodic solid see �g. 21 (taken from ref. [34]).

Figure 21: E�ective band structure of a periodic solid.

The green arrow shows how the complicated band structure is unfolded, resulting inthe same band structure as the primitive cell for the special case that the supercell isjust a repetition of the primitive cell. The last calculation is performed via projection

PKm (ki) =∑n

|〈Km|kin〉|2, (86)

PKm (ki) is the probability to �nd states |kin〉 contributing to the state |Km〉 alsocalled the amount of Bloch character ki in |Km〉 with the same energy En = Em.Also the spectral function can be derived from PKm (ki):

A (ki, E) =∑m

PKm (ki) δ (Em − E) . (87)

If the matrix M is diagonal and the supercell is merely a spatial repetition of theprimitive cell (�g. 21, taken from ref. [34]) there is only one |kin〉 state contributingto PKm (ki) for each |Km〉 state and the e�ective band structure is the same asthe one from the primitve cell. For a practical utility of this method please have alook at �g. 22 (taken from ref. [35]).

49

Figure 22: E�ective band structure of La doped Sr2IrO4.

The panel shows the e�ective band structure of La doped Sr2IrO4 as a function ofthe e�ective Hubbard interactions Ue� (eV). The vertical coloured bar indicates thespectral function (cf. eq. (87)). Even though a supercell was used to model dopingthe band structure is easy to interpret due to the unfolding method and can easilybe compared with results by means of ARPES.

50

7 NaOsO3

Even though the past sixty years the metal-to-insulator transition (MIT) of transitionmetal oxides were under scrutiny a lot of questions still remain open [36]. Thereexist some textbook examples for Mott-Hubbard solids like V2O3 and NdNiO3 wherean arti�cial MIT by applying pressure or temperature can be achieved but alsoborderline cases with MITs that are hard to pin down are common. For these casesdi�erent MITs than the Mott-Hubbard transition have to be conisdered.

On the one hand there is the Slater MIT introduced by Slater in the 1950s. Here, theMIT is independent from electron correlations and induced by the antiferromagneticorder alone. Below the Néel temperature TN, when the compound turns from theparamagnetic to the antiferromagnetic state a gap occurs due to the doubling ofthe unit cell. Fig. 23 (taken from ref. [37]) depicts the transition.

Figure 23: Slater MIT.

In the upper panel the unit cell with lattice constant a, let's say has one electronper unit cell and is therefore a metal. When the temperature gets reduced belowTN a long range antiferromagnetic order occurs. But as there are now two di�erentpotentials the unit cell is doubled which means that the Brillouin zone is bisceted.This is shown in the lower panel of �g. 23. As there are now two electrons per unitcell the lower band is full and the upper one empty resulting in a band gap. As thegap is due to exchange interactions and not correlation e�ects itinerant magnetismis expected. This itinerancy requires a magnetic moment away from half-integralvalues and as a result there exists hybridization.

On the other hand a di�erent MIT was proposed by Lifshitz in 1960 which is dueto the change of topology on the Fermi surface, that is the surface with constantenergy in momentum space. For certain energy values in regions of high pressures

51

there occurs a continuous deformation of the Fermi surface near the boundary. Toassert the electron transformation the variation of topology on the Fermi surfacecan be checked. This in turn has in�uence on the density of states. An examplecan be seen in �g. 24 (taken from ref. [38]) .

Figure 24: Topology and DOS of NaOsO3.

The lowest panel has no Fermi surface because it is an insulator with an energygap. The more the temperature is increased the more the topology changes. Thisgoes also along with a reduction of the magnetic moment. Therefore, the featuresa compound has to exhibit for a spin-driven Liftshitz transition are a small U, asmall magnetic moment and strong hybridization [38]. The variation of topologyhas nothing to do with a symmetry change of the lattice and is hence di�erent fromsecond order phase transitions. Unfortunately the concept of the Fermi surface cannot be used for solids with impurities because of the loss of translation symmetry[39]. The elusive Lifshitz transition was created with a strong transverse electric�eld to a h-BN encapsuled bilayer graphene in 2014 [40].

As the Mott-Hubbard transition is abruptly and not continuous this MIT is notsuitable for NaOsO3. Until recent investigations NaOsO3 was proposed to be the�rst found compound undergoing a Slater MIT and there are several reasons why toclassify it as such. S. Calder and al. (2012) found by neutron and x-ray experimentsthat the set in of the long-range magnetic order coincide with the temperature wherethe MIT occurs (TN=410 K). Also a reduced magnetic moment which has no morehalf-integral values was detected. This itinerant magnetism that goes along withstrong hybridization of Os d-orbitals and O p-orbitals is a de�nite sign for a SlaterMIT. Though, examinations by B. Kim et al. (2016) found inconsistencies for this

52

kind of MIT. First of all a bad metal behaviour within an intermediate temperatureregion was evaluated. In addition DFT calculations revealed that to open the gapquite a large U is required in the itinerant antiferromagnetic region. Instead, itturns out that SOC is strong and responsible for the W/U ratio which depends onthe amount of electrons in the t2g orbitals. For a system less than half �lled t2gstates SOC induces band splitting and also increases the magnitude of U, bringingthe system more into direction of relativistic Mott-physics. For a system with morethan half �lled orbitals SOC increases the bandwidth W leaving de�nitely the Mott-physics area. NaOsO3 represents exactly the half-�lled case and the conclusion byB. Kim et al. (2016) is that even though the orbital moment is quenched, Le� =0and no band splitting is happening SOC reduces the correlation energy which isquite an interesting observation! Three di�erent temperature regions are observed.At low temperature it is an insulator. Then there exists an intermediate region witha bad-metal behaviour. Above TN the compound turns into a paramagnetic metal.All in all the MIT in NaOsO3 seems to be rather a Lifshitz- than a Slater- transition[38]. For the change in topology with increasing temperature please see �g. 24again. Since NaOsO3 has peculiar properties and the properties are dependent fromthe �lled orbitals, doping is a possibility to give further insight into the interplay ofSOC and the correlation strength, revealing probably totally new transitions.

7.1 Structural features

NaOsO3 belongs to a family of compound called perovskites, named afer the Russianmineralogist Lev Aleksevich von Petrovski. It was �rst detected in 1839 in the uralmountains and has the formula CaTiO3. That is why the general formula is or is atleast close to ABX3 with an orthorhombic structure. A is usually a large cation, Bis medium sized and X is an anion. Fig. 25 (�g. (a) taken from [41] and (b) from[42]) shows the perovskite CaTiO3 in (a) and in (b) the orthorhombic structurewith formula ABX3. The dark brown and black colors are due to impurities but apure crystal has a refrective index of 2.38 [43].

53

(a) CaTiO3 (b) Orthorhombic structure

Figure 25: CaTiO3 and the according orthorhombic structure.

After the discovery of BaTiO3 in the 1950's which turned out to have dielectric andferroelectric properties that could be used for electronic devices like capacitors andtransducers, materials belonging to this family raised attention.

NaOsO3 is realizable under high pressure technique and the oxydation state isNa1+Os5+O3

2-. The tolerance factor is t = 0.87 using Shannon ionic radius. Theseorthorhombic distortions lead to a

√2× 2×

√2 quadrupled gadolinium orthoferrite

type cell that belongs to Pnma space goup no. 62 [44]. The lattice constants area = 5.3842 A , b = 7.5804 A , c = 5.3282 A respectively with the atomic positionsgiven in table 1. The four formula units per unit cell consist all in all of 20 atomsand can be seen in �gure 26 (created with VESTA [45])

Site symmetry x y z

Na 4c 0.0328 0.25 -0.0065Os 4b 0 0 0.5

Apical O 4c 0.4834 0.25 0.0808Planar O 8d 0.2881 0.03940 0.7112

Table 1: Direct coordinates of NaOsO3.

54

Figure 26: Primitive cell of NaOsO3 in the a-c plane.

Albeit there are three di�erent Os-O bond lenghts the aspect-ratio is almost ideal(∼1.002 [44]) because all bond lengths are close to 1.94 Å. Also the OsO6 octa-hedras are almost ideal with bond angles of 90.7°, 89.3° or 89.1° [44]. Somewhatdi�erent is the situation regarding the position of the octrahedras compared to thecell itself. There exists a tilting of the O-Os-O axis by 11° relative to the b axis anda rotation of 9° in the c (or a) axis [44].

7.2 Electronic features

Orthorhombic distortions inside the oxygen �cage� have to be considered within thetheory of ligand �eld splitting as the oxygen atoms are regarded as neutral. The t2gorbitals, dxy, dxz and dyz point in between the oxygen atoms whereas the eg orbitals,dx2-y2 and dz2 point directly to the oxygen atoms. Therefore Coulomb's force isdecreased and increased respectively, resulting in an energy di�erence between t2gand eg sub shells. The more distorted the cage the more splitting occurs, hence inNaOsO3 the degeneracy of the two eg and three t2g states themselves are also lifted,even though only by a small value. An additional shift in energy, also sourced fromthe distorted oxygen cage, are the overlapping oxygen p-orbitals with the osmiumt2g d-orbitals.

Anyhow, the speci�c structure of NaOsO3 is responsible for a comparative smallcrystal �eld splitting between the t2g and eg states. That is why the system is ahigh spin state with equally occupied t2g states and a total spin of S=3/2. Asthe primitve cell consists of 4 oxygen cages each one with an osmium atom insidethe t2g states are stabilizing themselves resulting in a G-type antiferromagnet withaltering spin directions in the c-plane at adjacent sites, please see �gure 27 (takenfrom reference [46]).

55

Figure 27: G-type antiferromagnet.

The total magnetic moment hence is zero in the ground-state and the orbital mo-mentum is quenched i.e. Le�=0. Experiments revealed that NaOsO3 is an antifer-romagnetic insulator with an on-site Coulomb repulsion U that should be chosenaround 0.6 eV [38] making the compound more a weakly correlated solid.

56

8 Computational setup

All calculations herein were performed with the Vienna Ab initio Simulation Package(VASP) including spin orbit interaction with 0 0 1 as quantization axis. Projectedaugmented wave pseudo potentials in combination with the generalized gradientapproximation were the chosen DFT parameters. In addition DFT+U after Dudarevwith U=0.6 eV was applied. All supercells had the experimental lattice parametersfrom table 1 consisting of 40 atoms and were fully relaxed. To sample the Brillouin-zone a Monkhorst-Pack mesh of 3x3x3 k points was used and the energy cut-o�was set to 400 eV.

After calculations on the undoped sample the replacement with Mg atoms wereperformed correspondent to Na+1

1-xMg+2xOs+5O-2

3. Accordingly, the doping con-centration was done gradually from x=0.125, 0.25, 0.375 up to 0.5 with di�erentcombinations for each concentration.

In order to obtain band structures re�ecting experiments as close as possible theunfolding technique implemented in VASP was applied [32]. To run the unfoldingboth cells, the supercell and primitive cell with lattice constants and atomic positionshave to be entered. For supercells that are not relaxed the primitive cell is known.But as my supercells were fully relaxed I had to �nd the underlying primitive one.To check if my program to calculate them works properly I chose from all mycon�gurations the one that is a merely repetition of the primitive one, the latticeconstants were found by applying eq. (78). Figure 28 (created with VESTA [45])depicts the situation. The left side is the relaxed supercell, the red balls are theoxygen atoms, the yellow ones sodium, the grey ones osmium and the orange onesare the replaced magnesium. On the right hand side is the calculated primitve cell.

Figure 28: Cells for the unfolding technique.

Figure 29 shows the according band structures without spin, the dotted lines arefrom the supercell, the regular lines are the primitive cell. Both of them are inexcellent agreement.

57

−3

−2

−1

0

1

2

S Γ T R Γ

E−

EF (

eV)

0

0.2

0.4

0.6

0.8

1

1.2

Figure 29: Unfolded and primitive band structure, the intensity is the amount ofBloch character PKm(ki).

58

9 Results

In the following sections the band structure of the undoped case is discussed, after-wards the doped cases as well as their interpretation will follow.

9.1 Undoped case

The undoped band structure (�gure 30) clearly shows the insulating antiferromag-netic ground-state.

−3

−2

−1

0

1

2

S Γ T R Γ

E−

EF

Figure 30: Band structure of the undoped supercell.

From the Hubbard point of view this situation is exactly the one described in chapter2.1.1. Due to the virtual hopping process the delocalization of the t2g electrons leadto a decrease in kinetic energy and therefore to a reduction of the total energy. Tothe same conclusion also comes the exchange energy, this was discussed in section3.1. The magnetic moment has therefore an exact value of 1.23 µB on each sitecompensating each other perfectly. The slightly reduced magnetic moment is byvirtue of the strong hybridization with the oxygen p-orbitals [47]. Within DFTthis insulating behaviour, even though the bands are half �lled, is managed byapplying the DFT+U after Dudarev scheme. The narrow bands below the Fermienergy represent the full t2g bands and above the Fermi energy are the empty egconduction bands also with a �at dispersion relation. The Os d-orbitals on adjacentatoms are hardly overlapping resulting in a gap of 0.27 eV.

59

9.2 Doped samples

As the dopants are in reality distributed randomly, di�erent atomic arrangementsfor each doping concentration were analyzed. The one with the lowest total energyis the most stable one and therefore also the most likely one. In the following �gures(all created with VESTA [45]) the chosen supercells are shown, all of them are in thea-c plane like in �g. 26 and also the colours of the balls are kept. In the followingparagraphs the di�erent cells that were calculated at certain doping concentrationlevels are depicted and their most stable con�guration named.

For a doping concentration of x=0.125 three di�erent cells were created which canbe seen in �g. 31.

(a)

(b)

Figure 31: Doping concentration x=0.125.

The most stable one was (c). The next four ones belong to x=0.25 and can beseen in �g. 32 with the most stable con�guation depicted in (c).

60

(a)

(b)

(c)

Figure 32: Doping concentration x=0.25.

For x=0.375 the two arrangements from �g. 33 were calculated and �gure (a) isthe one with the lowest total energy.

(a) (b)

Figure 33: Doping concentration x=0.375.

Last but not least four sodium were replaced, x=0.5 and can be seen in �g. 34.

61

Figure 34: Doping concentration x=0.5.

If localized electrons want to jump from site A to site B they must �pay� the energyU because of altering spins on adjacent sites and that is why they remain there.As long as this value is not compensated the compound will show insulting or badconducting behaviour. At the same time this localization costs kinetic energy. But,with itinerant delocalized electrons it is possible to gain back more and more ofkinetic energy because additional delocalized electrons and holes reduce the corre-lation energy (in DFT the exchange hole) due to their electrostatic shielding. Theenergy gain for each electron or hole is proportional to W/2 (cf. chapter 2.3) andthe total energy is decreasing eventually. In the case of NaOsO3 the additional neg-ative charged electron comes on the bottom of the eg conduction band. In terms ofelectron hopping this can be interpreted as following, the additional spin up electronis placed due to Pauli's exclusion principle into an eg orbital with parallel t2g elec-trons. A benchmark to which extent the electrons are delocalized is the magneticmoment on each site. The more its value is reduced the more delocalization isachieved and kinetic energy gained.

Another matter that e�ects conductivity is the reduction of the cell volume if sodiumis replaced by magnesium atoms.

The atomic radius depends on the e�ective nuclear charge, that is given by thefollowing equation:

Zeff = Z − EV . (88)

Z denotes the number of protons in the nucleus, the atomic number, and EV denotesthe number of non valence electrons. Na1+ has the electron con�guration 1s22s22p6

with atomic number 11. There are 8 electrons in the valence shell which meansthat EV=2, therefore Ze�=9+.

Magnesium Mg2+ has the same electron con�guration as Na1+ but with atomicnumber 12, therefore Ze�=10+. As a result the orbitals of the Mg atom are moreshrinking than the one from Na, the s orbitals shrink the most and so on. Thisreduction in cell size favors the electrostatic shielding which in turn reduces thecorrelation energy and also the d-orbitals are more overlapping which increases thetunneling probability.

62

414

416

418

420

422

424

426

0 0.125 0.25 0.375 0.5

Vol

ume

of c

ell (

Å3 )

x

x=0x=0.125

x=0.25x=0.375

x=0.5linear interpol.

Figure 35: Volume as a function of the doping concentration.

Figure 35 shows that the volume of cells are reduced almost linarly dependent fromthe doping concentration. The according lattice parameters are shown in �gure 36which depicts the average of each con�guration. There is also a linear decrease in cwhich was the shortest lattice constant and also the one mainly responsible for thereduction of the volume, in total the cell gets reduced by 2.76%.

63

7.4

7.42

7.44

7.46

7.48

7.5

7.52

7.54

0 0.125 0.25 0.375 0.5

Latti

ce g

eom

etry

(Å

)

x

abc

Figure 36: Lattice parameters as a function of the doping concentration.

The �uctuations in the a and b lattice constants are due to the relaxation, theratio of the lattice constants are always ∼1 keeping these two lattice parametersalmost equal. Even though cell shaping during the relaxation process was orderedthe orthorhombic distortions still exist in the doped cases.

9.3 Band structures

The following four �gures show the calculated band structures for the most stablecon�gurations. Figure 37 shows the �rst doping con�guration with one substitutedmagnesium atom.

64

−3

−2

−1

0

1

2

S Γ T R Γ

E−

EF (

eV)

0

0.2

0.4

0.6

0.8

1

1.2

Figure 37: Doping concentration of x=0.125, the intensity is the amount of Blochcharacter PKm(ki).

As SOC is enhanced here there exists an energy shift compared to the undoped case.A rule of thumb is that the spin orbit coupling constant λ is proportional to λnZ2,the atomic number. The reason for this dependence is that with an increase of Zone electron feels a higher e�ective nuclear charge which is in turn proportional tothe spin orbit coupling constant for an electron. The conduction band slips slightlyunder the Fermi energy which means that besides the additional electron also SOCreduced the correlation strength. Because there is still a gap, this is called a pseudogap. Its value is 0.18 eV and the conduction of the pseudo metal is poor. Thevolume is reduced from 425.99 Å3 to 424.27 Å3 and the magnetic moment is 1.18µB compared to 1.23 µB of the undoped case.

65

−3

−2

−1

0

1

2

S Γ T R Γ

En

erg

y (

eV

)

0

0.2

0.4

0.6

0.8

1

1.2

Figure 38: Doping concentration of x=0.25, the intensity is the amount of Blochcharacter PKm(ki).

Figure 38 depicts the next step of doping. Even though the volume is now only420.69 Å3 and the magnetic moment has a value of 1.13 µB it is still a bad metalwith a pseudo gap of 0.11 eV.

66

−3

−2

−1

0

1

2

S Γ T R Γ

E−

EF (

eV)

0

0.2

0.4

0.6

0.8

1

1.2

Figure 39: Doping concentration of x=0.375, the intensity is the amount of Blochcharacter PKm(ki).

In �gure 39 the band broadening is already obvious and the full metallic regime,manifested by the overlap between the partially �lled conduction bands and theunderlying valence band is almost reached. The pseudo gap is only 0.01 eV with amagnetic moment of 1.02 µB . The hopping is further increased due to a anothercell reduction to 420.1 Å3.

67

−3

−2

−1

0

1

2

S Γ T R Γ

E−

EF (

eV)

0

0.2

0.4

0.6

0.8

1

1.2

Figure 40: Doping concentration of x=0.5, the intensity is the amount of Blochcharacter PKm(ki).

Finally, at a doping concentration of 50% (�gure 40) the gap vanishes and thecompound is a full metal with a magnetic moment of 0.89 µB and a volume of414.66 Å3. This situation is exactly the half-�lled case cf. chapter 2.4 when thebands are touching each other. The correlation energy was gradually reduced bygaining kinetic energy due to the delocalization of electrons. Figure 41 illustratesthe continuing reduction of the gap sizes. The values of the gap are from the sumof each con�guration.

68

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.125 0.25 0.375 0.5

Ga

p (

eV

)

x

Figure 41: Gap size as a function of the doping concentration.

As now the delocalized d electrons are responsible for conductivity as well as mag-netism (the Os d-orbitals have unpaired electrons), the itinerancy does also e�ectmagnetism. Figure 42 shows the drop of the magentic moment which is the averageof each con�guration..

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

0 0.125 0.25 0.375 0.5

Magnetic m

om

en

t (µ

B)

x

Figure 42: Magnetic moment as a function of the doping concentration.

69

The more the correlation strength drops and metallicity is in sight the more themagnetic moment gets reduced on each Os site. At a doping concentration of x=0.5the bands are touching each other because the reduction of Coulomb's force and thegain in kinetic energy are equal. That is why the magnetic moments on adjacentOs sites are still compensating each other resulting in an itinerant antiferromagnetinstead of an itinerant ferromagnet.

9.4 Discussion

By chemical doping an insulator-to-metal transition was induced. The G-type anti-ferromagnetic insulating state has hardly overlapping Os d-orbitals with half �lled t2gstates. The magnetic moments are compensating each other perfectly on adjacentsites with a value of 1.23 µB . This reduced value is due to the strong hybridizationwith the oxygen p-orbitals. By virtue of substituting sodium with magnesium atomsan additional negative charged electron is placed in the lowest point of the conduc-tion band, the osmium eg orbital. Each additional electron reduces the correlationenergy because the shielding is enhanced, making a gain in kinetic energy possibleand bringing the system more and more into the itinerant regime. This process istightly linked to magnetism, i.e. the local magnetic moments. These were furtherreduced with an increased doping concentration, antiferromagnetism is maintainedthough.

The substitution of atoms does not only a�ect magnetism also the structure of thecompound changes. As magnesium has a higher atomic number the cell volumegets reduced equivalent to the contraction of the lattice parameters. This favoursshielding and higher overlapping orbitals increase the tunneling probability. Also,the atomic number is proportional to the SOC strength and therefore SOC is raisedif magnesium dopants are in the compound, shifting the conduction band beneaththe Fermi energy. This means that SOC reduces as well the correlation energy.

All in all the interplay of these e�ects are responsible for gradually closing the gapbetween the conduction and valence band. After the insulating state an intermediateregion, a pseudo metal with poor conduction appears until full metallicity is reached.These three regions were also found by B. Kim et al. (2016) where the insulator-to-metal transition was induced by increasing the temperature [38].

70

10 Conclusion

In summary, the role of doping of the perovskite structured transition metal oxideNaOsO3 with half �lled t2g orbitals was successfully examined at zero temperature.The chemical doping was carried out by substituting sodium with magnesium atomsaccording Na+1

1-xMg+2xOs+5O-2

3 until a doping concentration of x=0.5. Eventhough NaOsO3 belongs to the weakly correlated materials, to achieve results thatcoincide with experiments the correlation value of U=0.6 eV has to be factored in byapplying the DFT+U scheme after Dudarev. In addition, the unfolding procedureimplemented in VASP was performed. This has basically two reasons. First of all,the band structure can be compared with results from ARPES and secondly theband structure is far more easy to interpret because unwanted translation points areremoved. For each doping concentration di�erent arrangements were created andalso each cell was fully relaxed to get more realistic results. The band structures wereanalyzed in terms of magnetism, structural features and SOC. Besides magnetismSOC seems to play a special role because compounds with less than half �lledorbitals increase the correlation strength but according to the band structure SOCshifted the conduction band below the Fermi energy which means that it reducedthe correlation energy. This is for further examination.

All in all the doping induces a insulator-to-metal transition that evolves graduallyfrom the insulating ground-state over a poor conducting intermediate state until fullmetallicity is reached. This is certainly no Mott-Hubbard insulator-to-metal transi-tion and this behaviour is also contradictive to a Slater insulator. The continuousinsulator-to-metal transition in combination with a strong hybridization between theosmium d- and oxygen p-orbitals, the itinerant antiferromagnetic behaviour that in-creases with the doping concentration and the weak correlation strenghts U would�t a Lifshitz transition. Wheter the magnesium doped NaOsO3 is indeed classi�ableas undergoing a spin-driven Lifshitz insulator-to-metal transition and whether all 5ddoped half �lled t2g compounds like Cd2Os2O7 show the same behaviour is stillopen for research.

71

References

[1] MODERN PERIODIC TABLE, n.d., http://www.funscience.in/study-zone/Chemistry/ModernPeriodicTable/DBlocksOnPeriodicTable.php, ac-cessed 10th June 2018

[2] V. Celeconovic, J. of Optoelectronics and Advanced Materials, 11, 1135 (2009)

[3] M. Opel, 6, DIE AUSTAUSCHWECHSELWIRKUNG in �Lecture on Ma-gentism�, (WS 2004/2005), 17. Juli 2015, , http://www.wmi.badw-muenchen.de/teaching/Lecturenotes/magnetismus/Kapitel-6.pdf, accesses10th June 2018

[4] F. Duan, J. Guojun, �Introduction to Condensed Matter Physics�, World Sci-enti�c Publishing Co. Pte. Ltd (2005)

[5] F. Essler, H.Frahm, F. Göhmann, A. Klümper, V.E. Korepin, �The One-Dimensional Hubbar Model�, Cambridge Universtiy Press (2009)

[6] K. Held, Adv. Phy., 56, 829 (2007)

[7] D. D. Sarma, Lecture �Electronic and magnetic properties of Transition MetalOxides�, University of Vienna, 2017

[8] P. Liu et al., Physical Rev. B 94, 195145 (2016)

[9] C. Franchini, �Electronic Structure of Transition Metal Oxides�, Lecture (2016)

[10] R. M. Martin, �Electronic Structure�, Cambridge University Press (2010)

[11] Self consistent Kohn-Sham scheme, n. d.,http://www.iue.tuwien.ac.at/phd/goes/dissse14.html, accessed 9th Au-gust 2018

[12] ETH Zürich, Kapitel 17 - Moleküle in "PhysikIV - Einführung in die Quantenmechanik", n.d.,https://qudev.phys.ethz.ch/content/science/BuchPhysikIV/PhysikIVch17.html,accessed 10th June 2018

[13] The Rayleigh-Ritz principle, n.d., https://www.etsf.eu/rayleigh-ritz_principle,accessed 10th June 2018

[14] J. G. Lee �Computational Materials Science, An Introduction�, CRC Press(2012)

[15] J. P. Perdew et al.,Phys. Rev. Lett. 77 3865, 1996

[16] E. �a³�o§lu et al., Phys. Rev. B 83, 121101 (2011)

[17] G. Kresse, M. Marsman, J. Furthmüller, �Vasp the Guide� ,https://cms.mpi.univie.ac.at/vasp/vasp/vasp.html, University of Vienna,2016

72

[18] C. C. J. Roothaan, Rev. of Modern Physics 23, 69-89 (1951)

[19] G. Kresse, Pseudopotentials (Part I), n.d., https://www.vasp.at/vasp-workshop/slides/pseudopp1.pdf, accessed 10th June 2018

[20] G. Kresse, Pseudopotentials (Part II) and PAW, n.d.,https://www.vasp.at/vasp-workshop/slides/pseudopp2.pdf, accessed 10thJune 2018

[21] P. E. Blöchl, Phys. Rev. B 50, 17953 (1994)

[22] J.-J. Mortensen, The Projector Augmented Wave method, n.d.,https://wiki.fysik.dtu.dk/gpaw/_downloads/mortensen_paw.pdf, accessed10th June 2018

[23] P. Blaha et al., Computer Phys. Commun. 59, 399

[24] K. Dewhurst, S. Sharma, Development ofthe Elk LAPW Code, n.d., http://www2.mpi-halle.mpg.de/theory_department/research/elk_code_development/, ac-cessed 10th June 2018

[25] Pseudopotential,n.d., https://de.wikipedia.org/wiki/Pseudopotential, ac-cessed 10th June 2018

[26] N. W. Ashcroft, N. D. Mermin, D. Wei, �Solid State Physics�, CENGAGELearning (2016)

[27] C. Kittel, �Einführung in die Festkörperphysik�, Oldenbourg Verlag (2013)

[28] B. E. Murphy, 2.6 Density functional theory in 'The physico-chemical prop-erties of fullerenes and porphyrin derivatives deposited on conducting sur-faces', 2014, https://maths.tcd.ie/~bmurphy/thesis/thesisse6.html, accessed10th June 2018

[29] O. Madelung, �Festkörpertheorie 1�, Springer-Verlag (1972)

[30] W. Setyawan, S. Curtarolo, Comp. Mat. Sci., 49, 299 (2010)

[31] P. Wisesa et al. Phys. Rev. B 93, 155109 (2016)

[32] M. Reticcioli et al., J. Phys.: Conf. Ser. 689 012027 (2016)

[33] W. Ku et al., Phys. Rev. Let. 104, 216401 (2010)

[34] V. Popescu, A. Zunger, Phys. Rev. B 65, 085201 (2012)

[35] P. Liu et al. Physical Rev. B 94, 195145 (2016)

[36] I. Lo Vechio et al., Scienti�c Reports 3, 2990 (2013)

[37] S. Calder et al., Phys. Rev. Let. 108, 257209 (2012)

73

[38] B. Kim et al., Phys. Rev. B 94, 241113(R) (2016)

[39] I. M. Lifshitz, J. Exptl. Theoret. Phys. 38, 1569-1576 (1960)

[40] A. Varlet et al., Phys. Rev. Let. 113, 116602 (2014)

[41] Perowskit, n.d., https://de.wikipedia.org/wiki/Perowskit, accessed 10th June2018

[42] N. Nampalli, Perovskites: The next PV revolution?, 24th July2014, https://www.solarchoice.net.au/blog/news/perovskites-the-next-solar-pv-revolution-240714, accessed 10th June 2018

[43] R. J. D. Tilley �Perovskites�, WILEY (2016)

[44] M-C. Jung et al., Phys. Rev. B 87, 115119 (2013)

[45] F. Izumi, J. Appl. Crystrallogr. 41, 653 (2008)

[46] G-type antiferromagnet, n.d. https://www.google.at/search?q=G-type+antiferromagnet&tbm=isch&tbo=u&source=univ&sa=X&ved=0ahUKEwjPvc_85NDbAhUKLsAKHXDLDqQQsAQIMQ&biw=1252&bih=581#imgrc=N6vxNGiziHInXM:,accessed 10th June 2018

[47] Y. Du et al., Phys. Rev. B 85, 174424 (2012)

74