Magnetization dynamics ofcomplex magnetic materials byatomistic spin dynamics simulations

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UNIVERSITATISACTA UPSALIENSIS

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1456

Magnetization dynamics of

complex magnetic materials by

atomistic spin dynamics simulations

RAGHUVEER CHIMATA

ISSN 1651-6214

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Dissertation presented at Uppsala University to be publicly examined in Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 24 February 2017 at 09:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Prof Koopmans Bert (Technische Universiteit Eindhoven, Department of Applied Physics).

Abstract

Chimata, R. 2017. Magnetization dynamics of complex magnetic materials by atomistic spin dynamics simulations. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1456. 89 pp. Uppsala: Acta Universitatis Upsaliensis.

ISBN 978-91-554-9763-7.

In recent years, there has been an intense interest in understanding the microscopic mechanism of laser induced ultrafast magnetization dynamics in picosecond time scales. Magnetization switching on such a time scale has potential to be a significant boost for the data storage industry.

It is expected that the writing process will become ~1000 times faster by this technology, compared to existing techniques. Understanding the microscopic mechanisms and controlling the magnetization in such a time scale is of paramount importance at present.

In this thesis, laser induced ultrafast magnetization dynamics has been studied for Fe, Co, GdFe, CoMn and Heusler alloys. A multiscale approach has been used, i.e., first-principles density functional theory combined with atomistic spin dynamics utilizing the Landau –Lifshitz- Gilbert equation, along with a three-temperature phenomenological model to obtain the spin temperature. Special attention has been paid to the calculations of exchange interaction and Gilbert damping parameters. These parameters play a crucial role in determining the ultrafast magnetization dynamics under laser fluence of the considered materials.

The role of longitudinal and transversal excitations was studied for elemental ferromagnets, such as Fe and Co. A variety of complex temporal behavior of the magnetic properties was observed, which can be understood from the interplay between electron, spin, and lattice subsystems. The very intricate structural and magnetic nature of amorphous Gd-Fe alloys for a wide range of Gd and Fe atomic concentrations at the nanoscale was studied. We have shown that the ultrafast thermal switching process can happen above the compensation temperature in GdFe alloys. It is demonstrated that the exchange frustration via Dzyaloshinskii- Moriya interaction between the atomic Gd moments, in Gd rich area of these alloys, leads to Gd demagnetization faster than the Fe sublattice. In addition, we show that Co is a perfect Heisenberg system. Both Co and CoMn alloys have been investigated with respect to ultrafast magnetization dynamics. Also, it is predicted that ultrafast switching process can happen in the Heulser alloys when they are doped with heavy elements. Finally, we studied multiferroic CoCr2O4 and Ca3CoMnO4 systems by using the multiscale approach to study magnetization dynamics. In summary, our approach is able to capture crucial details of ultrafast magnetization dynamics in technologically important materials.

Raghuveer Chimata, Department of Physics and Astronomy, Box 516, Uppsala University, SE-751 20 Uppsala, Sweden.

ISBN 978-91-554-9763-7

urn:nbn:se:uu:diva-308329 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-308329)

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To all teachers

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List of papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Microscopic Model for Ultrafast Remagnetization Dynamics Raghuveer Chimata, Anders Bergman, Lars Bergqvist, Biplab Sanyal, and Olle Eriksson

Phys. Rev. Lett. 109, 157201 (2012).

II All-thermal switching of amorphous Gd-Fe alloys: Analysis of structural properties and magnetization dynamics

Raghuveer Chimata, Leyla Isaeva, Krisztina Kádas, Anders Bergman, Biplab Sanyal, Johan H. Mentink, Mikhail I. Katsnelson, Theo Rasing, Andrei Kirilyuk, Alexey Kimel, Olle Eriksson, and Manuel Pereiro Phys. Rev. B 92, 094411 (2015).

III Magnetism and ultra-fast magnetization dynamics of Co and CoMn alloys at finite temperature

R. Chimata, E. K. Delczeg-Czirjak, A. Szilva, R. Almeida, Y.

Kvashnin, M. Pereiro, S. Mankovsky, H. Ebert, D. Thonig, B. Sanyal, A. B. Klautau, and O. Eriksson

Submitted to Phys. Rev. B.

IV Ultrafast magnetization dynamics in pure and doped Heusler alloys R. Chimata, J. Chico, E. K. Delczeg-Czirjak, M. Pereiro, D. Thonig, B. Sanyal, and O. Eriksson

In manuscript.

V Overcoming magnetic frustration and promoting half-metallicity in spinel CoCr2O4 by doping with Fe

Shreemoyee Ganguly, Raghuveer Chimata, and Biplab Sanyal Phys. Rev. B 92, 224417 (2015).

VI Temperature dependence of dielectric behavior in Ca₃CoMnO₆: insights from density functional theory and atomistic spin dynamics simulations

R. Chimata, J. Ganguli, S. Bhattacharjee, T. Saha-Dasgupta, O.

Eriksson, I. Dasgupta, and B. Sanyal In manuscript.

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Reprints were made with permission from the publishers.

The following papers are co-authored by me but are not included in this Thesis.

• Magnetization dynamics in disordered FexCo_1−xalloys : A first-principles augmented space approach and atomistic spin dynamics simulations Banasree Sadhukhan, Raghuveer Chimata, Biplab Sanyal and Abhijit Mookerjee

Submitted to Phys. Rev. B.

• Magnetic asymmetry around 3p absorption edge in Fe and Ni

Martina Ahlberg, Marco Battiato, Raghuveer Chimata, Olle Eriksson, Somnath Jana, Olof Karis, R. Knut, Yaroslav O. Kvashnin, Inka L. M.

Locht, Igor Di Marco, R. S. Malik, Hans T. Nembach, Justin M. Shaw, Thomas J. Silva, R. Stefanuik, Johan Åkerman, J. Söderström

In manuscript.

Comments on the contributions of the author

In all the papers listed above, I participated in discussions and contributed to the writing process. In Paper I, I performed ultrafast magnetisation part. In Paper II, I performed the ultrafast part and constructed supercells using molecular dynamics code. In Paper III, I performed the ultrafast magnetisation part.

In Paper IV, I performed the SPR-KKR calculations, ultrafast part and I involved writing the paper. In Paper V, I performed the magnetisation dynamics calculations. In Paper VI, I performed magnetism calculations and some part of paper written by me.

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1. Introduction

In the era of information technology, the need to store exponentially growing amounts of digital information is one of the biggest challenges faced by soci- ety. The storage medium has evolved from floppy discs to solid state drives and in fact about 70% of all data is currently stored on magnetic media. In magnetic storage devices, the digital data is stored on magnetic materials in the form of bits “0" and “1" each bit being the smallest unit within a collec- tion of magnetic grains. The bits are created, read and erased using magnetic fields created by very tiny electromagnets called writer (reader) heads. After the discovery of the giant magnetoresistance (GMR) effect in spintronics by Albert Fert and Peter Gr¨unberg in 1988, it became possible to shrink electronic devices to micrometer scales. For example, present reader heads have sizes of a few micrometers. Compared to the reader head, the writer faces more challenges. At present, the external field created by the writer heads is limited to 2.4 Tesla, thereby reducing the number of possible recording materials. The two most successful hard disk recording technologies are the longitudinal magnetic recording (LMR) [1] and the perpendicular magnetic recording (PMR) [2]. In the former technology, the magnetic grains have in-plane magnetisation, while in the later one the magnetic grains have perpendicular anisotropy, allowing a step change improvement in areal density. This improvement led to the LMR suppression and the success of PMR technology from 2005-2006. The present hard disk drives with PMR are able to store up to ∼1 Terabits/inch², but the digital data generated globally is keeping up with the demand of storage devices. The Global demand for data storage capacity will reach 35 Zettabyte (10²¹ bytes) by 2020 according to the international data corporation.

Recent progress of the heat assisted magnetic recording (HAMR) technology on high anisotropy magnetic materials shows promise to make high density data storage a possible way to increase the storage capacity by a factor of ten compared to conventional hard disk drives of the same size. In HAMR, an optical pulse is used to heat up the magnetic grain to the point that the external magnetic field created by the conventional magnetic writing heads can change the orientation of bits [3]. The development of HAMR technology is very challenging because it requires to control both the heat and magnetism of small nano-grains. A schematic diagram of HAMR technology is shown in Fig.1.1.

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0 1 0 1

Magnetic storage mediuam

Laser head

Electric coil

Writer head

Figure 1.1.Schematic representation of Heat Assisted Magnetic Recording (HAMR), where the writer head includes a laser used to heat the magnetic grains up to a param- agnetic state. This reduces the coercivity of the material which allows to write information by using small external magnetic fields created by the tiny electromagnets in the writer head.

An alternative high storage technology, which could be implemented in the near future is the bit patterned magnetic recording (BPMR) [4]. This technology improves the signal-to-noise ratio, while maintaining thermal stability. This can be achieved by lithographically patterned magnetic islands which are larger and more thermally stable than the conventional media grains.

Up to now we discussed the storage capacity, but another important problem is the speed with which information can be written and read.

At present, the magnetic field created by tiny electromagnets can switch the magnetic bits by order of ∼1 ns in conventional magnetic devices. Spin- transfer torque effect using spin-polarised currents [5], ultrafast magnetic vor- tex switching [6], microwave-assisted magnetic recording [7] and all-optical magnetic recording switching [8] have the great potential to reduce the switching times to 10ns, 150ps and 1-3ps, respectively.

From a scientific point of view, it is therefore interesting to study simple ferromagnets as well as complex magnets, magnetisation dynamics under external stimuli, micro and macro mechanisms, miniaturisation of components and to study new ways of manipulating bits by ultra short laser pulses and mechanisms behind ultrafast de- and re-magnetisation.

Another wide research area, where magnetism plays an important role is multiferroics. Usually multiferroics materials have more than one order parameter. In this thesis, we are dealing with materials which exhibit simultaneously ferromagnetism and ferroelectricity in the same phase. The

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magnetoelectric multiferroics, the magnetic and electric orders are linearly coupled to each other. Having two order parameters in the same phase, these materials exhibit a bifunctional property. The ferro-electric polarisation can be controlled by applying magnetic fields and the spins by electric fields. This makes it possible to use such materials in multifunctional devices. The materials allow an additional degree of freedom in the design of novel materials such as actuators, transducers and storage devices [9, 10]. In addition, they can be used as multi-state memory devices [11]. Here we considered two systems, the CoCr2O4 spinel system and the Ca3CoMnO₆ (CCMO) system. Both systems exhibit the magneto-electric effect. Both compounds are studied at zero and elevated temperatures in order to understand their magnetic properties.

The thesis is organised in the following manner: in Chapter 1, we in- troduce the origin of magnetism, ultrafast magnetisation dynamics in magnetic materials and the the origin of multiferrocity. In Chapter 2, the background of density functional theory will be introduced while paying more attention to the Korringa-Kohn-Rostoker methodology [12, 13]. Also, we used Liechtenstein, Katsnelsson, Antropov and Gubanov (LKAG) formalism to calculate the exchange interaction [14, 15]. A detailed description of atomistic spin dynamics (ASD) is presented in Chapter 3. In Chapter 4, we show the key results from the included publications. Lastly in Chapter 5, some conclusions and outlook for this thesis will be summarised.

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1.1 Origin of magnetism

Magnetism has been an interesting and intensely investigated topic in condensed matter physics over the last century. In order to understand magnetism in materials, two physical models were proposed, i.e., the localised electron model and the itinerant electron model. The itinerant electron model succeeded to explain magnetism in 3d transition metals and their alloys while the former model, has become successful in insulators and rare-earth metals.

These models capture two extreme cases and are complementary to each other.

However, some systems exhibit both the localised and the itinerant nature of magnetisation, for example rare-earth transition metals. In order to explain the magnetism in such case one should combine both models in a unified theory which has been proposed in Ref. [16].

1.1.1 Itinerant electron model

The itinerant electron model has been successful in explaining the saturation behaviour of magnetism in 3d transition metals and non-integer moments. For the first time, Weiss tried to understand ferromagnetism in materials and proposed a molecular field theory in 1907 [17]. This theory is quiet successful in explaining many properties of ferromagnets above or below the Curie temperature. However, the equivalent field responsible for the magnetic order is predicted by the Weiss theory to be orders of magnitude lower than estimated from the experimental Curie temperature. Later, Bloch described ferromagnetism in an electron gas on the basis of the Hartree-Fock approximation, where the single-occupied electron orbitals in the vicinity of the Fermi surface were considered to be the origin of ferromagnetism [18]. Finally, Heisen- berg showed that the exchange interaction due to Coulomb repulsion between the electrons is the origin of ferromagnetism and the large Coulomb energy is responsible for the large molecular field [19]. The collective electron ferromagnetism has been explained by Stoner, Wohlfarth, Lidiard and others.

Stoner explained the mechanism of ferromagnetism in itinerant electron systems (3d transition metals) based on the spontaneous splitting of spin-up and spin-down density of states (DOS) at the Fermi level. The splitting happens especially in systems with high DOS at the Fermi level [20]. The instability of non-magnetic states favouring ferromagnetism is given by the Stoner criterion which is defined as,

IN(EF) > 1 (1.1)

where I is the intra-atomic exchange integral and N(EF) is the DOS at Fermi energy. Only three systems (Fe, Co and Ni) in the Periodic table fulfil the Stoner criterion[21]. The Stoner model succeeded to describe the origin of

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magnetism via the band structure model. However, the estimated Curie temperature from this model failed to reproduce experimental values. An improvement of the Stoner model was achieved by considering spin fluctuations in a self consistent renormalised (SCR) theory [22, 23].

1.1.2 Localised electron model

Most ferromagnets posses electronic magnetic moments which are localised at atomic sites. Such behaviour can been seen in the lanthanide series, where the 4 f states are deeply shifted down in energy near to atomic cores. In 1928 Heisenberg showed that molecular field emerges as a consequence of the exchange interaction between two magnetic moments formulated as,

H=

∑

i, j

Ji jmimj (1.2)

where Ji j is the exchange interaction between the moments (mi) and (mj) where i and j refer to the lattice sites. The interatomic exchange interactions between different sites can lead to ferro, antiferromagnetic or non-collinear spin arrangements depending on the sign and the strength of the Ji j parameters. The exchange mechanisms can be classified as direct or indirect (kinetic) exchange. A renowned example for the direct exchange in the hydro- gen molecule is the singlet-triplet electronic states while an indirect or kinetic mechanism can stabilise further the ground state due to the hopping of electrons to the same sites with opposite spins.

Figure 1.2.Different types of exchange interactions.

Different mechanisms give rise to different behaviours of the Heisenberg exchange interaction. Some have been investigated in great detail and are de-

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scribed below.

Direct exchange: Direct exchange is the direct consequence of the Pauli exclusion principle and it is strongly dependent on the overlap of the wave functions. As a consequence, the value of the direct exchange interaction varies rapidly with the interatomic distances between the atomic sites and the coupling vanishes for long range separations. The direct exchange is obtained from first order perturbation theory. Short interatomic distances lead to antiferromagnetism in some systems. In simple words, the cost of energy to flip one atomic moment in a ferromagnetic unit cell with two atoms is equal to the exchange energy, as shown in Fig.1.2 (a).

Indirect exchange:Due to long range separation of the magnetic ions in a host, the direct exchange can not be accepted as a coupling mechanism in many solid-state materials. The strength of the overlap integral is too small due to large separations between the ions and is not sufficiently strong to mediate the coupling. Therefore, one should consider an indirect exchange mechanism which is mostly obtained from the second order perturbation theory.

• Superexchange: Superexchange is mainly found in insulators, where the magnetic moments are separated by diamagnetic atoms. In super exchange the electrons remain in their respective orbits and transmit spin information according to symmetry. In MnO for example, the spin information is mainly shared in between the 3d states from Mn via diamagnetic O ions with an overlap of 3d and p states as shown in Fig.1.2 (c). The involvement of 3d orbitals (eg or t_2g) in an insulator and bond angles between the atomic bonds, decides the strength and sign of the exchange.

To explain superexchange on a qualitative level, John B. Goodenough and Junjiro Kanamori developed a set of semi-empirical rules known as the Goodenough-Kanamori rules [24, 25, 26]. Basically ferromagnetic and antiferromagnetic superexchange is explained by checking the oc- cupation of the magnetic ions. For example, the superexchange coupling between two half-filled orbitals is postulated as strongly antiferromagnetic. If one orbital is half-filled on one ion and the other one is completely filled the coupling is instead ferromagnetic. Although very successful for simple cases, the rules fail for more complex cases, where direct exchange competes with superexchange, spin-orbit coupling is strong, orbital occupancy is dynamic or the bond angle between the two cations is not equal to 180^◦.

• Double exchange: One can see double exchange in oxides, where the magnetic ions display mixed valence states with different oxidation states.

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In double exchange, two electrons are simultaneously transfered. A spin-up electron hops from the mediating oxide to an ion (Mn⁺⁴) and is spontaneously replaced by another spin-up electron from (Mn⁺³) as shown in Fig.1.2 (d) . The hopping of electrons happens without changing their spin direction and by conforming to Hund’s rule. Overall the hopping of electrons reduces the kinetic energy and finally favours ferromagnetic alignment of the neighbouring ions. The ferromagnetic alignment easily allows the eg electron to hop through the crystal and the material becomes metallic.

• RKKY: This type of exchange interaction was first proposed by Ruder- man and Kittel and later extended by Kasuya and Yosida (RKKY) and in general it is known as RKKY interaction. It is an indirect coupling between moments separated over a relatively large distance. It is mediated by conduction electrons. Mainly the RKKY type interaction can be seen in metals where there is weaker interaction between the magnetic moments (or magnetic layers). The RKKY interaction shows an oscillating behaviour; the exchange parameter vs the separation between moments is shown in Fig.1.2 (b). For rare-earth elements and compounds, as well as for transition metal elements, the long ranged exchange interaction is of RKKY form.

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1.2 Ultrafast magnetisation dynamics in magnetic materials

The first part of the thesis contains the study of ultrafast magnetisation processes in Fe, Co, CoMn, GdFe and Heusler alloys under the influence of ultra short laser pulses. The influence of a femtosecond laser pulse on ordered magnetic materials has opened a new era in modern magnetisation dynamics. Con- trolling the magnetisation dynamics at femtosecond timescales will be useful for future data storage techniques. The advantage of such techniques is that they would escalate the speed of data storage by orders of magnitude. The first ultrafast light induced demagnetisation was demonstrated by Beaurepaire et alin 1996 [27], where a Ni film was irradiated with an intense femtosecond (60fs) light pulse. The pulse creates a spontaneous decrease in the magnetisation within several hundreds of femtoseconds. The effect is known as ultrafast demagnetisation and is observed with pump-probe experiments based on the time-resolved magneto-optical Kerr effect (TRMOKE). In the pioneering

Figure 1.3. The observed TRMOKE signal of a Ni thin film after irradiation with a femtosecond pulse. Reproduced with permission from [27].

work of Beaurepaire et al, the measured time of the evolution of demagnetisation after laser excitation was compared with analytical solutions obtained from a phenomenological three temperature model (3TM). The 3TM model has been built on the physical grounds of the dynamics of electron-electron, electron-phonon, electron-magnon and magnon-phonon scattering processes in the femtosecond time regime. The three reservoirs, electron, spin and lattice, have different temperatures and are coupled to each other by a set of coupling parameters. The schematic diagram of the 3TM with the three thermalised systems exchanging energy between each other is presented in Fig.1.3. A large effort was made to investigate the ultrafast demagnetisation

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experimentally and theoretically. On the one side, several experiments reproduced Beaurepaire’s work; on the other side, a lot of theoretical suggestions have been made to explain the ultrafast demagnetisation effect in simple ferromagnetic materials (Fe, Co and Ni).

To date, several mechanisms have been proposed, but the microscopic picture of ultrafast demagnetisation has remained elusive and is not well understood. Of fundamental interest is the question about angular momentum dissipation in ultrafast demagnetisation experiments as well as the different dynamical behaviours of spin and orbital angular momentum during an ultrafast demagnetisation process.

• Photo-quenching effect: The laser induced loss of the moment is explained through different theories. The direct transfer of the photon angular momentum to the spin momentum may lead to ultrafast demagnetisation. But the photo-quenching effect showed by experiments is negligible compared to the total demagnetisation rates. Some experiments have concluded that laser photons alone cannot lead to ultrafast demagnetisation [28].

• Super diffusion theory: According to an alternative theory (i.e. the super-diffusive spin transport theory [29]), the excited spin-up electrons diffuse for longer distance and reach the substrate fast, in contrast to the spin-down electrons. The theory neglects any spin-flip processes and depends on band structures calculated from first-principles. Overall the experiments support the theory, but the effect of demagnetisation is found to be dependent on the substrate. Not all experimental work seems to be consistent with this theory.

• Electron-phonon interaction: In this model, the spin-flip is mediated by electron-phonon interactions or by impurities via the Elliott-Yafet (EY) mechanism [30]. The demagnetisation calculated from ab initio spin-flip electron-phonon scattering processes always underestimates the demagnetisation rate. This is due to the fact that the theory itself builds on rigid band structure calculations. For a complete description one would need to consider dynamical exchange splitting in the calculations [31]. An another way to improve the model is by introducing Coulomb interactions in the EY mechanism [32].

• Magnon-phonon interaction: Magnon-phonon scattering has a much slower rate than the electron-electron and electron-phonon scatterings.

This model may not directly explain ultrafast demagnetisation in transition metal ferromagnetic materials, but could be applied to rare-earth ferromagnetic materials (Gd, Tb), where the demagnetisation happens on a timescale of more than 50ps [33]. The magnetic response time of 3d transition metals (type I) is different from RE ferromagnets (Type II). In type I materials, a fast demagnetisation takes place within 100’s

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of femtoseconds. For type II ferromagnets, two demagnetisation phases exist: in phase I, the 5d electrons participate in ultrafast demagnetisation (during the first picosecond) and in phase II the slower demagnetisation takes place after 1ps and involves the 4f electrons. In the first picosecond, the energy is induced by the laser pulse fluence on the 5d electronic system. Afterwards the energy is transferred from 5d states to 4f localised states during tens of picoseconds.

Role of inverse Faraday effect in ultrafast dynamics

It has been shown that the inverse Faraday effect plays an important role in magnetisation reversal and switching. A full ab initio theory on inverse Faraday effect was presented recently where it shows that the circu- larly polarised light induces spin moment as well as orbital moment in non- magnetic as well as magnetic materials too [34]. For non magnetic materials (Cu, Au, Pt) and anti-ferromagnets the induced spin and orbital moment is anti-symmetric in the light helicity. However for ferromagnets the induced magnetisation is completely asymmetric in the light helicity. A relativistic formalism for describing spin-orbit driven inverse Faraday effect has also been investigated recently [35].

Ultrafast demagnetisation and optical switching in amorphous RE-TM alloys Regardless of the different behaviours of the ultrafast demagnetisation processes in transition ferromagnets and rare-earth metals, it was shown that the demagnetisation dynamics of RE-TM alloys induced by an optical laser pulse was an ultrafast process. In amorphous RE-TM alloys, the magnetic moments of the TM (Co, Fe) (3d) and RE (Gd, Tb) (5d6s-4f) couple antiparallel to each other via a strong inter sub-lattice 3d-5d6s-4f exchange interaction, which is responsible for the ferrimagnetism of the alloy. Due to the strong exchange interaction most of the amorphous alloys exhibit a high Curie temperature. However, for some concentrations the amorphous RExTM1−x

alloys exhibit the compensation temperature (where the sub-lattice moments cancel each other, leading to zero magnetism) near room temperature values.

This is one of the peculiar characters of these alloys [36].

In helicity dependent ultrafast experiments, an all-optical switching (AOS) behaviour was observed in amorphous alloys near the compensation temperature [36]. The two sub-lattices change their polarity in ∼ 1.0 ps [37].

The switching behaviour is much faster when compared to other technologies. A proposed mechanism for the AOS behaviour is the inverse Faraday effect [38]. However, a theoretical description of AOS behaviour of multi sub-lattice systems can be explained in three regimes; in regime I, below one picosecond, the relaxation of the sub-lattices is dominated by longitudinal relaxation. In regime II, an exchange interaction plays a role on picosecond time

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scale, where the angular momentum is transferred between the sub-lattices. In regime III, both the longitudinal and exchange relaxations are important. The system reaches the critical order, where the antiferromagnetic coupling leads to the demagnetisation of the sub-lattices. The ferromagnetic coupling may counteract the longitudinal and the exchange relaxations [39].

Ultrafast demagnetisation in Heusler compounds

Recently, ultrafast spin-dynamics in half-metallic and spin gapless systems have drawn major attention to study the optical excitations around the Fermi-surface. CrO2 is the best known material with a high spin polarisation of ≥ 95%. The features of ultrafast spin dynamics in the system are different from the simple transition metal ferromagnets due to the inability to perform optical excitations in one of the spin channels. The 3TM model was used to explain the ultrafast demagnetisation in this material by assuming that the electron-spin coupling (ges) is almost negligible (ges∼ 0). The demagnetisation time of the system is almost ∼125 ps and it is speculated that the spin-flip process is coming from the crystal field effects [40]. Andreas Mann et al. [41], attempted to explain the demagnetisation times in Heusler compounds (Co2MnSi, Co2MnGe, and Co2FeAl) which exhibit a degree of spin polarisation (P) between 59% and 86%. The demagnetisation time increased from 100 fs for P ∼ 40% to approximately 380 fs for P ∼ 86%. In their approach, the spin-flip rate is proportional to the order of τel−sp∼ (1 − P)⁻¹. The τel−sp is derived from a golden-rule approach without considering thermal effects [41].

Gilbert damping in ultrafast demagnetisation

The description of Gilbert relaxation in ultrafast demagnetisation is so far not complete. In particular, the role of the spin-orbit coupling, which connects the spin system to the crystal, remains unclear. The proposed mechanism is known as the Elliot-Yafet scattering, where the spin-orbit coupling enters by significantly changing the spin polarised band structure. Based on this mechanism, a microscopic model was recently developed by Koopmans to explain the ultrafast demagnetisation in ferromagnets (Ni, Co and Gd). The model extended the 3TM to describe the demagnetisation in fs at microscopic level [33]. In this model, the effects of band-structure variations after the laser pulse are neglected and the heat capacity of the spin system is considered as zero. The Koopman’s model tries to relate the microscopic (electron-spin relaxation time τel−sp) and the macroscopic variables (Gilbert damping parameter, α and Curie temperature TC) as follows,

τel−sp∝ ¯h kBTC

1

α (1.3)

Also, an increase of spin-orbit coupling is observed during ultrafast demagnetisation using femtosecond x-ray absorption spectroscopy experiments [42].

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This shows that an alteration in damping values can happen while the laser pulse is on.

1.2.1 Three Temperature Model

Beaurepaire et. al., [27] divided the particle system into electron, spin and lattice thermal reservoirs. The intense laser pulse affects directly the electron system and the electron system is thermalised nearly instantly due to electron- electron interactions which happen on sub-femtosecond timescale. Thereafter, as time elapses, energy from the electrons will be transferred to the spin and lattice system via electron-spin and electron-phonon interactions. The flow of energy between the thermal reservoirs can be described by the phenomenological three temperature model (3TM). The electron-electron, electron-magnon and electron-phonon interaction (coupling) strengths can be chosen freely.

Figure 1.4.The quasi particle systems (electrons, spins and phonons) are exchanging energy via coupling constants after laser excitation.

Electron-electron scattering

The ultrafast laser pulse creates non- thermal electrons instantly. Sub- sequently, the optically excited electrons scatter from the majority and minor- ity channels and this creates an ensemble of incoherent electron-hole pairs which do not obey the Fermi-Dirac statistics. A broad set of non-thermal electron distribution extends from the Fermi energy up to the energy of the incident photons. The life time of the excited electrons can be estimated using the Fermi-liquid theory(FLT) [43]

τe−e= Const × n^5/6

(E − EF)², (1.4)

where n is the density of the electron gas. The life time is inversely proportional to the intensity of the laser pulse. Hence, a more intense laser creates second generation electrons above the Fermi energy. It is followed by a re-

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distribution of electrons at the Fermi surface after some time. The electron- electron scattering is responsible for the thermalisation of the electron system.

Therefore, the electronic distribution can now be described by the Fermi-Dirac distribution f (E) with a well-defined electronic temperature Te.

f(εi,s) = 1

e^(ε^i,s^−µ)/k^B^T^e+ 1 (1.5)

where s = +, − refers to the spin index and µ is the chemical potential. The electron temperature may reach up to a few thousand Kelvin. The longitudinal atomic moments (spin density at the temperature Te) can be estimated from first-principles calculations via the Fermi distribution,

ρs(r) =

∑

i

|ψi,s(r)|²f(εi,s) (1.6) where ρs(r) = ρ₊(s) + ρ−(s) is the total charge density.

Electron-phonon scattering

The hot electrons transfer their energy to the phonon system via electron-phonon coupling. The lifetime of electron-phonon scattering can be estimated by using the Debye formula [44],

τe−ph= ¯h

2πλ kBT, (1.7)

where λ is the strength of coupling between the electron and phonon system and kB the Boltzmann constant. The estimated life time of electron-phonon scattering is of the order of a few hundred femtoseconds. The 3TM can only be applied to equilibrium state of electrons, spins and lattice. The energy equi- libration processes between the electron, spin and lattice baths is represented through the rate equations as,

Ce(Te)dT_e

dt = −Gel(Te− Tl) − Ges(Te− Ts) + P(t) C_s(Ts)dT_s

dt = −Ges(Ts− Te) − Gsl(Ts− Tl) C_l(Tl)dT_l

dt = −Gel(Tl− Te) − Gsl(Tl− Ts),

(1.8)

where Ce, Cs and Cl are specific heats of electron, spin and lattice systems.

Gel, Gesand Gslare the coupling constants between electron-phonon, electron- spin and spin-lattice. P(t) represents the laser pulse. Here Ce = γTe, where γ is a constant which is related to the density of states at the Fermi energy(EF).

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It is possible to replace Eq. 1.8 with the simplified equation, Ce(Te)dTe

dt = −Gel(Te− Tl) −Ce

Te− Troom

τth

+ P(t) C_l(Tl)dT_l

dt = −Gel(Te− Tl) dTs(t)

dt = τ_M⁻¹[Te(t) − Ts(t)],

(1.9)

where τth is a diffusion time which determine the rate of energy dissipation from the material to reach ambient temperature. The demagnetisation times, τMdetermines the rate of demagnetisation and can be obtained from the experiments. The coupling parameters can be chosen as free parameters. Finally, we further simplified the 3TM model into a simple analytical expression which has been used in the papers, that form the basis of this thesis

Ts = T₀+ (1.10)

(TP− T₀) × (1 − exp^(−t/τ^initial⁾) × exp^(−t/τ^final⁾+ (TF− T₀) × (1 − exp^(−t/τ^final⁾)

where T_sis the spin temperature, T₀is initial temperature of the system, T_Pis the peak temperature after the laser pulse is applied and TFis the final temperature. τ_initial and τ_final are exponential parameters. The calculated spin temperature from Eq. (1.10) is explicitly passed into the Landau-Lifshitz-Gilbert (LLG) equation via the stochastic magnetic field to be further discussed in Sec. 4.3.1, which takes into account thermal fluctuations of the system.

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2. Magnetism in multiferroics

The electric and magnetic orderings in solids are often considered separately.

The separation in space of the electronic charge away from the positive ionic charge leads to the formation of an electric dipole, while it is the electronic spins solely that create a magnetic dipole in solids. However, in some cases, the electric and magnetic orderings are strongly coupled to each other. In spintronics, for example, the spins affect the transport properties in solids, and, vice versa, the magnetic properties may be affected by charge currents.

The magnetoelectric effect (ME) is the phenomenon of inducing an electric (magnetic) polarisation by applying an external magnetic (electric) field. The linear magnetoelectric effect was first predicted by Dzyaloshinskii [45] and later measured by Antropov in Fe₂O₃ [46].

The systems exhibiting at least two so-called “ferroic" orders, for example ferromagnetic and ferroelectric, simultaneously and in the same phase, are denoted as multiferroics. Fig.2.1 shows a schematic diagram of a multiferroic system, exhibiting also a strong magnetoelectric effect. In a [ferro]magnet, the magnetisation displays a hysteresis under an external magnetic field (blue), while in ferroelectrics the electric polarisation has a hysteresis in the presence of an electric field (yellow). Multiferroics, which exhibit the [ferro]magnetic and [ferro]electric polarisation simultaneously, may exhibit a strong magnetoelectric effect, as represented in green. In principle, one can construct a 4-state logic state: (P+, M+), (+, -), (-, +), (-, -). The microscopic origin of

Figure 2.1.The magnetisation of a ferromagnet in an external magnetic field displays an usual hysteresis in blue. Ferroelectrics have a similar response of the polarisation to an external electric field in yellow. A combination of ferromagnetic and ferroelectric region is shown in green. The figure is adapted from [47].

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ordering in magnets has been explained in section 1.1. In general, multiferroics can be classified into two groups, depending on the microscopic mechanism responsible for the ferroelectricity. The first group, so-called type-I multiferroics, where the ferroelectricity and magnetism originate from different sources. This leads to weak coupling between the two order parameters.

Typically, in these materials, the ferroelectricity appears at higher temperatures than magnetic order. The induced spontaneous polarisation P is of the order 10 - 100 µC/cm². In the second group, so-called type-II multiferroics, the spontaneous ferroelectricity is created by the magnetic order, implying a strong coupling between the two. However, the polarisation in these materials is small, of the order of 10⁻²µ C/cm². Type-I multiferroics have magnetic and ferroelectric transitions ordered well above room temperature. Unfortu- nately, the coupling between magnetism and ferroelectricity in these materials is usually rather weak thereby reducing the possibility for real applications.

Very few type-II mulitferroic systems exist and most of them have magnetic and ferroelectricity orders below room temperature. In the literature there are several different subclasses of type-I multiferroics, broadly classified into four major subgroups, depending on the mechanism of ferroelectricity: multifer- roicity in perovskites, ferroelectricity due to lone pairs, ferroelectricity due to charge ordering, geometric ferroelectricity. Detailed information about the subclasses can be found in Ref. [47]

Generally the type-II multiferroics can be classified as follows:

1) Spiral Multiferroics:

In type-II multiferroics, ferroelectricity is associated with a spiral magnetic phase, mostly the cycloid type of spin spiral states contributing to the ferroelectricity in the solid (e.g., TbMnO₃, Ni₃V₂O₆ and MnWO₄). The microscopic mechanism of polarisation in these materials can be explained by spin-orbit coupling. The spin-orbit coupling creates a frustration in the magnetic system of an insulator with a finite polarisation. The P of the system can be expressed in the form of frustration between the neighbouring spins with a phenomenological expression as,

P∼ ri j× [Si× Sj] ∼ [Q × e] (2.1) where ri jis the vector between the two neighbouring spins Siand Sj, Q is the wave vector which describes the spiral state and Q ∼ [Si× SJ] is the spin rota- tion axis. Eq. 2.1 is only valid in cubic and tetragonal systems, whereas, if the spiral Q vector is perpendicular to the plane of spin rotations (proper screw) the polarisation becomes zero.

2) Multiferroics with collinear magnetic structures:

In the second group multiferroics, the polarisation appears in a collinear configuration where the spins are aligned in a particular direction. The microscopic origin of the ferroelectricity in this system appears as a consequence of the exchange-striction without the involvement of spin-orbit coupling. The

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atomic moments of the systems vary with position. Very few type-II multiferroics have a collinear spin configuration. In this thesis, we studied type-II multiferroics with both spiral spin configurations and with collinear configurations.

3) Multiferroics with spin dependent metal-ligand hybridisation:

In some systems, an electric polarisation appears due to metal-ligand hybridisation. For example, in the canted antiferromagnetic phase of Ba₂CoGe₂O₇, the magnetic Co⁺²ions occupy tetragonal sites and the magnetic ions are sur- rounded by four ligand-metal cations with a different direction, which creates an electric dipole around the magnetic moment. The magnetic moments are arranged cooperatively in the canted antiferromagnetic magnetic phase structure. The electric dipole moment in a tetrahedron can be computed from a second-order perturbation of spin-orbit coupling and formulated as,

4P = C⁰(Si· eMX)²eMX (2.2) where eMXis the unit vector connecting the magnetic Co⁺²cation with anions O⁻²[48].

2.0.1 Multiferroic cubic Spinel with magnetic frustration

CoCr₂O₄is a ferrimagnetic cubic spinel system (AB₂O₄), in which the magnetic Co²⁺ions occupy the A site and Cr³⁺occupy the B site. The magnetic moments in this system exhibit a three conical spin spiral structure with a propagation vector (q,q,0) and the electric polarisation along the [0,0,1] and [1,¯1,0] directions [49]. A short range order is found below Tt ∼ 86 K and this transforms into a long range order below Ts∼ 27 K. There is a transition from the incommensurate to the commensurate spiral state (Tf=14K) as well [50].

Most of the theoretical studies were done with collinear Néel configuration and less attention was paid to the conical behaviour in this system [51].

Experimentally as well as theoretically, CoCr₂O₄is an established insulator [51]. However, no experimental report on the band gap of CoFe2O4ex- ists in the literature and theoretically there is a debate regarding the conducting state of the system. In one study, it was shown that the Co ions at the B sites lead the system to be half-metallic for some compositions[52]. By Fe doping (from 0% to 100%) the system can change from the frustrated to the perfect Néel configuration. This transition makes doped CoCr2O4 very interesting to study.

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Figure 2.2. (color line) Schematic representation of CCMO structure along the ab plane and c axis.

2.0.2 Low dimensional spin chains

The study of magnetisation, magneto-striction and electric polarisation at elevated temperatures in Ca₃CoMnO₆ (CCMO) is one of the most challenging tasks from the point of view of ab initio theory. In this system, Co and Mn are in high spin states and the large single ion anisotropy on the magnetic sites makes the magnetic moments behave as a low dimensional 1D Ising chain. The moments are aligned in the up-up-down-down configuration (↑↑↓↓)[53, 54]. The change in magnetic ordering is strongly coupled with the displacement of ions, which spontaneously breaks the spatial-inversion symmetry and results in a net polarisation in the system. The CCMO system shows a net magnetisation M and a net polarisation P in the same direction. This is not common in multiferroic compounds, where the transverse components of spins coupled to P and the longitudinal M is insensitive to P. The promising features of multiferroic systems are very important from a fundamental point of view and therefore have been studied in this thesis.

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3. Electronic structure methods and magnetism

The Holy Grail of the 20th century condensed matter physics is the solution of the eigenvalue problem of the Hamiltonian ˆH for a system of interacting electrons coupled to the ions

Hˆ = ˆTn+ ˆTe+ ˆVnn+ ˆVee+ ˆVne. (3.1) The terms from left to right are the kinetic energy of the nuclei and the electrons, the repulsive nuclei interaction, the repulsive interaction between the electrons and the attractive nucleus-electron interaction, respectively. This requires to deal with the coupled Schrödinger (or Dirac) equations of about 10²⁶ particles. A first step to reduce the complexity of the many-body problem was done by Born and Oppenheimer in 1927 [55]. Since the nuclei are much heavier than the electrons, they reasoned that it would be possible to treat the electrons in a system with frozen nuclei and thus, decouple the electronic from the nuclear degrees of freedom. The electrons evolve in an effective nuclear potential. Based on this, the today’s common and popular technique known as density functional theory (DFT) was proposed by Hohenberg, Kohn, and Sham in 1965 [56].

3.1 Density Functional Theory

The main ideas in DFT are: i) the many-body wavefunction problem is re- duced to an effective one-electron problem by ii) considering the ground-state total energy as a functional of the system’s charge density n(r), which depends only on the three spatial coordinates. In particular, DFT is founded on two important theorems, established by Hohenberg and Kohn in 1964 [57, 58]: The first one states that the effective potential ˆV = ˆVnn+ ˆVee+ ˆVne of a system is determined uniquely by the ground-state electron density n(r). The second theorem says that the ground state electron density n⁰(r) minimises the total energy functional for a given potential V : E[n⁰] < E[n]. The two theorems propose that the ground state electron density is universal, however they make no predictions about the form of the functional. Further approximations are re- quired to obtain the ground state electron density. The realisation of a ground- state electron density was made by Kohn and Sham [58]. In the Kohn-Sham

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(KS) formulation of the DFT the interacting electron system (a real system) is replaced by a fictitious non-interacting system having the same ground state density n(r). The Kohn-Sham functional of the total energy can be written as,

E[n] = Ts[n] + Z

d³r V(r)n(r) +e² 2

Z

d³rd³r⁰n(r)n(r⁰)

|r − r⁰| + Exc[n], (3.2) where Ts is the kinetic energy of the non-interacting electrons. The potential V(r) is the superposition of an external potential and the Coulomb potential coming from the fixed nuclei at positions Ri in the Born-Oppenheimer approximation [59]. The last two terms are the Hartree term — the Coulomb interactions between the electron densities n(r) at r, r⁰ — and the exchange- correlation energy, respectively. The KS formulation shows that with the knowledge of the exchange-correlation functional the approximation is exact within the Born-Oppenheimer restrictions. By minimising the KS functional with respect to the electron density n(r), one gets a Schrödinger-like equation with the non-interacting one-electron wave function φi(r) and their eigenener- gies, εi.

[− ¯h² 2me

∇²+Ve f f(r)]φi(r) = εiφi(r) (3.3) with the effective potential Ve f f(r) = V (r) + e²^R _|r−r’|^n(r’)d³r⁰+Vxc(r), including the exchange-correlation potential Vxc(r) defined by Vxc(r) =^{∂ E}^xc^[n]/∂ n. The electron density, n(r) in terms of Kohn-Sham orbitals is

n(r) =

occ

∑

i

|φi(r)|² (3.4)

where the state index i runs over occupied states.

In numerical computations, the electron density is obtained self-consistently, where the self-consistent cycle is started by an initial guess. This allows to construct an effective potential Ve f f; the final effective potential can be obtained from the ground state electron density after self-consistency is reached.

In the Kohn-Sham equations, the unknown part is the exchange-correlation term. Even though its contribution to the total energy is small compared to the kinetic and Coulomb terms, exchange and correlation is a key ingre- dient to accurately describe materials properties. The accuracy of the DFT method relies on the exchange-correlation functionals. Since the exact form of the exchange-correlation potential Exc[n(r)] is not known, approximations are necessary. Exc[n(r)] represents the difference between the exact total energy of the real many-body system and the energy of the auxiliary non-interacting particle system. The Exc[n(r)] = Ex[n(r)] + Ec[n(r)] term can be further split into an exchange part Ex[n(r)] and a correlation part Ec[n(r)]. The exchange energy is due to the antisymmetric properties of the wave function and it con- stitutes the underlying cause of magnetism [60]. Correlation is a more difficult concept, related to the fact that the probability density of one electron at one

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location reduces the probability density of another electron at the same location, beyond the probability reduction due to exchange [61]. The exchange part can be solved exactly within the Hartree-Fock method. The exchange energy is about 5-10% or even less and the correlation energy is less than 1%

of the total energy respectively. Nevertheless, various hybrid approaches exist to approach the full exchange-correlation functional [62, 63] or only the correlation part. The two fundamental and widely used approximations will be explained in the following.

Local density approximation (LDA)

Deriving the exact exchange-correlation functional seems mathemat- ically impossible. However, if one assumes the electron density to be the electron density of free-moving electrons in an electron gas which is homogenous everywhere in space, the Kohn-Sham DFT problem can be solved. Thus, the exchange-correlation functional reads

E_xc^LDA[n] = Z

d³r n(r)ε_xc^Hom[n] (3.5) where ε_xc^Hom[n(r)] is the exchange-correlation energy density of the uniform electron gas. This approach is known as the Local Density approximation (LDA) [58, 64]. It is the simplest form of the exchange-correlation functional.

A generalisation in terms of electron spin-degrees of freedom is possible (local spin density approximation (LSDA)) and can successfully be applied to a large number of systems where the electron density varies slowly. To increase the accuracy within the LDA approximation, the exchange energy part can be derived from the exact analytical formula given by Dirac [65]. Even the correlation part e^hom_c (n) is discussed in the limits rs−→ 0 and rs−→ ∞, where rsis the Wigner-Seitz radius. For rsin between these limits the correlation energy can be obtained by Quantum Monte Carlo (QMC) methods [66]. Other analytical forms of the correlation energy ecare given by e.g. Vosko-Wilk-Nusair [67], Perdew-Zunger [68], Cole-Perdew [69] or Perdew-Wang [70, 71, 72].

LDA is able to reproduce details of band structures and describes the ground state properties in very good agreement with experiments[73, 74, 75], but fails to give proper band gaps [76], which are underestimated. It also fails for strongly correlated systems [77]. All these problems arise due to rapidly vary- ing electron densities, not coverable within the LDA approach.

Generalised gradient approximation (GGA)

One can overcome the difficulties in LDA by also considering the gradients ∇n of the electron density in the functional. This approach is the generalised gradient approximation (GGA). Thus, the integrand f in the GGA, that replaces n(r)ε_xc^Hom[n] from LDA, is a functional of the first derivative of

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the density and the density itself:

E_xc^GGA[n] = Z

d³r f[n, ∇n] (3.6)

The most used GGA functional is the one proposed by Perdew, Burke, and Ernzerhof (PBE). The PBE functional is mainly adapted for solids. Overall there is an improvement in the calculated materials properties compared to LDA: e.g. in atomic and molecular total energies, stability of bulk phases, binding energies, activation energy barriers in chemical reactions, dissocia- tion energies of molecules and cohesive energies of solids. An analytical form for the GGA exchange-correlation functional is expressed e.g. by Perdew- Burke-Ernzerhof [78]. Meta-GGA methods contain the derivative of the local density up to the second order [79]. Other exchange-correlation functional approaches and self-interaction corrections for calculating the self-energy of the many-body electron system are e.g. L(S)DA+U [80], self-interaction correction (SIC) [81, 68], the self-energy correction (GW) [82, 83, 84], and the dynamical mean field theory (DMFT) [85, 86], and have also been applied to first-principles combined calculations. These functionals improve the band gaps in insulators [87] or rare earth-metals [88], in better accordance with experiments.

Relativistic DFT

Up to now relativistic effects are neglected in the DFT theory, however, generalised by Rajagopal and Callaway [89] and applied to the systems where an external field is involved or magnetic order is present. Later, the theory was extended to many-body systems in the presence of an external potential or magnetic field by neglecting diamagnetic effects. In the proposed theory, the total energy, Etot[ j^ν] of the system is written as a functional of the four component current, j^ν = (n, j), where n is the electron density. As con- tained in the non-relativistic DFT total energy Etot(n), the vector current has been explicitly written by using Gordon decomposition as,

Etot[ j^ν] = Ts[ j^ν] + Eext[ j^ν] + EH[ j^ν] + Exc[ j^ν] (3.7) where Tsis the kinetic energy, Eext, an external potential is sum of ionic potential and a vector potential resulting from an external magnetic field, EH being the Hartree term and Exc is the exchange correlation potential. The minimi- sation of total energy Etot[ j^ν] with respect to the four component current j^ν, which give rise to the Dirac- type Kohn-Sham equations,

(c ˆα · p + β c²+ ve f f(r) − m(r) · Be f f(r))ψi= εiψi (3.8) where ve f f(r) is the effective potential,

ve f f(r) = v(r) +

Z n_σ(r)

|r − r⁰|dr+δ Exc[n_σ(r), m(r)]

δ nσ(r) , (3.9)

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Be f f(r) is an effective magnetic field,

B_{e f f}(r) = Bext(r) +δ Exc[nσ(r), m(r)]

δ m_σ(r) , (3.10)

and ˆαi(i = 1, 2, 3) and β are the Dirac matrices. In this thesis, most of the calculations are done either with the fully relativistic Dirac equation or with a scalar relativistic approach. The latter assumes that the spin-orbit coupling effects are small and can be treated as a perturbation. In this case the Pauli equation has to be solved instead [90]. The Dirac-Kohn-Sham equations as well as the Kohn-Sham equation has to be to be solved self-consistently. One of the main aims of the density functional theory is to characterise materials properties within a periodic arrangement of atoms — study materials properties —, where the effective Kohn-Sham potential is a translational invariant:

Ve f f(r + T) = Ve f f(r), T being the translation vector. This defines the electron band structure offering valuable clues regarding the electric and magnetic properties of materials. The periodicity of a crystal further reduces the single- electronic problem 3.3 from the whole bulk or surface to just one region: the effective Kohn-Sham equation has to be solved only within the Wigner-Seitz cell. Applying furthermore Bloch’s theorem [91],

φk(r) = e^ik·ruk(r), (3.11) one obtains the solution for the entire crystal. More precisely, the Bloch theorem states that the eigenstates φk(r) of a periodic system may be written as product of a phase factor and a lattice periodic Bloch function uk. In addition, the effective one-electron wave function for the bulk is composed of a linear combination of atomic orbitals χik(r) (LCAO approach):

φnk(r) =

∑

i

c_i,nkχik(r), (3.12)

Hence, ab initio methods set out to obtain the expansion coefficients ci,nk. Fi- nally, the atomic orbital wave function has to be constructed. Various method- ologies were developed to solve the electronic structure problem with a chosen format set of basis functions χi such as, the linearised augmented plane waves (LAPW), linear muffin-tin orbitals (LMTO), exact muffin-tin orbitals (EMTO), or Gaussian orbitals. All these methods make a presupposition for the shape of the potential field and evaluate and expand the atomic wave functions. The form of the inhomogeneous differential Eq. 3.3 also allows to analyse its pulse response, which is the Green’s function.

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3.2 Korringa-Kohn-Rostoker (KKR) Green’s function method

To solve the Kohn-Sham Eq. 3.3, one can represent it as an effective Schrödinger equation ˆHφkn= εknφkn. The linear form of the differential operator allows to apply Green’s function techniques. By definition, the Greens function ˆGof Eq. 3.3 is specified as

(z Ê− ˆH) ˆG= Ê or ˆG= (z Ê− ˆH)⁻¹. (3.13) z = ε + iΓ is a complex energy and Ê is the unit operator. Assuming the potential to be zero, the Kohn-Sham problem reduces to the non-relativistic free-electron problem [92], i.e electron that is moving at a speed which is small compared to the speed of light. Here, the Green’s function is

G₀(r, r⁰, z) = −2m

¯h² e⁻ⁱ

√ E|^r−r⁰|

4π |r − r⁰|. (3.14) This Green function plays an important role in the KKR theory. This is due to the fact that the free electron is the reference system used for calculating the Green function of the electron under the influence of a potential. A good approach is to assume the potential to be spherically symmetric Ve f f(r) = Ve f f(r) (atomic sphere approximations[ASA]). In order to avoid unsolvable boundary conditions between the Wigner-Seitz cells, the crystal potential is represented as a muffin-tin potential, where the radius rMT of the potential is determined through non-overlapping spheres. In between the spheres a constant value is considered. Within this approximation, the effective potential of the whole lattice is then a superposition of radial potentials, each measured from a particular ionic position and, consequently, fulfilling the Bloch theorem. Further- more, states of different angular momentum are scattered independently. It is therefore convenient to expand the wave function in terms of superposed partial waves with different angular momenta

φnk(r) =

∑

lm

c_lm,nkR_l(r)Ylm(ˆr). (3.15)

Here, Rl is the radial function, Ylmthe spherical harmonics, ˆr denotes the direction of the vector r, and l, m are the angular momentum and the magnetic quantum number, respectively. Thus, the free-electron Green’s function in angular momentum expansion is

G₀(r, r⁰, E) = −i√ E2m

¯h²

∑

lm

Y_lm(ˆr)Y_lm^∗(ˆr⁰) jl(√

Er_<)hl(√

Er_>), (3.16) expressed in Bessel jl and Hankel functions hl. The Lippmann-Schwinger equation connects finally this free electron case with the general case, i.e. with the solution of the Kohn-Sham Eq. 3.3 for a particle (electron) under the

Magnetization dynamics ofcomplex magnetic materials byatomistic spin dynamics simulations